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  1. Int[Times[Optional[Pattern[u, Blank[]]], Power[Plus[Pattern[a, Blank[]], Times[Optional[Pattern[b, Blank[]]], Power[Pattern[x, Blank[]], Optional[Pattern[n, Blank[]]]]]], Optional[Pattern[p, Blank[]]]]], Pattern[x, Blank[Symbol]]] := Int[u*(b*x^n)^p, x] /; FreeQ[{a, b, n, p}, x] && EqQ[a, 0]
  2. Int[Times[Optional[Pattern[u, Blank[]]], Power[Plus[Optional[Pattern[a, Blank[]]], Times[Optional[Pattern[b, Blank[]]], Power[Pattern[x, Blank[]], Optional[Pattern[n, Blank[]]]]]], Optional[Pattern[p, Blank[]]]]], Pattern[x, Blank[Symbol]]] := Int[u*a^p, x] /; FreeQ[{a, b, n, p}, x] && EqQ[b, 0]
  3. Int[Times[Optional[Pattern[u, Blank[]]], Power[Plus[Pattern[a, Blank[]], Times[Optional[Pattern[c, Blank[]]], Power[Pattern[x, Blank[]], Optional[Pattern[j, Blank[]]]]], Times[Optional[Pattern[b, Blank[]]], Power[Pattern[x, Blank[]], Optional[Pattern[n, Blank[]]]]]], Optional[Pattern[p, Blank[]]]]], Pattern[x, Blank[Symbol]]] := Int[u*(b*x^n + c*x^(2*n))^p, x] /; FreeQ[{a, b, c, n, p}, x] && EqQ[j, 2*n] && EqQ[a, 0]
  4. Int[Times[Optional[Pattern[u, Blank[]]], Power[Plus[Optional[Pattern[a, Blank[]]], Times[Optional[Pattern[c, Blank[]]], Power[Pattern[x, Blank[]], Optional[Pattern[j, Blank[]]]]], Times[Optional[Pattern[b, Blank[]]], Power[Pattern[x, Blank[]], Optional[Pattern[n, Blank[]]]]]], Optional[Pattern[p, Blank[]]]]], Pattern[x, Blank[Symbol]]] := Int[u*(a + c*x^(2*n))^p, x] /; FreeQ[{a, b, c, n, p}, x] && EqQ[j, 2*n] && EqQ[b, 0]
  5. Int[Times[Optional[Pattern[u, Blank[]]], Power[Plus[Optional[Pattern[a, Blank[]]], Times[Optional[Pattern[c, Blank[]]], Power[Pattern[x, Blank[]], Optional[Pattern[j, Blank[]]]]], Times[Optional[Pattern[b, Blank[]]], Power[Pattern[x, Blank[]], Optional[Pattern[n, Blank[]]]]]], Optional[Pattern[p, Blank[]]]]], Pattern[x, Blank[Symbol]]] := Int[u*(a + b*x^n)^p, x] /; FreeQ[{a, b, c, n, p}, x] && EqQ[j, 2*n] && EqQ[c, 0]
  6. Int[Times[Optional[Pattern[u, Blank[]]], Power[Plus[Optional[Pattern[w, Blank[]]], Times[Optional[Pattern[a, Blank[]]], Pattern[v, Blank[]]], Times[Optional[Pattern[b, Blank[]]], Pattern[v, Blank[]]]], Optional[Pattern[p, Blank[]]]]], Pattern[x, Blank[Symbol]]] := Int[u*((a + b)*v + w)^p, x] /; FreeQ[{a, b}, x] && !FreeQ[v, x]
  7. Int[Times[Optional[Pattern[u, Blank[]]], Power[Pattern[Px, Blank[]], Pattern[p, Blank[]]]], Pattern[x, Blank[Symbol]]] := Int[u*Px^Simplify[p], x] /; PolyQ[Px, x] && !RationalQ[p] && FreeQ[p, x] && RationalQ[Simplify[p]]
  8. Int[Pattern[a, Blank[]], Pattern[x, Blank[Symbol]]] := Simp[a*x, x] /; FreeQ[a, x]
  9. Int[Times[Pattern[a, Blank[]], Plus[Pattern[b, Blank[]], Times[Optional[Pattern[c, Blank[]]], Pattern[x, Blank[]]]]], Pattern[x, Blank[Symbol]]] := Simp[(a*(b + c*x)^2)/(2*c), x] /; FreeQ[{a, b, c}, x]
  10. Int[Times[-1, Pattern[u, Blank[]]], Pattern[x, Blank[Symbol]]] := Dist[Identity[-1], Int[u, x], x]
  11. Int[Times[Complex[0, Pattern[a, Blank[]]], Pattern[u, Blank[]]], Pattern[x, Blank[Symbol]]] := Dist[Complex[Identity[0], a], Int[u, x], x] /; FreeQ[a, x] && EqQ[a^2, 1]
  12. Int[Times[Pattern[a, Blank[]], Pattern[u, Blank[]]], Pattern[x, Blank[Symbol]]] := Dist[a, Int[u, x], x] /; FreeQ[a, x] && !MatchQ[u, (b_)*(v_) /; FreeQ[b, x]]
  13. Int[Pattern[u, Blank[]], Pattern[x, Blank[Symbol]]] := Simp[IntSum[u, x], x] /; SumQ[u]
  14. Int[Times[Pattern[u, Blank[]], Power[Times[Optional[Pattern[c, Blank[]]], Pattern[x, Blank[]]], Optional[Pattern[m, Blank[]]]]], Pattern[x, Blank[Symbol]]] := Int[ExpandIntegrand[(c*x)^m*u, x], x] /; FreeQ[{c, m}, x] && SumQ[u] && !LinearQ[u, x] && !MatchQ[u, (a_) + (b_.)*(v_) /; FreeQ[{a, b}, x] && InverseFunctionQ[v]]
  15. Int[Times[Optional[Pattern[u, Blank[]]], Power[Times[Optional[Pattern[a, Blank[]]], Power[Pattern[x, Blank[]], Pattern[n, Blank[]]]], Pattern[m, Blank[]]]], Pattern[x, Blank[Symbol]]] := Dist[(a^IntPart[m]*(a*x^n)^FracPart[m])/x^(n*FracPart[m]), Int[u*x^(m*n), x], x] /; FreeQ[{a, m, n}, x] && !IntegerQ[m]
  16. Int[Times[Optional[Pattern[u, Blank[]]], Power[Pattern[v, Blank[]], Optional[Pattern[m, Blank[]]]], Power[Times[Pattern[b, Blank[]], Pattern[v, Blank[]]], Pattern[n, Blank[]]]], Pattern[x, Blank[Symbol]]] := Dist[1/b^m, Int[u*(b*v)^(m + n), x], x] /; FreeQ[{b, n}, x] && IntegerQ[m]
  17. Int[Times[Optional[Pattern[u, Blank[]]], Power[Times[Optional[Pattern[a, Blank[]]], Pattern[v, Blank[]]], Pattern[m, Blank[]]], Power[Times[Optional[Pattern[b, Blank[]]], Pattern[v, Blank[]]], Pattern[n, Blank[]]]], Pattern[x, Blank[Symbol]]] := Dist[(a^(m + 1/2)*b^(n - 1/2)*Sqrt[b*v])/Sqrt[a*v], Int[u*v^(m + n), x], x] /; FreeQ[{a, b, m}, x] && !IntegerQ[m] && IGtQ[n + 1/2, 0] && IntegerQ[m + n]
  18. Int[Times[Optional[Pattern[u, Blank[]]], Power[Times[Optional[Pattern[a, Blank[]]], Pattern[v, Blank[]]], Pattern[m, Blank[]]], Power[Times[Optional[Pattern[b, Blank[]]], Pattern[v, Blank[]]], Pattern[n, Blank[]]]], Pattern[x, Blank[Symbol]]] := Dist[(a^(m - 1/2)*b^(n + 1/2)*Sqrt[a*v])/Sqrt[b*v], Int[u*v^(m + n), x], x] /; FreeQ[{a, b, m}, x] && !IntegerQ[m] && ILtQ[n - 1/2, 0] && IntegerQ[m + n]
  19. Int[Times[Optional[Pattern[u, Blank[]]], Power[Times[Optional[Pattern[a, Blank[]]], Pattern[v, Blank[]]], Pattern[m, Blank[]]], Power[Times[Optional[Pattern[b, Blank[]]], Pattern[v, Blank[]]], Pattern[n, Blank[]]]], Pattern[x, Blank[Symbol]]] := Dist[(a^(m + n)*(b*v)^n)/(a*v)^n, Int[u*v^(m + n), x], x] /; FreeQ[{a, b, m, n}, x] && !IntegerQ[m] && !IntegerQ[n] && IntegerQ[m + n]
  20. Int[Times[Optional[Pattern[u, Blank[]]], Power[Times[Optional[Pattern[a, Blank[]]], Pattern[v, Blank[]]], Pattern[m, Blank[]]], Power[Times[Optional[Pattern[b, Blank[]]], Pattern[v, Blank[]]], Pattern[n, Blank[]]]], Pattern[x, Blank[Symbol]]] := Dist[(b^IntPart[n]*(b*v)^FracPart[n])/(a^IntPart[n]*(a*v)^FracPart[n]), Int[u*(a*v)^(m + n), x], x] /; FreeQ[{a, b, m, n}, x] && !IntegerQ[m] && !IntegerQ[n] && !IntegerQ[m + n]
  21. Int[Times[Optional[Pattern[u, Blank[]]], Power[Plus[Pattern[a, Blank[]], Times[Optional[Pattern[b, Blank[]]], Pattern[v, Blank[]]]], Optional[Pattern[m, Blank[]]]], Power[Plus[Pattern[c, Blank[]], Times[Optional[Pattern[d, Blank[]]], Pattern[v, Blank[]]]], Optional[Pattern[n, Blank[]]]]], Pattern[x, Blank[Symbol]]] := Dist[(b/d)^m, Int[u*(c + d*v)^(m + n), x], x] /; FreeQ[{a, b, c, d, n}, x] && EqQ[b*c - a*d, 0] && IntegerQ[m] && ( !IntegerQ[n] || SimplerQ[c + d*x, a + b*x])
  22. Int[Times[Optional[Pattern[u, Blank[]]], Power[Plus[Pattern[a, Blank[]], Times[Optional[Pattern[b, Blank[]]], Pattern[v, Blank[]]]], Pattern[m, Blank[]]], Power[Plus[Pattern[c, Blank[]], Times[Optional[Pattern[d, Blank[]]], Pattern[v, Blank[]]]], Pattern[n, Blank[]]]], Pattern[x, Blank[Symbol]]] := Dist[(b/d)^m, Int[u*(c + d*v)^(m + n), x], x] /; FreeQ[{a, b, c, d, m, n}, x] && EqQ[b*c - a*d, 0] && GtQ[b/d, 0] && !(IntegerQ[m] || IntegerQ[n])
  23. Int[Times[Optional[Pattern[u, Blank[]]], Power[Plus[Pattern[a, Blank[]], Times[Optional[Pattern[b, Blank[]]], Pattern[v, Blank[]]]], Pattern[m, Blank[]]], Power[Plus[Pattern[c, Blank[]], Times[Optional[Pattern[d, Blank[]]], Pattern[v, Blank[]]]], Pattern[n, Blank[]]]], Pattern[x, Blank[Symbol]]] := Dist[(a + b*v)^m/(c + d*v)^m, Int[u*(c + d*v)^(m + n), x], x] /; FreeQ[{a, b, c, d, m, n}, x] && EqQ[b*c - a*d, 0] && !(IntegerQ[m] || IntegerQ[n] || GtQ[b/d, 0])
  24. Int[Times[Optional[Pattern[u, Blank[]]], Power[Plus[Pattern[a, Blank[]], Times[Optional[Pattern[b, Blank[]]], Pattern[v, Blank[]]]], Pattern[m, Blank[]]], Plus[Optional[Pattern[A, Blank[]]], Times[Optional[Pattern[B, Blank[]]], Pattern[v, Blank[]]], Times[Optional[Pattern[C, Blank[]]], Power[Pattern[v, Blank[]], 2]]]], Pattern[x, Blank[Symbol]]] := Dist[1/b^2, Int[u*(a + b*v)^(m + 1)*Simp[b*B - a*C + b*C*v, x], x], x] /; FreeQ[{a, b, A, B, C}, x] && EqQ[A*b^2 - a*b*B + a^2*C, 0] && LeQ[m, -1]
  25. Int[Times[Optional[Pattern[u, Blank[]]], Power[Plus[Pattern[a, Blank[]], Times[Optional[Pattern[b, Blank[]]], Power[Pattern[x, Blank[]], Optional[Pattern[n, Blank[]]]]]], Optional[Pattern[m, Blank[]]]], Power[Plus[Pattern[c, Blank[]], Times[Optional[Pattern[d, Blank[]]], Power[Pattern[x, Blank[]], Optional[Pattern[q, Blank[]]]]]], Optional[Pattern[p, Blank[]]]]], Pattern[x, Blank[Symbol]]] := Dist[(d/a)^p, Int[(u*(a + b*x^n)^(m + p))/x^(n*p), x], x] /; FreeQ[{a, b, c, d, m, n}, x] && EqQ[q, -n] && IntegerQ[p] && EqQ[a*c - b*d, 0] && !(IntegerQ[m] && NegQ[n])
  26. Int[Times[Optional[Pattern[u, Blank[]]], Power[Plus[Pattern[a, Blank[]], Times[Optional[Pattern[b, Blank[]]], Power[Pattern[x, Blank[]], Optional[Pattern[n, Blank[]]]]]], Optional[Pattern[m, Blank[]]]], Power[Plus[Pattern[c, Blank[]], Times[Optional[Pattern[d, Blank[]]], Power[Pattern[x, Blank[]], Pattern[j, Blank[]]]]], Optional[Pattern[p, Blank[]]]]], Pattern[x, Blank[Symbol]]] := Dist[(-(b^2/d))^m, Int[u/(a - b*x^n)^m, x], x] /; FreeQ[{a, b, c, d, m, n, p}, x] && EqQ[j, 2*n] && EqQ[p, -m] && EqQ[b^2*c + a^2*d, 0] && GtQ[a, 0] && LtQ[d, 0]
  27. Int[Times[Optional[Pattern[u, Blank[]]], Power[Plus[Pattern[a, Blank[]], Times[Optional[Pattern[b, Blank[]]], Pattern[x, Blank[]]], Times[Optional[Pattern[c, Blank[]]], Power[Pattern[x, Blank[]], 2]]], Optional[Pattern[p, Blank[]]]]], Pattern[x, Blank[Symbol]]] := Int[u*Cancel[(b/2 + c*x)^(2*p)/c^p], x] /; FreeQ[{a, b, c}, x] && EqQ[b^2 - 4*a*c, 0] && IntegerQ[p]
  28. Int[Times[Optional[Pattern[u, Blank[]]], Power[Plus[Pattern[a, Blank[]], Times[Optional[Pattern[c, Blank[]]], Power[Pattern[x, Blank[]], Optional[Pattern[n2, Blank[]]]]], Times[Optional[Pattern[b, Blank[]]], Power[Pattern[x, Blank[]], Pattern[n, Blank[]]]]], Optional[Pattern[p, Blank[]]]]], Pattern[x, Blank[Symbol]]] := Dist[1/c^p, Int[u*(b/2 + c*x^n)^(2*p), x], x] /; FreeQ[{a, b, c, n}, x] && EqQ[n2, 2*n] && EqQ[b^2 - 4*a*c, 0] && IntegerQ[p]
  29. Int[Power[Pattern[x, Blank[]], -1], Pattern[x, Blank[Symbol]]] := Simp[Log[x], x]
  30. Int[Power[Pattern[x, Blank[]], Optional[Pattern[m, Blank[]]]], Pattern[x, Blank[Symbol]]] := Simp[x^(m + 1)/(m + 1), x] /; FreeQ[m, x] && NeQ[m, -1]
  31. Int[Power[Plus[Pattern[a, Blank[]], Times[Optional[Pattern[b, Blank[]]], Pattern[x, Blank[]]]], -1], Pattern[x, Blank[Symbol]]] := Simp[Log[RemoveContent[a + b*x, x]]/b, x] /; FreeQ[{a, b}, x]
  32. Int[Power[Plus[Optional[Pattern[a, Blank[]]], Times[Optional[Pattern[b, Blank[]]], Pattern[x, Blank[]]]], Pattern[m, Blank[]]], Pattern[x, Blank[Symbol]]] := Simp[(a + b*x)^(m + 1)/(b*(m + 1)), x] /; FreeQ[{a, b, m}, x] && NeQ[m, -1]
  33. Int[Power[Plus[Optional[Pattern[a, Blank[]]], Times[Optional[Pattern[b, Blank[]]], Pattern[u, Blank[]]]], Pattern[m, Blank[]]], Pattern[x, Blank[Symbol]]] := Dist[1/Coefficient[u, x, 1], Subst[Int[(a + b*x)^m, x], x, u], x] /; FreeQ[{a, b, m}, x] && LinearQ[u, x] && NeQ[u, x]
  34. Int[Times[Power[Plus[Pattern[a, Blank[]], Times[Optional[Pattern[b, Blank[]]], Pattern[x, Blank[]]]], Optional[Pattern[m, Blank[]]]], Plus[Pattern[c, Blank[]], Times[Optional[Pattern[d, Blank[]]], Pattern[x, Blank[]]]]], Pattern[x, Blank[Symbol]]] := Simp[(d*x*(a + b*x)^(m + 1))/(b*(m + 2)), x] /; FreeQ[{a, b, c, d, m}, x] && EqQ[a*d - b*c*(m + 2), 0]
  35. Int[Times[Power[Plus[Pattern[a, Blank[]], Times[Optional[Pattern[b, Blank[]]], Pattern[x, Blank[]]]], -1], Power[Plus[Pattern[c, Blank[]], Times[Optional[Pattern[d, Blank[]]], Pattern[x, Blank[]]]], -1]], Pattern[x, Blank[Symbol]]] := Int[1/(a*c + b*d*x^2), x] /; FreeQ[{a, b, c, d}, x] && EqQ[b*c + a*d, 0]
  36. Int[Times[Power[Plus[Optional[Pattern[a, Blank[]]], Times[Optional[Pattern[b, Blank[]]], Pattern[x, Blank[]]]], -1], Power[Plus[Optional[Pattern[c, Blank[]]], Times[Optional[Pattern[d, Blank[]]], Pattern[x, Blank[]]]], -1]], Pattern[x, Blank[Symbol]]] := Dist[b/(b*c - a*d), Int[1/(a + b*x), x], x] - Dist[d/(b*c - a*d), Int[1/(c + d*x), x], x] /; FreeQ[{a, b, c, d}, x] && NeQ[b*c - a*d, 0]
  37. Int[Times[Power[Plus[Optional[Pattern[a, Blank[]]], Times[Optional[Pattern[b, Blank[]]], Pattern[x, Blank[]]]], Optional[Pattern[m, Blank[]]]], Power[Plus[Optional[Pattern[c, Blank[]]], Times[Optional[Pattern[d, Blank[]]], Pattern[x, Blank[]]]], Pattern[n, Blank[]]]], Pattern[x, Blank[Symbol]]] := Simp[((a + b*x)^(m + 1)*(c + d*x)^(n + 1))/((b*c - a*d)*(m + 1)), x] /; FreeQ[{a, b, c, d, m, n}, x] && NeQ[b*c - a*d, 0] && EqQ[m + n + 2, 0] && NeQ[m, -1]
  38. Int[Times[Power[Plus[Pattern[a, Blank[]], Times[Optional[Pattern[b, Blank[]]], Pattern[x, Blank[]]]], Pattern[m, Blank[]]], Power[Plus[Pattern[c, Blank[]], Times[Optional[Pattern[d, Blank[]]], Pattern[x, Blank[]]]], Pattern[m, Blank[]]]], Pattern[x, Blank[Symbol]]] := Simp[(x*(a + b*x)^m*(c + d*x)^m)/(2*m + 1), x] + Dist[(2*a*c*m)/(2*m + 1), Int[(a + b*x)^(m - 1)*(c + d*x)^(m - 1), x], x] /; FreeQ[{a, b, c, d}, x] && EqQ[b*c + a*d, 0] && IGtQ[m + 1/2, 0]
  39. Int[Times[Power[Plus[Pattern[a, Blank[]], Times[Optional[Pattern[b, Blank[]]], Pattern[x, Blank[]]]], Rational[-3, 2]], Power[Plus[Pattern[c, Blank[]], Times[Optional[Pattern[d, Blank[]]], Pattern[x, Blank[]]]], Rational[-3, 2]]], Pattern[x, Blank[Symbol]]] := Simp[x/(a*c*Sqrt[a + b*x]*Sqrt[c + d*x]), x] /; FreeQ[{a, b, c, d}, x] && EqQ[b*c + a*d, 0]
  40. Int[Times[Power[Plus[Pattern[a, Blank[]], Times[Optional[Pattern[b, Blank[]]], Pattern[x, Blank[]]]], Pattern[m, Blank[]]], Power[Plus[Pattern[c, Blank[]], Times[Optional[Pattern[d, Blank[]]], Pattern[x, Blank[]]]], Pattern[m, Blank[]]]], Pattern[x, Blank[Symbol]]] := -Simp[(x*(a + b*x)^(m + 1)*(c + d*x)^(m + 1))/(2*a*c*(m + 1)), x] + Dist[(2*m + 3)/(2*a*c*(m + 1)), Int[(a + b*x)^(m + 1)*(c + d*x)^(m + 1), x], x] /; FreeQ[{a, b, c, d}, x] && EqQ[b*c + a*d, 0] && ILtQ[m + 3/2, 0]
  41. Int[Times[Power[Plus[Pattern[a, Blank[]], Times[Optional[Pattern[b, Blank[]]], Pattern[x, Blank[]]]], Optional[Pattern[m, Blank[]]]], Power[Plus[Pattern[c, Blank[]], Times[Optional[Pattern[d, Blank[]]], Pattern[x, Blank[]]]], Optional[Pattern[m, Blank[]]]]], Pattern[x, Blank[Symbol]]] := Int[(a*c + b*d*x^2)^m, x] /; FreeQ[{a, b, c, d, m}, x] && EqQ[b*c + a*d, 0] && (IntegerQ[m] || (GtQ[a, 0] && GtQ[c, 0]))
  42. Int[Times[Power[Plus[Pattern[a, Blank[]], Times[Optional[Pattern[b, Blank[]]], Pattern[x, Blank[]]]], Pattern[m, Blank[]]], Power[Plus[Pattern[c, Blank[]], Times[Optional[Pattern[d, Blank[]]], Pattern[x, Blank[]]]], Pattern[m, Blank[]]]], Pattern[x, Blank[Symbol]]] := Dist[((a + b*x)^FracPart[m]*(c + d*x)^FracPart[m])/(a*c + b*d*x^2)^FracPart[m], Int[(a*c + b*d*x^2)^m, x], x] /; FreeQ[{a, b, c, d, m}, x] && EqQ[b*c + a*d, 0] && !IntegerQ[2*m]
  43. Int[Times[Power[Plus[Optional[Pattern[a, Blank[]]], Times[Optional[Pattern[b, Blank[]]], Pattern[x, Blank[]]]], Optional[Pattern[m, Blank[]]]], Power[Plus[Optional[Pattern[c, Blank[]]], Times[Optional[Pattern[d, Blank[]]], Pattern[x, Blank[]]]], Optional[Pattern[n, Blank[]]]]], Pattern[x, Blank[Symbol]]] := Int[ExpandIntegrand[(a + b*x)^m*(c + d*x)^n, x], x] /; FreeQ[{a, b, c, d, n}, x] && NeQ[b*c - a*d, 0] && IGtQ[m, 0] && ( !IntegerQ[n] || (EqQ[c, 0] && LeQ[7*m + 4*n + 4, 0]) || LtQ[9*m + 5*(n + 1), 0] || GtQ[m + n + 2, 0])
  44. Int[Times[Power[Plus[Pattern[a, Blank[]], Times[Optional[Pattern[b, Blank[]]], Pattern[x, Blank[]]]], Optional[Pattern[m, Blank[]]]], Power[Plus[Optional[Pattern[c, Blank[]]], Times[Optional[Pattern[d, Blank[]]], Pattern[x, Blank[]]]], Optional[Pattern[n, Blank[]]]]], Pattern[x, Blank[Symbol]]] := Int[ExpandIntegrand[(a + b*x)^m*(c + d*x)^n, x], x] /; FreeQ[{a, b, c, d}, x] && NeQ[b*c - a*d, 0] && ILtQ[m, 0] && IntegerQ[n] && !(IGtQ[n, 0] && LtQ[m + n + 2, 0])
  45. Int[Times[Power[Plus[Optional[Pattern[a, Blank[]]], Times[Optional[Pattern[b, Blank[]]], Pattern[x, Blank[]]]], Pattern[m, Blank[]]], Power[Plus[Optional[Pattern[c, Blank[]]], Times[Optional[Pattern[d, Blank[]]], Pattern[x, Blank[]]]], Pattern[n, Blank[]]]], Pattern[x, Blank[Symbol]]] := Simp[((a + b*x)^(m + 1)*(c + d*x)^(n + 1))/((b*c - a*d)*(m + 1)), x] - Dist[(d*Simplify[m + n + 2])/((b*c - a*d)*(m + 1)), Int[(a + b*x)^Simplify[m + 1]*(c + d*x)^n, x], x] /; FreeQ[{a, b, c, d, m, n}, x] && NeQ[b*c - a*d, 0] && ILtQ[Simplify[m + n + 2], 0] && NeQ[m, -1] && !(LtQ[m, -1] && LtQ[n, -1] && (EqQ[a, 0] || (NeQ[c, 0] && LtQ[m - n, 0] && IntegerQ[n]))) && (SumSimplerQ[m, 1] || !SumSimplerQ[n, 1])
  46. Int[Times[Power[Plus[Pattern[a, Blank[]], Times[Optional[Pattern[b, Blank[]]], Pattern[x, Blank[]]]], Rational[-9, 4]], Power[Plus[Pattern[c, Blank[]], Times[Optional[Pattern[d, Blank[]]], Pattern[x, Blank[]]]], Rational[-1, 4]]], Pattern[x, Blank[Symbol]]] := Simp[-4/(5*b*(a + b*x)^(5/4)*(c + d*x)^(1/4)), x] - Dist[d/(5*b), Int[1/((a + b*x)^(5/4)*(c + d*x)^(5/4)), x], x] /; FreeQ[{a, b, c, d}, x] && EqQ[b*c + a*d, 0] && NegQ[a^2*b^2]
  47. Int[Times[Power[Plus[Optional[Pattern[a, Blank[]]], Times[Optional[Pattern[b, Blank[]]], Pattern[x, Blank[]]]], Pattern[m, Blank[]]], Power[Plus[Optional[Pattern[c, Blank[]]], Times[Optional[Pattern[d, Blank[]]], Pattern[x, Blank[]]]], Pattern[n, Blank[]]]], Pattern[x, Blank[Symbol]]] := Simp[((a + b*x)^(m + 1)*(c + d*x)^n)/(b*(m + 1)), x] - Dist[(d*n)/(b*(m + 1)), Int[(a + b*x)^(m + 1)*(c + d*x)^(n - 1), x], x] /; FreeQ[{a, b, c, d}, x] && NeQ[b*c - a*d, 0] && GtQ[n, 0] && LtQ[m, -1] && !(IntegerQ[n] && !IntegerQ[m]) && !(ILeQ[m + n + 2, 0] && (FractionQ[m] || GeQ[2*n + m + 1, 0])) && IntLinearQ[a, b, c, d, m, n, x]
  48. Int[Times[Power[Plus[Pattern[a, Blank[]], Times[Optional[Pattern[b, Blank[]]], Pattern[x, Blank[]]]], Rational[-5, 4]], Power[Plus[Pattern[c, Blank[]], Times[Optional[Pattern[d, Blank[]]], Pattern[x, Blank[]]]], Rational[-1, 4]]], Pattern[x, Blank[Symbol]]] := Simp[-2/(b*(a + b*x)^(1/4)*(c + d*x)^(1/4)), x] + Dist[c, Int[1/((a + b*x)^(5/4)*(c + d*x)^(5/4)), x], x] /; FreeQ[{a, b, c, d}, x] && EqQ[b*c + a*d, 0] && NegQ[a^2*b^2]
  49. Int[Times[Power[Plus[Pattern[a, Blank[]], Times[Optional[Pattern[b, Blank[]]], Pattern[x, Blank[]]]], Pattern[m, Blank[]]], Power[Plus[Pattern[c, Blank[]], Times[Optional[Pattern[d, Blank[]]], Pattern[x, Blank[]]]], Pattern[n, Blank[]]]], Pattern[x, Blank[Symbol]]] := Simp[((a + b*x)^(m + 1)*(c + d*x)^n)/(b*(m + n + 1)), x] + Dist[(2*c*n)/(m + n + 1), Int[(a + b*x)^m*(c + d*x)^(n - 1), x], x] /; FreeQ[{a, b, c, d}, x] && EqQ[b*c + a*d, 0] && IGtQ[m + 1/2, 0] && IGtQ[n + 1/2, 0] && LtQ[m, n]
  50. Int[Times[Power[Plus[Optional[Pattern[a, Blank[]]], Times[Optional[Pattern[b, Blank[]]], Pattern[x, Blank[]]]], Pattern[m, Blank[]]], Power[Plus[Optional[Pattern[c, Blank[]]], Times[Optional[Pattern[d, Blank[]]], Pattern[x, Blank[]]]], Pattern[n, Blank[]]]], Pattern[x, Blank[Symbol]]] := Simp[((a + b*x)^(m + 1)*(c + d*x)^n)/(b*(m + n + 1)), x] + Dist[(n*(b*c - a*d))/(b*(m + n + 1)), Int[(a + b*x)^m*(c + d*x)^(n - 1), x], x] /; FreeQ[{a, b, c, d}, x] && NeQ[b*c - a*d, 0] && GtQ[n, 0] && NeQ[m + n + 1, 0] && !(IGtQ[m, 0] && ( !IntegerQ[n] || (GtQ[m, 0] && LtQ[m - n, 0]))) && !ILtQ[m + n + 2, 0] && IntLinearQ[a, b, c, d, m, n, x]
  51. Int[Times[Power[Plus[Optional[Pattern[a, Blank[]]], Times[Optional[Pattern[b, Blank[]]], Pattern[x, Blank[]]]], Pattern[m, Blank[]]], Power[Plus[Optional[Pattern[c, Blank[]]], Times[Optional[Pattern[d, Blank[]]], Pattern[x, Blank[]]]], Pattern[n, Blank[]]]], Pattern[x, Blank[Symbol]]] := Simp[((a + b*x)^(m + 1)*(c + d*x)^(n + 1))/((b*c - a*d)*(m + 1)), x] - Dist[(d*(m + n + 2))/((b*c - a*d)*(m + 1)), Int[(a + b*x)^(m + 1)*(c + d*x)^n, x], x] /; FreeQ[{a, b, c, d, n}, x] && NeQ[b*c - a*d, 0] && LtQ[m, -1] && !(LtQ[n, -1] && (EqQ[a, 0] || (NeQ[c, 0] && LtQ[m - n, 0] && IntegerQ[n]))) && IntLinearQ[a, b, c, d, m, n, x]
  52. Int[Times[Power[Plus[Pattern[a, Blank[]], Times[Optional[Pattern[b, Blank[]]], Pattern[x, Blank[]]]], Rational[-1, 2]], Power[Plus[Pattern[c, Blank[]], Times[Optional[Pattern[d, Blank[]]], Pattern[x, Blank[]]]], Rational[-1, 2]]], Pattern[x, Blank[Symbol]]] := Simp[ArcCosh[(b*x)/a]/b, x] /; FreeQ[{a, b, c, d}, x] && EqQ[a + c, 0] && EqQ[b - d, 0] && GtQ[a, 0]
  53. Int[Times[Power[Plus[Pattern[a, Blank[]], Times[Optional[Pattern[b, Blank[]]], Pattern[x, Blank[]]]], Rational[-1, 2]], Power[Plus[Optional[Pattern[c, Blank[]]], Times[Optional[Pattern[d, Blank[]]], Pattern[x, Blank[]]]], Rational[-1, 2]]], Pattern[x, Blank[Symbol]]] := Int[1/Sqrt[a*c - b*(a - c)*x - b^2*x^2], x] /; FreeQ[{a, b, c, d}, x] && EqQ[b + d, 0] && GtQ[a + c, 0]
  54. Int[Times[Power[Plus[Optional[Pattern[a, Blank[]]], Times[Optional[Pattern[b, Blank[]]], Pattern[x, Blank[]]]], Rational[-1, 2]], Power[Plus[Optional[Pattern[c, Blank[]]], Times[Optional[Pattern[d, Blank[]]], Pattern[x, Blank[]]]], Rational[-1, 2]]], Pattern[x, Blank[Symbol]]] := Dist[2/Sqrt[b], Subst[Int[1/Sqrt[b*c - a*d + d*x^2], x], x, Sqrt[a + b*x]], x] /; FreeQ[{a, b, c, d}, x] && GtQ[b*c - a*d, 0] && GtQ[b, 0]
  55. Int[Times[Power[Plus[Optional[Pattern[a, Blank[]]], Times[Optional[Pattern[b, Blank[]]], Pattern[x, Blank[]]]], -1], Power[Plus[Optional[Pattern[c, Blank[]]], Times[Optional[Pattern[d, Blank[]]], Pattern[x, Blank[]]]], Rational[-1, 3]]], Pattern[x, Blank[Symbol]]] := With[{q = Rt[(b*c - a*d)/b, 3]}, -Simp[Log[RemoveContent[a + b*x, x]]/(2*b*q), x] + (Dist[3/(2*b), Subst[Int[1/(q^2 + q*x + x^2), x], x, (c + d*x)^(1/3)], x] - Dist[3/(2*b*q), Subst[Int[1/(q - x), x], x, (c + d*x)^(1/3)], x])] /; FreeQ[{a, b, c, d}, x] && PosQ[(b*c - a*d)/b]
  56. Int[Times[Power[Plus[Optional[Pattern[a, Blank[]]], Times[Optional[Pattern[b, Blank[]]], Pattern[x, Blank[]]]], -1], Power[Plus[Optional[Pattern[c, Blank[]]], Times[Optional[Pattern[d, Blank[]]], Pattern[x, Blank[]]]], Rational[-1, 3]]], Pattern[x, Blank[Symbol]]] := With[{q = Rt[-((b*c - a*d)/b), 3]}, Simp[Log[RemoveContent[a + b*x, x]]/(2*b*q), x] + (Dist[3/(2*b), Subst[Int[1/(q^2 - q*x + x^2), x], x, (c + d*x)^(1/3)], x] - Dist[3/(2*b*q), Subst[Int[1/(q + x), x], x, (c + d*x)^(1/3)], x])] /; FreeQ[{a, b, c, d}, x] && NegQ[(b*c - a*d)/b]
  57. Int[Times[Power[Plus[Optional[Pattern[a, Blank[]]], Times[Optional[Pattern[b, Blank[]]], Pattern[x, Blank[]]]], -1], Power[Plus[Optional[Pattern[c, Blank[]]], Times[Optional[Pattern[d, Blank[]]], Pattern[x, Blank[]]]], Rational[-2, 3]]], Pattern[x, Blank[Symbol]]] := With[{q = Rt[(b*c - a*d)/b, 3]}, -Simp[Log[RemoveContent[a + b*x, x]]/(2*b*q^2), x] + (-Dist[3/(2*b*q), Subst[Int[1/(q^2 + q*x + x^2), x], x, (c + d*x)^(1/3)], x] - Dist[3/(2*b*q^2), Subst[Int[1/(q - x), x], x, (c + d*x)^(1/3)], x])] /; FreeQ[{a, b, c, d}, x] && PosQ[(b*c - a*d)/b]
  58. Int[Times[Power[Plus[Optional[Pattern[a, Blank[]]], Times[Optional[Pattern[b, Blank[]]], Pattern[x, Blank[]]]], -1], Power[Plus[Optional[Pattern[c, Blank[]]], Times[Optional[Pattern[d, Blank[]]], Pattern[x, Blank[]]]], Rational[-2, 3]]], Pattern[x, Blank[Symbol]]] := With[{q = Rt[-((b*c - a*d)/b), 3]}, -Simp[Log[RemoveContent[a + b*x, x]]/(2*b*q^2), x] + (Dist[3/(2*b*q), Subst[Int[1/(q^2 - q*x + x^2), x], x, (c + d*x)^(1/3)], x] + Dist[3/(2*b*q^2), Subst[Int[1/(q + x), x], x, (c + d*x)^(1/3)], x])] /; FreeQ[{a, b, c, d}, x] && NegQ[(b*c - a*d)/b]
  59. Int[Times[Power[Plus[Optional[Pattern[a, Blank[]]], Times[Optional[Pattern[b, Blank[]]], Pattern[x, Blank[]]]], Rational[-1, 3]], Power[Plus[Optional[Pattern[c, Blank[]]], Times[Optional[Pattern[d, Blank[]]], Pattern[x, Blank[]]]], Rational[-2, 3]]], Pattern[x, Blank[Symbol]]] := With[{q = Rt[d/b, 3]}, -Simp[(Sqrt[3]*q*ArcTan[(2*q*(a + b*x)^(1/3))/(Sqrt[3]*(c + d*x)^(1/3)) + 1/Sqrt[3]])/d, x] + (-Simp[(3*q*Log[(q*(a + b*x)^(1/3))/(c + d*x)^(1/3) - 1])/(2*d), x] - Simp[(q*Log[c + d*x])/(2*d), x])] /; FreeQ[{a, b, c, d}, x] && NeQ[b*c - a*d, 0] && PosQ[d/b]
  60. Int[Times[Power[Plus[Optional[Pattern[a, Blank[]]], Times[Optional[Pattern[b, Blank[]]], Pattern[x, Blank[]]]], Rational[-1, 3]], Power[Plus[Optional[Pattern[c, Blank[]]], Times[Optional[Pattern[d, Blank[]]], Pattern[x, Blank[]]]], Rational[-2, 3]]], Pattern[x, Blank[Symbol]]] := With[{q = Rt[-(d/b), 3]}, Simp[(Sqrt[3]*q*ArcTan[1/Sqrt[3] - (2*q*(a + b*x)^(1/3))/(Sqrt[3]*(c + d*x)^(1/3))])/d, x] + (Simp[(3*q*Log[(q*(a + b*x)^(1/3))/(c + d*x)^(1/3) + 1])/(2*d), x] + Simp[(q*Log[c + d*x])/(2*d), x])] /; FreeQ[{a, b, c, d}, x] && NeQ[b*c - a*d, 0] && NegQ[d/b]
  61. Int[Times[Power[Plus[Optional[Pattern[a, Blank[]]], Times[Optional[Pattern[b, Blank[]]], Pattern[x, Blank[]]]], Pattern[m, Blank[]]], Power[Plus[Pattern[c, Blank[]], Times[Optional[Pattern[d, Blank[]]], Pattern[x, Blank[]]]], Pattern[m, Blank[]]]], Pattern[x, Blank[Symbol]]] := Dist[((a + b*x)^m*(c + d*x)^m)/(a*c + (b*c + a*d)*x + b*d*x^2)^m, Int[(a*c + (b*c + a*d)*x + b*d*x^2)^m, x], x] /; FreeQ[{a, b, c, d}, x] && NeQ[b*c - a*d, 0] && LtQ[-1, m, 0] && LeQ[3, Denominator[m], 4] && AtomQ[b*c + a*d]
  62. Int[Times[Power[Plus[Optional[Pattern[a, Blank[]]], Times[Optional[Pattern[b, Blank[]]], Pattern[x, Blank[]]]], Pattern[m, Blank[]]], Power[Plus[Pattern[c, Blank[]], Times[Optional[Pattern[d, Blank[]]], Pattern[x, Blank[]]]], Pattern[m, Blank[]]]], Pattern[x, Blank[Symbol]]] := Dist[((a + b*x)^m*(c + d*x)^m)/((a + b*x)*(c + d*x))^m, Int[(a*c + (b*c + a*d)*x + b*d*x^2)^m, x], x] /; FreeQ[{a, b, c, d}, x] && NeQ[b*c - a*d, 0] && LtQ[-1, m, 0] && LeQ[3, Denominator[m], 4]
  63. Int[Times[Power[Plus[Optional[Pattern[a, Blank[]]], Times[Optional[Pattern[b, Blank[]]], Pattern[x, Blank[]]]], Pattern[m, Blank[]]], Power[Plus[Optional[Pattern[c, Blank[]]], Times[Optional[Pattern[d, Blank[]]], Pattern[x, Blank[]]]], Pattern[n, Blank[]]]], Pattern[x, Blank[Symbol]]] := With[{p = Denominator[m]}, Dist[p/b, Subst[Int[x^(p*(m + 1) - 1)*(c - (a*d)/b + (d*x^p)/b)^n, x], x, (a + b*x)^(1/p)], x]] /; FreeQ[{a, b, c, d}, x] && NeQ[b*c - a*d, 0] && LtQ[-1, m, 0] && LeQ[-1, n, 0] && LeQ[Denominator[n], Denominator[m]] && IntLinearQ[a, b, c, d, m, n, x]
  64. Int[Times[Power[Times[Optional[Pattern[b, Blank[]]], Pattern[x, Blank[]]], Pattern[m, Blank[]]], Power[Plus[Pattern[c, Blank[]], Times[Optional[Pattern[d, Blank[]]], Pattern[x, Blank[]]]], Pattern[n, Blank[]]]], Pattern[x, Blank[Symbol]]] := Simp[(c^n*(b*x)^(m + 1)*Hypergeometric2F1[-n, m + 1, m + 2, -((d*x)/c)])/(b*(m + 1)), x] /; FreeQ[{b, c, d, m, n}, x] && !IntegerQ[m] && (IntegerQ[n] || (GtQ[c, 0] && !(EqQ[n, -2^(-1)] && EqQ[c^2 - d^2, 0] && GtQ[-(d/(b*c)), 0])))
  65. Int[Times[Power[Times[Optional[Pattern[b, Blank[]]], Pattern[x, Blank[]]], Pattern[m, Blank[]]], Power[Plus[Pattern[c, Blank[]], Times[Optional[Pattern[d, Blank[]]], Pattern[x, Blank[]]]], Pattern[n, Blank[]]]], Pattern[x, Blank[Symbol]]] := Simp[((c + d*x)^(n + 1)*Hypergeometric2F1[-m, n + 1, n + 2, 1 + (d*x)/c])/(d*(n + 1)*(-(d/(b*c)))^m), x] /; FreeQ[{b, c, d, m, n}, x] && !IntegerQ[n] && (IntegerQ[m] || GtQ[-(d/(b*c)), 0])
  66. Int[Times[Power[Times[Optional[Pattern[b, Blank[]]], Pattern[x, Blank[]]], Pattern[m, Blank[]]], Power[Plus[Pattern[c, Blank[]], Times[Optional[Pattern[d, Blank[]]], Pattern[x, Blank[]]]], Pattern[n, Blank[]]]], Pattern[x, Blank[Symbol]]] := Dist[(c^IntPart[n]*(c + d*x)^FracPart[n])/(1 + (d*x)/c)^FracPart[n], Int[(b*x)^m*(1 + (d*x)/c)^n, x], x] /; FreeQ[{b, c, d, m, n}, x] && !IntegerQ[m] && !IntegerQ[n] && !GtQ[c, 0] && !GtQ[-(d/(b*c)), 0] && ((RationalQ[m] && !(EqQ[n, -2^(-1)] && EqQ[c^2 - d^2, 0])) || !RationalQ[n])
  67. Int[Times[Power[Times[Optional[Pattern[b, Blank[]]], Pattern[x, Blank[]]], Pattern[m, Blank[]]], Power[Plus[Pattern[c, Blank[]], Times[Optional[Pattern[d, Blank[]]], Pattern[x, Blank[]]]], Pattern[n, Blank[]]]], Pattern[x, Blank[Symbol]]] := Dist[((-((b*c)/d))^IntPart[m]*(b*x)^FracPart[m])/(-((d*x)/c))^FracPart[m], Int[(-((d*x)/c))^m*(c + d*x)^n, x], x] /; FreeQ[{b, c, d, m, n}, x] && !IntegerQ[m] && !IntegerQ[n] && !GtQ[c, 0] && !GtQ[-(d/(b*c)), 0]
  68. Int[Times[Power[Plus[Pattern[a, Blank[]], Times[Optional[Pattern[b, Blank[]]], Pattern[x, Blank[]]]], Pattern[m, Blank[]]], Power[Plus[Pattern[c, Blank[]], Times[Optional[Pattern[d, Blank[]]], Pattern[x, Blank[]]]], Pattern[n, Blank[]]]], Pattern[x, Blank[Symbol]]] := Simp[((b*c - a*d)^n*(a + b*x)^(m + 1)*Hypergeometric2F1[-n, m + 1, m + 2, -((d*(a + b*x))/(b*c - a*d))])/(b^(n + 1)*(m + 1)), x] /; FreeQ[{a, b, c, d, m}, x] && NeQ[b*c - a*d, 0] && !IntegerQ[m] && IntegerQ[n]
  69. Int[Times[Power[Plus[Pattern[a, Blank[]], Times[Optional[Pattern[b, Blank[]]], Pattern[x, Blank[]]]], Pattern[m, Blank[]]], Power[Plus[Pattern[c, Blank[]], Times[Optional[Pattern[d, Blank[]]], Pattern[x, Blank[]]]], Pattern[n, Blank[]]]], Pattern[x, Blank[Symbol]]] := Simp[((a + b*x)^(m + 1)*Hypergeometric2F1[-n, m + 1, m + 2, -((d*(a + b*x))/(b*c - a*d))])/(b*(m + 1)*(b/(b*c - a*d))^n), x] /; FreeQ[{a, b, c, d, m, n}, x] && NeQ[b*c - a*d, 0] && !IntegerQ[m] && !IntegerQ[n] && GtQ[b/(b*c - a*d), 0] && (RationalQ[m] || !(RationalQ[n] && GtQ[-(d/(b*c - a*d)), 0]))
  70. Int[Times[Power[Plus[Pattern[a, Blank[]], Times[Optional[Pattern[b, Blank[]]], Pattern[x, Blank[]]]], Pattern[m, Blank[]]], Power[Plus[Pattern[c, Blank[]], Times[Optional[Pattern[d, Blank[]]], Pattern[x, Blank[]]]], Pattern[n, Blank[]]]], Pattern[x, Blank[Symbol]]] := Dist[(c + d*x)^FracPart[n]/((b/(b*c - a*d))^IntPart[n]*((b*(c + d*x))/(b*c - a*d))^FracPart[n]), Int[(a + b*x)^m*Simp[(b*c)/(b*c - a*d) + (b*d*x)/(b*c - a*d), x]^n, x], x] /; FreeQ[{a, b, c, d, m, n}, x] && NeQ[b*c - a*d, 0] && !IntegerQ[m] && !IntegerQ[n] && (RationalQ[m] || !SimplerQ[n + 1, m + 1])
  71. Int[Times[Power[Plus[Optional[Pattern[a, Blank[]]], Times[Optional[Pattern[b, Blank[]]], Pattern[u, Blank[]]]], Optional[Pattern[m, Blank[]]]], Power[Plus[Optional[Pattern[c, Blank[]]], Times[Optional[Pattern[d, Blank[]]], Pattern[u, Blank[]]]], Optional[Pattern[n, Blank[]]]]], Pattern[x, Blank[Symbol]]] := Dist[1/Coefficient[u, x, 1], Subst[Int[(a + b*x)^m*(c + d*x)^n, x], x, u], x] /; FreeQ[{a, b, c, d, m, n}, x] && LinearQ[u, x] && NeQ[Coefficient[u, x, 0], 0]
  72. Int[Times[Power[Plus[Optional[Pattern[a, Blank[]]], Times[Optional[Pattern[b, Blank[]]], Pattern[x, Blank[]]]], -1], Power[Plus[Optional[Pattern[c, Blank[]]], Times[Optional[Pattern[d, Blank[]]], Pattern[x, Blank[]]]], -1], Power[Plus[Optional[Pattern[e, Blank[]]], Times[Optional[Pattern[f, Blank[]]], Pattern[x, Blank[]]]], Optional[Pattern[p, Blank[]]]]], Pattern[x, Blank[Symbol]]] := Int[ExpandIntegrand[(e + f*x)^p/((a + b*x)*(c + d*x)), x], x] /; FreeQ[{a, b, c, d, e, f}, x] && IntegerQ[p]
  73. Int[Times[Power[Plus[Pattern[a, Blank[]], Times[Optional[Pattern[b, Blank[]]], Pattern[x, Blank[]]]], Optional[Pattern[m, Blank[]]]], Power[Plus[Pattern[c, Blank[]], Times[Optional[Pattern[d, Blank[]]], Pattern[x, Blank[]]]], Optional[Pattern[n, Blank[]]]], Power[Plus[Optional[Pattern[e, Blank[]]], Times[Optional[Pattern[f, Blank[]]], Pattern[x, Blank[]]]], Optional[Pattern[p, Blank[]]]]], Pattern[x, Blank[Symbol]]] := Int[(a*c + b*d*x^2)^m*(e + f*x)^p, x] /; FreeQ[{a, b, c, d, e, f, m, n, p}, x] && EqQ[b*c + a*d, 0] && EqQ[n, m] && IntegerQ[m]
  74. Int[Times[Plus[Optional[Pattern[a, Blank[]]], Times[Optional[Pattern[b, Blank[]]], Pattern[x, Blank[]]]], Power[Plus[Optional[Pattern[c, Blank[]]], Times[Optional[Pattern[d, Blank[]]], Pattern[x, Blank[]]]], Optional[Pattern[n, Blank[]]]], Power[Plus[Optional[Pattern[e, Blank[]]], Times[Optional[Pattern[f, Blank[]]], Pattern[x, Blank[]]]], Optional[Pattern[p, Blank[]]]]], Pattern[x, Blank[Symbol]]] := Simp[(b*(c + d*x)^(n + 1)*(e + f*x)^(p + 1))/(d*f*(n + p + 2)), x] /; FreeQ[{a, b, c, d, e, f, n, p}, x] && NeQ[n + p + 2, 0] && EqQ[a*d*f*(n + p + 2) - b*(d*e*(n + 1) + c*f*(p + 1)), 0]
  75. Int[Times[Power[Times[Optional[Pattern[d, Blank[]]], Pattern[x, Blank[]]], Optional[Pattern[n, Blank[]]]], Plus[Pattern[a, Blank[]], Times[Optional[Pattern[b, Blank[]]], Pattern[x, Blank[]]]], Power[Plus[Pattern[e, Blank[]], Times[Optional[Pattern[f, Blank[]]], Pattern[x, Blank[]]]], Optional[Pattern[p, Blank[]]]]], Pattern[x, Blank[Symbol]]] := Int[ExpandIntegrand[(a + b*x)*(d*x)^n*(e + f*x)^p, x], x] /; FreeQ[{a, b, d, e, f, n}, x] && IGtQ[p, 0] && EqQ[b*e + a*f, 0] && !(ILtQ[n + p + 2, 0] && GtQ[n + 2*p, 0])
  76. Int[Times[Power[Times[Optional[Pattern[d, Blank[]]], Pattern[x, Blank[]]], Optional[Pattern[n, Blank[]]]], Plus[Pattern[a, Blank[]], Times[Optional[Pattern[b, Blank[]]], Pattern[x, Blank[]]]], Power[Plus[Pattern[e, Blank[]], Times[Optional[Pattern[f, Blank[]]], Pattern[x, Blank[]]]], Optional[Pattern[p, Blank[]]]]], Pattern[x, Blank[Symbol]]] := Int[ExpandIntegrand[(a + b*x)*(d*x)^n*(e + f*x)^p, x], x] /; FreeQ[{a, b, d, e, f, n}, x] && IGtQ[p, 0] && (NeQ[n, -1] || EqQ[p, 1]) && NeQ[b*e + a*f, 0] && ( !IntegerQ[n] || LtQ[9*p + 5*n, 0] || GeQ[n + p + 1, 0] || (GeQ[n + p + 2, 0] && RationalQ[a, b, d, e, f])) && (NeQ[n + p + 3, 0] || EqQ[p, 1])
  77. Int[Times[Plus[Optional[Pattern[a, Blank[]]], Times[Optional[Pattern[b, Blank[]]], Pattern[x, Blank[]]]], Power[Plus[Pattern[c, Blank[]], Times[Optional[Pattern[d, Blank[]]], Pattern[x, Blank[]]]], Optional[Pattern[n, Blank[]]]], Power[Plus[Optional[Pattern[e, Blank[]]], Times[Optional[Pattern[f, Blank[]]], Pattern[x, Blank[]]]], Optional[Pattern[p, Blank[]]]]], Pattern[x, Blank[Symbol]]] := Int[ExpandIntegrand[(a + b*x)*(c + d*x)^n*(e + f*x)^p, x], x] /; FreeQ[{a, b, c, d, e, f, n}, x] && NeQ[b*c - a*d, 0] && ((ILtQ[n, 0] && ILtQ[p, 0]) || EqQ[p, 1] || (IGtQ[p, 0] && ( !IntegerQ[n] || LeQ[9*p + 5*(n + 2), 0] || GeQ[n + p + 1, 0] || (GeQ[n + p + 2, 0] && RationalQ[a, b, c, d, e, f]))))
  78. Int[Times[Plus[Optional[Pattern[a, Blank[]]], Times[Optional[Pattern[b, Blank[]]], Pattern[x, Blank[]]]], Power[Plus[Optional[Pattern[c, Blank[]]], Times[Optional[Pattern[d, Blank[]]], Pattern[x, Blank[]]]], Optional[Pattern[n, Blank[]]]], Power[Plus[Optional[Pattern[e, Blank[]]], Times[Optional[Pattern[f, Blank[]]], Pattern[x, Blank[]]]], Optional[Pattern[p, Blank[]]]]], Pattern[x, Blank[Symbol]]] := -Simp[((b*e - a*f)*(c + d*x)^(n + 1)*(e + f*x)^(p + 1))/(f*(p + 1)*(c*f - d*e)), x] - Dist[(a*d*f*(n + p + 2) - b*(d*e*(n + 1) + c*f*(p + 1)))/(f*(p + 1)*(c*f - d*e)), Int[(c + d*x)^n*(e + f*x)^(p + 1), x], x] /; FreeQ[{a, b, c, d, e, f, n}, x] && LtQ[p, -1] && ( !LtQ[n, -1] || IntegerQ[p] || !(IntegerQ[n] || !(EqQ[e, 0] || !(EqQ[c, 0] || LtQ[p, n]))))
  79. Int[Times[Plus[Optional[Pattern[a, Blank[]]], Times[Optional[Pattern[b, Blank[]]], Pattern[x, Blank[]]]], Power[Plus[Optional[Pattern[c, Blank[]]], Times[Optional[Pattern[d, Blank[]]], Pattern[x, Blank[]]]], Optional[Pattern[n, Blank[]]]], Power[Plus[Optional[Pattern[e, Blank[]]], Times[Optional[Pattern[f, Blank[]]], Pattern[x, Blank[]]]], Optional[Pattern[p, Blank[]]]]], Pattern[x, Blank[Symbol]]] := -Simp[((b*e - a*f)*(c + d*x)^(n + 1)*(e + f*x)^(p + 1))/(f*(p + 1)*(c*f - d*e)), x] - Dist[(a*d*f*(n + p + 2) - b*(d*e*(n + 1) + c*f*(p + 1)))/(f*(p + 1)*(c*f - d*e)), Int[(c + d*x)^n*(e + f*x)^Simplify[p + 1], x], x] /; FreeQ[{a, b, c, d, e, f, n, p}, x] && !RationalQ[p] && SumSimplerQ[p, 1]
  80. Int[Times[Plus[Optional[Pattern[a, Blank[]]], Times[Optional[Pattern[b, Blank[]]], Pattern[x, Blank[]]]], Power[Plus[Optional[Pattern[c, Blank[]]], Times[Optional[Pattern[d, Blank[]]], Pattern[x, Blank[]]]], Optional[Pattern[n, Blank[]]]], Power[Plus[Optional[Pattern[e, Blank[]]], Times[Optional[Pattern[f, Blank[]]], Pattern[x, Blank[]]]], Optional[Pattern[p, Blank[]]]]], Pattern[x, Blank[Symbol]]] := Simp[(b*(c + d*x)^(n + 1)*(e + f*x)^(p + 1))/(d*f*(n + p + 2)), x] + Dist[(a*d*f*(n + p + 2) - b*(d*e*(n + 1) + c*f*(p + 1)))/(d*f*(n + p + 2)), Int[(c + d*x)^n*(e + f*x)^p, x], x] /; FreeQ[{a, b, c, d, e, f, n, p}, x] && NeQ[n + p + 2, 0]
  81. Int[Times[Power[Plus[Optional[Pattern[a, Blank[]]], Times[Optional[Pattern[b, Blank[]]], Pattern[x, Blank[]]]], 2], Power[Plus[Optional[Pattern[c, Blank[]]], Times[Optional[Pattern[d, Blank[]]], Pattern[x, Blank[]]]], Optional[Pattern[n, Blank[]]]], Power[Plus[Optional[Pattern[e, Blank[]]], Times[Optional[Pattern[f, Blank[]]], Pattern[x, Blank[]]]], Optional[Pattern[p, Blank[]]]]], Pattern[x, Blank[Symbol]]] := Simp[(b*(c + d*x)^(n + 1)*(e + f*x)^(p + 1)*(2*a*d*f*(n + p + 3) - b*(d*e*(n + 2) + c*f*(p + 2)) + b*d*f*(n + p + 2)*x))/(d^2*f^2*(n + p + 2)*(n + p + 3)), x] /; FreeQ[{a, b, c, d, e, f, n, p}, x] && NeQ[n + p + 2, 0] && NeQ[n + p + 3, 0] && EqQ[d*f*(n + p + 2)*(a^2*d*f*(n + p + 3) - b*(b*c*e + a*(d*e*(n + 1) + c*f*(p + 1)))) - b*(d*e*(n + 1) + c*f*(p + 1))*(a*d*f*(n + p + 4) - b*(d*e*(n + 2) + c*f*(p + 2))), 0]
  82. Int[Times[Power[Times[Optional[Pattern[f, Blank[]]], Pattern[x, Blank[]]], Optional[Pattern[p, Blank[]]]], Power[Plus[Optional[Pattern[a, Blank[]]], Times[Optional[Pattern[b, Blank[]]], Pattern[x, Blank[]]]], Optional[Pattern[m, Blank[]]]], Power[Plus[Optional[Pattern[c, Blank[]]], Times[Optional[Pattern[d, Blank[]]], Pattern[x, Blank[]]]], Optional[Pattern[n, Blank[]]]]], Pattern[x, Blank[Symbol]]] := Dist[a, Int[(a + b*x)^n*(c + d*x)^n*(f*x)^p, x], x] + Dist[b/f, Int[(a + b*x)^n*(c + d*x)^n*(f*x)^(p + 1), x], x] /; FreeQ[{a, b, c, d, f, m, n, p}, x] && EqQ[b*c + a*d, 0] && EqQ[m - n - 1, 0] && !RationalQ[p] && !IGtQ[m, 0] && NeQ[m + n + p + 2, 0]
  83. Int[Times[Power[Plus[Optional[Pattern[a, Blank[]]], Times[Optional[Pattern[b, Blank[]]], Pattern[x, Blank[]]]], -1], Power[Plus[Optional[Pattern[c, Blank[]]], Times[Optional[Pattern[d, Blank[]]], Pattern[x, Blank[]]]], -1], Power[Plus[Optional[Pattern[e, Blank[]]], Times[Optional[Pattern[f, Blank[]]], Pattern[x, Blank[]]]], Optional[Pattern[p, Blank[]]]]], Pattern[x, Blank[Symbol]]] := Dist[(b*e - a*f)/(b*c - a*d), Int[(e + f*x)^(p - 1)/(a + b*x), x], x] - Dist[(d*e - c*f)/(b*c - a*d), Int[(e + f*x)^(p - 1)/(c + d*x), x], x] /; FreeQ[{a, b, c, d, e, f}, x] && LtQ[0, p, 1]
  84. Int[Times[Power[Plus[Optional[Pattern[a, Blank[]]], Times[Optional[Pattern[b, Blank[]]], Pattern[x, Blank[]]]], -1], Power[Plus[Optional[Pattern[c, Blank[]]], Times[Optional[Pattern[d, Blank[]]], Pattern[x, Blank[]]]], -1], Power[Plus[Optional[Pattern[e, Blank[]]], Times[Optional[Pattern[f, Blank[]]], Pattern[x, Blank[]]]], Pattern[p, Blank[]]]], Pattern[x, Blank[Symbol]]] := Simp[(f*(e + f*x)^(p - 1))/(b*d*(p - 1)), x] + Dist[1/(b*d), Int[((b*d*e^2 - a*c*f^2 + f*(2*b*d*e - b*c*f - a*d*f)*x)*(e + f*x)^(p - 2))/((a + b*x)*(c + d*x)), x], x] /; FreeQ[{a, b, c, d, e, f}, x] && GtQ[p, 1]
  85. Int[Times[Power[Plus[Optional[Pattern[a, Blank[]]], Times[Optional[Pattern[b, Blank[]]], Pattern[x, Blank[]]]], -1], Power[Plus[Optional[Pattern[c, Blank[]]], Times[Optional[Pattern[d, Blank[]]], Pattern[x, Blank[]]]], -1], Power[Plus[Optional[Pattern[e, Blank[]]], Times[Optional[Pattern[f, Blank[]]], Pattern[x, Blank[]]]], Pattern[p, Blank[]]]], Pattern[x, Blank[Symbol]]] := Simp[(f*(e + f*x)^(p + 1))/((p + 1)*(b*e - a*f)*(d*e - c*f)), x] + Dist[1/((b*e - a*f)*(d*e - c*f)), Int[((b*d*e - b*c*f - a*d*f - b*d*f*x)*(e + f*x)^(p + 1))/((a + b*x)*(c + d*x)), x], x] /; FreeQ[{a, b, c, d, e, f}, x] && LtQ[p, -1]
  86. Int[Times[Power[Plus[Optional[Pattern[a, Blank[]]], Times[Optional[Pattern[b, Blank[]]], Pattern[x, Blank[]]]], -1], Power[Plus[Optional[Pattern[c, Blank[]]], Times[Optional[Pattern[d, Blank[]]], Pattern[x, Blank[]]]], -1], Power[Plus[Optional[Pattern[e, Blank[]]], Times[Optional[Pattern[f, Blank[]]], Pattern[x, Blank[]]]], Pattern[p, Blank[]]]], Pattern[x, Blank[Symbol]]] := Dist[b/(b*c - a*d), Int[(e + f*x)^p/(a + b*x), x], x] - Dist[d/(b*c - a*d), Int[(e + f*x)^p/(c + d*x), x], x] /; FreeQ[{a, b, c, d, e, f, p}, x] && !IntegerQ[p]
  87. Int[Times[Power[Plus[Optional[Pattern[a, Blank[]]], Times[Optional[Pattern[b, Blank[]]], Pattern[x, Blank[]]]], -1], Power[Plus[Optional[Pattern[c, Blank[]]], Times[Optional[Pattern[d, Blank[]]], Pattern[x, Blank[]]]], Optional[Pattern[n, Blank[]]]], Power[Plus[Optional[Pattern[e, Blank[]]], Times[Optional[Pattern[f, Blank[]]], Pattern[x, Blank[]]]], Pattern[p, Blank[]]]], Pattern[x, Blank[Symbol]]] := Int[ExpandIntegrand[(e + f*x)^FractionalPart[p], ((c + d*x)^n*(e + f*x)^IntegerPart[p])/(a + b*x), x], x] /; FreeQ[{a, b, c, d, e, f}, x] && IGtQ[n, 0] && LtQ[p, -1] && FractionQ[p]
  88. Int[Times[Power[Plus[Optional[Pattern[a, Blank[]]], Times[Optional[Pattern[b, Blank[]]], Pattern[x, Blank[]]]], Optional[Pattern[m, Blank[]]]], Power[Plus[Optional[Pattern[c, Blank[]]], Times[Optional[Pattern[d, Blank[]]], Pattern[x, Blank[]]]], Optional[Pattern[n, Blank[]]]], Power[Plus[Optional[Pattern[e, Blank[]]], Times[Optional[Pattern[f, Blank[]]], Pattern[x, Blank[]]]], Optional[Pattern[p, Blank[]]]]], Pattern[x, Blank[Symbol]]] := Int[ExpandIntegrand[(a + b*x)^m*(c + d*x)^n*(e + f*x)^p, x], x] /; FreeQ[{a, b, c, d, e, f, p}, x] && IntegersQ[m, n] && (IntegerQ[p] || (GtQ[m, 0] && GeQ[n, -1]))
  89. Int[Times[Power[Plus[Optional[Pattern[a, Blank[]]], Times[Optional[Pattern[b, Blank[]]], Pattern[x, Blank[]]]], 2], Power[Plus[Optional[Pattern[c, Blank[]]], Times[Optional[Pattern[d, Blank[]]], Pattern[x, Blank[]]]], Optional[Pattern[n, Blank[]]]], Power[Plus[Optional[Pattern[e, Blank[]]], Times[Optional[Pattern[f, Blank[]]], Pattern[x, Blank[]]]], Optional[Pattern[p, Blank[]]]]], Pattern[x, Blank[Symbol]]] := Simp[((b*c - a*d)^2*(c + d*x)^(n + 1)*(e + f*x)^(p + 1))/(d^2*(d*e - c*f)*(n + 1)), x] - Dist[1/(d^2*(d*e - c*f)*(n + 1)), Int[(c + d*x)^(n + 1)*(e + f*x)^p*Simp[a^2*d^2*f*(n + p + 2) + b^2*c*(d*e*(n + 1) + c*f*(p + 1)) - 2*a*b*d*(d*e*(n + 1) + c*f*(p + 1)) - b^2*d*(d*e - c*f)*(n + 1)*x, x], x], x] /; FreeQ[{a, b, c, d, e, f, n, p}, x] && (LtQ[n, -1] || (EqQ[n + p + 3, 0] && NeQ[n, -1] && (SumSimplerQ[n, 1] || !SumSimplerQ[p, 1])))
  90. Int[Times[Power[Plus[Optional[Pattern[a, Blank[]]], Times[Optional[Pattern[b, Blank[]]], Pattern[x, Blank[]]]], 2], Power[Plus[Optional[Pattern[c, Blank[]]], Times[Optional[Pattern[d, Blank[]]], Pattern[x, Blank[]]]], Optional[Pattern[n, Blank[]]]], Power[Plus[Optional[Pattern[e, Blank[]]], Times[Optional[Pattern[f, Blank[]]], Pattern[x, Blank[]]]], Optional[Pattern[p, Blank[]]]]], Pattern[x, Blank[Symbol]]] := Simp[(b*(a + b*x)*(c + d*x)^(n + 1)*(e + f*x)^(p + 1))/(d*f*(n + p + 3)), x] + Dist[1/(d*f*(n + p + 3)), Int[(c + d*x)^n*(e + f*x)^p*Simp[a^2*d*f*(n + p + 3) - b*(b*c*e + a*(d*e*(n + 1) + c*f*(p + 1))) + b*(a*d*f*(n + p + 4) - b*(d*e*(n + 2) + c*f*(p + 2)))*x, x], x], x] /; FreeQ[{a, b, c, d, e, f, n, p}, x] && NeQ[n + p + 3, 0]
  91. Int[Times[Power[Plus[Optional[Pattern[a, Blank[]]], Times[Optional[Pattern[b, Blank[]]], Pattern[x, Blank[]]]], Rational[-1, 3]], Power[Plus[Optional[Pattern[c, Blank[]]], Times[Optional[Pattern[d, Blank[]]], Pattern[x, Blank[]]]], Rational[-2, 3]], Power[Plus[Optional[Pattern[e, Blank[]]], Times[Optional[Pattern[f, Blank[]]], Pattern[x, Blank[]]]], -1]], Pattern[x, Blank[Symbol]]] := With[{q = Rt[(d*e - c*f)/(b*e - a*f), 3]}, -Simp[(Sqrt[3]*q*ArcTan[1/Sqrt[3] + (2*q*(a + b*x)^(1/3))/(Sqrt[3]*(c + d*x)^(1/3))])/(d*e - c*f), x] + (Simp[(q*Log[e + f*x])/(2*(d*e - c*f)), x] - Simp[(3*q*Log[q*(a + b*x)^(1/3) - (c + d*x)^(1/3)])/(2*(d*e - c*f)), x])] /; FreeQ[{a, b, c, d, e, f}, x]
  92. Int[Times[Power[Plus[Optional[Pattern[a, Blank[]]], Times[Optional[Pattern[b, Blank[]]], Pattern[x, Blank[]]]], Rational[-1, 2]], Power[Plus[Optional[Pattern[c, Blank[]]], Times[Optional[Pattern[d, Blank[]]], Pattern[x, Blank[]]]], Rational[-1, 2]], Power[Plus[Optional[Pattern[e, Blank[]]], Times[Optional[Pattern[f, Blank[]]], Pattern[x, Blank[]]]], -1]], Pattern[x, Blank[Symbol]]] := Dist[b*f, Subst[Int[1/(d*(b*e - a*f)^2 + b*f^2*x^2), x], x, Sqrt[a + b*x]*Sqrt[c + d*x]], x] /; FreeQ[{a, b, c, d, e, f}, x] && EqQ[2*b*d*e - f*(b*c + a*d), 0]
  93. Int[Times[Power[Plus[Optional[Pattern[a, Blank[]]], Times[Optional[Pattern[b, Blank[]]], Pattern[x, Blank[]]]], Pattern[m, Blank[]]], Power[Plus[Optional[Pattern[c, Blank[]]], Times[Optional[Pattern[d, Blank[]]], Pattern[x, Blank[]]]], Pattern[n, Blank[]]], Power[Plus[Optional[Pattern[e, Blank[]]], Times[Optional[Pattern[f, Blank[]]], Pattern[x, Blank[]]]], -1]], Pattern[x, Blank[Symbol]]] := With[{q = Denominator[m]}, Dist[q, Subst[Int[x^(q*(m + 1) - 1)/(b*e - a*f - (d*e - c*f)*x^q), x], x, (a + b*x)^(1/q)/(c + d*x)^(1/q)], x]] /; FreeQ[{a, b, c, d, e, f}, x] && EqQ[m + n + 1, 0] && RationalQ[n] && LtQ[-1, m, 0] && SimplerQ[a + b*x, c + d*x]
  94. Int[Times[Power[Plus[Optional[Pattern[a, Blank[]]], Times[Optional[Pattern[b, Blank[]]], Pattern[x, Blank[]]]], Pattern[m, Blank[]]], Power[Plus[Optional[Pattern[c, Blank[]]], Times[Optional[Pattern[d, Blank[]]], Pattern[x, Blank[]]]], Optional[Pattern[n, Blank[]]]], Power[Plus[Optional[Pattern[e, Blank[]]], Times[Optional[Pattern[f, Blank[]]], Pattern[x, Blank[]]]], Optional[Pattern[p, Blank[]]]]], Pattern[x, Blank[Symbol]]] := Simp[((a + b*x)^(m + 1)*(c + d*x)^n*(e + f*x)^(p + 1))/((m + 1)*(b*e - a*f)), x] - Dist[(n*(d*e - c*f))/((m + 1)*(b*e - a*f)), Int[(a + b*x)^(m + 1)*(c + d*x)^(n - 1)*(e + f*x)^p, x], x] /; FreeQ[{a, b, c, d, e, f, m, p}, x] && EqQ[m + n + p + 2, 0] && GtQ[n, 0] && !(SumSimplerQ[p, 1] && !SumSimplerQ[m, 1])
  95. Int[Times[Power[Plus[Optional[Pattern[a, Blank[]]], Times[Optional[Pattern[b, Blank[]]], Pattern[x, Blank[]]]], Pattern[m, Blank[]]], Power[Plus[Optional[Pattern[c, Blank[]]], Times[Optional[Pattern[d, Blank[]]], Pattern[x, Blank[]]]], Optional[Pattern[n, Blank[]]]], Power[Plus[Optional[Pattern[e, Blank[]]], Times[Optional[Pattern[f, Blank[]]], Pattern[x, Blank[]]]], Optional[Pattern[p, Blank[]]]]], Pattern[x, Blank[Symbol]]] := Simp[(b*(a + b*x)^(m + 1)*(c + d*x)^(n + 1)*(e + f*x)^(p + 1))/((m + 1)*(b*c - a*d)*(b*e - a*f)), x] /; FreeQ[{a, b, c, d, e, f, m, n, p}, x] && EqQ[Simplify[m + n + p + 3], 0] && EqQ[a*d*f*(m + 1) + b*c*f*(n + 1) + b*d*e*(p + 1), 0] && NeQ[m, -1]
  96. Int[Times[Power[Plus[Optional[Pattern[a, Blank[]]], Times[Optional[Pattern[b, Blank[]]], Pattern[x, Blank[]]]], Pattern[m, Blank[]]], Power[Plus[Optional[Pattern[c, Blank[]]], Times[Optional[Pattern[d, Blank[]]], Pattern[x, Blank[]]]], Optional[Pattern[n, Blank[]]]], Power[Plus[Optional[Pattern[e, Blank[]]], Times[Optional[Pattern[f, Blank[]]], Pattern[x, Blank[]]]], Optional[Pattern[p, Blank[]]]]], Pattern[x, Blank[Symbol]]] := Simp[(b*(a + b*x)^(m + 1)*(c + d*x)^(n + 1)*(e + f*x)^(p + 1))/((m + 1)*(b*c - a*d)*(b*e - a*f)), x] + Dist[(a*d*f*(m + 1) + b*c*f*(n + 1) + b*d*e*(p + 1))/((m + 1)*(b*c - a*d)*(b*e - a*f)), Int[(a + b*x)^(m + 1)*(c + d*x)^n*(e + f*x)^p, x], x] /; FreeQ[{a, b, c, d, e, f, m, n, p}, x] && EqQ[Simplify[m + n + p + 3], 0] && (LtQ[m, -1] || SumSimplerQ[m, 1])
  97. Int[Times[Power[Plus[Optional[Pattern[a, Blank[]]], Times[Optional[Pattern[b, Blank[]]], Pattern[x, Blank[]]]], Pattern[m, Blank[]]], Power[Plus[Optional[Pattern[c, Blank[]]], Times[Optional[Pattern[d, Blank[]]], Pattern[x, Blank[]]]], Optional[Pattern[n, Blank[]]]], Power[Plus[Optional[Pattern[e, Blank[]]], Times[Optional[Pattern[f, Blank[]]], Pattern[x, Blank[]]]], Optional[Pattern[p, Blank[]]]]], Pattern[x, Blank[Symbol]]] := Simp[((a + b*x)^(m + 1)*(c + d*x)^n*(e + f*x)^p)/(b*(m + 1)), x] - Dist[1/(b*(m + 1)), Int[(a + b*x)^(m + 1)*(c + d*x)^(n - 1)*(e + f*x)^(p - 1)*Simp[d*e*n + c*f*p + d*f*(n + p)*x, x], x], x] /; FreeQ[{a, b, c, d, e, f}, x] && LtQ[m, -1] && GtQ[n, 0] && GtQ[p, 0] && (IntegersQ[2*m, 2*n, 2*p] || IntegersQ[m, n + p] || IntegersQ[p, m + n])
  98. Int[Times[Power[Plus[Optional[Pattern[a, Blank[]]], Times[Optional[Pattern[b, Blank[]]], Pattern[x, Blank[]]]], Pattern[m, Blank[]]], Power[Plus[Optional[Pattern[c, Blank[]]], Times[Optional[Pattern[d, Blank[]]], Pattern[x, Blank[]]]], Optional[Pattern[n, Blank[]]]], Power[Plus[Optional[Pattern[e, Blank[]]], Times[Optional[Pattern[f, Blank[]]], Pattern[x, Blank[]]]], Optional[Pattern[p, Blank[]]]]], Pattern[x, Blank[Symbol]]] := Simp[((b*c - a*d)*(a + b*x)^(m + 1)*(c + d*x)^(n - 1)*(e + f*x)^(p + 1))/(b*(b*e - a*f)*(m + 1)), x] + Dist[1/(b*(b*e - a*f)*(m + 1)), Int[(a + b*x)^(m + 1)*(c + d*x)^(n - 2)*(e + f*x)^p*Simp[a*d*(d*e*(n - 1) + c*f*(p + 1)) + b*c*(d*e*(m - n + 2) - c*f*(m + p + 2)) + d*(a*d*f*(n + p) + b*(d*e*(m + 1) - c*f*(m + n + p + 1)))*x, x], x], x] /; FreeQ[{a, b, c, d, e, f, p}, x] && LtQ[m, -1] && GtQ[n, 1] && (IntegersQ[2*m, 2*n, 2*p] || IntegersQ[m, n + p] || IntegersQ[p, m + n])
  99. Int[Times[Power[Plus[Optional[Pattern[a, Blank[]]], Times[Optional[Pattern[b, Blank[]]], Pattern[x, Blank[]]]], Pattern[m, Blank[]]], Power[Plus[Optional[Pattern[c, Blank[]]], Times[Optional[Pattern[d, Blank[]]], Pattern[x, Blank[]]]], Optional[Pattern[n, Blank[]]]], Power[Plus[Optional[Pattern[e, Blank[]]], Times[Optional[Pattern[f, Blank[]]], Pattern[x, Blank[]]]], Optional[Pattern[p, Blank[]]]]], Pattern[x, Blank[Symbol]]] := Simp[((a + b*x)^(m + 1)*(c + d*x)^n*(e + f*x)^(p + 1))/((m + 1)*(b*e - a*f)), x] - Dist[1/((m + 1)*(b*e - a*f)), Int[(a + b*x)^(m + 1)*(c + d*x)^(n - 1)*(e + f*x)^p*Simp[d*e*n + c*f*(m + p + 2) + d*f*(m + n + p + 2)*x, x], x], x] /; FreeQ[{a, b, c, d, e, f, p}, x] && LtQ[m, -1] && GtQ[n, 0] && (IntegersQ[2*m, 2*n, 2*p] || IntegersQ[m, n + p] || IntegersQ[p, m + n])
  100. Int[Times[Power[Plus[Optional[Pattern[a, Blank[]]], Times[Optional[Pattern[b, Blank[]]], Pattern[x, Blank[]]]], Pattern[m, Blank[]]], Power[Plus[Optional[Pattern[c, Blank[]]], Times[Optional[Pattern[d, Blank[]]], Pattern[x, Blank[]]]], Optional[Pattern[n, Blank[]]]], Power[Plus[Optional[Pattern[e, Blank[]]], Times[Optional[Pattern[f, Blank[]]], Pattern[x, Blank[]]]], Optional[Pattern[p, Blank[]]]]], Pattern[x, Blank[Symbol]]] := Simp[(b*(a + b*x)^(m - 1)*(c + d*x)^(n + 1)*(e + f*x)^(p + 1))/(d*f*(m + n + p + 1)), x] + Dist[1/(d*f*(m + n + p + 1)), Int[(a + b*x)^(m - 2)*(c + d*x)^n*(e + f*x)^p*Simp[a^2*d*f*(m + n + p + 1) - b*(b*c*e*(m - 1) + a*(d*e*(n + 1) + c*f*(p + 1))) + b*(a*d*f*(2*m + n + p) - b*(d*e*(m + n) + c*f*(m + p)))*x, x], x], x] /; FreeQ[{a, b, c, d, e, f, n, p}, x] && GtQ[m, 1] && NeQ[m + n + p + 1, 0] && IntegerQ[m]
  101. Int[Times[Power[Plus[Optional[Pattern[a, Blank[]]], Times[Optional[Pattern[b, Blank[]]], Pattern[x, Blank[]]]], Optional[Pattern[m, Blank[]]]], Power[Plus[Optional[Pattern[c, Blank[]]], Times[Optional[Pattern[d, Blank[]]], Pattern[x, Blank[]]]], Optional[Pattern[n, Blank[]]]], Power[Plus[Optional[Pattern[e, Blank[]]], Times[Optional[Pattern[f, Blank[]]], Pattern[x, Blank[]]]], Optional[Pattern[p, Blank[]]]]], Pattern[x, Blank[Symbol]]] := Simp[((a + b*x)^m*(c + d*x)^n*(e + f*x)^(p + 1))/(f*(m + n + p + 1)), x] - Dist[1/(f*(m + n + p + 1)), Int[(a + b*x)^(m - 1)*(c + d*x)^(n - 1)*(e + f*x)^p*Simp[c*m*(b*e - a*f) + a*n*(d*e - c*f) + (d*m*(b*e - a*f) + b*n*(d*e - c*f))*x, x], x], x] /; FreeQ[{a, b, c, d, e, f, p}, x] && GtQ[m, 0] && GtQ[n, 0] && NeQ[m + n + p + 1, 0] && (IntegersQ[2*m, 2*n, 2*p] || (IntegersQ[m, n + p] || IntegersQ[p, m + n]))
  102. Int[Times[Power[Plus[Optional[Pattern[a, Blank[]]], Times[Optional[Pattern[b, Blank[]]], Pattern[x, Blank[]]]], Pattern[m, Blank[]]], Power[Plus[Optional[Pattern[c, Blank[]]], Times[Optional[Pattern[d, Blank[]]], Pattern[x, Blank[]]]], Optional[Pattern[n, Blank[]]]], Power[Plus[Optional[Pattern[e, Blank[]]], Times[Optional[Pattern[f, Blank[]]], Pattern[x, Blank[]]]], Optional[Pattern[p, Blank[]]]]], Pattern[x, Blank[Symbol]]] := Simp[(b*(a + b*x)^(m - 1)*(c + d*x)^(n + 1)*(e + f*x)^(p + 1))/(d*f*(m + n + p + 1)), x] + Dist[1/(d*f*(m + n + p + 1)), Int[(a + b*x)^(m - 2)*(c + d*x)^n*(e + f*x)^p*Simp[a^2*d*f*(m + n + p + 1) - b*(b*c*e*(m - 1) + a*(d*e*(n + 1) + c*f*(p + 1))) + b*(a*d*f*(2*m + n + p) - b*(d*e*(m + n) + c*f*(m + p)))*x, x], x], x] /; FreeQ[{a, b, c, d, e, f, n, p}, x] && GtQ[m, 1] && NeQ[m + n + p + 1, 0] && IntegersQ[2*m, 2*n, 2*p]
  103. Int[Times[Power[Plus[Optional[Pattern[a, Blank[]]], Times[Optional[Pattern[b, Blank[]]], Pattern[x, Blank[]]]], Pattern[m, Blank[]]], Power[Plus[Optional[Pattern[c, Blank[]]], Times[Optional[Pattern[d, Blank[]]], Pattern[x, Blank[]]]], Optional[Pattern[n, Blank[]]]], Power[Plus[Optional[Pattern[e, Blank[]]], Times[Optional[Pattern[f, Blank[]]], Pattern[x, Blank[]]]], Optional[Pattern[p, Blank[]]]]], Pattern[x, Blank[Symbol]]] := Simp[(b*(a + b*x)^(m + 1)*(c + d*x)^(n + 1)*(e + f*x)^(p + 1))/((m + 1)*(b*c - a*d)*(b*e - a*f)), x] + Dist[1/((m + 1)*(b*c - a*d)*(b*e - a*f)), Int[(a + b*x)^(m + 1)*(c + d*x)^n*(e + f*x)^p*Simp[a*d*f*(m + 1) - b*(d*e*(m + n + 2) + c*f*(m + p + 2)) - b*d*f*(m + n + p + 3)*x, x], x], x] /; FreeQ[{a, b, c, d, e, f, n, p}, x] && LtQ[m, -1] && IntegerQ[m] && (IntegerQ[n] || IntegersQ[2*n, 2*p])
  104. Int[Times[Power[Plus[Optional[Pattern[a, Blank[]]], Times[Optional[Pattern[b, Blank[]]], Pattern[x, Blank[]]]], Pattern[m, Blank[]]], Power[Plus[Optional[Pattern[c, Blank[]]], Times[Optional[Pattern[d, Blank[]]], Pattern[x, Blank[]]]], Optional[Pattern[n, Blank[]]]], Power[Plus[Optional[Pattern[e, Blank[]]], Times[Optional[Pattern[f, Blank[]]], Pattern[x, Blank[]]]], Optional[Pattern[p, Blank[]]]]], Pattern[x, Blank[Symbol]]] := Simp[(b*(a + b*x)^(m + 1)*(c + d*x)^(n + 1)*(e + f*x)^(p + 1))/((m + 1)*(b*c - a*d)*(b*e - a*f)), x] + Dist[1/((m + 1)*(b*c - a*d)*(b*e - a*f)), Int[(a + b*x)^(m + 1)*(c + d*x)^n*(e + f*x)^p*Simp[a*d*f*(m + 1) - b*(d*e*(m + n + 2) + c*f*(m + p + 2)) - b*d*f*(m + n + p + 3)*x, x], x], x] /; FreeQ[{a, b, c, d, e, f, n, p}, x] && LtQ[m, -1] && IntegersQ[2*m, 2*n, 2*p]
  105. Int[Times[Power[Plus[Optional[Pattern[a, Blank[]]], Times[Optional[Pattern[b, Blank[]]], Pattern[x, Blank[]]]], Pattern[m, Blank[]]], Power[Plus[Optional[Pattern[c, Blank[]]], Times[Optional[Pattern[d, Blank[]]], Pattern[x, Blank[]]]], Pattern[n, Blank[]]], Power[Plus[Optional[Pattern[e, Blank[]]], Times[Optional[Pattern[f, Blank[]]], Pattern[x, Blank[]]]], -1]], Pattern[x, Blank[Symbol]]] := Dist[b/f, Int[(a + b*x)^(m - 1)*(c + d*x)^n, x], x] - Dist[(b*e - a*f)/f, Int[((a + b*x)^(m - 1)*(c + d*x)^n)/(e + f*x), x], x] /; FreeQ[{a, b, c, d, e, f, m, n}, x] && IGtQ[Simplify[m + n + 1], 0] && (GtQ[m, 0] || ( !RationalQ[m] && (SumSimplerQ[m, -1] || !SumSimplerQ[n, -1])))
  106. Int[Times[Power[Plus[Optional[Pattern[a, Blank[]]], Times[Optional[Pattern[b, Blank[]]], Pattern[x, Blank[]]]], -1], Power[Plus[Optional[Pattern[c, Blank[]]], Times[Optional[Pattern[d, Blank[]]], Pattern[x, Blank[]]]], Rational[-1, 2]], Power[Plus[Optional[Pattern[e, Blank[]]], Times[Optional[Pattern[f, Blank[]]], Pattern[x, Blank[]]]], Rational[-1, 4]]], Pattern[x, Blank[Symbol]]] := Dist[-4, Subst[Int[x^2/((b*e - a*f - b*x^4)*Sqrt[c - (d*e)/f + (d*x^4)/f]), x], x, (e + f*x)^(1/4)], x] /; FreeQ[{a, b, c, d, e, f}, x] && GtQ[-(f/(d*e - c*f)), 0]
  107. Int[Times[Power[Plus[Optional[Pattern[a, Blank[]]], Times[Optional[Pattern[b, Blank[]]], Pattern[x, Blank[]]]], -1], Power[Plus[Optional[Pattern[c, Blank[]]], Times[Optional[Pattern[d, Blank[]]], Pattern[x, Blank[]]]], Rational[-1, 2]], Power[Plus[Optional[Pattern[e, Blank[]]], Times[Optional[Pattern[f, Blank[]]], Pattern[x, Blank[]]]], Rational[-1, 4]]], Pattern[x, Blank[Symbol]]] := Dist[Sqrt[-((f*(c + d*x))/(d*e - c*f))]/Sqrt[c + d*x], Int[1/((a + b*x)*Sqrt[-((c*f)/(d*e - c*f)) - (d*f*x)/(d*e - c*f)]*(e + f*x)^(1/4)), x], x] /; FreeQ[{a, b, c, d, e, f}, x] && !GtQ[-(f/(d*e - c*f)), 0]
  108. Int[Times[Power[Plus[Optional[Pattern[a, Blank[]]], Times[Optional[Pattern[b, Blank[]]], Pattern[x, Blank[]]]], -1], Power[Plus[Optional[Pattern[c, Blank[]]], Times[Optional[Pattern[d, Blank[]]], Pattern[x, Blank[]]]], Rational[-1, 2]], Power[Plus[Optional[Pattern[e, Blank[]]], Times[Optional[Pattern[f, Blank[]]], Pattern[x, Blank[]]]], Rational[-3, 4]]], Pattern[x, Blank[Symbol]]] := Dist[-4, Subst[Int[1/((b*e - a*f - b*x^4)*Sqrt[c - (d*e)/f + (d*x^4)/f]), x], x, (e + f*x)^(1/4)], x] /; FreeQ[{a, b, c, d, e, f}, x] && GtQ[-(f/(d*e - c*f)), 0]
  109. Int[Times[Power[Plus[Optional[Pattern[a, Blank[]]], Times[Optional[Pattern[b, Blank[]]], Pattern[x, Blank[]]]], -1], Power[Plus[Optional[Pattern[c, Blank[]]], Times[Optional[Pattern[d, Blank[]]], Pattern[x, Blank[]]]], Rational[-1, 2]], Power[Plus[Optional[Pattern[e, Blank[]]], Times[Optional[Pattern[f, Blank[]]], Pattern[x, Blank[]]]], Rational[-3, 4]]], Pattern[x, Blank[Symbol]]] := Dist[Sqrt[-((f*(c + d*x))/(d*e - c*f))]/Sqrt[c + d*x], Int[1/((a + b*x)*Sqrt[-((c*f)/(d*e - c*f)) - (d*f*x)/(d*e - c*f)]*(e + f*x)^(3/4)), x], x] /; FreeQ[{a, b, c, d, e, f}, x] && !GtQ[-(f/(d*e - c*f)), 0]
  110. Int[Times[Power[Times[Optional[Pattern[b, Blank[]]], Pattern[x, Blank[]]], Rational[-1, 2]], Power[Plus[Pattern[c, Blank[]], Times[Optional[Pattern[d, Blank[]]], Pattern[x, Blank[]]]], Rational[-1, 2]], Power[Plus[Pattern[e, Blank[]], Times[Optional[Pattern[f, Blank[]]], Pattern[x, Blank[]]]], Rational[1, 2]]], Pattern[x, Blank[Symbol]]] := Simp[(2*Sqrt[e]*Rt[-(b/d), 2]*EllipticE[ArcSin[Sqrt[b*x]/(Sqrt[c]*Rt[-(b/d), 2])], (c*f)/(d*e)])/b, x] /; FreeQ[{b, c, d, e, f}, x] && NeQ[d*e - c*f, 0] && GtQ[c, 0] && GtQ[e, 0] && !LtQ[-(b/d), 0]
  111. Int[Times[Power[Times[Optional[Pattern[b, Blank[]]], Pattern[x, Blank[]]], Rational[-1, 2]], Power[Plus[Pattern[c, Blank[]], Times[Optional[Pattern[d, Blank[]]], Pattern[x, Blank[]]]], Rational[-1, 2]], Power[Plus[Pattern[e, Blank[]], Times[Optional[Pattern[f, Blank[]]], Pattern[x, Blank[]]]], Rational[1, 2]]], Pattern[x, Blank[Symbol]]] := Dist[Sqrt[-(b*x)]/Sqrt[b*x], Int[Sqrt[e + f*x]/(Sqrt[-(b*x)]*Sqrt[c + d*x]), x], x] /; FreeQ[{b, c, d, e, f}, x] && NeQ[d*e - c*f, 0] && GtQ[c, 0] && GtQ[e, 0] && LtQ[-(b/d), 0]
  112. Int[Times[Power[Times[Optional[Pattern[b, Blank[]]], Pattern[x, Blank[]]], Rational[-1, 2]], Power[Plus[Pattern[c, Blank[]], Times[Optional[Pattern[d, Blank[]]], Pattern[x, Blank[]]]], Rational[-1, 2]], Power[Plus[Pattern[e, Blank[]], Times[Optional[Pattern[f, Blank[]]], Pattern[x, Blank[]]]], Rational[1, 2]]], Pattern[x, Blank[Symbol]]] := Dist[(Sqrt[e + f*x]*Sqrt[1 + (d*x)/c])/(Sqrt[c + d*x]*Sqrt[1 + (f*x)/e]), Int[Sqrt[1 + (f*x)/e]/(Sqrt[b*x]*Sqrt[1 + (d*x)/c]), x], x] /; FreeQ[{b, c, d, e, f}, x] && NeQ[d*e - c*f, 0] && !(GtQ[c, 0] && GtQ[e, 0])
  113. Int[Times[Power[Plus[Pattern[a, Blank[]], Times[Optional[Pattern[b, Blank[]]], Pattern[x, Blank[]]]], Rational[-1, 2]], Power[Plus[Pattern[c, Blank[]], Times[Optional[Pattern[d, Blank[]]], Pattern[x, Blank[]]]], Rational[-1, 2]], Power[Plus[Optional[Pattern[e, Blank[]]], Times[Optional[Pattern[f, Blank[]]], Pattern[x, Blank[]]]], Rational[1, 2]]], Pattern[x, Blank[Symbol]]] := Simp[(2*Rt[-((b*e - a*f)/d), 2]*EllipticE[ArcSin[Sqrt[a + b*x]/Rt[-((b*c - a*d)/d), 2]], (f*(b*c - a*d))/(d*(b*e - a*f))])/b, x] /; FreeQ[{a, b, c, d, e, f}, x] && GtQ[b/(b*c - a*d), 0] && GtQ[b/(b*e - a*f), 0] && !LtQ[-((b*c - a*d)/d), 0] && !(SimplerQ[c + d*x, a + b*x] && GtQ[-(d/(b*c - a*d)), 0] && GtQ[d/(d*e - c*f), 0] && !LtQ[(b*c - a*d)/b, 0])
  114. Int[Times[Power[Plus[Pattern[a, Blank[]], Times[Optional[Pattern[b, Blank[]]], Pattern[x, Blank[]]]], Rational[-1, 2]], Power[Plus[Pattern[c, Blank[]], Times[Optional[Pattern[d, Blank[]]], Pattern[x, Blank[]]]], Rational[-1, 2]], Power[Plus[Optional[Pattern[e, Blank[]]], Times[Optional[Pattern[f, Blank[]]], Pattern[x, Blank[]]]], Rational[1, 2]]], Pattern[x, Blank[Symbol]]] := Dist[(Sqrt[e + f*x]*Sqrt[(b*(c + d*x))/(b*c - a*d)])/(Sqrt[c + d*x]*Sqrt[(b*(e + f*x))/(b*e - a*f)]), Int[Sqrt[(b*e)/(b*e - a*f) + (b*f*x)/(b*e - a*f)]/(Sqrt[a + b*x]*Sqrt[(b*c)/(b*c - a*d) + (b*d*x)/(b*c - a*d)]), x], x] /; FreeQ[{a, b, c, d, e, f}, x] && !(GtQ[b/(b*c - a*d), 0] && GtQ[b/(b*e - a*f), 0]) && !LtQ[-((b*c - a*d)/d), 0]
  115. Int[Times[Power[Times[Optional[Pattern[b, Blank[]]], Pattern[x, Blank[]]], Rational[-1, 2]], Power[Plus[Pattern[c, Blank[]], Times[Optional[Pattern[d, Blank[]]], Pattern[x, Blank[]]]], Rational[-1, 2]], Power[Plus[Pattern[e, Blank[]], Times[Optional[Pattern[f, Blank[]]], Pattern[x, Blank[]]]], Rational[-1, 2]]], Pattern[x, Blank[Symbol]]] := Simp[(2*Rt[-(b/d), 2]*EllipticF[ArcSin[Sqrt[b*x]/(Sqrt[c]*Rt[-(b/d), 2])], (c*f)/(d*e)])/(b*Sqrt[e]), x] /; FreeQ[{b, c, d, e, f}, x] && GtQ[c, 0] && GtQ[e, 0] && (GtQ[-(b/d), 0] || LtQ[-(b/f), 0])
  116. Int[Times[Power[Times[Optional[Pattern[b, Blank[]]], Pattern[x, Blank[]]], Rational[-1, 2]], Power[Plus[Pattern[c, Blank[]], Times[Optional[Pattern[d, Blank[]]], Pattern[x, Blank[]]]], Rational[-1, 2]], Power[Plus[Pattern[e, Blank[]], Times[Optional[Pattern[f, Blank[]]], Pattern[x, Blank[]]]], Rational[-1, 2]]], Pattern[x, Blank[Symbol]]] := Simp[(2*Rt[-(b/d), 2]*EllipticF[ArcSin[Sqrt[b*x]/(Sqrt[c]*Rt[-(b/d), 2])], (c*f)/(d*e)])/(b*Sqrt[e]), x] /; FreeQ[{b, c, d, e, f}, x] && GtQ[c, 0] && GtQ[e, 0] && (PosQ[-(b/d)] || NegQ[-(b/f)])
  117. Int[Times[Power[Times[Optional[Pattern[b, Blank[]]], Pattern[x, Blank[]]], Rational[-1, 2]], Power[Plus[Pattern[c, Blank[]], Times[Optional[Pattern[d, Blank[]]], Pattern[x, Blank[]]]], Rational[-1, 2]], Power[Plus[Pattern[e, Blank[]], Times[Optional[Pattern[f, Blank[]]], Pattern[x, Blank[]]]], Rational[-1, 2]]], Pattern[x, Blank[Symbol]]] := Dist[(Sqrt[1 + (d*x)/c]*Sqrt[1 + (f*x)/e])/(Sqrt[c + d*x]*Sqrt[e + f*x]), Int[1/(Sqrt[b*x]*Sqrt[1 + (d*x)/c]*Sqrt[1 + (f*x)/e]), x], x] /; FreeQ[{b, c, d, e, f}, x] && !(GtQ[c, 0] && GtQ[e, 0])
  118. Int[Times[Power[Plus[Pattern[a, Blank[]], Times[Optional[Pattern[b, Blank[]]], Pattern[x, Blank[]]]], Rational[-1, 2]], Power[Plus[Pattern[c, Blank[]], Times[Optional[Pattern[d, Blank[]]], Pattern[x, Blank[]]]], Rational[-1, 2]], Power[Plus[Pattern[e, Blank[]], Times[Optional[Pattern[f, Blank[]]], Pattern[x, Blank[]]]], Rational[-1, 2]]], Pattern[x, Blank[Symbol]]] := Simp[(-2*Sqrt[d/f]*EllipticF[ArcSin[Rt[-((b*e - a*f)/f), 2]/Sqrt[a + b*x]], (f*(b*c - a*d))/(d*(b*e - a*f))])/(d*Rt[-((b*e - a*f)/f), 2]), x] /; FreeQ[{a, b, c, d, e, f}, x] && GtQ[d/b, 0] && GtQ[f/b, 0] && LeQ[c, (a*d)/b] && LeQ[e, (a*f)/b]
  119. Int[Times[Power[Plus[Pattern[a, Blank[]], Times[Optional[Pattern[b, Blank[]]], Pattern[x, Blank[]]]], Rational[-1, 2]], Power[Plus[Pattern[c, Blank[]], Times[Optional[Pattern[d, Blank[]]], Pattern[x, Blank[]]]], Rational[-1, 2]], Power[Plus[Pattern[e, Blank[]], Times[Optional[Pattern[f, Blank[]]], Pattern[x, Blank[]]]], Rational[-1, 2]]], Pattern[x, Blank[Symbol]]] := Simp[(2*Rt[-(b/d), 2]*EllipticF[ArcSin[Sqrt[a + b*x]/(Rt[-(b/d), 2]*Sqrt[(b*c - a*d)/b])], (f*(b*c - a*d))/(d*(b*e - a*f))])/(b*Sqrt[(b*e - a*f)/b]), x] /; FreeQ[{a, b, c, d, e, f}, x] && GtQ[(b*c - a*d)/b, 0] && GtQ[(b*e - a*f)/b, 0] && PosQ[-(b/d)] && !(SimplerQ[c + d*x, a + b*x] && GtQ[(d*e - c*f)/d, 0] && GtQ[-(d/b), 0]) && !(SimplerQ[c + d*x, a + b*x] && GtQ[(-(b*e) + a*f)/f, 0] && GtQ[-(f/b), 0]) && !(SimplerQ[e + f*x, a + b*x] && GtQ[(-(d*e) + c*f)/f, 0] && GtQ[(-(b*e) + a*f)/f, 0] && (PosQ[-(f/d)] || PosQ[-(f/b)]))
  120. Int[Times[Power[Plus[Pattern[a, Blank[]], Times[Optional[Pattern[b, Blank[]]], Pattern[x, Blank[]]]], Rational[-1, 2]], Power[Plus[Pattern[c, Blank[]], Times[Optional[Pattern[d, Blank[]]], Pattern[x, Blank[]]]], Rational[-1, 2]], Power[Plus[Pattern[e, Blank[]], Times[Optional[Pattern[f, Blank[]]], Pattern[x, Blank[]]]], Rational[-1, 2]]], Pattern[x, Blank[Symbol]]] := Simp[(2*Rt[-(b/d), 2]*EllipticF[ArcSin[Sqrt[a + b*x]/(Rt[-(b/d), 2]*Sqrt[(b*c - a*d)/b])], (f*(b*c - a*d))/(d*(b*e - a*f))])/(b*Sqrt[(b*e - a*f)/b]), x] /; FreeQ[{a, b, c, d, e, f}, x] && GtQ[b/(b*c - a*d), 0] && GtQ[b/(b*e - a*f), 0] && SimplerQ[a + b*x, c + d*x] && SimplerQ[a + b*x, e + f*x] && (PosQ[-((b*c - a*d)/d)] || NegQ[-((b*e - a*f)/f)])
  121. Int[Times[Power[Plus[Pattern[a, Blank[]], Times[Optional[Pattern[b, Blank[]]], Pattern[x, Blank[]]]], Rational[-1, 2]], Power[Plus[Pattern[c, Blank[]], Times[Optional[Pattern[d, Blank[]]], Pattern[x, Blank[]]]], Rational[-1, 2]], Power[Plus[Pattern[e, Blank[]], Times[Optional[Pattern[f, Blank[]]], Pattern[x, Blank[]]]], Rational[-1, 2]]], Pattern[x, Blank[Symbol]]] := Dist[Sqrt[(b*(c + d*x))/(b*c - a*d)]/Sqrt[c + d*x], Int[1/(Sqrt[a + b*x]*Sqrt[(b*c)/(b*c - a*d) + (b*d*x)/(b*c - a*d)]*Sqrt[e + f*x]), x], x] /; FreeQ[{a, b, c, d, e, f}, x] && !GtQ[(b*c - a*d)/b, 0] && SimplerQ[a + b*x, c + d*x] && SimplerQ[a + b*x, e + f*x]
  122. Int[Times[Power[Plus[Pattern[a, Blank[]], Times[Optional[Pattern[b, Blank[]]], Pattern[x, Blank[]]]], Rational[-1, 2]], Power[Plus[Pattern[c, Blank[]], Times[Optional[Pattern[d, Blank[]]], Pattern[x, Blank[]]]], Rational[-1, 2]], Power[Plus[Pattern[e, Blank[]], Times[Optional[Pattern[f, Blank[]]], Pattern[x, Blank[]]]], Rational[-1, 2]]], Pattern[x, Blank[Symbol]]] := Dist[Sqrt[(b*(e + f*x))/(b*e - a*f)]/Sqrt[e + f*x], Int[1/(Sqrt[a + b*x]*Sqrt[c + d*x]*Sqrt[(b*e)/(b*e - a*f) + (b*f*x)/(b*e - a*f)]), x], x] /; FreeQ[{a, b, c, d, e, f}, x] && !GtQ[(b*e - a*f)/b, 0]
  123. Int[Times[Power[Plus[Optional[Pattern[a, Blank[]]], Times[Optional[Pattern[b, Blank[]]], Pattern[x, Blank[]]]], -1], Power[Plus[Optional[Pattern[c, Blank[]]], Times[Optional[Pattern[d, Blank[]]], Pattern[x, Blank[]]]], Rational[-1, 3]], Power[Plus[Optional[Pattern[e, Blank[]]], Times[Optional[Pattern[f, Blank[]]], Pattern[x, Blank[]]]], Rational[-1, 3]]], Pattern[x, Blank[Symbol]]] := With[{q = Rt[(b*(b*e - a*f))/(b*c - a*d)^2, 3]}, -Simp[Log[a + b*x]/(2*q*(b*c - a*d)), x] + (-Simp[(Sqrt[3]*ArcTan[1/Sqrt[3] + (2*q*(c + d*x)^(2/3))/(Sqrt[3]*(e + f*x)^(1/3))])/(2*q*(b*c - a*d)), x] + Simp[(3*Log[q*(c + d*x)^(2/3) - (e + f*x)^(1/3)])/(4*q*(b*c - a*d)), x])] /; FreeQ[{a, b, c, d, e, f}, x] && EqQ[2*b*d*e - b*c*f - a*d*f, 0]
  124. Int[Times[Power[Plus[Optional[Pattern[a, Blank[]]], Times[Optional[Pattern[b, Blank[]]], Pattern[x, Blank[]]]], Pattern[m, Blank[]]], Power[Plus[Optional[Pattern[c, Blank[]]], Times[Optional[Pattern[d, Blank[]]], Pattern[x, Blank[]]]], Rational[-1, 3]], Power[Plus[Optional[Pattern[e, Blank[]]], Times[Optional[Pattern[f, Blank[]]], Pattern[x, Blank[]]]], Rational[-1, 3]]], Pattern[x, Blank[Symbol]]] := Simp[(b*(a + b*x)^(m + 1)*(c + d*x)^(2/3)*(e + f*x)^(2/3))/((m + 1)*(b*c - a*d)*(b*e - a*f)), x] + Dist[f/(6*(m + 1)*(b*c - a*d)*(b*e - a*f)), Int[((a + b*x)^(m + 1)*(a*d*(3*m + 1) - 3*b*c*(3*m + 5) - 2*b*d*(3*m + 7)*x))/((c + d*x)^(1/3)*(e + f*x)^(1/3)), x], x] /; FreeQ[{a, b, c, d, e, f}, x] && EqQ[2*b*d*e - b*c*f - a*d*f, 0] && ILtQ[m, -1]
  125. Int[Times[Power[Times[Optional[Pattern[f, Blank[]]], Pattern[x, Blank[]]], Optional[Pattern[p, Blank[]]]], Power[Plus[Optional[Pattern[a, Blank[]]], Times[Optional[Pattern[b, Blank[]]], Pattern[x, Blank[]]]], Optional[Pattern[m, Blank[]]]], Power[Plus[Optional[Pattern[c, Blank[]]], Times[Optional[Pattern[d, Blank[]]], Pattern[x, Blank[]]]], Optional[Pattern[n, Blank[]]]]], Pattern[x, Blank[Symbol]]] := Int[(a*c + b*d*x^2)^m*(f*x)^p, x] /; FreeQ[{a, b, c, d, f, m, n, p}, x] && EqQ[b*c + a*d, 0] && EqQ[m - n, 0] && GtQ[a, 0] && GtQ[c, 0]
  126. Int[Times[Power[Times[Optional[Pattern[f, Blank[]]], Pattern[x, Blank[]]], Optional[Pattern[p, Blank[]]]], Power[Plus[Optional[Pattern[a, Blank[]]], Times[Optional[Pattern[b, Blank[]]], Pattern[x, Blank[]]]], Optional[Pattern[m, Blank[]]]], Power[Plus[Optional[Pattern[c, Blank[]]], Times[Optional[Pattern[d, Blank[]]], Pattern[x, Blank[]]]], Optional[Pattern[n, Blank[]]]]], Pattern[x, Blank[Symbol]]] := Dist[((a + b*x)^FracPart[m]*(c + d*x)^FracPart[m])/(a*c + b*d*x^2)^FracPart[m], Int[(a*c + b*d*x^2)^m*(f*x)^p, x], x] /; FreeQ[{a, b, c, d, f, m, n, p}, x] && EqQ[b*c + a*d, 0] && EqQ[m - n, 0]
  127. Int[Times[Power[Times[Optional[Pattern[f, Blank[]]], Pattern[x, Blank[]]], Optional[Pattern[p, Blank[]]]], Power[Plus[Optional[Pattern[a, Blank[]]], Times[Optional[Pattern[b, Blank[]]], Pattern[x, Blank[]]]], Optional[Pattern[m, Blank[]]]], Power[Plus[Optional[Pattern[c, Blank[]]], Times[Optional[Pattern[d, Blank[]]], Pattern[x, Blank[]]]], Optional[Pattern[n, Blank[]]]]], Pattern[x, Blank[Symbol]]] := Int[ExpandIntegrand[(a + b*x)^n*(c + d*x)^n*(f*x)^p, (a + b*x)^(m - n), x], x] /; FreeQ[{a, b, c, d, f, m, n, p}, x] && EqQ[b*c + a*d, 0] && IGtQ[m - n, 0] && NeQ[m + n + p + 2, 0]
  128. Int[Times[Power[Plus[Optional[Pattern[a, Blank[]]], Times[Optional[Pattern[b, Blank[]]], Pattern[x, Blank[]]]], Optional[Pattern[m, Blank[]]]], Power[Plus[Optional[Pattern[c, Blank[]]], Times[Optional[Pattern[d, Blank[]]], Pattern[x, Blank[]]]], Optional[Pattern[n, Blank[]]]], Power[Plus[Optional[Pattern[e, Blank[]]], Times[Optional[Pattern[f, Blank[]]], Pattern[x, Blank[]]]], Optional[Pattern[p, Blank[]]]]], Pattern[x, Blank[Symbol]]] := Int[ExpandIntegrand[(a + b*x)^m*(c + d*x)^n*(e + f*x)^p, x], x] /; FreeQ[{a, b, c, d, e, f, n, p}, x] && (IGtQ[m, 0] || (ILtQ[m, 0] && ILtQ[n, 0]))
  129. Int[Times[Power[Plus[Optional[Pattern[a, Blank[]]], Times[Optional[Pattern[b, Blank[]]], Pattern[x, Blank[]]]], Pattern[m, Blank[]]], Power[Plus[Optional[Pattern[c, Blank[]]], Times[Optional[Pattern[d, Blank[]]], Pattern[x, Blank[]]]], Optional[Pattern[n, Blank[]]]], Power[Plus[Optional[Pattern[e, Blank[]]], Times[Optional[Pattern[f, Blank[]]], Pattern[x, Blank[]]]], Optional[Pattern[p, Blank[]]]]], Pattern[x, Blank[Symbol]]] := Simp[(b*(a + b*x)^(m + 1)*(c + d*x)^(n + 1)*(e + f*x)^(p + 1))/((m + 1)*(b*c - a*d)*(b*e - a*f)), x] + Dist[1/((m + 1)*(b*c - a*d)*(b*e - a*f)), Int[(a + b*x)^(m + 1)*(c + d*x)^n*(e + f*x)^p*Simp[a*d*f*(m + 1) - b*(d*e*(m + n + 2) + c*f*(m + p + 2)) - b*d*f*(m + n + p + 3)*x, x], x], x] /; FreeQ[{a, b, c, d, e, f, m, n, p}, x] && ILtQ[m + n + p + 2, 0] && NeQ[m, -1] && (SumSimplerQ[m, 1] || ( !(NeQ[n, -1] && SumSimplerQ[n, 1]) && !(NeQ[p, -1] && SumSimplerQ[p, 1])))
  130. Int[Times[Power[Times[Optional[Pattern[e, Blank[]]], Pattern[x, Blank[]]], Pattern[p, Blank[]]], Power[Plus[Pattern[a, Blank[]], Times[Optional[Pattern[b, Blank[]]], Pattern[x, Blank[]]]], Pattern[m, Blank[]]], Power[Plus[Pattern[c, Blank[]], Times[Optional[Pattern[d, Blank[]]], Pattern[x, Blank[]]]], Pattern[n, Blank[]]]], Pattern[x, Blank[Symbol]]] := With[{k = Denominator[p]}, Dist[k/e, Subst[Int[x^(k*(p + 1) - 1)*(a + (b*x^k)/e)^m*(c + (d*x^k)/e)^n, x], x, (e*x)^(1/k)], x]] /; FreeQ[{a, b, c, d, e, m, n}, x] && NeQ[b*c - a*d, 0] && FractionQ[p] && IntegerQ[m]
  131. Int[Times[Power[Plus[Optional[Pattern[a, Blank[]]], Times[Optional[Pattern[b, Blank[]]], Pattern[x, Blank[]]]], Pattern[m, Blank[]]], Power[Plus[Optional[Pattern[c, Blank[]]], Times[Optional[Pattern[d, Blank[]]], Pattern[x, Blank[]]]], Optional[Pattern[n, Blank[]]]], Power[Plus[Optional[Pattern[e, Blank[]]], Times[Optional[Pattern[f, Blank[]]], Pattern[x, Blank[]]]], Pattern[p, Blank[]]]], Pattern[x, Blank[Symbol]]] := Simp[((b*c - a*d)^n*(a + b*x)^(m + 1)*Hypergeometric2F1[m + 1, -n, m + 2, -(((d*e - c*f)*(a + b*x))/((b*c - a*d)*(e + f*x)))])/((m + 1)*(b*e - a*f)^(n + 1)*(e + f*x)^(m + 1)), x] /; FreeQ[{a, b, c, d, e, f, m, p}, x] && EqQ[m + n + p + 2, 0] && ILtQ[n, 0]
  132. Int[Times[Power[Plus[Optional[Pattern[a, Blank[]]], Times[Optional[Pattern[b, Blank[]]], Pattern[x, Blank[]]]], Pattern[m, Blank[]]], Power[Plus[Optional[Pattern[c, Blank[]]], Times[Optional[Pattern[d, Blank[]]], Pattern[x, Blank[]]]], Pattern[n, Blank[]]], Power[Plus[Optional[Pattern[e, Blank[]]], Times[Optional[Pattern[f, Blank[]]], Pattern[x, Blank[]]]], Pattern[p, Blank[]]]], Pattern[x, Blank[Symbol]]] := Simp[((a + b*x)^(m + 1)*(c + d*x)^n*(e + f*x)^(p + 1)*Hypergeometric2F1[m + 1, -n, m + 2, -(((d*e - c*f)*(a + b*x))/((b*c - a*d)*(e + f*x)))])/(((b*e - a*f)*(m + 1))*(((b*e - a*f)*(c + d*x))/((b*c - a*d)*(e + f*x)))^n), x] /; FreeQ[{a, b, c, d, e, f, m, n, p}, x] && EqQ[m + n + p + 2, 0] && !IntegerQ[n]
  133. Int[Times[Power[Times[Optional[Pattern[b, Blank[]]], Pattern[x, Blank[]]], Pattern[m, Blank[]]], Power[Plus[Pattern[c, Blank[]], Times[Optional[Pattern[d, Blank[]]], Pattern[x, Blank[]]]], Pattern[n, Blank[]]], Power[Plus[Pattern[e, Blank[]], Times[Optional[Pattern[f, Blank[]]], Pattern[x, Blank[]]]], Pattern[p, Blank[]]]], Pattern[x, Blank[Symbol]]] := Simp[(c^n*e^p*(b*x)^(m + 1)*AppellF1[m + 1, -n, -p, m + 2, -((d*x)/c), -((f*x)/e)])/(b*(m + 1)), x] /; FreeQ[{b, c, d, e, f, m, n, p}, x] && !IntegerQ[m] && !IntegerQ[n] && GtQ[c, 0] && (IntegerQ[p] || GtQ[e, 0])
  134. Int[Times[Power[Times[Optional[Pattern[b, Blank[]]], Pattern[x, Blank[]]], Pattern[m, Blank[]]], Power[Plus[Pattern[c, Blank[]], Times[Optional[Pattern[d, Blank[]]], Pattern[x, Blank[]]]], Pattern[n, Blank[]]], Power[Plus[Pattern[e, Blank[]], Times[Optional[Pattern[f, Blank[]]], Pattern[x, Blank[]]]], Pattern[p, Blank[]]]], Pattern[x, Blank[Symbol]]] := Simp[((c + d*x)^(n + 1)*AppellF1[n + 1, -m, -p, n + 2, 1 + (d*x)/c, -((f*(c + d*x))/(d*e - c*f))])/(d*(n + 1)*(-(d/(b*c)))^m*(d/(d*e - c*f))^p), x] /; FreeQ[{b, c, d, e, f, m, n, p}, x] && !IntegerQ[m] && !IntegerQ[n] && GtQ[-(d/(b*c)), 0] && (IntegerQ[p] || GtQ[d/(d*e - c*f), 0])
  135. Int[Times[Power[Times[Optional[Pattern[b, Blank[]]], Pattern[x, Blank[]]], Pattern[m, Blank[]]], Power[Plus[Pattern[c, Blank[]], Times[Optional[Pattern[d, Blank[]]], Pattern[x, Blank[]]]], Pattern[n, Blank[]]], Power[Plus[Pattern[e, Blank[]], Times[Optional[Pattern[f, Blank[]]], Pattern[x, Blank[]]]], Pattern[p, Blank[]]]], Pattern[x, Blank[Symbol]]] := Dist[(c^IntPart[n]*(c + d*x)^FracPart[n])/(1 + (d*x)/c)^FracPart[n], Int[(b*x)^m*(1 + (d*x)/c)^n*(e + f*x)^p, x], x] /; FreeQ[{b, c, d, e, f, m, n, p}, x] && !IntegerQ[m] && !IntegerQ[n] && !GtQ[c, 0]
  136. Int[Times[Power[Plus[Pattern[a, Blank[]], Times[Optional[Pattern[b, Blank[]]], Pattern[x, Blank[]]]], Pattern[m, Blank[]]], Power[Plus[Optional[Pattern[c, Blank[]]], Times[Optional[Pattern[d, Blank[]]], Pattern[x, Blank[]]]], Pattern[n, Blank[]]], Power[Plus[Optional[Pattern[e, Blank[]]], Times[Optional[Pattern[f, Blank[]]], Pattern[x, Blank[]]]], Pattern[p, Blank[]]]], Pattern[x, Blank[Symbol]]] := Simp[((b*e - a*f)^p*(a + b*x)^(m + 1)*AppellF1[m + 1, -n, -p, m + 2, -((d*(a + b*x))/(b*c - a*d)), -((f*(a + b*x))/(b*e - a*f))])/(b^(p + 1)*(m + 1)*(b/(b*c - a*d))^n), x] /; FreeQ[{a, b, c, d, e, f, m, n}, x] && !IntegerQ[m] && !IntegerQ[n] && IntegerQ[p] && GtQ[b/(b*c - a*d), 0] && !(GtQ[d/(d*a - c*b), 0] && SimplerQ[c + d*x, a + b*x])
  137. Int[Times[Power[Plus[Pattern[a, Blank[]], Times[Optional[Pattern[b, Blank[]]], Pattern[x, Blank[]]]], Pattern[m, Blank[]]], Power[Plus[Optional[Pattern[c, Blank[]]], Times[Optional[Pattern[d, Blank[]]], Pattern[x, Blank[]]]], Pattern[n, Blank[]]], Power[Plus[Optional[Pattern[e, Blank[]]], Times[Optional[Pattern[f, Blank[]]], Pattern[x, Blank[]]]], Pattern[p, Blank[]]]], Pattern[x, Blank[Symbol]]] := Dist[(c + d*x)^FracPart[n]/((b/(b*c - a*d))^IntPart[n]*((b*(c + d*x))/(b*c - a*d))^FracPart[n]), Int[(a + b*x)^m*((b*c)/(b*c - a*d) + (b*d*x)/(b*c - a*d))^n*(e + f*x)^p, x], x] /; FreeQ[{a, b, c, d, e, f, m, n}, x] && !IntegerQ[m] && !IntegerQ[n] && IntegerQ[p] && !GtQ[b/(b*c - a*d), 0] && !SimplerQ[c + d*x, a + b*x]
  138. Int[Times[Power[Plus[Pattern[a, Blank[]], Times[Optional[Pattern[b, Blank[]]], Pattern[x, Blank[]]]], Pattern[m, Blank[]]], Power[Plus[Optional[Pattern[c, Blank[]]], Times[Optional[Pattern[d, Blank[]]], Pattern[x, Blank[]]]], Pattern[n, Blank[]]], Power[Plus[Optional[Pattern[e, Blank[]]], Times[Optional[Pattern[f, Blank[]]], Pattern[x, Blank[]]]], Pattern[p, Blank[]]]], Pattern[x, Blank[Symbol]]] := Simp[((a + b*x)^(m + 1)*AppellF1[m + 1, -n, -p, m + 2, -((d*(a + b*x))/(b*c - a*d)), -((f*(a + b*x))/(b*e - a*f))])/(b*(m + 1)*(b/(b*c - a*d))^n*(b/(b*e - a*f))^p), x] /; FreeQ[{a, b, c, d, e, f, m, n, p}, x] && !IntegerQ[m] && !IntegerQ[n] && !IntegerQ[p] && GtQ[b/(b*c - a*d), 0] && GtQ[b/(b*e - a*f), 0] && !(GtQ[d/(d*a - c*b), 0] && GtQ[d/(d*e - c*f), 0] && SimplerQ[c + d*x, a + b*x]) && !(GtQ[f/(f*a - e*b), 0] && GtQ[f/(f*c - e*d), 0] && SimplerQ[e + f*x, a + b*x])
  139. Int[Times[Power[Plus[Pattern[a, Blank[]], Times[Optional[Pattern[b, Blank[]]], Pattern[x, Blank[]]]], Pattern[m, Blank[]]], Power[Plus[Optional[Pattern[c, Blank[]]], Times[Optional[Pattern[d, Blank[]]], Pattern[x, Blank[]]]], Pattern[n, Blank[]]], Power[Plus[Optional[Pattern[e, Blank[]]], Times[Optional[Pattern[f, Blank[]]], Pattern[x, Blank[]]]], Pattern[p, Blank[]]]], Pattern[x, Blank[Symbol]]] := Dist[(e + f*x)^FracPart[p]/((b/(b*e - a*f))^IntPart[p]*((b*(e + f*x))/(b*e - a*f))^FracPart[p]), Int[(a + b*x)^m*(c + d*x)^n*((b*e)/(b*e - a*f) + (b*f*x)/(b*e - a*f))^p, x], x] /; FreeQ[{a, b, c, d, e, f, m, n, p}, x] && !IntegerQ[m] && !IntegerQ[n] && !IntegerQ[p] && GtQ[b/(b*c - a*d), 0] && !GtQ[b/(b*e - a*f), 0]
  140. Int[Times[Power[Plus[Pattern[a, Blank[]], Times[Optional[Pattern[b, Blank[]]], Pattern[x, Blank[]]]], Pattern[m, Blank[]]], Power[Plus[Optional[Pattern[c, Blank[]]], Times[Optional[Pattern[d, Blank[]]], Pattern[x, Blank[]]]], Pattern[n, Blank[]]], Power[Plus[Optional[Pattern[e, Blank[]]], Times[Optional[Pattern[f, Blank[]]], Pattern[x, Blank[]]]], Pattern[p, Blank[]]]], Pattern[x, Blank[Symbol]]] := Dist[(c + d*x)^FracPart[n]/((b/(b*c - a*d))^IntPart[n]*((b*(c + d*x))/(b*c - a*d))^FracPart[n]), Int[(a + b*x)^m*((b*c)/(b*c - a*d) + (b*d*x)/(b*c - a*d))^n*(e + f*x)^p, x], x] /; FreeQ[{a, b, c, d, e, f, m, n, p}, x] && !IntegerQ[m] && !IntegerQ[n] && !IntegerQ[p] && !GtQ[b/(b*c - a*d), 0] && !SimplerQ[c + d*x, a + b*x] && !SimplerQ[e + f*x, a + b*x]
  141. Int[Times[Power[Plus[Optional[Pattern[a, Blank[]]], Times[Optional[Pattern[b, Blank[]]], Pattern[u, Blank[]]]], Optional[Pattern[m, Blank[]]]], Power[Plus[Optional[Pattern[c, Blank[]]], Times[Optional[Pattern[d, Blank[]]], Pattern[u, Blank[]]]], Optional[Pattern[n, Blank[]]]], Power[Plus[Pattern[e, Blank[]], Times[Optional[Pattern[f, Blank[]]], Pattern[u, Blank[]]]], Optional[Pattern[p, Blank[]]]]], Pattern[x, Blank[Symbol]]] := Dist[1/Coefficient[u, x, 1], Subst[Int[(a + b*x)^m*(c + d*x)^n*(e + f*x)^p, x], x, u], x] /; FreeQ[{a, b, c, d, e, f, m, n, p}, x] && LinearQ[u, x] && NeQ[u, x]
  142. Int[Times[Power[Plus[Optional[Pattern[a, Blank[]]], Times[Optional[Pattern[b, Blank[]]], Pattern[x, Blank[]]]], Optional[Pattern[m, Blank[]]]], Power[Plus[Optional[Pattern[c, Blank[]]], Times[Optional[Pattern[d, Blank[]]], Pattern[x, Blank[]]]], Optional[Pattern[n, Blank[]]]], Plus[Pattern[e, Blank[]], Times[Optional[Pattern[f, Blank[]]], Pattern[x, Blank[]]]], Plus[Optional[Pattern[g, Blank[]]], Times[Optional[Pattern[h, Blank[]]], Pattern[x, Blank[]]]]], Pattern[x, Blank[Symbol]]] := Int[ExpandIntegrand[(a + b*x)^m*(c + d*x)^n*(e + f*x)*(g + h*x), x], x] /; FreeQ[{a, b, c, d, e, f, g, h}, x] && (IGtQ[m, 0] || IntegersQ[m, n])
  143. Int[Times[Power[Plus[Optional[Pattern[a, Blank[]]], Times[Optional[Pattern[b, Blank[]]], Pattern[x, Blank[]]]], Pattern[m, Blank[]]], Power[Plus[Optional[Pattern[c, Blank[]]], Times[Optional[Pattern[d, Blank[]]], Pattern[x, Blank[]]]], Optional[Pattern[n, Blank[]]]], Plus[Pattern[e, Blank[]], Times[Optional[Pattern[f, Blank[]]], Pattern[x, Blank[]]]], Plus[Optional[Pattern[g, Blank[]]], Times[Optional[Pattern[h, Blank[]]], Pattern[x, Blank[]]]]], Pattern[x, Blank[Symbol]]] := Simp[((b^2*d*e*g - a^2*d*f*h*m - a*b*(d*(f*g + e*h) - c*f*h*(m + 1)) + b*f*h*(b*c - a*d)*(m + 1)*x)*(a + b*x)^(m + 1)*(c + d*x)^(n + 1))/(b^2*d*(b*c - a*d)*(m + 1)), x] + Dist[(a*d*f*h*m + b*(d*(f*g + e*h) - c*f*h*(m + 2)))/(b^2*d), Int[(a + b*x)^(m + 1)*(c + d*x)^n, x], x] /; FreeQ[{a, b, c, d, e, f, g, h, m, n}, x] && EqQ[m + n + 2, 0] && NeQ[m, -1] && !(SumSimplerQ[n, 1] && !SumSimplerQ[m, 1])
  144. Int[Times[Power[Plus[Optional[Pattern[a, Blank[]]], Times[Optional[Pattern[b, Blank[]]], Pattern[x, Blank[]]]], Pattern[m, Blank[]]], Power[Plus[Optional[Pattern[c, Blank[]]], Times[Optional[Pattern[d, Blank[]]], Pattern[x, Blank[]]]], Pattern[n, Blank[]]], Plus[Pattern[e, Blank[]], Times[Optional[Pattern[f, Blank[]]], Pattern[x, Blank[]]]], Plus[Optional[Pattern[g, Blank[]]], Times[Optional[Pattern[h, Blank[]]], Pattern[x, Blank[]]]]], Pattern[x, Blank[Symbol]]] := Simp[((b^2*c*d*e*g*(n + 1) + a^2*c*d*f*h*(n + 1) + a*b*(d^2*e*g*(m + 1) + c^2*f*h*(m + 1) - c*d*(f*g + e*h)*(m + n + 2)) + (a^2*d^2*f*h*(n + 1) - a*b*d^2*(f*g + e*h)*(n + 1) + b^2*(c^2*f*h*(m + 1) - c*d*(f*g + e*h)*(m + 1) + d^2*e*g*(m + n + 2)))*x)*(a + b*x)^(m + 1)*(c + d*x)^(n + 1))/(b*d*(b*c - a*d)^2*(m + 1)*(n + 1)), x] - Dist[(a^2*d^2*f*h*(2 + 3*n + n^2) + a*b*d*(n + 1)*(2*c*f*h*(m + 1) - d*(f*g + e*h)*(m + n + 3)) + b^2*(c^2*f*h*(2 + 3*m + m^2) - c*d*(f*g + e*h)*(m + 1)*(m + n + 3) + d^2*e*g*(6 + m^2 + 5*n + n^2 + m*(2*n + 5))))/(b*d*(b*c - a*d)^2*(m + 1)*(n + 1)), Int[(a + b*x)^(m + 1)*(c + d*x)^(n + 1), x], x] /; FreeQ[{a, b, c, d, e, f, g, h}, x] && LtQ[m, -1] && LtQ[n, -1]
  145. Int[Times[Power[Plus[Optional[Pattern[a, Blank[]]], Times[Optional[Pattern[b, Blank[]]], Pattern[x, Blank[]]]], Pattern[m, Blank[]]], Power[Plus[Optional[Pattern[c, Blank[]]], Times[Optional[Pattern[d, Blank[]]], Pattern[x, Blank[]]]], Optional[Pattern[n, Blank[]]]], Plus[Pattern[e, Blank[]], Times[Optional[Pattern[f, Blank[]]], Pattern[x, Blank[]]]], Plus[Optional[Pattern[g, Blank[]]], Times[Optional[Pattern[h, Blank[]]], Pattern[x, Blank[]]]]], Pattern[x, Blank[Symbol]]] := Simp[((b^3*c*e*g*(m + 2) - a^3*d*f*h*(n + 2) - a^2*b*(c*f*h*m - d*(f*g + e*h)*(m + n + 3)) - a*b^2*(c*(f*g + e*h) + d*e*g*(2*m + n + 4)) + b*(a^2*d*f*h*(m - n) - a*b*(2*c*f*h*(m + 1) - d*(f*g + e*h)*(n + 1)) + b^2*(c*(f*g + e*h)*(m + 1) - d*e*g*(m + n + 2)))*x)*(a + b*x)^(m + 1)*(c + d*x)^(n + 1))/(b^2*(b*c - a*d)^2*(m + 1)*(m + 2)), x] + Dist[(f*h)/b^2 - (d*(m + n + 3)*(a^2*d*f*h*(m - n) - a*b*(2*c*f*h*(m + 1) - d*(f*g + e*h)*(n + 1)) + b^2*(c*(f*g + e*h)*(m + 1) - d*e*g*(m + n + 2))))/(b^2*(b*c - a*d)^2*(m + 1)*(m + 2)), Int[(a + b*x)^(m + 2)*(c + d*x)^n, x], x] /; FreeQ[{a, b, c, d, e, f, g, h, m, n}, x] && (LtQ[m, -2] || (EqQ[m + n + 3, 0] && !LtQ[n, -2]))
  146. Int[Times[Power[Plus[Optional[Pattern[a, Blank[]]], Times[Optional[Pattern[b, Blank[]]], Pattern[x, Blank[]]]], Pattern[m, Blank[]]], Power[Plus[Optional[Pattern[c, Blank[]]], Times[Optional[Pattern[d, Blank[]]], Pattern[x, Blank[]]]], Optional[Pattern[n, Blank[]]]], Plus[Pattern[e, Blank[]], Times[Optional[Pattern[f, Blank[]]], Pattern[x, Blank[]]]], Plus[Optional[Pattern[g, Blank[]]], Times[Optional[Pattern[h, Blank[]]], Pattern[x, Blank[]]]]], Pattern[x, Blank[Symbol]]] := Simp[((a^2*d*f*h*(n + 2) + b^2*d*e*g*(m + n + 3) + a*b*(c*f*h*(m + 1) - d*(f*g + e*h)*(m + n + 3)) + b*f*h*(b*c - a*d)*(m + 1)*x)*(a + b*x)^(m + 1)*(c + d*x)^(n + 1))/(b^2*d*(b*c - a*d)*(m + 1)*(m + n + 3)), x] - Dist[(a^2*d^2*f*h*(n + 1)*(n + 2) + a*b*d*(n + 1)*(2*c*f*h*(m + 1) - d*(f*g + e*h)*(m + n + 3)) + b^2*(c^2*f*h*(m + 1)*(m + 2) - c*d*(f*g + e*h)*(m + 1)*(m + n + 3) + d^2*e*g*(m + n + 2)*(m + n + 3)))/(b^2*d*(b*c - a*d)*(m + 1)*(m + n + 3)), Int[(a + b*x)^(m + 1)*(c + d*x)^n, x], x] /; FreeQ[{a, b, c, d, e, f, g, h, m, n}, x] && ((GeQ[m, -2] && LtQ[m, -1]) || SumSimplerQ[m, 1]) && NeQ[m, -1] && NeQ[m + n + 3, 0]
  147. Int[Times[Power[Plus[Optional[Pattern[a, Blank[]]], Times[Optional[Pattern[b, Blank[]]], Pattern[x, Blank[]]]], Optional[Pattern[m, Blank[]]]], Power[Plus[Optional[Pattern[c, Blank[]]], Times[Optional[Pattern[d, Blank[]]], Pattern[x, Blank[]]]], Optional[Pattern[n, Blank[]]]], Plus[Pattern[e, Blank[]], Times[Optional[Pattern[f, Blank[]]], Pattern[x, Blank[]]]], Plus[Optional[Pattern[g, Blank[]]], Times[Optional[Pattern[h, Blank[]]], Pattern[x, Blank[]]]]], Pattern[x, Blank[Symbol]]] := -Simp[((a*d*f*h*(n + 2) + b*c*f*h*(m + 2) - b*d*(f*g + e*h)*(m + n + 3) - b*d*f*h*(m + n + 2)*x)*(a + b*x)^(m + 1)*(c + d*x)^(n + 1))/(b^2*d^2*(m + n + 2)*(m + n + 3)), x] + Dist[(a^2*d^2*f*h*(n + 1)*(n + 2) + a*b*d*(n + 1)*(2*c*f*h*(m + 1) - d*(f*g + e*h)*(m + n + 3)) + b^2*(c^2*f*h*(m + 1)*(m + 2) - c*d*(f*g + e*h)*(m + 1)*(m + n + 3) + d^2*e*g*(m + n + 2)*(m + n + 3)))/(b^2*d^2*(m + n + 2)*(m + n + 3)), Int[(a + b*x)^m*(c + d*x)^n, x], x] /; FreeQ[{a, b, c, d, e, f, g, h, m, n}, x] && NeQ[m + n + 2, 0] && NeQ[m + n + 3, 0]
  148. Int[Times[Power[Plus[Optional[Pattern[a, Blank[]]], Times[Optional[Pattern[b, Blank[]]], Pattern[x, Blank[]]]], Pattern[m, Blank[]]], Power[Plus[Optional[Pattern[c, Blank[]]], Times[Optional[Pattern[d, Blank[]]], Pattern[x, Blank[]]]], Pattern[n, Blank[]]], Power[Plus[Optional[Pattern[e, Blank[]]], Times[Optional[Pattern[f, Blank[]]], Pattern[x, Blank[]]]], Pattern[p, Blank[]]], Plus[Optional[Pattern[g, Blank[]]], Times[Optional[Pattern[h, Blank[]]], Pattern[x, Blank[]]]]], Pattern[x, Blank[Symbol]]] := Int[ExpandIntegrand[(a + b*x)^m*(c + d*x)^n*(e + f*x)^p*(g + h*x), x], x] /; FreeQ[{a, b, c, d, e, f, g, h, m}, x] && (IntegersQ[m, n, p] || (IGtQ[n, 0] && IGtQ[p, 0]))
  149. Int[Times[Power[Plus[Optional[Pattern[a, Blank[]]], Times[Optional[Pattern[b, Blank[]]], Pattern[x, Blank[]]]], Pattern[m, Blank[]]], Power[Plus[Optional[Pattern[c, Blank[]]], Times[Optional[Pattern[d, Blank[]]], Pattern[x, Blank[]]]], Pattern[n, Blank[]]], Power[Plus[Optional[Pattern[e, Blank[]]], Times[Optional[Pattern[f, Blank[]]], Pattern[x, Blank[]]]], Pattern[p, Blank[]]], Plus[Optional[Pattern[g, Blank[]]], Times[Optional[Pattern[h, Blank[]]], Pattern[x, Blank[]]]]], Pattern[x, Blank[Symbol]]] := Simp[((b*g - a*h)*(a + b*x)^(m + 1)*(c + d*x)^n*(e + f*x)^(p + 1))/(b*(b*e - a*f)*(m + 1)), x] - Dist[1/(b*(b*e - a*f)*(m + 1)), Int[(a + b*x)^(m + 1)*(c + d*x)^(n - 1)*(e + f*x)^p*Simp[b*c*(f*g - e*h)*(m + 1) + (b*g - a*h)*(d*e*n + c*f*(p + 1)) + d*(b*(f*g - e*h)*(m + 1) + f*(b*g - a*h)*(n + p + 1))*x, x], x], x] /; FreeQ[{a, b, c, d, e, f, g, h, p}, x] && LtQ[m, -1] && GtQ[n, 0] && IntegerQ[m]
  150. Int[Times[Power[Plus[Optional[Pattern[a, Blank[]]], Times[Optional[Pattern[b, Blank[]]], Pattern[x, Blank[]]]], Pattern[m, Blank[]]], Power[Plus[Optional[Pattern[c, Blank[]]], Times[Optional[Pattern[d, Blank[]]], Pattern[x, Blank[]]]], Pattern[n, Blank[]]], Power[Plus[Optional[Pattern[e, Blank[]]], Times[Optional[Pattern[f, Blank[]]], Pattern[x, Blank[]]]], Pattern[p, Blank[]]], Plus[Optional[Pattern[g, Blank[]]], Times[Optional[Pattern[h, Blank[]]], Pattern[x, Blank[]]]]], Pattern[x, Blank[Symbol]]] := Simp[((b*g - a*h)*(a + b*x)^(m + 1)*(c + d*x)^n*(e + f*x)^(p + 1))/(b*(b*e - a*f)*(m + 1)), x] - Dist[1/(b*(b*e - a*f)*(m + 1)), Int[(a + b*x)^(m + 1)*(c + d*x)^(n - 1)*(e + f*x)^p*Simp[b*c*(f*g - e*h)*(m + 1) + (b*g - a*h)*(d*e*n + c*f*(p + 1)) + d*(b*(f*g - e*h)*(m + 1) + f*(b*g - a*h)*(n + p + 1))*x, x], x], x] /; FreeQ[{a, b, c, d, e, f, g, h, p}, x] && LtQ[m, -1] && GtQ[n, 0] && IntegersQ[2*m, 2*n, 2*p]
  151. Int[Times[Power[Plus[Optional[Pattern[a, Blank[]]], Times[Optional[Pattern[b, Blank[]]], Pattern[x, Blank[]]]], Pattern[m, Blank[]]], Power[Plus[Optional[Pattern[c, Blank[]]], Times[Optional[Pattern[d, Blank[]]], Pattern[x, Blank[]]]], Pattern[n, Blank[]]], Power[Plus[Optional[Pattern[e, Blank[]]], Times[Optional[Pattern[f, Blank[]]], Pattern[x, Blank[]]]], Pattern[p, Blank[]]], Plus[Optional[Pattern[g, Blank[]]], Times[Optional[Pattern[h, Blank[]]], Pattern[x, Blank[]]]]], Pattern[x, Blank[Symbol]]] := Simp[((b*g - a*h)*(a + b*x)^(m + 1)*(c + d*x)^(n + 1)*(e + f*x)^(p + 1))/((m + 1)*(b*c - a*d)*(b*e - a*f)), x] + Dist[1/((m + 1)*(b*c - a*d)*(b*e - a*f)), Int[(a + b*x)^(m + 1)*(c + d*x)^n*(e + f*x)^p*Simp[(a*d*f*g - b*(d*e + c*f)*g + b*c*e*h)*(m + 1) - (b*g - a*h)*(d*e*(n + 1) + c*f*(p + 1)) - d*f*(b*g - a*h)*(m + n + p + 3)*x, x], x], x] /; FreeQ[{a, b, c, d, e, f, g, h, n, p}, x] && LtQ[m, -1] && IntegerQ[m]
  152. Int[Times[Power[Plus[Optional[Pattern[a, Blank[]]], Times[Optional[Pattern[b, Blank[]]], Pattern[x, Blank[]]]], Pattern[m, Blank[]]], Power[Plus[Optional[Pattern[c, Blank[]]], Times[Optional[Pattern[d, Blank[]]], Pattern[x, Blank[]]]], Pattern[n, Blank[]]], Power[Plus[Optional[Pattern[e, Blank[]]], Times[Optional[Pattern[f, Blank[]]], Pattern[x, Blank[]]]], Pattern[p, Blank[]]], Plus[Optional[Pattern[g, Blank[]]], Times[Optional[Pattern[h, Blank[]]], Pattern[x, Blank[]]]]], Pattern[x, Blank[Symbol]]] := Simp[((b*g - a*h)*(a + b*x)^(m + 1)*(c + d*x)^(n + 1)*(e + f*x)^(p + 1))/((m + 1)*(b*c - a*d)*(b*e - a*f)), x] + Dist[1/((m + 1)*(b*c - a*d)*(b*e - a*f)), Int[(a + b*x)^(m + 1)*(c + d*x)^n*(e + f*x)^p*Simp[(a*d*f*g - b*(d*e + c*f)*g + b*c*e*h)*(m + 1) - (b*g - a*h)*(d*e*(n + 1) + c*f*(p + 1)) - d*f*(b*g - a*h)*(m + n + p + 3)*x, x], x], x] /; FreeQ[{a, b, c, d, e, f, g, h, n, p}, x] && LtQ[m, -1] && IntegersQ[2*m, 2*n, 2*p]
  153. Int[Times[Power[Plus[Optional[Pattern[a, Blank[]]], Times[Optional[Pattern[b, Blank[]]], Pattern[x, Blank[]]]], Pattern[m, Blank[]]], Power[Plus[Optional[Pattern[c, Blank[]]], Times[Optional[Pattern[d, Blank[]]], Pattern[x, Blank[]]]], Pattern[n, Blank[]]], Power[Plus[Optional[Pattern[e, Blank[]]], Times[Optional[Pattern[f, Blank[]]], Pattern[x, Blank[]]]], Pattern[p, Blank[]]], Plus[Optional[Pattern[g, Blank[]]], Times[Optional[Pattern[h, Blank[]]], Pattern[x, Blank[]]]]], Pattern[x, Blank[Symbol]]] := Simp[(h*(a + b*x)^m*(c + d*x)^(n + 1)*(e + f*x)^(p + 1))/(d*f*(m + n + p + 2)), x] + Dist[1/(d*f*(m + n + p + 2)), Int[(a + b*x)^(m - 1)*(c + d*x)^n*(e + f*x)^p*Simp[a*d*f*g*(m + n + p + 2) - h*(b*c*e*m + a*(d*e*(n + 1) + c*f*(p + 1))) + (b*d*f*g*(m + n + p + 2) + h*(a*d*f*m - b*(d*e*(m + n + 1) + c*f*(m + p + 1))))*x, x], x], x] /; FreeQ[{a, b, c, d, e, f, g, h, n, p}, x] && GtQ[m, 0] && NeQ[m + n + p + 2, 0] && IntegerQ[m]
  154. Int[Times[Power[Plus[Optional[Pattern[a, Blank[]]], Times[Optional[Pattern[b, Blank[]]], Pattern[x, Blank[]]]], Pattern[m, Blank[]]], Power[Plus[Optional[Pattern[c, Blank[]]], Times[Optional[Pattern[d, Blank[]]], Pattern[x, Blank[]]]], Pattern[n, Blank[]]], Power[Plus[Optional[Pattern[e, Blank[]]], Times[Optional[Pattern[f, Blank[]]], Pattern[x, Blank[]]]], Pattern[p, Blank[]]], Plus[Optional[Pattern[g, Blank[]]], Times[Optional[Pattern[h, Blank[]]], Pattern[x, Blank[]]]]], Pattern[x, Blank[Symbol]]] := Simp[(h*(a + b*x)^m*(c + d*x)^(n + 1)*(e + f*x)^(p + 1))/(d*f*(m + n + p + 2)), x] + Dist[1/(d*f*(m + n + p + 2)), Int[(a + b*x)^(m - 1)*(c + d*x)^n*(e + f*x)^p*Simp[a*d*f*g*(m + n + p + 2) - h*(b*c*e*m + a*(d*e*(n + 1) + c*f*(p + 1))) + (b*d*f*g*(m + n + p + 2) + h*(a*d*f*m - b*(d*e*(m + n + 1) + c*f*(m + p + 1))))*x, x], x], x] /; FreeQ[{a, b, c, d, e, f, g, h, n, p}, x] && GtQ[m, 0] && NeQ[m + n + p + 2, 0] && IntegersQ[2*m, 2*n, 2*p]
  155. Int[Times[Power[Plus[Optional[Pattern[a, Blank[]]], Times[Optional[Pattern[b, Blank[]]], Pattern[x, Blank[]]]], Pattern[m, Blank[]]], Power[Plus[Optional[Pattern[c, Blank[]]], Times[Optional[Pattern[d, Blank[]]], Pattern[x, Blank[]]]], Pattern[n, Blank[]]], Power[Plus[Optional[Pattern[e, Blank[]]], Times[Optional[Pattern[f, Blank[]]], Pattern[x, Blank[]]]], Pattern[p, Blank[]]], Plus[Optional[Pattern[g, Blank[]]], Times[Optional[Pattern[h, Blank[]]], Pattern[x, Blank[]]]]], Pattern[x, Blank[Symbol]]] := Simp[((b*g - a*h)*(a + b*x)^(m + 1)*(c + d*x)^(n + 1)*(e + f*x)^(p + 1))/((m + 1)*(b*c - a*d)*(b*e - a*f)), x] + Dist[1/((m + 1)*(b*c - a*d)*(b*e - a*f)), Int[(a + b*x)^(m + 1)*(c + d*x)^n*(e + f*x)^p*Simp[(a*d*f*g - b*(d*e + c*f)*g + b*c*e*h)*(m + 1) - (b*g - a*h)*(d*e*(n + 1) + c*f*(p + 1)) - d*f*(b*g - a*h)*(m + n + p + 3)*x, x], x], x] /; FreeQ[{a, b, c, d, e, f, g, h, n, p}, x] && ILtQ[m + n + p + 2, 0] && NeQ[m, -1] && (SumSimplerQ[m, 1] || ( !(NeQ[n, -1] && SumSimplerQ[n, 1]) && !(NeQ[p, -1] && SumSimplerQ[p, 1])))
  156. Int[Times[Power[Plus[Optional[Pattern[a, Blank[]]], Times[Optional[Pattern[b, Blank[]]], Pattern[x, Blank[]]]], -1], Power[Plus[Optional[Pattern[c, Blank[]]], Times[Optional[Pattern[d, Blank[]]], Pattern[x, Blank[]]]], -1], Power[Plus[Optional[Pattern[e, Blank[]]], Times[Optional[Pattern[f, Blank[]]], Pattern[x, Blank[]]]], Pattern[p, Blank[]]], Plus[Optional[Pattern[g, Blank[]]], Times[Optional[Pattern[h, Blank[]]], Pattern[x, Blank[]]]]], Pattern[x, Blank[Symbol]]] := Dist[(b*g - a*h)/(b*c - a*d), Int[(e + f*x)^p/(a + b*x), x], x] - Dist[(d*g - c*h)/(b*c - a*d), Int[(e + f*x)^p/(c + d*x), x], x] /; FreeQ[{a, b, c, d, e, f, g, h}, x]
  157. Int[Times[Power[Plus[Optional[Pattern[a, Blank[]]], Times[Optional[Pattern[b, Blank[]]], Pattern[x, Blank[]]]], -1], Power[Plus[Optional[Pattern[c, Blank[]]], Times[Optional[Pattern[d, Blank[]]], Pattern[x, Blank[]]]], Pattern[n, Blank[]]], Power[Plus[Optional[Pattern[e, Blank[]]], Times[Optional[Pattern[f, Blank[]]], Pattern[x, Blank[]]]], Pattern[p, Blank[]]], Plus[Optional[Pattern[g, Blank[]]], Times[Optional[Pattern[h, Blank[]]], Pattern[x, Blank[]]]]], Pattern[x, Blank[Symbol]]] := Dist[h/b, Int[(c + d*x)^n*(e + f*x)^p, x], x] + Dist[(b*g - a*h)/b, Int[((c + d*x)^n*(e + f*x)^p)/(a + b*x), x], x] /; FreeQ[{a, b, c, d, e, f, g, h, n, p}, x]
  158. Int[Times[Power[Plus[Optional[Pattern[a, Blank[]]], Times[Optional[Pattern[b, Blank[]]], Pattern[x, Blank[]]]], Rational[-1, 2]], Power[Plus[Pattern[c, Blank[]], Times[Optional[Pattern[d, Blank[]]], Pattern[x, Blank[]]]], Rational[-1, 2]], Power[Plus[Pattern[e, Blank[]], Times[Optional[Pattern[f, Blank[]]], Pattern[x, Blank[]]]], Rational[-1, 2]], Plus[Optional[Pattern[g, Blank[]]], Times[Optional[Pattern[h, Blank[]]], Pattern[x, Blank[]]]]], Pattern[x, Blank[Symbol]]] := Dist[h/f, Int[Sqrt[e + f*x]/(Sqrt[a + b*x]*Sqrt[c + d*x]), x], x] + Dist[(f*g - e*h)/f, Int[1/(Sqrt[a + b*x]*Sqrt[c + d*x]*Sqrt[e + f*x]), x], x] /; FreeQ[{a, b, c, d, e, f, g, h}, x] && SimplerQ[a + b*x, e + f*x] && SimplerQ[c + d*x, e + f*x]
  159. Int[Times[Power[Plus[Optional[Pattern[a, Blank[]]], Times[Optional[Pattern[b, Blank[]]], Pattern[x, Blank[]]]], Pattern[m, Blank[]]], Power[Plus[Optional[Pattern[c, Blank[]]], Times[Optional[Pattern[d, Blank[]]], Pattern[x, Blank[]]]], Pattern[n, Blank[]]], Power[Plus[Optional[Pattern[e, Blank[]]], Times[Optional[Pattern[f, Blank[]]], Pattern[x, Blank[]]]], Pattern[p, Blank[]]], Plus[Optional[Pattern[g, Blank[]]], Times[Optional[Pattern[h, Blank[]]], Pattern[x, Blank[]]]]], Pattern[x, Blank[Symbol]]] := Dist[h/b, Int[(a + b*x)^(m + 1)*(c + d*x)^n*(e + f*x)^p, x], x] + Dist[(b*g - a*h)/b, Int[(a + b*x)^m*(c + d*x)^n*(e + f*x)^p, x], x] /; FreeQ[{a, b, c, d, e, f, g, h, m, n, p}, x] && (SumSimplerQ[m, 1] || ( !SumSimplerQ[n, 1] && !SumSimplerQ[p, 1]))
  160. Int[Times[Power[Plus[Optional[Pattern[a, Blank[]]], Times[Optional[Pattern[b, Blank[]]], Pattern[x, Blank[]]]], Pattern[m, Blank[]]], Power[Plus[Optional[Pattern[c, Blank[]]], Times[Optional[Pattern[d, Blank[]]], Pattern[x, Blank[]]]], Rational[1, 2]], Power[Plus[Optional[Pattern[e, Blank[]]], Times[Optional[Pattern[f, Blank[]]], Pattern[x, Blank[]]]], Rational[1, 2]], Power[Plus[Optional[Pattern[g, Blank[]]], Times[Optional[Pattern[h, Blank[]]], Pattern[x, Blank[]]]], Rational[1, 2]]], Pattern[x, Blank[Symbol]]] := Simp[((a + b*x)^(m + 1)*Sqrt[c + d*x]*Sqrt[e + f*x]*Sqrt[g + h*x])/(b*(m + 1)), x] - Dist[1/(2*b*(m + 1)), Int[((a + b*x)^(m + 1)*Simp[d*e*g + c*f*g + c*e*h + 2*(d*f*g + d*e*h + c*f*h)*x + 3*d*f*h*x^2, x])/(Sqrt[c + d*x]*Sqrt[e + f*x]*Sqrt[g + h*x]), x], x] /; FreeQ[{a, b, c, d, e, f, g, h, m}, x] && IntegerQ[2*m] && LtQ[m, -1]
  161. Int[Times[Power[Plus[Optional[Pattern[a, Blank[]]], Times[Optional[Pattern[b, Blank[]]], Pattern[x, Blank[]]]], Pattern[m, Blank[]]], Power[Plus[Optional[Pattern[c, Blank[]]], Times[Optional[Pattern[d, Blank[]]], Pattern[x, Blank[]]]], Rational[1, 2]], Power[Plus[Optional[Pattern[e, Blank[]]], Times[Optional[Pattern[f, Blank[]]], Pattern[x, Blank[]]]], Rational[1, 2]], Power[Plus[Optional[Pattern[g, Blank[]]], Times[Optional[Pattern[h, Blank[]]], Pattern[x, Blank[]]]], Rational[1, 2]]], Pattern[x, Blank[Symbol]]] := Simp[(2*(a + b*x)^(m + 1)*Sqrt[c + d*x]*Sqrt[e + f*x]*Sqrt[g + h*x])/(b*(2*m + 5)), x] + Dist[1/(b*(2*m + 5)), Int[((a + b*x)^m*Simp[3*b*c*e*g - a*(d*e*g + c*f*g + c*e*h) + 2*(b*(d*e*g + c*f*g + c*e*h) - a*(d*f*g + d*e*h + c*f*h))*x - (3*a*d*f*h - b*(d*f*g + d*e*h + c*f*h))*x^2, x])/(Sqrt[c + d*x]*Sqrt[e + f*x]*Sqrt[g + h*x]), x], x] /; FreeQ[{a, b, c, d, e, f, g, h, m}, x] && IntegerQ[2*m] && !LtQ[m, -1]
  162. Int[Times[Power[Plus[Optional[Pattern[a, Blank[]]], Times[Optional[Pattern[b, Blank[]]], Pattern[x, Blank[]]]], Pattern[m, Blank[]]], Power[Plus[Optional[Pattern[c, Blank[]]], Times[Optional[Pattern[d, Blank[]]], Pattern[x, Blank[]]]], Rational[-1, 2]], Power[Plus[Optional[Pattern[e, Blank[]]], Times[Optional[Pattern[f, Blank[]]], Pattern[x, Blank[]]]], Rational[1, 2]], Power[Plus[Optional[Pattern[g, Blank[]]], Times[Optional[Pattern[h, Blank[]]], Pattern[x, Blank[]]]], Rational[1, 2]]], Pattern[x, Blank[Symbol]]] := Simp[(2*(a + b*x)^m*Sqrt[c + d*x]*Sqrt[e + f*x]*Sqrt[g + h*x])/(d*(2*m + 3)), x] - Dist[1/(d*(2*m + 3)), Int[((a + b*x)^(m - 1)/(Sqrt[c + d*x]*Sqrt[e + f*x]*Sqrt[g + h*x]))*Simp[2*b*c*e*g*m + a*(c*(f*g + e*h) - 2*d*e*g*(m + 1)) - (b*(2*d*e*g - c*(f*g + e*h)*(2*m + 1)) - a*(2*c*f*h - d*(2*m + 1)*(f*g + e*h)))*x - (2*a*d*f*h*m + b*(d*(f*g + e*h) - 2*c*f*h*(m + 1)))*x^2, x], x], x] /; FreeQ[{a, b, c, d, e, f, g, h, m}, x] && IntegerQ[2*m] && GtQ[m, 0]
  163. Int[Times[Power[Plus[Optional[Pattern[a, Blank[]]], Times[Optional[Pattern[b, Blank[]]], Pattern[x, Blank[]]]], -1], Power[Plus[Optional[Pattern[c, Blank[]]], Times[Optional[Pattern[d, Blank[]]], Pattern[x, Blank[]]]], Rational[-1, 2]], Power[Plus[Optional[Pattern[e, Blank[]]], Times[Optional[Pattern[f, Blank[]]], Pattern[x, Blank[]]]], Rational[1, 2]], Power[Plus[Optional[Pattern[g, Blank[]]], Times[Optional[Pattern[h, Blank[]]], Pattern[x, Blank[]]]], Rational[1, 2]]], Pattern[x, Blank[Symbol]]] := Dist[((b*e - a*f)*(b*g - a*h))/b^2, Int[1/((a + b*x)*Sqrt[c + d*x]*Sqrt[e + f*x]*Sqrt[g + h*x]), x], x] + Dist[1/b^2, Int[Simp[b*f*g + b*e*h - a*f*h + b*f*h*x, x]/(Sqrt[c + d*x]*Sqrt[e + f*x]*Sqrt[g + h*x]), x], x] /; FreeQ[{a, b, c, d, e, f, g, h}, x]
  164. Int[Times[Power[Plus[Optional[Pattern[a, Blank[]]], Times[Optional[Pattern[b, Blank[]]], Pattern[x, Blank[]]]], Pattern[m, Blank[]]], Power[Plus[Optional[Pattern[c, Blank[]]], Times[Optional[Pattern[d, Blank[]]], Pattern[x, Blank[]]]], Rational[-1, 2]], Power[Plus[Optional[Pattern[e, Blank[]]], Times[Optional[Pattern[f, Blank[]]], Pattern[x, Blank[]]]], Rational[1, 2]], Power[Plus[Optional[Pattern[g, Blank[]]], Times[Optional[Pattern[h, Blank[]]], Pattern[x, Blank[]]]], Rational[1, 2]]], Pattern[x, Blank[Symbol]]] := Simp[((a + b*x)^(m + 1)*Sqrt[c + d*x]*Sqrt[e + f*x]*Sqrt[g + h*x])/((m + 1)*(b*c - a*d)), x] - Dist[1/(2*(m + 1)*(b*c - a*d)), Int[((a + b*x)^(m + 1)/(Sqrt[c + d*x]*Sqrt[e + f*x]*Sqrt[g + h*x]))*Simp[c*(f*g + e*h) + d*e*g*(2*m + 3) + 2*(c*f*h + d*(m + 2)*(f*g + e*h))*x + d*f*h*(2*m + 5)*x^2, x], x], x] /; FreeQ[{a, b, c, d, e, f, g, h, m}, x] && IntegerQ[2*m] && LtQ[m, -1]
  165. Int[Times[Power[Plus[Optional[Pattern[a, Blank[]]], Times[Optional[Pattern[b, Blank[]]], Pattern[x, Blank[]]]], Rational[1, 2]], Power[Plus[Optional[Pattern[c, Blank[]]], Times[Optional[Pattern[d, Blank[]]], Pattern[x, Blank[]]]], Rational[-1, 2]], Power[Plus[Optional[Pattern[e, Blank[]]], Times[Optional[Pattern[f, Blank[]]], Pattern[x, Blank[]]]], Rational[-1, 2]], Power[Plus[Optional[Pattern[g, Blank[]]], Times[Optional[Pattern[h, Blank[]]], Pattern[x, Blank[]]]], Rational[-1, 2]]], Pattern[x, Blank[Symbol]]] := Dist[(2*(a + b*x)*Sqrt[((b*g - a*h)*(c + d*x))/((d*g - c*h)*(a + b*x))]*Sqrt[((b*g - a*h)*(e + f*x))/((f*g - e*h)*(a + b*x))])/(Sqrt[c + d*x]*Sqrt[e + f*x]), Subst[Int[1/((h - b*x^2)*Sqrt[1 + ((b*c - a*d)*x^2)/(d*g - c*h)]*Sqrt[1 + ((b*e - a*f)*x^2)/(f*g - e*h)]), x], x, Sqrt[g + h*x]/Sqrt[a + b*x]], x] /; FreeQ[{a, b, c, d, e, f, g, h}, x]
  166. Int[Times[Power[Plus[Optional[Pattern[a, Blank[]]], Times[Optional[Pattern[b, Blank[]]], Pattern[x, Blank[]]]], Rational[3, 2]], Power[Plus[Optional[Pattern[c, Blank[]]], Times[Optional[Pattern[d, Blank[]]], Pattern[x, Blank[]]]], Rational[-1, 2]], Power[Plus[Optional[Pattern[e, Blank[]]], Times[Optional[Pattern[f, Blank[]]], Pattern[x, Blank[]]]], Rational[-1, 2]], Power[Plus[Optional[Pattern[g, Blank[]]], Times[Optional[Pattern[h, Blank[]]], Pattern[x, Blank[]]]], Rational[-1, 2]]], Pattern[x, Blank[Symbol]]] := Dist[b/d, Int[(Sqrt[a + b*x]*Sqrt[c + d*x])/(Sqrt[e + f*x]*Sqrt[g + h*x]), x], x] - Dist[(b*c - a*d)/d, Int[Sqrt[a + b*x]/(Sqrt[c + d*x]*Sqrt[e + f*x]*Sqrt[g + h*x]), x], x] /; FreeQ[{a, b, c, d, e, f, g, h}, x]
  167. Int[Times[Power[Plus[Optional[Pattern[a, Blank[]]], Times[Optional[Pattern[b, Blank[]]], Pattern[x, Blank[]]]], Pattern[m, Blank[]]], Power[Plus[Optional[Pattern[c, Blank[]]], Times[Optional[Pattern[d, Blank[]]], Pattern[x, Blank[]]]], Rational[-1, 2]], Power[Plus[Optional[Pattern[e, Blank[]]], Times[Optional[Pattern[f, Blank[]]], Pattern[x, Blank[]]]], Rational[-1, 2]], Power[Plus[Optional[Pattern[g, Blank[]]], Times[Optional[Pattern[h, Blank[]]], Pattern[x, Blank[]]]], Rational[-1, 2]]], Pattern[x, Blank[Symbol]]] := Simp[(2*b^2*(a + b*x)^(m - 2)*Sqrt[c + d*x]*Sqrt[e + f*x]*Sqrt[g + h*x])/(d*f*h*(2*m - 1)), x] - Dist[1/(d*f*h*(2*m - 1)), Int[((a + b*x)^(m - 3)/(Sqrt[c + d*x]*Sqrt[e + f*x]*Sqrt[g + h*x]))*Simp[a*b^2*(d*e*g + c*f*g + c*e*h) + 2*b^3*c*e*g*(m - 2) - a^3*d*f*h*(2*m - 1) + b*(2*a*b*(d*f*g + d*e*h + c*f*h) + b^2*(2*m - 3)*(d*e*g + c*f*g + c*e*h) - 3*a^2*d*f*h*(2*m - 1))*x - 2*b^2*(m - 1)*(3*a*d*f*h - b*(d*f*g + d*e*h + c*f*h))*x^2, x], x], x] /; FreeQ[{a, b, c, d, e, f, g, h}, x] && IntegerQ[2*m] && GeQ[m, 2]
  168. Int[Times[Power[Plus[Optional[Pattern[a, Blank[]]], Times[Optional[Pattern[b, Blank[]]], Pattern[x, Blank[]]]], -1], Power[Plus[Optional[Pattern[c, Blank[]]], Times[Optional[Pattern[d, Blank[]]], Pattern[x, Blank[]]]], Rational[-1, 2]], Power[Plus[Optional[Pattern[e, Blank[]]], Times[Optional[Pattern[f, Blank[]]], Pattern[x, Blank[]]]], Rational[-1, 2]], Power[Plus[Optional[Pattern[g, Blank[]]], Times[Optional[Pattern[h, Blank[]]], Pattern[x, Blank[]]]], Rational[-1, 2]]], Pattern[x, Blank[Symbol]]] := Dist[-2, Subst[Int[1/(Simp[b*c - a*d - b*x^2, x]*Sqrt[Simp[(d*e - c*f)/d + (f*x^2)/d, x]]*Sqrt[Simp[(d*g - c*h)/d + (h*x^2)/d, x]]), x], x, Sqrt[c + d*x]], x] /; FreeQ[{a, b, c, d, e, f, g, h}, x] && GtQ[(d*e - c*f)/d, 0]
  169. Int[Times[Power[Plus[Optional[Pattern[a, Blank[]]], Times[Optional[Pattern[b, Blank[]]], Pattern[x, Blank[]]]], -1], Power[Plus[Optional[Pattern[c, Blank[]]], Times[Optional[Pattern[d, Blank[]]], Pattern[x, Blank[]]]], Rational[-1, 2]], Power[Plus[Optional[Pattern[e, Blank[]]], Times[Optional[Pattern[f, Blank[]]], Pattern[x, Blank[]]]], Rational[-1, 2]], Power[Plus[Optional[Pattern[g, Blank[]]], Times[Optional[Pattern[h, Blank[]]], Pattern[x, Blank[]]]], Rational[-1, 2]]], Pattern[x, Blank[Symbol]]] := Dist[-2, Subst[Int[1/(Simp[b*c - a*d - b*x^2, x]*Sqrt[Simp[(d*e - c*f)/d + (f*x^2)/d, x]]*Sqrt[Simp[(d*g - c*h)/d + (h*x^2)/d, x]]), x], x, Sqrt[c + d*x]], x] /; FreeQ[{a, b, c, d, e, f, g, h}, x] && !SimplerQ[e + f*x, c + d*x] && !SimplerQ[g + h*x, c + d*x]
  170. Int[Times[Power[Plus[Optional[Pattern[a, Blank[]]], Times[Optional[Pattern[b, Blank[]]], Pattern[x, Blank[]]]], Rational[-1, 2]], Power[Plus[Optional[Pattern[c, Blank[]]], Times[Optional[Pattern[d, Blank[]]], Pattern[x, Blank[]]]], Rational[-1, 2]], Power[Plus[Optional[Pattern[e, Blank[]]], Times[Optional[Pattern[f, Blank[]]], Pattern[x, Blank[]]]], Rational[-1, 2]], Power[Plus[Optional[Pattern[g, Blank[]]], Times[Optional[Pattern[h, Blank[]]], Pattern[x, Blank[]]]], Rational[-1, 2]]], Pattern[x, Blank[Symbol]]] := Dist[(2*Sqrt[g + h*x]*Sqrt[((b*e - a*f)*(c + d*x))/((d*e - c*f)*(a + b*x))])/((f*g - e*h)*Sqrt[c + d*x]*Sqrt[-(((b*e - a*f)*(g + h*x))/((f*g - e*h)*(a + b*x)))]), Subst[Int[1/(Sqrt[1 + ((b*c - a*d)*x^2)/(d*e - c*f)]*Sqrt[1 - ((b*g - a*h)*x^2)/(f*g - e*h)]), x], x, Sqrt[e + f*x]/Sqrt[a + b*x]], x] /; FreeQ[{a, b, c, d, e, f, g, h}, x]
  171. Int[Times[Power[Plus[Optional[Pattern[a, Blank[]]], Times[Optional[Pattern[b, Blank[]]], Pattern[x, Blank[]]]], Rational[-3, 2]], Power[Plus[Optional[Pattern[c, Blank[]]], Times[Optional[Pattern[d, Blank[]]], Pattern[x, Blank[]]]], Rational[-1, 2]], Power[Plus[Optional[Pattern[e, Blank[]]], Times[Optional[Pattern[f, Blank[]]], Pattern[x, Blank[]]]], Rational[-1, 2]], Power[Plus[Optional[Pattern[g, Blank[]]], Times[Optional[Pattern[h, Blank[]]], Pattern[x, Blank[]]]], Rational[-1, 2]]], Pattern[x, Blank[Symbol]]] := -Dist[d/(b*c - a*d), Int[1/(Sqrt[a + b*x]*Sqrt[c + d*x]*Sqrt[e + f*x]*Sqrt[g + h*x]), x], x] + Dist[b/(b*c - a*d), Int[Sqrt[c + d*x]/((a + b*x)^(3/2)*Sqrt[e + f*x]*Sqrt[g + h*x]), x], x] /; FreeQ[{a, b, c, d, e, f, g, h}, x]
  172. Int[Times[Power[Plus[Optional[Pattern[a, Blank[]]], Times[Optional[Pattern[b, Blank[]]], Pattern[x, Blank[]]]], Pattern[m, Blank[]]], Power[Plus[Optional[Pattern[c, Blank[]]], Times[Optional[Pattern[d, Blank[]]], Pattern[x, Blank[]]]], Rational[-1, 2]], Power[Plus[Optional[Pattern[e, Blank[]]], Times[Optional[Pattern[f, Blank[]]], Pattern[x, Blank[]]]], Rational[-1, 2]], Power[Plus[Optional[Pattern[g, Blank[]]], Times[Optional[Pattern[h, Blank[]]], Pattern[x, Blank[]]]], Rational[-1, 2]]], Pattern[x, Blank[Symbol]]] := Simp[(b^2*(a + b*x)^(m + 1)*Sqrt[c + d*x]*Sqrt[e + f*x]*Sqrt[g + h*x])/((m + 1)*(b*c - a*d)*(b*e - a*f)*(b*g - a*h)), x] - Dist[1/(2*(m + 1)*(b*c - a*d)*(b*e - a*f)*(b*g - a*h)), Int[((a + b*x)^(m + 1)/(Sqrt[c + d*x]*Sqrt[e + f*x]*Sqrt[g + h*x]))*Simp[2*a^2*d*f*h*(m + 1) - 2*a*b*(m + 1)*(d*f*g + d*e*h + c*f*h) + b^2*(2*m + 3)*(d*e*g + c*f*g + c*e*h) - 2*b*(a*d*f*h*(m + 1) - b*(m + 2)*(d*f*g + d*e*h + c*f*h))*x + d*f*h*(2*m + 5)*b^2*x^2, x], x], x] /; FreeQ[{a, b, c, d, e, f, g, h}, x] && IntegerQ[2*m] && LeQ[m, -2]
  173. Int[Times[Power[Plus[Optional[Pattern[a, Blank[]]], Times[Optional[Pattern[b, Blank[]]], Pattern[x, Blank[]]]], Rational[1, 2]], Power[Plus[Optional[Pattern[c, Blank[]]], Times[Optional[Pattern[d, Blank[]]], Pattern[x, Blank[]]]], Rational[1, 2]], Power[Plus[Optional[Pattern[e, Blank[]]], Times[Optional[Pattern[f, Blank[]]], Pattern[x, Blank[]]]], Rational[-1, 2]], Power[Plus[Optional[Pattern[g, Blank[]]], Times[Optional[Pattern[h, Blank[]]], Pattern[x, Blank[]]]], Rational[-1, 2]]], Pattern[x, Blank[Symbol]]] := Simp[(Sqrt[a + b*x]*Sqrt[c + d*x]*Sqrt[g + h*x])/(h*Sqrt[e + f*x]), x] + (-Dist[((d*e - c*f)*(f*g - e*h))/(2*f*h), Int[Sqrt[a + b*x]/(Sqrt[c + d*x]*(e + f*x)^(3/2)*Sqrt[g + h*x]), x], x] + Dist[((d*e - c*f)*(b*f*g + b*e*h - 2*a*f*h))/(2*f^2*h), Int[1/(Sqrt[a + b*x]*Sqrt[c + d*x]*Sqrt[e + f*x]*Sqrt[g + h*x]), x], x] + Dist[(a*d*f*h - b*(d*f*g + d*e*h - c*f*h))/(2*f^2*h), Int[Sqrt[e + f*x]/(Sqrt[a + b*x]*Sqrt[c + d*x]*Sqrt[g + h*x]), x], x]) /; FreeQ[{a, b, c, d, e, f, g, h}, x]
  174. Int[Times[Power[Plus[Optional[Pattern[a, Blank[]]], Times[Optional[Pattern[b, Blank[]]], Pattern[x, Blank[]]]], Pattern[m, Blank[]]], Power[Plus[Optional[Pattern[c, Blank[]]], Times[Optional[Pattern[d, Blank[]]], Pattern[x, Blank[]]]], Rational[1, 2]], Power[Plus[Optional[Pattern[e, Blank[]]], Times[Optional[Pattern[f, Blank[]]], Pattern[x, Blank[]]]], Rational[-1, 2]], Power[Plus[Optional[Pattern[g, Blank[]]], Times[Optional[Pattern[h, Blank[]]], Pattern[x, Blank[]]]], Rational[-1, 2]]], Pattern[x, Blank[Symbol]]] := Simp[(2*b*(a + b*x)^(m - 1)*Sqrt[c + d*x]*Sqrt[e + f*x]*Sqrt[g + h*x])/(f*h*(2*m + 1)), x] - Dist[1/(f*h*(2*m + 1)), Int[((a + b*x)^(m - 2)/(Sqrt[c + d*x]*Sqrt[e + f*x]*Sqrt[g + h*x]))*Simp[a*b*(d*e*g + c*(f*g + e*h)) + 2*b^2*c*e*g*(m - 1) - a^2*c*f*h*(2*m + 1) + (b^2*(2*m - 1)*(d*e*g + c*(f*g + e*h)) - a^2*d*f*h*(2*m + 1) + 2*a*b*(d*f*g + d*e*h - 2*c*f*h*m))*x - b*(a*d*f*h*(4*m - 1) + b*(c*f*h - 2*d*(f*g + e*h)*m))*x^2, x], x], x] /; FreeQ[{a, b, c, d, e, f, g, h, m}, x] && IntegerQ[2*m] && GtQ[m, 1]
  175. Int[Times[Power[Plus[Optional[Pattern[a, Blank[]]], Times[Optional[Pattern[b, Blank[]]], Pattern[x, Blank[]]]], -1], Power[Plus[Optional[Pattern[c, Blank[]]], Times[Optional[Pattern[d, Blank[]]], Pattern[x, Blank[]]]], Rational[1, 2]], Power[Plus[Optional[Pattern[e, Blank[]]], Times[Optional[Pattern[f, Blank[]]], Pattern[x, Blank[]]]], Rational[-1, 2]], Power[Plus[Optional[Pattern[g, Blank[]]], Times[Optional[Pattern[h, Blank[]]], Pattern[x, Blank[]]]], Rational[-1, 2]]], Pattern[x, Blank[Symbol]]] := Dist[d/b, Int[1/(Sqrt[c + d*x]*Sqrt[e + f*x]*Sqrt[g + h*x]), x], x] + Dist[(b*c - a*d)/b, Int[1/((a + b*x)*Sqrt[c + d*x]*Sqrt[e + f*x]*Sqrt[g + h*x]), x], x] /; FreeQ[{a, b, c, d, e, f, g, h}, x]
  176. Int[Times[Power[Plus[Optional[Pattern[a, Blank[]]], Times[Optional[Pattern[b, Blank[]]], Pattern[x, Blank[]]]], Rational[-3, 2]], Power[Plus[Optional[Pattern[c, Blank[]]], Times[Optional[Pattern[d, Blank[]]], Pattern[x, Blank[]]]], Rational[1, 2]], Power[Plus[Optional[Pattern[e, Blank[]]], Times[Optional[Pattern[f, Blank[]]], Pattern[x, Blank[]]]], Rational[-1, 2]], Power[Plus[Optional[Pattern[g, Blank[]]], Times[Optional[Pattern[h, Blank[]]], Pattern[x, Blank[]]]], Rational[-1, 2]]], Pattern[x, Blank[Symbol]]] := Dist[(-2*Sqrt[c + d*x]*Sqrt[-(((b*e - a*f)*(g + h*x))/((f*g - e*h)*(a + b*x)))])/((b*e - a*f)*Sqrt[g + h*x]*Sqrt[((b*e - a*f)*(c + d*x))/((d*e - c*f)*(a + b*x))]), Subst[Int[Sqrt[1 + ((b*c - a*d)*x^2)/(d*e - c*f)]/Sqrt[1 - ((b*g - a*h)*x^2)/(f*g - e*h)], x], x, Sqrt[e + f*x]/Sqrt[a + b*x]], x] /; FreeQ[{a, b, c, d, e, f, g, h}, x]
  177. Int[Times[Power[Plus[Optional[Pattern[a, Blank[]]], Times[Optional[Pattern[b, Blank[]]], Pattern[x, Blank[]]]], Pattern[m, Blank[]]], Power[Plus[Optional[Pattern[c, Blank[]]], Times[Optional[Pattern[d, Blank[]]], Pattern[x, Blank[]]]], Rational[1, 2]], Power[Plus[Optional[Pattern[e, Blank[]]], Times[Optional[Pattern[f, Blank[]]], Pattern[x, Blank[]]]], Rational[-1, 2]], Power[Plus[Optional[Pattern[g, Blank[]]], Times[Optional[Pattern[h, Blank[]]], Pattern[x, Blank[]]]], Rational[-1, 2]]], Pattern[x, Blank[Symbol]]] := Simp[(b*(a + b*x)^(m + 1)*Sqrt[c + d*x]*Sqrt[e + f*x]*Sqrt[g + h*x])/((m + 1)*(b*e - a*f)*(b*g - a*h)), x] + Dist[1/(2*(m + 1)*(b*e - a*f)*(b*g - a*h)), Int[((a + b*x)^(m + 1)/(Sqrt[c + d*x]*Sqrt[e + f*x]*Sqrt[g + h*x]))*Simp[2*a*c*f*h*(m + 1) - b*(d*e*g + c*(2*m + 3)*(f*g + e*h)) + 2*(a*d*f*h*(m + 1) - b*(m + 2)*(d*f*g + d*e*h + c*f*h))*x - b*d*f*h*(2*m + 5)*x^2, x], x], x] /; FreeQ[{a, b, c, d, e, f, g, h, m}, x] && IntegerQ[2*m] && LeQ[m, -2]
  178. Int[Times[Power[Plus[Optional[Pattern[a, Blank[]]], Times[Optional[Pattern[b, Blank[]]], Pattern[x, Blank[]]]], -1], Power[Plus[Optional[Pattern[c, Blank[]]], Times[Optional[Pattern[d, Blank[]]], Pattern[x, Blank[]]]], -1], Power[Plus[Optional[Pattern[e, Blank[]]], Times[Optional[Pattern[f, Blank[]]], Pattern[x, Blank[]]]], Pattern[p, Blank[]]], Power[Plus[Optional[Pattern[g, Blank[]]], Times[Optional[Pattern[h, Blank[]]], Pattern[x, Blank[]]]], Pattern[q, Blank[]]]], Pattern[x, Blank[Symbol]]] := Dist[(b*e - a*f)/(b*c - a*d), Int[((e + f*x)^(p - 1)*(g + h*x)^q)/(a + b*x), x], x] - Dist[(d*e - c*f)/(b*c - a*d), Int[((e + f*x)^(p - 1)*(g + h*x)^q)/(c + d*x), x], x] /; FreeQ[{a, b, c, d, e, f, g, h, q}, x] && LtQ[0, p, 1]
  179. Int[Times[Power[Plus[Optional[Pattern[a, Blank[]]], Times[Optional[Pattern[b, Blank[]]], Pattern[x, Blank[]]]], Optional[Pattern[m, Blank[]]]], Power[Plus[Optional[Pattern[c, Blank[]]], Times[Optional[Pattern[d, Blank[]]], Pattern[x, Blank[]]]], Pattern[n, Blank[]]], Power[Plus[Optional[Pattern[e, Blank[]]], Times[Optional[Pattern[f, Blank[]]], Pattern[x, Blank[]]]], Rational[-1, 2]], Power[Plus[Optional[Pattern[g, Blank[]]], Times[Optional[Pattern[h, Blank[]]], Pattern[x, Blank[]]]], Rational[-1, 2]]], Pattern[x, Blank[Symbol]]] := Int[ExpandIntegrand[1/(Sqrt[c + d*x]*Sqrt[e + f*x]*Sqrt[g + h*x]), (a + b*x)^m*(c + d*x)^(n + 1/2), x], x] /; FreeQ[{a, b, c, d, e, f, g, h}, x] && IntegerQ[m] && IntegerQ[n + 1/2]
  180. Int[Times[Power[Plus[Optional[Pattern[a, Blank[]]], Times[Optional[Pattern[b, Blank[]]], Pattern[x, Blank[]]]], Pattern[m, Blank[]]], Power[Plus[Optional[Pattern[c, Blank[]]], Times[Optional[Pattern[d, Blank[]]], Pattern[x, Blank[]]]], Pattern[n, Blank[]]], Power[Plus[Optional[Pattern[e, Blank[]]], Times[Optional[Pattern[f, Blank[]]], Pattern[x, Blank[]]]], Pattern[p, Blank[]]], Power[Plus[Optional[Pattern[g, Blank[]]], Times[Optional[Pattern[h, Blank[]]], Pattern[x, Blank[]]]], Pattern[q, Blank[]]]], Pattern[x, Blank[Symbol]]] := Int[ExpandIntegrand[(a + b*x)^m*(c + d*x)^n*(e + f*x)^p*(g + h*x)^q, x], x] /; FreeQ[{a, b, c, d, e, f, g, h, m, n}, x] && IntegersQ[p, q]
  181. Int[Times[Power[Plus[Optional[Pattern[a, Blank[]]], Times[Optional[Pattern[b, Blank[]]], Pattern[x, Blank[]]]], Pattern[m, Blank[]]], Power[Plus[Optional[Pattern[c, Blank[]]], Times[Optional[Pattern[d, Blank[]]], Pattern[x, Blank[]]]], Pattern[n, Blank[]]], Power[Plus[Optional[Pattern[e, Blank[]]], Times[Optional[Pattern[f, Blank[]]], Pattern[x, Blank[]]]], Pattern[p, Blank[]]], Power[Plus[Optional[Pattern[g, Blank[]]], Times[Optional[Pattern[h, Blank[]]], Pattern[x, Blank[]]]], Pattern[q, Blank[]]]], Pattern[x, Blank[Symbol]]] := Dist[h/b, Int[(a + b*x)^(m + 1)*(c + d*x)^n*(e + f*x)^p*(g + h*x)^(q - 1), x], x] + Dist[(b*g - a*h)/b, Int[(a + b*x)^m*(c + d*x)^n*(e + f*x)^p*(g + h*x)^(q - 1), x], x] /; FreeQ[{a, b, c, d, e, f, g, h, m, n, p}, x] && IGtQ[q, 0] && (SumSimplerQ[m, 1] || ( !SumSimplerQ[n, 1] && !SumSimplerQ[p, 1]))
  182. Int[Times[Power[Plus[Optional[Pattern[a, Blank[]]], Times[Optional[Pattern[b, Blank[]]], Pattern[x, Blank[]]]], Optional[Pattern[m, Blank[]]]], Power[Plus[Optional[Pattern[c, Blank[]]], Times[Optional[Pattern[d, Blank[]]], Pattern[x, Blank[]]]], Optional[Pattern[n, Blank[]]]], Power[Plus[Optional[Pattern[e, Blank[]]], Times[Optional[Pattern[f, Blank[]]], Pattern[x, Blank[]]]], Optional[Pattern[p, Blank[]]]], Power[Plus[Optional[Pattern[g, Blank[]]], Times[Optional[Pattern[h, Blank[]]], Pattern[x, Blank[]]]], Optional[Pattern[q, Blank[]]]]], Pattern[x, Blank[Symbol]]] := CannotIntegrate[(a + b*x)^m*(c + d*x)^n*(e + f*x)^p*(g + h*x)^q, x] /; FreeQ[{a, b, c, d, e, f, g, h, m, n, p, q}, x]
  183. Int[Times[Power[Plus[Optional[Pattern[a, Blank[]]], Times[Optional[Pattern[b, Blank[]]], Pattern[u, Blank[]]]], Optional[Pattern[m, Blank[]]]], Power[Plus[Optional[Pattern[c, Blank[]]], Times[Optional[Pattern[d, Blank[]]], Pattern[u, Blank[]]]], Optional[Pattern[n, Blank[]]]], Power[Plus[Optional[Pattern[e, Blank[]]], Times[Optional[Pattern[f, Blank[]]], Pattern[u, Blank[]]]], Optional[Pattern[p, Blank[]]]], Power[Plus[Optional[Pattern[g, Blank[]]], Times[Optional[Pattern[h, Blank[]]], Pattern[u, Blank[]]]], Optional[Pattern[q, Blank[]]]]], Pattern[x, Blank[Symbol]]] := Dist[1/Coefficient[u, x, 1], Subst[Int[(a + b*x)^m*(c + d*x)^n*(e + f*x)^p*(g + h*x)^q, x], x, u], x] /; FreeQ[{a, b, c, d, e, f, g, h, m, n, p, q}, x] && LinearQ[u, x] && NeQ[u, x]
  184. Int[Power[Times[Optional[Pattern[i, Blank[]]], Power[Plus[Optional[Pattern[a, Blank[]]], Times[Optional[Pattern[b, Blank[]]], Pattern[x, Blank[]]]], Pattern[m, Blank[]]], Power[Plus[Optional[Pattern[c, Blank[]]], Times[Optional[Pattern[d, Blank[]]], Pattern[x, Blank[]]]], Pattern[n, Blank[]]], Power[Plus[Optional[Pattern[e, Blank[]]], Times[Optional[Pattern[f, Blank[]]], Pattern[x, Blank[]]]], Pattern[p, Blank[]]], Power[Plus[Optional[Pattern[g, Blank[]]], Times[Optional[Pattern[h, Blank[]]], Pattern[x, Blank[]]]], Pattern[q, Blank[]]]], Pattern[r, Blank[]]], Pattern[x, Blank[Symbol]]] := Dist[(i*(a + b*x)^m*(c + d*x)^n*(e + f*x)^p*(g + h*x)^q)^r/((a + b*x)^(m*r)*(c + d*x)^(n*r)*(e + f*x)^(p*r)*(g + h*x)^(q*r)), Int[(a + b*x)^(m*r)*(c + d*x)^(n*r)*(e + f*x)^(p*r)*(g + h*x)^(q*r), x], x] /; FreeQ[{a, b, c, d, e, f, g, h, i, m, n, p, q, r}, x]
  185. Int[Power[Pattern[u, Blank[]], Pattern[m, Blank[]]], Pattern[x, Blank[Symbol]]] := Int[ExpandToSum[u, x]^m, x] /; FreeQ[m, x] && LinearQ[u, x] && !LinearMatchQ[u, x]
  186. Int[Times[Power[Pattern[u, Blank[]], Optional[Pattern[m, Blank[]]]], Power[Pattern[v, Blank[]], Optional[Pattern[n, Blank[]]]]], Pattern[x, Blank[Symbol]]] := Int[ExpandToSum[u, x]^m*ExpandToSum[v, x]^n, x] /; FreeQ[{m, n}, x] && LinearQ[{u, v}, x] && !LinearMatchQ[{u, v}, x]
  187. Int[Times[Power[Pattern[u, Blank[]], Optional[Pattern[m, Blank[]]]], Power[Pattern[v, Blank[]], Optional[Pattern[n, Blank[]]]], Power[Pattern[w, Blank[]], Optional[Pattern[p, Blank[]]]]], Pattern[x, Blank[Symbol]]] := Int[ExpandToSum[u, x]^m*ExpandToSum[v, x]^n*ExpandToSum[w, x]^p, x] /; FreeQ[{m, n, p}, x] && LinearQ[{u, v, w}, x] && !LinearMatchQ[{u, v, w}, x]
  188. Int[Times[Power[Pattern[u, Blank[]], Optional[Pattern[m, Blank[]]]], Power[Pattern[v, Blank[]], Optional[Pattern[n, Blank[]]]], Power[Pattern[w, Blank[]], Optional[Pattern[p, Blank[]]]], Power[Pattern[z, Blank[]], Optional[Pattern[q, Blank[]]]]], Pattern[x, Blank[Symbol]]] := Int[ExpandToSum[u, x]^m*ExpandToSum[v, x]^n*ExpandToSum[w, x]^p*ExpandToSum[z, x]^q, x] /; FreeQ[{m, n, p, q}, x] && LinearQ[{u, v, w, z}, x] && !LinearMatchQ[{u, v, w, z}, x]
  189. Int[Power[Times[Optional[Pattern[b, Blank[]]], Power[Pattern[x, Blank[]], Pattern[n, Blank[]]]], Pattern[p, Blank[]]], Pattern[x, Blank[Symbol]]] := Dist[(b^IntPart[p]*(b*x^n)^FracPart[p])/x^(n*FracPart[p]), Int[x^(n*p), x], x] /; FreeQ[{b, n, p}, x]
  190. Int[Power[Plus[Pattern[a, Blank[]], Times[Optional[Pattern[b, Blank[]]], Power[Pattern[x, Blank[]], Pattern[n, Blank[]]]]], Pattern[p, Blank[]]], Pattern[x, Blank[Symbol]]] := Dist[1/n, Subst[Int[x^(1/n - 1)*(a + b*x)^p, x], x, x^n], x] /; FreeQ[{a, b, p}, x] && FractionQ[n] && IntegerQ[1/n]
  191. Int[Power[Plus[Pattern[a, Blank[]], Times[Optional[Pattern[b, Blank[]]], Power[Pattern[x, Blank[]], Pattern[n, Blank[]]]]], Pattern[p, Blank[]]], Pattern[x, Blank[Symbol]]] := Simp[(x*(a + b*x^n)^(p + 1))/a, x] /; FreeQ[{a, b, n, p}, x] && EqQ[1/n + p + 1, 0]
  192. Int[Power[Plus[Pattern[a, Blank[]], Times[Optional[Pattern[b, Blank[]]], Power[Pattern[x, Blank[]], Pattern[n, Blank[]]]]], Pattern[p, Blank[]]], Pattern[x, Blank[Symbol]]] := -Simp[(x*(a + b*x^n)^(p + 1))/(a*n*(p + 1)), x] + Dist[(n*(p + 1) + 1)/(a*n*(p + 1)), Int[(a + b*x^n)^(p + 1), x], x] /; FreeQ[{a, b, n, p}, x] && ILtQ[Simplify[1/n + p + 1], 0] && NeQ[p, -1]
  193. Int[Power[Plus[Pattern[a, Blank[]], Times[Optional[Pattern[b, Blank[]]], Power[Pattern[x, Blank[]], Pattern[n, Blank[]]]]], Pattern[p, Blank[]]], Pattern[x, Blank[Symbol]]] := Int[x^(n*p)*(b + a/x^n)^p, x] /; FreeQ[{a, b}, x] && LtQ[n, 0] && IntegerQ[p]
  194. Int[Power[Plus[Pattern[a, Blank[]], Times[Optional[Pattern[b, Blank[]]], Power[Pattern[x, Blank[]], Pattern[n, Blank[]]]]], Pattern[p, Blank[]]], Pattern[x, Blank[Symbol]]] := Int[ExpandIntegrand[(a + b*x^n)^p, x], x] /; FreeQ[{a, b}, x] && IGtQ[n, 0] && IGtQ[p, 0]
  195. Int[Power[Plus[Pattern[a, Blank[]], Times[Optional[Pattern[b, Blank[]]], Power[Pattern[x, Blank[]], Pattern[n, Blank[]]]]], Pattern[p, Blank[]]], Pattern[x, Blank[Symbol]]] := Simp[(x*(a + b*x^n)^p)/(n*p + 1), x] + Dist[(a*n*p)/(n*p + 1), Int[(a + b*x^n)^(p - 1), x], x] /; FreeQ[{a, b}, x] && IGtQ[n, 0] && GtQ[p, 0] && (IntegerQ[2*p] || (EqQ[n, 2] && IntegerQ[4*p]) || (EqQ[n, 2] && IntegerQ[3*p]) || LtQ[Denominator[p + 1/n], Denominator[p]])
  196. Int[Power[Plus[Pattern[a, Blank[]], Times[Optional[Pattern[b, Blank[]]], Power[Pattern[x, Blank[]], 2]]], Rational[-5, 4]], Pattern[x, Blank[Symbol]]] := Simp[(2*EllipticE[(1*ArcTan[Rt[b/a, 2]*x])/2, 2])/(a^(5/4)*Rt[b/a, 2]), x] /; FreeQ[{a, b}, x] && GtQ[a, 0] && PosQ[b/a]
  197. Int[Power[Plus[Pattern[a, Blank[]], Times[Optional[Pattern[b, Blank[]]], Power[Pattern[x, Blank[]], 2]]], Rational[-5, 4]], Pattern[x, Blank[Symbol]]] := Dist[(1 + (b*x^2)/a)^(1/4)/(a*(a + b*x^2)^(1/4)), Int[1/(1 + (b*x^2)/a)^(5/4), x], x] /; FreeQ[{a, b}, x] && PosQ[a] && PosQ[b/a]
  198. Int[Power[Plus[Pattern[a, Blank[]], Times[Optional[Pattern[b, Blank[]]], Power[Pattern[x, Blank[]], 2]]], Rational[-7, 6]], Pattern[x, Blank[Symbol]]] := Dist[1/((a + b*x^2)^(2/3)*(a/(a + b*x^2))^(2/3)), Subst[Int[1/(1 - b*x^2)^(1/3), x], x, x/Sqrt[a + b*x^2]], x] /; FreeQ[{a, b}, x]
  199. Int[Power[Plus[Pattern[a, Blank[]], Times[Optional[Pattern[b, Blank[]]], Power[Pattern[x, Blank[]], Pattern[n, Blank[]]]]], Pattern[p, Blank[]]], Pattern[x, Blank[Symbol]]] := -Simp[(x*(a + b*x^n)^(p + 1))/(a*n*(p + 1)), x] + Dist[(n*(p + 1) + 1)/(a*n*(p + 1)), Int[(a + b*x^n)^(p + 1), x], x] /; FreeQ[{a, b}, x] && IGtQ[n, 0] && LtQ[p, -1] && (IntegerQ[2*p] || (n == 2 && IntegerQ[4*p]) || (n == 2 && IntegerQ[3*p]) || Denominator[p + 1/n] < Denominator[p])
  200. Int[Power[Plus[Pattern[a, Blank[]], Times[Optional[Pattern[b, Blank[]]], Power[Pattern[x, Blank[]], 3]]], -1], Pattern[x, Blank[Symbol]]] := Dist[1/(3*Rt[a, 3]^2), Int[1/(Rt[a, 3] + Rt[b, 3]*x), x], x] + Dist[1/(3*Rt[a, 3]^2), Int[(2*Rt[a, 3] - Rt[b, 3]*x)/(Rt[a, 3]^2 - Rt[a, 3]*Rt[b, 3]*x + Rt[b, 3]^2*x^2), x], x] /; FreeQ[{a, b}, x]
  201. Int[Power[Plus[Pattern[a, Blank[]], Times[Optional[Pattern[b, Blank[]]], Power[Pattern[x, Blank[]], Pattern[n, Blank[]]]]], -1], Pattern[x, Blank[Symbol]]] := Module[{r = Numerator[Rt[a/b, n]], s = Denominator[Rt[a/b, n]], k, u}, Simp[u = Int[(r - s*Cos[((2*k - 1)*Pi)/n]*x)/(r^2 - 2*r*s*Cos[((2*k - 1)*Pi)/n]*x + s^2*x^2), x]; (r*Int[1/(r + s*x), x])/(a*n) + Dist[(2*r)/(a*n), Sum[u, {k, 1, (n - 1)/2}], x], x]] /; FreeQ[{a, b}, x] && IGtQ[(n - 3)/2, 0] && PosQ[a/b]
  202. Int[Power[Plus[Pattern[a, Blank[]], Times[Optional[Pattern[b, Blank[]]], Power[Pattern[x, Blank[]], Pattern[n, Blank[]]]]], -1], Pattern[x, Blank[Symbol]]] := Module[{r = Numerator[Rt[-(a/b), n]], s = Denominator[Rt[-(a/b), n]], k, u}, Simp[u = Int[(r + s*Cos[((2*k - 1)*Pi)/n]*x)/(r^2 + 2*r*s*Cos[((2*k - 1)*Pi)/n]*x + s^2*x^2), x]; (r*Int[1/(r - s*x), x])/(a*n) + Dist[(2*r)/(a*n), Sum[u, {k, 1, (n - 1)/2}], x], x]] /; FreeQ[{a, b}, x] && IGtQ[(n - 3)/2, 0] && NegQ[a/b]
  203. Int[Power[Plus[Pattern[a, Blank[]], Times[Optional[Pattern[b, Blank[]]], Power[Pattern[x, Blank[]], 2]]], -1], Pattern[x, Blank[Symbol]]] := Simp[(1*ArcTan[(Rt[b, 2]*x)/Rt[a, 2]])/(Rt[a, 2]*Rt[b, 2]), x] /; FreeQ[{a, b}, x] && PosQ[a/b] && (GtQ[a, 0] || GtQ[b, 0])
  204. Int[Power[Plus[Pattern[a, Blank[]], Times[Optional[Pattern[b, Blank[]]], Power[Pattern[x, Blank[]], 2]]], -1], Pattern[x, Blank[Symbol]]] := -Simp[ArcTan[(Rt[-b, 2]*x)/Rt[-a, 2]]/(Rt[-a, 2]*Rt[-b, 2]), x] /; FreeQ[{a, b}, x] && PosQ[a/b] && (LtQ[a, 0] || LtQ[b, 0])
  205. Int[Power[Plus[Pattern[a, Blank[]], Times[Optional[Pattern[b, Blank[]]], Power[Pattern[x, Blank[]], 2]]], -1], Pattern[x, Blank[Symbol]]] := Simp[(Rt[a/b, 2]*ArcTan[x/Rt[a/b, 2]])/a, x] /; FreeQ[{a, b}, x] && PosQ[a/b]
  206. Int[Power[Plus[Pattern[a, Blank[]], Times[Optional[Pattern[b, Blank[]]], Power[Pattern[x, Blank[]], 2]]], -1], Pattern[x, Blank[Symbol]]] := Simp[(1*ArcTanh[(Rt[-b, 2]*x)/Rt[a, 2]])/(Rt[a, 2]*Rt[-b, 2]), x] /; FreeQ[{a, b}, x] && NegQ[a/b] && (GtQ[a, 0] || LtQ[b, 0])
  207. Int[Power[Plus[Pattern[a, Blank[]], Times[Optional[Pattern[b, Blank[]]], Power[Pattern[x, Blank[]], 2]]], -1], Pattern[x, Blank[Symbol]]] := -Simp[ArcTanh[(Rt[b, 2]*x)/Rt[-a, 2]]/(Rt[-a, 2]*Rt[b, 2]), x] /; FreeQ[{a, b}, x] && NegQ[a/b] && (LtQ[a, 0] || GtQ[b, 0])
  208. Int[Power[Plus[Pattern[a, Blank[]], Times[Optional[Pattern[b, Blank[]]], Power[Pattern[x, Blank[]], 2]]], -1], Pattern[x, Blank[Symbol]]] := Simp[(Rt[-(a/b), 2]*ArcTanh[x/Rt[-(a/b), 2]])/a, x] /; FreeQ[{a, b}, x] && NegQ[a/b]
  209. Int[Power[Plus[Pattern[a, Blank[]], Times[Optional[Pattern[b, Blank[]]], Power[Pattern[x, Blank[]], Pattern[n, Blank[]]]]], -1], Pattern[x, Blank[Symbol]]] := Module[{r = Numerator[Rt[a/b, n]], s = Denominator[Rt[a/b, n]], k, u, v}, Simp[u = Int[(r - s*Cos[((2*k - 1)*Pi)/n]*x)/(r^2 - 2*r*s*Cos[((2*k - 1)*Pi)/n]*x + s^2*x^2), x] + Int[(r + s*Cos[((2*k - 1)*Pi)/n]*x)/(r^2 + 2*r*s*Cos[((2*k - 1)*Pi)/n]*x + s^2*x^2), x]; (2*r^2*Int[1/(r^2 + s^2*x^2), x])/(a*n) + Dist[(2*r)/(a*n), Sum[u, {k, 1, (n - 2)/4}], x], x]] /; FreeQ[{a, b}, x] && IGtQ[(n - 2)/4, 0] && PosQ[a/b]
  210. Int[Power[Plus[Pattern[a, Blank[]], Times[Optional[Pattern[b, Blank[]]], Power[Pattern[x, Blank[]], Pattern[n, Blank[]]]]], -1], Pattern[x, Blank[Symbol]]] := Module[{r = Numerator[Rt[-(a/b), n]], s = Denominator[Rt[-(a/b), n]], k, u}, Simp[u = Int[(r - s*Cos[(2*k*Pi)/n]*x)/(r^2 - 2*r*s*Cos[(2*k*Pi)/n]*x + s^2*x^2), x] + Int[(r + s*Cos[(2*k*Pi)/n]*x)/(r^2 + 2*r*s*Cos[(2*k*Pi)/n]*x + s^2*x^2), x]; (2*r^2*Int[1/(r^2 - s^2*x^2), x])/(a*n) + Dist[(2*r)/(a*n), Sum[u, {k, 1, (n - 2)/4}], x], x]] /; FreeQ[{a, b}, x] && IGtQ[(n - 2)/4, 0] && NegQ[a/b]
  211. Int[Power[Plus[Pattern[a, Blank[]], Times[Optional[Pattern[b, Blank[]]], Power[Pattern[x, Blank[]], 4]]], -1], Pattern[x, Blank[Symbol]]] := With[{r = Numerator[Rt[a/b, 2]], s = Denominator[Rt[a/b, 2]]}, Dist[1/(2*r), Int[(r - s*x^2)/(a + b*x^4), x], x] + Dist[1/(2*r), Int[(r + s*x^2)/(a + b*x^4), x], x]] /; FreeQ[{a, b}, x] && (GtQ[a/b, 0] || (PosQ[a/b] && AtomQ[SplitProduct[SumBaseQ, a]] && AtomQ[SplitProduct[SumBaseQ, b]]))
  212. Int[Power[Plus[Pattern[a, Blank[]], Times[Optional[Pattern[b, Blank[]]], Power[Pattern[x, Blank[]], 4]]], -1], Pattern[x, Blank[Symbol]]] := With[{r = Numerator[Rt[-(a/b), 2]], s = Denominator[Rt[-(a/b), 2]]}, Dist[r/(2*a), Int[1/(r - s*x^2), x], x] + Dist[r/(2*a), Int[1/(r + s*x^2), x], x]] /; FreeQ[{a, b}, x] && !GtQ[a/b, 0]
  213. Int[Power[Plus[Pattern[a, Blank[]], Times[Optional[Pattern[b, Blank[]]], Power[Pattern[x, Blank[]], Pattern[n, Blank[]]]]], -1], Pattern[x, Blank[Symbol]]] := With[{r = Numerator[Rt[a/b, 4]], s = Denominator[Rt[a/b, 4]]}, Dist[r/(2*Sqrt[2]*a), Int[(Sqrt[2]*r - s*x^(n/4))/(r^2 - Sqrt[2]*r*s*x^(n/4) + s^2*x^(n/2)), x], x] + Dist[r/(2*Sqrt[2]*a), Int[(Sqrt[2]*r + s*x^(n/4))/(r^2 + Sqrt[2]*r*s*x^(n/4) + s^2*x^(n/2)), x], x]] /; FreeQ[{a, b}, x] && IGtQ[n/4, 1] && GtQ[a/b, 0]
  214. Int[Power[Plus[Pattern[a, Blank[]], Times[Optional[Pattern[b, Blank[]]], Power[Pattern[x, Blank[]], Pattern[n, Blank[]]]]], -1], Pattern[x, Blank[Symbol]]] := With[{r = Numerator[Rt[-(a/b), 2]], s = Denominator[Rt[-(a/b), 2]]}, Dist[r/(2*a), Int[1/(r - s*x^(n/2)), x], x] + Dist[r/(2*a), Int[1/(r + s*x^(n/2)), x], x]] /; FreeQ[{a, b}, x] && IGtQ[n/4, 1] && !GtQ[a/b, 0]
  215. Int[Power[Plus[Pattern[a, Blank[]], Times[Optional[Pattern[b, Blank[]]], Power[Pattern[x, Blank[]], 2]]], Rational[-1, 2]], Pattern[x, Blank[Symbol]]] := Simp[ArcSinh[(Rt[b, 2]*x)/Sqrt[a]]/Rt[b, 2], x] /; FreeQ[{a, b}, x] && GtQ[a, 0] && PosQ[b]
  216. Int[Power[Plus[Pattern[a, Blank[]], Times[Optional[Pattern[b, Blank[]]], Power[Pattern[x, Blank[]], 2]]], Rational[-1, 2]], Pattern[x, Blank[Symbol]]] := Simp[ArcSin[(Rt[-b, 2]*x)/Sqrt[a]]/Rt[-b, 2], x] /; FreeQ[{a, b}, x] && GtQ[a, 0] && NegQ[b]
  217. Int[Power[Plus[Pattern[a, Blank[]], Times[Optional[Pattern[b, Blank[]]], Power[Pattern[x, Blank[]], 2]]], Rational[-1, 2]], Pattern[x, Blank[Symbol]]] := Subst[Int[1/(1 - b*x^2), x], x, x/Sqrt[a + b*x^2]] /; FreeQ[{a, b}, x] && !GtQ[a, 0]
  218. Int[Power[Plus[Pattern[a, Blank[]], Times[Optional[Pattern[b, Blank[]]], Power[Pattern[x, Blank[]], 3]]], Rational[-1, 2]], Pattern[x, Blank[Symbol]]] := With[{r = Numer[Rt[b/a, 3]], s = Denom[Rt[b/a, 3]]}, Simp[(2*Sqrt[2 + Sqrt[3]]*(s + r*x)*Sqrt[(s^2 - r*s*x + r^2*x^2)/((1 + Sqrt[3])*s + r*x)^2]*EllipticF[ArcSin[((1 - Sqrt[3])*s + r*x)/((1 + Sqrt[3])*s + r*x)], -7 - 4*Sqrt[3]])/(3^(1/4)*r*Sqrt[a + b*x^3]*Sqrt[(s*(s + r*x))/((1 + Sqrt[3])*s + r*x)^2]), x]] /; FreeQ[{a, b}, x] && PosQ[a]
  219. Int[Power[Plus[Pattern[a, Blank[]], Times[Optional[Pattern[b, Blank[]]], Power[Pattern[x, Blank[]], 3]]], Rational[-1, 2]], Pattern[x, Blank[Symbol]]] := With[{r = Numer[Rt[b/a, 3]], s = Denom[Rt[b/a, 3]]}, Simp[(2*Sqrt[2 - Sqrt[3]]*(s + r*x)*Sqrt[(s^2 - r*s*x + r^2*x^2)/((1 - Sqrt[3])*s + r*x)^2]*EllipticF[ArcSin[((1 + Sqrt[3])*s + r*x)/((1 - Sqrt[3])*s + r*x)], -7 + 4*Sqrt[3]])/(3^(1/4)*r*Sqrt[a + b*x^3]*Sqrt[-((s*(s + r*x))/((1 - Sqrt[3])*s + r*x)^2)]), x]] /; FreeQ[{a, b}, x] && NegQ[a]
  220. Int[Power[Plus[Pattern[a, Blank[]], Times[Optional[Pattern[b, Blank[]]], Power[Pattern[x, Blank[]], 4]]], Rational[-1, 2]], Pattern[x, Blank[Symbol]]] := With[{q = Rt[b/a, 4]}, Simp[((1 + q^2*x^2)*Sqrt[(a + b*x^4)/(a*(1 + q^2*x^2)^2)]*EllipticF[2*ArcTan[q*x], 1/2])/(2*q*Sqrt[a + b*x^4]), x]] /; FreeQ[{a, b}, x] && PosQ[b/a]
  221. Int[Power[Plus[Pattern[a, Blank[]], Times[Optional[Pattern[b, Blank[]]], Power[Pattern[x, Blank[]], 4]]], Rational[-1, 2]], Pattern[x, Blank[Symbol]]] := Simp[EllipticF[ArcSin[(Rt[-b, 4]*x)/Rt[a, 4]], -1]/(Rt[a, 4]*Rt[-b, 4]), x] /; FreeQ[{a, b}, x] && NegQ[b/a] && GtQ[a, 0]
  222. Int[Power[Plus[Pattern[a, Blank[]], Times[Optional[Pattern[b, Blank[]]], Power[Pattern[x, Blank[]], 4]]], Rational[-1, 2]], Pattern[x, Blank[Symbol]]] := With[{q = Rt[-(a*b), 2]}, Simp[(Sqrt[-a + q*x^2]*Sqrt[(a + q*x^2)/q]*EllipticF[ArcSin[x/Sqrt[(a + q*x^2)/(2*q)]], 1/2])/(Sqrt[2]*Sqrt[-a]*Sqrt[a + b*x^4]), x] /; IntegerQ[q]] /; FreeQ[{a, b}, x] && LtQ[a, 0] && GtQ[b, 0]
  223. Int[Power[Plus[Pattern[a, Blank[]], Times[Optional[Pattern[b, Blank[]]], Power[Pattern[x, Blank[]], 4]]], Rational[-1, 2]], Pattern[x, Blank[Symbol]]] := With[{q = Rt[-(a*b), 2]}, Simp[(Sqrt[(a - q*x^2)/(a + q*x^2)]*Sqrt[(a + q*x^2)/q]*EllipticF[ArcSin[x/Sqrt[(a + q*x^2)/(2*q)]], 1/2])/(Sqrt[2]*Sqrt[a + b*x^4]*Sqrt[a/(a + q*x^2)]), x]] /; FreeQ[{a, b}, x] && LtQ[a, 0] && GtQ[b, 0]
  224. Int[Power[Plus[Pattern[a, Blank[]], Times[Optional[Pattern[b, Blank[]]], Power[Pattern[x, Blank[]], 4]]], Rational[-1, 2]], Pattern[x, Blank[Symbol]]] := Dist[Sqrt[1 + (b*x^4)/a]/Sqrt[a + b*x^4], Int[1/Sqrt[1 + (b*x^4)/a], x], x] /; FreeQ[{a, b}, x] && NegQ[b/a] && !GtQ[a, 0]
  225. Int[Power[Plus[Pattern[a, Blank[]], Times[Optional[Pattern[b, Blank[]]], Power[Pattern[x, Blank[]], 6]]], Rational[-1, 2]], Pattern[x, Blank[Symbol]]] := With[{r = Numer[Rt[b/a, 3]], s = Denom[Rt[b/a, 3]]}, Simp[(x*(s + r*x^2)*Sqrt[(s^2 - r*s*x^2 + r^2*x^4)/(s + (1 + Sqrt[3])*r*x^2)^2]*EllipticF[ArcCos[(s + (1 - Sqrt[3])*r*x^2)/(s + (1 + Sqrt[3])*r*x^2)], (2 + Sqrt[3])/4])/(2*3^(1/4)*s*Sqrt[a + b*x^6]*Sqrt[(r*x^2*(s + r*x^2))/(s + (1 + Sqrt[3])*r*x^2)^2]), x]] /; FreeQ[{a, b}, x]
  226. Int[Power[Plus[Pattern[a, Blank[]], Times[Optional[Pattern[b, Blank[]]], Power[Pattern[x, Blank[]], 8]]], Rational[-1, 2]], Pattern[x, Blank[Symbol]]] := Dist[1/2, Int[(1 - Rt[b/a, 4]*x^2)/Sqrt[a + b*x^8], x], x] + Dist[1/2, Int[(1 + Rt[b/a, 4]*x^2)/Sqrt[a + b*x^8], x], x] /; FreeQ[{a, b}, x]
  227. Int[Power[Plus[Pattern[a, Blank[]], Times[Optional[Pattern[b, Blank[]]], Power[Pattern[x, Blank[]], 2]]], Rational[-1, 4]], Pattern[x, Blank[Symbol]]] := Simp[(2*x)/(a + b*x^2)^(1/4), x] - Dist[a, Int[1/(a + b*x^2)^(5/4), x], x] /; FreeQ[{a, b}, x] && GtQ[a, 0] && PosQ[b/a]
  228. Int[Power[Plus[Pattern[a, Blank[]], Times[Optional[Pattern[b, Blank[]]], Power[Pattern[x, Blank[]], 2]]], Rational[-1, 4]], Pattern[x, Blank[Symbol]]] := Simp[(2*EllipticE[(1*ArcSin[Rt[-(b/a), 2]*x])/2, 2])/(a^(1/4)*Rt[-(b/a), 2]), x] /; FreeQ[{a, b}, x] && GtQ[a, 0] && NegQ[b/a]
  229. Int[Power[Plus[Pattern[a, Blank[]], Times[Optional[Pattern[b, Blank[]]], Power[Pattern[x, Blank[]], 2]]], Rational[-1, 4]], Pattern[x, Blank[Symbol]]] := Dist[(1 + (b*x^2)/a)^(1/4)/(a + b*x^2)^(1/4), Int[1/(1 + (b*x^2)/a)^(1/4), x], x] /; FreeQ[{a, b}, x] && PosQ[a]
  230. Int[Power[Plus[Pattern[a, Blank[]], Times[Optional[Pattern[b, Blank[]]], Power[Pattern[x, Blank[]], 2]]], Rational[-1, 4]], Pattern[x, Blank[Symbol]]] := Dist[(2*Sqrt[-((b*x^2)/a)])/(b*x), Subst[Int[x^2/Sqrt[1 - x^4/a], x], x, (a + b*x^2)^(1/4)], x] /; FreeQ[{a, b}, x] && NegQ[a]
  231. Int[Power[Plus[Pattern[a, Blank[]], Times[Optional[Pattern[b, Blank[]]], Power[Pattern[x, Blank[]], 2]]], Rational[-3, 4]], Pattern[x, Blank[Symbol]]] := Simp[(2*EllipticF[(1*ArcTan[Rt[b/a, 2]*x])/2, 2])/(a^(3/4)*Rt[b/a, 2]), x] /; FreeQ[{a, b}, x] && GtQ[a, 0] && PosQ[b/a]
  232. Int[Power[Plus[Pattern[a, Blank[]], Times[Optional[Pattern[b, Blank[]]], Power[Pattern[x, Blank[]], 2]]], Rational[-3, 4]], Pattern[x, Blank[Symbol]]] := Simp[(2*EllipticF[(1*ArcSin[Rt[-(b/a), 2]*x])/2, 2])/(a^(3/4)*Rt[-(b/a), 2]), x] /; FreeQ[{a, b}, x] && GtQ[a, 0] && NegQ[b/a]
  233. Int[Power[Plus[Pattern[a, Blank[]], Times[Optional[Pattern[b, Blank[]]], Power[Pattern[x, Blank[]], 2]]], Rational[-3, 4]], Pattern[x, Blank[Symbol]]] := Dist[(1 + (b*x^2)/a)^(3/4)/(a + b*x^2)^(3/4), Int[1/(1 + (b*x^2)/a)^(3/4), x], x] /; FreeQ[{a, b}, x] && PosQ[a]
  234. Int[Power[Plus[Pattern[a, Blank[]], Times[Optional[Pattern[b, Blank[]]], Power[Pattern[x, Blank[]], 2]]], Rational[-3, 4]], Pattern[x, Blank[Symbol]]] := Dist[(2*Sqrt[-((b*x^2)/a)])/(b*x), Subst[Int[1/Sqrt[1 - x^4/a], x], x, (a + b*x^2)^(1/4)], x] /; FreeQ[{a, b}, x] && NegQ[a]
  235. Int[Power[Plus[Pattern[a, Blank[]], Times[Optional[Pattern[b, Blank[]]], Power[Pattern[x, Blank[]], 2]]], Rational[-1, 3]], Pattern[x, Blank[Symbol]]] := Dist[(3*Sqrt[b*x^2])/(2*b*x), Subst[Int[x/Sqrt[-a + x^3], x], x, (a + b*x^2)^(1/3)], x] /; FreeQ[{a, b}, x]
  236. Int[Power[Plus[Pattern[a, Blank[]], Times[Optional[Pattern[b, Blank[]]], Power[Pattern[x, Blank[]], 2]]], Rational[-2, 3]], Pattern[x, Blank[Symbol]]] := Dist[(3*Sqrt[b*x^2])/(2*b*x), Subst[Int[1/Sqrt[-a + x^3], x], x, (a + b*x^2)^(1/3)], x] /; FreeQ[{a, b}, x]
  237. Int[Power[Plus[Pattern[a, Blank[]], Times[Optional[Pattern[b, Blank[]]], Power[Pattern[x, Blank[]], 4]]], Rational[-3, 4]], Pattern[x, Blank[Symbol]]] := Dist[(x^3*(1 + a/(b*x^4))^(3/4))/(a + b*x^4)^(3/4), Int[1/(x^3*(1 + a/(b*x^4))^(3/4)), x], x] /; FreeQ[{a, b}, x]
  238. Int[Power[Plus[Pattern[a, Blank[]], Times[Optional[Pattern[b, Blank[]]], Power[Pattern[x, Blank[]], 2]]], Rational[-1, 6]], Pattern[x, Blank[Symbol]]] := Simp[(3*x)/(2*(a + b*x^2)^(1/6)), x] - Dist[a/2, Int[1/(a + b*x^2)^(7/6), x], x] /; FreeQ[{a, b}, x]
  239. Int[Power[Plus[Pattern[a, Blank[]], Times[Optional[Pattern[b, Blank[]]], Power[Pattern[x, Blank[]], 3]]], Rational[-1, 3]], Pattern[x, Blank[Symbol]]] := Simp[ArcTan[(1 + (2*Rt[b, 3]*x)/(a + b*x^3)^(1/3))/Sqrt[3]]/(Sqrt[3]*Rt[b, 3]), x] - Simp[Log[(a + b*x^3)^(1/3) - Rt[b, 3]*x]/(2*Rt[b, 3]), x] /; FreeQ[{a, b}, x]
  240. Int[Power[Plus[Pattern[a, Blank[]], Times[Optional[Pattern[b, Blank[]]], Power[Pattern[x, Blank[]], Pattern[n, Blank[]]]]], Pattern[p, Blank[]]], Pattern[x, Blank[Symbol]]] := Dist[a^(p + 1/n), Subst[Int[1/(1 - b*x^n)^(p + 1/n + 1), x], x, x/(a + b*x^n)^(1/n)], x] /; FreeQ[{a, b}, x] && IGtQ[n, 0] && LtQ[-1, p, 0] && NeQ[p, -2^(-1)] && IntegerQ[p + 1/n]
  241. Int[Power[Plus[Pattern[a, Blank[]], Times[Optional[Pattern[b, Blank[]]], Power[Pattern[x, Blank[]], Pattern[n, Blank[]]]]], Pattern[p, Blank[]]], Pattern[x, Blank[Symbol]]] := Dist[(a/(a + b*x^n))^(p + 1/n)*(a + b*x^n)^(p + 1/n), Subst[Int[1/(1 - b*x^n)^(p + 1/n + 1), x], x, x/(a + b*x^n)^(1/n)], x] /; FreeQ[{a, b}, x] && IGtQ[n, 0] && LtQ[-1, p, 0] && NeQ[p, -2^(-1)] && LtQ[Denominator[p + 1/n], Denominator[p]]
  242. Int[Power[Plus[Pattern[a, Blank[]], Times[Optional[Pattern[b, Blank[]]], Power[Pattern[x, Blank[]], Pattern[n, Blank[]]]]], Pattern[p, Blank[]]], Pattern[x, Blank[Symbol]]] := -Subst[Int[(a + b/x^n)^p/x^2, x], x, 1/x] /; FreeQ[{a, b, p}, x] && ILtQ[n, 0]
  243. Int[Power[Plus[Pattern[a, Blank[]], Times[Optional[Pattern[b, Blank[]]], Power[Pattern[x, Blank[]], Pattern[n, Blank[]]]]], Pattern[p, Blank[]]], Pattern[x, Blank[Symbol]]] := With[{k = Denominator[n]}, Dist[k, Subst[Int[x^(k - 1)*(a + b*x^(k*n))^p, x], x, x^(1/k)], x]] /; FreeQ[{a, b, p}, x] && FractionQ[n]
  244. Int[Power[Plus[Pattern[a, Blank[]], Times[Optional[Pattern[b, Blank[]]], Power[Pattern[x, Blank[]], Pattern[n, Blank[]]]]], Pattern[p, Blank[]]], Pattern[x, Blank[Symbol]]] := Int[ExpandIntegrand[(a + b*x^n)^p, x], x] /; FreeQ[{a, b, n}, x] && IGtQ[p, 0]
  245. Int[Power[Plus[Pattern[a, Blank[]], Times[Optional[Pattern[b, Blank[]]], Power[Pattern[x, Blank[]], Pattern[n, Blank[]]]]], Pattern[p, Blank[]]], Pattern[x, Blank[Symbol]]] := Simp[a^p*x*Hypergeometric2F1[-p, 1/n, 1/n + 1, -((b*x^n)/a)], x] /; FreeQ[{a, b, n, p}, x] && !IGtQ[p, 0] && !IntegerQ[1/n] && !ILtQ[Simplify[1/n + p], 0] && (IntegerQ[p] || GtQ[a, 0])
  246. Int[Power[Plus[Pattern[a, Blank[]], Times[Optional[Pattern[b, Blank[]]], Power[Pattern[x, Blank[]], Pattern[n, Blank[]]]]], Pattern[p, Blank[]]], Pattern[x, Blank[Symbol]]] := Dist[(a^IntPart[p]*(a + b*x^n)^FracPart[p])/(1 + (b*x^n)/a)^FracPart[p], Int[(1 + (b*x^n)/a)^p, x], x] /; FreeQ[{a, b, n, p}, x] && !IGtQ[p, 0] && !IntegerQ[1/n] && !ILtQ[Simplify[1/n + p], 0] && !(IntegerQ[p] || GtQ[a, 0])
  247. Int[Power[Plus[Optional[Pattern[a, Blank[]]], Times[Optional[Pattern[b, Blank[]]], Power[Pattern[v, Blank[]], Pattern[n, Blank[]]]]], Pattern[p, Blank[]]], Pattern[x, Blank[Symbol]]] := Dist[1/Coefficient[v, x, 1], Subst[Int[(a + b*x^n)^p, x], x, v], x] /; FreeQ[{a, b, n, p}, x] && LinearQ[v, x] && NeQ[v, x]
  248. Int[Times[Power[Plus[Optional[Pattern[a1, Blank[]]], Times[Optional[Pattern[b1, Blank[]]], Power[Pattern[x, Blank[]], Pattern[n, Blank[]]]]], Optional[Pattern[p, Blank[]]]], Power[Plus[Optional[Pattern[a2, Blank[]]], Times[Optional[Pattern[b2, Blank[]]], Power[Pattern[x, Blank[]], Pattern[n, Blank[]]]]], Optional[Pattern[p, Blank[]]]]], Pattern[x, Blank[Symbol]]] := Int[(a1*a2 + b1*b2*x^(2*n))^p, x] /; FreeQ[{a1, b1, a2, b2, n, p}, x] && EqQ[a2*b1 + a1*b2, 0] && (IntegerQ[p] || (GtQ[a1, 0] && GtQ[a2, 0]))
  249. Int[Times[Power[Plus[Pattern[a1, Blank[]], Times[Optional[Pattern[b1, Blank[]]], Power[Pattern[x, Blank[]], Optional[Pattern[n, Blank[]]]]]], Optional[Pattern[p, Blank[]]]], Power[Plus[Pattern[a2, Blank[]], Times[Optional[Pattern[b2, Blank[]]], Power[Pattern[x, Blank[]], Optional[Pattern[n, Blank[]]]]]], Optional[Pattern[p, Blank[]]]]], Pattern[x, Blank[Symbol]]] := Simp[(x*(a1 + b1*x^n)^p*(a2 + b2*x^n)^p)/(2*n*p + 1), x] + Dist[(2*a1*a2*n*p)/(2*n*p + 1), Int[(a1 + b1*x^n)^(p - 1)*(a2 + b2*x^n)^(p - 1), x], x] /; FreeQ[{a1, b1, a2, b2}, x] && EqQ[a2*b1 + a1*b2, 0] && IGtQ[2*n, 0] && GtQ[p, 0] && (IntegerQ[2*p] || Denominator[p + 1/n] < Denominator[p])
  250. Int[Times[Power[Plus[Pattern[a1, Blank[]], Times[Optional[Pattern[b1, Blank[]]], Power[Pattern[x, Blank[]], Optional[Pattern[n, Blank[]]]]]], Pattern[p, Blank[]]], Power[Plus[Pattern[a2, Blank[]], Times[Optional[Pattern[b2, Blank[]]], Power[Pattern[x, Blank[]], Optional[Pattern[n, Blank[]]]]]], Pattern[p, Blank[]]]], Pattern[x, Blank[Symbol]]] := -Simp[(x*(a1 + b1*x^n)^(p + 1)*(a2 + b2*x^n)^(p + 1))/(2*a1*a2*n*(p + 1)), x] + Dist[(2*n*(p + 1) + 1)/(2*a1*a2*n*(p + 1)), Int[(a1 + b1*x^n)^(p + 1)*(a2 + b2*x^n)^(p + 1), x], x] /; FreeQ[{a1, b1, a2, b2}, x] && EqQ[a2*b1 + a1*b2, 0] && IGtQ[2*n, 0] && LtQ[p, -1] && (IntegerQ[2*p] || Denominator[p + 1/n] < Denominator[p])
  251. Int[Times[Power[Plus[Pattern[a1, Blank[]], Times[Optional[Pattern[b1, Blank[]]], Power[Pattern[x, Blank[]], Pattern[n, Blank[]]]]], Pattern[p, Blank[]]], Power[Plus[Pattern[a2, Blank[]], Times[Optional[Pattern[b2, Blank[]]], Power[Pattern[x, Blank[]], Pattern[n, Blank[]]]]], Pattern[p, Blank[]]]], Pattern[x, Blank[Symbol]]] := -Subst[Int[((a1 + b1/x^n)^p*(a2 + b2/x^n)^p)/x^2, x], x, 1/x] /; FreeQ[{a1, b1, a2, b2, p}, x] && EqQ[a2*b1 + a1*b2, 0] && ILtQ[2*n, 0]
  252. Int[Times[Power[Plus[Pattern[a1, Blank[]], Times[Optional[Pattern[b1, Blank[]]], Power[Pattern[x, Blank[]], Pattern[n, Blank[]]]]], Pattern[p, Blank[]]], Power[Plus[Pattern[a2, Blank[]], Times[Optional[Pattern[b2, Blank[]]], Power[Pattern[x, Blank[]], Pattern[n, Blank[]]]]], Pattern[p, Blank[]]]], Pattern[x, Blank[Symbol]]] := With[{k = Denominator[2*n]}, Dist[k, Subst[Int[x^(k - 1)*(a1 + b1*x^(k*n))^p*(a2 + b2*x^(k*n))^p, x], x, x^(1/k)], x]] /; FreeQ[{a1, b1, a2, b2, p}, x] && EqQ[a2*b1 + a1*b2, 0] && FractionQ[2*n]
  253. Int[Times[Power[Plus[Optional[Pattern[a1, Blank[]]], Times[Optional[Pattern[b1, Blank[]]], Power[Pattern[x, Blank[]], Pattern[n, Blank[]]]]], Pattern[p, Blank[]]], Power[Plus[Optional[Pattern[a2, Blank[]]], Times[Optional[Pattern[b2, Blank[]]], Power[Pattern[x, Blank[]], Pattern[n, Blank[]]]]], Pattern[p, Blank[]]]], Pattern[x, Blank[Symbol]]] := Dist[((a1 + b1*x^n)^FracPart[p]*(a2 + b2*x^n)^FracPart[p])/(a1*a2 + b1*b2*x^(2*n))^FracPart[p], Int[(a1*a2 + b1*b2*x^(2*n))^p, x], x] /; FreeQ[{a1, b1, a2, b2, n, p}, x] && EqQ[a2*b1 + a1*b2, 0] && !IntegerQ[p]
  254. Int[Power[Plus[Pattern[a, Blank[]], Times[Optional[Pattern[b, Blank[]]], Power[Times[Optional[Pattern[c, Blank[]]], Power[Pattern[x, Blank[]], Optional[Pattern[q, Blank[]]]]], Pattern[n, Blank[]]]]], Optional[Pattern[p, Blank[]]]], Pattern[x, Blank[Symbol]]] := Dist[x/(c*x^q)^(1/q), Subst[Int[(a + b*x^(n*q))^p, x], x, (c*x^q)^(1/q)], x] /; FreeQ[{a, b, c, n, p, q}, x] && IntegerQ[n*q] && NeQ[x, (c*x^q)^(1/q)]
  255. Int[Power[Plus[Pattern[a, Blank[]], Times[Optional[Pattern[b, Blank[]]], Power[Times[Optional[Pattern[c, Blank[]]], Power[Pattern[x, Blank[]], Optional[Pattern[q, Blank[]]]]], Pattern[n, Blank[]]]]], Optional[Pattern[p, Blank[]]]], Pattern[x, Blank[Symbol]]] := With[{k = Denominator[n]}, Subst[Int[(a + b*c^n*x^(n*q))^p, x], x^(1/k), (c*x^q)^(1/k)/(c^(1/k)*(x^(1/k))^(q - 1))]] /; FreeQ[{a, b, c, p, q}, x] && FractionQ[n]
  256. Int[Power[Plus[Pattern[a, Blank[]], Times[Optional[Pattern[b, Blank[]]], Power[Times[Optional[Pattern[c, Blank[]]], Power[Pattern[x, Blank[]], Optional[Pattern[q, Blank[]]]]], Pattern[n, Blank[]]]]], Optional[Pattern[p, Blank[]]]], Pattern[x, Blank[Symbol]]] := Subst[Int[(a + b*c^n*x^(n*q))^p, x], x^(n*q), (c*x^q)^n/c^n] /; FreeQ[{a, b, c, n, p, q}, x] && !RationalQ[n]
  257. Int[Power[Plus[Pattern[a, Blank[]], Times[Optional[Pattern[b, Blank[]]], Power[Times[Optional[Pattern[d, Blank[]]], Power[Pattern[x, Blank[]], Optional[Pattern[q, Blank[]]]]], Pattern[n, Blank[]]]]], Optional[Pattern[p, Blank[]]]], Pattern[x, Blank[Symbol]]] := -Subst[Int[(a + b*(d/x^q)^n)^p/x^2, x], x, 1/x] /; FreeQ[{a, b, d, n, p}, x] && ILtQ[q, 0]
  258. Int[Power[Plus[Pattern[a, Blank[]], Times[Optional[Pattern[b, Blank[]]], Power[Times[Optional[Pattern[d, Blank[]]], Power[Pattern[x, Blank[]], Optional[Pattern[q, Blank[]]]]], Pattern[n, Blank[]]]]], Optional[Pattern[p, Blank[]]]], Pattern[x, Blank[Symbol]]] := With[{s = Denominator[q]}, Dist[s, Subst[Int[x^(s - 1)*(a + b*(d*x^(q*s))^n)^p, x], x, x^(1/s)], x]] /; FreeQ[{a, b, d, n, p}, x] && FractionQ[q]
  259. Int[Times[Power[Times[Optional[Pattern[c, Blank[]]], Pattern[x, Blank[]]], Optional[Pattern[m, Blank[]]]], Power[Plus[Pattern[a1, Blank[]], Times[Optional[Pattern[b1, Blank[]]], Power[Pattern[x, Blank[]], Pattern[n, Blank[]]]]], Pattern[p, Blank[]]], Power[Plus[Pattern[a2, Blank[]], Times[Optional[Pattern[b2, Blank[]]], Power[Pattern[x, Blank[]], Pattern[n, Blank[]]]]], Pattern[p, Blank[]]]], Pattern[x, Blank[Symbol]]] := Int[(c*x)^m*(a1*a2 + b1*b2*x^(2*n))^p, x] /; FreeQ[{a1, b1, a2, b2, c, m, n, p}, x] && EqQ[a2*b1 + a1*b2, 0] && (IntegerQ[p] || (GtQ[a1, 0] && GtQ[a2, 0]))
  260. Int[Times[Power[Pattern[x, Blank[]], Optional[Pattern[m, Blank[]]]], Power[Plus[Pattern[a, Blank[]], Times[Optional[Pattern[b, Blank[]]], Power[Pattern[x, Blank[]], Pattern[n, Blank[]]]]], -1]], Pattern[x, Blank[Symbol]]] := Simp[Log[RemoveContent[a + b*x^n, x]]/(b*n), x] /; FreeQ[{a, b, m, n}, x] && EqQ[m, n - 1]
  261. Int[Times[Power[Pattern[x, Blank[]], Optional[Pattern[m, Blank[]]]], Power[Plus[Pattern[a, Blank[]], Times[Optional[Pattern[b, Blank[]]], Power[Pattern[x, Blank[]], Pattern[n, Blank[]]]]], Pattern[p, Blank[]]]], Pattern[x, Blank[Symbol]]] := Simp[(a + b*x^n)^(p + 1)/(b*n*(p + 1)), x] /; FreeQ[{a, b, m, n, p}, x] && EqQ[m, n - 1] && NeQ[p, -1]
  262. Int[Times[Power[Pattern[x, Blank[]], Optional[Pattern[m, Blank[]]]], Power[Plus[Pattern[a1, Blank[]], Times[Optional[Pattern[b1, Blank[]]], Power[Pattern[x, Blank[]], Optional[Pattern[n, Blank[]]]]]], Pattern[p, Blank[]]], Power[Plus[Pattern[a2, Blank[]], Times[Optional[Pattern[b2, Blank[]]], Power[Pattern[x, Blank[]], Optional[Pattern[n, Blank[]]]]]], Pattern[p, Blank[]]]], Pattern[x, Blank[Symbol]]] := Simp[((a1 + b1*x^n)^(p + 1)*(a2 + b2*x^n)^(p + 1))/(2*b1*b2*n*(p + 1)), x] /; FreeQ[{a1, b1, a2, b2, m, n, p}, x] && EqQ[a2*b1 + a1*b2, 0] && EqQ[m, 2*n - 1] && NeQ[p, -1]
  263. Int[Times[Power[Pattern[x, Blank[]], Optional[Pattern[m, Blank[]]]], Power[Plus[Pattern[a, Blank[]], Times[Optional[Pattern[b, Blank[]]], Power[Pattern[x, Blank[]], Pattern[n, Blank[]]]]], Pattern[p, Blank[]]]], Pattern[x, Blank[Symbol]]] := Int[x^(m + n*p)*(b + a/x^n)^p, x] /; FreeQ[{a, b, m, n}, x] && IntegerQ[p] && NegQ[n]
  264. Int[Times[Power[Times[Optional[Pattern[c, Blank[]]], Pattern[x, Blank[]]], Optional[Pattern[m, Blank[]]]], Power[Plus[Pattern[a, Blank[]], Times[Optional[Pattern[b, Blank[]]], Power[Pattern[x, Blank[]], Pattern[n, Blank[]]]]], Pattern[p, Blank[]]]], Pattern[x, Blank[Symbol]]] := Simp[((c*x)^(m + 1)*(a + b*x^n)^(p + 1))/(a*c*(m + 1)), x] /; FreeQ[{a, b, c, m, n, p}, x] && EqQ[(m + 1)/n + p + 1, 0] && NeQ[m, -1]
  265. Int[Times[Power[Times[Optional[Pattern[c, Blank[]]], Pattern[x, Blank[]]], Optional[Pattern[m, Blank[]]]], Power[Plus[Pattern[a1, Blank[]], Times[Optional[Pattern[b1, Blank[]]], Power[Pattern[x, Blank[]], Pattern[n, Blank[]]]]], Pattern[p, Blank[]]], Power[Plus[Pattern[a2, Blank[]], Times[Optional[Pattern[b2, Blank[]]], Power[Pattern[x, Blank[]], Pattern[n, Blank[]]]]], Pattern[p, Blank[]]]], Pattern[x, Blank[Symbol]]] := Simp[((c*x)^(m + 1)*(a1 + b1*x^n)^(p + 1)*(a2 + b2*x^n)^(p + 1))/(a1*a2*c*(m + 1)), x] /; FreeQ[{a1, b1, a2, b2, c, m, n, p}, x] && EqQ[a2*b1 + a1*b2, 0] && EqQ[(m + 1)/(2*n) + p + 1, 0] && NeQ[m, -1]
  266. Int[Times[Power[Pattern[x, Blank[]], Optional[Pattern[m, Blank[]]]], Power[Plus[Pattern[a, Blank[]], Times[Optional[Pattern[b, Blank[]]], Power[Pattern[x, Blank[]], Pattern[n, Blank[]]]]], Pattern[p, Blank[]]]], Pattern[x, Blank[Symbol]]] := Dist[1/n, Subst[Int[x^(Simplify[(m + 1)/n] - 1)*(a + b*x)^p, x], x, x^n], x] /; FreeQ[{a, b, m, n, p}, x] && IntegerQ[Simplify[(m + 1)/n]]
  267. Int[Times[Power[Pattern[x, Blank[]], Optional[Pattern[m, Blank[]]]], Power[Plus[Pattern[a1, Blank[]], Times[Optional[Pattern[b1, Blank[]]], Power[Pattern[x, Blank[]], Pattern[n, Blank[]]]]], Pattern[p, Blank[]]], Power[Plus[Pattern[a2, Blank[]], Times[Optional[Pattern[b2, Blank[]]], Power[Pattern[x, Blank[]], Pattern[n, Blank[]]]]], Pattern[p, Blank[]]]], Pattern[x, Blank[Symbol]]] := Dist[1/n, Subst[Int[x^(Simplify[(m + 1)/n] - 1)*(a1 + b1*x)^p*(a2 + b2*x)^p, x], x, x^n], x] /; FreeQ[{a1, b1, a2, b2, m, n, p}, x] && EqQ[a2*b1 + a1*b2, 0] && IntegerQ[Simplify[(m + 1)/(2*n)]]
  268. Int[Times[Power[Times[Pattern[c, Blank[]], Pattern[x, Blank[]]], Pattern[m, Blank[]]], Power[Plus[Pattern[a, Blank[]], Times[Optional[Pattern[b, Blank[]]], Power[Pattern[x, Blank[]], Pattern[n, Blank[]]]]], Pattern[p, Blank[]]]], Pattern[x, Blank[Symbol]]] := Dist[(c^IntPart[m]*(c*x)^FracPart[m])/x^FracPart[m], Int[x^m*(a + b*x^n)^p, x], x] /; FreeQ[{a, b, c, m, n, p}, x] && IntegerQ[Simplify[(m + 1)/n]]
  269. Int[Times[Power[Times[Pattern[c, Blank[]], Pattern[x, Blank[]]], Pattern[m, Blank[]]], Power[Plus[Pattern[a1, Blank[]], Times[Optional[Pattern[b1, Blank[]]], Power[Pattern[x, Blank[]], Pattern[n, Blank[]]]]], Pattern[p, Blank[]]], Power[Plus[Pattern[a2, Blank[]], Times[Optional[Pattern[b2, Blank[]]], Power[Pattern[x, Blank[]], Pattern[n, Blank[]]]]], Pattern[p, Blank[]]]], Pattern[x, Blank[Symbol]]] := Dist[(c^IntPart[m]*(c*x)^FracPart[m])/x^FracPart[m], Int[x^m*(a1 + b1*x^n)^p*(a2 + b2*x^n)^p, x], x] /; FreeQ[{a1, b1, a2, b2, c, m, n, p}, x] && EqQ[a2*b1 + a1*b2, 0] && IntegerQ[Simplify[(m + 1)/(2*n)]]
  270. Int[Times[Power[Times[Optional[Pattern[c, Blank[]]], Pattern[x, Blank[]]], Optional[Pattern[m, Blank[]]]], Power[Plus[Pattern[a, Blank[]], Times[Optional[Pattern[b, Blank[]]], Power[Pattern[x, Blank[]], Pattern[n, Blank[]]]]], Optional[Pattern[p, Blank[]]]]], Pattern[x, Blank[Symbol]]] := Int[ExpandIntegrand[(c*x)^m*(a + b*x^n)^p, x], x] /; FreeQ[{a, b, c, m, n}, x] && IGtQ[p, 0]
  271. Int[Times[Power[Pattern[x, Blank[]], Pattern[m, Blank[]]], Power[Plus[Pattern[a, Blank[]], Times[Optional[Pattern[b, Blank[]]], Power[Pattern[x, Blank[]], Pattern[n, Blank[]]]]], Pattern[p, Blank[]]]], Pattern[x, Blank[Symbol]]] := Simp[(x^(m + 1)*(a + b*x^n)^(p + 1))/(a*(m + 1)), x] - Dist[(b*(m + n*(p + 1) + 1))/(a*(m + 1)), Int[x^(m + n)*(a + b*x^n)^p, x], x] /; FreeQ[{a, b, m, n, p}, x] && ILtQ[Simplify[(m + 1)/n + p + 1], 0] && NeQ[m, -1]
  272. Int[Times[Power[Pattern[x, Blank[]], Pattern[m, Blank[]]], Power[Plus[Pattern[a1, Blank[]], Times[Optional[Pattern[b1, Blank[]]], Power[Pattern[x, Blank[]], Pattern[n, Blank[]]]]], Pattern[p, Blank[]]], Power[Plus[Pattern[a2, Blank[]], Times[Optional[Pattern[b2, Blank[]]], Power[Pattern[x, Blank[]], Pattern[n, Blank[]]]]], Pattern[p, Blank[]]]], Pattern[x, Blank[Symbol]]] := Simp[(x^(m + 1)*(a1 + b1*x^n)^(p + 1)*(a2 + b2*x^n)^(p + 1))/(a1*a2*(m + 1)), x] - Dist[(b1*b2*(m + 2*n*(p + 1) + 1))/(a1*a2*(m + 1)), Int[x^(m + 2*n)*(a1 + b1*x^n)^p*(a2 + b2*x^n)^p, x], x] /; FreeQ[{a1, b1, a2, b2, m, n, p}, x] && EqQ[a2*b1 + a1*b2, 0] && ILtQ[Simplify[(m + 1)/(2*n) + p + 1], 0] && NeQ[m, -1]
  273. Int[Times[Power[Times[Optional[Pattern[c, Blank[]]], Pattern[x, Blank[]]], Optional[Pattern[m, Blank[]]]], Power[Plus[Pattern[a, Blank[]], Times[Optional[Pattern[b, Blank[]]], Power[Pattern[x, Blank[]], Pattern[n, Blank[]]]]], Pattern[p, Blank[]]]], Pattern[x, Blank[Symbol]]] := -Simp[((c*x)^(m + 1)*(a + b*x^n)^(p + 1))/(a*c*n*(p + 1)), x] + Dist[(m + n*(p + 1) + 1)/(a*n*(p + 1)), Int[(c*x)^m*(a + b*x^n)^(p + 1), x], x] /; FreeQ[{a, b, c, m, n, p}, x] && ILtQ[Simplify[(m + 1)/n + p + 1], 0] && NeQ[p, -1]
  274. Int[Times[Power[Times[Optional[Pattern[c, Blank[]]], Pattern[x, Blank[]]], Optional[Pattern[m, Blank[]]]], Power[Plus[Pattern[a1, Blank[]], Times[Optional[Pattern[b1, Blank[]]], Power[Pattern[x, Blank[]], Pattern[n, Blank[]]]]], Pattern[p, Blank[]]], Power[Plus[Pattern[a2, Blank[]], Times[Optional[Pattern[b2, Blank[]]], Power[Pattern[x, Blank[]], Pattern[n, Blank[]]]]], Pattern[p, Blank[]]]], Pattern[x, Blank[Symbol]]] := -Simp[((c*x)^(m + 1)*(a1 + b1*x^n)^(p + 1)*(a2 + b2*x^n)^(p + 1))/(2*a1*a2*c*n*(p + 1)), x] + Dist[(m + 2*n*(p + 1) + 1)/(2*a1*a2*n*(p + 1)), Int[(c*x)^m*(a1 + b1*x^n)^(p + 1)*(a2 + b2*x^n)^(p + 1), x], x] /; FreeQ[{a1, b1, a2, b2, c, m, n, p}, x] && EqQ[a2*b1 + a1*b2, 0] && ILtQ[Simplify[(m + 1)/(2*n) + p + 1], 0] && NeQ[p, -1]
  275. Int[Times[Power[Pattern[x, Blank[]], Optional[Pattern[m, Blank[]]]], Power[Plus[Pattern[a, Blank[]], Times[Optional[Pattern[b, Blank[]]], Power[Pattern[x, Blank[]], Pattern[n, Blank[]]]]], Pattern[p, Blank[]]]], Pattern[x, Blank[Symbol]]] := With[{k = GCD[m + 1, n]}, Dist[1/k, Subst[Int[x^((m + 1)/k - 1)*(a + b*x^(n/k))^p, x], x, x^k], x] /; k != 1] /; FreeQ[{a, b, p}, x] && IGtQ[n, 0] && IntegerQ[m]
  276. Int[Times[Power[Pattern[x, Blank[]], Optional[Pattern[m, Blank[]]]], Power[Plus[Pattern[a1, Blank[]], Times[Optional[Pattern[b1, Blank[]]], Power[Pattern[x, Blank[]], Pattern[n, Blank[]]]]], Pattern[p, Blank[]]], Power[Plus[Pattern[a2, Blank[]], Times[Optional[Pattern[b2, Blank[]]], Power[Pattern[x, Blank[]], Pattern[n, Blank[]]]]], Pattern[p, Blank[]]]], Pattern[x, Blank[Symbol]]] := With[{k = GCD[m + 1, 2*n]}, Dist[1/k, Subst[Int[x^((m + 1)/k - 1)*(a1 + b1*x^(n/k))^p*(a2 + b2*x^(n/k))^p, x], x, x^k], x] /; k != 1] /; FreeQ[{a1, b1, a2, b2, p}, x] && EqQ[a2*b1 + a1*b2, 0] && IGtQ[2*n, 0] && IntegerQ[m]
  277. Int[Times[Power[Times[Optional[Pattern[c, Blank[]]], Pattern[x, Blank[]]], Optional[Pattern[m, Blank[]]]], Power[Plus[Pattern[a, Blank[]], Times[Optional[Pattern[b, Blank[]]], Power[Pattern[x, Blank[]], Pattern[n, Blank[]]]]], Pattern[p, Blank[]]]], Pattern[x, Blank[Symbol]]] := Simp[((c*x)^(m + 1)*(a + b*x^n)^p)/(c*(m + 1)), x] - Dist[(b*n*p)/(c^n*(m + 1)), Int[(c*x)^(m + n)*(a + b*x^n)^(p - 1), x], x] /; FreeQ[{a, b, c}, x] && IGtQ[n, 0] && GtQ[p, 0] && LtQ[m, -1] && !ILtQ[(m + n*p + n + 1)/n, 0] && IntBinomialQ[a, b, c, n, m, p, x]
  278. Int[Times[Power[Times[Optional[Pattern[c, Blank[]]], Pattern[x, Blank[]]], Optional[Pattern[m, Blank[]]]], Power[Plus[Pattern[a1, Blank[]], Times[Optional[Pattern[b1, Blank[]]], Power[Pattern[x, Blank[]], Pattern[n, Blank[]]]]], Pattern[p, Blank[]]], Power[Plus[Pattern[a2, Blank[]], Times[Optional[Pattern[b2, Blank[]]], Power[Pattern[x, Blank[]], Pattern[n, Blank[]]]]], Pattern[p, Blank[]]]], Pattern[x, Blank[Symbol]]] := Simp[((c*x)^(m + 1)*(a1 + b1*x^n)^p*(a2 + b2*x^n)^p)/(c*(m + 1)), x] - Dist[(2*b1*b2*n*p)/(c^(2*n)*(m + 1)), Int[(c*x)^(m + 2*n)*(a1 + b1*x^n)^(p - 1)*(a2 + b2*x^n)^(p - 1), x], x] /; FreeQ[{a1, b1, a2, b2, c, m}, x] && EqQ[a2*b1 + a1*b2, 0] && IGtQ[2*n, 0] && GtQ[p, 0] && LtQ[m, -1] && NeQ[m + 2*n*p + 1, 0] && IntBinomialQ[a1*a2, b1*b2, c, 2*n, m, p, x]
  279. Int[Times[Power[Times[Optional[Pattern[c, Blank[]]], Pattern[x, Blank[]]], Optional[Pattern[m, Blank[]]]], Power[Plus[Pattern[a, Blank[]], Times[Optional[Pattern[b, Blank[]]], Power[Pattern[x, Blank[]], Pattern[n, Blank[]]]]], Pattern[p, Blank[]]]], Pattern[x, Blank[Symbol]]] := Simp[((c*x)^(m + 1)*(a + b*x^n)^p)/(c*(m + n*p + 1)), x] + Dist[(a*n*p)/(m + n*p + 1), Int[(c*x)^m*(a + b*x^n)^(p - 1), x], x] /; FreeQ[{a, b, c, m}, x] && IGtQ[n, 0] && GtQ[p, 0] && NeQ[m + n*p + 1, 0] && IntBinomialQ[a, b, c, n, m, p, x]
  280. Int[Times[Power[Times[Optional[Pattern[c, Blank[]]], Pattern[x, Blank[]]], Optional[Pattern[m, Blank[]]]], Power[Plus[Pattern[a1, Blank[]], Times[Optional[Pattern[b1, Blank[]]], Power[Pattern[x, Blank[]], Pattern[n, Blank[]]]]], Pattern[p, Blank[]]], Power[Plus[Pattern[a2, Blank[]], Times[Optional[Pattern[b2, Blank[]]], Power[Pattern[x, Blank[]], Pattern[n, Blank[]]]]], Pattern[p, Blank[]]]], Pattern[x, Blank[Symbol]]] := Simp[((c*x)^(m + 1)*(a1 + b1*x^n)^p*(a2 + b2*x^n)^p)/(c*(m + 2*n*p + 1)), x] + Dist[(2*a1*a2*n*p)/(m + 2*n*p + 1), Int[(c*x)^m*(a1 + b1*x^n)^(p - 1)*(a2 + b2*x^n)^(p - 1), x], x] /; FreeQ[{a1, b1, a2, b2, c, m}, x] && EqQ[a2*b1 + a1*b2, 0] && IGtQ[2*n, 0] && GtQ[p, 0] && NeQ[m + 2*n*p + 1, 0] && IntBinomialQ[a1*a2, b1*b2, c, 2*n, m, p, x]
  281. Int[Times[Power[Pattern[x, Blank[]], 2], Power[Plus[Pattern[a, Blank[]], Times[Optional[Pattern[b, Blank[]]], Power[Pattern[x, Blank[]], 4]]], Rational[-5, 4]]], Pattern[x, Blank[Symbol]]] := Dist[(x*(1 + a/(b*x^4))^(1/4))/(b*(a + b*x^4)^(1/4)), Int[1/(x^3*(1 + a/(b*x^4))^(5/4)), x], x] /; FreeQ[{a, b}, x] && PosQ[b/a]
  282. Int[Times[Power[Pattern[x, Blank[]], Pattern[m, Blank[]]], Power[Plus[Pattern[a, Blank[]], Times[Optional[Pattern[b, Blank[]]], Power[Pattern[x, Blank[]], 4]]], Rational[-5, 4]]], Pattern[x, Blank[Symbol]]] := Simp[x^(m - 3)/(b*(m - 4)*(a + b*x^4)^(1/4)), x] - Dist[(a*(m - 3))/(b*(m - 4)), Int[x^(m - 4)/(a + b*x^4)^(5/4), x], x] /; FreeQ[{a, b}, x] && PosQ[b/a] && IGtQ[(m - 2)/4, 0]
  283. Int[Times[Power[Pattern[x, Blank[]], Pattern[m, Blank[]]], Power[Plus[Pattern[a, Blank[]], Times[Optional[Pattern[b, Blank[]]], Power[Pattern[x, Blank[]], 4]]], Rational[-5, 4]]], Pattern[x, Blank[Symbol]]] := Simp[x^(m + 1)/(a*(m + 1)*(a + b*x^4)^(1/4)), x] - Dist[(b*m)/(a*(m + 1)), Int[x^(m + 4)/(a + b*x^4)^(5/4), x], x] /; FreeQ[{a, b}, x] && PosQ[b/a] && ILtQ[(m - 2)/4, 0]
  284. Int[Times[Power[Times[Optional[Pattern[c, Blank[]]], Pattern[x, Blank[]]], Rational[1, 2]], Power[Plus[Pattern[a, Blank[]], Times[Optional[Pattern[b, Blank[]]], Power[Pattern[x, Blank[]], 2]]], Rational[-5, 4]]], Pattern[x, Blank[Symbol]]] := Dist[(Sqrt[c*x]*(1 + a/(b*x^2))^(1/4))/(b*(a + b*x^2)^(1/4)), Int[1/(x^2*(1 + a/(b*x^2))^(5/4)), x], x] /; FreeQ[{a, b, c}, x] && PosQ[b/a]
  285. Int[Times[Power[Times[Optional[Pattern[c, Blank[]]], Pattern[x, Blank[]]], Pattern[m, Blank[]]], Power[Plus[Pattern[a, Blank[]], Times[Optional[Pattern[b, Blank[]]], Power[Pattern[x, Blank[]], 2]]], Rational[-5, 4]]], Pattern[x, Blank[Symbol]]] := Simp[(2*c*(c*x)^(m - 1))/(b*(2*m - 3)*(a + b*x^2)^(1/4)), x] - Dist[(2*a*c^2*(m - 1))/(b*(2*m - 3)), Int[(c*x)^(m - 2)/(a + b*x^2)^(5/4), x], x] /; FreeQ[{a, b, c}, x] && PosQ[b/a] && IntegerQ[2*m] && GtQ[m, 3/2]
  286. Int[Times[Power[Times[Optional[Pattern[c, Blank[]]], Pattern[x, Blank[]]], Pattern[m, Blank[]]], Power[Plus[Pattern[a, Blank[]], Times[Optional[Pattern[b, Blank[]]], Power[Pattern[x, Blank[]], 2]]], Rational[-5, 4]]], Pattern[x, Blank[Symbol]]] := Simp[(c*x)^(m + 1)/(a*c*(m + 1)*(a + b*x^2)^(1/4)), x] - Dist[(b*(2*m + 1))/(2*a*c^2*(m + 1)), Int[(c*x)^(m + 2)/(a + b*x^2)^(5/4), x], x] /; FreeQ[{a, b, c}, x] && PosQ[b/a] && IntegerQ[2*m] && LtQ[m, -1]
  287. Int[Times[Power[Pattern[x, Blank[]], 2], Power[Plus[Pattern[a, Blank[]], Times[Optional[Pattern[b, Blank[]]], Power[Pattern[x, Blank[]], 4]]], Rational[-5, 4]]], Pattern[x, Blank[Symbol]]] := -Simp[(b*x*(a + b*x^4)^(1/4))^(-1), x] - Dist[1/b, Int[1/(x^2*(a + b*x^4)^(1/4)), x], x] /; FreeQ[{a, b}, x] && NegQ[b/a]
  288. Int[Times[Power[Times[Optional[Pattern[c, Blank[]]], Pattern[x, Blank[]]], Optional[Pattern[m, Blank[]]]], Power[Plus[Pattern[a, Blank[]], Times[Optional[Pattern[b, Blank[]]], Power[Pattern[x, Blank[]], Pattern[n, Blank[]]]]], Pattern[p, Blank[]]]], Pattern[x, Blank[Symbol]]] := Simp[(c^(n - 1)*(c*x)^(m - n + 1)*(a + b*x^n)^(p + 1))/(b*n*(p + 1)), x] - Dist[(c^n*(m - n + 1))/(b*n*(p + 1)), Int[(c*x)^(m - n)*(a + b*x^n)^(p + 1), x], x] /; FreeQ[{a, b, c}, x] && IGtQ[n, 0] && LtQ[p, -1] && GtQ[m + 1, n] && !ILtQ[(m + n*(p + 1) + 1)/n, 0] && IntBinomialQ[a, b, c, n, m, p, x]
  289. Int[Times[Power[Times[Optional[Pattern[c, Blank[]]], Pattern[x, Blank[]]], Optional[Pattern[m, Blank[]]]], Power[Plus[Pattern[a1, Blank[]], Times[Optional[Pattern[b1, Blank[]]], Power[Pattern[x, Blank[]], Pattern[n, Blank[]]]]], Pattern[p, Blank[]]], Power[Plus[Pattern[a2, Blank[]], Times[Optional[Pattern[b2, Blank[]]], Power[Pattern[x, Blank[]], Pattern[n, Blank[]]]]], Pattern[p, Blank[]]]], Pattern[x, Blank[Symbol]]] := Simp[(c^(2*n - 1)*(c*x)^(m - 2*n + 1)*(a1 + b1*x^n)^(p + 1)*(a2 + b2*x^n)^(p + 1))/(2*b1*b2*n*(p + 1)), x] - Dist[(c^(2*n)*(m - 2*n + 1))/(2*b1*b2*n*(p + 1)), Int[(c*x)^(m - 2*n)*(a1 + b1*x^n)^(p + 1)*(a2 + b2*x^n)^(p + 1), x], x] /; FreeQ[{a1, b1, a2, b2, c}, x] && EqQ[a2*b1 + a1*b2, 0] && IGtQ[2*n, 0] && LtQ[p, -1] && m + 1 > 2*n && !ILtQ[(m + 2*n*(p + 1) + 1)/(2*n), 0] && IntBinomialQ[a1*a2, b1*b2, c, 2*n, m, p, x]
  290. Int[Times[Power[Times[Optional[Pattern[c, Blank[]]], Pattern[x, Blank[]]], Optional[Pattern[m, Blank[]]]], Power[Plus[Pattern[a, Blank[]], Times[Optional[Pattern[b, Blank[]]], Power[Pattern[x, Blank[]], Pattern[n, Blank[]]]]], Pattern[p, Blank[]]]], Pattern[x, Blank[Symbol]]] := -Simp[((c*x)^(m + 1)*(a + b*x^n)^(p + 1))/(a*c*n*(p + 1)), x] + Dist[(m + n*(p + 1) + 1)/(a*n*(p + 1)), Int[(c*x)^m*(a + b*x^n)^(p + 1), x], x] /; FreeQ[{a, b, c, m}, x] && IGtQ[n, 0] && LtQ[p, -1] && IntBinomialQ[a, b, c, n, m, p, x]
  291. Int[Times[Power[Times[Optional[Pattern[c, Blank[]]], Pattern[x, Blank[]]], Optional[Pattern[m, Blank[]]]], Power[Plus[Pattern[a1, Blank[]], Times[Optional[Pattern[b1, Blank[]]], Power[Pattern[x, Blank[]], Pattern[n, Blank[]]]]], Pattern[p, Blank[]]], Power[Plus[Pattern[a2, Blank[]], Times[Optional[Pattern[b2, Blank[]]], Power[Pattern[x, Blank[]], Pattern[n, Blank[]]]]], Pattern[p, Blank[]]]], Pattern[x, Blank[Symbol]]] := -Simp[((c*x)^(m + 1)*(a1 + b1*x^n)^(p + 1)*(a2 + b2*x^n)^(p + 1))/(2*a1*a2*c*n*(p + 1)), x] + Dist[(m + 2*n*(p + 1) + 1)/(2*a1*a2*n*(p + 1)), Int[(c*x)^m*(a1 + b1*x^n)^(p + 1)*(a2 + b2*x^n)^(p + 1), x], x] /; FreeQ[{a1, b1, a2, b2, c, m}, x] && EqQ[a2*b1 + a1*b2, 0] && IGtQ[2*n, 0] && LtQ[p, -1] && IntBinomialQ[a1*a2, b1*b2, c, 2*n, m, p, x]
  292. Int[Times[Pattern[x, Blank[]], Power[Plus[Pattern[a, Blank[]], Times[Optional[Pattern[b, Blank[]]], Power[Pattern[x, Blank[]], 3]]], -1]], Pattern[x, Blank[Symbol]]] := -Dist[(3*Rt[a, 3]*Rt[b, 3])^(-1), Int[1/(Rt[a, 3] + Rt[b, 3]*x), x], x] + Dist[1/(3*Rt[a, 3]*Rt[b, 3]), Int[(Rt[a, 3] + Rt[b, 3]*x)/(Rt[a, 3]^2 - Rt[a, 3]*Rt[b, 3]*x + Rt[b, 3]^2*x^2), x], x] /; FreeQ[{a, b}, x]
  293. Int[Times[Power[Pattern[x, Blank[]], Optional[Pattern[m, Blank[]]]], Power[Plus[Pattern[a, Blank[]], Times[Optional[Pattern[b, Blank[]]], Power[Pattern[x, Blank[]], Pattern[n, Blank[]]]]], -1]], Pattern[x, Blank[Symbol]]] := Module[{r = Numerator[Rt[a/b, n]], s = Denominator[Rt[a/b, n]], k, u}, Simp[u = Int[(r*Cos[((2*k - 1)*m*Pi)/n] - s*Cos[((2*k - 1)*(m + 1)*Pi)/n]*x)/(r^2 - 2*r*s*Cos[((2*k - 1)*Pi)/n]*x + s^2*x^2), x]; -(((-r)^(m + 1)*Int[1/(r + s*x), x])/(a*n*s^m)) + Dist[(2*r^(m + 1))/(a*n*s^m), Sum[u, {k, 1, (n - 1)/2}], x], x]] /; FreeQ[{a, b}, x] && IGtQ[(n - 1)/2, 0] && IGtQ[m, 0] && LtQ[m, n - 1] && PosQ[a/b]
  294. Int[Times[Power[Pattern[x, Blank[]], Optional[Pattern[m, Blank[]]]], Power[Plus[Pattern[a, Blank[]], Times[Optional[Pattern[b, Blank[]]], Power[Pattern[x, Blank[]], Pattern[n, Blank[]]]]], -1]], Pattern[x, Blank[Symbol]]] := Module[{r = Numerator[Rt[-(a/b), n]], s = Denominator[Rt[-(a/b), n]], k, u}, Simp[u = Int[(r*Cos[((2*k - 1)*m*Pi)/n] + s*Cos[((2*k - 1)*(m + 1)*Pi)/n]*x)/(r^2 + 2*r*s*Cos[((2*k - 1)*Pi)/n]*x + s^2*x^2), x]; (r^(m + 1)*Int[1/(r - s*x), x])/(a*n*s^m) - Dist[(2*(-r)^(m + 1))/(a*n*s^m), Sum[u, {k, 1, (n - 1)/2}], x], x]] /; FreeQ[{a, b}, x] && IGtQ[(n - 1)/2, 0] && IGtQ[m, 0] && LtQ[m, n - 1] && NegQ[a/b]
  295. Int[Times[Power[Pattern[x, Blank[]], Optional[Pattern[m, Blank[]]]], Power[Plus[Pattern[a, Blank[]], Times[Optional[Pattern[b, Blank[]]], Power[Pattern[x, Blank[]], Pattern[n, Blank[]]]]], -1]], Pattern[x, Blank[Symbol]]] := Module[{r = Numerator[Rt[a/b, n]], s = Denominator[Rt[a/b, n]], k, u}, Simp[u = Int[(r*Cos[((2*k - 1)*m*Pi)/n] - s*Cos[((2*k - 1)*(m + 1)*Pi)/n]*x)/(r^2 - 2*r*s*Cos[((2*k - 1)*Pi)/n]*x + s^2*x^2), x] + Int[(r*Cos[((2*k - 1)*m*Pi)/n] + s*Cos[((2*k - 1)*(m + 1)*Pi)/n]*x)/(r^2 + 2*r*s*Cos[((2*k - 1)*Pi)/n]*x + s^2*x^2), x]; (2*(-1)^(m/2)*r^(m + 2)*Int[1/(r^2 + s^2*x^2), x])/(a*n*s^m) + Dist[(2*r^(m + 1))/(a*n*s^m), Sum[u, {k, 1, (n - 2)/4}], x], x]] /; FreeQ[{a, b}, x] && IGtQ[(n - 2)/4, 0] && IGtQ[m, 0] && LtQ[m, n - 1] && PosQ[a/b]
  296. Int[Times[Power[Pattern[x, Blank[]], Optional[Pattern[m, Blank[]]]], Power[Plus[Pattern[a, Blank[]], Times[Optional[Pattern[b, Blank[]]], Power[Pattern[x, Blank[]], Pattern[n, Blank[]]]]], -1]], Pattern[x, Blank[Symbol]]] := Module[{r = Numerator[Rt[-(a/b), n]], s = Denominator[Rt[-(a/b), n]], k, u}, Simp[u = Int[(r*Cos[(2*k*m*Pi)/n] - s*Cos[(2*k*(m + 1)*Pi)/n]*x)/(r^2 - 2*r*s*Cos[(2*k*Pi)/n]*x + s^2*x^2), x] + Int[(r*Cos[(2*k*m*Pi)/n] + s*Cos[(2*k*(m + 1)*Pi)/n]*x)/(r^2 + 2*r*s*Cos[(2*k*Pi)/n]*x + s^2*x^2), x]; (2*r^(m + 2)*Int[1/(r^2 - s^2*x^2), x])/(a*n*s^m) + Dist[(2*r^(m + 1))/(a*n*s^m), Sum[u, {k, 1, (n - 2)/4}], x], x]] /; FreeQ[{a, b}, x] && IGtQ[(n - 2)/4, 0] && IGtQ[m, 0] && LtQ[m, n - 1] && NegQ[a/b]
  297. Int[Times[Power[Pattern[x, Blank[]], 2], Power[Plus[Pattern[a, Blank[]], Times[Optional[Pattern[b, Blank[]]], Power[Pattern[x, Blank[]], 4]]], -1]], Pattern[x, Blank[Symbol]]] := With[{r = Numerator[Rt[a/b, 2]], s = Denominator[Rt[a/b, 2]]}, Dist[1/(2*s), Int[(r + s*x^2)/(a + b*x^4), x], x] - Dist[1/(2*s), Int[(r - s*x^2)/(a + b*x^4), x], x]] /; FreeQ[{a, b}, x] && (GtQ[a/b, 0] || (PosQ[a/b] && AtomQ[SplitProduct[SumBaseQ, a]] && AtomQ[SplitProduct[SumBaseQ, b]]))
  298. Int[Times[Power[Pattern[x, Blank[]], 2], Power[Plus[Pattern[a, Blank[]], Times[Optional[Pattern[b, Blank[]]], Power[Pattern[x, Blank[]], 4]]], -1]], Pattern[x, Blank[Symbol]]] := With[{r = Numerator[Rt[-(a/b), 2]], s = Denominator[Rt[-(a/b), 2]]}, Dist[s/(2*b), Int[1/(r + s*x^2), x], x] - Dist[s/(2*b), Int[1/(r - s*x^2), x], x]] /; FreeQ[{a, b}, x] && !GtQ[a/b, 0]
  299. Int[Times[Power[Pattern[x, Blank[]], Optional[Pattern[m, Blank[]]]], Power[Plus[Pattern[a, Blank[]], Times[Optional[Pattern[b, Blank[]]], Power[Pattern[x, Blank[]], Pattern[n, Blank[]]]]], -1]], Pattern[x, Blank[Symbol]]] := With[{r = Numerator[Rt[a/b, 4]], s = Denominator[Rt[a/b, 4]]}, Dist[s^3/(2*Sqrt[2]*b*r), Int[x^(m - n/4)/(r^2 - Sqrt[2]*r*s*x^(n/4) + s^2*x^(n/2)), x], x] - Dist[s^3/(2*Sqrt[2]*b*r), Int[x^(m - n/4)/(r^2 + Sqrt[2]*r*s*x^(n/4) + s^2*x^(n/2)), x], x]] /; FreeQ[{a, b}, x] && IGtQ[n/4, 0] && IGtQ[m, 0] && LtQ[m, n - 1] && GtQ[a/b, 0]
  300. Int[Times[Power[Pattern[x, Blank[]], Pattern[m, Blank[]]], Power[Plus[Pattern[a, Blank[]], Times[Optional[Pattern[b, Blank[]]], Power[Pattern[x, Blank[]], Pattern[n, Blank[]]]]], -1]], Pattern[x, Blank[Symbol]]] := With[{r = Numerator[Rt[-(a/b), 2]], s = Denominator[Rt[-(a/b), 2]]}, Dist[r/(2*a), Int[x^m/(r + s*x^(n/2)), x], x] + Dist[r/(2*a), Int[x^m/(r - s*x^(n/2)), x], x]] /; FreeQ[{a, b}, x] && IGtQ[n/4, 0] && IGtQ[m, 0] && LtQ[m, n/2] && !GtQ[a/b, 0]
  301. Int[Times[Power[Pattern[x, Blank[]], Pattern[m, Blank[]]], Power[Plus[Pattern[a, Blank[]], Times[Optional[Pattern[b, Blank[]]], Power[Pattern[x, Blank[]], Pattern[n, Blank[]]]]], -1]], Pattern[x, Blank[Symbol]]] := With[{r = Numerator[Rt[-(a/b), 2]], s = Denominator[Rt[-(a/b), 2]]}, Dist[s/(2*b), Int[x^(m - n/2)/(r + s*x^(n/2)), x], x] - Dist[s/(2*b), Int[x^(m - n/2)/(r - s*x^(n/2)), x], x]] /; FreeQ[{a, b}, x] && IGtQ[n/4, 0] && IGtQ[m, 0] && LeQ[n/2, m] && LtQ[m, n] && !GtQ[a/b, 0]
  302. Int[Times[Power[Pattern[x, Blank[]], Pattern[m, Blank[]]], Power[Plus[Pattern[a, Blank[]], Times[Optional[Pattern[b, Blank[]]], Power[Pattern[x, Blank[]], Pattern[n, Blank[]]]]], -1]], Pattern[x, Blank[Symbol]]] := Int[PolynomialDivide[x^m, a + b*x^n, x], x] /; FreeQ[{a, b}, x] && IGtQ[m, 0] && IGtQ[n, 0] && GtQ[m, 2*n - 1]
  303. Int[Times[Pattern[x, Blank[]], Power[Plus[Pattern[a, Blank[]], Times[Optional[Pattern[b, Blank[]]], Power[Pattern[x, Blank[]], 3]]], Rational[-1, 2]]], Pattern[x, Blank[Symbol]]] := With[{r = Numer[Rt[b/a, 3]], s = Denom[Rt[b/a, 3]]}, Dist[(Sqrt[2]*s)/(Sqrt[2 + Sqrt[3]]*r), Int[1/Sqrt[a + b*x^3], x], x] + Dist[1/r, Int[((1 - Sqrt[3])*s + r*x)/Sqrt[a + b*x^3], x], x]] /; FreeQ[{a, b}, x] && PosQ[a]
  304. Int[Times[Pattern[x, Blank[]], Power[Plus[Pattern[a, Blank[]], Times[Optional[Pattern[b, Blank[]]], Power[Pattern[x, Blank[]], 3]]], Rational[-1, 2]]], Pattern[x, Blank[Symbol]]] := With[{r = Numer[Rt[b/a, 3]], s = Denom[Rt[b/a, 3]]}, -Dist[(Sqrt[2]*s)/(Sqrt[2 - Sqrt[3]]*r), Int[1/Sqrt[a + b*x^3], x], x] + Dist[1/r, Int[((1 + Sqrt[3])*s + r*x)/Sqrt[a + b*x^3], x], x]] /; FreeQ[{a, b}, x] && NegQ[a]
  305. Int[Times[Power[Pattern[x, Blank[]], 2], Power[Plus[Pattern[a, Blank[]], Times[Optional[Pattern[b, Blank[]]], Power[Pattern[x, Blank[]], 4]]], Rational[-1, 2]]], Pattern[x, Blank[Symbol]]] := With[{q = Rt[b/a, 2]}, Dist[1/q, Int[1/Sqrt[a + b*x^4], x], x] - Dist[1/q, Int[(1 - q*x^2)/Sqrt[a + b*x^4], x], x]] /; FreeQ[{a, b}, x] && PosQ[b/a]
  306. Int[Times[Power[Pattern[x, Blank[]], 2], Power[Plus[Pattern[a, Blank[]], Times[Optional[Pattern[b, Blank[]]], Power[Pattern[x, Blank[]], 4]]], Rational[-1, 2]]], Pattern[x, Blank[Symbol]]] := With[{q = Rt[-(b/a), 2]}, Dist[1/q, Int[1/Sqrt[a + b*x^4], x], x] - Dist[1/q, Int[(1 - q*x^2)/Sqrt[a + b*x^4], x], x]] /; FreeQ[{a, b}, x] && LtQ[a, 0] && GtQ[b, 0]
  307. Int[Times[Power[Pattern[x, Blank[]], 2], Power[Plus[Pattern[a, Blank[]], Times[Optional[Pattern[b, Blank[]]], Power[Pattern[x, Blank[]], 4]]], Rational[-1, 2]]], Pattern[x, Blank[Symbol]]] := With[{q = Rt[-(b/a), 2]}, -Dist[q^(-1), Int[1/Sqrt[a + b*x^4], x], x] + Dist[1/q, Int[(1 + q*x^2)/Sqrt[a + b*x^4], x], x]] /; FreeQ[{a, b}, x] && NegQ[b/a]
  308. Int[Times[Power[Pattern[x, Blank[]], 4], Power[Plus[Pattern[a, Blank[]], Times[Optional[Pattern[b, Blank[]]], Power[Pattern[x, Blank[]], 6]]], Rational[-1, 2]]], Pattern[x, Blank[Symbol]]] := With[{r = Numer[Rt[b/a, 3]], s = Denom[Rt[b/a, 3]]}, Dist[((Sqrt[3] - 1)*s^2)/(2*r^2), Int[1/Sqrt[a + b*x^6], x], x] - Dist[1/(2*r^2), Int[((Sqrt[3] - 1)*s^2 - 2*r^2*x^4)/Sqrt[a + b*x^6], x], x]] /; FreeQ[{a, b}, x]
  309. Int[Times[Power[Pattern[x, Blank[]], 2], Power[Plus[Pattern[a, Blank[]], Times[Optional[Pattern[b, Blank[]]], Power[Pattern[x, Blank[]], 8]]], Rational[-1, 2]]], Pattern[x, Blank[Symbol]]] := Dist[1/(2*Rt[b/a, 4]), Int[(1 + Rt[b/a, 4]*x^2)/Sqrt[a + b*x^8], x], x] - Dist[1/(2*Rt[b/a, 4]), Int[(1 - Rt[b/a, 4]*x^2)/Sqrt[a + b*x^8], x], x] /; FreeQ[{a, b}, x]
  310. Int[Times[Power[Pattern[x, Blank[]], 2], Power[Plus[Pattern[a, Blank[]], Times[Optional[Pattern[b, Blank[]]], Power[Pattern[x, Blank[]], 4]]], Rational[-1, 4]]], Pattern[x, Blank[Symbol]]] := Simp[x^3/(2*(a + b*x^4)^(1/4)), x] - Dist[a/2, Int[x^2/(a + b*x^4)^(5/4), x], x] /; FreeQ[{a, b}, x] && PosQ[b/a]
  311. Int[Times[Power[Pattern[x, Blank[]], 2], Power[Plus[Pattern[a, Blank[]], Times[Optional[Pattern[b, Blank[]]], Power[Pattern[x, Blank[]], 4]]], Rational[-1, 4]]], Pattern[x, Blank[Symbol]]] := Simp[(a + b*x^4)^(3/4)/(2*b*x), x] + Dist[a/(2*b), Int[1/(x^2*(a + b*x^4)^(1/4)), x], x] /; FreeQ[{a, b}, x] && NegQ[b/a]
  312. Int[Times[Power[Pattern[x, Blank[]], -2], Power[Plus[Pattern[a, Blank[]], Times[Optional[Pattern[b, Blank[]]], Power[Pattern[x, Blank[]], 4]]], Rational[-1, 4]]], Pattern[x, Blank[Symbol]]] := -Simp[(x*(a + b*x^4)^(1/4))^(-1), x] - Dist[b, Int[x^2/(a + b*x^4)^(5/4), x], x] /; FreeQ[{a, b}, x] && PosQ[b/a]
  313. Int[Times[Power[Pattern[x, Blank[]], -2], Power[Plus[Pattern[a, Blank[]], Times[Optional[Pattern[b, Blank[]]], Power[Pattern[x, Blank[]], 4]]], Rational[-1, 4]]], Pattern[x, Blank[Symbol]]] := Dist[(x*(1 + a/(b*x^4))^(1/4))/(a + b*x^4)^(1/4), Int[1/(x^3*(1 + a/(b*x^4))^(1/4)), x], x] /; FreeQ[{a, b}, x] && NegQ[b/a]
  314. Int[Times[Power[Times[Pattern[c, Blank[]], Pattern[x, Blank[]]], Rational[1, 2]], Power[Plus[Pattern[a, Blank[]], Times[Optional[Pattern[b, Blank[]]], Power[Pattern[x, Blank[]], 2]]], Rational[-1, 4]]], Pattern[x, Blank[Symbol]]] := Simp[(x*Sqrt[c*x])/(a + b*x^2)^(1/4), x] - Dist[a/2, Int[Sqrt[c*x]/(a + b*x^2)^(5/4), x], x] /; FreeQ[{a, b, c}, x] && PosQ[b/a]
  315. Int[Times[Power[Times[Pattern[c, Blank[]], Pattern[x, Blank[]]], Rational[1, 2]], Power[Plus[Pattern[a, Blank[]], Times[Optional[Pattern[b, Blank[]]], Power[Pattern[x, Blank[]], 2]]], Rational[-1, 4]]], Pattern[x, Blank[Symbol]]] := Simp[(c*(a + b*x^2)^(3/4))/(b*Sqrt[c*x]), x] + Dist[(a*c^2)/(2*b), Int[1/((c*x)^(3/2)*(a + b*x^2)^(1/4)), x], x] /; FreeQ[{a, b, c}, x] && NegQ[b/a]
  316. Int[Times[Power[Times[Optional[Pattern[c, Blank[]]], Pattern[x, Blank[]]], Rational[-3, 2]], Power[Plus[Pattern[a, Blank[]], Times[Optional[Pattern[b, Blank[]]], Power[Pattern[x, Blank[]], 2]]], Rational[-1, 4]]], Pattern[x, Blank[Symbol]]] := Simp[-2/(c*Sqrt[c*x]*(a + b*x^2)^(1/4)), x] - Dist[b/c^2, Int[Sqrt[c*x]/(a + b*x^2)^(5/4), x], x] /; FreeQ[{a, b, c}, x] && PosQ[b/a]
  317. Int[Times[Power[Times[Optional[Pattern[c, Blank[]]], Pattern[x, Blank[]]], Rational[-3, 2]], Power[Plus[Pattern[a, Blank[]], Times[Optional[Pattern[b, Blank[]]], Power[Pattern[x, Blank[]], 2]]], Rational[-1, 4]]], Pattern[x, Blank[Symbol]]] := Dist[(Sqrt[c*x]*(1 + a/(b*x^2))^(1/4))/(c^2*(a + b*x^2)^(1/4)), Int[1/(x^2*(1 + a/(b*x^2))^(1/4)), x], x] /; FreeQ[{a, b, c}, x] && NegQ[b/a]
  318. Int[Times[Power[Pattern[x, Blank[]], Rational[1, 2]], Power[Plus[Pattern[a, Blank[]], Times[Optional[Pattern[b, Blank[]]], Power[Pattern[x, Blank[]], 2]]], Rational[-1, 2]]], Pattern[x, Blank[Symbol]]] := Dist[-2/(Sqrt[a]*(-(b/a))^(3/4)), Subst[Int[Sqrt[1 - 2*x^2]/Sqrt[1 - x^2], x], x, Sqrt[1 - Sqrt[-(b/a)]*x]/Sqrt[2]], x] /; FreeQ[{a, b}, x] && GtQ[-(b/a), 0] && GtQ[a, 0]
  319. Int[Times[Power[Pattern[x, Blank[]], Rational[1, 2]], Power[Plus[Pattern[a, Blank[]], Times[Optional[Pattern[b, Blank[]]], Power[Pattern[x, Blank[]], 2]]], Rational[-1, 2]]], Pattern[x, Blank[Symbol]]] := Dist[Sqrt[1 + (b*x^2)/a]/Sqrt[a + b*x^2], Int[Sqrt[x]/Sqrt[1 + (b*x^2)/a], x], x] /; FreeQ[{a, b}, x] && GtQ[-(b/a), 0] && !GtQ[a, 0]
  320. Int[Times[Power[Times[Pattern[c, Blank[]], Pattern[x, Blank[]]], Rational[1, 2]], Power[Plus[Pattern[a, Blank[]], Times[Optional[Pattern[b, Blank[]]], Power[Pattern[x, Blank[]], 2]]], Rational[-1, 2]]], Pattern[x, Blank[Symbol]]] := Dist[Sqrt[c*x]/Sqrt[x], Int[Sqrt[x]/Sqrt[a + b*x^2], x], x] /; FreeQ[{a, b, c}, x] && GtQ[-(b/a), 0]
  321. Int[Times[Power[Times[Optional[Pattern[c, Blank[]]], Pattern[x, Blank[]]], Pattern[m, Blank[]]], Power[Plus[Pattern[a, Blank[]], Times[Optional[Pattern[b, Blank[]]], Power[Pattern[x, Blank[]], Pattern[n, Blank[]]]]], Pattern[p, Blank[]]]], Pattern[x, Blank[Symbol]]] := Simp[(c^(n - 1)*(c*x)^(m - n + 1)*(a + b*x^n)^(p + 1))/(b*(m + n*p + 1)), x] - Dist[(a*c^n*(m - n + 1))/(b*(m + n*p + 1)), Int[(c*x)^(m - n)*(a + b*x^n)^p, x], x] /; FreeQ[{a, b, c, p}, x] && IGtQ[n, 0] && GtQ[m, n - 1] && NeQ[m + n*p + 1, 0] && IntBinomialQ[a, b, c, n, m, p, x]
  322. Int[Times[Power[Times[Optional[Pattern[c, Blank[]]], Pattern[x, Blank[]]], Pattern[m, Blank[]]], Power[Plus[Pattern[a, Blank[]], Times[Optional[Pattern[b, Blank[]]], Power[Pattern[x, Blank[]], Pattern[n, Blank[]]]]], Pattern[p, Blank[]]]], Pattern[x, Blank[Symbol]]] := Simp[(c^(n - 1)*(c*x)^(m - n + 1)*(a + b*x^n)^(p + 1))/(b*(m + n*p + 1)), x] - Dist[(a*c^n*(m - n + 1))/(b*(m + n*p + 1)), Int[(c*x)^(m - n)*(a + b*x^n)^p, x], x] /; FreeQ[{a, b, c, m, p}, x] && IGtQ[n, 0] && SumSimplerQ[m, -n] && NeQ[m + n*p + 1, 0] && ILtQ[Simplify[(m + 1)/n + p], 0]
  323. Int[Times[Power[Times[Optional[Pattern[c, Blank[]]], Pattern[x, Blank[]]], Pattern[m, Blank[]]], Power[Plus[Pattern[a1, Blank[]], Times[Optional[Pattern[b1, Blank[]]], Power[Pattern[x, Blank[]], Pattern[n, Blank[]]]]], Pattern[p, Blank[]]], Power[Plus[Pattern[a2, Blank[]], Times[Optional[Pattern[b2, Blank[]]], Power[Pattern[x, Blank[]], Pattern[n, Blank[]]]]], Pattern[p, Blank[]]]], Pattern[x, Blank[Symbol]]] := Simp[(c^(2*n - 1)*(c*x)^(m - 2*n + 1)*(a1 + b1*x^n)^(p + 1)*(a2 + b2*x^n)^(p + 1))/(b1*b2*(m + 2*n*p + 1)), x] - Dist[(a1*a2*c^(2*n)*(m - 2*n + 1))/(b1*b2*(m + 2*n*p + 1)), Int[(c*x)^(m - 2*n)*(a1 + b1*x^n)^p*(a2 + b2*x^n)^p, x], x] /; FreeQ[{a1, b1, a2, b2, c, p}, x] && EqQ[a2*b1 + a1*b2, 0] && IGtQ[2*n, 0] && GtQ[m, 2*n - 1] && NeQ[m + 2*n*p + 1, 0] && IntBinomialQ[a1*a2, b1*b2, c, 2*n, m, p, x]
  324. Int[Times[Power[Times[Optional[Pattern[c, Blank[]]], Pattern[x, Blank[]]], Pattern[m, Blank[]]], Power[Plus[Pattern[a1, Blank[]], Times[Optional[Pattern[b1, Blank[]]], Power[Pattern[x, Blank[]], Pattern[n, Blank[]]]]], Pattern[p, Blank[]]], Power[Plus[Pattern[a2, Blank[]], Times[Optional[Pattern[b2, Blank[]]], Power[Pattern[x, Blank[]], Pattern[n, Blank[]]]]], Pattern[p, Blank[]]]], Pattern[x, Blank[Symbol]]] := Simp[(c^(2*n - 1)*(c*x)^(m - 2*n + 1)*(a1 + b1*x^n)^(p + 1)*(a2 + b2*x^n)^(p + 1))/(b1*b2*(m + 2*n*p + 1)), x] - Dist[(a1*a2*c^(2*n)*(m - 2*n + 1))/(b1*b2*(m + 2*n*p + 1)), Int[(c*x)^(m - 2*n)*(a1 + b1*x^n)^p*(a2 + b2*x^n)^p, x], x] /; FreeQ[{a1, b1, a2, b2, c, m, p}, x] && EqQ[a2*b1 + a1*b2, 0] && IGtQ[2*n, 0] && SumSimplerQ[m, -2*n] && NeQ[m + 2*n*p + 1, 0] && ILtQ[Simplify[(m + 1)/(2*n) + p], 0]
  325. Int[Times[Power[Times[Optional[Pattern[c, Blank[]]], Pattern[x, Blank[]]], Pattern[m, Blank[]]], Power[Plus[Pattern[a, Blank[]], Times[Optional[Pattern[b, Blank[]]], Power[Pattern[x, Blank[]], Pattern[n, Blank[]]]]], Pattern[p, Blank[]]]], Pattern[x, Blank[Symbol]]] := Simp[((c*x)^(m + 1)*(a + b*x^n)^(p + 1))/(a*c*(m + 1)), x] - Dist[(b*(m + n*(p + 1) + 1))/(a*c^n*(m + 1)), Int[(c*x)^(m + n)*(a + b*x^n)^p, x], x] /; FreeQ[{a, b, c, p}, x] && IGtQ[n, 0] && LtQ[m, -1] && IntBinomialQ[a, b, c, n, m, p, x]
  326. Int[Times[Power[Times[Optional[Pattern[c, Blank[]]], Pattern[x, Blank[]]], Pattern[m, Blank[]]], Power[Plus[Pattern[a, Blank[]], Times[Optional[Pattern[b, Blank[]]], Power[Pattern[x, Blank[]], Pattern[n, Blank[]]]]], Pattern[p, Blank[]]]], Pattern[x, Blank[Symbol]]] := Simp[((c*x)^(m + 1)*(a + b*x^n)^(p + 1))/(a*c*(m + 1)), x] - Dist[(b*(m + n*(p + 1) + 1))/(a*c^n*(m + 1)), Int[(c*x)^(m + n)*(a + b*x^n)^p, x], x] /; FreeQ[{a, b, c, m, p}, x] && IGtQ[n, 0] && SumSimplerQ[m, n] && ILtQ[Simplify[(m + 1)/n + p], 0]
  327. Int[Times[Power[Times[Optional[Pattern[c, Blank[]]], Pattern[x, Blank[]]], Pattern[m, Blank[]]], Power[Plus[Pattern[a1, Blank[]], Times[Optional[Pattern[b1, Blank[]]], Power[Pattern[x, Blank[]], Pattern[n, Blank[]]]]], Pattern[p, Blank[]]], Power[Plus[Pattern[a2, Blank[]], Times[Optional[Pattern[b2, Blank[]]], Power[Pattern[x, Blank[]], Pattern[n, Blank[]]]]], Pattern[p, Blank[]]]], Pattern[x, Blank[Symbol]]] := Simp[((c*x)^(m + 1)*(a1 + b1*x^n)^(p + 1)*(a2 + b2*x^n)^(p + 1))/(a1*a2*c*(m + 1)), x] - Dist[(b1*b2*(m + 2*n*(p + 1) + 1))/(a1*a2*c^(2*n)*(m + 1)), Int[(c*x)^(m + 2*n)*(a1 + b1*x^n)^p*(a2 + b2*x^n)^p, x], x] /; FreeQ[{a1, b1, a2, b2, c, p}, x] && EqQ[a2*b1 + a1*b2, 0] && IGtQ[2*n, 0] && LtQ[m, -1] && IntBinomialQ[a1*a2, b1*b2, c, 2*n, m, p, x]
  328. Int[Times[Power[Times[Optional[Pattern[c, Blank[]]], Pattern[x, Blank[]]], Pattern[m, Blank[]]], Power[Plus[Pattern[a1, Blank[]], Times[Optional[Pattern[b1, Blank[]]], Power[Pattern[x, Blank[]], Pattern[n, Blank[]]]]], Pattern[p, Blank[]]], Power[Plus[Pattern[a2, Blank[]], Times[Optional[Pattern[b2, Blank[]]], Power[Pattern[x, Blank[]], Pattern[n, Blank[]]]]], Pattern[p, Blank[]]]], Pattern[x, Blank[Symbol]]] := Simp[((c*x)^(m + 1)*(a1 + b1*x^n)^(p + 1)*(a2 + b2*x^n)^(p + 1))/(a1*a2*c*(m + 1)), x] - Dist[(b1*b2*(m + 2*n*(p + 1) + 1))/(a1*a2*c^(2*n)*(m + 1)), Int[(c*x)^(m + 2*n)*(a1 + b1*x^n)^p*(a2 + b2*x^n)^p, x], x] /; FreeQ[{a1, b1, a2, b2, c, m, p}, x] && EqQ[a2*b1 + a1*b2, 0] && IGtQ[2*n, 0] && SumSimplerQ[m, 2*n] && ILtQ[Simplify[(m + 1)/(2*n) + p], 0]
  329. Int[Times[Power[Times[Optional[Pattern[c, Blank[]]], Pattern[x, Blank[]]], Pattern[m, Blank[]]], Power[Plus[Pattern[a, Blank[]], Times[Optional[Pattern[b, Blank[]]], Power[Pattern[x, Blank[]], Pattern[n, Blank[]]]]], Pattern[p, Blank[]]]], Pattern[x, Blank[Symbol]]] := With[{k = Denominator[m]}, Dist[k/c, Subst[Int[x^(k*(m + 1) - 1)*(a + (b*x^(k*n))/c^n)^p, x], x, (c*x)^(1/k)], x]] /; FreeQ[{a, b, c, p}, x] && IGtQ[n, 0] && FractionQ[m] && IntBinomialQ[a, b, c, n, m, p, x]
  330. Int[Times[Power[Times[Optional[Pattern[c, Blank[]]], Pattern[x, Blank[]]], Pattern[m, Blank[]]], Power[Plus[Pattern[a1, Blank[]], Times[Optional[Pattern[b1, Blank[]]], Power[Pattern[x, Blank[]], Pattern[n, Blank[]]]]], Pattern[p, Blank[]]], Power[Plus[Pattern[a2, Blank[]], Times[Optional[Pattern[b2, Blank[]]], Power[Pattern[x, Blank[]], Pattern[n, Blank[]]]]], Pattern[p, Blank[]]]], Pattern[x, Blank[Symbol]]] := With[{k = Denominator[m]}, Dist[k/c, Subst[Int[x^(k*(m + 1) - 1)*(a1 + (b1*x^(k*n))/c^n)^p*(a2 + (b2*x^(k*n))/c^n)^p, x], x, (c*x)^(1/k)], x]] /; FreeQ[{a1, b1, a2, b2, c, p}, x] && EqQ[a2*b1 + a1*b2, 0] && IGtQ[2*n, 0] && FractionQ[m] && IntBinomialQ[a1*a2, b1*b2, c, 2*n, m, p, x]
  331. Int[Times[Power[Pattern[x, Blank[]], Optional[Pattern[m, Blank[]]]], Power[Plus[Pattern[a, Blank[]], Times[Optional[Pattern[b, Blank[]]], Power[Pattern[x, Blank[]], Pattern[n, Blank[]]]]], Pattern[p, Blank[]]]], Pattern[x, Blank[Symbol]]] := Dist[a^(p + (m + 1)/n), Subst[Int[x^m/(1 - b*x^n)^(p + (m + 1)/n + 1), x], x, x/(a + b*x^n)^(1/n)], x] /; FreeQ[{a, b}, x] && IGtQ[n, 0] && LtQ[-1, p, 0] && NeQ[p, -2^(-1)] && IntegersQ[m, p + (m + 1)/n]
  332. Int[Times[Power[Pattern[x, Blank[]], Optional[Pattern[m, Blank[]]]], Power[Plus[Pattern[a1, Blank[]], Times[Optional[Pattern[b1, Blank[]]], Power[Pattern[x, Blank[]], Pattern[n, Blank[]]]]], Pattern[p, Blank[]]], Power[Plus[Pattern[a2, Blank[]], Times[Optional[Pattern[b2, Blank[]]], Power[Pattern[x, Blank[]], Pattern[n, Blank[]]]]], Pattern[p, Blank[]]]], Pattern[x, Blank[Symbol]]] := Dist[(a1*a2)^(p + (m + 1)/(2*n)), Subst[Int[x^m/((1 - b1*x^n)^(p + (m + 1)/(2*n) + 1)*(1 - b2*x^n)^(p + (m + 1)/(2*n) + 1)), x], x, x/((a1 + b1*x^n)^(1/(2*n))*(a2 + b2*x^n)^(1/(2*n)))], x] /; FreeQ[{a1, b1, a2, b2}, x] && EqQ[a2*b1 + a1*b2, 0] && IGtQ[2*n, 0] && LtQ[-1, p, 0] && NeQ[p, -2^(-1)] && IntegersQ[m, p + (m + 1)/(2*n)]
  333. Int[Times[Power[Pattern[x, Blank[]], Optional[Pattern[m, Blank[]]]], Power[Plus[Pattern[a, Blank[]], Times[Optional[Pattern[b, Blank[]]], Power[Pattern[x, Blank[]], Pattern[n, Blank[]]]]], Pattern[p, Blank[]]]], Pattern[x, Blank[Symbol]]] := Dist[(a/(a + b*x^n))^(p + (m + 1)/n)*(a + b*x^n)^(p + (m + 1)/n), Subst[Int[x^m/(1 - b*x^n)^(p + (m + 1)/n + 1), x], x, x/(a + b*x^n)^(1/n)], x] /; FreeQ[{a, b}, x] && IGtQ[n, 0] && LtQ[-1, p, 0] && NeQ[p, -2^(-1)] && IntegerQ[m] && LtQ[Denominator[p + (m + 1)/n], Denominator[p]]
  334. Int[Times[Power[Pattern[x, Blank[]], Optional[Pattern[m, Blank[]]]], Power[Plus[Pattern[a1, Blank[]], Times[Optional[Pattern[b1, Blank[]]], Power[Pattern[x, Blank[]], Pattern[n, Blank[]]]]], Pattern[p, Blank[]]], Power[Plus[Pattern[a2, Blank[]], Times[Optional[Pattern[b2, Blank[]]], Power[Pattern[x, Blank[]], Pattern[n, Blank[]]]]], Pattern[p, Blank[]]]], Pattern[x, Blank[Symbol]]] := Dist[(a1/(a1 + b1*x^n))^(p + (m + 1)/(2*n))*(a1 + b1*x^n)^(p + (m + 1)/(2*n))*(a2/(a2 + b2*x^n))^(p + (m + 1)/(2*n))*(a2 + b2*x^n)^(p + (m + 1)/(2*n)), Subst[Int[x^m/((1 - b1*x^n)^(p + (m + 1)/(2*n) + 1)*(1 - b2*x^n)^(p + (m + 1)/(2*n) + 1)), x], x, x/((a1 + b1*x^n)^(1/(2*n))*(a2 + b2*x^n)^(1/(2*n)))], x] /; FreeQ[{a1, b1, a2, b2}, x] && EqQ[a2*b1 + a1*b2, 0] && IGtQ[2*n, 0] && LtQ[-1, p, 0] && NeQ[p, -2^(-1)] && IntegerQ[m] && LtQ[Denominator[p + (m + 1)/(2*n)], Denominator[p]]
  335. Int[Times[Power[Pattern[x, Blank[]], Optional[Pattern[m, Blank[]]]], Power[Plus[Pattern[a, Blank[]], Times[Optional[Pattern[b, Blank[]]], Power[Pattern[x, Blank[]], Pattern[n, Blank[]]]]], Pattern[p, Blank[]]]], Pattern[x, Blank[Symbol]]] := -Subst[Int[(a + b/x^n)^p/x^(m + 2), x], x, 1/x] /; FreeQ[{a, b, p}, x] && ILtQ[n, 0] && IntegerQ[m]
  336. Int[Times[Power[Pattern[x, Blank[]], Optional[Pattern[m, Blank[]]]], Power[Plus[Pattern[a1, Blank[]], Times[Optional[Pattern[b1, Blank[]]], Power[Pattern[x, Blank[]], Pattern[n, Blank[]]]]], Pattern[p, Blank[]]], Power[Plus[Pattern[a2, Blank[]], Times[Optional[Pattern[b2, Blank[]]], Power[Pattern[x, Blank[]], Pattern[n, Blank[]]]]], Pattern[p, Blank[]]]], Pattern[x, Blank[Symbol]]] := -Subst[Int[((a1 + b1/x^n)^p*(a2 + b2/x^n)^p)/x^(m + 2), x], x, 1/x] /; FreeQ[{a1, b1, a2, b2, p}, x] && EqQ[a2*b1 + a1*b2, 0] && ILtQ[2*n, 0] && IntegerQ[m]
  337. Int[Times[Power[Times[Optional[Pattern[c, Blank[]]], Pattern[x, Blank[]]], Pattern[m, Blank[]]], Power[Plus[Pattern[a, Blank[]], Times[Optional[Pattern[b, Blank[]]], Power[Pattern[x, Blank[]], Pattern[n, Blank[]]]]], Pattern[p, Blank[]]]], Pattern[x, Blank[Symbol]]] := With[{k = Denominator[m]}, -Dist[k/c, Subst[Int[(a + b/(c^n*x^(k*n)))^p/x^(k*(m + 1) + 1), x], x, 1/(c*x)^(1/k)], x]] /; FreeQ[{a, b, c, p}, x] && ILtQ[n, 0] && FractionQ[m]
  338. Int[Times[Power[Times[Optional[Pattern[c, Blank[]]], Pattern[x, Blank[]]], Pattern[m, Blank[]]], Power[Plus[Pattern[a1, Blank[]], Times[Optional[Pattern[b1, Blank[]]], Power[Pattern[x, Blank[]], Pattern[n, Blank[]]]]], Pattern[p, Blank[]]], Power[Plus[Pattern[a2, Blank[]], Times[Optional[Pattern[b2, Blank[]]], Power[Pattern[x, Blank[]], Pattern[n, Blank[]]]]], Pattern[p, Blank[]]]], Pattern[x, Blank[Symbol]]] := With[{k = Denominator[m]}, -Dist[k/c, Subst[Int[((a1 + b1/(c^n*x^(k*n)))^p*(a2 + b2/(c^n*x^(k*n)))^p)/x^(k*(m + 1) + 1), x], x, 1/(c*x)^(1/k)], x]] /; FreeQ[{a1, b1, a2, b2, c, p}, x] && EqQ[a2*b1 + a1*b2, 0] && ILtQ[2*n, 0] && FractionQ[m]
  339. Int[Times[Power[Times[Optional[Pattern[c, Blank[]]], Pattern[x, Blank[]]], Pattern[m, Blank[]]], Power[Plus[Pattern[a, Blank[]], Times[Optional[Pattern[b, Blank[]]], Power[Pattern[x, Blank[]], Pattern[n, Blank[]]]]], Pattern[p, Blank[]]]], Pattern[x, Blank[Symbol]]] := -Dist[((c*x)^(m + 1)*(1/x)^(m + 1))/c, Subst[Int[(a + b/x^n)^p/x^(m + 2), x], x, 1/x], x] /; FreeQ[{a, b, c, m, p}, x] && ILtQ[n, 0] && !RationalQ[m]
  340. Int[Times[Power[Times[Optional[Pattern[c, Blank[]]], Pattern[x, Blank[]]], Pattern[m, Blank[]]], Power[Plus[Pattern[a1, Blank[]], Times[Optional[Pattern[b1, Blank[]]], Power[Pattern[x, Blank[]], Pattern[n, Blank[]]]]], Pattern[p, Blank[]]], Power[Plus[Pattern[a2, Blank[]], Times[Optional[Pattern[b2, Blank[]]], Power[Pattern[x, Blank[]], Pattern[n, Blank[]]]]], Pattern[p, Blank[]]]], Pattern[x, Blank[Symbol]]] := -Dist[((c*x)^(m + 1)*(1/x)^(m + 1))/c, Subst[Int[((a1 + b1/x^n)^p*(a2 + b2/x^n)^p)/x^(m + 2), x], x, 1/x], x] /; FreeQ[{a1, b1, a2, b2, c, m, p}, x] && EqQ[a2*b1 + a1*b2, 0] && ILtQ[2*n, 0] && !RationalQ[m]
  341. Int[Times[Power[Pattern[x, Blank[]], Optional[Pattern[m, Blank[]]]], Power[Plus[Pattern[a, Blank[]], Times[Optional[Pattern[b, Blank[]]], Power[Pattern[x, Blank[]], Pattern[n, Blank[]]]]], Pattern[p, Blank[]]]], Pattern[x, Blank[Symbol]]] := With[{k = Denominator[n]}, Dist[k, Subst[Int[x^(k*(m + 1) - 1)*(a + b*x^(k*n))^p, x], x, x^(1/k)], x]] /; FreeQ[{a, b, m, p}, x] && FractionQ[n]
  342. Int[Times[Power[Pattern[x, Blank[]], Optional[Pattern[m, Blank[]]]], Power[Plus[Pattern[a1, Blank[]], Times[Optional[Pattern[b1, Blank[]]], Power[Pattern[x, Blank[]], Pattern[n, Blank[]]]]], Pattern[p, Blank[]]], Power[Plus[Pattern[a2, Blank[]], Times[Optional[Pattern[b2, Blank[]]], Power[Pattern[x, Blank[]], Pattern[n, Blank[]]]]], Pattern[p, Blank[]]]], Pattern[x, Blank[Symbol]]] := With[{k = Denominator[2*n]}, Dist[k, Subst[Int[x^(k*(m + 1) - 1)*(a1 + b1*x^(k*n))^p*(a2 + b2*x^(k*n))^p, x], x, x^(1/k)], x]] /; FreeQ[{a1, b1, a2, b2, m, p}, x] && EqQ[a2*b1 + a1*b2, 0] && FractionQ[2*n]
  343. Int[Times[Power[Times[Pattern[c, Blank[]], Pattern[x, Blank[]]], Pattern[m, Blank[]]], Power[Plus[Pattern[a, Blank[]], Times[Optional[Pattern[b, Blank[]]], Power[Pattern[x, Blank[]], Pattern[n, Blank[]]]]], Pattern[p, Blank[]]]], Pattern[x, Blank[Symbol]]] := Dist[(c^IntPart[m]*(c*x)^FracPart[m])/x^FracPart[m], Int[x^m*(a + b*x^n)^p, x], x] /; FreeQ[{a, b, c, m, p}, x] && FractionQ[n]
  344. Int[Times[Power[Times[Pattern[c, Blank[]], Pattern[x, Blank[]]], Pattern[m, Blank[]]], Power[Plus[Pattern[a1, Blank[]], Times[Optional[Pattern[b1, Blank[]]], Power[Pattern[x, Blank[]], Pattern[n, Blank[]]]]], Pattern[p, Blank[]]], Power[Plus[Pattern[a2, Blank[]], Times[Optional[Pattern[b2, Blank[]]], Power[Pattern[x, Blank[]], Pattern[n, Blank[]]]]], Pattern[p, Blank[]]]], Pattern[x, Blank[Symbol]]] := Dist[(c^IntPart[m]*(c*x)^FracPart[m])/x^FracPart[m], Int[x^m*(a1 + b1*x^n)^p*(a2 + b2*x^n)^p, x], x] /; FreeQ[{a1, b1, a2, b2, c, m, p}, x] && EqQ[a2*b1 + a1*b2, 0] && FractionQ[2*n]
  345. Int[Times[Power[Pattern[x, Blank[]], Optional[Pattern[m, Blank[]]]], Power[Plus[Pattern[a, Blank[]], Times[Optional[Pattern[b, Blank[]]], Power[Pattern[x, Blank[]], Pattern[n, Blank[]]]]], Pattern[p, Blank[]]]], Pattern[x, Blank[Symbol]]] := Dist[1/(m + 1), Subst[Int[(a + b*x^Simplify[n/(m + 1)])^p, x], x, x^(m + 1)], x] /; FreeQ[{a, b, m, n, p}, x] && IntegerQ[Simplify[n/(m + 1)]] && !IntegerQ[n]
  346. Int[Times[Power[Pattern[x, Blank[]], Optional[Pattern[m, Blank[]]]], Power[Plus[Pattern[a1, Blank[]], Times[Optional[Pattern[b1, Blank[]]], Power[Pattern[x, Blank[]], Pattern[n, Blank[]]]]], Pattern[p, Blank[]]], Power[Plus[Pattern[a2, Blank[]], Times[Optional[Pattern[b2, Blank[]]], Power[Pattern[x, Blank[]], Pattern[n, Blank[]]]]], Pattern[p, Blank[]]]], Pattern[x, Blank[Symbol]]] := Dist[1/(m + 1), Subst[Int[(a1 + b1*x^Simplify[n/(m + 1)])^p*(a2 + b2*x^Simplify[n/(m + 1)])^p, x], x, x^(m + 1)], x] /; FreeQ[{a1, b1, a2, b2, m, n, p}, x] && EqQ[a2*b1 + a1*b2, 0] && IntegerQ[Simplify[(2*n)/(m + 1)]] && !IntegerQ[2*n]
  347. Int[Times[Power[Times[Pattern[c, Blank[]], Pattern[x, Blank[]]], Pattern[m, Blank[]]], Power[Plus[Pattern[a, Blank[]], Times[Optional[Pattern[b, Blank[]]], Power[Pattern[x, Blank[]], Pattern[n, Blank[]]]]], Pattern[p, Blank[]]]], Pattern[x, Blank[Symbol]]] := Dist[(c^IntPart[m]*(c*x)^FracPart[m])/x^FracPart[m], Int[x^m*(a + b*x^n)^p, x], x] /; FreeQ[{a, b, c, m, n, p}, x] && IntegerQ[Simplify[n/(m + 1)]] && !IntegerQ[n]
  348. Int[Times[Power[Times[Pattern[c, Blank[]], Pattern[x, Blank[]]], Pattern[m, Blank[]]], Power[Plus[Pattern[a1, Blank[]], Times[Optional[Pattern[b1, Blank[]]], Power[Pattern[x, Blank[]], Pattern[n, Blank[]]]]], Pattern[p, Blank[]]], Power[Plus[Pattern[a2, Blank[]], Times[Optional[Pattern[b2, Blank[]]], Power[Pattern[x, Blank[]], Pattern[n, Blank[]]]]], Pattern[p, Blank[]]]], Pattern[x, Blank[Symbol]]] := Dist[(c^IntPart[m]*(c*x)^FracPart[m])/x^FracPart[m], Int[x^m*(a1 + b1*x^n)^p*(a2 + b2*x^n)^p, x], x] /; FreeQ[{a1, b1, a2, b2, c, m, n, p}, x] && EqQ[a2*b1 + a1*b2, 0] && IntegerQ[Simplify[(2*n)/(m + 1)]] && !IntegerQ[2*n]
  349. Int[Times[Power[Pattern[x, Blank[]], Optional[Pattern[m, Blank[]]]], Power[Plus[Pattern[a, Blank[]], Times[Optional[Pattern[b, Blank[]]], Power[Pattern[x, Blank[]], Pattern[n, Blank[]]]]], Pattern[p, Blank[]]]], Pattern[x, Blank[Symbol]]] := Simp[(x^(m + 1)*(a + b*x^n)^p)/(m + 1), x] - Dist[(b*n*p)/(m + 1), Int[x^(m + n)*(a + b*x^n)^(p - 1), x], x] /; FreeQ[{a, b, m, n}, x] && EqQ[(m + 1)/n + p, 0] && GtQ[p, 0]
  350. Int[Times[Power[Pattern[x, Blank[]], Optional[Pattern[m, Blank[]]]], Power[Plus[Pattern[a1, Blank[]], Times[Optional[Pattern[b1, Blank[]]], Power[Pattern[x, Blank[]], Pattern[n, Blank[]]]]], Pattern[p, Blank[]]], Power[Plus[Pattern[a2, Blank[]], Times[Optional[Pattern[b2, Blank[]]], Power[Pattern[x, Blank[]], Pattern[n, Blank[]]]]], Pattern[p, Blank[]]]], Pattern[x, Blank[Symbol]]] := Simp[(x^(m + 1)*(a1 + b1*x^n)^p*(a2 + b2*x^n)^p)/(m + 1), x] - Dist[(2*b1*b2*n*p)/(m + 1), Int[x^(m + 2*n)*(a1 + b1*x^n)^(p - 1)*(a2 + b2*x^n)^(p - 1), x], x] /; FreeQ[{a1, b1, a2, b2, m, n}, x] && EqQ[a2*b1 + a1*b2, 0] && EqQ[(m + 1)/(2*n) + p, 0] && GtQ[p, 0]
  351. Int[Times[Power[Times[Pattern[c, Blank[]], Pattern[x, Blank[]]], Pattern[m, Blank[]]], Power[Plus[Pattern[a, Blank[]], Times[Optional[Pattern[b, Blank[]]], Power[Pattern[x, Blank[]], Pattern[n, Blank[]]]]], Pattern[p, Blank[]]]], Pattern[x, Blank[Symbol]]] := Dist[(c^IntPart[m]*(c*x)^FracPart[m])/x^FracPart[m], Int[x^m*(a + b*x^n)^p, x], x] /; FreeQ[{a, b, c, m, n}, x] && EqQ[(m + 1)/n + p, 0] && GtQ[p, 0]
  352. Int[Times[Power[Times[Pattern[c, Blank[]], Pattern[x, Blank[]]], Pattern[m, Blank[]]], Power[Plus[Pattern[a1, Blank[]], Times[Optional[Pattern[b1, Blank[]]], Power[Pattern[x, Blank[]], Pattern[n, Blank[]]]]], Pattern[p, Blank[]]], Power[Plus[Pattern[a2, Blank[]], Times[Optional[Pattern[b2, Blank[]]], Power[Pattern[x, Blank[]], Pattern[n, Blank[]]]]], Pattern[p, Blank[]]]], Pattern[x, Blank[Symbol]]] := Dist[(c^IntPart[m]*(c*x)^FracPart[m])/x^FracPart[m], Int[x^m*(a1 + b1*x^n)^p*(a2 + b2*x^n)^p, x], x] /; FreeQ[{a1, b1, a2, b2, c, m, n}, x] && EqQ[a2*b1 + a1*b2, 0] && EqQ[(m + 1)/(2*n) + p, 0] && GtQ[p, 0]
  353. Int[Times[Power[Times[Optional[Pattern[c, Blank[]]], Pattern[x, Blank[]]], Optional[Pattern[m, Blank[]]]], Power[Plus[Pattern[a, Blank[]], Times[Optional[Pattern[b, Blank[]]], Power[Pattern[x, Blank[]], Pattern[n, Blank[]]]]], Pattern[p, Blank[]]]], Pattern[x, Blank[Symbol]]] := Simp[((c*x)^(m + 1)*(a + b*x^n)^p)/(c*(m + n*p + 1)), x] + Dist[(a*n*p)/(m + n*p + 1), Int[(c*x)^m*(a + b*x^n)^(p - 1), x], x] /; FreeQ[{a, b, c, m, n}, x] && IntegerQ[p + Simplify[(m + 1)/n]] && GtQ[p, 0] && NeQ[m + n*p + 1, 0]
  354. Int[Times[Power[Times[Optional[Pattern[c, Blank[]]], Pattern[x, Blank[]]], Optional[Pattern[m, Blank[]]]], Power[Plus[Pattern[a1, Blank[]], Times[Optional[Pattern[b1, Blank[]]], Power[Pattern[x, Blank[]], Pattern[n, Blank[]]]]], Pattern[p, Blank[]]], Power[Plus[Pattern[a2, Blank[]], Times[Optional[Pattern[b2, Blank[]]], Power[Pattern[x, Blank[]], Pattern[n, Blank[]]]]], Pattern[p, Blank[]]]], Pattern[x, Blank[Symbol]]] := Simp[((c*x)^(m + 1)*(a1 + b1*x^n)^p*(a2 + b2*x^n)^p)/(c*(m + 2*n*p + 1)), x] + Dist[(2*a1*a2*n*p)/(m + 2*n*p + 1), Int[(c*x)^m*(a1 + b1*x^n)^(p - 1)*(a2 + b2*x^n)^(p - 1), x], x] /; FreeQ[{a1, b1, a2, b2, c, m, n}, x] && EqQ[a2*b1 + a1*b2, 0] && IntegerQ[p + Simplify[(m + 1)/(2*n)]] && GtQ[p, 0] && NeQ[m + 2*n*p + 1, 0]
  355. Int[Times[Power[Pattern[x, Blank[]], Optional[Pattern[m, Blank[]]]], Power[Plus[Pattern[a, Blank[]], Times[Optional[Pattern[b, Blank[]]], Power[Pattern[x, Blank[]], Pattern[n, Blank[]]]]], Pattern[p, Blank[]]]], Pattern[x, Blank[Symbol]]] := With[{k = Denominator[p]}, Dist[(k*a^(p + Simplify[(m + 1)/n]))/n, Subst[Int[x^(k*Simplify[(m + 1)/n] - 1)/(1 - b*x^k)^(p + Simplify[(m + 1)/n] + 1), x], x, x^(n/k)/(a + b*x^n)^(1/k)], x]] /; FreeQ[{a, b, m, n}, x] && IntegerQ[p + Simplify[(m + 1)/n]] && LtQ[-1, p, 0]
  356. Int[Times[Power[Pattern[x, Blank[]], Optional[Pattern[m, Blank[]]]], Power[Plus[Pattern[a1, Blank[]], Times[Optional[Pattern[b1, Blank[]]], Power[Pattern[x, Blank[]], Pattern[n, Blank[]]]]], Pattern[p, Blank[]]], Power[Plus[Pattern[a2, Blank[]], Times[Optional[Pattern[b2, Blank[]]], Power[Pattern[x, Blank[]], Pattern[n, Blank[]]]]], Pattern[p, Blank[]]]], Pattern[x, Blank[Symbol]]] := With[{k = Denominator[p]}, Dist[(k*(a1*a2)^(p + Simplify[(m + 1)/(2*n)]))/(2*n), Subst[Int[x^(k*Simplify[(m + 1)/(2*n)] - 1)/(1 - b1*b2*x^k)^(p + Simplify[(m + 1)/(2*n)] + 1), x], x, x^((2*n)/k)/((a1 + b1*x^n)^(1/k)*(a2 + b2*x^n)^(1/k))], x]] /; FreeQ[{a1, b1, a2, b2, m, n}, x] && EqQ[a2*b1 + a1*b2, 0] && IntegerQ[p + Simplify[(m + 1)/(2*n)]] && LtQ[-1, p, 0]
  357. Int[Times[Power[Times[Pattern[c, Blank[]], Pattern[x, Blank[]]], Pattern[m, Blank[]]], Power[Plus[Pattern[a, Blank[]], Times[Optional[Pattern[b, Blank[]]], Power[Pattern[x, Blank[]], Pattern[n, Blank[]]]]], Pattern[p, Blank[]]]], Pattern[x, Blank[Symbol]]] := Dist[(c^IntPart[m]*(c*x)^FracPart[m])/x^FracPart[m], Int[x^m*(a + b*x^n)^p, x], x] /; FreeQ[{a, b, c, m, n}, x] && IntegerQ[p + Simplify[(m + 1)/n]] && LtQ[-1, p, 0]
  358. Int[Times[Power[Times[Pattern[c, Blank[]], Pattern[x, Blank[]]], Pattern[m, Blank[]]], Power[Plus[Pattern[a1, Blank[]], Times[Optional[Pattern[b1, Blank[]]], Power[Pattern[x, Blank[]], Pattern[n, Blank[]]]]], Pattern[p, Blank[]]], Power[Plus[Pattern[a2, Blank[]], Times[Optional[Pattern[b2, Blank[]]], Power[Pattern[x, Blank[]], Pattern[n, Blank[]]]]], Pattern[p, Blank[]]]], Pattern[x, Blank[Symbol]]] := Dist[(c^IntPart[m]*(c*x)^FracPart[m])/x^FracPart[m], Int[x^m*(a1 + b1*x^n)^p*(a2 + b2*x^n)^p, x], x] /; FreeQ[{a1, b1, a2, b2, c, m, n}, x] && EqQ[a2*b1 + a1*b2, 0] && IntegerQ[p + Simplify[(m + 1)/(2*n)]] && LtQ[-1, p, 0]
  359. Int[Times[Power[Times[Optional[Pattern[c, Blank[]]], Pattern[x, Blank[]]], Optional[Pattern[m, Blank[]]]], Power[Plus[Pattern[a, Blank[]], Times[Optional[Pattern[b, Blank[]]], Power[Pattern[x, Blank[]], Pattern[n, Blank[]]]]], Pattern[p, Blank[]]]], Pattern[x, Blank[Symbol]]] := -Simp[((c*x)^(m + 1)*(a + b*x^n)^(p + 1))/(a*c*n*(p + 1)), x] + Dist[(m + n*(p + 1) + 1)/(a*n*(p + 1)), Int[(c*x)^m*(a + b*x^n)^(p + 1), x], x] /; FreeQ[{a, b, c, m, n}, x] && IntegerQ[p + Simplify[(m + 1)/n]] && LtQ[p, -1]
  360. Int[Times[Power[Times[Optional[Pattern[c, Blank[]]], Pattern[x, Blank[]]], Optional[Pattern[m, Blank[]]]], Power[Plus[Pattern[a1, Blank[]], Times[Optional[Pattern[b1, Blank[]]], Power[Pattern[x, Blank[]], Pattern[n, Blank[]]]]], Pattern[p, Blank[]]], Power[Plus[Pattern[a2, Blank[]], Times[Optional[Pattern[b2, Blank[]]], Power[Pattern[x, Blank[]], Pattern[n, Blank[]]]]], Pattern[p, Blank[]]]], Pattern[x, Blank[Symbol]]] := -Simp[((c*x)^(m + 1)*(a1 + b1*x^n)^(p + 1)*(a2 + b2*x^n)^(p + 1))/(2*a1*a2*c*n*(p + 1)), x] + Dist[(m + 2*n*(p + 1) + 1)/(2*a1*a2*n*(p + 1)), Int[(c*x)^m*(a1 + b1*x^n)^(p + 1)*(a2 + b2*x^n)^(p + 1), x], x] /; FreeQ[{a1, b1, a2, b2, c, m, n}, x] && EqQ[a2*b1 + a1*b2, 0] && IntegerQ[p + Simplify[(m + 1)/(2*n)]] && LtQ[p, -1]
  361. Int[Times[Power[Pattern[x, Blank[]], Optional[Pattern[m, Blank[]]]], Power[Plus[Pattern[a, Blank[]], Times[Optional[Pattern[b, Blank[]]], Power[Pattern[x, Blank[]], Pattern[n, Blank[]]]]], -1]], Pattern[x, Blank[Symbol]]] := With[{mn = Simplify[m - n]}, Simp[x^(mn + 1)/(b*(mn + 1)), x] - Dist[a/b, Int[x^mn/(a + b*x^n), x], x]] /; FreeQ[{a, b, m, n}, x] && FractionQ[Simplify[(m + 1)/n]] && SumSimplerQ[m, -n]
  362. Int[Times[Power[Pattern[x, Blank[]], Pattern[m, Blank[]]], Power[Plus[Pattern[a, Blank[]], Times[Optional[Pattern[b, Blank[]]], Power[Pattern[x, Blank[]], Pattern[n, Blank[]]]]], -1]], Pattern[x, Blank[Symbol]]] := Simp[x^(m + 1)/(a*(m + 1)), x] - Dist[b/a, Int[x^Simplify[m + n]/(a + b*x^n), x], x] /; FreeQ[{a, b, m, n}, x] && FractionQ[Simplify[(m + 1)/n]] && SumSimplerQ[m, n]
  363. Int[Times[Power[Times[Pattern[c, Blank[]], Pattern[x, Blank[]]], Pattern[m, Blank[]]], Power[Plus[Pattern[a, Blank[]], Times[Optional[Pattern[b, Blank[]]], Power[Pattern[x, Blank[]], Pattern[n, Blank[]]]]], -1]], Pattern[x, Blank[Symbol]]] := Dist[(c^IntPart[m]*(c*x)^FracPart[m])/x^FracPart[m], Int[x^m/(a + b*x^n), x], x] /; FreeQ[{a, b, c, m, n}, x] && FractionQ[Simplify[(m + 1)/n]] && (SumSimplerQ[m, n] || SumSimplerQ[m, -n])
  364. Int[Times[Power[Times[Optional[Pattern[c, Blank[]]], Pattern[x, Blank[]]], Optional[Pattern[m, Blank[]]]], Power[Plus[Pattern[a, Blank[]], Times[Optional[Pattern[b, Blank[]]], Power[Pattern[x, Blank[]], Pattern[n, Blank[]]]]], Pattern[p, Blank[]]]], Pattern[x, Blank[Symbol]]] := Simp[(a^p*(c*x)^(m + 1)*Hypergeometric2F1[-p, (m + 1)/n, (m + 1)/n + 1, -((b*x^n)/a)])/(c*(m + 1)), x] /; FreeQ[{a, b, c, m, n, p}, x] && !IGtQ[p, 0] && (ILtQ[p, 0] || GtQ[a, 0])
  365. Int[Times[Power[Times[Optional[Pattern[c, Blank[]]], Pattern[x, Blank[]]], Optional[Pattern[m, Blank[]]]], Power[Plus[Pattern[a, Blank[]], Times[Optional[Pattern[b, Blank[]]], Power[Pattern[x, Blank[]], Pattern[n, Blank[]]]]], Pattern[p, Blank[]]]], Pattern[x, Blank[Symbol]]] := Dist[(a^IntPart[p]*(a + b*x^n)^FracPart[p])/(1 + (b*x^n)/a)^FracPart[p], Int[(c*x)^m*(1 + (b*x^n)/a)^p, x], x] /; FreeQ[{a, b, c, m, n, p}, x] && !IGtQ[p, 0] && !(ILtQ[p, 0] || GtQ[a, 0])
  366. Int[Times[Power[Times[Optional[Pattern[c, Blank[]]], Pattern[x, Blank[]]], Optional[Pattern[m, Blank[]]]], Power[Plus[Pattern[a1, Blank[]], Times[Optional[Pattern[b1, Blank[]]], Power[Pattern[x, Blank[]], Pattern[n, Blank[]]]]], Pattern[p, Blank[]]], Power[Plus[Pattern[a2, Blank[]], Times[Optional[Pattern[b2, Blank[]]], Power[Pattern[x, Blank[]], Pattern[n, Blank[]]]]], Pattern[p, Blank[]]]], Pattern[x, Blank[Symbol]]] := Dist[((a1 + b1*x^n)^FracPart[p]*(a2 + b2*x^n)^FracPart[p])/(a1*a2 + b1*b2*x^(2*n))^FracPart[p], Int[(c*x)^m*(a1*a2 + b1*b2*x^(2*n))^p, x], x] /; FreeQ[{a1, b1, a2, b2, c, m, n, p}, x] && EqQ[a2*b1 + a1*b2, 0] && !IntegerQ[p]
  367. Int[Times[Power[Times[Optional[Pattern[d, Blank[]]], Pattern[x, Blank[]]], Optional[Pattern[m, Blank[]]]], Power[Plus[Pattern[a, Blank[]], Times[Optional[Pattern[b, Blank[]]], Power[Times[Pattern[c, Blank[]], Pattern[x, Blank[]]], Pattern[n, Blank[]]]]], Optional[Pattern[p, Blank[]]]]], Pattern[x, Blank[Symbol]]] := Dist[1/c, Subst[Int[((d*x)/c)^m*(a + b*x^n)^p, x], x, c*x], x] /; FreeQ[{a, b, c, d, m, n, p}, x]
  368. Int[Times[Power[Times[Optional[Pattern[d, Blank[]]], Pattern[x, Blank[]]], Optional[Pattern[m, Blank[]]]], Power[Plus[Pattern[a, Blank[]], Times[Optional[Pattern[b, Blank[]]], Power[Times[Optional[Pattern[c, Blank[]]], Power[Pattern[x, Blank[]], Pattern[q, Blank[]]]], Pattern[n, Blank[]]]]], Optional[Pattern[p, Blank[]]]]], Pattern[x, Blank[Symbol]]] := Dist[(d*x)^(m + 1)/(d*((c*x^q)^(1/q))^(m + 1)), Subst[Int[x^m*(a + b*x^(n*q))^p, x], x, (c*x^q)^(1/q)], x] /; FreeQ[{a, b, c, d, m, n, p, q}, x] && IntegerQ[n*q] && NeQ[x, (c*x^q)^(1/q)]
  369. Int[Times[Power[Times[Optional[Pattern[d, Blank[]]], Pattern[x, Blank[]]], Optional[Pattern[m, Blank[]]]], Power[Plus[Pattern[a, Blank[]], Times[Optional[Pattern[b, Blank[]]], Power[Times[Optional[Pattern[c, Blank[]]], Power[Pattern[x, Blank[]], Pattern[q, Blank[]]]], Pattern[n, Blank[]]]]], Optional[Pattern[p, Blank[]]]]], Pattern[x, Blank[Symbol]]] := With[{k = Denominator[n]}, Subst[Int[(d*x)^m*(a + b*c^n*x^(n*q))^p, x], x^(1/k), (c*x^q)^(1/k)/(c^(1/k)*(x^(1/k))^(q - 1))]] /; FreeQ[{a, b, c, d, m, p, q}, x] && FractionQ[n]
  370. Int[Times[Power[Times[Optional[Pattern[d, Blank[]]], Pattern[x, Blank[]]], Optional[Pattern[m, Blank[]]]], Power[Plus[Pattern[a, Blank[]], Times[Optional[Pattern[b, Blank[]]], Power[Times[Optional[Pattern[c, Blank[]]], Power[Pattern[x, Blank[]], Pattern[q, Blank[]]]], Pattern[n, Blank[]]]]], Optional[Pattern[p, Blank[]]]]], Pattern[x, Blank[Symbol]]] := Subst[Int[(d*x)^m*(a + b*c^n*x^(n*q))^p, x], x^(n*q), (c*x^q)^n/c^n] /; FreeQ[{a, b, c, d, m, n, p, q}, x] && !RationalQ[n]
  371. Int[Times[Power[Plus[Pattern[a, Blank[]], Times[Optional[Pattern[b, Blank[]]], Power[Pattern[v, Blank[]], Pattern[n, Blank[]]]]], Optional[Pattern[p, Blank[]]]], Power[Pattern[x, Blank[]], Optional[Pattern[m, Blank[]]]]], Pattern[x, Blank[Symbol]]] := With[{c = Coefficient[v, x, 0], d = Coefficient[v, x, 1]}, Dist[1/d^(m + 1), Subst[Int[SimplifyIntegrand[(x - c)^m*(a + b*x^n)^p, x], x], x, v], x] /; NeQ[c, 0]] /; FreeQ[{a, b, n, p}, x] && LinearQ[v, x] && IntegerQ[m]
  372. Int[Times[Power[Pattern[u, Blank[]], Optional[Pattern[m, Blank[]]]], Power[Plus[Pattern[a, Blank[]], Times[Optional[Pattern[b, Blank[]]], Power[Pattern[v, Blank[]], Pattern[n, Blank[]]]]], Optional[Pattern[p, Blank[]]]]], Pattern[x, Blank[Symbol]]] := Dist[u^m/(Coefficient[v, x, 1]*v^m), Subst[Int[x^m*(a + b*x^n)^p, x], x, v], x] /; FreeQ[{a, b, m, n, p}, x] && LinearPairQ[u, v, x]
  373. Int[Times[Power[Plus[Pattern[a, Blank[]], Times[Optional[Pattern[b, Blank[]]], Power[Pattern[x, Blank[]], Pattern[n, Blank[]]]]], Optional[Pattern[p, Blank[]]]], Power[Plus[Pattern[c, Blank[]], Times[Optional[Pattern[d, Blank[]]], Power[Pattern[x, Blank[]], Pattern[n, Blank[]]]]], Optional[Pattern[q, Blank[]]]]], Pattern[x, Blank[Symbol]]] := Int[ExpandIntegrand[(a + b*x^n)^p*(c + d*x^n)^q, x], x] /; FreeQ[{a, b, c, d, n}, x] && NeQ[b*c - a*d, 0] && IGtQ[p, 0] && IGtQ[q, 0]
  374. Int[Times[Power[Plus[Pattern[a, Blank[]], Times[Optional[Pattern[b, Blank[]]], Power[Pattern[x, Blank[]], Pattern[n, Blank[]]]]], Optional[Pattern[p, Blank[]]]], Power[Plus[Pattern[c, Blank[]], Times[Optional[Pattern[d, Blank[]]], Power[Pattern[x, Blank[]], Pattern[n, Blank[]]]]], Optional[Pattern[q, Blank[]]]]], Pattern[x, Blank[Symbol]]] := Int[x^(n*(p + q))*(b + a/x^n)^p*(d + c/x^n)^q, x] /; FreeQ[{a, b, c, d, n}, x] && NeQ[b*c - a*d, 0] && IntegersQ[p, q] && NegQ[n]
  375. Int[Times[Power[Plus[Pattern[a, Blank[]], Times[Optional[Pattern[b, Blank[]]], Power[Pattern[x, Blank[]], Pattern[n, Blank[]]]]], Optional[Pattern[p, Blank[]]]], Power[Plus[Pattern[c, Blank[]], Times[Optional[Pattern[d, Blank[]]], Power[Pattern[x, Blank[]], Pattern[n, Blank[]]]]], Optional[Pattern[q, Blank[]]]]], Pattern[x, Blank[Symbol]]] := -Subst[Int[((a + b/x^n)^p*(c + d/x^n)^q)/x^2, x], x, 1/x] /; FreeQ[{a, b, c, d, p, q}, x] && NeQ[b*c - a*d, 0] && ILtQ[n, 0]
  376. Int[Times[Power[Plus[Pattern[a, Blank[]], Times[Optional[Pattern[b, Blank[]]], Power[Pattern[x, Blank[]], Pattern[n, Blank[]]]]], Optional[Pattern[p, Blank[]]]], Power[Plus[Pattern[c, Blank[]], Times[Optional[Pattern[d, Blank[]]], Power[Pattern[x, Blank[]], Pattern[n, Blank[]]]]], Optional[Pattern[q, Blank[]]]]], Pattern[x, Blank[Symbol]]] := With[{g = Denominator[n]}, Dist[g, Subst[Int[x^(g - 1)*(a + b*x^(g*n))^p*(c + d*x^(g*n))^q, x], x, x^(1/g)], x]] /; FreeQ[{a, b, c, d, p, q}, x] && NeQ[b*c - a*d, 0] && FractionQ[n]
  377. Int[Times[Power[Plus[Pattern[a, Blank[]], Times[Optional[Pattern[b, Blank[]]], Power[Pattern[x, Blank[]], Pattern[n, Blank[]]]]], Pattern[p, Blank[]]], Power[Plus[Pattern[c, Blank[]], Times[Optional[Pattern[d, Blank[]]], Power[Pattern[x, Blank[]], Pattern[n, Blank[]]]]], -1]], Pattern[x, Blank[Symbol]]] := Subst[Int[1/(c - (b*c - a*d)*x^n), x], x, x/(a + b*x^n)^(1/n)] /; FreeQ[{a, b, c, d}, x] && NeQ[b*c - a*d, 0] && EqQ[n*p + 1, 0] && IntegerQ[n]
  378. Int[Times[Power[Plus[Pattern[a, Blank[]], Times[Optional[Pattern[b, Blank[]]], Power[Pattern[x, Blank[]], Pattern[n, Blank[]]]]], Pattern[p, Blank[]]], Power[Plus[Pattern[c, Blank[]], Times[Optional[Pattern[d, Blank[]]], Power[Pattern[x, Blank[]], Pattern[n, Blank[]]]]], Optional[Pattern[q, Blank[]]]]], Pattern[x, Blank[Symbol]]] := -Simp[(x*(a + b*x^n)^(p + 1)*(c + d*x^n)^q)/(a*n*(p + 1)), x] - Dist[(c*q)/(a*(p + 1)), Int[(a + b*x^n)^(p + 1)*(c + d*x^n)^(q - 1), x], x] /; FreeQ[{a, b, c, d, n, p}, x] && NeQ[b*c - a*d, 0] && EqQ[n*(p + q + 1) + 1, 0] && GtQ[q, 0] && NeQ[p, -1]
  379. Int[Times[Power[Plus[Pattern[a, Blank[]], Times[Optional[Pattern[b, Blank[]]], Power[Pattern[x, Blank[]], Pattern[n, Blank[]]]]], Pattern[p, Blank[]]], Power[Plus[Pattern[c, Blank[]], Times[Optional[Pattern[d, Blank[]]], Power[Pattern[x, Blank[]], Pattern[n, Blank[]]]]], Pattern[q, Blank[]]]], Pattern[x, Blank[Symbol]]] := Simp[(a^p*x*Hypergeometric2F1[1/n, -p, 1 + 1/n, -(((b*c - a*d)*x^n)/(a*(c + d*x^n)))])/(c^(p + 1)*(c + d*x^n)^(1/n)), x] /; FreeQ[{a, b, c, d, n, q}, x] && NeQ[b*c - a*d, 0] && EqQ[n*(p + q + 1) + 1, 0] && ILtQ[p, 0]
  380. Int[Times[Power[Plus[Pattern[a, Blank[]], Times[Optional[Pattern[b, Blank[]]], Power[Pattern[x, Blank[]], Pattern[n, Blank[]]]]], Pattern[p, Blank[]]], Power[Plus[Pattern[c, Blank[]], Times[Optional[Pattern[d, Blank[]]], Power[Pattern[x, Blank[]], Pattern[n, Blank[]]]]], Pattern[q, Blank[]]]], Pattern[x, Blank[Symbol]]] := Simp[(x*(a + b*x^n)^p*Hypergeometric2F1[1/n, -p, 1 + 1/n, -(((b*c - a*d)*x^n)/(a*(c + d*x^n)))])/(c*((c*(a + b*x^n))/(a*(c + d*x^n)))^p*(c + d*x^n)^(1/n + p)), x] /; FreeQ[{a, b, c, d, n, p, q}, x] && NeQ[b*c - a*d, 0] && EqQ[n*(p + q + 1) + 1, 0]
  381. Int[Times[Power[Plus[Pattern[a, Blank[]], Times[Optional[Pattern[b, Blank[]]], Power[Pattern[x, Blank[]], Pattern[n, Blank[]]]]], Pattern[p, Blank[]]], Power[Plus[Pattern[c, Blank[]], Times[Optional[Pattern[d, Blank[]]], Power[Pattern[x, Blank[]], Pattern[n, Blank[]]]]], Pattern[q, Blank[]]]], Pattern[x, Blank[Symbol]]] := Simp[(x*(a + b*x^n)^(p + 1)*(c + d*x^n)^(q + 1))/(a*c), x] /; FreeQ[{a, b, c, d, n, p, q}, x] && NeQ[b*c - a*d, 0] && EqQ[n*(p + q + 2) + 1, 0] && EqQ[a*d*(p + 1) + b*c*(q + 1), 0]
  382. Int[Times[Power[Plus[Pattern[a, Blank[]], Times[Optional[Pattern[b, Blank[]]], Power[Pattern[x, Blank[]], Pattern[n, Blank[]]]]], Pattern[p, Blank[]]], Power[Plus[Pattern[c, Blank[]], Times[Optional[Pattern[d, Blank[]]], Power[Pattern[x, Blank[]], Pattern[n, Blank[]]]]], Pattern[q, Blank[]]]], Pattern[x, Blank[Symbol]]] := -Simp[(b*x*(a + b*x^n)^(p + 1)*(c + d*x^n)^(q + 1))/(a*n*(p + 1)*(b*c - a*d)), x] + Dist[(b*c + n*(p + 1)*(b*c - a*d))/(a*n*(p + 1)*(b*c - a*d)), Int[(a + b*x^n)^(p + 1)*(c + d*x^n)^q, x], x] /; FreeQ[{a, b, c, d, n, q}, x] && NeQ[b*c - a*d, 0] && EqQ[n*(p + q + 2) + 1, 0] && (LtQ[p, -1] || !LtQ[q, -1]) && NeQ[p, -1]
  383. Int[Times[Power[Plus[Pattern[a, Blank[]], Times[Optional[Pattern[b, Blank[]]], Power[Pattern[x, Blank[]], Pattern[n, Blank[]]]]], Optional[Pattern[p, Blank[]]]], Plus[Pattern[c, Blank[]], Times[Optional[Pattern[d, Blank[]]], Power[Pattern[x, Blank[]], Pattern[n, Blank[]]]]]], Pattern[x, Blank[Symbol]]] := Simp[(c*x*(a + b*x^n)^(p + 1))/a, x] /; FreeQ[{a, b, c, d, n, p}, x] && NeQ[b*c - a*d, 0] && EqQ[a*d - b*c*(n*(p + 1) + 1), 0]
  384. Int[Times[Power[Plus[Pattern[a1, Blank[]], Times[Optional[Pattern[b1, Blank[]]], Power[Pattern[x, Blank[]], Optional[Pattern[non2, Blank[]]]]]], Optional[Pattern[p, Blank[]]]], Power[Plus[Pattern[a2, Blank[]], Times[Optional[Pattern[b2, Blank[]]], Power[Pattern[x, Blank[]], Optional[Pattern[non2, Blank[]]]]]], Optional[Pattern[p, Blank[]]]], Plus[Pattern[c, Blank[]], Times[Optional[Pattern[d, Blank[]]], Power[Pattern[x, Blank[]], Pattern[n, Blank[]]]]]], Pattern[x, Blank[Symbol]]] := Simp[(c*x*(a1 + b1*x^(n/2))^(p + 1)*(a2 + b2*x^(n/2))^(p + 1))/(a1*a2), x] /; FreeQ[{a1, b1, a2, b2, c, d, n, p}, x] && EqQ[non2, n/2] && EqQ[a2*b1 + a1*b2, 0] && EqQ[a1*a2*d - b1*b2*c*(n*(p + 1) + 1), 0]
  385. Int[Times[Power[Plus[Pattern[a, Blank[]], Times[Optional[Pattern[b, Blank[]]], Power[Pattern[x, Blank[]], Pattern[n, Blank[]]]]], Pattern[p, Blank[]]], Plus[Pattern[c, Blank[]], Times[Optional[Pattern[d, Blank[]]], Power[Pattern[x, Blank[]], Pattern[n, Blank[]]]]]], Pattern[x, Blank[Symbol]]] := -Simp[((b*c - a*d)*x*(a + b*x^n)^(p + 1))/(a*b*n*(p + 1)), x] - Dist[(a*d - b*c*(n*(p + 1) + 1))/(a*b*n*(p + 1)), Int[(a + b*x^n)^(p + 1), x], x] /; FreeQ[{a, b, c, d, n, p}, x] && NeQ[b*c - a*d, 0] && (LtQ[p, -1] || ILtQ[1/n + p, 0])
  386. Int[Times[Power[Plus[Pattern[a1, Blank[]], Times[Optional[Pattern[b1, Blank[]]], Power[Pattern[x, Blank[]], Optional[Pattern[non2, Blank[]]]]]], Optional[Pattern[p, Blank[]]]], Power[Plus[Pattern[a2, Blank[]], Times[Optional[Pattern[b2, Blank[]]], Power[Pattern[x, Blank[]], Optional[Pattern[non2, Blank[]]]]]], Optional[Pattern[p, Blank[]]]], Plus[Pattern[c, Blank[]], Times[Optional[Pattern[d, Blank[]]], Power[Pattern[x, Blank[]], Pattern[n, Blank[]]]]]], Pattern[x, Blank[Symbol]]] := -Simp[((b1*b2*c - a1*a2*d)*x*(a1 + b1*x^(n/2))^(p + 1)*(a2 + b2*x^(n/2))^(p + 1))/(a1*a2*b1*b2*n*(p + 1)), x] - Dist[(a1*a2*d - b1*b2*c*(n*(p + 1) + 1))/(a1*a2*b1*b2*n*(p + 1)), Int[(a1 + b1*x^(n/2))^(p + 1)*(a2 + b2*x^(n/2))^(p + 1), x], x] /; FreeQ[{a1, b1, a2, b2, c, d, n}, x] && EqQ[non2, n/2] && EqQ[a2*b1 + a1*b2, 0] && (LtQ[p, -1] || ILtQ[1/n + p, 0])
  387. Int[Times[Power[Plus[Pattern[a, Blank[]], Times[Optional[Pattern[b, Blank[]]], Power[Pattern[x, Blank[]], Pattern[n, Blank[]]]]], -1], Plus[Pattern[c, Blank[]], Times[Optional[Pattern[d, Blank[]]], Power[Pattern[x, Blank[]], Pattern[n, Blank[]]]]]], Pattern[x, Blank[Symbol]]] := Simp[(c*x)/a, x] - Dist[(b*c - a*d)/a, Int[1/(b + a/x^n), x], x] /; FreeQ[{a, b, c, d, n}, x] && NeQ[b*c - a*d, 0] && LtQ[n, 0]
  388. Int[Times[Power[Plus[Pattern[a, Blank[]], Times[Optional[Pattern[b, Blank[]]], Power[Pattern[x, Blank[]], Pattern[n, Blank[]]]]], Pattern[p, Blank[]]], Plus[Pattern[c, Blank[]], Times[Optional[Pattern[d, Blank[]]], Power[Pattern[x, Blank[]], Pattern[n, Blank[]]]]]], Pattern[x, Blank[Symbol]]] := Simp[(d*x*(a + b*x^n)^(p + 1))/(b*(n*(p + 1) + 1)), x] - Dist[(a*d - b*c*(n*(p + 1) + 1))/(b*(n*(p + 1) + 1)), Int[(a + b*x^n)^p, x], x] /; FreeQ[{a, b, c, d, n}, x] && NeQ[b*c - a*d, 0] && NeQ[n*(p + 1) + 1, 0]
  389. Int[Times[Power[Plus[Pattern[a1, Blank[]], Times[Optional[Pattern[b1, Blank[]]], Power[Pattern[x, Blank[]], Optional[Pattern[non2, Blank[]]]]]], Optional[Pattern[p, Blank[]]]], Power[Plus[Pattern[a2, Blank[]], Times[Optional[Pattern[b2, Blank[]]], Power[Pattern[x, Blank[]], Optional[Pattern[non2, Blank[]]]]]], Optional[Pattern[p, Blank[]]]], Plus[Pattern[c, Blank[]], Times[Optional[Pattern[d, Blank[]]], Power[Pattern[x, Blank[]], Pattern[n, Blank[]]]]]], Pattern[x, Blank[Symbol]]] := Simp[(d*x*(a1 + b1*x^(n/2))^(p + 1)*(a2 + b2*x^(n/2))^(p + 1))/(b1*b2*(n*(p + 1) + 1)), x] - Dist[(a1*a2*d - b1*b2*c*(n*(p + 1) + 1))/(b1*b2*(n*(p + 1) + 1)), Int[(a1 + b1*x^(n/2))^p*(a2 + b2*x^(n/2))^p, x], x] /; FreeQ[{a1, b1, a2, b2, c, d, n, p}, x] && EqQ[non2, n/2] && EqQ[a2*b1 + a1*b2, 0] && NeQ[n*(p + 1) + 1, 0]
  390. Int[Times[Power[Plus[Pattern[a, Blank[]], Times[Optional[Pattern[b, Blank[]]], Power[Pattern[x, Blank[]], Pattern[n, Blank[]]]]], Pattern[p, Blank[]]], Power[Plus[Pattern[c, Blank[]], Times[Optional[Pattern[d, Blank[]]], Power[Pattern[x, Blank[]], Pattern[n, Blank[]]]]], Pattern[q, Blank[]]]], Pattern[x, Blank[Symbol]]] := Int[PolynomialDivide[(a + b*x^n)^p, (c + d*x^n)^(-q), x], x] /; FreeQ[{a, b, c, d}, x] && NeQ[b*c - a*d, 0] && IGtQ[n, 0] && IGtQ[p, 0] && ILtQ[q, 0] && GeQ[p, -q]
  391. Int[Times[Power[Plus[Pattern[a, Blank[]], Times[Optional[Pattern[b, Blank[]]], Power[Pattern[x, Blank[]], Pattern[n, Blank[]]]]], -1], Power[Plus[Pattern[c, Blank[]], Times[Optional[Pattern[d, Blank[]]], Power[Pattern[x, Blank[]], Pattern[n, Blank[]]]]], -1]], Pattern[x, Blank[Symbol]]] := Dist[b/(b*c - a*d), Int[1/(a + b*x^n), x], x] - Dist[d/(b*c - a*d), Int[1/(c + d*x^n), x], x] /; FreeQ[{a, b, c, d, n}, x] && NeQ[b*c - a*d, 0]
  392. Int[Times[Power[Plus[Pattern[a, Blank[]], Times[Optional[Pattern[b, Blank[]]], Power[Pattern[x, Blank[]], 2]]], Rational[-1, 3]], Power[Plus[Pattern[c, Blank[]], Times[Optional[Pattern[d, Blank[]]], Power[Pattern[x, Blank[]], 2]]], -1]], Pattern[x, Blank[Symbol]]] := With[{q = Rt[b/a, 2]}, Simp[(q*ArcTanh[Sqrt[3]/(q*x)])/(2*2^(2/3)*Sqrt[3]*a^(1/3)*d), x] + (-Simp[(q*ArcTan[(a^(1/3)*q*x)/(a^(1/3) + 2^(1/3)*(a + b*x^2)^(1/3))])/(2*2^(2/3)*a^(1/3)*d), x] + Simp[(q*ArcTan[q*x])/(6*2^(2/3)*a^(1/3)*d), x] + Simp[(q*ArcTanh[(Sqrt[3]*(a^(1/3) - 2^(1/3)*(a + b*x^2)^(1/3)))/(a^(1/3)*q*x)])/(2*2^(2/3)*Sqrt[3]*a^(1/3)*d), x])] /; FreeQ[{a, b, c, d}, x] && NeQ[b*c - a*d, 0] && EqQ[b*c + 3*a*d, 0] && PosQ[b/a]
  393. Int[Times[Power[Plus[Pattern[a, Blank[]], Times[Optional[Pattern[b, Blank[]]], Power[Pattern[x, Blank[]], 2]]], Rational[-1, 3]], Power[Plus[Pattern[c, Blank[]], Times[Optional[Pattern[d, Blank[]]], Power[Pattern[x, Blank[]], 2]]], -1]], Pattern[x, Blank[Symbol]]] := With[{q = Rt[-(b/a), 2]}, Simp[(q*ArcTan[Sqrt[3]/(q*x)])/(2*2^(2/3)*Sqrt[3]*a^(1/3)*d), x] + (Simp[(q*ArcTanh[(a^(1/3)*q*x)/(a^(1/3) + 2^(1/3)*(a + b*x^2)^(1/3))])/(2*2^(2/3)*a^(1/3)*d), x] - Simp[(q*ArcTanh[q*x])/(6*2^(2/3)*a^(1/3)*d), x] + Simp[(q*ArcTan[(Sqrt[3]*(a^(1/3) - 2^(1/3)*(a + b*x^2)^(1/3)))/(a^(1/3)*q*x)])/(2*2^(2/3)*Sqrt[3]*a^(1/3)*d), x])] /; FreeQ[{a, b, c, d}, x] && NeQ[b*c - a*d, 0] && EqQ[b*c + 3*a*d, 0] && NegQ[b/a]
  394. Int[Times[Power[Plus[Pattern[a, Blank[]], Times[Optional[Pattern[b, Blank[]]], Power[Pattern[x, Blank[]], 2]]], Rational[-1, 3]], Power[Plus[Pattern[c, Blank[]], Times[Optional[Pattern[d, Blank[]]], Power[Pattern[x, Blank[]], 2]]], -1]], Pattern[x, Blank[Symbol]]] := With[{q = Rt[b/a, 2]}, Simp[(q*ArcTan[(q*x)/3])/(12*Rt[a, 3]*d), x] + (Simp[(q*ArcTan[(Rt[a, 3] - (a + b*x^2)^(1/3))^2/(3*Rt[a, 3]^2*q*x)])/(12*Rt[a, 3]*d), x] - Simp[(q*ArcTanh[(Sqrt[3]*(Rt[a, 3] - (a + b*x^2)^(1/3)))/(Rt[a, 3]*q*x)])/(4*Sqrt[3]*Rt[a, 3]*d), x])] /; FreeQ[{a, b, c, d}, x] && NeQ[b*c - a*d, 0] && EqQ[b*c - 9*a*d, 0] && PosQ[b/a]
  395. Int[Times[Power[Plus[Pattern[a, Blank[]], Times[Optional[Pattern[b, Blank[]]], Power[Pattern[x, Blank[]], 2]]], Rational[-1, 3]], Power[Plus[Pattern[c, Blank[]], Times[Optional[Pattern[d, Blank[]]], Power[Pattern[x, Blank[]], 2]]], -1]], Pattern[x, Blank[Symbol]]] := With[{q = Rt[-(b/a), 2]}, -Simp[(q*ArcTanh[(q*x)/3])/(12*Rt[a, 3]*d), x] + (Simp[(q*ArcTanh[(Rt[a, 3] - (a + b*x^2)^(1/3))^2/(3*Rt[a, 3]^2*q*x)])/(12*Rt[a, 3]*d), x] - Simp[(q*ArcTan[(Sqrt[3]*(Rt[a, 3] - (a + b*x^2)^(1/3)))/(Rt[a, 3]*q*x)])/(4*Sqrt[3]*Rt[a, 3]*d), x])] /; FreeQ[{a, b, c, d}, x] && NeQ[b*c - a*d, 0] && EqQ[b*c - 9*a*d, 0] && NegQ[b/a]
  396. Int[Times[Power[Plus[Pattern[a, Blank[]], Times[Optional[Pattern[b, Blank[]]], Power[Pattern[x, Blank[]], 2]]], Rational[2, 3]], Power[Plus[Pattern[c, Blank[]], Times[Optional[Pattern[d, Blank[]]], Power[Pattern[x, Blank[]], 2]]], -1]], Pattern[x, Blank[Symbol]]] := Dist[b/d, Int[1/(a + b*x^2)^(1/3), x], x] - Dist[(b*c - a*d)/d, Int[1/((a + b*x^2)^(1/3)*(c + d*x^2)), x], x] /; FreeQ[{a, b, c, d}, x] && NeQ[b*c - a*d, 0] && EqQ[b*c + 3*a*d, 0]
  397. Int[Times[Power[Plus[Pattern[a, Blank[]], Times[Optional[Pattern[b, Blank[]]], Power[Pattern[x, Blank[]], 2]]], Rational[-1, 4]], Power[Plus[Pattern[c, Blank[]], Times[Optional[Pattern[d, Blank[]]], Power[Pattern[x, Blank[]], 2]]], -1]], Pattern[x, Blank[Symbol]]] := With[{q = Rt[b^2/a, 4]}, -Simp[(b*ArcTan[(b + q^2*Sqrt[a + b*x^2])/(q^3*x*(a + b*x^2)^(1/4))])/(2*a*d*q), x] - Simp[(b*ArcTanh[(b - q^2*Sqrt[a + b*x^2])/(q^3*x*(a + b*x^2)^(1/4))])/(2*a*d*q), x]] /; FreeQ[{a, b, c, d}, x] && EqQ[b*c - 2*a*d, 0] && PosQ[b^2/a]
  398. Int[Times[Power[Plus[Pattern[a, Blank[]], Times[Optional[Pattern[b, Blank[]]], Power[Pattern[x, Blank[]], 2]]], Rational[-1, 4]], Power[Plus[Pattern[c, Blank[]], Times[Optional[Pattern[d, Blank[]]], Power[Pattern[x, Blank[]], 2]]], -1]], Pattern[x, Blank[Symbol]]] := With[{q = Rt[-(b^2/a), 4]}, Simp[(b*ArcTan[(q*x)/(Sqrt[2]*(a + b*x^2)^(1/4))])/(2*Sqrt[2]*a*d*q), x] + Simp[(b*ArcTanh[(q*x)/(Sqrt[2]*(a + b*x^2)^(1/4))])/(2*Sqrt[2]*a*d*q), x]] /; FreeQ[{a, b, c, d}, x] && EqQ[b*c - 2*a*d, 0] && NegQ[b^2/a]
  399. Int[Times[Power[Plus[Pattern[a, Blank[]], Times[Optional[Pattern[b, Blank[]]], Power[Pattern[x, Blank[]], 2]]], Rational[-1, 4]], Power[Plus[Pattern[c, Blank[]], Times[Optional[Pattern[d, Blank[]]], Power[Pattern[x, Blank[]], 2]]], -1]], Pattern[x, Blank[Symbol]]] := Dist[(2*Sqrt[-((b*x^2)/a)])/x, Subst[Int[x^2/(Sqrt[1 - x^4/a]*(b*c - a*d + d*x^4)), x], x, (a + b*x^2)^(1/4)], x] /; FreeQ[{a, b, c, d}, x] && NeQ[b*c - a*d, 0]
  400. Int[Times[Power[Plus[Pattern[a, Blank[]], Times[Optional[Pattern[b, Blank[]]], Power[Pattern[x, Blank[]], 2]]], Rational[-3, 4]], Power[Plus[Pattern[c, Blank[]], Times[Optional[Pattern[d, Blank[]]], Power[Pattern[x, Blank[]], 2]]], -1]], Pattern[x, Blank[Symbol]]] := Dist[1/c, Int[1/(a + b*x^2)^(3/4), x], x] - Dist[d/c, Int[x^2/((a + b*x^2)^(3/4)*(c + d*x^2)), x], x] /; FreeQ[{a, b, c, d}, x] && EqQ[b*c - 2*a*d, 0]
  401. Int[Times[Power[Plus[Pattern[a, Blank[]], Times[Optional[Pattern[b, Blank[]]], Power[Pattern[x, Blank[]], 2]]], Rational[-3, 4]], Power[Plus[Pattern[c, Blank[]], Times[Optional[Pattern[d, Blank[]]], Power[Pattern[x, Blank[]], 2]]], -1]], Pattern[x, Blank[Symbol]]] := Dist[Sqrt[-((b*x^2)/a)]/(2*x), Subst[Int[1/(Sqrt[-((b*x)/a)]*(a + b*x)^(3/4)*(c + d*x)), x], x, x^2], x] /; FreeQ[{a, b, c, d}, x] && NeQ[b*c - a*d, 0]
  402. Int[Times[Power[Plus[Pattern[a, Blank[]], Times[Optional[Pattern[b, Blank[]]], Power[Pattern[x, Blank[]], 2]]], Optional[Pattern[p, Blank[]]]], Power[Plus[Pattern[c, Blank[]], Times[Optional[Pattern[d, Blank[]]], Power[Pattern[x, Blank[]], 2]]], -1]], Pattern[x, Blank[Symbol]]] := Dist[b/d, Int[(a + b*x^2)^(p - 1), x], x] - Dist[(b*c - a*d)/d, Int[(a + b*x^2)^(p - 1)/(c + d*x^2), x], x] /; FreeQ[{a, b, c, d}, x] && NeQ[b*c - a*d, 0] && GtQ[p, 0] && (EqQ[p, 1/2] || EqQ[Denominator[p], 4])
  403. Int[Times[Power[Plus[Pattern[a, Blank[]], Times[Optional[Pattern[b, Blank[]]], Power[Pattern[x, Blank[]], 2]]], Pattern[p, Blank[]]], Power[Plus[Pattern[c, Blank[]], Times[Optional[Pattern[d, Blank[]]], Power[Pattern[x, Blank[]], 2]]], -1]], Pattern[x, Blank[Symbol]]] := Dist[b/(b*c - a*d), Int[(a + b*x^2)^p, x], x] - Dist[d/(b*c - a*d), Int[(a + b*x^2)^(p + 1)/(c + d*x^2), x], x] /; FreeQ[{a, b, c, d}, x] && NeQ[b*c - a*d, 0] && LtQ[p, -1] && EqQ[Denominator[p], 4] && (EqQ[p, -5/4] || EqQ[p, -7/4])
  404. Int[Times[Power[Plus[Pattern[a, Blank[]], Times[Optional[Pattern[b, Blank[]]], Power[Pattern[x, Blank[]], 4]]], Rational[1, 2]], Power[Plus[Pattern[c, Blank[]], Times[Optional[Pattern[d, Blank[]]], Power[Pattern[x, Blank[]], 4]]], -1]], Pattern[x, Blank[Symbol]]] := Dist[a/c, Subst[Int[1/(1 - 4*a*b*x^4), x], x, x/Sqrt[a + b*x^4]], x] /; FreeQ[{a, b, c, d}, x] && EqQ[b*c + a*d, 0] && PosQ[a*b]
  405. Int[Times[Power[Plus[Pattern[a, Blank[]], Times[Optional[Pattern[b, Blank[]]], Power[Pattern[x, Blank[]], 4]]], Rational[1, 2]], Power[Plus[Pattern[c, Blank[]], Times[Optional[Pattern[d, Blank[]]], Power[Pattern[x, Blank[]], 4]]], -1]], Pattern[x, Blank[Symbol]]] := With[{q = Rt[-(a*b), 4]}, Simp[(a*ArcTan[(q*x*(a + q^2*x^2))/(a*Sqrt[a + b*x^4])])/(2*c*q), x] + Simp[(a*ArcTanh[(q*x*(a - q^2*x^2))/(a*Sqrt[a + b*x^4])])/(2*c*q), x]] /; FreeQ[{a, b, c, d}, x] && EqQ[b*c + a*d, 0] && NegQ[a*b]
  406. Int[Times[Power[Plus[Pattern[a, Blank[]], Times[Optional[Pattern[b, Blank[]]], Power[Pattern[x, Blank[]], 4]]], Rational[1, 2]], Power[Plus[Pattern[c, Blank[]], Times[Optional[Pattern[d, Blank[]]], Power[Pattern[x, Blank[]], 4]]], -1]], Pattern[x, Blank[Symbol]]] := Dist[b/d, Int[1/Sqrt[a + b*x^4], x], x] - Dist[(b*c - a*d)/d, Int[1/(Sqrt[a + b*x^4]*(c + d*x^4)), x], x] /; FreeQ[{a, b, c, d}, x] && NeQ[b*c - a*d, 0]
  407. Int[Times[Power[Plus[Pattern[a, Blank[]], Times[Optional[Pattern[b, Blank[]]], Power[Pattern[x, Blank[]], 4]]], Rational[1, 4]], Power[Plus[Pattern[c, Blank[]], Times[Optional[Pattern[d, Blank[]]], Power[Pattern[x, Blank[]], 4]]], -1]], Pattern[x, Blank[Symbol]]] := Dist[Sqrt[a + b*x^4]*Sqrt[a/(a + b*x^4)], Subst[Int[1/(Sqrt[1 - b*x^4]*(c - (b*c - a*d)*x^4)), x], x, x/(a + b*x^4)^(1/4)], x] /; FreeQ[{a, b, c, d}, x] && NeQ[b*c - a*d, 0]
  408. Int[Times[Power[Plus[Pattern[a, Blank[]], Times[Optional[Pattern[b, Blank[]]], Power[Pattern[x, Blank[]], 4]]], Pattern[p, Blank[]]], Power[Plus[Pattern[c, Blank[]], Times[Optional[Pattern[d, Blank[]]], Power[Pattern[x, Blank[]], 4]]], -1]], Pattern[x, Blank[Symbol]]] := Dist[b/d, Int[(a + b*x^4)^(p - 1), x], x] - Dist[(b*c - a*d)/d, Int[(a + b*x^4)^(p - 1)/(c + d*x^4), x], x] /; FreeQ[{a, b, c, d}, x] && NeQ[b*c - a*d, 0] && (EqQ[p, 3/4] || EqQ[p, 5/4])
  409. Int[Times[Power[Plus[Pattern[a, Blank[]], Times[Optional[Pattern[b, Blank[]]], Power[Pattern[x, Blank[]], 4]]], Rational[-1, 2]], Power[Plus[Pattern[c, Blank[]], Times[Optional[Pattern[d, Blank[]]], Power[Pattern[x, Blank[]], 4]]], -1]], Pattern[x, Blank[Symbol]]] := Dist[1/(2*c), Int[1/(Sqrt[a + b*x^4]*(1 - Rt[-(d/c), 2]*x^2)), x], x] + Dist[1/(2*c), Int[1/(Sqrt[a + b*x^4]*(1 + Rt[-(d/c), 2]*x^2)), x], x] /; FreeQ[{a, b, c, d}, x] && NeQ[b*c - a*d, 0]
  410. Int[Times[Power[Plus[Pattern[a, Blank[]], Times[Optional[Pattern[b, Blank[]]], Power[Pattern[x, Blank[]], 4]]], Rational[-3, 4]], Power[Plus[Pattern[c, Blank[]], Times[Optional[Pattern[d, Blank[]]], Power[Pattern[x, Blank[]], 4]]], -1]], Pattern[x, Blank[Symbol]]] := Dist[b/(b*c - a*d), Int[1/(a + b*x^4)^(3/4), x], x] - Dist[d/(b*c - a*d), Int[(a + b*x^4)^(1/4)/(c + d*x^4), x], x] /; FreeQ[{a, b, c, d}, x] && NeQ[b*c - a*d, 0]
  411. Int[Times[Power[Plus[Pattern[a, Blank[]], Times[Optional[Pattern[b, Blank[]]], Power[Pattern[x, Blank[]], 2]]], Rational[1, 2]], Power[Plus[Pattern[c, Blank[]], Times[Optional[Pattern[d, Blank[]]], Power[Pattern[x, Blank[]], 2]]], Rational[-3, 2]]], Pattern[x, Blank[Symbol]]] := Simp[(Sqrt[a + b*x^2]*EllipticE[ArcTan[Rt[d/c, 2]*x], 1 - (b*c)/(a*d)])/(c*Rt[d/c, 2]*Sqrt[c + d*x^2]*Sqrt[(c*(a + b*x^2))/(a*(c + d*x^2))]), x] /; FreeQ[{a, b, c, d}, x] && PosQ[b/a] && PosQ[d/c]
  412. Int[Times[Power[Plus[Pattern[a, Blank[]], Times[Optional[Pattern[b, Blank[]]], Power[Pattern[x, Blank[]], Pattern[n, Blank[]]]]], Pattern[p, Blank[]]], Power[Plus[Pattern[c, Blank[]], Times[Optional[Pattern[d, Blank[]]], Power[Pattern[x, Blank[]], Pattern[n, Blank[]]]]], Pattern[q, Blank[]]]], Pattern[x, Blank[Symbol]]] := -Simp[(x*(a + b*x^n)^(p + 1)*(c + d*x^n)^q)/(a*n*(p + 1)), x] + Dist[1/(a*n*(p + 1)), Int[(a + b*x^n)^(p + 1)*(c + d*x^n)^(q - 1)*Simp[c*(n*(p + 1) + 1) + d*(n*(p + q + 1) + 1)*x^n, x], x], x] /; FreeQ[{a, b, c, d, n}, x] && NeQ[b*c - a*d, 0] && LtQ[p, -1] && LtQ[0, q, 1] && IntBinomialQ[a, b, c, d, n, p, q, x]
  413. Int[Times[Power[Plus[Pattern[a, Blank[]], Times[Optional[Pattern[b, Blank[]]], Power[Pattern[x, Blank[]], Pattern[n, Blank[]]]]], Pattern[p, Blank[]]], Power[Plus[Pattern[c, Blank[]], Times[Optional[Pattern[d, Blank[]]], Power[Pattern[x, Blank[]], Pattern[n, Blank[]]]]], Pattern[q, Blank[]]]], Pattern[x, Blank[Symbol]]] := Simp[((a*d - c*b)*x*(a + b*x^n)^(p + 1)*(c + d*x^n)^(q - 1))/(a*b*n*(p + 1)), x] - Dist[1/(a*b*n*(p + 1)), Int[(a + b*x^n)^(p + 1)*(c + d*x^n)^(q - 2)*Simp[c*(a*d - c*b*(n*(p + 1) + 1)) + d*(a*d*(n*(q - 1) + 1) - b*c*(n*(p + q) + 1))*x^n, x], x], x] /; FreeQ[{a, b, c, d, n}, x] && NeQ[b*c - a*d, 0] && LtQ[p, -1] && GtQ[q, 1] && IntBinomialQ[a, b, c, d, n, p, q, x]
  414. Int[Times[Power[Plus[Pattern[a, Blank[]], Times[Optional[Pattern[b, Blank[]]], Power[Pattern[x, Blank[]], Pattern[n, Blank[]]]]], Pattern[p, Blank[]]], Power[Plus[Pattern[c, Blank[]], Times[Optional[Pattern[d, Blank[]]], Power[Pattern[x, Blank[]], Pattern[n, Blank[]]]]], Pattern[q, Blank[]]]], Pattern[x, Blank[Symbol]]] := -Simp[(b*x*(a + b*x^n)^(p + 1)*(c + d*x^n)^(q + 1))/(a*n*(p + 1)*(b*c - a*d)), x] + Dist[1/(a*n*(p + 1)*(b*c - a*d)), Int[(a + b*x^n)^(p + 1)*(c + d*x^n)^q*Simp[b*c + n*(p + 1)*(b*c - a*d) + d*b*(n*(p + q + 2) + 1)*x^n, x], x], x] /; FreeQ[{a, b, c, d, n, q}, x] && NeQ[b*c - a*d, 0] && LtQ[p, -1] && !( !IntegerQ[p] && IntegerQ[q] && LtQ[q, -1]) && IntBinomialQ[a, b, c, d, n, p, q, x]
  415. Int[Times[Power[Plus[Pattern[a, Blank[]], Times[Optional[Pattern[b, Blank[]]], Power[Pattern[x, Blank[]], Pattern[n, Blank[]]]]], Pattern[p, Blank[]]], Power[Plus[Pattern[c, Blank[]], Times[Optional[Pattern[d, Blank[]]], Power[Pattern[x, Blank[]], Pattern[n, Blank[]]]]], Pattern[q, Blank[]]]], Pattern[x, Blank[Symbol]]] := Int[ExpandIntegrand[(a + b*x^n)^p*(c + d*x^n)^q, x], x] /; FreeQ[{a, b, c, d}, x] && NeQ[b*c - a*d, 0] && IGtQ[n, 0] && IntegersQ[p, q] && GtQ[p + q, 0]
  416. Int[Times[Power[Plus[Pattern[a, Blank[]], Times[Optional[Pattern[b, Blank[]]], Power[Pattern[x, Blank[]], Pattern[n, Blank[]]]]], Pattern[p, Blank[]]], Power[Plus[Pattern[c, Blank[]], Times[Optional[Pattern[d, Blank[]]], Power[Pattern[x, Blank[]], Pattern[n, Blank[]]]]], Pattern[q, Blank[]]]], Pattern[x, Blank[Symbol]]] := Simp[(d*x*(a + b*x^n)^(p + 1)*(c + d*x^n)^(q - 1))/(b*(n*(p + q) + 1)), x] + Dist[1/(b*(n*(p + q) + 1)), Int[(a + b*x^n)^p*(c + d*x^n)^(q - 2)*Simp[c*(b*c*(n*(p + q) + 1) - a*d) + d*(b*c*(n*(p + 2*q - 1) + 1) - a*d*(n*(q - 1) + 1))*x^n, x], x], x] /; FreeQ[{a, b, c, d, n, p}, x] && NeQ[b*c - a*d, 0] && GtQ[q, 1] && NeQ[n*(p + q) + 1, 0] && !IGtQ[p, 1] && IntBinomialQ[a, b, c, d, n, p, q, x]
  417. Int[Times[Power[Plus[Pattern[a, Blank[]], Times[Optional[Pattern[b, Blank[]]], Power[Pattern[x, Blank[]], Pattern[n, Blank[]]]]], Pattern[p, Blank[]]], Power[Plus[Pattern[c, Blank[]], Times[Optional[Pattern[d, Blank[]]], Power[Pattern[x, Blank[]], Pattern[n, Blank[]]]]], Pattern[q, Blank[]]]], Pattern[x, Blank[Symbol]]] := Simp[(x*(a + b*x^n)^p*(c + d*x^n)^q)/(n*(p + q) + 1), x] + Dist[n/(n*(p + q) + 1), Int[(a + b*x^n)^(p - 1)*(c + d*x^n)^(q - 1)*Simp[a*c*(p + q) + (q*(b*c - a*d) + a*d*(p + q))*x^n, x], x], x] /; FreeQ[{a, b, c, d, n}, x] && NeQ[b*c - a*d, 0] && GtQ[q, 0] && GtQ[p, 0] && IntBinomialQ[a, b, c, d, n, p, q, x]
  418. Int[Times[Power[Plus[Pattern[a, Blank[]], Times[Optional[Pattern[b, Blank[]]], Power[Pattern[x, Blank[]], 2]]], Rational[-1, 2]], Power[Plus[Pattern[c, Blank[]], Times[Optional[Pattern[d, Blank[]]], Power[Pattern[x, Blank[]], 2]]], Rational[-1, 2]]], Pattern[x, Blank[Symbol]]] := Simp[(Sqrt[a + b*x^2]*EllipticF[ArcTan[Rt[d/c, 2]*x], 1 - (b*c)/(a*d)])/(a*Rt[d/c, 2]*Sqrt[c + d*x^2]*Sqrt[(c*(a + b*x^2))/(a*(c + d*x^2))]), x] /; FreeQ[{a, b, c, d}, x] && PosQ[d/c] && PosQ[b/a] && !SimplerSqrtQ[b/a, d/c]
  419. Int[Times[Power[Plus[Pattern[a, Blank[]], Times[Optional[Pattern[b, Blank[]]], Power[Pattern[x, Blank[]], 2]]], Rational[-1, 2]], Power[Plus[Pattern[c, Blank[]], Times[Optional[Pattern[d, Blank[]]], Power[Pattern[x, Blank[]], 2]]], Rational[-1, 2]]], Pattern[x, Blank[Symbol]]] := Simp[(1*EllipticF[ArcSin[Rt[-(d/c), 2]*x], (b*c)/(a*d)])/(Sqrt[a]*Sqrt[c]*Rt[-(d/c), 2]), x] /; FreeQ[{a, b, c, d}, x] && NegQ[d/c] && GtQ[c, 0] && GtQ[a, 0] && !(NegQ[b/a] && SimplerSqrtQ[-(b/a), -(d/c)])
  420. Int[Times[Power[Plus[Pattern[a, Blank[]], Times[Optional[Pattern[b, Blank[]]], Power[Pattern[x, Blank[]], 2]]], Rational[-1, 2]], Power[Plus[Pattern[c, Blank[]], Times[Optional[Pattern[d, Blank[]]], Power[Pattern[x, Blank[]], 2]]], Rational[-1, 2]]], Pattern[x, Blank[Symbol]]] := -Simp[EllipticF[ArcCos[Rt[-(d/c), 2]*x], (b*c)/(b*c - a*d)]/(Sqrt[c]*Rt[-(d/c), 2]*Sqrt[a - (b*c)/d]), x] /; FreeQ[{a, b, c, d}, x] && NegQ[d/c] && GtQ[c, 0] && GtQ[a - (b*c)/d, 0]
  421. Int[Times[Power[Plus[Pattern[a, Blank[]], Times[Optional[Pattern[b, Blank[]]], Power[Pattern[x, Blank[]], 2]]], Rational[-1, 2]], Power[Plus[Pattern[c, Blank[]], Times[Optional[Pattern[d, Blank[]]], Power[Pattern[x, Blank[]], 2]]], Rational[-1, 2]]], Pattern[x, Blank[Symbol]]] := Dist[Sqrt[1 + (d*x^2)/c]/Sqrt[c + d*x^2], Int[1/(Sqrt[a + b*x^2]*Sqrt[1 + (d*x^2)/c]), x], x] /; FreeQ[{a, b, c, d}, x] && !GtQ[c, 0]
  422. Int[Times[Power[Plus[Pattern[a, Blank[]], Times[Optional[Pattern[b, Blank[]]], Power[Pattern[x, Blank[]], 2]]], Rational[1, 2]], Power[Plus[Pattern[c, Blank[]], Times[Optional[Pattern[d, Blank[]]], Power[Pattern[x, Blank[]], 2]]], Rational[-1, 2]]], Pattern[x, Blank[Symbol]]] := Dist[a, Int[1/(Sqrt[a + b*x^2]*Sqrt[c + d*x^2]), x], x] + Dist[b, Int[x^2/(Sqrt[a + b*x^2]*Sqrt[c + d*x^2]), x], x] /; FreeQ[{a, b, c, d}, x] && PosQ[d/c] && PosQ[b/a]
  423. Int[Times[Power[Plus[Pattern[a, Blank[]], Times[Optional[Pattern[b, Blank[]]], Power[Pattern[x, Blank[]], 2]]], Rational[1, 2]], Power[Plus[Pattern[c, Blank[]], Times[Optional[Pattern[d, Blank[]]], Power[Pattern[x, Blank[]], 2]]], Rational[-1, 2]]], Pattern[x, Blank[Symbol]]] := Dist[b/d, Int[Sqrt[c + d*x^2]/Sqrt[a + b*x^2], x], x] - Dist[(b*c - a*d)/d, Int[1/(Sqrt[a + b*x^2]*Sqrt[c + d*x^2]), x], x] /; FreeQ[{a, b, c, d}, x] && PosQ[d/c] && NegQ[b/a]
  424. Int[Times[Power[Plus[Pattern[a, Blank[]], Times[Optional[Pattern[b, Blank[]]], Power[Pattern[x, Blank[]], 2]]], Rational[1, 2]], Power[Plus[Pattern[c, Blank[]], Times[Optional[Pattern[d, Blank[]]], Power[Pattern[x, Blank[]], 2]]], Rational[-1, 2]]], Pattern[x, Blank[Symbol]]] := Simp[(Sqrt[a]*EllipticE[ArcSin[Rt[-(d/c), 2]*x], (b*c)/(a*d)])/(Sqrt[c]*Rt[-(d/c), 2]), x] /; FreeQ[{a, b, c, d}, x] && NegQ[d/c] && GtQ[c, 0] && GtQ[a, 0]
  425. Int[Times[Power[Plus[Pattern[a, Blank[]], Times[Optional[Pattern[b, Blank[]]], Power[Pattern[x, Blank[]], 2]]], Rational[1, 2]], Power[Plus[Pattern[c, Blank[]], Times[Optional[Pattern[d, Blank[]]], Power[Pattern[x, Blank[]], 2]]], Rational[-1, 2]]], Pattern[x, Blank[Symbol]]] := -Simp[(Sqrt[a - (b*c)/d]*EllipticE[ArcCos[Rt[-(d/c), 2]*x], (b*c)/(b*c - a*d)])/(Sqrt[c]*Rt[-(d/c), 2]), x] /; FreeQ[{a, b, c, d}, x] && NegQ[d/c] && GtQ[c, 0] && GtQ[a - (b*c)/d, 0]
  426. Int[Times[Power[Plus[Pattern[a, Blank[]], Times[Optional[Pattern[b, Blank[]]], Power[Pattern[x, Blank[]], 2]]], Rational[1, 2]], Power[Plus[Pattern[c, Blank[]], Times[Optional[Pattern[d, Blank[]]], Power[Pattern[x, Blank[]], 2]]], Rational[-1, 2]]], Pattern[x, Blank[Symbol]]] := Dist[Sqrt[a + b*x^2]/Sqrt[1 + (b*x^2)/a], Int[Sqrt[1 + (b*x^2)/a]/Sqrt[c + d*x^2], x], x] /; FreeQ[{a, b, c, d}, x] && NegQ[d/c] && GtQ[c, 0] && !GtQ[a, 0]
  427. Int[Times[Power[Plus[Pattern[a, Blank[]], Times[Optional[Pattern[b, Blank[]]], Power[Pattern[x, Blank[]], 2]]], Rational[1, 2]], Power[Plus[Pattern[c, Blank[]], Times[Optional[Pattern[d, Blank[]]], Power[Pattern[x, Blank[]], 2]]], Rational[-1, 2]]], Pattern[x, Blank[Symbol]]] := Dist[Sqrt[1 + (d*x^2)/c]/Sqrt[c + d*x^2], Int[Sqrt[a + b*x^2]/Sqrt[1 + (d*x^2)/c], x], x] /; FreeQ[{a, b, c, d}, x] && NegQ[d/c] && !GtQ[c, 0]
  428. Int[Times[Power[Plus[Pattern[a, Blank[]], Times[Optional[Pattern[b, Blank[]]], Power[Pattern[x, Blank[]], Pattern[n, Blank[]]]]], Optional[Pattern[p, Blank[]]]], Power[Plus[Pattern[c, Blank[]], Times[Optional[Pattern[d, Blank[]]], Power[Pattern[x, Blank[]], Pattern[n, Blank[]]]]], Pattern[q, Blank[]]]], Pattern[x, Blank[Symbol]]] := Int[ExpandIntegrand[(a + b*x^n)^p*(c + d*x^n)^q, x], x] /; FreeQ[{a, b, c, d, n, q}, x] && NeQ[b*c - a*d, 0] && IGtQ[p, 0]
  429. Int[Times[Power[Plus[Pattern[a, Blank[]], Times[Optional[Pattern[b, Blank[]]], Power[Pattern[x, Blank[]], Pattern[n, Blank[]]]]], Pattern[p, Blank[]]], Power[Plus[Pattern[c, Blank[]], Times[Optional[Pattern[d, Blank[]]], Power[Pattern[x, Blank[]], Pattern[n, Blank[]]]]], Pattern[q, Blank[]]]], Pattern[x, Blank[Symbol]]] := Simp[a^p*c^q*x*AppellF1[1/n, -p, -q, 1 + 1/n, -((b*x^n)/a), -((d*x^n)/c)], x] /; FreeQ[{a, b, c, d, n, p, q}, x] && NeQ[b*c - a*d, 0] && NeQ[n, -1] && (IntegerQ[p] || GtQ[a, 0]) && (IntegerQ[q] || GtQ[c, 0])
  430. Int[Times[Power[Plus[Pattern[a, Blank[]], Times[Optional[Pattern[b, Blank[]]], Power[Pattern[x, Blank[]], Pattern[n, Blank[]]]]], Pattern[p, Blank[]]], Power[Plus[Pattern[c, Blank[]], Times[Optional[Pattern[d, Blank[]]], Power[Pattern[x, Blank[]], Pattern[n, Blank[]]]]], Pattern[q, Blank[]]]], Pattern[x, Blank[Symbol]]] := Dist[(a^IntPart[p]*(a + b*x^n)^FracPart[p])/(1 + (b*x^n)/a)^FracPart[p], Int[(1 + (b*x^n)/a)^p*(c + d*x^n)^q, x], x] /; FreeQ[{a, b, c, d, n, p, q}, x] && NeQ[b*c - a*d, 0] && NeQ[n, -1] && !(IntegerQ[p] || GtQ[a, 0])
  431. Int[Times[Power[Plus[Optional[Pattern[a, Blank[]]], Times[Optional[Pattern[b, Blank[]]], Power[Pattern[u, Blank[]], Pattern[n, Blank[]]]]], Optional[Pattern[p, Blank[]]]], Power[Plus[Optional[Pattern[c, Blank[]]], Times[Optional[Pattern[d, Blank[]]], Power[Pattern[u, Blank[]], Pattern[n, Blank[]]]]], Optional[Pattern[q, Blank[]]]]], Pattern[x, Blank[Symbol]]] := Dist[1/Coefficient[u, x, 1], Subst[Int[(a + b*x^n)^p*(c + d*x^n)^q, x], x, u], x] /; FreeQ[{a, b, c, d, n, p, q}, x] && LinearQ[u, x] && NeQ[u, x]
  432. Int[Times[Power[Pattern[u, Blank[]], Optional[Pattern[p, Blank[]]]], Power[Pattern[v, Blank[]], Optional[Pattern[q, Blank[]]]]], Pattern[x, Blank[Symbol]]] := Int[NormalizePseudoBinomial[u, x]^p*NormalizePseudoBinomial[v, x]^q, x] /; FreeQ[{p, q}, x] && PseudoBinomialPairQ[u, v, x]
  433. Int[Times[Power[Pattern[u, Blank[]], Optional[Pattern[p, Blank[]]]], Power[Pattern[v, Blank[]], Optional[Pattern[q, Blank[]]]], Power[Pattern[x, Blank[]], Optional[Pattern[m, Blank[]]]]], Pattern[x, Blank[Symbol]]] := Int[NormalizePseudoBinomial[x^(m/p)*u, x]^p*NormalizePseudoBinomial[v, x]^q, x] /; FreeQ[{p, q}, x] && IntegersQ[p, m/p] && PseudoBinomialPairQ[x^(m/p)*u, v, x]
  434. Int[Times[Power[Plus[Pattern[c, Blank[]], Times[Optional[Pattern[d, Blank[]]], Power[Pattern[x, Blank[]], Optional[Pattern[mn, Blank[]]]]]], Optional[Pattern[q, Blank[]]]], Power[Plus[Pattern[a, Blank[]], Times[Optional[Pattern[b, Blank[]]], Power[Pattern[x, Blank[]], Optional[Pattern[n, Blank[]]]]]], Optional[Pattern[p, Blank[]]]]], Pattern[x, Blank[Symbol]]] := Int[((a + b*x^n)^p*(d + c*x^n)^q)/x^(n*q), x] /; FreeQ[{a, b, c, d, n, p}, x] && EqQ[mn, -n] && IntegerQ[q] && (PosQ[n] || !IntegerQ[p])
  435. Int[Times[Power[Plus[Pattern[c, Blank[]], Times[Optional[Pattern[d, Blank[]]], Power[Pattern[x, Blank[]], Optional[Pattern[mn, Blank[]]]]]], Pattern[q, Blank[]]], Power[Plus[Pattern[a, Blank[]], Times[Optional[Pattern[b, Blank[]]], Power[Pattern[x, Blank[]], Optional[Pattern[n, Blank[]]]]]], Pattern[p, Blank[]]]], Pattern[x, Blank[Symbol]]] := Dist[(x^(n*FracPart[q])*(c + d/x^n)^FracPart[q])/(d + c*x^n)^FracPart[q], Int[((a + b*x^n)^p*(d + c*x^n)^q)/x^(n*q), x], x] /; FreeQ[{a, b, c, d, n, p, q}, x] && EqQ[mn, -n] && !IntegerQ[q] && !IntegerQ[p]
  436. Int[Times[Power[Times[Optional[Pattern[e, Blank[]]], Pattern[x, Blank[]]], Optional[Pattern[m, Blank[]]]], Power[Times[Optional[Pattern[b, Blank[]]], Power[Pattern[x, Blank[]], Pattern[n, Blank[]]]], Pattern[p, Blank[]]], Power[Plus[Pattern[c, Blank[]], Times[Optional[Pattern[d, Blank[]]], Power[Pattern[x, Blank[]], Pattern[n, Blank[]]]]], Optional[Pattern[q, Blank[]]]]], Pattern[x, Blank[Symbol]]] := Dist[e^m/(n*b^(Simplify[(m + 1)/n] - 1)), Subst[Int[(b*x)^(p + Simplify[(m + 1)/n] - 1)*(c + d*x)^q, x], x, x^n], x] /; FreeQ[{b, c, d, e, m, n, p, q}, x] && (IntegerQ[m] || GtQ[e, 0]) && IntegerQ[Simplify[(m + 1)/n]]
  437. Int[Times[Power[Times[Optional[Pattern[e, Blank[]]], Pattern[x, Blank[]]], Optional[Pattern[m, Blank[]]]], Power[Times[Optional[Pattern[b, Blank[]]], Power[Pattern[x, Blank[]], Optional[Pattern[n, Blank[]]]]], Pattern[p, Blank[]]], Power[Plus[Pattern[c, Blank[]], Times[Optional[Pattern[d, Blank[]]], Power[Pattern[x, Blank[]], Pattern[n, Blank[]]]]], Optional[Pattern[q, Blank[]]]]], Pattern[x, Blank[Symbol]]] := Dist[(e^m*b^IntPart[p]*(b*x^n)^FracPart[p])/x^(n*FracPart[p]), Int[x^(m + n*p)*(c + d*x^n)^q, x], x] /; FreeQ[{b, c, d, e, m, n, p, q}, x] && (IntegerQ[m] || GtQ[e, 0]) && !IntegerQ[Simplify[(m + 1)/n]]
  438. Int[Times[Power[Times[Pattern[e, Blank[]], Pattern[x, Blank[]]], Pattern[m, Blank[]]], Power[Times[Optional[Pattern[b, Blank[]]], Power[Pattern[x, Blank[]], Optional[Pattern[n, Blank[]]]]], Pattern[p, Blank[]]], Power[Plus[Pattern[c, Blank[]], Times[Optional[Pattern[d, Blank[]]], Power[Pattern[x, Blank[]], Pattern[n, Blank[]]]]], Optional[Pattern[q, Blank[]]]]], Pattern[x, Blank[Symbol]]] := Dist[(e^IntPart[m]*(e*x)^FracPart[m])/x^FracPart[m], Int[x^m*(b*x^n)^p*(c + d*x^n)^q, x], x] /; FreeQ[{b, c, d, e, m, n, p, q}, x] && !IntegerQ[m]
  439. Int[Times[Pattern[x, Blank[]], Power[Plus[Pattern[a, Blank[]], Times[Optional[Pattern[b, Blank[]]], Power[Pattern[x, Blank[]], 2]]], Rational[-1, 4]], Power[Plus[Pattern[c, Blank[]], Times[Optional[Pattern[d, Blank[]]], Power[Pattern[x, Blank[]], 2]]], -1]], Pattern[x, Blank[Symbol]]] := -Simp[ArcTan[(Rt[a, 4]^2 - Sqrt[a + b*x^2])/(Sqrt[2]*Rt[a, 4]*(a + b*x^2)^(1/4))]/(Sqrt[2]*Rt[a, 4]*d), x] - Simp[(1*ArcTanh[(Rt[a, 4]^2 + Sqrt[a + b*x^2])/(Sqrt[2]*Rt[a, 4]*(a + b*x^2)^(1/4))])/(Sqrt[2]*Rt[a, 4]*d), x] /; FreeQ[{a, b, c, d}, x] && EqQ[b*c - 2*a*d, 0] && PosQ[a]
  440. Int[Times[Power[Pattern[x, Blank[]], Pattern[m, Blank[]]], Power[Plus[Pattern[a, Blank[]], Times[Optional[Pattern[b, Blank[]]], Power[Pattern[x, Blank[]], 2]]], Rational[-1, 4]], Power[Plus[Pattern[c, Blank[]], Times[Optional[Pattern[d, Blank[]]], Power[Pattern[x, Blank[]], 2]]], -1]], Pattern[x, Blank[Symbol]]] := Int[ExpandIntegrand[x^m/((a + b*x^2)^(1/4)*(c + d*x^2)), x], x] /; FreeQ[{a, b, c, d}, x] && EqQ[b*c - 2*a*d, 0] && IntegerQ[m] && (PosQ[a] || IntegerQ[m/2])
  441. Int[Times[Power[Pattern[x, Blank[]], 2], Power[Plus[Pattern[a, Blank[]], Times[Optional[Pattern[b, Blank[]]], Power[Pattern[x, Blank[]], 2]]], Rational[-3, 4]], Power[Plus[Pattern[c, Blank[]], Times[Optional[Pattern[d, Blank[]]], Power[Pattern[x, Blank[]], 2]]], -1]], Pattern[x, Blank[Symbol]]] := -Simp[(b*ArcTan[(b + Rt[b^2/a, 4]^2*Sqrt[a + b*x^2])/(Rt[b^2/a, 4]^3*x*(a + b*x^2)^(1/4))])/(a*d*Rt[b^2/a, 4]^3), x] + Simp[(b*ArcTanh[(b - Rt[b^2/a, 4]^2*Sqrt[a + b*x^2])/(Rt[b^2/a, 4]^3*x*(a + b*x^2)^(1/4))])/(a*d*Rt[b^2/a, 4]^3), x] /; FreeQ[{a, b, c, d}, x] && EqQ[b*c - 2*a*d, 0] && PosQ[b^2/a]
  442. Int[Times[Power[Pattern[x, Blank[]], 2], Power[Plus[Pattern[a, Blank[]], Times[Optional[Pattern[b, Blank[]]], Power[Pattern[x, Blank[]], 2]]], Rational[-3, 4]], Power[Plus[Pattern[c, Blank[]], Times[Optional[Pattern[d, Blank[]]], Power[Pattern[x, Blank[]], 2]]], -1]], Pattern[x, Blank[Symbol]]] := -Simp[(b*ArcTan[(Rt[-(b^2/a), 4]*x)/(Sqrt[2]*(a + b*x^2)^(1/4))])/(Sqrt[2]*a*d*Rt[-(b^2/a), 4]^3), x] + Simp[(b*ArcTanh[(Rt[-(b^2/a), 4]*x)/(Sqrt[2]*(a + b*x^2)^(1/4))])/(Sqrt[2]*a*d*Rt[-(b^2/a), 4]^3), x] /; FreeQ[{a, b, c, d}, x] && EqQ[b*c - 2*a*d, 0] && NegQ[b^2/a]
  443. Int[Times[Power[Pattern[x, Blank[]], Pattern[m, Blank[]]], Power[Plus[Pattern[a, Blank[]], Times[Optional[Pattern[b, Blank[]]], Power[Pattern[x, Blank[]], 2]]], Rational[-3, 4]], Power[Plus[Pattern[c, Blank[]], Times[Optional[Pattern[d, Blank[]]], Power[Pattern[x, Blank[]], 2]]], -1]], Pattern[x, Blank[Symbol]]] := Int[ExpandIntegrand[x^m/((a + b*x^2)^(3/4)*(c + d*x^2)), x], x] /; FreeQ[{a, b, c, d}, x] && EqQ[b*c - 2*a*d, 0] && IntegerQ[m] && (PosQ[a] || IntegerQ[m/2])
  444. Int[Times[Power[Pattern[x, Blank[]], Optional[Pattern[m, Blank[]]]], Power[Plus[Pattern[a, Blank[]], Times[Optional[Pattern[b, Blank[]]], Power[Pattern[x, Blank[]], Pattern[n, Blank[]]]]], Optional[Pattern[p, Blank[]]]], Power[Plus[Pattern[c, Blank[]], Times[Optional[Pattern[d, Blank[]]], Power[Pattern[x, Blank[]], Pattern[n, Blank[]]]]], Optional[Pattern[q, Blank[]]]]], Pattern[x, Blank[Symbol]]] := Dist[1/n, Subst[Int[(a + b*x)^p*(c + d*x)^q, x], x, x^n], x] /; FreeQ[{a, b, c, d, m, n, p, q}, x] && NeQ[b*c - a*d, 0] && EqQ[m - n + 1, 0]
  445. Int[Times[Power[Pattern[x, Blank[]], Optional[Pattern[m, Blank[]]]], Power[Plus[Pattern[a, Blank[]], Times[Optional[Pattern[b, Blank[]]], Power[Pattern[x, Blank[]], Pattern[n, Blank[]]]]], Optional[Pattern[p, Blank[]]]], Power[Plus[Pattern[c, Blank[]], Times[Optional[Pattern[d, Blank[]]], Power[Pattern[x, Blank[]], Pattern[n, Blank[]]]]], Optional[Pattern[q, Blank[]]]]], Pattern[x, Blank[Symbol]]] := Int[x^(m + n*(p + q))*(b + a/x^n)^p*(d + c/x^n)^q, x] /; FreeQ[{a, b, c, d, m, n}, x] && NeQ[b*c - a*d, 0] && IntegersQ[p, q] && NegQ[n]
  446. Int[Times[Power[Pattern[x, Blank[]], Optional[Pattern[m, Blank[]]]], Power[Plus[Pattern[a, Blank[]], Times[Optional[Pattern[b, Blank[]]], Power[Pattern[x, Blank[]], Pattern[n, Blank[]]]]], Optional[Pattern[p, Blank[]]]], Power[Plus[Pattern[c, Blank[]], Times[Optional[Pattern[d, Blank[]]], Power[Pattern[x, Blank[]], Pattern[n, Blank[]]]]], Optional[Pattern[q, Blank[]]]]], Pattern[x, Blank[Symbol]]] := Dist[1/n, Subst[Int[x^(Simplify[(m + 1)/n] - 1)*(a + b*x)^p*(c + d*x)^q, x], x, x^n], x] /; FreeQ[{a, b, c, d, m, n, p, q}, x] && NeQ[b*c - a*d, 0] && IntegerQ[Simplify[(m + 1)/n]]
  447. Int[Times[Power[Times[Pattern[e, Blank[]], Pattern[x, Blank[]]], Optional[Pattern[m, Blank[]]]], Power[Plus[Pattern[a, Blank[]], Times[Optional[Pattern[b, Blank[]]], Power[Pattern[x, Blank[]], Pattern[n, Blank[]]]]], Optional[Pattern[p, Blank[]]]], Power[Plus[Pattern[c, Blank[]], Times[Optional[Pattern[d, Blank[]]], Power[Pattern[x, Blank[]], Pattern[n, Blank[]]]]], Optional[Pattern[q, Blank[]]]]], Pattern[x, Blank[Symbol]]] := Dist[(e^IntPart[m]*(e*x)^FracPart[m])/x^FracPart[m], Int[x^m*(a + b*x^n)^p*(c + d*x^n)^q, x], x] /; FreeQ[{a, b, c, d, e, m, n, p, q}, x] && NeQ[b*c - a*d, 0] && IntegerQ[Simplify[(m + 1)/n]]
  448. Int[Times[Power[Times[Optional[Pattern[e, Blank[]]], Pattern[x, Blank[]]], Optional[Pattern[m, Blank[]]]], Power[Plus[Pattern[a, Blank[]], Times[Optional[Pattern[b, Blank[]]], Power[Pattern[x, Blank[]], Pattern[n, Blank[]]]]], Optional[Pattern[p, Blank[]]]], Power[Plus[Pattern[c, Blank[]], Times[Optional[Pattern[d, Blank[]]], Power[Pattern[x, Blank[]], Pattern[n, Blank[]]]]], Optional[Pattern[q, Blank[]]]]], Pattern[x, Blank[Symbol]]] := Int[ExpandIntegrand[(e*x)^m*(a + b*x^n)^p*(c + d*x^n)^q, x], x] /; FreeQ[{a, b, c, d, e, m, n}, x] && NeQ[b*c - a*d, 0] && IGtQ[p, 0] && IGtQ[q, 0]
  449. Int[Times[Power[Times[Optional[Pattern[e, Blank[]]], Pattern[x, Blank[]]], Optional[Pattern[m, Blank[]]]], Power[Plus[Pattern[a, Blank[]], Times[Optional[Pattern[b, Blank[]]], Power[Pattern[x, Blank[]], Pattern[n, Blank[]]]]], Optional[Pattern[p, Blank[]]]], Plus[Pattern[c, Blank[]], Times[Optional[Pattern[d, Blank[]]], Power[Pattern[x, Blank[]], Pattern[n, Blank[]]]]]], Pattern[x, Blank[Symbol]]] := Simp[(c*(e*x)^(m + 1)*(a + b*x^n)^(p + 1))/(a*e*(m + 1)), x] /; FreeQ[{a, b, c, d, e, m, n, p}, x] && NeQ[b*c - a*d, 0] && EqQ[a*d*(m + 1) - b*c*(m + n*(p + 1) + 1), 0] && NeQ[m, -1]
  450. Int[Times[Power[Times[Optional[Pattern[e, Blank[]]], Pattern[x, Blank[]]], Optional[Pattern[m, Blank[]]]], Power[Plus[Pattern[a1, Blank[]], Times[Optional[Pattern[b1, Blank[]]], Power[Pattern[x, Blank[]], Optional[Pattern[non2, Blank[]]]]]], Optional[Pattern[p, Blank[]]]], Power[Plus[Pattern[a2, Blank[]], Times[Optional[Pattern[b2, Blank[]]], Power[Pattern[x, Blank[]], Optional[Pattern[non2, Blank[]]]]]], Optional[Pattern[p, Blank[]]]], Plus[Pattern[c, Blank[]], Times[Optional[Pattern[d, Blank[]]], Power[Pattern[x, Blank[]], Pattern[n, Blank[]]]]]], Pattern[x, Blank[Symbol]]] := Simp[(c*(e*x)^(m + 1)*(a1 + b1*x^(n/2))^(p + 1)*(a2 + b2*x^(n/2))^(p + 1))/(a1*a2*e*(m + 1)), x] /; FreeQ[{a1, b1, a2, b2, c, d, e, m, n, p}, x] && EqQ[non2, n/2] && EqQ[a2*b1 + a1*b2, 0] && EqQ[a1*a2*d*(m + 1) - b1*b2*c*(m + n*(p + 1) + 1), 0] && NeQ[m, -1]
  451. Int[Times[Power[Times[Optional[Pattern[e, Blank[]]], Pattern[x, Blank[]]], Optional[Pattern[m, Blank[]]]], Power[Plus[Pattern[a, Blank[]], Times[Optional[Pattern[b, Blank[]]], Power[Pattern[x, Blank[]], Pattern[n, Blank[]]]]], Optional[Pattern[p, Blank[]]]], Plus[Pattern[c, Blank[]], Times[Optional[Pattern[d, Blank[]]], Power[Pattern[x, Blank[]], Pattern[n, Blank[]]]]]], Pattern[x, Blank[Symbol]]] := Simp[(c*(e*x)^(m + 1)*(a + b*x^n)^(p + 1))/(a*e*(m + 1)), x] + Dist[d/e^n, Int[(e*x)^(m + n)*(a + b*x^n)^p, x], x] /; FreeQ[{a, b, c, d, e, m, n, p}, x] && NeQ[b*c - a*d, 0] && EqQ[m + n*(p + 1) + 1, 0] && (IntegerQ[n] || GtQ[e, 0]) && ((GtQ[n, 0] && LtQ[m, -1]) || (LtQ[n, 0] && GtQ[m + n, -1]))
  452. Int[Times[Power[Times[Optional[Pattern[e, Blank[]]], Pattern[x, Blank[]]], Optional[Pattern[m, Blank[]]]], Power[Plus[Pattern[a, Blank[]], Times[Optional[Pattern[b, Blank[]]], Power[Pattern[x, Blank[]], Pattern[n, Blank[]]]]], Optional[Pattern[p, Blank[]]]], Plus[Pattern[c, Blank[]], Times[Optional[Pattern[d, Blank[]]], Power[Pattern[x, Blank[]], Pattern[n, Blank[]]]]]], Pattern[x, Blank[Symbol]]] := Simp[((b*c - a*d)*(e*x)^(m + 1)*(a + b*x^n)^(p + 1))/(a*b*e*(m + 1)), x] + Dist[d/b, Int[(e*x)^m*(a + b*x^n)^(p + 1), x], x] /; FreeQ[{a, b, c, d, e, m, n, p}, x] && NeQ[b*c - a*d, 0] && EqQ[m + n*(p + 1) + 1, 0] && NeQ[m, -1]
  453. Int[Times[Power[Times[Optional[Pattern[e, Blank[]]], Pattern[x, Blank[]]], Optional[Pattern[m, Blank[]]]], Power[Plus[Pattern[a, Blank[]], Times[Optional[Pattern[b, Blank[]]], Power[Pattern[x, Blank[]], Pattern[n, Blank[]]]]], Optional[Pattern[p, Blank[]]]], Plus[Pattern[c, Blank[]], Times[Optional[Pattern[d, Blank[]]], Power[Pattern[x, Blank[]], Pattern[n, Blank[]]]]]], Pattern[x, Blank[Symbol]]] := Simp[(c*(e*x)^(m + 1)*(a + b*x^n)^(p + 1))/(a*e*(m + 1)), x] + Dist[(a*d*(m + 1) - b*c*(m + n*(p + 1) + 1))/(a*e^n*(m + 1)), Int[(e*x)^(m + n)*(a + b*x^n)^p, x], x] /; FreeQ[{a, b, c, d, e, p}, x] && NeQ[b*c - a*d, 0] && (IntegerQ[n] || GtQ[e, 0]) && ((GtQ[n, 0] && LtQ[m, -1]) || (LtQ[n, 0] && GtQ[m + n, -1])) && !ILtQ[p, -1]
  454. Int[Times[Power[Times[Optional[Pattern[e, Blank[]]], Pattern[x, Blank[]]], Optional[Pattern[m, Blank[]]]], Power[Plus[Pattern[a1, Blank[]], Times[Optional[Pattern[b1, Blank[]]], Power[Pattern[x, Blank[]], Optional[Pattern[non2, Blank[]]]]]], Optional[Pattern[p, Blank[]]]], Power[Plus[Pattern[a2, Blank[]], Times[Optional[Pattern[b2, Blank[]]], Power[Pattern[x, Blank[]], Optional[Pattern[non2, Blank[]]]]]], Optional[Pattern[p, Blank[]]]], Plus[Pattern[c, Blank[]], Times[Optional[Pattern[d, Blank[]]], Power[Pattern[x, Blank[]], Pattern[n, Blank[]]]]]], Pattern[x, Blank[Symbol]]] := Simp[(c*(e*x)^(m + 1)*(a1 + b1*x^(n/2))^(p + 1)*(a2 + b2*x^(n/2))^(p + 1))/(a1*a2*e*(m + 1)), x] + Dist[(a1*a2*d*(m + 1) - b1*b2*c*(m + n*(p + 1) + 1))/(a1*a2*e^n*(m + 1)), Int[(e*x)^(m + n)*(a1 + b1*x^(n/2))^p*(a2 + b2*x^(n/2))^p, x], x] /; FreeQ[{a1, b1, a2, b2, c, d, e, p}, x] && EqQ[non2, n/2] && EqQ[a2*b1 + a1*b2, 0] && (IntegerQ[n] || GtQ[e, 0]) && ((GtQ[n, 0] && LtQ[m, -1]) || (LtQ[n, 0] && GtQ[m + n, -1])) && !ILtQ[p, -1]
  455. Int[Times[Power[Pattern[x, Blank[]], Pattern[m, Blank[]]], Power[Plus[Pattern[a, Blank[]], Times[Optional[Pattern[b, Blank[]]], Power[Pattern[x, Blank[]], 2]]], Pattern[p, Blank[]]], Plus[Pattern[c, Blank[]], Times[Optional[Pattern[d, Blank[]]], Power[Pattern[x, Blank[]], 2]]]], Pattern[x, Blank[Symbol]]] := Simp[((-a)^(m/2 - 1)*(b*c - a*d)*x*(a + b*x^2)^(p + 1))/(2*b^(m/2 + 1)*(p + 1)), x] + Dist[1/(2*b^(m/2 + 1)*(p + 1)), Int[(a + b*x^2)^(p + 1)*ExpandToSum[2*b*(p + 1)*x^2*Together[(b^(m/2)*x^(m - 2)*(c + d*x^2) - (-a)^(m/2 - 1)*(b*c - a*d))/(a + b*x^2)] - (-a)^(m/2 - 1)*(b*c - a*d), x], x], x] /; FreeQ[{a, b, c, d}, x] && NeQ[b*c - a*d, 0] && LtQ[p, -1] && IGtQ[m/2, 0] && (IntegerQ[p] || EqQ[m + 2*p + 1, 0])
  456. Int[Times[Power[Pattern[x, Blank[]], Pattern[m, Blank[]]], Power[Plus[Pattern[a, Blank[]], Times[Optional[Pattern[b, Blank[]]], Power[Pattern[x, Blank[]], 2]]], Pattern[p, Blank[]]], Plus[Pattern[c, Blank[]], Times[Optional[Pattern[d, Blank[]]], Power[Pattern[x, Blank[]], 2]]]], Pattern[x, Blank[Symbol]]] := Simp[((-a)^(m/2 - 1)*(b*c - a*d)*x*(a + b*x^2)^(p + 1))/(2*b^(m/2 + 1)*(p + 1)), x] + Dist[1/(2*b^(m/2 + 1)*(p + 1)), Int[x^m*(a + b*x^2)^(p + 1)*ExpandToSum[2*b*(p + 1)*Together[(b^(m/2)*(c + d*x^2) - (-a)^(m/2 - 1)*(b*c - a*d)*x^(-m + 2))/(a + b*x^2)] - ((-a)^(m/2 - 1)*(b*c - a*d))/x^m, x], x], x] /; FreeQ[{a, b, c, d}, x] && NeQ[b*c - a*d, 0] && LtQ[p, -1] && ILtQ[m/2, 0] && (IntegerQ[p] || EqQ[m + 2*p + 1, 0])
  457. Int[Times[Power[Times[Optional[Pattern[e, Blank[]]], Pattern[x, Blank[]]], Optional[Pattern[m, Blank[]]]], Power[Plus[Pattern[a, Blank[]], Times[Optional[Pattern[b, Blank[]]], Power[Pattern[x, Blank[]], Pattern[n, Blank[]]]]], Optional[Pattern[p, Blank[]]]], Plus[Pattern[c, Blank[]], Times[Optional[Pattern[d, Blank[]]], Power[Pattern[x, Blank[]], Pattern[n, Blank[]]]]]], Pattern[x, Blank[Symbol]]] := -Simp[((b*c - a*d)*(e*x)^(m + 1)*(a + b*x^n)^(p + 1))/(a*b*e*n*(p + 1)), x] - Dist[(a*d*(m + 1) - b*c*(m + n*(p + 1) + 1))/(a*b*n*(p + 1)), Int[(e*x)^m*(a + b*x^n)^(p + 1), x], x] /; FreeQ[{a, b, c, d, e, m, n}, x] && NeQ[b*c - a*d, 0] && LtQ[p, -1] && (( !IntegerQ[p + 1/2] && NeQ[p, -5/4]) || !RationalQ[m] || (IGtQ[n, 0] && ILtQ[p + 1/2, 0] && LeQ[-1, m, -(n*(p + 1))]))
  458. Int[Times[Power[Times[Optional[Pattern[e, Blank[]]], Pattern[x, Blank[]]], Optional[Pattern[m, Blank[]]]], Power[Plus[Pattern[a1, Blank[]], Times[Optional[Pattern[b1, Blank[]]], Power[Pattern[x, Blank[]], Optional[Pattern[non2, Blank[]]]]]], Optional[Pattern[p, Blank[]]]], Power[Plus[Pattern[a2, Blank[]], Times[Optional[Pattern[b2, Blank[]]], Power[Pattern[x, Blank[]], Optional[Pattern[non2, Blank[]]]]]], Optional[Pattern[p, Blank[]]]], Plus[Pattern[c, Blank[]], Times[Optional[Pattern[d, Blank[]]], Power[Pattern[x, Blank[]], Pattern[n, Blank[]]]]]], Pattern[x, Blank[Symbol]]] := -Simp[((b1*b2*c - a1*a2*d)*(e*x)^(m + 1)*(a1 + b1*x^(n/2))^(p + 1)*(a2 + b2*x^(n/2))^(p + 1))/(a1*a2*b1*b2*e*n*(p + 1)), x] - Dist[(a1*a2*d*(m + 1) - b1*b2*c*(m + n*(p + 1) + 1))/(a1*a2*b1*b2*n*(p + 1)), Int[(e*x)^m*(a1 + b1*x^(n/2))^(p + 1)*(a2 + b2*x^(n/2))^(p + 1), x], x] /; FreeQ[{a1, b1, a2, b2, c, d, e, m, n}, x] && EqQ[non2, n/2] && EqQ[a2*b1 + a1*b2, 0] && LtQ[p, -1] && (( !IntegerQ[p + 1/2] && NeQ[p, -5/4]) || !RationalQ[m] || (IGtQ[n, 0] && ILtQ[p + 1/2, 0] && LeQ[-1, m, -(n*(p + 1))]))
  459. Int[Times[Power[Times[Optional[Pattern[e, Blank[]]], Pattern[x, Blank[]]], Optional[Pattern[m, Blank[]]]], Power[Plus[Pattern[a, Blank[]], Times[Optional[Pattern[b, Blank[]]], Power[Pattern[x, Blank[]], Pattern[n, Blank[]]]]], Optional[Pattern[p, Blank[]]]], Plus[Pattern[c, Blank[]], Times[Optional[Pattern[d, Blank[]]], Power[Pattern[x, Blank[]], Pattern[n, Blank[]]]]]], Pattern[x, Blank[Symbol]]] := Simp[(d*(e*x)^(m + 1)*(a + b*x^n)^(p + 1))/(b*e*(m + n*(p + 1) + 1)), x] - Dist[(a*d*(m + 1) - b*c*(m + n*(p + 1) + 1))/(b*(m + n*(p + 1) + 1)), Int[(e*x)^m*(a + b*x^n)^p, x], x] /; FreeQ[{a, b, c, d, e, m, n, p}, x] && NeQ[b*c - a*d, 0] && NeQ[m + n*(p + 1) + 1, 0]
  460. Int[Times[Power[Times[Optional[Pattern[e, Blank[]]], Pattern[x, Blank[]]], Optional[Pattern[m, Blank[]]]], Power[Plus[Pattern[a1, Blank[]], Times[Optional[Pattern[b1, Blank[]]], Power[Pattern[x, Blank[]], Optional[Pattern[non2, Blank[]]]]]], Optional[Pattern[p, Blank[]]]], Power[Plus[Pattern[a2, Blank[]], Times[Optional[Pattern[b2, Blank[]]], Power[Pattern[x, Blank[]], Optional[Pattern[non2, Blank[]]]]]], Optional[Pattern[p, Blank[]]]], Plus[Pattern[c, Blank[]], Times[Optional[Pattern[d, Blank[]]], Power[Pattern[x, Blank[]], Pattern[n, Blank[]]]]]], Pattern[x, Blank[Symbol]]] := Simp[(d*(e*x)^(m + 1)*(a1 + b1*x^(n/2))^(p + 1)*(a2 + b2*x^(n/2))^(p + 1))/(b1*b2*e*(m + n*(p + 1) + 1)), x] - Dist[(a1*a2*d*(m + 1) - b1*b2*c*(m + n*(p + 1) + 1))/(b1*b2*(m + n*(p + 1) + 1)), Int[(e*x)^m*(a1 + b1*x^(n/2))^p*(a2 + b2*x^(n/2))^p, x], x] /; FreeQ[{a1, b1, a2, b2, c, d, e, m, n, p}, x] && EqQ[non2, n/2] && EqQ[a2*b1 + a1*b2, 0] && NeQ[m + n*(p + 1) + 1, 0]
  461. Int[Times[Power[Times[Optional[Pattern[e, Blank[]]], Pattern[x, Blank[]]], Optional[Pattern[m, Blank[]]]], Power[Plus[Pattern[a, Blank[]], Times[Optional[Pattern[b, Blank[]]], Power[Pattern[x, Blank[]], Pattern[n, Blank[]]]]], Pattern[p, Blank[]]], Power[Plus[Pattern[c, Blank[]], Times[Optional[Pattern[d, Blank[]]], Power[Pattern[x, Blank[]], Pattern[n, Blank[]]]]], -1]], Pattern[x, Blank[Symbol]]] := Int[ExpandIntegrand[((e*x)^m*(a + b*x^n)^p)/(c + d*x^n), x], x] /; FreeQ[{a, b, c, d, e, m}, x] && NeQ[b*c - a*d, 0] && IGtQ[n, 0] && IGtQ[p, 0] && (IntegerQ[m] || IGtQ[2*(m + 1), 0] || !RationalQ[m])
  462. Int[Times[Power[Times[Optional[Pattern[e, Blank[]]], Pattern[x, Blank[]]], Pattern[m, Blank[]]], Power[Plus[Pattern[a, Blank[]], Times[Optional[Pattern[b, Blank[]]], Power[Pattern[x, Blank[]], Pattern[n, Blank[]]]]], Pattern[p, Blank[]]], Power[Plus[Pattern[c, Blank[]], Times[Optional[Pattern[d, Blank[]]], Power[Pattern[x, Blank[]], Pattern[n, Blank[]]]]], 2]], Pattern[x, Blank[Symbol]]] := Simp[(c^2*(e*x)^(m + 1)*(a + b*x^n)^(p + 1))/(a*e*(m + 1)), x] - Dist[1/(a*e^n*(m + 1)), Int[(e*x)^(m + n)*(a + b*x^n)^p*Simp[b*c^2*n*(p + 1) + c*(b*c - 2*a*d)*(m + 1) - a*(m + 1)*d^2*x^n, x], x], x] /; FreeQ[{a, b, c, d, e, p}, x] && NeQ[b*c - a*d, 0] && IGtQ[n, 0] && LtQ[m, -1] && GtQ[n, 0]
  463. Int[Times[Power[Times[Optional[Pattern[e, Blank[]]], Pattern[x, Blank[]]], Optional[Pattern[m, Blank[]]]], Power[Plus[Pattern[a, Blank[]], Times[Optional[Pattern[b, Blank[]]], Power[Pattern[x, Blank[]], Pattern[n, Blank[]]]]], Pattern[p, Blank[]]], Power[Plus[Pattern[c, Blank[]], Times[Optional[Pattern[d, Blank[]]], Power[Pattern[x, Blank[]], Pattern[n, Blank[]]]]], 2]], Pattern[x, Blank[Symbol]]] := -Simp[((b*c - a*d)^2*(e*x)^(m + 1)*(a + b*x^n)^(p + 1))/(a*b^2*e*n*(p + 1)), x] + Dist[1/(a*b^2*n*(p + 1)), Int[(e*x)^m*(a + b*x^n)^(p + 1)*Simp[(b*c - a*d)^2*(m + 1) + b^2*c^2*n*(p + 1) + a*b*d^2*n*(p + 1)*x^n, x], x], x] /; FreeQ[{a, b, c, d, e, m, n}, x] && NeQ[b*c - a*d, 0] && IGtQ[n, 0] && LtQ[p, -1]
  464. Int[Times[Power[Times[Optional[Pattern[e, Blank[]]], Pattern[x, Blank[]]], Optional[Pattern[m, Blank[]]]], Power[Plus[Pattern[a, Blank[]], Times[Optional[Pattern[b, Blank[]]], Power[Pattern[x, Blank[]], Pattern[n, Blank[]]]]], Pattern[p, Blank[]]], Power[Plus[Pattern[c, Blank[]], Times[Optional[Pattern[d, Blank[]]], Power[Pattern[x, Blank[]], Pattern[n, Blank[]]]]], 2]], Pattern[x, Blank[Symbol]]] := Simp[(d^2*(e*x)^(m + n + 1)*(a + b*x^n)^(p + 1))/(b*e^(n + 1)*(m + n*(p + 2) + 1)), x] + Dist[1/(b*(m + n*(p + 2) + 1)), Int[(e*x)^m*(a + b*x^n)^p*Simp[b*c^2*(m + n*(p + 2) + 1) + d*((2*b*c - a*d)*(m + n + 1) + 2*b*c*n*(p + 1))*x^n, x], x], x] /; FreeQ[{a, b, c, d, e, m, n, p}, x] && NeQ[b*c - a*d, 0] && IGtQ[n, 0] && NeQ[m + n*(p + 2) + 1, 0]
  465. Int[Times[Power[Pattern[x, Blank[]], Optional[Pattern[m, Blank[]]]], Power[Plus[Pattern[a, Blank[]], Times[Optional[Pattern[b, Blank[]]], Power[Pattern[x, Blank[]], Pattern[n, Blank[]]]]], Pattern[p, Blank[]]], Power[Plus[Pattern[c, Blank[]], Times[Optional[Pattern[d, Blank[]]], Power[Pattern[x, Blank[]], Pattern[n, Blank[]]]]], Pattern[q, Blank[]]]], Pattern[x, Blank[Symbol]]] := With[{k = GCD[m + 1, n]}, Dist[1/k, Subst[Int[x^((m + 1)/k - 1)*(a + b*x^(n/k))^p*(c + d*x^(n/k))^q, x], x, x^k], x] /; k != 1] /; FreeQ[{a, b, c, d, p, q}, x] && NeQ[b*c - a*d, 0] && IGtQ[n, 0] && IntegerQ[m]
  466. Int[Times[Power[Times[Optional[Pattern[e, Blank[]]], Pattern[x, Blank[]]], Pattern[m, Blank[]]], Power[Plus[Pattern[a, Blank[]], Times[Optional[Pattern[b, Blank[]]], Power[Pattern[x, Blank[]], Pattern[n, Blank[]]]]], Pattern[p, Blank[]]], Power[Plus[Pattern[c, Blank[]], Times[Optional[Pattern[d, Blank[]]], Power[Pattern[x, Blank[]], Pattern[n, Blank[]]]]], Pattern[q, Blank[]]]], Pattern[x, Blank[Symbol]]] := With[{k = Denominator[m]}, Dist[k/e, Subst[Int[x^(k*(m + 1) - 1)*(a + (b*x^(k*n))/e^n)^p*(c + (d*x^(k*n))/e^n)^q, x], x, (e*x)^(1/k)], x]] /; FreeQ[{a, b, c, d, e, p, q}, x] && NeQ[b*c - a*d, 0] && IGtQ[n, 0] && FractionQ[m] && IntegerQ[p]
  467. Int[Times[Power[Times[Optional[Pattern[e, Blank[]]], Pattern[x, Blank[]]], Optional[Pattern[m, Blank[]]]], Power[Plus[Pattern[a, Blank[]], Times[Optional[Pattern[b, Blank[]]], Power[Pattern[x, Blank[]], Pattern[n, Blank[]]]]], Pattern[p, Blank[]]], Power[Plus[Pattern[c, Blank[]], Times[Optional[Pattern[d, Blank[]]], Power[Pattern[x, Blank[]], Pattern[n, Blank[]]]]], Pattern[q, Blank[]]]], Pattern[x, Blank[Symbol]]] := Simp[(e^(n - 1)*(e*x)^(m - n + 1)*(a + b*x^n)^(p + 1)*(c + d*x^n)^q)/(b*n*(p + 1)), x] - Dist[e^n/(b*n*(p + 1)), Int[(e*x)^(m - n)*(a + b*x^n)^(p + 1)*(c + d*x^n)^(q - 1)*Simp[c*(m - n + 1) + d*(m + n*(q - 1) + 1)*x^n, x], x], x] /; FreeQ[{a, b, c, d, e}, x] && NeQ[b*c - a*d, 0] && IGtQ[n, 0] && LtQ[p, -1] && GtQ[q, 0] && GtQ[m - n + 1, 0] && IntBinomialQ[a, b, c, d, e, m, n, p, q, x]
  468. Int[Times[Power[Times[Optional[Pattern[e, Blank[]]], Pattern[x, Blank[]]], Optional[Pattern[m, Blank[]]]], Power[Plus[Pattern[a, Blank[]], Times[Optional[Pattern[b, Blank[]]], Power[Pattern[x, Blank[]], Pattern[n, Blank[]]]]], Pattern[p, Blank[]]], Power[Plus[Pattern[c, Blank[]], Times[Optional[Pattern[d, Blank[]]], Power[Pattern[x, Blank[]], Pattern[n, Blank[]]]]], Pattern[q, Blank[]]]], Pattern[x, Blank[Symbol]]] := -Simp[((c*b - a*d)*(e*x)^(m + 1)*(a + b*x^n)^(p + 1)*(c + d*x^n)^(q - 1))/(a*b*e*n*(p + 1)), x] + Dist[1/(a*b*n*(p + 1)), Int[(e*x)^m*(a + b*x^n)^(p + 1)*(c + d*x^n)^(q - 2)*Simp[c*(c*b*n*(p + 1) + (c*b - a*d)*(m + 1)) + d*(c*b*n*(p + 1) + (c*b - a*d)*(m + n*(q - 1) + 1))*x^n, x], x], x] /; FreeQ[{a, b, c, d, e, m}, x] && NeQ[b*c - a*d, 0] && IGtQ[n, 0] && LtQ[p, -1] && GtQ[q, 1] && IntBinomialQ[a, b, c, d, e, m, n, p, q, x]
  469. Int[Times[Power[Times[Optional[Pattern[e, Blank[]]], Pattern[x, Blank[]]], Optional[Pattern[m, Blank[]]]], Power[Plus[Pattern[a, Blank[]], Times[Optional[Pattern[b, Blank[]]], Power[Pattern[x, Blank[]], Pattern[n, Blank[]]]]], Pattern[p, Blank[]]], Power[Plus[Pattern[c, Blank[]], Times[Optional[Pattern[d, Blank[]]], Power[Pattern[x, Blank[]], Pattern[n, Blank[]]]]], Pattern[q, Blank[]]]], Pattern[x, Blank[Symbol]]] := -Simp[((e*x)^(m + 1)*(a + b*x^n)^(p + 1)*(c + d*x^n)^q)/(a*e*n*(p + 1)), x] + Dist[1/(a*n*(p + 1)), Int[(e*x)^m*(a + b*x^n)^(p + 1)*(c + d*x^n)^(q - 1)*Simp[c*(m + n*(p + 1) + 1) + d*(m + n*(p + q + 1) + 1)*x^n, x], x], x] /; FreeQ[{a, b, c, d, e, m}, x] && NeQ[b*c - a*d, 0] && IGtQ[n, 0] && LtQ[p, -1] && LtQ[0, q, 1] && IntBinomialQ[a, b, c, d, e, m, n, p, q, x]
  470. Int[Times[Power[Times[Optional[Pattern[e, Blank[]]], Pattern[x, Blank[]]], Optional[Pattern[m, Blank[]]]], Power[Plus[Pattern[a, Blank[]], Times[Optional[Pattern[b, Blank[]]], Power[Pattern[x, Blank[]], Pattern[n, Blank[]]]]], Pattern[p, Blank[]]], Power[Plus[Pattern[c, Blank[]], Times[Optional[Pattern[d, Blank[]]], Power[Pattern[x, Blank[]], Pattern[n, Blank[]]]]], Pattern[q, Blank[]]]], Pattern[x, Blank[Symbol]]] := -Simp[(a*e^(2*n - 1)*(e*x)^(m - 2*n + 1)*(a + b*x^n)^(p + 1)*(c + d*x^n)^(q + 1))/(b*n*(b*c - a*d)*(p + 1)), x] + Dist[e^(2*n)/(b*n*(b*c - a*d)*(p + 1)), Int[(e*x)^(m - 2*n)*(a + b*x^n)^(p + 1)*(c + d*x^n)^q*Simp[a*c*(m - 2*n + 1) + (a*d*(m - n + n*q + 1) + b*c*n*(p + 1))*x^n, x], x], x] /; FreeQ[{a, b, c, d, e, q}, x] && NeQ[b*c - a*d, 0] && IGtQ[n, 0] && LtQ[p, -1] && GtQ[m - n + 1, n] && IntBinomialQ[a, b, c, d, e, m, n, p, q, x]
  471. Int[Times[Power[Times[Optional[Pattern[e, Blank[]]], Pattern[x, Blank[]]], Optional[Pattern[m, Blank[]]]], Power[Plus[Pattern[a, Blank[]], Times[Optional[Pattern[b, Blank[]]], Power[Pattern[x, Blank[]], Pattern[n, Blank[]]]]], Pattern[p, Blank[]]], Power[Plus[Pattern[c, Blank[]], Times[Optional[Pattern[d, Blank[]]], Power[Pattern[x, Blank[]], Pattern[n, Blank[]]]]], Pattern[q, Blank[]]]], Pattern[x, Blank[Symbol]]] := Simp[(e^(n - 1)*(e*x)^(m - n + 1)*(a + b*x^n)^(p + 1)*(c + d*x^n)^(q + 1))/(n*(b*c - a*d)*(p + 1)), x] - Dist[e^n/(n*(b*c - a*d)*(p + 1)), Int[(e*x)^(m - n)*(a + b*x^n)^(p + 1)*(c + d*x^n)^q*Simp[c*(m - n + 1) + d*(m + n*(p + q + 1) + 1)*x^n, x], x], x] /; FreeQ[{a, b, c, d, e, q}, x] && NeQ[b*c - a*d, 0] && IGtQ[n, 0] && LtQ[p, -1] && GeQ[n, m - n + 1] && GtQ[m - n + 1, 0] && IntBinomialQ[a, b, c, d, e, m, n, p, q, x]
  472. Int[Times[Power[Times[Optional[Pattern[e, Blank[]]], Pattern[x, Blank[]]], Optional[Pattern[m, Blank[]]]], Power[Plus[Pattern[a, Blank[]], Times[Optional[Pattern[b, Blank[]]], Power[Pattern[x, Blank[]], Pattern[n, Blank[]]]]], Pattern[p, Blank[]]], Power[Plus[Pattern[c, Blank[]], Times[Optional[Pattern[d, Blank[]]], Power[Pattern[x, Blank[]], Pattern[n, Blank[]]]]], Pattern[q, Blank[]]]], Pattern[x, Blank[Symbol]]] := -Simp[(b*(e*x)^(m + 1)*(a + b*x^n)^(p + 1)*(c + d*x^n)^(q + 1))/(a*e*n*(b*c - a*d)*(p + 1)), x] + Dist[1/(a*n*(b*c - a*d)*(p + 1)), Int[(e*x)^m*(a + b*x^n)^(p + 1)*(c + d*x^n)^q*Simp[c*b*(m + 1) + n*(b*c - a*d)*(p + 1) + d*b*(m + n*(p + q + 2) + 1)*x^n, x], x], x] /; FreeQ[{a, b, c, d, e, m, q}, x] && NeQ[b*c - a*d, 0] && IGtQ[n, 0] && LtQ[p, -1] && IntBinomialQ[a, b, c, d, e, m, n, p, q, x]
  473. Int[Times[Power[Times[Optional[Pattern[e, Blank[]]], Pattern[x, Blank[]]], Pattern[m, Blank[]]], Power[Plus[Pattern[a, Blank[]], Times[Optional[Pattern[b, Blank[]]], Power[Pattern[x, Blank[]], Pattern[n, Blank[]]]]], Pattern[p, Blank[]]], Power[Plus[Pattern[c, Blank[]], Times[Optional[Pattern[d, Blank[]]], Power[Pattern[x, Blank[]], Pattern[n, Blank[]]]]], Pattern[q, Blank[]]]], Pattern[x, Blank[Symbol]]] := Simp[((e*x)^(m + 1)*(a + b*x^n)^p*(c + d*x^n)^q)/(e*(m + 1)), x] - Dist[n/(e^n*(m + 1)), Int[(e*x)^(m + n)*(a + b*x^n)^(p - 1)*(c + d*x^n)^(q - 1)*Simp[b*c*p + a*d*q + b*d*(p + q)*x^n, x], x], x] /; FreeQ[{a, b, c, d, e}, x] && NeQ[b*c - a*d, 0] && IGtQ[n, 0] && GtQ[q, 0] && LtQ[m, -1] && GtQ[p, 0] && IntBinomialQ[a, b, c, d, e, m, n, p, q, x]
  474. Int[Times[Power[Times[Optional[Pattern[e, Blank[]]], Pattern[x, Blank[]]], Pattern[m, Blank[]]], Power[Plus[Pattern[a, Blank[]], Times[Optional[Pattern[b, Blank[]]], Power[Pattern[x, Blank[]], Pattern[n, Blank[]]]]], Pattern[p, Blank[]]], Power[Plus[Pattern[c, Blank[]], Times[Optional[Pattern[d, Blank[]]], Power[Pattern[x, Blank[]], Pattern[n, Blank[]]]]], Pattern[q, Blank[]]]], Pattern[x, Blank[Symbol]]] := Simp[(c*(e*x)^(m + 1)*(a + b*x^n)^(p + 1)*(c + d*x^n)^(q - 1))/(a*e*(m + 1)), x] - Dist[1/(a*e^n*(m + 1)), Int[(e*x)^(m + n)*(a + b*x^n)^p*(c + d*x^n)^(q - 2)*Simp[c*(c*b - a*d)*(m + 1) + c*n*(b*c*(p + 1) + a*d*(q - 1)) + d*((c*b - a*d)*(m + 1) + c*b*n*(p + q))*x^n, x], x], x] /; FreeQ[{a, b, c, d, e, p}, x] && NeQ[b*c - a*d, 0] && IGtQ[n, 0] && GtQ[q, 1] && LtQ[m, -1] && IntBinomialQ[a, b, c, d, e, m, n, p, q, x]
  475. Int[Times[Power[Times[Optional[Pattern[e, Blank[]]], Pattern[x, Blank[]]], Pattern[m, Blank[]]], Power[Plus[Pattern[a, Blank[]], Times[Optional[Pattern[b, Blank[]]], Power[Pattern[x, Blank[]], Pattern[n, Blank[]]]]], Pattern[p, Blank[]]], Power[Plus[Pattern[c, Blank[]], Times[Optional[Pattern[d, Blank[]]], Power[Pattern[x, Blank[]], Pattern[n, Blank[]]]]], Pattern[q, Blank[]]]], Pattern[x, Blank[Symbol]]] := Simp[((e*x)^(m + 1)*(a + b*x^n)^(p + 1)*(c + d*x^n)^q)/(a*e*(m + 1)), x] - Dist[1/(a*e^n*(m + 1)), Int[(e*x)^(m + n)*(a + b*x^n)^p*(c + d*x^n)^(q - 1)*Simp[c*b*(m + 1) + n*(b*c*(p + 1) + a*d*q) + d*(b*(m + 1) + b*n*(p + q + 1))*x^n, x], x], x] /; FreeQ[{a, b, c, d, e, p}, x] && NeQ[b*c - a*d, 0] && IGtQ[n, 0] && LtQ[0, q, 1] && LtQ[m, -1] && IntBinomialQ[a, b, c, d, e, m, n, p, q, x]
  476. Int[Times[Power[Times[Optional[Pattern[e, Blank[]]], Pattern[x, Blank[]]], Optional[Pattern[m, Blank[]]]], Power[Plus[Pattern[a, Blank[]], Times[Optional[Pattern[b, Blank[]]], Power[Pattern[x, Blank[]], Pattern[n, Blank[]]]]], Pattern[p, Blank[]]], Power[Plus[Pattern[c, Blank[]], Times[Optional[Pattern[d, Blank[]]], Power[Pattern[x, Blank[]], Pattern[n, Blank[]]]]], Pattern[q, Blank[]]]], Pattern[x, Blank[Symbol]]] := Simp[((e*x)^(m + 1)*(a + b*x^n)^p*(c + d*x^n)^q)/(e*(m + n*(p + q) + 1)), x] + Dist[n/(m + n*(p + q) + 1), Int[(e*x)^m*(a + b*x^n)^(p - 1)*(c + d*x^n)^(q - 1)*Simp[a*c*(p + q) + (q*(b*c - a*d) + a*d*(p + q))*x^n, x], x], x] /; FreeQ[{a, b, c, d, e, m}, x] && NeQ[b*c - a*d, 0] && IGtQ[n, 0] && GtQ[q, 0] && GtQ[p, 0] && IntBinomialQ[a, b, c, d, e, m, n, p, q, x]
  477. Int[Times[Power[Times[Optional[Pattern[e, Blank[]]], Pattern[x, Blank[]]], Optional[Pattern[m, Blank[]]]], Power[Plus[Pattern[a, Blank[]], Times[Optional[Pattern[b, Blank[]]], Power[Pattern[x, Blank[]], Pattern[n, Blank[]]]]], Pattern[p, Blank[]]], Power[Plus[Pattern[c, Blank[]], Times[Optional[Pattern[d, Blank[]]], Power[Pattern[x, Blank[]], Pattern[n, Blank[]]]]], Pattern[q, Blank[]]]], Pattern[x, Blank[Symbol]]] := Simp[(d*(e*x)^(m + 1)*(a + b*x^n)^(p + 1)*(c + d*x^n)^(q - 1))/(b*e*(m + n*(p + q) + 1)), x] + Dist[1/(b*(m + n*(p + q) + 1)), Int[(e*x)^m*(a + b*x^n)^p*(c + d*x^n)^(q - 2)*Simp[c*((c*b - a*d)*(m + 1) + c*b*n*(p + q)) + (d*(c*b - a*d)*(m + 1) + d*n*(q - 1)*(b*c - a*d) + c*b*d*n*(p + q))*x^n, x], x], x] /; FreeQ[{a, b, c, d, e, m, p}, x] && NeQ[b*c - a*d, 0] && IGtQ[n, 0] && GtQ[q, 1] && IntBinomialQ[a, b, c, d, e, m, n, p, q, x]
  478. Int[Times[Power[Times[Optional[Pattern[e, Blank[]]], Pattern[x, Blank[]]], Optional[Pattern[m, Blank[]]]], Power[Plus[Pattern[a, Blank[]], Times[Optional[Pattern[b, Blank[]]], Power[Pattern[x, Blank[]], Pattern[n, Blank[]]]]], Pattern[p, Blank[]]], Power[Plus[Pattern[c, Blank[]], Times[Optional[Pattern[d, Blank[]]], Power[Pattern[x, Blank[]], Pattern[n, Blank[]]]]], Pattern[q, Blank[]]]], Pattern[x, Blank[Symbol]]] := Simp[(e^(n - 1)*(e*x)^(m - n + 1)*(a + b*x^n)^(p + 1)*(c + d*x^n)^q)/(b*(m + n*(p + q) + 1)), x] - Dist[e^n/(b*(m + n*(p + q) + 1)), Int[(e*x)^(m - n)*(a + b*x^n)^p*(c + d*x^n)^(q - 1)*Simp[a*c*(m - n + 1) + (a*d*(m - n + 1) - n*q*(b*c - a*d))*x^n, x], x], x] /; FreeQ[{a, b, c, d, e, p}, x] && NeQ[b*c - a*d, 0] && IGtQ[n, 0] && GtQ[q, 0] && GtQ[m - n + 1, 0] && IntBinomialQ[a, b, c, d, e, m, n, p, q, x]
  479. Int[Times[Power[Times[Optional[Pattern[e, Blank[]]], Pattern[x, Blank[]]], Optional[Pattern[m, Blank[]]]], Power[Plus[Pattern[a, Blank[]], Times[Optional[Pattern[b, Blank[]]], Power[Pattern[x, Blank[]], Pattern[n, Blank[]]]]], Pattern[p, Blank[]]], Power[Plus[Pattern[c, Blank[]], Times[Optional[Pattern[d, Blank[]]], Power[Pattern[x, Blank[]], Pattern[n, Blank[]]]]], Pattern[q, Blank[]]]], Pattern[x, Blank[Symbol]]] := Simp[(e^(2*n - 1)*(e*x)^(m - 2*n + 1)*(a + b*x^n)^(p + 1)*(c + d*x^n)^(q + 1))/(b*d*(m + n*(p + q) + 1)), x] - Dist[e^(2*n)/(b*d*(m + n*(p + q) + 1)), Int[(e*x)^(m - 2*n)*(a + b*x^n)^p*(c + d*x^n)^q*Simp[a*c*(m - 2*n + 1) + (a*d*(m + n*(q - 1) + 1) + b*c*(m + n*(p - 1) + 1))*x^n, x], x], x] /; FreeQ[{a, b, c, d, e, p, q}, x] && NeQ[b*c - a*d, 0] && IGtQ[n, 0] && GtQ[m - n + 1, n] && IntBinomialQ[a, b, c, d, e, m, n, p, q, x]
  480. Int[Times[Power[Times[Optional[Pattern[e, Blank[]]], Pattern[x, Blank[]]], Pattern[m, Blank[]]], Power[Plus[Pattern[a, Blank[]], Times[Optional[Pattern[b, Blank[]]], Power[Pattern[x, Blank[]], Pattern[n, Blank[]]]]], Pattern[p, Blank[]]], Power[Plus[Pattern[c, Blank[]], Times[Optional[Pattern[d, Blank[]]], Power[Pattern[x, Blank[]], Pattern[n, Blank[]]]]], Pattern[q, Blank[]]]], Pattern[x, Blank[Symbol]]] := Simp[((e*x)^(m + 1)*(a + b*x^n)^(p + 1)*(c + d*x^n)^(q + 1))/(a*c*e*(m + 1)), x] - Dist[1/(a*c*e^n*(m + 1)), Int[(e*x)^(m + n)*(a + b*x^n)^p*(c + d*x^n)^q*Simp[(b*c + a*d)*(m + n + 1) + n*(b*c*p + a*d*q) + b*d*(m + n*(p + q + 2) + 1)*x^n, x], x], x] /; FreeQ[{a, b, c, d, e, p, q}, x] && NeQ[b*c - a*d, 0] && IGtQ[n, 0] && LtQ[m, -1] && IntBinomialQ[a, b, c, d, e, m, n, p, q, x]
  481. Int[Times[Power[Times[Optional[Pattern[e, Blank[]]], Pattern[x, Blank[]]], Optional[Pattern[m, Blank[]]]], Power[Plus[Pattern[a, Blank[]], Times[Optional[Pattern[b, Blank[]]], Power[Pattern[x, Blank[]], Pattern[n, Blank[]]]]], -1], Power[Plus[Pattern[c, Blank[]], Times[Optional[Pattern[d, Blank[]]], Power[Pattern[x, Blank[]], Pattern[n, Blank[]]]]], -1]], Pattern[x, Blank[Symbol]]] := -Dist[(a*e^n)/(b*c - a*d), Int[(e*x)^(m - n)/(a + b*x^n), x], x] + Dist[(c*e^n)/(b*c - a*d), Int[(e*x)^(m - n)/(c + d*x^n), x], x] /; FreeQ[{a, b, c, d, e, m}, x] && NeQ[b*c - a*d, 0] && IGtQ[n, 0] && LeQ[n, m, 2*n - 1]
  482. Int[Times[Power[Times[Optional[Pattern[e, Blank[]]], Pattern[x, Blank[]]], Optional[Pattern[m, Blank[]]]], Power[Plus[Pattern[a, Blank[]], Times[Optional[Pattern[b, Blank[]]], Power[Pattern[x, Blank[]], Pattern[n, Blank[]]]]], -1], Power[Plus[Pattern[c, Blank[]], Times[Optional[Pattern[d, Blank[]]], Power[Pattern[x, Blank[]], Pattern[n, Blank[]]]]], -1]], Pattern[x, Blank[Symbol]]] := Dist[b/(b*c - a*d), Int[(e*x)^m/(a + b*x^n), x], x] - Dist[d/(b*c - a*d), Int[(e*x)^m/(c + d*x^n), x], x] /; FreeQ[{a, b, c, d, e, m}, x] && NeQ[b*c - a*d, 0] && IGtQ[n, 0]
  483. Int[Times[Power[Times[Optional[Pattern[e, Blank[]]], Pattern[x, Blank[]]], Pattern[m, Blank[]]], Power[Plus[Pattern[a, Blank[]], Times[Optional[Pattern[b, Blank[]]], Power[Pattern[x, Blank[]], Pattern[n, Blank[]]]]], -1], Power[Plus[Pattern[c, Blank[]], Times[Optional[Pattern[d, Blank[]]], Power[Pattern[x, Blank[]], Pattern[n, Blank[]]]]], Optional[Pattern[q, Blank[]]]]], Pattern[x, Blank[Symbol]]] := Dist[e^n/b, Int[(e*x)^(m - n)*(c + d*x^n)^q, x], x] - Dist[(a*e^n)/b, Int[((e*x)^(m - n)*(c + d*x^n)^q)/(a + b*x^n), x], x] /; FreeQ[{a, b, c, d, e, m, q}, x] && NeQ[b*c - a*d, 0] && IGtQ[n, 0] && LeQ[n, m, 2*n - 1] && IntBinomialQ[a, b, c, d, e, m, n, -1, q, x]
  484. Int[Times[Pattern[x, Blank[]], Power[Plus[Pattern[a, Blank[]], Times[Optional[Pattern[b, Blank[]]], Power[Pattern[x, Blank[]], 3]]], -1], Power[Plus[Pattern[c, Blank[]], Times[Optional[Pattern[d, Blank[]]], Power[Pattern[x, Blank[]], 3]]], Rational[-1, 2]]], Pattern[x, Blank[Symbol]]] := With[{q = Rt[d/c, 3]}, Simp[(q*ArcTanh[Sqrt[c + d*x^3]/Rt[c, 2]])/(9*2^(2/3)*b*Rt[c, 2]), x] + (-Simp[(q*ArcTanh[(Rt[c, 2]*(1 - 2^(1/3)*q*x))/Sqrt[c + d*x^3]])/(3*2^(2/3)*b*Rt[c, 2]), x] + Simp[(q*ArcTan[Sqrt[c + d*x^3]/(Sqrt[3]*Rt[c, 2])])/(3*2^(2/3)*Sqrt[3]*b*Rt[c, 2]), x] - Simp[(q*ArcTan[(Sqrt[3]*Rt[c, 2]*(1 + 2^(1/3)*q*x))/Sqrt[c + d*x^3]])/(3*2^(2/3)*Sqrt[3]*b*Rt[c, 2]), x])] /; FreeQ[{a, b, c, d}, x] && NeQ[b*c - a*d, 0] && EqQ[4*b*c - a*d, 0] && PosQ[c]
  485. Int[Times[Pattern[x, Blank[]], Power[Plus[Pattern[a, Blank[]], Times[Optional[Pattern[b, Blank[]]], Power[Pattern[x, Blank[]], 3]]], -1], Power[Plus[Pattern[c, Blank[]], Times[Optional[Pattern[d, Blank[]]], Power[Pattern[x, Blank[]], 3]]], Rational[-1, 2]]], Pattern[x, Blank[Symbol]]] := With[{q = Rt[d/c, 3]}, -Simp[(q*ArcTan[Sqrt[c + d*x^3]/Rt[-c, 2]])/(9*2^(2/3)*b*Rt[-c, 2]), x] + (-Simp[(q*ArcTan[(Rt[-c, 2]*(1 - 2^(1/3)*q*x))/Sqrt[c + d*x^3]])/(3*2^(2/3)*b*Rt[-c, 2]), x] - Simp[(q*ArcTanh[Sqrt[c + d*x^3]/(Sqrt[3]*Rt[-c, 2])])/(3*2^(2/3)*Sqrt[3]*b*Rt[-c, 2]), x] - Simp[(q*ArcTanh[(Sqrt[3]*Rt[-c, 2]*(1 + 2^(1/3)*q*x))/Sqrt[c + d*x^3]])/(3*2^(2/3)*Sqrt[3]*b*Rt[-c, 2]), x])] /; FreeQ[{a, b, c, d}, x] && NeQ[b*c - a*d, 0] && EqQ[4*b*c - a*d, 0] && NegQ[c]
  486. Int[Times[Pattern[x, Blank[]], Power[Plus[Pattern[a, Blank[]], Times[Optional[Pattern[b, Blank[]]], Power[Pattern[x, Blank[]], 3]]], -1], Power[Plus[Pattern[c, Blank[]], Times[Optional[Pattern[d, Blank[]]], Power[Pattern[x, Blank[]], 3]]], Rational[-1, 2]]], Pattern[x, Blank[Symbol]]] := With[{q = Rt[d/c, 3]}, Dist[(d*q)/(4*b), Int[x^2/((8*c - d*x^3)*Sqrt[c + d*x^3]), x], x] + (-Dist[q^2/(12*b), Int[(1 + q*x)/((2 - q*x)*Sqrt[c + d*x^3]), x], x] + Dist[1/(12*b*c), Int[(2*c*q^2 - 2*d*x - d*q*x^2)/((4 + 2*q*x + q^2*x^2)*Sqrt[c + d*x^3]), x], x])] /; FreeQ[{a, b, c, d}, x] && NeQ[b*c - a*d, 0] && EqQ[8*b*c + a*d, 0]
  487. Int[Times[Pattern[x, Blank[]], Power[Plus[Pattern[a, Blank[]], Times[Optional[Pattern[b, Blank[]]], Power[Pattern[x, Blank[]], 3]]], Rational[-1, 2]], Power[Plus[Pattern[c, Blank[]], Times[Optional[Pattern[d, Blank[]]], Power[Pattern[x, Blank[]], 3]]], -1]], Pattern[x, Blank[Symbol]]] := With[{q = Rt[b/a, 3], r = Simplify[(b*c - 10*a*d)/(6*a*d)]}, -Simp[(q*(2 - r)*ArcTan[((1 - r)*Sqrt[a + b*x^3])/(Sqrt[2]*Rt[a, 2]*r^(3/2))])/(3*Sqrt[2]*Rt[a, 2]*d*r^(3/2)), x] + (-Simp[(q*(2 - r)*ArcTan[(Rt[a, 2]*Sqrt[r]*(1 + r)*(1 + q*x))/(Sqrt[2]*Sqrt[a + b*x^3])])/(2*Sqrt[2]*Rt[a, 2]*d*r^(3/2)), x] - Simp[(q*(2 - r)*ArcTanh[(Rt[a, 2]*Sqrt[r]*(1 + r - 2*q*x))/(Sqrt[2]*Sqrt[a + b*x^3])])/(3*Sqrt[2]*Rt[a, 2]*d*Sqrt[r]), x] - Simp[(q*(2 - r)*ArcTanh[(Rt[a, 2]*(1 - r)*Sqrt[r]*(1 + q*x))/(Sqrt[2]*Sqrt[a + b*x^3])])/(6*Sqrt[2]*Rt[a, 2]*d*Sqrt[r]), x])] /; FreeQ[{a, b, c, d}, x] && NeQ[b*c - a*d, 0] && EqQ[b^2*c^2 - 20*a*b*c*d - 8*a^2*d^2, 0] && PosQ[a]
  488. Int[Times[Pattern[x, Blank[]], Power[Plus[Pattern[a, Blank[]], Times[Optional[Pattern[b, Blank[]]], Power[Pattern[x, Blank[]], 3]]], Rational[-1, 2]], Power[Plus[Pattern[c, Blank[]], Times[Optional[Pattern[d, Blank[]]], Power[Pattern[x, Blank[]], 3]]], -1]], Pattern[x, Blank[Symbol]]] := With[{q = Rt[b/a, 3], r = Simplify[(b*c - 10*a*d)/(6*a*d)]}, Simp[(q*(2 - r)*ArcTanh[((1 - r)*Sqrt[a + b*x^3])/(Sqrt[2]*Rt[-a, 2]*r^(3/2))])/(3*Sqrt[2]*Rt[-a, 2]*d*r^(3/2)), x] + (-Simp[(q*(2 - r)*ArcTanh[(Rt[-a, 2]*Sqrt[r]*(1 + r)*(1 + q*x))/(Sqrt[2]*Sqrt[a + b*x^3])])/(2*Sqrt[2]*Rt[-a, 2]*d*r^(3/2)), x] - Simp[(q*(2 - r)*ArcTan[(Rt[-a, 2]*Sqrt[r]*(1 + r - 2*q*x))/(Sqrt[2]*Sqrt[a + b*x^3])])/(3*Sqrt[2]*Rt[-a, 2]*d*Sqrt[r]), x] - Simp[(q*(2 - r)*ArcTan[(Rt[-a, 2]*(1 - r)*Sqrt[r]*(1 + q*x))/(Sqrt[2]*Sqrt[a + b*x^3])])/(6*Sqrt[2]*Rt[-a, 2]*d*Sqrt[r]), x])] /; FreeQ[{a, b, c, d}, x] && NeQ[b*c - a*d, 0] && EqQ[b^2*c^2 - 20*a*b*c*d - 8*a^2*d^2, 0] && NegQ[a]
  489. Int[Times[Pattern[x, Blank[]], Power[Plus[Pattern[a, Blank[]], Times[Optional[Pattern[b, Blank[]]], Power[Pattern[x, Blank[]], 3]]], Rational[1, 2]], Power[Plus[Pattern[c, Blank[]], Times[Optional[Pattern[d, Blank[]]], Power[Pattern[x, Blank[]], 3]]], -1]], Pattern[x, Blank[Symbol]]] := Dist[b/d, Int[x/Sqrt[a + b*x^3], x], x] - Dist[(b*c - a*d)/d, Int[x/((c + d*x^3)*Sqrt[a + b*x^3]), x], x] /; FreeQ[{c, d, a, b}, x] && NeQ[b*c - a*d, 0] && (EqQ[b*c - 4*a*d, 0] || EqQ[b*c + 8*a*d, 0] || EqQ[b^2*c^2 - 20*a*b*c*d - 8*a^2*d^2, 0])
  490. Int[Times[Power[Pattern[x, Blank[]], 2], Power[Plus[Pattern[a, Blank[]], Times[Optional[Pattern[b, Blank[]]], Power[Pattern[x, Blank[]], 4]]], -1], Power[Plus[Pattern[c, Blank[]], Times[Optional[Pattern[d, Blank[]]], Power[Pattern[x, Blank[]], 4]]], Rational[-1, 2]]], Pattern[x, Blank[Symbol]]] := With[{r = Numerator[Rt[-(a/b), 2]], s = Denominator[Rt[-(a/b), 2]]}, Dist[s/(2*b), Int[1/((r + s*x^2)*Sqrt[c + d*x^4]), x], x] - Dist[s/(2*b), Int[1/((r - s*x^2)*Sqrt[c + d*x^4]), x], x]] /; FreeQ[{a, b, c, d}, x] && NeQ[b*c - a*d, 0]
  491. Int[Times[Power[Pattern[x, Blank[]], 2], Power[Plus[Pattern[a, Blank[]], Times[Optional[Pattern[b, Blank[]]], Power[Pattern[x, Blank[]], 4]]], -1], Power[Plus[Pattern[c, Blank[]], Times[Optional[Pattern[d, Blank[]]], Power[Pattern[x, Blank[]], 4]]], Rational[1, 2]]], Pattern[x, Blank[Symbol]]] := Dist[d/b, Int[x^2/Sqrt[c + d*x^4], x], x] + Dist[(b*c - a*d)/b, Int[x^2/((a + b*x^4)*Sqrt[c + d*x^4]), x], x] /; FreeQ[{a, b, c, d}, x] && NeQ[b*c - a*d, 0]
  492. Int[Times[Power[Pattern[x, Blank[]], 2], Power[Plus[Pattern[a, Blank[]], Times[Optional[Pattern[b, Blank[]]], Power[Pattern[x, Blank[]], 2]]], Rational[-1, 2]], Power[Plus[Pattern[c, Blank[]], Times[Optional[Pattern[d, Blank[]]], Power[Pattern[x, Blank[]], 2]]], Rational[-1, 2]]], Pattern[x, Blank[Symbol]]] := Simp[(x*Sqrt[a + b*x^2])/(b*Sqrt[c + d*x^2]), x] - Dist[c/b, Int[Sqrt[a + b*x^2]/(c + d*x^2)^(3/2), x], x] /; FreeQ[{a, b, c, d}, x] && NeQ[b*c - a*d, 0] && PosQ[b/a] && PosQ[d/c] && !SimplerSqrtQ[b/a, d/c]
  493. Int[Times[Power[Pattern[x, Blank[]], Pattern[n, Blank[]]], Power[Plus[Pattern[a, Blank[]], Times[Optional[Pattern[b, Blank[]]], Power[Pattern[x, Blank[]], Pattern[n, Blank[]]]]], Rational[-1, 2]], Power[Plus[Pattern[c, Blank[]], Times[Optional[Pattern[d, Blank[]]], Power[Pattern[x, Blank[]], Pattern[n, Blank[]]]]], Rational[-1, 2]]], Pattern[x, Blank[Symbol]]] := Dist[1/b, Int[Sqrt[a + b*x^n]/Sqrt[c + d*x^n], x], x] - Dist[a/b, Int[1/(Sqrt[a + b*x^n]*Sqrt[c + d*x^n]), x], x] /; FreeQ[{a, b, c, d}, x] && NeQ[b*c - a*d, 0] && (EqQ[n, 2] || EqQ[n, 4]) && !(EqQ[n, 2] && SimplerSqrtQ[-(b/a), -(d/c)])
  494. Int[Times[Power[Pattern[x, Blank[]], Optional[Pattern[m, Blank[]]]], Power[Plus[Pattern[a, Blank[]], Times[Optional[Pattern[b, Blank[]]], Power[Pattern[x, Blank[]], Pattern[n, Blank[]]]]], Pattern[p, Blank[]]], Power[Plus[Pattern[c, Blank[]], Times[Optional[Pattern[d, Blank[]]], Power[Pattern[x, Blank[]], Pattern[n, Blank[]]]]], Optional[Pattern[q, Blank[]]]]], Pattern[x, Blank[Symbol]]] := With[{k = Denominator[p]}, Dist[(k*a^(p + (m + 1)/n))/n, Subst[Int[(x^((k*(m + 1))/n - 1)*(c - (b*c - a*d)*x^k)^q)/(1 - b*x^k)^(p + q + (m + 1)/n + 1), x], x, x^(n/k)/(a + b*x^n)^(1/k)], x]] /; FreeQ[{a, b, c, d}, x] && IGtQ[n, 0] && RationalQ[m, p] && IntegersQ[p + (m + 1)/n, q] && LtQ[-1, p, 0]
  495. Int[Times[Power[Pattern[x, Blank[]], Optional[Pattern[m, Blank[]]]], Power[Plus[Pattern[a, Blank[]], Times[Optional[Pattern[b, Blank[]]], Power[Pattern[x, Blank[]], Pattern[n, Blank[]]]]], Pattern[p, Blank[]]], Power[Plus[Pattern[c, Blank[]], Times[Optional[Pattern[d, Blank[]]], Power[Pattern[x, Blank[]], Pattern[n, Blank[]]]]], Pattern[q, Blank[]]]], Pattern[x, Blank[Symbol]]] := -Subst[Int[((a + b/x^n)^p*(c + d/x^n)^q)/x^(m + 2), x], x, 1/x] /; FreeQ[{a, b, c, d, p, q}, x] && NeQ[b*c - a*d, 0] && ILtQ[n, 0] && IntegerQ[m]
  496. Int[Times[Power[Times[Optional[Pattern[e, Blank[]]], Pattern[x, Blank[]]], Pattern[m, Blank[]]], Power[Plus[Pattern[a, Blank[]], Times[Optional[Pattern[b, Blank[]]], Power[Pattern[x, Blank[]], Pattern[n, Blank[]]]]], Pattern[p, Blank[]]], Power[Plus[Pattern[c, Blank[]], Times[Optional[Pattern[d, Blank[]]], Power[Pattern[x, Blank[]], Pattern[n, Blank[]]]]], Pattern[q, Blank[]]]], Pattern[x, Blank[Symbol]]] := With[{g = Denominator[m]}, -Dist[g/e, Subst[Int[((a + b/(e^n*x^(g*n)))^p*(c + d/(e^n*x^(g*n)))^q)/x^(g*(m + 1) + 1), x], x, 1/(e*x)^(1/g)], x]] /; FreeQ[{a, b, c, d, e, p, q}, x] && ILtQ[n, 0] && FractionQ[m]
  497. Int[Times[Power[Times[Optional[Pattern[e, Blank[]]], Pattern[x, Blank[]]], Pattern[m, Blank[]]], Power[Plus[Pattern[a, Blank[]], Times[Optional[Pattern[b, Blank[]]], Power[Pattern[x, Blank[]], Pattern[n, Blank[]]]]], Pattern[p, Blank[]]], Power[Plus[Pattern[c, Blank[]], Times[Optional[Pattern[d, Blank[]]], Power[Pattern[x, Blank[]], Pattern[n, Blank[]]]]], Pattern[q, Blank[]]]], Pattern[x, Blank[Symbol]]] := -Dist[(e*x)^m*(x^(-1))^m, Subst[Int[((a + b/x^n)^p*(c + d/x^n)^q)/x^(m + 2), x], x, 1/x], x] /; FreeQ[{a, b, c, d, e, m, p, q}, x] && NeQ[b*c - a*d, 0] && ILtQ[n, 0] && !RationalQ[m]
  498. Int[Times[Power[Pattern[x, Blank[]], Optional[Pattern[m, Blank[]]]], Power[Plus[Pattern[a, Blank[]], Times[Optional[Pattern[b, Blank[]]], Power[Pattern[x, Blank[]], Pattern[n, Blank[]]]]], Pattern[p, Blank[]]], Power[Plus[Pattern[c, Blank[]], Times[Optional[Pattern[d, Blank[]]], Power[Pattern[x, Blank[]], Pattern[n, Blank[]]]]], Pattern[q, Blank[]]]], Pattern[x, Blank[Symbol]]] := With[{g = Denominator[n]}, Dist[g, Subst[Int[x^(g*(m + 1) - 1)*(a + b*x^(g*n))^p*(c + d*x^(g*n))^q, x], x, x^(1/g)], x]] /; FreeQ[{a, b, c, d, m, p, q}, x] && NeQ[b*c - a*d, 0] && FractionQ[n]
  499. Int[Times[Power[Times[Pattern[e, Blank[]], Pattern[x, Blank[]]], Pattern[m, Blank[]]], Power[Plus[Pattern[a, Blank[]], Times[Optional[Pattern[b, Blank[]]], Power[Pattern[x, Blank[]], Pattern[n, Blank[]]]]], Pattern[p, Blank[]]], Power[Plus[Pattern[c, Blank[]], Times[Optional[Pattern[d, Blank[]]], Power[Pattern[x, Blank[]], Pattern[n, Blank[]]]]], Pattern[q, Blank[]]]], Pattern[x, Blank[Symbol]]] := Dist[(e^IntPart[m]*(e*x)^FracPart[m])/x^FracPart[m], Int[x^m*(a + b*x^n)^p*(c + d*x^n)^q, x], x] /; FreeQ[{a, b, c, d, e, m, p, q}, x] && NeQ[b*c - a*d, 0] && FractionQ[n]
  500. Int[Times[Power[Pattern[x, Blank[]], Optional[Pattern[m, Blank[]]]], Power[Plus[Pattern[a, Blank[]], Times[Optional[Pattern[b, Blank[]]], Power[Pattern[x, Blank[]], Pattern[n, Blank[]]]]], Pattern[p, Blank[]]], Power[Plus[Pattern[c, Blank[]], Times[Optional[Pattern[d, Blank[]]], Power[Pattern[x, Blank[]], Pattern[n, Blank[]]]]], Pattern[q, Blank[]]]], Pattern[x, Blank[Symbol]]] := Dist[1/(m + 1), Subst[Int[(a + b*x^Simplify[n/(m + 1)])^p*(c + d*x^Simplify[n/(m + 1)])^q, x], x, x^(m + 1)], x] /; FreeQ[{a, b, c, d, m, n, p, q}, x] && NeQ[b*c - a*d, 0] && IntegerQ[Simplify[n/(m + 1)]] && !IntegerQ[n]
  501. Int[Times[Power[Times[Pattern[e, Blank[]], Pattern[x, Blank[]]], Optional[Pattern[m, Blank[]]]], Power[Plus[Pattern[a, Blank[]], Times[Optional[Pattern[b, Blank[]]], Power[Pattern[x, Blank[]], Pattern[n, Blank[]]]]], Pattern[p, Blank[]]], Power[Plus[Pattern[c, Blank[]], Times[Optional[Pattern[d, Blank[]]], Power[Pattern[x, Blank[]], Pattern[n, Blank[]]]]], Pattern[q, Blank[]]]], Pattern[x, Blank[Symbol]]] := Dist[(e^IntPart[m]*(e*x)^FracPart[m])/x^FracPart[m], Int[x^m*(a + b*x^n)^p*(c + d*x^n)^q, x], x] /; FreeQ[{a, b, c, d, e, m, n, p, q}, x] && NeQ[b*c - a*d, 0] && IntegerQ[Simplify[n/(m + 1)]] && !IntegerQ[n]
  502. Int[Times[Power[Times[Optional[Pattern[e, Blank[]]], Pattern[x, Blank[]]], Optional[Pattern[m, Blank[]]]], Power[Plus[Pattern[a, Blank[]], Times[Optional[Pattern[b, Blank[]]], Power[Pattern[x, Blank[]], Pattern[n, Blank[]]]]], Pattern[p, Blank[]]], Power[Plus[Pattern[c, Blank[]], Times[Optional[Pattern[d, Blank[]]], Power[Pattern[x, Blank[]], Pattern[n, Blank[]]]]], Pattern[q, Blank[]]]], Pattern[x, Blank[Symbol]]] := -Simp[((c*b - a*d)*(e*x)^(m + 1)*(a + b*x^n)^(p + 1)*(c + d*x^n)^(q - 1))/(a*b*e*n*(p + 1)), x] + Dist[1/(a*b*n*(p + 1)), Int[(e*x)^m*(a + b*x^n)^(p + 1)*(c + d*x^n)^(q - 2)*Simp[c*(c*b*n*(p + 1) + (c*b - a*d)*(m + 1)) + d*(c*b*n*(p + 1) + (c*b - a*d)*(m + n*(q - 1) + 1))*x^n, x], x], x] /; FreeQ[{a, b, c, d, e, m, n}, x] && NeQ[b*c - a*d, 0] && LtQ[p, -1] && GtQ[q, 1] && IntBinomialQ[a, b, c, d, e, m, n, p, q, x]
  503. Int[Times[Power[Times[Optional[Pattern[e, Blank[]]], Pattern[x, Blank[]]], Optional[Pattern[m, Blank[]]]], Power[Plus[Pattern[a, Blank[]], Times[Optional[Pattern[b, Blank[]]], Power[Pattern[x, Blank[]], Pattern[n, Blank[]]]]], Pattern[p, Blank[]]], Power[Plus[Pattern[c, Blank[]], Times[Optional[Pattern[d, Blank[]]], Power[Pattern[x, Blank[]], Pattern[n, Blank[]]]]], Pattern[q, Blank[]]]], Pattern[x, Blank[Symbol]]] := -Simp[((e*x)^(m + 1)*(a + b*x^n)^(p + 1)*(c + d*x^n)^q)/(a*e*n*(p + 1)), x] + Dist[1/(a*n*(p + 1)), Int[(e*x)^m*(a + b*x^n)^(p + 1)*(c + d*x^n)^(q - 1)*Simp[c*(m + n*(p + 1) + 1) + d*(m + n*(p + q + 1) + 1)*x^n, x], x], x] /; FreeQ[{a, b, c, d, e, m, n}, x] && NeQ[b*c - a*d, 0] && LtQ[p, -1] && LtQ[0, q, 1] && IntBinomialQ[a, b, c, d, e, m, n, p, q, x]
  504. Int[Times[Power[Times[Optional[Pattern[e, Blank[]]], Pattern[x, Blank[]]], Optional[Pattern[m, Blank[]]]], Power[Plus[Pattern[a, Blank[]], Times[Optional[Pattern[b, Blank[]]], Power[Pattern[x, Blank[]], Pattern[n, Blank[]]]]], Pattern[p, Blank[]]], Power[Plus[Pattern[c, Blank[]], Times[Optional[Pattern[d, Blank[]]], Power[Pattern[x, Blank[]], Pattern[n, Blank[]]]]], Pattern[q, Blank[]]]], Pattern[x, Blank[Symbol]]] := -Simp[(b*(e*x)^(m + 1)*(a + b*x^n)^(p + 1)*(c + d*x^n)^(q + 1))/(a*e*n*(b*c - a*d)*(p + 1)), x] + Dist[1/(a*n*(b*c - a*d)*(p + 1)), Int[(e*x)^m*(a + b*x^n)^(p + 1)*(c + d*x^n)^q*Simp[c*b*(m + 1) + n*(b*c - a*d)*(p + 1) + d*b*(m + n*(p + q + 2) + 1)*x^n, x], x], x] /; FreeQ[{a, b, c, d, e, m, n, q}, x] && NeQ[b*c - a*d, 0] && LtQ[p, -1] && IntBinomialQ[a, b, c, d, e, m, n, p, q, x]
  505. Int[Times[Power[Times[Optional[Pattern[e, Blank[]]], Pattern[x, Blank[]]], Optional[Pattern[m, Blank[]]]], Power[Plus[Pattern[a, Blank[]], Times[Optional[Pattern[b, Blank[]]], Power[Pattern[x, Blank[]], Pattern[n, Blank[]]]]], Pattern[p, Blank[]]], Power[Plus[Pattern[c, Blank[]], Times[Optional[Pattern[d, Blank[]]], Power[Pattern[x, Blank[]], Pattern[n, Blank[]]]]], Pattern[q, Blank[]]]], Pattern[x, Blank[Symbol]]] := Simp[((e*x)^(m + 1)*(a + b*x^n)^p*(c + d*x^n)^q)/(e*(m + n*(p + q) + 1)), x] + Dist[n/(m + n*(p + q) + 1), Int[(e*x)^m*(a + b*x^n)^(p - 1)*(c + d*x^n)^(q - 1)*Simp[a*c*(p + q) + (q*(b*c - a*d) + a*d*(p + q))*x^n, x], x], x] /; FreeQ[{a, b, c, d, e, m, n}, x] && NeQ[b*c - a*d, 0] && GtQ[q, 0] && GtQ[p, 0] && IntBinomialQ[a, b, c, d, e, m, n, p, q, x]
  506. Int[Times[Power[Times[Optional[Pattern[e, Blank[]]], Pattern[x, Blank[]]], Optional[Pattern[m, Blank[]]]], Power[Plus[Pattern[a, Blank[]], Times[Optional[Pattern[b, Blank[]]], Power[Pattern[x, Blank[]], Pattern[n, Blank[]]]]], Pattern[p, Blank[]]], Power[Plus[Pattern[c, Blank[]], Times[Optional[Pattern[d, Blank[]]], Power[Pattern[x, Blank[]], Pattern[n, Blank[]]]]], Pattern[q, Blank[]]]], Pattern[x, Blank[Symbol]]] := Simp[(d*(e*x)^(m + 1)*(a + b*x^n)^(p + 1)*(c + d*x^n)^(q - 1))/(b*e*(m + n*(p + q) + 1)), x] + Dist[1/(b*(m + n*(p + q) + 1)), Int[(e*x)^m*(a + b*x^n)^p*(c + d*x^n)^(q - 2)*Simp[c*((c*b - a*d)*(m + 1) + c*b*n*(p + q)) + (d*(c*b - a*d)*(m + 1) + d*n*(q - 1)*(b*c - a*d) + c*b*d*n*(p + q))*x^n, x], x], x] /; FreeQ[{a, b, c, d, e, m, n, p}, x] && NeQ[b*c - a*d, 0] && GtQ[q, 1] && IntBinomialQ[a, b, c, d, e, m, n, p, q, x]
  507. Int[Times[Power[Pattern[x, Blank[]], Pattern[m, Blank[]]], Power[Plus[Pattern[a, Blank[]], Times[Optional[Pattern[b, Blank[]]], Power[Pattern[x, Blank[]], Pattern[n, Blank[]]]]], -1], Power[Plus[Pattern[c, Blank[]], Times[Optional[Pattern[d, Blank[]]], Power[Pattern[x, Blank[]], Pattern[n, Blank[]]]]], -1]], Pattern[x, Blank[Symbol]]] := -Dist[a/(b*c - a*d), Int[x^(m - n)/(a + b*x^n), x], x] + Dist[c/(b*c - a*d), Int[x^(m - n)/(c + d*x^n), x], x] /; FreeQ[{a, b, c, d, m, n}, x] && NeQ[b*c - a*d, 0] && (EqQ[m, n] || EqQ[m, 2*n - 1])
  508. Int[Times[Power[Times[Optional[Pattern[e, Blank[]]], Pattern[x, Blank[]]], Optional[Pattern[m, Blank[]]]], Power[Plus[Pattern[a, Blank[]], Times[Optional[Pattern[b, Blank[]]], Power[Pattern[x, Blank[]], Pattern[n, Blank[]]]]], -1], Power[Plus[Pattern[c, Blank[]], Times[Optional[Pattern[d, Blank[]]], Power[Pattern[x, Blank[]], Pattern[n, Blank[]]]]], -1]], Pattern[x, Blank[Symbol]]] := Dist[b/(b*c - a*d), Int[(e*x)^m/(a + b*x^n), x], x] - Dist[d/(b*c - a*d), Int[(e*x)^m/(c + d*x^n), x], x] /; FreeQ[{a, b, c, d, e, n, m}, x] && NeQ[b*c - a*d, 0]
  509. Int[Times[Power[Times[Optional[Pattern[e, Blank[]]], Pattern[x, Blank[]]], Optional[Pattern[m, Blank[]]]], Power[Plus[Pattern[a, Blank[]], Times[Optional[Pattern[b, Blank[]]], Power[Pattern[x, Blank[]], Pattern[n, Blank[]]]]], Pattern[p, Blank[]]], Power[Plus[Pattern[c, Blank[]], Times[Optional[Pattern[d, Blank[]]], Power[Pattern[x, Blank[]], Pattern[n, Blank[]]]]], Pattern[q, Blank[]]]], Pattern[x, Blank[Symbol]]] := Int[ExpandIntegrand[(e*x)^m*(a + b*x^n)^p*(c + d*x^n)^q, x], x] /; FreeQ[{a, b, c, d, e, m}, x] && NeQ[b*c - a*d, 0] && IGtQ[p, -2] && (IGtQ[q, -2] || (EqQ[q, -3] && IntegerQ[(m - 1)/2]))
  510. Int[Times[Power[Times[Optional[Pattern[e, Blank[]]], Pattern[x, Blank[]]], Optional[Pattern[m, Blank[]]]], Power[Plus[Pattern[a, Blank[]], Times[Optional[Pattern[b, Blank[]]], Power[Pattern[x, Blank[]], Pattern[n, Blank[]]]]], Pattern[p, Blank[]]], Power[Plus[Pattern[c, Blank[]], Times[Optional[Pattern[d, Blank[]]], Power[Pattern[x, Blank[]], Pattern[n, Blank[]]]]], Pattern[q, Blank[]]]], Pattern[x, Blank[Symbol]]] := Simp[(a^p*c^q*(e*x)^(m + 1)*AppellF1[(m + 1)/n, -p, -q, 1 + (m + 1)/n, -((b*x^n)/a), -((d*x^n)/c)])/(e*(m + 1)), x] /; FreeQ[{a, b, c, d, e, m, n, p, q}, x] && NeQ[b*c - a*d, 0] && NeQ[m, -1] && NeQ[m, n - 1] && (IntegerQ[p] || GtQ[a, 0]) && (IntegerQ[q] || GtQ[c, 0])
  511. Int[Times[Power[Times[Optional[Pattern[e, Blank[]]], Pattern[x, Blank[]]], Optional[Pattern[m, Blank[]]]], Power[Plus[Pattern[a, Blank[]], Times[Optional[Pattern[b, Blank[]]], Power[Pattern[x, Blank[]], Pattern[n, Blank[]]]]], Pattern[p, Blank[]]], Power[Plus[Pattern[c, Blank[]], Times[Optional[Pattern[d, Blank[]]], Power[Pattern[x, Blank[]], Pattern[n, Blank[]]]]], Pattern[q, Blank[]]]], Pattern[x, Blank[Symbol]]] := Dist[(a^IntPart[p]*(a + b*x^n)^FracPart[p])/(1 + (b*x^n)/a)^FracPart[p], Int[(e*x)^m*(1 + (b*x^n)/a)^p*(c + d*x^n)^q, x], x] /; FreeQ[{a, b, c, d, e, m, n, p, q}, x] && NeQ[b*c - a*d, 0] && NeQ[m, -1] && NeQ[m, n - 1] && !(IntegerQ[p] || GtQ[a, 0])
  512. Int[Times[Power[Plus[Optional[Pattern[a, Blank[]]], Times[Optional[Pattern[b, Blank[]]], Power[Pattern[v, Blank[]], Pattern[n, Blank[]]]]], Optional[Pattern[p, Blank[]]]], Power[Plus[Optional[Pattern[c, Blank[]]], Times[Optional[Pattern[d, Blank[]]], Power[Pattern[v, Blank[]], Pattern[n, Blank[]]]]], Optional[Pattern[q, Blank[]]]], Power[Pattern[x, Blank[]], Optional[Pattern[m, Blank[]]]]], Pattern[x, Blank[Symbol]]] := Dist[1/Coefficient[v, x, 1]^(m + 1), Subst[Int[SimplifyIntegrand[(x - Coefficient[v, x, 0])^m*(a + b*x^n)^p*(c + d*x^n)^q, x], x], x, v], x] /; FreeQ[{a, b, c, d, n, p, q}, x] && LinearQ[v, x] && IntegerQ[m] && NeQ[v, x]
  513. Int[Times[Power[Pattern[u, Blank[]], Optional[Pattern[m, Blank[]]]], Power[Plus[Optional[Pattern[a, Blank[]]], Times[Optional[Pattern[b, Blank[]]], Power[Pattern[v, Blank[]], Pattern[n, Blank[]]]]], Optional[Pattern[p, Blank[]]]], Power[Plus[Optional[Pattern[c, Blank[]]], Times[Optional[Pattern[d, Blank[]]], Power[Pattern[v, Blank[]], Pattern[n, Blank[]]]]], Optional[Pattern[q, Blank[]]]]], Pattern[x, Blank[Symbol]]] := Dist[u^m/(Coefficient[v, x, 1]*v^m), Subst[Int[x^m*(a + b*x^n)^p*(c + d*x^n)^q, x], x, v], x] /; FreeQ[{a, b, c, d, m, n, p, q}, x] && LinearPairQ[u, v, x]
  514. Int[Times[Power[Pattern[x, Blank[]], Optional[Pattern[m, Blank[]]]], Power[Plus[Pattern[c, Blank[]], Times[Optional[Pattern[d, Blank[]]], Power[Pattern[x, Blank[]], Optional[Pattern[mn, Blank[]]]]]], Optional[Pattern[q, Blank[]]]], Power[Plus[Pattern[a, Blank[]], Times[Optional[Pattern[b, Blank[]]], Power[Pattern[x, Blank[]], Optional[Pattern[n, Blank[]]]]]], Optional[Pattern[p, Blank[]]]]], Pattern[x, Blank[Symbol]]] := Int[x^(m - n*q)*(a + b*x^n)^p*(d + c*x^n)^q, x] /; FreeQ[{a, b, c, d, m, n, p}, x] && EqQ[mn, -n] && IntegerQ[q] && (PosQ[n] || !IntegerQ[p])
  515. Int[Times[Power[Pattern[x, Blank[]], Optional[Pattern[m, Blank[]]]], Power[Plus[Pattern[c, Blank[]], Times[Optional[Pattern[d, Blank[]]], Power[Pattern[x, Blank[]], Optional[Pattern[mn, Blank[]]]]]], Pattern[q, Blank[]]], Power[Plus[Pattern[a, Blank[]], Times[Optional[Pattern[b, Blank[]]], Power[Pattern[x, Blank[]], Optional[Pattern[n, Blank[]]]]]], Optional[Pattern[p, Blank[]]]]], Pattern[x, Blank[Symbol]]] := Dist[(x^(n*FracPart[q])*(c + d/x^n)^FracPart[q])/(d + c*x^n)^FracPart[q], Int[x^(m - n*q)*(a + b*x^n)^p*(d + c*x^n)^q, x], x] /; FreeQ[{a, b, c, d, m, n, p, q}, x] && EqQ[mn, -n] && !IntegerQ[q] && !IntegerQ[p]
  516. Int[Times[Power[Times[Pattern[e, Blank[]], Pattern[x, Blank[]]], Pattern[m, Blank[]]], Power[Plus[Pattern[c, Blank[]], Times[Optional[Pattern[d, Blank[]]], Power[Pattern[x, Blank[]], Optional[Pattern[mn, Blank[]]]]]], Optional[Pattern[q, Blank[]]]], Power[Plus[Optional[Pattern[a, Blank[]]], Times[Optional[Pattern[b, Blank[]]], Power[Pattern[x, Blank[]], Optional[Pattern[n, Blank[]]]]]], Optional[Pattern[p, Blank[]]]]], Pattern[x, Blank[Symbol]]] := Dist[(e^IntPart[m]*(e*x)^FracPart[m])/x^FracPart[m], Int[x^m*(a + b*x^n)^p*(c + d/x^n)^q, x], x] /; FreeQ[{a, b, c, d, e, m, n, p, q}, x] && EqQ[mn, -n]
  517. Int[Times[Optional[Pattern[u, Blank[]]], Power[Plus[Pattern[c, Blank[]], Times[Optional[Pattern[d, Blank[]]], Power[Pattern[x, Blank[]], Optional[Pattern[n, Blank[]]]]]], Optional[Pattern[q, Blank[]]]], Power[Plus[Pattern[a1, Blank[]], Times[Optional[Pattern[b1, Blank[]]], Power[Pattern[x, Blank[]], Optional[Pattern[non2, Blank[]]]]]], Optional[Pattern[p, Blank[]]]], Power[Plus[Pattern[a2, Blank[]], Times[Optional[Pattern[b2, Blank[]]], Power[Pattern[x, Blank[]], Optional[Pattern[non2, Blank[]]]]]], Optional[Pattern[p, Blank[]]]]], Pattern[x, Blank[Symbol]]] := Int[u*(a1*a2 + b1*b2*x^n)^p*(c + d*x^n)^q, x] /; FreeQ[{a1, b1, a2, b2, c, d, n, p, q}, x] && EqQ[non2, n/2] && EqQ[a2*b1 + a1*b2, 0] && (IntegerQ[p] || (GtQ[a1, 0] && GtQ[a2, 0]))
  518. Int[Times[Optional[Pattern[u, Blank[]]], Power[Plus[Pattern[c, Blank[]], Times[Optional[Pattern[d, Blank[]]], Power[Pattern[x, Blank[]], Optional[Pattern[n, Blank[]]]]], Times[Optional[Pattern[e, Blank[]]], Power[Pattern[x, Blank[]], Optional[Pattern[n2, Blank[]]]]]], Optional[Pattern[q, Blank[]]]], Power[Plus[Pattern[a1, Blank[]], Times[Optional[Pattern[b1, Blank[]]], Power[Pattern[x, Blank[]], Optional[Pattern[non2, Blank[]]]]]], Optional[Pattern[p, Blank[]]]], Power[Plus[Pattern[a2, Blank[]], Times[Optional[Pattern[b2, Blank[]]], Power[Pattern[x, Blank[]], Optional[Pattern[non2, Blank[]]]]]], Optional[Pattern[p, Blank[]]]]], Pattern[x, Blank[Symbol]]] := Int[u*(a1*a2 + b1*b2*x^n)^p*(c + d*x^n + e*x^(2*n))^q, x] /; FreeQ[{a1, b1, a2, b2, c, d, e, n, p, q}, x] && EqQ[non2, n/2] && EqQ[n2, 2*n] && EqQ[a2*b1 + a1*b2, 0] && (IntegerQ[p] || (GtQ[a1, 0] && GtQ[a2, 0]))
  519. Int[Times[Optional[Pattern[u, Blank[]]], Power[Plus[Pattern[c, Blank[]], Times[Optional[Pattern[d, Blank[]]], Power[Pattern[x, Blank[]], Optional[Pattern[n, Blank[]]]]]], Optional[Pattern[q, Blank[]]]], Power[Plus[Pattern[a1, Blank[]], Times[Optional[Pattern[b1, Blank[]]], Power[Pattern[x, Blank[]], Optional[Pattern[non2, Blank[]]]]]], Pattern[p, Blank[]]], Power[Plus[Pattern[a2, Blank[]], Times[Optional[Pattern[b2, Blank[]]], Power[Pattern[x, Blank[]], Optional[Pattern[non2, Blank[]]]]]], Pattern[p, Blank[]]]], Pattern[x, Blank[Symbol]]] := Dist[((a1 + b1*x^(n/2))^FracPart[p]*(a2 + b2*x^(n/2))^FracPart[p])/(a1*a2 + b1*b2*x^n)^FracPart[p], Int[u*(a1*a2 + b1*b2*x^n)^p*(c + d*x^n)^q, x], x] /; FreeQ[{a1, b1, a2, b2, c, d, n, p, q}, x] && EqQ[non2, n/2] && EqQ[a2*b1 + a1*b2, 0] && !(EqQ[n, 2] && IGtQ[q, 0])
  520. Int[Times[Optional[Pattern[u, Blank[]]], Power[Plus[Pattern[c, Blank[]], Times[Optional[Pattern[d, Blank[]]], Power[Pattern[x, Blank[]], Optional[Pattern[n, Blank[]]]]], Times[Optional[Pattern[e, Blank[]]], Power[Pattern[x, Blank[]], Optional[Pattern[n2, Blank[]]]]]], Optional[Pattern[q, Blank[]]]], Power[Plus[Pattern[a1, Blank[]], Times[Optional[Pattern[b1, Blank[]]], Power[Pattern[x, Blank[]], Optional[Pattern[non2, Blank[]]]]]], Optional[Pattern[p, Blank[]]]], Power[Plus[Pattern[a2, Blank[]], Times[Optional[Pattern[b2, Blank[]]], Power[Pattern[x, Blank[]], Optional[Pattern[non2, Blank[]]]]]], Optional[Pattern[p, Blank[]]]]], Pattern[x, Blank[Symbol]]] := Dist[((a1 + b1*x^(n/2))^FracPart[p]*(a2 + b2*x^(n/2))^FracPart[p])/(a1*a2 + b1*b2*x^n)^FracPart[p], Int[u*(a1*a2 + b1*b2*x^n)^p*(c + d*x^n + e*x^(2*n))^q, x], x] /; FreeQ[{a1, b1, a2, b2, c, d, e, n, p, q}, x] && EqQ[non2, n/2] && EqQ[n2, 2*n] && EqQ[a2*b1 + a1*b2, 0]
  521. Int[Times[Power[Plus[Pattern[a, Blank[]], Times[Optional[Pattern[b, Blank[]]], Power[Pattern[x, Blank[]], Pattern[n, Blank[]]]]], Optional[Pattern[p, Blank[]]]], Power[Plus[Pattern[c, Blank[]], Times[Optional[Pattern[d, Blank[]]], Power[Pattern[x, Blank[]], Pattern[n, Blank[]]]]], Optional[Pattern[q, Blank[]]]], Power[Plus[Pattern[e, Blank[]], Times[Optional[Pattern[f, Blank[]]], Power[Pattern[x, Blank[]], Pattern[n, Blank[]]]]], Optional[Pattern[r, Blank[]]]]], Pattern[x, Blank[Symbol]]] := Int[ExpandIntegrand[(a + b*x^n)^p*(c + d*x^n)^q*(e + f*x^n)^r, x], x] /; FreeQ[{a, b, c, d, e, f, n}, x] && IGtQ[p, 0] && IGtQ[q, 0] && IGtQ[r, 0]
  522. Int[Times[Power[Plus[Pattern[a, Blank[]], Times[Optional[Pattern[b, Blank[]]], Power[Pattern[x, Blank[]], Pattern[n, Blank[]]]]], -1], Power[Plus[Pattern[c, Blank[]], Times[Optional[Pattern[d, Blank[]]], Power[Pattern[x, Blank[]], Pattern[n, Blank[]]]]], -1], Plus[Pattern[e, Blank[]], Times[Optional[Pattern[f, Blank[]]], Power[Pattern[x, Blank[]], Pattern[n, Blank[]]]]]], Pattern[x, Blank[Symbol]]] := Dist[(b*e - a*f)/(b*c - a*d), Int[1/(a + b*x^n), x], x] - Dist[(d*e - c*f)/(b*c - a*d), Int[1/(c + d*x^n), x], x] /; FreeQ[{a, b, c, d, e, f, n}, x]
  523. Int[Times[Power[Plus[Pattern[a, Blank[]], Times[Optional[Pattern[b, Blank[]]], Power[Pattern[x, Blank[]], Pattern[n, Blank[]]]]], -1], Power[Plus[Pattern[c, Blank[]], Times[Optional[Pattern[d, Blank[]]], Power[Pattern[x, Blank[]], Pattern[n, Blank[]]]]], Rational[-1, 2]], Plus[Pattern[e, Blank[]], Times[Optional[Pattern[f, Blank[]]], Power[Pattern[x, Blank[]], Pattern[n, Blank[]]]]]], Pattern[x, Blank[Symbol]]] := Dist[f/b, Int[1/Sqrt[c + d*x^n], x], x] + Dist[(b*e - a*f)/b, Int[1/((a + b*x^n)*Sqrt[c + d*x^n]), x], x] /; FreeQ[{a, b, c, d, e, f, n}, x]
  524. Int[Times[Power[Plus[Pattern[a, Blank[]], Times[Optional[Pattern[b, Blank[]]], Power[Pattern[x, Blank[]], Pattern[n, Blank[]]]]], Rational[-1, 2]], Power[Plus[Pattern[c, Blank[]], Times[Optional[Pattern[d, Blank[]]], Power[Pattern[x, Blank[]], Pattern[n, Blank[]]]]], Rational[-1, 2]], Plus[Pattern[e, Blank[]], Times[Optional[Pattern[f, Blank[]]], Power[Pattern[x, Blank[]], Pattern[n, Blank[]]]]]], Pattern[x, Blank[Symbol]]] := Dist[f/b, Int[Sqrt[a + b*x^n]/Sqrt[c + d*x^n], x], x] + Dist[(b*e - a*f)/b, Int[1/(Sqrt[a + b*x^n]*Sqrt[c + d*x^n]), x], x] /; FreeQ[{a, b, c, d, e, f, n}, x] && !(EqQ[n, 2] && ((PosQ[b/a] && PosQ[d/c]) || (NegQ[b/a] && (PosQ[d/c] || (GtQ[a, 0] && ( !GtQ[c, 0] || SimplerSqrtQ[-(b/a), -(d/c)]))))))
  525. Int[Times[Power[Plus[Pattern[a, Blank[]], Times[Optional[Pattern[b, Blank[]]], Power[Pattern[x, Blank[]], 2]]], Rational[-1, 2]], Power[Plus[Pattern[c, Blank[]], Times[Optional[Pattern[d, Blank[]]], Power[Pattern[x, Blank[]], 2]]], Rational[-3, 2]], Plus[Pattern[e, Blank[]], Times[Optional[Pattern[f, Blank[]]], Power[Pattern[x, Blank[]], 2]]]], Pattern[x, Blank[Symbol]]] := Dist[(b*e - a*f)/(b*c - a*d), Int[1/(Sqrt[a + b*x^2]*Sqrt[c + d*x^2]), x], x] - Dist[(d*e - c*f)/(b*c - a*d), Int[Sqrt[a + b*x^2]/(c + d*x^2)^(3/2), x], x] /; FreeQ[{a, b, c, d, e, f}, x] && PosQ[b/a] && PosQ[d/c]
  526. Int[Times[Power[Plus[Pattern[a, Blank[]], Times[Optional[Pattern[b, Blank[]]], Power[Pattern[x, Blank[]], Pattern[n, Blank[]]]]], Pattern[p, Blank[]]], Power[Plus[Pattern[c, Blank[]], Times[Optional[Pattern[d, Blank[]]], Power[Pattern[x, Blank[]], Pattern[n, Blank[]]]]], Optional[Pattern[q, Blank[]]]], Plus[Pattern[e, Blank[]], Times[Optional[Pattern[f, Blank[]]], Power[Pattern[x, Blank[]], Pattern[n, Blank[]]]]]], Pattern[x, Blank[Symbol]]] := -Simp[((b*e - a*f)*x*(a + b*x^n)^(p + 1)*(c + d*x^n)^q)/(a*b*n*(p + 1)), x] + Dist[1/(a*b*n*(p + 1)), Int[(a + b*x^n)^(p + 1)*(c + d*x^n)^(q - 1)*Simp[c*(b*e*n*(p + 1) + b*e - a*f) + d*(b*e*n*(p + 1) + (b*e - a*f)*(n*q + 1))*x^n, x], x], x] /; FreeQ[{a, b, c, d, e, f, n}, x] && LtQ[p, -1] && GtQ[q, 0]
  527. Int[Times[Power[Plus[Pattern[a, Blank[]], Times[Optional[Pattern[b, Blank[]]], Power[Pattern[x, Blank[]], Pattern[n, Blank[]]]]], Pattern[p, Blank[]]], Power[Plus[Pattern[c, Blank[]], Times[Optional[Pattern[d, Blank[]]], Power[Pattern[x, Blank[]], Pattern[n, Blank[]]]]], Optional[Pattern[q, Blank[]]]], Plus[Pattern[e, Blank[]], Times[Optional[Pattern[f, Blank[]]], Power[Pattern[x, Blank[]], Pattern[n, Blank[]]]]]], Pattern[x, Blank[Symbol]]] := -Simp[((b*e - a*f)*x*(a + b*x^n)^(p + 1)*(c + d*x^n)^(q + 1))/(a*n*(b*c - a*d)*(p + 1)), x] + Dist[1/(a*n*(b*c - a*d)*(p + 1)), Int[(a + b*x^n)^(p + 1)*(c + d*x^n)^q*Simp[c*(b*e - a*f) + e*n*(b*c - a*d)*(p + 1) + d*(b*e - a*f)*(n*(p + q + 2) + 1)*x^n, x], x], x] /; FreeQ[{a, b, c, d, e, f, n, q}, x] && LtQ[p, -1]
  528. Int[Times[Power[Plus[Pattern[a, Blank[]], Times[Optional[Pattern[b, Blank[]]], Power[Pattern[x, Blank[]], Pattern[n, Blank[]]]]], Optional[Pattern[p, Blank[]]]], Power[Plus[Pattern[c, Blank[]], Times[Optional[Pattern[d, Blank[]]], Power[Pattern[x, Blank[]], Pattern[n, Blank[]]]]], Optional[Pattern[q, Blank[]]]], Plus[Pattern[e, Blank[]], Times[Optional[Pattern[f, Blank[]]], Power[Pattern[x, Blank[]], Pattern[n, Blank[]]]]]], Pattern[x, Blank[Symbol]]] := Simp[(f*x*(a + b*x^n)^(p + 1)*(c + d*x^n)^q)/(b*(n*(p + q + 1) + 1)), x] + Dist[1/(b*(n*(p + q + 1) + 1)), Int[(a + b*x^n)^p*(c + d*x^n)^(q - 1)*Simp[c*(b*e - a*f + b*e*n*(p + q + 1)) + (d*(b*e - a*f) + f*n*q*(b*c - a*d) + b*d*e*n*(p + q + 1))*x^n, x], x], x] /; FreeQ[{a, b, c, d, e, f, n, p}, x] && GtQ[q, 0] && NeQ[n*(p + q + 1) + 1, 0]
  529. Int[Times[Power[Plus[Pattern[a, Blank[]], Times[Optional[Pattern[b, Blank[]]], Power[Pattern[x, Blank[]], 4]]], Rational[-3, 4]], Power[Plus[Pattern[c, Blank[]], Times[Optional[Pattern[d, Blank[]]], Power[Pattern[x, Blank[]], 4]]], -1], Plus[Pattern[e, Blank[]], Times[Optional[Pattern[f, Blank[]]], Power[Pattern[x, Blank[]], 4]]]], Pattern[x, Blank[Symbol]]] := Dist[(b*e - a*f)/(b*c - a*d), Int[1/(a + b*x^4)^(3/4), x], x] - Dist[(d*e - c*f)/(b*c - a*d), Int[(a + b*x^4)^(1/4)/(c + d*x^4), x], x] /; FreeQ[{a, b, c, d, e, f}, x]
  530. Int[Times[Power[Plus[Pattern[a, Blank[]], Times[Optional[Pattern[b, Blank[]]], Power[Pattern[x, Blank[]], Pattern[n, Blank[]]]]], Pattern[p, Blank[]]], Power[Plus[Pattern[c, Blank[]], Times[Optional[Pattern[d, Blank[]]], Power[Pattern[x, Blank[]], Pattern[n, Blank[]]]]], -1], Plus[Pattern[e, Blank[]], Times[Optional[Pattern[f, Blank[]]], Power[Pattern[x, Blank[]], Pattern[n, Blank[]]]]]], Pattern[x, Blank[Symbol]]] := Dist[f/d, Int[(a + b*x^n)^p, x], x] + Dist[(d*e - c*f)/d, Int[(a + b*x^n)^p/(c + d*x^n), x], x] /; FreeQ[{a, b, c, d, e, f, p, n}, x]
  531. Int[Times[Power[Plus[Pattern[a, Blank[]], Times[Optional[Pattern[b, Blank[]]], Power[Pattern[x, Blank[]], Pattern[n, Blank[]]]]], Optional[Pattern[p, Blank[]]]], Power[Plus[Pattern[c, Blank[]], Times[Optional[Pattern[d, Blank[]]], Power[Pattern[x, Blank[]], Pattern[n, Blank[]]]]], Optional[Pattern[q, Blank[]]]], Plus[Pattern[e, Blank[]], Times[Optional[Pattern[f, Blank[]]], Power[Pattern[x, Blank[]], Pattern[n, Blank[]]]]]], Pattern[x, Blank[Symbol]]] := Dist[e, Int[(a + b*x^n)^p*(c + d*x^n)^q, x], x] + Dist[f, Int[x^n*(a + b*x^n)^p*(c + d*x^n)^q, x], x] /; FreeQ[{a, b, c, d, e, f, n, p, q}, x]
  532. Int[Times[Power[Plus[Pattern[a, Blank[]], Times[Optional[Pattern[b, Blank[]]], Power[Pattern[x, Blank[]], 2]]], -1], Power[Plus[Pattern[c, Blank[]], Times[Optional[Pattern[d, Blank[]]], Power[Pattern[x, Blank[]], 2]]], -1], Power[Plus[Pattern[e, Blank[]], Times[Optional[Pattern[f, Blank[]]], Power[Pattern[x, Blank[]], 2]]], Rational[-1, 2]]], Pattern[x, Blank[Symbol]]] := Dist[b/(b*c - a*d), Int[1/((a + b*x^2)*Sqrt[e + f*x^2]), x], x] - Dist[d/(b*c - a*d), Int[1/((c + d*x^2)*Sqrt[e + f*x^2]), x], x] /; FreeQ[{a, b, c, d, e, f}, x]
  533. Int[Times[Power[Pattern[x, Blank[]], -2], Power[Plus[Pattern[c, Blank[]], Times[Optional[Pattern[d, Blank[]]], Power[Pattern[x, Blank[]], 2]]], -1], Power[Plus[Pattern[e, Blank[]], Times[Optional[Pattern[f, Blank[]]], Power[Pattern[x, Blank[]], 2]]], Rational[-1, 2]]], Pattern[x, Blank[Symbol]]] := Dist[1/c, Int[1/(x^2*Sqrt[e + f*x^2]), x], x] - Dist[d/c, Int[1/((c + d*x^2)*Sqrt[e + f*x^2]), x], x] /; FreeQ[{c, d, e, f}, x] && NeQ[d*e - c*f, 0]
  534. Int[Times[Power[Plus[Pattern[a, Blank[]], Times[Optional[Pattern[b, Blank[]]], Power[Pattern[x, Blank[]], 2]]], -1], Power[Plus[Pattern[c, Blank[]], Times[Optional[Pattern[d, Blank[]]], Power[Pattern[x, Blank[]], 2]]], Rational[1, 2]], Power[Plus[Pattern[e, Blank[]], Times[Optional[Pattern[f, Blank[]]], Power[Pattern[x, Blank[]], 2]]], Rational[1, 2]]], Pattern[x, Blank[Symbol]]] := Dist[d/b, Int[Sqrt[e + f*x^2]/Sqrt[c + d*x^2], x], x] + Dist[(b*c - a*d)/b, Int[Sqrt[e + f*x^2]/((a + b*x^2)*Sqrt[c + d*x^2]), x], x] /; FreeQ[{a, b, c, d, e, f}, x] && GtQ[d/c, 0] && GtQ[f/e, 0] && !SimplerSqrtQ[d/c, f/e]
  535. Int[Times[Power[Plus[Pattern[a, Blank[]], Times[Optional[Pattern[b, Blank[]]], Power[Pattern[x, Blank[]], 2]]], -1], Power[Plus[Pattern[c, Blank[]], Times[Optional[Pattern[d, Blank[]]], Power[Pattern[x, Blank[]], 2]]], Rational[1, 2]], Power[Plus[Pattern[e, Blank[]], Times[Optional[Pattern[f, Blank[]]], Power[Pattern[x, Blank[]], 2]]], Rational[1, 2]]], Pattern[x, Blank[Symbol]]] := Dist[d/b, Int[Sqrt[e + f*x^2]/Sqrt[c + d*x^2], x], x] + Dist[(b*c - a*d)/b, Int[Sqrt[e + f*x^2]/((a + b*x^2)*Sqrt[c + d*x^2]), x], x] /; FreeQ[{a, b, c, d, e, f}, x] && !SimplerSqrtQ[-(f/e), -(d/c)]
  536. Int[Times[Power[Plus[Pattern[a, Blank[]], Times[Optional[Pattern[b, Blank[]]], Power[Pattern[x, Blank[]], 2]]], -1], Power[Plus[Pattern[c, Blank[]], Times[Optional[Pattern[d, Blank[]]], Power[Pattern[x, Blank[]], 2]]], Rational[-1, 2]], Power[Plus[Pattern[e, Blank[]], Times[Optional[Pattern[f, Blank[]]], Power[Pattern[x, Blank[]], 2]]], Rational[-1, 2]]], Pattern[x, Blank[Symbol]]] := -Dist[f/(b*e - a*f), Int[1/(Sqrt[c + d*x^2]*Sqrt[e + f*x^2]), x], x] + Dist[b/(b*e - a*f), Int[Sqrt[e + f*x^2]/((a + b*x^2)*Sqrt[c + d*x^2]), x], x] /; FreeQ[{a, b, c, d, e, f}, x] && GtQ[d/c, 0] && GtQ[f/e, 0] && !SimplerSqrtQ[d/c, f/e]
  537. Int[Times[Power[Plus[Pattern[a, Blank[]], Times[Optional[Pattern[b, Blank[]]], Power[Pattern[x, Blank[]], 2]]], -1], Power[Plus[Pattern[c, Blank[]], Times[Optional[Pattern[d, Blank[]]], Power[Pattern[x, Blank[]], 2]]], Rational[-1, 2]], Power[Plus[Pattern[e, Blank[]], Times[Optional[Pattern[f, Blank[]]], Power[Pattern[x, Blank[]], 2]]], Rational[-1, 2]]], Pattern[x, Blank[Symbol]]] := Simp[(1*EllipticPi[(b*c)/(a*d), ArcSin[Rt[-(d/c), 2]*x], (c*f)/(d*e)])/(a*Sqrt[c]*Sqrt[e]*Rt[-(d/c), 2]), x] /; FreeQ[{a, b, c, d, e, f}, x] && !GtQ[d/c, 0] && GtQ[c, 0] && GtQ[e, 0] && !( !GtQ[f/e, 0] && SimplerSqrtQ[-(f/e), -(d/c)])
  538. Int[Times[Power[Plus[Pattern[a, Blank[]], Times[Optional[Pattern[b, Blank[]]], Power[Pattern[x, Blank[]], 2]]], -1], Power[Plus[Pattern[c, Blank[]], Times[Optional[Pattern[d, Blank[]]], Power[Pattern[x, Blank[]], 2]]], Rational[-1, 2]], Power[Plus[Pattern[e, Blank[]], Times[Optional[Pattern[f, Blank[]]], Power[Pattern[x, Blank[]], 2]]], Rational[-1, 2]]], Pattern[x, Blank[Symbol]]] := Dist[Sqrt[1 + (d*x^2)/c]/Sqrt[c + d*x^2], Int[1/((a + b*x^2)*Sqrt[1 + (d*x^2)/c]*Sqrt[e + f*x^2]), x], x] /; FreeQ[{a, b, c, d, e, f}, x] && !GtQ[c, 0]
  539. Int[Times[Power[Plus[Pattern[a, Blank[]], Times[Optional[Pattern[b, Blank[]]], Power[Pattern[x, Blank[]], 2]]], -1], Power[Plus[Pattern[c, Blank[]], Times[Optional[Pattern[d, Blank[]]], Power[Pattern[x, Blank[]], 2]]], Rational[1, 2]], Power[Plus[Pattern[e, Blank[]], Times[Optional[Pattern[f, Blank[]]], Power[Pattern[x, Blank[]], 2]]], Rational[-1, 2]]], Pattern[x, Blank[Symbol]]] := Simp[(c*Sqrt[e + f*x^2]*EllipticPi[1 - (b*c)/(a*d), ArcTan[Rt[d/c, 2]*x], 1 - (c*f)/(d*e)])/(a*e*Rt[d/c, 2]*Sqrt[c + d*x^2]*Sqrt[(c*(e + f*x^2))/(e*(c + d*x^2))]), x] /; FreeQ[{a, b, c, d, e, f}, x] && PosQ[d/c]
  540. Int[Times[Power[Plus[Pattern[a, Blank[]], Times[Optional[Pattern[b, Blank[]]], Power[Pattern[x, Blank[]], 2]]], -1], Power[Plus[Pattern[c, Blank[]], Times[Optional[Pattern[d, Blank[]]], Power[Pattern[x, Blank[]], 2]]], Rational[1, 2]], Power[Plus[Pattern[e, Blank[]], Times[Optional[Pattern[f, Blank[]]], Power[Pattern[x, Blank[]], 2]]], Rational[-1, 2]]], Pattern[x, Blank[Symbol]]] := Dist[d/b, Int[1/(Sqrt[c + d*x^2]*Sqrt[e + f*x^2]), x], x] + Dist[(b*c - a*d)/b, Int[1/((a + b*x^2)*Sqrt[c + d*x^2]*Sqrt[e + f*x^2]), x], x] /; FreeQ[{a, b, c, d, e, f}, x] && NegQ[d/c]
  541. Int[Times[Power[Plus[Pattern[a, Blank[]], Times[Optional[Pattern[b, Blank[]]], Power[Pattern[x, Blank[]], 2]]], -1], Power[Plus[Pattern[c, Blank[]], Times[Optional[Pattern[d, Blank[]]], Power[Pattern[x, Blank[]], 2]]], Rational[-3, 2]], Power[Plus[Pattern[e, Blank[]], Times[Optional[Pattern[f, Blank[]]], Power[Pattern[x, Blank[]], 2]]], Rational[1, 2]]], Pattern[x, Blank[Symbol]]] := Dist[b/(b*c - a*d), Int[Sqrt[e + f*x^2]/((a + b*x^2)*Sqrt[c + d*x^2]), x], x] - Dist[d/(b*c - a*d), Int[Sqrt[e + f*x^2]/(c + d*x^2)^(3/2), x], x] /; FreeQ[{a, b, c, d, e, f}, x] && PosQ[d/c] && PosQ[f/e]
  542. Int[Times[Power[Plus[Pattern[a, Blank[]], Times[Optional[Pattern[b, Blank[]]], Power[Pattern[x, Blank[]], 2]]], -1], Power[Plus[Pattern[c, Blank[]], Times[Optional[Pattern[d, Blank[]]], Power[Pattern[x, Blank[]], 2]]], Rational[-3, 2]], Power[Plus[Pattern[e, Blank[]], Times[Optional[Pattern[f, Blank[]]], Power[Pattern[x, Blank[]], 2]]], Rational[3, 2]]], Pattern[x, Blank[Symbol]]] := Dist[(b*e - a*f)/(b*c - a*d), Int[Sqrt[e + f*x^2]/((a + b*x^2)*Sqrt[c + d*x^2]), x], x] - Dist[(d*e - c*f)/(b*c - a*d), Int[Sqrt[e + f*x^2]/(c + d*x^2)^(3/2), x], x] /; FreeQ[{a, b, c, d, e, f}, x] && PosQ[d/c] && PosQ[f/e]
  543. Int[Times[Power[Plus[Pattern[a, Blank[]], Times[Optional[Pattern[b, Blank[]]], Power[Pattern[x, Blank[]], 2]]], -1], Power[Plus[Pattern[c, Blank[]], Times[Optional[Pattern[d, Blank[]]], Power[Pattern[x, Blank[]], 2]]], Rational[3, 2]], Power[Plus[Pattern[e, Blank[]], Times[Optional[Pattern[f, Blank[]]], Power[Pattern[x, Blank[]], 2]]], Rational[1, 2]]], Pattern[x, Blank[Symbol]]] := Dist[(b*c - a*d)^2/b^2, Int[Sqrt[e + f*x^2]/((a + b*x^2)*Sqrt[c + d*x^2]), x], x] + Dist[d/b^2, Int[((2*b*c - a*d + b*d*x^2)*Sqrt[e + f*x^2])/Sqrt[c + d*x^2], x], x] /; FreeQ[{a, b, c, d, e, f}, x] && PosQ[d/c] && PosQ[f/e]
  544. Int[Times[Power[Plus[Pattern[a, Blank[]], Times[Optional[Pattern[b, Blank[]]], Power[Pattern[x, Blank[]], 2]]], -1], Power[Plus[Pattern[c, Blank[]], Times[Optional[Pattern[d, Blank[]]], Power[Pattern[x, Blank[]], 2]]], Pattern[q, Blank[]]], Power[Plus[Pattern[e, Blank[]], Times[Optional[Pattern[f, Blank[]]], Power[Pattern[x, Blank[]], 2]]], Pattern[r, Blank[]]]], Pattern[x, Blank[Symbol]]] := Dist[(b*(b*e - a*f))/(b*c - a*d)^2, Int[((c + d*x^2)^(q + 2)*(e + f*x^2)^(r - 1))/(a + b*x^2), x], x] - Dist[1/(b*c - a*d)^2, Int[(c + d*x^2)^q*(e + f*x^2)^(r - 1)*(2*b*c*d*e - a*d^2*e - b*c^2*f + d^2*(b*e - a*f)*x^2), x], x] /; FreeQ[{a, b, c, d, e, f}, x] && LtQ[q, -1] && GtQ[r, 1]
  545. Int[Times[Power[Plus[Pattern[a, Blank[]], Times[Optional[Pattern[b, Blank[]]], Power[Pattern[x, Blank[]], 2]]], -1], Power[Plus[Pattern[c, Blank[]], Times[Optional[Pattern[d, Blank[]]], Power[Pattern[x, Blank[]], 2]]], Pattern[q, Blank[]]], Power[Plus[Pattern[e, Blank[]], Times[Optional[Pattern[f, Blank[]]], Power[Pattern[x, Blank[]], 2]]], Pattern[r, Blank[]]]], Pattern[x, Blank[Symbol]]] := Dist[d/b, Int[(c + d*x^2)^(q - 1)*(e + f*x^2)^r, x], x] + Dist[(b*c - a*d)/b, Int[((c + d*x^2)^(q - 1)*(e + f*x^2)^r)/(a + b*x^2), x], x] /; FreeQ[{a, b, c, d, e, f, r}, x] && GtQ[q, 1]
  546. Int[Times[Power[Plus[Pattern[a, Blank[]], Times[Optional[Pattern[b, Blank[]]], Power[Pattern[x, Blank[]], 2]]], -1], Power[Plus[Pattern[c, Blank[]], Times[Optional[Pattern[d, Blank[]]], Power[Pattern[x, Blank[]], 2]]], Pattern[q, Blank[]]], Power[Plus[Pattern[e, Blank[]], Times[Optional[Pattern[f, Blank[]]], Power[Pattern[x, Blank[]], 2]]], Pattern[r, Blank[]]]], Pattern[x, Blank[Symbol]]] := Dist[b^2/(b*c - a*d)^2, Int[((c + d*x^2)^(q + 2)*(e + f*x^2)^r)/(a + b*x^2), x], x] - Dist[d/(b*c - a*d)^2, Int[(c + d*x^2)^q*(e + f*x^2)^r*(2*b*c - a*d + b*d*x^2), x], x] /; FreeQ[{a, b, c, d, e, f, r}, x] && LtQ[q, -1]
  547. Int[Times[Power[Plus[Pattern[a, Blank[]], Times[Optional[Pattern[b, Blank[]]], Power[Pattern[x, Blank[]], 2]]], -1], Power[Plus[Pattern[c, Blank[]], Times[Optional[Pattern[d, Blank[]]], Power[Pattern[x, Blank[]], 2]]], Pattern[q, Blank[]]], Power[Plus[Pattern[e, Blank[]], Times[Optional[Pattern[f, Blank[]]], Power[Pattern[x, Blank[]], 2]]], Pattern[r, Blank[]]]], Pattern[x, Blank[Symbol]]] := -Dist[d/(b*c - a*d), Int[(c + d*x^2)^q*(e + f*x^2)^r, x], x] + Dist[b/(b*c - a*d), Int[((c + d*x^2)^(q + 1)*(e + f*x^2)^r)/(a + b*x^2), x], x] /; FreeQ[{a, b, c, d, e, f, r}, x] && LeQ[q, -1]
  548. Int[Times[Power[Plus[Pattern[a, Blank[]], Times[Optional[Pattern[b, Blank[]]], Power[Pattern[x, Blank[]], 2]]], -2], Power[Plus[Pattern[c, Blank[]], Times[Optional[Pattern[d, Blank[]]], Power[Pattern[x, Blank[]], 2]]], Rational[1, 2]], Power[Plus[Pattern[e, Blank[]], Times[Optional[Pattern[f, Blank[]]], Power[Pattern[x, Blank[]], 2]]], Rational[1, 2]]], Pattern[x, Blank[Symbol]]] := Simp[(x*Sqrt[c + d*x^2]*Sqrt[e + f*x^2])/(2*a*(a + b*x^2)), x] + (Dist[(d*f)/(2*a*b^2), Int[(a - b*x^2)/(Sqrt[c + d*x^2]*Sqrt[e + f*x^2]), x], x] + Dist[(b^2*c*e - a^2*d*f)/(2*a*b^2), Int[1/((a + b*x^2)*Sqrt[c + d*x^2]*Sqrt[e + f*x^2]), x], x]) /; FreeQ[{a, b, c, d, e, f}, x]
  549. Int[Times[Power[Plus[Pattern[a, Blank[]], Times[Optional[Pattern[b, Blank[]]], Power[Pattern[x, Blank[]], 2]]], -2], Power[Plus[Pattern[c, Blank[]], Times[Optional[Pattern[d, Blank[]]], Power[Pattern[x, Blank[]], 2]]], Rational[-1, 2]], Power[Plus[Pattern[e, Blank[]], Times[Optional[Pattern[f, Blank[]]], Power[Pattern[x, Blank[]], 2]]], Rational[-1, 2]]], Pattern[x, Blank[Symbol]]] := Simp[(b^2*x*Sqrt[c + d*x^2]*Sqrt[e + f*x^2])/(2*a*(b*c - a*d)*(b*e - a*f)*(a + b*x^2)), x] + (-Dist[(d*f)/(2*a*(b*c - a*d)*(b*e - a*f)), Int[(a + b*x^2)/(Sqrt[c + d*x^2]*Sqrt[e + f*x^2]), x], x] + Dist[(b^2*c*e + 3*a^2*d*f - 2*a*b*(d*e + c*f))/(2*a*(b*c - a*d)*(b*e - a*f)), Int[1/((a + b*x^2)*Sqrt[c + d*x^2]*Sqrt[e + f*x^2]), x], x]) /; FreeQ[{a, b, c, d, e, f}, x]
  550. Int[Times[Power[Plus[Pattern[a, Blank[]], Times[Optional[Pattern[b, Blank[]]], Power[Pattern[x, Blank[]], Pattern[n, Blank[]]]]], Pattern[p, Blank[]]], Power[Plus[Pattern[c, Blank[]], Times[Optional[Pattern[d, Blank[]]], Power[Pattern[x, Blank[]], Pattern[n, Blank[]]]]], Pattern[q, Blank[]]], Power[Plus[Pattern[e, Blank[]], Times[Optional[Pattern[f, Blank[]]], Power[Pattern[x, Blank[]], Pattern[n, Blank[]]]]], Pattern[r, Blank[]]]], Pattern[x, Blank[Symbol]]] := Dist[d/b, Int[(a + b*x^n)^(p + 1)*(c + d*x^n)^(q - 1)*(e + f*x^n)^r, x], x] + Dist[(b*c - a*d)/b, Int[(a + b*x^n)^p*(c + d*x^n)^(q - 1)*(e + f*x^n)^r, x], x] /; FreeQ[{a, b, c, d, e, f, n, r}, x] && ILtQ[p, 0] && GtQ[q, 0]
  551. Int[Times[Power[Plus[Pattern[a, Blank[]], Times[Optional[Pattern[b, Blank[]]], Power[Pattern[x, Blank[]], Pattern[n, Blank[]]]]], Pattern[p, Blank[]]], Power[Plus[Pattern[c, Blank[]], Times[Optional[Pattern[d, Blank[]]], Power[Pattern[x, Blank[]], Pattern[n, Blank[]]]]], Pattern[q, Blank[]]], Power[Plus[Pattern[e, Blank[]], Times[Optional[Pattern[f, Blank[]]], Power[Pattern[x, Blank[]], Pattern[n, Blank[]]]]], Pattern[r, Blank[]]]], Pattern[x, Blank[Symbol]]] := Dist[b/(b*c - a*d), Int[(a + b*x^n)^p*(c + d*x^n)^(q + 1)*(e + f*x^n)^r, x], x] - Dist[d/(b*c - a*d), Int[(a + b*x^n)^(p + 1)*(c + d*x^n)^q*(e + f*x^n)^r, x], x] /; FreeQ[{a, b, c, d, e, f, n, q}, x] && ILtQ[p, 0] && LeQ[q, -1]
  552. Int[Times[Power[Plus[Pattern[a, Blank[]], Times[Optional[Pattern[b, Blank[]]], Power[Pattern[x, Blank[]], 2]]], Rational[-1, 2]], Power[Plus[Pattern[c, Blank[]], Times[Optional[Pattern[d, Blank[]]], Power[Pattern[x, Blank[]], 2]]], Rational[-1, 2]], Power[Plus[Pattern[e, Blank[]], Times[Optional[Pattern[f, Blank[]]], Power[Pattern[x, Blank[]], 2]]], Rational[-1, 2]]], Pattern[x, Blank[Symbol]]] := Dist[(Sqrt[c + d*x^2]*Sqrt[(a*(e + f*x^2))/(e*(a + b*x^2))])/(c*Sqrt[e + f*x^2]*Sqrt[(a*(c + d*x^2))/(c*(a + b*x^2))]), Subst[Int[1/(Sqrt[1 - ((b*c - a*d)*x^2)/c]*Sqrt[1 - ((b*e - a*f)*x^2)/e]), x], x, x/Sqrt[a + b*x^2]], x] /; FreeQ[{a, b, c, d, e, f}, x]
  553. Int[Times[Power[Plus[Pattern[a, Blank[]], Times[Optional[Pattern[b, Blank[]]], Power[Pattern[x, Blank[]], 2]]], Rational[1, 2]], Power[Plus[Pattern[c, Blank[]], Times[Optional[Pattern[d, Blank[]]], Power[Pattern[x, Blank[]], 2]]], Rational[-1, 2]], Power[Plus[Pattern[e, Blank[]], Times[Optional[Pattern[f, Blank[]]], Power[Pattern[x, Blank[]], 2]]], Rational[-1, 2]]], Pattern[x, Blank[Symbol]]] := Dist[(a*Sqrt[c + d*x^2]*Sqrt[(a*(e + f*x^2))/(e*(a + b*x^2))])/(c*Sqrt[e + f*x^2]*Sqrt[(a*(c + d*x^2))/(c*(a + b*x^2))]), Subst[Int[1/((1 - b*x^2)*Sqrt[1 - ((b*c - a*d)*x^2)/c]*Sqrt[1 - ((b*e - a*f)*x^2)/e]), x], x, x/Sqrt[a + b*x^2]], x] /; FreeQ[{a, b, c, d, e, f}, x]
  554. Int[Times[Power[Plus[Pattern[a, Blank[]], Times[Optional[Pattern[b, Blank[]]], Power[Pattern[x, Blank[]], 2]]], Rational[-3, 2]], Power[Plus[Pattern[c, Blank[]], Times[Optional[Pattern[d, Blank[]]], Power[Pattern[x, Blank[]], 2]]], Rational[1, 2]], Power[Plus[Pattern[e, Blank[]], Times[Optional[Pattern[f, Blank[]]], Power[Pattern[x, Blank[]], 2]]], Rational[-1, 2]]], Pattern[x, Blank[Symbol]]] := Dist[(Sqrt[c + d*x^2]*Sqrt[(a*(e + f*x^2))/(e*(a + b*x^2))])/(a*Sqrt[e + f*x^2]*Sqrt[(a*(c + d*x^2))/(c*(a + b*x^2))]), Subst[Int[Sqrt[1 - ((b*c - a*d)*x^2)/c]/Sqrt[1 - ((b*e - a*f)*x^2)/e], x], x, x/Sqrt[a + b*x^2]], x] /; FreeQ[{a, b, c, d, e, f}, x]
  555. Int[Times[Power[Plus[Pattern[a, Blank[]], Times[Optional[Pattern[b, Blank[]]], Power[Pattern[x, Blank[]], 2]]], Rational[1, 2]], Power[Plus[Pattern[c, Blank[]], Times[Optional[Pattern[d, Blank[]]], Power[Pattern[x, Blank[]], 2]]], Rational[1, 2]], Power[Plus[Pattern[e, Blank[]], Times[Optional[Pattern[f, Blank[]]], Power[Pattern[x, Blank[]], 2]]], Rational[-1, 2]]], Pattern[x, Blank[Symbol]]] := Simp[(d*x*Sqrt[a + b*x^2]*Sqrt[e + f*x^2])/(2*f*Sqrt[c + d*x^2]), x] + (-Dist[(c*(d*e - c*f))/(2*f), Int[Sqrt[a + b*x^2]/((c + d*x^2)^(3/2)*Sqrt[e + f*x^2]), x], x] + Dist[(b*c*(d*e - c*f))/(2*d*f), Int[1/(Sqrt[a + b*x^2]*Sqrt[c + d*x^2]*Sqrt[e + f*x^2]), x], x] - Dist[(b*d*e - b*c*f - a*d*f)/(2*d*f), Int[Sqrt[c + d*x^2]/(Sqrt[a + b*x^2]*Sqrt[e + f*x^2]), x], x]) /; FreeQ[{a, b, c, d, e, f}, x] && PosQ[(d*e - c*f)/c]
  556. Int[Times[Power[Plus[Pattern[a, Blank[]], Times[Optional[Pattern[b, Blank[]]], Power[Pattern[x, Blank[]], 2]]], Rational[1, 2]], Power[Plus[Pattern[c, Blank[]], Times[Optional[Pattern[d, Blank[]]], Power[Pattern[x, Blank[]], 2]]], Rational[1, 2]], Power[Plus[Pattern[e, Blank[]], Times[Optional[Pattern[f, Blank[]]], Power[Pattern[x, Blank[]], 2]]], Rational[-1, 2]]], Pattern[x, Blank[Symbol]]] := Simp[(x*Sqrt[a + b*x^2]*Sqrt[c + d*x^2])/(2*Sqrt[e + f*x^2]), x] + (Dist[(e*(b*e - a*f))/(2*f), Int[Sqrt[c + d*x^2]/(Sqrt[a + b*x^2]*(e + f*x^2)^(3/2)), x], x] + Dist[((b*e - a*f)*(d*e - 2*c*f))/(2*f^2), Int[1/(Sqrt[a + b*x^2]*Sqrt[c + d*x^2]*Sqrt[e + f*x^2]), x], x] - Dist[(b*d*e - b*c*f - a*d*f)/(2*f^2), Int[Sqrt[e + f*x^2]/(Sqrt[a + b*x^2]*Sqrt[c + d*x^2]), x], x]) /; FreeQ[{a, b, c, d, e, f}, x] && NegQ[(d*e - c*f)/c]
  557. Int[Times[Power[Plus[Pattern[a, Blank[]], Times[Optional[Pattern[b, Blank[]]], Power[Pattern[x, Blank[]], 2]]], Rational[1, 2]], Power[Plus[Pattern[c, Blank[]], Times[Optional[Pattern[d, Blank[]]], Power[Pattern[x, Blank[]], 2]]], Rational[1, 2]], Power[Plus[Pattern[e, Blank[]], Times[Optional[Pattern[f, Blank[]]], Power[Pattern[x, Blank[]], 2]]], Rational[-3, 2]]], Pattern[x, Blank[Symbol]]] := Dist[b/f, Int[Sqrt[c + d*x^2]/(Sqrt[a + b*x^2]*Sqrt[e + f*x^2]), x], x] - Dist[(b*e - a*f)/f, Int[Sqrt[c + d*x^2]/(Sqrt[a + b*x^2]*(e + f*x^2)^(3/2)), x], x] /; FreeQ[{a, b, c, d, e, f}, x]
  558. Int[Times[Power[Plus[Pattern[a, Blank[]], Times[Optional[Pattern[b, Blank[]]], Power[Pattern[x, Blank[]], Pattern[n, Blank[]]]]], Pattern[p, Blank[]]], Power[Plus[Pattern[c, Blank[]], Times[Optional[Pattern[d, Blank[]]], Power[Pattern[x, Blank[]], Pattern[n, Blank[]]]]], Pattern[q, Blank[]]], Power[Plus[Pattern[e, Blank[]], Times[Optional[Pattern[f, Blank[]]], Power[Pattern[x, Blank[]], Pattern[n, Blank[]]]]], Pattern[r, Blank[]]]], Pattern[x, Blank[Symbol]]] := With[{u = ExpandIntegrand[(a + b*x^n)^p*(c + d*x^n)^q*(e + f*x^n)^r, x]}, Int[u, x] /; SumQ[u]] /; FreeQ[{a, b, c, d, e, f, p, q, r}, x] && IGtQ[n, 0]
  559. Int[Times[Power[Plus[Pattern[a, Blank[]], Times[Optional[Pattern[b, Blank[]]], Power[Pattern[x, Blank[]], Pattern[n, Blank[]]]]], Pattern[p, Blank[]]], Power[Plus[Pattern[c, Blank[]], Times[Optional[Pattern[d, Blank[]]], Power[Pattern[x, Blank[]], Pattern[n, Blank[]]]]], Pattern[q, Blank[]]], Power[Plus[Pattern[e, Blank[]], Times[Optional[Pattern[f, Blank[]]], Power[Pattern[x, Blank[]], Pattern[n, Blank[]]]]], Pattern[r, Blank[]]]], Pattern[x, Blank[Symbol]]] := -Subst[Int[((a + b/x^n)^p*(c + d/x^n)^q*(e + f/x^n)^r)/x^2, x], x, 1/x] /; FreeQ[{a, b, c, d, e, f, p, q, r}, x] && ILtQ[n, 0]
  560. Int[Times[Power[Plus[Pattern[a, Blank[]], Times[Optional[Pattern[b, Blank[]]], Power[Pattern[x, Blank[]], Pattern[n, Blank[]]]]], Optional[Pattern[p, Blank[]]]], Power[Plus[Pattern[c, Blank[]], Times[Optional[Pattern[d, Blank[]]], Power[Pattern[x, Blank[]], Pattern[n, Blank[]]]]], Optional[Pattern[q, Blank[]]]], Power[Plus[Pattern[e, Blank[]], Times[Optional[Pattern[f, Blank[]]], Power[Pattern[x, Blank[]], Pattern[n, Blank[]]]]], Optional[Pattern[r, Blank[]]]]], Pattern[x, Blank[Symbol]]] := Unintegrable[(a + b*x^n)^p*(c + d*x^n)^q*(e + f*x^n)^r, x] /; FreeQ[{a, b, c, d, e, f, n, p, q, r}, x]
  561. Int[Times[Power[Plus[Optional[Pattern[a, Blank[]]], Times[Optional[Pattern[b, Blank[]]], Power[Pattern[u, Blank[]], Pattern[n, Blank[]]]]], Optional[Pattern[p, Blank[]]]], Power[Plus[Optional[Pattern[c, Blank[]]], Times[Optional[Pattern[d, Blank[]]], Power[Pattern[v, Blank[]], Pattern[n, Blank[]]]]], Optional[Pattern[q, Blank[]]]], Power[Plus[Optional[Pattern[e, Blank[]]], Times[Optional[Pattern[f, Blank[]]], Power[Pattern[w, Blank[]], Pattern[n, Blank[]]]]], Optional[Pattern[r, Blank[]]]]], Pattern[x, Blank[Symbol]]] := Dist[1/Coefficient[u, x, 1], Subst[Int[(a + b*x^n)^p*(c + d*x^n)^q*(e + f*x^n)^r, x], x, u], x] /; FreeQ[{a, b, c, d, e, f, p, n, q, r}, x] && EqQ[u, v] && EqQ[u, w] && LinearQ[u, x] && NeQ[u, x]
  562. Int[Times[Power[Plus[Pattern[c, Blank[]], Times[Optional[Pattern[d, Blank[]]], Power[Pattern[x, Blank[]], Optional[Pattern[mn, Blank[]]]]]], Optional[Pattern[q, Blank[]]]], Power[Plus[Optional[Pattern[a, Blank[]]], Times[Optional[Pattern[b, Blank[]]], Power[Pattern[x, Blank[]], Optional[Pattern[n, Blank[]]]]]], Optional[Pattern[p, Blank[]]]], Power[Plus[Pattern[e, Blank[]], Times[Optional[Pattern[f, Blank[]]], Power[Pattern[x, Blank[]], Optional[Pattern[n, Blank[]]]]]], Optional[Pattern[r, Blank[]]]]], Pattern[x, Blank[Symbol]]] := Int[((a + b*x^n)^p*(d + c*x^n)^q*(e + f*x^n)^r)/x^(n*q), x] /; FreeQ[{a, b, c, d, e, f, n, p, r}, x] && EqQ[mn, -n] && IntegerQ[q]
  563. Int[Times[Power[Plus[Pattern[c, Blank[]], Times[Optional[Pattern[d, Blank[]]], Power[Pattern[x, Blank[]], Optional[Pattern[mn, Blank[]]]]]], Optional[Pattern[q, Blank[]]]], Power[Plus[Optional[Pattern[a, Blank[]]], Times[Optional[Pattern[b, Blank[]]], Power[Pattern[x, Blank[]], Optional[Pattern[n, Blank[]]]]]], Optional[Pattern[p, Blank[]]]], Power[Plus[Pattern[e, Blank[]], Times[Optional[Pattern[f, Blank[]]], Power[Pattern[x, Blank[]], Optional[Pattern[n, Blank[]]]]]], Optional[Pattern[r, Blank[]]]]], Pattern[x, Blank[Symbol]]] := Int[x^(n*(p + r))*(b + a/x^n)^p*(c + d/x^n)^q*(f + e/x^n)^r, x] /; FreeQ[{a, b, c, d, e, f, n, q}, x] && EqQ[mn, -n] && IntegerQ[p] && IntegerQ[r]
  564. Int[Times[Power[Plus[Pattern[c, Blank[]], Times[Optional[Pattern[d, Blank[]]], Power[Pattern[x, Blank[]], Optional[Pattern[mn, Blank[]]]]]], Pattern[q, Blank[]]], Power[Plus[Optional[Pattern[a, Blank[]]], Times[Optional[Pattern[b, Blank[]]], Power[Pattern[x, Blank[]], Optional[Pattern[n, Blank[]]]]]], Optional[Pattern[p, Blank[]]]], Power[Plus[Pattern[e, Blank[]], Times[Optional[Pattern[f, Blank[]]], Power[Pattern[x, Blank[]], Optional[Pattern[n, Blank[]]]]]], Optional[Pattern[r, Blank[]]]]], Pattern[x, Blank[Symbol]]] := Dist[(x^(n*FracPart[q])*(c + d/x^n)^FracPart[q])/(d + c*x^n)^FracPart[q], Int[((a + b*x^n)^p*(d + c*x^n)^q*(e + f*x^n)^r)/x^(n*q), x], x] /; FreeQ[{a, b, c, d, e, f, n, p, q, r}, x] && EqQ[mn, -n] && !IntegerQ[q]
  565. Int[Times[Power[Plus[Pattern[e1, Blank[]], Times[Optional[Pattern[f1, Blank[]]], Power[Pattern[x, Blank[]], Optional[Pattern[n2, Blank[]]]]]], Optional[Pattern[r, Blank[]]]], Power[Plus[Pattern[e2, Blank[]], Times[Optional[Pattern[f2, Blank[]]], Power[Pattern[x, Blank[]], Optional[Pattern[n2, Blank[]]]]]], Optional[Pattern[r, Blank[]]]], Power[Plus[Pattern[a, Blank[]], Times[Optional[Pattern[b, Blank[]]], Power[Pattern[x, Blank[]], Pattern[n, Blank[]]]]], Optional[Pattern[p, Blank[]]]], Power[Plus[Pattern[c, Blank[]], Times[Optional[Pattern[d, Blank[]]], Power[Pattern[x, Blank[]], Pattern[n, Blank[]]]]], Optional[Pattern[q, Blank[]]]]], Pattern[x, Blank[Symbol]]] := Int[(a + b*x^n)^p*(c + d*x^n)^q*(e1*e2 + f1*f2*x^n)^r, x] /; FreeQ[{a, b, c, d, e1, f1, e2, f2, n, p, q, r}, x] && EqQ[n2, n/2] && EqQ[e2*f1 + e1*f2, 0] && (IntegerQ[r] || (GtQ[e1, 0] && GtQ[e2, 0]))
  566. Int[Times[Power[Plus[Pattern[e1, Blank[]], Times[Optional[Pattern[f1, Blank[]]], Power[Pattern[x, Blank[]], Optional[Pattern[n2, Blank[]]]]]], Optional[Pattern[r, Blank[]]]], Power[Plus[Pattern[e2, Blank[]], Times[Optional[Pattern[f2, Blank[]]], Power[Pattern[x, Blank[]], Optional[Pattern[n2, Blank[]]]]]], Optional[Pattern[r, Blank[]]]], Power[Plus[Pattern[a, Blank[]], Times[Optional[Pattern[b, Blank[]]], Power[Pattern[x, Blank[]], Pattern[n, Blank[]]]]], Optional[Pattern[p, Blank[]]]], Power[Plus[Pattern[c, Blank[]], Times[Optional[Pattern[d, Blank[]]], Power[Pattern[x, Blank[]], Pattern[n, Blank[]]]]], Optional[Pattern[q, Blank[]]]]], Pattern[x, Blank[Symbol]]] := Dist[((e1 + f1*x^(n/2))^FracPart[r]*(e2 + f2*x^(n/2))^FracPart[r])/(e1*e2 + f1*f2*x^n)^FracPart[r], Int[(a + b*x^n)^p*(c + d*x^n)^q*(e1*e2 + f1*f2*x^n)^r, x], x] /; FreeQ[{a, b, c, d, e1, f1, e2, f2, n, p, q, r}, x] && EqQ[n2, n/2] && EqQ[e2*f1 + e1*f2, 0]
  567. Int[Times[Power[Times[Optional[Pattern[g, Blank[]]], Pattern[x, Blank[]]], Optional[Pattern[m, Blank[]]]], Power[Times[Optional[Pattern[b, Blank[]]], Power[Pattern[x, Blank[]], Pattern[n, Blank[]]]], Pattern[p, Blank[]]], Power[Plus[Pattern[c, Blank[]], Times[Optional[Pattern[d, Blank[]]], Power[Pattern[x, Blank[]], Pattern[n, Blank[]]]]], Optional[Pattern[q, Blank[]]]], Power[Plus[Pattern[e, Blank[]], Times[Optional[Pattern[f, Blank[]]], Power[Pattern[x, Blank[]], Pattern[n, Blank[]]]]], Optional[Pattern[r, Blank[]]]]], Pattern[x, Blank[Symbol]]] := Dist[g^m/(n*b^(Simplify[(m + 1)/n] - 1)), Subst[Int[(b*x)^(p + Simplify[(m + 1)/n] - 1)*(c + d*x)^q*(e + f*x)^r, x], x, x^n], x] /; FreeQ[{b, c, d, e, f, g, m, n, p, q, r}, x] && (IntegerQ[m] || GtQ[g, 0]) && IntegerQ[Simplify[(m + 1)/n]]
  568. Int[Times[Power[Times[Optional[Pattern[g, Blank[]]], Pattern[x, Blank[]]], Optional[Pattern[m, Blank[]]]], Power[Times[Optional[Pattern[b, Blank[]]], Power[Pattern[x, Blank[]], Optional[Pattern[n, Blank[]]]]], Pattern[p, Blank[]]], Power[Plus[Pattern[c, Blank[]], Times[Optional[Pattern[d, Blank[]]], Power[Pattern[x, Blank[]], Pattern[n, Blank[]]]]], Optional[Pattern[q, Blank[]]]], Power[Plus[Pattern[e, Blank[]], Times[Optional[Pattern[f, Blank[]]], Power[Pattern[x, Blank[]], Pattern[n, Blank[]]]]], Optional[Pattern[r, Blank[]]]]], Pattern[x, Blank[Symbol]]] := Dist[(g^m*b^IntPart[p]*(b*x^n)^FracPart[p])/x^(n*FracPart[p]), Int[x^(m + n*p)*(c + d*x^n)^q*(e + f*x^n)^r, x], x] /; FreeQ[{b, c, d, e, f, g, m, n, p, q, r}, x] && (IntegerQ[m] || GtQ[g, 0]) && !IntegerQ[Simplify[(m + 1)/n]]
  569. Int[Times[Power[Times[Pattern[g, Blank[]], Pattern[x, Blank[]]], Pattern[m, Blank[]]], Power[Times[Optional[Pattern[b, Blank[]]], Power[Pattern[x, Blank[]], Optional[Pattern[n, Blank[]]]]], Pattern[p, Blank[]]], Power[Plus[Pattern[c, Blank[]], Times[Optional[Pattern[d, Blank[]]], Power[Pattern[x, Blank[]], Pattern[n, Blank[]]]]], Optional[Pattern[q, Blank[]]]], Power[Plus[Pattern[e, Blank[]], Times[Optional[Pattern[f, Blank[]]], Power[Pattern[x, Blank[]], Pattern[n, Blank[]]]]], Optional[Pattern[r, Blank[]]]]], Pattern[x, Blank[Symbol]]] := Dist[(g^IntPart[m]*(g*x)^FracPart[m])/x^FracPart[m], Int[x^m*(b*x^n)^p*(c + d*x^n)^q*(e + f*x^n)^r, x], x] /; FreeQ[{b, c, d, e, f, g, m, n, p, q, r}, x] && !IntegerQ[m]
  570. Int[Times[Power[Times[Optional[Pattern[g, Blank[]]], Pattern[x, Blank[]]], Optional[Pattern[m, Blank[]]]], Power[Plus[Pattern[a, Blank[]], Times[Optional[Pattern[b, Blank[]]], Power[Pattern[x, Blank[]], Pattern[n, Blank[]]]]], Optional[Pattern[p, Blank[]]]], Power[Plus[Pattern[c, Blank[]], Times[Optional[Pattern[d, Blank[]]], Power[Pattern[x, Blank[]], Pattern[n, Blank[]]]]], Optional[Pattern[q, Blank[]]]], Power[Plus[Pattern[e, Blank[]], Times[Optional[Pattern[f, Blank[]]], Power[Pattern[x, Blank[]], Pattern[n, Blank[]]]]], Optional[Pattern[r, Blank[]]]]], Pattern[x, Blank[Symbol]]] := Int[ExpandIntegrand[(g*x)^m*(a + b*x^n)^p*(c + d*x^n)^q*(e + f*x^n)^r, x], x] /; FreeQ[{a, b, c, d, e, f, g, m, n}, x] && IGtQ[p, -2] && IGtQ[q, 0] && IGtQ[r, 0]
  571. Int[Times[Power[Pattern[x, Blank[]], Optional[Pattern[m, Blank[]]]], Power[Plus[Pattern[a, Blank[]], Times[Optional[Pattern[b, Blank[]]], Power[Pattern[x, Blank[]], Pattern[n, Blank[]]]]], Optional[Pattern[p, Blank[]]]], Power[Plus[Pattern[c, Blank[]], Times[Optional[Pattern[d, Blank[]]], Power[Pattern[x, Blank[]], Pattern[n, Blank[]]]]], Optional[Pattern[q, Blank[]]]], Power[Plus[Pattern[e, Blank[]], Times[Optional[Pattern[f, Blank[]]], Power[Pattern[x, Blank[]], Pattern[n, Blank[]]]]], Optional[Pattern[r, Blank[]]]]], Pattern[x, Blank[Symbol]]] := Dist[1/n, Subst[Int[(a + b*x)^p*(c + d*x)^q*(e + f*x)^r, x], x, x^n], x] /; FreeQ[{a, b, c, d, e, f, m, n, p, q, r}, x] && EqQ[m - n + 1, 0]
  572. Int[Times[Power[Pattern[x, Blank[]], Optional[Pattern[m, Blank[]]]], Power[Plus[Pattern[a, Blank[]], Times[Optional[Pattern[b, Blank[]]], Power[Pattern[x, Blank[]], Pattern[n, Blank[]]]]], Optional[Pattern[p, Blank[]]]], Power[Plus[Pattern[c, Blank[]], Times[Optional[Pattern[d, Blank[]]], Power[Pattern[x, Blank[]], Pattern[n, Blank[]]]]], Optional[Pattern[q, Blank[]]]], Power[Plus[Pattern[e, Blank[]], Times[Optional[Pattern[f, Blank[]]], Power[Pattern[x, Blank[]], Pattern[n, Blank[]]]]], Optional[Pattern[r, Blank[]]]]], Pattern[x, Blank[Symbol]]] := Int[x^(m + n*(p + q + r))*(b + a/x^n)^p*(d + c/x^n)^q*(f + e/x^n)^r, x] /; FreeQ[{a, b, c, d, e, f, m, n}, x] && IntegersQ[p, q, r] && NegQ[n]
  573. Int[Times[Power[Pattern[x, Blank[]], Optional[Pattern[m, Blank[]]]], Power[Plus[Pattern[a, Blank[]], Times[Optional[Pattern[b, Blank[]]], Power[Pattern[x, Blank[]], Pattern[n, Blank[]]]]], Optional[Pattern[p, Blank[]]]], Power[Plus[Pattern[c, Blank[]], Times[Optional[Pattern[d, Blank[]]], Power[Pattern[x, Blank[]], Pattern[n, Blank[]]]]], Optional[Pattern[q, Blank[]]]], Power[Plus[Pattern[e, Blank[]], Times[Optional[Pattern[f, Blank[]]], Power[Pattern[x, Blank[]], Pattern[n, Blank[]]]]], Optional[Pattern[r, Blank[]]]]], Pattern[x, Blank[Symbol]]] := Dist[1/n, Subst[Int[x^(Simplify[(m + 1)/n] - 1)*(a + b*x)^p*(c + d*x)^q*(e + f*x)^r, x], x, x^n], x] /; FreeQ[{a, b, c, d, e, f, m, n, p, q, r}, x] && IntegerQ[Simplify[(m + 1)/n]]
  574. Int[Times[Power[Times[Pattern[g, Blank[]], Pattern[x, Blank[]]], Optional[Pattern[m, Blank[]]]], Power[Plus[Pattern[a, Blank[]], Times[Optional[Pattern[b, Blank[]]], Power[Pattern[x, Blank[]], Pattern[n, Blank[]]]]], Optional[Pattern[p, Blank[]]]], Power[Plus[Pattern[c, Blank[]], Times[Optional[Pattern[d, Blank[]]], Power[Pattern[x, Blank[]], Pattern[n, Blank[]]]]], Optional[Pattern[q, Blank[]]]], Power[Plus[Pattern[e, Blank[]], Times[Optional[Pattern[f, Blank[]]], Power[Pattern[x, Blank[]], Pattern[n, Blank[]]]]], Optional[Pattern[r, Blank[]]]]], Pattern[x, Blank[Symbol]]] := Dist[(g^IntPart[m]*(g*x)^FracPart[m])/x^FracPart[m], Int[x^m*(a + b*x^n)^p*(c + d*x^n)^q*(e + f*x^n)^r, x], x] /; FreeQ[{a, b, c, d, e, f, g, m, n, p, q, r}, x] && IntegerQ[Simplify[(m + 1)/n]]
  575. Int[Times[Power[Pattern[x, Blank[]], Optional[Pattern[m, Blank[]]]], Power[Plus[Pattern[a, Blank[]], Times[Optional[Pattern[b, Blank[]]], Power[Pattern[x, Blank[]], Pattern[n, Blank[]]]]], Optional[Pattern[p, Blank[]]]], Power[Plus[Pattern[c, Blank[]], Times[Optional[Pattern[d, Blank[]]], Power[Pattern[x, Blank[]], Pattern[n, Blank[]]]]], Optional[Pattern[q, Blank[]]]], Power[Plus[Pattern[e, Blank[]], Times[Optional[Pattern[f, Blank[]]], Power[Pattern[x, Blank[]], Pattern[n, Blank[]]]]], Optional[Pattern[r, Blank[]]]]], Pattern[x, Blank[Symbol]]] := With[{k = GCD[m + 1, n]}, Dist[1/k, Subst[Int[x^((m + 1)/k - 1)*(a + b*x^(n/k))^p*(c + d*x^(n/k))^q*(e + f*x^(n/k))^r, x], x, x^k], x] /; k != 1] /; FreeQ[{a, b, c, d, e, f, p, q, r}, x] && IGtQ[n, 0] && IntegerQ[m]
  576. Int[Times[Power[Times[Optional[Pattern[g, Blank[]]], Pattern[x, Blank[]]], Pattern[m, Blank[]]], Power[Plus[Pattern[a, Blank[]], Times[Optional[Pattern[b, Blank[]]], Power[Pattern[x, Blank[]], Pattern[n, Blank[]]]]], Pattern[p, Blank[]]], Power[Plus[Pattern[c, Blank[]], Times[Optional[Pattern[d, Blank[]]], Power[Pattern[x, Blank[]], Pattern[n, Blank[]]]]], Pattern[q, Blank[]]], Power[Plus[Pattern[e, Blank[]], Times[Optional[Pattern[f, Blank[]]], Power[Pattern[x, Blank[]], Pattern[n, Blank[]]]]], Pattern[r, Blank[]]]], Pattern[x, Blank[Symbol]]] := With[{k = Denominator[m]}, Dist[k/g, Subst[Int[x^(k*(m + 1) - 1)*(a + (b*x^(k*n))/g^n)^p*(c + (d*x^(k*n))/g^n)^q*(e + (f*x^(k*n))/g^n)^r, x], x, (g*x)^(1/k)], x]] /; FreeQ[{a, b, c, d, e, f, g, p, q, r}, x] && IGtQ[n, 0] && FractionQ[m]
  577. Int[Times[Power[Times[Optional[Pattern[g, Blank[]]], Pattern[x, Blank[]]], Optional[Pattern[m, Blank[]]]], Power[Plus[Pattern[a, Blank[]], Times[Optional[Pattern[b, Blank[]]], Power[Pattern[x, Blank[]], Pattern[n, Blank[]]]]], Pattern[p, Blank[]]], Power[Plus[Pattern[c, Blank[]], Times[Optional[Pattern[d, Blank[]]], Power[Pattern[x, Blank[]], Pattern[n, Blank[]]]]], Optional[Pattern[q, Blank[]]]], Plus[Pattern[e, Blank[]], Times[Optional[Pattern[f, Blank[]]], Power[Pattern[x, Blank[]], Pattern[n, Blank[]]]]]], Pattern[x, Blank[Symbol]]] := -Simp[((b*e - a*f)*(g*x)^(m + 1)*(a + b*x^n)^(p + 1)*(c + d*x^n)^q)/(a*b*g*n*(p + 1)), x] + Dist[1/(a*b*n*(p + 1)), Int[(g*x)^m*(a + b*x^n)^(p + 1)*(c + d*x^n)^(q - 1)*Simp[c*(b*e*n*(p + 1) + (b*e - a*f)*(m + 1)) + d*(b*e*n*(p + 1) + (b*e - a*f)*(m + n*q + 1))*x^n, x], x], x] /; FreeQ[{a, b, c, d, e, f, g, m}, x] && IGtQ[n, 0] && LtQ[p, -1] && GtQ[q, 0] && !(EqQ[q, 1] && SimplerQ[b*c - a*d, b*e - a*f])
  578. Int[Times[Power[Times[Optional[Pattern[g, Blank[]]], Pattern[x, Blank[]]], Optional[Pattern[m, Blank[]]]], Power[Plus[Pattern[a, Blank[]], Times[Optional[Pattern[b, Blank[]]], Power[Pattern[x, Blank[]], Pattern[n, Blank[]]]]], Pattern[p, Blank[]]], Power[Plus[Pattern[c, Blank[]], Times[Optional[Pattern[d, Blank[]]], Power[Pattern[x, Blank[]], Pattern[n, Blank[]]]]], Pattern[q, Blank[]]], Plus[Pattern[e, Blank[]], Times[Optional[Pattern[f, Blank[]]], Power[Pattern[x, Blank[]], Pattern[n, Blank[]]]]]], Pattern[x, Blank[Symbol]]] := Simp[(g^(n - 1)*(b*e - a*f)*(g*x)^(m - n + 1)*(a + b*x^n)^(p + 1)*(c + d*x^n)^(q + 1))/(b*n*(b*c - a*d)*(p + 1)), x] - Dist[g^n/(b*n*(b*c - a*d)*(p + 1)), Int[(g*x)^(m - n)*(a + b*x^n)^(p + 1)*(c + d*x^n)^q*Simp[c*(b*e - a*f)*(m - n + 1) + (d*(b*e - a*f)*(m + n*q + 1) - b*n*(c*f - d*e)*(p + 1))*x^n, x], x], x] /; FreeQ[{a, b, c, d, e, f, g, q}, x] && IGtQ[n, 0] && LtQ[p, -1] && GtQ[m - n + 1, 0]
  579. Int[Times[Power[Times[Optional[Pattern[g, Blank[]]], Pattern[x, Blank[]]], Optional[Pattern[m, Blank[]]]], Power[Plus[Pattern[a, Blank[]], Times[Optional[Pattern[b, Blank[]]], Power[Pattern[x, Blank[]], Pattern[n, Blank[]]]]], Pattern[p, Blank[]]], Power[Plus[Pattern[c, Blank[]], Times[Optional[Pattern[d, Blank[]]], Power[Pattern[x, Blank[]], Pattern[n, Blank[]]]]], Pattern[q, Blank[]]], Plus[Pattern[e, Blank[]], Times[Optional[Pattern[f, Blank[]]], Power[Pattern[x, Blank[]], Pattern[n, Blank[]]]]]], Pattern[x, Blank[Symbol]]] := -Simp[((b*e - a*f)*(g*x)^(m + 1)*(a + b*x^n)^(p + 1)*(c + d*x^n)^(q + 1))/(a*g*n*(b*c - a*d)*(p + 1)), x] + Dist[1/(a*n*(b*c - a*d)*(p + 1)), Int[(g*x)^m*(a + b*x^n)^(p + 1)*(c + d*x^n)^q*Simp[c*(b*e - a*f)*(m + 1) + e*n*(b*c - a*d)*(p + 1) + d*(b*e - a*f)*(m + n*(p + q + 2) + 1)*x^n, x], x], x] /; FreeQ[{a, b, c, d, e, f, g, m, q}, x] && IGtQ[n, 0] && LtQ[p, -1]
  580. Int[Times[Power[Times[Optional[Pattern[g, Blank[]]], Pattern[x, Blank[]]], Pattern[m, Blank[]]], Power[Plus[Pattern[a, Blank[]], Times[Optional[Pattern[b, Blank[]]], Power[Pattern[x, Blank[]], Pattern[n, Blank[]]]]], Optional[Pattern[p, Blank[]]]], Power[Plus[Pattern[c, Blank[]], Times[Optional[Pattern[d, Blank[]]], Power[Pattern[x, Blank[]], Pattern[n, Blank[]]]]], Optional[Pattern[q, Blank[]]]], Plus[Pattern[e, Blank[]], Times[Optional[Pattern[f, Blank[]]], Power[Pattern[x, Blank[]], Pattern[n, Blank[]]]]]], Pattern[x, Blank[Symbol]]] := Simp[(e*(g*x)^(m + 1)*(a + b*x^n)^(p + 1)*(c + d*x^n)^q)/(a*g*(m + 1)), x] - Dist[1/(a*g^n*(m + 1)), Int[(g*x)^(m + n)*(a + b*x^n)^p*(c + d*x^n)^(q - 1)*Simp[c*(b*e - a*f)*(m + 1) + e*n*(b*c*(p + 1) + a*d*q) + d*((b*e - a*f)*(m + 1) + b*e*n*(p + q + 1))*x^n, x], x], x] /; FreeQ[{a, b, c, d, e, f, g, p}, x] && IGtQ[n, 0] && GtQ[q, 0] && LtQ[m, -1] && !(EqQ[q, 1] && SimplerQ[e + f*x^n, c + d*x^n])
  581. Int[Times[Power[Times[Optional[Pattern[g, Blank[]]], Pattern[x, Blank[]]], Optional[Pattern[m, Blank[]]]], Power[Plus[Pattern[a, Blank[]], Times[Optional[Pattern[b, Blank[]]], Power[Pattern[x, Blank[]], Pattern[n, Blank[]]]]], Optional[Pattern[p, Blank[]]]], Power[Plus[Pattern[c, Blank[]], Times[Optional[Pattern[d, Blank[]]], Power[Pattern[x, Blank[]], Pattern[n, Blank[]]]]], Optional[Pattern[q, Blank[]]]], Plus[Pattern[e, Blank[]], Times[Optional[Pattern[f, Blank[]]], Power[Pattern[x, Blank[]], Pattern[n, Blank[]]]]]], Pattern[x, Blank[Symbol]]] := Simp[(f*(g*x)^(m + 1)*(a + b*x^n)^(p + 1)*(c + d*x^n)^q)/(b*g*(m + n*(p + q + 1) + 1)), x] + Dist[1/(b*(m + n*(p + q + 1) + 1)), Int[(g*x)^m*(a + b*x^n)^p*(c + d*x^n)^(q - 1)*Simp[c*((b*e - a*f)*(m + 1) + b*e*n*(p + q + 1)) + (d*(b*e - a*f)*(m + 1) + f*n*q*(b*c - a*d) + b*e*d*n*(p + q + 1))*x^n, x], x], x] /; FreeQ[{a, b, c, d, e, f, g, m, p}, x] && IGtQ[n, 0] && GtQ[q, 0] && !(EqQ[q, 1] && SimplerQ[e + f*x^n, c + d*x^n])
  582. Int[Times[Power[Times[Optional[Pattern[g, Blank[]]], Pattern[x, Blank[]]], Optional[Pattern[m, Blank[]]]], Power[Plus[Pattern[a, Blank[]], Times[Optional[Pattern[b, Blank[]]], Power[Pattern[x, Blank[]], Pattern[n, Blank[]]]]], Optional[Pattern[p, Blank[]]]], Power[Plus[Pattern[c, Blank[]], Times[Optional[Pattern[d, Blank[]]], Power[Pattern[x, Blank[]], Pattern[n, Blank[]]]]], Optional[Pattern[q, Blank[]]]], Plus[Pattern[e, Blank[]], Times[Optional[Pattern[f, Blank[]]], Power[Pattern[x, Blank[]], Pattern[n, Blank[]]]]]], Pattern[x, Blank[Symbol]]] := Simp[(f*g^(n - 1)*(g*x)^(m - n + 1)*(a + b*x^n)^(p + 1)*(c + d*x^n)^(q + 1))/(b*d*(m + n*(p + q + 1) + 1)), x] - Dist[g^n/(b*d*(m + n*(p + q + 1) + 1)), Int[(g*x)^(m - n)*(a + b*x^n)^p*(c + d*x^n)^q*Simp[a*f*c*(m - n + 1) + (a*f*d*(m + n*q + 1) + b*(f*c*(m + n*p + 1) - e*d*(m + n*(p + q + 1) + 1)))*x^n, x], x], x] /; FreeQ[{a, b, c, d, e, f, g, p, q}, x] && IGtQ[n, 0] && GtQ[m, n - 1]
  583. Int[Times[Power[Times[Optional[Pattern[g, Blank[]]], Pattern[x, Blank[]]], Pattern[m, Blank[]]], Power[Plus[Pattern[a, Blank[]], Times[Optional[Pattern[b, Blank[]]], Power[Pattern[x, Blank[]], Pattern[n, Blank[]]]]], Optional[Pattern[p, Blank[]]]], Power[Plus[Pattern[c, Blank[]], Times[Optional[Pattern[d, Blank[]]], Power[Pattern[x, Blank[]], Pattern[n, Blank[]]]]], Optional[Pattern[q, Blank[]]]], Plus[Pattern[e, Blank[]], Times[Optional[Pattern[f, Blank[]]], Power[Pattern[x, Blank[]], Pattern[n, Blank[]]]]]], Pattern[x, Blank[Symbol]]] := Simp[(e*(g*x)^(m + 1)*(a + b*x^n)^(p + 1)*(c + d*x^n)^(q + 1))/(a*c*g*(m + 1)), x] + Dist[1/(a*c*g^n*(m + 1)), Int[(g*x)^(m + n)*(a + b*x^n)^p*(c + d*x^n)^q*Simp[a*f*c*(m + 1) - e*(b*c + a*d)*(m + n + 1) - e*n*(b*c*p + a*d*q) - b*e*d*(m + n*(p + q + 2) + 1)*x^n, x], x], x] /; FreeQ[{a, b, c, d, e, f, g, p, q}, x] && IGtQ[n, 0] && LtQ[m, -1]
  584. Int[Times[Power[Times[Optional[Pattern[g, Blank[]]], Pattern[x, Blank[]]], Optional[Pattern[m, Blank[]]]], Power[Plus[Pattern[a, Blank[]], Times[Optional[Pattern[b, Blank[]]], Power[Pattern[x, Blank[]], Pattern[n, Blank[]]]]], Pattern[p, Blank[]]], Power[Plus[Pattern[c, Blank[]], Times[Optional[Pattern[d, Blank[]]], Power[Pattern[x, Blank[]], Pattern[n, Blank[]]]]], -1], Plus[Pattern[e, Blank[]], Times[Optional[Pattern[f, Blank[]]], Power[Pattern[x, Blank[]], Pattern[n, Blank[]]]]]], Pattern[x, Blank[Symbol]]] := Int[ExpandIntegrand[((g*x)^m*(a + b*x^n)^p*(e + f*x^n))/(c + d*x^n), x], x] /; FreeQ[{a, b, c, d, e, f, g, m, p}, x] && IGtQ[n, 0]
  585. Int[Times[Power[Times[Optional[Pattern[g, Blank[]]], Pattern[x, Blank[]]], Optional[Pattern[m, Blank[]]]], Power[Plus[Pattern[a, Blank[]], Times[Optional[Pattern[b, Blank[]]], Power[Pattern[x, Blank[]], Pattern[n, Blank[]]]]], Optional[Pattern[p, Blank[]]]], Power[Plus[Pattern[c, Blank[]], Times[Optional[Pattern[d, Blank[]]], Power[Pattern[x, Blank[]], Pattern[n, Blank[]]]]], Optional[Pattern[q, Blank[]]]], Plus[Pattern[e, Blank[]], Times[Optional[Pattern[f, Blank[]]], Power[Pattern[x, Blank[]], Pattern[n, Blank[]]]]]], Pattern[x, Blank[Symbol]]] := Dist[e, Int[(g*x)^m*(a + b*x^n)^p*(c + d*x^n)^q, x], x] + Dist[f/e^n, Int[(g*x)^(m + n)*(a + b*x^n)^p*(c + d*x^n)^q, x], x] /; FreeQ[{a, b, c, d, e, f, g, m, p, q}, x] && IGtQ[n, 0]
  586. Int[Times[Power[Times[Optional[Pattern[g, Blank[]]], Pattern[x, Blank[]]], Optional[Pattern[m, Blank[]]]], Power[Plus[Pattern[a, Blank[]], Times[Optional[Pattern[b, Blank[]]], Power[Pattern[x, Blank[]], Pattern[n, Blank[]]]]], Optional[Pattern[p, Blank[]]]], Power[Plus[Pattern[c, Blank[]], Times[Optional[Pattern[d, Blank[]]], Power[Pattern[x, Blank[]], Pattern[n, Blank[]]]]], Optional[Pattern[q, Blank[]]]], Power[Plus[Pattern[e, Blank[]], Times[Optional[Pattern[f, Blank[]]], Power[Pattern[x, Blank[]], Pattern[n, Blank[]]]]], Optional[Pattern[r, Blank[]]]]], Pattern[x, Blank[Symbol]]] := Dist[e, Int[(g*x)^m*(a + b*x^n)^p*(c + d*x^n)^q*(e + f*x^n)^(r - 1), x], x] + Dist[f/e^n, Int[(g*x)^(m + n)*(a + b*x^n)^p*(c + d*x^n)^q*(e + f*x^n)^(r - 1), x], x] /; FreeQ[{a, b, c, d, e, f, g, m, p, q}, x] && IGtQ[n, 0] && IGtQ[r, 0]
  587. Int[Times[Power[Pattern[x, Blank[]], Optional[Pattern[m, Blank[]]]], Power[Plus[Pattern[a, Blank[]], Times[Optional[Pattern[b, Blank[]]], Power[Pattern[x, Blank[]], Pattern[n, Blank[]]]]], Optional[Pattern[p, Blank[]]]], Power[Plus[Pattern[c, Blank[]], Times[Optional[Pattern[d, Blank[]]], Power[Pattern[x, Blank[]], Pattern[n, Blank[]]]]], Optional[Pattern[q, Blank[]]]], Power[Plus[Pattern[e, Blank[]], Times[Optional[Pattern[f, Blank[]]], Power[Pattern[x, Blank[]], Pattern[n, Blank[]]]]], Optional[Pattern[r, Blank[]]]]], Pattern[x, Blank[Symbol]]] := -Subst[Int[((a + b/x^n)^p*(c + d/x^n)^q*(e + f/x^n)^r)/x^(m + 2), x], x, 1/x] /; FreeQ[{a, b, c, d, e, f, p, q, r}, x] && ILtQ[n, 0] && IntegerQ[m]
  588. Int[Times[Power[Times[Optional[Pattern[g, Blank[]]], Pattern[x, Blank[]]], Pattern[m, Blank[]]], Power[Plus[Pattern[a, Blank[]], Times[Optional[Pattern[b, Blank[]]], Power[Pattern[x, Blank[]], Pattern[n, Blank[]]]]], Optional[Pattern[p, Blank[]]]], Power[Plus[Pattern[c, Blank[]], Times[Optional[Pattern[d, Blank[]]], Power[Pattern[x, Blank[]], Pattern[n, Blank[]]]]], Optional[Pattern[q, Blank[]]]], Power[Plus[Pattern[e, Blank[]], Times[Optional[Pattern[f, Blank[]]], Power[Pattern[x, Blank[]], Pattern[n, Blank[]]]]], Optional[Pattern[r, Blank[]]]]], Pattern[x, Blank[Symbol]]] := With[{k = Denominator[m]}, -Dist[k/g, Subst[Int[((a + b/(g^n*x^(k*n)))^p*(c + d/(g^n*x^(k*n)))^q*(e + f/(g^n*x^(k*n)))^r)/x^(k*(m + 1) + 1), x], x, 1/(g*x)^(1/k)], x]] /; FreeQ[{a, b, c, d, e, f, g, p, q, r}, x] && ILtQ[n, 0] && FractionQ[m]
  589. Int[Times[Power[Times[Optional[Pattern[g, Blank[]]], Pattern[x, Blank[]]], Pattern[m, Blank[]]], Power[Plus[Pattern[a, Blank[]], Times[Optional[Pattern[b, Blank[]]], Power[Pattern[x, Blank[]], Pattern[n, Blank[]]]]], Optional[Pattern[p, Blank[]]]], Power[Plus[Pattern[c, Blank[]], Times[Optional[Pattern[d, Blank[]]], Power[Pattern[x, Blank[]], Pattern[n, Blank[]]]]], Optional[Pattern[q, Blank[]]]], Power[Plus[Pattern[e, Blank[]], Times[Optional[Pattern[f, Blank[]]], Power[Pattern[x, Blank[]], Pattern[n, Blank[]]]]], Optional[Pattern[r, Blank[]]]]], Pattern[x, Blank[Symbol]]] := -Dist[(g*x)^m*(x^(-1))^m, Subst[Int[((a + b/x^n)^p*(c + d/x^n)^q*(e + f/x^n)^r)/x^(m + 2), x], x, 1/x], x] /; FreeQ[{a, b, c, d, e, f, g, m, p, q, r}, x] && ILtQ[n, 0] && !RationalQ[m]
  590. Int[Times[Power[Pattern[x, Blank[]], Optional[Pattern[m, Blank[]]]], Power[Plus[Pattern[a, Blank[]], Times[Optional[Pattern[b, Blank[]]], Power[Pattern[x, Blank[]], Pattern[n, Blank[]]]]], Optional[Pattern[p, Blank[]]]], Power[Plus[Pattern[c, Blank[]], Times[Optional[Pattern[d, Blank[]]], Power[Pattern[x, Blank[]], Pattern[n, Blank[]]]]], Optional[Pattern[q, Blank[]]]], Power[Plus[Pattern[e, Blank[]], Times[Optional[Pattern[f, Blank[]]], Power[Pattern[x, Blank[]], Pattern[n, Blank[]]]]], Optional[Pattern[r, Blank[]]]]], Pattern[x, Blank[Symbol]]] := With[{k = Denominator[n]}, Dist[k, Subst[Int[x^(k*(m + 1) - 1)*(a + b*x^(k*n))^p*(c + d*x^(k*n))^q*(e + f*x^(k*n))^r, x], x, x^(1/k)], x]] /; FreeQ[{a, b, c, d, e, f, m, p, q, r}, x] && FractionQ[n]
  591. Int[Times[Power[Times[Pattern[g, Blank[]], Pattern[x, Blank[]]], Pattern[m, Blank[]]], Power[Plus[Pattern[a, Blank[]], Times[Optional[Pattern[b, Blank[]]], Power[Pattern[x, Blank[]], Pattern[n, Blank[]]]]], Optional[Pattern[p, Blank[]]]], Power[Plus[Pattern[c, Blank[]], Times[Optional[Pattern[d, Blank[]]], Power[Pattern[x, Blank[]], Pattern[n, Blank[]]]]], Optional[Pattern[q, Blank[]]]], Power[Plus[Pattern[e, Blank[]], Times[Optional[Pattern[f, Blank[]]], Power[Pattern[x, Blank[]], Pattern[n, Blank[]]]]], Optional[Pattern[r, Blank[]]]]], Pattern[x, Blank[Symbol]]] := Dist[(g^IntPart[m]*(g*x)^FracPart[m])/x^FracPart[m], Int[x^m*(a + b*x^n)^p*(c + d*x^n)^q*(e + f*x^n)^r, x], x] /; FreeQ[{a, b, c, d, e, f, g, m, p, q, r}, x] && FractionQ[n]
  592. Int[Times[Power[Pattern[x, Blank[]], Optional[Pattern[m, Blank[]]]], Power[Plus[Pattern[a, Blank[]], Times[Optional[Pattern[b, Blank[]]], Power[Pattern[x, Blank[]], Pattern[n, Blank[]]]]], Optional[Pattern[p, Blank[]]]], Power[Plus[Pattern[c, Blank[]], Times[Optional[Pattern[d, Blank[]]], Power[Pattern[x, Blank[]], Pattern[n, Blank[]]]]], Optional[Pattern[q, Blank[]]]], Power[Plus[Pattern[e, Blank[]], Times[Optional[Pattern[f, Blank[]]], Power[Pattern[x, Blank[]], Pattern[n, Blank[]]]]], Optional[Pattern[r, Blank[]]]]], Pattern[x, Blank[Symbol]]] := Dist[1/(m + 1), Subst[Int[(a + b*x^Simplify[n/(m + 1)])^p*(c + d*x^Simplify[n/(m + 1)])^q*(e + f*x^Simplify[n/(m + 1)])^r, x], x, x^(m + 1)], x] /; FreeQ[{a, b, c, d, e, f, m, n, p, q, r}, x] && IntegerQ[Simplify[n/(m + 1)]]
  593. Int[Times[Power[Times[Pattern[g, Blank[]], Pattern[x, Blank[]]], Optional[Pattern[m, Blank[]]]], Power[Plus[Pattern[a, Blank[]], Times[Optional[Pattern[b, Blank[]]], Power[Pattern[x, Blank[]], Pattern[n, Blank[]]]]], Optional[Pattern[p, Blank[]]]], Power[Plus[Pattern[c, Blank[]], Times[Optional[Pattern[d, Blank[]]], Power[Pattern[x, Blank[]], Pattern[n, Blank[]]]]], Optional[Pattern[q, Blank[]]]], Power[Plus[Pattern[e, Blank[]], Times[Optional[Pattern[f, Blank[]]], Power[Pattern[x, Blank[]], Pattern[n, Blank[]]]]], Optional[Pattern[r, Blank[]]]]], Pattern[x, Blank[Symbol]]] := Dist[(g^IntPart[m]*(g*x)^FracPart[m])/x^FracPart[m], Int[x^m*(a + b*x^n)^p*(c + d*x^n)^q*(e + f*x^n)^r, x], x] /; FreeQ[{a, b, c, d, e, f, g, m, n, p, q, r}, x] && IntegerQ[Simplify[n/(m + 1)]]
  594. Int[Times[Power[Times[Optional[Pattern[g, Blank[]]], Pattern[x, Blank[]]], Optional[Pattern[m, Blank[]]]], Power[Plus[Pattern[a, Blank[]], Times[Optional[Pattern[b, Blank[]]], Power[Pattern[x, Blank[]], Pattern[n, Blank[]]]]], Pattern[p, Blank[]]], Power[Plus[Pattern[c, Blank[]], Times[Optional[Pattern[d, Blank[]]], Power[Pattern[x, Blank[]], Pattern[n, Blank[]]]]], Optional[Pattern[q, Blank[]]]], Plus[Pattern[e, Blank[]], Times[Optional[Pattern[f, Blank[]]], Power[Pattern[x, Blank[]], Pattern[n, Blank[]]]]]], Pattern[x, Blank[Symbol]]] := -Simp[((b*e - a*f)*(g*x)^(m + 1)*(a + b*x^n)^(p + 1)*(c + d*x^n)^q)/(a*b*g*n*(p + 1)), x] + Dist[1/(a*b*n*(p + 1)), Int[(g*x)^m*(a + b*x^n)^(p + 1)*(c + d*x^n)^(q - 1)*Simp[c*(b*e*n*(p + 1) + (b*e - a*f)*(m + 1)) + d*(b*e*n*(p + 1) + (b*e - a*f)*(m + n*q + 1))*x^n, x], x], x] /; FreeQ[{a, b, c, d, e, f, g, m, n}, x] && LtQ[p, -1] && GtQ[q, 0] && !(EqQ[q, 1] && SimplerQ[b*c - a*d, b*e - a*f])
  595. Int[Times[Power[Times[Optional[Pattern[g, Blank[]]], Pattern[x, Blank[]]], Optional[Pattern[m, Blank[]]]], Power[Plus[Pattern[a, Blank[]], Times[Optional[Pattern[b, Blank[]]], Power[Pattern[x, Blank[]], Pattern[n, Blank[]]]]], Pattern[p, Blank[]]], Power[Plus[Pattern[c, Blank[]], Times[Optional[Pattern[d, Blank[]]], Power[Pattern[x, Blank[]], Pattern[n, Blank[]]]]], Pattern[q, Blank[]]], Plus[Pattern[e, Blank[]], Times[Optional[Pattern[f, Blank[]]], Power[Pattern[x, Blank[]], Pattern[n, Blank[]]]]]], Pattern[x, Blank[Symbol]]] := -Simp[((b*e - a*f)*(g*x)^(m + 1)*(a + b*x^n)^(p + 1)*(c + d*x^n)^(q + 1))/(a*g*n*(b*c - a*d)*(p + 1)), x] + Dist[1/(a*n*(b*c - a*d)*(p + 1)), Int[(g*x)^m*(a + b*x^n)^(p + 1)*(c + d*x^n)^q*Simp[c*(b*e - a*f)*(m + 1) + e*n*(b*c - a*d)*(p + 1) + d*(b*e - a*f)*(m + n*(p + q + 2) + 1)*x^n, x], x], x] /; FreeQ[{a, b, c, d, e, f, g, m, n, q}, x] && LtQ[p, -1]
  596. Int[Times[Power[Times[Optional[Pattern[g, Blank[]]], Pattern[x, Blank[]]], Optional[Pattern[m, Blank[]]]], Power[Plus[Pattern[a, Blank[]], Times[Optional[Pattern[b, Blank[]]], Power[Pattern[x, Blank[]], Pattern[n, Blank[]]]]], Optional[Pattern[p, Blank[]]]], Power[Plus[Pattern[c, Blank[]], Times[Optional[Pattern[d, Blank[]]], Power[Pattern[x, Blank[]], Pattern[n, Blank[]]]]], Optional[Pattern[q, Blank[]]]], Plus[Pattern[e, Blank[]], Times[Optional[Pattern[f, Blank[]]], Power[Pattern[x, Blank[]], Pattern[n, Blank[]]]]]], Pattern[x, Blank[Symbol]]] := Simp[(f*(g*x)^(m + 1)*(a + b*x^n)^(p + 1)*(c + d*x^n)^q)/(b*g*(m + n*(p + q + 1) + 1)), x] + Dist[1/(b*(m + n*(p + q + 1) + 1)), Int[(g*x)^m*(a + b*x^n)^p*(c + d*x^n)^(q - 1)*Simp[c*((b*e - a*f)*(m + 1) + b*e*n*(p + q + 1)) + (d*(b*e - a*f)*(m + 1) + f*n*q*(b*c - a*d) + b*e*d*n*(p + q + 1))*x^n, x], x], x] /; FreeQ[{a, b, c, d, e, f, g, m, n, p}, x] && GtQ[q, 0] && !(EqQ[q, 1] && SimplerQ[e + f*x^n, c + d*x^n])
  597. Int[Times[Power[Times[Optional[Pattern[g, Blank[]]], Pattern[x, Blank[]]], Optional[Pattern[m, Blank[]]]], Power[Plus[Pattern[a, Blank[]], Times[Optional[Pattern[b, Blank[]]], Power[Pattern[x, Blank[]], Pattern[n, Blank[]]]]], Pattern[p, Blank[]]], Power[Plus[Pattern[c, Blank[]], Times[Optional[Pattern[d, Blank[]]], Power[Pattern[x, Blank[]], Pattern[n, Blank[]]]]], -1], Plus[Pattern[e, Blank[]], Times[Optional[Pattern[f, Blank[]]], Power[Pattern[x, Blank[]], Pattern[n, Blank[]]]]]], Pattern[x, Blank[Symbol]]] := Int[ExpandIntegrand[((g*x)^m*(a + b*x^n)^p*(e + f*x^n))/(c + d*x^n), x], x] /; FreeQ[{a, b, c, d, e, f, g, m, n, p}, x]
  598. Int[Times[Power[Times[Optional[Pattern[g, Blank[]]], Pattern[x, Blank[]]], Optional[Pattern[m, Blank[]]]], Power[Plus[Pattern[a, Blank[]], Times[Optional[Pattern[b, Blank[]]], Power[Pattern[x, Blank[]], Pattern[n, Blank[]]]]], Pattern[p, Blank[]]], Power[Plus[Pattern[c, Blank[]], Times[Optional[Pattern[d, Blank[]]], Power[Pattern[x, Blank[]], Pattern[n, Blank[]]]]], Pattern[q, Blank[]]], Plus[Pattern[e, Blank[]], Times[Optional[Pattern[f, Blank[]]], Power[Pattern[x, Blank[]], Pattern[n, Blank[]]]]]], Pattern[x, Blank[Symbol]]] := Dist[e, Int[(g*x)^m*(a + b*x^n)^p*(c + d*x^n)^q, x], x] + Dist[(f*(g*x)^m)/x^m, Int[x^(m + n)*(a + b*x^n)^p*(c + d*x^n)^q, x], x] /; FreeQ[{a, b, c, d, e, f, g, m, n, p, q}, x]
  599. Int[Times[Power[Pattern[x, Blank[]], Optional[Pattern[m, Blank[]]]], Power[Plus[Pattern[c, Blank[]], Times[Optional[Pattern[d, Blank[]]], Power[Pattern[x, Blank[]], Optional[Pattern[mn, Blank[]]]]]], Optional[Pattern[q, Blank[]]]], Power[Plus[Pattern[a, Blank[]], Times[Optional[Pattern[b, Blank[]]], Power[Pattern[x, Blank[]], Optional[Pattern[n, Blank[]]]]]], Optional[Pattern[p, Blank[]]]], Power[Plus[Pattern[e, Blank[]], Times[Optional[Pattern[f, Blank[]]], Power[Pattern[x, Blank[]], Optional[Pattern[n, Blank[]]]]]], Optional[Pattern[r, Blank[]]]]], Pattern[x, Blank[Symbol]]] := Int[x^(m - n*q)*(a + b*x^n)^p*(d + c*x^n)^q*(e + f*x^n)^r, x] /; FreeQ[{a, b, c, d, e, f, m, n, p, r}, x] && EqQ[mn, -n] && IntegerQ[q]
  600. Int[Times[Power[Pattern[x, Blank[]], Optional[Pattern[m, Blank[]]]], Power[Plus[Pattern[c, Blank[]], Times[Optional[Pattern[d, Blank[]]], Power[Pattern[x, Blank[]], Optional[Pattern[mn, Blank[]]]]]], Optional[Pattern[q, Blank[]]]], Power[Plus[Optional[Pattern[a, Blank[]]], Times[Optional[Pattern[b, Blank[]]], Power[Pattern[x, Blank[]], Optional[Pattern[n, Blank[]]]]]], Optional[Pattern[p, Blank[]]]], Power[Plus[Pattern[e, Blank[]], Times[Optional[Pattern[f, Blank[]]], Power[Pattern[x, Blank[]], Optional[Pattern[n, Blank[]]]]]], Optional[Pattern[r, Blank[]]]]], Pattern[x, Blank[Symbol]]] := Int[x^(m + n*(p + r))*(b + a/x^n)^p*(c + d/x^n)^q*(f + e/x^n)^r, x] /; FreeQ[{a, b, c, d, e, f, m, n, q}, x] && EqQ[mn, -n] && IntegerQ[p] && IntegerQ[r]
  601. Int[Times[Power[Pattern[x, Blank[]], Optional[Pattern[m, Blank[]]]], Power[Plus[Pattern[c, Blank[]], Times[Optional[Pattern[d, Blank[]]], Power[Pattern[x, Blank[]], Optional[Pattern[mn, Blank[]]]]]], Pattern[q, Blank[]]], Power[Plus[Optional[Pattern[a, Blank[]]], Times[Optional[Pattern[b, Blank[]]], Power[Pattern[x, Blank[]], Optional[Pattern[n, Blank[]]]]]], Optional[Pattern[p, Blank[]]]], Power[Plus[Pattern[e, Blank[]], Times[Optional[Pattern[f, Blank[]]], Power[Pattern[x, Blank[]], Optional[Pattern[n, Blank[]]]]]], Optional[Pattern[r, Blank[]]]]], Pattern[x, Blank[Symbol]]] := Dist[(x^(n*FracPart[q])*(c + d/x^n)^FracPart[q])/(d + c*x^n)^FracPart[q], Int[x^(m - n*q)*(a + b*x^n)^p*(d + c*x^n)^q*(e + f*x^n)^r, x], x] /; FreeQ[{a, b, c, d, e, f, m, n, p, q, r}, x] && EqQ[mn, -n] && !IntegerQ[q]
  602. Int[Times[Power[Times[Pattern[g, Blank[]], Pattern[x, Blank[]]], Pattern[m, Blank[]]], Power[Plus[Pattern[c, Blank[]], Times[Optional[Pattern[d, Blank[]]], Power[Pattern[x, Blank[]], Optional[Pattern[mn, Blank[]]]]]], Optional[Pattern[q, Blank[]]]], Power[Plus[Pattern[a, Blank[]], Times[Optional[Pattern[b, Blank[]]], Power[Pattern[x, Blank[]], Optional[Pattern[n, Blank[]]]]]], Optional[Pattern[p, Blank[]]]], Power[Plus[Pattern[e, Blank[]], Times[Optional[Pattern[f, Blank[]]], Power[Pattern[x, Blank[]], Optional[Pattern[n, Blank[]]]]]], Optional[Pattern[r, Blank[]]]]], Pattern[x, Blank[Symbol]]] := Dist[(g^IntPart[m]*(g*x)^FracPart[m])/x^FracPart[m], Int[x^m*(a + b*x^n)^p*(c + d/x^n)^q*(e + f*x^n)^r, x], x] /; FreeQ[{a, b, c, d, e, f, g, m, n, p, q, r}, x] && EqQ[mn, -n]
  603. Int[Times[Power[Times[Optional[Pattern[g, Blank[]]], Pattern[x, Blank[]]], Optional[Pattern[m, Blank[]]]], Power[Plus[Pattern[a, Blank[]], Times[Optional[Pattern[b, Blank[]]], Power[Pattern[x, Blank[]], Pattern[n, Blank[]]]]], Optional[Pattern[p, Blank[]]]], Power[Plus[Pattern[c, Blank[]], Times[Optional[Pattern[d, Blank[]]], Power[Pattern[x, Blank[]], Pattern[n, Blank[]]]]], Optional[Pattern[q, Blank[]]]], Power[Plus[Pattern[e, Blank[]], Times[Optional[Pattern[f, Blank[]]], Power[Pattern[x, Blank[]], Pattern[n, Blank[]]]]], Optional[Pattern[r, Blank[]]]]], Pattern[x, Blank[Symbol]]] := Unintegrable[(g*x)^m*(a + b*x^n)^p*(c + d*x^n)^q*(e + f*x^n)^r, x] /; FreeQ[{a, b, c, d, e, f, g, m, n, p, q, r}, x]
  604. Int[Times[Power[Pattern[u, Blank[]], Optional[Pattern[m, Blank[]]]], Power[Plus[Optional[Pattern[a, Blank[]]], Times[Optional[Pattern[b, Blank[]]], Power[Pattern[v, Blank[]], Pattern[n, Blank[]]]]], Optional[Pattern[p, Blank[]]]], Power[Plus[Optional[Pattern[c, Blank[]]], Times[Optional[Pattern[d, Blank[]]], Power[Pattern[v, Blank[]], Pattern[n, Blank[]]]]], Optional[Pattern[q, Blank[]]]], Power[Plus[Pattern[e, Blank[]], Times[Optional[Pattern[f, Blank[]]], Power[Pattern[v, Blank[]], Pattern[n, Blank[]]]]], Optional[Pattern[r, Blank[]]]]], Pattern[x, Blank[Symbol]]] := Dist[u^m/(Coefficient[v, x, 1]*v^m), Subst[Int[x^m*(a + b*x^n)^p*(c + d*x^n)^q*(e + f*x^n)^r, x], x, v], x] /; FreeQ[{a, b, c, d, e, f, m, n, p, q, r}, x] && LinearPairQ[u, v, x]
  605. Int[Times[Power[Times[Optional[Pattern[g, Blank[]]], Pattern[x, Blank[]]], Optional[Pattern[m, Blank[]]]], Power[Plus[Pattern[e1, Blank[]], Times[Optional[Pattern[f1, Blank[]]], Power[Pattern[x, Blank[]], Optional[Pattern[n2, Blank[]]]]]], Optional[Pattern[r, Blank[]]]], Power[Plus[Pattern[e2, Blank[]], Times[Optional[Pattern[f2, Blank[]]], Power[Pattern[x, Blank[]], Optional[Pattern[n2, Blank[]]]]]], Optional[Pattern[r, Blank[]]]], Power[Plus[Pattern[a, Blank[]], Times[Optional[Pattern[b, Blank[]]], Power[Pattern[x, Blank[]], Pattern[n, Blank[]]]]], Optional[Pattern[p, Blank[]]]], Power[Plus[Pattern[c, Blank[]], Times[Optional[Pattern[d, Blank[]]], Power[Pattern[x, Blank[]], Pattern[n, Blank[]]]]], Optional[Pattern[q, Blank[]]]]], Pattern[x, Blank[Symbol]]] := Int[(g*x)^m*(a + b*x^n)^p*(c + d*x^n)^q*(e1*e2 + f1*f2*x^n)^r, x] /; FreeQ[{a, b, c, d, e1, f1, e2, f2, g, m, n, p, q, r}, x] && EqQ[n2, n/2] && EqQ[e2*f1 + e1*f2, 0] && (IntegerQ[r] || (GtQ[e1, 0] && GtQ[e2, 0]))
  606. Int[Times[Power[Times[Optional[Pattern[g, Blank[]]], Pattern[x, Blank[]]], Optional[Pattern[m, Blank[]]]], Power[Plus[Pattern[e1, Blank[]], Times[Optional[Pattern[f1, Blank[]]], Power[Pattern[x, Blank[]], Optional[Pattern[n2, Blank[]]]]]], Optional[Pattern[r, Blank[]]]], Power[Plus[Pattern[e2, Blank[]], Times[Optional[Pattern[f2, Blank[]]], Power[Pattern[x, Blank[]], Optional[Pattern[n2, Blank[]]]]]], Optional[Pattern[r, Blank[]]]], Power[Plus[Pattern[a, Blank[]], Times[Optional[Pattern[b, Blank[]]], Power[Pattern[x, Blank[]], Pattern[n, Blank[]]]]], Optional[Pattern[p, Blank[]]]], Power[Plus[Pattern[c, Blank[]], Times[Optional[Pattern[d, Blank[]]], Power[Pattern[x, Blank[]], Pattern[n, Blank[]]]]], Optional[Pattern[q, Blank[]]]]], Pattern[x, Blank[Symbol]]] := Dist[((e1 + f1*x^(n/2))^FracPart[r]*(e2 + f2*x^(n/2))^FracPart[r])/(e1*e2 + f1*f2*x^n)^FracPart[r], Int[(g*x)^m*(a + b*x^n)^p*(c + d*x^n)^q*(e1*e2 + f1*f2*x^n)^r, x], x] /; FreeQ[{a, b, c, d, e1, f1, e2, f2, g, m, n, p, q, r}, x] && EqQ[n2, n/2] && EqQ[e2*f1 + e1*f2, 0]
  607. Int[Power[Plus[Pattern[a, Blank[]], Times[Optional[Pattern[b, Blank[]]], Pattern[x, Blank[]]], Times[Optional[Pattern[c, Blank[]]], Power[Pattern[x, Blank[]], 2]]], Pattern[p, Blank[]]], Pattern[x, Blank[Symbol]]] := Simp[(2*(a + b*x + c*x^2)^(p + 1))/((2*p + 1)*(b + 2*c*x)), x] /; FreeQ[{a, b, c, p}, x] && EqQ[b^2 - 4*a*c, 0] && LtQ[p, -1]
  608. Int[Power[Plus[Pattern[a, Blank[]], Times[Optional[Pattern[b, Blank[]]], Pattern[x, Blank[]]], Times[Optional[Pattern[c, Blank[]]], Power[Pattern[x, Blank[]], 2]]], Rational[-1, 2]], Pattern[x, Blank[Symbol]]] := Dist[(b/2 + c*x)/Sqrt[a + b*x + c*x^2], Int[1/(b/2 + c*x), x], x] /; FreeQ[{a, b, c}, x] && EqQ[b^2 - 4*a*c, 0]
  609. Int[Power[Plus[Pattern[a, Blank[]], Times[Optional[Pattern[b, Blank[]]], Pattern[x, Blank[]]], Times[Optional[Pattern[c, Blank[]]], Power[Pattern[x, Blank[]], 2]]], Pattern[p, Blank[]]], Pattern[x, Blank[Symbol]]] := Simp[((b + 2*c*x)*(a + b*x + c*x^2)^p)/(2*c*(2*p + 1)), x] /; FreeQ[{a, b, c, p}, x] && EqQ[b^2 - 4*a*c, 0] && NeQ[p, -2^(-1)]
  610. Int[Power[Plus[Pattern[a, Blank[]], Times[Optional[Pattern[b, Blank[]]], Pattern[x, Blank[]]], Times[Optional[Pattern[c, Blank[]]], Power[Pattern[x, Blank[]], 2]]], Pattern[p, Blank[]]], Pattern[x, Blank[Symbol]]] := With[{q = Rt[b^2 - 4*a*c, 2]}, Dist[1/c^p, Int[Simp[b/2 - q/2 + c*x, x]^p*Simp[b/2 + q/2 + c*x, x]^p, x], x]] /; FreeQ[{a, b, c}, x] && NeQ[b^2 - 4*a*c, 0] && IGtQ[p, 0] && PerfectSquareQ[b^2 - 4*a*c]
  611. Int[Power[Plus[Optional[Pattern[a, Blank[]]], Times[Optional[Pattern[b, Blank[]]], Pattern[x, Blank[]]], Times[Optional[Pattern[c, Blank[]]], Power[Pattern[x, Blank[]], 2]]], Pattern[p, Blank[]]], Pattern[x, Blank[Symbol]]] := Int[ExpandIntegrand[(a + b*x + c*x^2)^p, x], x] /; FreeQ[{a, b, c}, x] && NeQ[b^2 - 4*a*c, 0] && IGtQ[p, 0] && (EqQ[a, 0] || !PerfectSquareQ[b^2 - 4*a*c])
  612. Int[Power[Plus[Optional[Pattern[a, Blank[]]], Times[Optional[Pattern[b, Blank[]]], Pattern[x, Blank[]]], Times[Optional[Pattern[c, Blank[]]], Power[Pattern[x, Blank[]], 2]]], Pattern[p, Blank[]]], Pattern[x, Blank[Symbol]]] := Simp[((b + 2*c*x)*(a + b*x + c*x^2)^p)/(2*c*(2*p + 1)), x] - Dist[(p*(b^2 - 4*a*c))/(2*c*(2*p + 1)), Int[(a + b*x + c*x^2)^(p - 1), x], x] /; FreeQ[{a, b, c}, x] && NeQ[b^2 - 4*a*c, 0] && GtQ[p, 0] && IntegerQ[4*p]
  613. Int[Power[Plus[Optional[Pattern[a, Blank[]]], Times[Optional[Pattern[b, Blank[]]], Pattern[x, Blank[]]], Times[Optional[Pattern[c, Blank[]]], Power[Pattern[x, Blank[]], 2]]], Rational[-3, 2]], Pattern[x, Blank[Symbol]]] := Simp[(-2*(b + 2*c*x))/((b^2 - 4*a*c)*Sqrt[a + b*x + c*x^2]), x] /; FreeQ[{a, b, c}, x] && NeQ[b^2 - 4*a*c, 0]
  614. Int[Power[Plus[Optional[Pattern[a, Blank[]]], Times[Optional[Pattern[b, Blank[]]], Pattern[x, Blank[]]], Times[Optional[Pattern[c, Blank[]]], Power[Pattern[x, Blank[]], 2]]], Pattern[p, Blank[]]], Pattern[x, Blank[Symbol]]] := Simp[((b + 2*c*x)*(a + b*x + c*x^2)^(p + 1))/((p + 1)*(b^2 - 4*a*c)), x] - Dist[(2*c*(2*p + 3))/((p + 1)*(b^2 - 4*a*c)), Int[(a + b*x + c*x^2)^(p + 1), x], x] /; FreeQ[{a, b, c}, x] && NeQ[b^2 - 4*a*c, 0] && LtQ[p, -1] && NeQ[p, -3/2] && IntegerQ[4*p]
  615. Int[Power[Plus[Times[Optional[Pattern[b, Blank[]]], Pattern[x, Blank[]]], Times[Optional[Pattern[c, Blank[]]], Power[Pattern[x, Blank[]], 2]]], -1], Pattern[x, Blank[Symbol]]] := Simp[Log[x]/b, x] - Simp[Log[RemoveContent[b + c*x, x]]/b, x] /; FreeQ[{b, c}, x]
  616. Int[Power[Plus[Optional[Pattern[a, Blank[]]], Times[Optional[Pattern[b, Blank[]]], Pattern[x, Blank[]]], Times[Optional[Pattern[c, Blank[]]], Power[Pattern[x, Blank[]], 2]]], -1], Pattern[x, Blank[Symbol]]] := With[{q = Rt[b^2 - 4*a*c, 2]}, Dist[c/q, Int[1/Simp[b/2 - q/2 + c*x, x], x], x] - Dist[c/q, Int[1/Simp[b/2 + q/2 + c*x, x], x], x]] /; FreeQ[{a, b, c}, x] && NeQ[b^2 - 4*a*c, 0] && PosQ[b^2 - 4*a*c] && PerfectSquareQ[b^2 - 4*a*c]
  617. Int[Power[Plus[Pattern[a, Blank[]], Times[Optional[Pattern[b, Blank[]]], Pattern[x, Blank[]]], Times[Optional[Pattern[c, Blank[]]], Power[Pattern[x, Blank[]], 2]]], -1], Pattern[x, Blank[Symbol]]] := With[{q = 1 - 4*Simplify[(a*c)/b^2]}, Dist[-2/b, Subst[Int[1/(q - x^2), x], x, 1 + (2*c*x)/b], x] /; RationalQ[q] && (EqQ[q^2, 1] || !RationalQ[b^2 - 4*a*c])] /; FreeQ[{a, b, c}, x] && NeQ[b^2 - 4*a*c, 0]
  618. Int[Power[Plus[Optional[Pattern[a, Blank[]]], Times[Optional[Pattern[b, Blank[]]], Pattern[x, Blank[]]], Times[Optional[Pattern[c, Blank[]]], Power[Pattern[x, Blank[]], 2]]], -1], Pattern[x, Blank[Symbol]]] := Dist[-2, Subst[Int[1/Simp[b^2 - 4*a*c - x^2, x], x], x, b + 2*c*x], x] /; FreeQ[{a, b, c}, x] && NeQ[b^2 - 4*a*c, 0]
  619. Int[Power[Plus[Optional[Pattern[a, Blank[]]], Times[Optional[Pattern[b, Blank[]]], Pattern[x, Blank[]]], Times[Optional[Pattern[c, Blank[]]], Power[Pattern[x, Blank[]], 2]]], Pattern[p, Blank[]]], Pattern[x, Blank[Symbol]]] := Dist[1/(2*c*((-4*c)/(b^2 - 4*a*c))^p), Subst[Int[Simp[1 - x^2/(b^2 - 4*a*c), x]^p, x], x, b + 2*c*x], x] /; FreeQ[{a, b, c, p}, x] && GtQ[4*a - b^2/c, 0]
  620. Int[Power[Plus[Times[Optional[Pattern[b, Blank[]]], Pattern[x, Blank[]]], Times[Optional[Pattern[c, Blank[]]], Power[Pattern[x, Blank[]], 2]]], Rational[-1, 2]], Pattern[x, Blank[Symbol]]] := Dist[2, Subst[Int[1/(1 - c*x^2), x], x, x/Sqrt[b*x + c*x^2]], x] /; FreeQ[{b, c}, x]
  621. Int[Power[Plus[Pattern[a, Blank[]], Times[Optional[Pattern[b, Blank[]]], Pattern[x, Blank[]]], Times[Optional[Pattern[c, Blank[]]], Power[Pattern[x, Blank[]], 2]]], Rational[-1, 2]], Pattern[x, Blank[Symbol]]] := Dist[2, Subst[Int[1/(4*c - x^2), x], x, (b + 2*c*x)/Sqrt[a + b*x + c*x^2]], x] /; FreeQ[{a, b, c}, x] && NeQ[b^2 - 4*a*c, 0]
  622. Int[Power[Plus[Times[Optional[Pattern[b, Blank[]]], Pattern[x, Blank[]]], Times[Optional[Pattern[c, Blank[]]], Power[Pattern[x, Blank[]], 2]]], Pattern[p, Blank[]]], Pattern[x, Blank[Symbol]]] := Dist[(b*x + c*x^2)^p/(-((c*(b*x + c*x^2))/b^2))^p, Int[(-((c*x)/b) - (c^2*x^2)/b^2)^p, x], x] /; FreeQ[{b, c}, x] && RationalQ[p] && 3 <= Denominator[p] <= 4
  623. Int[Power[Plus[Optional[Pattern[a, Blank[]]], Times[Optional[Pattern[b, Blank[]]], Pattern[x, Blank[]]], Times[Optional[Pattern[c, Blank[]]], Power[Pattern[x, Blank[]], 2]]], Pattern[p, Blank[]]], Pattern[x, Blank[Symbol]]] := With[{d = Denominator[p]}, Dist[(d*Sqrt[(b + 2*c*x)^2])/(b + 2*c*x), Subst[Int[x^(d*(p + 1) - 1)/Sqrt[b^2 - 4*a*c + 4*c*x^d], x], x, (a + b*x + c*x^2)^(1/d)], x] /; 3 <= d <= 4] /; FreeQ[{a, b, c}, x] && NeQ[b^2 - 4*a*c, 0] && RationalQ[p]
  624. Int[Power[Plus[Optional[Pattern[a, Blank[]]], Times[Optional[Pattern[b, Blank[]]], Pattern[x, Blank[]]], Times[Optional[Pattern[c, Blank[]]], Power[Pattern[x, Blank[]], 2]]], Pattern[p, Blank[]]], Pattern[x, Blank[Symbol]]] := With[{q = Rt[b^2 - 4*a*c, 2]}, -Simp[((a + b*x + c*x^2)^(p + 1)*Hypergeometric2F1[-p, p + 1, p + 2, (b + q + 2*c*x)/(2*q)])/(q*(p + 1)*((q - b - 2*c*x)/(2*q))^(p + 1)), x]] /; FreeQ[{a, b, c, p}, x] && NeQ[b^2 - 4*a*c, 0] && !IntegerQ[4*p]
  625. Int[Power[Plus[Optional[Pattern[a, Blank[]]], Times[Optional[Pattern[b, Blank[]]], Pattern[u, Blank[]]], Times[Optional[Pattern[c, Blank[]]], Power[Pattern[u, Blank[]], 2]]], Pattern[p, Blank[]]], Pattern[x, Blank[Symbol]]] := Dist[1/Coefficient[u, x, 1], Subst[Int[(a + b*x + c*x^2)^p, x], x, u], x] /; FreeQ[{a, b, c, p}, x] && LinearQ[u, x] && NeQ[u, x]
  626. Int[Times[Power[Plus[Pattern[d, Blank[]], Times[Optional[Pattern[e, Blank[]]], Pattern[x, Blank[]]]], Optional[Pattern[m, Blank[]]]], Power[Plus[Optional[Pattern[a, Blank[]]], Times[Optional[Pattern[b, Blank[]]], Pattern[x, Blank[]]], Times[Optional[Pattern[c, Blank[]]], Power[Pattern[x, Blank[]], 2]]], Optional[Pattern[p, Blank[]]]]], Pattern[x, Blank[Symbol]]] := Int[(d + e*x)^(m + p)*(a/d + (c*x)/e)^p, x] /; FreeQ[{a, b, c, d, e, m}, x] && NeQ[b^2 - 4*a*c, 0] && EqQ[c*d^2 - b*d*e + a*e^2, 0] && IntegerQ[p]
  627. Int[Times[Power[Plus[Pattern[d, Blank[]], Times[Optional[Pattern[e, Blank[]]], Pattern[x, Blank[]]]], Optional[Pattern[m, Blank[]]]], Power[Plus[Pattern[a, Blank[]], Times[Optional[Pattern[c, Blank[]]], Power[Pattern[x, Blank[]], 2]]], Optional[Pattern[p, Blank[]]]]], Pattern[x, Blank[Symbol]]] := Int[(d + e*x)^(m + p)*(a/d + (c*x)/e)^p, x] /; FreeQ[{a, c, d, e, m, p}, x] && EqQ[c*d^2 + a*e^2, 0] && (IntegerQ[p] || (GtQ[a, 0] && GtQ[d, 0] && IntegerQ[m + p]))
  628. Int[Times[Plus[Pattern[d, Blank[]], Times[Optional[Pattern[e, Blank[]]], Pattern[x, Blank[]]]], Power[Plus[Optional[Pattern[a, Blank[]]], Times[Optional[Pattern[b, Blank[]]], Pattern[x, Blank[]]], Times[Optional[Pattern[c, Blank[]]], Power[Pattern[x, Blank[]], 2]]], -1]], Pattern[x, Blank[Symbol]]] := Simp[(d*Log[RemoveContent[a + b*x + c*x^2, x]])/b, x] /; FreeQ[{a, b, c, d, e}, x] && EqQ[2*c*d - b*e, 0]
  629. Int[Times[Plus[Pattern[d, Blank[]], Times[Optional[Pattern[e, Blank[]]], Pattern[x, Blank[]]]], Power[Plus[Optional[Pattern[a, Blank[]]], Times[Optional[Pattern[b, Blank[]]], Pattern[x, Blank[]]], Times[Optional[Pattern[c, Blank[]]], Power[Pattern[x, Blank[]], 2]]], Optional[Pattern[p, Blank[]]]]], Pattern[x, Blank[Symbol]]] := Simp[(d*(a + b*x + c*x^2)^(p + 1))/(b*(p + 1)), x] /; FreeQ[{a, b, c, d, e, p}, x] && EqQ[2*c*d - b*e, 0] && NeQ[p, -1]
  630. Int[Times[Plus[Pattern[d, Blank[]], Times[Optional[Pattern[e, Blank[]]], Pattern[x, Blank[]]]], Power[Plus[Optional[Pattern[a, Blank[]]], Times[Optional[Pattern[b, Blank[]]], Pattern[x, Blank[]]], Times[Optional[Pattern[c, Blank[]]], Power[Pattern[x, Blank[]], 2]]], Optional[Pattern[p, Blank[]]]]], Pattern[x, Blank[Symbol]]] := Int[(d + e*x)^(p + 1)*(a/d + (c*x)/e)^p, x] /; FreeQ[{a, b, c, d, e}, x] && NeQ[2*c*d - b*e, 0] && IGtQ[p, 0] && EqQ[c*d^2 - b*d*e + a*e^2, 0]
  631. Int[Times[Plus[Optional[Pattern[d, Blank[]]], Times[Optional[Pattern[e, Blank[]]], Pattern[x, Blank[]]]], Power[Plus[Optional[Pattern[a, Blank[]]], Times[Optional[Pattern[b, Blank[]]], Pattern[x, Blank[]]], Times[Optional[Pattern[c, Blank[]]], Power[Pattern[x, Blank[]], 2]]], Optional[Pattern[p, Blank[]]]]], Pattern[x, Blank[Symbol]]] := Int[ExpandIntegrand[(d + e*x)*(a + b*x + c*x^2)^p, x], x] /; FreeQ[{a, b, c, d, e}, x] && NeQ[2*c*d - b*e, 0] && IntegerQ[p] && (GtQ[p, 0] || EqQ[a, 0])
  632. Int[Times[Plus[Optional[Pattern[d, Blank[]]], Times[Optional[Pattern[e, Blank[]]], Pattern[x, Blank[]]]], Power[Plus[Pattern[a, Blank[]], Times[Optional[Pattern[b, Blank[]]], Pattern[x, Blank[]]], Times[Optional[Pattern[c, Blank[]]], Power[Pattern[x, Blank[]], 2]]], -1]], Pattern[x, Blank[Symbol]]] := With[{q = Rt[b^2 - 4*a*c, 2]}, Dist[(c*d - e*(b/2 - q/2))/q, Int[1/(b/2 - q/2 + c*x), x], x] - Dist[(c*d - e*(b/2 + q/2))/q, Int[1/(b/2 + q/2 + c*x), x], x]] /; FreeQ[{a, b, c, d, e}, x] && NeQ[2*c*d - b*e, 0] && NeQ[b^2 - 4*a*c, 0] && NiceSqrtQ[b^2 - 4*a*c]
  633. Int[Times[Plus[Pattern[d, Blank[]], Times[Optional[Pattern[e, Blank[]]], Pattern[x, Blank[]]]], Power[Plus[Pattern[a, Blank[]], Times[Optional[Pattern[c, Blank[]]], Power[Pattern[x, Blank[]], 2]]], -1]], Pattern[x, Blank[Symbol]]] := With[{q = Rt[-(a*c), 2]}, Dist[e/2 + (c*d)/(2*q), Int[1/(-q + c*x), x], x] + Dist[e/2 - (c*d)/(2*q), Int[1/(q + c*x), x], x]] /; FreeQ[{a, c, d, e}, x] && NiceSqrtQ[-(a*c)]
  634. Int[Times[Plus[Optional[Pattern[d, Blank[]]], Times[Optional[Pattern[e, Blank[]]], Pattern[x, Blank[]]]], Power[Plus[Pattern[a, Blank[]], Times[Optional[Pattern[b, Blank[]]], Pattern[x, Blank[]]], Times[Optional[Pattern[c, Blank[]]], Power[Pattern[x, Blank[]], 2]]], -1]], Pattern[x, Blank[Symbol]]] := Dist[(2*c*d - b*e)/(2*c), Int[1/(a + b*x + c*x^2), x], x] + Dist[e/(2*c), Int[(b + 2*c*x)/(a + b*x + c*x^2), x], x] /; FreeQ[{a, b, c, d, e}, x] && NeQ[2*c*d - b*e, 0] && NeQ[b^2 - 4*a*c, 0] && !NiceSqrtQ[b^2 - 4*a*c]
  635. Int[Times[Plus[Pattern[d, Blank[]], Times[Optional[Pattern[e, Blank[]]], Pattern[x, Blank[]]]], Power[Plus[Pattern[a, Blank[]], Times[Optional[Pattern[c, Blank[]]], Power[Pattern[x, Blank[]], 2]]], -1]], Pattern[x, Blank[Symbol]]] := Dist[d, Int[1/(a + c*x^2), x], x] + Dist[e, Int[x/(a + c*x^2), x], x] /; FreeQ[{a, c, d, e}, x] && !NiceSqrtQ[-(a*c)]
  636. Int[Times[Plus[Optional[Pattern[d, Blank[]]], Times[Optional[Pattern[e, Blank[]]], Pattern[x, Blank[]]]], Power[Plus[Optional[Pattern[a, Blank[]]], Times[Optional[Pattern[b, Blank[]]], Pattern[x, Blank[]]], Times[Optional[Pattern[c, Blank[]]], Power[Pattern[x, Blank[]], 2]]], Rational[-3, 2]]], Pattern[x, Blank[Symbol]]] := Simp[(-2*(b*d - 2*a*e + (2*c*d - b*e)*x))/((b^2 - 4*a*c)*Sqrt[a + b*x + c*x^2]), x] /; FreeQ[{a, b, c, d, e}, x] && NeQ[2*c*d - b*e, 0] && NeQ[b^2 - 4*a*c, 0]
  637. Int[Times[Plus[Pattern[d, Blank[]], Times[Optional[Pattern[e, Blank[]]], Pattern[x, Blank[]]]], Power[Plus[Pattern[a, Blank[]], Times[Optional[Pattern[c, Blank[]]], Power[Pattern[x, Blank[]], 2]]], Rational[-3, 2]]], Pattern[x, Blank[Symbol]]] := Simp[(-(a*e) + c*d*x)/(a*c*Sqrt[a + c*x^2]), x] /; FreeQ[{a, c, d, e}, x]
  638. Int[Times[Plus[Optional[Pattern[d, Blank[]]], Times[Optional[Pattern[e, Blank[]]], Pattern[x, Blank[]]]], Power[Plus[Optional[Pattern[a, Blank[]]], Times[Optional[Pattern[b, Blank[]]], Pattern[x, Blank[]]], Times[Optional[Pattern[c, Blank[]]], Power[Pattern[x, Blank[]], 2]]], Pattern[p, Blank[]]]], Pattern[x, Blank[Symbol]]] := Simp[((b*d - 2*a*e + (2*c*d - b*e)*x)*(a + b*x + c*x^2)^(p + 1))/((p + 1)*(b^2 - 4*a*c)), x] - Dist[((2*p + 3)*(2*c*d - b*e))/((p + 1)*(b^2 - 4*a*c)), Int[(a + b*x + c*x^2)^(p + 1), x], x] /; FreeQ[{a, b, c, d, e}, x] && NeQ[2*c*d - b*e, 0] && NeQ[b^2 - 4*a*c, 0] && LtQ[p, -1] && NeQ[p, -3/2]
  639. Int[Times[Plus[Pattern[d, Blank[]], Times[Optional[Pattern[e, Blank[]]], Pattern[x, Blank[]]]], Power[Plus[Pattern[a, Blank[]], Times[Optional[Pattern[c, Blank[]]], Power[Pattern[x, Blank[]], 2]]], Pattern[p, Blank[]]]], Pattern[x, Blank[Symbol]]] := Simp[((a*e - c*d*x)*(a + c*x^2)^(p + 1))/(2*a*c*(p + 1)), x] + Dist[(d*(2*p + 3))/(2*a*(p + 1)), Int[(a + c*x^2)^(p + 1), x], x] /; FreeQ[{a, c, d, e}, x] && LtQ[p, -1] && NeQ[p, -3/2]
  640. Int[Times[Plus[Optional[Pattern[d, Blank[]]], Times[Optional[Pattern[e, Blank[]]], Pattern[x, Blank[]]]], Power[Plus[Optional[Pattern[a, Blank[]]], Times[Optional[Pattern[b, Blank[]]], Pattern[x, Blank[]]], Times[Optional[Pattern[c, Blank[]]], Power[Pattern[x, Blank[]], 2]]], Pattern[p, Blank[]]]], Pattern[x, Blank[Symbol]]] := Simp[(e*(a + b*x + c*x^2)^(p + 1))/(2*c*(p + 1)), x] + Dist[(2*c*d - b*e)/(2*c), Int[(a + b*x + c*x^2)^p, x], x] /; FreeQ[{a, b, c, d, e, p}, x] && NeQ[2*c*d - b*e, 0] && NeQ[p, -1]
  641. Int[Times[Plus[Pattern[d, Blank[]], Times[Optional[Pattern[e, Blank[]]], Pattern[x, Blank[]]]], Power[Plus[Pattern[a, Blank[]], Times[Optional[Pattern[c, Blank[]]], Power[Pattern[x, Blank[]], 2]]], Optional[Pattern[p, Blank[]]]]], Pattern[x, Blank[Symbol]]] := Simp[(e*(a + c*x^2)^(p + 1))/(2*c*(p + 1)), x] + Dist[d, Int[(a + c*x^2)^p, x], x] /; FreeQ[{a, c, d, e, p}, x] && NeQ[p, -1]
  642. Int[Times[Power[Plus[Pattern[d, Blank[]], Times[Optional[Pattern[e, Blank[]]], Pattern[x, Blank[]]]], Pattern[m, Blank[]]], Power[Plus[Pattern[a, Blank[]], Times[Optional[Pattern[b, Blank[]]], Pattern[x, Blank[]]], Times[Optional[Pattern[c, Blank[]]], Power[Pattern[x, Blank[]], 2]]], Pattern[p, Blank[]]]], Pattern[x, Blank[Symbol]]] := Dist[e^m/c^(m/2), Int[(a + b*x + c*x^2)^(p + m/2), x], x] /; FreeQ[{a, b, c, d, e, p}, x] && EqQ[b^2 - 4*a*c, 0] && !IntegerQ[p] && EqQ[2*c*d - b*e, 0] && IntegerQ[m/2]
  643. Int[Times[Power[Plus[Pattern[d, Blank[]], Times[Optional[Pattern[e, Blank[]]], Pattern[x, Blank[]]]], Pattern[m, Blank[]]], Power[Plus[Pattern[a, Blank[]], Times[Optional[Pattern[b, Blank[]]], Pattern[x, Blank[]]], Times[Optional[Pattern[c, Blank[]]], Power[Pattern[x, Blank[]], 2]]], Pattern[p, Blank[]]]], Pattern[x, Blank[Symbol]]] := Dist[e^(m - 1)/c^((m - 1)/2), Int[(d + e*x)*(a + b*x + c*x^2)^(p + (m - 1)/2), x], x] /; FreeQ[{a, b, c, d, e, p}, x] && EqQ[b^2 - 4*a*c, 0] && !IntegerQ[p] && EqQ[2*c*d - b*e, 0] && IntegerQ[(m - 1)/2]
  644. Int[Times[Power[Plus[Pattern[d, Blank[]], Times[Optional[Pattern[e, Blank[]]], Pattern[x, Blank[]]]], Pattern[m, Blank[]]], Power[Plus[Pattern[a, Blank[]], Times[Optional[Pattern[b, Blank[]]], Pattern[x, Blank[]]], Times[Optional[Pattern[c, Blank[]]], Power[Pattern[x, Blank[]], 2]]], Pattern[p, Blank[]]]], Pattern[x, Blank[Symbol]]] := Dist[(a + b*x + c*x^2)^p/(d + e*x)^(2*p), Int[(d + e*x)^(m + 2*p), x], x] /; FreeQ[{a, b, c, d, e, m, p}, x] && EqQ[b^2 - 4*a*c, 0] && !IntegerQ[p] && EqQ[2*c*d - b*e, 0] && !IntegerQ[m]
  645. Int[Times[Power[Plus[Optional[Pattern[d, Blank[]]], Times[Optional[Pattern[e, Blank[]]], Pattern[x, Blank[]]]], Pattern[m, Blank[]]], Power[Plus[Pattern[a, Blank[]], Times[Optional[Pattern[b, Blank[]]], Pattern[x, Blank[]]], Times[Optional[Pattern[c, Blank[]]], Power[Pattern[x, Blank[]], 2]]], Pattern[p, Blank[]]]], Pattern[x, Blank[Symbol]]] := Dist[(a + b*x + c*x^2)^FracPart[p]/(c^IntPart[p]*(b/2 + c*x)^(2*FracPart[p])), Int[ExpandLinearProduct[(b/2 + c*x)^(2*p), (d + e*x)^m, b/2, c, x], x], x] /; FreeQ[{a, b, c, d, e, m, p}, x] && EqQ[b^2 - 4*a*c, 0] && !IntegerQ[p] && NeQ[2*c*d - b*e, 0] && IGtQ[m, 0] && EqQ[m - 2*p + 1, 0]
  646. Int[Times[Power[Plus[Optional[Pattern[d, Blank[]]], Times[Optional[Pattern[e, Blank[]]], Pattern[x, Blank[]]]], Pattern[m, Blank[]]], Power[Plus[Pattern[a, Blank[]], Times[Optional[Pattern[b, Blank[]]], Pattern[x, Blank[]]], Times[Optional[Pattern[c, Blank[]]], Power[Pattern[x, Blank[]], 2]]], Pattern[p, Blank[]]]], Pattern[x, Blank[Symbol]]] := Dist[(a + b*x + c*x^2)^FracPart[p]/(c^IntPart[p]*(b/2 + c*x)^(2*FracPart[p])), Int[(d + e*x)^m*(b/2 + c*x)^(2*p), x], x] /; FreeQ[{a, b, c, d, e, m, p}, x] && EqQ[b^2 - 4*a*c, 0] && !IntegerQ[p] && NeQ[2*c*d - b*e, 0]
  647. Int[Times[Power[Times[Optional[Pattern[e, Blank[]]], Pattern[x, Blank[]]], Optional[Pattern[m, Blank[]]]], Power[Plus[Times[Optional[Pattern[b, Blank[]]], Pattern[x, Blank[]]], Times[Optional[Pattern[c, Blank[]]], Power[Pattern[x, Blank[]], 2]]], Optional[Pattern[p, Blank[]]]]], Pattern[x, Blank[Symbol]]] := Dist[1/e^p, Int[(e*x)^(m + p)*(b + c*x)^p, x], x] /; FreeQ[{b, c, e, m}, x] && IntegerQ[p]
  648. Int[Times[Power[Plus[Optional[Pattern[d, Blank[]]], Times[Optional[Pattern[e, Blank[]]], Pattern[x, Blank[]]]], Pattern[m, Blank[]]], Power[Plus[Optional[Pattern[a, Blank[]]], Times[Optional[Pattern[b, Blank[]]], Pattern[x, Blank[]]], Times[Optional[Pattern[c, Blank[]]], Power[Pattern[x, Blank[]], 2]]], Pattern[p, Blank[]]]], Pattern[x, Blank[Symbol]]] := Simp[(e*(d + e*x)^(m - 1)*(a + b*x + c*x^2)^(p + 1))/(c*(p + 1)), x] /; FreeQ[{a, b, c, d, e, m, p}, x] && NeQ[b^2 - 4*a*c, 0] && EqQ[c*d^2 - b*d*e + a*e^2, 0] && !IntegerQ[p] && EqQ[m + p, 0]
  649. Int[Times[Power[Plus[Pattern[d, Blank[]], Times[Optional[Pattern[e, Blank[]]], Pattern[x, Blank[]]]], Pattern[m, Blank[]]], Power[Plus[Pattern[a, Blank[]], Times[Optional[Pattern[c, Blank[]]], Power[Pattern[x, Blank[]], 2]]], Pattern[p, Blank[]]]], Pattern[x, Blank[Symbol]]] := Simp[(e*(d + e*x)^(m - 1)*(a + c*x^2)^(p + 1))/(c*(p + 1)), x] /; FreeQ[{a, c, d, e, m, p}, x] && EqQ[c*d^2 + a*e^2, 0] && !IntegerQ[p] && EqQ[m + p, 0]
  650. Int[Times[Power[Plus[Optional[Pattern[d, Blank[]]], Times[Optional[Pattern[e, Blank[]]], Pattern[x, Blank[]]]], Pattern[m, Blank[]]], Power[Plus[Optional[Pattern[a, Blank[]]], Times[Optional[Pattern[b, Blank[]]], Pattern[x, Blank[]]], Times[Optional[Pattern[c, Blank[]]], Power[Pattern[x, Blank[]], 2]]], Pattern[p, Blank[]]]], Pattern[x, Blank[Symbol]]] := Simp[(e*(d + e*x)^m*(a + b*x + c*x^2)^(p + 1))/((p + 1)*(2*c*d - b*e)), x] /; FreeQ[{a, b, c, d, e, m, p}, x] && NeQ[b^2 - 4*a*c, 0] && EqQ[c*d^2 - b*d*e + a*e^2, 0] && !IntegerQ[p] && EqQ[m + 2*p + 2, 0]
  651. Int[Times[Power[Plus[Pattern[d, Blank[]], Times[Optional[Pattern[e, Blank[]]], Pattern[x, Blank[]]]], Pattern[m, Blank[]]], Power[Plus[Pattern[a, Blank[]], Times[Optional[Pattern[c, Blank[]]], Power[Pattern[x, Blank[]], 2]]], Pattern[p, Blank[]]]], Pattern[x, Blank[Symbol]]] := Simp[(e*(d + e*x)^m*(a + c*x^2)^(p + 1))/(2*c*d*(p + 1)), x] /; FreeQ[{a, c, d, e, m, p}, x] && EqQ[c*d^2 + a*e^2, 0] && !IntegerQ[p] && EqQ[m + 2*p + 2, 0]
  652. Int[Times[Power[Plus[Optional[Pattern[d, Blank[]]], Times[Optional[Pattern[e, Blank[]]], Pattern[x, Blank[]]]], 2], Power[Plus[Optional[Pattern[a, Blank[]]], Times[Optional[Pattern[b, Blank[]]], Pattern[x, Blank[]]], Times[Optional[Pattern[c, Blank[]]], Power[Pattern[x, Blank[]], 2]]], Pattern[p, Blank[]]]], Pattern[x, Blank[Symbol]]] := Simp[(e*(d + e*x)*(a + b*x + c*x^2)^(p + 1))/(c*(p + 1)), x] - Dist[(e^2*(p + 2))/(c*(p + 1)), Int[(a + b*x + c*x^2)^(p + 1), x], x] /; FreeQ[{a, b, c, d, e, p}, x] && NeQ[b^2 - 4*a*c, 0] && EqQ[c*d^2 - b*d*e + a*e^2, 0] && !IntegerQ[p] && LtQ[p, -1]
  653. Int[Times[Power[Plus[Pattern[d, Blank[]], Times[Optional[Pattern[e, Blank[]]], Pattern[x, Blank[]]]], 2], Power[Plus[Pattern[a, Blank[]], Times[Optional[Pattern[c, Blank[]]], Power[Pattern[x, Blank[]], 2]]], Pattern[p, Blank[]]]], Pattern[x, Blank[Symbol]]] := Simp[(e*(d + e*x)*(a + c*x^2)^(p + 1))/(c*(p + 1)), x] - Dist[(e^2*(p + 2))/(c*(p + 1)), Int[(a + c*x^2)^(p + 1), x], x] /; FreeQ[{a, c, d, e, p}, x] && EqQ[c*d^2 + a*e^2, 0] && !IntegerQ[p] && LtQ[p, -1]
  654. Int[Times[Power[Plus[Pattern[d, Blank[]], Times[Optional[Pattern[e, Blank[]]], Pattern[x, Blank[]]]], Pattern[m, Blank[]]], Power[Plus[Optional[Pattern[a, Blank[]]], Times[Optional[Pattern[b, Blank[]]], Pattern[x, Blank[]]], Times[Optional[Pattern[c, Blank[]]], Power[Pattern[x, Blank[]], 2]]], Pattern[p, Blank[]]]], Pattern[x, Blank[Symbol]]] := Int[(a + b*x + c*x^2)^(m + p)/(a/d + (c*x)/e)^m, x] /; FreeQ[{a, b, c, d, e}, x] && NeQ[b^2 - 4*a*c, 0] && EqQ[c*d^2 - b*d*e + a*e^2, 0] && !IntegerQ[p] && IntegerQ[m] && RationalQ[p] && (LtQ[0, -m, p] || LtQ[p, -m, 0]) && NeQ[m, 2] && NeQ[m, -1]
  655. Int[Times[Power[Plus[Pattern[d, Blank[]], Times[Optional[Pattern[e, Blank[]]], Pattern[x, Blank[]]]], Pattern[m, Blank[]]], Power[Plus[Pattern[a, Blank[]], Times[Optional[Pattern[c, Blank[]]], Power[Pattern[x, Blank[]], 2]]], Pattern[p, Blank[]]]], Pattern[x, Blank[Symbol]]] := Dist[d^(2*m)/a^m, Int[(a + c*x^2)^(m + p)/(d - e*x)^m, x], x] /; FreeQ[{a, c, d, e, m, p}, x] && EqQ[c*d^2 + a*e^2, 0] && !IntegerQ[p] && IntegerQ[m] && RationalQ[p] && (LtQ[0, -m, p] || LtQ[p, -m, 0]) && NeQ[m, 2] && NeQ[m, -1]
  656. Int[Times[Power[Plus[Optional[Pattern[d, Blank[]]], Times[Optional[Pattern[e, Blank[]]], Pattern[x, Blank[]]]], Pattern[m, Blank[]]], Power[Plus[Optional[Pattern[a, Blank[]]], Times[Optional[Pattern[b, Blank[]]], Pattern[x, Blank[]]], Times[Optional[Pattern[c, Blank[]]], Power[Pattern[x, Blank[]], 2]]], Pattern[p, Blank[]]]], Pattern[x, Blank[Symbol]]] := Simp[(e*(d + e*x)^(m - 1)*(a + b*x + c*x^2)^(p + 1))/(c*(m + 2*p + 1)), x] + Dist[(Simplify[m + p]*(2*c*d - b*e))/(c*(m + 2*p + 1)), Int[(d + e*x)^(m - 1)*(a + b*x + c*x^2)^p, x], x] /; FreeQ[{a, b, c, d, e, m, p}, x] && NeQ[b^2 - 4*a*c, 0] && EqQ[c*d^2 - b*d*e + a*e^2, 0] && !IntegerQ[p] && IGtQ[Simplify[m + p], 0]
  657. Int[Times[Power[Plus[Pattern[d, Blank[]], Times[Optional[Pattern[e, Blank[]]], Pattern[x, Blank[]]]], Pattern[m, Blank[]]], Power[Plus[Pattern[a, Blank[]], Times[Optional[Pattern[c, Blank[]]], Power[Pattern[x, Blank[]], 2]]], Pattern[p, Blank[]]]], Pattern[x, Blank[Symbol]]] := Simp[(e*(d + e*x)^(m - 1)*(a + c*x^2)^(p + 1))/(c*(m + 2*p + 1)), x] + Dist[(2*c*d*Simplify[m + p])/(c*(m + 2*p + 1)), Int[(d + e*x)^(m - 1)*(a + c*x^2)^p, x], x] /; FreeQ[{a, c, d, e, m, p}, x] && EqQ[c*d^2 + a*e^2, 0] && !IntegerQ[p] && IGtQ[Simplify[m + p], 0]
  658. Int[Times[Power[Plus[Optional[Pattern[d, Blank[]]], Times[Optional[Pattern[e, Blank[]]], Pattern[x, Blank[]]]], Pattern[m, Blank[]]], Power[Plus[Optional[Pattern[a, Blank[]]], Times[Optional[Pattern[b, Blank[]]], Pattern[x, Blank[]]], Times[Optional[Pattern[c, Blank[]]], Power[Pattern[x, Blank[]], 2]]], Pattern[p, Blank[]]]], Pattern[x, Blank[Symbol]]] := -Simp[(e*(d + e*x)^m*(a + b*x + c*x^2)^(p + 1))/((m + p + 1)*(2*c*d - b*e)), x] + Dist[(c*Simplify[m + 2*p + 2])/((m + p + 1)*(2*c*d - b*e)), Int[(d + e*x)^(m + 1)*(a + b*x + c*x^2)^p, x], x] /; FreeQ[{a, b, c, d, e, m, p}, x] && NeQ[b^2 - 4*a*c, 0] && EqQ[c*d^2 - b*d*e + a*e^2, 0] && !IntegerQ[p] && ILtQ[Simplify[m + 2*p + 2], 0]
  659. Int[Times[Power[Plus[Pattern[d, Blank[]], Times[Optional[Pattern[e, Blank[]]], Pattern[x, Blank[]]]], Pattern[m, Blank[]]], Power[Plus[Pattern[a, Blank[]], Times[Optional[Pattern[c, Blank[]]], Power[Pattern[x, Blank[]], 2]]], Pattern[p, Blank[]]]], Pattern[x, Blank[Symbol]]] := -Simp[(e*(d + e*x)^m*(a + c*x^2)^(p + 1))/(2*c*d*(m + p + 1)), x] + Dist[Simplify[m + 2*p + 2]/(2*d*(m + p + 1)), Int[(d + e*x)^(m + 1)*(a + c*x^2)^p, x], x] /; FreeQ[{a, c, d, e, m, p}, x] && EqQ[c*d^2 + a*e^2, 0] && !IntegerQ[p] && ILtQ[Simplify[m + 2*p + 2], 0]
  660. Int[Times[Power[Plus[Optional[Pattern[d, Blank[]]], Times[Optional[Pattern[e, Blank[]]], Pattern[x, Blank[]]]], Rational[-1, 2]], Power[Plus[Optional[Pattern[a, Blank[]]], Times[Optional[Pattern[b, Blank[]]], Pattern[x, Blank[]]], Times[Optional[Pattern[c, Blank[]]], Power[Pattern[x, Blank[]], 2]]], Rational[-1, 2]]], Pattern[x, Blank[Symbol]]] := Dist[2*e, Subst[Int[1/(2*c*d - b*e + e^2*x^2), x], x, Sqrt[a + b*x + c*x^2]/Sqrt[d + e*x]], x] /; FreeQ[{a, b, c, d, e}, x] && NeQ[b^2 - 4*a*c, 0] && EqQ[c*d^2 - b*d*e + a*e^2, 0]
  661. Int[Times[Power[Plus[Pattern[d, Blank[]], Times[Optional[Pattern[e, Blank[]]], Pattern[x, Blank[]]]], Rational[-1, 2]], Power[Plus[Pattern[a, Blank[]], Times[Optional[Pattern[c, Blank[]]], Power[Pattern[x, Blank[]], 2]]], Rational[-1, 2]]], Pattern[x, Blank[Symbol]]] := Dist[2*e, Subst[Int[1/(2*c*d + e^2*x^2), x], x, Sqrt[a + c*x^2]/Sqrt[d + e*x]], x] /; FreeQ[{a, c, d, e}, x] && EqQ[c*d^2 + a*e^2, 0]
  662. Int[Times[Power[Plus[Optional[Pattern[d, Blank[]]], Times[Optional[Pattern[e, Blank[]]], Pattern[x, Blank[]]]], Pattern[m, Blank[]]], Power[Plus[Optional[Pattern[a, Blank[]]], Times[Optional[Pattern[b, Blank[]]], Pattern[x, Blank[]]], Times[Optional[Pattern[c, Blank[]]], Power[Pattern[x, Blank[]], 2]]], Pattern[p, Blank[]]]], Pattern[x, Blank[Symbol]]] := Simp[((d + e*x)^(m + 1)*(a + b*x + c*x^2)^p)/(e*(m + p + 1)), x] - Dist[(c*p)/(e^2*(m + p + 1)), Int[(d + e*x)^(m + 2)*(a + b*x + c*x^2)^(p - 1), x], x] /; FreeQ[{a, b, c, d, e}, x] && NeQ[b^2 - 4*a*c, 0] && EqQ[c*d^2 - b*d*e + a*e^2, 0] && GtQ[p, 0] && (LtQ[m, -2] || EqQ[m + 2*p + 1, 0]) && NeQ[m + p + 1, 0] && IntegerQ[2*p]
  663. Int[Times[Power[Plus[Pattern[d, Blank[]], Times[Optional[Pattern[e, Blank[]]], Pattern[x, Blank[]]]], Pattern[m, Blank[]]], Power[Plus[Pattern[a, Blank[]], Times[Optional[Pattern[c, Blank[]]], Power[Pattern[x, Blank[]], 2]]], Pattern[p, Blank[]]]], Pattern[x, Blank[Symbol]]] := Simp[((d + e*x)^(m + 1)*(a + c*x^2)^p)/(e*(m + p + 1)), x] - Dist[(c*p)/(e^2*(m + p + 1)), Int[(d + e*x)^(m + 2)*(a + c*x^2)^(p - 1), x], x] /; FreeQ[{a, c, d, e}, x] && EqQ[c*d^2 + a*e^2, 0] && GtQ[p, 0] && (LtQ[m, -2] || EqQ[m + 2*p + 1, 0]) && NeQ[m + p + 1, 0] && IntegerQ[2*p]
  664. Int[Times[Power[Plus[Optional[Pattern[d, Blank[]]], Times[Optional[Pattern[e, Blank[]]], Pattern[x, Blank[]]]], Pattern[m, Blank[]]], Power[Plus[Optional[Pattern[a, Blank[]]], Times[Optional[Pattern[b, Blank[]]], Pattern[x, Blank[]]], Times[Optional[Pattern[c, Blank[]]], Power[Pattern[x, Blank[]], 2]]], Pattern[p, Blank[]]]], Pattern[x, Blank[Symbol]]] := Simp[((d + e*x)^(m + 1)*(a + b*x + c*x^2)^p)/(e*(m + 2*p + 1)), x] - Dist[(p*(2*c*d - b*e))/(e^2*(m + 2*p + 1)), Int[(d + e*x)^(m + 1)*(a + b*x + c*x^2)^(p - 1), x], x] /; FreeQ[{a, b, c, d, e}, x] && NeQ[b^2 - 4*a*c, 0] && EqQ[c*d^2 - b*d*e + a*e^2, 0] && GtQ[p, 0] && (LeQ[-2, m, 0] || EqQ[m + p + 1, 0]) && NeQ[m + 2*p + 1, 0] && IntegerQ[2*p]
  665. Int[Times[Power[Plus[Pattern[d, Blank[]], Times[Optional[Pattern[e, Blank[]]], Pattern[x, Blank[]]]], Pattern[m, Blank[]]], Power[Plus[Pattern[a, Blank[]], Times[Optional[Pattern[c, Blank[]]], Power[Pattern[x, Blank[]], 2]]], Pattern[p, Blank[]]]], Pattern[x, Blank[Symbol]]] := Simp[((d + e*x)^(m + 1)*(a + c*x^2)^p)/(e*(m + 2*p + 1)), x] - Dist[(2*c*d*p)/(e^2*(m + 2*p + 1)), Int[(d + e*x)^(m + 1)*(a + c*x^2)^(p - 1), x], x] /; FreeQ[{a, c, d, e}, x] && EqQ[c*d^2 + a*e^2, 0] && GtQ[p, 0] && (LeQ[-2, m, 0] || EqQ[m + p + 1, 0]) && NeQ[m + 2*p + 1, 0] && IntegerQ[2*p]
  666. Int[Times[Power[Plus[Optional[Pattern[d, Blank[]]], Times[Optional[Pattern[e, Blank[]]], Pattern[x, Blank[]]]], Pattern[m, Blank[]]], Power[Plus[Optional[Pattern[a, Blank[]]], Times[Optional[Pattern[b, Blank[]]], Pattern[x, Blank[]]], Times[Optional[Pattern[c, Blank[]]], Power[Pattern[x, Blank[]], 2]]], Pattern[p, Blank[]]]], Pattern[x, Blank[Symbol]]] := Simp[((2*c*d - b*e)*(d + e*x)^m*(a + b*x + c*x^2)^(p + 1))/(e*(p + 1)*(b^2 - 4*a*c)), x] - Dist[((2*c*d - b*e)*(m + 2*p + 2))/((p + 1)*(b^2 - 4*a*c)), Int[(d + e*x)^(m - 1)*(a + b*x + c*x^2)^(p + 1), x], x] /; FreeQ[{a, b, c, d, e}, x] && NeQ[b^2 - 4*a*c, 0] && EqQ[c*d^2 - b*d*e + a*e^2, 0] && LtQ[p, -1] && LtQ[0, m, 1] && IntegerQ[2*p]
  667. Int[Times[Power[Plus[Pattern[d, Blank[]], Times[Optional[Pattern[e, Blank[]]], Pattern[x, Blank[]]]], Pattern[m, Blank[]]], Power[Plus[Pattern[a, Blank[]], Times[Optional[Pattern[c, Blank[]]], Power[Pattern[x, Blank[]], 2]]], Pattern[p, Blank[]]]], Pattern[x, Blank[Symbol]]] := -Simp[(d*(d + e*x)^m*(a + c*x^2)^(p + 1))/(2*a*e*(p + 1)), x] + Dist[(d*(m + 2*p + 2))/(2*a*(p + 1)), Int[(d + e*x)^(m - 1)*(a + c*x^2)^(p + 1), x], x] /; FreeQ[{a, c, d, e}, x] && EqQ[c*d^2 + a*e^2, 0] && LtQ[p, -1] && LtQ[0, m, 1] && IntegerQ[2*p]
  668. Int[Times[Power[Plus[Optional[Pattern[d, Blank[]]], Times[Optional[Pattern[e, Blank[]]], Pattern[x, Blank[]]]], Pattern[m, Blank[]]], Power[Plus[Optional[Pattern[a, Blank[]]], Times[Optional[Pattern[b, Blank[]]], Pattern[x, Blank[]]], Times[Optional[Pattern[c, Blank[]]], Power[Pattern[x, Blank[]], 2]]], Pattern[p, Blank[]]]], Pattern[x, Blank[Symbol]]] := Simp[(e*(d + e*x)^(m - 1)*(a + b*x + c*x^2)^(p + 1))/(c*(p + 1)), x] - Dist[(e^2*(m + p))/(c*(p + 1)), Int[(d + e*x)^(m - 2)*(a + b*x + c*x^2)^(p + 1), x], x] /; FreeQ[{a, b, c, d, e}, x] && NeQ[b^2 - 4*a*c, 0] && EqQ[c*d^2 - b*d*e + a*e^2, 0] && LtQ[p, -1] && GtQ[m, 1] && IntegerQ[2*p]
  669. Int[Times[Power[Plus[Pattern[d, Blank[]], Times[Optional[Pattern[e, Blank[]]], Pattern[x, Blank[]]]], Pattern[m, Blank[]]], Power[Plus[Pattern[a, Blank[]], Times[Optional[Pattern[c, Blank[]]], Power[Pattern[x, Blank[]], 2]]], Pattern[p, Blank[]]]], Pattern[x, Blank[Symbol]]] := Simp[(e*(d + e*x)^(m - 1)*(a + c*x^2)^(p + 1))/(c*(p + 1)), x] - Dist[(e^2*(m + p))/(c*(p + 1)), Int[(d + e*x)^(m - 2)*(a + c*x^2)^(p + 1), x], x] /; FreeQ[{a, c, d, e}, x] && EqQ[c*d^2 + a*e^2, 0] && LtQ[p, -1] && GtQ[m, 1] && IntegerQ[2*p]
  670. Int[Times[Power[Plus[Optional[Pattern[d, Blank[]]], Times[Optional[Pattern[e, Blank[]]], Pattern[x, Blank[]]]], Pattern[m, Blank[]]], Power[Plus[Optional[Pattern[a, Blank[]]], Times[Optional[Pattern[b, Blank[]]], Pattern[x, Blank[]]], Times[Optional[Pattern[c, Blank[]]], Power[Pattern[x, Blank[]], 2]]], Pattern[p, Blank[]]]], Pattern[x, Blank[Symbol]]] := Simp[(e*(d + e*x)^(m - 1)*(a + b*x + c*x^2)^(p + 1))/(c*(m + 2*p + 1)), x] + Dist[((m + p)*(2*c*d - b*e))/(c*(m + 2*p + 1)), Int[(d + e*x)^(m - 1)*(a + b*x + c*x^2)^p, x], x] /; FreeQ[{a, b, c, d, e, p}, x] && NeQ[b^2 - 4*a*c, 0] && EqQ[c*d^2 - b*d*e + a*e^2, 0] && GtQ[m, 1] && NeQ[m + 2*p + 1, 0] && IntegerQ[2*p]
  671. Int[Times[Power[Plus[Pattern[d, Blank[]], Times[Optional[Pattern[e, Blank[]]], Pattern[x, Blank[]]]], Pattern[m, Blank[]]], Power[Plus[Pattern[a, Blank[]], Times[Optional[Pattern[c, Blank[]]], Power[Pattern[x, Blank[]], 2]]], Pattern[p, Blank[]]]], Pattern[x, Blank[Symbol]]] := Simp[(e*(d + e*x)^(m - 1)*(a + c*x^2)^(p + 1))/(c*(m + 2*p + 1)), x] + Dist[(2*c*d*(m + p))/(c*(m + 2*p + 1)), Int[(d + e*x)^(m - 1)*(a + c*x^2)^p, x], x] /; FreeQ[{a, c, d, e, p}, x] && EqQ[c*d^2 + a*e^2, 0] && GtQ[m, 1] && NeQ[m + 2*p + 1, 0] && IntegerQ[2*p]
  672. Int[Times[Power[Plus[Optional[Pattern[d, Blank[]]], Times[Optional[Pattern[e, Blank[]]], Pattern[x, Blank[]]]], Pattern[m, Blank[]]], Power[Plus[Optional[Pattern[a, Blank[]]], Times[Optional[Pattern[b, Blank[]]], Pattern[x, Blank[]]], Times[Optional[Pattern[c, Blank[]]], Power[Pattern[x, Blank[]], 2]]], Pattern[p, Blank[]]]], Pattern[x, Blank[Symbol]]] := -Simp[(e*(d + e*x)^m*(a + b*x + c*x^2)^(p + 1))/((m + p + 1)*(2*c*d - b*e)), x] + Dist[(c*(m + 2*p + 2))/((m + p + 1)*(2*c*d - b*e)), Int[(d + e*x)^(m + 1)*(a + b*x + c*x^2)^p, x], x] /; FreeQ[{a, b, c, d, e, p}, x] && NeQ[b^2 - 4*a*c, 0] && EqQ[c*d^2 - b*d*e + a*e^2, 0] && LtQ[m, 0] && NeQ[m + p + 1, 0] && IntegerQ[2*p]
  673. Int[Times[Power[Plus[Pattern[d, Blank[]], Times[Optional[Pattern[e, Blank[]]], Pattern[x, Blank[]]]], Pattern[m, Blank[]]], Power[Plus[Pattern[a, Blank[]], Times[Optional[Pattern[c, Blank[]]], Power[Pattern[x, Blank[]], 2]]], Pattern[p, Blank[]]]], Pattern[x, Blank[Symbol]]] := -Simp[(e*(d + e*x)^m*(a + c*x^2)^(p + 1))/(2*c*d*(m + p + 1)), x] + Dist[(m + 2*p + 2)/(2*d*(m + p + 1)), Int[(d + e*x)^(m + 1)*(a + c*x^2)^p, x], x] /; FreeQ[{a, c, d, e, p}, x] && EqQ[c*d^2 + a*e^2, 0] && LtQ[m, 0] && NeQ[m + p + 1, 0] && IntegerQ[2*p]
  674. Int[Times[Power[Times[Optional[Pattern[e, Blank[]]], Pattern[x, Blank[]]], Pattern[m, Blank[]]], Power[Plus[Times[Optional[Pattern[b, Blank[]]], Pattern[x, Blank[]]], Times[Optional[Pattern[c, Blank[]]], Power[Pattern[x, Blank[]], 2]]], Pattern[p, Blank[]]]], Pattern[x, Blank[Symbol]]] := Dist[((e*x)^m*(b*x + c*x^2)^p)/(x^(m + p)*(b + c*x)^p), Int[x^(m + p)*(b + c*x)^p, x], x] /; FreeQ[{b, c, e, m}, x] && !IntegerQ[p]
  675. Int[Times[Power[Plus[Pattern[d, Blank[]], Times[Optional[Pattern[e, Blank[]]], Pattern[x, Blank[]]]], Pattern[m, Blank[]]], Power[Plus[Pattern[a, Blank[]], Times[Optional[Pattern[c, Blank[]]], Power[Pattern[x, Blank[]], 2]]], Pattern[p, Blank[]]]], Pattern[x, Blank[Symbol]]] := Int[(d + e*x)^(m + p)*(a/d + (c*x)/e)^p, x] /; FreeQ[{a, c, d, e, m, p}, x] && EqQ[c*d^2 + a*e^2, 0] && !IntegerQ[p] && GtQ[a, 0] && GtQ[d, 0] && !IGtQ[m, 0]
  676. Int[Times[Power[Plus[Pattern[d, Blank[]], Times[Optional[Pattern[e, Blank[]]], Pattern[x, Blank[]]]], Pattern[m, Blank[]]], Power[Plus[Pattern[a, Blank[]], Times[Optional[Pattern[c, Blank[]]], Power[Pattern[x, Blank[]], 2]]], Pattern[p, Blank[]]]], Pattern[x, Blank[Symbol]]] := Dist[(a^(p + 1)*d^(m - 1)*((d - e*x)/d)^(p + 1))/(a/d + (c*x)/e)^(p + 1), Int[(1 + (e*x)/d)^(m + p)*(a/d + (c*x)/e)^p, x], x] /; FreeQ[{a, c, d, e, m}, x] && EqQ[c*d^2 + a*e^2, 0] && !IntegerQ[p] && (IntegerQ[m] || GtQ[d, 0]) && GtQ[a, 0] && !(IGtQ[m, 0] && (IntegerQ[3*p] || IntegerQ[4*p]))
  677. Int[Times[Power[Plus[Pattern[d, Blank[]], Times[Optional[Pattern[e, Blank[]]], Pattern[x, Blank[]]]], Pattern[m, Blank[]]], Power[Plus[Optional[Pattern[a, Blank[]]], Times[Optional[Pattern[b, Blank[]]], Pattern[x, Blank[]]], Times[Optional[Pattern[c, Blank[]]], Power[Pattern[x, Blank[]], 2]]], Pattern[p, Blank[]]]], Pattern[x, Blank[Symbol]]] := Dist[(d^m*(a + b*x + c*x^2)^FracPart[p])/((1 + (e*x)/d)^FracPart[p]*(a/d + (c*x)/e)^FracPart[p]), Int[(1 + (e*x)/d)^(m + p)*(a/d + (c*x)/e)^p, x], x] /; FreeQ[{a, b, c, d, e, m}, x] && NeQ[b^2 - 4*a*c, 0] && EqQ[c*d^2 - b*d*e + a*e^2, 0] && !IntegerQ[p] && (IntegerQ[m] || GtQ[d, 0]) && !(IGtQ[m, 0] && (IntegerQ[3*p] || IntegerQ[4*p]))
  678. Int[Times[Power[Plus[Pattern[d, Blank[]], Times[Optional[Pattern[e, Blank[]]], Pattern[x, Blank[]]]], Pattern[m, Blank[]]], Power[Plus[Pattern[a, Blank[]], Times[Optional[Pattern[c, Blank[]]], Power[Pattern[x, Blank[]], 2]]], Pattern[p, Blank[]]]], Pattern[x, Blank[Symbol]]] := Dist[(d^(m - 1)*(a + c*x^2)^(p + 1))/((1 + (e*x)/d)^(p + 1)*(a/d + (c*x)/e)^(p + 1)), Int[(1 + (e*x)/d)^(m + p)*(a/d + (c*x)/e)^p, x], x] /; FreeQ[{a, c, d, e, m}, x] && EqQ[c*d^2 + a*e^2, 0] && !IntegerQ[p] && (IntegerQ[m] || GtQ[d, 0]) && !(IGtQ[m, 0] && (IntegerQ[3*p] || IntegerQ[4*p]))
  679. Int[Times[Power[Plus[Pattern[d, Blank[]], Times[Optional[Pattern[e, Blank[]]], Pattern[x, Blank[]]]], Pattern[m, Blank[]]], Power[Plus[Optional[Pattern[a, Blank[]]], Times[Optional[Pattern[b, Blank[]]], Pattern[x, Blank[]]], Times[Optional[Pattern[c, Blank[]]], Power[Pattern[x, Blank[]], 2]]], Pattern[p, Blank[]]]], Pattern[x, Blank[Symbol]]] := Dist[(d^IntPart[m]*(d + e*x)^FracPart[m])/(1 + (e*x)/d)^FracPart[m], Int[(1 + (e*x)/d)^m*(a + b*x + c*x^2)^p, x], x] /; FreeQ[{a, b, c, d, e, m}, x] && NeQ[b^2 - 4*a*c, 0] && EqQ[c*d^2 - b*d*e + a*e^2, 0] && !IntegerQ[p] && !(IntegerQ[m] || GtQ[d, 0])
  680. Int[Times[Power[Plus[Pattern[d, Blank[]], Times[Optional[Pattern[e, Blank[]]], Pattern[x, Blank[]]]], Pattern[m, Blank[]]], Power[Plus[Pattern[a, Blank[]], Times[Optional[Pattern[c, Blank[]]], Power[Pattern[x, Blank[]], 2]]], Pattern[p, Blank[]]]], Pattern[x, Blank[Symbol]]] := Dist[(d^IntPart[m]*(d + e*x)^FracPart[m])/(1 + (e*x)/d)^FracPart[m], Int[(1 + (e*x)/d)^m*(a + c*x^2)^p, x], x] /; FreeQ[{a, c, d, e, m}, x] && EqQ[c*d^2 + a*e^2, 0] && !IntegerQ[p] && !(IntegerQ[m] || GtQ[d, 0])
  681. Int[Times[Power[Plus[Pattern[d, Blank[]], Times[Optional[Pattern[e, Blank[]]], Pattern[x, Blank[]]]], -1], Power[Plus[Optional[Pattern[a, Blank[]]], Times[Optional[Pattern[b, Blank[]]], Pattern[x, Blank[]]], Times[Optional[Pattern[c, Blank[]]], Power[Pattern[x, Blank[]], 2]]], -1]], Pattern[x, Blank[Symbol]]] := Dist[(-4*b*c)/(d*(b^2 - 4*a*c)), Int[1/(b + 2*c*x), x], x] + Dist[b^2/(d^2*(b^2 - 4*a*c)), Int[(d + e*x)/(a + b*x + c*x^2), x], x] /; FreeQ[{a, b, c, d, e}, x] && NeQ[b^2 - 4*a*c, 0] && EqQ[2*c*d - b*e, 0]
  682. Int[Times[Power[Plus[Pattern[d, Blank[]], Times[Optional[Pattern[e, Blank[]]], Pattern[x, Blank[]]]], Pattern[m, Blank[]]], Power[Plus[Optional[Pattern[a, Blank[]]], Times[Optional[Pattern[b, Blank[]]], Pattern[x, Blank[]]], Times[Optional[Pattern[c, Blank[]]], Power[Pattern[x, Blank[]], 2]]], Optional[Pattern[p, Blank[]]]]], Pattern[x, Blank[Symbol]]] := Simp[(2*c*(d + e*x)^(m + 1)*(a + b*x + c*x^2)^(p + 1))/(e*(p + 1)*(b^2 - 4*a*c)), x] /; FreeQ[{a, b, c, d, e, m, p}, x] && NeQ[b^2 - 4*a*c, 0] && EqQ[2*c*d - b*e, 0] && EqQ[m + 2*p + 3, 0] && NeQ[p, -1]
  683. Int[Times[Power[Plus[Optional[Pattern[d, Blank[]]], Times[Optional[Pattern[e, Blank[]]], Pattern[x, Blank[]]]], Pattern[m, Blank[]]], Power[Plus[Optional[Pattern[a, Blank[]]], Times[Optional[Pattern[b, Blank[]]], Pattern[x, Blank[]]], Times[Optional[Pattern[c, Blank[]]], Power[Pattern[x, Blank[]], 2]]], Optional[Pattern[p, Blank[]]]]], Pattern[x, Blank[Symbol]]] := Int[ExpandIntegrand[(d + e*x)^m*(a + b*x + c*x^2)^p, x], x] /; FreeQ[{a, b, c, d, e, m}, x] && NeQ[b^2 - 4*a*c, 0] && EqQ[2*c*d - b*e, 0] && IGtQ[p, 0] && !(EqQ[m, 3] && NeQ[p, 1])
  684. Int[Times[Power[Plus[Pattern[d, Blank[]], Times[Optional[Pattern[e, Blank[]]], Pattern[x, Blank[]]]], Pattern[m, Blank[]]], Power[Plus[Optional[Pattern[a, Blank[]]], Times[Optional[Pattern[b, Blank[]]], Pattern[x, Blank[]]], Times[Optional[Pattern[c, Blank[]]], Power[Pattern[x, Blank[]], 2]]], Optional[Pattern[p, Blank[]]]]], Pattern[x, Blank[Symbol]]] := Simp[((d + e*x)^(m + 1)*(a + b*x + c*x^2)^p)/(e*(m + 1)), x] - Dist[(b*p)/(d*e*(m + 1)), Int[(d + e*x)^(m + 2)*(a + b*x + c*x^2)^(p - 1), x], x] /; FreeQ[{a, b, c, d, e}, x] && NeQ[b^2 - 4*a*c, 0] && EqQ[2*c*d - b*e, 0] && NeQ[m + 2*p + 3, 0] && GtQ[p, 0] && LtQ[m, -1] && !(IntegerQ[m/2] && LtQ[m + 2*p + 3, 0]) && IntegerQ[2*p]
  685. Int[Times[Power[Plus[Pattern[d, Blank[]], Times[Optional[Pattern[e, Blank[]]], Pattern[x, Blank[]]]], Pattern[m, Blank[]]], Power[Plus[Optional[Pattern[a, Blank[]]], Times[Optional[Pattern[b, Blank[]]], Pattern[x, Blank[]]], Times[Optional[Pattern[c, Blank[]]], Power[Pattern[x, Blank[]], 2]]], Optional[Pattern[p, Blank[]]]]], Pattern[x, Blank[Symbol]]] := Simp[((d + e*x)^(m + 1)*(a + b*x + c*x^2)^p)/(e*(m + 2*p + 1)), x] - Dist[(d*p*(b^2 - 4*a*c))/(b*e*(m + 2*p + 1)), Int[(d + e*x)^m*(a + b*x + c*x^2)^(p - 1), x], x] /; FreeQ[{a, b, c, d, e, m}, x] && NeQ[b^2 - 4*a*c, 0] && EqQ[2*c*d - b*e, 0] && NeQ[m + 2*p + 3, 0] && GtQ[p, 0] && !LtQ[m, -1] && !(IGtQ[(m - 1)/2, 0] && ( !IntegerQ[p] || LtQ[m, 2*p])) && RationalQ[m] && IntegerQ[2*p]
  686. Int[Times[Power[Plus[Pattern[d, Blank[]], Times[Optional[Pattern[e, Blank[]]], Pattern[x, Blank[]]]], Pattern[m, Blank[]]], Power[Plus[Optional[Pattern[a, Blank[]]], Times[Optional[Pattern[b, Blank[]]], Pattern[x, Blank[]]], Times[Optional[Pattern[c, Blank[]]], Power[Pattern[x, Blank[]], 2]]], Pattern[p, Blank[]]]], Pattern[x, Blank[Symbol]]] := Simp[(d*(d + e*x)^(m - 1)*(a + b*x + c*x^2)^(p + 1))/(b*(p + 1)), x] - Dist[(d*e*(m - 1))/(b*(p + 1)), Int[(d + e*x)^(m - 2)*(a + b*x + c*x^2)^(p + 1), x], x] /; FreeQ[{a, b, c, d, e}, x] && NeQ[b^2 - 4*a*c, 0] && EqQ[2*c*d - b*e, 0] && NeQ[m + 2*p + 3, 0] && LtQ[p, -1] && GtQ[m, 1] && IntegerQ[2*p]
  687. Int[Times[Power[Plus[Pattern[d, Blank[]], Times[Optional[Pattern[e, Blank[]]], Pattern[x, Blank[]]]], Pattern[m, Blank[]]], Power[Plus[Optional[Pattern[a, Blank[]]], Times[Optional[Pattern[b, Blank[]]], Pattern[x, Blank[]]], Times[Optional[Pattern[c, Blank[]]], Power[Pattern[x, Blank[]], 2]]], Optional[Pattern[p, Blank[]]]]], Pattern[x, Blank[Symbol]]] := Simp[(2*c*(d + e*x)^(m + 1)*(a + b*x + c*x^2)^(p + 1))/(e*(p + 1)*(b^2 - 4*a*c)), x] - Dist[(2*c*e*(m + 2*p + 3))/(e*(p + 1)*(b^2 - 4*a*c)), Int[(d + e*x)^m*(a + b*x + c*x^2)^(p + 1), x], x] /; FreeQ[{a, b, c, d, e, m}, x] && NeQ[b^2 - 4*a*c, 0] && EqQ[2*c*d - b*e, 0] && NeQ[m + 2*p + 3, 0] && LtQ[p, -1] && !GtQ[m, 1] && RationalQ[m] && IntegerQ[2*p]
  688. Int[Times[Power[Plus[Pattern[d, Blank[]], Times[Optional[Pattern[e, Blank[]]], Pattern[x, Blank[]]]], -1], Power[Plus[Optional[Pattern[a, Blank[]]], Times[Optional[Pattern[b, Blank[]]], Pattern[x, Blank[]]], Times[Optional[Pattern[c, Blank[]]], Power[Pattern[x, Blank[]], 2]]], Rational[-1, 2]]], Pattern[x, Blank[Symbol]]] := Dist[4*c, Subst[Int[1/(b^2*e - 4*a*c*e + 4*c*e*x^2), x], x, Sqrt[a + b*x + c*x^2]], x] /; FreeQ[{a, b, c, d, e}, x] && NeQ[b^2 - 4*a*c, 0] && EqQ[2*c*d - b*e, 0]
  689. Int[Times[Power[Plus[Pattern[d, Blank[]], Times[Optional[Pattern[e, Blank[]]], Pattern[x, Blank[]]]], Rational[-1, 2]], Power[Plus[Optional[Pattern[a, Blank[]]], Times[Optional[Pattern[b, Blank[]]], Pattern[x, Blank[]]], Times[Optional[Pattern[c, Blank[]]], Power[Pattern[x, Blank[]], 2]]], Rational[-1, 2]]], Pattern[x, Blank[Symbol]]] := Dist[(4*Sqrt[-(c/(b^2 - 4*a*c))])/e, Subst[Int[1/Sqrt[Simp[1 - (b^2*x^4)/(d^2*(b^2 - 4*a*c)), x]], x], x, Sqrt[d + e*x]], x] /; FreeQ[{a, b, c, d, e}, x] && NeQ[b^2 - 4*a*c, 0] && EqQ[2*c*d - b*e, 0] && LtQ[c/(b^2 - 4*a*c), 0]
  690. Int[Times[Power[Plus[Pattern[d, Blank[]], Times[Optional[Pattern[e, Blank[]]], Pattern[x, Blank[]]]], Rational[1, 2]], Power[Plus[Optional[Pattern[a, Blank[]]], Times[Optional[Pattern[b, Blank[]]], Pattern[x, Blank[]]], Times[Optional[Pattern[c, Blank[]]], Power[Pattern[x, Blank[]], 2]]], Rational[-1, 2]]], Pattern[x, Blank[Symbol]]] := Dist[(4*Sqrt[-(c/(b^2 - 4*a*c))])/e, Subst[Int[x^2/Sqrt[Simp[1 - (b^2*x^4)/(d^2*(b^2 - 4*a*c)), x]], x], x, Sqrt[d + e*x]], x] /; FreeQ[{a, b, c, d, e}, x] && NeQ[b^2 - 4*a*c, 0] && EqQ[2*c*d - b*e, 0] && LtQ[c/(b^2 - 4*a*c), 0]
  691. Int[Times[Power[Plus[Pattern[d, Blank[]], Times[Optional[Pattern[e, Blank[]]], Pattern[x, Blank[]]]], Pattern[m, Blank[]]], Power[Plus[Optional[Pattern[a, Blank[]]], Times[Optional[Pattern[b, Blank[]]], Pattern[x, Blank[]]], Times[Optional[Pattern[c, Blank[]]], Power[Pattern[x, Blank[]], 2]]], Rational[-1, 2]]], Pattern[x, Blank[Symbol]]] := Dist[Sqrt[-((c*(a + b*x + c*x^2))/(b^2 - 4*a*c))]/Sqrt[a + b*x + c*x^2], Int[(d + e*x)^m/Sqrt[-((a*c)/(b^2 - 4*a*c)) - (b*c*x)/(b^2 - 4*a*c) - (c^2*x^2)/(b^2 - 4*a*c)], x], x] /; FreeQ[{a, b, c, d, e}, x] && NeQ[b^2 - 4*a*c, 0] && EqQ[2*c*d - b*e, 0] && EqQ[m^2, 1/4]
  692. Int[Times[Power[Plus[Pattern[d, Blank[]], Times[Optional[Pattern[e, Blank[]]], Pattern[x, Blank[]]]], Pattern[m, Blank[]]], Power[Plus[Optional[Pattern[a, Blank[]]], Times[Optional[Pattern[b, Blank[]]], Pattern[x, Blank[]]], Times[Optional[Pattern[c, Blank[]]], Power[Pattern[x, Blank[]], 2]]], Optional[Pattern[p, Blank[]]]]], Pattern[x, Blank[Symbol]]] := Simp[(2*d*(d + e*x)^(m - 1)*(a + b*x + c*x^2)^(p + 1))/(b*(m + 2*p + 1)), x] + Dist[(d^2*(m - 1)*(b^2 - 4*a*c))/(b^2*(m + 2*p + 1)), Int[(d + e*x)^(m - 2)*(a + b*x + c*x^2)^p, x], x] /; FreeQ[{a, b, c, d, e, p}, x] && NeQ[b^2 - 4*a*c, 0] && EqQ[2*c*d - b*e, 0] && NeQ[m + 2*p + 3, 0] && GtQ[m, 1] && NeQ[m + 2*p + 1, 0] && (IntegerQ[2*p] || (IntegerQ[m] && RationalQ[p]) || OddQ[m])
  693. Int[Times[Power[Plus[Pattern[d, Blank[]], Times[Optional[Pattern[e, Blank[]]], Pattern[x, Blank[]]]], Pattern[m, Blank[]]], Power[Plus[Optional[Pattern[a, Blank[]]], Times[Optional[Pattern[b, Blank[]]], Pattern[x, Blank[]]], Times[Optional[Pattern[c, Blank[]]], Power[Pattern[x, Blank[]], 2]]], Optional[Pattern[p, Blank[]]]]], Pattern[x, Blank[Symbol]]] := Simp[(-2*b*d*(d + e*x)^(m + 1)*(a + b*x + c*x^2)^(p + 1))/(d^2*(m + 1)*(b^2 - 4*a*c)), x] + Dist[(b^2*(m + 2*p + 3))/(d^2*(m + 1)*(b^2 - 4*a*c)), Int[(d + e*x)^(m + 2)*(a + b*x + c*x^2)^p, x], x] /; FreeQ[{a, b, c, d, e, p}, x] && NeQ[b^2 - 4*a*c, 0] && EqQ[2*c*d - b*e, 0] && NeQ[m + 2*p + 3, 0] && LtQ[m, -1] && (IntegerQ[2*p] || (IntegerQ[m] && RationalQ[p]) || IntegerQ[(m + 2*p + 3)/2])
  694. Int[Times[Power[Plus[Pattern[d, Blank[]], Times[Optional[Pattern[e, Blank[]]], Pattern[x, Blank[]]]], Pattern[m, Blank[]]], Power[Plus[Optional[Pattern[a, Blank[]]], Times[Optional[Pattern[b, Blank[]]], Pattern[x, Blank[]]], Times[Optional[Pattern[c, Blank[]]], Power[Pattern[x, Blank[]], 2]]], Optional[Pattern[p, Blank[]]]]], Pattern[x, Blank[Symbol]]] := Dist[1/e, Subst[Int[x^m*(a - b^2/(4*c) + (c*x^2)/e^2)^p, x], x, d + e*x], x] /; FreeQ[{a, b, c, d, e, m, p}, x] && NeQ[b^2 - 4*a*c, 0] && EqQ[2*c*d - b*e, 0]
  695. Int[Times[Power[Plus[Pattern[d, Blank[]], Times[Optional[Pattern[e, Blank[]]], Pattern[x, Blank[]]]], -1], Power[Plus[Optional[Pattern[a, Blank[]]], Times[Optional[Pattern[c, Blank[]]], Power[Pattern[x, Blank[]], 2]]], Rational[-1, 4]]], Pattern[x, Blank[Symbol]]] := Simp[(1*ArcTan[(-1 - (c*x^2)/a)^(1/4)/(1 - (c*d*x)/(2*a*e) - Sqrt[-1 - (c*x^2)/a])])/(2*(-a)^(1/4)*e), x] + Simp[(1*Log[(1 - (c*d*x)/(2*a*e) + Sqrt[-1 - (c*x^2)/a] - (-1 - (c*x^2)/a)^(1/4))/(1 - (c*d*x)/(2*a*e) + Sqrt[-1 - (c*x^2)/a] + (-1 - (c*x^2)/a)^(1/4))])/(4*(-a)^(1/4)*e), x] /; FreeQ[{a, c, d, e}, x] && EqQ[c*d^2 + 2*a*e^2, 0] && LtQ[a, 0]
  696. Int[Times[Power[Plus[Pattern[d, Blank[]], Times[Optional[Pattern[e, Blank[]]], Pattern[x, Blank[]]]], Pattern[m, Blank[]]], Power[Plus[Pattern[a, Blank[]], Times[Optional[Pattern[c, Blank[]]], Power[Pattern[x, Blank[]], 2]]], Pattern[p, Blank[]]]], Pattern[x, Blank[Symbol]]] := Simp[(e*m*d^(m - 1)*(a + c*x^2)^(p + 1))/(2*c*(p + 1)), x] + Int[((d + e*x)^m - e*m*d^(m - 1)*x)*(a + c*x^2)^p, x] /; FreeQ[{a, c, d, e}, x] && NeQ[c*d^2 + a*e^2, 0] && IGtQ[p, 1] && IGtQ[m, 0] && LeQ[m, p]
  697. Int[Times[Power[Plus[Pattern[d, Blank[]], Times[Optional[Pattern[e, Blank[]]], Pattern[x, Blank[]]]], Pattern[m, Blank[]]], Power[Plus[Pattern[a, Blank[]], Times[Optional[Pattern[c, Blank[]]], Power[Pattern[x, Blank[]], 2]]], Optional[Pattern[p, Blank[]]]]], Pattern[x, Blank[Symbol]]] := Int[ExpandIntegrand[(d + e*x)^m*(a + c*x^2)^p, x], x] /; FreeQ[{a, c, d, e, m}, x] && NeQ[c*d^2 + a*e^2, 0] && IGtQ[p, 0]
  698. Int[Times[Power[Plus[Optional[Pattern[d, Blank[]]], Times[Optional[Pattern[e, Blank[]]], Pattern[x, Blank[]]]], Pattern[m, Blank[]]], Power[Plus[Optional[Pattern[a, Blank[]]], Times[Optional[Pattern[b, Blank[]]], Pattern[x, Blank[]]], Times[Optional[Pattern[c, Blank[]]], Power[Pattern[x, Blank[]], 2]]], Optional[Pattern[p, Blank[]]]]], Pattern[x, Blank[Symbol]]] := Int[ExpandIntegrand[(d + e*x)^m*(a + b*x + c*x^2)^p, x], x] /; FreeQ[{a, b, c, d, e, m}, x] && NeQ[b^2 - 4*a*c, 0] && NeQ[c*d^2 - b*d*e + a*e^2, 0] && NeQ[2*c*d - b*e, 0] && IntegerQ[p] && (GtQ[p, 0] || (EqQ[a, 0] && IntegerQ[m]))
  699. Int[Times[Power[Plus[Optional[Pattern[d, Blank[]]], Times[Optional[Pattern[e, Blank[]]], Pattern[x, Blank[]]]], Rational[1, 2]], Power[Plus[Optional[Pattern[a, Blank[]]], Times[Optional[Pattern[b, Blank[]]], Pattern[x, Blank[]]], Times[Optional[Pattern[c, Blank[]]], Power[Pattern[x, Blank[]], 2]]], -1]], Pattern[x, Blank[Symbol]]] := Dist[2*e, Subst[Int[x^2/(c*d^2 - b*d*e + a*e^2 - (2*c*d - b*e)*x^2 + c*x^4), x], x, Sqrt[d + e*x]], x] /; FreeQ[{a, b, c, d, e}, x] && NeQ[b^2 - 4*a*c, 0] && NeQ[c*d^2 - b*d*e + a*e^2, 0] && NeQ[2*c*d - b*e, 0]
  700. Int[Times[Power[Plus[Pattern[d, Blank[]], Times[Optional[Pattern[e, Blank[]]], Pattern[x, Blank[]]]], Rational[1, 2]], Power[Plus[Pattern[a, Blank[]], Times[Optional[Pattern[c, Blank[]]], Power[Pattern[x, Blank[]], 2]]], -1]], Pattern[x, Blank[Symbol]]] := Dist[2*e, Subst[Int[x^2/(c*d^2 + a*e^2 - 2*c*d*x^2 + c*x^4), x], x, Sqrt[d + e*x]], x] /; FreeQ[{a, c, d, e}, x] && NeQ[c*d^2 + a*e^2, 0]
  701. Int[Times[Power[Plus[Optional[Pattern[d, Blank[]]], Times[Optional[Pattern[e, Blank[]]], Pattern[x, Blank[]]]], Pattern[m, Blank[]]], Power[Plus[Optional[Pattern[a, Blank[]]], Times[Optional[Pattern[b, Blank[]]], Pattern[x, Blank[]]], Times[Optional[Pattern[c, Blank[]]], Power[Pattern[x, Blank[]], 2]]], -1]], Pattern[x, Blank[Symbol]]] := Int[PolynomialDivide[(d + e*x)^m, a + b*x + c*x^2, x], x] /; FreeQ[{a, b, c, d, e}, x] && NeQ[b^2 - 4*a*c, 0] && NeQ[c*d^2 - b*d*e + a*e^2, 0] && NeQ[2*c*d - b*e, 0] && IGtQ[m, 1] && (NeQ[d, 0] || GtQ[m, 2])
  702. Int[Times[Power[Plus[Pattern[d, Blank[]], Times[Optional[Pattern[e, Blank[]]], Pattern[x, Blank[]]]], Pattern[m, Blank[]]], Power[Plus[Pattern[a, Blank[]], Times[Optional[Pattern[c, Blank[]]], Power[Pattern[x, Blank[]], 2]]], -1]], Pattern[x, Blank[Symbol]]] := Int[PolynomialDivide[(d + e*x)^m, a + c*x^2, x], x] /; FreeQ[{a, c, d, e}, x] && NeQ[c*d^2 + a*e^2, 0] && IGtQ[m, 1] && (NeQ[d, 0] || GtQ[m, 2])
  703. Int[Times[Power[Plus[Optional[Pattern[d, Blank[]]], Times[Optional[Pattern[e, Blank[]]], Pattern[x, Blank[]]]], Pattern[m, Blank[]]], Power[Plus[Optional[Pattern[a, Blank[]]], Times[Optional[Pattern[b, Blank[]]], Pattern[x, Blank[]]], Times[Optional[Pattern[c, Blank[]]], Power[Pattern[x, Blank[]], 2]]], -1]], Pattern[x, Blank[Symbol]]] := Simp[(e*(d + e*x)^(m - 1))/(c*(m - 1)), x] + Dist[1/c, Int[((d + e*x)^(m - 2)*Simp[c*d^2 - a*e^2 + e*(2*c*d - b*e)*x, x])/(a + b*x + c*x^2), x], x] /; FreeQ[{a, b, c, d, e}, x] && NeQ[b^2 - 4*a*c, 0] && NeQ[c*d^2 - b*d*e + a*e^2, 0] && NeQ[2*c*d - b*e, 0] && GtQ[m, 1]
  704. Int[Times[Power[Plus[Pattern[d, Blank[]], Times[Optional[Pattern[e, Blank[]]], Pattern[x, Blank[]]]], Pattern[m, Blank[]]], Power[Plus[Pattern[a, Blank[]], Times[Optional[Pattern[c, Blank[]]], Power[Pattern[x, Blank[]], 2]]], -1]], Pattern[x, Blank[Symbol]]] := Simp[(e*(d + e*x)^(m - 1))/(c*(m - 1)), x] + Dist[1/c, Int[((d + e*x)^(m - 2)*Simp[c*d^2 - a*e^2 + 2*c*d*e*x, x])/(a + c*x^2), x], x] /; FreeQ[{a, c, d, e}, x] && NeQ[c*d^2 + a*e^2, 0] && GtQ[m, 1]
  705. Int[Times[Power[Plus[Optional[Pattern[d, Blank[]]], Times[Optional[Pattern[e, Blank[]]], Pattern[x, Blank[]]]], -1], Power[Plus[Optional[Pattern[a, Blank[]]], Times[Optional[Pattern[b, Blank[]]], Pattern[x, Blank[]]], Times[Optional[Pattern[c, Blank[]]], Power[Pattern[x, Blank[]], 2]]], -1]], Pattern[x, Blank[Symbol]]] := Dist[e^2/(c*d^2 - b*d*e + a*e^2), Int[1/(d + e*x), x], x] + Dist[1/(c*d^2 - b*d*e + a*e^2), Int[(c*d - b*e - c*e*x)/(a + b*x + c*x^2), x], x] /; FreeQ[{a, b, c, d, e}, x] && NeQ[b^2 - 4*a*c, 0] && NeQ[c*d^2 - b*d*e + a*e^2, 0] && NeQ[2*c*d - b*e, 0]
  706. Int[Times[Power[Plus[Pattern[d, Blank[]], Times[Optional[Pattern[e, Blank[]]], Pattern[x, Blank[]]]], -1], Power[Plus[Pattern[a, Blank[]], Times[Optional[Pattern[c, Blank[]]], Power[Pattern[x, Blank[]], 2]]], -1]], Pattern[x, Blank[Symbol]]] := Dist[e^2/(c*d^2 + a*e^2), Int[1/(d + e*x), x], x] + Dist[1/(c*d^2 + a*e^2), Int[(c*d - c*e*x)/(a + c*x^2), x], x] /; FreeQ[{a, c, d, e}, x] && NeQ[c*d^2 + a*e^2, 0]
  707. Int[Times[Power[Plus[Optional[Pattern[d, Blank[]]], Times[Optional[Pattern[e, Blank[]]], Pattern[x, Blank[]]]], Rational[-1, 2]], Power[Plus[Optional[Pattern[a, Blank[]]], Times[Optional[Pattern[b, Blank[]]], Pattern[x, Blank[]]], Times[Optional[Pattern[c, Blank[]]], Power[Pattern[x, Blank[]], 2]]], -1]], Pattern[x, Blank[Symbol]]] := Dist[2*e, Subst[Int[1/(c*d^2 - b*d*e + a*e^2 - (2*c*d - b*e)*x^2 + c*x^4), x], x, Sqrt[d + e*x]], x] /; FreeQ[{a, b, c, d, e}, x] && NeQ[b^2 - 4*a*c, 0] && NeQ[c*d^2 - b*d*e + a*e^2, 0] && NeQ[2*c*d - b*e, 0]
  708. Int[Times[Power[Plus[Pattern[d, Blank[]], Times[Optional[Pattern[e, Blank[]]], Pattern[x, Blank[]]]], Rational[-1, 2]], Power[Plus[Pattern[a, Blank[]], Times[Optional[Pattern[c, Blank[]]], Power[Pattern[x, Blank[]], 2]]], -1]], Pattern[x, Blank[Symbol]]] := Dist[2*e, Subst[Int[1/(c*d^2 + a*e^2 - 2*c*d*x^2 + c*x^4), x], x, Sqrt[d + e*x]], x] /; FreeQ[{a, c, d, e}, x] && NeQ[c*d^2 + a*e^2, 0]
  709. Int[Times[Power[Plus[Optional[Pattern[d, Blank[]]], Times[Optional[Pattern[e, Blank[]]], Pattern[x, Blank[]]]], Pattern[m, Blank[]]], Power[Plus[Optional[Pattern[a, Blank[]]], Times[Optional[Pattern[b, Blank[]]], Pattern[x, Blank[]]], Times[Optional[Pattern[c, Blank[]]], Power[Pattern[x, Blank[]], 2]]], -1]], Pattern[x, Blank[Symbol]]] := Simp[(e*(d + e*x)^(m + 1))/((m + 1)*(c*d^2 - b*d*e + a*e^2)), x] + Dist[1/(c*d^2 - b*d*e + a*e^2), Int[((d + e*x)^(m + 1)*Simp[c*d - b*e - c*e*x, x])/(a + b*x + c*x^2), x], x] /; FreeQ[{a, b, c, d, e, m}, x] && NeQ[b^2 - 4*a*c, 0] && NeQ[c*d^2 - b*d*e + a*e^2, 0] && NeQ[2*c*d - b*e, 0] && LtQ[m, -1]
  710. Int[Times[Power[Plus[Pattern[d, Blank[]], Times[Optional[Pattern[e, Blank[]]], Pattern[x, Blank[]]]], Pattern[m, Blank[]]], Power[Plus[Pattern[a, Blank[]], Times[Optional[Pattern[c, Blank[]]], Power[Pattern[x, Blank[]], 2]]], -1]], Pattern[x, Blank[Symbol]]] := Simp[(e*(d + e*x)^(m + 1))/((m + 1)*(c*d^2 + a*e^2)), x] + Dist[c/(c*d^2 + a*e^2), Int[((d + e*x)^(m + 1)*(d - e*x))/(a + c*x^2), x], x] /; FreeQ[{a, c, d, e, m}, x] && NeQ[c*d^2 + a*e^2, 0] && LtQ[m, -1]
  711. Int[Times[Power[Plus[Optional[Pattern[d, Blank[]]], Times[Optional[Pattern[e, Blank[]]], Pattern[x, Blank[]]]], Pattern[m, Blank[]]], Power[Plus[Optional[Pattern[a, Blank[]]], Times[Optional[Pattern[b, Blank[]]], Pattern[x, Blank[]]], Times[Optional[Pattern[c, Blank[]]], Power[Pattern[x, Blank[]], 2]]], -1]], Pattern[x, Blank[Symbol]]] := Int[ExpandIntegrand[(d + e*x)^m, 1/(a + b*x + c*x^2), x], x] /; FreeQ[{a, b, c, d, e, m}, x] && NeQ[b^2 - 4*a*c, 0] && NeQ[c*d^2 - b*d*e + a*e^2, 0] && NeQ[2*c*d - b*e, 0] && !IntegerQ[m]
  712. Int[Times[Power[Plus[Pattern[d, Blank[]], Times[Optional[Pattern[e, Blank[]]], Pattern[x, Blank[]]]], Pattern[m, Blank[]]], Power[Plus[Pattern[a, Blank[]], Times[Optional[Pattern[c, Blank[]]], Power[Pattern[x, Blank[]], 2]]], -1]], Pattern[x, Blank[Symbol]]] := Int[ExpandIntegrand[(d + e*x)^m, 1/(a + c*x^2), x], x] /; FreeQ[{a, c, d, e, m}, x] && NeQ[c*d^2 + a*e^2, 0] && !IntegerQ[m]
  713. Int[Times[Power[Plus[Optional[Pattern[d, Blank[]]], Times[Optional[Pattern[e, Blank[]]], Pattern[x, Blank[]]]], Pattern[m, Blank[]]], Power[Plus[Pattern[a, Blank[]], Times[Optional[Pattern[b, Blank[]]], Pattern[x, Blank[]]], Times[Optional[Pattern[c, Blank[]]], Power[Pattern[x, Blank[]], 2]]], Pattern[p, Blank[]]]], Pattern[x, Blank[Symbol]]] := Dist[((d + e*x)^FracPart[p]*(a + b*x + c*x^2)^FracPart[p])/(a*d + c*e*x^3)^FracPart[p], Int[(d + e*x)^(m - p)*(a*d + c*e*x^3)^p, x], x] /; FreeQ[{a, b, c, d, e, m, p}, x] && EqQ[b*d + a*e, 0] && EqQ[c*d + b*e, 0] && IGtQ[m - p + 1, 0] && !IntegerQ[p]
  714. Int[Times[Power[Plus[Optional[Pattern[d, Blank[]]], Times[Optional[Pattern[e, Blank[]]], Pattern[x, Blank[]]]], Pattern[m, Blank[]]], Power[Plus[Times[Optional[Pattern[b, Blank[]]], Pattern[x, Blank[]]], Times[Optional[Pattern[c, Blank[]]], Power[Pattern[x, Blank[]], 2]]], Rational[-1, 2]]], Pattern[x, Blank[Symbol]]] := Int[(d + e*x)^m/(Sqrt[b*x]*Sqrt[1 + (c*x)/b]), x] /; FreeQ[{b, c, d, e}, x] && NeQ[c*d - b*e, 0] && NeQ[2*c*d - b*e, 0] && EqQ[m^2, 1/4] && LtQ[c, 0] && RationalQ[b]
  715. Int[Times[Power[Plus[Optional[Pattern[d, Blank[]]], Times[Optional[Pattern[e, Blank[]]], Pattern[x, Blank[]]]], Pattern[m, Blank[]]], Power[Plus[Times[Optional[Pattern[b, Blank[]]], Pattern[x, Blank[]]], Times[Optional[Pattern[c, Blank[]]], Power[Pattern[x, Blank[]], 2]]], Rational[-1, 2]]], Pattern[x, Blank[Symbol]]] := Dist[(Sqrt[x]*Sqrt[b + c*x])/Sqrt[b*x + c*x^2], Int[(d + e*x)^m/(Sqrt[x]*Sqrt[b + c*x]), x], x] /; FreeQ[{b, c, d, e}, x] && NeQ[c*d - b*e, 0] && NeQ[2*c*d - b*e, 0] && EqQ[m^2, 1/4]
  716. Int[Times[Power[Pattern[x, Blank[]], Pattern[m, Blank[]]], Power[Plus[Pattern[a, Blank[]], Times[Optional[Pattern[b, Blank[]]], Pattern[x, Blank[]]], Times[Optional[Pattern[c, Blank[]]], Power[Pattern[x, Blank[]], 2]]], Rational[-1, 2]]], Pattern[x, Blank[Symbol]]] := Dist[2, Subst[Int[x^(2*m + 1)/Sqrt[a + b*x^2 + c*x^4], x], x, Sqrt[x]], x] /; FreeQ[{a, b, c}, x] && NeQ[b^2 - 4*a*c, 0] && EqQ[m^2, 1/4]
  717. Int[Times[Power[Times[Pattern[e, Blank[]], Pattern[x, Blank[]]], Pattern[m, Blank[]]], Power[Plus[Pattern[a, Blank[]], Times[Optional[Pattern[b, Blank[]]], Pattern[x, Blank[]]], Times[Optional[Pattern[c, Blank[]]], Power[Pattern[x, Blank[]], 2]]], Rational[-1, 2]]], Pattern[x, Blank[Symbol]]] := Dist[(e*x)^m/x^m, Int[x^m/Sqrt[a + b*x + c*x^2], x], x] /; FreeQ[{a, b, c, e}, x] && NeQ[b^2 - 4*a*c, 0] && EqQ[m^2, 1/4]
  718. Int[Times[Power[Plus[Optional[Pattern[d, Blank[]]], Times[Optional[Pattern[e, Blank[]]], Pattern[x, Blank[]]]], Pattern[m, Blank[]]], Power[Plus[Optional[Pattern[a, Blank[]]], Times[Optional[Pattern[b, Blank[]]], Pattern[x, Blank[]]], Times[Optional[Pattern[c, Blank[]]], Power[Pattern[x, Blank[]], 2]]], Rational[-1, 2]]], Pattern[x, Blank[Symbol]]] := Dist[(2*Rt[b^2 - 4*a*c, 2]*(d + e*x)^m*Sqrt[-((c*(a + b*x + c*x^2))/(b^2 - 4*a*c))])/(c*Sqrt[a + b*x + c*x^2]*((2*c*(d + e*x))/(2*c*d - b*e - e*Rt[b^2 - 4*a*c, 2]))^m), Subst[Int[(1 + (2*e*Rt[b^2 - 4*a*c, 2]*x^2)/(2*c*d - b*e - e*Rt[b^2 - 4*a*c, 2]))^m/Sqrt[1 - x^2], x], x, Sqrt[(b + Rt[b^2 - 4*a*c, 2] + 2*c*x)/(2*Rt[b^2 - 4*a*c, 2])]], x] /; FreeQ[{a, b, c, d, e}, x] && NeQ[b^2 - 4*a*c, 0] && NeQ[c*d^2 - b*d*e + a*e^2, 0] && NeQ[2*c*d - b*e, 0] && EqQ[m^2, 1/4]
  719. Int[Times[Power[Plus[Pattern[d, Blank[]], Times[Optional[Pattern[e, Blank[]]], Pattern[x, Blank[]]]], Pattern[m, Blank[]]], Power[Plus[Pattern[a, Blank[]], Times[Optional[Pattern[c, Blank[]]], Power[Pattern[x, Blank[]], 2]]], Rational[-1, 2]]], Pattern[x, Blank[Symbol]]] := Dist[(2*a*Rt[-(c/a), 2]*(d + e*x)^m*Sqrt[1 + (c*x^2)/a])/(c*Sqrt[a + c*x^2]*((c*(d + e*x))/(c*d - a*e*Rt[-(c/a), 2]))^m), Subst[Int[(1 + (2*a*e*Rt[-(c/a), 2]*x^2)/(c*d - a*e*Rt[-(c/a), 2]))^m/Sqrt[1 - x^2], x], x, Sqrt[(1 - Rt[-(c/a), 2]*x)/2]], x] /; FreeQ[{a, c, d, e}, x] && NeQ[c*d^2 + a*e^2, 0] && EqQ[m^2, 1/4]
  720. Int[Times[Power[Plus[Optional[Pattern[d, Blank[]]], Times[Optional[Pattern[e, Blank[]]], Pattern[x, Blank[]]]], Pattern[m, Blank[]]], Power[Plus[Optional[Pattern[a, Blank[]]], Times[Optional[Pattern[b, Blank[]]], Pattern[x, Blank[]]], Times[Optional[Pattern[c, Blank[]]], Power[Pattern[x, Blank[]], 2]]], Pattern[p, Blank[]]]], Pattern[x, Blank[Symbol]]] := -Simp[((d + e*x)^(m + 1)*(d*b - 2*a*e + (2*c*d - b*e)*x)*(a + b*x + c*x^2)^p)/(2*(m + 1)*(c*d^2 - b*d*e + a*e^2)), x] + Dist[(p*(b^2 - 4*a*c))/(2*(m + 1)*(c*d^2 - b*d*e + a*e^2)), Int[(d + e*x)^(m + 2)*(a + b*x + c*x^2)^(p - 1), x], x] /; FreeQ[{a, b, c, d, e}, x] && NeQ[b^2 - 4*a*c, 0] && NeQ[c*d^2 - b*d*e + a*e^2, 0] && NeQ[2*c*d - b*e, 0] && EqQ[m + 2*p + 2, 0] && GtQ[p, 0]
  721. Int[Times[Power[Plus[Pattern[d, Blank[]], Times[Optional[Pattern[e, Blank[]]], Pattern[x, Blank[]]]], Pattern[m, Blank[]]], Power[Plus[Pattern[a, Blank[]], Times[Optional[Pattern[c, Blank[]]], Power[Pattern[x, Blank[]], 2]]], Pattern[p, Blank[]]]], Pattern[x, Blank[Symbol]]] := -Simp[((d + e*x)^(m + 1)*(-2*a*e + (2*c*d)*x)*(a + c*x^2)^p)/(2*(m + 1)*(c*d^2 + a*e^2)), x] - Dist[(4*a*c*p)/(2*(m + 1)*(c*d^2 + a*e^2)), Int[(d + e*x)^(m + 2)*(a + c*x^2)^(p - 1), x], x] /; FreeQ[{a, c, d, e}, x] && NeQ[c*d^2 + a*e^2, 0] && EqQ[m + 2*p + 2, 0] && GtQ[p, 0]
  722. Int[Times[Power[Plus[Optional[Pattern[d, Blank[]]], Times[Optional[Pattern[e, Blank[]]], Pattern[x, Blank[]]]], Pattern[m, Blank[]]], Power[Plus[Optional[Pattern[a, Blank[]]], Times[Optional[Pattern[b, Blank[]]], Pattern[x, Blank[]]], Times[Optional[Pattern[c, Blank[]]], Power[Pattern[x, Blank[]], 2]]], Pattern[p, Blank[]]]], Pattern[x, Blank[Symbol]]] := Simp[((d + e*x)^(m - 1)*(d*b - 2*a*e + (2*c*d - b*e)*x)*(a + b*x + c*x^2)^(p + 1))/((p + 1)*(b^2 - 4*a*c)), x] - Dist[(2*(2*p + 3)*(c*d^2 - b*d*e + a*e^2))/((p + 1)*(b^2 - 4*a*c)), Int[(d + e*x)^(m - 2)*(a + b*x + c*x^2)^(p + 1), x], x] /; FreeQ[{a, b, c, d, e}, x] && NeQ[b^2 - 4*a*c, 0] && NeQ[c*d^2 - b*d*e + a*e^2, 0] && NeQ[2*c*d - b*e, 0] && EqQ[m + 2*p + 2, 0] && LtQ[p, -1]
  723. Int[Times[Power[Plus[Pattern[d, Blank[]], Times[Optional[Pattern[e, Blank[]]], Pattern[x, Blank[]]]], Pattern[m, Blank[]]], Power[Plus[Pattern[a, Blank[]], Times[Optional[Pattern[c, Blank[]]], Power[Pattern[x, Blank[]], 2]]], Pattern[p, Blank[]]]], Pattern[x, Blank[Symbol]]] := Simp[((d + e*x)^(m - 1)*(a*e - c*d*x)*(a + c*x^2)^(p + 1))/(2*a*c*(p + 1)), x] + Dist[((2*p + 3)*(c*d^2 + a*e^2))/(2*a*c*(p + 1)), Int[(d + e*x)^(m - 2)*(a + c*x^2)^(p + 1), x], x] /; FreeQ[{a, c, d, e}, x] && NeQ[c*d^2 + a*e^2, 0] && EqQ[m + 2*p + 2, 0] && LtQ[p, -1]
  724. Int[Times[Power[Plus[Optional[Pattern[d, Blank[]]], Times[Optional[Pattern[e, Blank[]]], Pattern[x, Blank[]]]], -1], Power[Plus[Optional[Pattern[a, Blank[]]], Times[Optional[Pattern[b, Blank[]]], Pattern[x, Blank[]]], Times[Optional[Pattern[c, Blank[]]], Power[Pattern[x, Blank[]], 2]]], Rational[-1, 2]]], Pattern[x, Blank[Symbol]]] := Dist[-2, Subst[Int[1/(4*c*d^2 - 4*b*d*e + 4*a*e^2 - x^2), x], x, (2*a*e - b*d - (2*c*d - b*e)*x)/Sqrt[a + b*x + c*x^2]], x] /; FreeQ[{a, b, c, d, e}, x] && NeQ[b^2 - 4*a*c, 0] && NeQ[2*c*d - b*e, 0]
  725. Int[Times[Power[Plus[Pattern[d, Blank[]], Times[Optional[Pattern[e, Blank[]]], Pattern[x, Blank[]]]], -1], Power[Plus[Pattern[a, Blank[]], Times[Optional[Pattern[c, Blank[]]], Power[Pattern[x, Blank[]], 2]]], Rational[-1, 2]]], Pattern[x, Blank[Symbol]]] := -Subst[Int[1/(c*d^2 + a*e^2 - x^2), x], x, (a*e - c*d*x)/Sqrt[a + c*x^2]] /; FreeQ[{a, c, d, e}, x]
  726. Int[Times[Power[Plus[Optional[Pattern[d, Blank[]]], Times[Optional[Pattern[e, Blank[]]], Pattern[x, Blank[]]]], Pattern[m, Blank[]]], Power[Plus[Optional[Pattern[a, Blank[]]], Times[Optional[Pattern[b, Blank[]]], Pattern[x, Blank[]]], Times[Optional[Pattern[c, Blank[]]], Power[Pattern[x, Blank[]], 2]]], Pattern[p, Blank[]]]], Pattern[x, Blank[Symbol]]] := -Simp[((b - Rt[b^2 - 4*a*c, 2] + 2*c*x)*(d + e*x)^(m + 1)*(a + b*x + c*x^2)^p*Hypergeometric2F1[m + 1, -p, m + 2, (-4*c*Rt[b^2 - 4*a*c, 2]*(d + e*x))/((2*c*d - b*e - e*Rt[b^2 - 4*a*c, 2])*(b - Rt[b^2 - 4*a*c, 2] + 2*c*x))])/((m + 1)*(2*c*d - b*e + e*Rt[b^2 - 4*a*c, 2])*(((2*c*d - b*e + e*Rt[b^2 - 4*a*c, 2])*(b + Rt[b^2 - 4*a*c, 2] + 2*c*x))/((2*c*d - b*e - e*Rt[b^2 - 4*a*c, 2])*(b - Rt[b^2 - 4*a*c, 2] + 2*c*x)))^p), x] /; FreeQ[{a, b, c, d, e, m, p}, x] && NeQ[b^2 - 4*a*c, 0] && NeQ[c*d^2 - b*d*e + a*e^2, 0] && NeQ[2*c*d - b*e, 0] && !IntegerQ[p] && EqQ[m + 2*p + 2, 0]
  727. Int[Times[Power[Plus[Pattern[d, Blank[]], Times[Optional[Pattern[e, Blank[]]], Pattern[x, Blank[]]]], Pattern[m, Blank[]]], Power[Plus[Pattern[a, Blank[]], Times[Optional[Pattern[c, Blank[]]], Power[Pattern[x, Blank[]], 2]]], Pattern[p, Blank[]]]], Pattern[x, Blank[Symbol]]] := Simp[((Rt[-(a*c), 2] - c*x)*(d + e*x)^(m + 1)*(a + c*x^2)^p*Hypergeometric2F1[m + 1, -p, m + 2, (2*c*Rt[-(a*c), 2]*(d + e*x))/((c*d - e*Rt[-(a*c), 2])*(Rt[-(a*c), 2] - c*x))])/((m + 1)*(c*d + e*Rt[-(a*c), 2])*(((c*d + e*Rt[-(a*c), 2])*(Rt[-(a*c), 2] + c*x))/((c*d - e*Rt[-(a*c), 2])*(-Rt[-(a*c), 2] + c*x)))^p), x] /; FreeQ[{a, c, d, e, m, p}, x] && NeQ[c*d^2 + a*e^2, 0] && !IntegerQ[p] && EqQ[m + 2*p + 2, 0]
  728. Int[Times[Power[Plus[Optional[Pattern[d, Blank[]]], Times[Optional[Pattern[e, Blank[]]], Pattern[x, Blank[]]]], Pattern[m, Blank[]]], Power[Plus[Optional[Pattern[a, Blank[]]], Times[Optional[Pattern[b, Blank[]]], Pattern[x, Blank[]]], Times[Optional[Pattern[c, Blank[]]], Power[Pattern[x, Blank[]], 2]]], Pattern[p, Blank[]]]], Pattern[x, Blank[Symbol]]] := Simp[((d + e*x)^m*(b + 2*c*x)*(a + b*x + c*x^2)^(p + 1))/((p + 1)*(b^2 - 4*a*c)), x] + Dist[(m*(2*c*d - b*e))/((p + 1)*(b^2 - 4*a*c)), Int[(d + e*x)^(m - 1)*(a + b*x + c*x^2)^(p + 1), x], x] /; FreeQ[{a, b, c, d, e, m, p}, x] && NeQ[b^2 - 4*a*c, 0] && NeQ[c*d^2 - b*d*e + a*e^2, 0] && NeQ[2*c*d - b*e, 0] && EqQ[m + 2*p + 3, 0] && LtQ[p, -1]
  729. Int[Times[Power[Plus[Pattern[d, Blank[]], Times[Optional[Pattern[e, Blank[]]], Pattern[x, Blank[]]]], Pattern[m, Blank[]]], Power[Plus[Pattern[a, Blank[]], Times[Optional[Pattern[c, Blank[]]], Power[Pattern[x, Blank[]], 2]]], Pattern[p, Blank[]]]], Pattern[x, Blank[Symbol]]] := -Simp[((d + e*x)^m*(2*c*x)*(a + c*x^2)^(p + 1))/(4*a*c*(p + 1)), x] - Dist[(m*(2*c*d))/(4*a*c*(p + 1)), Int[(d + e*x)^(m - 1)*(a + c*x^2)^(p + 1), x], x] /; FreeQ[{a, c, d, e, m, p}, x] && NeQ[c*d^2 + a*e^2, 0] && EqQ[m + 2*p + 3, 0] && LtQ[p, -1]
  730. Int[Times[Power[Plus[Optional[Pattern[d, Blank[]]], Times[Optional[Pattern[e, Blank[]]], Pattern[x, Blank[]]]], Pattern[m, Blank[]]], Power[Plus[Optional[Pattern[a, Blank[]]], Times[Optional[Pattern[b, Blank[]]], Pattern[x, Blank[]]], Times[Optional[Pattern[c, Blank[]]], Power[Pattern[x, Blank[]], 2]]], Pattern[p, Blank[]]]], Pattern[x, Blank[Symbol]]] := Simp[(e*(d + e*x)^(m + 1)*(a + b*x + c*x^2)^(p + 1))/((m + 1)*(c*d^2 - b*d*e + a*e^2)), x] + Dist[(2*c*d - b*e)/(2*(c*d^2 - b*d*e + a*e^2)), Int[(d + e*x)^(m + 1)*(a + b*x + c*x^2)^p, x], x] /; FreeQ[{a, b, c, d, e, m, p}, x] && NeQ[b^2 - 4*a*c, 0] && NeQ[c*d^2 - b*d*e + a*e^2, 0] && NeQ[2*c*d - b*e, 0] && EqQ[m + 2*p + 3, 0]
  731. Int[Times[Power[Plus[Pattern[d, Blank[]], Times[Optional[Pattern[e, Blank[]]], Pattern[x, Blank[]]]], Pattern[m, Blank[]]], Power[Plus[Pattern[a, Blank[]], Times[Optional[Pattern[c, Blank[]]], Power[Pattern[x, Blank[]], 2]]], Pattern[p, Blank[]]]], Pattern[x, Blank[Symbol]]] := Simp[(e*(d + e*x)^(m + 1)*(a + c*x^2)^(p + 1))/((m + 1)*(c*d^2 + a*e^2)), x] + Dist[(c*d)/(c*d^2 + a*e^2), Int[(d + e*x)^(m + 1)*(a + c*x^2)^p, x], x] /; FreeQ[{a, c, d, e, m, p}, x] && NeQ[c*d^2 + a*e^2, 0] && EqQ[m + 2*p + 3, 0]
  732. Int[Times[Power[Plus[Optional[Pattern[d, Blank[]]], Times[Optional[Pattern[e, Blank[]]], Pattern[x, Blank[]]]], Pattern[m, Blank[]]], Power[Plus[Optional[Pattern[a, Blank[]]], Times[Optional[Pattern[b, Blank[]]], Pattern[x, Blank[]]], Times[Optional[Pattern[c, Blank[]]], Power[Pattern[x, Blank[]], 2]]], Pattern[p, Blank[]]]], Pattern[x, Blank[Symbol]]] := Simp[((d + e*x)^(m + 1)*(a + b*x + c*x^2)^p)/(e*(m + 1)), x] - Dist[p/(e*(m + 1)), Int[(d + e*x)^(m + 1)*(b + 2*c*x)*(a + b*x + c*x^2)^(p - 1), x], x] /; FreeQ[{a, b, c, d, e, m}, x] && NeQ[b^2 - 4*a*c, 0] && NeQ[c*d^2 - b*d*e + a*e^2, 0] && NeQ[2*c*d - b*e, 0] && GtQ[p, 0] && (IntegerQ[p] || LtQ[m, -1]) && NeQ[m, -1] && !ILtQ[m + 2*p + 1, 0] && IntQuadraticQ[a, b, c, d, e, m, p, x]
  733. Int[Times[Power[Plus[Pattern[d, Blank[]], Times[Optional[Pattern[e, Blank[]]], Pattern[x, Blank[]]]], Pattern[m, Blank[]]], Power[Plus[Pattern[a, Blank[]], Times[Optional[Pattern[c, Blank[]]], Power[Pattern[x, Blank[]], 2]]], Pattern[p, Blank[]]]], Pattern[x, Blank[Symbol]]] := Simp[((d + e*x)^(m + 1)*(a + c*x^2)^p)/(e*(m + 1)), x] - Dist[(2*c*p)/(e*(m + 1)), Int[x*(d + e*x)^(m + 1)*(a + c*x^2)^(p - 1), x], x] /; FreeQ[{a, c, d, e, m}, x] && NeQ[c*d^2 + a*e^2, 0] && GtQ[p, 0] && (IntegerQ[p] || LtQ[m, -1]) && NeQ[m, -1] && !ILtQ[m + 2*p + 1, 0] && IntQuadraticQ[a, 0, c, d, e, m, p, x]
  734. Int[Times[Power[Plus[Optional[Pattern[d, Blank[]]], Times[Optional[Pattern[e, Blank[]]], Pattern[x, Blank[]]]], Pattern[m, Blank[]]], Power[Plus[Optional[Pattern[a, Blank[]]], Times[Optional[Pattern[b, Blank[]]], Pattern[x, Blank[]]], Times[Optional[Pattern[c, Blank[]]], Power[Pattern[x, Blank[]], 2]]], Pattern[p, Blank[]]]], Pattern[x, Blank[Symbol]]] := Simp[((d + e*x)^(m + 1)*(a + b*x + c*x^2)^p)/(e*(m + 2*p + 1)), x] - Dist[p/(e*(m + 2*p + 1)), Int[(d + e*x)^m*Simp[b*d - 2*a*e + (2*c*d - b*e)*x, x]*(a + b*x + c*x^2)^(p - 1), x], x] /; FreeQ[{a, b, c, d, e, m}, x] && NeQ[b^2 - 4*a*c, 0] && NeQ[c*d^2 - b*d*e + a*e^2, 0] && NeQ[2*c*d - b*e, 0] && GtQ[p, 0] && NeQ[m + 2*p + 1, 0] && ( !RationalQ[m] || LtQ[m, 1]) && !ILtQ[m + 2*p, 0] && IntQuadraticQ[a, b, c, d, e, m, p, x]
  735. Int[Times[Power[Plus[Pattern[d, Blank[]], Times[Optional[Pattern[e, Blank[]]], Pattern[x, Blank[]]]], Pattern[m, Blank[]]], Power[Plus[Pattern[a, Blank[]], Times[Optional[Pattern[c, Blank[]]], Power[Pattern[x, Blank[]], 2]]], Pattern[p, Blank[]]]], Pattern[x, Blank[Symbol]]] := Simp[((d + e*x)^(m + 1)*(a + c*x^2)^p)/(e*(m + 2*p + 1)), x] + Dist[(2*p)/(e*(m + 2*p + 1)), Int[(d + e*x)^m*Simp[a*e - c*d*x, x]*(a + c*x^2)^(p - 1), x], x] /; FreeQ[{a, c, d, e, m}, x] && NeQ[c*d^2 + a*e^2, 0] && GtQ[p, 0] && NeQ[m + 2*p + 1, 0] && ( !RationalQ[m] || LtQ[m, 1]) && !ILtQ[m + 2*p, 0] && IntQuadraticQ[a, 0, c, d, e, m, p, x]
  736. Int[Times[Power[Plus[Optional[Pattern[d, Blank[]]], Times[Optional[Pattern[e, Blank[]]], Pattern[x, Blank[]]]], Pattern[m, Blank[]]], Power[Plus[Optional[Pattern[a, Blank[]]], Times[Optional[Pattern[b, Blank[]]], Pattern[x, Blank[]]], Times[Optional[Pattern[c, Blank[]]], Power[Pattern[x, Blank[]], 2]]], Pattern[p, Blank[]]]], Pattern[x, Blank[Symbol]]] := Simp[((d + e*x)^m*(b + 2*c*x)*(a + b*x + c*x^2)^(p + 1))/((p + 1)*(b^2 - 4*a*c)), x] - Dist[1/((p + 1)*(b^2 - 4*a*c)), Int[(d + e*x)^(m - 1)*(b*e*m + 2*c*d*(2*p + 3) + 2*c*e*(m + 2*p + 3)*x)*(a + b*x + c*x^2)^(p + 1), x], x] /; FreeQ[{a, b, c, d, e}, x] && NeQ[b^2 - 4*a*c, 0] && NeQ[c*d^2 - b*d*e + a*e^2, 0] && NeQ[2*c*d - b*e, 0] && LtQ[p, -1] && GtQ[m, 0] && (LtQ[m, 1] || (ILtQ[m + 2*p + 3, 0] && NeQ[m, 2])) && IntQuadraticQ[a, b, c, d, e, m, p, x]
  737. Int[Times[Power[Plus[Pattern[d, Blank[]], Times[Optional[Pattern[e, Blank[]]], Pattern[x, Blank[]]]], Pattern[m, Blank[]]], Power[Plus[Pattern[a, Blank[]], Times[Optional[Pattern[c, Blank[]]], Power[Pattern[x, Blank[]], 2]]], Pattern[p, Blank[]]]], Pattern[x, Blank[Symbol]]] := -Simp[(x*(d + e*x)^m*(a + c*x^2)^(p + 1))/(2*a*(p + 1)), x] + Dist[1/(2*a*(p + 1)), Int[(d + e*x)^(m - 1)*(d*(2*p + 3) + e*(m + 2*p + 3)*x)*(a + c*x^2)^(p + 1), x], x] /; FreeQ[{a, c, d, e}, x] && NeQ[c*d^2 + a*e^2, 0] && LtQ[p, -1] && GtQ[m, 0] && (LtQ[m, 1] || (ILtQ[m + 2*p + 3, 0] && NeQ[m, 2])) && IntQuadraticQ[a, 0, c, d, e, m, p, x]
  738. Int[Times[Power[Plus[Optional[Pattern[d, Blank[]]], Times[Optional[Pattern[e, Blank[]]], Pattern[x, Blank[]]]], Pattern[m, Blank[]]], Power[Plus[Optional[Pattern[a, Blank[]]], Times[Optional[Pattern[b, Blank[]]], Pattern[x, Blank[]]], Times[Optional[Pattern[c, Blank[]]], Power[Pattern[x, Blank[]], 2]]], Pattern[p, Blank[]]]], Pattern[x, Blank[Symbol]]] := Simp[((d + e*x)^(m - 1)*(d*b - 2*a*e + (2*c*d - b*e)*x)*(a + b*x + c*x^2)^(p + 1))/((p + 1)*(b^2 - 4*a*c)), x] + Dist[1/((p + 1)*(b^2 - 4*a*c)), Int[(d + e*x)^(m - 2)*Simp[e*(2*a*e*(m - 1) + b*d*(2*p - m + 4)) - 2*c*d^2*(2*p + 3) + e*(b*e - 2*d*c)*(m + 2*p + 2)*x, x]*(a + b*x + c*x^2)^(p + 1), x], x] /; FreeQ[{a, b, c, d, e}, x] && NeQ[b^2 - 4*a*c, 0] && NeQ[c*d^2 - b*d*e + a*e^2, 0] && NeQ[2*c*d - b*e, 0] && LtQ[p, -1] && GtQ[m, 1] && IntQuadraticQ[a, b, c, d, e, m, p, x]
  739. Int[Times[Power[Plus[Pattern[d, Blank[]], Times[Optional[Pattern[e, Blank[]]], Pattern[x, Blank[]]]], Pattern[m, Blank[]]], Power[Plus[Pattern[a, Blank[]], Times[Optional[Pattern[c, Blank[]]], Power[Pattern[x, Blank[]], 2]]], Pattern[p, Blank[]]]], Pattern[x, Blank[Symbol]]] := Simp[((d + e*x)^(m - 1)*(a*e - c*d*x)*(a + c*x^2)^(p + 1))/(2*a*c*(p + 1)), x] + Dist[1/((p + 1)*(-2*a*c)), Int[(d + e*x)^(m - 2)*Simp[a*e^2*(m - 1) - c*d^2*(2*p + 3) - d*c*e*(m + 2*p + 2)*x, x]*(a + c*x^2)^(p + 1), x], x] /; FreeQ[{a, c, d, e}, x] && NeQ[c*d^2 + a*e^2, 0] && LtQ[p, -1] && GtQ[m, 1] && IntQuadraticQ[a, 0, c, d, e, m, p, x]
  740. Int[Times[Power[Plus[Optional[Pattern[d, Blank[]]], Times[Optional[Pattern[e, Blank[]]], Pattern[x, Blank[]]]], Pattern[m, Blank[]]], Power[Plus[Optional[Pattern[a, Blank[]]], Times[Optional[Pattern[b, Blank[]]], Pattern[x, Blank[]]], Times[Optional[Pattern[c, Blank[]]], Power[Pattern[x, Blank[]], 2]]], Pattern[p, Blank[]]]], Pattern[x, Blank[Symbol]]] := Simp[((d + e*x)^(m + 1)*(b*c*d - b^2*e + 2*a*c*e + c*(2*c*d - b*e)*x)*(a + b*x + c*x^2)^(p + 1))/((p + 1)*(b^2 - 4*a*c)*(c*d^2 - b*d*e + a*e^2)), x] + Dist[1/((p + 1)*(b^2 - 4*a*c)*(c*d^2 - b*d*e + a*e^2)), Int[(d + e*x)^m*Simp[b*c*d*e*(2*p - m + 2) + b^2*e^2*(m + p + 2) - 2*c^2*d^2*(2*p + 3) - 2*a*c*e^2*(m + 2*p + 3) - c*e*(2*c*d - b*e)*(m + 2*p + 4)*x, x]*(a + b*x + c*x^2)^(p + 1), x], x] /; FreeQ[{a, b, c, d, e, m}, x] && NeQ[b^2 - 4*a*c, 0] && NeQ[c*d^2 - b*d*e + a*e^2, 0] && NeQ[2*c*d - b*e, 0] && LtQ[p, -1] && IntQuadraticQ[a, b, c, d, e, m, p, x]
  741. Int[Times[Power[Plus[Pattern[d, Blank[]], Times[Optional[Pattern[e, Blank[]]], Pattern[x, Blank[]]]], Pattern[m, Blank[]]], Power[Plus[Pattern[a, Blank[]], Times[Optional[Pattern[c, Blank[]]], Power[Pattern[x, Blank[]], 2]]], Pattern[p, Blank[]]]], Pattern[x, Blank[Symbol]]] := -Simp[((d + e*x)^(m + 1)*(a*e + c*d*x)*(a + c*x^2)^(p + 1))/(2*a*(p + 1)*(c*d^2 + a*e^2)), x] + Dist[1/(2*a*(p + 1)*(c*d^2 + a*e^2)), Int[(d + e*x)^m*Simp[c*d^2*(2*p + 3) + a*e^2*(m + 2*p + 3) + c*e*d*(m + 2*p + 4)*x, x]*(a + c*x^2)^(p + 1), x], x] /; FreeQ[{a, c, d, e, m}, x] && NeQ[c*d^2 + a*e^2, 0] && LtQ[p, -1] && IntQuadraticQ[a, 0, c, d, e, m, p, x]
  742. Int[Times[Power[Plus[Optional[Pattern[d, Blank[]]], Times[Optional[Pattern[e, Blank[]]], Pattern[x, Blank[]]]], Pattern[m, Blank[]]], Power[Plus[Optional[Pattern[a, Blank[]]], Times[Optional[Pattern[b, Blank[]]], Pattern[x, Blank[]]], Times[Optional[Pattern[c, Blank[]]], Power[Pattern[x, Blank[]], 2]]], Pattern[p, Blank[]]]], Pattern[x, Blank[Symbol]]] := Simp[(e*(d + e*x)^(m - 1)*(a + b*x + c*x^2)^(p + 1))/(c*(m + 2*p + 1)), x] + Dist[1/(c*(m + 2*p + 1)), Int[(d + e*x)^(m - 2)*Simp[c*d^2*(m + 2*p + 1) - e*(a*e*(m - 1) + b*d*(p + 1)) + e*(2*c*d - b*e)*(m + p)*x, x]*(a + b*x + c*x^2)^p, x], x] /; FreeQ[{a, b, c, d, e, m, p}, x] && NeQ[b^2 - 4*a*c, 0] && NeQ[c*d^2 - b*d*e + a*e^2, 0] && NeQ[2*c*d - b*e, 0] && If[RationalQ[m], GtQ[m, 1], SumSimplerQ[m, -2]] && NeQ[m + 2*p + 1, 0] && IntQuadraticQ[a, b, c, d, e, m, p, x]
  743. Int[Times[Power[Plus[Pattern[d, Blank[]], Times[Optional[Pattern[e, Blank[]]], Pattern[x, Blank[]]]], Pattern[m, Blank[]]], Power[Plus[Pattern[a, Blank[]], Times[Optional[Pattern[c, Blank[]]], Power[Pattern[x, Blank[]], 2]]], Pattern[p, Blank[]]]], Pattern[x, Blank[Symbol]]] := Simp[(e*(d + e*x)^(m - 1)*(a + c*x^2)^(p + 1))/(c*(m + 2*p + 1)), x] + Dist[1/(c*(m + 2*p + 1)), Int[(d + e*x)^(m - 2)*Simp[c*d^2*(m + 2*p + 1) - a*e^2*(m - 1) + 2*c*d*e*(m + p)*x, x]*(a + c*x^2)^p, x], x] /; FreeQ[{a, c, d, e, m, p}, x] && NeQ[c*d^2 + a*e^2, 0] && If[RationalQ[m], GtQ[m, 1], SumSimplerQ[m, -2]] && NeQ[m + 2*p + 1, 0] && IntQuadraticQ[a, 0, c, d, e, m, p, x]
  744. Int[Times[Power[Plus[Optional[Pattern[d, Blank[]]], Times[Optional[Pattern[e, Blank[]]], Pattern[x, Blank[]]]], Pattern[m, Blank[]]], Power[Plus[Optional[Pattern[a, Blank[]]], Times[Optional[Pattern[b, Blank[]]], Pattern[x, Blank[]]], Times[Optional[Pattern[c, Blank[]]], Power[Pattern[x, Blank[]], 2]]], Pattern[p, Blank[]]]], Pattern[x, Blank[Symbol]]] := Simp[(e*(d + e*x)^(m + 1)*(a + b*x + c*x^2)^(p + 1))/((m + 1)*(c*d^2 - b*d*e + a*e^2)), x] + Dist[1/((m + 1)*(c*d^2 - b*d*e + a*e^2)), Int[(d + e*x)^(m + 1)*Simp[c*d*(m + 1) - b*e*(m + p + 2) - c*e*(m + 2*p + 3)*x, x]*(a + b*x + c*x^2)^p, x], x] /; FreeQ[{a, b, c, d, e, m, p}, x] && NeQ[b^2 - 4*a*c, 0] && NeQ[c*d^2 - b*d*e + a*e^2, 0] && NeQ[2*c*d - b*e, 0] && NeQ[m, -1] && ((LtQ[m, -1] && IntQuadraticQ[a, b, c, d, e, m, p, x]) || (SumSimplerQ[m, 1] && IntegerQ[p]) || ILtQ[Simplify[m + 2*p + 3], 0])
  745. Int[Times[Power[Plus[Pattern[d, Blank[]], Times[Optional[Pattern[e, Blank[]]], Pattern[x, Blank[]]]], Pattern[m, Blank[]]], Power[Plus[Pattern[a, Blank[]], Times[Optional[Pattern[c, Blank[]]], Power[Pattern[x, Blank[]], 2]]], Pattern[p, Blank[]]]], Pattern[x, Blank[Symbol]]] := Simp[(e*(d + e*x)^(m + 1)*(a + c*x^2)^(p + 1))/((m + 1)*(c*d^2 + a*e^2)), x] + Dist[c/((m + 1)*(c*d^2 + a*e^2)), Int[(d + e*x)^(m + 1)*Simp[d*(m + 1) - e*(m + 2*p + 3)*x, x]*(a + c*x^2)^p, x], x] /; FreeQ[{a, c, d, e, m, p}, x] && NeQ[c*d^2 + a*e^2, 0] && NeQ[m, -1] && ((LtQ[m, -1] && IntQuadraticQ[a, 0, c, d, e, m, p, x]) || (SumSimplerQ[m, 1] && IntegerQ[p]) || ILtQ[Simplify[m + 2*p + 3], 0])
  746. Int[Times[Power[Plus[Pattern[d, Blank[]], Times[Optional[Pattern[e, Blank[]]], Pattern[x, Blank[]]]], -1], Power[Plus[Pattern[a, Blank[]], Times[Optional[Pattern[c, Blank[]]], Power[Pattern[x, Blank[]], 2]]], Rational[-1, 4]]], Pattern[x, Blank[Symbol]]] := Dist[d, Int[1/((d^2 - e^2*x^2)*(a + c*x^2)^(1/4)), x], x] - Dist[e, Int[x/((d^2 - e^2*x^2)*(a + c*x^2)^(1/4)), x], x] /; FreeQ[{a, c, d, e}, x] && NeQ[c*d^2 + a*e^2, 0]
  747. Int[Times[Power[Plus[Pattern[d, Blank[]], Times[Optional[Pattern[e, Blank[]]], Pattern[x, Blank[]]]], -1], Power[Plus[Pattern[a, Blank[]], Times[Optional[Pattern[c, Blank[]]], Power[Pattern[x, Blank[]], 2]]], Rational[-3, 4]]], Pattern[x, Blank[Symbol]]] := Dist[d, Int[1/((d^2 - e^2*x^2)*(a + c*x^2)^(3/4)), x], x] - Dist[e, Int[x/((d^2 - e^2*x^2)*(a + c*x^2)^(3/4)), x], x] /; FreeQ[{a, c, d, e}, x] && NeQ[c*d^2 + a*e^2, 0]
  748. Int[Times[Power[Plus[Optional[Pattern[d, Blank[]]], Times[Optional[Pattern[e, Blank[]]], Pattern[x, Blank[]]]], -1], Power[Plus[Optional[Pattern[a, Blank[]]], Times[Optional[Pattern[b, Blank[]]], Pattern[x, Blank[]]], Times[Optional[Pattern[c, Blank[]]], Power[Pattern[x, Blank[]], 2]]], Pattern[p, Blank[]]]], Pattern[x, Blank[Symbol]]] := Dist[1/((-4*c)/(b^2 - 4*a*c))^p, Subst[Int[Simp[1 - x^2/(b^2 - 4*a*c), x]^p/Simp[2*c*d - b*e + e*x, x], x], x, b + 2*c*x], x] /; FreeQ[{a, b, c, d, e, p}, x] && GtQ[4*a - b^2/c, 0] && IntegerQ[4*p]
  749. Int[Times[Power[Plus[Optional[Pattern[d, Blank[]]], Times[Optional[Pattern[e, Blank[]]], Pattern[x, Blank[]]]], -1], Power[Plus[Optional[Pattern[a, Blank[]]], Times[Optional[Pattern[b, Blank[]]], Pattern[x, Blank[]]], Times[Optional[Pattern[c, Blank[]]], Power[Pattern[x, Blank[]], 2]]], Pattern[p, Blank[]]]], Pattern[x, Blank[Symbol]]] := Dist[(a + b*x + c*x^2)^p/(-((c*(a + b*x + c*x^2))/(b^2 - 4*a*c)))^p, Int[(-((a*c)/(b^2 - 4*a*c)) - (b*c*x)/(b^2 - 4*a*c) - (c^2*x^2)/(b^2 - 4*a*c))^p/(d + e*x), x], x] /; FreeQ[{a, b, c, d, e, p}, x] && !GtQ[4*a - b^2/c, 0] && IntegerQ[4*p]
  750. Int[Times[Power[Plus[Optional[Pattern[d, Blank[]]], Times[Optional[Pattern[e, Blank[]]], Pattern[x, Blank[]]]], -1], Power[Plus[Pattern[a, Blank[]], Times[Optional[Pattern[b, Blank[]]], Pattern[x, Blank[]]], Times[Optional[Pattern[c, Blank[]]], Power[Pattern[x, Blank[]], 2]]], Rational[-1, 3]]], Pattern[x, Blank[Symbol]]] := With[{q = Rt[3*c*e^2*(2*c*d - b*e), 3]}, -Simp[(Sqrt[3]*c*e*ArcTan[1/Sqrt[3] + (2*(c*d - b*e - c*e*x))/(Sqrt[3]*q*(a + b*x + c*x^2)^(1/3))])/q^2, x] + (-Simp[(3*c*e*Log[d + e*x])/(2*q^2), x] + Simp[(3*c*e*Log[c*d - b*e - c*e*x - q*(a + b*x + c*x^2)^(1/3)])/(2*q^2), x])] /; FreeQ[{a, b, c, d, e}, x] && NeQ[2*c*d - b*e, 0] && EqQ[c^2*d^2 - b*c*d*e + b^2*e^2 - 3*a*c*e^2, 0] && PosQ[c*e^2*(2*c*d - b*e)]
  751. Int[Times[Power[Plus[Pattern[d, Blank[]], Times[Optional[Pattern[e, Blank[]]], Pattern[x, Blank[]]]], -1], Power[Plus[Pattern[a, Blank[]], Times[Optional[Pattern[c, Blank[]]], Power[Pattern[x, Blank[]], 2]]], Rational[-1, 3]]], Pattern[x, Blank[Symbol]]] := With[{q = Rt[(6*c^2*e^2)/d^2, 3]}, -Simp[(Sqrt[3]*c*e*ArcTan[1/Sqrt[3] + (2*c*(d - e*x))/(Sqrt[3]*d*q*(a + c*x^2)^(1/3))])/(d^2*q^2), x] + (-Simp[(3*c*e*Log[d + e*x])/(2*d^2*q^2), x] + Simp[(3*c*e*Log[c*d - c*e*x - d*q*(a + c*x^2)^(1/3)])/(2*d^2*q^2), x])] /; FreeQ[{a, c, d, e}, x] && EqQ[c*d^2 - 3*a*e^2, 0]
  752. Int[Times[Power[Plus[Optional[Pattern[d, Blank[]]], Times[Optional[Pattern[e, Blank[]]], Pattern[x, Blank[]]]], -1], Power[Plus[Pattern[a, Blank[]], Times[Optional[Pattern[b, Blank[]]], Pattern[x, Blank[]]], Times[Optional[Pattern[c, Blank[]]], Power[Pattern[x, Blank[]], 2]]], Rational[-1, 3]]], Pattern[x, Blank[Symbol]]] := With[{q = Rt[-3*c*e^2*(2*c*d - b*e), 3]}, -Simp[(Sqrt[3]*c*e*ArcTan[1/Sqrt[3] - (2*(c*d - b*e - c*e*x))/(Sqrt[3]*q*(a + b*x + c*x^2)^(1/3))])/q^2, x] + (-Simp[(3*c*e*Log[d + e*x])/(2*q^2), x] + Simp[(3*c*e*Log[c*d - b*e - c*e*x + q*(a + b*x + c*x^2)^(1/3)])/(2*q^2), x])] /; FreeQ[{a, b, c, d, e}, x] && NeQ[2*c*d - b*e, 0] && EqQ[c^2*d^2 - b*c*d*e + b^2*e^2 - 3*a*c*e^2, 0] && NegQ[c*e^2*(2*c*d - b*e)]
  753. Int[Times[Power[Plus[Pattern[d, Blank[]], Times[Optional[Pattern[e, Blank[]]], Pattern[x, Blank[]]]], -1], Power[Plus[Pattern[a, Blank[]], Times[Optional[Pattern[c, Blank[]]], Power[Pattern[x, Blank[]], 2]]], Rational[-1, 3]]], Pattern[x, Blank[Symbol]]] := Dist[a^(1/3), Int[1/((d + e*x)*(1 - (3*e*x)/d)^(1/3)*(1 + (3*e*x)/d)^(1/3)), x], x] /; FreeQ[{a, c, d, e}, x] && EqQ[c*d^2 + 9*a*e^2, 0] && GtQ[a, 0]
  754. Int[Times[Power[Plus[Pattern[d, Blank[]], Times[Optional[Pattern[e, Blank[]]], Pattern[x, Blank[]]]], -1], Power[Plus[Pattern[a, Blank[]], Times[Optional[Pattern[c, Blank[]]], Power[Pattern[x, Blank[]], 2]]], Rational[-1, 3]]], Pattern[x, Blank[Symbol]]] := Dist[(1 + (c*x^2)/a)^(1/3)/(a + c*x^2)^(1/3), Int[1/((d + e*x)*(1 + (c*x^2)/a)^(1/3)), x], x] /; FreeQ[{a, c, d, e}, x] && EqQ[c*d^2 + 9*a*e^2, 0] && !GtQ[a, 0]
  755. Int[Times[Power[Plus[Optional[Pattern[d, Blank[]]], Times[Optional[Pattern[e, Blank[]]], Pattern[x, Blank[]]]], -1], Power[Plus[Pattern[a, Blank[]], Times[Optional[Pattern[b, Blank[]]], Pattern[x, Blank[]]], Times[Optional[Pattern[c, Blank[]]], Power[Pattern[x, Blank[]], 2]]], Rational[-1, 3]]], Pattern[x, Blank[Symbol]]] := With[{q = Rt[b^2 - 4*a*c, 2]}, Dist[((b + q + 2*c*x)^(1/3)*(b - q + 2*c*x)^(1/3))/(a + b*x + c*x^2)^(1/3), Int[1/((d + e*x)*(b + q + 2*c*x)^(1/3)*(b - q + 2*c*x)^(1/3)), x], x]] /; FreeQ[{a, b, c, d, e}, x] && NeQ[b^2 - 4*a*c, 0] && EqQ[c^2*d^2 - b*c*d*e - 2*b^2*e^2 + 9*a*c*e^2, 0]
  756. Int[Times[Power[Plus[Pattern[d, Blank[]], Times[Optional[Pattern[e, Blank[]]], Pattern[x, Blank[]]]], Pattern[m, Blank[]]], Power[Plus[Pattern[a, Blank[]], Times[Optional[Pattern[c, Blank[]]], Power[Pattern[x, Blank[]], 2]]], Pattern[p, Blank[]]]], Pattern[x, Blank[Symbol]]] := Int[(d + e*x)^m*(Rt[a, 2] + Rt[-c, 2]*x)^p*(Rt[a, 2] - Rt[-c, 2]*x)^p, x] /; FreeQ[{a, c, d, e, m, p}, x] && NeQ[c*d^2 + a*e^2, 0] && !IntegerQ[p] && GtQ[a, 0] && LtQ[c, 0]
  757. Int[Times[Power[Plus[Pattern[d, Blank[]], Times[Optional[Pattern[e, Blank[]]], Pattern[x, Blank[]]]], Pattern[m, Blank[]]], Power[Plus[Pattern[a, Blank[]], Times[Optional[Pattern[c, Blank[]]], Power[Pattern[x, Blank[]], 2]]], Pattern[p, Blank[]]]], Pattern[x, Blank[Symbol]]] := Int[ExpandIntegrand[(a + c*x^2)^p, (d/(d^2 - e^2*x^2) - (e*x)/(d^2 - e^2*x^2))^(-m), x], x] /; FreeQ[{a, c, d, e, p}, x] && NeQ[c*d^2 + a*e^2, 0] && !IntegerQ[p] && ILtQ[m, 0]
  758. Int[Times[Power[Plus[Optional[Pattern[d, Blank[]]], Times[Optional[Pattern[e, Blank[]]], Pattern[x, Blank[]]]], Pattern[m, Blank[]]], Power[Plus[Optional[Pattern[a, Blank[]]], Times[Optional[Pattern[b, Blank[]]], Pattern[x, Blank[]]], Times[Optional[Pattern[c, Blank[]]], Power[Pattern[x, Blank[]], 2]]], Pattern[p, Blank[]]]], Pattern[x, Blank[Symbol]]] := With[{q = Rt[b^2 - 4*a*c, 2]}, -Dist[((1/(d + e*x))^(2*p)*(a + b*x + c*x^2)^p)/(e*((e*(b - q + 2*c*x))/(2*c*(d + e*x)))^p*((e*(b + q + 2*c*x))/(2*c*(d + e*x)))^p), Subst[Int[x^(-m - 2*(p + 1))*Simp[1 - (d - (e*(b - q))/(2*c))*x, x]^p*Simp[1 - (d - (e*(b + q))/(2*c))*x, x]^p, x], x, 1/(d + e*x)], x]] /; FreeQ[{a, b, c, d, e, p}, x] && NeQ[b^2 - 4*a*c, 0] && NeQ[c*d^2 - b*d*e + a*e^2, 0] && NeQ[2*c*d - b*e, 0] && !IntegerQ[p] && ILtQ[m, 0]
  759. Int[Times[Power[Plus[Optional[Pattern[d, Blank[]]], Times[Optional[Pattern[e, Blank[]]], Pattern[x, Blank[]]]], Pattern[m, Blank[]]], Power[Plus[Optional[Pattern[a, Blank[]]], Times[Optional[Pattern[b, Blank[]]], Pattern[x, Blank[]]], Times[Optional[Pattern[c, Blank[]]], Power[Pattern[x, Blank[]], 2]]], Pattern[p, Blank[]]]], Pattern[x, Blank[Symbol]]] := With[{q = Rt[b^2 - 4*a*c, 2]}, Dist[(a + b*x + c*x^2)^p/(e*(1 - (d + e*x)/(d - (e*(b - q))/(2*c)))^p*(1 - (d + e*x)/(d - (e*(b + q))/(2*c)))^p), Subst[Int[x^m*Simp[1 - x/(d - (e*(b - q))/(2*c)), x]^p*Simp[1 - x/(d - (e*(b + q))/(2*c)), x]^p, x], x, d + e*x], x]] /; FreeQ[{a, b, c, d, e, m, p}, x] && NeQ[b^2 - 4*a*c, 0] && NeQ[c*d^2 - b*d*e + a*e^2, 0] && NeQ[2*c*d - b*e, 0] && !IntegerQ[p]
  760. Int[Times[Power[Plus[Pattern[d, Blank[]], Times[Optional[Pattern[e, Blank[]]], Pattern[x, Blank[]]]], Pattern[m, Blank[]]], Power[Plus[Pattern[a, Blank[]], Times[Optional[Pattern[c, Blank[]]], Power[Pattern[x, Blank[]], 2]]], Pattern[p, Blank[]]]], Pattern[x, Blank[Symbol]]] := With[{q = Rt[-(a*c), 2]}, Dist[(a + c*x^2)^p/(e*(1 - (d + e*x)/(d + (e*q)/c))^p*(1 - (d + e*x)/(d - (e*q)/c))^p), Subst[Int[x^m*Simp[1 - x/(d + (e*q)/c), x]^p*Simp[1 - x/(d - (e*q)/c), x]^p, x], x, d + e*x], x]] /; FreeQ[{a, c, d, e, m, p}, x] && NeQ[c*d^2 + a*e^2, 0] && !IntegerQ[p]
  761. Int[Times[Power[Plus[Optional[Pattern[d, Blank[]]], Times[Optional[Pattern[e, Blank[]]], Pattern[u, Blank[]]]], Optional[Pattern[m, Blank[]]]], Power[Plus[Pattern[a, Blank[]], Times[Optional[Pattern[b, Blank[]]], Pattern[u, Blank[]]], Times[Optional[Pattern[c, Blank[]]], Power[Pattern[u, Blank[]], 2]]], Optional[Pattern[p, Blank[]]]]], Pattern[x, Blank[Symbol]]] := Dist[1/Coefficient[u, x, 1], Subst[Int[(d + e*x)^m*(a + b*x + c*x^2)^p, x], x, u], x] /; FreeQ[{a, b, c, d, e, m, p}, x] && LinearQ[u, x] && NeQ[u, x]
  762. Int[Times[Power[Plus[Optional[Pattern[d, Blank[]]], Times[Optional[Pattern[e, Blank[]]], Pattern[u, Blank[]]]], Optional[Pattern[m, Blank[]]]], Power[Plus[Pattern[a, Blank[]], Times[Optional[Pattern[c, Blank[]]], Power[Pattern[u, Blank[]], 2]]], Optional[Pattern[p, Blank[]]]]], Pattern[x, Blank[Symbol]]] := Dist[1/Coefficient[u, x, 1], Subst[Int[(d + e*x)^m*(a + c*x^2)^p, x], x, u], x] /; FreeQ[{a, c, d, e, m, p}, x] && LinearQ[u, x] && NeQ[u, x]
  763. Int[Times[Power[Times[Optional[Pattern[e, Blank[]]], Pattern[x, Blank[]]], Optional[Pattern[m, Blank[]]]], Plus[Pattern[f, Blank[]], Times[Optional[Pattern[g, Blank[]]], Pattern[x, Blank[]]]], Power[Plus[Times[Optional[Pattern[b, Blank[]]], Pattern[x, Blank[]]], Times[Optional[Pattern[c, Blank[]]], Power[Pattern[x, Blank[]], 2]]], Optional[Pattern[p, Blank[]]]]], Pattern[x, Blank[Symbol]]] := Simp[(g*(e*x)^m*(b*x + c*x^2)^(p + 1))/(c*(m + 2*p + 2)), x] /; FreeQ[{b, c, e, f, g, m, p}, x] && EqQ[b*g*(m + p + 1) - c*f*(m + 2*p + 2), 0] && NeQ[m + 2*p + 2, 0]
  764. Int[Times[Power[Pattern[x, Blank[]], Optional[Pattern[m, Blank[]]]], Plus[Pattern[f, Blank[]], Times[Optional[Pattern[g, Blank[]]], Pattern[x, Blank[]]]], Power[Plus[Pattern[a, Blank[]], Times[Optional[Pattern[c, Blank[]]], Power[Pattern[x, Blank[]], 2]]], Optional[Pattern[p, Blank[]]]]], Pattern[x, Blank[Symbol]]] := Dist[f, Int[x^m*(a + c*x^2)^p, x], x] + Dist[g, Int[x^(m + 1)*(a + c*x^2)^p, x], x] /; FreeQ[{a, c, f, g, p}, x] && IntegerQ[m] && !IntegerQ[2*p]
  765. Int[Times[Power[Times[Optional[Pattern[e, Blank[]]], Pattern[x, Blank[]]], Optional[Pattern[m, Blank[]]]], Plus[Optional[Pattern[f, Blank[]]], Times[Optional[Pattern[g, Blank[]]], Pattern[x, Blank[]]]], Power[Plus[Optional[Pattern[a, Blank[]]], Times[Optional[Pattern[b, Blank[]]], Pattern[x, Blank[]]], Times[Optional[Pattern[c, Blank[]]], Power[Pattern[x, Blank[]], 2]]], Optional[Pattern[p, Blank[]]]]], Pattern[x, Blank[Symbol]]] := Int[ExpandIntegrand[(e*x)^m*(f + g*x)*(a + b*x + c*x^2)^p, x], x] /; FreeQ[{a, b, c, e, f, g, m}, x] && IntegerQ[p] && (GtQ[p, 0] || (EqQ[a, 0] && IntegerQ[m]))
  766. Int[Times[Power[Times[Optional[Pattern[e, Blank[]]], Pattern[x, Blank[]]], Optional[Pattern[m, Blank[]]]], Plus[Optional[Pattern[f, Blank[]]], Times[Optional[Pattern[g, Blank[]]], Pattern[x, Blank[]]]], Power[Plus[Pattern[a, Blank[]], Times[Optional[Pattern[c, Blank[]]], Power[Pattern[x, Blank[]], 2]]], Optional[Pattern[p, Blank[]]]]], Pattern[x, Blank[Symbol]]] := Int[ExpandIntegrand[(e*x)^m*(f + g*x)*(a + c*x^2)^p, x], x] /; FreeQ[{a, c, e, f, g, m}, x] && IGtQ[p, 0]
  767. Int[Times[Power[Plus[Optional[Pattern[d, Blank[]]], Times[Optional[Pattern[e, Blank[]]], Pattern[x, Blank[]]]], Optional[Pattern[m, Blank[]]]], Plus[Pattern[f, Blank[]], Times[Optional[Pattern[g, Blank[]]], Pattern[x, Blank[]]]], Power[Plus[Pattern[a, Blank[]], Times[Optional[Pattern[b, Blank[]]], Pattern[x, Blank[]]], Times[Optional[Pattern[c, Blank[]]], Power[Pattern[x, Blank[]], 2]]], Optional[Pattern[p, Blank[]]]]], Pattern[x, Blank[Symbol]]] := -Simp[(f*g*(d + e*x)^(m + 1)*(a + b*x + c*x^2)^(p + 1))/(b*(p + 1)*(e*f - d*g)), x] /; FreeQ[{a, b, c, d, e, f, g, m, p}, x] && EqQ[b^2 - 4*a*c, 0] && EqQ[m + 2*p + 3, 0] && EqQ[2*c*f - b*g, 0]
  768. Int[Times[Power[Plus[Optional[Pattern[d, Blank[]]], Times[Optional[Pattern[e, Blank[]]], Pattern[x, Blank[]]]], Optional[Pattern[m, Blank[]]]], Plus[Optional[Pattern[f, Blank[]]], Times[Optional[Pattern[g, Blank[]]], Pattern[x, Blank[]]]], Power[Plus[Optional[Pattern[a, Blank[]]], Times[Optional[Pattern[b, Blank[]]], Pattern[x, Blank[]]], Times[Optional[Pattern[c, Blank[]]], Power[Pattern[x, Blank[]], 2]]], Pattern[p, Blank[]]]], Pattern[x, Blank[Symbol]]] := Simp[(g*(d + e*x)^m*(a + b*x + c*x^2)^(p + 1))/(2*c*(p + 1)), x] - Dist[(e*g*m)/(2*c*(p + 1)), Int[(d + e*x)^(m - 1)*(a + b*x + c*x^2)^(p + 1), x], x] /; FreeQ[{a, b, c, d, e, f, g}, x] && EqQ[2*c*f - b*g, 0] && LtQ[p, -1] && GtQ[m, 0]
  769. Int[Times[Power[Plus[Optional[Pattern[d, Blank[]]], Times[Optional[Pattern[e, Blank[]]], Pattern[x, Blank[]]]], Optional[Pattern[m, Blank[]]]], Plus[Optional[Pattern[f, Blank[]]], Times[Optional[Pattern[g, Blank[]]], Pattern[x, Blank[]]]], Power[Plus[Pattern[a, Blank[]], Times[Optional[Pattern[b, Blank[]]], Pattern[x, Blank[]]], Times[Optional[Pattern[c, Blank[]]], Power[Pattern[x, Blank[]], 2]]], Pattern[p, Blank[]]]], Pattern[x, Blank[Symbol]]] := Simp[(-2*c*(e*f - d*g)*(d + e*x)^(m + 1)*(a + b*x + c*x^2)^(p + 1))/((p + 1)*(2*c*d - b*e)^2), x] + Dist[(2*c*f - b*g)/(2*c*d - b*e), Int[(d + e*x)^(m + 1)*(a + b*x + c*x^2)^p, x], x] /; FreeQ[{a, b, c, d, e, f, g, m, p}, x] && EqQ[b^2 - 4*a*c, 0] && EqQ[m + 2*p + 3, 0] && NeQ[2*c*f - b*g, 0] && NeQ[2*c*d - b*e, 0]
  770. Int[Times[Power[Plus[Optional[Pattern[d, Blank[]]], Times[Optional[Pattern[e, Blank[]]], Pattern[x, Blank[]]]], Optional[Pattern[m, Blank[]]]], Plus[Optional[Pattern[f, Blank[]]], Times[Optional[Pattern[g, Blank[]]], Pattern[x, Blank[]]]], Power[Plus[Optional[Pattern[a, Blank[]]], Times[Optional[Pattern[b, Blank[]]], Pattern[x, Blank[]]], Times[Optional[Pattern[c, Blank[]]], Power[Pattern[x, Blank[]], 2]]], Pattern[p, Blank[]]]], Pattern[x, Blank[Symbol]]] := Dist[(a + b*x + c*x^2)^FracPart[p]/(c^IntPart[p]*(b/2 + c*x)^(2*FracPart[p])), Int[(d + e*x)^m*(f + g*x)*(b/2 + c*x)^(2*p), x], x] /; FreeQ[{a, b, c, d, e, f, g, m}, x] && EqQ[b^2 - 4*a*c, 0]
  771. Int[Times[Power[Plus[Optional[Pattern[d, Blank[]]], Times[Optional[Pattern[e, Blank[]]], Pattern[x, Blank[]]]], Optional[Pattern[m, Blank[]]]], Plus[Optional[Pattern[f, Blank[]]], Times[Optional[Pattern[g, Blank[]]], Pattern[x, Blank[]]]], Power[Plus[Optional[Pattern[a, Blank[]]], Times[Optional[Pattern[b, Blank[]]], Pattern[x, Blank[]]], Times[Optional[Pattern[c, Blank[]]], Power[Pattern[x, Blank[]], 2]]], Optional[Pattern[p, Blank[]]]]], Pattern[x, Blank[Symbol]]] := Int[ExpandIntegrand[(d + e*x)^m*(f + g*x)*(a + b*x + c*x^2)^p, x], x] /; FreeQ[{a, b, c, d, e, f, g, m}, x] && NeQ[b^2 - 4*a*c, 0] && IntegerQ[p] && (GtQ[p, 0] || (EqQ[a, 0] && IntegerQ[m]))
  772. Int[Times[Power[Plus[Optional[Pattern[d, Blank[]]], Times[Optional[Pattern[e, Blank[]]], Pattern[x, Blank[]]]], Optional[Pattern[m, Blank[]]]], Plus[Optional[Pattern[f, Blank[]]], Times[Optional[Pattern[g, Blank[]]], Pattern[x, Blank[]]]], Power[Plus[Pattern[a, Blank[]], Times[Optional[Pattern[c, Blank[]]], Power[Pattern[x, Blank[]], 2]]], Optional[Pattern[p, Blank[]]]]], Pattern[x, Blank[Symbol]]] := Int[ExpandIntegrand[(d + e*x)^m*(f + g*x)*(a + c*x^2)^p, x], x] /; FreeQ[{a, c, d, e, f, g, m}, x] && IGtQ[p, 0]
  773. Int[Times[Plus[Optional[Pattern[d, Blank[]]], Times[Optional[Pattern[e, Blank[]]], Pattern[x, Blank[]]]], Plus[Pattern[f, Blank[]], Times[Optional[Pattern[g, Blank[]]], Pattern[x, Blank[]]]], Power[Plus[Optional[Pattern[a, Blank[]]], Times[Optional[Pattern[b, Blank[]]], Pattern[x, Blank[]]], Times[Optional[Pattern[c, Blank[]]], Power[Pattern[x, Blank[]], 2]]], -1]], Pattern[x, Blank[Symbol]]] := Simp[(e*g*x)/c, x] + Dist[1/c, Int[(c*d*f - a*e*g + (c*e*f + c*d*g - b*e*g)*x)/(a + b*x + c*x^2), x], x] /; FreeQ[{a, b, c, d, e, f, g}, x] && NeQ[b^2 - 4*a*c, 0]
  774. Int[Times[Plus[Optional[Pattern[d, Blank[]]], Times[Optional[Pattern[e, Blank[]]], Pattern[x, Blank[]]]], Plus[Pattern[f, Blank[]], Times[Optional[Pattern[g, Blank[]]], Pattern[x, Blank[]]]], Power[Plus[Pattern[a, Blank[]], Times[Optional[Pattern[c, Blank[]]], Power[Pattern[x, Blank[]], 2]]], -1]], Pattern[x, Blank[Symbol]]] := Simp[(e*g*x)/c, x] + Dist[1/c, Int[(c*d*f - a*e*g + c*(e*f + d*g)*x)/(a + c*x^2), x], x] /; FreeQ[{a, c, d, e, f, g}, x]
  775. Int[Times[Plus[Optional[Pattern[d, Blank[]]], Times[Optional[Pattern[e, Blank[]]], Pattern[x, Blank[]]]], Plus[Optional[Pattern[f, Blank[]]], Times[Optional[Pattern[g, Blank[]]], Pattern[x, Blank[]]]], Power[Plus[Optional[Pattern[a, Blank[]]], Times[Optional[Pattern[b, Blank[]]], Pattern[x, Blank[]]], Times[Optional[Pattern[c, Blank[]]], Power[Pattern[x, Blank[]], 2]]], Pattern[p, Blank[]]]], Pattern[x, Blank[Symbol]]] := -Simp[((b*e*g*(p + 2) - c*(e*f + d*g)*(2*p + 3) - 2*c*e*g*(p + 1)*x)*(a + b*x + c*x^2)^(p + 1))/(2*c^2*(p + 1)*(2*p + 3)), x] /; FreeQ[{a, b, c, d, e, f, g, p}, x] && NeQ[b^2 - 4*a*c, 0] && EqQ[b^2*e*g*(p + 2) - 2*a*c*e*g + c*(2*c*d*f - b*(e*f + d*g))*(2*p + 3), 0] && NeQ[p, -1]
  776. Int[Times[Plus[Optional[Pattern[d, Blank[]]], Times[Optional[Pattern[e, Blank[]]], Pattern[x, Blank[]]]], Plus[Optional[Pattern[f, Blank[]]], Times[Optional[Pattern[g, Blank[]]], Pattern[x, Blank[]]]], Power[Plus[Pattern[a, Blank[]], Times[Optional[Pattern[c, Blank[]]], Power[Pattern[x, Blank[]], 2]]], Pattern[p, Blank[]]]], Pattern[x, Blank[Symbol]]] := Simp[(((e*f + d*g)*(2*p + 3) + 2*e*g*(p + 1)*x)*(a + c*x^2)^(p + 1))/(2*c*(p + 1)*(2*p + 3)), x] /; FreeQ[{a, c, d, e, f, g, p}, x] && EqQ[a*e*g - c*d*f*(2*p + 3), 0] && NeQ[p, -1]
  777. Int[Times[Plus[Optional[Pattern[d, Blank[]]], Times[Optional[Pattern[e, Blank[]]], Pattern[x, Blank[]]]], Plus[Optional[Pattern[f, Blank[]]], Times[Optional[Pattern[g, Blank[]]], Pattern[x, Blank[]]]], Power[Plus[Optional[Pattern[a, Blank[]]], Times[Optional[Pattern[b, Blank[]]], Pattern[x, Blank[]]], Times[Optional[Pattern[c, Blank[]]], Power[Pattern[x, Blank[]], 2]]], Pattern[p, Blank[]]]], Pattern[x, Blank[Symbol]]] := -Simp[((2*a*c*(e*f + d*g) - b*(c*d*f + a*e*g) - (b^2*e*g - b*c*(e*f + d*g) + 2*c*(c*d*f - a*e*g))*x)*(a + b*x + c*x^2)^(p + 1))/(c*(p + 1)*(b^2 - 4*a*c)), x] - Dist[(b^2*e*g*(p + 2) - 2*a*c*e*g + c*(2*c*d*f - b*(e*f + d*g))*(2*p + 3))/(c*(p + 1)*(b^2 - 4*a*c)), Int[(a + b*x + c*x^2)^(p + 1), x], x] /; FreeQ[{a, b, c, d, e, f, g}, x] && NeQ[b^2 - 4*a*c, 0] && LtQ[p, -1]
  778. Int[Times[Plus[Optional[Pattern[d, Blank[]]], Times[Optional[Pattern[e, Blank[]]], Pattern[x, Blank[]]]], Plus[Optional[Pattern[f, Blank[]]], Times[Optional[Pattern[g, Blank[]]], Pattern[x, Blank[]]]], Power[Plus[Pattern[a, Blank[]], Times[Optional[Pattern[c, Blank[]]], Power[Pattern[x, Blank[]], 2]]], Pattern[p, Blank[]]]], Pattern[x, Blank[Symbol]]] := Simp[((a*(e*f + d*g) - (c*d*f - a*e*g)*x)*(a + c*x^2)^(p + 1))/(2*a*c*(p + 1)), x] - Dist[(a*e*g - c*d*f*(2*p + 3))/(2*a*c*(p + 1)), Int[(a + c*x^2)^(p + 1), x], x] /; FreeQ[{a, c, d, e, f, g}, x] && LtQ[p, -1]
  779. Int[Times[Plus[Optional[Pattern[d, Blank[]]], Times[Optional[Pattern[e, Blank[]]], Pattern[x, Blank[]]]], Plus[Optional[Pattern[f, Blank[]]], Times[Optional[Pattern[g, Blank[]]], Pattern[x, Blank[]]]], Power[Plus[Optional[Pattern[a, Blank[]]], Times[Optional[Pattern[b, Blank[]]], Pattern[x, Blank[]]], Times[Optional[Pattern[c, Blank[]]], Power[Pattern[x, Blank[]], 2]]], Pattern[p, Blank[]]]], Pattern[x, Blank[Symbol]]] := -Simp[((b*e*g*(p + 2) - c*(e*f + d*g)*(2*p + 3) - 2*c*e*g*(p + 1)*x)*(a + b*x + c*x^2)^(p + 1))/(2*c^2*(p + 1)*(2*p + 3)), x] + Dist[(b^2*e*g*(p + 2) - 2*a*c*e*g + c*(2*c*d*f - b*(e*f + d*g))*(2*p + 3))/(2*c^2*(2*p + 3)), Int[(a + b*x + c*x^2)^p, x], x] /; FreeQ[{a, b, c, d, e, f, g, p}, x] && NeQ[b^2 - 4*a*c, 0] && !LeQ[p, -1]
  780. Int[Times[Plus[Optional[Pattern[d, Blank[]]], Times[Optional[Pattern[e, Blank[]]], Pattern[x, Blank[]]]], Plus[Optional[Pattern[f, Blank[]]], Times[Optional[Pattern[g, Blank[]]], Pattern[x, Blank[]]]], Power[Plus[Pattern[a, Blank[]], Times[Optional[Pattern[c, Blank[]]], Power[Pattern[x, Blank[]], 2]]], Pattern[p, Blank[]]]], Pattern[x, Blank[Symbol]]] := Simp[(((e*f + d*g)*(2*p + 3) + 2*e*g*(p + 1)*x)*(a + c*x^2)^(p + 1))/(2*c*(p + 1)*(2*p + 3)), x] - Dist[(a*e*g - c*d*f*(2*p + 3))/(c*(2*p + 3)), Int[(a + c*x^2)^p, x], x] /; FreeQ[{a, c, d, e, f, g, p}, x] && !LeQ[p, -1]
  781. Int[Times[Power[Times[Optional[Pattern[e, Blank[]]], Pattern[x, Blank[]]], Optional[Pattern[m, Blank[]]]], Plus[Optional[Pattern[f, Blank[]]], Times[Optional[Pattern[g, Blank[]]], Pattern[x, Blank[]]]], Power[Plus[Times[Optional[Pattern[b, Blank[]]], Pattern[x, Blank[]]], Times[Optional[Pattern[c, Blank[]]], Power[Pattern[x, Blank[]], 2]]], Optional[Pattern[p, Blank[]]]]], Pattern[x, Blank[Symbol]]] := Dist[1/e^p, Int[(e*x)^(m + p)*(f + g*x)*(b + c*x)^p, x], x] /; FreeQ[{b, c, e, f, g, m}, x] && IntegerQ[p]
  782. Int[Times[Power[Plus[Pattern[d, Blank[]], Times[Optional[Pattern[e, Blank[]]], Pattern[x, Blank[]]]], Pattern[m, Blank[]]], Plus[Optional[Pattern[f, Blank[]]], Times[Optional[Pattern[g, Blank[]]], Pattern[x, Blank[]]]], Power[Plus[Optional[Pattern[a, Blank[]]], Times[Optional[Pattern[b, Blank[]]], Pattern[x, Blank[]]], Times[Optional[Pattern[c, Blank[]]], Power[Pattern[x, Blank[]], 2]]], Optional[Pattern[p, Blank[]]]]], Pattern[x, Blank[Symbol]]] := Int[(d + e*x)^(m + p)*(f + g*x)*(a/d + (c*x)/e)^p, x] /; FreeQ[{a, b, c, d, e, f, g, m}, x] && NeQ[b^2 - 4*a*c, 0] && EqQ[c*d^2 - b*d*e + a*e^2, 0] && IntegerQ[p]
  783. Int[Times[Power[Plus[Pattern[d, Blank[]], Times[Optional[Pattern[e, Blank[]]], Pattern[x, Blank[]]]], Pattern[m, Blank[]]], Plus[Optional[Pattern[f, Blank[]]], Times[Optional[Pattern[g, Blank[]]], Pattern[x, Blank[]]]], Power[Plus[Pattern[a, Blank[]], Times[Optional[Pattern[c, Blank[]]], Power[Pattern[x, Blank[]], 2]]], Optional[Pattern[p, Blank[]]]]], Pattern[x, Blank[Symbol]]] := Int[(d + e*x)^(m + p)*(f + g*x)*(a/d + (c*x)/e)^p, x] /; FreeQ[{a, c, d, e, f, g, m}, x] && EqQ[c*d^2 + a*e^2, 0] && (IntegerQ[p] || (GtQ[a, 0] && GtQ[d, 0] && EqQ[m + p, 0]))
  784. Int[Times[Power[Plus[Pattern[d, Blank[]], Times[Optional[Pattern[e, Blank[]]], Pattern[x, Blank[]]]], Pattern[m, Blank[]]], Plus[Optional[Pattern[f, Blank[]]], Times[Optional[Pattern[g, Blank[]]], Pattern[x, Blank[]]]], Power[Plus[Optional[Pattern[a, Blank[]]], Times[Optional[Pattern[b, Blank[]]], Pattern[x, Blank[]]], Times[Optional[Pattern[c, Blank[]]], Power[Pattern[x, Blank[]], 2]]], Pattern[p, Blank[]]]], Pattern[x, Blank[Symbol]]] := Dist[d^m*e^m, Int[((f + g*x)*(a + b*x + c*x^2)^(m + p))/(a*e + c*d*x)^m, x], x] /; FreeQ[{a, b, c, d, e, f, g, p}, x] && NeQ[b^2 - 4*a*c, 0] && EqQ[c*d^2 - b*d*e + a*e^2, 0] && !IntegerQ[2*p] && ILtQ[m, 0]
  785. Int[Times[Pattern[x, Blank[]], Power[Plus[Pattern[d, Blank[]], Times[Optional[Pattern[e, Blank[]]], Pattern[x, Blank[]]]], Pattern[m, Blank[]]], Power[Plus[Pattern[a, Blank[]], Times[Optional[Pattern[c, Blank[]]], Power[Pattern[x, Blank[]], 2]]], Pattern[p, Blank[]]]], Pattern[x, Blank[Symbol]]] := Dist[d^m*e^m, Int[(x*(a + c*x^2)^(m + p))/(a*e + c*d*x)^m, x], x] /; FreeQ[{a, c, d, e, p}, x] && EqQ[c*d^2 + a*e^2, 0] && !IntegerQ[p] && ILtQ[m, 0] && EqQ[m, -1] && !ILtQ[p - 1/2, 0]
  786. Int[Times[Power[Plus[Optional[Pattern[d, Blank[]]], Times[Optional[Pattern[e, Blank[]]], Pattern[x, Blank[]]]], Pattern[m, Blank[]]], Plus[Optional[Pattern[f, Blank[]]], Times[Optional[Pattern[g, Blank[]]], Pattern[x, Blank[]]]], Power[Plus[Optional[Pattern[a, Blank[]]], Times[Optional[Pattern[b, Blank[]]], Pattern[x, Blank[]]], Times[Optional[Pattern[c, Blank[]]], Power[Pattern[x, Blank[]], 2]]], Pattern[p, Blank[]]]], Pattern[x, Blank[Symbol]]] := Simp[(g*(d + e*x)^m*(a + b*x + c*x^2)^(p + 1))/(c*(m + 2*p + 2)), x] /; FreeQ[{a, b, c, d, e, f, g, m, p}, x] && NeQ[b^2 - 4*a*c, 0] && EqQ[c*d^2 - b*d*e + a*e^2, 0] && EqQ[m*(g*(c*d - b*e) + c*e*f) + e*(p + 1)*(2*c*f - b*g), 0]
  787. Int[Times[Power[Plus[Pattern[d, Blank[]], Times[Optional[Pattern[e, Blank[]]], Pattern[x, Blank[]]]], Pattern[m, Blank[]]], Plus[Optional[Pattern[f, Blank[]]], Times[Optional[Pattern[g, Blank[]]], Pattern[x, Blank[]]]], Power[Plus[Pattern[a, Blank[]], Times[Optional[Pattern[c, Blank[]]], Power[Pattern[x, Blank[]], 2]]], Pattern[p, Blank[]]]], Pattern[x, Blank[Symbol]]] := Simp[(g*(d + e*x)^m*(a + c*x^2)^(p + 1))/(c*(m + 2*p + 2)), x] /; FreeQ[{a, c, d, e, f, g, m, p}, x] && EqQ[c*d^2 + a*e^2, 0] && EqQ[m*(d*g + e*f) + 2*e*f*(p + 1), 0]
  788. Int[Times[Power[Plus[Optional[Pattern[d, Blank[]]], Times[Optional[Pattern[e, Blank[]]], Pattern[x, Blank[]]]], Pattern[m, Blank[]]], Plus[Optional[Pattern[f, Blank[]]], Times[Optional[Pattern[g, Blank[]]], Pattern[x, Blank[]]]], Power[Plus[Optional[Pattern[a, Blank[]]], Times[Optional[Pattern[b, Blank[]]], Pattern[x, Blank[]]], Times[Optional[Pattern[c, Blank[]]], Power[Pattern[x, Blank[]], 2]]], Pattern[p, Blank[]]]], Pattern[x, Blank[Symbol]]] := Simp[((g*(c*d - b*e) + c*e*f)*(d + e*x)^m*(a + b*x + c*x^2)^(p + 1))/(c*(p + 1)*(2*c*d - b*e)), x] - Dist[(e*(m*(g*(c*d - b*e) + c*e*f) + e*(p + 1)*(2*c*f - b*g)))/(c*(p + 1)*(2*c*d - b*e)), Int[(d + e*x)^(m - 1)*(a + b*x + c*x^2)^(p + 1), x], x] /; FreeQ[{a, b, c, d, e, f, g}, x] && NeQ[b^2 - 4*a*c, 0] && EqQ[c*d^2 - b*d*e + a*e^2, 0] && LtQ[p, -1] && GtQ[m, 0]
  789. Int[Times[Power[Plus[Optional[Pattern[d, Blank[]]], Times[Optional[Pattern[e, Blank[]]], Pattern[x, Blank[]]]], Pattern[m, Blank[]]], Plus[Optional[Pattern[f, Blank[]]], Times[Optional[Pattern[g, Blank[]]], Pattern[x, Blank[]]]], Power[Plus[Pattern[a, Blank[]], Times[Optional[Pattern[c, Blank[]]], Power[Pattern[x, Blank[]], 2]]], Pattern[p, Blank[]]]], Pattern[x, Blank[Symbol]]] := Simp[((d*g + e*f)*(d + e*x)^m*(a + c*x^2)^(p + 1))/(2*c*d*(p + 1)), x] - Dist[(e*(m*(d*g + e*f) + 2*e*f*(p + 1)))/(2*c*d*(p + 1)), Int[(d + e*x)^(m - 1)*(a + c*x^2)^(p + 1), x], x] /; FreeQ[{a, c, d, e, f, g}, x] && EqQ[c*d^2 + a*e^2, 0] && LtQ[p, -1] && GtQ[m, 0]
  790. Int[Times[Power[Plus[Optional[Pattern[d, Blank[]]], Times[Optional[Pattern[e, Blank[]]], Pattern[x, Blank[]]]], Pattern[m, Blank[]]], Plus[Optional[Pattern[f, Blank[]]], Times[Optional[Pattern[g, Blank[]]], Pattern[x, Blank[]]]], Power[Plus[Optional[Pattern[a, Blank[]]], Times[Optional[Pattern[b, Blank[]]], Pattern[x, Blank[]]], Times[Optional[Pattern[c, Blank[]]], Power[Pattern[x, Blank[]], 2]]], Pattern[p, Blank[]]]], Pattern[x, Blank[Symbol]]] := Simp[((g*(c*d - b*e) + c*e*f)*(d + e*x)^m*(a + b*x + c*x^2)^(p + 1))/(c*(p + 1)*(2*c*d - b*e)), x] - Dist[(e*(m*(g*(c*d - b*e) + c*e*f) + e*(p + 1)*(2*c*f - b*g)))/(c*(p + 1)*(2*c*d - b*e)), Int[(d + e*x)^Simplify[m - 1]*(a + b*x + c*x^2)^Simplify[p + 1], x], x] /; FreeQ[{a, b, c, d, e, f, g, m, p}, x] && NeQ[b^2 - 4*a*c, 0] && EqQ[c*d^2 - b*d*e + a*e^2, 0] && SumSimplerQ[p, 1] && SumSimplerQ[m, -1] && NeQ[p, -1]
  791. Int[Times[Power[Plus[Pattern[d, Blank[]], Times[Optional[Pattern[e, Blank[]]], Pattern[x, Blank[]]]], Pattern[m, Blank[]]], Plus[Optional[Pattern[f, Blank[]]], Times[Optional[Pattern[g, Blank[]]], Pattern[x, Blank[]]]], Power[Plus[Pattern[a, Blank[]], Times[Optional[Pattern[c, Blank[]]], Power[Pattern[x, Blank[]], 2]]], Pattern[p, Blank[]]]], Pattern[x, Blank[Symbol]]] := Simp[((d*g + e*f)*(d + e*x)^m*(a + c*x^2)^(p + 1))/(2*c*d*(p + 1)), x] - Dist[(e*(m*(d*g + e*f) + 2*e*f*(p + 1)))/(2*c*d*(p + 1)), Int[(d + e*x)^Simplify[m - 1]*(a + c*x^2)^Simplify[p + 1], x], x] /; FreeQ[{a, c, d, e, f, g, m, p}, x] && EqQ[c*d^2 + a*e^2, 0] && SumSimplerQ[p, 1] && SumSimplerQ[m, -1] && NeQ[p, -1] && !IGtQ[m, 0]
  792. Int[Times[Power[Plus[Optional[Pattern[d, Blank[]]], Times[Optional[Pattern[e, Blank[]]], Pattern[x, Blank[]]]], Pattern[m, Blank[]]], Plus[Optional[Pattern[f, Blank[]]], Times[Optional[Pattern[g, Blank[]]], Pattern[x, Blank[]]]], Power[Plus[Optional[Pattern[a, Blank[]]], Times[Optional[Pattern[b, Blank[]]], Pattern[x, Blank[]]], Times[Optional[Pattern[c, Blank[]]], Power[Pattern[x, Blank[]], 2]]], Pattern[p, Blank[]]]], Pattern[x, Blank[Symbol]]] := Simp[((d*g - e*f)*(d + e*x)^m*(a + b*x + c*x^2)^(p + 1))/((2*c*d - b*e)*(m + p + 1)), x] + Dist[(m*(g*(c*d - b*e) + c*e*f) + e*(p + 1)*(2*c*f - b*g))/(e*(2*c*d - b*e)*(m + p + 1)), Int[(d + e*x)^(m + 1)*(a + b*x + c*x^2)^p, x], x] /; FreeQ[{a, b, c, d, e, f, g, m, p}, x] && NeQ[b^2 - 4*a*c, 0] && EqQ[c*d^2 - b*d*e + a*e^2, 0] && ((LtQ[m, -1] && !IGtQ[m + p + 1, 0]) || (LtQ[m, 0] && LtQ[p, -1]) || EqQ[m + 2*p + 2, 0]) && NeQ[m + p + 1, 0]
  793. Int[Times[Power[Plus[Pattern[d, Blank[]], Times[Optional[Pattern[e, Blank[]]], Pattern[x, Blank[]]]], Pattern[m, Blank[]]], Plus[Optional[Pattern[f, Blank[]]], Times[Optional[Pattern[g, Blank[]]], Pattern[x, Blank[]]]], Power[Plus[Pattern[a, Blank[]], Times[Optional[Pattern[c, Blank[]]], Power[Pattern[x, Blank[]], 2]]], Pattern[p, Blank[]]]], Pattern[x, Blank[Symbol]]] := Simp[((d*g - e*f)*(d + e*x)^m*(a + c*x^2)^(p + 1))/(2*c*d*(m + p + 1)), x] + Dist[(m*(g*c*d + c*e*f) + 2*e*c*f*(p + 1))/(e*(2*c*d)*(m + p + 1)), Int[(d + e*x)^(m + 1)*(a + c*x^2)^p, x], x] /; FreeQ[{a, c, d, e, f, g, m, p}, x] && EqQ[c*d^2 + a*e^2, 0] && ((LtQ[m, -1] && !IGtQ[m + p + 1, 0]) || (LtQ[m, 0] && LtQ[p, -1]) || EqQ[m + 2*p + 2, 0]) && NeQ[m + p + 1, 0]
  794. Int[Times[Power[Plus[Optional[Pattern[d, Blank[]]], Times[Optional[Pattern[e, Blank[]]], Pattern[x, Blank[]]]], Pattern[m, Blank[]]], Plus[Optional[Pattern[f, Blank[]]], Times[Optional[Pattern[g, Blank[]]], Pattern[x, Blank[]]]], Power[Plus[Optional[Pattern[a, Blank[]]], Times[Optional[Pattern[b, Blank[]]], Pattern[x, Blank[]]], Times[Optional[Pattern[c, Blank[]]], Power[Pattern[x, Blank[]], 2]]], Pattern[p, Blank[]]]], Pattern[x, Blank[Symbol]]] := Simp[(g*(d + e*x)^m*(a + b*x + c*x^2)^(p + 1))/(c*(m + 2*p + 2)), x] + Dist[(m*(g*(c*d - b*e) + c*e*f) + e*(p + 1)*(2*c*f - b*g))/(c*e*(m + 2*p + 2)), Int[(d + e*x)^m*(a + b*x + c*x^2)^p, x], x] /; FreeQ[{a, b, c, d, e, f, g, m, p}, x] && NeQ[b^2 - 4*a*c, 0] && EqQ[c*d^2 - b*d*e + a*e^2, 0] && NeQ[m + 2*p + 2, 0] && (NeQ[m, 2] || EqQ[d, 0])
  795. Int[Times[Power[Plus[Pattern[d, Blank[]], Times[Optional[Pattern[e, Blank[]]], Pattern[x, Blank[]]]], Pattern[m, Blank[]]], Plus[Optional[Pattern[f, Blank[]]], Times[Optional[Pattern[g, Blank[]]], Pattern[x, Blank[]]]], Power[Plus[Pattern[a, Blank[]], Times[Optional[Pattern[c, Blank[]]], Power[Pattern[x, Blank[]], 2]]], Pattern[p, Blank[]]]], Pattern[x, Blank[Symbol]]] := Simp[(g*(d + e*x)^m*(a + c*x^2)^(p + 1))/(c*(m + 2*p + 2)), x] + Dist[(m*(d*g + e*f) + 2*e*f*(p + 1))/(e*(m + 2*p + 2)), Int[(d + e*x)^m*(a + c*x^2)^p, x], x] /; FreeQ[{a, c, d, e, f, g, m, p}, x] && EqQ[c*d^2 + a*e^2, 0] && NeQ[m + 2*p + 2, 0] && NeQ[m, 2]
  796. Int[Times[Power[Pattern[x, Blank[]], 2], Plus[Pattern[f, Blank[]], Times[Optional[Pattern[g, Blank[]]], Pattern[x, Blank[]]]], Power[Plus[Pattern[a, Blank[]], Times[Optional[Pattern[c, Blank[]]], Power[Pattern[x, Blank[]], 2]]], Pattern[p, Blank[]]]], Pattern[x, Blank[Symbol]]] := Simp[(x^2*(a*g - c*f*x)*(a + c*x^2)^(p + 1))/(2*a*c*(p + 1)), x] - Dist[1/(2*a*c*(p + 1)), Int[x*Simp[2*a*g - c*f*(2*p + 5)*x, x]*(a + c*x^2)^(p + 1), x], x] /; FreeQ[{a, c, f, g}, x] && EqQ[a*g^2 + f^2*c, 0] && LtQ[p, -2]
  797. Int[Times[Power[Pattern[x, Blank[]], 2], Plus[Pattern[f, Blank[]], Times[Optional[Pattern[g, Blank[]]], Pattern[x, Blank[]]]], Power[Plus[Pattern[a, Blank[]], Times[Optional[Pattern[c, Blank[]]], Power[Pattern[x, Blank[]], 2]]], Pattern[p, Blank[]]]], Pattern[x, Blank[Symbol]]] := Dist[1/c, Int[(f + g*x)*(a + c*x^2)^(p + 1), x], x] - Dist[a/c, Int[(f + g*x)*(a + c*x^2)^p, x], x] /; FreeQ[{a, c, f, g, p}, x] && EqQ[a*g^2 + f^2*c, 0]
  798. Int[Times[Power[Plus[Pattern[d, Blank[]], Times[Optional[Pattern[e, Blank[]]], Pattern[x, Blank[]]]], Pattern[m, Blank[]]], Plus[Optional[Pattern[f, Blank[]]], Times[Optional[Pattern[g, Blank[]]], Pattern[x, Blank[]]]], Power[Plus[Optional[Pattern[a, Blank[]]], Times[Optional[Pattern[b, Blank[]]], Pattern[x, Blank[]]], Times[Optional[Pattern[c, Blank[]]], Power[Pattern[x, Blank[]], 2]]], Optional[Pattern[p, Blank[]]]]], Pattern[x, Blank[Symbol]]] := Int[(d + e*x)^m*(f + g*x)^(p + 1)*(a/f + (c*x)/g)^p, x] /; FreeQ[{a, b, c, d, e, f, g, m}, x] && NeQ[b^2 - 4*a*c, 0] && EqQ[c*f^2 - b*f*g + a*g^2, 0] && IntegerQ[p]
  799. Int[Times[Power[Plus[Pattern[d, Blank[]], Times[Optional[Pattern[e, Blank[]]], Pattern[x, Blank[]]]], Pattern[m, Blank[]]], Plus[Optional[Pattern[f, Blank[]]], Times[Optional[Pattern[g, Blank[]]], Pattern[x, Blank[]]]], Power[Plus[Optional[Pattern[a, Blank[]]], Times[Optional[Pattern[c, Blank[]]], Power[Pattern[x, Blank[]], 2]]], Optional[Pattern[p, Blank[]]]]], Pattern[x, Blank[Symbol]]] := Int[(d + e*x)^m*(f + g*x)^(p + 1)*(a/f + (c*x)/g)^p, x] /; FreeQ[{a, c, d, e, f, g, m}, x] && EqQ[c*f^2 + a*g^2, 0] && (IntegerQ[p] || (GtQ[a, 0] && GtQ[f, 0] && EqQ[p, -1]))
  800. Int[Times[Power[Plus[Optional[Pattern[d, Blank[]]], Times[Optional[Pattern[e, Blank[]]], Pattern[x, Blank[]]]], Pattern[m, Blank[]]], Plus[Optional[Pattern[f, Blank[]]], Times[Optional[Pattern[g, Blank[]]], Pattern[x, Blank[]]]], Power[Plus[Optional[Pattern[a, Blank[]]], Times[Optional[Pattern[b, Blank[]]], Pattern[x, Blank[]]], Times[Optional[Pattern[c, Blank[]]], Power[Pattern[x, Blank[]], 2]]], -1]], Pattern[x, Blank[Symbol]]] := Int[ExpandIntegrand[((d + e*x)^m*(f + g*x))/(a + b*x + c*x^2), x], x] /; FreeQ[{a, b, c, d, e, f, g}, x] && NeQ[b^2 - 4*a*c, 0] && NeQ[c*d^2 - b*d*e + a*e^2, 0] && IntegerQ[m]
  801. Int[Times[Power[Plus[Optional[Pattern[d, Blank[]]], Times[Optional[Pattern[e, Blank[]]], Pattern[x, Blank[]]]], Pattern[m, Blank[]]], Plus[Optional[Pattern[f, Blank[]]], Times[Optional[Pattern[g, Blank[]]], Pattern[x, Blank[]]]], Power[Plus[Pattern[a, Blank[]], Times[Optional[Pattern[c, Blank[]]], Power[Pattern[x, Blank[]], 2]]], -1]], Pattern[x, Blank[Symbol]]] := Int[ExpandIntegrand[((d + e*x)^m*(f + g*x))/(a + c*x^2), x], x] /; FreeQ[{a, c, d, e, f, g}, x] && NeQ[c*d^2 + a*e^2, 0] && IntegerQ[m]
  802. Int[Times[Power[Plus[Optional[Pattern[d, Blank[]]], Times[Optional[Pattern[e, Blank[]]], Pattern[x, Blank[]]]], Pattern[m, Blank[]]], Plus[Optional[Pattern[f, Blank[]]], Times[Optional[Pattern[g, Blank[]]], Pattern[x, Blank[]]]], Power[Plus[Optional[Pattern[a, Blank[]]], Times[Optional[Pattern[b, Blank[]]], Pattern[x, Blank[]]], Times[Optional[Pattern[c, Blank[]]], Power[Pattern[x, Blank[]], 2]]], Optional[Pattern[p, Blank[]]]]], Pattern[x, Blank[Symbol]]] := -Simp[((e*f - d*g)*(d + e*x)^(m + 1)*(a + b*x + c*x^2)^(p + 1))/(2*(p + 1)*(c*d^2 - b*d*e + a*e^2)), x] /; FreeQ[{a, b, c, d, e, f, g, m, p}, x] && NeQ[b^2 - 4*a*c, 0] && NeQ[c*d^2 - b*d*e + a*e^2, 0] && EqQ[Simplify[m + 2*p + 3], 0] && EqQ[b*(e*f + d*g) - 2*(c*d*f + a*e*g), 0]
  803. Int[Times[Power[Plus[Optional[Pattern[d, Blank[]]], Times[Optional[Pattern[e, Blank[]]], Pattern[x, Blank[]]]], Pattern[m, Blank[]]], Plus[Optional[Pattern[f, Blank[]]], Times[Optional[Pattern[g, Blank[]]], Pattern[x, Blank[]]]], Power[Plus[Pattern[a, Blank[]], Times[Optional[Pattern[c, Blank[]]], Power[Pattern[x, Blank[]], 2]]], Optional[Pattern[p, Blank[]]]]], Pattern[x, Blank[Symbol]]] := -Simp[((e*f - d*g)*(d + e*x)^(m + 1)*(a + c*x^2)^(p + 1))/(2*(p + 1)*(c*d^2 + a*e^2)), x] /; FreeQ[{a, c, d, e, f, g, m, p}, x] && NeQ[c*d^2 + a*e^2, 0] && EqQ[Simplify[m + 2*p + 3], 0] && EqQ[c*d*f + a*e*g, 0]
  804. Int[Times[Power[Plus[Optional[Pattern[d, Blank[]]], Times[Optional[Pattern[e, Blank[]]], Pattern[x, Blank[]]]], Pattern[m, Blank[]]], Plus[Optional[Pattern[f, Blank[]]], Times[Optional[Pattern[g, Blank[]]], Pattern[x, Blank[]]]], Power[Plus[Optional[Pattern[a, Blank[]]], Times[Optional[Pattern[b, Blank[]]], Pattern[x, Blank[]]], Times[Optional[Pattern[c, Blank[]]], Power[Pattern[x, Blank[]], 2]]], Pattern[p, Blank[]]]], Pattern[x, Blank[Symbol]]] := Simp[((d + e*x)^m*(a + b*x + c*x^2)^(p + 1)*(b*f - 2*a*g + (2*c*f - b*g)*x))/((p + 1)*(b^2 - 4*a*c)), x] - Dist[(m*(b*(e*f + d*g) - 2*(c*d*f + a*e*g)))/((p + 1)*(b^2 - 4*a*c)), Int[(d + e*x)^(m - 1)*(a + b*x + c*x^2)^(p + 1), x], x] /; FreeQ[{a, b, c, d, e, f, g}, x] && NeQ[b^2 - 4*a*c, 0] && NeQ[c*d^2 - b*d*e + a*e^2, 0] && EqQ[Simplify[m + 2*p + 3], 0] && LtQ[p, -1]
  805. Int[Times[Power[Plus[Optional[Pattern[d, Blank[]]], Times[Optional[Pattern[e, Blank[]]], Pattern[x, Blank[]]]], Pattern[m, Blank[]]], Plus[Optional[Pattern[f, Blank[]]], Times[Optional[Pattern[g, Blank[]]], Pattern[x, Blank[]]]], Power[Plus[Pattern[a, Blank[]], Times[Optional[Pattern[c, Blank[]]], Power[Pattern[x, Blank[]], 2]]], Pattern[p, Blank[]]]], Pattern[x, Blank[Symbol]]] := Simp[((d + e*x)^m*(a + c*x^2)^(p + 1)*(a*g - c*f*x))/(2*a*c*(p + 1)), x] - Dist[(m*(c*d*f + a*e*g))/(2*a*c*(p + 1)), Int[(d + e*x)^(m - 1)*(a + c*x^2)^(p + 1), x], x] /; FreeQ[{a, c, d, e, f, g}, x] && NeQ[c*d^2 + a*e^2, 0] && EqQ[Simplify[m + 2*p + 3], 0] && LtQ[p, -1]
  806. Int[Times[Power[Plus[Optional[Pattern[d, Blank[]]], Times[Optional[Pattern[e, Blank[]]], Pattern[x, Blank[]]]], Pattern[m, Blank[]]], Plus[Optional[Pattern[f, Blank[]]], Times[Optional[Pattern[g, Blank[]]], Pattern[x, Blank[]]]], Power[Plus[Optional[Pattern[a, Blank[]]], Times[Optional[Pattern[b, Blank[]]], Pattern[x, Blank[]]], Times[Optional[Pattern[c, Blank[]]], Power[Pattern[x, Blank[]], 2]]], Optional[Pattern[p, Blank[]]]]], Pattern[x, Blank[Symbol]]] := -Simp[((e*f - d*g)*(d + e*x)^(m + 1)*(a + b*x + c*x^2)^(p + 1))/(2*(p + 1)*(c*d^2 - b*d*e + a*e^2)), x] - Dist[(b*(e*f + d*g) - 2*(c*d*f + a*e*g))/(2*(c*d^2 - b*d*e + a*e^2)), Int[(d + e*x)^(m + 1)*(a + b*x + c*x^2)^p, x], x] /; FreeQ[{a, b, c, d, e, f, g, m, p}, x] && NeQ[b^2 - 4*a*c, 0] && NeQ[c*d^2 - b*d*e + a*e^2, 0] && EqQ[Simplify[m + 2*p + 3], 0]
  807. Int[Times[Power[Plus[Optional[Pattern[d, Blank[]]], Times[Optional[Pattern[e, Blank[]]], Pattern[x, Blank[]]]], Pattern[m, Blank[]]], Plus[Optional[Pattern[f, Blank[]]], Times[Optional[Pattern[g, Blank[]]], Pattern[x, Blank[]]]], Power[Plus[Pattern[a, Blank[]], Times[Optional[Pattern[c, Blank[]]], Power[Pattern[x, Blank[]], 2]]], Optional[Pattern[p, Blank[]]]]], Pattern[x, Blank[Symbol]]] := -Simp[((e*f - d*g)*(d + e*x)^(m + 1)*(a + c*x^2)^(p + 1))/(2*(p + 1)*(c*d^2 + a*e^2)), x] + Dist[(c*d*f + a*e*g)/(c*d^2 + a*e^2), Int[(d + e*x)^(m + 1)*(a + c*x^2)^p, x], x] /; FreeQ[{a, c, d, e, f, g, m, p}, x] && NeQ[c*d^2 + a*e^2, 0] && EqQ[Simplify[m + 2*p + 3], 0]
  808. Int[Times[Power[Times[Optional[Pattern[e, Blank[]]], Pattern[x, Blank[]]], Pattern[m, Blank[]]], Plus[Pattern[f, Blank[]], Times[Optional[Pattern[g, Blank[]]], Pattern[x, Blank[]]]], Power[Plus[Pattern[a, Blank[]], Times[Optional[Pattern[c, Blank[]]], Power[Pattern[x, Blank[]], 2]]], Pattern[p, Blank[]]]], Pattern[x, Blank[Symbol]]] := Dist[f, Int[(e*x)^m*(a + c*x^2)^p, x], x] + Dist[g/e, Int[(e*x)^(m + 1)*(a + c*x^2)^p, x], x] /; FreeQ[{a, c, e, f, g, p}, x] && !RationalQ[m] && !IGtQ[p, 0]
  809. Int[Times[Power[Plus[Optional[Pattern[d, Blank[]]], Times[Optional[Pattern[e, Blank[]]], Pattern[x, Blank[]]]], Pattern[m, Blank[]]], Plus[Optional[Pattern[f, Blank[]]], Times[Optional[Pattern[g, Blank[]]], Pattern[x, Blank[]]]], Power[Plus[Pattern[a, Blank[]], Times[Optional[Pattern[b, Blank[]]], Pattern[x, Blank[]]], Times[Optional[Pattern[c, Blank[]]], Power[Pattern[x, Blank[]], 2]]], Pattern[p, Blank[]]]], Pattern[x, Blank[Symbol]]] := Dist[((d + e*x)^FracPart[p]*(a + b*x + c*x^2)^FracPart[p])/(a*d + c*e*x^3)^FracPart[p], Int[(f + g*x)*(a*d + c*e*x^3)^p, x], x] /; FreeQ[{a, b, c, d, e, f, g, m, p}, x] && EqQ[m, p] && EqQ[b*d + a*e, 0] && EqQ[c*d + b*e, 0]
  810. Int[Times[Power[Plus[Optional[Pattern[d, Blank[]]], Times[Optional[Pattern[e, Blank[]]], Pattern[x, Blank[]]]], Pattern[m, Blank[]]], Plus[Optional[Pattern[f, Blank[]]], Times[Optional[Pattern[g, Blank[]]], Pattern[x, Blank[]]]], Power[Plus[Optional[Pattern[a, Blank[]]], Times[Optional[Pattern[b, Blank[]]], Pattern[x, Blank[]]], Times[Optional[Pattern[c, Blank[]]], Power[Pattern[x, Blank[]], 2]]], Optional[Pattern[p, Blank[]]]]], Pattern[x, Blank[Symbol]]] := -Simp[((d + e*x)^(m + 1)*(a + b*x + c*x^2)^p*((d*g - e*f*(m + 2))*(c*d^2 - b*d*e + a*e^2) - d*p*(2*c*d - b*e)*(e*f - d*g) - e*(g*(m + 1)*(c*d^2 - b*d*e + a*e^2) + p*(2*c*d - b*e)*(e*f - d*g))*x))/(e^2*(m + 1)*(m + 2)*(c*d^2 - b*d*e + a*e^2)), x] - Dist[p/(e^2*(m + 1)*(m + 2)*(c*d^2 - b*d*e + a*e^2)), Int[(d + e*x)^(m + 2)*(a + b*x + c*x^2)^(p - 1)*Simp[2*a*c*e*(e*f - d*g)*(m + 2) + b^2*e*(d*g*(p + 1) - e*f*(m + p + 2)) + b*(a*e^2*g*(m + 1) - c*d*(d*g*(2*p + 1) - e*f*(m + 2*p + 2))) - c*(2*c*d*(d*g*(2*p + 1) - e*f*(m + 2*p + 2)) - e*(2*a*e*g*(m + 1) - b*(d*g*(m - 2*p) + e*f*(m + 2*p + 2))))*x, x], x], x] /; FreeQ[{a, b, c, d, e, f, g}, x] && NeQ[b^2 - 4*a*c, 0] && NeQ[c*d^2 - b*d*e + a*e^2, 0] && GtQ[p, 0] && LtQ[m, -2] && LtQ[m + 2*p, 0] && !ILtQ[m + 2*p + 3, 0]
  811. Int[Times[Power[Plus[Optional[Pattern[d, Blank[]]], Times[Optional[Pattern[e, Blank[]]], Pattern[x, Blank[]]]], Pattern[m, Blank[]]], Plus[Optional[Pattern[f, Blank[]]], Times[Optional[Pattern[g, Blank[]]], Pattern[x, Blank[]]]], Power[Plus[Pattern[a, Blank[]], Times[Optional[Pattern[c, Blank[]]], Power[Pattern[x, Blank[]], 2]]], Optional[Pattern[p, Blank[]]]]], Pattern[x, Blank[Symbol]]] := -Simp[((d + e*x)^(m + 1)*(a + c*x^2)^p*((d*g - e*f*(m + 2))*(c*d^2 + a*e^2) - 2*c*d^2*p*(e*f - d*g) - e*(g*(m + 1)*(c*d^2 + a*e^2) + 2*c*d*p*(e*f - d*g))*x))/(e^2*(m + 1)*(m + 2)*(c*d^2 + a*e^2)), x] - Dist[p/(e^2*(m + 1)*(m + 2)*(c*d^2 + a*e^2)), Int[(d + e*x)^(m + 2)*(a + c*x^2)^(p - 1)*Simp[2*a*c*e*(e*f - d*g)*(m + 2) - c*(2*c*d*(d*g*(2*p + 1) - e*f*(m + 2*p + 2)) - 2*a*e^2*g*(m + 1))*x, x], x], x] /; FreeQ[{a, c, d, e, f, g}, x] && NeQ[c*d^2 + a*e^2, 0] && GtQ[p, 0] && LtQ[m, -2] && LtQ[m + 2*p, 0] && !ILtQ[m + 2*p + 3, 0]
  812. Int[Times[Power[Plus[Optional[Pattern[d, Blank[]]], Times[Optional[Pattern[e, Blank[]]], Pattern[x, Blank[]]]], Pattern[m, Blank[]]], Plus[Optional[Pattern[f, Blank[]]], Times[Optional[Pattern[g, Blank[]]], Pattern[x, Blank[]]]], Power[Plus[Optional[Pattern[a, Blank[]]], Times[Optional[Pattern[b, Blank[]]], Pattern[x, Blank[]]], Times[Optional[Pattern[c, Blank[]]], Power[Pattern[x, Blank[]], 2]]], Optional[Pattern[p, Blank[]]]]], Pattern[x, Blank[Symbol]]] := Simp[((d + e*x)^(m + 1)*(e*f*(m + 2*p + 2) - d*g*(2*p + 1) + e*g*(m + 1)*x)*(a + b*x + c*x^2)^p)/(e^2*(m + 1)*(m + 2*p + 2)), x] + Dist[p/(e^2*(m + 1)*(m + 2*p + 2)), Int[(d + e*x)^(m + 1)*(a + b*x + c*x^2)^(p - 1)*Simp[g*(b*d + 2*a*e + 2*a*e*m + 2*b*d*p) - f*b*e*(m + 2*p + 2) + (g*(2*c*d + b*e + b*e*m + 4*c*d*p) - 2*c*e*f*(m + 2*p + 2))*x, x], x], x] /; FreeQ[{a, b, c, d, e, f, g, m}, x] && NeQ[b^2 - 4*a*c, 0] && NeQ[c*d^2 - b*d*e + a*e^2, 0] && RationalQ[p] && p > 0 && (LtQ[m, -1] || EqQ[p, 1] || (IntegerQ[p] && !RationalQ[m])) && NeQ[m, -1] && !ILtQ[m + 2*p + 1, 0] && (IntegerQ[m] || IntegerQ[p] || IntegersQ[2*m, 2*p])
  813. Int[Times[Power[Plus[Optional[Pattern[d, Blank[]]], Times[Optional[Pattern[e, Blank[]]], Pattern[x, Blank[]]]], Pattern[m, Blank[]]], Plus[Optional[Pattern[f, Blank[]]], Times[Optional[Pattern[g, Blank[]]], Pattern[x, Blank[]]]], Power[Plus[Pattern[a, Blank[]], Times[Optional[Pattern[c, Blank[]]], Power[Pattern[x, Blank[]], 2]]], Optional[Pattern[p, Blank[]]]]], Pattern[x, Blank[Symbol]]] := Simp[((d + e*x)^(m + 1)*(e*f*(m + 2*p + 2) - d*g*(2*p + 1) + e*g*(m + 1)*x)*(a + c*x^2)^p)/(e^2*(m + 1)*(m + 2*p + 2)), x] + Dist[p/(e^2*(m + 1)*(m + 2*p + 2)), Int[(d + e*x)^(m + 1)*(a + c*x^2)^(p - 1)*Simp[g*(2*a*e + 2*a*e*m) + (g*(2*c*d + 4*c*d*p) - 2*c*e*f*(m + 2*p + 2))*x, x], x], x] /; FreeQ[{a, c, d, e, f, g, m}, x] && NeQ[c*d^2 + a*e^2, 0] && RationalQ[p] && p > 0 && (LtQ[m, -1] || EqQ[p, 1] || (IntegerQ[p] && !RationalQ[m])) && NeQ[m, -1] && !ILtQ[m + 2*p + 1, 0] && (IntegerQ[m] || IntegerQ[p] || IntegersQ[2*m, 2*p])
  814. Int[Times[Power[Plus[Optional[Pattern[d, Blank[]]], Times[Optional[Pattern[e, Blank[]]], Pattern[x, Blank[]]]], Pattern[m, Blank[]]], Plus[Optional[Pattern[f, Blank[]]], Times[Optional[Pattern[g, Blank[]]], Pattern[x, Blank[]]]], Power[Plus[Optional[Pattern[a, Blank[]]], Times[Optional[Pattern[b, Blank[]]], Pattern[x, Blank[]]], Times[Optional[Pattern[c, Blank[]]], Power[Pattern[x, Blank[]], 2]]], Optional[Pattern[p, Blank[]]]]], Pattern[x, Blank[Symbol]]] := Simp[((d + e*x)^(m + 1)*(c*e*f*(m + 2*p + 2) - g*(c*d + 2*c*d*p - b*e*p) + g*c*e*(m + 2*p + 1)*x)*(a + b*x + c*x^2)^p)/(c*e^2*(m + 2*p + 1)*(m + 2*p + 2)), x] - Dist[p/(c*e^2*(m + 2*p + 1)*(m + 2*p + 2)), Int[(d + e*x)^m*(a + b*x + c*x^2)^(p - 1)*Simp[c*e*f*(b*d - 2*a*e)*(m + 2*p + 2) + g*(a*e*(b*e - 2*c*d*m + b*e*m) + b*d*(b*e*p - c*d - 2*c*d*p)) + (c*e*f*(2*c*d - b*e)*(m + 2*p + 2) + g*(b^2*e^2*(p + m + 1) - 2*c^2*d^2*(1 + 2*p) - c*e*(b*d*(m - 2*p) + 2*a*e*(m + 2*p + 1))))*x, x], x], x] /; FreeQ[{a, b, c, d, e, f, g, m}, x] && NeQ[b^2 - 4*a*c, 0] && NeQ[c*d^2 - b*d*e + a*e^2, 0] && GtQ[p, 0] && (IntegerQ[p] || !RationalQ[m] || (GeQ[m, -1] && LtQ[m, 0])) && !ILtQ[m + 2*p, 0] && (IntegerQ[m] || IntegerQ[p] || IntegersQ[2*m, 2*p])
  815. Int[Times[Power[Plus[Optional[Pattern[d, Blank[]]], Times[Optional[Pattern[e, Blank[]]], Pattern[x, Blank[]]]], Pattern[m, Blank[]]], Plus[Optional[Pattern[f, Blank[]]], Times[Optional[Pattern[g, Blank[]]], Pattern[x, Blank[]]]], Power[Plus[Pattern[a, Blank[]], Times[Optional[Pattern[c, Blank[]]], Power[Pattern[x, Blank[]], 2]]], Optional[Pattern[p, Blank[]]]]], Pattern[x, Blank[Symbol]]] := Simp[((d + e*x)^(m + 1)*(c*e*f*(m + 2*p + 2) - g*c*d*(2*p + 1) + g*c*e*(m + 2*p + 1)*x)*(a + c*x^2)^p)/(c*e^2*(m + 2*p + 1)*(m + 2*p + 2)), x] + Dist[(2*p)/(c*e^2*(m + 2*p + 1)*(m + 2*p + 2)), Int[(d + e*x)^m*(a + c*x^2)^(p - 1)*Simp[f*a*c*e^2*(m + 2*p + 2) + a*c*d*e*g*m - (c^2*f*d*e*(m + 2*p + 2) - g*(c^2*d^2*(2*p + 1) + a*c*e^2*(m + 2*p + 1)))*x, x], x], x] /; FreeQ[{a, c, d, e, f, g, m}, x] && NeQ[c*d^2 + a*e^2, 0] && GtQ[p, 0] && (IntegerQ[p] || !RationalQ[m] || (GeQ[m, -1] && LtQ[m, 0])) && !ILtQ[m + 2*p, 0] && (IntegerQ[m] || IntegerQ[p] || IntegersQ[2*m, 2*p])
  816. Int[Times[Power[Plus[Pattern[d, Blank[]], Times[Optional[Pattern[e, Blank[]]], Pattern[x, Blank[]]]], Pattern[m, Blank[]]], Plus[Pattern[f, Blank[]], Times[Optional[Pattern[g, Blank[]]], Pattern[x, Blank[]]]], Power[Plus[Optional[Pattern[a, Blank[]]], Times[Optional[Pattern[b, Blank[]]], Pattern[x, Blank[]]], Times[Optional[Pattern[c, Blank[]]], Power[Pattern[x, Blank[]], 2]]], Pattern[p, Blank[]]]], Pattern[x, Blank[Symbol]]] := Int[(a + b*x + c*x^2)^p*ExpandIntegrand[(d + e*x)^m*(f + g*x), x], x] /; FreeQ[{a, b, c, d, e, f, g}, x] && NeQ[b^2 - 4*a*c, 0] && NeQ[c*d^2 - b*d*e + a*e^2, 0] && ILtQ[p, -1] && IGtQ[m, 0] && RationalQ[a, b, c, d, e, f, g]
  817. Int[Times[Power[Plus[Pattern[d, Blank[]], Times[Optional[Pattern[e, Blank[]]], Pattern[x, Blank[]]]], Pattern[m, Blank[]]], Plus[Pattern[f, Blank[]], Times[Optional[Pattern[g, Blank[]]], Pattern[x, Blank[]]]], Power[Plus[Pattern[a, Blank[]], Times[Optional[Pattern[c, Blank[]]], Power[Pattern[x, Blank[]], 2]]], Pattern[p, Blank[]]]], Pattern[x, Blank[Symbol]]] := Int[(a + c*x^2)^p*ExpandIntegrand[(d + e*x)^m*(f + g*x), x], x] /; FreeQ[{a, c, d, e, f, g}, x] && NeQ[c*d^2 + a*e^2, 0] && ILtQ[p, -1] && IGtQ[m, 0] && RationalQ[a, c, d, e, f, g]
  818. Int[Times[Power[Plus[Optional[Pattern[d, Blank[]]], Times[Optional[Pattern[e, Blank[]]], Pattern[x, Blank[]]]], Pattern[m, Blank[]]], Plus[Optional[Pattern[f, Blank[]]], Times[Optional[Pattern[g, Blank[]]], Pattern[x, Blank[]]]], Power[Plus[Optional[Pattern[a, Blank[]]], Times[Optional[Pattern[b, Blank[]]], Pattern[x, Blank[]]], Times[Optional[Pattern[c, Blank[]]], Power[Pattern[x, Blank[]], 2]]], Optional[Pattern[p, Blank[]]]]], Pattern[x, Blank[Symbol]]] := -Simp[((d + e*x)^(m - 1)*(a + b*x + c*x^2)^(p + 1)*(2*a*c*(e*f + d*g) - b*(c*d*f + a*e*g) - (2*c^2*d*f + b^2*e*g - c*(b*e*f + b*d*g + 2*a*e*g))*x))/(c*(p + 1)*(b^2 - 4*a*c)), x] - Dist[1/(c*(p + 1)*(b^2 - 4*a*c)), Int[(d + e*x)^(m - 2)*(a + b*x + c*x^2)^(p + 1)*Simp[2*c^2*d^2*f*(2*p + 3) + b*e*g*(a*e*(m - 1) + b*d*(p + 2)) - c*(2*a*e*(e*f*(m - 1) + d*g*m) + b*d*(d*g*(2*p + 3) - e*f*(m - 2*p - 4))) + e*(b^2*e*g*(m + p + 1) + 2*c^2*d*f*(m + 2*p + 2) - c*(2*a*e*g*m + b*(e*f + d*g)*(m + 2*p + 2)))*x, x], x], x] /; FreeQ[{a, b, c, d, e, f, g}, x] && NeQ[b^2 - 4*a*c, 0] && NeQ[c*d^2 - b*d*e + a*e^2, 0] && LtQ[p, -1] && GtQ[m, 1] && ((EqQ[m, 2] && EqQ[p, -3] && RationalQ[a, b, c, d, e, f, g]) || !ILtQ[m + 2*p + 3, 0])
  819. Int[Times[Power[Plus[Optional[Pattern[d, Blank[]]], Times[Optional[Pattern[e, Blank[]]], Pattern[x, Blank[]]]], Pattern[m, Blank[]]], Plus[Optional[Pattern[f, Blank[]]], Times[Optional[Pattern[g, Blank[]]], Pattern[x, Blank[]]]], Power[Plus[Pattern[a, Blank[]], Times[Optional[Pattern[c, Blank[]]], Power[Pattern[x, Blank[]], 2]]], Optional[Pattern[p, Blank[]]]]], Pattern[x, Blank[Symbol]]] := Simp[((d + e*x)^(m - 1)*(a + c*x^2)^(p + 1)*(a*(e*f + d*g) - (c*d*f - a*e*g)*x))/(2*a*c*(p + 1)), x] - Dist[1/(2*a*c*(p + 1)), Int[(d + e*x)^(m - 2)*(a + c*x^2)^(p + 1)*Simp[a*e*(e*f*(m - 1) + d*g*m) - c*d^2*f*(2*p + 3) + e*(a*e*g*m - c*d*f*(m + 2*p + 2))*x, x], x], x] /; FreeQ[{a, c, d, e, f, g}, x] && NeQ[c*d^2 + a*e^2, 0] && LtQ[p, -1] && GtQ[m, 1] && (EqQ[d, 0] || (EqQ[m, 2] && EqQ[p, -3] && RationalQ[a, c, d, e, f, g]) || !ILtQ[m + 2*p + 3, 0])
  820. Int[Times[Power[Plus[Optional[Pattern[d, Blank[]]], Times[Optional[Pattern[e, Blank[]]], Pattern[x, Blank[]]]], Pattern[m, Blank[]]], Plus[Optional[Pattern[f, Blank[]]], Times[Optional[Pattern[g, Blank[]]], Pattern[x, Blank[]]]], Power[Plus[Optional[Pattern[a, Blank[]]], Times[Optional[Pattern[b, Blank[]]], Pattern[x, Blank[]]], Times[Optional[Pattern[c, Blank[]]], Power[Pattern[x, Blank[]], 2]]], Pattern[p, Blank[]]]], Pattern[x, Blank[Symbol]]] := Simp[((d + e*x)^m*(a + b*x + c*x^2)^(p + 1)*(f*b - 2*a*g + (2*c*f - b*g)*x))/((p + 1)*(b^2 - 4*a*c)), x] + Dist[1/((p + 1)*(b^2 - 4*a*c)), Int[(d + e*x)^(m - 1)*(a + b*x + c*x^2)^(p + 1)*Simp[g*(2*a*e*m + b*d*(2*p + 3)) - f*(b*e*m + 2*c*d*(2*p + 3)) - e*(2*c*f - b*g)*(m + 2*p + 3)*x, x], x], x] /; FreeQ[{a, b, c, d, e, f, g}, x] && NeQ[b^2 - 4*a*c, 0] && NeQ[c*d^2 - b*d*e + a*e^2, 0] && LtQ[p, -1] && GtQ[m, 0] && (IntegerQ[m] || IntegerQ[p] || IntegersQ[2*m, 2*p])
  821. Int[Times[Power[Plus[Optional[Pattern[d, Blank[]]], Times[Optional[Pattern[e, Blank[]]], Pattern[x, Blank[]]]], Pattern[m, Blank[]]], Plus[Optional[Pattern[f, Blank[]]], Times[Optional[Pattern[g, Blank[]]], Pattern[x, Blank[]]]], Power[Plus[Pattern[a, Blank[]], Times[Optional[Pattern[c, Blank[]]], Power[Pattern[x, Blank[]], 2]]], Pattern[p, Blank[]]]], Pattern[x, Blank[Symbol]]] := Simp[((d + e*x)^m*(a + c*x^2)^(p + 1)*(a*g - c*f*x))/(2*a*c*(p + 1)), x] - Dist[1/(2*a*c*(p + 1)), Int[(d + e*x)^(m - 1)*(a + c*x^2)^(p + 1)*Simp[a*e*g*m - c*d*f*(2*p + 3) - c*e*f*(m + 2*p + 3)*x, x], x], x] /; FreeQ[{a, c, d, e, f, g}, x] && NeQ[c*d^2 + a*e^2, 0] && LtQ[p, -1] && GtQ[m, 0] && (IntegerQ[m] || IntegerQ[p] || IntegersQ[2*m, 2*p])
  822. Int[Times[Power[Plus[Optional[Pattern[d, Blank[]]], Times[Optional[Pattern[e, Blank[]]], Pattern[x, Blank[]]]], Pattern[m, Blank[]]], Plus[Optional[Pattern[f, Blank[]]], Times[Optional[Pattern[g, Blank[]]], Pattern[x, Blank[]]]], Power[Plus[Optional[Pattern[a, Blank[]]], Times[Optional[Pattern[b, Blank[]]], Pattern[x, Blank[]]], Times[Optional[Pattern[c, Blank[]]], Power[Pattern[x, Blank[]], 2]]], Pattern[p, Blank[]]]], Pattern[x, Blank[Symbol]]] := Simp[((d + e*x)^(m + 1)*(f*(b*c*d - b^2*e + 2*a*c*e) - a*g*(2*c*d - b*e) + c*(f*(2*c*d - b*e) - g*(b*d - 2*a*e))*x)*(a + b*x + c*x^2)^(p + 1))/((p + 1)*(b^2 - 4*a*c)*(c*d^2 - b*d*e + a*e^2)), x] + Dist[1/((p + 1)*(b^2 - 4*a*c)*(c*d^2 - b*d*e + a*e^2)), Int[(d + e*x)^m*(a + b*x + c*x^2)^(p + 1)*Simp[f*(b*c*d*e*(2*p - m + 2) + b^2*e^2*(p + m + 2) - 2*c^2*d^2*(2*p + 3) - 2*a*c*e^2*(m + 2*p + 3)) - g*(a*e*(b*e - 2*c*d*m + b*e*m) - b*d*(3*c*d - b*e + 2*c*d*p - b*e*p)) + c*e*(g*(b*d - 2*a*e) - f*(2*c*d - b*e))*(m + 2*p + 4)*x, x], x], x] /; FreeQ[{a, b, c, d, e, f, g, m}, x] && NeQ[b^2 - 4*a*c, 0] && NeQ[c*d^2 - b*d*e + a*e^2, 0] && LtQ[p, -1] && (IntegerQ[m] || IntegerQ[p] || IntegersQ[2*m, 2*p])
  823. Int[Times[Power[Plus[Optional[Pattern[d, Blank[]]], Times[Optional[Pattern[e, Blank[]]], Pattern[x, Blank[]]]], Pattern[m, Blank[]]], Plus[Optional[Pattern[f, Blank[]]], Times[Optional[Pattern[g, Blank[]]], Pattern[x, Blank[]]]], Power[Plus[Pattern[a, Blank[]], Times[Optional[Pattern[c, Blank[]]], Power[Pattern[x, Blank[]], 2]]], Pattern[p, Blank[]]]], Pattern[x, Blank[Symbol]]] := -Simp[((d + e*x)^(m + 1)*(f*a*c*e - a*g*c*d + c*(c*d*f + a*e*g)*x)*(a + c*x^2)^(p + 1))/(2*a*c*(p + 1)*(c*d^2 + a*e^2)), x] + Dist[1/(2*a*c*(p + 1)*(c*d^2 + a*e^2)), Int[(d + e*x)^m*(a + c*x^2)^(p + 1)*Simp[f*(c^2*d^2*(2*p + 3) + a*c*e^2*(m + 2*p + 3)) - a*c*d*e*g*m + c*e*(c*d*f + a*e*g)*(m + 2*p + 4)*x, x], x], x] /; FreeQ[{a, c, d, e, f, g}, x] && NeQ[c*d^2 + a*e^2, 0] && LtQ[p, -1] && (IntegerQ[m] || IntegerQ[p] || IntegersQ[2*m, 2*p])
  824. Int[Times[Power[Plus[Optional[Pattern[d, Blank[]]], Times[Optional[Pattern[e, Blank[]]], Pattern[x, Blank[]]]], Pattern[m, Blank[]]], Plus[Optional[Pattern[f, Blank[]]], Times[Optional[Pattern[g, Blank[]]], Pattern[x, Blank[]]]], Power[Plus[Optional[Pattern[a, Blank[]]], Times[Optional[Pattern[b, Blank[]]], Pattern[x, Blank[]]], Times[Optional[Pattern[c, Blank[]]], Power[Pattern[x, Blank[]], 2]]], -1]], Pattern[x, Blank[Symbol]]] := Simp[(g*(d + e*x)^m)/(c*m), x] + Dist[1/c, Int[((d + e*x)^(m - 1)*Simp[c*d*f - a*e*g + (g*c*d - b*e*g + c*e*f)*x, x])/(a + b*x + c*x^2), x], x] /; FreeQ[{a, b, c, d, e, f, g}, x] && NeQ[b^2 - 4*a*c, 0] && NeQ[c*d^2 - b*d*e + a*e^2, 0] && FractionQ[m] && GtQ[m, 0]
  825. Int[Times[Power[Plus[Optional[Pattern[d, Blank[]]], Times[Optional[Pattern[e, Blank[]]], Pattern[x, Blank[]]]], Pattern[m, Blank[]]], Plus[Optional[Pattern[f, Blank[]]], Times[Optional[Pattern[g, Blank[]]], Pattern[x, Blank[]]]], Power[Plus[Pattern[a, Blank[]], Times[Optional[Pattern[c, Blank[]]], Power[Pattern[x, Blank[]], 2]]], -1]], Pattern[x, Blank[Symbol]]] := Simp[(g*(d + e*x)^m)/(c*m), x] + Dist[1/c, Int[((d + e*x)^(m - 1)*Simp[c*d*f - a*e*g + (g*c*d + c*e*f)*x, x])/(a + c*x^2), x], x] /; FreeQ[{a, c, d, e, f, g}, x] && NeQ[c*d^2 + a*e^2, 0] && FractionQ[m] && GtQ[m, 0]
  826. Int[Times[Power[Plus[Optional[Pattern[d, Blank[]]], Times[Optional[Pattern[e, Blank[]]], Pattern[x, Blank[]]]], Rational[-1, 2]], Plus[Optional[Pattern[f, Blank[]]], Times[Optional[Pattern[g, Blank[]]], Pattern[x, Blank[]]]], Power[Plus[Optional[Pattern[a, Blank[]]], Times[Optional[Pattern[b, Blank[]]], Pattern[x, Blank[]]], Times[Optional[Pattern[c, Blank[]]], Power[Pattern[x, Blank[]], 2]]], -1]], Pattern[x, Blank[Symbol]]] := Dist[2, Subst[Int[(e*f - d*g + g*x^2)/(c*d^2 - b*d*e + a*e^2 - (2*c*d - b*e)*x^2 + c*x^4), x], x, Sqrt[d + e*x]], x] /; FreeQ[{a, b, c, d, e, f, g}, x] && NeQ[b^2 - 4*a*c, 0] && NeQ[c*d^2 - b*d*e + a*e^2, 0]
  827. Int[Times[Power[Plus[Optional[Pattern[d, Blank[]]], Times[Optional[Pattern[e, Blank[]]], Pattern[x, Blank[]]]], Rational[-1, 2]], Plus[Optional[Pattern[f, Blank[]]], Times[Optional[Pattern[g, Blank[]]], Pattern[x, Blank[]]]], Power[Plus[Pattern[a, Blank[]], Times[Optional[Pattern[c, Blank[]]], Power[Pattern[x, Blank[]], 2]]], -1]], Pattern[x, Blank[Symbol]]] := Dist[2, Subst[Int[(e*f - d*g + g*x^2)/(c*d^2 + a*e^2 - 2*c*d*x^2 + c*x^4), x], x, Sqrt[d + e*x]], x] /; FreeQ[{a, c, d, e, f, g}, x] && NeQ[c*d^2 + a*e^2, 0]
  828. Int[Times[Power[Plus[Optional[Pattern[d, Blank[]]], Times[Optional[Pattern[e, Blank[]]], Pattern[x, Blank[]]]], Pattern[m, Blank[]]], Plus[Optional[Pattern[f, Blank[]]], Times[Optional[Pattern[g, Blank[]]], Pattern[x, Blank[]]]], Power[Plus[Optional[Pattern[a, Blank[]]], Times[Optional[Pattern[b, Blank[]]], Pattern[x, Blank[]]], Times[Optional[Pattern[c, Blank[]]], Power[Pattern[x, Blank[]], 2]]], -1]], Pattern[x, Blank[Symbol]]] := Simp[((e*f - d*g)*(d + e*x)^(m + 1))/((m + 1)*(c*d^2 - b*d*e + a*e^2)), x] + Dist[1/(c*d^2 - b*d*e + a*e^2), Int[((d + e*x)^(m + 1)*Simp[c*d*f - f*b*e + a*e*g - c*(e*f - d*g)*x, x])/(a + b*x + c*x^2), x], x] /; FreeQ[{a, b, c, d, e, f, g, m}, x] && NeQ[b^2 - 4*a*c, 0] && NeQ[c*d^2 - b*d*e + a*e^2, 0] && FractionQ[m] && LtQ[m, -1]
  829. Int[Times[Power[Plus[Optional[Pattern[d, Blank[]]], Times[Optional[Pattern[e, Blank[]]], Pattern[x, Blank[]]]], Pattern[m, Blank[]]], Plus[Optional[Pattern[f, Blank[]]], Times[Optional[Pattern[g, Blank[]]], Pattern[x, Blank[]]]], Power[Plus[Pattern[a, Blank[]], Times[Optional[Pattern[c, Blank[]]], Power[Pattern[x, Blank[]], 2]]], -1]], Pattern[x, Blank[Symbol]]] := Simp[((e*f - d*g)*(d + e*x)^(m + 1))/((m + 1)*(c*d^2 + a*e^2)), x] + Dist[1/(c*d^2 + a*e^2), Int[((d + e*x)^(m + 1)*Simp[c*d*f + a*e*g - c*(e*f - d*g)*x, x])/(a + c*x^2), x], x] /; FreeQ[{a, c, d, e, f, g, m}, x] && NeQ[c*d^2 + a*e^2, 0] && FractionQ[m] && LtQ[m, -1]
  830. Int[Times[Power[Plus[Optional[Pattern[d, Blank[]]], Times[Optional[Pattern[e, Blank[]]], Pattern[x, Blank[]]]], Pattern[m, Blank[]]], Plus[Optional[Pattern[f, Blank[]]], Times[Optional[Pattern[g, Blank[]]], Pattern[x, Blank[]]]], Power[Plus[Optional[Pattern[a, Blank[]]], Times[Optional[Pattern[b, Blank[]]], Pattern[x, Blank[]]], Times[Optional[Pattern[c, Blank[]]], Power[Pattern[x, Blank[]], 2]]], -1]], Pattern[x, Blank[Symbol]]] := Int[ExpandIntegrand[(d + e*x)^m, (f + g*x)/(a + b*x + c*x^2), x], x] /; FreeQ[{a, b, c, d, e, f, g}, x] && NeQ[b^2 - 4*a*c, 0] && NeQ[c*d^2 - b*d*e + a*e^2, 0] && !RationalQ[m]
  831. Int[Times[Power[Plus[Optional[Pattern[d, Blank[]]], Times[Optional[Pattern[e, Blank[]]], Pattern[x, Blank[]]]], Pattern[m, Blank[]]], Plus[Optional[Pattern[f, Blank[]]], Times[Optional[Pattern[g, Blank[]]], Pattern[x, Blank[]]]], Power[Plus[Pattern[a, Blank[]], Times[Optional[Pattern[c, Blank[]]], Power[Pattern[x, Blank[]], 2]]], -1]], Pattern[x, Blank[Symbol]]] := Int[ExpandIntegrand[(d + e*x)^m, (f + g*x)/(a + c*x^2), x], x] /; FreeQ[{a, c, d, e, f, g}, x] && NeQ[c*d^2 + a*e^2, 0] && !RationalQ[m]
  832. Int[Times[Power[Plus[Optional[Pattern[d, Blank[]]], Times[Optional[Pattern[e, Blank[]]], Pattern[x, Blank[]]]], Pattern[m, Blank[]]], Plus[Optional[Pattern[f, Blank[]]], Times[Optional[Pattern[g, Blank[]]], Pattern[x, Blank[]]]], Power[Plus[Optional[Pattern[a, Blank[]]], Times[Optional[Pattern[b, Blank[]]], Pattern[x, Blank[]]], Times[Optional[Pattern[c, Blank[]]], Power[Pattern[x, Blank[]], 2]]], Optional[Pattern[p, Blank[]]]]], Pattern[x, Blank[Symbol]]] := Simp[(g*(d + e*x)^m*(a + b*x + c*x^2)^(p + 1))/(c*(m + 2*p + 2)), x] + Dist[1/(c*(m + 2*p + 2)), Int[(d + e*x)^(m - 1)*(a + b*x + c*x^2)^p*Simp[m*(c*d*f - a*e*g) + d*(2*c*f - b*g)*(p + 1) + (m*(c*e*f + c*d*g - b*e*g) + e*(p + 1)*(2*c*f - b*g))*x, x], x], x] /; FreeQ[{a, b, c, d, e, f, g, p}, x] && NeQ[b^2 - 4*a*c, 0] && NeQ[c*d^2 - b*d*e + a*e^2, 0] && GtQ[m, 0] && NeQ[m + 2*p + 2, 0] && (IntegerQ[m] || IntegerQ[p] || IntegersQ[2*m, 2*p]) && !(IGtQ[m, 0] && EqQ[f, 0])
  833. Int[Times[Power[Plus[Optional[Pattern[d, Blank[]]], Times[Optional[Pattern[e, Blank[]]], Pattern[x, Blank[]]]], Pattern[m, Blank[]]], Plus[Optional[Pattern[f, Blank[]]], Times[Optional[Pattern[g, Blank[]]], Pattern[x, Blank[]]]], Power[Plus[Pattern[a, Blank[]], Times[Optional[Pattern[c, Blank[]]], Power[Pattern[x, Blank[]], 2]]], Optional[Pattern[p, Blank[]]]]], Pattern[x, Blank[Symbol]]] := Simp[(g*(d + e*x)^m*(a + c*x^2)^(p + 1))/(c*(m + 2*p + 2)), x] + Dist[1/(c*(m + 2*p + 2)), Int[(d + e*x)^(m - 1)*(a + c*x^2)^p*Simp[c*d*f*(m + 2*p + 2) - a*e*g*m + c*(e*f*(m + 2*p + 2) + d*g*m)*x, x], x], x] /; FreeQ[{a, c, d, e, f, g, p}, x] && NeQ[c*d^2 + a*e^2, 0] && GtQ[m, 0] && NeQ[m + 2*p + 2, 0] && (IntegerQ[m] || IntegerQ[p] || IntegersQ[2*m, 2*p]) && !(IGtQ[m, 0] && EqQ[f, 0])
  834. Int[Times[Power[Plus[Optional[Pattern[d, Blank[]]], Times[Optional[Pattern[e, Blank[]]], Pattern[x, Blank[]]]], Pattern[m, Blank[]]], Plus[Optional[Pattern[f, Blank[]]], Times[Optional[Pattern[g, Blank[]]], Pattern[x, Blank[]]]], Power[Plus[Optional[Pattern[a, Blank[]]], Times[Optional[Pattern[b, Blank[]]], Pattern[x, Blank[]]], Times[Optional[Pattern[c, Blank[]]], Power[Pattern[x, Blank[]], 2]]], Optional[Pattern[p, Blank[]]]]], Pattern[x, Blank[Symbol]]] := Simp[((e*f - d*g)*(d + e*x)^(m + 1)*(a + b*x + c*x^2)^(p + 1))/((m + 1)*(c*d^2 - b*d*e + a*e^2)), x] + Dist[1/((m + 1)*(c*d^2 - b*d*e + a*e^2)), Int[(d + e*x)^(m + 1)*(a + b*x + c*x^2)^p*Simp[(c*d*f - f*b*e + a*e*g)*(m + 1) + b*(d*g - e*f)*(p + 1) - c*(e*f - d*g)*(m + 2*p + 3)*x, x], x], x] /; FreeQ[{a, b, c, d, e, f, g, p}, x] && NeQ[b^2 - 4*a*c, 0] && NeQ[c*d^2 - b*d*e + a*e^2, 0] && LtQ[m, -1] && (IntegerQ[m] || IntegerQ[p] || IntegersQ[2*m, 2*p])
  835. Int[Times[Power[Plus[Optional[Pattern[d, Blank[]]], Times[Optional[Pattern[e, Blank[]]], Pattern[x, Blank[]]]], Pattern[m, Blank[]]], Plus[Optional[Pattern[f, Blank[]]], Times[Optional[Pattern[g, Blank[]]], Pattern[x, Blank[]]]], Power[Plus[Pattern[a, Blank[]], Times[Optional[Pattern[c, Blank[]]], Power[Pattern[x, Blank[]], 2]]], Optional[Pattern[p, Blank[]]]]], Pattern[x, Blank[Symbol]]] := Simp[((e*f - d*g)*(d + e*x)^(m + 1)*(a + c*x^2)^(p + 1))/((m + 1)*(c*d^2 + a*e^2)), x] + Dist[1/((m + 1)*(c*d^2 + a*e^2)), Int[(d + e*x)^(m + 1)*(a + c*x^2)^p*Simp[(c*d*f + a*e*g)*(m + 1) - c*(e*f - d*g)*(m + 2*p + 3)*x, x], x], x] /; FreeQ[{a, c, d, e, f, g, p}, x] && NeQ[c*d^2 + a*e^2, 0] && LtQ[m, -1] && (IntegerQ[m] || IntegerQ[p] || IntegersQ[2*m, 2*p])
  836. Int[Times[Power[Plus[Optional[Pattern[d, Blank[]]], Times[Optional[Pattern[e, Blank[]]], Pattern[x, Blank[]]]], Pattern[m, Blank[]]], Plus[Optional[Pattern[f, Blank[]]], Times[Optional[Pattern[g, Blank[]]], Pattern[x, Blank[]]]], Power[Plus[Optional[Pattern[a, Blank[]]], Times[Optional[Pattern[b, Blank[]]], Pattern[x, Blank[]]], Times[Optional[Pattern[c, Blank[]]], Power[Pattern[x, Blank[]], 2]]], Optional[Pattern[p, Blank[]]]]], Pattern[x, Blank[Symbol]]] := Simp[((e*f - d*g)*(d + e*x)^(m + 1)*(a + b*x + c*x^2)^(p + 1))/((m + 1)*(c*d^2 - b*d*e + a*e^2)), x] + Dist[1/((m + 1)*(c*d^2 - b*d*e + a*e^2)), Int[(d + e*x)^(m + 1)*(a + b*x + c*x^2)^p*Simp[(c*d*f - f*b*e + a*e*g)*(m + 1) + b*(d*g - e*f)*(p + 1) - c*(e*f - d*g)*(m + 2*p + 3)*x, x], x], x] /; FreeQ[{a, b, c, d, e, f, g, m, p}, x] && NeQ[b^2 - 4*a*c, 0] && NeQ[c*d^2 - b*d*e + a*e^2, 0] && ILtQ[Simplify[m + 2*p + 3], 0] && NeQ[m, -1]
  837. Int[Times[Power[Plus[Optional[Pattern[d, Blank[]]], Times[Optional[Pattern[e, Blank[]]], Pattern[x, Blank[]]]], Pattern[m, Blank[]]], Plus[Optional[Pattern[f, Blank[]]], Times[Optional[Pattern[g, Blank[]]], Pattern[x, Blank[]]]], Power[Plus[Pattern[a, Blank[]], Times[Optional[Pattern[c, Blank[]]], Power[Pattern[x, Blank[]], 2]]], Optional[Pattern[p, Blank[]]]]], Pattern[x, Blank[Symbol]]] := Simp[((e*f - d*g)*(d + e*x)^(m + 1)*(a + c*x^2)^(p + 1))/((m + 1)*(c*d^2 + a*e^2)), x] + Dist[1/((m + 1)*(c*d^2 + a*e^2)), Int[(d + e*x)^(m + 1)*(a + c*x^2)^p*Simp[(c*d*f + a*e*g)*(m + 1) - c*(e*f - d*g)*(m + 2*p + 3)*x, x], x], x] /; FreeQ[{a, c, d, e, f, g, m, p}, x] && NeQ[c*d^2 + a*e^2, 0] && ILtQ[Simplify[m + 2*p + 3], 0] && NeQ[m, -1]
  838. Int[Times[Power[Plus[Optional[Pattern[d, Blank[]]], Times[Optional[Pattern[e, Blank[]]], Pattern[x, Blank[]]]], -1], Plus[Pattern[f, Blank[]], Times[Optional[Pattern[g, Blank[]]], Pattern[x, Blank[]]]], Power[Plus[Optional[Pattern[a, Blank[]]], Times[Optional[Pattern[b, Blank[]]], Pattern[x, Blank[]]], Times[Optional[Pattern[c, Blank[]]], Power[Pattern[x, Blank[]], 2]]], Rational[-1, 2]]], Pattern[x, Blank[Symbol]]] := Dist[(4*f*(a - d))/(b*d - a*e), Subst[Int[1/(4*(a - d) - x^2), x], x, (2*(a - d) + (b - e)*x)/Sqrt[a + b*x + c*x^2]], x] /; FreeQ[{a, b, c, d, e, f, g}, x] && EqQ[4*c*(a - d) - (b - e)^2, 0] && EqQ[e*f*(b - e) - 2*g*(b*d - a*e), 0] && NeQ[b*d - a*e, 0]
  839. Int[Times[Power[Pattern[x, Blank[]], Rational[-1, 2]], Plus[Pattern[f, Blank[]], Times[Optional[Pattern[g, Blank[]]], Pattern[x, Blank[]]]], Power[Plus[Pattern[a, Blank[]], Times[Optional[Pattern[b, Blank[]]], Pattern[x, Blank[]]], Times[Optional[Pattern[c, Blank[]]], Power[Pattern[x, Blank[]], 2]]], Rational[-1, 2]]], Pattern[x, Blank[Symbol]]] := Dist[2, Subst[Int[(f + g*x^2)/Sqrt[a + b*x^2 + c*x^4], x], x, Sqrt[x]], x] /; FreeQ[{a, b, c, f, g}, x] && NeQ[b^2 - 4*a*c, 0]
  840. Int[Times[Power[Pattern[x, Blank[]], Rational[-1, 2]], Plus[Pattern[f, Blank[]], Times[Optional[Pattern[g, Blank[]]], Pattern[x, Blank[]]]], Power[Plus[Pattern[a, Blank[]], Times[Optional[Pattern[c, Blank[]]], Power[Pattern[x, Blank[]], 2]]], Rational[-1, 2]]], Pattern[x, Blank[Symbol]]] := Dist[2, Subst[Int[(f + g*x^2)/Sqrt[a + c*x^4], x], x, Sqrt[x]], x] /; FreeQ[{a, c, f, g}, x]
  841. Int[Times[Power[Times[Pattern[e, Blank[]], Pattern[x, Blank[]]], Rational[-1, 2]], Plus[Pattern[f, Blank[]], Times[Optional[Pattern[g, Blank[]]], Pattern[x, Blank[]]]], Power[Plus[Pattern[a, Blank[]], Times[Optional[Pattern[b, Blank[]]], Pattern[x, Blank[]]], Times[Optional[Pattern[c, Blank[]]], Power[Pattern[x, Blank[]], 2]]], Rational[-1, 2]]], Pattern[x, Blank[Symbol]]] := Dist[Sqrt[x]/Sqrt[e*x], Int[(f + g*x)/(Sqrt[x]*Sqrt[a + b*x + c*x^2]), x], x] /; FreeQ[{a, b, c, e, f, g}, x] && NeQ[b^2 - 4*a*c, 0]
  842. Int[Times[Power[Times[Pattern[e, Blank[]], Pattern[x, Blank[]]], Rational[-1, 2]], Plus[Pattern[f, Blank[]], Times[Optional[Pattern[g, Blank[]]], Pattern[x, Blank[]]]], Power[Plus[Pattern[a, Blank[]], Times[Optional[Pattern[c, Blank[]]], Power[Pattern[x, Blank[]], 2]]], Rational[-1, 2]]], Pattern[x, Blank[Symbol]]] := Dist[Sqrt[x]/Sqrt[e*x], Int[(f + g*x)/(Sqrt[x]*Sqrt[a + c*x^2]), x], x] /; FreeQ[{a, c, e, f, g}, x]
  843. Int[Times[Power[Plus[Optional[Pattern[d, Blank[]]], Times[Optional[Pattern[e, Blank[]]], Pattern[x, Blank[]]]], Pattern[m, Blank[]]], Plus[Optional[Pattern[f, Blank[]]], Times[Optional[Pattern[g, Blank[]]], Pattern[x, Blank[]]]], Power[Plus[Optional[Pattern[a, Blank[]]], Times[Optional[Pattern[b, Blank[]]], Pattern[x, Blank[]]], Times[Optional[Pattern[c, Blank[]]], Power[Pattern[x, Blank[]], 2]]], Optional[Pattern[p, Blank[]]]]], Pattern[x, Blank[Symbol]]] := Dist[g/e, Int[(d + e*x)^(m + 1)*(a + b*x + c*x^2)^p, x], x] + Dist[(e*f - d*g)/e, Int[(d + e*x)^m*(a + b*x + c*x^2)^p, x], x] /; FreeQ[{a, b, c, d, e, f, g, m, p}, x] && NeQ[b^2 - 4*a*c, 0] && NeQ[c*d^2 - b*d*e + a*e^2, 0] && !IGtQ[m, 0]
  844. Int[Times[Power[Plus[Optional[Pattern[d, Blank[]]], Times[Optional[Pattern[e, Blank[]]], Pattern[x, Blank[]]]], Pattern[m, Blank[]]], Plus[Optional[Pattern[f, Blank[]]], Times[Optional[Pattern[g, Blank[]]], Pattern[x, Blank[]]]], Power[Plus[Pattern[a, Blank[]], Times[Optional[Pattern[c, Blank[]]], Power[Pattern[x, Blank[]], 2]]], Optional[Pattern[p, Blank[]]]]], Pattern[x, Blank[Symbol]]] := Dist[g/e, Int[(d + e*x)^(m + 1)*(a + c*x^2)^p, x], x] + Dist[(e*f - d*g)/e, Int[(d + e*x)^m*(a + c*x^2)^p, x], x] /; FreeQ[{a, c, d, e, f, g, m, p}, x] && NeQ[c*d^2 + a*e^2, 0] && !IGtQ[m, 0]
  845. Int[Times[Power[Pattern[x, Blank[]], Optional[Pattern[m, Blank[]]]], Power[Plus[Pattern[f, Blank[]], Times[Optional[Pattern[g, Blank[]]], Pattern[x, Blank[]]]], Optional[Pattern[n, Blank[]]]], Plus[Times[Optional[Pattern[b, Blank[]]], Pattern[x, Blank[]]], Times[Optional[Pattern[c, Blank[]]], Power[Pattern[x, Blank[]], 2]]]], Pattern[x, Blank[Symbol]]] := Simp[(c*x^(m + 2)*(f + g*x)^(n + 1))/(g*(m + n + 3)), x] /; FreeQ[{b, c, f, g, m, n}, x] && EqQ[c*f*(m + 2) - b*g*(m + n + 3), 0] && NeQ[m + n + 3, 0]
  846. Int[Times[Power[Plus[Optional[Pattern[d, Blank[]]], Times[Optional[Pattern[e, Blank[]]], Pattern[x, Blank[]]]], Pattern[m, Blank[]]], Power[Plus[Optional[Pattern[f, Blank[]]], Times[Optional[Pattern[g, Blank[]]], Pattern[x, Blank[]]]], Pattern[n, Blank[]]], Power[Plus[Optional[Pattern[a, Blank[]]], Times[Optional[Pattern[b, Blank[]]], Pattern[x, Blank[]]], Times[Optional[Pattern[c, Blank[]]], Power[Pattern[x, Blank[]], 2]]], Pattern[p, Blank[]]]], Pattern[x, Blank[Symbol]]] := Dist[(a + b*x + c*x^2)^FracPart[p]/(c^IntPart[p]*(b/2 + c*x)^(2*FracPart[p])), Int[(d + e*x)^m*(f + g*x)^n*(b/2 + c*x)^(2*p), x], x] /; FreeQ[{a, b, c, d, e, f, g, m, n}, x] && NeQ[e*f - d*g, 0] && EqQ[b^2 - 4*a*c, 0] && !IntegerQ[p]
  847. Int[Times[Power[Plus[Pattern[d, Blank[]], Times[Optional[Pattern[e, Blank[]]], Pattern[x, Blank[]]]], Pattern[m, Blank[]]], Power[Plus[Optional[Pattern[f, Blank[]]], Times[Optional[Pattern[g, Blank[]]], Pattern[x, Blank[]]]], Pattern[n, Blank[]]], Power[Plus[Optional[Pattern[a, Blank[]]], Times[Optional[Pattern[b, Blank[]]], Pattern[x, Blank[]]], Times[Optional[Pattern[c, Blank[]]], Power[Pattern[x, Blank[]], 2]]], Optional[Pattern[p, Blank[]]]]], Pattern[x, Blank[Symbol]]] := Int[(d + e*x)^(m + p)*(f + g*x)^n*(a/d + (c*x)/e)^p, x] /; FreeQ[{a, b, c, d, e, f, g, m, n}, x] && NeQ[e*f - d*g, 0] && NeQ[b^2 - 4*a*c, 0] && EqQ[c*d^2 - b*d*e + a*e^2, 0] && IntegerQ[p] && !IGtQ[n, 0]
  848. Int[Times[Power[Plus[Pattern[d, Blank[]], Times[Optional[Pattern[e, Blank[]]], Pattern[x, Blank[]]]], Pattern[m, Blank[]]], Power[Plus[Optional[Pattern[f, Blank[]]], Times[Optional[Pattern[g, Blank[]]], Pattern[x, Blank[]]]], Pattern[n, Blank[]]], Power[Plus[Pattern[a, Blank[]], Times[Optional[Pattern[c, Blank[]]], Power[Pattern[x, Blank[]], 2]]], Optional[Pattern[p, Blank[]]]]], Pattern[x, Blank[Symbol]]] := Int[(d + e*x)^(m + p)*(f + g*x)^n*(a/d + (c*x)/e)^p, x] /; FreeQ[{a, c, d, e, f, g, m, n}, x] && NeQ[e*f - d*g, 0] && EqQ[c*d^2 + a*e^2, 0] && (IntegerQ[p] || (GtQ[a, 0] && GtQ[d, 0] && EqQ[m + p, 0]))
  849. Int[Times[Power[Pattern[x, Blank[]], Optional[Pattern[n, Blank[]]]], Power[Plus[Pattern[d, Blank[]], Times[Optional[Pattern[e, Blank[]]], Pattern[x, Blank[]]]], -1], Power[Plus[Optional[Pattern[a, Blank[]]], Times[Optional[Pattern[b, Blank[]]], Pattern[x, Blank[]]], Times[Optional[Pattern[c, Blank[]]], Power[Pattern[x, Blank[]], 2]]], Pattern[p, Blank[]]]], Pattern[x, Blank[Symbol]]] := Int[x^n*(a/d + (c*x)/e)*(a + b*x + c*x^2)^(p - 1), x] /; FreeQ[{a, b, c, d, e, n, p}, x] && NeQ[b^2 - 4*a*c, 0] && EqQ[c*d^2 - b*d*e + a*e^2, 0] && !IntegerQ[p] && ( !IntegerQ[n] || !IntegerQ[2*p] || IGtQ[n, 2] || (GtQ[p, 0] && NeQ[n, 2]))
  850. Int[Times[Power[Pattern[x, Blank[]], Optional[Pattern[n, Blank[]]]], Power[Plus[Pattern[d, Blank[]], Times[Optional[Pattern[e, Blank[]]], Pattern[x, Blank[]]]], -1], Power[Plus[Pattern[a, Blank[]], Times[Optional[Pattern[c, Blank[]]], Power[Pattern[x, Blank[]], 2]]], Pattern[p, Blank[]]]], Pattern[x, Blank[Symbol]]] := Int[x^n*(a/d + (c*x)/e)*(a + c*x^2)^(p - 1), x] /; FreeQ[{a, c, d, e, n, p}, x] && EqQ[c*d^2 + a*e^2, 0] && !IntegerQ[p] && ( !IntegerQ[n] || !IntegerQ[2*p] || IGtQ[n, 2] || (GtQ[p, 0] && NeQ[n, 2]))
  851. Int[Times[Power[Plus[Pattern[d, Blank[]], Times[Optional[Pattern[e, Blank[]]], Pattern[x, Blank[]]]], Pattern[m, Blank[]]], Power[Plus[Optional[Pattern[f, Blank[]]], Times[Optional[Pattern[g, Blank[]]], Pattern[x, Blank[]]]], Pattern[n, Blank[]]], Power[Plus[Optional[Pattern[a, Blank[]]], Times[Optional[Pattern[b, Blank[]]], Pattern[x, Blank[]]], Times[Optional[Pattern[c, Blank[]]], Power[Pattern[x, Blank[]], 2]]], Pattern[p, Blank[]]]], Pattern[x, Blank[Symbol]]] := Int[((f + g*x)^n*(a + b*x + c*x^2)^(m + p))/(a/d + (c*x)/e)^m, x] /; FreeQ[{a, b, c, d, e, f, g, n, p}, x] && NeQ[e*f - d*g, 0] && NeQ[b^2 - 4*a*c, 0] && EqQ[c*d^2 - b*d*e + a*e^2, 0] && !IntegerQ[p] && ILtQ[m, 0] && IntegerQ[n] && (LtQ[n, 0] || GtQ[p, 0])
  852. Int[Times[Power[Plus[Pattern[d, Blank[]], Times[Optional[Pattern[e, Blank[]]], Pattern[x, Blank[]]]], Pattern[m, Blank[]]], Power[Plus[Optional[Pattern[f, Blank[]]], Times[Optional[Pattern[g, Blank[]]], Pattern[x, Blank[]]]], Pattern[n, Blank[]]], Power[Plus[Pattern[a, Blank[]], Times[Optional[Pattern[c, Blank[]]], Power[Pattern[x, Blank[]], 2]]], Pattern[p, Blank[]]]], Pattern[x, Blank[Symbol]]] := Dist[d^(2*m)/a^m, Int[((f + g*x)^n*(a + c*x^2)^(m + p))/(d - e*x)^m, x], x] /; FreeQ[{a, c, d, e, f, g, n, p}, x] && NeQ[e*f - d*g, 0] && EqQ[c*d^2 + a*e^2, 0] && !IntegerQ[p] && EqQ[f, 0] && ILtQ[m, -1] && !(IGtQ[n, 0] && ILtQ[m + n, 0] && !GtQ[p, 1])
  853. Int[Times[Power[Plus[Pattern[d, Blank[]], Times[Optional[Pattern[e, Blank[]]], Pattern[x, Blank[]]]], Pattern[m, Blank[]]], Power[Plus[Pattern[f, Blank[]], Times[Optional[Pattern[g, Blank[]]], Pattern[x, Blank[]]]], Pattern[n, Blank[]]], Power[Plus[Pattern[a, Blank[]], Times[Optional[Pattern[c, Blank[]]], Power[Pattern[x, Blank[]], 2]]], Pattern[p, Blank[]]]], Pattern[x, Blank[Symbol]]] := Dist[d^(2*m)/a^m, Int[((f + g*x)^n*(a + c*x^2)^(m + p))/(d - e*x)^m, x], x] /; FreeQ[{a, c, d, e, f, g, n, p}, x] && NeQ[e*f - d*g, 0] && EqQ[c*d^2 + a*e^2, 0] && !IntegerQ[p] && ILtQ[m, 0] && IntegerQ[n]
  854. Int[Times[Power[Plus[Pattern[d, Blank[]], Times[Optional[Pattern[e, Blank[]]], Pattern[x, Blank[]]]], -1], Power[Plus[Optional[Pattern[f, Blank[]]], Times[Optional[Pattern[g, Blank[]]], Pattern[x, Blank[]]]], Pattern[n, Blank[]]], Power[Plus[Optional[Pattern[a, Blank[]]], Times[Optional[Pattern[b, Blank[]]], Pattern[x, Blank[]]], Times[Optional[Pattern[c, Blank[]]], Power[Pattern[x, Blank[]], 2]]], Pattern[p, Blank[]]]], Pattern[x, Blank[Symbol]]] := -Simp[((2*c*d - b*e)*(f + g*x)^n*(a + b*x + c*x^2)^(p + 1))/(e*p*(b^2 - 4*a*c)*(d + e*x)), x] - Dist[1/(d*e*p*(b^2 - 4*a*c)), Int[(f + g*x)^(n - 1)*(a + b*x + c*x^2)^p*Simp[b*(a*e*g*n - c*d*f*(2*p + 1)) - 2*a*c*(d*g*n - e*f*(2*p + 1)) - c*g*(b*d - 2*a*e)*(n + 2*p + 1)*x, x], x], x] /; FreeQ[{a, b, c, d, e, f, g}, x] && NeQ[e*f - d*g, 0] && NeQ[b^2 - 4*a*c, 0] && EqQ[c*d^2 - b*d*e + a*e^2, 0] && !IntegerQ[p] && IGtQ[n, 0] && ILtQ[n + 2*p, 0]
  855. Int[Times[Power[Plus[Pattern[d, Blank[]], Times[Optional[Pattern[e, Blank[]]], Pattern[x, Blank[]]]], -1], Power[Plus[Optional[Pattern[f, Blank[]]], Times[Optional[Pattern[g, Blank[]]], Pattern[x, Blank[]]]], Pattern[n, Blank[]]], Power[Plus[Pattern[a, Blank[]], Times[Optional[Pattern[c, Blank[]]], Power[Pattern[x, Blank[]], 2]]], Pattern[p, Blank[]]]], Pattern[x, Blank[Symbol]]] := Simp[(d*(f + g*x)^n*(a + c*x^2)^(p + 1))/(2*a*e*p*(d + e*x)), x] - Dist[1/(2*d*e*p), Int[(f + g*x)^(n - 1)*(a + c*x^2)^p*Simp[d*g*n - e*f*(2*p + 1) - e*g*(n + 2*p + 1)*x, x], x], x] /; FreeQ[{a, c, d, e, f, g}, x] && NeQ[e*f - d*g, 0] && EqQ[c*d^2 + a*e^2, 0] && !IntegerQ[p] && IGtQ[n, 0] && ILtQ[n + 2*p, 0]
  856. Int[Times[Power[Plus[Pattern[d, Blank[]], Times[Optional[Pattern[e, Blank[]]], Pattern[x, Blank[]]]], -1], Power[Plus[Optional[Pattern[f, Blank[]]], Times[Optional[Pattern[g, Blank[]]], Pattern[x, Blank[]]]], Pattern[n, Blank[]]], Power[Plus[Optional[Pattern[a, Blank[]]], Times[Optional[Pattern[b, Blank[]]], Pattern[x, Blank[]]], Times[Optional[Pattern[c, Blank[]]], Power[Pattern[x, Blank[]], 2]]], Pattern[p, Blank[]]]], Pattern[x, Blank[Symbol]]] := Simp[((f + g*x)^(n + 1)*(a + b*x + c*x^2)^p*(c*d - b*e - c*e*x))/(p*(2*c*d - b*e)*(e*f - d*g)), x] + Dist[1/(p*(2*c*d - b*e)*(e*f - d*g)), Int[(f + g*x)^n*(a + b*x + c*x^2)^p*(b*e*g*(n + p + 1) + c*e*f*(2*p + 1) - c*d*g*(n + 2*p + 1) + c*e*g*(n + 2*p + 2)*x), x], x] /; FreeQ[{a, b, c, d, e, f, g}, x] && NeQ[e*f - d*g, 0] && NeQ[b^2 - 4*a*c, 0] && EqQ[c*d^2 - b*d*e + a*e^2, 0] && !IntegerQ[p] && ILtQ[n, 0] && ILtQ[n + 2*p, 0] && !IGtQ[n, 0]
  857. Int[Times[Power[Plus[Pattern[d, Blank[]], Times[Optional[Pattern[e, Blank[]]], Pattern[x, Blank[]]]], -1], Power[Plus[Optional[Pattern[f, Blank[]]], Times[Optional[Pattern[g, Blank[]]], Pattern[x, Blank[]]]], Pattern[n, Blank[]]], Power[Plus[Pattern[a, Blank[]], Times[Optional[Pattern[c, Blank[]]], Power[Pattern[x, Blank[]], 2]]], Pattern[p, Blank[]]]], Pattern[x, Blank[Symbol]]] := Simp[(d*(f + g*x)^(n + 1)*(a + c*x^2)^(p + 1))/(2*a*p*(e*f - d*g)*(d + e*x)), x] + Dist[1/(p*(2*c*d)*(e*f - d*g)), Int[(f + g*x)^n*(a + c*x^2)^p*(c*e*f*(2*p + 1) - c*d*g*(n + 2*p + 1) + c*e*g*(n + 2*p + 2)*x), x], x] /; FreeQ[{a, c, d, e, f, g}, x] && NeQ[e*f - d*g, 0] && EqQ[c*d^2 + a*e^2, 0] && !IntegerQ[p] && ILtQ[n, 0] && ILtQ[n + 2*p, 0] && !IGtQ[n, 0]
  858. Int[Times[Power[Plus[Pattern[d, Blank[]], Times[Optional[Pattern[e, Blank[]]], Pattern[x, Blank[]]]], Pattern[m, Blank[]]], Power[Plus[Optional[Pattern[f, Blank[]]], Times[Optional[Pattern[g, Blank[]]], Pattern[x, Blank[]]]], Pattern[n, Blank[]]], Power[Plus[Optional[Pattern[a, Blank[]]], Times[Optional[Pattern[b, Blank[]]], Pattern[x, Blank[]]], Times[Optional[Pattern[c, Blank[]]], Power[Pattern[x, Blank[]], 2]]], Pattern[p, Blank[]]]], Pattern[x, Blank[Symbol]]] := -Simp[(e*(d + e*x)^(m - 1)*(f + g*x)^n*(a + b*x + c*x^2)^(p + 1))/(c*(m - n - 1)), x] /; FreeQ[{a, b, c, d, e, f, g, m, n, p}, x] && NeQ[e*f - d*g, 0] && NeQ[b^2 - 4*a*c, 0] && EqQ[c*d^2 - b*d*e + a*e^2, 0] && !IntegerQ[p] && EqQ[m + p, 0] && EqQ[c*e*f + c*d*g - b*e*g, 0] && NeQ[m - n - 1, 0]
  859. Int[Times[Power[Plus[Pattern[d, Blank[]], Times[Optional[Pattern[e, Blank[]]], Pattern[x, Blank[]]]], Pattern[m, Blank[]]], Power[Plus[Optional[Pattern[f, Blank[]]], Times[Optional[Pattern[g, Blank[]]], Pattern[x, Blank[]]]], Pattern[n, Blank[]]], Power[Plus[Pattern[a, Blank[]], Times[Optional[Pattern[c, Blank[]]], Power[Pattern[x, Blank[]], 2]]], Pattern[p, Blank[]]]], Pattern[x, Blank[Symbol]]] := -Simp[(e*(d + e*x)^(m - 1)*(f + g*x)^n*(a + c*x^2)^(p + 1))/(c*(m - n - 1)), x] /; FreeQ[{a, c, d, e, f, g, m, n, p}, x] && NeQ[e*f - d*g, 0] && EqQ[c*d^2 + a*e^2, 0] && !IntegerQ[p] && EqQ[m + p, 0] && EqQ[e*f + d*g, 0] && NeQ[m - n - 1, 0]
  860. Int[Times[Power[Plus[Pattern[d, Blank[]], Times[Optional[Pattern[e, Blank[]]], Pattern[x, Blank[]]]], Pattern[m, Blank[]]], Power[Plus[Optional[Pattern[f, Blank[]]], Times[Optional[Pattern[g, Blank[]]], Pattern[x, Blank[]]]], Pattern[n, Blank[]]], Power[Plus[Optional[Pattern[a, Blank[]]], Times[Optional[Pattern[b, Blank[]]], Pattern[x, Blank[]]], Times[Optional[Pattern[c, Blank[]]], Power[Pattern[x, Blank[]], 2]]], Pattern[p, Blank[]]]], Pattern[x, Blank[Symbol]]] := -Simp[(e^2*(d + e*x)^(m - 1)*(f + g*x)^(n + 1)*(a + b*x + c*x^2)^(p + 1))/((n + 1)*(c*e*f + c*d*g - b*e*g)), x] /; FreeQ[{a, b, c, d, e, f, g, m, n, p}, x] && NeQ[e*f - d*g, 0] && NeQ[b^2 - 4*a*c, 0] && EqQ[c*d^2 - b*d*e + a*e^2, 0] && !IntegerQ[p] && EqQ[m + p, 0] && EqQ[m - n - 2, 0]
  861. Int[Times[Power[Plus[Pattern[d, Blank[]], Times[Optional[Pattern[e, Blank[]]], Pattern[x, Blank[]]]], Pattern[m, Blank[]]], Power[Plus[Optional[Pattern[f, Blank[]]], Times[Optional[Pattern[g, Blank[]]], Pattern[x, Blank[]]]], Pattern[n, Blank[]]], Power[Plus[Pattern[a, Blank[]], Times[Optional[Pattern[c, Blank[]]], Power[Pattern[x, Blank[]], 2]]], Pattern[p, Blank[]]]], Pattern[x, Blank[Symbol]]] := -Simp[(e^2*(d + e*x)^(m - 1)*(f + g*x)^(n + 1)*(a + c*x^2)^(p + 1))/(c*(n + 1)*(e*f + d*g)), x] /; FreeQ[{a, c, d, e, f, g, m, n, p}, x] && NeQ[e*f - d*g, 0] && EqQ[c*d^2 + a*e^2, 0] && !IntegerQ[p] && EqQ[m + p, 0] && EqQ[m - n - 2, 0]
  862. Int[Times[Power[Plus[Pattern[d, Blank[]], Times[Optional[Pattern[e, Blank[]]], Pattern[x, Blank[]]]], Pattern[m, Blank[]]], Power[Plus[Optional[Pattern[f, Blank[]]], Times[Optional[Pattern[g, Blank[]]], Pattern[x, Blank[]]]], Pattern[n, Blank[]]], Power[Plus[Optional[Pattern[a, Blank[]]], Times[Optional[Pattern[b, Blank[]]], Pattern[x, Blank[]]], Times[Optional[Pattern[c, Blank[]]], Power[Pattern[x, Blank[]], 2]]], Pattern[p, Blank[]]]], Pattern[x, Blank[Symbol]]] := Simp[((d + e*x)^m*(f + g*x)^(n + 1)*(a + b*x + c*x^2)^p)/(g*(n + 1)), x] + Dist[(c*m)/(e*g*(n + 1)), Int[(d + e*x)^(m + 1)*(f + g*x)^(n + 1)*(a + b*x + c*x^2)^(p - 1), x], x] /; FreeQ[{a, b, c, d, e, f, g}, x] && NeQ[e*f - d*g, 0] && NeQ[b^2 - 4*a*c, 0] && EqQ[c*d^2 - b*d*e + a*e^2, 0] && !IntegerQ[p] && EqQ[m + p, 0] && GtQ[p, 0] && LtQ[n, -1] && !(IntegerQ[n + p] && LeQ[n + p + 2, 0])
  863. Int[Times[Power[Plus[Pattern[d, Blank[]], Times[Optional[Pattern[e, Blank[]]], Pattern[x, Blank[]]]], Pattern[m, Blank[]]], Power[Plus[Optional[Pattern[f, Blank[]]], Times[Optional[Pattern[g, Blank[]]], Pattern[x, Blank[]]]], Pattern[n, Blank[]]], Power[Plus[Pattern[a, Blank[]], Times[Optional[Pattern[c, Blank[]]], Power[Pattern[x, Blank[]], 2]]], Pattern[p, Blank[]]]], Pattern[x, Blank[Symbol]]] := Simp[((d + e*x)^m*(f + g*x)^(n + 1)*(a + c*x^2)^p)/(g*(n + 1)), x] + Dist[(c*m)/(e*g*(n + 1)), Int[(d + e*x)^(m + 1)*(f + g*x)^(n + 1)*(a + c*x^2)^(p - 1), x], x] /; FreeQ[{a, c, d, e, f, g}, x] && NeQ[e*f - d*g, 0] && EqQ[c*d^2 + a*e^2, 0] && !IntegerQ[p] && EqQ[m + p, 0] && GtQ[p, 0] && LtQ[n, -1] && !(IntegerQ[n + p] && LeQ[n + p + 2, 0])
  864. Int[Times[Power[Plus[Pattern[d, Blank[]], Times[Optional[Pattern[e, Blank[]]], Pattern[x, Blank[]]]], Pattern[m, Blank[]]], Power[Plus[Optional[Pattern[f, Blank[]]], Times[Optional[Pattern[g, Blank[]]], Pattern[x, Blank[]]]], Pattern[n, Blank[]]], Power[Plus[Optional[Pattern[a, Blank[]]], Times[Optional[Pattern[b, Blank[]]], Pattern[x, Blank[]]], Times[Optional[Pattern[c, Blank[]]], Power[Pattern[x, Blank[]], 2]]], Pattern[p, Blank[]]]], Pattern[x, Blank[Symbol]]] := -Simp[((d + e*x)^m*(f + g*x)^(n + 1)*(a + b*x + c*x^2)^p)/(g*(m - n - 1)), x] - Dist[(m*(c*e*f + c*d*g - b*e*g))/(e^2*g*(m - n - 1)), Int[(d + e*x)^(m + 1)*(f + g*x)^n*(a + b*x + c*x^2)^(p - 1), x], x] /; FreeQ[{a, b, c, d, e, f, g, n}, x] && NeQ[e*f - d*g, 0] && NeQ[b^2 - 4*a*c, 0] && EqQ[c*d^2 - b*d*e + a*e^2, 0] && !IntegerQ[p] && EqQ[m + p, 0] && GtQ[p, 0] && NeQ[m - n - 1, 0] && !IGtQ[n, 0] && !(IntegerQ[n + p] && LtQ[n + p + 2, 0]) && RationalQ[n]
  865. Int[Times[Power[Plus[Pattern[d, Blank[]], Times[Optional[Pattern[e, Blank[]]], Pattern[x, Blank[]]]], Pattern[m, Blank[]]], Power[Plus[Optional[Pattern[f, Blank[]]], Times[Optional[Pattern[g, Blank[]]], Pattern[x, Blank[]]]], Pattern[n, Blank[]]], Power[Plus[Pattern[a, Blank[]], Times[Optional[Pattern[c, Blank[]]], Power[Pattern[x, Blank[]], 2]]], Pattern[p, Blank[]]]], Pattern[x, Blank[Symbol]]] := -Simp[((d + e*x)^m*(f + g*x)^(n + 1)*(a + c*x^2)^p)/(g*(m - n - 1)), x] - Dist[(c*m*(e*f + d*g))/(e^2*g*(m - n - 1)), Int[(d + e*x)^(m + 1)*(f + g*x)^n*(a + c*x^2)^(p - 1), x], x] /; FreeQ[{a, c, d, e, f, g, n}, x] && NeQ[e*f - d*g, 0] && EqQ[c*d^2 + a*e^2, 0] && !IntegerQ[p] && EqQ[m + p, 0] && GtQ[p, 0] && NeQ[m - n - 1, 0] && !IGtQ[n, 0] && !(IntegerQ[n + p] && LtQ[n + p + 2, 0]) && RationalQ[n]
  866. Int[Times[Power[Plus[Pattern[d, Blank[]], Times[Optional[Pattern[e, Blank[]]], Pattern[x, Blank[]]]], Pattern[m, Blank[]]], Power[Plus[Optional[Pattern[f, Blank[]]], Times[Optional[Pattern[g, Blank[]]], Pattern[x, Blank[]]]], Pattern[n, Blank[]]], Power[Plus[Optional[Pattern[a, Blank[]]], Times[Optional[Pattern[b, Blank[]]], Pattern[x, Blank[]]], Times[Optional[Pattern[c, Blank[]]], Power[Pattern[x, Blank[]], 2]]], Pattern[p, Blank[]]]], Pattern[x, Blank[Symbol]]] := Simp[(e*(d + e*x)^(m - 1)*(f + g*x)^n*(a + b*x + c*x^2)^(p + 1))/(c*(p + 1)), x] - Dist[(e*g*n)/(c*(p + 1)), Int[(d + e*x)^(m - 1)*(f + g*x)^(n - 1)*(a + b*x + c*x^2)^(p + 1), x], x] /; FreeQ[{a, b, c, d, e, f, g}, x] && NeQ[e*f - d*g, 0] && NeQ[b^2 - 4*a*c, 0] && EqQ[c*d^2 - b*d*e + a*e^2, 0] && !IntegerQ[p] && EqQ[m + p, 0] && LtQ[p, -1] && GtQ[n, 0]
  867. Int[Times[Power[Plus[Pattern[d, Blank[]], Times[Optional[Pattern[e, Blank[]]], Pattern[x, Blank[]]]], Pattern[m, Blank[]]], Power[Plus[Optional[Pattern[f, Blank[]]], Times[Optional[Pattern[g, Blank[]]], Pattern[x, Blank[]]]], Pattern[n, Blank[]]], Power[Plus[Pattern[a, Blank[]], Times[Optional[Pattern[c, Blank[]]], Power[Pattern[x, Blank[]], 2]]], Pattern[p, Blank[]]]], Pattern[x, Blank[Symbol]]] := Simp[(e*(d + e*x)^(m - 1)*(f + g*x)^n*(a + c*x^2)^(p + 1))/(c*(p + 1)), x] - Dist[(e*g*n)/(c*(p + 1)), Int[(d + e*x)^(m - 1)*(f + g*x)^(n - 1)*(a + c*x^2)^(p + 1), x], x] /; FreeQ[{a, c, d, e, f, g}, x] && NeQ[e*f - d*g, 0] && EqQ[c*d^2 + a*e^2, 0] && !IntegerQ[p] && EqQ[m + p, 0] && LtQ[p, -1] && GtQ[n, 0]
  868. Int[Times[Power[Plus[Pattern[d, Blank[]], Times[Optional[Pattern[e, Blank[]]], Pattern[x, Blank[]]]], Pattern[m, Blank[]]], Power[Plus[Optional[Pattern[f, Blank[]]], Times[Optional[Pattern[g, Blank[]]], Pattern[x, Blank[]]]], Pattern[n, Blank[]]], Power[Plus[Optional[Pattern[a, Blank[]]], Times[Optional[Pattern[b, Blank[]]], Pattern[x, Blank[]]], Times[Optional[Pattern[c, Blank[]]], Power[Pattern[x, Blank[]], 2]]], Pattern[p, Blank[]]]], Pattern[x, Blank[Symbol]]] := Simp[(e^2*(d + e*x)^(m - 1)*(f + g*x)^(n + 1)*(a + b*x + c*x^2)^(p + 1))/((p + 1)*(c*e*f + c*d*g - b*e*g)), x] + Dist[(e^2*g*(m - n - 2))/((p + 1)*(c*e*f + c*d*g - b*e*g)), Int[(d + e*x)^(m - 1)*(f + g*x)^n*(a + b*x + c*x^2)^(p + 1), x], x] /; FreeQ[{a, b, c, d, e, f, g, n}, x] && NeQ[e*f - d*g, 0] && NeQ[b^2 - 4*a*c, 0] && EqQ[c*d^2 - b*d*e + a*e^2, 0] && !IntegerQ[p] && EqQ[m + p, 0] && LtQ[p, -1] && RationalQ[n]
  869. Int[Times[Power[Plus[Pattern[d, Blank[]], Times[Optional[Pattern[e, Blank[]]], Pattern[x, Blank[]]]], Pattern[m, Blank[]]], Power[Plus[Optional[Pattern[f, Blank[]]], Times[Optional[Pattern[g, Blank[]]], Pattern[x, Blank[]]]], Pattern[n, Blank[]]], Power[Plus[Pattern[a, Blank[]], Times[Optional[Pattern[c, Blank[]]], Power[Pattern[x, Blank[]], 2]]], Pattern[p, Blank[]]]], Pattern[x, Blank[Symbol]]] := Simp[(e^2*(d + e*x)^(m - 1)*(f + g*x)^(n + 1)*(a + c*x^2)^(p + 1))/(c*(p + 1)*(e*f + d*g)), x] + Dist[(e^2*g*(m - n - 2))/(c*(p + 1)*(e*f + d*g)), Int[(d + e*x)^(m - 1)*(f + g*x)^n*(a + c*x^2)^(p + 1), x], x] /; FreeQ[{a, c, d, e, f, g, n}, x] && NeQ[e*f - d*g, 0] && EqQ[c*d^2 + a*e^2, 0] && !IntegerQ[p] && EqQ[m + p, 0] && LtQ[p, -1] && RationalQ[n]
  870. Int[Times[Power[Plus[Pattern[d, Blank[]], Times[Optional[Pattern[e, Blank[]]], Pattern[x, Blank[]]]], Pattern[m, Blank[]]], Power[Plus[Optional[Pattern[f, Blank[]]], Times[Optional[Pattern[g, Blank[]]], Pattern[x, Blank[]]]], Pattern[n, Blank[]]], Power[Plus[Optional[Pattern[a, Blank[]]], Times[Optional[Pattern[b, Blank[]]], Pattern[x, Blank[]]], Times[Optional[Pattern[c, Blank[]]], Power[Pattern[x, Blank[]], 2]]], Pattern[p, Blank[]]]], Pattern[x, Blank[Symbol]]] := -Simp[(e*(d + e*x)^(m - 1)*(f + g*x)^n*(a + b*x + c*x^2)^(p + 1))/(c*(m - n - 1)), x] - Dist[(n*(c*e*f + c*d*g - b*e*g))/(c*e*(m - n - 1)), Int[(d + e*x)^m*(f + g*x)^(n - 1)*(a + b*x + c*x^2)^p, x], x] /; FreeQ[{a, b, c, d, e, f, g, m, p}, x] && NeQ[e*f - d*g, 0] && NeQ[b^2 - 4*a*c, 0] && EqQ[c*d^2 - b*d*e + a*e^2, 0] && !IntegerQ[p] && EqQ[m + p, 0] && GtQ[n, 0] && NeQ[m - n - 1, 0] && (IntegerQ[2*p] || IntegerQ[n])
  871. Int[Times[Power[Plus[Pattern[d, Blank[]], Times[Optional[Pattern[e, Blank[]]], Pattern[x, Blank[]]]], Pattern[m, Blank[]]], Power[Plus[Optional[Pattern[f, Blank[]]], Times[Optional[Pattern[g, Blank[]]], Pattern[x, Blank[]]]], Pattern[n, Blank[]]], Power[Plus[Pattern[a, Blank[]], Times[Optional[Pattern[c, Blank[]]], Power[Pattern[x, Blank[]], 2]]], Pattern[p, Blank[]]]], Pattern[x, Blank[Symbol]]] := -Simp[(e*(d + e*x)^(m - 1)*(f + g*x)^n*(a + c*x^2)^(p + 1))/(c*(m - n - 1)), x] - Dist[(n*(e*f + d*g))/(e*(m - n - 1)), Int[(d + e*x)^m*(f + g*x)^(n - 1)*(a + c*x^2)^p, x], x] /; FreeQ[{a, c, d, e, f, g, m, p}, x] && NeQ[e*f - d*g, 0] && EqQ[c*d^2 + a*e^2, 0] && !IntegerQ[p] && EqQ[m + p, 0] && GtQ[n, 0] && NeQ[m - n - 1, 0] && (IntegerQ[2*p] || IntegerQ[n])
  872. Int[Times[Power[Plus[Pattern[d, Blank[]], Times[Optional[Pattern[e, Blank[]]], Pattern[x, Blank[]]]], Pattern[m, Blank[]]], Power[Plus[Optional[Pattern[f, Blank[]]], Times[Optional[Pattern[g, Blank[]]], Pattern[x, Blank[]]]], Pattern[n, Blank[]]], Power[Plus[Optional[Pattern[a, Blank[]]], Times[Optional[Pattern[b, Blank[]]], Pattern[x, Blank[]]], Times[Optional[Pattern[c, Blank[]]], Power[Pattern[x, Blank[]], 2]]], Pattern[p, Blank[]]]], Pattern[x, Blank[Symbol]]] := -Simp[(e^2*(d + e*x)^(m - 1)*(f + g*x)^(n + 1)*(a + b*x + c*x^2)^(p + 1))/((n + 1)*(c*e*f + c*d*g - b*e*g)), x] - Dist[(c*e*(m - n - 2))/((n + 1)*(c*e*f + c*d*g - b*e*g)), Int[(d + e*x)^m*(f + g*x)^(n + 1)*(a + b*x + c*x^2)^p, x], x] /; FreeQ[{a, b, c, d, e, f, g, m, p}, x] && NeQ[e*f - d*g, 0] && NeQ[b^2 - 4*a*c, 0] && EqQ[c*d^2 - b*d*e + a*e^2, 0] && !IntegerQ[p] && EqQ[m + p, 0] && LtQ[n, -1] && IntegerQ[2*p]
  873. Int[Times[Power[Plus[Pattern[d, Blank[]], Times[Optional[Pattern[e, Blank[]]], Pattern[x, Blank[]]]], Pattern[m, Blank[]]], Power[Plus[Optional[Pattern[f, Blank[]]], Times[Optional[Pattern[g, Blank[]]], Pattern[x, Blank[]]]], Pattern[n, Blank[]]], Power[Plus[Pattern[a, Blank[]], Times[Optional[Pattern[c, Blank[]]], Power[Pattern[x, Blank[]], 2]]], Pattern[p, Blank[]]]], Pattern[x, Blank[Symbol]]] := -Simp[(e^2*(d + e*x)^(m - 1)*(f + g*x)^(n + 1)*(a + c*x^2)^(p + 1))/((n + 1)*(c*e*f + c*d*g)), x] - Dist[(e*(m - n - 2))/((n + 1)*(e*f + d*g)), Int[(d + e*x)^m*(f + g*x)^(n + 1)*(a + c*x^2)^p, x], x] /; FreeQ[{a, c, d, e, f, g, m, p}, x] && NeQ[e*f - d*g, 0] && EqQ[c*d^2 + a*e^2, 0] && !IntegerQ[p] && EqQ[m + p, 0] && LtQ[n, -1] && IntegerQ[2*p]
  874. Int[Times[Power[Plus[Pattern[d, Blank[]], Times[Optional[Pattern[e, Blank[]]], Pattern[x, Blank[]]]], Rational[1, 2]], Power[Plus[Optional[Pattern[f, Blank[]]], Times[Optional[Pattern[g, Blank[]]], Pattern[x, Blank[]]]], -1], Power[Plus[Optional[Pattern[a, Blank[]]], Times[Optional[Pattern[b, Blank[]]], Pattern[x, Blank[]]], Times[Optional[Pattern[c, Blank[]]], Power[Pattern[x, Blank[]], 2]]], Rational[-1, 2]]], Pattern[x, Blank[Symbol]]] := Dist[2*e^2, Subst[Int[1/(c*(e*f + d*g) - b*e*g + e^2*g*x^2), x], x, Sqrt[a + b*x + c*x^2]/Sqrt[d + e*x]], x] /; FreeQ[{a, b, c, d, e, f, g}, x] && NeQ[e*f - d*g, 0] && NeQ[b^2 - 4*a*c, 0] && EqQ[c*d^2 - b*d*e + a*e^2, 0]
  875. Int[Times[Power[Plus[Pattern[d, Blank[]], Times[Optional[Pattern[e, Blank[]]], Pattern[x, Blank[]]]], Rational[1, 2]], Power[Plus[Optional[Pattern[f, Blank[]]], Times[Optional[Pattern[g, Blank[]]], Pattern[x, Blank[]]]], -1], Power[Plus[Pattern[a, Blank[]], Times[Optional[Pattern[c, Blank[]]], Power[Pattern[x, Blank[]], 2]]], Rational[-1, 2]]], Pattern[x, Blank[Symbol]]] := Dist[2*e^2, Subst[Int[1/(c*(e*f + d*g) + e^2*g*x^2), x], x, Sqrt[a + c*x^2]/Sqrt[d + e*x]], x] /; FreeQ[{a, c, d, e, f, g}, x] && NeQ[e*f - d*g, 0] && EqQ[c*d^2 + a*e^2, 0]
  876. Int[Times[Power[Plus[Pattern[d, Blank[]], Times[Optional[Pattern[e, Blank[]]], Pattern[x, Blank[]]]], Pattern[m, Blank[]]], Power[Plus[Optional[Pattern[f, Blank[]]], Times[Optional[Pattern[g, Blank[]]], Pattern[x, Blank[]]]], Pattern[n, Blank[]]], Power[Plus[Optional[Pattern[a, Blank[]]], Times[Optional[Pattern[b, Blank[]]], Pattern[x, Blank[]]], Times[Optional[Pattern[c, Blank[]]], Power[Pattern[x, Blank[]], 2]]], Pattern[p, Blank[]]]], Pattern[x, Blank[Symbol]]] := Simp[(e^2*(d + e*x)^(m - 2)*(f + g*x)^(n + 1)*(a + b*x + c*x^2)^(p + 1))/(c*g*(n + p + 2)), x] /; FreeQ[{a, b, c, d, e, f, g, m, n, p}, x] && NeQ[e*f - d*g, 0] && NeQ[b^2 - 4*a*c, 0] && EqQ[c*d^2 - b*d*e + a*e^2, 0] && !IntegerQ[p] && EqQ[m + p - 1, 0] && EqQ[b*e*g*(n + 1) + c*e*f*(p + 1) - c*d*g*(2*n + p + 3), 0] && NeQ[n + p + 2, 0]
  877. Int[Times[Power[Plus[Pattern[d, Blank[]], Times[Optional[Pattern[e, Blank[]]], Pattern[x, Blank[]]]], Pattern[m, Blank[]]], Power[Plus[Optional[Pattern[f, Blank[]]], Times[Optional[Pattern[g, Blank[]]], Pattern[x, Blank[]]]], Pattern[n, Blank[]]], Power[Plus[Pattern[a, Blank[]], Times[Optional[Pattern[c, Blank[]]], Power[Pattern[x, Blank[]], 2]]], Pattern[p, Blank[]]]], Pattern[x, Blank[Symbol]]] := Simp[(e^2*(d + e*x)^(m - 2)*(f + g*x)^(n + 1)*(a + c*x^2)^(p + 1))/(c*g*(n + p + 2)), x] /; FreeQ[{a, c, d, e, f, g, m, n, p}, x] && NeQ[e*f - d*g, 0] && EqQ[c*d^2 + a*e^2, 0] && !IntegerQ[p] && EqQ[m + p - 1, 0] && EqQ[e*f*(p + 1) - d*g*(2*n + p + 3), 0] && NeQ[n + p + 2, 0]
  878. Int[Times[Power[Plus[Pattern[d, Blank[]], Times[Optional[Pattern[e, Blank[]]], Pattern[x, Blank[]]]], Pattern[m, Blank[]]], Power[Plus[Optional[Pattern[f, Blank[]]], Times[Optional[Pattern[g, Blank[]]], Pattern[x, Blank[]]]], Pattern[n, Blank[]]], Power[Plus[Optional[Pattern[a, Blank[]]], Times[Optional[Pattern[b, Blank[]]], Pattern[x, Blank[]]], Times[Optional[Pattern[c, Blank[]]], Power[Pattern[x, Blank[]], 2]]], Pattern[p, Blank[]]]], Pattern[x, Blank[Symbol]]] := Simp[(e^2*(e*f - d*g)*(d + e*x)^(m - 2)*(f + g*x)^(n + 1)*(a + b*x + c*x^2)^(p + 1))/(g*(n + 1)*(c*e*f + c*d*g - b*e*g)), x] - Dist[(e*(b*e*g*(n + 1) + c*e*f*(p + 1) - c*d*g*(2*n + p + 3)))/(g*(n + 1)*(c*e*f + c*d*g - b*e*g)), Int[(d + e*x)^(m - 1)*(f + g*x)^(n + 1)*(a + b*x + c*x^2)^p, x], x] /; FreeQ[{a, b, c, d, e, f, g, m, p}, x] && NeQ[e*f - d*g, 0] && NeQ[b^2 - 4*a*c, 0] && EqQ[c*d^2 - b*d*e + a*e^2, 0] && !IntegerQ[p] && EqQ[m + p - 1, 0] && LtQ[n, -1] && IntegerQ[2*p]
  879. Int[Times[Power[Plus[Pattern[d, Blank[]], Times[Optional[Pattern[e, Blank[]]], Pattern[x, Blank[]]]], Pattern[m, Blank[]]], Power[Plus[Optional[Pattern[f, Blank[]]], Times[Optional[Pattern[g, Blank[]]], Pattern[x, Blank[]]]], Pattern[n, Blank[]]], Power[Plus[Pattern[a, Blank[]], Times[Optional[Pattern[c, Blank[]]], Power[Pattern[x, Blank[]], 2]]], Pattern[p, Blank[]]]], Pattern[x, Blank[Symbol]]] := Simp[(e^2*(e*f - d*g)*(d + e*x)^(m - 2)*(f + g*x)^(n + 1)*(a + c*x^2)^(p + 1))/(c*g*(n + 1)*(e*f + d*g)), x] - Dist[(e*(e*f*(p + 1) - d*g*(2*n + p + 3)))/(g*(n + 1)*(e*f + d*g)), Int[(d + e*x)^(m - 1)*(f + g*x)^(n + 1)*(a + c*x^2)^p, x], x] /; FreeQ[{a, c, d, e, f, g, m, p}, x] && NeQ[e*f - d*g, 0] && EqQ[c*d^2 + a*e^2, 0] && !IntegerQ[p] && EqQ[m + p - 1, 0] && LtQ[n, -1] && IntegerQ[2*p]
  880. Int[Times[Power[Plus[Pattern[d, Blank[]], Times[Optional[Pattern[e, Blank[]]], Pattern[x, Blank[]]]], Pattern[m, Blank[]]], Power[Plus[Optional[Pattern[f, Blank[]]], Times[Optional[Pattern[g, Blank[]]], Pattern[x, Blank[]]]], Pattern[n, Blank[]]], Power[Plus[Optional[Pattern[a, Blank[]]], Times[Optional[Pattern[b, Blank[]]], Pattern[x, Blank[]]], Times[Optional[Pattern[c, Blank[]]], Power[Pattern[x, Blank[]], 2]]], Pattern[p, Blank[]]]], Pattern[x, Blank[Symbol]]] := Simp[(e^2*(d + e*x)^(m - 2)*(f + g*x)^(n + 1)*(a + b*x + c*x^2)^(p + 1))/(c*g*(n + p + 2)), x] - Dist[(b*e*g*(n + 1) + c*e*f*(p + 1) - c*d*g*(2*n + p + 3))/(c*g*(n + p + 2)), Int[(d + e*x)^(m - 1)*(f + g*x)^n*(a + b*x + c*x^2)^p, x], x] /; FreeQ[{a, b, c, d, e, f, g, m, n, p}, x] && NeQ[e*f - d*g, 0] && NeQ[b^2 - 4*a*c, 0] && EqQ[c*d^2 - b*d*e + a*e^2, 0] && !IntegerQ[p] && EqQ[m + p - 1, 0] && !LtQ[n, -1] && IntegerQ[2*p]
  881. Int[Times[Power[Plus[Pattern[d, Blank[]], Times[Optional[Pattern[e, Blank[]]], Pattern[x, Blank[]]]], Pattern[m, Blank[]]], Power[Plus[Optional[Pattern[f, Blank[]]], Times[Optional[Pattern[g, Blank[]]], Pattern[x, Blank[]]]], Pattern[n, Blank[]]], Power[Plus[Pattern[a, Blank[]], Times[Optional[Pattern[c, Blank[]]], Power[Pattern[x, Blank[]], 2]]], Pattern[p, Blank[]]]], Pattern[x, Blank[Symbol]]] := Simp[(e^2*(d + e*x)^(m - 2)*(f + g*x)^(n + 1)*(a + c*x^2)^(p + 1))/(c*g*(n + p + 2)), x] - Dist[(e*f*(p + 1) - d*g*(2*n + p + 3))/(g*(n + p + 2)), Int[(d + e*x)^(m - 1)*(f + g*x)^n*(a + c*x^2)^p, x], x] /; FreeQ[{a, c, d, e, f, g, m, n, p}, x] && NeQ[e*f - d*g, 0] && EqQ[c*d^2 + a*e^2, 0] && !IntegerQ[p] && EqQ[m + p - 1, 0] && !LtQ[n, -1] && IntegerQ[2*p]
  882. Int[Times[Power[Plus[Pattern[d, Blank[]], Times[Optional[Pattern[e, Blank[]]], Pattern[x, Blank[]]]], Pattern[m, Blank[]]], Power[Plus[Optional[Pattern[f, Blank[]]], Times[Optional[Pattern[g, Blank[]]], Pattern[x, Blank[]]]], Pattern[n, Blank[]]], Power[Plus[Optional[Pattern[a, Blank[]]], Times[Optional[Pattern[b, Blank[]]], Pattern[x, Blank[]]], Times[Optional[Pattern[c, Blank[]]], Power[Pattern[x, Blank[]], 2]]], Pattern[p, Blank[]]]], Pattern[x, Blank[Symbol]]] := Int[ExpandIntegrand[(d + e*x)^m*(f + g*x)^n*(a + b*x + c*x^2)^p, x], x] /; FreeQ[{a, b, c, d, e, f, g, n, p}, x] && NeQ[e*f - d*g, 0] && NeQ[b^2 - 4*a*c, 0] && EqQ[c*d^2 - b*d*e + a*e^2, 0] && !IntegerQ[p] && ILtQ[m, 0] && (ILtQ[n, 0] || (IGtQ[n, 0] && ILtQ[p + 1/2, 0])) && !IGtQ[n, 0]
  883. Int[Times[Power[Plus[Pattern[d, Blank[]], Times[Optional[Pattern[e, Blank[]]], Pattern[x, Blank[]]]], Pattern[m, Blank[]]], Power[Plus[Optional[Pattern[f, Blank[]]], Times[Optional[Pattern[g, Blank[]]], Pattern[x, Blank[]]]], Pattern[n, Blank[]]], Power[Plus[Pattern[a, Blank[]], Times[Optional[Pattern[c, Blank[]]], Power[Pattern[x, Blank[]], 2]]], Pattern[p, Blank[]]]], Pattern[x, Blank[Symbol]]] := Int[ExpandIntegrand[1/Sqrt[a + c*x^2], (d + e*x)^m*(f + g*x)^n*(a + c*x^2)^(p + 1/2), x], x] /; FreeQ[{a, c, d, e, f, g, n, p}, x] && NeQ[e*f - d*g, 0] && EqQ[c*d^2 + a*e^2, 0] && IntegerQ[p - 1/2] && ILtQ[m, 0] && ILtQ[n, 0] && !IGtQ[n, 0]
  884. Int[Times[Power[Plus[Pattern[d, Blank[]], Times[Optional[Pattern[e, Blank[]]], Pattern[x, Blank[]]]], Pattern[m, Blank[]]], Power[Plus[Optional[Pattern[f, Blank[]]], Times[Optional[Pattern[g, Blank[]]], Pattern[x, Blank[]]]], Pattern[n, Blank[]]], Power[Plus[Pattern[a, Blank[]], Times[Optional[Pattern[c, Blank[]]], Power[Pattern[x, Blank[]], 2]]], Pattern[p, Blank[]]]], Pattern[x, Blank[Symbol]]] := Int[ExpandIntegrand[(d + e*x)^m*(f + g*x)^n*(a + c*x^2)^p, x], x] /; FreeQ[{a, c, d, e, f, g, n, p}, x] && NeQ[e*f - d*g, 0] && EqQ[c*d^2 + a*e^2, 0] && !IntegerQ[p] && ILtQ[m, 0] && (ILtQ[n, 0] || (IGtQ[n, 0] && ILtQ[p + 1/2, 0])) && !IGtQ[n, 0]
  885. Int[Times[Power[Plus[Optional[Pattern[d, Blank[]]], Times[Optional[Pattern[e, Blank[]]], Pattern[x, Blank[]]]], Optional[Pattern[m, Blank[]]]], Power[Plus[Optional[Pattern[f, Blank[]]], Times[Optional[Pattern[g, Blank[]]], Pattern[x, Blank[]]]], Pattern[n, Blank[]]], Power[Plus[Optional[Pattern[a, Blank[]]], Times[Optional[Pattern[b, Blank[]]], Pattern[x, Blank[]]], Times[Optional[Pattern[c, Blank[]]], Power[Pattern[x, Blank[]], 2]]], Pattern[p, Blank[]]]], Pattern[x, Blank[Symbol]]] := With[{Q = PolynomialQuotient[(f + g*x)^n, a*e + c*d*x, x], h = PolynomialRemainder[(f + g*x)^n, a*e + c*d*x, x]}, Simp[(h*(2*c*d - b*e)*(d + e*x)^m*(a + b*x + c*x^2)^(p + 1))/(e*(p + 1)*(b^2 - 4*a*c)), x] + Dist[1/((p + 1)*(b^2 - 4*a*c)), Int[(d + e*x)^(m - 1)*(a + b*x + c*x^2)^(p + 1)*ExpandToSum[d*e*(p + 1)*(b^2 - 4*a*c)*Q - h*(2*c*d - b*e)*(m + 2*p + 2), x], x], x]] /; FreeQ[{a, b, c, d, e, f, g}, x] && NeQ[e*f - d*g, 0] && NeQ[b^2 - 4*a*c, 0] && EqQ[c*d^2 - b*d*e + a*e^2, 0] && ILtQ[p + 1/2, 0] && IGtQ[m, 0] && IGtQ[n, 0] && !IGtQ[n, 0]
  886. Int[Times[Power[Plus[Optional[Pattern[d, Blank[]]], Times[Optional[Pattern[e, Blank[]]], Pattern[x, Blank[]]]], Optional[Pattern[m, Blank[]]]], Power[Plus[Optional[Pattern[f, Blank[]]], Times[Optional[Pattern[g, Blank[]]], Pattern[x, Blank[]]]], Pattern[n, Blank[]]], Power[Plus[Pattern[a, Blank[]], Times[Optional[Pattern[c, Blank[]]], Power[Pattern[x, Blank[]], 2]]], Pattern[p, Blank[]]]], Pattern[x, Blank[Symbol]]] := With[{Q = PolynomialQuotient[(f + g*x)^n, a*e + c*d*x, x], h = PolynomialRemainder[(f + g*x)^n, a*e + c*d*x, x]}, -Simp[(d*h*(d + e*x)^m*(a + c*x^2)^(p + 1))/(2*a*e*(p + 1)), x] + Dist[d/(2*a*(p + 1)), Int[(d + e*x)^(m - 1)*(a + c*x^2)^(p + 1)*ExpandToSum[2*a*e*(p + 1)*Q + h*(m + 2*p + 2), x], x], x]] /; FreeQ[{a, c, d, e, f, g}, x] && NeQ[e*f - d*g, 0] && EqQ[c*d^2 + a*e^2, 0] && ILtQ[p + 1/2, 0] && IGtQ[m, 0] && IGtQ[n, 0] && !IGtQ[n, 0]
  887. Int[Times[Power[Plus[Optional[Pattern[d, Blank[]]], Times[Optional[Pattern[e, Blank[]]], Pattern[x, Blank[]]]], Pattern[m, Blank[]]], Power[Plus[Optional[Pattern[f, Blank[]]], Times[Optional[Pattern[g, Blank[]]], Pattern[x, Blank[]]]], Pattern[n, Blank[]]], Power[Plus[Optional[Pattern[a, Blank[]]], Times[Optional[Pattern[b, Blank[]]], Pattern[x, Blank[]]], Times[Optional[Pattern[c, Blank[]]], Power[Pattern[x, Blank[]], 2]]], Pattern[p, Blank[]]]], Pattern[x, Blank[Symbol]]] := Int[ExpandIntegrand[(a + b*x + c*x^2)^p, (d + e*x)^m*(f + g*x)^n, x], x] /; FreeQ[{a, b, c, d, e, f, g}, x] && NeQ[e*f - d*g, 0] && NeQ[b^2 - 4*a*c, 0] && EqQ[c*d^2 - b*d*e + a*e^2, 0] && !IntegerQ[p] && EqQ[m + n + 2*p + 1, 0] && ILtQ[m, 0] && ILtQ[n, 0]
  888. Int[Times[Power[Plus[Optional[Pattern[d, Blank[]]], Times[Optional[Pattern[e, Blank[]]], Pattern[x, Blank[]]]], Pattern[m, Blank[]]], Power[Plus[Optional[Pattern[f, Blank[]]], Times[Optional[Pattern[g, Blank[]]], Pattern[x, Blank[]]]], Pattern[n, Blank[]]], Power[Plus[Pattern[a, Blank[]], Times[Optional[Pattern[c, Blank[]]], Power[Pattern[x, Blank[]], 2]]], Pattern[p, Blank[]]]], Pattern[x, Blank[Symbol]]] := Int[ExpandIntegrand[(a + c*x^2)^p, (d + e*x)^m*(f + g*x)^n, x], x] /; FreeQ[{a, c, d, e, f, g}, x] && NeQ[e*f - d*g, 0] && EqQ[c*d^2 + a*e^2, 0] && !IntegerQ[p] && EqQ[m + n + 2*p + 1, 0] && ILtQ[m, 0] && ILtQ[n, 0]
  889. Int[Times[Power[Times[Optional[Pattern[e, Blank[]]], Pattern[x, Blank[]]], Pattern[m, Blank[]]], Power[Plus[Optional[Pattern[f, Blank[]]], Times[Optional[Pattern[g, Blank[]]], Pattern[x, Blank[]]]], Pattern[n, Blank[]]], Power[Plus[Times[Optional[Pattern[b, Blank[]]], Pattern[x, Blank[]]], Times[Optional[Pattern[c, Blank[]]], Power[Pattern[x, Blank[]], 2]]], Pattern[p, Blank[]]]], Pattern[x, Blank[Symbol]]] := Dist[((e*x)^m*(b*x + c*x^2)^p)/(x^(m + p)*(b + c*x)^p), Int[x^(m + p)*(f + g*x)^n*(b + c*x)^p, x], x] /; FreeQ[{b, c, e, f, g, m, n}, x] && !IntegerQ[p] && !IGtQ[n, 0]
  890. Int[Times[Power[Plus[Pattern[d, Blank[]], Times[Optional[Pattern[e, Blank[]]], Pattern[x, Blank[]]]], Pattern[m, Blank[]]], Power[Plus[Optional[Pattern[f, Blank[]]], Times[Optional[Pattern[g, Blank[]]], Pattern[x, Blank[]]]], Pattern[n, Blank[]]], Power[Plus[Pattern[a, Blank[]], Times[Optional[Pattern[c, Blank[]]], Power[Pattern[x, Blank[]], 2]]], Pattern[p, Blank[]]]], Pattern[x, Blank[Symbol]]] := Int[(d + e*x)^(m + p)*(f + g*x)^n*(a/d + (c*x)/e)^p, x] /; FreeQ[{a, c, d, e, f, g, m, n}, x] && NeQ[e*f - d*g, 0] && EqQ[c*d^2 + a*e^2, 0] && !IntegerQ[p] && GtQ[a, 0] && GtQ[d, 0] && !IGtQ[m, 0] && !IGtQ[n, 0]
  891. Int[Times[Power[Plus[Pattern[d, Blank[]], Times[Optional[Pattern[e, Blank[]]], Pattern[x, Blank[]]]], Pattern[m, Blank[]]], Power[Plus[Optional[Pattern[f, Blank[]]], Times[Optional[Pattern[g, Blank[]]], Pattern[x, Blank[]]]], Pattern[n, Blank[]]], Power[Plus[Optional[Pattern[a, Blank[]]], Times[Optional[Pattern[b, Blank[]]], Pattern[x, Blank[]]], Times[Optional[Pattern[c, Blank[]]], Power[Pattern[x, Blank[]], 2]]], Pattern[p, Blank[]]]], Pattern[x, Blank[Symbol]]] := Dist[(a + b*x + c*x^2)^FracPart[p]/((d + e*x)^FracPart[p]*(a/d + (c*x)/e)^FracPart[p]), Int[(d + e*x)^(m + p)*(f + g*x)^n*(a/d + (c*x)/e)^p, x], x] /; FreeQ[{a, b, c, d, e, f, g, m, n}, x] && NeQ[e*f - d*g, 0] && NeQ[b^2 - 4*a*c, 0] && EqQ[c*d^2 - b*d*e + a*e^2, 0] && !IntegerQ[p] && !IGtQ[m, 0] && !IGtQ[n, 0]
  892. Int[Times[Power[Plus[Pattern[d, Blank[]], Times[Optional[Pattern[e, Blank[]]], Pattern[x, Blank[]]]], Pattern[m, Blank[]]], Power[Plus[Optional[Pattern[f, Blank[]]], Times[Optional[Pattern[g, Blank[]]], Pattern[x, Blank[]]]], Pattern[n, Blank[]]], Power[Plus[Pattern[a, Blank[]], Times[Optional[Pattern[c, Blank[]]], Power[Pattern[x, Blank[]], 2]]], Pattern[p, Blank[]]]], Pattern[x, Blank[Symbol]]] := Dist[(a + c*x^2)^FracPart[p]/((d + e*x)^FracPart[p]*(a/d + (c*x)/e)^FracPart[p]), Int[(d + e*x)^(m + p)*(f + g*x)^n*(a/d + (c*x)/e)^p, x], x] /; FreeQ[{a, c, d, e, f, g, m, n}, x] && NeQ[e*f - d*g, 0] && EqQ[c*d^2 + a*e^2, 0] && !IntegerQ[p] && !IGtQ[m, 0] && !IGtQ[n, 0]
  893. Int[Times[Power[Plus[Optional[Pattern[d, Blank[]]], Times[Optional[Pattern[e, Blank[]]], Pattern[x, Blank[]]]], Pattern[m, Blank[]]], Power[Plus[Optional[Pattern[f, Blank[]]], Times[Optional[Pattern[g, Blank[]]], Pattern[x, Blank[]]]], Pattern[n, Blank[]]], Power[Plus[Optional[Pattern[a, Blank[]]], Times[Optional[Pattern[b, Blank[]]], Pattern[x, Blank[]]], Times[Optional[Pattern[c, Blank[]]], Power[Pattern[x, Blank[]], 2]]], Optional[Pattern[p, Blank[]]]]], Pattern[x, Blank[Symbol]]] := Int[ExpandIntegrand[(d + e*x)^m*(f + g*x)^n*(a + b*x + c*x^2)^p, x], x] /; FreeQ[{a, b, c, d, e, f, g}, x] && NeQ[e*f - d*g, 0] && NeQ[b^2 - 4*a*c, 0] && NeQ[c*d^2 - b*d*e + a*e^2, 0] && IntegerQ[p] && ((EqQ[p, 1] && IntegersQ[m, n]) || (ILtQ[m, 0] && ILtQ[n, 0]))
  894. Int[Times[Power[Plus[Optional[Pattern[d, Blank[]]], Times[Optional[Pattern[e, Blank[]]], Pattern[x, Blank[]]]], Pattern[m, Blank[]]], Power[Plus[Optional[Pattern[f, Blank[]]], Times[Optional[Pattern[g, Blank[]]], Pattern[x, Blank[]]]], Pattern[n, Blank[]]], Power[Plus[Pattern[a, Blank[]], Times[Optional[Pattern[c, Blank[]]], Power[Pattern[x, Blank[]], 2]]], Optional[Pattern[p, Blank[]]]]], Pattern[x, Blank[Symbol]]] := Int[ExpandIntegrand[(d + e*x)^m*(f + g*x)^n*(a + c*x^2)^p, x], x] /; FreeQ[{a, c, d, e, f, g}, x] && NeQ[e*f - d*g, 0] && NeQ[c*d^2 + a*e^2, 0] && IntegerQ[p] && ((EqQ[p, 1] && IntegersQ[m, n]) || (ILtQ[m, 0] && ILtQ[n, 0]))
  895. Int[Times[Power[Plus[Optional[Pattern[d, Blank[]]], Times[Optional[Pattern[e, Blank[]]], Pattern[x, Blank[]]]], -1], Power[Plus[Optional[Pattern[f, Blank[]]], Times[Optional[Pattern[g, Blank[]]], Pattern[x, Blank[]]]], -1], Power[Plus[Optional[Pattern[a, Blank[]]], Times[Optional[Pattern[b, Blank[]]], Pattern[x, Blank[]]], Times[Optional[Pattern[c, Blank[]]], Power[Pattern[x, Blank[]], 2]]], Pattern[p, Blank[]]]], Pattern[x, Blank[Symbol]]] := Dist[(c*d^2 - b*d*e + a*e^2)/(e*(e*f - d*g)), Int[(a + b*x + c*x^2)^(p - 1)/(d + e*x), x], x] - Dist[1/(e*(e*f - d*g)), Int[(Simp[c*d*f - b*e*f + a*e*g - c*(e*f - d*g)*x, x]*(a + b*x + c*x^2)^(p - 1))/(f + g*x), x], x] /; FreeQ[{a, b, c, d, e, f, g}, x] && NeQ[e*f - d*g, 0] && NeQ[b^2 - 4*a*c, 0] && NeQ[c*d^2 - b*d*e + a*e^2, 0] && FractionQ[p] && GtQ[p, 0]
  896. Int[Times[Power[Plus[Optional[Pattern[d, Blank[]]], Times[Optional[Pattern[e, Blank[]]], Pattern[x, Blank[]]]], -1], Power[Plus[Optional[Pattern[f, Blank[]]], Times[Optional[Pattern[g, Blank[]]], Pattern[x, Blank[]]]], -1], Power[Plus[Pattern[a, Blank[]], Times[Optional[Pattern[c, Blank[]]], Power[Pattern[x, Blank[]], 2]]], Pattern[p, Blank[]]]], Pattern[x, Blank[Symbol]]] := Dist[(c*d^2 + a*e^2)/(e*(e*f - d*g)), Int[(a + c*x^2)^(p - 1)/(d + e*x), x], x] - Dist[1/(e*(e*f - d*g)), Int[(Simp[c*d*f + a*e*g - c*(e*f - d*g)*x, x]*(a + c*x^2)^(p - 1))/(f + g*x), x], x] /; FreeQ[{a, c, d, e, f, g}, x] && NeQ[e*f - d*g, 0] && NeQ[c*d^2 + a*e^2, 0] && FractionQ[p] && GtQ[p, 0]
  897. Int[Times[Power[Plus[Optional[Pattern[d, Blank[]]], Times[Optional[Pattern[e, Blank[]]], Pattern[x, Blank[]]]], Pattern[m, Blank[]]], Power[Plus[Optional[Pattern[f, Blank[]]], Times[Optional[Pattern[g, Blank[]]], Pattern[x, Blank[]]]], Pattern[n, Blank[]]], Power[Plus[Optional[Pattern[a, Blank[]]], Times[Optional[Pattern[b, Blank[]]], Pattern[x, Blank[]]], Times[Optional[Pattern[c, Blank[]]], Power[Pattern[x, Blank[]], 2]]], Optional[Pattern[p, Blank[]]]]], Pattern[x, Blank[Symbol]]] := With[{q = Denominator[m]}, Dist[q/e, Subst[Int[x^(q*(m + 1) - 1)*((e*f - d*g)/e + (g*x^q)/e)^n*((c*d^2 - b*d*e + a*e^2)/e^2 - ((2*c*d - b*e)*x^q)/e^2 + (c*x^(2*q))/e^2)^p, x], x, (d + e*x)^(1/q)], x]] /; FreeQ[{a, b, c, d, e, f, g}, x] && NeQ[e*f - d*g, 0] && NeQ[b^2 - 4*a*c, 0] && NeQ[c*d^2 - b*d*e + a*e^2, 0] && IntegersQ[n, p] && FractionQ[m]
  898. Int[Times[Power[Plus[Optional[Pattern[d, Blank[]]], Times[Optional[Pattern[e, Blank[]]], Pattern[x, Blank[]]]], Pattern[m, Blank[]]], Power[Plus[Optional[Pattern[f, Blank[]]], Times[Optional[Pattern[g, Blank[]]], Pattern[x, Blank[]]]], Pattern[n, Blank[]]], Power[Plus[Pattern[a, Blank[]], Times[Optional[Pattern[c, Blank[]]], Power[Pattern[x, Blank[]], 2]]], Optional[Pattern[p, Blank[]]]]], Pattern[x, Blank[Symbol]]] := With[{q = Denominator[m]}, Dist[q/e, Subst[Int[x^(q*(m + 1) - 1)*((e*f - d*g)/e + (g*x^q)/e)^n*((c*d^2 + a*e^2)/e^2 - (2*c*d*x^q)/e^2 + (c*x^(2*q))/e^2)^p, x], x, (d + e*x)^(1/q)], x]] /; FreeQ[{a, c, d, e, f, g}, x] && NeQ[e*f - d*g, 0] && NeQ[c*d^2 + a*e^2, 0] && IntegersQ[n, p] && FractionQ[m]
  899. Int[Times[Power[Plus[Pattern[d, Blank[]], Times[Optional[Pattern[e, Blank[]]], Pattern[x, Blank[]]]], Pattern[m, Blank[]]], Power[Plus[Pattern[f, Blank[]], Times[Optional[Pattern[g, Blank[]]], Pattern[x, Blank[]]]], Pattern[n, Blank[]]], Power[Plus[Optional[Pattern[a, Blank[]]], Times[Optional[Pattern[b, Blank[]]], Pattern[x, Blank[]]], Times[Optional[Pattern[c, Blank[]]], Power[Pattern[x, Blank[]], 2]]], Optional[Pattern[p, Blank[]]]]], Pattern[x, Blank[Symbol]]] := Int[(d*f + e*g*x^2)^m*(a + b*x + c*x^2)^p, x] /; FreeQ[{a, b, c, d, e, f, g, m, n, p}, x] && EqQ[m - n, 0] && EqQ[e*f + d*g, 0] && (IntegerQ[m] || (GtQ[d, 0] && GtQ[f, 0]))
  900. Int[Times[Power[Plus[Pattern[d, Blank[]], Times[Optional[Pattern[e, Blank[]]], Pattern[x, Blank[]]]], Pattern[m, Blank[]]], Power[Plus[Pattern[f, Blank[]], Times[Optional[Pattern[g, Blank[]]], Pattern[x, Blank[]]]], Pattern[n, Blank[]]], Power[Plus[Optional[Pattern[a, Blank[]]], Times[Optional[Pattern[c, Blank[]]], Power[Pattern[x, Blank[]], 2]]], Optional[Pattern[p, Blank[]]]]], Pattern[x, Blank[Symbol]]] := Int[(d*f + e*g*x^2)^m*(a + c*x^2)^p, x] /; FreeQ[{a, c, d, e, f, g, m, n, p}, x] && EqQ[m - n, 0] && EqQ[e*f + d*g, 0] && (IntegerQ[m] || (GtQ[d, 0] && GtQ[f, 0]))
  901. Int[Times[Power[Plus[Pattern[d, Blank[]], Times[Optional[Pattern[e, Blank[]]], Pattern[x, Blank[]]]], Pattern[m, Blank[]]], Power[Plus[Pattern[f, Blank[]], Times[Optional[Pattern[g, Blank[]]], Pattern[x, Blank[]]]], Pattern[n, Blank[]]], Power[Plus[Optional[Pattern[a, Blank[]]], Times[Optional[Pattern[b, Blank[]]], Pattern[x, Blank[]]], Times[Optional[Pattern[c, Blank[]]], Power[Pattern[x, Blank[]], 2]]], Optional[Pattern[p, Blank[]]]]], Pattern[x, Blank[Symbol]]] := Dist[((d + e*x)^FracPart[m]*(f + g*x)^FracPart[m])/(d*f + e*g*x^2)^FracPart[m], Int[(d*f + e*g*x^2)^m*(a + b*x + c*x^2)^p, x], x] /; FreeQ[{a, b, c, d, e, f, g, m, n, p}, x] && EqQ[m - n, 0] && EqQ[e*f + d*g, 0]
  902. Int[Times[Power[Plus[Pattern[d, Blank[]], Times[Optional[Pattern[e, Blank[]]], Pattern[x, Blank[]]]], Pattern[m, Blank[]]], Power[Plus[Pattern[f, Blank[]], Times[Optional[Pattern[g, Blank[]]], Pattern[x, Blank[]]]], Pattern[n, Blank[]]], Power[Plus[Optional[Pattern[a, Blank[]]], Times[Optional[Pattern[c, Blank[]]], Power[Pattern[x, Blank[]], 2]]], Pattern[p, Blank[]]]], Pattern[x, Blank[Symbol]]] := Dist[((d + e*x)^FracPart[m]*(f + g*x)^FracPart[m])/(d*f + e*g*x^2)^FracPart[m], Int[(d*f + e*g*x^2)^m*(a + c*x^2)^p, x], x] /; FreeQ[{a, c, d, e, f, g, m, n, p}, x] && EqQ[m - n, 0] && EqQ[e*f + d*g, 0]
  903. Int[Times[Power[Plus[Optional[Pattern[d, Blank[]]], Times[Optional[Pattern[e, Blank[]]], Pattern[x, Blank[]]]], Pattern[m, Blank[]]], Power[Plus[Optional[Pattern[f, Blank[]]], Times[Optional[Pattern[g, Blank[]]], Pattern[x, Blank[]]]], Pattern[n, Blank[]]], Power[Plus[Optional[Pattern[a, Blank[]]], Times[Optional[Pattern[b, Blank[]]], Pattern[x, Blank[]]], Times[Optional[Pattern[c, Blank[]]], Power[Pattern[x, Blank[]], 2]]], -1]], Pattern[x, Blank[Symbol]]] := Dist[g/c^2, Int[Simp[2*c*e*f + c*d*g - b*e*g + c*e*g*x, x]*(d + e*x)^(m - 1)*(f + g*x)^(n - 2), x], x] + Dist[1/c^2, Int[(Simp[c^2*d*f^2 - 2*a*c*e*f*g - a*c*d*g^2 + a*b*e*g^2 + (c^2*e*f^2 + 2*c^2*d*f*g - 2*b*c*e*f*g - b*c*d*g^2 + b^2*e*g^2 - a*c*e*g^2)*x, x]*(d + e*x)^(m - 1)*(f + g*x)^(n - 2))/(a + b*x + c*x^2), x], x] /; FreeQ[{a, b, c, d, e, f, g}, x] && NeQ[b^2 - 4*a*c, 0] && NeQ[c*d^2 - b*d*e + a*e^2, 0] && !IntegerQ[m] && !IntegerQ[n] && GtQ[m, 0] && GtQ[n, 1]
  904. Int[Times[Power[Plus[Optional[Pattern[d, Blank[]]], Times[Optional[Pattern[e, Blank[]]], Pattern[x, Blank[]]]], Pattern[m, Blank[]]], Power[Plus[Optional[Pattern[f, Blank[]]], Times[Optional[Pattern[g, Blank[]]], Pattern[x, Blank[]]]], Pattern[n, Blank[]]], Power[Plus[Pattern[a, Blank[]], Times[Optional[Pattern[c, Blank[]]], Power[Pattern[x, Blank[]], 2]]], -1]], Pattern[x, Blank[Symbol]]] := Dist[g/c, Int[Simp[2*e*f + d*g + e*g*x, x]*(d + e*x)^(m - 1)*(f + g*x)^(n - 2), x], x] + Dist[1/c, Int[(Simp[c*d*f^2 - 2*a*e*f*g - a*d*g^2 + (c*e*f^2 + 2*c*d*f*g - a*e*g^2)*x, x]*(d + e*x)^(m - 1)*(f + g*x)^(n - 2))/(a + c*x^2), x], x] /; FreeQ[{a, c, d, e, f, g}, x] && NeQ[c*d^2 + a*e^2, 0] && !IntegerQ[m] && !IntegerQ[n] && GtQ[m, 0] && GtQ[n, 1]
  905. Int[Times[Power[Plus[Optional[Pattern[d, Blank[]]], Times[Optional[Pattern[e, Blank[]]], Pattern[x, Blank[]]]], Pattern[m, Blank[]]], Power[Plus[Optional[Pattern[f, Blank[]]], Times[Optional[Pattern[g, Blank[]]], Pattern[x, Blank[]]]], Pattern[n, Blank[]]], Power[Plus[Optional[Pattern[a, Blank[]]], Times[Optional[Pattern[b, Blank[]]], Pattern[x, Blank[]]], Times[Optional[Pattern[c, Blank[]]], Power[Pattern[x, Blank[]], 2]]], -1]], Pattern[x, Blank[Symbol]]] := Dist[(e*g)/c, Int[(d + e*x)^(m - 1)*(f + g*x)^(n - 1), x], x] + Dist[1/c, Int[(Simp[c*d*f - a*e*g + (c*e*f + c*d*g - b*e*g)*x, x]*(d + e*x)^(m - 1)*(f + g*x)^(n - 1))/(a + b*x + c*x^2), x], x] /; FreeQ[{a, b, c, d, e, f, g}, x] && NeQ[b^2 - 4*a*c, 0] && NeQ[c*d^2 - b*d*e + a*e^2, 0] && !IntegerQ[m] && !IntegerQ[n] && GtQ[m, 0] && GtQ[n, 0]
  906. Int[Times[Power[Plus[Optional[Pattern[d, Blank[]]], Times[Optional[Pattern[e, Blank[]]], Pattern[x, Blank[]]]], Pattern[m, Blank[]]], Power[Plus[Optional[Pattern[f, Blank[]]], Times[Optional[Pattern[g, Blank[]]], Pattern[x, Blank[]]]], Pattern[n, Blank[]]], Power[Plus[Pattern[a, Blank[]], Times[Optional[Pattern[c, Blank[]]], Power[Pattern[x, Blank[]], 2]]], -1]], Pattern[x, Blank[Symbol]]] := Dist[(e*g)/c, Int[(d + e*x)^(m - 1)*(f + g*x)^(n - 1), x], x] + Dist[1/c, Int[(Simp[c*d*f - a*e*g + (c*e*f + c*d*g)*x, x]*(d + e*x)^(m - 1)*(f + g*x)^(n - 1))/(a + c*x^2), x], x] /; FreeQ[{a, c, d, e, f, g}, x] && NeQ[c*d^2 + a*e^2, 0] && !IntegerQ[m] && !IntegerQ[n] && GtQ[m, 0] && GtQ[n, 0]
  907. Int[Times[Power[Plus[Optional[Pattern[d, Blank[]]], Times[Optional[Pattern[e, Blank[]]], Pattern[x, Blank[]]]], Pattern[m, Blank[]]], Power[Plus[Optional[Pattern[f, Blank[]]], Times[Optional[Pattern[g, Blank[]]], Pattern[x, Blank[]]]], Pattern[n, Blank[]]], Power[Plus[Optional[Pattern[a, Blank[]]], Times[Optional[Pattern[b, Blank[]]], Pattern[x, Blank[]]], Times[Optional[Pattern[c, Blank[]]], Power[Pattern[x, Blank[]], 2]]], -1]], Pattern[x, Blank[Symbol]]] := -Dist[(g*(e*f - d*g))/(c*f^2 - b*f*g + a*g^2), Int[(d + e*x)^(m - 1)*(f + g*x)^n, x], x] + Dist[1/(c*f^2 - b*f*g + a*g^2), Int[(Simp[c*d*f - b*d*g + a*e*g + c*(e*f - d*g)*x, x]*(d + e*x)^(m - 1)*(f + g*x)^(n + 1))/(a + b*x + c*x^2), x], x] /; FreeQ[{a, b, c, d, e, f, g}, x] && NeQ[b^2 - 4*a*c, 0] && NeQ[c*d^2 - b*d*e + a*e^2, 0] && !IntegerQ[m] && !IntegerQ[n] && GtQ[m, 0] && LtQ[n, -1]
  908. Int[Times[Power[Plus[Optional[Pattern[d, Blank[]]], Times[Optional[Pattern[e, Blank[]]], Pattern[x, Blank[]]]], Pattern[m, Blank[]]], Power[Plus[Optional[Pattern[f, Blank[]]], Times[Optional[Pattern[g, Blank[]]], Pattern[x, Blank[]]]], Pattern[n, Blank[]]], Power[Plus[Pattern[a, Blank[]], Times[Optional[Pattern[c, Blank[]]], Power[Pattern[x, Blank[]], 2]]], -1]], Pattern[x, Blank[Symbol]]] := -Dist[(g*(e*f - d*g))/(c*f^2 + a*g^2), Int[(d + e*x)^(m - 1)*(f + g*x)^n, x], x] + Dist[1/(c*f^2 + a*g^2), Int[(Simp[c*d*f + a*e*g + c*(e*f - d*g)*x, x]*(d + e*x)^(m - 1)*(f + g*x)^(n + 1))/(a + c*x^2), x], x] /; FreeQ[{a, c, d, e, f, g}, x] && NeQ[c*d^2 + a*e^2, 0] && !IntegerQ[m] && !IntegerQ[n] && GtQ[m, 0] && LtQ[n, -1]
  909. Int[Times[Power[Plus[Optional[Pattern[d, Blank[]]], Times[Optional[Pattern[e, Blank[]]], Pattern[x, Blank[]]]], Pattern[m, Blank[]]], Power[Plus[Optional[Pattern[f, Blank[]]], Times[Optional[Pattern[g, Blank[]]], Pattern[x, Blank[]]]], Rational[-1, 2]], Power[Plus[Optional[Pattern[a, Blank[]]], Times[Optional[Pattern[b, Blank[]]], Pattern[x, Blank[]]], Times[Optional[Pattern[c, Blank[]]], Power[Pattern[x, Blank[]], 2]]], -1]], Pattern[x, Blank[Symbol]]] := Int[ExpandIntegrand[1/(Sqrt[d + e*x]*Sqrt[f + g*x]), (d + e*x)^(m + 1/2)/(a + b*x + c*x^2), x], x] /; FreeQ[{a, b, c, d, e, f, g}, x] && NeQ[b^2 - 4*a*c, 0] && NeQ[c*d^2 - b*d*e + a*e^2, 0] && IGtQ[m + 1/2, 0]
  910. Int[Times[Power[Plus[Optional[Pattern[d, Blank[]]], Times[Optional[Pattern[e, Blank[]]], Pattern[x, Blank[]]]], Pattern[m, Blank[]]], Power[Plus[Optional[Pattern[f, Blank[]]], Times[Optional[Pattern[g, Blank[]]], Pattern[x, Blank[]]]], Rational[-1, 2]], Power[Plus[Optional[Pattern[a, Blank[]]], Times[Optional[Pattern[c, Blank[]]], Power[Pattern[x, Blank[]], 2]]], -1]], Pattern[x, Blank[Symbol]]] := Int[ExpandIntegrand[1/(Sqrt[d + e*x]*Sqrt[f + g*x]), (d + e*x)^(m + 1/2)/(a + c*x^2), x], x] /; FreeQ[{a, c, d, e, f, g}, x] && NeQ[c*d^2 + a*e^2, 0] && IGtQ[m + 1/2, 0]
  911. Int[Times[Power[Plus[Optional[Pattern[d, Blank[]]], Times[Optional[Pattern[e, Blank[]]], Pattern[x, Blank[]]]], Pattern[m, Blank[]]], Power[Plus[Optional[Pattern[f, Blank[]]], Times[Optional[Pattern[g, Blank[]]], Pattern[x, Blank[]]]], Pattern[n, Blank[]]], Power[Plus[Optional[Pattern[a, Blank[]]], Times[Optional[Pattern[b, Blank[]]], Pattern[x, Blank[]]], Times[Optional[Pattern[c, Blank[]]], Power[Pattern[x, Blank[]], 2]]], -1]], Pattern[x, Blank[Symbol]]] := Int[ExpandIntegrand[(d + e*x)^m*(f + g*x)^n, 1/(a + b*x + c*x^2), x], x] /; FreeQ[{a, b, c, d, e, f, g, m, n}, x] && NeQ[b^2 - 4*a*c, 0] && NeQ[c*d^2 - b*d*e + a*e^2, 0] && !IntegerQ[m] && !IntegerQ[n]
  912. Int[Times[Power[Plus[Optional[Pattern[d, Blank[]]], Times[Optional[Pattern[e, Blank[]]], Pattern[x, Blank[]]]], Pattern[m, Blank[]]], Power[Plus[Optional[Pattern[f, Blank[]]], Times[Optional[Pattern[g, Blank[]]], Pattern[x, Blank[]]]], Pattern[n, Blank[]]], Power[Plus[Pattern[a, Blank[]], Times[Optional[Pattern[c, Blank[]]], Power[Pattern[x, Blank[]], 2]]], -1]], Pattern[x, Blank[Symbol]]] := Int[ExpandIntegrand[(d + e*x)^m*(f + g*x)^n, 1/(a + c*x^2), x], x] /; FreeQ[{a, c, d, e, f, g, m, n}, x] && NeQ[c*d^2 + a*e^2, 0] && !IntegerQ[m] && !IntegerQ[n]
  913. Int[Times[Power[Pattern[x, Blank[]], 2], Power[Plus[Optional[Pattern[d, Blank[]]], Times[Optional[Pattern[e, Blank[]]], Pattern[x, Blank[]]]], Optional[Pattern[m, Blank[]]]], Power[Plus[Optional[Pattern[a, Blank[]]], Times[Optional[Pattern[b, Blank[]]], Pattern[x, Blank[]]], Times[Optional[Pattern[c, Blank[]]], Power[Pattern[x, Blank[]], 2]]], Optional[Pattern[p, Blank[]]]]], Pattern[x, Blank[Symbol]]] := Simp[((d + e*x)^(m + 1)*(a + b*x + c*x^2)^(p + 1))/(c*e*(m + 2*p + 3)), x] /; FreeQ[{a, b, c, d, e, m, p}, x] && EqQ[b*e*(m + p + 2) + 2*c*d*(p + 1), 0] && EqQ[b*d*(p + 1) + a*e*(m + 1), 0] && NeQ[m + 2*p + 3, 0]
  914. Int[Times[Power[Pattern[x, Blank[]], 2], Power[Plus[Optional[Pattern[d, Blank[]]], Times[Optional[Pattern[e, Blank[]]], Pattern[x, Blank[]]]], Optional[Pattern[m, Blank[]]]], Power[Plus[Optional[Pattern[a, Blank[]]], Times[Optional[Pattern[c, Blank[]]], Power[Pattern[x, Blank[]], 2]]], Optional[Pattern[p, Blank[]]]]], Pattern[x, Blank[Symbol]]] := Simp[((d + e*x)^(m + 1)*(a + c*x^2)^(p + 1))/(c*e*(m + 2*p + 3)), x] /; FreeQ[{a, c, d, e, m, p}, x] && EqQ[d*(p + 1), 0] && EqQ[a*(m + 1), 0] && NeQ[m + 2*p + 3, 0]
  915. Int[Times[Power[Times[Optional[Pattern[g, Blank[]]], Pattern[x, Blank[]]], Pattern[n, Blank[]]], Power[Plus[Optional[Pattern[d, Blank[]]], Times[Optional[Pattern[e, Blank[]]], Pattern[x, Blank[]]]], Pattern[m, Blank[]]], Power[Plus[Pattern[a, Blank[]], Times[Optional[Pattern[b, Blank[]]], Pattern[x, Blank[]]], Times[Optional[Pattern[c, Blank[]]], Power[Pattern[x, Blank[]], 2]]], Pattern[p, Blank[]]]], Pattern[x, Blank[Symbol]]] := Dist[((d + e*x)^FracPart[p]*(a + b*x + c*x^2)^FracPart[p])/(a*d + c*e*x^3)^FracPart[p], Int[(g*x)^n*(a*d + c*e*x^3)^p, x], x] /; FreeQ[{a, b, c, d, e, g, m, n, p}, x] && EqQ[m - p, 0] && EqQ[b*d + a*e, 0] && EqQ[c*d + b*e, 0]
  916. Int[Times[Power[Plus[Optional[Pattern[d, Blank[]]], Times[Optional[Pattern[e, Blank[]]], Pattern[x, Blank[]]]], Optional[Pattern[m, Blank[]]]], Power[Plus[Optional[Pattern[f, Blank[]]], Times[Optional[Pattern[g, Blank[]]], Pattern[x, Blank[]]]], Rational[1, 2]], Power[Plus[Optional[Pattern[a, Blank[]]], Times[Optional[Pattern[b, Blank[]]], Pattern[x, Blank[]]], Times[Optional[Pattern[c, Blank[]]], Power[Pattern[x, Blank[]], 2]]], Rational[1, 2]]], Pattern[x, Blank[Symbol]]] := Simp[((d + e*x)^(m + 1)*Sqrt[f + g*x]*Sqrt[a + b*x + c*x^2])/(e*(m + 1)), x] - Dist[1/(2*e*(m + 1)), Int[((d + e*x)^(m + 1)*Simp[b*f + a*g + 2*(c*f + b*g)*x + 3*c*g*x^2, x])/(Sqrt[f + g*x]*Sqrt[a + b*x + c*x^2]), x], x] /; FreeQ[{a, b, c, d, e, f, g}, x] && NeQ[e*f - d*g, 0] && NeQ[b^2 - 4*a*c, 0] && NeQ[c*d^2 - b*d*e + a*e^2, 0] && IntegerQ[2*m] && LtQ[m, -1]
  917. Int[Times[Power[Plus[Optional[Pattern[d, Blank[]]], Times[Optional[Pattern[e, Blank[]]], Pattern[x, Blank[]]]], Optional[Pattern[m, Blank[]]]], Power[Plus[Optional[Pattern[f, Blank[]]], Times[Optional[Pattern[g, Blank[]]], Pattern[x, Blank[]]]], Rational[1, 2]], Power[Plus[Pattern[a, Blank[]], Times[Optional[Pattern[c, Blank[]]], Power[Pattern[x, Blank[]], 2]]], Rational[1, 2]]], Pattern[x, Blank[Symbol]]] := Simp[((d + e*x)^(m + 1)*Sqrt[f + g*x]*Sqrt[a + c*x^2])/(e*(m + 1)), x] - Dist[1/(2*e*(m + 1)), Int[((d + e*x)^(m + 1)*Simp[a*g + 2*c*f*x + 3*c*g*x^2, x])/(Sqrt[f + g*x]*Sqrt[a + c*x^2]), x], x] /; FreeQ[{a, c, d, e, f, g}, x] && NeQ[e*f - d*g, 0] && NeQ[c*d^2 + a*e^2, 0] && IntegerQ[2*m] && LtQ[m, -1]
  918. Int[Times[Power[Plus[Optional[Pattern[d, Blank[]]], Times[Optional[Pattern[e, Blank[]]], Pattern[x, Blank[]]]], Optional[Pattern[m, Blank[]]]], Power[Plus[Optional[Pattern[f, Blank[]]], Times[Optional[Pattern[g, Blank[]]], Pattern[x, Blank[]]]], Rational[1, 2]], Power[Plus[Optional[Pattern[a, Blank[]]], Times[Optional[Pattern[b, Blank[]]], Pattern[x, Blank[]]], Times[Optional[Pattern[c, Blank[]]], Power[Pattern[x, Blank[]], 2]]], Rational[1, 2]]], Pattern[x, Blank[Symbol]]] := Simp[(2*(d + e*x)^(m + 1)*Sqrt[f + g*x]*Sqrt[a + b*x + c*x^2])/(e*(2*m + 5)), x] - Dist[1/(e*(2*m + 5)), Int[((d + e*x)^m*Simp[b*d*f - 3*a*e*f + a*d*g + 2*(c*d*f - b*e*f + b*d*g - a*e*g)*x - (c*e*f - 3*c*d*g + b*e*g)*x^2, x])/(Sqrt[f + g*x]*Sqrt[a + b*x + c*x^2]), x], x] /; FreeQ[{a, b, c, d, e, f, g, m}, x] && NeQ[e*f - d*g, 0] && NeQ[b^2 - 4*a*c, 0] && NeQ[c*d^2 - b*d*e + a*e^2, 0] && IntegerQ[2*m] && !LtQ[m, -1]
  919. Int[Times[Power[Plus[Optional[Pattern[d, Blank[]]], Times[Optional[Pattern[e, Blank[]]], Pattern[x, Blank[]]]], Optional[Pattern[m, Blank[]]]], Power[Plus[Optional[Pattern[f, Blank[]]], Times[Optional[Pattern[g, Blank[]]], Pattern[x, Blank[]]]], Rational[1, 2]], Power[Plus[Pattern[a, Blank[]], Times[Optional[Pattern[c, Blank[]]], Power[Pattern[x, Blank[]], 2]]], Rational[1, 2]]], Pattern[x, Blank[Symbol]]] := Simp[(2*(d + e*x)^(m + 1)*Sqrt[f + g*x]*Sqrt[a + c*x^2])/(e*(2*m + 5)), x] + Dist[1/(e*(2*m + 5)), Int[((d + e*x)^m*Simp[3*a*e*f - a*d*g - 2*(c*d*f - a*e*g)*x + (c*e*f - 3*c*d*g)*x^2, x])/(Sqrt[f + g*x]*Sqrt[a + c*x^2]), x], x] /; FreeQ[{a, c, d, e, f, g, m}, x] && NeQ[e*f - d*g, 0] && NeQ[c*d^2 + a*e^2, 0] && IntegerQ[2*m] && !LtQ[m, -1]
  920. Int[Times[Power[Plus[Optional[Pattern[d, Blank[]]], Times[Optional[Pattern[e, Blank[]]], Pattern[x, Blank[]]]], Optional[Pattern[m, Blank[]]]], Power[Plus[Optional[Pattern[f, Blank[]]], Times[Optional[Pattern[g, Blank[]]], Pattern[x, Blank[]]]], Rational[-1, 2]], Power[Plus[Optional[Pattern[a, Blank[]]], Times[Optional[Pattern[b, Blank[]]], Pattern[x, Blank[]]], Times[Optional[Pattern[c, Blank[]]], Power[Pattern[x, Blank[]], 2]]], Rational[1, 2]]], Pattern[x, Blank[Symbol]]] := Simp[(2*(d + e*x)^m*Sqrt[f + g*x]*Sqrt[a + b*x + c*x^2])/(g*(2*m + 3)), x] - Dist[1/(g*(2*m + 3)), Int[((d + e*x)^(m - 1)*Simp[b*d*f + 2*a*(e*f*m - d*g*(m + 1)) + (2*c*d*f - 2*a*e*g + b*(e*f - d*g)*(2*m + 1))*x - (b*e*g + 2*c*(d*g*m - e*f*(m + 1)))*x^2, x])/(Sqrt[f + g*x]*Sqrt[a + b*x + c*x^2]), x], x] /; FreeQ[{a, b, c, d, e, f, g}, x] && NeQ[e*f - d*g, 0] && NeQ[b^2 - 4*a*c, 0] && NeQ[c*d^2 - b*d*e + a*e^2, 0] && IntegerQ[2*m] && GtQ[m, 0]
  921. Int[Times[Power[Plus[Optional[Pattern[d, Blank[]]], Times[Optional[Pattern[e, Blank[]]], Pattern[x, Blank[]]]], Optional[Pattern[m, Blank[]]]], Power[Plus[Optional[Pattern[f, Blank[]]], Times[Optional[Pattern[g, Blank[]]], Pattern[x, Blank[]]]], Rational[-1, 2]], Power[Plus[Pattern[a, Blank[]], Times[Optional[Pattern[c, Blank[]]], Power[Pattern[x, Blank[]], 2]]], Rational[1, 2]]], Pattern[x, Blank[Symbol]]] := Simp[(2*(d + e*x)^m*Sqrt[f + g*x]*Sqrt[a + c*x^2])/(g*(2*m + 3)), x] - Dist[1/(g*(2*m + 3)), Int[((d + e*x)^(m - 1)*Simp[2*a*(e*f*m - d*g*(m + 1)) + (2*c*d*f - 2*a*e*g)*x - (2*c*(d*g*m - e*f*(m + 1)))*x^2, x])/(Sqrt[f + g*x]*Sqrt[a + c*x^2]), x], x] /; FreeQ[{a, c, d, e, f, g}, x] && NeQ[e*f - d*g, 0] && NeQ[c*d^2 + a*e^2, 0] && IntegerQ[2*m] && GtQ[m, 0]
  922. Int[Times[Power[Plus[Optional[Pattern[d, Blank[]]], Times[Optional[Pattern[e, Blank[]]], Pattern[x, Blank[]]]], -1], Power[Plus[Optional[Pattern[f, Blank[]]], Times[Optional[Pattern[g, Blank[]]], Pattern[x, Blank[]]]], Rational[-1, 2]], Power[Plus[Optional[Pattern[a, Blank[]]], Times[Optional[Pattern[b, Blank[]]], Pattern[x, Blank[]]], Times[Optional[Pattern[c, Blank[]]], Power[Pattern[x, Blank[]], 2]]], Rational[1, 2]]], Pattern[x, Blank[Symbol]]] := Dist[(c*d^2 - b*d*e + a*e^2)/e^2, Int[1/((d + e*x)*Sqrt[f + g*x]*Sqrt[a + b*x + c*x^2]), x], x] - Dist[1/e^2, Int[(c*d - b*e - c*e*x)/(Sqrt[f + g*x]*Sqrt[a + b*x + c*x^2]), x], x] /; FreeQ[{a, b, c, d, e, f, g}, x] && NeQ[e*f - d*g, 0] && NeQ[b^2 - 4*a*c, 0] && NeQ[c*d^2 - b*d*e + a*e^2, 0]
  923. Int[Times[Power[Plus[Optional[Pattern[d, Blank[]]], Times[Optional[Pattern[e, Blank[]]], Pattern[x, Blank[]]]], -1], Power[Plus[Optional[Pattern[f, Blank[]]], Times[Optional[Pattern[g, Blank[]]], Pattern[x, Blank[]]]], Rational[-1, 2]], Power[Plus[Pattern[a, Blank[]], Times[Optional[Pattern[c, Blank[]]], Power[Pattern[x, Blank[]], 2]]], Rational[1, 2]]], Pattern[x, Blank[Symbol]]] := Dist[(c*d^2 + a*e^2)/e^2, Int[1/((d + e*x)*Sqrt[f + g*x]*Sqrt[a + c*x^2]), x], x] - Dist[1/e^2, Int[(c*d - c*e*x)/(Sqrt[f + g*x]*Sqrt[a + c*x^2]), x], x] /; FreeQ[{a, c, d, e, f, g}, x] && NeQ[e*f - d*g, 0] && NeQ[c*d^2 + a*e^2, 0]
  924. Int[Times[Power[Plus[Optional[Pattern[d, Blank[]]], Times[Optional[Pattern[e, Blank[]]], Pattern[x, Blank[]]]], Optional[Pattern[m, Blank[]]]], Power[Plus[Optional[Pattern[f, Blank[]]], Times[Optional[Pattern[g, Blank[]]], Pattern[x, Blank[]]]], Rational[-1, 2]], Power[Plus[Optional[Pattern[a, Blank[]]], Times[Optional[Pattern[b, Blank[]]], Pattern[x, Blank[]]], Times[Optional[Pattern[c, Blank[]]], Power[Pattern[x, Blank[]], 2]]], Rational[1, 2]]], Pattern[x, Blank[Symbol]]] := Simp[((d + e*x)^(m + 1)*Sqrt[f + g*x]*Sqrt[a + b*x + c*x^2])/((m + 1)*(e*f - d*g)), x] - Dist[1/(2*(m + 1)*(e*f - d*g)), Int[((d + e*x)^(m + 1)*Simp[b*f + a*g*(2*m + 3) + 2*(c*f + b*g*(m + 2))*x + c*g*(2*m + 5)*x^2, x])/(Sqrt[f + g*x]*Sqrt[a + b*x + c*x^2]), x], x] /; FreeQ[{a, b, c, d, e, f, g}, x] && NeQ[e*f - d*g, 0] && NeQ[b^2 - 4*a*c, 0] && NeQ[c*d^2 - b*d*e + a*e^2, 0] && IntegerQ[2*m] && LtQ[m, -1]
  925. Int[Times[Power[Plus[Optional[Pattern[d, Blank[]]], Times[Optional[Pattern[e, Blank[]]], Pattern[x, Blank[]]]], Optional[Pattern[m, Blank[]]]], Power[Plus[Optional[Pattern[f, Blank[]]], Times[Optional[Pattern[g, Blank[]]], Pattern[x, Blank[]]]], Rational[-1, 2]], Power[Plus[Pattern[a, Blank[]], Times[Optional[Pattern[c, Blank[]]], Power[Pattern[x, Blank[]], 2]]], Rational[1, 2]]], Pattern[x, Blank[Symbol]]] := Simp[((d + e*x)^(m + 1)*Sqrt[f + g*x]*Sqrt[a + c*x^2])/((m + 1)*(e*f - d*g)), x] - Dist[1/(2*(m + 1)*(e*f - d*g)), Int[((d + e*x)^(m + 1)*Simp[a*g*(2*m + 3) + 2*(c*f)*x + c*g*(2*m + 5)*x^2, x])/(Sqrt[f + g*x]*Sqrt[a + c*x^2]), x], x] /; FreeQ[{a, c, d, e, f, g}, x] && NeQ[e*f - d*g, 0] && NeQ[c*d^2 + a*e^2, 0] && IntegerQ[2*m] && LtQ[m, -1]
  926. Int[Times[Power[Plus[Optional[Pattern[d, Blank[]]], Times[Optional[Pattern[e, Blank[]]], Pattern[x, Blank[]]]], Rational[1, 2]], Power[Plus[Optional[Pattern[f, Blank[]]], Times[Optional[Pattern[g, Blank[]]], Pattern[x, Blank[]]]], Rational[-1, 2]], Power[Plus[Optional[Pattern[a, Blank[]]], Times[Optional[Pattern[b, Blank[]]], Pattern[x, Blank[]]], Times[Optional[Pattern[c, Blank[]]], Power[Pattern[x, Blank[]], 2]]], Rational[-1, 2]]], Pattern[x, Blank[Symbol]]] := With[{q = Rt[b^2 - 4*a*c, 2]}, Simp[(Sqrt[2]*Sqrt[2*c*f - g*(b + q)]*Sqrt[b - q + 2*c*x]*(d + e*x)*Sqrt[((e*f - d*g)*(b + q + 2*c*x))/((2*c*f - g*(b + q))*(d + e*x))]*Sqrt[((e*f - d*g)*(2*a + (b + q)*x))/((b*f + q*f - 2*a*g)*(d + e*x))]*EllipticPi[(e*(2*c*f - g*(b + q)))/(g*(2*c*d - e*(b + q))), ArcSin[(Sqrt[2*c*d - e*(b + q)]*Sqrt[f + g*x])/(Sqrt[2*c*f - g*(b + q)]*Sqrt[d + e*x])], ((b*d + q*d - 2*a*e)*(2*c*f - g*(b + q)))/((b*f + q*f - 2*a*g)*(2*c*d - e*(b + q)))])/(g*Sqrt[2*c*d - e*(b + q)]*Sqrt[(2*a*c)/(b + q) + c*x]*Sqrt[a + b*x + c*x^2]), x]] /; FreeQ[{a, b, c, d, e, f, g}, x] && NeQ[e*f - d*g, 0] && NeQ[b^2 - 4*a*c, 0] && NeQ[c*d^2 - b*d*e + a*e^2, 0]
  927. Int[Times[Power[Plus[Optional[Pattern[d, Blank[]]], Times[Optional[Pattern[e, Blank[]]], Pattern[x, Blank[]]]], Rational[1, 2]], Power[Plus[Optional[Pattern[f, Blank[]]], Times[Optional[Pattern[g, Blank[]]], Pattern[x, Blank[]]]], Rational[-1, 2]], Power[Plus[Pattern[a, Blank[]], Times[Optional[Pattern[c, Blank[]]], Power[Pattern[x, Blank[]], 2]]], Rational[-1, 2]]], Pattern[x, Blank[Symbol]]] := With[{q = Rt[-4*a*c, 2]}, Simp[(Sqrt[2]*Sqrt[2*c*f - g*q]*Sqrt[-q + 2*c*x]*(d + e*x)*Sqrt[((e*f - d*g)*(q + 2*c*x))/((2*c*f - g*q)*(d + e*x))]*Sqrt[((e*f - d*g)*(2*a + q*x))/((q*f - 2*a*g)*(d + e*x))]*EllipticPi[(e*(2*c*f - g*q))/(g*(2*c*d - e*q)), ArcSin[(Sqrt[2*c*d - e*q]*Sqrt[f + g*x])/(Sqrt[2*c*f - g*q]*Sqrt[d + e*x])], ((q*d - 2*a*e)*(2*c*f - g*q))/((q*f - 2*a*g)*(2*c*d - e*q))])/(g*Sqrt[2*c*d - e*q]*Sqrt[(2*a*c)/q + c*x]*Sqrt[a + c*x^2]), x]] /; FreeQ[{a, c, d, e, f, g}, x] && NeQ[e*f - d*g, 0] && NeQ[c*d^2 + a*e^2, 0]
  928. Int[Times[Power[Plus[Optional[Pattern[d, Blank[]]], Times[Optional[Pattern[e, Blank[]]], Pattern[x, Blank[]]]], Rational[3, 2]], Power[Plus[Optional[Pattern[f, Blank[]]], Times[Optional[Pattern[g, Blank[]]], Pattern[x, Blank[]]]], Rational[-1, 2]], Power[Plus[Optional[Pattern[a, Blank[]]], Times[Optional[Pattern[b, Blank[]]], Pattern[x, Blank[]]], Times[Optional[Pattern[c, Blank[]]], Power[Pattern[x, Blank[]], 2]]], Rational[-1, 2]]], Pattern[x, Blank[Symbol]]] := Dist[e/g, Int[(Sqrt[d + e*x]*Sqrt[f + g*x])/Sqrt[a + b*x + c*x^2], x], x] - Dist[(e*f - d*g)/g, Int[Sqrt[d + e*x]/(Sqrt[f + g*x]*Sqrt[a + b*x + c*x^2]), x], x] /; FreeQ[{a, b, c, d, e, f, g}, x] && NeQ[e*f - d*g, 0] && NeQ[b^2 - 4*a*c, 0] && NeQ[c*d^2 - b*d*e + a*e^2, 0]
  929. Int[Times[Power[Plus[Optional[Pattern[d, Blank[]]], Times[Optional[Pattern[e, Blank[]]], Pattern[x, Blank[]]]], Rational[3, 2]], Power[Plus[Optional[Pattern[f, Blank[]]], Times[Optional[Pattern[g, Blank[]]], Pattern[x, Blank[]]]], Rational[-1, 2]], Power[Plus[Pattern[a, Blank[]], Times[Optional[Pattern[c, Blank[]]], Power[Pattern[x, Blank[]], 2]]], Rational[-1, 2]]], Pattern[x, Blank[Symbol]]] := Dist[e/g, Int[(Sqrt[d + e*x]*Sqrt[f + g*x])/Sqrt[a + c*x^2], x], x] - Dist[(e*f - d*g)/g, Int[Sqrt[d + e*x]/(Sqrt[f + g*x]*Sqrt[a + c*x^2]), x], x] /; FreeQ[{a, c, d, e, f, g}, x] && NeQ[e*f - d*g, 0] && NeQ[c*d^2 + a*e^2, 0]
  930. Int[Times[Power[Plus[Optional[Pattern[d, Blank[]]], Times[Optional[Pattern[e, Blank[]]], Pattern[x, Blank[]]]], Pattern[m, Blank[]]], Power[Plus[Optional[Pattern[f, Blank[]]], Times[Optional[Pattern[g, Blank[]]], Pattern[x, Blank[]]]], Rational[-1, 2]], Power[Plus[Optional[Pattern[a, Blank[]]], Times[Optional[Pattern[b, Blank[]]], Pattern[x, Blank[]]], Times[Optional[Pattern[c, Blank[]]], Power[Pattern[x, Blank[]], 2]]], Rational[-1, 2]]], Pattern[x, Blank[Symbol]]] := Simp[(2*e^2*(d + e*x)^(m - 2)*Sqrt[f + g*x]*Sqrt[a + b*x + c*x^2])/(c*g*(2*m - 1)), x] - Dist[1/(c*g*(2*m - 1)), Int[((d + e*x)^(m - 3)*Simp[b*d*e^2*f + a*e^2*(d*g + 2*e*f*(m - 2)) - c*d^3*g*(2*m - 1) + e*(e*(2*b*d*g + e*(b*f + a*g)*(2*m - 3)) + c*d*(2*e*f - 3*d*g*(2*m - 1)))*x + 2*e^2*(c*e*f - 3*c*d*g + b*e*g)*(m - 1)*x^2, x])/(Sqrt[f + g*x]*Sqrt[a + b*x + c*x^2]), x], x] /; FreeQ[{a, b, c, d, e, f, g}, x] && NeQ[e*f - d*g, 0] && NeQ[b^2 - 4*a*c, 0] && NeQ[c*d^2 - b*d*e + a*e^2, 0] && IntegerQ[2*m] && GeQ[m, 2]
  931. Int[Times[Power[Plus[Optional[Pattern[d, Blank[]]], Times[Optional[Pattern[e, Blank[]]], Pattern[x, Blank[]]]], Pattern[m, Blank[]]], Power[Plus[Optional[Pattern[f, Blank[]]], Times[Optional[Pattern[g, Blank[]]], Pattern[x, Blank[]]]], Rational[-1, 2]], Power[Plus[Pattern[a, Blank[]], Times[Optional[Pattern[c, Blank[]]], Power[Pattern[x, Blank[]], 2]]], Rational[-1, 2]]], Pattern[x, Blank[Symbol]]] := Simp[(2*e^2*(d + e*x)^(m - 2)*Sqrt[f + g*x]*Sqrt[a + c*x^2])/(c*g*(2*m - 1)), x] - Dist[1/(c*g*(2*m - 1)), Int[((d + e*x)^(m - 3)*Simp[a*e^2*(d*g + 2*e*f*(m - 2)) - c*d^3*g*(2*m - 1) + e*(e*(a*e*g*(2*m - 3)) + c*d*(2*e*f - 3*d*g*(2*m - 1)))*x + 2*e^2*(c*e*f - 3*c*d*g)*(m - 1)*x^2, x])/(Sqrt[f + g*x]*Sqrt[a + c*x^2]), x], x] /; FreeQ[{a, c, d, e, f, g}, x] && NeQ[e*f - d*g, 0] && NeQ[c*d^2 + a*e^2, 0] && IntegerQ[2*m] && GeQ[m, 2]
  932. Int[Times[Power[Plus[Optional[Pattern[d, Blank[]]], Times[Optional[Pattern[e, Blank[]]], Pattern[x, Blank[]]]], -1], Power[Plus[Optional[Pattern[f, Blank[]]], Times[Optional[Pattern[g, Blank[]]], Pattern[x, Blank[]]]], Rational[-1, 2]], Power[Plus[Pattern[a, Blank[]], Times[Optional[Pattern[c, Blank[]]], Power[Pattern[x, Blank[]], 2]]], Rational[-1, 2]]], Pattern[x, Blank[Symbol]]] := With[{q = Rt[-(c/a), 2]}, Dist[1/Sqrt[a], Int[1/((d + e*x)*Sqrt[f + g*x]*Sqrt[1 - q*x]*Sqrt[1 + q*x]), x], x]] /; FreeQ[{a, c, d, e, f, g}, x] && NeQ[e*f - d*g, 0] && NeQ[c*d^2 + a*e^2, 0] && GtQ[a, 0]
  933. Int[Times[Power[Plus[Optional[Pattern[d, Blank[]]], Times[Optional[Pattern[e, Blank[]]], Pattern[x, Blank[]]]], -1], Power[Plus[Optional[Pattern[f, Blank[]]], Times[Optional[Pattern[g, Blank[]]], Pattern[x, Blank[]]]], Rational[-1, 2]], Power[Plus[Pattern[a, Blank[]], Times[Optional[Pattern[c, Blank[]]], Power[Pattern[x, Blank[]], 2]]], Rational[-1, 2]]], Pattern[x, Blank[Symbol]]] := With[{q = Rt[-(c/a), 2]}, Dist[Sqrt[1 + (c*x^2)/a]/Sqrt[a + c*x^2], Int[1/((d + e*x)*Sqrt[f + g*x]*Sqrt[1 - q*x]*Sqrt[1 + q*x]), x], x]] /; FreeQ[{a, c, d, e, f, g}, x] && NeQ[e*f - d*g, 0] && NeQ[c*d^2 + a*e^2, 0] && !GtQ[a, 0]
  934. Int[Times[Power[Plus[Optional[Pattern[d, Blank[]]], Times[Optional[Pattern[e, Blank[]]], Pattern[x, Blank[]]]], -1], Power[Plus[Optional[Pattern[f, Blank[]]], Times[Optional[Pattern[g, Blank[]]], Pattern[x, Blank[]]]], Rational[-1, 2]], Power[Plus[Optional[Pattern[a, Blank[]]], Times[Optional[Pattern[b, Blank[]]], Pattern[x, Blank[]]], Times[Optional[Pattern[c, Blank[]]], Power[Pattern[x, Blank[]], 2]]], Rational[-1, 2]]], Pattern[x, Blank[Symbol]]] := With[{q = Rt[b^2 - 4*a*c, 2]}, Dist[(Sqrt[b - q + 2*c*x]*Sqrt[b + q + 2*c*x])/Sqrt[a + b*x + c*x^2], Int[1/((d + e*x)*Sqrt[f + g*x]*Sqrt[b - q + 2*c*x]*Sqrt[b + q + 2*c*x]), x], x]] /; FreeQ[{a, b, c, d, e, f, g}, x] && NeQ[e*f - d*g, 0] && NeQ[b^2 - 4*a*c, 0] && NeQ[c*d^2 - b*d*e + a*e^2, 0]
  935. Int[Times[Power[Plus[Optional[Pattern[d, Blank[]]], Times[Optional[Pattern[e, Blank[]]], Pattern[x, Blank[]]]], Rational[-1, 2]], Power[Plus[Optional[Pattern[f, Blank[]]], Times[Optional[Pattern[g, Blank[]]], Pattern[x, Blank[]]]], Rational[-1, 2]], Power[Plus[Optional[Pattern[a, Blank[]]], Times[Optional[Pattern[b, Blank[]]], Pattern[x, Blank[]]], Times[Optional[Pattern[c, Blank[]]], Power[Pattern[x, Blank[]], 2]]], Rational[-1, 2]]], Pattern[x, Blank[Symbol]]] := Dist[(-2*(d + e*x)*Sqrt[((e*f - d*g)^2*(a + b*x + c*x^2))/((c*f^2 - b*f*g + a*g^2)*(d + e*x)^2)])/((e*f - d*g)*Sqrt[a + b*x + c*x^2]), Subst[Int[1/Sqrt[1 - ((2*c*d*f - b*e*f - b*d*g + 2*a*e*g)*x^2)/(c*f^2 - b*f*g + a*g^2) + ((c*d^2 - b*d*e + a*e^2)*x^4)/(c*f^2 - b*f*g + a*g^2)], x], x, Sqrt[f + g*x]/Sqrt[d + e*x]], x] /; FreeQ[{a, b, c, d, e, f, g}, x] && NeQ[e*f - d*g, 0] && NeQ[b^2 - 4*a*c, 0] && NeQ[c*d^2 - b*d*e + a*e^2, 0]
  936. Int[Times[Power[Plus[Optional[Pattern[d, Blank[]]], Times[Optional[Pattern[e, Blank[]]], Pattern[x, Blank[]]]], Rational[-1, 2]], Power[Plus[Optional[Pattern[f, Blank[]]], Times[Optional[Pattern[g, Blank[]]], Pattern[x, Blank[]]]], Rational[-1, 2]], Power[Plus[Pattern[a, Blank[]], Times[Optional[Pattern[c, Blank[]]], Power[Pattern[x, Blank[]], 2]]], Rational[-1, 2]]], Pattern[x, Blank[Symbol]]] := Dist[(-2*(d + e*x)*Sqrt[((e*f - d*g)^2*(a + c*x^2))/((c*f^2 + a*g^2)*(d + e*x)^2)])/((e*f - d*g)*Sqrt[a + c*x^2]), Subst[Int[1/Sqrt[1 - ((2*c*d*f + 2*a*e*g)*x^2)/(c*f^2 + a*g^2) + ((c*d^2 + a*e^2)*x^4)/(c*f^2 + a*g^2)], x], x, Sqrt[f + g*x]/Sqrt[d + e*x]], x] /; FreeQ[{a, c, d, e, f, g}, x] && NeQ[e*f - d*g, 0] && NeQ[c*d^2 + a*e^2, 0]
  937. Int[Times[Power[Plus[Optional[Pattern[d, Blank[]]], Times[Optional[Pattern[e, Blank[]]], Pattern[x, Blank[]]]], Rational[-3, 2]], Power[Plus[Optional[Pattern[f, Blank[]]], Times[Optional[Pattern[g, Blank[]]], Pattern[x, Blank[]]]], Rational[-1, 2]], Power[Plus[Optional[Pattern[a, Blank[]]], Times[Optional[Pattern[b, Blank[]]], Pattern[x, Blank[]]], Times[Optional[Pattern[c, Blank[]]], Power[Pattern[x, Blank[]], 2]]], Rational[-1, 2]]], Pattern[x, Blank[Symbol]]] := -Dist[g/(e*f - d*g), Int[1/(Sqrt[d + e*x]*Sqrt[f + g*x]*Sqrt[a + b*x + c*x^2]), x], x] + Dist[e/(e*f - d*g), Int[Sqrt[f + g*x]/((d + e*x)^(3/2)*Sqrt[a + b*x + c*x^2]), x], x] /; FreeQ[{a, b, c, d, e, f, g}, x] && NeQ[e*f - d*g, 0] && NeQ[b^2 - 4*a*c, 0] && NeQ[c*d^2 - b*d*e + a*e^2, 0]
  938. Int[Times[Power[Plus[Optional[Pattern[d, Blank[]]], Times[Optional[Pattern[e, Blank[]]], Pattern[x, Blank[]]]], Rational[-3, 2]], Power[Plus[Optional[Pattern[f, Blank[]]], Times[Optional[Pattern[g, Blank[]]], Pattern[x, Blank[]]]], Rational[-1, 2]], Power[Plus[Pattern[a, Blank[]], Times[Optional[Pattern[c, Blank[]]], Power[Pattern[x, Blank[]], 2]]], Rational[-1, 2]]], Pattern[x, Blank[Symbol]]] := -Dist[g/(e*f - d*g), Int[1/(Sqrt[d + e*x]*Sqrt[f + g*x]*Sqrt[a + c*x^2]), x], x] + Dist[e/(e*f - d*g), Int[Sqrt[f + g*x]/((d + e*x)^(3/2)*Sqrt[a + c*x^2]), x], x] /; FreeQ[{a, c, d, e, f, g}, x] && NeQ[e*f - d*g, 0] && NeQ[c*d^2 + a*e^2, 0]
  939. Int[Times[Power[Plus[Optional[Pattern[d, Blank[]]], Times[Optional[Pattern[e, Blank[]]], Pattern[x, Blank[]]]], Pattern[m, Blank[]]], Power[Plus[Optional[Pattern[f, Blank[]]], Times[Optional[Pattern[g, Blank[]]], Pattern[x, Blank[]]]], Rational[-1, 2]], Power[Plus[Optional[Pattern[a, Blank[]]], Times[Optional[Pattern[b, Blank[]]], Pattern[x, Blank[]]], Times[Optional[Pattern[c, Blank[]]], Power[Pattern[x, Blank[]], 2]]], Rational[-1, 2]]], Pattern[x, Blank[Symbol]]] := Simp[(e^2*(d + e*x)^(m + 1)*Sqrt[f + g*x]*Sqrt[a + b*x + c*x^2])/((m + 1)*(e*f - d*g)*(c*d^2 - b*d*e + a*e^2)), x] + Dist[1/(2*(m + 1)*(e*f - d*g)*(c*d^2 - b*d*e + a*e^2)), Int[((d + e*x)^(m + 1)*Simp[2*d*(c*e*f - c*d*g + b*e*g)*(m + 1) - e^2*(b*f + a*g)*(2*m + 3) + 2*e*(c*d*g*(m + 1) - e*(c*f + b*g)*(m + 2))*x - c*e^2*g*(2*m + 5)*x^2, x])/(Sqrt[f + g*x]*Sqrt[a + b*x + c*x^2]), x], x] /; FreeQ[{a, b, c, d, e, f, g}, x] && NeQ[e*f - d*g, 0] && NeQ[b^2 - 4*a*c, 0] && NeQ[c*d^2 - b*d*e + a*e^2, 0] && IntegerQ[2*m] && LeQ[m, -2]
  940. Int[Times[Power[Plus[Optional[Pattern[d, Blank[]]], Times[Optional[Pattern[e, Blank[]]], Pattern[x, Blank[]]]], Pattern[m, Blank[]]], Power[Plus[Optional[Pattern[f, Blank[]]], Times[Optional[Pattern[g, Blank[]]], Pattern[x, Blank[]]]], Rational[-1, 2]], Power[Plus[Pattern[a, Blank[]], Times[Optional[Pattern[c, Blank[]]], Power[Pattern[x, Blank[]], 2]]], Rational[-1, 2]]], Pattern[x, Blank[Symbol]]] := Simp[(e^2*(d + e*x)^(m + 1)*Sqrt[f + g*x]*Sqrt[a + c*x^2])/((m + 1)*(e*f - d*g)*(c*d^2 + a*e^2)), x] + Dist[1/(2*(m + 1)*(e*f - d*g)*(c*d^2 + a*e^2)), Int[((d + e*x)^(m + 1)*Simp[2*d*(c*e*f - c*d*g)*(m + 1) - a*e^2*g*(2*m + 3) + 2*e*(c*d*g*(m + 1) - c*e*f*(m + 2))*x - c*e^2*g*(2*m + 5)*x^2, x])/(Sqrt[f + g*x]*Sqrt[a + c*x^2]), x], x] /; FreeQ[{a, c, d, e, f, g}, x] && NeQ[e*f - d*g, 0] && NeQ[c*d^2 + a*e^2, 0] && IntegerQ[2*m] && LeQ[m, -2]
  941. Int[Times[Power[Plus[Optional[Pattern[d, Blank[]]], Times[Optional[Pattern[e, Blank[]]], Pattern[x, Blank[]]]], Pattern[m, Blank[]]], Power[Plus[Optional[Pattern[f, Blank[]]], Times[Optional[Pattern[g, Blank[]]], Pattern[x, Blank[]]]], Rational[1, 2]], Power[Plus[Optional[Pattern[a, Blank[]]], Times[Optional[Pattern[b, Blank[]]], Pattern[x, Blank[]]], Times[Optional[Pattern[c, Blank[]]], Power[Pattern[x, Blank[]], 2]]], Rational[-1, 2]]], Pattern[x, Blank[Symbol]]] := Simp[(2*e*(d + e*x)^(m - 1)*Sqrt[f + g*x]*Sqrt[a + b*x + c*x^2])/(c*(2*m + 1)), x] - Dist[1/(c*(2*m + 1)), Int[((d + e*x)^(m - 2)*Simp[e*(b*d*f + a*(d*g + 2*e*f*(m - 1))) - c*d^2*f*(2*m + 1) + (a*e^2*g*(2*m - 1) - c*d*(4*e*f*m + d*g*(2*m + 1)) + b*e*(2*d*g + e*f*(2*m - 1)))*x + e*(2*b*e*g*m - c*(e*f + d*g*(4*m - 1)))*x^2, x])/(Sqrt[f + g*x]*Sqrt[a + b*x + c*x^2]), x], x] /; FreeQ[{a, b, c, d, e, f, g}, x] && NeQ[e*f - d*g, 0] && NeQ[b^2 - 4*a*c, 0] && NeQ[c*d^2 - b*d*e + a*e^2, 0] && IntegerQ[2*m] && GtQ[m, 1]
  942. Int[Times[Power[Plus[Optional[Pattern[d, Blank[]]], Times[Optional[Pattern[e, Blank[]]], Pattern[x, Blank[]]]], Pattern[m, Blank[]]], Power[Plus[Optional[Pattern[f, Blank[]]], Times[Optional[Pattern[g, Blank[]]], Pattern[x, Blank[]]]], Rational[1, 2]], Power[Plus[Pattern[a, Blank[]], Times[Optional[Pattern[c, Blank[]]], Power[Pattern[x, Blank[]], 2]]], Rational[-1, 2]]], Pattern[x, Blank[Symbol]]] := Simp[(2*e*(d + e*x)^(m - 1)*Sqrt[f + g*x]*Sqrt[a + c*x^2])/(c*(2*m + 1)), x] - Dist[1/(c*(2*m + 1)), Int[((d + e*x)^(m - 2)*Simp[a*e*(d*g + 2*e*f*(m - 1)) - c*d^2*f*(2*m + 1) + (a*e^2*g*(2*m - 1) - c*d*(4*e*f*m + d*g*(2*m + 1)))*x - c*e*(e*f + d*g*(4*m - 1))*x^2, x])/(Sqrt[f + g*x]*Sqrt[a + c*x^2]), x], x] /; FreeQ[{a, c, d, e, f, g}, x] && NeQ[e*f - d*g, 0] && NeQ[c*d^2 + a*e^2, 0] && IntegerQ[2*m] && GtQ[m, 1]
  943. Int[Times[Power[Plus[Optional[Pattern[d, Blank[]]], Times[Optional[Pattern[e, Blank[]]], Pattern[x, Blank[]]]], -1], Power[Plus[Optional[Pattern[f, Blank[]]], Times[Optional[Pattern[g, Blank[]]], Pattern[x, Blank[]]]], Rational[1, 2]], Power[Plus[Optional[Pattern[a, Blank[]]], Times[Optional[Pattern[b, Blank[]]], Pattern[x, Blank[]]], Times[Optional[Pattern[c, Blank[]]], Power[Pattern[x, Blank[]], 2]]], Rational[-1, 2]]], Pattern[x, Blank[Symbol]]] := Dist[g/e, Int[1/(Sqrt[f + g*x]*Sqrt[a + b*x + c*x^2]), x], x] + Dist[(e*f - d*g)/e, Int[1/((d + e*x)*Sqrt[f + g*x]*Sqrt[a + b*x + c*x^2]), x], x] /; FreeQ[{a, b, c, d, e, f, g}, x] && NeQ[e*f - d*g, 0] && NeQ[b^2 - 4*a*c, 0] && NeQ[c*d^2 - b*d*e + a*e^2, 0]
  944. Int[Times[Power[Plus[Optional[Pattern[d, Blank[]]], Times[Optional[Pattern[e, Blank[]]], Pattern[x, Blank[]]]], -1], Power[Plus[Optional[Pattern[f, Blank[]]], Times[Optional[Pattern[g, Blank[]]], Pattern[x, Blank[]]]], Rational[1, 2]], Power[Plus[Pattern[a, Blank[]], Times[Optional[Pattern[c, Blank[]]], Power[Pattern[x, Blank[]], 2]]], Rational[-1, 2]]], Pattern[x, Blank[Symbol]]] := Dist[g/e, Int[1/(Sqrt[f + g*x]*Sqrt[a + c*x^2]), x], x] + Dist[(e*f - d*g)/e, Int[1/((d + e*x)*Sqrt[f + g*x]*Sqrt[a + c*x^2]), x], x] /; FreeQ[{a, c, d, e, f, g}, x] && NeQ[e*f - d*g, 0] && NeQ[c*d^2 + a*e^2, 0]
  945. Int[Times[Power[Plus[Optional[Pattern[d, Blank[]]], Times[Optional[Pattern[e, Blank[]]], Pattern[x, Blank[]]]], Pattern[m, Blank[]]], Power[Plus[Optional[Pattern[f, Blank[]]], Times[Optional[Pattern[g, Blank[]]], Pattern[x, Blank[]]]], Rational[1, 2]], Power[Plus[Optional[Pattern[a, Blank[]]], Times[Optional[Pattern[b, Blank[]]], Pattern[x, Blank[]]], Times[Optional[Pattern[c, Blank[]]], Power[Pattern[x, Blank[]], 2]]], Rational[-1, 2]]], Pattern[x, Blank[Symbol]]] := Simp[(e*(d + e*x)^(m + 1)*Sqrt[f + g*x]*Sqrt[a + b*x + c*x^2])/((m + 1)*(c*d^2 - b*d*e + a*e^2)), x] + Dist[1/(2*(m + 1)*(c*d^2 - b*d*e + a*e^2)), Int[((d + e*x)^(m + 1)*Simp[2*c*d*f*(m + 1) - e*(a*g + b*f*(2*m + 3)) - 2*(b*e*g*(2 + m) - c*(d*g*(m + 1) - e*f*(m + 2)))*x - c*e*g*(2*m + 5)*x^2, x])/(Sqrt[f + g*x]*Sqrt[a + b*x + c*x^2]), x], x] /; FreeQ[{a, b, c, d, e, f, g}, x] && NeQ[e*f - d*g, 0] && NeQ[b^2 - 4*a*c, 0] && NeQ[c*d^2 - b*d*e + a*e^2, 0] && IntegerQ[2*m] && LeQ[m, -2]
  946. Int[Times[Power[Plus[Optional[Pattern[d, Blank[]]], Times[Optional[Pattern[e, Blank[]]], Pattern[x, Blank[]]]], Pattern[m, Blank[]]], Power[Plus[Optional[Pattern[f, Blank[]]], Times[Optional[Pattern[g, Blank[]]], Pattern[x, Blank[]]]], Rational[1, 2]], Power[Plus[Pattern[a, Blank[]], Times[Optional[Pattern[c, Blank[]]], Power[Pattern[x, Blank[]], 2]]], Rational[-1, 2]]], Pattern[x, Blank[Symbol]]] := Simp[(e*(d + e*x)^(m + 1)*Sqrt[f + g*x]*Sqrt[a + c*x^2])/((m + 1)*(c*d^2 + a*e^2)), x] + Dist[1/(2*(m + 1)*(c*d^2 + a*e^2)), Int[((d + e*x)^(m + 1)*Simp[2*c*d*f*(m + 1) - e*(a*g) + 2*c*(d*g*(m + 1) - e*f*(m + 2))*x - c*e*g*(2*m + 5)*x^2, x])/(Sqrt[f + g*x]*Sqrt[a + c*x^2]), x], x] /; FreeQ[{a, c, d, e, f, g}, x] && NeQ[e*f - d*g, 0] && NeQ[c*d^2 + a*e^2, 0] && IntegerQ[2*m] && LeQ[m, -2]
  947. Int[Times[Power[Plus[Optional[Pattern[d, Blank[]]], Times[Optional[Pattern[e, Blank[]]], Pattern[x, Blank[]]]], Pattern[m, Blank[]]], Power[Plus[Optional[Pattern[f, Blank[]]], Times[Optional[Pattern[g, Blank[]]], Pattern[x, Blank[]]]], Pattern[n, Blank[]]], Power[Plus[Optional[Pattern[a, Blank[]]], Times[Optional[Pattern[b, Blank[]]], Pattern[x, Blank[]]], Times[Optional[Pattern[c, Blank[]]], Power[Pattern[x, Blank[]], 2]]], Optional[Pattern[p, Blank[]]]]], Pattern[x, Blank[Symbol]]] := Int[ExpandIntegrand[(d + e*x)^m*(f + g*x)^n*(a + b*x + c*x^2)^p, x], x] /; FreeQ[{a, b, c, d, e, f, g}, x] && NeQ[e*f - d*g, 0] && NeQ[b^2 - 4*a*c, 0] && NeQ[c*d^2 - b*d*e + a*e^2, 0] && IGtQ[p, 0] && (IGtQ[m, 0] || (EqQ[m, -2] && EqQ[p, 1] && EqQ[2*c*d - b*e, 0]))
  948. Int[Times[Power[Plus[Optional[Pattern[d, Blank[]]], Times[Optional[Pattern[e, Blank[]]], Pattern[x, Blank[]]]], Pattern[m, Blank[]]], Power[Plus[Optional[Pattern[f, Blank[]]], Times[Optional[Pattern[g, Blank[]]], Pattern[x, Blank[]]]], Pattern[n, Blank[]]], Power[Plus[Pattern[a, Blank[]], Times[Optional[Pattern[c, Blank[]]], Power[Pattern[x, Blank[]], 2]]], Optional[Pattern[p, Blank[]]]]], Pattern[x, Blank[Symbol]]] := Int[ExpandIntegrand[(d + e*x)^m*(f + g*x)^n*(a + c*x^2)^p, x], x] /; FreeQ[{a, c, d, e, f, g}, x] && NeQ[e*f - d*g, 0] && NeQ[c*d^2 + a*e^2, 0] && IGtQ[p, 0] && (IGtQ[m, 0] || (EqQ[m, -2] && EqQ[p, 1] && EqQ[d, 0]))
  949. Int[Times[Power[Plus[Optional[Pattern[d, Blank[]]], Times[Optional[Pattern[e, Blank[]]], Pattern[x, Blank[]]]], Pattern[m, Blank[]]], Power[Plus[Optional[Pattern[f, Blank[]]], Times[Optional[Pattern[g, Blank[]]], Pattern[x, Blank[]]]], Pattern[n, Blank[]]], Power[Plus[Optional[Pattern[a, Blank[]]], Times[Optional[Pattern[b, Blank[]]], Pattern[x, Blank[]]], Times[Optional[Pattern[c, Blank[]]], Power[Pattern[x, Blank[]], 2]]], Optional[Pattern[p, Blank[]]]]], Pattern[x, Blank[Symbol]]] := With[{Qx = PolynomialQuotient[(a + b*x + c*x^2)^p, d + e*x, x], R = PolynomialRemainder[(a + b*x + c*x^2)^p, d + e*x, x]}, Simp[(R*(d + e*x)^(m + 1)*(f + g*x)^(n + 1))/((m + 1)*(e*f - d*g)), x] + Dist[1/((m + 1)*(e*f - d*g)), Int[(d + e*x)^(m + 1)*(f + g*x)^n*ExpandToSum[(m + 1)*(e*f - d*g)*Qx - g*R*(m + n + 2), x], x], x]] /; FreeQ[{a, b, c, d, e, f, g}, x] && NeQ[e*f - d*g, 0] && NeQ[b^2 - 4*a*c, 0] && NeQ[c*d^2 - b*d*e + a*e^2, 0] && IGtQ[p, 0] && LtQ[m, -1]
  950. Int[Times[Power[Plus[Optional[Pattern[d, Blank[]]], Times[Optional[Pattern[e, Blank[]]], Pattern[x, Blank[]]]], Pattern[m, Blank[]]], Power[Plus[Optional[Pattern[f, Blank[]]], Times[Optional[Pattern[g, Blank[]]], Pattern[x, Blank[]]]], Pattern[n, Blank[]]], Power[Plus[Pattern[a, Blank[]], Times[Optional[Pattern[c, Blank[]]], Power[Pattern[x, Blank[]], 2]]], Optional[Pattern[p, Blank[]]]]], Pattern[x, Blank[Symbol]]] := With[{Qx = PolynomialQuotient[(a + c*x^2)^p, d + e*x, x], R = PolynomialRemainder[(a + c*x^2)^p, d + e*x, x]}, Simp[(R*(d + e*x)^(m + 1)*(f + g*x)^(n + 1))/((m + 1)*(e*f - d*g)), x] + Dist[1/((m + 1)*(e*f - d*g)), Int[(d + e*x)^(m + 1)*(f + g*x)^n*ExpandToSum[(m + 1)*(e*f - d*g)*Qx - g*R*(m + n + 2), x], x], x]] /; FreeQ[{a, c, d, e, f, g}, x] && NeQ[e*f - d*g, 0] && NeQ[c*d^2 + a*e^2, 0] && IGtQ[p, 0] && LtQ[m, -1]
  951. Int[Times[Power[Plus[Optional[Pattern[d, Blank[]]], Times[Optional[Pattern[e, Blank[]]], Pattern[x, Blank[]]]], Pattern[m, Blank[]]], Power[Plus[Optional[Pattern[f, Blank[]]], Times[Optional[Pattern[g, Blank[]]], Pattern[x, Blank[]]]], Pattern[n, Blank[]]], Power[Plus[Optional[Pattern[a, Blank[]]], Times[Optional[Pattern[b, Blank[]]], Pattern[x, Blank[]]], Times[Optional[Pattern[c, Blank[]]], Power[Pattern[x, Blank[]], 2]]], Optional[Pattern[p, Blank[]]]]], Pattern[x, Blank[Symbol]]] := Simp[(c^p*(d + e*x)^(m + 2*p)*(f + g*x)^(n + 1))/(g*e^(2*p)*(m + n + 2*p + 1)), x] + Dist[1/(g*e^(2*p)*(m + n + 2*p + 1)), Int[(d + e*x)^m*(f + g*x)^n*ExpandToSum[g*(m + n + 2*p + 1)*(e^(2*p)*(a + b*x + c*x^2)^p - c^p*(d + e*x)^(2*p)) - c^p*(e*f - d*g)*(m + 2*p)*(d + e*x)^(2*p - 1), x], x], x] /; FreeQ[{a, b, c, d, e, f, g}, x] && NeQ[e*f - d*g, 0] && NeQ[b^2 - 4*a*c, 0] && NeQ[c*d^2 - b*d*e + a*e^2, 0] && IGtQ[p, 0] && NeQ[m + n + 2*p + 1, 0] && (IntegerQ[n] || !IntegerQ[m])
  952. Int[Times[Power[Plus[Optional[Pattern[d, Blank[]]], Times[Optional[Pattern[e, Blank[]]], Pattern[x, Blank[]]]], Pattern[m, Blank[]]], Power[Plus[Optional[Pattern[f, Blank[]]], Times[Optional[Pattern[g, Blank[]]], Pattern[x, Blank[]]]], Pattern[n, Blank[]]], Power[Plus[Pattern[a, Blank[]], Times[Optional[Pattern[c, Blank[]]], Power[Pattern[x, Blank[]], 2]]], Optional[Pattern[p, Blank[]]]]], Pattern[x, Blank[Symbol]]] := Simp[(c^p*(d + e*x)^(m + 2*p)*(f + g*x)^(n + 1))/(g*e^(2*p)*(m + n + 2*p + 1)), x] + Dist[1/(g*e^(2*p)*(m + n + 2*p + 1)), Int[(d + e*x)^m*(f + g*x)^n*ExpandToSum[g*(m + n + 2*p + 1)*(e^(2*p)*(a + c*x^2)^p - c^p*(d + e*x)^(2*p)) - c^p*(e*f - d*g)*(m + 2*p)*(d + e*x)^(2*p - 1), x], x], x] /; FreeQ[{a, c, d, e, f, g}, x] && NeQ[e*f - d*g, 0] && NeQ[c*d^2 + a*e^2, 0] && IGtQ[p, 0] && NeQ[m + n + 2*p + 1, 0] && (IntegerQ[n] || !IntegerQ[m])
  953. Int[Times[Power[Plus[Optional[Pattern[d, Blank[]]], Times[Optional[Pattern[e, Blank[]]], Pattern[x, Blank[]]]], -1], Power[Plus[Optional[Pattern[f, Blank[]]], Times[Optional[Pattern[g, Blank[]]], Pattern[x, Blank[]]]], Pattern[n, Blank[]]], Power[Plus[Optional[Pattern[a, Blank[]]], Times[Optional[Pattern[b, Blank[]]], Pattern[x, Blank[]]], Times[Optional[Pattern[c, Blank[]]], Power[Pattern[x, Blank[]], 2]]], Pattern[p, Blank[]]]], Pattern[x, Blank[Symbol]]] := Dist[(c*d^2 - b*d*e + a*e^2)/(e*(e*f - d*g)), Int[((f + g*x)^(n + 1)*(a + b*x + c*x^2)^(p - 1))/(d + e*x), x], x] - Dist[1/(e*(e*f - d*g)), Int[(f + g*x)^n*(c*d*f - b*e*f + a*e*g - c*(e*f - d*g)*x)*(a + b*x + c*x^2)^(p - 1), x], x] /; FreeQ[{a, b, c, d, e, f, g}, x] && NeQ[e*f - d*g, 0] && NeQ[b^2 - 4*a*c, 0] && NeQ[c*d^2 - b*d*e + a*e^2, 0] && !IntegerQ[n] && !IntegerQ[p] && GtQ[p, 0] && LtQ[n, -1]
  954. Int[Times[Power[Plus[Optional[Pattern[d, Blank[]]], Times[Optional[Pattern[e, Blank[]]], Pattern[x, Blank[]]]], -1], Power[Plus[Optional[Pattern[f, Blank[]]], Times[Optional[Pattern[g, Blank[]]], Pattern[x, Blank[]]]], Pattern[n, Blank[]]], Power[Plus[Pattern[a, Blank[]], Times[Optional[Pattern[c, Blank[]]], Power[Pattern[x, Blank[]], 2]]], Pattern[p, Blank[]]]], Pattern[x, Blank[Symbol]]] := Dist[(c*d^2 + a*e^2)/(e*(e*f - d*g)), Int[((f + g*x)^(n + 1)*(a + c*x^2)^(p - 1))/(d + e*x), x], x] - Dist[1/(e*(e*f - d*g)), Int[(f + g*x)^n*(c*d*f + a*e*g - c*(e*f - d*g)*x)*(a + c*x^2)^(p - 1), x], x] /; FreeQ[{a, c, d, e, f, g}, x] && NeQ[e*f - d*g, 0] && NeQ[c*d^2 + a*e^2, 0] && !IntegerQ[n] && !IntegerQ[p] && GtQ[p, 0] && LtQ[n, -1]
  955. Int[Times[Power[Plus[Optional[Pattern[d, Blank[]]], Times[Optional[Pattern[e, Blank[]]], Pattern[x, Blank[]]]], -1], Power[Plus[Optional[Pattern[f, Blank[]]], Times[Optional[Pattern[g, Blank[]]], Pattern[x, Blank[]]]], Pattern[n, Blank[]]], Power[Plus[Optional[Pattern[a, Blank[]]], Times[Optional[Pattern[b, Blank[]]], Pattern[x, Blank[]]], Times[Optional[Pattern[c, Blank[]]], Power[Pattern[x, Blank[]], 2]]], Pattern[p, Blank[]]]], Pattern[x, Blank[Symbol]]] := Dist[(e*(e*f - d*g))/(c*d^2 - b*d*e + a*e^2), Int[((f + g*x)^(n - 1)*(a + b*x + c*x^2)^(p + 1))/(d + e*x), x], x] + Dist[1/(c*d^2 - b*d*e + a*e^2), Int[(f + g*x)^(n - 1)*(c*d*f - b*e*f + a*e*g - c*(e*f - d*g)*x)*(a + b*x + c*x^2)^p, x], x] /; FreeQ[{a, b, c, d, e, f, g}, x] && NeQ[e*f - d*g, 0] && NeQ[b^2 - 4*a*c, 0] && NeQ[c*d^2 - b*d*e + a*e^2, 0] && !IntegerQ[n] && !IntegerQ[p] && LtQ[p, -1] && GtQ[n, 0]
  956. Int[Times[Power[Plus[Optional[Pattern[d, Blank[]]], Times[Optional[Pattern[e, Blank[]]], Pattern[x, Blank[]]]], -1], Power[Plus[Optional[Pattern[f, Blank[]]], Times[Optional[Pattern[g, Blank[]]], Pattern[x, Blank[]]]], Pattern[n, Blank[]]], Power[Plus[Pattern[a, Blank[]], Times[Optional[Pattern[c, Blank[]]], Power[Pattern[x, Blank[]], 2]]], Pattern[p, Blank[]]]], Pattern[x, Blank[Symbol]]] := Dist[(e*(e*f - d*g))/(c*d^2 + a*e^2), Int[((f + g*x)^(n - 1)*(a + c*x^2)^(p + 1))/(d + e*x), x], x] + Dist[1/(c*d^2 + a*e^2), Int[(f + g*x)^(n - 1)*(c*d*f + a*e*g - c*(e*f - d*g)*x)*(a + c*x^2)^p, x], x] /; FreeQ[{a, c, d, e, f, g}, x] && NeQ[e*f - d*g, 0] && NeQ[c*d^2 + a*e^2, 0] && !IntegerQ[n] && !IntegerQ[p] && LtQ[p, -1] && GtQ[n, 0]
  957. Int[Times[Power[Plus[Optional[Pattern[d, Blank[]]], Times[Optional[Pattern[e, Blank[]]], Pattern[x, Blank[]]]], -1], Power[Plus[Optional[Pattern[f, Blank[]]], Times[Optional[Pattern[g, Blank[]]], Pattern[x, Blank[]]]], Pattern[n, Blank[]]], Power[Plus[Optional[Pattern[a, Blank[]]], Times[Optional[Pattern[b, Blank[]]], Pattern[x, Blank[]]], Times[Optional[Pattern[c, Blank[]]], Power[Pattern[x, Blank[]], 2]]], Rational[-1, 2]]], Pattern[x, Blank[Symbol]]] := Int[ExpandIntegrand[1/(Sqrt[f + g*x]*Sqrt[a + b*x + c*x^2]), (f + g*x)^(n + 1/2)/(d + e*x), x], x] /; FreeQ[{a, b, c, d, e, f, g}, x] && NeQ[e*f - d*g, 0] && NeQ[b^2 - 4*a*c, 0] && NeQ[c*d^2 - b*d*e + a*e^2, 0] && IntegerQ[n + 1/2]
  958. Int[Times[Power[Plus[Optional[Pattern[d, Blank[]]], Times[Optional[Pattern[e, Blank[]]], Pattern[x, Blank[]]]], -1], Power[Plus[Optional[Pattern[f, Blank[]]], Times[Optional[Pattern[g, Blank[]]], Pattern[x, Blank[]]]], Pattern[n, Blank[]]], Power[Plus[Pattern[a, Blank[]], Times[Optional[Pattern[c, Blank[]]], Power[Pattern[x, Blank[]], 2]]], Rational[-1, 2]]], Pattern[x, Blank[Symbol]]] := Int[ExpandIntegrand[1/(Sqrt[f + g*x]*Sqrt[a + c*x^2]), (f + g*x)^(n + 1/2)/(d + e*x), x], x] /; FreeQ[{a, c, d, e, f, g}, x] && NeQ[e*f - d*g, 0] && NeQ[c*d^2 + a*e^2, 0] && IntegerQ[n + 1/2]
  959. Int[Times[Power[Times[Optional[Pattern[g, Blank[]]], Pattern[x, Blank[]]], Optional[Pattern[n, Blank[]]]], Power[Plus[Pattern[d, Blank[]], Times[Optional[Pattern[e, Blank[]]], Pattern[x, Blank[]]]], -1], Power[Plus[Pattern[a, Blank[]], Times[Optional[Pattern[c, Blank[]]], Power[Pattern[x, Blank[]], 2]]], Pattern[p, Blank[]]]], Pattern[x, Blank[Symbol]]] := Dist[(d*(g*x)^n)/x^n, Int[(x^n*(a + c*x^2)^p)/(d^2 - e^2*x^2), x], x] - Dist[(e*(g*x)^n)/x^n, Int[(x^(n + 1)*(a + c*x^2)^p)/(d^2 - e^2*x^2), x], x] /; FreeQ[{a, c, d, e, g, n, p}, x] && NeQ[c*d^2 + a*e^2, 0] && !IntegerQ[p] && !IntegersQ[n, 2*p]
  960. Int[Times[Power[Plus[Optional[Pattern[d, Blank[]]], Times[Optional[Pattern[e, Blank[]]], Pattern[x, Blank[]]]], Pattern[m, Blank[]]], Power[Plus[Optional[Pattern[f, Blank[]]], Times[Optional[Pattern[g, Blank[]]], Pattern[x, Blank[]]]], Pattern[n, Blank[]]], Power[Plus[Optional[Pattern[a, Blank[]]], Times[Optional[Pattern[b, Blank[]]], Pattern[x, Blank[]]], Times[Optional[Pattern[c, Blank[]]], Power[Pattern[x, Blank[]], 2]]], Optional[Pattern[p, Blank[]]]]], Pattern[x, Blank[Symbol]]] := Int[ExpandIntegrand[(d + e*x)^m*(f + g*x)^n*(a + b*x + c*x^2)^p, x], x] /; FreeQ[{a, b, c, d, e, f, g}, x] && NeQ[e*f - d*g, 0] && NeQ[b^2 - 4*a*c, 0] && NeQ[c*d^2 - b*d*e + a*e^2, 0] && (IntegerQ[p] || (ILtQ[m, 0] && ILtQ[n, 0])) && !(IGtQ[m, 0] || IGtQ[n, 0])
  961. Int[Times[Power[Plus[Optional[Pattern[d, Blank[]]], Times[Optional[Pattern[e, Blank[]]], Pattern[x, Blank[]]]], Pattern[m, Blank[]]], Power[Plus[Optional[Pattern[f, Blank[]]], Times[Optional[Pattern[g, Blank[]]], Pattern[x, Blank[]]]], Pattern[n, Blank[]]], Power[Plus[Pattern[a, Blank[]], Times[Optional[Pattern[c, Blank[]]], Power[Pattern[x, Blank[]], 2]]], Optional[Pattern[p, Blank[]]]]], Pattern[x, Blank[Symbol]]] := Int[ExpandIntegrand[(d + e*x)^m*(f + g*x)^n*(a + c*x^2)^p, x], x] /; FreeQ[{a, c, d, e, f, g}, x] && NeQ[e*f - d*g, 0] && NeQ[c*d^2 + a*e^2, 0] && (IntegerQ[p] || (ILtQ[m, 0] && ILtQ[n, 0])) && !(IGtQ[m, 0] || IGtQ[n, 0])
  962. Int[Times[Power[Times[Optional[Pattern[g, Blank[]]], Pattern[x, Blank[]]], Optional[Pattern[n, Blank[]]]], Power[Plus[Pattern[d, Blank[]], Times[Optional[Pattern[e, Blank[]]], Pattern[x, Blank[]]]], Pattern[m, Blank[]]], Power[Plus[Pattern[a, Blank[]], Times[Optional[Pattern[c, Blank[]]], Power[Pattern[x, Blank[]], 2]]], Pattern[p, Blank[]]]], Pattern[x, Blank[Symbol]]] := Dist[(g*x)^n/x^n, Int[ExpandIntegrand[x^n*(a + c*x^2)^p, (d/(d^2 - e^2*x^2) - (e*x)/(d^2 - e^2*x^2))^(-m), x], x], x] /; FreeQ[{a, c, d, e, g, n, p}, x] && NeQ[c*d^2 + a*e^2, 0] && ILtQ[m, 0] && !IntegerQ[p] && !IntegerQ[n]
  963. Int[Times[Power[Plus[Optional[Pattern[d, Blank[]]], Times[Optional[Pattern[e, Blank[]]], Pattern[x, Blank[]]]], Pattern[m, Blank[]]], Power[Plus[Optional[Pattern[f, Blank[]]], Times[Optional[Pattern[g, Blank[]]], Pattern[x, Blank[]]]], Pattern[n, Blank[]]], Power[Plus[Optional[Pattern[a, Blank[]]], Times[Optional[Pattern[b, Blank[]]], Pattern[x, Blank[]]], Times[Optional[Pattern[c, Blank[]]], Power[Pattern[x, Blank[]], 2]]], Pattern[p, Blank[]]]], Pattern[x, Blank[Symbol]]] := Unintegrable[(d + e*x)^m*(f + g*x)^n*(a + b*x + c*x^2)^p, x] /; FreeQ[{a, b, c, d, e, f, g, m, n, p}, x] && !(IGtQ[m, 0] || IGtQ[n, 0])
  964. Int[Times[Power[Plus[Optional[Pattern[d, Blank[]]], Times[Optional[Pattern[e, Blank[]]], Pattern[x, Blank[]]]], Pattern[m, Blank[]]], Power[Plus[Optional[Pattern[f, Blank[]]], Times[Optional[Pattern[g, Blank[]]], Pattern[x, Blank[]]]], Pattern[n, Blank[]]], Power[Plus[Pattern[a, Blank[]], Times[Optional[Pattern[c, Blank[]]], Power[Pattern[x, Blank[]], 2]]], Pattern[p, Blank[]]]], Pattern[x, Blank[Symbol]]] := Unintegrable[(d + e*x)^m*(f + g*x)^n*(a + c*x^2)^p, x] /; FreeQ[{a, c, d, e, f, g, m, n, p}, x] && !(IGtQ[m, 0] || IGtQ[n, 0])
  965. Int[Times[Power[Plus[Optional[Pattern[d, Blank[]]], Times[Optional[Pattern[e, Blank[]]], Pattern[u, Blank[]]]], Optional[Pattern[m, Blank[]]]], Power[Plus[Optional[Pattern[f, Blank[]]], Times[Optional[Pattern[g, Blank[]]], Pattern[u, Blank[]]]], Optional[Pattern[n, Blank[]]]], Power[Plus[Pattern[a, Blank[]], Times[Optional[Pattern[b, Blank[]]], Pattern[u, Blank[]]], Times[Optional[Pattern[c, Blank[]]], Power[Pattern[u, Blank[]], 2]]], Optional[Pattern[p, Blank[]]]]], Pattern[x, Blank[Symbol]]] := Dist[1/Coefficient[u, x, 1], Subst[Int[(d + e*x)^m*(f + g*x)^n*(a + b*x + c*x^2)^p, x], x, u], x] /; FreeQ[{a, b, c, d, e, f, g, m, n, p}, x] && LinearQ[u, x] && NeQ[u, x]
  966. Int[Times[Power[Plus[Optional[Pattern[d, Blank[]]], Times[Optional[Pattern[e, Blank[]]], Pattern[u, Blank[]]]], Optional[Pattern[m, Blank[]]]], Power[Plus[Optional[Pattern[f, Blank[]]], Times[Optional[Pattern[g, Blank[]]], Pattern[u, Blank[]]]], Optional[Pattern[n, Blank[]]]], Power[Plus[Pattern[a, Blank[]], Times[Optional[Pattern[c, Blank[]]], Power[Pattern[u, Blank[]], 2]]], Optional[Pattern[p, Blank[]]]]], Pattern[x, Blank[Symbol]]] := Dist[1/Coefficient[u, x, 1], Subst[Int[(d + e*x)^m*(f + g*x)^n*(a + c*x^2)^p, x], x, u], x] /; FreeQ[{a, c, d, e, f, g, m, n, p}, x] && LinearQ[u, x] && NeQ[u, x]
  967. Int[Times[Power[Plus[Pattern[a, Blank[]], Times[Optional[Pattern[b, Blank[]]], Pattern[x, Blank[]]], Times[Optional[Pattern[c, Blank[]]], Power[Pattern[x, Blank[]], 2]]], Optional[Pattern[p, Blank[]]]], Power[Plus[Pattern[d, Blank[]], Times[Optional[Pattern[e, Blank[]]], Pattern[x, Blank[]]], Times[Optional[Pattern[f, Blank[]]], Power[Pattern[x, Blank[]], 2]]], Optional[Pattern[q, Blank[]]]]], Pattern[x, Blank[Symbol]]] := Dist[(c/f)^p, Int[(d + e*x + f*x^2)^(p + q), x], x] /; FreeQ[{a, b, c, d, e, f, p, q}, x] && EqQ[c*d - a*f, 0] && EqQ[b*d - a*e, 0] && (IntegerQ[p] || GtQ[c/f, 0]) && ( !IntegerQ[q] || LeafCount[d + e*x + f*x^2] <= LeafCount[a + b*x + c*x^2])
  968. Int[Times[Power[Plus[Pattern[a, Blank[]], Times[Optional[Pattern[b, Blank[]]], Pattern[x, Blank[]]], Times[Optional[Pattern[c, Blank[]]], Power[Pattern[x, Blank[]], 2]]], Pattern[p, Blank[]]], Power[Plus[Pattern[d, Blank[]], Times[Optional[Pattern[e, Blank[]]], Pattern[x, Blank[]]], Times[Optional[Pattern[f, Blank[]]], Power[Pattern[x, Blank[]], 2]]], Optional[Pattern[q, Blank[]]]]], Pattern[x, Blank[Symbol]]] := Dist[(a^IntPart[p]*(a + b*x + c*x^2)^FracPart[p])/(d^IntPart[p]*(d + e*x + f*x^2)^FracPart[p]), Int[(d + e*x + f*x^2)^(p + q), x], x] /; FreeQ[{a, b, c, d, e, f, p, q}, x] && EqQ[c*d - a*f, 0] && EqQ[b*d - a*e, 0] && !IntegerQ[p] && !IntegerQ[q] && !GtQ[c/f, 0]
  969. Int[Times[Power[Plus[Pattern[a, Blank[]], Times[Optional[Pattern[b, Blank[]]], Pattern[x, Blank[]]], Times[Optional[Pattern[c, Blank[]]], Power[Pattern[x, Blank[]], 2]]], Pattern[p, Blank[]]], Power[Plus[Pattern[d, Blank[]], Times[Optional[Pattern[e, Blank[]]], Pattern[x, Blank[]]], Times[Optional[Pattern[f, Blank[]]], Power[Pattern[x, Blank[]], 2]]], Optional[Pattern[q, Blank[]]]]], Pattern[x, Blank[Symbol]]] := Dist[(a + b*x + c*x^2)^FracPart[p]/((4*c)^IntPart[p]*(b + 2*c*x)^(2*FracPart[p])), Int[(b + 2*c*x)^(2*p)*(d + e*x + f*x^2)^q, x], x] /; FreeQ[{a, b, c, d, e, f, p, q}, x] && EqQ[b^2 - 4*a*c, 0] && !IntegerQ[p]
  970. Int[Times[Power[Plus[Pattern[a, Blank[]], Times[Optional[Pattern[b, Blank[]]], Pattern[x, Blank[]]], Times[Optional[Pattern[c, Blank[]]], Power[Pattern[x, Blank[]], 2]]], Pattern[p, Blank[]]], Power[Plus[Pattern[d, Blank[]], Times[Optional[Pattern[f, Blank[]]], Power[Pattern[x, Blank[]], 2]]], Optional[Pattern[q, Blank[]]]]], Pattern[x, Blank[Symbol]]] := Dist[(a + b*x + c*x^2)^FracPart[p]/((4*c)^IntPart[p]*(b + 2*c*x)^(2*FracPart[p])), Int[(b + 2*c*x)^(2*p)*(d + f*x^2)^q, x], x] /; FreeQ[{a, b, c, d, f, p, q}, x] && EqQ[b^2 - 4*a*c, 0] && !IntegerQ[p]
  971. Int[Times[Power[Plus[Optional[Pattern[a, Blank[]]], Times[Optional[Pattern[b, Blank[]]], Pattern[x, Blank[]]], Times[Optional[Pattern[c, Blank[]]], Power[Pattern[x, Blank[]], 2]]], Pattern[p, Blank[]]], Power[Plus[Optional[Pattern[d, Blank[]]], Times[Optional[Pattern[e, Blank[]]], Pattern[x, Blank[]]], Times[Optional[Pattern[f, Blank[]]], Power[Pattern[x, Blank[]], 2]]], Pattern[q, Blank[]]]], Pattern[x, Blank[Symbol]]] := Simp[((b + 2*c*x)*(a + b*x + c*x^2)^(p + 1)*(d + e*x + f*x^2)^q)/((b^2 - 4*a*c)*(p + 1)), x] - Dist[1/((b^2 - 4*a*c)*(p + 1)), Int[(a + b*x + c*x^2)^(p + 1)*(d + e*x + f*x^2)^(q - 1)*Simp[2*c*d*(2*p + 3) + b*e*q + (2*b*f*q + 2*c*e*(2*p + q + 3))*x + 2*c*f*(2*p + 2*q + 3)*x^2, x], x], x] /; FreeQ[{a, b, c, d, e, f}, x] && NeQ[b^2 - 4*a*c, 0] && NeQ[e^2 - 4*d*f, 0] && LtQ[p, -1] && GtQ[q, 0] && !IGtQ[q, 0]
  972. Int[Times[Power[Plus[Optional[Pattern[a, Blank[]]], Times[Optional[Pattern[b, Blank[]]], Pattern[x, Blank[]]], Times[Optional[Pattern[c, Blank[]]], Power[Pattern[x, Blank[]], 2]]], Pattern[p, Blank[]]], Power[Plus[Optional[Pattern[d, Blank[]]], Times[Optional[Pattern[f, Blank[]]], Power[Pattern[x, Blank[]], 2]]], Pattern[q, Blank[]]]], Pattern[x, Blank[Symbol]]] := Simp[((b + 2*c*x)*(a + b*x + c*x^2)^(p + 1)*(d + f*x^2)^q)/((b^2 - 4*a*c)*(p + 1)), x] - Dist[1/((b^2 - 4*a*c)*(p + 1)), Int[(a + b*x + c*x^2)^(p + 1)*(d + f*x^2)^(q - 1)*Simp[2*c*d*(2*p + 3) + (2*b*f*q)*x + 2*c*f*(2*p + 2*q + 3)*x^2, x], x], x] /; FreeQ[{a, b, c, d, f}, x] && NeQ[b^2 - 4*a*c, 0] && LtQ[p, -1] && GtQ[q, 0] && !IGtQ[q, 0]
  973. Int[Times[Power[Plus[Optional[Pattern[a, Blank[]]], Times[Optional[Pattern[c, Blank[]]], Power[Pattern[x, Blank[]], 2]]], Pattern[p, Blank[]]], Power[Plus[Optional[Pattern[d, Blank[]]], Times[Optional[Pattern[e, Blank[]]], Pattern[x, Blank[]]], Times[Optional[Pattern[f, Blank[]]], Power[Pattern[x, Blank[]], 2]]], Pattern[q, Blank[]]]], Pattern[x, Blank[Symbol]]] := Simp[((2*c*x)*(a + c*x^2)^(p + 1)*(d + e*x + f*x^2)^q)/((-4*a*c)*(p + 1)), x] - Dist[1/((-4*a*c)*(p + 1)), Int[(a + c*x^2)^(p + 1)*(d + e*x + f*x^2)^(q - 1)*Simp[2*c*d*(2*p + 3) + (2*c*e*(2*p + q + 3))*x + 2*c*f*(2*p + 2*q + 3)*x^2, x], x], x] /; FreeQ[{a, c, d, e, f}, x] && NeQ[e^2 - 4*d*f] && LtQ[p, -1] && GtQ[q, 0] && !IGtQ[q, 0]
  974. Int[Times[Power[Plus[Optional[Pattern[a, Blank[]]], Times[Optional[Pattern[b, Blank[]]], Pattern[x, Blank[]]], Times[Optional[Pattern[c, Blank[]]], Power[Pattern[x, Blank[]], 2]]], Pattern[p, Blank[]]], Power[Plus[Optional[Pattern[d, Blank[]]], Times[Optional[Pattern[e, Blank[]]], Pattern[x, Blank[]]], Times[Optional[Pattern[f, Blank[]]], Power[Pattern[x, Blank[]], 2]]], Pattern[q, Blank[]]]], Pattern[x, Blank[Symbol]]] := Simp[((2*a*c^2*e - b^2*c*e + b^3*f + b*c*(c*d - 3*a*f) + c*(2*c^2*d + b^2*f - c*(b*e + 2*a*f))*x)*(a + b*x + c*x^2)^(p + 1)*(d + e*x + f*x^2)^(q + 1))/((b^2 - 4*a*c)*((c*d - a*f)^2 - (b*d - a*e)*(c*e - b*f))*(p + 1)), x] - Dist[1/((b^2 - 4*a*c)*((c*d - a*f)^2 - (b*d - a*e)*(c*e - b*f))*(p + 1)), Int[(a + b*x + c*x^2)^(p + 1)*(d + e*x + f*x^2)^q*Simp[2*c*((c*d - a*f)^2 - (b*d - a*e)*(c*e - b*f))*(p + 1) - (2*c^2*d + b^2*f - c*(b*e + 2*a*f))*(a*f*(p + 1) - c*d*(p + 2)) - e*(b^2*c*e - 2*a*c^2*e - b^3*f - b*c*(c*d - 3*a*f))*(p + q + 2) + (2*f*(2*a*c^2*e - b^2*c*e + b^3*f + b*c*(c*d - 3*a*f))*(p + q + 2) - (2*c^2*d + b^2*f - c*(b*e + 2*a*f))*(b*f*(p + 1) - c*e*(2*p + q + 4)))*x + c*f*(2*c^2*d + b^2*f - c*(b*e + 2*a*f))*(2*p + 2*q + 5)*x^2, x], x], x] /; FreeQ[{a, b, c, d, e, f, q}, x] && NeQ[b^2 - 4*a*c, 0] && NeQ[e^2 - 4*d*f, 0] && LtQ[p, -1] && NeQ[(c*d - a*f)^2 - (b*d - a*e)*(c*e - b*f), 0] && !( !IntegerQ[p] && ILtQ[q, -1]) && !IGtQ[q, 0]
  975. Int[Times[Power[Plus[Optional[Pattern[a, Blank[]]], Times[Optional[Pattern[b, Blank[]]], Pattern[x, Blank[]]], Times[Optional[Pattern[c, Blank[]]], Power[Pattern[x, Blank[]], 2]]], Pattern[p, Blank[]]], Power[Plus[Optional[Pattern[d, Blank[]]], Times[Optional[Pattern[f, Blank[]]], Power[Pattern[x, Blank[]], 2]]], Pattern[q, Blank[]]]], Pattern[x, Blank[Symbol]]] := Simp[((b^3*f + b*c*(c*d - 3*a*f) + c*(2*c^2*d + b^2*f - c*(2*a*f))*x)*(a + b*x + c*x^2)^(p + 1)*(d + f*x^2)^(q + 1))/((b^2 - 4*a*c)*(b^2*d*f + (c*d - a*f)^2)*(p + 1)), x] - Dist[1/((b^2 - 4*a*c)*(b^2*d*f + (c*d - a*f)^2)*(p + 1)), Int[(a + b*x + c*x^2)^(p + 1)*(d + f*x^2)^q*Simp[2*c*(b^2*d*f + (c*d - a*f)^2)*(p + 1) - (2*c^2*d + b^2*f - c*(2*a*f))*(a*f*(p + 1) - c*d*(p + 2)) + (2*f*(b^3*f + b*c*(c*d - 3*a*f))*(p + q + 2) - (2*c^2*d + b^2*f - c*(2*a*f))*(b*f*(p + 1)))*x + c*f*(2*c^2*d + b^2*f - c*(2*a*f))*(2*p + 2*q + 5)*x^2, x], x], x] /; FreeQ[{a, b, c, d, f, q}, x] && NeQ[b^2 - 4*a*c, 0] && LtQ[p, -1] && NeQ[b^2*d*f + (c*d - a*f)^2, 0] && !( !IntegerQ[p] && ILtQ[q, -1]) && !IGtQ[q, 0]
  976. Int[Times[Power[Plus[Optional[Pattern[a, Blank[]]], Times[Optional[Pattern[c, Blank[]]], Power[Pattern[x, Blank[]], 2]]], Pattern[p, Blank[]]], Power[Plus[Optional[Pattern[d, Blank[]]], Times[Optional[Pattern[e, Blank[]]], Pattern[x, Blank[]]], Times[Optional[Pattern[f, Blank[]]], Power[Pattern[x, Blank[]], 2]]], Pattern[q, Blank[]]]], Pattern[x, Blank[Symbol]]] := Simp[((2*a*c^2*e + c*(2*c^2*d - c*(2*a*f))*x)*(a + c*x^2)^(p + 1)*(d + e*x + f*x^2)^(q + 1))/((-4*a*c)*(a*c*e^2 + (c*d - a*f)^2)*(p + 1)), x] - Dist[1/((-4*a*c)*(a*c*e^2 + (c*d - a*f)^2)*(p + 1)), Int[(a + c*x^2)^(p + 1)*(d + e*x + f*x^2)^q*Simp[2*c*((c*d - a*f)^2 - (-(a*e))*(c*e))*(p + 1) - (2*c^2*d - c*(2*a*f))*(a*f*(p + 1) - c*d*(p + 2)) - e*(-2*a*c^2*e)*(p + q + 2) + (2*f*(2*a*c^2*e)*(p + q + 2) - (2*c^2*d - c*(2*a*f))*(-(c*e*(2*p + q + 4))))*x + c*f*(2*c^2*d - c*(2*a*f))*(2*p + 2*q + 5)*x^2, x], x], x] /; FreeQ[{a, c, d, e, f, q}, x] && NeQ[e^2 - 4*d*f, 0] && LtQ[p, -1] && NeQ[a*c*e^2 + (c*d - a*f)^2, 0] && !( !IntegerQ[p] && ILtQ[q, -1]) && !IGtQ[q, 0]
  977. Int[Times[Power[Plus[Optional[Pattern[a, Blank[]]], Times[Optional[Pattern[b, Blank[]]], Pattern[x, Blank[]]], Times[Optional[Pattern[c, Blank[]]], Power[Pattern[x, Blank[]], 2]]], Pattern[p, Blank[]]], Power[Plus[Optional[Pattern[d, Blank[]]], Times[Optional[Pattern[e, Blank[]]], Pattern[x, Blank[]]], Times[Optional[Pattern[f, Blank[]]], Power[Pattern[x, Blank[]], 2]]], Pattern[q, Blank[]]]], Pattern[x, Blank[Symbol]]] := Simp[((b*f*(3*p + 2*q) - c*e*(2*p + q) + 2*c*f*(p + q)*x)*(a + b*x + c*x^2)^(p - 1)*(d + e*x + f*x^2)^(q + 1))/(2*f^2*(p + q)*(2*p + 2*q + 1)), x] - Dist[1/(2*f^2*(p + q)*(2*p + 2*q + 1)), Int[(a + b*x + c*x^2)^(p - 2)*(d + e*x + f*x^2)^q*Simp[(b*d - a*e)*(c*e - b*f)*(1 - p)*(2*p + q) - (p + q)*(b^2*d*f*(1 - p) - a*(f*(b*e - 2*a*f)*(2*p + 2*q + 1) + c*(2*d*f - e^2*(2*p + q)))) + (2*(c*d - a*f)*(c*e - b*f)*(1 - p)*(2*p + q) - (p + q)*((b^2 - 4*a*c)*e*f*(1 - p) + b*(c*(e^2 - 4*d*f)*(2*p + q) + f*(2*c*d - b*e + 2*a*f)*(2*p + 2*q + 1))))*x + ((c*e - b*f)^2*(1 - p)*p + c*(p + q)*(f*(b*e - 2*a*f)*(4*p + 2*q - 1) - c*(2*d*f*(1 - 2*p) + e^2*(3*p + q - 1))))*x^2, x], x], x] /; FreeQ[{a, b, c, d, e, f, q}, x] && NeQ[b^2 - 4*a*c, 0] && NeQ[e^2 - 4*d*f, 0] && GtQ[p, 1] && NeQ[p + q, 0] && NeQ[2*p + 2*q + 1, 0] && !IGtQ[p, 0] && !IGtQ[q, 0]
  978. Int[Times[Power[Plus[Optional[Pattern[a, Blank[]]], Times[Optional[Pattern[b, Blank[]]], Pattern[x, Blank[]]], Times[Optional[Pattern[c, Blank[]]], Power[Pattern[x, Blank[]], 2]]], Pattern[p, Blank[]]], Power[Plus[Optional[Pattern[d, Blank[]]], Times[Optional[Pattern[f, Blank[]]], Power[Pattern[x, Blank[]], 2]]], Pattern[q, Blank[]]]], Pattern[x, Blank[Symbol]]] := Simp[((b*(3*p + 2*q) + 2*c*(p + q)*x)*(a + b*x + c*x^2)^(p - 1)*(d + f*x^2)^(q + 1))/(2*f*(p + q)*(2*p + 2*q + 1)), x] - Dist[1/(2*f*(p + q)*(2*p + 2*q + 1)), Int[(a + b*x + c*x^2)^(p - 2)*(d + f*x^2)^q*Simp[b^2*d*(p - 1)*(2*p + q) - (p + q)*(b^2*d*(1 - p) - 2*a*(c*d - a*f*(2*p + 2*q + 1))) - (2*b*(c*d - a*f)*(1 - p)*(2*p + q) - 2*(p + q)*b*(2*c*d*(2*p + q) - (c*d + a*f)*(2*p + 2*q + 1)))*x + (b^2*f*p*(1 - p) + 2*c*(p + q)*(c*d*(2*p - 1) - a*f*(4*p + 2*q - 1)))*x^2, x], x], x] /; FreeQ[{a, b, c, d, f, q}, x] && NeQ[b^2 - 4*a*c, 0] && GtQ[p, 1] && NeQ[p + q, 0] && NeQ[2*p + 2*q + 1, 0] && !IGtQ[p, 0] && !IGtQ[q, 0]
  979. Int[Times[Power[Plus[Optional[Pattern[a, Blank[]]], Times[Optional[Pattern[c, Blank[]]], Power[Pattern[x, Blank[]], 2]]], Pattern[p, Blank[]]], Power[Plus[Optional[Pattern[d, Blank[]]], Times[Optional[Pattern[e, Blank[]]], Pattern[x, Blank[]]], Times[Optional[Pattern[f, Blank[]]], Power[Pattern[x, Blank[]], 2]]], Pattern[q, Blank[]]]], Pattern[x, Blank[Symbol]]] := -Simp[(c*(e*(2*p + q) - 2*f*(p + q)*x)*(a + c*x^2)^(p - 1)*(d + e*x + f*x^2)^(q + 1))/(2*f^2*(p + q)*(2*p + 2*q + 1)), x] - Dist[1/(2*f^2*(p + q)*(2*p + 2*q + 1)), Int[(a + c*x^2)^(p - 2)*(d + e*x + f*x^2)^q*Simp[-(a*c*e^2*(1 - p)*(2*p + q)) + a*(p + q)*(-2*a*f^2*(2*p + 2*q + 1) + c*(2*d*f - e^2*(2*p + q))) + (2*(c*d - a*f)*(c*e)*(1 - p)*(2*p + q) + 4*a*c*e*f*(1 - p)*(p + q))*x + (p*c^2*e^2*(1 - p) - c*(p + q)*(2*a*f^2*(4*p + 2*q - 1) + c*(2*d*f*(1 - 2*p) + e^2*(3*p + q - 1))))*x^2, x], x], x] /; FreeQ[{a, c, d, e, f, q}, x] && NeQ[e^2 - 4*d*f, 0] && GtQ[p, 1] && NeQ[p + q, 0] && NeQ[2*p + 2*q + 1, 0] && !IGtQ[p, 0] && !IGtQ[q, 0]
  980. Int[Times[Power[Plus[Pattern[a, Blank[]], Times[Optional[Pattern[b, Blank[]]], Pattern[x, Blank[]]], Times[Optional[Pattern[c, Blank[]]], Power[Pattern[x, Blank[]], 2]]], -1], Power[Plus[Pattern[d, Blank[]], Times[Optional[Pattern[e, Blank[]]], Pattern[x, Blank[]]], Times[Optional[Pattern[f, Blank[]]], Power[Pattern[x, Blank[]], 2]]], -1]], Pattern[x, Blank[Symbol]]] := With[{q = c^2*d^2 - b*c*d*e + a*c*e^2 + b^2*d*f - 2*a*c*d*f - a*b*e*f + a^2*f^2}, Dist[1/q, Int[(c^2*d - b*c*e + b^2*f - a*c*f - (c^2*e - b*c*f)*x)/(a + b*x + c*x^2), x], x] + Dist[1/q, Int[(c*e^2 - c*d*f - b*e*f + a*f^2 + (c*e*f - b*f^2)*x)/(d + e*x + f*x^2), x], x] /; NeQ[q, 0]] /; FreeQ[{a, b, c, d, e, f}, x] && NeQ[b^2 - 4*a*c, 0] && NeQ[e^2 - 4*d*f, 0]
  981. Int[Times[Power[Plus[Pattern[a, Blank[]], Times[Optional[Pattern[b, Blank[]]], Pattern[x, Blank[]]], Times[Optional[Pattern[c, Blank[]]], Power[Pattern[x, Blank[]], 2]]], -1], Power[Plus[Pattern[d, Blank[]], Times[Optional[Pattern[f, Blank[]]], Power[Pattern[x, Blank[]], 2]]], -1]], Pattern[x, Blank[Symbol]]] := With[{q = c^2*d^2 + b^2*d*f - 2*a*c*d*f + a^2*f^2}, Dist[1/q, Int[(c^2*d + b^2*f - a*c*f + b*c*f*x)/(a + b*x + c*x^2), x], x] - Dist[1/q, Int[(c*d*f - a*f^2 + b*f^2*x)/(d + f*x^2), x], x] /; NeQ[q, 0]] /; FreeQ[{a, b, c, d, f}, x] && NeQ[b^2 - 4*a*c, 0]
  982. Int[Times[Power[Plus[Pattern[a, Blank[]], Times[Optional[Pattern[b, Blank[]]], Pattern[x, Blank[]]], Times[Optional[Pattern[c, Blank[]]], Power[Pattern[x, Blank[]], 2]]], -1], Power[Plus[Optional[Pattern[d, Blank[]]], Times[Optional[Pattern[e, Blank[]]], Pattern[x, Blank[]]], Times[Optional[Pattern[f, Blank[]]], Power[Pattern[x, Blank[]], 2]]], Rational[-1, 2]]], Pattern[x, Blank[Symbol]]] := Dist[-2*e, Subst[Int[1/(e*(b*e - 4*a*f) - (b*d - a*e)*x^2), x], x, (e + 2*f*x)/Sqrt[d + e*x + f*x^2]], x] /; FreeQ[{a, b, c, d, e, f}, x] && NeQ[b^2 - 4*a*c, 0] && NeQ[e^2 - 4*d*f, 0] && EqQ[c*e - b*f, 0]
  983. Int[Times[Power[Plus[Pattern[a, Blank[]], Times[Optional[Pattern[b, Blank[]]], Pattern[x, Blank[]]], Times[Optional[Pattern[c, Blank[]]], Power[Pattern[x, Blank[]], 2]]], -1], Power[Plus[Optional[Pattern[d, Blank[]]], Times[Optional[Pattern[e, Blank[]]], Pattern[x, Blank[]]], Times[Optional[Pattern[f, Blank[]]], Power[Pattern[x, Blank[]], 2]]], Rational[-1, 2]]], Pattern[x, Blank[Symbol]]] := With[{q = Rt[b^2 - 4*a*c, 2]}, Dist[(2*c)/q, Int[1/((b - q + 2*c*x)*Sqrt[d + e*x + f*x^2]), x], x] - Dist[(2*c)/q, Int[1/((b + q + 2*c*x)*Sqrt[d + e*x + f*x^2]), x], x]] /; FreeQ[{a, b, c, d, e, f}, x] && NeQ[b^2 - 4*a*c, 0] && NeQ[e^2 - 4*d*f, 0] && NeQ[c*e - b*f, 0] && PosQ[b^2 - 4*a*c]
  984. Int[Times[Power[Plus[Pattern[a, Blank[]], Times[Optional[Pattern[c, Blank[]]], Power[Pattern[x, Blank[]], 2]]], -1], Power[Plus[Optional[Pattern[d, Blank[]]], Times[Optional[Pattern[e, Blank[]]], Pattern[x, Blank[]]], Times[Optional[Pattern[f, Blank[]]], Power[Pattern[x, Blank[]], 2]]], Rational[-1, 2]]], Pattern[x, Blank[Symbol]]] := Dist[1/2, Int[1/((a - Rt[-(a*c), 2]*x)*Sqrt[d + e*x + f*x^2]), x], x] + Dist[1/2, Int[1/((a + Rt[-(a*c), 2]*x)*Sqrt[d + e*x + f*x^2]), x], x] /; FreeQ[{a, c, d, e, f}, x] && NeQ[e^2 - 4*d*f, 0] && PosQ[-(a*c)]
  985. Int[Times[Power[Plus[Pattern[a, Blank[]], Times[Optional[Pattern[b, Blank[]]], Pattern[x, Blank[]]], Times[Optional[Pattern[c, Blank[]]], Power[Pattern[x, Blank[]], 2]]], -1], Power[Plus[Pattern[d, Blank[]], Times[Optional[Pattern[f, Blank[]]], Power[Pattern[x, Blank[]], 2]]], Rational[-1, 2]]], Pattern[x, Blank[Symbol]]] := With[{q = Rt[b^2 - 4*a*c, 2]}, Dist[(2*c)/q, Int[1/((b - q + 2*c*x)*Sqrt[d + f*x^2]), x], x] - Dist[(2*c)/q, Int[1/((b + q + 2*c*x)*Sqrt[d + f*x^2]), x], x]] /; FreeQ[{a, b, c, d, f}, x] && NeQ[b^2 - 4*a*c, 0] && PosQ[b^2 - 4*a*c]
  986. Int[Times[Power[Plus[Optional[Pattern[a, Blank[]]], Times[Optional[Pattern[b, Blank[]]], Pattern[x, Blank[]]], Times[Optional[Pattern[c, Blank[]]], Power[Pattern[x, Blank[]], 2]]], -1], Power[Plus[Optional[Pattern[d, Blank[]]], Times[Optional[Pattern[e, Blank[]]], Pattern[x, Blank[]]], Times[Optional[Pattern[f, Blank[]]], Power[Pattern[x, Blank[]], 2]]], Rational[-1, 2]]], Pattern[x, Blank[Symbol]]] := With[{q = Rt[(c*d - a*f)^2 - (b*d - a*e)*(c*e - b*f), 2]}, Dist[1/(2*q), Int[(c*d - a*f + q + (c*e - b*f)*x)/((a + b*x + c*x^2)*Sqrt[d + e*x + f*x^2]), x], x] - Dist[1/(2*q), Int[(c*d - a*f - q + (c*e - b*f)*x)/((a + b*x + c*x^2)*Sqrt[d + e*x + f*x^2]), x], x]] /; FreeQ[{a, b, c, d, e, f}, x] && NeQ[b^2 - 4*a*c, 0] && NeQ[e^2 - 4*d*f, 0] && NeQ[c*e - b*f, 0] && NegQ[b^2 - 4*a*c]
  987. Int[Times[Power[Plus[Optional[Pattern[a, Blank[]]], Times[Optional[Pattern[c, Blank[]]], Power[Pattern[x, Blank[]], 2]]], -1], Power[Plus[Optional[Pattern[d, Blank[]]], Times[Optional[Pattern[e, Blank[]]], Pattern[x, Blank[]]], Times[Optional[Pattern[f, Blank[]]], Power[Pattern[x, Blank[]], 2]]], Rational[-1, 2]]], Pattern[x, Blank[Symbol]]] := With[{q = Rt[(c*d - a*f)^2 + a*c*e^2, 2]}, Dist[1/(2*q), Int[(c*d - a*f + q + c*e*x)/((a + c*x^2)*Sqrt[d + e*x + f*x^2]), x], x] - Dist[1/(2*q), Int[(c*d - a*f - q + c*e*x)/((a + c*x^2)*Sqrt[d + e*x + f*x^2]), x], x]] /; FreeQ[{a, c, d, e, f}, x] && NeQ[e^2 - 4*d*f, 0] && NegQ[-(a*c)]
  988. Int[Times[Power[Plus[Optional[Pattern[a, Blank[]]], Times[Optional[Pattern[b, Blank[]]], Pattern[x, Blank[]]], Times[Optional[Pattern[c, Blank[]]], Power[Pattern[x, Blank[]], 2]]], -1], Power[Plus[Optional[Pattern[d, Blank[]]], Times[Optional[Pattern[f, Blank[]]], Power[Pattern[x, Blank[]], 2]]], Rational[-1, 2]]], Pattern[x, Blank[Symbol]]] := With[{q = Rt[(c*d - a*f)^2 + b^2*d*f, 2]}, Dist[1/(2*q), Int[(c*d - a*f + q + (-(b*f))*x)/((a + b*x + c*x^2)*Sqrt[d + f*x^2]), x], x] - Dist[1/(2*q), Int[(c*d - a*f - q + (-(b*f))*x)/((a + b*x + c*x^2)*Sqrt[d + f*x^2]), x], x]] /; FreeQ[{a, b, c, d, f}, x] && NeQ[b^2 - 4*a*c, 0] && NegQ[b^2 - 4*a*c]
  989. Int[Times[Power[Plus[Pattern[a, Blank[]], Times[Optional[Pattern[b, Blank[]]], Pattern[x, Blank[]]], Times[Optional[Pattern[c, Blank[]]], Power[Pattern[x, Blank[]], 2]]], Rational[1, 2]], Power[Plus[Pattern[d, Blank[]], Times[Optional[Pattern[e, Blank[]]], Pattern[x, Blank[]]], Times[Optional[Pattern[f, Blank[]]], Power[Pattern[x, Blank[]], 2]]], -1]], Pattern[x, Blank[Symbol]]] := Dist[c/f, Int[1/Sqrt[a + b*x + c*x^2], x], x] - Dist[1/f, Int[(c*d - a*f + (c*e - b*f)*x)/(Sqrt[a + b*x + c*x^2]*(d + e*x + f*x^2)), x], x] /; FreeQ[{a, b, c, d, e, f}, x] && NeQ[b^2 - 4*a*c, 0] && NeQ[e^2 - 4*d*f, 0]
  990. Int[Times[Power[Plus[Pattern[a, Blank[]], Times[Optional[Pattern[b, Blank[]]], Pattern[x, Blank[]]], Times[Optional[Pattern[c, Blank[]]], Power[Pattern[x, Blank[]], 2]]], Rational[1, 2]], Power[Plus[Pattern[d, Blank[]], Times[Optional[Pattern[f, Blank[]]], Power[Pattern[x, Blank[]], 2]]], -1]], Pattern[x, Blank[Symbol]]] := Dist[c/f, Int[1/Sqrt[a + b*x + c*x^2], x], x] - Dist[1/f, Int[(c*d - a*f - b*f*x)/(Sqrt[a + b*x + c*x^2]*(d + f*x^2)), x], x] /; FreeQ[{a, b, c, d, f}, x] && NeQ[b^2 - 4*a*c, 0]
  991. Int[Times[Power[Plus[Pattern[a, Blank[]], Times[Optional[Pattern[c, Blank[]]], Power[Pattern[x, Blank[]], 2]]], Rational[1, 2]], Power[Plus[Pattern[d, Blank[]], Times[Optional[Pattern[e, Blank[]]], Pattern[x, Blank[]]], Times[Optional[Pattern[f, Blank[]]], Power[Pattern[x, Blank[]], 2]]], -1]], Pattern[x, Blank[Symbol]]] := Dist[c/f, Int[1/Sqrt[a + c*x^2], x], x] - Dist[1/f, Int[(c*d - a*f + c*e*x)/(Sqrt[a + c*x^2]*(d + e*x + f*x^2)), x], x] /; FreeQ[{a, c, d, e, f}, x] && NeQ[e^2 - 4*d*f, 0]
  992. Int[Times[Power[Plus[Pattern[a, Blank[]], Times[Optional[Pattern[b, Blank[]]], Pattern[x, Blank[]]], Times[Optional[Pattern[c, Blank[]]], Power[Pattern[x, Blank[]], 2]]], Rational[-1, 2]], Power[Plus[Pattern[d, Blank[]], Times[Optional[Pattern[e, Blank[]]], Pattern[x, Blank[]]], Times[Optional[Pattern[f, Blank[]]], Power[Pattern[x, Blank[]], 2]]], Rational[-1, 2]]], Pattern[x, Blank[Symbol]]] := With[{r = Rt[b^2 - 4*a*c, 2]}, Dist[(Sqrt[b + r + 2*c*x]*Sqrt[2*a + (b + r)*x])/Sqrt[a + b*x + c*x^2], Int[1/(Sqrt[b + r + 2*c*x]*Sqrt[2*a + (b + r)*x]*Sqrt[d + e*x + f*x^2]), x], x]] /; FreeQ[{a, b, c, d, e, f}, x] && NeQ[b^2 - 4*a*c, 0] && NeQ[e^2 - 4*d*f, 0]
  993. Int[Times[Power[Plus[Pattern[a, Blank[]], Times[Optional[Pattern[b, Blank[]]], Pattern[x, Blank[]]], Times[Optional[Pattern[c, Blank[]]], Power[Pattern[x, Blank[]], 2]]], Rational[-1, 2]], Power[Plus[Pattern[d, Blank[]], Times[Optional[Pattern[f, Blank[]]], Power[Pattern[x, Blank[]], 2]]], Rational[-1, 2]]], Pattern[x, Blank[Symbol]]] := With[{r = Rt[b^2 - 4*a*c, 2]}, Dist[(Sqrt[b + r + 2*c*x]*Sqrt[2*a + (b + r)*x])/Sqrt[a + b*x + c*x^2], Int[1/(Sqrt[b + r + 2*c*x]*Sqrt[2*a + (b + r)*x]*Sqrt[d + f*x^2]), x], x]] /; FreeQ[{a, b, c, d, f}, x] && NeQ[b^2 - 4*a*c, 0]
  994. Int[Times[Power[Plus[Optional[Pattern[a, Blank[]]], Times[Optional[Pattern[b, Blank[]]], Pattern[x, Blank[]]], Times[Optional[Pattern[c, Blank[]]], Power[Pattern[x, Blank[]], 2]]], Pattern[p, Blank[]]], Power[Plus[Optional[Pattern[d, Blank[]]], Times[Optional[Pattern[e, Blank[]]], Pattern[x, Blank[]]], Times[Optional[Pattern[f, Blank[]]], Power[Pattern[x, Blank[]], 2]]], Pattern[q, Blank[]]]], Pattern[x, Blank[Symbol]]] := Unintegrable[(a + b*x + c*x^2)^p*(d + e*x + f*x^2)^q, x] /; FreeQ[{a, b, c, d, e, f, p, q}, x] && !IGtQ[p, 0] && !IGtQ[q, 0]
  995. Int[Times[Power[Plus[Pattern[a, Blank[]], Times[Optional[Pattern[c, Blank[]]], Power[Pattern[x, Blank[]], 2]]], Pattern[p, Blank[]]], Power[Plus[Optional[Pattern[d, Blank[]]], Times[Optional[Pattern[e, Blank[]]], Pattern[x, Blank[]]], Times[Optional[Pattern[f, Blank[]]], Power[Pattern[x, Blank[]], 2]]], Pattern[q, Blank[]]]], Pattern[x, Blank[Symbol]]] := Unintegrable[(a + c*x^2)^p*(d + e*x + f*x^2)^q, x] /; FreeQ[{a, c, d, e, f, p, q}, x] && !IGtQ[p, 0] && !IGtQ[q, 0]
  996. Int[Times[Power[Plus[Optional[Pattern[a, Blank[]]], Times[Optional[Pattern[b, Blank[]]], Pattern[u, Blank[]]], Times[Optional[Pattern[c, Blank[]]], Power[Pattern[u, Blank[]], 2]]], Optional[Pattern[p, Blank[]]]], Power[Plus[Optional[Pattern[d, Blank[]]], Times[Optional[Pattern[e, Blank[]]], Pattern[u, Blank[]]], Times[Optional[Pattern[f, Blank[]]], Power[Pattern[u, Blank[]], 2]]], Optional[Pattern[q, Blank[]]]]], Pattern[x, Blank[Symbol]]] := Dist[1/Coefficient[u, x, 1], Subst[Int[(a + b*x + c*x^2)^p*(d + e*x + f*x^2)^q, x], x, u], x] /; FreeQ[{a, b, c, d, e, f, p, q}, x] && LinearQ[u, x] && NeQ[u, x]
  997. Int[Times[Power[Plus[Optional[Pattern[a, Blank[]]], Times[Optional[Pattern[c, Blank[]]], Power[Pattern[u, Blank[]], 2]]], Optional[Pattern[p, Blank[]]]], Power[Plus[Optional[Pattern[d, Blank[]]], Times[Optional[Pattern[e, Blank[]]], Pattern[u, Blank[]]], Times[Optional[Pattern[f, Blank[]]], Power[Pattern[u, Blank[]], 2]]], Optional[Pattern[q, Blank[]]]]], Pattern[x, Blank[Symbol]]] := Dist[1/Coefficient[u, x, 1], Subst[Int[(a + c*x^2)^p*(d + e*x + f*x^2)^q, x], x, u], x] /; FreeQ[{a, c, d, e, f, p, q}, x] && LinearQ[u, x] && NeQ[u, x]
  998. Int[Times[Power[Plus[Optional[Pattern[g, Blank[]]], Times[Optional[Pattern[h, Blank[]]], Pattern[x, Blank[]]]], Optional[Pattern[m, Blank[]]]], Power[Plus[Pattern[a, Blank[]], Times[Optional[Pattern[b, Blank[]]], Pattern[x, Blank[]]], Times[Optional[Pattern[c, Blank[]]], Power[Pattern[x, Blank[]], 2]]], Pattern[p, Blank[]]], Power[Plus[Pattern[d, Blank[]], Times[Optional[Pattern[e, Blank[]]], Pattern[x, Blank[]]], Times[Optional[Pattern[f, Blank[]]], Power[Pattern[x, Blank[]], 2]]], Pattern[q, Blank[]]]], Pattern[x, Blank[Symbol]]] := Dist[(c/f)^p, Int[(g + h*x)^m*(d + e*x + f*x^2)^(p + q), x], x] /; FreeQ[{a, b, c, d, e, f, g, h, p, q}, x] && EqQ[c*d - a*f, 0] && EqQ[b*d - a*e, 0] && (IntegerQ[p] || GtQ[c/f, 0]) && ( !IntegerQ[q] || LeafCount[d + e*x + f*x^2] <= LeafCount[a + b*x + c*x^2])
  999. Int[Times[Power[Plus[Optional[Pattern[g, Blank[]]], Times[Optional[Pattern[h, Blank[]]], Pattern[x, Blank[]]]], Optional[Pattern[m, Blank[]]]], Power[Plus[Pattern[a, Blank[]], Times[Optional[Pattern[b, Blank[]]], Pattern[x, Blank[]]], Times[Optional[Pattern[c, Blank[]]], Power[Pattern[x, Blank[]], 2]]], Pattern[p, Blank[]]], Power[Plus[Pattern[d, Blank[]], Times[Optional[Pattern[e, Blank[]]], Pattern[x, Blank[]]], Times[Optional[Pattern[f, Blank[]]], Power[Pattern[x, Blank[]], 2]]], Pattern[q, Blank[]]]], Pattern[x, Blank[Symbol]]] := Dist[(a^IntPart[p]*(a + b*x + c*x^2)^FracPart[p])/(d^IntPart[p]*(d + e*x + f*x^2)^FracPart[p]), Int[(g + h*x)^m*(d + e*x + f*x^2)^(p + q), x], x] /; FreeQ[{a, b, c, d, e, f, g, h, p, q}, x] && EqQ[c*d - a*f, 0] && EqQ[b*d - a*e, 0] && !IntegerQ[p] && !IntegerQ[q] && !GtQ[c/f, 0]
  1000. Int[Times[Power[Plus[Optional[Pattern[g, Blank[]]], Times[Optional[Pattern[h, Blank[]]], Pattern[x, Blank[]]]], Optional[Pattern[m, Blank[]]]], Power[Plus[Pattern[a, Blank[]], Times[Optional[Pattern[b, Blank[]]], Pattern[x, Blank[]]], Times[Optional[Pattern[c, Blank[]]], Power[Pattern[x, Blank[]], 2]]], Pattern[p, Blank[]]], Power[Plus[Pattern[d, Blank[]], Times[Optional[Pattern[e, Blank[]]], Pattern[x, Blank[]]], Times[Optional[Pattern[f, Blank[]]], Power[Pattern[x, Blank[]], 2]]], Pattern[q, Blank[]]]], Pattern[x, Blank[Symbol]]] := Dist[(a + b*x + c*x^2)^FracPart[p]/((4*c)^IntPart[p]*(b + 2*c*x)^(2*FracPart[p])), Int[(g + h*x)^m*(b + 2*c*x)^(2*p)*(d + e*x + f*x^2)^q, x], x] /; FreeQ[{a, b, c, d, e, f, g, h, m, p, q}, x] && EqQ[b^2 - 4*a*c, 0]
  1001. Int[Times[Power[Plus[Optional[Pattern[g, Blank[]]], Times[Optional[Pattern[h, Blank[]]], Pattern[x, Blank[]]]], Optional[Pattern[m, Blank[]]]], Power[Plus[Pattern[a, Blank[]], Times[Optional[Pattern[b, Blank[]]], Pattern[x, Blank[]]], Times[Optional[Pattern[c, Blank[]]], Power[Pattern[x, Blank[]], 2]]], Pattern[p, Blank[]]], Power[Plus[Pattern[d, Blank[]], Times[Optional[Pattern[f, Blank[]]], Power[Pattern[x, Blank[]], 2]]], Pattern[q, Blank[]]]], Pattern[x, Blank[Symbol]]] := Dist[(a + b*x + c*x^2)^FracPart[p]/((4*c)^IntPart[p]*(b + 2*c*x)^(2*FracPart[p])), Int[(g + h*x)^m*(b + 2*c*x)^(2*p)*(d + f*x^2)^q, x], x] /; FreeQ[{a, b, c, d, f, g, h, m, p, q}, x] && EqQ[b^2 - 4*a*c, 0]
  1002. Int[Times[Power[Plus[Pattern[g, Blank[]], Times[Optional[Pattern[h, Blank[]]], Pattern[x, Blank[]]]], Optional[Pattern[m, Blank[]]]], Power[Plus[Pattern[a, Blank[]], Times[Optional[Pattern[b, Blank[]]], Pattern[x, Blank[]]], Times[Optional[Pattern[c, Blank[]]], Power[Pattern[x, Blank[]], 2]]], Pattern[p, Blank[]]], Power[Plus[Optional[Pattern[d, Blank[]]], Times[Optional[Pattern[e, Blank[]]], Pattern[x, Blank[]]], Times[Optional[Pattern[f, Blank[]]], Power[Pattern[x, Blank[]], 2]]], Optional[Pattern[m, Blank[]]]]], Pattern[x, Blank[Symbol]]] := Int[((d*g)/a + (f*h*x)/c)^m*(a + b*x + c*x^2)^(m + p), x] /; FreeQ[{a, b, c, d, e, f, g, h, p}, x] && EqQ[c*g^2 - b*g*h + a*h^2, 0] && EqQ[c^2*d*g^2 - a*c*e*g*h + a^2*f*h^2, 0] && IntegerQ[m]
  1003. Int[Times[Power[Plus[Pattern[g, Blank[]], Times[Optional[Pattern[h, Blank[]]], Pattern[x, Blank[]]]], Optional[Pattern[m, Blank[]]]], Power[Plus[Pattern[a, Blank[]], Times[Optional[Pattern[c, Blank[]]], Power[Pattern[x, Blank[]], 2]]], Pattern[p, Blank[]]], Power[Plus[Optional[Pattern[d, Blank[]]], Times[Optional[Pattern[e, Blank[]]], Pattern[x, Blank[]]], Times[Optional[Pattern[f, Blank[]]], Power[Pattern[x, Blank[]], 2]]], Optional[Pattern[m, Blank[]]]]], Pattern[x, Blank[Symbol]]] := Int[((d*g)/a + (f*h*x)/c)^m*(a + c*x^2)^(m + p), x] /; FreeQ[{a, c, d, e, f, g, h, p}, x] && EqQ[c*g^2 + a*h^2, 0] && EqQ[c^2*d*g^2 - a*c*e*g*h + a^2*f*h^2, 0] && IntegerQ[m]
  1004. Int[Times[Power[Plus[Pattern[g, Blank[]], Times[Optional[Pattern[h, Blank[]]], Pattern[x, Blank[]]]], Optional[Pattern[m, Blank[]]]], Power[Plus[Pattern[a, Blank[]], Times[Optional[Pattern[b, Blank[]]], Pattern[x, Blank[]]], Times[Optional[Pattern[c, Blank[]]], Power[Pattern[x, Blank[]], 2]]], Pattern[p, Blank[]]], Power[Plus[Optional[Pattern[d, Blank[]]], Times[Optional[Pattern[f, Blank[]]], Power[Pattern[x, Blank[]], 2]]], Optional[Pattern[m, Blank[]]]]], Pattern[x, Blank[Symbol]]] := Int[((d*g)/a + (f*h*x)/c)^m*(a + b*x + c*x^2)^(m + p), x] /; FreeQ[{a, b, c, d, f, g, h, p}, x] && EqQ[c*g^2 - b*g*h + a*h^2, 0] && EqQ[c^2*d*g^2 + a^2*f*h^2, 0] && IntegerQ[m]
  1005. Int[Times[Power[Plus[Pattern[g, Blank[]], Times[Optional[Pattern[h, Blank[]]], Pattern[x, Blank[]]]], Optional[Pattern[m, Blank[]]]], Power[Plus[Pattern[a, Blank[]], Times[Optional[Pattern[c, Blank[]]], Power[Pattern[x, Blank[]], 2]]], Pattern[p, Blank[]]], Power[Plus[Optional[Pattern[d, Blank[]]], Times[Optional[Pattern[f, Blank[]]], Power[Pattern[x, Blank[]], 2]]], Optional[Pattern[m, Blank[]]]]], Pattern[x, Blank[Symbol]]] := Int[((d*g)/a + (f*h*x)/c)^m*(a + c*x^2)^(m + p), x] /; FreeQ[{a, c, d, f, g, h, p}, x] && EqQ[c*g^2 + a*h^2, 0] && EqQ[c^2*d*g^2 + a^2*f*h^2, 0] && IntegerQ[m]
  1006. Int[Times[Power[Pattern[x, Blank[]], Pattern[p, Blank[]]], Power[Plus[Optional[Pattern[a, Blank[]]], Times[Optional[Pattern[b, Blank[]]], Pattern[x, Blank[]]], Times[Optional[Pattern[c, Blank[]]], Power[Pattern[x, Blank[]], 2]]], Pattern[p, Blank[]]], Power[Plus[Times[Optional[Pattern[e, Blank[]]], Pattern[x, Blank[]]], Times[Optional[Pattern[f, Blank[]]], Power[Pattern[x, Blank[]], 2]]], Pattern[q, Blank[]]]], Pattern[x, Blank[Symbol]]] := Int[(a/e + (c*x)/f)^p*(e*x + f*x^2)^(p + q), x] /; FreeQ[{a, b, c, e, f, q}, x] && NeQ[b^2 - 4*a*c, 0] && EqQ[c*e^2 - b*e*f + a*f^2, 0] && IntegerQ[p]
  1007. Int[Times[Power[Pattern[x, Blank[]], Pattern[p, Blank[]]], Power[Plus[Pattern[a, Blank[]], Times[Optional[Pattern[c, Blank[]]], Power[Pattern[x, Blank[]], 2]]], Pattern[p, Blank[]]], Power[Plus[Times[Optional[Pattern[e, Blank[]]], Pattern[x, Blank[]]], Times[Optional[Pattern[f, Blank[]]], Power[Pattern[x, Blank[]], 2]]], Pattern[q, Blank[]]]], Pattern[x, Blank[Symbol]]] := Int[(a/e + (c*x)/f)^p*(e*x + f*x^2)^(p + q), x] /; FreeQ[{a, c, e, f, q}, x] && EqQ[c*e^2 + a*f^2, 0] && IntegerQ[p]
  1008. Int[Times[Plus[Pattern[g, Blank[]], Times[Optional[Pattern[h, Blank[]]], Pattern[x, Blank[]]]], Power[Plus[Pattern[a, Blank[]], Times[Optional[Pattern[c, Blank[]]], Power[Pattern[x, Blank[]], 2]]], Rational[-1, 3]], Power[Plus[Pattern[d, Blank[]], Times[Optional[Pattern[f, Blank[]]], Power[Pattern[x, Blank[]], 2]]], -1]], Pattern[x, Blank[Symbol]]] := Simp[(Sqrt[3]*h*ArcTan[1/Sqrt[3] - (2^(2/3)*(1 - (3*h*x)/g)^(2/3))/(Sqrt[3]*(1 + (3*h*x)/g)^(1/3))])/(2^(2/3)*a^(1/3)*f), x] + (-Simp[(3*h*Log[(1 - (3*h*x)/g)^(2/3) + 2^(1/3)*(1 + (3*h*x)/g)^(1/3)])/(2^(5/3)*a^(1/3)*f), x] + Simp[(h*Log[d + f*x^2])/(2^(5/3)*a^(1/3)*f), x]) /; FreeQ[{a, c, d, f, g, h}, x] && EqQ[c*d + 3*a*f, 0] && EqQ[c*g^2 + 9*a*h^2, 0] && GtQ[a, 0]
  1009. Int[Times[Plus[Pattern[g, Blank[]], Times[Optional[Pattern[h, Blank[]]], Pattern[x, Blank[]]]], Power[Plus[Pattern[a, Blank[]], Times[Optional[Pattern[c, Blank[]]], Power[Pattern[x, Blank[]], 2]]], Rational[-1, 3]], Power[Plus[Pattern[d, Blank[]], Times[Optional[Pattern[f, Blank[]]], Power[Pattern[x, Blank[]], 2]]], -1]], Pattern[x, Blank[Symbol]]] := Dist[(1 + (c*x^2)/a)^(1/3)/(a + c*x^2)^(1/3), Int[(g + h*x)/((1 + (c*x^2)/a)^(1/3)*(d + f*x^2)), x], x] /; FreeQ[{a, c, d, f, g, h}, x] && EqQ[c*d + 3*a*f, 0] && EqQ[c*g^2 + 9*a*h^2, 0] && !GtQ[a, 0]
  1010. Int[Times[Plus[Pattern[g, Blank[]], Times[Optional[Pattern[h, Blank[]]], Pattern[x, Blank[]]]], Power[Plus[Pattern[a, Blank[]], Times[Optional[Pattern[c, Blank[]]], Power[Pattern[x, Blank[]], 2]]], Pattern[p, Blank[]]], Power[Plus[Pattern[d, Blank[]], Times[Optional[Pattern[f, Blank[]]], Power[Pattern[x, Blank[]], 2]]], Pattern[q, Blank[]]]], Pattern[x, Blank[Symbol]]] := Dist[g, Int[(a + c*x^2)^p*(d + f*x^2)^q, x], x] + Dist[h, Int[x*(a + c*x^2)^p*(d + f*x^2)^q, x], x] /; FreeQ[{a, c, d, f, g, h, p, q}, x]
  1011. Int[Times[Plus[Optional[Pattern[g, Blank[]]], Times[Optional[Pattern[h, Blank[]]], Pattern[x, Blank[]]]], Power[Plus[Pattern[a, Blank[]], Times[Optional[Pattern[b, Blank[]]], Pattern[x, Blank[]]], Times[Optional[Pattern[c, Blank[]]], Power[Pattern[x, Blank[]], 2]]], Pattern[p, Blank[]]], Power[Plus[Pattern[d, Blank[]], Times[Optional[Pattern[e, Blank[]]], Pattern[x, Blank[]]], Times[Optional[Pattern[f, Blank[]]], Power[Pattern[x, Blank[]], 2]]], Pattern[q, Blank[]]]], Pattern[x, Blank[Symbol]]] := Int[ExpandIntegrand[(a + b*x + c*x^2)^p*(d + e*x + f*x^2)^q*(g + h*x), x], x] /; FreeQ[{a, b, c, d, e, f, g, h}, x] && NeQ[b^2 - 4*a*c, 0] && NeQ[e^2 - 4*d*f, 0] && IGtQ[p, 0] && IntegerQ[q]
  1012. Int[Times[Plus[Optional[Pattern[g, Blank[]]], Times[Optional[Pattern[h, Blank[]]], Pattern[x, Blank[]]]], Power[Plus[Pattern[a, Blank[]], Times[Optional[Pattern[c, Blank[]]], Power[Pattern[x, Blank[]], 2]]], Pattern[p, Blank[]]], Power[Plus[Pattern[d, Blank[]], Times[Optional[Pattern[e, Blank[]]], Pattern[x, Blank[]]], Times[Optional[Pattern[f, Blank[]]], Power[Pattern[x, Blank[]], 2]]], Pattern[q, Blank[]]]], Pattern[x, Blank[Symbol]]] := Int[ExpandIntegrand[(a + c*x^2)^p*(d + e*x + f*x^2)^q*(g + h*x), x], x] /; FreeQ[{a, c, d, e, f, g, h}, x] && NeQ[e^2 - 4*d*f, 0] && IntegersQ[p, q] && (GtQ[p, 0] || GtQ[q, 0])
  1013. Int[Times[Plus[Optional[Pattern[g, Blank[]]], Times[Optional[Pattern[h, Blank[]]], Pattern[x, Blank[]]]], Power[Plus[Pattern[a, Blank[]], Times[Optional[Pattern[b, Blank[]]], Pattern[x, Blank[]]], Times[Optional[Pattern[c, Blank[]]], Power[Pattern[x, Blank[]], 2]]], Pattern[p, Blank[]]], Power[Plus[Pattern[d, Blank[]], Times[Optional[Pattern[e, Blank[]]], Pattern[x, Blank[]]], Times[Optional[Pattern[f, Blank[]]], Power[Pattern[x, Blank[]], 2]]], Pattern[q, Blank[]]]], Pattern[x, Blank[Symbol]]] := Simp[((g*b - 2*a*h - (b*h - 2*g*c)*x)*(a + b*x + c*x^2)^(p + 1)*(d + e*x + f*x^2)^q)/((b^2 - 4*a*c)*(p + 1)), x] - Dist[1/((b^2 - 4*a*c)*(p + 1)), Int[(a + b*x + c*x^2)^(p + 1)*(d + e*x + f*x^2)^(q - 1)*Simp[e*q*(g*b - 2*a*h) - d*(b*h - 2*g*c)*(2*p + 3) + (2*f*q*(g*b - 2*a*h) - e*(b*h - 2*g*c)*(2*p + q + 3))*x - f*(b*h - 2*g*c)*(2*p + 2*q + 3)*x^2, x], x], x] /; FreeQ[{a, b, c, d, e, f, g, h}, x] && NeQ[b^2 - 4*a*c, 0] && NeQ[e^2 - 4*d*f, 0] && LtQ[p, -1] && GtQ[q, 0]
  1014. Int[Times[Plus[Optional[Pattern[g, Blank[]]], Times[Optional[Pattern[h, Blank[]]], Pattern[x, Blank[]]]], Power[Plus[Pattern[a, Blank[]], Times[Optional[Pattern[c, Blank[]]], Power[Pattern[x, Blank[]], 2]]], Pattern[p, Blank[]]], Power[Plus[Pattern[d, Blank[]], Times[Optional[Pattern[e, Blank[]]], Pattern[x, Blank[]]], Times[Optional[Pattern[f, Blank[]]], Power[Pattern[x, Blank[]], 2]]], Pattern[q, Blank[]]]], Pattern[x, Blank[Symbol]]] := Simp[((a*h - g*c*x)*(a + c*x^2)^(p + 1)*(d + e*x + f*x^2)^q)/(2*a*c*(p + 1)), x] + Dist[2/(4*a*c*(p + 1)), Int[(a + c*x^2)^(p + 1)*(d + e*x + f*x^2)^(q - 1)*Simp[g*c*d*(2*p + 3) - a*(h*e*q) + (g*c*e*(2*p + q + 3) - a*(2*h*f*q))*x + g*c*f*(2*p + 2*q + 3)*x^2, x], x], x] /; FreeQ[{a, c, d, e, f, g, h}, x] && NeQ[e^2 - 4*d*f, 0] && LtQ[p, -1] && GtQ[q, 0]
  1015. Int[Times[Plus[Optional[Pattern[g, Blank[]]], Times[Optional[Pattern[h, Blank[]]], Pattern[x, Blank[]]]], Power[Plus[Pattern[a, Blank[]], Times[Optional[Pattern[b, Blank[]]], Pattern[x, Blank[]]], Times[Optional[Pattern[c, Blank[]]], Power[Pattern[x, Blank[]], 2]]], Pattern[p, Blank[]]], Power[Plus[Pattern[d, Blank[]], Times[Optional[Pattern[f, Blank[]]], Power[Pattern[x, Blank[]], 2]]], Pattern[q, Blank[]]]], Pattern[x, Blank[Symbol]]] := Simp[((g*b - 2*a*h - (b*h - 2*g*c)*x)*(a + b*x + c*x^2)^(p + 1)*(d + f*x^2)^q)/((b^2 - 4*a*c)*(p + 1)), x] - Dist[1/((b^2 - 4*a*c)*(p + 1)), Int[(a + b*x + c*x^2)^(p + 1)*(d + f*x^2)^(q - 1)*Simp[-(d*(b*h - 2*g*c)*(2*p + 3)) + (2*f*q*(g*b - 2*a*h))*x - f*(b*h - 2*g*c)*(2*p + 2*q + 3)*x^2, x], x], x] /; FreeQ[{a, b, c, d, f, g, h}, x] && NeQ[b^2 - 4*a*c, 0] && LtQ[p, -1] && GtQ[q, 0]
  1016. Int[Times[Plus[Optional[Pattern[g, Blank[]]], Times[Optional[Pattern[h, Blank[]]], Pattern[x, Blank[]]]], Power[Plus[Pattern[a, Blank[]], Times[Optional[Pattern[b, Blank[]]], Pattern[x, Blank[]]], Times[Optional[Pattern[c, Blank[]]], Power[Pattern[x, Blank[]], 2]]], Pattern[p, Blank[]]], Power[Plus[Pattern[d, Blank[]], Times[Optional[Pattern[e, Blank[]]], Pattern[x, Blank[]]], Times[Optional[Pattern[f, Blank[]]], Power[Pattern[x, Blank[]], 2]]], Pattern[q, Blank[]]]], Pattern[x, Blank[Symbol]]] := Simp[((a + b*x + c*x^2)^(p + 1)*(d + e*x + f*x^2)^(q + 1)*(g*c*(2*a*c*e - b*(c*d + a*f)) + (g*b - a*h)*(2*c^2*d + b^2*f - c*(b*e + 2*a*f)) + c*(g*(2*c^2*d + b^2*f - c*(b*e + 2*a*f)) - h*(b*c*d - 2*a*c*e + a*b*f))*x))/((b^2 - 4*a*c)*((c*d - a*f)^2 - (b*d - a*e)*(c*e - b*f))*(p + 1)), x] + Dist[1/((b^2 - 4*a*c)*((c*d - a*f)^2 - (b*d - a*e)*(c*e - b*f))*(p + 1)), Int[(a + b*x + c*x^2)^(p + 1)*(d + e*x + f*x^2)^q*Simp[(b*h - 2*g*c)*((c*d - a*f)^2 - (b*d - a*e)*(c*e - b*f))*(p + 1) + (b^2*(g*f) - b*(h*c*d + g*c*e + a*h*f) + 2*(g*c*(c*d - a*f) - a*(-(h*c*e))))*(a*f*(p + 1) - c*d*(p + 2)) - e*((g*c)*(2*a*c*e - b*(c*d + a*f)) + (g*b - a*h)*(2*c^2*d + b^2*f - c*(b*e + 2*a*f)))*(p + q + 2) - (2*f*((g*c)*(2*a*c*e - b*(c*d + a*f)) + (g*b - a*h)*(2*c^2*d + b^2*f - c*(b*e + 2*a*f)))*(p + q + 2) - (b^2*g*f - b*(h*c*d + g*c*e + a*h*f) + 2*(g*c*(c*d - a*f) - a*(-(h*c*e))))*(b*f*(p + 1) - c*e*(2*p + q + 4)))*x - c*f*(b^2*(g*f) - b*(h*c*d + g*c*e + a*h*f) + 2*(g*c*(c*d - a*f) + a*h*c*e))*(2*p + 2*q + 5)*x^2, x], x], x] /; FreeQ[{a, b, c, d, e, f, g, h, q}, x] && NeQ[b^2 - 4*a*c, 0] && NeQ[e^2 - 4*d*f, 0] && LtQ[p, -1] && NeQ[(c*d - a*f)^2 - (b*d - a*e)*(c*e - b*f), 0] && !( !IntegerQ[p] && ILtQ[q, -1])
  1017. Int[Times[Plus[Optional[Pattern[g, Blank[]]], Times[Optional[Pattern[h, Blank[]]], Pattern[x, Blank[]]]], Power[Plus[Pattern[a, Blank[]], Times[Optional[Pattern[c, Blank[]]], Power[Pattern[x, Blank[]], 2]]], Pattern[p, Blank[]]], Power[Plus[Pattern[d, Blank[]], Times[Optional[Pattern[e, Blank[]]], Pattern[x, Blank[]]], Times[Optional[Pattern[f, Blank[]]], Power[Pattern[x, Blank[]], 2]]], Pattern[q, Blank[]]]], Pattern[x, Blank[Symbol]]] := Simp[((a + c*x^2)^(p + 1)*(d + e*x + f*x^2)^(q + 1)*(g*c*(2*a*c*e) + (-(a*h))*(2*c^2*d - c*(2*a*f)) + c*(g*(2*c^2*d - c*(2*a*f)) - h*(-2*a*c*e))*x))/((-4*a*c)*(a*c*e^2 + (c*d - a*f)^2)*(p + 1)), x] + Dist[1/((-4*a*c)*(a*c*e^2 + (c*d - a*f)^2)*(p + 1)), Int[(a + c*x^2)^(p + 1)*(d + e*x + f*x^2)^q*Simp[(-2*g*c)*((c*d - a*f)^2 - (-(a*e))*(c*e))*(p + 1) + (2*(g*c*(c*d - a*f) - a*(-(h*c*e))))*(a*f*(p + 1) - c*d*(p + 2)) - e*((g*c)*(2*a*c*e) + (-(a*h))*(2*c^2*d - c*((Plus[2])*a*f)))*(p + q + 2) - (2*f*((g*c)*(2*a*c*e) + (-(a*h))*(2*c^2*d - c*((Plus[2])*a*f)))*(p + q + 2) - (2*(g*c*(c*d - a*f) - a*(-(h*c*e))))*(-(c*e*(2*p + q + 4))))*x - c*f*(2*(g*c*(c*d - a*f) - a*(-(h*c*e))))*(2*p + 2*q + 5)*x^2, x], x], x] /; FreeQ[{a, c, d, e, f, g, h, q}, x] && NeQ[e^2 - 4*d*f, 0] && LtQ[p, -1] && NeQ[a*c*e^2 + (c*d - a*f)^2, 0] && !( !IntegerQ[p] && ILtQ[q, -1])
  1018. Int[Times[Plus[Optional[Pattern[g, Blank[]]], Times[Optional[Pattern[h, Blank[]]], Pattern[x, Blank[]]]], Power[Plus[Pattern[a, Blank[]], Times[Optional[Pattern[b, Blank[]]], Pattern[x, Blank[]]], Times[Optional[Pattern[c, Blank[]]], Power[Pattern[x, Blank[]], 2]]], Pattern[p, Blank[]]], Power[Plus[Pattern[d, Blank[]], Times[Optional[Pattern[f, Blank[]]], Power[Pattern[x, Blank[]], 2]]], Pattern[q, Blank[]]]], Pattern[x, Blank[Symbol]]] := Simp[((a + b*x + c*x^2)^(p + 1)*(d + f*x^2)^(q + 1)*((g*c)*(-(b*(c*d + a*f))) + (g*b - a*h)*(2*c^2*d + b^2*f - c*(2*a*f)) + c*(g*(2*c^2*d + b^2*f - c*(2*a*f)) - h*(b*c*d + a*b*f))*x))/((b^2 - 4*a*c)*(b^2*d*f + (c*d - a*f)^2)*(p + 1)), x] + Dist[1/((b^2 - 4*a*c)*(b^2*d*f + (c*d - a*f)^2)*(p + 1)), Int[(a + b*x + c*x^2)^(p + 1)*(d + f*x^2)^q*Simp[(b*h - 2*g*c)*((c*d - a*f)^2 - (b*d)*(-(b*f)))*(p + 1) + (b^2*(g*f) - b*(h*c*d + a*h*f) + 2*(g*c*(c*d - a*f)))*(a*f*(p + 1) - c*d*(p + 2)) - (2*f*((g*c)*(-(b*(c*d + a*f))) + (g*b - a*h)*(2*c^2*d + b^2*f - c*(2*a*f)))*(p + q + 2) - (b^2*(g*f) - b*(h*c*d + a*h*f) + 2*(g*c*(c*d - a*f)))*(b*f*(p + 1)))*x - c*f*(b^2*(g*f) - b*(h*c*d + a*h*f) + 2*(g*c*(c*d - a*f)))*(2*p + 2*q + 5)*x^2, x], x], x] /; FreeQ[{a, b, c, d, f, g, h, q}, x] && NeQ[b^2 - 4*a*c, 0] && LtQ[p, -1] && NeQ[b^2*d*f + (c*d - a*f)^2, 0] && !( !IntegerQ[p] && ILtQ[q, -1])
  1019. Int[Times[Plus[Optional[Pattern[g, Blank[]]], Times[Optional[Pattern[h, Blank[]]], Pattern[x, Blank[]]]], Power[Plus[Pattern[a, Blank[]], Times[Optional[Pattern[b, Blank[]]], Pattern[x, Blank[]]], Times[Optional[Pattern[c, Blank[]]], Power[Pattern[x, Blank[]], 2]]], Pattern[p, Blank[]]], Power[Plus[Pattern[d, Blank[]], Times[Optional[Pattern[e, Blank[]]], Pattern[x, Blank[]]], Times[Optional[Pattern[f, Blank[]]], Power[Pattern[x, Blank[]], 2]]], Pattern[q, Blank[]]]], Pattern[x, Blank[Symbol]]] := Simp[(h*(a + b*x + c*x^2)^p*(d + e*x + f*x^2)^(q + 1))/(2*f*(p + q + 1)), x] - Dist[1/(2*f*(p + q + 1)), Int[(a + b*x + c*x^2)^(p - 1)*(d + e*x + f*x^2)^q*Simp[h*p*(b*d - a*e) + a*(h*e - 2*g*f)*(p + q + 1) + (2*h*p*(c*d - a*f) + b*(h*e - 2*g*f)*(p + q + 1))*x + (h*p*(c*e - b*f) + c*(h*e - 2*g*f)*(p + q + 1))*x^2, x], x], x] /; FreeQ[{a, b, c, d, e, f, g, h, q}, x] && NeQ[b^2 - 4*a*c, 0] && NeQ[e^2 - 4*d*f, 0] && GtQ[p, 0] && NeQ[p + q + 1, 0]
  1020. Int[Times[Plus[Optional[Pattern[g, Blank[]]], Times[Optional[Pattern[h, Blank[]]], Pattern[x, Blank[]]]], Power[Plus[Pattern[a, Blank[]], Times[Optional[Pattern[c, Blank[]]], Power[Pattern[x, Blank[]], 2]]], Pattern[p, Blank[]]], Power[Plus[Pattern[d, Blank[]], Times[Optional[Pattern[e, Blank[]]], Pattern[x, Blank[]]], Times[Optional[Pattern[f, Blank[]]], Power[Pattern[x, Blank[]], 2]]], Pattern[q, Blank[]]]], Pattern[x, Blank[Symbol]]] := Simp[(h*(a + c*x^2)^p*(d + e*x + f*x^2)^(q + 1))/(2*f*(p + q + 1)), x] + Dist[1/(2*f*(p + q + 1)), Int[(a + c*x^2)^(p - 1)*(d + e*x + f*x^2)^q*Simp[a*h*e*p - a*(h*e - 2*g*f)*(p + q + 1) - 2*h*p*(c*d - a*f)*x - (h*c*e*p + c*(h*e - 2*g*f)*(p + q + 1))*x^2, x], x], x] /; FreeQ[{a, c, d, e, f, g, h, q}, x] && NeQ[e^2 - 4*d*f, 0] && GtQ[p, 0] && NeQ[p + q + 1, 0]
  1021. Int[Times[Plus[Optional[Pattern[g, Blank[]]], Times[Optional[Pattern[h, Blank[]]], Pattern[x, Blank[]]]], Power[Plus[Pattern[a, Blank[]], Times[Optional[Pattern[b, Blank[]]], Pattern[x, Blank[]]], Times[Optional[Pattern[c, Blank[]]], Power[Pattern[x, Blank[]], 2]]], Pattern[p, Blank[]]], Power[Plus[Pattern[d, Blank[]], Times[Optional[Pattern[f, Blank[]]], Power[Pattern[x, Blank[]], 2]]], Pattern[q, Blank[]]]], Pattern[x, Blank[Symbol]]] := Simp[(h*(a + b*x + c*x^2)^p*(d + f*x^2)^(q + 1))/(2*f*(p + q + 1)), x] - Dist[1/(2*f*(p + q + 1)), Int[(a + b*x + c*x^2)^(p - 1)*(d + f*x^2)^q*Simp[h*p*(b*d) + a*(-2*g*f)*(p + q + 1) + (2*h*p*(c*d - a*f) + b*(-2*g*f)*(p + q + 1))*x + (h*p*(-(b*f)) + c*(-2*g*f)*(p + q + 1))*x^2, x], x], x] /; FreeQ[{a, b, c, d, f, g, h, q}, x] && NeQ[b^2 - 4*a*c, 0] && GtQ[p, 0] && NeQ[p + q + 1, 0]
  1022. Int[Times[Plus[Optional[Pattern[g, Blank[]]], Times[Optional[Pattern[h, Blank[]]], Pattern[x, Blank[]]]], Power[Plus[Pattern[a, Blank[]], Times[Optional[Pattern[b, Blank[]]], Pattern[x, Blank[]]], Times[Optional[Pattern[c, Blank[]]], Power[Pattern[x, Blank[]], 2]]], -1], Power[Plus[Pattern[d, Blank[]], Times[Optional[Pattern[e, Blank[]]], Pattern[x, Blank[]]], Times[Optional[Pattern[f, Blank[]]], Power[Pattern[x, Blank[]], 2]]], -1]], Pattern[x, Blank[Symbol]]] := With[{q = Simplify[c^2*d^2 - b*c*d*e + a*c*e^2 + b^2*d*f - 2*a*c*d*f - a*b*e*f + a^2*f^2]}, Dist[1/q, Int[Simp[g*c^2*d - g*b*c*e + a*h*c*e + g*b^2*f - a*b*h*f - a*g*c*f + c*(h*c*d - g*c*e + g*b*f - a*h*f)*x, x]/(a + b*x + c*x^2), x], x] + Dist[1/q, Int[Simp[-(h*c*d*e) + g*c*e^2 + b*h*d*f - g*c*d*f - g*b*e*f + a*g*f^2 - f*(h*c*d - g*c*e + g*b*f - a*h*f)*x, x]/(d + e*x + f*x^2), x], x] /; NeQ[q, 0]] /; FreeQ[{a, b, c, d, e, f, g, h}, x] && NeQ[b^2 - 4*a*c, 0] && NeQ[e^2 - 4*d*f, 0]
  1023. Int[Times[Plus[Optional[Pattern[g, Blank[]]], Times[Optional[Pattern[h, Blank[]]], Pattern[x, Blank[]]]], Power[Plus[Pattern[a, Blank[]], Times[Optional[Pattern[b, Blank[]]], Pattern[x, Blank[]]], Times[Optional[Pattern[c, Blank[]]], Power[Pattern[x, Blank[]], 2]]], -1], Power[Plus[Pattern[d, Blank[]], Times[Optional[Pattern[f, Blank[]]], Power[Pattern[x, Blank[]], 2]]], -1]], Pattern[x, Blank[Symbol]]] := With[{q = Simplify[c^2*d^2 + b^2*d*f - 2*a*c*d*f + a^2*f^2]}, Dist[1/q, Int[Simp[g*c^2*d + g*b^2*f - a*b*h*f - a*g*c*f + c*(h*c*d + g*b*f - a*h*f)*x, x]/(a + b*x + c*x^2), x], x] + Dist[1/q, Int[Simp[b*h*d*f - g*c*d*f + a*g*f^2 - f*(h*c*d + g*b*f - a*h*f)*x, x]/(d + f*x^2), x], x] /; NeQ[q, 0]] /; FreeQ[{a, b, c, d, f, g, h}, x] && NeQ[b^2 - 4*a*c, 0]
  1024. Int[Times[Plus[Pattern[g, Blank[]], Times[Optional[Pattern[h, Blank[]]], Pattern[x, Blank[]]]], Power[Plus[Pattern[a, Blank[]], Times[Optional[Pattern[b, Blank[]]], Pattern[x, Blank[]]], Times[Optional[Pattern[c, Blank[]]], Power[Pattern[x, Blank[]], 2]]], -1], Power[Plus[Optional[Pattern[d, Blank[]]], Times[Optional[Pattern[e, Blank[]]], Pattern[x, Blank[]]], Times[Optional[Pattern[f, Blank[]]], Power[Pattern[x, Blank[]], 2]]], Rational[-1, 2]]], Pattern[x, Blank[Symbol]]] := Dist[-2*g, Subst[Int[1/(b*d - a*e - b*x^2), x], x, Sqrt[d + e*x + f*x^2]], x] /; FreeQ[{a, b, c, d, e, f, g, h}, x] && NeQ[b^2 - 4*a*c, 0] && NeQ[e^2 - 4*d*f, 0] && EqQ[c*e - b*f, 0] && EqQ[h*e - 2*g*f, 0]
  1025. Int[Times[Plus[Optional[Pattern[g, Blank[]]], Times[Optional[Pattern[h, Blank[]]], Pattern[x, Blank[]]]], Power[Plus[Pattern[a, Blank[]], Times[Optional[Pattern[b, Blank[]]], Pattern[x, Blank[]]], Times[Optional[Pattern[c, Blank[]]], Power[Pattern[x, Blank[]], 2]]], -1], Power[Plus[Optional[Pattern[d, Blank[]]], Times[Optional[Pattern[e, Blank[]]], Pattern[x, Blank[]]], Times[Optional[Pattern[f, Blank[]]], Power[Pattern[x, Blank[]], 2]]], Rational[-1, 2]]], Pattern[x, Blank[Symbol]]] := -Dist[(h*e - 2*g*f)/(2*f), Int[1/((a + b*x + c*x^2)*Sqrt[d + e*x + f*x^2]), x], x] + Dist[h/(2*f), Int[(e + 2*f*x)/((a + b*x + c*x^2)*Sqrt[d + e*x + f*x^2]), x], x] /; FreeQ[{a, b, c, d, e, f, g, h}, x] && NeQ[b^2 - 4*a*c, 0] && NeQ[e^2 - 4*d*f, 0] && EqQ[c*e - b*f, 0] && NeQ[h*e - 2*g*f, 0]
  1026. Int[Times[Pattern[x, Blank[]], Power[Plus[Pattern[a, Blank[]], Times[Optional[Pattern[b, Blank[]]], Pattern[x, Blank[]]], Times[Optional[Pattern[c, Blank[]]], Power[Pattern[x, Blank[]], 2]]], -1], Power[Plus[Pattern[d, Blank[]], Times[Optional[Pattern[e, Blank[]]], Pattern[x, Blank[]]], Times[Optional[Pattern[f, Blank[]]], Power[Pattern[x, Blank[]], 2]]], Rational[-1, 2]]], Pattern[x, Blank[Symbol]]] := Dist[-2*e, Subst[Int[(1 - d*x^2)/(c*e - b*f - e*(2*c*d - b*e + 2*a*f)*x^2 + d^2*(c*e - b*f)*x^4), x], x, (1 + ((e + Sqrt[e^2 - 4*d*f])*x)/(2*d))/Sqrt[d + e*x + f*x^2]], x] /; FreeQ[{a, b, c, d, e, f}, x] && NeQ[b^2 - 4*a*c, 0] && NeQ[e^2 - 4*d*f, 0] && EqQ[b*d - a*e, 0]
  1027. Int[Times[Plus[Pattern[g, Blank[]], Times[Optional[Pattern[h, Blank[]]], Pattern[x, Blank[]]]], Power[Plus[Pattern[a, Blank[]], Times[Optional[Pattern[b, Blank[]]], Pattern[x, Blank[]]], Times[Optional[Pattern[c, Blank[]]], Power[Pattern[x, Blank[]], 2]]], -1], Power[Plus[Pattern[d, Blank[]], Times[Optional[Pattern[e, Blank[]]], Pattern[x, Blank[]]], Times[Optional[Pattern[f, Blank[]]], Power[Pattern[x, Blank[]], 2]]], Rational[-1, 2]]], Pattern[x, Blank[Symbol]]] := Dist[g, Subst[Int[1/(a + (c*d - a*f)*x^2), x], x, x/Sqrt[d + e*x + f*x^2]], x] /; FreeQ[{a, b, c, d, e, f, g, h}, x] && NeQ[b^2 - 4*a*c, 0] && NeQ[e^2 - 4*d*f, 0] && EqQ[b*d - a*e, 0] && EqQ[2*h*d - g*e, 0]
  1028. Int[Times[Plus[Pattern[g, Blank[]], Times[Optional[Pattern[h, Blank[]]], Pattern[x, Blank[]]]], Power[Plus[Pattern[a, Blank[]], Times[Optional[Pattern[b, Blank[]]], Pattern[x, Blank[]]], Times[Optional[Pattern[c, Blank[]]], Power[Pattern[x, Blank[]], 2]]], -1], Power[Plus[Pattern[d, Blank[]], Times[Optional[Pattern[e, Blank[]]], Pattern[x, Blank[]]], Times[Optional[Pattern[f, Blank[]]], Power[Pattern[x, Blank[]], 2]]], Rational[-1, 2]]], Pattern[x, Blank[Symbol]]] := -Dist[(2*h*d - g*e)/e, Int[1/((a + b*x + c*x^2)*Sqrt[d + e*x + f*x^2]), x], x] + Dist[h/e, Int[(2*d + e*x)/((a + b*x + c*x^2)*Sqrt[d + e*x + f*x^2]), x], x] /; FreeQ[{a, b, c, d, e, f, g, h}, x] && NeQ[b^2 - 4*a*c, 0] && NeQ[e^2 - 4*d*f, 0] && EqQ[b*d - a*e, 0] && NeQ[2*h*d - g*e, 0]
  1029. Int[Times[Plus[Optional[Pattern[g, Blank[]]], Times[Optional[Pattern[h, Blank[]]], Pattern[x, Blank[]]]], Power[Plus[Optional[Pattern[a, Blank[]]], Times[Optional[Pattern[b, Blank[]]], Pattern[x, Blank[]]], Times[Optional[Pattern[c, Blank[]]], Power[Pattern[x, Blank[]], 2]]], -1], Power[Plus[Optional[Pattern[d, Blank[]]], Times[Optional[Pattern[e, Blank[]]], Pattern[x, Blank[]]], Times[Optional[Pattern[f, Blank[]]], Power[Pattern[x, Blank[]], 2]]], Rational[-1, 2]]], Pattern[x, Blank[Symbol]]] := Dist[-2*g*(g*b - 2*a*h), Subst[Int[1/Simp[g*(g*b - 2*a*h)*(b^2 - 4*a*c) - (b*d - a*e)*x^2, x], x], x, Simp[g*b - 2*a*h - (b*h - 2*g*c)*x, x]/Sqrt[d + e*x + f*x^2]], x] /; FreeQ[{a, b, c, d, e, f, g, h}, x] && NeQ[b^2 - 4*a*c, 0] && NeQ[e^2 - 4*d*f, 0] && NeQ[b*d - a*e, 0] && EqQ[h^2*(b*d - a*e) - 2*g*h*(c*d - a*f) + g^2*(c*e - b*f), 0]
  1030. Int[Times[Plus[Pattern[g, Blank[]], Times[Optional[Pattern[h, Blank[]]], Pattern[x, Blank[]]]], Power[Plus[Pattern[a, Blank[]], Times[Optional[Pattern[c, Blank[]]], Power[Pattern[x, Blank[]], 2]]], -1], Power[Plus[Optional[Pattern[d, Blank[]]], Times[Optional[Pattern[e, Blank[]]], Pattern[x, Blank[]]], Times[Optional[Pattern[f, Blank[]]], Power[Pattern[x, Blank[]], 2]]], Rational[-1, 2]]], Pattern[x, Blank[Symbol]]] := Dist[-2*a*g*h, Subst[Int[1/Simp[2*a^2*g*h*c + a*e*x^2, x], x], x, Simp[a*h - g*c*x, x]/Sqrt[d + e*x + f*x^2]], x] /; FreeQ[{a, c, d, e, f, g, h}, x] && EqQ[a*h^2*e + 2*g*h*(c*d - a*f) - g^2*c*e, 0]
  1031. Int[Times[Plus[Pattern[g, Blank[]], Times[Optional[Pattern[h, Blank[]]], Pattern[x, Blank[]]]], Power[Plus[Optional[Pattern[a, Blank[]]], Times[Optional[Pattern[b, Blank[]]], Pattern[x, Blank[]]], Times[Optional[Pattern[c, Blank[]]], Power[Pattern[x, Blank[]], 2]]], -1], Power[Plus[Pattern[d, Blank[]], Times[Optional[Pattern[f, Blank[]]], Power[Pattern[x, Blank[]], 2]]], Rational[-1, 2]]], Pattern[x, Blank[Symbol]]] := Dist[-2*g*(g*b - 2*a*h), Subst[Int[1/Simp[g*(g*b - 2*a*h)*(b^2 - 4*a*c) - b*d*x^2, x], x], x, Simp[g*b - 2*a*h - (b*h - 2*g*c)*x, x]/Sqrt[d + f*x^2]], x] /; FreeQ[{a, b, c, d, f, g, h}, x] && NeQ[b^2 - 4*a*c, 0] && EqQ[b*h^2*d - 2*g*h*(c*d - a*f) - g^2*b*f, 0]
  1032. Int[Times[Plus[Optional[Pattern[g, Blank[]]], Times[Optional[Pattern[h, Blank[]]], Pattern[x, Blank[]]]], Power[Plus[Pattern[a, Blank[]], Times[Optional[Pattern[b, Blank[]]], Pattern[x, Blank[]]], Times[Optional[Pattern[c, Blank[]]], Power[Pattern[x, Blank[]], 2]]], -1], Power[Plus[Optional[Pattern[d, Blank[]]], Times[Optional[Pattern[e, Blank[]]], Pattern[x, Blank[]]], Times[Optional[Pattern[f, Blank[]]], Power[Pattern[x, Blank[]], 2]]], Rational[-1, 2]]], Pattern[x, Blank[Symbol]]] := With[{q = Rt[b^2 - 4*a*c, 2]}, Dist[(2*c*g - h*(b - q))/q, Int[1/((b - q + 2*c*x)*Sqrt[d + e*x + f*x^2]), x], x] - Dist[(2*c*g - h*(b + q))/q, Int[1/((b + q + 2*c*x)*Sqrt[d + e*x + f*x^2]), x], x]] /; FreeQ[{a, b, c, d, e, f, g, h}, x] && NeQ[b^2 - 4*a*c, 0] && NeQ[e^2 - 4*d*f, 0] && PosQ[b^2 - 4*a*c]
  1033. Int[Times[Plus[Optional[Pattern[g, Blank[]]], Times[Optional[Pattern[h, Blank[]]], Pattern[x, Blank[]]]], Power[Plus[Pattern[a, Blank[]], Times[Optional[Pattern[c, Blank[]]], Power[Pattern[x, Blank[]], 2]]], -1], Power[Plus[Optional[Pattern[d, Blank[]]], Times[Optional[Pattern[e, Blank[]]], Pattern[x, Blank[]]], Times[Optional[Pattern[f, Blank[]]], Power[Pattern[x, Blank[]], 2]]], Rational[-1, 2]]], Pattern[x, Blank[Symbol]]] := With[{q = Rt[-(a*c), 2]}, Dist[h/2 + (c*g)/(2*q), Int[1/((-q + c*x)*Sqrt[d + e*x + f*x^2]), x], x] + Dist[h/2 - (c*g)/(2*q), Int[1/((q + c*x)*Sqrt[d + e*x + f*x^2]), x], x]] /; FreeQ[{a, c, d, e, f, g, h}, x] && NeQ[e^2 - 4*d*f, 0] && PosQ[-(a*c)]
  1034. Int[Times[Plus[Optional[Pattern[g, Blank[]]], Times[Optional[Pattern[h, Blank[]]], Pattern[x, Blank[]]]], Power[Plus[Pattern[a, Blank[]], Times[Optional[Pattern[b, Blank[]]], Pattern[x, Blank[]]], Times[Optional[Pattern[c, Blank[]]], Power[Pattern[x, Blank[]], 2]]], -1], Power[Plus[Pattern[d, Blank[]], Times[Optional[Pattern[f, Blank[]]], Power[Pattern[x, Blank[]], 2]]], Rational[-1, 2]]], Pattern[x, Blank[Symbol]]] := With[{q = Rt[b^2 - 4*a*c, 2]}, Dist[(2*c*g - h*(b - q))/q, Int[1/((b - q + 2*c*x)*Sqrt[d + f*x^2]), x], x] - Dist[(2*c*g - h*(b + q))/q, Int[1/((b + q + 2*c*x)*Sqrt[d + f*x^2]), x], x]] /; FreeQ[{a, b, c, d, f, g, h}, x] && NeQ[b^2 - 4*a*c, 0] && PosQ[b^2 - 4*a*c]
  1035. Int[Times[Plus[Optional[Pattern[g, Blank[]]], Times[Optional[Pattern[h, Blank[]]], Pattern[x, Blank[]]]], Power[Plus[Optional[Pattern[a, Blank[]]], Times[Optional[Pattern[b, Blank[]]], Pattern[x, Blank[]]], Times[Optional[Pattern[c, Blank[]]], Power[Pattern[x, Blank[]], 2]]], -1], Power[Plus[Optional[Pattern[d, Blank[]]], Times[Optional[Pattern[e, Blank[]]], Pattern[x, Blank[]]], Times[Optional[Pattern[f, Blank[]]], Power[Pattern[x, Blank[]], 2]]], Rational[-1, 2]]], Pattern[x, Blank[Symbol]]] := With[{q = Rt[(c*d - a*f)^2 - (b*d - a*e)*(c*e - b*f), 2]}, Dist[1/(2*q), Int[Simp[h*(b*d - a*e) - g*(c*d - a*f - q) - (g*(c*e - b*f) - h*(c*d - a*f + q))*x, x]/((a + b*x + c*x^2)*Sqrt[d + e*x + f*x^2]), x], x] - Dist[1/(2*q), Int[Simp[h*(b*d - a*e) - g*(c*d - a*f + q) - (g*(c*e - b*f) - h*(c*d - a*f - q))*x, x]/((a + b*x + c*x^2)*Sqrt[d + e*x + f*x^2]), x], x]] /; FreeQ[{a, b, c, d, e, f, g, h}, x] && NeQ[b^2 - 4*a*c, 0] && NeQ[e^2 - 4*d*f, 0] && NeQ[b*d - a*e, 0] && NegQ[b^2 - 4*a*c]
  1036. Int[Times[Plus[Optional[Pattern[g, Blank[]]], Times[Optional[Pattern[h, Blank[]]], Pattern[x, Blank[]]]], Power[Plus[Pattern[a, Blank[]], Times[Optional[Pattern[c, Blank[]]], Power[Pattern[x, Blank[]], 2]]], -1], Power[Plus[Optional[Pattern[d, Blank[]]], Times[Optional[Pattern[e, Blank[]]], Pattern[x, Blank[]]], Times[Optional[Pattern[f, Blank[]]], Power[Pattern[x, Blank[]], 2]]], Rational[-1, 2]]], Pattern[x, Blank[Symbol]]] := With[{q = Rt[(c*d - a*f)^2 + a*c*e^2, 2]}, Dist[1/(2*q), Int[Simp[-(a*h*e) - g*(c*d - a*f - q) + (h*(c*d - a*f + q) - g*c*e)*x, x]/((a + c*x^2)*Sqrt[d + e*x + f*x^2]), x], x] - Dist[1/(2*q), Int[Simp[-(a*h*e) - g*(c*d - a*f + q) + (h*(c*d - a*f - q) - g*c*e)*x, x]/((a + c*x^2)*Sqrt[d + e*x + f*x^2]), x], x]] /; FreeQ[{a, c, d, e, f, g, h}, x] && NeQ[e^2 - 4*d*f, 0] && NegQ[-(a*c)]
  1037. Int[Times[Plus[Optional[Pattern[g, Blank[]]], Times[Optional[Pattern[h, Blank[]]], Pattern[x, Blank[]]]], Power[Plus[Optional[Pattern[a, Blank[]]], Times[Optional[Pattern[b, Blank[]]], Pattern[x, Blank[]]], Times[Optional[Pattern[c, Blank[]]], Power[Pattern[x, Blank[]], 2]]], -1], Power[Plus[Pattern[d, Blank[]], Times[Optional[Pattern[f, Blank[]]], Power[Pattern[x, Blank[]], 2]]], Rational[-1, 2]]], Pattern[x, Blank[Symbol]]] := With[{q = Rt[(c*d - a*f)^2 + b^2*d*f, 2]}, Dist[1/(2*q), Int[Simp[h*b*d - g*(c*d - a*f - q) + (h*(c*d - a*f + q) + g*b*f)*x, x]/((a + b*x + c*x^2)*Sqrt[d + f*x^2]), x], x] - Dist[1/(2*q), Int[Simp[h*b*d - g*(c*d - a*f + q) + (h*(c*d - a*f - q) + g*b*f)*x, x]/((a + b*x + c*x^2)*Sqrt[d + f*x^2]), x], x]] /; FreeQ[{a, b, c, d, f, g, h}, x] && NeQ[b^2 - 4*a*c, 0] && NegQ[b^2 - 4*a*c]
  1038. Int[Times[Plus[Optional[Pattern[g, Blank[]]], Times[Optional[Pattern[h, Blank[]]], Pattern[x, Blank[]]]], Power[Plus[Pattern[a, Blank[]], Times[Optional[Pattern[b, Blank[]]], Pattern[x, Blank[]]], Times[Optional[Pattern[c, Blank[]]], Power[Pattern[x, Blank[]], 2]]], Rational[-1, 2]], Power[Plus[Pattern[d, Blank[]], Times[Optional[Pattern[e, Blank[]]], Pattern[x, Blank[]]], Times[Optional[Pattern[f, Blank[]]], Power[Pattern[x, Blank[]], 2]]], Rational[-1, 2]]], Pattern[x, Blank[Symbol]]] := With[{s = Rt[b^2 - 4*a*c, 2], t = Rt[e^2 - 4*d*f, 2]}, Dist[(Sqrt[b + s + 2*c*x]*Sqrt[2*a + (b + s)*x]*Sqrt[e + t + 2*f*x]*Sqrt[2*d + (e + t)*x])/(Sqrt[a + b*x + c*x^2]*Sqrt[d + e*x + f*x^2]), Int[(g + h*x)/(Sqrt[b + s + 2*c*x]*Sqrt[2*a + (b + s)*x]*Sqrt[e + t + 2*f*x]*Sqrt[2*d + (e + t)*x]), x], x]] /; FreeQ[{a, b, c, d, e, f, g, h}, x] && NeQ[b^2 - 4*a*c, 0] && NeQ[e^2 - 4*d*f, 0]
  1039. Int[Times[Plus[Optional[Pattern[g, Blank[]]], Times[Optional[Pattern[h, Blank[]]], Pattern[x, Blank[]]]], Power[Plus[Pattern[a, Blank[]], Times[Optional[Pattern[b, Blank[]]], Pattern[x, Blank[]]], Times[Optional[Pattern[c, Blank[]]], Power[Pattern[x, Blank[]], 2]]], Rational[-1, 2]], Power[Plus[Pattern[d, Blank[]], Times[Optional[Pattern[f, Blank[]]], Power[Pattern[x, Blank[]], 2]]], Rational[-1, 2]]], Pattern[x, Blank[Symbol]]] := With[{s = Rt[b^2 - 4*a*c, 2], t = Rt[-4*d*f, 2]}, Dist[(Sqrt[b + s + 2*c*x]*Sqrt[2*a + (b + s)*x]*Sqrt[t + 2*f*x]*Sqrt[2*d + t*x])/(Sqrt[a + b*x + c*x^2]*Sqrt[d + f*x^2]), Int[(g + h*x)/(Sqrt[b + s + 2*c*x]*Sqrt[2*a + (b + s)*x]*Sqrt[t + 2*f*x]*Sqrt[2*d + t*x]), x], x]] /; FreeQ[{a, b, c, d, f, g, h}, x] && NeQ[b^2 - 4*a*c, 0]
  1040. Int[Times[Plus[Optional[Pattern[g, Blank[]]], Times[Optional[Pattern[h, Blank[]]], Pattern[x, Blank[]]]], Power[Plus[Optional[Pattern[a, Blank[]]], Times[Optional[Pattern[b, Blank[]]], Pattern[x, Blank[]]], Times[Optional[Pattern[c, Blank[]]], Power[Pattern[x, Blank[]], 2]]], Rational[-1, 3]], Power[Plus[Optional[Pattern[d, Blank[]]], Times[Optional[Pattern[e, Blank[]]], Pattern[x, Blank[]]], Times[Optional[Pattern[f, Blank[]]], Power[Pattern[x, Blank[]], 2]]], -1]], Pattern[x, Blank[Symbol]]] := With[{q = ((-9*c*h^2)/(2*c*g - b*h)^2)^(1/3)}, Simp[(Sqrt[3]*h*q*ArcTan[1/Sqrt[3] - (2^(2/3)*(1 - (3*h*(b + 2*c*x))/(2*c*g - b*h))^(2/3))/(Sqrt[3]*(1 + (3*h*(b + 2*c*x))/(2*c*g - b*h))^(1/3))])/f, x] + (-Simp[(3*h*q*Log[(1 - (3*h*(b + 2*c*x))/(2*c*g - b*h))^(2/3) + 2^(1/3)*(1 + (3*h*(b + 2*c*x))/(2*c*g - b*h))^(1/3)])/(2*f), x] + Simp[(h*q*Log[d + e*x + f*x^2])/(2*f), x])] /; FreeQ[{a, b, c, d, e, f, g, h}, x] && EqQ[c*e - b*f, 0] && EqQ[c^2*d - f*(b^2 - 3*a*c), 0] && EqQ[c^2*g^2 - b*c*g*h - 2*b^2*h^2 + 9*a*c*h^2, 0] && GtQ[(-9*c*h^2)/(2*c*g - b*h)^2, 0]
  1041. Int[Times[Plus[Optional[Pattern[g, Blank[]]], Times[Optional[Pattern[h, Blank[]]], Pattern[x, Blank[]]]], Power[Plus[Optional[Pattern[a, Blank[]]], Times[Optional[Pattern[b, Blank[]]], Pattern[x, Blank[]]], Times[Optional[Pattern[c, Blank[]]], Power[Pattern[x, Blank[]], 2]]], Rational[-1, 3]], Power[Plus[Optional[Pattern[d, Blank[]]], Times[Optional[Pattern[e, Blank[]]], Pattern[x, Blank[]]], Times[Optional[Pattern[f, Blank[]]], Power[Pattern[x, Blank[]], 2]]], -1]], Pattern[x, Blank[Symbol]]] := With[{q = -(c/(b^2 - 4*a*c))}, Dist[(q*(a + b*x + c*x^2))^(1/3)/(a + b*x + c*x^2)^(1/3), Int[(g + h*x)/((q*a + b*q*x + c*q*x^2)^(1/3)*(d + e*x + f*x^2)), x], x]] /; FreeQ[{a, b, c, d, e, f, g, h}, x] && EqQ[c*e - b*f, 0] && EqQ[c^2*d - f*(b^2 - 3*a*c), 0] && EqQ[c^2*g^2 - b*c*g*h - 2*b^2*h^2 + 9*a*c*h^2, 0] && !GtQ[4*a - b^2/c, 0]
  1042. Int[Times[Plus[Optional[Pattern[g, Blank[]]], Times[Optional[Pattern[h, Blank[]]], Pattern[x, Blank[]]]], Power[Plus[Optional[Pattern[a, Blank[]]], Times[Optional[Pattern[b, Blank[]]], Pattern[x, Blank[]]], Times[Optional[Pattern[c, Blank[]]], Power[Pattern[x, Blank[]], 2]]], Pattern[p, Blank[]]], Power[Plus[Optional[Pattern[d, Blank[]]], Times[Optional[Pattern[e, Blank[]]], Pattern[x, Blank[]]], Times[Optional[Pattern[f, Blank[]]], Power[Pattern[x, Blank[]], 2]]], Pattern[q, Blank[]]]], Pattern[x, Blank[Symbol]]] := Unintegrable[(g + h*x)*(a + b*x + c*x^2)^p*(d + e*x + f*x^2)^q, x] /; FreeQ[{a, b, c, d, e, f, g, h, p, q}, x]
  1043. Int[Times[Plus[Optional[Pattern[g, Blank[]]], Times[Optional[Pattern[h, Blank[]]], Pattern[x, Blank[]]]], Power[Plus[Optional[Pattern[a, Blank[]]], Times[Optional[Pattern[c, Blank[]]], Power[Pattern[x, Blank[]], 2]]], Pattern[p, Blank[]]], Power[Plus[Optional[Pattern[d, Blank[]]], Times[Optional[Pattern[e, Blank[]]], Pattern[x, Blank[]]], Times[Optional[Pattern[f, Blank[]]], Power[Pattern[x, Blank[]], 2]]], Pattern[q, Blank[]]]], Pattern[x, Blank[Symbol]]] := Unintegrable[(g + h*x)*(a + c*x^2)^p*(d + e*x + f*x^2)^q, x] /; FreeQ[{a, c, d, e, f, g, h, p, q}, x]
  1044. Int[Times[Power[Plus[Optional[Pattern[g, Blank[]]], Times[Optional[Pattern[h, Blank[]]], Pattern[u, Blank[]]]], Optional[Pattern[m, Blank[]]]], Power[Plus[Optional[Pattern[a, Blank[]]], Times[Optional[Pattern[b, Blank[]]], Pattern[u, Blank[]]], Times[Optional[Pattern[c, Blank[]]], Power[Pattern[u, Blank[]], 2]]], Optional[Pattern[p, Blank[]]]], Power[Plus[Optional[Pattern[d, Blank[]]], Times[Optional[Pattern[e, Blank[]]], Pattern[u, Blank[]]], Times[Optional[Pattern[f, Blank[]]], Power[Pattern[u, Blank[]], 2]]], Optional[Pattern[q, Blank[]]]]], Pattern[x, Blank[Symbol]]] := Dist[1/Coefficient[u, x, 1], Subst[Int[(g + h*x)^m*(a + b*x + c*x^2)^p*(d + e*x + f*x^2)^q, x], x, u], x] /; FreeQ[{a, b, c, d, e, f, g, h, m, p, q}, x] && LinearQ[u, x] && NeQ[u, x]
  1045. Int[Times[Power[Plus[Optional[Pattern[g, Blank[]]], Times[Optional[Pattern[h, Blank[]]], Pattern[u, Blank[]]]], Optional[Pattern[m, Blank[]]]], Power[Plus[Optional[Pattern[a, Blank[]]], Times[Optional[Pattern[c, Blank[]]], Power[Pattern[u, Blank[]], 2]]], Optional[Pattern[p, Blank[]]]], Power[Plus[Optional[Pattern[d, Blank[]]], Times[Optional[Pattern[e, Blank[]]], Pattern[u, Blank[]]], Times[Optional[Pattern[f, Blank[]]], Power[Pattern[u, Blank[]], 2]]], Optional[Pattern[q, Blank[]]]]], Pattern[x, Blank[Symbol]]] := Dist[1/Coefficient[u, x, 1], Subst[Int[(g + h*x)^m*(a + c*x^2)^p*(d + e*x + f*x^2)^q, x], x, u], x] /; FreeQ[{a, c, d, e, f, g, h, m, p, q}, x] && LinearQ[u, x] && NeQ[u, x]
  1046. Int[Times[Power[Plus[Pattern[a, Blank[]], Times[Optional[Pattern[b, Blank[]]], Pattern[x, Blank[]]], Times[Optional[Pattern[c, Blank[]]], Power[Pattern[x, Blank[]], 2]]], Optional[Pattern[p, Blank[]]]], Plus[Optional[Pattern[A, Blank[]]], Times[Optional[Pattern[B, Blank[]]], Pattern[x, Blank[]]], Times[Optional[Pattern[C, Blank[]]], Power[Pattern[x, Blank[]], 2]]], Power[Plus[Pattern[d, Blank[]], Times[Optional[Pattern[e, Blank[]]], Pattern[x, Blank[]]], Times[Optional[Pattern[f, Blank[]]], Power[Pattern[x, Blank[]], 2]]], Optional[Pattern[q, Blank[]]]]], Pattern[x, Blank[Symbol]]] := Dist[(c/f)^p, Int[(d + e*x + f*x^2)^(p + q)*(A + B*x + C*x^2), x], x] /; FreeQ[{a, b, c, d, e, f, A, B, C, p, q}, x] && EqQ[c*d - a*f, 0] && EqQ[b*d - a*e, 0] && (IntegerQ[p] || GtQ[c/f, 0]) && ( !IntegerQ[q] || LeafCount[d + e*x + f*x^2] <= LeafCount[a + b*x + c*x^2])
  1047. Int[Times[Power[Plus[Pattern[a, Blank[]], Times[Optional[Pattern[b, Blank[]]], Pattern[x, Blank[]]], Times[Optional[Pattern[c, Blank[]]], Power[Pattern[x, Blank[]], 2]]], Optional[Pattern[p, Blank[]]]], Plus[Optional[Pattern[A, Blank[]]], Times[Optional[Pattern[C, Blank[]]], Power[Pattern[x, Blank[]], 2]]], Power[Plus[Pattern[d, Blank[]], Times[Optional[Pattern[e, Blank[]]], Pattern[x, Blank[]]], Times[Optional[Pattern[f, Blank[]]], Power[Pattern[x, Blank[]], 2]]], Optional[Pattern[q, Blank[]]]]], Pattern[x, Blank[Symbol]]] := Dist[(c/f)^p, Int[(d + e*x + f*x^2)^(p + q)*(A + C*x^2), x], x] /; FreeQ[{a, b, c, d, e, f, A, C, p, q}, x] && EqQ[c*d - a*f, 0] && EqQ[b*d - a*e, 0] && (IntegerQ[p] || GtQ[c/f, 0]) && ( !IntegerQ[q] || LeafCount[d + e*x + f*x^2] <= LeafCount[a + b*x + c*x^2])
  1048. Int[Times[Power[Plus[Pattern[a, Blank[]], Times[Optional[Pattern[b, Blank[]]], Pattern[x, Blank[]]], Times[Optional[Pattern[c, Blank[]]], Power[Pattern[x, Blank[]], 2]]], Optional[Pattern[p, Blank[]]]], Plus[Optional[Pattern[A, Blank[]]], Times[Optional[Pattern[B, Blank[]]], Pattern[x, Blank[]]], Times[Optional[Pattern[C, Blank[]]], Power[Pattern[x, Blank[]], 2]]], Power[Plus[Pattern[d, Blank[]], Times[Optional[Pattern[e, Blank[]]], Pattern[x, Blank[]]], Times[Optional[Pattern[f, Blank[]]], Power[Pattern[x, Blank[]], 2]]], Optional[Pattern[q, Blank[]]]]], Pattern[x, Blank[Symbol]]] := Dist[(a^IntPart[p]*(a + b*x + c*x^2)^FracPart[p])/(d^IntPart[p]*(d + e*x + f*x^2)^FracPart[p]), Int[(d + e*x + f*x^2)^(p + q)*(A + B*x + C*x^2), x], x] /; FreeQ[{a, b, c, d, e, f, A, B, C, p, q}, x] && EqQ[c*d - a*f, 0] && EqQ[b*d - a*e, 0] && !IntegerQ[p] && !IntegerQ[q] && !GtQ[c/f, 0]
  1049. Int[Times[Power[Plus[Pattern[a, Blank[]], Times[Optional[Pattern[b, Blank[]]], Pattern[x, Blank[]]], Times[Optional[Pattern[c, Blank[]]], Power[Pattern[x, Blank[]], 2]]], Optional[Pattern[p, Blank[]]]], Plus[Optional[Pattern[A, Blank[]]], Times[Optional[Pattern[C, Blank[]]], Power[Pattern[x, Blank[]], 2]]], Power[Plus[Pattern[d, Blank[]], Times[Optional[Pattern[e, Blank[]]], Pattern[x, Blank[]]], Times[Optional[Pattern[f, Blank[]]], Power[Pattern[x, Blank[]], 2]]], Optional[Pattern[q, Blank[]]]]], Pattern[x, Blank[Symbol]]] := Dist[(a^IntPart[p]*(a + b*x + c*x^2)^FracPart[p])/(d^IntPart[p]*(d + e*x + f*x^2)^FracPart[p]), Int[(d + e*x + f*x^2)^(p + q)*(A + C*x^2), x], x] /; FreeQ[{a, b, c, d, e, f, A, C, p, q}, x] && EqQ[c*d - a*f, 0] && EqQ[b*d - a*e, 0] && !IntegerQ[p] && !IntegerQ[q] && !GtQ[c/f, 0]
  1050. Int[Times[Power[Plus[Pattern[a, Blank[]], Times[Optional[Pattern[b, Blank[]]], Pattern[x, Blank[]]], Times[Optional[Pattern[c, Blank[]]], Power[Pattern[x, Blank[]], 2]]], Optional[Pattern[p, Blank[]]]], Plus[Optional[Pattern[A, Blank[]]], Times[Optional[Pattern[B, Blank[]]], Pattern[x, Blank[]]], Times[Optional[Pattern[C, Blank[]]], Power[Pattern[x, Blank[]], 2]]], Power[Plus[Optional[Pattern[d, Blank[]]], Times[Optional[Pattern[e, Blank[]]], Pattern[x, Blank[]]], Times[Optional[Pattern[f, Blank[]]], Power[Pattern[x, Blank[]], 2]]], Optional[Pattern[q, Blank[]]]]], Pattern[x, Blank[Symbol]]] := Dist[(a + b*x + c*x^2)^FracPart[p]/((4*c)^IntPart[p]*(b + 2*c*x)^(2*FracPart[p])), Int[(b + 2*c*x)^(2*p)*(d + e*x + f*x^2)^q*(A + B*x + C*x^2), x], x] /; FreeQ[{a, b, c, d, e, f, A, B, C, p, q}, x] && EqQ[b^2 - 4*a*c, 0]
  1051. Int[Times[Power[Plus[Pattern[a, Blank[]], Times[Optional[Pattern[b, Blank[]]], Pattern[x, Blank[]]], Times[Optional[Pattern[c, Blank[]]], Power[Pattern[x, Blank[]], 2]]], Optional[Pattern[p, Blank[]]]], Plus[Optional[Pattern[A, Blank[]]], Times[Optional[Pattern[C, Blank[]]], Power[Pattern[x, Blank[]], 2]]], Power[Plus[Optional[Pattern[d, Blank[]]], Times[Optional[Pattern[e, Blank[]]], Pattern[x, Blank[]]], Times[Optional[Pattern[f, Blank[]]], Power[Pattern[x, Blank[]], 2]]], Optional[Pattern[q, Blank[]]]]], Pattern[x, Blank[Symbol]]] := Dist[(a + b*x + c*x^2)^FracPart[p]/((4*c)^IntPart[p]*(b + 2*c*x)^(2*FracPart[p])), Int[(b + 2*c*x)^(2*p)*(d + e*x + f*x^2)^q*(A + C*x^2), x], x] /; FreeQ[{a, b, c, d, e, f, A, C, p, q}, x] && EqQ[b^2 - 4*a*c, 0]
  1052. Int[Times[Power[Plus[Pattern[a, Blank[]], Times[Optional[Pattern[b, Blank[]]], Pattern[x, Blank[]]], Times[Optional[Pattern[c, Blank[]]], Power[Pattern[x, Blank[]], 2]]], Optional[Pattern[p, Blank[]]]], Plus[Optional[Pattern[A, Blank[]]], Times[Optional[Pattern[B, Blank[]]], Pattern[x, Blank[]]], Times[Optional[Pattern[C, Blank[]]], Power[Pattern[x, Blank[]], 2]]], Power[Plus[Optional[Pattern[d, Blank[]]], Times[Optional[Pattern[f, Blank[]]], Power[Pattern[x, Blank[]], 2]]], Optional[Pattern[q, Blank[]]]]], Pattern[x, Blank[Symbol]]] := Dist[(a + b*x + c*x^2)^FracPart[p]/((4*c)^IntPart[p]*(b + 2*c*x)^(2*FracPart[p])), Int[(b + 2*c*x)^(2*p)*(d + f*x^2)^q*(A + B*x + C*x^2), x], x] /; FreeQ[{a, b, c, d, f, A, B, C, p, q}, x] && EqQ[b^2 - 4*a*c, 0]
  1053. Int[Times[Power[Plus[Pattern[a, Blank[]], Times[Optional[Pattern[b, Blank[]]], Pattern[x, Blank[]]], Times[Optional[Pattern[c, Blank[]]], Power[Pattern[x, Blank[]], 2]]], Optional[Pattern[p, Blank[]]]], Plus[Optional[Pattern[A, Blank[]]], Times[Optional[Pattern[C, Blank[]]], Power[Pattern[x, Blank[]], 2]]], Power[Plus[Optional[Pattern[d, Blank[]]], Times[Optional[Pattern[f, Blank[]]], Power[Pattern[x, Blank[]], 2]]], Optional[Pattern[q, Blank[]]]]], Pattern[x, Blank[Symbol]]] := Dist[(a + b*x + c*x^2)^FracPart[p]/((4*c)^IntPart[p]*(b + 2*c*x)^(2*FracPart[p])), Int[(b + 2*c*x)^(2*p)*(d + f*x^2)^q*(A + C*x^2), x], x] /; FreeQ[{a, b, c, d, f, A, C, p, q}, x] && EqQ[b^2 - 4*a*c, 0]
  1054. Int[Times[Power[Plus[Pattern[a, Blank[]], Times[Optional[Pattern[b, Blank[]]], Pattern[x, Blank[]]], Times[Optional[Pattern[c, Blank[]]], Power[Pattern[x, Blank[]], 2]]], Pattern[p, Blank[]]], Plus[Optional[Pattern[A, Blank[]]], Times[Optional[Pattern[B, Blank[]]], Pattern[x, Blank[]]], Times[Optional[Pattern[C, Blank[]]], Power[Pattern[x, Blank[]], 2]]], Power[Plus[Pattern[d, Blank[]], Times[Optional[Pattern[e, Blank[]]], Pattern[x, Blank[]]], Times[Optional[Pattern[f, Blank[]]], Power[Pattern[x, Blank[]], 2]]], Pattern[q, Blank[]]]], Pattern[x, Blank[Symbol]]] := Simp[((A*b*c - 2*a*B*c + a*b*C - (c*(b*B - 2*A*c) - C*(b^2 - 2*a*c))*x)*(a + b*x + c*x^2)^(p + 1)*(d + e*x + f*x^2)^q)/(c*(b^2 - 4*a*c)*(p + 1)), x] - Dist[1/(c*(b^2 - 4*a*c)*(p + 1)), Int[(a + b*x + c*x^2)^(p + 1)*(d + e*x + f*x^2)^(q - 1)*Simp[e*q*(A*b*c - 2*a*B*c + a*b*C) - d*(c*(b*B - 2*A*c)*(2*p + 3) + C*(2*a*c - b^2*(p + 2))) + (2*f*q*(A*b*c - 2*a*B*c + a*b*C) - e*(c*(b*B - 2*A*c)*(2*p + q + 3) + C*(2*a*c*(q + 1) - b^2*(p + q + 2))))*x - f*(c*(b*B - 2*A*c)*(2*p + 2*q + 3) + C*(2*a*c*(2*q + 1) - b^2*(p + 2*q + 2)))*x^2, x], x], x] /; FreeQ[{a, b, c, d, e, f, A, B, C}, x] && NeQ[b^2 - 4*a*c, 0] && NeQ[e^2 - 4*d*f, 0] && LtQ[p, -1] && GtQ[q, 0] && !IGtQ[q, 0]
  1055. Int[Times[Power[Plus[Pattern[a, Blank[]], Times[Optional[Pattern[b, Blank[]]], Pattern[x, Blank[]]], Times[Optional[Pattern[c, Blank[]]], Power[Pattern[x, Blank[]], 2]]], Pattern[p, Blank[]]], Plus[Optional[Pattern[A, Blank[]]], Times[Optional[Pattern[C, Blank[]]], Power[Pattern[x, Blank[]], 2]]], Power[Plus[Pattern[d, Blank[]], Times[Optional[Pattern[e, Blank[]]], Pattern[x, Blank[]]], Times[Optional[Pattern[f, Blank[]]], Power[Pattern[x, Blank[]], 2]]], Pattern[q, Blank[]]]], Pattern[x, Blank[Symbol]]] := Simp[((A*b*c + a*b*C + (2*A*c^2 + C*(b^2 - 2*a*c))*x)*(a + b*x + c*x^2)^(p + 1)*(d + e*x + f*x^2)^q)/(c*(b^2 - 4*a*c)*(p + 1)), x] - Dist[1/(c*(b^2 - 4*a*c)*(p + 1)), Int[(a + b*x + c*x^2)^(p + 1)*(d + e*x + f*x^2)^(q - 1)*Simp[A*c*(2*c*d*(2*p + 3) + b*e*q) - C*(2*a*c*d - b^2*d*(p + 2) - a*b*e*q) + (C*(2*a*b*f*q - 2*a*c*e*(q + 1) + b^2*e*(p + q + 2)) + 2*A*c*(b*f*q + c*e*(2*p + q + 3)))*x - f*(-2*A*c^2*(2*p + 2*q + 3) + C*(2*a*c*(2*q + 1) - b^2*(p + 2*q + 2)))*x^2, x], x], x] /; FreeQ[{a, b, c, d, e, f, A, C}, x] && NeQ[b^2 - 4*a*c, 0] && NeQ[e^2 - 4*d*f, 0] && LtQ[p, -1] && GtQ[q, 0] && !IGtQ[q, 0]
  1056. Int[Times[Power[Plus[Pattern[a, Blank[]], Times[Optional[Pattern[c, Blank[]]], Power[Pattern[x, Blank[]], 2]]], Pattern[p, Blank[]]], Plus[Optional[Pattern[A, Blank[]]], Times[Optional[Pattern[B, Blank[]]], Pattern[x, Blank[]]], Times[Optional[Pattern[C, Blank[]]], Power[Pattern[x, Blank[]], 2]]], Power[Plus[Pattern[d, Blank[]], Times[Optional[Pattern[e, Blank[]]], Pattern[x, Blank[]]], Times[Optional[Pattern[f, Blank[]]], Power[Pattern[x, Blank[]], 2]]], Pattern[q, Blank[]]]], Pattern[x, Blank[Symbol]]] := Simp[((a*B - (A*c - a*C)*x)*(a + c*x^2)^(p + 1)*(d + e*x + f*x^2)^q)/(2*a*c*(p + 1)), x] - Dist[2/((-4*a*c)*(p + 1)), Int[(a + c*x^2)^(p + 1)*(d + e*x + f*x^2)^(q - 1)*Simp[A*c*d*(2*p + 3) - a*(C*d + B*e*q) + (A*c*e*(2*p + q + 3) - a*(2*B*f*q + C*e*(q + 1)))*x - f*(a*C*(2*q + 1) - A*c*(2*p + 2*q + 3))*x^2, x], x], x] /; FreeQ[{a, c, d, e, f, A, B, C}, x] && NeQ[e^2 - 4*d*f, 0] && LtQ[p, -1] && GtQ[q, 0] && !IGtQ[q, 0]
  1057. Int[Times[Power[Plus[Pattern[a, Blank[]], Times[Optional[Pattern[c, Blank[]]], Power[Pattern[x, Blank[]], 2]]], Pattern[p, Blank[]]], Plus[Optional[Pattern[A, Blank[]]], Times[Optional[Pattern[C, Blank[]]], Power[Pattern[x, Blank[]], 2]]], Power[Plus[Pattern[d, Blank[]], Times[Optional[Pattern[e, Blank[]]], Pattern[x, Blank[]]], Times[Optional[Pattern[f, Blank[]]], Power[Pattern[x, Blank[]], 2]]], Pattern[q, Blank[]]]], Pattern[x, Blank[Symbol]]] := -Simp[((A*c - a*C)*x*(a + c*x^2)^(p + 1)*(d + e*x + f*x^2)^q)/(2*a*c*(p + 1)), x] + Dist[2/(4*a*c*(p + 1)), Int[(a + c*x^2)^(p + 1)*(d + e*x + f*x^2)^(q - 1)*Simp[A*c*d*(2*p + 3) - a*C*d + (A*c*e*(2*p + q + 3) - a*C*e*(q + 1))*x - f*(a*C*(2*q + 1) - A*c*(2*p + 2*q + 3))*x^2, x], x], x] /; FreeQ[{a, c, d, e, f, A, C}, x] && NeQ[e^2 - 4*d*f, 0] && LtQ[p, -1] && GtQ[q, 0] && !IGtQ[q, 0]
  1058. Int[Times[Power[Plus[Pattern[a, Blank[]], Times[Optional[Pattern[b, Blank[]]], Pattern[x, Blank[]]], Times[Optional[Pattern[c, Blank[]]], Power[Pattern[x, Blank[]], 2]]], Pattern[p, Blank[]]], Plus[Optional[Pattern[A, Blank[]]], Times[Optional[Pattern[B, Blank[]]], Pattern[x, Blank[]]], Times[Optional[Pattern[C, Blank[]]], Power[Pattern[x, Blank[]], 2]]], Power[Plus[Pattern[d, Blank[]], Times[Optional[Pattern[f, Blank[]]], Power[Pattern[x, Blank[]], 2]]], Pattern[q, Blank[]]]], Pattern[x, Blank[Symbol]]] := Simp[((A*b*c - 2*a*B*c + a*b*C - (c*(b*B - 2*A*c) - C*(b^2 - 2*a*c))*x)*(a + b*x + c*x^2)^(p + 1)*(d + f*x^2)^q)/(c*(b^2 - 4*a*c)*(p + 1)), x] - Dist[1/(c*(b^2 - 4*a*c)*(p + 1)), Int[(a + b*x + c*x^2)^(p + 1)*(d + f*x^2)^(q - 1)*Simp[-(d*(c*(b*B - 2*A*c)*(2*p + 3) + C*(2*a*c - b^2*(p + 2)))) + (2*f*q*(A*b*c - 2*a*B*c + a*b*C))*x - f*(c*(b*B - 2*A*c)*(2*p + 2*q + 3) + C*(2*a*c*(2*q + 1) - b^2*(p + 2*q + 2)))*x^2, x], x], x] /; FreeQ[{a, b, c, d, f, A, B, C}, x] && NeQ[b^2 - 4*a*c, 0] && LtQ[p, -1] && GtQ[q, 0] && !IGtQ[q, 0]
  1059. Int[Times[Power[Plus[Pattern[a, Blank[]], Times[Optional[Pattern[b, Blank[]]], Pattern[x, Blank[]]], Times[Optional[Pattern[c, Blank[]]], Power[Pattern[x, Blank[]], 2]]], Pattern[p, Blank[]]], Plus[Optional[Pattern[A, Blank[]]], Times[Optional[Pattern[C, Blank[]]], Power[Pattern[x, Blank[]], 2]]], Power[Plus[Pattern[d, Blank[]], Times[Optional[Pattern[f, Blank[]]], Power[Pattern[x, Blank[]], 2]]], Pattern[q, Blank[]]]], Pattern[x, Blank[Symbol]]] := Simp[((A*b*c + a*b*C + (2*A*c^2 + C*(b^2 - 2*a*c))*x)*(a + b*x + c*x^2)^(p + 1)*(d + f*x^2)^q)/(c*(b^2 - 4*a*c)*(p + 1)), x] - Dist[1/(c*(b^2 - 4*a*c)*(p + 1)), Int[(a + b*x + c*x^2)^(p + 1)*(d + f*x^2)^(q - 1)*Simp[A*c*(2*c*d*(2*p + 3)) - C*(2*a*c*d - b^2*d*(p + 2)) + (C*(2*a*b*f*q) + 2*A*c*(b*f*q))*x - f*(-2*A*c^2*(2*p + 2*q + 3) + C*(2*a*c*(2*q + 1) - b^2*(p + 2*q + 2)))*x^2, x], x], x] /; FreeQ[{a, b, c, d, f, A, C}, x] && NeQ[b^2 - 4*a*c, 0] && LtQ[p, -1] && GtQ[q, 0] && !IGtQ[q, 0]
  1060. Int[Times[Power[Plus[Pattern[a, Blank[]], Times[Optional[Pattern[b, Blank[]]], Pattern[x, Blank[]]], Times[Optional[Pattern[c, Blank[]]], Power[Pattern[x, Blank[]], 2]]], Pattern[p, Blank[]]], Plus[Optional[Pattern[A, Blank[]]], Times[Optional[Pattern[B, Blank[]]], Pattern[x, Blank[]]], Times[Optional[Pattern[C, Blank[]]], Power[Pattern[x, Blank[]], 2]]], Power[Plus[Pattern[d, Blank[]], Times[Optional[Pattern[e, Blank[]]], Pattern[x, Blank[]]], Times[Optional[Pattern[f, Blank[]]], Power[Pattern[x, Blank[]], 2]]], Pattern[q, Blank[]]]], Pattern[x, Blank[Symbol]]] := Simp[((a + b*x + c*x^2)^(p + 1)*(d + e*x + f*x^2)^(q + 1)*((A*c - a*C)*(2*a*c*e - b*(c*d + a*f)) + (A*b - a*B)*(2*c^2*d + b^2*f - c*(b*e + 2*a*f)) + c*(A*(2*c^2*d + b^2*f - c*(b*e + 2*a*f)) - B*(b*c*d - 2*a*c*e + a*b*f) + C*(b^2*d - a*b*e - 2*a*(c*d - a*f)))*x))/((b^2 - 4*a*c)*((c*d - a*f)^2 - (b*d - a*e)*(c*e - b*f))*(p + 1)), x] + Dist[1/((b^2 - 4*a*c)*((c*d - a*f)^2 - (b*d - a*e)*(c*e - b*f))*(p + 1)), Int[(a + b*x + c*x^2)^(p + 1)*(d + e*x + f*x^2)^q*Simp[(b*B - 2*A*c - 2*a*C)*((c*d - a*f)^2 - (b*d - a*e)*(c*e - b*f))*(p + 1) + (b^2*(C*d + A*f) - b*(B*c*d + A*c*e + a*C*e + a*B*f) + 2*(A*c*(c*d - a*f) - a*(c*C*d - B*c*e - a*C*f)))*(a*f*(p + 1) - c*d*(p + 2)) - e*((A*c - a*C)*(2*a*c*e - b*(c*d + a*f)) + (A*b - a*B)*(2*c^2*d + b^2*f - c*(b*e + 2*a*f)))*(p + q + 2) - (2*f*((A*c - a*C)*(2*a*c*e - b*(c*d + a*f)) + (A*b - a*B)*(2*c^2*d + b^2*f - c*(b*e + 2*a*f)))*(p + q + 2) - (b^2*(C*d + A*f) - b*(B*c*d + A*c*e + a*C*e + a*B*f) + 2*(A*c*(c*d - a*f) - a*(c*C*d - B*c*e - a*C*f)))*(b*f*(p + 1) - c*e*(2*p + q + 4)))*x - c*f*(b^2*(C*d + A*f) - b*(B*c*d + A*c*e + a*C*e + a*B*f) + 2*(A*c*(c*d - a*f) - a*(c*C*d - B*c*e - a*C*f)))*(2*p + 2*q + 5)*x^2, x], x], x] /; FreeQ[{a, b, c, d, e, f, A, B, C, q}, x] && NeQ[b^2 - 4*a*c, 0] && NeQ[e^2 - 4*d*f, 0] && LtQ[p, -1] && NeQ[(c*d - a*f)^2 - (b*d - a*e)*(c*e - b*f), 0] && !( !IntegerQ[p] && ILtQ[q, -1]) && !IGtQ[q, 0]
  1061. Int[Times[Power[Plus[Pattern[a, Blank[]], Times[Optional[Pattern[b, Blank[]]], Pattern[x, Blank[]]], Times[Optional[Pattern[c, Blank[]]], Power[Pattern[x, Blank[]], 2]]], Pattern[p, Blank[]]], Plus[Optional[Pattern[A, Blank[]]], Times[Optional[Pattern[C, Blank[]]], Power[Pattern[x, Blank[]], 2]]], Power[Plus[Pattern[d, Blank[]], Times[Optional[Pattern[e, Blank[]]], Pattern[x, Blank[]]], Times[Optional[Pattern[f, Blank[]]], Power[Pattern[x, Blank[]], 2]]], Pattern[q, Blank[]]]], Pattern[x, Blank[Symbol]]] := Simp[((a + b*x + c*x^2)^(p + 1)*(d + e*x + f*x^2)^(q + 1)*((A*c - a*C)*(2*a*c*e - b*(c*d + a*f)) + (A*b)*(2*c^2*d + b^2*f - c*(b*e + 2*a*f)) + c*(A*(2*c^2*d + b^2*f - c*(b*e + 2*a*f)) + C*(b^2*d - a*b*e - 2*a*(c*d - a*f)))*x))/((b^2 - 4*a*c)*((c*d - a*f)^2 - (b*d - a*e)*(c*e - b*f))*(p + 1)), x] + Dist[1/((b^2 - 4*a*c)*((c*d - a*f)^2 - (b*d - a*e)*(c*e - b*f))*(p + 1)), Int[(a + b*x + c*x^2)^(p + 1)*(d + e*x + f*x^2)^q*Simp[(-2*A*c - 2*a*C)*((c*d - a*f)^2 - (b*d - a*e)*(c*e - b*f))*(p + 1) + (b^2*(C*d + A*f) - b*((Plus[A])*c*e + a*C*e) + 2*(A*c*(c*d - a*f) - a*(c*C*d - a*C*f)))*(a*f*(p + 1) - c*d*(p + 2)) - e*((A*c - a*C)*(2*a*c*e - b*(c*d + a*f)) + (A*b)*(2*c^2*d + b^2*f - c*(b*e + 2*a*f)))*(p + q + 2) - (2*f*((A*c - a*C)*(2*a*c*e - b*(c*d + a*f)) + (A*b)*(2*c^2*d + b^2*f - c*(b*e + 2*a*f)))*(p + q + 2) - (b^2*(C*d + A*f) - b*(A*c*e + a*C*e) + 2*(A*c*(c*d - a*f) - a*(c*C*d - a*C*f)))*(b*f*(p + 1) - c*e*(2*p + q + 4)))*x - c*f*(b^2*(C*d + A*f) - b*(A*c*e + a*C*e) + 2*(A*c*(c*d - a*f) - a*(c*C*d - a*C*f)))*(2*p + 2*q + 5)*x^2, x], x], x] /; FreeQ[{a, b, c, d, e, f, A, C, q}, x] && NeQ[b^2 - 4*a*c, 0] && NeQ[e^2 - 4*d*f, 0] && LtQ[p, -1] && NeQ[(c*d - a*f)^2 - (b*d - a*e)*(c*e - b*f), 0] && !( !IntegerQ[p] && ILtQ[q, -1]) && !IGtQ[q, 0]
  1062. Int[Times[Power[Plus[Pattern[a, Blank[]], Times[Optional[Pattern[c, Blank[]]], Power[Pattern[x, Blank[]], 2]]], Pattern[p, Blank[]]], Plus[Optional[Pattern[A, Blank[]]], Times[Optional[Pattern[B, Blank[]]], Pattern[x, Blank[]]], Times[Optional[Pattern[C, Blank[]]], Power[Pattern[x, Blank[]], 2]]], Power[Plus[Pattern[d, Blank[]], Times[Optional[Pattern[e, Blank[]]], Pattern[x, Blank[]]], Times[Optional[Pattern[f, Blank[]]], Power[Pattern[x, Blank[]], 2]]], Pattern[q, Blank[]]]], Pattern[x, Blank[Symbol]]] := Simp[((a + c*x^2)^(p + 1)*(d + e*x + f*x^2)^(q + 1)*((A*c - a*C)*(2*a*c*e) + (-(a*B))*(2*c^2*d - c*(2*a*f)) + c*(A*(2*c^2*d - c*(2*a*f)) - B*(-2*a*c*e) + C*(-2*a*(c*d - a*f)))*x))/((-4*a*c)*(a*c*e^2 + (c*d - a*f)^2)*(p + 1)), x] + Dist[1/((-4*a*c)*(a*c*e^2 + (c*d - a*f)^2)*(p + 1)), Int[(a + c*x^2)^(p + 1)*(d + e*x + f*x^2)^q*Simp[(-2*A*c - 2*a*C)*((c*d - a*f)^2 - (-(a*e))*(c*e))*(p + 1) + (2*(A*c*(c*d - a*f) - a*(c*C*d - B*c*e - a*C*f)))*(a*f*(p + 1) - c*d*(p + 2)) - e*((A*c - a*C)*(2*a*c*e) + (-(a*B))*(2*c^2*d - c*((Plus[2])*a*f)))*(p + q + 2) - (2*f*((A*c - a*C)*(2*a*c*e) + (-(a*B))*(2*c^2*d - c*((Plus[2])*a*f)))*(p + q + 2) - (2*(A*c*(c*d - a*f) - a*(c*C*d - B*c*e - a*C*f)))*(-(c*e*(2*p + q + 4))))*x - c*f*(2*(A*c*(c*d - a*f) - a*(c*C*d - B*c*e - a*C*f)))*(2*p + 2*q + 5)*x^2, x], x], x] /; FreeQ[{a, c, d, e, f, A, B, C, q}, x] && NeQ[e^2 - 4*d*f, 0] && LtQ[p, -1] && NeQ[a*c*e^2 + (c*d - a*f)^2, 0] && !( !IntegerQ[p] && ILtQ[q, -1]) && !IGtQ[q, 0]
  1063. Int[Times[Power[Plus[Pattern[a, Blank[]], Times[Optional[Pattern[c, Blank[]]], Power[Pattern[x, Blank[]], 2]]], Pattern[p, Blank[]]], Plus[Optional[Pattern[A, Blank[]]], Times[Optional[Pattern[C, Blank[]]], Power[Pattern[x, Blank[]], 2]]], Power[Plus[Pattern[d, Blank[]], Times[Optional[Pattern[e, Blank[]]], Pattern[x, Blank[]]], Times[Optional[Pattern[f, Blank[]]], Power[Pattern[x, Blank[]], 2]]], Pattern[q, Blank[]]]], Pattern[x, Blank[Symbol]]] := Simp[((a + c*x^2)^(p + 1)*(d + e*x + f*x^2)^(q + 1)*((A*c - a*C)*(2*a*c*e) + c*(A*(2*c^2*d - c*(2*a*f)) + C*(-2*a*(c*d - a*f)))*x))/((-4*a*c)*(a*c*e^2 + (c*d - a*f)^2)*(p + 1)), x] + Dist[1/((-4*a*c)*(a*c*e^2 + (c*d - a*f)^2)*(p + 1)), Int[(a + c*x^2)^(p + 1)*(d + e*x + f*x^2)^q*Simp[(-2*A*c - 2*a*C)*((c*d - a*f)^2 - (-(a*e))*(c*e))*(p + 1) + (2*(A*c*(c*d - a*f) - a*(c*C*d - a*C*f)))*(a*f*(p + 1) - c*d*(p + 2)) - e*((A*c - a*C)*(2*a*c*e))*(p + q + 2) - (2*f*((A*c - a*C)*(2*a*c*e))*(p + q + 2) - (2*(A*c*(c*d - a*f) - a*(c*C*d - a*C*f)))*(-(c*e*(2*p + q + 4))))*x - c*f*(2*(A*c*(c*d - a*f) - a*(c*C*d - a*C*f)))*(2*p + 2*q + 5)*x^2, x], x], x] /; FreeQ[{a, c, d, e, f, A, C, q}, x] && NeQ[e^2 - 4*d*f, 0] && LtQ[p, -1] && NeQ[a*c*e^2 + (c*d - a*f)^2, 0] && !( !IntegerQ[p] && ILtQ[q, -1]) && !IGtQ[q, 0]
  1064. Int[Times[Power[Plus[Pattern[a, Blank[]], Times[Optional[Pattern[b, Blank[]]], Pattern[x, Blank[]]], Times[Optional[Pattern[c, Blank[]]], Power[Pattern[x, Blank[]], 2]]], Pattern[p, Blank[]]], Plus[Optional[Pattern[A, Blank[]]], Times[Optional[Pattern[B, Blank[]]], Pattern[x, Blank[]]], Times[Optional[Pattern[C, Blank[]]], Power[Pattern[x, Blank[]], 2]]], Power[Plus[Pattern[d, Blank[]], Times[Optional[Pattern[f, Blank[]]], Power[Pattern[x, Blank[]], 2]]], Pattern[q, Blank[]]]], Pattern[x, Blank[Symbol]]] := Simp[((a + b*x + c*x^2)^(p + 1)*(d + f*x^2)^(q + 1)*((A*c - a*C)*(-(b*(c*d + a*f))) + (A*b - a*B)*(2*c^2*d + b^2*f - c*(2*a*f)) + c*(A*(2*c^2*d + b^2*f - c*(2*a*f)) - B*(b*c*d + a*b*f) + C*(b^2*d - 2*a*(c*d - a*f)))*x))/((b^2 - 4*a*c)*(b^2*d*f + (c*d - a*f)^2)*(p + 1)), x] + Dist[1/((b^2 - 4*a*c)*(b^2*d*f + (c*d - a*f)^2)*(p + 1)), Int[(a + b*x + c*x^2)^(p + 1)*(d + f*x^2)^q*Simp[(b*B - 2*A*c - 2*a*C)*((c*d - a*f)^2 - (b*d)*(-(b*f)))*(p + 1) + (b^2*(C*d + A*f) - b*(B*c*d + a*B*f) + 2*(A*c*(c*d - a*f) - a*(c*C*d - a*C*f)))*(a*f*(p + 1) - c*d*(p + 2)) - (2*f*((A*c - a*C)*(-(b*(c*d + a*f))) + (A*b - a*B)*(2*c^2*d + b^2*f - c*(2*a*f)))*(p + q + 2) - (b^2*(C*d + A*f) - b*(B*c*d + a*B*f) + 2*(A*c*(c*d - a*f) - a*(c*C*d - a*C*f)))*(b*f*(p + 1)))*x - c*f*(b^2*(C*d + A*f) - b*(B*c*d + a*B*f) + 2*(A*c*(c*d - a*f) - a*(c*C*d - a*C*f)))*(2*p + 2*q + 5)*x^2, x], x], x] /; FreeQ[{a, b, c, d, f, A, B, C, q}, x] && NeQ[b^2 - 4*a*c, 0] && LtQ[p, -1] && NeQ[b^2*d*f + (c*d - a*f)^2, 0] && !( !IntegerQ[p] && ILtQ[q, -1]) && !IGtQ[q, 0]
  1065. Int[Times[Power[Plus[Pattern[a, Blank[]], Times[Optional[Pattern[b, Blank[]]], Pattern[x, Blank[]]], Times[Optional[Pattern[c, Blank[]]], Power[Pattern[x, Blank[]], 2]]], Pattern[p, Blank[]]], Plus[Optional[Pattern[A, Blank[]]], Times[Optional[Pattern[C, Blank[]]], Power[Pattern[x, Blank[]], 2]]], Power[Plus[Pattern[d, Blank[]], Times[Optional[Pattern[f, Blank[]]], Power[Pattern[x, Blank[]], 2]]], Pattern[q, Blank[]]]], Pattern[x, Blank[Symbol]]] := Simp[((a + b*x + c*x^2)^(p + 1)*(d + f*x^2)^(q + 1)*((A*c - a*C)*(-(b*(c*d + a*f))) + (A*b)*(2*c^2*d + b^2*f - c*(2*a*f)) + c*(A*(2*c^2*d + b^2*f - c*(2*a*f)) + C*(b^2*d - 2*a*(c*d - a*f)))*x))/((b^2 - 4*a*c)*(b^2*d*f + (c*d - a*f)^2)*(p + 1)), x] + Dist[1/((b^2 - 4*a*c)*(b^2*d*f + (c*d - a*f)^2)*(p + 1)), Int[(a + b*x + c*x^2)^(p + 1)*(d + f*x^2)^q*Simp[(-2*A*c - 2*a*C)*((c*d - a*f)^2 - (b*d)*(-(b*f)))*(p + 1) + (b^2*(C*d + A*f) + 2*(A*c*(c*d - a*f) - a*(c*C*d - a*C*f)))*(a*f*(p + 1) - c*d*(p + 2)) - (2*f*((A*c - a*C)*(-(b*(c*d + a*f))) + (A*b)*(2*c^2*d + b^2*f - c*(2*a*f)))*(p + q + 2) - (b^2*(C*d + A*f) + 2*(A*c*(c*d - a*f) - a*(c*C*d - a*C*f)))*(b*f*(p + 1)))*x - c*f*(b^2*(C*d + A*f) + 2*(A*c*(c*d - a*f) - a*(c*C*d - a*C*f)))*(2*p + 2*q + 5)*x^2, x], x], x] /; FreeQ[{a, b, c, d, f, A, C, q}, x] && NeQ[b^2 - 4*a*c, 0] && LtQ[p, -1] && NeQ[b^2*d*f + (c*d - a*f)^2, 0] && !( !IntegerQ[p] && ILtQ[q, -1]) && !IGtQ[q, 0]
  1066. Int[Times[Power[Plus[Pattern[a, Blank[]], Times[Optional[Pattern[b, Blank[]]], Pattern[x, Blank[]]], Times[Optional[Pattern[c, Blank[]]], Power[Pattern[x, Blank[]], 2]]], Pattern[p, Blank[]]], Plus[Optional[Pattern[A, Blank[]]], Times[Optional[Pattern[B, Blank[]]], Pattern[x, Blank[]]], Times[Optional[Pattern[C, Blank[]]], Power[Pattern[x, Blank[]], 2]]], Power[Plus[Pattern[d, Blank[]], Times[Optional[Pattern[e, Blank[]]], Pattern[x, Blank[]]], Times[Optional[Pattern[f, Blank[]]], Power[Pattern[x, Blank[]], 2]]], Pattern[q, Blank[]]]], Pattern[x, Blank[Symbol]]] := Simp[((B*c*f*(2*p + 2*q + 3) + C*(b*f*p - c*e*(2*p + q + 2)) + 2*c*C*f*(p + q + 1)*x)*(a + b*x + c*x^2)^p*(d + e*x + f*x^2)^(q + 1))/(2*c*f^2*(p + q + 1)*(2*p + 2*q + 3)), x] - Dist[1/(2*c*f^2*(p + q + 1)*(2*p + 2*q + 3)), Int[(a + b*x + c*x^2)^(p - 1)*(d + e*x + f*x^2)^q*Simp[p*(b*d - a*e)*(C*(c*e - b*f)*(q + 1) - c*(C*e - B*f)*(2*p + 2*q + 3)) + (p + q + 1)*(b^2*C*d*f*p + a*c*(C*(2*d*f - e^2*(2*p + q + 2)) + f*(B*e - 2*A*f)*(2*p + 2*q + 3))) + (2*p*(c*d - a*f)*(C*(c*e - b*f)*(q + 1) - c*(C*e - B*f)*(2*p + 2*q + 3)) + (p + q + 1)*(C*e*f*p*(b^2 - 4*a*c) - b*c*(C*(e^2 - 4*d*f)*(2*p + q + 2) + f*(2*C*d - B*e + 2*A*f)*(2*p + 2*q + 3))))*x + (p*(c*e - b*f)*(C*(c*e - b*f)*(q + 1) - c*(C*e - B*f)*(2*p + 2*q + 3)) + (p + q + 1)*(C*f^2*p*(b^2 - 4*a*c) - c^2*(C*(e^2 - 4*d*f)*(2*p + q + 2) + f*(2*C*d - B*e + 2*A*f)*(2*p + 2*q + 3))))*x^2, x], x], x] /; FreeQ[{a, b, c, d, e, f, A, B, C, q}, x] && NeQ[b^2 - 4*a*c, 0] && NeQ[e^2 - 4*d*f, 0] && GtQ[p, 0] && NeQ[p + q + 1, 0] && NeQ[2*p + 2*q + 3, 0] && !IGtQ[p, 0] && !IGtQ[q, 0]
  1067. Int[Times[Power[Plus[Pattern[a, Blank[]], Times[Optional[Pattern[b, Blank[]]], Pattern[x, Blank[]]], Times[Optional[Pattern[c, Blank[]]], Power[Pattern[x, Blank[]], 2]]], Pattern[p, Blank[]]], Plus[Optional[Pattern[A, Blank[]]], Times[Optional[Pattern[C, Blank[]]], Power[Pattern[x, Blank[]], 2]]], Power[Plus[Pattern[d, Blank[]], Times[Optional[Pattern[e, Blank[]]], Pattern[x, Blank[]]], Times[Optional[Pattern[f, Blank[]]], Power[Pattern[x, Blank[]], 2]]], Pattern[q, Blank[]]]], Pattern[x, Blank[Symbol]]] := Simp[((C*(b*f*p - c*e*(2*p + q + 2)) + 2*c*C*f*(p + q + 1)*x)*(a + b*x + c*x^2)^p*(d + e*x + f*x^2)^(q + 1))/(2*c*f^2*(p + q + 1)*(2*p + 2*q + 3)), x] - Dist[1/(2*c*f^2*(p + q + 1)*(2*p + 2*q + 3)), Int[(a + b*x + c*x^2)^(p - 1)*(d + e*x + f*x^2)^q*Simp[p*(b*d - a*e)*(C*(c*e - b*f)*(q + 1) - c*(C*e)*(2*p + 2*q + 3)) + (p + q + 1)*(b^2*C*d*f*p + a*c*(C*(2*d*f - e^2*(2*p + q + 2)) + f*(-2*A*f)*(2*p + 2*q + 3))) + (2*p*(c*d - a*f)*(C*(c*e - b*f)*(q + 1) - c*(C*e)*(2*p + 2*q + 3)) + (p + q + 1)*(C*e*f*p*(b^2 - 4*a*c) - b*c*(C*(e^2 - 4*d*f)*(2*p + q + 2) + f*(2*C*d + 2*A*f)*(2*p + 2*q + 3))))*x + (p*(c*e - b*f)*(C*(c*e - b*f)*(q + 1) - c*(C*e)*(2*p + 2*q + 3)) + (p + q + 1)*(C*f^2*p*(b^2 - 4*a*c) - c^2*(C*(e^2 - 4*d*f)*(2*p + q + 2) + f*(2*C*d + 2*A*f)*(2*p + 2*q + 3))))*x^2, x], x], x] /; FreeQ[{a, b, c, d, e, f, A, C, q}, x] && NeQ[b^2 - 4*a*c, 0] && NeQ[e^2 - 4*d*f, 0] && GtQ[p, 0] && NeQ[p + q + 1, 0] && NeQ[2*p + 2*q + 3, 0] && !IGtQ[p, 0] && !IGtQ[q, 0]
  1068. Int[Times[Power[Plus[Pattern[a, Blank[]], Times[Optional[Pattern[c, Blank[]]], Power[Pattern[x, Blank[]], 2]]], Pattern[p, Blank[]]], Plus[Optional[Pattern[A, Blank[]]], Times[Optional[Pattern[B, Blank[]]], Pattern[x, Blank[]]], Times[Optional[Pattern[C, Blank[]]], Power[Pattern[x, Blank[]], 2]]], Power[Plus[Pattern[d, Blank[]], Times[Optional[Pattern[e, Blank[]]], Pattern[x, Blank[]]], Times[Optional[Pattern[f, Blank[]]], Power[Pattern[x, Blank[]], 2]]], Pattern[q, Blank[]]]], Pattern[x, Blank[Symbol]]] := Simp[((B*c*f*(2*p + 2*q + 3) + C*(-(c*e*(2*p + q + 2))) + 2*c*C*f*(p + q + 1)*x)*(a + c*x^2)^p*(d + e*x + f*x^2)^(q + 1))/(2*c*f^2*(p + q + 1)*(2*p + 2*q + 3)), x] - Dist[1/(2*c*f^2*(p + q + 1)*(2*p + 2*q + 3)), Int[(a + c*x^2)^(p - 1)*(d + e*x + f*x^2)^q*Simp[p*(-(a*e))*(C*(c*e)*(q + 1) - c*(C*e - B*f)*(2*p + 2*q + 3)) + (p + q + 1)*(a*c*(C*(2*d*f - e^2*(2*p + q + 2)) + f*(B*e - 2*A*f)*(2*p + 2*q + 3))) + (2*p*(c*d - a*f)*(C*(c*e)*(q + 1) - c*(C*e - B*f)*(2*p + 2*q + 3)) + (p + q + 1)*(C*e*f*p*(-4*a*c)))*x + (p*(c*e)*(C*(c*e)*(q + 1) - c*(C*e - B*f)*(2*p + 2*q + 3)) + (p + q + 1)*(C*f^2*p*(-4*a*c) - c^2*(C*(e^2 - 4*d*f)*(2*p + q + 2) + f*(2*C*d - B*e + 2*A*f)*(2*p + 2*q + 3))))*x^2, x], x], x] /; FreeQ[{a, c, d, e, f, A, B, C, q}, x] && NeQ[e^2 - 4*d*f, 0] && GtQ[p, 0] && NeQ[p + q + 1, 0] && NeQ[2*p + 2*q + 3, 0] && !IGtQ[p, 0] && !IGtQ[q, 0]
  1069. Int[Times[Power[Plus[Pattern[a, Blank[]], Times[Optional[Pattern[c, Blank[]]], Power[Pattern[x, Blank[]], 2]]], Pattern[p, Blank[]]], Plus[Optional[Pattern[A, Blank[]]], Times[Optional[Pattern[C, Blank[]]], Power[Pattern[x, Blank[]], 2]]], Power[Plus[Pattern[d, Blank[]], Times[Optional[Pattern[e, Blank[]]], Pattern[x, Blank[]]], Times[Optional[Pattern[f, Blank[]]], Power[Pattern[x, Blank[]], 2]]], Pattern[q, Blank[]]]], Pattern[x, Blank[Symbol]]] := Simp[((C*(-(c*e*(2*p + q + 2))) + 2*c*C*f*(p + q + 1)*x)*(a + c*x^2)^p*(d + e*x + f*x^2)^(q + 1))/(2*c*f^2*(p + q + 1)*(2*p + 2*q + 3)), x] - Dist[1/(2*c*f^2*(p + q + 1)*(2*p + 2*q + 3)), Int[(a + c*x^2)^(p - 1)*(d + e*x + f*x^2)^q*Simp[p*(-(a*e))*(C*(c*e)*(q + 1) - c*(C*e)*(2*p + 2*q + 3)) + (p + q + 1)*(a*c*(C*(2*d*f - e^2*(2*p + q + 2)) + f*(-2*A*f)*(2*p + 2*q + 3))) + (2*p*(c*d - a*f)*(C*(c*e)*(q + 1) - c*(C*e)*(2*p + 2*q + 3)) + (p + q + 1)*(C*e*f*p*(-4*a*c)))*x + (p*(c*e)*(C*(c*e)*(q + 1) - c*(C*e)*(2*p + 2*q + 3)) + (p + q + 1)*(C*f^2*p*(-4*a*c) - c^2*(C*(e^2 - 4*d*f)*(2*p + q + 2) + f*(2*C*d + 2*A*f)*(2*p + 2*q + 3))))*x^2, x], x], x] /; FreeQ[{a, c, d, e, f, A, C, q}, x] && NeQ[e^2 - 4*d*f, 0] && GtQ[p, 0] && NeQ[p + q + 1, 0] && NeQ[2*p + 2*q + 3, 0] && !IGtQ[p, 0] && !IGtQ[q, 0]
  1070. Int[Times[Power[Plus[Pattern[a, Blank[]], Times[Optional[Pattern[b, Blank[]]], Pattern[x, Blank[]]], Times[Optional[Pattern[c, Blank[]]], Power[Pattern[x, Blank[]], 2]]], Pattern[p, Blank[]]], Plus[Optional[Pattern[A, Blank[]]], Times[Optional[Pattern[B, Blank[]]], Pattern[x, Blank[]]], Times[Optional[Pattern[C, Blank[]]], Power[Pattern[x, Blank[]], 2]]], Power[Plus[Pattern[d, Blank[]], Times[Optional[Pattern[f, Blank[]]], Power[Pattern[x, Blank[]], 2]]], Pattern[q, Blank[]]]], Pattern[x, Blank[Symbol]]] := Simp[((B*c*f*(2*p + 2*q + 3) + C*(b*f*p) + 2*c*C*f*(p + q + 1)*x)*(a + b*x + c*x^2)^p*(d + f*x^2)^(q + 1))/(2*c*f^2*(p + q + 1)*(2*p + 2*q + 3)), x] - Dist[1/(2*c*f^2*(p + q + 1)*(2*p + 2*q + 3)), Int[(a + b*x + c*x^2)^(p - 1)*(d + f*x^2)^q*Simp[p*(b*d)*(C*(-(b*f))*(q + 1) - c*(-(B*f))*(2*p + 2*q + 3)) + (p + q + 1)*(b^2*C*d*f*p + a*c*(C*(2*d*f) + f*(-2*A*f)*(2*p + 2*q + 3))) + (2*p*(c*d - a*f)*(C*(-(b*f))*(q + 1) - c*(-(B*f))*(2*p + 2*q + 3)) + (p + q + 1)*(-(b*c*(C*(-4*d*f)*(2*p + q + 2) + f*(2*C*d + 2*A*f)*(2*p + 2*q + 3)))))*x + (p*(-(b*f))*(C*(-(b*f))*(q + 1) - c*(-(B*f))*(2*p + 2*q + 3)) + (p + q + 1)*(C*f^2*p*(b^2 - 4*a*c) - c^2*(C*(-4*d*f)*(2*p + q + 2) + f*(2*C*d + 2*A*f)*(2*p + 2*q + 3))))*x^2, x], x], x] /; FreeQ[{a, b, c, d, f, A, B, C, q}, x] && NeQ[b^2 - 4*a*c, 0] && GtQ[p, 0] && NeQ[p + q + 1, 0] && NeQ[2*p + 2*q + 3, 0] && !IGtQ[p, 0] && !IGtQ[q, 0]
  1071. Int[Times[Power[Plus[Pattern[a, Blank[]], Times[Optional[Pattern[b, Blank[]]], Pattern[x, Blank[]]], Times[Optional[Pattern[c, Blank[]]], Power[Pattern[x, Blank[]], 2]]], Pattern[p, Blank[]]], Plus[Optional[Pattern[A, Blank[]]], Times[Optional[Pattern[C, Blank[]]], Power[Pattern[x, Blank[]], 2]]], Power[Plus[Pattern[d, Blank[]], Times[Optional[Pattern[f, Blank[]]], Power[Pattern[x, Blank[]], 2]]], Pattern[q, Blank[]]]], Pattern[x, Blank[Symbol]]] := Simp[((C*(b*f*p) + 2*c*C*f*(p + q + 1)*x)*(a + b*x + c*x^2)^p*(d + f*x^2)^(q + 1))/(2*c*f^2*(p + q + 1)*(2*p + 2*q + 3)), x] - Dist[1/(2*c*f^2*(p + q + 1)*(2*p + 2*q + 3)), Int[(a + b*x + c*x^2)^(p - 1)*(d + f*x^2)^q*Simp[p*(b*d)*(C*(-(b*f))*(q + 1)) + (p + q + 1)*(b^2*C*d*f*p + a*c*(C*(2*d*f) + f*(-2*A*f)*(2*p + 2*q + 3))) + (2*p*(c*d - a*f)*(C*(-(b*f))*(q + 1)) + (p + q + 1)*(-(b*c*(C*(-4*d*f)*(2*p + q + 2) + f*(2*C*d + 2*A*f)*(2*p + 2*q + 3)))))*x + (p*(-(b*f))*(C*(-(b*f))*(q + 1)) + (p + q + 1)*(C*f^2*p*(b^2 - 4*a*c) - c^2*(C*(-4*d*f)*(2*p + q + 2) + f*(2*C*d + 2*A*f)*(2*p + 2*q + 3))))*x^2, x], x], x] /; FreeQ[{a, b, c, d, f, A, C, q}, x] && NeQ[b^2 - 4*a*c, 0] && GtQ[p, 0] && NeQ[p + q + 1, 0] && NeQ[2*p + 2*q + 3, 0] && !IGtQ[p, 0] && !IGtQ[q, 0]
  1072. Int[Times[Power[Plus[Pattern[a, Blank[]], Times[Optional[Pattern[b, Blank[]]], Pattern[x, Blank[]]], Times[Optional[Pattern[c, Blank[]]], Power[Pattern[x, Blank[]], 2]]], -1], Plus[Optional[Pattern[A, Blank[]]], Times[Optional[Pattern[B, Blank[]]], Pattern[x, Blank[]]], Times[Optional[Pattern[C, Blank[]]], Power[Pattern[x, Blank[]], 2]]], Power[Plus[Pattern[d, Blank[]], Times[Optional[Pattern[e, Blank[]]], Pattern[x, Blank[]]], Times[Optional[Pattern[f, Blank[]]], Power[Pattern[x, Blank[]], 2]]], -1]], Pattern[x, Blank[Symbol]]] := With[{q = c^2*d^2 - b*c*d*e + a*c*e^2 + b^2*d*f - 2*a*c*d*f - a*b*e*f + a^2*f^2}, Dist[1/q, Int[(A*c^2*d - a*c*C*d - A*b*c*e + a*B*c*e + A*b^2*f - a*b*B*f - a*A*c*f + a^2*C*f + c*(B*c*d - b*C*d - A*c*e + a*C*e + A*b*f - a*B*f)*x)/(a + b*x + c*x^2), x], x] + Dist[1/q, Int[(c*C*d^2 - B*c*d*e + A*c*e^2 + b*B*d*f - A*c*d*f - a*C*d*f - A*b*e*f + a*A*f^2 - f*(B*c*d - b*C*d - A*c*e + a*C*e + A*b*f - a*B*f)*x)/(d + e*x + f*x^2), x], x] /; NeQ[q, 0]] /; FreeQ[{a, b, c, d, e, f, A, B, C}, x] && NeQ[b^2 - 4*a*c, 0] && NeQ[e^2 - 4*d*f, 0]
  1073. Int[Times[Power[Plus[Pattern[a, Blank[]], Times[Optional[Pattern[b, Blank[]]], Pattern[x, Blank[]]], Times[Optional[Pattern[c, Blank[]]], Power[Pattern[x, Blank[]], 2]]], -1], Plus[Optional[Pattern[A, Blank[]]], Times[Optional[Pattern[C, Blank[]]], Power[Pattern[x, Blank[]], 2]]], Power[Plus[Pattern[d, Blank[]], Times[Optional[Pattern[e, Blank[]]], Pattern[x, Blank[]]], Times[Optional[Pattern[f, Blank[]]], Power[Pattern[x, Blank[]], 2]]], -1]], Pattern[x, Blank[Symbol]]] := With[{q = c^2*d^2 - b*c*d*e + a*c*e^2 + b^2*d*f - 2*a*c*d*f - a*b*e*f + a^2*f^2}, Dist[1/q, Int[(A*c^2*d - a*c*C*d - A*b*c*e + A*b^2*f - a*A*c*f + a^2*C*f + c*(-(b*C*d) - A*c*e + a*C*e + A*b*f)*x)/(a + b*x + c*x^2), x], x] + Dist[1/q, Int[(c*C*d^2 + A*c*e^2 - A*c*d*f - a*C*d*f - A*b*e*f + a*A*f^2 - f*(-(b*C*d) - A*c*e + a*C*e + A*b*f)*x)/(d + e*x + f*x^2), x], x] /; NeQ[q, 0]] /; FreeQ[{a, b, c, d, e, f, A, C}, x] && NeQ[b^2 - 4*a*c, 0] && NeQ[e^2 - 4*d*f, 0]
  1074. Int[Times[Power[Plus[Pattern[a, Blank[]], Times[Optional[Pattern[b, Blank[]]], Pattern[x, Blank[]]], Times[Optional[Pattern[c, Blank[]]], Power[Pattern[x, Blank[]], 2]]], -1], Plus[Optional[Pattern[A, Blank[]]], Times[Optional[Pattern[B, Blank[]]], Pattern[x, Blank[]]], Times[Optional[Pattern[C, Blank[]]], Power[Pattern[x, Blank[]], 2]]], Power[Plus[Pattern[d, Blank[]], Times[Optional[Pattern[f, Blank[]]], Power[Pattern[x, Blank[]], 2]]], -1]], Pattern[x, Blank[Symbol]]] := With[{q = c^2*d^2 + b^2*d*f - 2*a*c*d*f + a^2*f^2}, Dist[1/q, Int[(A*c^2*d - a*c*C*d + A*b^2*f - a*b*B*f - a*A*c*f + a^2*C*f + c*(B*c*d - b*C*d + A*b*f - a*B*f)*x)/(a + b*x + c*x^2), x], x] + Dist[1/q, Int[(c*C*d^2 + b*B*d*f - A*c*d*f - a*C*d*f + a*A*f^2 - f*(B*c*d - b*C*d + A*b*f - a*B*f)*x)/(d + f*x^2), x], x] /; NeQ[q, 0]] /; FreeQ[{a, b, c, d, f, A, B, C}, x] && NeQ[b^2 - 4*a*c, 0]
  1075. Int[Times[Power[Plus[Pattern[a, Blank[]], Times[Optional[Pattern[b, Blank[]]], Pattern[x, Blank[]]], Times[Optional[Pattern[c, Blank[]]], Power[Pattern[x, Blank[]], 2]]], -1], Plus[Optional[Pattern[A, Blank[]]], Times[Optional[Pattern[C, Blank[]]], Power[Pattern[x, Blank[]], 2]]], Power[Plus[Pattern[d, Blank[]], Times[Optional[Pattern[f, Blank[]]], Power[Pattern[x, Blank[]], 2]]], -1]], Pattern[x, Blank[Symbol]]] := With[{q = c^2*d^2 + b^2*d*f - 2*a*c*d*f + a^2*f^2}, Dist[1/q, Int[(A*c^2*d - a*c*C*d + A*b^2*f - a*A*c*f + a^2*C*f + c*(-(b*C*d) + A*b*f)*x)/(a + b*x + c*x^2), x], x] + Dist[1/q, Int[(c*C*d^2 - A*c*d*f - a*C*d*f + a*A*f^2 - f*(-(b*C*d) + A*b*f)*x)/(d + f*x^2), x], x] /; NeQ[q, 0]] /; FreeQ[{a, b, c, d, f, A, C}, x] && NeQ[b^2 - 4*a*c, 0]
  1076. Int[Times[Power[Plus[Pattern[a, Blank[]], Times[Optional[Pattern[b, Blank[]]], Pattern[x, Blank[]]], Times[Optional[Pattern[c, Blank[]]], Power[Pattern[x, Blank[]], 2]]], -1], Plus[Optional[Pattern[A, Blank[]]], Times[Optional[Pattern[B, Blank[]]], Pattern[x, Blank[]]], Times[Optional[Pattern[C, Blank[]]], Power[Pattern[x, Blank[]], 2]]], Power[Plus[Optional[Pattern[d, Blank[]]], Times[Optional[Pattern[e, Blank[]]], Pattern[x, Blank[]]], Times[Optional[Pattern[f, Blank[]]], Power[Pattern[x, Blank[]], 2]]], Rational[-1, 2]]], Pattern[x, Blank[Symbol]]] := Dist[C/c, Int[1/Sqrt[d + e*x + f*x^2], x], x] + Dist[1/c, Int[(A*c - a*C + (B*c - b*C)*x)/((a + b*x + c*x^2)*Sqrt[d + e*x + f*x^2]), x], x] /; FreeQ[{a, b, c, d, e, f, A, B, C}, x] && NeQ[b^2 - 4*a*c, 0] && NeQ[e^2 - 4*d*f, 0]
  1077. Int[Times[Power[Plus[Pattern[a, Blank[]], Times[Optional[Pattern[b, Blank[]]], Pattern[x, Blank[]]], Times[Optional[Pattern[c, Blank[]]], Power[Pattern[x, Blank[]], 2]]], -1], Plus[Optional[Pattern[A, Blank[]]], Times[Optional[Pattern[C, Blank[]]], Power[Pattern[x, Blank[]], 2]]], Power[Plus[Optional[Pattern[d, Blank[]]], Times[Optional[Pattern[e, Blank[]]], Pattern[x, Blank[]]], Times[Optional[Pattern[f, Blank[]]], Power[Pattern[x, Blank[]], 2]]], Rational[-1, 2]]], Pattern[x, Blank[Symbol]]] := Dist[C/c, Int[1/Sqrt[d + e*x + f*x^2], x], x] + Dist[1/c, Int[(A*c - a*C - b*C*x)/((a + b*x + c*x^2)*Sqrt[d + e*x + f*x^2]), x], x] /; FreeQ[{a, b, c, d, e, f, A, C}, x] && NeQ[b^2 - 4*a*c, 0] && NeQ[e^2 - 4*d*f, 0]
  1078. Int[Times[Power[Plus[Pattern[a, Blank[]], Times[Optional[Pattern[c, Blank[]]], Power[Pattern[x, Blank[]], 2]]], -1], Plus[Optional[Pattern[A, Blank[]]], Times[Optional[Pattern[B, Blank[]]], Pattern[x, Blank[]]], Times[Optional[Pattern[C, Blank[]]], Power[Pattern[x, Blank[]], 2]]], Power[Plus[Optional[Pattern[d, Blank[]]], Times[Optional[Pattern[e, Blank[]]], Pattern[x, Blank[]]], Times[Optional[Pattern[f, Blank[]]], Power[Pattern[x, Blank[]], 2]]], Rational[-1, 2]]], Pattern[x, Blank[Symbol]]] := Dist[C/c, Int[1/Sqrt[d + e*x + f*x^2], x], x] + Dist[1/c, Int[(A*c - a*C + B*c*x)/((a + c*x^2)*Sqrt[d + e*x + f*x^2]), x], x] /; FreeQ[{a, c, d, e, f, A, B, C}, x] && NeQ[e^2 - 4*d*f, 0]
  1079. Int[Times[Power[Plus[Pattern[a, Blank[]], Times[Optional[Pattern[c, Blank[]]], Power[Pattern[x, Blank[]], 2]]], -1], Plus[Optional[Pattern[A, Blank[]]], Times[Optional[Pattern[C, Blank[]]], Power[Pattern[x, Blank[]], 2]]], Power[Plus[Optional[Pattern[d, Blank[]]], Times[Optional[Pattern[e, Blank[]]], Pattern[x, Blank[]]], Times[Optional[Pattern[f, Blank[]]], Power[Pattern[x, Blank[]], 2]]], Rational[-1, 2]]], Pattern[x, Blank[Symbol]]] := Dist[C/c, Int[1/Sqrt[d + e*x + f*x^2], x], x] + Dist[(A*c - a*C)/c, Int[1/((a + c*x^2)*Sqrt[d + e*x + f*x^2]), x], x] /; FreeQ[{a, c, d, e, f, A, C}, x] && NeQ[e^2 - 4*d*f, 0]
  1080. Int[Times[Power[Plus[Pattern[a, Blank[]], Times[Optional[Pattern[b, Blank[]]], Pattern[x, Blank[]]], Times[Optional[Pattern[c, Blank[]]], Power[Pattern[x, Blank[]], 2]]], -1], Plus[Optional[Pattern[A, Blank[]]], Times[Optional[Pattern[B, Blank[]]], Pattern[x, Blank[]]], Times[Optional[Pattern[C, Blank[]]], Power[Pattern[x, Blank[]], 2]]], Power[Plus[Optional[Pattern[d, Blank[]]], Times[Optional[Pattern[f, Blank[]]], Power[Pattern[x, Blank[]], 2]]], Rational[-1, 2]]], Pattern[x, Blank[Symbol]]] := Dist[C/c, Int[1/Sqrt[d + f*x^2], x], x] + Dist[1/c, Int[(A*c - a*C + (B*c - b*C)*x)/((a + b*x + c*x^2)*Sqrt[d + f*x^2]), x], x] /; FreeQ[{a, b, c, d, f, A, B, C}, x] && NeQ[b^2 - 4*a*c, 0]
  1081. Int[Times[Power[Plus[Pattern[a, Blank[]], Times[Optional[Pattern[b, Blank[]]], Pattern[x, Blank[]]], Times[Optional[Pattern[c, Blank[]]], Power[Pattern[x, Blank[]], 2]]], -1], Plus[Optional[Pattern[A, Blank[]]], Times[Optional[Pattern[C, Blank[]]], Power[Pattern[x, Blank[]], 2]]], Power[Plus[Optional[Pattern[d, Blank[]]], Times[Optional[Pattern[f, Blank[]]], Power[Pattern[x, Blank[]], 2]]], Rational[-1, 2]]], Pattern[x, Blank[Symbol]]] := Dist[C/c, Int[1/Sqrt[d + f*x^2], x], x] + Dist[1/c, Int[(A*c - a*C - b*C*x)/((a + b*x + c*x^2)*Sqrt[d + f*x^2]), x], x] /; FreeQ[{a, b, c, d, f, A, C}, x] && NeQ[b^2 - 4*a*c, 0]
  1082. Int[Times[Power[Plus[Optional[Pattern[a, Blank[]]], Times[Optional[Pattern[b, Blank[]]], Pattern[u, Blank[]]], Times[Optional[Pattern[c, Blank[]]], Power[Pattern[u, Blank[]], 2]]], Optional[Pattern[p, Blank[]]]], Plus[Optional[Pattern[A, Blank[]]], Times[Optional[Pattern[B, Blank[]]], Pattern[u, Blank[]]], Times[Optional[Pattern[C, Blank[]]], Power[Pattern[u, Blank[]], 2]]], Power[Plus[Optional[Pattern[d, Blank[]]], Times[Optional[Pattern[e, Blank[]]], Pattern[u, Blank[]]], Times[Optional[Pattern[f, Blank[]]], Power[Pattern[u, Blank[]], 2]]], Optional[Pattern[q, Blank[]]]]], Pattern[x, Blank[Symbol]]] := Dist[1/Coefficient[u, x, 1], Subst[Int[(a + b*x + c*x^2)^p*(d + e*x + f*x^2)^q*(A + B*x + C*x^2), x], x, u], x] /; FreeQ[{a, b, c, d, e, f, A, B, C, p, q}, x] && LinearQ[u, x] && NeQ[u, x]
  1083. Int[Times[Plus[Optional[Pattern[A, Blank[]]], Times[Optional[Pattern[B, Blank[]]], Pattern[u, Blank[]]]], Power[Plus[Optional[Pattern[a, Blank[]]], Times[Optional[Pattern[b, Blank[]]], Pattern[u, Blank[]]], Times[Optional[Pattern[c, Blank[]]], Power[Pattern[u, Blank[]], 2]]], Optional[Pattern[p, Blank[]]]], Power[Plus[Optional[Pattern[d, Blank[]]], Times[Optional[Pattern[e, Blank[]]], Pattern[u, Blank[]]], Times[Optional[Pattern[f, Blank[]]], Power[Pattern[u, Blank[]], 2]]], Optional[Pattern[q, Blank[]]]]], Pattern[x, Blank[Symbol]]] := Dist[1/Coefficient[u, x, 1], Subst[Int[(a + b*x + c*x^2)^p*(d + e*x + f*x^2)^q*(A + B*x), x], x, u], x] /; FreeQ[{a, b, c, d, e, f, A, B, C, p, q}, x] && LinearQ[u, x] && NeQ[u, x]
  1084. Int[Times[Power[Plus[Optional[Pattern[a, Blank[]]], Times[Optional[Pattern[b, Blank[]]], Pattern[u, Blank[]]], Times[Optional[Pattern[c, Blank[]]], Power[Pattern[u, Blank[]], 2]]], Optional[Pattern[p, Blank[]]]], Plus[Optional[Pattern[A, Blank[]]], Times[Optional[Pattern[C, Blank[]]], Power[Pattern[u, Blank[]], 2]]], Power[Plus[Optional[Pattern[d, Blank[]]], Times[Optional[Pattern[e, Blank[]]], Pattern[u, Blank[]]], Times[Optional[Pattern[f, Blank[]]], Power[Pattern[u, Blank[]], 2]]], Optional[Pattern[q, Blank[]]]]], Pattern[x, Blank[Symbol]]] := Dist[1/Coefficient[u, x, 1], Subst[Int[(a + b*x + c*x^2)^p*(d + e*x + f*x^2)^q*(A + C*x^2), x], x, u], x] /; FreeQ[{a, b, c, d, e, f, A, C, p, q}, x] && LinearQ[u, x] && NeQ[u, x]
  1085. Int[Times[Power[Plus[Optional[Pattern[a, Blank[]]], Times[Optional[Pattern[c, Blank[]]], Power[Pattern[u, Blank[]], 2]]], Optional[Pattern[p, Blank[]]]], Plus[Optional[Pattern[A, Blank[]]], Times[Optional[Pattern[B, Blank[]]], Pattern[u, Blank[]]], Times[Optional[Pattern[C, Blank[]]], Power[Pattern[u, Blank[]], 2]]], Power[Plus[Optional[Pattern[d, Blank[]]], Times[Optional[Pattern[e, Blank[]]], Pattern[u, Blank[]]], Times[Optional[Pattern[f, Blank[]]], Power[Pattern[u, Blank[]], 2]]], Optional[Pattern[q, Blank[]]]]], Pattern[x, Blank[Symbol]]] := Dist[1/Coefficient[u, x, 1], Subst[Int[(a + c*x^2)^p*(d + e*x + f*x^2)^q*(A + B*x + C*x^2), x], x, u], x] /; FreeQ[{a, c, d, e, f, A, B, C, p, q}, x] && LinearQ[u, x] && NeQ[u, x]
  1086. Int[Times[Plus[Optional[Pattern[A, Blank[]]], Times[Optional[Pattern[B, Blank[]]], Pattern[u, Blank[]]]], Power[Plus[Optional[Pattern[a, Blank[]]], Times[Optional[Pattern[c, Blank[]]], Power[Pattern[u, Blank[]], 2]]], Optional[Pattern[p, Blank[]]]], Power[Plus[Optional[Pattern[d, Blank[]]], Times[Optional[Pattern[e, Blank[]]], Pattern[u, Blank[]]], Times[Optional[Pattern[f, Blank[]]], Power[Pattern[u, Blank[]], 2]]], Optional[Pattern[q, Blank[]]]]], Pattern[x, Blank[Symbol]]] := Dist[1/Coefficient[u, x, 1], Subst[Int[(a + c*x^2)^p*(d + e*x + f*x^2)^q*(A + B*x), x], x, u], x] /; FreeQ[{a, c, d, e, f, A, B, C, p, q}, x] && LinearQ[u, x] && NeQ[u, x]
  1087. Int[Times[Power[Plus[Optional[Pattern[a, Blank[]]], Times[Optional[Pattern[c, Blank[]]], Power[Pattern[u, Blank[]], 2]]], Optional[Pattern[p, Blank[]]]], Plus[Optional[Pattern[A, Blank[]]], Times[Optional[Pattern[C, Blank[]]], Power[Pattern[u, Blank[]], 2]]], Power[Plus[Optional[Pattern[d, Blank[]]], Times[Optional[Pattern[e, Blank[]]], Pattern[u, Blank[]]], Times[Optional[Pattern[f, Blank[]]], Power[Pattern[u, Blank[]], 2]]], Optional[Pattern[q, Blank[]]]]], Pattern[x, Blank[Symbol]]] := Dist[1/Coefficient[u, x, 1], Subst[Int[(a + c*x^2)^p*(d + e*x + f*x^2)^q*(A + C*x^2), x], x, u], x] /; FreeQ[{a, c, d, e, f, A, C, p, q}, x] && LinearQ[u, x] && NeQ[u, x]
  1088. Int[Power[Plus[Pattern[a, Blank[]], Times[Optional[Pattern[b, Blank[]]], Power[Pattern[x, Blank[]], 2]], Times[Optional[Pattern[c, Blank[]]], Power[Pattern[x, Blank[]], 4]]], Pattern[p, Blank[]]], Pattern[x, Blank[Symbol]]] := Dist[(a + b*x^2 + c*x^4)^p/(b + 2*c*x^2)^(2*p), Int[(b + 2*c*x^2)^(2*p), x], x] /; FreeQ[{a, b, c, p}, x] && EqQ[b^2 - 4*a*c, 0] && IntegerQ[p - 1/2]
  1089. Int[Power[Plus[Pattern[a, Blank[]], Times[Optional[Pattern[b, Blank[]]], Power[Pattern[x, Blank[]], 2]], Times[Optional[Pattern[c, Blank[]]], Power[Pattern[x, Blank[]], 4]]], Pattern[p, Blank[]]], Pattern[x, Blank[Symbol]]] := Dist[(a^IntPart[p]*(a + b*x^2 + c*x^4)^FracPart[p])/(1 + (2*c*x^2)/b)^(2*FracPart[p]), Int[(1 + (2*c*x^2)/b)^(2*p), x], x] /; FreeQ[{a, b, c, p}, x] && EqQ[b^2 - 4*a*c, 0] && !IntegerQ[2*p]
  1090. Int[Power[Plus[Pattern[a, Blank[]], Times[Optional[Pattern[b, Blank[]]], Power[Pattern[x, Blank[]], 2]], Times[Optional[Pattern[c, Blank[]]], Power[Pattern[x, Blank[]], 4]]], Pattern[p, Blank[]]], Pattern[x, Blank[Symbol]]] := Int[ExpandIntegrand[(a + b*x^2 + c*x^4)^p, x], x] /; FreeQ[{a, b, c}, x] && NeQ[b^2 - 4*a*c, 0] && IGtQ[p, 0]
  1091. Int[Power[Plus[Pattern[a, Blank[]], Times[Optional[Pattern[b, Blank[]]], Power[Pattern[x, Blank[]], 2]], Times[Optional[Pattern[c, Blank[]]], Power[Pattern[x, Blank[]], 4]]], Pattern[p, Blank[]]], Pattern[x, Blank[Symbol]]] := Simp[(x*(a + b*x^2 + c*x^4)^p)/(4*p + 1), x] + Dist[(2*p)/(4*p + 1), Int[(2*a + b*x^2)*(a + b*x^2 + c*x^4)^(p - 1), x], x] /; FreeQ[{a, b, c}, x] && NeQ[b^2 - 4*a*c, 0] && GtQ[p, 0] && IntegerQ[2*p]
  1092. Int[Power[Plus[Pattern[a, Blank[]], Times[Optional[Pattern[b, Blank[]]], Power[Pattern[x, Blank[]], 2]], Times[Optional[Pattern[c, Blank[]]], Power[Pattern[x, Blank[]], 4]]], Pattern[p, Blank[]]], Pattern[x, Blank[Symbol]]] := -Simp[(x*(b^2 - 2*a*c + b*c*x^2)*(a + b*x^2 + c*x^4)^(p + 1))/(2*a*(p + 1)*(b^2 - 4*a*c)), x] + Dist[1/(2*a*(p + 1)*(b^2 - 4*a*c)), Int[(b^2 - 2*a*c + 2*(p + 1)*(b^2 - 4*a*c) + b*c*(4*p + 7)*x^2)*(a + b*x^2 + c*x^4)^(p + 1), x], x] /; FreeQ[{a, b, c}, x] && NeQ[b^2 - 4*a*c, 0] && LtQ[p, -1] && IntegerQ[2*p]
  1093. Int[Power[Plus[Pattern[a, Blank[]], Times[Optional[Pattern[b, Blank[]]], Power[Pattern[x, Blank[]], 2]], Times[Optional[Pattern[c, Blank[]]], Power[Pattern[x, Blank[]], 4]]], -1], Pattern[x, Blank[Symbol]]] := With[{q = Rt[b^2 - 4*a*c, 2]}, Dist[c/q, Int[1/(b/2 - q/2 + c*x^2), x], x] - Dist[c/q, Int[1/(b/2 + q/2 + c*x^2), x], x]] /; FreeQ[{a, b, c}, x] && NeQ[b^2 - 4*a*c, 0] && PosQ[b^2 - 4*a*c]
  1094. Int[Power[Plus[Pattern[a, Blank[]], Times[Optional[Pattern[b, Blank[]]], Power[Pattern[x, Blank[]], 2]], Times[Optional[Pattern[c, Blank[]]], Power[Pattern[x, Blank[]], 4]]], -1], Pattern[x, Blank[Symbol]]] := With[{q = Rt[a/c, 2]}, With[{r = Rt[2*q - b/c, 2]}, Dist[1/(2*c*q*r), Int[(r - x)/(q - r*x + x^2), x], x] + Dist[1/(2*c*q*r), Int[(r + x)/(q + r*x + x^2), x], x]]] /; FreeQ[{a, b, c}, x] && NeQ[b^2 - 4*a*c, 0] && NegQ[b^2 - 4*a*c]
  1095. Int[Power[Plus[Pattern[a, Blank[]], Times[Optional[Pattern[b, Blank[]]], Power[Pattern[x, Blank[]], 2]], Times[Optional[Pattern[c, Blank[]]], Power[Pattern[x, Blank[]], 4]]], Rational[-1, 2]], Pattern[x, Blank[Symbol]]] := With[{q = Rt[b^2 - 4*a*c, 2]}, Dist[2*Sqrt[-c], Int[1/(Sqrt[b + q + 2*c*x^2]*Sqrt[-b + q - 2*c*x^2]), x], x]] /; FreeQ[{a, b, c}, x] && GtQ[b^2 - 4*a*c, 0] && LtQ[c, 0]
  1096. Int[Power[Plus[Pattern[a, Blank[]], Times[Optional[Pattern[b, Blank[]]], Power[Pattern[x, Blank[]], 2]], Times[Optional[Pattern[c, Blank[]]], Power[Pattern[x, Blank[]], 4]]], Rational[-1, 2]], Pattern[x, Blank[Symbol]]] := With[{q = Rt[c/a, 4]}, Simp[((1 + q^2*x^2)*Sqrt[(a + b*x^2 + c*x^4)/(a*(1 + q^2*x^2)^2)]*EllipticF[2*ArcTan[q*x], 1/2 - (b*q^2)/(4*c)])/(2*q*Sqrt[a + b*x^2 + c*x^4]), x]] /; FreeQ[{a, b, c}, x] && GtQ[b^2 - 4*a*c, 0] && GtQ[c/a, 0] && LtQ[b/a, 0]
  1097. Int[Power[Plus[Pattern[a, Blank[]], Times[Optional[Pattern[b, Blank[]]], Power[Pattern[x, Blank[]], 2]], Times[Optional[Pattern[c, Blank[]]], Power[Pattern[x, Blank[]], 4]]], Rational[-1, 2]], Pattern[x, Blank[Symbol]]] := With[{q = Rt[b^2 - 4*a*c, 2]}, Simp[(Sqrt[-2*a - (b - q)*x^2]*Sqrt[(2*a + (b + q)*x^2)/q]*EllipticF[ArcSin[x/Sqrt[(2*a + (b + q)*x^2)/(2*q)]], (b + q)/(2*q)])/(2*Sqrt[-a]*Sqrt[a + b*x^2 + c*x^4]), x] /; IntegerQ[q]] /; FreeQ[{a, b, c}, x] && GtQ[b^2 - 4*a*c, 0] && LtQ[a, 0] && GtQ[c, 0]
  1098. Int[Power[Plus[Pattern[a, Blank[]], Times[Optional[Pattern[b, Blank[]]], Power[Pattern[x, Blank[]], 2]], Times[Optional[Pattern[c, Blank[]]], Power[Pattern[x, Blank[]], 4]]], Rational[-1, 2]], Pattern[x, Blank[Symbol]]] := With[{q = Rt[b^2 - 4*a*c, 2]}, Simp[(Sqrt[(2*a + (b - q)*x^2)/(2*a + (b + q)*x^2)]*Sqrt[(2*a + (b + q)*x^2)/q]*EllipticF[ArcSin[x/Sqrt[(2*a + (b + q)*x^2)/(2*q)]], (b + q)/(2*q)])/(2*Sqrt[a + b*x^2 + c*x^4]*Sqrt[a/(2*a + (b + q)*x^2)]), x]] /; FreeQ[{a, b, c}, x] && GtQ[b^2 - 4*a*c, 0] && LtQ[a, 0] && GtQ[c, 0]
  1099. Int[Power[Plus[Pattern[a, Blank[]], Times[Optional[Pattern[b, Blank[]]], Power[Pattern[x, Blank[]], 2]], Times[Optional[Pattern[c, Blank[]]], Power[Pattern[x, Blank[]], 4]]], Rational[-1, 2]], Pattern[x, Blank[Symbol]]] := With[{q = Rt[b^2 - 4*a*c, 2]}, Simp[((2*a + (b + q)*x^2)*Sqrt[(2*a + (b - q)*x^2)/(2*a + (b + q)*x^2)]*EllipticF[ArcTan[Rt[(b + q)/(2*a), 2]*x], (2*q)/(b + q)])/(2*a*Rt[(b + q)/(2*a), 2]*Sqrt[a + b*x^2 + c*x^4]), x] /; PosQ[(b + q)/a] && !(PosQ[(b - q)/a] && SimplerSqrtQ[(b - q)/(2*a), (b + q)/(2*a)])] /; FreeQ[{a, b, c}, x] && GtQ[b^2 - 4*a*c, 0]
  1100. Int[Power[Plus[Pattern[a, Blank[]], Times[Optional[Pattern[b, Blank[]]], Power[Pattern[x, Blank[]], 2]], Times[Optional[Pattern[c, Blank[]]], Power[Pattern[x, Blank[]], 4]]], Rational[-1, 2]], Pattern[x, Blank[Symbol]]] := With[{q = Rt[b^2 - 4*a*c, 2]}, Simp[((2*a + (b - q)*x^2)*Sqrt[(2*a + (b + q)*x^2)/(2*a + (b - q)*x^2)]*EllipticF[ArcTan[Rt[(b - q)/(2*a), 2]*x], (-2*q)/(b - q)])/(2*a*Rt[(b - q)/(2*a), 2]*Sqrt[a + b*x^2 + c*x^4]), x] /; PosQ[(b - q)/a]] /; FreeQ[{a, b, c}, x] && GtQ[b^2 - 4*a*c, 0]
  1101. Int[Power[Plus[Pattern[a, Blank[]], Times[Optional[Pattern[b, Blank[]]], Power[Pattern[x, Blank[]], 2]], Times[Optional[Pattern[c, Blank[]]], Power[Pattern[x, Blank[]], 4]]], Rational[-1, 2]], Pattern[x, Blank[Symbol]]] := With[{q = Rt[b^2 - 4*a*c, 2]}, Simp[(Sqrt[1 + ((b + q)*x^2)/(2*a)]*Sqrt[1 + ((b - q)*x^2)/(2*a)]*EllipticF[ArcSin[Rt[-((b + q)/(2*a)), 2]*x], (b - q)/(b + q)])/(Rt[-((b + q)/(2*a)), 2]*Sqrt[a + b*x^2 + c*x^4]), x] /; NegQ[(b + q)/a] && !(NegQ[(b - q)/a] && SimplerSqrtQ[-((b - q)/(2*a)), -((b + q)/(2*a))])] /; FreeQ[{a, b, c}, x] && GtQ[b^2 - 4*a*c, 0]
  1102. Int[Power[Plus[Pattern[a, Blank[]], Times[Optional[Pattern[b, Blank[]]], Power[Pattern[x, Blank[]], 2]], Times[Optional[Pattern[c, Blank[]]], Power[Pattern[x, Blank[]], 4]]], Rational[-1, 2]], Pattern[x, Blank[Symbol]]] := With[{q = Rt[b^2 - 4*a*c, 2]}, Simp[(Sqrt[1 + ((b - q)*x^2)/(2*a)]*Sqrt[1 + ((b + q)*x^2)/(2*a)]*EllipticF[ArcSin[Rt[-((b - q)/(2*a)), 2]*x], (b + q)/(b - q)])/(Rt[-((b - q)/(2*a)), 2]*Sqrt[a + b*x^2 + c*x^4]), x] /; NegQ[(b - q)/a]] /; FreeQ[{a, b, c}, x] && GtQ[b^2 - 4*a*c, 0]
  1103. Int[Power[Plus[Pattern[a, Blank[]], Times[Optional[Pattern[b, Blank[]]], Power[Pattern[x, Blank[]], 2]], Times[Optional[Pattern[c, Blank[]]], Power[Pattern[x, Blank[]], 4]]], Rational[-1, 2]], Pattern[x, Blank[Symbol]]] := With[{q = Rt[c/a, 4]}, Simp[((1 + q^2*x^2)*Sqrt[(a + b*x^2 + c*x^4)/(a*(1 + q^2*x^2)^2)]*EllipticF[2*ArcTan[q*x], 1/2 - (b*q^2)/(4*c)])/(2*q*Sqrt[a + b*x^2 + c*x^4]), x]] /; FreeQ[{a, b, c}, x] && NeQ[b^2 - 4*a*c, 0] && PosQ[c/a]
  1104. Int[Power[Plus[Pattern[a, Blank[]], Times[Optional[Pattern[b, Blank[]]], Power[Pattern[x, Blank[]], 2]], Times[Optional[Pattern[c, Blank[]]], Power[Pattern[x, Blank[]], 4]]], Rational[-1, 2]], Pattern[x, Blank[Symbol]]] := With[{q = Rt[b^2 - 4*a*c, 2]}, Dist[(Sqrt[1 + (2*c*x^2)/(b - q)]*Sqrt[1 + (2*c*x^2)/(b + q)])/Sqrt[a + b*x^2 + c*x^4], Int[1/(Sqrt[1 + (2*c*x^2)/(b - q)]*Sqrt[1 + (2*c*x^2)/(b + q)]), x], x]] /; FreeQ[{a, b, c}, x] && NeQ[b^2 - 4*a*c, 0] && NegQ[c/a]
  1105. Int[Power[Plus[Pattern[a, Blank[]], Times[Optional[Pattern[b, Blank[]]], Power[Pattern[x, Blank[]], 2]], Times[Optional[Pattern[c, Blank[]]], Power[Pattern[x, Blank[]], 4]]], Pattern[p, Blank[]]], Pattern[x, Blank[Symbol]]] := With[{q = Rt[b^2 - 4*a*c, 2]}, Dist[(a^IntPart[p]*(a + b*x^2 + c*x^4)^FracPart[p])/((1 + (2*c*x^2)/(b + q))^FracPart[p]*(1 + (2*c*x^2)/(b - q))^FracPart[p]), Int[(1 + (2*c*x^2)/(b + q))^p*(1 + (2*c*x^2)/(b - q))^p, x], x]] /; FreeQ[{a, b, c, p}, x] && NeQ[b^2 - 4*a*c, 0]
  1106. Int[Power[Pattern[P4, Blank[]], Pattern[p, Blank[]]], Pattern[x, Blank[Symbol]]] := With[{a = Coeff[P4, x, 0], b = Coeff[P4, x, 1], c = Coeff[P4, x, 2], d = Coeff[P4, x, 3], e = Coeff[P4, x, 4]}, Subst[Int[SimplifyIntegrand[(a + d^4/(256*e^3) - (b*d)/(8*e) + (c - (3*d^2)/(8*e))*x^2 + e*x^4)^p, x], x], x, d/(4*e) + x] /; EqQ[d^3 - 4*c*d*e + 8*b*e^2, 0] && NeQ[d, 0]] /; FreeQ[p, x] && PolyQ[P4, x, 4] && NeQ[p, 2] && NeQ[p, 3]
  1107. Int[Times[Pattern[x, Blank[]], Power[Plus[Pattern[a, Blank[]], Times[Optional[Pattern[b, Blank[]]], Power[Pattern[x, Blank[]], 2]], Times[Optional[Pattern[c, Blank[]]], Power[Pattern[x, Blank[]], 4]]], Optional[Pattern[p, Blank[]]]]], Pattern[x, Blank[Symbol]]] := Dist[1/2, Subst[Int[(a + b*x + c*x^2)^p, x], x, x^2], x] /; FreeQ[{a, b, c, p}, x]
  1108. Int[Times[Power[Times[Optional[Pattern[d, Blank[]]], Pattern[x, Blank[]]], Optional[Pattern[m, Blank[]]]], Power[Plus[Pattern[a, Blank[]], Times[Optional[Pattern[b, Blank[]]], Power[Pattern[x, Blank[]], 2]], Times[Optional[Pattern[c, Blank[]]], Power[Pattern[x, Blank[]], 4]]], Optional[Pattern[p, Blank[]]]]], Pattern[x, Blank[Symbol]]] := Int[ExpandIntegrand[(d*x)^m*(a + b*x^2 + c*x^4)^p, x], x] /; FreeQ[{a, b, c, d, m}, x] && IGtQ[p, 0] && !IntegerQ[(m + 1)/2]
  1109. Int[Times[Power[Times[Optional[Pattern[d, Blank[]]], Pattern[x, Blank[]]], Optional[Pattern[m, Blank[]]]], Power[Plus[Pattern[a, Blank[]], Times[Optional[Pattern[b, Blank[]]], Power[Pattern[x, Blank[]], 2]], Times[Optional[Pattern[c, Blank[]]], Power[Pattern[x, Blank[]], 4]]], Pattern[p, Blank[]]]], Pattern[x, Blank[Symbol]]] := Simp[(2*(d*x)^(m + 1)*(a + b*x^2 + c*x^4)^(p + 1))/(d*(m + 3)*(2*a + b*x^2)), x] - Simp[((d*x)^(m + 1)*(a + b*x^2 + c*x^4)^(p + 1))/(2*a*d*(m + 3)*(p + 1)), x] /; FreeQ[{a, b, c, d, m, p}, x] && EqQ[b^2 - 4*a*c, 0] && !IntegerQ[p] && EqQ[m + 4*p + 5, 0] && LtQ[p, -1]
  1110. Int[Times[Power[Times[Optional[Pattern[d, Blank[]]], Pattern[x, Blank[]]], Optional[Pattern[m, Blank[]]]], Power[Plus[Pattern[a, Blank[]], Times[Optional[Pattern[b, Blank[]]], Power[Pattern[x, Blank[]], 2]], Times[Optional[Pattern[c, Blank[]]], Power[Pattern[x, Blank[]], 4]]], Pattern[p, Blank[]]]], Pattern[x, Blank[Symbol]]] := Simp[((d*x)^(m + 1)*(a + b*x^2 + c*x^4)^(p + 1))/(4*a*d*(p + 1)*(2*p + 1)), x] - Simp[((d*x)^(m + 1)*(2*a + b*x^2)*(a + b*x^2 + c*x^4)^p)/(4*a*d*(2*p + 1)), x] /; FreeQ[{a, b, c, d, m, p}, x] && EqQ[b^2 - 4*a*c, 0] && !IntegerQ[p] && EqQ[m + 4*p + 5, 0] && NeQ[p, -2^(-1)]
  1111. Int[Times[Power[Pattern[x, Blank[]], Optional[Pattern[m, Blank[]]]], Power[Plus[Pattern[a, Blank[]], Times[Optional[Pattern[b, Blank[]]], Power[Pattern[x, Blank[]], 2]], Times[Optional[Pattern[c, Blank[]]], Power[Pattern[x, Blank[]], 4]]], Pattern[p, Blank[]]]], Pattern[x, Blank[Symbol]]] := Dist[1/2, Subst[Int[x^((m - 1)/2)*(a + b*x + c*x^2)^p, x], x, x^2], x] /; FreeQ[{a, b, c, p}, x] && EqQ[b^2 - 4*a*c, 0] && IntegerQ[p - 1/2] && IntegerQ[(m - 1)/2] && (GtQ[m, 0] || LtQ[0, 4*p, -m - 1])
  1112. Int[Times[Power[Times[Optional[Pattern[d, Blank[]]], Pattern[x, Blank[]]], Optional[Pattern[m, Blank[]]]], Power[Plus[Pattern[a, Blank[]], Times[Optional[Pattern[b, Blank[]]], Power[Pattern[x, Blank[]], 2]], Times[Optional[Pattern[c, Blank[]]], Power[Pattern[x, Blank[]], 4]]], Pattern[p, Blank[]]]], Pattern[x, Blank[Symbol]]] := Dist[(a + b*x^2 + c*x^4)^FracPart[p]/(c^IntPart[p]*(b/2 + c*x^2)^(2*FracPart[p])), Int[(d*x)^m*(b/2 + c*x^2)^(2*p), x], x] /; FreeQ[{a, b, c, d, m, p}, x] && EqQ[b^2 - 4*a*c, 0] && IntegerQ[p - 1/2]
  1113. Int[Times[Power[Times[Optional[Pattern[d, Blank[]]], Pattern[x, Blank[]]], Optional[Pattern[m, Blank[]]]], Power[Plus[Pattern[a, Blank[]], Times[Optional[Pattern[b, Blank[]]], Power[Pattern[x, Blank[]], 2]], Times[Optional[Pattern[c, Blank[]]], Power[Pattern[x, Blank[]], 4]]], Pattern[p, Blank[]]]], Pattern[x, Blank[Symbol]]] := Dist[(a^IntPart[p]*(a + b*x^2 + c*x^4)^FracPart[p])/(1 + (2*c*x^2)/b)^(2*FracPart[p]), Int[(d*x)^m*(1 + (2*c*x^2)/b)^(2*p), x], x] /; FreeQ[{a, b, c, d, m, p}, x] && EqQ[b^2 - 4*a*c, 0] && !IntegerQ[2*p]
  1114. Int[Times[Power[Pattern[x, Blank[]], Optional[Pattern[m, Blank[]]]], Power[Plus[Pattern[a, Blank[]], Times[Optional[Pattern[b, Blank[]]], Power[Pattern[x, Blank[]], 2]], Times[Optional[Pattern[c, Blank[]]], Power[Pattern[x, Blank[]], 4]]], Optional[Pattern[p, Blank[]]]]], Pattern[x, Blank[Symbol]]] := Dist[1/2, Subst[Int[x^((m - 1)/2)*(a + b*x + c*x^2)^p, x], x, x^2], x] /; FreeQ[{a, b, c, p}, x] && IntegerQ[(m - 1)/2]
  1115. Int[Times[Power[Times[Optional[Pattern[d, Blank[]]], Pattern[x, Blank[]]], Pattern[m, Blank[]]], Power[Plus[Pattern[a, Blank[]], Times[Optional[Pattern[b, Blank[]]], Power[Pattern[x, Blank[]], 2]], Times[Optional[Pattern[c, Blank[]]], Power[Pattern[x, Blank[]], 4]]], Pattern[p, Blank[]]]], Pattern[x, Blank[Symbol]]] := With[{k = Denominator[m]}, Dist[k/d, Subst[Int[x^(k*(m + 1) - 1)*(a + (b*x^(2*k))/d^2 + (c*x^(4*k))/d^4)^p, x], x, (d*x)^(1/k)], x]] /; FreeQ[{a, b, c, d, p}, x] && NeQ[b^2 - 4*a*c, 0] && FractionQ[m] && IntegerQ[p]
  1116. Int[Times[Power[Times[Optional[Pattern[d, Blank[]]], Pattern[x, Blank[]]], Optional[Pattern[m, Blank[]]]], Power[Plus[Pattern[a, Blank[]], Times[Optional[Pattern[b, Blank[]]], Power[Pattern[x, Blank[]], 2]], Times[Optional[Pattern[c, Blank[]]], Power[Pattern[x, Blank[]], 4]]], Pattern[p, Blank[]]]], Pattern[x, Blank[Symbol]]] := Simp[(d*(d*x)^(m - 1)*(a + b*x^2 + c*x^4)^p*(2*b*p + c*(m + 4*p - 1)*x^2))/(c*(m + 4*p + 1)*(m + 4*p - 1)), x] - Dist[(2*p*d^2)/(c*(m + 4*p + 1)*(m + 4*p - 1)), Int[(d*x)^(m - 2)*(a + b*x^2 + c*x^4)^(p - 1)*Simp[a*b*(m - 1) - (2*a*c*(m + 4*p - 1) - b^2*(m + 2*p - 1))*x^2, x], x], x] /; FreeQ[{a, b, c, d}, x] && NeQ[b^2 - 4*a*c, 0] && GtQ[p, 0] && GtQ[m, 1] && IntegerQ[2*p] && (IntegerQ[p] || IntegerQ[m])
  1117. Int[Times[Power[Times[Optional[Pattern[d, Blank[]]], Pattern[x, Blank[]]], Optional[Pattern[m, Blank[]]]], Power[Plus[Pattern[a, Blank[]], Times[Optional[Pattern[b, Blank[]]], Power[Pattern[x, Blank[]], 2]], Times[Optional[Pattern[c, Blank[]]], Power[Pattern[x, Blank[]], 4]]], Pattern[p, Blank[]]]], Pattern[x, Blank[Symbol]]] := Simp[((d*x)^(m + 1)*(a + b*x^2 + c*x^4)^p)/(d*(m + 1)), x] - Dist[(2*p)/(d^2*(m + 1)), Int[(d*x)^(m + 2)*(b + 2*c*x^2)*(a + b*x^2 + c*x^4)^(p - 1), x], x] /; FreeQ[{a, b, c, d}, x] && NeQ[b^2 - 4*a*c, 0] && GtQ[p, 0] && LtQ[m, -1] && IntegerQ[2*p] && (IntegerQ[p] || IntegerQ[m])
  1118. Int[Times[Power[Times[Optional[Pattern[d, Blank[]]], Pattern[x, Blank[]]], Optional[Pattern[m, Blank[]]]], Power[Plus[Pattern[a, Blank[]], Times[Optional[Pattern[b, Blank[]]], Power[Pattern[x, Blank[]], 2]], Times[Optional[Pattern[c, Blank[]]], Power[Pattern[x, Blank[]], 4]]], Pattern[p, Blank[]]]], Pattern[x, Blank[Symbol]]] := Simp[((d*x)^(m + 1)*(a + b*x^2 + c*x^4)^p)/(d*(m + 4*p + 1)), x] + Dist[(2*p)/(m + 4*p + 1), Int[(d*x)^m*(2*a + b*x^2)*(a + b*x^2 + c*x^4)^(p - 1), x], x] /; FreeQ[{a, b, c, d, m}, x] && NeQ[b^2 - 4*a*c, 0] && GtQ[p, 0] && NeQ[m + 4*p + 1, 0] && IntegerQ[2*p] && (IntegerQ[p] || IntegerQ[m])
  1119. Int[Times[Power[Times[Optional[Pattern[d, Blank[]]], Pattern[x, Blank[]]], Optional[Pattern[m, Blank[]]]], Power[Plus[Pattern[a, Blank[]], Times[Optional[Pattern[b, Blank[]]], Power[Pattern[x, Blank[]], 2]], Times[Optional[Pattern[c, Blank[]]], Power[Pattern[x, Blank[]], 4]]], Pattern[p, Blank[]]]], Pattern[x, Blank[Symbol]]] := Simp[(d*(d*x)^(m - 1)*(b + 2*c*x^2)*(a + b*x^2 + c*x^4)^(p + 1))/(2*(p + 1)*(b^2 - 4*a*c)), x] - Dist[d^2/(2*(p + 1)*(b^2 - 4*a*c)), Int[(d*x)^(m - 2)*(b*(m - 1) + 2*c*(m + 4*p + 5)*x^2)*(a + b*x^2 + c*x^4)^(p + 1), x], x] /; FreeQ[{a, b, c, d}, x] && NeQ[b^2 - 4*a*c, 0] && LtQ[p, -1] && GtQ[m, 1] && LeQ[m, 3] && IntegerQ[2*p] && (IntegerQ[p] || IntegerQ[m])
  1120. Int[Times[Power[Times[Optional[Pattern[d, Blank[]]], Pattern[x, Blank[]]], Optional[Pattern[m, Blank[]]]], Power[Plus[Pattern[a, Blank[]], Times[Optional[Pattern[b, Blank[]]], Power[Pattern[x, Blank[]], 2]], Times[Optional[Pattern[c, Blank[]]], Power[Pattern[x, Blank[]], 4]]], Pattern[p, Blank[]]]], Pattern[x, Blank[Symbol]]] := -Simp[(d^3*(d*x)^(m - 3)*(2*a + b*x^2)*(a + b*x^2 + c*x^4)^(p + 1))/(2*(p + 1)*(b^2 - 4*a*c)), x] + Dist[d^4/(2*(p + 1)*(b^2 - 4*a*c)), Int[(d*x)^(m - 4)*(2*a*(m - 3) + b*(m + 4*p + 3)*x^2)*(a + b*x^2 + c*x^4)^(p + 1), x], x] /; FreeQ[{a, b, c, d}, x] && NeQ[b^2 - 4*a*c, 0] && LtQ[p, -1] && GtQ[m, 3] && IntegerQ[2*p] && (IntegerQ[p] || IntegerQ[m])
  1121. Int[Times[Power[Times[Optional[Pattern[d, Blank[]]], Pattern[x, Blank[]]], Optional[Pattern[m, Blank[]]]], Power[Plus[Pattern[a, Blank[]], Times[Optional[Pattern[b, Blank[]]], Power[Pattern[x, Blank[]], 2]], Times[Optional[Pattern[c, Blank[]]], Power[Pattern[x, Blank[]], 4]]], Pattern[p, Blank[]]]], Pattern[x, Blank[Symbol]]] := -Simp[((d*x)^(m + 1)*(b^2 - 2*a*c + b*c*x^2)*(a + b*x^2 + c*x^4)^(p + 1))/(2*a*d*(p + 1)*(b^2 - 4*a*c)), x] + Dist[1/(2*a*(p + 1)*(b^2 - 4*a*c)), Int[(d*x)^m*(a + b*x^2 + c*x^4)^(p + 1)*Simp[b^2*(m + 2*p + 3) - 2*a*c*(m + 4*p + 5) + b*c*(m + 4*p + 7)*x^2, x], x], x] /; FreeQ[{a, b, c, d, m}, x] && NeQ[b^2 - 4*a*c, 0] && LtQ[p, -1] && IntegerQ[2*p] && (IntegerQ[p] || IntegerQ[m])
  1122. Int[Times[Power[Times[Optional[Pattern[d, Blank[]]], Pattern[x, Blank[]]], Optional[Pattern[m, Blank[]]]], Power[Plus[Pattern[a, Blank[]], Times[Optional[Pattern[b, Blank[]]], Power[Pattern[x, Blank[]], 2]], Times[Optional[Pattern[c, Blank[]]], Power[Pattern[x, Blank[]], 4]]], Pattern[p, Blank[]]]], Pattern[x, Blank[Symbol]]] := Simp[(d^3*(d*x)^(m - 3)*(a + b*x^2 + c*x^4)^(p + 1))/(c*(m + 4*p + 1)), x] - Dist[d^4/(c*(m + 4*p + 1)), Int[(d*x)^(m - 4)*Simp[a*(m - 3) + b*(m + 2*p - 1)*x^2, x]*(a + b*x^2 + c*x^4)^p, x], x] /; FreeQ[{a, b, c, d, p}, x] && NeQ[b^2 - 4*a*c, 0] && GtQ[m, 3] && NeQ[m + 4*p + 1, 0] && IntegerQ[2*p] && (IntegerQ[p] || IntegerQ[m])
  1123. Int[Times[Power[Times[Optional[Pattern[d, Blank[]]], Pattern[x, Blank[]]], Pattern[m, Blank[]]], Power[Plus[Pattern[a, Blank[]], Times[Optional[Pattern[b, Blank[]]], Power[Pattern[x, Blank[]], 2]], Times[Optional[Pattern[c, Blank[]]], Power[Pattern[x, Blank[]], 4]]], Pattern[p, Blank[]]]], Pattern[x, Blank[Symbol]]] := Simp[((d*x)^(m + 1)*(a + b*x^2 + c*x^4)^(p + 1))/(a*d*(m + 1)), x] - Dist[1/(a*d^2*(m + 1)), Int[(d*x)^(m + 2)*(b*(m + 2*p + 3) + c*(m + 4*p + 5)*x^2)*(a + b*x^2 + c*x^4)^p, x], x] /; FreeQ[{a, b, c, d, p}, x] && NeQ[b^2 - 4*a*c, 0] && LtQ[m, -1] && IntegerQ[2*p] && (IntegerQ[p] || IntegerQ[m])
  1124. Int[Times[Power[Times[Optional[Pattern[d, Blank[]]], Pattern[x, Blank[]]], Pattern[m, Blank[]]], Power[Plus[Pattern[a, Blank[]], Times[Optional[Pattern[b, Blank[]]], Power[Pattern[x, Blank[]], 2]], Times[Optional[Pattern[c, Blank[]]], Power[Pattern[x, Blank[]], 4]]], -1]], Pattern[x, Blank[Symbol]]] := Simp[(d*x)^(m + 1)/(a*d*(m + 1)), x] - Dist[1/(a*d^2), Int[((d*x)^(m + 2)*(b + c*x^2))/(a + b*x^2 + c*x^4), x], x] /; FreeQ[{a, b, c, d}, x] && NeQ[b^2 - 4*a*c, 0] && LtQ[m, -1]
  1125. Int[Times[Power[Pattern[x, Blank[]], Pattern[m, Blank[]]], Power[Plus[Pattern[a, Blank[]], Times[Optional[Pattern[b, Blank[]]], Power[Pattern[x, Blank[]], 2]], Times[Optional[Pattern[c, Blank[]]], Power[Pattern[x, Blank[]], 4]]], -1]], Pattern[x, Blank[Symbol]]] := Int[PolynomialDivide[x^m, a + b*x^2 + c*x^4, x], x] /; FreeQ[{a, b, c}, x] && NeQ[b^2 - 4*a*c, 0] && IGtQ[m, 5]
  1126. Int[Times[Power[Times[Optional[Pattern[d, Blank[]]], Pattern[x, Blank[]]], Pattern[m, Blank[]]], Power[Plus[Pattern[a, Blank[]], Times[Optional[Pattern[b, Blank[]]], Power[Pattern[x, Blank[]], 2]], Times[Optional[Pattern[c, Blank[]]], Power[Pattern[x, Blank[]], 4]]], -1]], Pattern[x, Blank[Symbol]]] := Simp[(d^3*(d*x)^(m - 3))/(c*(m - 3)), x] - Dist[d^4/c, Int[((d*x)^(m - 4)*(a + b*x^2))/(a + b*x^2 + c*x^4), x], x] /; FreeQ[{a, b, c, d}, x] && NeQ[b^2 - 4*a*c, 0] && GtQ[m, 3]
  1127. Int[Times[Power[Pattern[x, Blank[]], 2], Power[Plus[Pattern[a, Blank[]], Times[Optional[Pattern[b, Blank[]]], Power[Pattern[x, Blank[]], 2]], Times[Optional[Pattern[c, Blank[]]], Power[Pattern[x, Blank[]], 4]]], -1]], Pattern[x, Blank[Symbol]]] := With[{q = Rt[a/c, 2]}, Dist[1/2, Int[(q + x^2)/(a + b*x^2 + c*x^4), x], x] - Dist[1/2, Int[(q - x^2)/(a + b*x^2 + c*x^4), x], x]] /; FreeQ[{a, b, c}, x] && LtQ[b^2 - 4*a*c, 0] && PosQ[a*c]
  1128. Int[Times[Power[Pattern[x, Blank[]], Optional[Pattern[m, Blank[]]]], Power[Plus[Pattern[a, Blank[]], Times[Optional[Pattern[b, Blank[]]], Power[Pattern[x, Blank[]], 2]], Times[Optional[Pattern[c, Blank[]]], Power[Pattern[x, Blank[]], 4]]], -1]], Pattern[x, Blank[Symbol]]] := With[{q = Rt[a/c, 2]}, With[{r = Rt[2*q - b/c, 2]}, -Dist[1/(2*c*r), Int[(x^(m - 3)*(q - r*x))/(q - r*x + x^2), x], x] + Dist[1/(2*c*r), Int[(x^(m - 3)*(q + r*x))/(q + r*x + x^2), x], x]]] /; FreeQ[{a, b, c}, x] && NeQ[b^2 - 4*a*c, 0] && GeQ[m, 3] && LtQ[m, 4] && NegQ[b^2 - 4*a*c]
  1129. Int[Times[Power[Pattern[x, Blank[]], Optional[Pattern[m, Blank[]]]], Power[Plus[Pattern[a, Blank[]], Times[Optional[Pattern[b, Blank[]]], Power[Pattern[x, Blank[]], 2]], Times[Optional[Pattern[c, Blank[]]], Power[Pattern[x, Blank[]], 4]]], -1]], Pattern[x, Blank[Symbol]]] := With[{q = Rt[a/c, 2]}, With[{r = Rt[2*q - b/c, 2]}, Dist[1/(2*c*r), Int[x^(m - 1)/(q - r*x + x^2), x], x] - Dist[1/(2*c*r), Int[x^(m - 1)/(q + r*x + x^2), x], x]]] /; FreeQ[{a, b, c}, x] && NeQ[b^2 - 4*a*c, 0] && GeQ[m, 1] && LtQ[m, 3] && NegQ[b^2 - 4*a*c]
  1130. Int[Times[Power[Times[Optional[Pattern[d, Blank[]]], Pattern[x, Blank[]]], Pattern[m, Blank[]]], Power[Plus[Pattern[a, Blank[]], Times[Optional[Pattern[b, Blank[]]], Power[Pattern[x, Blank[]], 2]], Times[Optional[Pattern[c, Blank[]]], Power[Pattern[x, Blank[]], 4]]], -1]], Pattern[x, Blank[Symbol]]] := With[{q = Rt[b^2 - 4*a*c, 2]}, Dist[(d^2*(b/q + 1))/2, Int[(d*x)^(m - 2)/(b/2 + q/2 + c*x^2), x], x] - Dist[(d^2*(b/q - 1))/2, Int[(d*x)^(m - 2)/(b/2 - q/2 + c*x^2), x], x]] /; FreeQ[{a, b, c, d}, x] && NeQ[b^2 - 4*a*c, 0] && GeQ[m, 2]
  1131. Int[Times[Power[Times[Optional[Pattern[d, Blank[]]], Pattern[x, Blank[]]], Optional[Pattern[m, Blank[]]]], Power[Plus[Pattern[a, Blank[]], Times[Optional[Pattern[b, Blank[]]], Power[Pattern[x, Blank[]], 2]], Times[Optional[Pattern[c, Blank[]]], Power[Pattern[x, Blank[]], 4]]], -1]], Pattern[x, Blank[Symbol]]] := With[{q = Rt[b^2 - 4*a*c, 2]}, Dist[c/q, Int[(d*x)^m/(b/2 - q/2 + c*x^2), x], x] - Dist[c/q, Int[(d*x)^m/(b/2 + q/2 + c*x^2), x], x]] /; FreeQ[{a, b, c, d, m}, x] && NeQ[b^2 - 4*a*c, 0]
  1132. Int[Times[Power[Pattern[x, Blank[]], 2], Power[Plus[Pattern[a, Blank[]], Times[Optional[Pattern[b, Blank[]]], Power[Pattern[x, Blank[]], 2]], Times[Optional[Pattern[c, Blank[]]], Power[Pattern[x, Blank[]], 4]]], Rational[-1, 2]]], Pattern[x, Blank[Symbol]]] := With[{q = Rt[b^2 - 4*a*c, 2]}, Dist[2*Sqrt[-c], Int[x^2/(Sqrt[b + q + 2*c*x^2]*Sqrt[-b + q - 2*c*x^2]), x], x]] /; FreeQ[{a, b, c}, x] && GtQ[b^2 - 4*a*c, 0] && LtQ[c, 0]
  1133. Int[Times[Power[Pattern[x, Blank[]], 2], Power[Plus[Pattern[a, Blank[]], Times[Optional[Pattern[b, Blank[]]], Power[Pattern[x, Blank[]], 2]], Times[Optional[Pattern[c, Blank[]]], Power[Pattern[x, Blank[]], 4]]], Rational[-1, 2]]], Pattern[x, Blank[Symbol]]] := With[{q = Rt[c/a, 2]}, Dist[1/q, Int[1/Sqrt[a + b*x^2 + c*x^4], x], x] - Dist[1/q, Int[(1 - q*x^2)/Sqrt[a + b*x^2 + c*x^4], x], x]] /; FreeQ[{a, b, c}, x] && GtQ[b^2 - 4*a*c, 0] && GtQ[c/a, 0] && LtQ[b/a, 0]
  1134. Int[Times[Power[Pattern[x, Blank[]], 2], Power[Plus[Pattern[a, Blank[]], Times[Optional[Pattern[b, Blank[]]], Power[Pattern[x, Blank[]], 2]], Times[Optional[Pattern[c, Blank[]]], Power[Pattern[x, Blank[]], 4]]], Rational[-1, 2]]], Pattern[x, Blank[Symbol]]] := With[{q = Rt[b^2 - 4*a*c, 2]}, -Dist[(b - q)/(2*c), Int[1/Sqrt[a + b*x^2 + c*x^4], x], x] + Dist[1/(2*c), Int[(b - q + 2*c*x^2)/Sqrt[a + b*x^2 + c*x^4], x], x]] /; FreeQ[{a, b, c}, x] && GtQ[b^2 - 4*a*c, 0] && LtQ[a, 0] && GtQ[c, 0]
  1135. Int[Times[Power[Pattern[x, Blank[]], 2], Power[Plus[Pattern[a, Blank[]], Times[Optional[Pattern[b, Blank[]]], Power[Pattern[x, Blank[]], 2]], Times[Optional[Pattern[c, Blank[]]], Power[Pattern[x, Blank[]], 4]]], Rational[-1, 2]]], Pattern[x, Blank[Symbol]]] := With[{q = Rt[b^2 - 4*a*c, 2]}, Simp[(x*(b + q + 2*c*x^2))/(2*c*Sqrt[a + b*x^2 + c*x^4]), x] - Simp[(Rt[(b + q)/(2*a), 2]*(2*a + (b + q)*x^2)*Sqrt[(2*a + (b - q)*x^2)/(2*a + (b + q)*x^2)]*EllipticE[ArcTan[Rt[(b + q)/(2*a), 2]*x], (2*q)/(b + q)])/(2*c*Sqrt[a + b*x^2 + c*x^4]), x] /; PosQ[(b + q)/a] && !(PosQ[(b - q)/a] && SimplerSqrtQ[(b - q)/(2*a), (b + q)/(2*a)])] /; FreeQ[{a, b, c}, x] && GtQ[b^2 - 4*a*c, 0]
  1136. Int[Times[Power[Pattern[x, Blank[]], 2], Power[Plus[Pattern[a, Blank[]], Times[Optional[Pattern[b, Blank[]]], Power[Pattern[x, Blank[]], 2]], Times[Optional[Pattern[c, Blank[]]], Power[Pattern[x, Blank[]], 4]]], Rational[-1, 2]]], Pattern[x, Blank[Symbol]]] := With[{q = Rt[b^2 - 4*a*c, 2]}, Simp[(x*(b - q + 2*c*x^2))/(2*c*Sqrt[a + b*x^2 + c*x^4]), x] - Simp[(Rt[(b - q)/(2*a), 2]*(2*a + (b - q)*x^2)*Sqrt[(2*a + (b + q)*x^2)/(2*a + (b - q)*x^2)]*EllipticE[ArcTan[Rt[(b - q)/(2*a), 2]*x], (-2*q)/(b - q)])/(2*c*Sqrt[a + b*x^2 + c*x^4]), x] /; PosQ[(b - q)/a]] /; FreeQ[{a, b, c}, x] && GtQ[b^2 - 4*a*c, 0]
  1137. Int[Times[Power[Pattern[x, Blank[]], 2], Power[Plus[Pattern[a, Blank[]], Times[Optional[Pattern[b, Blank[]]], Power[Pattern[x, Blank[]], 2]], Times[Optional[Pattern[c, Blank[]]], Power[Pattern[x, Blank[]], 4]]], Rational[-1, 2]]], Pattern[x, Blank[Symbol]]] := With[{q = Rt[b^2 - 4*a*c, 2]}, -Dist[(b + q)/(2*c), Int[1/Sqrt[a + b*x^2 + c*x^4], x], x] + Dist[1/(2*c), Int[(b + q + 2*c*x^2)/Sqrt[a + b*x^2 + c*x^4], x], x] /; NegQ[(b + q)/a] && !(NegQ[(b - q)/a] && SimplerSqrtQ[-((b - q)/(2*a)), -((b + q)/(2*a))])] /; FreeQ[{a, b, c}, x] && GtQ[b^2 - 4*a*c, 0]
  1138. Int[Times[Power[Pattern[x, Blank[]], 2], Power[Plus[Pattern[a, Blank[]], Times[Optional[Pattern[b, Blank[]]], Power[Pattern[x, Blank[]], 2]], Times[Optional[Pattern[c, Blank[]]], Power[Pattern[x, Blank[]], 4]]], Rational[-1, 2]]], Pattern[x, Blank[Symbol]]] := With[{q = Rt[b^2 - 4*a*c, 2]}, -Dist[(b - q)/(2*c), Int[1/Sqrt[a + b*x^2 + c*x^4], x], x] + Dist[1/(2*c), Int[(b - q + 2*c*x^2)/Sqrt[a + b*x^2 + c*x^4], x], x] /; NegQ[(b - q)/a]] /; FreeQ[{a, b, c}, x] && GtQ[b^2 - 4*a*c, 0]
  1139. Int[Times[Power[Pattern[x, Blank[]], 2], Power[Plus[Pattern[a, Blank[]], Times[Optional[Pattern[b, Blank[]]], Power[Pattern[x, Blank[]], 2]], Times[Optional[Pattern[c, Blank[]]], Power[Pattern[x, Blank[]], 4]]], Rational[-1, 2]]], Pattern[x, Blank[Symbol]]] := With[{q = Rt[c/a, 2]}, Dist[1/q, Int[1/Sqrt[a + b*x^2 + c*x^4], x], x] - Dist[1/q, Int[(1 - q*x^2)/Sqrt[a + b*x^2 + c*x^4], x], x]] /; FreeQ[{a, b, c}, x] && NeQ[b^2 - 4*a*c, 0] && PosQ[c/a]
  1140. Int[Times[Power[Pattern[x, Blank[]], 2], Power[Plus[Pattern[a, Blank[]], Times[Optional[Pattern[b, Blank[]]], Power[Pattern[x, Blank[]], 2]], Times[Optional[Pattern[c, Blank[]]], Power[Pattern[x, Blank[]], 4]]], Rational[-1, 2]]], Pattern[x, Blank[Symbol]]] := With[{q = Rt[b^2 - 4*a*c, 2]}, Dist[(Sqrt[1 + (2*c*x^2)/(b - q)]*Sqrt[1 + (2*c*x^2)/(b + q)])/Sqrt[a + b*x^2 + c*x^4], Int[x^2/(Sqrt[1 + (2*c*x^2)/(b - q)]*Sqrt[1 + (2*c*x^2)/(b + q)]), x], x]] /; FreeQ[{a, b, c}, x] && NeQ[b^2 - 4*a*c, 0] && NegQ[c/a]
  1141. Int[Times[Power[Times[Optional[Pattern[d, Blank[]]], Pattern[x, Blank[]]], Optional[Pattern[m, Blank[]]]], Power[Plus[Pattern[a, Blank[]], Times[Optional[Pattern[b, Blank[]]], Power[Pattern[x, Blank[]], 2]], Times[Optional[Pattern[c, Blank[]]], Power[Pattern[x, Blank[]], 4]]], Pattern[p, Blank[]]]], Pattern[x, Blank[Symbol]]] := Dist[(a^IntPart[p]*(a + b*x^2 + c*x^4)^FracPart[p])/((1 + (2*c*x^2)/(b + Rt[b^2 - 4*a*c, 2]))^FracPart[p]*(1 + (2*c*x^2)/(b - Rt[b^2 - 4*a*c, 2]))^FracPart[p]), Int[(d*x)^m*(1 + (2*c*x^2)/(b + Sqrt[b^2 - 4*a*c]))^p*(1 + (2*c*x^2)/(b - Sqrt[b^2 - 4*a*c]))^p, x], x] /; FreeQ[{a, b, c, d, m, p}, x]
  1142. Int[Times[Power[Pattern[u, Blank[]], Optional[Pattern[m, Blank[]]]], Power[Plus[Optional[Pattern[a, Blank[]]], Times[Optional[Pattern[b, Blank[]]], Power[Pattern[v, Blank[]], 2]], Times[Optional[Pattern[c, Blank[]]], Power[Pattern[v, Blank[]], 4]]], Optional[Pattern[p, Blank[]]]]], Pattern[x, Blank[Symbol]]] := Dist[u^m/(Coefficient[v, x, 1]*v^m), Subst[Int[x^m*(a + b*x^2 + c*x^(2*2))^p, x], x, v], x] /; FreeQ[{a, b, c, m, p}, x] && LinearPairQ[u, v, x]
  1143. Int[Times[Plus[Pattern[d, Blank[]], Times[Optional[Pattern[e, Blank[]]], Power[Pattern[x, Blank[]], 2]]], Power[Plus[Times[Optional[Pattern[b, Blank[]]], Power[Pattern[x, Blank[]], 2]], Times[Optional[Pattern[c, Blank[]]], Power[Pattern[x, Blank[]], 4]]], Rational[-3, 4]]], Pattern[x, Blank[Symbol]]] := Simp[(-2*(c*d - b*e)*(b*x^2 + c*x^4)^(1/4))/(b*c*x), x] + Dist[e/c, Int[(b*x^2 + c*x^4)^(1/4)/x^2, x], x] /; FreeQ[{b, c, d, e}, x]
  1144. Int[Times[Plus[Pattern[d, Blank[]], Times[Optional[Pattern[e, Blank[]]], Power[Pattern[x, Blank[]], 2]]], Power[Plus[Times[Optional[Pattern[b, Blank[]]], Power[Pattern[x, Blank[]], 2]], Times[Optional[Pattern[c, Blank[]]], Power[Pattern[x, Blank[]], 4]]], Pattern[p, Blank[]]]], Pattern[x, Blank[Symbol]]] := Simp[(e*(b*x^2 + c*x^4)^(p + 1))/(c*(4*p + 3)*x), x] /; FreeQ[{b, c, d, e, p}, x] && !IntegerQ[p] && NeQ[4*p + 3, 0] && EqQ[b*e*(2*p + 1) - c*d*(4*p + 3), 0]
  1145. Int[Times[Plus[Pattern[d, Blank[]], Times[Optional[Pattern[e, Blank[]]], Power[Pattern[x, Blank[]], 2]]], Power[Plus[Times[Optional[Pattern[b, Blank[]]], Power[Pattern[x, Blank[]], 2]], Times[Optional[Pattern[c, Blank[]]], Power[Pattern[x, Blank[]], 4]]], Pattern[p, Blank[]]]], Pattern[x, Blank[Symbol]]] := Simp[(e*(b*x^2 + c*x^4)^(p + 1))/(c*(4*p + 3)*x), x] - Dist[(b*e*(2*p + 1) - c*d*(4*p + 3))/(c*(4*p + 3)), Int[(b*x^2 + c*x^4)^p, x], x] /; FreeQ[{b, c, d, e, p}, x] && !IntegerQ[p] && NeQ[4*p + 3, 0] && NeQ[b*e*(2*p + 1) - c*d*(4*p + 3), 0]
  1146. Int[Times[Power[Plus[Pattern[d, Blank[]], Times[Optional[Pattern[e, Blank[]]], Power[Pattern[x, Blank[]], 2]]], Optional[Pattern[q, Blank[]]]], Power[Plus[Times[Optional[Pattern[b, Blank[]]], Power[Pattern[x, Blank[]], 2]], Times[Optional[Pattern[c, Blank[]]], Power[Pattern[x, Blank[]], 4]]], Pattern[p, Blank[]]]], Pattern[x, Blank[Symbol]]] := Dist[(b*x^2 + c*x^4)^FracPart[p]/(x^(2*FracPart[p])*(b + c*x^2)^FracPart[p]), Int[x^(2*p)*(d + e*x^2)^q*(b + c*x^2)^p, x], x] /; FreeQ[{b, c, d, e, p, q}, x] && !IntegerQ[p]
  1147. Int[Times[Power[Plus[Pattern[d, Blank[]], Times[Optional[Pattern[e, Blank[]]], Power[Pattern[x, Blank[]], 2]]], Optional[Pattern[q, Blank[]]]], Power[Plus[Pattern[a, Blank[]], Times[Optional[Pattern[b, Blank[]]], Power[Pattern[x, Blank[]], 2]], Times[Optional[Pattern[c, Blank[]]], Power[Pattern[x, Blank[]], 4]]], Pattern[p, Blank[]]]], Pattern[x, Blank[Symbol]]] := Dist[(a + b*x^2 + c*x^4)^p/(d + e*x^2)^(2*p), Int[(d + e*x^2)^(q + 2*p), x], x] /; FreeQ[{a, b, c, d, e, p, q}, x] && EqQ[b^2 - 4*a*c, 0] && !IntegerQ[p] && EqQ[2*c*d - b*e, 0]
  1148. Int[Times[Power[Plus[Pattern[d, Blank[]], Times[Optional[Pattern[e, Blank[]]], Power[Pattern[x, Blank[]], 2]]], Optional[Pattern[q, Blank[]]]], Power[Plus[Pattern[a, Blank[]], Times[Optional[Pattern[b, Blank[]]], Power[Pattern[x, Blank[]], 2]], Times[Optional[Pattern[c, Blank[]]], Power[Pattern[x, Blank[]], 4]]], Pattern[p, Blank[]]]], Pattern[x, Blank[Symbol]]] := Dist[(a + b*x^2 + c*x^4)^FracPart[p]/(c^IntPart[p]*(b/2 + c*x^2)^(2*FracPart[p])), Int[(d + e*x^2)^q*(b/2 + c*x^2)^(2*p), x], x] /; FreeQ[{a, b, c, d, e, p, q}, x] && EqQ[b^2 - 4*a*c, 0] && !IntegerQ[p]
  1149. Int[Times[Power[Plus[Pattern[d, Blank[]], Times[Optional[Pattern[e, Blank[]]], Power[Pattern[x, Blank[]], 2]]], Optional[Pattern[q, Blank[]]]], Power[Plus[Pattern[a, Blank[]], Times[Optional[Pattern[b, Blank[]]], Power[Pattern[x, Blank[]], 2]], Times[Optional[Pattern[c, Blank[]]], Power[Pattern[x, Blank[]], 4]]], Optional[Pattern[p, Blank[]]]]], Pattern[x, Blank[Symbol]]] := Int[(d + e*x^2)^(p + q)*(a/d + (c*x^2)/e)^p, x] /; FreeQ[{a, b, c, d, e, q}, x] && NeQ[b^2 - 4*a*c, 0] && EqQ[c*d^2 - b*d*e + a*e^2, 0] && IntegerQ[p]
  1150. Int[Times[Power[Plus[Pattern[d, Blank[]], Times[Optional[Pattern[e, Blank[]]], Power[Pattern[x, Blank[]], 2]]], Optional[Pattern[q, Blank[]]]], Power[Plus[Pattern[a, Blank[]], Times[Optional[Pattern[c, Blank[]]], Power[Pattern[x, Blank[]], 4]]], Optional[Pattern[p, Blank[]]]]], Pattern[x, Blank[Symbol]]] := Int[(d + e*x^2)^(p + q)*(a/d + (c*x^2)/e)^p, x] /; FreeQ[{a, c, d, e, q}, x] && EqQ[c*d^2 + a*e^2, 0] && IntegerQ[p]
  1151. Int[Times[Power[Plus[Pattern[d, Blank[]], Times[Optional[Pattern[e, Blank[]]], Power[Pattern[x, Blank[]], 2]]], Pattern[q, Blank[]]], Power[Plus[Pattern[a, Blank[]], Times[Optional[Pattern[b, Blank[]]], Power[Pattern[x, Blank[]], 2]], Times[Optional[Pattern[c, Blank[]]], Power[Pattern[x, Blank[]], 4]]], Pattern[p, Blank[]]]], Pattern[x, Blank[Symbol]]] := Dist[(a + b*x^2 + c*x^4)^FracPart[p]/((d + e*x^2)^FracPart[p]*(a/d + (c*x^2)/e)^FracPart[p]), Int[(d + e*x^2)^(p + q)*(a/d + (c*x^2)/e)^p, x], x] /; FreeQ[{a, b, c, d, e, p, q}, x] && NeQ[b^2 - 4*a*c, 0] && EqQ[c*d^2 - b*d*e + a*e^2, 0] && !IntegerQ[p]
  1152. Int[Times[Power[Plus[Pattern[d, Blank[]], Times[Optional[Pattern[e, Blank[]]], Power[Pattern[x, Blank[]], 2]]], Pattern[q, Blank[]]], Power[Plus[Pattern[a, Blank[]], Times[Optional[Pattern[c, Blank[]]], Power[Pattern[x, Blank[]], 4]]], Pattern[p, Blank[]]]], Pattern[x, Blank[Symbol]]] := Dist[(a + c*x^4)^FracPart[p]/((d + e*x^2)^FracPart[p]*(a/d + (c*x^2)/e)^FracPart[p]), Int[(d + e*x^2)^(p + q)*(a/d + (c*x^2)/e)^p, x], x] /; FreeQ[{a, c, d, e, p, q}, x] && EqQ[c*d^2 + a*e^2, 0] && !IntegerQ[p]
  1153. Int[Times[Power[Plus[Pattern[d, Blank[]], Times[Optional[Pattern[e, Blank[]]], Power[Pattern[x, Blank[]], 2]]], Optional[Pattern[q, Blank[]]]], Power[Plus[Pattern[a, Blank[]], Times[Optional[Pattern[b, Blank[]]], Power[Pattern[x, Blank[]], 2]], Times[Optional[Pattern[c, Blank[]]], Power[Pattern[x, Blank[]], 4]]], Optional[Pattern[p, Blank[]]]]], Pattern[x, Blank[Symbol]]] := Int[ExpandIntegrand[(d + e*x^2)^q*(a + b*x^2 + c*x^4)^p, x], x] /; FreeQ[{a, b, c, d, e}, x] && NeQ[b^2 - 4*a*c, 0] && NeQ[c*d^2 - b*d*e + a*e^2, 0] && IGtQ[p, 0] && IGtQ[q, -2]
  1154. Int[Times[Power[Plus[Pattern[d, Blank[]], Times[Optional[Pattern[e, Blank[]]], Power[Pattern[x, Blank[]], 2]]], Optional[Pattern[q, Blank[]]]], Power[Plus[Pattern[a, Blank[]], Times[Optional[Pattern[c, Blank[]]], Power[Pattern[x, Blank[]], 4]]], Optional[Pattern[p, Blank[]]]]], Pattern[x, Blank[Symbol]]] := Int[ExpandIntegrand[(d + e*x^2)^q*(a + c*x^4)^p, x], x] /; FreeQ[{a, c, d, e}, x] && NeQ[c*d^2 + a*e^2, 0] && IGtQ[p, 0] && IGtQ[q, -2]
  1155. Int[Times[Power[Plus[Pattern[d, Blank[]], Times[Optional[Pattern[e, Blank[]]], Power[Pattern[x, Blank[]], 2]]], Pattern[q, Blank[]]], Power[Plus[Pattern[a, Blank[]], Times[Optional[Pattern[b, Blank[]]], Power[Pattern[x, Blank[]], 2]], Times[Optional[Pattern[c, Blank[]]], Power[Pattern[x, Blank[]], 4]]], Optional[Pattern[p, Blank[]]]]], Pattern[x, Blank[Symbol]]] := Simp[(a^p*x*(d + e*x^2)^(q + 1))/d, x] + Dist[1/d, Int[x^2*(d + e*x^2)^q*(d*PolynomialQuotient[(a + b*x^2 + c*x^4)^p - a^p, x^2, x] - e*a^p*(2*q + 3)), x], x] /; FreeQ[{a, b, c, d, e}, x] && NeQ[b^2 - 4*a*c, 0] && NeQ[c*d^2 - b*d*e + a*e^2, 0] && IGtQ[p, 0] && ILtQ[q + 1/2, 0] && LtQ[4*p + 2*q + 1, 0]
  1156. Int[Times[Power[Plus[Pattern[d, Blank[]], Times[Optional[Pattern[e, Blank[]]], Power[Pattern[x, Blank[]], 2]]], Pattern[q, Blank[]]], Power[Plus[Pattern[a, Blank[]], Times[Optional[Pattern[c, Blank[]]], Power[Pattern[x, Blank[]], 4]]], Optional[Pattern[p, Blank[]]]]], Pattern[x, Blank[Symbol]]] := Simp[(a^p*x*(d + e*x^2)^(q + 1))/d, x] + Dist[1/d, Int[x^2*(d + e*x^2)^q*(d*PolynomialQuotient[(a + c*x^4)^p - a^p, x^2, x] - e*a^p*(2*q + 3)), x], x] /; FreeQ[{a, c, d, e}, x] && NeQ[c*d^2 + a*e^2, 0] && IGtQ[p, 0] && ILtQ[q + 1/2, 0] && LtQ[4*p + 2*q + 1, 0]
  1157. Int[Times[Power[Plus[Pattern[d, Blank[]], Times[Optional[Pattern[e, Blank[]]], Power[Pattern[x, Blank[]], 2]]], Pattern[q, Blank[]]], Power[Plus[Pattern[a, Blank[]], Times[Optional[Pattern[b, Blank[]]], Power[Pattern[x, Blank[]], 2]], Times[Optional[Pattern[c, Blank[]]], Power[Pattern[x, Blank[]], 4]]], Optional[Pattern[p, Blank[]]]]], Pattern[x, Blank[Symbol]]] := With[{Qx = PolynomialQuotient[(a + b*x^2 + c*x^4)^p, d + e*x^2, x], R = Coeff[PolynomialRemainder[(a + b*x^2 + c*x^4)^p, d + e*x^2, x], x, 0]}, -Simp[(R*x*(d + e*x^2)^(q + 1))/(2*d*(q + 1)), x] + Dist[1/(2*d*(q + 1)), Int[(d + e*x^2)^(q + 1)*ExpandToSum[2*d*(q + 1)*Qx + R*(2*q + 3), x], x], x]] /; FreeQ[{a, b, c, d, e}, x] && NeQ[b^2 - 4*a*c, 0] && NeQ[c*d^2 - b*d*e + a*e^2, 0] && IGtQ[p, 0] && LtQ[q, -1]
  1158. Int[Times[Power[Plus[Pattern[d, Blank[]], Times[Optional[Pattern[e, Blank[]]], Power[Pattern[x, Blank[]], 2]]], Pattern[q, Blank[]]], Power[Plus[Pattern[a, Blank[]], Times[Optional[Pattern[c, Blank[]]], Power[Pattern[x, Blank[]], 4]]], Optional[Pattern[p, Blank[]]]]], Pattern[x, Blank[Symbol]]] := With[{Qx = PolynomialQuotient[(a + c*x^4)^p, d + e*x^2, x], R = Coeff[PolynomialRemainder[(a + c*x^4)^p, d + e*x^2, x], x, 0]}, -Simp[(R*x*(d + e*x^2)^(q + 1))/(2*d*(q + 1)), x] + Dist[1/(2*d*(q + 1)), Int[(d + e*x^2)^(q + 1)*ExpandToSum[2*d*(q + 1)*Qx + R*(2*q + 3), x], x], x]] /; FreeQ[{a, c, d, e}, x] && NeQ[c*d^2 + a*e^2, 0] && IGtQ[p, 0] && LtQ[q, -1]
  1159. Int[Times[Power[Plus[Pattern[d, Blank[]], Times[Optional[Pattern[e, Blank[]]], Power[Pattern[x, Blank[]], 2]]], Pattern[q, Blank[]]], Power[Plus[Pattern[a, Blank[]], Times[Optional[Pattern[b, Blank[]]], Power[Pattern[x, Blank[]], 2]], Times[Optional[Pattern[c, Blank[]]], Power[Pattern[x, Blank[]], 4]]], Optional[Pattern[p, Blank[]]]]], Pattern[x, Blank[Symbol]]] := Simp[(c^p*x^(4*p - 1)*(d + e*x^2)^(q + 1))/(e*(4*p + 2*q + 1)), x] + Dist[1/(e*(4*p + 2*q + 1)), Int[(d + e*x^2)^q*ExpandToSum[e*(4*p + 2*q + 1)*(a + b*x^2 + c*x^4)^p - d*c^p*(4*p - 1)*x^(4*p - 2) - e*c^p*(4*p + 2*q + 1)*x^(4*p), x], x], x] /; FreeQ[{a, b, c, d, e, q}, x] && NeQ[b^2 - 4*a*c, 0] && NeQ[c*d^2 - b*d*e + a*e^2, 0] && IGtQ[p, 0] && !LtQ[q, -1]
  1160. Int[Times[Power[Plus[Pattern[d, Blank[]], Times[Optional[Pattern[e, Blank[]]], Power[Pattern[x, Blank[]], 2]]], Pattern[q, Blank[]]], Power[Plus[Pattern[a, Blank[]], Times[Optional[Pattern[c, Blank[]]], Power[Pattern[x, Blank[]], 4]]], Optional[Pattern[p, Blank[]]]]], Pattern[x, Blank[Symbol]]] := Simp[(c^p*x^(4*p - 1)*(d + e*x^2)^(q + 1))/(e*(4*p + 2*q + 1)), x] + Dist[1/(e*(4*p + 2*q + 1)), Int[(d + e*x^2)^q*ExpandToSum[e*(4*p + 2*q + 1)*(a + c*x^4)^p - d*c^p*(4*p - 1)*x^(4*p - 2) - e*c^p*(4*p + 2*q + 1)*x^(4*p), x], x], x] /; FreeQ[{a, c, d, e, q}, x] && NeQ[c*d^2 + a*e^2, 0] && IGtQ[p, 0] && !LtQ[q, -1]
  1161. Int[Times[Plus[Pattern[d, Blank[]], Times[Optional[Pattern[e, Blank[]]], Power[Pattern[x, Blank[]], 2]]], Power[Plus[Pattern[a, Blank[]], Times[Optional[Pattern[b, Blank[]]], Power[Pattern[x, Blank[]], 2]], Times[Optional[Pattern[c, Blank[]]], Power[Pattern[x, Blank[]], 4]]], -1]], Pattern[x, Blank[Symbol]]] := With[{q = Rt[(2*d)/e - b/c, 2]}, Dist[e/(2*c), Int[1/Simp[d/e + q*x + x^2, x], x], x] + Dist[e/(2*c), Int[1/Simp[d/e - q*x + x^2, x], x], x]] /; FreeQ[{a, b, c, d, e}, x] && NeQ[b^2 - 4*a*c, 0] && EqQ[c*d^2 - a*e^2, 0] && (GtQ[(2*d)/e - b/c, 0] || ( !LtQ[(2*d)/e - b/c, 0] && EqQ[d - e*Rt[a/c, 2], 0]))
  1162. Int[Times[Plus[Pattern[d, Blank[]], Times[Optional[Pattern[e, Blank[]]], Power[Pattern[x, Blank[]], 2]]], Power[Plus[Pattern[a, Blank[]], Times[Optional[Pattern[c, Blank[]]], Power[Pattern[x, Blank[]], 4]]], -1]], Pattern[x, Blank[Symbol]]] := With[{q = Rt[(2*d)/e, 2]}, Dist[e/(2*c), Int[1/Simp[d/e + q*x + x^2, x], x], x] + Dist[e/(2*c), Int[1/Simp[d/e - q*x + x^2, x], x], x]] /; FreeQ[{a, c, d, e}, x] && EqQ[c*d^2 - a*e^2, 0] && PosQ[d*e]
  1163. Int[Times[Plus[Pattern[d, Blank[]], Times[Optional[Pattern[e, Blank[]]], Power[Pattern[x, Blank[]], 2]]], Power[Plus[Pattern[a, Blank[]], Times[Optional[Pattern[b, Blank[]]], Power[Pattern[x, Blank[]], 2]], Times[Optional[Pattern[c, Blank[]]], Power[Pattern[x, Blank[]], 4]]], -1]], Pattern[x, Blank[Symbol]]] := With[{q = Rt[b^2 - 4*a*c, 2]}, Dist[e/2 + (2*c*d - b*e)/(2*q), Int[1/(b/2 - q/2 + c*x^2), x], x] + Dist[e/2 - (2*c*d - b*e)/(2*q), Int[1/(b/2 + q/2 + c*x^2), x], x]] /; FreeQ[{a, b, c, d, e}, x] && NeQ[b^2 - 4*a*c, 0] && EqQ[c*d^2 - a*e^2, 0] && GtQ[b^2 - 4*a*c, 0]
  1164. Int[Times[Plus[Pattern[d, Blank[]], Times[Optional[Pattern[e, Blank[]]], Power[Pattern[x, Blank[]], 2]]], Power[Plus[Pattern[a, Blank[]], Times[Optional[Pattern[b, Blank[]]], Power[Pattern[x, Blank[]], 2]], Times[Optional[Pattern[c, Blank[]]], Power[Pattern[x, Blank[]], 4]]], -1]], Pattern[x, Blank[Symbol]]] := With[{q = Rt[(-2*d)/e - b/c, 2]}, Dist[e/(2*c*q), Int[(q - 2*x)/Simp[d/e + q*x - x^2, x], x], x] + Dist[e/(2*c*q), Int[(q + 2*x)/Simp[d/e - q*x - x^2, x], x], x]] /; FreeQ[{a, b, c, d, e}, x] && NeQ[b^2 - 4*a*c, 0] && EqQ[c*d^2 - a*e^2, 0] && !GtQ[b^2 - 4*a*c, 0]
  1165. Int[Times[Plus[Pattern[d, Blank[]], Times[Optional[Pattern[e, Blank[]]], Power[Pattern[x, Blank[]], 2]]], Power[Plus[Pattern[a, Blank[]], Times[Optional[Pattern[c, Blank[]]], Power[Pattern[x, Blank[]], 4]]], -1]], Pattern[x, Blank[Symbol]]] := With[{q = Rt[(-2*d)/e, 2]}, Dist[e/(2*c*q), Int[(q - 2*x)/Simp[d/e + q*x - x^2, x], x], x] + Dist[e/(2*c*q), Int[(q + 2*x)/Simp[d/e - q*x - x^2, x], x], x]] /; FreeQ[{a, c, d, e}, x] && EqQ[c*d^2 - a*e^2, 0] && NegQ[d*e]
  1166. Int[Times[Plus[Pattern[d, Blank[]], Times[Optional[Pattern[e, Blank[]]], Power[Pattern[x, Blank[]], 2]]], Power[Plus[Pattern[a, Blank[]], Times[Optional[Pattern[b, Blank[]]], Power[Pattern[x, Blank[]], 2]], Times[Optional[Pattern[c, Blank[]]], Power[Pattern[x, Blank[]], 4]]], -1]], Pattern[x, Blank[Symbol]]] := With[{q = Rt[b^2 - 4*a*c, 2]}, Dist[e/2 + (2*c*d - b*e)/(2*q), Int[1/(b/2 - q/2 + c*x^2), x], x] + Dist[e/2 - (2*c*d - b*e)/(2*q), Int[1/(b/2 + q/2 + c*x^2), x], x]] /; FreeQ[{a, b, c, d, e}, x] && NeQ[b^2 - 4*a*c, 0] && NeQ[c*d^2 - a*e^2, 0] && PosQ[b^2 - 4*a*c]
  1167. Int[Times[Plus[Pattern[d, Blank[]], Times[Optional[Pattern[e, Blank[]]], Power[Pattern[x, Blank[]], 2]]], Power[Plus[Pattern[a, Blank[]], Times[Optional[Pattern[c, Blank[]]], Power[Pattern[x, Blank[]], 4]]], -1]], Pattern[x, Blank[Symbol]]] := With[{q = Rt[-(a*c), 2]}, Dist[e/2 + (c*d)/(2*q), Int[1/(-q + c*x^2), x], x] + Dist[e/2 - (c*d)/(2*q), Int[1/(q + c*x^2), x], x]] /; FreeQ[{a, c, d, e}, x] && NeQ[c*d^2 - a*e^2, 0] && PosQ[-(a*c)]
  1168. Int[Times[Plus[Pattern[d, Blank[]], Times[Optional[Pattern[e, Blank[]]], Power[Pattern[x, Blank[]], 2]]], Power[Plus[Pattern[a, Blank[]], Times[Optional[Pattern[c, Blank[]]], Power[Pattern[x, Blank[]], 4]]], -1]], Pattern[x, Blank[Symbol]]] := With[{q = Rt[a*c, 2]}, Dist[(d*q + a*e)/(2*a*c), Int[(q + c*x^2)/(a + c*x^4), x], x] + Dist[(d*q - a*e)/(2*a*c), Int[(q - c*x^2)/(a + c*x^4), x], x]] /; FreeQ[{a, c, d, e}, x] && NeQ[c*d^2 + a*e^2, 0] && NeQ[c*d^2 - a*e^2, 0] && NegQ[-(a*c)]
  1169. Int[Times[Plus[Pattern[d, Blank[]], Times[Optional[Pattern[e, Blank[]]], Power[Pattern[x, Blank[]], 2]]], Power[Plus[Pattern[a, Blank[]], Times[Optional[Pattern[b, Blank[]]], Power[Pattern[x, Blank[]], 2]], Times[Optional[Pattern[c, Blank[]]], Power[Pattern[x, Blank[]], 4]]], -1]], Pattern[x, Blank[Symbol]]] := With[{q = Rt[a/c, 2]}, With[{r = Rt[2*q - b/c, 2]}, Dist[1/(2*c*q*r), Int[(d*r - (d - e*q)*x)/(q - r*x + x^2), x], x] + Dist[1/(2*c*q*r), Int[(d*r + (d - e*q)*x)/(q + r*x + x^2), x], x]]] /; FreeQ[{a, b, c, d, e}, x] && NeQ[b^2 - 4*a*c, 0] && NeQ[c*d^2 - b*d*e + a*e^2, 0] && NegQ[b^2 - 4*a*c]
  1170. Int[Times[Power[Plus[Pattern[d, Blank[]], Times[Optional[Pattern[e, Blank[]]], Power[Pattern[x, Blank[]], 2]]], Pattern[q, Blank[]]], Power[Plus[Pattern[a, Blank[]], Times[Optional[Pattern[b, Blank[]]], Power[Pattern[x, Blank[]], 2]], Times[Optional[Pattern[c, Blank[]]], Power[Pattern[x, Blank[]], 4]]], -1]], Pattern[x, Blank[Symbol]]] := Int[ExpandIntegrand[(d + e*x^2)^q/(a + b*x^2 + c*x^4), x], x] /; FreeQ[{a, b, c, d, e}, x] && NeQ[b^2 - 4*a*c, 0] && NeQ[c*d^2 - b*d*e + a*e^2, 0] && IntegerQ[q]
  1171. Int[Times[Power[Plus[Pattern[d, Blank[]], Times[Optional[Pattern[e, Blank[]]], Power[Pattern[x, Blank[]], 2]]], Pattern[q, Blank[]]], Power[Plus[Pattern[a, Blank[]], Times[Optional[Pattern[c, Blank[]]], Power[Pattern[x, Blank[]], 4]]], -1]], Pattern[x, Blank[Symbol]]] := Int[ExpandIntegrand[(d + e*x^2)^q/(a + c*x^4), x], x] /; FreeQ[{a, c, d, e}, x] && NeQ[c*d^2 + a*e^2, 0] && IntegerQ[q]
  1172. Int[Times[Power[Plus[Pattern[d, Blank[]], Times[Optional[Pattern[e, Blank[]]], Power[Pattern[x, Blank[]], 2]]], Pattern[q, Blank[]]], Power[Plus[Pattern[a, Blank[]], Times[Optional[Pattern[b, Blank[]]], Power[Pattern[x, Blank[]], 2]], Times[Optional[Pattern[c, Blank[]]], Power[Pattern[x, Blank[]], 4]]], -1]], Pattern[x, Blank[Symbol]]] := Dist[e^2/(c*d^2 - b*d*e + a*e^2), Int[(d + e*x^2)^q, x], x] + Dist[1/(c*d^2 - b*d*e + a*e^2), Int[((d + e*x^2)^(q + 1)*(c*d - b*e - c*e*x^2))/(a + b*x^2 + c*x^4), x], x] /; FreeQ[{a, b, c, d, e}, x] && NeQ[b^2 - 4*a*c, 0] && NeQ[c*d^2 - b*d*e + a*e^2, 0] && !IntegerQ[q] && LtQ[q, -1]
  1173. Int[Times[Power[Plus[Pattern[d, Blank[]], Times[Optional[Pattern[e, Blank[]]], Power[Pattern[x, Blank[]], 2]]], Pattern[q, Blank[]]], Power[Plus[Pattern[a, Blank[]], Times[Optional[Pattern[c, Blank[]]], Power[Pattern[x, Blank[]], 4]]], -1]], Pattern[x, Blank[Symbol]]] := Dist[e^2/(c*d^2 + a*e^2), Int[(d + e*x^2)^q, x], x] + Dist[c/(c*d^2 + a*e^2), Int[((d + e*x^2)^(q + 1)*(d - e*x^2))/(a + c*x^4), x], x] /; FreeQ[{a, c, d, e}, x] && NeQ[c*d^2 + a*e^2, 0] && !IntegerQ[q] && LtQ[q, -1]
  1174. Int[Times[Power[Plus[Pattern[d, Blank[]], Times[Optional[Pattern[e, Blank[]]], Power[Pattern[x, Blank[]], 2]]], Pattern[q, Blank[]]], Power[Plus[Pattern[a, Blank[]], Times[Optional[Pattern[b, Blank[]]], Power[Pattern[x, Blank[]], 2]], Times[Optional[Pattern[c, Blank[]]], Power[Pattern[x, Blank[]], 4]]], -1]], Pattern[x, Blank[Symbol]]] := With[{r = Rt[b^2 - 4*a*c, 2]}, Dist[(2*c)/r, Int[(d + e*x^2)^q/(b - r + 2*c*x^2), x], x] - Dist[(2*c)/r, Int[(d + e*x^2)^q/(b + r + 2*c*x^2), x], x]] /; FreeQ[{a, b, c, d, e, q}, x] && NeQ[b^2 - 4*a*c, 0] && NeQ[c*d^2 - b*d*e + a*e^2, 0] && !IntegerQ[q]
  1175. Int[Times[Power[Plus[Pattern[d, Blank[]], Times[Optional[Pattern[e, Blank[]]], Power[Pattern[x, Blank[]], 2]]], Pattern[q, Blank[]]], Power[Plus[Pattern[a, Blank[]], Times[Optional[Pattern[c, Blank[]]], Power[Pattern[x, Blank[]], 4]]], -1]], Pattern[x, Blank[Symbol]]] := With[{r = Rt[-(a*c), 2]}, -Dist[c/(2*r), Int[(d + e*x^2)^q/(r - c*x^2), x], x] - Dist[c/(2*r), Int[(d + e*x^2)^q/(r + c*x^2), x], x]] /; FreeQ[{a, c, d, e, q}, x] && NeQ[c*d^2 + a*e^2, 0] && !IntegerQ[q]
  1176. Int[Times[Plus[Pattern[d, Blank[]], Times[Optional[Pattern[e, Blank[]]], Power[Pattern[x, Blank[]], 2]]], Power[Plus[Pattern[a, Blank[]], Times[Optional[Pattern[b, Blank[]]], Power[Pattern[x, Blank[]], 2]], Times[Optional[Pattern[c, Blank[]]], Power[Pattern[x, Blank[]], 4]]], Pattern[p, Blank[]]]], Pattern[x, Blank[Symbol]]] := Simp[(x*(2*b*e*p + c*d*(4*p + 3) + c*e*(4*p + 1)*x^2)*(a + b*x^2 + c*x^4)^p)/(c*(4*p + 1)*(4*p + 3)), x] + Dist[(2*p)/(c*(4*p + 1)*(4*p + 3)), Int[Simp[2*a*c*d*(4*p + 3) - a*b*e + (2*a*c*e*(4*p + 1) + b*c*d*(4*p + 3) - b^2*e*(2*p + 1))*x^2, x]*(a + b*x^2 + c*x^4)^(p - 1), x], x] /; FreeQ[{a, b, c, d, e}, x] && NeQ[b^2 - 4*a*c, 0] && NeQ[c*d^2 - b*d*e + a*e^2, 0] && GtQ[p, 0] && FractionQ[p] && IntegerQ[2*p]
  1177. Int[Times[Plus[Pattern[d, Blank[]], Times[Optional[Pattern[e, Blank[]]], Power[Pattern[x, Blank[]], 2]]], Power[Plus[Pattern[a, Blank[]], Times[Optional[Pattern[c, Blank[]]], Power[Pattern[x, Blank[]], 4]]], Pattern[p, Blank[]]]], Pattern[x, Blank[Symbol]]] := Simp[(x*(d*(4*p + 3) + e*(4*p + 1)*x^2)*(a + c*x^4)^p)/((4*p + 1)*(4*p + 3)), x] + Dist[(2*p)/((4*p + 1)*(4*p + 3)), Int[Simp[2*a*d*(4*p + 3) + (2*a*e*(4*p + 1))*x^2, x]*(a + c*x^4)^(p - 1), x], x] /; FreeQ[{a, c, d, e}, x] && NeQ[c*d^2 + a*e^2, 0] && GtQ[p, 0] && FractionQ[p] && IntegerQ[2*p]
  1178. Int[Times[Plus[Pattern[d, Blank[]], Times[Optional[Pattern[e, Blank[]]], Power[Pattern[x, Blank[]], 2]]], Power[Plus[Pattern[a, Blank[]], Times[Optional[Pattern[b, Blank[]]], Power[Pattern[x, Blank[]], 2]], Times[Optional[Pattern[c, Blank[]]], Power[Pattern[x, Blank[]], 4]]], Pattern[p, Blank[]]]], Pattern[x, Blank[Symbol]]] := Simp[(x*(a*b*e - d*(b^2 - 2*a*c) - c*(b*d - 2*a*e)*x^2)*(a + b*x^2 + c*x^4)^(p + 1))/(2*a*(p + 1)*(b^2 - 4*a*c)), x] + Dist[1/(2*a*(p + 1)*(b^2 - 4*a*c)), Int[Simp[(2*p + 3)*d*b^2 - a*b*e - 2*a*c*d*(4*p + 5) + (4*p + 7)*(d*b - 2*a*e)*c*x^2, x]*(a + b*x^2 + c*x^4)^(p + 1), x], x] /; FreeQ[{a, b, c, d, e}, x] && NeQ[b^2 - 4*a*c, 0] && NeQ[c*d^2 - b*d*e + a*e^2, 0] && LtQ[p, -1] && IntegerQ[2*p]
  1179. Int[Times[Plus[Pattern[d, Blank[]], Times[Optional[Pattern[e, Blank[]]], Power[Pattern[x, Blank[]], 2]]], Power[Plus[Pattern[a, Blank[]], Times[Optional[Pattern[c, Blank[]]], Power[Pattern[x, Blank[]], 4]]], Pattern[p, Blank[]]]], Pattern[x, Blank[Symbol]]] := -Simp[(x*(d + e*x^2)*(a + c*x^4)^(p + 1))/(4*a*(p + 1)), x] + Dist[1/(4*a*(p + 1)), Int[Simp[d*(4*p + 5) + e*(4*p + 7)*x^2, x]*(a + c*x^4)^(p + 1), x], x] /; FreeQ[{a, c, d, e}, x] && NeQ[c*d^2 + a*e^2, 0] && LtQ[p, -1] && IntegerQ[2*p]
  1180. Int[Times[Plus[Pattern[d, Blank[]], Times[Optional[Pattern[e, Blank[]]], Power[Pattern[x, Blank[]], 2]]], Power[Plus[Pattern[a, Blank[]], Times[Optional[Pattern[b, Blank[]]], Power[Pattern[x, Blank[]], 2]], Times[Optional[Pattern[c, Blank[]]], Power[Pattern[x, Blank[]], 4]]], Rational[-1, 2]]], Pattern[x, Blank[Symbol]]] := With[{q = Rt[b^2 - 4*a*c, 2]}, Dist[2*Sqrt[-c], Int[(d + e*x^2)/(Sqrt[b + q + 2*c*x^2]*Sqrt[-b + q - 2*c*x^2]), x], x]] /; FreeQ[{a, b, c, d, e}, x] && GtQ[b^2 - 4*a*c, 0] && LtQ[c, 0]
  1181. Int[Times[Plus[Pattern[d, Blank[]], Times[Optional[Pattern[e, Blank[]]], Power[Pattern[x, Blank[]], 2]]], Power[Plus[Pattern[a, Blank[]], Times[Optional[Pattern[c, Blank[]]], Power[Pattern[x, Blank[]], 4]]], Rational[-1, 2]]], Pattern[x, Blank[Symbol]]] := With[{q = Rt[-(a*c), 2]}, Dist[Sqrt[-c], Int[(d + e*x^2)/(Sqrt[q + c*x^2]*Sqrt[q - c*x^2]), x], x]] /; FreeQ[{a, c, d, e}, x] && GtQ[a, 0] && LtQ[c, 0]
  1182. Int[Times[Plus[Pattern[d, Blank[]], Times[Optional[Pattern[e, Blank[]]], Power[Pattern[x, Blank[]], 2]]], Power[Plus[Pattern[a, Blank[]], Times[Optional[Pattern[b, Blank[]]], Power[Pattern[x, Blank[]], 2]], Times[Optional[Pattern[c, Blank[]]], Power[Pattern[x, Blank[]], 4]]], Rational[-1, 2]]], Pattern[x, Blank[Symbol]]] := With[{q = Rt[c/a, 4]}, -Simp[(d*x*Sqrt[a + b*x^2 + c*x^4])/(a*(1 + q^2*x^2)), x] + Simp[(d*(1 + q^2*x^2)*Sqrt[(a + b*x^2 + c*x^4)/(a*(1 + q^2*x^2)^2)]*EllipticE[2*ArcTan[q*x], 1/2 - (b*q^2)/(4*c)])/(q*Sqrt[a + b*x^2 + c*x^4]), x] /; EqQ[e + d*q^2, 0]] /; FreeQ[{a, b, c, d, e}, x] && GtQ[b^2 - 4*a*c, 0] && GtQ[c/a, 0] && LtQ[b/a, 0]
  1183. Int[Times[Plus[Pattern[d, Blank[]], Times[Optional[Pattern[e, Blank[]]], Power[Pattern[x, Blank[]], 2]]], Power[Plus[Pattern[a, Blank[]], Times[Optional[Pattern[b, Blank[]]], Power[Pattern[x, Blank[]], 2]], Times[Optional[Pattern[c, Blank[]]], Power[Pattern[x, Blank[]], 4]]], Rational[-1, 2]]], Pattern[x, Blank[Symbol]]] := With[{q = Rt[c/a, 2]}, Dist[(e + d*q)/q, Int[1/Sqrt[a + b*x^2 + c*x^4], x], x] - Dist[e/q, Int[(1 - q*x^2)/Sqrt[a + b*x^2 + c*x^4], x], x] /; NeQ[e + d*q, 0]] /; FreeQ[{a, b, c, d, e}, x] && GtQ[b^2 - 4*a*c, 0] && GtQ[c/a, 0] && LtQ[b/a, 0]
  1184. Int[Times[Plus[Pattern[d, Blank[]], Times[Optional[Pattern[e, Blank[]]], Power[Pattern[x, Blank[]], 2]]], Power[Plus[Pattern[a, Blank[]], Times[Optional[Pattern[b, Blank[]]], Power[Pattern[x, Blank[]], 2]], Times[Optional[Pattern[c, Blank[]]], Power[Pattern[x, Blank[]], 4]]], Rational[-1, 2]]], Pattern[x, Blank[Symbol]]] := With[{q = Rt[b^2 - 4*a*c, 2]}, Simp[(e*x*(b + q + 2*c*x^2))/(2*c*Sqrt[a + b*x^2 + c*x^4]), x] - Simp[(e*q*Sqrt[(2*a + (b - q)*x^2)/(2*a + (b + q)*x^2)]*Sqrt[(2*a + (b + q)*x^2)/q]*EllipticE[ArcSin[x/Sqrt[(2*a + (b + q)*x^2)/(2*q)]], (b + q)/(2*q)])/(2*c*Sqrt[a + b*x^2 + c*x^4]*Sqrt[a/(2*a + (b + q)*x^2)]), x] /; EqQ[2*c*d - e*(b - q), 0]] /; FreeQ[{a, b, c, d, e}, x] && GtQ[b^2 - 4*a*c, 0] && LtQ[a, 0] && GtQ[c, 0]
  1185. Int[Times[Plus[Pattern[d, Blank[]], Times[Optional[Pattern[e, Blank[]]], Power[Pattern[x, Blank[]], 2]]], Power[Plus[Pattern[a, Blank[]], Times[Optional[Pattern[c, Blank[]]], Power[Pattern[x, Blank[]], 4]]], Rational[-1, 2]]], Pattern[x, Blank[Symbol]]] := With[{q = Rt[-(a*c), 2]}, Simp[(e*x*(q + c*x^2))/(c*Sqrt[a + c*x^4]), x] - Simp[(Sqrt[2]*e*q*Sqrt[-a + q*x^2]*Sqrt[(a + q*x^2)/q]*EllipticE[ArcSin[x/Sqrt[(a + q*x^2)/(2*q)]], 1/2])/(Sqrt[-a]*c*Sqrt[a + c*x^4]), x] /; EqQ[c*d + e*q, 0] && IntegerQ[q]] /; FreeQ[{a, c, d, e}, x] && LtQ[a, 0] && GtQ[c, 0]
  1186. Int[Times[Plus[Pattern[d, Blank[]], Times[Optional[Pattern[e, Blank[]]], Power[Pattern[x, Blank[]], 2]]], Power[Plus[Pattern[a, Blank[]], Times[Optional[Pattern[c, Blank[]]], Power[Pattern[x, Blank[]], 4]]], Rational[-1, 2]]], Pattern[x, Blank[Symbol]]] := With[{q = Rt[-(a*c), 2]}, Simp[(e*x*(q + c*x^2))/(c*Sqrt[a + c*x^4]), x] - Simp[(Sqrt[2]*e*q*Sqrt[(a - q*x^2)/(a + q*x^2)]*Sqrt[(a + q*x^2)/q]*EllipticE[ArcSin[x/Sqrt[(a + q*x^2)/(2*q)]], 1/2])/(c*Sqrt[a + c*x^4]*Sqrt[a/(a + q*x^2)]), x] /; EqQ[c*d + e*q, 0]] /; FreeQ[{a, c, d, e}, x] && LtQ[a, 0] && GtQ[c, 0]
  1187. Int[Times[Plus[Pattern[d, Blank[]], Times[Optional[Pattern[e, Blank[]]], Power[Pattern[x, Blank[]], 2]]], Power[Plus[Pattern[a, Blank[]], Times[Optional[Pattern[b, Blank[]]], Power[Pattern[x, Blank[]], 2]], Times[Optional[Pattern[c, Blank[]]], Power[Pattern[x, Blank[]], 4]]], Rational[-1, 2]]], Pattern[x, Blank[Symbol]]] := With[{q = Rt[b^2 - 4*a*c, 2]}, Dist[(2*c*d - e*(b - q))/(2*c), Int[1/Sqrt[a + b*x^2 + c*x^4], x], x] + Dist[e/(2*c), Int[(b - q + 2*c*x^2)/Sqrt[a + b*x^2 + c*x^4], x], x] /; NeQ[2*c*d - e*(b - q), 0]] /; FreeQ[{a, b, c, d, e}, x] && GtQ[b^2 - 4*a*c, 0] && LtQ[a, 0] && GtQ[c, 0]
  1188. Int[Times[Plus[Pattern[d, Blank[]], Times[Optional[Pattern[e, Blank[]]], Power[Pattern[x, Blank[]], 2]]], Power[Plus[Pattern[a, Blank[]], Times[Optional[Pattern[c, Blank[]]], Power[Pattern[x, Blank[]], 4]]], Rational[-1, 2]]], Pattern[x, Blank[Symbol]]] := With[{q = Rt[-(a*c), 2]}, Dist[(c*d + e*q)/c, Int[1/Sqrt[a + c*x^4], x], x] - Dist[e/c, Int[(q - c*x^2)/Sqrt[a + c*x^4], x], x] /; NeQ[c*d + e*q, 0]] /; FreeQ[{a, c, d, e}, x] && LtQ[a, 0] && GtQ[c, 0]
  1189. Int[Times[Plus[Pattern[d, Blank[]], Times[Optional[Pattern[e, Blank[]]], Power[Pattern[x, Blank[]], 2]]], Power[Plus[Pattern[a, Blank[]], Times[Optional[Pattern[b, Blank[]]], Power[Pattern[x, Blank[]], 2]], Times[Optional[Pattern[c, Blank[]]], Power[Pattern[x, Blank[]], 4]]], Rational[-1, 2]]], Pattern[x, Blank[Symbol]]] := With[{q = Rt[b^2 - 4*a*c, 2]}, Dist[d, Int[1/Sqrt[a + b*x^2 + c*x^4], x], x] + Dist[e, Int[x^2/Sqrt[a + b*x^2 + c*x^4], x], x] /; PosQ[(b + q)/a] || PosQ[(b - q)/a]] /; FreeQ[{a, b, c, d, e}, x] && GtQ[b^2 - 4*a*c, 0]
  1190. Int[Times[Plus[Pattern[d, Blank[]], Times[Optional[Pattern[e, Blank[]]], Power[Pattern[x, Blank[]], 2]]], Power[Plus[Pattern[a, Blank[]], Times[Optional[Pattern[c, Blank[]]], Power[Pattern[x, Blank[]], 4]]], Rational[-1, 2]]], Pattern[x, Blank[Symbol]]] := Dist[d, Int[1/Sqrt[a + c*x^4], x], x] + Dist[e, Int[x^2/Sqrt[a + c*x^4], x], x] /; FreeQ[{a, c, d, e}, x] && GtQ[-(a*c), 0]
  1191. Int[Times[Plus[Pattern[d, Blank[]], Times[Optional[Pattern[e, Blank[]]], Power[Pattern[x, Blank[]], 2]]], Power[Plus[Pattern[a, Blank[]], Times[Optional[Pattern[b, Blank[]]], Power[Pattern[x, Blank[]], 2]], Times[Optional[Pattern[c, Blank[]]], Power[Pattern[x, Blank[]], 4]]], Rational[-1, 2]]], Pattern[x, Blank[Symbol]]] := With[{q = Rt[b^2 - 4*a*c, 2]}, -Simp[(a*e*Rt[-((b + q)/(2*a)), 2]*Sqrt[1 + ((b + q)*x^2)/(2*a)]*Sqrt[1 + ((b - q)*x^2)/(2*a)]*EllipticE[ArcSin[Rt[-((b + q)/(2*a)), 2]*x], (b - q)/(b + q)])/(c*Sqrt[a + b*x^2 + c*x^4]), x] /; NegQ[(b + q)/a] && EqQ[2*c*d - e*(b + q), 0] && !SimplerSqrtQ[-((b - q)/(2*a)), -((b + q)/(2*a))]] /; FreeQ[{a, b, c, d, e}, x] && GtQ[b^2 - 4*a*c, 0]
  1192. Int[Times[Plus[Pattern[d, Blank[]], Times[Optional[Pattern[e, Blank[]]], Power[Pattern[x, Blank[]], 2]]], Power[Plus[Pattern[a, Blank[]], Times[Optional[Pattern[b, Blank[]]], Power[Pattern[x, Blank[]], 2]], Times[Optional[Pattern[c, Blank[]]], Power[Pattern[x, Blank[]], 4]]], Rational[-1, 2]]], Pattern[x, Blank[Symbol]]] := With[{q = Rt[b^2 - 4*a*c, 2]}, Dist[(2*c*d - e*(b + q))/(2*c), Int[1/Sqrt[a + b*x^2 + c*x^4], x], x] + Dist[e/(2*c), Int[(b + q + 2*c*x^2)/Sqrt[a + b*x^2 + c*x^4], x], x] /; NegQ[(b + q)/a] && NeQ[2*c*d - e*(b + q), 0] && !SimplerSqrtQ[-((b - q)/(2*a)), -((b + q)/(2*a))]] /; FreeQ[{a, b, c, d, e}, x] && GtQ[b^2 - 4*a*c, 0]
  1193. Int[Times[Plus[Pattern[d, Blank[]], Times[Optional[Pattern[e, Blank[]]], Power[Pattern[x, Blank[]], 2]]], Power[Plus[Pattern[a, Blank[]], Times[Optional[Pattern[b, Blank[]]], Power[Pattern[x, Blank[]], 2]], Times[Optional[Pattern[c, Blank[]]], Power[Pattern[x, Blank[]], 4]]], Rational[-1, 2]]], Pattern[x, Blank[Symbol]]] := With[{q = Rt[b^2 - 4*a*c, 2]}, -Simp[(a*e*Rt[-((b - q)/(2*a)), 2]*Sqrt[1 + ((b - q)*x^2)/(2*a)]*Sqrt[1 + ((b + q)*x^2)/(2*a)]*EllipticE[ArcSin[Rt[-((b - q)/(2*a)), 2]*x], (b + q)/(b - q)])/(c*Sqrt[a + b*x^2 + c*x^4]), x] /; NegQ[(b - q)/a] && EqQ[2*c*d - e*(b - q), 0]] /; FreeQ[{a, b, c, d, e}, x] && GtQ[b^2 - 4*a*c, 0]
  1194. Int[Times[Plus[Pattern[d, Blank[]], Times[Optional[Pattern[e, Blank[]]], Power[Pattern[x, Blank[]], 2]]], Power[Plus[Pattern[a, Blank[]], Times[Optional[Pattern[b, Blank[]]], Power[Pattern[x, Blank[]], 2]], Times[Optional[Pattern[c, Blank[]]], Power[Pattern[x, Blank[]], 4]]], Rational[-1, 2]]], Pattern[x, Blank[Symbol]]] := With[{q = Rt[b^2 - 4*a*c, 2]}, Dist[(2*c*d - e*(b - q))/(2*c), Int[1/Sqrt[a + b*x^2 + c*x^4], x], x] + Dist[e/(2*c), Int[(b - q + 2*c*x^2)/Sqrt[a + b*x^2 + c*x^4], x], x] /; NegQ[(b - q)/a] && NeQ[2*c*d - e*(b - q), 0]] /; FreeQ[{a, b, c, d, e}, x] && GtQ[b^2 - 4*a*c, 0]
  1195. Int[Times[Plus[Pattern[d, Blank[]], Times[Optional[Pattern[e, Blank[]]], Power[Pattern[x, Blank[]], 2]]], Power[Plus[Pattern[a, Blank[]], Times[Optional[Pattern[b, Blank[]]], Power[Pattern[x, Blank[]], 2]], Times[Optional[Pattern[c, Blank[]]], Power[Pattern[x, Blank[]], 4]]], Rational[-1, 2]]], Pattern[x, Blank[Symbol]]] := With[{q = Rt[c/a, 4]}, -Simp[(d*x*Sqrt[a + b*x^2 + c*x^4])/(a*(1 + q^2*x^2)), x] + Simp[(d*(1 + q^2*x^2)*Sqrt[(a + b*x^2 + c*x^4)/(a*(1 + q^2*x^2)^2)]*EllipticE[2*ArcTan[q*x], 1/2 - (b*q^2)/(4*c)])/(q*Sqrt[a + b*x^2 + c*x^4]), x] /; EqQ[e + d*q^2, 0]] /; FreeQ[{a, b, c, d, e}, x] && NeQ[b^2 - 4*a*c, 0] && PosQ[c/a]
  1196. Int[Times[Plus[Pattern[d, Blank[]], Times[Optional[Pattern[e, Blank[]]], Power[Pattern[x, Blank[]], 2]]], Power[Plus[Pattern[a, Blank[]], Times[Optional[Pattern[c, Blank[]]], Power[Pattern[x, Blank[]], 4]]], Rational[-1, 2]]], Pattern[x, Blank[Symbol]]] := With[{q = Rt[c/a, 4]}, -Simp[(d*x*Sqrt[a + c*x^4])/(a*(1 + q^2*x^2)), x] + Simp[(d*(1 + q^2*x^2)*Sqrt[(a + c*x^4)/(a*(1 + q^2*x^2)^2)]*EllipticE[2*ArcTan[q*x], 1/2])/(q*Sqrt[a + c*x^4]), x] /; EqQ[e + d*q^2, 0]] /; FreeQ[{a, c, d, e}, x] && PosQ[c/a]
  1197. Int[Times[Plus[Pattern[d, Blank[]], Times[Optional[Pattern[e, Blank[]]], Power[Pattern[x, Blank[]], 2]]], Power[Plus[Pattern[a, Blank[]], Times[Optional[Pattern[b, Blank[]]], Power[Pattern[x, Blank[]], 2]], Times[Optional[Pattern[c, Blank[]]], Power[Pattern[x, Blank[]], 4]]], Rational[-1, 2]]], Pattern[x, Blank[Symbol]]] := With[{q = Rt[c/a, 2]}, Dist[(e + d*q)/q, Int[1/Sqrt[a + b*x^2 + c*x^4], x], x] - Dist[e/q, Int[(1 - q*x^2)/Sqrt[a + b*x^2 + c*x^4], x], x] /; NeQ[e + d*q, 0]] /; FreeQ[{a, b, c, d, e}, x] && NeQ[b^2 - 4*a*c, 0] && PosQ[c/a]
  1198. Int[Times[Plus[Pattern[d, Blank[]], Times[Optional[Pattern[e, Blank[]]], Power[Pattern[x, Blank[]], 2]]], Power[Plus[Pattern[a, Blank[]], Times[Optional[Pattern[c, Blank[]]], Power[Pattern[x, Blank[]], 4]]], Rational[-1, 2]]], Pattern[x, Blank[Symbol]]] := With[{q = Rt[c/a, 2]}, Dist[(e + d*q)/q, Int[1/Sqrt[a + c*x^4], x], x] - Dist[e/q, Int[(1 - q*x^2)/Sqrt[a + c*x^4], x], x] /; NeQ[e + d*q, 0]] /; FreeQ[{a, c, d, e}, x] && PosQ[c/a]
  1199. Int[Times[Plus[Pattern[d, Blank[]], Times[Optional[Pattern[e, Blank[]]], Power[Pattern[x, Blank[]], 2]]], Power[Plus[Pattern[a, Blank[]], Times[Optional[Pattern[c, Blank[]]], Power[Pattern[x, Blank[]], 4]]], Rational[-1, 2]]], Pattern[x, Blank[Symbol]]] := Dist[d/Sqrt[a], Int[Sqrt[1 + (e*x^2)/d]/Sqrt[1 - (e*x^2)/d], x], x] /; FreeQ[{a, c, d, e}, x] && NegQ[c/a] && EqQ[c*d^2 + a*e^2, 0] && GtQ[a, 0]
  1200. Int[Times[Plus[Pattern[d, Blank[]], Times[Optional[Pattern[e, Blank[]]], Power[Pattern[x, Blank[]], 2]]], Power[Plus[Pattern[a, Blank[]], Times[Optional[Pattern[c, Blank[]]], Power[Pattern[x, Blank[]], 4]]], Rational[-1, 2]]], Pattern[x, Blank[Symbol]]] := Dist[Sqrt[1 + (c*x^4)/a]/Sqrt[a + c*x^4], Int[(d + e*x^2)/Sqrt[1 + (c*x^4)/a], x], x] /; FreeQ[{a, c, d, e}, x] && NegQ[c/a] && EqQ[c*d^2 + a*e^2, 0] && !GtQ[a, 0]
  1201. Int[Times[Plus[Pattern[d, Blank[]], Times[Optional[Pattern[e, Blank[]]], Power[Pattern[x, Blank[]], 2]]], Power[Plus[Pattern[a, Blank[]], Times[Optional[Pattern[c, Blank[]]], Power[Pattern[x, Blank[]], 4]]], Rational[-1, 2]]], Pattern[x, Blank[Symbol]]] := With[{q = Rt[-(c/a), 2]}, Dist[(d*q - e)/q, Int[1/Sqrt[a + c*x^4], x], x] + Dist[e/q, Int[(1 + q*x^2)/Sqrt[a + c*x^4], x], x]] /; FreeQ[{a, c, d, e}, x] && NegQ[c/a] && NeQ[c*d^2 + a*e^2, 0]
  1202. Int[Times[Plus[Pattern[d, Blank[]], Times[Optional[Pattern[e, Blank[]]], Power[Pattern[x, Blank[]], 2]]], Power[Plus[Pattern[a, Blank[]], Times[Optional[Pattern[b, Blank[]]], Power[Pattern[x, Blank[]], 2]], Times[Optional[Pattern[c, Blank[]]], Power[Pattern[x, Blank[]], 4]]], Rational[-1, 2]]], Pattern[x, Blank[Symbol]]] := With[{q = Rt[b^2 - 4*a*c, 2]}, Dist[(Sqrt[1 + (2*c*x^2)/(b - q)]*Sqrt[1 + (2*c*x^2)/(b + q)])/Sqrt[a + b*x^2 + c*x^4], Int[(d + e*x^2)/(Sqrt[1 + (2*c*x^2)/(b - q)]*Sqrt[1 + (2*c*x^2)/(b + q)]), x], x]] /; FreeQ[{a, b, c, d, e}, x] && NeQ[b^2 - 4*a*c, 0] && NegQ[c/a]
  1203. Int[Times[Plus[Pattern[d, Blank[]], Times[Optional[Pattern[e, Blank[]]], Power[Pattern[x, Blank[]], 2]]], Power[Plus[Pattern[a, Blank[]], Times[Optional[Pattern[b, Blank[]]], Power[Pattern[x, Blank[]], 2]], Times[Optional[Pattern[c, Blank[]]], Power[Pattern[x, Blank[]], 4]]], Pattern[p, Blank[]]]], Pattern[x, Blank[Symbol]]] := Int[ExpandIntegrand[(d + e*x^2)*(a + b*x^2 + c*x^4)^p, x], x] /; FreeQ[{a, b, c, d, e}, x] && NeQ[b^2 - 4*a*c, 0] && NeQ[c*d^2 - b*d*e + a*e^2, 0]
  1204. Int[Times[Plus[Pattern[d, Blank[]], Times[Optional[Pattern[e, Blank[]]], Power[Pattern[x, Blank[]], 2]]], Power[Plus[Pattern[a, Blank[]], Times[Optional[Pattern[c, Blank[]]], Power[Pattern[x, Blank[]], 4]]], Pattern[p, Blank[]]]], Pattern[x, Blank[Symbol]]] := Int[ExpandIntegrand[(d + e*x^2)*(a + c*x^4)^p, x], x] /; FreeQ[{a, c, d, e}, x] && NeQ[c*d^2 + a*e^2, 0]
  1205. Int[Times[Power[Plus[Pattern[d, Blank[]], Times[Optional[Pattern[e, Blank[]]], Power[Pattern[x, Blank[]], 2]]], Pattern[q, Blank[]]], Power[Plus[Pattern[a, Blank[]], Times[Optional[Pattern[b, Blank[]]], Power[Pattern[x, Blank[]], 2]], Times[Optional[Pattern[c, Blank[]]], Power[Pattern[x, Blank[]], 4]]], Pattern[p, Blank[]]]], Pattern[x, Blank[Symbol]]] := With[{f = Coeff[PolynomialRemainder[(d + e*x^2)^q, a + b*x^2 + c*x^4, x], x, 0], g = Coeff[PolynomialRemainder[(d + e*x^2)^q, a + b*x^2 + c*x^4, x], x, 2]}, Simp[(x*(a + b*x^2 + c*x^4)^(p + 1)*(a*b*g - f*(b^2 - 2*a*c) - c*(b*f - 2*a*g)*x^2))/(2*a*(p + 1)*(b^2 - 4*a*c)), x] + Dist[1/(2*a*(p + 1)*(b^2 - 4*a*c)), Int[(a + b*x^2 + c*x^4)^(p + 1)*ExpandToSum[2*a*(p + 1)*(b^2 - 4*a*c)*PolynomialQuotient[(d + e*x^2)^q, a + b*x^2 + c*x^4, x] + b^2*f*(2*p + 3) - 2*a*c*f*(4*p + 5) - a*b*g + c*(4*p + 7)*(b*f - 2*a*g)*x^2, x], x], x]] /; FreeQ[{a, b, c, d, e}, x] && NeQ[b^2 - 4*a*c, 0] && NeQ[c*d^2 - b*d*e + a*e^2, 0] && IGtQ[q, 1] && LtQ[p, -1]
  1206. Int[Times[Power[Plus[Pattern[d, Blank[]], Times[Optional[Pattern[e, Blank[]]], Power[Pattern[x, Blank[]], 2]]], Pattern[q, Blank[]]], Power[Plus[Pattern[a, Blank[]], Times[Optional[Pattern[b, Blank[]]], Power[Pattern[x, Blank[]], 2]], Times[Optional[Pattern[c, Blank[]]], Power[Pattern[x, Blank[]], 4]]], Pattern[p, Blank[]]]], Pattern[x, Blank[Symbol]]] := Simp[(e^q*x^(2*q - 3)*(a + b*x^2 + c*x^4)^(p + 1))/(c*(4*p + 2*q + 1)), x] + Dist[1/(c*(4*p + 2*q + 1)), Int[(a + b*x^2 + c*x^4)^p*ExpandToSum[c*(4*p + 2*q + 1)*(d + e*x^2)^q - a*(2*q - 3)*e^q*x^(2*q - 4) - b*(2*p + 2*q - 1)*e^q*x^(2*q - 2) - c*(4*p + 2*q + 1)*e^q*x^(2*q), x], x], x] /; FreeQ[{a, b, c, d, e, p}, x] && NeQ[b^2 - 4*a*c, 0] && NeQ[c*d^2 - b*d*e + a*e^2, 0] && IGtQ[q, 1]
  1207. Int[Times[Power[Plus[Pattern[d, Blank[]], Times[Optional[Pattern[e, Blank[]]], Power[Pattern[x, Blank[]], 2]]], Pattern[q, Blank[]]], Power[Plus[Pattern[a, Blank[]], Times[Optional[Pattern[c, Blank[]]], Power[Pattern[x, Blank[]], 4]]], Pattern[p, Blank[]]]], Pattern[x, Blank[Symbol]]] := Simp[(e^q*x^(2*q - 3)*(a + c*x^4)^(p + 1))/(c*(4*p + 2*q + 1)), x] + Dist[1/(c*(4*p + 2*q + 1)), Int[(a + c*x^4)^p*ExpandToSum[c*(4*p + 2*q + 1)*(d + e*x^2)^q - a*(2*q - 3)*e^q*x^(2*q - 4) - c*(4*p + 2*q + 1)*e^q*x^(2*q), x], x], x] /; FreeQ[{a, c, d, e, p}, x] && NeQ[c*d^2 + a*e^2, 0] && IGtQ[q, 1]
  1208. Int[Times[Power[Plus[Pattern[d, Blank[]], Times[Optional[Pattern[e, Blank[]]], Power[Pattern[x, Blank[]], 2]]], -1], Power[Plus[Pattern[a, Blank[]], Times[Optional[Pattern[b, Blank[]]], Power[Pattern[x, Blank[]], 2]], Times[Optional[Pattern[c, Blank[]]], Power[Pattern[x, Blank[]], 4]]], Pattern[p, Blank[]]]], Pattern[x, Blank[Symbol]]] := -Dist[(e^2)^(-1), Int[(c*d - b*e - c*e*x^2)*(a + b*x^2 + c*x^4)^(p - 1), x], x] + Dist[(c*d^2 - b*d*e + a*e^2)/e^2, Int[(a + b*x^2 + c*x^4)^(p - 1)/(d + e*x^2), x], x] /; FreeQ[{a, b, c, d, e}, x] && NeQ[b^2 - 4*a*c, 0] && NeQ[c*d^2 - b*d*e + a*e^2, 0] && IGtQ[p + 1/2, 0]
  1209. Int[Times[Power[Plus[Pattern[d, Blank[]], Times[Optional[Pattern[e, Blank[]]], Power[Pattern[x, Blank[]], 2]]], -1], Power[Plus[Pattern[a, Blank[]], Times[Optional[Pattern[c, Blank[]]], Power[Pattern[x, Blank[]], 4]]], Pattern[p, Blank[]]]], Pattern[x, Blank[Symbol]]] := -Dist[(e^2)^(-1), Int[(c*d - c*e*x^2)*(a + c*x^4)^(p - 1), x], x] + Dist[(c*d^2 + a*e^2)/e^2, Int[(a + c*x^4)^(p - 1)/(d + e*x^2), x], x] /; FreeQ[{a, c, d, e}, x] && NeQ[c*d^2 + a*e^2, 0] && IGtQ[p + 1/2, 0]
  1210. Int[Times[Power[Plus[Pattern[d, Blank[]], Times[Optional[Pattern[e, Blank[]]], Power[Pattern[x, Blank[]], 2]]], -1], Power[Plus[Pattern[a, Blank[]], Times[Optional[Pattern[b, Blank[]]], Power[Pattern[x, Blank[]], 2]], Times[Optional[Pattern[c, Blank[]]], Power[Pattern[x, Blank[]], 4]]], Rational[-1, 2]]], Pattern[x, Blank[Symbol]]] := Dist[1/(2*d), Int[1/Sqrt[a + b*x^2 + c*x^4], x], x] + Dist[1/(2*d), Int[(d - e*x^2)/((d + e*x^2)*Sqrt[a + b*x^2 + c*x^4]), x], x] /; FreeQ[{a, b, c, d, e}, x] && NeQ[b^2 - 4*a*c, 0] && NeQ[c*d^2 - b*d*e + a*e^2, 0] && EqQ[c*d^2 - a*e^2, 0]
  1211. Int[Times[Power[Plus[Pattern[d, Blank[]], Times[Optional[Pattern[e, Blank[]]], Power[Pattern[x, Blank[]], 2]]], -1], Power[Plus[Pattern[a, Blank[]], Times[Optional[Pattern[c, Blank[]]], Power[Pattern[x, Blank[]], 4]]], Rational[-1, 2]]], Pattern[x, Blank[Symbol]]] := Dist[1/(2*d), Int[1/Sqrt[a + c*x^4], x], x] + Dist[1/(2*d), Int[(d - e*x^2)/((d + e*x^2)*Sqrt[a + c*x^4]), x], x] /; FreeQ[{a, c, d, e}, x] && NeQ[c*d^2 + a*e^2, 0] && EqQ[c*d^2 - a*e^2, 0]
  1212. Int[Times[Power[Plus[Pattern[d, Blank[]], Times[Optional[Pattern[e, Blank[]]], Power[Pattern[x, Blank[]], 2]]], -1], Power[Plus[Pattern[a, Blank[]], Times[Optional[Pattern[b, Blank[]]], Power[Pattern[x, Blank[]], 2]], Times[Optional[Pattern[c, Blank[]]], Power[Pattern[x, Blank[]], 4]]], Rational[-1, 2]]], Pattern[x, Blank[Symbol]]] := With[{q = Rt[b^2 - 4*a*c, 2]}, Dist[2*Sqrt[-c], Int[1/((d + e*x^2)*Sqrt[b + q + 2*c*x^2]*Sqrt[-b + q - 2*c*x^2]), x], x]] /; FreeQ[{a, b, c, d, e}, x] && GtQ[b^2 - 4*a*c, 0] && LtQ[c, 0]
  1213. Int[Times[Power[Plus[Pattern[d, Blank[]], Times[Optional[Pattern[e, Blank[]]], Power[Pattern[x, Blank[]], 2]]], -1], Power[Plus[Pattern[a, Blank[]], Times[Optional[Pattern[c, Blank[]]], Power[Pattern[x, Blank[]], 4]]], Rational[-1, 2]]], Pattern[x, Blank[Symbol]]] := With[{q = Rt[-(a*c), 2]}, Dist[Sqrt[-c], Int[1/((d + e*x^2)*Sqrt[q + c*x^2]*Sqrt[q - c*x^2]), x], x]] /; FreeQ[{a, c, d, e}, x] && GtQ[a, 0] && LtQ[c, 0]
  1214. Int[Times[Power[Plus[Pattern[d, Blank[]], Times[Optional[Pattern[e, Blank[]]], Power[Pattern[x, Blank[]], 2]]], -1], Power[Plus[Pattern[a, Blank[]], Times[Optional[Pattern[b, Blank[]]], Power[Pattern[x, Blank[]], 2]], Times[Optional[Pattern[c, Blank[]]], Power[Pattern[x, Blank[]], 4]]], Rational[-1, 2]]], Pattern[x, Blank[Symbol]]] := With[{q = Rt[b^2 - 4*a*c, 2]}, Dist[(2*c)/(2*c*d - e*(b - q)), Int[1/Sqrt[a + b*x^2 + c*x^4], x], x] - Dist[e/(2*c*d - e*(b - q)), Int[(b - q + 2*c*x^2)/((d + e*x^2)*Sqrt[a + b*x^2 + c*x^4]), x], x]] /; FreeQ[{a, b, c, d, e}, x] && GtQ[b^2 - 4*a*c, 0] && !LtQ[c, 0]
  1215. Int[Times[Power[Plus[Pattern[d, Blank[]], Times[Optional[Pattern[e, Blank[]]], Power[Pattern[x, Blank[]], 2]]], -1], Power[Plus[Pattern[a, Blank[]], Times[Optional[Pattern[c, Blank[]]], Power[Pattern[x, Blank[]], 4]]], Rational[-1, 2]]], Pattern[x, Blank[Symbol]]] := With[{q = Rt[-(a*c), 2]}, Dist[c/(c*d + e*q), Int[1/Sqrt[a + c*x^4], x], x] + Dist[e/(c*d + e*q), Int[(q - c*x^2)/((d + e*x^2)*Sqrt[a + c*x^4]), x], x]] /; FreeQ[{a, c, d, e}, x] && GtQ[-(a*c), 0] && !LtQ[c, 0]
  1216. Int[Times[Power[Plus[Pattern[d, Blank[]], Times[Optional[Pattern[e, Blank[]]], Power[Pattern[x, Blank[]], 2]]], -1], Power[Plus[Pattern[a, Blank[]], Times[Optional[Pattern[b, Blank[]]], Power[Pattern[x, Blank[]], 2]], Times[Optional[Pattern[c, Blank[]]], Power[Pattern[x, Blank[]], 4]]], Rational[-1, 2]]], Pattern[x, Blank[Symbol]]] := With[{q = Rt[c/a, 2]}, Dist[(c*d + a*e*q)/(c*d^2 - a*e^2), Int[1/Sqrt[a + b*x^2 + c*x^4], x], x] - Dist[(a*e*(e + d*q))/(c*d^2 - a*e^2), Int[(1 + q*x^2)/((d + e*x^2)*Sqrt[a + b*x^2 + c*x^4]), x], x]] /; FreeQ[{a, b, c, d, e}, x] && NeQ[b^2 - 4*a*c, 0] && NeQ[c*d^2 - b*d*e + a*e^2, 0] && NeQ[c*d^2 - a*e^2, 0] && PosQ[c/a]
  1217. Int[Times[Power[Plus[Pattern[d, Blank[]], Times[Optional[Pattern[e, Blank[]]], Power[Pattern[x, Blank[]], 2]]], -1], Power[Plus[Pattern[a, Blank[]], Times[Optional[Pattern[c, Blank[]]], Power[Pattern[x, Blank[]], 4]]], Rational[-1, 2]]], Pattern[x, Blank[Symbol]]] := With[{q = Rt[c/a, 2]}, Dist[(c*d + a*e*q)/(c*d^2 - a*e^2), Int[1/Sqrt[a + c*x^4], x], x] - Dist[(a*e*(e + d*q))/(c*d^2 - a*e^2), Int[(1 + q*x^2)/((d + e*x^2)*Sqrt[a + c*x^4]), x], x]] /; FreeQ[{a, c, d, e}, x] && NeQ[c*d^2 + a*e^2, 0] && NeQ[c*d^2 - a*e^2, 0] && PosQ[c/a]
  1218. Int[Times[Power[Plus[Pattern[d, Blank[]], Times[Optional[Pattern[e, Blank[]]], Power[Pattern[x, Blank[]], 2]]], -1], Power[Plus[Pattern[a, Blank[]], Times[Optional[Pattern[c, Blank[]]], Power[Pattern[x, Blank[]], 4]]], Rational[-1, 2]]], Pattern[x, Blank[Symbol]]] := With[{q = Rt[-(c/a), 4]}, Simp[(1*EllipticPi[-(e/(d*q^2)), ArcSin[q*x], -1])/(d*Sqrt[a]*q), x]] /; FreeQ[{a, c, d, e}, x] && NegQ[c/a] && GtQ[a, 0]
  1219. Int[Times[Power[Plus[Pattern[d, Blank[]], Times[Optional[Pattern[e, Blank[]]], Power[Pattern[x, Blank[]], 2]]], -1], Power[Plus[Pattern[a, Blank[]], Times[Optional[Pattern[c, Blank[]]], Power[Pattern[x, Blank[]], 4]]], Rational[-1, 2]]], Pattern[x, Blank[Symbol]]] := Dist[Sqrt[1 + (c*x^4)/a]/Sqrt[a + c*x^4], Int[1/((d + e*x^2)*Sqrt[1 + (c*x^4)/a]), x], x] /; FreeQ[{a, c, d, e}, x] && NegQ[c/a] && !GtQ[a, 0]
  1220. Int[Times[Power[Plus[Pattern[d, Blank[]], Times[Optional[Pattern[e, Blank[]]], Power[Pattern[x, Blank[]], 2]]], -1], Power[Plus[Pattern[a, Blank[]], Times[Optional[Pattern[b, Blank[]]], Power[Pattern[x, Blank[]], 2]], Times[Optional[Pattern[c, Blank[]]], Power[Pattern[x, Blank[]], 4]]], Rational[-1, 2]]], Pattern[x, Blank[Symbol]]] := With[{q = Rt[b^2 - 4*a*c, 2]}, Dist[(Sqrt[1 + (2*c*x^2)/(b - q)]*Sqrt[1 + (2*c*x^2)/(b + q)])/Sqrt[a + b*x^2 + c*x^4], Int[1/((d + e*x^2)*Sqrt[1 + (2*c*x^2)/(b - q)]*Sqrt[1 + (2*c*x^2)/(b + q)]), x], x]] /; FreeQ[{a, b, c, d, e}, x] && NeQ[b^2 - 4*a*c, 0] && NegQ[c/a]
  1221. Int[Times[Power[Plus[Pattern[d, Blank[]], Times[Optional[Pattern[e, Blank[]]], Power[Pattern[x, Blank[]], 2]]], -1], Power[Plus[Pattern[a, Blank[]], Times[Optional[Pattern[b, Blank[]]], Power[Pattern[x, Blank[]], 2]], Times[Optional[Pattern[c, Blank[]]], Power[Pattern[x, Blank[]], 4]]], Pattern[p, Blank[]]]], Pattern[x, Blank[Symbol]]] := Dist[1/(c*d^2 - b*d*e + a*e^2), Int[(c*d - b*e - c*e*x^2)*(a + b*x^2 + c*x^4)^p, x], x] + Dist[e^2/(c*d^2 - b*d*e + a*e^2), Int[(a + b*x^2 + c*x^4)^(p + 1)/(d + e*x^2), x], x] /; FreeQ[{a, b, c, d, e}, x] && NeQ[b^2 - 4*a*c, 0] && NeQ[c*d^2 - b*d*e + a*e^2, 0] && ILtQ[p + 1/2, 0]
  1222. Int[Times[Power[Plus[Pattern[d, Blank[]], Times[Optional[Pattern[e, Blank[]]], Power[Pattern[x, Blank[]], 2]]], -1], Power[Plus[Pattern[a, Blank[]], Times[Optional[Pattern[c, Blank[]]], Power[Pattern[x, Blank[]], 4]]], Pattern[p, Blank[]]]], Pattern[x, Blank[Symbol]]] := Dist[1/(c*d^2 + a*e^2), Int[(c*d - c*e*x^2)*(a + c*x^4)^p, x], x] + Dist[e^2/(c*d^2 + a*e^2), Int[(a + c*x^4)^(p + 1)/(d + e*x^2), x], x] /; FreeQ[{a, c, d, e}, x] && NeQ[c*d^2 + a*e^2, 0] && ILtQ[p + 1/2, 0]
  1223. Int[Times[Power[Plus[Pattern[d, Blank[]], Times[Optional[Pattern[e, Blank[]]], Power[Pattern[x, Blank[]], 2]]], Pattern[q, Blank[]]], Power[Plus[Pattern[a, Blank[]], Times[Optional[Pattern[b, Blank[]]], Power[Pattern[x, Blank[]], 2]], Times[Optional[Pattern[c, Blank[]]], Power[Pattern[x, Blank[]], 4]]], Rational[-1, 2]]], Pattern[x, Blank[Symbol]]] := -Simp[(e^2*x*(d + e*x^2)^(q + 1)*Sqrt[a + b*x^2 + c*x^4])/(2*d*(q + 1)*(c*d^2 - b*d*e + a*e^2)), x] + Dist[1/(2*d*(q + 1)*(c*d^2 - b*d*e + a*e^2)), Int[((d + e*x^2)^(q + 1)*Simp[a*e^2*(2*q + 3) + 2*d*(c*d - b*e)*(q + 1) - 2*e*(c*d*(q + 1) - b*e*(q + 2))*x^2 + c*e^2*(2*q + 5)*x^4, x])/Sqrt[a + b*x^2 + c*x^4], x], x] /; FreeQ[{a, b, c, d, e}, x] && NeQ[b^2 - 4*a*c, 0] && ILtQ[q, -1]
  1224. Int[Times[Power[Plus[Pattern[d, Blank[]], Times[Optional[Pattern[e, Blank[]]], Power[Pattern[x, Blank[]], 2]]], Pattern[q, Blank[]]], Power[Plus[Pattern[a, Blank[]], Times[Optional[Pattern[c, Blank[]]], Power[Pattern[x, Blank[]], 4]]], Rational[-1, 2]]], Pattern[x, Blank[Symbol]]] := -Simp[(e^2*x*(d + e*x^2)^(q + 1)*Sqrt[a + c*x^4])/(2*d*(q + 1)*(c*d^2 + a*e^2)), x] + Dist[1/(2*d*(q + 1)*(c*d^2 + a*e^2)), Int[((d + e*x^2)^(q + 1)*Simp[a*e^2*(2*q + 3) + 2*c*d^2*(q + 1) - 2*e*c*d*(q + 1)*x^2 + c*e^2*(2*q + 5)*x^4, x])/Sqrt[a + c*x^4], x], x] /; FreeQ[{a, c, d, e}, x] && ILtQ[q, -1]
  1225. Int[Times[Power[Plus[Pattern[d, Blank[]], Times[Optional[Pattern[e, Blank[]]], Power[Pattern[x, Blank[]], 2]]], -2], Power[Plus[Pattern[a, Blank[]], Times[Optional[Pattern[b, Blank[]]], Power[Pattern[x, Blank[]], 2]], Times[Optional[Pattern[c, Blank[]]], Power[Pattern[x, Blank[]], 4]]], Rational[1, 2]]], Pattern[x, Blank[Symbol]]] := With[{q = Rt[e/d, 2]}, Simp[(c*(d + e*x^2)*Sqrt[(e^2*(a + b*x^2 + c*x^4))/(c*(d + e*x^2)^2)]*EllipticE[2*ArcTan[q*x], (2*c*d - b*e)/(4*c*d)])/(2*d*e^2*q*Sqrt[a + b*x^2 + c*x^4]), x]] /; FreeQ[{a, b, c, d, e}, x] && NeQ[b^2 - 4*a*c, 0] && NeQ[c*d^2 - b*d*e + a*e^2, 0] && EqQ[c*d^2 - a*e^2, 0] && PosQ[e/d]
  1226. Int[Times[Power[Plus[Pattern[d, Blank[]], Times[Optional[Pattern[e, Blank[]]], Power[Pattern[x, Blank[]], 2]]], -2], Power[Plus[Pattern[a, Blank[]], Times[Optional[Pattern[b, Blank[]]], Power[Pattern[x, Blank[]], 2]], Times[Optional[Pattern[c, Blank[]]], Power[Pattern[x, Blank[]], 4]]], Rational[1, 2]]], Pattern[x, Blank[Symbol]]] := Simp[(x*Sqrt[a + b*x^2 + c*x^4])/(2*d*(d + e*x^2)), x] + (Dist[c/(2*d*e^2), Int[(d - e*x^2)/Sqrt[a + b*x^2 + c*x^4], x], x] - Dist[(c*d^2 - a*e^2)/(2*d*e^2), Int[1/((d + e*x^2)*Sqrt[a + b*x^2 + c*x^4]), x], x]) /; FreeQ[{a, b, c, d, e}, x] && NeQ[b^2 - 4*a*c, 0] && NeQ[c*d^2 - b*d*e + a*e^2, 0]
  1227. Int[Times[Power[Plus[Pattern[d, Blank[]], Times[Optional[Pattern[e, Blank[]]], Power[Pattern[x, Blank[]], 2]]], -2], Power[Plus[Pattern[a, Blank[]], Times[Optional[Pattern[c, Blank[]]], Power[Pattern[x, Blank[]], 4]]], Rational[1, 2]]], Pattern[x, Blank[Symbol]]] := Simp[(x*Sqrt[a + c*x^4])/(2*d*(d + e*x^2)), x] + (Dist[c/(2*d*e^2), Int[(d - e*x^2)/Sqrt[a + c*x^4], x], x] - Dist[(c*d^2 - a*e^2)/(2*d*e^2), Int[1/((d + e*x^2)*Sqrt[a + c*x^4]), x], x]) /; FreeQ[{a, c, d, e}, x] && NeQ[c*d^2 + a*e^2, 0]
  1228. Int[Times[Power[Plus[Pattern[d, Blank[]], Times[Optional[Pattern[e, Blank[]]], Power[Pattern[x, Blank[]], 2]]], Pattern[q, Blank[]]], Power[Plus[Pattern[a, Blank[]], Times[Optional[Pattern[b, Blank[]]], Power[Pattern[x, Blank[]], 2]], Times[Optional[Pattern[c, Blank[]]], Power[Pattern[x, Blank[]], 4]]], Pattern[p, Blank[]]]], Pattern[x, Blank[Symbol]]] := Module[{aa, bb, cc}, Int[ExpandIntegrand[1/Sqrt[aa + bb*x^2 + cc*x^4], (d + e*x^2)^q*(aa + bb*x^2 + cc*x^4)^(p + 1/2), x] /. {aa -> a, bb -> b, cc -> c}, x]] /; FreeQ[{a, b, c, d, e}, x] && NeQ[b^2 - 4*a*c, 0] && NeQ[c*d^2 - b*d*e + a*e^2, 0] && ILtQ[q, 0] && IntegerQ[p + 1/2]
  1229. Int[Times[Power[Plus[Pattern[d, Blank[]], Times[Optional[Pattern[e, Blank[]]], Power[Pattern[x, Blank[]], 2]]], Pattern[q, Blank[]]], Power[Plus[Pattern[a, Blank[]], Times[Optional[Pattern[c, Blank[]]], Power[Pattern[x, Blank[]], 4]]], Pattern[p, Blank[]]]], Pattern[x, Blank[Symbol]]] := Module[{aa, cc}, Int[ExpandIntegrand[1/Sqrt[aa + cc*x^4], (d + e*x^2)^q*(aa + cc*x^4)^(p + 1/2), x] /. {aa -> a, cc -> c}, x]] /; FreeQ[{a, c, d, e}, x] && NeQ[c*d^2 + a*e^2, 0] && ILtQ[q, 0] && IntegerQ[p + 1/2]
  1230. Int[Times[Power[Plus[Pattern[d, Blank[]], Times[Optional[Pattern[e, Blank[]]], Power[Pattern[x, Blank[]], 2]]], Rational[-1, 2]], Power[Plus[Pattern[a, Blank[]], Times[Optional[Pattern[b, Blank[]]], Power[Pattern[x, Blank[]], 2]], Times[Optional[Pattern[c, Blank[]]], Power[Pattern[x, Blank[]], 4]]], Rational[-1, 2]]], Pattern[x, Blank[Symbol]]] := Simp[(1*EllipticF[2*ArcSin[Rt[-(e/d), 2]*x], (b*d)/(4*a*e)])/(2*Sqrt[a]*Sqrt[d]*Rt[-(e/d), 2]), x] /; FreeQ[{a, b, c, d, e}, x] && EqQ[c*d - b*e, 0] && GtQ[a, 0] && GtQ[d, 0]
  1231. Int[Times[Power[Plus[Pattern[d, Blank[]], Times[Optional[Pattern[e, Blank[]]], Power[Pattern[x, Blank[]], 2]]], Rational[-1, 2]], Power[Plus[Pattern[a, Blank[]], Times[Optional[Pattern[b, Blank[]]], Power[Pattern[x, Blank[]], 2]], Times[Optional[Pattern[c, Blank[]]], Power[Pattern[x, Blank[]], 4]]], Rational[-1, 2]]], Pattern[x, Blank[Symbol]]] := Dist[(Sqrt[(d + e*x^2)/d]*Sqrt[(a + b*x^2 + c*x^4)/a])/(Sqrt[d + e*x^2]*Sqrt[a + b*x^2 + c*x^4]), Int[1/(Sqrt[1 + (e*x^2)/d]*Sqrt[1 + (b*x^2)/a + (c*x^4)/a]), x], x] /; FreeQ[{a, b, c, d, e}, x] && EqQ[c*d - b*e, 0] && !(GtQ[a, 0] && GtQ[d, 0])
  1232. Int[Times[Power[Plus[Pattern[d, Blank[]], Times[Optional[Pattern[e, Blank[]]], Power[Pattern[x, Blank[]], 2]]], Rational[-1, 2]], Power[Plus[Pattern[a, Blank[]], Times[Optional[Pattern[b, Blank[]]], Power[Pattern[x, Blank[]], 2]], Times[Optional[Pattern[c, Blank[]]], Power[Pattern[x, Blank[]], 4]]], Rational[-1, 2]]], Pattern[x, Blank[Symbol]]] := Dist[(x^3*Sqrt[e + d/x^2]*Sqrt[c + b/x^2 + a/x^4])/(Sqrt[d + e*x^2]*Sqrt[a + b*x^2 + c*x^4]), Int[1/(x^3*Sqrt[e + d/x^2]*Sqrt[c + b/x^2 + a/x^4]), x], x] /; FreeQ[{a, b, c, d, e}, x] && NeQ[b^2 - 4*a*c, 0] && NeQ[c*d^2 - b*d*e + a*e^2, 0]
  1233. Int[Times[Power[Plus[Pattern[d, Blank[]], Times[Optional[Pattern[e, Blank[]]], Power[Pattern[x, Blank[]], 2]]], Rational[-1, 2]], Power[Plus[Pattern[a, Blank[]], Times[Optional[Pattern[c, Blank[]]], Power[Pattern[x, Blank[]], 4]]], Rational[-1, 2]]], Pattern[x, Blank[Symbol]]] := Dist[(x^3*Sqrt[e + d/x^2]*Sqrt[c + a/x^4])/(Sqrt[d + e*x^2]*Sqrt[a + c*x^4]), Int[1/(x^3*Sqrt[e + d/x^2]*Sqrt[c + a/x^4]), x], x] /; FreeQ[{a, c, d, e}, x] && NeQ[c*d^2 + a*e^2, 0]
  1234. Int[Times[Power[Plus[Pattern[d, Blank[]], Times[Optional[Pattern[e, Blank[]]], Power[Pattern[x, Blank[]], 2]]], Rational[-1, 2]], Power[Plus[Pattern[a, Blank[]], Times[Optional[Pattern[b, Blank[]]], Power[Pattern[x, Blank[]], 2]], Times[Optional[Pattern[c, Blank[]]], Power[Pattern[x, Blank[]], 4]]], Rational[1, 2]]], Pattern[x, Blank[Symbol]]] := Simp[(Sqrt[a]*EllipticE[2*ArcSin[Rt[-(e/d), 2]*x], (b*d)/(4*a*e)])/(2*Sqrt[d]*Rt[-(e/d), 2]), x] /; FreeQ[{a, b, c, d, e}, x] && EqQ[c*d - b*e, 0] && GtQ[a, 0] && GtQ[d, 0]
  1235. Int[Times[Power[Plus[Pattern[d, Blank[]], Times[Optional[Pattern[e, Blank[]]], Power[Pattern[x, Blank[]], 2]]], Rational[-1, 2]], Power[Plus[Pattern[a, Blank[]], Times[Optional[Pattern[b, Blank[]]], Power[Pattern[x, Blank[]], 2]], Times[Optional[Pattern[c, Blank[]]], Power[Pattern[x, Blank[]], 4]]], Rational[1, 2]]], Pattern[x, Blank[Symbol]]] := Dist[(Sqrt[a + b*x^2 + c*x^4]*Sqrt[(d + e*x^2)/d])/(Sqrt[d + e*x^2]*Sqrt[(a + b*x^2 + c*x^4)/a]), Int[Sqrt[1 + (b*x^2)/a + (c*x^4)/a]/Sqrt[1 + (e*x^2)/d], x], x] /; FreeQ[{a, b, c, d, e}, x] && EqQ[c*d - b*e, 0] && !(GtQ[a, 0] && GtQ[d, 0])
  1236. Int[Times[Power[Plus[Pattern[d, Blank[]], Times[Optional[Pattern[e, Blank[]]], Power[Pattern[x, Blank[]], 2]]], Rational[-1, 2]], Power[Plus[Pattern[a, Blank[]], Times[Optional[Pattern[b, Blank[]]], Power[Pattern[x, Blank[]], 2]], Times[Optional[Pattern[c, Blank[]]], Power[Pattern[x, Blank[]], 4]]], Rational[1, 2]]], Pattern[x, Blank[Symbol]]] := Dist[(Sqrt[e + d/x^2]*Sqrt[a + b*x^2 + c*x^4])/(x*Sqrt[d + e*x^2]*Sqrt[c + b/x^2 + a/x^4]), Int[(x*Sqrt[c + b/x^2 + a/x^4])/Sqrt[e + d/x^2], x], x] /; FreeQ[{a, b, c, d, e}, x] && NeQ[b^2 - 4*a*c, 0] && NeQ[c*d^2 - b*d*e + a*e^2, 0]
  1237. Int[Times[Power[Plus[Pattern[d, Blank[]], Times[Optional[Pattern[e, Blank[]]], Power[Pattern[x, Blank[]], 2]]], Rational[-1, 2]], Power[Plus[Pattern[a, Blank[]], Times[Optional[Pattern[c, Blank[]]], Power[Pattern[x, Blank[]], 4]]], Rational[1, 2]]], Pattern[x, Blank[Symbol]]] := Dist[(Sqrt[e + d/x^2]*Sqrt[a + c*x^4])/(x*Sqrt[d + e*x^2]*Sqrt[c + a/x^4]), Int[(x*Sqrt[c + a/x^4])/Sqrt[e + d/x^2], x], x] /; FreeQ[{a, c, d, e}, x] && NeQ[c*d^2 + a*e^2, 0]
  1238. Int[Times[Power[Plus[Pattern[d, Blank[]], Times[Optional[Pattern[e, Blank[]]], Power[Pattern[x, Blank[]], 2]]], Pattern[q, Blank[]]], Power[Plus[Pattern[a, Blank[]], Times[Optional[Pattern[b, Blank[]]], Power[Pattern[x, Blank[]], 2]], Times[Optional[Pattern[c, Blank[]]], Power[Pattern[x, Blank[]], 4]]], Pattern[p, Blank[]]]], Pattern[x, Blank[Symbol]]] := Int[ExpandIntegrand[(d + e*x^2)^q*(a + b*x^2 + c*x^4)^p, x], x] /; FreeQ[{a, b, c, d, e, p, q}, x] && NeQ[b^2 - 4*a*c, 0] && ((IntegerQ[p] && IntegerQ[q]) || IGtQ[p, 0] || IGtQ[q, 0])
  1239. Int[Times[Power[Plus[Pattern[d, Blank[]], Times[Optional[Pattern[e, Blank[]]], Power[Pattern[x, Blank[]], 2]]], Pattern[q, Blank[]]], Power[Plus[Pattern[a, Blank[]], Times[Optional[Pattern[c, Blank[]]], Power[Pattern[x, Blank[]], 4]]], Pattern[p, Blank[]]]], Pattern[x, Blank[Symbol]]] := Int[ExpandIntegrand[(d + e*x^2)^q*(a + c*x^4)^p, x], x] /; FreeQ[{a, c, d, e, p, q}, x] && ((IntegerQ[p] && IntegerQ[q]) || IGtQ[p, 0])
  1240. Int[Times[Power[Plus[Pattern[d, Blank[]], Times[Optional[Pattern[e, Blank[]]], Power[Pattern[x, Blank[]], 2]]], Pattern[q, Blank[]]], Power[Plus[Pattern[a, Blank[]], Times[Optional[Pattern[c, Blank[]]], Power[Pattern[x, Blank[]], 4]]], Pattern[p, Blank[]]]], Pattern[x, Blank[Symbol]]] := Int[ExpandIntegrand[(a + c*x^4)^p, (d/(d^2 - e^2*x^4) - (e*x^2)/(d^2 - e^2*x^4))^(-q), x], x] /; FreeQ[{a, c, d, e, p}, x] && NeQ[c*d^2 + a*e^2, 0] && !IntegerQ[p] && ILtQ[q, 0]
  1241. Int[Times[Power[Plus[Pattern[d, Blank[]], Times[Optional[Pattern[e, Blank[]]], Power[Pattern[x, Blank[]], 2]]], Optional[Pattern[q, Blank[]]]], Power[Plus[Pattern[a, Blank[]], Times[Optional[Pattern[b, Blank[]]], Power[Pattern[x, Blank[]], 2]], Times[Optional[Pattern[c, Blank[]]], Power[Pattern[x, Blank[]], 4]]], Optional[Pattern[p, Blank[]]]]], Pattern[x, Blank[Symbol]]] := Unintegrable[(d + e*x^2)^q*(a + b*x^2 + c*x^4)^p, x] /; FreeQ[{a, b, c, d, e, p, q}, x]
  1242. Int[Times[Power[Plus[Pattern[d, Blank[]], Times[Optional[Pattern[e, Blank[]]], Power[Pattern[x, Blank[]], 2]]], Optional[Pattern[q, Blank[]]]], Power[Plus[Pattern[a, Blank[]], Times[Optional[Pattern[c, Blank[]]], Power[Pattern[x, Blank[]], 4]]], Optional[Pattern[p, Blank[]]]]], Pattern[x, Blank[Symbol]]] := Unintegrable[(d + e*x^2)^q*(a + c*x^4)^p, x] /; FreeQ[{a, c, d, e, p, q}, x]
  1243. Int[Times[Power[Pattern[x, Blank[]], Optional[Pattern[m, Blank[]]]], Power[Times[Optional[Pattern[e, Blank[]]], Power[Pattern[x, Blank[]], 2]], Pattern[q, Blank[]]], Power[Plus[Pattern[a, Blank[]], Times[Optional[Pattern[b, Blank[]]], Power[Pattern[x, Blank[]], 2]], Times[Optional[Pattern[c, Blank[]]], Power[Pattern[x, Blank[]], 4]]], Optional[Pattern[p, Blank[]]]]], Pattern[x, Blank[Symbol]]] := Dist[1/(2*e^((m - 1)/2)), Subst[Int[(e*x)^(q + (m - 1)/2)*(a + b*x + c*x^2)^p, x], x, x^2], x] /; FreeQ[{a, b, c, e, p, q}, x] && !IntegerQ[q] && IntegerQ[(m - 1)/2]
  1244. Int[Times[Power[Pattern[x, Blank[]], Optional[Pattern[m, Blank[]]]], Power[Times[Optional[Pattern[e, Blank[]]], Power[Pattern[x, Blank[]], 2]], Pattern[q, Blank[]]], Power[Plus[Pattern[a, Blank[]], Times[Optional[Pattern[c, Blank[]]], Power[Pattern[x, Blank[]], 4]]], Optional[Pattern[p, Blank[]]]]], Pattern[x, Blank[Symbol]]] := Dist[1/(2*e^((m - 1)/2)), Subst[Int[(e*x)^(q + (m - 1)/2)*(a + c*x^2)^p, x], x, x^2], x] /; FreeQ[{a, c, e, p, q}, x] && !IntegerQ[q] && IntegerQ[(m - 1)/2]
  1245. Int[Times[Power[Times[Optional[Pattern[f, Blank[]]], Pattern[x, Blank[]]], Optional[Pattern[m, Blank[]]]], Power[Times[Optional[Pattern[e, Blank[]]], Power[Pattern[x, Blank[]], 2]], Pattern[q, Blank[]]], Power[Plus[Pattern[a, Blank[]], Times[Optional[Pattern[b, Blank[]]], Power[Pattern[x, Blank[]], 2]], Times[Optional[Pattern[c, Blank[]]], Power[Pattern[x, Blank[]], 4]]], Optional[Pattern[p, Blank[]]]]], Pattern[x, Blank[Symbol]]] := Dist[(e^IntPart[q]*(e*x^2)^FracPart[q])/(f^(2*IntPart[q])*(f*x)^(2*FracPart[q])), Int[(f*x)^(m + 2*q)*(a + b*x^2 + c*x^4)^p, x], x] /; FreeQ[{a, b, c, e, f, m, p, q}, x] && !IntegerQ[q]
  1246. Int[Times[Power[Times[Optional[Pattern[f, Blank[]]], Pattern[x, Blank[]]], Optional[Pattern[m, Blank[]]]], Power[Times[Optional[Pattern[e, Blank[]]], Power[Pattern[x, Blank[]], 2]], Pattern[q, Blank[]]], Power[Plus[Pattern[a, Blank[]], Times[Optional[Pattern[c, Blank[]]], Power[Pattern[x, Blank[]], 4]]], Optional[Pattern[p, Blank[]]]]], Pattern[x, Blank[Symbol]]] := Dist[(e^IntPart[q]*(e*x^2)^FracPart[q])/(f^(2*IntPart[q])*(f*x)^(2*FracPart[q])), Int[(f*x)^(m + 2*q)*(a + c*x^4)^p, x], x] /; FreeQ[{a, c, e, f, m, p, q}, x] && !IntegerQ[q]
  1247. Int[Times[Pattern[x, Blank[]], Power[Plus[Pattern[d, Blank[]], Times[Optional[Pattern[e, Blank[]]], Power[Pattern[x, Blank[]], 2]]], Optional[Pattern[q, Blank[]]]], Power[Plus[Pattern[a, Blank[]], Times[Optional[Pattern[b, Blank[]]], Power[Pattern[x, Blank[]], 2]], Times[Optional[Pattern[c, Blank[]]], Power[Pattern[x, Blank[]], 4]]], Optional[Pattern[p, Blank[]]]]], Pattern[x, Blank[Symbol]]] := Dist[1/2, Subst[Int[(d + e*x)^q*(a + b*x + c*x^2)^p, x], x, x^2], x] /; FreeQ[{a, b, c, d, e, p, q}, x]
  1248. Int[Times[Pattern[x, Blank[]], Power[Plus[Pattern[d, Blank[]], Times[Optional[Pattern[e, Blank[]]], Power[Pattern[x, Blank[]], 2]]], Optional[Pattern[q, Blank[]]]], Power[Plus[Pattern[a, Blank[]], Times[Optional[Pattern[c, Blank[]]], Power[Pattern[x, Blank[]], 4]]], Optional[Pattern[p, Blank[]]]]], Pattern[x, Blank[Symbol]]] := Dist[1/2, Subst[Int[(d + e*x)^q*(a + c*x^2)^p, x], x, x^2], x] /; FreeQ[{a, c, d, e, p, q}, x]
  1249. Int[Times[Power[Pattern[x, Blank[]], Optional[Pattern[m, Blank[]]]], Power[Plus[Pattern[d, Blank[]], Times[Optional[Pattern[e, Blank[]]], Power[Pattern[x, Blank[]], 2]]], Optional[Pattern[q, Blank[]]]], Power[Plus[Pattern[a, Blank[]], Times[Optional[Pattern[b, Blank[]]], Power[Pattern[x, Blank[]], 2]], Times[Optional[Pattern[c, Blank[]]], Power[Pattern[x, Blank[]], 4]]], Pattern[p, Blank[]]]], Pattern[x, Blank[Symbol]]] := Dist[1/2, Subst[Int[x^((m - 1)/2)*(d + e*x)^q*(a + b*x + c*x^2)^p, x], x, x^2], x] /; FreeQ[{a, b, c, d, e, p, q}, x] && EqQ[b^2 - 4*a*c, 0] && !IntegerQ[p] && IGtQ[(m + 1)/2, 0]
  1250. Int[Times[Power[Times[Optional[Pattern[f, Blank[]]], Pattern[x, Blank[]]], Optional[Pattern[m, Blank[]]]], Power[Plus[Pattern[d, Blank[]], Times[Optional[Pattern[e, Blank[]]], Power[Pattern[x, Blank[]], 2]]], Optional[Pattern[q, Blank[]]]], Power[Plus[Pattern[a, Blank[]], Times[Optional[Pattern[b, Blank[]]], Power[Pattern[x, Blank[]], 2]], Times[Optional[Pattern[c, Blank[]]], Power[Pattern[x, Blank[]], 4]]], Pattern[p, Blank[]]]], Pattern[x, Blank[Symbol]]] := Dist[(a + b*x^2 + c*x^4)^FracPart[p]/(c^IntPart[p]*(b/2 + c*x^2)^(2*FracPart[p])), Int[(f*x)^m*(d + e*x^2)^q*(b/2 + c*x^2)^(2*p), x], x] /; FreeQ[{a, b, c, d, e, f, m, p, q}, x] && EqQ[b^2 - 4*a*c, 0] && !IntegerQ[p]
  1251. Int[Times[Power[Pattern[x, Blank[]], Optional[Pattern[m, Blank[]]]], Power[Plus[Pattern[d, Blank[]], Times[Optional[Pattern[e, Blank[]]], Power[Pattern[x, Blank[]], 2]]], Optional[Pattern[q, Blank[]]]], Power[Plus[Pattern[a, Blank[]], Times[Optional[Pattern[b, Blank[]]], Power[Pattern[x, Blank[]], 2]], Times[Optional[Pattern[c, Blank[]]], Power[Pattern[x, Blank[]], 4]]], Optional[Pattern[p, Blank[]]]]], Pattern[x, Blank[Symbol]]] := Dist[1/2, Subst[Int[x^((m - 1)/2)*(d + e*x)^q*(a + b*x + c*x^2)^p, x], x, x^2], x] /; FreeQ[{a, b, c, d, e, p, q}, x] && IntegerQ[(m - 1)/2]
  1252. Int[Times[Power[Pattern[x, Blank[]], Optional[Pattern[m, Blank[]]]], Power[Plus[Pattern[d, Blank[]], Times[Optional[Pattern[e, Blank[]]], Power[Pattern[x, Blank[]], 2]]], Optional[Pattern[q, Blank[]]]], Power[Plus[Pattern[a, Blank[]], Times[Optional[Pattern[c, Blank[]]], Power[Pattern[x, Blank[]], 4]]], Optional[Pattern[p, Blank[]]]]], Pattern[x, Blank[Symbol]]] := Dist[1/2, Subst[Int[x^((m - 1)/2)*(d + e*x)^q*(a + c*x^2)^p, x], x, x^2], x] /; FreeQ[{a, c, d, e, p, q}, x] && IntegerQ[(m + 1)/2]
  1253. Int[Times[Power[Times[Optional[Pattern[f, Blank[]]], Pattern[x, Blank[]]], Optional[Pattern[m, Blank[]]]], Power[Plus[Pattern[d, Blank[]], Times[Optional[Pattern[e, Blank[]]], Power[Pattern[x, Blank[]], 2]]], Optional[Pattern[q, Blank[]]]], Power[Plus[Pattern[a, Blank[]], Times[Optional[Pattern[b, Blank[]]], Power[Pattern[x, Blank[]], 2]], Times[Optional[Pattern[c, Blank[]]], Power[Pattern[x, Blank[]], 4]]], Optional[Pattern[p, Blank[]]]]], Pattern[x, Blank[Symbol]]] := Int[(f*x)^m*(d + e*x^2)^(q + p)*(a/d + (c*x^2)/e)^p, x] /; FreeQ[{a, b, c, d, e, f, m, q}, x] && NeQ[b^2 - 4*a*c, 0] && EqQ[c*d^2 - b*d*e + a*e^2, 0] && IntegerQ[p]
  1254. Int[Times[Power[Times[Optional[Pattern[f, Blank[]]], Pattern[x, Blank[]]], Optional[Pattern[m, Blank[]]]], Power[Plus[Pattern[d, Blank[]], Times[Optional[Pattern[e, Blank[]]], Power[Pattern[x, Blank[]], 2]]], Optional[Pattern[q, Blank[]]]], Power[Plus[Pattern[a, Blank[]], Times[Optional[Pattern[c, Blank[]]], Power[Pattern[x, Blank[]], 4]]], Optional[Pattern[p, Blank[]]]]], Pattern[x, Blank[Symbol]]] := Int[(f*x)^m*(d + e*x^2)^(q + p)*(a/d + (c*x^2)/e)^p, x] /; FreeQ[{a, c, d, e, f, q, m, q}, x] && EqQ[c*d^2 + a*e^2, 0] && IntegerQ[p]
  1255. Int[Times[Power[Times[Optional[Pattern[f, Blank[]]], Pattern[x, Blank[]]], Optional[Pattern[m, Blank[]]]], Power[Plus[Pattern[d, Blank[]], Times[Optional[Pattern[e, Blank[]]], Power[Pattern[x, Blank[]], 2]]], Pattern[q, Blank[]]], Power[Plus[Pattern[a, Blank[]], Times[Optional[Pattern[b, Blank[]]], Power[Pattern[x, Blank[]], 2]], Times[Optional[Pattern[c, Blank[]]], Power[Pattern[x, Blank[]], 4]]], Pattern[p, Blank[]]]], Pattern[x, Blank[Symbol]]] := Dist[(a + b*x^2 + c*x^4)^FracPart[p]/((d + e*x^2)^FracPart[p]*(a/d + (c*x^2)/e)^FracPart[p]), Int[(f*x)^m*(d + e*x^2)^(q + p)*(a/d + (c*x^2)/e)^p, x], x] /; FreeQ[{a, b, c, d, e, f, m, p, q}, x] && NeQ[b^2 - 4*a*c, 0] && EqQ[c*d^2 - b*d*e + a*e^2, 0] && !IntegerQ[p]
  1256. Int[Times[Power[Times[Optional[Pattern[f, Blank[]]], Pattern[x, Blank[]]], Optional[Pattern[m, Blank[]]]], Power[Plus[Pattern[d, Blank[]], Times[Optional[Pattern[e, Blank[]]], Power[Pattern[x, Blank[]], 2]]], Pattern[q, Blank[]]], Power[Plus[Pattern[a, Blank[]], Times[Optional[Pattern[c, Blank[]]], Power[Pattern[x, Blank[]], 4]]], Pattern[p, Blank[]]]], Pattern[x, Blank[Symbol]]] := Dist[(a + c*x^4)^FracPart[p]/((d + e*x^2)^FracPart[p]*(a/d + (c*x^2)/e)^FracPart[p]), Int[(f*x)^m*(d + e*x^2)^(q + p)*(a/d + (c*x^2)/e)^p, x], x] /; FreeQ[{a, c, d, e, f, m, p, q}, x] && EqQ[c*d^2 + a*e^2, 0] && !IntegerQ[p]
  1257. Int[Times[Power[Pattern[x, Blank[]], Optional[Pattern[m, Blank[]]]], Power[Plus[Pattern[d, Blank[]], Times[Optional[Pattern[e, Blank[]]], Power[Pattern[x, Blank[]], 2]]], Pattern[q, Blank[]]], Power[Plus[Pattern[a, Blank[]], Times[Optional[Pattern[b, Blank[]]], Power[Pattern[x, Blank[]], 2]], Times[Optional[Pattern[c, Blank[]]], Power[Pattern[x, Blank[]], 4]]], Optional[Pattern[p, Blank[]]]]], Pattern[x, Blank[Symbol]]] := Simp[((-d)^(m/2 - 1)*(c*d^2 - b*d*e + a*e^2)^p*x*(d + e*x^2)^(q + 1))/(2*e^(2*p + m/2)*(q + 1)), x] + Dist[1/(2*e^(2*p + m/2)*(q + 1)), Int[(d + e*x^2)^(q + 1)*ExpandToSum[Together[(1*(2*e^(2*p + m/2)*(q + 1)*x^m*(a + b*x^2 + c*x^4)^p - (-d)^(m/2 - 1)*(c*d^2 - b*d*e + a*e^2)^p*(d + e*(2*q + 3)*x^2)))/(d + e*x^2)], x], x], x] /; FreeQ[{a, b, c, d, e}, x] && NeQ[b^2 - 4*a*c, 0] && IGtQ[p, 0] && ILtQ[q, -1] && IGtQ[m/2, 0]
  1258. Int[Times[Power[Pattern[x, Blank[]], Optional[Pattern[m, Blank[]]]], Power[Plus[Pattern[d, Blank[]], Times[Optional[Pattern[e, Blank[]]], Power[Pattern[x, Blank[]], 2]]], Pattern[q, Blank[]]], Power[Plus[Pattern[a, Blank[]], Times[Optional[Pattern[c, Blank[]]], Power[Pattern[x, Blank[]], 4]]], Optional[Pattern[p, Blank[]]]]], Pattern[x, Blank[Symbol]]] := Simp[((-d)^(m/2 - 1)*(c*d^2 + a*e^2)^p*x*(d + e*x^2)^(q + 1))/(2*e^(2*p + m/2)*(q + 1)), x] + Dist[1/(2*e^(2*p + m/2)*(q + 1)), Int[(d + e*x^2)^(q + 1)*ExpandToSum[Together[(1*(2*e^(2*p + m/2)*(q + 1)*x^m*(a + c*x^4)^p - (-d)^(m/2 - 1)*(c*d^2 + a*e^2)^p*(d + e*(2*q + 3)*x^2)))/(d + e*x^2)], x], x], x] /; FreeQ[{a, c, d, e}, x] && IGtQ[p, 0] && ILtQ[q, -1] && IGtQ[m/2, 0]
  1259. Int[Times[Power[Pattern[x, Blank[]], Pattern[m, Blank[]]], Power[Plus[Pattern[d, Blank[]], Times[Optional[Pattern[e, Blank[]]], Power[Pattern[x, Blank[]], 2]]], Pattern[q, Blank[]]], Power[Plus[Pattern[a, Blank[]], Times[Optional[Pattern[b, Blank[]]], Power[Pattern[x, Blank[]], 2]], Times[Optional[Pattern[c, Blank[]]], Power[Pattern[x, Blank[]], 4]]], Optional[Pattern[p, Blank[]]]]], Pattern[x, Blank[Symbol]]] := Simp[((-d)^(m/2 - 1)*(c*d^2 - b*d*e + a*e^2)^p*x*(d + e*x^2)^(q + 1))/(2*e^(2*p + m/2)*(q + 1)), x] + Dist[(-d)^(m/2 - 1)/(2*e^(2*p)*(q + 1)), Int[x^m*(d + e*x^2)^(q + 1)*ExpandToSum[Together[(1*(2*(-d)^(-(m/2) + 1)*e^(2*p)*(q + 1)*(a + b*x^2 + c*x^4)^p - ((c*d^2 - b*d*e + a*e^2)^p/(e^(m/2)*x^m))*(d + e*(2*q + 3)*x^2)))/(d + e*x^2)], x], x], x] /; FreeQ[{a, b, c, d, e}, x] && NeQ[b^2 - 4*a*c, 0] && IGtQ[p, 0] && ILtQ[q, -1] && ILtQ[m/2, 0]
  1260. Int[Times[Power[Pattern[x, Blank[]], Pattern[m, Blank[]]], Power[Plus[Pattern[d, Blank[]], Times[Optional[Pattern[e, Blank[]]], Power[Pattern[x, Blank[]], 2]]], Pattern[q, Blank[]]], Power[Plus[Pattern[a, Blank[]], Times[Optional[Pattern[c, Blank[]]], Power[Pattern[x, Blank[]], 4]]], Optional[Pattern[p, Blank[]]]]], Pattern[x, Blank[Symbol]]] := Simp[((-d)^(m/2 - 1)*(c*d^2 + a*e^2)^p*x*(d + e*x^2)^(q + 1))/(2*e^(2*p + m/2)*(q + 1)), x] + Dist[(-d)^(m/2 - 1)/(2*e^(2*p)*(q + 1)), Int[x^m*(d + e*x^2)^(q + 1)*ExpandToSum[Together[(1*(2*(-d)^(-(m/2) + 1)*e^(2*p)*(q + 1)*(a + c*x^4)^p - ((c*d^2 + a*e^2)^p/(e^(m/2)*x^m))*(d + e*(2*q + 3)*x^2)))/(d + e*x^2)], x], x], x] /; FreeQ[{a, c, d, e}, x] && IGtQ[p, 0] && ILtQ[q, -1] && ILtQ[m/2, 0]
  1261. Int[Times[Power[Times[Optional[Pattern[f, Blank[]]], Pattern[x, Blank[]]], Optional[Pattern[m, Blank[]]]], Power[Plus[Pattern[d, Blank[]], Times[Optional[Pattern[e, Blank[]]], Power[Pattern[x, Blank[]], 2]]], Optional[Pattern[q, Blank[]]]], Power[Plus[Pattern[a, Blank[]], Times[Optional[Pattern[b, Blank[]]], Power[Pattern[x, Blank[]], 2]], Times[Optional[Pattern[c, Blank[]]], Power[Pattern[x, Blank[]], 4]]], Optional[Pattern[p, Blank[]]]]], Pattern[x, Blank[Symbol]]] := Int[ExpandIntegrand[(f*x)^m*(d + e*x^2)^q*(a + b*x^2 + c*x^4)^p, x], x] /; FreeQ[{a, b, c, d, e, f, m, q}, x] && NeQ[b^2 - 4*a*c, 0] && IGtQ[p, 0] && IGtQ[q, -2]
  1262. Int[Times[Power[Times[Optional[Pattern[f, Blank[]]], Pattern[x, Blank[]]], Optional[Pattern[m, Blank[]]]], Power[Plus[Pattern[d, Blank[]], Times[Optional[Pattern[e, Blank[]]], Power[Pattern[x, Blank[]], 2]]], Optional[Pattern[q, Blank[]]]], Power[Plus[Pattern[a, Blank[]], Times[Optional[Pattern[c, Blank[]]], Power[Pattern[x, Blank[]], 4]]], Optional[Pattern[p, Blank[]]]]], Pattern[x, Blank[Symbol]]] := Int[ExpandIntegrand[(f*x)^m*(d + e*x^2)^q*(a + c*x^4)^p, x], x] /; FreeQ[{a, c, d, e, f, m, q}, x] && IGtQ[p, 0] && IGtQ[q, -2]
  1263. Int[Times[Power[Times[Optional[Pattern[f, Blank[]]], Pattern[x, Blank[]]], Optional[Pattern[m, Blank[]]]], Power[Plus[Pattern[d, Blank[]], Times[Optional[Pattern[e, Blank[]]], Power[Pattern[x, Blank[]], 2]]], Pattern[q, Blank[]]], Power[Plus[Pattern[a, Blank[]], Times[Optional[Pattern[b, Blank[]]], Power[Pattern[x, Blank[]], 2]], Times[Optional[Pattern[c, Blank[]]], Power[Pattern[x, Blank[]], 4]]], Optional[Pattern[p, Blank[]]]]], Pattern[x, Blank[Symbol]]] := With[{Qx = PolynomialQuotient[(a + b*x^2 + c*x^4)^p, d + e*x^2, x], R = Coeff[PolynomialRemainder[(a + b*x^2 + c*x^4)^p, d + e*x^2, x], x, 0]}, -Simp[(R*(f*x)^(m + 1)*(d + e*x^2)^(q + 1))/(2*d*f*(q + 1)), x] + Dist[f/(2*d*(q + 1)), Int[(f*x)^(m - 1)*(d + e*x^2)^(q + 1)*ExpandToSum[2*d*(q + 1)*x*Qx + R*(m + 2*q + 3)*x, x], x], x]] /; FreeQ[{a, b, c, d, e, f}, x] && NeQ[b^2 - 4*a*c, 0] && IGtQ[p, 0] && LtQ[q, -1] && GtQ[m, 0]
  1264. Int[Times[Power[Times[Optional[Pattern[f, Blank[]]], Pattern[x, Blank[]]], Optional[Pattern[m, Blank[]]]], Power[Plus[Pattern[d, Blank[]], Times[Optional[Pattern[e, Blank[]]], Power[Pattern[x, Blank[]], 2]]], Pattern[q, Blank[]]], Power[Plus[Pattern[a, Blank[]], Times[Optional[Pattern[c, Blank[]]], Power[Pattern[x, Blank[]], 4]]], Optional[Pattern[p, Blank[]]]]], Pattern[x, Blank[Symbol]]] := With[{Qx = PolynomialQuotient[(a + c*x^4)^p, d + e*x^2, x], R = Coeff[PolynomialRemainder[(a + c*x^4)^p, d + e*x^2, x], x, 0]}, -Simp[(R*(f*x)^(m + 1)*(d + e*x^2)^(q + 1))/(2*d*f*(q + 1)), x] + Dist[f/(2*d*(q + 1)), Int[(f*x)^(m - 1)*(d + e*x^2)^(q + 1)*ExpandToSum[2*d*(q + 1)*x*Qx + R*(m + 2*q + 3)*x, x], x], x]] /; FreeQ[{a, c, d, e, f}, x] && IGtQ[p, 0] && LtQ[q, -1] && GtQ[m, 0]
  1265. Int[Times[Power[Times[Optional[Pattern[f, Blank[]]], Pattern[x, Blank[]]], Pattern[m, Blank[]]], Power[Plus[Pattern[d, Blank[]], Times[Optional[Pattern[e, Blank[]]], Power[Pattern[x, Blank[]], 2]]], Optional[Pattern[q, Blank[]]]], Power[Plus[Pattern[a, Blank[]], Times[Optional[Pattern[b, Blank[]]], Power[Pattern[x, Blank[]], 2]], Times[Optional[Pattern[c, Blank[]]], Power[Pattern[x, Blank[]], 4]]], Optional[Pattern[p, Blank[]]]]], Pattern[x, Blank[Symbol]]] := With[{Qx = PolynomialQuotient[(a + b*x^2 + c*x^4)^p, f*x, x], R = PolynomialRemainder[(a + b*x^2 + c*x^4)^p, f*x, x]}, Simp[(R*(f*x)^(m + 1)*(d + e*x^2)^(q + 1))/(d*f*(m + 1)), x] + Dist[1/(d*f^2*(m + 1)), Int[(f*x)^(m + 2)*(d + e*x^2)^q*ExpandToSum[(d*f*(m + 1)*Qx)/x - e*R*(m + 2*q + 3), x], x], x]] /; FreeQ[{a, b, c, d, e, f, q}, x] && NeQ[b^2 - 4*a*c, 0] && IGtQ[p, 0] && LtQ[m, -1]
  1266. Int[Times[Power[Times[Optional[Pattern[f, Blank[]]], Pattern[x, Blank[]]], Pattern[m, Blank[]]], Power[Plus[Pattern[d, Blank[]], Times[Optional[Pattern[e, Blank[]]], Power[Pattern[x, Blank[]], 2]]], Optional[Pattern[q, Blank[]]]], Power[Plus[Pattern[a, Blank[]], Times[Optional[Pattern[c, Blank[]]], Power[Pattern[x, Blank[]], 4]]], Optional[Pattern[p, Blank[]]]]], Pattern[x, Blank[Symbol]]] := With[{Qx = PolynomialQuotient[(a + c*x^4)^p, f*x, x], R = PolynomialRemainder[(a + c*x^4)^p, f*x, x]}, Simp[(R*(f*x)^(m + 1)*(d + e*x^2)^(q + 1))/(d*f*(m + 1)), x] + Dist[1/(d*f^2*(m + 1)), Int[(f*x)^(m + 2)*(d + e*x^2)^q*ExpandToSum[(d*f*(m + 1)*Qx)/x - e*R*(m + 2*q + 3), x], x], x]] /; FreeQ[{a, c, d, e, f, q}, x] && IGtQ[p, 0] && LtQ[m, -1]
  1267. Int[Times[Power[Times[Optional[Pattern[f, Blank[]]], Pattern[x, Blank[]]], Optional[Pattern[m, Blank[]]]], Power[Plus[Pattern[d, Blank[]], Times[Optional[Pattern[e, Blank[]]], Power[Pattern[x, Blank[]], 2]]], Optional[Pattern[q, Blank[]]]], Power[Plus[Pattern[a, Blank[]], Times[Optional[Pattern[b, Blank[]]], Power[Pattern[x, Blank[]], 2]], Times[Optional[Pattern[c, Blank[]]], Power[Pattern[x, Blank[]], 4]]], Optional[Pattern[p, Blank[]]]]], Pattern[x, Blank[Symbol]]] := Simp[(c^p*(f*x)^(m + 4*p - 1)*(d + e*x^2)^(q + 1))/(e*f^(4*p - 1)*(m + 4*p + 2*q + 1)), x] + Dist[1/(e*(m + 4*p + 2*q + 1)), Int[(f*x)^m*(d + e*x^2)^q*ExpandToSum[e*(m + 4*p + 2*q + 1)*((a + b*x^2 + c*x^4)^p - c^p*x^(4*p)) - d*c^p*(m + 4*p - 1)*x^(4*p - 2), x], x], x] /; FreeQ[{a, b, c, d, e, f, m, q}, x] && NeQ[b^2 - 4*a*c, 0] && IGtQ[p, 0] && !IntegerQ[q] && NeQ[m + 4*p + 2*q + 1, 0]
  1268. Int[Times[Power[Times[Optional[Pattern[f, Blank[]]], Pattern[x, Blank[]]], Optional[Pattern[m, Blank[]]]], Power[Plus[Pattern[d, Blank[]], Times[Optional[Pattern[e, Blank[]]], Power[Pattern[x, Blank[]], 2]]], Optional[Pattern[q, Blank[]]]], Power[Plus[Pattern[a, Blank[]], Times[Optional[Pattern[c, Blank[]]], Power[Pattern[x, Blank[]], 4]]], Optional[Pattern[p, Blank[]]]]], Pattern[x, Blank[Symbol]]] := Simp[(c^p*(f*x)^(m + 4*p - 1)*(d + e*x^2)^(q + 1))/(e*f^(4*p - 1)*(m + 4*p + 2*q + 1)), x] + Dist[1/(e*(m + 4*p + 2*q + 1)), Int[(f*x)^m*(d + e*x^2)^q*ExpandToSum[e*(m + 4*p + 2*q + 1)*((a + c*x^4)^p - c^p*x^(4*p)) - d*c^p*(m + 4*p - 1)*x^(4*p - 2), x], x], x] /; FreeQ[{a, c, d, e, f, m, q}, x] && IGtQ[p, 0] && !IntegerQ[q] && NeQ[m + 4*p + 2*q + 1, 0]
  1269. Int[Times[Power[Times[Optional[Pattern[f, Blank[]]], Pattern[x, Blank[]]], Pattern[m, Blank[]]], Power[Plus[Pattern[d, Blank[]], Times[Optional[Pattern[e, Blank[]]], Power[Pattern[x, Blank[]], 2]]], Optional[Pattern[q, Blank[]]]], Power[Plus[Pattern[a, Blank[]], Times[Optional[Pattern[b, Blank[]]], Power[Pattern[x, Blank[]], 2]], Times[Optional[Pattern[c, Blank[]]], Power[Pattern[x, Blank[]], 4]]], Pattern[p, Blank[]]]], Pattern[x, Blank[Symbol]]] := With[{k = Denominator[m]}, Dist[k/f, Subst[Int[x^(k*(m + 1) - 1)*(d + (e*x^(2*k))/f^2)^q*(a + (b*x^(2*k))/f^k + (c*x^(4*k))/f^4)^p, x], x, (f*x)^(1/k)], x]] /; FreeQ[{a, b, c, d, e, f, p, q}, x] && NeQ[b^2 - 4*a*c, 0] && FractionQ[m] && IntegerQ[p]
  1270. Int[Times[Power[Times[Optional[Pattern[f, Blank[]]], Pattern[x, Blank[]]], Pattern[m, Blank[]]], Power[Plus[Pattern[d, Blank[]], Times[Optional[Pattern[e, Blank[]]], Power[Pattern[x, Blank[]], 2]]], Optional[Pattern[q, Blank[]]]], Power[Plus[Pattern[a, Blank[]], Times[Optional[Pattern[c, Blank[]]], Power[Pattern[x, Blank[]], 4]]], Pattern[p, Blank[]]]], Pattern[x, Blank[Symbol]]] := With[{k = Denominator[m]}, Dist[k/f, Subst[Int[x^(k*(m + 1) - 1)*(d + (e*x^(2*k))/f)^q*(a + (c*x^(4*k))/f)^p, x], x, (f*x)^(1/k)], x]] /; FreeQ[{a, c, d, e, f, p, q}, x] && FractionQ[m] && IntegerQ[p]
  1271. Int[Times[Power[Times[Optional[Pattern[f, Blank[]]], Pattern[x, Blank[]]], Optional[Pattern[m, Blank[]]]], Plus[Pattern[d, Blank[]], Times[Optional[Pattern[e, Blank[]]], Power[Pattern[x, Blank[]], 2]]], Power[Plus[Pattern[a, Blank[]], Times[Optional[Pattern[b, Blank[]]], Power[Pattern[x, Blank[]], 2]], Times[Optional[Pattern[c, Blank[]]], Power[Pattern[x, Blank[]], 4]]], Optional[Pattern[p, Blank[]]]]], Pattern[x, Blank[Symbol]]] := Simp[((f*x)^(m + 1)*(a + b*x^2 + c*x^4)^p*(d*(m + 4*p + 3) + e*(m + 1)*x^2))/(f*(m + 1)*(m + 4*p + 3)), x] + Dist[(2*p)/(f^2*(m + 1)*(m + 4*p + 3)), Int[(f*x)^(m + 2)*(a + b*x^2 + c*x^4)^(p - 1)*Simp[2*a*e*(m + 1) - b*d*(m + 4*p + 3) + (b*e*(m + 1) - 2*c*d*(m + 4*p + 3))*x^2, x], x], x] /; FreeQ[{a, b, c, d, e, f}, x] && NeQ[b^2 - 4*a*c, 0] && GtQ[p, 0] && LtQ[m, -1] && m + 4*p + 3 != 0 && IntegerQ[2*p] && (IntegerQ[p] || IntegerQ[m])
  1272. Int[Times[Power[Times[Optional[Pattern[f, Blank[]]], Pattern[x, Blank[]]], Optional[Pattern[m, Blank[]]]], Plus[Pattern[d, Blank[]], Times[Optional[Pattern[e, Blank[]]], Power[Pattern[x, Blank[]], 2]]], Power[Plus[Pattern[a, Blank[]], Times[Optional[Pattern[c, Blank[]]], Power[Pattern[x, Blank[]], 4]]], Optional[Pattern[p, Blank[]]]]], Pattern[x, Blank[Symbol]]] := Simp[((f*x)^(m + 1)*(a + c*x^4)^p*(d*(m + 4*p + 3) + e*(m + 1)*x^2))/(f*(m + 1)*(m + 4*p + 3)), x] + Dist[(4*p)/(f^2*(m + 1)*(m + 4*p + 3)), Int[(f*x)^(m + 2)*(a + c*x^4)^(p - 1)*(a*e*(m + 1) - c*d*(m + 4*p + 3)*x^2), x], x] /; FreeQ[{a, c, d, e, f}, x] && GtQ[p, 0] && LtQ[m, -1] && m + 4*p + 3 != 0 && IntegerQ[2*p] && (IntegerQ[p] || IntegerQ[m])
  1273. Int[Times[Power[Times[Optional[Pattern[f, Blank[]]], Pattern[x, Blank[]]], Optional[Pattern[m, Blank[]]]], Plus[Pattern[d, Blank[]], Times[Optional[Pattern[e, Blank[]]], Power[Pattern[x, Blank[]], 2]]], Power[Plus[Pattern[a, Blank[]], Times[Optional[Pattern[b, Blank[]]], Power[Pattern[x, Blank[]], 2]], Times[Optional[Pattern[c, Blank[]]], Power[Pattern[x, Blank[]], 4]]], Optional[Pattern[p, Blank[]]]]], Pattern[x, Blank[Symbol]]] := Simp[((f*x)^(m + 1)*(a + b*x^2 + c*x^4)^p*(b*e*2*p + c*d*(m + 4*p + 3) + c*e*(4*p + m + 1)*x^2))/(c*f*(4*p + m + 1)*(m + 4*p + 3)), x] + Dist[(2*p)/(c*(4*p + m + 1)*(m + 4*p + 3)), Int[(f*x)^m*(a + b*x^2 + c*x^4)^(p - 1)*Simp[2*a*c*d*(m + 4*p + 3) - a*b*e*(m + 1) + (2*a*c*e*(4*p + m + 1) + b*c*d*(m + 4*p + 3) - b^2*e*(m + 2*p + 1))*x^2, x], x], x] /; FreeQ[{a, b, c, d, e, f, m}, x] && NeQ[b^2 - 4*a*c, 0] && GtQ[p, 0] && NeQ[4*p + m + 1, 0] && NeQ[m + 4*p + 3, 0] && IntegerQ[2*p] && (IntegerQ[p] || IntegerQ[m])
  1274. Int[Times[Power[Times[Optional[Pattern[f, Blank[]]], Pattern[x, Blank[]]], Optional[Pattern[m, Blank[]]]], Plus[Pattern[d, Blank[]], Times[Optional[Pattern[e, Blank[]]], Power[Pattern[x, Blank[]], 2]]], Power[Plus[Pattern[a, Blank[]], Times[Optional[Pattern[c, Blank[]]], Power[Pattern[x, Blank[]], 4]]], Optional[Pattern[p, Blank[]]]]], Pattern[x, Blank[Symbol]]] := Simp[((f*x)^(m + 1)*(a + c*x^4)^p*(c*d*(m + 4*p + 3) + c*e*(4*p + m + 1)*x^2))/(c*f*(4*p + m + 1)*(m + 4*p + 3)), x] + Dist[(4*a*p)/((4*p + m + 1)*(m + 4*p + 3)), Int[(f*x)^m*(a + c*x^4)^(p - 1)*Simp[d*(m + 4*p + 3) + e*(4*p + m + 1)*x^2, x], x], x] /; FreeQ[{a, c, d, e, f, m}, x] && GtQ[p, 0] && NeQ[4*p + m + 1, 0] && NeQ[m + 4*p + 3, 0] && IntegerQ[2*p] && (IntegerQ[p] || IntegerQ[m])
  1275. Int[Times[Power[Times[Optional[Pattern[f, Blank[]]], Pattern[x, Blank[]]], Optional[Pattern[m, Blank[]]]], Plus[Pattern[d, Blank[]], Times[Optional[Pattern[e, Blank[]]], Power[Pattern[x, Blank[]], 2]]], Power[Plus[Pattern[a, Blank[]], Times[Optional[Pattern[b, Blank[]]], Power[Pattern[x, Blank[]], 2]], Times[Optional[Pattern[c, Blank[]]], Power[Pattern[x, Blank[]], 4]]], Optional[Pattern[p, Blank[]]]]], Pattern[x, Blank[Symbol]]] := Simp[(f*(f*x)^(m - 1)*(a + b*x^2 + c*x^4)^(p + 1)*(b*d - 2*a*e - (b*e - 2*c*d)*x^2))/(2*(p + 1)*(b^2 - 4*a*c)), x] - Dist[f^2/(2*(p + 1)*(b^2 - 4*a*c)), Int[(f*x)^(m - 2)*(a + b*x^2 + c*x^4)^(p + 1)*Simp[(m - 1)*(b*d - 2*a*e) - (4*p + 4 + m + 1)*(b*e - 2*c*d)*x^2, x], x], x] /; FreeQ[{a, b, c, d, e, f}, x] && NeQ[b^2 - 4*a*c, 0] && LtQ[p, -1] && GtQ[m, 1] && IntegerQ[2*p] && (IntegerQ[p] || IntegerQ[m])
  1276. Int[Times[Power[Times[Optional[Pattern[f, Blank[]]], Pattern[x, Blank[]]], Optional[Pattern[m, Blank[]]]], Plus[Pattern[d, Blank[]], Times[Optional[Pattern[e, Blank[]]], Power[Pattern[x, Blank[]], 2]]], Power[Plus[Pattern[a, Blank[]], Times[Optional[Pattern[c, Blank[]]], Power[Pattern[x, Blank[]], 4]]], Optional[Pattern[p, Blank[]]]]], Pattern[x, Blank[Symbol]]] := Simp[(f*(f*x)^(m - 1)*(a + c*x^4)^(p + 1)*(a*e - c*d*x^2))/(4*a*c*(p + 1)), x] - Dist[f^2/(4*a*c*(p + 1)), Int[(f*x)^(m - 2)*(a + c*x^4)^(p + 1)*(a*e*(m - 1) - c*d*(4*p + 4 + m + 1)*x^2), x], x] /; FreeQ[{a, c, d, e, f}, x] && LtQ[p, -1] && GtQ[m, 1] && IntegerQ[2*p] && (IntegerQ[p] || IntegerQ[m])
  1277. Int[Times[Power[Times[Optional[Pattern[f, Blank[]]], Pattern[x, Blank[]]], Optional[Pattern[m, Blank[]]]], Plus[Pattern[d, Blank[]], Times[Optional[Pattern[e, Blank[]]], Power[Pattern[x, Blank[]], 2]]], Power[Plus[Pattern[a, Blank[]], Times[Optional[Pattern[b, Blank[]]], Power[Pattern[x, Blank[]], 2]], Times[Optional[Pattern[c, Blank[]]], Power[Pattern[x, Blank[]], 4]]], Pattern[p, Blank[]]]], Pattern[x, Blank[Symbol]]] := -Simp[((f*x)^(m + 1)*(a + b*x^2 + c*x^4)^(p + 1)*(d*(b^2 - 2*a*c) - a*b*e + (b*d - 2*a*e)*c*x^2))/(2*a*f*(p + 1)*(b^2 - 4*a*c)), x] + Dist[1/(2*a*(p + 1)*(b^2 - 4*a*c)), Int[(f*x)^m*(a + b*x^2 + c*x^4)^(p + 1)*Simp[d*(b^2*(m + 2*(p + 1) + 1) - 2*a*c*(m + 4*(p + 1) + 1)) - a*b*e*(m + 1) + c*(m + 2*(2*p + 3) + 1)*(b*d - 2*a*e)*x^2, x], x], x] /; FreeQ[{a, b, c, d, e, f, m}, x] && NeQ[b^2 - 4*a*c, 0] && LtQ[p, -1] && IntegerQ[2*p] && (IntegerQ[p] || IntegerQ[m])
  1278. Int[Times[Power[Times[Optional[Pattern[f, Blank[]]], Pattern[x, Blank[]]], Optional[Pattern[m, Blank[]]]], Plus[Pattern[d, Blank[]], Times[Optional[Pattern[e, Blank[]]], Power[Pattern[x, Blank[]], 2]]], Power[Plus[Pattern[a, Blank[]], Times[Optional[Pattern[c, Blank[]]], Power[Pattern[x, Blank[]], 4]]], Pattern[p, Blank[]]]], Pattern[x, Blank[Symbol]]] := -Simp[((f*x)^(m + 1)*(a + c*x^4)^(p + 1)*(d + e*x^2))/(4*a*f*(p + 1)), x] + Dist[1/(4*a*(p + 1)), Int[(f*x)^m*(a + c*x^4)^(p + 1)*Simp[d*(m + 4*(p + 1) + 1) + e*(m + 2*(2*p + 3) + 1)*x^2, x], x], x] /; FreeQ[{a, c, d, e, f, m}, x] && LtQ[p, -1] && IntegerQ[2*p] && (IntegerQ[p] || IntegerQ[m])
  1279. Int[Times[Power[Times[Optional[Pattern[f, Blank[]]], Pattern[x, Blank[]]], Optional[Pattern[m, Blank[]]]], Plus[Pattern[d, Blank[]], Times[Optional[Pattern[e, Blank[]]], Power[Pattern[x, Blank[]], 2]]], Power[Plus[Pattern[a, Blank[]], Times[Optional[Pattern[b, Blank[]]], Power[Pattern[x, Blank[]], 2]], Times[Optional[Pattern[c, Blank[]]], Power[Pattern[x, Blank[]], 4]]], Pattern[p, Blank[]]]], Pattern[x, Blank[Symbol]]] := Simp[(e*f*(f*x)^(m - 1)*(a + b*x^2 + c*x^4)^(p + 1))/(c*(m + 4*p + 3)), x] - Dist[f^2/(c*(m + 4*p + 3)), Int[(f*x)^(m - 2)*(a + b*x^2 + c*x^4)^p*Simp[a*e*(m - 1) + (b*e*(m + 2*p + 1) - c*d*(m + 4*p + 3))*x^2, x], x], x] /; FreeQ[{a, b, c, d, e, f, p}, x] && NeQ[b^2 - 4*a*c, 0] && GtQ[m, 1] && NeQ[m + 4*p + 3, 0] && IntegerQ[2*p] && (IntegerQ[p] || IntegerQ[m])
  1280. Int[Times[Power[Times[Optional[Pattern[f, Blank[]]], Pattern[x, Blank[]]], Optional[Pattern[m, Blank[]]]], Plus[Pattern[d, Blank[]], Times[Optional[Pattern[e, Blank[]]], Power[Pattern[x, Blank[]], 2]]], Power[Plus[Pattern[a, Blank[]], Times[Optional[Pattern[c, Blank[]]], Power[Pattern[x, Blank[]], 4]]], Pattern[p, Blank[]]]], Pattern[x, Blank[Symbol]]] := Simp[(e*f*(f*x)^(m - 1)*(a + c*x^4)^(p + 1))/(c*(m + 4*p + 3)), x] - Dist[f^2/(c*(m + 4*p + 3)), Int[(f*x)^(m - 2)*(a + c*x^4)^p*(a*e*(m - 1) - c*d*(m + 4*p + 3)*x^2), x], x] /; FreeQ[{a, c, d, e, f, p}, x] && GtQ[m, 1] && NeQ[m + 4*p + 3, 0] && IntegerQ[2*p] && (IntegerQ[p] || IntegerQ[m])
  1281. Int[Times[Power[Times[Optional[Pattern[f, Blank[]]], Pattern[x, Blank[]]], Optional[Pattern[m, Blank[]]]], Plus[Pattern[d, Blank[]], Times[Optional[Pattern[e, Blank[]]], Power[Pattern[x, Blank[]], 2]]], Power[Plus[Pattern[a, Blank[]], Times[Optional[Pattern[b, Blank[]]], Power[Pattern[x, Blank[]], 2]], Times[Optional[Pattern[c, Blank[]]], Power[Pattern[x, Blank[]], 4]]], Pattern[p, Blank[]]]], Pattern[x, Blank[Symbol]]] := Simp[(d*(f*x)^(m + 1)*(a + b*x^2 + c*x^4)^(p + 1))/(a*f*(m + 1)), x] + Dist[1/(a*f^2*(m + 1)), Int[(f*x)^(m + 2)*(a + b*x^2 + c*x^4)^p*Simp[a*e*(m + 1) - b*d*(m + 2*p + 3) - c*d*(m + 4*p + 5)*x^2, x], x], x] /; FreeQ[{a, b, c, d, e, f, p}, x] && NeQ[b^2 - 4*a*c, 0] && LtQ[m, -1] && IntegerQ[2*p] && (IntegerQ[p] || IntegerQ[m])
  1282. Int[Times[Power[Times[Optional[Pattern[f, Blank[]]], Pattern[x, Blank[]]], Optional[Pattern[m, Blank[]]]], Plus[Pattern[d, Blank[]], Times[Optional[Pattern[e, Blank[]]], Power[Pattern[x, Blank[]], 2]]], Power[Plus[Pattern[a, Blank[]], Times[Optional[Pattern[c, Blank[]]], Power[Pattern[x, Blank[]], 4]]], Pattern[p, Blank[]]]], Pattern[x, Blank[Symbol]]] := Simp[(d*(f*x)^(m + 1)*(a + c*x^4)^(p + 1))/(a*f*(m + 1)), x] + Dist[1/(a*f^2*(m + 1)), Int[(f*x)^(m + 2)*(a + c*x^4)^p*(a*e*(m + 1) - c*d*(m + 4*p + 5)*x^2), x], x] /; FreeQ[{a, c, d, e, f, p}, x] && LtQ[m, -1] && IntegerQ[2*p] && (IntegerQ[p] || IntegerQ[m])
  1283. Int[Times[Power[Times[Optional[Pattern[f, Blank[]]], Pattern[x, Blank[]]], Optional[Pattern[m, Blank[]]]], Plus[Pattern[d, Blank[]], Times[Optional[Pattern[e, Blank[]]], Power[Pattern[x, Blank[]], 2]]], Power[Plus[Pattern[a, Blank[]], Times[Optional[Pattern[b, Blank[]]], Power[Pattern[x, Blank[]], 2]], Times[Optional[Pattern[c, Blank[]]], Power[Pattern[x, Blank[]], 4]]], -1]], Pattern[x, Blank[Symbol]]] := With[{r = Rt[(c*(2*c*d - b*e))/e, 2]}, Dist[e/2, Int[(f*x)^m/((c*d)/e - r*x + c*x^2), x], x] + Dist[e/2, Int[(f*x)^m/((c*d)/e + r*x + c*x^2), x], x]] /; FreeQ[{a, b, c, d, e, f, m}, x] && NeQ[b^2 - 4*a*c, 0] && EqQ[c*d^2 - a*e^2, 0] && GtQ[d/e, 0] && PosQ[(c*(2*c*d - b*e))/e]
  1284. Int[Times[Power[Times[Optional[Pattern[f, Blank[]]], Pattern[x, Blank[]]], Optional[Pattern[m, Blank[]]]], Plus[Pattern[d, Blank[]], Times[Optional[Pattern[e, Blank[]]], Power[Pattern[x, Blank[]], 2]]], Power[Plus[Pattern[a, Blank[]], Times[Optional[Pattern[c, Blank[]]], Power[Pattern[x, Blank[]], 4]]], -1]], Pattern[x, Blank[Symbol]]] := With[{r = Rt[(2*c^2*d)/e, 2]}, Dist[e/2, Int[(f*x)^m/((c*d)/e - r*x + c*x^2), x], x] + Dist[e/2, Int[(f*x)^m/((c*d)/e + r*x + c*x^2), x], x]] /; FreeQ[{a, c, d, e, f, m}, x] && EqQ[c*d^2 - a*e^2, 0] && GtQ[d/e, 0]
  1285. Int[Times[Power[Times[Optional[Pattern[f, Blank[]]], Pattern[x, Blank[]]], Optional[Pattern[m, Blank[]]]], Plus[Pattern[d, Blank[]], Times[Optional[Pattern[e, Blank[]]], Power[Pattern[x, Blank[]], 2]]], Power[Plus[Pattern[a, Blank[]], Times[Optional[Pattern[b, Blank[]]], Power[Pattern[x, Blank[]], 2]], Times[Optional[Pattern[c, Blank[]]], Power[Pattern[x, Blank[]], 4]]], -1]], Pattern[x, Blank[Symbol]]] := With[{q = Rt[b^2 - 4*a*c, 2]}, Dist[e/2 + (2*c*d - b*e)/(2*q), Int[(f*x)^m/(b/2 - q/2 + c*x^2), x], x] + Dist[e/2 - (2*c*d - b*e)/(2*q), Int[(f*x)^m/(b/2 + q/2 + c*x^2), x], x]] /; FreeQ[{a, b, c, d, e, f, m}, x] && NeQ[b^2 - 4*a*c, 0]
  1286. Int[Times[Power[Times[Optional[Pattern[f, Blank[]]], Pattern[x, Blank[]]], Optional[Pattern[m, Blank[]]]], Plus[Pattern[d, Blank[]], Times[Optional[Pattern[e, Blank[]]], Power[Pattern[x, Blank[]], 2]]], Power[Plus[Pattern[a, Blank[]], Times[Optional[Pattern[c, Blank[]]], Power[Pattern[x, Blank[]], 4]]], -1]], Pattern[x, Blank[Symbol]]] := With[{q = Rt[-(a*c), 2]}, -Dist[e/2 + (c*d)/(2*q), Int[(f*x)^m/(q - c*x^2), x], x] + Dist[e/2 - (c*d)/(2*q), Int[(f*x)^m/(q + c*x^2), x], x]] /; FreeQ[{a, c, d, e, f, m}, x]
  1287. Int[Times[Power[Times[Optional[Pattern[f, Blank[]]], Pattern[x, Blank[]]], Optional[Pattern[m, Blank[]]]], Power[Plus[Pattern[d, Blank[]], Times[Optional[Pattern[e, Blank[]]], Power[Pattern[x, Blank[]], 2]]], Optional[Pattern[q, Blank[]]]], Power[Plus[Pattern[a, Blank[]], Times[Optional[Pattern[b, Blank[]]], Power[Pattern[x, Blank[]], 2]], Times[Optional[Pattern[c, Blank[]]], Power[Pattern[x, Blank[]], 4]]], -1]], Pattern[x, Blank[Symbol]]] := Int[ExpandIntegrand[((f*x)^m*(d + e*x^2)^q)/(a + b*x^2 + c*x^4), x], x] /; FreeQ[{a, b, c, d, e, f, m}, x] && NeQ[b^2 - 4*a*c, 0] && IntegerQ[q] && IntegerQ[m]
  1288. Int[Times[Power[Times[Optional[Pattern[f, Blank[]]], Pattern[x, Blank[]]], Optional[Pattern[m, Blank[]]]], Power[Plus[Pattern[d, Blank[]], Times[Optional[Pattern[e, Blank[]]], Power[Pattern[x, Blank[]], 2]]], Optional[Pattern[q, Blank[]]]], Power[Plus[Pattern[a, Blank[]], Times[Optional[Pattern[c, Blank[]]], Power[Pattern[x, Blank[]], 4]]], -1]], Pattern[x, Blank[Symbol]]] := Int[ExpandIntegrand[((f*x)^m*(d + e*x^2)^q)/(a + c*x^4), x], x] /; FreeQ[{a, c, d, e, f, m}, x] && IntegerQ[q] && IntegerQ[m]
  1289. Int[Times[Power[Times[Optional[Pattern[f, Blank[]]], Pattern[x, Blank[]]], Optional[Pattern[m, Blank[]]]], Power[Plus[Pattern[d, Blank[]], Times[Optional[Pattern[e, Blank[]]], Power[Pattern[x, Blank[]], 2]]], Optional[Pattern[q, Blank[]]]], Power[Plus[Pattern[a, Blank[]], Times[Optional[Pattern[b, Blank[]]], Power[Pattern[x, Blank[]], 2]], Times[Optional[Pattern[c, Blank[]]], Power[Pattern[x, Blank[]], 4]]], -1]], Pattern[x, Blank[Symbol]]] := Int[ExpandIntegrand[(f*x)^m, (d + e*x^2)^q/(a + b*x^2 + c*x^4), x], x] /; FreeQ[{a, b, c, d, e, f, m}, x] && NeQ[b^2 - 4*a*c, 0] && IntegerQ[q] && !IntegerQ[m]
  1290. Int[Times[Power[Times[Optional[Pattern[f, Blank[]]], Pattern[x, Blank[]]], Optional[Pattern[m, Blank[]]]], Power[Plus[Pattern[d, Blank[]], Times[Optional[Pattern[e, Blank[]]], Power[Pattern[x, Blank[]], 2]]], Optional[Pattern[q, Blank[]]]], Power[Plus[Pattern[a, Blank[]], Times[Optional[Pattern[c, Blank[]]], Power[Pattern[x, Blank[]], 4]]], -1]], Pattern[x, Blank[Symbol]]] := Int[ExpandIntegrand[(f*x)^m, (d + e*x^2)^q/(a + c*x^4), x], x] /; FreeQ[{a, c, d, e, f, m}, x] && IntegerQ[q] && !IntegerQ[m]
  1291. Int[Times[Power[Times[Optional[Pattern[f, Blank[]]], Pattern[x, Blank[]]], Optional[Pattern[m, Blank[]]]], Power[Plus[Optional[Pattern[d, Blank[]]], Times[Optional[Pattern[e, Blank[]]], Power[Pattern[x, Blank[]], 2]]], Pattern[q, Blank[]]], Power[Plus[Pattern[a, Blank[]], Times[Optional[Pattern[b, Blank[]]], Power[Pattern[x, Blank[]], 2]], Times[Optional[Pattern[c, Blank[]]], Power[Pattern[x, Blank[]], 4]]], -1]], Pattern[x, Blank[Symbol]]] := Dist[f^4/c^2, Int[(f*x)^(m - 4)*(c*d - b*e + c*e*x^2)*(d + e*x^2)^(q - 1), x], x] - Dist[f^4/c^2, Int[((f*x)^(m - 4)*(d + e*x^2)^(q - 1)*Simp[a*(c*d - b*e) + (b*c*d - b^2*e + a*c*e)*x^2, x])/(a + b*x^2 + c*x^4), x], x] /; FreeQ[{a, b, c, d, e, f}, x] && NeQ[b^2 - 4*a*c, 0] && !IntegerQ[q] && GtQ[q, 0] && GtQ[m, 3]
  1292. Int[Times[Power[Times[Optional[Pattern[f, Blank[]]], Pattern[x, Blank[]]], Optional[Pattern[m, Blank[]]]], Power[Plus[Optional[Pattern[d, Blank[]]], Times[Optional[Pattern[e, Blank[]]], Power[Pattern[x, Blank[]], 2]]], Pattern[q, Blank[]]], Power[Plus[Pattern[a, Blank[]], Times[Optional[Pattern[c, Blank[]]], Power[Pattern[x, Blank[]], 4]]], -1]], Pattern[x, Blank[Symbol]]] := Dist[f^4/c, Int[(f*x)^(m - 4)*(d + e*x^2)^q, x], x] - Dist[(a*f^4)/c, Int[((f*x)^(m - 4)*(d + e*x^2)^q)/(a + c*x^4), x], x] /; FreeQ[{a, c, d, e, f, q}, x] && !IntegerQ[q] && GtQ[m, 3]
  1293. Int[Times[Power[Times[Optional[Pattern[f, Blank[]]], Pattern[x, Blank[]]], Optional[Pattern[m, Blank[]]]], Power[Plus[Optional[Pattern[d, Blank[]]], Times[Optional[Pattern[e, Blank[]]], Power[Pattern[x, Blank[]], 2]]], Pattern[q, Blank[]]], Power[Plus[Pattern[a, Blank[]], Times[Optional[Pattern[b, Blank[]]], Power[Pattern[x, Blank[]], 2]], Times[Optional[Pattern[c, Blank[]]], Power[Pattern[x, Blank[]], 4]]], -1]], Pattern[x, Blank[Symbol]]] := Dist[(e*f^2)/c, Int[(f*x)^(m - 2)*(d + e*x^2)^(q - 1), x], x] - Dist[f^2/c, Int[((f*x)^(m - 2)*(d + e*x^2)^(q - 1)*Simp[a*e - (c*d - b*e)*x^2, x])/(a + b*x^2 + c*x^4), x], x] /; FreeQ[{a, b, c, d, e, f}, x] && NeQ[b^2 - 4*a*c, 0] && !IntegerQ[q] && GtQ[q, 0] && GtQ[m, 1] && LeQ[m, 3]
  1294. Int[Times[Power[Times[Optional[Pattern[f, Blank[]]], Pattern[x, Blank[]]], Optional[Pattern[m, Blank[]]]], Power[Plus[Optional[Pattern[d, Blank[]]], Times[Optional[Pattern[e, Blank[]]], Power[Pattern[x, Blank[]], 2]]], Pattern[q, Blank[]]], Power[Plus[Pattern[a, Blank[]], Times[Optional[Pattern[c, Blank[]]], Power[Pattern[x, Blank[]], 4]]], -1]], Pattern[x, Blank[Symbol]]] := Dist[(e*f^2)/c, Int[(f*x)^(m - 2)*(d + e*x^2)^(q - 1), x], x] - Dist[f^2/c, Int[((f*x)^(m - 2)*(d + e*x^2)^(q - 1)*Simp[a*e - c*d*x^2, x])/(a + c*x^4), x], x] /; FreeQ[{a, c, d, e, f}, x] && !IntegerQ[q] && GtQ[q, 0] && GtQ[m, 1] && LeQ[m, 3]
  1295. Int[Times[Power[Times[Optional[Pattern[f, Blank[]]], Pattern[x, Blank[]]], Pattern[m, Blank[]]], Power[Plus[Optional[Pattern[d, Blank[]]], Times[Optional[Pattern[e, Blank[]]], Power[Pattern[x, Blank[]], 2]]], Pattern[q, Blank[]]], Power[Plus[Pattern[a, Blank[]], Times[Optional[Pattern[b, Blank[]]], Power[Pattern[x, Blank[]], 2]], Times[Optional[Pattern[c, Blank[]]], Power[Pattern[x, Blank[]], 4]]], -1]], Pattern[x, Blank[Symbol]]] := Dist[d/a, Int[(f*x)^m*(d + e*x^2)^(q - 1), x], x] - Dist[1/(a*f^2), Int[((f*x)^(m + 2)*(d + e*x^2)^(q - 1)*Simp[b*d - a*e + c*d*x^2, x])/(a + b*x^2 + c*x^4), x], x] /; FreeQ[{a, b, c, d, e, f}, x] && NeQ[b^2 - 4*a*c, 0] && !IntegerQ[q] && GtQ[q, 0] && LtQ[m, 0]
  1296. Int[Times[Power[Times[Optional[Pattern[f, Blank[]]], Pattern[x, Blank[]]], Pattern[m, Blank[]]], Power[Plus[Optional[Pattern[d, Blank[]]], Times[Optional[Pattern[e, Blank[]]], Power[Pattern[x, Blank[]], 2]]], Pattern[q, Blank[]]], Power[Plus[Pattern[a, Blank[]], Times[Optional[Pattern[c, Blank[]]], Power[Pattern[x, Blank[]], 4]]], -1]], Pattern[x, Blank[Symbol]]] := Dist[d/a, Int[(f*x)^m*(d + e*x^2)^(q - 1), x], x] + Dist[1/(a*f^2), Int[((f*x)^(m + 2)*(d + e*x^2)^(q - 1)*Simp[a*e - c*d*x^2, x])/(a + c*x^4), x], x] /; FreeQ[{a, c, d, e, f}, x] && !IntegerQ[q] && GtQ[q, 0] && LtQ[m, 0]
  1297. Int[Times[Power[Times[Optional[Pattern[f, Blank[]]], Pattern[x, Blank[]]], Optional[Pattern[m, Blank[]]]], Power[Plus[Optional[Pattern[d, Blank[]]], Times[Optional[Pattern[e, Blank[]]], Power[Pattern[x, Blank[]], 2]]], Pattern[q, Blank[]]], Power[Plus[Pattern[a, Blank[]], Times[Optional[Pattern[b, Blank[]]], Power[Pattern[x, Blank[]], 2]], Times[Optional[Pattern[c, Blank[]]], Power[Pattern[x, Blank[]], 4]]], -1]], Pattern[x, Blank[Symbol]]] := Dist[(d^2*f^4)/(c*d^2 - b*d*e + a*e^2), Int[(f*x)^(m - 4)*(d + e*x^2)^q, x], x] - Dist[f^4/(c*d^2 - b*d*e + a*e^2), Int[((f*x)^(m - 4)*(d + e*x^2)^(q + 1)*Simp[a*d + (b*d - a*e)*x^2, x])/(a + b*x^2 + c*x^4), x], x] /; FreeQ[{a, b, c, d, e, f}, x] && NeQ[b^2 - 4*a*c, 0] && !IntegerQ[q] && LtQ[q, -1] && GtQ[m, 3]
  1298. Int[Times[Power[Times[Optional[Pattern[f, Blank[]]], Pattern[x, Blank[]]], Optional[Pattern[m, Blank[]]]], Power[Plus[Optional[Pattern[d, Blank[]]], Times[Optional[Pattern[e, Blank[]]], Power[Pattern[x, Blank[]], 2]]], Pattern[q, Blank[]]], Power[Plus[Pattern[a, Blank[]], Times[Optional[Pattern[c, Blank[]]], Power[Pattern[x, Blank[]], 4]]], -1]], Pattern[x, Blank[Symbol]]] := Dist[(d^2*f^4)/(c*d^2 + a*e^2), Int[(f*x)^(m - 4)*(d + e*x^2)^q, x], x] - Dist[(a*f^4)/(c*d^2 + a*e^2), Int[((f*x)^(m - 4)*(d + e*x^2)^(q + 1)*(d - e*x^2))/(a + c*x^4), x], x] /; FreeQ[{a, c, d, e, f}, x] && !IntegerQ[q] && LtQ[q, -1] && GtQ[m, 3]
  1299. Int[Times[Power[Times[Optional[Pattern[f, Blank[]]], Pattern[x, Blank[]]], Optional[Pattern[m, Blank[]]]], Power[Plus[Optional[Pattern[d, Blank[]]], Times[Optional[Pattern[e, Blank[]]], Power[Pattern[x, Blank[]], 2]]], Pattern[q, Blank[]]], Power[Plus[Pattern[a, Blank[]], Times[Optional[Pattern[b, Blank[]]], Power[Pattern[x, Blank[]], 2]], Times[Optional[Pattern[c, Blank[]]], Power[Pattern[x, Blank[]], 4]]], -1]], Pattern[x, Blank[Symbol]]] := -Dist[(d*e*f^2)/(c*d^2 - b*d*e + a*e^2), Int[(f*x)^(m - 2)*(d + e*x^2)^q, x], x] + Dist[f^2/(c*d^2 - b*d*e + a*e^2), Int[((f*x)^(m - 2)*(d + e*x^2)^(q + 1)*Simp[a*e + c*d*x^2, x])/(a + b*x^2 + c*x^4), x], x] /; FreeQ[{a, b, c, d, e, f}, x] && NeQ[b^2 - 4*a*c, 0] && !IntegerQ[q] && LtQ[q, -1] && GtQ[m, 1] && LeQ[m, 3]
  1300. Int[Times[Power[Times[Optional[Pattern[f, Blank[]]], Pattern[x, Blank[]]], Optional[Pattern[m, Blank[]]]], Power[Plus[Optional[Pattern[d, Blank[]]], Times[Optional[Pattern[e, Blank[]]], Power[Pattern[x, Blank[]], 2]]], Pattern[q, Blank[]]], Power[Plus[Pattern[a, Blank[]], Times[Optional[Pattern[c, Blank[]]], Power[Pattern[x, Blank[]], 4]]], -1]], Pattern[x, Blank[Symbol]]] := -Dist[(d*e*f^2)/(c*d^2 + a*e^2), Int[(f*x)^(m - 2)*(d + e*x^2)^q, x], x] + Dist[f^2/(c*d^2 + a*e^2), Int[((f*x)^(m - 2)*(d + e*x^2)^(q + 1)*Simp[a*e + c*d*x^2, x])/(a + c*x^4), x], x] /; FreeQ[{a, c, d, e, f}, x] && !IntegerQ[q] && LtQ[q, -1] && GtQ[m, 1] && LeQ[m, 3]
  1301. Int[Times[Power[Times[Optional[Pattern[f, Blank[]]], Pattern[x, Blank[]]], Optional[Pattern[m, Blank[]]]], Power[Plus[Pattern[d, Blank[]], Times[Optional[Pattern[e, Blank[]]], Power[Pattern[x, Blank[]], 2]]], Pattern[q, Blank[]]], Power[Plus[Pattern[a, Blank[]], Times[Optional[Pattern[b, Blank[]]], Power[Pattern[x, Blank[]], 2]], Times[Optional[Pattern[c, Blank[]]], Power[Pattern[x, Blank[]], 4]]], -1]], Pattern[x, Blank[Symbol]]] := Dist[e^2/(c*d^2 - b*d*e + a*e^2), Int[(f*x)^m*(d + e*x^2)^q, x], x] + Dist[1/(c*d^2 - b*d*e + a*e^2), Int[((f*x)^m*(d + e*x^2)^(q + 1)*Simp[c*d - b*e - c*e*x^2, x])/(a + b*x^2 + c*x^4), x], x] /; FreeQ[{a, b, c, d, e, f, m}, x] && NeQ[b^2 - 4*a*c, 0] && !IntegerQ[q] && LtQ[q, -1]
  1302. Int[Times[Power[Times[Optional[Pattern[f, Blank[]]], Pattern[x, Blank[]]], Optional[Pattern[m, Blank[]]]], Power[Plus[Pattern[d, Blank[]], Times[Optional[Pattern[e, Blank[]]], Power[Pattern[x, Blank[]], 2]]], Pattern[q, Blank[]]], Power[Plus[Pattern[a, Blank[]], Times[Optional[Pattern[c, Blank[]]], Power[Pattern[x, Blank[]], 4]]], -1]], Pattern[x, Blank[Symbol]]] := Dist[e^2/(c*d^2 + a*e^2), Int[(f*x)^m*(d + e*x^2)^q, x], x] + Dist[c/(c*d^2 + a*e^2), Int[((f*x)^m*(d + e*x^2)^(q + 1)*(d - e*x^2))/(a + c*x^4), x], x] /; FreeQ[{a, c, d, e, f, m}, x] && !IntegerQ[q] && LtQ[q, -1]
  1303. Int[Times[Power[Times[Optional[Pattern[f, Blank[]]], Pattern[x, Blank[]]], Optional[Pattern[m, Blank[]]]], Power[Plus[Pattern[d, Blank[]], Times[Optional[Pattern[e, Blank[]]], Power[Pattern[x, Blank[]], 2]]], Pattern[q, Blank[]]], Power[Plus[Pattern[a, Blank[]], Times[Optional[Pattern[b, Blank[]]], Power[Pattern[x, Blank[]], 2]], Times[Optional[Pattern[c, Blank[]]], Power[Pattern[x, Blank[]], 4]]], -1]], Pattern[x, Blank[Symbol]]] := Int[ExpandIntegrand[(d + e*x^2)^q, (f*x)^m/(a + b*x^2 + c*x^4), x], x] /; FreeQ[{a, b, c, d, e, f, q}, x] && NeQ[b^2 - 4*a*c, 0] && !IntegerQ[q] && IntegerQ[m]
  1304. Int[Times[Power[Times[Optional[Pattern[f, Blank[]]], Pattern[x, Blank[]]], Optional[Pattern[m, Blank[]]]], Power[Plus[Pattern[d, Blank[]], Times[Optional[Pattern[e, Blank[]]], Power[Pattern[x, Blank[]], 2]]], Pattern[q, Blank[]]], Power[Plus[Pattern[a, Blank[]], Times[Optional[Pattern[c, Blank[]]], Power[Pattern[x, Blank[]], 4]]], -1]], Pattern[x, Blank[Symbol]]] := Int[ExpandIntegrand[(d + e*x^2)^q, (f*x)^m/(a + c*x^4), x], x] /; FreeQ[{a, c, d, e, f, q}, x] && !IntegerQ[q] && IntegerQ[m]
  1305. Int[Times[Power[Times[Optional[Pattern[f, Blank[]]], Pattern[x, Blank[]]], Optional[Pattern[m, Blank[]]]], Power[Plus[Pattern[d, Blank[]], Times[Optional[Pattern[e, Blank[]]], Power[Pattern[x, Blank[]], 2]]], Pattern[q, Blank[]]], Power[Plus[Pattern[a, Blank[]], Times[Optional[Pattern[b, Blank[]]], Power[Pattern[x, Blank[]], 2]], Times[Optional[Pattern[c, Blank[]]], Power[Pattern[x, Blank[]], 4]]], -1]], Pattern[x, Blank[Symbol]]] := Int[ExpandIntegrand[(f*x)^m*(d + e*x^2)^q, 1/(a + b*x^2 + c*x^4), x], x] /; FreeQ[{a, b, c, d, e, f, m, q}, x] && NeQ[b^2 - 4*a*c, 0] && !IntegerQ[q] && !IntegerQ[m]
  1306. Int[Times[Power[Times[Optional[Pattern[f, Blank[]]], Pattern[x, Blank[]]], Optional[Pattern[m, Blank[]]]], Power[Plus[Pattern[d, Blank[]], Times[Optional[Pattern[e, Blank[]]], Power[Pattern[x, Blank[]], 2]]], Pattern[q, Blank[]]], Power[Plus[Pattern[a, Blank[]], Times[Optional[Pattern[c, Blank[]]], Power[Pattern[x, Blank[]], 4]]], -1]], Pattern[x, Blank[Symbol]]] := Int[ExpandIntegrand[(f*x)^m*(d + e*x^2)^q, 1/(a + c*x^4), x], x] /; FreeQ[{a, c, d, e, f, m, q}, x] && !IntegerQ[q] && !IntegerQ[m]
  1307. Int[Times[Power[Times[Optional[Pattern[f, Blank[]]], Pattern[x, Blank[]]], Optional[Pattern[m, Blank[]]]], Power[Plus[Pattern[d, Blank[]], Times[Optional[Pattern[e, Blank[]]], Power[Pattern[x, Blank[]], 2]]], Pattern[q, Blank[]]], Power[Plus[Pattern[a, Blank[]], Times[Optional[Pattern[b, Blank[]]], Power[Pattern[x, Blank[]], 2]], Times[Optional[Pattern[c, Blank[]]], Power[Pattern[x, Blank[]], 4]]], -1]], Pattern[x, Blank[Symbol]]] := With[{r = Rt[b^2 - 4*a*c, 2]}, Dist[(2*c)/r, Int[((f*x)^m*(d + e*x^2)^q)/(b - r + 2*c*x^2), x], x] - Dist[(2*c)/r, Int[((f*x)^m*(d + e*x^2)^q)/(b + r + 2*c*x^2), x], x]] /; FreeQ[{a, b, c, d, e, f, m, q}, x] && NeQ[b^2 - 4*a*c, 0]
  1308. Int[Times[Power[Times[Optional[Pattern[f, Blank[]]], Pattern[x, Blank[]]], Optional[Pattern[m, Blank[]]]], Power[Plus[Pattern[d, Blank[]], Times[Optional[Pattern[e, Blank[]]], Power[Pattern[x, Blank[]], 2]]], Pattern[q, Blank[]]], Power[Plus[Pattern[a, Blank[]], Times[Optional[Pattern[c, Blank[]]], Power[Pattern[x, Blank[]], 4]]], -1]], Pattern[x, Blank[Symbol]]] := With[{r = Rt[-(a*c), 2]}, -Dist[c/(2*r), Int[((f*x)^m*(d + e*x^2)^q)/(r - c*x^2), x], x] - Dist[c/(2*r), Int[((f*x)^m*(d + e*x^2)^q)/(r + c*x^2), x], x]] /; FreeQ[{a, c, d, e, f, m, q}, x]
  1309. Int[Times[Power[Times[Optional[Pattern[f, Blank[]]], Pattern[x, Blank[]]], Pattern[m, Blank[]]], Power[Plus[Optional[Pattern[d, Blank[]]], Times[Optional[Pattern[e, Blank[]]], Power[Pattern[x, Blank[]], 2]]], -1], Power[Plus[Optional[Pattern[a, Blank[]]], Times[Optional[Pattern[b, Blank[]]], Power[Pattern[x, Blank[]], 2]], Times[Optional[Pattern[c, Blank[]]], Power[Pattern[x, Blank[]], 4]]], Optional[Pattern[p, Blank[]]]]], Pattern[x, Blank[Symbol]]] := Dist[1/d^2, Int[(f*x)^m*(a*d + (b*d - a*e)*x^2)*(a + b*x^2 + c*x^4)^(p - 1), x], x] + Dist[(c*d^2 - b*d*e + a*e^2)/(d^2*f^4), Int[((f*x)^(m + 4)*(a + b*x^2 + c*x^4)^(p - 1))/(d + e*x^2), x], x] /; FreeQ[{a, b, c, d, e, f}, x] && NeQ[b^2 - 4*a*c, 0] && GtQ[p, 0] && LtQ[m, -2]
  1310. Int[Times[Power[Times[Optional[Pattern[f, Blank[]]], Pattern[x, Blank[]]], Pattern[m, Blank[]]], Power[Plus[Optional[Pattern[d, Blank[]]], Times[Optional[Pattern[e, Blank[]]], Power[Pattern[x, Blank[]], 2]]], -1], Power[Plus[Pattern[a, Blank[]], Times[Optional[Pattern[c, Blank[]]], Power[Pattern[x, Blank[]], 4]]], Optional[Pattern[p, Blank[]]]]], Pattern[x, Blank[Symbol]]] := Dist[a/d^2, Int[(f*x)^m*(d - e*x^2)*(a + c*x^4)^(p - 1), x], x] + Dist[(c*d^2 + a*e^2)/(d^2*f^4), Int[((f*x)^(m + 4)*(a + c*x^4)^(p - 1))/(d + e*x^2), x], x] /; FreeQ[{a, c, d, e, f}, x] && GtQ[p, 0] && LtQ[m, -2]
  1311. Int[Times[Power[Times[Optional[Pattern[f, Blank[]]], Pattern[x, Blank[]]], Pattern[m, Blank[]]], Power[Plus[Optional[Pattern[d, Blank[]]], Times[Optional[Pattern[e, Blank[]]], Power[Pattern[x, Blank[]], 2]]], -1], Power[Plus[Optional[Pattern[a, Blank[]]], Times[Optional[Pattern[b, Blank[]]], Power[Pattern[x, Blank[]], 2]], Times[Optional[Pattern[c, Blank[]]], Power[Pattern[x, Blank[]], 4]]], Optional[Pattern[p, Blank[]]]]], Pattern[x, Blank[Symbol]]] := Dist[1/(d*e), Int[(f*x)^m*(a*e + c*d*x^2)*(a + b*x^2 + c*x^4)^(p - 1), x], x] - Dist[(c*d^2 - b*d*e + a*e^2)/(d*e*f^2), Int[((f*x)^(m + 2)*(a + b*x^2 + c*x^4)^(p - 1))/(d + e*x^2), x], x] /; FreeQ[{a, b, c, d, e, f}, x] && NeQ[b^2 - 4*a*c, 0] && GtQ[p, 0] && LtQ[m, 0]
  1312. Int[Times[Power[Times[Optional[Pattern[f, Blank[]]], Pattern[x, Blank[]]], Pattern[m, Blank[]]], Power[Plus[Optional[Pattern[d, Blank[]]], Times[Optional[Pattern[e, Blank[]]], Power[Pattern[x, Blank[]], 2]]], -1], Power[Plus[Pattern[a, Blank[]], Times[Optional[Pattern[c, Blank[]]], Power[Pattern[x, Blank[]], 4]]], Optional[Pattern[p, Blank[]]]]], Pattern[x, Blank[Symbol]]] := Dist[1/(d*e), Int[(f*x)^m*(a*e + c*d*x^2)*(a + c*x^4)^(p - 1), x], x] - Dist[(c*d^2 + a*e^2)/(d*e*f^2), Int[((f*x)^(m + 2)*(a + c*x^4)^(p - 1))/(d + e*x^2), x], x] /; FreeQ[{a, c, d, e, f}, x] && GtQ[p, 0] && LtQ[m, 0]
  1313. Int[Times[Power[Times[Optional[Pattern[f, Blank[]]], Pattern[x, Blank[]]], Optional[Pattern[m, Blank[]]]], Power[Plus[Optional[Pattern[d, Blank[]]], Times[Optional[Pattern[e, Blank[]]], Power[Pattern[x, Blank[]], 2]]], -1], Power[Plus[Optional[Pattern[a, Blank[]]], Times[Optional[Pattern[b, Blank[]]], Power[Pattern[x, Blank[]], 2]], Times[Optional[Pattern[c, Blank[]]], Power[Pattern[x, Blank[]], 4]]], Pattern[p, Blank[]]]], Pattern[x, Blank[Symbol]]] := -Dist[f^4/(c*d^2 - b*d*e + a*e^2), Int[(f*x)^(m - 4)*(a*d + (b*d - a*e)*x^2)*(a + b*x^2 + c*x^4)^p, x], x] + Dist[(d^2*f^4)/(c*d^2 - b*d*e + a*e^2), Int[((f*x)^(m - 4)*(a + b*x^2 + c*x^4)^(p + 1))/(d + e*x^2), x], x] /; FreeQ[{a, b, c, d, e, f}, x] && NeQ[b^2 - 4*a*c, 0] && LtQ[p, -1] && GtQ[m, 2]
  1314. Int[Times[Power[Times[Optional[Pattern[f, Blank[]]], Pattern[x, Blank[]]], Optional[Pattern[m, Blank[]]]], Power[Plus[Optional[Pattern[d, Blank[]]], Times[Optional[Pattern[e, Blank[]]], Power[Pattern[x, Blank[]], 2]]], -1], Power[Plus[Pattern[a, Blank[]], Times[Optional[Pattern[c, Blank[]]], Power[Pattern[x, Blank[]], 4]]], Pattern[p, Blank[]]]], Pattern[x, Blank[Symbol]]] := -Dist[(a*f^4)/(c*d^2 + a*e^2), Int[(f*x)^(m - 4)*(d - e*x^2)*(a + c*x^4)^p, x], x] + Dist[(d^2*f^4)/(c*d^2 + a*e^2), Int[((f*x)^(m - 4)*(a + c*x^4)^(p + 1))/(d + e*x^2), x], x] /; FreeQ[{a, c, d, e, f}, x] && LtQ[p, -1] && GtQ[m, 2]
  1315. Int[Times[Power[Times[Optional[Pattern[f, Blank[]]], Pattern[x, Blank[]]], Optional[Pattern[m, Blank[]]]], Power[Plus[Optional[Pattern[d, Blank[]]], Times[Optional[Pattern[e, Blank[]]], Power[Pattern[x, Blank[]], 2]]], -1], Power[Plus[Optional[Pattern[a, Blank[]]], Times[Optional[Pattern[b, Blank[]]], Power[Pattern[x, Blank[]], 2]], Times[Optional[Pattern[c, Blank[]]], Power[Pattern[x, Blank[]], 4]]], Pattern[p, Blank[]]]], Pattern[x, Blank[Symbol]]] := Dist[f^2/(c*d^2 - b*d*e + a*e^2), Int[(f*x)^(m - 2)*(a*e + c*d*x^2)*(a + b*x^2 + c*x^4)^p, x], x] - Dist[(d*e*f^2)/(c*d^2 - b*d*e + a*e^2), Int[((f*x)^(m - 2)*(a + b*x^2 + c*x^4)^(p + 1))/(d + e*x^2), x], x] /; FreeQ[{a, b, c, d, e, f}, x] && NeQ[b^2 - 4*a*c, 0] && LtQ[p, -1] && GtQ[m, 0]
  1316. Int[Times[Power[Times[Optional[Pattern[f, Blank[]]], Pattern[x, Blank[]]], Optional[Pattern[m, Blank[]]]], Power[Plus[Optional[Pattern[d, Blank[]]], Times[Optional[Pattern[e, Blank[]]], Power[Pattern[x, Blank[]], 2]]], -1], Power[Plus[Pattern[a, Blank[]], Times[Optional[Pattern[c, Blank[]]], Power[Pattern[x, Blank[]], 4]]], Pattern[p, Blank[]]]], Pattern[x, Blank[Symbol]]] := Dist[f^2/(c*d^2 + a*e^2), Int[(f*x)^(m - 2)*(a*e + c*d*x^2)*(a + c*x^4)^p, x], x] - Dist[(d*e*f^2)/(c*d^2 + a*e^2), Int[((f*x)^(m - 2)*(a + c*x^4)^(p + 1))/(d + e*x^2), x], x] /; FreeQ[{a, c, d, e, f}, x] && LtQ[p, -1] && GtQ[m, 0]
  1317. Int[Times[Power[Pattern[x, Blank[]], 2], Power[Plus[Pattern[d, Blank[]], Times[Optional[Pattern[e, Blank[]]], Power[Pattern[x, Blank[]], 2]]], -1], Power[Plus[Pattern[a, Blank[]], Times[Optional[Pattern[b, Blank[]]], Power[Pattern[x, Blank[]], 2]], Times[Optional[Pattern[c, Blank[]]], Power[Pattern[x, Blank[]], 4]]], Rational[-1, 2]]], Pattern[x, Blank[Symbol]]] := Dist[d/(2*d*e), Int[1/Sqrt[a + b*x^2 + c*x^4], x], x] - Dist[d/(2*d*e), Int[(d - e*x^2)/((d + e*x^2)*Sqrt[a + b*x^2 + c*x^4]), x], x] /; FreeQ[{a, b, c, d, e}, x] && NeQ[b^2 - 4*a*c, 0] && NeQ[c*d^2 - b*d*e + a*e^2, 0] && PosQ[c/a] && EqQ[c*d^2 - a*e^2, 0]
  1318. Int[Times[Power[Pattern[x, Blank[]], 2], Power[Plus[Pattern[d, Blank[]], Times[Optional[Pattern[e, Blank[]]], Power[Pattern[x, Blank[]], 2]]], -1], Power[Plus[Pattern[a, Blank[]], Times[Optional[Pattern[c, Blank[]]], Power[Pattern[x, Blank[]], 4]]], Rational[-1, 2]]], Pattern[x, Blank[Symbol]]] := Dist[d/(2*d*e), Int[1/Sqrt[a + c*x^4], x], x] - Dist[d/(2*d*e), Int[(d - e*x^2)/((d + e*x^2)*Sqrt[a + c*x^4]), x], x] /; FreeQ[{a, c, d, e}, x] && NeQ[c*d^2 + a*e^2, 0] && PosQ[c/a] && EqQ[c*d^2 - a*e^2, 0]
  1319. Int[Times[Power[Pattern[x, Blank[]], 2], Power[Plus[Pattern[d, Blank[]], Times[Optional[Pattern[e, Blank[]]], Power[Pattern[x, Blank[]], 2]]], -1], Power[Plus[Pattern[a, Blank[]], Times[Optional[Pattern[b, Blank[]]], Power[Pattern[x, Blank[]], 2]], Times[Optional[Pattern[c, Blank[]]], Power[Pattern[x, Blank[]], 4]]], Rational[-1, 2]]], Pattern[x, Blank[Symbol]]] := With[{q = Rt[c/a, 2]}, -Dist[(a*(e + d*q))/(c*d^2 - a*e^2), Int[1/Sqrt[a + b*x^2 + c*x^4], x], x] + Dist[(a*d*(e + d*q))/(c*d^2 - a*e^2), Int[(1 + q*x^2)/((d + e*x^2)*Sqrt[a + b*x^2 + c*x^4]), x], x]] /; FreeQ[{a, b, c, d, e}, x] && NeQ[b^2 - 4*a*c, 0] && NeQ[c*d^2 - b*d*e + a*e^2, 0] && PosQ[c/a] && NeQ[c*d^2 - a*e^2, 0]
  1320. Int[Times[Power[Pattern[x, Blank[]], 2], Power[Plus[Pattern[d, Blank[]], Times[Optional[Pattern[e, Blank[]]], Power[Pattern[x, Blank[]], 2]]], -1], Power[Plus[Pattern[a, Blank[]], Times[Optional[Pattern[c, Blank[]]], Power[Pattern[x, Blank[]], 4]]], Rational[-1, 2]]], Pattern[x, Blank[Symbol]]] := With[{q = Rt[c/a, 2]}, -Dist[(a*(e + d*q))/(c*d^2 - a*e^2), Int[1/Sqrt[a + c*x^4], x], x] + Dist[(a*d*(e + d*q))/(c*d^2 - a*e^2), Int[(1 + q*x^2)/((d + e*x^2)*Sqrt[a + c*x^4]), x], x]] /; FreeQ[{a, c, d, e}, x] && NeQ[c*d^2 + a*e^2, 0] && PosQ[c/a] && NeQ[c*d^2 - a*e^2, 0]
  1321. Int[Times[Power[Pattern[x, Blank[]], 4], Power[Plus[Pattern[d, Blank[]], Times[Optional[Pattern[e, Blank[]]], Power[Pattern[x, Blank[]], 2]]], -1], Power[Plus[Pattern[a, Blank[]], Times[Optional[Pattern[b, Blank[]]], Power[Pattern[x, Blank[]], 2]], Times[Optional[Pattern[c, Blank[]]], Power[Pattern[x, Blank[]], 4]]], Rational[-1, 2]]], Pattern[x, Blank[Symbol]]] := -Dist[(e^2)^(-1), Int[(d - e*x^2)/Sqrt[a + b*x^2 + c*x^4], x], x] + Dist[d^2/e^2, Int[1/((d + e*x^2)*Sqrt[a + b*x^2 + c*x^4]), x], x] /; FreeQ[{a, b, c, d, e}, x] && NeQ[b^2 - 4*a*c, 0] && PosQ[c/a] && EqQ[c*d^2 - a*e^2, 0]
  1322. Int[Times[Power[Pattern[x, Blank[]], 4], Power[Plus[Pattern[d, Blank[]], Times[Optional[Pattern[e, Blank[]]], Power[Pattern[x, Blank[]], 2]]], -1], Power[Plus[Pattern[a, Blank[]], Times[Optional[Pattern[c, Blank[]]], Power[Pattern[x, Blank[]], 4]]], Rational[-1, 2]]], Pattern[x, Blank[Symbol]]] := -Dist[(e^2)^(-1), Int[(d - e*x^2)/Sqrt[a + c*x^4], x], x] + Dist[d^2/e^2, Int[1/((d + e*x^2)*Sqrt[a + c*x^4]), x], x] /; FreeQ[{a, c, d, e}, x] && PosQ[c/a] && EqQ[c*d^2 - a*e^2, 0]
  1323. Int[Times[Power[Pattern[x, Blank[]], 4], Power[Plus[Pattern[d, Blank[]], Times[Optional[Pattern[e, Blank[]]], Power[Pattern[x, Blank[]], 2]]], -1], Power[Plus[Pattern[a, Blank[]], Times[Optional[Pattern[b, Blank[]]], Power[Pattern[x, Blank[]], 2]], Times[Optional[Pattern[c, Blank[]]], Power[Pattern[x, Blank[]], 4]]], Rational[-1, 2]]], Pattern[x, Blank[Symbol]]] := With[{q = Rt[c/a, 2]}, -Dist[(e*q)^(-1), Int[(1 - q*x^2)/Sqrt[a + b*x^2 + c*x^4], x], x] + Dist[d^2/(e*(e - d*q)), Int[(1 + q*x^2)/((d + e*x^2)*Sqrt[a + b*x^2 + c*x^4]), x], x] /; EqQ[2*c*d - a*e*q, 0]] /; FreeQ[{a, b, c, d, e}, x] && NeQ[b^2 - 4*a*c, 0] && PosQ[c/a] && NeQ[c*d^2 - a*e^2, 0]
  1324. Int[Times[Power[Pattern[x, Blank[]], 4], Power[Plus[Pattern[d, Blank[]], Times[Optional[Pattern[e, Blank[]]], Power[Pattern[x, Blank[]], 2]]], -1], Power[Plus[Pattern[a, Blank[]], Times[Optional[Pattern[c, Blank[]]], Power[Pattern[x, Blank[]], 4]]], Rational[-1, 2]]], Pattern[x, Blank[Symbol]]] := With[{q = Rt[c/a, 2]}, -Dist[(e*q)^(-1), Int[(1 - q*x^2)/Sqrt[a + c*x^4], x], x] + Dist[d^2/(e*(e - d*q)), Int[(1 + q*x^2)/((d + e*x^2)*Sqrt[a + c*x^4]), x], x] /; EqQ[2*c*d - a*e*q, 0]] /; FreeQ[{a, c, d, e}, x] && PosQ[c/a] && NeQ[c*d^2 - a*e^2, 0]
  1325. Int[Times[Power[Pattern[x, Blank[]], 4], Power[Plus[Pattern[d, Blank[]], Times[Optional[Pattern[e, Blank[]]], Power[Pattern[x, Blank[]], 2]]], -1], Power[Plus[Pattern[a, Blank[]], Times[Optional[Pattern[b, Blank[]]], Power[Pattern[x, Blank[]], 2]], Times[Optional[Pattern[c, Blank[]]], Power[Pattern[x, Blank[]], 4]]], Rational[-1, 2]]], Pattern[x, Blank[Symbol]]] := With[{q = Rt[c/a, 2]}, -Dist[(2*c*d - a*e*q)/(c*e*(e - d*q)), Int[1/Sqrt[a + b*x^2 + c*x^4], x], x] + (-Dist[1/(e*q), Int[(1 - q*x^2)/Sqrt[a + b*x^2 + c*x^4], x], x] + Dist[d^2/(e*(e - d*q)), Int[(1 + q*x^2)/((d + e*x^2)*Sqrt[a + b*x^2 + c*x^4]), x], x])] /; FreeQ[{a, b, c, d, e}, x] && NeQ[b^2 - 4*a*c, 0] && PosQ[c/a] && NeQ[c*d^2 - a*e^2, 0]
  1326. Int[Times[Power[Pattern[x, Blank[]], 4], Power[Plus[Pattern[d, Blank[]], Times[Optional[Pattern[e, Blank[]]], Power[Pattern[x, Blank[]], 2]]], -1], Power[Plus[Pattern[a, Blank[]], Times[Optional[Pattern[c, Blank[]]], Power[Pattern[x, Blank[]], 4]]], Rational[-1, 2]]], Pattern[x, Blank[Symbol]]] := With[{q = Rt[c/a, 2]}, -Dist[(2*c*d - a*e*q)/(c*e*(e - d*q)), Int[1/Sqrt[a + c*x^4], x], x] + (-Dist[1/(e*q), Int[(1 - q*x^2)/Sqrt[a + c*x^4], x], x] + Dist[d^2/(e*(e - d*q)), Int[(1 + q*x^2)/((d + e*x^2)*Sqrt[a + c*x^4]), x], x])] /; FreeQ[{a, c, d, e}, x] && PosQ[c/a] && NeQ[c*d^2 - a*e^2, 0]
  1327. Int[Times[Power[Pattern[x, Blank[]], Pattern[m, Blank[]]], Power[Plus[Pattern[d, Blank[]], Times[Optional[Pattern[e, Blank[]]], Power[Pattern[x, Blank[]], 2]]], -1], Power[Plus[Pattern[a, Blank[]], Times[Optional[Pattern[b, Blank[]]], Power[Pattern[x, Blank[]], 2]], Times[Optional[Pattern[c, Blank[]]], Power[Pattern[x, Blank[]], 4]]], Rational[-1, 2]]], Pattern[x, Blank[Symbol]]] := Simp[(x^(m - 5)*Sqrt[a + b*x^2 + c*x^4])/(c*e*(m - 3)), x] - Dist[1/(c*e*(m - 3)), Int[(x^(m - 6)*Simp[a*d*(m - 5) + (a*e*(m - 5) + b*d*(m - 4))*x^2 + (b*e*(m - 4) + c*d*(m - 3))*x^4, x])/((d + e*x^2)*Sqrt[a + b*x^2 + c*x^4]), x], x] /; FreeQ[{a, b, c, d, e}, x] && NeQ[b^2 - 4*a*c, 0] && IGtQ[m/2, 2]
  1328. Int[Times[Power[Pattern[x, Blank[]], Pattern[m, Blank[]]], Power[Plus[Pattern[d, Blank[]], Times[Optional[Pattern[e, Blank[]]], Power[Pattern[x, Blank[]], 2]]], -1], Power[Plus[Pattern[a, Blank[]], Times[Optional[Pattern[c, Blank[]]], Power[Pattern[x, Blank[]], 4]]], Rational[-1, 2]]], Pattern[x, Blank[Symbol]]] := Simp[(x^(m - 5)*Sqrt[a + c*x^4])/(c*e*(m - 3)), x] - Dist[1/(c*e*(m - 3)), Int[(x^(m - 6)*Simp[a*d*(m - 5) + a*e*(m - 5)*x^2 + c*d*(m - 3)*x^4, x])/((d + e*x^2)*Sqrt[a + c*x^4]), x], x] /; FreeQ[{a, c, d, e}, x] && IGtQ[m/2, 2]
  1329. Int[Times[Power[Pattern[x, Blank[]], Pattern[m, Blank[]]], Power[Plus[Pattern[d, Blank[]], Times[Optional[Pattern[e, Blank[]]], Power[Pattern[x, Blank[]], 2]]], -1], Power[Plus[Pattern[a, Blank[]], Times[Optional[Pattern[b, Blank[]]], Power[Pattern[x, Blank[]], 2]], Times[Optional[Pattern[c, Blank[]]], Power[Pattern[x, Blank[]], 4]]], Rational[-1, 2]]], Pattern[x, Blank[Symbol]]] := Simp[(x^(m + 1)*Sqrt[a + b*x^2 + c*x^4])/(a*d*(m + 1)), x] - Dist[1/(a*d*(m + 1)), Int[(x^(m + 2)*Simp[a*e*(m + 1) + b*d*(m + 2) + (b*e*(m + 2) + c*d*(m + 3))*x^2 + c*e*(m + 3)*x^4, x])/((d + e*x^2)*Sqrt[a + b*x^2 + c*x^4]), x], x] /; FreeQ[{a, b, c, d, e}, x] && NeQ[b^2 - 4*a*c, 0] && ILtQ[m/2, 0]
  1330. Int[Times[Power[Pattern[x, Blank[]], Pattern[m, Blank[]]], Power[Plus[Pattern[d, Blank[]], Times[Optional[Pattern[e, Blank[]]], Power[Pattern[x, Blank[]], 2]]], -1], Power[Plus[Pattern[a, Blank[]], Times[Optional[Pattern[c, Blank[]]], Power[Pattern[x, Blank[]], 4]]], Rational[-1, 2]]], Pattern[x, Blank[Symbol]]] := Simp[(x^(m + 1)*Sqrt[a + c*x^4])/(a*d*(m + 1)), x] - Dist[1/(a*d*(m + 1)), Int[(x^(m + 2)*Simp[a*e*(m + 1) + c*d*(m + 3)*x^2 + c*e*(m + 3)*x^4, x])/((d + e*x^2)*Sqrt[a + c*x^4]), x], x] /; FreeQ[{a, c, d, e}, x] && ILtQ[m/2, 0]
  1331. Int[Times[Power[Pattern[x, Blank[]], Pattern[m, Blank[]]], Power[Plus[Pattern[d, Blank[]], Times[Optional[Pattern[e, Blank[]]], Power[Pattern[x, Blank[]], 2]]], Rational[-1, 2]], Power[Plus[Pattern[a, Blank[]], Times[Optional[Pattern[b, Blank[]]], Power[Pattern[x, Blank[]], 2]], Times[Optional[Pattern[c, Blank[]]], Power[Pattern[x, Blank[]], 4]]], Rational[-1, 2]]], Pattern[x, Blank[Symbol]]] := Dist[(x^3*Sqrt[e + d/x^2]*Sqrt[c + b/x^2 + a/x^4])/(Sqrt[d + e*x^2]*Sqrt[a + b*x^2 + c*x^4]), Int[x^(m - 3)/(Sqrt[e + d/x^2]*Sqrt[c + b/x^2 + a/x^4]), x], x] /; FreeQ[{a, b, c, d, e}, x] && NeQ[b^2 - 4*a*c, 0] && IntegerQ[m/2]
  1332. Int[Times[Power[Pattern[x, Blank[]], Pattern[m, Blank[]]], Power[Plus[Pattern[d, Blank[]], Times[Optional[Pattern[e, Blank[]]], Power[Pattern[x, Blank[]], 2]]], Rational[-1, 2]], Power[Plus[Pattern[a, Blank[]], Times[Optional[Pattern[c, Blank[]]], Power[Pattern[x, Blank[]], 4]]], Rational[-1, 2]]], Pattern[x, Blank[Symbol]]] := Dist[(x^3*Sqrt[e + d/x^2]*Sqrt[c + a/x^4])/(Sqrt[d + e*x^2]*Sqrt[a + c*x^4]), Int[x^(m - 3)/(Sqrt[e + d/x^2]*Sqrt[c + a/x^4]), x], x] /; FreeQ[{a, c, d, e}, x] && IntegerQ[m/2]
  1333. Int[Times[Power[Pattern[x, Blank[]], Pattern[m, Blank[]]], Power[Plus[Pattern[d, Blank[]], Times[Optional[Pattern[e, Blank[]]], Power[Pattern[x, Blank[]], 2]]], Pattern[q, Blank[]]], Power[Plus[Pattern[a, Blank[]], Times[Optional[Pattern[b, Blank[]]], Power[Pattern[x, Blank[]], 2]], Times[Optional[Pattern[c, Blank[]]], Power[Pattern[x, Blank[]], 4]]], Pattern[p, Blank[]]]], Pattern[x, Blank[Symbol]]] := With[{f = Coeff[PolynomialRemainder[x^m*(d + e*x^2)^q, a + b*x^2 + c*x^4, x], x, 0], g = Coeff[PolynomialRemainder[x^m*(d + e*x^2)^q, a + b*x^2 + c*x^4, x], x, 2]}, Simp[(x*(a + b*x^2 + c*x^4)^(p + 1)*(a*b*g - f*(b^2 - 2*a*c) - c*(b*f - 2*a*g)*x^2))/(2*a*(p + 1)*(b^2 - 4*a*c)), x] + Dist[1/(2*a*(p + 1)*(b^2 - 4*a*c)), Int[(a + b*x^2 + c*x^4)^(p + 1)*Simp[ExpandToSum[2*a*(p + 1)*(b^2 - 4*a*c)*PolynomialQuotient[x^m*(d + e*x^2)^q, a + b*x^2 + c*x^4, x] + b^2*f*(2*p + 3) - 2*a*c*f*(4*p + 5) - a*b*g + c*(4*p + 7)*(b*f - 2*a*g)*x^2, x], x], x], x]] /; FreeQ[{a, b, c, d, e}, x] && NeQ[b^2 - 4*a*c, 0] && LtQ[p, -1] && IGtQ[q, 1] && IGtQ[m/2, 0]
  1334. Int[Times[Power[Pattern[x, Blank[]], Pattern[m, Blank[]]], Power[Plus[Pattern[d, Blank[]], Times[Optional[Pattern[e, Blank[]]], Power[Pattern[x, Blank[]], 2]]], Pattern[q, Blank[]]], Power[Plus[Pattern[a, Blank[]], Times[Optional[Pattern[b, Blank[]]], Power[Pattern[x, Blank[]], 2]], Times[Optional[Pattern[c, Blank[]]], Power[Pattern[x, Blank[]], 4]]], Pattern[p, Blank[]]]], Pattern[x, Blank[Symbol]]] := With[{f = Coeff[PolynomialRemainder[x^m*(d + e*x^2)^q, a + b*x^2 + c*x^4, x], x, 0], g = Coeff[PolynomialRemainder[x^m*(d + e*x^2)^q, a + b*x^2 + c*x^4, x], x, 2]}, Simp[(x*(a + b*x^2 + c*x^4)^(p + 1)*(a*b*g - f*(b^2 - 2*a*c) - c*(b*f - 2*a*g)*x^2))/(2*a*(p + 1)*(b^2 - 4*a*c)), x] + Dist[1/(2*a*(p + 1)*(b^2 - 4*a*c)), Int[x^m*(a + b*x^2 + c*x^4)^(p + 1)*Simp[ExpandToSum[(2*a*(p + 1)*(b^2 - 4*a*c)*PolynomialQuotient[x^m*(d + e*x^2)^q, a + b*x^2 + c*x^4, x])/x^m + (b^2*f*(2*p + 3) - 2*a*c*f*(4*p + 5) - a*b*g)/x^m + c*(4*p + 7)*(b*f - 2*a*g)*x^(2 - m), x], x], x], x]] /; FreeQ[{a, b, c, d, e}, x] && NeQ[b^2 - 4*a*c, 0] && LtQ[p, -1] && IGtQ[q, 1] && ILtQ[m/2, 0]
  1335. Int[Times[Power[Times[Optional[Pattern[f, Blank[]]], Pattern[x, Blank[]]], Optional[Pattern[m, Blank[]]]], Power[Plus[Pattern[d, Blank[]], Times[Optional[Pattern[e, Blank[]]], Power[Pattern[x, Blank[]], 2]]], Optional[Pattern[q, Blank[]]]], Power[Plus[Pattern[a, Blank[]], Times[Optional[Pattern[b, Blank[]]], Power[Pattern[x, Blank[]], 2]], Times[Optional[Pattern[c, Blank[]]], Power[Pattern[x, Blank[]], 4]]], Optional[Pattern[p, Blank[]]]]], Pattern[x, Blank[Symbol]]] := Int[ExpandIntegrand[(f*x)^m*(d + e*x^2)^q*(a + b*x^2 + c*x^4)^p, x], x] /; FreeQ[{a, b, c, d, e, f, m, p, q}, x] && NeQ[b^2 - 4*a*c, 0] && (IGtQ[p, 0] || IGtQ[q, 0] || IntegersQ[m, q])
  1336. Int[Times[Power[Times[Optional[Pattern[f, Blank[]]], Pattern[x, Blank[]]], Optional[Pattern[m, Blank[]]]], Power[Plus[Pattern[d, Blank[]], Times[Optional[Pattern[e, Blank[]]], Power[Pattern[x, Blank[]], 2]]], Optional[Pattern[q, Blank[]]]], Power[Plus[Pattern[a, Blank[]], Times[Optional[Pattern[c, Blank[]]], Power[Pattern[x, Blank[]], 4]]], Optional[Pattern[p, Blank[]]]]], Pattern[x, Blank[Symbol]]] := Int[ExpandIntegrand[(f*x)^m*(d + e*x^2)^q*(a + c*x^4)^p, x], x] /; FreeQ[{a, c, d, e, f, m, p, q}, x] && (IGtQ[p, 0] || IGtQ[q, 0] || IntegersQ[m, q])
  1337. Int[Times[Power[Times[Optional[Pattern[f, Blank[]]], Pattern[x, Blank[]]], Optional[Pattern[m, Blank[]]]], Power[Plus[Pattern[d, Blank[]], Times[Optional[Pattern[e, Blank[]]], Power[Pattern[x, Blank[]], 2]]], Pattern[q, Blank[]]], Power[Plus[Pattern[a, Blank[]], Times[Optional[Pattern[c, Blank[]]], Power[Pattern[x, Blank[]], 4]]], Pattern[p, Blank[]]]], Pattern[x, Blank[Symbol]]] := Dist[(f*x)^m/x^m, Int[ExpandIntegrand[x^m*(a + c*x^4)^p, (d/(d^2 - e^2*x^4) - (e*x^2)/(d^2 - e^2*x^4))^(-q), x], x], x] /; FreeQ[{a, c, d, e, f, m, p}, x] && !IntegerQ[p] && ILtQ[q, 0]
  1338. Int[Times[Power[Times[Optional[Pattern[f, Blank[]]], Pattern[x, Blank[]]], Optional[Pattern[m, Blank[]]]], Power[Plus[Pattern[d, Blank[]], Times[Optional[Pattern[e, Blank[]]], Power[Pattern[x, Blank[]], 2]]], Optional[Pattern[q, Blank[]]]], Power[Plus[Pattern[a, Blank[]], Times[Optional[Pattern[b, Blank[]]], Power[Pattern[x, Blank[]], 2]], Times[Optional[Pattern[c, Blank[]]], Power[Pattern[x, Blank[]], 4]]], Optional[Pattern[p, Blank[]]]]], Pattern[x, Blank[Symbol]]] := Unintegrable[(f*x)^m*(d + e*x^2)^q*(a + b*x^2 + c*x^4)^p, x] /; FreeQ[{a, b, c, d, e, f, m, p, q}, x]
  1339. Int[Times[Power[Times[Optional[Pattern[f, Blank[]]], Pattern[x, Blank[]]], Optional[Pattern[m, Blank[]]]], Power[Plus[Pattern[d, Blank[]], Times[Optional[Pattern[e, Blank[]]], Power[Pattern[x, Blank[]], 2]]], Optional[Pattern[q, Blank[]]]], Power[Plus[Pattern[a, Blank[]], Times[Optional[Pattern[c, Blank[]]], Power[Pattern[x, Blank[]], 4]]], Optional[Pattern[p, Blank[]]]]], Pattern[x, Blank[Symbol]]] := Unintegrable[(f*x)^m*(d + e*x^2)^q*(a + c*x^4)^p, x] /; FreeQ[{a, c, d, e, f, m, p, q}, x]
  1340. Int[Power[Plus[Pattern[a, Blank[]], Times[Optional[Pattern[c, Blank[]]], Power[Pattern[x, Blank[]], Optional[Pattern[n2, Blank[]]]]], Times[Optional[Pattern[b, Blank[]]], Power[Pattern[x, Blank[]], Pattern[n, Blank[]]]]], Optional[Pattern[p, Blank[]]]], Pattern[x, Blank[Symbol]]] := Int[x^(2*n*p)*(c + b/x^n + a/x^(2*n))^p, x] /; FreeQ[{a, b, c}, x] && EqQ[n2, 2*n] && LtQ[n, 0] && IntegerQ[p]
  1341. Int[Power[Plus[Pattern[a, Blank[]], Times[Optional[Pattern[c, Blank[]]], Power[Pattern[x, Blank[]], Optional[Pattern[n2, Blank[]]]]], Times[Optional[Pattern[b, Blank[]]], Power[Pattern[x, Blank[]], Pattern[n, Blank[]]]]], Pattern[p, Blank[]]], Pattern[x, Blank[Symbol]]] := With[{k = Denominator[n]}, Dist[k, Subst[Int[x^(k - 1)*(a + b*x^(k*n) + c*x^(2*k*n))^p, x], x, x^(1/k)], x]] /; FreeQ[{a, b, c, p}, x] && EqQ[n2, 2*n] && FractionQ[n]
  1342. Int[Power[Plus[Pattern[a, Blank[]], Times[Optional[Pattern[c, Blank[]]], Power[Pattern[x, Blank[]], Optional[Pattern[n2, Blank[]]]]], Times[Optional[Pattern[b, Blank[]]], Power[Pattern[x, Blank[]], Pattern[n, Blank[]]]]], Pattern[p, Blank[]]], Pattern[x, Blank[Symbol]]] := -Subst[Int[(a + b/x^n + c/x^(2*n))^p/x^2, x], x, 1/x] /; FreeQ[{a, b, c, p}, x] && EqQ[n2, 2*n] && ILtQ[n, 0]
  1343. Int[Power[Plus[Pattern[a, Blank[]], Times[Optional[Pattern[b, Blank[]]], Power[Pattern[x, Blank[]], Optional[Pattern[n, Blank[]]]]], Times[Optional[Pattern[c, Blank[]]], Power[Pattern[x, Blank[]], Optional[Pattern[n2, Blank[]]]]]], Pattern[p, Blank[]]], Pattern[x, Blank[Symbol]]] := Dist[(a + b*x^n + c*x^(2*n))^p/(b + 2*c*x^n)^(2*p), Int[(b + 2*c*x^n)^(2*p), x], x] /; FreeQ[{a, b, c, n, p}, x] && EqQ[n2, 2*n] && EqQ[b^2 - 4*a*c, 0]
  1344. Int[Power[Plus[Pattern[a, Blank[]], Times[Optional[Pattern[c, Blank[]]], Power[Pattern[x, Blank[]], Optional[Pattern[n2, Blank[]]]]], Times[Optional[Pattern[b, Blank[]]], Power[Pattern[x, Blank[]], Pattern[n, Blank[]]]]], Pattern[p, Blank[]]], Pattern[x, Blank[Symbol]]] := Int[ExpandIntegrand[(a + b*x^n + c*x^(2*n))^p, x], x] /; FreeQ[{a, b, c, n}, x] && EqQ[n2, 2*n] && NeQ[b^2 - 4*a*c, 0] && IGtQ[p, 0]
  1345. Int[Power[Plus[Pattern[a, Blank[]], Times[Optional[Pattern[c, Blank[]]], Power[Pattern[x, Blank[]], Optional[Pattern[n2, Blank[]]]]], Times[Optional[Pattern[b, Blank[]]], Power[Pattern[x, Blank[]], Pattern[n, Blank[]]]]], Pattern[p, Blank[]]], Pattern[x, Blank[Symbol]]] := -Simp[(x*(b^2 - 2*a*c + b*c*x^n)*(a + b*x^n + c*x^(2*n))^(p + 1))/(a*n*(p + 1)*(b^2 - 4*a*c)), x] + Dist[1/(a*n*(p + 1)*(b^2 - 4*a*c)), Int[(b^2 - 2*a*c + n*(p + 1)*(b^2 - 4*a*c) + b*c*(n*(2*p + 3) + 1)*x^n)*(a + b*x^n + c*x^(2*n))^(p + 1), x], x] /; FreeQ[{a, b, c, n}, x] && EqQ[n2, 2*n] && NeQ[b^2 - 4*a*c, 0] && ILtQ[p, -1]
  1346. Int[Power[Plus[Pattern[a, Blank[]], Times[Optional[Pattern[b, Blank[]]], Power[Pattern[x, Blank[]], Pattern[n, Blank[]]]], Times[Optional[Pattern[c, Blank[]]], Power[Pattern[x, Blank[]], Pattern[n2, Blank[]]]]], -1], Pattern[x, Blank[Symbol]]] := With[{q = Rt[a/c, 2]}, With[{r = Rt[2*q - b/c, 2]}, Dist[1/(2*c*q*r), Int[(r - x^(n/2))/(q - r*x^(n/2) + x^n), x], x] + Dist[1/(2*c*q*r), Int[(r + x^(n/2))/(q + r*x^(n/2) + x^n), x], x]]] /; FreeQ[{a, b, c}, x] && EqQ[n2, 2*n] && NeQ[b^2 - 4*a*c, 0] && IGtQ[n/2, 0] && NegQ[b^2 - 4*a*c]
  1347. Int[Power[Plus[Pattern[a, Blank[]], Times[Optional[Pattern[b, Blank[]]], Power[Pattern[x, Blank[]], Pattern[n, Blank[]]]], Times[Optional[Pattern[c, Blank[]]], Power[Pattern[x, Blank[]], Pattern[n2, Blank[]]]]], -1], Pattern[x, Blank[Symbol]]] := With[{q = Rt[b^2 - 4*a*c, 2]}, Dist[c/q, Int[1/(b/2 - q/2 + c*x^n), x], x] - Dist[c/q, Int[1/(b/2 + q/2 + c*x^n), x], x]] /; FreeQ[{a, b, c}, x] && EqQ[n2, 2*n] && NeQ[b^2 - 4*a*c, 0]
  1348. Int[Power[Plus[Pattern[a, Blank[]], Times[Optional[Pattern[c, Blank[]]], Power[Pattern[x, Blank[]], Optional[Pattern[n2, Blank[]]]]], Times[Optional[Pattern[b, Blank[]]], Power[Pattern[x, Blank[]], Pattern[n, Blank[]]]]], Pattern[p, Blank[]]], Pattern[x, Blank[Symbol]]] := Dist[(a^IntPart[p]*(a + b*x^n + c*x^(2*n))^FracPart[p])/((1 + (2*c*x^n)/(b + Rt[b^2 - 4*a*c, 2]))^FracPart[p]*(1 + (2*c*x^n)/(b - Rt[b^2 - 4*a*c, 2]))^FracPart[p]), Int[(1 + (2*c*x^n)/(b + Sqrt[b^2 - 4*a*c]))^p*(1 + (2*c*x^n)/(b - Sqrt[b^2 - 4*a*c]))^p, x], x] /; FreeQ[{a, b, c, n, p}, x] && EqQ[n2, 2*n] && NeQ[b^2 - 4*a*c, 0] && !IntegerQ[p]
  1349. Int[Power[Plus[Pattern[a, Blank[]], Times[Optional[Pattern[c, Blank[]]], Power[Pattern[u, Blank[]], Optional[Pattern[n2, Blank[]]]]], Times[Optional[Pattern[b, Blank[]]], Power[Pattern[u, Blank[]], Pattern[n, Blank[]]]]], Pattern[p, Blank[]]], Pattern[x, Blank[Symbol]]] := Dist[1/Coefficient[u, x, 1], Subst[Int[(a + b*x^n + c*x^(2*n))^p, x], x, u], x] /; FreeQ[{a, b, c, n, p}, x] && EqQ[n2, 2*n] && LinearQ[u, x] && NeQ[u, x]
  1350. Int[Power[Plus[Pattern[a, Blank[]], Times[Optional[Pattern[c, Blank[]]], Power[Pattern[x, Blank[]], Optional[Pattern[n, Blank[]]]]], Times[Optional[Pattern[b, Blank[]]], Power[Pattern[x, Blank[]], Pattern[mn, Blank[]]]]], Optional[Pattern[p, Blank[]]]], Pattern[x, Blank[Symbol]]] := Int[(b + a*x^n + c*x^(2*n))^p/x^(n*p), x] /; FreeQ[{a, b, c, n}, x] && EqQ[mn, -n] && IntegerQ[p] && PosQ[n]
  1351. Int[Power[Plus[Pattern[a, Blank[]], Times[Optional[Pattern[c, Blank[]]], Power[Pattern[x, Blank[]], Optional[Pattern[n, Blank[]]]]], Times[Optional[Pattern[b, Blank[]]], Power[Pattern[x, Blank[]], Pattern[mn, Blank[]]]]], Pattern[p, Blank[]]], Pattern[x, Blank[Symbol]]] := Dist[(x^(n*FracPart[p])*(a + b/x^n + c*x^n)^FracPart[p])/(b + a*x^n + c*x^(2*n))^FracPart[p], Int[(b + a*x^n + c*x^(2*n))^p/x^(n*p), x], x] /; FreeQ[{a, b, c, n, p}, x] && EqQ[mn, -n] && !IntegerQ[p] && PosQ[n]
  1352. Int[Times[Power[Pattern[x, Blank[]], Optional[Pattern[m, Blank[]]]], Power[Plus[Pattern[a, Blank[]], Times[Optional[Pattern[c, Blank[]]], Power[Pattern[x, Blank[]], Optional[Pattern[n2, Blank[]]]]], Times[Optional[Pattern[b, Blank[]]], Power[Pattern[x, Blank[]], Pattern[n, Blank[]]]]], Optional[Pattern[p, Blank[]]]]], Pattern[x, Blank[Symbol]]] := Dist[1/n, Subst[Int[(a + b*x + c*x^2)^p, x], x, x^n], x] /; FreeQ[{a, b, c, m, n, p}, x] && EqQ[n2, 2*n] && EqQ[Simplify[m - n + 1], 0]
  1353. Int[Times[Power[Times[Optional[Pattern[d, Blank[]]], Pattern[x, Blank[]]], Optional[Pattern[m, Blank[]]]], Power[Plus[Pattern[a, Blank[]], Times[Optional[Pattern[c, Blank[]]], Power[Pattern[x, Blank[]], Optional[Pattern[n2, Blank[]]]]], Times[Optional[Pattern[b, Blank[]]], Power[Pattern[x, Blank[]], Pattern[n, Blank[]]]]], Optional[Pattern[p, Blank[]]]]], Pattern[x, Blank[Symbol]]] := Int[ExpandIntegrand[(d*x)^m*(a + b*x^n + c*x^(2*n))^p, x], x] /; FreeQ[{a, b, c, d, m, n}, x] && EqQ[n2, 2*n] && IGtQ[p, 0] && !IntegerQ[Simplify[(m + 1)/n]]
  1354. Int[Times[Power[Pattern[x, Blank[]], Optional[Pattern[m, Blank[]]]], Power[Plus[Pattern[a, Blank[]], Times[Optional[Pattern[c, Blank[]]], Power[Pattern[x, Blank[]], Optional[Pattern[n2, Blank[]]]]], Times[Optional[Pattern[b, Blank[]]], Power[Pattern[x, Blank[]], Pattern[n, Blank[]]]]], Pattern[p, Blank[]]]], Pattern[x, Blank[Symbol]]] := Int[x^(m + 2*n*p)*(c + b/x^n + a/x^(2*n))^p, x] /; FreeQ[{a, b, c, m, n}, x] && EqQ[n2, 2*n] && ILtQ[p, 0] && NegQ[n]
  1355. Int[Times[Power[Times[Optional[Pattern[d, Blank[]]], Pattern[x, Blank[]]], Optional[Pattern[m, Blank[]]]], Power[Plus[Pattern[a, Blank[]], Times[Optional[Pattern[b, Blank[]]], Power[Pattern[x, Blank[]], Optional[Pattern[n, Blank[]]]]], Times[Optional[Pattern[c, Blank[]]], Power[Pattern[x, Blank[]], Optional[Pattern[n2, Blank[]]]]]], Pattern[p, Blank[]]]], Pattern[x, Blank[Symbol]]] := Dist[(a + b*x^n + c*x^(2*n))^FracPart[p]/(c^IntPart[p]*(b/2 + c*x^n)^(2*FracPart[p])), Int[(d*x)^m*(b/2 + c*x^n)^(2*p), x], x] /; FreeQ[{a, b, c, d, m, n, p}, x] && EqQ[n2, 2*n] && EqQ[b^2 - 4*a*c, 0] && IntegerQ[p - 1/2]
  1356. Int[Times[Power[Times[Optional[Pattern[d, Blank[]]], Pattern[x, Blank[]]], Optional[Pattern[m, Blank[]]]], Power[Plus[Pattern[a, Blank[]], Times[Optional[Pattern[b, Blank[]]], Power[Pattern[x, Blank[]], Optional[Pattern[n, Blank[]]]]], Times[Optional[Pattern[c, Blank[]]], Power[Pattern[x, Blank[]], Optional[Pattern[n2, Blank[]]]]]], Pattern[p, Blank[]]]], Pattern[x, Blank[Symbol]]] := Dist[(a^IntPart[p]*(a + b*x^n + c*x^(2*n))^FracPart[p])/(1 + (2*c*x^n)/b)^(2*FracPart[p]), Int[(d*x)^m*(1 + (2*c*x^n)/b)^(2*p), x], x] /; FreeQ[{a, b, c, d, m, n, p}, x] && EqQ[n2, 2*n] && EqQ[b^2 - 4*a*c, 0] && !IntegerQ[2*p]
  1357. Int[Times[Power[Pattern[x, Blank[]], Optional[Pattern[m, Blank[]]]], Power[Plus[Pattern[a, Blank[]], Times[Optional[Pattern[c, Blank[]]], Power[Pattern[x, Blank[]], Optional[Pattern[n2, Blank[]]]]], Times[Optional[Pattern[b, Blank[]]], Power[Pattern[x, Blank[]], Pattern[n, Blank[]]]]], Optional[Pattern[p, Blank[]]]]], Pattern[x, Blank[Symbol]]] := Dist[1/n, Subst[Int[x^(Simplify[(m + 1)/n] - 1)*(a + b*x + c*x^2)^p, x], x, x^n], x] /; FreeQ[{a, b, c, m, n, p}, x] && EqQ[n2, 2*n] && NeQ[b^2 - 4*a*c, 0] && IntegerQ[Simplify[(m + 1)/n]]
  1358. Int[Times[Power[Times[Pattern[d, Blank[]], Pattern[x, Blank[]]], Optional[Pattern[m, Blank[]]]], Power[Plus[Pattern[a, Blank[]], Times[Optional[Pattern[c, Blank[]]], Power[Pattern[x, Blank[]], Optional[Pattern[n2, Blank[]]]]], Times[Optional[Pattern[b, Blank[]]], Power[Pattern[x, Blank[]], Pattern[n, Blank[]]]]], Optional[Pattern[p, Blank[]]]]], Pattern[x, Blank[Symbol]]] := Dist[(d^IntPart[m]*(d*x)^FracPart[m])/x^FracPart[m], Int[x^m*(a + b*x^n + c*x^(2*n))^p, x], x] /; FreeQ[{a, b, c, d, m, n, p}, x] && EqQ[n2, 2*n] && NeQ[b^2 - 4*a*c, 0] && IntegerQ[Simplify[(m + 1)/n]]
  1359. Int[Times[Power[Pattern[x, Blank[]], Optional[Pattern[m, Blank[]]]], Power[Plus[Pattern[a, Blank[]], Times[Optional[Pattern[c, Blank[]]], Power[Pattern[x, Blank[]], Optional[Pattern[n2, Blank[]]]]], Times[Optional[Pattern[b, Blank[]]], Power[Pattern[x, Blank[]], Pattern[n, Blank[]]]]], Pattern[p, Blank[]]]], Pattern[x, Blank[Symbol]]] := With[{k = GCD[m + 1, n]}, Dist[1/k, Subst[Int[x^((m + 1)/k - 1)*(a + b*x^(n/k) + c*x^((2*n)/k))^p, x], x, x^k], x] /; k != 1] /; FreeQ[{a, b, c, p}, x] && EqQ[n2, 2*n] && NeQ[b^2 - 4*a*c, 0] && IGtQ[n, 0] && IntegerQ[m]
  1360. Int[Times[Power[Times[Optional[Pattern[d, Blank[]]], Pattern[x, Blank[]]], Pattern[m, Blank[]]], Power[Plus[Pattern[a, Blank[]], Times[Optional[Pattern[c, Blank[]]], Power[Pattern[x, Blank[]], Optional[Pattern[n2, Blank[]]]]], Times[Optional[Pattern[b, Blank[]]], Power[Pattern[x, Blank[]], Pattern[n, Blank[]]]]], Pattern[p, Blank[]]]], Pattern[x, Blank[Symbol]]] := With[{k = Denominator[m]}, Dist[k/d, Subst[Int[x^(k*(m + 1) - 1)*(a + (b*x^(k*n))/d^n + (c*x^(2*k*n))/d^(2*n))^p, x], x, (d*x)^(1/k)], x]] /; FreeQ[{a, b, c, d, p}, x] && EqQ[n2, 2*n] && NeQ[b^2 - 4*a*c, 0] && IGtQ[n, 0] && FractionQ[m] && IntegerQ[p]
  1361. Int[Times[Power[Times[Optional[Pattern[d, Blank[]]], Pattern[x, Blank[]]], Optional[Pattern[m, Blank[]]]], Power[Plus[Pattern[a, Blank[]], Times[Optional[Pattern[c, Blank[]]], Power[Pattern[x, Blank[]], Optional[Pattern[n2, Blank[]]]]], Times[Optional[Pattern[b, Blank[]]], Power[Pattern[x, Blank[]], Pattern[n, Blank[]]]]], Pattern[p, Blank[]]]], Pattern[x, Blank[Symbol]]] := Simp[(d^(n - 1)*(d*x)^(m - n + 1)*(a + b*x^n + c*x^(2*n))^p*(b*n*p + c*(m + n*(2*p - 1) + 1)*x^n))/(c*(m + 2*n*p + 1)*(m + n*(2*p - 1) + 1)), x] - Dist[(n*p*d^n)/(c*(m + 2*n*p + 1)*(m + n*(2*p - 1) + 1)), Int[(d*x)^(m - n)*(a + b*x^n + c*x^(2*n))^(p - 1)*Simp[a*b*(m - n + 1) - (2*a*c*(m + n*(2*p - 1) + 1) - b^2*(m + n*(p - 1) + 1))*x^n, x], x], x] /; FreeQ[{a, b, c, d}, x] && EqQ[n2, 2*n] && NeQ[b^2 - 4*a*c, 0] && IGtQ[n, 0] && IGtQ[p, 0] && GtQ[m, n - 1] && NeQ[m + 2*n*p + 1, 0] && NeQ[m + n*(2*p - 1) + 1, 0]
  1362. Int[Times[Power[Times[Optional[Pattern[d, Blank[]]], Pattern[x, Blank[]]], Optional[Pattern[m, Blank[]]]], Power[Plus[Pattern[a, Blank[]], Times[Optional[Pattern[c, Blank[]]], Power[Pattern[x, Blank[]], Optional[Pattern[n2, Blank[]]]]], Times[Optional[Pattern[b, Blank[]]], Power[Pattern[x, Blank[]], Pattern[n, Blank[]]]]], Pattern[p, Blank[]]]], Pattern[x, Blank[Symbol]]] := Simp[((d*x)^(m + 1)*(a + b*x^n + c*x^(2*n))^p)/(d*(m + 1)), x] - Dist[(n*p)/(d^n*(m + 1)), Int[(d*x)^(m + n)*(b + 2*c*x^n)*(a + b*x^n + c*x^(2*n))^(p - 1), x], x] /; FreeQ[{a, b, c, d}, x] && EqQ[n2, 2*n] && NeQ[b^2 - 4*a*c, 0] && IGtQ[n, 0] && IGtQ[p, 0] && LtQ[m, -1]
  1363. Int[Times[Power[Times[Optional[Pattern[d, Blank[]]], Pattern[x, Blank[]]], Optional[Pattern[m, Blank[]]]], Power[Plus[Pattern[a, Blank[]], Times[Optional[Pattern[c, Blank[]]], Power[Pattern[x, Blank[]], Optional[Pattern[n2, Blank[]]]]], Times[Optional[Pattern[b, Blank[]]], Power[Pattern[x, Blank[]], Pattern[n, Blank[]]]]], Pattern[p, Blank[]]]], Pattern[x, Blank[Symbol]]] := Simp[((d*x)^(m + 1)*(a + b*x^n + c*x^(2*n))^p)/(d*(m + 2*n*p + 1)), x] + Dist[(n*p)/(m + 2*n*p + 1), Int[(d*x)^m*(2*a + b*x^n)*(a + b*x^n + c*x^(2*n))^(p - 1), x], x] /; FreeQ[{a, b, c, d, m}, x] && EqQ[n2, 2*n] && NeQ[b^2 - 4*a*c, 0] && IGtQ[n, 0] && IGtQ[p, 0] && NeQ[m + 2*n*p + 1, 0]
  1364. Int[Times[Power[Times[Optional[Pattern[d, Blank[]]], Pattern[x, Blank[]]], Optional[Pattern[m, Blank[]]]], Power[Plus[Pattern[a, Blank[]], Times[Optional[Pattern[c, Blank[]]], Power[Pattern[x, Blank[]], Optional[Pattern[n2, Blank[]]]]], Times[Optional[Pattern[b, Blank[]]], Power[Pattern[x, Blank[]], Pattern[n, Blank[]]]]], Pattern[p, Blank[]]]], Pattern[x, Blank[Symbol]]] := Simp[(d^(n - 1)*(d*x)^(m - n + 1)*(b + 2*c*x^n)*(a + b*x^n + c*x^(2*n))^(p + 1))/(n*(p + 1)*(b^2 - 4*a*c)), x] - Dist[d^n/(n*(p + 1)*(b^2 - 4*a*c)), Int[(d*x)^(m - n)*(b*(m - n + 1) + 2*c*(m + 2*n*(p + 1) + 1)*x^n)*(a + b*x^n + c*x^(2*n))^(p + 1), x], x] /; FreeQ[{a, b, c, d}, x] && EqQ[n2, 2*n] && NeQ[b^2 - 4*a*c, 0] && IGtQ[n, 0] && ILtQ[p, -1] && GtQ[m, n - 1] && LeQ[m, 2*n - 1]
  1365. Int[Times[Power[Times[Optional[Pattern[d, Blank[]]], Pattern[x, Blank[]]], Optional[Pattern[m, Blank[]]]], Power[Plus[Pattern[a, Blank[]], Times[Optional[Pattern[c, Blank[]]], Power[Pattern[x, Blank[]], Optional[Pattern[n2, Blank[]]]]], Times[Optional[Pattern[b, Blank[]]], Power[Pattern[x, Blank[]], Pattern[n, Blank[]]]]], Pattern[p, Blank[]]]], Pattern[x, Blank[Symbol]]] := -Simp[(d^(2*n - 1)*(d*x)^(m - 2*n + 1)*(2*a + b*x^n)*(a + b*x^n + c*x^(2*n))^(p + 1))/(n*(p + 1)*(b^2 - 4*a*c)), x] + Dist[d^(2*n)/(n*(p + 1)*(b^2 - 4*a*c)), Int[(d*x)^(m - 2*n)*(2*a*(m - 2*n + 1) + b*(m + n*(2*p + 1) + 1)*x^n)*(a + b*x^n + c*x^(2*n))^(p + 1), x], x] /; FreeQ[{a, b, c, d}, x] && EqQ[n2, 2*n] && NeQ[b^2 - 4*a*c, 0] && IGtQ[n, 0] && ILtQ[p, -1] && GtQ[m, 2*n - 1]
  1366. Int[Times[Power[Times[Optional[Pattern[d, Blank[]]], Pattern[x, Blank[]]], Optional[Pattern[m, Blank[]]]], Power[Plus[Pattern[a, Blank[]], Times[Optional[Pattern[c, Blank[]]], Power[Pattern[x, Blank[]], Optional[Pattern[n2, Blank[]]]]], Times[Optional[Pattern[b, Blank[]]], Power[Pattern[x, Blank[]], Pattern[n, Blank[]]]]], Pattern[p, Blank[]]]], Pattern[x, Blank[Symbol]]] := -Simp[((d*x)^(m + 1)*(b^2 - 2*a*c + b*c*x^n)*(a + b*x^n + c*x^(2*n))^(p + 1))/(a*d*n*(p + 1)*(b^2 - 4*a*c)), x] + Dist[1/(a*n*(p + 1)*(b^2 - 4*a*c)), Int[(d*x)^m*(a + b*x^n + c*x^(2*n))^(p + 1)*Simp[b^2*(m + n*(p + 1) + 1) - 2*a*c*(m + 2*n*(p + 1) + 1) + b*c*(m + n*(2*p + 3) + 1)*x^n, x], x], x] /; FreeQ[{a, b, c, d, m}, x] && EqQ[n2, 2*n] && NeQ[b^2 - 4*a*c, 0] && IGtQ[n, 0] && ILtQ[p, -1]
  1367. Int[Times[Power[Times[Optional[Pattern[d, Blank[]]], Pattern[x, Blank[]]], Optional[Pattern[m, Blank[]]]], Power[Plus[Pattern[a, Blank[]], Times[Optional[Pattern[c, Blank[]]], Power[Pattern[x, Blank[]], Optional[Pattern[n2, Blank[]]]]], Times[Optional[Pattern[b, Blank[]]], Power[Pattern[x, Blank[]], Pattern[n, Blank[]]]]], Pattern[p, Blank[]]]], Pattern[x, Blank[Symbol]]] := Simp[(d^(2*n - 1)*(d*x)^(m - 2*n + 1)*(a + b*x^n + c*x^(2*n))^(p + 1))/(c*(m + 2*n*p + 1)), x] - Dist[d^(2*n)/(c*(m + 2*n*p + 1)), Int[(d*x)^(m - 2*n)*Simp[a*(m - 2*n + 1) + b*(m + n*(p - 1) + 1)*x^n, x]*(a + b*x^n + c*x^(2*n))^p, x], x] /; FreeQ[{a, b, c, d, p}, x] && EqQ[n2, 2*n] && NeQ[b^2 - 4*a*c, 0] && IGtQ[n, 0] && GtQ[m, 2*n - 1] && NeQ[m + 2*n*p + 1, 0] && IntegerQ[p]
  1368. Int[Times[Power[Times[Optional[Pattern[d, Blank[]]], Pattern[x, Blank[]]], Pattern[m, Blank[]]], Power[Plus[Pattern[a, Blank[]], Times[Optional[Pattern[c, Blank[]]], Power[Pattern[x, Blank[]], Optional[Pattern[n2, Blank[]]]]], Times[Optional[Pattern[b, Blank[]]], Power[Pattern[x, Blank[]], Pattern[n, Blank[]]]]], Pattern[p, Blank[]]]], Pattern[x, Blank[Symbol]]] := Simp[((d*x)^(m + 1)*(a + b*x^n + c*x^(2*n))^(p + 1))/(a*d*(m + 1)), x] - Dist[1/(a*d^n*(m + 1)), Int[(d*x)^(m + n)*(b*(m + n*(p + 1) + 1) + c*(m + 2*n*(p + 1) + 1)*x^n)*(a + b*x^n + c*x^(2*n))^p, x], x] /; FreeQ[{a, b, c, d, p}, x] && EqQ[n2, 2*n] && NeQ[b^2 - 4*a*c, 0] && IGtQ[n, 0] && LtQ[m, -1] && IntegerQ[p]
  1369. Int[Times[Power[Times[Optional[Pattern[d, Blank[]]], Pattern[x, Blank[]]], Pattern[m, Blank[]]], Power[Plus[Pattern[a, Blank[]], Times[Optional[Pattern[c, Blank[]]], Power[Pattern[x, Blank[]], Optional[Pattern[n2, Blank[]]]]], Times[Optional[Pattern[b, Blank[]]], Power[Pattern[x, Blank[]], Pattern[n, Blank[]]]]], -1]], Pattern[x, Blank[Symbol]]] := Simp[(d*x)^(m + 1)/(a*d*(m + 1)), x] - Dist[1/(a*d^n), Int[((d*x)^(m + n)*(b + c*x^n))/(a + b*x^n + c*x^(2*n)), x], x] /; FreeQ[{a, b, c, d}, x] && EqQ[n2, 2*n] && NeQ[b^2 - 4*a*c, 0] && IGtQ[n, 0] && LtQ[m, -1]
  1370. Int[Times[Power[Pattern[x, Blank[]], Pattern[m, Blank[]]], Power[Plus[Pattern[a, Blank[]], Times[Optional[Pattern[c, Blank[]]], Power[Pattern[x, Blank[]], Optional[Pattern[n2, Blank[]]]]], Times[Optional[Pattern[b, Blank[]]], Power[Pattern[x, Blank[]], Pattern[n, Blank[]]]]], -1]], Pattern[x, Blank[Symbol]]] := Int[PolynomialDivide[x^m, a + b*x^n + c*x^(2*n), x], x] /; FreeQ[{a, b, c}, x] && EqQ[n2, 2*n] && NeQ[b^2 - 4*a*c, 0] && IGtQ[n, 0] && IGtQ[m, 3*n - 1]
  1371. Int[Times[Power[Times[Optional[Pattern[d, Blank[]]], Pattern[x, Blank[]]], Pattern[m, Blank[]]], Power[Plus[Pattern[a, Blank[]], Times[Optional[Pattern[c, Blank[]]], Power[Pattern[x, Blank[]], Optional[Pattern[n2, Blank[]]]]], Times[Optional[Pattern[b, Blank[]]], Power[Pattern[x, Blank[]], Pattern[n, Blank[]]]]], -1]], Pattern[x, Blank[Symbol]]] := Simp[(d^(2*n - 1)*(d*x)^(m - 2*n + 1))/(c*(m - 2*n + 1)), x] - Dist[d^(2*n)/c, Int[((d*x)^(m - 2*n)*(a + b*x^n))/(a + b*x^n + c*x^(2*n)), x], x] /; FreeQ[{a, b, c, d}, x] && EqQ[n2, 2*n] && NeQ[b^2 - 4*a*c, 0] && IGtQ[n, 0] && GtQ[m, 2*n - 1]
  1372. Int[Times[Power[Pattern[x, Blank[]], Optional[Pattern[m, Blank[]]]], Power[Plus[Pattern[a, Blank[]], Times[Optional[Pattern[c, Blank[]]], Power[Pattern[x, Blank[]], Optional[Pattern[n2, Blank[]]]]], Times[Optional[Pattern[b, Blank[]]], Power[Pattern[x, Blank[]], Pattern[n, Blank[]]]]], -1]], Pattern[x, Blank[Symbol]]] := With[{q = Rt[a/c, 2]}, With[{r = Rt[2*q - b/c, 2]}, -Dist[1/(2*c*r), Int[(x^(m - 3*(n/2))*(q - r*x^(n/2)))/(q - r*x^(n/2) + x^n), x], x] + Dist[1/(2*c*r), Int[(x^(m - 3*(n/2))*(q + r*x^(n/2)))/(q + r*x^(n/2) + x^n), x], x]]] /; FreeQ[{a, b, c}, x] && EqQ[n2, 2*n] && NeQ[b^2 - 4*a*c, 0] && IGtQ[n/2, 0] && IGtQ[m, 0] && GeQ[m, (3*n)/2] && LtQ[m, 2*n] && NegQ[b^2 - 4*a*c]
  1373. Int[Times[Power[Pattern[x, Blank[]], Optional[Pattern[m, Blank[]]]], Power[Plus[Pattern[a, Blank[]], Times[Optional[Pattern[c, Blank[]]], Power[Pattern[x, Blank[]], Optional[Pattern[n2, Blank[]]]]], Times[Optional[Pattern[b, Blank[]]], Power[Pattern[x, Blank[]], Pattern[n, Blank[]]]]], -1]], Pattern[x, Blank[Symbol]]] := With[{q = Rt[a/c, 2]}, With[{r = Rt[2*q - b/c, 2]}, Dist[1/(2*c*r), Int[x^(m - n/2)/(q - r*x^(n/2) + x^n), x], x] - Dist[1/(2*c*r), Int[x^(m - n/2)/(q + r*x^(n/2) + x^n), x], x]]] /; FreeQ[{a, b, c}, x] && EqQ[n2, 2*n] && NeQ[b^2 - 4*a*c, 0] && IGtQ[n/2, 0] && IGtQ[m, 0] && GeQ[m, n/2] && LtQ[m, (3*n)/2] && NegQ[b^2 - 4*a*c]
  1374. Int[Times[Power[Times[Optional[Pattern[d, Blank[]]], Pattern[x, Blank[]]], Pattern[m, Blank[]]], Power[Plus[Pattern[a, Blank[]], Times[Optional[Pattern[c, Blank[]]], Power[Pattern[x, Blank[]], Optional[Pattern[n2, Blank[]]]]], Times[Optional[Pattern[b, Blank[]]], Power[Pattern[x, Blank[]], Pattern[n, Blank[]]]]], -1]], Pattern[x, Blank[Symbol]]] := With[{q = Rt[b^2 - 4*a*c, 2]}, Dist[(d^n*(b/q + 1))/2, Int[(d*x)^(m - n)/(b/2 + q/2 + c*x^n), x], x] - Dist[(d^n*(b/q - 1))/2, Int[(d*x)^(m - n)/(b/2 - q/2 + c*x^n), x], x]] /; FreeQ[{a, b, c, d}, x] && EqQ[n2, 2*n] && NeQ[b^2 - 4*a*c, 0] && IGtQ[n, 0] && GeQ[m, n]
  1375. Int[Times[Power[Times[Optional[Pattern[d, Blank[]]], Pattern[x, Blank[]]], Optional[Pattern[m, Blank[]]]], Power[Plus[Pattern[a, Blank[]], Times[Optional[Pattern[c, Blank[]]], Power[Pattern[x, Blank[]], Optional[Pattern[n2, Blank[]]]]], Times[Optional[Pattern[b, Blank[]]], Power[Pattern[x, Blank[]], Pattern[n, Blank[]]]]], -1]], Pattern[x, Blank[Symbol]]] := With[{q = Rt[b^2 - 4*a*c, 2]}, Dist[c/q, Int[(d*x)^m/(b/2 - q/2 + c*x^n), x], x] - Dist[c/q, Int[(d*x)^m/(b/2 + q/2 + c*x^n), x], x]] /; FreeQ[{a, b, c, d, m}, x] && EqQ[n2, 2*n] && NeQ[b^2 - 4*a*c, 0] && IGtQ[n, 0]
  1376. Int[Times[Power[Pattern[x, Blank[]], Optional[Pattern[m, Blank[]]]], Power[Plus[Pattern[a, Blank[]], Times[Optional[Pattern[c, Blank[]]], Power[Pattern[x, Blank[]], Optional[Pattern[n2, Blank[]]]]], Times[Optional[Pattern[b, Blank[]]], Power[Pattern[x, Blank[]], Pattern[n, Blank[]]]]], Pattern[p, Blank[]]]], Pattern[x, Blank[Symbol]]] := -Subst[Int[(a + b/x^n + c/x^(2*n))^p/x^(m + 2), x], x, 1/x] /; FreeQ[{a, b, c, p}, x] && EqQ[n2, 2*n] && NeQ[b^2 - 4*a*c, 0] && ILtQ[n, 0] && IntegerQ[m]
  1377. Int[Times[Power[Times[Optional[Pattern[d, Blank[]]], Pattern[x, Blank[]]], Optional[Pattern[m, Blank[]]]], Power[Plus[Pattern[a, Blank[]], Times[Optional[Pattern[c, Blank[]]], Power[Pattern[x, Blank[]], Optional[Pattern[n2, Blank[]]]]], Times[Optional[Pattern[b, Blank[]]], Power[Pattern[x, Blank[]], Pattern[n, Blank[]]]]], Pattern[p, Blank[]]]], Pattern[x, Blank[Symbol]]] := With[{k = Denominator[m]}, -Dist[k/d, Subst[Int[(a + b/(d^n*x^(k*n)) + c/(d^(2*n)*x^(2*k*n)))^p/x^(k*(m + 1) + 1), x], x, 1/(d*x)^(1/k)], x]] /; FreeQ[{a, b, c, d, p}, x] && EqQ[n2, 2*n] && NeQ[b^2 - 4*a*c, 0] && ILtQ[n, 0] && FractionQ[m]
  1378. Int[Times[Power[Times[Optional[Pattern[d, Blank[]]], Pattern[x, Blank[]]], Pattern[m, Blank[]]], Power[Plus[Pattern[a, Blank[]], Times[Optional[Pattern[c, Blank[]]], Power[Pattern[x, Blank[]], Optional[Pattern[n2, Blank[]]]]], Times[Optional[Pattern[b, Blank[]]], Power[Pattern[x, Blank[]], Pattern[n, Blank[]]]]], Pattern[p, Blank[]]]], Pattern[x, Blank[Symbol]]] := -Dist[d^IntPart[m]*(d*x)^FracPart[m]*(x^(-1))^FracPart[m], Subst[Int[(a + b/x^n + c/x^(2*n))^p/x^(m + 2), x], x, 1/x], x] /; FreeQ[{a, b, c, d, m, p}, x] && EqQ[n2, 2*n] && NeQ[b^2 - 4*a*c, 0] && ILtQ[n, 0] && !RationalQ[m]
  1379. Int[Times[Power[Pattern[x, Blank[]], Optional[Pattern[m, Blank[]]]], Power[Plus[Pattern[a, Blank[]], Times[Optional[Pattern[c, Blank[]]], Power[Pattern[x, Blank[]], Optional[Pattern[n2, Blank[]]]]], Times[Optional[Pattern[b, Blank[]]], Power[Pattern[x, Blank[]], Pattern[n, Blank[]]]]], Pattern[p, Blank[]]]], Pattern[x, Blank[Symbol]]] := With[{k = Denominator[n]}, Dist[k, Subst[Int[x^(k*(m + 1) - 1)*(a + b*x^(k*n) + c*x^(2*k*n))^p, x], x, x^(1/k)], x]] /; FreeQ[{a, b, c, m, p}, x] && EqQ[n2, 2*n] && NeQ[b^2 - 4*a*c, 0] && FractionQ[n]
  1380. Int[Times[Power[Times[Pattern[d, Blank[]], Pattern[x, Blank[]]], Pattern[m, Blank[]]], Power[Plus[Pattern[a, Blank[]], Times[Optional[Pattern[c, Blank[]]], Power[Pattern[x, Blank[]], Optional[Pattern[n2, Blank[]]]]], Times[Optional[Pattern[b, Blank[]]], Power[Pattern[x, Blank[]], Pattern[n, Blank[]]]]], Pattern[p, Blank[]]]], Pattern[x, Blank[Symbol]]] := Dist[(d^IntPart[m]*(d*x)^FracPart[m])/x^FracPart[m], Int[x^m*(a + b*x^n + c*x^(2*n))^p, x], x] /; FreeQ[{a, b, c, d, m, p}, x] && EqQ[n2, 2*n] && NeQ[b^2 - 4*a*c, 0] && FractionQ[n]
  1381. Int[Times[Power[Pattern[x, Blank[]], Optional[Pattern[m, Blank[]]]], Power[Plus[Pattern[a, Blank[]], Times[Optional[Pattern[c, Blank[]]], Power[Pattern[x, Blank[]], Optional[Pattern[n2, Blank[]]]]], Times[Optional[Pattern[b, Blank[]]], Power[Pattern[x, Blank[]], Pattern[n, Blank[]]]]], Pattern[p, Blank[]]]], Pattern[x, Blank[Symbol]]] := Dist[1/(m + 1), Subst[Int[(a + b*x^Simplify[n/(m + 1)] + c*x^Simplify[(2*n)/(m + 1)])^p, x], x, x^(m + 1)], x] /; FreeQ[{a, b, c, m, n, p}, x] && EqQ[n2, 2*n] && NeQ[b^2 - 4*a*c, 0] && IntegerQ[Simplify[n/(m + 1)]] && !IntegerQ[n]
  1382. Int[Times[Power[Times[Pattern[d, Blank[]], Pattern[x, Blank[]]], Pattern[m, Blank[]]], Power[Plus[Pattern[a, Blank[]], Times[Optional[Pattern[c, Blank[]]], Power[Pattern[x, Blank[]], Optional[Pattern[n2, Blank[]]]]], Times[Optional[Pattern[b, Blank[]]], Power[Pattern[x, Blank[]], Pattern[n, Blank[]]]]], Pattern[p, Blank[]]]], Pattern[x, Blank[Symbol]]] := Dist[(d^IntPart[m]*(d*x)^FracPart[m])/x^FracPart[m], Int[x^m*(a + b*x^n + c*x^(2*n))^p, x], x] /; FreeQ[{a, b, c, d, m, n, p}, x] && EqQ[n2, 2*n] && NeQ[b^2 - 4*a*c, 0] && IntegerQ[Simplify[n/(m + 1)]] && !IntegerQ[n]
  1383. Int[Times[Power[Times[Optional[Pattern[d, Blank[]]], Pattern[x, Blank[]]], Optional[Pattern[m, Blank[]]]], Power[Plus[Pattern[a, Blank[]], Times[Optional[Pattern[c, Blank[]]], Power[Pattern[x, Blank[]], Optional[Pattern[n2, Blank[]]]]], Times[Optional[Pattern[b, Blank[]]], Power[Pattern[x, Blank[]], Pattern[n, Blank[]]]]], -1]], Pattern[x, Blank[Symbol]]] := With[{q = Rt[b^2 - 4*a*c, 2]}, Dist[(2*c)/q, Int[(d*x)^m/(b - q + 2*c*x^n), x], x] - Dist[(2*c)/q, Int[(d*x)^m/(b + q + 2*c*x^n), x], x]] /; FreeQ[{a, b, c, d, m, n}, x] && EqQ[n2, 2*n] && NeQ[b^2 - 4*a*c, 0]
  1384. Int[Times[Power[Times[Optional[Pattern[d, Blank[]]], Pattern[x, Blank[]]], Optional[Pattern[m, Blank[]]]], Power[Plus[Pattern[a, Blank[]], Times[Optional[Pattern[c, Blank[]]], Power[Pattern[x, Blank[]], Optional[Pattern[n2, Blank[]]]]], Times[Optional[Pattern[b, Blank[]]], Power[Pattern[x, Blank[]], Pattern[n, Blank[]]]]], Pattern[p, Blank[]]]], Pattern[x, Blank[Symbol]]] := -Simp[((d*x)^(m + 1)*(b^2 - 2*a*c + b*c*x^n)*(a + b*x^n + c*x^(2*n))^(p + 1))/(a*d*n*(p + 1)*(b^2 - 4*a*c)), x] + Dist[1/(a*n*(p + 1)*(b^2 - 4*a*c)), Int[(d*x)^m*(a + b*x^n + c*x^(2*n))^(p + 1)*Simp[b^2*(n*(p + 1) + m + 1) - 2*a*c*(m + 2*n*(p + 1) + 1) + b*c*(2*n*p + 3*n + m + 1)*x^n, x], x], x] /; FreeQ[{a, b, c, d, m, n}, x] && EqQ[n2, 2*n] && NeQ[b^2 - 4*a*c, 0] && ILtQ[p + 1, 0]
  1385. Int[Times[Power[Times[Optional[Pattern[d, Blank[]]], Pattern[x, Blank[]]], Optional[Pattern[m, Blank[]]]], Power[Plus[Pattern[a, Blank[]], Times[Optional[Pattern[c, Blank[]]], Power[Pattern[x, Blank[]], Optional[Pattern[n2, Blank[]]]]], Times[Optional[Pattern[b, Blank[]]], Power[Pattern[x, Blank[]], Pattern[n, Blank[]]]]], Pattern[p, Blank[]]]], Pattern[x, Blank[Symbol]]] := Dist[(a^IntPart[p]*(a + b*x^n + c*x^(2*n))^FracPart[p])/((1 + (2*c*x^n)/(b + Rt[b^2 - 4*a*c, 2]))^FracPart[p]*(1 + (2*c*x^n)/(b - Rt[b^2 - 4*a*c, 2]))^FracPart[p]), Int[(d*x)^m*(1 + (2*c*x^n)/(b + Sqrt[b^2 - 4*a*c]))^p*(1 + (2*c*x^n)/(b - Sqrt[b^2 - 4*a*c]))^p, x], x] /; FreeQ[{a, b, c, d, m, n, p}, x] && EqQ[n2, 2*n]
  1386. Int[Times[Power[Pattern[x, Blank[]], Optional[Pattern[m, Blank[]]]], Power[Plus[Pattern[a, Blank[]], Times[Optional[Pattern[c, Blank[]]], Power[Pattern[x, Blank[]], Optional[Pattern[n, Blank[]]]]], Times[Optional[Pattern[b, Blank[]]], Power[Pattern[x, Blank[]], Pattern[mn, Blank[]]]]], Optional[Pattern[p, Blank[]]]]], Pattern[x, Blank[Symbol]]] := Int[x^(m - n*p)*(b + a*x^n + c*x^(2*n))^p, x] /; FreeQ[{a, b, c, m, n}, x] && EqQ[mn, -n] && IntegerQ[p] && PosQ[n]
  1387. Int[Times[Power[Pattern[x, Blank[]], Optional[Pattern[m, Blank[]]]], Power[Plus[Pattern[a, Blank[]], Times[Optional[Pattern[c, Blank[]]], Power[Pattern[x, Blank[]], Optional[Pattern[n, Blank[]]]]], Times[Optional[Pattern[b, Blank[]]], Power[Pattern[x, Blank[]], Pattern[mn, Blank[]]]]], Optional[Pattern[p, Blank[]]]]], Pattern[x, Blank[Symbol]]] := Dist[(x^(n*FracPart[p])*(a + b/x^n + c*x^n)^FracPart[p])/(b + a*x^n + c*x^(2*n))^FracPart[p], Int[x^(m - n*p)*(b + a*x^n + c*x^(2*n))^p, x], x] /; FreeQ[{a, b, c, m, n, p}, x] && EqQ[mn, -n] && !IntegerQ[p] && PosQ[n]
  1388. Int[Times[Power[Times[Pattern[d, Blank[]], Pattern[x, Blank[]]], Optional[Pattern[m, Blank[]]]], Power[Plus[Pattern[a, Blank[]], Times[Optional[Pattern[c, Blank[]]], Power[Pattern[x, Blank[]], Optional[Pattern[n, Blank[]]]]], Times[Optional[Pattern[b, Blank[]]], Power[Pattern[x, Blank[]], Pattern[mn, Blank[]]]]], Optional[Pattern[p, Blank[]]]]], Pattern[x, Blank[Symbol]]] := Dist[(d^IntPart[m]*(d*x)^FracPart[m])/x^FracPart[m], Int[x^m*(a + b/x^n + c*x^n)^p, x], x] /; FreeQ[{a, b, c, d, m, n, p}, x] && EqQ[mn, -n]
  1389. Int[Times[Power[Plus[Optional[Pattern[a, Blank[]]], Times[Optional[Pattern[c, Blank[]]], Power[Pattern[v, Blank[]], Optional[Pattern[n2, Blank[]]]]], Times[Optional[Pattern[b, Blank[]]], Power[Pattern[v, Blank[]], Pattern[n, Blank[]]]]], Optional[Pattern[p, Blank[]]]], Power[Pattern[x, Blank[]], Optional[Pattern[m, Blank[]]]]], Pattern[x, Blank[Symbol]]] := Dist[1/Coefficient[v, x, 1]^(m + 1), Subst[Int[SimplifyIntegrand[(x - Coefficient[v, x, 0])^m*(a + b*x^n + c*x^(2*n))^p, x], x], x, v], x] /; FreeQ[{a, b, c, n, p}, x] && EqQ[n2, 2*n] && LinearQ[v, x] && IntegerQ[m] && NeQ[v, x]
  1390. Int[Times[Power[Pattern[u, Blank[]], Optional[Pattern[m, Blank[]]]], Power[Plus[Optional[Pattern[a, Blank[]]], Times[Optional[Pattern[c, Blank[]]], Power[Pattern[v, Blank[]], Optional[Pattern[n2, Blank[]]]]], Times[Optional[Pattern[b, Blank[]]], Power[Pattern[v, Blank[]], Pattern[n, Blank[]]]]], Optional[Pattern[p, Blank[]]]]], Pattern[x, Blank[Symbol]]] := Dist[u^m/(Coefficient[v, x, 1]*v^m), Subst[Int[x^m*(a + b*x^n + c*x^(2*n))^p, x], x, v], x] /; FreeQ[{a, b, c, m, n, p}, x] && EqQ[n2, 2*n] && LinearPairQ[u, v, x]
  1391. Int[Times[Power[Plus[Pattern[d, Blank[]], Times[Optional[Pattern[e, Blank[]]], Power[Pattern[x, Blank[]], Optional[Pattern[n, Blank[]]]]]], Optional[Pattern[q, Blank[]]]], Power[Plus[Pattern[a, Blank[]], Times[Optional[Pattern[b, Blank[]]], Power[Pattern[x, Blank[]], Optional[Pattern[n, Blank[]]]]], Times[Optional[Pattern[c, Blank[]]], Power[Pattern[x, Blank[]], Optional[Pattern[n2, Blank[]]]]]], Pattern[p, Blank[]]]], Pattern[x, Blank[Symbol]]] := Dist[(a + b*x^n + c*x^(2*n))^p/(d + e*x^n)^(2*p), Int[(d + e*x^n)^(q + 2*p), x], x] /; FreeQ[{a, b, c, d, e, n, p, q}, x] && EqQ[n2, 2*n] && EqQ[b^2 - 4*a*c, 0] && !IntegerQ[p] && EqQ[2*c*d - b*e, 0]
  1392. Int[Times[Power[Plus[Pattern[d, Blank[]], Times[Optional[Pattern[e, Blank[]]], Power[Pattern[x, Blank[]], Optional[Pattern[n, Blank[]]]]]], Optional[Pattern[q, Blank[]]]], Power[Plus[Pattern[a, Blank[]], Times[Optional[Pattern[b, Blank[]]], Power[Pattern[x, Blank[]], Optional[Pattern[n, Blank[]]]]], Times[Optional[Pattern[c, Blank[]]], Power[Pattern[x, Blank[]], Optional[Pattern[n2, Blank[]]]]]], Pattern[p, Blank[]]]], Pattern[x, Blank[Symbol]]] := Dist[(a + b*x^n + c*x^(2*n))^FracPart[p]/(c^IntPart[p]*(b/2 + c*x^n)^(2*FracPart[p])), Int[(d + e*x^n)^q*(b/2 + c*x^n)^(2*p), x], x] /; FreeQ[{a, b, c, d, e, n, p, q}, x] && EqQ[n2, 2*n] && EqQ[b^2 - 4*a*c, 0] && !IntegerQ[p]
  1393. Int[Times[Power[Plus[Pattern[a, Blank[]], Times[Optional[Pattern[c, Blank[]]], Power[Pattern[x, Blank[]], Optional[Pattern[n2, Blank[]]]]], Times[Optional[Pattern[b, Blank[]]], Power[Pattern[x, Blank[]], Pattern[n, Blank[]]]]], Optional[Pattern[p, Blank[]]]], Power[Plus[Pattern[d, Blank[]], Times[Optional[Pattern[e, Blank[]]], Power[Pattern[x, Blank[]], Pattern[n, Blank[]]]]], Optional[Pattern[q, Blank[]]]]], Pattern[x, Blank[Symbol]]] := Int[x^(n*(2*p + q))*(e + d/x^n)^q*(c + b/x^n + a/x^(2*n))^p, x] /; FreeQ[{a, b, c, d, e, n}, x] && EqQ[n2, 2*n] && IntegersQ[p, q] && NegQ[n]
  1394. Int[Times[Power[Plus[Pattern[a, Blank[]], Times[Optional[Pattern[c, Blank[]]], Power[Pattern[x, Blank[]], Optional[Pattern[n2, Blank[]]]]]], Optional[Pattern[p, Blank[]]]], Power[Plus[Pattern[d, Blank[]], Times[Optional[Pattern[e, Blank[]]], Power[Pattern[x, Blank[]], Pattern[n, Blank[]]]]], Optional[Pattern[q, Blank[]]]]], Pattern[x, Blank[Symbol]]] := Int[x^(n*(2*p + q))*(e + d/x^n)^q*(c + a/x^(2*n))^p, x] /; FreeQ[{a, c, d, e, n}, x] && EqQ[n2, 2*n] && IntegersQ[p, q] && NegQ[n]
  1395. Int[Times[Power[Plus[Pattern[d, Blank[]], Times[Optional[Pattern[e, Blank[]]], Power[Pattern[x, Blank[]], Pattern[n, Blank[]]]]], Optional[Pattern[q, Blank[]]]], Power[Plus[Optional[Pattern[a, Blank[]]], Times[Optional[Pattern[b, Blank[]]], Power[Pattern[x, Blank[]], Pattern[n, Blank[]]]], Times[Optional[Pattern[c, Blank[]]], Power[Pattern[x, Blank[]], Pattern[n2, Blank[]]]]], Pattern[p, Blank[]]]], Pattern[x, Blank[Symbol]]] := -Subst[Int[((d + e/x^n)^q*(a + b/x^n + c/x^(2*n))^p)/x^2, x], x, 1/x] /; FreeQ[{a, b, c, d, e, p, q}, x] && EqQ[n2, 2*n] && ILtQ[n, 0]
  1396. Int[Times[Power[Plus[Pattern[d, Blank[]], Times[Optional[Pattern[e, Blank[]]], Power[Pattern[x, Blank[]], Pattern[n, Blank[]]]]], Optional[Pattern[q, Blank[]]]], Power[Plus[Pattern[a, Blank[]], Times[Optional[Pattern[c, Blank[]]], Power[Pattern[x, Blank[]], Pattern[n2, Blank[]]]]], Pattern[p, Blank[]]]], Pattern[x, Blank[Symbol]]] := -Subst[Int[((d + e/x^n)^q*(a + c/x^(2*n))^p)/x^2, x], x, 1/x] /; FreeQ[{a, c, d, e, p, q}, x] && EqQ[n2, 2*n] && ILtQ[n, 0]
  1397. Int[Times[Power[Plus[Optional[Pattern[a, Blank[]]], Times[Optional[Pattern[c, Blank[]]], Power[Pattern[x, Blank[]], Optional[Pattern[n2, Blank[]]]]], Times[Optional[Pattern[b, Blank[]]], Power[Pattern[x, Blank[]], Pattern[n, Blank[]]]]], Optional[Pattern[p, Blank[]]]], Power[Plus[Pattern[d, Blank[]], Times[Optional[Pattern[e, Blank[]]], Power[Pattern[x, Blank[]], Pattern[n, Blank[]]]]], Optional[Pattern[q, Blank[]]]]], Pattern[x, Blank[Symbol]]] := With[{g = Denominator[n]}, Dist[g, Subst[Int[x^(g - 1)*(d + e*x^(g*n))^q*(a + b*x^(g*n) + c*x^(2*g*n))^p, x], x, x^(1/g)], x]] /; FreeQ[{a, b, c, d, e, p, q}, x] && EqQ[n2, 2*n] && FractionQ[n]
  1398. Int[Times[Power[Plus[Pattern[a, Blank[]], Times[Optional[Pattern[c, Blank[]]], Power[Pattern[x, Blank[]], Optional[Pattern[n2, Blank[]]]]]], Optional[Pattern[p, Blank[]]]], Power[Plus[Pattern[d, Blank[]], Times[Optional[Pattern[e, Blank[]]], Power[Pattern[x, Blank[]], Pattern[n, Blank[]]]]], Optional[Pattern[q, Blank[]]]]], Pattern[x, Blank[Symbol]]] := With[{g = Denominator[n]}, Dist[g, Subst[Int[x^(g - 1)*(d + e*x^(g*n))^q*(a + c*x^(2*g*n))^p, x], x, x^(1/g)], x]] /; FreeQ[{a, c, d, e, p, q}, x] && EqQ[n2, 2*n] && FractionQ[n]
  1399. Int[Times[Plus[Pattern[d, Blank[]], Times[Optional[Pattern[e, Blank[]]], Power[Pattern[x, Blank[]], Pattern[n, Blank[]]]]], Power[Plus[Times[Optional[Pattern[b, Blank[]]], Power[Pattern[x, Blank[]], Pattern[n, Blank[]]]], Times[Optional[Pattern[c, Blank[]]], Power[Pattern[x, Blank[]], Pattern[n2, Blank[]]]]], Pattern[p, Blank[]]]], Pattern[x, Blank[Symbol]]] := Simp[((b*e - d*c)*(b*x^n + c*x^(2*n))^(p + 1))/(b*c*n*(p + 1)*x^(2*n*(p + 1))), x] + Dist[e/c, Int[(b*x^n + c*x^(2*n))^(p + 1)/x^n, x], x] /; FreeQ[{b, c, d, e, n, p}, x] && EqQ[n2, 2*n] && !IntegerQ[p] && EqQ[n*(2*p + 1) + 1, 0]
  1400. Int[Times[Plus[Pattern[d, Blank[]], Times[Optional[Pattern[e, Blank[]]], Power[Pattern[x, Blank[]], Pattern[n, Blank[]]]]], Power[Plus[Times[Optional[Pattern[b, Blank[]]], Power[Pattern[x, Blank[]], Pattern[n, Blank[]]]], Times[Optional[Pattern[c, Blank[]]], Power[Pattern[x, Blank[]], Pattern[n2, Blank[]]]]], Pattern[p, Blank[]]]], Pattern[x, Blank[Symbol]]] := Simp[(e*x^(-n + 1)*(b*x^n + c*x^(2*n))^(p + 1))/(c*(n*(2*p + 1) + 1)), x] /; FreeQ[{b, c, d, e, n, p}, x] && EqQ[n2, 2*n] && !IntegerQ[p] && NeQ[n*(2*p + 1) + 1, 0] && EqQ[b*e*(n*p + 1) - c*d*(n*(2*p + 1) + 1), 0]
  1401. Int[Times[Plus[Pattern[d, Blank[]], Times[Optional[Pattern[e, Blank[]]], Power[Pattern[x, Blank[]], Pattern[n, Blank[]]]]], Power[Plus[Times[Optional[Pattern[b, Blank[]]], Power[Pattern[x, Blank[]], Pattern[n, Blank[]]]], Times[Optional[Pattern[c, Blank[]]], Power[Pattern[x, Blank[]], Pattern[n2, Blank[]]]]], Pattern[p, Blank[]]]], Pattern[x, Blank[Symbol]]] := Simp[(e*x^(-n + 1)*(b*x^n + c*x^(2*n))^(p + 1))/(c*(n*(2*p + 1) + 1)), x] - Dist[(b*e*(n*p + 1) - c*d*(n*(2*p + 1) + 1))/(c*(n*(2*p + 1) + 1)), Int[(b*x^n + c*x^(2*n))^p, x], x] /; FreeQ[{b, c, d, e, n, p}, x] && EqQ[n2, 2*n] && !IntegerQ[p] && NeQ[n*(2*p + 1) + 1, 0] && NeQ[b*e*(n*p + 1) - c*d*(n*(2*p + 1) + 1), 0]
  1402. Int[Times[Power[Plus[Pattern[d, Blank[]], Times[Optional[Pattern[e, Blank[]]], Power[Pattern[x, Blank[]], Pattern[n, Blank[]]]]], Optional[Pattern[q, Blank[]]]], Power[Plus[Times[Optional[Pattern[b, Blank[]]], Power[Pattern[x, Blank[]], Pattern[n, Blank[]]]], Times[Optional[Pattern[c, Blank[]]], Power[Pattern[x, Blank[]], Pattern[n2, Blank[]]]]], Pattern[p, Blank[]]]], Pattern[x, Blank[Symbol]]] := Dist[(b*x^n + c*x^(2*n))^FracPart[p]/(x^(n*FracPart[p])*(b + c*x^n)^FracPart[p]), Int[x^(n*p)*(d + e*x^n)^q*(b + c*x^n)^p, x], x] /; FreeQ[{b, c, d, e, n, p, q}, x] && EqQ[n2, 2*n] && !IntegerQ[p]
  1403. Int[Times[Power[Plus[Pattern[d, Blank[]], Times[Optional[Pattern[e, Blank[]]], Power[Pattern[x, Blank[]], Pattern[n, Blank[]]]]], Optional[Pattern[q, Blank[]]]], Power[Plus[Pattern[a, Blank[]], Times[Optional[Pattern[b, Blank[]]], Power[Pattern[x, Blank[]], Pattern[n, Blank[]]]], Times[Optional[Pattern[c, Blank[]]], Power[Pattern[x, Blank[]], Pattern[n2, Blank[]]]]], Optional[Pattern[p, Blank[]]]]], Pattern[x, Blank[Symbol]]] := Int[(d + e*x^n)^(p + q)*(a/d + (c*x^n)/e)^p, x] /; FreeQ[{a, b, c, d, e, n, q}, x] && EqQ[n2, 2*n] && NeQ[b^2 - 4*a*c, 0] && EqQ[c*d^2 - b*d*e + a*e^2, 0] && IntegerQ[p]
  1404. Int[Times[Power[Plus[Pattern[d, Blank[]], Times[Optional[Pattern[e, Blank[]]], Power[Pattern[x, Blank[]], Pattern[n, Blank[]]]]], Optional[Pattern[q, Blank[]]]], Power[Plus[Pattern[a, Blank[]], Times[Optional[Pattern[c, Blank[]]], Power[Pattern[x, Blank[]], Pattern[n2, Blank[]]]]], Optional[Pattern[p, Blank[]]]]], Pattern[x, Blank[Symbol]]] := Int[(d + e*x^n)^(p + q)*(a/d + (c*x^n)/e)^p, x] /; FreeQ[{a, c, d, e, n, q}, x] && EqQ[n2, 2*n] && EqQ[c*d^2 + a*e^2, 0] && IntegerQ[p]
  1405. Int[Times[Power[Plus[Pattern[d, Blank[]], Times[Optional[Pattern[e, Blank[]]], Power[Pattern[x, Blank[]], Pattern[n, Blank[]]]]], Pattern[q, Blank[]]], Power[Plus[Pattern[a, Blank[]], Times[Optional[Pattern[b, Blank[]]], Power[Pattern[x, Blank[]], Pattern[n, Blank[]]]], Times[Optional[Pattern[c, Blank[]]], Power[Pattern[x, Blank[]], Pattern[n2, Blank[]]]]], Pattern[p, Blank[]]]], Pattern[x, Blank[Symbol]]] := Dist[(a + b*x^n + c*x^(2*n))^FracPart[p]/((d + e*x^n)^FracPart[p]*(a/d + (c*x^n)/e)^FracPart[p]), Int[(d + e*x^n)^(p + q)*(a/d + (c*x^n)/e)^p, x], x] /; FreeQ[{a, b, c, d, e, n, p, q}, x] && EqQ[n2, 2*n] && NeQ[b^2 - 4*a*c, 0] && EqQ[c*d^2 - b*d*e + a*e^2, 0] && !IntegerQ[p]
  1406. Int[Times[Power[Plus[Pattern[d, Blank[]], Times[Optional[Pattern[e, Blank[]]], Power[Pattern[x, Blank[]], Pattern[n, Blank[]]]]], Pattern[q, Blank[]]], Power[Plus[Pattern[a, Blank[]], Times[Optional[Pattern[c, Blank[]]], Power[Pattern[x, Blank[]], Pattern[n2, Blank[]]]]], Pattern[p, Blank[]]]], Pattern[x, Blank[Symbol]]] := Dist[(a + c*x^(2*n))^FracPart[p]/((d + e*x^n)^FracPart[p]*(a/d + (c*x^n)/e)^FracPart[p]), Int[(d + e*x^n)^(p + q)*(a/d + (c*x^n)/e)^p, x], x] /; FreeQ[{a, c, d, e, n, p, q}, x] && EqQ[n2, 2*n] && EqQ[c*d^2 + a*e^2, 0] && !IntegerQ[p]
  1407. Int[Times[Power[Plus[Pattern[d, Blank[]], Times[Optional[Pattern[e, Blank[]]], Power[Pattern[x, Blank[]], Pattern[n, Blank[]]]]], Optional[Pattern[q, Blank[]]]], Plus[Pattern[a, Blank[]], Times[Optional[Pattern[b, Blank[]]], Power[Pattern[x, Blank[]], Pattern[n, Blank[]]]], Times[Optional[Pattern[c, Blank[]]], Power[Pattern[x, Blank[]], Pattern[n2, Blank[]]]]]], Pattern[x, Blank[Symbol]]] := Int[ExpandIntegrand[(d + e*x^n)^q*(a + b*x^n + c*x^(2*n)), x], x] /; FreeQ[{a, b, c, d, e, n}, x] && EqQ[n2, 2*n] && NeQ[b^2 - 4*a*c, 0] && NeQ[c*d^2 - b*d*e + a*e^2, 0] && IGtQ[q, 0]
  1408. Int[Times[Power[Plus[Pattern[d, Blank[]], Times[Optional[Pattern[e, Blank[]]], Power[Pattern[x, Blank[]], Pattern[n, Blank[]]]]], Optional[Pattern[q, Blank[]]]], Plus[Pattern[a, Blank[]], Times[Optional[Pattern[c, Blank[]]], Power[Pattern[x, Blank[]], Pattern[n2, Blank[]]]]]], Pattern[x, Blank[Symbol]]] := Int[ExpandIntegrand[(d + e*x^n)^q*(a + c*x^(2*n)), x], x] /; FreeQ[{a, c, d, e, n}, x] && EqQ[n2, 2*n] && NeQ[c*d^2 + a*e^2, 0] && IGtQ[q, 0]
  1409. Int[Times[Power[Plus[Pattern[d, Blank[]], Times[Optional[Pattern[e, Blank[]]], Power[Pattern[x, Blank[]], Pattern[n, Blank[]]]]], Pattern[q, Blank[]]], Plus[Pattern[a, Blank[]], Times[Optional[Pattern[b, Blank[]]], Power[Pattern[x, Blank[]], Pattern[n, Blank[]]]], Times[Optional[Pattern[c, Blank[]]], Power[Pattern[x, Blank[]], Pattern[n2, Blank[]]]]]], Pattern[x, Blank[Symbol]]] := -Simp[((c*d^2 - b*d*e + a*e^2)*x*(d + e*x^n)^(q + 1))/(d*e^2*n*(q + 1)), x] + Dist[1/(n*(q + 1)*d*e^2), Int[(d + e*x^n)^(q + 1)*Simp[c*d^2 - b*d*e + a*e^2*(n*(q + 1) + 1) + c*d*e*n*(q + 1)*x^n, x], x], x] /; FreeQ[{a, b, c, d, e, n}, x] && EqQ[n2, 2*n] && NeQ[b^2 - 4*a*c, 0] && NeQ[c*d^2 - b*d*e + a*e^2, 0] && LtQ[q, -1]
  1410. Int[Times[Power[Plus[Pattern[d, Blank[]], Times[Optional[Pattern[e, Blank[]]], Power[Pattern[x, Blank[]], Pattern[n, Blank[]]]]], Pattern[q, Blank[]]], Plus[Pattern[a, Blank[]], Times[Optional[Pattern[c, Blank[]]], Power[Pattern[x, Blank[]], Pattern[n2, Blank[]]]]]], Pattern[x, Blank[Symbol]]] := -Simp[((c*d^2 + a*e^2)*x*(d + e*x^n)^(q + 1))/(d*e^2*n*(q + 1)), x] + Dist[1/(n*(q + 1)*d*e^2), Int[(d + e*x^n)^(q + 1)*Simp[c*d^2 + a*e^2*(n*(q + 1) + 1) + c*d*e*n*(q + 1)*x^n, x], x], x] /; FreeQ[{a, c, d, e, n}, x] && EqQ[n2, 2*n] && NeQ[c*d^2 + a*e^2, 0] && LtQ[q, -1]
  1411. Int[Times[Power[Plus[Pattern[d, Blank[]], Times[Optional[Pattern[e, Blank[]]], Power[Pattern[x, Blank[]], Pattern[n, Blank[]]]]], Pattern[q, Blank[]]], Plus[Pattern[a, Blank[]], Times[Optional[Pattern[b, Blank[]]], Power[Pattern[x, Blank[]], Pattern[n, Blank[]]]], Times[Optional[Pattern[c, Blank[]]], Power[Pattern[x, Blank[]], Pattern[n2, Blank[]]]]]], Pattern[x, Blank[Symbol]]] := Simp[(c*x^(n + 1)*(d + e*x^n)^(q + 1))/(e*(n*(q + 2) + 1)), x] + Dist[1/(e*(n*(q + 2) + 1)), Int[(d + e*x^n)^q*(a*e*(n*(q + 2) + 1) - (c*d*(n + 1) - b*e*(n*(q + 2) + 1))*x^n), x], x] /; FreeQ[{a, b, c, d, e, n, q}, x] && EqQ[n2, 2*n] && NeQ[b^2 - 4*a*c, 0] && NeQ[c*d^2 - b*d*e + a*e^2, 0]
  1412. Int[Times[Power[Plus[Pattern[d, Blank[]], Times[Optional[Pattern[e, Blank[]]], Power[Pattern[x, Blank[]], Pattern[n, Blank[]]]]], Pattern[q, Blank[]]], Plus[Pattern[a, Blank[]], Times[Optional[Pattern[c, Blank[]]], Power[Pattern[x, Blank[]], Pattern[n2, Blank[]]]]]], Pattern[x, Blank[Symbol]]] := Simp[(c*x^(n + 1)*(d + e*x^n)^(q + 1))/(e*(n*(q + 2) + 1)), x] + Dist[1/(e*(n*(q + 2) + 1)), Int[(d + e*x^n)^q*(a*e*(n*(q + 2) + 1) - c*d*(n + 1)*x^n), x], x] /; FreeQ[{a, c, d, e, n, q}, x] && EqQ[n2, 2*n] && NeQ[c*d^2 + a*e^2, 0]
  1413. Int[Times[Plus[Pattern[d, Blank[]], Times[Optional[Pattern[e, Blank[]]], Power[Pattern[x, Blank[]], Pattern[n, Blank[]]]]], Power[Plus[Pattern[a, Blank[]], Times[Optional[Pattern[c, Blank[]]], Power[Pattern[x, Blank[]], Pattern[n2, Blank[]]]]], -1]], Pattern[x, Blank[Symbol]]] := With[{q = Rt[2*d*e, 2]}, Dist[e^2/(2*c), Int[1/(d + q*x^(n/2) + e*x^n), x], x] + Dist[e^2/(2*c), Int[1/(d - q*x^(n/2) + e*x^n), x], x]] /; FreeQ[{a, c, d, e}, x] && EqQ[n2, 2*n] && EqQ[c*d^2 - a*e^2, 0] && IGtQ[n/2, 0] && PosQ[d*e]
  1414. Int[Times[Plus[Pattern[d, Blank[]], Times[Optional[Pattern[e, Blank[]]], Power[Pattern[x, Blank[]], Pattern[n, Blank[]]]]], Power[Plus[Pattern[a, Blank[]], Times[Optional[Pattern[c, Blank[]]], Power[Pattern[x, Blank[]], Pattern[n2, Blank[]]]]], -1]], Pattern[x, Blank[Symbol]]] := With[{q = Rt[-2*d*e, 2]}, Dist[d/(2*a), Int[(d - q*x^(n/2))/(d - q*x^(n/2) - e*x^n), x], x] + Dist[d/(2*a), Int[(d + q*x^(n/2))/(d + q*x^(n/2) - e*x^n), x], x]] /; FreeQ[{a, c, d, e}, x] && EqQ[n2, 2*n] && EqQ[c*d^2 - a*e^2, 0] && IGtQ[n/2, 0] && NegQ[d*e]
  1415. Int[Times[Plus[Pattern[d, Blank[]], Times[Optional[Pattern[e, Blank[]]], Power[Pattern[x, Blank[]], Pattern[n, Blank[]]]]], Power[Plus[Pattern[a, Blank[]], Times[Optional[Pattern[c, Blank[]]], Power[Pattern[x, Blank[]], Pattern[n2, Blank[]]]]], -1]], Pattern[x, Blank[Symbol]]] := With[{q = Rt[a/c, 4]}, Dist[1/(2*Sqrt[2]*c*q^3), Int[(Sqrt[2]*d*q - (d - e*q^2)*x^(n/2))/(q^2 - Sqrt[2]*q*x^(n/2) + x^n), x], x] + Dist[1/(2*Sqrt[2]*c*q^3), Int[(Sqrt[2]*d*q + (d - e*q^2)*x^(n/2))/(q^2 + Sqrt[2]*q*x^(n/2) + x^n), x], x]] /; FreeQ[{a, c, d, e}, x] && EqQ[n2, 2*n] && NeQ[c*d^2 + a*e^2, 0] && NeQ[c*d^2 - a*e^2, 0] && IGtQ[n/2, 0] && PosQ[a*c]
  1416. Int[Times[Plus[Pattern[d, Blank[]], Times[Optional[Pattern[e, Blank[]]], Power[Pattern[x, Blank[]], 3]]], Power[Plus[Pattern[a, Blank[]], Times[Optional[Pattern[c, Blank[]]], Power[Pattern[x, Blank[]], 6]]], -1]], Pattern[x, Blank[Symbol]]] := With[{q = Rt[c/a, 6]}, Dist[1/(3*a*q^2), Int[(q^2*d - e*x)/(1 + q^2*x^2), x], x] + (Dist[1/(6*a*q^2), Int[(2*q^2*d - (Sqrt[3]*q^3*d - e)*x)/(1 - Sqrt[3]*q*x + q^2*x^2), x], x] + Dist[1/(6*a*q^2), Int[(2*q^2*d + (Sqrt[3]*q^3*d + e)*x)/(1 + Sqrt[3]*q*x + q^2*x^2), x], x])] /; FreeQ[{a, c, d, e}, x] && NeQ[c*d^2 + a*e^2, 0] && PosQ[c/a]
  1417. Int[Times[Plus[Pattern[d, Blank[]], Times[Optional[Pattern[e, Blank[]]], Power[Pattern[x, Blank[]], Pattern[n, Blank[]]]]], Power[Plus[Pattern[a, Blank[]], Times[Optional[Pattern[c, Blank[]]], Power[Pattern[x, Blank[]], Pattern[n2, Blank[]]]]], -1]], Pattern[x, Blank[Symbol]]] := With[{q = Rt[-(a/c), 2]}, Dist[(d + e*q)/2, Int[1/(a + c*q*x^n), x], x] + Dist[(d - e*q)/2, Int[1/(a - c*q*x^n), x], x]] /; FreeQ[{a, c, d, e, n}, x] && EqQ[n2, 2*n] && NeQ[c*d^2 + a*e^2, 0] && NegQ[a*c] && IntegerQ[n]
  1418. Int[Times[Plus[Pattern[d, Blank[]], Times[Optional[Pattern[e, Blank[]]], Power[Pattern[x, Blank[]], Pattern[n, Blank[]]]]], Power[Plus[Pattern[a, Blank[]], Times[Optional[Pattern[c, Blank[]]], Power[Pattern[x, Blank[]], Pattern[n2, Blank[]]]]], -1]], Pattern[x, Blank[Symbol]]] := Dist[d, Int[1/(a + c*x^(2*n)), x], x] + Dist[e, Int[x^n/(a + c*x^(2*n)), x], x] /; FreeQ[{a, c, d, e, n}, x] && EqQ[n2, 2*n] && NeQ[c*d^2 + a*e^2, 0] && (PosQ[a*c] || !IntegerQ[n])
  1419. Int[Times[Plus[Pattern[d, Blank[]], Times[Optional[Pattern[e, Blank[]]], Power[Pattern[x, Blank[]], Pattern[n, Blank[]]]]], Power[Plus[Pattern[a, Blank[]], Times[Optional[Pattern[b, Blank[]]], Power[Pattern[x, Blank[]], Pattern[n, Blank[]]]], Times[Optional[Pattern[c, Blank[]]], Power[Pattern[x, Blank[]], Pattern[n2, Blank[]]]]], -1]], Pattern[x, Blank[Symbol]]] := With[{q = Rt[(2*d)/e - b/c, 2]}, Dist[e/(2*c), Int[1/Simp[d/e + q*x^(n/2) + x^n, x], x], x] + Dist[e/(2*c), Int[1/Simp[d/e - q*x^(n/2) + x^n, x], x], x]] /; FreeQ[{a, b, c, d, e}, x] && EqQ[n2, 2*n] && NeQ[b^2 - 4*a*c, 0] && EqQ[c*d^2 - a*e^2, 0] && IGtQ[n/2, 0] && (GtQ[(2*d)/e - b/c, 0] || ( !LtQ[(2*d)/e - b/c, 0] && EqQ[d, e*Rt[a/c, 2]]))
  1420. Int[Times[Plus[Pattern[d, Blank[]], Times[Optional[Pattern[e, Blank[]]], Power[Pattern[x, Blank[]], Pattern[n, Blank[]]]]], Power[Plus[Pattern[a, Blank[]], Times[Optional[Pattern[b, Blank[]]], Power[Pattern[x, Blank[]], Pattern[n, Blank[]]]], Times[Optional[Pattern[c, Blank[]]], Power[Pattern[x, Blank[]], Pattern[n2, Blank[]]]]], -1]], Pattern[x, Blank[Symbol]]] := With[{q = Rt[b^2 - 4*a*c, 2]}, Dist[e/2 + (2*c*d - b*e)/(2*q), Int[1/(b/2 - q/2 + c*x^n), x], x] + Dist[e/2 - (2*c*d - b*e)/(2*q), Int[1/(b/2 + q/2 + c*x^n), x], x]] /; FreeQ[{a, b, c, d, e, n}, x] && EqQ[n2, 2*n] && NeQ[b^2 - 4*a*c, 0] && EqQ[c*d^2 - a*e^2, 0] && IGtQ[n/2, 0] && GtQ[b^2 - 4*a*c, 0]
  1421. Int[Times[Plus[Pattern[d, Blank[]], Times[Optional[Pattern[e, Blank[]]], Power[Pattern[x, Blank[]], Pattern[n, Blank[]]]]], Power[Plus[Pattern[a, Blank[]], Times[Optional[Pattern[b, Blank[]]], Power[Pattern[x, Blank[]], Pattern[n, Blank[]]]], Times[Optional[Pattern[c, Blank[]]], Power[Pattern[x, Blank[]], Pattern[n2, Blank[]]]]], -1]], Pattern[x, Blank[Symbol]]] := With[{q = Rt[(-2*d)/e - b/c, 2]}, Dist[e/(2*c*q), Int[(q - 2*x^(n/2))/Simp[d/e + q*x^(n/2) - x^n, x], x], x] + Dist[e/(2*c*q), Int[(q + 2*x^(n/2))/Simp[d/e - q*x^(n/2) - x^n, x], x], x]] /; FreeQ[{a, b, c, d, e}, x] && EqQ[n2, 2*n] && NeQ[b^2 - 4*a*c, 0] && EqQ[c*d^2 - a*e^2, 0] && IGtQ[n/2, 0] && !GtQ[b^2 - 4*a*c, 0]
  1422. Int[Times[Plus[Pattern[d, Blank[]], Times[Optional[Pattern[e, Blank[]]], Power[Pattern[x, Blank[]], Pattern[n, Blank[]]]]], Power[Plus[Pattern[a, Blank[]], Times[Optional[Pattern[b, Blank[]]], Power[Pattern[x, Blank[]], Pattern[n, Blank[]]]], Times[Optional[Pattern[c, Blank[]]], Power[Pattern[x, Blank[]], Pattern[n2, Blank[]]]]], -1]], Pattern[x, Blank[Symbol]]] := With[{q = Rt[b^2 - 4*a*c, 2]}, Dist[e/2 + (2*c*d - b*e)/(2*q), Int[1/(b/2 - q/2 + c*x^n), x], x] + Dist[e/2 - (2*c*d - b*e)/(2*q), Int[1/(b/2 + q/2 + c*x^n), x], x]] /; FreeQ[{a, b, c, d, e, n}, x] && EqQ[n2, 2*n] && NeQ[b^2 - 4*a*c, 0] && NeQ[c*d^2 - b*d*e + a*e^2, 0] && (PosQ[b^2 - 4*a*c] || !IGtQ[n/2, 0])
  1423. Int[Times[Plus[Pattern[d, Blank[]], Times[Optional[Pattern[e, Blank[]]], Power[Pattern[x, Blank[]], Pattern[n, Blank[]]]]], Power[Plus[Pattern[a, Blank[]], Times[Optional[Pattern[b, Blank[]]], Power[Pattern[x, Blank[]], Pattern[n, Blank[]]]], Times[Optional[Pattern[c, Blank[]]], Power[Pattern[x, Blank[]], Pattern[n2, Blank[]]]]], -1]], Pattern[x, Blank[Symbol]]] := With[{q = Rt[a/c, 2]}, With[{r = Rt[2*q - b/c, 2]}, Dist[1/(2*c*q*r), Int[(d*r - (d - e*q)*x^(n/2))/(q - r*x^(n/2) + x^n), x], x] + Dist[1/(2*c*q*r), Int[(d*r + (d - e*q)*x^(n/2))/(q + r*x^(n/2) + x^n), x], x]]] /; FreeQ[{a, b, c, d, e}, x] && EqQ[n2, 2*n] && NeQ[b^2 - 4*a*c, 0] && NeQ[c*d^2 - b*d*e + a*e^2, 0] && IGtQ[n/2, 0] && NegQ[b^2 - 4*a*c]
  1424. Int[Times[Power[Plus[Pattern[d, Blank[]], Times[Optional[Pattern[e, Blank[]]], Power[Pattern[x, Blank[]], Pattern[n, Blank[]]]]], Pattern[q, Blank[]]], Power[Plus[Pattern[a, Blank[]], Times[Optional[Pattern[b, Blank[]]], Power[Pattern[x, Blank[]], Pattern[n, Blank[]]]], Times[Optional[Pattern[c, Blank[]]], Power[Pattern[x, Blank[]], Pattern[n2, Blank[]]]]], -1]], Pattern[x, Blank[Symbol]]] := Int[ExpandIntegrand[(d + e*x^n)^q/(a + b*x^n + c*x^(2*n)), x], x] /; FreeQ[{a, b, c, d, e, n}, x] && EqQ[n2, 2*n] && NeQ[b^2 - 4*a*c, 0] && NeQ[c*d^2 - b*d*e + a*e^2, 0] && IntegerQ[q]
  1425. Int[Times[Power[Plus[Pattern[d, Blank[]], Times[Optional[Pattern[e, Blank[]]], Power[Pattern[x, Blank[]], Pattern[n, Blank[]]]]], Pattern[q, Blank[]]], Power[Plus[Pattern[a, Blank[]], Times[Optional[Pattern[c, Blank[]]], Power[Pattern[x, Blank[]], Pattern[n2, Blank[]]]]], -1]], Pattern[x, Blank[Symbol]]] := Int[ExpandIntegrand[(d + e*x^n)^q/(a + c*x^(2*n)), x], x] /; FreeQ[{a, c, d, e, n}, x] && EqQ[n2, 2*n] && NeQ[c*d^2 + a*e^2, 0] && IntegerQ[q]
  1426. Int[Times[Power[Plus[Pattern[d, Blank[]], Times[Optional[Pattern[e, Blank[]]], Power[Pattern[x, Blank[]], Pattern[n, Blank[]]]]], Pattern[q, Blank[]]], Power[Plus[Pattern[a, Blank[]], Times[Optional[Pattern[b, Blank[]]], Power[Pattern[x, Blank[]], Pattern[n, Blank[]]]], Times[Optional[Pattern[c, Blank[]]], Power[Pattern[x, Blank[]], Pattern[n2, Blank[]]]]], -1]], Pattern[x, Blank[Symbol]]] := Dist[e^2/(c*d^2 - b*d*e + a*e^2), Int[(d + e*x^n)^q, x], x] + Dist[1/(c*d^2 - b*d*e + a*e^2), Int[((d + e*x^n)^(q + 1)*(c*d - b*e - c*e*x^n))/(a + b*x^n + c*x^(2*n)), x], x] /; FreeQ[{a, b, c, d, e, n}, x] && EqQ[n2, 2*n] && NeQ[b^2 - 4*a*c, 0] && NeQ[c*d^2 - b*d*e + a*e^2, 0] && !IntegerQ[q] && LtQ[q, -1]
  1427. Int[Times[Power[Plus[Pattern[d, Blank[]], Times[Optional[Pattern[e, Blank[]]], Power[Pattern[x, Blank[]], Pattern[n, Blank[]]]]], Pattern[q, Blank[]]], Power[Plus[Pattern[a, Blank[]], Times[Optional[Pattern[c, Blank[]]], Power[Pattern[x, Blank[]], Pattern[n2, Blank[]]]]], -1]], Pattern[x, Blank[Symbol]]] := Dist[e^2/(c*d^2 + a*e^2), Int[(d + e*x^n)^q, x], x] + Dist[c/(c*d^2 + a*e^2), Int[((d + e*x^n)^(q + 1)*(d - e*x^n))/(a + c*x^(2*n)), x], x] /; FreeQ[{a, c, d, e, n}, x] && EqQ[n2, 2*n] && NeQ[c*d^2 + a*e^2, 0] && !IntegerQ[q] && LtQ[q, -1]
  1428. Int[Times[Power[Plus[Pattern[d, Blank[]], Times[Optional[Pattern[e, Blank[]]], Power[Pattern[x, Blank[]], Pattern[n, Blank[]]]]], Pattern[q, Blank[]]], Power[Plus[Pattern[a, Blank[]], Times[Optional[Pattern[b, Blank[]]], Power[Pattern[x, Blank[]], Pattern[n, Blank[]]]], Times[Optional[Pattern[c, Blank[]]], Power[Pattern[x, Blank[]], Pattern[n2, Blank[]]]]], -1]], Pattern[x, Blank[Symbol]]] := With[{r = Rt[b^2 - 4*a*c, 2]}, Dist[(2*c)/r, Int[(d + e*x^n)^q/(b - r + 2*c*x^n), x], x] - Dist[(2*c)/r, Int[(d + e*x^n)^q/(b + r + 2*c*x^n), x], x]] /; FreeQ[{a, b, c, d, e, n, q}, x] && EqQ[n2, 2*n] && NeQ[b^2 - 4*a*c, 0] && NeQ[c*d^2 - b*d*e + a*e^2, 0] && !IntegerQ[q]
  1429. Int[Times[Power[Plus[Pattern[d, Blank[]], Times[Optional[Pattern[e, Blank[]]], Power[Pattern[x, Blank[]], Pattern[n, Blank[]]]]], Pattern[q, Blank[]]], Power[Plus[Pattern[a, Blank[]], Times[Optional[Pattern[c, Blank[]]], Power[Pattern[x, Blank[]], Pattern[n2, Blank[]]]]], -1]], Pattern[x, Blank[Symbol]]] := With[{r = Rt[-(a*c), 2]}, -Dist[c/(2*r), Int[(d + e*x^n)^q/(r - c*x^n), x], x] - Dist[c/(2*r), Int[(d + e*x^n)^q/(r + c*x^n), x], x]] /; FreeQ[{a, c, d, e, n, q}, x] && EqQ[n2, 2*n] && NeQ[c*d^2 + a*e^2, 0] && !IntegerQ[q]
  1430. Int[Times[Plus[Pattern[d, Blank[]], Times[Optional[Pattern[e, Blank[]]], Power[Pattern[x, Blank[]], Pattern[n, Blank[]]]]], Power[Plus[Pattern[a, Blank[]], Times[Optional[Pattern[b, Blank[]]], Power[Pattern[x, Blank[]], Pattern[n, Blank[]]]], Times[Optional[Pattern[c, Blank[]]], Power[Pattern[x, Blank[]], Pattern[n2, Blank[]]]]], Pattern[p, Blank[]]]], Pattern[x, Blank[Symbol]]] := -Simp[(x*(d*b^2 - a*b*e - 2*a*c*d + (b*d - 2*a*e)*c*x^n)*(a + b*x^n + c*x^(2*n))^(p + 1))/(a*n*(p + 1)*(b^2 - 4*a*c)), x] + Dist[1/(a*n*(p + 1)*(b^2 - 4*a*c)), Int[Simp[(n*p + n + 1)*d*b^2 - a*b*e - 2*a*c*d*(2*n*p + 2*n + 1) + (2*n*p + 3*n + 1)*(d*b - 2*a*e)*c*x^n, x]*(a + b*x^n + c*x^(2*n))^(p + 1), x], x] /; FreeQ[{a, b, c, d, e, n}, x] && EqQ[n2, 2*n] && NeQ[b^2 - 4*a*c, 0] && ILtQ[p, -1]
  1431. Int[Times[Plus[Pattern[d, Blank[]], Times[Optional[Pattern[e, Blank[]]], Power[Pattern[x, Blank[]], Pattern[n, Blank[]]]]], Power[Plus[Pattern[a, Blank[]], Times[Optional[Pattern[c, Blank[]]], Power[Pattern[x, Blank[]], Pattern[n2, Blank[]]]]], Pattern[p, Blank[]]]], Pattern[x, Blank[Symbol]]] := -Simp[(x*(d + e*x^n)*(a + c*x^(2*n))^(p + 1))/(2*a*n*(p + 1)), x] + Dist[1/(2*a*n*(p + 1)), Int[(d*(2*n*p + 2*n + 1) + e*(2*n*p + 3*n + 1)*x^n)*(a + c*x^(2*n))^(p + 1), x], x] /; FreeQ[{a, c, d, e, n}, x] && EqQ[n2, 2*n] && ILtQ[p, -1]
  1432. Int[Times[Plus[Pattern[d, Blank[]], Times[Optional[Pattern[e, Blank[]]], Power[Pattern[x, Blank[]], Pattern[n, Blank[]]]]], Power[Plus[Pattern[a, Blank[]], Times[Optional[Pattern[b, Blank[]]], Power[Pattern[x, Blank[]], Pattern[n, Blank[]]]], Times[Optional[Pattern[c, Blank[]]], Power[Pattern[x, Blank[]], Pattern[n2, Blank[]]]]], Pattern[p, Blank[]]]], Pattern[x, Blank[Symbol]]] := Int[ExpandIntegrand[(d + e*x^n)*(a + b*x^n + c*x^(2*n))^p, x], x] /; FreeQ[{a, b, c, d, e, n}, x] && EqQ[n2, 2*n] && NeQ[b^2 - 4*a*c, 0]
  1433. Int[Times[Plus[Pattern[d, Blank[]], Times[Optional[Pattern[e, Blank[]]], Power[Pattern[x, Blank[]], Pattern[n, Blank[]]]]], Power[Plus[Pattern[a, Blank[]], Times[Optional[Pattern[c, Blank[]]], Power[Pattern[x, Blank[]], Pattern[n2, Blank[]]]]], Pattern[p, Blank[]]]], Pattern[x, Blank[Symbol]]] := Int[ExpandIntegrand[(d + e*x^n)*(a + c*x^(2*n))^p, x], x] /; FreeQ[{a, c, d, e, n}, x] && EqQ[n2, 2*n]
  1434. Int[Times[Power[Plus[Pattern[d, Blank[]], Times[Optional[Pattern[e, Blank[]]], Power[Pattern[x, Blank[]], Pattern[n, Blank[]]]]], Pattern[q, Blank[]]], Power[Plus[Pattern[a, Blank[]], Times[Optional[Pattern[b, Blank[]]], Power[Pattern[x, Blank[]], Pattern[n, Blank[]]]], Times[Optional[Pattern[c, Blank[]]], Power[Pattern[x, Blank[]], Pattern[n2, Blank[]]]]], Pattern[p, Blank[]]]], Pattern[x, Blank[Symbol]]] := Simp[(c^p*x^(2*n*p - n + 1)*(d + e*x^n)^(q + 1))/(e*(2*n*p + n*q + 1)), x] + Int[(d + e*x^n)^q*ExpandToSum[(a + b*x^n + c*x^(2*n))^p - c^p*x^(2*n*p) - (d*c^p*(2*n*p - n + 1)*x^(2*n*p - n))/(e*(2*n*p + n*q + 1)), x], x] /; FreeQ[{a, b, c, d, e, n, q}, x] && EqQ[n2, 2*n] && NeQ[b^2 - 4*a*c, 0] && IGtQ[p, 0] && NeQ[2*n*p + n*q + 1, 0] && IGtQ[n, 0] && !IGtQ[q, 0]
  1435. Int[Times[Power[Plus[Pattern[d, Blank[]], Times[Optional[Pattern[e, Blank[]]], Power[Pattern[x, Blank[]], Pattern[n, Blank[]]]]], Pattern[q, Blank[]]], Power[Plus[Pattern[a, Blank[]], Times[Optional[Pattern[c, Blank[]]], Power[Pattern[x, Blank[]], Pattern[n2, Blank[]]]]], Pattern[p, Blank[]]]], Pattern[x, Blank[Symbol]]] := Simp[(c^p*x^(2*n*p - n + 1)*(d + e*x^n)^(q + 1))/(e*(2*n*p + n*q + 1)), x] + Int[(d + e*x^n)^q*ExpandToSum[(a + c*x^(2*n))^p - c^p*x^(2*n*p) - (d*c^p*(2*n*p - n + 1)*x^(2*n*p - n))/(e*(2*n*p + n*q + 1)), x], x] /; FreeQ[{a, c, d, e, n, q}, x] && EqQ[n2, 2*n] && IGtQ[p, 0] && NeQ[2*n*p + n*q + 1, 0] && IGtQ[n, 0] && !IGtQ[q, 0]
  1436. Int[Times[Power[Plus[Pattern[d, Blank[]], Times[Optional[Pattern[e, Blank[]]], Power[Pattern[x, Blank[]], Pattern[n, Blank[]]]]], Pattern[q, Blank[]]], Power[Plus[Pattern[a, Blank[]], Times[Optional[Pattern[b, Blank[]]], Power[Pattern[x, Blank[]], Pattern[n, Blank[]]]], Times[Optional[Pattern[c, Blank[]]], Power[Pattern[x, Blank[]], Pattern[n2, Blank[]]]]], Pattern[p, Blank[]]]], Pattern[x, Blank[Symbol]]] := Int[ExpandIntegrand[(d + e*x^n)^q*(a + b*x^n + c*x^(2*n))^p, x], x] /; FreeQ[{a, b, c, d, e, n, p, q}, x] && EqQ[n2, 2*n] && NeQ[b^2 - 4*a*c, 0] && NeQ[c*d^2 - b*d*e + a*e^2, 0] && ((IntegersQ[p, q] && !IntegerQ[n]) || IGtQ[p, 0] || (IGtQ[q, 0] && !IntegerQ[n]))
  1437. Int[Times[Power[Plus[Pattern[d, Blank[]], Times[Optional[Pattern[e, Blank[]]], Power[Pattern[x, Blank[]], Pattern[n, Blank[]]]]], Pattern[q, Blank[]]], Power[Plus[Pattern[a, Blank[]], Times[Optional[Pattern[c, Blank[]]], Power[Pattern[x, Blank[]], Pattern[n2, Blank[]]]]], Pattern[p, Blank[]]]], Pattern[x, Blank[Symbol]]] := Int[ExpandIntegrand[(d + e*x^n)^q*(a + c*x^(2*n))^p, x], x] /; FreeQ[{a, c, d, e, n, p, q}, x] && EqQ[n2, 2*n] && NeQ[c*d^2 + a*e^2, 0] && ((IntegersQ[p, q] && !IntegerQ[n]) || IGtQ[p, 0] || (IGtQ[q, 0] && !IntegerQ[n]))
  1438. Int[Times[Power[Plus[Pattern[d, Blank[]], Times[Optional[Pattern[e, Blank[]]], Power[Pattern[x, Blank[]], Pattern[n, Blank[]]]]], Pattern[q, Blank[]]], Power[Plus[Pattern[a, Blank[]], Times[Optional[Pattern[c, Blank[]]], Power[Pattern[x, Blank[]], Pattern[n2, Blank[]]]]], Pattern[p, Blank[]]]], Pattern[x, Blank[Symbol]]] := Int[ExpandIntegrand[(a + c*x^(2*n))^p, (d/(d^2 - e^2*x^(2*n)) - (e*x^n)/(d^2 - e^2*x^(2*n)))^(-q), x], x] /; FreeQ[{a, c, d, e, n, p}, x] && EqQ[n2, 2*n] && NeQ[c*d^2 + a*e^2, 0] && !IntegerQ[p] && ILtQ[q, 0]
  1439. Int[Times[Power[Plus[Pattern[d, Blank[]], Times[Optional[Pattern[e, Blank[]]], Power[Pattern[x, Blank[]], Pattern[n, Blank[]]]]], Pattern[q, Blank[]]], Power[Plus[Pattern[a, Blank[]], Times[Optional[Pattern[b, Blank[]]], Power[Pattern[x, Blank[]], Pattern[n, Blank[]]]], Times[Optional[Pattern[c, Blank[]]], Power[Pattern[x, Blank[]], Pattern[n2, Blank[]]]]], Pattern[p, Blank[]]]], Pattern[x, Blank[Symbol]]] := Unintegrable[(d + e*x^n)^q*(a + b*x^n + c*x^(2*n))^p, x] /; FreeQ[{a, b, c, d, e, n, p, q}, x] && EqQ[n2, 2*n]
  1440. Int[Times[Power[Plus[Pattern[d, Blank[]], Times[Optional[Pattern[e, Blank[]]], Power[Pattern[x, Blank[]], Pattern[n, Blank[]]]]], Pattern[q, Blank[]]], Power[Plus[Pattern[a, Blank[]], Times[Optional[Pattern[c, Blank[]]], Power[Pattern[x, Blank[]], Pattern[n2, Blank[]]]]], Pattern[p, Blank[]]]], Pattern[x, Blank[Symbol]]] := Unintegrable[(d + e*x^n)^q*(a + c*x^(2*n))^p, x] /; FreeQ[{a, c, d, e, n, p, q}, x] && EqQ[n2, 2*n]
  1441. Int[Times[Power[Plus[Pattern[d, Blank[]], Times[Optional[Pattern[e, Blank[]]], Power[Pattern[u, Blank[]], Pattern[n, Blank[]]]]], Optional[Pattern[q, Blank[]]]], Power[Plus[Pattern[a, Blank[]], Times[Optional[Pattern[b, Blank[]]], Power[Pattern[u, Blank[]], Pattern[n, Blank[]]]], Times[Optional[Pattern[c, Blank[]]], Power[Pattern[u, Blank[]], Pattern[n2, Blank[]]]]], Optional[Pattern[p, Blank[]]]]], Pattern[x, Blank[Symbol]]] := Dist[1/Coefficient[u, x, 1], Subst[Int[(d + e*x^n)^q*(a + b*x^n + c*x^(2*n))^p, x], x, u], x] /; FreeQ[{a, b, c, d, e, n, p, q}, x] && EqQ[n2, 2*n] && LinearQ[u, x] && NeQ[u, x]
  1442. Int[Times[Power[Plus[Pattern[d, Blank[]], Times[Optional[Pattern[e, Blank[]]], Power[Pattern[u, Blank[]], Pattern[n, Blank[]]]]], Optional[Pattern[q, Blank[]]]], Power[Plus[Pattern[a, Blank[]], Times[Optional[Pattern[c, Blank[]]], Power[Pattern[u, Blank[]], Pattern[n2, Blank[]]]]], Optional[Pattern[p, Blank[]]]]], Pattern[x, Blank[Symbol]]] := Dist[1/Coefficient[u, x, 1], Subst[Int[(d + e*x^n)^q*(a + c*x^(2*n))^p, x], x, u], x] /; FreeQ[{a, c, d, e, n, p, q}, x] && EqQ[n2, 2*n] && LinearQ[u, x] && NeQ[u, x]
  1443. Int[Times[Power[Plus[Pattern[d, Blank[]], Times[Optional[Pattern[e, Blank[]]], Power[Pattern[x, Blank[]], Optional[Pattern[mn, Blank[]]]]]], Optional[Pattern[q, Blank[]]]], Power[Plus[Optional[Pattern[a, Blank[]]], Times[Optional[Pattern[b, Blank[]]], Power[Pattern[x, Blank[]], Optional[Pattern[n, Blank[]]]]], Times[Optional[Pattern[c, Blank[]]], Power[Pattern[x, Blank[]], Optional[Pattern[n2, Blank[]]]]]], Optional[Pattern[p, Blank[]]]]], Pattern[x, Blank[Symbol]]] := Int[((e + d*x^n)^q*(a + b*x^n + c*x^(2*n))^p)/x^(n*q), x] /; FreeQ[{a, b, c, d, e, n, p}, x] && EqQ[n2, 2*n] && EqQ[mn, -n] && IntegerQ[q] && (PosQ[n] || !IntegerQ[p])
  1444. Int[Times[Power[Plus[Pattern[d, Blank[]], Times[Optional[Pattern[e, Blank[]]], Power[Pattern[x, Blank[]], Optional[Pattern[mn, Blank[]]]]]], Optional[Pattern[q, Blank[]]]], Power[Plus[Pattern[a, Blank[]], Times[Optional[Pattern[c, Blank[]]], Power[Pattern[x, Blank[]], Optional[Pattern[n2, Blank[]]]]]], Optional[Pattern[p, Blank[]]]]], Pattern[x, Blank[Symbol]]] := Int[x^(mn*q)*(e + d/x^mn)^q*(a + c*x^n2)^p, x] /; FreeQ[{a, c, d, e, mn, p}, x] && EqQ[n2, -2*mn] && IntegerQ[q] && (PosQ[n2] || !IntegerQ[p])
  1445. Int[Times[Power[Plus[Optional[Pattern[a, Blank[]]], Times[Optional[Pattern[b, Blank[]]], Power[Pattern[x, Blank[]], Optional[Pattern[mn, Blank[]]]]], Times[Optional[Pattern[c, Blank[]]], Power[Pattern[x, Blank[]], Optional[Pattern[mn2, Blank[]]]]]], Optional[Pattern[p, Blank[]]]], Power[Plus[Pattern[d, Blank[]], Times[Optional[Pattern[e, Blank[]]], Power[Pattern[x, Blank[]], Optional[Pattern[n, Blank[]]]]]], Optional[Pattern[q, Blank[]]]]], Pattern[x, Blank[Symbol]]] := Int[((d + e*x^n)^q*(c + b*x^n + a*x^(2*n))^p)/x^(2*n*p), x] /; FreeQ[{a, b, c, d, e, n, q}, x] && EqQ[mn, -n] && EqQ[mn2, 2*mn] && IntegerQ[p]
  1446. Int[Times[Power[Plus[Optional[Pattern[a, Blank[]]], Times[Optional[Pattern[c, Blank[]]], Power[Pattern[x, Blank[]], Optional[Pattern[mn2, Blank[]]]]]], Optional[Pattern[p, Blank[]]]], Power[Plus[Pattern[d, Blank[]], Times[Optional[Pattern[e, Blank[]]], Power[Pattern[x, Blank[]], Optional[Pattern[n, Blank[]]]]]], Optional[Pattern[q, Blank[]]]]], Pattern[x, Blank[Symbol]]] := Int[((d + e*x^n)^q*(c + a*x^(2*n))^p)/x^(2*n*p), x] /; FreeQ[{a, c, d, e, n, q}, x] && EqQ[mn2, -2*n] && IntegerQ[p]
  1447. Int[Times[Power[Plus[Pattern[d, Blank[]], Times[Optional[Pattern[e, Blank[]]], Power[Pattern[x, Blank[]], Optional[Pattern[mn, Blank[]]]]]], Pattern[q, Blank[]]], Power[Plus[Optional[Pattern[a, Blank[]]], Times[Optional[Pattern[b, Blank[]]], Power[Pattern[x, Blank[]], Optional[Pattern[n, Blank[]]]]], Times[Optional[Pattern[c, Blank[]]], Power[Pattern[x, Blank[]], Optional[Pattern[n2, Blank[]]]]]], Optional[Pattern[p, Blank[]]]]], Pattern[x, Blank[Symbol]]] := Dist[(e^IntPart[q]*x^(n*FracPart[q])*(d + e/x^n)^FracPart[q])/(1 + (d*x^n)/e)^FracPart[q], Int[((1 + (d*x^n)/e)^q*(a + b*x^n + c*x^(2*n))^p)/x^(n*q), x], x] /; FreeQ[{a, b, c, d, e, n, p, q}, x] && EqQ[n2, 2*n] && EqQ[mn, -n] && !IntegerQ[p] && !IntegerQ[q] && PosQ[n]
  1448. Int[Times[Power[Plus[Pattern[d, Blank[]], Times[Optional[Pattern[e, Blank[]]], Power[Pattern[x, Blank[]], Optional[Pattern[mn, Blank[]]]]]], Pattern[q, Blank[]]], Power[Plus[Pattern[a, Blank[]], Times[Optional[Pattern[c, Blank[]]], Power[Pattern[x, Blank[]], Optional[Pattern[n2, Blank[]]]]]], Optional[Pattern[p, Blank[]]]]], Pattern[x, Blank[Symbol]]] := Dist[(e^IntPart[q]*(d + e*x^mn)^FracPart[q])/(x^(mn*FracPart[q])*(1 + d/(x^mn*e))^FracPart[q]), Int[x^(mn*q)*(1 + d/(x^mn*e))^q*(a + c*x^n2)^p, x], x] /; FreeQ[{a, c, d, e, mn, p, q}, x] && EqQ[n2, -2*mn] && !IntegerQ[p] && !IntegerQ[q] && PosQ[n2]
  1449. Int[Times[Power[Plus[Optional[Pattern[a, Blank[]]], Times[Optional[Pattern[b, Blank[]]], Power[Pattern[x, Blank[]], Optional[Pattern[mn, Blank[]]]]], Times[Optional[Pattern[c, Blank[]]], Power[Pattern[x, Blank[]], Optional[Pattern[mn2, Blank[]]]]]], Pattern[p, Blank[]]], Power[Plus[Pattern[d, Blank[]], Times[Optional[Pattern[e, Blank[]]], Power[Pattern[x, Blank[]], Optional[Pattern[n, Blank[]]]]]], Optional[Pattern[q, Blank[]]]]], Pattern[x, Blank[Symbol]]] := Dist[(x^(2*n*FracPart[p])*(a + b/x^n + c/x^(2*n))^FracPart[p])/(c + b*x^n + a*x^(2*n))^FracPart[p], Int[((d + e*x^n)^q*(c + b*x^n + a*x^(2*n))^p)/x^(2*n*p), x], x] /; FreeQ[{a, b, c, d, e, n, p, q}, x] && EqQ[mn, -n] && EqQ[mn2, 2*mn] && !IntegerQ[p] && !IntegerQ[q] && PosQ[n]
  1450. Int[Times[Power[Plus[Optional[Pattern[a, Blank[]]], Times[Optional[Pattern[c, Blank[]]], Power[Pattern[x, Blank[]], Optional[Pattern[mn2, Blank[]]]]]], Pattern[p, Blank[]]], Power[Plus[Pattern[d, Blank[]], Times[Optional[Pattern[e, Blank[]]], Power[Pattern[x, Blank[]], Optional[Pattern[n, Blank[]]]]]], Optional[Pattern[q, Blank[]]]]], Pattern[x, Blank[Symbol]]] := Dist[(x^(2*n*FracPart[p])*(a + c/x^(2*n))^FracPart[p])/(c + a*x^(2*n))^FracPart[p], Int[((d + e*x^n)^q*(c + a*x^(2*n))^p)/x^(2*n*p), x], x] /; FreeQ[{a, c, d, e, n, p, q}, x] && EqQ[mn2, -2*n] && !IntegerQ[p] && !IntegerQ[q] && PosQ[n]
  1451. Int[Times[Power[Plus[Pattern[a, Blank[]], Times[Optional[Pattern[c, Blank[]]], Power[Pattern[x, Blank[]], Optional[Pattern[n, Blank[]]]]], Times[Optional[Pattern[b, Blank[]]], Power[Pattern[x, Blank[]], Pattern[mn, Blank[]]]]], Optional[Pattern[p, Blank[]]]], Power[Plus[Pattern[d, Blank[]], Times[Optional[Pattern[e, Blank[]]], Power[Pattern[x, Blank[]], Pattern[n, Blank[]]]]], Optional[Pattern[q, Blank[]]]]], Pattern[x, Blank[Symbol]]] := Int[((d + e*x^n)^q*(b + a*x^n + c*x^(2*n))^p)/x^(n*p), x] /; FreeQ[{a, b, c, d, e, n, q}, x] && EqQ[mn, -n] && IntegerQ[p]
  1452. Int[Times[Power[Plus[Pattern[a, Blank[]], Times[Optional[Pattern[c, Blank[]]], Power[Pattern[x, Blank[]], Optional[Pattern[n, Blank[]]]]], Times[Optional[Pattern[b, Blank[]]], Power[Pattern[x, Blank[]], Pattern[mn, Blank[]]]]], Optional[Pattern[p, Blank[]]]], Power[Plus[Pattern[d, Blank[]], Times[Optional[Pattern[e, Blank[]]], Power[Pattern[x, Blank[]], Pattern[n, Blank[]]]]], Optional[Pattern[q, Blank[]]]]], Pattern[x, Blank[Symbol]]] := Dist[(x^(n*FracPart[p])*(a + b/x^n + c*x^n)^FracPart[p])/(b + a*x^n + c*x^(2*n))^FracPart[p], Int[((d + e*x^n)^q*(b + a*x^n + c*x^(2*n))^p)/x^(n*p), x], x] /; FreeQ[{a, b, c, d, e, n, p, q}, x] && EqQ[mn, -n] && !IntegerQ[p]
  1453. Int[Times[Power[Plus[Pattern[d, Blank[]], Times[Optional[Pattern[e, Blank[]]], Power[Pattern[x, Blank[]], Pattern[n, Blank[]]]]], Optional[Pattern[q, Blank[]]]], Power[Plus[Pattern[f, Blank[]], Times[Optional[Pattern[g, Blank[]]], Power[Pattern[x, Blank[]], Pattern[n, Blank[]]]]], Optional[Pattern[r, Blank[]]]], Power[Plus[Pattern[a, Blank[]], Times[Optional[Pattern[b, Blank[]]], Power[Pattern[x, Blank[]], Pattern[n, Blank[]]]], Times[Optional[Pattern[c, Blank[]]], Power[Pattern[x, Blank[]], Pattern[n2, Blank[]]]]], Pattern[p, Blank[]]]], Pattern[x, Blank[Symbol]]] := Dist[(a + b*x^n + c*x^(2*n))^FracPart[p]/((4*c)^IntPart[p]*(b + 2*c*x^n)^(2*FracPart[p])), Int[(d + e*x^n)^q*(f + g*x^n)^r*(b + 2*c*x^n)^(2*p), x], x] /; FreeQ[{a, b, c, d, e, f, g, n, p, q, r}, x] && EqQ[n2, 2*n] && EqQ[b^2 - 4*a*c, 0] && !IntegerQ[p]
  1454. Int[Times[Power[Plus[Pattern[d, Blank[]], Times[Optional[Pattern[e, Blank[]]], Power[Pattern[x, Blank[]], Pattern[n, Blank[]]]]], Optional[Pattern[q, Blank[]]]], Power[Plus[Pattern[f, Blank[]], Times[Optional[Pattern[g, Blank[]]], Power[Pattern[x, Blank[]], Pattern[n, Blank[]]]]], Optional[Pattern[r, Blank[]]]], Power[Plus[Pattern[a, Blank[]], Times[Optional[Pattern[b, Blank[]]], Power[Pattern[x, Blank[]], Pattern[n, Blank[]]]], Times[Optional[Pattern[c, Blank[]]], Power[Pattern[x, Blank[]], Pattern[n2, Blank[]]]]], Optional[Pattern[p, Blank[]]]]], Pattern[x, Blank[Symbol]]] := Int[(d + e*x^n)^(p + q)*(f + g*x^n)^r*(a/d + (c*x^n)/e)^p, x] /; FreeQ[{a, b, c, d, e, f, g, n, q, r}, x] && EqQ[n2, 2*n] && NeQ[b^2 - 4*a*c, 0] && EqQ[c*d^2 - b*d*e + a*e^2, 0] && IntegerQ[p]
  1455. Int[Times[Power[Plus[Pattern[d, Blank[]], Times[Optional[Pattern[e, Blank[]]], Power[Pattern[x, Blank[]], Pattern[n, Blank[]]]]], Optional[Pattern[q, Blank[]]]], Power[Plus[Pattern[f, Blank[]], Times[Optional[Pattern[g, Blank[]]], Power[Pattern[x, Blank[]], Pattern[n, Blank[]]]]], Optional[Pattern[r, Blank[]]]], Power[Plus[Pattern[a, Blank[]], Times[Optional[Pattern[c, Blank[]]], Power[Pattern[x, Blank[]], Pattern[n2, Blank[]]]]], Optional[Pattern[p, Blank[]]]]], Pattern[x, Blank[Symbol]]] := Int[(d + e*x^n)^(p + q)*(f + g*x^n)^r*(a/d + (c*x^n)/e)^p, x] /; FreeQ[{a, c, d, e, f, g, n, q, r}, x] && EqQ[n2, 2*n] && EqQ[c*d^2 + a*e^2, 0] && IntegerQ[p]
  1456. Int[Times[Power[Plus[Pattern[d, Blank[]], Times[Optional[Pattern[e, Blank[]]], Power[Pattern[x, Blank[]], Pattern[n, Blank[]]]]], Optional[Pattern[q, Blank[]]]], Power[Plus[Pattern[f, Blank[]], Times[Optional[Pattern[g, Blank[]]], Power[Pattern[x, Blank[]], Pattern[n, Blank[]]]]], Optional[Pattern[r, Blank[]]]], Power[Plus[Pattern[a, Blank[]], Times[Optional[Pattern[b, Blank[]]], Power[Pattern[x, Blank[]], Pattern[n, Blank[]]]], Times[Optional[Pattern[c, Blank[]]], Power[Pattern[x, Blank[]], Pattern[n2, Blank[]]]]], Pattern[p, Blank[]]]], Pattern[x, Blank[Symbol]]] := Dist[(a + b*x^n + c*x^(2*n))^FracPart[p]/((d + e*x^n)^FracPart[p]*(a/d + (c*x^n)/e)^FracPart[p]), Int[(d + e*x^n)^(p + q)*(f + g*x^n)^r*(a/d + (c*x^n)/e)^p, x], x] /; FreeQ[{a, b, c, d, e, f, g, n, p, q, r}, x] && EqQ[n2, 2*n] && NeQ[b^2 - 4*a*c, 0] && EqQ[c*d^2 - b*d*e + a*e^2, 0] && !IntegerQ[p]
  1457. Int[Times[Power[Plus[Pattern[d, Blank[]], Times[Optional[Pattern[e, Blank[]]], Power[Pattern[x, Blank[]], Pattern[n, Blank[]]]]], Optional[Pattern[q, Blank[]]]], Power[Plus[Pattern[f, Blank[]], Times[Optional[Pattern[g, Blank[]]], Power[Pattern[x, Blank[]], Pattern[n, Blank[]]]]], Optional[Pattern[r, Blank[]]]], Power[Plus[Pattern[a, Blank[]], Times[Optional[Pattern[c, Blank[]]], Power[Pattern[x, Blank[]], Pattern[n2, Blank[]]]]], Pattern[p, Blank[]]]], Pattern[x, Blank[Symbol]]] := Dist[(a + c*x^(2*n))^FracPart[p]/((d + e*x^n)^FracPart[p]*(a/d + (c*x^n)/e)^FracPart[p]), Int[(d + e*x^n)^(p + q)*(f + g*x^n)^r*(a/d + (c*x^n)/e)^p, x], x] /; FreeQ[{a, c, d, e, f, g, n, p, q, r}, x] && EqQ[n2, 2*n] && EqQ[c*d^2 + a*e^2, 0] && !IntegerQ[p]
  1458. Int[Times[Power[Plus[Pattern[d1, Blank[]], Times[Optional[Pattern[e1, Blank[]]], Power[Pattern[x, Blank[]], Optional[Pattern[non2, Blank[]]]]]], Optional[Pattern[q, Blank[]]]], Power[Plus[Pattern[d2, Blank[]], Times[Optional[Pattern[e2, Blank[]]], Power[Pattern[x, Blank[]], Optional[Pattern[non2, Blank[]]]]]], Optional[Pattern[q, Blank[]]]], Power[Plus[Optional[Pattern[a, Blank[]]], Times[Optional[Pattern[b, Blank[]]], Power[Pattern[x, Blank[]], Pattern[n, Blank[]]]], Times[Optional[Pattern[c, Blank[]]], Power[Pattern[x, Blank[]], Pattern[n2, Blank[]]]]], Optional[Pattern[p, Blank[]]]]], Pattern[x, Blank[Symbol]]] := Int[(d1*d2 + e1*e2*x^n)^q*(a + b*x^n + c*x^(2*n))^p, x] /; FreeQ[{a, b, c, d1, e1, d2, e2, n, p, q}, x] && EqQ[n2, 2*n] && EqQ[non2, n/2] && EqQ[d2*e1 + d1*e2, 0] && (IntegerQ[q] || (GtQ[d1, 0] && GtQ[d2, 0]))
  1459. Int[Times[Power[Plus[Pattern[d1, Blank[]], Times[Optional[Pattern[e1, Blank[]]], Power[Pattern[x, Blank[]], Optional[Pattern[non2, Blank[]]]]]], Optional[Pattern[q, Blank[]]]], Power[Plus[Pattern[d2, Blank[]], Times[Optional[Pattern[e2, Blank[]]], Power[Pattern[x, Blank[]], Optional[Pattern[non2, Blank[]]]]]], Optional[Pattern[q, Blank[]]]], Power[Plus[Optional[Pattern[a, Blank[]]], Times[Optional[Pattern[b, Blank[]]], Power[Pattern[x, Blank[]], Pattern[n, Blank[]]]], Times[Optional[Pattern[c, Blank[]]], Power[Pattern[x, Blank[]], Pattern[n2, Blank[]]]]], Optional[Pattern[p, Blank[]]]]], Pattern[x, Blank[Symbol]]] := Dist[((d1 + e1*x^(n/2))^FracPart[q]*(d2 + e2*x^(n/2))^FracPart[q])/(d1*d2 + e1*e2*x^n)^FracPart[q], Int[(d1*d2 + e1*e2*x^n)^q*(a + b*x^n + c*x^(2*n))^p, x], x] /; FreeQ[{a, b, c, d1, e1, d2, e2, n, p, q}, x] && EqQ[n2, 2*n] && EqQ[non2, n/2] && EqQ[d2*e1 + d1*e2, 0]
  1460. Int[Times[Plus[Pattern[A, Blank[]], Times[Optional[Pattern[B, Blank[]]], Power[Pattern[x, Blank[]], Optional[Pattern[m, Blank[]]]]]], Power[Plus[Pattern[d, Blank[]], Times[Optional[Pattern[e, Blank[]]], Power[Pattern[x, Blank[]], Pattern[n, Blank[]]]]], Optional[Pattern[q, Blank[]]]], Power[Plus[Pattern[a, Blank[]], Times[Optional[Pattern[b, Blank[]]], Power[Pattern[x, Blank[]], Pattern[n, Blank[]]]], Times[Optional[Pattern[c, Blank[]]], Power[Pattern[x, Blank[]], Pattern[n2, Blank[]]]]], Optional[Pattern[p, Blank[]]]]], Pattern[x, Blank[Symbol]]] := Dist[A, Int[(d + e*x^n)^q*(a + b*x^n + c*x^(2*n))^p, x], x] + Dist[B, Int[x^m*(d + e*x^n)^q*(a + b*x^n + c*x^(2*n))^p, x], x] /; FreeQ[{a, b, c, d, e, A, B, m, n, p, q}, x] && EqQ[n2, 2*n] && EqQ[m - n + 1, 0]
  1461. Int[Times[Plus[Pattern[A, Blank[]], Times[Optional[Pattern[B, Blank[]]], Power[Pattern[x, Blank[]], Optional[Pattern[m, Blank[]]]]]], Power[Plus[Pattern[d, Blank[]], Times[Optional[Pattern[e, Blank[]]], Power[Pattern[x, Blank[]], Pattern[n, Blank[]]]]], Optional[Pattern[q, Blank[]]]], Power[Plus[Pattern[a, Blank[]], Times[Optional[Pattern[c, Blank[]]], Power[Pattern[x, Blank[]], Pattern[n2, Blank[]]]]], Optional[Pattern[p, Blank[]]]]], Pattern[x, Blank[Symbol]]] := Dist[A, Int[(d + e*x^n)^q*(a + c*x^(2*n))^p, x], x] + Dist[B, Int[x^m*(d + e*x^n)^q*(a + c*x^(2*n))^p, x], x] /; FreeQ[{a, c, d, e, A, B, m, n, p, q}, x] && EqQ[n2, 2*n] && EqQ[m - n + 1, 0]
  1462. Int[Times[Power[Times[Optional[Pattern[f, Blank[]]], Pattern[x, Blank[]]], Optional[Pattern[m, Blank[]]]], Power[Times[Optional[Pattern[e, Blank[]]], Power[Pattern[x, Blank[]], Pattern[n, Blank[]]]], Pattern[q, Blank[]]], Power[Plus[Pattern[a, Blank[]], Times[Optional[Pattern[c, Blank[]]], Power[Pattern[x, Blank[]], Optional[Pattern[n2, Blank[]]]]], Times[Optional[Pattern[b, Blank[]]], Power[Pattern[x, Blank[]], Pattern[n, Blank[]]]]], Optional[Pattern[p, Blank[]]]]], Pattern[x, Blank[Symbol]]] := Dist[f^m/(n*e^((m + 1)/n - 1)), Subst[Int[(e*x)^(q + (m + 1)/n - 1)*(a + b*x + c*x^2)^p, x], x, x^n], x] /; FreeQ[{a, b, c, e, f, m, n, p, q}, x] && EqQ[n2, 2*n] && (IntegerQ[m] || GtQ[f, 0]) && IntegerQ[Simplify[(m + 1)/n]]
  1463. Int[Times[Power[Times[Optional[Pattern[f, Blank[]]], Pattern[x, Blank[]]], Optional[Pattern[m, Blank[]]]], Power[Times[Optional[Pattern[e, Blank[]]], Power[Pattern[x, Blank[]], Pattern[n, Blank[]]]], Pattern[q, Blank[]]], Power[Plus[Pattern[a, Blank[]], Times[Optional[Pattern[c, Blank[]]], Power[Pattern[x, Blank[]], Optional[Pattern[n2, Blank[]]]]]], Optional[Pattern[p, Blank[]]]]], Pattern[x, Blank[Symbol]]] := Dist[f^m/(n*e^((m + 1)/n - 1)), Subst[Int[(e*x)^(q + (m + 1)/n - 1)*(a + c*x^2)^p, x], x, x^n], x] /; FreeQ[{a, c, e, f, m, n, p, q}, x] && EqQ[n2, 2*n] && (IntegerQ[m] || GtQ[f, 0]) && IntegerQ[Simplify[(m + 1)/n]]
  1464. Int[Times[Power[Times[Optional[Pattern[f, Blank[]]], Pattern[x, Blank[]]], Optional[Pattern[m, Blank[]]]], Power[Times[Optional[Pattern[e, Blank[]]], Power[Pattern[x, Blank[]], Pattern[n, Blank[]]]], Pattern[q, Blank[]]], Power[Plus[Pattern[a, Blank[]], Times[Optional[Pattern[c, Blank[]]], Power[Pattern[x, Blank[]], Optional[Pattern[n2, Blank[]]]]], Times[Optional[Pattern[b, Blank[]]], Power[Pattern[x, Blank[]], Pattern[n, Blank[]]]]], Optional[Pattern[p, Blank[]]]]], Pattern[x, Blank[Symbol]]] := Dist[(f^m*e^IntPart[q]*(e*x^n)^FracPart[q])/x^(n*FracPart[q]), Int[x^(m + n*q)*(a + b*x^n + c*x^(2*n))^p, x], x] /; FreeQ[{a, b, c, e, f, m, n, p, q}, x] && EqQ[n2, 2*n] && (IntegerQ[m] || GtQ[f, 0]) && !IntegerQ[Simplify[(m + 1)/n]]
  1465. Int[Times[Power[Times[Optional[Pattern[f, Blank[]]], Pattern[x, Blank[]]], Optional[Pattern[m, Blank[]]]], Power[Times[Optional[Pattern[e, Blank[]]], Power[Pattern[x, Blank[]], Pattern[n, Blank[]]]], Pattern[q, Blank[]]], Power[Plus[Pattern[a, Blank[]], Times[Optional[Pattern[c, Blank[]]], Power[Pattern[x, Blank[]], Optional[Pattern[n2, Blank[]]]]]], Optional[Pattern[p, Blank[]]]]], Pattern[x, Blank[Symbol]]] := Dist[(f^m*e^IntPart[q]*(e*x^n)^FracPart[q])/x^(n*FracPart[q]), Int[x^(m + n*q)*(a + c*x^(2*n))^p, x], x] /; FreeQ[{a, c, e, f, m, n, p, q}, x] && EqQ[n2, 2*n] && (IntegerQ[m] || GtQ[f, 0]) && !IntegerQ[Simplify[(m + 1)/n]]
  1466. Int[Times[Power[Times[Pattern[f, Blank[]], Pattern[x, Blank[]]], Optional[Pattern[m, Blank[]]]], Power[Times[Optional[Pattern[e, Blank[]]], Power[Pattern[x, Blank[]], Pattern[n, Blank[]]]], Pattern[q, Blank[]]], Power[Plus[Pattern[a, Blank[]], Times[Optional[Pattern[c, Blank[]]], Power[Pattern[x, Blank[]], Optional[Pattern[n2, Blank[]]]]], Times[Optional[Pattern[b, Blank[]]], Power[Pattern[x, Blank[]], Pattern[n, Blank[]]]]], Optional[Pattern[p, Blank[]]]]], Pattern[x, Blank[Symbol]]] := Dist[(f^IntPart[m]*(f*x)^FracPart[m])/x^FracPart[m], Int[x^m*(e*x^n)^q*(a + b*x^n + c*x^(2*n))^p, x], x] /; FreeQ[{a, b, c, e, f, m, n, p, q}, x] && EqQ[n2, 2*n] && !IntegerQ[m]
  1467. Int[Times[Power[Times[Pattern[f, Blank[]], Pattern[x, Blank[]]], Optional[Pattern[m, Blank[]]]], Power[Times[Optional[Pattern[e, Blank[]]], Power[Pattern[x, Blank[]], Pattern[n, Blank[]]]], Pattern[q, Blank[]]], Power[Plus[Pattern[a, Blank[]], Times[Optional[Pattern[c, Blank[]]], Power[Pattern[x, Blank[]], Optional[Pattern[n2, Blank[]]]]]], Optional[Pattern[p, Blank[]]]]], Pattern[x, Blank[Symbol]]] := Dist[(f^IntPart[m]*(f*x)^FracPart[m])/x^FracPart[m], Int[x^m*(e*x^n)^q*(a + c*x^(2*n))^p, x], x] /; FreeQ[{a, c, e, f, m, n, p, q}, x] && EqQ[n2, 2*n] && !IntegerQ[m]
  1468. Int[Times[Power[Pattern[x, Blank[]], Optional[Pattern[m, Blank[]]]], Power[Plus[Pattern[a, Blank[]], Times[Optional[Pattern[c, Blank[]]], Power[Pattern[x, Blank[]], Optional[Pattern[n2, Blank[]]]]], Times[Optional[Pattern[b, Blank[]]], Power[Pattern[x, Blank[]], Pattern[n, Blank[]]]]], Optional[Pattern[p, Blank[]]]], Power[Plus[Pattern[d, Blank[]], Times[Optional[Pattern[e, Blank[]]], Power[Pattern[x, Blank[]], Pattern[n, Blank[]]]]], Optional[Pattern[q, Blank[]]]]], Pattern[x, Blank[Symbol]]] := Dist[1/n, Subst[Int[(d + e*x)^q*(a + b*x + c*x^2)^p, x], x, x^n], x] /; FreeQ[{a, b, c, d, e, m, n, p, q}, x] && EqQ[n2, 2*n] && EqQ[Simplify[m - n + 1], 0]
  1469. Int[Times[Power[Pattern[x, Blank[]], Optional[Pattern[m, Blank[]]]], Power[Plus[Pattern[a, Blank[]], Times[Optional[Pattern[c, Blank[]]], Power[Pattern[x, Blank[]], Optional[Pattern[n2, Blank[]]]]]], Optional[Pattern[p, Blank[]]]], Power[Plus[Pattern[d, Blank[]], Times[Optional[Pattern[e, Blank[]]], Power[Pattern[x, Blank[]], Pattern[n, Blank[]]]]], Optional[Pattern[q, Blank[]]]]], Pattern[x, Blank[Symbol]]] := Dist[1/n, Subst[Int[(d + e*x)^q*(a + c*x^2)^p, x], x, x^n], x] /; FreeQ[{a, c, d, e, m, n, p, q}, x] && EqQ[n2, 2*n] && EqQ[Simplify[m - n + 1], 0]
  1470. Int[Times[Power[Pattern[x, Blank[]], Optional[Pattern[m, Blank[]]]], Power[Plus[Pattern[a, Blank[]], Times[Optional[Pattern[c, Blank[]]], Power[Pattern[x, Blank[]], Optional[Pattern[n2, Blank[]]]]], Times[Optional[Pattern[b, Blank[]]], Power[Pattern[x, Blank[]], Pattern[n, Blank[]]]]], Optional[Pattern[p, Blank[]]]], Power[Plus[Pattern[d, Blank[]], Times[Optional[Pattern[e, Blank[]]], Power[Pattern[x, Blank[]], Pattern[n, Blank[]]]]], Optional[Pattern[q, Blank[]]]]], Pattern[x, Blank[Symbol]]] := Int[x^(m + n*(2*p + q))*(e + d/x^n)^q*(c + b/x^n + a/x^(2*n))^p, x] /; FreeQ[{a, b, c, d, e, m, n}, x] && EqQ[n2, 2*n] && IntegersQ[p, q] && NegQ[n]
  1471. Int[Times[Power[Pattern[x, Blank[]], Optional[Pattern[m, Blank[]]]], Power[Plus[Pattern[a, Blank[]], Times[Optional[Pattern[c, Blank[]]], Power[Pattern[x, Blank[]], Optional[Pattern[n2, Blank[]]]]]], Optional[Pattern[p, Blank[]]]], Power[Plus[Pattern[d, Blank[]], Times[Optional[Pattern[e, Blank[]]], Power[Pattern[x, Blank[]], Pattern[n, Blank[]]]]], Optional[Pattern[q, Blank[]]]]], Pattern[x, Blank[Symbol]]] := Int[x^(m + n*(2*p + q))*(e + d/x^n)^q*(c + a/x^(2*n))^p, x] /; FreeQ[{a, c, d, e, m, n}, x] && EqQ[n2, 2*n] && IntegersQ[p, q] && NegQ[n]
  1472. Int[Times[Power[Pattern[x, Blank[]], Optional[Pattern[m, Blank[]]]], Power[Plus[Pattern[a, Blank[]], Times[Optional[Pattern[c, Blank[]]], Power[Pattern[x, Blank[]], Optional[Pattern[n2, Blank[]]]]], Times[Optional[Pattern[b, Blank[]]], Power[Pattern[x, Blank[]], Pattern[n, Blank[]]]]], Pattern[p, Blank[]]], Power[Plus[Pattern[d, Blank[]], Times[Optional[Pattern[e, Blank[]]], Power[Pattern[x, Blank[]], Pattern[n, Blank[]]]]], Optional[Pattern[q, Blank[]]]]], Pattern[x, Blank[Symbol]]] := Dist[1/n, Subst[Int[x^((m + 1)/n - 1)*(d + e*x)^q*(a + b*x + c*x^2)^p, x], x, x^n], x] /; FreeQ[{a, b, c, d, e, p, q}, x] && EqQ[n2, 2*n] && EqQ[b^2 - 4*a*c, 0] && !IntegerQ[p] && IGtQ[m, 0] && IGtQ[n, 0] && IGtQ[(m + 1)/n, 0]
  1473. Int[Times[Power[Times[Optional[Pattern[f, Blank[]]], Pattern[x, Blank[]]], Optional[Pattern[m, Blank[]]]], Power[Plus[Pattern[a, Blank[]], Times[Optional[Pattern[c, Blank[]]], Power[Pattern[x, Blank[]], Optional[Pattern[n2, Blank[]]]]], Times[Optional[Pattern[b, Blank[]]], Power[Pattern[x, Blank[]], Pattern[n, Blank[]]]]], Pattern[p, Blank[]]], Power[Plus[Pattern[d, Blank[]], Times[Optional[Pattern[e, Blank[]]], Power[Pattern[x, Blank[]], Pattern[n, Blank[]]]]], Optional[Pattern[q, Blank[]]]]], Pattern[x, Blank[Symbol]]] := Dist[(a + b*x^n + c*x^(2*n))^FracPart[p]/(c^IntPart[p]*(b/2 + c*x^n)^(2*FracPart[p])), Int[(f*x)^m*(d + e*x^n)^q*(b/2 + c*x^n)^(2*p), x], x] /; FreeQ[{a, b, c, d, e, f, m, n, p, q}, x] && EqQ[n2, 2*n] && EqQ[b^2 - 4*a*c, 0] && !IntegerQ[p]
  1474. Int[Times[Power[Pattern[x, Blank[]], Optional[Pattern[m, Blank[]]]], Power[Plus[Pattern[a, Blank[]], Times[Optional[Pattern[c, Blank[]]], Power[Pattern[x, Blank[]], Optional[Pattern[n2, Blank[]]]]], Times[Optional[Pattern[b, Blank[]]], Power[Pattern[x, Blank[]], Pattern[n, Blank[]]]]], Optional[Pattern[p, Blank[]]]], Power[Plus[Pattern[d, Blank[]], Times[Optional[Pattern[e, Blank[]]], Power[Pattern[x, Blank[]], Pattern[n, Blank[]]]]], Optional[Pattern[q, Blank[]]]]], Pattern[x, Blank[Symbol]]] := Dist[1/n, Subst[Int[x^(Simplify[(m + 1)/n] - 1)*(d + e*x)^q*(a + b*x + c*x^2)^p, x], x, x^n], x] /; FreeQ[{a, b, c, d, e, m, n, p, q}, x] && EqQ[n2, 2*n] && IntegerQ[Simplify[(m + 1)/n]]
  1475. Int[Times[Power[Pattern[x, Blank[]], Optional[Pattern[m, Blank[]]]], Power[Plus[Pattern[a, Blank[]], Times[Optional[Pattern[c, Blank[]]], Power[Pattern[x, Blank[]], Optional[Pattern[n2, Blank[]]]]]], Optional[Pattern[p, Blank[]]]], Power[Plus[Pattern[d, Blank[]], Times[Optional[Pattern[e, Blank[]]], Power[Pattern[x, Blank[]], Pattern[n, Blank[]]]]], Optional[Pattern[q, Blank[]]]]], Pattern[x, Blank[Symbol]]] := Dist[1/n, Subst[Int[x^(Simplify[(m + 1)/n] - 1)*(d + e*x)^q*(a + c*x^2)^p, x], x, x^n], x] /; FreeQ[{a, c, d, e, m, n, p, q}, x] && EqQ[n2, 2*n] && IntegerQ[Simplify[(m + 1)/n]]
  1476. Int[Times[Power[Times[Pattern[f, Blank[]], Pattern[x, Blank[]]], Optional[Pattern[m, Blank[]]]], Power[Plus[Pattern[a, Blank[]], Times[Optional[Pattern[c, Blank[]]], Power[Pattern[x, Blank[]], Optional[Pattern[n2, Blank[]]]]], Times[Optional[Pattern[b, Blank[]]], Power[Pattern[x, Blank[]], Pattern[n, Blank[]]]]], Optional[Pattern[p, Blank[]]]], Power[Plus[Pattern[d, Blank[]], Times[Optional[Pattern[e, Blank[]]], Power[Pattern[x, Blank[]], Pattern[n, Blank[]]]]], Optional[Pattern[q, Blank[]]]]], Pattern[x, Blank[Symbol]]] := Dist[(f^IntPart[m]*(f*x)^FracPart[m])/x^FracPart[m], Int[x^m*(d + e*x^n)^q*(a + b*x^n + c*x^(2*n))^p, x], x] /; FreeQ[{a, b, c, d, e, f, m, n, p, q}, x] && EqQ[n2, 2*n] && IntegerQ[Simplify[(m + 1)/n]]
  1477. Int[Times[Power[Times[Pattern[f, Blank[]], Pattern[x, Blank[]]], Optional[Pattern[m, Blank[]]]], Power[Plus[Pattern[a, Blank[]], Times[Optional[Pattern[c, Blank[]]], Power[Pattern[x, Blank[]], Optional[Pattern[n2, Blank[]]]]]], Optional[Pattern[p, Blank[]]]], Power[Plus[Pattern[d, Blank[]], Times[Optional[Pattern[e, Blank[]]], Power[Pattern[x, Blank[]], Pattern[n, Blank[]]]]], Optional[Pattern[q, Blank[]]]]], Pattern[x, Blank[Symbol]]] := Dist[(f^IntPart[m]*(f*x)^FracPart[m])/x^FracPart[m], Int[x^m*(d + e*x^n)^q*(a + c*x^(2*n))^p, x], x] /; FreeQ[{a, c, d, e, f, m, n, p, q}, x] && EqQ[n2, 2*n] && IntegerQ[Simplify[(m + 1)/n]]
  1478. Int[Times[Power[Times[Optional[Pattern[f, Blank[]]], Pattern[x, Blank[]]], Optional[Pattern[m, Blank[]]]], Power[Plus[Pattern[d, Blank[]], Times[Optional[Pattern[e, Blank[]]], Power[Pattern[x, Blank[]], Pattern[n, Blank[]]]]], Optional[Pattern[q, Blank[]]]], Power[Plus[Pattern[a, Blank[]], Times[Optional[Pattern[b, Blank[]]], Power[Pattern[x, Blank[]], Pattern[n, Blank[]]]], Times[Optional[Pattern[c, Blank[]]], Power[Pattern[x, Blank[]], Pattern[n2, Blank[]]]]], Optional[Pattern[p, Blank[]]]]], Pattern[x, Blank[Symbol]]] := Int[(f*x)^m*(d + e*x^n)^(q + p)*(a/d + (c*x^n)/e)^p, x] /; FreeQ[{a, b, c, d, e, f, m, n, q}, x] && EqQ[n2, 2*n] && NeQ[b^2 - 4*a*c, 0] && EqQ[c*d^2 - b*d*e + a*e^2, 0] && IntegerQ[p]
  1479. Int[Times[Power[Times[Optional[Pattern[f, Blank[]]], Pattern[x, Blank[]]], Optional[Pattern[m, Blank[]]]], Power[Plus[Pattern[d, Blank[]], Times[Optional[Pattern[e, Blank[]]], Power[Pattern[x, Blank[]], Pattern[n, Blank[]]]]], Optional[Pattern[q, Blank[]]]], Power[Plus[Pattern[a, Blank[]], Times[Optional[Pattern[c, Blank[]]], Power[Pattern[x, Blank[]], Pattern[n2, Blank[]]]]], Optional[Pattern[p, Blank[]]]]], Pattern[x, Blank[Symbol]]] := Int[(f*x)^m*(d + e*x^n)^(q + p)*(a/d + (c*x^n)/e)^p, x] /; FreeQ[{a, c, d, e, f, q, m, n, q}, x] && EqQ[n2, 2*n] && EqQ[c*d^2 + a*e^2, 0] && IntegerQ[p]
  1480. Int[Times[Power[Times[Optional[Pattern[f, Blank[]]], Pattern[x, Blank[]]], Optional[Pattern[m, Blank[]]]], Power[Plus[Pattern[d, Blank[]], Times[Optional[Pattern[e, Blank[]]], Power[Pattern[x, Blank[]], Pattern[n, Blank[]]]]], Pattern[q, Blank[]]], Power[Plus[Pattern[a, Blank[]], Times[Optional[Pattern[b, Blank[]]], Power[Pattern[x, Blank[]], Pattern[n, Blank[]]]], Times[Optional[Pattern[c, Blank[]]], Power[Pattern[x, Blank[]], Pattern[n2, Blank[]]]]], Pattern[p, Blank[]]]], Pattern[x, Blank[Symbol]]] := Dist[(a + b*x^n + c*x^(2*n))^FracPart[p]/((d + e*x^n)^FracPart[p]*(a/d + (c*x^n)/e)^FracPart[p]), Int[(f*x)^m*(d + e*x^n)^(q + p)*(a/d + (c*x^n)/e)^p, x], x] /; FreeQ[{a, b, c, d, e, f, m, n, p, q}, x] && EqQ[n2, 2*n] && NeQ[b^2 - 4*a*c, 0] && EqQ[c*d^2 - b*d*e + a*e^2, 0] && !IntegerQ[p]
  1481. Int[Times[Power[Times[Optional[Pattern[f, Blank[]]], Pattern[x, Blank[]]], Optional[Pattern[m, Blank[]]]], Power[Plus[Pattern[d, Blank[]], Times[Optional[Pattern[e, Blank[]]], Power[Pattern[x, Blank[]], Pattern[n, Blank[]]]]], Pattern[q, Blank[]]], Power[Plus[Pattern[a, Blank[]], Times[Optional[Pattern[c, Blank[]]], Power[Pattern[x, Blank[]], Pattern[n2, Blank[]]]]], Pattern[p, Blank[]]]], Pattern[x, Blank[Symbol]]] := Dist[(a + c*x^(2*n))^FracPart[p]/((d + e*x^n)^FracPart[p]*(a/d + (c*x^n)/e)^FracPart[p]), Int[(f*x)^m*(d + e*x^n)^(q + p)*(a/d + (c*x^n)/e)^p, x], x] /; FreeQ[{a, c, d, e, f, m, n, p, q}, x] && EqQ[n2, 2*n] && EqQ[c*d^2 + a*e^2, 0] && !IntegerQ[p]
  1482. Int[Times[Power[Pattern[x, Blank[]], Optional[Pattern[m, Blank[]]]], Power[Plus[Pattern[a, Blank[]], Times[Optional[Pattern[c, Blank[]]], Power[Pattern[x, Blank[]], Optional[Pattern[n2, Blank[]]]]], Times[Optional[Pattern[b, Blank[]]], Power[Pattern[x, Blank[]], Pattern[n, Blank[]]]]], Optional[Pattern[p, Blank[]]]], Power[Plus[Pattern[d, Blank[]], Times[Optional[Pattern[e, Blank[]]], Power[Pattern[x, Blank[]], Pattern[n, Blank[]]]]], Pattern[q, Blank[]]]], Pattern[x, Blank[Symbol]]] := Simp[((-d)^((m - Mod[m, n])/n - 1)*(c*d^2 - b*d*e + a*e^2)^p*x^(Mod[m, n] + 1)*(d + e*x^n)^(q + 1))/(n*e^(2*p + (m - Mod[m, n])/n)*(q + 1)), x] + Dist[1/(n*e^(2*p + (m - Mod[m, n])/n)*(q + 1)), Int[x^Mod[m, n]*(d + e*x^n)^(q + 1)*ExpandToSum[Together[(1*(n*e^(2*p + (m - Mod[m, n])/n)*(q + 1)*x^(m - Mod[m, n])*(a + b*x^n + c*x^(2*n))^p - (-d)^((m - Mod[m, n])/n - 1)*(c*d^2 - b*d*e + a*e^2)^p*(d*(Mod[m, n] + 1) + e*(Mod[m, n] + n*(q + 1) + 1)*x^n)))/(d + e*x^n)], x], x], x] /; FreeQ[{a, b, c, d, e}, x] && EqQ[n2, 2*n] && NeQ[b^2 - 4*a*c, 0] && IGtQ[n, 0] && IGtQ[p, 0] && ILtQ[q, -1] && IGtQ[m, 0]
  1483. Int[Times[Power[Pattern[x, Blank[]], Optional[Pattern[m, Blank[]]]], Power[Plus[Pattern[a, Blank[]], Times[Optional[Pattern[c, Blank[]]], Power[Pattern[x, Blank[]], Optional[Pattern[n2, Blank[]]]]]], Optional[Pattern[p, Blank[]]]], Power[Plus[Pattern[d, Blank[]], Times[Optional[Pattern[e, Blank[]]], Power[Pattern[x, Blank[]], Pattern[n, Blank[]]]]], Pattern[q, Blank[]]]], Pattern[x, Blank[Symbol]]] := Simp[((-d)^((m - Mod[m, n])/n - 1)*(c*d^2 + a*e^2)^p*x^(Mod[m, n] + 1)*(d + e*x^n)^(q + 1))/(n*e^(2*p + (m - Mod[m, n])/n)*(q + 1)), x] + Dist[1/(n*e^(2*p + (m - Mod[m, n])/n)*(q + 1)), Int[x^Mod[m, n]*(d + e*x^n)^(q + 1)*ExpandToSum[Together[(1*(n*e^(2*p + (m - Mod[m, n])/n)*(q + 1)*x^(m - Mod[m, n])*(a + c*x^(2*n))^p - (-d)^((m - Mod[m, n])/n - 1)*(c*d^2 + a*e^2)^p*(d*(Mod[m, n] + 1) + e*(Mod[m, n] + n*(q + 1) + 1)*x^n)))/(d + e*x^n)], x], x], x] /; FreeQ[{a, c, d, e}, x] && EqQ[n2, 2*n] && IGtQ[n, 0] && IGtQ[p, 0] && ILtQ[q, -1] && IGtQ[m, 0]
  1484. Int[Times[Power[Pattern[x, Blank[]], Pattern[m, Blank[]]], Power[Plus[Pattern[a, Blank[]], Times[Optional[Pattern[c, Blank[]]], Power[Pattern[x, Blank[]], Optional[Pattern[n2, Blank[]]]]], Times[Optional[Pattern[b, Blank[]]], Power[Pattern[x, Blank[]], Pattern[n, Blank[]]]]], Optional[Pattern[p, Blank[]]]], Power[Plus[Pattern[d, Blank[]], Times[Optional[Pattern[e, Blank[]]], Power[Pattern[x, Blank[]], Pattern[n, Blank[]]]]], Pattern[q, Blank[]]]], Pattern[x, Blank[Symbol]]] := Simp[((-d)^((m - Mod[m, n])/n - 1)*(c*d^2 - b*d*e + a*e^2)^p*x^(Mod[m, n] + 1)*(d + e*x^n)^(q + 1))/(n*e^(2*p + (m - Mod[m, n])/n)*(q + 1)), x] + Dist[(-d)^((m - Mod[m, n])/n - 1)/(n*e^(2*p)*(q + 1)), Int[x^m*(d + e*x^n)^(q + 1)*ExpandToSum[Together[(1*(n*(-d)^(-((m - Mod[m, n])/n) + 1)*e^(2*p)*(q + 1)*(a + b*x^n + c*x^(2*n))^p - ((c*d^2 - b*d*e + a*e^2)^p/(e^((m - Mod[m, n])/n)*x^(m - Mod[m, n])))*(d*(Mod[m, n] + 1) + e*(Mod[m, n] + n*(q + 1) + 1)*x^n)))/(d + e*x^n)], x], x], x] /; FreeQ[{a, b, c, d, e}, x] && EqQ[n2, 2*n] && NeQ[b^2 - 4*a*c, 0] && IGtQ[n, 0] && IGtQ[p, 0] && ILtQ[q, -1] && ILtQ[m, 0]
  1485. Int[Times[Power[Pattern[x, Blank[]], Pattern[m, Blank[]]], Power[Plus[Pattern[a, Blank[]], Times[Optional[Pattern[c, Blank[]]], Power[Pattern[x, Blank[]], Optional[Pattern[n2, Blank[]]]]]], Optional[Pattern[p, Blank[]]]], Power[Plus[Pattern[d, Blank[]], Times[Optional[Pattern[e, Blank[]]], Power[Pattern[x, Blank[]], Pattern[n, Blank[]]]]], Pattern[q, Blank[]]]], Pattern[x, Blank[Symbol]]] := Simp[((-d)^((m - Mod[m, n])/n - 1)*(c*d^2 + a*e^2)^p*x^(Mod[m, n] + 1)*(d + e*x^n)^(q + 1))/(n*e^(2*p + (m - Mod[m, n])/n)*(q + 1)), x] + Dist[(-d)^((m - Mod[m, n])/n - 1)/(n*e^(2*p)*(q + 1)), Int[x^m*(d + e*x^n)^(q + 1)*ExpandToSum[Together[(1*(n*(-d)^(-((m - Mod[m, n])/n) + 1)*e^(2*p)*(q + 1)*(a + c*x^(2*n))^p - ((c*d^2 + a*e^2)^p/(e^((m - Mod[m, n])/n)*x^(m - Mod[m, n])))*(d*(Mod[m, n] + 1) + e*(Mod[m, n] + n*(q + 1) + 1)*x^n)))/(d + e*x^n)], x], x], x] /; FreeQ[{a, c, d, e}, x] && EqQ[n2, 2*n] && IGtQ[n, 0] && IGtQ[p, 0] && IntegersQ[m, q] && ILtQ[q, -1] && ILtQ[m, 0]
  1486. Int[Times[Power[Times[Optional[Pattern[f, Blank[]]], Pattern[x, Blank[]]], Optional[Pattern[m, Blank[]]]], Power[Plus[Pattern[a, Blank[]], Times[Optional[Pattern[c, Blank[]]], Power[Pattern[x, Blank[]], Optional[Pattern[n2, Blank[]]]]], Times[Optional[Pattern[b, Blank[]]], Power[Pattern[x, Blank[]], Pattern[n, Blank[]]]]], Optional[Pattern[p, Blank[]]]], Power[Plus[Pattern[d, Blank[]], Times[Optional[Pattern[e, Blank[]]], Power[Pattern[x, Blank[]], Pattern[n, Blank[]]]]], Optional[Pattern[q, Blank[]]]]], Pattern[x, Blank[Symbol]]] := Simp[(c^p*(f*x)^(m + 2*n*p - n + 1)*(d + e*x^n)^(q + 1))/(e*f^(2*n*p - n + 1)*(m + 2*n*p + n*q + 1)), x] + Dist[1/(e*(m + 2*n*p + n*q + 1)), Int[(f*x)^m*(d + e*x^n)^q*ExpandToSum[e*(m + 2*n*p + n*q + 1)*((a + b*x^n + c*x^(2*n))^p - c^p*x^(2*n*p)) - d*c^p*(m + 2*n*p - n + 1)*x^(2*n*p - n), x], x], x] /; FreeQ[{a, b, c, d, e, f, m, q}, x] && EqQ[n2, 2*n] && NeQ[b^2 - 4*a*c, 0] && IGtQ[n, 0] && IGtQ[p, 0] && GtQ[2*n*p, n - 1] && !IntegerQ[q] && NeQ[m + 2*n*p + n*q + 1, 0]
  1487. Int[Times[Power[Times[Optional[Pattern[f, Blank[]]], Pattern[x, Blank[]]], Optional[Pattern[m, Blank[]]]], Power[Plus[Pattern[a, Blank[]], Times[Optional[Pattern[c, Blank[]]], Power[Pattern[x, Blank[]], Optional[Pattern[n2, Blank[]]]]]], Optional[Pattern[p, Blank[]]]], Power[Plus[Pattern[d, Blank[]], Times[Optional[Pattern[e, Blank[]]], Power[Pattern[x, Blank[]], Pattern[n, Blank[]]]]], Optional[Pattern[q, Blank[]]]]], Pattern[x, Blank[Symbol]]] := Simp[(c^p*(f*x)^(m + 2*n*p - n + 1)*(d + e*x^n)^(q + 1))/(e*f^(2*n*p - n + 1)*(m + 2*n*p + n*q + 1)), x] + Dist[1/(e*(m + 2*n*p + n*q + 1)), Int[(f*x)^m*(d + e*x^n)^q*ExpandToSum[e*(m + 2*n*p + n*q + 1)*((a + c*x^(2*n))^p - c^p*x^(2*n*p)) - d*c^p*(m + 2*n*p - n + 1)*x^(2*n*p - n), x], x], x] /; FreeQ[{a, c, d, e, f, m, q}, x] && EqQ[n2, 2*n] && IGtQ[n, 0] && IGtQ[p, 0] && GtQ[2*n*p, n - 1] && !IntegerQ[q] && NeQ[m + 2*n*p + n*q + 1, 0]
  1488. Int[Times[Power[Times[Optional[Pattern[f, Blank[]]], Pattern[x, Blank[]]], Optional[Pattern[m, Blank[]]]], Power[Plus[Pattern[a, Blank[]], Times[Optional[Pattern[c, Blank[]]], Power[Pattern[x, Blank[]], Optional[Pattern[n2, Blank[]]]]], Times[Optional[Pattern[b, Blank[]]], Power[Pattern[x, Blank[]], Pattern[n, Blank[]]]]], Optional[Pattern[p, Blank[]]]], Power[Plus[Pattern[d, Blank[]], Times[Optional[Pattern[e, Blank[]]], Power[Pattern[x, Blank[]], Pattern[n, Blank[]]]]], Optional[Pattern[q, Blank[]]]]], Pattern[x, Blank[Symbol]]] := Int[ExpandIntegrand[(f*x)^m*(d + e*x^n)^q*(a + b*x^n + c*x^(2*n))^p, x], x] /; FreeQ[{a, b, c, d, e, f, m, q}, x] && EqQ[n2, 2*n] && IGtQ[n, 0] && IGtQ[p, 0]
  1489. Int[Times[Power[Times[Optional[Pattern[f, Blank[]]], Pattern[x, Blank[]]], Optional[Pattern[m, Blank[]]]], Power[Plus[Pattern[a, Blank[]], Times[Optional[Pattern[c, Blank[]]], Power[Pattern[x, Blank[]], Optional[Pattern[n2, Blank[]]]]]], Optional[Pattern[p, Blank[]]]], Power[Plus[Pattern[d, Blank[]], Times[Optional[Pattern[e, Blank[]]], Power[Pattern[x, Blank[]], Pattern[n, Blank[]]]]], Optional[Pattern[q, Blank[]]]]], Pattern[x, Blank[Symbol]]] := Int[ExpandIntegrand[(f*x)^m*(d + e*x^n)^q*(a + c*x^(2*n))^p, x], x] /; FreeQ[{a, c, d, e, f, m, q}, x] && EqQ[n2, 2*n] && IGtQ[n, 0] && IGtQ[p, 0]
  1490. Int[Times[Power[Pattern[x, Blank[]], Optional[Pattern[m, Blank[]]]], Power[Plus[Pattern[a, Blank[]], Times[Optional[Pattern[c, Blank[]]], Power[Pattern[x, Blank[]], Optional[Pattern[n2, Blank[]]]]], Times[Optional[Pattern[b, Blank[]]], Power[Pattern[x, Blank[]], Pattern[n, Blank[]]]]], Pattern[p, Blank[]]], Power[Plus[Pattern[d, Blank[]], Times[Optional[Pattern[e, Blank[]]], Power[Pattern[x, Blank[]], Pattern[n, Blank[]]]]], Optional[Pattern[q, Blank[]]]]], Pattern[x, Blank[Symbol]]] := With[{k = GCD[m + 1, n]}, Dist[1/k, Subst[Int[x^((m + 1)/k - 1)*(d + e*x^(n/k))^q*(a + b*x^(n/k) + c*x^((2*n)/k))^p, x], x, x^k], x] /; k != 1] /; FreeQ[{a, b, c, d, e, p, q}, x] && EqQ[n2, 2*n] && NeQ[b^2 - 4*a*c, 0] && IGtQ[n, 0] && IntegerQ[m]
  1491. Int[Times[Power[Pattern[x, Blank[]], Optional[Pattern[m, Blank[]]]], Power[Plus[Pattern[a, Blank[]], Times[Optional[Pattern[c, Blank[]]], Power[Pattern[x, Blank[]], Optional[Pattern[n2, Blank[]]]]]], Pattern[p, Blank[]]], Power[Plus[Pattern[d, Blank[]], Times[Optional[Pattern[e, Blank[]]], Power[Pattern[x, Blank[]], Pattern[n, Blank[]]]]], Optional[Pattern[q, Blank[]]]]], Pattern[x, Blank[Symbol]]] := With[{k = GCD[m + 1, n]}, Dist[1/k, Subst[Int[x^((m + 1)/k - 1)*(d + e*x^(n/k))^q*(a + c*x^((2*n)/k))^p, x], x, x^k], x] /; k != 1] /; FreeQ[{a, c, d, e, p, q}, x] && EqQ[n2, 2*n] && IGtQ[n, 0] && IntegerQ[m]
  1492. Int[Times[Power[Times[Optional[Pattern[f, Blank[]]], Pattern[x, Blank[]]], Pattern[m, Blank[]]], Power[Plus[Pattern[a, Blank[]], Times[Optional[Pattern[c, Blank[]]], Power[Pattern[x, Blank[]], Optional[Pattern[n2, Blank[]]]]], Times[Optional[Pattern[b, Blank[]]], Power[Pattern[x, Blank[]], Pattern[n, Blank[]]]]], Pattern[p, Blank[]]], Power[Plus[Pattern[d, Blank[]], Times[Optional[Pattern[e, Blank[]]], Power[Pattern[x, Blank[]], Pattern[n, Blank[]]]]], Optional[Pattern[q, Blank[]]]]], Pattern[x, Blank[Symbol]]] := With[{k = Denominator[m]}, Dist[k/f, Subst[Int[x^(k*(m + 1) - 1)*(d + (e*x^(k*n))/f^n)^q*(a + (b*x^(k*n))/f^n + (c*x^(2*k*n))/f^(2*n))^p, x], x, (f*x)^(1/k)], x]] /; FreeQ[{a, b, c, d, e, f, p, q}, x] && EqQ[n2, 2*n] && NeQ[b^2 - 4*a*c, 0] && IGtQ[n, 0] && FractionQ[m] && IntegerQ[p]
  1493. Int[Times[Power[Times[Optional[Pattern[f, Blank[]]], Pattern[x, Blank[]]], Pattern[m, Blank[]]], Power[Plus[Pattern[a, Blank[]], Times[Optional[Pattern[c, Blank[]]], Power[Pattern[x, Blank[]], Optional[Pattern[n2, Blank[]]]]]], Pattern[p, Blank[]]], Power[Plus[Pattern[d, Blank[]], Times[Optional[Pattern[e, Blank[]]], Power[Pattern[x, Blank[]], Pattern[n, Blank[]]]]], Optional[Pattern[q, Blank[]]]]], Pattern[x, Blank[Symbol]]] := With[{k = Denominator[m]}, Dist[k/f, Subst[Int[x^(k*(m + 1) - 1)*(d + (e*x^(k*n))/f)^q*(a + (c*x^(2*k*n))/f)^p, x], x, (f*x)^(1/k)], x]] /; FreeQ[{a, c, d, e, f, p, q}, x] && EqQ[n2, 2*n] && IGtQ[n, 0] && FractionQ[m] && IntegerQ[p]
  1494. Int[Times[Power[Times[Optional[Pattern[f, Blank[]]], Pattern[x, Blank[]]], Optional[Pattern[m, Blank[]]]], Plus[Pattern[d, Blank[]], Times[Optional[Pattern[e, Blank[]]], Power[Pattern[x, Blank[]], Pattern[n, Blank[]]]]], Power[Plus[Pattern[a, Blank[]], Times[Optional[Pattern[b, Blank[]]], Power[Pattern[x, Blank[]], Pattern[n, Blank[]]]], Times[Optional[Pattern[c, Blank[]]], Power[Pattern[x, Blank[]], Pattern[n2, Blank[]]]]], Optional[Pattern[p, Blank[]]]]], Pattern[x, Blank[Symbol]]] := Simp[((f*x)^(m + 1)*(a + b*x^n + c*x^(2*n))^p*(d*(m + n*(2*p + 1) + 1) + e*(m + 1)*x^n))/(f*(m + 1)*(m + n*(2*p + 1) + 1)), x] + Dist[(n*p)/(f^n*(m + 1)*(m + n*(2*p + 1) + 1)), Int[(f*x)^(m + n)*(a + b*x^n + c*x^(2*n))^(p - 1)*Simp[2*a*e*(m + 1) - b*d*(m + n*(2*p + 1) + 1) + (b*e*(m + 1) - 2*c*d*(m + n*(2*p + 1) + 1))*x^n, x], x], x] /; FreeQ[{a, b, c, d, e, f}, x] && EqQ[n2, 2*n] && NeQ[b^2 - 4*a*c, 0] && IGtQ[n, 0] && GtQ[p, 0] && LtQ[m, -1] && NeQ[m + n*(2*p + 1) + 1, 0] && IntegerQ[p]
  1495. Int[Times[Power[Times[Optional[Pattern[f, Blank[]]], Pattern[x, Blank[]]], Optional[Pattern[m, Blank[]]]], Plus[Pattern[d, Blank[]], Times[Optional[Pattern[e, Blank[]]], Power[Pattern[x, Blank[]], Pattern[n, Blank[]]]]], Power[Plus[Pattern[a, Blank[]], Times[Optional[Pattern[c, Blank[]]], Power[Pattern[x, Blank[]], Pattern[n2, Blank[]]]]], Optional[Pattern[p, Blank[]]]]], Pattern[x, Blank[Symbol]]] := Simp[((f*x)^(m + 1)*(a + c*x^(2*n))^p*(d*(m + n*(2*p + 1) + 1) + e*(m + 1)*x^n))/(f*(m + 1)*(m + n*(2*p + 1) + 1)), x] + Dist[(2*n*p)/(f^n*(m + 1)*(m + n*(2*p + 1) + 1)), Int[(f*x)^(m + n)*(a + c*x^(2*n))^(p - 1)*(a*e*(m + 1) - c*d*(m + n*(2*p + 1) + 1)*x^n), x], x] /; FreeQ[{a, c, d, e, f}, x] && EqQ[n2, 2*n] && IGtQ[n, 0] && GtQ[p, 0] && LtQ[m, -1] && NeQ[m + n*(2*p + 1) + 1, 0] && IntegerQ[p]
  1496. Int[Times[Power[Times[Optional[Pattern[f, Blank[]]], Pattern[x, Blank[]]], Optional[Pattern[m, Blank[]]]], Plus[Pattern[d, Blank[]], Times[Optional[Pattern[e, Blank[]]], Power[Pattern[x, Blank[]], Pattern[n, Blank[]]]]], Power[Plus[Pattern[a, Blank[]], Times[Optional[Pattern[b, Blank[]]], Power[Pattern[x, Blank[]], Pattern[n, Blank[]]]], Times[Optional[Pattern[c, Blank[]]], Power[Pattern[x, Blank[]], Pattern[n2, Blank[]]]]], Optional[Pattern[p, Blank[]]]]], Pattern[x, Blank[Symbol]]] := Simp[((f*x)^(m + 1)*(a + b*x^n + c*x^(2*n))^p*(b*e*n*p + c*d*(m + n*(2*p + 1) + 1) + c*e*(2*n*p + m + 1)*x^n))/(c*f*(2*n*p + m + 1)*(m + n*(2*p + 1) + 1)), x] + Dist[(n*p)/(c*(2*n*p + m + 1)*(m + n*(2*p + 1) + 1)), Int[(f*x)^m*(a + b*x^n + c*x^(2*n))^(p - 1)*Simp[2*a*c*d*(m + n*(2*p + 1) + 1) - a*b*e*(m + 1) + (2*a*c*e*(2*n*p + m + 1) + b*c*d*(m + n*(2*p + 1) + 1) - b^2*e*(m + n*p + 1))*x^n, x], x], x] /; FreeQ[{a, b, c, d, e, f, m}, x] && EqQ[n2, 2*n] && NeQ[b^2 - 4*a*c, 0] && IGtQ[n, 0] && GtQ[p, 0] && NeQ[2*n*p + m + 1, 0] && NeQ[m + n*(2*p + 1) + 1, 0] && IntegerQ[p]
  1497. Int[Times[Power[Times[Optional[Pattern[f, Blank[]]], Pattern[x, Blank[]]], Optional[Pattern[m, Blank[]]]], Plus[Pattern[d, Blank[]], Times[Optional[Pattern[e, Blank[]]], Power[Pattern[x, Blank[]], Pattern[n, Blank[]]]]], Power[Plus[Pattern[a, Blank[]], Times[Optional[Pattern[c, Blank[]]], Power[Pattern[x, Blank[]], Pattern[n2, Blank[]]]]], Optional[Pattern[p, Blank[]]]]], Pattern[x, Blank[Symbol]]] := Simp[((f*x)^(m + 1)*(a + c*x^(2*n))^p*(c*d*(m + n*(2*p + 1) + 1) + c*e*(2*n*p + m + 1)*x^n))/(c*f*(2*n*p + m + 1)*(m + n*(2*p + 1) + 1)), x] + Dist[(2*a*n*p)/((2*n*p + m + 1)*(m + n*(2*p + 1) + 1)), Int[(f*x)^m*(a + c*x^(2*n))^(p - 1)*Simp[d*(m + n*(2*p + 1) + 1) + e*(2*n*p + m + 1)*x^n, x], x], x] /; FreeQ[{a, c, d, e, f, m}, x] && EqQ[n2, 2*n] && IGtQ[n, 0] && GtQ[p, 0] && NeQ[2*n*p + m + 1, 0] && NeQ[m + n*(2*p + 1) + 1, 0] && IntegerQ[p]
  1498. Int[Times[Power[Times[Optional[Pattern[f, Blank[]]], Pattern[x, Blank[]]], Optional[Pattern[m, Blank[]]]], Plus[Pattern[d, Blank[]], Times[Optional[Pattern[e, Blank[]]], Power[Pattern[x, Blank[]], Pattern[n, Blank[]]]]], Power[Plus[Pattern[a, Blank[]], Times[Optional[Pattern[b, Blank[]]], Power[Pattern[x, Blank[]], Pattern[n, Blank[]]]], Times[Optional[Pattern[c, Blank[]]], Power[Pattern[x, Blank[]], Pattern[n2, Blank[]]]]], Optional[Pattern[p, Blank[]]]]], Pattern[x, Blank[Symbol]]] := Simp[(f^(n - 1)*(f*x)^(m - n + 1)*(a + b*x^n + c*x^(2*n))^(p + 1)*(b*d - 2*a*e - (b*e - 2*c*d)*x^n))/(n*(p + 1)*(b^2 - 4*a*c)), x] + Dist[f^n/(n*(p + 1)*(b^2 - 4*a*c)), Int[(f*x)^(m - n)*(a + b*x^n + c*x^(2*n))^(p + 1)*Simp[(n - m - 1)*(b*d - 2*a*e) + (2*n*p + 2*n + m + 1)*(b*e - 2*c*d)*x^n, x], x], x] /; FreeQ[{a, b, c, d, e, f}, x] && EqQ[n2, 2*n] && NeQ[b^2 - 4*a*c, 0] && IGtQ[n, 0] && LtQ[p, -1] && GtQ[m, n - 1] && IntegerQ[p]
  1499. Int[Times[Power[Times[Optional[Pattern[f, Blank[]]], Pattern[x, Blank[]]], Optional[Pattern[m, Blank[]]]], Plus[Pattern[d, Blank[]], Times[Optional[Pattern[e, Blank[]]], Power[Pattern[x, Blank[]], Pattern[n, Blank[]]]]], Power[Plus[Pattern[a, Blank[]], Times[Optional[Pattern[c, Blank[]]], Power[Pattern[x, Blank[]], Pattern[n2, Blank[]]]]], Optional[Pattern[p, Blank[]]]]], Pattern[x, Blank[Symbol]]] := Simp[(f^(n - 1)*(f*x)^(m - n + 1)*(a + c*x^(2*n))^(p + 1)*(a*e - c*d*x^n))/(2*a*c*n*(p + 1)), x] + Dist[f^n/(2*a*c*n*(p + 1)), Int[(f*x)^(m - n)*(a + c*x^(2*n))^(p + 1)*(a*e*(n - m - 1) + c*d*(2*n*p + 2*n + m + 1)*x^n), x], x] /; FreeQ[{a, c, d, e, f}, x] && EqQ[n2, 2*n] && IGtQ[n, 0] && LtQ[p, -1] && GtQ[m, n - 1] && IntegerQ[p]
  1500. Int[Times[Power[Times[Optional[Pattern[f, Blank[]]], Pattern[x, Blank[]]], Optional[Pattern[m, Blank[]]]], Plus[Pattern[d, Blank[]], Times[Optional[Pattern[e, Blank[]]], Power[Pattern[x, Blank[]], Pattern[n, Blank[]]]]], Power[Plus[Pattern[a, Blank[]], Times[Optional[Pattern[b, Blank[]]], Power[Pattern[x, Blank[]], Pattern[n, Blank[]]]], Times[Optional[Pattern[c, Blank[]]], Power[Pattern[x, Blank[]], Pattern[n2, Blank[]]]]], Pattern[p, Blank[]]]], Pattern[x, Blank[Symbol]]] := -Simp[((f*x)^(m + 1)*(a + b*x^n + c*x^(2*n))^(p + 1)*(d*(b^2 - 2*a*c) - a*b*e + (b*d - 2*a*e)*c*x^n))/(a*f*n*(p + 1)*(b^2 - 4*a*c)), x] + Dist[1/(a*n*(p + 1)*(b^2 - 4*a*c)), Int[(f*x)^m*(a + b*x^n + c*x^(2*n))^(p + 1)*Simp[d*(b^2*(m + n*(p + 1) + 1) - 2*a*c*(m + 2*n*(p + 1) + 1)) - a*b*e*(m + 1) + c*(m + n*(2*p + 3) + 1)*(b*d - 2*a*e)*x^n, x], x], x] /; FreeQ[{a, b, c, d, e, f, m}, x] && EqQ[n2, 2*n] && NeQ[b^2 - 4*a*c, 0] && IGtQ[n, 0] && LtQ[p, -1] && IntegerQ[p]
  1501. Int[Times[Power[Times[Optional[Pattern[f, Blank[]]], Pattern[x, Blank[]]], Optional[Pattern[m, Blank[]]]], Plus[Pattern[d, Blank[]], Times[Optional[Pattern[e, Blank[]]], Power[Pattern[x, Blank[]], Pattern[n, Blank[]]]]], Power[Plus[Pattern[a, Blank[]], Times[Optional[Pattern[c, Blank[]]], Power[Pattern[x, Blank[]], Pattern[n2, Blank[]]]]], Pattern[p, Blank[]]]], Pattern[x, Blank[Symbol]]] := -Simp[((f*x)^(m + 1)*(a + c*x^(2*n))^(p + 1)*(d + e*x^n))/(2*a*f*n*(p + 1)), x] + Dist[1/(2*a*n*(p + 1)), Int[(f*x)^m*(a + c*x^(2*n))^(p + 1)*Simp[d*(m + 2*n*(p + 1) + 1) + e*(m + n*(2*p + 3) + 1)*x^n, x], x], x] /; FreeQ[{a, c, d, e, f, m}, x] && EqQ[n2, 2*n] && IGtQ[n, 0] && LtQ[p, -1] && IntegerQ[p]
  1502. Int[Times[Power[Times[Optional[Pattern[f, Blank[]]], Pattern[x, Blank[]]], Optional[Pattern[m, Blank[]]]], Plus[Pattern[d, Blank[]], Times[Optional[Pattern[e, Blank[]]], Power[Pattern[x, Blank[]], Pattern[n, Blank[]]]]], Power[Plus[Pattern[a, Blank[]], Times[Optional[Pattern[b, Blank[]]], Power[Pattern[x, Blank[]], Pattern[n, Blank[]]]], Times[Optional[Pattern[c, Blank[]]], Power[Pattern[x, Blank[]], Pattern[n2, Blank[]]]]], Pattern[p, Blank[]]]], Pattern[x, Blank[Symbol]]] := Simp[(e*f^(n - 1)*(f*x)^(m - n + 1)*(a + b*x^n + c*x^(2*n))^(p + 1))/(c*(m + n*(2*p + 1) + 1)), x] - Dist[f^n/(c*(m + n*(2*p + 1) + 1)), Int[(f*x)^(m - n)*(a + b*x^n + c*x^(2*n))^p*Simp[a*e*(m - n + 1) + (b*e*(m + n*p + 1) - c*d*(m + n*(2*p + 1) + 1))*x^n, x], x], x] /; FreeQ[{a, b, c, d, e, f, p}, x] && EqQ[n2, 2*n] && NeQ[b^2 - 4*a*c, 0] && IGtQ[n, 0] && GtQ[m, n - 1] && NeQ[m + n*(2*p + 1) + 1, 0] && IntegerQ[p]
  1503. Int[Times[Power[Times[Optional[Pattern[f, Blank[]]], Pattern[x, Blank[]]], Optional[Pattern[m, Blank[]]]], Plus[Pattern[d, Blank[]], Times[Optional[Pattern[e, Blank[]]], Power[Pattern[x, Blank[]], Pattern[n, Blank[]]]]], Power[Plus[Pattern[a, Blank[]], Times[Optional[Pattern[c, Blank[]]], Power[Pattern[x, Blank[]], Pattern[n2, Blank[]]]]], Pattern[p, Blank[]]]], Pattern[x, Blank[Symbol]]] := Simp[(e*f^(n - 1)*(f*x)^(m - n + 1)*(a + c*x^(2*n))^(p + 1))/(c*(m + n*(2*p + 1) + 1)), x] - Dist[f^n/(c*(m + n*(2*p + 1) + 1)), Int[(f*x)^(m - n)*(a + c*x^(2*n))^p*(a*e*(m - n + 1) - c*d*(m + n*(2*p + 1) + 1)*x^n), x], x] /; FreeQ[{a, c, d, e, f, p}, x] && EqQ[n2, 2*n] && IGtQ[n, 0] && GtQ[m, n - 1] && NeQ[m + n*(2*p + 1) + 1, 0] && IntegerQ[p]
  1504. Int[Times[Power[Times[Optional[Pattern[f, Blank[]]], Pattern[x, Blank[]]], Optional[Pattern[m, Blank[]]]], Plus[Pattern[d, Blank[]], Times[Optional[Pattern[e, Blank[]]], Power[Pattern[x, Blank[]], Pattern[n, Blank[]]]]], Power[Plus[Pattern[a, Blank[]], Times[Optional[Pattern[b, Blank[]]], Power[Pattern[x, Blank[]], Pattern[n, Blank[]]]], Times[Optional[Pattern[c, Blank[]]], Power[Pattern[x, Blank[]], Pattern[n2, Blank[]]]]], Pattern[p, Blank[]]]], Pattern[x, Blank[Symbol]]] := Simp[(d*(f*x)^(m + 1)*(a + b*x^n + c*x^(2*n))^(p + 1))/(a*f*(m + 1)), x] + Dist[1/(a*f^n*(m + 1)), Int[(f*x)^(m + n)*(a + b*x^n + c*x^(2*n))^p*Simp[a*e*(m + 1) - b*d*(m + n*(p + 1) + 1) - c*d*(m + 2*n*(p + 1) + 1)*x^n, x], x], x] /; FreeQ[{a, b, c, d, e, f, p}, x] && EqQ[n2, 2*n] && NeQ[b^2 - 4*a*c, 0] && IGtQ[n, 0] && LtQ[m, -1] && IntegerQ[p]
  1505. Int[Times[Power[Times[Optional[Pattern[f, Blank[]]], Pattern[x, Blank[]]], Optional[Pattern[m, Blank[]]]], Plus[Pattern[d, Blank[]], Times[Optional[Pattern[e, Blank[]]], Power[Pattern[x, Blank[]], Pattern[n, Blank[]]]]], Power[Plus[Pattern[a, Blank[]], Times[Optional[Pattern[c, Blank[]]], Power[Pattern[x, Blank[]], Pattern[n2, Blank[]]]]], Pattern[p, Blank[]]]], Pattern[x, Blank[Symbol]]] := Simp[(d*(f*x)^(m + 1)*(a + c*x^(2*n))^(p + 1))/(a*f*(m + 1)), x] + Dist[1/(a*f^n*(m + 1)), Int[(f*x)^(m + n)*(a + c*x^(2*n))^p*(a*e*(m + 1) - c*d*(m + 2*n*(p + 1) + 1)*x^n), x], x] /; FreeQ[{a, c, d, e, f, p}, x] && EqQ[n2, 2*n] && IGtQ[n, 0] && LtQ[m, -1] && IntegerQ[p]
  1506. Int[Times[Power[Times[Optional[Pattern[f, Blank[]]], Pattern[x, Blank[]]], Pattern[m, Blank[]]], Plus[Pattern[d, Blank[]], Times[Optional[Pattern[e, Blank[]]], Power[Pattern[x, Blank[]], Pattern[n, Blank[]]]]], Power[Plus[Pattern[a, Blank[]], Times[Optional[Pattern[b, Blank[]]], Power[Pattern[x, Blank[]], Pattern[n, Blank[]]]], Times[Optional[Pattern[c, Blank[]]], Power[Pattern[x, Blank[]], Pattern[n2, Blank[]]]]], -1]], Pattern[x, Blank[Symbol]]] := With[{q = Rt[a*c, 2]}, With[{r = Rt[2*c*q - b*c, 2]}, Dist[c/(2*q*r), Int[((f*x)^m*Simp[d*r - (c*d - e*q)*x^(n/2), x])/(q - r*x^(n/2) + c*x^n), x], x] + Dist[c/(2*q*r), Int[((f*x)^m*Simp[d*r + (c*d - e*q)*x^(n/2), x])/(q + r*x^(n/2) + c*x^n), x], x]] /; !LtQ[2*c*q - b*c, 0]] /; FreeQ[{a, b, c, d, e, f}, x] && EqQ[n2, 2*n] && LtQ[b^2 - 4*a*c, 0] && IntegersQ[m, n/2] && LtQ[0, m, n] && PosQ[a*c]
  1507. Int[Times[Power[Times[Optional[Pattern[f, Blank[]]], Pattern[x, Blank[]]], Pattern[m, Blank[]]], Plus[Pattern[d, Blank[]], Times[Optional[Pattern[e, Blank[]]], Power[Pattern[x, Blank[]], Pattern[n, Blank[]]]]], Power[Plus[Pattern[a, Blank[]], Times[Optional[Pattern[c, Blank[]]], Power[Pattern[x, Blank[]], Pattern[n2, Blank[]]]]], -1]], Pattern[x, Blank[Symbol]]] := With[{q = Rt[a*c, 2]}, With[{r = Rt[2*c*q, 2]}, Dist[c/(2*q*r), Int[((f*x)^m*Simp[d*r - (c*d - e*q)*x^(n/2), x])/(q - r*x^(n/2) + c*x^n), x], x] + Dist[c/(2*q*r), Int[((f*x)^m*Simp[d*r + (c*d - e*q)*x^(n/2), x])/(q + r*x^(n/2) + c*x^n), x], x]] /; !LtQ[2*c*q, 0]] /; FreeQ[{a, c, d, e, f}, x] && EqQ[n2, 2*n] && GtQ[a*c, 0] && IntegersQ[m, n/2] && LtQ[0, m, n]
  1508. Int[Times[Power[Times[Optional[Pattern[f, Blank[]]], Pattern[x, Blank[]]], Optional[Pattern[m, Blank[]]]], Plus[Pattern[d, Blank[]], Times[Optional[Pattern[e, Blank[]]], Power[Pattern[x, Blank[]], Pattern[n, Blank[]]]]], Power[Plus[Pattern[a, Blank[]], Times[Optional[Pattern[b, Blank[]]], Power[Pattern[x, Blank[]], Pattern[n, Blank[]]]], Times[Optional[Pattern[c, Blank[]]], Power[Pattern[x, Blank[]], Pattern[n2, Blank[]]]]], -1]], Pattern[x, Blank[Symbol]]] := With[{q = Rt[a*c, 2]}, With[{r = Rt[2*c*q - b*c, 2]}, Dist[c/(2*q*r), Int[((f*x)^m*(d*r - (c*d - e*q)*x^(n/2)))/(q - r*x^(n/2) + c*x^n), x], x] + Dist[c/(2*q*r), Int[((f*x)^m*(d*r + (c*d - e*q)*x^(n/2)))/(q + r*x^(n/2) + c*x^n), x], x]] /; !LtQ[2*c*q - b*c, 0]] /; FreeQ[{a, b, c, d, e, f, m}, x] && EqQ[n2, 2*n] && LtQ[b^2 - 4*a*c, 0] && IGtQ[n/2, 1] && PosQ[a*c]
  1509. Int[Times[Power[Times[Optional[Pattern[f, Blank[]]], Pattern[x, Blank[]]], Optional[Pattern[m, Blank[]]]], Plus[Pattern[d, Blank[]], Times[Optional[Pattern[e, Blank[]]], Power[Pattern[x, Blank[]], Pattern[n, Blank[]]]]], Power[Plus[Pattern[a, Blank[]], Times[Optional[Pattern[c, Blank[]]], Power[Pattern[x, Blank[]], Pattern[n2, Blank[]]]]], -1]], Pattern[x, Blank[Symbol]]] := With[{q = Rt[a*c, 2]}, With[{r = Rt[2*c*q, 2]}, Dist[c/(2*q*r), Int[((f*x)^m*(d*r - (c*d - e*q)*x^(n/2)))/(q - r*x^(n/2) + c*x^n), x], x] + Dist[c/(2*q*r), Int[((f*x)^m*(d*r + (c*d - e*q)*x^(n/2)))/(q + r*x^(n/2) + c*x^n), x], x]] /; !LtQ[2*c*q, 0]] /; FreeQ[{a, c, d, e, f, m}, x] && EqQ[n2, 2*n] && IGtQ[n/2, 1] && GtQ[a*c, 0]
  1510. Int[Times[Power[Times[Optional[Pattern[f, Blank[]]], Pattern[x, Blank[]]], Optional[Pattern[m, Blank[]]]], Plus[Pattern[d, Blank[]], Times[Optional[Pattern[e, Blank[]]], Power[Pattern[x, Blank[]], Pattern[n, Blank[]]]]], Power[Plus[Pattern[a, Blank[]], Times[Optional[Pattern[b, Blank[]]], Power[Pattern[x, Blank[]], Pattern[n, Blank[]]]], Times[Optional[Pattern[c, Blank[]]], Power[Pattern[x, Blank[]], Pattern[n2, Blank[]]]]], -1]], Pattern[x, Blank[Symbol]]] := With[{q = Rt[b^2 - 4*a*c, 2]}, Dist[e/2 + (2*c*d - b*e)/(2*q), Int[(f*x)^m/(b/2 - q/2 + c*x^n), x], x] + Dist[e/2 - (2*c*d - b*e)/(2*q), Int[(f*x)^m/(b/2 + q/2 + c*x^n), x], x]] /; FreeQ[{a, b, c, d, e, f, m}, x] && EqQ[n2, 2*n] && NeQ[b^2 - 4*a*c, 0] && IGtQ[n, 0]
  1511. Int[Times[Power[Times[Optional[Pattern[f, Blank[]]], Pattern[x, Blank[]]], Optional[Pattern[m, Blank[]]]], Plus[Pattern[d, Blank[]], Times[Optional[Pattern[e, Blank[]]], Power[Pattern[x, Blank[]], Pattern[n, Blank[]]]]], Power[Plus[Pattern[a, Blank[]], Times[Optional[Pattern[c, Blank[]]], Power[Pattern[x, Blank[]], Pattern[n2, Blank[]]]]], -1]], Pattern[x, Blank[Symbol]]] := With[{q = Rt[-(a*c), 2]}, -Dist[e/2 + (c*d)/(2*q), Int[(f*x)^m/(q - c*x^n), x], x] + Dist[e/2 - (c*d)/(2*q), Int[(f*x)^m/(q + c*x^n), x], x]] /; FreeQ[{a, c, d, e, f, m}, x] && EqQ[n2, 2*n] && IGtQ[n, 0]
  1512. Int[Times[Power[Times[Optional[Pattern[f, Blank[]]], Pattern[x, Blank[]]], Optional[Pattern[m, Blank[]]]], Power[Plus[Pattern[a, Blank[]], Times[Optional[Pattern[c, Blank[]]], Power[Pattern[x, Blank[]], Optional[Pattern[n2, Blank[]]]]], Times[Optional[Pattern[b, Blank[]]], Power[Pattern[x, Blank[]], Pattern[n, Blank[]]]]], -1], Power[Plus[Pattern[d, Blank[]], Times[Optional[Pattern[e, Blank[]]], Power[Pattern[x, Blank[]], Pattern[n, Blank[]]]]], Optional[Pattern[q, Blank[]]]]], Pattern[x, Blank[Symbol]]] := Int[ExpandIntegrand[((f*x)^m*(d + e*x^n)^q)/(a + b*x^n + c*x^(2*n)), x], x] /; FreeQ[{a, b, c, d, e, f, m}, x] && EqQ[n2, 2*n] && NeQ[b^2 - 4*a*c, 0] && IGtQ[n, 0] && IntegerQ[q] && IntegerQ[m]
  1513. Int[Times[Power[Times[Optional[Pattern[f, Blank[]]], Pattern[x, Blank[]]], Optional[Pattern[m, Blank[]]]], Power[Plus[Pattern[a, Blank[]], Times[Optional[Pattern[c, Blank[]]], Power[Pattern[x, Blank[]], Optional[Pattern[n2, Blank[]]]]]], -1], Power[Plus[Pattern[d, Blank[]], Times[Optional[Pattern[e, Blank[]]], Power[Pattern[x, Blank[]], Pattern[n, Blank[]]]]], Optional[Pattern[q, Blank[]]]]], Pattern[x, Blank[Symbol]]] := Int[ExpandIntegrand[((f*x)^m*(d + e*x^n)^q)/(a + c*x^(2*n)), x], x] /; FreeQ[{a, c, d, e, f, m}, x] && EqQ[n2, 2*n] && IGtQ[n, 0] && IntegerQ[q] && IntegerQ[m]
  1514. Int[Times[Power[Times[Optional[Pattern[f, Blank[]]], Pattern[x, Blank[]]], Optional[Pattern[m, Blank[]]]], Power[Plus[Pattern[a, Blank[]], Times[Optional[Pattern[c, Blank[]]], Power[Pattern[x, Blank[]], Optional[Pattern[n2, Blank[]]]]], Times[Optional[Pattern[b, Blank[]]], Power[Pattern[x, Blank[]], Pattern[n, Blank[]]]]], -1], Power[Plus[Pattern[d, Blank[]], Times[Optional[Pattern[e, Blank[]]], Power[Pattern[x, Blank[]], Pattern[n, Blank[]]]]], Optional[Pattern[q, Blank[]]]]], Pattern[x, Blank[Symbol]]] := Int[ExpandIntegrand[(f*x)^m, (d + e*x^n)^q/(a + b*x^n + c*x^(2*n)), x], x] /; FreeQ[{a, b, c, d, e, f, m}, x] && EqQ[n2, 2*n] && NeQ[b^2 - 4*a*c, 0] && IGtQ[n, 0] && IntegerQ[q] && !IntegerQ[m]
  1515. Int[Times[Power[Times[Optional[Pattern[f, Blank[]]], Pattern[x, Blank[]]], Optional[Pattern[m, Blank[]]]], Power[Plus[Pattern[a, Blank[]], Times[Optional[Pattern[c, Blank[]]], Power[Pattern[x, Blank[]], Optional[Pattern[n2, Blank[]]]]]], -1], Power[Plus[Pattern[d, Blank[]], Times[Optional[Pattern[e, Blank[]]], Power[Pattern[x, Blank[]], Pattern[n, Blank[]]]]], Optional[Pattern[q, Blank[]]]]], Pattern[x, Blank[Symbol]]] := Int[ExpandIntegrand[(f*x)^m, (d + e*x^n)^q/(a + c*x^(2*n)), x], x] /; FreeQ[{a, c, d, e, f, m}, x] && EqQ[n2, 2*n] && IGtQ[n, 0] && IntegerQ[q] && !IntegerQ[m]
  1516. Int[Times[Power[Times[Optional[Pattern[f, Blank[]]], Pattern[x, Blank[]]], Optional[Pattern[m, Blank[]]]], Power[Plus[Pattern[a, Blank[]], Times[Optional[Pattern[c, Blank[]]], Power[Pattern[x, Blank[]], Optional[Pattern[n2, Blank[]]]]], Times[Optional[Pattern[b, Blank[]]], Power[Pattern[x, Blank[]], Pattern[n, Blank[]]]]], -1], Power[Plus[Optional[Pattern[d, Blank[]]], Times[Optional[Pattern[e, Blank[]]], Power[Pattern[x, Blank[]], Pattern[n, Blank[]]]]], Pattern[q, Blank[]]]], Pattern[x, Blank[Symbol]]] := Dist[f^(2*n)/c^2, Int[(f*x)^(m - 2*n)*(c*d - b*e + c*e*x^n)*(d + e*x^n)^(q - 1), x], x] - Dist[f^(2*n)/c^2, Int[((f*x)^(m - 2*n)*(d + e*x^n)^(q - 1)*Simp[a*(c*d - b*e) + (b*c*d - b^2*e + a*c*e)*x^n, x])/(a + b*x^n + c*x^(2*n)), x], x] /; FreeQ[{a, b, c, d, e, f}, x] && EqQ[n2, 2*n] && NeQ[b^2 - 4*a*c, 0] && IGtQ[n, 0] && !IntegerQ[q] && GtQ[q, 0] && GtQ[m, 2*n - 1]
  1517. Int[Times[Power[Times[Optional[Pattern[f, Blank[]]], Pattern[x, Blank[]]], Optional[Pattern[m, Blank[]]]], Power[Plus[Pattern[a, Blank[]], Times[Optional[Pattern[c, Blank[]]], Power[Pattern[x, Blank[]], Optional[Pattern[n2, Blank[]]]]]], -1], Power[Plus[Optional[Pattern[d, Blank[]]], Times[Optional[Pattern[e, Blank[]]], Power[Pattern[x, Blank[]], Pattern[n, Blank[]]]]], Pattern[q, Blank[]]]], Pattern[x, Blank[Symbol]]] := Dist[f^(2*n)/c, Int[(f*x)^(m - 2*n)*(d + e*x^n)^q, x], x] - Dist[(a*f^(2*n))/c, Int[((f*x)^(m - 2*n)*(d + e*x^n)^q)/(a + c*x^(2*n)), x], x] /; FreeQ[{a, c, d, e, f, q}, x] && EqQ[n2, 2*n] && IGtQ[n, 0] && !IntegerQ[q] && GtQ[m, 2*n - 1]
  1518. Int[Times[Power[Times[Optional[Pattern[f, Blank[]]], Pattern[x, Blank[]]], Optional[Pattern[m, Blank[]]]], Power[Plus[Pattern[a, Blank[]], Times[Optional[Pattern[c, Blank[]]], Power[Pattern[x, Blank[]], Optional[Pattern[n2, Blank[]]]]], Times[Optional[Pattern[b, Blank[]]], Power[Pattern[x, Blank[]], Pattern[n, Blank[]]]]], -1], Power[Plus[Optional[Pattern[d, Blank[]]], Times[Optional[Pattern[e, Blank[]]], Power[Pattern[x, Blank[]], Pattern[n, Blank[]]]]], Pattern[q, Blank[]]]], Pattern[x, Blank[Symbol]]] := Dist[(e*f^n)/c, Int[(f*x)^(m - n)*(d + e*x^n)^(q - 1), x], x] - Dist[f^n/c, Int[((f*x)^(m - n)*(d + e*x^n)^(q - 1)*Simp[a*e - (c*d - b*e)*x^n, x])/(a + b*x^n + c*x^(2*n)), x], x] /; FreeQ[{a, b, c, d, e, f}, x] && EqQ[n2, 2*n] && NeQ[b^2 - 4*a*c, 0] && IGtQ[n, 0] && !IntegerQ[q] && GtQ[q, 0] && GtQ[m, n - 1] && LeQ[m, 2*n - 1]
  1519. Int[Times[Power[Times[Optional[Pattern[f, Blank[]]], Pattern[x, Blank[]]], Optional[Pattern[m, Blank[]]]], Power[Plus[Pattern[a, Blank[]], Times[Optional[Pattern[c, Blank[]]], Power[Pattern[x, Blank[]], Optional[Pattern[n2, Blank[]]]]]], -1], Power[Plus[Optional[Pattern[d, Blank[]]], Times[Optional[Pattern[e, Blank[]]], Power[Pattern[x, Blank[]], Pattern[n, Blank[]]]]], Pattern[q, Blank[]]]], Pattern[x, Blank[Symbol]]] := Dist[(e*f^n)/c, Int[(f*x)^(m - n)*(d + e*x^n)^(q - 1), x], x] - Dist[f^n/c, Int[((f*x)^(m - n)*(d + e*x^n)^(q - 1)*Simp[a*e - c*d*x^n, x])/(a + c*x^(2*n)), x], x] /; FreeQ[{a, c, d, e, f}, x] && EqQ[n2, 2*n] && IGtQ[n, 0] && !IntegerQ[q] && GtQ[q, 0] && GtQ[m, n - 1] && LeQ[m, 2*n - 1]
  1520. Int[Times[Power[Times[Optional[Pattern[f, Blank[]]], Pattern[x, Blank[]]], Pattern[m, Blank[]]], Power[Plus[Pattern[a, Blank[]], Times[Optional[Pattern[c, Blank[]]], Power[Pattern[x, Blank[]], Optional[Pattern[n2, Blank[]]]]], Times[Optional[Pattern[b, Blank[]]], Power[Pattern[x, Blank[]], Pattern[n, Blank[]]]]], -1], Power[Plus[Optional[Pattern[d, Blank[]]], Times[Optional[Pattern[e, Blank[]]], Power[Pattern[x, Blank[]], Pattern[n, Blank[]]]]], Pattern[q, Blank[]]]], Pattern[x, Blank[Symbol]]] := Dist[d/a, Int[(f*x)^m*(d + e*x^n)^(q - 1), x], x] - Dist[1/(a*f^n), Int[((f*x)^(m + n)*(d + e*x^n)^(q - 1)*Simp[b*d - a*e + c*d*x^n, x])/(a + b*x^n + c*x^(2*n)), x], x] /; FreeQ[{a, b, c, d, e, f}, x] && EqQ[n2, 2*n] && NeQ[b^2 - 4*a*c, 0] && IGtQ[n, 0] && !IntegerQ[q] && GtQ[q, 0] && LtQ[m, 0]
  1521. Int[Times[Power[Times[Optional[Pattern[f, Blank[]]], Pattern[x, Blank[]]], Pattern[m, Blank[]]], Power[Plus[Pattern[a, Blank[]], Times[Optional[Pattern[c, Blank[]]], Power[Pattern[x, Blank[]], Optional[Pattern[n2, Blank[]]]]]], -1], Power[Plus[Optional[Pattern[d, Blank[]]], Times[Optional[Pattern[e, Blank[]]], Power[Pattern[x, Blank[]], Pattern[n, Blank[]]]]], Pattern[q, Blank[]]]], Pattern[x, Blank[Symbol]]] := Dist[d/a, Int[(f*x)^m*(d + e*x^n)^(q - 1), x], x] + Dist[1/(a*f^n), Int[((f*x)^(m + n)*(d + e*x^n)^(q - 1)*Simp[a*e - c*d*x^n, x])/(a + c*x^(2*n)), x], x] /; FreeQ[{a, c, d, e, f}, x] && EqQ[n2, 2*n] && IGtQ[n, 0] && !IntegerQ[q] && GtQ[q, 0] && LtQ[m, 0]
  1522. Int[Times[Power[Times[Optional[Pattern[f, Blank[]]], Pattern[x, Blank[]]], Optional[Pattern[m, Blank[]]]], Power[Plus[Pattern[a, Blank[]], Times[Optional[Pattern[c, Blank[]]], Power[Pattern[x, Blank[]], Optional[Pattern[n2, Blank[]]]]], Times[Optional[Pattern[b, Blank[]]], Power[Pattern[x, Blank[]], Pattern[n, Blank[]]]]], -1], Power[Plus[Optional[Pattern[d, Blank[]]], Times[Optional[Pattern[e, Blank[]]], Power[Pattern[x, Blank[]], Pattern[n, Blank[]]]]], Pattern[q, Blank[]]]], Pattern[x, Blank[Symbol]]] := Dist[(d^2*f^(2*n))/(c*d^2 - b*d*e + a*e^2), Int[(f*x)^(m - 2*n)*(d + e*x^n)^q, x], x] - Dist[f^(2*n)/(c*d^2 - b*d*e + a*e^2), Int[((f*x)^(m - 2*n)*(d + e*x^n)^(q + 1)*Simp[a*d + (b*d - a*e)*x^n, x])/(a + b*x^n + c*x^(2*n)), x], x] /; FreeQ[{a, b, c, d, e, f}, x] && EqQ[n2, 2*n] && NeQ[b^2 - 4*a*c, 0] && IGtQ[n, 0] && !IntegerQ[q] && LtQ[q, -1] && GtQ[m, 2*n - 1]
  1523. Int[Times[Power[Times[Optional[Pattern[f, Blank[]]], Pattern[x, Blank[]]], Optional[Pattern[m, Blank[]]]], Power[Plus[Pattern[a, Blank[]], Times[Optional[Pattern[c, Blank[]]], Power[Pattern[x, Blank[]], Optional[Pattern[n2, Blank[]]]]]], -1], Power[Plus[Optional[Pattern[d, Blank[]]], Times[Optional[Pattern[e, Blank[]]], Power[Pattern[x, Blank[]], Pattern[n, Blank[]]]]], Pattern[q, Blank[]]]], Pattern[x, Blank[Symbol]]] := Dist[(d^2*f^(2*n))/(c*d^2 + a*e^2), Int[(f*x)^(m - 2*n)*(d + e*x^n)^q, x], x] - Dist[(a*f^(2*n))/(c*d^2 + a*e^2), Int[((f*x)^(m - 2*n)*(d + e*x^n)^(q + 1)*(d - e*x^n))/(a + c*x^(2*n)), x], x] /; FreeQ[{a, c, d, e, f}, x] && EqQ[n2, 2*n] && IGtQ[n, 0] && !IntegerQ[q] && LtQ[q, -1] && GtQ[m, 2*n - 1]
  1524. Int[Times[Power[Times[Optional[Pattern[f, Blank[]]], Pattern[x, Blank[]]], Optional[Pattern[m, Blank[]]]], Power[Plus[Pattern[a, Blank[]], Times[Optional[Pattern[c, Blank[]]], Power[Pattern[x, Blank[]], Optional[Pattern[n2, Blank[]]]]], Times[Optional[Pattern[b, Blank[]]], Power[Pattern[x, Blank[]], Pattern[n, Blank[]]]]], -1], Power[Plus[Optional[Pattern[d, Blank[]]], Times[Optional[Pattern[e, Blank[]]], Power[Pattern[x, Blank[]], Pattern[n, Blank[]]]]], Pattern[q, Blank[]]]], Pattern[x, Blank[Symbol]]] := -Dist[(d*e*f^n)/(c*d^2 - b*d*e + a*e^2), Int[(f*x)^(m - n)*(d + e*x^n)^q, x], x] + Dist[f^n/(c*d^2 - b*d*e + a*e^2), Int[((f*x)^(m - n)*(d + e*x^n)^(q + 1)*Simp[a*e + c*d*x^n, x])/(a + b*x^n + c*x^(2*n)), x], x] /; FreeQ[{a, b, c, d, e, f}, x] && EqQ[n2, 2*n] && NeQ[b^2 - 4*a*c, 0] && IGtQ[n, 0] && !IntegerQ[q] && LtQ[q, -1] && GtQ[m, n - 1] && LeQ[m, 2*n - 1]
  1525. Int[Times[Power[Times[Optional[Pattern[f, Blank[]]], Pattern[x, Blank[]]], Optional[Pattern[m, Blank[]]]], Power[Plus[Pattern[a, Blank[]], Times[Optional[Pattern[c, Blank[]]], Power[Pattern[x, Blank[]], Optional[Pattern[n2, Blank[]]]]]], -1], Power[Plus[Optional[Pattern[d, Blank[]]], Times[Optional[Pattern[e, Blank[]]], Power[Pattern[x, Blank[]], Pattern[n, Blank[]]]]], Pattern[q, Blank[]]]], Pattern[x, Blank[Symbol]]] := -Dist[(d*e*f^n)/(c*d^2 + a*e^2), Int[(f*x)^(m - n)*(d + e*x^n)^q, x], x] + Dist[f^n/(c*d^2 + a*e^2), Int[((f*x)^(m - n)*(d + e*x^n)^(q + 1)*Simp[a*e + c*d*x^n, x])/(a + c*x^(2*n)), x], x] /; FreeQ[{a, c, d, e, f}, x] && EqQ[n2, 2*n] && IGtQ[n, 0] && !IntegerQ[q] && LtQ[q, -1] && GtQ[m, n - 1] && LeQ[m, 2*n - 1]
  1526. Int[Times[Power[Times[Optional[Pattern[f, Blank[]]], Pattern[x, Blank[]]], Optional[Pattern[m, Blank[]]]], Power[Plus[Pattern[d, Blank[]], Times[Optional[Pattern[e, Blank[]]], Power[Pattern[x, Blank[]], Pattern[n, Blank[]]]]], Pattern[q, Blank[]]], Power[Plus[Pattern[a, Blank[]], Times[Optional[Pattern[b, Blank[]]], Power[Pattern[x, Blank[]], Pattern[n, Blank[]]]], Times[Optional[Pattern[c, Blank[]]], Power[Pattern[x, Blank[]], Pattern[n2, Blank[]]]]], -1]], Pattern[x, Blank[Symbol]]] := Dist[e^2/(c*d^2 - b*d*e + a*e^2), Int[(f*x)^m*(d + e*x^n)^q, x], x] + Dist[1/(c*d^2 - b*d*e + a*e^2), Int[((f*x)^m*(d + e*x^n)^(q + 1)*Simp[c*d - b*e - c*e*x^n, x])/(a + b*x^n + c*x^(2*n)), x], x] /; FreeQ[{a, b, c, d, e, f, m}, x] && EqQ[n2, 2*n] && NeQ[b^2 - 4*a*c, 0] && IGtQ[n, 0] && !IntegerQ[q] && LtQ[q, -1]
  1527. Int[Times[Power[Times[Optional[Pattern[f, Blank[]]], Pattern[x, Blank[]]], Optional[Pattern[m, Blank[]]]], Power[Plus[Pattern[d, Blank[]], Times[Optional[Pattern[e, Blank[]]], Power[Pattern[x, Blank[]], Pattern[n, Blank[]]]]], Pattern[q, Blank[]]], Power[Plus[Pattern[a, Blank[]], Times[Optional[Pattern[c, Blank[]]], Power[Pattern[x, Blank[]], Pattern[n2, Blank[]]]]], -1]], Pattern[x, Blank[Symbol]]] := Dist[e^2/(c*d^2 + a*e^2), Int[(f*x)^m*(d + e*x^n)^q, x], x] + Dist[c/(c*d^2 + a*e^2), Int[((f*x)^m*(d + e*x^n)^(q + 1)*(d - e*x^n))/(a + c*x^(2*n)), x], x] /; FreeQ[{a, c, d, e, f, m}, x] && EqQ[n2, 2*n] && IGtQ[n, 0] && !IntegerQ[q] && LtQ[q, -1]
  1528. Int[Times[Power[Times[Optional[Pattern[f, Blank[]]], Pattern[x, Blank[]]], Optional[Pattern[m, Blank[]]]], Power[Plus[Pattern[a, Blank[]], Times[Optional[Pattern[c, Blank[]]], Power[Pattern[x, Blank[]], Optional[Pattern[n2, Blank[]]]]], Times[Optional[Pattern[b, Blank[]]], Power[Pattern[x, Blank[]], Pattern[n, Blank[]]]]], -1], Power[Plus[Pattern[d, Blank[]], Times[Optional[Pattern[e, Blank[]]], Power[Pattern[x, Blank[]], Pattern[n, Blank[]]]]], Pattern[q, Blank[]]]], Pattern[x, Blank[Symbol]]] := Int[ExpandIntegrand[(d + e*x^n)^q, (f*x)^m/(a + b*x^n + c*x^(2*n)), x], x] /; FreeQ[{a, b, c, d, e, f, q, n}, x] && EqQ[n2, 2*n] && NeQ[b^2 - 4*a*c, 0] && IGtQ[n, 0] && !IntegerQ[q] && IntegerQ[m]
  1529. Int[Times[Power[Times[Optional[Pattern[f, Blank[]]], Pattern[x, Blank[]]], Optional[Pattern[m, Blank[]]]], Power[Plus[Pattern[a, Blank[]], Times[Optional[Pattern[c, Blank[]]], Power[Pattern[x, Blank[]], Optional[Pattern[n2, Blank[]]]]]], -1], Power[Plus[Pattern[d, Blank[]], Times[Optional[Pattern[e, Blank[]]], Power[Pattern[x, Blank[]], Pattern[n, Blank[]]]]], Pattern[q, Blank[]]]], Pattern[x, Blank[Symbol]]] := Int[ExpandIntegrand[(d + e*x^n)^q, (f*x)^m/(a + c*x^(2*n)), x], x] /; FreeQ[{a, c, d, e, f, q, n}, x] && EqQ[n2, 2*n] && IGtQ[n, 0] && !IntegerQ[q] && IntegerQ[m]
  1530. Int[Times[Power[Times[Optional[Pattern[f, Blank[]]], Pattern[x, Blank[]]], Optional[Pattern[m, Blank[]]]], Power[Plus[Pattern[a, Blank[]], Times[Optional[Pattern[c, Blank[]]], Power[Pattern[x, Blank[]], Optional[Pattern[n2, Blank[]]]]], Times[Optional[Pattern[b, Blank[]]], Power[Pattern[x, Blank[]], Pattern[n, Blank[]]]]], -1], Power[Plus[Pattern[d, Blank[]], Times[Optional[Pattern[e, Blank[]]], Power[Pattern[x, Blank[]], Pattern[n, Blank[]]]]], Pattern[q, Blank[]]]], Pattern[x, Blank[Symbol]]] := Int[ExpandIntegrand[(f*x)^m*(d + e*x^n)^q, 1/(a + b*x^n + c*x^(2*n)), x], x] /; FreeQ[{a, b, c, d, e, f, m, q, n}, x] && EqQ[n2, 2*n] && NeQ[b^2 - 4*a*c, 0] && IGtQ[n, 0] && !IntegerQ[q] && !IntegerQ[m]
  1531. Int[Times[Power[Times[Optional[Pattern[f, Blank[]]], Pattern[x, Blank[]]], Optional[Pattern[m, Blank[]]]], Power[Plus[Pattern[a, Blank[]], Times[Optional[Pattern[c, Blank[]]], Power[Pattern[x, Blank[]], Optional[Pattern[n2, Blank[]]]]]], -1], Power[Plus[Pattern[d, Blank[]], Times[Optional[Pattern[e, Blank[]]], Power[Pattern[x, Blank[]], Pattern[n, Blank[]]]]], Pattern[q, Blank[]]]], Pattern[x, Blank[Symbol]]] := Int[ExpandIntegrand[(f*x)^m*(d + e*x^n)^q, 1/(a + c*x^(2*n)), x], x] /; FreeQ[{a, c, d, e, f, m, q, n}, x] && EqQ[n2, 2*n] && IGtQ[n, 0] && !IntegerQ[q] && !IntegerQ[m]
  1532. Int[Times[Power[Times[Optional[Pattern[f, Blank[]]], Pattern[x, Blank[]]], Pattern[m, Blank[]]], Power[Plus[Optional[Pattern[a, Blank[]]], Times[Optional[Pattern[c, Blank[]]], Power[Pattern[x, Blank[]], Optional[Pattern[n2, Blank[]]]]], Times[Optional[Pattern[b, Blank[]]], Power[Pattern[x, Blank[]], Pattern[n, Blank[]]]]], Optional[Pattern[p, Blank[]]]], Power[Plus[Optional[Pattern[d, Blank[]]], Times[Optional[Pattern[e, Blank[]]], Power[Pattern[x, Blank[]], Pattern[n, Blank[]]]]], -1]], Pattern[x, Blank[Symbol]]] := Dist[1/d^2, Int[(f*x)^m*(a*d + (b*d - a*e)*x^n)*(a + b*x^n + c*x^(2*n))^(p - 1), x], x] + Dist[(c*d^2 - b*d*e + a*e^2)/(d^2*f^(2*n)), Int[((f*x)^(m + 2*n)*(a + b*x^n + c*x^(2*n))^(p - 1))/(d + e*x^n), x], x] /; FreeQ[{a, b, c, d, e, f}, x] && EqQ[n2, 2*n] && NeQ[b^2 - 4*a*c, 0] && IGtQ[n, 0] && GtQ[p, 0] && LtQ[m, -n]
  1533. Int[Times[Power[Times[Optional[Pattern[f, Blank[]]], Pattern[x, Blank[]]], Pattern[m, Blank[]]], Power[Plus[Pattern[a, Blank[]], Times[Optional[Pattern[c, Blank[]]], Power[Pattern[x, Blank[]], Optional[Pattern[n2, Blank[]]]]]], Optional[Pattern[p, Blank[]]]], Power[Plus[Optional[Pattern[d, Blank[]]], Times[Optional[Pattern[e, Blank[]]], Power[Pattern[x, Blank[]], Pattern[n, Blank[]]]]], -1]], Pattern[x, Blank[Symbol]]] := Dist[a/d^2, Int[(f*x)^m*(d - e*x^n)*(a + c*x^(2*n))^(p - 1), x], x] + Dist[(c*d^2 + a*e^2)/(d^2*f^(2*n)), Int[((f*x)^(m + 2*n)*(a + c*x^(2*n))^(p - 1))/(d + e*x^n), x], x] /; FreeQ[{a, c, d, e, f}, x] && EqQ[n2, 2*n] && IGtQ[n, 0] && GtQ[p, 0] && LtQ[m, -n]
  1534. Int[Times[Power[Times[Optional[Pattern[f, Blank[]]], Pattern[x, Blank[]]], Pattern[m, Blank[]]], Power[Plus[Optional[Pattern[a, Blank[]]], Times[Optional[Pattern[c, Blank[]]], Power[Pattern[x, Blank[]], Optional[Pattern[n2, Blank[]]]]], Times[Optional[Pattern[b, Blank[]]], Power[Pattern[x, Blank[]], Pattern[n, Blank[]]]]], Optional[Pattern[p, Blank[]]]], Power[Plus[Optional[Pattern[d, Blank[]]], Times[Optional[Pattern[e, Blank[]]], Power[Pattern[x, Blank[]], Pattern[n, Blank[]]]]], -1]], Pattern[x, Blank[Symbol]]] := Dist[1/(d*e), Int[(f*x)^m*(a*e + c*d*x^n)*(a + b*x^n + c*x^(2*n))^(p - 1), x], x] - Dist[(c*d^2 - b*d*e + a*e^2)/(d*e*f^n), Int[((f*x)^(m + n)*(a + b*x^n + c*x^(2*n))^(p - 1))/(d + e*x^n), x], x] /; FreeQ[{a, b, c, d, e, f}, x] && EqQ[n2, 2*n] && NeQ[b^2 - 4*a*c, 0] && IGtQ[n, 0] && GtQ[p, 0] && LtQ[m, 0]
  1535. Int[Times[Power[Times[Optional[Pattern[f, Blank[]]], Pattern[x, Blank[]]], Pattern[m, Blank[]]], Power[Plus[Pattern[a, Blank[]], Times[Optional[Pattern[c, Blank[]]], Power[Pattern[x, Blank[]], Optional[Pattern[n2, Blank[]]]]]], Optional[Pattern[p, Blank[]]]], Power[Plus[Optional[Pattern[d, Blank[]]], Times[Optional[Pattern[e, Blank[]]], Power[Pattern[x, Blank[]], Pattern[n, Blank[]]]]], -1]], Pattern[x, Blank[Symbol]]] := Dist[1/(d*e), Int[(f*x)^m*(a*e + c*d*x^n)*(a + c*x^(2*n))^(p - 1), x], x] - Dist[(c*d^2 + a*e^2)/(d*e*f^n), Int[((f*x)^(m + n)*(a + c*x^(2*n))^(p - 1))/(d + e*x^n), x], x] /; FreeQ[{a, c, d, e, f}, x] && EqQ[n2, 2*n] && IGtQ[n, 0] && GtQ[p, 0] && LtQ[m, 0]
  1536. Int[Times[Power[Times[Optional[Pattern[f, Blank[]]], Pattern[x, Blank[]]], Optional[Pattern[m, Blank[]]]], Power[Plus[Optional[Pattern[a, Blank[]]], Times[Optional[Pattern[c, Blank[]]], Power[Pattern[x, Blank[]], Optional[Pattern[n2, Blank[]]]]], Times[Optional[Pattern[b, Blank[]]], Power[Pattern[x, Blank[]], Pattern[n, Blank[]]]]], Pattern[p, Blank[]]], Power[Plus[Optional[Pattern[d, Blank[]]], Times[Optional[Pattern[e, Blank[]]], Power[Pattern[x, Blank[]], Pattern[n, Blank[]]]]], -1]], Pattern[x, Blank[Symbol]]] := -Dist[f^(2*n)/(c*d^2 - b*d*e + a*e^2), Int[(f*x)^(m - 2*n)*(a*d + (b*d - a*e)*x^n)*(a + b*x^n + c*x^(2*n))^p, x], x] + Dist[(d^2*f^(2*n))/(c*d^2 - b*d*e + a*e^2), Int[((f*x)^(m - 2*n)*(a + b*x^n + c*x^(2*n))^(p + 1))/(d + e*x^n), x], x] /; FreeQ[{a, b, c, d, e, f}, x] && EqQ[n2, 2*n] && NeQ[b^2 - 4*a*c, 0] && IGtQ[n, 0] && LtQ[p, -1] && GtQ[m, n]
  1537. Int[Times[Power[Times[Optional[Pattern[f, Blank[]]], Pattern[x, Blank[]]], Optional[Pattern[m, Blank[]]]], Power[Plus[Pattern[a, Blank[]], Times[Optional[Pattern[c, Blank[]]], Power[Pattern[x, Blank[]], Optional[Pattern[n2, Blank[]]]]]], Pattern[p, Blank[]]], Power[Plus[Optional[Pattern[d, Blank[]]], Times[Optional[Pattern[e, Blank[]]], Power[Pattern[x, Blank[]], Pattern[n, Blank[]]]]], -1]], Pattern[x, Blank[Symbol]]] := -Dist[(a*f^(2*n))/(c*d^2 + a*e^2), Int[(f*x)^(m - 2*n)*(d - e*x^n)*(a + c*x^(2*n))^p, x], x] + Dist[(d^2*f^(2*n))/(c*d^2 + a*e^2), Int[((f*x)^(m - 2*n)*(a + c*x^(2*n))^(p + 1))/(d + e*x^n), x], x] /; FreeQ[{a, c, d, e, f}, x] && EqQ[n2, 2*n] && IGtQ[n, 0] && LtQ[p, -1] && GtQ[m, n]
  1538. Int[Times[Power[Times[Optional[Pattern[f, Blank[]]], Pattern[x, Blank[]]], Optional[Pattern[m, Blank[]]]], Power[Plus[Optional[Pattern[a, Blank[]]], Times[Optional[Pattern[c, Blank[]]], Power[Pattern[x, Blank[]], Optional[Pattern[n2, Blank[]]]]], Times[Optional[Pattern[b, Blank[]]], Power[Pattern[x, Blank[]], Pattern[n, Blank[]]]]], Pattern[p, Blank[]]], Power[Plus[Optional[Pattern[d, Blank[]]], Times[Optional[Pattern[e, Blank[]]], Power[Pattern[x, Blank[]], Pattern[n, Blank[]]]]], -1]], Pattern[x, Blank[Symbol]]] := Dist[f^n/(c*d^2 - b*d*e + a*e^2), Int[(f*x)^(m - n)*(a*e + c*d*x^n)*(a + b*x^n + c*x^(2*n))^p, x], x] - Dist[(d*e*f^n)/(c*d^2 - b*d*e + a*e^2), Int[((f*x)^(m - n)*(a + b*x^n + c*x^(2*n))^(p + 1))/(d + e*x^n), x], x] /; FreeQ[{a, b, c, d, e, f}, x] && EqQ[n2, 2*n] && NeQ[b^2 - 4*a*c, 0] && IGtQ[n, 0] && LtQ[p, -1] && GtQ[m, 0]
  1539. Int[Times[Power[Times[Optional[Pattern[f, Blank[]]], Pattern[x, Blank[]]], Optional[Pattern[m, Blank[]]]], Power[Plus[Pattern[a, Blank[]], Times[Optional[Pattern[c, Blank[]]], Power[Pattern[x, Blank[]], Optional[Pattern[n2, Blank[]]]]]], Pattern[p, Blank[]]], Power[Plus[Optional[Pattern[d, Blank[]]], Times[Optional[Pattern[e, Blank[]]], Power[Pattern[x, Blank[]], Pattern[n, Blank[]]]]], -1]], Pattern[x, Blank[Symbol]]] := Dist[f^n/(c*d^2 + a*e^2), Int[(f*x)^(m - n)*(a*e + c*d*x^n)*(a + c*x^(2*n))^p, x], x] - Dist[(d*e*f^n)/(c*d^2 + a*e^2), Int[((f*x)^(m - n)*(a + c*x^(2*n))^(p + 1))/(d + e*x^n), x], x] /; FreeQ[{a, c, d, e, f}, x] && EqQ[n2, 2*n] && IGtQ[n, 0] && LtQ[p, -1] && GtQ[m, 0]
  1540. Int[Times[Power[Times[Optional[Pattern[f, Blank[]]], Pattern[x, Blank[]]], Optional[Pattern[m, Blank[]]]], Power[Plus[Pattern[a, Blank[]], Times[Optional[Pattern[c, Blank[]]], Power[Pattern[x, Blank[]], Optional[Pattern[n2, Blank[]]]]], Times[Optional[Pattern[b, Blank[]]], Power[Pattern[x, Blank[]], Pattern[n, Blank[]]]]], Optional[Pattern[p, Blank[]]]], Power[Plus[Pattern[d, Blank[]], Times[Optional[Pattern[e, Blank[]]], Power[Pattern[x, Blank[]], Pattern[n, Blank[]]]]], Optional[Pattern[q, Blank[]]]]], Pattern[x, Blank[Symbol]]] := Int[ExpandIntegrand[(a + b*x^n + c*x^(2*n))^p, (f*x)^m*(d + e*x^n)^q, x], x] /; FreeQ[{a, b, c, d, e, f, m, q}, x] && EqQ[n2, 2*n] && NeQ[b^2 - 4*a*c, 0] && IGtQ[n, 0] && (IGtQ[q, 0] || IntegersQ[m, q])
  1541. Int[Times[Power[Times[Optional[Pattern[f, Blank[]]], Pattern[x, Blank[]]], Optional[Pattern[m, Blank[]]]], Power[Plus[Pattern[a, Blank[]], Times[Optional[Pattern[c, Blank[]]], Power[Pattern[x, Blank[]], Optional[Pattern[n2, Blank[]]]]]], Optional[Pattern[p, Blank[]]]], Power[Plus[Pattern[d, Blank[]], Times[Optional[Pattern[e, Blank[]]], Power[Pattern[x, Blank[]], Pattern[n, Blank[]]]]], Optional[Pattern[q, Blank[]]]]], Pattern[x, Blank[Symbol]]] := Int[ExpandIntegrand[(a + c*x^(2*n))^p, (f*x)^m*(d + e*x^n)^q, x], x] /; FreeQ[{a, c, d, e, f, m, q}, x] && EqQ[n2, 2*n] && IGtQ[n, 0] && IGtQ[q, 0]
  1542. Int[Times[Power[Pattern[x, Blank[]], Optional[Pattern[m, Blank[]]]], Power[Plus[Pattern[a, Blank[]], Times[Optional[Pattern[c, Blank[]]], Power[Pattern[x, Blank[]], Optional[Pattern[n2, Blank[]]]]], Times[Optional[Pattern[b, Blank[]]], Power[Pattern[x, Blank[]], Pattern[n, Blank[]]]]], Pattern[p, Blank[]]], Power[Plus[Pattern[d, Blank[]], Times[Optional[Pattern[e, Blank[]]], Power[Pattern[x, Blank[]], Pattern[n, Blank[]]]]], Optional[Pattern[q, Blank[]]]]], Pattern[x, Blank[Symbol]]] := -Subst[Int[((d + e/x^n)^q*(a + b/x^n + c/x^(2*n))^p)/x^(m + 2), x], x, 1/x] /; FreeQ[{a, b, c, d, e, p, q}, x] && EqQ[n2, 2*n] && NeQ[b^2 - 4*a*c, 0] && ILtQ[n, 0] && IntegerQ[m]
  1543. Int[Times[Power[Pattern[x, Blank[]], Optional[Pattern[m, Blank[]]]], Power[Plus[Pattern[a, Blank[]], Times[Optional[Pattern[c, Blank[]]], Power[Pattern[x, Blank[]], Optional[Pattern[n2, Blank[]]]]]], Pattern[p, Blank[]]], Power[Plus[Pattern[d, Blank[]], Times[Optional[Pattern[e, Blank[]]], Power[Pattern[x, Blank[]], Pattern[n, Blank[]]]]], Optional[Pattern[q, Blank[]]]]], Pattern[x, Blank[Symbol]]] := -Subst[Int[((d + e/x^n)^q*(a + c/x^(2*n))^p)/x^(m + 2), x], x, 1/x] /; FreeQ[{a, c, d, e, p, q}, x] && EqQ[n2, 2*n] && ILtQ[n, 0] && IntegerQ[m]
  1544. Int[Times[Power[Times[Optional[Pattern[f, Blank[]]], Pattern[x, Blank[]]], Optional[Pattern[m, Blank[]]]], Power[Plus[Pattern[a, Blank[]], Times[Optional[Pattern[c, Blank[]]], Power[Pattern[x, Blank[]], Optional[Pattern[n2, Blank[]]]]], Times[Optional[Pattern[b, Blank[]]], Power[Pattern[x, Blank[]], Pattern[n, Blank[]]]]], Pattern[p, Blank[]]], Power[Plus[Pattern[d, Blank[]], Times[Optional[Pattern[e, Blank[]]], Power[Pattern[x, Blank[]], Pattern[n, Blank[]]]]], Optional[Pattern[q, Blank[]]]]], Pattern[x, Blank[Symbol]]] := With[{g = Denominator[m]}, -Dist[g/f, Subst[Int[((d + e/(f^n*x^(g*n)))^q*(a + b/(f^n*x^(g*n)) + c/(f^(2*n)*x^(2*g*n)))^p)/x^(g*(m + 1) + 1), x], x, 1/(f*x)^(1/g)], x]] /; FreeQ[{a, b, c, d, e, f, p, q}, x] && EqQ[n2, 2*n] && NeQ[b^2 - 4*a*c, 0] && ILtQ[n, 0] && FractionQ[m]
  1545. Int[Times[Power[Times[Optional[Pattern[f, Blank[]]], Pattern[x, Blank[]]], Optional[Pattern[m, Blank[]]]], Power[Plus[Pattern[a, Blank[]], Times[Optional[Pattern[c, Blank[]]], Power[Pattern[x, Blank[]], Optional[Pattern[n2, Blank[]]]]]], Pattern[p, Blank[]]], Power[Plus[Pattern[d, Blank[]], Times[Optional[Pattern[e, Blank[]]], Power[Pattern[x, Blank[]], Pattern[n, Blank[]]]]], Optional[Pattern[q, Blank[]]]]], Pattern[x, Blank[Symbol]]] := With[{g = Denominator[m]}, -Dist[g/f, Subst[Int[((d + e/(f^n*x^(g*n)))^q*(a + c/(f^(2*n)*x^(2*g*n)))^p)/x^(g*(m + 1) + 1), x], x, 1/(f*x)^(1/g)], x]] /; FreeQ[{a, c, d, e, f, p, q}, x] && EqQ[n2, 2*n] && ILtQ[n, 0] && FractionQ[m]
  1546. Int[Times[Power[Times[Optional[Pattern[f, Blank[]]], Pattern[x, Blank[]]], Pattern[m, Blank[]]], Power[Plus[Pattern[a, Blank[]], Times[Optional[Pattern[c, Blank[]]], Power[Pattern[x, Blank[]], Optional[Pattern[n2, Blank[]]]]], Times[Optional[Pattern[b, Blank[]]], Power[Pattern[x, Blank[]], Pattern[n, Blank[]]]]], Pattern[p, Blank[]]], Power[Plus[Pattern[d, Blank[]], Times[Optional[Pattern[e, Blank[]]], Power[Pattern[x, Blank[]], Pattern[n, Blank[]]]]], Optional[Pattern[q, Blank[]]]]], Pattern[x, Blank[Symbol]]] := -Dist[f^IntPart[m]*(f*x)^FracPart[m]*(x^(-1))^FracPart[m], Subst[Int[((d + e/x^n)^q*(a + b/x^n + c/x^(2*n))^p)/x^(m + 2), x], x, 1/x], x] /; FreeQ[{a, b, c, d, e, f, m, p, q}, x] && EqQ[n2, 2*n] && NeQ[b^2 - 4*a*c, 0] && ILtQ[n, 0] && !RationalQ[m]
  1547. Int[Times[Power[Times[Optional[Pattern[f, Blank[]]], Pattern[x, Blank[]]], Pattern[m, Blank[]]], Power[Plus[Pattern[a, Blank[]], Times[Optional[Pattern[c, Blank[]]], Power[Pattern[x, Blank[]], Optional[Pattern[n2, Blank[]]]]]], Pattern[p, Blank[]]], Power[Plus[Pattern[d, Blank[]], Times[Optional[Pattern[e, Blank[]]], Power[Pattern[x, Blank[]], Pattern[n, Blank[]]]]], Optional[Pattern[q, Blank[]]]]], Pattern[x, Blank[Symbol]]] := -Dist[f^IntPart[m]*(f*x)^FracPart[m]*(x^(-1))^FracPart[m], Subst[Int[((d + e/x^n)^q*(a + c/x^(2*n))^p)/x^(m + 2), x], x, 1/x], x] /; FreeQ[{a, c, d, e, f, m, p, q}, x] && EqQ[n2, 2*n] && ILtQ[n, 0] && !RationalQ[m]
  1548. Int[Times[Power[Pattern[x, Blank[]], Optional[Pattern[m, Blank[]]]], Power[Plus[Pattern[a, Blank[]], Times[Optional[Pattern[c, Blank[]]], Power[Pattern[x, Blank[]], Optional[Pattern[n2, Blank[]]]]], Times[Optional[Pattern[b, Blank[]]], Power[Pattern[x, Blank[]], Pattern[n, Blank[]]]]], Pattern[p, Blank[]]], Power[Plus[Pattern[d, Blank[]], Times[Optional[Pattern[e, Blank[]]], Power[Pattern[x, Blank[]], Pattern[n, Blank[]]]]], Optional[Pattern[q, Blank[]]]]], Pattern[x, Blank[Symbol]]] := With[{g = Denominator[n]}, Dist[g, Subst[Int[x^(g*(m + 1) - 1)*(d + e*x^(g*n))^q*(a + b*x^(g*n) + c*x^(2*g*n))^p, x], x, x^(1/g)], x]] /; FreeQ[{a, b, c, d, e, m, p, q}, x] && EqQ[n2, 2*n] && NeQ[b^2 - 4*a*c, 0] && FractionQ[n]
  1549. Int[Times[Power[Pattern[x, Blank[]], Optional[Pattern[m, Blank[]]]], Power[Plus[Pattern[a, Blank[]], Times[Optional[Pattern[c, Blank[]]], Power[Pattern[x, Blank[]], Optional[Pattern[n2, Blank[]]]]]], Pattern[p, Blank[]]], Power[Plus[Pattern[d, Blank[]], Times[Optional[Pattern[e, Blank[]]], Power[Pattern[x, Blank[]], Pattern[n, Blank[]]]]], Optional[Pattern[q, Blank[]]]]], Pattern[x, Blank[Symbol]]] := With[{g = Denominator[n]}, Dist[g, Subst[Int[x^(g*(m + 1) - 1)*(d + e*x^(g*n))^q*(a + c*x^(2*g*n))^p, x], x, x^(1/g)], x]] /; FreeQ[{a, c, d, e, m, p, q}, x] && EqQ[n2, 2*n] && FractionQ[n]
  1550. Int[Times[Power[Times[Pattern[f, Blank[]], Pattern[x, Blank[]]], Pattern[m, Blank[]]], Power[Plus[Pattern[a, Blank[]], Times[Optional[Pattern[c, Blank[]]], Power[Pattern[x, Blank[]], Optional[Pattern[n2, Blank[]]]]], Times[Optional[Pattern[b, Blank[]]], Power[Pattern[x, Blank[]], Pattern[n, Blank[]]]]], Pattern[p, Blank[]]], Power[Plus[Pattern[d, Blank[]], Times[Optional[Pattern[e, Blank[]]], Power[Pattern[x, Blank[]], Pattern[n, Blank[]]]]], Optional[Pattern[q, Blank[]]]]], Pattern[x, Blank[Symbol]]] := Dist[(f^IntPart[m]*(f*x)^FracPart[m])/x^FracPart[m], Int[x^m*(d + e*x^n)^q*(a + b*x^n + c*x^(2*n))^p, x], x] /; FreeQ[{a, b, c, d, e, f, m, p, q}, x] && EqQ[n2, 2*n] && NeQ[b^2 - 4*a*c, 0] && FractionQ[n]
  1551. Int[Times[Power[Times[Pattern[f, Blank[]], Pattern[x, Blank[]]], Pattern[m, Blank[]]], Power[Plus[Pattern[a, Blank[]], Times[Optional[Pattern[c, Blank[]]], Power[Pattern[x, Blank[]], Optional[Pattern[n2, Blank[]]]]]], Pattern[p, Blank[]]], Power[Plus[Pattern[d, Blank[]], Times[Optional[Pattern[e, Blank[]]], Power[Pattern[x, Blank[]], Pattern[n, Blank[]]]]], Optional[Pattern[q, Blank[]]]]], Pattern[x, Blank[Symbol]]] := Dist[(f^IntPart[m]*(f*x)^FracPart[m])/x^FracPart[m], Int[x^m*(d + e*x^n)^q*(a + c*x^(2*n))^p, x], x] /; FreeQ[{a, c, d, e, f, m, p, q}, x] && EqQ[n2, 2*n] && FractionQ[n]
  1552. Int[Times[Power[Pattern[x, Blank[]], Optional[Pattern[m, Blank[]]]], Power[Plus[Pattern[a, Blank[]], Times[Optional[Pattern[c, Blank[]]], Power[Pattern[x, Blank[]], Optional[Pattern[n2, Blank[]]]]], Times[Optional[Pattern[b, Blank[]]], Power[Pattern[x, Blank[]], Pattern[n, Blank[]]]]], Pattern[p, Blank[]]], Power[Plus[Pattern[d, Blank[]], Times[Optional[Pattern[e, Blank[]]], Power[Pattern[x, Blank[]], Pattern[n, Blank[]]]]], Optional[Pattern[q, Blank[]]]]], Pattern[x, Blank[Symbol]]] := Dist[1/(m + 1), Subst[Int[(d + e*x^Simplify[n/(m + 1)])^q*(a + b*x^Simplify[n/(m + 1)] + c*x^Simplify[(2*n)/(m + 1)])^p, x], x, x^(m + 1)], x] /; FreeQ[{a, b, c, d, e, m, n, p, q}, x] && EqQ[n2, 2*n] && NeQ[b^2 - 4*a*c, 0] && IntegerQ[Simplify[n/(m + 1)]] && !IntegerQ[n]
  1553. Int[Times[Power[Pattern[x, Blank[]], Optional[Pattern[m, Blank[]]]], Power[Plus[Pattern[a, Blank[]], Times[Optional[Pattern[c, Blank[]]], Power[Pattern[x, Blank[]], Optional[Pattern[n2, Blank[]]]]]], Pattern[p, Blank[]]], Power[Plus[Pattern[d, Blank[]], Times[Optional[Pattern[e, Blank[]]], Power[Pattern[x, Blank[]], Pattern[n, Blank[]]]]], Optional[Pattern[q, Blank[]]]]], Pattern[x, Blank[Symbol]]] := Dist[1/(m + 1), Subst[Int[(d + e*x^Simplify[n/(m + 1)])^q*(a + c*x^Simplify[(2*n)/(m + 1)])^p, x], x, x^(m + 1)], x] /; FreeQ[{a, c, d, e, m, n, p, q}, x] && EqQ[n2, 2*n] && IntegerQ[Simplify[n/(m + 1)]] && !IntegerQ[n]
  1554. Int[Times[Power[Times[Pattern[f, Blank[]], Pattern[x, Blank[]]], Pattern[m, Blank[]]], Power[Plus[Pattern[a, Blank[]], Times[Optional[Pattern[c, Blank[]]], Power[Pattern[x, Blank[]], Optional[Pattern[n2, Blank[]]]]], Times[Optional[Pattern[b, Blank[]]], Power[Pattern[x, Blank[]], Pattern[n, Blank[]]]]], Pattern[p, Blank[]]], Power[Plus[Pattern[d, Blank[]], Times[Optional[Pattern[e, Blank[]]], Power[Pattern[x, Blank[]], Pattern[n, Blank[]]]]], Optional[Pattern[q, Blank[]]]]], Pattern[x, Blank[Symbol]]] := Dist[(f^IntPart[m]*(f*x)^FracPart[m])/x^FracPart[m], Int[x^m*(d + e*x^n)^q*(a + b*x^n + c*x^(2*n))^p, x], x] /; FreeQ[{a, b, c, d, e, f, m, p, q}, x] && EqQ[n2, 2*n] && NeQ[b^2 - 4*a*c, 0] && IntegerQ[Simplify[n/(m + 1)]] && !IntegerQ[n]
  1555. Int[Times[Power[Times[Pattern[f, Blank[]], Pattern[x, Blank[]]], Pattern[m, Blank[]]], Power[Plus[Pattern[a, Blank[]], Times[Optional[Pattern[c, Blank[]]], Power[Pattern[x, Blank[]], Optional[Pattern[n2, Blank[]]]]]], Pattern[p, Blank[]]], Power[Plus[Pattern[d, Blank[]], Times[Optional[Pattern[e, Blank[]]], Power[Pattern[x, Blank[]], Pattern[n, Blank[]]]]], Optional[Pattern[q, Blank[]]]]], Pattern[x, Blank[Symbol]]] := Dist[(f^IntPart[m]*(f*x)^FracPart[m])/x^FracPart[m], Int[x^m*(d + e*x^n)^q*(a + c*x^(2*n))^p, x], x] /; FreeQ[{a, c, d, e, f, m, p, q}, x] && EqQ[n2, 2*n] && IntegerQ[Simplify[n/(m + 1)]] && !IntegerQ[n]
  1556. Int[Times[Power[Times[Optional[Pattern[f, Blank[]]], Pattern[x, Blank[]]], Optional[Pattern[m, Blank[]]]], Power[Plus[Pattern[a, Blank[]], Times[Optional[Pattern[c, Blank[]]], Power[Pattern[x, Blank[]], Optional[Pattern[n2, Blank[]]]]], Times[Optional[Pattern[b, Blank[]]], Power[Pattern[x, Blank[]], Pattern[n, Blank[]]]]], -1], Power[Plus[Pattern[d, Blank[]], Times[Optional[Pattern[e, Blank[]]], Power[Pattern[x, Blank[]], Pattern[n, Blank[]]]]], Pattern[q, Blank[]]]], Pattern[x, Blank[Symbol]]] := With[{r = Rt[b^2 - 4*a*c, 2]}, Dist[(2*c)/r, Int[((f*x)^m*(d + e*x^n)^q)/(b - r + 2*c*x^n), x], x] - Dist[(2*c)/r, Int[((f*x)^m*(d + e*x^n)^q)/(b + r + 2*c*x^n), x], x]] /; FreeQ[{a, b, c, d, e, f, m, n, q}, x] && EqQ[n2, 2*n] && NeQ[b^2 - 4*a*c, 0]
  1557. Int[Times[Power[Times[Optional[Pattern[f, Blank[]]], Pattern[x, Blank[]]], Optional[Pattern[m, Blank[]]]], Power[Plus[Pattern[a, Blank[]], Times[Optional[Pattern[c, Blank[]]], Power[Pattern[x, Blank[]], Optional[Pattern[n2, Blank[]]]]]], -1], Power[Plus[Pattern[d, Blank[]], Times[Optional[Pattern[e, Blank[]]], Power[Pattern[x, Blank[]], Pattern[n, Blank[]]]]], Pattern[q, Blank[]]]], Pattern[x, Blank[Symbol]]] := With[{r = Rt[-(a*c), 2]}, -Dist[c/(2*r), Int[((f*x)^m*(d + e*x^n)^q)/(r - c*x^n), x], x] - Dist[c/(2*r), Int[((f*x)^m*(d + e*x^n)^q)/(r + c*x^n), x], x]] /; FreeQ[{a, c, d, e, f, m, n, q}, x] && EqQ[n2, 2*n]
  1558. Int[Times[Power[Times[Optional[Pattern[f, Blank[]]], Pattern[x, Blank[]]], Optional[Pattern[m, Blank[]]]], Plus[Pattern[d, Blank[]], Times[Optional[Pattern[e, Blank[]]], Power[Pattern[x, Blank[]], Pattern[n, Blank[]]]]], Power[Plus[Pattern[a, Blank[]], Times[Optional[Pattern[b, Blank[]]], Power[Pattern[x, Blank[]], Pattern[n, Blank[]]]], Times[Optional[Pattern[c, Blank[]]], Power[Pattern[x, Blank[]], Pattern[n2, Blank[]]]]], Pattern[p, Blank[]]]], Pattern[x, Blank[Symbol]]] := -Simp[((f*x)^(m + 1)*(a + b*x^n + c*x^(2*n))^(p + 1)*(d*(b^2 - 2*a*c) - a*b*e + (b*d - 2*a*e)*c*x^n))/(a*f*n*(p + 1)*(b^2 - 4*a*c)), x] + Dist[1/(a*n*(p + 1)*(b^2 - 4*a*c)), Int[(f*x)^m*(a + b*x^n + c*x^(2*n))^(p + 1)*Simp[d*(b^2*(m + n*(p + 1) + 1) - 2*a*c*(m + 2*n*(p + 1) + 1)) - a*b*e*(m + 1) + (m + n*(2*p + 3) + 1)*(b*d - 2*a*e)*c*x^n, x], x], x] /; FreeQ[{a, b, c, d, e, f, m, n}, x] && EqQ[n2, 2*n] && NeQ[b^2 - 4*a*c, 0] && ILtQ[p + 1, 0]
  1559. Int[Times[Power[Times[Optional[Pattern[f, Blank[]]], Pattern[x, Blank[]]], Optional[Pattern[m, Blank[]]]], Plus[Pattern[d, Blank[]], Times[Optional[Pattern[e, Blank[]]], Power[Pattern[x, Blank[]], Pattern[n, Blank[]]]]], Power[Plus[Pattern[a, Blank[]], Times[Optional[Pattern[c, Blank[]]], Power[Pattern[x, Blank[]], Pattern[n2, Blank[]]]]], Pattern[p, Blank[]]]], Pattern[x, Blank[Symbol]]] := -Simp[((f*x)^(m + 1)*(a + c*x^(2*n))^(p + 1)*(d + e*x^n))/(2*a*f*n*(p + 1)), x] + Dist[1/(2*a*n*(p + 1)), Int[(f*x)^m*(a + c*x^(2*n))^(p + 1)*Simp[d*(m + 2*n*(p + 1) + 1) + e*(m + n*(2*p + 3) + 1)*x^n, x], x], x] /; FreeQ[{a, c, d, e, f, m, n}, x] && EqQ[n2, 2*n] && ILtQ[p + 1, 0]
  1560. Int[Times[Power[Times[Optional[Pattern[f, Blank[]]], Pattern[x, Blank[]]], Optional[Pattern[m, Blank[]]]], Power[Plus[Pattern[a, Blank[]], Times[Optional[Pattern[c, Blank[]]], Power[Pattern[x, Blank[]], Optional[Pattern[n2, Blank[]]]]], Times[Optional[Pattern[b, Blank[]]], Power[Pattern[x, Blank[]], Pattern[n, Blank[]]]]], Optional[Pattern[p, Blank[]]]], Power[Plus[Pattern[d, Blank[]], Times[Optional[Pattern[e, Blank[]]], Power[Pattern[x, Blank[]], Pattern[n, Blank[]]]]], Optional[Pattern[q, Blank[]]]]], Pattern[x, Blank[Symbol]]] := Int[ExpandIntegrand[(f*x)^m*(d + e*x^n)^q*(a + b*x^n + c*x^(2*n))^p, x], x] /; FreeQ[{a, b, c, d, e, f, m, n, p, q}, x] && EqQ[n2, 2*n] && NeQ[b^2 - 4*a*c, 0] && (IGtQ[p, 0] || IGtQ[q, 0])
  1561. Int[Times[Power[Times[Optional[Pattern[f, Blank[]]], Pattern[x, Blank[]]], Optional[Pattern[m, Blank[]]]], Power[Plus[Pattern[a, Blank[]], Times[Optional[Pattern[c, Blank[]]], Power[Pattern[x, Blank[]], Optional[Pattern[n2, Blank[]]]]]], Optional[Pattern[p, Blank[]]]], Power[Plus[Pattern[d, Blank[]], Times[Optional[Pattern[e, Blank[]]], Power[Pattern[x, Blank[]], Pattern[n, Blank[]]]]], Optional[Pattern[q, Blank[]]]]], Pattern[x, Blank[Symbol]]] := Int[ExpandIntegrand[(f*x)^m*(d + e*x^n)^q*(a + c*x^(2*n))^p, x], x] /; FreeQ[{a, c, d, e, f, m, n, p, q}, x] && EqQ[n2, 2*n] && (IGtQ[p, 0] || IGtQ[q, 0])
  1562. Int[Times[Power[Times[Optional[Pattern[f, Blank[]]], Pattern[x, Blank[]]], Optional[Pattern[m, Blank[]]]], Power[Plus[Pattern[d, Blank[]], Times[Optional[Pattern[e, Blank[]]], Power[Pattern[x, Blank[]], Pattern[n, Blank[]]]]], Pattern[q, Blank[]]], Power[Plus[Pattern[a, Blank[]], Times[Optional[Pattern[c, Blank[]]], Power[Pattern[x, Blank[]], Pattern[n2, Blank[]]]]], Pattern[p, Blank[]]]], Pattern[x, Blank[Symbol]]] := Dist[(f*x)^m/x^m, Int[ExpandIntegrand[x^m*(a + c*x^(2*n))^p, (d/(d^2 - e^2*x^(2*n)) - (e*x^n)/(d^2 - e^2*x^(2*n)))^(-q), x], x], x] /; FreeQ[{a, c, d, e, f, m, n, p}, x] && EqQ[n2, 2*n] && !IntegerQ[p] && ILtQ[q, 0]
  1563. Int[Times[Power[Times[Optional[Pattern[f, Blank[]]], Pattern[x, Blank[]]], Optional[Pattern[m, Blank[]]]], Power[Plus[Pattern[a, Blank[]], Times[Optional[Pattern[c, Blank[]]], Power[Pattern[x, Blank[]], Optional[Pattern[n2, Blank[]]]]], Times[Optional[Pattern[b, Blank[]]], Power[Pattern[x, Blank[]], Pattern[n, Blank[]]]]], Optional[Pattern[p, Blank[]]]], Power[Plus[Pattern[d, Blank[]], Times[Optional[Pattern[e, Blank[]]], Power[Pattern[x, Blank[]], Pattern[n, Blank[]]]]], Optional[Pattern[q, Blank[]]]]], Pattern[x, Blank[Symbol]]] := Unintegrable[(f*x)^m*(d + e*x^n)^q*(a + b*x^n + c*x^(2*n))^p, x] /; FreeQ[{a, b, c, d, e, f, m, n, p, q}, x] && EqQ[n2, 2*n]
  1564. Int[Times[Power[Times[Optional[Pattern[f, Blank[]]], Pattern[x, Blank[]]], Optional[Pattern[m, Blank[]]]], Power[Plus[Pattern[a, Blank[]], Times[Optional[Pattern[c, Blank[]]], Power[Pattern[x, Blank[]], Optional[Pattern[n2, Blank[]]]]]], Optional[Pattern[p, Blank[]]]], Power[Plus[Pattern[d, Blank[]], Times[Optional[Pattern[e, Blank[]]], Power[Pattern[x, Blank[]], Pattern[n, Blank[]]]]], Optional[Pattern[q, Blank[]]]]], Pattern[x, Blank[Symbol]]] := Unintegrable[(f*x)^m*(d + e*x^n)^q*(a + c*x^(2*n))^p, x] /; FreeQ[{a, c, d, e, f, m, n, p, q}, x] && EqQ[n2, 2*n]
  1565. Int[Times[Power[Pattern[u, Blank[]], Optional[Pattern[m, Blank[]]]], Power[Plus[Pattern[a, Blank[]], Times[Optional[Pattern[c, Blank[]]], Power[Pattern[v, Blank[]], Optional[Pattern[n2, Blank[]]]]], Times[Optional[Pattern[b, Blank[]]], Power[Pattern[v, Blank[]], Pattern[n, Blank[]]]]], Optional[Pattern[p, Blank[]]]], Power[Plus[Pattern[d, Blank[]], Times[Optional[Pattern[e, Blank[]]], Power[Pattern[v, Blank[]], Pattern[n, Blank[]]]]], Optional[Pattern[q, Blank[]]]]], Pattern[x, Blank[Symbol]]] := Dist[u^m/(Coefficient[v, x, 1]*v^m), Subst[Int[x^m*(d + e*x^n)^q*(a + b*x^n + c*x^(2*n))^p, x], x, v], x] /; FreeQ[{a, b, c, d, e, m, n, p, q}, x] && EqQ[n2, 2*n] && LinearPairQ[u, v, x] && NeQ[v, x]
  1566. Int[Times[Power[Pattern[u, Blank[]], Optional[Pattern[m, Blank[]]]], Power[Plus[Pattern[a, Blank[]], Times[Optional[Pattern[c, Blank[]]], Power[Pattern[v, Blank[]], Optional[Pattern[n2, Blank[]]]]]], Optional[Pattern[p, Blank[]]]], Power[Plus[Pattern[d, Blank[]], Times[Optional[Pattern[e, Blank[]]], Power[Pattern[v, Blank[]], Pattern[n, Blank[]]]]], Optional[Pattern[q, Blank[]]]]], Pattern[x, Blank[Symbol]]] := Dist[u^m/(Coefficient[v, x, 1]*v^m), Subst[Int[x^m*(d + e*x^n)^q*(a + c*x^(2*n))^p, x], x, v], x] /; FreeQ[{a, c, d, e, m, n, p}, x] && EqQ[n2, 2*n] && LinearPairQ[u, v, x] && NeQ[v, x]
  1567. Int[Times[Power[Pattern[x, Blank[]], Optional[Pattern[m, Blank[]]]], Power[Plus[Pattern[d, Blank[]], Times[Optional[Pattern[e, Blank[]]], Power[Pattern[x, Blank[]], Optional[Pattern[mn, Blank[]]]]]], Optional[Pattern[q, Blank[]]]], Power[Plus[Optional[Pattern[a, Blank[]]], Times[Optional[Pattern[b, Blank[]]], Power[Pattern[x, Blank[]], Optional[Pattern[n, Blank[]]]]], Times[Optional[Pattern[c, Blank[]]], Power[Pattern[x, Blank[]], Optional[Pattern[n2, Blank[]]]]]], Optional[Pattern[p, Blank[]]]]], Pattern[x, Blank[Symbol]]] := Int[x^(m - n*q)*(e + d*x^n)^q*(a + b*x^n + c*x^(2*n))^p, x] /; FreeQ[{a, b, c, d, e, m, n, p}, x] && EqQ[n2, 2*n] && EqQ[mn, -n] && IntegerQ[q] && (PosQ[n] || !IntegerQ[p])
  1568. Int[Times[Power[Pattern[x, Blank[]], Optional[Pattern[m, Blank[]]]], Power[Plus[Pattern[d, Blank[]], Times[Optional[Pattern[e, Blank[]]], Power[Pattern[x, Blank[]], Optional[Pattern[mn, Blank[]]]]]], Optional[Pattern[q, Blank[]]]], Power[Plus[Pattern[a, Blank[]], Times[Optional[Pattern[c, Blank[]]], Power[Pattern[x, Blank[]], Optional[Pattern[n2, Blank[]]]]]], Optional[Pattern[p, Blank[]]]]], Pattern[x, Blank[Symbol]]] := Int[x^(m + mn*q)*(e + d/x^mn)^q*(a + c*x^n2)^p, x] /; FreeQ[{a, c, d, e, m, mn, p}, x] && EqQ[n2, -2*mn] && IntegerQ[q] && (PosQ[n2] || !IntegerQ[p])
  1569. Int[Times[Power[Pattern[x, Blank[]], Optional[Pattern[m, Blank[]]]], Power[Plus[Optional[Pattern[a, Blank[]]], Times[Optional[Pattern[b, Blank[]]], Power[Pattern[x, Blank[]], Optional[Pattern[mn, Blank[]]]]], Times[Optional[Pattern[c, Blank[]]], Power[Pattern[x, Blank[]], Optional[Pattern[mn2, Blank[]]]]]], Optional[Pattern[p, Blank[]]]], Power[Plus[Pattern[d, Blank[]], Times[Optional[Pattern[e, Blank[]]], Power[Pattern[x, Blank[]], Optional[Pattern[n, Blank[]]]]]], Optional[Pattern[q, Blank[]]]]], Pattern[x, Blank[Symbol]]] := Int[x^(m - 2*n*p)*(d + e*x^n)^q*(c + b*x^n + a*x^(2*n))^p, x] /; FreeQ[{a, b, c, d, e, m, n, q}, x] && EqQ[mn, -n] && EqQ[mn2, 2*mn] && IntegerQ[p]
  1570. Int[Times[Power[Pattern[x, Blank[]], Optional[Pattern[m, Blank[]]]], Power[Plus[Optional[Pattern[a, Blank[]]], Times[Optional[Pattern[c, Blank[]]], Power[Pattern[x, Blank[]], Optional[Pattern[mn2, Blank[]]]]]], Optional[Pattern[p, Blank[]]]], Power[Plus[Pattern[d, Blank[]], Times[Optional[Pattern[e, Blank[]]], Power[Pattern[x, Blank[]], Optional[Pattern[n, Blank[]]]]]], Optional[Pattern[q, Blank[]]]]], Pattern[x, Blank[Symbol]]] := Int[x^(m - 2*n*p)*(d + e*x^n)^q*(c + a*x^(2*n))^p, x] /; FreeQ[{a, c, d, e, m, n, q}, x] && EqQ[mn2, -2*n] && IntegerQ[p]
  1571. Int[Times[Power[Pattern[x, Blank[]], Optional[Pattern[m, Blank[]]]], Power[Plus[Pattern[d, Blank[]], Times[Optional[Pattern[e, Blank[]]], Power[Pattern[x, Blank[]], Optional[Pattern[mn, Blank[]]]]]], Pattern[q, Blank[]]], Power[Plus[Optional[Pattern[a, Blank[]]], Times[Optional[Pattern[b, Blank[]]], Power[Pattern[x, Blank[]], Optional[Pattern[n, Blank[]]]]], Times[Optional[Pattern[c, Blank[]]], Power[Pattern[x, Blank[]], Optional[Pattern[n2, Blank[]]]]]], Optional[Pattern[p, Blank[]]]]], Pattern[x, Blank[Symbol]]] := Dist[(e^IntPart[q]*x^(n*FracPart[q])*(d + e/x^n)^FracPart[q])/(1 + (d*x^n)/e)^FracPart[q], Int[x^(m - n*q)*(1 + (d*x^n)/e)^q*(a + b*x^n + c*x^(2*n))^p, x], x] /; FreeQ[{a, b, c, d, e, m, n, p, q}, x] && EqQ[n2, 2*n] && EqQ[mn, -n] && !IntegerQ[p] && !IntegerQ[q] && PosQ[n]
  1572. Int[Times[Power[Pattern[x, Blank[]], Optional[Pattern[m, Blank[]]]], Power[Plus[Pattern[d, Blank[]], Times[Optional[Pattern[e, Blank[]]], Power[Pattern[x, Blank[]], Optional[Pattern[mn, Blank[]]]]]], Pattern[q, Blank[]]], Power[Plus[Pattern[a, Blank[]], Times[Optional[Pattern[c, Blank[]]], Power[Pattern[x, Blank[]], Optional[Pattern[n2, Blank[]]]]]], Optional[Pattern[p, Blank[]]]]], Pattern[x, Blank[Symbol]]] := Dist[(e^IntPart[q]*(d + e*x^mn)^FracPart[q])/(x^(mn*FracPart[q])*(1 + d/(x^mn*e))^FracPart[q]), Int[x^(m + mn*q)*(1 + d/(x^mn*e))^q*(a + c*x^n2)^p, x], x] /; FreeQ[{a, c, d, e, m, mn, p, q}, x] && EqQ[n2, -2*mn] && !IntegerQ[p] && !IntegerQ[q] && PosQ[n2]
  1573. Int[Times[Power[Pattern[x, Blank[]], Optional[Pattern[m, Blank[]]]], Power[Plus[Optional[Pattern[a, Blank[]]], Times[Optional[Pattern[b, Blank[]]], Power[Pattern[x, Blank[]], Optional[Pattern[mn, Blank[]]]]], Times[Optional[Pattern[c, Blank[]]], Power[Pattern[x, Blank[]], Optional[Pattern[mn2, Blank[]]]]]], Pattern[p, Blank[]]], Power[Plus[Pattern[d, Blank[]], Times[Optional[Pattern[e, Blank[]]], Power[Pattern[x, Blank[]], Optional[Pattern[n, Blank[]]]]]], Optional[Pattern[q, Blank[]]]]], Pattern[x, Blank[Symbol]]] := Dist[(x^(2*n*FracPart[p])*(a + b/x^n + c/x^(2*n))^FracPart[p])/(c + b*x^n + a*x^(2*n))^FracPart[p], Int[x^(m - 2*n*p)*(d + e*x^n)^q*(c + b*x^n + a*x^(2*n))^p, x], x] /; FreeQ[{a, b, c, d, e, m, n, p, q}, x] && EqQ[mn, -n] && EqQ[mn2, 2*mn] && !IntegerQ[p] && !IntegerQ[q] && PosQ[n]
  1574. Int[Times[Power[Pattern[x, Blank[]], Optional[Pattern[m, Blank[]]]], Power[Plus[Optional[Pattern[a, Blank[]]], Times[Optional[Pattern[c, Blank[]]], Power[Pattern[x, Blank[]], Optional[Pattern[mn2, Blank[]]]]]], Pattern[p, Blank[]]], Power[Plus[Pattern[d, Blank[]], Times[Optional[Pattern[e, Blank[]]], Power[Pattern[x, Blank[]], Optional[Pattern[n, Blank[]]]]]], Optional[Pattern[q, Blank[]]]]], Pattern[x, Blank[Symbol]]] := Dist[(x^(2*n*FracPart[p])*(a + c/x^(2*n))^FracPart[p])/(c + a*x^(2*n))^FracPart[p], Int[x^(m - 2*n*p)*(d + e*x^n)^q*(c + a*x^(2*n))^p, x], x] /; FreeQ[{a, c, d, e, m, n, p, q}, x] && EqQ[mn2, -2*n] && !IntegerQ[p] && !IntegerQ[q] && PosQ[n]
  1575. Int[Times[Power[Times[Pattern[f, Blank[]], Pattern[x, Blank[]]], Pattern[m, Blank[]]], Power[Plus[Pattern[d, Blank[]], Times[Optional[Pattern[e, Blank[]]], Power[Pattern[x, Blank[]], Optional[Pattern[mn, Blank[]]]]]], Optional[Pattern[q, Blank[]]]], Power[Plus[Optional[Pattern[a, Blank[]]], Times[Optional[Pattern[b, Blank[]]], Power[Pattern[x, Blank[]], Optional[Pattern[n, Blank[]]]]], Times[Optional[Pattern[c, Blank[]]], Power[Pattern[x, Blank[]], Optional[Pattern[n2, Blank[]]]]]], Optional[Pattern[p, Blank[]]]]], Pattern[x, Blank[Symbol]]] := Dist[(f^IntPart[m]*(f*x)^FracPart[m])/x^FracPart[m], Int[x^m*(d + e*x^mn)^q*(a + b*x^n + c*x^(2*n))^p, x], x] /; FreeQ[{a, b, c, d, e, f, m, n, p, q}, x] && EqQ[n2, 2*n] && EqQ[mn, -n]
  1576. Int[Times[Power[Times[Pattern[f, Blank[]], Pattern[x, Blank[]]], Pattern[m, Blank[]]], Power[Plus[Pattern[d, Blank[]], Times[Optional[Pattern[e, Blank[]]], Power[Pattern[x, Blank[]], Optional[Pattern[mn, Blank[]]]]]], Optional[Pattern[q, Blank[]]]], Power[Plus[Pattern[a, Blank[]], Times[Optional[Pattern[c, Blank[]]], Power[Pattern[x, Blank[]], Optional[Pattern[n2, Blank[]]]]]], Optional[Pattern[p, Blank[]]]]], Pattern[x, Blank[Symbol]]] := Dist[(f^IntPart[m]*(f*x)^FracPart[m])/x^FracPart[m], Int[x^m*(d + e*x^mn)^q*(a + c*x^n2)^p, x], x] /; FreeQ[{a, c, d, e, f, m, mn, p, q}, x] && EqQ[n2, -2*mn]
  1577. Int[Times[Power[Pattern[x, Blank[]], Optional[Pattern[m, Blank[]]]], Power[Plus[Pattern[a, Blank[]], Times[Optional[Pattern[c, Blank[]]], Power[Pattern[x, Blank[]], Optional[Pattern[n, Blank[]]]]], Times[Optional[Pattern[b, Blank[]]], Power[Pattern[x, Blank[]], Pattern[mn, Blank[]]]]], Optional[Pattern[p, Blank[]]]], Power[Plus[Pattern[d, Blank[]], Times[Optional[Pattern[e, Blank[]]], Power[Pattern[x, Blank[]], Pattern[n, Blank[]]]]], Optional[Pattern[q, Blank[]]]]], Pattern[x, Blank[Symbol]]] := Int[x^(m - n*p)*(d + e*x^n)^q*(b + a*x^n + c*x^(2*n))^p, x] /; FreeQ[{a, b, c, d, e, m, n, q}, x] && EqQ[mn, -n] && IntegerQ[p]
  1578. Int[Times[Power[Pattern[x, Blank[]], Optional[Pattern[m, Blank[]]]], Power[Plus[Pattern[a, Blank[]], Times[Optional[Pattern[c, Blank[]]], Power[Pattern[x, Blank[]], Optional[Pattern[n, Blank[]]]]], Times[Optional[Pattern[b, Blank[]]], Power[Pattern[x, Blank[]], Pattern[mn, Blank[]]]]], Optional[Pattern[p, Blank[]]]], Power[Plus[Pattern[d, Blank[]], Times[Optional[Pattern[e, Blank[]]], Power[Pattern[x, Blank[]], Pattern[n, Blank[]]]]], Optional[Pattern[q, Blank[]]]]], Pattern[x, Blank[Symbol]]] := Dist[(x^(n*FracPart[p])*(a + b/x^n + c*x^n)^FracPart[p])/(b + a*x^n + c*x^(2*n))^FracPart[p], Int[x^(m - n*p)*(d + e*x^n)^q*(b + a*x^n + c*x^(2*n))^p, x], x] /; FreeQ[{a, b, c, d, e, m, n, p, q}, x] && EqQ[mn, -n] && !IntegerQ[p]
  1579. Int[Times[Power[Times[Pattern[f, Blank[]], Pattern[x, Blank[]]], Optional[Pattern[m, Blank[]]]], Power[Plus[Pattern[a, Blank[]], Times[Optional[Pattern[c, Blank[]]], Power[Pattern[x, Blank[]], Optional[Pattern[n, Blank[]]]]], Times[Optional[Pattern[b, Blank[]]], Power[Pattern[x, Blank[]], Pattern[mn, Blank[]]]]], Optional[Pattern[p, Blank[]]]], Power[Plus[Pattern[d, Blank[]], Times[Optional[Pattern[e, Blank[]]], Power[Pattern[x, Blank[]], Pattern[n, Blank[]]]]], Optional[Pattern[q, Blank[]]]]], Pattern[x, Blank[Symbol]]] := Dist[(f^IntPart[m]*(f*x)^FracPart[m])/x^FracPart[m], Int[x^m*(d + e*x^n)^q*(a + b/x^n + c*x^n)^p, x], x] /; FreeQ[{a, b, c, d, e, f, m, n, p, q}, x] && EqQ[mn, -n]
  1580. Int[Times[Power[Times[Optional[Pattern[f, Blank[]]], Pattern[x, Blank[]]], Optional[Pattern[m, Blank[]]]], Power[Plus[Pattern[d1, Blank[]], Times[Optional[Pattern[e1, Blank[]]], Power[Pattern[x, Blank[]], Optional[Pattern[non2, Blank[]]]]]], Optional[Pattern[q, Blank[]]]], Power[Plus[Pattern[d2, Blank[]], Times[Optional[Pattern[e2, Blank[]]], Power[Pattern[x, Blank[]], Optional[Pattern[non2, Blank[]]]]]], Optional[Pattern[q, Blank[]]]], Power[Plus[Optional[Pattern[a, Blank[]]], Times[Optional[Pattern[b, Blank[]]], Power[Pattern[x, Blank[]], Pattern[n, Blank[]]]], Times[Optional[Pattern[c, Blank[]]], Power[Pattern[x, Blank[]], Pattern[n2, Blank[]]]]], Optional[Pattern[p, Blank[]]]]], Pattern[x, Blank[Symbol]]] := Int[(f*x)^m*(d1*d2 + e1*e2*x^n)^q*(a + b*x^n + c*x^(2*n))^p, x] /; FreeQ[{a, b, c, d1, e1, d2, e2, f, n, p, q}, x] && EqQ[n2, 2*n] && EqQ[non2, n/2] && EqQ[d2*e1 + d1*e2, 0] && (IntegerQ[q] || (GtQ[d1, 0] && GtQ[d2, 0]))
  1581. Int[Times[Power[Times[Optional[Pattern[f, Blank[]]], Pattern[x, Blank[]]], Optional[Pattern[m, Blank[]]]], Power[Plus[Pattern[d1, Blank[]], Times[Optional[Pattern[e1, Blank[]]], Power[Pattern[x, Blank[]], Optional[Pattern[non2, Blank[]]]]]], Optional[Pattern[q, Blank[]]]], Power[Plus[Pattern[d2, Blank[]], Times[Optional[Pattern[e2, Blank[]]], Power[Pattern[x, Blank[]], Optional[Pattern[non2, Blank[]]]]]], Optional[Pattern[q, Blank[]]]], Power[Plus[Optional[Pattern[a, Blank[]]], Times[Optional[Pattern[b, Blank[]]], Power[Pattern[x, Blank[]], Pattern[n, Blank[]]]], Times[Optional[Pattern[c, Blank[]]], Power[Pattern[x, Blank[]], Pattern[n2, Blank[]]]]], Optional[Pattern[p, Blank[]]]]], Pattern[x, Blank[Symbol]]] := Dist[((d1 + e1*x^(n/2))^FracPart[q]*(d2 + e2*x^(n/2))^FracPart[q])/(d1*d2 + e1*e2*x^n)^FracPart[q], Int[(f*x)^m*(d1*d2 + e1*e2*x^n)^q*(a + b*x^n + c*x^(2*n))^p, x], x] /; FreeQ[{a, b, c, d1, e1, d2, e2, f, n, p, q}, x] && EqQ[n2, 2*n] && EqQ[non2, n/2] && EqQ[d2*e1 + d1*e2, 0]
  1582. Int[Times[Pattern[Px, Blank[]], Power[Plus[Pattern[a, Blank[]], Times[Optional[Pattern[b, Blank[]]], Power[Pattern[x, Blank[]], Pattern[n, Blank[]]]]], Pattern[p, Blank[]]]], Pattern[x, Blank[Symbol]]] := Simp[(Coeff[Px, x, n - 1]*(a + b*x^n)^(p + 1))/(b*n*(p + 1)), x] + Int[(Px - Coeff[Px, x, n - 1]*x^(n - 1))*(a + b*x^n)^p, x] /; FreeQ[{a, b}, x] && PolyQ[Px, x] && IGtQ[p, 1] && IGtQ[n, 1] && NeQ[Coeff[Px, x, n - 1], 0] && NeQ[Px, Coeff[Px, x, n - 1]*x^(n - 1)] && !MatchQ[Px, (Qx_.)*((c_) + (d_.)*x^(m_))^(q_) /; FreeQ[{c, d}, x] && PolyQ[Qx, x] && IGtQ[q, 1] && IGtQ[m, 1] && NeQ[Coeff[Qx*(a + b*x^n)^p, x, m - 1], 0] && GtQ[m*q, n*p]]
  1583. Int[Times[Pattern[Px, Blank[]], Power[Pattern[x, Blank[]], Optional[Pattern[m, Blank[]]]], Power[Plus[Pattern[a, Blank[]], Times[Optional[Pattern[b, Blank[]]], Power[Pattern[x, Blank[]], Optional[Pattern[n, Blank[]]]]]], Pattern[p, Blank[]]]], Pattern[x, Blank[Symbol]]] := Simp[(Coeff[Px, x, n - m - 1]*(a + b*x^n)^(p + 1))/(b*n*(p + 1)), x] + Int[(Px - Coeff[Px, x, n - m - 1]*x^(n - m - 1))*x^m*(a + b*x^n)^p, x] /; FreeQ[{a, b, m, n}, x] && PolyQ[Px, x] && IGtQ[p, 1] && IGtQ[n - m, 0] && NeQ[Coeff[Px, x, n - m - 1], 0]
  1584. Int[Times[Optional[Pattern[u, Blank[]]], Power[Pattern[x, Blank[]], Optional[Pattern[m, Blank[]]]], Power[Plus[Times[Optional[Pattern[a, Blank[]]], Power[Pattern[x, Blank[]], Optional[Pattern[p, Blank[]]]]], Times[Optional[Pattern[b, Blank[]]], Power[Pattern[x, Blank[]], Optional[Pattern[q, Blank[]]]]]], Optional[Pattern[n, Blank[]]]]], Pattern[x, Blank[Symbol]]] := Int[u*x^(m + n*p)*(a + b*x^(q - p))^n, x] /; FreeQ[{a, b, m, p, q}, x] && IntegerQ[n] && PosQ[q - p]
  1585. Int[Times[Optional[Pattern[u, Blank[]]], Power[Pattern[x, Blank[]], Optional[Pattern[m, Blank[]]]], Power[Plus[Times[Optional[Pattern[a, Blank[]]], Power[Pattern[x, Blank[]], Optional[Pattern[p, Blank[]]]]], Times[Optional[Pattern[b, Blank[]]], Power[Pattern[x, Blank[]], Optional[Pattern[q, Blank[]]]]], Times[Optional[Pattern[c, Blank[]]], Power[Pattern[x, Blank[]], Optional[Pattern[r, Blank[]]]]]], Optional[Pattern[n, Blank[]]]]], Pattern[x, Blank[Symbol]]] := Int[u*x^(m + n*p)*(a + b*x^(q - p) + c*x^(r - p))^n, x] /; FreeQ[{a, b, c, m, p, q, r}, x] && IntegerQ[n] && PosQ[q - p] && PosQ[r - p]
  1586. Int[Times[Optional[Pattern[u, Blank[]]], Power[Pattern[Px, Blank[]], Optional[Pattern[p, Blank[]]]], Power[Pattern[Qx, Blank[]], Optional[Pattern[q, Blank[]]]]], Pattern[x, Blank[Symbol]]] := Int[u*PolynomialQuotient[Px, Qx, x]^p*Qx^(p + q), x] /; FreeQ[q, x] && PolyQ[Px, x] && PolyQ[Qx, x] && EqQ[PolynomialRemainder[Px, Qx, x], 0] && IntegerQ[p] && LtQ[p*q, 0]
  1587. Int[Times[Pattern[Pp, Blank[]], Power[Pattern[Qq, Blank[]], -1]], Pattern[x, Blank[Symbol]]] := With[{p = Expon[Pp, x], q = Expon[Qq, x]}, Simp[(Coeff[Pp, x, p]*Log[RemoveContent[Qq, x]])/(q*Coeff[Qq, x, q]), x] /; EqQ[p, q - 1] && EqQ[Pp, Simplify[(Coeff[Pp, x, p]*D[Qq, x])/(q*Coeff[Qq, x, q])]]] /; PolyQ[Pp, x] && PolyQ[Qq, x]
  1588. Int[Times[Pattern[Pp, Blank[]], Power[Pattern[Qq, Blank[]], Optional[Pattern[m, Blank[]]]]], Pattern[x, Blank[Symbol]]] := With[{p = Expon[Pp, x], q = Expon[Qq, x]}, Simp[(Coeff[Pp, x, p]*x^(p - q + 1)*Qq^(m + 1))/((p + m*q + 1)*Coeff[Qq, x, q]), x] /; NeQ[p + m*q + 1, 0] && EqQ[(p + m*q + 1)*Coeff[Qq, x, q]*Pp, Coeff[Pp, x, p]*x^(p - q)*((p - q + 1)*Qq + (m + 1)*x*D[Qq, x])]] /; FreeQ[m, x] && PolyQ[Pp, x] && PolyQ[Qq, x] && NeQ[m, -1]
  1589. Int[Times[Power[Pattern[x, Blank[]], Optional[Pattern[m, Blank[]]]], Power[Plus[Pattern[a1, Blank[]], Times[Optional[Pattern[b1, Blank[]]], Power[Pattern[x, Blank[]], Optional[Pattern[n, Blank[]]]]]], Pattern[p, Blank[]]], Power[Plus[Pattern[a2, Blank[]], Times[Optional[Pattern[b2, Blank[]]], Power[Pattern[x, Blank[]], Optional[Pattern[n, Blank[]]]]]], Pattern[p, Blank[]]]], Pattern[x, Blank[Symbol]]] := Simp[((a1 + b1*x^n)^(p + 1)*(a2 + b2*x^n)^(p + 1))/(2*b1*b2*n*(p + 1)), x] /; FreeQ[{a1, b1, a2, b2, m, n, p}, x] && EqQ[a2*b1 + a1*b2, 0] && EqQ[m - 2*n + 1, 0] && NeQ[p, -1]
  1590. Int[Times[Pattern[Pp, Blank[]], Power[Pattern[Qq, Blank[]], Optional[Pattern[m, Blank[]]]], Power[Pattern[Rr, Blank[]], Optional[Pattern[n, Blank[]]]]], Pattern[x, Blank[Symbol]]] := With[{p = Expon[Pp, x], q = Expon[Qq, x], r = Expon[Rr, x]}, Simp[(Coeff[Pp, x, p]*x^(p - q - r + 1)*Qq^(m + 1)*Rr^(n + 1))/((p + m*q + n*r + 1)*Coeff[Qq, x, q]*Coeff[Rr, x, r]), x] /; NeQ[p + m*q + n*r + 1, 0] && EqQ[(p + m*q + n*r + 1)*Coeff[Qq, x, q]*Coeff[Rr, x, r]*Pp, Coeff[Pp, x, p]*x^(p - q - r)*((p - q - r + 1)*Qq*Rr + (m + 1)*x*Rr*D[Qq, x] + (n + 1)*x*Qq*D[Rr, x])]] /; FreeQ[{m, n}, x] && PolyQ[Pp, x] && PolyQ[Qq, x] && PolyQ[Rr, x] && NeQ[m, -1] && NeQ[n, -1]
  1591. Int[Times[Power[Plus[Optional[Pattern[a, Blank[]]], Times[Optional[Pattern[b, Blank[]]], Power[Pattern[Pq, Blank[]], Optional[Pattern[n, Blank[]]]]]], Optional[Pattern[p, Blank[]]]], Pattern[Qr, Blank[]]], Pattern[x, Blank[Symbol]]] := With[{q = Expon[Pq, x], r = Expon[Qr, x]}, Dist[Coeff[Qr, x, r]/(q*Coeff[Pq, x, q]), Subst[Int[(a + b*x^n)^p, x], x, Pq], x] /; EqQ[r, q - 1] && EqQ[Coeff[Qr, x, r]*D[Pq, x], q*Coeff[Pq, x, q]*Qr]] /; FreeQ[{a, b, n, p}, x] && PolyQ[Pq, x] && PolyQ[Qr, x]
  1592. Int[Times[Power[Plus[Optional[Pattern[a, Blank[]]], Times[Optional[Pattern[b, Blank[]]], Power[Pattern[Pq, Blank[]], Optional[Pattern[n, Blank[]]]]], Times[Optional[Pattern[c, Blank[]]], Power[Pattern[Pq, Blank[]], Optional[Pattern[n2, Blank[]]]]]], Optional[Pattern[p, Blank[]]]], Pattern[Qr, Blank[]]], Pattern[x, Blank[Symbol]]] := Module[{q = Expon[Pq, x], r = Expon[Qr, x]}, Dist[Coeff[Qr, x, r]/(q*Coeff[Pq, x, q]), Subst[Int[(a + b*x^n + c*x^(2*n))^p, x], x, Pq], x] /; EqQ[r, q - 1] && EqQ[Coeff[Qr, x, r]*D[Pq, x], q*Coeff[Pq, x, q]*Qr]] /; FreeQ[{a, b, c, n, p}, x] && EqQ[n2, 2*n] && PolyQ[Pq, x] && PolyQ[Qr, x]
  1593. Int[Times[Optional[Pattern[u, Blank[]]], Power[Plus[Times[Optional[Pattern[a, Blank[]]], Power[Pattern[x, Blank[]], Optional[Pattern[p, Blank[]]]]], Times[Optional[Pattern[b, Blank[]]], Power[Pattern[x, Blank[]], Optional[Pattern[q, Blank[]]]]]], Optional[Pattern[n, Blank[]]]]], Pattern[x, Blank[Symbol]]] := Int[u*x^(n*p)*(a + b*x^(q - p))^n, x] /; FreeQ[{a, b, p, q}, x] && IntegerQ[n] && PosQ[q - p]
  1594. Int[Times[Optional[Pattern[u, Blank[]]], Power[Plus[Times[Optional[Pattern[a, Blank[]]], Power[Pattern[x, Blank[]], Optional[Pattern[p, Blank[]]]]], Times[Optional[Pattern[b, Blank[]]], Power[Pattern[x, Blank[]], Optional[Pattern[q, Blank[]]]]], Times[Optional[Pattern[c, Blank[]]], Power[Pattern[x, Blank[]], Optional[Pattern[r, Blank[]]]]]], Optional[Pattern[n, Blank[]]]]], Pattern[x, Blank[Symbol]]] := Int[u*x^(n*p)*(a + b*x^(q - p) + c*x^(r - p))^n, x] /; FreeQ[{a, b, c, p, q, r}, x] && IntegerQ[n] && PosQ[q - p] && PosQ[r - p]
  1595. Int[Times[Power[Plus[Optional[Pattern[a, Blank[]]], Times[Optional[Pattern[b, Blank[]]], Pattern[x, Blank[]]]], Rational[1, 2]], Plus[Optional[Pattern[A, Blank[]]], Times[Optional[Pattern[B, Blank[]]], Pattern[x, Blank[]]]], Power[Plus[Optional[Pattern[c, Blank[]]], Times[Optional[Pattern[d, Blank[]]], Pattern[x, Blank[]]]], Rational[-1, 2]], Power[Plus[Optional[Pattern[e, Blank[]]], Times[Optional[Pattern[f, Blank[]]], Pattern[x, Blank[]]]], Rational[-1, 2]], Power[Plus[Optional[Pattern[g, Blank[]]], Times[Optional[Pattern[h, Blank[]]], Pattern[x, Blank[]]]], Rational[-1, 2]]], Pattern[x, Blank[Symbol]]] := Simp[(b*B*Sqrt[c + d*x]*Sqrt[e + f*x]*Sqrt[g + h*x])/(d*f*h*Sqrt[a + b*x]), x] + (-Dist[(B*(b*g - a*h))/(2*f*h), Int[Sqrt[e + f*x]/(Sqrt[a + b*x]*Sqrt[c + d*x]*Sqrt[g + h*x]), x], x] + Dist[(B*(b*e - a*f)*(b*g - a*h))/(2*d*f*h), Int[Sqrt[c + d*x]/((a + b*x)^(3/2)*Sqrt[e + f*x]*Sqrt[g + h*x]), x], x]) /; FreeQ[{a, b, c, d, e, f, g, h, A, B}, x] && EqQ[2*A*d*f - B*(d*e + c*f), 0]
  1596. Int[Times[Power[Plus[Optional[Pattern[a, Blank[]]], Times[Optional[Pattern[b, Blank[]]], Pattern[x, Blank[]]]], Rational[1, 2]], Plus[Optional[Pattern[A, Blank[]]], Times[Optional[Pattern[B, Blank[]]], Pattern[x, Blank[]]]], Power[Plus[Optional[Pattern[c, Blank[]]], Times[Optional[Pattern[d, Blank[]]], Pattern[x, Blank[]]]], Rational[-1, 2]], Power[Plus[Optional[Pattern[e, Blank[]]], Times[Optional[Pattern[f, Blank[]]], Pattern[x, Blank[]]]], Rational[-1, 2]], Power[Plus[Optional[Pattern[g, Blank[]]], Times[Optional[Pattern[h, Blank[]]], Pattern[x, Blank[]]]], Rational[-1, 2]]], Pattern[x, Blank[Symbol]]] := Simp[(B*Sqrt[a + b*x]*Sqrt[e + f*x]*Sqrt[g + h*x])/(f*h*Sqrt[c + d*x]), x] + (-Dist[(B*(b*e - a*f)*(b*g - a*h))/(2*b*f*h), Int[1/(Sqrt[a + b*x]*Sqrt[c + d*x]*Sqrt[e + f*x]*Sqrt[g + h*x]), x], x] + Dist[(B*(d*e - c*f)*(d*g - c*h))/(2*d*f*h), Int[Sqrt[a + b*x]/((c + d*x)^(3/2)*Sqrt[e + f*x]*Sqrt[g + h*x]), x], x] + Dist[(2*A*b*d*f*h + B*(a*d*f*h - b*(d*f*g + d*e*h + c*f*h)))/(2*b*d*f*h), Int[Sqrt[a + b*x]/(Sqrt[c + d*x]*Sqrt[e + f*x]*Sqrt[g + h*x]), x], x]) /; FreeQ[{a, b, c, d, e, f, g, h, A, B}, x] && NeQ[2*A*d*f - B*(d*e + c*f), 0]
  1597. Int[Times[Power[Plus[Optional[Pattern[a, Blank[]]], Times[Optional[Pattern[b, Blank[]]], Pattern[x, Blank[]]]], Optional[Pattern[m, Blank[]]]], Plus[Optional[Pattern[A, Blank[]]], Times[Optional[Pattern[B, Blank[]]], Pattern[x, Blank[]]]], Power[Plus[Optional[Pattern[c, Blank[]]], Times[Optional[Pattern[d, Blank[]]], Pattern[x, Blank[]]]], Rational[-1, 2]], Power[Plus[Optional[Pattern[e, Blank[]]], Times[Optional[Pattern[f, Blank[]]], Pattern[x, Blank[]]]], Rational[-1, 2]], Power[Plus[Optional[Pattern[g, Blank[]]], Times[Optional[Pattern[h, Blank[]]], Pattern[x, Blank[]]]], Rational[-1, 2]]], Pattern[x, Blank[Symbol]]] := Dist[1/(d*f*h*(2*m + 3)), Int[((a + b*x)^(m - 1)/(Sqrt[c + d*x]*Sqrt[e + f*x]*Sqrt[g + h*x]))*Simp[a*A*d*f*h*(2*m + 3) + (A*b + a*B)*d*f*h*(2*m + 3)*x + b*B*d*f*h*(2*m + 3)*x^2, x], x], x] /; FreeQ[{a, b, c, d, e, f, g, h, A, B}, x] && IntegerQ[2*m] && GtQ[m, 0]
  1598. Int[Times[Power[Plus[Optional[Pattern[a, Blank[]]], Times[Optional[Pattern[b, Blank[]]], Pattern[x, Blank[]]]], Rational[-1, 2]], Plus[Optional[Pattern[A, Blank[]]], Times[Optional[Pattern[B, Blank[]]], Pattern[x, Blank[]]]], Power[Plus[Optional[Pattern[c, Blank[]]], Times[Optional[Pattern[d, Blank[]]], Pattern[x, Blank[]]]], Rational[-1, 2]], Power[Plus[Optional[Pattern[e, Blank[]]], Times[Optional[Pattern[f, Blank[]]], Pattern[x, Blank[]]]], Rational[-1, 2]], Power[Plus[Optional[Pattern[g, Blank[]]], Times[Optional[Pattern[h, Blank[]]], Pattern[x, Blank[]]]], Rational[-1, 2]]], Pattern[x, Blank[Symbol]]] := Dist[(A*b - a*B)/b, Int[1/(Sqrt[a + b*x]*Sqrt[c + d*x]*Sqrt[e + f*x]*Sqrt[g + h*x]), x], x] + Dist[B/b, Int[Sqrt[a + b*x]/(Sqrt[c + d*x]*Sqrt[e + f*x]*Sqrt[g + h*x]), x], x] /; FreeQ[{a, b, c, d, e, f, g, h, A, B}, x]
  1599. Int[Times[Power[Plus[Optional[Pattern[a, Blank[]]], Times[Optional[Pattern[b, Blank[]]], Pattern[x, Blank[]]]], Pattern[m, Blank[]]], Plus[Optional[Pattern[A, Blank[]]], Times[Optional[Pattern[B, Blank[]]], Pattern[x, Blank[]]]], Power[Plus[Optional[Pattern[c, Blank[]]], Times[Optional[Pattern[d, Blank[]]], Pattern[x, Blank[]]]], Rational[-1, 2]], Power[Plus[Optional[Pattern[e, Blank[]]], Times[Optional[Pattern[f, Blank[]]], Pattern[x, Blank[]]]], Rational[-1, 2]], Power[Plus[Optional[Pattern[g, Blank[]]], Times[Optional[Pattern[h, Blank[]]], Pattern[x, Blank[]]]], Rational[-1, 2]]], Pattern[x, Blank[Symbol]]] := Simp[((A*b^2 - a*b*B)*(a + b*x)^(m + 1)*Sqrt[c + d*x]*Sqrt[e + f*x]*Sqrt[g + h*x])/((m + 1)*(b*c - a*d)*(b*e - a*f)*(b*g - a*h)), x] - Dist[1/(2*(m + 1)*(b*c - a*d)*(b*e - a*f)*(b*g - a*h)), Int[((a + b*x)^(m + 1)/(Sqrt[c + d*x]*Sqrt[e + f*x]*Sqrt[g + h*x]))*Simp[A*(2*a^2*d*f*h*(m + 1) - 2*a*b*(m + 1)*(d*f*g + d*e*h + c*f*h) + b^2*(2*m + 3)*(d*e*g + c*f*g + c*e*h)) - b*B*(a*(d*e*g + c*f*g + c*e*h) + 2*b*c*e*g*(m + 1)) - 2*((A*b - a*B)*(a*d*f*h*(m + 1) - b*(m + 2)*(d*f*g + d*e*h + c*f*h)))*x + d*f*h*(2*m + 5)*(A*b^2 - a*b*B)*x^2, x], x], x] /; FreeQ[{a, b, c, d, e, f, g, h, A, B}, x] && IntegerQ[2*m] && LtQ[m, -1]
  1600. Int[Times[Power[Plus[Optional[Pattern[a, Blank[]]], Times[Optional[Pattern[b, Blank[]]], Pattern[x, Blank[]]]], Optional[Pattern[m, Blank[]]]], Power[Plus[Optional[Pattern[c, Blank[]]], Times[Optional[Pattern[d, Blank[]]], Pattern[x, Blank[]]]], Rational[-1, 2]], Power[Plus[Optional[Pattern[e, Blank[]]], Times[Optional[Pattern[f, Blank[]]], Pattern[x, Blank[]]]], Rational[-1, 2]], Power[Plus[Optional[Pattern[g, Blank[]]], Times[Optional[Pattern[h, Blank[]]], Pattern[x, Blank[]]]], Rational[-1, 2]], Plus[Optional[Pattern[A, Blank[]]], Times[Optional[Pattern[B, Blank[]]], Pattern[x, Blank[]]], Times[Optional[Pattern[C, Blank[]]], Power[Pattern[x, Blank[]], 2]]]], Pattern[x, Blank[Symbol]]] := Simp[(2*C*(a + b*x)^m*Sqrt[c + d*x]*Sqrt[e + f*x]*Sqrt[g + h*x])/(d*f*h*(2*m + 3)), x] + Dist[1/(d*f*h*(2*m + 3)), Int[((a + b*x)^(m - 1)/(Sqrt[c + d*x]*Sqrt[e + f*x]*Sqrt[g + h*x]))*Simp[a*A*d*f*h*(2*m + 3) - C*(a*(d*e*g + c*f*g + c*e*h) + 2*b*c*e*g*m) + ((A*b + a*B)*d*f*h*(2*m + 3) - C*(2*a*(d*f*g + d*e*h + c*f*h) + b*(2*m + 1)*(d*e*g + c*f*g + c*e*h)))*x + (b*B*d*f*h*(2*m + 3) + 2*C*(a*d*f*h*m - b*(m + 1)*(d*f*g + d*e*h + c*f*h)))*x^2, x], x], x] /; FreeQ[{a, b, c, d, e, f, g, h, A, B, C}, x] && IntegerQ[2*m] && GtQ[m, 0]
  1601. Int[Times[Power[Plus[Optional[Pattern[a, Blank[]]], Times[Optional[Pattern[b, Blank[]]], Pattern[x, Blank[]]]], Optional[Pattern[m, Blank[]]]], Power[Plus[Optional[Pattern[c, Blank[]]], Times[Optional[Pattern[d, Blank[]]], Pattern[x, Blank[]]]], Rational[-1, 2]], Power[Plus[Optional[Pattern[e, Blank[]]], Times[Optional[Pattern[f, Blank[]]], Pattern[x, Blank[]]]], Rational[-1, 2]], Power[Plus[Optional[Pattern[g, Blank[]]], Times[Optional[Pattern[h, Blank[]]], Pattern[x, Blank[]]]], Rational[-1, 2]], Plus[Optional[Pattern[A, Blank[]]], Times[Optional[Pattern[C, Blank[]]], Power[Pattern[x, Blank[]], 2]]]], Pattern[x, Blank[Symbol]]] := Simp[(2*C*(a + b*x)^m*Sqrt[c + d*x]*Sqrt[e + f*x]*Sqrt[g + h*x])/(d*f*h*(2*m + 3)), x] + Dist[1/(d*f*h*(2*m + 3)), Int[((a + b*x)^(m - 1)/(Sqrt[c + d*x]*Sqrt[e + f*x]*Sqrt[g + h*x]))*Simp[a*A*d*f*h*(2*m + 3) - C*(a*(d*e*g + c*f*g + c*e*h) + 2*b*c*e*g*m) + (A*b*d*f*h*(2*m + 3) - C*(2*a*(d*f*g + d*e*h + c*f*h) + b*(2*m + 1)*(d*e*g + c*f*g + c*e*h)))*x + 2*C*(a*d*f*h*m - b*(m + 1)*(d*f*g + d*e*h + c*f*h))*x^2, x], x], x] /; FreeQ[{a, b, c, d, e, f, g, h, A, C}, x] && IntegerQ[2*m] && GtQ[m, 0]
  1602. Int[Times[Power[Plus[Optional[Pattern[a, Blank[]]], Times[Optional[Pattern[b, Blank[]]], Pattern[x, Blank[]]]], Rational[-1, 2]], Power[Plus[Optional[Pattern[c, Blank[]]], Times[Optional[Pattern[d, Blank[]]], Pattern[x, Blank[]]]], Rational[-1, 2]], Power[Plus[Optional[Pattern[e, Blank[]]], Times[Optional[Pattern[f, Blank[]]], Pattern[x, Blank[]]]], Rational[-1, 2]], Power[Plus[Optional[Pattern[g, Blank[]]], Times[Optional[Pattern[h, Blank[]]], Pattern[x, Blank[]]]], Rational[-1, 2]], Plus[Optional[Pattern[A, Blank[]]], Times[Optional[Pattern[B, Blank[]]], Pattern[x, Blank[]]], Times[Optional[Pattern[C, Blank[]]], Power[Pattern[x, Blank[]], 2]]]], Pattern[x, Blank[Symbol]]] := Simp[(C*Sqrt[a + b*x]*Sqrt[e + f*x]*Sqrt[g + h*x])/(b*f*h*Sqrt[c + d*x]), x] + (Dist[1/(2*b*d*f*h), Int[(1*Simp[2*A*b*d*f*h - C*(b*d*e*g + a*c*f*h) + (2*b*B*d*f*h - C*(a*d*f*h + b*(d*f*g + d*e*h + c*f*h)))*x, x])/(Sqrt[a + b*x]*Sqrt[c + d*x]*Sqrt[e + f*x]*Sqrt[g + h*x]), x], x] + Dist[(C*(d*e - c*f)*(d*g - c*h))/(2*b*d*f*h), Int[Sqrt[a + b*x]/((c + d*x)^(3/2)*Sqrt[e + f*x]*Sqrt[g + h*x]), x], x]) /; FreeQ[{a, b, c, d, e, f, g, h, A, B, C}, x]
  1603. Int[Times[Power[Plus[Optional[Pattern[a, Blank[]]], Times[Optional[Pattern[b, Blank[]]], Pattern[x, Blank[]]]], Rational[-1, 2]], Power[Plus[Optional[Pattern[c, Blank[]]], Times[Optional[Pattern[d, Blank[]]], Pattern[x, Blank[]]]], Rational[-1, 2]], Power[Plus[Optional[Pattern[e, Blank[]]], Times[Optional[Pattern[f, Blank[]]], Pattern[x, Blank[]]]], Rational[-1, 2]], Power[Plus[Optional[Pattern[g, Blank[]]], Times[Optional[Pattern[h, Blank[]]], Pattern[x, Blank[]]]], Rational[-1, 2]], Plus[Optional[Pattern[A, Blank[]]], Times[Optional[Pattern[C, Blank[]]], Power[Pattern[x, Blank[]], 2]]]], Pattern[x, Blank[Symbol]]] := Simp[(C*Sqrt[a + b*x]*Sqrt[e + f*x]*Sqrt[g + h*x])/(b*f*h*Sqrt[c + d*x]), x] + (Dist[1/(2*b*d*f*h), Int[(1*Simp[2*A*b*d*f*h - C*(b*d*e*g + a*c*f*h) - C*(a*d*f*h + b*(d*f*g + d*e*h + c*f*h))*x, x])/(Sqrt[a + b*x]*Sqrt[c + d*x]*Sqrt[e + f*x]*Sqrt[g + h*x]), x], x] + Dist[(C*(d*e - c*f)*(d*g - c*h))/(2*b*d*f*h), Int[Sqrt[a + b*x]/((c + d*x)^(3/2)*Sqrt[e + f*x]*Sqrt[g + h*x]), x], x]) /; FreeQ[{a, b, c, d, e, f, g, h, A, C}, x]
  1604. Int[Times[Power[Plus[Optional[Pattern[a, Blank[]]], Times[Optional[Pattern[b, Blank[]]], Pattern[x, Blank[]]]], Pattern[m, Blank[]]], Power[Plus[Optional[Pattern[c, Blank[]]], Times[Optional[Pattern[d, Blank[]]], Pattern[x, Blank[]]]], Rational[-1, 2]], Power[Plus[Optional[Pattern[e, Blank[]]], Times[Optional[Pattern[f, Blank[]]], Pattern[x, Blank[]]]], Rational[-1, 2]], Power[Plus[Optional[Pattern[g, Blank[]]], Times[Optional[Pattern[h, Blank[]]], Pattern[x, Blank[]]]], Rational[-1, 2]], Plus[Optional[Pattern[A, Blank[]]], Times[Optional[Pattern[B, Blank[]]], Pattern[x, Blank[]]], Times[Optional[Pattern[C, Blank[]]], Power[Pattern[x, Blank[]], 2]]]], Pattern[x, Blank[Symbol]]] := Simp[((A*b^2 - a*b*B + a^2*C)*(a + b*x)^(m + 1)*Sqrt[c + d*x]*Sqrt[e + f*x]*Sqrt[g + h*x])/((m + 1)*(b*c - a*d)*(b*e - a*f)*(b*g - a*h)), x] - Dist[1/(2*(m + 1)*(b*c - a*d)*(b*e - a*f)*(b*g - a*h)), Int[((a + b*x)^(m + 1)/(Sqrt[c + d*x]*Sqrt[e + f*x]*Sqrt[g + h*x]))*Simp[A*(2*a^2*d*f*h*(m + 1) - 2*a*b*(m + 1)*(d*f*g + d*e*h + c*f*h) + b^2*(2*m + 3)*(d*e*g + c*f*g + c*e*h)) - (b*B - a*C)*(a*(d*e*g + c*f*g + c*e*h) + 2*b*c*e*g*(m + 1)) - 2*((A*b - a*B)*(a*d*f*h*(m + 1) - b*(m + 2)*(d*f*g + d*e*h + c*f*h)) - C*(a^2*(d*f*g + d*e*h + c*f*h) - b^2*c*e*g*(m + 1) + a*b*(m + 1)*(d*e*g + c*f*g + c*e*h)))*x + d*f*h*(2*m + 5)*(A*b^2 - a*b*B + a^2*C)*x^2, x], x], x] /; FreeQ[{a, b, c, d, e, f, g, h, A, B, C}, x] && IntegerQ[2*m] && LtQ[m, -1]
  1605. Int[Times[Power[Plus[Optional[Pattern[a, Blank[]]], Times[Optional[Pattern[b, Blank[]]], Pattern[x, Blank[]]]], Pattern[m, Blank[]]], Power[Plus[Optional[Pattern[c, Blank[]]], Times[Optional[Pattern[d, Blank[]]], Pattern[x, Blank[]]]], Rational[-1, 2]], Power[Plus[Optional[Pattern[e, Blank[]]], Times[Optional[Pattern[f, Blank[]]], Pattern[x, Blank[]]]], Rational[-1, 2]], Power[Plus[Optional[Pattern[g, Blank[]]], Times[Optional[Pattern[h, Blank[]]], Pattern[x, Blank[]]]], Rational[-1, 2]], Plus[Optional[Pattern[A, Blank[]]], Times[Optional[Pattern[C, Blank[]]], Power[Pattern[x, Blank[]], 2]]]], Pattern[x, Blank[Symbol]]] := Simp[((A*b^2 + a^2*C)*(a + b*x)^(m + 1)*Sqrt[c + d*x]*Sqrt[e + f*x]*Sqrt[g + h*x])/((m + 1)*(b*c - a*d)*(b*e - a*f)*(b*g - a*h)), x] - Dist[1/(2*(m + 1)*(b*c - a*d)*(b*e - a*f)*(b*g - a*h)), Int[((a + b*x)^(m + 1)/(Sqrt[c + d*x]*Sqrt[e + f*x]*Sqrt[g + h*x]))*Simp[A*(2*a^2*d*f*h*(m + 1) - 2*a*b*(m + 1)*(d*f*g + d*e*h + c*f*h) + b^2*(2*m + 3)*(d*e*g + c*f*g + c*e*h)) + a*C*(a*(d*e*g + c*f*g + c*e*h) + 2*b*c*e*g*(m + 1)) - 2*(A*b*(a*d*f*h*(m + 1) - b*(m + 2)*(d*f*g + d*e*h + c*f*h)) - C*(a^2*(d*f*g + d*e*h + c*f*h) - b^2*c*e*g*(m + 1) + a*b*(m + 1)*(d*e*g + c*f*g + c*e*h)))*x + d*f*h*(2*m + 5)*(A*b^2 + a^2*C)*x^2, x], x], x] /; FreeQ[{a, b, c, d, e, f, g, h, A, C}, x] && IntegerQ[2*m] && LtQ[m, -1]
  1606. Int[Times[Pattern[Px, Blank[]], Power[Plus[Optional[Pattern[a, Blank[]]], Times[Optional[Pattern[b, Blank[]]], Pattern[x, Blank[]]]], Optional[Pattern[m, Blank[]]]], Power[Plus[Optional[Pattern[c, Blank[]]], Times[Optional[Pattern[d, Blank[]]], Pattern[x, Blank[]]]], Optional[Pattern[n, Blank[]]]], Power[Plus[Optional[Pattern[e, Blank[]]], Times[Optional[Pattern[f, Blank[]]], Pattern[x, Blank[]]]], Optional[Pattern[p, Blank[]]]], Power[Plus[Optional[Pattern[g, Blank[]]], Times[Optional[Pattern[h, Blank[]]], Pattern[x, Blank[]]]], Optional[Pattern[q, Blank[]]]]], Pattern[x, Blank[Symbol]]] := Int[ExpandIntegrand[Px*(a + b*x)^m*(c + d*x)^n*(e + f*x)^p*(g + h*x)^q, x], x] /; FreeQ[{a, b, c, d, e, f, g, h, m, n, p, q}, x] && PolyQ[Px, x] && IntegersQ[m, n]
  1607. Int[Times[Pattern[Px, Blank[]], Power[Plus[Optional[Pattern[a, Blank[]]], Times[Optional[Pattern[b, Blank[]]], Pattern[x, Blank[]]]], Optional[Pattern[m, Blank[]]]], Power[Plus[Optional[Pattern[c, Blank[]]], Times[Optional[Pattern[d, Blank[]]], Pattern[x, Blank[]]]], Optional[Pattern[n, Blank[]]]], Power[Plus[Optional[Pattern[e, Blank[]]], Times[Optional[Pattern[f, Blank[]]], Pattern[x, Blank[]]]], Optional[Pattern[p, Blank[]]]], Power[Plus[Optional[Pattern[g, Blank[]]], Times[Optional[Pattern[h, Blank[]]], Pattern[x, Blank[]]]], Optional[Pattern[q, Blank[]]]]], Pattern[x, Blank[Symbol]]] := Dist[PolynomialRemainder[Px, a + b*x, x], Int[(a + b*x)^m*(c + d*x)^n*(e + f*x)^p*(g + h*x)^q, x], x] + Int[PolynomialQuotient[Px, a + b*x, x]*(a + b*x)^(m + 1)*(c + d*x)^n*(e + f*x)^p*(g + h*x)^q, x] /; FreeQ[{a, b, c, d, e, f, g, h, m, n, p, q}, x] && PolyQ[Px, x] && EqQ[m, -1]
  1608. Int[Times[Pattern[Px, Blank[]], Power[Plus[Optional[Pattern[a, Blank[]]], Times[Optional[Pattern[b, Blank[]]], Pattern[x, Blank[]]]], Optional[Pattern[m, Blank[]]]], Power[Plus[Optional[Pattern[c, Blank[]]], Times[Optional[Pattern[d, Blank[]]], Pattern[x, Blank[]]]], Optional[Pattern[n, Blank[]]]], Power[Plus[Optional[Pattern[e, Blank[]]], Times[Optional[Pattern[f, Blank[]]], Pattern[x, Blank[]]]], Optional[Pattern[p, Blank[]]]], Power[Plus[Optional[Pattern[g, Blank[]]], Times[Optional[Pattern[h, Blank[]]], Pattern[x, Blank[]]]], Optional[Pattern[q, Blank[]]]]], Pattern[x, Blank[Symbol]]] := Dist[PolynomialRemainder[Px, a + b*x, x], Int[(a + b*x)^m*(c + d*x)^n*(e + f*x)^p*(g + h*x)^q, x], x] + Int[PolynomialQuotient[Px, a + b*x, x]*(a + b*x)^(m + 1)*(c + d*x)^n*(e + f*x)^p*(g + h*x)^q, x] /; FreeQ[{a, b, c, d, e, f, g, h, m, n, p, q}, x] && PolyQ[Px, x]
  1609. Int[Times[Pattern[Px, Blank[]], Power[Plus[Optional[Pattern[a, Blank[]]], Times[Optional[Pattern[b, Blank[]]], Pattern[x, Blank[]]]], Optional[Pattern[m, Blank[]]]], Power[Plus[Optional[Pattern[c, Blank[]]], Times[Optional[Pattern[d, Blank[]]], Pattern[x, Blank[]]]], Optional[Pattern[n, Blank[]]]], Power[Plus[Optional[Pattern[e, Blank[]]], Times[Optional[Pattern[f, Blank[]]], Pattern[x, Blank[]]]], Optional[Pattern[p, Blank[]]]]], Pattern[x, Blank[Symbol]]] := Int[Px*(a*c + b*d*x^2)^m*(e + f*x)^p, x] /; FreeQ[{a, b, c, d, e, f, m, n, p}, x] && PolyQ[Px, x] && EqQ[b*c + a*d, 0] && EqQ[m, n] && (IntegerQ[m] || (GtQ[a, 0] && GtQ[c, 0]))
  1610. Int[Times[Pattern[Px, Blank[]], Power[Plus[Optional[Pattern[a, Blank[]]], Times[Optional[Pattern[b, Blank[]]], Pattern[x, Blank[]]]], Pattern[m, Blank[]]], Power[Plus[Optional[Pattern[c, Blank[]]], Times[Optional[Pattern[d, Blank[]]], Pattern[x, Blank[]]]], Pattern[n, Blank[]]], Power[Plus[Optional[Pattern[e, Blank[]]], Times[Optional[Pattern[f, Blank[]]], Pattern[x, Blank[]]]], Optional[Pattern[p, Blank[]]]]], Pattern[x, Blank[Symbol]]] := Dist[((a + b*x)^FracPart[m]*(c + d*x)^FracPart[m])/(a*c + b*d*x^2)^FracPart[m], Int[Px*(a*c + b*d*x^2)^m*(e + f*x)^p, x], x] /; FreeQ[{a, b, c, d, e, f, m, n, p}, x] && PolyQ[Px, x] && EqQ[b*c + a*d, 0] && EqQ[m, n] && !IntegerQ[m]
  1611. Int[Times[Pattern[Px, Blank[]], Power[Plus[Optional[Pattern[a, Blank[]]], Times[Optional[Pattern[b, Blank[]]], Pattern[x, Blank[]]]], Optional[Pattern[m, Blank[]]]], Power[Plus[Optional[Pattern[c, Blank[]]], Times[Optional[Pattern[d, Blank[]]], Pattern[x, Blank[]]]], Optional[Pattern[n, Blank[]]]], Power[Plus[Optional[Pattern[e, Blank[]]], Times[Optional[Pattern[f, Blank[]]], Pattern[x, Blank[]]]], Optional[Pattern[p, Blank[]]]]], Pattern[x, Blank[Symbol]]] := Int[PolynomialQuotient[Px, a + b*x, x]*(a + b*x)^(m + 1)*(c + d*x)^n*(e + f*x)^p, x] /; FreeQ[{a, b, c, d, e, f, m, n, p}, x] && PolyQ[Px, x] && EqQ[PolynomialRemainder[Px, a + b*x, x], 0]
  1612. Int[Times[Pattern[Px, Blank[]], Power[Plus[Optional[Pattern[a, Blank[]]], Times[Optional[Pattern[b, Blank[]]], Pattern[x, Blank[]]]], Optional[Pattern[m, Blank[]]]], Power[Plus[Optional[Pattern[c, Blank[]]], Times[Optional[Pattern[d, Blank[]]], Pattern[x, Blank[]]]], Optional[Pattern[n, Blank[]]]], Power[Plus[Optional[Pattern[e, Blank[]]], Times[Optional[Pattern[f, Blank[]]], Pattern[x, Blank[]]]], Optional[Pattern[p, Blank[]]]]], Pattern[x, Blank[Symbol]]] := Int[ExpandIntegrand[Px*(a + b*x)^m*(c + d*x)^n*(e + f*x)^p, x], x] /; FreeQ[{a, b, c, d, e, f, m, n, p}, x] && PolyQ[Px, x] && IntegersQ[m, n]
  1613. Int[Times[Pattern[Px, Blank[]], Power[Plus[Optional[Pattern[a, Blank[]]], Times[Optional[Pattern[b, Blank[]]], Pattern[x, Blank[]]]], Pattern[m, Blank[]]], Power[Plus[Optional[Pattern[c, Blank[]]], Times[Optional[Pattern[d, Blank[]]], Pattern[x, Blank[]]]], Optional[Pattern[n, Blank[]]]], Power[Plus[Optional[Pattern[e, Blank[]]], Times[Optional[Pattern[f, Blank[]]], Pattern[x, Blank[]]]], Optional[Pattern[p, Blank[]]]]], Pattern[x, Blank[Symbol]]] := With[{Qx = PolynomialQuotient[Px, a + b*x, x], R = PolynomialRemainder[Px, a + b*x, x]}, Simp[(b*R*(a + b*x)^(m + 1)*(c + d*x)^(n + 1)*(e + f*x)^(p + 1))/((m + 1)*(b*c - a*d)*(b*e - a*f)), x] + Dist[1/((m + 1)*(b*c - a*d)*(b*e - a*f)), Int[(a + b*x)^(m + 1)*(c + d*x)^n*(e + f*x)^p*ExpandToSum[(m + 1)*(b*c - a*d)*(b*e - a*f)*Qx + a*d*f*R*(m + 1) - b*R*(d*e*(m + n + 2) + c*f*(m + p + 2)) - b*d*f*R*(m + n + p + 3)*x, x], x], x]] /; FreeQ[{a, b, c, d, e, f, n, p}, x] && PolyQ[Px, x] && ILtQ[m, -1] && IntegersQ[2*m, 2*n, 2*p]
  1614. Int[Times[Pattern[Px, Blank[]], Power[Plus[Optional[Pattern[a, Blank[]]], Times[Optional[Pattern[b, Blank[]]], Pattern[x, Blank[]]]], Pattern[m, Blank[]]], Power[Plus[Optional[Pattern[c, Blank[]]], Times[Optional[Pattern[d, Blank[]]], Pattern[x, Blank[]]]], Optional[Pattern[n, Blank[]]]], Power[Plus[Optional[Pattern[e, Blank[]]], Times[Optional[Pattern[f, Blank[]]], Pattern[x, Blank[]]]], Optional[Pattern[p, Blank[]]]]], Pattern[x, Blank[Symbol]]] := With[{Qx = PolynomialQuotient[Px, a + b*x, x], R = PolynomialRemainder[Px, a + b*x, x]}, Simp[(b*R*(a + b*x)^(m + 1)*(c + d*x)^(n + 1)*(e + f*x)^(p + 1))/((m + 1)*(b*c - a*d)*(b*e - a*f)), x] + Dist[1/((m + 1)*(b*c - a*d)*(b*e - a*f)), Int[(a + b*x)^(m + 1)*(c + d*x)^n*(e + f*x)^p*ExpandToSum[(m + 1)*(b*c - a*d)*(b*e - a*f)*Qx + a*d*f*R*(m + 1) - b*R*(d*e*(m + n + 2) + c*f*(m + p + 2)) - b*d*f*R*(m + n + p + 3)*x, x], x], x]] /; FreeQ[{a, b, c, d, e, f, n, p}, x] && PolyQ[Px, x] && LtQ[m, -1] && IntegersQ[2*m, 2*n, 2*p]
  1615. Int[Times[Pattern[Px, Blank[]], Power[Plus[Optional[Pattern[a, Blank[]]], Times[Optional[Pattern[b, Blank[]]], Pattern[x, Blank[]]]], Optional[Pattern[m, Blank[]]]], Power[Plus[Optional[Pattern[c, Blank[]]], Times[Optional[Pattern[d, Blank[]]], Pattern[x, Blank[]]]], Optional[Pattern[n, Blank[]]]], Power[Plus[Optional[Pattern[e, Blank[]]], Times[Optional[Pattern[f, Blank[]]], Pattern[x, Blank[]]]], Optional[Pattern[p, Blank[]]]]], Pattern[x, Blank[Symbol]]] := With[{q = Expon[Px, x], k = Coeff[Px, x, Expon[Px, x]]}, Simp[(k*(a + b*x)^(m + q - 1)*(c + d*x)^(n + 1)*(e + f*x)^(p + 1))/(d*f*b^(q - 1)*(m + n + p + q + 1)), x] + Dist[1/(d*f*b^q*(m + n + p + q + 1)), Int[(a + b*x)^m*(c + d*x)^n*(e + f*x)^p*ExpandToSum[d*f*b^q*(m + n + p + q + 1)*Px - d*f*k*(m + n + p + q + 1)*(a + b*x)^q + k*(a + b*x)^(q - 2)*(a^2*d*f*(m + n + p + q + 1) - b*(b*c*e*(m + q - 1) + a*(d*e*(n + 1) + c*f*(p + 1))) + b*(a*d*f*(2*(m + q) + n + p) - b*(d*e*(m + q + n) + c*f*(m + q + p)))*x), x], x], x] /; NeQ[m + n + p + q + 1, 0]] /; FreeQ[{a, b, c, d, e, f, m, n, p}, x] && PolyQ[Px, x] && IntegersQ[2*m, 2*n, 2*p]
  1616. Int[Times[Pattern[Px, Blank[]], Power[Plus[Optional[Pattern[a, Blank[]]], Times[Optional[Pattern[b, Blank[]]], Pattern[x, Blank[]]]], Optional[Pattern[m, Blank[]]]], Power[Plus[Optional[Pattern[c, Blank[]]], Times[Optional[Pattern[d, Blank[]]], Pattern[x, Blank[]]]], Optional[Pattern[n, Blank[]]]]], Pattern[x, Blank[Symbol]]] := Int[Px*(a*c + b*d*x^2)^m, x] /; FreeQ[{a, b, c, d, m, n}, x] && PolyQ[Px, x] && EqQ[b*c + a*d, 0] && EqQ[m, n] && (IntegerQ[m] || (GtQ[a, 0] && GtQ[c, 0]))
  1617. Int[Times[Pattern[Px, Blank[]], Power[Plus[Optional[Pattern[a, Blank[]]], Times[Optional[Pattern[b, Blank[]]], Pattern[x, Blank[]]]], Pattern[m, Blank[]]], Power[Plus[Optional[Pattern[c, Blank[]]], Times[Optional[Pattern[d, Blank[]]], Pattern[x, Blank[]]]], Pattern[n, Blank[]]]], Pattern[x, Blank[Symbol]]] := Dist[((a + b*x)^FracPart[m]*(c + d*x)^FracPart[m])/(a*c + b*d*x^2)^FracPart[m], Int[Px*(a*c + b*d*x^2)^m, x], x] /; FreeQ[{a, b, c, d, m, n}, x] && PolyQ[Px, x] && EqQ[b*c + a*d, 0] && EqQ[m, n] && !IntegerQ[m]
  1618. Int[Times[Pattern[Px, Blank[]], Power[Plus[Optional[Pattern[a, Blank[]]], Times[Optional[Pattern[b, Blank[]]], Pattern[x, Blank[]]]], Optional[Pattern[m, Blank[]]]], Power[Plus[Optional[Pattern[c, Blank[]]], Times[Optional[Pattern[d, Blank[]]], Pattern[x, Blank[]]]], Optional[Pattern[n, Blank[]]]]], Pattern[x, Blank[Symbol]]] := Int[PolynomialQuotient[Px, a + b*x, x]*(a + b*x)^(m + 1)*(c + d*x)^n, x] /; FreeQ[{a, b, c, d, m, n}, x] && PolyQ[Px, x] && EqQ[PolynomialRemainder[Px, a + b*x, x], 0]
  1619. Int[Times[Pattern[Px, Blank[]], Power[Plus[Optional[Pattern[a, Blank[]]], Times[Optional[Pattern[b, Blank[]]], Pattern[x, Blank[]]]], -1], Power[Plus[Optional[Pattern[c, Blank[]]], Times[Optional[Pattern[d, Blank[]]], Pattern[x, Blank[]]]], Optional[Pattern[n, Blank[]]]]], Pattern[x, Blank[Symbol]]] := Int[ExpandIntegrand[1/Sqrt[c + d*x], (Px*(c + d*x)^(n + 1/2))/(a + b*x), x], x] /; FreeQ[{a, b, c, d, n}, x] && PolyQ[Px, x] && ILtQ[n + 1/2, 0] && GtQ[Expon[Px, x], 2]
  1620. Int[Times[Pattern[Px, Blank[]], Power[Plus[Optional[Pattern[a, Blank[]]], Times[Optional[Pattern[b, Blank[]]], Pattern[x, Blank[]]]], Optional[Pattern[m, Blank[]]]], Power[Plus[Optional[Pattern[c, Blank[]]], Times[Optional[Pattern[d, Blank[]]], Pattern[x, Blank[]]]], Optional[Pattern[n, Blank[]]]]], Pattern[x, Blank[Symbol]]] := Int[ExpandIntegrand[Px*(a + b*x)^m*(c + d*x)^n, x], x] /; FreeQ[{a, b, c, d, m, n}, x] && PolyQ[Px, x] && (IntegersQ[m, n] || IGtQ[m, -2]) && GtQ[Expon[Px, x], 2]
  1621. Int[Times[Pattern[Px, Blank[]], Power[Plus[Optional[Pattern[a, Blank[]]], Times[Optional[Pattern[b, Blank[]]], Pattern[x, Blank[]]]], Pattern[m, Blank[]]], Power[Plus[Optional[Pattern[c, Blank[]]], Times[Optional[Pattern[d, Blank[]]], Pattern[x, Blank[]]]], Optional[Pattern[n, Blank[]]]]], Pattern[x, Blank[Symbol]]] := With[{Qx = PolynomialQuotient[Px, a + b*x, x], R = PolynomialRemainder[Px, a + b*x, x]}, Simp[(R*(a + b*x)^(m + 1)*(c + d*x)^(n + 1))/((m + 1)*(b*c - a*d)), x] + Dist[1/((m + 1)*(b*c - a*d)), Int[(a + b*x)^(m + 1)*(c + d*x)^n*ExpandToSum[(m + 1)*(b*c - a*d)*Qx - d*R*(m + n + 2), x], x], x]] /; FreeQ[{a, b, c, d, n}, x] && PolyQ[Px, x] && ILtQ[m, -1] && GtQ[Expon[Px, x], 2]
  1622. Int[Times[Pattern[Px, Blank[]], Power[Plus[Optional[Pattern[a, Blank[]]], Times[Optional[Pattern[b, Blank[]]], Pattern[x, Blank[]]]], Pattern[m, Blank[]]], Power[Plus[Optional[Pattern[c, Blank[]]], Times[Optional[Pattern[d, Blank[]]], Pattern[x, Blank[]]]], Optional[Pattern[n, Blank[]]]]], Pattern[x, Blank[Symbol]]] := With[{Qx = PolynomialQuotient[Px, a + b*x, x], R = PolynomialRemainder[Px, a + b*x, x]}, Simp[(R*(a + b*x)^(m + 1)*(c + d*x)^(n + 1))/((m + 1)*(b*c - a*d)), x] + Dist[1/((m + 1)*(b*c - a*d)), Int[(a + b*x)^(m + 1)*(c + d*x)^n*ExpandToSum[(m + 1)*(b*c - a*d)*Qx - d*R*(m + n + 2), x], x], x]] /; FreeQ[{a, b, c, d, n}, x] && PolyQ[Px, x] && LtQ[m, -1] && GtQ[Expon[Px, x], 2]
  1623. Int[Times[Pattern[Px, Blank[]], Power[Plus[Optional[Pattern[a, Blank[]]], Times[Optional[Pattern[b, Blank[]]], Pattern[x, Blank[]]]], Optional[Pattern[m, Blank[]]]], Power[Plus[Optional[Pattern[c, Blank[]]], Times[Optional[Pattern[d, Blank[]]], Pattern[x, Blank[]]]], Optional[Pattern[n, Blank[]]]]], Pattern[x, Blank[Symbol]]] := With[{q = Expon[Px, x], k = Coeff[Px, x, Expon[Px, x]]}, Simp[(k*(a + b*x)^(m + q)*(c + d*x)^(n + 1))/(d*b^q*(m + n + q + 1)), x] + Dist[1/(d*b^q*(m + n + q + 1)), Int[(a + b*x)^m*(c + d*x)^n*ExpandToSum[d*b^q*(m + n + q + 1)*Px - d*k*(m + n + q + 1)*(a + b*x)^q - k*(b*c - a*d)*(m + q)*(a + b*x)^(q - 1), x], x], x] /; NeQ[m + n + q + 1, 0]] /; FreeQ[{a, b, c, d, m, n}, x] && PolyQ[Px, x] && GtQ[Expon[Px, x], 2]
  1624. Int[Times[Pattern[Pq, Blank[]], Power[Plus[Optional[Pattern[d, Blank[]]], Times[Optional[Pattern[e, Blank[]]], Pattern[x, Blank[]]]], Optional[Pattern[m, Blank[]]]], Power[Plus[Optional[Pattern[a, Blank[]]], Times[Optional[Pattern[b, Blank[]]], Pattern[x, Blank[]]], Times[Optional[Pattern[c, Blank[]]], Power[Pattern[x, Blank[]], 2]]], Optional[Pattern[p, Blank[]]]]], Pattern[x, Blank[Symbol]]] := Int[(d + e*x)^(m + 1)*PolynomialQuotient[Pq, d + e*x, x]*(a + b*x + c*x^2)^p, x] /; FreeQ[{a, b, c, d, e, m, p}, x] && PolyQ[Pq, x] && EqQ[PolynomialRemainder[Pq, d + e*x, x], 0]
  1625. Int[Times[Pattern[Pq, Blank[]], Power[Plus[Pattern[d, Blank[]], Times[Optional[Pattern[e, Blank[]]], Pattern[x, Blank[]]]], Optional[Pattern[m, Blank[]]]], Power[Plus[Pattern[a, Blank[]], Times[Optional[Pattern[c, Blank[]]], Power[Pattern[x, Blank[]], 2]]], Optional[Pattern[p, Blank[]]]]], Pattern[x, Blank[Symbol]]] := Int[(d + e*x)^(m + 1)*PolynomialQuotient[Pq, d + e*x, x]*(a + c*x^2)^p, x] /; FreeQ[{a, c, d, e, m, p}, x] && PolyQ[Pq, x] && EqQ[PolynomialRemainder[Pq, d + e*x, x], 0]
  1626. Int[Times[Pattern[P2, Blank[]], Power[Plus[Optional[Pattern[d, Blank[]]], Times[Optional[Pattern[e, Blank[]]], Pattern[x, Blank[]]]], Optional[Pattern[m, Blank[]]]], Power[Plus[Optional[Pattern[a, Blank[]]], Times[Optional[Pattern[b, Blank[]]], Pattern[x, Blank[]]], Times[Optional[Pattern[c, Blank[]]], Power[Pattern[x, Blank[]], 2]]], Optional[Pattern[p, Blank[]]]]], Pattern[x, Blank[Symbol]]] := With[{f = Coeff[P2, x, 0], g = Coeff[P2, x, 1], h = Coeff[P2, x, 2]}, Simp[(h*(d + e*x)^(m + 1)*(a + b*x + c*x^2)^(p + 1))/(c*e*(m + 2*p + 3)), x] /; EqQ[b*e*h*(m + p + 2) + 2*c*d*h*(p + 1) - c*e*g*(m + 2*p + 3), 0] && EqQ[b*d*h*(p + 1) + a*e*h*(m + 1) - c*e*f*(m + 2*p + 3), 0]] /; FreeQ[{a, b, c, d, e, m, p}, x] && PolyQ[P2, x, 2] && NeQ[m + 2*p + 3, 0]
  1627. Int[Times[Pattern[P2, Blank[]], Power[Plus[Pattern[d, Blank[]], Times[Optional[Pattern[e, Blank[]]], Pattern[x, Blank[]]]], Optional[Pattern[m, Blank[]]]], Power[Plus[Pattern[a, Blank[]], Times[Optional[Pattern[c, Blank[]]], Power[Pattern[x, Blank[]], 2]]], Optional[Pattern[p, Blank[]]]]], Pattern[x, Blank[Symbol]]] := With[{f = Coeff[P2, x, 0], g = Coeff[P2, x, 1], h = Coeff[P2, x, 2]}, Simp[(h*(d + e*x)^(m + 1)*(a + c*x^2)^(p + 1))/(c*e*(m + 2*p + 3)), x] /; EqQ[2*d*h*(p + 1) - e*g*(m + 2*p + 3), 0] && EqQ[a*h*(m + 1) - c*f*(m + 2*p + 3), 0]] /; FreeQ[{a, c, d, e, m, p}, x] && PolyQ[P2, x, 2] && NeQ[m + 2*p + 3, 0]
  1628. Int[Times[Pattern[Pq, Blank[]], Power[Plus[Optional[Pattern[d, Blank[]]], Times[Optional[Pattern[e, Blank[]]], Pattern[x, Blank[]]]], Optional[Pattern[m, Blank[]]]], Power[Plus[Optional[Pattern[a, Blank[]]], Times[Optional[Pattern[b, Blank[]]], Pattern[x, Blank[]]], Times[Optional[Pattern[c, Blank[]]], Power[Pattern[x, Blank[]], 2]]], Optional[Pattern[p, Blank[]]]]], Pattern[x, Blank[Symbol]]] := Int[ExpandIntegrand[(d + e*x)^m*Pq*(a + b*x + c*x^2)^p, x], x] /; FreeQ[{a, b, c, d, e, m}, x] && PolyQ[Pq, x] && IGtQ[p, -2]
  1629. Int[Times[Pattern[Pq, Blank[]], Power[Plus[Pattern[d, Blank[]], Times[Optional[Pattern[e, Blank[]]], Pattern[x, Blank[]]]], Optional[Pattern[m, Blank[]]]], Power[Plus[Pattern[a, Blank[]], Times[Optional[Pattern[c, Blank[]]], Power[Pattern[x, Blank[]], 2]]], Optional[Pattern[p, Blank[]]]]], Pattern[x, Blank[Symbol]]] := Int[ExpandIntegrand[(d + e*x)^m*Pq*(a + c*x^2)^p, x], x] /; FreeQ[{a, c, d, e, m}, x] && PolyQ[Pq, x] && IGtQ[p, -2]
  1630. Int[Times[Pattern[Pq, Blank[]], Power[Plus[Optional[Pattern[d, Blank[]]], Times[Optional[Pattern[e, Blank[]]], Pattern[x, Blank[]]]], Optional[Pattern[m, Blank[]]]], Power[Plus[Pattern[a, Blank[]], Times[Optional[Pattern[b, Blank[]]], Pattern[x, Blank[]]], Times[Optional[Pattern[c, Blank[]]], Power[Pattern[x, Blank[]], 2]]], Pattern[p, Blank[]]]], Pattern[x, Blank[Symbol]]] := Dist[(a + b*x + c*x^2)^FracPart[p]/((4*c)^IntPart[p]*(b + 2*c*x)^(2*FracPart[p])), Int[(d + e*x)^m*Pq*(b + 2*c*x)^(2*p), x], x] /; FreeQ[{a, b, c, d, e, m, p}, x] && PolyQ[Pq, x] && EqQ[b^2 - 4*a*c, 0]
  1631. Int[Times[Pattern[Pq, Blank[]], Power[Times[Optional[Pattern[e, Blank[]]], Pattern[x, Blank[]]], Optional[Pattern[m, Blank[]]]], Power[Plus[Times[Optional[Pattern[b, Blank[]]], Pattern[x, Blank[]]], Times[Optional[Pattern[c, Blank[]]], Power[Pattern[x, Blank[]], 2]]], Optional[Pattern[p, Blank[]]]]], Pattern[x, Blank[Symbol]]] := Dist[e, Int[(e*x)^(m - 1)*PolynomialQuotient[Pq, b + c*x, x]*(b*x + c*x^2)^(p + 1), x], x] /; FreeQ[{b, c, e, m, p}, x] && PolyQ[Pq, x] && EqQ[PolynomialRemainder[Pq, b + c*x, x], 0]
  1632. Int[Times[Pattern[Pq, Blank[]], Power[Plus[Pattern[d, Blank[]], Times[Optional[Pattern[e, Blank[]]], Pattern[x, Blank[]]]], Optional[Pattern[m, Blank[]]]], Power[Plus[Optional[Pattern[a, Blank[]]], Times[Optional[Pattern[b, Blank[]]], Pattern[x, Blank[]]], Times[Optional[Pattern[c, Blank[]]], Power[Pattern[x, Blank[]], 2]]], Optional[Pattern[p, Blank[]]]]], Pattern[x, Blank[Symbol]]] := Dist[d*e, Int[(d + e*x)^(m - 1)*PolynomialQuotient[Pq, a*e + c*d*x, x]*(a + b*x + c*x^2)^(p + 1), x], x] /; FreeQ[{a, b, c, d, e, m, p}, x] && PolyQ[Pq, x] && NeQ[b^2 - 4*a*c, 0] && EqQ[c*d^2 - b*d*e + a*e^2, 0] && EqQ[PolynomialRemainder[Pq, a*e + c*d*x, x], 0]
  1633. Int[Times[Pattern[Pq, Blank[]], Power[Plus[Pattern[d, Blank[]], Times[Optional[Pattern[e, Blank[]]], Pattern[x, Blank[]]]], Optional[Pattern[m, Blank[]]]], Power[Plus[Pattern[a, Blank[]], Times[Optional[Pattern[c, Blank[]]], Power[Pattern[x, Blank[]], 2]]], Optional[Pattern[p, Blank[]]]]], Pattern[x, Blank[Symbol]]] := Dist[d*e, Int[(d + e*x)^(m - 1)*PolynomialQuotient[Pq, a*e + c*d*x, x]*(a + c*x^2)^(p + 1), x], x] /; FreeQ[{a, c, d, e, m, p}, x] && PolyQ[Pq, x] && EqQ[c*d^2 + a*e^2, 0] && EqQ[PolynomialRemainder[Pq, a*e + c*d*x, x], 0]
  1634. Int[Times[Pattern[Pq, Blank[]], Power[Plus[Optional[Pattern[d, Blank[]]], Times[Optional[Pattern[e, Blank[]]], Pattern[x, Blank[]]]], Optional[Pattern[m, Blank[]]]], Power[Plus[Optional[Pattern[a, Blank[]]], Times[Optional[Pattern[b, Blank[]]], Pattern[x, Blank[]]], Times[Optional[Pattern[c, Blank[]]], Power[Pattern[x, Blank[]], 2]]], Pattern[p, Blank[]]]], Pattern[x, Blank[Symbol]]] := With[{Q = PolynomialQuotient[Pq, a*e + c*d*x, x], f = PolynomialRemainder[Pq, a*e + c*d*x, x]}, Simp[(f*(2*c*d - b*e)*(d + e*x)^m*(a + b*x + c*x^2)^(p + 1))/(e*(p + 1)*(b^2 - 4*a*c)), x] + Dist[1/((p + 1)*(b^2 - 4*a*c)), Int[(d + e*x)^(m - 1)*(a + b*x + c*x^2)^(p + 1)*ExpandToSum[d*e*(p + 1)*(b^2 - 4*a*c)*Q - f*(2*c*d - b*e)*(m + 2*p + 2), x], x], x]] /; FreeQ[{a, b, c, d, e}, x] && PolyQ[Pq, x] && NeQ[b^2 - 4*a*c, 0] && EqQ[c*d^2 - b*d*e + a*e^2, 0] && ILtQ[p + 1/2, 0] && GtQ[m, 0]
  1635. Int[Times[Pattern[Pq, Blank[]], Power[Plus[Pattern[d, Blank[]], Times[Optional[Pattern[e, Blank[]]], Pattern[x, Blank[]]]], Optional[Pattern[m, Blank[]]]], Power[Plus[Pattern[a, Blank[]], Times[Optional[Pattern[c, Blank[]]], Power[Pattern[x, Blank[]], 2]]], Pattern[p, Blank[]]]], Pattern[x, Blank[Symbol]]] := With[{Q = PolynomialQuotient[Pq, a*e + c*d*x, x], f = PolynomialRemainder[Pq, a*e + c*d*x, x]}, -Simp[(d*f*(d + e*x)^m*(a + c*x^2)^(p + 1))/(2*a*e*(p + 1)), x] + Dist[d/(2*a*(p + 1)), Int[(d + e*x)^(m - 1)*(a + c*x^2)^(p + 1)*ExpandToSum[2*a*e*(p + 1)*Q + f*(m + 2*p + 2), x], x], x]] /; FreeQ[{a, c, d, e}, x] && PolyQ[Pq, x] && EqQ[c*d^2 + a*e^2, 0] && ILtQ[p + 1/2, 0] && GtQ[m, 0]
  1636. Int[Times[Pattern[Pq, Blank[]], Power[Plus[Optional[Pattern[d, Blank[]]], Times[Optional[Pattern[e, Blank[]]], Pattern[x, Blank[]]]], Pattern[m, Blank[]]], Power[Plus[Optional[Pattern[a, Blank[]]], Times[Optional[Pattern[b, Blank[]]], Pattern[x, Blank[]]], Times[Optional[Pattern[c, Blank[]]], Power[Pattern[x, Blank[]], 2]]], Pattern[p, Blank[]]]], Pattern[x, Blank[Symbol]]] := Int[ExpandIntegrand[(a + b*x + c*x^2)^p, (d + e*x)^m*Pq, x], x] /; FreeQ[{a, b, c, d, e}, x] && PolyQ[Pq, x] && NeQ[b^2 - 4*a*c, 0] && EqQ[c*d^2 - b*d*e + a*e^2, 0] && EqQ[m + Expon[Pq, x] + 2*p + 1, 0] && ILtQ[m, 0]
  1637. Int[Times[Pattern[Pq, Blank[]], Power[Plus[Pattern[d, Blank[]], Times[Optional[Pattern[e, Blank[]]], Pattern[x, Blank[]]]], Pattern[m, Blank[]]], Power[Plus[Pattern[a, Blank[]], Times[Optional[Pattern[c, Blank[]]], Power[Pattern[x, Blank[]], 2]]], Pattern[p, Blank[]]]], Pattern[x, Blank[Symbol]]] := Int[ExpandIntegrand[(a + c*x^2)^p, (d + e*x)^m*Pq, x], x] /; FreeQ[{a, c, d, e}, x] && PolyQ[Pq, x] && EqQ[c*d^2 + a*e^2, 0] && EqQ[m + Expon[Pq, x] + 2*p + 1, 0] && ILtQ[m, 0]
  1638. Int[Times[Pattern[Pq, Blank[]], Power[Plus[Optional[Pattern[d, Blank[]]], Times[Optional[Pattern[e, Blank[]]], Pattern[x, Blank[]]]], Optional[Pattern[m, Blank[]]]], Power[Plus[Optional[Pattern[a, Blank[]]], Times[Optional[Pattern[b, Blank[]]], Pattern[x, Blank[]]], Times[Optional[Pattern[c, Blank[]]], Power[Pattern[x, Blank[]], 2]]], Pattern[p, Blank[]]]], Pattern[x, Blank[Symbol]]] := With[{q = Expon[Pq, x], f = Coeff[Pq, x, Expon[Pq, x]]}, Simp[(f*(d + e*x)^(m + q - 1)*(a + b*x + c*x^2)^(p + 1))/(c*e^(q - 1)*(m + q + 2*p + 1)), x] + Dist[1/(c*e^q*(m + q + 2*p + 1)), Int[(d + e*x)^m*(a + b*x + c*x^2)^p*ExpandToSum[c*e^q*(m + q + 2*p + 1)*Pq - c*f*(m + q + 2*p + 1)*(d + e*x)^q + e*f*(m + p + q)*(d + e*x)^(q - 2)*(b*d - 2*a*e + (2*c*d - b*e)*x), x], x], x] /; NeQ[m + q + 2*p + 1, 0]] /; FreeQ[{a, b, c, d, e, m, p}, x] && PolyQ[Pq, x] && NeQ[b^2 - 4*a*c, 0] && EqQ[c*d^2 - b*d*e + a*e^2, 0]
  1639. Int[Times[Pattern[Pq, Blank[]], Power[Plus[Pattern[d, Blank[]], Times[Optional[Pattern[e, Blank[]]], Pattern[x, Blank[]]]], Optional[Pattern[m, Blank[]]]], Power[Plus[Pattern[a, Blank[]], Times[Optional[Pattern[c, Blank[]]], Power[Pattern[x, Blank[]], 2]]], Pattern[p, Blank[]]]], Pattern[x, Blank[Symbol]]] := With[{q = Expon[Pq, x], f = Coeff[Pq, x, Expon[Pq, x]]}, Simp[(f*(d + e*x)^(m + q - 1)*(a + c*x^2)^(p + 1))/(c*e^(q - 1)*(m + q + 2*p + 1)), x] + Dist[1/(c*e^q*(m + q + 2*p + 1)), Int[(d + e*x)^m*(a + c*x^2)^p*ExpandToSum[c*e^q*(m + q + 2*p + 1)*Pq - c*f*(m + q + 2*p + 1)*(d + e*x)^q - 2*e*f*(m + p + q)*(d + e*x)^(q - 2)*(a*e - c*d*x), x], x], x] /; NeQ[m + q + 2*p + 1, 0]] /; FreeQ[{a, c, d, e, m, p}, x] && PolyQ[Pq, x] && EqQ[c*d^2 + a*e^2, 0] && !IGtQ[m, 0]
  1640. Int[Times[Pattern[Pq, Blank[]], Power[Plus[Optional[Pattern[d, Blank[]]], Times[Optional[Pattern[e, Blank[]]], Pattern[x, Blank[]]]], Optional[Pattern[m, Blank[]]]], Power[Plus[Optional[Pattern[a, Blank[]]], Times[Optional[Pattern[b, Blank[]]], Pattern[x, Blank[]]], Times[Optional[Pattern[c, Blank[]]], Power[Pattern[x, Blank[]], 2]]], Optional[Pattern[p, Blank[]]]]], Pattern[x, Blank[Symbol]]] := Int[(d + e*x)^(m + p)*(a/d + (c*x)/e)^p*Pq, x] /; FreeQ[{a, b, c, d, e, m}, x] && PolyQ[Pq, x] && NeQ[b^2 - 4*a*c, 0] && EqQ[c*d^2 - b*d*e + a*e^2, 0] && IntegerQ[p]
  1641. Int[Times[Pattern[Pq, Blank[]], Power[Plus[Pattern[d, Blank[]], Times[Optional[Pattern[e, Blank[]]], Pattern[x, Blank[]]]], Optional[Pattern[m, Blank[]]]], Power[Plus[Pattern[a, Blank[]], Times[Optional[Pattern[c, Blank[]]], Power[Pattern[x, Blank[]], 2]]], Optional[Pattern[p, Blank[]]]]], Pattern[x, Blank[Symbol]]] := Int[(d + e*x)^(m + p)*(a/d + (c*x)/e)^p*Pq, x] /; FreeQ[{a, c, d, e, m}, x] && PolyQ[Pq, x] && EqQ[c*d^2 + a*e^2, 0] && IntegerQ[p]
  1642. Int[Times[Pattern[Pq, Blank[]], Power[Plus[Optional[Pattern[d, Blank[]]], Times[Optional[Pattern[e, Blank[]]], Pattern[x, Blank[]]]], Optional[Pattern[m, Blank[]]]], Power[Plus[Optional[Pattern[a, Blank[]]], Times[Optional[Pattern[b, Blank[]]], Pattern[x, Blank[]]], Times[Optional[Pattern[c, Blank[]]], Power[Pattern[x, Blank[]], 2]]], Pattern[p, Blank[]]]], Pattern[x, Blank[Symbol]]] := Dist[(a + b*x + c*x^2)^FracPart[p]/((d + e*x)^FracPart[p]*(a/d + (c*x)/e)^FracPart[p]), Int[(d + e*x)^(m + p)*(a/d + (c*x)/e)^p*Pq, x], x] /; FreeQ[{a, b, c, d, e, m, p}, x] && PolyQ[Pq, x] && NeQ[b^2 - 4*a*c, 0] && EqQ[c*d^2 - b*d*e + a*e^2, 0] && !IntegerQ[p] && !IGtQ[m, 0]
  1643. Int[Times[Pattern[Pq, Blank[]], Power[Plus[Pattern[d, Blank[]], Times[Optional[Pattern[e, Blank[]]], Pattern[x, Blank[]]]], Optional[Pattern[m, Blank[]]]], Power[Plus[Pattern[a, Blank[]], Times[Optional[Pattern[c, Blank[]]], Power[Pattern[x, Blank[]], 2]]], Pattern[p, Blank[]]]], Pattern[x, Blank[Symbol]]] := Dist[(a + c*x^2)^FracPart[p]/((d + e*x)^FracPart[p]*(a/d + (c*x)/e)^FracPart[p]), Int[(d + e*x)^(m + p)*(a/d + (c*x)/e)^p*Pq, x], x] /; FreeQ[{a, c, d, e, m, p}, x] && PolyQ[Pq, x] && EqQ[c*d^2 + a*e^2, 0] && !IntegerQ[p] && !IGtQ[m, 0]
  1644. Int[Times[Pattern[Pq, Blank[]], Power[Plus[Optional[Pattern[d, Blank[]]], Times[Optional[Pattern[e, Blank[]]], Pattern[x, Blank[]]]], Optional[Pattern[m, Blank[]]]], Power[Plus[Optional[Pattern[a, Blank[]]], Times[Optional[Pattern[b, Blank[]]], Pattern[x, Blank[]]], Times[Optional[Pattern[c, Blank[]]], Power[Pattern[x, Blank[]], 2]]], Pattern[p, Blank[]]]], Pattern[x, Blank[Symbol]]] := With[{Q = PolynomialQuotient[Pq, a + b*x + c*x^2, x], f = Coeff[PolynomialRemainder[Pq, a + b*x + c*x^2, x], x, 0], g = Coeff[PolynomialRemainder[Pq, a + b*x + c*x^2, x], x, 1]}, Simp[((d + e*x)^m*(a + b*x + c*x^2)^(p + 1)*(f*b - 2*a*g + (2*c*f - b*g)*x))/((p + 1)*(b^2 - 4*a*c)), x] + Dist[1/((p + 1)*(b^2 - 4*a*c)), Int[(d + e*x)^(m - 1)*(a + b*x + c*x^2)^(p + 1)*ExpandToSum[(p + 1)*(b^2 - 4*a*c)*(d + e*x)*Q + g*(2*a*e*m + b*d*(2*p + 3)) - f*(b*e*m + 2*c*d*(2*p + 3)) - e*(2*c*f - b*g)*(m + 2*p + 3)*x, x], x], x]] /; FreeQ[{a, b, c, d, e}, x] && PolyQ[Pq, x] && NeQ[b^2 - 4*a*c, 0] && NeQ[c*d^2 - b*d*e + a*e^2, 0] && LtQ[p, -1] && GtQ[m, 0] && (IntegerQ[p] || !IntegerQ[m] || !RationalQ[a, b, c, d, e]) && !(IGtQ[m, 0] && RationalQ[a, b, c, d, e] && (IntegerQ[p] || ILtQ[p + 1/2, 0]))
  1645. Int[Times[Pattern[Pq, Blank[]], Power[Plus[Pattern[d, Blank[]], Times[Optional[Pattern[e, Blank[]]], Pattern[x, Blank[]]]], Optional[Pattern[m, Blank[]]]], Power[Plus[Pattern[a, Blank[]], Times[Optional[Pattern[c, Blank[]]], Power[Pattern[x, Blank[]], 2]]], Pattern[p, Blank[]]]], Pattern[x, Blank[Symbol]]] := With[{Q = PolynomialQuotient[Pq, a + c*x^2, x], f = Coeff[PolynomialRemainder[Pq, a + c*x^2, x], x, 0], g = Coeff[PolynomialRemainder[Pq, a + c*x^2, x], x, 1]}, Simp[((d + e*x)^m*(a + c*x^2)^(p + 1)*(a*g - c*f*x))/(2*a*c*(p + 1)), x] + Dist[1/(2*a*c*(p + 1)), Int[(d + e*x)^(m - 1)*(a + c*x^2)^(p + 1)*ExpandToSum[2*a*c*(p + 1)*(d + e*x)*Q - a*e*g*m + c*d*f*(2*p + 3) + c*e*f*(m + 2*p + 3)*x, x], x], x]] /; FreeQ[{a, c, d, e}, x] && PolyQ[Pq, x] && NeQ[c*d^2 + a*e^2, 0] && LtQ[p, -1] && GtQ[m, 0] && !(IGtQ[m, 0] && RationalQ[a, c, d, e] && (IntegerQ[p] || ILtQ[p + 1/2, 0]))
  1646. Int[Times[Pattern[Pq, Blank[]], Power[Plus[Optional[Pattern[d, Blank[]]], Times[Optional[Pattern[e, Blank[]]], Pattern[x, Blank[]]]], Optional[Pattern[m, Blank[]]]], Power[Plus[Optional[Pattern[a, Blank[]]], Times[Optional[Pattern[b, Blank[]]], Pattern[x, Blank[]]], Times[Optional[Pattern[c, Blank[]]], Power[Pattern[x, Blank[]], 2]]], Pattern[p, Blank[]]]], Pattern[x, Blank[Symbol]]] := With[{Q = PolynomialQuotient[(d + e*x)^m*Pq, a + b*x + c*x^2, x], f = Coeff[PolynomialRemainder[(d + e*x)^m*Pq, a + b*x + c*x^2, x], x, 0], g = Coeff[PolynomialRemainder[(d + e*x)^m*Pq, a + b*x + c*x^2, x], x, 1]}, Simp[((b*f - 2*a*g + (2*c*f - b*g)*x)*(a + b*x + c*x^2)^(p + 1))/((p + 1)*(b^2 - 4*a*c)), x] + Dist[1/((p + 1)*(b^2 - 4*a*c)), Int[(d + e*x)^m*(a + b*x + c*x^2)^(p + 1)*ExpandToSum[((p + 1)*(b^2 - 4*a*c)*Q)/(d + e*x)^m - ((2*p + 3)*(2*c*f - b*g))/(d + e*x)^m, x], x], x]] /; FreeQ[{a, b, c, d, e}, x] && PolyQ[Pq, x] && NeQ[b^2 - 4*a*c, 0] && NeQ[c*d^2 - b*d*e + a*e^2, 0] && LtQ[p, -1] && ILtQ[m, 0]
  1647. Int[Times[Pattern[Pq, Blank[]], Power[Plus[Pattern[d, Blank[]], Times[Optional[Pattern[e, Blank[]]], Pattern[x, Blank[]]]], Optional[Pattern[m, Blank[]]]], Power[Plus[Pattern[a, Blank[]], Times[Optional[Pattern[c, Blank[]]], Power[Pattern[x, Blank[]], 2]]], Pattern[p, Blank[]]]], Pattern[x, Blank[Symbol]]] := With[{Q = PolynomialQuotient[(d + e*x)^m*Pq, a + c*x^2, x], f = Coeff[PolynomialRemainder[(d + e*x)^m*Pq, a + c*x^2, x], x, 0], g = Coeff[PolynomialRemainder[(d + e*x)^m*Pq, a + c*x^2, x], x, 1]}, Simp[((a*g - c*f*x)*(a + c*x^2)^(p + 1))/(2*a*c*(p + 1)), x] + Dist[1/(2*a*c*(p + 1)), Int[(d + e*x)^m*(a + c*x^2)^(p + 1)*ExpandToSum[(2*a*c*(p + 1)*Q)/(d + e*x)^m + (c*f*(2*p + 3))/(d + e*x)^m, x], x], x]] /; FreeQ[{a, c, d, e}, x] && PolyQ[Pq, x] && NeQ[c*d^2 + a*e^2, 0] && LtQ[p, -1] && ILtQ[m, 0]
  1648. Int[Times[Pattern[Pq, Blank[]], Power[Plus[Optional[Pattern[d, Blank[]]], Times[Optional[Pattern[e, Blank[]]], Pattern[x, Blank[]]]], Optional[Pattern[m, Blank[]]]], Power[Plus[Optional[Pattern[a, Blank[]]], Times[Optional[Pattern[b, Blank[]]], Pattern[x, Blank[]]], Times[Optional[Pattern[c, Blank[]]], Power[Pattern[x, Blank[]], 2]]], Pattern[p, Blank[]]]], Pattern[x, Blank[Symbol]]] := With[{Q = PolynomialQuotient[Pq, a + b*x + c*x^2, x], f = Coeff[PolynomialRemainder[Pq, a + b*x + c*x^2, x], x, 0], g = Coeff[PolynomialRemainder[Pq, a + b*x + c*x^2, x], x, 1]}, Simp[((d + e*x)^(m + 1)*(a + b*x + c*x^2)^(p + 1)*(f*(b*c*d - b^2*e + 2*a*c*e) - a*g*(2*c*d - b*e) + c*(f*(2*c*d - b*e) - g*(b*d - 2*a*e))*x))/((p + 1)*(b^2 - 4*a*c)*(c*d^2 - b*d*e + a*e^2)), x] + Dist[1/((p + 1)*(b^2 - 4*a*c)*(c*d^2 - b*d*e + a*e^2)), Int[(d + e*x)^m*(a + b*x + c*x^2)^(p + 1)*ExpandToSum[(p + 1)*(b^2 - 4*a*c)*(c*d^2 - b*d*e + a*e^2)*Q + f*(b*c*d*e*(2*p - m + 2) + b^2*e^2*(p + m + 2) - 2*c^2*d^2*(2*p + 3) - 2*a*c*e^2*(m + 2*p + 3)) - g*(a*e*(b*e - 2*c*d*m + b*e*m) - b*d*(3*c*d - b*e + 2*c*d*p - b*e*p)) + c*e*(g*(b*d - 2*a*e) - f*(2*c*d - b*e))*(m + 2*p + 4)*x, x], x], x]] /; FreeQ[{a, b, c, d, e, m}, x] && PolyQ[Pq, x] && NeQ[b^2 - 4*a*c, 0] && NeQ[c*d^2 - b*d*e + a*e^2, 0] && LtQ[p, -1] && !(IGtQ[m, 0] && RationalQ[a, b, c, d, e] && (IntegerQ[p] || ILtQ[p + 1/2, 0]))
  1649. Int[Times[Pattern[Pq, Blank[]], Power[Plus[Pattern[d, Blank[]], Times[Optional[Pattern[e, Blank[]]], Pattern[x, Blank[]]]], Optional[Pattern[m, Blank[]]]], Power[Plus[Pattern[a, Blank[]], Times[Optional[Pattern[c, Blank[]]], Power[Pattern[x, Blank[]], 2]]], Pattern[p, Blank[]]]], Pattern[x, Blank[Symbol]]] := With[{Q = PolynomialQuotient[Pq, a + c*x^2, x], f = Coeff[PolynomialRemainder[Pq, a + c*x^2, x], x, 0], g = Coeff[PolynomialRemainder[Pq, a + c*x^2, x], x, 1]}, -Simp[((d + e*x)^(m + 1)*(a + c*x^2)^(p + 1)*(a*(e*f - d*g) + (c*d*f + a*e*g)*x))/(2*a*(p + 1)*(c*d^2 + a*e^2)), x] + Dist[1/(2*a*(p + 1)*(c*d^2 + a*e^2)), Int[(d + e*x)^m*(a + c*x^2)^(p + 1)*ExpandToSum[2*a*(p + 1)*(c*d^2 + a*e^2)*Q + c*d^2*f*(2*p + 3) - a*e*(d*g*m - e*f*(m + 2*p + 3)) + e*(c*d*f + a*e*g)*(m + 2*p + 4)*x, x], x], x]] /; FreeQ[{a, c, d, e, m}, x] && PolyQ[Pq, x] && NeQ[c*d^2 + a*e^2, 0] && LtQ[p, -1] && !(IGtQ[m, 0] && RationalQ[a, c, d, e] && (IntegerQ[p] || ILtQ[p + 1/2, 0]))
  1650. Int[Times[Pattern[Pq, Blank[]], Power[Plus[Optional[Pattern[d, Blank[]]], Times[Optional[Pattern[e, Blank[]]], Pattern[x, Blank[]]]], Pattern[m, Blank[]]], Power[Plus[Optional[Pattern[a, Blank[]]], Times[Optional[Pattern[b, Blank[]]], Pattern[x, Blank[]]], Times[Optional[Pattern[c, Blank[]]], Power[Pattern[x, Blank[]], 2]]], Pattern[p, Blank[]]]], Pattern[x, Blank[Symbol]]] := With[{Q = PolynomialQuotient[Pq, d + e*x, x], R = PolynomialRemainder[Pq, d + e*x, x]}, Simp[(e*R*(d + e*x)^(m + 1)*(a + b*x + c*x^2)^(p + 1))/((m + 1)*(c*d^2 - b*d*e + a*e^2)), x] + Dist[1/((m + 1)*(c*d^2 - b*d*e + a*e^2)), Int[(d + e*x)^(m + 1)*(a + b*x + c*x^2)^p*ExpandToSum[(m + 1)*(c*d^2 - b*d*e + a*e^2)*Q + c*d*R*(m + 1) - b*e*R*(m + p + 2) - c*e*R*(m + 2*p + 3)*x, x], x], x]] /; FreeQ[{a, b, c, d, e, p}, x] && PolyQ[Pq, x] && NeQ[b^2 - 4*a*c, 0] && NeQ[c*d^2 - b*d*e + a*e^2, 0] && LtQ[m, -1]
  1651. Int[Times[Pattern[Pq, Blank[]], Power[Plus[Pattern[d, Blank[]], Times[Optional[Pattern[e, Blank[]]], Pattern[x, Blank[]]]], Pattern[m, Blank[]]], Power[Plus[Pattern[a, Blank[]], Times[Optional[Pattern[c, Blank[]]], Power[Pattern[x, Blank[]], 2]]], Pattern[p, Blank[]]]], Pattern[x, Blank[Symbol]]] := With[{Q = PolynomialQuotient[Pq, d + e*x, x], R = PolynomialRemainder[Pq, d + e*x, x]}, Simp[(e*R*(d + e*x)^(m + 1)*(a + c*x^2)^(p + 1))/((m + 1)*(c*d^2 + a*e^2)), x] + Dist[1/((m + 1)*(c*d^2 + a*e^2)), Int[(d + e*x)^(m + 1)*(a + c*x^2)^p*ExpandToSum[(m + 1)*(c*d^2 + a*e^2)*Q + c*d*R*(m + 1) - c*e*R*(m + 2*p + 3)*x, x], x], x]] /; FreeQ[{a, c, d, e, p}, x] && PolyQ[Pq, x] && NeQ[c*d^2 + a*e^2, 0] && LtQ[m, -1]
  1652. Int[Times[Pattern[Pq, Blank[]], Power[Pattern[x, Blank[]], Optional[Pattern[m, Blank[]]]], Power[Plus[Pattern[a, Blank[]], Times[Optional[Pattern[b, Blank[]]], Power[Pattern[x, Blank[]], 2]]], Pattern[p, Blank[]]]], Pattern[x, Blank[Symbol]]] := Module[{q = Expon[Pq, x], k}, Int[x^m*Sum[Coeff[Pq, x, 2*k]*x^(2*k), {k, 0, q/2}]*(a + b*x^2)^p, x] + Int[x^(m + 1)*Sum[Coeff[Pq, x, 2*k + 1]*x^(2*k), {k, 0, (q - 1)/2}]*(a + b*x^2)^p, x]] /; FreeQ[{a, b, p}, x] && PolyQ[Pq, x] && !PolyQ[Pq, x^2] && IGtQ[m, -2] && !IntegerQ[2*p]
  1653. Int[Times[Pattern[Pq, Blank[]], Power[Plus[Optional[Pattern[d, Blank[]]], Times[Optional[Pattern[e, Blank[]]], Pattern[x, Blank[]]]], Optional[Pattern[m, Blank[]]]], Power[Plus[Optional[Pattern[a, Blank[]]], Times[Optional[Pattern[b, Blank[]]], Pattern[x, Blank[]]], Times[Optional[Pattern[c, Blank[]]], Power[Pattern[x, Blank[]], 2]]], Pattern[p, Blank[]]]], Pattern[x, Blank[Symbol]]] := With[{q = Expon[Pq, x], f = Coeff[Pq, x, Expon[Pq, x]]}, Simp[(f*(d + e*x)^(m + q - 1)*(a + b*x + c*x^2)^(p + 1))/(c*e^(q - 1)*(m + q + 2*p + 1)), x] + Dist[1/(c*e^q*(m + q + 2*p + 1)), Int[(d + e*x)^m*(a + b*x + c*x^2)^p*ExpandToSum[c*e^q*(m + q + 2*p + 1)*Pq - c*f*(m + q + 2*p + 1)*(d + e*x)^q - f*(d + e*x)^(q - 2)*(b*d*e*(p + 1) + a*e^2*(m + q - 1) - c*d^2*(m + q + 2*p + 1) - e*(2*c*d - b*e)*(m + q + p)*x), x], x], x] /; GtQ[q, 1] && NeQ[m + q + 2*p + 1, 0]] /; FreeQ[{a, b, c, d, e, m, p}, x] && PolyQ[Pq, x] && NeQ[b^2 - 4*a*c, 0] && NeQ[c*d^2 - b*d*e + a*e^2, 0] && !(IGtQ[m, 0] && RationalQ[a, b, c, d, e] && (IntegerQ[p] || ILtQ[p + 1/2, 0]))
  1654. Int[Times[Pattern[Pq, Blank[]], Power[Plus[Pattern[d, Blank[]], Times[Optional[Pattern[e, Blank[]]], Pattern[x, Blank[]]]], Optional[Pattern[m, Blank[]]]], Power[Plus[Pattern[a, Blank[]], Times[Optional[Pattern[c, Blank[]]], Power[Pattern[x, Blank[]], 2]]], Pattern[p, Blank[]]]], Pattern[x, Blank[Symbol]]] := With[{q = Expon[Pq, x], f = Coeff[Pq, x, Expon[Pq, x]]}, Simp[(f*(d + e*x)^(m + q - 1)*(a + c*x^2)^(p + 1))/(c*e^(q - 1)*(m + q + 2*p + 1)), x] + Dist[1/(c*e^q*(m + q + 2*p + 1)), Int[(d + e*x)^m*(a + c*x^2)^p*ExpandToSum[c*e^q*(m + q + 2*p + 1)*Pq - c*f*(m + q + 2*p + 1)*(d + e*x)^q - f*(d + e*x)^(q - 2)*(a*e^2*(m + q - 1) - c*d^2*(m + q + 2*p + 1) - 2*c*d*e*(m + q + p)*x), x], x], x] /; GtQ[q, 1] && NeQ[m + q + 2*p + 1, 0]] /; FreeQ[{a, c, d, e, m, p}, x] && PolyQ[Pq, x] && NeQ[c*d^2 + a*e^2, 0] && !(EqQ[d, 0] && True) && !(IGtQ[m, 0] && RationalQ[a, c, d, e] && (IntegerQ[p] || ILtQ[p + 1/2, 0]))
  1655. Int[Times[Pattern[Pq, Blank[]], Power[Plus[Optional[Pattern[d, Blank[]]], Times[Optional[Pattern[e, Blank[]]], Pattern[x, Blank[]]]], Optional[Pattern[m, Blank[]]]], Power[Plus[Optional[Pattern[a, Blank[]]], Times[Optional[Pattern[b, Blank[]]], Pattern[x, Blank[]]], Times[Optional[Pattern[c, Blank[]]], Power[Pattern[x, Blank[]], 2]]], Optional[Pattern[p, Blank[]]]]], Pattern[x, Blank[Symbol]]] := With[{q = Expon[Pq, x]}, Dist[Coeff[Pq, x, q]/e^q, Int[(d + e*x)^(m + q)*(a + b*x + c*x^2)^p, x], x] + Dist[1/e^q, Int[(d + e*x)^m*(a + b*x + c*x^2)^p*ExpandToSum[e^q*Pq - Coeff[Pq, x, q]*(d + e*x)^q, x], x], x]] /; FreeQ[{a, b, c, d, e, m, p}, x] && PolyQ[Pq, x] && NeQ[b^2 - 4*a*c, 0] && NeQ[c*d^2 - b*d*e + a*e^2, 0] && !(IGtQ[m, 0] && RationalQ[a, b, c, d, e] && (IntegerQ[p] || ILtQ[p + 1/2, 0]))
  1656. Int[Times[Pattern[Pq, Blank[]], Power[Plus[Pattern[d, Blank[]], Times[Optional[Pattern[e, Blank[]]], Pattern[x, Blank[]]]], Optional[Pattern[m, Blank[]]]], Power[Plus[Pattern[a, Blank[]], Times[Optional[Pattern[c, Blank[]]], Power[Pattern[x, Blank[]], 2]]], Optional[Pattern[p, Blank[]]]]], Pattern[x, Blank[Symbol]]] := With[{q = Expon[Pq, x]}, Dist[Coeff[Pq, x, q]/e^q, Int[(d + e*x)^(m + q)*(a + c*x^2)^p, x], x] + Dist[1/e^q, Int[(d + e*x)^m*(a + c*x^2)^p*ExpandToSum[e^q*Pq - Coeff[Pq, x, q]*(d + e*x)^q, x], x], x]] /; FreeQ[{a, c, d, e, m, p}, x] && PolyQ[Pq, x] && NeQ[c*d^2 + a*e^2, 0] && !(IGtQ[m, 0] && RationalQ[a, c, d, e] && (IntegerQ[p] || ILtQ[p + 1/2, 0]))
  1657. Int[Times[Pattern[Pq, Blank[]], Power[Plus[Pattern[a, Blank[]], Times[Optional[Pattern[b, Blank[]]], Pattern[x, Blank[]]], Times[Optional[Pattern[c, Blank[]]], Power[Pattern[x, Blank[]], 2]]], Optional[Pattern[p, Blank[]]]]], Pattern[x, Blank[Symbol]]] := Int[ExpandIntegrand[Pq*(a + b*x + c*x^2)^p, x], x] /; FreeQ[{a, b, c}, x] && PolyQ[Pq, x] && IGtQ[p, -2]
  1658. Int[Times[Pattern[Pq, Blank[]], Power[Plus[Optional[Pattern[a, Blank[]]], Times[Optional[Pattern[b, Blank[]]], Pattern[x, Blank[]]], Times[Optional[Pattern[c, Blank[]]], Power[Pattern[x, Blank[]], 2]]], Pattern[p, Blank[]]]], Pattern[x, Blank[Symbol]]] := Int[x*PolynomialQuotient[Pq, x, x]*(a + b*x + c*x^2)^p, x] /; FreeQ[{a, b, c, p}, x] && PolyQ[Pq, x] && EqQ[Coeff[Pq, x, 0], 0] && !MatchQ[Pq, x^(m_.)*(u_.) /; IntegerQ[m]]
  1659. Int[Times[Pattern[Pq, Blank[]], Power[Plus[Pattern[a, Blank[]], Times[Optional[Pattern[b, Blank[]]], Pattern[x, Blank[]]], Times[Optional[Pattern[c, Blank[]]], Power[Pattern[x, Blank[]], 2]]], Pattern[p, Blank[]]]], Pattern[x, Blank[Symbol]]] := Dist[(a + b*x + c*x^2)^FracPart[p]/((4*c)^IntPart[p]*(b + 2*c*x)^(2*FracPart[p])), Int[Pq*(b + 2*c*x)^(2*p), x], x] /; FreeQ[{a, b, c, p}, x] && PolyQ[Pq, x] && EqQ[b^2 - 4*a*c, 0]
  1660. Int[Times[Pattern[Pq, Blank[]], Power[Plus[Optional[Pattern[a, Blank[]]], Times[Optional[Pattern[b, Blank[]]], Pattern[x, Blank[]]], Times[Optional[Pattern[c, Blank[]]], Power[Pattern[x, Blank[]], 2]]], Pattern[p, Blank[]]]], Pattern[x, Blank[Symbol]]] := With[{Q = PolynomialQuotient[Pq, a + b*x + c*x^2, x], f = Coeff[PolynomialRemainder[Pq, a + b*x + c*x^2, x], x, 0], g = Coeff[PolynomialRemainder[Pq, a + b*x + c*x^2, x], x, 1]}, Simp[((b*f - 2*a*g + (2*c*f - b*g)*x)*(a + b*x + c*x^2)^(p + 1))/((p + 1)*(b^2 - 4*a*c)), x] + Dist[1/((p + 1)*(b^2 - 4*a*c)), Int[(a + b*x + c*x^2)^(p + 1)*ExpandToSum[(p + 1)*(b^2 - 4*a*c)*Q - (2*p + 3)*(2*c*f - b*g), x], x], x]] /; FreeQ[{a, b, c}, x] && PolyQ[Pq, x] && NeQ[b^2 - 4*a*c, 0] && LtQ[p, -1]
  1661. Int[Times[Pattern[Pq, Blank[]], Power[Plus[Optional[Pattern[a, Blank[]]], Times[Optional[Pattern[b, Blank[]]], Pattern[x, Blank[]]], Times[Optional[Pattern[c, Blank[]]], Power[Pattern[x, Blank[]], 2]]], Pattern[p, Blank[]]]], Pattern[x, Blank[Symbol]]] := With[{q = Expon[Pq, x], e = Coeff[Pq, x, Expon[Pq, x]]}, Simp[(e*x^(q - 1)*(a + b*x + c*x^2)^(p + 1))/(c*(q + 2*p + 1)), x] + Dist[1/(c*(q + 2*p + 1)), Int[(a + b*x + c*x^2)^p*ExpandToSum[c*(q + 2*p + 1)*Pq - a*e*(q - 1)*x^(q - 2) - b*e*(q + p)*x^(q - 1) - c*e*(q + 2*p + 1)*x^q, x], x], x]] /; FreeQ[{a, b, c, p}, x] && PolyQ[Pq, x] && NeQ[b^2 - 4*a*c, 0] && !LeQ[p, -1]
  1662. Int[Times[Pattern[Pq, Blank[]], Power[Times[Optional[Pattern[d, Blank[]]], Pattern[x, Blank[]]], Optional[Pattern[m, Blank[]]]], Power[Plus[Pattern[a, Blank[]], Times[Optional[Pattern[b, Blank[]]], Power[Pattern[x, Blank[]], 2]], Times[Optional[Pattern[c, Blank[]]], Power[Pattern[x, Blank[]], 4]]], Pattern[p, Blank[]]]], Pattern[x, Blank[Symbol]]] := Module[{q = Expon[Pq, x], k}, Int[(d*x)^m*Sum[Coeff[Pq, x, 2*k]*x^(2*k), {k, 0, q/2 + 1}]*(a + b*x^2 + c*x^4)^p, x] + Dist[1/d, Int[(d*x)^(m + 1)*Sum[Coeff[Pq, x, 2*k + 1]*x^(2*k), {k, 0, (q - 1)/2 + 1}]*(a + b*x^2 + c*x^4)^p, x], x]] /; FreeQ[{a, b, c, d, m, p}, x] && PolyQ[Pq, x] && !PolyQ[Pq, x^2]
  1663. Int[Times[Pattern[Pq, Blank[]], Power[Pattern[x, Blank[]], Optional[Pattern[m, Blank[]]]], Power[Plus[Pattern[a, Blank[]], Times[Optional[Pattern[b, Blank[]]], Power[Pattern[x, Blank[]], 2]], Times[Optional[Pattern[c, Blank[]]], Power[Pattern[x, Blank[]], 4]]], Pattern[p, Blank[]]]], Pattern[x, Blank[Symbol]]] := Dist[1/2, Subst[Int[x^((m - 1)/2)*SubstFor[x^2, Pq, x]*(a + b*x + c*x^2)^p, x], x, x^2], x] /; FreeQ[{a, b, c, p}, x] && PolyQ[Pq, x^2] && IntegerQ[(m - 1)/2]
  1664. Int[Times[Pattern[Pq, Blank[]], Power[Times[Optional[Pattern[d, Blank[]]], Pattern[x, Blank[]]], Optional[Pattern[m, Blank[]]]], Power[Plus[Pattern[a, Blank[]], Times[Optional[Pattern[b, Blank[]]], Power[Pattern[x, Blank[]], 2]], Times[Optional[Pattern[c, Blank[]]], Power[Pattern[x, Blank[]], 4]]], Optional[Pattern[p, Blank[]]]]], Pattern[x, Blank[Symbol]]] := Int[ExpandIntegrand[(d*x)^m*Pq*(a + b*x^2 + c*x^4)^p, x], x] /; FreeQ[{a, b, c, d, m}, x] && PolyQ[Pq, x^2] && IGtQ[p, -2]
  1665. Int[Times[Pattern[Pq, Blank[]], Power[Times[Optional[Pattern[d, Blank[]]], Pattern[x, Blank[]]], Optional[Pattern[m, Blank[]]]], Power[Plus[Pattern[a, Blank[]], Times[Optional[Pattern[b, Blank[]]], Power[Pattern[x, Blank[]], 2]], Times[Optional[Pattern[c, Blank[]]], Power[Pattern[x, Blank[]], 4]]], Pattern[p, Blank[]]]], Pattern[x, Blank[Symbol]]] := Dist[1/d^2, Int[(d*x)^(m + 2)*ExpandToSum[Pq/x^2, x]*(a + b*x^2 + c*x^4)^p, x], x] /; FreeQ[{a, b, c, d, m, p}, x] && PolyQ[Pq, x^2] && EqQ[Coeff[Pq, x, 0], 0]
  1666. Int[Times[Pattern[Pq, Blank[]], Power[Times[Optional[Pattern[d, Blank[]]], Pattern[x, Blank[]]], Optional[Pattern[m, Blank[]]]], Power[Plus[Pattern[a, Blank[]], Times[Optional[Pattern[b, Blank[]]], Power[Pattern[x, Blank[]], 2]], Times[Optional[Pattern[c, Blank[]]], Power[Pattern[x, Blank[]], 4]]], Optional[Pattern[p, Blank[]]]]], Pattern[x, Blank[Symbol]]] := With[{e = Coeff[Pq, x, 0], f = Coeff[Pq, x, 2], g = Coeff[Pq, x, 4]}, Simp[(e*(d*x)^(m + 1)*(a + b*x^2 + c*x^4)^(p + 1))/(a*d*(m + 1)), x] /; EqQ[a*f*(m + 1) - b*e*(m + 2*p + 3), 0] && EqQ[a*g*(m + 1) - c*e*(m + 4*p + 5), 0] && NeQ[m, -1]] /; FreeQ[{a, b, c, d, m, p}, x] && PolyQ[Pq, x^2] && EqQ[Expon[Pq, x], 4]
  1667. Int[Times[Pattern[Pq, Blank[]], Power[Times[Optional[Pattern[d, Blank[]]], Pattern[x, Blank[]]], Optional[Pattern[m, Blank[]]]], Power[Plus[Pattern[a, Blank[]], Times[Optional[Pattern[b, Blank[]]], Power[Pattern[x, Blank[]], 2]], Times[Optional[Pattern[c, Blank[]]], Power[Pattern[x, Blank[]], 4]]], Pattern[p, Blank[]]]], Pattern[x, Blank[Symbol]]] := Dist[(a + b*x^2 + c*x^4)^FracPart[p]/((4*c)^IntPart[p]*(b + 2*c*x^2)^(2*FracPart[p])), Int[(d*x)^m*Pq*(b + 2*c*x^2)^(2*p), x], x] /; FreeQ[{a, b, c, d, m, p}, x] && PolyQ[Pq, x^2] && GtQ[Expon[Pq, x^2], 1] && EqQ[b^2 - 4*a*c, 0]
  1668. Int[Times[Pattern[Pq, Blank[]], Power[Pattern[x, Blank[]], Pattern[m, Blank[]]], Power[Plus[Pattern[a, Blank[]], Times[Optional[Pattern[b, Blank[]]], Power[Pattern[x, Blank[]], 2]], Times[Optional[Pattern[c, Blank[]]], Power[Pattern[x, Blank[]], 4]]], Pattern[p, Blank[]]]], Pattern[x, Blank[Symbol]]] := With[{d = Coeff[PolynomialRemainder[x^m*Pq, a + b*x^2 + c*x^4, x], x, 0], e = Coeff[PolynomialRemainder[x^m*Pq, a + b*x^2 + c*x^4, x], x, 2]}, Simp[(x*(a + b*x^2 + c*x^4)^(p + 1)*(a*b*e - d*(b^2 - 2*a*c) - c*(b*d - 2*a*e)*x^2))/(2*a*(p + 1)*(b^2 - 4*a*c)), x] + Dist[1/(2*a*(p + 1)*(b^2 - 4*a*c)), Int[(a + b*x^2 + c*x^4)^(p + 1)*ExpandToSum[2*a*(p + 1)*(b^2 - 4*a*c)*PolynomialQuotient[x^m*Pq, a + b*x^2 + c*x^4, x] + b^2*d*(2*p + 3) - 2*a*c*d*(4*p + 5) - a*b*e + c*(4*p + 7)*(b*d - 2*a*e)*x^2, x], x], x]] /; FreeQ[{a, b, c}, x] && PolyQ[Pq, x^2] && GtQ[Expon[Pq, x^2], 1] && NeQ[b^2 - 4*a*c, 0] && LtQ[p, -1] && IGtQ[m/2, 0]
  1669. Int[Times[Pattern[Pq, Blank[]], Power[Pattern[x, Blank[]], Pattern[m, Blank[]]], Power[Plus[Pattern[a, Blank[]], Times[Optional[Pattern[b, Blank[]]], Power[Pattern[x, Blank[]], 2]], Times[Optional[Pattern[c, Blank[]]], Power[Pattern[x, Blank[]], 4]]], Pattern[p, Blank[]]]], Pattern[x, Blank[Symbol]]] := With[{d = Coeff[PolynomialRemainder[x^m*Pq, a + b*x^2 + c*x^4, x], x, 0], e = Coeff[PolynomialRemainder[x^m*Pq, a + b*x^2 + c*x^4, x], x, 2]}, Simp[(x*(a + b*x^2 + c*x^4)^(p + 1)*(a*b*e - d*(b^2 - 2*a*c) - c*(b*d - 2*a*e)*x^2))/(2*a*(p + 1)*(b^2 - 4*a*c)), x] + Dist[1/(2*a*(p + 1)*(b^2 - 4*a*c)), Int[x^m*(a + b*x^2 + c*x^4)^(p + 1)*ExpandToSum[(2*a*(p + 1)*(b^2 - 4*a*c)*PolynomialQuotient[x^m*Pq, a + b*x^2 + c*x^4, x])/x^m + (b^2*d*(2*p + 3) - 2*a*c*d*(4*p + 5) - a*b*e)/x^m + c*(4*p + 7)*(b*d - 2*a*e)*x^(2 - m), x], x], x]] /; FreeQ[{a, b, c}, x] && PolyQ[Pq, x^2] && GtQ[Expon[Pq, x^2], 1] && NeQ[b^2 - 4*a*c, 0] && LtQ[p, -1] && ILtQ[m/2, 0]
  1670. Int[Times[Pattern[Pq, Blank[]], Power[Times[Optional[Pattern[d, Blank[]]], Pattern[x, Blank[]]], Optional[Pattern[m, Blank[]]]], Power[Plus[Pattern[a, Blank[]], Times[Optional[Pattern[b, Blank[]]], Power[Pattern[x, Blank[]], 2]], Times[Optional[Pattern[c, Blank[]]], Power[Pattern[x, Blank[]], 4]]], Optional[Pattern[p, Blank[]]]]], Pattern[x, Blank[Symbol]]] := Unintegrable[Pq*(d*x)^m*(a + b*x^2 + c*x^4)^p, x] /; FreeQ[{a, b, c, d, m, p}, x] && PolyQ[Pq, x]
  1671. Int[Times[Pattern[Pq, Blank[]], Power[Plus[Pattern[a, Blank[]], Times[Optional[Pattern[b, Blank[]]], Power[Pattern[x, Blank[]], 2]], Times[Optional[Pattern[c, Blank[]]], Power[Pattern[x, Blank[]], 4]]], Optional[Pattern[p, Blank[]]]]], Pattern[x, Blank[Symbol]]] := Int[ExpandIntegrand[Pq*(a + b*x^2 + c*x^4)^p, x], x] /; FreeQ[{a, b, c}, x] && PolyQ[Pq, x] && IGtQ[p, 0]
  1672. Int[Times[Pattern[Pq, Blank[]], Power[Plus[Pattern[a, Blank[]], Times[Optional[Pattern[b, Blank[]]], Power[Pattern[x, Blank[]], 2]], Times[Optional[Pattern[c, Blank[]]], Power[Pattern[x, Blank[]], 4]]], Pattern[p, Blank[]]]], Pattern[x, Blank[Symbol]]] := Int[x*PolynomialQuotient[Pq, x, x]*(a + b*x^2 + c*x^4)^p, x] /; FreeQ[{a, b, c, p}, x] && PolyQ[Pq, x] && EqQ[Coeff[Pq, x, 0], 0] && !MatchQ[Pq, x^(m_.)*(u_.) /; IntegerQ[m]]
  1673. Int[Times[Pattern[Pq, Blank[]], Power[Plus[Pattern[a, Blank[]], Times[Optional[Pattern[b, Blank[]]], Power[Pattern[x, Blank[]], 2]], Times[Optional[Pattern[c, Blank[]]], Power[Pattern[x, Blank[]], 4]]], Pattern[p, Blank[]]]], Pattern[x, Blank[Symbol]]] := Module[{q = Expon[Pq, x], k}, Int[Sum[Coeff[Pq, x, 2*k]*x^(2*k), {k, 0, q/2}]*(a + b*x^2 + c*x^4)^p, x] + Int[x*Sum[Coeff[Pq, x, 2*k + 1]*x^(2*k), {k, 0, (q - 1)/2}]*(a + b*x^2 + c*x^4)^p, x]] /; FreeQ[{a, b, c, p}, x] && PolyQ[Pq, x] && !PolyQ[Pq, x^2]
  1674. Int[Times[Pattern[Pq, Blank[]], Power[Plus[Pattern[a, Blank[]], Times[Optional[Pattern[b, Blank[]]], Power[Pattern[x, Blank[]], 2]], Times[Optional[Pattern[c, Blank[]]], Power[Pattern[x, Blank[]], 4]]], Optional[Pattern[p, Blank[]]]]], Pattern[x, Blank[Symbol]]] := With[{d = Coeff[Pq, x, 0], e = Coeff[Pq, x, 2], f = Coeff[Pq, x, 4]}, Simp[(d*x*(a + b*x^2 + c*x^4)^(p + 1))/a, x] /; EqQ[a*e - b*d*(2*p + 3), 0] && EqQ[a*f - c*d*(4*p + 5), 0]] /; FreeQ[{a, b, c, p}, x] && PolyQ[Pq, x^2] && EqQ[Expon[Pq, x], 4]
  1675. Int[Times[Pattern[Pq, Blank[]], Power[Plus[Pattern[a, Blank[]], Times[Optional[Pattern[b, Blank[]]], Power[Pattern[x, Blank[]], 2]], Times[Optional[Pattern[c, Blank[]]], Power[Pattern[x, Blank[]], 4]]], Optional[Pattern[p, Blank[]]]]], Pattern[x, Blank[Symbol]]] := With[{d = Coeff[Pq, x, 0], e = Coeff[Pq, x, 2], f = Coeff[Pq, x, 4], g = Coeff[Pq, x, 6]}, Simp[(x*(3*a*d + (a*e - b*d*(2*p + 3))*x^2)*(a + b*x^2 + c*x^4)^(p + 1))/(3*a^2), x] /; EqQ[3*a^2*g - c*(4*p + 7)*(a*e - b*d*(2*p + 3)), 0] && EqQ[3*a^2*f - 3*a*c*d*(4*p + 5) - b*(2*p + 5)*(a*e - b*d*(2*p + 3)), 0]] /; FreeQ[{a, b, c, p}, x] && PolyQ[Pq, x^2] && EqQ[Expon[Pq, x], 6]
  1676. Int[Times[Pattern[Pq, Blank[]], Power[Plus[Pattern[a, Blank[]], Times[Optional[Pattern[b, Blank[]]], Power[Pattern[x, Blank[]], 2]], Times[Optional[Pattern[c, Blank[]]], Power[Pattern[x, Blank[]], 4]]], -1]], Pattern[x, Blank[Symbol]]] := Int[ExpandIntegrand[Pq/(a + b*x^2 + c*x^4), x], x] /; FreeQ[{a, b, c}, x] && PolyQ[Pq, x^2] && Expon[Pq, x^2] > 1
  1677. Int[Times[Pattern[Pq, Blank[]], Power[Plus[Pattern[a, Blank[]], Times[Optional[Pattern[b, Blank[]]], Power[Pattern[x, Blank[]], 2]], Times[Optional[Pattern[c, Blank[]]], Power[Pattern[x, Blank[]], 4]]], Pattern[p, Blank[]]]], Pattern[x, Blank[Symbol]]] := Dist[(a + b*x^2 + c*x^4)^FracPart[p]/((4*c)^IntPart[p]*(b + 2*c*x^2)^(2*FracPart[p])), Int[Pq*(b + 2*c*x^2)^(2*p), x], x] /; FreeQ[{a, b, c, p}, x] && PolyQ[Pq, x^2] && Expon[Pq, x^2] > 1 && EqQ[b^2 - 4*a*c, 0]
  1678. Int[Times[Pattern[Pq, Blank[]], Power[Plus[Pattern[a, Blank[]], Times[Optional[Pattern[b, Blank[]]], Power[Pattern[x, Blank[]], 2]], Times[Optional[Pattern[c, Blank[]]], Power[Pattern[x, Blank[]], 4]]], Pattern[p, Blank[]]]], Pattern[x, Blank[Symbol]]] := With[{d = Coeff[PolynomialRemainder[Pq, a + b*x^2 + c*x^4, x], x, 0], e = Coeff[PolynomialRemainder[Pq, a + b*x^2 + c*x^4, x], x, 2]}, Simp[(x*(a + b*x^2 + c*x^4)^(p + 1)*(a*b*e - d*(b^2 - 2*a*c) - c*(b*d - 2*a*e)*x^2))/(2*a*(p + 1)*(b^2 - 4*a*c)), x] + Dist[1/(2*a*(p + 1)*(b^2 - 4*a*c)), Int[(a + b*x^2 + c*x^4)^(p + 1)*ExpandToSum[2*a*(p + 1)*(b^2 - 4*a*c)*PolynomialQuotient[Pq, a + b*x^2 + c*x^4, x] + b^2*d*(2*p + 3) - 2*a*c*d*(4*p + 5) - a*b*e + c*(4*p + 7)*(b*d - 2*a*e)*x^2, x], x], x]] /; FreeQ[{a, b, c}, x] && PolyQ[Pq, x^2] && Expon[Pq, x^2] > 1 && NeQ[b^2 - 4*a*c, 0] && LtQ[p, -1]
  1679. Int[Times[Pattern[Pq, Blank[]], Power[Plus[Pattern[a, Blank[]], Times[Optional[Pattern[b, Blank[]]], Power[Pattern[x, Blank[]], 2]], Times[Optional[Pattern[c, Blank[]]], Power[Pattern[x, Blank[]], 4]]], Pattern[p, Blank[]]]], Pattern[x, Blank[Symbol]]] := With[{q = Expon[Pq, x^2], e = Coeff[Pq, x^2, Expon[Pq, x^2]]}, Simp[(e*x^(2*q - 3)*(a + b*x^2 + c*x^4)^(p + 1))/(c*(2*q + 4*p + 1)), x] + Dist[1/(c*(2*q + 4*p + 1)), Int[(a + b*x^2 + c*x^4)^p*ExpandToSum[c*(2*q + 4*p + 1)*Pq - a*e*(2*q - 3)*x^(2*q - 4) - b*e*(2*q + 2*p - 1)*x^(2*q - 2) - c*e*(2*q + 4*p + 1)*x^(2*q), x], x], x]] /; FreeQ[{a, b, c, p}, x] && PolyQ[Pq, x^2] && Expon[Pq, x^2] > 1 && NeQ[b^2 - 4*a*c, 0] && !LtQ[p, -1]
  1680. Int[Times[Pattern[Pq, Blank[]], Power[Pattern[Q4, Blank[]], Pattern[p, Blank[]]]], Pattern[x, Blank[Symbol]]] := With[{a = Coeff[Q4, x, 0], b = Coeff[Q4, x, 1], c = Coeff[Q4, x, 2], d = Coeff[Q4, x, 3], e = Coeff[Q4, x, 4]}, Subst[Int[SimplifyIntegrand[(Pq /. x -> -(d/(4*e)) + x)*(a + d^4/(256*e^3) - (b*d)/(8*e) + (c - (3*d^2)/(8*e))*x^2 + e*x^4)^p, x], x], x, d/(4*e) + x] /; EqQ[d^3 - 4*c*d*e + 8*b*e^2, 0] && NeQ[d, 0]] /; FreeQ[p, x] && PolyQ[Pq, x] && PolyQ[Q4, x, 4] && !IGtQ[p, 0]
  1681. Int[Times[Pattern[Px, Blank[]], Power[Pattern[x, Blank[]], Pattern[m, Blank[]]], Power[Plus[Pattern[d, Blank[]], Times[Optional[Pattern[e, Blank[]]], Power[Pattern[x, Blank[]], 2]]], -1], Power[Plus[Pattern[a, Blank[]], Times[Optional[Pattern[b, Blank[]]], Power[Pattern[x, Blank[]], 2]], Times[Optional[Pattern[c, Blank[]]], Power[Pattern[x, Blank[]], 4]]], Rational[-1, 2]]], Pattern[x, Blank[Symbol]]] := With[{A = Coeff[Px, x, 0], B = Coeff[Px, x, 2], C = Coeff[Px, x, 4]}, Simp[(C*x^(m - 1)*Sqrt[a + b*x^2 + c*x^4])/(c*e*(m + 1)), x] - Dist[1/(c*e*(m + 1)), Int[(x^(m - 2)/((d + e*x^2)*Sqrt[a + b*x^2 + c*x^4]))*Simp[a*C*d*(m - 1) - (A*c*e*(m + 1) - C*(a*e*(m - 1) + b*d*m))*x^2 - (B*c*e*(m + 1) - C*(b*e*m + c*d*(m + 1)))*x^4, x], x], x]] /; FreeQ[{a, b, c, d, e}, x] && PolyQ[Px, x^2, 2] && NeQ[b^2 - 4*a*c, 0] && IGtQ[m/2, 0]
  1682. Int[Times[Pattern[Px, Blank[]], Power[Pattern[x, Blank[]], Pattern[m, Blank[]]], Power[Plus[Pattern[d, Blank[]], Times[Optional[Pattern[e, Blank[]]], Power[Pattern[x, Blank[]], 2]]], -1], Power[Plus[Pattern[a, Blank[]], Times[Optional[Pattern[c, Blank[]]], Power[Pattern[x, Blank[]], 4]]], Rational[-1, 2]]], Pattern[x, Blank[Symbol]]] := With[{A = Coeff[Px, x, 0], B = Coeff[Px, x, 2], C = Coeff[Px, x, 4]}, Simp[(C*x^(m - 1)*Sqrt[a + c*x^4])/(c*e*(m + 1)), x] - Dist[1/(c*e*(m + 1)), Int[(x^(m - 2)/((d + e*x^2)*Sqrt[a + c*x^4]))*Simp[a*C*d*(m - 1) - (A*c*e*(m + 1) - C*a*e*(m - 1))*x^2 - (B*c*e*(m + 1) - C*c*d*(m + 1))*x^4, x], x], x]] /; FreeQ[{a, c, d, e}, x] && PolyQ[Px, x^2, 2] && IGtQ[m/2, 0]
  1683. Int[Times[Pattern[Px, Blank[]], Power[Pattern[x, Blank[]], Pattern[m, Blank[]]], Power[Plus[Pattern[d, Blank[]], Times[Optional[Pattern[e, Blank[]]], Power[Pattern[x, Blank[]], 2]]], -1], Power[Plus[Pattern[a, Blank[]], Times[Optional[Pattern[b, Blank[]]], Power[Pattern[x, Blank[]], 2]], Times[Optional[Pattern[c, Blank[]]], Power[Pattern[x, Blank[]], 4]]], Rational[-1, 2]]], Pattern[x, Blank[Symbol]]] := With[{A = Coeff[Px, x, 0], B = Coeff[Px, x, 2], C = Coeff[Px, x, 4]}, Simp[(A*x^(m + 1)*Sqrt[a + b*x^2 + c*x^4])/(a*d*(m + 1)), x] + Dist[1/(a*d*(m + 1)), Int[(x^(m + 2)/((d + e*x^2)*Sqrt[a + b*x^2 + c*x^4]))*Simp[a*B*d*(m + 1) - A*(a*e*(m + 1) + b*d*(m + 2)) + (a*C*d*(m + 1) - A*(b*e*(m + 2) + c*d*(m + 3)))*x^2 - A*c*e*(m + 3)*x^4, x], x], x]] /; FreeQ[{a, b, c, d, e}, x] && PolyQ[Px, x^2, 2] && NeQ[b^2 - 4*a*c, 0] && ILtQ[m/2, 0]
  1684. Int[Times[Pattern[Px, Blank[]], Power[Pattern[x, Blank[]], Pattern[m, Blank[]]], Power[Plus[Pattern[d, Blank[]], Times[Optional[Pattern[e, Blank[]]], Power[Pattern[x, Blank[]], 2]]], -1], Power[Plus[Pattern[a, Blank[]], Times[Optional[Pattern[c, Blank[]]], Power[Pattern[x, Blank[]], 4]]], Rational[-1, 2]]], Pattern[x, Blank[Symbol]]] := With[{A = Coeff[Px, x, 0], B = Coeff[Px, x, 2], C = Coeff[Px, x, 4]}, Simp[(A*x^(m + 1)*Sqrt[a + c*x^4])/(a*d*(m + 1)), x] + Dist[1/(a*d*(m + 1)), Int[(x^(m + 2)/((d + e*x^2)*Sqrt[a + c*x^4]))*Simp[a*B*d*(m + 1) - A*a*e*(m + 1) + (a*C*d*(m + 1) - A*c*d*(m + 3))*x^2 - A*c*e*(m + 3)*x^4, x], x], x]] /; FreeQ[{a, c, d, e}, x] && PolyQ[Px, x^2, 2] && ILtQ[m/2, 0]
  1685. Int[Times[Pattern[Px, Blank[]], Pattern[x, Blank[]], Power[Plus[Pattern[d, Blank[]], Times[Optional[Pattern[e, Blank[]]], Power[Pattern[x, Blank[]], 2]]], Optional[Pattern[q, Blank[]]]], Power[Plus[Pattern[a, Blank[]], Times[Optional[Pattern[b, Blank[]]], Power[Pattern[x, Blank[]], 2]], Times[Optional[Pattern[c, Blank[]]], Power[Pattern[x, Blank[]], 4]]], Optional[Pattern[p, Blank[]]]]], Pattern[x, Blank[Symbol]]] := Dist[1/2, Subst[Int[(Px /. x -> Sqrt[x])*(d + e*x)^q*(a + b*x + c*x^2)^p, x], x, x^2], x] /; FreeQ[{a, b, c, d, e, p, q}, x] && PolyQ[Px, x^2]
  1686. Int[Times[Pattern[Pr, Blank[]], Power[Plus[Pattern[d, Blank[]], Times[Optional[Pattern[e, Blank[]]], Power[Pattern[x, Blank[]], 2]]], Optional[Pattern[q, Blank[]]]], Power[Plus[Pattern[a, Blank[]], Times[Optional[Pattern[b, Blank[]]], Power[Pattern[x, Blank[]], 2]], Times[Optional[Pattern[c, Blank[]]], Power[Pattern[x, Blank[]], 4]]], Pattern[p, Blank[]]]], Pattern[x, Blank[Symbol]]] := Int[x*PolynomialQuotient[Pr, x, x]*(d + e*x^2)^q*(a + b*x^2 + c*x^4)^p, x] /; FreeQ[{a, b, c, d, e, p, q}, x] && PolyQ[Pr, x] && EqQ[PolynomialRemainder[Pr, x, x], 0] && !MatchQ[Pr, x^(m_.)*(u_.) /; IntegerQ[m]]
  1687. Int[Times[Pattern[Pr, Blank[]], Power[Plus[Pattern[d, Blank[]], Times[Optional[Pattern[e, Blank[]]], Power[Pattern[x, Blank[]], 2]]], Optional[Pattern[q, Blank[]]]], Power[Plus[Pattern[a, Blank[]], Times[Optional[Pattern[b, Blank[]]], Power[Pattern[x, Blank[]], 2]], Times[Optional[Pattern[c, Blank[]]], Power[Pattern[x, Blank[]], 4]]], Pattern[p, Blank[]]]], Pattern[x, Blank[Symbol]]] := Module[{r = Expon[Pr, x], k}, Int[Sum[Coeff[Pr, x, 2*k]*x^(2*k), {k, 0, r/2}]*(d + e*x^2)^q*(a + b*x^2 + c*x^4)^p, x] + Int[x*Sum[Coeff[Pr, x, 2*k + 1]*x^(2*k), {k, 0, (r - 1)/2}]*(d + e*x^2)^q*(a + b*x^2 + c*x^4)^p, x]] /; FreeQ[{a, b, c, d, e, p, q}, x] && PolyQ[Pr, x] && !PolyQ[Pr, x^2]
  1688. Int[Times[Pattern[Px, Blank[]], Power[Plus[Pattern[d, Blank[]], Times[Optional[Pattern[e, Blank[]]], Power[Pattern[x, Blank[]], 2]]], Optional[Pattern[q, Blank[]]]], Power[Plus[Pattern[a, Blank[]], Times[Optional[Pattern[b, Blank[]]], Power[Pattern[x, Blank[]], 2]], Times[Optional[Pattern[c, Blank[]]], Power[Pattern[x, Blank[]], 4]]], Optional[Pattern[p, Blank[]]]]], Pattern[x, Blank[Symbol]]] := Int[Px*(d + e*x^2)^(p + q)*(a/d + (c*x^2)/e)^p, x] /; FreeQ[{a, b, c, d, e, q}, x] && NeQ[b^2 - 4*a*c, 0] && EqQ[c*d^2 - b*d*e + a*e^2, 0] && IntegerQ[p] && (PolyQ[Px, x^2] || MatchQ[Px, ((f_) + (g_.)*x^2)^(r_.) /; FreeQ[{f, g, r}, x]])
  1689. Int[Times[Pattern[Px, Blank[]], Power[Plus[Pattern[d, Blank[]], Times[Optional[Pattern[e, Blank[]]], Power[Pattern[x, Blank[]], 2]]], Optional[Pattern[q, Blank[]]]], Power[Plus[Pattern[a, Blank[]], Times[Optional[Pattern[c, Blank[]]], Power[Pattern[x, Blank[]], 4]]], Optional[Pattern[p, Blank[]]]]], Pattern[x, Blank[Symbol]]] := Int[Px*(d + e*x^2)^(p + q)*(a/d + (c*x^2)/e)^p, x] /; FreeQ[{a, c, d, e, q}, x] && EqQ[c*d^2 + a*e^2, 0] && IntegerQ[p] && (PolyQ[Px, x^2] || MatchQ[Px, ((f_) + (g_.)*x^2)^(r_.) /; FreeQ[{f, g, r}, x]])
  1690. Int[Times[Pattern[Px, Blank[]], Power[Plus[Pattern[d, Blank[]], Times[Optional[Pattern[e, Blank[]]], Power[Pattern[x, Blank[]], 2]]], Optional[Pattern[q, Blank[]]]], Power[Plus[Pattern[a, Blank[]], Times[Optional[Pattern[b, Blank[]]], Power[Pattern[x, Blank[]], 2]], Times[Optional[Pattern[c, Blank[]]], Power[Pattern[x, Blank[]], 4]]], Pattern[p, Blank[]]]], Pattern[x, Blank[Symbol]]] := Dist[(a + b*x^2 + c*x^4)^FracPart[p]/((d + e*x^2)^FracPart[p]*(a/d + (c*x^2)/e)^FracPart[p]), Int[Px*(d + e*x^2)^(p + q)*(a/d + (c*x^2)/e)^p, x], x] /; FreeQ[{a, b, c, d, e, p, q}, x] && NeQ[b^2 - 4*a*c, 0] && EqQ[c*d^2 - b*d*e + a*e^2, 0] && !IntegerQ[p] && (PolyQ[Px, x^2] || MatchQ[Px, ((f_) + (g_.)*x^2)^(r_.) /; FreeQ[{f, g, r}, x]])
  1691. Int[Times[Pattern[Px, Blank[]], Power[Plus[Pattern[d, Blank[]], Times[Optional[Pattern[e, Blank[]]], Power[Pattern[x, Blank[]], 2]]], Optional[Pattern[q, Blank[]]]], Power[Plus[Pattern[a, Blank[]], Times[Optional[Pattern[c, Blank[]]], Power[Pattern[x, Blank[]], 4]]], Pattern[p, Blank[]]]], Pattern[x, Blank[Symbol]]] := Dist[(a + c*x^4)^FracPart[p]/((d + e*x^2)^FracPart[p]*(a/d + (c*x^2)/e)^FracPart[p]), Int[Px*(d + e*x^2)^(p + q)*(a/d + (c*x^2)/e)^p, x], x] /; FreeQ[{a, c, d, e, p, q}, x] && EqQ[c*d^2 + a*e^2, 0] && !IntegerQ[p] && (PolyQ[Px, x^2] || MatchQ[Px, ((f_) + (g_.)*x^2)^(r_.) /; FreeQ[{f, g, r}, x]])
  1692. Int[Times[Pattern[Px, Blank[]], Power[Plus[Pattern[d, Blank[]], Times[Optional[Pattern[e, Blank[]]], Power[Pattern[x, Blank[]], 2]]], Optional[Pattern[q, Blank[]]]], Power[Plus[Pattern[a, Blank[]], Times[Optional[Pattern[b, Blank[]]], Power[Pattern[x, Blank[]], 2]], Times[Optional[Pattern[c, Blank[]]], Power[Pattern[x, Blank[]], 4]]], Optional[Pattern[p, Blank[]]]]], Pattern[x, Blank[Symbol]]] := Int[ExpandIntegrand[Px*(d + e*x^2)^q*(a + b*x^2 + c*x^4)^p, x], x] /; FreeQ[{a, b, c, d, e, q}, x] && PolyQ[Px, x^2] && NeQ[b^2 - 4*a*c, 0] && NeQ[c*d^2 - b*d*e + a*e^2, 0] && IntegerQ[p]
  1693. Int[Times[Pattern[Px, Blank[]], Power[Plus[Pattern[d, Blank[]], Times[Optional[Pattern[e, Blank[]]], Power[Pattern[x, Blank[]], 2]]], Optional[Pattern[q, Blank[]]]], Power[Plus[Pattern[a, Blank[]], Times[Optional[Pattern[c, Blank[]]], Power[Pattern[x, Blank[]], 4]]], Optional[Pattern[p, Blank[]]]]], Pattern[x, Blank[Symbol]]] := Int[ExpandIntegrand[Px*(d + e*x^2)^q*(a + c*x^4)^p, x], x] /; FreeQ[{a, c, d, e, q}, x] && PolyQ[Px, x^2] && NeQ[c*d^2 + a*e^2, 0] && IntegerQ[p]
  1694. Int[Times[Pattern[P4x, Blank[]], Power[Plus[Pattern[d, Blank[]], Times[Optional[Pattern[e, Blank[]]], Power[Pattern[x, Blank[]], 2]]], Pattern[q, Blank[]]], Power[Plus[Pattern[a, Blank[]], Times[Optional[Pattern[b, Blank[]]], Power[Pattern[x, Blank[]], 2]], Times[Optional[Pattern[c, Blank[]]], Power[Pattern[x, Blank[]], 4]]], Rational[-1, 2]]], Pattern[x, Blank[Symbol]]] := With[{A = Coeff[P4x, x, 0], B = Coeff[P4x, x, 2], C = Coeff[P4x, x, 4]}, Simp[(C*x*(d + e*x^2)^q*Sqrt[a + b*x^2 + c*x^4])/(c*(2*q + 3)), x] + Dist[1/(c*(2*q + 3)), Int[((d + e*x^2)^(q - 1)*Simp[A*c*d*(2*q + 3) - a*C*d + (c*(B*d + A*e)*(2*q + 3) - C*(2*b*d + a*e + 2*a*e*q))*x^2 + (B*c*e*(2*q + 3) - 2*C*(b*e - c*d*q + b*e*q))*x^4, x])/Sqrt[a + b*x^2 + c*x^4], x], x]] /; FreeQ[{a, b, c, d, e}, x] && PolyQ[P4x, x^2] && EqQ[Expon[P4x, x], 4] && NeQ[b^2 - 4*a*c, 0] && NeQ[c*d^2 - b*d*e + a*e^2, 0] && IGtQ[q, 0]
  1695. Int[Times[Pattern[P4x, Blank[]], Power[Plus[Pattern[d, Blank[]], Times[Optional[Pattern[e, Blank[]]], Power[Pattern[x, Blank[]], 2]]], Pattern[q, Blank[]]], Power[Plus[Pattern[a, Blank[]], Times[Optional[Pattern[c, Blank[]]], Power[Pattern[x, Blank[]], 4]]], Rational[-1, 2]]], Pattern[x, Blank[Symbol]]] := With[{A = Coeff[P4x, x, 0], B = Coeff[P4x, x, 2], C = Coeff[P4x, x, 4]}, Simp[(C*x*(d + e*x^2)^q*Sqrt[a + c*x^4])/(c*(2*q + 3)), x] + Dist[1/(c*(2*q + 3)), Int[((d + e*x^2)^(q - 1)*Simp[A*c*d*(2*q + 3) - a*C*d + (c*(B*d + A*e)*(2*q + 3) - a*C*e*(2*q + 1))*x^2 + (B*c*e*(2*q + 3) + 2*c*C*d*q)*x^4, x])/Sqrt[a + c*x^4], x], x]] /; FreeQ[{a, c, d, e}, x] && PolyQ[P4x, x^2] && EqQ[Expon[P4x, x], 4] && NeQ[c*d^2 + a*e^2, 0] && IGtQ[q, 0]
  1696. Int[Times[Pattern[P4x, Blank[]], Power[Plus[Pattern[d, Blank[]], Times[Optional[Pattern[e, Blank[]]], Power[Pattern[x, Blank[]], 2]]], Pattern[q, Blank[]]], Power[Plus[Pattern[a, Blank[]], Times[Optional[Pattern[b, Blank[]]], Power[Pattern[x, Blank[]], 2]], Times[Optional[Pattern[c, Blank[]]], Power[Pattern[x, Blank[]], 4]]], Rational[-1, 2]]], Pattern[x, Blank[Symbol]]] := With[{A = Coeff[P4x, x, 0], B = Coeff[P4x, x, 2], C = Coeff[P4x, x, 4]}, -Simp[((C*d^2 - B*d*e + A*e^2)*x*(d + e*x^2)^(q + 1)*Sqrt[a + b*x^2 + c*x^4])/(2*d*(q + 1)*(c*d^2 - b*d*e + a*e^2)), x] + Dist[1/(2*d*(q + 1)*(c*d^2 - b*d*e + a*e^2)), Int[((d + e*x^2)^(q + 1)*Simp[a*d*(C*d - B*e) + A*(a*e^2*(2*q + 3) + 2*d*(c*d - b*e)*(q + 1)) - 2*((B*d - A*e)*(b*e*(q + 2) - c*d*(q + 1)) - C*d*(b*d + a*e*(q + 1)))*x^2 + c*(C*d^2 - B*d*e + A*e^2)*(2*q + 5)*x^4, x])/Sqrt[a + b*x^2 + c*x^4], x], x]] /; FreeQ[{a, b, c, d, e}, x] && PolyQ[P4x, x^2] && LeQ[Expon[P4x, x], 4] && NeQ[b^2 - 4*a*c, 0] && NeQ[c*d^2 - b*d*e + a*e^2, 0] && ILtQ[q, -1]
  1697. Int[Times[Pattern[P4x, Blank[]], Power[Plus[Pattern[d, Blank[]], Times[Optional[Pattern[e, Blank[]]], Power[Pattern[x, Blank[]], 2]]], Pattern[q, Blank[]]], Power[Plus[Pattern[a, Blank[]], Times[Optional[Pattern[c, Blank[]]], Power[Pattern[x, Blank[]], 4]]], Rational[-1, 2]]], Pattern[x, Blank[Symbol]]] := With[{A = Coeff[P4x, x, 0], B = Coeff[P4x, x, 2], C = Coeff[P4x, x, 4]}, -Simp[((C*d^2 - B*d*e + A*e^2)*x*(d + e*x^2)^(q + 1)*Sqrt[a + c*x^4])/(2*d*(q + 1)*(c*d^2 + a*e^2)), x] + Dist[1/(2*d*(q + 1)*(c*d^2 + a*e^2)), Int[((d + e*x^2)^(q + 1)*Simp[a*d*(C*d - B*e) + A*(a*e^2*(2*q + 3) + 2*c*d^2*(q + 1)) + 2*d*(B*c*d - A*c*e + a*C*e)*(q + 1)*x^2 + c*(C*d^2 - B*d*e + A*e^2)*(2*q + 5)*x^4, x])/Sqrt[a + c*x^4], x], x]] /; FreeQ[{a, c, d, e}, x] && PolyQ[P4x, x^2] && LeQ[Expon[P4x, x], 4] && NeQ[c*d^2 + a*e^2, 0] && ILtQ[q, -1]
  1698. Int[Times[Plus[Pattern[A, Blank[]], Times[Optional[Pattern[B, Blank[]]], Power[Pattern[x, Blank[]], 2]]], Power[Plus[Pattern[d, Blank[]], Times[Optional[Pattern[e, Blank[]]], Power[Pattern[x, Blank[]], 2]]], -1], Power[Plus[Pattern[a, Blank[]], Times[Optional[Pattern[b, Blank[]]], Power[Pattern[x, Blank[]], 2]], Times[Optional[Pattern[c, Blank[]]], Power[Pattern[x, Blank[]], 4]]], Rational[-1, 2]]], Pattern[x, Blank[Symbol]]] := Dist[A, Subst[Int[1/(d - (b*d - 2*a*e)*x^2), x], x, x/Sqrt[a + b*x^2 + c*x^4]], x] /; FreeQ[{a, b, c, d, e, A, B}, x] && NeQ[b^2 - 4*a*c, 0] && NeQ[c*d^2 - b*d*e + a*e^2, 0] && EqQ[c*d^2 - a*e^2, 0] && EqQ[B*d + A*e, 0]
  1699. Int[Times[Plus[Pattern[A, Blank[]], Times[Optional[Pattern[B, Blank[]]], Power[Pattern[x, Blank[]], 2]]], Power[Plus[Pattern[d, Blank[]], Times[Optional[Pattern[e, Blank[]]], Power[Pattern[x, Blank[]], 2]]], -1], Power[Plus[Pattern[a, Blank[]], Times[Optional[Pattern[c, Blank[]]], Power[Pattern[x, Blank[]], 4]]], Rational[-1, 2]]], Pattern[x, Blank[Symbol]]] := Dist[A, Subst[Int[1/(d + 2*a*e*x^2), x], x, x/Sqrt[a + c*x^4]], x] /; FreeQ[{a, c, d, e, A, B}, x] && NeQ[c*d^2 + a*e^2, 0] && EqQ[c*d^2 - a*e^2, 0] && EqQ[B*d + A*e, 0]
  1700. Int[Times[Plus[Pattern[A, Blank[]], Times[Optional[Pattern[B, Blank[]]], Power[Pattern[x, Blank[]], 2]]], Power[Plus[Pattern[d, Blank[]], Times[Optional[Pattern[e, Blank[]]], Power[Pattern[x, Blank[]], 2]]], -1], Power[Plus[Pattern[a, Blank[]], Times[Optional[Pattern[b, Blank[]]], Power[Pattern[x, Blank[]], 2]], Times[Optional[Pattern[c, Blank[]]], Power[Pattern[x, Blank[]], 4]]], Rational[-1, 2]]], Pattern[x, Blank[Symbol]]] := Dist[(B*d + A*e)/(2*d*e), Int[1/Sqrt[a + b*x^2 + c*x^4], x], x] - Dist[(B*d - A*e)/(2*d*e), Int[(d - e*x^2)/((d + e*x^2)*Sqrt[a + b*x^2 + c*x^4]), x], x] /; FreeQ[{a, b, c, d, e, A, B}, x] && NeQ[b^2 - 4*a*c, 0] && NeQ[c*d^2 - b*d*e + a*e^2, 0] && EqQ[c*d^2 - a*e^2, 0] && NeQ[B*d + A*e, 0]
  1701. Int[Times[Plus[Pattern[A, Blank[]], Times[Optional[Pattern[B, Blank[]]], Power[Pattern[x, Blank[]], 2]]], Power[Plus[Pattern[d, Blank[]], Times[Optional[Pattern[e, Blank[]]], Power[Pattern[x, Blank[]], 2]]], -1], Power[Plus[Pattern[a, Blank[]], Times[Optional[Pattern[c, Blank[]]], Power[Pattern[x, Blank[]], 4]]], Rational[-1, 2]]], Pattern[x, Blank[Symbol]]] := Dist[(B*d + A*e)/(2*d*e), Int[1/Sqrt[a + c*x^4], x], x] - Dist[(B*d - A*e)/(2*d*e), Int[(d - e*x^2)/((d + e*x^2)*Sqrt[a + c*x^4]), x], x] /; FreeQ[{a, c, d, e, A, B}, x] && NeQ[c*d^2 + a*e^2, 0] && EqQ[c*d^2 - a*e^2, 0] && NeQ[B*d + A*e, 0]
  1702. Int[Times[Plus[Pattern[A, Blank[]], Times[Optional[Pattern[B, Blank[]]], Power[Pattern[x, Blank[]], 2]]], Power[Plus[Pattern[d, Blank[]], Times[Optional[Pattern[e, Blank[]]], Power[Pattern[x, Blank[]], 2]]], -1], Power[Plus[Pattern[a, Blank[]], Times[Optional[Pattern[b, Blank[]]], Power[Pattern[x, Blank[]], 2]], Times[Optional[Pattern[c, Blank[]]], Power[Pattern[x, Blank[]], 4]]], Rational[-1, 2]]], Pattern[x, Blank[Symbol]]] := Dist[(Sqrt[A + B*x^2]*Sqrt[a/A + (c*x^2)/B])/Sqrt[a + b*x^2 + c*x^4], Int[Sqrt[A + B*x^2]/((d + e*x^2)*Sqrt[a/A + (c*x^2)/B]), x], x] /; FreeQ[{a, b, c, d, e, A, B}, x] && NeQ[b^2 - 4*a*c, 0] && NeQ[c*d^2 - b*d*e + a*e^2, 0] && EqQ[c*A^2 - b*A*B + a*B^2, 0]
  1703. Int[Times[Plus[Pattern[A, Blank[]], Times[Optional[Pattern[B, Blank[]]], Power[Pattern[x, Blank[]], 2]]], Power[Plus[Pattern[d, Blank[]], Times[Optional[Pattern[e, Blank[]]], Power[Pattern[x, Blank[]], 2]]], -1], Power[Plus[Pattern[a, Blank[]], Times[Optional[Pattern[c, Blank[]]], Power[Pattern[x, Blank[]], 4]]], Rational[-1, 2]]], Pattern[x, Blank[Symbol]]] := Dist[(Sqrt[A + B*x^2]*Sqrt[a/A + (c*x^2)/B])/Sqrt[a + c*x^4], Int[Sqrt[A + B*x^2]/((d + e*x^2)*Sqrt[a/A + (c*x^2)/B]), x], x] /; FreeQ[{a, c, d, e, A, B}, x] && NeQ[c*d^2 + a*e^2, 0] && EqQ[c*A^2 + a*B^2, 0]
  1704. Int[Times[Plus[Pattern[A, Blank[]], Times[Optional[Pattern[B, Blank[]]], Power[Pattern[x, Blank[]], 2]]], Power[Plus[Pattern[d, Blank[]], Times[Optional[Pattern[e, Blank[]]], Power[Pattern[x, Blank[]], 2]]], -1], Power[Plus[Pattern[a, Blank[]], Times[Optional[Pattern[b, Blank[]]], Power[Pattern[x, Blank[]], 2]], Times[Optional[Pattern[c, Blank[]]], Power[Pattern[x, Blank[]], 4]]], Rational[-1, 2]]], Pattern[x, Blank[Symbol]]] := With[{q = Sqrt[b^2 - 4*a*c]}, Dist[(2*a*B - A*(b + q))/(2*a*e - d*(b + q)), Int[1/Sqrt[a + b*x^2 + c*x^4], x], x] - Dist[(B*d - A*e)/(2*a*e - d*(b + q)), Int[(2*a + (b + q)*x^2)/((d + e*x^2)*Sqrt[a + b*x^2 + c*x^4]), x], x] /; RationalQ[q]] /; FreeQ[{a, b, c, d, e, A, B}, x] && GtQ[b^2 - 4*a*c, 0] && NeQ[c*d^2 - b*d*e + a*e^2, 0] && NeQ[c*A^2 - b*A*B + a*B^2, 0]
  1705. Int[Times[Plus[Pattern[A, Blank[]], Times[Optional[Pattern[B, Blank[]]], Power[Pattern[x, Blank[]], 2]]], Power[Plus[Pattern[d, Blank[]], Times[Optional[Pattern[e, Blank[]]], Power[Pattern[x, Blank[]], 2]]], -1], Power[Plus[Pattern[a, Blank[]], Times[Optional[Pattern[c, Blank[]]], Power[Pattern[x, Blank[]], 4]]], Rational[-1, 2]]], Pattern[x, Blank[Symbol]]] := With[{q = Sqrt[-(a*c)]}, Dist[(a*B - A*q)/(a*e - d*q), Int[1/Sqrt[a + c*x^4], x], x] - Dist[(B*d - A*e)/(a*e - d*q), Int[(a + q*x^2)/((d + e*x^2)*Sqrt[a + c*x^4]), x], x] /; RationalQ[q]] /; FreeQ[{a, c, d, e, A, B}, x] && GtQ[-(a*c), 0] && EqQ[c*d^2 + a*e^2, 0] && NeQ[c*A^2 + a*B^2, 0]
  1706. Int[Times[Plus[Pattern[A, Blank[]], Times[Optional[Pattern[B, Blank[]]], Power[Pattern[x, Blank[]], 2]]], Power[Plus[Pattern[d, Blank[]], Times[Optional[Pattern[e, Blank[]]], Power[Pattern[x, Blank[]], 2]]], -1], Power[Plus[Pattern[a, Blank[]], Times[Optional[Pattern[b, Blank[]]], Power[Pattern[x, Blank[]], 2]], Times[Optional[Pattern[c, Blank[]]], Power[Pattern[x, Blank[]], 4]]], Rational[-1, 2]]], Pattern[x, Blank[Symbol]]] := With[{q = Rt[B/A, 2]}, -Simp[((B*d - A*e)*ArcTan[(Rt[-b + (c*d)/e + (a*e)/d, 2]*x)/Sqrt[a + b*x^2 + c*x^4]])/(2*d*e*Rt[-b + (c*d)/e + (a*e)/d, 2]), x] + Simp[((B*d + A*e)*(A + B*x^2)*Sqrt[(A^2*(a + b*x^2 + c*x^4))/(a*(A + B*x^2)^2)]*EllipticPi[Cancel[-((B*d - A*e)^2/(4*d*e*A*B))], 2*ArcTan[q*x], 1/2 - (b*A)/(4*a*B)])/(4*d*e*A*q*Sqrt[a + b*x^2 + c*x^4]), x]] /; FreeQ[{a, b, c, d, e, A, B}, x] && NeQ[b^2 - 4*a*c, 0] && NeQ[c*d^2 - b*d*e + a*e^2, 0] && NeQ[c*d^2 - a*e^2, 0] && PosQ[c/a] && EqQ[c*A^2 - a*B^2, 0]
  1707. Int[Times[Plus[Pattern[A, Blank[]], Times[Optional[Pattern[B, Blank[]]], Power[Pattern[x, Blank[]], 2]]], Power[Plus[Pattern[d, Blank[]], Times[Optional[Pattern[e, Blank[]]], Power[Pattern[x, Blank[]], 2]]], -1], Power[Plus[Pattern[a, Blank[]], Times[Optional[Pattern[c, Blank[]]], Power[Pattern[x, Blank[]], 4]]], Rational[-1, 2]]], Pattern[x, Blank[Symbol]]] := With[{q = Rt[B/A, 2]}, -Simp[((B*d - A*e)*ArcTan[(Rt[(c*d)/e + (a*e)/d, 2]*x)/Sqrt[a + c*x^4]])/(2*d*e*Rt[(c*d)/e + (a*e)/d, 2]), x] + Simp[((B*d + A*e)*(A + B*x^2)*Sqrt[(A^2*(a + c*x^4))/(a*(A + B*x^2)^2)]*EllipticPi[Cancel[-((B*d - A*e)^2/(4*d*e*A*B))], 2*ArcTan[q*x], 1/2])/(4*d*e*A*q*Sqrt[a + c*x^4]), x]] /; FreeQ[{a, c, d, e, A, B}, x] && NeQ[c*d^2 + a*e^2, 0] && NeQ[c*d^2 - a*e^2, 0] && PosQ[c/a] && EqQ[c*A^2 - a*B^2, 0]
  1708. Int[Times[Plus[Optional[Pattern[A, Blank[]]], Times[Optional[Pattern[B, Blank[]]], Power[Pattern[x, Blank[]], 2]]], Power[Plus[Pattern[d, Blank[]], Times[Optional[Pattern[e, Blank[]]], Power[Pattern[x, Blank[]], 2]]], -1], Power[Plus[Pattern[a, Blank[]], Times[Optional[Pattern[b, Blank[]]], Power[Pattern[x, Blank[]], 2]], Times[Optional[Pattern[c, Blank[]]], Power[Pattern[x, Blank[]], 4]]], Rational[-1, 2]]], Pattern[x, Blank[Symbol]]] := With[{q = Rt[c/a, 2]}, Dist[(A*(c*d + a*e*q) - a*B*(e + d*q))/(c*d^2 - a*e^2), Int[1/Sqrt[a + b*x^2 + c*x^4], x], x] + Dist[(a*(B*d - A*e)*(e + d*q))/(c*d^2 - a*e^2), Int[(1 + q*x^2)/((d + e*x^2)*Sqrt[a + b*x^2 + c*x^4]), x], x]] /; FreeQ[{a, b, c, d, e, A, B}, x] && NeQ[b^2 - 4*a*c, 0] && NeQ[c*d^2 - b*d*e + a*e^2, 0] && NeQ[c*d^2 - a*e^2, 0] && PosQ[c/a] && NeQ[c*A^2 - a*B^2, 0]
  1709. Int[Times[Plus[Optional[Pattern[A, Blank[]]], Times[Optional[Pattern[B, Blank[]]], Power[Pattern[x, Blank[]], 2]]], Power[Plus[Pattern[d, Blank[]], Times[Optional[Pattern[e, Blank[]]], Power[Pattern[x, Blank[]], 2]]], -1], Power[Plus[Pattern[a, Blank[]], Times[Optional[Pattern[c, Blank[]]], Power[Pattern[x, Blank[]], 4]]], Rational[-1, 2]]], Pattern[x, Blank[Symbol]]] := With[{q = Rt[c/a, 2]}, Dist[(A*(c*d + a*e*q) - a*B*(e + d*q))/(c*d^2 - a*e^2), Int[1/Sqrt[a + c*x^4], x], x] + Dist[(a*(B*d - A*e)*(e + d*q))/(c*d^2 - a*e^2), Int[(1 + q*x^2)/((d + e*x^2)*Sqrt[a + c*x^4]), x], x]] /; FreeQ[{a, c, d, e, A, B}, x] && NeQ[c*d^2 + a*e^2, 0] && NeQ[c*d^2 - a*e^2, 0] && PosQ[c/a] && NeQ[c*A^2 - a*B^2, 0]
  1710. Int[Times[Plus[Optional[Pattern[A, Blank[]]], Times[Optional[Pattern[B, Blank[]]], Power[Pattern[x, Blank[]], 2]]], Power[Plus[Pattern[d, Blank[]], Times[Optional[Pattern[e, Blank[]]], Power[Pattern[x, Blank[]], 2]]], -1], Power[Plus[Pattern[a, Blank[]], Times[Optional[Pattern[b, Blank[]]], Power[Pattern[x, Blank[]], 2]], Times[Optional[Pattern[c, Blank[]]], Power[Pattern[x, Blank[]], 4]]], Rational[-1, 2]]], Pattern[x, Blank[Symbol]]] := Dist[B/e, Int[1/Sqrt[a + b*x^2 + c*x^4], x], x] + Dist[(e*A - d*B)/e, Int[1/((d + e*x^2)*Sqrt[a + b*x^2 + c*x^4]), x], x] /; FreeQ[{a, b, c, d, e, A, B}, x] && NeQ[b^2 - 4*a*c, 0] && NeQ[c*d^2 - b*d*e + a*e^2, 0] && NeQ[c*d^2 - a*e^2, 0] && NegQ[c/a]
  1711. Int[Times[Plus[Optional[Pattern[A, Blank[]]], Times[Optional[Pattern[B, Blank[]]], Power[Pattern[x, Blank[]], 2]]], Power[Plus[Pattern[d, Blank[]], Times[Optional[Pattern[e, Blank[]]], Power[Pattern[x, Blank[]], 2]]], -1], Power[Plus[Pattern[a, Blank[]], Times[Optional[Pattern[c, Blank[]]], Power[Pattern[x, Blank[]], 4]]], Rational[-1, 2]]], Pattern[x, Blank[Symbol]]] := Dist[B/e, Int[1/Sqrt[a + c*x^4], x], x] + Dist[(e*A - d*B)/e, Int[1/((d + e*x^2)*Sqrt[a + c*x^4]), x], x] /; FreeQ[{a, c, d, e, A, B}, x] && NeQ[c*d^2 + a*e^2, 0] && NeQ[c*d^2 - a*e^2, 0] && NegQ[c/a]
  1712. Int[Times[Pattern[P4x, Blank[]], Power[Plus[Pattern[d, Blank[]], Times[Optional[Pattern[e, Blank[]]], Power[Pattern[x, Blank[]], 2]]], -1], Power[Plus[Pattern[a, Blank[]], Times[Optional[Pattern[b, Blank[]]], Power[Pattern[x, Blank[]], 2]], Times[Optional[Pattern[c, Blank[]]], Power[Pattern[x, Blank[]], 4]]], Rational[-1, 2]]], Pattern[x, Blank[Symbol]]] := With[{A = Coeff[P4x, x, 0], B = Coeff[P4x, x, 2], C = Coeff[P4x, x, 4]}, -Dist[C/e^2, Int[(d - e*x^2)/Sqrt[a + b*x^2 + c*x^4], x], x] + Dist[1/e^2, Int[(C*d^2 + A*e^2 + B*e^2*x^2)/((d + e*x^2)*Sqrt[a + b*x^2 + c*x^4]), x], x]] /; FreeQ[{a, b, c, d, e}, x] && PolyQ[P4x, x^2, 2] && NeQ[b^2 - 4*a*c, 0] && NeQ[c*d^2 - b*d*e + a*e^2, 0] && EqQ[c*d^2 - a*e^2, 0]
  1713. Int[Times[Pattern[P4x, Blank[]], Power[Plus[Pattern[d, Blank[]], Times[Optional[Pattern[e, Blank[]]], Power[Pattern[x, Blank[]], 2]]], -1], Power[Plus[Pattern[a, Blank[]], Times[Optional[Pattern[c, Blank[]]], Power[Pattern[x, Blank[]], 4]]], Rational[-1, 2]]], Pattern[x, Blank[Symbol]]] := With[{A = Coeff[P4x, x, 0], B = Coeff[P4x, x, 2], C = Coeff[P4x, x, 4]}, -Dist[C/e^2, Int[(d - e*x^2)/Sqrt[a + c*x^4], x], x] + Dist[1/e^2, Int[(C*d^2 + A*e^2 + B*e^2*x^2)/((d + e*x^2)*Sqrt[a + c*x^4]), x], x]] /; FreeQ[{a, c, d, e}, x] && PolyQ[P4x, x^2, 2] && NeQ[c*d^2 + a*e^2, 0] && EqQ[c*d^2 - a*e^2, 0]
  1714. Int[Times[Pattern[P4x, Blank[]], Power[Plus[Pattern[d, Blank[]], Times[Optional[Pattern[e, Blank[]]], Power[Pattern[x, Blank[]], 2]]], -1], Power[Plus[Pattern[a, Blank[]], Times[Optional[Pattern[b, Blank[]]], Power[Pattern[x, Blank[]], 2]], Times[Optional[Pattern[c, Blank[]]], Power[Pattern[x, Blank[]], 4]]], Rational[-1, 2]]], Pattern[x, Blank[Symbol]]] := With[{q = Rt[c/a, 2], A = Coeff[P4x, x, 0], B = Coeff[P4x, x, 2], C = Coeff[P4x, x, 4]}, -Dist[C/(e*q), Int[(1 - q*x^2)/Sqrt[a + b*x^2 + c*x^4], x], x] + Dist[1/(c*e), Int[(A*c*e + a*C*d*q + (B*c*e - C*(c*d - a*e*q))*x^2)/((d + e*x^2)*Sqrt[a + b*x^2 + c*x^4]), x], x]] /; FreeQ[{a, b, c, d, e}, x] && PolyQ[P4x, x^2, 2] && NeQ[b^2 - 4*a*c, 0] && NeQ[c*d^2 - b*d*e + a*e^2, 0] && NeQ[c*d^2 - a*e^2, 0] && PosQ[c/a] && !GtQ[b^2 - 4*a*c, 0]
  1715. Int[Times[Pattern[P4x, Blank[]], Power[Plus[Pattern[d, Blank[]], Times[Optional[Pattern[e, Blank[]]], Power[Pattern[x, Blank[]], 2]]], -1], Power[Plus[Pattern[a, Blank[]], Times[Optional[Pattern[c, Blank[]]], Power[Pattern[x, Blank[]], 4]]], Rational[-1, 2]]], Pattern[x, Blank[Symbol]]] := With[{q = Rt[c/a, 2], A = Coeff[P4x, x, 0], B = Coeff[P4x, x, 2], C = Coeff[P4x, x, 4]}, -Dist[C/(e*q), Int[(1 - q*x^2)/Sqrt[a + c*x^4], x], x] + Dist[1/(c*e), Int[(A*c*e + a*C*d*q + (B*c*e - C*(c*d - a*e*q))*x^2)/((d + e*x^2)*Sqrt[a + c*x^4]), x], x]] /; FreeQ[{a, c, d, e}, x] && PolyQ[P4x, x^2, 2] && NeQ[c*d^2 + a*e^2, 0] && NeQ[c*d^2 - a*e^2, 0] && PosQ[c/a]
  1716. Int[Times[Pattern[P4x, Blank[]], Power[Plus[Pattern[d, Blank[]], Times[Optional[Pattern[e, Blank[]]], Power[Pattern[x, Blank[]], 2]]], -1], Power[Plus[Pattern[a, Blank[]], Times[Optional[Pattern[b, Blank[]]], Power[Pattern[x, Blank[]], 2]], Times[Optional[Pattern[c, Blank[]]], Power[Pattern[x, Blank[]], 4]]], Rational[-1, 2]]], Pattern[x, Blank[Symbol]]] := With[{A = Coeff[P4x, x, 0], B = Coeff[P4x, x, 2], C = Coeff[P4x, x, 4]}, -Dist[(e^2)^(-1), Int[(C*d - B*e - C*e*x^2)/Sqrt[a + b*x^2 + c*x^4], x], x] + Dist[(C*d^2 - B*d*e + A*e^2)/e^2, Int[1/((d + e*x^2)*Sqrt[a + b*x^2 + c*x^4]), x], x]] /; FreeQ[{a, b, c, d, e}, x] && PolyQ[P4x, x^2, 2] && NeQ[b^2 - 4*a*c, 0] && NeQ[c*d^2 - b*d*e + a*e^2, 0] && NeQ[c*d^2 - a*e^2, 0]
  1717. Int[Times[Pattern[P4x, Blank[]], Power[Plus[Pattern[d, Blank[]], Times[Optional[Pattern[e, Blank[]]], Power[Pattern[x, Blank[]], 2]]], -1], Power[Plus[Pattern[a, Blank[]], Times[Optional[Pattern[c, Blank[]]], Power[Pattern[x, Blank[]], 4]]], Rational[-1, 2]]], Pattern[x, Blank[Symbol]]] := With[{A = Coeff[P4x, x, 0], B = Coeff[P4x, x, 2], C = Coeff[P4x, x, 4]}, -Dist[(e^2)^(-1), Int[(C*d - B*e - C*e*x^2)/Sqrt[a + c*x^4], x], x] + Dist[(C*d^2 - B*d*e + A*e^2)/e^2, Int[1/((d + e*x^2)*Sqrt[a + c*x^4]), x], x]] /; FreeQ[{a, c, d, e}, x] && PolyQ[P4x, x^2, 2] && NeQ[c*d^2 + a*e^2, 0] && NeQ[c*d^2 - a*e^2, 0]
  1718. Int[Times[Pattern[Px, Blank[]], Power[Plus[Pattern[d, Blank[]], Times[Optional[Pattern[e, Blank[]]], Power[Pattern[x, Blank[]], 2]]], -1], Power[Plus[Pattern[a, Blank[]], Times[Optional[Pattern[b, Blank[]]], Power[Pattern[x, Blank[]], 2]], Times[Optional[Pattern[c, Blank[]]], Power[Pattern[x, Blank[]], 4]]], Rational[-1, 2]]], Pattern[x, Blank[Symbol]]] := With[{q = Expon[Px, x]}, Simp[(Coeff[Px, x, q]*x^(q - 5)*Sqrt[a + b*x^2 + c*x^4])/(c*e*(q - 3)), x] + Dist[1/(c*e*(q - 3)), Int[(c*e*(q - 3)*Px - Coeff[Px, x, q]*x^(q - 6)*(d + e*x^2)*(a*(q - 5) + b*(q - 4)*x^2 + c*(q - 3)*x^4))/((d + e*x^2)*Sqrt[a + b*x^2 + c*x^4]), x], x] /; GtQ[q, 4]] /; FreeQ[{a, b, c, d, e}, x] && PolyQ[Px, x] && NeQ[b^2 - 4*a*c, 0] && NeQ[c*d^2 - b*d*e + a*e^2, 0]
  1719. Int[Times[Pattern[Px, Blank[]], Power[Plus[Pattern[d, Blank[]], Times[Optional[Pattern[e, Blank[]]], Power[Pattern[x, Blank[]], 2]]], -1], Power[Plus[Pattern[a, Blank[]], Times[Optional[Pattern[c, Blank[]]], Power[Pattern[x, Blank[]], 4]]], Rational[-1, 2]]], Pattern[x, Blank[Symbol]]] := With[{q = Expon[Px, x]}, Simp[(Coeff[Px, x, q]*x^(q - 5)*Sqrt[a + c*x^4])/(c*e*(q - 3)), x] + Dist[1/(c*e*(q - 3)), Int[(c*e*(q - 3)*Px - Coeff[Px, x, q]*x^(q - 6)*(d + e*x^2)*(a*(q - 5) + c*(q - 3)*x^4))/((d + e*x^2)*Sqrt[a + c*x^4]), x], x] /; GtQ[q, 4]] /; FreeQ[{a, c, d, e}, x] && PolyQ[Px, x] && NeQ[c*d^2 + a*e^2, 0]
  1720. Int[Times[Pattern[Px, Blank[]], Power[Plus[Pattern[d, Blank[]], Times[Optional[Pattern[e, Blank[]]], Power[Pattern[x, Blank[]], 2]]], Optional[Pattern[q, Blank[]]]], Power[Plus[Pattern[a, Blank[]], Times[Optional[Pattern[b, Blank[]]], Power[Pattern[x, Blank[]], 2]], Times[Optional[Pattern[c, Blank[]]], Power[Pattern[x, Blank[]], 4]]], Pattern[p, Blank[]]]], Pattern[x, Blank[Symbol]]] := Int[ExpandIntegrand[1/Sqrt[a + b*x^2 + c*x^4], Px*(d + e*x^2)^q*(a + b*x^2 + c*x^4)^(p + 1/2), x], x] /; FreeQ[{a, b, c, d, e}, x] && PolyQ[Px, x^2] && NeQ[b^2 - 4*a*c, 0] && NeQ[c*d^2 - b*d*e + a*e^2, 0] && IntegerQ[p + 1/2] && IntegerQ[q]
  1721. Int[Times[Pattern[Px, Blank[]], Power[Plus[Pattern[d, Blank[]], Times[Optional[Pattern[e, Blank[]]], Power[Pattern[x, Blank[]], 2]]], Optional[Pattern[q, Blank[]]]], Power[Plus[Pattern[a, Blank[]], Times[Optional[Pattern[c, Blank[]]], Power[Pattern[x, Blank[]], 4]]], Pattern[p, Blank[]]]], Pattern[x, Blank[Symbol]]] := Int[ExpandIntegrand[1/Sqrt[a + c*x^4], Px*(d + e*x^2)^q*(a + c*x^4)^(p + 1/2), x], x] /; FreeQ[{a, c, d, e}, x] && PolyQ[Px, x^2] && NeQ[c*d^2 + a*e^2, 0] && IntegerQ[p + 1/2] && IntegerQ[q]
  1722. Int[Times[Pattern[Px, Blank[]], Power[Plus[Pattern[d, Blank[]], Times[Optional[Pattern[e, Blank[]]], Power[Pattern[x, Blank[]], 2]]], Optional[Pattern[q, Blank[]]]], Power[Plus[Pattern[a, Blank[]], Times[Optional[Pattern[b, Blank[]]], Power[Pattern[x, Blank[]], 2]], Times[Optional[Pattern[c, Blank[]]], Power[Pattern[x, Blank[]], 4]]], Optional[Pattern[p, Blank[]]]]], Pattern[x, Blank[Symbol]]] := Unintegrable[Px*(d + e*x^2)^q*(a + b*x^2 + c*x^4)^p, x] /; FreeQ[{a, b, c, d, e, p, q}, x] && PolyQ[Px, x]
  1723. Int[Times[Pattern[Px, Blank[]], Power[Plus[Pattern[d, Blank[]], Times[Optional[Pattern[e, Blank[]]], Power[Pattern[x, Blank[]], 2]]], Optional[Pattern[q, Blank[]]]], Power[Plus[Pattern[a, Blank[]], Times[Optional[Pattern[c, Blank[]]], Power[Pattern[x, Blank[]], 4]]], Optional[Pattern[p, Blank[]]]]], Pattern[x, Blank[Symbol]]] := Unintegrable[Px*(d + e*x^2)^q*(a + c*x^4)^p, x] /; FreeQ[{a, c, d, e, p, q}, x] && PolyQ[Px, x]
  1724. Int[Times[Power[Plus[Pattern[d, Blank[]], Times[Optional[Pattern[e, Blank[]]], Pattern[x, Blank[]]]], -1], Power[Plus[Pattern[a, Blank[]], Times[Optional[Pattern[b, Blank[]]], Power[Pattern[x, Blank[]], 2]], Times[Optional[Pattern[c, Blank[]]], Power[Pattern[x, Blank[]], 4]]], Rational[-1, 2]]], Pattern[x, Blank[Symbol]]] := Dist[d, Int[1/((d^2 - e^2*x^2)*Sqrt[a + b*x^2 + c*x^4]), x], x] - Dist[e, Int[x/((d^2 - e^2*x^2)*Sqrt[a + b*x^2 + c*x^4]), x], x] /; FreeQ[{a, b, c, d, e}, x]
  1725. Int[Times[Power[Plus[Pattern[d, Blank[]], Times[Optional[Pattern[e, Blank[]]], Pattern[x, Blank[]]]], -1], Power[Plus[Pattern[a, Blank[]], Times[Optional[Pattern[c, Blank[]]], Power[Pattern[x, Blank[]], 4]]], Rational[-1, 2]]], Pattern[x, Blank[Symbol]]] := Dist[d, Int[1/((d^2 - e^2*x^2)*Sqrt[a + c*x^4]), x], x] - Dist[e, Int[x/((d^2 - e^2*x^2)*Sqrt[a + c*x^4]), x], x] /; FreeQ[{a, c, d, e}, x]
  1726. Int[Times[Power[Plus[Pattern[d, Blank[]], Times[Optional[Pattern[e, Blank[]]], Pattern[x, Blank[]]]], Pattern[q, Blank[]]], Power[Plus[Pattern[a, Blank[]], Times[Optional[Pattern[b, Blank[]]], Power[Pattern[x, Blank[]], 2]], Times[Optional[Pattern[c, Blank[]]], Power[Pattern[x, Blank[]], 4]]], Rational[-1, 2]]], Pattern[x, Blank[Symbol]]] := Simp[(e^3*(d + e*x)^(q + 1)*Sqrt[a + b*x^2 + c*x^4])/((q + 1)*(c*d^4 + b*d^2*e^2 + a*e^4)), x] + Dist[1/((q + 1)*(c*d^4 + b*d^2*e^2 + a*e^4)), Int[((d + e*x)^(q + 1)*Simp[d*(q + 1)*(c*d^2 + b*e^2) - e*(c*d^2*(q + 1) + b*e^2*(q + 2))*x + c*d*e^2*(q + 1)*x^2 - c*e^3*(q + 3)*x^3, x])/Sqrt[a + b*x^2 + c*x^4], x], x] /; FreeQ[{a, b, c, d, e}, x] && NeQ[c*d^4 + b*d^2*e^2 + a*e^4, 0] && ILtQ[q, -1]
  1727. Int[Times[Power[Plus[Pattern[d, Blank[]], Times[Optional[Pattern[e, Blank[]]], Pattern[x, Blank[]]]], Pattern[q, Blank[]]], Power[Plus[Pattern[a, Blank[]], Times[Optional[Pattern[c, Blank[]]], Power[Pattern[x, Blank[]], 4]]], Rational[-1, 2]]], Pattern[x, Blank[Symbol]]] := Simp[(e^3*(d + e*x)^(q + 1)*Sqrt[a + c*x^4])/((q + 1)*(c*d^4 + a*e^4)), x] + Dist[c/((q + 1)*(c*d^4 + a*e^4)), Int[((d + e*x)^(q + 1)*Simp[d^3*(q + 1) - d^2*e*(q + 1)*x + d*e^2*(q + 1)*x^2 - e^3*(q + 3)*x^3, x])/Sqrt[a + c*x^4], x], x] /; FreeQ[{a, c, d, e}, x] && NeQ[c*d^4 + a*e^4, 0] && ILtQ[q, -1]
  1728. Int[Times[Power[Plus[Pattern[d, Blank[]], Times[Optional[Pattern[e, Blank[]]], Pattern[x, Blank[]]]], -1], Power[Plus[Pattern[a, Blank[]], Times[Optional[Pattern[b, Blank[]]], Power[Pattern[x, Blank[]], 2]], Times[Optional[Pattern[c, Blank[]]], Power[Pattern[x, Blank[]], 4]]], Optional[Pattern[p, Blank[]]]]], Pattern[x, Blank[Symbol]]] := Dist[d, Int[(a + b*x^2 + c*x^4)^p/(d^2 - e^2*x^2), x], x] - Dist[e, Int[(x*(a + b*x^2 + c*x^4)^p)/(d^2 - e^2*x^2), x], x] /; FreeQ[{a, b, c, d, e}, x] && IntegerQ[p + 1/2]
  1729. Int[Times[Power[Plus[Pattern[d, Blank[]], Times[Optional[Pattern[e, Blank[]]], Pattern[x, Blank[]]]], -1], Power[Plus[Pattern[a, Blank[]], Times[Optional[Pattern[c, Blank[]]], Power[Pattern[x, Blank[]], 4]]], Optional[Pattern[p, Blank[]]]]], Pattern[x, Blank[Symbol]]] := Dist[d, Int[(a + c*x^4)^p/(d^2 - e^2*x^2), x], x] - Dist[e, Int[(x*(a + c*x^4)^p)/(d^2 - e^2*x^2), x], x] /; FreeQ[{a, c, d, e}, x] && IntegerQ[p + 1/2]
  1730. Int[Times[Pattern[Px, Blank[]], Power[Plus[Pattern[d, Blank[]], Times[Optional[Pattern[e, Blank[]]], Pattern[x, Blank[]]]], Optional[Pattern[q, Blank[]]]], Power[Plus[Pattern[a, Blank[]], Times[Optional[Pattern[b, Blank[]]], Power[Pattern[x, Blank[]], 2]], Times[Optional[Pattern[c, Blank[]]], Power[Pattern[x, Blank[]], 4]]], Optional[Pattern[p, Blank[]]]]], Pattern[x, Blank[Symbol]]] := Int[PolynomialQuotient[Px, d + e*x, x]*(d + e*x)^(q + 1)*(a + b*x^2 + c*x^4)^p, x] /; FreeQ[{a, b, c, d, e, p, q}, x] && PolyQ[Px, x] && EqQ[PolynomialRemainder[Px, d + e*x, x], 0]
  1731. Int[Times[Pattern[Px, Blank[]], Power[Plus[Pattern[d, Blank[]], Times[Optional[Pattern[e, Blank[]]], Pattern[x, Blank[]]]], Optional[Pattern[q, Blank[]]]], Power[Plus[Pattern[a, Blank[]], Times[Optional[Pattern[c, Blank[]]], Power[Pattern[x, Blank[]], 4]]], Optional[Pattern[p, Blank[]]]]], Pattern[x, Blank[Symbol]]] := Int[PolynomialQuotient[Px, d + e*x, x]*(d + e*x)^(q + 1)*(a + c*x^4)^p, x] /; FreeQ[{a, c, d, e, p, q}, x] && PolyQ[Px, x] && EqQ[PolynomialRemainder[Px, d + e*x, x], 0]
  1732. Int[Times[Pattern[Px, Blank[]], Power[Plus[Pattern[d, Blank[]], Times[Optional[Pattern[e, Blank[]]], Pattern[x, Blank[]]]], Optional[Pattern[q, Blank[]]]], Power[Plus[Pattern[a, Blank[]], Times[Optional[Pattern[b, Blank[]]], Power[Pattern[x, Blank[]], 2]], Times[Optional[Pattern[c, Blank[]]], Power[Pattern[x, Blank[]], 4]]], Optional[Pattern[p, Blank[]]]]], Pattern[x, Blank[Symbol]]] := Int[PolynomialQuotient[Px, a + b*x^2 + c*x^4, x]*(d + e*x)^q*(a + b*x^2 + c*x^4)^(p + 1), x] /; FreeQ[{a, b, c, d, e, p, q}, x] && PolyQ[Px, x] && EqQ[PolynomialRemainder[Px, a + b*x^2 + c*x^4, x], 0]
  1733. Int[Times[Pattern[Px, Blank[]], Power[Plus[Pattern[d, Blank[]], Times[Optional[Pattern[e, Blank[]]], Pattern[x, Blank[]]]], Optional[Pattern[q, Blank[]]]], Power[Plus[Pattern[a, Blank[]], Times[Optional[Pattern[c, Blank[]]], Power[Pattern[x, Blank[]], 4]]], Optional[Pattern[p, Blank[]]]]], Pattern[x, Blank[Symbol]]] := Int[PolynomialQuotient[Px, a + c*x^4, x]*(d + e*x)^q*(a + c*x^4)^(p + 1), x] /; FreeQ[{a, c, d, e, p, q}, x] && PolyQ[Px, x] && EqQ[PolynomialRemainder[Px, a + c*x^4, x], 0]
  1734. Int[Times[Pattern[Px, Blank[]], Power[Plus[Pattern[d, Blank[]], Times[Optional[Pattern[e, Blank[]]], Pattern[x, Blank[]]]], Pattern[q, Blank[]]], Power[Plus[Pattern[a, Blank[]], Times[Optional[Pattern[b, Blank[]]], Power[Pattern[x, Blank[]], 2]], Times[Optional[Pattern[c, Blank[]]], Power[Pattern[x, Blank[]], 4]]], Rational[-1, 2]]], Pattern[x, Blank[Symbol]]] := With[{A = Coeff[Px, x, 0], B = Coeff[Px, x, 1], C = Coeff[Px, x, 2], D = Coeff[Px, x, 3]}, Int[((d + e*x)^(q - 1)*(A*d + (B*d + A*e)*x + (C*d + B*e)*x^2 + C*e*x^3))/Sqrt[a + b*x^2 + c*x^4], x]] /; FreeQ[{a, b, c, d, e}, x] && PolyQ[Px, x] && LeQ[Expon[Px, x], 2] && NeQ[c*d^4 + b*d^2*e^2 + a*e^4, 0] && GtQ[q, 0]
  1735. Int[Times[Pattern[Px, Blank[]], Power[Plus[Pattern[d, Blank[]], Times[Optional[Pattern[e, Blank[]]], Pattern[x, Blank[]]]], Pattern[q, Blank[]]], Power[Plus[Pattern[a, Blank[]], Times[Optional[Pattern[c, Blank[]]], Power[Pattern[x, Blank[]], 4]]], Rational[-1, 2]]], Pattern[x, Blank[Symbol]]] := With[{A = Coeff[Px, x, 0], B = Coeff[Px, x, 1], C = Coeff[Px, x, 2], D = Coeff[Px, x, 3]}, Int[((d + e*x)^(q - 1)*(A*d + (B*d + A*e)*x + (C*d + B*e)*x^2 + C*e*x^3))/Sqrt[a + c*x^4], x]] /; FreeQ[{a, c, d, e}, x] && PolyQ[Px, x] && LeQ[Expon[Px, x], 2] && NeQ[c*d^4 + a*e^4, 0] && GtQ[q, 0]
  1736. Int[Times[Pattern[Px, Blank[]], Power[Plus[Pattern[d, Blank[]], Times[Optional[Pattern[e, Blank[]]], Pattern[x, Blank[]]]], Pattern[q, Blank[]]], Power[Plus[Pattern[a, Blank[]], Times[Optional[Pattern[b, Blank[]]], Power[Pattern[x, Blank[]], 2]], Times[Optional[Pattern[c, Blank[]]], Power[Pattern[x, Blank[]], 4]]], Rational[-1, 2]]], Pattern[x, Blank[Symbol]]] := With[{A = Coeff[Px, x, 0], B = Coeff[Px, x, 1], C = Coeff[Px, x, 2], D = Coeff[Px, x, 3]}, Simp[(D*(d + e*x)^q*Sqrt[a + b*x^2 + c*x^4])/(c*(q + 2)), x] - Dist[1/(c*(q + 2)), Int[((d + e*x)^(q - 1)*Simp[a*D*e*q - A*c*d*(q + 2) + (b*d*D - B*c*d*(q + 2) - A*c*e*(q + 2))*x + (b*D*e*(q + 1) - c*(C*d + B*e)*(q + 2))*x^2 - c*(d*D*q + C*e*(q + 2))*x^3, x])/Sqrt[a + b*x^2 + c*x^4], x], x]] /; FreeQ[{a, b, c, d, e}, x] && PolyQ[Px, x, 3] && NeQ[c*d^4 + b*d^2*e^2 + a*e^4, 0] && GtQ[q, 0]
  1737. Int[Times[Pattern[Px, Blank[]], Power[Plus[Pattern[d, Blank[]], Times[Optional[Pattern[e, Blank[]]], Pattern[x, Blank[]]]], Pattern[q, Blank[]]], Power[Plus[Pattern[a, Blank[]], Times[Optional[Pattern[c, Blank[]]], Power[Pattern[x, Blank[]], 4]]], Rational[-1, 2]]], Pattern[x, Blank[Symbol]]] := With[{A = Coeff[Px, x, 0], B = Coeff[Px, x, 1], C = Coeff[Px, x, 2], D = Coeff[Px, x, 3]}, Simp[(D*(d + e*x)^q*Sqrt[a + c*x^4])/(c*(q + 2)), x] - Dist[1/(c*(q + 2)), Int[((d + e*x)^(q - 1)*Simp[a*D*e*q - A*c*d*(q + 2) - c*(B*d*(q + 2) + A*e*(q + 2))*x - c*(C*d + B*e)*(q + 2)*x^2 - c*(d*D*q + C*e*(q + 2))*x^3, x])/Sqrt[a + c*x^4], x], x]] /; FreeQ[{a, c, d, e}, x] && PolyQ[Px, x, 3] && NeQ[c*d^4 + a*e^4, 0] && GtQ[q, 0]
  1738. Int[Times[Pattern[Px, Blank[]], Power[Plus[Pattern[d, Blank[]], Times[Optional[Pattern[e, Blank[]]], Pattern[x, Blank[]]]], Pattern[q, Blank[]]], Power[Plus[Pattern[a, Blank[]], Times[Optional[Pattern[b, Blank[]]], Power[Pattern[x, Blank[]], 2]], Times[Optional[Pattern[c, Blank[]]], Power[Pattern[x, Blank[]], 4]]], Rational[-1, 2]]], Pattern[x, Blank[Symbol]]] := With[{A = Coeff[Px, x, 0], B = Coeff[Px, x, 1], C = Coeff[Px, x, 2], D = Coeff[Px, x, 3]}, -Simp[((d^3*D - C*d^2*e + B*d*e^2 - A*e^3)*(d + e*x)^(q + 1)*Sqrt[a + b*x^2 + c*x^4])/((q + 1)*(c*d^4 + b*d^2*e^2 + a*e^4)), x] + Dist[1/((q + 1)*(c*d^4 + b*d^2*e^2 + a*e^4)), Int[((d + e*x)^(q + 1)/Sqrt[a + b*x^2 + c*x^4])*Simp[(q + 1)*(a*e*(d^2*D - C*d*e + B*e^2) + A*d*(c*d^2 + b*e^2)) - (e*(q + 1)*(A*c*d^2 + a*e*(d*D - C*e)) - B*d*(c*d^2*(q + 1) + b*e^2*(q + 2)) - b*(d^3*D - C*d^2*e - A*e^3*(q + 2)))*x + (q + 1)*(D*e*(b*d^2 + a*e^2) + c*d*(C*d^2 - e*(B*d - A*e)))*x^2 + c*(q + 3)*(d^3*D - C*d^2*e + B*d*e^2 - A*e^3)*x^3, x], x], x]] /; FreeQ[{a, b, c, d, e}, x] && PolyQ[Px, x] && LeQ[Expon[Px, x], 3] && NeQ[c*d^4 + b*d^2*e^2 + a*e^4, 0] && LtQ[q, -1]
  1739. Int[Times[Pattern[Px, Blank[]], Power[Plus[Pattern[d, Blank[]], Times[Optional[Pattern[e, Blank[]]], Pattern[x, Blank[]]]], Pattern[q, Blank[]]], Power[Plus[Pattern[a, Blank[]], Times[Optional[Pattern[c, Blank[]]], Power[Pattern[x, Blank[]], 4]]], Rational[-1, 2]]], Pattern[x, Blank[Symbol]]] := With[{A = Coeff[Px, x, 0], B = Coeff[Px, x, 1], C = Coeff[Px, x, 2], D = Coeff[Px, x, 3]}, -Simp[((d^3*D - C*d^2*e + B*d*e^2 - A*e^3)*(d + e*x)^(q + 1)*Sqrt[a + c*x^4])/((q + 1)*(c*d^4 + a*e^4)), x] + Dist[1/((q + 1)*(c*d^4 + a*e^4)), Int[((d + e*x)^(q + 1)/Sqrt[a + c*x^4])*Simp[(q + 1)*(a*e*(d^2*D - C*d*e + B*e^2) + A*d*(c*d^2)) - (e*(q + 1)*(A*c*d^2 + a*e*(d*D - C*e)) - B*d*(c*d^2*(q + 1)))*x + (q + 1)*(D*e*(a*e^2) + c*d*(C*d^2 - e*(B*d - A*e)))*x^2 + c*(q + 3)*(d^3*D - C*d^2*e + B*d*e^2 - A*e^3)*x^3, x], x], x]] /; FreeQ[{a, c, d, e}, x] && PolyQ[Px, x] && LeQ[Expon[Px, x], 3] && NeQ[c*d^4 + a*e^4, 0] && LtQ[q, -1]
  1740. Int[Times[Plus[Pattern[A, Blank[]], Times[Optional[Pattern[B, Blank[]]], Pattern[x, Blank[]]]], Power[Plus[Pattern[d, Blank[]], Times[Optional[Pattern[e, Blank[]]], Pattern[x, Blank[]]]], -1], Power[Plus[Pattern[a, Blank[]], Times[Optional[Pattern[b, Blank[]]], Power[Pattern[x, Blank[]], 2]], Times[Optional[Pattern[c, Blank[]]], Power[Pattern[x, Blank[]], 4]]], Rational[-1, 2]]], Pattern[x, Blank[Symbol]]] := -Dist[(A^2*(B*d + A*e))/e, Subst[Int[1/(6*A^3*B*d + 3*A^4*e - a*e*x^2), x], x, (A + B*x)^2/Sqrt[a + b*x^2 + c*x^4]], x] /; FreeQ[{a, b, c, d, e, A, B}, x] && NeQ[B*d - A*e, 0] && EqQ[c^2*d^6 + a*e^4*(13*c*d^2 + b*e^2), 0] && EqQ[b^2*e^4 - 12*c*d^2*(c*d^2 - b*e^2), 0] && EqQ[4*A*c*d*e + B*(2*c*d^2 - b*e^2), 0]
  1741. Int[Times[Pattern[Px, Blank[]], Power[Plus[Pattern[d, Blank[]], Times[Optional[Pattern[e, Blank[]]], Pattern[x, Blank[]]]], -1], Power[Plus[Pattern[a, Blank[]], Times[Optional[Pattern[b, Blank[]]], Power[Pattern[x, Blank[]], 2]], Times[Optional[Pattern[c, Blank[]]], Power[Pattern[x, Blank[]], 4]]], Rational[-1, 2]]], Pattern[x, Blank[Symbol]]] := With[{A = Coeff[Px, x, 0], B = Coeff[Px, x, 1], C = Coeff[Px, x, 2], D = Coeff[Px, x, 3]}, Int[(x*(B*d - A*e + (d*D - C*e)*x^2))/((d^2 - e^2*x^2)*Sqrt[a + b*x^2 + c*x^4]), x] + Int[(A*d + (C*d - B*e)*x^2 - D*e*x^4)/((d^2 - e^2*x^2)*Sqrt[a + b*x^2 + c*x^4]), x]] /; FreeQ[{a, b, c, d, e}, x] && PolyQ[Px, x] && LeQ[Expon[Px, x], 3] && NeQ[c*d^4 + b*d^2*e^2 + a*e^4, 0]
  1742. Int[Times[Pattern[Px, Blank[]], Power[Plus[Pattern[d, Blank[]], Times[Optional[Pattern[e, Blank[]]], Pattern[x, Blank[]]]], -1], Power[Plus[Pattern[a, Blank[]], Times[Optional[Pattern[c, Blank[]]], Power[Pattern[x, Blank[]], 4]]], Rational[-1, 2]]], Pattern[x, Blank[Symbol]]] := With[{A = Coeff[Px, x, 0], B = Coeff[Px, x, 1], C = Coeff[Px, x, 2], D = Coeff[Px, x, 3]}, Int[(x*(B*d - A*e + (d*D - C*e)*x^2))/((d^2 - e^2*x^2)*Sqrt[a + c*x^4]), x] + Int[(A*d + (C*d - B*e)*x^2 - D*e*x^4)/((d^2 - e^2*x^2)*Sqrt[a + c*x^4]), x]] /; FreeQ[{a, c, d, e}, x] && PolyQ[Px, x] && LeQ[Expon[Px, x], 3] && NeQ[c*d^4 + a*e^4, 0]
  1743. Int[Times[Pattern[Px, Blank[]], Power[Plus[Pattern[d, Blank[]], Times[Optional[Pattern[e, Blank[]]], Pattern[x, Blank[]]]], -1], Power[Plus[Pattern[a, Blank[]], Times[Optional[Pattern[b, Blank[]]], Power[Pattern[x, Blank[]], 2]], Times[Optional[Pattern[c, Blank[]]], Power[Pattern[x, Blank[]], 4]]], Optional[Pattern[p, Blank[]]]]], Pattern[x, Blank[Symbol]]] := Dist[d, Int[(Px*(a + b*x^2 + c*x^4)^p)/(d^2 - e^2*x^2), x], x] - Dist[e, Int[(x*Px*(a + b*x^2 + c*x^4)^p)/(d^2 - e^2*x^2), x], x] /; FreeQ[{a, b, c, d, e}, x] && PolyQ[Px, x] && IntegerQ[p + 1/2]
  1744. Int[Times[Pattern[Px, Blank[]], Power[Plus[Pattern[d, Blank[]], Times[Optional[Pattern[e, Blank[]]], Pattern[x, Blank[]]]], -1], Power[Plus[Pattern[a, Blank[]], Times[Optional[Pattern[c, Blank[]]], Power[Pattern[x, Blank[]], 4]]], Optional[Pattern[p, Blank[]]]]], Pattern[x, Blank[Symbol]]] := Dist[d, Int[(Px*(a + c*x^4)^p)/(d^2 - e^2*x^2), x], x] - Dist[e, Int[(x*Px*(a + c*x^4)^p)/(d^2 - e^2*x^2), x], x] /; FreeQ[{a, c, d, e}, x] && PolyQ[Px, x] && IntegerQ[p + 1/2]
  1745. Int[Times[Pattern[Pq, Blank[]], Power[Pattern[x, Blank[]], Optional[Pattern[m, Blank[]]]], Power[Plus[Pattern[a, Blank[]], Times[Optional[Pattern[c, Blank[]]], Power[Pattern[x, Blank[]], Optional[Pattern[n2, Blank[]]]]], Times[Optional[Pattern[b, Blank[]]], Power[Pattern[x, Blank[]], Pattern[n, Blank[]]]]], Optional[Pattern[p, Blank[]]]]], Pattern[x, Blank[Symbol]]] := Dist[1/n, Subst[Int[SubstFor[x^n, Pq, x]*(a + b*x + c*x^2)^p, x], x, x^n], x] /; FreeQ[{a, b, c, m, n, p}, x] && EqQ[n2, 2*n] && PolyQ[Pq, x^n] && EqQ[Simplify[m - n + 1], 0]
  1746. Int[Times[Pattern[Pq, Blank[]], Power[Times[Optional[Pattern[d, Blank[]]], Pattern[x, Blank[]]], Optional[Pattern[m, Blank[]]]], Power[Plus[Pattern[a, Blank[]], Times[Optional[Pattern[b, Blank[]]], Power[Pattern[x, Blank[]], Optional[Pattern[n, Blank[]]]]], Times[Optional[Pattern[c, Blank[]]], Power[Pattern[x, Blank[]], Optional[Pattern[n2, Blank[]]]]]], Optional[Pattern[p, Blank[]]]]], Pattern[x, Blank[Symbol]]] := Int[ExpandIntegrand[(d*x)^m*Pq*(a + b*x^n + c*x^(2*n))^p, x], x] /; FreeQ[{a, b, c, d, m, n}, x] && EqQ[n2, 2*n] && PolyQ[Pq, x] && IGtQ[p, 0]
  1747. Int[Times[Power[Times[Optional[Pattern[g, Blank[]]], Pattern[x, Blank[]]], Optional[Pattern[m, Blank[]]]], Power[Plus[Pattern[a, Blank[]], Times[Optional[Pattern[b, Blank[]]], Power[Pattern[x, Blank[]], Optional[Pattern[n, Blank[]]]]], Times[Optional[Pattern[c, Blank[]]], Power[Pattern[x, Blank[]], Optional[Pattern[n2, Blank[]]]]]], Optional[Pattern[p, Blank[]]]], Plus[Pattern[d, Blank[]], Times[Optional[Pattern[e, Blank[]]], Power[Pattern[x, Blank[]], Optional[Pattern[n, Blank[]]]]], Times[Optional[Pattern[f, Blank[]]], Power[Pattern[x, Blank[]], Optional[Pattern[n2, Blank[]]]]]]], Pattern[x, Blank[Symbol]]] := Simp[(d*(g*x)^(m + 1)*(a + b*x^n + c*x^(2*n))^(p + 1))/(a*g*(m + 1)), x] /; FreeQ[{a, b, c, d, e, f, g, m, n, p}, x] && EqQ[n2, 2*n] && EqQ[a*e*(m + 1) - b*d*(m + n*(p + 1) + 1), 0] && EqQ[a*f*(m + 1) - c*d*(m + 2*n*(p + 1) + 1), 0] && NeQ[m, -1]
  1748. Int[Times[Power[Times[Optional[Pattern[g, Blank[]]], Pattern[x, Blank[]]], Optional[Pattern[m, Blank[]]]], Power[Plus[Pattern[a, Blank[]], Times[Optional[Pattern[b, Blank[]]], Power[Pattern[x, Blank[]], Optional[Pattern[n, Blank[]]]]], Times[Optional[Pattern[c, Blank[]]], Power[Pattern[x, Blank[]], Optional[Pattern[n2, Blank[]]]]]], Optional[Pattern[p, Blank[]]]], Plus[Pattern[d, Blank[]], Times[Optional[Pattern[f, Blank[]]], Power[Pattern[x, Blank[]], Optional[Pattern[n2, Blank[]]]]]]], Pattern[x, Blank[Symbol]]] := Simp[(d*(g*x)^(m + 1)*(a + b*x^n + c*x^(2*n))^(p + 1))/(a*g*(m + 1)), x] /; FreeQ[{a, b, c, d, f, g, m, n, p}, x] && EqQ[n2, 2*n] && EqQ[m + n*(p + 1) + 1, 0] && EqQ[c*d + a*f, 0] && NeQ[m, -1]
  1749. Int[Times[Pattern[Pq, Blank[]], Power[Times[Optional[Pattern[d, Blank[]]], Pattern[x, Blank[]]], Optional[Pattern[m, Blank[]]]], Power[Plus[Pattern[a, Blank[]], Times[Optional[Pattern[b, Blank[]]], Power[Pattern[x, Blank[]], Optional[Pattern[n, Blank[]]]]], Times[Optional[Pattern[c, Blank[]]], Power[Pattern[x, Blank[]], Optional[Pattern[n2, Blank[]]]]]], Pattern[p, Blank[]]]], Pattern[x, Blank[Symbol]]] := Dist[(a + b*x^n + c*x^(2*n))^FracPart[p]/((4*c)^IntPart[p]*(b + 2*c*x^n)^(2*FracPart[p])), Int[(d*x)^m*Pq*(b + 2*c*x^n)^(2*p), x], x] /; FreeQ[{a, b, c, d, m, n, p}, x] && EqQ[n2, 2*n] && PolyQ[Pq, x] && EqQ[b^2 - 4*a*c, 0] && !IntegerQ[2*p]
  1750. Int[Times[Pattern[Pq, Blank[]], Power[Pattern[x, Blank[]], Optional[Pattern[m, Blank[]]]], Power[Plus[Pattern[a, Blank[]], Times[Optional[Pattern[c, Blank[]]], Power[Pattern[x, Blank[]], Optional[Pattern[n2, Blank[]]]]], Times[Optional[Pattern[b, Blank[]]], Power[Pattern[x, Blank[]], Pattern[n, Blank[]]]]], Pattern[p, Blank[]]]], Pattern[x, Blank[Symbol]]] := Dist[1/n, Subst[Int[x^(Simplify[(m + 1)/n] - 1)*SubstFor[x^n, Pq, x]*(a + b*x + c*x^2)^p, x], x, x^n], x] /; FreeQ[{a, b, c, m, n, p}, x] && EqQ[n2, 2*n] && PolyQ[Pq, x^n] && NeQ[b^2 - 4*a*c, 0] && IntegerQ[Simplify[(m + 1)/n]]
  1751. Int[Times[Pattern[Pq, Blank[]], Power[Times[Pattern[d, Blank[]], Pattern[x, Blank[]]], Optional[Pattern[m, Blank[]]]], Power[Plus[Pattern[a, Blank[]], Times[Optional[Pattern[c, Blank[]]], Power[Pattern[x, Blank[]], Optional[Pattern[n2, Blank[]]]]], Times[Optional[Pattern[b, Blank[]]], Power[Pattern[x, Blank[]], Pattern[n, Blank[]]]]], Pattern[p, Blank[]]]], Pattern[x, Blank[Symbol]]] := Dist[(d*x)^m/x^m, Int[x^m*Pq*(a + b*x^n + c*x^(2*n))^p, x], x] /; FreeQ[{a, b, c, d, m, n, p}, x] && EqQ[n2, 2*n] && PolyQ[Pq, x^n] && NeQ[b^2 - 4*a*c, 0] && IntegerQ[Simplify[(m + 1)/n]]
  1752. Int[Times[Pattern[Pq, Blank[]], Power[Times[Optional[Pattern[d, Blank[]]], Pattern[x, Blank[]]], Optional[Pattern[m, Blank[]]]], Power[Plus[Pattern[a, Blank[]], Times[Optional[Pattern[b, Blank[]]], Power[Pattern[x, Blank[]], Optional[Pattern[n, Blank[]]]]], Times[Optional[Pattern[c, Blank[]]], Power[Pattern[x, Blank[]], Optional[Pattern[n2, Blank[]]]]]], Pattern[p, Blank[]]]], Pattern[x, Blank[Symbol]]] := Dist[1/d, Int[(d*x)^(m + 1)*PolynomialQuotient[Pq, x, x]*(a + b*x^n + c*x^(2*n))^p, x], x] /; FreeQ[{a, b, c, d, m, n, p}, x] && EqQ[n2, 2*n] && PolyQ[Pq, x] && EqQ[Coeff[Pq, x, 0], 0]
  1753. Int[Times[Power[Pattern[x, Blank[]], Optional[Pattern[m, Blank[]]]], Power[Plus[Pattern[a, Blank[]], Times[Optional[Pattern[b, Blank[]]], Power[Pattern[x, Blank[]], Optional[Pattern[n, Blank[]]]]], Times[Optional[Pattern[c, Blank[]]], Power[Pattern[x, Blank[]], Optional[Pattern[n2, Blank[]]]]]], Rational[-3, 2]], Plus[Pattern[e, Blank[]], Times[Optional[Pattern[f, Blank[]]], Power[Pattern[x, Blank[]], Optional[Pattern[q, Blank[]]]]], Times[Optional[Pattern[g, Blank[]]], Power[Pattern[x, Blank[]], Optional[Pattern[r, Blank[]]]]], Times[Optional[Pattern[h, Blank[]]], Power[Pattern[x, Blank[]], Optional[Pattern[s, Blank[]]]]]]], Pattern[x, Blank[Symbol]]] := -Simp[(2*c*(b*f - 2*a*g) + 2*h*(b^2 - 4*a*c)*x^(n/2) + 2*c*(2*c*f - b*g)*x^n)/(c*n*(b^2 - 4*a*c)*Sqrt[a + b*x^n + c*x^(2*n)]), x] /; FreeQ[{a, b, c, e, f, g, h, m, n}, x] && EqQ[n2, 2*n] && EqQ[q, n/2] && EqQ[r, (3*n)/2] && EqQ[s, 2*n] && NeQ[b^2 - 4*a*c, 0] && EqQ[2*m - n + 2, 0] && EqQ[c*e + a*h, 0]
  1754. Int[Times[Power[Times[Pattern[d, Blank[]], Pattern[x, Blank[]]], Optional[Pattern[m, Blank[]]]], Power[Plus[Pattern[a, Blank[]], Times[Optional[Pattern[b, Blank[]]], Power[Pattern[x, Blank[]], Optional[Pattern[n, Blank[]]]]], Times[Optional[Pattern[c, Blank[]]], Power[Pattern[x, Blank[]], Optional[Pattern[n2, Blank[]]]]]], Rational[-3, 2]], Plus[Pattern[e, Blank[]], Times[Optional[Pattern[f, Blank[]]], Power[Pattern[x, Blank[]], Optional[Pattern[q, Blank[]]]]], Times[Optional[Pattern[g, Blank[]]], Power[Pattern[x, Blank[]], Optional[Pattern[r, Blank[]]]]], Times[Optional[Pattern[h, Blank[]]], Power[Pattern[x, Blank[]], Optional[Pattern[s, Blank[]]]]]]], Pattern[x, Blank[Symbol]]] := Dist[(d*x)^m/x^m, Int[(x^m*(e + f*x^(n/2) + g*x^((3*n)/2) + h*x^(2*n)))/(a + b*x^n + c*x^(2*n))^(3/2), x], x] /; FreeQ[{a, b, c, d, e, f, g, h, m, n}, x] && EqQ[n2, 2*n] && EqQ[q, n/2] && EqQ[r, (3*n)/2] && EqQ[s, 2*n] && NeQ[b^2 - 4*a*c, 0] && EqQ[2*m - n + 2, 0] && EqQ[c*e + a*h, 0]
  1755. Int[Times[Pattern[Pq, Blank[]], Power[Pattern[x, Blank[]], Pattern[m, Blank[]]], Power[Plus[Pattern[a, Blank[]], Times[Optional[Pattern[b, Blank[]]], Power[Pattern[x, Blank[]], Pattern[n, Blank[]]]], Times[Optional[Pattern[c, Blank[]]], Power[Pattern[x, Blank[]], Pattern[n2, Blank[]]]]], Pattern[p, Blank[]]]], Pattern[x, Blank[Symbol]]] := Module[{q = Expon[Pq, x]}, Module[{Q = PolynomialQuotient[a*(b*c)^(Floor[(q - 1)/n] + 1)*x^m*Pq, a + b*x^n + c*x^(2*n), x], R = PolynomialRemainder[a*(b*c)^(Floor[(q - 1)/n] + 1)*x^m*Pq, a + b*x^n + c*x^(2*n), x], i}, Dist[1/(a*n*(p + 1)*(b^2 - 4*a*c)*(b*c)^(Floor[(q - 1)/n] + 1)), Int[x^m*(a + b*x^n + c*x^(2*n))^(p + 1)*ExpandToSum[(n*(p + 1)*(b^2 - 4*a*c)*Q)/x^m + Sum[(((b^2*(n*(p + 1) + i + 1))/a - 2*c*(2*n*(p + 1) + i + 1))*Coeff[R, x, i] - b*(i + 1)*Coeff[R, x, n + i])*x^(i - m) + c*(n*(2*p + 3) + i + 1)*((b*Coeff[R, x, i])/a - 2*Coeff[R, x, n + i])*x^(n + i - m), {i, 0, n - 1}], x], x], x] - Simp[(x*(a + b*x^n + c*x^(2*n))^(p + 1)*Sum[((b^2 - 2*a*c)*Coeff[R, x, i] - a*b*Coeff[R, x, n + i])*x^i + c*(b*Coeff[R, x, i] - 2*a*Coeff[R, x, n + i])*x^(n + i), {i, 0, n - 1}])/(a^2*n*(p + 1)*(b^2 - 4*a*c)*(b*c)^(Floor[(q - 1)/n] + 1)), x]] /; GeQ[q, 2*n]] /; FreeQ[{a, b, c}, x] && EqQ[n2, 2*n] && PolyQ[Pq, x] && NeQ[b^2 - 4*a*c, 0] && IGtQ[n, 0] && LtQ[p, -1] && ILtQ[m, 0]
  1756. Int[Times[Pattern[Pq, Blank[]], Power[Pattern[x, Blank[]], Optional[Pattern[m, Blank[]]]], Power[Plus[Pattern[a, Blank[]], Times[Optional[Pattern[b, Blank[]]], Power[Pattern[x, Blank[]], Pattern[n, Blank[]]]], Times[Optional[Pattern[c, Blank[]]], Power[Pattern[x, Blank[]], Pattern[n2, Blank[]]]]], Pattern[p, Blank[]]]], Pattern[x, Blank[Symbol]]] := With[{g = GCD[m + 1, n]}, Dist[1/g, Subst[Int[x^((m + 1)/g - 1)*(Pq /. x -> x^(1/g))*(a + b*x^(n/g) + c*x^((2*n)/g))^p, x], x, x^g], x] /; NeQ[g, 1]] /; FreeQ[{a, b, c, p}, x] && EqQ[n2, 2*n] && PolyQ[Pq, x^n] && NeQ[b^2 - 4*a*c, 0] && IGtQ[n, 0] && IntegerQ[m]
  1757. Int[Times[Pattern[Pq, Blank[]], Power[Times[Optional[Pattern[d, Blank[]]], Pattern[x, Blank[]]], Optional[Pattern[m, Blank[]]]], Power[Plus[Pattern[a, Blank[]], Times[Optional[Pattern[b, Blank[]]], Power[Pattern[x, Blank[]], Optional[Pattern[n, Blank[]]]]], Times[Optional[Pattern[c, Blank[]]], Power[Pattern[x, Blank[]], Pattern[n2, Blank[]]]]], -1]], Pattern[x, Blank[Symbol]]] := Int[ExpandIntegrand[((d*x)^m*Pq)/(a + b*x^n + c*x^(2*n)), x], x] /; FreeQ[{a, b, c, d, m}, x] && EqQ[n2, 2*n] && PolyQ[Pq, x^n] && NeQ[b^2 - 4*a*c, 0] && IGtQ[n, 0] && NiceSqrtQ[b^2 - 4*a*c]
  1758. Int[Times[Pattern[Pq, Blank[]], Power[Times[Optional[Pattern[d, Blank[]]], Pattern[x, Blank[]]], Optional[Pattern[m, Blank[]]]], Power[Plus[Pattern[a, Blank[]], Times[Optional[Pattern[b, Blank[]]], Power[Pattern[x, Blank[]], Optional[Pattern[n, Blank[]]]]], Times[Optional[Pattern[c, Blank[]]], Power[Pattern[x, Blank[]], Pattern[n2, Blank[]]]]], Pattern[p, Blank[]]]], Pattern[x, Blank[Symbol]]] := With[{q = Expon[Pq, x]}, With[{Pqq = Coeff[Pq, x, q]}, Int[(d*x)^m*ExpandToSum[Pq - Pqq*x^q - (Pqq*(a*(m + q - 2*n + 1)*x^(q - 2*n) + b*(m + q + n*(p - 1) + 1)*x^(q - n)))/(c*(m + q + 2*n*p + 1)), x]*(a + b*x^n + c*x^(2*n))^p, x] + Simp[(Pqq*(d*x)^(m + q - 2*n + 1)*(a + b*x^n + c*x^(2*n))^(p + 1))/(c*d^(q - 2*n + 1)*(m + q + 2*n*p + 1)), x]] /; GeQ[q, 2*n] && NeQ[m + q + 2*n*p + 1, 0] && (IntegerQ[2*p] || (EqQ[n, 1] && IntegerQ[4*p]) || IntegerQ[p + (q + 1)/(2*n)])] /; FreeQ[{a, b, c, d, m, p}, x] && EqQ[n2, 2*n] && PolyQ[Pq, x^n] && NeQ[b^2 - 4*a*c, 0] && IGtQ[n, 0]
  1759. Int[Times[Pattern[Pq, Blank[]], Power[Times[Optional[Pattern[d, Blank[]]], Pattern[x, Blank[]]], Optional[Pattern[m, Blank[]]]], Power[Plus[Pattern[a, Blank[]], Times[Optional[Pattern[b, Blank[]]], Power[Pattern[x, Blank[]], Pattern[n, Blank[]]]], Times[Optional[Pattern[c, Blank[]]], Power[Pattern[x, Blank[]], Pattern[n2, Blank[]]]]], Pattern[p, Blank[]]]], Pattern[x, Blank[Symbol]]] := Module[{q = Expon[Pq, x], j, k}, Int[Sum[(1*(d*x)^(m + j)*Sum[Coeff[Pq, x, j + k*n]*x^(k*n), {k, 0, (q - j)/n + 1}]*(a + b*x^n + c*x^(2*n))^p)/d^j, {j, 0, n - 1}], x]] /; FreeQ[{a, b, c, d, m, p}, x] && EqQ[n2, 2*n] && PolyQ[Pq, x] && NeQ[b^2 - 4*a*c, 0] && IGtQ[n, 0] && !PolyQ[Pq, x^n]
  1760. Int[Times[Pattern[Pq, Blank[]], Power[Times[Optional[Pattern[d, Blank[]]], Pattern[x, Blank[]]], Optional[Pattern[m, Blank[]]]], Power[Plus[Pattern[a, Blank[]], Times[Optional[Pattern[b, Blank[]]], Power[Pattern[x, Blank[]], Optional[Pattern[n, Blank[]]]]], Times[Optional[Pattern[c, Blank[]]], Power[Pattern[x, Blank[]], Optional[Pattern[n2, Blank[]]]]]], -1]], Pattern[x, Blank[Symbol]]] := Int[RationalFunctionExpand[((d*x)^m*Pq)/(a + b*x^n + c*x^(2*n)), x], x] /; FreeQ[{a, b, c, d, m}, x] && EqQ[n2, 2*n] && PolyQ[Pq, x] && NeQ[b^2 - 4*a*c, 0] && IGtQ[n, 0]
  1761. Int[Times[Pattern[Pq, Blank[]], Power[Pattern[x, Blank[]], Optional[Pattern[m, Blank[]]]], Power[Plus[Pattern[a, Blank[]], Times[Optional[Pattern[c, Blank[]]], Power[Pattern[x, Blank[]], Optional[Pattern[n2, Blank[]]]]], Times[Optional[Pattern[b, Blank[]]], Power[Pattern[x, Blank[]], Pattern[n, Blank[]]]]], Pattern[p, Blank[]]]], Pattern[x, Blank[Symbol]]] := With[{q = Expon[Pq, x]}, -Subst[Int[(ExpandToSum[x^q*(Pq /. x -> x^(-1)), x]*(a + b/x^n + c/x^(2*n))^p)/x^(m + q + 2), x], x, 1/x]] /; FreeQ[{a, b, c, p}, x] && EqQ[n2, 2*n] && PolyQ[Pq, x] && NeQ[b^2 - 4*a*c, 0] && ILtQ[n, 0] && IntegerQ[m]
  1762. Int[Times[Pattern[Pq, Blank[]], Power[Times[Optional[Pattern[d, Blank[]]], Pattern[x, Blank[]]], Optional[Pattern[m, Blank[]]]], Power[Plus[Pattern[a, Blank[]], Times[Optional[Pattern[c, Blank[]]], Power[Pattern[x, Blank[]], Optional[Pattern[n2, Blank[]]]]], Times[Optional[Pattern[b, Blank[]]], Power[Pattern[x, Blank[]], Pattern[n, Blank[]]]]], Pattern[p, Blank[]]]], Pattern[x, Blank[Symbol]]] := With[{g = Denominator[m], q = Expon[Pq, x]}, -Dist[g/d, Subst[Int[(ExpandToSum[x^(g*q)*(Pq /. x -> 1/(d*x^g)), x]*(a + b/(d^n*x^(g*n)) + c/(d^(2*n)*x^(2*g*n)))^p)/x^(g*(m + q + 1) + 1), x], x, 1/(d*x)^(1/g)], x]] /; FreeQ[{a, b, c, d, p}, x] && EqQ[n2, 2*n] && PolyQ[Pq, x] && NeQ[b^2 - 4*a*c, 0] && ILtQ[n, 0] && FractionQ[m]
  1763. Int[Times[Pattern[Pq, Blank[]], Power[Times[Optional[Pattern[d, Blank[]]], Pattern[x, Blank[]]], Pattern[m, Blank[]]], Power[Plus[Pattern[a, Blank[]], Times[Optional[Pattern[c, Blank[]]], Power[Pattern[x, Blank[]], Optional[Pattern[n2, Blank[]]]]], Times[Optional[Pattern[b, Blank[]]], Power[Pattern[x, Blank[]], Pattern[n, Blank[]]]]], Pattern[p, Blank[]]]], Pattern[x, Blank[Symbol]]] := With[{q = Expon[Pq, x]}, -Dist[(d*x)^m*(x^(-1))^m, Subst[Int[(ExpandToSum[x^q*(Pq /. x -> x^(-1)), x]*(a + b/x^n + c/x^(2*n))^p)/x^(m + q + 2), x], x, 1/x], x]] /; FreeQ[{a, b, c, d, m, p}, x] && EqQ[n2, 2*n] && PolyQ[Pq, x] && NeQ[b^2 - 4*a*c, 0] && ILtQ[n, 0] && !RationalQ[m]
  1764. Int[Times[Pattern[Pq, Blank[]], Power[Pattern[x, Blank[]], Optional[Pattern[m, Blank[]]]], Power[Plus[Pattern[a, Blank[]], Times[Optional[Pattern[c, Blank[]]], Power[Pattern[x, Blank[]], Optional[Pattern[n2, Blank[]]]]], Times[Optional[Pattern[b, Blank[]]], Power[Pattern[x, Blank[]], Pattern[n, Blank[]]]]], Pattern[p, Blank[]]]], Pattern[x, Blank[Symbol]]] := With[{g = Denominator[n]}, Dist[g, Subst[Int[x^(g*(m + 1) - 1)*(Pq /. x -> x^g)*(a + b*x^(g*n) + c*x^(2*g*n))^p, x], x, x^(1/g)], x]] /; FreeQ[{a, b, c, m, p}, x] && EqQ[n2, 2*n] && PolyQ[Pq, x] && NeQ[b^2 - 4*a*c, 0] && FractionQ[n]
  1765. Int[Times[Pattern[Pq, Blank[]], Power[Times[Pattern[d, Blank[]], Pattern[x, Blank[]]], Pattern[m, Blank[]]], Power[Plus[Pattern[a, Blank[]], Times[Optional[Pattern[c, Blank[]]], Power[Pattern[x, Blank[]], Optional[Pattern[n2, Blank[]]]]], Times[Optional[Pattern[b, Blank[]]], Power[Pattern[x, Blank[]], Pattern[n, Blank[]]]]], Pattern[p, Blank[]]]], Pattern[x, Blank[Symbol]]] := Dist[(d^(m - 1/2)*Sqrt[d*x])/Sqrt[x], Int[x^m*Pq*(a + b*x^n + c*x^(2*n))^p, x], x] /; FreeQ[{a, b, c, d, p}, x] && EqQ[n2, 2*n] && PolyQ[Pq, x] && NeQ[b^2 - 4*a*c, 0] && FractionQ[n] && IGtQ[m + 1/2, 0]
  1766. Int[Times[Pattern[Pq, Blank[]], Power[Times[Pattern[d, Blank[]], Pattern[x, Blank[]]], Pattern[m, Blank[]]], Power[Plus[Pattern[a, Blank[]], Times[Optional[Pattern[c, Blank[]]], Power[Pattern[x, Blank[]], Optional[Pattern[n2, Blank[]]]]], Times[Optional[Pattern[b, Blank[]]], Power[Pattern[x, Blank[]], Pattern[n, Blank[]]]]], Pattern[p, Blank[]]]], Pattern[x, Blank[Symbol]]] := Dist[(d^(m + 1/2)*Sqrt[x])/Sqrt[d*x], Int[x^m*Pq*(a + b*x^n + c*x^(2*n))^p, x], x] /; FreeQ[{a, b, c, d, p}, x] && EqQ[n2, 2*n] && PolyQ[Pq, x] && NeQ[b^2 - 4*a*c, 0] && FractionQ[n] && ILtQ[m - 1/2, 0]
  1767. Int[Times[Pattern[Pq, Blank[]], Power[Times[Pattern[d, Blank[]], Pattern[x, Blank[]]], Pattern[m, Blank[]]], Power[Plus[Pattern[a, Blank[]], Times[Optional[Pattern[c, Blank[]]], Power[Pattern[x, Blank[]], Optional[Pattern[n2, Blank[]]]]], Times[Optional[Pattern[b, Blank[]]], Power[Pattern[x, Blank[]], Pattern[n, Blank[]]]]], Pattern[p, Blank[]]]], Pattern[x, Blank[Symbol]]] := Dist[(d*x)^m/x^m, Int[x^m*Pq*(a + b*x^n + c*x^(2*n))^p, x], x] /; FreeQ[{a, b, c, d, m, p}, x] && EqQ[n2, 2*n] && PolyQ[Pq, x] && NeQ[b^2 - 4*a*c, 0] && FractionQ[n]
  1768. Int[Times[Pattern[Pq, Blank[]], Power[Pattern[x, Blank[]], Optional[Pattern[m, Blank[]]]], Power[Plus[Pattern[a, Blank[]], Times[Optional[Pattern[c, Blank[]]], Power[Pattern[x, Blank[]], Optional[Pattern[n2, Blank[]]]]], Times[Optional[Pattern[b, Blank[]]], Power[Pattern[x, Blank[]], Pattern[n, Blank[]]]]], Pattern[p, Blank[]]]], Pattern[x, Blank[Symbol]]] := Dist[1/(m + 1), Subst[Int[(SubstFor[x^n, Pq, x] /. x -> x^Simplify[n/(m + 1)])*(a + b*x^Simplify[n/(m + 1)] + c*x^Simplify[(2*n)/(m + 1)])^p, x], x, x^(m + 1)], x] /; FreeQ[{a, b, c, m, n, p}, x] && EqQ[n2, 2*n] && PolyQ[Pq, x^n] && NeQ[b^2 - 4*a*c, 0] && IntegerQ[Simplify[n/(m + 1)]] && !IntegerQ[n]
  1769. Int[Times[Pattern[Pq, Blank[]], Power[Times[Pattern[d, Blank[]], Pattern[x, Blank[]]], Pattern[m, Blank[]]], Power[Plus[Pattern[a, Blank[]], Times[Optional[Pattern[c, Blank[]]], Power[Pattern[x, Blank[]], Optional[Pattern[n2, Blank[]]]]], Times[Optional[Pattern[b, Blank[]]], Power[Pattern[x, Blank[]], Pattern[n, Blank[]]]]], Pattern[p, Blank[]]]], Pattern[x, Blank[Symbol]]] := Dist[(d*x)^m/x^m, Int[x^m*Pq*(a + b*x^n + c*x^(2*n))^p, x], x] /; FreeQ[{a, b, c, d, m, p}, x] && EqQ[n2, 2*n] && PolyQ[Pq, x^n] && NeQ[b^2 - 4*a*c, 0] && IntegerQ[Simplify[n/(m + 1)]] && !IntegerQ[n]
  1770. Int[Times[Pattern[Pq, Blank[]], Power[Times[Optional[Pattern[d, Blank[]]], Pattern[x, Blank[]]], Optional[Pattern[m, Blank[]]]], Power[Plus[Pattern[a, Blank[]], Times[Optional[Pattern[b, Blank[]]], Power[Pattern[x, Blank[]], Optional[Pattern[n, Blank[]]]]], Times[Optional[Pattern[c, Blank[]]], Power[Pattern[x, Blank[]], Optional[Pattern[n2, Blank[]]]]]], -1]], Pattern[x, Blank[Symbol]]] := With[{q = Rt[b^2 - 4*a*c, 2]}, Dist[(2*c)/q, Int[((d*x)^m*Pq)/(b - q + 2*c*x^n), x], x] - Dist[(2*c)/q, Int[((d*x)^m*Pq)/(b + q + 2*c*x^n), x], x]] /; FreeQ[{a, b, c, d, m, n}, x] && EqQ[n2, 2*n] && PolyQ[Pq, x] && NeQ[b^2 - 4*a*c, 0]
  1771. Int[Times[Pattern[Pq, Blank[]], Power[Times[Optional[Pattern[d, Blank[]]], Pattern[x, Blank[]]], Optional[Pattern[m, Blank[]]]], Power[Plus[Pattern[a, Blank[]], Times[Optional[Pattern[b, Blank[]]], Power[Pattern[x, Blank[]], Optional[Pattern[n, Blank[]]]]], Times[Optional[Pattern[c, Blank[]]], Power[Pattern[x, Blank[]], Optional[Pattern[n2, Blank[]]]]]], Pattern[p, Blank[]]]], Pattern[x, Blank[Symbol]]] := Int[ExpandIntegrand[(d*x)^m*Pq*(a + b*x^n + c*x^(2*n))^p, x], x] /; FreeQ[{a, b, c, d, m, n}, x] && EqQ[n2, 2*n] && PolyQ[Pq, x] && ILtQ[p + 1, 0]
  1772. Int[Times[Pattern[Pq, Blank[]], Power[Times[Optional[Pattern[d, Blank[]]], Pattern[x, Blank[]]], Optional[Pattern[m, Blank[]]]], Power[Plus[Pattern[a, Blank[]], Times[Optional[Pattern[b, Blank[]]], Power[Pattern[x, Blank[]], Optional[Pattern[n, Blank[]]]]], Times[Optional[Pattern[c, Blank[]]], Power[Pattern[x, Blank[]], Optional[Pattern[n2, Blank[]]]]]], Optional[Pattern[p, Blank[]]]]], Pattern[x, Blank[Symbol]]] := Unintegrable[Pq*(d*x)^m*(a + b*x^n + c*x^(2*n))^p, x] /; FreeQ[{a, b, c, d, m, n, p}, x] && EqQ[n2, 2*n] && (PolyQ[Pq, x] || PolyQ[Pq, x^n])
  1773. Int[Times[Pattern[Pq, Blank[]], Power[Pattern[u, Blank[]], Optional[Pattern[m, Blank[]]]], Power[Plus[Pattern[a, Blank[]], Times[Optional[Pattern[c, Blank[]]], Power[Pattern[v, Blank[]], Optional[Pattern[n2, Blank[]]]]], Times[Optional[Pattern[b, Blank[]]], Power[Pattern[v, Blank[]], Pattern[n, Blank[]]]]], Optional[Pattern[p, Blank[]]]]], Pattern[x, Blank[Symbol]]] := Dist[u^m/(Coefficient[v, x, 1]*v^m), Subst[Int[x^m*SubstFor[v, Pq, x]*(a + b*x^n + c*x^(2*n))^p, x], x, v], x] /; FreeQ[{a, b, c, m, n, p}, x] && EqQ[n2, 2*n] && LinearPairQ[u, v, x] && PolyQ[Pq, v^n]
  1774. Int[Times[Pattern[Pq, Blank[]], Power[Plus[Pattern[a, Blank[]], Times[Optional[Pattern[b, Blank[]]], Power[Pattern[x, Blank[]], Optional[Pattern[n, Blank[]]]]], Times[Optional[Pattern[c, Blank[]]], Power[Pattern[x, Blank[]], Optional[Pattern[n2, Blank[]]]]]], Optional[Pattern[p, Blank[]]]]], Pattern[x, Blank[Symbol]]] := Int[ExpandIntegrand[Pq*(a + b*x^n + c*x^(2*n))^p, x], x] /; FreeQ[{a, b, c, n}, x] && EqQ[n2, 2*n] && PolyQ[Pq, x] && IGtQ[p, 0]
  1775. Int[Times[Power[Plus[Pattern[a, Blank[]], Times[Optional[Pattern[b, Blank[]]], Power[Pattern[x, Blank[]], Optional[Pattern[n, Blank[]]]]], Times[Optional[Pattern[c, Blank[]]], Power[Pattern[x, Blank[]], Optional[Pattern[n2, Blank[]]]]]], Optional[Pattern[p, Blank[]]]], Plus[Pattern[d, Blank[]], Times[Optional[Pattern[e, Blank[]]], Power[Pattern[x, Blank[]], Optional[Pattern[n, Blank[]]]]], Times[Optional[Pattern[f, Blank[]]], Power[Pattern[x, Blank[]], Optional[Pattern[n2, Blank[]]]]]]], Pattern[x, Blank[Symbol]]] := Simp[(d*x*(a + b*x^n + c*x^(2*n))^(p + 1))/a, x] /; FreeQ[{a, b, c, d, e, f, n, p}, x] && EqQ[n2, 2*n] && EqQ[a*e - b*d*(n*(p + 1) + 1), 0] && EqQ[a*f - c*d*(2*n*(p + 1) + 1), 0]
  1776. Int[Times[Power[Plus[Pattern[a, Blank[]], Times[Optional[Pattern[b, Blank[]]], Power[Pattern[x, Blank[]], Optional[Pattern[n, Blank[]]]]], Times[Optional[Pattern[c, Blank[]]], Power[Pattern[x, Blank[]], Optional[Pattern[n2, Blank[]]]]]], Optional[Pattern[p, Blank[]]]], Plus[Pattern[d, Blank[]], Times[Optional[Pattern[f, Blank[]]], Power[Pattern[x, Blank[]], Optional[Pattern[n2, Blank[]]]]]]], Pattern[x, Blank[Symbol]]] := Simp[(d*x*(a + b*x^n + c*x^(2*n))^(p + 1))/a, x] /; FreeQ[{a, b, c, d, f, n, p}, x] && EqQ[n2, 2*n] && EqQ[n*(p + 1) + 1, 0] && EqQ[c*d + a*f, 0]
  1777. Int[Times[Pattern[Pq, Blank[]], Power[Plus[Pattern[a, Blank[]], Times[Optional[Pattern[b, Blank[]]], Power[Pattern[x, Blank[]], Optional[Pattern[n, Blank[]]]]], Times[Optional[Pattern[c, Blank[]]], Power[Pattern[x, Blank[]], Optional[Pattern[n2, Blank[]]]]]], Pattern[p, Blank[]]]], Pattern[x, Blank[Symbol]]] := Dist[(a + b*x^n + c*x^(2*n))^FracPart[p]/((4*c)^IntPart[p]*(b + 2*c*x^n)^(2*FracPart[p])), Int[Pq*(b + 2*c*x^n)^(2*p), x], x] /; FreeQ[{a, b, c, n, p}, x] && EqQ[n2, 2*n] && PolyQ[Pq, x] && EqQ[b^2 - 4*a*c, 0] && !IntegerQ[2*p]
  1778. Int[Times[Pattern[Pq, Blank[]], Power[Plus[Pattern[a, Blank[]], Times[Optional[Pattern[b, Blank[]]], Power[Pattern[x, Blank[]], Optional[Pattern[n, Blank[]]]]], Times[Optional[Pattern[c, Blank[]]], Power[Pattern[x, Blank[]], Optional[Pattern[n2, Blank[]]]]]], Pattern[p, Blank[]]]], Pattern[x, Blank[Symbol]]] := Int[x*PolynomialQuotient[Pq, x, x]*(a + b*x^n + c*x^(2*n))^p, x] /; FreeQ[{a, b, c, n, p}, x] && EqQ[n2, 2*n] && PolyQ[Pq, x] && EqQ[Coeff[Pq, x, 0], 0] && !MatchQ[Pq, x^(m_.)*(u_.) /; IntegerQ[m]]
  1779. Int[Times[Power[Plus[Pattern[a, Blank[]], Times[Optional[Pattern[c, Blank[]]], Power[Pattern[x, Blank[]], Optional[Pattern[n2, Blank[]]]]], Times[Optional[Pattern[b, Blank[]]], Power[Pattern[x, Blank[]], Pattern[n, Blank[]]]]], Optional[Pattern[p, Blank[]]]], Plus[Pattern[d, Blank[]], Times[Optional[Pattern[f, Blank[]]], Power[Pattern[x, Blank[]], Optional[Pattern[n2, Blank[]]]]], Times[Optional[Pattern[g, Blank[]]], Power[Pattern[x, Blank[]], Optional[Pattern[n3, Blank[]]]]], Times[Optional[Pattern[e, Blank[]]], Power[Pattern[x, Blank[]], Pattern[n, Blank[]]]]]], Pattern[x, Blank[Symbol]]] := Simp[(x*(a*d*(n + 1) + (a*e - b*d*(n*(p + 1) + 1))*x^n)*(a + b*x^n + c*x^(2*n))^(p + 1))/(a^2*(n + 1)), x] /; FreeQ[{a, b, c, d, e, f, g, n, p}, x] && EqQ[n2, 2*n] && EqQ[n3, 3*n] && NeQ[b^2 - 4*a*c, 0] && EqQ[a^2*g*(n + 1) - c*(n*(2*p + 3) + 1)*(a*e - b*d*(n*(p + 1) + 1)), 0] && EqQ[a^2*f*(n + 1) - a*c*d*(n + 1)*(2*n*(p + 1) + 1) - b*(n*(p + 2) + 1)*(a*e - b*d*(n*(p + 1) + 1)), 0]
  1780. Int[Times[Plus[Pattern[d, Blank[]], Times[Optional[Pattern[f, Blank[]]], Power[Pattern[x, Blank[]], Optional[Pattern[n2, Blank[]]]]], Times[Optional[Pattern[g, Blank[]]], Power[Pattern[x, Blank[]], Optional[Pattern[n3, Blank[]]]]]], Power[Plus[Pattern[a, Blank[]], Times[Optional[Pattern[c, Blank[]]], Power[Pattern[x, Blank[]], Optional[Pattern[n2, Blank[]]]]], Times[Optional[Pattern[b, Blank[]]], Power[Pattern[x, Blank[]], Pattern[n, Blank[]]]]], Optional[Pattern[p, Blank[]]]]], Pattern[x, Blank[Symbol]]] := Simp[(d*x*(a*(n + 1) - b*(n*(p + 1) + 1)*x^n)*(a + b*x^n + c*x^(2*n))^(p + 1))/(a^2*(n + 1)), x] /; FreeQ[{a, b, c, d, f, g, n, p}, x] && EqQ[n2, 2*n] && EqQ[n3, 3*n] && NeQ[b^2 - 4*a*c, 0] && EqQ[a^2*g*(n + 1) + c*b*d*(n*(2*p + 3) + 1)*(n*(p + 1) + 1), 0] && EqQ[a^2*f*(n + 1) - a*c*d*(n + 1)*(2*n*(p + 1) + 1) + b^2*d*(n*(p + 2) + 1)*(n*(p + 1) + 1), 0]
  1781. Int[Times[Power[Plus[Pattern[a, Blank[]], Times[Optional[Pattern[c, Blank[]]], Power[Pattern[x, Blank[]], Optional[Pattern[n2, Blank[]]]]], Times[Optional[Pattern[b, Blank[]]], Power[Pattern[x, Blank[]], Pattern[n, Blank[]]]]], Optional[Pattern[p, Blank[]]]], Plus[Pattern[d, Blank[]], Times[Optional[Pattern[g, Blank[]]], Power[Pattern[x, Blank[]], Optional[Pattern[n3, Blank[]]]]], Times[Optional[Pattern[e, Blank[]]], Power[Pattern[x, Blank[]], Pattern[n, Blank[]]]]]], Pattern[x, Blank[Symbol]]] := Simp[(x*(a*d*(n + 1) + (a*e - b*d*(n*(p + 1) + 1))*x^n)*(a + b*x^n + c*x^(2*n))^(p + 1))/(a^2*(n + 1)), x] /; FreeQ[{a, b, c, d, e, g, n, p}, x] && EqQ[n2, 2*n] && EqQ[n3, 3*n] && NeQ[b^2 - 4*a*c, 0] && EqQ[a^2*g*(n + 1) - c*(n*(2*p + 3) + 1)*(a*e - b*d*(n*(p + 1) + 1)), 0] && EqQ[a*c*d*(n + 1)*(2*n*(p + 1) + 1) + b*(n*(p + 2) + 1)*(a*e - b*d*(n*(p + 1) + 1)), 0]
  1782. Int[Times[Plus[Pattern[d, Blank[]], Times[Optional[Pattern[g, Blank[]]], Power[Pattern[x, Blank[]], Optional[Pattern[n3, Blank[]]]]]], Power[Plus[Pattern[a, Blank[]], Times[Optional[Pattern[c, Blank[]]], Power[Pattern[x, Blank[]], Optional[Pattern[n2, Blank[]]]]], Times[Optional[Pattern[b, Blank[]]], Power[Pattern[x, Blank[]], Pattern[n, Blank[]]]]], Optional[Pattern[p, Blank[]]]]], Pattern[x, Blank[Symbol]]] := Simp[(d*x*(a*(n + 1) - b*(n*(p + 1) + 1)*x^n)*(a + b*x^n + c*x^(2*n))^(p + 1))/(a^2*(n + 1)), x] /; FreeQ[{a, b, c, d, g, n, p}, x] && EqQ[n2, 2*n] && EqQ[n3, 3*n] && NeQ[b^2 - 4*a*c, 0] && EqQ[a^2*g*(n + 1) + c*b*d*(n*(2*p + 3) + 1)*(n*(p + 1) + 1), 0] && EqQ[a*c*d*(n + 1)*(2*n*(p + 1) + 1) - b^2*d*(n*(p + 2) + 1)*(n*(p + 1) + 1), 0]
  1783. Int[Times[Pattern[Pq, Blank[]], Power[Plus[Pattern[a, Blank[]], Times[Optional[Pattern[b, Blank[]]], Power[Pattern[x, Blank[]], Pattern[n, Blank[]]]], Times[Optional[Pattern[c, Blank[]]], Power[Pattern[x, Blank[]], Pattern[n2, Blank[]]]]], Pattern[p, Blank[]]]], Pattern[x, Blank[Symbol]]] := Module[{q = Expon[Pq, x], i}, -Simp[(x*(a + b*x^n + c*x^(2*n))^(p + 1)*Sum[((b^2 - 2*a*c)*Coeff[Pq, x, i] - a*b*Coeff[Pq, x, n + i])*x^i + c*(b*Coeff[Pq, x, i] - 2*a*Coeff[Pq, x, n + i])*x^(n + i), {i, 0, n - 1}])/(a*n*(p + 1)*(b^2 - 4*a*c)), x] + Dist[1/(a*n*(p + 1)*(b^2 - 4*a*c)), Int[(a + b*x^n + c*x^(2*n))^(p + 1)*Sum[((b^2*(n*(p + 1) + i + 1) - 2*a*c*(2*n*(p + 1) + i + 1))*Coeff[Pq, x, i] - a*b*(i + 1)*Coeff[Pq, x, n + i])*x^i + c*(n*(2*p + 3) + i + 1)*(b*Coeff[Pq, x, i] - 2*a*Coeff[Pq, x, n + i])*x^(n + i), {i, 0, n - 1}], x], x] /; LtQ[q, 2*n]] /; FreeQ[{a, b, c}, x] && EqQ[n2, 2*n] && PolyQ[Pq, x] && NeQ[b^2 - 4*a*c, 0] && IGtQ[n, 0] && LtQ[p, -1]
  1784. Int[Times[Pattern[Pq, Blank[]], Power[Plus[Pattern[a, Blank[]], Times[Optional[Pattern[b, Blank[]]], Power[Pattern[x, Blank[]], Pattern[n, Blank[]]]], Times[Optional[Pattern[c, Blank[]]], Power[Pattern[x, Blank[]], Pattern[n2, Blank[]]]]], Pattern[p, Blank[]]]], Pattern[x, Blank[Symbol]]] := With[{q = Expon[Pq, x]}, Module[{Q = PolynomialQuotient[(b*c)^(Floor[(q - 1)/n] + 1)*Pq, a + b*x^n + c*x^(2*n), x], R = PolynomialRemainder[(b*c)^(Floor[(q - 1)/n] + 1)*Pq, a + b*x^n + c*x^(2*n), x], i}, Dist[1/(a*n*(p + 1)*(b^2 - 4*a*c)*(b*c)^(Floor[(q - 1)/n] + 1)), Int[(a + b*x^n + c*x^(2*n))^(p + 1)*ExpandToSum[a*n*(p + 1)*(b^2 - 4*a*c)*Q + Sum[((b^2*(n*(p + 1) + i + 1) - 2*a*c*(2*n*(p + 1) + i + 1))*Coeff[R, x, i] - a*b*(i + 1)*Coeff[R, x, n + i])*x^i + c*(n*(2*p + 3) + i + 1)*(b*Coeff[R, x, i] - 2*a*Coeff[R, x, n + i])*x^(n + i), {i, 0, n - 1}], x], x], x] - Simp[(x*(a + b*x^n + c*x^(2*n))^(p + 1)*Sum[((b^2 - 2*a*c)*Coeff[R, x, i] - a*b*Coeff[R, x, n + i])*x^i + c*(b*Coeff[R, x, i] - 2*a*Coeff[R, x, n + i])*x^(n + i), {i, 0, n - 1}])/(a*n*(p + 1)*(b^2 - 4*a*c)*(b*c)^(Floor[(q - 1)/n] + 1)), x]] /; GeQ[q, 2*n]] /; FreeQ[{a, b, c}, x] && EqQ[n2, 2*n] && PolyQ[Pq, x] && NeQ[b^2 - 4*a*c, 0] && IGtQ[n, 0] && LtQ[p, -1]
  1785. Int[Times[Pattern[Pq, Blank[]], Power[Plus[Pattern[a, Blank[]], Times[Optional[Pattern[b, Blank[]]], Power[Pattern[x, Blank[]], Optional[Pattern[n, Blank[]]]]], Times[Optional[Pattern[c, Blank[]]], Power[Pattern[x, Blank[]], Pattern[n2, Blank[]]]]], -1]], Pattern[x, Blank[Symbol]]] := Int[ExpandIntegrand[Pq/(a + b*x^n + c*x^(2*n)), x], x] /; FreeQ[{a, b, c}, x] && EqQ[n2, 2*n] && PolyQ[Pq, x^n] && NeQ[b^2 - 4*a*c, 0] && IGtQ[n, 0] && (NiceSqrtQ[b^2 - 4*a*c] || LtQ[Expon[Pq, x], n])
  1786. Int[Times[Pattern[Pq, Blank[]], Power[Plus[Pattern[a, Blank[]], Times[Optional[Pattern[b, Blank[]]], Pattern[x, Blank[]]], Times[Optional[Pattern[c, Blank[]]], Power[Pattern[x, Blank[]], 2]]], Pattern[p, Blank[]]]], Pattern[x, Blank[Symbol]]] := With[{q = Expon[Pq, x]}, With[{Pqq = Coeff[Pq, x, q]}, Dist[1/2, Int[ExpandToSum[2*Pq - (c^p*Pqq*(b + 2*c*x))/(a + b*x + c*x^2)^(p + 1), x]*(a + b*x + c*x^2)^p, x], x] + Simp[(c^p*Pqq*Log[a + b*x + c*x^2])/2, x]] /; EqQ[q + 2*p + 1, 0]] /; FreeQ[{a, b, c}, x] && PolyQ[Pq, x] && NeQ[b^2 - 4*a*c, 0] && ILtQ[p, 0]
  1787. Int[Times[Pattern[Pq, Blank[]], Power[Plus[Pattern[a, Blank[]], Times[Optional[Pattern[b, Blank[]]], Pattern[x, Blank[]]], Times[Optional[Pattern[c, Blank[]]], Power[Pattern[x, Blank[]], 2]]], Pattern[p, Blank[]]]], Pattern[x, Blank[Symbol]]] := With[{q = Expon[Pq, x]}, With[{Pqq = Coeff[Pq, x, q]}, Int[ExpandToSum[Pq - (c^(p + 1/2)*Pqq)/(a + b*x + c*x^2)^(p + 1/2), x]*(a + b*x + c*x^2)^p, x] + Simp[c^p*Pqq*ArcTanh[(b + 2*c*x)/(2*Rt[c, 2]*Sqrt[a + b*x + c*x^2])], x]] /; EqQ[q + 2*p + 1, 0]] /; FreeQ[{a, b, c}, x] && PolyQ[Pq, x] && NeQ[b^2 - 4*a*c, 0] && ILtQ[p + 1/2, 0] && PosQ[c]
  1788. Int[Times[Pattern[Pq, Blank[]], Power[Plus[Pattern[a, Blank[]], Times[Optional[Pattern[b, Blank[]]], Pattern[x, Blank[]]], Times[Optional[Pattern[c, Blank[]]], Power[Pattern[x, Blank[]], 2]]], Pattern[p, Blank[]]]], Pattern[x, Blank[Symbol]]] := With[{q = Expon[Pq, x]}, With[{Pqq = Coeff[Pq, x, q]}, Int[ExpandToSum[Pq - ((-c)^(p + 1/2)*Pqq)/(a + b*x + c*x^2)^(p + 1/2), x]*(a + b*x + c*x^2)^p, x] - Simp[(-c)^p*Pqq*ArcTan[(b + 2*c*x)/(2*Rt[-c, 2]*Sqrt[a + b*x + c*x^2])], x]] /; EqQ[q + 2*p + 1, 0]] /; FreeQ[{a, b, c}, x] && PolyQ[Pq, x] && NeQ[b^2 - 4*a*c, 0] && ILtQ[p + 1/2, 0] && NegQ[c]
  1789. Int[Times[Pattern[Pq, Blank[]], Power[Plus[Pattern[a, Blank[]], Times[Optional[Pattern[b, Blank[]]], Power[Pattern[x, Blank[]], Optional[Pattern[n, Blank[]]]]], Times[Optional[Pattern[c, Blank[]]], Power[Pattern[x, Blank[]], Pattern[n2, Blank[]]]]], Pattern[p, Blank[]]]], Pattern[x, Blank[Symbol]]] := With[{q = Expon[Pq, x]}, With[{Pqq = Coeff[Pq, x, q]}, Int[ExpandToSum[Pq - Pqq*x^q - (Pqq*(a*(q - 2*n + 1)*x^(q - 2*n) + b*(q + n*(p - 1) + 1)*x^(q - n)))/(c*(q + 2*n*p + 1)), x]*(a + b*x^n + c*x^(2*n))^p, x] + Simp[(Pqq*x^(q - 2*n + 1)*(a + b*x^n + c*x^(2*n))^(p + 1))/(c*(q + 2*n*p + 1)), x]] /; GeQ[q, 2*n] && NeQ[q + 2*n*p + 1, 0] && (IntegerQ[2*p] || (EqQ[n, 1] && IntegerQ[4*p]) || IntegerQ[p + (q + 1)/(2*n)])] /; FreeQ[{a, b, c, p}, x] && EqQ[n2, 2*n] && PolyQ[Pq, x^n] && NeQ[b^2 - 4*a*c, 0] && IGtQ[n, 0]
  1790. Int[Times[Pattern[Pq, Blank[]], Power[Plus[Pattern[a, Blank[]], Times[Optional[Pattern[b, Blank[]]], Power[Pattern[x, Blank[]], Pattern[n, Blank[]]]], Times[Optional[Pattern[c, Blank[]]], Power[Pattern[x, Blank[]], Pattern[n2, Blank[]]]]], Pattern[p, Blank[]]]], Pattern[x, Blank[Symbol]]] := Module[{q = Expon[Pq, x], j, k}, Int[Sum[x^j*Sum[Coeff[Pq, x, j + k*n]*x^(k*n), {k, 0, (q - j)/n + 1}]*(a + b*x^n + c*x^(2*n))^p, {j, 0, n - 1}], x]] /; FreeQ[{a, b, c, p}, x] && EqQ[n2, 2*n] && PolyQ[Pq, x] && NeQ[b^2 - 4*a*c, 0] && IGtQ[n, 0] && !PolyQ[Pq, x^n]
  1791. Int[Times[Pattern[Pq, Blank[]], Power[Plus[Pattern[a, Blank[]], Times[Optional[Pattern[b, Blank[]]], Power[Pattern[x, Blank[]], Optional[Pattern[n, Blank[]]]]], Times[Optional[Pattern[c, Blank[]]], Power[Pattern[x, Blank[]], Optional[Pattern[n2, Blank[]]]]]], -1]], Pattern[x, Blank[Symbol]]] := Int[RationalFunctionExpand[Pq/(a + b*x^n + c*x^(2*n)), x], x] /; FreeQ[{a, b, c}, x] && EqQ[n2, 2*n] && PolyQ[Pq, x] && NeQ[b^2 - 4*a*c, 0] && IGtQ[n, 0]
  1792. Int[Times[Pattern[Pq, Blank[]], Power[Plus[Pattern[a, Blank[]], Times[Optional[Pattern[c, Blank[]]], Power[Pattern[x, Blank[]], Optional[Pattern[n2, Blank[]]]]], Times[Optional[Pattern[b, Blank[]]], Power[Pattern[x, Blank[]], Pattern[n, Blank[]]]]], Pattern[p, Blank[]]]], Pattern[x, Blank[Symbol]]] := With[{g = Denominator[n]}, Dist[g, Subst[Int[x^(g - 1)*(Pq /. x -> x^g)*(a + b*x^(g*n) + c*x^(2*g*n))^p, x], x, x^(1/g)], x]] /; FreeQ[{a, b, c, p}, x] && EqQ[n2, 2*n] && PolyQ[Pq, x] && NeQ[b^2 - 4*a*c, 0] && FractionQ[n]
  1793. Int[Times[Pattern[Pq, Blank[]], Power[Plus[Pattern[a, Blank[]], Times[Optional[Pattern[b, Blank[]]], Power[Pattern[x, Blank[]], Optional[Pattern[n, Blank[]]]]], Times[Optional[Pattern[c, Blank[]]], Power[Pattern[x, Blank[]], Optional[Pattern[n2, Blank[]]]]]], -1]], Pattern[x, Blank[Symbol]]] := With[{q = Rt[b^2 - 4*a*c, 2]}, Dist[(2*c)/q, Int[Pq/(b - q + 2*c*x^n), x], x] - Dist[(2*c)/q, Int[Pq/(b + q + 2*c*x^n), x], x]] /; FreeQ[{a, b, c, n}, x] && EqQ[n2, 2*n] && PolyQ[Pq, x] && NeQ[b^2 - 4*a*c, 0]
  1794. Int[Times[Pattern[P3, Blank[]], Power[Plus[Pattern[a, Blank[]], Times[Optional[Pattern[b, Blank[]]], Power[Pattern[x, Blank[]], Pattern[n, Blank[]]]], Times[Optional[Pattern[c, Blank[]]], Power[Pattern[x, Blank[]], Pattern[n2, Blank[]]]]], Pattern[p, Blank[]]]], Pattern[x, Blank[Symbol]]] := With[{d = Coeff[P3, x^n, 0], e = Coeff[P3, x^n, 1], f = Coeff[P3, x^n, 2], g = Coeff[P3, x^n, 3]}, -Simp[(x*(b^2*c*d - 2*a*c*(c*d - a*f) - a*b*(c*e + a*g) + (b*c*(c*d + a*f) - a*b^2*g - 2*a*c*(c*e - a*g))*x^n)*(a + b*x^n + c*x^(2*n))^(p + 1))/(a*c*n*(p + 1)*(b^2 - 4*a*c)), x] - Dist[1/(a*c*n*(p + 1)*(b^2 - 4*a*c)), Int[(a + b*x^n + c*x^(2*n))^(p + 1)*Simp[a*b*(c*e + a*g) - b^2*c*d*(n + n*p + 1) - 2*a*c*(a*f - c*d*(2*n*(p + 1) + 1)) + (a*b^2*g*(n*(p + 2) + 1) - b*c*(c*d + a*f)*(n*(2*p + 3) + 1) - 2*a*c*(a*g*(n + 1) - c*e*(n*(2*p + 3) + 1)))*x^n, x], x], x]] /; FreeQ[{a, b, c, n}, x] && EqQ[n2, 2*n] && PolyQ[P3, x^n, 3] && NeQ[b^2 - 4*a*c, 0] && ILtQ[p, -1]
  1795. Int[Times[Pattern[P2, Blank[]], Power[Plus[Pattern[a, Blank[]], Times[Optional[Pattern[b, Blank[]]], Power[Pattern[x, Blank[]], Pattern[n, Blank[]]]], Times[Optional[Pattern[c, Blank[]]], Power[Pattern[x, Blank[]], Pattern[n2, Blank[]]]]], Pattern[p, Blank[]]]], Pattern[x, Blank[Symbol]]] := With[{d = Coeff[P2, x^n, 0], e = Coeff[P2, x^n, 1], f = Coeff[P2, x^n, 2]}, -Simp[(x*(b^2*d - 2*a*(c*d - a*f) - a*b*e + (b*(c*d + a*f) - 2*a*c*e)*x^n)*(a + b*x^n + c*x^(2*n))^(p + 1))/(a*n*(p + 1)*(b^2 - 4*a*c)), x] - Dist[1/(a*n*(p + 1)*(b^2 - 4*a*c)), Int[(a + b*x^n + c*x^(2*n))^(p + 1)*Simp[a*b*e - b^2*d*(n + n*p + 1) - 2*a*(a*f - c*d*(2*n*(p + 1) + 1)) - (b*(c*d + a*f)*(n*(2*p + 3) + 1) - 2*a*c*e*(n*(2*p + 3) + 1))*x^n, x], x], x]] /; FreeQ[{a, b, c, n}, x] && EqQ[n2, 2*n] && PolyQ[P2, x^n, 2] && NeQ[b^2 - 4*a*c, 0] && ILtQ[p, -1]
  1796. Int[Times[Pattern[Pq, Blank[]], Power[Plus[Pattern[a, Blank[]], Times[Optional[Pattern[b, Blank[]]], Power[Pattern[x, Blank[]], Optional[Pattern[n, Blank[]]]]], Times[Optional[Pattern[c, Blank[]]], Power[Pattern[x, Blank[]], Optional[Pattern[n2, Blank[]]]]]], Pattern[p, Blank[]]]], Pattern[x, Blank[Symbol]]] := Int[ExpandIntegrand[Pq*(a + b*x^n + c*x^(2*n))^p, x], x] /; FreeQ[{a, b, c, n}, x] && EqQ[n2, 2*n] && PolyQ[Pq, x] && ILtQ[p, -1]
  1797. Int[Times[Pattern[Pq, Blank[]], Power[Plus[Pattern[a, Blank[]], Times[Optional[Pattern[b, Blank[]]], Power[Pattern[x, Blank[]], Optional[Pattern[n, Blank[]]]]], Times[Optional[Pattern[c, Blank[]]], Power[Pattern[x, Blank[]], Optional[Pattern[n2, Blank[]]]]]], Optional[Pattern[p, Blank[]]]]], Pattern[x, Blank[Symbol]]] := Unintegrable[Pq*(a + b*x^n + c*x^(2*n))^p, x] /; FreeQ[{a, b, c, n, p}, x] && EqQ[n2, 2*n] && (PolyQ[Pq, x] || PolyQ[Pq, x^n])
  1798. Int[Times[Pattern[Pq, Blank[]], Power[Plus[Pattern[a, Blank[]], Times[Optional[Pattern[c, Blank[]]], Power[Pattern[v, Blank[]], Optional[Pattern[n2, Blank[]]]]], Times[Optional[Pattern[b, Blank[]]], Power[Pattern[v, Blank[]], Pattern[n, Blank[]]]]], Optional[Pattern[p, Blank[]]]]], Pattern[x, Blank[Symbol]]] := Dist[1/Coefficient[v, x, 1], Subst[Int[SubstFor[v, Pq, x]*(a + b*x^n + c*x^(2*n))^p, x], x, v], x] /; FreeQ[{a, b, c, n, p}, x] && EqQ[n2, 2*n] && LinearQ[v, x] && PolyQ[Pq, v^n]
  1799. Int[Times[Pattern[Pq, Blank[]], Power[Pattern[x, Blank[]], Optional[Pattern[m, Blank[]]]], Power[Plus[Pattern[a, Blank[]], Times[Optional[Pattern[b, Blank[]]], Power[Pattern[x, Blank[]], 2]]], Optional[Pattern[p, Blank[]]]]], Pattern[x, Blank[Symbol]]] := Dist[1/2, Subst[Int[x^((m - 1)/2)*SubstFor[x^2, Pq, x]*(a + b*x)^p, x], x, x^2], x] /; FreeQ[{a, b, p}, x] && PolyQ[Pq, x^2] && IntegerQ[(m - 1)/2]
  1800. Int[Times[Pattern[Pq, Blank[]], Power[Times[Optional[Pattern[c, Blank[]]], Pattern[x, Blank[]]], Optional[Pattern[m, Blank[]]]], Power[Plus[Pattern[a, Blank[]], Times[Optional[Pattern[b, Blank[]]], Power[Pattern[x, Blank[]], 2]]], Optional[Pattern[p, Blank[]]]]], Pattern[x, Blank[Symbol]]] := Dist[1/c, Int[(c*x)^(m + 1)*PolynomialQuotient[Pq, x, x]*(a + b*x^2)^p, x], x] /; FreeQ[{a, b, c, m, p}, x] && PolyQ[Pq, x] && EqQ[Coeff[Pq, x, 0], 0]
  1801. Int[Times[Pattern[P2, Blank[]], Power[Times[Optional[Pattern[c, Blank[]]], Pattern[x, Blank[]]], Optional[Pattern[m, Blank[]]]], Power[Plus[Pattern[a, Blank[]], Times[Optional[Pattern[b, Blank[]]], Power[Pattern[x, Blank[]], 2]]], Optional[Pattern[p, Blank[]]]]], Pattern[x, Blank[Symbol]]] := With[{f = Coeff[P2, x, 0], g = Coeff[P2, x, 1], h = Coeff[P2, x, 2]}, Simp[(h*(c*x)^(m + 1)*(a + b*x^2)^(p + 1))/(b*c*(m + 2*p + 3)), x] /; EqQ[g, 0] && EqQ[a*h*(m + 1) - b*f*(m + 2*p + 3), 0]] /; FreeQ[{a, b, c, m, p}, x] && PolyQ[P2, x, 2] && NeQ[m, -1]
  1802. Int[Times[Pattern[Pq, Blank[]], Power[Times[Optional[Pattern[c, Blank[]]], Pattern[x, Blank[]]], Optional[Pattern[m, Blank[]]]], Power[Plus[Pattern[a, Blank[]], Times[Optional[Pattern[b, Blank[]]], Power[Pattern[x, Blank[]], 2]]], Optional[Pattern[p, Blank[]]]]], Pattern[x, Blank[Symbol]]] := Int[ExpandIntegrand[(c*x)^m*Pq*(a + b*x^2)^p, x], x] /; FreeQ[{a, b, c, m}, x] && PolyQ[Pq, x] && IGtQ[p, -2]
  1803. Int[Times[Pattern[Pq, Blank[]], Power[Pattern[x, Blank[]], Pattern[m, Blank[]]], Power[Plus[Pattern[a, Blank[]], Times[Optional[Pattern[b, Blank[]]], Power[Pattern[x, Blank[]], 2]]], Pattern[p, Blank[]]]], Pattern[x, Blank[Symbol]]] := With[{A = Coeff[Pq, x, 0], Q = PolynomialQuotient[Pq - Coeff[Pq, x, 0], x^2, x]}, Simp[(A*x^(m + 1)*(a + b*x^2)^(p + 1))/(a*(m + 1)), x] + Dist[1/(a*(m + 1)), Int[x^(m + 2)*(a + b*x^2)^p*(a*(m + 1)*Q - A*b*(m + 2*(p + 1) + 1)), x], x]] /; FreeQ[{a, b}, x] && PolyQ[Pq, x^2] && IntegerQ[m/2] && ILtQ[(m + 1)/2 + p, 0] && LtQ[m + Expon[Pq, x] + 2*p + 1, 0]
  1804. Int[Times[Pattern[Pq, Blank[]], Power[Times[Optional[Pattern[c, Blank[]]], Pattern[x, Blank[]]], Optional[Pattern[m, Blank[]]]], Power[Plus[Pattern[a, Blank[]], Times[Optional[Pattern[b, Blank[]]], Power[Pattern[x, Blank[]], 2]]], Pattern[p, Blank[]]]], Pattern[x, Blank[Symbol]]] := With[{Q = PolynomialQuotient[Pq, a + b*x^2, x], f = Coeff[PolynomialRemainder[Pq, a + b*x^2, x], x, 0], g = Coeff[PolynomialRemainder[Pq, a + b*x^2, x], x, 1]}, Simp[((c*x)^m*(a + b*x^2)^(p + 1)*(a*g - b*f*x))/(2*a*b*(p + 1)), x] + Dist[c/(2*a*b*(p + 1)), Int[(c*x)^(m - 1)*(a + b*x^2)^(p + 1)*ExpandToSum[2*a*b*(p + 1)*x*Q - a*g*m + b*f*(m + 2*p + 3)*x, x], x], x]] /; FreeQ[{a, b, c}, x] && PolyQ[Pq, x] && LtQ[p, -1] && GtQ[m, 0]
  1805. Int[Times[Pattern[Pq, Blank[]], Power[Times[Optional[Pattern[c, Blank[]]], Pattern[x, Blank[]]], Optional[Pattern[m, Blank[]]]], Power[Plus[Pattern[a, Blank[]], Times[Optional[Pattern[b, Blank[]]], Power[Pattern[x, Blank[]], 2]]], Pattern[p, Blank[]]]], Pattern[x, Blank[Symbol]]] := With[{Q = PolynomialQuotient[(c*x)^m*Pq, a + b*x^2, x], f = Coeff[PolynomialRemainder[(c*x)^m*Pq, a + b*x^2, x], x, 0], g = Coeff[PolynomialRemainder[(c*x)^m*Pq, a + b*x^2, x], x, 1]}, Simp[((a*g - b*f*x)*(a + b*x^2)^(p + 1))/(2*a*b*(p + 1)), x] + Dist[1/(2*a*(p + 1)), Int[(c*x)^m*(a + b*x^2)^(p + 1)*ExpandToSum[(2*a*(p + 1)*Q)/(c*x)^m + (f*(2*p + 3))/(c*x)^m, x], x], x]] /; FreeQ[{a, b, c}, x] && PolyQ[Pq, x] && LtQ[p, -1] && ILtQ[m, 0]
  1806. Int[Times[Pattern[Pq, Blank[]], Power[Times[Optional[Pattern[c, Blank[]]], Pattern[x, Blank[]]], Optional[Pattern[m, Blank[]]]], Power[Plus[Pattern[a, Blank[]], Times[Optional[Pattern[b, Blank[]]], Power[Pattern[x, Blank[]], 2]]], Pattern[p, Blank[]]]], Pattern[x, Blank[Symbol]]] := With[{Q = PolynomialQuotient[Pq, a + b*x^2, x], f = Coeff[PolynomialRemainder[Pq, a + b*x^2, x], x, 0], g = Coeff[PolynomialRemainder[Pq, a + b*x^2, x], x, 1]}, -Simp[((c*x)^(m + 1)*(f + g*x)*(a + b*x^2)^(p + 1))/(2*a*c*(p + 1)), x] + Dist[1/(2*a*(p + 1)), Int[(c*x)^m*(a + b*x^2)^(p + 1)*ExpandToSum[2*a*(p + 1)*Q + f*(m + 2*p + 3) + g*(m + 2*p + 4)*x, x], x], x]] /; FreeQ[{a, b, c, m}, x] && PolyQ[Pq, x] && LtQ[p, -1] && !GtQ[m, 0]
  1807. Int[Times[Pattern[Pq, Blank[]], Power[Times[Optional[Pattern[c, Blank[]]], Pattern[x, Blank[]]], Pattern[m, Blank[]]], Power[Plus[Pattern[a, Blank[]], Times[Optional[Pattern[b, Blank[]]], Power[Pattern[x, Blank[]], 2]]], Pattern[p, Blank[]]]], Pattern[x, Blank[Symbol]]] := With[{Q = PolynomialQuotient[Pq, c*x, x], R = PolynomialRemainder[Pq, c*x, x]}, Simp[(R*(c*x)^(m + 1)*(a + b*x^2)^(p + 1))/(a*c*(m + 1)), x] + Dist[1/(a*c*(m + 1)), Int[(c*x)^(m + 1)*(a + b*x^2)^p*ExpandToSum[a*c*(m + 1)*Q - b*R*(m + 2*p + 3)*x, x], x], x]] /; FreeQ[{a, b, c, p}, x] && PolyQ[Pq, x] && LtQ[m, -1] && (IntegerQ[2*p] || NeQ[Expon[Pq, x], 1])
  1808. Int[Times[Pattern[Pq, Blank[]], Power[Times[Optional[Pattern[c, Blank[]]], Pattern[x, Blank[]]], Optional[Pattern[m, Blank[]]]], Power[Plus[Pattern[a, Blank[]], Times[Optional[Pattern[b, Blank[]]], Power[Pattern[x, Blank[]], 2]]], Optional[Pattern[p, Blank[]]]]], Pattern[x, Blank[Symbol]]] := With[{q = Expon[Pq, x]}, Dist[Coeff[Pq, x, q]/c^q, Int[(c*x)^(m + q)*(a + b*x^2)^p, x], x] + Dist[1/c^q, Int[(c*x)^m*(a + b*x^2)^p*ExpandToSum[c^q*Pq - Coeff[Pq, x, q]*(c*x)^q, x], x], x] /; EqQ[q, 1] || EqQ[m + q + 2*p + 1, 0]] /; FreeQ[{a, b, c, m, p}, x] && PolyQ[Pq, x] && !(IGtQ[m, 0] && ILtQ[p + 1/2, 0])
  1809. Int[Times[Pattern[Pq, Blank[]], Power[Times[Optional[Pattern[c, Blank[]]], Pattern[x, Blank[]]], Optional[Pattern[m, Blank[]]]], Power[Plus[Pattern[a, Blank[]], Times[Optional[Pattern[b, Blank[]]], Power[Pattern[x, Blank[]], 2]]], Pattern[p, Blank[]]]], Pattern[x, Blank[Symbol]]] := With[{q = Expon[Pq, x], f = Coeff[Pq, x, Expon[Pq, x]]}, Simp[(f*(c*x)^(m + q - 1)*(a + b*x^2)^(p + 1))/(b*c^(q - 1)*(m + q + 2*p + 1)), x] + Dist[1/(b*(m + q + 2*p + 1)), Int[(c*x)^m*(a + b*x^2)^p*ExpandToSum[b*(m + q + 2*p + 1)*Pq - b*f*(m + q + 2*p + 1)*x^q - a*f*(m + q - 1)*x^(q - 2), x], x], x] /; GtQ[q, 1] && NeQ[m + q + 2*p + 1, 0]] /; FreeQ[{a, b, c, m, p}, x] && PolyQ[Pq, x] && ( !IGtQ[m, 0] || IGtQ[p + 1/2, -1])
  1810. Int[Times[Pattern[Pq, Blank[]], Power[Plus[Pattern[a, Blank[]], Times[Optional[Pattern[b, Blank[]]], Power[Pattern[x, Blank[]], 2]]], Optional[Pattern[p, Blank[]]]]], Pattern[x, Blank[Symbol]]] := Int[ExpandIntegrand[Pq*(a + b*x^2)^p, x], x] /; FreeQ[{a, b}, x] && PolyQ[Pq, x] && IGtQ[p, -2]
  1811. Int[Times[Pattern[Pq, Blank[]], Power[Plus[Pattern[a, Blank[]], Times[Optional[Pattern[b, Blank[]]], Power[Pattern[x, Blank[]], 2]]], Pattern[p, Blank[]]]], Pattern[x, Blank[Symbol]]] := Int[x*PolynomialQuotient[Pq, x, x]*(a + b*x^2)^p, x] /; FreeQ[{a, b, p}, x] && PolyQ[Pq, x] && EqQ[Coeff[Pq, x, 0], 0] && !MatchQ[Pq, x^(m_.)*(u_.) /; IntegerQ[m]]
  1812. Int[Times[Pattern[Px, Blank[]], Power[Plus[Pattern[a, Blank[]], Times[Optional[Pattern[b, Blank[]]], Power[Pattern[x, Blank[]], 2]]], Optional[Pattern[p, Blank[]]]]], Pattern[x, Blank[Symbol]]] := Int[PolynomialQuotient[Px, a + b*x^2, x]*(a + b*x^2)^(p + 1), x] /; FreeQ[{a, b, p}, x] && PolyQ[Px, x] && EqQ[PolynomialRemainder[Px, a + b*x^2, x], 0]
  1813. Int[Times[Pattern[Pq, Blank[]], Power[Plus[Pattern[a, Blank[]], Times[Optional[Pattern[b, Blank[]]], Power[Pattern[x, Blank[]], 2]]], Pattern[p, Blank[]]]], Pattern[x, Blank[Symbol]]] := With[{A = Coeff[Pq, x, 0], Q = PolynomialQuotient[Pq - Coeff[Pq, x, 0], x^2, x]}, Simp[(A*x*(a + b*x^2)^(p + 1))/a, x] + Dist[1/a, Int[x^2*(a + b*x^2)^p*(a*Q - A*b*(2*p + 3)), x], x]] /; FreeQ[{a, b}, x] && PolyQ[Pq, x^2] && ILtQ[p + 1/2, 0] && LtQ[Expon[Pq, x] + 2*p + 1, 0]
  1814. Int[Times[Pattern[Pq, Blank[]], Power[Plus[Pattern[a, Blank[]], Times[Optional[Pattern[b, Blank[]]], Power[Pattern[x, Blank[]], 2]]], Pattern[p, Blank[]]]], Pattern[x, Blank[Symbol]]] := With[{Q = PolynomialQuotient[Pq, a + b*x^2, x], f = Coeff[PolynomialRemainder[Pq, a + b*x^2, x], x, 0], g = Coeff[PolynomialRemainder[Pq, a + b*x^2, x], x, 1]}, Simp[((a*g - b*f*x)*(a + b*x^2)^(p + 1))/(2*a*b*(p + 1)), x] + Dist[1/(2*a*(p + 1)), Int[(a + b*x^2)^(p + 1)*ExpandToSum[2*a*(p + 1)*Q + f*(2*p + 3), x], x], x]] /; FreeQ[{a, b}, x] && PolyQ[Pq, x] && LtQ[p, -1]
  1815. Int[Times[Pattern[Pq, Blank[]], Power[Plus[Pattern[a, Blank[]], Times[Optional[Pattern[b, Blank[]]], Power[Pattern[x, Blank[]], 2]]], Pattern[p, Blank[]]]], Pattern[x, Blank[Symbol]]] := With[{q = Expon[Pq, x], e = Coeff[Pq, x, Expon[Pq, x]]}, Simp[(e*x^(q - 1)*(a + b*x^2)^(p + 1))/(b*(q + 2*p + 1)), x] + Dist[1/(b*(q + 2*p + 1)), Int[(a + b*x^2)^p*ExpandToSum[b*(q + 2*p + 1)*Pq - a*e*(q - 1)*x^(q - 2) - b*e*(q + 2*p + 1)*x^q, x], x], x]] /; FreeQ[{a, b, p}, x] && PolyQ[Pq, x] && !LeQ[p, -1]
  1816. Int[Times[Power[Pattern[x, Blank[]], Optional[Pattern[m, Blank[]]]], Power[Plus[Pattern[a, Blank[]], Times[Optional[Pattern[c, Blank[]]], Power[Pattern[x, Blank[]], Optional[Pattern[n, Blank[]]]]]], Rational[-3, 2]], Plus[Pattern[e, Blank[]], Times[Optional[Pattern[h, Blank[]]], Power[Pattern[x, Blank[]], Optional[Pattern[n, Blank[]]]]], Times[Optional[Pattern[f, Blank[]]], Power[Pattern[x, Blank[]], Optional[Pattern[q, Blank[]]]]], Times[Optional[Pattern[g, Blank[]]], Power[Pattern[x, Blank[]], Optional[Pattern[r, Blank[]]]]]]], Pattern[x, Blank[Symbol]]] := -Simp[(2*a*g + 4*a*h*x^(n/4) - 2*c*f*x^(n/2))/(a*c*n*Sqrt[a + c*x^n]), x] /; FreeQ[{a, c, e, f, g, h, m, n}, x] && EqQ[q, n/4] && EqQ[r, (3*n)/4] && EqQ[4*m - n + 4, 0] && EqQ[c*e + a*h, 0]
  1817. Int[Times[Power[Times[Pattern[d, Blank[]], Pattern[x, Blank[]]], Optional[Pattern[m, Blank[]]]], Power[Plus[Pattern[a, Blank[]], Times[Optional[Pattern[c, Blank[]]], Power[Pattern[x, Blank[]], Optional[Pattern[n, Blank[]]]]]], Rational[-3, 2]], Plus[Pattern[e, Blank[]], Times[Optional[Pattern[h, Blank[]]], Power[Pattern[x, Blank[]], Optional[Pattern[n, Blank[]]]]], Times[Optional[Pattern[f, Blank[]]], Power[Pattern[x, Blank[]], Optional[Pattern[q, Blank[]]]]], Times[Optional[Pattern[g, Blank[]]], Power[Pattern[x, Blank[]], Optional[Pattern[r, Blank[]]]]]]], Pattern[x, Blank[Symbol]]] := Dist[(d*x)^m/x^m, Int[(x^m*(e + f*x^(n/4) + g*x^((3*n)/4) + h*x^n))/(a + c*x^n)^(3/2), x], x] /; FreeQ[{a, c, d, e, f, g, h, m, n}, x] && EqQ[4*m - n + 4, 0] && EqQ[q, n/4] && EqQ[r, (3*n)/4] && EqQ[c*e + a*h, 0]
  1818. Int[Times[Pattern[Pq, Blank[]], Power[Times[Optional[Pattern[c, Blank[]]], Pattern[x, Blank[]]], Pattern[m, Blank[]]], Power[Plus[Pattern[a, Blank[]], Times[Optional[Pattern[b, Blank[]]], Pattern[x, Blank[]]]], Pattern[p, Blank[]]]], Pattern[x, Blank[Symbol]]] := With[{n = Denominator[p]}, Dist[n/b, Subst[Int[x^(n*p + n - 1)*(-((a*c)/b) + (c*x^n)/b)^m*(Pq /. x -> -(a/b) + x^n/b), x], x, (a + b*x)^(1/n)], x]] /; FreeQ[{a, b, c, m}, x] && PolyQ[Pq, x] && FractionQ[p] && ILtQ[m, -1]
  1819. Int[Times[Pattern[Pq, Blank[]], Power[Pattern[x, Blank[]], Optional[Pattern[m, Blank[]]]], Power[Plus[Pattern[a, Blank[]], Times[Optional[Pattern[b, Blank[]]], Power[Pattern[x, Blank[]], Pattern[n, Blank[]]]]], Optional[Pattern[p, Blank[]]]]], Pattern[x, Blank[Symbol]]] := Dist[1/(m + 1), Subst[Int[SubstFor[x^(m + 1), Pq, x]*(a + b*x^Simplify[n/(m + 1)])^p, x], x, x^(m + 1)], x] /; FreeQ[{a, b, m, n, p}, x] && NeQ[m, -1] && IGtQ[Simplify[n/(m + 1)], 0] && PolyQ[Pq, x^(m + 1)]
  1820. Int[Times[Pattern[Pq, Blank[]], Power[Times[Optional[Pattern[c, Blank[]]], Pattern[x, Blank[]]], Optional[Pattern[m, Blank[]]]], Power[Plus[Pattern[a, Blank[]], Times[Optional[Pattern[b, Blank[]]], Power[Pattern[x, Blank[]], Optional[Pattern[n, Blank[]]]]]], Optional[Pattern[p, Blank[]]]]], Pattern[x, Blank[Symbol]]] := Int[ExpandIntegrand[(c*x)^m*Pq*(a + b*x^n)^p, x], x] /; FreeQ[{a, b, c, m, n}, x] && PolyQ[Pq, x] && (IGtQ[p, 0] || EqQ[n, 1])
  1821. Int[Times[Pattern[Pq, Blank[]], Power[Pattern[x, Blank[]], Optional[Pattern[m, Blank[]]]], Power[Plus[Pattern[a, Blank[]], Times[Optional[Pattern[b, Blank[]]], Power[Pattern[x, Blank[]], Pattern[n, Blank[]]]]], Optional[Pattern[p, Blank[]]]]], Pattern[x, Blank[Symbol]]] := Dist[1/n, Subst[Int[x^(Simplify[(m + 1)/n] - 1)*SubstFor[x^n, Pq, x]*(a + b*x)^p, x], x, x^n], x] /; FreeQ[{a, b, m, n, p}, x] && PolyQ[Pq, x^n] && IntegerQ[Simplify[(m + 1)/n]]
  1822. Int[Times[Pattern[Pq, Blank[]], Power[Times[Pattern[c, Blank[]], Pattern[x, Blank[]]], Optional[Pattern[m, Blank[]]]], Power[Plus[Pattern[a, Blank[]], Times[Optional[Pattern[b, Blank[]]], Power[Pattern[x, Blank[]], Pattern[n, Blank[]]]]], Optional[Pattern[p, Blank[]]]]], Pattern[x, Blank[Symbol]]] := Dist[(c^IntPart[m]*(c*x)^FracPart[m])/x^FracPart[m], Int[x^m*Pq*(a + b*x^n)^p, x], x] /; FreeQ[{a, b, c, m, n, p}, x] && PolyQ[Pq, x^n] && IntegerQ[Simplify[(m + 1)/n]]
  1823. Int[Times[Pattern[Pq, Blank[]], Power[Pattern[x, Blank[]], Optional[Pattern[m, Blank[]]]], Power[Plus[Pattern[a, Blank[]], Times[Optional[Pattern[b, Blank[]]], Power[Pattern[x, Blank[]], Pattern[n, Blank[]]]]], Pattern[p, Blank[]]]], Pattern[x, Blank[Symbol]]] := Simp[(Pq*(a + b*x^n)^(p + 1))/(b*n*(p + 1)), x] - Dist[1/(b*n*(p + 1)), Int[D[Pq, x]*(a + b*x^n)^(p + 1), x], x] /; FreeQ[{a, b, m, n}, x] && PolyQ[Pq, x] && EqQ[m - n + 1, 0] && LtQ[p, -1]
  1824. Int[Times[Pattern[Pq, Blank[]], Power[Times[Optional[Pattern[d, Blank[]]], Pattern[x, Blank[]]], Optional[Pattern[m, Blank[]]]], Power[Plus[Pattern[a, Blank[]], Times[Optional[Pattern[b, Blank[]]], Power[Pattern[x, Blank[]], Optional[Pattern[n, Blank[]]]]]], Pattern[p, Blank[]]]], Pattern[x, Blank[Symbol]]] := Dist[1/d, Int[(d*x)^(m + 1)*PolynomialQuotient[Pq, x, x]*(a + b*x^n)^p, x], x] /; FreeQ[{a, b, d, m, n, p}, x] && PolyQ[Pq, x] && EqQ[Coeff[Pq, x, 0], 0]
  1825. Int[Times[Pattern[Pq, Blank[]], Power[Pattern[x, Blank[]], Optional[Pattern[m, Blank[]]]], Power[Plus[Pattern[a, Blank[]], Times[Optional[Pattern[b, Blank[]]], Power[Pattern[x, Blank[]], Optional[Pattern[n, Blank[]]]]]], Pattern[p, Blank[]]]], Pattern[x, Blank[Symbol]]] := Module[{u = IntHide[x^m*Pq, x]}, Simp[u*(a + b*x^n)^p, x] - Dist[b*n*p, Int[x^(m + n)*(a + b*x^n)^(p - 1)*ExpandToSum[u/x^(m + 1), x], x], x]] /; FreeQ[{a, b}, x] && PolyQ[Pq, x] && IGtQ[n, 0] && GtQ[p, 0] && LtQ[m + Expon[Pq, x] + 1, 0]
  1826. Int[Times[Pattern[Pq, Blank[]], Power[Times[Optional[Pattern[c, Blank[]]], Pattern[x, Blank[]]], Optional[Pattern[m, Blank[]]]], Power[Plus[Pattern[a, Blank[]], Times[Optional[Pattern[b, Blank[]]], Power[Pattern[x, Blank[]], Optional[Pattern[n, Blank[]]]]]], Pattern[p, Blank[]]]], Pattern[x, Blank[Symbol]]] := Module[{q = Expon[Pq, x], i}, Simp[(c*x)^m*(a + b*x^n)^p*Sum[(Coeff[Pq, x, i]*x^(i + 1))/(m + n*p + i + 1), {i, 0, q}], x] + Dist[a*n*p, Int[(c*x)^m*(a + b*x^n)^(p - 1)*Sum[(Coeff[Pq, x, i]*x^i)/(m + n*p + i + 1), {i, 0, q}], x], x]] /; FreeQ[{a, b, c, m}, x] && PolyQ[Pq, x] && IGtQ[(n - 1)/2, 0] && GtQ[p, 0]
  1827. Int[Times[Pattern[P4, Blank[]], Power[Pattern[x, Blank[]], 2], Power[Plus[Pattern[a, Blank[]], Times[Optional[Pattern[b, Blank[]]], Power[Pattern[x, Blank[]], 4]]], Rational[-3, 2]]], Pattern[x, Blank[Symbol]]] := With[{e = Coeff[P4, x, 0], f = Coeff[P4, x, 1], h = Coeff[P4, x, 4]}, -Simp[(f - 2*h*x^3)/(2*b*Sqrt[a + b*x^4]), x] /; EqQ[b*e - 3*a*h, 0]] /; FreeQ[{a, b}, x] && PolyQ[P4, x, 4] && EqQ[Coeff[P4, x, 2], 0] && EqQ[Coeff[P4, x, 3], 0]
  1828. Int[Times[Pattern[Pq, Blank[]], Power[Pattern[x, Blank[]], Optional[Pattern[m, Blank[]]]], Power[Plus[Pattern[a, Blank[]], Times[Optional[Pattern[b, Blank[]]], Power[Pattern[x, Blank[]], Optional[Pattern[n, Blank[]]]]]], Pattern[p, Blank[]]]], Pattern[x, Blank[Symbol]]] := With[{q = m + Expon[Pq, x]}, Module[{Q = PolynomialQuotient[b^(Floor[(q - 1)/n] + 1)*x^m*Pq, a + b*x^n, x], R = PolynomialRemainder[b^(Floor[(q - 1)/n] + 1)*x^m*Pq, a + b*x^n, x]}, Dist[1/(a*n*(p + 1)*b^(Floor[(q - 1)/n] + 1)), Int[(a + b*x^n)^(p + 1)*ExpandToSum[a*n*(p + 1)*Q + n*(p + 1)*R + D[x*R, x], x], x], x] - Simp[(x*R*(a + b*x^n)^(p + 1))/(a*n*(p + 1)*b^(Floor[(q - 1)/n] + 1)), x]] /; GeQ[q, n]] /; FreeQ[{a, b}, x] && PolyQ[Pq, x] && IGtQ[n, 0] && LtQ[p, -1] && IGtQ[m, 0]
  1829. Int[Times[Pattern[Pq, Blank[]], Power[Pattern[x, Blank[]], Pattern[m, Blank[]]], Power[Plus[Pattern[a, Blank[]], Times[Optional[Pattern[b, Blank[]]], Power[Pattern[x, Blank[]], Optional[Pattern[n, Blank[]]]]]], Pattern[p, Blank[]]]], Pattern[x, Blank[Symbol]]] := With[{q = Expon[Pq, x]}, Module[{Q = PolynomialQuotient[a*b^(Floor[(q - 1)/n] + 1)*x^m*Pq, a + b*x^n, x], R = PolynomialRemainder[a*b^(Floor[(q - 1)/n] + 1)*x^m*Pq, a + b*x^n, x], i}, Dist[1/(a*n*(p + 1)*b^(Floor[(q - 1)/n] + 1)), Int[x^m*(a + b*x^n)^(p + 1)*ExpandToSum[(n*(p + 1)*Q)/x^m + Sum[((n*(p + 1) + i + 1)*Coeff[R, x, i]*x^(i - m))/a, {i, 0, n - 1}], x], x], x] - Simp[(x*R*(a + b*x^n)^(p + 1))/(a^2*n*(p + 1)*b^(Floor[(q - 1)/n] + 1)), x]]] /; FreeQ[{a, b}, x] && PolyQ[Pq, x] && IGtQ[n, 0] && LtQ[p, -1] && ILtQ[m, 0]
  1830. Int[Times[Pattern[Pq, Blank[]], Power[Pattern[x, Blank[]], Optional[Pattern[m, Blank[]]]], Power[Plus[Pattern[a, Blank[]], Times[Optional[Pattern[b, Blank[]]], Power[Pattern[x, Blank[]], Pattern[n, Blank[]]]]], Pattern[p, Blank[]]]], Pattern[x, Blank[Symbol]]] := With[{g = GCD[m + 1, n]}, Dist[1/g, Subst[Int[x^((m + 1)/g - 1)*(Pq /. x -> x^(1/g))*(a + b*x^(n/g))^p, x], x, x^g], x] /; g != 1] /; FreeQ[{a, b, p}, x] && PolyQ[Pq, x^n] && IGtQ[n, 0] && IntegerQ[m]
  1831. Int[Times[Pattern[Pq, Blank[]], Power[Times[Optional[Pattern[c, Blank[]]], Pattern[x, Blank[]]], Optional[Pattern[m, Blank[]]]], Power[Plus[Pattern[a, Blank[]], Times[Optional[Pattern[b, Blank[]]], Power[Pattern[x, Blank[]], Pattern[n, Blank[]]]]], -1]], Pattern[x, Blank[Symbol]]] := With[{v = Sum[((c*x)^(m + ii)*(Coeff[Pq, x, ii] + Coeff[Pq, x, n/2 + ii]*x^(n/2)))/(c^ii*(a + b*x^n)), {ii, 0, n/2 - 1}]}, Int[v, x] /; SumQ[v]] /; FreeQ[{a, b, c, m}, x] && PolyQ[Pq, x] && IGtQ[n/2, 0] && Expon[Pq, x] < n
  1832. Int[Times[Pattern[Pq, Blank[]], Power[Pattern[x, Blank[]], -1], Power[Plus[Pattern[a, Blank[]], Times[Optional[Pattern[b, Blank[]]], Power[Pattern[x, Blank[]], Pattern[n, Blank[]]]]], Rational[-1, 2]]], Pattern[x, Blank[Symbol]]] := Dist[Coeff[Pq, x, 0], Int[1/(x*Sqrt[a + b*x^n]), x], x] + Int[ExpandToSum[(Pq - Coeff[Pq, x, 0])/x, x]/Sqrt[a + b*x^n], x] /; FreeQ[{a, b}, x] && PolyQ[Pq, x] && IGtQ[n, 0] && NeQ[Coeff[Pq, x, 0], 0]
  1833. Int[Times[Pattern[Pq, Blank[]], Power[Times[Optional[Pattern[c, Blank[]]], Pattern[x, Blank[]]], Optional[Pattern[m, Blank[]]]], Power[Plus[Pattern[a, Blank[]], Times[Optional[Pattern[b, Blank[]]], Power[Pattern[x, Blank[]], Pattern[n, Blank[]]]]], Pattern[p, Blank[]]]], Pattern[x, Blank[Symbol]]] := Module[{q = Expon[Pq, x], j, k}, Int[Sum[((c*x)^(m + j)*Sum[Coeff[Pq, x, j + (k*n)/2]*x^((k*n)/2), {k, 0, (2*(q - j))/n + 1}]*(a + b*x^n)^p)/c^j, {j, 0, n/2 - 1}], x]] /; FreeQ[{a, b, c, m, p}, x] && PolyQ[Pq, x] && IGtQ[n/2, 0] && !PolyQ[Pq, x^(n/2)]
  1834. Int[Times[Pattern[Pq, Blank[]], Power[Times[Optional[Pattern[c, Blank[]]], Pattern[x, Blank[]]], Optional[Pattern[m, Blank[]]]], Power[Plus[Pattern[a, Blank[]], Times[Optional[Pattern[b, Blank[]]], Power[Pattern[x, Blank[]], Pattern[n, Blank[]]]]], -1]], Pattern[x, Blank[Symbol]]] := Int[ExpandIntegrand[((c*x)^m*Pq)/(a + b*x^n), x], x] /; FreeQ[{a, b, c, m}, x] && PolyQ[Pq, x] && IntegerQ[n] && !IGtQ[m, 0]
  1835. Int[Times[Pattern[Pq, Blank[]], Power[Times[Optional[Pattern[c, Blank[]]], Pattern[x, Blank[]]], Pattern[m, Blank[]]], Power[Plus[Pattern[a, Blank[]], Times[Optional[Pattern[b, Blank[]]], Power[Pattern[x, Blank[]], Pattern[n, Blank[]]]]], Pattern[p, Blank[]]]], Pattern[x, Blank[Symbol]]] := With[{Pq0 = Coeff[Pq, x, 0]}, Simp[(Pq0*(c*x)^(m + 1)*(a + b*x^n)^(p + 1))/(a*c*(m + 1)), x] + Dist[1/(2*a*c*(m + 1)), Int[(c*x)^(m + 1)*ExpandToSum[(2*a*(m + 1)*(Pq - Pq0))/x - 2*b*Pq0*(m + n*(p + 1) + 1)*x^(n - 1), x]*(a + b*x^n)^p, x], x] /; NeQ[Pq0, 0]] /; FreeQ[{a, b, c, p}, x] && PolyQ[Pq, x] && IGtQ[n, 0] && LtQ[m, -1] && LeQ[n - 1, Expon[Pq, x]]
  1836. Int[Times[Pattern[Pq, Blank[]], Power[Times[Optional[Pattern[c, Blank[]]], Pattern[x, Blank[]]], Optional[Pattern[m, Blank[]]]], Power[Plus[Pattern[a, Blank[]], Times[Optional[Pattern[b, Blank[]]], Power[Pattern[x, Blank[]], Pattern[n, Blank[]]]]], Pattern[p, Blank[]]]], Pattern[x, Blank[Symbol]]] := With[{q = Expon[Pq, x]}, With[{Pqq = Coeff[Pq, x, q]}, Dist[1/(b*(m + q + n*p + 1)), Int[(c*x)^m*ExpandToSum[b*(m + q + n*p + 1)*(Pq - Pqq*x^q) - a*Pqq*(m + q - n + 1)*x^(q - n), x]*(a + b*x^n)^p, x], x] + Simp[(Pqq*(c*x)^(m + q - n + 1)*(a + b*x^n)^(p + 1))/(b*c^(q - n + 1)*(m + q + n*p + 1)), x]] /; NeQ[m + q + n*p + 1, 0] && q - n >= 0 && (IntegerQ[2*p] || IntegerQ[p + (q + 1)/(2*n)])] /; FreeQ[{a, b, c, m, p}, x] && PolyQ[Pq, x] && IGtQ[n, 0]
  1837. Int[Times[Pattern[Pq, Blank[]], Power[Pattern[x, Blank[]], Optional[Pattern[m, Blank[]]]], Power[Plus[Pattern[a, Blank[]], Times[Optional[Pattern[b, Blank[]]], Power[Pattern[x, Blank[]], Pattern[n, Blank[]]]]], Pattern[p, Blank[]]]], Pattern[x, Blank[Symbol]]] := With[{q = Expon[Pq, x]}, -Subst[Int[(ExpandToSum[x^q*(Pq /. x -> x^(-1)), x]*(a + b/x^n)^p)/x^(m + q + 2), x], x, 1/x]] /; FreeQ[{a, b, p}, x] && PolyQ[Pq, x] && ILtQ[n, 0] && IntegerQ[m]
  1838. Int[Times[Pattern[Pq, Blank[]], Power[Times[Optional[Pattern[c, Blank[]]], Pattern[x, Blank[]]], Optional[Pattern[m, Blank[]]]], Power[Plus[Pattern[a, Blank[]], Times[Optional[Pattern[b, Blank[]]], Power[Pattern[x, Blank[]], Pattern[n, Blank[]]]]], Pattern[p, Blank[]]]], Pattern[x, Blank[Symbol]]] := With[{g = Denominator[m], q = Expon[Pq, x]}, -Dist[g/c, Subst[Int[(ExpandToSum[x^(g*q)*(Pq /. x -> 1/(c*x^g)), x]*(a + b/(c^n*x^(g*n)))^p)/x^(g*(m + q + 1) + 1), x], x, 1/(c*x)^(1/g)], x]] /; FreeQ[{a, b, c, p}, x] && PolyQ[Pq, x] && ILtQ[n, 0] && FractionQ[m]
  1839. Int[Times[Pattern[Pq, Blank[]], Power[Times[Optional[Pattern[c, Blank[]]], Pattern[x, Blank[]]], Pattern[m, Blank[]]], Power[Plus[Pattern[a, Blank[]], Times[Optional[Pattern[b, Blank[]]], Power[Pattern[x, Blank[]], Pattern[n, Blank[]]]]], Pattern[p, Blank[]]]], Pattern[x, Blank[Symbol]]] := With[{q = Expon[Pq, x]}, -Dist[(c*x)^m*(x^(-1))^m, Subst[Int[(ExpandToSum[x^q*(Pq /. x -> x^(-1)), x]*(a + b/x^n)^p)/x^(m + q + 2), x], x, 1/x], x]] /; FreeQ[{a, b, c, m, p}, x] && PolyQ[Pq, x] && ILtQ[n, 0] && !RationalQ[m]
  1840. Int[Times[Pattern[Pq, Blank[]], Power[Pattern[x, Blank[]], Optional[Pattern[m, Blank[]]]], Power[Plus[Pattern[a, Blank[]], Times[Optional[Pattern[b, Blank[]]], Power[Pattern[x, Blank[]], Pattern[n, Blank[]]]]], Pattern[p, Blank[]]]], Pattern[x, Blank[Symbol]]] := With[{g = Denominator[n]}, Dist[g, Subst[Int[x^(g*(m + 1) - 1)*(Pq /. x -> x^g)*(a + b*x^(g*n))^p, x], x, x^(1/g)], x]] /; FreeQ[{a, b, m, p}, x] && PolyQ[Pq, x] && FractionQ[n]
  1841. Int[Times[Pattern[Pq, Blank[]], Power[Times[Pattern[c, Blank[]], Pattern[x, Blank[]]], Pattern[m, Blank[]]], Power[Plus[Pattern[a, Blank[]], Times[Optional[Pattern[b, Blank[]]], Power[Pattern[x, Blank[]], Pattern[n, Blank[]]]]], Pattern[p, Blank[]]]], Pattern[x, Blank[Symbol]]] := Dist[(c^IntPart[m]*(c*x)^FracPart[m])/x^FracPart[m], Int[x^m*Pq*(a + b*x^n)^p, x], x] /; FreeQ[{a, b, c, m, p}, x] && PolyQ[Pq, x] && FractionQ[n]
  1842. Int[Times[Pattern[Pq, Blank[]], Power[Pattern[x, Blank[]], Optional[Pattern[m, Blank[]]]], Power[Plus[Pattern[a, Blank[]], Times[Optional[Pattern[b, Blank[]]], Power[Pattern[x, Blank[]], Pattern[n, Blank[]]]]], Pattern[p, Blank[]]]], Pattern[x, Blank[Symbol]]] := Dist[1/(m + 1), Subst[Int[(SubstFor[x^n, Pq, x] /. x -> x^Simplify[n/(m + 1)])*(a + b*x^Simplify[n/(m + 1)])^p, x], x, x^(m + 1)], x] /; FreeQ[{a, b, m, n, p}, x] && PolyQ[Pq, x^n] && IntegerQ[Simplify[n/(m + 1)]] && !IntegerQ[n]
  1843. Int[Times[Pattern[Pq, Blank[]], Power[Times[Pattern[c, Blank[]], Pattern[x, Blank[]]], Pattern[m, Blank[]]], Power[Plus[Pattern[a, Blank[]], Times[Optional[Pattern[b, Blank[]]], Power[Pattern[x, Blank[]], Pattern[n, Blank[]]]]], Pattern[p, Blank[]]]], Pattern[x, Blank[Symbol]]] := Dist[(c^IntPart[m]*(c*x)^FracPart[m])/x^FracPart[m], Int[x^m*Pq*(a + b*x^n)^p, x], x] /; FreeQ[{a, b, c, m, n, p}, x] && PolyQ[Pq, x^n] && IntegerQ[Simplify[n/(m + 1)]] && !IntegerQ[n]
  1844. Int[Times[Pattern[Pq, Blank[]], Power[Times[Optional[Pattern[c, Blank[]]], Pattern[x, Blank[]]], Optional[Pattern[m, Blank[]]]], Power[Plus[Pattern[a, Blank[]], Times[Optional[Pattern[b, Blank[]]], Power[Pattern[x, Blank[]], Pattern[n, Blank[]]]]], Optional[Pattern[p, Blank[]]]]], Pattern[x, Blank[Symbol]]] := Int[ExpandIntegrand[(c*x)^m*Pq*(a + b*x^n)^p, x], x] /; FreeQ[{a, b, c, m, n, p}, x] && (PolyQ[Pq, x] || PolyQ[Pq, x^n]) && !IGtQ[m, 0]
  1845. Int[Times[Pattern[Pq, Blank[]], Power[Pattern[u, Blank[]], Optional[Pattern[m, Blank[]]]], Power[Plus[Pattern[a, Blank[]], Times[Optional[Pattern[b, Blank[]]], Power[Pattern[v, Blank[]], Optional[Pattern[n, Blank[]]]]]], Pattern[p, Blank[]]]], Pattern[x, Blank[Symbol]]] := Dist[u^m/(Coeff[v, x, 1]*v^m), Subst[Int[x^m*SubstFor[v, Pq, x]*(a + b*x^n)^p, x], x, v], x] /; FreeQ[{a, b, m, n, p}, x] && LinearPairQ[u, v, x] && PolyQ[Pq, v^n]
  1846. Int[Times[Pattern[Pq, Blank[]], Power[Times[Optional[Pattern[c, Blank[]]], Pattern[x, Blank[]]], Optional[Pattern[m, Blank[]]]], Power[Plus[Pattern[a1, Blank[]], Times[Optional[Pattern[b1, Blank[]]], Power[Pattern[x, Blank[]], Optional[Pattern[n, Blank[]]]]]], Optional[Pattern[p, Blank[]]]], Power[Plus[Pattern[a2, Blank[]], Times[Optional[Pattern[b2, Blank[]]], Power[Pattern[x, Blank[]], Optional[Pattern[n, Blank[]]]]]], Optional[Pattern[p, Blank[]]]]], Pattern[x, Blank[Symbol]]] := Int[(c*x)^m*Pq*(a1*a2 + b1*b2*x^(2*n))^p, x] /; FreeQ[{a1, b1, a2, b2, c, m, n, p}, x] && PolyQ[Pq, x] && EqQ[a2*b1 + a1*b2, 0] && (IntegerQ[p] || (GtQ[a1, 0] && GtQ[a2, 0]))
  1847. Int[Times[Pattern[Pq, Blank[]], Power[Times[Optional[Pattern[c, Blank[]]], Pattern[x, Blank[]]], Optional[Pattern[m, Blank[]]]], Power[Plus[Pattern[a1, Blank[]], Times[Optional[Pattern[b1, Blank[]]], Power[Pattern[x, Blank[]], Optional[Pattern[n, Blank[]]]]]], Optional[Pattern[p, Blank[]]]], Power[Plus[Pattern[a2, Blank[]], Times[Optional[Pattern[b2, Blank[]]], Power[Pattern[x, Blank[]], Optional[Pattern[n, Blank[]]]]]], Optional[Pattern[p, Blank[]]]]], Pattern[x, Blank[Symbol]]] := Dist[((a1 + b1*x^n)^FracPart[p]*(a2 + b2*x^n)^FracPart[p])/(a1*a2 + b1*b2*x^(2*n))^FracPart[p], Int[(c*x)^m*Pq*(a1*a2 + b1*b2*x^(2*n))^p, x], x] /; FreeQ[{a1, b1, a2, b2, c, m, n, p}, x] && PolyQ[Pq, x] && EqQ[a2*b1 + a1*b2, 0] && !(EqQ[n, 1] && LinearQ[Pq, x])
  1848. Int[Times[Power[Times[Optional[Pattern[h, Blank[]]], Pattern[x, Blank[]]], Optional[Pattern[m, Blank[]]]], Power[Plus[Pattern[a, Blank[]], Times[Optional[Pattern[b, Blank[]]], Power[Pattern[x, Blank[]], Optional[Pattern[n, Blank[]]]]]], Optional[Pattern[p, Blank[]]]], Power[Plus[Pattern[c, Blank[]], Times[Optional[Pattern[d, Blank[]]], Power[Pattern[x, Blank[]], Optional[Pattern[n, Blank[]]]]]], Optional[Pattern[p, Blank[]]]], Plus[Pattern[e, Blank[]], Times[Optional[Pattern[f, Blank[]]], Power[Pattern[x, Blank[]], Optional[Pattern[n, Blank[]]]]], Times[Optional[Pattern[g, Blank[]]], Power[Pattern[x, Blank[]], Optional[Pattern[n2, Blank[]]]]]]], Pattern[x, Blank[Symbol]]] := Simp[(e*(h*x)^(m + 1)*(a + b*x^n)^(p + 1)*(c + d*x^n)^(p + 1))/(a*c*h*(m + 1)), x] /; FreeQ[{a, b, c, d, e, f, g, h, m, n, p}, x] && EqQ[n2, 2*n] && EqQ[a*c*f*(m + 1) - e*(b*c + a*d)*(m + n*(p + 1) + 1), 0] && EqQ[a*c*g*(m + 1) - b*d*e*(m + 2*n*(p + 1) + 1), 0] && NeQ[m, -1]
  1849. Int[Times[Power[Times[Optional[Pattern[h, Blank[]]], Pattern[x, Blank[]]], Optional[Pattern[m, Blank[]]]], Power[Plus[Pattern[a, Blank[]], Times[Optional[Pattern[b, Blank[]]], Power[Pattern[x, Blank[]], Optional[Pattern[n, Blank[]]]]]], Optional[Pattern[p, Blank[]]]], Power[Plus[Pattern[c, Blank[]], Times[Optional[Pattern[d, Blank[]]], Power[Pattern[x, Blank[]], Optional[Pattern[n, Blank[]]]]]], Optional[Pattern[p, Blank[]]]], Plus[Pattern[e, Blank[]], Times[Optional[Pattern[g, Blank[]]], Power[Pattern[x, Blank[]], Optional[Pattern[n2, Blank[]]]]]]], Pattern[x, Blank[Symbol]]] := Simp[(e*(h*x)^(m + 1)*(a + b*x^n)^(p + 1)*(c + d*x^n)^(p + 1))/(a*c*h*(m + 1)), x] /; FreeQ[{a, b, c, d, e, g, h, m, n, p}, x] && EqQ[n2, 2*n] && EqQ[m + n*(p + 1) + 1, 0] && EqQ[a*c*g*(m + 1) - b*d*e*(m + 2*n*(p + 1) + 1), 0] && NeQ[m, -1]
  1850. Int[Times[Pattern[Pq, Blank[]], Power[Plus[Pattern[a, Blank[]], Times[Optional[Pattern[b, Blank[]]], Power[Pattern[x, Blank[]], Optional[Pattern[n, Blank[]]]]]], Optional[Pattern[p, Blank[]]]]], Pattern[x, Blank[Symbol]]] := Int[ExpandIntegrand[Pq*(a + b*x^n)^p, x], x] /; FreeQ[{a, b, n}, x] && PolyQ[Pq, x] && (IGtQ[p, 0] || EqQ[n, 1])
  1851. Int[Times[Pattern[Pq, Blank[]], Power[Plus[Pattern[a, Blank[]], Times[Optional[Pattern[b, Blank[]]], Power[Pattern[x, Blank[]], Optional[Pattern[n, Blank[]]]]]], Pattern[p, Blank[]]]], Pattern[x, Blank[Symbol]]] := Int[x*PolynomialQuotient[Pq, x, x]*(a + b*x^n)^p, x] /; FreeQ[{a, b, n, p}, x] && PolyQ[Pq, x] && EqQ[Coeff[Pq, x, 0], 0] && !MatchQ[Pq, x^(m_.)*(u_.) /; IntegerQ[m]]
  1852. Int[Times[Pattern[Pq, Blank[]], Power[Plus[Pattern[a, Blank[]], Times[Optional[Pattern[b, Blank[]]], Power[Pattern[x, Blank[]], Optional[Pattern[n, Blank[]]]]]], Optional[Pattern[p, Blank[]]]]], Pattern[x, Blank[Symbol]]] := Int[PolynomialQuotient[Pq, a + b*x^n, x]*(a + b*x^n)^(p + 1), x] /; FreeQ[{a, b, p}, x] && PolyQ[Pq, x] && IGtQ[n, 0] && GeQ[Expon[Pq, x], n] && EqQ[PolynomialRemainder[Pq, a + b*x^n, x], 0]
  1853. Int[Times[Pattern[Pq, Blank[]], Power[Plus[Pattern[a, Blank[]], Times[Optional[Pattern[b, Blank[]]], Power[Pattern[x, Blank[]], Optional[Pattern[n, Blank[]]]]]], Pattern[p, Blank[]]]], Pattern[x, Blank[Symbol]]] := Module[{q = Expon[Pq, x], i}, Simp[(a + b*x^n)^p*Sum[(Coeff[Pq, x, i]*x^(i + 1))/(n*p + i + 1), {i, 0, q}], x] + Dist[a*n*p, Int[(a + b*x^n)^(p - 1)*Sum[(Coeff[Pq, x, i]*x^i)/(n*p + i + 1), {i, 0, q}], x], x]] /; FreeQ[{a, b}, x] && PolyQ[Pq, x] && IGtQ[(n - 1)/2, 0] && GtQ[p, 0]
  1854. Int[Times[Pattern[Pq, Blank[]], Power[Plus[Pattern[a, Blank[]], Times[Optional[Pattern[b, Blank[]]], Power[Pattern[x, Blank[]], Pattern[n, Blank[]]]]], Pattern[p, Blank[]]]], Pattern[x, Blank[Symbol]]] := Module[{q = Expon[Pq, x], i}, Simp[((a*Coeff[Pq, x, q] - b*x*ExpandToSum[Pq - Coeff[Pq, x, q]*x^q, x])*(a + b*x^n)^(p + 1))/(a*b*n*(p + 1)), x] + Dist[1/(a*n*(p + 1)), Int[Sum[(n*(p + 1) + i + 1)*Coeff[Pq, x, i]*x^i, {i, 0, q - 1}]*(a + b*x^n)^(p + 1), x], x] /; q == n - 1] /; FreeQ[{a, b}, x] && PolyQ[Pq, x] && IGtQ[n, 0] && LtQ[p, -1]
  1855. Int[Times[Pattern[Pq, Blank[]], Power[Plus[Pattern[a, Blank[]], Times[Optional[Pattern[b, Blank[]]], Power[Pattern[x, Blank[]], Optional[Pattern[n, Blank[]]]]]], Pattern[p, Blank[]]]], Pattern[x, Blank[Symbol]]] := -Simp[(x*Pq*(a + b*x^n)^(p + 1))/(a*n*(p + 1)), x] + Dist[1/(a*n*(p + 1)), Int[ExpandToSum[n*(p + 1)*Pq + D[x*Pq, x], x]*(a + b*x^n)^(p + 1), x], x] /; FreeQ[{a, b}, x] && PolyQ[Pq, x] && IGtQ[n, 0] && LtQ[p, -1] && LtQ[Expon[Pq, x], n - 1]
  1856. Int[Times[Pattern[P4, Blank[]], Power[Plus[Pattern[a, Blank[]], Times[Optional[Pattern[b, Blank[]]], Power[Pattern[x, Blank[]], 4]]], Rational[-3, 2]]], Pattern[x, Blank[Symbol]]] := With[{d = Coeff[P4, x, 0], e = Coeff[P4, x, 1], f = Coeff[P4, x, 3], g = Coeff[P4, x, 4]}, -Simp[(a*f + 2*a*g*x - b*e*x^2)/(2*a*b*Sqrt[a + b*x^4]), x] /; EqQ[b*d + a*g, 0]] /; FreeQ[{a, b}, x] && PolyQ[P4, x, 4] && EqQ[Coeff[P4, x, 2], 0]
  1857. Int[Times[Pattern[P6, Blank[]], Power[Plus[Pattern[a, Blank[]], Times[Optional[Pattern[b, Blank[]]], Power[Pattern[x, Blank[]], 4]]], Rational[-3, 2]]], Pattern[x, Blank[Symbol]]] := With[{d = Coeff[P6, x, 0], e = Coeff[P6, x, 2], f = Coeff[P6, x, 3], g = Coeff[P6, x, 4], h = Coeff[P6, x, 6]}, -Simp[(a*f - 2*b*d*x - 2*a*h*x^3)/(2*a*b*Sqrt[a + b*x^4]), x] /; EqQ[b*e - 3*a*h, 0] && EqQ[b*d + a*g, 0]] /; FreeQ[{a, b}, x] && PolyQ[P6, x, 6] && EqQ[Coeff[P6, x, 1], 0] && EqQ[Coeff[P6, x, 5], 0]
  1858. Int[Times[Pattern[Pq, Blank[]], Power[Plus[Pattern[a, Blank[]], Times[Optional[Pattern[b, Blank[]]], Power[Pattern[x, Blank[]], Optional[Pattern[n, Blank[]]]]]], Pattern[p, Blank[]]]], Pattern[x, Blank[Symbol]]] := With[{q = Expon[Pq, x]}, Module[{Q = PolynomialQuotient[b^(Floor[(q - 1)/n] + 1)*Pq, a + b*x^n, x], R = PolynomialRemainder[b^(Floor[(q - 1)/n] + 1)*Pq, a + b*x^n, x]}, Dist[1/(a*n*(p + 1)*b^(Floor[(q - 1)/n] + 1)), Int[(a + b*x^n)^(p + 1)*ExpandToSum[a*n*(p + 1)*Q + n*(p + 1)*R + D[x*R, x], x], x], x] - Simp[(x*R*(a + b*x^n)^(p + 1))/(a*n*(p + 1)*b^(Floor[(q - 1)/n] + 1)), x]] /; GeQ[q, n]] /; FreeQ[{a, b}, x] && PolyQ[Pq, x] && IGtQ[n, 0] && LtQ[p, -1]
  1859. Int[Times[Plus[Pattern[A, Blank[]], Times[Optional[Pattern[B, Blank[]]], Pattern[x, Blank[]]]], Power[Plus[Pattern[a, Blank[]], Times[Optional[Pattern[b, Blank[]]], Power[Pattern[x, Blank[]], 3]]], -1]], Pattern[x, Blank[Symbol]]] := Dist[B^3/b, Int[1/(A^2 - A*B*x + B^2*x^2), x], x] /; FreeQ[{a, b, A, B}, x] && EqQ[a*B^3 - b*A^3, 0]
  1860. Int[Times[Plus[Pattern[A, Blank[]], Times[Optional[Pattern[B, Blank[]]], Pattern[x, Blank[]]]], Power[Plus[Pattern[a, Blank[]], Times[Optional[Pattern[b, Blank[]]], Power[Pattern[x, Blank[]], 3]]], -1]], Pattern[x, Blank[Symbol]]] := With[{r = Numerator[Rt[a/b, 3]], s = Denominator[Rt[a/b, 3]]}, -Dist[(r*(B*r - A*s))/(3*a*s), Int[1/(r + s*x), x], x] + Dist[r/(3*a*s), Int[(r*(B*r + 2*A*s) + s*(B*r - A*s)*x)/(r^2 - r*s*x + s^2*x^2), x], x]] /; FreeQ[{a, b, A, B}, x] && NeQ[a*B^3 - b*A^3, 0] && PosQ[a/b]
  1861. Int[Times[Plus[Pattern[A, Blank[]], Times[Optional[Pattern[B, Blank[]]], Pattern[x, Blank[]]]], Power[Plus[Pattern[a, Blank[]], Times[Optional[Pattern[b, Blank[]]], Power[Pattern[x, Blank[]], 3]]], -1]], Pattern[x, Blank[Symbol]]] := With[{r = Numerator[Rt[-(a/b), 3]], s = Denominator[Rt[-(a/b), 3]]}, Dist[(r*(B*r + A*s))/(3*a*s), Int[1/(r - s*x), x], x] - Dist[r/(3*a*s), Int[(r*(B*r - 2*A*s) - s*(B*r + A*s)*x)/(r^2 + r*s*x + s^2*x^2), x], x]] /; FreeQ[{a, b, A, B}, x] && NeQ[a*B^3 - b*A^3, 0] && NegQ[a/b]
  1862. Int[Times[Pattern[P2, Blank[]], Power[Plus[Pattern[a, Blank[]], Times[Optional[Pattern[b, Blank[]]], Power[Pattern[x, Blank[]], 3]]], -1]], Pattern[x, Blank[Symbol]]] := With[{A = Coeff[P2, x, 0], B = Coeff[P2, x, 1], C = Coeff[P2, x, 2]}, -Dist[C^2/b, Int[1/(B - C*x), x], x] /; EqQ[B^2 - A*C, 0] && EqQ[b*B^3 + a*C^3, 0]] /; FreeQ[{a, b}, x] && PolyQ[P2, x, 2]
  1863. Int[Times[Pattern[P2, Blank[]], Power[Plus[Pattern[a, Blank[]], Times[Optional[Pattern[b, Blank[]]], Power[Pattern[x, Blank[]], 3]]], -1]], Pattern[x, Blank[Symbol]]] := With[{A = Coeff[P2, x, 0], B = Coeff[P2, x, 1], C = Coeff[P2, x, 2]}, With[{q = a^(1/3)/b^(1/3)}, Dist[C/b, Int[1/(q + x), x], x] + Dist[(B + C*q)/b, Int[1/(q^2 - q*x + x^2), x], x]] /; EqQ[A*b^(2/3) - a^(1/3)*b^(1/3)*B - 2*a^(2/3)*C, 0]] /; FreeQ[{a, b}, x] && PolyQ[P2, x, 2]
  1864. Int[Times[Pattern[P2, Blank[]], Power[Plus[Pattern[a, Blank[]], Times[Optional[Pattern[b, Blank[]]], Power[Pattern[x, Blank[]], 3]]], -1]], Pattern[x, Blank[Symbol]]] := With[{A = Coeff[P2, x, 0], B = Coeff[P2, x, 1], C = Coeff[P2, x, 2]}, With[{q = (-a)^(1/3)/(-b)^(1/3)}, Dist[C/b, Int[1/(q + x), x], x] + Dist[(B + C*q)/b, Int[1/(q^2 - q*x + x^2), x], x]] /; EqQ[A*(-b)^(2/3) - (-a)^(1/3)*(-b)^(1/3)*B - 2*(-a)^(2/3)*C, 0]] /; FreeQ[{a, b}, x] && PolyQ[P2, x, 2]
  1865. Int[Times[Pattern[P2, Blank[]], Power[Plus[Pattern[a, Blank[]], Times[Optional[Pattern[b, Blank[]]], Power[Pattern[x, Blank[]], 3]]], -1]], Pattern[x, Blank[Symbol]]] := With[{A = Coeff[P2, x, 0], B = Coeff[P2, x, 1], C = Coeff[P2, x, 2]}, With[{q = (-a)^(1/3)/b^(1/3)}, -Dist[C/b, Int[1/(q - x), x], x] + Dist[(B - C*q)/b, Int[1/(q^2 + q*x + x^2), x], x]] /; EqQ[A*b^(2/3) + (-a)^(1/3)*b^(1/3)*B - 2*(-a)^(2/3)*C, 0]] /; FreeQ[{a, b}, x] && PolyQ[P2, x, 2]
  1866. Int[Times[Pattern[P2, Blank[]], Power[Plus[Pattern[a, Blank[]], Times[Optional[Pattern[b, Blank[]]], Power[Pattern[x, Blank[]], 3]]], -1]], Pattern[x, Blank[Symbol]]] := With[{A = Coeff[P2, x, 0], B = Coeff[P2, x, 1], C = Coeff[P2, x, 2]}, With[{q = a^(1/3)/(-b)^(1/3)}, -Dist[C/b, Int[1/(q - x), x], x] + Dist[(B - C*q)/b, Int[1/(q^2 + q*x + x^2), x], x]] /; EqQ[A*(-b)^(2/3) + a^(1/3)*(-b)^(1/3)*B - 2*a^(2/3)*C, 0]] /; FreeQ[{a, b}, x] && PolyQ[P2, x, 2]
  1867. Int[Times[Pattern[P2, Blank[]], Power[Plus[Pattern[a, Blank[]], Times[Optional[Pattern[b, Blank[]]], Power[Pattern[x, Blank[]], 3]]], -1]], Pattern[x, Blank[Symbol]]] := With[{A = Coeff[P2, x, 0], B = Coeff[P2, x, 1], C = Coeff[P2, x, 2]}, With[{q = (a/b)^(1/3)}, Dist[C/b, Int[1/(q + x), x], x] + Dist[(B + C*q)/b, Int[1/(q^2 - q*x + x^2), x], x]] /; EqQ[A - (a/b)^(1/3)*B - 2*(a/b)^(2/3)*C, 0]] /; FreeQ[{a, b}, x] && PolyQ[P2, x, 2]
  1868. Int[Times[Pattern[P2, Blank[]], Power[Plus[Pattern[a, Blank[]], Times[Optional[Pattern[b, Blank[]]], Power[Pattern[x, Blank[]], 3]]], -1]], Pattern[x, Blank[Symbol]]] := With[{A = Coeff[P2, x, 0], B = Coeff[P2, x, 1], C = Coeff[P2, x, 2]}, With[{q = Rt[a/b, 3]}, Dist[C/b, Int[1/(q + x), x], x] + Dist[(B + C*q)/b, Int[1/(q^2 - q*x + x^2), x], x]] /; EqQ[A - Rt[a/b, 3]*B - 2*Rt[a/b, 3]^2*C, 0]] /; FreeQ[{a, b}, x] && PolyQ[P2, x, 2]
  1869. Int[Times[Pattern[P2, Blank[]], Power[Plus[Pattern[a, Blank[]], Times[Optional[Pattern[b, Blank[]]], Power[Pattern[x, Blank[]], 3]]], -1]], Pattern[x, Blank[Symbol]]] := With[{A = Coeff[P2, x, 0], B = Coeff[P2, x, 1], C = Coeff[P2, x, 2]}, With[{q = (-(a/b))^(1/3)}, -Dist[C/b, Int[1/(q - x), x], x] + Dist[(B - C*q)/b, Int[1/(q^2 + q*x + x^2), x], x]] /; EqQ[A + (-(a/b))^(1/3)*B - 2*(-(a/b))^(2/3)*C, 0]] /; FreeQ[{a, b}, x] && PolyQ[P2, x, 2]
  1870. Int[Times[Pattern[P2, Blank[]], Power[Plus[Pattern[a, Blank[]], Times[Optional[Pattern[b, Blank[]]], Power[Pattern[x, Blank[]], 3]]], -1]], Pattern[x, Blank[Symbol]]] := With[{A = Coeff[P2, x, 0], B = Coeff[P2, x, 1], C = Coeff[P2, x, 2]}, With[{q = Rt[-(a/b), 3]}, -Dist[C/b, Int[1/(q - x), x], x] + Dist[(B - C*q)/b, Int[1/(q^2 + q*x + x^2), x], x]] /; EqQ[A + Rt[-(a/b), 3]*B - 2*Rt[-(a/b), 3]^2*C, 0]] /; FreeQ[{a, b}, x] && PolyQ[P2, x, 2]
  1871. Int[Times[Pattern[P2, Blank[]], Power[Plus[Pattern[a, Blank[]], Times[Optional[Pattern[b, Blank[]]], Power[Pattern[x, Blank[]], 3]]], -1]], Pattern[x, Blank[Symbol]]] := With[{A = Coeff[P2, x, 0], B = Coeff[P2, x, 1], C = Coeff[P2, x, 2]}, Int[(A + B*x)/(a + b*x^3), x] + Dist[C, Int[x^2/(a + b*x^3), x], x] /; EqQ[a*B^3 - b*A^3, 0] || !RationalQ[a/b]] /; FreeQ[{a, b}, x] && PolyQ[P2, x, 2]
  1872. Int[Times[Pattern[P2, Blank[]], Power[Plus[Pattern[a, Blank[]], Times[Optional[Pattern[b, Blank[]]], Power[Pattern[x, Blank[]], 3]]], -1]], Pattern[x, Blank[Symbol]]] := With[{A = Coeff[P2, x, 0], B = Coeff[P2, x, 1], C = Coeff[P2, x, 2]}, With[{q = (a/b)^(1/3)}, Dist[q^2/a, Int[(A + C*q*x)/(q^2 - q*x + x^2), x], x]] /; EqQ[A - B*(a/b)^(1/3) + C*(a/b)^(2/3), 0]] /; FreeQ[{a, b}, x] && PolyQ[P2, x, 2]
  1873. Int[Times[Pattern[P2, Blank[]], Power[Plus[Pattern[a, Blank[]], Times[Optional[Pattern[b, Blank[]]], Power[Pattern[x, Blank[]], 3]]], -1]], Pattern[x, Blank[Symbol]]] := With[{A = Coeff[P2, x, 0], B = Coeff[P2, x, 1], C = Coeff[P2, x, 2]}, With[{q = (-(a/b))^(1/3)}, Dist[q/a, Int[(A*q + (A + B*q)*x)/(q^2 + q*x + x^2), x], x]] /; EqQ[A + B*(-(a/b))^(1/3) + C*(-(a/b))^(2/3), 0]] /; FreeQ[{a, b}, x] && PolyQ[P2, x, 2]
  1874. Int[Times[Pattern[P2, Blank[]], Power[Plus[Pattern[a, Blank[]], Times[Optional[Pattern[b, Blank[]]], Power[Pattern[x, Blank[]], 3]]], -1]], Pattern[x, Blank[Symbol]]] := With[{A = Coeff[P2, x, 0], B = Coeff[P2, x, 1], C = Coeff[P2, x, 2], q = (a/b)^(1/3)}, Dist[(q*(A - B*q + C*q^2))/(3*a), Int[1/(q + x), x], x] + Dist[q/(3*a), Int[(q*(2*A + B*q - C*q^2) - (A - B*q - 2*C*q^2)*x)/(q^2 - q*x + x^2), x], x] /; NeQ[a*B^3 - b*A^3, 0] && NeQ[A - B*q + C*q^2, 0]] /; FreeQ[{a, b}, x] && PolyQ[P2, x, 2] && GtQ[a/b, 0]
  1875. Int[Times[Pattern[P2, Blank[]], Power[Plus[Pattern[a, Blank[]], Times[Optional[Pattern[b, Blank[]]], Power[Pattern[x, Blank[]], 3]]], -1]], Pattern[x, Blank[Symbol]]] := With[{A = Coeff[P2, x, 0], B = Coeff[P2, x, 1], C = Coeff[P2, x, 2], q = (-(a/b))^(1/3)}, Dist[(q*(A + B*q + C*q^2))/(3*a), Int[1/(q - x), x], x] + Dist[q/(3*a), Int[(q*(2*A - B*q - C*q^2) + (A + B*q - 2*C*q^2)*x)/(q^2 + q*x + x^2), x], x] /; NeQ[a*B^3 - b*A^3, 0] && NeQ[A + B*q + C*q^2, 0]] /; FreeQ[{a, b}, x] && PolyQ[P2, x, 2] && LtQ[a/b, 0]
  1876. Int[Times[Pattern[Pq, Blank[]], Power[Plus[Pattern[a, Blank[]], Times[Optional[Pattern[b, Blank[]]], Power[Pattern[x, Blank[]], Pattern[n, Blank[]]]]], -1]], Pattern[x, Blank[Symbol]]] := With[{v = Sum[(x^ii*(Coeff[Pq, x, ii] + Coeff[Pq, x, n/2 + ii]*x^(n/2)))/(a + b*x^n), {ii, 0, n/2 - 1}]}, Int[v, x] /; SumQ[v]] /; FreeQ[{a, b}, x] && PolyQ[Pq, x] && IGtQ[n/2, 0] && Expon[Pq, x] < n
  1877. Int[Times[Plus[Pattern[c, Blank[]], Times[Optional[Pattern[d, Blank[]]], Pattern[x, Blank[]]]], Power[Plus[Pattern[a, Blank[]], Times[Optional[Pattern[b, Blank[]]], Power[Pattern[x, Blank[]], 3]]], Rational[-1, 2]]], Pattern[x, Blank[Symbol]]] := With[{r = Numer[Simplify[((1 - Sqrt[3])*d)/c]], s = Denom[Simplify[((1 - Sqrt[3])*d)/c]]}, Simp[(2*d*s^3*Sqrt[a + b*x^3])/(a*r^2*((1 + Sqrt[3])*s + r*x)), x] - Simp[(3^(1/4)*Sqrt[2 - Sqrt[3]]*d*s*(s + r*x)*Sqrt[(s^2 - r*s*x + r^2*x^2)/((1 + Sqrt[3])*s + r*x)^2]*EllipticE[ArcSin[((1 - Sqrt[3])*s + r*x)/((1 + Sqrt[3])*s + r*x)], -7 - 4*Sqrt[3]])/(r^2*Sqrt[a + b*x^3]*Sqrt[(s*(s + r*x))/((1 + Sqrt[3])*s + r*x)^2]), x]] /; FreeQ[{a, b, c, d}, x] && PosQ[a] && EqQ[b*c^3 - 2*(5 - 3*Sqrt[3])*a*d^3, 0]
  1878. Int[Times[Plus[Pattern[c, Blank[]], Times[Optional[Pattern[d, Blank[]]], Pattern[x, Blank[]]]], Power[Plus[Pattern[a, Blank[]], Times[Optional[Pattern[b, Blank[]]], Power[Pattern[x, Blank[]], 3]]], Rational[-1, 2]]], Pattern[x, Blank[Symbol]]] := With[{r = Numer[Rt[b/a, 3]], s = Denom[Rt[b/a, 3]]}, Dist[(c*r - (1 - Sqrt[3])*d*s)/r, Int[1/Sqrt[a + b*x^3], x], x] + Dist[d/r, Int[((1 - Sqrt[3])*s + r*x)/Sqrt[a + b*x^3], x], x]] /; FreeQ[{a, b, c, d}, x] && PosQ[a] && NeQ[b*c^3 - 2*(5 - 3*Sqrt[3])*a*d^3, 0]
  1879. Int[Times[Plus[Pattern[c, Blank[]], Times[Optional[Pattern[d, Blank[]]], Pattern[x, Blank[]]]], Power[Plus[Pattern[a, Blank[]], Times[Optional[Pattern[b, Blank[]]], Power[Pattern[x, Blank[]], 3]]], Rational[-1, 2]]], Pattern[x, Blank[Symbol]]] := With[{r = Numer[Simplify[((1 + Sqrt[3])*d)/c]], s = Denom[Simplify[((1 + Sqrt[3])*d)/c]]}, Simp[(2*d*s^3*Sqrt[a + b*x^3])/(a*r^2*((1 - Sqrt[3])*s + r*x)), x] + Simp[(3^(1/4)*Sqrt[2 + Sqrt[3]]*d*s*(s + r*x)*Sqrt[(s^2 - r*s*x + r^2*x^2)/((1 - Sqrt[3])*s + r*x)^2]*EllipticE[ArcSin[((1 + Sqrt[3])*s + r*x)/((1 - Sqrt[3])*s + r*x)], -7 + 4*Sqrt[3]])/(r^2*Sqrt[a + b*x^3]*Sqrt[-((s*(s + r*x))/((1 - Sqrt[3])*s + r*x)^2)]), x]] /; FreeQ[{a, b, c, d}, x] && NegQ[a] && EqQ[b*c^3 - 2*(5 + 3*Sqrt[3])*a*d^3, 0]
  1880. Int[Times[Plus[Pattern[c, Blank[]], Times[Optional[Pattern[d, Blank[]]], Pattern[x, Blank[]]]], Power[Plus[Pattern[a, Blank[]], Times[Optional[Pattern[b, Blank[]]], Power[Pattern[x, Blank[]], 3]]], Rational[-1, 2]]], Pattern[x, Blank[Symbol]]] := With[{r = Numer[Rt[b/a, 3]], s = Denom[Rt[b/a, 3]]}, Dist[(c*r - (1 + Sqrt[3])*d*s)/r, Int[1/Sqrt[a + b*x^3], x], x] + Dist[d/r, Int[((1 + Sqrt[3])*s + r*x)/Sqrt[a + b*x^3], x], x]] /; FreeQ[{a, b, c, d}, x] && NegQ[a] && NeQ[b*c^3 - 2*(5 + 3*Sqrt[3])*a*d^3, 0]
  1881. Int[Times[Plus[Pattern[c, Blank[]], Times[Optional[Pattern[d, Blank[]]], Power[Pattern[x, Blank[]], 4]]], Power[Plus[Pattern[a, Blank[]], Times[Optional[Pattern[b, Blank[]]], Power[Pattern[x, Blank[]], 6]]], Rational[-1, 2]]], Pattern[x, Blank[Symbol]]] := With[{r = Numer[Rt[b/a, 3]], s = Denom[Rt[b/a, 3]]}, Simp[((1 + Sqrt[3])*d*s^3*x*Sqrt[a + b*x^6])/(2*a*r^2*(s + (1 + Sqrt[3])*r*x^2)), x] - Simp[(3^(1/4)*d*s*x*(s + r*x^2)*Sqrt[(s^2 - r*s*x^2 + r^2*x^4)/(s + (1 + Sqrt[3])*r*x^2)^2]*EllipticE[ArcCos[(s + (1 - Sqrt[3])*r*x^2)/(s + (1 + Sqrt[3])*r*x^2)], (2 + Sqrt[3])/4])/(2*r^2*Sqrt[(r*x^2*(s + r*x^2))/(s + (1 + Sqrt[3])*r*x^2)^2]*Sqrt[a + b*x^6]), x]] /; FreeQ[{a, b, c, d}, x] && EqQ[2*Rt[b/a, 3]^2*c - (1 - Sqrt[3])*d, 0]
  1882. Int[Times[Plus[Pattern[c, Blank[]], Times[Optional[Pattern[d, Blank[]]], Power[Pattern[x, Blank[]], 4]]], Power[Plus[Pattern[a, Blank[]], Times[Optional[Pattern[b, Blank[]]], Power[Pattern[x, Blank[]], 6]]], Rational[-1, 2]]], Pattern[x, Blank[Symbol]]] := With[{q = Rt[b/a, 3]}, Dist[(2*c*q^2 - (1 - Sqrt[3])*d)/(2*q^2), Int[1/Sqrt[a + b*x^6], x], x] + Dist[d/(2*q^2), Int[(1 - Sqrt[3] + 2*q^2*x^4)/Sqrt[a + b*x^6], x], x]] /; FreeQ[{a, b, c, d}, x] && NeQ[2*Rt[b/a, 3]^2*c - (1 - Sqrt[3])*d, 0]
  1883. Int[Times[Plus[Pattern[c, Blank[]], Times[Optional[Pattern[d, Blank[]]], Power[Pattern[x, Blank[]], 2]]], Power[Plus[Pattern[a, Blank[]], Times[Optional[Pattern[b, Blank[]]], Power[Pattern[x, Blank[]], 8]]], Rational[-1, 2]]], Pattern[x, Blank[Symbol]]] := -Simp[(c*d*x^3*Sqrt[-((c - d*x^2)^2/(c*d*x^2))]*Sqrt[-((d^2*(a + b*x^8))/(b*c^2*x^4))]*EllipticF[ArcSin[(1*Sqrt[(Sqrt[2]*c^2 + 2*c*d*x^2 + Sqrt[2]*d^2*x^4)/(c*d*x^2)])/2], -2*(1 - Sqrt[2])])/(Sqrt[2 + Sqrt[2]]*(c - d*x^2)*Sqrt[a + b*x^8]), x] /; FreeQ[{a, b, c, d}, x] && EqQ[b*c^4 - a*d^4, 0]
  1884. Int[Times[Plus[Pattern[c, Blank[]], Times[Optional[Pattern[d, Blank[]]], Power[Pattern[x, Blank[]], 2]]], Power[Plus[Pattern[a, Blank[]], Times[Optional[Pattern[b, Blank[]]], Power[Pattern[x, Blank[]], 8]]], Rational[-1, 2]]], Pattern[x, Blank[Symbol]]] := Dist[(d + Rt[b/a, 4]*c)/(2*Rt[b/a, 4]), Int[(1 + Rt[b/a, 4]*x^2)/Sqrt[a + b*x^8], x], x] - Dist[(d - Rt[b/a, 4]*c)/(2*Rt[b/a, 4]), Int[(1 - Rt[b/a, 4]*x^2)/Sqrt[a + b*x^8], x], x] /; FreeQ[{a, b, c, d}, x] && NeQ[b*c^4 - a*d^4, 0]
  1885. Int[Times[Pattern[Pq, Blank[]], Power[Plus[Pattern[a, Blank[]], Times[Optional[Pattern[b, Blank[]]], Power[Pattern[x, Blank[]], Pattern[n, Blank[]]]]], Pattern[p, Blank[]]]], Pattern[x, Blank[Symbol]]] := Module[{q = Expon[Pq, x], j, k}, Int[Sum[x^j*Sum[Coeff[Pq, x, j + (k*n)/2]*x^((k*n)/2), {k, 0, (2*(q - j))/n + 1}]*(a + b*x^n)^p, {j, 0, n/2 - 1}], x]] /; FreeQ[{a, b, p}, x] && PolyQ[Pq, x] && IGtQ[n/2, 0] && !PolyQ[Pq, x^(n/2)]
  1886. Int[Times[Pattern[Pq, Blank[]], Power[Plus[Pattern[a, Blank[]], Times[Optional[Pattern[b, Blank[]]], Power[Pattern[x, Blank[]], Pattern[n, Blank[]]]]], Pattern[p, Blank[]]]], Pattern[x, Blank[Symbol]]] := Dist[Coeff[Pq, x, n - 1], Int[x^(n - 1)*(a + b*x^n)^p, x], x] + Int[ExpandToSum[Pq - Coeff[Pq, x, n - 1]*x^(n - 1), x]*(a + b*x^n)^p, x] /; FreeQ[{a, b, p}, x] && PolyQ[Pq, x] && IGtQ[n, 0] && Expon[Pq, x] == n - 1
  1887. Int[Times[Pattern[Pq, Blank[]], Power[Plus[Pattern[a, Blank[]], Times[Optional[Pattern[b, Blank[]]], Power[Pattern[x, Blank[]], Pattern[n, Blank[]]]]], -1]], Pattern[x, Blank[Symbol]]] := Int[ExpandIntegrand[Pq/(a + b*x^n), x], x] /; FreeQ[{a, b}, x] && PolyQ[Pq, x] && IntegerQ[n]
  1888. Int[Times[Pattern[Pq, Blank[]], Power[Plus[Pattern[a, Blank[]], Times[Optional[Pattern[b, Blank[]]], Power[Pattern[x, Blank[]], Pattern[n, Blank[]]]]], Pattern[p, Blank[]]]], Pattern[x, Blank[Symbol]]] := With[{q = Expon[Pq, x]}, With[{Pqq = Coeff[Pq, x, q]}, Dist[1/(b*(q + n*p + 1)), Int[ExpandToSum[b*(q + n*p + 1)*(Pq - Pqq*x^q) - a*Pqq*(q - n + 1)*x^(q - n), x]*(a + b*x^n)^p, x], x] + Simp[(Pqq*x^(q - n + 1)*(a + b*x^n)^(p + 1))/(b*(q + n*p + 1)), x]] /; NeQ[q + n*p + 1, 0] && q - n >= 0 && (IntegerQ[2*p] || IntegerQ[p + (q + 1)/(2*n)])] /; FreeQ[{a, b, p}, x] && PolyQ[Pq, x] && IGtQ[n, 0]
  1889. Int[Times[Pattern[Pq, Blank[]], Power[Plus[Pattern[a, Blank[]], Times[Optional[Pattern[b, Blank[]]], Power[Pattern[x, Blank[]], Pattern[n, Blank[]]]]], Pattern[p, Blank[]]]], Pattern[x, Blank[Symbol]]] := With[{q = Expon[Pq, x]}, -Subst[Int[(ExpandToSum[x^q*(Pq /. x -> x^(-1)), x]*(a + b/x^n)^p)/x^(q + 2), x], x, 1/x]] /; FreeQ[{a, b, p}, x] && PolyQ[Pq, x] && ILtQ[n, 0]
  1890. Int[Times[Pattern[Pq, Blank[]], Power[Plus[Pattern[a, Blank[]], Times[Optional[Pattern[b, Blank[]]], Power[Pattern[x, Blank[]], Pattern[n, Blank[]]]]], Pattern[p, Blank[]]]], Pattern[x, Blank[Symbol]]] := With[{g = Denominator[n]}, Dist[g, Subst[Int[x^(g - 1)*(Pq /. x -> x^g)*(a + b*x^(g*n))^p, x], x, x^(1/g)], x]] /; FreeQ[{a, b, p}, x] && PolyQ[Pq, x] && FractionQ[n]
  1891. Int[Times[Plus[Pattern[A, Blank[]], Times[Optional[Pattern[B, Blank[]]], Power[Pattern[x, Blank[]], Optional[Pattern[m, Blank[]]]]]], Power[Plus[Pattern[a, Blank[]], Times[Optional[Pattern[b, Blank[]]], Power[Pattern[x, Blank[]], Pattern[n, Blank[]]]]], Optional[Pattern[p, Blank[]]]]], Pattern[x, Blank[Symbol]]] := Dist[A, Int[(a + b*x^n)^p, x], x] + Dist[B, Int[x^m*(a + b*x^n)^p, x], x] /; FreeQ[{a, b, A, B, m, n, p}, x] && EqQ[m - n + 1, 0]
  1892. Int[Times[Pattern[P3, Blank[]], Power[Plus[Pattern[a, Blank[]], Times[Optional[Pattern[b, Blank[]]], Power[Pattern[x, Blank[]], Pattern[n, Blank[]]]]], Pattern[p, Blank[]]]], Pattern[x, Blank[Symbol]]] := With[{A = Coeff[P3, x^(n/2), 0], B = Coeff[P3, x^(n/2), 1], C = Coeff[P3, x^(n/2), 2], D = Coeff[P3, x^(n/2), 3]}, -Simp[(x*(b*A - a*C + (b*B - a*D)*x^(n/2))*(a + b*x^n)^(p + 1))/(a*b*n*(p + 1)), x] - Dist[1/(2*a*b*n*(p + 1)), Int[(a + b*x^n)^(p + 1)*Simp[2*a*C - 2*b*A*(n*(p + 1) + 1) + (a*D*(n + 2) - b*B*(n*(2*p + 3) + 2))*x^(n/2), x], x], x]] /; FreeQ[{a, b, n}, x] && PolyQ[P3, x^(n/2), 3] && ILtQ[p, -1]
  1893. Int[Times[Pattern[Pq, Blank[]], Power[Plus[Pattern[a, Blank[]], Times[Optional[Pattern[b, Blank[]]], Power[Pattern[x, Blank[]], Pattern[n, Blank[]]]]], Optional[Pattern[p, Blank[]]]]], Pattern[x, Blank[Symbol]]] := Int[ExpandIntegrand[Pq*(a + b*x^n)^p, x], x] /; FreeQ[{a, b, n, p}, x] && (PolyQ[Pq, x] || PolyQ[Pq, x^n])
  1894. Int[Times[Pattern[Pq, Blank[]], Power[Plus[Pattern[a, Blank[]], Times[Optional[Pattern[b, Blank[]]], Power[Pattern[v, Blank[]], Optional[Pattern[n, Blank[]]]]]], Pattern[p, Blank[]]]], Pattern[x, Blank[Symbol]]] := Dist[1/Coeff[v, x, 1], Subst[Int[SubstFor[v, Pq, x]*(a + b*x^n)^p, x], x, v], x] /; FreeQ[{a, b, n, p}, x] && LinearQ[v, x] && PolyQ[Pq, v^n]
  1895. Int[Times[Pattern[Pq, Blank[]], Power[Plus[Pattern[a1, Blank[]], Times[Optional[Pattern[b1, Blank[]]], Power[Pattern[x, Blank[]], Optional[Pattern[n, Blank[]]]]]], Optional[Pattern[p, Blank[]]]], Power[Plus[Pattern[a2, Blank[]], Times[Optional[Pattern[b2, Blank[]]], Power[Pattern[x, Blank[]], Optional[Pattern[n, Blank[]]]]]], Optional[Pattern[p, Blank[]]]]], Pattern[x, Blank[Symbol]]] := Int[Pq*(a1*a2 + b1*b2*x^(2*n))^p, x] /; FreeQ[{a1, b1, a2, b2, n, p}, x] && PolyQ[Pq, x] && EqQ[a2*b1 + a1*b2, 0] && (IntegerQ[p] || (GtQ[a1, 0] && GtQ[a2, 0]))
  1896. Int[Times[Pattern[Pq, Blank[]], Power[Plus[Pattern[a1, Blank[]], Times[Optional[Pattern[b1, Blank[]]], Power[Pattern[x, Blank[]], Optional[Pattern[n, Blank[]]]]]], Optional[Pattern[p, Blank[]]]], Power[Plus[Pattern[a2, Blank[]], Times[Optional[Pattern[b2, Blank[]]], Power[Pattern[x, Blank[]], Optional[Pattern[n, Blank[]]]]]], Optional[Pattern[p, Blank[]]]]], Pattern[x, Blank[Symbol]]] := Dist[((a1 + b1*x^n)^FracPart[p]*(a2 + b2*x^n)^FracPart[p])/(a1*a2 + b1*b2*x^(2*n))^FracPart[p], Int[Pq*(a1*a2 + b1*b2*x^(2*n))^p, x], x] /; FreeQ[{a1, b1, a2, b2, n, p}, x] && PolyQ[Pq, x] && EqQ[a2*b1 + a1*b2, 0] && !(EqQ[n, 1] && LinearQ[Pq, x])
  1897. Int[Times[Power[Plus[Pattern[a, Blank[]], Times[Optional[Pattern[b, Blank[]]], Power[Pattern[x, Blank[]], Optional[Pattern[n, Blank[]]]]]], Optional[Pattern[p, Blank[]]]], Power[Plus[Pattern[c, Blank[]], Times[Optional[Pattern[d, Blank[]]], Power[Pattern[x, Blank[]], Optional[Pattern[n, Blank[]]]]]], Optional[Pattern[p, Blank[]]]], Plus[Pattern[e, Blank[]], Times[Optional[Pattern[f, Blank[]]], Power[Pattern[x, Blank[]], Optional[Pattern[n, Blank[]]]]], Times[Optional[Pattern[g, Blank[]]], Power[Pattern[x, Blank[]], Optional[Pattern[n2, Blank[]]]]]]], Pattern[x, Blank[Symbol]]] := Simp[(e*x*(a + b*x^n)^(p + 1)*(c + d*x^n)^(p + 1))/(a*c), x] /; FreeQ[{a, b, c, d, e, f, g, n, p}, x] && EqQ[n2, 2*n] && EqQ[a*c*f - e*(b*c + a*d)*(n*(p + 1) + 1), 0] && EqQ[a*c*g - b*d*e*(2*n*(p + 1) + 1), 0]
  1898. Int[Times[Power[Plus[Pattern[a, Blank[]], Times[Optional[Pattern[b, Blank[]]], Power[Pattern[x, Blank[]], Optional[Pattern[n, Blank[]]]]]], Optional[Pattern[p, Blank[]]]], Power[Plus[Pattern[c, Blank[]], Times[Optional[Pattern[d, Blank[]]], Power[Pattern[x, Blank[]], Optional[Pattern[n, Blank[]]]]]], Optional[Pattern[p, Blank[]]]], Plus[Pattern[e, Blank[]], Times[Optional[Pattern[g, Blank[]]], Power[Pattern[x, Blank[]], Optional[Pattern[n2, Blank[]]]]]]], Pattern[x, Blank[Symbol]]] := Simp[(e*x*(a + b*x^n)^(p + 1)*(c + d*x^n)^(p + 1))/(a*c), x] /; FreeQ[{a, b, c, d, e, g, n, p}, x] && EqQ[n2, 2*n] && EqQ[n*(p + 1) + 1, 0] && EqQ[a*c*g - b*d*e*(2*n*(p + 1) + 1), 0]
  1899. Int[Times[Plus[Pattern[A, Blank[]], Times[Optional[Pattern[B, Blank[]]], Power[Pattern[x, Blank[]], Optional[Pattern[m, Blank[]]]]]], Power[Plus[Optional[Pattern[a, Blank[]]], Times[Optional[Pattern[b, Blank[]]], Power[Pattern[x, Blank[]], Pattern[n, Blank[]]]]], Optional[Pattern[p, Blank[]]]], Power[Plus[Pattern[c, Blank[]], Times[Optional[Pattern[d, Blank[]]], Power[Pattern[x, Blank[]], Pattern[n, Blank[]]]]], Optional[Pattern[q, Blank[]]]]], Pattern[x, Blank[Symbol]]] := Dist[A, Int[(a + b*x^n)^p*(c + d*x^n)^q, x], x] + Dist[B, Int[x^m*(a + b*x^n)^p*(c + d*x^n)^q, x], x] /; FreeQ[{a, b, c, d, A, B, m, n, p, q}, x] && NeQ[b*c - a*d, 0] && EqQ[m - n + 1, 0]
  1900. Int[Times[Power[Pattern[Px, Blank[]], Optional[Pattern[q, Blank[]]]], Power[Plus[Optional[Pattern[a, Blank[]]], Times[Optional[Pattern[b, Blank[]]], Power[Plus[Pattern[c, Blank[]], Times[Optional[Pattern[d, Blank[]]], Pattern[x, Blank[]]]], Pattern[n, Blank[]]]]], Pattern[p, Blank[]]]], Pattern[x, Blank[Symbol]]] := With[{k = Denominator[n]}, Dist[k/d, Subst[Int[SimplifyIntegrand[x^(k - 1)*(Px /. x -> x^k/d - c/d)^q*(a + b*x^(k*n))^p, x], x], x, (c + d*x)^(1/k)], x]] /; FreeQ[{a, b, c, d, p}, x] && PolynomialQ[Px, x] && IntegerQ[q] && FractionQ[n]
  1901. Int[Power[Plus[Times[Optional[Pattern[b, Blank[]]], Power[Pattern[x, Blank[]], Optional[Pattern[n, Blank[]]]]], Times[Optional[Pattern[a, Blank[]]], Power[Pattern[x, Blank[]], Optional[Pattern[q, Blank[]]]]], Times[Optional[Pattern[c, Blank[]]], Power[Pattern[x, Blank[]], Optional[Pattern[r, Blank[]]]]]], Pattern[p, Blank[]]], Pattern[x, Blank[Symbol]]] := Int[((a + b + c)*x^n)^p, x] /; FreeQ[{a, b, c, n, p}, x] && EqQ[n, q] && EqQ[r, n]
  1902. Int[Power[Plus[Times[Optional[Pattern[b, Blank[]]], Power[Pattern[x, Blank[]], Optional[Pattern[n, Blank[]]]]], Times[Optional[Pattern[a, Blank[]]], Power[Pattern[x, Blank[]], Optional[Pattern[q, Blank[]]]]], Times[Optional[Pattern[c, Blank[]]], Power[Pattern[x, Blank[]], Optional[Pattern[r, Blank[]]]]]], Pattern[p, Blank[]]], Pattern[x, Blank[Symbol]]] := Int[x^(p*q)*(a + b*x^(n - q) + c*x^(2*(n - q)))^p, x] /; FreeQ[{a, b, c, n, q}, x] && EqQ[r, 2*n - q] && PosQ[n - q] && IntegerQ[p]
  1903. Int[Power[Plus[Times[Optional[Pattern[b, Blank[]]], Power[Pattern[x, Blank[]], Optional[Pattern[n, Blank[]]]]], Times[Optional[Pattern[a, Blank[]]], Power[Pattern[x, Blank[]], Optional[Pattern[q, Blank[]]]]], Times[Optional[Pattern[c, Blank[]]], Power[Pattern[x, Blank[]], Optional[Pattern[r, Blank[]]]]]], Rational[1, 2]], Pattern[x, Blank[Symbol]]] := Dist[Sqrt[a*x^q + b*x^n + c*x^(2*n - q)]/(x^(q/2)*Sqrt[a + b*x^(n - q) + c*x^(2*(n - q))]), Int[x^(q/2)*Sqrt[a + b*x^(n - q) + c*x^(2*(n - q))], x], x] /; FreeQ[{a, b, c, n, q}, x] && EqQ[r, 2*n - q] && PosQ[n - q]
  1904. Int[Power[Plus[Times[Optional[Pattern[a, Blank[]]], Power[Pattern[x, Blank[]], 2]], Times[Optional[Pattern[b, Blank[]]], Power[Pattern[x, Blank[]], Optional[Pattern[n, Blank[]]]]], Times[Optional[Pattern[c, Blank[]]], Power[Pattern[x, Blank[]], Optional[Pattern[r, Blank[]]]]]], Rational[-1, 2]], Pattern[x, Blank[Symbol]]] := Dist[-2/(n - 2), Subst[Int[1/(4*a - x^2), x], x, (x*(2*a + b*x^(n - 2)))/Sqrt[a*x^2 + b*x^n + c*x^r]], x] /; FreeQ[{a, b, c, n, r}, x] && EqQ[r, 2*n - 2] && PosQ[n - 2] && NeQ[b^2 - 4*a*c, 0]
  1905. Int[Power[Plus[Times[Optional[Pattern[b, Blank[]]], Power[Pattern[x, Blank[]], Optional[Pattern[n, Blank[]]]]], Times[Optional[Pattern[a, Blank[]]], Power[Pattern[x, Blank[]], Optional[Pattern[q, Blank[]]]]], Times[Optional[Pattern[c, Blank[]]], Power[Pattern[x, Blank[]], Optional[Pattern[r, Blank[]]]]]], Rational[-1, 2]], Pattern[x, Blank[Symbol]]] := Dist[(x^(q/2)*Sqrt[a + b*x^(n - q) + c*x^(2*(n - q))])/Sqrt[a*x^q + b*x^n + c*x^(2*n - q)], Int[1/(x^(q/2)*Sqrt[a + b*x^(n - q) + c*x^(2*(n - q))]), x], x] /; FreeQ[{a, b, c, n, q}, x] && EqQ[r, 2*n - q] && PosQ[n - q]
  1906. Int[Power[Plus[Times[Optional[Pattern[b, Blank[]]], Power[Pattern[x, Blank[]], Optional[Pattern[n, Blank[]]]]], Times[Optional[Pattern[a, Blank[]]], Power[Pattern[x, Blank[]], Optional[Pattern[q, Blank[]]]]], Times[Optional[Pattern[c, Blank[]]], Power[Pattern[x, Blank[]], Optional[Pattern[r, Blank[]]]]]], Pattern[p, Blank[]]], Pattern[x, Blank[Symbol]]] := Simp[(x*(a*x^q + b*x^n + c*x^(2*n - q))^p)/(p*(2*n - q) + 1), x] + Dist[((n - q)*p)/(p*(2*n - q) + 1), Int[x^q*(2*a + b*x^(n - q))*(a*x^q + b*x^n + c*x^(2*n - q))^(p - 1), x], x] /; FreeQ[{a, b, c, n, q}, x] && EqQ[r, 2*n - q] && PosQ[n - q] && !IntegerQ[p] && NeQ[b^2 - 4*a*c, 0] && GtQ[p, 0] && NeQ[p*(2*n - q) + 1, 0]
  1907. Int[Power[Plus[Times[Optional[Pattern[b, Blank[]]], Power[Pattern[x, Blank[]], Optional[Pattern[n, Blank[]]]]], Times[Optional[Pattern[a, Blank[]]], Power[Pattern[x, Blank[]], Optional[Pattern[q, Blank[]]]]], Times[Optional[Pattern[c, Blank[]]], Power[Pattern[x, Blank[]], Optional[Pattern[r, Blank[]]]]]], Pattern[p, Blank[]]], Pattern[x, Blank[Symbol]]] := -Simp[(x^(-q + 1)*(b^2 - 2*a*c + b*c*x^(n - q))*(a*x^q + b*x^n + c*x^(2*n - q))^(p + 1))/(a*(n - q)*(p + 1)*(b^2 - 4*a*c)), x] + Dist[1/(a*(n - q)*(p + 1)*(b^2 - 4*a*c)), Int[(((p*q + 1)*(b^2 - 2*a*c) + (n - q)*(p + 1)*(b^2 - 4*a*c) + b*c*(p*q + (n - q)*(2*p + 3) + 1)*x^(n - q))*(a*x^q + b*x^n + c*x^(2*n - q))^(p + 1))/x^q, x], x] /; FreeQ[{a, b, c, n, q}, x] && EqQ[r, 2*n - q] && PosQ[n - q] && !IntegerQ[p] && NeQ[b^2 - 4*a*c, 0] && LtQ[p, -1]
  1908. Int[Power[Plus[Times[Optional[Pattern[b, Blank[]]], Power[Pattern[x, Blank[]], Optional[Pattern[n, Blank[]]]]], Times[Optional[Pattern[a, Blank[]]], Power[Pattern[x, Blank[]], Optional[Pattern[q, Blank[]]]]], Times[Optional[Pattern[c, Blank[]]], Power[Pattern[x, Blank[]], Optional[Pattern[r, Blank[]]]]]], Pattern[p, Blank[]]], Pattern[x, Blank[Symbol]]] := Dist[(a*x^q + b*x^n + c*x^(2*n - q))^p/(x^(p*q)*(a + b*x^(n - q) + c*x^(2*(n - q)))^p), Int[x^(p*q)*(a + b*x^(n - q) + c*x^(2*(n - q)))^p, x], x] /; FreeQ[{a, b, c, n, p, q}, x] && EqQ[r, 2*n - q] && PosQ[n - q] && !IntegerQ[p]
  1909. Int[Power[Plus[Times[Optional[Pattern[b, Blank[]]], Power[Pattern[x, Blank[]], Optional[Pattern[n, Blank[]]]]], Times[Optional[Pattern[a, Blank[]]], Power[Pattern[x, Blank[]], Optional[Pattern[q, Blank[]]]]], Times[Optional[Pattern[c, Blank[]]], Power[Pattern[x, Blank[]], Optional[Pattern[r, Blank[]]]]]], Pattern[p, Blank[]]], Pattern[x, Blank[Symbol]]] := Unintegrable[(b*x^n + c*x^(2*n - q) + a*x^q)^p, x] /; FreeQ[{a, b, c, n, p, q}, x] && EqQ[r, 2*n - q]
  1910. Int[Power[Plus[Times[Optional[Pattern[b, Blank[]]], Power[Pattern[u, Blank[]], Optional[Pattern[n, Blank[]]]]], Times[Optional[Pattern[a, Blank[]]], Power[Pattern[u, Blank[]], Optional[Pattern[q, Blank[]]]]], Times[Optional[Pattern[c, Blank[]]], Power[Pattern[u, Blank[]], Optional[Pattern[r, Blank[]]]]]], Pattern[p, Blank[]]], Pattern[x, Blank[Symbol]]] := Dist[1/Coefficient[u, x, 1], Subst[Int[(a*x^q + b*x^n + c*x^(2*n - q))^p, x], x, u], x] /; FreeQ[{a, b, c, n, p, q}, x] && EqQ[r, 2*n - q] && LinearQ[u, x] && NeQ[u, x]
  1911. Int[Times[Power[Pattern[x, Blank[]], Optional[Pattern[m, Blank[]]]], Power[Plus[Times[Optional[Pattern[b, Blank[]]], Power[Pattern[x, Blank[]], Optional[Pattern[n, Blank[]]]]], Times[Optional[Pattern[a, Blank[]]], Power[Pattern[x, Blank[]], Optional[Pattern[q, Blank[]]]]], Times[Optional[Pattern[c, Blank[]]], Power[Pattern[x, Blank[]], Optional[Pattern[r, Blank[]]]]]], Optional[Pattern[p, Blank[]]]]], Pattern[x, Blank[Symbol]]] := Int[x^m*((a + b + c)*x^n)^p, x] /; FreeQ[{a, b, c, m, n, p}, x] && EqQ[q, n] && EqQ[r, n]
  1912. Int[Times[Power[Pattern[x, Blank[]], Optional[Pattern[m, Blank[]]]], Power[Plus[Times[Optional[Pattern[b, Blank[]]], Power[Pattern[x, Blank[]], Optional[Pattern[n, Blank[]]]]], Times[Optional[Pattern[a, Blank[]]], Power[Pattern[x, Blank[]], Optional[Pattern[q, Blank[]]]]], Times[Optional[Pattern[c, Blank[]]], Power[Pattern[x, Blank[]], Optional[Pattern[r, Blank[]]]]]], Optional[Pattern[p, Blank[]]]]], Pattern[x, Blank[Symbol]]] := Int[x^(m + p*q)*(a + b*x^(n - q) + c*x^(2*(n - q)))^p, x] /; FreeQ[{a, b, c, m, n, q}, x] && EqQ[r, 2*n - q] && IntegerQ[p] && PosQ[n - q]
  1913. Int[Times[Power[Pattern[x, Blank[]], Optional[Pattern[m, Blank[]]]], Power[Plus[Times[Optional[Pattern[b, Blank[]]], Power[Pattern[x, Blank[]], Optional[Pattern[n, Blank[]]]]], Times[Optional[Pattern[a, Blank[]]], Power[Pattern[x, Blank[]], Optional[Pattern[q, Blank[]]]]], Times[Optional[Pattern[c, Blank[]]], Power[Pattern[x, Blank[]], Optional[Pattern[r, Blank[]]]]]], Rational[-1, 2]]], Pattern[x, Blank[Symbol]]] := Dist[-2/(n - q), Subst[Int[1/(4*a - x^2), x], x, (x^(m + 1)*(2*a + b*x^(n - q)))/Sqrt[a*x^q + b*x^n + c*x^r]], x] /; FreeQ[{a, b, c, m, n, q, r}, x] && EqQ[r, 2*n - q] && PosQ[n - q] && NeQ[b^2 - 4*a*c, 0] && EqQ[m, q/2 - 1]
  1914. Int[Times[Power[Pattern[x, Blank[]], Optional[Pattern[m, Blank[]]]], Power[Plus[Times[Optional[Pattern[b, Blank[]]], Power[Pattern[x, Blank[]], Optional[Pattern[n, Blank[]]]]], Times[Optional[Pattern[a, Blank[]]], Power[Pattern[x, Blank[]], Optional[Pattern[q, Blank[]]]]], Times[Optional[Pattern[c, Blank[]]], Power[Pattern[x, Blank[]], Optional[Pattern[r, Blank[]]]]]], Rational[-1, 2]]], Pattern[x, Blank[Symbol]]] := Dist[(x^(q/2)*Sqrt[a + b*x^(n - q) + c*x^(2*(n - q))])/Sqrt[a*x^q + b*x^n + c*x^(2*n - q)], Int[x^(m - q/2)/Sqrt[a + b*x^(n - q) + c*x^(2*(n - q))], x], x] /; FreeQ[{a, b, c, m, n, q}, x] && EqQ[r, 2*n - q] && PosQ[n - q] && ((EqQ[m, 1] && EqQ[n, 3] && EqQ[q, 2]) || ((EqQ[m + 1/2] || EqQ[m, 3/2] || EqQ[m, 1/2] || EqQ[m, 5/2]) && EqQ[n, 3] && EqQ[q, 1]))
  1915. Int[Times[Power[Pattern[x, Blank[]], Optional[Pattern[m, Blank[]]]], Power[Plus[Times[Optional[Pattern[b, Blank[]]], Power[Pattern[x, Blank[]], Optional[Pattern[n, Blank[]]]]], Times[Optional[Pattern[a, Blank[]]], Power[Pattern[x, Blank[]], Optional[Pattern[q, Blank[]]]]], Times[Optional[Pattern[c, Blank[]]], Power[Pattern[x, Blank[]], Optional[Pattern[r, Blank[]]]]]], Rational[-3, 2]]], Pattern[x, Blank[Symbol]]] := Simp[(-2*x^((n - 1)/2)*(b + 2*c*x))/((b^2 - 4*a*c)*Sqrt[a*x^(n - 1) + b*x^n + c*x^(n + 1)]), x] /; FreeQ[{a, b, c, n}, x] && EqQ[m, (3*(n - 1))/2] && EqQ[q, n - 1] && EqQ[r, n + 1] && NeQ[b^2 - 4*a*c, 0]
  1916. Int[Times[Power[Pattern[x, Blank[]], Optional[Pattern[m, Blank[]]]], Power[Plus[Times[Optional[Pattern[b, Blank[]]], Power[Pattern[x, Blank[]], Optional[Pattern[n, Blank[]]]]], Times[Optional[Pattern[a, Blank[]]], Power[Pattern[x, Blank[]], Optional[Pattern[q, Blank[]]]]], Times[Optional[Pattern[c, Blank[]]], Power[Pattern[x, Blank[]], Optional[Pattern[r, Blank[]]]]]], Rational[-3, 2]]], Pattern[x, Blank[Symbol]]] := Simp[(x^((n - 1)/2)*(4*a + 2*b*x))/((b^2 - 4*a*c)*Sqrt[a*x^(n - 1) + b*x^n + c*x^(n + 1)]), x] /; FreeQ[{a, b, c, n}, x] && EqQ[m, (3*n - 1)/2] && EqQ[q, n - 1] && EqQ[r, n + 1] && NeQ[b^2 - 4*a*c, 0]
  1917. Int[Times[Power[Pattern[x, Blank[]], Optional[Pattern[m, Blank[]]]], Power[Plus[Times[Optional[Pattern[b, Blank[]]], Power[Pattern[x, Blank[]], Optional[Pattern[n, Blank[]]]]], Times[Optional[Pattern[a, Blank[]]], Power[Pattern[x, Blank[]], Optional[Pattern[q, Blank[]]]]], Times[Optional[Pattern[c, Blank[]]], Power[Pattern[x, Blank[]], Optional[Pattern[r, Blank[]]]]]], Pattern[p, Blank[]]]], Pattern[x, Blank[Symbol]]] := Simp[(x^(m - n)*(a*x^(n - 1) + b*x^n + c*x^(n + 1))^(p + 1))/(2*c*(p + 1)), x] - Dist[b/(2*c), Int[x^(m - 1)*(a*x^(n - 1) + b*x^n + c*x^(n + 1))^p, x], x] /; FreeQ[{a, b, c}, x] && EqQ[r, 2*n - q] && PosQ[n - q] && !IntegerQ[p] && NeQ[b^2 - 4*a*c, 0] && IGtQ[n, 0] && RationalQ[m, p, q] && EqQ[m + p*(n - 1) - 1, 0]
  1918. Int[Times[Power[Pattern[x, Blank[]], Optional[Pattern[m, Blank[]]]], Power[Plus[Times[Optional[Pattern[b, Blank[]]], Power[Pattern[x, Blank[]], Optional[Pattern[n, Blank[]]]]], Times[Optional[Pattern[a, Blank[]]], Power[Pattern[x, Blank[]], Optional[Pattern[q, Blank[]]]]], Times[Optional[Pattern[c, Blank[]]], Power[Pattern[x, Blank[]], Optional[Pattern[r, Blank[]]]]]], Pattern[p, Blank[]]]], Pattern[x, Blank[Symbol]]] := Simp[(x^(m - n + q + 1)*(b + 2*c*x^(n - q))*(a*x^q + b*x^n + c*x^(2*n - q))^p)/(2*c*(n - q)*(2*p + 1)), x] - Dist[(p*(b^2 - 4*a*c))/(2*c*(2*p + 1)), Int[x^(m + q)*(a*x^q + b*x^n + c*x^(2*n - q))^(p - 1), x], x] /; FreeQ[{a, b, c}, x] && EqQ[r, 2*n - q] && PosQ[n - q] && !IntegerQ[p] && NeQ[b^2 - 4*a*c, 0] && IGtQ[n, 0] && GtQ[p, 0] && RationalQ[m, q] && EqQ[m + p*q + 1, n - q]
  1919. Int[Times[Power[Pattern[x, Blank[]], Optional[Pattern[m, Blank[]]]], Power[Plus[Times[Optional[Pattern[b, Blank[]]], Power[Pattern[x, Blank[]], Optional[Pattern[n, Blank[]]]]], Times[Optional[Pattern[a, Blank[]]], Power[Pattern[x, Blank[]], Optional[Pattern[q, Blank[]]]]], Times[Optional[Pattern[c, Blank[]]], Power[Pattern[x, Blank[]], Optional[Pattern[r, Blank[]]]]]], Pattern[p, Blank[]]]], Pattern[x, Blank[Symbol]]] := Simp[(x^(m - n + q + 1)*(b*(n - q)*p + c*(m + p*q + (n - q)*(2*p - 1) + 1)*x^(n - q))*(a*x^q + b*x^n + c*x^(2*n - q))^p)/(c*(m + p*(2*n - q) + 1)*(m + p*q + (n - q)*(2*p - 1) + 1)), x] + Dist[((n - q)*p)/(c*(m + p*(2*n - q) + 1)*(m + p*q + (n - q)*(2*p - 1) + 1)), Int[x^(m - (n - 2*q))*Simp[-(a*b*(m + p*q - n + q + 1)) + (2*a*c*(m + p*q + (n - q)*(2*p - 1) + 1) - b^2*(m + p*q + (n - q)*(p - 1) + 1))*x^(n - q), x]*(a*x^q + b*x^n + c*x^(2*n - q))^(p - 1), x], x] /; FreeQ[{a, b, c}, x] && EqQ[r, 2*n - q] && PosQ[n - q] && !IntegerQ[p] && NeQ[b^2 - 4*a*c, 0] && IGtQ[n, 0] && GtQ[p, 0] && RationalQ[m, q] && GtQ[m + p*q + 1, n - q] && NeQ[m + p*(2*n - q) + 1, 0] && NeQ[m + p*q + (n - q)*(2*p - 1) + 1, 0]
  1920. Int[Times[Power[Pattern[x, Blank[]], Optional[Pattern[m, Blank[]]]], Power[Plus[Times[Optional[Pattern[b, Blank[]]], Power[Pattern[x, Blank[]], Optional[Pattern[n, Blank[]]]]], Times[Optional[Pattern[a, Blank[]]], Power[Pattern[x, Blank[]], Optional[Pattern[q, Blank[]]]]], Times[Optional[Pattern[c, Blank[]]], Power[Pattern[x, Blank[]], Optional[Pattern[r, Blank[]]]]]], Pattern[p, Blank[]]]], Pattern[x, Blank[Symbol]]] := Simp[(x^(m + 1)*(a*x^q + b*x^n + c*x^(2*n - q))^p)/(m + p*q + 1), x] - Dist[((n - q)*p)/(m + p*q + 1), Int[x^(m + n)*(b + 2*c*x^(n - q))*(a*x^q + b*x^n + c*x^(2*n - q))^(p - 1), x], x] /; FreeQ[{a, b, c}, x] && EqQ[r, 2*n - q] && PosQ[n - q] && !IntegerQ[p] && NeQ[b^2 - 4*a*c, 0] && IGtQ[n, 0] && GtQ[p, 0] && RationalQ[m, q] && LeQ[m + p*q + 1, -(n - q) + 1] && NeQ[m + p*q + 1, 0]
  1921. Int[Times[Power[Pattern[x, Blank[]], Optional[Pattern[m, Blank[]]]], Power[Plus[Times[Optional[Pattern[b, Blank[]]], Power[Pattern[x, Blank[]], Optional[Pattern[n, Blank[]]]]], Times[Optional[Pattern[a, Blank[]]], Power[Pattern[x, Blank[]], Optional[Pattern[q, Blank[]]]]], Times[Optional[Pattern[c, Blank[]]], Power[Pattern[x, Blank[]], Optional[Pattern[r, Blank[]]]]]], Pattern[p, Blank[]]]], Pattern[x, Blank[Symbol]]] := Simp[(x^(m + 1)*(a*x^q + b*x^n + c*x^(2*n - q))^p)/(m + p*(2*n - q) + 1), x] + Dist[((n - q)*p)/(m + p*(2*n - q) + 1), Int[x^(m + q)*(2*a + b*x^(n - q))*(a*x^q + b*x^n + c*x^(2*n - q))^(p - 1), x], x] /; FreeQ[{a, b, c}, x] && EqQ[r, 2*n - q] && PosQ[n - q] && !IntegerQ[p] && NeQ[b^2 - 4*a*c, 0] && IGtQ[n, 0] && GtQ[p, 0] && RationalQ[m, q] && GtQ[m + p*q + 1, -(n - q)] && NeQ[m + p*(2*n - q) + 1, 0]
  1922. Int[Times[Power[Pattern[x, Blank[]], Optional[Pattern[m, Blank[]]]], Power[Plus[Times[Optional[Pattern[b, Blank[]]], Power[Pattern[x, Blank[]], Optional[Pattern[n, Blank[]]]]], Times[Optional[Pattern[a, Blank[]]], Power[Pattern[x, Blank[]], Optional[Pattern[q, Blank[]]]]], Times[Optional[Pattern[c, Blank[]]], Power[Pattern[x, Blank[]], Optional[Pattern[r, Blank[]]]]]], Pattern[p, Blank[]]]], Pattern[x, Blank[Symbol]]] := -Simp[(x^(m - q + 1)*(b^2 - 2*a*c + b*c*x^(n - q))*(a*x^q + b*x^n + c*x^(2*n - q))^(p + 1))/(a*(n - q)*(p + 1)*(b^2 - 4*a*c)), x] + Dist[(2*a*c - b^2*(p + 2))/(a*(p + 1)*(b^2 - 4*a*c)), Int[x^(m - q)*(a*x^q + b*x^n + c*x^(2*n - q))^(p + 1), x], x] /; FreeQ[{a, b, c}, x] && EqQ[r, 2*n - q] && PosQ[n - q] && !IntegerQ[p] && NeQ[b^2 - 4*a*c, 0] && IGtQ[n, 0] && LtQ[p, -1] && RationalQ[m, p, q] && EqQ[m + p*q + 1, -((n - q)*(2*p + 3))]
  1923. Int[Times[Power[Pattern[x, Blank[]], Optional[Pattern[m, Blank[]]]], Power[Plus[Times[Optional[Pattern[b, Blank[]]], Power[Pattern[x, Blank[]], Optional[Pattern[n, Blank[]]]]], Times[Optional[Pattern[a, Blank[]]], Power[Pattern[x, Blank[]], Optional[Pattern[q, Blank[]]]]], Times[Optional[Pattern[c, Blank[]]], Power[Pattern[x, Blank[]], Optional[Pattern[r, Blank[]]]]]], Pattern[p, Blank[]]]], Pattern[x, Blank[Symbol]]] := -Simp[(x^(m - 2*n + q + 1)*(2*a + b*x^(n - q))*(a*x^q + b*x^n + c*x^(2*n - q))^(p + 1))/((n - q)*(p + 1)*(b^2 - 4*a*c)), x] + Dist[1/((n - q)*(p + 1)*(b^2 - 4*a*c)), Int[x^(m - 2*n + q)*(2*a*(m + p*q - 2*(n - q) + 1) + b*(m + p*q + (n - q)*(2*p + 1) + 1)*x^(n - q))*(a*x^q + b*x^n + c*x^(2*n - q))^(p + 1), x], x] /; FreeQ[{a, b, c}, x] && EqQ[r, 2*n - q] && PosQ[n - q] && !IntegerQ[p] && NeQ[b^2 - 4*a*c, 0] && IGtQ[n, 0] && LtQ[p, -1] && RationalQ[m, q] && GtQ[m + p*q + 1, 2*(n - q)]
  1924. Int[Times[Power[Pattern[x, Blank[]], Optional[Pattern[m, Blank[]]]], Power[Plus[Times[Optional[Pattern[b, Blank[]]], Power[Pattern[x, Blank[]], Optional[Pattern[n, Blank[]]]]], Times[Optional[Pattern[a, Blank[]]], Power[Pattern[x, Blank[]], Optional[Pattern[q, Blank[]]]]], Times[Optional[Pattern[c, Blank[]]], Power[Pattern[x, Blank[]], Optional[Pattern[r, Blank[]]]]]], Pattern[p, Blank[]]]], Pattern[x, Blank[Symbol]]] := -Simp[(x^(m - q + 1)*(b^2 - 2*a*c + b*c*x^(n - q))*(a*x^q + b*x^n + c*x^(2*n - q))^(p + 1))/(a*(n - q)*(p + 1)*(b^2 - 4*a*c)), x] + Dist[1/(a*(n - q)*(p + 1)*(b^2 - 4*a*c)), Int[x^(m - q)*(b^2*(m + p*q + (n - q)*(p + 1) + 1) - 2*a*c*(m + p*q + 2*(n - q)*(p + 1) + 1) + b*c*(m + p*q + (n - q)*(2*p + 3) + 1)*x^(n - q))*(a*x^q + b*x^n + c*x^(2*n - q))^(p + 1), x], x] /; FreeQ[{a, b, c}, x] && EqQ[r, 2*n - q] && PosQ[n - q] && !IntegerQ[p] && NeQ[b^2 - 4*a*c, 0] && IGtQ[n, 0] && LtQ[p, -1] && RationalQ[m, q] && LtQ[m + p*q + 1, n - q]
  1925. Int[Times[Power[Pattern[x, Blank[]], Optional[Pattern[m, Blank[]]]], Power[Plus[Times[Optional[Pattern[b, Blank[]]], Power[Pattern[x, Blank[]], Optional[Pattern[n, Blank[]]]]], Times[Optional[Pattern[a, Blank[]]], Power[Pattern[x, Blank[]], Optional[Pattern[q, Blank[]]]]], Times[Optional[Pattern[c, Blank[]]], Power[Pattern[x, Blank[]], Optional[Pattern[r, Blank[]]]]]], Pattern[p, Blank[]]]], Pattern[x, Blank[Symbol]]] := Simp[(x^(m - n + 1)*(b + 2*c*x^(n - q))*(a*x^q + b*x^n + c*x^(2*n - q))^(p + 1))/((n - q)*(p + 1)*(b^2 - 4*a*c)), x] - Dist[1/((n - q)*(p + 1)*(b^2 - 4*a*c)), Int[x^(m - n)*(b*(m + p*q - n + q + 1) + 2*c*(m + p*q + 2*(n - q)*(p + 1) + 1)*x^(n - q))*(a*x^q + b*x^n + c*x^(2*n - q))^(p + 1), x], x] /; FreeQ[{a, b, c}, x] && EqQ[r, 2*n - q] && PosQ[n - q] && !IntegerQ[p] && NeQ[b^2 - 4*a*c, 0] && IGtQ[n, 0] && LtQ[p, -1] && RationalQ[m, q] && LtQ[n - q, m + p*q + 1, 2*(n - q)]
  1926. Int[Times[Power[Pattern[x, Blank[]], Optional[Pattern[m, Blank[]]]], Power[Plus[Times[Optional[Pattern[b, Blank[]]], Power[Pattern[x, Blank[]], Optional[Pattern[n, Blank[]]]]], Times[Optional[Pattern[a, Blank[]]], Power[Pattern[x, Blank[]], Optional[Pattern[q, Blank[]]]]], Times[Optional[Pattern[c, Blank[]]], Power[Pattern[x, Blank[]], Optional[Pattern[r, Blank[]]]]]], Pattern[p, Blank[]]]], Pattern[x, Blank[Symbol]]] := Simp[(x^(m - 2*n + q + 1)*(a*x^q + b*x^n + c*x^(2*n - q))^(p + 1))/(2*c*(n - q)*(p + 1)), x] - Dist[b/(2*c), Int[x^(m - n + q)*(a*x^q + b*x^n + c*x^(2*n - q))^p, x], x] /; FreeQ[{a, b, c}, x] && EqQ[r, 2*n - q] && PosQ[n - q] && !IntegerQ[p] && NeQ[b^2 - 4*a*c, 0] && IGtQ[n, 0] && GeQ[p, -1] && LtQ[p, 0] && RationalQ[m, q] && EqQ[m + p*q + 1, 2*(n - q)]
  1927. Int[Times[Power[Pattern[x, Blank[]], Optional[Pattern[m, Blank[]]]], Power[Plus[Times[Optional[Pattern[b, Blank[]]], Power[Pattern[x, Blank[]], Optional[Pattern[n, Blank[]]]]], Times[Optional[Pattern[a, Blank[]]], Power[Pattern[x, Blank[]], Optional[Pattern[q, Blank[]]]]], Times[Optional[Pattern[c, Blank[]]], Power[Pattern[x, Blank[]], Optional[Pattern[r, Blank[]]]]]], Pattern[p, Blank[]]]], Pattern[x, Blank[Symbol]]] := -Simp[(x^(m - q + 1)*(a*x^q + b*x^n + c*x^(2*n - q))^(p + 1))/(2*a*(n - q)*(p + 1)), x] - Dist[b/(2*a), Int[x^(m + n - q)*(a*x^q + b*x^n + c*x^(2*n - q))^p, x], x] /; FreeQ[{a, b, c}, x] && EqQ[r, 2*n - q] && PosQ[n - q] && !IntegerQ[p] && NeQ[b^2 - 4*a*c, 0] && IGtQ[n, 0] && GeQ[p, -1] && LtQ[p, 0] && RationalQ[m, q] && EqQ[m + p*q + 1, -2*(n - q)*(p + 1)]
  1928. Int[Times[Power[Pattern[x, Blank[]], Optional[Pattern[m, Blank[]]]], Power[Plus[Times[Optional[Pattern[b, Blank[]]], Power[Pattern[x, Blank[]], Optional[Pattern[n, Blank[]]]]], Times[Optional[Pattern[a, Blank[]]], Power[Pattern[x, Blank[]], Optional[Pattern[q, Blank[]]]]], Times[Optional[Pattern[c, Blank[]]], Power[Pattern[x, Blank[]], Optional[Pattern[r, Blank[]]]]]], Pattern[p, Blank[]]]], Pattern[x, Blank[Symbol]]] := Simp[(x^(m - 2*n + q + 1)*(a*x^q + b*x^n + c*x^(2*n - q))^(p + 1))/(c*(m + p*q + 2*(n - q)*p + 1)), x] - Dist[1/(c*(m + p*q + 2*(n - q)*p + 1)), Int[x^(m - 2*(n - q))*(a*(m + p*q - 2*(n - q) + 1) + b*(m + p*q + (n - q)*(p - 1) + 1)*x^(n - q))*(a*x^q + b*x^n + c*x^(2*n - q))^p, x], x] /; FreeQ[{a, b, c}, x] && EqQ[r, 2*n - q] && PosQ[n - q] && !IntegerQ[p] && NeQ[b^2 - 4*a*c, 0] && IGtQ[n, 0] && GeQ[p, -1] && LtQ[p, 0] && RationalQ[m, q] && GtQ[m + p*q + 1, 2*(n - q)]
  1929. Int[Times[Power[Pattern[x, Blank[]], Optional[Pattern[m, Blank[]]]], Power[Plus[Times[Optional[Pattern[b, Blank[]]], Power[Pattern[x, Blank[]], Optional[Pattern[n, Blank[]]]]], Times[Optional[Pattern[a, Blank[]]], Power[Pattern[x, Blank[]], Optional[Pattern[q, Blank[]]]]], Times[Optional[Pattern[c, Blank[]]], Power[Pattern[x, Blank[]], Optional[Pattern[r, Blank[]]]]]], Pattern[p, Blank[]]]], Pattern[x, Blank[Symbol]]] := Simp[(x^(m - q + 1)*(a*x^q + b*x^n + c*x^(2*n - q))^(p + 1))/(a*(m + p*q + 1)), x] - Dist[1/(a*(m + p*q + 1)), Int[x^(m + n - q)*(b*(m + p*q + (n - q)*(p + 1) + 1) + c*(m + p*q + 2*(n - q)*(p + 1) + 1)*x^(n - q))*(a*x^q + b*x^n + c*x^(2*n - q))^p, x], x] /; FreeQ[{a, b, c}, x] && EqQ[r, 2*n - q] && PosQ[n - q] && !IntegerQ[p] && NeQ[b^2 - 4*a*c, 0] && IGtQ[n, 0] && GeQ[p, -1] && LtQ[p, 0] && RationalQ[m, q] && LtQ[m + p*q + 1, 0]
  1930. Int[Times[Power[Pattern[x, Blank[]], Optional[Pattern[m, Blank[]]]], Power[Plus[Times[Optional[Pattern[b, Blank[]]], Power[Pattern[x, Blank[]], Optional[Pattern[n, Blank[]]]]], Times[Optional[Pattern[a, Blank[]]], Power[Pattern[x, Blank[]], Optional[Pattern[q, Blank[]]]]], Times[Optional[Pattern[c, Blank[]]], Power[Pattern[x, Blank[]], Optional[Pattern[r, Blank[]]]]]], Pattern[p, Blank[]]]], Pattern[x, Blank[Symbol]]] := Dist[(a*x^q + b*x^n + c*x^(2*n - q))^p/(x^(p*q)*(a + b*x^(n - q) + c*x^(2*(n - q)))^p), Int[x^(m + p*q)*(a + b*x^(n - q) + c*x^(2*(n - q)))^p, x], x] /; FreeQ[{a, b, c, m, n, p, q}, x] && EqQ[r, 2*n - q] && !IntegerQ[p] && PosQ[n - q]
  1931. Int[Times[Power[Pattern[u, Blank[]], Optional[Pattern[m, Blank[]]]], Power[Plus[Times[Optional[Pattern[b, Blank[]]], Power[Pattern[u, Blank[]], Optional[Pattern[n, Blank[]]]]], Times[Optional[Pattern[a, Blank[]]], Power[Pattern[u, Blank[]], Optional[Pattern[q, Blank[]]]]], Times[Optional[Pattern[c, Blank[]]], Power[Pattern[u, Blank[]], Optional[Pattern[r, Blank[]]]]]], Optional[Pattern[p, Blank[]]]]], Pattern[x, Blank[Symbol]]] := Dist[1/Coefficient[u, x, 1], Subst[Int[x^m*(a*x^q + b*x^n + c*x^(2*n - q))^p, x], x, u], x] /; FreeQ[{a, b, c, m, n, p, q}, x] && EqQ[r, 2*n - q] && LinearQ[u, x] && NeQ[u, x]
  1932. Int[Times[Power[Plus[Times[Optional[Pattern[c, Blank[]]], Power[Pattern[x, Blank[]], Optional[Pattern[j, Blank[]]]]], Times[Optional[Pattern[b, Blank[]]], Power[Pattern[x, Blank[]], Optional[Pattern[n, Blank[]]]]], Times[Optional[Pattern[a, Blank[]]], Power[Pattern[x, Blank[]], Optional[Pattern[q, Blank[]]]]]], Optional[Pattern[p, Blank[]]]], Plus[Pattern[A, Blank[]], Times[Optional[Pattern[B, Blank[]]], Power[Pattern[x, Blank[]], Optional[Pattern[r, Blank[]]]]]]], Pattern[x, Blank[Symbol]]] := Int[x^(p*q)*(A + B*x^(n - q))*(a + b*x^(n - q) + c*x^(2*(n - q)))^p, x] /; FreeQ[{a, b, c, A, B, n, q}, x] && EqQ[r, n - q] && EqQ[j, 2*n - q] && IntegerQ[p] && PosQ[n - q]
  1933. Int[Times[Plus[Pattern[A, Blank[]], Times[Optional[Pattern[B, Blank[]]], Power[Pattern[x, Blank[]], Optional[Pattern[j, Blank[]]]]]], Power[Plus[Times[Optional[Pattern[b, Blank[]]], Power[Pattern[x, Blank[]], Optional[Pattern[n, Blank[]]]]], Times[Optional[Pattern[a, Blank[]]], Power[Pattern[x, Blank[]], Optional[Pattern[q, Blank[]]]]], Times[Optional[Pattern[c, Blank[]]], Power[Pattern[x, Blank[]], Optional[Pattern[r, Blank[]]]]]], Rational[-1, 2]]], Pattern[x, Blank[Symbol]]] := Dist[(x^(q/2)*Sqrt[a + b*x^(n - q) + c*x^(2*(n - q))])/Sqrt[a*x^q + b*x^n + c*x^(2*n - q)], Int[(A + B*x^(n - q))/(x^(q/2)*Sqrt[a + b*x^(n - q) + c*x^(2*(n - q))]), x], x] /; FreeQ[{a, b, c, A, B, n, q}, x] && EqQ[j, n - q] && EqQ[r, 2*n - q] && PosQ[n - q] && EqQ[n, 3] && EqQ[q, 2]
  1934. Int[Times[Power[Plus[Times[Optional[Pattern[c, Blank[]]], Power[Pattern[x, Blank[]], Optional[Pattern[j, Blank[]]]]], Times[Optional[Pattern[b, Blank[]]], Power[Pattern[x, Blank[]], Optional[Pattern[n, Blank[]]]]], Times[Optional[Pattern[a, Blank[]]], Power[Pattern[x, Blank[]], Optional[Pattern[q, Blank[]]]]]], Pattern[p, Blank[]]], Plus[Pattern[A, Blank[]], Times[Optional[Pattern[B, Blank[]]], Power[Pattern[x, Blank[]], Optional[Pattern[r, Blank[]]]]]]], Pattern[x, Blank[Symbol]]] := Simp[(x*(b*B*(n - q)*p + A*c*(p*q + (n - q)*(2*p + 1) + 1) + B*c*(p*(2*n - q) + 1)*x^(n - q))*(a*x^q + b*x^n + c*x^(2*n - q))^p)/(c*(p*(2*n - q) + 1)*(p*q + (n - q)*(2*p + 1) + 1)), x] + Dist[((n - q)*p)/(c*(p*(2*n - q) + 1)*(p*q + (n - q)*(2*p + 1) + 1)), Int[x^q*(2*a*A*c*(p*q + (n - q)*(2*p + 1) + 1) - a*b*B*(p*q + 1) + (2*a*B*c*(p*(2*n - q) + 1) + A*b*c*(p*q + (n - q)*(2*p + 1) + 1) - b^2*B*(p*q + (n - q)*p + 1))*x^(n - q))*(a*x^q + b*x^n + c*x^(2*n - q))^(p - 1), x], x] /; FreeQ[{a, b, c, A, B, n, q}, x] && EqQ[r, n - q] && EqQ[j, 2*n - q] && !IntegerQ[p] && NeQ[b^2 - 4*a*c, 0] && GtQ[p, 0] && NeQ[p*(2*n - q) + 1, 0] && NeQ[p*q + (n - q)*(2*p + 1) + 1, 0]
  1935. Int[Times[Power[Plus[Times[Optional[Pattern[c, Blank[]]], Power[Pattern[x, Blank[]], Optional[Pattern[j, Blank[]]]]], Times[Optional[Pattern[a, Blank[]]], Power[Pattern[x, Blank[]], Optional[Pattern[q, Blank[]]]]]], Pattern[p, Blank[]]], Plus[Pattern[A, Blank[]], Times[Optional[Pattern[B, Blank[]]], Power[Pattern[x, Blank[]], Optional[Pattern[r, Blank[]]]]]]], Pattern[x, Blank[Symbol]]] := With[{n = q + r}, Simp[(x*(A*(p*q + (n - q)*(2*p + 1) + 1) + B*(p*(2*n - q) + 1)*x^(n - q))*(a*x^q + c*x^(2*n - q))^p)/((p*(2*n - q) + 1)*(p*q + (n - q)*(2*p + 1) + 1)), x] + Dist[((n - q)*p)/((p*(2*n - q) + 1)*(p*q + (n - q)*(2*p + 1) + 1)), Int[x^q*(2*a*A*(p*q + (n - q)*(2*p + 1) + 1) + (2*a*B*(p*(2*n - q) + 1))*x^(n - q))*(a*x^q + c*x^(2*n - q))^(p - 1), x], x] /; EqQ[j, 2*n - q] && NeQ[p*(2*n - q) + 1, 0] && NeQ[p*q + (n - q)*(2*p + 1) + 1, 0]] /; FreeQ[{a, c, A, B, q}, x] && !IntegerQ[p] && GtQ[p, 0]
  1936. Int[Times[Power[Plus[Times[Optional[Pattern[c, Blank[]]], Power[Pattern[x, Blank[]], Optional[Pattern[j, Blank[]]]]], Times[Optional[Pattern[b, Blank[]]], Power[Pattern[x, Blank[]], Optional[Pattern[n, Blank[]]]]], Times[Optional[Pattern[a, Blank[]]], Power[Pattern[x, Blank[]], Optional[Pattern[q, Blank[]]]]]], Pattern[p, Blank[]]], Plus[Pattern[A, Blank[]], Times[Optional[Pattern[B, Blank[]]], Power[Pattern[x, Blank[]], Optional[Pattern[r, Blank[]]]]]]], Pattern[x, Blank[Symbol]]] := -Simp[(x^(-q + 1)*(A*b^2 - a*b*B - 2*a*A*c + (A*b - 2*a*B)*c*x^(n - q))*(a*x^q + b*x^n + c*x^(2*n - q))^(p + 1))/(a*(n - q)*(p + 1)*(b^2 - 4*a*c)), x] + Dist[1/(a*(n - q)*(p + 1)*(b^2 - 4*a*c)), Int[((A*b^2*(p*q + (n - q)*(p + 1) + 1) - a*b*B*(p*q + 1) - 2*a*A*c*(p*q + 2*(n - q)*(p + 1) + 1) + (p*q + (n - q)*(2*p + 3) + 1)*(A*b - 2*a*B)*c*x^(n - q))*(a*x^q + b*x^n + c*x^(2*n - q))^(p + 1))/x^q, x], x] /; FreeQ[{a, b, c, A, B, n, q}, x] && EqQ[r, n - q] && EqQ[j, 2*n - q] && !IntegerQ[p] && NeQ[b^2 - 4*a*c, 0] && LtQ[p, -1]
  1937. Int[Times[Power[Plus[Times[Optional[Pattern[c, Blank[]]], Power[Pattern[x, Blank[]], Optional[Pattern[j, Blank[]]]]], Times[Optional[Pattern[a, Blank[]]], Power[Pattern[x, Blank[]], Optional[Pattern[q, Blank[]]]]]], Pattern[p, Blank[]]], Plus[Pattern[A, Blank[]], Times[Optional[Pattern[B, Blank[]]], Power[Pattern[x, Blank[]], Optional[Pattern[r, Blank[]]]]]]], Pattern[x, Blank[Symbol]]] := With[{n = q + r}, -Simp[(x^(-q + 1)*(a*A*c + a*B*c*x^(n - q))*(a*x^q + c*x^(2*n - q))^(p + 1))/(a*(n - q)*(p + 1)*(2*a*c)), x] + Dist[1/(a*(n - q)*(p + 1)*(2*a*c)), Int[((a*A*c*(p*q + 2*(n - q)*(p + 1) + 1) + a*B*c*(p*q + (n - q)*(2*p + 3) + 1)*x^(n - q))*(a*x^q + c*x^(2*n - q))^(p + 1))/x^q, x], x] /; EqQ[j, 2*n - q]] /; FreeQ[{a, c, A, B, q}, x] && !IntegerQ[p] && LtQ[p, -1]
  1938. Int[Times[Plus[Pattern[A, Blank[]], Times[Optional[Pattern[B, Blank[]]], Power[Pattern[x, Blank[]], Optional[Pattern[j, Blank[]]]]]], Power[Plus[Times[Optional[Pattern[b, Blank[]]], Power[Pattern[x, Blank[]], Optional[Pattern[n, Blank[]]]]], Times[Optional[Pattern[a, Blank[]]], Power[Pattern[x, Blank[]], Optional[Pattern[q, Blank[]]]]], Times[Optional[Pattern[c, Blank[]]], Power[Pattern[x, Blank[]], Optional[Pattern[r, Blank[]]]]]], Optional[Pattern[p, Blank[]]]]], Pattern[x, Blank[Symbol]]] := Unintegrable[(A + B*x^(n - q))*(b*x^n + c*x^(2*n - q) + a*x^q)^p, x] /; FreeQ[{a, b, c, A, B, n, p, q}, x] && EqQ[j, n - q] && EqQ[r, 2*n - q]
  1939. Int[Times[Plus[Pattern[A, Blank[]], Times[Optional[Pattern[B, Blank[]]], Power[Pattern[u, Blank[]], Optional[Pattern[j, Blank[]]]]]], Power[Plus[Times[Optional[Pattern[b, Blank[]]], Power[Pattern[u, Blank[]], Optional[Pattern[n, Blank[]]]]], Times[Optional[Pattern[a, Blank[]]], Power[Pattern[u, Blank[]], Optional[Pattern[q, Blank[]]]]], Times[Optional[Pattern[c, Blank[]]], Power[Pattern[u, Blank[]], Optional[Pattern[r, Blank[]]]]]], Optional[Pattern[p, Blank[]]]]], Pattern[x, Blank[Symbol]]] := Dist[1/Coefficient[u, x, 1], Subst[Int[(A + B*x^(n - q))*(a*x^q + b*x^n + c*x^(2*n - q))^p, x], x, u], x] /; FreeQ[{a, b, c, A, B, n, p, q}, x] && EqQ[j, n - q] && EqQ[r, 2*n - q] && LinearQ[u, x] && NeQ[u, x]
  1940. Int[Times[Power[Pattern[x, Blank[]], Optional[Pattern[m, Blank[]]]], Power[Plus[Times[Optional[Pattern[c, Blank[]]], Power[Pattern[x, Blank[]], Optional[Pattern[j, Blank[]]]]], Times[Optional[Pattern[b, Blank[]]], Power[Pattern[x, Blank[]], Optional[Pattern[n, Blank[]]]]], Times[Optional[Pattern[a, Blank[]]], Power[Pattern[x, Blank[]], Optional[Pattern[q, Blank[]]]]]], Optional[Pattern[p, Blank[]]]], Plus[Pattern[A, Blank[]], Times[Optional[Pattern[B, Blank[]]], Power[Pattern[x, Blank[]], Optional[Pattern[r, Blank[]]]]]]], Pattern[x, Blank[Symbol]]] := Int[x^(m + p*q)*(A + B*x^(n - q))*(a + b*x^(n - q) + c*x^(2*(n - q)))^p, x] /; FreeQ[{a, b, c, A, B, m, n, q}, x] && EqQ[r, n - q] && EqQ[j, 2*n - q] && IntegerQ[p] && PosQ[n - q]
  1941. Int[Times[Power[Pattern[x, Blank[]], Optional[Pattern[m, Blank[]]]], Power[Plus[Times[Optional[Pattern[c, Blank[]]], Power[Pattern[x, Blank[]], Optional[Pattern[j, Blank[]]]]], Times[Optional[Pattern[b, Blank[]]], Power[Pattern[x, Blank[]], Optional[Pattern[n, Blank[]]]]], Times[Optional[Pattern[a, Blank[]]], Power[Pattern[x, Blank[]], Optional[Pattern[q, Blank[]]]]]], Optional[Pattern[p, Blank[]]]], Plus[Pattern[A, Blank[]], Times[Optional[Pattern[B, Blank[]]], Power[Pattern[x, Blank[]], Optional[Pattern[r, Blank[]]]]]]], Pattern[x, Blank[Symbol]]] := Simp[(x^(m + 1)*(A*(m + p*q + (n - q)*(2*p + 1) + 1) + B*(m + p*q + 1)*x^(n - q))*(a*x^q + b*x^n + c*x^(2*n - q))^p)/((m + p*q + 1)*(m + p*q + (n - q)*(2*p + 1) + 1)), x] + Dist[((n - q)*p)/((m + p*q + 1)*(m + p*q + (n - q)*(2*p + 1) + 1)), Int[x^(n + m)*Simp[2*a*B*(m + p*q + 1) - A*b*(m + p*q + (n - q)*(2*p + 1) + 1) + (b*B*(m + p*q + 1) - 2*A*c*(m + p*q + (n - q)*(2*p + 1) + 1))*x^(n - q), x]*(a*x^q + b*x^n + c*x^(2*n - q))^(p - 1), x], x] /; FreeQ[{a, b, c, A, B}, x] && EqQ[r, n - q] && EqQ[j, 2*n - q] && !IntegerQ[p] && NeQ[b^2 - 4*a*c, 0] && IGtQ[n, 0] && GtQ[p, 0] && RationalQ[m, q] && LeQ[m + p*q, -(n - q)] && NeQ[m + p*q + 1, 0] && NeQ[m + p*q + (n - q)*(2*p + 1) + 1, 0]
  1942. Int[Times[Power[Pattern[x, Blank[]], Optional[Pattern[m, Blank[]]]], Power[Plus[Times[Optional[Pattern[c, Blank[]]], Power[Pattern[x, Blank[]], Optional[Pattern[j, Blank[]]]]], Times[Optional[Pattern[a, Blank[]]], Power[Pattern[x, Blank[]], Optional[Pattern[q, Blank[]]]]]], Optional[Pattern[p, Blank[]]]], Plus[Pattern[A, Blank[]], Times[Optional[Pattern[B, Blank[]]], Power[Pattern[x, Blank[]], Optional[Pattern[r, Blank[]]]]]]], Pattern[x, Blank[Symbol]]] := With[{n = q + r}, Simp[(x^(m + 1)*(A*(m + p*q + (n - q)*(2*p + 1) + 1) + B*(m + p*q + 1)*x^(n - q))*(a*x^q + c*x^(2*n - q))^p)/((m + p*q + 1)*(m + p*q + (n - q)*(2*p + 1) + 1)), x] + Dist[(2*(n - q)*p)/((m + p*q + 1)*(m + p*q + (n - q)*(2*p + 1) + 1)), Int[x^(n + m)*Simp[a*B*(m + p*q + 1) - A*c*(m + p*q + (n - q)*(2*p + 1) + 1)*x^(n - q), x]*(a*x^q + c*x^(2*n - q))^(p - 1), x], x] /; EqQ[j, 2*n - q] && IGtQ[n, 0] && LeQ[m + p*q, -(n - q)] && NeQ[m + p*q + 1, 0] && NeQ[m + p*q + (n - q)*(2*p + 1) + 1, 0]] /; FreeQ[{a, c, A, B}, x] && !IntegerQ[p] && RationalQ[m, p, q] && GtQ[p, 0]
  1943. Int[Times[Power[Pattern[x, Blank[]], Optional[Pattern[m, Blank[]]]], Power[Plus[Times[Optional[Pattern[c, Blank[]]], Power[Pattern[x, Blank[]], Optional[Pattern[j, Blank[]]]]], Times[Optional[Pattern[b, Blank[]]], Power[Pattern[x, Blank[]], Optional[Pattern[n, Blank[]]]]], Times[Optional[Pattern[a, Blank[]]], Power[Pattern[x, Blank[]], Optional[Pattern[q, Blank[]]]]]], Optional[Pattern[p, Blank[]]]], Plus[Pattern[A, Blank[]], Times[Optional[Pattern[B, Blank[]]], Power[Pattern[x, Blank[]], Optional[Pattern[r, Blank[]]]]]]], Pattern[x, Blank[Symbol]]] := Simp[(x^(m - n + 1)*(A*b - 2*a*B - (b*B - 2*A*c)*x^(n - q))*(a*x^q + b*x^n + c*x^(2*n - q))^(p + 1))/((n - q)*(p + 1)*(b^2 - 4*a*c)), x] + Dist[1/((n - q)*(p + 1)*(b^2 - 4*a*c)), Int[x^(m - n)*Simp[(m + p*q - n + q + 1)*(2*a*B - A*b) + (m + p*q + 2*(n - q)*(p + 1) + 1)*(b*B - 2*A*c)*x^(n - q), x]*(a*x^q + b*x^n + c*x^(2*n - q))^(p + 1), x], x] /; FreeQ[{a, b, c, A, B}, x] && EqQ[r, n - q] && EqQ[j, 2*n - q] && !IntegerQ[p] && NeQ[b^2 - 4*a*c, 0] && IGtQ[n, 0] && LtQ[p, -1] && RationalQ[m, q] && GtQ[m + p*q, n - q - 1]
  1944. Int[Times[Power[Pattern[x, Blank[]], Optional[Pattern[m, Blank[]]]], Power[Plus[Times[Optional[Pattern[c, Blank[]]], Power[Pattern[x, Blank[]], Optional[Pattern[j, Blank[]]]]], Times[Optional[Pattern[a, Blank[]]], Power[Pattern[x, Blank[]], Optional[Pattern[q, Blank[]]]]]], Optional[Pattern[p, Blank[]]]], Plus[Pattern[A, Blank[]], Times[Optional[Pattern[B, Blank[]]], Power[Pattern[x, Blank[]], Optional[Pattern[r, Blank[]]]]]]], Pattern[x, Blank[Symbol]]] := With[{n = q + r}, Simp[(x^(m - n + 1)*(a*B - A*c*x^(n - q))*(a*x^q + c*x^(2*n - q))^(p + 1))/(2*a*c*(n - q)*(p + 1)), x] - Dist[1/(2*a*c*(n - q)*(p + 1)), Int[x^(m - n)*Simp[a*B*(m + p*q - n + q + 1) - A*c*(m + p*q + (n - q)*2*(p + 1) + 1)*x^(n - q), x]*(a*x^q + c*x^(2*n - q))^(p + 1), x], x] /; EqQ[j, 2*n - q] && IGtQ[n, 0] && m + p*q > n - q - 1] /; FreeQ[{a, c, A, B}, x] && !IntegerQ[p] && RationalQ[m, q] && LtQ[p, -1]
  1945. Int[Times[Power[Pattern[x, Blank[]], Optional[Pattern[m, Blank[]]]], Power[Plus[Times[Optional[Pattern[c, Blank[]]], Power[Pattern[x, Blank[]], Optional[Pattern[j, Blank[]]]]], Times[Optional[Pattern[b, Blank[]]], Power[Pattern[x, Blank[]], Optional[Pattern[n, Blank[]]]]], Times[Optional[Pattern[a, Blank[]]], Power[Pattern[x, Blank[]], Optional[Pattern[q, Blank[]]]]]], Optional[Pattern[p, Blank[]]]], Plus[Pattern[A, Blank[]], Times[Optional[Pattern[B, Blank[]]], Power[Pattern[x, Blank[]], Optional[Pattern[r, Blank[]]]]]]], Pattern[x, Blank[Symbol]]] := Simp[(x^(m + 1)*(b*B*(n - q)*p + A*c*(m + p*q + (n - q)*(2*p + 1) + 1) + B*c*(m + p*q + 2*(n - q)*p + 1)*x^(n - q))*(a*x^q + b*x^n + c*x^(2*n - q))^p)/(c*(m + p*(2*n - q) + 1)*(m + p*q + (n - q)*(2*p + 1) + 1)), x] + Dist[((n - q)*p)/(c*(m + p*(2*n - q) + 1)*(m + p*q + (n - q)*(2*p + 1) + 1)), Int[x^(m + q)*Simp[2*a*A*c*(m + p*q + (n - q)*(2*p + 1) + 1) - a*b*B*(m + p*q + 1) + (2*a*B*c*(m + p*q + 2*(n - q)*p + 1) + A*b*c*(m + p*q + (n - q)*(2*p + 1) + 1) - b^2*B*(m + p*q + (n - q)*p + 1))*x^(n - q), x]*(a*x^q + b*x^n + c*x^(2*n - q))^(p - 1), x], x] /; FreeQ[{a, b, c, A, B}, x] && EqQ[r, n - q] && EqQ[j, 2*n - q] && !IntegerQ[p] && NeQ[b^2 - 4*a*c, 0] && IGtQ[n, 0] && GtQ[p, 0] && RationalQ[m, q] && GtQ[m + p*q, -(n - q) - 1] && NeQ[m + p*(2*n - q) + 1, 0] && NeQ[m + p*q + (n - q)*(2*p + 1) + 1, 0]
  1946. Int[Times[Power[Pattern[x, Blank[]], Optional[Pattern[m, Blank[]]]], Power[Plus[Times[Optional[Pattern[c, Blank[]]], Power[Pattern[x, Blank[]], Optional[Pattern[j, Blank[]]]]], Times[Optional[Pattern[a, Blank[]]], Power[Pattern[x, Blank[]], Optional[Pattern[q, Blank[]]]]]], Optional[Pattern[p, Blank[]]]], Plus[Pattern[A, Blank[]], Times[Optional[Pattern[B, Blank[]]], Power[Pattern[x, Blank[]], Optional[Pattern[r, Blank[]]]]]]], Pattern[x, Blank[Symbol]]] := With[{n = q + r}, Simp[(x^(m + 1)*(A*(m + p*q + (n - q)*(2*p + 1) + 1) + B*(m + p*q + 2*(n - q)*p + 1)*x^(n - q))*(a*x^q + c*x^(2*n - q))^p)/((m + p*(2*n - q) + 1)*(m + p*q + (n - q)*(2*p + 1) + 1)), x] + Dist[((n - q)*p)/((m + p*(2*n - q) + 1)*(m + p*q + (n - q)*(2*p + 1) + 1)), Int[x^(m + q)*Simp[2*a*A*(m + p*q + (n - q)*(2*p + 1) + 1) + 2*a*B*(m + p*q + 2*(n - q)*p + 1)*x^(n - q), x]*(a*x^q + c*x^(2*n - q))^(p - 1), x], x] /; EqQ[j, 2*n - q] && IGtQ[n, 0] && GtQ[m + p*q, -(n - q)] && NeQ[m + p*q + 2*(n - q)*p + 1, 0] && NeQ[m + p*q + (n - q)*(2*p + 1) + 1, 0] && NeQ[m + 1, n]] /; FreeQ[{a, c, A, B}, x] && !IntegerQ[p] && RationalQ[m, q] && GtQ[p, 0]
  1947. Int[Times[Power[Pattern[x, Blank[]], Optional[Pattern[m, Blank[]]]], Power[Plus[Times[Optional[Pattern[c, Blank[]]], Power[Pattern[x, Blank[]], Optional[Pattern[j, Blank[]]]]], Times[Optional[Pattern[b, Blank[]]], Power[Pattern[x, Blank[]], Optional[Pattern[n, Blank[]]]]], Times[Optional[Pattern[a, Blank[]]], Power[Pattern[x, Blank[]], Optional[Pattern[q, Blank[]]]]]], Optional[Pattern[p, Blank[]]]], Plus[Pattern[A, Blank[]], Times[Optional[Pattern[B, Blank[]]], Power[Pattern[x, Blank[]], Optional[Pattern[r, Blank[]]]]]]], Pattern[x, Blank[Symbol]]] := -Simp[(x^(m - q + 1)*(A*b^2 - a*b*B - 2*a*A*c + (A*b - 2*a*B)*c*x^(n - q))*(a*x^q + b*x^n + c*x^(2*n - q))^(p + 1))/(a*(n - q)*(p + 1)*(b^2 - 4*a*c)), x] + Dist[1/(a*(n - q)*(p + 1)*(b^2 - 4*a*c)), Int[x^(m - q)*Simp[A*b^2*(m + p*q + (n - q)*(p + 1) + 1) - a*b*B*(m + p*q + 1) - 2*a*A*c*(m + p*q + 2*(n - q)*(p + 1) + 1) + (m + p*q + (n - q)*(2*p + 3) + 1)*(A*b - 2*a*B)*c*x^(n - q), x]*(a*x^q + b*x^n + c*x^(2*n - q))^(p + 1), x], x] /; FreeQ[{a, b, c, A, B}, x] && EqQ[r, n - q] && EqQ[j, 2*n - q] && !IntegerQ[p] && NeQ[b^2 - 4*a*c, 0] && IGtQ[n, 0] && LtQ[p, -1] && RationalQ[m, q] && m + p*q < n - q - 1
  1948. Int[Times[Power[Pattern[x, Blank[]], Optional[Pattern[m, Blank[]]]], Power[Plus[Times[Optional[Pattern[c, Blank[]]], Power[Pattern[x, Blank[]], Optional[Pattern[j, Blank[]]]]], Times[Optional[Pattern[a, Blank[]]], Power[Pattern[x, Blank[]], Optional[Pattern[q, Blank[]]]]]], Optional[Pattern[p, Blank[]]]], Plus[Pattern[A, Blank[]], Times[Optional[Pattern[B, Blank[]]], Power[Pattern[x, Blank[]], Optional[Pattern[r, Blank[]]]]]]], Pattern[x, Blank[Symbol]]] := With[{n = q + r}, -Simp[(x^(m - q + 1)*(A*c + B*c*x^(n - q))*(a*x^q + c*x^(2*n - q))^(p + 1))/(2*a*c*(n - q)*(p + 1)), x] + Dist[1/(2*a*c*(n - q)*(p + 1)), Int[x^(m - q)*Simp[A*c*(m + p*q + 2*(n - q)*(p + 1) + 1) + B*(m + p*q + (n - q)*(2*p + 3) + 1)*c*x^(n - q), x]*(a*x^q + c*x^(2*n - q))^(p + 1), x], x] /; EqQ[j, 2*n - q] && IGtQ[n, 0] && LtQ[m + p*q, n - q - 1]] /; FreeQ[{a, c, A, B}, x] && !IntegerQ[p] && RationalQ[m, q] && LtQ[p, -1]
  1949. Int[Times[Power[Pattern[x, Blank[]], Optional[Pattern[m, Blank[]]]], Power[Plus[Times[Optional[Pattern[c, Blank[]]], Power[Pattern[x, Blank[]], Optional[Pattern[j, Blank[]]]]], Times[Optional[Pattern[b, Blank[]]], Power[Pattern[x, Blank[]], Optional[Pattern[n, Blank[]]]]], Times[Optional[Pattern[a, Blank[]]], Power[Pattern[x, Blank[]], Optional[Pattern[q, Blank[]]]]]], Optional[Pattern[p, Blank[]]]], Plus[Pattern[A, Blank[]], Times[Optional[Pattern[B, Blank[]]], Power[Pattern[x, Blank[]], Optional[Pattern[r, Blank[]]]]]]], Pattern[x, Blank[Symbol]]] := Simp[(B*x^(m - n + 1)*(a*x^q + b*x^n + c*x^(2*n - q))^(p + 1))/(c*(m + p*q + (n - q)*(2*p + 1) + 1)), x] - Dist[1/(c*(m + p*q + (n - q)*(2*p + 1) + 1)), Int[x^(m - n + q)*Simp[a*B*(m + p*q - n + q + 1) + (b*B*(m + p*q + (n - q)*p + 1) - A*c*(m + p*q + (n - q)*(2*p + 1) + 1))*x^(n - q), x]*(a*x^q + b*x^n + c*x^(2*n - q))^p, x], x] /; FreeQ[{a, b, c, A, B}, x] && EqQ[r, n - q] && EqQ[j, 2*n - q] && !IntegerQ[p] && NeQ[b^2 - 4*a*c, 0] && IGtQ[n, 0] && GeQ[p, -1] && LtQ[p, 0] && RationalQ[m, q] && GeQ[m + p*q, n - q - 1] && NeQ[m + p*q + (n - q)*(2*p + 1) + 1, 0]
  1950. Int[Times[Power[Pattern[x, Blank[]], Optional[Pattern[m, Blank[]]]], Power[Plus[Times[Optional[Pattern[c, Blank[]]], Power[Pattern[x, Blank[]], Optional[Pattern[j, Blank[]]]]], Times[Optional[Pattern[a, Blank[]]], Power[Pattern[x, Blank[]], Optional[Pattern[q, Blank[]]]]]], Optional[Pattern[p, Blank[]]]], Plus[Pattern[A, Blank[]], Times[Optional[Pattern[B, Blank[]]], Power[Pattern[x, Blank[]], Optional[Pattern[r, Blank[]]]]]]], Pattern[x, Blank[Symbol]]] := With[{n = q + r}, Simp[(B*x^(m - n + 1)*(a*x^q + c*x^(2*n - q))^(p + 1))/(c*(m + p*q + (n - q)*(2*p + 1) + 1)), x] - Dist[1/(c*(m + p*q + (n - q)*(2*p + 1) + 1)), Int[x^(m - n + q)*Simp[a*B*(m + p*q - n + q + 1) - A*c*(m + p*q + (n - q)*(2*p + 1) + 1)*x^(n - q), x]*(a*x^q + c*x^(2*n - q))^p, x], x] /; EqQ[j, 2*n - q] && IGtQ[n, 0] && GeQ[m + p*q, n - q - 1] && NeQ[m + p*q + (n - q)*(2*p + 1) + 1, 0]] /; FreeQ[{a, c, A, B}, x] && !IntegerQ[p] && RationalQ[m, p, q] && GeQ[p, -1] && LtQ[p, 0]
  1951. Int[Times[Power[Pattern[x, Blank[]], Optional[Pattern[m, Blank[]]]], Power[Plus[Times[Optional[Pattern[c, Blank[]]], Power[Pattern[x, Blank[]], Optional[Pattern[j, Blank[]]]]], Times[Optional[Pattern[b, Blank[]]], Power[Pattern[x, Blank[]], Optional[Pattern[n, Blank[]]]]], Times[Optional[Pattern[a, Blank[]]], Power[Pattern[x, Blank[]], Optional[Pattern[q, Blank[]]]]]], Optional[Pattern[p, Blank[]]]], Plus[Pattern[A, Blank[]], Times[Optional[Pattern[B, Blank[]]], Power[Pattern[x, Blank[]], Optional[Pattern[r, Blank[]]]]]]], Pattern[x, Blank[Symbol]]] := Simp[(A*x^(m - q + 1)*(a*x^q + b*x^n + c*x^(2*n - q))^(p + 1))/(a*(m + p*q + 1)), x] + Dist[1/(a*(m + p*q + 1)), Int[x^(m + n - q)*Simp[a*B*(m + p*q + 1) - A*b*(m + p*q + (n - q)*(p + 1) + 1) - A*c*(m + p*q + 2*(n - q)*(p + 1) + 1)*x^(n - q), x]*(a*x^q + b*x^n + c*x^(2*n - q))^p, x], x] /; FreeQ[{a, b, c, A, B}, x] && EqQ[r, n - q] && EqQ[j, 2*n - q] && !IntegerQ[p] && NeQ[b^2 - 4*a*c, 0] && IGtQ[n, 0] && RationalQ[m, p, q] && ((GeQ[p, -1] && LtQ[p, 0]) || EqQ[m + p*q + (n - q)*(2*p + 1) + 1, 0]) && LeQ[m + p*q, -(n - q)] && NeQ[m + p*q + 1, 0]
  1952. Int[Times[Power[Pattern[x, Blank[]], Optional[Pattern[m, Blank[]]]], Power[Plus[Times[Optional[Pattern[c, Blank[]]], Power[Pattern[x, Blank[]], Optional[Pattern[j, Blank[]]]]], Times[Optional[Pattern[a, Blank[]]], Power[Pattern[x, Blank[]], Optional[Pattern[q, Blank[]]]]]], Optional[Pattern[p, Blank[]]]], Plus[Pattern[A, Blank[]], Times[Optional[Pattern[B, Blank[]]], Power[Pattern[x, Blank[]], Optional[Pattern[r, Blank[]]]]]]], Pattern[x, Blank[Symbol]]] := With[{n = q + r}, Simp[(A*x^(m - q + 1)*(a*x^q + c*x^(2*n - q))^(p + 1))/(a*(m + p*q + 1)), x] + Dist[1/(a*(m + p*q + 1)), Int[x^(m + n - q)*Simp[a*B*(m + p*q + 1) - A*c*(m + p*q + 2*(n - q)*(p + 1) + 1)*x^(n - q), x]*(a*x^q + c*x^(2*n - q))^p, x], x] /; EqQ[j, 2*n - q] && IGtQ[n, 0] && ((GeQ[p, -1] && LtQ[p, 0]) || EqQ[m + p*q + (n - q)*(2*p + 1) + 1, 0]) && LeQ[m + p*q, -(n - q)] && NeQ[m + p*q + 1, 0]] /; FreeQ[{a, c, A, B}, x] && !IntegerQ[p] && RationalQ[m, p, q]
  1953. Int[Times[Power[Pattern[x, Blank[]], Optional[Pattern[m, Blank[]]]], Plus[Pattern[A, Blank[]], Times[Optional[Pattern[B, Blank[]]], Power[Pattern[x, Blank[]], Optional[Pattern[j, Blank[]]]]]], Power[Plus[Times[Optional[Pattern[b, Blank[]]], Power[Pattern[x, Blank[]], Optional[Pattern[n, Blank[]]]]], Times[Optional[Pattern[a, Blank[]]], Power[Pattern[x, Blank[]], Optional[Pattern[q, Blank[]]]]], Times[Optional[Pattern[c, Blank[]]], Power[Pattern[x, Blank[]], Optional[Pattern[r, Blank[]]]]]], Rational[-1, 2]]], Pattern[x, Blank[Symbol]]] := Dist[(x^(q/2)*Sqrt[a + b*x^(n - q) + c*x^(2*(n - q))])/Sqrt[a*x^q + b*x^n + c*x^(2*n - q)], Int[(x^(m - q/2)*(A + B*x^(n - q)))/Sqrt[a + b*x^(n - q) + c*x^(2*(n - q))], x], x] /; FreeQ[{a, b, c, A, B, m, n, q}, x] && EqQ[j, n - q] && EqQ[r, 2*n - q] && PosQ[n - q] && (EqQ[m, 1/2] || EqQ[m, -2^(-1)]) && EqQ[n, 3] && EqQ[q, 1]
  1954. Int[Times[Power[Pattern[x, Blank[]], Optional[Pattern[m, Blank[]]]], Power[Plus[Times[Optional[Pattern[a, Blank[]]], Power[Pattern[x, Blank[]], Optional[Pattern[j, Blank[]]]]], Times[Optional[Pattern[b, Blank[]]], Power[Pattern[x, Blank[]], Optional[Pattern[k, Blank[]]]]], Times[Optional[Pattern[c, Blank[]]], Power[Pattern[x, Blank[]], Optional[Pattern[n, Blank[]]]]]], Pattern[p, Blank[]]], Plus[Pattern[A, Blank[]], Times[Optional[Pattern[B, Blank[]]], Power[Pattern[x, Blank[]], Pattern[q, Blank[]]]]]], Pattern[x, Blank[Symbol]]] := Dist[(a*x^j + b*x^k + c*x^n)^p/(x^(j*p)*(a + b*x^(k - j) + c*x^(2*(k - j)))^p), Int[x^(m + j*p)*(A + B*x^(k - j))*(a + b*x^(k - j) + c*x^(2*(k - j)))^p, x], x] /; FreeQ[{a, b, c, A, B, j, k, m, p}, x] && EqQ[q, k - j] && EqQ[n, 2*k - j] && !IntegerQ[p] && PosQ[k - j]
  1955. Int[Times[Power[Pattern[u, Blank[]], Optional[Pattern[m, Blank[]]]], Plus[Pattern[A, Blank[]], Times[Optional[Pattern[B, Blank[]]], Power[Pattern[u, Blank[]], Optional[Pattern[j, Blank[]]]]]], Power[Plus[Times[Optional[Pattern[b, Blank[]]], Power[Pattern[u, Blank[]], Optional[Pattern[n, Blank[]]]]], Times[Optional[Pattern[a, Blank[]]], Power[Pattern[u, Blank[]], Optional[Pattern[q, Blank[]]]]], Times[Optional[Pattern[c, Blank[]]], Power[Pattern[u, Blank[]], Optional[Pattern[r, Blank[]]]]]], Optional[Pattern[p, Blank[]]]]], Pattern[x, Blank[Symbol]]] := Dist[1/Coefficient[u, x, 1], Subst[Int[x^m*(A + B*x^(n - q))*(a*x^q + b*x^n + c*x^(2*n - q))^p, x], x, u], x] /; FreeQ[{a, b, c, A, B, m, n, p, q}, x] && EqQ[j, n - q] && EqQ[r, 2*n - q] && LinearQ[u, x] && NeQ[u, x]
  1956. Int[Times[Power[Pattern[x, Blank[]], Optional[Pattern[m, Blank[]]]], Power[Times[Optional[Pattern[e, Blank[]]], Power[Plus[Pattern[a, Blank[]], Times[Optional[Pattern[b, Blank[]]], Power[Pattern[x, Blank[]], Optional[Pattern[n, Blank[]]]]]], Optional[Pattern[r, Blank[]]]]], Pattern[p, Blank[]]], Power[Times[Optional[Pattern[f, Blank[]]], Power[Plus[Pattern[c, Blank[]], Times[Optional[Pattern[d, Blank[]]], Power[Pattern[x, Blank[]], Optional[Pattern[n, Blank[]]]]]], Pattern[s, Blank[]]]], Pattern[q, Blank[]]]], Pattern[x, Blank[Symbol]]] := Dist[((e*(a + b*x^n)^r)^p*(f*(c + d*x^n)^s)^q)/((a + b*x^n)^(p*r)*(c + d*x^n)^(q*s)), Int[x^m*(a + b*x^n)^(p*r)*(c + d*x^n)^(q*s), x], x] /; FreeQ[{a, b, c, d, e, f, m, n, p, q, r, s}, x]
  1957. Int[Times[Optional[Pattern[u, Blank[]]], Power[Times[Optional[Pattern[e, Blank[]]], Plus[Optional[Pattern[a, Blank[]]], Times[Optional[Pattern[b, Blank[]]], Power[Pattern[x, Blank[]], Optional[Pattern[n, Blank[]]]]]], Power[Plus[Pattern[c, Blank[]], Times[Optional[Pattern[d, Blank[]]], Power[Pattern[x, Blank[]], Optional[Pattern[n, Blank[]]]]]], -1]], Pattern[p, Blank[]]]], Pattern[x, Blank[Symbol]]] := Dist[((b*e)/d)^p, Int[u, x], x] /; FreeQ[{a, b, c, d, e, n, p}, x] && EqQ[b*c - a*d, 0]
  1958. Int[Times[Optional[Pattern[u, Blank[]]], Power[Times[Optional[Pattern[e, Blank[]]], Plus[Optional[Pattern[a, Blank[]]], Times[Optional[Pattern[b, Blank[]]], Power[Pattern[x, Blank[]], Optional[Pattern[n, Blank[]]]]]], Power[Plus[Pattern[c, Blank[]], Times[Optional[Pattern[d, Blank[]]], Power[Pattern[x, Blank[]], Optional[Pattern[n, Blank[]]]]]], -1]], Pattern[p, Blank[]]]], Pattern[x, Blank[Symbol]]] := Int[(u*(e*(a + b*x^n))^p)/(c + d*x^n)^p, x] /; FreeQ[{a, b, c, d, e, n, p}, x] && GtQ[b*d*e, 0] && GtQ[c - (a*d)/b, 0]
  1959. Int[Power[Times[Optional[Pattern[e, Blank[]]], Plus[Optional[Pattern[a, Blank[]]], Times[Optional[Pattern[b, Blank[]]], Power[Pattern[x, Blank[]], Optional[Pattern[n, Blank[]]]]]], Power[Plus[Pattern[c, Blank[]], Times[Optional[Pattern[d, Blank[]]], Power[Pattern[x, Blank[]], Optional[Pattern[n, Blank[]]]]]], -1]], Pattern[p, Blank[]]], Pattern[x, Blank[Symbol]]] := With[{q = Denominator[p]}, Dist[(q*e*(b*c - a*d))/n, Subst[Int[(x^(q*(p + 1) - 1)*(-(a*e) + c*x^q)^(1/n - 1))/(b*e - d*x^q)^(1/n + 1), x], x, ((e*(a + b*x^n))/(c + d*x^n))^(1/q)], x]] /; FreeQ[{a, b, c, d, e}, x] && FractionQ[p] && IntegerQ[1/n]
  1960. Int[Times[Power[Pattern[x, Blank[]], Optional[Pattern[m, Blank[]]]], Power[Times[Optional[Pattern[e, Blank[]]], Plus[Optional[Pattern[a, Blank[]]], Times[Optional[Pattern[b, Blank[]]], Power[Pattern[x, Blank[]], Optional[Pattern[n, Blank[]]]]]], Power[Plus[Pattern[c, Blank[]], Times[Optional[Pattern[d, Blank[]]], Power[Pattern[x, Blank[]], Optional[Pattern[n, Blank[]]]]]], -1]], Pattern[p, Blank[]]]], Pattern[x, Blank[Symbol]]] := With[{q = Denominator[p]}, Dist[(q*e*(b*c - a*d))/n, Subst[Int[(x^(q*(p + 1) - 1)*(-(a*e) + c*x^q)^(Simplify[(m + 1)/n] - 1))/(b*e - d*x^q)^(Simplify[(m + 1)/n] + 1), x], x, ((e*(a + b*x^n))/(c + d*x^n))^(1/q)], x]] /; FreeQ[{a, b, c, d, e, m, n}, x] && FractionQ[p] && IntegerQ[Simplify[(m + 1)/n]]
  1961. Int[Times[Power[Pattern[u, Blank[]], Optional[Pattern[r, Blank[]]]], Power[Times[Optional[Pattern[e, Blank[]]], Plus[Optional[Pattern[a, Blank[]]], Times[Optional[Pattern[b, Blank[]]], Power[Pattern[x, Blank[]], Optional[Pattern[n, Blank[]]]]]], Power[Plus[Pattern[c, Blank[]], Times[Optional[Pattern[d, Blank[]]], Power[Pattern[x, Blank[]], Optional[Pattern[n, Blank[]]]]]], -1]], Pattern[p, Blank[]]]], Pattern[x, Blank[Symbol]]] := With[{q = Denominator[p]}, Dist[(q*e*(b*c - a*d))/n, Subst[Int[SimplifyIntegrand[(x^(q*(p + 1) - 1)*(-(a*e) + c*x^q)^(1/n - 1)*(u /. x -> (-(a*e) + c*x^q)^(1/n)/(b*e - d*x^q)^(1/n))^r)/(b*e - d*x^q)^(1/n + 1), x], x], x, ((e*(a + b*x^n))/(c + d*x^n))^(1/q)], x]] /; FreeQ[{a, b, c, d, e}, x] && PolynomialQ[u, x] && FractionQ[p] && IntegerQ[1/n] && IntegerQ[r]
  1962. Int[Times[Power[Pattern[u, Blank[]], Optional[Pattern[r, Blank[]]]], Power[Pattern[x, Blank[]], Optional[Pattern[m, Blank[]]]], Power[Times[Optional[Pattern[e, Blank[]]], Plus[Optional[Pattern[a, Blank[]]], Times[Optional[Pattern[b, Blank[]]], Power[Pattern[x, Blank[]], Optional[Pattern[n, Blank[]]]]]], Power[Plus[Pattern[c, Blank[]], Times[Optional[Pattern[d, Blank[]]], Power[Pattern[x, Blank[]], Optional[Pattern[n, Blank[]]]]]], -1]], Pattern[p, Blank[]]]], Pattern[x, Blank[Symbol]]] := With[{q = Denominator[p]}, Dist[(q*e*(b*c - a*d))/n, Subst[Int[SimplifyIntegrand[(x^(q*(p + 1) - 1)*(-(a*e) + c*x^q)^((m + 1)/n - 1)*(u /. x -> (-(a*e) + c*x^q)^(1/n)/(b*e - d*x^q)^(1/n))^r)/(b*e - d*x^q)^((m + 1)/n + 1), x], x], x, ((e*(a + b*x^n))/(c + d*x^n))^(1/q)], x]] /; FreeQ[{a, b, c, d, e}, x] && PolynomialQ[u, x] && FractionQ[p] && IntegerQ[1/n] && IntegersQ[m, r]
  1963. Int[Power[Plus[Optional[Pattern[a, Blank[]]], Times[Optional[Pattern[b, Blank[]]], Power[Times[Optional[Pattern[c, Blank[]]], Power[Pattern[x, Blank[]], -1]], Pattern[n, Blank[]]]]], Pattern[p, Blank[]]], Pattern[x, Blank[Symbol]]] := -Dist[c, Subst[Int[(a + b*x^n)^p/x^2, x], x, c/x], x] /; FreeQ[{a, b, c, n, p}, x]
  1964. Int[Times[Power[Plus[Optional[Pattern[a, Blank[]]], Times[Optional[Pattern[b, Blank[]]], Power[Times[Optional[Pattern[c, Blank[]]], Power[Pattern[x, Blank[]], -1]], Pattern[n, Blank[]]]]], Optional[Pattern[p, Blank[]]]], Power[Pattern[x, Blank[]], Optional[Pattern[m, Blank[]]]]], Pattern[x, Blank[Symbol]]] := -Dist[c^(m + 1), Subst[Int[(a + b*x^n)^p/x^(m + 2), x], x, c/x], x] /; FreeQ[{a, b, c, n, p}, x] && IntegerQ[m]
  1965. Int[Times[Power[Plus[Optional[Pattern[a, Blank[]]], Times[Optional[Pattern[b, Blank[]]], Power[Times[Optional[Pattern[c, Blank[]]], Power[Pattern[x, Blank[]], -1]], Pattern[n, Blank[]]]]], Optional[Pattern[p, Blank[]]]], Power[Times[Optional[Pattern[d, Blank[]]], Pattern[x, Blank[]]], Pattern[m, Blank[]]]], Pattern[x, Blank[Symbol]]] := -Dist[c*(d*x)^m*(c/x)^m, Subst[Int[(a + b*x^n)^p/x^(m + 2), x], x, c/x], x] /; FreeQ[{a, b, c, d, m, n, p}, x] && !IntegerQ[m]
  1966. Int[Power[Plus[Optional[Pattern[a, Blank[]]], Times[Optional[Pattern[c, Blank[]]], Power[Times[Optional[Pattern[d, Blank[]]], Power[Pattern[x, Blank[]], -1]], Optional[Pattern[n2, Blank[]]]]], Times[Optional[Pattern[b, Blank[]]], Power[Times[Optional[Pattern[d, Blank[]]], Power[Pattern[x, Blank[]], -1]], Pattern[n, Blank[]]]]], Optional[Pattern[p, Blank[]]]], Pattern[x, Blank[Symbol]]] := -Dist[d, Subst[Int[(a + b*x^n + c*x^(2*n))^p/x^2, x], x, d/x], x] /; FreeQ[{a, b, c, d, n, p}, x] && EqQ[n2, 2*n]
  1967. Int[Times[Power[Plus[Pattern[a, Blank[]], Times[Optional[Pattern[c, Blank[]]], Power[Times[Optional[Pattern[d, Blank[]]], Power[Pattern[x, Blank[]], -1]], Optional[Pattern[n2, Blank[]]]]], Times[Optional[Pattern[b, Blank[]]], Power[Times[Optional[Pattern[d, Blank[]]], Power[Pattern[x, Blank[]], -1]], Pattern[n, Blank[]]]]], Optional[Pattern[p, Blank[]]]], Power[Pattern[x, Blank[]], Optional[Pattern[m, Blank[]]]]], Pattern[x, Blank[Symbol]]] := -Dist[d^(m + 1), Subst[Int[(a + b*x^n + c*x^(2*n))^p/x^(m + 2), x], x, d/x], x] /; FreeQ[{a, b, c, d, n, p}, x] && EqQ[n2, 2*n] && IntegerQ[m]
  1968. Int[Times[Power[Plus[Pattern[a, Blank[]], Times[Optional[Pattern[c, Blank[]]], Power[Times[Optional[Pattern[d, Blank[]]], Power[Pattern[x, Blank[]], -1]], Optional[Pattern[n2, Blank[]]]]], Times[Optional[Pattern[b, Blank[]]], Power[Times[Optional[Pattern[d, Blank[]]], Power[Pattern[x, Blank[]], -1]], Pattern[n, Blank[]]]]], Optional[Pattern[p, Blank[]]]], Power[Times[Optional[Pattern[e, Blank[]]], Pattern[x, Blank[]]], Pattern[m, Blank[]]]], Pattern[x, Blank[Symbol]]] := -Dist[d*(e*x)^m*(d/x)^m, Subst[Int[(a + b*x^n + c*x^(2*n))^p/x^(m + 2), x], x, d/x], x] /; FreeQ[{a, b, c, d, e, m, n, p}, x] && EqQ[n2, 2*n] && !IntegerQ[m]
  1969. Int[Power[Plus[Optional[Pattern[a, Blank[]]], Times[Optional[Pattern[b, Blank[]]], Power[Times[Optional[Pattern[d, Blank[]]], Power[Pattern[x, Blank[]], -1]], Pattern[n, Blank[]]]], Times[Optional[Pattern[c, Blank[]]], Power[Pattern[x, Blank[]], Optional[Pattern[n2, Blank[]]]]]], Optional[Pattern[p, Blank[]]]], Pattern[x, Blank[Symbol]]] := -Dist[d, Subst[Int[(a + b*x^n + (c*x^(2*n))/d^(2*n))^p/x^2, x], x, d/x], x] /; FreeQ[{a, b, c, d, n, p}, x] && EqQ[n2, -2*n] && IntegerQ[2*n]
  1970. Int[Times[Power[Pattern[x, Blank[]], Optional[Pattern[m, Blank[]]]], Power[Plus[Pattern[a, Blank[]], Times[Optional[Pattern[b, Blank[]]], Power[Times[Optional[Pattern[d, Blank[]]], Power[Pattern[x, Blank[]], -1]], Pattern[n, Blank[]]]], Times[Optional[Pattern[c, Blank[]]], Power[Pattern[x, Blank[]], Optional[Pattern[n2, Blank[]]]]]], Optional[Pattern[p, Blank[]]]]], Pattern[x, Blank[Symbol]]] := -Dist[d^(m + 1), Subst[Int[(a + b*x^n + (c*x^(2*n))/d^(2*n))^p/x^(m + 2), x], x, d/x], x] /; FreeQ[{a, b, c, d, n, p}, x] && EqQ[n2, -2*n] && IntegerQ[2*n] && IntegerQ[m]
  1971. Int[Times[Power[Times[Optional[Pattern[e, Blank[]]], Pattern[x, Blank[]]], Pattern[m, Blank[]]], Power[Plus[Pattern[a, Blank[]], Times[Optional[Pattern[b, Blank[]]], Power[Times[Optional[Pattern[d, Blank[]]], Power[Pattern[x, Blank[]], -1]], Pattern[n, Blank[]]]], Times[Optional[Pattern[c, Blank[]]], Power[Pattern[x, Blank[]], Optional[Pattern[n2, Blank[]]]]]], Optional[Pattern[p, Blank[]]]]], Pattern[x, Blank[Symbol]]] := -Dist[d*(e*x)^m*(d/x)^m, Subst[Int[(a + b*x^n + (c*x^(2*n))/d^(2*n))^p/x^(m + 2), x], x, d/x], x] /; FreeQ[{a, b, c, d, e, n, p}, x] && EqQ[n2, -2*n] && !IntegerQ[m] && IntegerQ[2*n]
  1972. Int[Power[Pattern[u, Blank[]], Pattern[p, Blank[]]], Pattern[x, Blank[Symbol]]] := Int[ExpandToSum[u, x]^p, x] /; FreeQ[p, x] && BinomialQ[u, x] && !BinomialMatchQ[u, x]
  1973. Int[Times[Power[Pattern[u, Blank[]], Optional[Pattern[p, Blank[]]]], Power[Times[Optional[Pattern[c, Blank[]]], Pattern[x, Blank[]]], Optional[Pattern[m, Blank[]]]]], Pattern[x, Blank[Symbol]]] := Int[(c*x)^m*ExpandToSum[u, x]^p, x] /; FreeQ[{c, m, p}, x] && BinomialQ[u, x] && !BinomialMatchQ[u, x]
  1974. Int[Times[Power[Pattern[u, Blank[]], Optional[Pattern[p, Blank[]]]], Power[Pattern[v, Blank[]], Optional[Pattern[q, Blank[]]]]], Pattern[x, Blank[Symbol]]] := Int[ExpandToSum[u, x]^p*ExpandToSum[v, x]^q, x] /; FreeQ[{p, q}, x] && BinomialQ[{u, v}, x] && EqQ[BinomialDegree[u, x] - BinomialDegree[v, x], 0] && !BinomialMatchQ[{u, v}, x]
  1975. Int[Times[Power[Pattern[u, Blank[]], Optional[Pattern[p, Blank[]]]], Power[Pattern[v, Blank[]], Optional[Pattern[q, Blank[]]]], Power[Times[Optional[Pattern[e, Blank[]]], Pattern[x, Blank[]]], Optional[Pattern[m, Blank[]]]]], Pattern[x, Blank[Symbol]]] := Int[(e*x)^m*ExpandToSum[u, x]^p*ExpandToSum[v, x]^q, x] /; FreeQ[{e, m, p, q}, x] && BinomialQ[{u, v}, x] && EqQ[BinomialDegree[u, x] - BinomialDegree[v, x], 0] && !BinomialMatchQ[{u, v}, x]
  1976. Int[Times[Power[Pattern[u, Blank[]], Optional[Pattern[m, Blank[]]]], Power[Pattern[v, Blank[]], Optional[Pattern[p, Blank[]]]], Power[Pattern[w, Blank[]], Optional[Pattern[q, Blank[]]]]], Pattern[x, Blank[Symbol]]] := Int[ExpandToSum[u, x]^m*ExpandToSum[v, x]^p*ExpandToSum[w, x]^q, x] /; FreeQ[{m, p, q}, x] && BinomialQ[{u, v, w}, x] && EqQ[BinomialDegree[u, x] - BinomialDegree[v, x], 0] && EqQ[BinomialDegree[u, x] - BinomialDegree[w, x], 0] && !BinomialMatchQ[{u, v, w}, x]
  1977. Int[Times[Power[Pattern[u, Blank[]], Optional[Pattern[p, Blank[]]]], Power[Pattern[v, Blank[]], Optional[Pattern[q, Blank[]]]], Power[Times[Optional[Pattern[g, Blank[]]], Pattern[x, Blank[]]], Optional[Pattern[m, Blank[]]]], Power[Pattern[z, Blank[]], Optional[Pattern[r, Blank[]]]]], Pattern[x, Blank[Symbol]]] := Int[(g*x)^m*ExpandToSum[u, x]^p*ExpandToSum[v, x]^q*ExpandToSum[z, x]^r, x] /; FreeQ[{g, m, p, q, r}, x] && BinomialQ[{u, v, z}, x] && EqQ[BinomialDegree[u, x] - BinomialDegree[v, x], 0] && EqQ[BinomialDegree[u, x] - BinomialDegree[z, x], 0] && !BinomialMatchQ[{u, v, z}, x]
  1978. Int[Times[Pattern[Pq, Blank[]], Power[Pattern[u, Blank[]], Optional[Pattern[p, Blank[]]]], Power[Times[Optional[Pattern[c, Blank[]]], Pattern[x, Blank[]]], Optional[Pattern[m, Blank[]]]]], Pattern[x, Blank[Symbol]]] := Int[(c*x)^m*Pq*ExpandToSum[u, x]^p, x] /; FreeQ[{c, m, p}, x] && PolyQ[Pq, x] && BinomialQ[u, x] && !BinomialMatchQ[u, x]
  1979. Int[Power[Pattern[u, Blank[]], Pattern[p, Blank[]]], Pattern[x, Blank[Symbol]]] := Int[ExpandToSum[u, x]^p, x] /; FreeQ[p, x] && GeneralizedBinomialQ[u, x] && !GeneralizedBinomialMatchQ[u, x]
  1980. Int[Times[Power[Pattern[u, Blank[]], Optional[Pattern[p, Blank[]]]], Power[Times[Optional[Pattern[c, Blank[]]], Pattern[x, Blank[]]], Optional[Pattern[m, Blank[]]]]], Pattern[x, Blank[Symbol]]] := Int[(c*x)^m*ExpandToSum[u, x]^p, x] /; FreeQ[{c, m, p}, x] && GeneralizedBinomialQ[u, x] && !GeneralizedBinomialMatchQ[u, x]
  1981. Int[Power[Pattern[u, Blank[]], Pattern[p, Blank[]]], Pattern[x, Blank[Symbol]]] := Int[ExpandToSum[u, x]^p, x] /; FreeQ[p, x] && QuadraticQ[u, x] && !QuadraticMatchQ[u, x]
  1982. Int[Times[Power[Pattern[u, Blank[]], Optional[Pattern[m, Blank[]]]], Power[Pattern[v, Blank[]], Optional[Pattern[p, Blank[]]]]], Pattern[x, Blank[Symbol]]] := Int[ExpandToSum[u, x]^m*ExpandToSum[v, x]^p, x] /; FreeQ[{m, p}, x] && LinearQ[u, x] && QuadraticQ[v, x] && !(LinearMatchQ[u, x] && QuadraticMatchQ[v, x])
  1983. Int[Times[Power[Pattern[u, Blank[]], Optional[Pattern[m, Blank[]]]], Power[Pattern[v, Blank[]], Optional[Pattern[n, Blank[]]]], Power[Pattern[w, Blank[]], Optional[Pattern[p, Blank[]]]]], Pattern[x, Blank[Symbol]]] := Int[ExpandToSum[u, x]^m*ExpandToSum[v, x]^n*ExpandToSum[w, x]^p, x] /; FreeQ[{m, n, p}, x] && LinearQ[{u, v}, x] && QuadraticQ[w, x] && !(LinearMatchQ[{u, v}, x] && QuadraticMatchQ[w, x])
  1984. Int[Times[Power[Pattern[u, Blank[]], Optional[Pattern[p, Blank[]]]], Power[Pattern[v, Blank[]], Optional[Pattern[q, Blank[]]]]], Pattern[x, Blank[Symbol]]] := Int[ExpandToSum[u, x]^p*ExpandToSum[v, x]^q, x] /; FreeQ[{p, q}, x] && QuadraticQ[{u, v}, x] && !QuadraticMatchQ[{u, v}, x]
  1985. Int[Times[Power[Pattern[u, Blank[]], Optional[Pattern[p, Blank[]]]], Power[Pattern[v, Blank[]], Optional[Pattern[q, Blank[]]]], Power[Pattern[z, Blank[]], Optional[Pattern[m, Blank[]]]]], Pattern[x, Blank[Symbol]]] := Int[ExpandToSum[z, x]^m*ExpandToSum[u, x]^p*ExpandToSum[v, x]^q, x] /; FreeQ[{m, p, q}, x] && LinearQ[z, x] && QuadraticQ[{u, v}, x] && !(LinearMatchQ[z, x] && QuadraticMatchQ[{u, v}, x])
  1986. Int[Times[Pattern[Pq, Blank[]], Power[Pattern[u, Blank[]], Optional[Pattern[p, Blank[]]]]], Pattern[x, Blank[Symbol]]] := Int[Pq*ExpandToSum[u, x]^p, x] /; FreeQ[p, x] && PolyQ[Pq, x] && QuadraticQ[u, x] && !QuadraticMatchQ[u, x]
  1987. Int[Times[Pattern[Pq, Blank[]], Power[Pattern[u, Blank[]], Optional[Pattern[m, Blank[]]]], Power[Pattern[v, Blank[]], Optional[Pattern[p, Blank[]]]]], Pattern[x, Blank[Symbol]]] := Int[ExpandToSum[u, x]^m*Pq*ExpandToSum[v, x]^p, x] /; FreeQ[{m, p}, x] && PolyQ[Pq, x] && LinearQ[u, x] && QuadraticQ[v, x] && !(LinearMatchQ[u, x] && QuadraticMatchQ[v, x])
  1988. Int[Power[Pattern[u, Blank[]], Pattern[p, Blank[]]], Pattern[x, Blank[Symbol]]] := Int[ExpandToSum[u, x]^p, x] /; FreeQ[p, x] && TrinomialQ[u, x] && !TrinomialMatchQ[u, x]
  1989. Int[Times[Power[Pattern[u, Blank[]], Optional[Pattern[p, Blank[]]]], Power[Times[Optional[Pattern[d, Blank[]]], Pattern[x, Blank[]]], Optional[Pattern[m, Blank[]]]]], Pattern[x, Blank[Symbol]]] := Int[(d*x)^m*ExpandToSum[u, x]^p, x] /; FreeQ[{d, m, p}, x] && TrinomialQ[u, x] && !TrinomialMatchQ[u, x]
  1990. Int[Times[Power[Pattern[u, Blank[]], Optional[Pattern[q, Blank[]]]], Power[Pattern[v, Blank[]], Optional[Pattern[p, Blank[]]]]], Pattern[x, Blank[Symbol]]] := Int[ExpandToSum[u, x]^q*ExpandToSum[v, x]^p, x] /; FreeQ[{p, q}, x] && BinomialQ[u, x] && TrinomialQ[v, x] && !(BinomialMatchQ[u, x] && TrinomialMatchQ[v, x])
  1991. Int[Times[Power[Pattern[u, Blank[]], Optional[Pattern[q, Blank[]]]], Power[Pattern[v, Blank[]], Optional[Pattern[p, Blank[]]]]], Pattern[x, Blank[Symbol]]] := Int[ExpandToSum[u, x]^q*ExpandToSum[v, x]^p, x] /; FreeQ[{p, q}, x] && BinomialQ[u, x] && BinomialQ[v, x] && !(BinomialMatchQ[u, x] && BinomialMatchQ[v, x])
  1992. Int[Times[Power[Pattern[u, Blank[]], Optional[Pattern[p, Blank[]]]], Power[Times[Optional[Pattern[f, Blank[]]], Pattern[x, Blank[]]], Optional[Pattern[m, Blank[]]]], Power[Pattern[z, Blank[]], Optional[Pattern[q, Blank[]]]]], Pattern[x, Blank[Symbol]]] := Int[(f*x)^m*ExpandToSum[z, x]^q*ExpandToSum[u, x]^p, x] /; FreeQ[{f, m, p, q}, x] && BinomialQ[z, x] && TrinomialQ[u, x] && !(BinomialMatchQ[z, x] && TrinomialMatchQ[u, x])
  1993. Int[Times[Power[Pattern[u, Blank[]], Optional[Pattern[p, Blank[]]]], Power[Times[Optional[Pattern[f, Blank[]]], Pattern[x, Blank[]]], Optional[Pattern[m, Blank[]]]], Power[Pattern[z, Blank[]], Optional[Pattern[q, Blank[]]]]], Pattern[x, Blank[Symbol]]] := Int[(f*x)^m*ExpandToSum[z, x]^q*ExpandToSum[u, x]^p, x] /; FreeQ[{f, m, p, q}, x] && BinomialQ[z, x] && BinomialQ[u, x] && !(BinomialMatchQ[z, x] && BinomialMatchQ[u, x])
  1994. Int[Times[Pattern[Pq, Blank[]], Power[Pattern[u, Blank[]], Optional[Pattern[p, Blank[]]]]], Pattern[x, Blank[Symbol]]] := Int[Pq*ExpandToSum[u, x]^p, x] /; FreeQ[p, x] && PolyQ[Pq, x] && TrinomialQ[u, x] && !TrinomialMatchQ[u, x]
  1995. Int[Times[Pattern[Pq, Blank[]], Power[Pattern[u, Blank[]], Optional[Pattern[p, Blank[]]]], Power[Times[Optional[Pattern[d, Blank[]]], Pattern[x, Blank[]]], Optional[Pattern[m, Blank[]]]]], Pattern[x, Blank[Symbol]]] := Int[(d*x)^m*Pq*ExpandToSum[u, x]^p, x] /; FreeQ[{d, m, p}, x] && PolyQ[Pq, x] && TrinomialQ[u, x] && !TrinomialMatchQ[u, x]
  1996. Int[Power[Pattern[u, Blank[]], Pattern[p, Blank[]]], Pattern[x, Blank[Symbol]]] := Int[ExpandToSum[u, x]^p, x] /; FreeQ[p, x] && GeneralizedTrinomialQ[u, x] && !GeneralizedTrinomialMatchQ[u, x]
  1997. Int[Times[Power[Pattern[u, Blank[]], Optional[Pattern[p, Blank[]]]], Power[Times[Optional[Pattern[d, Blank[]]], Pattern[x, Blank[]]], Optional[Pattern[m, Blank[]]]]], Pattern[x, Blank[Symbol]]] := Int[(d*x)^m*ExpandToSum[u, x]^p, x] /; FreeQ[{d, m, p}, x] && GeneralizedTrinomialQ[u, x] && !GeneralizedTrinomialMatchQ[u, x]
  1998. Int[Times[Power[Pattern[u, Blank[]], Optional[Pattern[p, Blank[]]]], Pattern[z, Blank[]]], Pattern[x, Blank[Symbol]]] := Int[ExpandToSum[z, x]*ExpandToSum[u, x]^p, x] /; FreeQ[p, x] && BinomialQ[z, x] && GeneralizedTrinomialQ[u, x] && EqQ[BinomialDegree[z, x] - GeneralizedTrinomialDegree[u, x], 0] && !(BinomialMatchQ[z, x] && GeneralizedTrinomialMatchQ[u, x])
  1999. Int[Times[Power[Pattern[u, Blank[]], Optional[Pattern[p, Blank[]]]], Power[Times[Optional[Pattern[f, Blank[]]], Pattern[x, Blank[]]], Optional[Pattern[m, Blank[]]]], Pattern[z, Blank[]]], Pattern[x, Blank[Symbol]]] := Int[(f*x)^m*ExpandToSum[z, x]*ExpandToSum[u, x]^p, x] /; FreeQ[{f, m, p}, x] && BinomialQ[z, x] && GeneralizedTrinomialQ[u, x] && EqQ[BinomialDegree[z, x] - GeneralizedTrinomialDegree[u, x], 0] && !(BinomialMatchQ[z, x] && GeneralizedTrinomialMatchQ[u, x])
  2000. Int[Power[Plus[Times[Optional[Pattern[a, Blank[]]], Power[Pattern[x, Blank[]], Optional[Pattern[j, Blank[]]]]], Times[Optional[Pattern[b, Blank[]]], Power[Pattern[x, Blank[]], Optional[Pattern[n, Blank[]]]]]], Pattern[p, Blank[]]], Pattern[x, Blank[Symbol]]] := Simp[(a*x^j + b*x^n)^(p + 1)/(b*(n - j)*(p + 1)*x^(n - 1)), x] /; FreeQ[{a, b, j, n, p}, x] && !IntegerQ[p] && NeQ[n, j] && EqQ[j*p - n + j + 1, 0]
  2001. Int[Power[Plus[Times[Optional[Pattern[a, Blank[]]], Power[Pattern[x, Blank[]], Optional[Pattern[j, Blank[]]]]], Times[Optional[Pattern[b, Blank[]]], Power[Pattern[x, Blank[]], Optional[Pattern[n, Blank[]]]]]], Pattern[p, Blank[]]], Pattern[x, Blank[Symbol]]] := -Simp[(a*x^j + b*x^n)^(p + 1)/(a*(n - j)*(p + 1)*x^(j - 1)), x] + Dist[(n*p + n - j + 1)/(a*(n - j)*(p + 1)), Int[(a*x^j + b*x^n)^(p + 1)/x^j, x], x] /; FreeQ[{a, b, j, n}, x] && !IntegerQ[p] && NeQ[n, j] && ILtQ[Simplify[(n*p + n - j + 1)/(n - j)], 0] && LtQ[p, -1]
  2002. Int[Power[Plus[Times[Optional[Pattern[a, Blank[]]], Power[Pattern[x, Blank[]], Optional[Pattern[j, Blank[]]]]], Times[Optional[Pattern[b, Blank[]]], Power[Pattern[x, Blank[]], Optional[Pattern[n, Blank[]]]]]], Pattern[p, Blank[]]], Pattern[x, Blank[Symbol]]] := Simp[(a*x^j + b*x^n)^(p + 1)/(a*(j*p + 1)*x^(j - 1)), x] - Dist[(b*(n*p + n - j + 1))/(a*(j*p + 1)), Int[x^(n - j)*(a*x^j + b*x^n)^p, x], x] /; FreeQ[{a, b, j, n, p}, x] && !IntegerQ[p] && NeQ[n, j] && ILtQ[Simplify[(n*p + n - j + 1)/(n - j)], 0] && NeQ[j*p + 1, 0]
  2003. Int[Power[Plus[Times[Optional[Pattern[a, Blank[]]], Power[Pattern[x, Blank[]], Optional[Pattern[j, Blank[]]]]], Times[Optional[Pattern[b, Blank[]]], Power[Pattern[x, Blank[]], Optional[Pattern[n, Blank[]]]]]], Pattern[p, Blank[]]], Pattern[x, Blank[Symbol]]] := Simp[(x*(a*x^j + b*x^n)^p)/(j*p + 1), x] - Dist[(b*(n - j)*p)/(j*p + 1), Int[x^n*(a*x^j + b*x^n)^(p - 1), x], x] /; FreeQ[{a, b}, x] && !IntegerQ[p] && LtQ[0, j, n] && GtQ[p, 0] && LtQ[j*p + 1, 0]
  2004. Int[Power[Plus[Times[Optional[Pattern[a, Blank[]]], Power[Pattern[x, Blank[]], Optional[Pattern[j, Blank[]]]]], Times[Optional[Pattern[b, Blank[]]], Power[Pattern[x, Blank[]], Optional[Pattern[n, Blank[]]]]]], Pattern[p, Blank[]]], Pattern[x, Blank[Symbol]]] := Simp[(x*(a*x^j + b*x^n)^p)/(n*p + 1), x] + Dist[(a*(n - j)*p)/(n*p + 1), Int[x^j*(a*x^j + b*x^n)^(p - 1), x], x] /; FreeQ[{a, b}, x] && !IntegerQ[p] && LtQ[0, j, n] && GtQ[p, 0] && NeQ[n*p + 1, 0]
  2005. Int[Power[Plus[Times[Optional[Pattern[a, Blank[]]], Power[Pattern[x, Blank[]], Optional[Pattern[j, Blank[]]]]], Times[Optional[Pattern[b, Blank[]]], Power[Pattern[x, Blank[]], Optional[Pattern[n, Blank[]]]]]], Pattern[p, Blank[]]], Pattern[x, Blank[Symbol]]] := Simp[(a*x^j + b*x^n)^(p + 1)/(b*(n - j)*(p + 1)*x^(n - 1)), x] - Dist[(j*p - n + j + 1)/(b*(n - j)*(p + 1)), Int[(a*x^j + b*x^n)^(p + 1)/x^n, x], x] /; FreeQ[{a, b}, x] && !IntegerQ[p] && LtQ[0, j, n] && LtQ[p, -1] && GtQ[j*p + 1, n - j]
  2006. Int[Power[Plus[Times[Optional[Pattern[a, Blank[]]], Power[Pattern[x, Blank[]], Optional[Pattern[j, Blank[]]]]], Times[Optional[Pattern[b, Blank[]]], Power[Pattern[x, Blank[]], Optional[Pattern[n, Blank[]]]]]], Pattern[p, Blank[]]], Pattern[x, Blank[Symbol]]] := -Simp[(a*x^j + b*x^n)^(p + 1)/(a*(n - j)*(p + 1)*x^(j - 1)), x] + Dist[(n*p + n - j + 1)/(a*(n - j)*(p + 1)), Int[(a*x^j + b*x^n)^(p + 1)/x^j, x], x] /; FreeQ[{a, b}, x] && !IntegerQ[p] && LtQ[0, j, n] && LtQ[p, -1]
  2007. Int[Power[Plus[Times[Optional[Pattern[a, Blank[]]], Power[Pattern[x, Blank[]], Optional[Pattern[j, Blank[]]]]], Times[Optional[Pattern[b, Blank[]]], Power[Pattern[x, Blank[]], Optional[Pattern[n, Blank[]]]]]], Pattern[p, Blank[]]], Pattern[x, Blank[Symbol]]] := Simp[(x*(a*x^j + b*x^n)^p)/(p*(n - j)), x] + Dist[a, Int[x^j*(a*x^j + b*x^n)^(p - 1), x], x] /; FreeQ[{a, b, j, n}, x] && IGtQ[p + 1/2, 0] && NeQ[n, j] && EqQ[Simplify[j*p + 1], 0]
  2008. Int[Power[Plus[Times[Optional[Pattern[a, Blank[]]], Power[Pattern[x, Blank[]], 2]], Times[Optional[Pattern[b, Blank[]]], Power[Pattern[x, Blank[]], Optional[Pattern[n, Blank[]]]]]], Rational[-1, 2]], Pattern[x, Blank[Symbol]]] := Dist[2/(2 - n), Subst[Int[1/(1 - a*x^2), x], x, x/Sqrt[a*x^2 + b*x^n]], x] /; FreeQ[{a, b, n}, x] && NeQ[n, 2]
  2009. Int[Power[Plus[Times[Optional[Pattern[a, Blank[]]], Power[Pattern[x, Blank[]], Optional[Pattern[j, Blank[]]]]], Times[Optional[Pattern[b, Blank[]]], Power[Pattern[x, Blank[]], Optional[Pattern[n, Blank[]]]]]], Pattern[p, Blank[]]], Pattern[x, Blank[Symbol]]] := -Simp[(a*x^j + b*x^n)^(p + 1)/(a*(n - j)*(p + 1)*x^(j - 1)), x] + Dist[(n*p + n - j + 1)/(a*(n - j)*(p + 1)), Int[(a*x^j + b*x^n)^(p + 1)/x^j, x], x] /; FreeQ[{a, b, j, n}, x] && ILtQ[p + 1/2, 0] && NeQ[n, j] && EqQ[Simplify[j*p + 1], 0]
  2010. Int[Power[Plus[Times[Optional[Pattern[a, Blank[]]], Power[Pattern[x, Blank[]], Optional[Pattern[j, Blank[]]]]], Times[Optional[Pattern[b, Blank[]]], Power[Pattern[x, Blank[]], Optional[Pattern[n, Blank[]]]]]], Rational[-1, 2]], Pattern[x, Blank[Symbol]]] := Simp[(-2*Sqrt[a*x^j + b*x^n])/(b*(n - 2)*x^(n - 1)), x] - Dist[(a*(2*n - j - 2))/(b*(n - 2)), Int[1/(x^(n - j)*Sqrt[a*x^j + b*x^n]), x], x] /; FreeQ[{a, b}, x] && LtQ[2*(n - 1), j, n]
  2011. Int[Power[Plus[Times[Optional[Pattern[a, Blank[]]], Power[Pattern[x, Blank[]], Optional[Pattern[j, Blank[]]]]], Times[Optional[Pattern[b, Blank[]]], Power[Pattern[x, Blank[]], Optional[Pattern[n, Blank[]]]]]], Pattern[p, Blank[]]], Pattern[x, Blank[Symbol]]] := Dist[(a*x^j + b*x^n)^FracPart[p]/(x^(j*FracPart[p])*(a + b*x^(n - j))^FracPart[p]), Int[x^(j*p)*(a + b*x^(n - j))^p, x], x] /; FreeQ[{a, b, j, n, p}, x] && !IntegerQ[p] && NeQ[n, j] && PosQ[n - j]
  2012. Int[Power[Plus[Times[Optional[Pattern[a, Blank[]]], Power[Pattern[u, Blank[]], Optional[Pattern[j, Blank[]]]]], Times[Optional[Pattern[b, Blank[]]], Power[Pattern[u, Blank[]], Optional[Pattern[n, Blank[]]]]]], Pattern[p, Blank[]]], Pattern[x, Blank[Symbol]]] := Dist[1/Coefficient[u, x, 1], Subst[Int[(a*x^j + b*x^n)^p, x], x, u], x] /; FreeQ[{a, b, j, n, p}, x] && LinearQ[u, x] && NeQ[u, x]
  2013. Int[Times[Power[Pattern[x, Blank[]], Optional[Pattern[m, Blank[]]]], Power[Plus[Times[Optional[Pattern[a, Blank[]]], Power[Pattern[x, Blank[]], Optional[Pattern[j, Blank[]]]]], Times[Optional[Pattern[b, Blank[]]], Power[Pattern[x, Blank[]], Pattern[n, Blank[]]]]], Pattern[p, Blank[]]]], Pattern[x, Blank[Symbol]]] := Dist[1/n, Subst[Int[(a*x^Simplify[j/n] + b*x)^p, x], x, x^n], x] /; FreeQ[{a, b, j, m, n, p}, x] && !IntegerQ[p] && NeQ[n, j] && IntegerQ[Simplify[j/n]] && EqQ[Simplify[m - n + 1], 0]
  2014. Int[Times[Power[Times[Optional[Pattern[c, Blank[]]], Pattern[x, Blank[]]], Optional[Pattern[m, Blank[]]]], Power[Plus[Times[Optional[Pattern[a, Blank[]]], Power[Pattern[x, Blank[]], Optional[Pattern[j, Blank[]]]]], Times[Optional[Pattern[b, Blank[]]], Power[Pattern[x, Blank[]], Optional[Pattern[n, Blank[]]]]]], Pattern[p, Blank[]]]], Pattern[x, Blank[Symbol]]] := -Simp[(c^(j - 1)*(c*x)^(m - j + 1)*(a*x^j + b*x^n)^(p + 1))/(a*(n - j)*(p + 1)), x] /; FreeQ[{a, b, c, j, m, n, p}, x] && !IntegerQ[p] && NeQ[n, j] && EqQ[m + n*p + n - j + 1, 0] && (IntegerQ[j] || GtQ[c, 0])
  2015. Int[Times[Power[Times[Optional[Pattern[c, Blank[]]], Pattern[x, Blank[]]], Optional[Pattern[m, Blank[]]]], Power[Plus[Times[Optional[Pattern[a, Blank[]]], Power[Pattern[x, Blank[]], Optional[Pattern[j, Blank[]]]]], Times[Optional[Pattern[b, Blank[]]], Power[Pattern[x, Blank[]], Optional[Pattern[n, Blank[]]]]]], Pattern[p, Blank[]]]], Pattern[x, Blank[Symbol]]] := -Simp[(c^(j - 1)*(c*x)^(m - j + 1)*(a*x^j + b*x^n)^(p + 1))/(a*(n - j)*(p + 1)), x] + Dist[(c^j*(m + n*p + n - j + 1))/(a*(n - j)*(p + 1)), Int[(c*x)^(m - j)*(a*x^j + b*x^n)^(p + 1), x], x] /; FreeQ[{a, b, c, j, m, n}, x] && !IntegerQ[p] && NeQ[n, j] && ILtQ[Simplify[(m + n*p + n - j + 1)/(n - j)], 0] && LtQ[p, -1] && (IntegerQ[j] || GtQ[c, 0])
  2016. Int[Times[Power[Times[Optional[Pattern[c, Blank[]]], Pattern[x, Blank[]]], Optional[Pattern[m, Blank[]]]], Power[Plus[Times[Optional[Pattern[a, Blank[]]], Power[Pattern[x, Blank[]], Optional[Pattern[j, Blank[]]]]], Times[Optional[Pattern[b, Blank[]]], Power[Pattern[x, Blank[]], Optional[Pattern[n, Blank[]]]]]], Pattern[p, Blank[]]]], Pattern[x, Blank[Symbol]]] := Simp[(c^(j - 1)*(c*x)^(m - j + 1)*(a*x^j + b*x^n)^(p + 1))/(a*(m + j*p + 1)), x] - Dist[(b*(m + n*p + n - j + 1))/(a*c^(n - j)*(m + j*p + 1)), Int[(c*x)^(m + n - j)*(a*x^j + b*x^n)^p, x], x] /; FreeQ[{a, b, c, j, m, n, p}, x] && !IntegerQ[p] && NeQ[n, j] && ILtQ[Simplify[(m + n*p + n - j + 1)/(n - j)], 0] && NeQ[m + j*p + 1, 0] && (IntegersQ[j, n] || GtQ[c, 0])
  2017. Int[Times[Power[Times[Pattern[c, Blank[]], Pattern[x, Blank[]]], Optional[Pattern[m, Blank[]]]], Power[Plus[Times[Optional[Pattern[a, Blank[]]], Power[Pattern[x, Blank[]], Optional[Pattern[j, Blank[]]]]], Times[Optional[Pattern[b, Blank[]]], Power[Pattern[x, Blank[]], Optional[Pattern[n, Blank[]]]]]], Pattern[p, Blank[]]]], Pattern[x, Blank[Symbol]]] := Dist[(c^IntPart[m]*(c*x)^FracPart[m])/x^FracPart[m], Int[x^m*(a*x^j + b*x^n)^p, x], x] /; FreeQ[{a, b, c, j, m, n, p}, x] && !IntegerQ[p] && NeQ[n, j] && ILtQ[Simplify[(m + n*p + n - j + 1)/(n - j)], 0]
  2018. Int[Times[Power[Pattern[x, Blank[]], Optional[Pattern[m, Blank[]]]], Power[Plus[Times[Optional[Pattern[a, Blank[]]], Power[Pattern[x, Blank[]], Optional[Pattern[j, Blank[]]]]], Times[Optional[Pattern[b, Blank[]]], Power[Pattern[x, Blank[]], Pattern[n, Blank[]]]]], Pattern[p, Blank[]]]], Pattern[x, Blank[Symbol]]] := Dist[1/n, Subst[Int[x^(Simplify[(m + 1)/n] - 1)*(a*x^Simplify[j/n] + b*x)^p, x], x, x^n], x] /; FreeQ[{a, b, j, m, n, p}, x] && !IntegerQ[p] && NeQ[n, j] && IntegerQ[Simplify[j/n]] && IntegerQ[Simplify[(m + 1)/n]] && NeQ[n^2, 1]
  2019. Int[Times[Power[Times[Pattern[c, Blank[]], Pattern[x, Blank[]]], Optional[Pattern[m, Blank[]]]], Power[Plus[Times[Optional[Pattern[a, Blank[]]], Power[Pattern[x, Blank[]], Optional[Pattern[j, Blank[]]]]], Times[Optional[Pattern[b, Blank[]]], Power[Pattern[x, Blank[]], Pattern[n, Blank[]]]]], Pattern[p, Blank[]]]], Pattern[x, Blank[Symbol]]] := Dist[(c^IntPart[m]*(c*x)^FracPart[m])/x^FracPart[m], Int[x^m*(a*x^j + b*x^n)^p, x], x] /; FreeQ[{a, b, c, j, m, n, p}, x] && !IntegerQ[p] && NeQ[n, j] && IntegerQ[Simplify[j/n]] && IntegerQ[Simplify[(m + 1)/n]] && NeQ[n^2, 1]
  2020. Int[Times[Power[Times[Optional[Pattern[c, Blank[]]], Pattern[x, Blank[]]], Pattern[m, Blank[]]], Power[Plus[Times[Optional[Pattern[a, Blank[]]], Power[Pattern[x, Blank[]], Optional[Pattern[j, Blank[]]]]], Times[Optional[Pattern[b, Blank[]]], Power[Pattern[x, Blank[]], Optional[Pattern[n, Blank[]]]]]], Pattern[p, Blank[]]]], Pattern[x, Blank[Symbol]]] := Simp[((c*x)^(m + 1)*(a*x^j + b*x^n)^p)/(c*(m + j*p + 1)), x] - Dist[(b*p*(n - j))/(c^n*(m + j*p + 1)), Int[(c*x)^(m + n)*(a*x^j + b*x^n)^(p - 1), x], x] /; FreeQ[{a, b, c}, x] && !IntegerQ[p] && LtQ[0, j, n] && (IntegersQ[j, n] || GtQ[c, 0]) && GtQ[p, 0] && LtQ[m + j*p + 1, 0]
  2021. Int[Times[Power[Times[Optional[Pattern[c, Blank[]]], Pattern[x, Blank[]]], Optional[Pattern[m, Blank[]]]], Power[Plus[Times[Optional[Pattern[a, Blank[]]], Power[Pattern[x, Blank[]], Optional[Pattern[j, Blank[]]]]], Times[Optional[Pattern[b, Blank[]]], Power[Pattern[x, Blank[]], Optional[Pattern[n, Blank[]]]]]], Pattern[p, Blank[]]]], Pattern[x, Blank[Symbol]]] := Simp[((c*x)^(m + 1)*(a*x^j + b*x^n)^p)/(c*(m + n*p + 1)), x] + Dist[(a*(n - j)*p)/(c^j*(m + n*p + 1)), Int[(c*x)^(m + j)*(a*x^j + b*x^n)^(p - 1), x], x] /; FreeQ[{a, b, c, m}, x] && !IntegerQ[p] && LtQ[0, j, n] && (IntegersQ[j, n] || GtQ[c, 0]) && GtQ[p, 0] && NeQ[m + n*p + 1, 0]
  2022. Int[Times[Power[Times[Optional[Pattern[c, Blank[]]], Pattern[x, Blank[]]], Optional[Pattern[m, Blank[]]]], Power[Plus[Times[Optional[Pattern[a, Blank[]]], Power[Pattern[x, Blank[]], Optional[Pattern[j, Blank[]]]]], Times[Optional[Pattern[b, Blank[]]], Power[Pattern[x, Blank[]], Optional[Pattern[n, Blank[]]]]]], Pattern[p, Blank[]]]], Pattern[x, Blank[Symbol]]] := Simp[(c^(n - 1)*(c*x)^(m - n + 1)*(a*x^j + b*x^n)^(p + 1))/(b*(n - j)*(p + 1)), x] - Dist[(c^n*(m + j*p - n + j + 1))/(b*(n - j)*(p + 1)), Int[(c*x)^(m - n)*(a*x^j + b*x^n)^(p + 1), x], x] /; FreeQ[{a, b, c}, x] && !IntegerQ[p] && LtQ[0, j, n] && (IntegersQ[j, n] || GtQ[c, 0]) && LtQ[p, -1] && GtQ[m + j*p + 1, n - j]
  2023. Int[Times[Power[Times[Optional[Pattern[c, Blank[]]], Pattern[x, Blank[]]], Optional[Pattern[m, Blank[]]]], Power[Plus[Times[Optional[Pattern[a, Blank[]]], Power[Pattern[x, Blank[]], Optional[Pattern[j, Blank[]]]]], Times[Optional[Pattern[b, Blank[]]], Power[Pattern[x, Blank[]], Optional[Pattern[n, Blank[]]]]]], Pattern[p, Blank[]]]], Pattern[x, Blank[Symbol]]] := -Simp[(c^(j - 1)*(c*x)^(m - j + 1)*(a*x^j + b*x^n)^(p + 1))/(a*(n - j)*(p + 1)), x] + Dist[(c^j*(m + n*p + n - j + 1))/(a*(n - j)*(p + 1)), Int[(c*x)^(m - j)*(a*x^j + b*x^n)^(p + 1), x], x] /; FreeQ[{a, b, c, m}, x] && !IntegerQ[p] && LtQ[0, j, n] && (IntegersQ[j, n] || GtQ[c, 0]) && LtQ[p, -1]
  2024. Int[Times[Power[Times[Optional[Pattern[c, Blank[]]], Pattern[x, Blank[]]], Optional[Pattern[m, Blank[]]]], Power[Plus[Times[Optional[Pattern[a, Blank[]]], Power[Pattern[x, Blank[]], Optional[Pattern[j, Blank[]]]]], Times[Optional[Pattern[b, Blank[]]], Power[Pattern[x, Blank[]], Optional[Pattern[n, Blank[]]]]]], Pattern[p, Blank[]]]], Pattern[x, Blank[Symbol]]] := Simp[(c^(n - 1)*(c*x)^(m - n + 1)*(a*x^j + b*x^n)^(p + 1))/(b*(m + n*p + 1)), x] - Dist[(a*c^(n - j)*(m + j*p - n + j + 1))/(b*(m + n*p + 1)), Int[(c*x)^(m - (n - j))*(a*x^j + b*x^n)^p, x], x] /; FreeQ[{a, b, c, m, p}, x] && !IntegerQ[p] && LtQ[0, j, n] && (IntegersQ[j, n] || GtQ[c, 0]) && GtQ[m + j*p + 1 - n + j, 0] && NeQ[m + n*p + 1, 0]
  2025. Int[Times[Power[Times[Optional[Pattern[c, Blank[]]], Pattern[x, Blank[]]], Optional[Pattern[m, Blank[]]]], Power[Plus[Times[Optional[Pattern[a, Blank[]]], Power[Pattern[x, Blank[]], Optional[Pattern[j, Blank[]]]]], Times[Optional[Pattern[b, Blank[]]], Power[Pattern[x, Blank[]], Optional[Pattern[n, Blank[]]]]]], Pattern[p, Blank[]]]], Pattern[x, Blank[Symbol]]] := Simp[(c^(j - 1)*(c*x)^(m - j + 1)*(a*x^j + b*x^n)^(p + 1))/(a*(m + j*p + 1)), x] - Dist[(b*(m + n*p + n - j + 1))/(a*c^(n - j)*(m + j*p + 1)), Int[(c*x)^(m + n - j)*(a*x^j + b*x^n)^p, x], x] /; FreeQ[{a, b, c, m, p}, x] && !IntegerQ[p] && LtQ[0, j, n] && (IntegersQ[j, n] || GtQ[c, 0]) && LtQ[m + j*p + 1, 0]
  2026. Int[Times[Power[Pattern[x, Blank[]], Optional[Pattern[m, Blank[]]]], Power[Plus[Times[Optional[Pattern[a, Blank[]]], Power[Pattern[x, Blank[]], Optional[Pattern[j, Blank[]]]]], Times[Optional[Pattern[b, Blank[]]], Power[Pattern[x, Blank[]], Pattern[n, Blank[]]]]], Pattern[p, Blank[]]]], Pattern[x, Blank[Symbol]]] := Dist[1/(m + 1), Subst[Int[(a*x^Simplify[j/(m + 1)] + b*x^Simplify[n/(m + 1)])^p, x], x, x^(m + 1)], x] /; FreeQ[{a, b, j, m, n, p}, x] && !IntegerQ[p] && NeQ[n, j] && IntegerQ[Simplify[j/n]] && NeQ[m, -1] && IntegerQ[Simplify[n/(m + 1)]] && !IntegerQ[n]
  2027. Int[Times[Power[Times[Pattern[c, Blank[]], Pattern[x, Blank[]]], Optional[Pattern[m, Blank[]]]], Power[Plus[Times[Optional[Pattern[a, Blank[]]], Power[Pattern[x, Blank[]], Optional[Pattern[j, Blank[]]]]], Times[Optional[Pattern[b, Blank[]]], Power[Pattern[x, Blank[]], Pattern[n, Blank[]]]]], Pattern[p, Blank[]]]], Pattern[x, Blank[Symbol]]] := Dist[(c^IntPart[m]*(c*x)^FracPart[m])/x^FracPart[m], Int[x^m*(a*x^j + b*x^n)^p, x], x] /; FreeQ[{a, b, c, j, m, n, p}, x] && !IntegerQ[p] && NeQ[n, j] && IntegerQ[Simplify[j/n]] && NeQ[m, -1] && IntegerQ[Simplify[n/(m + 1)]] && !IntegerQ[n]
  2028. Int[Times[Power[Times[Optional[Pattern[c, Blank[]]], Pattern[x, Blank[]]], Optional[Pattern[m, Blank[]]]], Power[Plus[Times[Optional[Pattern[a, Blank[]]], Power[Pattern[x, Blank[]], Optional[Pattern[j, Blank[]]]]], Times[Optional[Pattern[b, Blank[]]], Power[Pattern[x, Blank[]], Optional[Pattern[n, Blank[]]]]]], Pattern[p, Blank[]]]], Pattern[x, Blank[Symbol]]] := Simp[((c*x)^(m + 1)*(a*x^j + b*x^n)^p)/(c*p*(n - j)), x] + Dist[a/c^j, Int[(c*x)^(m + j)*(a*x^j + b*x^n)^(p - 1), x], x] /; FreeQ[{a, b, c, j, m, n}, x] && IGtQ[p + 1/2, 0] && NeQ[n, j] && EqQ[Simplify[m + j*p + 1], 0] && (IntegerQ[j] || GtQ[c, 0])
  2029. Int[Times[Power[Pattern[x, Blank[]], Optional[Pattern[m, Blank[]]]], Power[Plus[Times[Optional[Pattern[a, Blank[]]], Power[Pattern[x, Blank[]], Optional[Pattern[j, Blank[]]]]], Times[Optional[Pattern[b, Blank[]]], Power[Pattern[x, Blank[]], Optional[Pattern[n, Blank[]]]]]], Rational[-1, 2]]], Pattern[x, Blank[Symbol]]] := Dist[-2/(n - j), Subst[Int[1/(1 - a*x^2), x], x, x^(j/2)/Sqrt[a*x^j + b*x^n]], x] /; FreeQ[{a, b, j, n}, x] && EqQ[m, j/2 - 1] && NeQ[n, j]
  2030. Int[Times[Power[Times[Optional[Pattern[c, Blank[]]], Pattern[x, Blank[]]], Optional[Pattern[m, Blank[]]]], Power[Plus[Times[Optional[Pattern[a, Blank[]]], Power[Pattern[x, Blank[]], Optional[Pattern[j, Blank[]]]]], Times[Optional[Pattern[b, Blank[]]], Power[Pattern[x, Blank[]], Optional[Pattern[n, Blank[]]]]]], Pattern[p, Blank[]]]], Pattern[x, Blank[Symbol]]] := -Simp[(c^(j - 1)*(c*x)^(m - j + 1)*(a*x^j + b*x^n)^(p + 1))/(a*(n - j)*(p + 1)), x] + Dist[(c^j*(m + n*p + n - j + 1))/(a*(n - j)*(p + 1)), Int[(c*x)^(m - j)*(a*x^j + b*x^n)^(p + 1), x], x] /; FreeQ[{a, b, c, j, m, n}, x] && ILtQ[p + 1/2, 0] && NeQ[n, j] && EqQ[Simplify[m + j*p + 1], 0] && (IntegerQ[j] || GtQ[c, 0])
  2031. Int[Times[Power[Times[Pattern[c, Blank[]], Pattern[x, Blank[]]], Optional[Pattern[m, Blank[]]]], Power[Plus[Times[Optional[Pattern[a, Blank[]]], Power[Pattern[x, Blank[]], Optional[Pattern[j, Blank[]]]]], Times[Optional[Pattern[b, Blank[]]], Power[Pattern[x, Blank[]], Pattern[n, Blank[]]]]], Pattern[p, Blank[]]]], Pattern[x, Blank[Symbol]]] := Dist[(c^IntPart[m]*(c*x)^FracPart[m])/x^FracPart[m], Int[x^m*(a*x^j + b*x^n)^p, x], x] /; FreeQ[{a, b, c, j, m, n, p}, x] && IntegerQ[p + 1/2] && NeQ[n, j] && EqQ[Simplify[m + j*p + 1], 0]
  2032. Int[Times[Power[Times[Optional[Pattern[c, Blank[]]], Pattern[x, Blank[]]], Optional[Pattern[m, Blank[]]]], Power[Plus[Times[Optional[Pattern[a, Blank[]]], Power[Pattern[x, Blank[]], Optional[Pattern[j, Blank[]]]]], Times[Optional[Pattern[b, Blank[]]], Power[Pattern[x, Blank[]], Optional[Pattern[n, Blank[]]]]]], Pattern[p, Blank[]]]], Pattern[x, Blank[Symbol]]] := Dist[(c^IntPart[m]*(c*x)^FracPart[m]*(a*x^j + b*x^n)^FracPart[p])/(x^(FracPart[m] + j*FracPart[p])*(a + b*x^(n - j))^FracPart[p]), Int[x^(m + j*p)*(a + b*x^(n - j))^p, x], x] /; FreeQ[{a, b, c, j, m, n, p}, x] && !IntegerQ[p] && NeQ[n, j] && PosQ[n - j]
  2033. Int[Times[Power[Pattern[u, Blank[]], Optional[Pattern[m, Blank[]]]], Power[Plus[Times[Optional[Pattern[a, Blank[]]], Power[Pattern[v, Blank[]], Optional[Pattern[j, Blank[]]]]], Times[Optional[Pattern[b, Blank[]]], Power[Pattern[v, Blank[]], Optional[Pattern[n, Blank[]]]]]], Optional[Pattern[p, Blank[]]]]], Pattern[x, Blank[Symbol]]] := Dist[u^m/(Coefficient[v, x, 1]*v^m), Subst[Int[x^m*(a*x^j + b*x^n)^p, x], x, v], x] /; FreeQ[{a, b, j, m, n, p}, x] && LinearPairQ[u, v, x]
  2034. Int[Times[Power[Pattern[x, Blank[]], Optional[Pattern[m, Blank[]]]], Power[Plus[Times[Optional[Pattern[b, Blank[]]], Power[Pattern[x, Blank[]], Optional[Pattern[k, Blank[]]]]], Times[Optional[Pattern[a, Blank[]]], Power[Pattern[x, Blank[]], Pattern[j, Blank[]]]]], Pattern[p, Blank[]]], Power[Plus[Pattern[c, Blank[]], Times[Optional[Pattern[d, Blank[]]], Power[Pattern[x, Blank[]], Pattern[n, Blank[]]]]], Optional[Pattern[q, Blank[]]]]], Pattern[x, Blank[Symbol]]] := Dist[1/n, Subst[Int[x^(Simplify[(m + 1)/n] - 1)*(a*x^Simplify[j/n] + b*x^Simplify[k/n])^p*(c + d*x)^q, x], x, x^n], x] /; FreeQ[{a, b, c, d, j, k, m, n, p, q}, x] && !IntegerQ[p] && NeQ[k, j] && IntegerQ[Simplify[j/n]] && IntegerQ[Simplify[k/n]] && IntegerQ[Simplify[(m + 1)/n]] && NeQ[n^2, 1]
  2035. Int[Times[Power[Times[Pattern[e, Blank[]], Pattern[x, Blank[]]], Optional[Pattern[m, Blank[]]]], Power[Plus[Pattern[c, Blank[]], Times[Optional[Pattern[d, Blank[]]], Power[Pattern[x, Blank[]], Optional[Pattern[n, Blank[]]]]]], Optional[Pattern[q, Blank[]]]], Power[Plus[Times[Optional[Pattern[b, Blank[]]], Power[Pattern[x, Blank[]], Optional[Pattern[k, Blank[]]]]], Times[Optional[Pattern[a, Blank[]]], Power[Pattern[x, Blank[]], Pattern[j, Blank[]]]]], Pattern[p, Blank[]]]], Pattern[x, Blank[Symbol]]] := Dist[(e^IntPart[m]*(e*x)^FracPart[m])/x^FracPart[m], Int[x^m*(a*x^j + b*x^k)^p*(c + d*x^n)^q, x], x] /; FreeQ[{a, b, c, d, e, j, k, m, n, p, q}, x] && !IntegerQ[p] && NeQ[k, j] && IntegerQ[Simplify[j/n]] && IntegerQ[Simplify[k/n]] && IntegerQ[Simplify[(m + 1)/n]] && NeQ[n^2, 1]
  2036. Int[Times[Power[Times[Optional[Pattern[e, Blank[]]], Pattern[x, Blank[]]], Optional[Pattern[m, Blank[]]]], Power[Plus[Times[Optional[Pattern[a, Blank[]]], Power[Pattern[x, Blank[]], Optional[Pattern[j, Blank[]]]]], Times[Optional[Pattern[b, Blank[]]], Power[Pattern[x, Blank[]], Optional[Pattern[jn, Blank[]]]]]], Pattern[p, Blank[]]], Plus[Pattern[c, Blank[]], Times[Optional[Pattern[d, Blank[]]], Power[Pattern[x, Blank[]], Optional[Pattern[n, Blank[]]]]]]], Pattern[x, Blank[Symbol]]] := Simp[(c*e^(j - 1)*(e*x)^(m - j + 1)*(a*x^j + b*x^(j + n))^(p + 1))/(a*(m + j*p + 1)), x] /; FreeQ[{a, b, c, d, e, j, m, n, p}, x] && EqQ[jn, j + n] && !IntegerQ[p] && NeQ[b*c - a*d, 0] && EqQ[a*d*(m + j*p + 1) - b*c*(m + n + p*(j + n) + 1), 0] && (GtQ[e, 0] || IntegersQ[j]) && NeQ[m + j*p + 1, 0]
  2037. Int[Times[Power[Times[Optional[Pattern[e, Blank[]]], Pattern[x, Blank[]]], Optional[Pattern[m, Blank[]]]], Power[Plus[Times[Optional[Pattern[a, Blank[]]], Power[Pattern[x, Blank[]], Optional[Pattern[j, Blank[]]]]], Times[Optional[Pattern[b, Blank[]]], Power[Pattern[x, Blank[]], Optional[Pattern[jn, Blank[]]]]]], Pattern[p, Blank[]]], Plus[Pattern[c, Blank[]], Times[Optional[Pattern[d, Blank[]]], Power[Pattern[x, Blank[]], Optional[Pattern[n, Blank[]]]]]]], Pattern[x, Blank[Symbol]]] := -Simp[(e^(j - 1)*(b*c - a*d)*(e*x)^(m - j + 1)*(a*x^j + b*x^(j + n))^(p + 1))/(a*b*n*(p + 1)), x] - Dist[(e^j*(a*d*(m + j*p + 1) - b*c*(m + n + p*(j + n) + 1)))/(a*b*n*(p + 1)), Int[(e*x)^(m - j)*(a*x^j + b*x^(j + n))^(p + 1), x], x] /; FreeQ[{a, b, c, d, e, j, m, n}, x] && EqQ[jn, j + n] && !IntegerQ[p] && NeQ[b*c - a*d, 0] && LtQ[p, -1] && GtQ[j, 0] && LeQ[j, m] && (GtQ[e, 0] || IntegerQ[j])
  2038. Int[Times[Power[Times[Optional[Pattern[e, Blank[]]], Pattern[x, Blank[]]], Optional[Pattern[m, Blank[]]]], Power[Plus[Times[Optional[Pattern[a, Blank[]]], Power[Pattern[x, Blank[]], Optional[Pattern[j, Blank[]]]]], Times[Optional[Pattern[b, Blank[]]], Power[Pattern[x, Blank[]], Optional[Pattern[jn, Blank[]]]]]], Pattern[p, Blank[]]], Plus[Pattern[c, Blank[]], Times[Optional[Pattern[d, Blank[]]], Power[Pattern[x, Blank[]], Optional[Pattern[n, Blank[]]]]]]], Pattern[x, Blank[Symbol]]] := Simp[(c*e^(j - 1)*(e*x)^(m - j + 1)*(a*x^j + b*x^(j + n))^(p + 1))/(a*(m + j*p + 1)), x] + Dist[(a*d*(m + j*p + 1) - b*c*(m + n + p*(j + n) + 1))/(a*e^n*(m + j*p + 1)), Int[(e*x)^(m + n)*(a*x^j + b*x^(j + n))^p, x], x] /; FreeQ[{a, b, c, d, e, j, p}, x] && EqQ[jn, j + n] && !IntegerQ[p] && NeQ[b*c - a*d, 0] && GtQ[n, 0] && (LtQ[m + j*p, -1] || (IntegersQ[m - 1/2, p - 1/2] && LtQ[p, 0] && LtQ[m, -(n*p) - 1])) && (GtQ[e, 0] || IntegersQ[j, n]) && NeQ[m + j*p + 1, 0] && NeQ[m - n + j*p + 1, 0]
  2039. Int[Times[Power[Times[Optional[Pattern[e, Blank[]]], Pattern[x, Blank[]]], Optional[Pattern[m, Blank[]]]], Power[Plus[Times[Optional[Pattern[a, Blank[]]], Power[Pattern[x, Blank[]], Optional[Pattern[j, Blank[]]]]], Times[Optional[Pattern[b, Blank[]]], Power[Pattern[x, Blank[]], Optional[Pattern[jn, Blank[]]]]]], Pattern[p, Blank[]]], Plus[Pattern[c, Blank[]], Times[Optional[Pattern[d, Blank[]]], Power[Pattern[x, Blank[]], Optional[Pattern[n, Blank[]]]]]]], Pattern[x, Blank[Symbol]]] := Simp[(d*e^(j - 1)*(e*x)^(m - j + 1)*(a*x^j + b*x^(j + n))^(p + 1))/(b*(m + n + p*(j + n) + 1)), x] - Dist[(a*d*(m + j*p + 1) - b*c*(m + n + p*(j + n) + 1))/(b*(m + n + p*(j + n) + 1)), Int[(e*x)^m*(a*x^j + b*x^(j + n))^p, x], x] /; FreeQ[{a, b, c, d, e, j, m, n, p}, x] && EqQ[jn, j + n] && !IntegerQ[p] && NeQ[b*c - a*d, 0] && NeQ[m + n + p*(j + n) + 1, 0] && (GtQ[e, 0] || IntegerQ[j])
  2040. Int[Times[Power[Pattern[x, Blank[]], Optional[Pattern[m, Blank[]]]], Power[Plus[Pattern[c, Blank[]], Times[Optional[Pattern[d, Blank[]]], Power[Pattern[x, Blank[]], Optional[Pattern[n, Blank[]]]]]], Optional[Pattern[q, Blank[]]]], Power[Plus[Times[Optional[Pattern[b, Blank[]]], Power[Pattern[x, Blank[]], Optional[Pattern[k, Blank[]]]]], Times[Optional[Pattern[a, Blank[]]], Power[Pattern[x, Blank[]], Pattern[j, Blank[]]]]], Pattern[p, Blank[]]]], Pattern[x, Blank[Symbol]]] := Dist[1/(m + 1), Subst[Int[(a*x^Simplify[j/(m + 1)] + b*x^Simplify[k/(m + 1)])^p*(c + d*x^Simplify[n/(m + 1)])^q, x], x, x^(m + 1)], x] /; FreeQ[{a, b, c, d, j, k, m, n, p, q}, x] && !IntegerQ[p] && NeQ[k, j] && IntegerQ[Simplify[j/n]] && IntegerQ[Simplify[k/n]] && NeQ[m, -1] && IntegerQ[Simplify[n/(m + 1)]] && !IntegerQ[n]
  2041. Int[Times[Power[Times[Pattern[e, Blank[]], Pattern[x, Blank[]]], Optional[Pattern[m, Blank[]]]], Power[Plus[Pattern[c, Blank[]], Times[Optional[Pattern[d, Blank[]]], Power[Pattern[x, Blank[]], Optional[Pattern[n, Blank[]]]]]], Optional[Pattern[q, Blank[]]]], Power[Plus[Times[Optional[Pattern[b, Blank[]]], Power[Pattern[x, Blank[]], Optional[Pattern[k, Blank[]]]]], Times[Optional[Pattern[a, Blank[]]], Power[Pattern[x, Blank[]], Pattern[j, Blank[]]]]], Pattern[p, Blank[]]]], Pattern[x, Blank[Symbol]]] := Dist[(e^IntPart[m]*(e*x)^FracPart[m])/x^FracPart[m], Int[x^m*(a*x^j + b*x^k)^p*(c + d*x^n)^q, x], x] /; FreeQ[{a, b, c, d, e, j, k, m, n, p, q}, x] && !IntegerQ[p] && NeQ[k, j] && IntegerQ[Simplify[j/n]] && IntegerQ[Simplify[k/n]] && NeQ[m, -1] && IntegerQ[Simplify[n/(m + 1)]] && !IntegerQ[n]
  2042. Int[Times[Power[Times[Optional[Pattern[e, Blank[]]], Pattern[x, Blank[]]], Optional[Pattern[m, Blank[]]]], Power[Plus[Times[Optional[Pattern[a, Blank[]]], Power[Pattern[x, Blank[]], Optional[Pattern[j, Blank[]]]]], Times[Optional[Pattern[b, Blank[]]], Power[Pattern[x, Blank[]], Optional[Pattern[jn, Blank[]]]]]], Pattern[p, Blank[]]], Power[Plus[Pattern[c, Blank[]], Times[Optional[Pattern[d, Blank[]]], Power[Pattern[x, Blank[]], Optional[Pattern[n, Blank[]]]]]], Optional[Pattern[q, Blank[]]]]], Pattern[x, Blank[Symbol]]] := Dist[(e^IntPart[m]*(e*x)^FracPart[m]*(a*x^j + b*x^(j + n))^FracPart[p])/(x^(FracPart[m] + j*FracPart[p])*(a + b*x^n)^FracPart[p]), Int[x^(m + j*p)*(a + b*x^n)^p*(c + d*x^n)^q, x], x] /; FreeQ[{a, b, c, d, e, j, m, n, p, q}, x] && EqQ[jn, j + n] && !IntegerQ[p] && NeQ[b*c - a*d, 0] && !(EqQ[n, 1] && EqQ[j, 1])
  2043. Int[Times[Pattern[Pq, Blank[]], Power[Plus[Times[Optional[Pattern[a, Blank[]]], Power[Pattern[x, Blank[]], Optional[Pattern[j, Blank[]]]]], Times[Optional[Pattern[b, Blank[]]], Power[Pattern[x, Blank[]], Pattern[n, Blank[]]]]], Pattern[p, Blank[]]]], Pattern[x, Blank[Symbol]]] := With[{d = Denominator[n]}, Dist[d, Subst[Int[x^(d - 1)*(SubstFor[x^n, Pq, x] /. x -> x^(d*n))*(a*x^(d*j) + b*x^(d*n))^p, x], x, x^(1/d)], x]] /; FreeQ[{a, b, j, n, p}, x] && PolyQ[Pq, x^n] && !IntegerQ[p] && NeQ[n, j] && RationalQ[j, n] && IntegerQ[j/n] && LtQ[-1, n, 1]
  2044. Int[Times[Pattern[Pq, Blank[]], Power[Pattern[x, Blank[]], Optional[Pattern[m, Blank[]]]], Power[Plus[Times[Optional[Pattern[a, Blank[]]], Power[Pattern[x, Blank[]], Optional[Pattern[j, Blank[]]]]], Times[Optional[Pattern[b, Blank[]]], Power[Pattern[x, Blank[]], Pattern[n, Blank[]]]]], Pattern[p, Blank[]]]], Pattern[x, Blank[Symbol]]] := Dist[1/n, Subst[Int[x^(Simplify[(m + 1)/n] - 1)*SubstFor[x^n, Pq, x]*(a*x^Simplify[j/n] + b*x)^p, x], x, x^n], x] /; FreeQ[{a, b, j, m, n, p}, x] && PolyQ[Pq, x^n] && !IntegerQ[p] && NeQ[n, j] && IntegerQ[Simplify[j/n]] && IntegerQ[Simplify[(m + 1)/n]]
  2045. Int[Times[Pattern[Pq, Blank[]], Power[Times[Pattern[c, Blank[]], Pattern[x, Blank[]]], Optional[Pattern[m, Blank[]]]], Power[Plus[Times[Optional[Pattern[a, Blank[]]], Power[Pattern[x, Blank[]], Optional[Pattern[j, Blank[]]]]], Times[Optional[Pattern[b, Blank[]]], Power[Pattern[x, Blank[]], Pattern[n, Blank[]]]]], Pattern[p, Blank[]]]], Pattern[x, Blank[Symbol]]] := Dist[(c^(Sign[m]*Quotient[m, Sign[m]])*(c*x)^Mod[m, Sign[m]])/x^Mod[m, Sign[m]], Int[x^m*Pq*(a*x^j + b*x^n)^p, x], x] /; FreeQ[{a, b, c, j, n, p}, x] && PolyQ[Pq, x^n] && !IntegerQ[p] && NeQ[n, j] && IntegerQ[Simplify[j/n]] && IntegerQ[Simplify[(m + 1)/n]] && RationalQ[m] && GtQ[m^2, 1]
  2046. Int[Times[Pattern[Pq, Blank[]], Power[Times[Pattern[c, Blank[]], Pattern[x, Blank[]]], Optional[Pattern[m, Blank[]]]], Power[Plus[Times[Optional[Pattern[a, Blank[]]], Power[Pattern[x, Blank[]], Optional[Pattern[j, Blank[]]]]], Times[Optional[Pattern[b, Blank[]]], Power[Pattern[x, Blank[]], Pattern[n, Blank[]]]]], Pattern[p, Blank[]]]], Pattern[x, Blank[Symbol]]] := Dist[(c*x)^m/x^m, Int[x^m*Pq*(a*x^j + b*x^n)^p, x], x] /; FreeQ[{a, b, c, j, m, n, p}, x] && PolyQ[Pq, x^n] && !IntegerQ[p] && NeQ[n, j] && IntegerQ[Simplify[j/n]] && IntegerQ[Simplify[(m + 1)/n]]
  2047. Int[Times[Pattern[Pq, Blank[]], Power[Pattern[x, Blank[]], Optional[Pattern[m, Blank[]]]], Power[Plus[Times[Optional[Pattern[a, Blank[]]], Power[Pattern[x, Blank[]], Optional[Pattern[j, Blank[]]]]], Times[Optional[Pattern[b, Blank[]]], Power[Pattern[x, Blank[]], Pattern[n, Blank[]]]]], Pattern[p, Blank[]]]], Pattern[x, Blank[Symbol]]] := With[{g = GCD[m + 1, n]}, Dist[1/g, Subst[Int[x^((m + 1)/g - 1)*(Pq /. x -> x^(1/g))*(a*x^(j/g) + b*x^(n/g))^p, x], x, x^g], x] /; NeQ[g, 1]] /; FreeQ[{a, b, p}, x] && PolyQ[Pq, x^n] && !IntegerQ[p] && IGtQ[j, 0] && IGtQ[n, 0] && IGtQ[j/n, 0] && IntegerQ[m]
  2048. Int[Times[Pattern[Pq, Blank[]], Power[Times[Optional[Pattern[c, Blank[]]], Pattern[x, Blank[]]], Optional[Pattern[m, Blank[]]]], Power[Plus[Times[Optional[Pattern[a, Blank[]]], Power[Pattern[x, Blank[]], Optional[Pattern[j, Blank[]]]]], Times[Optional[Pattern[b, Blank[]]], Power[Pattern[x, Blank[]], Pattern[n, Blank[]]]]], Pattern[p, Blank[]]]], Pattern[x, Blank[Symbol]]] := With[{q = Expon[Pq, x]}, With[{Pqq = Coeff[Pq, x, q]}, Int[(c*x)^m*ExpandToSum[Pq - Pqq*x^q - (a*Pqq*(m + q - n + 1)*x^(q - n))/(b*(m + q + n*p + 1)), x]*(a*x^j + b*x^n)^p, x] + Simp[(Pqq*(c*x)^(m + q - n + 1)*(a*x^j + b*x^n)^(p + 1))/(b*c^(q - n + 1)*(m + q + n*p + 1)), x]] /; GtQ[q, n - 1] && NeQ[m + q + n*p + 1, 0] && (IntegerQ[2*p] || IntegerQ[p + (q + 1)/(2*n)])] /; FreeQ[{a, b, c, m, p}, x] && PolyQ[Pq, x] && !IntegerQ[p] && IGtQ[j, 0] && IGtQ[n, 0] && LtQ[j, n]
  2049. Int[Times[Pattern[Pq, Blank[]], Power[Pattern[x, Blank[]], Optional[Pattern[m, Blank[]]]], Power[Plus[Times[Optional[Pattern[a, Blank[]]], Power[Pattern[x, Blank[]], Optional[Pattern[j, Blank[]]]]], Times[Optional[Pattern[b, Blank[]]], Power[Pattern[x, Blank[]], Pattern[n, Blank[]]]]], Pattern[p, Blank[]]]], Pattern[x, Blank[Symbol]]] := Dist[1/(m + 1), Subst[Int[(SubstFor[x^n, Pq, x] /. x -> x^Simplify[n/(m + 1)])*(a*x^Simplify[j/(m + 1)] + b*x^Simplify[n/(m + 1)])^p, x], x, x^(m + 1)], x] /; FreeQ[{a, b, j, m, n, p}, x] && PolyQ[Pq, x^n] && !IntegerQ[p] && NeQ[n, j] && IntegerQ[Simplify[j/n]] && IntegerQ[Simplify[n/(m + 1)]] && !IntegerQ[n]
  2050. Int[Times[Pattern[Pq, Blank[]], Power[Times[Pattern[c, Blank[]], Pattern[x, Blank[]]], Pattern[m, Blank[]]], Power[Plus[Times[Optional[Pattern[a, Blank[]]], Power[Pattern[x, Blank[]], Optional[Pattern[j, Blank[]]]]], Times[Optional[Pattern[b, Blank[]]], Power[Pattern[x, Blank[]], Pattern[n, Blank[]]]]], Pattern[p, Blank[]]]], Pattern[x, Blank[Symbol]]] := Dist[(c^(Sign[m]*Quotient[m, Sign[m]])*(c*x)^Mod[m, Sign[m]])/x^Mod[m, Sign[m]], Int[x^m*Pq*(a*x^j + b*x^n)^p, x], x] /; FreeQ[{a, b, c, j, n, p}, x] && PolyQ[Pq, x^n] && !IntegerQ[p] && NeQ[n, j] && IntegerQ[Simplify[j/n]] && IntegerQ[Simplify[n/(m + 1)]] && !IntegerQ[n] && GtQ[m^2, 1]
  2051. Int[Times[Pattern[Pq, Blank[]], Power[Times[Pattern[c, Blank[]], Pattern[x, Blank[]]], Pattern[m, Blank[]]], Power[Plus[Times[Optional[Pattern[a, Blank[]]], Power[Pattern[x, Blank[]], Optional[Pattern[j, Blank[]]]]], Times[Optional[Pattern[b, Blank[]]], Power[Pattern[x, Blank[]], Pattern[n, Blank[]]]]], Pattern[p, Blank[]]]], Pattern[x, Blank[Symbol]]] := Dist[(c*x)^m/x^m, Int[x^m*Pq*(a*x^j + b*x^n)^p, x], x] /; FreeQ[{a, b, c, j, m, n, p}, x] && PolyQ[Pq, x^n] && !IntegerQ[p] && NeQ[n, j] && IntegerQ[Simplify[j/n]] && IntegerQ[Simplify[n/(m + 1)]] && !IntegerQ[n]
  2052. Int[Times[Pattern[Pq, Blank[]], Power[Times[Optional[Pattern[c, Blank[]]], Pattern[x, Blank[]]], Optional[Pattern[m, Blank[]]]], Power[Plus[Times[Optional[Pattern[a, Blank[]]], Power[Pattern[x, Blank[]], Optional[Pattern[j, Blank[]]]]], Times[Optional[Pattern[b, Blank[]]], Power[Pattern[x, Blank[]], Pattern[n, Blank[]]]]], Pattern[p, Blank[]]]], Pattern[x, Blank[Symbol]]] := Int[ExpandIntegrand[(c*x)^m*Pq*(a*x^j + b*x^n)^p, x], x] /; FreeQ[{a, b, c, j, m, n, p}, x] && (PolyQ[Pq, x] || PolyQ[Pq, x^n]) && !IntegerQ[p] && NeQ[n, j]
  2053. Int[Times[Pattern[Pq, Blank[]], Power[Plus[Times[Optional[Pattern[a, Blank[]]], Power[Pattern[x, Blank[]], Optional[Pattern[j, Blank[]]]]], Times[Optional[Pattern[b, Blank[]]], Power[Pattern[x, Blank[]], Pattern[n, Blank[]]]]], Pattern[p, Blank[]]]], Pattern[x, Blank[Symbol]]] := Int[ExpandIntegrand[Pq*(a*x^j + b*x^n)^p, x], x] /; FreeQ[{a, b, j, n, p}, x] && (PolyQ[Pq, x] || PolyQ[Pq, x^n]) && !IntegerQ[p] && NeQ[n, j]
  2054. Int[Times[Optional[Pattern[u, Blank[]]], Power[Pattern[P, Blank[]], Pattern[p, Blank[]]], Power[Pattern[Q, Blank[]], Pattern[q, Blank[]]]], Pattern[x, Blank[Symbol]]] := Module[{gcd = PolyGCD[P, Q, x]}, Int[u*gcd^(p + q)*PolynomialQuotient[P, gcd, x]^p*PolynomialQuotient[Q, gcd, x]^q, x] /; NeQ[gcd, 1]] /; IGtQ[p, 0] && ILtQ[q, 0] && PolyQ[P, x] && PolyQ[Q, x]
  2055. Int[Times[Optional[Pattern[u, Blank[]]], Pattern[P, Blank[]], Power[Pattern[Q, Blank[]], Pattern[q, Blank[]]]], Pattern[x, Blank[Symbol]]] := Module[{gcd = PolyGCD[P, Q, x]}, Int[u*gcd^(q + 1)*PolynomialQuotient[P, gcd, x]*PolynomialQuotient[Q, gcd, x]^q, x] /; NeQ[gcd, 1]] /; ILtQ[q, 0] && PolyQ[P, x] && PolyQ[Q, x]
  2056. Int[Times[Optional[Pattern[u, Blank[]]], Power[Pattern[P, Blank[]], Optional[Pattern[p, Blank[]]]]], Pattern[x, Blank[Symbol]]] := With[{m = MinimumMonomialExponent[P, x]}, Dist[P^FracPart[p]/(x^(m*FracPart[p])*Distrib[1/x^m, P]^FracPart[p]), Int[u*x^(m*p)*Distrib[1/x^m, P]^p, x], x]] /; FreeQ[p, x] && !IntegerQ[p] && SumQ[P] && EveryQ[BinomialQ[#1, x] & , P] && !PolyQ[P, x, 2]
  2057. Int[Power[Pattern[P, Blank[]], Pattern[p, Blank[]]], Pattern[x, Blank[Symbol]]] := With[{u = Factor[P /. x -> Sqrt[x]]}, Int[ExpandIntegrand[(u /. x -> x^2)^p, x], x] /; !SumQ[NonfreeFactors[u, x]]] /; PolyQ[P, x^2] && ILtQ[p, 0]
  2058. Int[Power[Pattern[P, Blank[]], Pattern[p, Blank[]]], Pattern[x, Blank[Symbol]]] := With[{u = Factor[P]}, Int[ExpandIntegrand[u^p, x], x] /; !SumQ[NonfreeFactors[u, x]]] /; PolyQ[P, x] && ILtQ[p, 0]
  2059. Int[Power[Pattern[P, Blank[]], Pattern[p, Blank[]]], Pattern[x, Blank[Symbol]]] := With[{u = Factor[P]}, Int[u^p, x] /; !SumQ[NonfreeFactors[u, x]]] /; PolyQ[P, x] && IntegerQ[p]
  2060. Int[Power[Plus[Optional[Pattern[a, Blank[]]], Times[Optional[Pattern[b, Blank[]]], Pattern[x, Blank[]]], Times[Optional[Pattern[c, Blank[]]], Power[Pattern[x, Blank[]], 2]], Times[Optional[Pattern[d, Blank[]]], Power[Pattern[x, Blank[]], 3]]], Pattern[p, Blank[]]], Pattern[x, Blank[Symbol]]] := Dist[1/3^p, Subst[Int[Simp[(3*a*c - b^2)/c + (c^2*x^3)/b, x]^p, x], x, c/(3*d) + x], x] /; FreeQ[{a, b, c, d}, x] && IGtQ[p, 0] && EqQ[c^2 - 3*b*d, 0]
  2061. Int[Power[Pattern[P, Blank[]], Pattern[p, Blank[]]], Pattern[x, Blank[Symbol]]] := Int[ExpandToSum[P^p, x], x] /; PolyQ[P, x] && IGtQ[p, 0]
  2062. Int[Power[Pattern[P, Blank[]], Pattern[p, Blank[]]], Pattern[x, Blank[Symbol]]] := Int[ExpandIntegrand[P^p, x], x] /; PolyQ[P, x] && IntegerQ[p] && QuadraticProductQ[Factor[P], x]
  2063. Int[Power[Plus[Optional[Pattern[a, Blank[]]], Times[Optional[Pattern[b, Blank[]]], Pattern[x, Blank[]]], Times[Optional[Pattern[d, Blank[]]], Power[Pattern[x, Blank[]], 3]]], Pattern[p, Blank[]]], Pattern[x, Blank[Symbol]]] := Dist[1/(3^(3*p)*a^(2*p)), Int[(3*a - b*x)^p*(3*a + 2*b*x)^(2*p), x], x] /; FreeQ[{a, b, d}, x] && EqQ[4*b^3 + 27*a^2*d, 0] && IntegerQ[p]
  2064. Int[Power[Plus[Optional[Pattern[a, Blank[]]], Times[Optional[Pattern[b, Blank[]]], Pattern[x, Blank[]]], Times[Optional[Pattern[d, Blank[]]], Power[Pattern[x, Blank[]], 3]]], Pattern[p, Blank[]]], Pattern[x, Blank[Symbol]]] := Dist[(a + b*x + d*x^3)^p/((3*a - b*x)^p*(3*a + 2*b*x)^(2*p)), Int[(3*a - b*x)^p*(3*a + 2*b*x)^(2*p), x], x] /; FreeQ[{a, b, d, p}, x] && EqQ[4*b^3 + 27*a^2*d, 0] && !IntegerQ[p]
  2065. Int[Power[Plus[Optional[Pattern[a, Blank[]]], Times[Optional[Pattern[b, Blank[]]], Pattern[x, Blank[]]], Times[Optional[Pattern[d, Blank[]]], Power[Pattern[x, Blank[]], 3]]], Pattern[p, Blank[]]], Pattern[x, Blank[Symbol]]] := With[{r = Rt[-9*a*d^2 + Sqrt[3]*d*Sqrt[4*b^3*d + 27*a^2*d^2], 3]}, Dist[1/d^(2*p), Int[Simp[(18^(1/3)*b*d)/(3*r) - r/18^(1/3) + d*x, x]^p*Simp[(b*d)/3 + (12^(1/3)*b^2*d^2)/(3*r^2) + r^2/(3*12^(1/3)) - d*((2^(1/3)*b*d)/(3^(1/3)*r) - r/18^(1/3))*x + d^2*x^2, x]^p, x], x]] /; FreeQ[{a, b, d}, x] && NeQ[4*b^3 + 27*a^2*d, 0] && IntegerQ[p]
  2066. Int[Power[Plus[Optional[Pattern[a, Blank[]]], Times[Optional[Pattern[b, Blank[]]], Pattern[x, Blank[]]], Times[Optional[Pattern[d, Blank[]]], Power[Pattern[x, Blank[]], 3]]], Pattern[p, Blank[]]], Pattern[x, Blank[Symbol]]] := With[{r = Rt[-9*a*d^2 + Sqrt[3]*d*Sqrt[4*b^3*d + 27*a^2*d^2], 3]}, Dist[(a + b*x + d*x^3)^p/(Simp[(18^(1/3)*b*d)/(3*r) - r/18^(1/3) + d*x, x]^p*Simp[(b*d)/3 + (12^(1/3)*b^2*d^2)/(3*r^2) + r^2/(3*12^(1/3)) - d*((2^(1/3)*b*d)/(3^(1/3)*r) - r/18^(1/3))*x + d^2*x^2, x]^p), Int[Simp[(18^(1/3)*b*d)/(3*r) - r/18^(1/3) + d*x, x]^p*Simp[(b*d)/3 + (12^(1/3)*b^2*d^2)/(3*r^2) + r^2/(3*12^(1/3)) - d*((2^(1/3)*b*d)/(3^(1/3)*r) - r/18^(1/3))*x + d^2*x^2, x]^p, x], x]] /; FreeQ[{a, b, d, p}, x] && NeQ[4*b^3 + 27*a^2*d, 0] && !IntegerQ[p]
  2067. Int[Power[Pattern[P3, Blank[]], Pattern[p, Blank[]]], Pattern[x, Blank[Symbol]]] := With[{a = Coeff[P3, x, 0], b = Coeff[P3, x, 1], c = Coeff[P3, x, 2], d = Coeff[P3, x, 3]}, Subst[Int[Simp[(2*c^3 - 9*b*c*d + 27*a*d^2)/(27*d^2) - ((c^2 - 3*b*d)*x)/(3*d) + d*x^3, x]^p, x], x, x + c/(3*d)] /; NeQ[c, 0]] /; FreeQ[p, x] && PolyQ[P3, x, 3]
  2068. Int[Power[Pattern[P4, Blank[]], Pattern[p, Blank[]]], Pattern[x, Blank[Symbol]]] := With[{a = Coeff[P4, x, 0], b = Coeff[P4, x, 1], c = Coeff[P4, x, 2], d = Coeff[P4, x, 3], e = Coeff[P4, x, 4]}, Dist[1/a^(3*p), Int[ExpandIntegrand[1/((a - b*x)^p/(a^5 - b^5*x^5)^p), x], x], x] /; NeQ[a, 0] && EqQ[c, b^2/a] && EqQ[d, b^3/a^2] && EqQ[e, b^4/a^3]] /; FreeQ[p, x] && PolyQ[P4, x, 4] && ILtQ[p, 0]
  2069. Int[Power[Pattern[P4, Blank[]], Pattern[p, Blank[]]], Pattern[x, Blank[Symbol]]] := With[{a = Coeff[P4, x, 0], b = Coeff[P4, x, 1], c = Coeff[P4, x, 2], d = Coeff[P4, x, 3], e = Coeff[P4, x, 4]}, Dist[-16*a^2, Subst[Int[(1*((a*(-3*b^4 + 16*a*b^2*c - 64*a^2*b*d + 256*a^3*e - 32*a^2*(3*b^2 - 8*a*c)*x^2 + 256*a^4*x^4))/(b - 4*a*x)^4)^p)/(b - 4*a*x)^2, x], x, b/(4*a) + 1/x], x] /; NeQ[a, 0] && NeQ[b, 0] && EqQ[b^3 - 4*a*b*c + 8*a^2*d, 0]] /; FreeQ[p, x] && PolyQ[P4, x, 4] && IntegerQ[2*p] && !IGtQ[p, 0]
  2070. Int[Power[Pattern[Q6, Blank[]], Pattern[p, Blank[]]], Pattern[x, Blank[Symbol]]] := With[{a = Coeff[Q6, x, 0], b = Coeff[Q6, x, 2], c = Coeff[Q6, x, 3], d = Coeff[Q6, x, 4], e = Coeff[Q6, x, 6]}, Dist[1/(3^(3*p)*a^(2*p)), Int[ExpandIntegrand[(3*a + 3*Rt[a, 3]^2*Rt[c, 3]*x + b*x^2)^p*(3*a - 3*(-1)^(1/3)*Rt[a, 3]^2*Rt[c, 3]*x + b*x^2)^p*(3*a + 3*(-1)^(2/3)*Rt[a, 3]^2*Rt[c, 3]*x + b*x^2)^p, x], x], x] /; EqQ[b^2 - 3*a*d, 0] && EqQ[b^3 - 27*a^2*e, 0]] /; ILtQ[p, 0] && PolyQ[Q6, x, 6] && EqQ[Coeff[Q6, x, 1], 0] && EqQ[Coeff[Q6, x, 5], 0] && RationalFunctionQ[u, x]
  2071. Int[Times[Power[Pattern[v, Blank[]], Rational[1, 2]], Power[Plus[Pattern[d, Blank[]], Times[Optional[Pattern[e, Blank[]]], Power[Pattern[x, Blank[]], 4]]], -1]], Pattern[x, Blank[Symbol]]] := With[{a = Coeff[v, x, 0], b = Coeff[v, x, 2], c = Coeff[v, x, 4]}, Dist[a/d, Subst[Int[1/(1 - 2*b*x^2 + (b^2 - 4*a*c)*x^4), x], x, x/Sqrt[v]], x] /; EqQ[c*d + a*e, 0] && PosQ[a*c]] /; FreeQ[{d, e}, x] && PolyQ[v, x^2, 2]
  2072. Int[Times[Power[Plus[Pattern[a, Blank[]], Times[Optional[Pattern[b, Blank[]]], Power[Pattern[x, Blank[]], 2]], Times[Optional[Pattern[c, Blank[]]], Power[Pattern[x, Blank[]], 4]]], Rational[1, 2]], Power[Plus[Pattern[d, Blank[]], Times[Optional[Pattern[e, Blank[]]], Power[Pattern[x, Blank[]], 4]]], -1]], Pattern[x, Blank[Symbol]]] := With[{q = Sqrt[b^2 - 4*a*c]}, -Simp[(a*Sqrt[b + q]*ArcTan[(Sqrt[b + q]*x*(b - q + 2*c*x^2))/(2*Sqrt[2]*Rt[-(a*c), 2]*Sqrt[a + b*x^2 + c*x^4])])/(2*Sqrt[2]*Rt[-(a*c), 2]*d), x] + Simp[(a*Sqrt[-b + q]*ArcTanh[(Sqrt[-b + q]*x*(b + q + 2*c*x^2))/(2*Sqrt[2]*Rt[-(a*c), 2]*Sqrt[a + b*x^2 + c*x^4])])/(2*Sqrt[2]*Rt[-(a*c), 2]*d), x]] /; FreeQ[{a, b, c, d, e}, x] && EqQ[c*d + a*e, 0] && NegQ[a*c]
  2073. Int[Times[Power[Pattern[P, Blank[]], Pattern[p, Blank[]]], Power[Pattern[Q, Blank[]], Optional[Pattern[q, Blank[]]]]], Pattern[x, Blank[Symbol]]] := With[{PP = Factor[P /. x -> Sqrt[x]]}, Int[ExpandIntegrand[(PP /. x -> x^2)^p*Q^q, x], x] /; !SumQ[NonfreeFactors[PP, x]]] /; FreeQ[q, x] && PolyQ[P, x^2] && PolyQ[Q, x] && ILtQ[p, 0]
  2074. Int[Times[Power[Pattern[P, Blank[]], Pattern[p, Blank[]]], Power[Pattern[Q, Blank[]], Optional[Pattern[q, Blank[]]]]], Pattern[x, Blank[Symbol]]] := With[{PP = Factor[P]}, Int[ExpandIntegrand[PP^p*Q^q, x], x] /; !SumQ[NonfreeFactors[PP, x]]] /; FreeQ[q, x] && PolyQ[P, x] && PolyQ[Q, x] && IntegerQ[p] && NeQ[P, x]
  2075. Int[Times[Power[Pattern[P, Blank[]], Pattern[p, Blank[]]], Pattern[Qm, Blank[]]], Pattern[x, Blank[Symbol]]] := With[{PP = Factor[P]}, Int[ExpandIntegrand[PP^p*Qm, x], x] /; QuadraticProductQ[PP, x]] /; PolyQ[Qm, x] && PolyQ[P, x] && ILtQ[p, 0]
  2076. Int[Times[Power[Plus[Optional[Pattern[e, Blank[]]], Times[Optional[Pattern[f, Blank[]]], Pattern[x, Blank[]]]], Optional[Pattern[m, Blank[]]]], Power[Plus[Pattern[a, Blank[]], Times[Optional[Pattern[b, Blank[]]], Pattern[x, Blank[]]], Times[Optional[Pattern[d, Blank[]]], Power[Pattern[x, Blank[]], 3]]], Optional[Pattern[p, Blank[]]]]], Pattern[x, Blank[Symbol]]] := Dist[1/(3^(3*p)*a^(2*p)), Int[(e + f*x)^m*(3*a - b*x)^p*(3*a + 2*b*x)^(2*p), x], x] /; FreeQ[{a, b, d, e, f, m}, x] && EqQ[4*b^3 + 27*a^2*d, 0] && IntegerQ[p]
  2077. Int[Times[Power[Plus[Optional[Pattern[e, Blank[]]], Times[Optional[Pattern[f, Blank[]]], Pattern[x, Blank[]]]], Optional[Pattern[m, Blank[]]]], Power[Plus[Pattern[a, Blank[]], Times[Optional[Pattern[b, Blank[]]], Pattern[x, Blank[]]], Times[Optional[Pattern[d, Blank[]]], Power[Pattern[x, Blank[]], 3]]], Pattern[p, Blank[]]]], Pattern[x, Blank[Symbol]]] := Dist[(a + b*x + d*x^3)^p/((3*a - b*x)^p*(3*a + 2*b*x)^(2*p)), Int[(e + f*x)^m*(3*a - b*x)^p*(3*a + 2*b*x)^(2*p), x], x] /; FreeQ[{a, b, d, e, f, m, p}, x] && EqQ[4*b^3 + 27*a^2*d, 0] && !IntegerQ[p]
  2078. Int[Times[Power[Plus[Optional[Pattern[e, Blank[]]], Times[Optional[Pattern[f, Blank[]]], Pattern[x, Blank[]]]], Optional[Pattern[m, Blank[]]]], Power[Plus[Pattern[a, Blank[]], Times[Optional[Pattern[b, Blank[]]], Pattern[x, Blank[]]], Times[Optional[Pattern[d, Blank[]]], Power[Pattern[x, Blank[]], 3]]], Optional[Pattern[p, Blank[]]]]], Pattern[x, Blank[Symbol]]] := Int[ExpandIntegrand[(e + f*x)^m*(a + b*x + d*x^3)^p, x], x] /; FreeQ[{a, b, d, e, f, m}, x] && NeQ[4*b^3 + 27*a^2*d, 0] && IGtQ[p, 0]
  2079. Int[Times[Power[Plus[Optional[Pattern[e, Blank[]]], Times[Optional[Pattern[f, Blank[]]], Pattern[x, Blank[]]]], Optional[Pattern[m, Blank[]]]], Power[Plus[Pattern[a, Blank[]], Times[Optional[Pattern[b, Blank[]]], Pattern[x, Blank[]]], Times[Optional[Pattern[d, Blank[]]], Power[Pattern[x, Blank[]], 3]]], Pattern[p, Blank[]]]], Pattern[x, Blank[Symbol]]] := With[{r = Rt[-9*a*d^2 + Sqrt[3]*d*Sqrt[4*b^3*d + 27*a^2*d^2], 3]}, Dist[1/d^(2*p), Int[(e + f*x)^m*Simp[(18^(1/3)*b*d)/(3*r) - r/18^(1/3) + d*x, x]^p*Simp[(b*d)/3 + (12^(1/3)*b^2*d^2)/(3*r^2) + r^2/(3*12^(1/3)) - d*((2^(1/3)*b*d)/(3^(1/3)*r) - r/18^(1/3))*x + d^2*x^2, x]^p, x], x]] /; FreeQ[{a, b, d, e, f, m}, x] && NeQ[4*b^3 + 27*a^2*d, 0] && ILtQ[p, 0]
  2080. Int[Times[Power[Plus[Optional[Pattern[e, Blank[]]], Times[Optional[Pattern[f, Blank[]]], Pattern[x, Blank[]]]], Optional[Pattern[m, Blank[]]]], Power[Plus[Pattern[a, Blank[]], Times[Optional[Pattern[b, Blank[]]], Pattern[x, Blank[]]], Times[Optional[Pattern[d, Blank[]]], Power[Pattern[x, Blank[]], 3]]], Pattern[p, Blank[]]]], Pattern[x, Blank[Symbol]]] := With[{r = Rt[-9*a*d^2 + Sqrt[3]*d*Sqrt[4*b^3*d + 27*a^2*d^2], 3]}, Dist[(a + b*x + d*x^3)^p/(Simp[(18^(1/3)*b*d)/(3*r) - r/18^(1/3) + d*x, x]^p*Simp[(b*d)/3 + (12^(1/3)*b^2*d^2)/(3*r^2) + r^2/(3*12^(1/3)) - d*((2^(1/3)*b*d)/(3^(1/3)*r) - r/18^(1/3))*x + d^2*x^2, x]^p), Int[(e + f*x)^m*Simp[(18^(1/3)*b*d)/(3*r) - r/18^(1/3) + d*x, x]^p*Simp[(b*d)/3 + (12^(1/3)*b^2*d^2)/(3*r^2) + r^2/(3*12^(1/3)) - d*((2^(1/3)*b*d)/(3^(1/3)*r) - r/18^(1/3))*x + d^2*x^2, x]^p, x], x]] /; FreeQ[{a, b, d, e, f, m, p}, x] && NeQ[4*b^3 + 27*a^2*d, 0] && !IntegerQ[p]
  2081. Int[Times[Power[Pattern[P3, Blank[]], Optional[Pattern[p, Blank[]]]], Power[Plus[Optional[Pattern[e, Blank[]]], Times[Optional[Pattern[f, Blank[]]], Pattern[x, Blank[]]]], Optional[Pattern[m, Blank[]]]]], Pattern[x, Blank[Symbol]]] := With[{a = Coeff[P3, x, 0], b = Coeff[P3, x, 1], c = Coeff[P3, x, 2], d = Coeff[P3, x, 3]}, Subst[Int[((3*d*e - c*f)/(3*d) + f*x)^m*Simp[(2*c^3 - 9*b*c*d + 27*a*d^2)/(27*d^2) - ((c^2 - 3*b*d)*x)/(3*d) + d*x^3, x]^p, x], x, x + c/(3*d)] /; NeQ[c, 0]] /; FreeQ[{e, f, m, p}, x] && PolyQ[P3, x, 3]
  2082. Int[Times[Pattern[x, Blank[]], Power[Plus[Pattern[a, Blank[]], Times[Optional[Pattern[b, Blank[]]], Pattern[x, Blank[]]], Times[Optional[Pattern[c, Blank[]]], Power[Pattern[x, Blank[]], 2]], Times[Optional[Pattern[e, Blank[]]], Power[Pattern[x, Blank[]], 4]]], Rational[-1, 2]]], Pattern[x, Blank[Symbol]]] := With[{Px = (1*(33*b^2*c + 6*a*c^2 + 40*a^2*e))/320 - (22*a*c*e*x^2)/5 + (22*b*c*e*x^3)/15 + (1*e*(5*c^2 + 4*a*e)*x^4)/4 + (4*b*e^2*x^5)/3 + 2*c*e^2*x^6 + e^3*x^8}, Simp[(1*Log[Px + Dist[1/(8*Rt[e, 2]*x), D[Px, x], x]*Sqrt[a + b*x + c*x^2 + e*x^4]])/(8*Rt[e, 2]), x]] /; FreeQ[{a, b, c, e}, x] && EqQ[71*c^2 + 100*a*e, 0] && EqQ[1152*c^3 - 125*b^2*e, 0]
  2083. Int[Times[Plus[Pattern[A, Blank[]], Times[Optional[Pattern[B, Blank[]]], Pattern[x, Blank[]]]], Power[Plus[Pattern[a, Blank[]], Times[Optional[Pattern[b, Blank[]]], Pattern[x, Blank[]]], Times[Optional[Pattern[c, Blank[]]], Power[Pattern[x, Blank[]], 2]], Times[Optional[Pattern[d, Blank[]]], Power[Pattern[x, Blank[]], 3]], Times[Optional[Pattern[e, Blank[]]], Power[Pattern[x, Blank[]], 4]]], Rational[-1, 2]]], Pattern[x, Blank[Symbol]]] := Dist[B, Subst[Int[x/Sqrt[(-3*d^4 + 16*c*d^2*e - 64*b*d*e^2 + 256*a*e^3)/(256*e^3) + ((d^3 - 4*c*d*e + 8*b*e^2)*x)/(8*e^2) - ((3*d^2 - 8*c*e)*x^2)/(8*e) + e*x^4], x], x, d/(4*e) + x], x] /; FreeQ[{a, b, c, d, e, A, B}, x] && EqQ[B*d - 4*A*e, 0] && EqQ[d*(141*d^3 - 752*c*d*e - 400*b*e^2) + 16*e^2*(71*c^2 + 100*a*e), 0] && EqQ[144*(3*d^2 - 8*c*e)^3 + 125*(d^3 - 4*c*d*e + 8*b*e^2)^2, 0]
  2084. Int[Times[Power[Plus[Pattern[d, Blank[]], Times[Optional[Pattern[e, Blank[]]], Pattern[x, Blank[]]], Times[Optional[Pattern[d, Blank[]]], Power[Pattern[x, Blank[]], 2]]], -1], Plus[Pattern[f, Blank[]], Times[Optional[Pattern[g, Blank[]]], Power[Pattern[x, Blank[]], 2]]], Power[Plus[Pattern[a, Blank[]], Times[Optional[Pattern[b, Blank[]]], Pattern[x, Blank[]]], Times[Optional[Pattern[c, Blank[]]], Power[Pattern[x, Blank[]], 2]], Times[Optional[Pattern[b, Blank[]]], Power[Pattern[x, Blank[]], 3]], Times[Optional[Pattern[a, Blank[]]], Power[Pattern[x, Blank[]], 4]]], Rational[-1, 2]]], Pattern[x, Blank[Symbol]]] := Simp[(a*f*ArcTan[(a*b + (4*a^2 + b^2 - 2*a*c)*x + a*b*x^2)/(2*Rt[a^2*(2*a - c), 2]*Sqrt[a + b*x + c*x^2 + b*x^3 + a*x^4])])/(d*Rt[a^2*(2*a - c), 2]), x] /; FreeQ[{a, b, c, d, e, f, g}, x] && EqQ[b*d - a*e, 0] && EqQ[f + g, 0] && PosQ[a^2*(2*a - c)]
  2085. Int[Times[Power[Plus[Pattern[d, Blank[]], Times[Optional[Pattern[e, Blank[]]], Pattern[x, Blank[]]], Times[Optional[Pattern[d, Blank[]]], Power[Pattern[x, Blank[]], 2]]], -1], Plus[Pattern[f, Blank[]], Times[Optional[Pattern[g, Blank[]]], Power[Pattern[x, Blank[]], 2]]], Power[Plus[Pattern[a, Blank[]], Times[Optional[Pattern[b, Blank[]]], Pattern[x, Blank[]]], Times[Optional[Pattern[c, Blank[]]], Power[Pattern[x, Blank[]], 2]], Times[Optional[Pattern[b, Blank[]]], Power[Pattern[x, Blank[]], 3]], Times[Optional[Pattern[a, Blank[]]], Power[Pattern[x, Blank[]], 4]]], Rational[-1, 2]]], Pattern[x, Blank[Symbol]]] := -Simp[(a*f*ArcTanh[(a*b + (4*a^2 + b^2 - 2*a*c)*x + a*b*x^2)/(2*Rt[-(a^2*(2*a - c)), 2]*Sqrt[a + b*x + c*x^2 + b*x^3 + a*x^4])])/(d*Rt[-(a^2*(2*a - c)), 2]), x] /; FreeQ[{a, b, c, d, e, f, g}, x] && EqQ[b*d - a*e, 0] && EqQ[f + g, 0] && NegQ[a^2*(2*a - c)]
  2086. Int[Times[Pattern[P3, Blank[]], Power[Plus[Pattern[a, Blank[]], Times[Optional[Pattern[b, Blank[]]], Pattern[x, Blank[]]], Times[Optional[Pattern[c, Blank[]]], Power[Pattern[x, Blank[]], 2]], Times[Optional[Pattern[d, Blank[]]], Power[Pattern[x, Blank[]], 3]], Times[Optional[Pattern[e, Blank[]]], Power[Pattern[x, Blank[]], 4]]], -1]], Pattern[x, Blank[Symbol]]] := With[{q = Sqrt[8*a^2 + b^2 - 4*a*c], A = Coeff[P3, x, 0], B = Coeff[P3, x, 1], C = Coeff[P3, x, 2], D = Coeff[P3, x, 3]}, Dist[1/q, Int[(b*A - 2*a*B + 2*a*D + A*q + (2*a*A - 2*a*C + b*D + D*q)*x)/(2*a + (b + q)*x + 2*a*x^2), x], x] - Dist[1/q, Int[(b*A - 2*a*B + 2*a*D - A*q + (2*a*A - 2*a*C + b*D - D*q)*x)/(2*a + (b - q)*x + 2*a*x^2), x], x]] /; FreeQ[{a, b, c}, x] && PolyQ[P3, x, 3] && EqQ[a, e] && EqQ[b, d]
  2087. Int[Times[Pattern[P3, Blank[]], Power[Pattern[x, Blank[]], Optional[Pattern[m, Blank[]]]], Power[Plus[Pattern[a, Blank[]], Times[Optional[Pattern[b, Blank[]]], Pattern[x, Blank[]]], Times[Optional[Pattern[c, Blank[]]], Power[Pattern[x, Blank[]], 2]], Times[Optional[Pattern[d, Blank[]]], Power[Pattern[x, Blank[]], 3]], Times[Optional[Pattern[e, Blank[]]], Power[Pattern[x, Blank[]], 4]]], -1]], Pattern[x, Blank[Symbol]]] := With[{q = Sqrt[8*a^2 + b^2 - 4*a*c], A = Coeff[P3, x, 0], B = Coeff[P3, x, 1], C = Coeff[P3, x, 2], D = Coeff[P3, x, 3]}, Dist[1/q, Int[(x^m*(b*A - 2*a*B + 2*a*D + A*q + (2*a*A - 2*a*C + b*D + D*q)*x))/(2*a + (b + q)*x + 2*a*x^2), x], x] - Dist[1/q, Int[(x^m*(b*A - 2*a*B + 2*a*D - A*q + (2*a*A - 2*a*C + b*D - D*q)*x))/(2*a + (b - q)*x + 2*a*x^2), x], x]] /; FreeQ[{a, b, c, m}, x] && PolyQ[P3, x, 3] && EqQ[a, e] && EqQ[b, d]
  2088. Int[Times[Plus[Optional[Pattern[A, Blank[]]], Times[Optional[Pattern[B, Blank[]]], Pattern[x, Blank[]]], Times[Optional[Pattern[C, Blank[]]], Power[Pattern[x, Blank[]], 2]]], Power[Plus[Pattern[a, Blank[]], Times[Optional[Pattern[b, Blank[]]], Pattern[x, Blank[]]], Times[Optional[Pattern[c, Blank[]]], Power[Pattern[x, Blank[]], 2]], Times[Optional[Pattern[d, Blank[]]], Power[Pattern[x, Blank[]], 3]], Times[Optional[Pattern[e, Blank[]]], Power[Pattern[x, Blank[]], 4]]], -1]], Pattern[x, Blank[Symbol]]] := With[{q = Rt[C*(2*e*(B*d - 4*A*e) + C*(d^2 - 4*c*e)), 2]}, Simp[(-2*C^2*ArcTanh[(C*d - B*e + 2*C*e*x)/q])/q, x] + Simp[(2*C^2*ArcTanh[(C*(4*B*c*C - 3*B^2*d - 4*A*C*d + 12*A*B*e + 4*C*(2*c*C - B*d + 2*A*e)*x + 4*C*(2*C*d - B*e)*x^2 + 8*C^2*e*x^3))/(q*(B^2 - 4*A*C))])/q, x]] /; FreeQ[{a, b, c, d, e, A, B, C}, x] && EqQ[B^2*d + 2*C*(b*C + A*d) - 2*B*(c*C + 2*A*e), 0] && EqQ[2*B^2*c*C - 8*a*C^3 - B^3*d - 4*A*B*C*d + 4*A*(B^2 + 2*A*C)*e, 0] && PosQ[C*(2*e*(B*d - 4*A*e) + C*(d^2 - 4*c*e))]
  2089. Int[Times[Plus[Optional[Pattern[A, Blank[]]], Times[Optional[Pattern[C, Blank[]]], Power[Pattern[x, Blank[]], 2]]], Power[Plus[Pattern[a, Blank[]], Times[Optional[Pattern[b, Blank[]]], Pattern[x, Blank[]]], Times[Optional[Pattern[c, Blank[]]], Power[Pattern[x, Blank[]], 2]], Times[Optional[Pattern[d, Blank[]]], Power[Pattern[x, Blank[]], 3]], Times[Optional[Pattern[e, Blank[]]], Power[Pattern[x, Blank[]], 4]]], -1]], Pattern[x, Blank[Symbol]]] := With[{q = Rt[C*(-8*A*e^2 + C*(d^2 - 4*c*e)), 2]}, Simp[(-2*C^2*ArcTanh[(C*(d + 2*e*x))/q])/q, x] + Simp[(2*C^2*ArcTanh[(C*(A*d - 2*(c*C + A*e)*x - 2*C*d*x^2 - 2*C*e*x^3))/(A*q)])/q, x]] /; FreeQ[{a, b, c, d, e, A, C}, x] && EqQ[b*C + A*d, 0] && EqQ[a*C^2 - A^2*e, 0] && PosQ[C*(-8*A*e^2 + C*(d^2 - 4*c*e))]
  2090. Int[Times[Plus[Optional[Pattern[A, Blank[]]], Times[Optional[Pattern[B, Blank[]]], Pattern[x, Blank[]]], Times[Optional[Pattern[C, Blank[]]], Power[Pattern[x, Blank[]], 2]]], Power[Plus[Pattern[a, Blank[]], Times[Optional[Pattern[b, Blank[]]], Pattern[x, Blank[]]], Times[Optional[Pattern[c, Blank[]]], Power[Pattern[x, Blank[]], 2]], Times[Optional[Pattern[d, Blank[]]], Power[Pattern[x, Blank[]], 3]], Times[Optional[Pattern[e, Blank[]]], Power[Pattern[x, Blank[]], 4]]], -1]], Pattern[x, Blank[Symbol]]] := With[{q = Rt[-(C*(2*e*(B*d - 4*A*e) + C*(d^2 - 4*c*e))), 2]}, Simp[(2*C^2*ArcTan[(C*d - B*e + 2*C*e*x)/q])/q, x] - Simp[(2*C^2*ArcTan[(C*(4*B*c*C - 3*B^2*d - 4*A*C*d + 12*A*B*e + 4*C*(2*c*C - B*d + 2*A*e)*x + 4*C*(2*C*d - B*e)*x^2 + 8*C^2*e*x^3))/(q*(B^2 - 4*A*C))])/q, x]] /; FreeQ[{a, b, c, d, e, A, B, C}, x] && EqQ[B^2*d + 2*C*(b*C + A*d) - 2*B*(c*C + 2*A*e), 0] && EqQ[2*B^2*c*C - 8*a*C^3 - B^3*d - 4*A*B*C*d + 4*A*(B^2 + 2*A*C)*e, 0] && NegQ[C*(2*e*(B*d - 4*A*e) + C*(d^2 - 4*c*e))]
  2091. Int[Times[Plus[Optional[Pattern[A, Blank[]]], Times[Optional[Pattern[C, Blank[]]], Power[Pattern[x, Blank[]], 2]]], Power[Plus[Pattern[a, Blank[]], Times[Optional[Pattern[b, Blank[]]], Pattern[x, Blank[]]], Times[Optional[Pattern[c, Blank[]]], Power[Pattern[x, Blank[]], 2]], Times[Optional[Pattern[d, Blank[]]], Power[Pattern[x, Blank[]], 3]], Times[Optional[Pattern[e, Blank[]]], Power[Pattern[x, Blank[]], 4]]], -1]], Pattern[x, Blank[Symbol]]] := With[{q = Rt[-(C*(-8*A*e^2 + C*(d^2 - 4*c*e))), 2]}, Simp[(2*C^2*ArcTan[(C*d + 2*C*e*x)/q])/q, x] - Simp[(2*C^2*ArcTan[-((C*(-(A*d) + 2*(c*C + A*e)*x + 2*C*d*x^2 + 2*C*e*x^3))/(A*q))])/q, x]] /; FreeQ[{a, b, c, d, e, A, C}, x] && EqQ[b*C + A*d, 0] && EqQ[a*C^2 - A^2*e, 0] && NegQ[C*(-8*A*e^2 + C*(d^2 - 4*c*e))]
  2092. Int[Times[Power[Pattern[P4, Blank[]], Pattern[p, Blank[]]], Pattern[Px, Blank[]]], Pattern[x, Blank[Symbol]]] := With[{a = Coeff[P4, x, 0], b = Coeff[P4, x, 1], c = Coeff[P4, x, 2], d = Coeff[P4, x, 3], e = Coeff[P4, x, 4]}, Dist[1/a^(3*p), Int[ExpandIntegrand[Px/((a - b*x)^p/(a^5 - b^5*x^5)^p), x], x], x] /; NeQ[a, 0] && EqQ[c, b^2/a] && EqQ[d, b^3/a^2] && EqQ[e, b^4/a^3]] /; FreeQ[p, x] && PolyQ[P4, x, 4] && PolyQ[Px, x] && ILtQ[p, 0]
  2093. Int[Times[Plus[Pattern[A, Blank[]], Times[Optional[Pattern[B, Blank[]]], Power[Pattern[x, Blank[]], Pattern[n, Blank[]]]]], Power[Plus[Pattern[a, Blank[]], Times[Optional[Pattern[b, Blank[]]], Power[Pattern[x, Blank[]], 2]], Times[Optional[Pattern[c, Blank[]]], Power[Pattern[x, Blank[]], Pattern[n, Blank[]]]], Times[Optional[Pattern[d, Blank[]]], Power[Pattern[x, Blank[]], Pattern[n2, Blank[]]]]], -1]], Pattern[x, Blank[Symbol]]] := Dist[A^2*(n - 1), Subst[Int[1/(a + A^2*b*(n - 1)^2*x^2), x], x, x/(A*(n - 1) - B*x^n)], x] /; FreeQ[{a, b, c, d, A, B, n}, x] && EqQ[n2, 2*n] && NeQ[n, 2] && EqQ[a*B^2 - A^2*d*(n - 1)^2, 0] && EqQ[B*c + 2*A*d*(n - 1), 0]
  2094. Int[Times[Power[Pattern[x, Blank[]], Optional[Pattern[m, Blank[]]]], Plus[Pattern[A, Blank[]], Times[Optional[Pattern[B, Blank[]]], Power[Pattern[x, Blank[]], Optional[Pattern[n, Blank[]]]]]], Power[Plus[Pattern[a, Blank[]], Times[Optional[Pattern[b, Blank[]]], Power[Pattern[x, Blank[]], Optional[Pattern[k, Blank[]]]]], Times[Optional[Pattern[c, Blank[]]], Power[Pattern[x, Blank[]], Optional[Pattern[n, Blank[]]]]], Times[Optional[Pattern[d, Blank[]]], Power[Pattern[x, Blank[]], Pattern[n2, Blank[]]]]], -1]], Pattern[x, Blank[Symbol]]] := Dist[(A^2*(m - n + 1))/(m + 1), Subst[Int[1/(a + A^2*b*(m - n + 1)^2*x^2), x], x, x^(m + 1)/(A*(m - n + 1) + B*(m + 1)*x^n)], x] /; FreeQ[{a, b, c, d, A, B, m, n}, x] && EqQ[n2, 2*n] && EqQ[k, 2*(m + 1)] && EqQ[a*B^2*(m + 1)^2 - A^2*d*(m - n + 1)^2, 0] && EqQ[B*c*(m + 1) - 2*A*d*(m - n + 1), 0]
  2095. Int[Times[Plus[Pattern[a, Blank[]], Times[Optional[Pattern[b, Blank[]]], Power[Pattern[x, Blank[]], 2]], Times[Optional[Pattern[c, Blank[]]], Power[Pattern[x, Blank[]], 4]]], Power[Plus[Pattern[d, Blank[]], Times[Optional[Pattern[e, Blank[]]], Power[Pattern[x, Blank[]], 2]], Times[Optional[Pattern[f, Blank[]]], Power[Pattern[x, Blank[]], 4]], Times[Optional[Pattern[g, Blank[]]], Power[Pattern[x, Blank[]], 6]]], -1]], Pattern[x, Blank[Symbol]]] := With[{q = Rt[(-(a*c*f^2) + 12*a^2*g^2 + f*(3*c^2*d - 2*a*b*g))/(c*g*(3*c*d - a*f)), 2], r = Rt[(a*c*f^2 + 4*g*(b*c*d + a^2*g) - f*(3*c^2*d + 2*a*b*g))/(c*g*(3*c*d - a*f)), 2]}, Simp[(c*ArcTan[(r + 2*x)/q])/(g*q), x] + (-Simp[(c*ArcTan[(r - 2*x)/q])/(g*q), x] - Simp[(c*ArcTan[((3*c*d - a*f)*x*(b*c^2*d*f - a*b^2*f*g - 2*a^2*c*f*g + 6*a^2*b*g^2 + c*(3*c^2*d*f - a*c*f^2 - b*c*d*g + 2*a^2*g^2)*x^2 + c^2*g*(3*c*d - a*f)*x^4))/(g*q*(b*c*d - 2*a^2*g)*(b*c*d - a*b*f + 4*a^2*g))])/(g*q), x])] /; FreeQ[{a, b, c, d, e, f, g}, x] && EqQ[9*c^3*d^2 - c*(b^2 + 6*a*c)*d*f + a^2*c*f^2 + 2*a*b*(3*c*d + a*f)*g - 12*a^3*g^2, 0] && EqQ[3*c^4*d^2*e - 3*a^2*c^2*d*f*g + a^3*c*f^2*g + 2*a^3*g^2*(b*f - 6*a*g) - c^3*d*(2*b*d*f + a*e*f - 12*a*d*g), 0] && NeQ[3*c*d - a*f, 0] && NeQ[b*c*d - 2*a^2*g, 0] && NeQ[b*c*d - a*b*f + 4*a^2*g, 0] && PosQ[(-(a*c*f^2) + 12*a^2*g^2 + f*(3*c^2*d - 2*a*b*g))/(c*g*(3*c*d - a*f))]
  2096. Int[Times[Plus[Pattern[a, Blank[]], Times[Optional[Pattern[c, Blank[]]], Power[Pattern[x, Blank[]], 4]]], Power[Plus[Pattern[d, Blank[]], Times[Optional[Pattern[e, Blank[]]], Power[Pattern[x, Blank[]], 2]], Times[Optional[Pattern[f, Blank[]]], Power[Pattern[x, Blank[]], 4]], Times[Optional[Pattern[g, Blank[]]], Power[Pattern[x, Blank[]], 6]]], -1]], Pattern[x, Blank[Symbol]]] := With[{q = Rt[(-(a*c*f^2) + 12*a^2*g^2 + 3*f*c^2*d)/(c*g*(3*c*d - a*f)), 2], r = Rt[(a*c*f^2 + 4*a^2*g^2 - 3*c^2*d*f)/(c*g*(3*c*d - a*f)), 2]}, Simp[(c*ArcTan[(r + 2*x)/q])/(g*q), x] + (-Simp[(c*ArcTan[(r - 2*x)/q])/(g*q), x] - Simp[(c*ArcTan[(c*(3*c*d - a*f)*x*(2*a^2*f*g - (3*c^2*d*f - a*c*f^2 + 2*a^2*g^2)*x^2 - c*(3*c*d - a*f)*g*x^4))/(8*a^4*g^3*q)])/(g*q), x])] /; FreeQ[{a, c, d, e, f, g}, x] && EqQ[9*c^3*d^2 - 6*a*c^2*d*f + a^2*c*f^2 - 12*a^3*g^2, 0] && EqQ[3*c^4*d^2*e - 3*a^2*c^2*d*f*g + a^3*c*f^2*g - 12*a^4*g^3 - a*c^3*d*(e*f - 12*d*g), 0] && NeQ[3*c*d - a*f, 0] && PosQ[(-(a*c*f^2) + 12*a^2*g^2 + 3*c^2*d*f)/(c*g*(3*c*d - a*f))]
  2097. Int[Times[Power[Pattern[Q6, Blank[]], Pattern[p, Blank[]]], Pattern[u, Blank[]]], Pattern[x, Blank[Symbol]]] := With[{a = Coeff[Q6, x, 0], b = Coeff[Q6, x, 2], c = Coeff[Q6, x, 3], d = Coeff[Q6, x, 4], e = Coeff[Q6, x, 6]}, Dist[1/(3^(3*p)*a^(2*p)), Int[ExpandIntegrand[u*(3*a + 3*Rt[a, 3]^2*Rt[c, 3]*x + b*x^2)^p*(3*a - 3*(-1)^(1/3)*Rt[a, 3]^2*Rt[c, 3]*x + b*x^2)^p*(3*a + 3*(-1)^(2/3)*Rt[a, 3]^2*Rt[c, 3]*x + b*x^2)^p, x], x], x] /; EqQ[b^2 - 3*a*d, 0] && EqQ[b^3 - 27*a^2*e, 0]] /; ILtQ[p, 0] && PolyQ[Q6, x, 6] && EqQ[Coeff[Q6, x, 1], 0] && EqQ[Coeff[Q6, x, 5], 0] && RationalFunctionQ[u, x]
  2098. Int[Times[Pattern[Pm, Blank[]], Power[Pattern[Qn, Blank[]], -1]], Pattern[x, Blank[Symbol]]] := With[{m = Expon[Pm, x], n = Expon[Qn, x]}, Simp[(Coeff[Pm, x, m]*Log[Qn])/(n*Coeff[Qn, x, n]), x] + Dist[Simplify[Pm - (Coeff[Pm, x, m]*D[Qn, x])/(n*Coeff[Qn, x, n])], Int[1/Qn, x], x] /; EqQ[m, n - 1] && EqQ[D[Simplify[Pm - (Coeff[Pm, x, m]*D[Qn, x])/(n*Coeff[Qn, x, n])], x], 0]] /; PolyQ[Pm, x] && PolyQ[Qn, x]
  2099. Int[Times[Pattern[Pm, Blank[]], Power[Pattern[Qn, Blank[]], Pattern[p, Blank[]]]], Pattern[x, Blank[Symbol]]] := With[{m = Expon[Pm, x], n = Expon[Qn, x]}, Simp[(Coeff[Pm, x, m]*Qn^(p + 1))/(n*(p + 1)*Coeff[Qn, x, n]), x] + Dist[Simplify[Pm - (Coeff[Pm, x, m]*D[Qn, x])/(n*Coeff[Qn, x, n])], Int[Qn^p, x], x] /; EqQ[m, n - 1] && EqQ[D[Simplify[Pm - (Coeff[Pm, x, m]*D[Qn, x])/(n*Coeff[Qn, x, n])], x], 0]] /; FreeQ[p, x] && PolyQ[Pm, x] && PolyQ[Qn, x] && NeQ[p, -1]
  2100. Int[Times[Pattern[Pm, Blank[]], Power[Pattern[Qn, Blank[]], -1]], Pattern[x, Blank[Symbol]]] := With[{m = Expon[Pm, x], n = Expon[Qn, x]}, Simp[(Coeff[Pm, x, m]*Log[Qn])/(n*Coeff[Qn, x, n]), x] + Dist[1/(n*Coeff[Qn, x, n]), Int[ExpandToSum[n*Coeff[Qn, x, n]*Pm - Coeff[Pm, x, m]*D[Qn, x], x]/Qn, x], x] /; EqQ[m, n - 1]] /; PolyQ[Pm, x] && PolyQ[Qn, x]
  2101. Int[Times[Pattern[Pm, Blank[]], Power[Pattern[Qn, Blank[]], Pattern[p, Blank[]]]], Pattern[x, Blank[Symbol]]] := With[{m = Expon[Pm, x], n = Expon[Qn, x]}, Simp[(Coeff[Pm, x, m]*Qn^(p + 1))/(n*(p + 1)*Coeff[Qn, x, n]), x] + Dist[1/(n*Coeff[Qn, x, n]), Int[ExpandToSum[n*Coeff[Qn, x, n]*Pm - Coeff[Pm, x, m]*D[Qn, x], x]*Qn^p, x], x] /; EqQ[m, n - 1]] /; FreeQ[p, x] && PolyQ[Pm, x] && PolyQ[Qn, x] && NeQ[p, -1]
  2102. Int[Times[Pattern[Pm, Blank[]], Power[Pattern[Qn, Blank[]], Optional[Pattern[p, Blank[]]]]], Pattern[x, Blank[Symbol]]] := With[{m = Expon[Pm, x], n = Expon[Qn, x]}, Simp[(Coeff[Pm, x, m]*x^(m - n + 1)*Qn^(p + 1))/((m + n*p + 1)*Coeff[Qn, x, n]), x] + Dist[1/((m + n*p + 1)*Coeff[Qn, x, n]), Int[ExpandToSum[(m + n*p + 1)*Coeff[Qn, x, n]*Pm - Coeff[Pm, x, m]*x^(m - n)*((m - n + 1)*Qn + (p + 1)*x*D[Qn, x]), x]*Qn^p, x], x] /; LtQ[1, n, m + 1] && m + n*p + 1 < 0] /; FreeQ[p, x] && PolyQ[Pm, x] && PolyQ[Qn, x] && LtQ[p, -1]
  2103. Int[Times[Pattern[u, Blank[]], Power[Plus[Times[Optional[Pattern[e, Blank[]]], Power[Plus[Optional[Pattern[a, Blank[]]], Times[Optional[Pattern[b, Blank[]]], Pattern[x, Blank[]]]], Rational[1, 2]]], Times[Optional[Pattern[f, Blank[]]], Power[Plus[Optional[Pattern[c, Blank[]]], Times[Optional[Pattern[d, Blank[]]], Pattern[x, Blank[]]]], Rational[1, 2]]]], -1]], Pattern[x, Blank[Symbol]]] := Dist[c/(e*(b*c - a*d)), Int[(u*Sqrt[a + b*x])/x, x], x] - Dist[a/(f*(b*c - a*d)), Int[(u*Sqrt[c + d*x])/x, x], x] /; FreeQ[{a, b, c, d, e, f}, x] && NeQ[b*c - a*d, 0] && EqQ[a*e^2 - c*f^2, 0]
  2104. Int[Times[Pattern[u, Blank[]], Power[Plus[Times[Optional[Pattern[e, Blank[]]], Power[Plus[Optional[Pattern[a, Blank[]]], Times[Optional[Pattern[b, Blank[]]], Pattern[x, Blank[]]]], Rational[1, 2]]], Times[Optional[Pattern[f, Blank[]]], Power[Plus[Optional[Pattern[c, Blank[]]], Times[Optional[Pattern[d, Blank[]]], Pattern[x, Blank[]]]], Rational[1, 2]]]], -1]], Pattern[x, Blank[Symbol]]] := -Dist[d/(e*(b*c - a*d)), Int[u*Sqrt[a + b*x], x], x] + Dist[b/(f*(b*c - a*d)), Int[u*Sqrt[c + d*x], x], x] /; FreeQ[{a, b, c, d, e, f}, x] && NeQ[b*c - a*d, 0] && EqQ[b*e^2 - d*f^2, 0]
  2105. Int[Times[Pattern[u, Blank[]], Power[Plus[Times[Optional[Pattern[e, Blank[]]], Power[Plus[Optional[Pattern[a, Blank[]]], Times[Optional[Pattern[b, Blank[]]], Pattern[x, Blank[]]]], Rational[1, 2]]], Times[Optional[Pattern[f, Blank[]]], Power[Plus[Optional[Pattern[c, Blank[]]], Times[Optional[Pattern[d, Blank[]]], Pattern[x, Blank[]]]], Rational[1, 2]]]], -1]], Pattern[x, Blank[Symbol]]] := Dist[e, Int[(u*Sqrt[a + b*x])/(a*e^2 - c*f^2 + (b*e^2 - d*f^2)*x), x], x] - Dist[f, Int[(u*Sqrt[c + d*x])/(a*e^2 - c*f^2 + (b*e^2 - d*f^2)*x), x], x] /; FreeQ[{a, b, c, d, e, f}, x] && NeQ[a*e^2 - c*f^2, 0] && NeQ[b*e^2 - d*f^2, 0]
  2106. Int[Times[Optional[Pattern[u, Blank[]]], Power[Plus[Times[Optional[Pattern[d, Blank[]]], Power[Pattern[x, Blank[]], Optional[Pattern[n, Blank[]]]]], Times[Optional[Pattern[c, Blank[]]], Power[Plus[Optional[Pattern[a, Blank[]]], Times[Optional[Pattern[b, Blank[]]], Power[Pattern[x, Blank[]], Optional[Pattern[p, Blank[]]]]]], Rational[1, 2]]]], -1]], Pattern[x, Blank[Symbol]]] := -Dist[b/(a*d), Int[u*x^n, x], x] + Dist[1/(a*c), Int[u*Sqrt[a + b*x^(2*n)], x], x] /; FreeQ[{a, b, c, d, n}, x] && EqQ[p, 2*n] && EqQ[b*c^2 - d^2, 0]
  2107. Int[Times[Power[Pattern[x, Blank[]], Optional[Pattern[m, Blank[]]]], Power[Plus[Times[Optional[Pattern[d, Blank[]]], Power[Pattern[x, Blank[]], Optional[Pattern[n, Blank[]]]]], Times[Optional[Pattern[c, Blank[]]], Power[Plus[Optional[Pattern[a, Blank[]]], Times[Optional[Pattern[b, Blank[]]], Power[Pattern[x, Blank[]], Optional[Pattern[p, Blank[]]]]]], Rational[1, 2]]]], -1]], Pattern[x, Blank[Symbol]]] := -Dist[d, Int[x^(m + n)/(a*c^2 + (b*c^2 - d^2)*x^(2*n)), x], x] + Dist[c, Int[(x^m*Sqrt[a + b*x^(2*n)])/(a*c^2 + (b*c^2 - d^2)*x^(2*n)), x], x] /; FreeQ[{a, b, c, d, m, n}, x] && EqQ[p, 2*n] && NeQ[b*c^2 - d^2, 0]
  2108. Int[Times[Power[Plus[Optional[Pattern[d, Blank[]]], Times[Optional[Pattern[e, Blank[]]], Pattern[x, Blank[]]], Times[Optional[Pattern[f, Blank[]]], Power[Pattern[x, Blank[]], 2]]], Rational[-1, 2]], Power[Plus[Pattern[a, Blank[]], Times[Optional[Pattern[b, Blank[]]], Power[Pattern[x, Blank[]], 3]]], -1]], Pattern[x, Blank[Symbol]]] := With[{r = Numerator[Rt[a/b, 3]], s = Denominator[Rt[a/b, 3]]}, Dist[r/(3*a), Int[1/((r + s*x)*Sqrt[d + e*x + f*x^2]), x], x] + Dist[r/(3*a), Int[(2*r - s*x)/((r^2 - r*s*x + s^2*x^2)*Sqrt[d + e*x + f*x^2]), x], x]] /; FreeQ[{a, b, d, e, f}, x] && PosQ[a/b]
  2109. Int[Times[Power[Plus[Optional[Pattern[d, Blank[]]], Times[Optional[Pattern[f, Blank[]]], Power[Pattern[x, Blank[]], 2]]], Rational[-1, 2]], Power[Plus[Pattern[a, Blank[]], Times[Optional[Pattern[b, Blank[]]], Power[Pattern[x, Blank[]], 3]]], -1]], Pattern[x, Blank[Symbol]]] := With[{r = Numerator[Rt[a/b, 3]], s = Denominator[Rt[a/b, 3]]}, Dist[r/(3*a), Int[1/((r + s*x)*Sqrt[d + f*x^2]), x], x] + Dist[r/(3*a), Int[(2*r - s*x)/((r^2 - r*s*x + s^2*x^2)*Sqrt[d + f*x^2]), x], x]] /; FreeQ[{a, b, d, f}, x] && PosQ[a/b]
  2110. Int[Times[Power[Plus[Optional[Pattern[d, Blank[]]], Times[Optional[Pattern[e, Blank[]]], Pattern[x, Blank[]]], Times[Optional[Pattern[f, Blank[]]], Power[Pattern[x, Blank[]], 2]]], Rational[-1, 2]], Power[Plus[Pattern[a, Blank[]], Times[Optional[Pattern[b, Blank[]]], Power[Pattern[x, Blank[]], 3]]], -1]], Pattern[x, Blank[Symbol]]] := With[{r = Numerator[Rt[-(a/b), 3]], s = Denominator[Rt[-(a/b), 3]]}, Dist[r/(3*a), Int[1/((r - s*x)*Sqrt[d + e*x + f*x^2]), x], x] + Dist[r/(3*a), Int[(2*r + s*x)/((r^2 + r*s*x + s^2*x^2)*Sqrt[d + e*x + f*x^2]), x], x]] /; FreeQ[{a, b, d, e, f}, x] && NegQ[a/b]
  2111. Int[Times[Power[Plus[Optional[Pattern[d, Blank[]]], Times[Optional[Pattern[f, Blank[]]], Power[Pattern[x, Blank[]], 2]]], Rational[-1, 2]], Power[Plus[Pattern[a, Blank[]], Times[Optional[Pattern[b, Blank[]]], Power[Pattern[x, Blank[]], 3]]], -1]], Pattern[x, Blank[Symbol]]] := With[{r = Numerator[Rt[-(a/b), 3]], s = Denominator[Rt[-(a/b), 3]]}, Dist[r/(3*a), Int[1/((r - s*x)*Sqrt[d + f*x^2]), x], x] + Dist[r/(3*a), Int[(2*r + s*x)/((r^2 + r*s*x + s^2*x^2)*Sqrt[d + f*x^2]), x], x]] /; FreeQ[{a, b, d, f}, x] && NegQ[a/b]
  2112. Int[Times[Pattern[u, Blank[]], Power[Pattern[v, Blank[]], Rational[-1, 2]], Plus[Pattern[A, Blank[]], Times[Optional[Pattern[B, Blank[]]], Power[Pattern[x, Blank[]], 4]]]], Pattern[x, Blank[Symbol]]] := With[{a = Coeff[v, x, 0], b = Coeff[v, x, 2], c = Coeff[v, x, 4], d = Coeff[1/u, x, 0], e = Coeff[1/u, x, 2], f = Coeff[1/u, x, 4]}, Dist[A, Subst[Int[1/(d - (b*d - a*e)*x^2), x], x, x/Sqrt[v]], x] /; EqQ[a*B + A*c, 0] && EqQ[c*d - a*f, 0]] /; FreeQ[{A, B}, x] && PolyQ[v, x^2, 2] && PolyQ[1/u, x^2, 2]
  2113. Int[Times[Power[Plus[Pattern[a, Blank[]], Times[Optional[Pattern[b, Blank[]]], Pattern[x, Blank[]]]], -1], Power[Plus[Pattern[c, Blank[]], Times[Optional[Pattern[d, Blank[]]], Power[Pattern[x, Blank[]], 2]]], Rational[-1, 2]], Power[Plus[Pattern[e, Blank[]], Times[Optional[Pattern[f, Blank[]]], Power[Pattern[x, Blank[]], 2]]], Rational[-1, 2]]], Pattern[x, Blank[Symbol]]] := Dist[a, Int[1/((a^2 - b^2*x^2)*Sqrt[c + d*x^2]*Sqrt[e + f*x^2]), x], x] - Dist[b, Int[x/((a^2 - b^2*x^2)*Sqrt[c + d*x^2]*Sqrt[e + f*x^2]), x], x] /; FreeQ[{a, b, c, d, e, f}, x]
  2114. Int[Times[Plus[Optional[Pattern[g, Blank[]]], Times[Optional[Pattern[h, Blank[]]], Pattern[x, Blank[]]]], Power[Plus[Optional[Pattern[d, Blank[]]], Times[Optional[Pattern[e, Blank[]]], Pattern[x, Blank[]]], Times[Optional[Pattern[f, Blank[]]], Power[Plus[Optional[Pattern[a, Blank[]]], Times[Optional[Pattern[b, Blank[]]], Pattern[x, Blank[]]], Times[Optional[Pattern[c, Blank[]]], Power[Pattern[x, Blank[]], 2]]], Rational[1, 2]]]], Rational[1, 2]]], Pattern[x, Blank[Symbol]]] := Simp[(2*(f*(5*b*c*g^2 - 2*b^2*g*h - 3*a*c*g*h + 2*a*b*h^2) + c*f*(10*c*g^2 - b*g*h + a*h^2)*x + 9*c^2*f*g*h*x^2 + 3*c^2*f*h^2*x^3 - (e*g - d*h)*(5*c*g - 2*b*h + c*h*x)*Sqrt[a + b*x + c*x^2])*Sqrt[d + e*x + f*Sqrt[a + b*x + c*x^2]])/(15*c^2*f*(g + h*x)), x] /; FreeQ[{a, b, c, d, e, f, g, h}, x] && EqQ[(e*g - d*h)^2 - f^2*(c*g^2 - b*g*h + a*h^2), 0] && EqQ[2*e^2*g - 2*d*e*h - f^2*(2*c*g - b*h), 0]
  2115. Int[Times[Power[Plus[Pattern[u, Blank[]], Times[Optional[Pattern[f, Blank[]]], Plus[Optional[Pattern[j, Blank[]]], Times[Optional[Pattern[k, Blank[]]], Power[Pattern[v, Blank[]], Rational[1, 2]]]]]], Optional[Pattern[n, Blank[]]]], Power[Plus[Optional[Pattern[g, Blank[]]], Times[Optional[Pattern[h, Blank[]]], Pattern[x, Blank[]]]], Optional[Pattern[m, Blank[]]]]], Pattern[x, Blank[Symbol]]] := Int[(g + h*x)^m*(ExpandToSum[u + f*j, x] + f*k*Sqrt[ExpandToSum[v, x]])^n, x] /; FreeQ[{f, g, h, j, k, m, n}, x] && LinearQ[u, x] && QuadraticQ[v, x] && !(LinearMatchQ[u, x] && QuadraticMatchQ[v, x] && (EqQ[j, 0] || EqQ[f, 1])) && EqQ[(Coefficient[u, x, 1]*g - h*(Coefficient[u, x, 0] + f*j))^2 - f^2*k^2*(Coefficient[v, x, 2]*g^2 - Coefficient[v, x, 1]*g*h + Coefficient[v, x, 0]*h^2), 0]
  2116. Int[Power[Plus[Optional[Pattern[g, Blank[]]], Times[Optional[Pattern[h, Blank[]]], Power[Plus[Optional[Pattern[d, Blank[]]], Times[Optional[Pattern[e, Blank[]]], Pattern[x, Blank[]]], Times[Optional[Pattern[f, Blank[]]], Power[Plus[Optional[Pattern[a, Blank[]]], Times[Optional[Pattern[b, Blank[]]], Pattern[x, Blank[]]], Times[Optional[Pattern[c, Blank[]]], Power[Pattern[x, Blank[]], 2]]], Rational[1, 2]]]], Pattern[n, Blank[]]]]], Optional[Pattern[p, Blank[]]]], Pattern[x, Blank[Symbol]]] := Dist[2, Subst[Int[((g + h*x^n)^p*(d^2*e - (b*d - a*e)*f^2 - (2*d*e - b*f^2)*x + e*x^2))/(-2*d*e + b*f^2 + 2*e*x)^2, x], x, d + e*x + f*Sqrt[a + b*x + c*x^2]], x] /; FreeQ[{a, b, c, d, e, f, g, h, n}, x] && EqQ[e^2 - c*f^2, 0] && IntegerQ[p]
  2117. Int[Power[Plus[Optional[Pattern[g, Blank[]]], Times[Optional[Pattern[h, Blank[]]], Power[Plus[Optional[Pattern[d, Blank[]]], Times[Optional[Pattern[e, Blank[]]], Pattern[x, Blank[]]], Times[Optional[Pattern[f, Blank[]]], Power[Plus[Pattern[a, Blank[]], Times[Optional[Pattern[c, Blank[]]], Power[Pattern[x, Blank[]], 2]]], Rational[1, 2]]]], Pattern[n, Blank[]]]]], Optional[Pattern[p, Blank[]]]], Pattern[x, Blank[Symbol]]] := Dist[1/(2*e), Subst[Int[((g + h*x^n)^p*(d^2 + a*f^2 - 2*d*x + x^2))/(d - x)^2, x], x, d + e*x + f*Sqrt[a + c*x^2]], x] /; FreeQ[{a, c, d, e, f, g, h, n}, x] && EqQ[e^2 - c*f^2, 0] && IntegerQ[p]
  2118. Int[Power[Plus[Optional[Pattern[g, Blank[]]], Times[Optional[Pattern[h, Blank[]]], Power[Plus[Pattern[u, Blank[]], Times[Optional[Pattern[f, Blank[]]], Power[Pattern[v, Blank[]], Rational[1, 2]]]], Pattern[n, Blank[]]]]], Optional[Pattern[p, Blank[]]]], Pattern[x, Blank[Symbol]]] := Int[(g + h*(ExpandToSum[u, x] + f*Sqrt[ExpandToSum[v, x]])^n)^p, x] /; FreeQ[{f, g, h, n}, x] && LinearQ[u, x] && QuadraticQ[v, x] && !(LinearMatchQ[u, x] && QuadraticMatchQ[v, x]) && EqQ[Coefficient[u, x, 1]^2 - Coefficient[v, x, 2]*f^2, 0] && IntegerQ[p]
  2119. Int[Times[Power[Plus[Optional[Pattern[g, Blank[]]], Times[Optional[Pattern[h, Blank[]]], Pattern[x, Blank[]]]], Optional[Pattern[m, Blank[]]]], Power[Plus[Times[Optional[Pattern[e, Blank[]]], Pattern[x, Blank[]]], Times[Optional[Pattern[f, Blank[]]], Power[Plus[Optional[Pattern[a, Blank[]]], Times[Optional[Pattern[c, Blank[]]], Power[Pattern[x, Blank[]], 2]]], Rational[1, 2]]]], Optional[Pattern[n, Blank[]]]]], Pattern[x, Blank[Symbol]]] := Dist[1/(2^(m + 1)*e^(m + 1)), Subst[Int[x^(n - m - 2)*(a*f^2 + x^2)*(-(a*f^2*h) + 2*e*g*x + h*x^2)^m, x], x, e*x + f*Sqrt[a + c*x^2]], x] /; FreeQ[{a, c, e, f, g, h, n}, x] && EqQ[e^2 - c*f^2, 0] && IntegerQ[m]
  2120. Int[Times[Power[Pattern[x, Blank[]], Optional[Pattern[p, Blank[]]]], Power[Plus[Pattern[g, Blank[]], Times[Optional[Pattern[i, Blank[]]], Power[Pattern[x, Blank[]], 2]]], Optional[Pattern[m, Blank[]]]], Power[Plus[Times[Optional[Pattern[e, Blank[]]], Pattern[x, Blank[]]], Times[Optional[Pattern[f, Blank[]]], Power[Plus[Pattern[a, Blank[]], Times[Optional[Pattern[c, Blank[]]], Power[Pattern[x, Blank[]], 2]]], Rational[1, 2]]]], Optional[Pattern[n, Blank[]]]]], Pattern[x, Blank[Symbol]]] := Dist[(1*(i/c)^m)/(2^(2*m + p + 1)*e^(p + 1)*f^(2*m)), Subst[Int[x^(n - 2*m - p - 2)*(-(a*f^2) + x^2)^p*(a*f^2 + x^2)^(2*m + 1), x], x, e*x + f*Sqrt[a + c*x^2]], x] /; FreeQ[{a, c, e, f, g, i, n}, x] && EqQ[e^2 - c*f^2, 0] && EqQ[c*g - a*i, 0] && IntegersQ[p, 2*m] && (IntegerQ[m] || GtQ[i/c, 0])
  2121. Int[Times[Power[Plus[Optional[Pattern[g, Blank[]]], Times[Optional[Pattern[h, Blank[]]], Pattern[x, Blank[]]], Times[Optional[Pattern[i, Blank[]]], Power[Pattern[x, Blank[]], 2]]], Optional[Pattern[m, Blank[]]]], Power[Plus[Optional[Pattern[d, Blank[]]], Times[Optional[Pattern[e, Blank[]]], Pattern[x, Blank[]]], Times[Optional[Pattern[f, Blank[]]], Power[Plus[Optional[Pattern[a, Blank[]]], Times[Optional[Pattern[b, Blank[]]], Pattern[x, Blank[]]], Times[Optional[Pattern[c, Blank[]]], Power[Pattern[x, Blank[]], 2]]], Rational[1, 2]]]], Optional[Pattern[n, Blank[]]]]], Pattern[x, Blank[Symbol]]] := Dist[(2*(i/c)^m)/f^(2*m), Subst[Int[(x^n*(d^2*e - (b*d - a*e)*f^2 - (2*d*e - b*f^2)*x + e*x^2)^(2*m + 1))/(-2*d*e + b*f^2 + 2*e*x)^(2*(m + 1)), x], x, d + e*x + f*Sqrt[a + b*x + c*x^2]], x] /; FreeQ[{a, b, c, d, e, f, g, h, i, n}, x] && EqQ[e^2 - c*f^2, 0] && EqQ[c*g - a*i, 0] && EqQ[c*h - b*i, 0] && IntegerQ[2*m] && (IntegerQ[m] || GtQ[i/c, 0])
  2122. Int[Times[Power[Plus[Pattern[g, Blank[]], Times[Optional[Pattern[i, Blank[]]], Power[Pattern[x, Blank[]], 2]]], Optional[Pattern[m, Blank[]]]], Power[Plus[Optional[Pattern[d, Blank[]]], Times[Optional[Pattern[e, Blank[]]], Pattern[x, Blank[]]], Times[Optional[Pattern[f, Blank[]]], Power[Plus[Pattern[a, Blank[]], Times[Optional[Pattern[c, Blank[]]], Power[Pattern[x, Blank[]], 2]]], Rational[1, 2]]]], Optional[Pattern[n, Blank[]]]]], Pattern[x, Blank[Symbol]]] := Dist[(1*(i/c)^m)/(2^(2*m + 1)*e*f^(2*m)), Subst[Int[(x^n*(d^2 + a*f^2 - 2*d*x + x^2)^(2*m + 1))/(-d + x)^(2*(m + 1)), x], x, d + e*x + f*Sqrt[a + c*x^2]], x] /; FreeQ[{a, c, d, e, f, g, i, n}, x] && EqQ[e^2 - c*f^2, 0] && EqQ[c*g - a*i, 0] && IntegerQ[2*m] && (IntegerQ[m] || GtQ[i/c, 0])
  2123. Int[Times[Power[Plus[Optional[Pattern[g, Blank[]]], Times[Optional[Pattern[h, Blank[]]], Pattern[x, Blank[]]], Times[Optional[Pattern[i, Blank[]]], Power[Pattern[x, Blank[]], 2]]], Optional[Pattern[m, Blank[]]]], Power[Plus[Optional[Pattern[d, Blank[]]], Times[Optional[Pattern[e, Blank[]]], Pattern[x, Blank[]]], Times[Optional[Pattern[f, Blank[]]], Power[Plus[Optional[Pattern[a, Blank[]]], Times[Optional[Pattern[b, Blank[]]], Pattern[x, Blank[]]], Times[Optional[Pattern[c, Blank[]]], Power[Pattern[x, Blank[]], 2]]], Rational[1, 2]]]], Optional[Pattern[n, Blank[]]]]], Pattern[x, Blank[Symbol]]] := Dist[((i/c)^(m - 1/2)*Sqrt[g + h*x + i*x^2])/Sqrt[a + b*x + c*x^2], Int[(a + b*x + c*x^2)^m*(d + e*x + f*Sqrt[a + b*x + c*x^2])^n, x], x] /; FreeQ[{a, b, c, d, e, f, g, h, i, n}, x] && EqQ[e^2 - c*f^2, 0] && EqQ[c*g - a*i, 0] && EqQ[c*h - b*i, 0] && IGtQ[m + 1/2, 0] && !GtQ[i/c, 0]
  2124. Int[Times[Power[Plus[Pattern[g, Blank[]], Times[Optional[Pattern[i, Blank[]]], Power[Pattern[x, Blank[]], 2]]], Optional[Pattern[m, Blank[]]]], Power[Plus[Optional[Pattern[d, Blank[]]], Times[Optional[Pattern[e, Blank[]]], Pattern[x, Blank[]]], Times[Optional[Pattern[f, Blank[]]], Power[Plus[Pattern[a, Blank[]], Times[Optional[Pattern[c, Blank[]]], Power[Pattern[x, Blank[]], 2]]], Rational[1, 2]]]], Optional[Pattern[n, Blank[]]]]], Pattern[x, Blank[Symbol]]] := Dist[((i/c)^(m - 1/2)*Sqrt[g + i*x^2])/Sqrt[a + c*x^2], Int[(a + c*x^2)^m*(d + e*x + f*Sqrt[a + c*x^2])^n, x], x] /; FreeQ[{a, c, d, e, f, g, i, n}, x] && EqQ[e^2 - c*f^2, 0] && EqQ[c*g - a*i, 0] && IGtQ[m + 1/2, 0] && !GtQ[i/c, 0]
  2125. Int[Times[Power[Plus[Optional[Pattern[g, Blank[]]], Times[Optional[Pattern[h, Blank[]]], Pattern[x, Blank[]]], Times[Optional[Pattern[i, Blank[]]], Power[Pattern[x, Blank[]], 2]]], Optional[Pattern[m, Blank[]]]], Power[Plus[Optional[Pattern[d, Blank[]]], Times[Optional[Pattern[e, Blank[]]], Pattern[x, Blank[]]], Times[Optional[Pattern[f, Blank[]]], Power[Plus[Optional[Pattern[a, Blank[]]], Times[Optional[Pattern[b, Blank[]]], Pattern[x, Blank[]]], Times[Optional[Pattern[c, Blank[]]], Power[Pattern[x, Blank[]], 2]]], Rational[1, 2]]]], Optional[Pattern[n, Blank[]]]]], Pattern[x, Blank[Symbol]]] := Dist[((i/c)^(m + 1/2)*Sqrt[a + b*x + c*x^2])/Sqrt[g + h*x + i*x^2], Int[(a + b*x + c*x^2)^m*(d + e*x + f*Sqrt[a + b*x + c*x^2])^n, x], x] /; FreeQ[{a, b, c, d, e, f, g, h, i, n}, x] && EqQ[e^2 - c*f^2, 0] && EqQ[c*g - a*i, 0] && EqQ[c*h - b*i, 0] && ILtQ[m - 1/2, 0] && !GtQ[i/c, 0]
  2126. Int[Times[Power[Plus[Pattern[g, Blank[]], Times[Optional[Pattern[i, Blank[]]], Power[Pattern[x, Blank[]], 2]]], Optional[Pattern[m, Blank[]]]], Power[Plus[Optional[Pattern[d, Blank[]]], Times[Optional[Pattern[e, Blank[]]], Pattern[x, Blank[]]], Times[Optional[Pattern[f, Blank[]]], Power[Plus[Pattern[a, Blank[]], Times[Optional[Pattern[c, Blank[]]], Power[Pattern[x, Blank[]], 2]]], Rational[1, 2]]]], Optional[Pattern[n, Blank[]]]]], Pattern[x, Blank[Symbol]]] := Dist[((i/c)^(m + 1/2)*Sqrt[a + c*x^2])/Sqrt[g + i*x^2], Int[(a + c*x^2)^m*(d + e*x + f*Sqrt[a + c*x^2])^n, x], x] /; FreeQ[{a, c, d, e, f, g, i, n}, x] && EqQ[e^2 - c*f^2, 0] && EqQ[c*g - a*i, 0] && ILtQ[m - 1/2, 0] && !GtQ[i/c, 0]
  2127. Int[Times[Power[Plus[Pattern[u, Blank[]], Times[Optional[Pattern[f, Blank[]]], Plus[Optional[Pattern[j, Blank[]]], Times[Optional[Pattern[k, Blank[]]], Power[Pattern[v, Blank[]], Rational[1, 2]]]]]], Optional[Pattern[n, Blank[]]]], Power[Pattern[w, Blank[]], Optional[Pattern[m, Blank[]]]]], Pattern[x, Blank[Symbol]]] := Int[ExpandToSum[w, x]^m*(ExpandToSum[u + f*j, x] + f*k*Sqrt[ExpandToSum[v, x]])^n, x] /; FreeQ[{f, j, k, m, n}, x] && LinearQ[u, x] && QuadraticQ[{v, w}, x] && !(LinearMatchQ[u, x] && QuadraticMatchQ[{v, w}, x] && (EqQ[j, 0] || EqQ[f, 1])) && EqQ[Coefficient[u, x, 1]^2 - Coefficient[v, x, 2]*f^2*k^2, 0]
  2128. Int[Times[Power[Plus[Pattern[a, Blank[]], Times[Optional[Pattern[b, Blank[]]], Power[Pattern[x, Blank[]], Optional[Pattern[n, Blank[]]]]]], -1], Power[Plus[Times[Optional[Pattern[c, Blank[]]], Power[Pattern[x, Blank[]], 2]], Times[Optional[Pattern[d, Blank[]]], Power[Plus[Pattern[a, Blank[]], Times[Optional[Pattern[b, Blank[]]], Power[Pattern[x, Blank[]], Optional[Pattern[n, Blank[]]]]]], Optional[Pattern[p, Blank[]]]]]], Rational[-1, 2]]], Pattern[x, Blank[Symbol]]] := Dist[1/a, Subst[Int[1/(1 - c*x^2), x], x, x/Sqrt[c*x^2 + d*(a + b*x^n)^(2/n)]], x] /; FreeQ[{a, b, c, d, n}, x] && EqQ[p, 2/n]
  2129. Int[Power[Plus[Pattern[a, Blank[]], Times[Optional[Pattern[b, Blank[]]], Power[Plus[Pattern[c, Blank[]], Times[Optional[Pattern[d, Blank[]]], Power[Pattern[x, Blank[]], 2]]], Rational[1, 2]]]], Rational[1, 2]], Pattern[x, Blank[Symbol]]] := Simp[(2*b^2*d*x^3)/(3*(a + b*Sqrt[c + d*x^2])^(3/2)), x] + Simp[(2*a*x)/Sqrt[a + b*Sqrt[c + d*x^2]], x] /; FreeQ[{a, b, c, d}, x] && EqQ[a^2 - b^2*c, 0]
  2130. Int[Times[Power[Pattern[x, Blank[]], -1], Power[Plus[Pattern[c, Blank[]], Times[Optional[Pattern[d, Blank[]]], Power[Pattern[x, Blank[]], 2]]], Rational[-1, 2]], Power[Plus[Times[Optional[Pattern[a, Blank[]]], Power[Pattern[x, Blank[]], 2]], Times[Optional[Pattern[b, Blank[]]], Pattern[x, Blank[]], Power[Plus[Pattern[c, Blank[]], Times[Optional[Pattern[d, Blank[]]], Power[Pattern[x, Blank[]], 2]]], Rational[1, 2]]]], Rational[1, 2]]], Pattern[x, Blank[Symbol]]] := Dist[(Sqrt[2]*b)/a, Subst[Int[1/Sqrt[1 + x^2/a], x], x, a*x + b*Sqrt[c + d*x^2]], x] /; FreeQ[{a, b, c, d}, x] && EqQ[a^2 - b^2*d, 0] && EqQ[b^2*c + a, 0]
  2131. Int[Times[Power[Pattern[x, Blank[]], -1], Power[Plus[Pattern[c, Blank[]], Times[Optional[Pattern[d, Blank[]]], Power[Pattern[x, Blank[]], 2]]], Rational[-1, 2]], Power[Times[Optional[Pattern[e, Blank[]]], Pattern[x, Blank[]], Plus[Times[Optional[Pattern[a, Blank[]]], Pattern[x, Blank[]]], Times[Optional[Pattern[b, Blank[]]], Power[Plus[Pattern[c, Blank[]], Times[Optional[Pattern[d, Blank[]]], Power[Pattern[x, Blank[]], 2]]], Rational[1, 2]]]]], Rational[1, 2]]], Pattern[x, Blank[Symbol]]] := Int[Sqrt[a*e*x^2 + b*e*x*Sqrt[c + d*x^2]]/(x*Sqrt[c + d*x^2]), x] /; FreeQ[{a, b, c, d, e}, x] && EqQ[a^2 - b^2*d, 0] && EqQ[b^2*c*e + a, 0]
  2132. Int[Times[Power[Plus[Pattern[a, Blank[]], Times[Optional[Pattern[b, Blank[]]], Power[Pattern[x, Blank[]], 4]]], Rational[-1, 2]], Power[Plus[Times[Optional[Pattern[c, Blank[]]], Power[Pattern[x, Blank[]], 2]], Times[Optional[Pattern[d, Blank[]]], Power[Plus[Pattern[a, Blank[]], Times[Optional[Pattern[b, Blank[]]], Power[Pattern[x, Blank[]], 4]]], Rational[1, 2]]]], Rational[1, 2]]], Pattern[x, Blank[Symbol]]] := Dist[d, Subst[Int[1/(1 - 2*c*x^2), x], x, x/Sqrt[c*x^2 + d*Sqrt[a + b*x^4]]], x] /; FreeQ[{a, b, c, d}, x] && EqQ[c^2 - b*d^2, 0]
  2133. Int[Times[Power[Plus[Optional[Pattern[c, Blank[]]], Times[Optional[Pattern[d, Blank[]]], Pattern[x, Blank[]]]], Optional[Pattern[m, Blank[]]]], Power[Plus[Pattern[a, Blank[]], Times[Optional[Pattern[e, Blank[]]], Power[Pattern[x, Blank[]], 4]]], Rational[-1, 2]], Power[Plus[Times[Optional[Pattern[b, Blank[]]], Power[Pattern[x, Blank[]], 2]], Power[Plus[Pattern[a, Blank[]], Times[Optional[Pattern[e, Blank[]]], Power[Pattern[x, Blank[]], 4]]], Rational[1, 2]]], Rational[1, 2]]], Pattern[x, Blank[Symbol]]] := Dist[(1 - I)/2, Int[(c + d*x)^m/Sqrt[Sqrt[a] - I*b*x^2], x], x] + Dist[(1 + I)/2, Int[(c + d*x)^m/Sqrt[Sqrt[a] + I*b*x^2], x], x] /; FreeQ[{a, b, c, d, m}, x] && EqQ[e, b^2] && GtQ[a, 0]
  2134. Int[Times[Power[Plus[Pattern[c, Blank[]], Times[Optional[Pattern[d, Blank[]]], Pattern[x, Blank[]]]], -1], Power[Plus[Pattern[a, Blank[]], Times[Optional[Pattern[b, Blank[]]], Power[Pattern[x, Blank[]], 3]]], Rational[-1, 2]]], Pattern[x, Blank[Symbol]]] := Dist[2/(3*c), Int[1/Sqrt[a + b*x^3], x], x] + Dist[1/(3*c), Int[(c - 2*d*x)/((c + d*x)*Sqrt[a + b*x^3]), x], x] /; FreeQ[{a, b, c, d}, x] && EqQ[b*c^3 - 4*a*d^3, 0]
  2135. Int[Times[Power[Plus[Pattern[c, Blank[]], Times[Optional[Pattern[d, Blank[]]], Pattern[x, Blank[]]]], -1], Power[Plus[Pattern[a, Blank[]], Times[Optional[Pattern[b, Blank[]]], Power[Pattern[x, Blank[]], 3]]], Rational[-1, 2]]], Pattern[x, Blank[Symbol]]] := Dist[(-6*a*d^3)/(c*(b*c^3 - 28*a*d^3)), Int[1/Sqrt[a + b*x^3], x], x] + Dist[1/(c*(b*c^3 - 28*a*d^3)), Int[Simp[c*(b*c^3 - 22*a*d^3) + 6*a*d^4*x, x]/((c + d*x)*Sqrt[a + b*x^3]), x], x] /; FreeQ[{a, b, c, d}, x] && EqQ[b^2*c^6 - 20*a*b*c^3*d^3 - 8*a^2*d^6, 0]
  2136. Int[Times[Power[Plus[Pattern[c, Blank[]], Times[Optional[Pattern[d, Blank[]]], Pattern[x, Blank[]]]], -1], Power[Plus[Pattern[a, Blank[]], Times[Optional[Pattern[b, Blank[]]], Power[Pattern[x, Blank[]], 3]]], Rational[-1, 2]]], Pattern[x, Blank[Symbol]]] := With[{q = Rt[b/a, 3]}, -Dist[q/((1 + Sqrt[3])*d - c*q), Int[1/Sqrt[a + b*x^3], x], x] + Dist[d/((1 + Sqrt[3])*d - c*q), Int[(1 + Sqrt[3] + q*x)/((c + d*x)*Sqrt[a + b*x^3]), x], x]] /; FreeQ[{a, b, c, d}, x] && NeQ[b^2*c^6 - 20*a*b*c^3*d^3 - 8*a^2*d^6, 0]
  2137. Int[Times[Power[Plus[Pattern[c, Blank[]], Times[Optional[Pattern[d, Blank[]]], Pattern[x, Blank[]]]], -1], Plus[Pattern[e, Blank[]], Times[Optional[Pattern[f, Blank[]]], Pattern[x, Blank[]]]], Power[Plus[Pattern[a, Blank[]], Times[Optional[Pattern[b, Blank[]]], Power[Pattern[x, Blank[]], 3]]], Rational[-1, 2]]], Pattern[x, Blank[Symbol]]] := Dist[(2*e)/d, Subst[Int[1/(1 + 3*a*x^2), x], x, (1 + (2*d*x)/c)/Sqrt[a + b*x^3]], x] /; FreeQ[{a, b, c, d, e, f}, x] && NeQ[d*e - c*f, 0] && EqQ[b*c^3 - 4*a*d^3, 0] && EqQ[2*d*e + c*f, 0]
  2138. Int[Times[Power[Plus[Pattern[c, Blank[]], Times[Optional[Pattern[d, Blank[]]], Pattern[x, Blank[]]]], -1], Plus[Pattern[e, Blank[]], Times[Optional[Pattern[f, Blank[]]], Pattern[x, Blank[]]]], Power[Plus[Pattern[a, Blank[]], Times[Optional[Pattern[b, Blank[]]], Power[Pattern[x, Blank[]], 3]]], Rational[-1, 2]]], Pattern[x, Blank[Symbol]]] := Dist[(-2*e)/d, Subst[Int[1/(9 - a*x^2), x], x, (1 + (f*x)/e)^2/Sqrt[a + b*x^3]], x] /; FreeQ[{a, b, c, d, e, f}, x] && NeQ[d*e - c*f, 0] && EqQ[b*c^3 + 8*a*d^3, 0] && EqQ[2*d*e + c*f, 0]
  2139. Int[Times[Power[Plus[Pattern[c, Blank[]], Times[Optional[Pattern[d, Blank[]]], Pattern[x, Blank[]]]], -1], Plus[Optional[Pattern[e, Blank[]]], Times[Optional[Pattern[f, Blank[]]], Pattern[x, Blank[]]]], Power[Plus[Pattern[a, Blank[]], Times[Optional[Pattern[b, Blank[]]], Power[Pattern[x, Blank[]], 3]]], Rational[-1, 2]]], Pattern[x, Blank[Symbol]]] := Dist[(2*d*e + c*f)/(3*c*d), Int[1/Sqrt[a + b*x^3], x], x] + Dist[(d*e - c*f)/(3*c*d), Int[(c - 2*d*x)/((c + d*x)*Sqrt[a + b*x^3]), x], x] /; FreeQ[{a, b, c, d, e, f}, x] && NeQ[d*e - c*f, 0] && (EqQ[b*c^3 - 4*a*d^3, 0] || EqQ[b*c^3 + 8*a*d^3, 0]) && NeQ[2*d*e + c*f, 0]
  2140. Int[Times[Power[Plus[Pattern[c, Blank[]], Times[Optional[Pattern[d, Blank[]]], Pattern[x, Blank[]]]], -1], Plus[Pattern[e, Blank[]], Times[Optional[Pattern[f, Blank[]]], Pattern[x, Blank[]]]], Power[Plus[Pattern[a, Blank[]], Times[Optional[Pattern[b, Blank[]]], Power[Pattern[x, Blank[]], 3]]], Rational[-1, 2]]], Pattern[x, Blank[Symbol]]] := With[{k = Simplify[(d*e + 2*c*f)/(c*f)]}, Dist[((1 + k)*e)/d, Subst[Int[1/(1 + (3 + 2*k)*a*x^2), x], x, (1 + ((1 + k)*d*x)/c)/Sqrt[a + b*x^3]], x]] /; FreeQ[{a, b, c, d, e, f}, x] && NeQ[d*e - c*f, 0] && EqQ[b^2*c^6 - 20*a*b*c^3*d^3 - 8*a^2*d^6, 0] && EqQ[6*a*d^4*e - c*f*(b*c^3 - 22*a*d^3), 0]
  2141. Int[Times[Power[Plus[Pattern[c, Blank[]], Times[Optional[Pattern[d, Blank[]]], Pattern[x, Blank[]]]], -1], Plus[Optional[Pattern[e, Blank[]]], Times[Optional[Pattern[f, Blank[]]], Pattern[x, Blank[]]]], Power[Plus[Pattern[a, Blank[]], Times[Optional[Pattern[b, Blank[]]], Power[Pattern[x, Blank[]], 3]]], Rational[-1, 2]]], Pattern[x, Blank[Symbol]]] := -Dist[(6*a*d^4*e - c*f*(b*c^3 - 22*a*d^3))/(c*d*(b*c^3 - 28*a*d^3)), Int[1/Sqrt[a + b*x^3], x], x] + Dist[(d*e - c*f)/(c*d*(b*c^3 - 28*a*d^3)), Int[(c*(b*c^3 - 22*a*d^3) + 6*a*d^4*x)/((c + d*x)*Sqrt[a + b*x^3]), x], x] /; FreeQ[{a, b, c, d, e, f}, x] && NeQ[d*e - c*f, 0] && EqQ[b^2*c^6 - 20*a*b*c^3*d^3 - 8*a^2*d^6, 0] && NeQ[6*a*d^4*e - c*f*(b*c^3 - 22*a*d^3), 0]
  2142. Int[Times[Power[Plus[Pattern[c, Blank[]], Times[Optional[Pattern[d, Blank[]]], Pattern[x, Blank[]]]], -1], Plus[Pattern[e, Blank[]], Times[Optional[Pattern[f, Blank[]]], Pattern[x, Blank[]]]], Power[Plus[Pattern[a, Blank[]], Times[Optional[Pattern[b, Blank[]]], Power[Pattern[x, Blank[]], 3]]], Rational[-1, 2]]], Pattern[x, Blank[Symbol]]] := With[{q = Simplify[((1 + Sqrt[3])*f)/e]}, Dist[(4*3^(1/4)*Sqrt[2 - Sqrt[3]]*f*(1 + q*x)*Sqrt[(1 - q*x + q^2*x^2)/(1 + Sqrt[3] + q*x)^2])/(q*Sqrt[a + b*x^3]*Sqrt[(1 + q*x)/(1 + Sqrt[3] + q*x)^2]), Subst[Int[1/(((1 - Sqrt[3])*d - c*q + ((1 + Sqrt[3])*d - c*q)*x)*Sqrt[1 - x^2]*Sqrt[7 - 4*Sqrt[3] + x^2]), x], x, (-1 + Sqrt[3] - q*x)/(1 + Sqrt[3] + q*x)], x]] /; FreeQ[{a, b, c, d, e, f}, x] && NeQ[d*e - c*f, 0] && EqQ[b*e^3 - 2*(5 + 3*Sqrt[3])*a*f^3, 0] && NeQ[b*c^3 - 2*(5 - 3*Sqrt[3])*a*d^3, 0]
  2143. Int[Times[Power[Plus[Pattern[c, Blank[]], Times[Optional[Pattern[d, Blank[]]], Pattern[x, Blank[]]]], -1], Plus[Pattern[e, Blank[]], Times[Optional[Pattern[f, Blank[]]], Pattern[x, Blank[]]]], Power[Plus[Pattern[a, Blank[]], Times[Optional[Pattern[b, Blank[]]], Power[Pattern[x, Blank[]], 3]]], Rational[-1, 2]]], Pattern[x, Blank[Symbol]]] := With[{q = Simplify[((-1 + Sqrt[3])*f)/e]}, Dist[(4*3^(1/4)*Sqrt[2 + Sqrt[3]]*f*(1 - q*x)*Sqrt[(1 + q*x + q^2*x^2)/(1 - Sqrt[3] - q*x)^2])/(q*Sqrt[a + b*x^3]*Sqrt[-((1 - q*x)/(1 - Sqrt[3] - q*x)^2)]), Subst[Int[1/(((1 + Sqrt[3])*d + c*q + ((1 - Sqrt[3])*d + c*q)*x)*Sqrt[1 - x^2]*Sqrt[7 + 4*Sqrt[3] + x^2]), x], x, (1 + Sqrt[3] - q*x)/(-1 + Sqrt[3] + q*x)], x]] /; FreeQ[{a, b, c, d, e, f}, x] && NeQ[d*e - c*f, 0] && EqQ[b*e^3 - 2*(5 - 3*Sqrt[3])*a*f^3, 0] && NeQ[b*c^3 - 2*(5 + 3*Sqrt[3])*a*d^3, 0]
  2144. Int[Times[Power[Plus[Pattern[c, Blank[]], Times[Optional[Pattern[d, Blank[]]], Pattern[x, Blank[]]]], -1], Plus[Optional[Pattern[e, Blank[]]], Times[Optional[Pattern[f, Blank[]]], Pattern[x, Blank[]]]], Power[Plus[Pattern[a, Blank[]], Times[Optional[Pattern[b, Blank[]]], Power[Pattern[x, Blank[]], 3]]], Rational[-1, 2]]], Pattern[x, Blank[Symbol]]] := With[{q = Rt[b/a, 3]}, Dist[((1 + Sqrt[3])*f - e*q)/((1 + Sqrt[3])*d - c*q), Int[1/Sqrt[a + b*x^3], x], x] + Dist[(d*e - c*f)/((1 + Sqrt[3])*d - c*q), Int[(1 + Sqrt[3] + q*x)/((c + d*x)*Sqrt[a + b*x^3]), x], x]] /; FreeQ[{a, b, c, d, e, f}, x] && NeQ[d*e - c*f, 0] && NeQ[b^2*c^6 - 20*a*b*c^3*d^3 - 8*a^2*d^6, 0] && NeQ[b^2*e^6 - 20*a*b*e^3*f^3 - 8*a^2*f^6, 0]
  2145. Int[Times[Power[Plus[Pattern[c, Blank[]], Times[Optional[Pattern[d, Blank[]]], Pattern[x, Blank[]]], Times[Optional[Pattern[e, Blank[]]], Power[Pattern[x, Blank[]], 2]]], -1], Plus[Pattern[f, Blank[]], Times[Optional[Pattern[g, Blank[]]], Pattern[x, Blank[]]], Times[Optional[Pattern[h, Blank[]]], Power[Pattern[x, Blank[]], 2]]], Power[Plus[Pattern[a, Blank[]], Times[Optional[Pattern[b, Blank[]]], Power[Pattern[x, Blank[]], 3]]], Rational[-1, 2]]], Pattern[x, Blank[Symbol]]] := Dist[-2*g*h, Subst[Int[1/(2*e*h - (b*d*f - 2*a*e*h)*x^2), x], x, (1 + (2*h*x)/g)/Sqrt[a + b*x^3]], x] /; FreeQ[{a, b, c, d, e, f, g, h}, x] && NeQ[b*d*f - 2*a*e*h, 0] && EqQ[b*g^3 - 8*a*h^3, 0] && EqQ[g^2 + 2*f*h, 0] && EqQ[b*d*f + b*c*g - 4*a*e*h, 0]
  2146. Int[Times[Power[Plus[Pattern[c, Blank[]], Times[Optional[Pattern[e, Blank[]]], Power[Pattern[x, Blank[]], 2]]], -1], Plus[Pattern[f, Blank[]], Times[Optional[Pattern[g, Blank[]]], Pattern[x, Blank[]]], Times[Optional[Pattern[h, Blank[]]], Power[Pattern[x, Blank[]], 2]]], Power[Plus[Pattern[a, Blank[]], Times[Optional[Pattern[b, Blank[]]], Power[Pattern[x, Blank[]], 3]]], Rational[-1, 2]]], Pattern[x, Blank[Symbol]]] := -Dist[g/e, Subst[Int[1/(1 + a*x^2), x], x, (1 + (2*h*x)/g)/Sqrt[a + b*x^3]], x] /; FreeQ[{a, b, c, e, f, g, h}, x] && EqQ[b*g^3 - 8*a*h^3, 0] && EqQ[g^2 + 2*f*h, 0] && EqQ[b*c*g - 4*a*e*h, 0]
  2147. Int[Times[Power[Plus[Pattern[c, Blank[]], Times[Optional[Pattern[d, Blank[]]], Pattern[x, Blank[]]]], -1], Power[Plus[Pattern[a, Blank[]], Times[Optional[Pattern[b, Blank[]]], Power[Pattern[x, Blank[]], 3]]], Rational[1, 2]]], Pattern[x, Blank[Symbol]]] := Dist[b/d, Int[x^2/Sqrt[a + b*x^3], x], x] + (Dist[(b*c)/d^3, Int[(c - d*x)/Sqrt[a + b*x^3], x], x] - Dist[(b*c^3 - a*d^3)/d^3, Int[1/((c + d*x)*Sqrt[a + b*x^3]), x], x]) /; FreeQ[{a, b, c, d}, x]
  2148. Int[Times[Power[Plus[Pattern[c, Blank[]], Times[Optional[Pattern[d, Blank[]]], Pattern[x, Blank[]]]], -1], Power[Plus[Pattern[a, Blank[]], Times[Optional[Pattern[b, Blank[]]], Power[Pattern[x, Blank[]], 3]]], Rational[-1, 3]]], Pattern[x, Blank[Symbol]]] := Simp[(Sqrt[3]*ArcTan[(1 - (2^(1/3)*Rt[b, 3]*(c - d*x))/(d*(a + b*x^3)^(1/3)))/Sqrt[3]])/(2^(4/3)*Rt[b, 3]*c), x] + (Simp[Log[(c + d*x)^2*(c - d*x)]/(2^(7/3)*Rt[b, 3]*c), x] - Simp[(3*Log[Rt[b, 3]*(c - d*x) + 2^(2/3)*d*(a + b*x^3)^(1/3)])/(2^(7/3)*Rt[b, 3]*c), x]) /; FreeQ[{a, b, c, d}, x] && EqQ[b*c^3 + a*d^3, 0]
  2149. Int[Times[Power[Plus[Pattern[c, Blank[]], Times[Optional[Pattern[d, Blank[]]], Pattern[x, Blank[]]]], -1], Power[Plus[Pattern[a, Blank[]], Times[Optional[Pattern[b, Blank[]]], Power[Pattern[x, Blank[]], 3]]], Rational[-1, 3]]], Pattern[x, Blank[Symbol]]] := Dist[1/(2*c), Int[1/(a + b*x^3)^(1/3), x], x] + Dist[1/(2*c), Int[(c - d*x)/((c + d*x)*(a + b*x^3)^(1/3)), x], x] /; FreeQ[{a, b, c, d}, x] && EqQ[2*b*c^3 - a*d^3, 0]
  2150. Int[Times[Power[Plus[Pattern[c, Blank[]], Times[Optional[Pattern[d, Blank[]]], Pattern[x, Blank[]]]], -1], Power[Plus[Pattern[a, Blank[]], Times[Optional[Pattern[b, Blank[]]], Power[Pattern[x, Blank[]], 3]]], Rational[-1, 3]]], Pattern[x, Blank[Symbol]]] := Unintegrable[1/((c + d*x)*(a + b*x^3)^(1/3)), x] /; FreeQ[{a, b, c, d}, x]
  2151. Int[Times[Power[Plus[Pattern[c, Blank[]], Times[Optional[Pattern[d, Blank[]]], Pattern[x, Blank[]]]], -1], Plus[Pattern[e, Blank[]], Times[Optional[Pattern[f, Blank[]]], Pattern[x, Blank[]]]], Power[Plus[Pattern[a, Blank[]], Times[Optional[Pattern[b, Blank[]]], Power[Pattern[x, Blank[]], 3]]], Rational[-1, 3]]], Pattern[x, Blank[Symbol]]] := Simp[(Sqrt[3]*f*ArcTan[(1 + (2*Rt[b, 3]*(2*c + d*x))/(d*(a + b*x^3)^(1/3)))/Sqrt[3]])/(Rt[b, 3]*d), x] + (Simp[(f*Log[c + d*x])/(Rt[b, 3]*d), x] - Simp[(3*f*Log[Rt[b, 3]*(2*c + d*x) - d*(a + b*x^3)^(1/3)])/(2*Rt[b, 3]*d), x]) /; FreeQ[{a, b, c, d, e, f}, x] && EqQ[d*e + c*f, 0] && EqQ[2*b*c^3 - a*d^3, 0]
  2152. Int[Times[Power[Plus[Optional[Pattern[c, Blank[]]], Times[Optional[Pattern[d, Blank[]]], Pattern[x, Blank[]]]], -1], Plus[Optional[Pattern[e, Blank[]]], Times[Optional[Pattern[f, Blank[]]], Pattern[x, Blank[]]]], Power[Plus[Pattern[a, Blank[]], Times[Optional[Pattern[b, Blank[]]], Power[Pattern[x, Blank[]], 3]]], Rational[-1, 3]]], Pattern[x, Blank[Symbol]]] := Dist[f/d, Int[1/(a + b*x^3)^(1/3), x], x] + Dist[(d*e - c*f)/d, Int[1/((c + d*x)*(a + b*x^3)^(1/3)), x], x] /; FreeQ[{a, b, c, d, e, f}, x]
  2153. Int[Times[Power[Plus[Pattern[c, Blank[]], Times[Optional[Pattern[d, Blank[]]], Power[Pattern[x, Blank[]], Optional[Pattern[n, Blank[]]]]]], Pattern[q, Blank[]]], Power[Plus[Pattern[a, Blank[]], Times[Optional[Pattern[b, Blank[]]], Power[Pattern[x, Blank[]], Optional[Pattern[nn, Blank[]]]]]], Pattern[p, Blank[]]]], Pattern[x, Blank[Symbol]]] := Int[ExpandIntegrand[(a + b*x^nn)^p, (c/(c^2 - d^2*x^(2*n)) - (d*x^n)/(c^2 - d^2*x^(2*n)))^(-q), x], x] /; FreeQ[{a, b, c, d, n, nn, p}, x] && !IntegerQ[p] && ILtQ[q, 0] && IGtQ[Log[2, nn/n], 0]
  2154. Int[Times[Power[Times[Optional[Pattern[e, Blank[]]], Pattern[x, Blank[]]], Optional[Pattern[m, Blank[]]]], Power[Plus[Pattern[c, Blank[]], Times[Optional[Pattern[d, Blank[]]], Power[Pattern[x, Blank[]], Optional[Pattern[n, Blank[]]]]]], Pattern[q, Blank[]]], Power[Plus[Pattern[a, Blank[]], Times[Optional[Pattern[b, Blank[]]], Power[Pattern[x, Blank[]], Optional[Pattern[nn, Blank[]]]]]], Pattern[p, Blank[]]]], Pattern[x, Blank[Symbol]]] := Dist[(e*x)^m/x^m, Int[ExpandIntegrand[x^m*(a + b*x^nn)^p, (c/(c^2 - d^2*x^(2*n)) - (d*x^n)/(c^2 - d^2*x^(2*n)))^(-q), x], x], x] /; FreeQ[{a, b, c, d, e, m, n, nn, p}, x] && !IntegerQ[p] && ILtQ[q, 0] && IGtQ[Log[2, nn/n], 0]
  2155. Int[Times[Power[Pattern[x, Blank[]], Optional[Pattern[m, Blank[]]]], Power[Plus[Pattern[c, Blank[]], Times[Optional[Pattern[d, Blank[]]], Power[Pattern[x, Blank[]], Pattern[n, Blank[]]]], Times[Optional[Pattern[e, Blank[]]], Power[Plus[Pattern[a, Blank[]], Times[Optional[Pattern[b, Blank[]]], Power[Pattern[x, Blank[]], Pattern[n, Blank[]]]]], Rational[1, 2]]]], -1]], Pattern[x, Blank[Symbol]]] := Dist[1/n, Subst[Int[x^((m + 1)/n - 1)/(c + d*x + e*Sqrt[a + b*x]), x], x, x^n], x] /; FreeQ[{a, b, c, d, e, m, n}, x] && EqQ[b*c - a*d, 0] && IntegerQ[(m + 1)/n]
  2156. Int[Times[Optional[Pattern[u, Blank[]]], Power[Plus[Pattern[c, Blank[]], Times[Optional[Pattern[d, Blank[]]], Power[Pattern[x, Blank[]], Pattern[n, Blank[]]]], Times[Optional[Pattern[e, Blank[]]], Power[Plus[Pattern[a, Blank[]], Times[Optional[Pattern[b, Blank[]]], Power[Pattern[x, Blank[]], Pattern[n, Blank[]]]]], Rational[1, 2]]]], -1]], Pattern[x, Blank[Symbol]]] := Dist[c, Int[u/(c^2 - a*e^2 + c*d*x^n), x], x] - Dist[a*e, Int[u/((c^2 - a*e^2 + c*d*x^n)*Sqrt[a + b*x^n]), x], x] /; FreeQ[{a, b, c, d, e, n}, x] && EqQ[b*c - a*d, 0]
  2157. Int[Power[Pattern[u, Blank[]], Optional[Pattern[m, Blank[]]]], Pattern[x, Blank[Symbol]]] := With[{c = Simplify[D[u, x]]}, Dist[1/c, Subst[Int[x^m, x], x, u], x]] /; FreeQ[m, x] && PiecewiseLinearQ[u, x]
  2158. Int[Times[Power[Pattern[u, Blank[]], -1], Pattern[v, Blank[]]], Pattern[x, Blank[Symbol]]] := With[{a = Simplify[D[u, x]], b = Simplify[D[v, x]]}, Simp[(b*x)/a, x] - Dist[(b*u - a*v)/a, Int[1/u, x], x] /; NeQ[b*u - a*v, 0]] /; PiecewiseLinearQ[u, v, x]
  2159. Int[Times[Power[Pattern[u, Blank[]], -1], Power[Pattern[v, Blank[]], Pattern[n, Blank[]]]], Pattern[x, Blank[Symbol]]] := With[{a = Simplify[D[u, x]], b = Simplify[D[v, x]]}, Simp[v^n/(a*n), x] - Dist[(b*u - a*v)/a, Int[v^(n - 1)/u, x], x] /; NeQ[b*u - a*v, 0]] /; PiecewiseLinearQ[u, v, x] && GtQ[n, 0] && NeQ[n, 1]
  2160. Int[Times[Power[Pattern[u, Blank[]], -1], Power[Pattern[v, Blank[]], -1]], Pattern[x, Blank[Symbol]]] := With[{a = Simplify[D[u, x]], b = Simplify[D[v, x]]}, Dist[b/(b*u - a*v), Int[1/v, x], x] - Dist[a/(b*u - a*v), Int[1/u, x], x] /; NeQ[b*u - a*v, 0]] /; PiecewiseLinearQ[u, v, x]
  2161. Int[Times[Power[Pattern[u, Blank[]], -1], Power[Pattern[v, Blank[]], Rational[-1, 2]]], Pattern[x, Blank[Symbol]]] := With[{a = Simplify[D[u, x]], b = Simplify[D[v, x]]}, Simp[(2*ArcTan[Sqrt[v]/Rt[(b*u - a*v)/a, 2]])/(a*Rt[(b*u - a*v)/a, 2]), x] /; NeQ[b*u - a*v, 0] && PosQ[(b*u - a*v)/a]] /; PiecewiseLinearQ[u, v, x]
  2162. Int[Times[Power[Pattern[u, Blank[]], -1], Power[Pattern[v, Blank[]], Rational[-1, 2]]], Pattern[x, Blank[Symbol]]] := With[{a = Simplify[D[u, x]], b = Simplify[D[v, x]]}, Simp[(-2*ArcTanh[Sqrt[v]/Rt[-((b*u - a*v)/a), 2]])/(a*Rt[-((b*u - a*v)/a), 2]), x] /; NeQ[b*u - a*v, 0] && NegQ[(b*u - a*v)/a]] /; PiecewiseLinearQ[u, v, x]
  2163. Int[Times[Power[Pattern[u, Blank[]], -1], Power[Pattern[v, Blank[]], Pattern[n, Blank[]]]], Pattern[x, Blank[Symbol]]] := With[{a = Simplify[D[u, x]], b = Simplify[D[v, x]]}, Simp[v^(n + 1)/((n + 1)*(b*u - a*v)), x] - Dist[(a*(n + 1))/((n + 1)*(b*u - a*v)), Int[v^(n + 1)/u, x], x] /; NeQ[b*u - a*v, 0]] /; PiecewiseLinearQ[u, v, x] && LtQ[n, -1]
  2164. Int[Times[Power[Pattern[u, Blank[]], -1], Power[Pattern[v, Blank[]], Pattern[n, Blank[]]]], Pattern[x, Blank[Symbol]]] := With[{a = Simplify[D[u, x]], b = Simplify[D[v, x]]}, Simp[(v^(n + 1)*Hypergeometric2F1[1, n + 1, n + 2, -((a*v)/(b*u - a*v))])/((n + 1)*(b*u - a*v)), x] /; NeQ[b*u - a*v, 0]] /; PiecewiseLinearQ[u, v, x] && !IntegerQ[n]
  2165. Int[Times[Power[Pattern[u, Blank[]], Rational[-1, 2]], Power[Pattern[v, Blank[]], Rational[-1, 2]]], Pattern[x, Blank[Symbol]]] := With[{a = Simplify[D[u, x]], b = Simplify[D[v, x]]}, Simp[(2*ArcTanh[(Rt[a*b, 2]*Sqrt[u])/(a*Sqrt[v])])/Rt[a*b, 2], x] /; NeQ[b*u - a*v, 0] && PosQ[a*b]] /; PiecewiseLinearQ[u, v, x]
  2166. Int[Times[Power[Pattern[u, Blank[]], Rational[-1, 2]], Power[Pattern[v, Blank[]], Rational[-1, 2]]], Pattern[x, Blank[Symbol]]] := With[{a = Simplify[D[u, x]], b = Simplify[D[v, x]]}, Simp[(2*ArcTan[(Rt[-(a*b), 2]*Sqrt[u])/(a*Sqrt[v])])/Rt[-(a*b), 2], x] /; NeQ[b*u - a*v, 0] && NegQ[a*b]] /; PiecewiseLinearQ[u, v, x]
  2167. Int[Times[Power[Pattern[u, Blank[]], Pattern[m, Blank[]]], Power[Pattern[v, Blank[]], Pattern[n, Blank[]]]], Pattern[x, Blank[Symbol]]] := With[{a = Simplify[D[u, x]], b = Simplify[D[v, x]]}, -Simp[(u^(m + 1)*v^(n + 1))/((m + 1)*(b*u - a*v)), x] /; NeQ[b*u - a*v, 0]] /; FreeQ[{m, n}, x] && PiecewiseLinearQ[u, v, x] && EqQ[m + n + 2, 0] && NeQ[m, -1]
  2168. Int[Times[Power[Pattern[u, Blank[]], Pattern[m, Blank[]]], Power[Pattern[v, Blank[]], Optional[Pattern[n, Blank[]]]]], Pattern[x, Blank[Symbol]]] := With[{a = Simplify[D[u, x]], b = Simplify[D[v, x]]}, Simp[(u^(m + 1)*v^n)/(a*(m + 1)), x] - Dist[(b*n)/(a*(m + 1)), Int[u^(m + 1)*v^(n - 1), x], x] /; NeQ[b*u - a*v, 0]] /; FreeQ[{m, n}, x] && PiecewiseLinearQ[u, v, x] && NeQ[m, -1] && ((LtQ[m, -1] && GtQ[n, 0] && !(ILtQ[m + n, -2] && (FractionQ[m] || GeQ[2*n + m + 1, 0]))) || (IGtQ[n, 0] && IGtQ[m, 0] && LeQ[n, m]) || (IGtQ[n, 0] && !IntegerQ[m]) || (ILtQ[m, 0] && !IntegerQ[n]))
  2169. Int[Times[Power[Pattern[u, Blank[]], Pattern[m, Blank[]]], Power[Pattern[v, Blank[]], Optional[Pattern[n, Blank[]]]]], Pattern[x, Blank[Symbol]]] := With[{a = Simplify[D[u, x]], b = Simplify[D[v, x]]}, Simp[(u^(m + 1)*v^n)/(a*(m + n + 1)), x] - Dist[(n*(b*u - a*v))/(a*(m + n + 1)), Int[u^m*v^(n - 1), x], x] /; NeQ[b*u - a*v, 0]] /; PiecewiseLinearQ[u, v, x] && NeQ[m + n + 2, 0] && GtQ[n, 0] && NeQ[m + n + 1, 0] && !(IGtQ[m, 0] && ( !IntegerQ[n] || LtQ[0, m, n])) && !ILtQ[m + n, -2]
  2170. Int[Times[Power[Pattern[u, Blank[]], Pattern[m, Blank[]]], Power[Pattern[v, Blank[]], Pattern[n, Blank[]]]], Pattern[x, Blank[Symbol]]] := With[{a = Simplify[D[u, x]], b = Simplify[D[v, x]]}, Simp[(u^(m + 1)*v^n)/(a*(m + n + 1)), x] - Dist[(n*(b*u - a*v))/(a*(m + n + 1)), Int[u^m*v^Simplify[n - 1], x], x] /; NeQ[b*u - a*v, 0]] /; PiecewiseLinearQ[u, v, x] && NeQ[m + n + 1, 0] && !RationalQ[n] && SumSimplerQ[n, -1]
  2171. Int[Times[Power[Pattern[u, Blank[]], Pattern[m, Blank[]]], Power[Pattern[v, Blank[]], Pattern[n, Blank[]]]], Pattern[x, Blank[Symbol]]] := With[{a = Simplify[D[u, x]], b = Simplify[D[v, x]]}, -Simp[(u^(m + 1)*v^(n + 1))/((m + 1)*(b*u - a*v)), x] + Dist[(b*(m + n + 2))/((m + 1)*(b*u - a*v)), Int[u^(m + 1)*v^n, x], x] /; NeQ[b*u - a*v, 0]] /; PiecewiseLinearQ[u, v, x] && NeQ[m + n + 2, 0] && LtQ[m, -1]
  2172. Int[Times[Power[Pattern[u, Blank[]], Pattern[m, Blank[]]], Power[Pattern[v, Blank[]], Pattern[n, Blank[]]]], Pattern[x, Blank[Symbol]]] := With[{a = Simplify[D[u, x]], b = Simplify[D[v, x]]}, -Simp[(u^(m + 1)*v^(n + 1))/((m + 1)*(b*u - a*v)), x] + Dist[(b*(m + n + 2))/((m + 1)*(b*u - a*v)), Int[u^Simplify[m + 1]*v^n, x], x] /; NeQ[b*u - a*v, 0]] /; PiecewiseLinearQ[u, v, x] && !RationalQ[m] && SumSimplerQ[m, 1]
  2173. Int[Times[Power[Pattern[u, Blank[]], Pattern[m, Blank[]]], Power[Pattern[v, Blank[]], Pattern[n, Blank[]]]], Pattern[x, Blank[Symbol]]] := With[{a = Simplify[D[u, x]], b = Simplify[D[v, x]]}, Simp[(u^m*v^(n + 1)*Hypergeometric2F1[-m, n + 1, n + 2, -((a*v)/(b*u - a*v))])/(b*(n + 1)*((b*u)/(b*u - a*v))^m), x] /; NeQ[b*u - a*v, 0]] /; PiecewiseLinearQ[u, v, x] && !IntegerQ[m] && !IntegerQ[n]
  2174. Int[Times[Log[Plus[Optional[Pattern[a, Blank[]]], Times[Optional[Pattern[b, Blank[]]], Pattern[x, Blank[]]]]], Power[Pattern[u, Blank[]], Optional[Pattern[n, Blank[]]]]], Pattern[x, Blank[Symbol]]] := With[{c = Simplify[D[u, x]]}, Simp[(u^n*(a + b*x)*Log[a + b*x])/b, x] + (-Dist[(c*n)/b, Int[u^(n - 1)*(a + b*x)*Log[a + b*x], x], x] - Int[u^n, x])] /; FreeQ[{a, b}, x] && PiecewiseLinearQ[u, x] && !LinearQ[u, x] && GtQ[n, 0]
  2175. Int[Times[Log[Plus[Optional[Pattern[a, Blank[]]], Times[Optional[Pattern[b, Blank[]]], Pattern[x, Blank[]]]]], Power[Pattern[u, Blank[]], Optional[Pattern[n, Blank[]]]], Power[Plus[Optional[Pattern[a, Blank[]]], Times[Optional[Pattern[b, Blank[]]], Pattern[x, Blank[]]]], Optional[Pattern[m, Blank[]]]]], Pattern[x, Blank[Symbol]]] := With[{c = Simplify[D[u, x]]}, Simp[(u^n*(a + b*x)^(m + 1)*Log[a + b*x])/(b*(m + 1)), x] + (-Dist[1/(m + 1), Int[u^n*(a + b*x)^m, x], x] - Dist[(c*n)/(b*(m + 1)), Int[u^(n - 1)*(a + b*x)^(m + 1)*Log[a + b*x], x], x])] /; FreeQ[{a, b, m}, x] && PiecewiseLinearQ[u, x] && !LinearQ[u, x] && GtQ[n, 0] && NeQ[m, -1]
  2176. Int[Times[Power[Times[Optional[Pattern[b, Blank[]]], Power[Pattern[F, Blank[]], Times[Optional[Pattern[g, Blank[]]], Plus[Optional[Pattern[e, Blank[]]], Times[Optional[Pattern[f, Blank[]]], Pattern[x, Blank[]]]]]]], Optional[Pattern[n, Blank[]]]], Power[Plus[Optional[Pattern[c, Blank[]]], Times[Optional[Pattern[d, Blank[]]], Pattern[x, Blank[]]]], Optional[Pattern[m, Blank[]]]]], Pattern[x, Blank[Symbol]]] := Simp[((c + d*x)^m*(b*F^(g*(e + f*x)))^n)/(f*g*n*Log[F]), x] - Dist[(d*m)/(f*g*n*Log[F]), Int[(c + d*x)^(m - 1)*(b*F^(g*(e + f*x)))^n, x], x] /; FreeQ[{F, b, c, d, e, f, g, n}, x] && GtQ[m, 0] && IntegerQ[2*m] && !False === True
  2177. Int[Times[Power[Times[Optional[Pattern[b, Blank[]]], Power[Pattern[F, Blank[]], Times[Optional[Pattern[g, Blank[]]], Plus[Optional[Pattern[e, Blank[]]], Times[Optional[Pattern[f, Blank[]]], Pattern[x, Blank[]]]]]]], Optional[Pattern[n, Blank[]]]], Power[Plus[Optional[Pattern[c, Blank[]]], Times[Optional[Pattern[d, Blank[]]], Pattern[x, Blank[]]]], Pattern[m, Blank[]]]], Pattern[x, Blank[Symbol]]] := Simp[((c + d*x)^(m + 1)*(b*F^(g*(e + f*x)))^n)/(d*(m + 1)), x] - Dist[(f*g*n*Log[F])/(d*(m + 1)), Int[(c + d*x)^(m + 1)*(b*F^(g*(e + f*x)))^n, x], x] /; FreeQ[{F, b, c, d, e, f, g, n}, x] && LtQ[m, -1] && IntegerQ[2*m] && !False === True
  2178. Int[Times[Power[Pattern[F, Blank[]], Times[Optional[Pattern[g, Blank[]]], Plus[Optional[Pattern[e, Blank[]]], Times[Optional[Pattern[f, Blank[]]], Pattern[x, Blank[]]]]]], Power[Plus[Optional[Pattern[c, Blank[]]], Times[Optional[Pattern[d, Blank[]]], Pattern[x, Blank[]]]], -1]], Pattern[x, Blank[Symbol]]] := Simp[(F^(g*(e - (c*f)/d))*ExpIntegralEi[(f*g*(c + d*x)*Log[F])/d])/d, x] /; FreeQ[{F, c, d, e, f, g}, x] && !False === True
  2179. Int[Times[Power[Pattern[F, Blank[]], Times[Optional[Pattern[g, Blank[]]], Plus[Optional[Pattern[e, Blank[]]], Times[Optional[Pattern[f, Blank[]]], Pattern[x, Blank[]]]]]], Power[Plus[Optional[Pattern[c, Blank[]]], Times[Optional[Pattern[d, Blank[]]], Pattern[x, Blank[]]]], Optional[Pattern[m, Blank[]]]]], Pattern[x, Blank[Symbol]]] := Simp[((-d)^m*F^(g*(e - (c*f)/d))*Gamma[m + 1, -((f*g*Log[F]*(c + d*x))/d)])/(f^(m + 1)*g^(m + 1)*Log[F]^(m + 1)), x] /; FreeQ[{F, c, d, e, f, g}, x] && IntegerQ[m]
  2180. Int[Times[Power[Pattern[F, Blank[]], Times[Optional[Pattern[g, Blank[]]], Plus[Optional[Pattern[e, Blank[]]], Times[Optional[Pattern[f, Blank[]]], Pattern[x, Blank[]]]]]], Power[Plus[Optional[Pattern[c, Blank[]]], Times[Optional[Pattern[d, Blank[]]], Pattern[x, Blank[]]]], Rational[-1, 2]]], Pattern[x, Blank[Symbol]]] := Dist[2/d, Subst[Int[F^(g*(e - (c*f)/d) + (f*g*x^2)/d), x], x, Sqrt[c + d*x]], x] /; FreeQ[{F, c, d, e, f, g}, x] && !False === True
  2181. Int[Times[Power[Pattern[F, Blank[]], Times[Optional[Pattern[g, Blank[]]], Plus[Optional[Pattern[e, Blank[]]], Times[Optional[Pattern[f, Blank[]]], Pattern[x, Blank[]]]]]], Power[Plus[Optional[Pattern[c, Blank[]]], Times[Optional[Pattern[d, Blank[]]], Pattern[x, Blank[]]]], Pattern[m, Blank[]]]], Pattern[x, Blank[Symbol]]] := -Simp[(F^(g*(e - (c*f)/d))*(c + d*x)^FracPart[m]*Gamma[m + 1, (-((f*g*Log[F])/d))*(c + d*x)])/(d*(-((f*g*Log[F])/d))^(IntPart[m] + 1)*(-((f*g*Log[F]*(c + d*x))/d))^FracPart[m]), x] /; FreeQ[{F, c, d, e, f, g, m}, x] && !IntegerQ[m]
  2182. Int[Times[Power[Times[Optional[Pattern[b, Blank[]]], Power[Pattern[F, Blank[]], Times[Optional[Pattern[g, Blank[]]], Plus[Optional[Pattern[e, Blank[]]], Times[Optional[Pattern[f, Blank[]]], Pattern[x, Blank[]]]]]]], Pattern[n, Blank[]]], Power[Plus[Optional[Pattern[c, Blank[]]], Times[Optional[Pattern[d, Blank[]]], Pattern[x, Blank[]]]], Optional[Pattern[m, Blank[]]]]], Pattern[x, Blank[Symbol]]] := Dist[(b*F^(g*(e + f*x)))^n/F^(g*n*(e + f*x)), Int[(c + d*x)^m*F^(g*n*(e + f*x)), x], x] /; FreeQ[{F, b, c, d, e, f, g, m, n}, x]
  2183. Int[Times[Power[Plus[Pattern[a, Blank[]], Times[Optional[Pattern[b, Blank[]]], Power[Power[Pattern[F, Blank[]], Times[Optional[Pattern[g, Blank[]]], Plus[Optional[Pattern[e, Blank[]]], Times[Optional[Pattern[f, Blank[]]], Pattern[x, Blank[]]]]]], Optional[Pattern[n, Blank[]]]]]], Optional[Pattern[p, Blank[]]]], Power[Plus[Optional[Pattern[c, Blank[]]], Times[Optional[Pattern[d, Blank[]]], Pattern[x, Blank[]]]], Optional[Pattern[m, Blank[]]]]], Pattern[x, Blank[Symbol]]] := Int[ExpandIntegrand[(c + d*x)^m, (a + b*(F^(g*(e + f*x)))^n)^p, x], x] /; FreeQ[{F, a, b, c, d, e, f, g, m, n}, x] && IGtQ[p, 0]
  2184. Int[Times[Power[Plus[Pattern[a, Blank[]], Times[Optional[Pattern[b, Blank[]]], Power[Power[Pattern[F, Blank[]], Times[Optional[Pattern[g, Blank[]]], Plus[Optional[Pattern[e, Blank[]]], Times[Optional[Pattern[f, Blank[]]], Pattern[x, Blank[]]]]]], Optional[Pattern[n, Blank[]]]]]], -1], Power[Plus[Optional[Pattern[c, Blank[]]], Times[Optional[Pattern[d, Blank[]]], Pattern[x, Blank[]]]], Optional[Pattern[m, Blank[]]]]], Pattern[x, Blank[Symbol]]] := Simp[(c + d*x)^(m + 1)/(a*d*(m + 1)), x] - Dist[b/a, Int[((c + d*x)^m*(F^(g*(e + f*x)))^n)/(a + b*(F^(g*(e + f*x)))^n), x], x] /; FreeQ[{F, a, b, c, d, e, f, g, n}, x] && IGtQ[m, 0]
  2185. Int[Times[Power[Plus[Pattern[a, Blank[]], Times[Optional[Pattern[b, Blank[]]], Power[Power[Pattern[F, Blank[]], Times[Optional[Pattern[g, Blank[]]], Plus[Optional[Pattern[e, Blank[]]], Times[Optional[Pattern[f, Blank[]]], Pattern[x, Blank[]]]]]], Optional[Pattern[n, Blank[]]]]]], Pattern[p, Blank[]]], Power[Plus[Optional[Pattern[c, Blank[]]], Times[Optional[Pattern[d, Blank[]]], Pattern[x, Blank[]]]], Optional[Pattern[m, Blank[]]]]], Pattern[x, Blank[Symbol]]] := Dist[1/a, Int[(c + d*x)^m*(a + b*(F^(g*(e + f*x)))^n)^(p + 1), x], x] - Dist[b/a, Int[(c + d*x)^m*(F^(g*(e + f*x)))^n*(a + b*(F^(g*(e + f*x)))^n)^p, x], x] /; FreeQ[{F, a, b, c, d, e, f, g, n}, x] && ILtQ[p, 0] && IGtQ[m, 0]
  2186. Int[Times[Power[Plus[Pattern[a, Blank[]], Times[Optional[Pattern[b, Blank[]]], Power[Power[Pattern[F, Blank[]], Times[Optional[Pattern[g, Blank[]]], Plus[Optional[Pattern[e, Blank[]]], Times[Optional[Pattern[f, Blank[]]], Pattern[x, Blank[]]]]]], Optional[Pattern[n, Blank[]]]]]], Pattern[p, Blank[]]], Power[Plus[Optional[Pattern[c, Blank[]]], Times[Optional[Pattern[d, Blank[]]], Pattern[x, Blank[]]]], Optional[Pattern[m, Blank[]]]]], Pattern[x, Blank[Symbol]]] := With[{u = IntHide[(a + b*(F^(g*(e + f*x)))^n)^p, x]}, Dist[(c + d*x)^m, u, x] - Dist[d*m, Int[(c + d*x)^(m - 1)*u, x], x]] /; FreeQ[{F, a, b, c, d, e, f, g, n}, x] && IGtQ[m, 0] && LtQ[p, -1]
  2187. Int[Times[Power[Plus[Optional[Pattern[a, Blank[]]], Times[Optional[Pattern[b, Blank[]]], Power[Power[Pattern[F, Blank[]], Times[Optional[Pattern[g, Blank[]]], Pattern[v, Blank[]]]], Optional[Pattern[n, Blank[]]]]]], Optional[Pattern[p, Blank[]]]], Power[Pattern[u, Blank[]], Optional[Pattern[m, Blank[]]]]], Pattern[x, Blank[Symbol]]] := Int[NormalizePowerOfLinear[u, x]^m*(a + b*(F^(g*ExpandToSum[v, x]))^n)^p, x] /; FreeQ[{F, a, b, g, n, p}, x] && LinearQ[v, x] && PowerOfLinearQ[u, x] && !(LinearMatchQ[v, x] && PowerOfLinearMatchQ[u, x]) && IntegerQ[m]
  2188. Int[Times[Power[Plus[Optional[Pattern[a, Blank[]]], Times[Optional[Pattern[b, Blank[]]], Power[Power[Pattern[F, Blank[]], Times[Optional[Pattern[g, Blank[]]], Pattern[v, Blank[]]]], Optional[Pattern[n, Blank[]]]]]], Optional[Pattern[p, Blank[]]]], Power[Pattern[u, Blank[]], Optional[Pattern[m, Blank[]]]]], Pattern[x, Blank[Symbol]]] := Module[{uu = NormalizePowerOfLinear[u, x], z}, Simp[z = If[PowerQ[uu] && FreeQ[uu[[2]], x], uu[[1]]^(m*uu[[2]]), uu^m]; (uu^m*Int[z*(a + b*(F^(g*ExpandToSum[v, x]))^n)^p, x])/z, x]] /; FreeQ[{F, a, b, g, m, n, p}, x] && LinearQ[v, x] && PowerOfLinearQ[u, x] && !(LinearMatchQ[v, x] && PowerOfLinearMatchQ[u, x]) && !IntegerQ[m]
  2189. Int[Times[Power[Plus[Pattern[a, Blank[]], Times[Optional[Pattern[b, Blank[]]], Power[Power[Pattern[F, Blank[]], Times[Optional[Pattern[g, Blank[]]], Plus[Optional[Pattern[e, Blank[]]], Times[Optional[Pattern[f, Blank[]]], Pattern[x, Blank[]]]]]], Optional[Pattern[n, Blank[]]]]]], Optional[Pattern[p, Blank[]]]], Power[Plus[Optional[Pattern[c, Blank[]]], Times[Optional[Pattern[d, Blank[]]], Pattern[x, Blank[]]]], Optional[Pattern[m, Blank[]]]]], Pattern[x, Blank[Symbol]]] := Unintegrable[(a + b*(F^(g*(e + f*x)))^n)^p*(c + d*x)^m, x] /; FreeQ[{a, b, c, d, e, f, g, m, n, p}, x]
  2190. Int[Times[Power[Power[Pattern[F, Blank[]], Times[Optional[Pattern[g, Blank[]]], Plus[Optional[Pattern[e, Blank[]]], Times[Optional[Pattern[f, Blank[]]], Pattern[x, Blank[]]]]]], Optional[Pattern[n, Blank[]]]], Power[Plus[Pattern[a, Blank[]], Times[Optional[Pattern[b, Blank[]]], Power[Power[Pattern[F, Blank[]], Times[Optional[Pattern[g, Blank[]]], Plus[Optional[Pattern[e, Blank[]]], Times[Optional[Pattern[f, Blank[]]], Pattern[x, Blank[]]]]]], Optional[Pattern[n, Blank[]]]]]], -1], Power[Plus[Optional[Pattern[c, Blank[]]], Times[Optional[Pattern[d, Blank[]]], Pattern[x, Blank[]]]], Optional[Pattern[m, Blank[]]]]], Pattern[x, Blank[Symbol]]] := Simp[((c + d*x)^m*Log[1 + (b*(F^(g*(e + f*x)))^n)/a])/(b*f*g*n*Log[F]), x] - Dist[(d*m)/(b*f*g*n*Log[F]), Int[(c + d*x)^(m - 1)*Log[1 + (b*(F^(g*(e + f*x)))^n)/a], x], x] /; FreeQ[{F, a, b, c, d, e, f, g, n}, x] && IGtQ[m, 0]
  2191. Int[Times[Power[Power[Pattern[F, Blank[]], Times[Optional[Pattern[g, Blank[]]], Plus[Optional[Pattern[e, Blank[]]], Times[Optional[Pattern[f, Blank[]]], Pattern[x, Blank[]]]]]], Optional[Pattern[n, Blank[]]]], Power[Plus[Optional[Pattern[a, Blank[]]], Times[Optional[Pattern[b, Blank[]]], Power[Power[Pattern[F, Blank[]], Times[Optional[Pattern[g, Blank[]]], Plus[Optional[Pattern[e, Blank[]]], Times[Optional[Pattern[f, Blank[]]], Pattern[x, Blank[]]]]]], Optional[Pattern[n, Blank[]]]]]], Optional[Pattern[p, Blank[]]]], Power[Plus[Optional[Pattern[c, Blank[]]], Times[Optional[Pattern[d, Blank[]]], Pattern[x, Blank[]]]], Optional[Pattern[m, Blank[]]]]], Pattern[x, Blank[Symbol]]] := Simp[((c + d*x)^m*(a + b*(F^(g*(e + f*x)))^n)^(p + 1))/(b*f*g*n*(p + 1)*Log[F]), x] - Dist[(d*m)/(b*f*g*n*(p + 1)*Log[F]), Int[(c + d*x)^(m - 1)*(a + b*(F^(g*(e + f*x)))^n)^(p + 1), x], x] /; FreeQ[{F, a, b, c, d, e, f, g, m, n, p}, x] && NeQ[p, -1]
  2192. Int[Times[Power[Power[Pattern[F, Blank[]], Times[Optional[Pattern[g, Blank[]]], Plus[Optional[Pattern[e, Blank[]]], Times[Optional[Pattern[f, Blank[]]], Pattern[x, Blank[]]]]]], Optional[Pattern[n, Blank[]]]], Power[Plus[Optional[Pattern[a, Blank[]]], Times[Optional[Pattern[b, Blank[]]], Power[Power[Pattern[F, Blank[]], Times[Optional[Pattern[g, Blank[]]], Plus[Optional[Pattern[e, Blank[]]], Times[Optional[Pattern[f, Blank[]]], Pattern[x, Blank[]]]]]], Optional[Pattern[n, Blank[]]]]]], Optional[Pattern[p, Blank[]]]], Power[Plus[Optional[Pattern[c, Blank[]]], Times[Optional[Pattern[d, Blank[]]], Pattern[x, Blank[]]]], Optional[Pattern[m, Blank[]]]]], Pattern[x, Blank[Symbol]]] := Unintegrable[(F^(g*(e + f*x)))^n*(a + b*(F^(g*(e + f*x)))^n)^p*(c + d*x)^m, x] /; FreeQ[{F, a, b, c, d, e, f, g, m, n, p}, x]
  2193. Int[Times[Power[Plus[Optional[Pattern[a, Blank[]]], Times[Optional[Pattern[b, Blank[]]], Power[Power[Pattern[F, Blank[]], Times[Optional[Pattern[g, Blank[]]], Plus[Optional[Pattern[e, Blank[]]], Times[Optional[Pattern[f, Blank[]]], Pattern[x, Blank[]]]]]], Optional[Pattern[n, Blank[]]]]]], Optional[Pattern[p, Blank[]]]], Power[Times[Optional[Pattern[k, Blank[]]], Power[Pattern[G, Blank[]], Times[Optional[Pattern[j, Blank[]]], Plus[Optional[Pattern[h, Blank[]]], Times[Optional[Pattern[i, Blank[]]], Pattern[x, Blank[]]]]]]], Optional[Pattern[q, Blank[]]]], Power[Plus[Optional[Pattern[c, Blank[]]], Times[Optional[Pattern[d, Blank[]]], Pattern[x, Blank[]]]], Optional[Pattern[m, Blank[]]]]], Pattern[x, Blank[Symbol]]] := Dist[(k*G^(j*(h + i*x)))^q/(F^(g*(e + f*x)))^n, Int[(c + d*x)^m*(F^(g*(e + f*x)))^n*(a + b*(F^(g*(e + f*x)))^n)^p, x], x] /; FreeQ[{F, a, b, c, d, e, f, g, h, i, j, k, m, n, p, q}, x] && EqQ[f*g*n*Log[F] - i*j*q*Log[G], 0] && NeQ[(k*G^(j*(h + i*x)))^q - (F^(g*(e + f*x)))^n, 0]
  2194. Int[Power[Power[Pattern[F, Blank[]], Times[Optional[Pattern[c, Blank[]]], Plus[Optional[Pattern[a, Blank[]]], Times[Optional[Pattern[b, Blank[]]], Pattern[x, Blank[]]]]]], Optional[Pattern[n, Blank[]]]], Pattern[x, Blank[Symbol]]] := Simp[(F^(c*(a + b*x)))^n/(b*c*n*Log[F]), x] /; FreeQ[{F, a, b, c, n}, x]
  2195. Int[Times[Power[Pattern[F, Blank[]], Times[Optional[Pattern[c, Blank[]]], Pattern[v, Blank[]]]], Pattern[u, Blank[]]], Pattern[x, Blank[Symbol]]] := Int[ExpandIntegrand[u*F^(c*ExpandToSum[v, x]), x], x] /; FreeQ[{F, c}, x] && PolynomialQ[u, x] && LinearQ[v, x] && False === True
  2196. Int[Times[Power[Pattern[F, Blank[]], Times[Optional[Pattern[c, Blank[]]], Pattern[v, Blank[]]]], Pattern[u, Blank[]]], Pattern[x, Blank[Symbol]]] := Int[ExpandIntegrand[F^(c*ExpandToSum[v, x]), u, x], x] /; FreeQ[{F, c}, x] && PolynomialQ[u, x] && LinearQ[v, x] && !False === True
  2197. Int[Times[Power[Pattern[F, Blank[]], Times[Optional[Pattern[c, Blank[]]], Pattern[v, Blank[]]]], Power[Pattern[u, Blank[]], Optional[Pattern[m, Blank[]]]], Pattern[w, Blank[]]], Pattern[x, Blank[Symbol]]] := With[{b = Coefficient[v, x, 1], d = Coefficient[u, x, 0], e = Coefficient[u, x, 1], f = Coefficient[w, x, 0], g = Coefficient[w, x, 1]}, Simp[(g*u^(m + 1)*F^(c*v))/(b*c*e*Log[F]), x] /; EqQ[e*g*(m + 1) - b*c*(e*f - d*g)*Log[F], 0]] /; FreeQ[{F, c, m}, x] && LinearQ[{u, v, w}, x]
  2198. Int[Times[Power[Pattern[F, Blank[]], Times[Optional[Pattern[c, Blank[]]], Pattern[v, Blank[]]]], Power[Pattern[u, Blank[]], Optional[Pattern[m, Blank[]]]], Pattern[w, Blank[]]], Pattern[x, Blank[Symbol]]] := Int[ExpandIntegrand[w*NormalizePowerOfLinear[u, x]^m*F^(c*ExpandToSum[v, x]), x], x] /; FreeQ[{F, c}, x] && PolynomialQ[w, x] && LinearQ[v, x] && PowerOfLinearQ[u, x] && IntegerQ[m] && False === True
  2199. Int[Times[Power[Pattern[F, Blank[]], Times[Optional[Pattern[c, Blank[]]], Pattern[v, Blank[]]]], Power[Pattern[u, Blank[]], Optional[Pattern[m, Blank[]]]], Pattern[w, Blank[]]], Pattern[x, Blank[Symbol]]] := Int[ExpandIntegrand[F^(c*ExpandToSum[v, x]), w*NormalizePowerOfLinear[u, x]^m, x], x] /; FreeQ[{F, c}, x] && PolynomialQ[w, x] && LinearQ[v, x] && PowerOfLinearQ[u, x] && IntegerQ[m] && !False === True
  2200. Int[Times[Power[Pattern[F, Blank[]], Times[Optional[Pattern[c, Blank[]]], Pattern[v, Blank[]]]], Power[Pattern[u, Blank[]], Optional[Pattern[m, Blank[]]]], Pattern[w, Blank[]]], Pattern[x, Blank[Symbol]]] := Module[{uu = NormalizePowerOfLinear[u, x], z}, Simp[z = If[PowerQ[uu] && FreeQ[uu[[2]], x], uu[[1]]^(m*uu[[2]]), uu^m]; (uu^m*Int[ExpandIntegrand[w*z*F^(c*ExpandToSum[v, x]), x], x])/z, x]] /; FreeQ[{F, c, m}, x] && PolynomialQ[w, x] && LinearQ[v, x] && PowerOfLinearQ[u, x] && !IntegerQ[m]
  2201. Int[Times[Power[Log[Times[Optional[Pattern[d, Blank[]]], Pattern[x, Blank[]]]], Optional[Pattern[n, Blank[]]]], Power[Pattern[F, Blank[]], Times[Optional[Pattern[c, Blank[]]], Plus[Optional[Pattern[a, Blank[]]], Times[Optional[Pattern[b, Blank[]]], Pattern[x, Blank[]]]]]], Plus[Pattern[e, Blank[]], Times[Log[Times[Optional[Pattern[d, Blank[]]], Pattern[x, Blank[]]]], Optional[Pattern[h, Blank[]]], Plus[Optional[Pattern[f, Blank[]]], Times[Optional[Pattern[g, Blank[]]], Pattern[x, Blank[]]]]]]], Pattern[x, Blank[Symbol]]] := Simp[(e*x*F^(c*(a + b*x))*Log[d*x]^(n + 1))/(n + 1), x] /; FreeQ[{F, a, b, c, d, e, f, g, h, n}, x] && EqQ[e - f*h*(n + 1), 0] && EqQ[g*h*(n + 1) - b*c*e*Log[F], 0] && NeQ[n, -1]
  2202. Int[Times[Power[Log[Times[Optional[Pattern[d, Blank[]]], Pattern[x, Blank[]]]], Optional[Pattern[n, Blank[]]]], Power[Pattern[F, Blank[]], Times[Optional[Pattern[c, Blank[]]], Plus[Optional[Pattern[a, Blank[]]], Times[Optional[Pattern[b, Blank[]]], Pattern[x, Blank[]]]]]], Power[Pattern[x, Blank[]], Optional[Pattern[m, Blank[]]]], Plus[Pattern[e, Blank[]], Times[Log[Times[Optional[Pattern[d, Blank[]]], Pattern[x, Blank[]]]], Optional[Pattern[h, Blank[]]], Plus[Optional[Pattern[f, Blank[]]], Times[Optional[Pattern[g, Blank[]]], Pattern[x, Blank[]]]]]]], Pattern[x, Blank[Symbol]]] := Simp[(e*x^(m + 1)*F^(c*(a + b*x))*Log[d*x]^(n + 1))/(n + 1), x] /; FreeQ[{F, a, b, c, d, e, f, g, h, m, n}, x] && EqQ[e*(m + 1) - f*h*(n + 1), 0] && EqQ[g*h*(n + 1) - b*c*e*Log[F], 0] && NeQ[n, -1]
  2203. Int[Power[Pattern[F, Blank[]], Plus[Optional[Pattern[a, Blank[]]], Times[Optional[Pattern[b, Blank[]]], Plus[Optional[Pattern[c, Blank[]]], Times[Optional[Pattern[d, Blank[]]], Pattern[x, Blank[]]]]]]], Pattern[x, Blank[Symbol]]] := Simp[F^(a + b*(c + d*x))/(b*d*Log[F]), x] /; FreeQ[{F, a, b, c, d}, x]
  2204. Int[Power[Pattern[F, Blank[]], Plus[Optional[Pattern[a, Blank[]]], Times[Optional[Pattern[b, Blank[]]], Power[Plus[Optional[Pattern[c, Blank[]]], Times[Optional[Pattern[d, Blank[]]], Pattern[x, Blank[]]]], 2]]]], Pattern[x, Blank[Symbol]]] := Simp[(F^a*Sqrt[Pi]*Erfi[(c + d*x)*Rt[b*Log[F], 2]])/(2*d*Rt[b*Log[F], 2]), x] /; FreeQ[{F, a, b, c, d}, x] && PosQ[b]
  2205. Int[Power[Pattern[F, Blank[]], Plus[Optional[Pattern[a, Blank[]]], Times[Optional[Pattern[b, Blank[]]], Power[Plus[Optional[Pattern[c, Blank[]]], Times[Optional[Pattern[d, Blank[]]], Pattern[x, Blank[]]]], 2]]]], Pattern[x, Blank[Symbol]]] := Simp[(F^a*Sqrt[Pi]*Erf[(c + d*x)*Rt[-(b*Log[F]), 2]])/(2*d*Rt[-(b*Log[F]), 2]), x] /; FreeQ[{F, a, b, c, d}, x] && NegQ[b]
  2206. Int[Power[Pattern[F, Blank[]], Plus[Optional[Pattern[a, Blank[]]], Times[Optional[Pattern[b, Blank[]]], Power[Plus[Optional[Pattern[c, Blank[]]], Times[Optional[Pattern[d, Blank[]]], Pattern[x, Blank[]]]], Pattern[n, Blank[]]]]]], Pattern[x, Blank[Symbol]]] := Simp[((c + d*x)*F^(a + b*(c + d*x)^n))/d, x] - Dist[b*n*Log[F], Int[(c + d*x)^n*F^(a + b*(c + d*x)^n), x], x] /; FreeQ[{F, a, b, c, d}, x] && IntegerQ[2/n] && ILtQ[n, 0]
  2207. Int[Power[Pattern[F, Blank[]], Plus[Optional[Pattern[a, Blank[]]], Times[Optional[Pattern[b, Blank[]]], Power[Plus[Optional[Pattern[c, Blank[]]], Times[Optional[Pattern[d, Blank[]]], Pattern[x, Blank[]]]], Pattern[n, Blank[]]]]]], Pattern[x, Blank[Symbol]]] := With[{k = Denominator[n]}, Dist[k/d, Subst[Int[x^(k - 1)*F^(a + b*x^(k*n)), x], x, (c + d*x)^(1/k)], x]] /; FreeQ[{F, a, b, c, d}, x] && IntegerQ[2/n] && !IntegerQ[n]
  2208. Int[Power[Pattern[F, Blank[]], Plus[Optional[Pattern[a, Blank[]]], Times[Optional[Pattern[b, Blank[]]], Power[Plus[Optional[Pattern[c, Blank[]]], Times[Optional[Pattern[d, Blank[]]], Pattern[x, Blank[]]]], Pattern[n, Blank[]]]]]], Pattern[x, Blank[Symbol]]] := -Simp[(F^a*(c + d*x)*Gamma[1/n, -(b*(c + d*x)^n*Log[F])])/(d*n*(-(b*(c + d*x)^n*Log[F]))^(1/n)), x] /; FreeQ[{F, a, b, c, d, n}, x] && !IntegerQ[2/n]
  2209. Int[Times[Power[Pattern[F, Blank[]], Plus[Optional[Pattern[a, Blank[]]], Times[Optional[Pattern[b, Blank[]]], Power[Plus[Optional[Pattern[c, Blank[]]], Times[Optional[Pattern[d, Blank[]]], Pattern[x, Blank[]]]], Pattern[n, Blank[]]]]]], Power[Plus[Optional[Pattern[e, Blank[]]], Times[Optional[Pattern[f, Blank[]]], Pattern[x, Blank[]]]], Optional[Pattern[m, Blank[]]]]], Pattern[x, Blank[Symbol]]] := Simp[((e + f*x)^n*F^(a + b*(c + d*x)^n))/(b*f*n*(c + d*x)^n*Log[F]), x] /; FreeQ[{F, a, b, c, d, e, f, n}, x] && EqQ[m, n - 1] && EqQ[d*e - c*f, 0]
  2210. Int[Times[Power[Pattern[F, Blank[]], Plus[Optional[Pattern[a, Blank[]]], Times[Optional[Pattern[b, Blank[]]], Power[Plus[Optional[Pattern[c, Blank[]]], Times[Optional[Pattern[d, Blank[]]], Pattern[x, Blank[]]]], Pattern[n, Blank[]]]]]], Power[Plus[Optional[Pattern[e, Blank[]]], Times[Optional[Pattern[f, Blank[]]], Pattern[x, Blank[]]]], -1]], Pattern[x, Blank[Symbol]]] := Simp[(F^a*ExpIntegralEi[b*(c + d*x)^n*Log[F]])/(f*n), x] /; FreeQ[{F, a, b, c, d, e, f, n}, x] && EqQ[d*e - c*f, 0]
  2211. Int[Times[Power[Pattern[F, Blank[]], Plus[Optional[Pattern[a, Blank[]]], Times[Optional[Pattern[b, Blank[]]], Power[Plus[Optional[Pattern[c, Blank[]]], Times[Optional[Pattern[d, Blank[]]], Pattern[x, Blank[]]]], Pattern[n, Blank[]]]]]], Power[Plus[Optional[Pattern[c, Blank[]]], Times[Optional[Pattern[d, Blank[]]], Pattern[x, Blank[]]]], Optional[Pattern[m, Blank[]]]]], Pattern[x, Blank[Symbol]]] := Dist[1/(d*(m + 1)), Subst[Int[F^(a + b*x^2), x], x, (c + d*x)^(m + 1)], x] /; FreeQ[{F, a, b, c, d, m, n}, x] && EqQ[n, 2*(m + 1)]
  2212. Int[Times[Power[Pattern[F, Blank[]], Plus[Optional[Pattern[a, Blank[]]], Times[Optional[Pattern[b, Blank[]]], Power[Plus[Optional[Pattern[c, Blank[]]], Times[Optional[Pattern[d, Blank[]]], Pattern[x, Blank[]]]], Pattern[n, Blank[]]]]]], Power[Plus[Optional[Pattern[c, Blank[]]], Times[Optional[Pattern[d, Blank[]]], Pattern[x, Blank[]]]], Optional[Pattern[m, Blank[]]]]], Pattern[x, Blank[Symbol]]] := Simp[((c + d*x)^(m - n + 1)*F^(a + b*(c + d*x)^n))/(b*d*n*Log[F]), x] - Dist[(m - n + 1)/(b*n*Log[F]), Int[(c + d*x)^(m - n)*F^(a + b*(c + d*x)^n), x], x] /; FreeQ[{F, a, b, c, d}, x] && IntegerQ[(2*(m + 1))/n] && LtQ[0, (m + 1)/n, 5] && IntegerQ[n] && (LtQ[0, n, m + 1] || LtQ[m, n, 0])
  2213. Int[Times[Power[Pattern[F, Blank[]], Plus[Optional[Pattern[a, Blank[]]], Times[Optional[Pattern[b, Blank[]]], Power[Plus[Optional[Pattern[c, Blank[]]], Times[Optional[Pattern[d, Blank[]]], Pattern[x, Blank[]]]], Pattern[n, Blank[]]]]]], Power[Plus[Optional[Pattern[c, Blank[]]], Times[Optional[Pattern[d, Blank[]]], Pattern[x, Blank[]]]], Optional[Pattern[m, Blank[]]]]], Pattern[x, Blank[Symbol]]] := Simp[((c + d*x)^(m - n + 1)*F^(a + b*(c + d*x)^n))/(b*d*n*Log[F]), x] - Dist[(m - n + 1)/(b*n*Log[F]), Int[(c + d*x)^Simplify[m - n]*F^(a + b*(c + d*x)^n), x], x] /; FreeQ[{F, a, b, c, d, m, n}, x] && IntegerQ[2*Simplify[(m + 1)/n]] && LtQ[0, Simplify[(m + 1)/n], 5] && !RationalQ[m] && SumSimplerQ[m, -n]
  2214. Int[Times[Power[Pattern[F, Blank[]], Plus[Optional[Pattern[a, Blank[]]], Times[Optional[Pattern[b, Blank[]]], Power[Plus[Optional[Pattern[c, Blank[]]], Times[Optional[Pattern[d, Blank[]]], Pattern[x, Blank[]]]], Pattern[n, Blank[]]]]]], Power[Plus[Optional[Pattern[c, Blank[]]], Times[Optional[Pattern[d, Blank[]]], Pattern[x, Blank[]]]], Optional[Pattern[m, Blank[]]]]], Pattern[x, Blank[Symbol]]] := Simp[((c + d*x)^(m + 1)*F^(a + b*(c + d*x)^n))/(d*(m + 1)), x] - Dist[(b*n*Log[F])/(m + 1), Int[(c + d*x)^(m + n)*F^(a + b*(c + d*x)^n), x], x] /; FreeQ[{F, a, b, c, d}, x] && IntegerQ[(2*(m + 1))/n] && LtQ[-4, (m + 1)/n, 5] && IntegerQ[n] && ((GtQ[n, 0] && LtQ[m, -1]) || (GtQ[-n, 0] && LeQ[-n, m + 1]))
  2215. Int[Times[Power[Pattern[F, Blank[]], Plus[Optional[Pattern[a, Blank[]]], Times[Optional[Pattern[b, Blank[]]], Power[Plus[Optional[Pattern[c, Blank[]]], Times[Optional[Pattern[d, Blank[]]], Pattern[x, Blank[]]]], Pattern[n, Blank[]]]]]], Power[Plus[Optional[Pattern[c, Blank[]]], Times[Optional[Pattern[d, Blank[]]], Pattern[x, Blank[]]]], Optional[Pattern[m, Blank[]]]]], Pattern[x, Blank[Symbol]]] := Simp[((c + d*x)^(m + 1)*F^(a + b*(c + d*x)^n))/(d*(m + 1)), x] - Dist[(b*n*Log[F])/(m + 1), Int[(c + d*x)^Simplify[m + n]*F^(a + b*(c + d*x)^n), x], x] /; FreeQ[{F, a, b, c, d, m, n}, x] && IntegerQ[2*Simplify[(m + 1)/n]] && LtQ[-4, Simplify[(m + 1)/n], 5] && !RationalQ[m] && SumSimplerQ[m, n]
  2216. Int[Times[Power[Pattern[F, Blank[]], Plus[Optional[Pattern[a, Blank[]]], Times[Optional[Pattern[b, Blank[]]], Power[Plus[Optional[Pattern[c, Blank[]]], Times[Optional[Pattern[d, Blank[]]], Pattern[x, Blank[]]]], Pattern[n, Blank[]]]]]], Power[Plus[Optional[Pattern[c, Blank[]]], Times[Optional[Pattern[d, Blank[]]], Pattern[x, Blank[]]]], Optional[Pattern[m, Blank[]]]]], Pattern[x, Blank[Symbol]]] := With[{k = Denominator[n]}, Dist[k/d, Subst[Int[x^(k*(m + 1) - 1)*F^(a + b*x^(k*n)), x], x, (c + d*x)^(1/k)], x]] /; FreeQ[{F, a, b, c, d, m, n}, x] && IntegerQ[(2*(m + 1))/n] && LtQ[0, (m + 1)/n, 5] && !IntegerQ[n]
  2217. Int[Times[Power[Pattern[F, Blank[]], Plus[Optional[Pattern[a, Blank[]]], Times[Optional[Pattern[b, Blank[]]], Power[Plus[Optional[Pattern[c, Blank[]]], Times[Optional[Pattern[d, Blank[]]], Pattern[x, Blank[]]]], Pattern[n, Blank[]]]]]], Power[Plus[Optional[Pattern[e, Blank[]]], Times[Optional[Pattern[f, Blank[]]], Pattern[x, Blank[]]]], Optional[Pattern[m, Blank[]]]]], Pattern[x, Blank[Symbol]]] := Dist[(e + f*x)^m/(c + d*x)^m, Int[(c + d*x)^m*F^(a + b*(c + d*x)^n), x], x] /; FreeQ[{F, a, b, c, d, e, f, m, n}, x] && EqQ[d*e - c*f, 0] && IntegerQ[2*Simplify[(m + 1)/n]] && NeQ[f, d] && !IntegerQ[m] && NeQ[c*e, 0]
  2218. Int[Times[Power[Pattern[F, Blank[]], Plus[Optional[Pattern[a, Blank[]]], Times[Optional[Pattern[b, Blank[]]], Power[Plus[Optional[Pattern[c, Blank[]]], Times[Optional[Pattern[d, Blank[]]], Pattern[x, Blank[]]]], Pattern[n, Blank[]]]]]], Power[Plus[Optional[Pattern[e, Blank[]]], Times[Optional[Pattern[f, Blank[]]], Pattern[x, Blank[]]]], Optional[Pattern[m, Blank[]]]]], Pattern[x, Blank[Symbol]]] := -Simp[(F^a*(e + f*x)^(m + 1)*Gamma[(m + 1)/n, -(b*(c + d*x)^n*Log[F])])/(f*n*(-(b*(c + d*x)^n*Log[F]))^((m + 1)/n)), x] /; FreeQ[{F, a, b, c, d, e, f, m, n}, x] && EqQ[d*e - c*f, 0]
  2219. Int[Times[Power[Pattern[F, Blank[]], Plus[Optional[Pattern[a, Blank[]]], Times[Optional[Pattern[b, Blank[]]], Power[Plus[Optional[Pattern[c, Blank[]]], Times[Optional[Pattern[d, Blank[]]], Pattern[x, Blank[]]]], 2]]]], Power[Plus[Optional[Pattern[e, Blank[]]], Times[Optional[Pattern[f, Blank[]]], Pattern[x, Blank[]]]], Pattern[m, Blank[]]]], Pattern[x, Blank[Symbol]]] := Simp[(f*(e + f*x)^(m - 1)*F^(a + b*(c + d*x)^2))/(2*b*d^2*Log[F]), x] + (Dist[(d*e - c*f)/d, Int[(e + f*x)^(m - 1)*F^(a + b*(c + d*x)^2), x], x] - Dist[((m - 1)*f^2)/(2*b*d^2*Log[F]), Int[(e + f*x)^(m - 2)*F^(a + b*(c + d*x)^2), x], x]) /; FreeQ[{F, a, b, c, d, e, f}, x] && NeQ[d*e - c*f, 0] && FractionQ[m] && GtQ[m, 1]
  2220. Int[Times[Power[Pattern[F, Blank[]], Plus[Optional[Pattern[a, Blank[]]], Times[Optional[Pattern[b, Blank[]]], Power[Plus[Optional[Pattern[c, Blank[]]], Times[Optional[Pattern[d, Blank[]]], Pattern[x, Blank[]]]], 2]]]], Power[Plus[Optional[Pattern[e, Blank[]]], Times[Optional[Pattern[f, Blank[]]], Pattern[x, Blank[]]]], Pattern[m, Blank[]]]], Pattern[x, Blank[Symbol]]] := Simp[(f*(e + f*x)^(m + 1)*F^(a + b*(c + d*x)^2))/((m + 1)*f^2), x] + (-Dist[(2*b*d^2*Log[F])/(f^2*(m + 1)), Int[(e + f*x)^(m + 2)*F^(a + b*(c + d*x)^2), x], x] + Dist[(2*b*d*(d*e - c*f)*Log[F])/(f^2*(m + 1)), Int[(e + f*x)^(m + 1)*F^(a + b*(c + d*x)^2), x], x]) /; FreeQ[{F, a, b, c, d, e, f}, x] && NeQ[d*e - c*f, 0] && LtQ[m, -1]
  2221. Int[Times[Power[Pattern[F, Blank[]], Plus[Optional[Pattern[a, Blank[]]], Times[Optional[Pattern[b, Blank[]]], Power[Plus[Optional[Pattern[c, Blank[]]], Times[Optional[Pattern[d, Blank[]]], Pattern[x, Blank[]]]], Pattern[n, Blank[]]]]]], Power[Plus[Optional[Pattern[e, Blank[]]], Times[Optional[Pattern[f, Blank[]]], Pattern[x, Blank[]]]], Pattern[m, Blank[]]]], Pattern[x, Blank[Symbol]]] := Simp[((e + f*x)^(m + 1)*F^(a + b*(c + d*x)^n))/(f*(m + 1)), x] - Dist[(b*d*n*Log[F])/(f*(m + 1)), Int[(e + f*x)^(m + 1)*(c + d*x)^(n - 1)*F^(a + b*(c + d*x)^n), x], x] /; FreeQ[{F, a, b, c, d, e, f}, x] && NeQ[d*e - c*f, 0] && IGtQ[n, 2] && LtQ[m, -1]
  2222. Int[Times[Power[Pattern[F, Blank[]], Plus[Optional[Pattern[a, Blank[]]], Times[Optional[Pattern[b, Blank[]]], Power[Plus[Optional[Pattern[c, Blank[]]], Times[Optional[Pattern[d, Blank[]]], Pattern[x, Blank[]]]], -1]]]], Power[Plus[Optional[Pattern[e, Blank[]]], Times[Optional[Pattern[f, Blank[]]], Pattern[x, Blank[]]]], -1]], Pattern[x, Blank[Symbol]]] := Dist[d/f, Int[F^(a + b/(c + d*x))/(c + d*x), x], x] - Dist[(d*e - c*f)/f, Int[F^(a + b/(c + d*x))/((c + d*x)*(e + f*x)), x], x] /; FreeQ[{F, a, b, c, d, e, f}, x] && NeQ[d*e - c*f, 0]
  2223. Int[Times[Power[Pattern[F, Blank[]], Plus[Optional[Pattern[a, Blank[]]], Times[Optional[Pattern[b, Blank[]]], Power[Plus[Optional[Pattern[c, Blank[]]], Times[Optional[Pattern[d, Blank[]]], Pattern[x, Blank[]]]], -1]]]], Power[Plus[Optional[Pattern[e, Blank[]]], Times[Optional[Pattern[f, Blank[]]], Pattern[x, Blank[]]]], Pattern[m, Blank[]]]], Pattern[x, Blank[Symbol]]] := Simp[((e + f*x)^(m + 1)*F^(a + b/(c + d*x)))/(f*(m + 1)), x] + Dist[(b*d*Log[F])/(f*(m + 1)), Int[((e + f*x)^(m + 1)*F^(a + b/(c + d*x)))/(c + d*x)^2, x], x] /; FreeQ[{F, a, b, c, d, e, f}, x] && NeQ[d*e - c*f, 0] && ILtQ[m, -1]
  2224. Int[Times[Power[Pattern[F, Blank[]], Plus[Optional[Pattern[a, Blank[]]], Times[Optional[Pattern[b, Blank[]]], Power[Plus[Optional[Pattern[c, Blank[]]], Times[Optional[Pattern[d, Blank[]]], Pattern[x, Blank[]]]], Pattern[n, Blank[]]]]]], Power[Plus[Optional[Pattern[e, Blank[]]], Times[Optional[Pattern[f, Blank[]]], Pattern[x, Blank[]]]], -1]], Pattern[x, Blank[Symbol]]] := Unintegrable[F^(a + b*(c + d*x)^n)/(e + f*x), x] /; FreeQ[{F, a, b, c, d, e, f, n}, x] && NeQ[d*e - c*f, 0]
  2225. Int[Times[Power[Pattern[F, Blank[]], Pattern[v, Blank[]]], Power[Pattern[u, Blank[]], Optional[Pattern[m, Blank[]]]]], Pattern[x, Blank[Symbol]]] := Int[ExpandToSum[u, x]^m*F^ExpandToSum[v, x], x] /; FreeQ[{F, m}, x] && LinearQ[u, x] && BinomialQ[v, x] && !(LinearMatchQ[u, x] && BinomialMatchQ[v, x])
  2226. Int[Times[Power[Pattern[F, Blank[]], Plus[Optional[Pattern[a, Blank[]]], Times[Optional[Pattern[b, Blank[]]], Power[Plus[Optional[Pattern[c, Blank[]]], Times[Optional[Pattern[d, Blank[]]], Pattern[x, Blank[]]]], Pattern[n, Blank[]]]]]], Pattern[u, Blank[]]], Pattern[x, Blank[Symbol]]] := Int[ExpandLinearProduct[F^(a + b*(c + d*x)^n), u, c, d, x], x] /; FreeQ[{F, a, b, c, d, n}, x] && PolynomialQ[u, x]
  2227. Int[Times[Optional[Pattern[u, Blank[]]], Power[Pattern[F, Blank[]], Plus[Optional[Pattern[a, Blank[]]], Times[Optional[Pattern[b, Blank[]]], Pattern[v, Blank[]]]]]], Pattern[x, Blank[Symbol]]] := Int[u*F^(a + b*NormalizePowerOfLinear[v, x]), x] /; FreeQ[{F, a, b}, x] && PolynomialQ[u, x] && PowerOfLinearQ[v, x] && !PowerOfLinearMatchQ[v, x]
  2228. Int[Times[Power[Pattern[F, Blank[]], Plus[Optional[Pattern[a, Blank[]]], Times[Optional[Pattern[b, Blank[]]], Power[Plus[Optional[Pattern[c, Blank[]]], Times[Optional[Pattern[d, Blank[]]], Pattern[x, Blank[]]]], -1]]]], Power[Plus[Optional[Pattern[e, Blank[]]], Times[Optional[Pattern[f, Blank[]]], Pattern[x, Blank[]]]], -1], Power[Plus[Optional[Pattern[g, Blank[]]], Times[Optional[Pattern[h, Blank[]]], Pattern[x, Blank[]]]], -1]], Pattern[x, Blank[Symbol]]] := -Dist[d/(f*(d*g - c*h)), Subst[Int[F^(a - (b*h)/(d*g - c*h) + (d*b*x)/(d*g - c*h))/x, x], x, (g + h*x)/(c + d*x)], x] /; FreeQ[{F, a, b, c, d, e, f}, x] && EqQ[d*e - c*f, 0]
  2229. Int[Times[Power[Pattern[F, Blank[]], Plus[Optional[Pattern[e, Blank[]]], Times[Optional[Pattern[f, Blank[]]], Plus[Optional[Pattern[a, Blank[]]], Times[Optional[Pattern[b, Blank[]]], Pattern[x, Blank[]]]], Power[Plus[Optional[Pattern[c, Blank[]]], Times[Optional[Pattern[d, Blank[]]], Pattern[x, Blank[]]]], -1]]]], Power[Plus[Optional[Pattern[g, Blank[]]], Times[Optional[Pattern[h, Blank[]]], Pattern[x, Blank[]]]], Optional[Pattern[m, Blank[]]]]], Pattern[x, Blank[Symbol]]] := Dist[F^(e + (f*b)/d), Int[(g + h*x)^m, x], x] /; FreeQ[{F, a, b, c, d, e, f, g, h, m}, x] && EqQ[b*c - a*d, 0]
  2230. Int[Times[Power[Pattern[F, Blank[]], Plus[Optional[Pattern[e, Blank[]]], Times[Optional[Pattern[f, Blank[]]], Plus[Optional[Pattern[a, Blank[]]], Times[Optional[Pattern[b, Blank[]]], Pattern[x, Blank[]]]], Power[Plus[Optional[Pattern[c, Blank[]]], Times[Optional[Pattern[d, Blank[]]], Pattern[x, Blank[]]]], -1]]]], Power[Plus[Optional[Pattern[g, Blank[]]], Times[Optional[Pattern[h, Blank[]]], Pattern[x, Blank[]]]], Optional[Pattern[m, Blank[]]]]], Pattern[x, Blank[Symbol]]] := Int[(g + h*x)^m*F^((d*e + b*f)/d - (f*(b*c - a*d))/(d*(c + d*x))), x] /; FreeQ[{F, a, b, c, d, e, f, g, h, m}, x] && NeQ[b*c - a*d, 0] && EqQ[d*g - c*h, 0]
  2231. Int[Times[Power[Pattern[F, Blank[]], Plus[Optional[Pattern[e, Blank[]]], Times[Optional[Pattern[f, Blank[]]], Plus[Optional[Pattern[a, Blank[]]], Times[Optional[Pattern[b, Blank[]]], Pattern[x, Blank[]]]], Power[Plus[Optional[Pattern[c, Blank[]]], Times[Optional[Pattern[d, Blank[]]], Pattern[x, Blank[]]]], -1]]]], Power[Plus[Optional[Pattern[g, Blank[]]], Times[Optional[Pattern[h, Blank[]]], Pattern[x, Blank[]]]], -1]], Pattern[x, Blank[Symbol]]] := Dist[d/h, Int[F^(e + (f*(a + b*x))/(c + d*x))/(c + d*x), x], x] - Dist[(d*g - c*h)/h, Int[F^(e + (f*(a + b*x))/(c + d*x))/((c + d*x)*(g + h*x)), x], x] /; FreeQ[{F, a, b, c, d, e, f, g, h}, x] && NeQ[b*c - a*d, 0] && NeQ[d*g - c*h, 0]
  2232. Int[Times[Power[Pattern[F, Blank[]], Plus[Optional[Pattern[e, Blank[]]], Times[Optional[Pattern[f, Blank[]]], Plus[Optional[Pattern[a, Blank[]]], Times[Optional[Pattern[b, Blank[]]], Pattern[x, Blank[]]]], Power[Plus[Optional[Pattern[c, Blank[]]], Times[Optional[Pattern[d, Blank[]]], Pattern[x, Blank[]]]], -1]]]], Power[Plus[Optional[Pattern[g, Blank[]]], Times[Optional[Pattern[h, Blank[]]], Pattern[x, Blank[]]]], Pattern[m, Blank[]]]], Pattern[x, Blank[Symbol]]] := Simp[((g + h*x)^(m + 1)*F^(e + (f*(a + b*x))/(c + d*x)))/(h*(m + 1)), x] - Dist[(f*(b*c - a*d)*Log[F])/(h*(m + 1)), Int[((g + h*x)^(m + 1)*F^(e + (f*(a + b*x))/(c + d*x)))/(c + d*x)^2, x], x] /; FreeQ[{F, a, b, c, d, e, f, g, h}, x] && NeQ[b*c - a*d, 0] && NeQ[d*g - c*h, 0] && ILtQ[m, -1]
  2233. Int[Times[Power[Pattern[F, Blank[]], Plus[Optional[Pattern[e, Blank[]]], Times[Optional[Pattern[f, Blank[]]], Plus[Optional[Pattern[a, Blank[]]], Times[Optional[Pattern[b, Blank[]]], Pattern[x, Blank[]]]], Power[Plus[Optional[Pattern[c, Blank[]]], Times[Optional[Pattern[d, Blank[]]], Pattern[x, Blank[]]]], -1]]]], Power[Plus[Optional[Pattern[g, Blank[]]], Times[Optional[Pattern[h, Blank[]]], Pattern[x, Blank[]]]], -1], Power[Plus[Optional[Pattern[i, Blank[]]], Times[Optional[Pattern[j, Blank[]]], Pattern[x, Blank[]]]], -1]], Pattern[x, Blank[Symbol]]] := -Dist[d/(h*(d*i - c*j)), Subst[Int[F^(e + (f*(b*i - a*j))/(d*i - c*j) - ((b*c - a*d)*f*x)/(d*i - c*j))/x, x], x, (i + j*x)/(c + d*x)], x] /; FreeQ[{F, a, b, c, d, e, f, g, h}, x] && EqQ[d*g - c*h, 0]
  2234. Int[Power[Pattern[F, Blank[]], Plus[Optional[Pattern[a, Blank[]]], Times[Optional[Pattern[b, Blank[]]], Pattern[x, Blank[]]], Times[Optional[Pattern[c, Blank[]]], Power[Pattern[x, Blank[]], 2]]]], Pattern[x, Blank[Symbol]]] := Dist[F^(a - b^2/(4*c)), Int[F^((b + 2*c*x)^2/(4*c)), x], x] /; FreeQ[{F, a, b, c}, x]
  2235. Int[Power[Pattern[F, Blank[]], Pattern[v, Blank[]]], Pattern[x, Blank[Symbol]]] := Int[F^ExpandToSum[v, x], x] /; FreeQ[F, x] && QuadraticQ[v, x] && !QuadraticMatchQ[v, x]
  2236. Int[Times[Power[Pattern[F, Blank[]], Plus[Optional[Pattern[a, Blank[]]], Times[Optional[Pattern[b, Blank[]]], Pattern[x, Blank[]]], Times[Optional[Pattern[c, Blank[]]], Power[Pattern[x, Blank[]], 2]]]], Plus[Optional[Pattern[d, Blank[]]], Times[Optional[Pattern[e, Blank[]]], Pattern[x, Blank[]]]]], Pattern[x, Blank[Symbol]]] := Simp[(e*F^(a + b*x + c*x^2))/(2*c*Log[F]), x] /; FreeQ[{F, a, b, c, d, e}, x] && EqQ[b*e - 2*c*d, 0]
  2237. Int[Times[Power[Pattern[F, Blank[]], Plus[Optional[Pattern[a, Blank[]]], Times[Optional[Pattern[b, Blank[]]], Pattern[x, Blank[]]], Times[Optional[Pattern[c, Blank[]]], Power[Pattern[x, Blank[]], 2]]]], Power[Plus[Optional[Pattern[d, Blank[]]], Times[Optional[Pattern[e, Blank[]]], Pattern[x, Blank[]]]], Pattern[m, Blank[]]]], Pattern[x, Blank[Symbol]]] := Simp[(e*(d + e*x)^(m - 1)*F^(a + b*x + c*x^2))/(2*c*Log[F]), x] - Dist[((m - 1)*e^2)/(2*c*Log[F]), Int[(d + e*x)^(m - 2)*F^(a + b*x + c*x^2), x], x] /; FreeQ[{F, a, b, c, d, e}, x] && EqQ[b*e - 2*c*d, 0] && GtQ[m, 1]
  2238. Int[Times[Power[Pattern[F, Blank[]], Plus[Optional[Pattern[a, Blank[]]], Times[Optional[Pattern[b, Blank[]]], Pattern[x, Blank[]]], Times[Optional[Pattern[c, Blank[]]], Power[Pattern[x, Blank[]], 2]]]], Power[Plus[Optional[Pattern[d, Blank[]]], Times[Optional[Pattern[e, Blank[]]], Pattern[x, Blank[]]]], -1]], Pattern[x, Blank[Symbol]]] := Simp[(1*F^(a - b^2/(4*c))*ExpIntegralEi[((b + 2*c*x)^2*Log[F])/(4*c)])/(2*e), x] /; FreeQ[{F, a, b, c, d, e}, x] && EqQ[b*e - 2*c*d, 0]
  2239. Int[Times[Power[Pattern[F, Blank[]], Plus[Optional[Pattern[a, Blank[]]], Times[Optional[Pattern[b, Blank[]]], Pattern[x, Blank[]]], Times[Optional[Pattern[c, Blank[]]], Power[Pattern[x, Blank[]], 2]]]], Power[Plus[Optional[Pattern[d, Blank[]]], Times[Optional[Pattern[e, Blank[]]], Pattern[x, Blank[]]]], Pattern[m, Blank[]]]], Pattern[x, Blank[Symbol]]] := Simp[((d + e*x)^(m + 1)*F^(a + b*x + c*x^2))/(e*(m + 1)), x] - Dist[(2*c*Log[F])/(e^2*(m + 1)), Int[(d + e*x)^(m + 2)*F^(a + b*x + c*x^2), x], x] /; FreeQ[{F, a, b, c, d, e}, x] && EqQ[b*e - 2*c*d, 0] && LtQ[m, -1]
  2240. Int[Times[Power[Pattern[F, Blank[]], Plus[Optional[Pattern[a, Blank[]]], Times[Optional[Pattern[b, Blank[]]], Pattern[x, Blank[]]], Times[Optional[Pattern[c, Blank[]]], Power[Pattern[x, Blank[]], 2]]]], Plus[Optional[Pattern[d, Blank[]]], Times[Optional[Pattern[e, Blank[]]], Pattern[x, Blank[]]]]], Pattern[x, Blank[Symbol]]] := Simp[(e*F^(a + b*x + c*x^2))/(2*c*Log[F]), x] - Dist[(b*e - 2*c*d)/(2*c), Int[F^(a + b*x + c*x^2), x], x] /; FreeQ[{F, a, b, c, d, e}, x] && NeQ[b*e - 2*c*d, 0]
  2241. Int[Times[Power[Pattern[F, Blank[]], Plus[Optional[Pattern[a, Blank[]]], Times[Optional[Pattern[b, Blank[]]], Pattern[x, Blank[]]], Times[Optional[Pattern[c, Blank[]]], Power[Pattern[x, Blank[]], 2]]]], Power[Plus[Optional[Pattern[d, Blank[]]], Times[Optional[Pattern[e, Blank[]]], Pattern[x, Blank[]]]], Pattern[m, Blank[]]]], Pattern[x, Blank[Symbol]]] := Simp[(e*(d + e*x)^(m - 1)*F^(a + b*x + c*x^2))/(2*c*Log[F]), x] + (-Dist[(b*e - 2*c*d)/(2*c), Int[(d + e*x)^(m - 1)*F^(a + b*x + c*x^2), x], x] - Dist[((m - 1)*e^2)/(2*c*Log[F]), Int[(d + e*x)^(m - 2)*F^(a + b*x + c*x^2), x], x]) /; FreeQ[{F, a, b, c, d, e}, x] && NeQ[b*e - 2*c*d, 0] && GtQ[m, 1]
  2242. Int[Times[Power[Pattern[F, Blank[]], Plus[Optional[Pattern[a, Blank[]]], Times[Optional[Pattern[b, Blank[]]], Pattern[x, Blank[]]], Times[Optional[Pattern[c, Blank[]]], Power[Pattern[x, Blank[]], 2]]]], Power[Plus[Optional[Pattern[d, Blank[]]], Times[Optional[Pattern[e, Blank[]]], Pattern[x, Blank[]]]], Pattern[m, Blank[]]]], Pattern[x, Blank[Symbol]]] := Simp[((d + e*x)^(m + 1)*F^(a + b*x + c*x^2))/(e*(m + 1)), x] + (-Dist[(2*c*Log[F])/(e^2*(m + 1)), Int[(d + e*x)^(m + 2)*F^(a + b*x + c*x^2), x], x] - Dist[((b*e - 2*c*d)*Log[F])/(e^2*(m + 1)), Int[(d + e*x)^(m + 1)*F^(a + b*x + c*x^2), x], x]) /; FreeQ[{F, a, b, c, d, e}, x] && NeQ[b*e - 2*c*d, 0] && LtQ[m, -1]
  2243. Int[Times[Power[Pattern[F, Blank[]], Plus[Optional[Pattern[a, Blank[]]], Times[Optional[Pattern[b, Blank[]]], Pattern[x, Blank[]]], Times[Optional[Pattern[c, Blank[]]], Power[Pattern[x, Blank[]], 2]]]], Power[Plus[Optional[Pattern[d, Blank[]]], Times[Optional[Pattern[e, Blank[]]], Pattern[x, Blank[]]]], Optional[Pattern[m, Blank[]]]]], Pattern[x, Blank[Symbol]]] := Unintegrable[F^(a + b*x + c*x^2)*(d + e*x)^m, x] /; FreeQ[{F, a, b, c, d, e, m}, x]
  2244. Int[Times[Power[Pattern[F, Blank[]], Pattern[v, Blank[]]], Power[Pattern[u, Blank[]], Optional[Pattern[m, Blank[]]]]], Pattern[x, Blank[Symbol]]] := Int[ExpandToSum[u, x]^m*F^ExpandToSum[v, x], x] /; FreeQ[{F, m}, x] && LinearQ[u, x] && QuadraticQ[v, x] && !(LinearMatchQ[u, x] && QuadraticMatchQ[v, x])
  2245. Int[Times[Power[Pattern[F, Blank[]], Times[Optional[Pattern[e, Blank[]]], Plus[Optional[Pattern[c, Blank[]]], Times[Optional[Pattern[d, Blank[]]], Pattern[x, Blank[]]]]]], Power[Plus[Optional[Pattern[a, Blank[]]], Times[Optional[Pattern[b, Blank[]]], Power[Pattern[F, Blank[]], Pattern[v, Blank[]]]]], Pattern[p, Blank[]]], Power[Pattern[x, Blank[]], Optional[Pattern[m, Blank[]]]]], Pattern[x, Blank[Symbol]]] := With[{u = IntHide[F^(e*(c + d*x))*(a + b*F^v)^p, x]}, Dist[x^m, u, x] - Dist[m, Int[x^(m - 1)*u, x], x]] /; FreeQ[{F, a, b, c, d, e}, x] && EqQ[v, 2*e*(c + d*x)] && GtQ[m, 0] && ILtQ[p, 0]
  2246. Int[Times[Power[Power[Pattern[F, Blank[]], Times[Optional[Pattern[e, Blank[]]], Plus[Optional[Pattern[c, Blank[]]], Times[Optional[Pattern[d, Blank[]]], Pattern[x, Blank[]]]]]], Optional[Pattern[n, Blank[]]]], Power[Plus[Pattern[a, Blank[]], Times[Optional[Pattern[b, Blank[]]], Power[Power[Pattern[F, Blank[]], Times[Optional[Pattern[e, Blank[]]], Plus[Optional[Pattern[c, Blank[]]], Times[Optional[Pattern[d, Blank[]]], Pattern[x, Blank[]]]]]], Optional[Pattern[n, Blank[]]]]]], Optional[Pattern[p, Blank[]]]]], Pattern[x, Blank[Symbol]]] := Dist[1/(d*e*n*Log[F]), Subst[Int[(a + b*x)^p, x], x, (F^(e*(c + d*x)))^n], x] /; FreeQ[{F, a, b, c, d, e, n, p}, x]
  2247. Int[Times[Power[Plus[Pattern[a, Blank[]], Times[Optional[Pattern[b, Blank[]]], Power[Power[Pattern[F, Blank[]], Times[Optional[Pattern[e, Blank[]]], Plus[Optional[Pattern[c, Blank[]]], Times[Optional[Pattern[d, Blank[]]], Pattern[x, Blank[]]]]]], Optional[Pattern[n, Blank[]]]]]], Optional[Pattern[p, Blank[]]]], Power[Power[Pattern[G, Blank[]], Times[Optional[Pattern[h, Blank[]]], Plus[Optional[Pattern[f, Blank[]]], Times[Optional[Pattern[g, Blank[]]], Pattern[x, Blank[]]]]]], Optional[Pattern[m, Blank[]]]]], Pattern[x, Blank[Symbol]]] := Dist[(G^(h*(f + g*x)))^m/(F^(e*(c + d*x)))^n, Int[(F^(e*(c + d*x)))^n*(a + b*(F^(e*(c + d*x)))^n)^p, x], x] /; FreeQ[{F, G, a, b, c, d, e, f, g, h, m, n, p}, x] && EqQ[d*e*n*Log[F], g*h*m*Log[G]]
  2248. Int[Times[Power[Plus[Pattern[a, Blank[]], Times[Optional[Pattern[b, Blank[]]], Power[Pattern[F, Blank[]], Times[Optional[Pattern[e, Blank[]]], Plus[Optional[Pattern[c, Blank[]]], Times[Optional[Pattern[d, Blank[]]], Pattern[x, Blank[]]]]]]]], Optional[Pattern[p, Blank[]]]], Power[Pattern[G, Blank[]], Times[Optional[Pattern[h, Blank[]]], Plus[Optional[Pattern[f, Blank[]]], Times[Optional[Pattern[g, Blank[]]], Pattern[x, Blank[]]]]]]], Pattern[x, Blank[Symbol]]] := With[{m = FullSimplify[(g*h*Log[G])/(d*e*Log[F])]}, Dist[(Denominator[m]*G^(f*h - (c*g*h)/d))/(d*e*Log[F]), Subst[Int[x^(Numerator[m] - 1)*(a + b*x^Denominator[m])^p, x], x, F^((e*(c + d*x))/Denominator[m])], x] /; LeQ[m, -1] || GeQ[m, 1]] /; FreeQ[{F, G, a, b, c, d, e, f, g, h, p}, x]
  2249. Int[Times[Power[Plus[Pattern[a, Blank[]], Times[Optional[Pattern[b, Blank[]]], Power[Pattern[F, Blank[]], Times[Optional[Pattern[e, Blank[]]], Plus[Optional[Pattern[c, Blank[]]], Times[Optional[Pattern[d, Blank[]]], Pattern[x, Blank[]]]]]]]], Optional[Pattern[p, Blank[]]]], Power[Pattern[G, Blank[]], Times[Optional[Pattern[h, Blank[]]], Plus[Optional[Pattern[f, Blank[]]], Times[Optional[Pattern[g, Blank[]]], Pattern[x, Blank[]]]]]]], Pattern[x, Blank[Symbol]]] := With[{m = FullSimplify[(d*e*Log[F])/(g*h*Log[G])]}, Dist[Denominator[m]/(g*h*Log[G]), Subst[Int[x^(Denominator[m] - 1)*(a + b*F^(c*e - (d*e*f)/g)*x^Numerator[m])^p, x], x, G^((h*(f + g*x))/Denominator[m])], x] /; LtQ[m, -1] || GtQ[m, 1]] /; FreeQ[{F, G, a, b, c, d, e, f, g, h, p}, x]
  2250. Int[Times[Power[Plus[Pattern[a, Blank[]], Times[Optional[Pattern[b, Blank[]]], Power[Pattern[F, Blank[]], Times[Optional[Pattern[e, Blank[]]], Plus[Optional[Pattern[c, Blank[]]], Times[Optional[Pattern[d, Blank[]]], Pattern[x, Blank[]]]]]]]], Optional[Pattern[p, Blank[]]]], Power[Pattern[G, Blank[]], Times[Optional[Pattern[h, Blank[]]], Plus[Optional[Pattern[f, Blank[]]], Times[Optional[Pattern[g, Blank[]]], Pattern[x, Blank[]]]]]]], Pattern[x, Blank[Symbol]]] := Int[Expand[G^(h*(f + g*x))*(a + b*F^(e*(c + d*x)))^p, x], x] /; FreeQ[{F, G, a, b, c, d, e, f, g, h}, x] && IGtQ[p, 0]
  2251. Int[Times[Power[Plus[Pattern[a, Blank[]], Times[Optional[Pattern[b, Blank[]]], Power[Pattern[F, Blank[]], Times[Optional[Pattern[e, Blank[]]], Plus[Optional[Pattern[c, Blank[]]], Times[Optional[Pattern[d, Blank[]]], Pattern[x, Blank[]]]]]]]], Pattern[p, Blank[]]], Power[Pattern[G, Blank[]], Times[Optional[Pattern[h, Blank[]]], Plus[Optional[Pattern[f, Blank[]]], Times[Optional[Pattern[g, Blank[]]], Pattern[x, Blank[]]]]]]], Pattern[x, Blank[Symbol]]] := Simp[(a^p*G^(h*(f + g*x))*Hypergeometric2F1[-p, (g*h*Log[G])/(d*e*Log[F]), (g*h*Log[G])/(d*e*Log[F]) + 1, Simplify[-((b*F^(e*(c + d*x)))/a)]])/(g*h*Log[G]), x] /; FreeQ[{F, G, a, b, c, d, e, f, g, h, p}, x] && (ILtQ[p, 0] || GtQ[a, 0])
  2252. Int[Times[Power[Plus[Pattern[a, Blank[]], Times[Optional[Pattern[b, Blank[]]], Power[Pattern[F, Blank[]], Times[Optional[Pattern[e, Blank[]]], Plus[Optional[Pattern[c, Blank[]]], Times[Optional[Pattern[d, Blank[]]], Pattern[x, Blank[]]]]]]]], Pattern[p, Blank[]]], Power[Pattern[G, Blank[]], Times[Optional[Pattern[h, Blank[]]], Plus[Optional[Pattern[f, Blank[]]], Times[Optional[Pattern[g, Blank[]]], Pattern[x, Blank[]]]]]]], Pattern[x, Blank[Symbol]]] := Dist[(a + b*F^(e*(c + d*x)))^p/(1 + (b/a)*F^(e*(c + d*x)))^p, Int[G^(h*(f + g*x))*(1 + (b*F^(e*(c + d*x)))/a)^p, x], x] /; FreeQ[{F, G, a, b, c, d, e, f, g, h, p}, x] && !(ILtQ[p, 0] || GtQ[a, 0])
  2253. Int[Times[Power[Plus[Pattern[a, Blank[]], Times[Optional[Pattern[b, Blank[]]], Power[Pattern[F, Blank[]], Times[Optional[Pattern[e, Blank[]]], Pattern[v, Blank[]]]]]], Pattern[p, Blank[]]], Power[Pattern[G, Blank[]], Times[Optional[Pattern[h, Blank[]]], Pattern[u, Blank[]]]]], Pattern[x, Blank[Symbol]]] := Int[G^(h*ExpandToSum[u, x])*(a + b*F^(e*ExpandToSum[v, x]))^p, x] /; FreeQ[{F, G, a, b, e, h, p}, x] && LinearQ[{u, v}, x] && !LinearMatchQ[{u, v}, x]
  2254. Int[Times[Power[Plus[Optional[Pattern[a, Blank[]]], Times[Optional[Pattern[b, Blank[]]], Power[Pattern[F, Blank[]], Pattern[u, Blank[]]]]], Optional[Pattern[p, Blank[]]]], Power[Plus[Optional[Pattern[c, Blank[]]], Times[Optional[Pattern[d, Blank[]]], Power[Pattern[F, Blank[]], Pattern[v, Blank[]]]]], Optional[Pattern[q, Blank[]]]], Power[Plus[Optional[Pattern[e, Blank[]]], Times[Optional[Pattern[f, Blank[]]], Pattern[x, Blank[]]]], Optional[Pattern[m, Blank[]]]]], Pattern[x, Blank[Symbol]]] := With[{w = ExpandIntegrand[(e + f*x)^m, (a + b*F^u)^p*(c + d*F^v)^q, x]}, Int[w, x] /; SumQ[w]] /; FreeQ[{F, a, b, c, d, e, f, m}, x] && IntegersQ[p, q] && LinearQ[{u, v}, x] && RationalQ[Simplify[u/v]]
  2255. Int[Times[Power[Plus[Pattern[a, Blank[]], Times[Optional[Pattern[b, Blank[]]], Power[Pattern[F, Blank[]], Times[Optional[Pattern[e, Blank[]]], Plus[Optional[Pattern[c, Blank[]]], Times[Optional[Pattern[d, Blank[]]], Pattern[x, Blank[]]]]]]]], Optional[Pattern[p, Blank[]]]], Power[Pattern[G, Blank[]], Times[Optional[Pattern[h, Blank[]]], Plus[Optional[Pattern[f, Blank[]]], Times[Optional[Pattern[g, Blank[]]], Pattern[x, Blank[]]]]]], Power[Pattern[H, Blank[]], Times[Optional[Pattern[t, Blank[]]], Plus[Optional[Pattern[r, Blank[]]], Times[Optional[Pattern[s, Blank[]]], Pattern[x, Blank[]]]]]]], Pattern[x, Blank[Symbol]]] := With[{m = FullSimplify[(g*h*Log[G] + s*t*Log[H])/(d*e*Log[F])]}, Dist[(Denominator[m]*G^(f*h - (c*g*h)/d)*H^(r*t - (c*s*t)/d))/(d*e*Log[F]), Subst[Int[x^(Numerator[m] - 1)*(a + b*x^Denominator[m])^p, x], x, F^((e*(c + d*x))/Denominator[m])], x] /; RationalQ[m]] /; FreeQ[{F, G, H, a, b, c, d, e, f, g, h, r, s, t, p}, x]
  2256. Int[Times[Power[Plus[Pattern[a, Blank[]], Times[Optional[Pattern[b, Blank[]]], Power[Pattern[F, Blank[]], Times[Optional[Pattern[e, Blank[]]], Plus[Optional[Pattern[c, Blank[]]], Times[Optional[Pattern[d, Blank[]]], Pattern[x, Blank[]]]]]]]], Optional[Pattern[p, Blank[]]]], Power[Pattern[G, Blank[]], Times[Optional[Pattern[h, Blank[]]], Plus[Optional[Pattern[f, Blank[]]], Times[Optional[Pattern[g, Blank[]]], Pattern[x, Blank[]]]]]], Power[Pattern[H, Blank[]], Times[Optional[Pattern[t, Blank[]]], Plus[Optional[Pattern[r, Blank[]]], Times[Optional[Pattern[s, Blank[]]], Pattern[x, Blank[]]]]]]], Pattern[x, Blank[Symbol]]] := Dist[G^((f - (c*g)/d)*h), Int[H^(t*(r + s*x))*(b + a/F^(e*(c + d*x)))^p, x], x] /; FreeQ[{F, G, H, a, b, c, d, e, f, g, h, r, s, t}, x] && EqQ[d*e*p*Log[F] + g*h*Log[G], 0] && IntegerQ[p]
  2257. Int[Times[Power[Plus[Pattern[a, Blank[]], Times[Optional[Pattern[b, Blank[]]], Power[Pattern[F, Blank[]], Times[Optional[Pattern[e, Blank[]]], Plus[Optional[Pattern[c, Blank[]]], Times[Optional[Pattern[d, Blank[]]], Pattern[x, Blank[]]]]]]]], Optional[Pattern[p, Blank[]]]], Power[Pattern[G, Blank[]], Times[Optional[Pattern[h, Blank[]]], Plus[Optional[Pattern[f, Blank[]]], Times[Optional[Pattern[g, Blank[]]], Pattern[x, Blank[]]]]]], Power[Pattern[H, Blank[]], Times[Optional[Pattern[t, Blank[]]], Plus[Optional[Pattern[r, Blank[]]], Times[Optional[Pattern[s, Blank[]]], Pattern[x, Blank[]]]]]]], Pattern[x, Blank[Symbol]]] := Int[Expand[G^(h*(f + g*x))*H^(t*(r + s*x))*(a + b*F^(e*(c + d*x)))^p, x], x] /; FreeQ[{F, G, H, a, b, c, d, e, f, g, h, r, s, t}, x] && IGtQ[p, 0]
  2258. Int[Times[Power[Plus[Pattern[a, Blank[]], Times[Optional[Pattern[b, Blank[]]], Power[Pattern[F, Blank[]], Times[Optional[Pattern[e, Blank[]]], Plus[Optional[Pattern[c, Blank[]]], Times[Optional[Pattern[d, Blank[]]], Pattern[x, Blank[]]]]]]]], Pattern[p, Blank[]]], Power[Pattern[G, Blank[]], Times[Optional[Pattern[h, Blank[]]], Plus[Optional[Pattern[f, Blank[]]], Times[Optional[Pattern[g, Blank[]]], Pattern[x, Blank[]]]]]], Power[Pattern[H, Blank[]], Times[Optional[Pattern[t, Blank[]]], Plus[Optional[Pattern[r, Blank[]]], Times[Optional[Pattern[s, Blank[]]], Pattern[x, Blank[]]]]]]], Pattern[x, Blank[Symbol]]] := Simp[(a^p*G^(h*(f + g*x))*H^(t*(r + s*x))*Hypergeometric2F1[-p, (g*h*Log[G] + s*t*Log[H])/(d*e*Log[F]), (g*h*Log[G] + s*t*Log[H])/(d*e*Log[F]) + 1, Simplify[-((b*F^(e*(c + d*x)))/a)]])/(g*h*Log[G] + s*t*Log[H]), x] /; FreeQ[{F, G, H, a, b, c, d, e, f, g, h, r, s, t}, x] && ILtQ[p, 0]
  2259. Int[Times[Power[Plus[Pattern[a, Blank[]], Times[Optional[Pattern[b, Blank[]]], Power[Pattern[F, Blank[]], Times[Optional[Pattern[e, Blank[]]], Plus[Optional[Pattern[c, Blank[]]], Times[Optional[Pattern[d, Blank[]]], Pattern[x, Blank[]]]]]]]], Pattern[p, Blank[]]], Power[Pattern[G, Blank[]], Times[Optional[Pattern[h, Blank[]]], Plus[Optional[Pattern[f, Blank[]]], Times[Optional[Pattern[g, Blank[]]], Pattern[x, Blank[]]]]]], Power[Pattern[H, Blank[]], Times[Optional[Pattern[t, Blank[]]], Plus[Optional[Pattern[r, Blank[]]], Times[Optional[Pattern[s, Blank[]]], Pattern[x, Blank[]]]]]]], Pattern[x, Blank[Symbol]]] := Simp[(G^(h*(f + g*x))*H^(t*(r + s*x))*(a + b*F^(e*(c + d*x)))^p*Hypergeometric2F1[-p, (g*h*Log[G] + s*t*Log[H])/(d*e*Log[F]), (g*h*Log[G] + s*t*Log[H])/(d*e*Log[F]) + 1, Simplify[-((b*F^(e*(c + d*x)))/a)]])/((g*h*Log[G] + s*t*Log[H])*((a + b*F^(e*(c + d*x)))/a)^p), x] /; FreeQ[{F, G, H, a, b, c, d, e, f, g, h, r, s, t, p}, x] && !IntegerQ[p]
  2260. Int[Times[Power[Plus[Pattern[a, Blank[]], Times[Optional[Pattern[b, Blank[]]], Power[Pattern[F, Blank[]], Times[Optional[Pattern[e, Blank[]]], Pattern[v, Blank[]]]]]], Pattern[p, Blank[]]], Power[Pattern[G, Blank[]], Times[Optional[Pattern[h, Blank[]]], Pattern[u, Blank[]]]], Power[Pattern[H, Blank[]], Times[Optional[Pattern[t, Blank[]]], Pattern[w, Blank[]]]]], Pattern[x, Blank[Symbol]]] := Int[G^(h*ExpandToSum[u, x])*H^(t*ExpandToSum[w, x])*(a + b*F^(e*ExpandToSum[v, x]))^p, x] /; FreeQ[{F, G, H, a, b, e, h, t, p}, x] && LinearQ[{u, v, w}, x] && !LinearMatchQ[{u, v, w}, x]
  2261. Int[Times[Power[Pattern[F, Blank[]], Times[Optional[Pattern[e, Blank[]]], Plus[Optional[Pattern[c, Blank[]]], Times[Optional[Pattern[d, Blank[]]], Pattern[x, Blank[]]]]]], Power[Plus[Times[Optional[Pattern[b, Blank[]]], Power[Pattern[F, Blank[]], Times[Optional[Pattern[e, Blank[]]], Plus[Optional[Pattern[c, Blank[]]], Times[Optional[Pattern[d, Blank[]]], Pattern[x, Blank[]]]]]]], Times[Optional[Pattern[a, Blank[]]], Power[Pattern[x, Blank[]], Optional[Pattern[n, Blank[]]]]]], Optional[Pattern[p, Blank[]]]]], Pattern[x, Blank[Symbol]]] := Simp[(a*x^n + b*F^(e*(c + d*x)))^(p + 1)/(b*d*e*(p + 1)*Log[F]), x] - Dist[(a*n)/(b*d*e*Log[F]), Int[x^(n - 1)*(a*x^n + b*F^(e*(c + d*x)))^p, x], x] /; FreeQ[{F, a, b, c, d, e, n, p}, x] && NeQ[p, -1]
  2262. Int[Times[Power[Pattern[F, Blank[]], Times[Optional[Pattern[e, Blank[]]], Plus[Optional[Pattern[c, Blank[]]], Times[Optional[Pattern[d, Blank[]]], Pattern[x, Blank[]]]]]], Power[Pattern[x, Blank[]], Optional[Pattern[m, Blank[]]]], Power[Plus[Times[Optional[Pattern[b, Blank[]]], Power[Pattern[F, Blank[]], Times[Optional[Pattern[e, Blank[]]], Plus[Optional[Pattern[c, Blank[]]], Times[Optional[Pattern[d, Blank[]]], Pattern[x, Blank[]]]]]]], Times[Optional[Pattern[a, Blank[]]], Power[Pattern[x, Blank[]], Optional[Pattern[n, Blank[]]]]]], Optional[Pattern[p, Blank[]]]]], Pattern[x, Blank[Symbol]]] := Simp[(x^m*(a*x^n + b*F^(e*(c + d*x)))^(p + 1))/(b*d*e*(p + 1)*Log[F]), x] + (-Dist[m/(b*d*e*(p + 1)*Log[F]), Int[x^(m - 1)*(a*x^n + b*F^(e*(c + d*x)))^(p + 1), x], x] - Dist[(a*n)/(b*d*e*Log[F]), Int[x^(m + n - 1)*(a*x^n + b*F^(e*(c + d*x)))^p, x], x]) /; FreeQ[{F, a, b, c, d, e, m, n, p}, x] && NeQ[p, -1]
  2263. Int[Times[Power[Plus[Optional[Pattern[a, Blank[]]], Times[Optional[Pattern[b, Blank[]]], Power[Pattern[F, Blank[]], Pattern[u, Blank[]]]], Times[Optional[Pattern[c, Blank[]]], Power[Pattern[F, Blank[]], Pattern[v, Blank[]]]]], -1], Power[Plus[Optional[Pattern[f, Blank[]]], Times[Optional[Pattern[g, Blank[]]], Pattern[x, Blank[]]]], Optional[Pattern[m, Blank[]]]]], Pattern[x, Blank[Symbol]]] := With[{q = Rt[b^2 - 4*a*c, 2]}, Dist[(2*c)/q, Int[(f + g*x)^m/(b - q + 2*c*F^u), x], x] - Dist[(2*c)/q, Int[(f + g*x)^m/(b + q + 2*c*F^u), x], x]] /; FreeQ[{F, a, b, c, f, g}, x] && EqQ[v, 2*u] && LinearQ[u, x] && NeQ[b^2 - 4*a*c, 0] && IGtQ[m, 0]
  2264. Int[Times[Power[Pattern[F, Blank[]], Pattern[u, Blank[]]], Power[Plus[Optional[Pattern[a, Blank[]]], Times[Optional[Pattern[b, Blank[]]], Power[Pattern[F, Blank[]], Pattern[u, Blank[]]]], Times[Optional[Pattern[c, Blank[]]], Power[Pattern[F, Blank[]], Pattern[v, Blank[]]]]], -1], Power[Plus[Optional[Pattern[f, Blank[]]], Times[Optional[Pattern[g, Blank[]]], Pattern[x, Blank[]]]], Optional[Pattern[m, Blank[]]]]], Pattern[x, Blank[Symbol]]] := With[{q = Rt[b^2 - 4*a*c, 2]}, Dist[(2*c)/q, Int[((f + g*x)^m*F^u)/(b - q + 2*c*F^u), x], x] - Dist[(2*c)/q, Int[((f + g*x)^m*F^u)/(b + q + 2*c*F^u), x], x]] /; FreeQ[{F, a, b, c, f, g}, x] && EqQ[v, 2*u] && LinearQ[u, x] && NeQ[b^2 - 4*a*c, 0] && IGtQ[m, 0]
  2265. Int[Times[Power[Plus[Optional[Pattern[a, Blank[]]], Times[Optional[Pattern[b, Blank[]]], Power[Pattern[F, Blank[]], Pattern[u, Blank[]]]], Times[Optional[Pattern[c, Blank[]]], Power[Pattern[F, Blank[]], Pattern[v, Blank[]]]]], -1], Plus[Times[Optional[Pattern[i, Blank[]]], Power[Pattern[F, Blank[]], Pattern[u, Blank[]]]], Pattern[h, Blank[]]], Power[Plus[Optional[Pattern[f, Blank[]]], Times[Optional[Pattern[g, Blank[]]], Pattern[x, Blank[]]]], Optional[Pattern[m, Blank[]]]]], Pattern[x, Blank[Symbol]]] := With[{q = Rt[b^2 - 4*a*c, 2]}, Dist[Simplify[(2*c*h - b*i)/q] + i, Int[(f + g*x)^m/(b - q + 2*c*F^u), x], x] - Dist[Simplify[(2*c*h - b*i)/q] - i, Int[(f + g*x)^m/(b + q + 2*c*F^u), x], x]] /; FreeQ[{F, a, b, c, f, g, h, i}, x] && EqQ[v, 2*u] && LinearQ[u, x] && NeQ[b^2 - 4*a*c, 0] && IGtQ[m, 0]
  2266. Int[Times[Power[Plus[Times[Optional[Pattern[b, Blank[]]], Power[Pattern[F, Blank[]], Pattern[v, Blank[]]]], Times[Optional[Pattern[a, Blank[]]], Power[Pattern[F, Blank[]], Plus[Optional[Pattern[c, Blank[]]], Times[Optional[Pattern[d, Blank[]]], Pattern[x, Blank[]]]]]]], -1], Power[Pattern[x, Blank[]], Optional[Pattern[m, Blank[]]]]], Pattern[x, Blank[Symbol]]] := With[{u = IntHide[1/(a*F^(c + d*x) + b*F^v), x]}, Simp[x^m*u, x] - Dist[m, Int[x^(m - 1)*u, x], x]] /; FreeQ[{F, a, b, c, d}, x] && EqQ[v, -(c + d*x)] && GtQ[m, 0]
  2267. Int[Times[Power[Plus[Pattern[a, Blank[]], Times[Optional[Pattern[b, Blank[]]], Power[Pattern[F, Blank[]], Pattern[v, Blank[]]]], Times[Optional[Pattern[c, Blank[]]], Power[Pattern[F, Blank[]], Pattern[w, Blank[]]]]], -1], Pattern[u, Blank[]]], Pattern[x, Blank[Symbol]]] := Int[(u*F^v)/(c + a*F^v + b*F^(2*v)), x] /; FreeQ[{F, a, b, c}, x] && EqQ[w, -v] && LinearQ[v, x] && If[RationalQ[Coefficient[v, x, 1]], GtQ[Coefficient[v, x, 1], 0], LtQ[LeafCount[v], LeafCount[w]]]
  2268. Int[Times[Power[Pattern[F, Blank[]], Times[Optional[Pattern[g, Blank[]]], Power[Plus[Optional[Pattern[d, Blank[]]], Times[Optional[Pattern[e, Blank[]]], Pattern[x, Blank[]]]], Optional[Pattern[n, Blank[]]]]]], Power[Plus[Optional[Pattern[a, Blank[]]], Times[Optional[Pattern[b, Blank[]]], Pattern[x, Blank[]]], Times[Optional[Pattern[c, Blank[]]], Power[Pattern[x, Blank[]], 2]]], -1]], Pattern[x, Blank[Symbol]]] := Int[ExpandIntegrand[F^(g*(d + e*x)^n), 1/(a + b*x + c*x^2), x], x] /; FreeQ[{F, a, b, c, d, e, g, n}, x]
  2269. Int[Times[Power[Pattern[F, Blank[]], Times[Optional[Pattern[g, Blank[]]], Power[Plus[Optional[Pattern[d, Blank[]]], Times[Optional[Pattern[e, Blank[]]], Pattern[x, Blank[]]]], Optional[Pattern[n, Blank[]]]]]], Power[Plus[Pattern[a, Blank[]], Times[Optional[Pattern[c, Blank[]]], Power[Pattern[x, Blank[]], 2]]], -1]], Pattern[x, Blank[Symbol]]] := Int[ExpandIntegrand[F^(g*(d + e*x)^n), 1/(a + c*x^2), x], x] /; FreeQ[{F, a, c, d, e, g, n}, x]
  2270. Int[Times[Power[Pattern[F, Blank[]], Times[Optional[Pattern[g, Blank[]]], Power[Plus[Optional[Pattern[d, Blank[]]], Times[Optional[Pattern[e, Blank[]]], Pattern[x, Blank[]]]], Optional[Pattern[n, Blank[]]]]]], Power[Pattern[u, Blank[]], Optional[Pattern[m, Blank[]]]], Power[Plus[Optional[Pattern[a, Blank[]]], Times[Optional[Pattern[b, Blank[]]], Pattern[x, Blank[]]], Times[Pattern[c, Blank[]], Power[Pattern[x, Blank[]], 2]]], -1]], Pattern[x, Blank[Symbol]]] := Int[ExpandIntegrand[F^(g*(d + e*x)^n), u^m/(a + b*x + c*x^2), x], x] /; FreeQ[{F, a, b, c, d, e, g, n}, x] && PolynomialQ[u, x] && IntegerQ[m]
  2271. Int[Times[Power[Pattern[F, Blank[]], Times[Optional[Pattern[g, Blank[]]], Power[Plus[Optional[Pattern[d, Blank[]]], Times[Optional[Pattern[e, Blank[]]], Pattern[x, Blank[]]]], Optional[Pattern[n, Blank[]]]]]], Power[Pattern[u, Blank[]], Optional[Pattern[m, Blank[]]]], Power[Plus[Pattern[a, Blank[]], Times[Pattern[c, Blank[]], Power[Pattern[x, Blank[]], 2]]], -1]], Pattern[x, Blank[Symbol]]] := Int[ExpandIntegrand[F^(g*(d + e*x)^n), u^m/(a + c*x^2), x], x] /; FreeQ[{F, a, c, d, e, g, n}, x] && PolynomialQ[u, x] && IntegerQ[m]
  2272. Int[Power[Pattern[F, Blank[]], Times[Power[Pattern[x, Blank[]], -2], Plus[Optional[Pattern[a, Blank[]]], Times[Optional[Pattern[b, Blank[]]], Power[Pattern[x, Blank[]], 4]]]]], Pattern[x, Blank[Symbol]]] := Simp[(Sqrt[Pi]*Exp[2*Sqrt[-(a*Log[F])]*Sqrt[-(b*Log[F])]]*Erf[(Sqrt[-(a*Log[F])] + Sqrt[-(b*Log[F])]*x^2)/x])/(4*Sqrt[-(b*Log[F])]), x] - Simp[(Sqrt[Pi]*Exp[-2*Sqrt[-(a*Log[F])]*Sqrt[-(b*Log[F])]]*Erf[(Sqrt[-(a*Log[F])] - Sqrt[-(b*Log[F])]*x^2)/x])/(4*Sqrt[-(b*Log[F])]), x] /; FreeQ[{F, a, b}, x]
  2273. Int[Times[Power[Pattern[x, Blank[]], Optional[Pattern[m, Blank[]]]], Power[Plus[Power[E, Pattern[x, Blank[]]], Power[Pattern[x, Blank[]], Optional[Pattern[m, Blank[]]]]], Pattern[n, Blank[]]]], Pattern[x, Blank[Symbol]]] := -Simp[(E^x + x^m)^(n + 1)/(n + 1), x] + (Dist[m, Int[x^(m - 1)*(E^x + x^m)^n, x], x] + Int[(E^x + x^m)^(n + 1), x]) /; RationalQ[m, n] && GtQ[m, 0] && LtQ[n, 0] && NeQ[n, -1]
  2274. Int[Times[Optional[Pattern[u, Blank[]]], Power[Pattern[F, Blank[]], Times[Optional[Pattern[a, Blank[]]], Plus[Times[Log[Pattern[z, Blank[]]], Optional[Pattern[b, Blank[]]]], Optional[Pattern[v, Blank[]]]]]]], Pattern[x, Blank[Symbol]]] := Int[u*F^(a*v)*z^(a*b*Log[F]), x] /; FreeQ[{F, a, b}, x]
  2275. Int[Power[Pattern[F, Blank[]], Times[Plus[Optional[Pattern[a, Blank[]]], Times[Power[Log[Times[Optional[Pattern[c, Blank[]]], Power[Pattern[x, Blank[]], Optional[Pattern[n, Blank[]]]]]], 2], Optional[Pattern[b, Blank[]]]]], Optional[Pattern[d, Blank[]]]]], Pattern[x, Blank[Symbol]]] := Dist[x/(n*(c*x^n)^(1/n)), Subst[Int[E^(a*d*Log[F] + x/n + b*d*Log[F]*x^2), x], x, Log[c*x^n]], x] /; FreeQ[{F, a, b, c, d, n}, x]
  2276. Int[Times[Power[Pattern[F, Blank[]], Times[Plus[Optional[Pattern[a, Blank[]]], Times[Power[Log[Times[Optional[Pattern[c, Blank[]]], Power[Pattern[x, Blank[]], Optional[Pattern[n, Blank[]]]]]], 2], Optional[Pattern[b, Blank[]]]]], Optional[Pattern[d, Blank[]]]]], Power[Times[Optional[Pattern[e, Blank[]]], Pattern[x, Blank[]]], Optional[Pattern[m, Blank[]]]]], Pattern[x, Blank[Symbol]]] := Dist[(e*x)^(m + 1)/(e*n*(c*x^n)^((m + 1)/n)), Subst[Int[E^(a*d*Log[F] + ((m + 1)*x)/n + b*d*Log[F]*x^2), x], x, Log[c*x^n]], x] /; FreeQ[{F, a, b, c, d, e, m, n}, x]
  2277. Int[Power[Pattern[F, Blank[]], Times[Power[Plus[Optional[Pattern[a, Blank[]]], Times[Log[Times[Optional[Pattern[c, Blank[]]], Power[Pattern[x, Blank[]], Optional[Pattern[n, Blank[]]]]]], Optional[Pattern[b, Blank[]]]]], 2], Optional[Pattern[d, Blank[]]]]], Pattern[x, Blank[Symbol]]] := Int[F^(a^2*d + 2*a*b*d*Log[c*x^n] + b^2*d*Log[c*x^n]^2), x] /; FreeQ[{F, a, b, c, d, n}, x]
  2278. Int[Times[Power[Pattern[F, Blank[]], Times[Power[Plus[Optional[Pattern[a, Blank[]]], Times[Log[Times[Optional[Pattern[c, Blank[]]], Power[Pattern[x, Blank[]], Optional[Pattern[n, Blank[]]]]]], Optional[Pattern[b, Blank[]]]]], 2], Optional[Pattern[d, Blank[]]]]], Power[Times[Optional[Pattern[e, Blank[]]], Pattern[x, Blank[]]], Optional[Pattern[m, Blank[]]]]], Pattern[x, Blank[Symbol]]] := Int[(e*x)^m*F^(a^2*d + 2*a*b*d*Log[c*x^n] + b^2*d*Log[c*x^n]^2), x] /; FreeQ[{F, a, b, c, d, e, m, n}, x]
  2279. Int[Log[Plus[Pattern[a, Blank[]], Times[Optional[Pattern[b, Blank[]]], Power[Power[Pattern[F, Blank[]], Times[Optional[Pattern[e, Blank[]]], Plus[Optional[Pattern[c, Blank[]]], Times[Optional[Pattern[d, Blank[]]], Pattern[x, Blank[]]]]]], Optional[Pattern[n, Blank[]]]]]]], Pattern[x, Blank[Symbol]]] := Dist[1/(d*e*n*Log[F]), Subst[Int[Log[a + b*x]/x, x], x, (F^(e*(c + d*x)))^n], x] /; FreeQ[{F, a, b, c, d, e, n}, x] && GtQ[a, 0]
  2280. Int[Log[Plus[Pattern[a, Blank[]], Times[Optional[Pattern[b, Blank[]]], Power[Power[Pattern[F, Blank[]], Times[Optional[Pattern[e, Blank[]]], Plus[Optional[Pattern[c, Blank[]]], Times[Optional[Pattern[d, Blank[]]], Pattern[x, Blank[]]]]]], Optional[Pattern[n, Blank[]]]]]]], Pattern[x, Blank[Symbol]]] := Simp[x*Log[a + b*(F^(e*(c + d*x)))^n], x] - Dist[b*d*e*n*Log[F], Int[(x*(F^(e*(c + d*x)))^n)/(a + b*(F^(e*(c + d*x)))^n), x], x] /; FreeQ[{F, a, b, c, d, e, n}, x] && !GtQ[a, 0]
  2281. Int[Times[Optional[Pattern[u, Blank[]]], Power[Times[Optional[Pattern[a, Blank[]]], Power[Pattern[F, Blank[]], Pattern[v, Blank[]]]], Pattern[n, Blank[]]]], Pattern[x, Blank[Symbol]]] := Dist[(a*F^v)^n/F^(n*v), Int[u*F^(n*v), x], x] /; FreeQ[{F, a, n}, x] && !IntegerQ[n]
  2282. Int[Pattern[u, Blank[]], Pattern[x, Blank[Symbol]]] := With[{v = FunctionOfExponential[u, x]}, Dist[v/D[v, x], Subst[Int[FunctionOfExponentialFunction[u, x]/x, x], x, v], x]] /; FunctionOfExponentialQ[u, x] && !MatchQ[u, (w_)*((a_.)*(v_)^(n_))^(m_) /; FreeQ[{a, m, n}, x] && IntegerQ[m*n]] && !MatchQ[u, E^((c_.)*((a_.) + (b_.)*x))*(F_)[v_] /; FreeQ[{a, b, c}, x] && InverseFunctionQ[F[x]]]
  2283. Int[Times[Optional[Pattern[u, Blank[]]], Power[Plus[Times[Optional[Pattern[a, Blank[]]], Power[Pattern[F, Blank[]], Pattern[v, Blank[]]]], Times[Optional[Pattern[b, Blank[]]], Power[Pattern[F, Blank[]], Pattern[w, Blank[]]]]], Pattern[n, Blank[]]]], Pattern[x, Blank[Symbol]]] := Int[u*F^(n*v)*(a + b*F^ExpandToSum[w - v, x])^n, x] /; FreeQ[{F, a, b, n}, x] && ILtQ[n, 0] && LinearQ[{v, w}, x]
  2284. Int[Times[Optional[Pattern[u, Blank[]]], Power[Plus[Times[Optional[Pattern[a, Blank[]]], Power[Pattern[F, Blank[]], Pattern[v, Blank[]]]], Times[Optional[Pattern[b, Blank[]]], Power[Pattern[G, Blank[]], Pattern[w, Blank[]]]]], Pattern[n, Blank[]]]], Pattern[x, Blank[Symbol]]] := Int[u*F^(n*v)*(a + b*E^ExpandToSum[Log[G]*w - Log[F]*v, x])^n, x] /; FreeQ[{F, G, a, b, n}, x] && ILtQ[n, 0] && LinearQ[{v, w}, x]
  2285. Int[Times[Optional[Pattern[u, Blank[]]], Power[Plus[Times[Optional[Pattern[a, Blank[]]], Power[Pattern[F, Blank[]], Pattern[v, Blank[]]]], Times[Optional[Pattern[b, Blank[]]], Power[Pattern[F, Blank[]], Pattern[w, Blank[]]]]], Pattern[n, Blank[]]]], Pattern[x, Blank[Symbol]]] := Dist[(a*F^v + b*F^w)^n/(F^(n*v)*(a + b*F^ExpandToSum[w - v, x])^n), Int[u*F^(n*v)*(a + b*F^ExpandToSum[w - v, x])^n, x], x] /; FreeQ[{F, a, b, n}, x] && !IntegerQ[n] && LinearQ[{v, w}, x]
  2286. Int[Times[Optional[Pattern[u, Blank[]]], Power[Plus[Times[Optional[Pattern[a, Blank[]]], Power[Pattern[F, Blank[]], Pattern[v, Blank[]]]], Times[Optional[Pattern[b, Blank[]]], Power[Pattern[G, Blank[]], Pattern[w, Blank[]]]]], Pattern[n, Blank[]]]], Pattern[x, Blank[Symbol]]] := Dist[(a*F^v + b*G^w)^n/(F^(n*v)*(a + b*E^ExpandToSum[Log[G]*w - Log[F]*v, x])^n), Int[u*F^(n*v)*(a + b*E^ExpandToSum[Log[G]*w - Log[F]*v, x])^n, x], x] /; FreeQ[{F, G, a, b, n}, x] && !IntegerQ[n] && LinearQ[{v, w}, x]
  2287. Int[Times[Optional[Pattern[u, Blank[]]], Power[Pattern[F, Blank[]], Pattern[v, Blank[]]], Power[Pattern[G, Blank[]], Pattern[w, Blank[]]]], Pattern[x, Blank[Symbol]]] := With[{z = v*Log[F] + w*Log[G]}, Int[u*NormalizeIntegrand[E^z, x], x] /; BinomialQ[z, x] || (PolynomialQ[z, x] && LeQ[Exponent[z, x], 2])] /; FreeQ[{F, G}, x]
  2288. Int[Times[Optional[Pattern[y, Blank[]]], Power[Pattern[F, Blank[]], Pattern[u, Blank[]]], Plus[Pattern[v, Blank[]], Pattern[w, Blank[]]]], Pattern[x, Blank[Symbol]]] := With[{z = (v*y)/(Log[F]*D[u, x])}, Simp[F^u*z, x] /; EqQ[D[z, x], w*y]] /; FreeQ[F, x]
  2289. Int[Times[Power[Pattern[F, Blank[]], Pattern[u, Blank[]]], Power[Pattern[v, Blank[]], Optional[Pattern[n, Blank[]]]], Pattern[w, Blank[]]], Pattern[x, Blank[Symbol]]] := With[{z = Log[F]*v*D[u, x] + (n + 1)*D[v, x]}, Simp[(Coefficient[w, x, Exponent[w, x]]*F^u*v^(n + 1))/Coefficient[z, x, Exponent[z, x]], x] /; EqQ[Exponent[w, x], Exponent[z, x]] && EqQ[w*Coefficient[z, x, Exponent[z, x]], z*Coefficient[w, x, Exponent[w, x]]]] /; FreeQ[{F, n}, x] && PolynomialQ[u, x] && PolynomialQ[v, x] && PolynomialQ[w, x]
  2290. Int[Times[Power[Plus[Optional[Pattern[a, Blank[]]], Times[Optional[Pattern[b, Blank[]]], Power[Pattern[F, Blank[]], Times[Optional[Pattern[c, Blank[]]], Power[Plus[Optional[Pattern[d, Blank[]]], Times[Optional[Pattern[e, Blank[]]], Pattern[x, Blank[]]]], Rational[1, 2]], Power[Plus[Optional[Pattern[f, Blank[]]], Times[Optional[Pattern[g, Blank[]]], Pattern[x, Blank[]]]], Rational[-1, 2]]]]]], Optional[Pattern[n, Blank[]]]], Power[Plus[Optional[Pattern[A, Blank[]]], Times[Optional[Pattern[B, Blank[]]], Pattern[x, Blank[]]], Times[Optional[Pattern[C, Blank[]]], Power[Pattern[x, Blank[]], 2]]], -1]], Pattern[x, Blank[Symbol]]] := Dist[(2*e*g)/(C*(e*f - d*g)), Subst[Int[(a + b*F^(c*x))^n/x, x], x, Sqrt[d + e*x]/Sqrt[f + g*x]], x] /; FreeQ[{a, b, c, d, e, f, g, A, B, C, F}, x] && EqQ[C*d*f - A*e*g, 0] && EqQ[B*e*g - C*(e*f + d*g), 0] && IGtQ[n, 0]
  2291. Int[Times[Power[Plus[Optional[Pattern[a, Blank[]]], Times[Optional[Pattern[b, Blank[]]], Power[Pattern[F, Blank[]], Times[Optional[Pattern[c, Blank[]]], Power[Plus[Optional[Pattern[d, Blank[]]], Times[Optional[Pattern[e, Blank[]]], Pattern[x, Blank[]]]], Rational[1, 2]], Power[Plus[Optional[Pattern[f, Blank[]]], Times[Optional[Pattern[g, Blank[]]], Pattern[x, Blank[]]]], Rational[-1, 2]]]]]], Optional[Pattern[n, Blank[]]]], Power[Plus[Pattern[A, Blank[]], Times[Optional[Pattern[C, Blank[]]], Power[Pattern[x, Blank[]], 2]]], -1]], Pattern[x, Blank[Symbol]]] := Dist[(2*e*g)/(C*(e*f - d*g)), Subst[Int[(a + b*F^(c*x))^n/x, x], x, Sqrt[d + e*x]/Sqrt[f + g*x]], x] /; FreeQ[{a, b, c, d, e, f, g, A, C, F}, x] && EqQ[C*d*f - A*e*g, 0] && EqQ[e*f + d*g, 0] && IGtQ[n, 0]
  2292. Int[Times[Power[Plus[Optional[Pattern[a, Blank[]]], Times[Optional[Pattern[b, Blank[]]], Power[Pattern[F, Blank[]], Times[Optional[Pattern[c, Blank[]]], Power[Plus[Optional[Pattern[d, Blank[]]], Times[Optional[Pattern[e, Blank[]]], Pattern[x, Blank[]]]], Rational[1, 2]], Power[Plus[Optional[Pattern[f, Blank[]]], Times[Optional[Pattern[g, Blank[]]], Pattern[x, Blank[]]]], Rational[-1, 2]]]]]], Pattern[n, Blank[]]], Power[Plus[Optional[Pattern[A, Blank[]]], Times[Optional[Pattern[B, Blank[]]], Pattern[x, Blank[]]], Times[Optional[Pattern[C, Blank[]]], Power[Pattern[x, Blank[]], 2]]], -1]], Pattern[x, Blank[Symbol]]] := Unintegrable[(a + b*F^((c*Sqrt[d + e*x])/Sqrt[f + g*x]))^n/(A + B*x + C*x^2), x] /; FreeQ[{a, b, c, d, e, f, g, A, B, C, F, n}, x] && EqQ[C*d*f - A*e*g, 0] && EqQ[B*e*g - C*(e*f + d*g), 0] && !IGtQ[n, 0]
  2293. Int[Times[Power[Plus[Optional[Pattern[a, Blank[]]], Times[Optional[Pattern[b, Blank[]]], Power[Pattern[F, Blank[]], Times[Optional[Pattern[c, Blank[]]], Power[Plus[Optional[Pattern[d, Blank[]]], Times[Optional[Pattern[e, Blank[]]], Pattern[x, Blank[]]]], Rational[1, 2]], Power[Plus[Optional[Pattern[f, Blank[]]], Times[Optional[Pattern[g, Blank[]]], Pattern[x, Blank[]]]], Rational[-1, 2]]]]]], Pattern[n, Blank[]]], Power[Plus[Pattern[A, Blank[]], Times[Optional[Pattern[C, Blank[]]], Power[Pattern[x, Blank[]], 2]]], -1]], Pattern[x, Blank[Symbol]]] := Unintegrable[(a + b*F^((c*Sqrt[d + e*x])/Sqrt[f + g*x]))^n/(A + C*x^2), x] /; FreeQ[{a, b, c, d, e, f, g, A, C, F, n}, x] && EqQ[C*d*f - A*e*g, 0] && EqQ[e*f + d*g, 0] && !IGtQ[n, 0]
  2294. Int[Times[Plus[Optional[Pattern[A, Blank[]]], Times[Log[Times[Optional[Pattern[c, Blank[]]], Power[Plus[Optional[Pattern[d, Blank[]]], Times[Optional[Pattern[e, Blank[]]], Pattern[x, Blank[]]]], Optional[Pattern[n, Blank[]]]]]], Optional[Pattern[B, Blank[]]]]], Power[Plus[Times[Log[Times[Optional[Pattern[c, Blank[]]], Power[Plus[Optional[Pattern[d, Blank[]]], Times[Optional[Pattern[e, Blank[]]], Pattern[x, Blank[]]]], Optional[Pattern[n, Blank[]]]]]], Optional[Pattern[b, Blank[]]]], Pattern[a, Blank[]]], Rational[-1, 2]]], Pattern[x, Blank[Symbol]]] := Simp[(B*(d + e*x)*Sqrt[a + b*Log[c*(d + e*x)^n]])/(b*e), x] + Dist[(2*A*b - B*(2*a + b*n))/(2*b), Int[1/Sqrt[a + b*Log[c*(d + e*x)^n]], x], x] /; FreeQ[{a, b, c, d, e, A, B, n}, x]
  2295. Int[Log[Times[Optional[Pattern[c, Blank[]]], Power[Pattern[x, Blank[]], Optional[Pattern[n, Blank[]]]]]], Pattern[x, Blank[Symbol]]] := Simp[x*Log[c*x^n], x] - Simp[n*x, x] /; FreeQ[{c, n}, x]
  2296. Int[Power[Plus[Optional[Pattern[a, Blank[]]], Times[Log[Times[Optional[Pattern[c, Blank[]]], Power[Pattern[x, Blank[]], Optional[Pattern[n, Blank[]]]]]], Optional[Pattern[b, Blank[]]]]], Optional[Pattern[p, Blank[]]]], Pattern[x, Blank[Symbol]]] := Simp[x*(a + b*Log[c*x^n])^p, x] - Dist[b*n*p, Int[(a + b*Log[c*x^n])^(p - 1), x], x] /; FreeQ[{a, b, c, n}, x] && GtQ[p, 0] && IntegerQ[2*p]
  2297. Int[Power[Plus[Optional[Pattern[a, Blank[]]], Times[Log[Times[Optional[Pattern[c, Blank[]]], Power[Pattern[x, Blank[]], Optional[Pattern[n, Blank[]]]]]], Optional[Pattern[b, Blank[]]]]], Pattern[p, Blank[]]], Pattern[x, Blank[Symbol]]] := Simp[(x*(a + b*Log[c*x^n])^(p + 1))/(b*n*(p + 1)), x] - Dist[1/(b*n*(p + 1)), Int[(a + b*Log[c*x^n])^(p + 1), x], x] /; FreeQ[{a, b, c, n}, x] && LtQ[p, -1] && IntegerQ[2*p]
  2298. Int[Power[Log[Times[Optional[Pattern[c, Blank[]]], Pattern[x, Blank[]]]], -1], Pattern[x, Blank[Symbol]]] := Simp[LogIntegral[c*x]/c, x] /; FreeQ[c, x]
  2299. Int[Power[Plus[Optional[Pattern[a, Blank[]]], Times[Log[Times[Optional[Pattern[c, Blank[]]], Power[Pattern[x, Blank[]], Optional[Pattern[n, Blank[]]]]]], Optional[Pattern[b, Blank[]]]]], Pattern[p, Blank[]]], Pattern[x, Blank[Symbol]]] := Dist[1/(n*c^(1/n)), Subst[Int[E^(x/n)*(a + b*x)^p, x], x, Log[c*x^n]], x] /; FreeQ[{a, b, c, p}, x] && IntegerQ[1/n]
  2300. Int[Power[Plus[Optional[Pattern[a, Blank[]]], Times[Log[Times[Optional[Pattern[c, Blank[]]], Power[Pattern[x, Blank[]], Optional[Pattern[n, Blank[]]]]]], Optional[Pattern[b, Blank[]]]]], Pattern[p, Blank[]]], Pattern[x, Blank[Symbol]]] := Dist[x/(n*(c*x^n)^(1/n)), Subst[Int[E^(x/n)*(a + b*x)^p, x], x, Log[c*x^n]], x] /; FreeQ[{a, b, c, n, p}, x]
  2301. Int[Times[Plus[Optional[Pattern[a, Blank[]]], Times[Log[Times[Optional[Pattern[c, Blank[]]], Power[Pattern[x, Blank[]], Optional[Pattern[n, Blank[]]]]]], Optional[Pattern[b, Blank[]]]]], Power[Pattern[x, Blank[]], -1]], Pattern[x, Blank[Symbol]]] := Simp[(a + b*Log[c*x^n])^2/(2*b*n), x] /; FreeQ[{a, b, c, n}, x]
  2302. Int[Times[Power[Plus[Optional[Pattern[a, Blank[]]], Times[Log[Times[Optional[Pattern[c, Blank[]]], Power[Pattern[x, Blank[]], Optional[Pattern[n, Blank[]]]]]], Optional[Pattern[b, Blank[]]]]], Optional[Pattern[p, Blank[]]]], Power[Pattern[x, Blank[]], -1]], Pattern[x, Blank[Symbol]]] := Dist[1/(b*n), Subst[Int[x^p, x], x, a + b*Log[c*x^n]], x] /; FreeQ[{a, b, c, n, p}, x]
  2303. Int[Times[Plus[Optional[Pattern[a, Blank[]]], Times[Log[Times[Optional[Pattern[c, Blank[]]], Power[Pattern[x, Blank[]], Optional[Pattern[n, Blank[]]]]]], Optional[Pattern[b, Blank[]]]]], Power[Times[Optional[Pattern[d, Blank[]]], Pattern[x, Blank[]]], Optional[Pattern[m, Blank[]]]]], Pattern[x, Blank[Symbol]]] := Simp[(b*(d*x)^(m + 1)*Log[c*x^n])/(d*(m + 1)), x] /; FreeQ[{a, b, c, d, m, n}, x] && NeQ[m, -1] && EqQ[a*(m + 1) - b*n, 0]
  2304. Int[Times[Plus[Optional[Pattern[a, Blank[]]], Times[Log[Times[Optional[Pattern[c, Blank[]]], Power[Pattern[x, Blank[]], Optional[Pattern[n, Blank[]]]]]], Optional[Pattern[b, Blank[]]]]], Power[Times[Optional[Pattern[d, Blank[]]], Pattern[x, Blank[]]], Optional[Pattern[m, Blank[]]]]], Pattern[x, Blank[Symbol]]] := Simp[((d*x)^(m + 1)*(a + b*Log[c*x^n]))/(d*(m + 1)), x] - Simp[(b*n*(d*x)^(m + 1))/(d*(m + 1)^2), x] /; FreeQ[{a, b, c, d, m, n}, x] && NeQ[m, -1]
  2305. Int[Times[Power[Plus[Optional[Pattern[a, Blank[]]], Times[Log[Times[Optional[Pattern[c, Blank[]]], Power[Pattern[x, Blank[]], Optional[Pattern[n, Blank[]]]]]], Optional[Pattern[b, Blank[]]]]], Optional[Pattern[p, Blank[]]]], Power[Times[Optional[Pattern[d, Blank[]]], Pattern[x, Blank[]]], Optional[Pattern[m, Blank[]]]]], Pattern[x, Blank[Symbol]]] := Simp[((d*x)^(m + 1)*(a + b*Log[c*x^n])^p)/(d*(m + 1)), x] - Dist[(b*n*p)/(m + 1), Int[(d*x)^m*(a + b*Log[c*x^n])^(p - 1), x], x] /; FreeQ[{a, b, c, d, m, n}, x] && NeQ[m, -1] && GtQ[p, 0]
  2306. Int[Times[Power[Plus[Optional[Pattern[a, Blank[]]], Times[Log[Times[Optional[Pattern[c, Blank[]]], Power[Pattern[x, Blank[]], Optional[Pattern[n, Blank[]]]]]], Optional[Pattern[b, Blank[]]]]], Pattern[p, Blank[]]], Power[Times[Optional[Pattern[d, Blank[]]], Pattern[x, Blank[]]], Optional[Pattern[m, Blank[]]]]], Pattern[x, Blank[Symbol]]] := Simp[((d*x)^(m + 1)*(a + b*Log[c*x^n])^(p + 1))/(b*d*n*(p + 1)), x] - Dist[(m + 1)/(b*n*(p + 1)), Int[(d*x)^m*(a + b*Log[c*x^n])^(p + 1), x], x] /; FreeQ[{a, b, c, d, m, n}, x] && NeQ[m, -1] && LtQ[p, -1]
  2307. Int[Times[Power[Log[Times[Optional[Pattern[c, Blank[]]], Power[Pattern[x, Blank[]], Pattern[n, Blank[]]]]], -1], Power[Pattern[x, Blank[]], Optional[Pattern[m, Blank[]]]]], Pattern[x, Blank[Symbol]]] := Dist[1/n, Subst[Int[1/Log[c*x], x], x, x^n], x] /; FreeQ[{c, m, n}, x] && EqQ[m, n - 1]
  2308. Int[Times[Power[Log[Times[Optional[Pattern[c, Blank[]]], Power[Pattern[x, Blank[]], Pattern[n, Blank[]]]]], -1], Power[Times[Pattern[d, Blank[]], Pattern[x, Blank[]]], Optional[Pattern[m, Blank[]]]]], Pattern[x, Blank[Symbol]]] := Dist[(d*x)^m/x^m, Int[x^m/Log[c*x^n], x], x] /; FreeQ[{c, d, m, n}, x] && EqQ[m, n - 1]
  2309. Int[Times[Power[Plus[Optional[Pattern[a, Blank[]]], Times[Log[Times[Optional[Pattern[c, Blank[]]], Pattern[x, Blank[]]]], Optional[Pattern[b, Blank[]]]]], Pattern[p, Blank[]]], Power[Pattern[x, Blank[]], Optional[Pattern[m, Blank[]]]]], Pattern[x, Blank[Symbol]]] := Dist[1/c^(m + 1), Subst[Int[E^((m + 1)*x)*(a + b*x)^p, x], x, Log[c*x]], x] /; FreeQ[{a, b, c, p}, x] && IntegerQ[m]
  2310. Int[Times[Power[Plus[Optional[Pattern[a, Blank[]]], Times[Log[Times[Optional[Pattern[c, Blank[]]], Power[Pattern[x, Blank[]], Optional[Pattern[n, Blank[]]]]]], Optional[Pattern[b, Blank[]]]]], Pattern[p, Blank[]]], Power[Times[Optional[Pattern[d, Blank[]]], Pattern[x, Blank[]]], Optional[Pattern[m, Blank[]]]]], Pattern[x, Blank[Symbol]]] := Dist[(d*x)^(m + 1)/(d*n*(c*x^n)^((m + 1)/n)), Subst[Int[E^(((m + 1)*x)/n)*(a + b*x)^p, x], x, Log[c*x^n]], x] /; FreeQ[{a, b, c, d, m, n, p}, x]
  2311. Int[Times[Power[Plus[Optional[Pattern[a, Blank[]]], Times[Log[Times[Optional[Pattern[c, Blank[]]], Power[Pattern[x, Blank[]], Optional[Pattern[n, Blank[]]]]]], Optional[Pattern[b, Blank[]]]]], Optional[Pattern[p, Blank[]]]], Power[Times[Optional[Pattern[d, Blank[]]], Power[Pattern[x, Blank[]], Pattern[q, Blank[]]]], Pattern[m, Blank[]]]], Pattern[x, Blank[Symbol]]] := Dist[(d*x^q)^m/x^(m*q), Int[x^(m*q)*(a + b*Log[c*x^n])^p, x], x] /; FreeQ[{a, b, c, d, m, n, p, q}, x]
  2312. Int[Times[Power[Plus[Optional[Pattern[a, Blank[]]], Times[Log[Times[Optional[Pattern[c, Blank[]]], Power[Pattern[x, Blank[]], Optional[Pattern[n, Blank[]]]]]], Optional[Pattern[b, Blank[]]]]], Optional[Pattern[p, Blank[]]]], Power[Times[Optional[Pattern[d1, Blank[]]], Power[Pattern[x, Blank[]], Pattern[q1, Blank[]]]], Pattern[m1, Blank[]]], Power[Times[Optional[Pattern[d2, Blank[]]], Power[Pattern[x, Blank[]], Pattern[q2, Blank[]]]], Pattern[m2, Blank[]]]], Pattern[x, Blank[Symbol]]] := Dist[((d1*x^q1)^m1*(d2*x^q2)^m2)/x^(m1*q1 + m2*q2), Int[x^(m1*q1 + m2*q2)*(a + b*Log[c*x^n])^p, x], x] /; FreeQ[{a, b, c, d1, d2, m1, m2, n, p, q1, q2}, x]
  2313. Int[Times[Plus[Optional[Pattern[a, Blank[]]], Times[Log[Times[Optional[Pattern[c, Blank[]]], Power[Pattern[x, Blank[]], Optional[Pattern[n, Blank[]]]]]], Optional[Pattern[b, Blank[]]]]], Power[Plus[Pattern[d, Blank[]], Times[Optional[Pattern[e, Blank[]]], Power[Pattern[x, Blank[]], Optional[Pattern[r, Blank[]]]]]], Optional[Pattern[q, Blank[]]]]], Pattern[x, Blank[Symbol]]] := With[{u = IntHide[(d + e*x^r)^q, x]}, Simp[u*(a + b*Log[c*x^n]), x] - Dist[b*n, Int[SimplifyIntegrand[u/x, x], x], x]] /; FreeQ[{a, b, c, d, e, n, r}, x] && IGtQ[q, 0]
  2314. Int[Times[Plus[Optional[Pattern[a, Blank[]]], Times[Log[Times[Optional[Pattern[c, Blank[]]], Power[Pattern[x, Blank[]], Optional[Pattern[n, Blank[]]]]]], Optional[Pattern[b, Blank[]]]]], Power[Plus[Pattern[d, Blank[]], Times[Optional[Pattern[e, Blank[]]], Power[Pattern[x, Blank[]], Optional[Pattern[r, Blank[]]]]]], Pattern[q, Blank[]]]], Pattern[x, Blank[Symbol]]] := Simp[(x*(d + e*x^r)^(q + 1)*(a + b*Log[c*x^n]))/d, x] - Dist[(b*n)/d, Int[(d + e*x^r)^(q + 1), x], x] /; FreeQ[{a, b, c, d, e, n, q, r}, x] && EqQ[r*(q + 1) + 1, 0]
  2315. Int[Times[Log[Times[Optional[Pattern[c, Blank[]]], Pattern[x, Blank[]]]], Power[Plus[Pattern[d, Blank[]], Times[Optional[Pattern[e, Blank[]]], Pattern[x, Blank[]]]], -1]], Pattern[x, Blank[Symbol]]] := -Simp[PolyLog[2, 1 - c*x]/e, x] /; FreeQ[{c, d, e}, x] && EqQ[e + c*d, 0]
  2316. Int[Times[Plus[Optional[Pattern[a, Blank[]]], Times[Log[Times[Optional[Pattern[c, Blank[]]], Pattern[x, Blank[]]]], Optional[Pattern[b, Blank[]]]]], Power[Plus[Pattern[d, Blank[]], Times[Optional[Pattern[e, Blank[]]], Pattern[x, Blank[]]]], -1]], Pattern[x, Blank[Symbol]]] := Simp[((a + b*Log[-((c*d)/e)])*Log[d + e*x])/e, x] + Dist[b, Int[Log[-((e*x)/d)]/(d + e*x), x], x] /; FreeQ[{a, b, c, d, e}, x] && GtQ[-((c*d)/e), 0]
  2317. Int[Times[Power[Plus[Optional[Pattern[a, Blank[]]], Times[Log[Times[Optional[Pattern[c, Blank[]]], Power[Pattern[x, Blank[]], Optional[Pattern[n, Blank[]]]]]], Optional[Pattern[b, Blank[]]]]], Optional[Pattern[p, Blank[]]]], Power[Plus[Pattern[d, Blank[]], Times[Optional[Pattern[e, Blank[]]], Pattern[x, Blank[]]]], -1]], Pattern[x, Blank[Symbol]]] := Simp[(Log[1 + (e*x)/d]*(a + b*Log[c*x^n])^p)/e, x] - Dist[(b*n*p)/e, Int[(Log[1 + (e*x)/d]*(a + b*Log[c*x^n])^(p - 1))/x, x], x] /; FreeQ[{a, b, c, d, e, n}, x] && IGtQ[p, 0]
  2318. Int[Times[Power[Plus[Optional[Pattern[a, Blank[]]], Times[Log[Times[Optional[Pattern[c, Blank[]]], Power[Pattern[x, Blank[]], Optional[Pattern[n, Blank[]]]]]], Optional[Pattern[b, Blank[]]]]], Optional[Pattern[p, Blank[]]]], Power[Plus[Pattern[d, Blank[]], Times[Optional[Pattern[e, Blank[]]], Pattern[x, Blank[]]]], -2]], Pattern[x, Blank[Symbol]]] := Simp[(x*(a + b*Log[c*x^n])^p)/(d*(d + e*x)), x] - Dist[(b*n*p)/d, Int[(a + b*Log[c*x^n])^(p - 1)/(d + e*x), x], x] /; FreeQ[{a, b, c, d, e, n, p}, x] && GtQ[p, 0]
  2319. Int[Times[Power[Plus[Optional[Pattern[a, Blank[]]], Times[Log[Times[Optional[Pattern[c, Blank[]]], Power[Pattern[x, Blank[]], Optional[Pattern[n, Blank[]]]]]], Optional[Pattern[b, Blank[]]]]], Optional[Pattern[p, Blank[]]]], Power[Plus[Pattern[d, Blank[]], Times[Optional[Pattern[e, Blank[]]], Pattern[x, Blank[]]]], Optional[Pattern[q, Blank[]]]]], Pattern[x, Blank[Symbol]]] := Simp[((d + e*x)^(q + 1)*(a + b*Log[c*x^n])^p)/(e*(q + 1)), x] - Dist[(b*n*p)/(e*(q + 1)), Int[((d + e*x)^(q + 1)*(a + b*Log[c*x^n])^(p - 1))/x, x], x] /; FreeQ[{a, b, c, d, e, n, p, q}, x] && GtQ[p, 0] && NeQ[q, -1] && (EqQ[p, 1] || (IntegersQ[2*p, 2*q] && !IGtQ[q, 0]) || (EqQ[p, 2] && NeQ[q, 1]))
  2320. Int[Times[Power[Plus[Optional[Pattern[a, Blank[]]], Times[Log[Times[Optional[Pattern[c, Blank[]]], Power[Pattern[x, Blank[]], Optional[Pattern[n, Blank[]]]]]], Optional[Pattern[b, Blank[]]]]], Pattern[p, Blank[]]], Power[Plus[Pattern[d, Blank[]], Times[Optional[Pattern[e, Blank[]]], Pattern[x, Blank[]]]], Optional[Pattern[q, Blank[]]]]], Pattern[x, Blank[Symbol]]] := Simp[(x*(d + e*x)^q*(a + b*Log[c*x^n])^(p + 1))/(b*n*(p + 1)), x] + (-Dist[(q + 1)/(b*n*(p + 1)), Int[(d + e*x)^q*(a + b*Log[c*x^n])^(p + 1), x], x] + Dist[(d*q)/(b*n*(p + 1)), Int[(d + e*x)^(q - 1)*(a + b*Log[c*x^n])^(p + 1), x], x]) /; FreeQ[{a, b, c, d, e, n}, x] && LtQ[p, -1] && GtQ[q, 0]
  2321. Int[Times[Plus[Optional[Pattern[a, Blank[]]], Times[Log[Times[Optional[Pattern[c, Blank[]]], Power[Pattern[x, Blank[]], Optional[Pattern[n, Blank[]]]]]], Optional[Pattern[b, Blank[]]]]], Power[Plus[Pattern[d, Blank[]], Times[Optional[Pattern[e, Blank[]]], Power[Pattern[x, Blank[]], 2]]], Optional[Pattern[q, Blank[]]]]], Pattern[x, Blank[Symbol]]] := Simp[(x*(d + e*x^2)^q*(a + b*Log[c*x^n]))/(2*q + 1), x] + (-Dist[(b*n)/(2*q + 1), Int[(d + e*x^2)^q, x], x] + Dist[(2*d*q)/(2*q + 1), Int[(d + e*x^2)^(q - 1)*(a + b*Log[c*x^n]), x], x]) /; FreeQ[{a, b, c, d, e, n}, x] && GtQ[q, 0]
  2322. Int[Times[Plus[Optional[Pattern[a, Blank[]]], Times[Log[Times[Optional[Pattern[c, Blank[]]], Power[Pattern[x, Blank[]], Optional[Pattern[n, Blank[]]]]]], Optional[Pattern[b, Blank[]]]]], Power[Plus[Pattern[d, Blank[]], Times[Optional[Pattern[e, Blank[]]], Power[Pattern[x, Blank[]], 2]]], Rational[-3, 2]]], Pattern[x, Blank[Symbol]]] := Simp[(x*(a + b*Log[c*x^n]))/(d*Sqrt[d + e*x^2]), x] - Dist[(b*n)/d, Int[1/Sqrt[d + e*x^2], x], x] /; FreeQ[{a, b, c, d, e, n}, x]
  2323. Int[Times[Plus[Optional[Pattern[a, Blank[]]], Times[Log[Times[Optional[Pattern[c, Blank[]]], Power[Pattern[x, Blank[]], Optional[Pattern[n, Blank[]]]]]], Optional[Pattern[b, Blank[]]]]], Power[Plus[Pattern[d, Blank[]], Times[Optional[Pattern[e, Blank[]]], Power[Pattern[x, Blank[]], 2]]], Pattern[q, Blank[]]]], Pattern[x, Blank[Symbol]]] := -Simp[(x*(d + e*x^2)^(q + 1)*(a + b*Log[c*x^n]))/(2*d*(q + 1)), x] + (Dist[(2*q + 3)/(2*d*(q + 1)), Int[(d + e*x^2)^(q + 1)*(a + b*Log[c*x^n]), x], x] + Dist[(b*n)/(2*d*(q + 1)), Int[(d + e*x^2)^(q + 1), x], x]) /; FreeQ[{a, b, c, d, e, n}, x] && LtQ[q, -1]
  2324. Int[Times[Plus[Optional[Pattern[a, Blank[]]], Times[Log[Times[Optional[Pattern[c, Blank[]]], Power[Pattern[x, Blank[]], Optional[Pattern[n, Blank[]]]]]], Optional[Pattern[b, Blank[]]]]], Power[Plus[Pattern[d, Blank[]], Times[Optional[Pattern[e, Blank[]]], Power[Pattern[x, Blank[]], 2]]], -1]], Pattern[x, Blank[Symbol]]] := With[{u = IntHide[1/(d + e*x^2), x]}, Simp[u*(a + b*Log[c*x^n]), x] - Dist[b*n, Int[u/x, x], x]] /; FreeQ[{a, b, c, d, e, n}, x]
  2325. Int[Times[Plus[Optional[Pattern[a, Blank[]]], Times[Log[Times[Optional[Pattern[c, Blank[]]], Power[Pattern[x, Blank[]], Optional[Pattern[n, Blank[]]]]]], Optional[Pattern[b, Blank[]]]]], Power[Plus[Pattern[d, Blank[]], Times[Optional[Pattern[e, Blank[]]], Power[Pattern[x, Blank[]], 2]]], Rational[-1, 2]]], Pattern[x, Blank[Symbol]]] := Simp[(ArcSinh[(Rt[e, 2]*x)/Sqrt[d]]*(a + b*Log[c*x^n]))/Rt[e, 2], x] - Dist[(b*n)/Rt[e, 2], Int[ArcSinh[(Rt[e, 2]*x)/Sqrt[d]]/x, x], x] /; FreeQ[{a, b, c, d, e, n}, x] && GtQ[d, 0] && PosQ[e]
  2326. Int[Times[Plus[Optional[Pattern[a, Blank[]]], Times[Log[Times[Optional[Pattern[c, Blank[]]], Power[Pattern[x, Blank[]], Optional[Pattern[n, Blank[]]]]]], Optional[Pattern[b, Blank[]]]]], Power[Plus[Pattern[d, Blank[]], Times[Optional[Pattern[e, Blank[]]], Power[Pattern[x, Blank[]], 2]]], Rational[-1, 2]]], Pattern[x, Blank[Symbol]]] := Simp[(ArcSin[(Rt[-e, 2]*x)/Sqrt[d]]*(a + b*Log[c*x^n]))/Rt[-e, 2], x] - Dist[(b*n)/Rt[-e, 2], Int[ArcSin[(Rt[-e, 2]*x)/Sqrt[d]]/x, x], x] /; FreeQ[{a, b, c, d, e, n}, x] && GtQ[d, 0] && NegQ[e]
  2327. Int[Times[Plus[Optional[Pattern[a, Blank[]]], Times[Log[Times[Optional[Pattern[c, Blank[]]], Power[Pattern[x, Blank[]], Optional[Pattern[n, Blank[]]]]]], Optional[Pattern[b, Blank[]]]]], Power[Plus[Pattern[d, Blank[]], Times[Optional[Pattern[e, Blank[]]], Power[Pattern[x, Blank[]], 2]]], Rational[-1, 2]]], Pattern[x, Blank[Symbol]]] := Dist[Sqrt[1 + (e*x^2)/d]/Sqrt[d + e*x^2], Int[(a + b*Log[c*x^n])/Sqrt[1 + (e*x^2)/d], x], x] /; FreeQ[{a, b, c, d, e, n}, x] && !GtQ[d, 0]
  2328. Int[Times[Plus[Optional[Pattern[a, Blank[]]], Times[Log[Times[Optional[Pattern[c, Blank[]]], Power[Pattern[x, Blank[]], Optional[Pattern[n, Blank[]]]]]], Optional[Pattern[b, Blank[]]]]], Power[Plus[Pattern[d1, Blank[]], Times[Optional[Pattern[e1, Blank[]]], Pattern[x, Blank[]]]], Rational[-1, 2]], Power[Plus[Pattern[d2, Blank[]], Times[Optional[Pattern[e2, Blank[]]], Pattern[x, Blank[]]]], Rational[-1, 2]]], Pattern[x, Blank[Symbol]]] := Dist[Sqrt[1 + (e1*e2*x^2)/(d1*d2)]/(Sqrt[d1 + e1*x]*Sqrt[d2 + e2*x]), Int[(a + b*Log[c*x^n])/Sqrt[1 + (e1*e2*x^2)/(d1*d2)], x], x] /; FreeQ[{a, b, c, d1, e1, d2, e2, n}, x] && EqQ[d2*e1 + d1*e2, 0]
  2329. Int[Times[Plus[Optional[Pattern[a, Blank[]]], Times[Log[Times[Optional[Pattern[c, Blank[]]], Power[Pattern[x, Blank[]], Optional[Pattern[n, Blank[]]]]]], Optional[Pattern[b, Blank[]]]]], Power[Plus[Pattern[d, Blank[]], Times[Optional[Pattern[e, Blank[]]], Power[Pattern[x, Blank[]], Optional[Pattern[r, Blank[]]]]]], Optional[Pattern[q, Blank[]]]]], Pattern[x, Blank[Symbol]]] := With[{u = IntHide[(d + e*x^r)^q, x]}, Dist[a + b*Log[c*x^n], u, x] - Dist[b*n, Int[SimplifyIntegrand[u/x, x], x], x] /; (EqQ[r, 1] && IntegerQ[q - 1/2]) || (EqQ[r, 2] && EqQ[q, -1]) || InverseFunctionFreeQ[u, x]] /; FreeQ[{a, b, c, d, e, n, q, r}, x] && IntegerQ[2*q] && IntegerQ[r]
  2330. Int[Times[Power[Plus[Optional[Pattern[a, Blank[]]], Times[Log[Times[Optional[Pattern[c, Blank[]]], Power[Pattern[x, Blank[]], Optional[Pattern[n, Blank[]]]]]], Optional[Pattern[b, Blank[]]]]], Optional[Pattern[p, Blank[]]]], Power[Plus[Pattern[d, Blank[]], Times[Optional[Pattern[e, Blank[]]], Power[Pattern[x, Blank[]], Optional[Pattern[r, Blank[]]]]]], Optional[Pattern[q, Blank[]]]]], Pattern[x, Blank[Symbol]]] := With[{u = ExpandIntegrand[(a + b*Log[c*x^n])^p, (d + e*x^r)^q, x]}, Int[u, x] /; SumQ[u]] /; FreeQ[{a, b, c, d, e, n, p, q, r}, x] && IntegerQ[q] && (GtQ[q, 0] || (IGtQ[p, 0] && IntegerQ[r]))
  2331. Int[Times[Power[Plus[Optional[Pattern[a, Blank[]]], Times[Log[Times[Optional[Pattern[c, Blank[]]], Power[Pattern[x, Blank[]], Optional[Pattern[n, Blank[]]]]]], Optional[Pattern[b, Blank[]]]]], Optional[Pattern[p, Blank[]]]], Power[Plus[Pattern[d, Blank[]], Times[Optional[Pattern[e, Blank[]]], Power[Pattern[x, Blank[]], Optional[Pattern[r, Blank[]]]]]], Optional[Pattern[q, Blank[]]]]], Pattern[x, Blank[Symbol]]] := Unintegrable[(d + e*x^r)^q*(a + b*Log[c*x^n])^p, x] /; FreeQ[{a, b, c, d, e, n, p, q, r}, x]
  2332. Int[Times[Power[Plus[Optional[Pattern[a, Blank[]]], Times[Log[Times[Optional[Pattern[c, Blank[]]], Power[Pattern[x, Blank[]], Optional[Pattern[n, Blank[]]]]]], Optional[Pattern[b, Blank[]]]]], Optional[Pattern[p, Blank[]]]], Power[Pattern[u, Blank[]], Optional[Pattern[q, Blank[]]]]], Pattern[x, Blank[Symbol]]] := Int[ExpandToSum[u, x]^q*(a + b*Log[c*x^n])^p, x] /; FreeQ[{a, b, c, n, p, q}, x] && BinomialQ[u, x] && !BinomialMatchQ[u, x]
  2333. Int[Times[Power[Plus[Optional[Pattern[a, Blank[]]], Times[Log[Times[Optional[Pattern[c, Blank[]]], Power[Pattern[x, Blank[]], Optional[Pattern[n, Blank[]]]]]], Optional[Pattern[b, Blank[]]]]], Optional[Pattern[p, Blank[]]]], Power[Plus[Pattern[d, Blank[]], Times[Optional[Pattern[e, Blank[]]], Power[Pattern[x, Blank[]], -1]]], Optional[Pattern[q, Blank[]]]], Power[Pattern[x, Blank[]], Optional[Pattern[m, Blank[]]]]], Pattern[x, Blank[Symbol]]] := Int[(e + d*x)^q*(a + b*Log[c*x^n])^p, x] /; FreeQ[{a, b, c, d, e, m, n, p}, x] && EqQ[m, q] && IntegerQ[q]
  2334. Int[Times[Plus[Optional[Pattern[a, Blank[]]], Times[Log[Times[Optional[Pattern[c, Blank[]]], Power[Pattern[x, Blank[]], Optional[Pattern[n, Blank[]]]]]], Optional[Pattern[b, Blank[]]]]], Power[Pattern[x, Blank[]], Optional[Pattern[m, Blank[]]]], Power[Plus[Pattern[d, Blank[]], Times[Optional[Pattern[e, Blank[]]], Power[Pattern[x, Blank[]], Optional[Pattern[r, Blank[]]]]]], Optional[Pattern[q, Blank[]]]]], Pattern[x, Blank[Symbol]]] := With[{u = IntHide[x^m*(d + e*x^r)^q, x]}, Simp[u*(a + b*Log[c*x^n]), x] - Dist[b*n, Int[SimplifyIntegrand[u/x, x], x], x]] /; FreeQ[{a, b, c, d, e, n, r}, x] && IGtQ[q, 0] && IntegerQ[m] && !(EqQ[q, 1] && EqQ[m, -1])
  2335. Int[Times[Plus[Optional[Pattern[a, Blank[]]], Times[Log[Times[Optional[Pattern[c, Blank[]]], Power[Pattern[x, Blank[]], Optional[Pattern[n, Blank[]]]]]], Optional[Pattern[b, Blank[]]]]], Power[Times[Optional[Pattern[f, Blank[]]], Pattern[x, Blank[]]], Optional[Pattern[m, Blank[]]]], Power[Plus[Pattern[d, Blank[]], Times[Optional[Pattern[e, Blank[]]], Power[Pattern[x, Blank[]], Optional[Pattern[r, Blank[]]]]]], Pattern[q, Blank[]]]], Pattern[x, Blank[Symbol]]] := Simp[((f*x)^(m + 1)*(d + e*x^r)^(q + 1)*(a + b*Log[c*x^n]))/(d*f*(m + 1)), x] - Dist[(b*n)/(d*(m + 1)), Int[(f*x)^m*(d + e*x^r)^(q + 1), x], x] /; FreeQ[{a, b, c, d, e, f, m, n, q, r}, x] && EqQ[m + r*(q + 1) + 1, 0] && NeQ[m, -1]
  2336. Int[Times[Power[Plus[Optional[Pattern[a, Blank[]]], Times[Log[Times[Optional[Pattern[c, Blank[]]], Power[Pattern[x, Blank[]], Pattern[n, Blank[]]]]], Optional[Pattern[b, Blank[]]]]], Optional[Pattern[p, Blank[]]]], Power[Times[Optional[Pattern[f, Blank[]]], Pattern[x, Blank[]]], Optional[Pattern[m, Blank[]]]], Power[Plus[Pattern[d, Blank[]], Times[Optional[Pattern[e, Blank[]]], Power[Pattern[x, Blank[]], Pattern[r, Blank[]]]]], Optional[Pattern[q, Blank[]]]]], Pattern[x, Blank[Symbol]]] := Dist[f^m/n, Subst[Int[(d + e*x)^q*(a + b*Log[c*x])^p, x], x, x^n], x] /; FreeQ[{a, b, c, d, e, f, m, n, q, r}, x] && EqQ[m, r - 1] && IGtQ[p, 0] && (IntegerQ[m] || GtQ[f, 0]) && EqQ[r, n]
  2337. Int[Times[Power[Plus[Optional[Pattern[a, Blank[]]], Times[Log[Times[Optional[Pattern[c, Blank[]]], Power[Pattern[x, Blank[]], Optional[Pattern[n, Blank[]]]]]], Optional[Pattern[b, Blank[]]]]], Optional[Pattern[p, Blank[]]]], Power[Times[Optional[Pattern[f, Blank[]]], Pattern[x, Blank[]]], Optional[Pattern[m, Blank[]]]], Power[Plus[Pattern[d, Blank[]], Times[Optional[Pattern[e, Blank[]]], Power[Pattern[x, Blank[]], Pattern[r, Blank[]]]]], -1]], Pattern[x, Blank[Symbol]]] := Simp[(f^m*Log[1 + (e*x^r)/d]*(a + b*Log[c*x^n])^p)/(e*r), x] - Dist[(b*f^m*n*p)/(e*r), Int[(Log[1 + (e*x^r)/d]*(a + b*Log[c*x^n])^(p - 1))/x, x], x] /; FreeQ[{a, b, c, d, e, f, m, n, r}, x] && EqQ[m, r - 1] && IGtQ[p, 0] && (IntegerQ[m] || GtQ[f, 0]) && NeQ[r, n]
  2338. Int[Times[Power[Plus[Optional[Pattern[a, Blank[]]], Times[Log[Times[Optional[Pattern[c, Blank[]]], Power[Pattern[x, Blank[]], Optional[Pattern[n, Blank[]]]]]], Optional[Pattern[b, Blank[]]]]], Optional[Pattern[p, Blank[]]]], Power[Times[Optional[Pattern[f, Blank[]]], Pattern[x, Blank[]]], Optional[Pattern[m, Blank[]]]], Power[Plus[Pattern[d, Blank[]], Times[Optional[Pattern[e, Blank[]]], Power[Pattern[x, Blank[]], Pattern[r, Blank[]]]]], Optional[Pattern[q, Blank[]]]]], Pattern[x, Blank[Symbol]]] := Simp[(f^m*(d + e*x^r)^(q + 1)*(a + b*Log[c*x^n])^p)/(e*r*(q + 1)), x] - Dist[(b*f^m*n*p)/(e*r*(q + 1)), Int[((d + e*x^r)^(q + 1)*(a + b*Log[c*x^n])^(p - 1))/x, x], x] /; FreeQ[{a, b, c, d, e, f, m, n, q, r}, x] && EqQ[m, r - 1] && IGtQ[p, 0] && (IntegerQ[m] || GtQ[f, 0]) && NeQ[r, n] && NeQ[q, -1]
  2339. Int[Times[Power[Plus[Optional[Pattern[a, Blank[]]], Times[Log[Times[Optional[Pattern[c, Blank[]]], Power[Pattern[x, Blank[]], Optional[Pattern[n, Blank[]]]]]], Optional[Pattern[b, Blank[]]]]], Optional[Pattern[p, Blank[]]]], Power[Times[Pattern[f, Blank[]], Pattern[x, Blank[]]], Optional[Pattern[m, Blank[]]]], Power[Plus[Pattern[d, Blank[]], Times[Optional[Pattern[e, Blank[]]], Power[Pattern[x, Blank[]], Pattern[r, Blank[]]]]], Optional[Pattern[q, Blank[]]]]], Pattern[x, Blank[Symbol]]] := Dist[(f*x)^m/x^m, Int[x^m*(d + e*x^r)^q*(a + b*Log[c*x^n])^p, x], x] /; FreeQ[{a, b, c, d, e, f, m, n, q, r}, x] && EqQ[m, r - 1] && IGtQ[p, 0] && !(IntegerQ[m] || GtQ[f, 0])
  2340. Int[Times[Plus[Optional[Pattern[a, Blank[]]], Times[Log[Times[Optional[Pattern[c, Blank[]]], Power[Pattern[x, Blank[]], Optional[Pattern[n, Blank[]]]]]], Optional[Pattern[b, Blank[]]]]], Power[Times[Optional[Pattern[f, Blank[]]], Pattern[x, Blank[]]], Optional[Pattern[m, Blank[]]]], Power[Plus[Pattern[d, Blank[]], Times[Optional[Pattern[e, Blank[]]], Power[Pattern[x, Blank[]], 2]]], Optional[Pattern[q, Blank[]]]]], Pattern[x, Blank[Symbol]]] := -Simp[((f*x)^(m + 1)*(d + e*x^2)^(q + 1)*(a + b*Log[c*x^n]))/(2*d*f*(q + 1)), x] + Dist[1/(2*d*(q + 1)), Int[(f*x)^m*(d + e*x^2)^(q + 1)*(a*(m + 2*q + 3) + b*n + b*(m + 2*q + 3)*Log[c*x^n]), x], x] /; FreeQ[{a, b, c, d, e, f, m, n}, x] && ILtQ[q, -1] && ILtQ[m, 0]
  2341. Int[Times[Plus[Optional[Pattern[a, Blank[]]], Times[Log[Times[Optional[Pattern[c, Blank[]]], Power[Pattern[x, Blank[]], Optional[Pattern[n, Blank[]]]]]], Optional[Pattern[b, Blank[]]]]], Power[Pattern[x, Blank[]], Optional[Pattern[m, Blank[]]]], Power[Plus[Pattern[d, Blank[]], Times[Optional[Pattern[e, Blank[]]], Power[Pattern[x, Blank[]], 2]]], Pattern[q, Blank[]]]], Pattern[x, Blank[Symbol]]] := Dist[(d^IntPart[q]*(d + e*x^2)^FracPart[q])/(1 + (e*x^2)/d)^FracPart[q], Int[x^m*(1 + (e*x^2)/d)^q*(a + b*Log[c*x^n]), x], x] /; FreeQ[{a, b, c, d, e, n}, x] && IntegerQ[m/2] && IntegerQ[q - 1/2] && !(LtQ[m + 2*q, -2] || GtQ[d, 0])
  2342. Int[Times[Plus[Optional[Pattern[a, Blank[]]], Times[Log[Times[Optional[Pattern[c, Blank[]]], Power[Pattern[x, Blank[]], Optional[Pattern[n, Blank[]]]]]], Optional[Pattern[b, Blank[]]]]], Power[Pattern[x, Blank[]], Optional[Pattern[m, Blank[]]]], Power[Plus[Pattern[d1, Blank[]], Times[Optional[Pattern[e1, Blank[]]], Pattern[x, Blank[]]]], Pattern[q, Blank[]]], Power[Plus[Pattern[d2, Blank[]], Times[Optional[Pattern[e2, Blank[]]], Pattern[x, Blank[]]]], Pattern[q, Blank[]]]], Pattern[x, Blank[Symbol]]] := Dist[((d1 + e1*x)^q*(d2 + e2*x)^q)/(1 + (e1*e2*x^2)/(d1*d2))^q, Int[x^m*(1 + (e1*e2*x^2)/(d1*d2))^q*(a + b*Log[c*x^n]), x], x] /; FreeQ[{a, b, c, d1, e1, d2, e2, n}, x] && EqQ[d2*e1 + d1*e2, 0] && IntegerQ[m] && IntegerQ[q - 1/2]
  2343. Int[Times[Plus[Optional[Pattern[a, Blank[]]], Times[Log[Times[Optional[Pattern[c, Blank[]]], Power[Pattern[x, Blank[]], Pattern[n, Blank[]]]]], Optional[Pattern[b, Blank[]]]]], Power[Pattern[x, Blank[]], -1], Power[Plus[Pattern[d, Blank[]], Times[Optional[Pattern[e, Blank[]]], Power[Pattern[x, Blank[]], Optional[Pattern[r, Blank[]]]]]], -1]], Pattern[x, Blank[Symbol]]] := Dist[1/n, Subst[Int[(a + b*Log[c*x])/(x*(d + e*x^(r/n))), x], x, x^n], x] /; FreeQ[{a, b, c, d, e, n, r}, x] && IntegerQ[r/n]
  2344. Int[Times[Power[Plus[Optional[Pattern[a, Blank[]]], Times[Log[Times[Optional[Pattern[c, Blank[]]], Power[Pattern[x, Blank[]], Optional[Pattern[n, Blank[]]]]]], Optional[Pattern[b, Blank[]]]]], Optional[Pattern[p, Blank[]]]], Power[Pattern[x, Blank[]], -1], Power[Plus[Pattern[d, Blank[]], Times[Optional[Pattern[e, Blank[]]], Pattern[x, Blank[]]]], -1]], Pattern[x, Blank[Symbol]]] := Dist[1/d, Int[(a + b*Log[c*x^n])^p/x, x], x] - Dist[e/d, Int[(a + b*Log[c*x^n])^p/(d + e*x), x], x] /; FreeQ[{a, b, c, d, e, n}, x] && IGtQ[p, 0]
  2345. Int[Times[Power[Plus[Optional[Pattern[a, Blank[]]], Times[Log[Times[Optional[Pattern[c, Blank[]]], Power[Pattern[x, Blank[]], Optional[Pattern[n, Blank[]]]]]], Optional[Pattern[b, Blank[]]]]], Optional[Pattern[p, Blank[]]]], Power[Pattern[x, Blank[]], -1], Power[Plus[Pattern[d, Blank[]], Times[Optional[Pattern[e, Blank[]]], Power[Pattern[x, Blank[]], Optional[Pattern[r, Blank[]]]]]], -1]], Pattern[x, Blank[Symbol]]] := -Simp[(Log[1 + d/(e*x^r)]*(a + b*Log[c*x^n])^p)/(d*r), x] + Dist[(b*n*p)/(d*r), Int[(Log[1 + d/(e*x^r)]*(a + b*Log[c*x^n])^(p - 1))/x, x], x] /; FreeQ[{a, b, c, d, e, n, r}, x] && IGtQ[p, 0]
  2346. Int[Times[Power[Plus[Optional[Pattern[a, Blank[]]], Times[Log[Times[Optional[Pattern[c, Blank[]]], Power[Pattern[x, Blank[]], Optional[Pattern[n, Blank[]]]]]], Optional[Pattern[b, Blank[]]]]], Optional[Pattern[p, Blank[]]]], Power[Pattern[x, Blank[]], -1], Power[Plus[Pattern[d, Blank[]], Times[Optional[Pattern[e, Blank[]]], Pattern[x, Blank[]]]], Optional[Pattern[q, Blank[]]]]], Pattern[x, Blank[Symbol]]] := Dist[d, Int[((d + e*x)^(q - 1)*(a + b*Log[c*x^n])^p)/x, x], x] + Dist[e, Int[(d + e*x)^(q - 1)*(a + b*Log[c*x^n])^p, x], x] /; FreeQ[{a, b, c, d, e, n}, x] && IGtQ[p, 0] && GtQ[q, 0] && IntegerQ[2*q]
  2347. Int[Times[Power[Plus[Optional[Pattern[a, Blank[]]], Times[Log[Times[Optional[Pattern[c, Blank[]]], Power[Pattern[x, Blank[]], Optional[Pattern[n, Blank[]]]]]], Optional[Pattern[b, Blank[]]]]], Optional[Pattern[p, Blank[]]]], Power[Pattern[x, Blank[]], -1], Power[Plus[Pattern[d, Blank[]], Times[Optional[Pattern[e, Blank[]]], Pattern[x, Blank[]]]], Pattern[q, Blank[]]]], Pattern[x, Blank[Symbol]]] := Dist[1/d, Int[((d + e*x)^(q + 1)*(a + b*Log[c*x^n])^p)/x, x], x] - Dist[e/d, Int[(d + e*x)^q*(a + b*Log[c*x^n])^p, x], x] /; FreeQ[{a, b, c, d, e, n}, x] && IGtQ[p, 0] && LtQ[q, -1] && IntegerQ[2*q]
  2348. Int[Times[Plus[Optional[Pattern[a, Blank[]]], Times[Log[Times[Optional[Pattern[c, Blank[]]], Power[Pattern[x, Blank[]], Optional[Pattern[n, Blank[]]]]]], Optional[Pattern[b, Blank[]]]]], Power[Pattern[x, Blank[]], -1], Power[Plus[Pattern[d, Blank[]], Times[Optional[Pattern[e, Blank[]]], Power[Pattern[x, Blank[]], Optional[Pattern[r, Blank[]]]]]], Optional[Pattern[q, Blank[]]]]], Pattern[x, Blank[Symbol]]] := With[{u = IntHide[(d + e*x^r)^q/x, x]}, Simp[u*(a + b*Log[c*x^n]), x] - Dist[b*n, Int[Dist[1/x, u, x], x], x]] /; FreeQ[{a, b, c, d, e, n, r}, x] && IntegerQ[q - 1/2]
  2349. Int[Times[Power[Plus[Optional[Pattern[a, Blank[]]], Times[Log[Times[Optional[Pattern[c, Blank[]]], Power[Pattern[x, Blank[]], Optional[Pattern[n, Blank[]]]]]], Optional[Pattern[b, Blank[]]]]], Optional[Pattern[p, Blank[]]]], Power[Pattern[x, Blank[]], -1], Power[Plus[Pattern[d, Blank[]], Times[Optional[Pattern[e, Blank[]]], Power[Pattern[x, Blank[]], Optional[Pattern[r, Blank[]]]]]], Pattern[q, Blank[]]]], Pattern[x, Blank[Symbol]]] := Dist[1/d, Int[((d + e*x^r)^(q + 1)*(a + b*Log[c*x^n])^p)/x, x], x] - Dist[e/d, Int[x^(r - 1)*(d + e*x^r)^q*(a + b*Log[c*x^n])^p, x], x] /; FreeQ[{a, b, c, d, e, n, r}, x] && IGtQ[p, 0] && ILtQ[q, -1]
  2350. Int[Times[Plus[Optional[Pattern[a, Blank[]]], Times[Log[Times[Optional[Pattern[c, Blank[]]], Power[Pattern[x, Blank[]], Optional[Pattern[n, Blank[]]]]]], Optional[Pattern[b, Blank[]]]]], Power[Times[Optional[Pattern[f, Blank[]]], Pattern[x, Blank[]]], Optional[Pattern[m, Blank[]]]], Power[Plus[Pattern[d, Blank[]], Times[Optional[Pattern[e, Blank[]]], Power[Pattern[x, Blank[]], Optional[Pattern[r, Blank[]]]]]], Optional[Pattern[q, Blank[]]]]], Pattern[x, Blank[Symbol]]] := With[{u = IntHide[(f*x)^m*(d + e*x^r)^q, x]}, Dist[a + b*Log[c*x^n], u, x] - Dist[b*n, Int[SimplifyIntegrand[u/x, x], x], x] /; ((EqQ[r, 1] || EqQ[r, 2]) && IntegerQ[m] && IntegerQ[q - 1/2]) || InverseFunctionFreeQ[u, x]] /; FreeQ[{a, b, c, d, e, f, m, n, q, r}, x] && IntegerQ[2*q] && ((IntegerQ[m] && IntegerQ[r]) || IGtQ[q, 0])
  2351. Int[Times[Plus[Optional[Pattern[a, Blank[]]], Times[Log[Times[Optional[Pattern[c, Blank[]]], Power[Pattern[x, Blank[]], Optional[Pattern[n, Blank[]]]]]], Optional[Pattern[b, Blank[]]]]], Power[Times[Optional[Pattern[f, Blank[]]], Pattern[x, Blank[]]], Optional[Pattern[m, Blank[]]]], Power[Plus[Pattern[d, Blank[]], Times[Optional[Pattern[e, Blank[]]], Power[Pattern[x, Blank[]], Optional[Pattern[r, Blank[]]]]]], Optional[Pattern[q, Blank[]]]]], Pattern[x, Blank[Symbol]]] := With[{u = ExpandIntegrand[a + b*Log[c*x^n], (f*x)^m*(d + e*x^r)^q, x]}, Int[u, x] /; SumQ[u]] /; FreeQ[{a, b, c, d, e, f, m, n, q, r}, x] && IntegerQ[q] && (GtQ[q, 0] || (IntegerQ[m] && IntegerQ[r]))
  2352. Int[Times[Power[Plus[Optional[Pattern[a, Blank[]]], Times[Log[Times[Optional[Pattern[c, Blank[]]], Power[Pattern[x, Blank[]], Pattern[n, Blank[]]]]], Optional[Pattern[b, Blank[]]]]], Optional[Pattern[p, Blank[]]]], Power[Pattern[x, Blank[]], Optional[Pattern[m, Blank[]]]], Power[Plus[Pattern[d, Blank[]], Times[Optional[Pattern[e, Blank[]]], Power[Pattern[x, Blank[]], Optional[Pattern[r, Blank[]]]]]], Optional[Pattern[q, Blank[]]]]], Pattern[x, Blank[Symbol]]] := Dist[1/n, Subst[Int[x^(Simplify[(m + 1)/n] - 1)*(d + e*x^(r/n))^q*(a + b*Log[c*x])^p, x], x, x^n], x] /; FreeQ[{a, b, c, d, e, m, n, p, q, r}, x] && IntegerQ[q] && IntegerQ[r/n] && IntegerQ[Simplify[(m + 1)/n]] && (GtQ[(m + 1)/n, 0] || IGtQ[p, 0])
  2353. Int[Times[Power[Plus[Optional[Pattern[a, Blank[]]], Times[Log[Times[Optional[Pattern[c, Blank[]]], Power[Pattern[x, Blank[]], Optional[Pattern[n, Blank[]]]]]], Optional[Pattern[b, Blank[]]]]], Optional[Pattern[p, Blank[]]]], Power[Times[Optional[Pattern[f, Blank[]]], Pattern[x, Blank[]]], Optional[Pattern[m, Blank[]]]], Power[Plus[Pattern[d, Blank[]], Times[Optional[Pattern[e, Blank[]]], Power[Pattern[x, Blank[]], Optional[Pattern[r, Blank[]]]]]], Optional[Pattern[q, Blank[]]]]], Pattern[x, Blank[Symbol]]] := With[{u = ExpandIntegrand[(a + b*Log[c*x^n])^p, (f*x)^m*(d + e*x^r)^q, x]}, Int[u, x] /; SumQ[u]] /; FreeQ[{a, b, c, d, e, f, m, n, p, q, r}, x] && IntegerQ[q] && (GtQ[q, 0] || (IGtQ[p, 0] && IntegerQ[m] && IntegerQ[r]))
  2354. Int[Times[Power[Plus[Optional[Pattern[a, Blank[]]], Times[Log[Times[Optional[Pattern[c, Blank[]]], Power[Pattern[x, Blank[]], Optional[Pattern[n, Blank[]]]]]], Optional[Pattern[b, Blank[]]]]], Optional[Pattern[p, Blank[]]]], Power[Times[Optional[Pattern[f, Blank[]]], Pattern[x, Blank[]]], Optional[Pattern[m, Blank[]]]], Power[Plus[Pattern[d, Blank[]], Times[Optional[Pattern[e, Blank[]]], Power[Pattern[x, Blank[]], Optional[Pattern[r, Blank[]]]]]], Optional[Pattern[q, Blank[]]]]], Pattern[x, Blank[Symbol]]] := Unintegrable[(f*x)^m*(d + e*x^r)^q*(a + b*Log[c*x^n])^p, x] /; FreeQ[{a, b, c, d, e, f, m, n, p, q, r}, x]
  2355. Int[Times[Power[Plus[Optional[Pattern[a, Blank[]]], Times[Log[Times[Optional[Pattern[c, Blank[]]], Power[Pattern[x, Blank[]], Optional[Pattern[n, Blank[]]]]]], Optional[Pattern[b, Blank[]]]]], Optional[Pattern[p, Blank[]]]], Power[Pattern[u, Blank[]], Optional[Pattern[q, Blank[]]]], Power[Times[Optional[Pattern[f, Blank[]]], Pattern[x, Blank[]]], Optional[Pattern[m, Blank[]]]]], Pattern[x, Blank[Symbol]]] := Int[(f*x)^m*ExpandToSum[u, x]^q*(a + b*Log[c*x^n])^p, x] /; FreeQ[{a, b, c, f, m, n, p, q}, x] && BinomialQ[u, x] && !BinomialMatchQ[u, x]
  2356. Int[Times[Power[Plus[Optional[Pattern[a, Blank[]]], Times[Log[Times[Optional[Pattern[c, Blank[]]], Power[Pattern[x, Blank[]], Optional[Pattern[n, Blank[]]]]]], Optional[Pattern[b, Blank[]]]]], Optional[Pattern[p, Blank[]]]], Pattern[Polyx, Blank[]]], Pattern[x, Blank[Symbol]]] := Int[ExpandIntegrand[Polyx*(a + b*Log[c*x^n])^p, x], x] /; FreeQ[{a, b, c, n, p}, x] && PolynomialQ[Polyx, x]
  2357. Int[Times[Power[Plus[Optional[Pattern[a, Blank[]]], Times[Log[Times[Optional[Pattern[c, Blank[]]], Power[Pattern[x, Blank[]], Optional[Pattern[n, Blank[]]]]]], Optional[Pattern[b, Blank[]]]]], Optional[Pattern[p, Blank[]]]], Pattern[RFx, Blank[]]], Pattern[x, Blank[Symbol]]] := With[{u = ExpandIntegrand[(a + b*Log[c*x^n])^p, RFx, x]}, Int[u, x] /; SumQ[u]] /; FreeQ[{a, b, c, n}, x] && RationalFunctionQ[RFx, x] && IGtQ[p, 0]
  2358. Int[Times[Power[Plus[Optional[Pattern[a, Blank[]]], Times[Log[Times[Optional[Pattern[c, Blank[]]], Power[Pattern[x, Blank[]], Optional[Pattern[n, Blank[]]]]]], Optional[Pattern[b, Blank[]]]]], Optional[Pattern[p, Blank[]]]], Pattern[RFx, Blank[]]], Pattern[x, Blank[Symbol]]] := With[{u = ExpandIntegrand[RFx*(a + b*Log[c*x^n])^p, x]}, Int[u, x] /; SumQ[u]] /; FreeQ[{a, b, c, n}, x] && RationalFunctionQ[RFx, x] && IGtQ[p, 0]
  2359. Int[Times[Power[Plus[Optional[Pattern[a, Blank[]]], Times[Log[Times[Optional[Pattern[c, Blank[]]], Power[Pattern[x, Blank[]], Optional[Pattern[n, Blank[]]]]]], Optional[Pattern[b, Blank[]]]]], Optional[Pattern[p, Blank[]]]], Pattern[AFx, Blank[]]], Pattern[x, Blank[Symbol]]] := Unintegrable[AFx*(a + b*Log[c*x^n])^p, x] /; FreeQ[{a, b, c, n, p}, x] && AlgebraicFunctionQ[AFx, x, True]
  2360. Int[Times[Power[Plus[Optional[Pattern[a, Blank[]]], Times[Log[Times[Optional[Pattern[c, Blank[]]], Power[Pattern[x, Blank[]], Optional[Pattern[n, Blank[]]]]]], Optional[Pattern[b, Blank[]]]]], Optional[Pattern[p, Blank[]]]], Power[Plus[Times[Log[Times[Optional[Pattern[c, Blank[]]], Power[Pattern[x, Blank[]], Optional[Pattern[n, Blank[]]]]]], Optional[Pattern[e, Blank[]]]], Pattern[d, Blank[]]], Optional[Pattern[q, Blank[]]]]], Pattern[x, Blank[Symbol]]] := Int[ExpandIntegrand[(a + b*Log[c*x^n])^p*(d + e*Log[c*x^n])^q, x], x] /; FreeQ[{a, b, c, d, e, n}, x] && IntegerQ[p] && IntegerQ[q]
  2361. Int[Times[Power[Plus[Optional[Pattern[a, Blank[]]], Times[Log[Times[Optional[Pattern[c, Blank[]]], Power[Pattern[x, Blank[]], Optional[Pattern[n, Blank[]]]]]], Optional[Pattern[b, Blank[]]]]], Optional[Pattern[p, Blank[]]]], Plus[Optional[Pattern[d, Blank[]]], Times[Log[Times[Optional[Pattern[f, Blank[]]], Power[Pattern[x, Blank[]], Optional[Pattern[r, Blank[]]]]]], Optional[Pattern[e, Blank[]]]]]], Pattern[x, Blank[Symbol]]] := With[{u = IntHide[(a + b*Log[c*x^n])^p, x]}, Dist[d + e*Log[f*x^r], u, x] - Dist[e*r, Int[SimplifyIntegrand[u/x, x], x], x]] /; FreeQ[{a, b, c, d, e, f, n, p, r}, x]
  2362. Int[Times[Power[Plus[Optional[Pattern[a, Blank[]]], Times[Log[Times[Optional[Pattern[c, Blank[]]], Power[Pattern[x, Blank[]], Optional[Pattern[n, Blank[]]]]]], Optional[Pattern[b, Blank[]]]]], Optional[Pattern[p, Blank[]]]], Power[Plus[Optional[Pattern[d, Blank[]]], Times[Log[Times[Optional[Pattern[f, Blank[]]], Power[Pattern[x, Blank[]], Optional[Pattern[r, Blank[]]]]]], Optional[Pattern[e, Blank[]]]]], Optional[Pattern[q, Blank[]]]]], Pattern[x, Blank[Symbol]]] := Simp[x*(a + b*Log[c*x^n])^p*(d + e*Log[f*x^r])^q, x] + (-Dist[b*n*p, Int[(a + b*Log[c*x^n])^(p - 1)*(d + e*Log[f*x^r])^q, x], x] - Dist[e*q*r, Int[(a + b*Log[c*x^n])^p*(d + e*Log[f*x^r])^(q - 1), x], x]) /; FreeQ[{a, b, c, d, e, f, n, r}, x] && IGtQ[p, 0] && IGtQ[q, 0]
  2363. Int[Times[Power[Plus[Optional[Pattern[a, Blank[]]], Times[Log[Times[Optional[Pattern[c, Blank[]]], Power[Pattern[x, Blank[]], Optional[Pattern[n, Blank[]]]]]], Optional[Pattern[b, Blank[]]]]], Optional[Pattern[p, Blank[]]]], Power[Plus[Optional[Pattern[d, Blank[]]], Times[Log[Times[Optional[Pattern[f, Blank[]]], Power[Pattern[x, Blank[]], Optional[Pattern[r, Blank[]]]]]], Optional[Pattern[e, Blank[]]]]], Optional[Pattern[q, Blank[]]]]], Pattern[x, Blank[Symbol]]] := Unintegrable[(a + b*Log[c*x^n])^p*(d + e*Log[f*x^r])^q, x] /; FreeQ[{a, b, c, d, e, f, n, p, q, r}, x]
  2364. Int[Times[Power[Plus[Optional[Pattern[a, Blank[]]], Times[Log[Pattern[v, Blank[]]], Optional[Pattern[b, Blank[]]]]], Optional[Pattern[p, Blank[]]]], Power[Plus[Optional[Pattern[c, Blank[]]], Times[Log[Pattern[v, Blank[]]], Optional[Pattern[d, Blank[]]]]], Optional[Pattern[q, Blank[]]]]], Pattern[x, Blank[Symbol]]] := Dist[1/Coeff[v, x, 1], Subst[Int[(a + b*Log[x])^p*(c + d*Log[x])^q, x], x, v], x] /; FreeQ[{a, b, c, d, p, q}, x] && LinearQ[v, x] && NeQ[Coeff[v, x, 0], 0]
  2365. Int[Times[Power[Plus[Optional[Pattern[a, Blank[]]], Times[Log[Times[Optional[Pattern[c, Blank[]]], Power[Pattern[x, Blank[]], Optional[Pattern[n, Blank[]]]]]], Optional[Pattern[b, Blank[]]]]], Optional[Pattern[p, Blank[]]]], Power[Plus[Optional[Pattern[d, Blank[]]], Times[Log[Times[Optional[Pattern[c, Blank[]]], Power[Pattern[x, Blank[]], Optional[Pattern[n, Blank[]]]]]], Optional[Pattern[e, Blank[]]]]], Optional[Pattern[q, Blank[]]]], Power[Pattern[x, Blank[]], -1]], Pattern[x, Blank[Symbol]]] := Dist[1/n, Subst[Int[(a + b*x)^p*(d + e*x)^q, x], x, Log[c*x^n]], x] /; FreeQ[{a, b, c, d, e, n, p, q}, x]
  2366. Int[Times[Power[Plus[Optional[Pattern[a, Blank[]]], Times[Log[Times[Optional[Pattern[c, Blank[]]], Power[Pattern[x, Blank[]], Optional[Pattern[n, Blank[]]]]]], Optional[Pattern[b, Blank[]]]]], Optional[Pattern[p, Blank[]]]], Plus[Optional[Pattern[d, Blank[]]], Times[Log[Times[Optional[Pattern[f, Blank[]]], Power[Pattern[x, Blank[]], Optional[Pattern[r, Blank[]]]]]], Optional[Pattern[e, Blank[]]]]], Power[Times[Optional[Pattern[g, Blank[]]], Pattern[x, Blank[]]], Optional[Pattern[m, Blank[]]]]], Pattern[x, Blank[Symbol]]] := With[{u = IntHide[(g*x)^m*(a + b*Log[c*x^n])^p, x]}, Dist[d + e*Log[f*x^r], u, x] - Dist[e*r, Int[SimplifyIntegrand[u/x, x], x], x]] /; FreeQ[{a, b, c, d, e, f, g, m, n, p, r}, x] && !(EqQ[p, 1] && EqQ[a, 0] && NeQ[d, 0])
  2367. Int[Times[Power[Plus[Optional[Pattern[a, Blank[]]], Times[Log[Times[Optional[Pattern[c, Blank[]]], Power[Pattern[x, Blank[]], Optional[Pattern[n, Blank[]]]]]], Optional[Pattern[b, Blank[]]]]], Optional[Pattern[p, Blank[]]]], Power[Plus[Optional[Pattern[d, Blank[]]], Times[Log[Times[Optional[Pattern[f, Blank[]]], Power[Pattern[x, Blank[]], Optional[Pattern[r, Blank[]]]]]], Optional[Pattern[e, Blank[]]]]], Optional[Pattern[q, Blank[]]]], Power[Times[Optional[Pattern[g, Blank[]]], Pattern[x, Blank[]]], Optional[Pattern[m, Blank[]]]]], Pattern[x, Blank[Symbol]]] := Simp[((g*x)^(m + 1)*(a + b*Log[c*x^n])^p*(d + e*Log[f*x^r])^q)/(g*(m + 1)), x] + (-Dist[(b*n*p)/(m + 1), Int[(g*x)^m*(a + b*Log[c*x^n])^(p - 1)*(d + e*Log[f*x^r])^q, x], x] - Dist[(e*q*r)/(m + 1), Int[(g*x)^m*(a + b*Log[c*x^n])^p*(d + e*Log[f*x^r])^(q - 1), x], x]) /; FreeQ[{a, b, c, d, e, f, g, m, n, r}, x] && IGtQ[p, 0] && IGtQ[q, 0] && NeQ[m, -1]
  2368. Int[Times[Power[Plus[Optional[Pattern[a, Blank[]]], Times[Log[Times[Optional[Pattern[c, Blank[]]], Power[Pattern[x, Blank[]], Optional[Pattern[n, Blank[]]]]]], Optional[Pattern[b, Blank[]]]]], Optional[Pattern[p, Blank[]]]], Power[Plus[Optional[Pattern[d, Blank[]]], Times[Log[Times[Optional[Pattern[f, Blank[]]], Power[Pattern[x, Blank[]], Optional[Pattern[r, Blank[]]]]]], Optional[Pattern[e, Blank[]]]]], Optional[Pattern[q, Blank[]]]], Power[Times[Optional[Pattern[g, Blank[]]], Pattern[x, Blank[]]], Optional[Pattern[m, Blank[]]]]], Pattern[x, Blank[Symbol]]] := Unintegrable[(g*x)^m*(a + b*Log[c*x^n])^p*(d + e*Log[f*x^r])^q, x] /; FreeQ[{a, b, c, d, e, f, g, m, n, p, q, r}, x]
  2369. Int[Times[Power[Plus[Optional[Pattern[a, Blank[]]], Times[Log[Pattern[v, Blank[]]], Optional[Pattern[b, Blank[]]]]], Optional[Pattern[p, Blank[]]]], Power[Plus[Optional[Pattern[c, Blank[]]], Times[Log[Pattern[v, Blank[]]], Optional[Pattern[d, Blank[]]]]], Optional[Pattern[q, Blank[]]]], Power[Pattern[u, Blank[]], Optional[Pattern[m, Blank[]]]]], Pattern[x, Blank[Symbol]]] := With[{e = Coeff[u, x, 0], f = Coeff[u, x, 1], g = Coeff[v, x, 0], h = Coeff[v, x, 1]}, Dist[1/h, Subst[Int[((f*x)/h)^m*(a + b*Log[x])^p*(c + d*Log[x])^q, x], x, v], x] /; EqQ[f*g - e*h, 0] && NeQ[g, 0]] /; FreeQ[{a, b, c, d, m, p, q}, x] && LinearQ[{u, v}, x]
  2370. Int[Times[Log[Times[Optional[Pattern[d, Blank[]]], Power[Plus[Pattern[e, Blank[]], Times[Optional[Pattern[f, Blank[]]], Power[Pattern[x, Blank[]], Optional[Pattern[m, Blank[]]]]]], Optional[Pattern[r, Blank[]]]]]], Power[Plus[Optional[Pattern[a, Blank[]]], Times[Log[Times[Optional[Pattern[c, Blank[]]], Power[Pattern[x, Blank[]], Optional[Pattern[n, Blank[]]]]]], Optional[Pattern[b, Blank[]]]]], Optional[Pattern[p, Blank[]]]]], Pattern[x, Blank[Symbol]]] := With[{u = IntHide[Log[d*(e + f*x^m)^r], x]}, Dist[(a + b*Log[c*x^n])^p, u, x] - Dist[b*n*p, Int[Dist[(a + b*Log[c*x^n])^(p - 1)/x, u, x], x], x]] /; FreeQ[{a, b, c, d, e, f, r, m, n}, x] && IGtQ[p, 0] && RationalQ[m] && (EqQ[p, 1] || (FractionQ[m] && IntegerQ[1/m]) || (EqQ[r, 1] && EqQ[m, 1] && EqQ[d*e, 1]))
  2371. Int[Times[Log[Times[Optional[Pattern[d, Blank[]]], Power[Plus[Pattern[e, Blank[]], Times[Optional[Pattern[f, Blank[]]], Power[Pattern[x, Blank[]], Optional[Pattern[m, Blank[]]]]]], Optional[Pattern[r, Blank[]]]]]], Power[Plus[Optional[Pattern[a, Blank[]]], Times[Log[Times[Optional[Pattern[c, Blank[]]], Power[Pattern[x, Blank[]], Optional[Pattern[n, Blank[]]]]]], Optional[Pattern[b, Blank[]]]]], Optional[Pattern[p, Blank[]]]]], Pattern[x, Blank[Symbol]]] := With[{u = IntHide[(a + b*Log[c*x^n])^p, x]}, Dist[Log[d*(e + f*x^m)^r], u, x] - Dist[f*m*r, Int[Dist[x^(m - 1)/(e + f*x^m), u, x], x], x]] /; FreeQ[{a, b, c, d, e, f, r, m, n}, x] && IGtQ[p, 0] && IntegerQ[m]
  2372. Int[Times[Log[Times[Optional[Pattern[d, Blank[]]], Power[Plus[Pattern[e, Blank[]], Times[Optional[Pattern[f, Blank[]]], Power[Pattern[x, Blank[]], Optional[Pattern[m, Blank[]]]]]], Optional[Pattern[r, Blank[]]]]]], Power[Plus[Optional[Pattern[a, Blank[]]], Times[Log[Times[Optional[Pattern[c, Blank[]]], Power[Pattern[x, Blank[]], Optional[Pattern[n, Blank[]]]]]], Optional[Pattern[b, Blank[]]]]], Optional[Pattern[p, Blank[]]]]], Pattern[x, Blank[Symbol]]] := Unintegrable[(a + b*Log[c*x^n])^p*Log[d*(e + f*x^m)^r], x] /; FreeQ[{a, b, c, d, e, f, r, m, n, p}, x]
  2373. Int[Times[Log[Times[Optional[Pattern[d, Blank[]]], Power[Pattern[u, Blank[]], Optional[Pattern[r, Blank[]]]]]], Power[Plus[Optional[Pattern[a, Blank[]]], Times[Log[Times[Optional[Pattern[c, Blank[]]], Power[Pattern[x, Blank[]], Optional[Pattern[n, Blank[]]]]]], Optional[Pattern[b, Blank[]]]]], Optional[Pattern[p, Blank[]]]]], Pattern[x, Blank[Symbol]]] := Int[Log[d*ExpandToSum[u, x]^r]*(a + b*Log[c*x^n])^p, x] /; FreeQ[{a, b, c, d, r, n, p}, x] && BinomialQ[u, x] && !BinomialMatchQ[u, x]
  2374. Int[Times[Log[Times[Optional[Pattern[d, Blank[]]], Plus[Pattern[e, Blank[]], Times[Optional[Pattern[f, Blank[]]], Power[Pattern[x, Blank[]], Optional[Pattern[m, Blank[]]]]]]]], Power[Plus[Optional[Pattern[a, Blank[]]], Times[Log[Times[Optional[Pattern[c, Blank[]]], Power[Pattern[x, Blank[]], Optional[Pattern[n, Blank[]]]]]], Optional[Pattern[b, Blank[]]]]], Optional[Pattern[p, Blank[]]]], Power[Pattern[x, Blank[]], -1]], Pattern[x, Blank[Symbol]]] := -Simp[(PolyLog[2, -(d*f*x^m)]*(a + b*Log[c*x^n])^p)/m, x] + Dist[(b*n*p)/m, Int[(PolyLog[2, -(d*f*x^m)]*(a + b*Log[c*x^n])^(p - 1))/x, x], x] /; FreeQ[{a, b, c, d, e, f, m, n}, x] && IGtQ[p, 0] && EqQ[d*e, 1]
  2375. Int[Times[Log[Times[Optional[Pattern[d, Blank[]]], Power[Plus[Pattern[e, Blank[]], Times[Optional[Pattern[f, Blank[]]], Power[Pattern[x, Blank[]], Optional[Pattern[m, Blank[]]]]]], Optional[Pattern[r, Blank[]]]]]], Power[Plus[Optional[Pattern[a, Blank[]]], Times[Log[Times[Optional[Pattern[c, Blank[]]], Power[Pattern[x, Blank[]], Optional[Pattern[n, Blank[]]]]]], Optional[Pattern[b, Blank[]]]]], Optional[Pattern[p, Blank[]]]], Power[Pattern[x, Blank[]], -1]], Pattern[x, Blank[Symbol]]] := Simp[(Log[d*(e + f*x^m)^r]*(a + b*Log[c*x^n])^(p + 1))/(b*n*(p + 1)), x] - Dist[(f*m*r)/(b*n*(p + 1)), Int[(x^(m - 1)*(a + b*Log[c*x^n])^(p + 1))/(e + f*x^m), x], x] /; FreeQ[{a, b, c, d, e, f, r, m, n}, x] && IGtQ[p, 0] && NeQ[d*e, 1]
  2376. Int[Times[Log[Times[Optional[Pattern[d, Blank[]]], Power[Plus[Pattern[e, Blank[]], Times[Optional[Pattern[f, Blank[]]], Power[Pattern[x, Blank[]], Optional[Pattern[m, Blank[]]]]]], Optional[Pattern[r, Blank[]]]]]], Plus[Optional[Pattern[a, Blank[]]], Times[Log[Times[Optional[Pattern[c, Blank[]]], Power[Pattern[x, Blank[]], Optional[Pattern[n, Blank[]]]]]], Optional[Pattern[b, Blank[]]]]], Power[Times[Optional[Pattern[g, Blank[]]], Pattern[x, Blank[]]], Optional[Pattern[q, Blank[]]]]], Pattern[x, Blank[Symbol]]] := With[{u = IntHide[(g*x)^q*Log[d*(e + f*x^m)^r], x]}, Dist[a + b*Log[c*x^n], u, x] - Dist[b*n, Int[Dist[1/x, u, x], x], x]] /; FreeQ[{a, b, c, d, e, f, g, r, m, n, q}, x] && (IntegerQ[(q + 1)/m] || (RationalQ[m] && RationalQ[q])) && NeQ[q, -1]
  2377. Int[Times[Log[Times[Optional[Pattern[d, Blank[]]], Plus[Pattern[e, Blank[]], Times[Optional[Pattern[f, Blank[]]], Power[Pattern[x, Blank[]], Optional[Pattern[m, Blank[]]]]]]]], Power[Plus[Optional[Pattern[a, Blank[]]], Times[Log[Times[Optional[Pattern[c, Blank[]]], Power[Pattern[x, Blank[]], Optional[Pattern[n, Blank[]]]]]], Optional[Pattern[b, Blank[]]]]], Optional[Pattern[p, Blank[]]]], Power[Times[Optional[Pattern[g, Blank[]]], Pattern[x, Blank[]]], Optional[Pattern[q, Blank[]]]]], Pattern[x, Blank[Symbol]]] := With[{u = IntHide[(g*x)^q*Log[d*(e + f*x^m)], x]}, Dist[(a + b*Log[c*x^n])^p, u, x] - Dist[b*n*p, Int[Dist[(a + b*Log[c*x^n])^(p - 1)/x, u, x], x], x]] /; FreeQ[{a, b, c, d, e, f, g, m, n, q}, x] && IGtQ[p, 0] && RationalQ[m] && RationalQ[q] && NeQ[q, -1] && (EqQ[p, 1] || (FractionQ[m] && IntegerQ[(q + 1)/m]) || (IGtQ[q, 0] && IntegerQ[(q + 1)/m] && EqQ[d*e, 1]))
  2378. Int[Times[Log[Times[Optional[Pattern[d, Blank[]]], Power[Plus[Pattern[e, Blank[]], Times[Optional[Pattern[f, Blank[]]], Power[Pattern[x, Blank[]], Optional[Pattern[m, Blank[]]]]]], Optional[Pattern[r, Blank[]]]]]], Power[Plus[Optional[Pattern[a, Blank[]]], Times[Log[Times[Optional[Pattern[c, Blank[]]], Power[Pattern[x, Blank[]], Optional[Pattern[n, Blank[]]]]]], Optional[Pattern[b, Blank[]]]]], Optional[Pattern[p, Blank[]]]], Power[Times[Optional[Pattern[g, Blank[]]], Pattern[x, Blank[]]], Optional[Pattern[q, Blank[]]]]], Pattern[x, Blank[Symbol]]] := With[{u = IntHide[(g*x)^q*(a + b*Log[c*x^n])^p, x]}, Dist[Log[d*(e + f*x^m)^r], u, x] - Dist[f*m*r, Int[Dist[x^(m - 1)/(e + f*x^m), u, x], x], x]] /; FreeQ[{a, b, c, d, e, f, g, r, m, n, q}, x] && IGtQ[p, 0] && RationalQ[m] && RationalQ[q]
  2379. Int[Times[Log[Times[Optional[Pattern[d, Blank[]]], Power[Plus[Pattern[e, Blank[]], Times[Optional[Pattern[f, Blank[]]], Power[Pattern[x, Blank[]], Optional[Pattern[m, Blank[]]]]]], Optional[Pattern[r, Blank[]]]]]], Power[Plus[Optional[Pattern[a, Blank[]]], Times[Log[Times[Optional[Pattern[c, Blank[]]], Power[Pattern[x, Blank[]], Optional[Pattern[n, Blank[]]]]]], Optional[Pattern[b, Blank[]]]]], Optional[Pattern[p, Blank[]]]], Power[Times[Optional[Pattern[g, Blank[]]], Pattern[x, Blank[]]], Optional[Pattern[q, Blank[]]]]], Pattern[x, Blank[Symbol]]] := Unintegrable[(g*x)^q*(a + b*Log[c*x^n])^p*Log[d*(e + f*x^m)^r], x] /; FreeQ[{a, b, c, d, e, f, g, r, m, n, p, q}, x]
  2380. Int[Times[Log[Times[Optional[Pattern[d, Blank[]]], Power[Pattern[u, Blank[]], Optional[Pattern[r, Blank[]]]]]], Power[Plus[Optional[Pattern[a, Blank[]]], Times[Log[Times[Optional[Pattern[c, Blank[]]], Power[Pattern[x, Blank[]], Optional[Pattern[n, Blank[]]]]]], Optional[Pattern[b, Blank[]]]]], Optional[Pattern[p, Blank[]]]], Power[Times[Optional[Pattern[g, Blank[]]], Pattern[x, Blank[]]], Optional[Pattern[q, Blank[]]]]], Pattern[x, Blank[Symbol]]] := Int[(g*x)^q*Log[d*ExpandToSum[u, x]^r]*(a + b*Log[c*x^n])^p, x] /; FreeQ[{a, b, c, d, g, r, n, p, q}, x] && BinomialQ[u, x] && !BinomialMatchQ[u, x]
  2381. Int[Times[Plus[Optional[Pattern[a, Blank[]]], Times[Log[Times[Optional[Pattern[c, Blank[]]], Power[Pattern[x, Blank[]], Optional[Pattern[n, Blank[]]]]]], Optional[Pattern[b, Blank[]]]]], PolyLog[Pattern[k, Blank[]], Times[Optional[Pattern[e, Blank[]]], Power[Pattern[x, Blank[]], Optional[Pattern[q, Blank[]]]]]]], Pattern[x, Blank[Symbol]]] := -Simp[b*n*x*PolyLog[k, e*x^q], x] + (-Dist[q, Int[PolyLog[k - 1, e*x^q]*(a + b*Log[c*x^n]), x], x] + Dist[b*n*q, Int[PolyLog[k - 1, e*x^q], x], x] + Simp[x*PolyLog[k, e*x^q]*(a + b*Log[c*x^n]), x]) /; FreeQ[{a, b, c, e, n, q}, x] && IGtQ[k, 0]
  2382. Int[Times[Power[Plus[Optional[Pattern[a, Blank[]]], Times[Log[Times[Optional[Pattern[c, Blank[]]], Power[Pattern[x, Blank[]], Optional[Pattern[n, Blank[]]]]]], Optional[Pattern[b, Blank[]]]]], Optional[Pattern[p, Blank[]]]], PolyLog[Pattern[k, Blank[]], Times[Optional[Pattern[e, Blank[]]], Power[Pattern[x, Blank[]], Optional[Pattern[q, Blank[]]]]]]], Pattern[x, Blank[Symbol]]] := Unintegrable[(a + b*Log[c*x^n])^p*PolyLog[k, e*x^q], x] /; FreeQ[{a, b, c, e, n, p, q}, x]
  2383. Int[Times[Power[Plus[Optional[Pattern[a, Blank[]]], Times[Log[Times[Optional[Pattern[c, Blank[]]], Power[Pattern[x, Blank[]], Optional[Pattern[n, Blank[]]]]]], Optional[Pattern[b, Blank[]]]]], Optional[Pattern[p, Blank[]]]], Power[Pattern[x, Blank[]], -1], PolyLog[Pattern[k, Blank[]], Times[Optional[Pattern[e, Blank[]]], Power[Pattern[x, Blank[]], Optional[Pattern[q, Blank[]]]]]]], Pattern[x, Blank[Symbol]]] := Simp[(PolyLog[k + 1, e*x^q]*(a + b*Log[c*x^n])^p)/q, x] - Dist[(b*n*p)/q, Int[(PolyLog[k + 1, e*x^q]*(a + b*Log[c*x^n])^(p - 1))/x, x], x] /; FreeQ[{a, b, c, e, k, n, q}, x] && GtQ[p, 0]
  2384. Int[Times[Power[Plus[Optional[Pattern[a, Blank[]]], Times[Log[Times[Optional[Pattern[c, Blank[]]], Power[Pattern[x, Blank[]], Optional[Pattern[n, Blank[]]]]]], Optional[Pattern[b, Blank[]]]]], Optional[Pattern[p, Blank[]]]], Power[Pattern[x, Blank[]], -1], PolyLog[Pattern[k, Blank[]], Times[Optional[Pattern[e, Blank[]]], Power[Pattern[x, Blank[]], Optional[Pattern[q, Blank[]]]]]]], Pattern[x, Blank[Symbol]]] := Simp[(PolyLog[k, e*x^q]*(a + b*Log[c*x^n])^(p + 1))/(b*n*(p + 1)), x] - Dist[q/(b*n*(p + 1)), Int[(PolyLog[k - 1, e*x^q]*(a + b*Log[c*x^n])^(p + 1))/x, x], x] /; FreeQ[{a, b, c, e, k, n, q}, x] && LtQ[p, -1]
  2385. Int[Times[Plus[Optional[Pattern[a, Blank[]]], Times[Log[Times[Optional[Pattern[c, Blank[]]], Power[Pattern[x, Blank[]], Optional[Pattern[n, Blank[]]]]]], Optional[Pattern[b, Blank[]]]]], Power[Times[Optional[Pattern[d, Blank[]]], Pattern[x, Blank[]]], Optional[Pattern[m, Blank[]]]], PolyLog[Pattern[k, Blank[]], Times[Optional[Pattern[e, Blank[]]], Power[Pattern[x, Blank[]], Optional[Pattern[q, Blank[]]]]]]], Pattern[x, Blank[Symbol]]] := -Simp[(b*n*(d*x)^(m + 1)*PolyLog[k, e*x^q])/(d*(m + 1)^2), x] + (-Dist[q/(m + 1), Int[(d*x)^m*PolyLog[k - 1, e*x^q]*(a + b*Log[c*x^n]), x], x] + Dist[(b*n*q)/(m + 1)^2, Int[(d*x)^m*PolyLog[k - 1, e*x^q], x], x] + Simp[((d*x)^(m + 1)*PolyLog[k, e*x^q]*(a + b*Log[c*x^n]))/(d*(m + 1)), x]) /; FreeQ[{a, b, c, d, e, m, n, q}, x] && IGtQ[k, 0]
  2386. Int[Times[Power[Plus[Optional[Pattern[a, Blank[]]], Times[Log[Times[Optional[Pattern[c, Blank[]]], Power[Pattern[x, Blank[]], Optional[Pattern[n, Blank[]]]]]], Optional[Pattern[b, Blank[]]]]], Optional[Pattern[p, Blank[]]]], Power[Times[Optional[Pattern[d, Blank[]]], Pattern[x, Blank[]]], Optional[Pattern[m, Blank[]]]], PolyLog[Pattern[k, Blank[]], Times[Optional[Pattern[e, Blank[]]], Power[Pattern[x, Blank[]], Optional[Pattern[q, Blank[]]]]]]], Pattern[x, Blank[Symbol]]] := Unintegrable[(d*x)^m*(a + b*Log[c*x^n])^p*PolyLog[k, e*x^q], x] /; FreeQ[{a, b, c, d, e, m, n, p, q}, x]
  2387. Int[Times[Plus[Optional[Pattern[a, Blank[]]], Times[Log[Times[Optional[Pattern[c, Blank[]]], Power[Pattern[x, Blank[]], Optional[Pattern[n, Blank[]]]]]], Optional[Pattern[b, Blank[]]]]], Optional[Pattern[Px, Blank[]]], Power[Pattern[F, Blank[]][Times[Optional[Pattern[d, Blank[]]], Plus[Optional[Pattern[e, Blank[]]], Times[Optional[Pattern[f, Blank[]]], Pattern[x, Blank[]]]]]], Optional[Pattern[m, Blank[]]]]], Pattern[x, Blank[Symbol]]] := With[{u = IntHide[Px*F[d*(e + f*x)]^m, x]}, Dist[a + b*Log[c*x^n], u, x] - Dist[b*n, Int[Dist[1/x, u, x], x], x]] /; FreeQ[{a, b, c, d, e, f, n}, x] && PolynomialQ[Px, x] && IGtQ[m, 0] && MemberQ[{ArcSin, ArcCos, ArcSinh, ArcCosh}, F]
  2388. Int[Times[Plus[Optional[Pattern[a, Blank[]]], Times[Log[Times[Optional[Pattern[c, Blank[]]], Power[Pattern[x, Blank[]], Optional[Pattern[n, Blank[]]]]]], Optional[Pattern[b, Blank[]]]]], Optional[Pattern[Px, Blank[]]], Pattern[F, Blank[]][Times[Optional[Pattern[d, Blank[]]], Plus[Optional[Pattern[e, Blank[]]], Times[Optional[Pattern[f, Blank[]]], Pattern[x, Blank[]]]]]]], Pattern[x, Blank[Symbol]]] := With[{u = IntHide[Px*F[d*(e + f*x)], x]}, Dist[a + b*Log[c*x^n], u, x] - Dist[b*n, Int[Dist[1/x, u, x], x], x]] /; FreeQ[{a, b, c, d, e, f, n}, x] && PolynomialQ[Px, x] && MemberQ[{ArcTan, ArcCot, ArcTanh, ArcCoth}, F]
  2389. Int[Power[Plus[Optional[Pattern[a, Blank[]]], Times[Log[Times[Optional[Pattern[c, Blank[]]], Power[Plus[Pattern[d, Blank[]], Times[Optional[Pattern[e, Blank[]]], Pattern[x, Blank[]]]], Optional[Pattern[n, Blank[]]]]]], Optional[Pattern[b, Blank[]]]]], Optional[Pattern[p, Blank[]]]], Pattern[x, Blank[Symbol]]] := Dist[1/e, Subst[Int[(a + b*Log[c*x^n])^p, x], x, d + e*x], x] /; FreeQ[{a, b, c, d, e, n, p}, x]
  2390. Int[Times[Power[Plus[Optional[Pattern[a, Blank[]]], Times[Log[Times[Optional[Pattern[c, Blank[]]], Power[Plus[Pattern[d, Blank[]], Times[Optional[Pattern[e, Blank[]]], Pattern[x, Blank[]]]], Optional[Pattern[n, Blank[]]]]]], Optional[Pattern[b, Blank[]]]]], Optional[Pattern[p, Blank[]]]], Power[Plus[Pattern[f, Blank[]], Times[Optional[Pattern[g, Blank[]]], Pattern[x, Blank[]]]], Optional[Pattern[q, Blank[]]]]], Pattern[x, Blank[Symbol]]] := Dist[1/e, Subst[Int[((f*x)/d)^q*(a + b*Log[c*x^n])^p, x], x, d + e*x], x] /; FreeQ[{a, b, c, d, e, f, g, n, p, q}, x] && EqQ[e*f - d*g, 0]
  2391. Int[Times[Log[Times[Optional[Pattern[c, Blank[]]], Plus[Pattern[d, Blank[]], Times[Optional[Pattern[e, Blank[]]], Power[Pattern[x, Blank[]], Optional[Pattern[n, Blank[]]]]]]]], Power[Pattern[x, Blank[]], -1]], Pattern[x, Blank[Symbol]]] := -Simp[PolyLog[2, -(c*e*x^n)]/n, x] /; FreeQ[{c, d, e, n}, x] && EqQ[c*d, 1]
  2392. Int[Times[Plus[Optional[Pattern[a, Blank[]]], Times[Log[Times[Optional[Pattern[c, Blank[]]], Plus[Pattern[d, Blank[]], Times[Optional[Pattern[e, Blank[]]], Pattern[x, Blank[]]]]]], Optional[Pattern[b, Blank[]]]]], Power[Pattern[x, Blank[]], -1]], Pattern[x, Blank[Symbol]]] := Simp[(a + b*Log[c*d])*Log[x], x] + Dist[b, Int[Log[1 + (e*x)/d]/x, x], x] /; FreeQ[{a, b, c, d, e}, x] && GtQ[c*d, 0]
  2393. Int[Times[Plus[Optional[Pattern[a, Blank[]]], Times[Log[Times[Optional[Pattern[c, Blank[]]], Plus[Pattern[d, Blank[]], Times[Optional[Pattern[e, Blank[]]], Pattern[x, Blank[]]]]]], Optional[Pattern[b, Blank[]]]]], Power[Plus[Optional[Pattern[f, Blank[]]], Times[Optional[Pattern[g, Blank[]]], Pattern[x, Blank[]]]], -1]], Pattern[x, Blank[Symbol]]] := Dist[1/g, Subst[Int[(a + b*Log[1 + (c*e*x)/g])/x, x], x, f + g*x], x] /; FreeQ[{a, b, c, d, e, f, g}, x] && NeQ[e*f - d*g, 0] && EqQ[g + c*(e*f - d*g), 0]
  2394. Int[Times[Plus[Optional[Pattern[a, Blank[]]], Times[Log[Times[Optional[Pattern[c, Blank[]]], Power[Plus[Pattern[d, Blank[]], Times[Optional[Pattern[e, Blank[]]], Pattern[x, Blank[]]]], Optional[Pattern[n, Blank[]]]]]], Optional[Pattern[b, Blank[]]]]], Power[Plus[Optional[Pattern[f, Blank[]]], Times[Optional[Pattern[g, Blank[]]], Pattern[x, Blank[]]]], -1]], Pattern[x, Blank[Symbol]]] := Simp[(Log[(e*(f + g*x))/(e*f - d*g)]*(a + b*Log[c*(d + e*x)^n]))/g, x] - Dist[(b*e*n)/g, Int[Log[(e*(f + g*x))/(e*f - d*g)]/(d + e*x), x], x] /; FreeQ[{a, b, c, d, e, f, g, n}, x] && NeQ[e*f - d*g, 0]
  2395. Int[Times[Plus[Optional[Pattern[a, Blank[]]], Times[Log[Times[Optional[Pattern[c, Blank[]]], Power[Plus[Pattern[d, Blank[]], Times[Optional[Pattern[e, Blank[]]], Pattern[x, Blank[]]]], Optional[Pattern[n, Blank[]]]]]], Optional[Pattern[b, Blank[]]]]], Power[Plus[Optional[Pattern[f, Blank[]]], Times[Optional[Pattern[g, Blank[]]], Pattern[x, Blank[]]]], Optional[Pattern[q, Blank[]]]]], Pattern[x, Blank[Symbol]]] := Simp[((f + g*x)^(q + 1)*(a + b*Log[c*(d + e*x)^n]))/(g*(q + 1)), x] - Dist[(b*e*n)/(g*(q + 1)), Int[(f + g*x)^(q + 1)/(d + e*x), x], x] /; FreeQ[{a, b, c, d, e, f, g, n, q}, x] && NeQ[e*f - d*g, 0] && NeQ[q, -1]
  2396. Int[Times[Power[Plus[Optional[Pattern[a, Blank[]]], Times[Log[Times[Optional[Pattern[c, Blank[]]], Power[Plus[Pattern[d, Blank[]], Times[Optional[Pattern[e, Blank[]]], Pattern[x, Blank[]]]], Optional[Pattern[n, Blank[]]]]]], Optional[Pattern[b, Blank[]]]]], Pattern[p, Blank[]]], Power[Plus[Optional[Pattern[f, Blank[]]], Times[Optional[Pattern[g, Blank[]]], Pattern[x, Blank[]]]], -1]], Pattern[x, Blank[Symbol]]] := Simp[(Log[(e*(f + g*x))/(e*f - d*g)]*(a + b*Log[c*(d + e*x)^n])^p)/g, x] - Dist[(b*e*n*p)/g, Int[(Log[(e*(f + g*x))/(e*f - d*g)]*(a + b*Log[c*(d + e*x)^n])^(p - 1))/(d + e*x), x], x] /; FreeQ[{a, b, c, d, e, f, g, n, p}, x] && NeQ[e*f - d*g, 0] && IGtQ[p, 1]
  2397. Int[Times[Power[Plus[Optional[Pattern[a, Blank[]]], Times[Log[Times[Optional[Pattern[c, Blank[]]], Power[Plus[Pattern[d, Blank[]], Times[Optional[Pattern[e, Blank[]]], Pattern[x, Blank[]]]], Optional[Pattern[n, Blank[]]]]]], Optional[Pattern[b, Blank[]]]]], Pattern[p, Blank[]]], Power[Plus[Optional[Pattern[f, Blank[]]], Times[Optional[Pattern[g, Blank[]]], Pattern[x, Blank[]]]], -2]], Pattern[x, Blank[Symbol]]] := Simp[((d + e*x)*(a + b*Log[c*(d + e*x)^n])^p)/((e*f - d*g)*(f + g*x)), x] - Dist[(b*e*n*p)/(e*f - d*g), Int[(a + b*Log[c*(d + e*x)^n])^(p - 1)/(f + g*x), x], x] /; FreeQ[{a, b, c, d, e, f, g, n}, x] && NeQ[e*f - d*g, 0] && GtQ[p, 0]
  2398. Int[Times[Power[Plus[Optional[Pattern[a, Blank[]]], Times[Log[Times[Optional[Pattern[c, Blank[]]], Power[Plus[Pattern[d, Blank[]], Times[Optional[Pattern[e, Blank[]]], Pattern[x, Blank[]]]], Optional[Pattern[n, Blank[]]]]]], Optional[Pattern[b, Blank[]]]]], Pattern[p, Blank[]]], Power[Plus[Optional[Pattern[f, Blank[]]], Times[Optional[Pattern[g, Blank[]]], Pattern[x, Blank[]]]], Optional[Pattern[q, Blank[]]]]], Pattern[x, Blank[Symbol]]] := Simp[((f + g*x)^(q + 1)*(a + b*Log[c*(d + e*x)^n])^p)/(g*(q + 1)), x] - Dist[(b*e*n*p)/(g*(q + 1)), Int[((f + g*x)^(q + 1)*(a + b*Log[c*(d + e*x)^n])^(p - 1))/(d + e*x), x], x] /; FreeQ[{a, b, c, d, e, f, g, n, q}, x] && NeQ[e*f - d*g, 0] && GtQ[p, 0] && NeQ[q, -1] && IntegersQ[2*p, 2*q] && ( !IGtQ[q, 0] || (EqQ[p, 2] && NeQ[q, 1]))
  2399. Int[Times[Power[Plus[Optional[Pattern[a, Blank[]]], Times[Log[Times[Optional[Pattern[c, Blank[]]], Power[Plus[Pattern[d, Blank[]], Times[Optional[Pattern[e, Blank[]]], Pattern[x, Blank[]]]], Optional[Pattern[n, Blank[]]]]]], Optional[Pattern[b, Blank[]]]]], -1], Power[Plus[Optional[Pattern[f, Blank[]]], Times[Optional[Pattern[g, Blank[]]], Pattern[x, Blank[]]]], Optional[Pattern[q, Blank[]]]]], Pattern[x, Blank[Symbol]]] := Int[ExpandIntegrand[(f + g*x)^q/(a + b*Log[c*(d + e*x)^n]), x], x] /; FreeQ[{a, b, c, d, e, f, g, n}, x] && NeQ[e*f - d*g, 0] && IGtQ[q, 0]
  2400. Int[Times[Power[Plus[Optional[Pattern[a, Blank[]]], Times[Log[Times[Optional[Pattern[c, Blank[]]], Power[Plus[Pattern[d, Blank[]], Times[Optional[Pattern[e, Blank[]]], Pattern[x, Blank[]]]], Optional[Pattern[n, Blank[]]]]]], Optional[Pattern[b, Blank[]]]]], Pattern[p, Blank[]]], Power[Plus[Optional[Pattern[f, Blank[]]], Times[Optional[Pattern[g, Blank[]]], Pattern[x, Blank[]]]], Optional[Pattern[q, Blank[]]]]], Pattern[x, Blank[Symbol]]] := Simp[((d + e*x)*(f + g*x)^q*(a + b*Log[c*(d + e*x)^n])^(p + 1))/(b*e*n*(p + 1)), x] + (-Dist[(q + 1)/(b*n*(p + 1)), Int[(f + g*x)^q*(a + b*Log[c*(d + e*x)^n])^(p + 1), x], x] + Dist[(q*(e*f - d*g))/(b*e*n*(p + 1)), Int[(f + g*x)^(q - 1)*(a + b*Log[c*(d + e*x)^n])^(p + 1), x], x]) /; FreeQ[{a, b, c, d, e, f, g, n}, x] && NeQ[e*f - d*g, 0] && LtQ[p, -1] && GtQ[q, 0]
  2401. Int[Times[Power[Plus[Optional[Pattern[a, Blank[]]], Times[Log[Times[Optional[Pattern[c, Blank[]]], Power[Plus[Pattern[d, Blank[]], Times[Optional[Pattern[e, Blank[]]], Pattern[x, Blank[]]]], Optional[Pattern[n, Blank[]]]]]], Optional[Pattern[b, Blank[]]]]], Pattern[p, Blank[]]], Power[Plus[Optional[Pattern[f, Blank[]]], Times[Optional[Pattern[g, Blank[]]], Pattern[x, Blank[]]]], Optional[Pattern[q, Blank[]]]]], Pattern[x, Blank[Symbol]]] := Int[ExpandIntegrand[(f + g*x)^q*(a + b*Log[c*(d + e*x)^n])^p, x], x] /; FreeQ[{a, b, c, d, e, f, g, n, p}, x] && NeQ[e*f - d*g, 0] && IGtQ[q, 0]
  2402. Int[Times[Log[Times[Optional[Pattern[c, Blank[]]], Power[Plus[Pattern[d, Blank[]], Times[Optional[Pattern[e, Blank[]]], Pattern[x, Blank[]]]], -1]]], Power[Plus[Pattern[f, Blank[]], Times[Optional[Pattern[g, Blank[]]], Power[Pattern[x, Blank[]], 2]]], -1]], Pattern[x, Blank[Symbol]]] := -Dist[e/g, Subst[Int[Log[2*d*x]/(1 - 2*d*x), x], x, 1/(d + e*x)], x] /; FreeQ[{c, d, e, f, g}, x] && EqQ[c, 2*d] && EqQ[e^2*f + d^2*g, 0]
  2403. Int[Times[Plus[Optional[Pattern[a, Blank[]]], Times[Log[Times[Optional[Pattern[c, Blank[]]], Power[Plus[Pattern[d, Blank[]], Times[Optional[Pattern[e, Blank[]]], Pattern[x, Blank[]]]], -1]]], Optional[Pattern[b, Blank[]]]]], Power[Plus[Pattern[f, Blank[]], Times[Optional[Pattern[g, Blank[]]], Power[Pattern[x, Blank[]], 2]]], -1]], Pattern[x, Blank[Symbol]]] := Dist[a + b*Log[c/(2*d)], Int[1/(f + g*x^2), x], x] + Dist[b, Int[Log[(2*d)/(d + e*x)]/(f + g*x^2), x], x] /; FreeQ[{a, b, c, d, e, f, g}, x] && EqQ[e^2*f + d^2*g, 0] && GtQ[c/(2*d), 0]
  2404. Int[Times[Plus[Optional[Pattern[a, Blank[]]], Times[Log[Times[Optional[Pattern[c, Blank[]]], Power[Plus[Pattern[d, Blank[]], Times[Optional[Pattern[e, Blank[]]], Pattern[x, Blank[]]]], Optional[Pattern[n, Blank[]]]]]], Optional[Pattern[b, Blank[]]]]], Power[Plus[Pattern[f, Blank[]], Times[Optional[Pattern[g, Blank[]]], Power[Pattern[x, Blank[]], 2]]], Rational[-1, 2]]], Pattern[x, Blank[Symbol]]] := With[{u = IntHide[1/Sqrt[f + g*x^2], x]}, Simp[u*(a + b*Log[c*(d + e*x)^n]), x] - Dist[b*e*n, Int[SimplifyIntegrand[u/(d + e*x), x], x], x]] /; FreeQ[{a, b, c, d, e, f, g, n}, x] && GtQ[f, 0]
  2405. Int[Times[Plus[Optional[Pattern[a, Blank[]]], Times[Log[Times[Optional[Pattern[c, Blank[]]], Power[Plus[Pattern[d, Blank[]], Times[Optional[Pattern[e, Blank[]]], Pattern[x, Blank[]]]], Optional[Pattern[n, Blank[]]]]]], Optional[Pattern[b, Blank[]]]]], Power[Plus[Pattern[f1, Blank[]], Times[Optional[Pattern[g1, Blank[]]], Pattern[x, Blank[]]]], Rational[-1, 2]], Power[Plus[Pattern[f2, Blank[]], Times[Optional[Pattern[g2, Blank[]]], Pattern[x, Blank[]]]], Rational[-1, 2]]], Pattern[x, Blank[Symbol]]] := With[{u = IntHide[1/Sqrt[f1*f2 + g1*g2*x^2], x]}, Simp[u*(a + b*Log[c*(d + e*x)^n]), x] - Dist[b*e*n, Int[SimplifyIntegrand[u/(d + e*x), x], x], x]] /; FreeQ[{a, b, c, d, e, f1, g1, f2, g2, n}, x] && EqQ[f2*g1 + f1*g2, 0] && GtQ[f1, 0] && GtQ[f2, 0]
  2406. Int[Times[Plus[Optional[Pattern[a, Blank[]]], Times[Log[Times[Optional[Pattern[c, Blank[]]], Power[Plus[Pattern[d, Blank[]], Times[Optional[Pattern[e, Blank[]]], Pattern[x, Blank[]]]], Optional[Pattern[n, Blank[]]]]]], Optional[Pattern[b, Blank[]]]]], Power[Plus[Pattern[f, Blank[]], Times[Optional[Pattern[g, Blank[]]], Power[Pattern[x, Blank[]], 2]]], Rational[-1, 2]]], Pattern[x, Blank[Symbol]]] := Dist[Sqrt[1 + (g*x^2)/f]/Sqrt[f + g*x^2], Int[(a + b*Log[c*(d + e*x)^n])/Sqrt[1 + (g*x^2)/f], x], x] /; FreeQ[{a, b, c, d, e, f, g, n}, x] && !GtQ[f, 0]
  2407. Int[Times[Plus[Optional[Pattern[a, Blank[]]], Times[Log[Times[Optional[Pattern[c, Blank[]]], Power[Plus[Pattern[d, Blank[]], Times[Optional[Pattern[e, Blank[]]], Pattern[x, Blank[]]]], Optional[Pattern[n, Blank[]]]]]], Optional[Pattern[b, Blank[]]]]], Power[Plus[Pattern[f1, Blank[]], Times[Optional[Pattern[g1, Blank[]]], Pattern[x, Blank[]]]], Rational[-1, 2]], Power[Plus[Pattern[f2, Blank[]], Times[Optional[Pattern[g2, Blank[]]], Pattern[x, Blank[]]]], Rational[-1, 2]]], Pattern[x, Blank[Symbol]]] := Dist[Sqrt[1 + (g1*g2*x^2)/(f1*f2)]/(Sqrt[f1 + g1*x]*Sqrt[f2 + g2*x]), Int[(a + b*Log[c*(d + e*x)^n])/Sqrt[1 + (g1*g2*x^2)/(f1*f2)], x], x] /; FreeQ[{a, b, c, d, e, f1, g1, f2, g2, n}, x] && EqQ[f2*g1 + f1*g2, 0]
  2408. Int[Times[Power[Plus[Optional[Pattern[a, Blank[]]], Times[Log[Times[Optional[Pattern[c, Blank[]]], Power[Plus[Pattern[d, Blank[]], Times[Optional[Pattern[e, Blank[]]], Pattern[x, Blank[]]]], Optional[Pattern[n, Blank[]]]]]], Optional[Pattern[b, Blank[]]]]], Optional[Pattern[p, Blank[]]]], Power[Plus[Optional[Pattern[f, Blank[]]], Times[Optional[Pattern[g, Blank[]]], Power[Pattern[x, Blank[]], Pattern[r, Blank[]]]]], Optional[Pattern[q, Blank[]]]]], Pattern[x, Blank[Symbol]]] := With[{k = Denominator[r]}, Dist[k, Subst[Int[x^(k - 1)*(f + g*x^(k*r))^q*(a + b*Log[c*(d + e*x^k)^n])^p, x], x, x^(1/k)], x]] /; FreeQ[{a, b, c, d, e, f, g, n, p, q}, x] && FractionQ[r] && IGtQ[p, 0]
  2409. Int[Times[Power[Plus[Optional[Pattern[a, Blank[]]], Times[Log[Times[Optional[Pattern[c, Blank[]]], Power[Plus[Pattern[d, Blank[]], Times[Optional[Pattern[e, Blank[]]], Pattern[x, Blank[]]]], Optional[Pattern[n, Blank[]]]]]], Optional[Pattern[b, Blank[]]]]], Optional[Pattern[p, Blank[]]]], Power[Plus[Pattern[f, Blank[]], Times[Optional[Pattern[g, Blank[]]], Power[Pattern[x, Blank[]], Pattern[r, Blank[]]]]], Optional[Pattern[q, Blank[]]]]], Pattern[x, Blank[Symbol]]] := Int[ExpandIntegrand[(a + b*Log[c*(d + e*x)^n])^p, (f + g*x^r)^q, x], x] /; FreeQ[{a, b, c, d, e, f, g, n, r}, x] && IGtQ[p, 0] && IntegerQ[q] && (GtQ[q, 0] || (IntegerQ[r] && NeQ[r, 1]))
  2410. Int[Times[Log[Times[Optional[Pattern[c, Blank[]]], Plus[Pattern[d, Blank[]], Times[Optional[Pattern[e, Blank[]]], Pattern[x, Blank[]]]]]], Power[Pattern[x, Blank[]], Optional[Pattern[m, Blank[]]]], Power[Plus[Pattern[f, Blank[]], Times[Optional[Pattern[g, Blank[]]], Pattern[x, Blank[]]]], -1]], Pattern[x, Blank[Symbol]]] := Int[ExpandIntegrand[Log[c*(d + e*x)], x^m/(f + g*x), x], x] /; FreeQ[{c, d, e, f, g}, x] && EqQ[e*f - d*g, 0] && EqQ[c*d, 1] && IntegerQ[m]
  2411. Int[Times[Power[Plus[Optional[Pattern[a, Blank[]]], Times[Log[Times[Optional[Pattern[c, Blank[]]], Power[Plus[Pattern[d, Blank[]], Times[Optional[Pattern[e, Blank[]]], Pattern[x, Blank[]]]], Optional[Pattern[n, Blank[]]]]]], Optional[Pattern[b, Blank[]]]]], Optional[Pattern[p, Blank[]]]], Power[Plus[Optional[Pattern[f, Blank[]]], Times[Optional[Pattern[g, Blank[]]], Pattern[x, Blank[]]]], Optional[Pattern[q, Blank[]]]], Power[Plus[Optional[Pattern[h, Blank[]]], Times[Optional[Pattern[i, Blank[]]], Pattern[x, Blank[]]]], Optional[Pattern[r, Blank[]]]]], Pattern[x, Blank[Symbol]]] := Dist[1/e, Subst[Int[((g*x)/e)^q*((e*h - d*i)/e + (i*x)/e)^r*(a + b*Log[c*x^n])^p, x], x, d + e*x], x] /; FreeQ[{a, b, c, d, e, f, g, h, i, n, p, q, r}, x] && EqQ[e*f - d*g, 0] && (IGtQ[p, 0] || IGtQ[r, 0]) && IntegerQ[2*r]
  2412. Int[Times[Power[Plus[Optional[Pattern[a, Blank[]]], Times[Log[Times[Optional[Pattern[c, Blank[]]], Power[Plus[Pattern[d, Blank[]], Times[Optional[Pattern[e, Blank[]]], Pattern[x, Blank[]]]], Optional[Pattern[n, Blank[]]]]]], Optional[Pattern[b, Blank[]]]]], Optional[Pattern[p, Blank[]]]], Power[Plus[Pattern[f, Blank[]], Times[Optional[Pattern[g, Blank[]]], Power[Pattern[x, Blank[]], -1]]], Optional[Pattern[q, Blank[]]]], Power[Pattern[x, Blank[]], Optional[Pattern[m, Blank[]]]]], Pattern[x, Blank[Symbol]]] := Int[(g + f*x)^q*(a + b*Log[c*(d + e*x)^n])^p, x] /; FreeQ[{a, b, c, d, e, f, g, m, n, p, q}, x] && EqQ[m, q] && IntegerQ[q]
  2413. Int[Times[Power[Plus[Optional[Pattern[a, Blank[]]], Times[Log[Times[Optional[Pattern[c, Blank[]]], Power[Plus[Pattern[d, Blank[]], Times[Optional[Pattern[e, Blank[]]], Pattern[x, Blank[]]]], Optional[Pattern[n, Blank[]]]]]], Optional[Pattern[b, Blank[]]]]], Optional[Pattern[p, Blank[]]]], Power[Pattern[x, Blank[]], Optional[Pattern[m, Blank[]]]], Power[Plus[Optional[Pattern[f, Blank[]]], Times[Optional[Pattern[g, Blank[]]], Power[Pattern[x, Blank[]], Optional[Pattern[r, Blank[]]]]]], Optional[Pattern[q, Blank[]]]]], Pattern[x, Blank[Symbol]]] := Simp[((f + g*x^r)^(q + 1)*(a + b*Log[c*(d + e*x)^n])^p)/(g*r*(q + 1)), x] - Dist[(b*e*n*p)/(g*r*(q + 1)), Int[((f + g*x^r)^(q + 1)*(a + b*Log[c*(d + e*x)^n])^(p - 1))/(d + e*x), x], x] /; FreeQ[{a, b, c, d, e, f, g, m, n, q, r}, x] && EqQ[m, r - 1] && NeQ[q, -1] && IGtQ[p, 0]
  2414. Int[Times[Plus[Optional[Pattern[a, Blank[]]], Times[Log[Times[Optional[Pattern[c, Blank[]]], Power[Plus[Pattern[d, Blank[]], Times[Optional[Pattern[e, Blank[]]], Pattern[x, Blank[]]]], Optional[Pattern[n, Blank[]]]]]], Optional[Pattern[b, Blank[]]]]], Power[Pattern[x, Blank[]], Optional[Pattern[m, Blank[]]]], Power[Plus[Pattern[f, Blank[]], Times[Optional[Pattern[g, Blank[]]], Power[Pattern[x, Blank[]], Optional[Pattern[r, Blank[]]]]]], Optional[Pattern[q, Blank[]]]]], Pattern[x, Blank[Symbol]]] := With[{u = IntHide[x^m*(f + g*x^r)^q, x]}, Dist[a + b*Log[c*(d + e*x)^n], u, x] - Dist[b*e*n, Int[SimplifyIntegrand[u/(d + e*x), x], x], x] /; InverseFunctionFreeQ[u, x]] /; FreeQ[{a, b, c, d, e, f, g, m, n, q, r}, x] && IntegerQ[m] && IntegerQ[q] && IntegerQ[r]
  2415. Int[Times[Power[Plus[Optional[Pattern[a, Blank[]]], Times[Log[Times[Optional[Pattern[c, Blank[]]], Power[Plus[Pattern[d, Blank[]], Times[Optional[Pattern[e, Blank[]]], Pattern[x, Blank[]]]], Optional[Pattern[n, Blank[]]]]]], Optional[Pattern[b, Blank[]]]]], Optional[Pattern[p, Blank[]]]], Power[Pattern[x, Blank[]], Optional[Pattern[m, Blank[]]]], Power[Plus[Optional[Pattern[f, Blank[]]], Times[Optional[Pattern[g, Blank[]]], Power[Pattern[x, Blank[]], Pattern[r, Blank[]]]]], Optional[Pattern[q, Blank[]]]]], Pattern[x, Blank[Symbol]]] := With[{k = Denominator[r]}, Dist[k, Subst[Int[x^(k*(m + 1) - 1)*(f + g*x^(k*r))^q*(a + b*Log[c*(d + e*x^k)^n])^p, x], x, x^(1/k)], x]] /; FreeQ[{a, b, c, d, e, f, g, n, p, q}, x] && FractionQ[r] && IGtQ[p, 0] && IntegerQ[m]
  2416. Int[Times[Power[Plus[Optional[Pattern[a, Blank[]]], Times[Log[Times[Optional[Pattern[c, Blank[]]], Power[Plus[Pattern[d, Blank[]], Times[Optional[Pattern[e, Blank[]]], Pattern[x, Blank[]]]], Optional[Pattern[n, Blank[]]]]]], Optional[Pattern[b, Blank[]]]]], Optional[Pattern[p, Blank[]]]], Power[Times[Optional[Pattern[h, Blank[]]], Pattern[x, Blank[]]], Optional[Pattern[m, Blank[]]]], Power[Plus[Pattern[f, Blank[]], Times[Optional[Pattern[g, Blank[]]], Power[Pattern[x, Blank[]], Optional[Pattern[r, Blank[]]]]]], Optional[Pattern[q, Blank[]]]]], Pattern[x, Blank[Symbol]]] := Int[ExpandIntegrand[(a + b*Log[c*(d + e*x)^n])^p, (h*x)^m*(f + g*x^r)^q, x], x] /; FreeQ[{a, b, c, d, e, f, g, h, m, n, p, q, r}, x] && IntegerQ[m] && IntegerQ[q]
  2417. Int[Times[Power[Plus[Optional[Pattern[a, Blank[]]], Times[Log[Times[Optional[Pattern[c, Blank[]]], Power[Plus[Pattern[d, Blank[]], Times[Optional[Pattern[e, Blank[]]], Pattern[x, Blank[]]]], Optional[Pattern[n, Blank[]]]]]], Optional[Pattern[b, Blank[]]]]], Optional[Pattern[p, Blank[]]]], Pattern[Polyx, Blank[]]], Pattern[x, Blank[Symbol]]] := Int[ExpandIntegrand[Polyx*(a + b*Log[c*(d + e*x)^n])^p, x], x] /; FreeQ[{a, b, c, d, e, n, p}, x] && PolynomialQ[Polyx, x]
  2418. Int[Times[Power[Plus[Optional[Pattern[a, Blank[]]], Times[Log[Times[Optional[Pattern[c, Blank[]]], Power[Plus[Pattern[d, Blank[]], Times[Optional[Pattern[e, Blank[]]], Pattern[x, Blank[]]]], Optional[Pattern[n, Blank[]]]]]], Optional[Pattern[b, Blank[]]]]], Optional[Pattern[p, Blank[]]]], Pattern[RFx, Blank[]]], Pattern[x, Blank[Symbol]]] := With[{u = ExpandIntegrand[(a + b*Log[c*(d + e*x)^n])^p, RFx, x]}, Int[u, x] /; SumQ[u]] /; FreeQ[{a, b, c, d, e, n}, x] && RationalFunctionQ[RFx, x] && IntegerQ[p]
  2419. Int[Times[Power[Plus[Optional[Pattern[a, Blank[]]], Times[Log[Times[Optional[Pattern[c, Blank[]]], Power[Plus[Pattern[d, Blank[]], Times[Optional[Pattern[e, Blank[]]], Pattern[x, Blank[]]]], Optional[Pattern[n, Blank[]]]]]], Optional[Pattern[b, Blank[]]]]], Optional[Pattern[p, Blank[]]]], Pattern[RFx, Blank[]]], Pattern[x, Blank[Symbol]]] := With[{u = ExpandIntegrand[RFx*(a + b*Log[c*(d + e*x)^n])^p, x]}, Int[u, x] /; SumQ[u]] /; FreeQ[{a, b, c, d, e, n}, x] && RationalFunctionQ[RFx, x] && IntegerQ[p]
  2420. Int[Times[Power[Plus[Optional[Pattern[a, Blank[]]], Times[Log[Times[Optional[Pattern[c, Blank[]]], Power[Plus[Pattern[d, Blank[]], Times[Optional[Pattern[e, Blank[]]], Pattern[x, Blank[]]]], Optional[Pattern[n, Blank[]]]]]], Optional[Pattern[b, Blank[]]]]], Optional[Pattern[p, Blank[]]]], Pattern[AFx, Blank[]]], Pattern[x, Blank[Symbol]]] := Unintegrable[AFx*(a + b*Log[c*(d + e*x)^n])^p, x] /; FreeQ[{a, b, c, d, e, n, p}, x] && AlgebraicFunctionQ[AFx, x, True]
  2421. Int[Times[Power[Plus[Optional[Pattern[a, Blank[]]], Times[Log[Times[Optional[Pattern[c, Blank[]]], Power[Pattern[v, Blank[]], Optional[Pattern[n, Blank[]]]]]], Optional[Pattern[b, Blank[]]]]], Optional[Pattern[p, Blank[]]]], Power[Pattern[u, Blank[]], Optional[Pattern[q, Blank[]]]]], Pattern[x, Blank[Symbol]]] := Int[ExpandToSum[u, x]^q*(a + b*Log[c*ExpandToSum[v, x]^n])^p, x] /; FreeQ[{a, b, c, n, p, q}, x] && BinomialQ[u, x] && LinearQ[v, x] && !(BinomialMatchQ[u, x] && LinearMatchQ[v, x])
  2422. Int[Times[Log[Times[Optional[Pattern[f, Blank[]]], Power[Pattern[x, Blank[]], Optional[Pattern[m, Blank[]]]]]], Plus[Optional[Pattern[a, Blank[]]], Times[Log[Times[Optional[Pattern[c, Blank[]]], Power[Plus[Pattern[d, Blank[]], Times[Optional[Pattern[e, Blank[]]], Pattern[x, Blank[]]]], Optional[Pattern[n, Blank[]]]]]], Optional[Pattern[b, Blank[]]]]]], Pattern[x, Blank[Symbol]]] := -Simp[x*(m - Log[f*x^m])*(a + b*Log[c*(d + e*x)^n]), x] + (-Dist[b*e*n, Int[(x*Log[f*x^m])/(d + e*x), x], x] + Dist[b*e*m*n, Int[x/(d + e*x), x], x]) /; FreeQ[{a, b, c, d, e, f, m, n}, x]
  2423. Int[Times[Log[Times[Optional[Pattern[f, Blank[]]], Power[Pattern[x, Blank[]], Optional[Pattern[m, Blank[]]]]]], Power[Plus[Optional[Pattern[a, Blank[]]], Times[Log[Times[Optional[Pattern[c, Blank[]]], Power[Plus[Pattern[d, Blank[]], Times[Optional[Pattern[e, Blank[]]], Pattern[x, Blank[]]]], Optional[Pattern[n, Blank[]]]]]], Optional[Pattern[b, Blank[]]]]], Pattern[p, Blank[]]]], Pattern[x, Blank[Symbol]]] := With[{u = IntHide[(a + b*Log[c*(d + e*x)^n])^p, x]}, Dist[Log[f*x^m], u, x] - Dist[m, Int[Dist[1/x, u, x], x], x]] /; FreeQ[{a, b, c, d, e, f, m, n}, x] && IGtQ[p, 1]
  2424. Int[Times[Log[Times[Optional[Pattern[f, Blank[]]], Power[Pattern[x, Blank[]], Optional[Pattern[m, Blank[]]]]]], Power[Plus[Optional[Pattern[a, Blank[]]], Times[Log[Times[Optional[Pattern[c, Blank[]]], Power[Plus[Pattern[d, Blank[]], Times[Optional[Pattern[e, Blank[]]], Pattern[x, Blank[]]]], Optional[Pattern[n, Blank[]]]]]], Optional[Pattern[b, Blank[]]]]], Optional[Pattern[p, Blank[]]]]], Pattern[x, Blank[Symbol]]] := Unintegrable[Log[f*x^m]*(a + b*Log[c*(d + e*x)^n])^p, x] /; FreeQ[{a, b, c, d, e, f, m, n, p}, x]
  2425. Int[Times[Log[Times[Optional[Pattern[f, Blank[]]], Power[Pattern[x, Blank[]], Optional[Pattern[m, Blank[]]]]]], Plus[Optional[Pattern[a, Blank[]]], Times[Log[Times[Optional[Pattern[c, Blank[]]], Power[Plus[Pattern[d, Blank[]], Times[Optional[Pattern[e, Blank[]]], Pattern[x, Blank[]]]], Optional[Pattern[n, Blank[]]]]]], Optional[Pattern[b, Blank[]]]]], Power[Pattern[x, Blank[]], -1]], Pattern[x, Blank[Symbol]]] := Simp[(Log[f*x^m]^2*(a + b*Log[c*(d + e*x)^n]))/(2*m), x] - Dist[(b*e*n)/(2*m), Int[Log[f*x^m]^2/(d + e*x), x], x] /; FreeQ[{a, b, c, d, e, f, m, n}, x]
  2426. Int[Times[Log[Times[Optional[Pattern[f, Blank[]]], Power[Pattern[x, Blank[]], Optional[Pattern[m, Blank[]]]]]], Plus[Optional[Pattern[a, Blank[]]], Times[Log[Times[Optional[Pattern[c, Blank[]]], Power[Plus[Pattern[d, Blank[]], Times[Optional[Pattern[e, Blank[]]], Pattern[x, Blank[]]]], Optional[Pattern[n, Blank[]]]]]], Optional[Pattern[b, Blank[]]]]], Power[Times[Optional[Pattern[g, Blank[]]], Pattern[x, Blank[]]], Optional[Pattern[q, Blank[]]]]], Pattern[x, Blank[Symbol]]] := -Simp[(((m*(g*x)^(q + 1))/(q + 1) - (g*x)^(q + 1)*Log[f*x^m])*(a + b*Log[c*(d + e*x)^n]))/(g*(q + 1)), x] + (-Dist[(b*e*n)/(g*(q + 1)), Int[((g*x)^(q + 1)*Log[f*x^m])/(d + e*x), x], x] + Dist[(b*e*m*n)/(g*(q + 1)^2), Int[(g*x)^(q + 1)/(d + e*x), x], x]) /; FreeQ[{a, b, c, d, e, f, g, m, n, q}, x] && NeQ[q, -1]
  2427. Int[Times[Log[Times[Optional[Pattern[f, Blank[]]], Power[Pattern[x, Blank[]], Optional[Pattern[m, Blank[]]]]]], Power[Plus[Optional[Pattern[a, Blank[]]], Times[Log[Times[Optional[Pattern[c, Blank[]]], Power[Plus[Pattern[d, Blank[]], Times[Optional[Pattern[e, Blank[]]], Pattern[x, Blank[]]]], Optional[Pattern[n, Blank[]]]]]], Optional[Pattern[b, Blank[]]]]], Optional[Pattern[p, Blank[]]]], Power[Pattern[x, Blank[]], -1]], Pattern[x, Blank[Symbol]]] := Simp[(Log[f*x^m]^2*(a + b*Log[c*(d + e*x)^n])^p)/(2*m), x] - Dist[(b*e*n*p)/(2*m), Int[(Log[f*x^m]^2*(a + b*Log[c*(d + e*x)^n])^(p - 1))/(d + e*x), x], x] /; FreeQ[{a, b, c, d, e, f, m, n}, x] && IGtQ[p, 0]
  2428. Int[Times[Log[Times[Optional[Pattern[f, Blank[]]], Power[Pattern[x, Blank[]], Optional[Pattern[m, Blank[]]]]]], Power[Plus[Optional[Pattern[a, Blank[]]], Times[Log[Times[Optional[Pattern[c, Blank[]]], Power[Plus[Pattern[d, Blank[]], Times[Optional[Pattern[e, Blank[]]], Pattern[x, Blank[]]]], Optional[Pattern[n, Blank[]]]]]], Optional[Pattern[b, Blank[]]]]], Pattern[p, Blank[]]], Power[Times[Optional[Pattern[g, Blank[]]], Pattern[x, Blank[]]], Optional[Pattern[q, Blank[]]]]], Pattern[x, Blank[Symbol]]] := With[{u = IntHide[(g*x)^q*(a + b*Log[c*(d + e*x)^n])^p, x]}, Dist[Log[f*x^m], u, x] - Dist[m, Int[Dist[1/x, u, x], x], x]] /; FreeQ[{a, b, c, d, e, f, g, m, n, q}, x] && IGtQ[p, 1] && IGtQ[q, 0]
  2429. Int[Times[Log[Times[Optional[Pattern[f, Blank[]]], Power[Pattern[x, Blank[]], Optional[Pattern[m, Blank[]]]]]], Power[Plus[Optional[Pattern[a, Blank[]]], Times[Log[Times[Optional[Pattern[c, Blank[]]], Power[Plus[Pattern[d, Blank[]], Times[Optional[Pattern[e, Blank[]]], Pattern[x, Blank[]]]], Optional[Pattern[n, Blank[]]]]]], Optional[Pattern[b, Blank[]]]]], Optional[Pattern[p, Blank[]]]], Power[Times[Optional[Pattern[g, Blank[]]], Pattern[x, Blank[]]], Optional[Pattern[q, Blank[]]]]], Pattern[x, Blank[Symbol]]] := Unintegrable[(g*x)^q*Log[f*x^m]*(a + b*Log[c*(d + e*x)^n])^p, x] /; FreeQ[{a, b, c, d, e, f, g, m, n, p, q}, x]
  2430. Int[Times[Power[Plus[Optional[Pattern[a, Blank[]]], Times[Log[Times[Optional[Pattern[c, Blank[]]], Power[Plus[Pattern[d, Blank[]], Times[Optional[Pattern[e, Blank[]]], Pattern[x, Blank[]]]], Optional[Pattern[n, Blank[]]]]]], Optional[Pattern[b, Blank[]]]]], Optional[Pattern[p, Blank[]]]], Plus[Optional[Pattern[f, Blank[]]], Times[Log[Times[Optional[Pattern[h, Blank[]]], Power[Plus[Optional[Pattern[i, Blank[]]], Times[Optional[Pattern[j, Blank[]]], Pattern[x, Blank[]]]], Optional[Pattern[m, Blank[]]]]]], Optional[Pattern[g, Blank[]]]]]], Pattern[x, Blank[Symbol]]] := Simp[x*(a + b*Log[c*(d + e*x)^n])^p*(f + g*Log[h*(i + j*x)^m]), x] + (-Dist[g*j*m, Int[(x*(a + b*Log[c*(d + e*x)^n])^p)/(i + j*x), x], x] - Dist[b*e*n*p, Int[(x*(a + b*Log[c*(d + e*x)^n])^(p - 1)*(f + g*Log[h*(i + j*x)^m]))/(d + e*x), x], x]) /; FreeQ[{a, b, c, d, e, f, g, h, i, j, m, n}, x] && IGtQ[p, 0]
  2431. Int[Times[Log[Times[Optional[Pattern[f, Blank[]]], Power[Plus[Optional[Pattern[g, Blank[]]], Times[Optional[Pattern[h, Blank[]]], Pattern[x, Blank[]]]], Optional[Pattern[m, Blank[]]]]]], Power[Plus[Optional[Pattern[a, Blank[]]], Times[Log[Times[Optional[Pattern[c, Blank[]]], Power[Plus[Pattern[d, Blank[]], Times[Optional[Pattern[e, Blank[]]], Pattern[x, Blank[]]]], Optional[Pattern[n, Blank[]]]]]], Optional[Pattern[b, Blank[]]]]], Optional[Pattern[p, Blank[]]]]], Pattern[x, Blank[Symbol]]] := Dist[1/e, Subst[Int[Log[f*((g*x)/d)^m]*(a + b*Log[c*x^n])^p, x], x, d + e*x], x] /; FreeQ[{a, b, c, d, e, f, g, h, m, n, p}, x] && EqQ[e*f - d*g, 0]
  2432. Int[Times[Power[Plus[Optional[Pattern[a, Blank[]]], Times[Log[Times[Optional[Pattern[c, Blank[]]], Power[Plus[Pattern[d, Blank[]], Times[Optional[Pattern[e, Blank[]]], Pattern[x, Blank[]]]], Optional[Pattern[n, Blank[]]]]]], Optional[Pattern[b, Blank[]]]]], Optional[Pattern[p, Blank[]]]], Power[Plus[Optional[Pattern[f, Blank[]]], Times[Log[Times[Optional[Pattern[h, Blank[]]], Power[Plus[Optional[Pattern[i, Blank[]]], Times[Optional[Pattern[j, Blank[]]], Pattern[x, Blank[]]]], Optional[Pattern[m, Blank[]]]]]], Optional[Pattern[g, Blank[]]]]], Optional[Pattern[q, Blank[]]]]], Pattern[x, Blank[Symbol]]] := Unintegrable[(a + b*Log[c*(d + e*x)^n])^p*(f + g*Log[h*(i + j*x)^m])^q, x] /; FreeQ[{a, b, c, d, e, f, g, h, i, j, m, n, p}, x]
  2433. Int[Times[Power[Plus[Optional[Pattern[a, Blank[]]], Times[Log[Times[Optional[Pattern[c, Blank[]]], Power[Plus[Pattern[d, Blank[]], Times[Optional[Pattern[e, Blank[]]], Pattern[x, Blank[]]]], Optional[Pattern[n, Blank[]]]]]], Optional[Pattern[b, Blank[]]]]], Optional[Pattern[p, Blank[]]]], Plus[Optional[Pattern[f, Blank[]]], Times[Log[Times[Optional[Pattern[h, Blank[]]], Power[Plus[Optional[Pattern[i, Blank[]]], Times[Optional[Pattern[j, Blank[]]], Pattern[x, Blank[]]]], Optional[Pattern[m, Blank[]]]]]], Optional[Pattern[g, Blank[]]]]], Power[Plus[Optional[Pattern[k, Blank[]]], Times[Optional[Pattern[l, Blank[]]], Pattern[x, Blank[]]]], Optional[Pattern[r, Blank[]]]]], Pattern[x, Blank[Symbol]]] := Dist[1/e, Subst[Int[((k*x)/d)^r*(a + b*Log[c*x^n])^p*(f + g*Log[h*((e*i - d*j)/e + (j*x)/e)^m]), x], x, d + e*x], x] /; FreeQ[{a, b, c, d, e, f, g, h, i, j, k, l, n, p, r}, x] && EqQ[e*k - d*l, 0]
  2434. Int[Times[Plus[Optional[Pattern[a, Blank[]]], Times[Log[Times[Optional[Pattern[c, Blank[]]], Power[Plus[Pattern[d, Blank[]], Times[Optional[Pattern[e, Blank[]]], Pattern[x, Blank[]]]], Optional[Pattern[n, Blank[]]]]]], Optional[Pattern[b, Blank[]]]]], Plus[Optional[Pattern[f, Blank[]]], Times[Log[Times[Optional[Pattern[h, Blank[]]], Power[Plus[Optional[Pattern[i, Blank[]]], Times[Optional[Pattern[j, Blank[]]], Pattern[x, Blank[]]]], Optional[Pattern[m, Blank[]]]]]], Optional[Pattern[g, Blank[]]]]], Power[Pattern[x, Blank[]], -1]], Pattern[x, Blank[Symbol]]] := Simp[Log[x]*(a + b*Log[c*(d + e*x)^n])*(f + g*Log[h*(i + j*x)^m]), x] + (-Dist[e*g*m, Int[(Log[x]*(a + b*Log[c*(d + e*x)^n]))/(d + e*x), x], x] - Dist[b*j*n, Int[(Log[x]*(f + g*Log[h*(i + j*x)^m]))/(i + j*x), x], x]) /; FreeQ[{a, b, c, d, e, f, g, h, i, j, m, n}, x] && EqQ[e*i - d*j, 0]
  2435. Int[Times[Log[Plus[Pattern[a, Blank[]], Times[Optional[Pattern[b, Blank[]]], Pattern[x, Blank[]]]]], Log[Plus[Pattern[c, Blank[]], Times[Optional[Pattern[d, Blank[]]], Pattern[x, Blank[]]]]], Power[Pattern[x, Blank[]], -1]], Pattern[x, Blank[Symbol]]] := Simp[Log[-((b*x)/a)]*Log[a + b*x]*Log[c + d*x], x] + (Simp[(1*(Log[-((b*x)/a)] - Log[-(((b*c - a*d)*x)/(a*(c + d*x)))] + Log[(b*c - a*d)/(b*(c + d*x))])*Log[(a*(c + d*x))/(c*(a + b*x))]^2)/2, x] - Simp[(1*(Log[-((b*x)/a)] - Log[-((d*x)/c)])*(Log[a + b*x] + Log[(a*(c + d*x))/(c*(a + b*x))])^2)/2, x] + Simp[(Log[c + d*x] - Log[(a*(c + d*x))/(c*(a + b*x))])*PolyLog[2, 1 + (b*x)/a], x] + Simp[(Log[a + b*x] + Log[(a*(c + d*x))/(c*(a + b*x))])*PolyLog[2, 1 + (d*x)/c], x] + Simp[Log[(a*(c + d*x))/(c*(a + b*x))]*PolyLog[2, (c*(a + b*x))/(a*(c + d*x))], x] - Simp[Log[(a*(c + d*x))/(c*(a + b*x))]*PolyLog[2, (d*(a + b*x))/(b*(c + d*x))], x] - Simp[PolyLog[3, 1 + (b*x)/a], x] - Simp[PolyLog[3, 1 + (d*x)/c], x] + Simp[PolyLog[3, (c*(a + b*x))/(a*(c + d*x))], x] - Simp[PolyLog[3, (d*(a + b*x))/(b*(c + d*x))], x]) /; FreeQ[{a, b, c, d}, x] && NeQ[b*c - a*d, 0]
  2436. Int[Times[Log[Pattern[v, Blank[]]], Log[Pattern[w, Blank[]]], Power[Pattern[x, Blank[]], -1]], Pattern[x, Blank[Symbol]]] := Int[(Log[ExpandToSum[v, x]]*Log[ExpandToSum[w, x]])/x, x] /; LinearQ[{v, w}, x] && !LinearMatchQ[{v, w}, x]
  2437. Int[Times[Log[Times[Optional[Pattern[c, Blank[]]], Power[Plus[Pattern[d, Blank[]], Times[Optional[Pattern[e, Blank[]]], Pattern[x, Blank[]]]], Optional[Pattern[n, Blank[]]]]]], Log[Times[Optional[Pattern[h, Blank[]]], Power[Plus[Optional[Pattern[i, Blank[]]], Times[Optional[Pattern[j, Blank[]]], Pattern[x, Blank[]]]], Optional[Pattern[m, Blank[]]]]]], Power[Pattern[x, Blank[]], -1]], Pattern[x, Blank[Symbol]]] := Dist[m, Int[(Log[i + j*x]*Log[c*(d + e*x)^n])/x, x], x] - Dist[m*Log[i + j*x] - Log[h*(i + j*x)^m], Int[Log[c*(d + e*x)^n]/x, x], x] /; FreeQ[{c, d, e, h, i, j, m, n}, x] && NeQ[e*i - d*j, 0] && NeQ[i + j*x, h*(i + j*x)^m]
  2438. Int[Times[Plus[Optional[Pattern[a, Blank[]]], Times[Log[Times[Optional[Pattern[c, Blank[]]], Power[Plus[Pattern[d, Blank[]], Times[Optional[Pattern[e, Blank[]]], Pattern[x, Blank[]]]], Optional[Pattern[n, Blank[]]]]]], Optional[Pattern[b, Blank[]]]]], Plus[Times[Log[Times[Optional[Pattern[h, Blank[]]], Power[Plus[Optional[Pattern[i, Blank[]]], Times[Optional[Pattern[j, Blank[]]], Pattern[x, Blank[]]]], Optional[Pattern[m, Blank[]]]]]], Optional[Pattern[g, Blank[]]]], Pattern[f, Blank[]]], Power[Pattern[x, Blank[]], -1]], Pattern[x, Blank[Symbol]]] := Dist[f, Int[(a + b*Log[c*(d + e*x)^n])/x, x], x] + Dist[g, Int[(Log[h*(i + j*x)^m]*(a + b*Log[c*(d + e*x)^n]))/x, x], x] /; FreeQ[{a, b, c, d, e, f, g, h, i, j, m, n}, x] && NeQ[e*i - d*j, 0]
  2439. Int[Times[Power[Plus[Optional[Pattern[a, Blank[]]], Times[Log[Times[Optional[Pattern[c, Blank[]]], Power[Plus[Pattern[d, Blank[]], Times[Optional[Pattern[e, Blank[]]], Pattern[x, Blank[]]]], Optional[Pattern[n, Blank[]]]]]], Optional[Pattern[b, Blank[]]]]], Optional[Pattern[p, Blank[]]]], Plus[Optional[Pattern[f, Blank[]]], Times[Log[Times[Optional[Pattern[h, Blank[]]], Power[Plus[Optional[Pattern[i, Blank[]]], Times[Optional[Pattern[j, Blank[]]], Pattern[x, Blank[]]]], Optional[Pattern[m, Blank[]]]]]], Optional[Pattern[g, Blank[]]]]], Power[Pattern[x, Blank[]], Optional[Pattern[r, Blank[]]]]], Pattern[x, Blank[Symbol]]] := Simp[(x^(r + 1)*(a + b*Log[c*(d + e*x)^n])^p*(f + g*Log[h*(i + j*x)^m]))/(r + 1), x] + (-Dist[(g*j*m)/(r + 1), Int[(x^(r + 1)*(a + b*Log[c*(d + e*x)^n])^p)/(i + j*x), x], x] - Dist[(b*e*n*p)/(r + 1), Int[(x^(r + 1)*(a + b*Log[c*(d + e*x)^n])^(p - 1)*(f + g*Log[h*(i + j*x)^m]))/(d + e*x), x], x]) /; FreeQ[{a, b, c, d, e, f, g, h, i, j, m, n}, x] && IGtQ[p, 0] && IntegerQ[r] && (EqQ[p, 1] || GtQ[r, 0]) && NeQ[r, -1]
  2440. Int[Times[Plus[Optional[Pattern[a, Blank[]]], Times[Log[Times[Optional[Pattern[c, Blank[]]], Power[Plus[Pattern[d, Blank[]], Times[Optional[Pattern[e, Blank[]]], Pattern[x, Blank[]]]], Optional[Pattern[n, Blank[]]]]]], Optional[Pattern[b, Blank[]]]]], Plus[Optional[Pattern[f, Blank[]]], Times[Log[Times[Optional[Pattern[h, Blank[]]], Power[Plus[Optional[Pattern[i, Blank[]]], Times[Optional[Pattern[j, Blank[]]], Pattern[x, Blank[]]]], Optional[Pattern[m, Blank[]]]]]], Optional[Pattern[g, Blank[]]]]], Power[Plus[Pattern[k, Blank[]], Times[Optional[Pattern[l, Blank[]]], Pattern[x, Blank[]]]], Optional[Pattern[r, Blank[]]]]], Pattern[x, Blank[Symbol]]] := Dist[1/l, Subst[Int[x^r*(a + b*Log[c*(-((e*k - d*l)/l) + (e*x)/l)^n])*(f + g*Log[h*(-((j*k - i*l)/l) + (j*x)/l)^m]), x], x, k + l*x], x] /; FreeQ[{a, b, c, d, e, f, g, h, i, j, k, l, m, n}, x] && IntegerQ[r]
  2441. Int[Times[Power[Plus[Optional[Pattern[a, Blank[]]], Times[Log[Times[Optional[Pattern[c, Blank[]]], Power[Plus[Pattern[d, Blank[]], Times[Optional[Pattern[e, Blank[]]], Pattern[x, Blank[]]]], Optional[Pattern[n, Blank[]]]]]], Optional[Pattern[b, Blank[]]]]], Optional[Pattern[p, Blank[]]]], Power[Plus[Optional[Pattern[f, Blank[]]], Times[Log[Times[Optional[Pattern[h, Blank[]]], Power[Plus[Optional[Pattern[i, Blank[]]], Times[Optional[Pattern[j, Blank[]]], Pattern[x, Blank[]]]], Optional[Pattern[m, Blank[]]]]]], Optional[Pattern[g, Blank[]]]]], Optional[Pattern[q, Blank[]]]], Power[Plus[Optional[Pattern[k, Blank[]]], Times[Optional[Pattern[l, Blank[]]], Pattern[x, Blank[]]]], Optional[Pattern[r, Blank[]]]]], Pattern[x, Blank[Symbol]]] := Unintegrable[(k + l*x)^r*(a + b*Log[c*(d + e*x)^n])^p*(f + g*Log[h*(i + j*x)^m])^q, x] /; FreeQ[{a, b, c, d, e, f, g, h, i, j, k, l, m, n, p, q, r}, x]
  2442. Int[Times[Power[Plus[Optional[Pattern[a, Blank[]]], Times[Log[Times[Optional[Pattern[c, Blank[]]], Power[Plus[Pattern[d, Blank[]], Times[Optional[Pattern[e, Blank[]]], Pattern[x, Blank[]]]], Optional[Pattern[n, Blank[]]]]]], Optional[Pattern[b, Blank[]]]]], Optional[Pattern[p, Blank[]]]], Power[Plus[Pattern[f, Blank[]], Times[Optional[Pattern[g, Blank[]]], Pattern[x, Blank[]]]], -1], PolyLog[Pattern[k, Blank[]], Plus[Pattern[h, Blank[]], Times[Optional[Pattern[i, Blank[]]], Pattern[x, Blank[]]]]]], Pattern[x, Blank[Symbol]]] := Dist[1/g, Subst[Int[(PolyLog[k, (h*x)/d]*(a + b*Log[c*x^n])^p)/x, x], x, d + e*x], x] /; FreeQ[{a, b, c, d, e, f, g, h, i, k, n}, x] && EqQ[e*f - d*g, 0] && EqQ[g*h - f*i, 0] && IGtQ[p, 0]
  2443. Int[Times[Plus[Optional[Pattern[a, Blank[]]], Times[Log[Times[Optional[Pattern[c, Blank[]]], Power[Plus[Pattern[d, Blank[]], Times[Optional[Pattern[e, Blank[]]], Pattern[x, Blank[]]]], Optional[Pattern[n, Blank[]]]]]], Optional[Pattern[b, Blank[]]]]], Optional[Pattern[Px, Blank[]]], Pattern[F, Blank[]][Times[Optional[Pattern[f, Blank[]]], Plus[Optional[Pattern[g, Blank[]]], Times[Optional[Pattern[h, Blank[]]], Pattern[x, Blank[]]]]]]], Pattern[x, Blank[Symbol]]] := With[{u = IntHide[Px*F[f*(g + h*x)], x]}, Dist[a + b*Log[c*(d + e*x)^n], u, x] - Dist[b*e*n, Int[SimplifyIntegrand[u/(d + e*x), x], x], x]] /; FreeQ[{a, b, c, d, e, f, g, h, n}, x] && PolynomialQ[Px, x] && MemberQ[{ArcSin, ArcCos, ArcTan, ArcCot, ArcSinh, ArcCosh, ArcTanh, ArcCoth}, F]
  2444. Int[Times[Power[Plus[Optional[Pattern[a, Blank[]]], Times[Log[Times[Optional[Pattern[c, Blank[]]], Power[Pattern[v, Blank[]], Optional[Pattern[n, Blank[]]]]]], Optional[Pattern[b, Blank[]]]]], Optional[Pattern[p, Blank[]]]], Optional[Pattern[u, Blank[]]]], Pattern[x, Blank[Symbol]]] := Int[u*(a + b*Log[c*ExpandToSum[v, x]^n])^p, x] /; FreeQ[{a, b, c, n, p}, x] && LinearQ[v, x] && !LinearMatchQ[v, x] && !(EqQ[n, 1] && MatchQ[c*v, (e_.)*((f_) + (g_.)*x) /; FreeQ[{e, f, g}, x]])
  2445. Int[Times[Power[Plus[Optional[Pattern[a, Blank[]]], Times[Log[Times[Optional[Pattern[c, Blank[]]], Power[Times[Optional[Pattern[d, Blank[]]], Power[Plus[Optional[Pattern[e, Blank[]]], Times[Optional[Pattern[f, Blank[]]], Pattern[x, Blank[]]]], Optional[Pattern[m, Blank[]]]]], Pattern[n, Blank[]]]]], Optional[Pattern[b, Blank[]]]]], Optional[Pattern[p, Blank[]]]], Optional[Pattern[u, Blank[]]]], Pattern[x, Blank[Symbol]]] := Subst[Int[u*(a + b*Log[c*d^n*(e + f*x)^(m*n)])^p, x], c*d^n*(e + f*x)^(m*n), c*(d*(e + f*x)^m)^n] /; FreeQ[{a, b, c, d, e, f, m, n, p}, x] && !IntegerQ[n] && !(EqQ[d, 1] && EqQ[m, 1]) && IntegralFreeQ[IntHide[u*(a + b*Log[c*d^n*(e + f*x)^(m*n)])^p, x]]
  2446. Int[Times[Power[Plus[Optional[Pattern[a, Blank[]]], Times[Log[Times[Optional[Pattern[c, Blank[]]], Power[Times[Optional[Pattern[d, Blank[]]], Power[Plus[Optional[Pattern[e, Blank[]]], Times[Optional[Pattern[f, Blank[]]], Pattern[x, Blank[]]]], Optional[Pattern[m, Blank[]]]]], Pattern[n, Blank[]]]]], Optional[Pattern[b, Blank[]]]]], Optional[Pattern[p, Blank[]]]], Pattern[AFx, Blank[]]], Pattern[x, Blank[Symbol]]] := Unintegrable[AFx*(a + b*Log[c*(d*(e + f*x)^m)^n])^p, x] /; FreeQ[{a, b, c, d, e, f, m, n, p}, x] && AlgebraicFunctionQ[AFx, x, True]
  2447. Int[Times[Log[Pattern[u, Blank[]]], Power[Pattern[Pq, Blank[]], Optional[Pattern[m, Blank[]]]]], Pattern[x, Blank[Symbol]]] := With[{C = FullSimplify[(Pq^m*(1 - u))/D[u, x]]}, Simp[C*PolyLog[2, 1 - u], x] /; FreeQ[C, x]] /; IntegerQ[m] && PolyQ[Pq, x] && RationalFunctionQ[u, x] && LeQ[RationalFunctionExponents[u, x][[2]], Expon[Pq, x]]
  2448. Int[Log[Times[Optional[Pattern[c, Blank[]]], Power[Plus[Pattern[d, Blank[]], Times[Optional[Pattern[e, Blank[]]], Power[Pattern[x, Blank[]], Pattern[n, Blank[]]]]], Optional[Pattern[p, Blank[]]]]]], Pattern[x, Blank[Symbol]]] := Simp[x*Log[c*(d + e*x^n)^p], x] - Dist[e*n*p, Int[x^n/(d + e*x^n), x], x] /; FreeQ[{c, d, e, n, p}, x]
  2449. Int[Power[Plus[Optional[Pattern[a, Blank[]]], Times[Log[Times[Optional[Pattern[c, Blank[]]], Power[Plus[Pattern[d, Blank[]], Times[Optional[Pattern[e, Blank[]]], Power[Pattern[x, Blank[]], -1]]], Optional[Pattern[p, Blank[]]]]]], Optional[Pattern[b, Blank[]]]]], Pattern[q, Blank[]]], Pattern[x, Blank[Symbol]]] := Simp[((e + d*x)*(a + b*Log[c*(d + e/x)^p])^q)/d, x] + Dist[(b*e*p*q)/d, Int[(a + b*Log[c*(d + e/x)^p])^(q - 1)/x, x], x] /; FreeQ[{a, b, c, d, e, p}, x] && IGtQ[q, 0]
  2450. Int[Power[Plus[Optional[Pattern[a, Blank[]]], Times[Log[Times[Optional[Pattern[c, Blank[]]], Power[Plus[Pattern[d, Blank[]], Times[Optional[Pattern[e, Blank[]]], Power[Pattern[x, Blank[]], Pattern[n, Blank[]]]]], Optional[Pattern[p, Blank[]]]]]], Optional[Pattern[b, Blank[]]]]], Pattern[q, Blank[]]], Pattern[x, Blank[Symbol]]] := Simp[x*(a + b*Log[c*(d + e*x^n)^p])^q, x] - Dist[b*e*n*p*q, Int[(x^n*(a + b*Log[c*(d + e*x^n)^p])^(q - 1))/(d + e*x^n), x], x] /; FreeQ[{a, b, c, d, e, n, p}, x] && IGtQ[q, 0] && (EqQ[q, 1] || IntegerQ[n])
  2451. Int[Power[Plus[Optional[Pattern[a, Blank[]]], Times[Log[Times[Optional[Pattern[c, Blank[]]], Power[Plus[Pattern[d, Blank[]], Times[Optional[Pattern[e, Blank[]]], Power[Pattern[x, Blank[]], Pattern[n, Blank[]]]]], Optional[Pattern[p, Blank[]]]]]], Optional[Pattern[b, Blank[]]]]], Pattern[q, Blank[]]], Pattern[x, Blank[Symbol]]] := With[{k = Denominator[n]}, Dist[k, Subst[Int[x^(k - 1)*(a + b*Log[c*(d + e*x^(k*n))^p])^q, x], x, x^(1/k)], x]] /; FreeQ[{a, b, c, d, e, p, q}, x] && FractionQ[n]
  2452. Int[Power[Plus[Optional[Pattern[a, Blank[]]], Times[Log[Times[Optional[Pattern[c, Blank[]]], Power[Plus[Pattern[d, Blank[]], Times[Optional[Pattern[e, Blank[]]], Power[Pattern[x, Blank[]], Pattern[n, Blank[]]]]], Optional[Pattern[p, Blank[]]]]]], Optional[Pattern[b, Blank[]]]]], Pattern[q, Blank[]]], Pattern[x, Blank[Symbol]]] := Unintegrable[(a + b*Log[c*(d + e*x^n)^p])^q, x] /; FreeQ[{a, b, c, d, e, n, p, q}, x]
  2453. Int[Power[Plus[Optional[Pattern[a, Blank[]]], Times[Log[Times[Optional[Pattern[c, Blank[]]], Power[Pattern[v, Blank[]], Optional[Pattern[p, Blank[]]]]]], Optional[Pattern[b, Blank[]]]]], Optional[Pattern[q, Blank[]]]], Pattern[x, Blank[Symbol]]] := Int[(a + b*Log[c*ExpandToSum[v, x]^p])^q, x] /; FreeQ[{a, b, c, p, q}, x] && BinomialQ[v, x] && !BinomialMatchQ[v, x]
  2454. Int[Times[Power[Plus[Optional[Pattern[a, Blank[]]], Times[Log[Times[Optional[Pattern[c, Blank[]]], Power[Plus[Pattern[d, Blank[]], Times[Optional[Pattern[e, Blank[]]], Power[Pattern[x, Blank[]], Pattern[n, Blank[]]]]], Optional[Pattern[p, Blank[]]]]]], Optional[Pattern[b, Blank[]]]]], Optional[Pattern[q, Blank[]]]], Power[Pattern[x, Blank[]], Optional[Pattern[m, Blank[]]]]], Pattern[x, Blank[Symbol]]] := Dist[1/n, Subst[Int[x^(Simplify[(m + 1)/n] - 1)*(a + b*Log[c*(d + e*x)^p])^q, x], x, x^n], x] /; FreeQ[{a, b, c, d, e, m, n, p, q}, x] && IntegerQ[Simplify[(m + 1)/n]] && (GtQ[(m + 1)/n, 0] || IGtQ[q, 0]) && !(EqQ[q, 1] && ILtQ[n, 0] && IGtQ[m, 0])
  2455. Int[Times[Plus[Optional[Pattern[a, Blank[]]], Times[Log[Times[Optional[Pattern[c, Blank[]]], Power[Plus[Pattern[d, Blank[]], Times[Optional[Pattern[e, Blank[]]], Power[Pattern[x, Blank[]], Pattern[n, Blank[]]]]], Optional[Pattern[p, Blank[]]]]]], Optional[Pattern[b, Blank[]]]]], Power[Times[Optional[Pattern[f, Blank[]]], Pattern[x, Blank[]]], Optional[Pattern[m, Blank[]]]]], Pattern[x, Blank[Symbol]]] := Simp[((f*x)^(m + 1)*(a + b*Log[c*(d + e*x^n)^p]))/(f*(m + 1)), x] - Dist[(b*e*n*p)/(f*(m + 1)), Int[(x^(n - 1)*(f*x)^(m + 1))/(d + e*x^n), x], x] /; FreeQ[{a, b, c, d, e, f, m, n, p}, x] && NeQ[m, -1]
  2456. Int[Times[Power[Plus[Optional[Pattern[a, Blank[]]], Times[Log[Times[Optional[Pattern[c, Blank[]]], Power[Plus[Pattern[d, Blank[]], Times[Optional[Pattern[e, Blank[]]], Power[Pattern[x, Blank[]], Pattern[n, Blank[]]]]], Optional[Pattern[p, Blank[]]]]]], Optional[Pattern[b, Blank[]]]]], Optional[Pattern[q, Blank[]]]], Power[Times[Pattern[f, Blank[]], Pattern[x, Blank[]]], Pattern[m, Blank[]]]], Pattern[x, Blank[Symbol]]] := Dist[(f*x)^m/x^m, Int[x^m*(a + b*Log[c*(d + e*x^n)^p])^q, x], x] /; FreeQ[{a, b, c, d, e, f, m, n, p, q}, x] && IntegerQ[Simplify[(m + 1)/n]] && (GtQ[(m + 1)/n, 0] || IGtQ[q, 0])
  2457. Int[Times[Power[Plus[Optional[Pattern[a, Blank[]]], Times[Log[Times[Optional[Pattern[c, Blank[]]], Power[Plus[Pattern[d, Blank[]], Times[Optional[Pattern[e, Blank[]]], Power[Pattern[x, Blank[]], Pattern[n, Blank[]]]]], Optional[Pattern[p, Blank[]]]]]], Optional[Pattern[b, Blank[]]]]], Pattern[q, Blank[]]], Power[Times[Optional[Pattern[f, Blank[]]], Pattern[x, Blank[]]], Optional[Pattern[m, Blank[]]]]], Pattern[x, Blank[Symbol]]] := Simp[((f*x)^(m + 1)*(a + b*Log[c*(d + e*x^n)^p])^q)/(f*(m + 1)), x] - Dist[(b*e*n*p*q)/(f^n*(m + 1)), Int[((f*x)^(m + n)*(a + b*Log[c*(d + e*x^n)^p])^(q - 1))/(d + e*x^n), x], x] /; FreeQ[{a, b, c, d, e, f, m, p}, x] && IGtQ[q, 1] && IntegerQ[n] && NeQ[m, -1]
  2458. Int[Times[Power[Plus[Optional[Pattern[a, Blank[]]], Times[Log[Times[Optional[Pattern[c, Blank[]]], Power[Plus[Pattern[d, Blank[]], Times[Optional[Pattern[e, Blank[]]], Power[Pattern[x, Blank[]], Pattern[n, Blank[]]]]], Optional[Pattern[p, Blank[]]]]]], Optional[Pattern[b, Blank[]]]]], Pattern[q, Blank[]]], Power[Pattern[x, Blank[]], Optional[Pattern[m, Blank[]]]]], Pattern[x, Blank[Symbol]]] := With[{k = Denominator[n]}, Dist[k, Subst[Int[x^(k*(m + 1) - 1)*(a + b*Log[c*(d + e*x^(k*n))^p])^q, x], x, x^(1/k)], x]] /; FreeQ[{a, b, c, d, e, m, p, q}, x] && FractionQ[n]
  2459. Int[Times[Power[Plus[Optional[Pattern[a, Blank[]]], Times[Log[Times[Optional[Pattern[c, Blank[]]], Power[Plus[Pattern[d, Blank[]], Times[Optional[Pattern[e, Blank[]]], Power[Pattern[x, Blank[]], Pattern[n, Blank[]]]]], Optional[Pattern[p, Blank[]]]]]], Optional[Pattern[b, Blank[]]]]], Optional[Pattern[q, Blank[]]]], Power[Times[Pattern[f, Blank[]], Pattern[x, Blank[]]], Pattern[m, Blank[]]]], Pattern[x, Blank[Symbol]]] := Dist[(f*x)^m/x^m, Int[x^m*(a + b*Log[c*(d + e*x^n)^p])^q, x], x] /; FreeQ[{a, b, c, d, e, f, m, p, q}, x] && FractionQ[n]
  2460. Int[Times[Power[Plus[Optional[Pattern[a, Blank[]]], Times[Log[Times[Optional[Pattern[c, Blank[]]], Power[Plus[Pattern[d, Blank[]], Times[Optional[Pattern[e, Blank[]]], Power[Pattern[x, Blank[]], Pattern[n, Blank[]]]]], Optional[Pattern[p, Blank[]]]]]], Optional[Pattern[b, Blank[]]]]], Optional[Pattern[q, Blank[]]]], Power[Times[Optional[Pattern[f, Blank[]]], Pattern[x, Blank[]]], Optional[Pattern[m, Blank[]]]]], Pattern[x, Blank[Symbol]]] := Unintegrable[(f*x)^m*(a + b*Log[c*(d + e*x^n)^p])^q, x] /; FreeQ[{a, b, c, d, e, f, m, n, p, q}, x]
  2461. Int[Times[Power[Plus[Optional[Pattern[a, Blank[]]], Times[Log[Times[Optional[Pattern[c, Blank[]]], Power[Pattern[v, Blank[]], Optional[Pattern[p, Blank[]]]]]], Optional[Pattern[b, Blank[]]]]], Optional[Pattern[q, Blank[]]]], Power[Times[Optional[Pattern[f, Blank[]]], Pattern[x, Blank[]]], Optional[Pattern[m, Blank[]]]]], Pattern[x, Blank[Symbol]]] := Int[(f*x)^m*(a + b*Log[c*ExpandToSum[v, x]^p])^q, x] /; FreeQ[{a, b, c, f, m, p, q}, x] && BinomialQ[v, x] && !BinomialMatchQ[v, x]
  2462. Int[Times[Plus[Optional[Pattern[a, Blank[]]], Times[Log[Times[Optional[Pattern[c, Blank[]]], Power[Plus[Pattern[d, Blank[]], Times[Optional[Pattern[e, Blank[]]], Power[Pattern[x, Blank[]], Pattern[n, Blank[]]]]], Optional[Pattern[p, Blank[]]]]]], Optional[Pattern[b, Blank[]]]]], Power[Plus[Optional[Pattern[f, Blank[]]], Times[Optional[Pattern[g, Blank[]]], Pattern[x, Blank[]]]], -1]], Pattern[x, Blank[Symbol]]] := Simp[(Log[f + g*x]*(a + b*Log[c*(d + e*x^n)^p]))/g, x] - Dist[(b*e*n*p)/g, Int[(x^(n - 1)*Log[f + g*x])/(d + e*x^n), x], x] /; FreeQ[{a, b, c, d, e, f, g, n, p}, x] && RationalQ[n]
  2463. Int[Times[Plus[Optional[Pattern[a, Blank[]]], Times[Log[Times[Optional[Pattern[c, Blank[]]], Power[Plus[Pattern[d, Blank[]], Times[Optional[Pattern[e, Blank[]]], Power[Pattern[x, Blank[]], Pattern[n, Blank[]]]]], Optional[Pattern[p, Blank[]]]]]], Optional[Pattern[b, Blank[]]]]], Power[Plus[Optional[Pattern[f, Blank[]]], Times[Optional[Pattern[g, Blank[]]], Pattern[x, Blank[]]]], Optional[Pattern[r, Blank[]]]]], Pattern[x, Blank[Symbol]]] := Simp[((f + g*x)^(r + 1)*(a + b*Log[c*(d + e*x^n)^p]))/(g*(r + 1)), x] - Dist[(b*e*n*p)/(g*(r + 1)), Int[(x^(n - 1)*(f + g*x)^(r + 1))/(d + e*x^n), x], x] /; FreeQ[{a, b, c, d, e, f, g, n, p, r}, x] && (IGtQ[r, 0] || RationalQ[n]) && NeQ[r, -1]
  2464. Int[Times[Power[Plus[Optional[Pattern[a, Blank[]]], Times[Log[Times[Optional[Pattern[c, Blank[]]], Power[Plus[Pattern[d, Blank[]], Times[Optional[Pattern[e, Blank[]]], Power[Pattern[x, Blank[]], Pattern[n, Blank[]]]]], Optional[Pattern[p, Blank[]]]]]], Optional[Pattern[b, Blank[]]]]], Optional[Pattern[q, Blank[]]]], Power[Plus[Optional[Pattern[f, Blank[]]], Times[Optional[Pattern[g, Blank[]]], Pattern[x, Blank[]]]], Optional[Pattern[r, Blank[]]]]], Pattern[x, Blank[Symbol]]] := Unintegrable[(f + g*x)^r*(a + b*Log[c*(d + e*x^n)^p])^q, x] /; FreeQ[{a, b, c, d, e, f, g, n, p, q, r}, x]
  2465. Int[Times[Power[Plus[Optional[Pattern[a, Blank[]]], Times[Log[Times[Optional[Pattern[c, Blank[]]], Power[Pattern[v, Blank[]], Optional[Pattern[p, Blank[]]]]]], Optional[Pattern[b, Blank[]]]]], Optional[Pattern[q, Blank[]]]], Power[Pattern[u, Blank[]], Optional[Pattern[r, Blank[]]]]], Pattern[x, Blank[Symbol]]] := Int[ExpandToSum[u, x]^r*(a + b*Log[c*ExpandToSum[v, x]^p])^q, x] /; FreeQ[{a, b, c, p, q, r}, x] && LinearQ[u, x] && BinomialQ[v, x] && !(LinearMatchQ[u, x] && BinomialMatchQ[v, x])
  2466. Int[Times[Power[Plus[Optional[Pattern[a, Blank[]]], Times[Log[Times[Optional[Pattern[c, Blank[]]], Power[Plus[Pattern[d, Blank[]], Times[Optional[Pattern[e, Blank[]]], Power[Pattern[x, Blank[]], Pattern[n, Blank[]]]]], Optional[Pattern[p, Blank[]]]]]], Optional[Pattern[b, Blank[]]]]], Optional[Pattern[q, Blank[]]]], Power[Pattern[x, Blank[]], Optional[Pattern[m, Blank[]]]], Power[Plus[Optional[Pattern[f, Blank[]]], Times[Optional[Pattern[g, Blank[]]], Pattern[x, Blank[]]]], Optional[Pattern[r, Blank[]]]]], Pattern[x, Blank[Symbol]]] := Int[ExpandIntegrand[(a + b*Log[c*(d + e*x^n)^p])^q, x^m*(f + g*x)^r, x], x] /; FreeQ[{a, b, c, d, e, f, g, n, p, q}, x] && IntegerQ[m] && IntegerQ[r]
  2467. Int[Times[Power[Plus[Optional[Pattern[a, Blank[]]], Times[Log[Times[Optional[Pattern[c, Blank[]]], Power[Plus[Pattern[d, Blank[]], Times[Optional[Pattern[e, Blank[]]], Power[Pattern[x, Blank[]], Optional[Pattern[n, Blank[]]]]]], Optional[Pattern[p, Blank[]]]]]], Optional[Pattern[b, Blank[]]]]], Optional[Pattern[q, Blank[]]]], Power[Times[Optional[Pattern[h, Blank[]]], Pattern[x, Blank[]]], Pattern[m, Blank[]]], Power[Plus[Optional[Pattern[f, Blank[]]], Times[Optional[Pattern[g, Blank[]]], Pattern[x, Blank[]]]], Optional[Pattern[r, Blank[]]]]], Pattern[x, Blank[Symbol]]] := With[{k = Denominator[m]}, Dist[k/h, Subst[Int[x^(k*(m + 1) - 1)*(f + (g*x^k)/h)^r*(a + b*Log[c*(d + (e*x^(k*n))/h^n)^p])^q, x], x, (h*x)^(1/k)], x]] /; FreeQ[{a, b, c, d, e, f, g, h, p, r}, x] && FractionQ[m] && IntegerQ[n] && IntegerQ[r]
  2468. Int[Times[Power[Plus[Optional[Pattern[a, Blank[]]], Times[Log[Times[Optional[Pattern[c, Blank[]]], Power[Plus[Pattern[d, Blank[]], Times[Optional[Pattern[e, Blank[]]], Power[Pattern[x, Blank[]], Pattern[n, Blank[]]]]], Optional[Pattern[p, Blank[]]]]]], Optional[Pattern[b, Blank[]]]]], Optional[Pattern[q, Blank[]]]], Power[Times[Optional[Pattern[h, Blank[]]], Pattern[x, Blank[]]], Optional[Pattern[m, Blank[]]]], Power[Plus[Optional[Pattern[f, Blank[]]], Times[Optional[Pattern[g, Blank[]]], Pattern[x, Blank[]]]], Optional[Pattern[r, Blank[]]]]], Pattern[x, Blank[Symbol]]] := Unintegrable[(h*x)^m*(f + g*x)^r*(a + b*Log[c*(d + e*x^n)^p])^q, x] /; FreeQ[{a, b, c, d, e, f, g, h, m, n, p, q, r}, x]
  2469. Int[Times[Power[Plus[Optional[Pattern[a, Blank[]]], Times[Log[Times[Optional[Pattern[c, Blank[]]], Power[Pattern[v, Blank[]], Optional[Pattern[p, Blank[]]]]]], Optional[Pattern[b, Blank[]]]]], Optional[Pattern[q, Blank[]]]], Power[Pattern[u, Blank[]], Optional[Pattern[r, Blank[]]]], Power[Times[Optional[Pattern[h, Blank[]]], Pattern[x, Blank[]]], Optional[Pattern[m, Blank[]]]]], Pattern[x, Blank[Symbol]]] := Int[(h*x)^m*ExpandToSum[u, x]^r*(a + b*Log[c*ExpandToSum[v, x]^p])^q, x] /; FreeQ[{a, b, c, h, m, p, q, r}, x] && LinearQ[u, x] && BinomialQ[v, x] && !(LinearMatchQ[u, x] && BinomialMatchQ[v, x])
  2470. Int[Times[Plus[Optional[Pattern[a, Blank[]]], Times[Log[Times[Optional[Pattern[c, Blank[]]], Power[Plus[Pattern[d, Blank[]], Times[Optional[Pattern[e, Blank[]]], Power[Pattern[x, Blank[]], Pattern[n, Blank[]]]]], Optional[Pattern[p, Blank[]]]]]], Optional[Pattern[b, Blank[]]]]], Power[Plus[Pattern[f, Blank[]], Times[Optional[Pattern[g, Blank[]]], Power[Pattern[x, Blank[]], 2]]], -1]], Pattern[x, Blank[Symbol]]] := With[{u = IntHide[1/(f + g*x^2), x]}, Simp[u*(a + b*Log[c*(d + e*x^n)^p]), x] - Dist[b*e*n*p, Int[(u*x^(n - 1))/(d + e*x^n), x], x]] /; FreeQ[{a, b, c, d, e, f, g, n, p}, x] && IntegerQ[n]
  2471. Int[Times[Power[Plus[Optional[Pattern[a, Blank[]]], Times[Log[Times[Optional[Pattern[c, Blank[]]], Power[Plus[Pattern[d, Blank[]], Times[Optional[Pattern[e, Blank[]]], Power[Pattern[x, Blank[]], Pattern[n, Blank[]]]]], Optional[Pattern[p, Blank[]]]]]], Optional[Pattern[b, Blank[]]]]], Optional[Pattern[q, Blank[]]]], Power[Plus[Pattern[f, Blank[]], Times[Optional[Pattern[g, Blank[]]], Power[Pattern[x, Blank[]], Pattern[s, Blank[]]]]], Optional[Pattern[r, Blank[]]]]], Pattern[x, Blank[Symbol]]] := With[{t = ExpandIntegrand[(a + b*Log[c*(d + e*x^n)^p])^q, (f + g*x^s)^r, x]}, Int[t, x] /; SumQ[t]] /; FreeQ[{a, b, c, d, e, f, g, n, p, q, r, s}, x] && IntegerQ[n] && IGtQ[q, 0] && IntegerQ[r] && IntegerQ[s] && (EqQ[q, 1] || (GtQ[r, 0] && GtQ[s, 1]) || (LtQ[s, 0] && LtQ[r, 0]))
  2472. Int[Times[Power[Plus[Optional[Pattern[a, Blank[]]], Times[Log[Times[Optional[Pattern[c, Blank[]]], Power[Plus[Pattern[d, Blank[]], Times[Optional[Pattern[e, Blank[]]], Power[Pattern[x, Blank[]], Pattern[n, Blank[]]]]], Optional[Pattern[p, Blank[]]]]]], Optional[Pattern[b, Blank[]]]]], Optional[Pattern[q, Blank[]]]], Power[Plus[Pattern[f, Blank[]], Times[Optional[Pattern[g, Blank[]]], Power[Pattern[x, Blank[]], Pattern[s, Blank[]]]]], Optional[Pattern[r, Blank[]]]]], Pattern[x, Blank[Symbol]]] := With[{k = Denominator[n]}, Dist[k, Subst[Int[x^(k - 1)*(f + g*x^(k*s))^r*(a + b*Log[c*(d + e*x^(k*n))^p])^q, x], x, x^(1/k)], x] /; IntegerQ[k*s]] /; FreeQ[{a, b, c, d, e, f, g, n, p, q, r, s}, x] && FractionQ[n]
  2473. Int[Times[Power[Plus[Optional[Pattern[a, Blank[]]], Times[Log[Times[Optional[Pattern[c, Blank[]]], Power[Plus[Pattern[d, Blank[]], Times[Optional[Pattern[e, Blank[]]], Power[Pattern[x, Blank[]], Pattern[n, Blank[]]]]], Optional[Pattern[p, Blank[]]]]]], Optional[Pattern[b, Blank[]]]]], Optional[Pattern[q, Blank[]]]], Power[Plus[Pattern[f, Blank[]], Times[Optional[Pattern[g, Blank[]]], Power[Pattern[x, Blank[]], Pattern[s, Blank[]]]]], Optional[Pattern[r, Blank[]]]]], Pattern[x, Blank[Symbol]]] := Unintegrable[(f + g*x^s)^r*(a + b*Log[c*(d + e*x^n)^p])^q, x] /; FreeQ[{a, b, c, d, e, f, g, n, p, q, r, s}, x]
  2474. Int[Times[Power[Plus[Optional[Pattern[a, Blank[]]], Times[Log[Times[Optional[Pattern[c, Blank[]]], Power[Pattern[v, Blank[]], Optional[Pattern[p, Blank[]]]]]], Optional[Pattern[b, Blank[]]]]], Optional[Pattern[q, Blank[]]]], Power[Pattern[u, Blank[]], Optional[Pattern[r, Blank[]]]]], Pattern[x, Blank[Symbol]]] := Int[ExpandToSum[u, x]^r*(a + b*Log[c*ExpandToSum[v, x]^p])^q, x] /; FreeQ[{a, b, c, p, q, r}, x] && BinomialQ[{u, v}, x] && !BinomialMatchQ[{u, v}, x]
  2475. Int[Times[Power[Plus[Optional[Pattern[a, Blank[]]], Times[Log[Times[Optional[Pattern[c, Blank[]]], Power[Plus[Pattern[d, Blank[]], Times[Optional[Pattern[e, Blank[]]], Power[Pattern[x, Blank[]], Pattern[n, Blank[]]]]], Optional[Pattern[p, Blank[]]]]]], Optional[Pattern[b, Blank[]]]]], Optional[Pattern[q, Blank[]]]], Power[Pattern[x, Blank[]], Optional[Pattern[m, Blank[]]]], Power[Plus[Pattern[f, Blank[]], Times[Optional[Pattern[g, Blank[]]], Power[Pattern[x, Blank[]], Pattern[s, Blank[]]]]], Optional[Pattern[r, Blank[]]]]], Pattern[x, Blank[Symbol]]] := Dist[1/n, Subst[Int[x^(Simplify[(m + 1)/n] - 1)*(f + g*x^(s/n))^r*(a + b*Log[c*(d + e*x)^p])^q, x], x, x^n], x] /; FreeQ[{a, b, c, d, e, f, g, m, n, p, q, r, s}, x] && IntegerQ[r] && IntegerQ[s/n] && IntegerQ[Simplify[(m + 1)/n]] && (GtQ[(m + 1)/n, 0] || IGtQ[q, 0])
  2476. Int[Times[Power[Plus[Optional[Pattern[a, Blank[]]], Times[Log[Times[Optional[Pattern[c, Blank[]]], Power[Plus[Pattern[d, Blank[]], Times[Optional[Pattern[e, Blank[]]], Power[Pattern[x, Blank[]], Pattern[n, Blank[]]]]], Optional[Pattern[p, Blank[]]]]]], Optional[Pattern[b, Blank[]]]]], Optional[Pattern[q, Blank[]]]], Power[Pattern[x, Blank[]], Optional[Pattern[m, Blank[]]]], Power[Plus[Pattern[f, Blank[]], Times[Optional[Pattern[g, Blank[]]], Power[Pattern[x, Blank[]], Pattern[s, Blank[]]]]], Optional[Pattern[r, Blank[]]]]], Pattern[x, Blank[Symbol]]] := Int[ExpandIntegrand[(a + b*Log[c*(d + e*x^n)^p])^q, x^m*(f + g*x^s)^r, x], x] /; FreeQ[{a, b, c, d, e, f, g, m, n, p, q, r, s}, x] && IGtQ[q, 0] && IntegerQ[m] && IntegerQ[r] && IntegerQ[s]
  2477. Int[Times[Power[Plus[Optional[Pattern[a, Blank[]]], Times[Log[Times[Optional[Pattern[c, Blank[]]], Power[Plus[Pattern[d, Blank[]], Times[Optional[Pattern[e, Blank[]]], Power[Pattern[x, Blank[]], Pattern[n, Blank[]]]]], Optional[Pattern[p, Blank[]]]]]], Optional[Pattern[b, Blank[]]]]], Optional[Pattern[q, Blank[]]]], Power[Pattern[x, Blank[]], Optional[Pattern[m, Blank[]]]], Power[Plus[Pattern[f, Blank[]], Times[Optional[Pattern[g, Blank[]]], Power[Pattern[x, Blank[]], Pattern[s, Blank[]]]]], Optional[Pattern[r, Blank[]]]]], Pattern[x, Blank[Symbol]]] := Dist[1/n, Subst[Int[x^(m + 1/n - 1)*(f + g*x^(s/n))^r*(a + b*Log[c*(d + e*x)^p])^q, x], x, x^n], x] /; FreeQ[{a, b, c, d, e, f, g, m, n, p, q, r, s}, x] && FractionQ[n] && IntegerQ[1/n] && IntegerQ[s/n]
  2478. Int[Times[Power[Plus[Optional[Pattern[a, Blank[]]], Times[Log[Times[Optional[Pattern[c, Blank[]]], Power[Plus[Pattern[d, Blank[]], Times[Optional[Pattern[e, Blank[]]], Power[Pattern[x, Blank[]], Optional[Pattern[n, Blank[]]]]]], Optional[Pattern[p, Blank[]]]]]], Optional[Pattern[b, Blank[]]]]], Optional[Pattern[q, Blank[]]]], Power[Times[Optional[Pattern[h, Blank[]]], Pattern[x, Blank[]]], Pattern[m, Blank[]]], Power[Plus[Optional[Pattern[f, Blank[]]], Times[Optional[Pattern[g, Blank[]]], Power[Pattern[x, Blank[]], Optional[Pattern[s, Blank[]]]]]], Optional[Pattern[r, Blank[]]]]], Pattern[x, Blank[Symbol]]] := With[{k = Denominator[m]}, Dist[k/h, Subst[Int[x^(k*(m + 1) - 1)*(f + (g*x^(k*s))/h^s)^r*(a + b*Log[c*(d + (e*x^(k*n))/h^n)^p])^q, x], x, (h*x)^(1/k)], x]] /; FreeQ[{a, b, c, d, e, f, g, h, p, r}, x] && FractionQ[m] && IntegerQ[n] && IntegerQ[s]
  2479. Int[Times[Power[Plus[Optional[Pattern[a, Blank[]]], Times[Log[Times[Optional[Pattern[c, Blank[]]], Power[Plus[Pattern[d, Blank[]], Times[Optional[Pattern[e, Blank[]]], Power[Pattern[x, Blank[]], Pattern[n, Blank[]]]]], Optional[Pattern[p, Blank[]]]]]], Optional[Pattern[b, Blank[]]]]], Optional[Pattern[q, Blank[]]]], Power[Times[Optional[Pattern[h, Blank[]]], Pattern[x, Blank[]]], Optional[Pattern[m, Blank[]]]], Power[Plus[Pattern[f, Blank[]], Times[Optional[Pattern[g, Blank[]]], Power[Pattern[x, Blank[]], Pattern[s, Blank[]]]]], Optional[Pattern[r, Blank[]]]]], Pattern[x, Blank[Symbol]]] := Unintegrable[(h*x)^m*(f + g*x^s)^r*(a + b*Log[c*(d + e*x^n)^p])^q, x] /; FreeQ[{a, b, c, d, e, f, g, h, m, n, p, q, r, s}, x]
  2480. Int[Times[Power[Plus[Optional[Pattern[a, Blank[]]], Times[Log[Times[Optional[Pattern[c, Blank[]]], Power[Pattern[v, Blank[]], Optional[Pattern[p, Blank[]]]]]], Optional[Pattern[b, Blank[]]]]], Optional[Pattern[q, Blank[]]]], Power[Pattern[u, Blank[]], Optional[Pattern[r, Blank[]]]], Power[Times[Optional[Pattern[h, Blank[]]], Pattern[x, Blank[]]], Optional[Pattern[m, Blank[]]]]], Pattern[x, Blank[Symbol]]] := Int[(h*x)^m*ExpandToSum[u, x]^r*(a + b*Log[c*ExpandToSum[v, x]^p])^q, x] /; FreeQ[{a, b, c, h, m, p, q, r}, x] && BinomialQ[{u, v}, x] && !BinomialMatchQ[{u, v}, x]
  2481. Int[Times[Power[Log[Times[Optional[Pattern[f, Blank[]]], Power[Pattern[x, Blank[]], Optional[Pattern[q, Blank[]]]]]], Optional[Pattern[m, Blank[]]]], Plus[Optional[Pattern[a, Blank[]]], Times[Log[Times[Optional[Pattern[c, Blank[]]], Power[Plus[Pattern[d, Blank[]], Times[Optional[Pattern[e, Blank[]]], Power[Pattern[x, Blank[]], Pattern[n, Blank[]]]]], Optional[Pattern[p, Blank[]]]]]], Optional[Pattern[b, Blank[]]]]], Power[Pattern[x, Blank[]], -1]], Pattern[x, Blank[Symbol]]] := Simp[(Log[f*x^q]^(m + 1)*(a + b*Log[c*(d + e*x^n)^p]))/(q*(m + 1)), x] - Dist[(b*e*n*p)/(q*(m + 1)), Int[(x^(n - 1)*Log[f*x^q]^(m + 1))/(d + e*x^n), x], x] /; FreeQ[{a, b, c, d, e, f, m, n, p, q}, x] && NeQ[m, -1]
  2482. Int[Times[Plus[Optional[Pattern[a, Blank[]]], Times[Log[Times[Optional[Pattern[c, Blank[]]], Power[Plus[Pattern[d, Blank[]], Times[Optional[Pattern[e, Blank[]]], Power[Pattern[x, Blank[]], Pattern[n, Blank[]]]]], Optional[Pattern[p, Blank[]]]]]], Optional[Pattern[b, Blank[]]]]], Power[Pattern[F, Blank[]][Times[Optional[Pattern[f, Blank[]]], Pattern[x, Blank[]]]], Optional[Pattern[m, Blank[]]]]], Pattern[x, Blank[Symbol]]] := With[{u = IntHide[F[f*x]^m, x]}, Dist[a + b*Log[c*(d + e*x^n)^p], u, x] - Dist[b*e*n*p, Int[SimplifyIntegrand[(u*x^(n - 1))/(d + e*x^n), x], x], x]] /; FreeQ[{a, b, c, d, e, f, p}, x] && MemberQ[{ArcSin, ArcCos, ArcSinh, ArcCosh}, F] && IGtQ[m, 0] && IGtQ[n, 1]
  2483. Int[Power[Plus[Optional[Pattern[a, Blank[]]], Times[Log[Times[Optional[Pattern[c, Blank[]]], Power[Plus[Pattern[d, Blank[]], Times[Optional[Pattern[e, Blank[]]], Power[Plus[Optional[Pattern[f, Blank[]]], Times[Optional[Pattern[g, Blank[]]], Pattern[x, Blank[]]]], Pattern[n, Blank[]]]]], Optional[Pattern[p, Blank[]]]]]], Optional[Pattern[b, Blank[]]]]], Optional[Pattern[q, Blank[]]]], Pattern[x, Blank[Symbol]]] := Dist[1/g, Subst[Int[(a + b*Log[c*(d + e*x^n)^p])^q, x], x, f + g*x], x] /; FreeQ[{a, b, c, d, e, f, g, n, p}, x] && IGtQ[q, 0] && (EqQ[q, 1] || IntegerQ[n])
  2484. Int[Power[Plus[Optional[Pattern[a, Blank[]]], Times[Log[Times[Optional[Pattern[c, Blank[]]], Power[Plus[Pattern[d, Blank[]], Times[Optional[Pattern[e, Blank[]]], Power[Plus[Optional[Pattern[f, Blank[]]], Times[Optional[Pattern[g, Blank[]]], Pattern[x, Blank[]]]], Pattern[n, Blank[]]]]], Optional[Pattern[p, Blank[]]]]]], Optional[Pattern[b, Blank[]]]]], Optional[Pattern[q, Blank[]]]], Pattern[x, Blank[Symbol]]] := Unintegrable[(a + b*Log[c*(d + e*(f + g*x)^n)^p])^q, x] /; FreeQ[{a, b, c, d, e, f, g, n, p, q}, x]
  2485. Int[Times[Power[Log[Times[Optional[Pattern[e, Blank[]]], Power[Times[Optional[Pattern[f, Blank[]]], Power[Plus[Optional[Pattern[a, Blank[]]], Times[Optional[Pattern[b, Blank[]]], Pattern[x, Blank[]]]], Optional[Pattern[p, Blank[]]]], Power[Plus[Optional[Pattern[c, Blank[]]], Times[Optional[Pattern[d, Blank[]]], Pattern[x, Blank[]]]], Optional[Pattern[q, Blank[]]]]], Optional[Pattern[r, Blank[]]]]]], Optional[Pattern[s, Blank[]]]], Optional[Pattern[u, Blank[]]]], Pattern[x, Blank[Symbol]]] := Int[u*Log[e*((b^p*f*(c + d*x)^(p + q))/d^p)^r]^s, x] /; FreeQ[{a, b, c, d, e, f, p, q, r, s}, x] && EqQ[b*c - a*d, 0] && IntegerQ[p]
  2486. Int[Power[Log[Times[Optional[Pattern[e, Blank[]]], Power[Times[Optional[Pattern[f, Blank[]]], Power[Plus[Optional[Pattern[a, Blank[]]], Times[Optional[Pattern[b, Blank[]]], Pattern[x, Blank[]]]], Optional[Pattern[p, Blank[]]]], Power[Plus[Optional[Pattern[c, Blank[]]], Times[Optional[Pattern[d, Blank[]]], Pattern[x, Blank[]]]], Optional[Pattern[q, Blank[]]]]], Optional[Pattern[r, Blank[]]]]]], Optional[Pattern[s, Blank[]]]], Pattern[x, Blank[Symbol]]] := Simp[((a + b*x)*Log[e*(f*(a + b*x)^p*(c + d*x)^q)^r]^s)/b, x] + Dist[(q*r*s*(b*c - a*d))/b, Int[Log[e*(f*(a + b*x)^p*(c + d*x)^q)^r]^(s - 1)/(c + d*x), x], x] /; FreeQ[{a, b, c, d, e, f, p, q, r, s}, x] && NeQ[b*c - a*d, 0] && EqQ[p + q, 0] && IGtQ[s, 0]
  2487. Int[Power[Log[Times[Optional[Pattern[e, Blank[]]], Power[Times[Optional[Pattern[f, Blank[]]], Power[Plus[Optional[Pattern[a, Blank[]]], Times[Optional[Pattern[b, Blank[]]], Pattern[x, Blank[]]]], Optional[Pattern[p, Blank[]]]], Power[Plus[Optional[Pattern[c, Blank[]]], Times[Optional[Pattern[d, Blank[]]], Pattern[x, Blank[]]]], Optional[Pattern[q, Blank[]]]]], Optional[Pattern[r, Blank[]]]]]], Optional[Pattern[s, Blank[]]]], Pattern[x, Blank[Symbol]]] := Simp[((a + b*x)*Log[e*(f*(a + b*x)^p*(c + d*x)^q)^r]^s)/b, x] + (Dist[(q*r*s*(b*c - a*d))/b, Int[Log[e*(f*(a + b*x)^p*(c + d*x)^q)^r]^(s - 1)/(c + d*x), x], x] - Dist[r*s*(p + q), Int[Log[e*(f*(a + b*x)^p*(c + d*x)^q)^r]^(s - 1), x], x]) /; FreeQ[{a, b, c, d, e, f, p, q, r, s}, x] && NeQ[b*c - a*d, 0] && NeQ[p + q, 0] && IGtQ[s, 0] && LtQ[s, 4]
  2488. Int[Times[Power[Log[Times[Optional[Pattern[e, Blank[]]], Power[Times[Optional[Pattern[f, Blank[]]], Power[Plus[Optional[Pattern[a, Blank[]]], Times[Optional[Pattern[b, Blank[]]], Pattern[x, Blank[]]]], Optional[Pattern[p, Blank[]]]], Power[Plus[Optional[Pattern[c, Blank[]]], Times[Optional[Pattern[d, Blank[]]], Pattern[x, Blank[]]]], Optional[Pattern[q, Blank[]]]]], Optional[Pattern[r, Blank[]]]]]], Optional[Pattern[s, Blank[]]]], Power[Plus[Optional[Pattern[g, Blank[]]], Times[Optional[Pattern[h, Blank[]]], Pattern[x, Blank[]]]], -1]], Pattern[x, Blank[Symbol]]] := -Simp[(Log[-((b*c - a*d)/(d*(a + b*x)))]*Log[e*(f*(a + b*x)^p*(c + d*x)^q)^r]^s)/h, x] + Dist[(p*r*s*(b*c - a*d))/h, Int[(Log[-((b*c - a*d)/(d*(a + b*x)))]*Log[e*(f*(a + b*x)^p*(c + d*x)^q)^r]^(s - 1))/((a + b*x)*(c + d*x)), x], x] /; FreeQ[{a, b, c, d, e, f, g, h, p, q, r, s}, x] && NeQ[b*c - a*d, 0] && EqQ[p + q, 0] && EqQ[b*g - a*h, 0] && IGtQ[s, 0]
  2489. Int[Times[Power[Log[Times[Optional[Pattern[e, Blank[]]], Power[Times[Optional[Pattern[f, Blank[]]], Power[Plus[Optional[Pattern[a, Blank[]]], Times[Optional[Pattern[b, Blank[]]], Pattern[x, Blank[]]]], Optional[Pattern[p, Blank[]]]], Power[Plus[Optional[Pattern[c, Blank[]]], Times[Optional[Pattern[d, Blank[]]], Pattern[x, Blank[]]]], Optional[Pattern[q, Blank[]]]]], Optional[Pattern[r, Blank[]]]]]], Pattern[s, Blank[]]], Power[Plus[Optional[Pattern[g, Blank[]]], Times[Optional[Pattern[h, Blank[]]], Pattern[x, Blank[]]]], -1]], Pattern[x, Blank[Symbol]]] := Dist[d/h, Int[Log[e*(f*(a + b*x)^p*(c + d*x)^q)^r]^s/(c + d*x), x], x] - Dist[(d*g - c*h)/h, Int[Log[e*(f*(a + b*x)^p*(c + d*x)^q)^r]^s/((c + d*x)*(g + h*x)), x], x] /; FreeQ[{a, b, c, d, e, f, g, h, p, q, r, s}, x] && NeQ[b*c - a*d, 0] && EqQ[p + q, 0] && NeQ[b*g - a*h, 0] && NeQ[d*g - c*h, 0] && IGtQ[s, 1]
  2490. Int[Times[Power[Log[Times[Optional[Pattern[e, Blank[]]], Power[Times[Optional[Pattern[f, Blank[]]], Power[Plus[Optional[Pattern[a, Blank[]]], Times[Optional[Pattern[b, Blank[]]], Pattern[x, Blank[]]]], Optional[Pattern[p, Blank[]]]], Power[Plus[Optional[Pattern[c, Blank[]]], Times[Optional[Pattern[d, Blank[]]], Pattern[x, Blank[]]]], Optional[Pattern[q, Blank[]]]]], Optional[Pattern[r, Blank[]]]]]], Optional[Pattern[s, Blank[]]]], Power[Plus[Optional[Pattern[g, Blank[]]], Times[Optional[Pattern[h, Blank[]]], Pattern[x, Blank[]]]], -2]], Pattern[x, Blank[Symbol]]] := Simp[((a + b*x)*Log[e*(f*(a + b*x)^p*(c + d*x)^q)^r]^s)/((b*g - a*h)*(g + h*x)), x] - Dist[(p*r*s*(b*c - a*d))/(b*g - a*h), Int[Log[e*(f*(a + b*x)^p*(c + d*x)^q)^r]^(s - 1)/((c + d*x)*(g + h*x)), x], x] /; FreeQ[{a, b, c, d, e, f, g, h, p, q, r, s}, x] && NeQ[b*c - a*d, 0] && EqQ[p + q, 0] && NeQ[b*g - a*h, 0] && IGtQ[s, 0]
  2491. Int[Times[Power[Log[Times[Optional[Pattern[e, Blank[]]], Power[Times[Optional[Pattern[f, Blank[]]], Power[Plus[Optional[Pattern[a, Blank[]]], Times[Optional[Pattern[b, Blank[]]], Pattern[x, Blank[]]]], Optional[Pattern[p, Blank[]]]], Power[Plus[Optional[Pattern[c, Blank[]]], Times[Optional[Pattern[d, Blank[]]], Pattern[x, Blank[]]]], Optional[Pattern[q, Blank[]]]]], Optional[Pattern[r, Blank[]]]]]], Pattern[s, Blank[]]], Power[Plus[Optional[Pattern[g, Blank[]]], Times[Optional[Pattern[h, Blank[]]], Pattern[x, Blank[]]]], -3]], Pattern[x, Blank[Symbol]]] := Dist[d/(d*g - c*h), Int[Log[e*(f*(a + b*x)^p*(c + d*x)^q)^r]^s/(g + h*x)^2, x], x] - Dist[h/(d*g - c*h), Int[((c + d*x)*Log[e*(f*(a + b*x)^p*(c + d*x)^q)^r]^s)/(g + h*x)^3, x], x] /; FreeQ[{a, b, c, d, e, f, g, h, p, q, r, s}, x] && NeQ[b*c - a*d, 0] && EqQ[p + q, 0] && EqQ[b*g - a*h, 0] && NeQ[d*g - c*h, 0] && IGtQ[s, 0]
  2492. Int[Times[Power[Log[Times[Optional[Pattern[e, Blank[]]], Power[Times[Optional[Pattern[f, Blank[]]], Power[Plus[Optional[Pattern[a, Blank[]]], Times[Optional[Pattern[b, Blank[]]], Pattern[x, Blank[]]]], Optional[Pattern[p, Blank[]]]], Power[Plus[Optional[Pattern[c, Blank[]]], Times[Optional[Pattern[d, Blank[]]], Pattern[x, Blank[]]]], Optional[Pattern[q, Blank[]]]]], Optional[Pattern[r, Blank[]]]]]], Optional[Pattern[s, Blank[]]]], Power[Plus[Optional[Pattern[g, Blank[]]], Times[Optional[Pattern[h, Blank[]]], Pattern[x, Blank[]]]], Optional[Pattern[m, Blank[]]]]], Pattern[x, Blank[Symbol]]] := Simp[((g + h*x)^(m + 1)*Log[e*(f*(a + b*x)^p*(c + d*x)^q)^r]^s)/(h*(m + 1)), x] - Dist[(p*r*s*(b*c - a*d))/(h*(m + 1)), Int[((g + h*x)^(m + 1)*Log[e*(f*(a + b*x)^p*(c + d*x)^q)^r]^(s - 1))/((a + b*x)*(c + d*x)), x], x] /; FreeQ[{a, b, c, d, e, f, g, h, m, p, q, r, s}, x] && NeQ[b*c - a*d, 0] && EqQ[p + q, 0] && IGtQ[s, 0] && NeQ[m, -1]
  2493. Int[Times[Power[Log[Times[Optional[Pattern[e, Blank[]]], Power[Times[Optional[Pattern[f, Blank[]]], Power[Plus[Optional[Pattern[a, Blank[]]], Times[Optional[Pattern[b, Blank[]]], Pattern[x, Blank[]]]], Optional[Pattern[p, Blank[]]]], Power[Plus[Optional[Pattern[c, Blank[]]], Times[Optional[Pattern[d, Blank[]]], Pattern[x, Blank[]]]], Optional[Pattern[q, Blank[]]]]], Optional[Pattern[r, Blank[]]]]]], -1], Power[Plus[Optional[Pattern[g, Blank[]]], Times[Optional[Pattern[h, Blank[]]], Pattern[x, Blank[]]]], -2]], Pattern[x, Blank[Symbol]]] := Simp[(b*(c + d*x)*(e*(f*(a + b*x)^p*(c + d*x)^q)^r)^(1/(p*r))*ExpIntegralEi[-(Log[e*(f*(a + b*x)^p*(c + d*x)^q)^r]/(p*r))])/(h*p*r*(b*c - a*d)*(g + h*x)), x] /; FreeQ[{a, b, c, d, e, f, g, h, p, q, r}, x] && NeQ[b*c - a*d, 0] && EqQ[p + q, 0] && EqQ[b*g - a*h, 0]
  2494. Int[Times[Log[Times[Optional[Pattern[e, Blank[]]], Power[Times[Optional[Pattern[f, Blank[]]], Power[Plus[Optional[Pattern[a, Blank[]]], Times[Optional[Pattern[b, Blank[]]], Pattern[x, Blank[]]]], Optional[Pattern[p, Blank[]]]], Power[Plus[Optional[Pattern[c, Blank[]]], Times[Optional[Pattern[d, Blank[]]], Pattern[x, Blank[]]]], Optional[Pattern[q, Blank[]]]]], Optional[Pattern[r, Blank[]]]]]], Power[Plus[Optional[Pattern[g, Blank[]]], Times[Optional[Pattern[h, Blank[]]], Pattern[x, Blank[]]]], -1]], Pattern[x, Blank[Symbol]]] := Simp[(Log[g + h*x]*Log[e*(f*(a + b*x)^p*(c + d*x)^q)^r])/h, x] + (-Dist[(b*p*r)/h, Int[Log[g + h*x]/(a + b*x), x], x] - Dist[(d*q*r)/h, Int[Log[g + h*x]/(c + d*x), x], x]) /; FreeQ[{a, b, c, d, e, f, g, h, p, q, r}, x] && NeQ[b*c - a*d, 0]
  2495. Int[Times[Log[Times[Optional[Pattern[e, Blank[]]], Power[Times[Optional[Pattern[f, Blank[]]], Power[Plus[Optional[Pattern[a, Blank[]]], Times[Optional[Pattern[b, Blank[]]], Pattern[x, Blank[]]]], Optional[Pattern[p, Blank[]]]], Power[Plus[Optional[Pattern[c, Blank[]]], Times[Optional[Pattern[d, Blank[]]], Pattern[x, Blank[]]]], Optional[Pattern[q, Blank[]]]]], Optional[Pattern[r, Blank[]]]]]], Power[Plus[Optional[Pattern[g, Blank[]]], Times[Optional[Pattern[h, Blank[]]], Pattern[x, Blank[]]]], Optional[Pattern[m, Blank[]]]]], Pattern[x, Blank[Symbol]]] := Simp[((g + h*x)^(m + 1)*Log[e*(f*(a + b*x)^p*(c + d*x)^q)^r])/(h*(m + 1)), x] + (-Dist[(b*p*r)/(h*(m + 1)), Int[(g + h*x)^(m + 1)/(a + b*x), x], x] - Dist[(d*q*r)/(h*(m + 1)), Int[(g + h*x)^(m + 1)/(c + d*x), x], x]) /; FreeQ[{a, b, c, d, e, f, g, h, m, p, q, r}, x] && NeQ[b*c - a*d, 0] && NeQ[m, -1]
  2496. Int[Times[Power[Log[Times[Optional[Pattern[e, Blank[]]], Power[Times[Optional[Pattern[f, Blank[]]], Power[Plus[Optional[Pattern[a, Blank[]]], Times[Optional[Pattern[b, Blank[]]], Pattern[x, Blank[]]]], Optional[Pattern[p, Blank[]]]], Power[Plus[Optional[Pattern[c, Blank[]]], Times[Optional[Pattern[d, Blank[]]], Pattern[x, Blank[]]]], Optional[Pattern[q, Blank[]]]]], Optional[Pattern[r, Blank[]]]]]], 2], Power[Plus[Optional[Pattern[g, Blank[]]], Times[Optional[Pattern[h, Blank[]]], Pattern[x, Blank[]]]], -1]], Pattern[x, Blank[Symbol]]] := Int[(Log[(a + b*x)^(p*r)] + Log[(c + d*x)^(q*r)])^2/(g + h*x), x] + Simp[(Log[e*(f*(a + b*x)^p*(c + d*x)^q)^r] - Log[(a + b*x)^(p*r)] - Log[(c + d*x)^(q*r)])*(2*Int[Log[(c + d*x)^(q*r)]/(g + h*x), x] + Int[(Log[(a + b*x)^(p*r)] - Log[(c + d*x)^(q*r)] + Log[e*(f*(a + b*x)^p*(c + d*x)^q)^r])/(g + h*x), x]), x] /; FreeQ[{a, b, c, d, e, f, g, h, p, q, r}, x] && NeQ[b*c - a*d, 0] && EqQ[b*g - a*h, 0]
  2497. Int[Times[Power[Log[Times[Optional[Pattern[e, Blank[]]], Power[Times[Optional[Pattern[f, Blank[]]], Power[Plus[Optional[Pattern[a, Blank[]]], Times[Optional[Pattern[b, Blank[]]], Pattern[x, Blank[]]]], Optional[Pattern[p, Blank[]]]], Power[Plus[Optional[Pattern[c, Blank[]]], Times[Optional[Pattern[d, Blank[]]], Pattern[x, Blank[]]]], Optional[Pattern[q, Blank[]]]]], Optional[Pattern[r, Blank[]]]]]], 2], Power[Plus[Optional[Pattern[g, Blank[]]], Times[Optional[Pattern[h, Blank[]]], Pattern[x, Blank[]]]], -1]], Pattern[x, Blank[Symbol]]] := Simp[(Log[g + h*x]*Log[e*(f*(a + b*x)^p*(c + d*x)^q)^r]^2)/h, x] + (-Dist[(2*b*p*r)/h, Int[(Log[g + h*x]*Log[e*(f*(a + b*x)^p*(c + d*x)^q)^r])/(a + b*x), x], x] - Dist[(2*d*q*r)/h, Int[(Log[g + h*x]*Log[e*(f*(a + b*x)^p*(c + d*x)^q)^r])/(c + d*x), x], x]) /; FreeQ[{a, b, c, d, e, f, g, h, p, q, r}, x] && NeQ[b*c - a*d, 0]
  2498. Int[Times[Power[Log[Times[Optional[Pattern[e, Blank[]]], Power[Times[Optional[Pattern[f, Blank[]]], Power[Plus[Optional[Pattern[a, Blank[]]], Times[Optional[Pattern[b, Blank[]]], Pattern[x, Blank[]]]], Optional[Pattern[p, Blank[]]]], Power[Plus[Optional[Pattern[c, Blank[]]], Times[Optional[Pattern[d, Blank[]]], Pattern[x, Blank[]]]], Optional[Pattern[q, Blank[]]]]], Optional[Pattern[r, Blank[]]]]]], Pattern[s, Blank[]]], Power[Plus[Optional[Pattern[g, Blank[]]], Times[Optional[Pattern[h, Blank[]]], Pattern[x, Blank[]]]], Optional[Pattern[m, Blank[]]]]], Pattern[x, Blank[Symbol]]] := Simp[((g + h*x)^(m + 1)*Log[e*(f*(a + b*x)^p*(c + d*x)^q)^r]^s)/(h*(m + 1)), x] + (-Dist[(b*p*r*s)/(h*(m + 1)), Int[((g + h*x)^(m + 1)*Log[e*(f*(a + b*x)^p*(c + d*x)^q)^r]^(s - 1))/(a + b*x), x], x] - Dist[(d*q*r*s)/(h*(m + 1)), Int[((g + h*x)^(m + 1)*Log[e*(f*(a + b*x)^p*(c + d*x)^q)^r]^(s - 1))/(c + d*x), x], x]) /; FreeQ[{a, b, c, d, e, f, g, h, m, p, q, r, s}, x] && NeQ[b*c - a*d, 0] && IGtQ[s, 0] && NeQ[m, -1]
  2499. Int[Times[Log[Times[Optional[Pattern[e, Blank[]]], Power[Times[Optional[Pattern[f, Blank[]]], Power[Plus[Optional[Pattern[a, Blank[]]], Times[Optional[Pattern[b, Blank[]]], Pattern[x, Blank[]]]], Optional[Pattern[p, Blank[]]]], Power[Plus[Optional[Pattern[c, Blank[]]], Times[Optional[Pattern[d, Blank[]]], Pattern[x, Blank[]]]], Optional[Pattern[q, Blank[]]]]], Optional[Pattern[r, Blank[]]]]]], Power[Plus[Optional[Pattern[s, Blank[]]], Times[Log[Times[Optional[Pattern[i, Blank[]]], Power[Plus[Optional[Pattern[g, Blank[]]], Times[Optional[Pattern[h, Blank[]]], Pattern[x, Blank[]]]], Optional[Pattern[n, Blank[]]]]]], Optional[Pattern[t, Blank[]]]]], Optional[Pattern[m, Blank[]]]], Power[Plus[Optional[Pattern[j, Blank[]]], Times[Optional[Pattern[k, Blank[]]], Pattern[x, Blank[]]]], -1]], Pattern[x, Blank[Symbol]]] := Simp[((s + t*Log[i*(g + h*x)^n])^(m + 1)*Log[e*(f*(a + b*x)^p*(c + d*x)^q)^r])/(k*n*t*(m + 1)), x] + (-Dist[(b*p*r)/(k*n*t*(m + 1)), Int[(s + t*Log[i*(g + h*x)^n])^(m + 1)/(a + b*x), x], x] - Dist[(d*q*r)/(k*n*t*(m + 1)), Int[(s + t*Log[i*(g + h*x)^n])^(m + 1)/(c + d*x), x], x]) /; FreeQ[{a, b, c, d, e, f, g, h, i, j, k, s, t, m, n, p, q, r}, x] && NeQ[b*c - a*d, 0] && EqQ[h*j - g*k, 0] && IGtQ[m, 0]
  2500. Int[Times[Log[Times[Optional[Pattern[e, Blank[]]], Power[Times[Optional[Pattern[f, Blank[]]], Power[Plus[Optional[Pattern[a, Blank[]]], Times[Optional[Pattern[b, Blank[]]], Pattern[x, Blank[]]]], Optional[Pattern[p, Blank[]]]], Power[Plus[Optional[Pattern[c, Blank[]]], Times[Optional[Pattern[d, Blank[]]], Pattern[x, Blank[]]]], Optional[Pattern[q, Blank[]]]]], Optional[Pattern[r, Blank[]]]]]], Plus[Optional[Pattern[s, Blank[]]], Times[Log[Times[Optional[Pattern[i, Blank[]]], Power[Plus[Optional[Pattern[g, Blank[]]], Times[Optional[Pattern[h, Blank[]]], Pattern[x, Blank[]]]], Optional[Pattern[n, Blank[]]]]]], Optional[Pattern[t, Blank[]]]]], Power[Plus[Optional[Pattern[j, Blank[]]], Times[Optional[Pattern[k, Blank[]]], Pattern[x, Blank[]]]], -1]], Pattern[x, Blank[Symbol]]] := Dist[Log[e*(f*(a + b*x)^p*(c + d*x)^q)^r] - Log[(a + b*x)^(p*r)] - Log[(c + d*x)^(q*r)], Int[(s + t*Log[i*(g + h*x)^n])/(j + k*x), x], x] + (Int[(Log[(a + b*x)^(p*r)]*(s + t*Log[i*(g + h*x)^n]))/(j + k*x), x] + Int[(Log[(c + d*x)^(q*r)]*(s + t*Log[i*(g + h*x)^n]))/(j + k*x), x]) /; FreeQ[{a, b, c, d, e, f, g, h, i, j, k, s, t, n, p, q, r}, x] && NeQ[b*c - a*d, 0]
  2501. Int[Times[Power[Log[Times[Optional[Pattern[e, Blank[]]], Power[Times[Optional[Pattern[f, Blank[]]], Power[Plus[Optional[Pattern[a, Blank[]]], Times[Optional[Pattern[b, Blank[]]], Pattern[x, Blank[]]]], Optional[Pattern[p, Blank[]]]], Power[Plus[Optional[Pattern[c, Blank[]]], Times[Optional[Pattern[d, Blank[]]], Pattern[x, Blank[]]]], Optional[Pattern[q, Blank[]]]]], Optional[Pattern[r, Blank[]]]]]], Optional[Pattern[u, Blank[]]]], Power[Plus[Optional[Pattern[s, Blank[]]], Times[Log[Times[Optional[Pattern[i, Blank[]]], Power[Plus[Optional[Pattern[g, Blank[]]], Times[Optional[Pattern[h, Blank[]]], Pattern[x, Blank[]]]], Optional[Pattern[n, Blank[]]]]]], Optional[Pattern[t, Blank[]]]]], Optional[Pattern[m, Blank[]]]], Power[Plus[Optional[Pattern[j, Blank[]]], Times[Optional[Pattern[k, Blank[]]], Pattern[x, Blank[]]]], -1]], Pattern[x, Blank[Symbol]]] := Unintegrable[(Log[e*(f*(a + b*x)^p*(c + d*x)^q)^r]^u*(s + t*Log[i*(g + h*x)^n])^m)/(j + k*x), x] /; FreeQ[{a, b, c, d, e, f, g, h, i, j, k, s, t, m, n, p, q, r, u}, x] && NeQ[b*c - a*d, 0]
  2502. Int[Times[Log[Times[Optional[Pattern[e, Blank[]]], Power[Plus[Optional[Pattern[a, Blank[]]], Times[Optional[Pattern[b, Blank[]]], Pattern[x, Blank[]]]], -1], Plus[Optional[Pattern[c, Blank[]]], Times[Optional[Pattern[d, Blank[]]], Pattern[x, Blank[]]]]]], Pattern[u, Blank[]]], Pattern[x, Blank[Symbol]]] := With[{g = Coeff[Simplify[1/(u*(a + b*x))], x, 0], h = Coeff[Simplify[1/(u*(a + b*x))], x, 1]}, -Dist[(b - d*e)/(h*(b*c - a*d)), Subst[Int[Log[e*x]/(1 - e*x), x], x, (c + d*x)/(a + b*x)], x] /; EqQ[g*(b - d*e) - h*(a - c*e), 0]] /; FreeQ[{a, b, c, d, e}, x] && NeQ[b*c - a*d, 0] && LinearQ[Simplify[1/(u*(a + b*x))], x]
  2503. Int[Times[Power[Log[Times[Optional[Pattern[e, Blank[]]], Power[Times[Optional[Pattern[f, Blank[]]], Power[Plus[Optional[Pattern[a, Blank[]]], Times[Optional[Pattern[b, Blank[]]], Pattern[x, Blank[]]]], Optional[Pattern[p, Blank[]]]], Power[Plus[Optional[Pattern[c, Blank[]]], Times[Optional[Pattern[d, Blank[]]], Pattern[x, Blank[]]]], Optional[Pattern[q, Blank[]]]]], Optional[Pattern[r, Blank[]]]]]], Optional[Pattern[s, Blank[]]]], Pattern[u, Blank[]]], Pattern[x, Blank[Symbol]]] := With[{g = Coeff[Simplify[1/(u*(a + b*x))], x, 0], h = Coeff[Simplify[1/(u*(a + b*x))], x, 1]}, -Simp[(Log[e*(f*(a + b*x)^p*(c + d*x)^q)^r]^s*Log[-(((b*c - a*d)*(g + h*x))/((d*g - c*h)*(a + b*x)))])/(b*g - a*h), x] + Dist[(p*r*s*(b*c - a*d))/(b*g - a*h), Int[(Log[e*(f*(a + b*x)^p*(c + d*x)^q)^r]^(s - 1)*Log[-(((b*c - a*d)*(g + h*x))/((d*g - c*h)*(a + b*x)))])/((a + b*x)*(c + d*x)), x], x] /; NeQ[b*g - a*h, 0] && NeQ[d*g - c*h, 0]] /; FreeQ[{a, b, c, d, e, f, p, q, r, s}, x] && NeQ[b*c - a*d, 0] && IGtQ[s, 0] && EqQ[p + q, 0] && LinearQ[Simplify[1/(u*(a + b*x))], x]
  2504. Int[Times[Power[Log[Times[Optional[Pattern[e, Blank[]]], Power[Times[Optional[Pattern[f, Blank[]]], Power[Plus[Optional[Pattern[a, Blank[]]], Times[Optional[Pattern[b, Blank[]]], Pattern[x, Blank[]]]], Optional[Pattern[p, Blank[]]]], Power[Plus[Optional[Pattern[c, Blank[]]], Times[Optional[Pattern[d, Blank[]]], Pattern[x, Blank[]]]], Optional[Pattern[q, Blank[]]]]], Optional[Pattern[r, Blank[]]]]]], -1], Pattern[u, Blank[]]], Pattern[x, Blank[Symbol]]] := With[{h = Simplify[u*(a + b*x)*(c + d*x)]}, Simp[(h*Log[Log[e*(f*(a + b*x)^p*(c + d*x)^q)^r]])/(p*r*(b*c - a*d)), x] /; FreeQ[h, x]] /; FreeQ[{a, b, c, d, e, f, p, q, r}, x] && NeQ[b*c - a*d, 0] && EqQ[p + q, 0]
  2505. Int[Times[Power[Log[Times[Optional[Pattern[e, Blank[]]], Power[Times[Optional[Pattern[f, Blank[]]], Power[Plus[Optional[Pattern[a, Blank[]]], Times[Optional[Pattern[b, Blank[]]], Pattern[x, Blank[]]]], Optional[Pattern[p, Blank[]]]], Power[Plus[Optional[Pattern[c, Blank[]]], Times[Optional[Pattern[d, Blank[]]], Pattern[x, Blank[]]]], Optional[Pattern[q, Blank[]]]]], Optional[Pattern[r, Blank[]]]]]], Optional[Pattern[s, Blank[]]]], Pattern[u, Blank[]]], Pattern[x, Blank[Symbol]]] := With[{h = Simplify[u*(a + b*x)*(c + d*x)]}, Simp[(h*Log[e*(f*(a + b*x)^p*(c + d*x)^q)^r]^(s + 1))/(p*r*(s + 1)*(b*c - a*d)), x] /; FreeQ[h, x]] /; FreeQ[{a, b, c, d, e, f, p, q, r, s}, x] && NeQ[b*c - a*d, 0] && EqQ[p + q, 0] && NeQ[s, -1]
  2506. Int[Times[Log[Pattern[v, Blank[]]], Power[Log[Times[Optional[Pattern[e, Blank[]]], Power[Times[Optional[Pattern[f, Blank[]]], Power[Plus[Optional[Pattern[a, Blank[]]], Times[Optional[Pattern[b, Blank[]]], Pattern[x, Blank[]]]], Optional[Pattern[p, Blank[]]]], Power[Plus[Optional[Pattern[c, Blank[]]], Times[Optional[Pattern[d, Blank[]]], Pattern[x, Blank[]]]], Optional[Pattern[q, Blank[]]]]], Optional[Pattern[r, Blank[]]]]]], Optional[Pattern[s, Blank[]]]], Pattern[u, Blank[]]], Pattern[x, Blank[Symbol]]] := With[{g = Simplify[((v - 1)*(c + d*x))/(a + b*x)], h = Simplify[u*(a + b*x)*(c + d*x)]}, -Simp[(h*PolyLog[2, 1 - v]*Log[e*(f*(a + b*x)^p*(c + d*x)^q)^r]^s)/(b*c - a*d), x] + Dist[h*p*r*s, Int[(PolyLog[2, 1 - v]*Log[e*(f*(a + b*x)^p*(c + d*x)^q)^r]^(s - 1))/((a + b*x)*(c + d*x)), x], x] /; FreeQ[{g, h}, x]] /; FreeQ[{a, b, c, d, e, f, p, q, r, s}, x] && NeQ[b*c - a*d, 0] && IGtQ[s, 0] && EqQ[p + q, 0]
  2507. Int[Times[Power[Log[Times[Optional[Pattern[e, Blank[]]], Power[Times[Optional[Pattern[f, Blank[]]], Power[Plus[Optional[Pattern[a, Blank[]]], Times[Optional[Pattern[b, Blank[]]], Pattern[x, Blank[]]]], Optional[Pattern[p, Blank[]]]], Power[Plus[Optional[Pattern[c, Blank[]]], Times[Optional[Pattern[d, Blank[]]], Pattern[x, Blank[]]]], Optional[Pattern[q, Blank[]]]]], Optional[Pattern[r, Blank[]]]]]], Optional[Pattern[s, Blank[]]]], Log[Times[Optional[Pattern[i, Blank[]]], Power[Times[Optional[Pattern[j, Blank[]]], Power[Plus[Optional[Pattern[g, Blank[]]], Times[Optional[Pattern[h, Blank[]]], Pattern[x, Blank[]]]], Optional[Pattern[t, Blank[]]]]], Optional[Pattern[u, Blank[]]]]]], Pattern[v, Blank[]]], Pattern[x, Blank[Symbol]]] := With[{k = Simplify[v*(a + b*x)*(c + d*x)]}, Simp[(k*Log[i*(j*(g + h*x)^t)^u]*Log[e*(f*(a + b*x)^p*(c + d*x)^q)^r]^(s + 1))/(p*r*(s + 1)*(b*c - a*d)), x] - Dist[(k*h*t*u)/(p*r*(s + 1)*(b*c - a*d)), Int[Log[e*(f*(a + b*x)^p*(c + d*x)^q)^r]^(s + 1)/(g + h*x), x], x] /; FreeQ[k, x]] /; FreeQ[{a, b, c, d, e, f, g, h, i, j, p, q, r, s, t, u}, x] && NeQ[b*c - a*d, 0] && EqQ[p + q, 0] && NeQ[s, -1]
  2508. Int[Times[Power[Log[Times[Optional[Pattern[e, Blank[]]], Power[Times[Optional[Pattern[f, Blank[]]], Power[Plus[Optional[Pattern[a, Blank[]]], Times[Optional[Pattern[b, Blank[]]], Pattern[x, Blank[]]]], Optional[Pattern[p, Blank[]]]], Power[Plus[Optional[Pattern[c, Blank[]]], Times[Optional[Pattern[d, Blank[]]], Pattern[x, Blank[]]]], Optional[Pattern[q, Blank[]]]]], Optional[Pattern[r, Blank[]]]]]], Optional[Pattern[s, Blank[]]]], Pattern[u, Blank[]], PolyLog[Pattern[n, Blank[]], Pattern[v, Blank[]]]], Pattern[x, Blank[Symbol]]] := With[{g = Simplify[(v*(c + d*x))/(a + b*x)], h = Simplify[u*(a + b*x)*(c + d*x)]}, Simp[(h*PolyLog[n + 1, v]*Log[e*(f*(a + b*x)^p*(c + d*x)^q)^r]^s)/(b*c - a*d), x] - Dist[h*p*r*s, Int[(PolyLog[n + 1, v]*Log[e*(f*(a + b*x)^p*(c + d*x)^q)^r]^(s - 1))/((a + b*x)*(c + d*x)), x], x] /; FreeQ[{g, h}, x]] /; FreeQ[{a, b, c, d, e, f, n, p, q, r, s}, x] && NeQ[b*c - a*d, 0] && IGtQ[s, 0] && EqQ[p + q, 0]
  2509. Int[Times[Power[Log[Times[Optional[Pattern[e, Blank[]]], Power[Times[Optional[Pattern[f, Blank[]]], Power[Plus[Optional[Pattern[a, Blank[]]], Times[Optional[Pattern[b, Blank[]]], Pattern[x, Blank[]]]], Optional[Pattern[p, Blank[]]]], Power[Plus[Optional[Pattern[c, Blank[]]], Times[Optional[Pattern[d, Blank[]]], Pattern[x, Blank[]]]], Optional[Pattern[q, Blank[]]]]], Optional[Pattern[r, Blank[]]]]]], Optional[Pattern[s, Blank[]]]], Power[Plus[Optional[Pattern[a, Blank[]]], Times[Optional[Pattern[b, Blank[]]], Pattern[x, Blank[]]]], Optional[Pattern[m, Blank[]]]], Power[Plus[Optional[Pattern[c, Blank[]]], Times[Optional[Pattern[d, Blank[]]], Pattern[x, Blank[]]]], Optional[Pattern[n, Blank[]]]]], Pattern[x, Blank[Symbol]]] := Simp[((a + b*x)^(m + 1)*(c + d*x)^(n + 1)*Log[e*(f*(a + b*x)^p*(c + d*x)^q)^r]^s)/((m + 1)*(b*c - a*d)), x] - Dist[(p*r*s*(b*c - a*d))/((m + 1)*(b*c - a*d)), Int[(a + b*x)^m*(c + d*x)^n*Log[e*(f*(a + b*x)^p*(c + d*x)^q)^r]^(s - 1), x], x] /; FreeQ[{a, b, c, d, e, f, m, n, p, q, r, s}, x] && NeQ[b*c - a*d, 0] && EqQ[p + q, 0] && EqQ[m + n + 2, 0] && NeQ[m, -1] && IGtQ[s, 0]
  2510. Int[Times[Power[Log[Times[Optional[Pattern[e, Blank[]]], Power[Times[Optional[Pattern[f, Blank[]]], Power[Plus[Optional[Pattern[a, Blank[]]], Times[Optional[Pattern[b, Blank[]]], Pattern[x, Blank[]]]], Optional[Pattern[p, Blank[]]]], Power[Plus[Optional[Pattern[c, Blank[]]], Times[Optional[Pattern[d, Blank[]]], Pattern[x, Blank[]]]], Optional[Pattern[q, Blank[]]]]], Optional[Pattern[r, Blank[]]]]]], -1], Power[Plus[Optional[Pattern[a, Blank[]]], Times[Optional[Pattern[b, Blank[]]], Pattern[x, Blank[]]]], Optional[Pattern[m, Blank[]]]], Power[Plus[Optional[Pattern[c, Blank[]]], Times[Optional[Pattern[d, Blank[]]], Pattern[x, Blank[]]]], Optional[Pattern[n, Blank[]]]]], Pattern[x, Blank[Symbol]]] := Simp[((a + b*x)^(m + 1)*(c + d*x)^(n + 1)*ExpIntegralEi[((m + 1)*Log[e*(f*(a + b*x)^p*(c + d*x)^q)^r])/(p*r)])/(p*r*(b*c - a*d)*(e*(f*(a + b*x)^p*(c + d*x)^q)^r)^((m + 1)/(p*r))), x] /; FreeQ[{a, b, c, d, e, f, m, n, p, q, r}, x] && NeQ[b*c - a*d, 0] && EqQ[p + q, 0] && EqQ[m + n + 2, 0] && NeQ[m, -1]
  2511. Int[Times[Power[Plus[Optional[Pattern[a, Blank[]]], Times[Log[Times[Optional[Pattern[c, Blank[]]], Power[Plus[Optional[Pattern[d, Blank[]]], Times[Optional[Pattern[e, Blank[]]], Pattern[x, Blank[]]]], Rational[1, 2]], Power[Plus[Optional[Pattern[f, Blank[]]], Times[Optional[Pattern[g, Blank[]]], Pattern[x, Blank[]]]], Rational[-1, 2]]]], Optional[Pattern[b, Blank[]]]]], Optional[Pattern[n, Blank[]]]], Power[Plus[Optional[Pattern[A, Blank[]]], Times[Optional[Pattern[B, Blank[]]], Pattern[x, Blank[]]], Times[Optional[Pattern[C, Blank[]]], Power[Pattern[x, Blank[]], 2]]], -1]], Pattern[x, Blank[Symbol]]] := Dist[(2*e*g)/(C*(e*f - d*g)), Subst[Int[(a + b*Log[c*x])^n/x, x], x, Sqrt[d + e*x]/Sqrt[f + g*x]], x] /; FreeQ[{a, b, c, d, e, f, g, A, B, C, n}, x] && EqQ[C*d*f - A*e*g, 0] && EqQ[B*e*g - C*(e*f + d*g), 0]
  2512. Int[Times[Power[Plus[Optional[Pattern[a, Blank[]]], Times[Log[Times[Optional[Pattern[c, Blank[]]], Power[Plus[Optional[Pattern[d, Blank[]]], Times[Optional[Pattern[e, Blank[]]], Pattern[x, Blank[]]]], Rational[1, 2]], Power[Plus[Optional[Pattern[f, Blank[]]], Times[Optional[Pattern[g, Blank[]]], Pattern[x, Blank[]]]], Rational[-1, 2]]]], Optional[Pattern[b, Blank[]]]]], Optional[Pattern[n, Blank[]]]], Power[Plus[Optional[Pattern[A, Blank[]]], Times[Optional[Pattern[C, Blank[]]], Power[Pattern[x, Blank[]], 2]]], -1]], Pattern[x, Blank[Symbol]]] := Dist[g/(C*f), Subst[Int[(a + b*Log[c*x])^n/x, x], x, Sqrt[d + e*x]/Sqrt[f + g*x]], x] /; FreeQ[{a, b, c, d, e, f, g, A, C, n}, x] && EqQ[C*d*f - A*e*g, 0] && EqQ[e*f + d*g, 0]
  2513. Int[Times[Log[Times[Optional[Pattern[e, Blank[]]], Power[Times[Optional[Pattern[f, Blank[]]], Power[Plus[Optional[Pattern[a, Blank[]]], Times[Optional[Pattern[b, Blank[]]], Pattern[x, Blank[]]]], Optional[Pattern[p, Blank[]]]], Power[Plus[Optional[Pattern[c, Blank[]]], Times[Optional[Pattern[d, Blank[]]], Pattern[x, Blank[]]]], Optional[Pattern[q, Blank[]]]]], Optional[Pattern[r, Blank[]]]]]], Optional[Pattern[RFx, Blank[]]]], Pattern[x, Blank[Symbol]]] := Dist[p*r, Int[RFx*Log[a + b*x], x], x] + (Dist[q*r, Int[RFx*Log[c + d*x], x], x] - Dist[p*r*Log[a + b*x] + q*r*Log[c + d*x] - Log[e*(f*(a + b*x)^p*(c + d*x)^q)^r], Int[RFx, x], x]) /; FreeQ[{a, b, c, d, e, f, p, q, r}, x] && RationalFunctionQ[RFx, x] && NeQ[b*c - a*d, 0] && !MatchQ[RFx, (u_.)*(a + b*x)^(m_.)*(c + d*x)^(n_.) /; IntegersQ[m, n]]
  2514. Int[Times[Power[Log[Times[Optional[Pattern[e, Blank[]]], Power[Times[Optional[Pattern[f, Blank[]]], Power[Plus[Optional[Pattern[a, Blank[]]], Times[Optional[Pattern[b, Blank[]]], Pattern[x, Blank[]]]], Optional[Pattern[p, Blank[]]]], Power[Plus[Optional[Pattern[c, Blank[]]], Times[Optional[Pattern[d, Blank[]]], Pattern[x, Blank[]]]], Optional[Pattern[q, Blank[]]]]], Optional[Pattern[r, Blank[]]]]]], Optional[Pattern[s, Blank[]]]], Pattern[RFx, Blank[]]], Pattern[x, Blank[Symbol]]] := With[{u = ExpandIntegrand[Log[e*(f*(a + b*x)^p*(c + d*x)^q)^r]^s, RFx, x]}, Int[u, x] /; SumQ[u]] /; FreeQ[{a, b, c, d, e, f, p, q, r, s}, x] && RationalFunctionQ[RFx, x] && IGtQ[s, 0]
  2515. Int[Times[Power[Log[Times[Optional[Pattern[e, Blank[]]], Power[Times[Optional[Pattern[f, Blank[]]], Power[Plus[Optional[Pattern[a, Blank[]]], Times[Optional[Pattern[b, Blank[]]], Pattern[x, Blank[]]]], Optional[Pattern[p, Blank[]]]], Power[Plus[Optional[Pattern[c, Blank[]]], Times[Optional[Pattern[d, Blank[]]], Pattern[x, Blank[]]]], Optional[Pattern[q, Blank[]]]]], Optional[Pattern[r, Blank[]]]]]], Optional[Pattern[s, Blank[]]]], Pattern[RFx, Blank[]]], Pattern[x, Blank[Symbol]]] := Unintegrable[RFx*Log[e*(f*(a + b*x)^p*(c + d*x)^q)^r]^s, x] /; FreeQ[{a, b, c, d, e, f, p, q, r, s}, x] && RationalFunctionQ[RFx, x]
  2516. Int[Times[Power[Log[Times[Optional[Pattern[e, Blank[]]], Power[Times[Optional[Pattern[f, Blank[]]], Power[Pattern[v, Blank[]], Optional[Pattern[p, Blank[]]]], Power[Pattern[w, Blank[]], Optional[Pattern[q, Blank[]]]]], Optional[Pattern[r, Blank[]]]]]], Optional[Pattern[s, Blank[]]]], Optional[Pattern[u, Blank[]]]], Pattern[x, Blank[Symbol]]] := Int[u*Log[e*(f*ExpandToSum[v, x]^p*ExpandToSum[w, x]^q)^r]^s, x] /; FreeQ[{e, f, p, q, r, s}, x] && LinearQ[{v, w}, x] && !LinearMatchQ[{v, w}, x] && AlgebraicFunctionQ[u, x]
  2517. Int[Times[Power[Log[Times[Optional[Pattern[e, Blank[]]], Power[Times[Optional[Pattern[f, Blank[]]], Plus[Pattern[g, Blank[]], Times[Optional[Pattern[v, Blank[]]], Power[Pattern[w, Blank[]], -1]]]], Optional[Pattern[r, Blank[]]]]]], Optional[Pattern[s, Blank[]]]], Optional[Pattern[u, Blank[]]]], Pattern[x, Blank[Symbol]]] := Int[u*Log[e*((f*ExpandToSum[v + g*w, x])/ExpandToSum[w, x])^r]^s, x] /; FreeQ[{e, f, g, r, s}, x] && LinearQ[w, x] && (FreeQ[v, x] || LinearQ[v, x]) && AlgebraicFunctionQ[u, x]
  2518. Int[Times[Log[Pattern[v, Blank[]]], Pattern[u, Blank[]]], Pattern[x, Blank[Symbol]]] := With[{w = DerivativeDivides[v, u*(1 - v), x]}, Simp[w*PolyLog[2, 1 - v], x] /; !FalseQ[w]]
  2519. Int[Times[Log[Pattern[v, Blank[]]], Plus[Optional[Pattern[a, Blank[]]], Times[Log[Pattern[u, Blank[]]], Optional[Pattern[b, Blank[]]]]], Pattern[w, Blank[]]], Pattern[x, Blank[Symbol]]] := With[{z = DerivativeDivides[v, w*(1 - v), x]}, Simp[z*(a + b*Log[u])*PolyLog[2, 1 - v], x] - Dist[b, Int[SimplifyIntegrand[(z*PolyLog[2, 1 - v]*D[u, x])/u, x], x], x] /; !FalseQ[z]] /; FreeQ[{a, b}, x] && InverseFunctionFreeQ[u, x]
  2520. Int[Log[Times[Power[Log[Times[Optional[Pattern[d, Blank[]]], Power[Pattern[x, Blank[]], Optional[Pattern[n, Blank[]]]]]], Optional[Pattern[p, Blank[]]]], Optional[Pattern[c, Blank[]]]]], Pattern[x, Blank[Symbol]]] := Simp[x*Log[c*Log[d*x^n]^p], x] - Dist[n*p, Int[1/Log[d*x^n], x], x] /; FreeQ[{c, d, n, p}, x]
  2521. Int[Times[Plus[Optional[Pattern[a, Blank[]]], Times[Log[Times[Power[Log[Times[Optional[Pattern[d, Blank[]]], Power[Pattern[x, Blank[]], Optional[Pattern[n, Blank[]]]]]], Optional[Pattern[p, Blank[]]]], Optional[Pattern[c, Blank[]]]]], Optional[Pattern[b, Blank[]]]]], Power[Pattern[x, Blank[]], -1]], Pattern[x, Blank[Symbol]]] := Simp[(Log[d*x^n]*(a + b*Log[c*Log[d*x^n]^p]))/n, x] - Simp[b*p*Log[x], x] /; FreeQ[{a, b, c, d, n, p}, x]
  2522. Int[Times[Plus[Optional[Pattern[a, Blank[]]], Times[Log[Times[Power[Log[Times[Optional[Pattern[d, Blank[]]], Power[Pattern[x, Blank[]], Optional[Pattern[n, Blank[]]]]]], Optional[Pattern[p, Blank[]]]], Optional[Pattern[c, Blank[]]]]], Optional[Pattern[b, Blank[]]]]], Power[Times[Optional[Pattern[e, Blank[]]], Pattern[x, Blank[]]], Optional[Pattern[m, Blank[]]]]], Pattern[x, Blank[Symbol]]] := Simp[((e*x)^(m + 1)*(a + b*Log[c*Log[d*x^n]^p]))/(e*(m + 1)), x] - Dist[(b*n*p)/(m + 1), Int[(e*x)^m/Log[d*x^n], x], x] /; FreeQ[{a, b, c, d, e, m, n, p}, x] && NeQ[m, -1]
  2523. Int[Power[Plus[Optional[Pattern[a, Blank[]]], Times[Log[Times[Optional[Pattern[c, Blank[]]], Power[Pattern[RFx, Blank[]], Optional[Pattern[p, Blank[]]]]]], Optional[Pattern[b, Blank[]]]]], Optional[Pattern[n, Blank[]]]], Pattern[x, Blank[Symbol]]] := Simp[x*(a + b*Log[c*RFx^p])^n, x] - Dist[b*n*p, Int[SimplifyIntegrand[(x*(a + b*Log[c*RFx^p])^(n - 1)*D[RFx, x])/RFx, x], x], x] /; FreeQ[{a, b, c, p}, x] && RationalFunctionQ[RFx, x] && IGtQ[n, 0]
  2524. Int[Times[Power[Plus[Optional[Pattern[a, Blank[]]], Times[Log[Times[Optional[Pattern[c, Blank[]]], Power[Pattern[RFx, Blank[]], Optional[Pattern[p, Blank[]]]]]], Optional[Pattern[b, Blank[]]]]], Optional[Pattern[n, Blank[]]]], Power[Plus[Optional[Pattern[d, Blank[]]], Times[Optional[Pattern[e, Blank[]]], Pattern[x, Blank[]]]], -1]], Pattern[x, Blank[Symbol]]] := Simp[(Log[d + e*x]*(a + b*Log[c*RFx^p])^n)/e, x] - Dist[(b*n*p)/e, Int[(Log[d + e*x]*(a + b*Log[c*RFx^p])^(n - 1)*D[RFx, x])/RFx, x], x] /; FreeQ[{a, b, c, d, e, p}, x] && RationalFunctionQ[RFx, x] && IGtQ[n, 0]
  2525. Int[Times[Power[Plus[Optional[Pattern[a, Blank[]]], Times[Log[Times[Optional[Pattern[c, Blank[]]], Power[Pattern[RFx, Blank[]], Optional[Pattern[p, Blank[]]]]]], Optional[Pattern[b, Blank[]]]]], Optional[Pattern[n, Blank[]]]], Power[Plus[Optional[Pattern[d, Blank[]]], Times[Optional[Pattern[e, Blank[]]], Pattern[x, Blank[]]]], Optional[Pattern[m, Blank[]]]]], Pattern[x, Blank[Symbol]]] := Simp[((d + e*x)^(m + 1)*(a + b*Log[c*RFx^p])^n)/(e*(m + 1)), x] - Dist[(b*n*p)/(e*(m + 1)), Int[SimplifyIntegrand[((d + e*x)^(m + 1)*(a + b*Log[c*RFx^p])^(n - 1)*D[RFx, x])/RFx, x], x], x] /; FreeQ[{a, b, c, d, e, m, p}, x] && RationalFunctionQ[RFx, x] && IGtQ[n, 0] && (EqQ[n, 1] || IntegerQ[m]) && NeQ[m, -1]
  2526. Int[Times[Log[Times[Optional[Pattern[c, Blank[]]], Power[Pattern[RFx, Blank[]], Optional[Pattern[n, Blank[]]]]]], Power[Plus[Pattern[d, Blank[]], Times[Optional[Pattern[e, Blank[]]], Power[Pattern[x, Blank[]], 2]]], -1]], Pattern[x, Blank[Symbol]]] := With[{u = IntHide[1/(d + e*x^2), x]}, Simp[u*Log[c*RFx^n], x] - Dist[n, Int[SimplifyIntegrand[(u*D[RFx, x])/RFx, x], x], x]] /; FreeQ[{c, d, e, n}, x] && RationalFunctionQ[RFx, x] && !PolynomialQ[RFx, x]
  2527. Int[Times[Log[Times[Optional[Pattern[c, Blank[]]], Power[Pattern[Px, Blank[]], Optional[Pattern[n, Blank[]]]]]], Power[Pattern[Qx, Blank[]], -1]], Pattern[x, Blank[Symbol]]] := With[{u = IntHide[1/Qx, x]}, Simp[u*Log[c*Px^n], x] - Dist[n, Int[SimplifyIntegrand[(u*D[Px, x])/Px, x], x], x]] /; FreeQ[{c, n}, x] && QuadraticQ[{Qx, Px}, x] && EqQ[D[Px/Qx, x], 0]
  2528. Int[Times[Power[Plus[Optional[Pattern[a, Blank[]]], Times[Log[Times[Optional[Pattern[c, Blank[]]], Power[Pattern[RFx, Blank[]], Optional[Pattern[p, Blank[]]]]]], Optional[Pattern[b, Blank[]]]]], Optional[Pattern[n, Blank[]]]], Pattern[RGx, Blank[]]], Pattern[x, Blank[Symbol]]] := With[{u = ExpandIntegrand[(a + b*Log[c*RFx^p])^n, RGx, x]}, Int[u, x] /; SumQ[u]] /; FreeQ[{a, b, c, p}, x] && RationalFunctionQ[RFx, x] && RationalFunctionQ[RGx, x] && IGtQ[n, 0]
  2529. Int[Times[Power[Plus[Optional[Pattern[a, Blank[]]], Times[Log[Times[Optional[Pattern[c, Blank[]]], Power[Pattern[RFx, Blank[]], Optional[Pattern[p, Blank[]]]]]], Optional[Pattern[b, Blank[]]]]], Optional[Pattern[n, Blank[]]]], Pattern[RGx, Blank[]]], Pattern[x, Blank[Symbol]]] := With[{u = ExpandIntegrand[RGx*(a + b*Log[c*RFx^p])^n, x]}, Int[u, x] /; SumQ[u]] /; FreeQ[{a, b, c, p}, x] && RationalFunctionQ[RFx, x] && RationalFunctionQ[RGx, x] && IGtQ[n, 0]
  2530. Int[Times[Plus[Optional[Pattern[a, Blank[]]], Times[Log[Pattern[u, Blank[]]], Optional[Pattern[b, Blank[]]]]], Pattern[RFx, Blank[]]], Pattern[x, Blank[Symbol]]] := With[{lst = SubstForFractionalPowerOfLinear[RFx*(a + b*Log[u]), x]}, Dist[lst[[2]]*lst[[4]], Subst[Int[lst[[1]], x], x, lst[[3]]^(1/lst[[2]])], x] /; !FalseQ[lst]] /; FreeQ[{a, b}, x] && RationalFunctionQ[RFx, x]
  2531. Int[Times[Log[Plus[1, Times[Optional[Pattern[e, Blank[]]], Power[Power[Pattern[F, Blank[]], Times[Optional[Pattern[c, Blank[]]], Plus[Optional[Pattern[a, Blank[]]], Times[Optional[Pattern[b, Blank[]]], Pattern[x, Blank[]]]]]], Optional[Pattern[n, Blank[]]]]]]], Power[Plus[Optional[Pattern[f, Blank[]]], Times[Optional[Pattern[g, Blank[]]], Pattern[x, Blank[]]]], Optional[Pattern[m, Blank[]]]]], Pattern[x, Blank[Symbol]]] := -Simp[((f + g*x)^m*PolyLog[2, -(e*(F^(c*(a + b*x)))^n)])/(b*c*n*Log[F]), x] + Dist[(g*m)/(b*c*n*Log[F]), Int[(f + g*x)^(m - 1)*PolyLog[2, -(e*(F^(c*(a + b*x)))^n)], x], x] /; FreeQ[{F, a, b, c, e, f, g, n}, x] && GtQ[m, 0]
  2532. Int[Times[Log[Plus[Pattern[d, Blank[]], Times[Optional[Pattern[e, Blank[]]], Power[Power[Pattern[F, Blank[]], Times[Optional[Pattern[c, Blank[]]], Plus[Optional[Pattern[a, Blank[]]], Times[Optional[Pattern[b, Blank[]]], Pattern[x, Blank[]]]]]], Optional[Pattern[n, Blank[]]]]]]], Power[Plus[Optional[Pattern[f, Blank[]]], Times[Optional[Pattern[g, Blank[]]], Pattern[x, Blank[]]]], Optional[Pattern[m, Blank[]]]]], Pattern[x, Blank[Symbol]]] := Simp[((f + g*x)^(m + 1)*Log[d + e*(F^(c*(a + b*x)))^n])/(g*(m + 1)), x] + (Int[(f + g*x)^m*Log[1 + (e*(F^(c*(a + b*x)))^n)/d], x] - Simp[((f + g*x)^(m + 1)*Log[1 + (e*(F^(c*(a + b*x)))^n)/d])/(g*(m + 1)), x]) /; FreeQ[{F, a, b, c, d, e, f, g, n}, x] && GtQ[m, 0] && NeQ[d, 1]
  2533. Int[Log[Plus[Optional[Pattern[d, Blank[]]], Times[Optional[Pattern[e, Blank[]]], Pattern[x, Blank[]]], Times[Optional[Pattern[f, Blank[]]], Power[Plus[Optional[Pattern[a, Blank[]]], Times[Optional[Pattern[b, Blank[]]], Pattern[x, Blank[]]], Times[Optional[Pattern[c, Blank[]]], Power[Pattern[x, Blank[]], 2]]], Rational[1, 2]]]]], Pattern[x, Blank[Symbol]]] := Simp[x*Log[d + e*x + f*Sqrt[a + b*x + c*x^2]], x] + Dist[(f^2*(b^2 - 4*a*c))/2, Int[x/((2*d*e - b*f^2)*(a + b*x + c*x^2) - f*(b*d - 2*a*e + (2*c*d - b*e)*x)*Sqrt[a + b*x + c*x^2]), x], x] /; FreeQ[{a, b, c, d, e, f}, x] && EqQ[e^2 - c*f^2, 0]
  2534. Int[Log[Plus[Optional[Pattern[d, Blank[]]], Times[Optional[Pattern[e, Blank[]]], Pattern[x, Blank[]]], Times[Optional[Pattern[f, Blank[]]], Power[Plus[Optional[Pattern[a, Blank[]]], Times[Optional[Pattern[c, Blank[]]], Power[Pattern[x, Blank[]], 2]]], Rational[1, 2]]]]], Pattern[x, Blank[Symbol]]] := Simp[x*Log[d + e*x + f*Sqrt[a + c*x^2]], x] - Dist[a*c*f^2, Int[x/(d*e*(a + c*x^2) + f*(a*e - c*d*x)*Sqrt[a + c*x^2]), x], x] /; FreeQ[{a, c, d, e, f}, x] && EqQ[e^2 - c*f^2, 0]
  2535. Int[Times[Log[Plus[Optional[Pattern[d, Blank[]]], Times[Optional[Pattern[e, Blank[]]], Pattern[x, Blank[]]], Times[Optional[Pattern[f, Blank[]]], Power[Plus[Optional[Pattern[a, Blank[]]], Times[Optional[Pattern[b, Blank[]]], Pattern[x, Blank[]]], Times[Optional[Pattern[c, Blank[]]], Power[Pattern[x, Blank[]], 2]]], Rational[1, 2]]]]], Power[Times[Optional[Pattern[g, Blank[]]], Pattern[x, Blank[]]], Optional[Pattern[m, Blank[]]]]], Pattern[x, Blank[Symbol]]] := Simp[((g*x)^(m + 1)*Log[d + e*x + f*Sqrt[a + b*x + c*x^2]])/(g*(m + 1)), x] + Dist[(f^2*(b^2 - 4*a*c))/(2*g*(m + 1)), Int[(g*x)^(m + 1)/((2*d*e - b*f^2)*(a + b*x + c*x^2) - f*(b*d - 2*a*e + (2*c*d - b*e)*x)*Sqrt[a + b*x + c*x^2]), x], x] /; FreeQ[{a, b, c, d, e, f, g, m}, x] && EqQ[e^2 - c*f^2, 0] && NeQ[m, -1] && IntegerQ[2*m]
  2536. Int[Times[Log[Plus[Optional[Pattern[d, Blank[]]], Times[Optional[Pattern[e, Blank[]]], Pattern[x, Blank[]]], Times[Optional[Pattern[f, Blank[]]], Power[Plus[Optional[Pattern[a, Blank[]]], Times[Optional[Pattern[c, Blank[]]], Power[Pattern[x, Blank[]], 2]]], Rational[1, 2]]]]], Power[Times[Optional[Pattern[g, Blank[]]], Pattern[x, Blank[]]], Optional[Pattern[m, Blank[]]]]], Pattern[x, Blank[Symbol]]] := Simp[((g*x)^(m + 1)*Log[d + e*x + f*Sqrt[a + c*x^2]])/(g*(m + 1)), x] - Dist[(a*c*f^2)/(g*(m + 1)), Int[(g*x)^(m + 1)/(d*e*(a + c*x^2) + f*(a*e - c*d*x)*Sqrt[a + c*x^2]), x], x] /; FreeQ[{a, c, d, e, f, g, m}, x] && EqQ[e^2 - c*f^2, 0] && NeQ[m, -1] && IntegerQ[2*m]
  2537. Int[Times[Log[Plus[Optional[Pattern[d, Blank[]]], Times[Optional[Pattern[f, Blank[]]], Power[Pattern[u, Blank[]], Rational[1, 2]]], Times[Optional[Pattern[e, Blank[]]], Pattern[x, Blank[]]]]], Optional[Pattern[v, Blank[]]]], Pattern[x, Blank[Symbol]]] := Int[v*Log[d + e*x + f*Sqrt[ExpandToSum[u, x]]], x] /; FreeQ[{d, e, f}, x] && QuadraticQ[u, x] && !QuadraticMatchQ[u, x] && (EqQ[v, 1] || MatchQ[v, ((g_.)*x)^(m_.) /; FreeQ[{g, m}, x]])
  2538. Int[Times[Power[Log[Times[Optional[Pattern[c, Blank[]]], Power[Pattern[x, Blank[]], Optional[Pattern[n, Blank[]]]]]], Optional[Pattern[r, Blank[]]]], Power[Pattern[x, Blank[]], -1], Power[Plus[Times[Power[Log[Times[Optional[Pattern[c, Blank[]]], Power[Pattern[x, Blank[]], Optional[Pattern[n, Blank[]]]]]], Pattern[q, Blank[]]], Optional[Pattern[b, Blank[]]]], Times[Optional[Pattern[a, Blank[]]], Power[Pattern[x, Blank[]], Optional[Pattern[m, Blank[]]]]]], -1]], Pattern[x, Blank[Symbol]]] := Simp[Log[a*x^m + b*Log[c*x^n]^q]/(b*n*q), x] - Dist[(a*m)/(b*n*q), Int[x^(m - 1)/(a*x^m + b*Log[c*x^n]^q), x], x] /; FreeQ[{a, b, c, m, n, q, r}, x] && EqQ[r, q - 1]
  2539. Int[Times[Power[Log[Times[Optional[Pattern[c, Blank[]]], Power[Pattern[x, Blank[]], Optional[Pattern[n, Blank[]]]]]], Optional[Pattern[r, Blank[]]]], Power[Pattern[x, Blank[]], -1], Power[Plus[Times[Power[Log[Times[Optional[Pattern[c, Blank[]]], Power[Pattern[x, Blank[]], Optional[Pattern[n, Blank[]]]]]], Pattern[q, Blank[]]], Optional[Pattern[b, Blank[]]]], Times[Optional[Pattern[a, Blank[]]], Power[Pattern[x, Blank[]], Optional[Pattern[m, Blank[]]]]]], Optional[Pattern[p, Blank[]]]]], Pattern[x, Blank[Symbol]]] := Int[ExpandIntegrand[Log[c*x^n]^r/x, (a*x^m + b*Log[c*x^n]^q)^p, x], x] /; FreeQ[{a, b, c, m, n, p, q, r}, x] && EqQ[r, q - 1] && IGtQ[p, 0]
  2540. Int[Times[Power[Log[Times[Optional[Pattern[c, Blank[]]], Power[Pattern[x, Blank[]], Optional[Pattern[n, Blank[]]]]]], Optional[Pattern[r, Blank[]]]], Power[Pattern[x, Blank[]], -1], Power[Plus[Times[Power[Log[Times[Optional[Pattern[c, Blank[]]], Power[Pattern[x, Blank[]], Optional[Pattern[n, Blank[]]]]]], Pattern[q, Blank[]]], Optional[Pattern[b, Blank[]]]], Times[Optional[Pattern[a, Blank[]]], Power[Pattern[x, Blank[]], Optional[Pattern[m, Blank[]]]]]], Optional[Pattern[p, Blank[]]]]], Pattern[x, Blank[Symbol]]] := Simp[(a*x^m + b*Log[c*x^n]^q)^(p + 1)/(b*n*q*(p + 1)), x] - Dist[(a*m)/(b*n*q), Int[x^(m - 1)*(a*x^m + b*Log[c*x^n]^q)^p, x], x] /; FreeQ[{a, b, c, m, n, p, q, r}, x] && EqQ[r, q - 1] && NeQ[p, -1]
  2541. Int[Times[Power[Pattern[x, Blank[]], -1], Power[Plus[Times[Power[Log[Times[Optional[Pattern[c, Blank[]]], Power[Pattern[x, Blank[]], Optional[Pattern[n, Blank[]]]]]], Pattern[q, Blank[]]], Optional[Pattern[b, Blank[]]]], Times[Optional[Pattern[a, Blank[]]], Power[Pattern[x, Blank[]], Optional[Pattern[m, Blank[]]]]]], -1], Plus[Times[Power[Log[Times[Optional[Pattern[c, Blank[]]], Power[Pattern[x, Blank[]], Optional[Pattern[n, Blank[]]]]]], Optional[Pattern[r, Blank[]]]], Optional[Pattern[e, Blank[]]]], Times[Optional[Pattern[d, Blank[]]], Power[Pattern[x, Blank[]], Optional[Pattern[m, Blank[]]]]]]], Pattern[x, Blank[Symbol]]] := Simp[(e*Log[a*x^m + b*Log[c*x^n]^q])/(b*n*q), x] /; FreeQ[{a, b, c, d, e, m, n, q, r}, x] && EqQ[r, q - 1] && EqQ[a*e*m - b*d*n*q, 0]
  2542. Int[Times[Power[Pattern[x, Blank[]], -1], Power[Plus[Times[Power[Log[Times[Optional[Pattern[c, Blank[]]], Power[Pattern[x, Blank[]], Optional[Pattern[n, Blank[]]]]]], Pattern[q, Blank[]]], Optional[Pattern[b, Blank[]]]], Times[Optional[Pattern[a, Blank[]]], Power[Pattern[x, Blank[]], Optional[Pattern[m, Blank[]]]]]], -1], Plus[Times[Power[Log[Times[Optional[Pattern[c, Blank[]]], Power[Pattern[x, Blank[]], Optional[Pattern[n, Blank[]]]]]], Optional[Pattern[r, Blank[]]]], Optional[Pattern[e, Blank[]]]], Pattern[u, Blank[]], Times[Optional[Pattern[d, Blank[]]], Power[Pattern[x, Blank[]], Optional[Pattern[m, Blank[]]]]]]], Pattern[x, Blank[Symbol]]] := Simp[(e*Log[a*x^m + b*Log[c*x^n]^q])/(b*n*q), x] + Int[u/(x*(a*x^m + b*Log[c*x^n]^q)), x] /; FreeQ[{a, b, c, d, e, m, n, q, r}, x] && EqQ[r, q - 1] && EqQ[a*e*m - b*d*n*q, 0]
  2543. Int[Times[Power[Pattern[x, Blank[]], -1], Power[Plus[Times[Power[Log[Times[Optional[Pattern[c, Blank[]]], Power[Pattern[x, Blank[]], Optional[Pattern[n, Blank[]]]]]], Pattern[q, Blank[]]], Optional[Pattern[b, Blank[]]]], Times[Optional[Pattern[a, Blank[]]], Power[Pattern[x, Blank[]], Optional[Pattern[m, Blank[]]]]]], -1], Plus[Times[Power[Log[Times[Optional[Pattern[c, Blank[]]], Power[Pattern[x, Blank[]], Optional[Pattern[n, Blank[]]]]]], Optional[Pattern[r, Blank[]]]], Optional[Pattern[e, Blank[]]]], Times[Optional[Pattern[d, Blank[]]], Power[Pattern[x, Blank[]], Optional[Pattern[m, Blank[]]]]]]], Pattern[x, Blank[Symbol]]] := Simp[(e*Log[a*x^m + b*Log[c*x^n]^q])/(b*n*q), x] - Dist[(a*e*m - b*d*n*q)/(b*n*q), Int[x^(m - 1)/(a*x^m + b*Log[c*x^n]^q), x], x] /; FreeQ[{a, b, c, d, e, m, n, q, r}, x] && EqQ[r, q - 1] && NeQ[a*e*m - b*d*n*q, 0]
  2544. Int[Times[Power[Pattern[x, Blank[]], -1], Power[Plus[Times[Power[Log[Times[Optional[Pattern[c, Blank[]]], Power[Pattern[x, Blank[]], Optional[Pattern[n, Blank[]]]]]], Pattern[q, Blank[]]], Optional[Pattern[b, Blank[]]]], Times[Optional[Pattern[a, Blank[]]], Power[Pattern[x, Blank[]], Optional[Pattern[m, Blank[]]]]]], Optional[Pattern[p, Blank[]]]], Plus[Times[Power[Log[Times[Optional[Pattern[c, Blank[]]], Power[Pattern[x, Blank[]], Optional[Pattern[n, Blank[]]]]]], Optional[Pattern[r, Blank[]]]], Optional[Pattern[e, Blank[]]]], Times[Optional[Pattern[d, Blank[]]], Power[Pattern[x, Blank[]], Optional[Pattern[m, Blank[]]]]]]], Pattern[x, Blank[Symbol]]] := Simp[(e*(a*x^m + b*Log[c*x^n]^q)^(p + 1))/(b*n*q*(p + 1)), x] /; FreeQ[{a, b, c, d, e, m, n, p, q, r}, x] && EqQ[r, q - 1] && NeQ[p, -1] && EqQ[a*e*m - b*d*n*q, 0]
  2545. Int[Times[Power[Pattern[x, Blank[]], -1], Power[Plus[Times[Power[Log[Times[Optional[Pattern[c, Blank[]]], Power[Pattern[x, Blank[]], Optional[Pattern[n, Blank[]]]]]], Pattern[q, Blank[]]], Optional[Pattern[b, Blank[]]]], Times[Optional[Pattern[a, Blank[]]], Power[Pattern[x, Blank[]], Optional[Pattern[m, Blank[]]]]]], Optional[Pattern[p, Blank[]]]], Plus[Times[Power[Log[Times[Optional[Pattern[c, Blank[]]], Power[Pattern[x, Blank[]], Optional[Pattern[n, Blank[]]]]]], Optional[Pattern[r, Blank[]]]], Optional[Pattern[e, Blank[]]]], Times[Optional[Pattern[d, Blank[]]], Power[Pattern[x, Blank[]], Optional[Pattern[m, Blank[]]]]]]], Pattern[x, Blank[Symbol]]] := Simp[(e*(a*x^m + b*Log[c*x^n]^q)^(p + 1))/(b*n*q*(p + 1)), x] - Dist[(a*e*m - b*d*n*q)/(b*n*q), Int[x^(m - 1)*(a*x^m + b*Log[c*x^n]^q)^p, x], x] /; FreeQ[{a, b, c, d, e, m, n, p, q, r}, x] && EqQ[r, q - 1] && NeQ[p, -1] && NeQ[a*e*m - b*d*n*q, 0]
  2546. Int[Times[Power[Pattern[x, Blank[]], -1], Power[Plus[Times[Power[Log[Times[Optional[Pattern[c, Blank[]]], Power[Pattern[x, Blank[]], Optional[Pattern[n, Blank[]]]]]], Pattern[q, Blank[]]], Optional[Pattern[b, Blank[]]]], Times[Optional[Pattern[a, Blank[]]], Power[Pattern[x, Blank[]], Optional[Pattern[m, Blank[]]]]]], -2], Plus[Times[Power[Log[Times[Optional[Pattern[c, Blank[]]], Power[Pattern[x, Blank[]], Optional[Pattern[n, Blank[]]]]]], Optional[Pattern[q, Blank[]]]], Optional[Pattern[f, Blank[]]]], Times[Optional[Pattern[d, Blank[]]], Power[Pattern[x, Blank[]], Optional[Pattern[m, Blank[]]]]], Times[Log[Times[Optional[Pattern[c, Blank[]]], Power[Pattern[x, Blank[]], Optional[Pattern[n, Blank[]]]]]], Optional[Pattern[e, Blank[]]], Power[Pattern[x, Blank[]], Optional[Pattern[m, Blank[]]]]]]], Pattern[x, Blank[Symbol]]] := Simp[(d*Log[c*x^n])/(a*n*(a*x^m + b*Log[c*x^n]^q)), x] /; FreeQ[{a, b, c, d, e, f, m, n, q}, x] && EqQ[e*n + d*m, 0] && EqQ[a*f + b*d*(q - 1), 0]
  2547. Int[Times[Plus[Times[Log[Times[Optional[Pattern[c, Blank[]]], Power[Pattern[x, Blank[]], Optional[Pattern[n, Blank[]]]]]], Optional[Pattern[e, Blank[]]]], Pattern[d, Blank[]]], Power[Plus[Times[Power[Log[Times[Optional[Pattern[c, Blank[]]], Power[Pattern[x, Blank[]], Optional[Pattern[n, Blank[]]]]]], Pattern[q, Blank[]]], Optional[Pattern[b, Blank[]]]], Times[Optional[Pattern[a, Blank[]]], Pattern[x, Blank[]]]], -2]], Pattern[x, Blank[Symbol]]] := -Simp[(e*Log[c*x^n])/(a*(a*x + b*Log[c*x^n]^q)), x] + Dist[(d + e*n)/a, Int[1/(x*(a*x + b*Log[c*x^n]^q)), x], x] /; FreeQ[{a, b, c, d, e, n, q}, x] && EqQ[d + e*n*q, 0]
  2548. Int[Log[Pattern[u, Blank[]]], Pattern[x, Blank[Symbol]]] := Simp[x*Log[u], x] - Int[SimplifyIntegrand[(x*D[u, x])/u, x], x] /; InverseFunctionFreeQ[u, x]
  2549. Int[Log[Pattern[u, Blank[]]], Pattern[x, Blank[Symbol]]] := Simp[x*Log[u], x] - Int[SimplifyIntegrand[x*Simplify[D[u, x]/u], x], x] /; ProductQ[u]
  2550. Int[Times[Log[Pattern[u, Blank[]]], Power[Plus[Optional[Pattern[a, Blank[]]], Times[Optional[Pattern[b, Blank[]]], Pattern[x, Blank[]]]], -1]], Pattern[x, Blank[Symbol]]] := Simp[(Log[a + b*x]*Log[u])/b, x] - Dist[1/b, Int[SimplifyIntegrand[(Log[a + b*x]*D[u, x])/u, x], x], x] /; FreeQ[{a, b}, x] && RationalFunctionQ[D[u, x]/u, x] && (NeQ[a, 0] || !(BinomialQ[u, x] && EqQ[BinomialDegree[u, x]^2, 1]))
  2551. Int[Times[Log[Pattern[u, Blank[]]], Power[Plus[Optional[Pattern[a, Blank[]]], Times[Optional[Pattern[b, Blank[]]], Pattern[x, Blank[]]]], Optional[Pattern[m, Blank[]]]]], Pattern[x, Blank[Symbol]]] := Simp[((a + b*x)^(m + 1)*Log[u])/(b*(m + 1)), x] - Dist[1/(b*(m + 1)), Int[SimplifyIntegrand[((a + b*x)^(m + 1)*D[u, x])/u, x], x], x] /; FreeQ[{a, b, m}, x] && InverseFunctionFreeQ[u, x] && NeQ[m, -1]
  2552. Int[Times[Log[Pattern[u, Blank[]]], Power[Pattern[Qx, Blank[]], -1]], Pattern[x, Blank[Symbol]]] := With[{v = IntHide[1/Qx, x]}, Simp[v*Log[u], x] - Int[SimplifyIntegrand[(v*D[u, x])/u, x], x]] /; QuadraticQ[Qx, x] && InverseFunctionFreeQ[u, x]
  2553. Int[Times[Log[Pattern[u, Blank[]]], Power[Pattern[u, Blank[]], Times[Optional[Pattern[a, Blank[]]], Pattern[x, Blank[]]]]], Pattern[x, Blank[Symbol]]] := Simp[u^(a*x)/a, x] - Int[SimplifyIntegrand[x*u^(a*x - 1)*D[u, x], x], x] /; FreeQ[a, x] && InverseFunctionFreeQ[u, x]
  2554. Int[Times[Log[Pattern[u, Blank[]]], Pattern[v, Blank[]]], Pattern[x, Blank[Symbol]]] := With[{w = IntHide[v, x]}, Dist[Log[u], w, x] - Int[SimplifyIntegrand[(w*D[u, x])/u, x], x] /; InverseFunctionFreeQ[w, x]] /; InverseFunctionFreeQ[u, x]
  2555. Int[Times[Log[Pattern[u, Blank[]]], Pattern[v, Blank[]]], Pattern[x, Blank[Symbol]]] := With[{w = IntHide[v, x]}, Dist[Log[u], w, x] - Int[SimplifyIntegrand[w*Simplify[D[u, x]/u], x], x] /; InverseFunctionFreeQ[w, x]] /; ProductQ[u]
  2556. Int[Times[Log[Pattern[v, Blank[]]], Log[Pattern[w, Blank[]]]], Pattern[x, Blank[Symbol]]] := Simp[x*Log[v]*Log[w], x] + (-Int[SimplifyIntegrand[(x*Log[w]*D[v, x])/v, x], x] - Int[SimplifyIntegrand[(x*Log[v]*D[w, x])/w, x], x]) /; InverseFunctionFreeQ[v, x] && InverseFunctionFreeQ[w, x]
  2557. Int[Times[Log[Pattern[v, Blank[]]], Log[Pattern[w, Blank[]]], Pattern[u, Blank[]]], Pattern[x, Blank[Symbol]]] := With[{z = IntHide[u, x]}, Dist[Log[v]*Log[w], z, x] + (-Int[SimplifyIntegrand[(z*Log[w]*D[v, x])/v, x], x] - Int[SimplifyIntegrand[(z*Log[v]*D[w, x])/w, x], x]) /; InverseFunctionFreeQ[z, x]] /; InverseFunctionFreeQ[v, x] && InverseFunctionFreeQ[w, x]
  2558. Int[Power[Pattern[f, Blank[]], Times[Log[Pattern[u, Blank[]]], Optional[Pattern[a, Blank[]]]]], Pattern[x, Blank[Symbol]]] := Int[u^(a*Log[f]), x] /; FreeQ[{a, f}, x]
  2559. Int[Pattern[u, Blank[]], Pattern[x, Blank[Symbol]]] := With[{lst = FunctionOfLog[Cancel[x*u], x]}, Dist[1/lst[[3]], Subst[Int[lst[[1]], x], x, Log[lst[[2]]]], x] /; !FalseQ[lst]] /; NonsumQ[u]
  2560. Int[Times[Log[Gamma[Pattern[v, Blank[]]]], Optional[Pattern[u, Blank[]]]], Pattern[x, Blank[Symbol]]] := Dist[Log[Gamma[v]] - LogGamma[v], Int[u, x], x] + Int[u*LogGamma[v], x]
  2561. Int[Times[Optional[Pattern[u, Blank[]]], Power[Plus[Times[Optional[Pattern[a, Blank[]]], Power[Pattern[x, Blank[]], Optional[Pattern[m, Blank[]]]]], Times[Power[Log[Times[Optional[Pattern[c, Blank[]]], Power[Pattern[x, Blank[]], Optional[Pattern[n, Blank[]]]]]], Optional[Pattern[q, Blank[]]]], Optional[Pattern[b, Blank[]]], Power[Pattern[x, Blank[]], Optional[Pattern[r, Blank[]]]]]], Optional[Pattern[p, Blank[]]]]], Pattern[x, Blank[Symbol]]] := Int[u*x^(p*r)*(a*x^(m - r) + b*Log[c*x^n]^q)^p, x] /; FreeQ[{a, b, c, m, n, p, q, r}, x] && IntegerQ[p]
  2562. Int[Pattern[u, Blank[]], Pattern[x, Blank[Symbol]]] := Int[DeactivateTrig[u, x], x] /; FunctionOfTrigOfLinearQ[u, x]
  2563. Int[Times[Power[Times[cos[Plus[Optional[Pattern[e, Blank[]]], Times[Optional[Pattern[f, Blank[]]], Pattern[x, Blank[]]]]], Optional[Pattern[b, Blank[]]]], Optional[Pattern[n, Blank[]]]], Power[Times[Optional[Pattern[a, Blank[]]], sin[Plus[Optional[Pattern[e, Blank[]]], Times[Optional[Pattern[f, Blank[]]], Pattern[x, Blank[]]]]]], Optional[Pattern[m, Blank[]]]]], Pattern[x, Blank[Symbol]]] := Simp[((a*Sin[e + f*x])^(m + 1)*(b*Cos[e + f*x])^(n + 1))/(a*b*f*(m + 1)), x] /; FreeQ[{a, b, e, f, m, n}, x] && EqQ[m + n + 2, 0] && NeQ[m, -1]
  2564. Int[Times[Power[cos[Plus[Optional[Pattern[e, Blank[]]], Times[Optional[Pattern[f, Blank[]]], Pattern[x, Blank[]]]]], Optional[Pattern[n, Blank[]]]], Power[Times[Optional[Pattern[a, Blank[]]], sin[Plus[Optional[Pattern[e, Blank[]]], Times[Optional[Pattern[f, Blank[]]], Pattern[x, Blank[]]]]]], Optional[Pattern[m, Blank[]]]]], Pattern[x, Blank[Symbol]]] := Dist[1/(a*f), Subst[Int[x^m*(1 - x^2/a^2)^((n - 1)/2), x], x, a*Sin[e + f*x]], x] /; FreeQ[{a, e, f, m}, x] && IntegerQ[(n - 1)/2] && !(IntegerQ[(m - 1)/2] && LtQ[0, m, n])
  2565. Int[Times[Power[Times[cos[Plus[Optional[Pattern[e, Blank[]]], Times[Optional[Pattern[f, Blank[]]], Pattern[x, Blank[]]]]], Optional[Pattern[a, Blank[]]]], Optional[Pattern[m, Blank[]]]], Power[sin[Plus[Optional[Pattern[e, Blank[]]], Times[Optional[Pattern[f, Blank[]]], Pattern[x, Blank[]]]]], Optional[Pattern[n, Blank[]]]]], Pattern[x, Blank[Symbol]]] := -Dist[(a*f)^(-1), Subst[Int[x^m*(1 - x^2/a^2)^((n - 1)/2), x], x, a*Cos[e + f*x]], x] /; FreeQ[{a, e, f, m}, x] && IntegerQ[(n - 1)/2] && !(IntegerQ[(m - 1)/2] && GtQ[m, 0] && LeQ[m, n])
  2566. Int[Times[Power[Times[cos[Plus[Optional[Pattern[e, Blank[]]], Times[Optional[Pattern[f, Blank[]]], Pattern[x, Blank[]]]]], Optional[Pattern[b, Blank[]]]], Pattern[n, Blank[]]], Power[Times[Optional[Pattern[a, Blank[]]], sin[Plus[Optional[Pattern[e, Blank[]]], Times[Optional[Pattern[f, Blank[]]], Pattern[x, Blank[]]]]]], Pattern[m, Blank[]]]], Pattern[x, Blank[Symbol]]] := -Simp[(a*(a*Sin[e + f*x])^(m - 1)*(b*Cos[e + f*x])^(n + 1))/(b*f*(n + 1)), x] + Dist[(a^2*(m - 1))/(b^2*(n + 1)), Int[(a*Sin[e + f*x])^(m - 2)*(b*Cos[e + f*x])^(n + 2), x], x] /; FreeQ[{a, b, e, f}, x] && GtQ[m, 1] && LtQ[n, -1] && (IntegersQ[2*m, 2*n] || EqQ[m + n, 0])
  2567. Int[Times[Power[Times[cos[Plus[Optional[Pattern[e, Blank[]]], Times[Optional[Pattern[f, Blank[]]], Pattern[x, Blank[]]]]], Optional[Pattern[a, Blank[]]]], Pattern[m, Blank[]]], Power[Times[Optional[Pattern[b, Blank[]]], sin[Plus[Optional[Pattern[e, Blank[]]], Times[Optional[Pattern[f, Blank[]]], Pattern[x, Blank[]]]]]], Pattern[n, Blank[]]]], Pattern[x, Blank[Symbol]]] := Simp[(a*(a*Cos[e + f*x])^(m - 1)*(b*Sin[e + f*x])^(n + 1))/(b*f*(n + 1)), x] + Dist[(a^2*(m - 1))/(b^2*(n + 1)), Int[(a*Cos[e + f*x])^(m - 2)*(b*Sin[e + f*x])^(n + 2), x], x] /; FreeQ[{a, b, e, f}, x] && GtQ[m, 1] && LtQ[n, -1] && (IntegersQ[2*m, 2*n] || EqQ[m + n, 0])
  2568. Int[Times[Power[Times[cos[Plus[Optional[Pattern[e, Blank[]]], Times[Optional[Pattern[f, Blank[]]], Pattern[x, Blank[]]]]], Optional[Pattern[b, Blank[]]]], Pattern[n, Blank[]]], Power[Times[Optional[Pattern[a, Blank[]]], sin[Plus[Optional[Pattern[e, Blank[]]], Times[Optional[Pattern[f, Blank[]]], Pattern[x, Blank[]]]]]], Pattern[m, Blank[]]]], Pattern[x, Blank[Symbol]]] := -Simp[(a*(b*Cos[e + f*x])^(n + 1)*(a*Sin[e + f*x])^(m - 1))/(b*f*(m + n)), x] + Dist[(a^2*(m - 1))/(m + n), Int[(b*Cos[e + f*x])^n*(a*Sin[e + f*x])^(m - 2), x], x] /; FreeQ[{a, b, e, f, n}, x] && GtQ[m, 1] && NeQ[m + n, 0] && IntegersQ[2*m, 2*n]
  2569. Int[Times[Power[Times[cos[Plus[Optional[Pattern[e, Blank[]]], Times[Optional[Pattern[f, Blank[]]], Pattern[x, Blank[]]]]], Optional[Pattern[a, Blank[]]]], Pattern[m, Blank[]]], Power[Times[Optional[Pattern[b, Blank[]]], sin[Plus[Optional[Pattern[e, Blank[]]], Times[Optional[Pattern[f, Blank[]]], Pattern[x, Blank[]]]]]], Pattern[n, Blank[]]]], Pattern[x, Blank[Symbol]]] := Simp[(a*(b*Sin[e + f*x])^(n + 1)*(a*Cos[e + f*x])^(m - 1))/(b*f*(m + n)), x] + Dist[(a^2*(m - 1))/(m + n), Int[(b*Sin[e + f*x])^n*(a*Cos[e + f*x])^(m - 2), x], x] /; FreeQ[{a, b, e, f, n}, x] && GtQ[m, 1] && NeQ[m + n, 0] && IntegersQ[2*m, 2*n]
  2570. Int[Times[Power[Times[cos[Plus[Optional[Pattern[e, Blank[]]], Times[Optional[Pattern[f, Blank[]]], Pattern[x, Blank[]]]]], Optional[Pattern[b, Blank[]]]], Pattern[n, Blank[]]], Power[Times[Optional[Pattern[a, Blank[]]], sin[Plus[Optional[Pattern[e, Blank[]]], Times[Optional[Pattern[f, Blank[]]], Pattern[x, Blank[]]]]]], Pattern[m, Blank[]]]], Pattern[x, Blank[Symbol]]] := Simp[((b*Cos[e + f*x])^(n + 1)*(a*Sin[e + f*x])^(m + 1))/(a*b*f*(m + 1)), x] + Dist[(m + n + 2)/(a^2*(m + 1)), Int[(b*Cos[e + f*x])^n*(a*Sin[e + f*x])^(m + 2), x], x] /; FreeQ[{a, b, e, f, n}, x] && LtQ[m, -1] && IntegersQ[2*m, 2*n]
  2571. Int[Times[Power[Times[cos[Plus[Optional[Pattern[e, Blank[]]], Times[Optional[Pattern[f, Blank[]]], Pattern[x, Blank[]]]]], Optional[Pattern[a, Blank[]]]], Pattern[m, Blank[]]], Power[Times[Optional[Pattern[b, Blank[]]], sin[Plus[Optional[Pattern[e, Blank[]]], Times[Optional[Pattern[f, Blank[]]], Pattern[x, Blank[]]]]]], Pattern[n, Blank[]]]], Pattern[x, Blank[Symbol]]] := -Simp[((b*Sin[e + f*x])^(n + 1)*(a*Cos[e + f*x])^(m + 1))/(a*b*f*(m + 1)), x] + Dist[(m + n + 2)/(a^2*(m + 1)), Int[(b*Sin[e + f*x])^n*(a*Cos[e + f*x])^(m + 2), x], x] /; FreeQ[{a, b, e, f, n}, x] && LtQ[m, -1] && IntegersQ[2*m, 2*n]
  2572. Int[Times[Power[Times[cos[Plus[Optional[Pattern[e, Blank[]]], Times[Optional[Pattern[f, Blank[]]], Pattern[x, Blank[]]]]], Optional[Pattern[b, Blank[]]]], Rational[1, 2]], Power[Times[Optional[Pattern[a, Blank[]]], sin[Plus[Optional[Pattern[e, Blank[]]], Times[Optional[Pattern[f, Blank[]]], Pattern[x, Blank[]]]]]], Rational[1, 2]]], Pattern[x, Blank[Symbol]]] := Dist[(Sqrt[a*Sin[e + f*x]]*Sqrt[b*Cos[e + f*x]])/Sqrt[Sin[2*e + 2*f*x]], Int[Sqrt[Sin[2*e + 2*f*x]], x], x] /; FreeQ[{a, b, e, f}, x]
  2573. Int[Times[Power[Times[cos[Plus[Optional[Pattern[e, Blank[]]], Times[Optional[Pattern[f, Blank[]]], Pattern[x, Blank[]]]]], Optional[Pattern[b, Blank[]]]], Rational[-1, 2]], Power[Times[Optional[Pattern[a, Blank[]]], sin[Plus[Optional[Pattern[e, Blank[]]], Times[Optional[Pattern[f, Blank[]]], Pattern[x, Blank[]]]]]], Rational[-1, 2]]], Pattern[x, Blank[Symbol]]] := Dist[Sqrt[Sin[2*e + 2*f*x]]/(Sqrt[a*Sin[e + f*x]]*Sqrt[b*Cos[e + f*x]]), Int[1/Sqrt[Sin[2*e + 2*f*x]], x], x] /; FreeQ[{a, b, e, f}, x]
  2574. Int[Times[Power[Times[cos[Plus[Optional[Pattern[e, Blank[]]], Times[Optional[Pattern[f, Blank[]]], Pattern[x, Blank[]]]]], Optional[Pattern[b, Blank[]]]], Pattern[n, Blank[]]], Power[Times[Optional[Pattern[a, Blank[]]], sin[Plus[Optional[Pattern[e, Blank[]]], Times[Optional[Pattern[f, Blank[]]], Pattern[x, Blank[]]]]]], Pattern[m, Blank[]]]], Pattern[x, Blank[Symbol]]] := With[{k = Denominator[m]}, Dist[(k*a*b)/f, Subst[Int[x^(k*(m + 1) - 1)/(a^2 + b^2*x^(2*k)), x], x, (a*Sin[e + f*x])^(1/k)/(b*Cos[e + f*x])^(1/k)], x]] /; FreeQ[{a, b, e, f}, x] && EqQ[m + n, 0] && GtQ[m, 0] && LtQ[m, 1]
  2575. Int[Times[Power[Times[cos[Plus[Optional[Pattern[e, Blank[]]], Times[Optional[Pattern[f, Blank[]]], Pattern[x, Blank[]]]]], Optional[Pattern[a, Blank[]]]], Pattern[m, Blank[]]], Power[Times[Optional[Pattern[b, Blank[]]], sin[Plus[Optional[Pattern[e, Blank[]]], Times[Optional[Pattern[f, Blank[]]], Pattern[x, Blank[]]]]]], Pattern[n, Blank[]]]], Pattern[x, Blank[Symbol]]] := With[{k = Denominator[m]}, -Dist[(k*a*b)/f, Subst[Int[x^(k*(m + 1) - 1)/(a^2 + b^2*x^(2*k)), x], x, (a*Cos[e + f*x])^(1/k)/(b*Sin[e + f*x])^(1/k)], x]] /; FreeQ[{a, b, e, f}, x] && EqQ[m + n, 0] && GtQ[m, 0] && LtQ[m, 1]
  2576. Int[Times[Power[Times[cos[Plus[Optional[Pattern[e, Blank[]]], Times[Optional[Pattern[f, Blank[]]], Pattern[x, Blank[]]]]], Optional[Pattern[a, Blank[]]]], Pattern[m, Blank[]]], Power[Times[Optional[Pattern[b, Blank[]]], sin[Plus[Optional[Pattern[e, Blank[]]], Times[Optional[Pattern[f, Blank[]]], Pattern[x, Blank[]]]]]], Pattern[n, Blank[]]]], Pattern[x, Blank[Symbol]]] := -Simp[(b^(2*IntPart[(n - 1)/2] + 1)*(b*Sin[e + f*x])^(2*FracPart[(n - 1)/2])*(a*Cos[e + f*x])^(m + 1)*Hypergeometric2F1[(1 + m)/2, (1 - n)/2, (3 + m)/2, Cos[e + f*x]^2])/(a*f*(m + 1)*(Sin[e + f*x]^2)^FracPart[(n - 1)/2]), x] /; FreeQ[{a, b, e, f, m, n}, x] && SimplerQ[n, m]
  2577. Int[Times[Power[Times[cos[Plus[Optional[Pattern[e, Blank[]]], Times[Optional[Pattern[f, Blank[]]], Pattern[x, Blank[]]]]], Optional[Pattern[b, Blank[]]]], Pattern[n, Blank[]]], Power[Times[Optional[Pattern[a, Blank[]]], sin[Plus[Optional[Pattern[e, Blank[]]], Times[Optional[Pattern[f, Blank[]]], Pattern[x, Blank[]]]]]], Pattern[m, Blank[]]]], Pattern[x, Blank[Symbol]]] := Simp[(b^(2*IntPart[(n - 1)/2] + 1)*(b*Cos[e + f*x])^(2*FracPart[(n - 1)/2])*(a*Sin[e + f*x])^(m + 1)*Hypergeometric2F1[(1 + m)/2, (1 - n)/2, (3 + m)/2, Sin[e + f*x]^2])/(a*f*(m + 1)*(Cos[e + f*x]^2)^FracPart[(n - 1)/2]), x] /; FreeQ[{a, b, e, f, m, n}, x]
  2578. Int[Times[Power[Times[Optional[Pattern[b, Blank[]]], sec[Plus[Optional[Pattern[e, Blank[]]], Times[Optional[Pattern[f, Blank[]]], Pattern[x, Blank[]]]]]], Optional[Pattern[n, Blank[]]]], Power[Times[Optional[Pattern[a, Blank[]]], sin[Plus[Optional[Pattern[e, Blank[]]], Times[Optional[Pattern[f, Blank[]]], Pattern[x, Blank[]]]]]], Optional[Pattern[m, Blank[]]]]], Pattern[x, Blank[Symbol]]] := Simp[(b*(a*Sin[e + f*x])^(m + 1)*(b*Sec[e + f*x])^(n - 1))/(a*f*(m + 1)), x] /; FreeQ[{a, b, e, f, m, n}, x] && EqQ[m - n + 2, 0] && NeQ[m, -1]
  2579. Int[Times[Power[Times[Optional[Pattern[b, Blank[]]], sec[Plus[Optional[Pattern[e, Blank[]]], Times[Optional[Pattern[f, Blank[]]], Pattern[x, Blank[]]]]]], Pattern[n, Blank[]]], Power[Times[Optional[Pattern[a, Blank[]]], sin[Plus[Optional[Pattern[e, Blank[]]], Times[Optional[Pattern[f, Blank[]]], Pattern[x, Blank[]]]]]], Pattern[m, Blank[]]]], Pattern[x, Blank[Symbol]]] := Simp[(a*b*(a*Sin[e + f*x])^(m - 1)*(b*Sec[e + f*x])^(n - 1))/(f*(n - 1)), x] - Dist[(a^2*b^2*(m - 1))/(n - 1), Int[(a*Sin[e + f*x])^(m - 2)*(b*Sec[e + f*x])^(n - 2), x], x] /; FreeQ[{a, b, e, f}, x] && GtQ[n, 1] && GtQ[m, 1] && IntegersQ[2*m, 2*n]
  2580. Int[Times[Power[Times[Optional[Pattern[b, Blank[]]], sec[Plus[Optional[Pattern[e, Blank[]]], Times[Optional[Pattern[f, Blank[]]], Pattern[x, Blank[]]]]]], Pattern[n, Blank[]]], Power[Times[Optional[Pattern[a, Blank[]]], sin[Plus[Optional[Pattern[e, Blank[]]], Times[Optional[Pattern[f, Blank[]]], Pattern[x, Blank[]]]]]], Pattern[m, Blank[]]]], Pattern[x, Blank[Symbol]]] := Simp[((a*Sin[e + f*x])^(m + 1)*(b*Sec[e + f*x])^(n + 1))/(a*b*f*(m - n)), x] - Dist[(n + 1)/(b^2*(m - n)), Int[(a*Sin[e + f*x])^m*(b*Sec[e + f*x])^(n + 2), x], x] /; FreeQ[{a, b, e, f, m}, x] && GtQ[n, 1] && IntegersQ[2*m, 2*n]
  2581. Int[Times[Power[Times[Optional[Pattern[b, Blank[]]], sec[Plus[Optional[Pattern[e, Blank[]]], Times[Optional[Pattern[f, Blank[]]], Pattern[x, Blank[]]]]]], Pattern[n, Blank[]]], Power[Times[Optional[Pattern[a, Blank[]]], sin[Plus[Optional[Pattern[e, Blank[]]], Times[Optional[Pattern[f, Blank[]]], Pattern[x, Blank[]]]]]], Pattern[m, Blank[]]]], Pattern[x, Blank[Symbol]]] := Simp[((a*Sin[e + f*x])^(m + 1)*(b*Sec[e + f*x])^(n + 1))/(a*b*f*(m + 1)), x] - Dist[(n + 1)/(a^2*b^2*(m + 1)), Int[(a*Sin[e + f*x])^(m + 2)*(b*Sec[e + f*x])^(n + 2), x], x] /; FreeQ[{a, b, e, f}, x] && LtQ[n, -1] && LtQ[m, -1] && IntegersQ[2*m, 2*n]
  2582. Int[Times[Power[Times[Optional[Pattern[b, Blank[]]], sec[Plus[Optional[Pattern[e, Blank[]]], Times[Optional[Pattern[f, Blank[]]], Pattern[x, Blank[]]]]]], Pattern[n, Blank[]]], Power[Times[Optional[Pattern[a, Blank[]]], sin[Plus[Optional[Pattern[e, Blank[]]], Times[Optional[Pattern[f, Blank[]]], Pattern[x, Blank[]]]]]], Pattern[m, Blank[]]]], Pattern[x, Blank[Symbol]]] := Simp[((a*Sin[e + f*x])^(m + 1)*(b*Sec[e + f*x])^(n + 1))/(a*b*f*(m - n)), x] - Dist[(n + 1)/(b^2*(m - n)), Int[(a*Sin[e + f*x])^m*(b*Sec[e + f*x])^(n + 2), x], x] /; FreeQ[{a, b, e, f, m}, x] && LtQ[n, -1] && NeQ[m - n, 0] && IntegersQ[2*m, 2*n]
  2583. Int[Times[Power[Times[Optional[Pattern[b, Blank[]]], sec[Plus[Optional[Pattern[e, Blank[]]], Times[Optional[Pattern[f, Blank[]]], Pattern[x, Blank[]]]]]], Pattern[n, Blank[]]], Power[Times[Optional[Pattern[a, Blank[]]], sin[Plus[Optional[Pattern[e, Blank[]]], Times[Optional[Pattern[f, Blank[]]], Pattern[x, Blank[]]]]]], Pattern[m, Blank[]]]], Pattern[x, Blank[Symbol]]] := -Simp[(a*b*(a*Sin[e + f*x])^(m - 1)*(b*Sec[e + f*x])^(n - 1))/(f*(m - n)), x] + Dist[(a^2*(m - 1))/(m - n), Int[(a*Sin[e + f*x])^(m - 2)*(b*Sec[e + f*x])^n, x], x] /; FreeQ[{a, b, e, f, n}, x] && GtQ[m, 1] && NeQ[m - n, 0] && IntegersQ[2*m, 2*n]
  2584. Int[Times[Power[Times[Optional[Pattern[b, Blank[]]], sec[Plus[Optional[Pattern[e, Blank[]]], Times[Optional[Pattern[f, Blank[]]], Pattern[x, Blank[]]]]]], Pattern[n, Blank[]]], Power[Times[Optional[Pattern[a, Blank[]]], sin[Plus[Optional[Pattern[e, Blank[]]], Times[Optional[Pattern[f, Blank[]]], Pattern[x, Blank[]]]]]], Pattern[m, Blank[]]]], Pattern[x, Blank[Symbol]]] := Simp[(b*(a*Sin[e + f*x])^(m + 1)*(b*Sec[e + f*x])^(n - 1))/(a*f*(m + 1)), x] + Dist[(m - n + 2)/(a^2*(m + 1)), Int[(a*Sin[e + f*x])^(m + 2)*(b*Sec[e + f*x])^n, x], x] /; FreeQ[{a, b, e, f, n}, x] && LtQ[m, -1] && IntegersQ[2*m, 2*n]
  2585. Int[Times[Power[Times[Optional[Pattern[b, Blank[]]], sec[Plus[Optional[Pattern[e, Blank[]]], Times[Optional[Pattern[f, Blank[]]], Pattern[x, Blank[]]]]]], Pattern[n, Blank[]]], Power[Times[Optional[Pattern[a, Blank[]]], sin[Plus[Optional[Pattern[e, Blank[]]], Times[Optional[Pattern[f, Blank[]]], Pattern[x, Blank[]]]]]], Pattern[m, Blank[]]]], Pattern[x, Blank[Symbol]]] := Dist[(b*Cos[e + f*x])^n*(b*Sec[e + f*x])^n, Int[(a*Sin[e + f*x])^m/(b*Cos[e + f*x])^n, x], x] /; FreeQ[{a, b, e, f, m, n}, x] && IntegerQ[m - 1/2] && IntegerQ[n - 1/2]
  2586. Int[Times[Power[Times[Optional[Pattern[b, Blank[]]], sec[Plus[Optional[Pattern[e, Blank[]]], Times[Optional[Pattern[f, Blank[]]], Pattern[x, Blank[]]]]]], Pattern[n, Blank[]]], Power[Times[Optional[Pattern[a, Blank[]]], sin[Plus[Optional[Pattern[e, Blank[]]], Times[Optional[Pattern[f, Blank[]]], Pattern[x, Blank[]]]]]], Pattern[m, Blank[]]]], Pattern[x, Blank[Symbol]]] := Dist[(1*(b*Cos[e + f*x])^(n + 1)*(b*Sec[e + f*x])^(n + 1))/b^2, Int[(a*Sin[e + f*x])^m/(b*Cos[e + f*x])^n, x], x] /; FreeQ[{a, b, e, f, m, n}, x] && !IntegerQ[m] && !IntegerQ[n] && LtQ[n, 1]
  2587. Int[Times[Power[Times[Optional[Pattern[b, Blank[]]], sec[Plus[Optional[Pattern[e, Blank[]]], Times[Optional[Pattern[f, Blank[]]], Pattern[x, Blank[]]]]]], Pattern[n, Blank[]]], Power[Times[Optional[Pattern[a, Blank[]]], sin[Plus[Optional[Pattern[e, Blank[]]], Times[Optional[Pattern[f, Blank[]]], Pattern[x, Blank[]]]]]], Pattern[m, Blank[]]]], Pattern[x, Blank[Symbol]]] := Dist[b^2*(b*Cos[e + f*x])^(n - 1)*(b*Sec[e + f*x])^(n - 1), Int[(a*Sin[e + f*x])^m/(b*Cos[e + f*x])^n, x], x] /; FreeQ[{a, b, e, f, m, n}, x] && !IntegerQ[m] && !IntegerQ[n]
  2588. Int[Times[Power[Times[csc[Plus[Optional[Pattern[e, Blank[]]], Times[Optional[Pattern[f, Blank[]]], Pattern[x, Blank[]]]]], Optional[Pattern[b, Blank[]]]], Pattern[n, Blank[]]], Power[Times[Optional[Pattern[a, Blank[]]], sin[Plus[Optional[Pattern[e, Blank[]]], Times[Optional[Pattern[f, Blank[]]], Pattern[x, Blank[]]]]]], Optional[Pattern[m, Blank[]]]]], Pattern[x, Blank[Symbol]]] := Dist[(a*b)^IntPart[n]*(a*Sin[e + f*x])^FracPart[n]*(b*Csc[e + f*x])^FracPart[n], Int[(a*Sin[e + f*x])^(m - n), x], x] /; FreeQ[{a, b, e, f, m, n}, x] && !IntegerQ[m] && !IntegerQ[n]
  2589. Int[Times[Power[Times[Optional[Pattern[a, Blank[]]], sin[Plus[Optional[Pattern[e, Blank[]]], Times[Optional[Pattern[f, Blank[]]], Pattern[x, Blank[]]]]]], Pattern[m, Blank[]]], Power[Times[Optional[Pattern[b, Blank[]]], tan[Plus[Optional[Pattern[e, Blank[]]], Times[Optional[Pattern[f, Blank[]]], Pattern[x, Blank[]]]]]], Pattern[n, Blank[]]]], Pattern[x, Blank[Symbol]]] := -Simp[(b*(a*Sin[e + f*x])^m*(b*Tan[e + f*x])^(n - 1))/(f*m), x] /; FreeQ[{a, b, e, f, m, n}, x] && EqQ[m + n - 1, 0]
  2590. Int[Times[Power[sin[Plus[Optional[Pattern[e, Blank[]]], Times[Optional[Pattern[f, Blank[]]], Pattern[x, Blank[]]]]], Optional[Pattern[m, Blank[]]]], Power[tan[Plus[Optional[Pattern[e, Blank[]]], Times[Optional[Pattern[f, Blank[]]], Pattern[x, Blank[]]]]], Optional[Pattern[n, Blank[]]]]], Pattern[x, Blank[Symbol]]] := -Dist[f^(-1), Subst[Int[(1 - x^2)^((m + n - 1)/2)/x^n, x], x, Cos[e + f*x]], x] /; FreeQ[{e, f}, x] && IntegersQ[m, n, (m + n - 1)/2]
  2591. Int[Times[Power[sin[Plus[Optional[Pattern[e, Blank[]]], Times[Optional[Pattern[f, Blank[]]], Pattern[x, Blank[]]]]], Pattern[m, Blank[]]], Power[Times[Optional[Pattern[b, Blank[]]], tan[Plus[Optional[Pattern[e, Blank[]]], Times[Optional[Pattern[f, Blank[]]], Pattern[x, Blank[]]]]]], Optional[Pattern[n, Blank[]]]]], Pattern[x, Blank[Symbol]]] := With[{ff = FreeFactors[Tan[e + f*x], x]}, Dist[(b*ff)/f, Subst[Int[(ff*x)^(m + n)/(b^2 + ff^2*x^2)^(m/2 + 1), x], x, (b*Tan[e + f*x])/ff], x]] /; FreeQ[{b, e, f, n}, x] && IntegerQ[m/2]
  2592. Int[Times[Power[Times[Optional[Pattern[a, Blank[]]], sin[Plus[Optional[Pattern[e, Blank[]]], Times[Optional[Pattern[f, Blank[]]], Pattern[x, Blank[]]]]]], Optional[Pattern[m, Blank[]]]], Power[tan[Plus[Optional[Pattern[e, Blank[]]], Times[Optional[Pattern[f, Blank[]]], Pattern[x, Blank[]]]]], Optional[Pattern[n, Blank[]]]]], Pattern[x, Blank[Symbol]]] := With[{ff = FreeFactors[Sin[e + f*x], x]}, Dist[ff/f, Subst[Int[(ff*x)^(m + n)/(a^2 - ff^2*x^2)^((n + 1)/2), x], x, (a*Sin[e + f*x])/ff], x]] /; FreeQ[{a, e, f, m}, x] && IntegerQ[(n + 1)/2]
  2593. Int[Times[Power[Times[Optional[Pattern[a, Blank[]]], sin[Plus[Optional[Pattern[e, Blank[]]], Times[Optional[Pattern[f, Blank[]]], Pattern[x, Blank[]]]]]], Pattern[m, Blank[]]], Power[Times[Optional[Pattern[b, Blank[]]], tan[Plus[Optional[Pattern[e, Blank[]]], Times[Optional[Pattern[f, Blank[]]], Pattern[x, Blank[]]]]]], Pattern[n, Blank[]]]], Pattern[x, Blank[Symbol]]] := Simp[(b*(a*Sin[e + f*x])^(m + 2)*(b*Tan[e + f*x])^(n - 1))/(a^2*f*(n - 1)), x] - Dist[(b^2*(m + 2))/(a^2*(n - 1)), Int[(a*Sin[e + f*x])^(m + 2)*(b*Tan[e + f*x])^(n - 2), x], x] /; FreeQ[{a, b, e, f}, x] && GtQ[n, 1] && (LtQ[m, -1] || (EqQ[m, -1] && EqQ[n, 3/2])) && IntegersQ[2*m, 2*n]
  2594. Int[Times[Power[Times[Optional[Pattern[a, Blank[]]], sin[Plus[Optional[Pattern[e, Blank[]]], Times[Optional[Pattern[f, Blank[]]], Pattern[x, Blank[]]]]]], Optional[Pattern[m, Blank[]]]], Power[Times[Optional[Pattern[b, Blank[]]], tan[Plus[Optional[Pattern[e, Blank[]]], Times[Optional[Pattern[f, Blank[]]], Pattern[x, Blank[]]]]]], Pattern[n, Blank[]]]], Pattern[x, Blank[Symbol]]] := Simp[(b*(a*Sin[e + f*x])^m*(b*Tan[e + f*x])^(n - 1))/(f*(n - 1)), x] - Dist[(b^2*(m + n - 1))/(n - 1), Int[(a*Sin[e + f*x])^m*(b*Tan[e + f*x])^(n - 2), x], x] /; FreeQ[{a, b, e, f, m}, x] && GtQ[n, 1] && IntegersQ[2*m, 2*n] && !(GtQ[m, 1] && !IntegerQ[(m - 1)/2])
  2595. Int[Times[Power[Times[Optional[Pattern[a, Blank[]]], sin[Plus[Optional[Pattern[e, Blank[]]], Times[Optional[Pattern[f, Blank[]]], Pattern[x, Blank[]]]]]], Rational[1, 2]], Power[Times[Optional[Pattern[b, Blank[]]], tan[Plus[Optional[Pattern[e, Blank[]]], Times[Optional[Pattern[f, Blank[]]], Pattern[x, Blank[]]]]]], Rational[-3, 2]]], Pattern[x, Blank[Symbol]]] := Simp[(2*Sqrt[a*Sin[e + f*x]])/(b*f*Sqrt[b*Tan[e + f*x]]), x] + Dist[a^2/b^2, Int[Sqrt[b*Tan[e + f*x]]/(a*Sin[e + f*x])^(3/2), x], x] /; FreeQ[{a, b, e, f}, x]
  2596. Int[Times[Power[Times[Optional[Pattern[a, Blank[]]], sin[Plus[Optional[Pattern[e, Blank[]]], Times[Optional[Pattern[f, Blank[]]], Pattern[x, Blank[]]]]]], Pattern[m, Blank[]]], Power[Times[Optional[Pattern[b, Blank[]]], tan[Plus[Optional[Pattern[e, Blank[]]], Times[Optional[Pattern[f, Blank[]]], Pattern[x, Blank[]]]]]], Pattern[n, Blank[]]]], Pattern[x, Blank[Symbol]]] := Simp[((a*Sin[e + f*x])^m*(b*Tan[e + f*x])^(n + 1))/(b*f*m), x] - Dist[(a^2*(n + 1))/(b^2*m), Int[(a*Sin[e + f*x])^(m - 2)*(b*Tan[e + f*x])^(n + 2), x], x] /; FreeQ[{a, b, e, f}, x] && LtQ[n, -1] && GtQ[m, 1] && IntegersQ[2*m, 2*n]
  2597. Int[Times[Power[Times[Optional[Pattern[a, Blank[]]], sin[Plus[Optional[Pattern[e, Blank[]]], Times[Optional[Pattern[f, Blank[]]], Pattern[x, Blank[]]]]]], Optional[Pattern[m, Blank[]]]], Power[Times[Optional[Pattern[b, Blank[]]], tan[Plus[Optional[Pattern[e, Blank[]]], Times[Optional[Pattern[f, Blank[]]], Pattern[x, Blank[]]]]]], Pattern[n, Blank[]]]], Pattern[x, Blank[Symbol]]] := Simp[((a*Sin[e + f*x])^m*(b*Tan[e + f*x])^(n + 1))/(b*f*(m + n + 1)), x] - Dist[(n + 1)/(b^2*(m + n + 1)), Int[(a*Sin[e + f*x])^m*(b*Tan[e + f*x])^(n + 2), x], x] /; FreeQ[{a, b, e, f, m}, x] && LtQ[n, -1] && NeQ[m + n + 1, 0] && IntegersQ[2*m, 2*n] && !(EqQ[n, -3/2] && EqQ[m, 1])
  2598. Int[Times[Power[Times[Optional[Pattern[a, Blank[]]], sin[Plus[Optional[Pattern[e, Blank[]]], Times[Optional[Pattern[f, Blank[]]], Pattern[x, Blank[]]]]]], Optional[Pattern[m, Blank[]]]], Power[Times[Optional[Pattern[b, Blank[]]], tan[Plus[Optional[Pattern[e, Blank[]]], Times[Optional[Pattern[f, Blank[]]], Pattern[x, Blank[]]]]]], Optional[Pattern[n, Blank[]]]]], Pattern[x, Blank[Symbol]]] := -Simp[(b*(a*Sin[e + f*x])^m*(b*Tan[e + f*x])^(n - 1))/(f*m), x] + Dist[(a^2*(m + n - 1))/m, Int[(a*Sin[e + f*x])^(m - 2)*(b*Tan[e + f*x])^n, x], x] /; FreeQ[{a, b, e, f, n}, x] && (GtQ[m, 1] || (EqQ[m, 1] && EqQ[n, 1/2])) && IntegersQ[2*m, 2*n]
  2599. Int[Times[Power[Times[Optional[Pattern[a, Blank[]]], sin[Plus[Optional[Pattern[e, Blank[]]], Times[Optional[Pattern[f, Blank[]]], Pattern[x, Blank[]]]]]], Pattern[m, Blank[]]], Power[Times[Optional[Pattern[b, Blank[]]], tan[Plus[Optional[Pattern[e, Blank[]]], Times[Optional[Pattern[f, Blank[]]], Pattern[x, Blank[]]]]]], Optional[Pattern[n, Blank[]]]]], Pattern[x, Blank[Symbol]]] := Simp[(b*(a*Sin[e + f*x])^(m + 2)*(b*Tan[e + f*x])^(n - 1))/(a^2*f*(m + n + 1)), x] + Dist[(m + 2)/(a^2*(m + n + 1)), Int[(a*Sin[e + f*x])^(m + 2)*(b*Tan[e + f*x])^n, x], x] /; FreeQ[{a, b, e, f, n}, x] && LtQ[m, -1] && NeQ[m + n + 1, 0] && IntegersQ[2*m, 2*n]
  2600. Int[Times[Power[Times[Optional[Pattern[a, Blank[]]], sin[Plus[Optional[Pattern[e, Blank[]]], Times[Optional[Pattern[f, Blank[]]], Pattern[x, Blank[]]]]]], Pattern[m, Blank[]]], Power[tan[Plus[Optional[Pattern[e, Blank[]]], Times[Optional[Pattern[f, Blank[]]], Pattern[x, Blank[]]]]], Pattern[n, Blank[]]]], Pattern[x, Blank[Symbol]]] := Dist[1/a^n, Int[(a*Sin[e + f*x])^(m + n)/Cos[e + f*x]^n, x], x] /; FreeQ[{a, e, f, m}, x] && IntegerQ[n] && !IntegerQ[m]
  2601. Int[Times[Power[Times[Optional[Pattern[a, Blank[]]], sin[Plus[Optional[Pattern[e, Blank[]]], Times[Optional[Pattern[f, Blank[]]], Pattern[x, Blank[]]]]]], Optional[Pattern[m, Blank[]]]], Power[Times[Optional[Pattern[b, Blank[]]], tan[Plus[Optional[Pattern[e, Blank[]]], Times[Optional[Pattern[f, Blank[]]], Pattern[x, Blank[]]]]]], Pattern[n, Blank[]]]], Pattern[x, Blank[Symbol]]] := Dist[(Cos[e + f*x]^n*(b*Tan[e + f*x])^n)/(a*Sin[e + f*x])^n, Int[(a*Sin[e + f*x])^(m + n)/Cos[e + f*x]^n, x], x] /; FreeQ[{a, b, e, f, m, n}, x] && !IntegerQ[n] && (ILtQ[m, 0] || (EqQ[m, 1] && EqQ[n, -2^(-1)]) || IntegersQ[m - 1/2, n - 1/2])
  2602. Int[Times[Power[Times[Optional[Pattern[a, Blank[]]], sin[Plus[Optional[Pattern[e, Blank[]]], Times[Optional[Pattern[f, Blank[]]], Pattern[x, Blank[]]]]]], Optional[Pattern[m, Blank[]]]], Power[Times[Optional[Pattern[b, Blank[]]], tan[Plus[Optional[Pattern[e, Blank[]]], Times[Optional[Pattern[f, Blank[]]], Pattern[x, Blank[]]]]]], Pattern[n, Blank[]]]], Pattern[x, Blank[Symbol]]] := Dist[(a*Cos[e + f*x]^(n + 1)*(b*Tan[e + f*x])^(n + 1))/(b*(a*Sin[e + f*x])^(n + 1)), Int[(a*Sin[e + f*x])^(m + n)/Cos[e + f*x]^n, x], x] /; FreeQ[{a, b, e, f, m, n}, x] && !IntegerQ[n]
  2603. Int[Times[Power[Times[cos[Plus[Optional[Pattern[e, Blank[]]], Times[Optional[Pattern[f, Blank[]]], Pattern[x, Blank[]]]]], Optional[Pattern[a, Blank[]]]], Pattern[m, Blank[]]], Power[Times[Optional[Pattern[b, Blank[]]], tan[Plus[Optional[Pattern[e, Blank[]]], Times[Optional[Pattern[f, Blank[]]], Pattern[x, Blank[]]]]]], Pattern[n, Blank[]]]], Pattern[x, Blank[Symbol]]] := Dist[(a*Cos[e + f*x])^FracPart[m]*(Sec[e + f*x]/a)^FracPart[m], Int[(b*Tan[e + f*x])^n/(Sec[e + f*x]/a)^m, x], x] /; FreeQ[{a, b, e, f, m, n}, x] && !IntegerQ[m] && !IntegerQ[n]
  2604. Int[Times[Power[Times[cot[Plus[Optional[Pattern[e, Blank[]]], Times[Optional[Pattern[f, Blank[]]], Pattern[x, Blank[]]]]], Optional[Pattern[a, Blank[]]]], Pattern[m, Blank[]]], Power[Times[Optional[Pattern[b, Blank[]]], tan[Plus[Optional[Pattern[e, Blank[]]], Times[Optional[Pattern[f, Blank[]]], Pattern[x, Blank[]]]]]], Pattern[n, Blank[]]]], Pattern[x, Blank[Symbol]]] := Dist[(a*Cot[e + f*x])^m*(b*Tan[e + f*x])^m, Int[(b*Tan[e + f*x])^(n - m), x], x] /; FreeQ[{a, b, e, f, m, n}, x] && !IntegerQ[m] && !IntegerQ[n]
  2605. Int[Times[Power[Times[Optional[Pattern[a, Blank[]]], sec[Plus[Optional[Pattern[e, Blank[]]], Times[Optional[Pattern[f, Blank[]]], Pattern[x, Blank[]]]]]], Optional[Pattern[m, Blank[]]]], Power[Times[Optional[Pattern[b, Blank[]]], tan[Plus[Optional[Pattern[e, Blank[]]], Times[Optional[Pattern[f, Blank[]]], Pattern[x, Blank[]]]]]], Optional[Pattern[n, Blank[]]]]], Pattern[x, Blank[Symbol]]] := -Simp[((a*Sec[e + f*x])^m*(b*Tan[e + f*x])^(n + 1))/(b*f*m), x] /; FreeQ[{a, b, e, f, m, n}, x] && EqQ[m + n + 1, 0]
  2606. Int[Times[Power[Times[Optional[Pattern[a, Blank[]]], sec[Plus[Optional[Pattern[e, Blank[]]], Times[Optional[Pattern[f, Blank[]]], Pattern[x, Blank[]]]]]], Optional[Pattern[m, Blank[]]]], Power[Times[Optional[Pattern[b, Blank[]]], tan[Plus[Optional[Pattern[e, Blank[]]], Times[Optional[Pattern[f, Blank[]]], Pattern[x, Blank[]]]]]], Optional[Pattern[n, Blank[]]]]], Pattern[x, Blank[Symbol]]] := Dist[a/f, Subst[Int[(a*x)^(m - 1)*(-1 + x^2)^((n - 1)/2), x], x, Sec[e + f*x]], x] /; FreeQ[{a, e, f, m}, x] && IntegerQ[(n - 1)/2] && !(IntegerQ[m/2] && LtQ[0, m, n + 1])
  2607. Int[Times[Power[sec[Plus[Optional[Pattern[e, Blank[]]], Times[Optional[Pattern[f, Blank[]]], Pattern[x, Blank[]]]]], Pattern[m, Blank[]]], Power[Times[Optional[Pattern[b, Blank[]]], tan[Plus[Optional[Pattern[e, Blank[]]], Times[Optional[Pattern[f, Blank[]]], Pattern[x, Blank[]]]]]], Optional[Pattern[n, Blank[]]]]], Pattern[x, Blank[Symbol]]] := Dist[1/f, Subst[Int[(b*x)^n*(1 + x^2)^(m/2 - 1), x], x, Tan[e + f*x]], x] /; FreeQ[{b, e, f, n}, x] && IntegerQ[m/2] && !(IntegerQ[(n - 1)/2] && LtQ[0, n, m - 1])
  2608. Int[Times[Power[Times[Optional[Pattern[a, Blank[]]], sec[Plus[Optional[Pattern[e, Blank[]]], Times[Optional[Pattern[f, Blank[]]], Pattern[x, Blank[]]]]]], Optional[Pattern[m, Blank[]]]], Power[Times[Optional[Pattern[b, Blank[]]], tan[Plus[Optional[Pattern[e, Blank[]]], Times[Optional[Pattern[f, Blank[]]], Pattern[x, Blank[]]]]]], Pattern[n, Blank[]]]], Pattern[x, Blank[Symbol]]] := Simp[(a^2*(a*Sec[e + f*x])^(m - 2)*(b*Tan[e + f*x])^(n + 1))/(b*f*(n + 1)), x] - Dist[(a^2*(m - 2))/(b^2*(n + 1)), Int[(a*Sec[e + f*x])^(m - 2)*(b*Tan[e + f*x])^(n + 2), x], x] /; FreeQ[{a, b, e, f}, x] && LtQ[n, -1] && (GtQ[m, 1] || (EqQ[m, 1] && EqQ[n, -3/2])) && IntegersQ[2*m, 2*n]
  2609. Int[Times[Power[Times[Optional[Pattern[a, Blank[]]], sec[Plus[Optional[Pattern[e, Blank[]]], Times[Optional[Pattern[f, Blank[]]], Pattern[x, Blank[]]]]]], Optional[Pattern[m, Blank[]]]], Power[Times[Optional[Pattern[b, Blank[]]], tan[Plus[Optional[Pattern[e, Blank[]]], Times[Optional[Pattern[f, Blank[]]], Pattern[x, Blank[]]]]]], Pattern[n, Blank[]]]], Pattern[x, Blank[Symbol]]] := Simp[((a*Sec[e + f*x])^m*(b*Tan[e + f*x])^(n + 1))/(b*f*(n + 1)), x] - Dist[(m + n + 1)/(b^2*(n + 1)), Int[(a*Sec[e + f*x])^m*(b*Tan[e + f*x])^(n + 2), x], x] /; FreeQ[{a, b, e, f, m}, x] && LtQ[n, -1] && IntegersQ[2*m, 2*n]
  2610. Int[Times[Power[Times[Optional[Pattern[a, Blank[]]], sec[Plus[Optional[Pattern[e, Blank[]]], Times[Optional[Pattern[f, Blank[]]], Pattern[x, Blank[]]]]]], Pattern[m, Blank[]]], Power[Times[Optional[Pattern[b, Blank[]]], tan[Plus[Optional[Pattern[e, Blank[]]], Times[Optional[Pattern[f, Blank[]]], Pattern[x, Blank[]]]]]], Pattern[n, Blank[]]]], Pattern[x, Blank[Symbol]]] := Simp[(b*(a*Sec[e + f*x])^m*(b*Tan[e + f*x])^(n - 1))/(f*m), x] - Dist[(b^2*(n - 1))/(a^2*m), Int[(a*Sec[e + f*x])^(m + 2)*(b*Tan[e + f*x])^(n - 2), x], x] /; FreeQ[{a, b, e, f}, x] && GtQ[n, 1] && (LtQ[m, -1] || (EqQ[m, -1] && EqQ[n, 3/2])) && IntegersQ[2*m, 2*n]
  2611. Int[Times[Power[Times[Optional[Pattern[a, Blank[]]], sec[Plus[Optional[Pattern[e, Blank[]]], Times[Optional[Pattern[f, Blank[]]], Pattern[x, Blank[]]]]]], Optional[Pattern[m, Blank[]]]], Power[Times[Optional[Pattern[b, Blank[]]], tan[Plus[Optional[Pattern[e, Blank[]]], Times[Optional[Pattern[f, Blank[]]], Pattern[x, Blank[]]]]]], Pattern[n, Blank[]]]], Pattern[x, Blank[Symbol]]] := Simp[(b*(a*Sec[e + f*x])^m*(b*Tan[e + f*x])^(n - 1))/(f*(m + n - 1)), x] - Dist[(b^2*(n - 1))/(m + n - 1), Int[(a*Sec[e + f*x])^m*(b*Tan[e + f*x])^(n - 2), x], x] /; FreeQ[{a, b, e, f, m}, x] && GtQ[n, 1] && NeQ[m + n - 1, 0] && IntegersQ[2*m, 2*n]
  2612. Int[Times[Power[Times[Optional[Pattern[a, Blank[]]], sec[Plus[Optional[Pattern[e, Blank[]]], Times[Optional[Pattern[f, Blank[]]], Pattern[x, Blank[]]]]]], Pattern[m, Blank[]]], Power[Times[Optional[Pattern[b, Blank[]]], tan[Plus[Optional[Pattern[e, Blank[]]], Times[Optional[Pattern[f, Blank[]]], Pattern[x, Blank[]]]]]], Pattern[n, Blank[]]]], Pattern[x, Blank[Symbol]]] := -Simp[((a*Sec[e + f*x])^m*(b*Tan[e + f*x])^(n + 1))/(b*f*m), x] + Dist[(m + n + 1)/(a^2*m), Int[(a*Sec[e + f*x])^(m + 2)*(b*Tan[e + f*x])^n, x], x] /; FreeQ[{a, b, e, f, n}, x] && (LtQ[m, -1] || (EqQ[m, -1] && EqQ[n, -2^(-1)])) && IntegersQ[2*m, 2*n]
  2613. Int[Times[Power[Times[Optional[Pattern[a, Blank[]]], sec[Plus[Optional[Pattern[e, Blank[]]], Times[Optional[Pattern[f, Blank[]]], Pattern[x, Blank[]]]]]], Optional[Pattern[m, Blank[]]]], Power[Times[Optional[Pattern[b, Blank[]]], tan[Plus[Optional[Pattern[e, Blank[]]], Times[Optional[Pattern[f, Blank[]]], Pattern[x, Blank[]]]]]], Pattern[n, Blank[]]]], Pattern[x, Blank[Symbol]]] := Simp[(a^2*(a*Sec[e + f*x])^(m - 2)*(b*Tan[e + f*x])^(n + 1))/(b*f*(m + n - 1)), x] + Dist[(a^2*(m - 2))/(m + n - 1), Int[(a*Sec[e + f*x])^(m - 2)*(b*Tan[e + f*x])^n, x], x] /; FreeQ[{a, b, e, f, n}, x] && (GtQ[m, 1] || (EqQ[m, 1] && EqQ[n, 1/2])) && NeQ[m + n - 1, 0] && IntegersQ[2*m, 2*n]
  2614. Int[Times[sec[Plus[Optional[Pattern[e, Blank[]]], Times[Optional[Pattern[f, Blank[]]], Pattern[x, Blank[]]]]], Power[Times[Optional[Pattern[b, Blank[]]], tan[Plus[Optional[Pattern[e, Blank[]]], Times[Optional[Pattern[f, Blank[]]], Pattern[x, Blank[]]]]]], Rational[-1, 2]]], Pattern[x, Blank[Symbol]]] := Dist[Sqrt[Sin[e + f*x]]/(Sqrt[Cos[e + f*x]]*Sqrt[b*Tan[e + f*x]]), Int[1/(Sqrt[Cos[e + f*x]]*Sqrt[Sin[e + f*x]]), x], x] /; FreeQ[{b, e, f}, x]
  2615. Int[Times[Power[sec[Plus[Optional[Pattern[e, Blank[]]], Times[Optional[Pattern[f, Blank[]]], Pattern[x, Blank[]]]]], -1], Power[Times[Optional[Pattern[b, Blank[]]], tan[Plus[Optional[Pattern[e, Blank[]]], Times[Optional[Pattern[f, Blank[]]], Pattern[x, Blank[]]]]]], Rational[1, 2]]], Pattern[x, Blank[Symbol]]] := Dist[(Sqrt[Cos[e + f*x]]*Sqrt[b*Tan[e + f*x]])/Sqrt[Sin[e + f*x]], Int[Sqrt[Cos[e + f*x]]*Sqrt[Sin[e + f*x]], x], x] /; FreeQ[{b, e, f}, x]
  2616. Int[Times[Power[Times[Optional[Pattern[a, Blank[]]], sec[Plus[Optional[Pattern[e, Blank[]]], Times[Optional[Pattern[f, Blank[]]], Pattern[x, Blank[]]]]]], Pattern[m, Blank[]]], Power[Times[Optional[Pattern[b, Blank[]]], tan[Plus[Optional[Pattern[e, Blank[]]], Times[Optional[Pattern[f, Blank[]]], Pattern[x, Blank[]]]]]], Pattern[n, Blank[]]]], Pattern[x, Blank[Symbol]]] := Dist[(a^(m + n)*(b*Tan[e + f*x])^n)/((a*Sec[e + f*x])^n*(b*Sin[e + f*x])^n), Int[(b*Sin[e + f*x])^n/Cos[e + f*x]^(m + n), x], x] /; FreeQ[{a, b, e, f, m, n}, x] && IntegerQ[n + 1/2] && IntegerQ[m + 1/2]
  2617. Int[Times[Power[Times[Optional[Pattern[a, Blank[]]], sec[Plus[Optional[Pattern[e, Blank[]]], Times[Optional[Pattern[f, Blank[]]], Pattern[x, Blank[]]]]]], Optional[Pattern[m, Blank[]]]], Power[Times[Optional[Pattern[b, Blank[]]], tan[Plus[Optional[Pattern[e, Blank[]]], Times[Optional[Pattern[f, Blank[]]], Pattern[x, Blank[]]]]]], Pattern[n, Blank[]]]], Pattern[x, Blank[Symbol]]] := Simp[((a*Sec[e + f*x])^m*(b*Tan[e + f*x])^(n + 1)*(Cos[e + f*x]^2)^((m + n + 1)/2)*Hypergeometric2F1[(n + 1)/2, (m + n + 1)/2, (n + 3)/2, Sin[e + f*x]^2])/(b*f*(n + 1)), x] /; FreeQ[{a, b, e, f, m, n}, x] && !IntegerQ[(n - 1)/2] && !IntegerQ[m/2]
  2618. Int[Times[Power[Times[csc[Plus[Optional[Pattern[e, Blank[]]], Times[Optional[Pattern[f, Blank[]]], Pattern[x, Blank[]]]]], Optional[Pattern[a, Blank[]]]], Pattern[m, Blank[]]], Power[Times[Optional[Pattern[b, Blank[]]], tan[Plus[Optional[Pattern[e, Blank[]]], Times[Optional[Pattern[f, Blank[]]], Pattern[x, Blank[]]]]]], Pattern[n, Blank[]]]], Pattern[x, Blank[Symbol]]] := Dist[(a*Csc[e + f*x])^FracPart[m]*(Sin[e + f*x]/a)^FracPart[m], Int[(b*Tan[e + f*x])^n/(Sin[e + f*x]/a)^m, x], x] /; FreeQ[{a, b, e, f, m, n}, x] && !IntegerQ[m] && !IntegerQ[n]
  2619. Int[Times[Power[Times[csc[Plus[Optional[Pattern[e, Blank[]]], Times[Optional[Pattern[f, Blank[]]], Pattern[x, Blank[]]]]], Optional[Pattern[a, Blank[]]]], Pattern[m, Blank[]]], Power[Times[Optional[Pattern[b, Blank[]]], sec[Plus[Optional[Pattern[e, Blank[]]], Times[Optional[Pattern[f, Blank[]]], Pattern[x, Blank[]]]]]], Pattern[n, Blank[]]]], Pattern[x, Blank[Symbol]]] := Simp[(a*b*(a*Csc[e + f*x])^(m - 1)*(b*Sec[e + f*x])^(n - 1))/(f*(n - 1)), x] /; FreeQ[{a, b, e, f, m, n}, x] && EqQ[m + n - 2, 0] && NeQ[n, 1]
  2620. Int[Times[Power[csc[Plus[Optional[Pattern[e, Blank[]]], Times[Optional[Pattern[f, Blank[]]], Pattern[x, Blank[]]]]], Optional[Pattern[m, Blank[]]]], Power[sec[Plus[Optional[Pattern[e, Blank[]]], Times[Optional[Pattern[f, Blank[]]], Pattern[x, Blank[]]]]], Optional[Pattern[n, Blank[]]]]], Pattern[x, Blank[Symbol]]] := Dist[1/f, Subst[Int[(1 + x^2)^((m + n)/2 - 1)/x^m, x], x, Tan[e + f*x]], x] /; FreeQ[{e, f}, x] && IntegersQ[m, n, (m + n)/2]
  2621. Int[Times[Power[Times[csc[Plus[Optional[Pattern[e, Blank[]]], Times[Optional[Pattern[f, Blank[]]], Pattern[x, Blank[]]]]], Optional[Pattern[a, Blank[]]]], Pattern[m, Blank[]]], Power[sec[Plus[Optional[Pattern[e, Blank[]]], Times[Optional[Pattern[f, Blank[]]], Pattern[x, Blank[]]]]], Optional[Pattern[n, Blank[]]]]], Pattern[x, Blank[Symbol]]] := -Dist[(f*a^n)^(-1), Subst[Int[x^(m + n - 1)/(-1 + x^2/a^2)^((n + 1)/2), x], x, a*Csc[e + f*x]], x] /; FreeQ[{a, e, f, m}, x] && IntegerQ[(n + 1)/2] && !(IntegerQ[(m + 1)/2] && LtQ[0, m, n])
  2622. Int[Times[Power[csc[Plus[Optional[Pattern[e, Blank[]]], Times[Optional[Pattern[f, Blank[]]], Pattern[x, Blank[]]]]], Optional[Pattern[n, Blank[]]]], Power[Times[Optional[Pattern[a, Blank[]]], sec[Plus[Optional[Pattern[e, Blank[]]], Times[Optional[Pattern[f, Blank[]]], Pattern[x, Blank[]]]]]], Pattern[m, Blank[]]]], Pattern[x, Blank[Symbol]]] := Dist[1/(f*a^n), Subst[Int[x^(m + n - 1)/(-1 + x^2/a^2)^((n + 1)/2), x], x, a*Sec[e + f*x]], x] /; FreeQ[{a, e, f, m}, x] && IntegerQ[(n + 1)/2] && !(IntegerQ[(m + 1)/2] && LtQ[0, m, n])
  2623. Int[Times[Power[Times[csc[Plus[Optional[Pattern[e, Blank[]]], Times[Optional[Pattern[f, Blank[]]], Pattern[x, Blank[]]]]], Optional[Pattern[a, Blank[]]]], Pattern[m, Blank[]]], Power[Times[Optional[Pattern[b, Blank[]]], sec[Plus[Optional[Pattern[e, Blank[]]], Times[Optional[Pattern[f, Blank[]]], Pattern[x, Blank[]]]]]], Pattern[n, Blank[]]]], Pattern[x, Blank[Symbol]]] := -Simp[(a*(a*Csc[e + f*x])^(m - 1)*(b*Sec[e + f*x])^(n + 1))/(f*b*(m - 1)), x] + Dist[(a^2*(n + 1))/(b^2*(m - 1)), Int[(a*Csc[e + f*x])^(m - 2)*(b*Sec[e + f*x])^(n + 2), x], x] /; FreeQ[{a, b, e, f}, x] && GtQ[m, 1] && LtQ[n, -1] && IntegersQ[2*m, 2*n]
  2624. Int[Times[Power[Times[csc[Plus[Optional[Pattern[e, Blank[]]], Times[Optional[Pattern[f, Blank[]]], Pattern[x, Blank[]]]]], Optional[Pattern[a, Blank[]]]], Pattern[m, Blank[]]], Power[Times[Optional[Pattern[b, Blank[]]], sec[Plus[Optional[Pattern[e, Blank[]]], Times[Optional[Pattern[f, Blank[]]], Pattern[x, Blank[]]]]]], Pattern[n, Blank[]]]], Pattern[x, Blank[Symbol]]] := Simp[(b*(a*Csc[e + f*x])^(m + 1)*(b*Sec[e + f*x])^(n - 1))/(f*a*(n - 1)), x] + Dist[(b^2*(m + 1))/(a^2*(n - 1)), Int[(a*Csc[e + f*x])^(m + 2)*(b*Sec[e + f*x])^(n - 2), x], x] /; FreeQ[{a, b, e, f}, x] && GtQ[n, 1] && LtQ[m, -1] && IntegersQ[2*m, 2*n]
  2625. Int[Times[Power[Times[csc[Plus[Optional[Pattern[e, Blank[]]], Times[Optional[Pattern[f, Blank[]]], Pattern[x, Blank[]]]]], Optional[Pattern[a, Blank[]]]], Pattern[m, Blank[]]], Power[Times[Optional[Pattern[b, Blank[]]], sec[Plus[Optional[Pattern[e, Blank[]]], Times[Optional[Pattern[f, Blank[]]], Pattern[x, Blank[]]]]]], Optional[Pattern[n, Blank[]]]]], Pattern[x, Blank[Symbol]]] := -Simp[(a*b*(a*Csc[e + f*x])^(m - 1)*(b*Sec[e + f*x])^(n - 1))/(f*(m - 1)), x] + Dist[(a^2*(m + n - 2))/(m - 1), Int[(a*Csc[e + f*x])^(m - 2)*(b*Sec[e + f*x])^n, x], x] /; FreeQ[{a, b, e, f, n}, x] && GtQ[m, 1] && IntegersQ[2*m, 2*n] && !GtQ[n, m]
  2626. Int[Times[Power[Times[csc[Plus[Optional[Pattern[e, Blank[]]], Times[Optional[Pattern[f, Blank[]]], Pattern[x, Blank[]]]]], Optional[Pattern[a, Blank[]]]], Optional[Pattern[m, Blank[]]]], Power[Times[Optional[Pattern[b, Blank[]]], sec[Plus[Optional[Pattern[e, Blank[]]], Times[Optional[Pattern[f, Blank[]]], Pattern[x, Blank[]]]]]], Pattern[n, Blank[]]]], Pattern[x, Blank[Symbol]]] := Simp[(a*b*(a*Csc[e + f*x])^(m - 1)*(b*Sec[e + f*x])^(n - 1))/(f*(n - 1)), x] + Dist[(b^2*(m + n - 2))/(n - 1), Int[(a*Csc[e + f*x])^m*(b*Sec[e + f*x])^(n - 2), x], x] /; FreeQ[{a, b, e, f, m}, x] && GtQ[n, 1] && IntegersQ[2*m, 2*n]
  2627. Int[Times[Power[Times[csc[Plus[Optional[Pattern[e, Blank[]]], Times[Optional[Pattern[f, Blank[]]], Pattern[x, Blank[]]]]], Optional[Pattern[a, Blank[]]]], Pattern[m, Blank[]]], Power[Times[Optional[Pattern[b, Blank[]]], sec[Plus[Optional[Pattern[e, Blank[]]], Times[Optional[Pattern[f, Blank[]]], Pattern[x, Blank[]]]]]], Optional[Pattern[n, Blank[]]]]], Pattern[x, Blank[Symbol]]] := Simp[(b*(a*Csc[e + f*x])^(m + 1)*(b*Sec[e + f*x])^(n - 1))/(a*f*(m + n)), x] + Dist[(m + 1)/(a^2*(m + n)), Int[(a*Csc[e + f*x])^(m + 2)*(b*Sec[e + f*x])^n, x], x] /; FreeQ[{a, b, e, f, n}, x] && LtQ[m, -1] && NeQ[m + n, 0] && IntegersQ[2*m, 2*n]
  2628. Int[Times[Power[Times[csc[Plus[Optional[Pattern[e, Blank[]]], Times[Optional[Pattern[f, Blank[]]], Pattern[x, Blank[]]]]], Optional[Pattern[a, Blank[]]]], Optional[Pattern[m, Blank[]]]], Power[Times[Optional[Pattern[b, Blank[]]], sec[Plus[Optional[Pattern[e, Blank[]]], Times[Optional[Pattern[f, Blank[]]], Pattern[x, Blank[]]]]]], Pattern[n, Blank[]]]], Pattern[x, Blank[Symbol]]] := -Simp[(a*(a*Csc[e + f*x])^(m - 1)*(b*Sec[e + f*x])^(n + 1))/(b*f*(m + n)), x] + Dist[(n + 1)/(b^2*(m + n)), Int[(a*Csc[e + f*x])^m*(b*Sec[e + f*x])^(n + 2), x], x] /; FreeQ[{a, b, e, f, m}, x] && LtQ[n, -1] && NeQ[m + n, 0] && IntegersQ[2*m, 2*n]
  2629. Int[Times[Power[Times[csc[Plus[Optional[Pattern[e, Blank[]]], Times[Optional[Pattern[f, Blank[]]], Pattern[x, Blank[]]]]], Optional[Pattern[a, Blank[]]]], Pattern[m, Blank[]]], Power[Times[Optional[Pattern[b, Blank[]]], sec[Plus[Optional[Pattern[e, Blank[]]], Times[Optional[Pattern[f, Blank[]]], Pattern[x, Blank[]]]]]], Pattern[n, Blank[]]]], Pattern[x, Blank[Symbol]]] := Dist[((a*Csc[e + f*x])^m*(b*Sec[e + f*x])^n)/Tan[e + f*x]^n, Int[Tan[e + f*x]^n, x], x] /; FreeQ[{a, b, e, f, m, n}, x] && !IntegerQ[n] && EqQ[m + n, 0]
  2630. Int[Times[Power[Times[csc[Plus[Optional[Pattern[e, Blank[]]], Times[Optional[Pattern[f, Blank[]]], Pattern[x, Blank[]]]]], Optional[Pattern[a, Blank[]]]], Pattern[m, Blank[]]], Power[Times[Optional[Pattern[b, Blank[]]], sec[Plus[Optional[Pattern[e, Blank[]]], Times[Optional[Pattern[f, Blank[]]], Pattern[x, Blank[]]]]]], Pattern[n, Blank[]]]], Pattern[x, Blank[Symbol]]] := Dist[(a*Csc[e + f*x])^m*(b*Sec[e + f*x])^n*(a*Sin[e + f*x])^m*(b*Cos[e + f*x])^n, Int[1/((a*Sin[e + f*x])^m*(b*Cos[e + f*x])^n), x], x] /; FreeQ[{a, b, e, f, m, n}, x] && IntegerQ[m - 1/2] && IntegerQ[n - 1/2]
  2631. Int[Times[Power[Times[csc[Plus[Optional[Pattern[e, Blank[]]], Times[Optional[Pattern[f, Blank[]]], Pattern[x, Blank[]]]]], Optional[Pattern[a, Blank[]]]], Pattern[m, Blank[]]], Power[Times[Optional[Pattern[b, Blank[]]], sec[Plus[Optional[Pattern[e, Blank[]]], Times[Optional[Pattern[f, Blank[]]], Pattern[x, Blank[]]]]]], Pattern[n, Blank[]]]], Pattern[x, Blank[Symbol]]] := Dist[(a^2*(a*Csc[e + f*x])^(m - 1)*(b*Sec[e + f*x])^(n + 1)*(a*Sin[e + f*x])^(m - 1)*(b*Cos[e + f*x])^(n + 1))/b^2, Int[1/((a*Sin[e + f*x])^m*(b*Cos[e + f*x])^n), x], x] /; FreeQ[{a, b, e, f, m, n}, x] && !SimplerQ[-m, -n]
  2632. Int[Times[Power[Times[csc[Plus[Optional[Pattern[e, Blank[]]], Times[Optional[Pattern[f, Blank[]]], Pattern[x, Blank[]]]]], Optional[Pattern[b, Blank[]]]], Pattern[n, Blank[]]], Power[Times[Optional[Pattern[a, Blank[]]], sec[Plus[Optional[Pattern[e, Blank[]]], Times[Optional[Pattern[f, Blank[]]], Pattern[x, Blank[]]]]]], Pattern[m, Blank[]]]], Pattern[x, Blank[Symbol]]] := Dist[(a^2*(a*Sec[e + f*x])^(m - 1)*(b*Csc[e + f*x])^(n + 1)*(a*Cos[e + f*x])^(m - 1)*(b*Sin[e + f*x])^(n + 1))/b^2, Int[1/((a*Cos[e + f*x])^m*(b*Sin[e + f*x])^n), x], x] /; FreeQ[{a, b, e, f, m, n}, x]
  2633. Int[Power[sin[Plus[Optional[Pattern[c, Blank[]]], Times[Optional[Pattern[d, Blank[]]], Pattern[x, Blank[]]]]], Pattern[n, Blank[]]], Pattern[x, Blank[Symbol]]] := -Dist[d^(-1), Subst[Int[Expand[(1 - x^2)^((n - 1)/2), x], x], x, Cos[c + d*x]], x] /; FreeQ[{c, d}, x] && IGtQ[(n - 1)/2, 0]
  2634. Int[Power[sin[Plus[Optional[Pattern[c, Blank[]]], Times[Rational[1, 2], Optional[Pattern[d, Blank[]]], Pattern[x, Blank[]]]]], 2], Pattern[x, Blank[Symbol]]] := Simp[x/2, x] - Simp[Sin[2*c + d*x]/(2*d), x] /; FreeQ[{c, d}, x]
  2635. Int[Power[Times[Optional[Pattern[b, Blank[]]], sin[Plus[Optional[Pattern[c, Blank[]]], Times[Optional[Pattern[d, Blank[]]], Pattern[x, Blank[]]]]]], Pattern[n, Blank[]]], Pattern[x, Blank[Symbol]]] := -Simp[(b*Cos[c + d*x]*(b*Sin[c + d*x])^(n - 1))/(d*n), x] + Dist[(b^2*(n - 1))/n, Int[(b*Sin[c + d*x])^(n - 2), x], x] /; FreeQ[{b, c, d}, x] && GtQ[n, 1] && IntegerQ[2*n]
  2636. Int[Power[Times[Optional[Pattern[b, Blank[]]], sin[Plus[Optional[Pattern[c, Blank[]]], Times[Optional[Pattern[d, Blank[]]], Pattern[x, Blank[]]]]]], Pattern[n, Blank[]]], Pattern[x, Blank[Symbol]]] := Simp[(Cos[c + d*x]*(b*Sin[c + d*x])^(n + 1))/(b*d*(n + 1)), x] + Dist[(n + 2)/(b^2*(n + 1)), Int[(b*Sin[c + d*x])^(n + 2), x], x] /; FreeQ[{b, c, d}, x] && LtQ[n, -1] && IntegerQ[2*n]
  2637. Int[sin[Plus[Times[Rational[1, 2], Pi], Optional[Pattern[c, Blank[]]], Times[Optional[Pattern[d, Blank[]]], Pattern[x, Blank[]]]]], Pattern[x, Blank[Symbol]]] := Simp[Sin[c + d*x]/d, x] /; FreeQ[{c, d}, x]
  2638. Int[sin[Plus[Optional[Pattern[c, Blank[]]], Times[Optional[Pattern[d, Blank[]]], Pattern[x, Blank[]]]]], Pattern[x, Blank[Symbol]]] := -Simp[Cos[c + d*x]/d, x] /; FreeQ[{c, d}, x]
  2639. Int[Power[sin[Plus[Optional[Pattern[c, Blank[]]], Times[Optional[Pattern[d, Blank[]]], Pattern[x, Blank[]]]]], Rational[1, 2]], Pattern[x, Blank[Symbol]]] := Simp[(2*EllipticE[(1*(c - Pi/2 + d*x))/2, 2])/d, x] /; FreeQ[{c, d}, x]
  2640. Int[Power[Times[Pattern[b, Blank[]], sin[Plus[Optional[Pattern[c, Blank[]]], Times[Optional[Pattern[d, Blank[]]], Pattern[x, Blank[]]]]]], Rational[1, 2]], Pattern[x, Blank[Symbol]]] := Dist[Sqrt[b*Sin[c + d*x]]/Sqrt[Sin[c + d*x]], Int[Sqrt[Sin[c + d*x]], x], x] /; FreeQ[{b, c, d}, x]
  2641. Int[Power[sin[Plus[Optional[Pattern[c, Blank[]]], Times[Optional[Pattern[d, Blank[]]], Pattern[x, Blank[]]]]], Rational[-1, 2]], Pattern[x, Blank[Symbol]]] := Simp[(2*EllipticF[(1*(c - Pi/2 + d*x))/2, 2])/d, x] /; FreeQ[{c, d}, x]
  2642. Int[Power[Times[Pattern[b, Blank[]], sin[Plus[Optional[Pattern[c, Blank[]]], Times[Optional[Pattern[d, Blank[]]], Pattern[x, Blank[]]]]]], Rational[-1, 2]], Pattern[x, Blank[Symbol]]] := Dist[Sqrt[Sin[c + d*x]]/Sqrt[b*Sin[c + d*x]], Int[1/Sqrt[Sin[c + d*x]], x], x] /; FreeQ[{b, c, d}, x]
  2643. Int[Power[Times[Optional[Pattern[b, Blank[]]], sin[Plus[Optional[Pattern[c, Blank[]]], Times[Optional[Pattern[d, Blank[]]], Pattern[x, Blank[]]]]]], Pattern[n, Blank[]]], Pattern[x, Blank[Symbol]]] := Simp[(Cos[c + d*x]*(b*Sin[c + d*x])^(n + 1)*Hypergeometric2F1[1/2, (n + 1)/2, (n + 3)/2, Sin[c + d*x]^2])/(b*d*(n + 1)*Sqrt[Cos[c + d*x]^2]), x] /; FreeQ[{b, c, d, n}, x] && !IntegerQ[2*n]
  2644. Int[Power[Plus[Pattern[a, Blank[]], Times[Optional[Pattern[b, Blank[]]], sin[Plus[Optional[Pattern[c, Blank[]]], Times[Optional[Pattern[d, Blank[]]], Pattern[x, Blank[]]]]]]], 2], Pattern[x, Blank[Symbol]]] := Simp[((2*a^2 + b^2)*x)/2, x] + (-Simp[(2*a*b*Cos[c + d*x])/d, x] - Simp[(b^2*Cos[c + d*x]*Sin[c + d*x])/(2*d), x]) /; FreeQ[{a, b, c, d}, x]
  2645. Int[Power[Plus[Pattern[a, Blank[]], Times[Optional[Pattern[b, Blank[]]], sin[Plus[Optional[Pattern[c, Blank[]]], Times[Optional[Pattern[d, Blank[]]], Pattern[x, Blank[]]]]]]], Pattern[n, Blank[]]], Pattern[x, Blank[Symbol]]] := Int[ExpandTrig[(a + b*sin[c + d*x])^n, x], x] /; FreeQ[{a, b, c, d, n}, x] && EqQ[a^2 - b^2, 0] && IGtQ[n, 0]
  2646. Int[Power[Plus[Pattern[a, Blank[]], Times[Optional[Pattern[b, Blank[]]], sin[Plus[Optional[Pattern[c, Blank[]]], Times[Optional[Pattern[d, Blank[]]], Pattern[x, Blank[]]]]]]], Rational[1, 2]], Pattern[x, Blank[Symbol]]] := Simp[(-2*b*Cos[c + d*x])/(d*Sqrt[a + b*Sin[c + d*x]]), x] /; FreeQ[{a, b, c, d}, x] && EqQ[a^2 - b^2, 0]
  2647. Int[Power[Plus[Pattern[a, Blank[]], Times[Optional[Pattern[b, Blank[]]], sin[Plus[Optional[Pattern[c, Blank[]]], Times[Optional[Pattern[d, Blank[]]], Pattern[x, Blank[]]]]]]], Pattern[n, Blank[]]], Pattern[x, Blank[Symbol]]] := -Simp[(b*Cos[c + d*x]*(a + b*Sin[c + d*x])^(n - 1))/(d*n), x] + Dist[(a*(2*n - 1))/n, Int[(a + b*Sin[c + d*x])^(n - 1), x], x] /; FreeQ[{a, b, c, d}, x] && EqQ[a^2 - b^2, 0] && IGtQ[n - 1/2, 0]
  2648. Int[Power[Plus[Pattern[a, Blank[]], Times[Optional[Pattern[b, Blank[]]], sin[Plus[Optional[Pattern[c, Blank[]]], Times[Optional[Pattern[d, Blank[]]], Pattern[x, Blank[]]]]]]], -1], Pattern[x, Blank[Symbol]]] := -Simp[Cos[c + d*x]/(d*(b + a*Sin[c + d*x])), x] /; FreeQ[{a, b, c, d}, x] && EqQ[a^2 - b^2, 0]
  2649. Int[Power[Plus[Pattern[a, Blank[]], Times[Optional[Pattern[b, Blank[]]], sin[Plus[Optional[Pattern[c, Blank[]]], Times[Optional[Pattern[d, Blank[]]], Pattern[x, Blank[]]]]]]], Rational[-1, 2]], Pattern[x, Blank[Symbol]]] := Dist[-2/d, Subst[Int[1/(2*a - x^2), x], x, (b*Cos[c + d*x])/Sqrt[a + b*Sin[c + d*x]]], x] /; FreeQ[{a, b, c, d}, x] && EqQ[a^2 - b^2, 0]
  2650. Int[Power[Plus[Pattern[a, Blank[]], Times[Optional[Pattern[b, Blank[]]], sin[Plus[Optional[Pattern[c, Blank[]]], Times[Optional[Pattern[d, Blank[]]], Pattern[x, Blank[]]]]]]], Pattern[n, Blank[]]], Pattern[x, Blank[Symbol]]] := Simp[(b*Cos[c + d*x]*(a + b*Sin[c + d*x])^n)/(a*d*(2*n + 1)), x] + Dist[(n + 1)/(a*(2*n + 1)), Int[(a + b*Sin[c + d*x])^(n + 1), x], x] /; FreeQ[{a, b, c, d}, x] && EqQ[a^2 - b^2, 0] && LtQ[n, -1] && IntegerQ[2*n]
  2651. Int[Power[Plus[Pattern[a, Blank[]], Times[Optional[Pattern[b, Blank[]]], sin[Plus[Optional[Pattern[c, Blank[]]], Times[Optional[Pattern[d, Blank[]]], Pattern[x, Blank[]]]]]]], Pattern[n, Blank[]]], Pattern[x, Blank[Symbol]]] := -Simp[(2^(n + 1/2)*a^(n - 1/2)*b*Cos[c + d*x]*Hypergeometric2F1[1/2, 1/2 - n, 3/2, (1*(1 - (b*Sin[c + d*x])/a))/2])/(d*Sqrt[a + b*Sin[c + d*x]]), x] /; FreeQ[{a, b, c, d, n}, x] && EqQ[a^2 - b^2, 0] && !IntegerQ[2*n] && GtQ[a, 0]
  2652. Int[Power[Plus[Pattern[a, Blank[]], Times[Optional[Pattern[b, Blank[]]], sin[Plus[Optional[Pattern[c, Blank[]]], Times[Optional[Pattern[d, Blank[]]], Pattern[x, Blank[]]]]]]], Pattern[n, Blank[]]], Pattern[x, Blank[Symbol]]] := Dist[(a^IntPart[n]*(a + b*Sin[c + d*x])^FracPart[n])/(1 + (b*Sin[c + d*x])/a)^FracPart[n], Int[(1 + (b*Sin[c + d*x])/a)^n, x], x] /; FreeQ[{a, b, c, d, n}, x] && EqQ[a^2 - b^2, 0] && !IntegerQ[2*n] && !GtQ[a, 0]
  2653. Int[Power[Plus[Pattern[a, Blank[]], Times[Optional[Pattern[b, Blank[]]], sin[Plus[Optional[Pattern[c, Blank[]]], Times[Optional[Pattern[d, Blank[]]], Pattern[x, Blank[]]]]]]], Rational[1, 2]], Pattern[x, Blank[Symbol]]] := Simp[(2*Sqrt[a + b]*EllipticE[(1*(c - Pi/2 + d*x))/2, (2*b)/(a + b)])/d, x] /; FreeQ[{a, b, c, d}, x] && NeQ[a^2 - b^2, 0] && GtQ[a + b, 0]
  2654. Int[Power[Plus[Pattern[a, Blank[]], Times[Optional[Pattern[b, Blank[]]], sin[Plus[Optional[Pattern[c, Blank[]]], Times[Optional[Pattern[d, Blank[]]], Pattern[x, Blank[]]]]]]], Rational[1, 2]], Pattern[x, Blank[Symbol]]] := Simp[(2*Sqrt[a - b]*EllipticE[(1*(c + Pi/2 + d*x))/2, (-2*b)/(a - b)])/d, x] /; FreeQ[{a, b, c, d}, x] && NeQ[a^2 - b^2, 0] && GtQ[a - b, 0]
  2655. Int[Power[Plus[Pattern[a, Blank[]], Times[Optional[Pattern[b, Blank[]]], sin[Plus[Optional[Pattern[c, Blank[]]], Times[Optional[Pattern[d, Blank[]]], Pattern[x, Blank[]]]]]]], Rational[1, 2]], Pattern[x, Blank[Symbol]]] := Dist[Sqrt[a + b*Sin[c + d*x]]/Sqrt[(a + b*Sin[c + d*x])/(a + b)], Int[Sqrt[a/(a + b) + (b*Sin[c + d*x])/(a + b)], x], x] /; FreeQ[{a, b, c, d}, x] && NeQ[a^2 - b^2, 0] && !GtQ[a + b, 0]
  2656. Int[Power[Plus[Pattern[a, Blank[]], Times[Optional[Pattern[b, Blank[]]], sin[Plus[Optional[Pattern[c, Blank[]]], Times[Optional[Pattern[d, Blank[]]], Pattern[x, Blank[]]]]]]], Pattern[n, Blank[]]], Pattern[x, Blank[Symbol]]] := -Simp[(b*Cos[c + d*x]*(a + b*Sin[c + d*x])^(n - 1))/(d*n), x] + Dist[1/n, Int[(a + b*Sin[c + d*x])^(n - 2)*Simp[a^2*n + b^2*(n - 1) + a*b*(2*n - 1)*Sin[c + d*x], x], x], x] /; FreeQ[{a, b, c, d}, x] && NeQ[a^2 - b^2, 0] && GtQ[n, 1] && IntegerQ[2*n]
  2657. Int[Power[Plus[Pattern[a, Blank[]], Times[Optional[Pattern[b, Blank[]]], sin[Plus[Optional[Pattern[c, Blank[]]], Times[Optional[Pattern[d, Blank[]]], Pattern[x, Blank[]]]]]]], -1], Pattern[x, Blank[Symbol]]] := With[{q = Rt[a^2 - b^2, 2]}, Simp[x/q, x] + Simp[(2*ArcTan[(b*Cos[c + d*x])/(a + q + b*Sin[c + d*x])])/(d*q), x]] /; FreeQ[{a, b, c, d}, x] && GtQ[a^2 - b^2, 0] && PosQ[a]
  2658. Int[Power[Plus[Pattern[a, Blank[]], Times[Optional[Pattern[b, Blank[]]], sin[Plus[Optional[Pattern[c, Blank[]]], Times[Optional[Pattern[d, Blank[]]], Pattern[x, Blank[]]]]]]], -1], Pattern[x, Blank[Symbol]]] := With[{q = Rt[a^2 - b^2, 2]}, -Simp[x/q, x] - Simp[(2*ArcTan[(b*Cos[c + d*x])/(a - q + b*Sin[c + d*x])])/(d*q), x]] /; FreeQ[{a, b, c, d}, x] && GtQ[a^2 - b^2, 0] && NegQ[a]
  2659. Int[Power[Plus[Pattern[a, Blank[]], Times[Optional[Pattern[b, Blank[]]], sin[Plus[Times[Rational[1, 2], Pi], Optional[Pattern[c, Blank[]]], Times[Optional[Pattern[d, Blank[]]], Pattern[x, Blank[]]]]]]], -1], Pattern[x, Blank[Symbol]]] := With[{e = FreeFactors[Tan[(c + d*x)/2], x]}, Dist[(2*e)/d, Subst[Int[1/(a + b + (a - b)*e^2*x^2), x], x, Tan[(c + d*x)/2]/e], x]] /; FreeQ[{a, b, c, d}, x] && NeQ[a^2 - b^2, 0]
  2660. Int[Power[Plus[Pattern[a, Blank[]], Times[Optional[Pattern[b, Blank[]]], sin[Plus[Optional[Pattern[c, Blank[]]], Times[Optional[Pattern[d, Blank[]]], Pattern[x, Blank[]]]]]]], -1], Pattern[x, Blank[Symbol]]] := With[{e = FreeFactors[Tan[(c + d*x)/2], x]}, Dist[(2*e)/d, Subst[Int[1/(a + 2*b*e*x + a*e^2*x^2), x], x, Tan[(c + d*x)/2]/e], x]] /; FreeQ[{a, b, c, d}, x] && NeQ[a^2 - b^2, 0]
  2661. Int[Power[Plus[Pattern[a, Blank[]], Times[Optional[Pattern[b, Blank[]]], sin[Plus[Optional[Pattern[c, Blank[]]], Times[Optional[Pattern[d, Blank[]]], Pattern[x, Blank[]]]]]]], Rational[-1, 2]], Pattern[x, Blank[Symbol]]] := Simp[(2*EllipticF[(1*(c - Pi/2 + d*x))/2, (2*b)/(a + b)])/(d*Sqrt[a + b]), x] /; FreeQ[{a, b, c, d}, x] && NeQ[a^2 - b^2, 0] && GtQ[a + b, 0]
  2662. Int[Power[Plus[Pattern[a, Blank[]], Times[Optional[Pattern[b, Blank[]]], sin[Plus[Optional[Pattern[c, Blank[]]], Times[Optional[Pattern[d, Blank[]]], Pattern[x, Blank[]]]]]]], Rational[-1, 2]], Pattern[x, Blank[Symbol]]] := Simp[(2*EllipticF[(1*(c + Pi/2 + d*x))/2, (-2*b)/(a - b)])/(d*Sqrt[a - b]), x] /; FreeQ[{a, b, c, d}, x] && NeQ[a^2 - b^2, 0] && GtQ[a - b, 0]
  2663. Int[Power[Plus[Pattern[a, Blank[]], Times[Optional[Pattern[b, Blank[]]], sin[Plus[Optional[Pattern[c, Blank[]]], Times[Optional[Pattern[d, Blank[]]], Pattern[x, Blank[]]]]]]], Rational[-1, 2]], Pattern[x, Blank[Symbol]]] := Dist[Sqrt[(a + b*Sin[c + d*x])/(a + b)]/Sqrt[a + b*Sin[c + d*x]], Int[1/Sqrt[a/(a + b) + (b*Sin[c + d*x])/(a + b)], x], x] /; FreeQ[{a, b, c, d}, x] && NeQ[a^2 - b^2, 0] && !GtQ[a + b, 0]
  2664. Int[Power[Plus[Pattern[a, Blank[]], Times[Optional[Pattern[b, Blank[]]], sin[Plus[Optional[Pattern[c, Blank[]]], Times[Optional[Pattern[d, Blank[]]], Pattern[x, Blank[]]]]]]], Pattern[n, Blank[]]], Pattern[x, Blank[Symbol]]] := -Simp[(b*Cos[c + d*x]*(a + b*Sin[c + d*x])^(n + 1))/(d*(n + 1)*(a^2 - b^2)), x] + Dist[1/((n + 1)*(a^2 - b^2)), Int[(a + b*Sin[c + d*x])^(n + 1)*Simp[a*(n + 1) - b*(n + 2)*Sin[c + d*x], x], x], x] /; FreeQ[{a, b, c, d}, x] && NeQ[a^2 - b^2, 0] && LtQ[n, -1] && IntegerQ[2*n]
  2665. Int[Power[Plus[Pattern[a, Blank[]], Times[Optional[Pattern[b, Blank[]]], sin[Plus[Optional[Pattern[c, Blank[]]], Times[Optional[Pattern[d, Blank[]]], Pattern[x, Blank[]]]]]]], Pattern[n, Blank[]]], Pattern[x, Blank[Symbol]]] := Dist[Cos[c + d*x]/(d*Sqrt[1 + Sin[c + d*x]]*Sqrt[1 - Sin[c + d*x]]), Subst[Int[(a + b*x)^n/(Sqrt[1 + x]*Sqrt[1 - x]), x], x, Sin[c + d*x]], x] /; FreeQ[{a, b, c, d, n}, x] && NeQ[a^2 - b^2, 0] && !IntegerQ[2*n]
  2666. Int[Power[Plus[Pattern[a, Blank[]], Times[cos[Plus[Optional[Pattern[c, Blank[]]], Times[Optional[Pattern[d, Blank[]]], Pattern[x, Blank[]]]]], Optional[Pattern[b, Blank[]]], sin[Plus[Optional[Pattern[c, Blank[]]], Times[Optional[Pattern[d, Blank[]]], Pattern[x, Blank[]]]]]]], Pattern[n, Blank[]]], Pattern[x, Blank[Symbol]]] := Int[(a + (b*Sin[2*c + 2*d*x])/2)^n, x] /; FreeQ[{a, b, c, d, n}, x]
  2667. Int[Times[Power[cos[Plus[Optional[Pattern[e, Blank[]]], Times[Optional[Pattern[f, Blank[]]], Pattern[x, Blank[]]]]], Optional[Pattern[p, Blank[]]]], Power[Plus[Pattern[a, Blank[]], Times[Optional[Pattern[b, Blank[]]], sin[Plus[Optional[Pattern[e, Blank[]]], Times[Optional[Pattern[f, Blank[]]], Pattern[x, Blank[]]]]]]], Optional[Pattern[m, Blank[]]]]], Pattern[x, Blank[Symbol]]] := Dist[1/(b^p*f), Subst[Int[(a + x)^(m + (p - 1)/2)*(a - x)^((p - 1)/2), x], x, b*Sin[e + f*x]], x] /; FreeQ[{a, b, e, f, m}, x] && IntegerQ[(p - 1)/2] && EqQ[a^2 - b^2, 0] && (GeQ[p, -1] || !IntegerQ[m + 1/2])
  2668. Int[Times[Power[cos[Plus[Optional[Pattern[e, Blank[]]], Times[Optional[Pattern[f, Blank[]]], Pattern[x, Blank[]]]]], Optional[Pattern[p, Blank[]]]], Power[Plus[Pattern[a, Blank[]], Times[Optional[Pattern[b, Blank[]]], sin[Plus[Optional[Pattern[e, Blank[]]], Times[Optional[Pattern[f, Blank[]]], Pattern[x, Blank[]]]]]]], Optional[Pattern[m, Blank[]]]]], Pattern[x, Blank[Symbol]]] := Dist[1/(b^p*f), Subst[Int[(a + x)^m*(b^2 - x^2)^((p - 1)/2), x], x, b*Sin[e + f*x]], x] /; FreeQ[{a, b, e, f, m}, x] && IntegerQ[(p - 1)/2] && NeQ[a^2 - b^2, 0]
  2669. Int[Times[Power[Times[cos[Plus[Optional[Pattern[e, Blank[]]], Times[Optional[Pattern[f, Blank[]]], Pattern[x, Blank[]]]]], Optional[Pattern[g, Blank[]]]], Pattern[p, Blank[]]], Plus[Pattern[a, Blank[]], Times[Optional[Pattern[b, Blank[]]], sin[Plus[Optional[Pattern[e, Blank[]]], Times[Optional[Pattern[f, Blank[]]], Pattern[x, Blank[]]]]]]]], Pattern[x, Blank[Symbol]]] := -Simp[(b*(g*Cos[e + f*x])^(p + 1))/(f*g*(p + 1)), x] + Dist[a, Int[(g*Cos[e + f*x])^p, x], x] /; FreeQ[{a, b, e, f, g, p}, x] && (IntegerQ[2*p] || NeQ[a^2 - b^2, 0])
  2670. Int[Times[Power[Times[cos[Plus[Optional[Pattern[e, Blank[]]], Times[Optional[Pattern[f, Blank[]]], Pattern[x, Blank[]]]]], Optional[Pattern[g, Blank[]]]], Pattern[p, Blank[]]], Power[Plus[Pattern[a, Blank[]], Times[Optional[Pattern[b, Blank[]]], sin[Plus[Optional[Pattern[e, Blank[]]], Times[Optional[Pattern[f, Blank[]]], Pattern[x, Blank[]]]]]]], Pattern[m, Blank[]]]], Pattern[x, Blank[Symbol]]] := Dist[(a/g)^(2*m), Int[(g*Cos[e + f*x])^(2*m + p)/(a - b*Sin[e + f*x])^m, x], x] /; FreeQ[{a, b, e, f, g}, x] && EqQ[a^2 - b^2, 0] && IntegerQ[m] && LtQ[p, -1] && GeQ[2*m + p, 0]
  2671. Int[Times[Power[Times[cos[Plus[Optional[Pattern[e, Blank[]]], Times[Optional[Pattern[f, Blank[]]], Pattern[x, Blank[]]]]], Optional[Pattern[g, Blank[]]]], Pattern[p, Blank[]]], Power[Plus[Pattern[a, Blank[]], Times[Optional[Pattern[b, Blank[]]], sin[Plus[Optional[Pattern[e, Blank[]]], Times[Optional[Pattern[f, Blank[]]], Pattern[x, Blank[]]]]]]], Pattern[m, Blank[]]]], Pattern[x, Blank[Symbol]]] := Simp[(b*(g*Cos[e + f*x])^(p + 1)*(a + b*Sin[e + f*x])^m)/(a*f*g*m), x] /; FreeQ[{a, b, e, f, g, m, p}, x] && EqQ[a^2 - b^2, 0] && EqQ[Simplify[m + p + 1], 0] && !ILtQ[p, 0]
  2672. Int[Times[Power[Times[cos[Plus[Optional[Pattern[e, Blank[]]], Times[Optional[Pattern[f, Blank[]]], Pattern[x, Blank[]]]]], Optional[Pattern[g, Blank[]]]], Pattern[p, Blank[]]], Power[Plus[Pattern[a, Blank[]], Times[Optional[Pattern[b, Blank[]]], sin[Plus[Optional[Pattern[e, Blank[]]], Times[Optional[Pattern[f, Blank[]]], Pattern[x, Blank[]]]]]]], Pattern[m, Blank[]]]], Pattern[x, Blank[Symbol]]] := Simp[(b*(g*Cos[e + f*x])^(p + 1)*(a + b*Sin[e + f*x])^m)/(a*f*g*Simplify[2*m + p + 1]), x] + Dist[Simplify[m + p + 1]/(a*Simplify[2*m + p + 1]), Int[(g*Cos[e + f*x])^p*(a + b*Sin[e + f*x])^(m + 1), x], x] /; FreeQ[{a, b, e, f, g, m, p}, x] && EqQ[a^2 - b^2, 0] && ILtQ[Simplify[m + p + 1], 0] && NeQ[2*m + p + 1, 0] && !IGtQ[m, 0]
  2673. Int[Times[Power[Times[cos[Plus[Optional[Pattern[e, Blank[]]], Times[Optional[Pattern[f, Blank[]]], Pattern[x, Blank[]]]]], Optional[Pattern[g, Blank[]]]], Pattern[p, Blank[]]], Power[Plus[Pattern[a, Blank[]], Times[Optional[Pattern[b, Blank[]]], sin[Plus[Optional[Pattern[e, Blank[]]], Times[Optional[Pattern[f, Blank[]]], Pattern[x, Blank[]]]]]]], Pattern[m, Blank[]]]], Pattern[x, Blank[Symbol]]] := Simp[(b*(g*Cos[e + f*x])^(p + 1)*(a + b*Sin[e + f*x])^(m - 1))/(f*g*(m - 1)), x] /; FreeQ[{a, b, e, f, g, m, p}, x] && EqQ[a^2 - b^2, 0] && EqQ[2*m + p - 1, 0] && NeQ[m, 1]
  2674. Int[Times[Power[Times[cos[Plus[Optional[Pattern[e, Blank[]]], Times[Optional[Pattern[f, Blank[]]], Pattern[x, Blank[]]]]], Optional[Pattern[g, Blank[]]]], Pattern[p, Blank[]]], Power[Plus[Pattern[a, Blank[]], Times[Optional[Pattern[b, Blank[]]], sin[Plus[Optional[Pattern[e, Blank[]]], Times[Optional[Pattern[f, Blank[]]], Pattern[x, Blank[]]]]]]], Pattern[m, Blank[]]]], Pattern[x, Blank[Symbol]]] := -Simp[(b*(g*Cos[e + f*x])^(p + 1)*(a + b*Sin[e + f*x])^(m - 1))/(f*g*(m + p)), x] + Dist[(a*(2*m + p - 1))/(m + p), Int[(g*Cos[e + f*x])^p*(a + b*Sin[e + f*x])^(m - 1), x], x] /; FreeQ[{a, b, e, f, g, m, p}, x] && EqQ[a^2 - b^2, 0] && IGtQ[Simplify[(2*m + p - 1)/2], 0] && NeQ[m + p, 0]
  2675. Int[Times[Power[Times[cos[Plus[Optional[Pattern[e, Blank[]]], Times[Optional[Pattern[f, Blank[]]], Pattern[x, Blank[]]]]], Optional[Pattern[g, Blank[]]]], Pattern[p, Blank[]]], Power[Plus[Pattern[a, Blank[]], Times[Optional[Pattern[b, Blank[]]], sin[Plus[Optional[Pattern[e, Blank[]]], Times[Optional[Pattern[f, Blank[]]], Pattern[x, Blank[]]]]]]], Pattern[m, Blank[]]]], Pattern[x, Blank[Symbol]]] := -Simp[(b*(g*Cos[e + f*x])^(p + 1)*(a + b*Sin[e + f*x])^m)/(a*f*g*(p + 1)), x] + Dist[(a*(m + p + 1))/(g^2*(p + 1)), Int[(g*Cos[e + f*x])^(p + 2)*(a + b*Sin[e + f*x])^(m - 1), x], x] /; FreeQ[{a, b, e, f, g}, x] && EqQ[a^2 - b^2, 0] && GtQ[m, 0] && LeQ[p, -2*m] && IntegersQ[m + 1/2, 2*p]
  2676. Int[Times[Power[Times[cos[Plus[Optional[Pattern[e, Blank[]]], Times[Optional[Pattern[f, Blank[]]], Pattern[x, Blank[]]]]], Optional[Pattern[g, Blank[]]]], Pattern[p, Blank[]]], Power[Plus[Pattern[a, Blank[]], Times[Optional[Pattern[b, Blank[]]], sin[Plus[Optional[Pattern[e, Blank[]]], Times[Optional[Pattern[f, Blank[]]], Pattern[x, Blank[]]]]]]], Pattern[m, Blank[]]]], Pattern[x, Blank[Symbol]]] := Simp[(-2*b*(g*Cos[e + f*x])^(p + 1)*(a + b*Sin[e + f*x])^(m - 1))/(f*g*(p + 1)), x] + Dist[(b^2*(2*m + p - 1))/(g^2*(p + 1)), Int[(g*Cos[e + f*x])^(p + 2)*(a + b*Sin[e + f*x])^(m - 2), x], x] /; FreeQ[{a, b, e, f, g}, x] && EqQ[a^2 - b^2, 0] && GtQ[m, 1] && LtQ[p, -1] && IntegersQ[2*m, 2*p]
  2677. Int[Times[Power[Times[cos[Plus[Optional[Pattern[e, Blank[]]], Times[Optional[Pattern[f, Blank[]]], Pattern[x, Blank[]]]]], Optional[Pattern[g, Blank[]]]], Rational[-1, 2]], Power[Plus[Pattern[a, Blank[]], Times[Optional[Pattern[b, Blank[]]], sin[Plus[Optional[Pattern[e, Blank[]]], Times[Optional[Pattern[f, Blank[]]], Pattern[x, Blank[]]]]]]], Rational[1, 2]]], Pattern[x, Blank[Symbol]]] := Dist[(a*Sqrt[1 + Cos[e + f*x]]*Sqrt[a + b*Sin[e + f*x]])/(a + a*Cos[e + f*x] + b*Sin[e + f*x]), Int[Sqrt[1 + Cos[e + f*x]]/Sqrt[g*Cos[e + f*x]], x], x] + Dist[(b*Sqrt[1 + Cos[e + f*x]]*Sqrt[a + b*Sin[e + f*x]])/(a + a*Cos[e + f*x] + b*Sin[e + f*x]), Int[Sin[e + f*x]/(Sqrt[g*Cos[e + f*x]]*Sqrt[1 + Cos[e + f*x]]), x], x] /; FreeQ[{a, b, e, f, g}, x] && EqQ[a^2 - b^2, 0]
  2678. Int[Times[Power[Times[cos[Plus[Optional[Pattern[e, Blank[]]], Times[Optional[Pattern[f, Blank[]]], Pattern[x, Blank[]]]]], Optional[Pattern[g, Blank[]]]], Pattern[p, Blank[]]], Power[Plus[Pattern[a, Blank[]], Times[Optional[Pattern[b, Blank[]]], sin[Plus[Optional[Pattern[e, Blank[]]], Times[Optional[Pattern[f, Blank[]]], Pattern[x, Blank[]]]]]]], Pattern[m, Blank[]]]], Pattern[x, Blank[Symbol]]] := -Simp[(b*(g*Cos[e + f*x])^(p + 1)*(a + b*Sin[e + f*x])^(m - 1))/(f*g*(m + p)), x] + Dist[(a*(2*m + p - 1))/(m + p), Int[(g*Cos[e + f*x])^p*(a + b*Sin[e + f*x])^(m - 1), x], x] /; FreeQ[{a, b, e, f, g, m, p}, x] && EqQ[a^2 - b^2, 0] && GtQ[m, 0] && NeQ[m + p, 0] && IntegersQ[2*m, 2*p]
  2679. Int[Times[Power[Times[cos[Plus[Optional[Pattern[e, Blank[]]], Times[Optional[Pattern[f, Blank[]]], Pattern[x, Blank[]]]]], Optional[Pattern[g, Blank[]]]], Pattern[p, Blank[]]], Power[Plus[Pattern[a, Blank[]], Times[Optional[Pattern[b, Blank[]]], sin[Plus[Optional[Pattern[e, Blank[]]], Times[Optional[Pattern[f, Blank[]]], Pattern[x, Blank[]]]]]]], Pattern[m, Blank[]]]], Pattern[x, Blank[Symbol]]] := Simp[(g*(g*Cos[e + f*x])^(p - 1)*(a + b*Sin[e + f*x])^(m + 1))/(b*f*(m + p)), x] + Dist[(g^2*(p - 1))/(a*(m + p)), Int[(g*Cos[e + f*x])^(p - 2)*(a + b*Sin[e + f*x])^(m + 1), x], x] /; FreeQ[{a, b, e, f, g}, x] && EqQ[a^2 - b^2, 0] && LtQ[m, -1] && GtQ[p, 1] && (GtQ[m, -2] || EqQ[2*m + p + 1, 0] || (EqQ[m, -2] && IntegerQ[p])) && NeQ[m + p, 0] && IntegersQ[2*m, 2*p]
  2680. Int[Times[Power[Times[cos[Plus[Optional[Pattern[e, Blank[]]], Times[Optional[Pattern[f, Blank[]]], Pattern[x, Blank[]]]]], Optional[Pattern[g, Blank[]]]], Pattern[p, Blank[]]], Power[Plus[Pattern[a, Blank[]], Times[Optional[Pattern[b, Blank[]]], sin[Plus[Optional[Pattern[e, Blank[]]], Times[Optional[Pattern[f, Blank[]]], Pattern[x, Blank[]]]]]]], Pattern[m, Blank[]]]], Pattern[x, Blank[Symbol]]] := Simp[(2*g*(g*Cos[e + f*x])^(p - 1)*(a + b*Sin[e + f*x])^(m + 1))/(b*f*(2*m + p + 1)), x] + Dist[(g^2*(p - 1))/(b^2*(2*m + p + 1)), Int[(g*Cos[e + f*x])^(p - 2)*(a + b*Sin[e + f*x])^(m + 2), x], x] /; FreeQ[{a, b, e, f, g}, x] && EqQ[a^2 - b^2, 0] && LeQ[m, -2] && GtQ[p, 1] && NeQ[2*m + p + 1, 0] && !ILtQ[m + p + 1, 0] && IntegersQ[2*m, 2*p]
  2681. Int[Times[Power[Times[cos[Plus[Optional[Pattern[e, Blank[]]], Times[Optional[Pattern[f, Blank[]]], Pattern[x, Blank[]]]]], Optional[Pattern[g, Blank[]]]], Pattern[p, Blank[]]], Power[Plus[Pattern[a, Blank[]], Times[Optional[Pattern[b, Blank[]]], sin[Plus[Optional[Pattern[e, Blank[]]], Times[Optional[Pattern[f, Blank[]]], Pattern[x, Blank[]]]]]]], Pattern[m, Blank[]]]], Pattern[x, Blank[Symbol]]] := Simp[(b*(g*Cos[e + f*x])^(p + 1)*(a + b*Sin[e + f*x])^m)/(a*f*g*(2*m + p + 1)), x] + Dist[(m + p + 1)/(a*(2*m + p + 1)), Int[(g*Cos[e + f*x])^p*(a + b*Sin[e + f*x])^(m + 1), x], x] /; FreeQ[{a, b, e, f, g, m, p}, x] && EqQ[a^2 - b^2, 0] && LtQ[m, -1] && NeQ[2*m + p + 1, 0] && IntegersQ[2*m, 2*p]
  2682. Int[Times[Power[Times[cos[Plus[Optional[Pattern[e, Blank[]]], Times[Optional[Pattern[f, Blank[]]], Pattern[x, Blank[]]]]], Optional[Pattern[g, Blank[]]]], Pattern[p, Blank[]]], Power[Plus[Pattern[a, Blank[]], Times[Optional[Pattern[b, Blank[]]], sin[Plus[Optional[Pattern[e, Blank[]]], Times[Optional[Pattern[f, Blank[]]], Pattern[x, Blank[]]]]]]], -1]], Pattern[x, Blank[Symbol]]] := Simp[(g*(g*Cos[e + f*x])^(p - 1))/(b*f*(p - 1)), x] + Dist[g^2/a, Int[(g*Cos[e + f*x])^(p - 2), x], x] /; FreeQ[{a, b, e, f, g}, x] && EqQ[a^2 - b^2, 0] && GtQ[p, 1] && IntegerQ[2*p]
  2683. Int[Times[Power[Times[cos[Plus[Optional[Pattern[e, Blank[]]], Times[Optional[Pattern[f, Blank[]]], Pattern[x, Blank[]]]]], Optional[Pattern[g, Blank[]]]], Pattern[p, Blank[]]], Power[Plus[Pattern[a, Blank[]], Times[Optional[Pattern[b, Blank[]]], sin[Plus[Optional[Pattern[e, Blank[]]], Times[Optional[Pattern[f, Blank[]]], Pattern[x, Blank[]]]]]]], -1]], Pattern[x, Blank[Symbol]]] := Simp[(b*(g*Cos[e + f*x])^(p + 1))/(a*f*g*(p - 1)*(a + b*Sin[e + f*x])), x] + Dist[p/(a*(p - 1)), Int[(g*Cos[e + f*x])^p, x], x] /; FreeQ[{a, b, e, f, g, p}, x] && EqQ[a^2 - b^2, 0] && !GeQ[p, 1] && IntegerQ[2*p]
  2684. Int[Times[Power[Times[cos[Plus[Optional[Pattern[e, Blank[]]], Times[Optional[Pattern[f, Blank[]]], Pattern[x, Blank[]]]]], Optional[Pattern[g, Blank[]]]], Rational[1, 2]], Power[Plus[Pattern[a, Blank[]], Times[Optional[Pattern[b, Blank[]]], sin[Plus[Optional[Pattern[e, Blank[]]], Times[Optional[Pattern[f, Blank[]]], Pattern[x, Blank[]]]]]]], Rational[-1, 2]]], Pattern[x, Blank[Symbol]]] := Dist[(g*Sqrt[1 + Cos[e + f*x]]*Sqrt[a + b*Sin[e + f*x]])/(a + a*Cos[e + f*x] + b*Sin[e + f*x]), Int[Sqrt[1 + Cos[e + f*x]]/Sqrt[g*Cos[e + f*x]], x], x] - Dist[(g*Sqrt[1 + Cos[e + f*x]]*Sqrt[a + b*Sin[e + f*x]])/(b + b*Cos[e + f*x] + a*Sin[e + f*x]), Int[Sin[e + f*x]/(Sqrt[g*Cos[e + f*x]]*Sqrt[1 + Cos[e + f*x]]), x], x] /; FreeQ[{a, b, e, f, g}, x] && EqQ[a^2 - b^2, 0]
  2685. Int[Times[Power[Times[cos[Plus[Optional[Pattern[e, Blank[]]], Times[Optional[Pattern[f, Blank[]]], Pattern[x, Blank[]]]]], Optional[Pattern[g, Blank[]]]], Rational[3, 2]], Power[Plus[Pattern[a, Blank[]], Times[Optional[Pattern[b, Blank[]]], sin[Plus[Optional[Pattern[e, Blank[]]], Times[Optional[Pattern[f, Blank[]]], Pattern[x, Blank[]]]]]]], Rational[-1, 2]]], Pattern[x, Blank[Symbol]]] := Simp[(g*Sqrt[g*Cos[e + f*x]]*Sqrt[a + b*Sin[e + f*x]])/(b*f), x] + Dist[g^2/(2*a), Int[Sqrt[a + b*Sin[e + f*x]]/Sqrt[g*Cos[e + f*x]], x], x] /; FreeQ[{a, b, e, f, g}, x] && EqQ[a^2 - b^2, 0]
  2686. Int[Times[Power[Times[cos[Plus[Optional[Pattern[e, Blank[]]], Times[Optional[Pattern[f, Blank[]]], Pattern[x, Blank[]]]]], Optional[Pattern[g, Blank[]]]], Pattern[p, Blank[]]], Power[Plus[Pattern[a, Blank[]], Times[Optional[Pattern[b, Blank[]]], sin[Plus[Optional[Pattern[e, Blank[]]], Times[Optional[Pattern[f, Blank[]]], Pattern[x, Blank[]]]]]]], Rational[-1, 2]]], Pattern[x, Blank[Symbol]]] := Simp[(-2*b*(g*Cos[e + f*x])^(p + 1))/(f*g*(2*p - 1)*(a + b*Sin[e + f*x])^(3/2)), x] + Dist[(2*a*(p - 2))/(2*p - 1), Int[(g*Cos[e + f*x])^p/(a + b*Sin[e + f*x])^(3/2), x], x] /; FreeQ[{a, b, e, f, g}, x] && EqQ[a^2 - b^2, 0] && GtQ[p, 2] && IntegerQ[2*p]
  2687. Int[Times[Power[Times[cos[Plus[Optional[Pattern[e, Blank[]]], Times[Optional[Pattern[f, Blank[]]], Pattern[x, Blank[]]]]], Optional[Pattern[g, Blank[]]]], Pattern[p, Blank[]]], Power[Plus[Pattern[a, Blank[]], Times[Optional[Pattern[b, Blank[]]], sin[Plus[Optional[Pattern[e, Blank[]]], Times[Optional[Pattern[f, Blank[]]], Pattern[x, Blank[]]]]]]], Rational[-1, 2]]], Pattern[x, Blank[Symbol]]] := -Simp[(b*(g*Cos[e + f*x])^(p + 1))/(a*f*g*(p + 1)*Sqrt[a + b*Sin[e + f*x]]), x] + Dist[(a*(2*p + 1))/(2*g^2*(p + 1)), Int[(g*Cos[e + f*x])^(p + 2)/(a + b*Sin[e + f*x])^(3/2), x], x] /; FreeQ[{a, b, e, f, g}, x] && EqQ[a^2 - b^2, 0] && LtQ[p, -1] && IntegerQ[2*p]
  2688. Int[Times[Power[Times[cos[Plus[Optional[Pattern[e, Blank[]]], Times[Optional[Pattern[f, Blank[]]], Pattern[x, Blank[]]]]], Optional[Pattern[g, Blank[]]]], Pattern[p, Blank[]]], Power[Plus[Pattern[a, Blank[]], Times[Optional[Pattern[b, Blank[]]], sin[Plus[Optional[Pattern[e, Blank[]]], Times[Optional[Pattern[f, Blank[]]], Pattern[x, Blank[]]]]]]], Optional[Pattern[m, Blank[]]]]], Pattern[x, Blank[Symbol]]] := Dist[(a^m*(g*Cos[e + f*x])^(p + 1))/(f*g*(1 + Sin[e + f*x])^((p + 1)/2)*(1 - Sin[e + f*x])^((p + 1)/2)), Subst[Int[(1 + (b*x)/a)^(m + (p - 1)/2)*(1 - (b*x)/a)^((p - 1)/2), x], x, Sin[e + f*x]], x] /; FreeQ[{a, b, e, f, g, p}, x] && EqQ[a^2 - b^2, 0] && IntegerQ[m]
  2689. Int[Times[Power[Times[cos[Plus[Optional[Pattern[e, Blank[]]], Times[Optional[Pattern[f, Blank[]]], Pattern[x, Blank[]]]]], Optional[Pattern[g, Blank[]]]], Pattern[p, Blank[]]], Power[Plus[Pattern[a, Blank[]], Times[Optional[Pattern[b, Blank[]]], sin[Plus[Optional[Pattern[e, Blank[]]], Times[Optional[Pattern[f, Blank[]]], Pattern[x, Blank[]]]]]]], Optional[Pattern[m, Blank[]]]]], Pattern[x, Blank[Symbol]]] := Dist[(a^2*(g*Cos[e + f*x])^(p + 1))/(f*g*(a + b*Sin[e + f*x])^((p + 1)/2)*(a - b*Sin[e + f*x])^((p + 1)/2)), Subst[Int[(a + b*x)^(m + (p - 1)/2)*(a - b*x)^((p - 1)/2), x], x, Sin[e + f*x]], x] /; FreeQ[{a, b, e, f, g, m, p}, x] && EqQ[a^2 - b^2, 0] && !IntegerQ[m]
  2690. Int[Times[Power[Times[cos[Plus[Optional[Pattern[e, Blank[]]], Times[Optional[Pattern[f, Blank[]]], Pattern[x, Blank[]]]]], Optional[Pattern[g, Blank[]]]], Pattern[p, Blank[]]], Power[Plus[Pattern[a, Blank[]], Times[Optional[Pattern[b, Blank[]]], sin[Plus[Optional[Pattern[e, Blank[]]], Times[Optional[Pattern[f, Blank[]]], Pattern[x, Blank[]]]]]]], Pattern[m, Blank[]]]], Pattern[x, Blank[Symbol]]] := -Simp[((g*Cos[e + f*x])^(p + 1)*(a + b*Sin[e + f*x])^m*Sin[e + f*x])/(f*g*(p + 1)), x] + Dist[1/(g^2*(p + 1)), Int[(g*Cos[e + f*x])^(p + 2)*(a + b*Sin[e + f*x])^(m - 1)*(a*(p + 2) + b*(m + p + 2)*Sin[e + f*x]), x], x] /; FreeQ[{a, b, e, f, g}, x] && NeQ[a^2 - b^2, 0] && LtQ[0, m, 1] && LtQ[p, -1] && (IntegersQ[2*m, 2*p] || IntegerQ[m])
  2691. Int[Times[Power[Times[cos[Plus[Optional[Pattern[e, Blank[]]], Times[Optional[Pattern[f, Blank[]]], Pattern[x, Blank[]]]]], Optional[Pattern[g, Blank[]]]], Pattern[p, Blank[]]], Power[Plus[Pattern[a, Blank[]], Times[Optional[Pattern[b, Blank[]]], sin[Plus[Optional[Pattern[e, Blank[]]], Times[Optional[Pattern[f, Blank[]]], Pattern[x, Blank[]]]]]]], Pattern[m, Blank[]]]], Pattern[x, Blank[Symbol]]] := -Simp[((g*Cos[e + f*x])^(p + 1)*(a + b*Sin[e + f*x])^(m - 1)*(b + a*Sin[e + f*x]))/(f*g*(p + 1)), x] + Dist[1/(g^2*(p + 1)), Int[(g*Cos[e + f*x])^(p + 2)*(a + b*Sin[e + f*x])^(m - 2)*(b^2*(m - 1) + a^2*(p + 2) + a*b*(m + p + 1)*Sin[e + f*x]), x], x] /; FreeQ[{a, b, e, f, g}, x] && NeQ[a^2 - b^2, 0] && GtQ[m, 1] && LtQ[p, -1] && (IntegersQ[2*m, 2*p] || IntegerQ[m])
  2692. Int[Times[Power[Times[cos[Plus[Optional[Pattern[e, Blank[]]], Times[Optional[Pattern[f, Blank[]]], Pattern[x, Blank[]]]]], Optional[Pattern[g, Blank[]]]], Pattern[p, Blank[]]], Power[Plus[Pattern[a, Blank[]], Times[Optional[Pattern[b, Blank[]]], sin[Plus[Optional[Pattern[e, Blank[]]], Times[Optional[Pattern[f, Blank[]]], Pattern[x, Blank[]]]]]]], Pattern[m, Blank[]]]], Pattern[x, Blank[Symbol]]] := -Simp[(b*(g*Cos[e + f*x])^(p + 1)*(a + b*Sin[e + f*x])^(m - 1))/(f*g*(m + p)), x] + Dist[1/(m + p), Int[(g*Cos[e + f*x])^p*(a + b*Sin[e + f*x])^(m - 2)*(b^2*(m - 1) + a^2*(m + p) + a*b*(2*m + p - 1)*Sin[e + f*x]), x], x] /; FreeQ[{a, b, e, f, g, p}, x] && NeQ[a^2 - b^2, 0] && GtQ[m, 1] && NeQ[m + p, 0] && (IntegersQ[2*m, 2*p] || IntegerQ[m])
  2693. Int[Times[Power[Times[cos[Plus[Optional[Pattern[e, Blank[]]], Times[Optional[Pattern[f, Blank[]]], Pattern[x, Blank[]]]]], Optional[Pattern[g, Blank[]]]], Pattern[p, Blank[]]], Power[Plus[Pattern[a, Blank[]], Times[Optional[Pattern[b, Blank[]]], sin[Plus[Optional[Pattern[e, Blank[]]], Times[Optional[Pattern[f, Blank[]]], Pattern[x, Blank[]]]]]]], Pattern[m, Blank[]]]], Pattern[x, Blank[Symbol]]] := Simp[(g*(g*Cos[e + f*x])^(p - 1)*(a + b*Sin[e + f*x])^(m + 1))/(b*f*(m + 1)), x] + Dist[(g^2*(p - 1))/(b*(m + 1)), Int[(g*Cos[e + f*x])^(p - 2)*(a + b*Sin[e + f*x])^(m + 1)*Sin[e + f*x], x], x] /; FreeQ[{a, b, e, f, g}, x] && NeQ[a^2 - b^2, 0] && LtQ[m, -1] && GtQ[p, 1] && IntegersQ[2*m, 2*p]
  2694. Int[Times[Power[Times[cos[Plus[Optional[Pattern[e, Blank[]]], Times[Optional[Pattern[f, Blank[]]], Pattern[x, Blank[]]]]], Optional[Pattern[g, Blank[]]]], Pattern[p, Blank[]]], Power[Plus[Pattern[a, Blank[]], Times[Optional[Pattern[b, Blank[]]], sin[Plus[Optional[Pattern[e, Blank[]]], Times[Optional[Pattern[f, Blank[]]], Pattern[x, Blank[]]]]]]], Pattern[m, Blank[]]]], Pattern[x, Blank[Symbol]]] := -Simp[(b*(g*Cos[e + f*x])^(p + 1)*(a + b*Sin[e + f*x])^(m + 1))/(f*g*(a^2 - b^2)*(m + 1)), x] + Dist[1/((a^2 - b^2)*(m + 1)), Int[(g*Cos[e + f*x])^p*(a + b*Sin[e + f*x])^(m + 1)*(a*(m + 1) - b*(m + p + 2)*Sin[e + f*x]), x], x] /; FreeQ[{a, b, e, f, g, p}, x] && NeQ[a^2 - b^2, 0] && LtQ[m, -1] && IntegersQ[2*m, 2*p]
  2695. Int[Times[Power[Times[cos[Plus[Optional[Pattern[e, Blank[]]], Times[Optional[Pattern[f, Blank[]]], Pattern[x, Blank[]]]]], Optional[Pattern[g, Blank[]]]], Pattern[p, Blank[]]], Power[Plus[Pattern[a, Blank[]], Times[Optional[Pattern[b, Blank[]]], sin[Plus[Optional[Pattern[e, Blank[]]], Times[Optional[Pattern[f, Blank[]]], Pattern[x, Blank[]]]]]]], Pattern[m, Blank[]]]], Pattern[x, Blank[Symbol]]] := Simp[(g*(g*Cos[e + f*x])^(p - 1)*(a + b*Sin[e + f*x])^(m + 1))/(b*f*(m + p)), x] + Dist[(g^2*(p - 1))/(b*(m + p)), Int[(g*Cos[e + f*x])^(p - 2)*(a + b*Sin[e + f*x])^m*(b + a*Sin[e + f*x]), x], x] /; FreeQ[{a, b, e, f, g, m}, x] && NeQ[a^2 - b^2, 0] && GtQ[p, 1] && NeQ[m + p, 0] && IntegersQ[2*m, 2*p]
  2696. Int[Times[Power[Times[cos[Plus[Optional[Pattern[e, Blank[]]], Times[Optional[Pattern[f, Blank[]]], Pattern[x, Blank[]]]]], Optional[Pattern[g, Blank[]]]], Pattern[p, Blank[]]], Power[Plus[Pattern[a, Blank[]], Times[Optional[Pattern[b, Blank[]]], sin[Plus[Optional[Pattern[e, Blank[]]], Times[Optional[Pattern[f, Blank[]]], Pattern[x, Blank[]]]]]]], Pattern[m, Blank[]]]], Pattern[x, Blank[Symbol]]] := Simp[((g*Cos[e + f*x])^(p + 1)*(a + b*Sin[e + f*x])^(m + 1)*(b - a*Sin[e + f*x]))/(f*g*(a^2 - b^2)*(p + 1)), x] + Dist[1/(g^2*(a^2 - b^2)*(p + 1)), Int[(g*Cos[e + f*x])^(p + 2)*(a + b*Sin[e + f*x])^m*(a^2*(p + 2) - b^2*(m + p + 2) + a*b*(m + p + 3)*Sin[e + f*x]), x], x] /; FreeQ[{a, b, e, f, g, m}, x] && NeQ[a^2 - b^2, 0] && LtQ[p, -1] && IntegersQ[2*m, 2*p]
  2697. Int[Times[Power[Times[cos[Plus[Optional[Pattern[e, Blank[]]], Times[Optional[Pattern[f, Blank[]]], Pattern[x, Blank[]]]]], Optional[Pattern[g, Blank[]]]], Rational[-1, 2]], Power[Plus[Pattern[a, Blank[]], Times[Optional[Pattern[b, Blank[]]], sin[Plus[Optional[Pattern[e, Blank[]]], Times[Optional[Pattern[f, Blank[]]], Pattern[x, Blank[]]]]]]], Rational[-1, 2]]], Pattern[x, Blank[Symbol]]] := Dist[(2*Sqrt[2]*Sqrt[g*Cos[e + f*x]]*Sqrt[(a + b*Sin[e + f*x])/((a - b)*(1 - Sin[e + f*x]))])/(f*g*Sqrt[a + b*Sin[e + f*x]]*Sqrt[(1 + Cos[e + f*x] + Sin[e + f*x])/(1 + Cos[e + f*x] - Sin[e + f*x])]), Subst[Int[1/Sqrt[1 + ((a + b)*x^4)/(a - b)], x], x, Sqrt[(1 + Cos[e + f*x] + Sin[e + f*x])/(1 + Cos[e + f*x] - Sin[e + f*x])]], x] /; FreeQ[{a, b, e, f, g}, x] && NeQ[a^2 - b^2, 0]
  2698. Int[Times[Power[Times[cos[Plus[Optional[Pattern[e, Blank[]]], Times[Optional[Pattern[f, Blank[]]], Pattern[x, Blank[]]]]], Optional[Pattern[g, Blank[]]]], Pattern[p, Blank[]]], Power[Plus[Pattern[a, Blank[]], Times[Optional[Pattern[b, Blank[]]], sin[Plus[Optional[Pattern[e, Blank[]]], Times[Optional[Pattern[f, Blank[]]], Pattern[x, Blank[]]]]]]], Pattern[m, Blank[]]]], Pattern[x, Blank[Symbol]]] := Simp[(g*(g*Cos[e + f*x])^(p - 1)*(1 - Sin[e + f*x])*(a + b*Sin[e + f*x])^(m + 1)*(-(((a - b)*(1 - Sin[e + f*x]))/((a + b)*(1 + Sin[e + f*x]))))^(m/2)*Hypergeometric2F1[m + 1, m/2 + 1, m + 2, (2*(a + b*Sin[e + f*x]))/((a + b)*(1 + Sin[e + f*x]))])/(f*(a + b)*(m + 1)), x] /; FreeQ[{a, b, e, f, g, m, p}, x] && NeQ[a^2 - b^2, 0] && EqQ[m + p + 1, 0]
  2699. Int[Times[Power[Times[cos[Plus[Optional[Pattern[e, Blank[]]], Times[Optional[Pattern[f, Blank[]]], Pattern[x, Blank[]]]]], Optional[Pattern[g, Blank[]]]], Pattern[p, Blank[]]], Power[Plus[Pattern[a, Blank[]], Times[Optional[Pattern[b, Blank[]]], sin[Plus[Optional[Pattern[e, Blank[]]], Times[Optional[Pattern[f, Blank[]]], Pattern[x, Blank[]]]]]]], Pattern[m, Blank[]]]], Pattern[x, Blank[Symbol]]] := Simp[((g*Cos[e + f*x])^(p + 1)*(a + b*Sin[e + f*x])^(m + 1))/(f*g*(a - b)*(p + 1)), x] + Dist[a/(g^2*(a - b)), Int[((g*Cos[e + f*x])^(p + 2)*(a + b*Sin[e + f*x])^m)/(1 - Sin[e + f*x]), x], x] /; FreeQ[{a, b, e, f, g, m, p}, x] && NeQ[a^2 - b^2, 0] && EqQ[m + p + 2, 0]
  2700. Int[Times[Power[Times[cos[Plus[Optional[Pattern[e, Blank[]]], Times[Optional[Pattern[f, Blank[]]], Pattern[x, Blank[]]]]], Optional[Pattern[g, Blank[]]]], Pattern[p, Blank[]]], Power[Plus[Pattern[a, Blank[]], Times[Optional[Pattern[b, Blank[]]], sin[Plus[Optional[Pattern[e, Blank[]]], Times[Optional[Pattern[f, Blank[]]], Pattern[x, Blank[]]]]]]], Pattern[m, Blank[]]]], Pattern[x, Blank[Symbol]]] := Simp[((g*Cos[e + f*x])^(p + 1)*(a + b*Sin[e + f*x])^(m + 1))/(f*g*(a - b)*(p + 1)), x] + (-Dist[(b*(m + p + 2))/(g^2*(a - b)*(p + 1)), Int[(g*Cos[e + f*x])^(p + 2)*(a + b*Sin[e + f*x])^m, x], x] + Dist[a/(g^2*(a - b)), Int[((g*Cos[e + f*x])^(p + 2)*(a + b*Sin[e + f*x])^m)/(1 - Sin[e + f*x]), x], x]) /; FreeQ[{a, b, e, f, g, m, p}, x] && NeQ[a^2 - b^2, 0] && ILtQ[m + p + 2, 0]
  2701. Int[Times[Power[Times[cos[Plus[Optional[Pattern[e, Blank[]]], Times[Optional[Pattern[f, Blank[]]], Pattern[x, Blank[]]]]], Optional[Pattern[g, Blank[]]]], Rational[1, 2]], Power[Plus[Pattern[a, Blank[]], Times[Optional[Pattern[b, Blank[]]], sin[Plus[Optional[Pattern[e, Blank[]]], Times[Optional[Pattern[f, Blank[]]], Pattern[x, Blank[]]]]]]], -1]], Pattern[x, Blank[Symbol]]] := With[{q = Rt[-a^2 + b^2, 2]}, Dist[(a*g)/(2*b), Int[1/(Sqrt[g*Cos[e + f*x]]*(q + b*Cos[e + f*x])), x], x] + (-Dist[(a*g)/(2*b), Int[1/(Sqrt[g*Cos[e + f*x]]*(q - b*Cos[e + f*x])), x], x] + Dist[(b*g)/f, Subst[Int[Sqrt[x]/(g^2*(a^2 - b^2) + b^2*x^2), x], x, g*Cos[e + f*x]], x])] /; FreeQ[{a, b, e, f, g}, x] && NeQ[a^2 - b^2, 0]
  2702. Int[Times[Power[Times[cos[Plus[Optional[Pattern[e, Blank[]]], Times[Optional[Pattern[f, Blank[]]], Pattern[x, Blank[]]]]], Optional[Pattern[g, Blank[]]]], Rational[-1, 2]], Power[Plus[Pattern[a, Blank[]], Times[Optional[Pattern[b, Blank[]]], sin[Plus[Optional[Pattern[e, Blank[]]], Times[Optional[Pattern[f, Blank[]]], Pattern[x, Blank[]]]]]]], -1]], Pattern[x, Blank[Symbol]]] := With[{q = Rt[-a^2 + b^2, 2]}, -Dist[a/(2*q), Int[1/(Sqrt[g*Cos[e + f*x]]*(q + b*Cos[e + f*x])), x], x] + (Dist[(b*g)/f, Subst[Int[1/(Sqrt[x]*(g^2*(a^2 - b^2) + b^2*x^2)), x], x, g*Cos[e + f*x]], x] - Dist[a/(2*q), Int[1/(Sqrt[g*Cos[e + f*x]]*(q - b*Cos[e + f*x])), x], x])] /; FreeQ[{a, b, e, f, g}, x] && NeQ[a^2 - b^2, 0]
  2703. Int[Times[Power[Times[cos[Plus[Optional[Pattern[e, Blank[]]], Times[Optional[Pattern[f, Blank[]]], Pattern[x, Blank[]]]]], Optional[Pattern[g, Blank[]]]], Pattern[p, Blank[]]], Power[Plus[Pattern[a, Blank[]], Times[Optional[Pattern[b, Blank[]]], sin[Plus[Optional[Pattern[e, Blank[]]], Times[Optional[Pattern[f, Blank[]]], Pattern[x, Blank[]]]]]]], Pattern[m, Blank[]]]], Pattern[x, Blank[Symbol]]] := Simp[(g*(g*Cos[e + f*x])^(p - 1)*(a + b*Sin[e + f*x])^(m + 1)*AppellF1[-p - m, (1 - p)/2, (1 - p)/2, 1 - p - m, (a + b)/(a + b*Sin[e + f*x]), (a - b)/(a + b*Sin[e + f*x])])/(b*f*(m + p)*(-((b*(1 - Sin[e + f*x]))/(a + b*Sin[e + f*x])))^((p - 1)/2)*((b*(1 + Sin[e + f*x]))/(a + b*Sin[e + f*x]))^((p - 1)/2)), x] /; FreeQ[{a, b, e, f, g, p}, x] && NeQ[a^2 - b^2, 0] && ILtQ[m, 0] && !IGtQ[m + p + 1, 0]
  2704. Int[Times[Power[Times[cos[Plus[Optional[Pattern[e, Blank[]]], Times[Optional[Pattern[f, Blank[]]], Pattern[x, Blank[]]]]], Optional[Pattern[g, Blank[]]]], Pattern[p, Blank[]]], Power[Plus[Pattern[a, Blank[]], Times[Optional[Pattern[b, Blank[]]], sin[Plus[Optional[Pattern[e, Blank[]]], Times[Optional[Pattern[f, Blank[]]], Pattern[x, Blank[]]]]]]], Pattern[m, Blank[]]]], Pattern[x, Blank[Symbol]]] := Dist[(g*(g*Cos[e + f*x])^(p - 1))/(f*(1 - (a + b*Sin[e + f*x])/(a - b))^((p - 1)/2)*(1 - (a + b*Sin[e + f*x])/(a + b))^((p - 1)/2)), Subst[Int[(-(b/(a - b)) - (b*x)/(a - b))^((p - 1)/2)*(b/(a + b) - (b*x)/(a + b))^((p - 1)/2)*(a + b*x)^m, x], x, Sin[e + f*x]], x] /; FreeQ[{a, b, e, f, g, m, p}, x] && NeQ[a^2 - b^2, 0] && !IGtQ[m, 0]
  2705. Int[Times[Power[Times[Optional[Pattern[g, Blank[]]], sec[Plus[Optional[Pattern[e, Blank[]]], Times[Optional[Pattern[f, Blank[]]], Pattern[x, Blank[]]]]]], Pattern[p, Blank[]]], Power[Plus[Pattern[a, Blank[]], Times[Optional[Pattern[b, Blank[]]], sin[Plus[Optional[Pattern[e, Blank[]]], Times[Optional[Pattern[f, Blank[]]], Pattern[x, Blank[]]]]]]], Optional[Pattern[m, Blank[]]]]], Pattern[x, Blank[Symbol]]] := Dist[g^(2*IntPart[p])*(g*Cos[e + f*x])^FracPart[p]*(g*Sec[e + f*x])^FracPart[p], Int[(a + b*Sin[e + f*x])^m/(g*Cos[e + f*x])^p, x], x] /; FreeQ[{a, b, e, f, g, m, p}, x] && !IntegerQ[p]
  2706. Int[Times[Power[Plus[Pattern[a, Blank[]], Times[Optional[Pattern[b, Blank[]]], sin[Plus[Optional[Pattern[e, Blank[]]], Times[Optional[Pattern[f, Blank[]]], Pattern[x, Blank[]]]]]]], -1], Power[Times[Optional[Pattern[g, Blank[]]], tan[Plus[Optional[Pattern[e, Blank[]]], Times[Optional[Pattern[f, Blank[]]], Pattern[x, Blank[]]]]]], Optional[Pattern[p, Blank[]]]]], Pattern[x, Blank[Symbol]]] := Dist[1/a, Int[Sec[e + f*x]^2*(g*Tan[e + f*x])^p, x], x] - Dist[1/(b*g), Int[Sec[e + f*x]*(g*Tan[e + f*x])^(p + 1), x], x] /; FreeQ[{a, b, e, f, g, p}, x] && EqQ[a^2 - b^2, 0] && NeQ[p, -1]
  2707. Int[Times[Power[Plus[Pattern[a, Blank[]], Times[Optional[Pattern[b, Blank[]]], sin[Plus[Optional[Pattern[e, Blank[]]], Times[Optional[Pattern[f, Blank[]]], Pattern[x, Blank[]]]]]]], Optional[Pattern[m, Blank[]]]], Power[tan[Plus[Optional[Pattern[e, Blank[]]], Times[Optional[Pattern[f, Blank[]]], Pattern[x, Blank[]]]]], Optional[Pattern[p, Blank[]]]]], Pattern[x, Blank[Symbol]]] := Dist[1/f, Subst[Int[(x^p*(a + x)^(m - (p + 1)/2))/(a - x)^((p + 1)/2), x], x, b*Sin[e + f*x]], x] /; FreeQ[{a, b, e, f, m}, x] && EqQ[a^2 - b^2, 0] && IntegerQ[(p + 1)/2]
  2708. Int[Times[Power[Plus[Pattern[a, Blank[]], Times[Optional[Pattern[b, Blank[]]], sin[Plus[Optional[Pattern[e, Blank[]]], Times[Optional[Pattern[f, Blank[]]], Pattern[x, Blank[]]]]]]], Optional[Pattern[m, Blank[]]]], Power[tan[Plus[Optional[Pattern[e, Blank[]]], Times[Optional[Pattern[f, Blank[]]], Pattern[x, Blank[]]]]], Pattern[p, Blank[]]]], Pattern[x, Blank[Symbol]]] := Dist[a^p, Int[Sin[e + f*x]^p/(a - b*Sin[e + f*x])^m, x], x] /; FreeQ[{a, b, e, f}, x] && EqQ[a^2 - b^2, 0] && IntegersQ[m, p] && EqQ[p, 2*m]
  2709. Int[Times[Power[Plus[Pattern[a, Blank[]], Times[Optional[Pattern[b, Blank[]]], sin[Plus[Optional[Pattern[e, Blank[]]], Times[Optional[Pattern[f, Blank[]]], Pattern[x, Blank[]]]]]]], Pattern[m, Blank[]]], Power[tan[Plus[Optional[Pattern[e, Blank[]]], Times[Optional[Pattern[f, Blank[]]], Pattern[x, Blank[]]]]], Pattern[p, Blank[]]]], Pattern[x, Blank[Symbol]]] := Dist[a^p, Int[ExpandIntegrand[(Sin[e + f*x]^p*(a + b*Sin[e + f*x])^(m - p/2))/(a - b*Sin[e + f*x])^(p/2), x], x], x] /; FreeQ[{a, b, e, f}, x] && EqQ[a^2 - b^2, 0] && IntegersQ[m, p/2] && (LtQ[p, 0] || GtQ[m - p/2, 0])
  2710. Int[Times[Power[Plus[Pattern[a, Blank[]], Times[Optional[Pattern[b, Blank[]]], sin[Plus[Optional[Pattern[e, Blank[]]], Times[Optional[Pattern[f, Blank[]]], Pattern[x, Blank[]]]]]]], Optional[Pattern[m, Blank[]]]], Power[Times[Optional[Pattern[g, Blank[]]], tan[Plus[Optional[Pattern[e, Blank[]]], Times[Optional[Pattern[f, Blank[]]], Pattern[x, Blank[]]]]]], Optional[Pattern[p, Blank[]]]]], Pattern[x, Blank[Symbol]]] := Int[ExpandIntegrand[(g*Tan[e + f*x])^p, (a + b*Sin[e + f*x])^m, x], x] /; FreeQ[{a, b, e, f, g, p}, x] && EqQ[a^2 - b^2, 0] && IGtQ[m, 0]
  2711. Int[Times[Power[Plus[Pattern[a, Blank[]], Times[Optional[Pattern[b, Blank[]]], sin[Plus[Optional[Pattern[e, Blank[]]], Times[Optional[Pattern[f, Blank[]]], Pattern[x, Blank[]]]]]]], Pattern[m, Blank[]]], Power[Times[Optional[Pattern[g, Blank[]]], tan[Plus[Optional[Pattern[e, Blank[]]], Times[Optional[Pattern[f, Blank[]]], Pattern[x, Blank[]]]]]], Optional[Pattern[p, Blank[]]]]], Pattern[x, Blank[Symbol]]] := Dist[a^(2*m), Int[ExpandIntegrand[(g*Tan[e + f*x])^p/Sec[e + f*x]^m, (a*Sec[e + f*x] - b*Tan[e + f*x])^(-m), x], x], x] /; FreeQ[{a, b, e, f, g, p}, x] && EqQ[a^2 - b^2, 0] && ILtQ[m, 0]
  2712. Int[Times[Power[Plus[Pattern[a, Blank[]], Times[Optional[Pattern[b, Blank[]]], sin[Plus[Optional[Pattern[e, Blank[]]], Times[Optional[Pattern[f, Blank[]]], Pattern[x, Blank[]]]]]]], Pattern[m, Blank[]]], Power[tan[Plus[Optional[Pattern[e, Blank[]]], Times[Optional[Pattern[f, Blank[]]], Pattern[x, Blank[]]]]], 2]], Pattern[x, Blank[Symbol]]] := Simp[(b*(a + b*Sin[e + f*x])^m)/(a*f*(2*m - 1)*Cos[e + f*x]), x] - Dist[1/(a^2*(2*m - 1)), Int[((a + b*Sin[e + f*x])^(m + 1)*(a*m - b*(2*m - 1)*Sin[e + f*x]))/Cos[e + f*x]^2, x], x] /; FreeQ[{a, b, e, f}, x] && EqQ[a^2 - b^2, 0] && !IntegerQ[m] && LtQ[m, 0]
  2713. Int[Times[Power[Plus[Pattern[a, Blank[]], Times[Optional[Pattern[b, Blank[]]], sin[Plus[Optional[Pattern[e, Blank[]]], Times[Optional[Pattern[f, Blank[]]], Pattern[x, Blank[]]]]]]], Pattern[m, Blank[]]], Power[tan[Plus[Optional[Pattern[e, Blank[]]], Times[Optional[Pattern[f, Blank[]]], Pattern[x, Blank[]]]]], 2]], Pattern[x, Blank[Symbol]]] := -Simp[(a + b*Sin[e + f*x])^(m + 1)/(b*f*m*Cos[e + f*x]), x] + Dist[1/(b*m), Int[((a + b*Sin[e + f*x])^m*(b*(m + 1) + a*Sin[e + f*x]))/Cos[e + f*x]^2, x], x] /; FreeQ[{a, b, e, f, m}, x] && EqQ[a^2 - b^2, 0] && !IntegerQ[m] && !LtQ[m, 0]
  2714. Int[Times[Power[Plus[Pattern[a, Blank[]], Times[Optional[Pattern[b, Blank[]]], sin[Plus[Optional[Pattern[e, Blank[]]], Times[Optional[Pattern[f, Blank[]]], Pattern[x, Blank[]]]]]]], Pattern[m, Blank[]]], Power[tan[Plus[Optional[Pattern[e, Blank[]]], Times[Optional[Pattern[f, Blank[]]], Pattern[x, Blank[]]]]], 4]], Pattern[x, Blank[Symbol]]] := Int[(a + b*Sin[e + f*x])^m, x] - Int[((a + b*Sin[e + f*x])^m*(1 - 2*Sin[e + f*x]^2))/Cos[e + f*x]^4, x] /; FreeQ[{a, b, e, f, m}, x] && EqQ[a^2 - b^2, 0] && IntegerQ[m - 1/2]
  2715. Int[Times[Power[Plus[Pattern[a, Blank[]], Times[Optional[Pattern[b, Blank[]]], sin[Plus[Optional[Pattern[e, Blank[]]], Times[Optional[Pattern[f, Blank[]]], Pattern[x, Blank[]]]]]]], Pattern[m, Blank[]]], Power[tan[Plus[Optional[Pattern[e, Blank[]]], Times[Optional[Pattern[f, Blank[]]], Pattern[x, Blank[]]]]], -2]], Pattern[x, Blank[Symbol]]] := -Simp[(a + b*Sin[e + f*x])^(m + 1)/(a*f*Tan[e + f*x]), x] + Dist[1/b^2, Int[((a + b*Sin[e + f*x])^(m + 1)*(b*m - a*(m + 1)*Sin[e + f*x]))/Sin[e + f*x], x], x] /; FreeQ[{a, b, e, f}, x] && EqQ[a^2 - b^2, 0] && IntegerQ[m - 1/2] && LtQ[m, -1]
  2716. Int[Times[Power[Plus[Pattern[a, Blank[]], Times[Optional[Pattern[b, Blank[]]], sin[Plus[Optional[Pattern[e, Blank[]]], Times[Optional[Pattern[f, Blank[]]], Pattern[x, Blank[]]]]]]], Optional[Pattern[m, Blank[]]]], Power[tan[Plus[Optional[Pattern[e, Blank[]]], Times[Optional[Pattern[f, Blank[]]], Pattern[x, Blank[]]]]], -2]], Pattern[x, Blank[Symbol]]] := -Simp[(a + b*Sin[e + f*x])^m/(f*Tan[e + f*x]), x] + Dist[1/a, Int[((a + b*Sin[e + f*x])^m*(b*m - a*(m + 1)*Sin[e + f*x]))/Sin[e + f*x], x], x] /; FreeQ[{a, b, e, f, m}, x] && EqQ[a^2 - b^2, 0] && IntegerQ[m - 1/2] && !LtQ[m, -1]
  2717. Int[Times[Power[Plus[Pattern[a, Blank[]], Times[Optional[Pattern[b, Blank[]]], sin[Plus[Optional[Pattern[e, Blank[]]], Times[Optional[Pattern[f, Blank[]]], Pattern[x, Blank[]]]]]]], Pattern[m, Blank[]]], Power[tan[Plus[Optional[Pattern[e, Blank[]]], Times[Optional[Pattern[f, Blank[]]], Pattern[x, Blank[]]]]], -4]], Pattern[x, Blank[Symbol]]] := Dist[-2/(a*b), Int[(a + b*Sin[e + f*x])^(m + 2)/Sin[e + f*x]^3, x], x] + Dist[1/a^2, Int[((a + b*Sin[e + f*x])^(m + 2)*(1 + Sin[e + f*x]^2))/Sin[e + f*x]^4, x], x] /; FreeQ[{a, b, e, f}, x] && EqQ[a^2 - b^2, 0] && IntegerQ[m - 1/2] && LtQ[m, -1]
  2718. Int[Times[Power[Plus[Pattern[a, Blank[]], Times[Optional[Pattern[b, Blank[]]], sin[Plus[Optional[Pattern[e, Blank[]]], Times[Optional[Pattern[f, Blank[]]], Pattern[x, Blank[]]]]]]], Pattern[m, Blank[]]], Power[tan[Plus[Optional[Pattern[e, Blank[]]], Times[Optional[Pattern[f, Blank[]]], Pattern[x, Blank[]]]]], -4]], Pattern[x, Blank[Symbol]]] := Int[(a + b*Sin[e + f*x])^m, x] + Int[((a + b*Sin[e + f*x])^m*(1 - 2*Sin[e + f*x]^2))/Sin[e + f*x]^4, x] /; FreeQ[{a, b, e, f, m}, x] && EqQ[a^2 - b^2, 0] && IntegerQ[m - 1/2] && !LtQ[m, -1]
  2719. Int[Times[Power[Plus[Pattern[a, Blank[]], Times[Optional[Pattern[b, Blank[]]], sin[Plus[Optional[Pattern[e, Blank[]]], Times[Optional[Pattern[f, Blank[]]], Pattern[x, Blank[]]]]]]], Pattern[m, Blank[]]], Power[tan[Plus[Optional[Pattern[e, Blank[]]], Times[Optional[Pattern[f, Blank[]]], Pattern[x, Blank[]]]]], Pattern[p, Blank[]]]], Pattern[x, Blank[Symbol]]] := Dist[(Sqrt[a + b*Sin[e + f*x]]*Sqrt[a - b*Sin[e + f*x]])/(b*f*Cos[e + f*x]), Subst[Int[(x^p*(a + x)^(m - (p + 1)/2))/(a - x)^((p + 1)/2), x], x, b*Sin[e + f*x]], x] /; FreeQ[{a, b, e, f, m}, x] && EqQ[a^2 - b^2, 0] && !IntegerQ[m] && IntegerQ[p/2]
  2720. Int[Times[Power[Plus[Pattern[a, Blank[]], Times[Optional[Pattern[b, Blank[]]], sin[Plus[Optional[Pattern[e, Blank[]]], Times[Optional[Pattern[f, Blank[]]], Pattern[x, Blank[]]]]]]], Pattern[m, Blank[]]], Power[Times[Optional[Pattern[g, Blank[]]], tan[Plus[Optional[Pattern[e, Blank[]]], Times[Optional[Pattern[f, Blank[]]], Pattern[x, Blank[]]]]]], Pattern[p, Blank[]]]], Pattern[x, Blank[Symbol]]] := Dist[((g*Tan[e + f*x])^(p + 1)*(a - b*Sin[e + f*x])^((p + 1)/2)*(a + b*Sin[e + f*x])^((p + 1)/2))/(f*g*(b*Sin[e + f*x])^(p + 1)), Subst[Int[(x^p*(a + x)^(m - (p + 1)/2))/(a - x)^((p + 1)/2), x], x, b*Sin[e + f*x]], x] /; FreeQ[{a, b, e, f, g, m, p}, x] && EqQ[a^2 - b^2, 0] && !IntegerQ[m] && !IntegerQ[p]
  2721. Int[Times[Power[Plus[Pattern[a, Blank[]], Times[Optional[Pattern[b, Blank[]]], sin[Plus[Optional[Pattern[e, Blank[]]], Times[Optional[Pattern[f, Blank[]]], Pattern[x, Blank[]]]]]]], Optional[Pattern[m, Blank[]]]], Power[tan[Plus[Optional[Pattern[e, Blank[]]], Times[Optional[Pattern[f, Blank[]]], Pattern[x, Blank[]]]]], Optional[Pattern[p, Blank[]]]]], Pattern[x, Blank[Symbol]]] := Dist[1/f, Subst[Int[(x^p*(a + x)^m)/(b^2 - x^2)^((p + 1)/2), x], x, b*Sin[e + f*x]], x] /; FreeQ[{a, b, e, f, m}, x] && NeQ[a^2 - b^2, 0] && IntegerQ[(p + 1)/2]
  2722. Int[Times[Power[Plus[Pattern[a, Blank[]], Times[Optional[Pattern[b, Blank[]]], sin[Plus[Optional[Pattern[e, Blank[]]], Times[Optional[Pattern[f, Blank[]]], Pattern[x, Blank[]]]]]]], Optional[Pattern[m, Blank[]]]], Power[Times[Optional[Pattern[g, Blank[]]], tan[Plus[Optional[Pattern[e, Blank[]]], Times[Optional[Pattern[f, Blank[]]], Pattern[x, Blank[]]]]]], Optional[Pattern[p, Blank[]]]]], Pattern[x, Blank[Symbol]]] := Int[ExpandIntegrand[(g*Tan[e + f*x])^p, (a + b*Sin[e + f*x])^m, x], x] /; FreeQ[{a, b, e, f, g, p}, x] && NeQ[a^2 - b^2, 0] && IGtQ[m, 0]
  2723. Int[Times[Power[Plus[Pattern[a, Blank[]], Times[Optional[Pattern[b, Blank[]]], sin[Plus[Optional[Pattern[e, Blank[]]], Times[Optional[Pattern[f, Blank[]]], Pattern[x, Blank[]]]]]]], Pattern[m, Blank[]]], Power[tan[Plus[Optional[Pattern[e, Blank[]]], Times[Optional[Pattern[f, Blank[]]], Pattern[x, Blank[]]]]], -2]], Pattern[x, Blank[Symbol]]] := Int[((a + b*Sin[e + f*x])^m*(1 - Sin[e + f*x]^2))/Sin[e + f*x]^2, x] /; FreeQ[{a, b, e, f, m}, x] && NeQ[a^2 - b^2, 0]
  2724. Int[Times[Power[Plus[Pattern[a, Blank[]], Times[Optional[Pattern[b, Blank[]]], sin[Plus[Optional[Pattern[e, Blank[]]], Times[Optional[Pattern[f, Blank[]]], Pattern[x, Blank[]]]]]]], Pattern[m, Blank[]]], Power[tan[Plus[Optional[Pattern[e, Blank[]]], Times[Optional[Pattern[f, Blank[]]], Pattern[x, Blank[]]]]], -4]], Pattern[x, Blank[Symbol]]] := -Simp[(Cos[e + f*x]*(a + b*Sin[e + f*x])^(m + 1))/(3*a*f*Sin[e + f*x]^3), x] + (-Dist[1/(3*a^2*b*(m + 1)), Int[((a + b*Sin[e + f*x])^(m + 1)*Simp[6*a^2 - b^2*(m - 1)*(m - 2) + a*b*(m + 1)*Sin[e + f*x] - (3*a^2 - b^2*m*(m - 2))*Sin[e + f*x]^2, x])/Sin[e + f*x]^3, x], x] - Simp[((3*a^2 + b^2*(m - 2))*Cos[e + f*x]*(a + b*Sin[e + f*x])^(m + 1))/(3*a^2*b*f*(m + 1)*Sin[e + f*x]^2), x]) /; FreeQ[{a, b, e, f}, x] && NeQ[a^2 - b^2, 0] && LtQ[m, -1] && IntegerQ[2*m]
  2725. Int[Times[Power[Plus[Pattern[a, Blank[]], Times[Optional[Pattern[b, Blank[]]], sin[Plus[Optional[Pattern[e, Blank[]]], Times[Optional[Pattern[f, Blank[]]], Pattern[x, Blank[]]]]]]], Pattern[m, Blank[]]], Power[tan[Plus[Optional[Pattern[e, Blank[]]], Times[Optional[Pattern[f, Blank[]]], Pattern[x, Blank[]]]]], -4]], Pattern[x, Blank[Symbol]]] := -Simp[(Cos[e + f*x]*(a + b*Sin[e + f*x])^(m + 1))/(3*a*f*Sin[e + f*x]^3), x] + (-Dist[1/(6*a^2), Int[((a + b*Sin[e + f*x])^m*Simp[8*a^2 - b^2*(m - 1)*(m - 2) + a*b*m*Sin[e + f*x] - (6*a^2 - b^2*m*(m - 2))*Sin[e + f*x]^2, x])/Sin[e + f*x]^2, x], x] - Simp[(b*(m - 2)*Cos[e + f*x]*(a + b*Sin[e + f*x])^(m + 1))/(6*a^2*f*Sin[e + f*x]^2), x]) /; FreeQ[{a, b, e, f, m}, x] && NeQ[a^2 - b^2, 0] && !LtQ[m, -1] && IntegerQ[2*m]
  2726. Int[Times[Power[Plus[Pattern[a, Blank[]], Times[Optional[Pattern[b, Blank[]]], sin[Plus[Optional[Pattern[e, Blank[]]], Times[Optional[Pattern[f, Blank[]]], Pattern[x, Blank[]]]]]]], Pattern[m, Blank[]]], Power[tan[Plus[Optional[Pattern[e, Blank[]]], Times[Optional[Pattern[f, Blank[]]], Pattern[x, Blank[]]]]], -6]], Pattern[x, Blank[Symbol]]] := -Simp[(Cos[e + f*x]*(a + b*Sin[e + f*x])^(m + 1))/(5*a*f*Sin[e + f*x]^5), x] + (Dist[1/(20*a^2*b^2*m*(m - 1)), Int[((a + b*Sin[e + f*x])^m*Simp[60*a^4 - 44*a^2*b^2*(m - 1)*m + b^4*m*(m - 1)*(m - 3)*(m - 4) + a*b*m*(20*a^2 - b^2*m*(m - 1))*Sin[e + f*x] - (40*a^4 + b^4*m*(m - 1)*(m - 2)*(m - 4) - 20*a^2*b^2*(m - 1)*(2*m + 1))*Sin[e + f*x]^2, x])/Sin[e + f*x]^4, x], x] + Simp[(Cos[e + f*x]*(a + b*Sin[e + f*x])^(m + 1))/(b*f*m*Sin[e + f*x]^2), x] + Simp[(a*Cos[e + f*x]*(a + b*Sin[e + f*x])^(m + 1))/(b^2*f*m*(m - 1)*Sin[e + f*x]^3), x] - Simp[(b*(m - 4)*Cos[e + f*x]*(a + b*Sin[e + f*x])^(m + 1))/(20*a^2*f*Sin[e + f*x]^4), x]) /; FreeQ[{a, b, e, f, m}, x] && NeQ[a^2 - b^2, 0] && NeQ[m, 1] && IntegerQ[2*m]
  2727. Int[Times[Power[Plus[Pattern[a, Blank[]], Times[Optional[Pattern[b, Blank[]]], sin[Plus[Optional[Pattern[e, Blank[]]], Times[Optional[Pattern[f, Blank[]]], Pattern[x, Blank[]]]]]]], -1], Power[Times[Optional[Pattern[g, Blank[]]], tan[Plus[Optional[Pattern[e, Blank[]]], Times[Optional[Pattern[f, Blank[]]], Pattern[x, Blank[]]]]]], Pattern[p, Blank[]]]], Pattern[x, Blank[Symbol]]] := Dist[a/(a^2 - b^2), Int[(g*Tan[e + f*x])^p/Sin[e + f*x]^2, x], x] + (-Dist[(b*g)/(a^2 - b^2), Int[(g*Tan[e + f*x])^(p - 1)/Cos[e + f*x], x], x] - Dist[(a^2*g^2)/(a^2 - b^2), Int[(g*Tan[e + f*x])^(p - 2)/(a + b*Sin[e + f*x]), x], x]) /; FreeQ[{a, b, e, f, g}, x] && NeQ[a^2 - b^2, 0] && IntegersQ[2*p] && GtQ[p, 1]
  2728. Int[Times[Power[Plus[Pattern[a, Blank[]], Times[Optional[Pattern[b, Blank[]]], sin[Plus[Optional[Pattern[e, Blank[]]], Times[Optional[Pattern[f, Blank[]]], Pattern[x, Blank[]]]]]]], -1], Power[Times[Optional[Pattern[g, Blank[]]], tan[Plus[Optional[Pattern[e, Blank[]]], Times[Optional[Pattern[f, Blank[]]], Pattern[x, Blank[]]]]]], Pattern[p, Blank[]]]], Pattern[x, Blank[Symbol]]] := Dist[1/a, Int[(g*Tan[e + f*x])^p/Cos[e + f*x]^2, x], x] + (-Dist[b/(a^2*g), Int[(g*Tan[e + f*x])^(p + 1)/Cos[e + f*x], x], x] - Dist[(a^2 - b^2)/(a^2*g^2), Int[(g*Tan[e + f*x])^(p + 2)/(a + b*Sin[e + f*x]), x], x]) /; FreeQ[{a, b, e, f, g}, x] && NeQ[a^2 - b^2, 0] && IntegersQ[2*p] && LtQ[p, -1]
  2729. Int[Times[Power[Plus[Pattern[a, Blank[]], Times[Optional[Pattern[b, Blank[]]], sin[Plus[Optional[Pattern[e, Blank[]]], Times[Optional[Pattern[f, Blank[]]], Pattern[x, Blank[]]]]]]], -1], Power[Times[Optional[Pattern[g, Blank[]]], tan[Plus[Optional[Pattern[e, Blank[]]], Times[Optional[Pattern[f, Blank[]]], Pattern[x, Blank[]]]]]], Rational[1, 2]]], Pattern[x, Blank[Symbol]]] := Dist[(Sqrt[Cos[e + f*x]]*Sqrt[g*Tan[e + f*x]])/Sqrt[Sin[e + f*x]], Int[Sqrt[Sin[e + f*x]]/(Sqrt[Cos[e + f*x]]*(a + b*Sin[e + f*x])), x], x] /; FreeQ[{a, b, e, f, g}, x] && NeQ[a^2 - b^2, 0]
  2730. Int[Times[Power[Plus[Pattern[a, Blank[]], Times[Optional[Pattern[b, Blank[]]], sin[Plus[Optional[Pattern[e, Blank[]]], Times[Optional[Pattern[f, Blank[]]], Pattern[x, Blank[]]]]]]], -1], Power[Times[Pattern[g, Blank[]], tan[Plus[Optional[Pattern[e, Blank[]]], Times[Optional[Pattern[f, Blank[]]], Pattern[x, Blank[]]]]]], Rational[-1, 2]]], Pattern[x, Blank[Symbol]]] := Dist[Sqrt[Sin[e + f*x]]/(Sqrt[Cos[e + f*x]]*Sqrt[g*Tan[e + f*x]]), Int[Sqrt[Cos[e + f*x]]/(Sqrt[Sin[e + f*x]]*(a + b*Sin[e + f*x])), x], x] /; FreeQ[{a, b, e, f, g}, x] && NeQ[a^2 - b^2, 0]
  2731. Int[Times[Power[Plus[Pattern[a, Blank[]], Times[Optional[Pattern[b, Blank[]]], sin[Plus[Optional[Pattern[e, Blank[]]], Times[Optional[Pattern[f, Blank[]]], Pattern[x, Blank[]]]]]]], Pattern[m, Blank[]]], Power[tan[Plus[Optional[Pattern[e, Blank[]]], Times[Optional[Pattern[f, Blank[]]], Pattern[x, Blank[]]]]], Pattern[p, Blank[]]]], Pattern[x, Blank[Symbol]]] := Int[ExpandIntegrand[(Sin[e + f*x]^p*(a + b*Sin[e + f*x])^m)/(1 - Sin[e + f*x]^2)^(p/2), x], x] /; FreeQ[{a, b, e, f}, x] && NeQ[a^2 - b^2, 0] && IntegersQ[m, p/2]
  2732. Int[Times[Power[Plus[Pattern[a, Blank[]], Times[Optional[Pattern[b, Blank[]]], sin[Plus[Optional[Pattern[e, Blank[]]], Times[Optional[Pattern[f, Blank[]]], Pattern[x, Blank[]]]]]]], Optional[Pattern[m, Blank[]]]], Power[Times[Optional[Pattern[g, Blank[]]], tan[Plus[Optional[Pattern[e, Blank[]]], Times[Optional[Pattern[f, Blank[]]], Pattern[x, Blank[]]]]]], Optional[Pattern[p, Blank[]]]]], Pattern[x, Blank[Symbol]]] := Unintegrable[(a + b*Sin[e + f*x])^m*(g*Tan[e + f*x])^p, x] /; FreeQ[{a, b, e, f, g, m, p}, x]
  2733. Int[Times[Power[Times[cot[Plus[Optional[Pattern[e, Blank[]]], Times[Optional[Pattern[f, Blank[]]], Pattern[x, Blank[]]]]], Optional[Pattern[g, Blank[]]]], Pattern[p, Blank[]]], Power[Plus[Pattern[a, Blank[]], Times[Optional[Pattern[b, Blank[]]], sin[Plus[Optional[Pattern[e, Blank[]]], Times[Optional[Pattern[f, Blank[]]], Pattern[x, Blank[]]]]]]], Optional[Pattern[m, Blank[]]]]], Pattern[x, Blank[Symbol]]] := Dist[g^(2*IntPart[p])*(g*Cot[e + f*x])^FracPart[p]*(g*Tan[e + f*x])^FracPart[p], Int[(a + b*Sin[e + f*x])^m/(g*Tan[e + f*x])^p, x], x] /; FreeQ[{a, b, e, f, g, m, p}, x] && !IntegerQ[p]
  2734. Int[Times[Plus[Pattern[a, Blank[]], Times[Optional[Pattern[b, Blank[]]], sin[Plus[Optional[Pattern[e, Blank[]]], Times[Optional[Pattern[f, Blank[]]], Pattern[x, Blank[]]]]]]], Plus[Optional[Pattern[c, Blank[]]], Times[Optional[Pattern[d, Blank[]]], sin[Plus[Optional[Pattern[e, Blank[]]], Times[Optional[Pattern[f, Blank[]]], Pattern[x, Blank[]]]]]]]], Pattern[x, Blank[Symbol]]] := Simp[((2*a*c + b*d)*x)/2, x] + (-Simp[((b*c + a*d)*Cos[e + f*x])/f, x] - Simp[(b*d*Cos[e + f*x]*Sin[e + f*x])/(2*f), x]) /; FreeQ[{a, b, c, d, e, f}, x] && NeQ[b*c - a*d, 0]
  2735. Int[Times[Plus[Optional[Pattern[a, Blank[]]], Times[Optional[Pattern[b, Blank[]]], sin[Plus[Optional[Pattern[e, Blank[]]], Times[Optional[Pattern[f, Blank[]]], Pattern[x, Blank[]]]]]]], Power[Plus[Optional[Pattern[c, Blank[]]], Times[Optional[Pattern[d, Blank[]]], sin[Plus[Optional[Pattern[e, Blank[]]], Times[Optional[Pattern[f, Blank[]]], Pattern[x, Blank[]]]]]]], -1]], Pattern[x, Blank[Symbol]]] := Simp[(b*x)/d, x] - Dist[(b*c - a*d)/d, Int[1/(c + d*Sin[e + f*x]), x], x] /; FreeQ[{a, b, c, d, e, f}, x] && NeQ[b*c - a*d, 0]
  2736. Int[Times[Power[Plus[Pattern[a, Blank[]], Times[Optional[Pattern[b, Blank[]]], sin[Plus[Optional[Pattern[e, Blank[]]], Times[Optional[Pattern[f, Blank[]]], Pattern[x, Blank[]]]]]]], Optional[Pattern[m, Blank[]]]], Power[Plus[Pattern[c, Blank[]], Times[Optional[Pattern[d, Blank[]]], sin[Plus[Optional[Pattern[e, Blank[]]], Times[Optional[Pattern[f, Blank[]]], Pattern[x, Blank[]]]]]]], Optional[Pattern[n, Blank[]]]]], Pattern[x, Blank[Symbol]]] := Dist[a^m*c^m, Int[Cos[e + f*x]^(2*m)*(c + d*Sin[e + f*x])^(n - m), x], x] /; FreeQ[{a, b, c, d, e, f, n}, x] && EqQ[b*c + a*d, 0] && EqQ[a^2 - b^2, 0] && IntegerQ[m] && !(IntegerQ[n] && ((LtQ[m, 0] && GtQ[n, 0]) || LtQ[0, n, m] || LtQ[m, n, 0]))
  2737. Int[Times[Power[Plus[Pattern[a, Blank[]], Times[Optional[Pattern[b, Blank[]]], sin[Plus[Optional[Pattern[e, Blank[]]], Times[Optional[Pattern[f, Blank[]]], Pattern[x, Blank[]]]]]]], Rational[1, 2]], Power[Plus[Pattern[c, Blank[]], Times[Optional[Pattern[d, Blank[]]], sin[Plus[Optional[Pattern[e, Blank[]]], Times[Optional[Pattern[f, Blank[]]], Pattern[x, Blank[]]]]]]], Rational[-1, 2]]], Pattern[x, Blank[Symbol]]] := Dist[(a*c*Cos[e + f*x])/(Sqrt[a + b*Sin[e + f*x]]*Sqrt[c + d*Sin[e + f*x]]), Int[Cos[e + f*x]/(c + d*Sin[e + f*x]), x], x] /; FreeQ[{a, b, c, d, e, f}, x] && EqQ[b*c + a*d, 0] && EqQ[a^2 - b^2, 0]
  2738. Int[Times[Power[Plus[Pattern[a, Blank[]], Times[Optional[Pattern[b, Blank[]]], sin[Plus[Optional[Pattern[e, Blank[]]], Times[Optional[Pattern[f, Blank[]]], Pattern[x, Blank[]]]]]]], Rational[1, 2]], Power[Plus[Pattern[c, Blank[]], Times[Optional[Pattern[d, Blank[]]], sin[Plus[Optional[Pattern[e, Blank[]]], Times[Optional[Pattern[f, Blank[]]], Pattern[x, Blank[]]]]]]], Pattern[n, Blank[]]]], Pattern[x, Blank[Symbol]]] := Simp[(-2*b*Cos[e + f*x]*(c + d*Sin[e + f*x])^n)/(f*(2*n + 1)*Sqrt[a + b*Sin[e + f*x]]), x] /; FreeQ[{a, b, c, d, e, f, n}, x] && EqQ[b*c + a*d, 0] && EqQ[a^2 - b^2, 0] && NeQ[n, -2^(-1)]
  2739. Int[Times[Power[Plus[Pattern[a, Blank[]], Times[Optional[Pattern[b, Blank[]]], sin[Plus[Optional[Pattern[e, Blank[]]], Times[Optional[Pattern[f, Blank[]]], Pattern[x, Blank[]]]]]]], Pattern[m, Blank[]]], Power[Plus[Pattern[c, Blank[]], Times[Optional[Pattern[d, Blank[]]], sin[Plus[Optional[Pattern[e, Blank[]]], Times[Optional[Pattern[f, Blank[]]], Pattern[x, Blank[]]]]]]], Pattern[n, Blank[]]]], Pattern[x, Blank[Symbol]]] := Simp[(-2*b*Cos[e + f*x]*(a + b*Sin[e + f*x])^(m - 1)*(c + d*Sin[e + f*x])^n)/(f*(2*n + 1)), x] - Dist[(b*(2*m - 1))/(d*(2*n + 1)), Int[(a + b*Sin[e + f*x])^(m - 1)*(c + d*Sin[e + f*x])^(n + 1), x], x] /; FreeQ[{a, b, c, d, e, f}, x] && EqQ[b*c + a*d, 0] && EqQ[a^2 - b^2, 0] && IGtQ[m - 1/2, 0] && LtQ[n, -1] && !(ILtQ[m + n, 0] && GtQ[2*m + n + 1, 0])
  2740. Int[Times[Power[Plus[Pattern[a, Blank[]], Times[Optional[Pattern[b, Blank[]]], sin[Plus[Optional[Pattern[e, Blank[]]], Times[Optional[Pattern[f, Blank[]]], Pattern[x, Blank[]]]]]]], Pattern[m, Blank[]]], Power[Plus[Pattern[c, Blank[]], Times[Optional[Pattern[d, Blank[]]], sin[Plus[Optional[Pattern[e, Blank[]]], Times[Optional[Pattern[f, Blank[]]], Pattern[x, Blank[]]]]]]], Pattern[n, Blank[]]]], Pattern[x, Blank[Symbol]]] := -Simp[(b*Cos[e + f*x]*(a + b*Sin[e + f*x])^(m - 1)*(c + d*Sin[e + f*x])^n)/(f*(m + n)), x] + Dist[(a*(2*m - 1))/(m + n), Int[(a + b*Sin[e + f*x])^(m - 1)*(c + d*Sin[e + f*x])^n, x], x] /; FreeQ[{a, b, c, d, e, f, n}, x] && EqQ[b*c + a*d, 0] && EqQ[a^2 - b^2, 0] && IGtQ[m - 1/2, 0] && !LtQ[n, -1] && !(IGtQ[n - 1/2, 0] && LtQ[n, m]) && !(ILtQ[m + n, 0] && GtQ[2*m + n + 1, 0])
  2741. Int[Times[Power[Plus[Pattern[a, Blank[]], Times[Optional[Pattern[b, Blank[]]], sin[Plus[Optional[Pattern[e, Blank[]]], Times[Optional[Pattern[f, Blank[]]], Pattern[x, Blank[]]]]]]], Rational[-1, 2]], Power[Plus[Pattern[c, Blank[]], Times[Optional[Pattern[d, Blank[]]], sin[Plus[Optional[Pattern[e, Blank[]]], Times[Optional[Pattern[f, Blank[]]], Pattern[x, Blank[]]]]]]], Rational[-1, 2]]], Pattern[x, Blank[Symbol]]] := Dist[Cos[e + f*x]/(Sqrt[a + b*Sin[e + f*x]]*Sqrt[c + d*Sin[e + f*x]]), Int[1/Cos[e + f*x], x], x] /; FreeQ[{a, b, c, d, e, f}, x] && EqQ[b*c + a*d, 0] && EqQ[a^2 - b^2, 0]
  2742. Int[Times[Power[Plus[Pattern[a, Blank[]], Times[Optional[Pattern[b, Blank[]]], sin[Plus[Optional[Pattern[e, Blank[]]], Times[Optional[Pattern[f, Blank[]]], Pattern[x, Blank[]]]]]]], Pattern[m, Blank[]]], Power[Plus[Pattern[c, Blank[]], Times[Optional[Pattern[d, Blank[]]], sin[Plus[Optional[Pattern[e, Blank[]]], Times[Optional[Pattern[f, Blank[]]], Pattern[x, Blank[]]]]]]], Pattern[n, Blank[]]]], Pattern[x, Blank[Symbol]]] := Simp[(b*Cos[e + f*x]*(a + b*Sin[e + f*x])^m*(c + d*Sin[e + f*x])^n)/(a*f*(2*m + 1)), x] /; FreeQ[{a, b, c, d, e, f, m, n}, x] && EqQ[b*c + a*d, 0] && EqQ[a^2 - b^2, 0] && EqQ[m + n + 1, 0] && NeQ[m, -2^(-1)]
  2743. Int[Times[Power[Plus[Pattern[a, Blank[]], Times[Optional[Pattern[b, Blank[]]], sin[Plus[Optional[Pattern[e, Blank[]]], Times[Optional[Pattern[f, Blank[]]], Pattern[x, Blank[]]]]]]], Pattern[m, Blank[]]], Power[Plus[Pattern[c, Blank[]], Times[Optional[Pattern[d, Blank[]]], sin[Plus[Optional[Pattern[e, Blank[]]], Times[Optional[Pattern[f, Blank[]]], Pattern[x, Blank[]]]]]]], Pattern[n, Blank[]]]], Pattern[x, Blank[Symbol]]] := Simp[(b*Cos[e + f*x]*(a + b*Sin[e + f*x])^m*(c + d*Sin[e + f*x])^n)/(a*f*(2*m + 1)), x] + Dist[(m + n + 1)/(a*(2*m + 1)), Int[(a + b*Sin[e + f*x])^(m + 1)*(c + d*Sin[e + f*x])^n, x], x] /; FreeQ[{a, b, c, d, e, f, m, n}, x] && EqQ[b*c + a*d, 0] && EqQ[a^2 - b^2, 0] && ILtQ[Simplify[m + n + 1], 0] && NeQ[m, -2^(-1)] && (SumSimplerQ[m, 1] || !SumSimplerQ[n, 1])
  2744. Int[Times[Power[Plus[Pattern[a, Blank[]], Times[Optional[Pattern[b, Blank[]]], sin[Plus[Optional[Pattern[e, Blank[]]], Times[Optional[Pattern[f, Blank[]]], Pattern[x, Blank[]]]]]]], Pattern[m, Blank[]]], Power[Plus[Pattern[c, Blank[]], Times[Optional[Pattern[d, Blank[]]], sin[Plus[Optional[Pattern[e, Blank[]]], Times[Optional[Pattern[f, Blank[]]], Pattern[x, Blank[]]]]]]], Pattern[n, Blank[]]]], Pattern[x, Blank[Symbol]]] := Simp[(b*Cos[e + f*x]*(a + b*Sin[e + f*x])^m*(c + d*Sin[e + f*x])^n)/(a*f*(2*m + 1)), x] + Dist[(m + n + 1)/(a*(2*m + 1)), Int[(a + b*Sin[e + f*x])^(m + 1)*(c + d*Sin[e + f*x])^n, x], x] /; FreeQ[{a, b, c, d, e, f, n}, x] && EqQ[b*c + a*d, 0] && EqQ[a^2 - b^2, 0] && LtQ[m, -1] && !LtQ[m, n, -1] && IntegersQ[2*m, 2*n]
  2745. Int[Times[Power[Plus[Pattern[a, Blank[]], Times[Optional[Pattern[b, Blank[]]], sin[Plus[Optional[Pattern[e, Blank[]]], Times[Optional[Pattern[f, Blank[]]], Pattern[x, Blank[]]]]]]], Pattern[m, Blank[]]], Power[Plus[Pattern[c, Blank[]], Times[Optional[Pattern[d, Blank[]]], sin[Plus[Optional[Pattern[e, Blank[]]], Times[Optional[Pattern[f, Blank[]]], Pattern[x, Blank[]]]]]]], Pattern[n, Blank[]]]], Pattern[x, Blank[Symbol]]] := Dist[(a^IntPart[m]*c^IntPart[m]*(a + b*Sin[e + f*x])^FracPart[m]*(c + d*Sin[e + f*x])^FracPart[m])/Cos[e + f*x]^(2*FracPart[m]), Int[Cos[e + f*x]^(2*m)*(c + d*Sin[e + f*x])^(n - m), x], x] /; FreeQ[{a, b, c, d, e, f, m, n}, x] && EqQ[b*c + a*d, 0] && EqQ[a^2 - b^2, 0] && (FractionQ[m] || !FractionQ[n])
  2746. Int[Times[Power[Plus[Optional[Pattern[a, Blank[]]], Times[Optional[Pattern[b, Blank[]]], sin[Plus[Optional[Pattern[e, Blank[]]], Times[Optional[Pattern[f, Blank[]]], Pattern[x, Blank[]]]]]]], 2], Power[Plus[Optional[Pattern[c, Blank[]]], Times[Optional[Pattern[d, Blank[]]], sin[Plus[Optional[Pattern[e, Blank[]]], Times[Optional[Pattern[f, Blank[]]], Pattern[x, Blank[]]]]]]], -1]], Pattern[x, Blank[Symbol]]] := -Simp[(b^2*Cos[e + f*x])/(d*f), x] + Dist[1/d, Int[Simp[a^2*d - b*(b*c - 2*a*d)*Sin[e + f*x], x]/(c + d*Sin[e + f*x]), x], x] /; FreeQ[{a, b, c, d, e, f}, x] && NeQ[b*c - a*d, 0]
  2747. Int[Times[Power[Plus[Optional[Pattern[a, Blank[]]], Times[Optional[Pattern[b, Blank[]]], sin[Plus[Optional[Pattern[e, Blank[]]], Times[Optional[Pattern[f, Blank[]]], Pattern[x, Blank[]]]]]]], -1], Power[Plus[Optional[Pattern[c, Blank[]]], Times[Optional[Pattern[d, Blank[]]], sin[Plus[Optional[Pattern[e, Blank[]]], Times[Optional[Pattern[f, Blank[]]], Pattern[x, Blank[]]]]]]], -1]], Pattern[x, Blank[Symbol]]] := Dist[b/(b*c - a*d), Int[1/(a + b*Sin[e + f*x]), x], x] - Dist[d/(b*c - a*d), Int[1/(c + d*Sin[e + f*x]), x], x] /; FreeQ[{a, b, c, d, e, f}, x] && NeQ[b*c - a*d, 0]
  2748. Int[Times[Power[Times[Optional[Pattern[b, Blank[]]], sin[Plus[Optional[Pattern[e, Blank[]]], Times[Optional[Pattern[f, Blank[]]], Pattern[x, Blank[]]]]]], Pattern[m, Blank[]]], Plus[Pattern[c, Blank[]], Times[Optional[Pattern[d, Blank[]]], sin[Plus[Optional[Pattern[e, Blank[]]], Times[Optional[Pattern[f, Blank[]]], Pattern[x, Blank[]]]]]]]], Pattern[x, Blank[Symbol]]] := Dist[c, Int[(b*Sin[e + f*x])^m, x], x] + Dist[d/b, Int[(b*Sin[e + f*x])^(m + 1), x], x] /; FreeQ[{b, c, d, e, f, m}, x]
  2749. Int[Times[Power[Plus[Pattern[a, Blank[]], Times[Optional[Pattern[b, Blank[]]], sin[Plus[Optional[Pattern[e, Blank[]]], Times[Optional[Pattern[f, Blank[]]], Pattern[x, Blank[]]]]]]], Pattern[m, Blank[]]], Plus[Pattern[c, Blank[]], Times[Optional[Pattern[d, Blank[]]], sin[Plus[Optional[Pattern[e, Blank[]]], Times[Optional[Pattern[f, Blank[]]], Pattern[x, Blank[]]]]]]]], Pattern[x, Blank[Symbol]]] := -Simp[(d*Cos[e + f*x]*(a + b*Sin[e + f*x])^m)/(f*(m + 1)), x] /; FreeQ[{a, b, c, d, e, f, m}, x] && NeQ[b*c - a*d, 0] && EqQ[a^2 - b^2, 0] && EqQ[a*d*m + b*c*(m + 1), 0]
  2750. Int[Times[Power[Plus[Pattern[a, Blank[]], Times[Optional[Pattern[b, Blank[]]], sin[Plus[Optional[Pattern[e, Blank[]]], Times[Optional[Pattern[f, Blank[]]], Pattern[x, Blank[]]]]]]], Pattern[m, Blank[]]], Plus[Optional[Pattern[c, Blank[]]], Times[Optional[Pattern[d, Blank[]]], sin[Plus[Optional[Pattern[e, Blank[]]], Times[Optional[Pattern[f, Blank[]]], Pattern[x, Blank[]]]]]]]], Pattern[x, Blank[Symbol]]] := Simp[((b*c - a*d)*Cos[e + f*x]*(a + b*Sin[e + f*x])^m)/(a*f*(2*m + 1)), x] + Dist[(a*d*m + b*c*(m + 1))/(a*b*(2*m + 1)), Int[(a + b*Sin[e + f*x])^(m + 1), x], x] /; FreeQ[{a, b, c, d, e, f}, x] && NeQ[b*c - a*d, 0] && EqQ[a^2 - b^2, 0] && LtQ[m, -2^(-1)]
  2751. Int[Times[Power[Plus[Pattern[a, Blank[]], Times[Optional[Pattern[b, Blank[]]], sin[Plus[Optional[Pattern[e, Blank[]]], Times[Optional[Pattern[f, Blank[]]], Pattern[x, Blank[]]]]]]], Pattern[m, Blank[]]], Plus[Optional[Pattern[c, Blank[]]], Times[Optional[Pattern[d, Blank[]]], sin[Plus[Optional[Pattern[e, Blank[]]], Times[Optional[Pattern[f, Blank[]]], Pattern[x, Blank[]]]]]]]], Pattern[x, Blank[Symbol]]] := -Simp[(d*Cos[e + f*x]*(a + b*Sin[e + f*x])^m)/(f*(m + 1)), x] + Dist[(a*d*m + b*c*(m + 1))/(b*(m + 1)), Int[(a + b*Sin[e + f*x])^m, x], x] /; FreeQ[{a, b, c, d, e, f, m}, x] && NeQ[b*c - a*d, 0] && EqQ[a^2 - b^2, 0] && !LtQ[m, -2^(-1)]
  2752. Int[Times[Power[Plus[Pattern[a, Blank[]], Times[Optional[Pattern[b, Blank[]]], sin[Plus[Optional[Pattern[e, Blank[]]], Times[Optional[Pattern[f, Blank[]]], Pattern[x, Blank[]]]]]]], Rational[-1, 2]], Plus[Optional[Pattern[c, Blank[]]], Times[Optional[Pattern[d, Blank[]]], sin[Plus[Optional[Pattern[e, Blank[]]], Times[Optional[Pattern[f, Blank[]]], Pattern[x, Blank[]]]]]]]], Pattern[x, Blank[Symbol]]] := Dist[(b*c - a*d)/b, Int[1/Sqrt[a + b*Sin[e + f*x]], x], x] + Dist[d/b, Int[Sqrt[a + b*Sin[e + f*x]], x], x] /; FreeQ[{a, b, c, d, e, f}, x] && NeQ[b*c - a*d, 0] && NeQ[a^2 - b^2, 0]
  2753. Int[Times[Power[Plus[Pattern[a, Blank[]], Times[Optional[Pattern[b, Blank[]]], sin[Plus[Optional[Pattern[e, Blank[]]], Times[Optional[Pattern[f, Blank[]]], Pattern[x, Blank[]]]]]]], Pattern[m, Blank[]]], Plus[Optional[Pattern[c, Blank[]]], Times[Optional[Pattern[d, Blank[]]], sin[Plus[Optional[Pattern[e, Blank[]]], Times[Optional[Pattern[f, Blank[]]], Pattern[x, Blank[]]]]]]]], Pattern[x, Blank[Symbol]]] := -Simp[(d*Cos[e + f*x]*(a + b*Sin[e + f*x])^m)/(f*(m + 1)), x] + Dist[1/(m + 1), Int[(a + b*Sin[e + f*x])^(m - 1)*Simp[b*d*m + a*c*(m + 1) + (a*d*m + b*c*(m + 1))*Sin[e + f*x], x], x], x] /; FreeQ[{a, b, c, d, e, f}, x] && NeQ[b*c - a*d, 0] && NeQ[a^2 - b^2, 0] && GtQ[m, 0] && IntegerQ[2*m]
  2754. Int[Times[Power[Plus[Pattern[a, Blank[]], Times[Optional[Pattern[b, Blank[]]], sin[Plus[Optional[Pattern[e, Blank[]]], Times[Optional[Pattern[f, Blank[]]], Pattern[x, Blank[]]]]]]], Pattern[m, Blank[]]], Plus[Optional[Pattern[c, Blank[]]], Times[Optional[Pattern[d, Blank[]]], sin[Plus[Optional[Pattern[e, Blank[]]], Times[Optional[Pattern[f, Blank[]]], Pattern[x, Blank[]]]]]]]], Pattern[x, Blank[Symbol]]] := -Simp[((b*c - a*d)*Cos[e + f*x]*(a + b*Sin[e + f*x])^(m + 1))/(f*(m + 1)*(a^2 - b^2)), x] + Dist[1/((m + 1)*(a^2 - b^2)), Int[(a + b*Sin[e + f*x])^(m + 1)*Simp[(a*c - b*d)*(m + 1) - (b*c - a*d)*(m + 2)*Sin[e + f*x], x], x], x] /; FreeQ[{a, b, c, d, e, f}, x] && NeQ[b*c - a*d, 0] && NeQ[a^2 - b^2, 0] && LtQ[m, -1] && IntegerQ[2*m]
  2755. Int[Times[Power[Plus[Pattern[a, Blank[]], Times[Optional[Pattern[b, Blank[]]], sin[Plus[Optional[Pattern[e, Blank[]]], Times[Optional[Pattern[f, Blank[]]], Pattern[x, Blank[]]]]]]], Pattern[m, Blank[]]], Plus[Pattern[c, Blank[]], Times[Optional[Pattern[d, Blank[]]], sin[Plus[Optional[Pattern[e, Blank[]]], Times[Optional[Pattern[f, Blank[]]], Pattern[x, Blank[]]]]]]]], Pattern[x, Blank[Symbol]]] := Dist[(c*Cos[e + f*x])/(f*Sqrt[1 + Sin[e + f*x]]*Sqrt[1 - Sin[e + f*x]]), Subst[Int[((a + b*x)^m*Sqrt[1 + (d*x)/c])/Sqrt[1 - (d*x)/c], x], x, Sin[e + f*x]], x] /; FreeQ[{a, b, c, d, e, f, m}, x] && NeQ[b*c - a*d, 0] && NeQ[a^2 - b^2, 0] && !IntegerQ[2*m] && EqQ[c^2 - d^2, 0]
  2756. Int[Times[Power[Plus[Pattern[a, Blank[]], Times[Optional[Pattern[b, Blank[]]], sin[Plus[Optional[Pattern[e, Blank[]]], Times[Optional[Pattern[f, Blank[]]], Pattern[x, Blank[]]]]]]], Pattern[m, Blank[]]], Plus[Optional[Pattern[c, Blank[]]], Times[Optional[Pattern[d, Blank[]]], sin[Plus[Optional[Pattern[e, Blank[]]], Times[Optional[Pattern[f, Blank[]]], Pattern[x, Blank[]]]]]]]], Pattern[x, Blank[Symbol]]] := Dist[(b*c - a*d)/b, Int[(a + b*Sin[e + f*x])^m, x], x] + Dist[d/b, Int[(a + b*Sin[e + f*x])^(m + 1), x], x] /; FreeQ[{a, b, c, d, e, f, m}, x] && NeQ[b*c - a*d, 0] && NeQ[a^2 - b^2, 0]
  2757. Int[Times[Power[Times[Optional[Pattern[d, Blank[]]], sin[Plus[Optional[Pattern[e, Blank[]]], Times[Optional[Pattern[f, Blank[]]], Pattern[x, Blank[]]]]]], Optional[Pattern[n, Blank[]]]], Power[Plus[Pattern[a, Blank[]], Times[Optional[Pattern[b, Blank[]]], sin[Plus[Optional[Pattern[e, Blank[]]], Times[Optional[Pattern[f, Blank[]]], Pattern[x, Blank[]]]]]]], Optional[Pattern[m, Blank[]]]]], Pattern[x, Blank[Symbol]]] := Int[ExpandTrig[(a + b*sin[e + f*x])^m*(d*sin[e + f*x])^n, x], x] /; FreeQ[{a, b, d, e, f, n}, x] && EqQ[a^2 - b^2, 0] && IGtQ[m, 0] && RationalQ[n]
  2758. Int[Times[Power[sin[Plus[Optional[Pattern[e, Blank[]]], Times[Optional[Pattern[f, Blank[]]], Pattern[x, Blank[]]]]], 2], Power[Plus[Pattern[a, Blank[]], Times[Optional[Pattern[b, Blank[]]], sin[Plus[Optional[Pattern[e, Blank[]]], Times[Optional[Pattern[f, Blank[]]], Pattern[x, Blank[]]]]]]], Pattern[m, Blank[]]]], Pattern[x, Blank[Symbol]]] := Simp[(b*Cos[e + f*x]*(a + b*Sin[e + f*x])^m)/(a*f*(2*m + 1)), x] - Dist[1/(a^2*(2*m + 1)), Int[(a + b*Sin[e + f*x])^(m + 1)*(a*m - b*(2*m + 1)*Sin[e + f*x]), x], x] /; FreeQ[{a, b, e, f}, x] && EqQ[a^2 - b^2, 0] && LtQ[m, -2^(-1)]
  2759. Int[Times[Power[sin[Plus[Optional[Pattern[e, Blank[]]], Times[Optional[Pattern[f, Blank[]]], Pattern[x, Blank[]]]]], 2], Power[Plus[Pattern[a, Blank[]], Times[Optional[Pattern[b, Blank[]]], sin[Plus[Optional[Pattern[e, Blank[]]], Times[Optional[Pattern[f, Blank[]]], Pattern[x, Blank[]]]]]]], Pattern[m, Blank[]]]], Pattern[x, Blank[Symbol]]] := -Simp[(Cos[e + f*x]*(a + b*Sin[e + f*x])^(m + 1))/(b*f*(m + 2)), x] + Dist[1/(b*(m + 2)), Int[(a + b*Sin[e + f*x])^m*(b*(m + 1) - a*Sin[e + f*x]), x], x] /; FreeQ[{a, b, e, f, m}, x] && EqQ[a^2 - b^2, 0] && !LtQ[m, -2^(-1)]
  2760. Int[Times[Power[Plus[Pattern[a, Blank[]], Times[Optional[Pattern[b, Blank[]]], sin[Plus[Optional[Pattern[e, Blank[]]], Times[Optional[Pattern[f, Blank[]]], Pattern[x, Blank[]]]]]]], Pattern[m, Blank[]]], Power[Plus[Pattern[c, Blank[]], Times[Optional[Pattern[d, Blank[]]], sin[Plus[Optional[Pattern[e, Blank[]]], Times[Optional[Pattern[f, Blank[]]], Pattern[x, Blank[]]]]]]], 2]], Pattern[x, Blank[Symbol]]] := Simp[((b*c - a*d)*Cos[e + f*x]*(a + b*Sin[e + f*x])^m*(c + d*Sin[e + f*x]))/(a*f*(2*m + 1)), x] + Dist[1/(a*b*(2*m + 1)), Int[(a + b*Sin[e + f*x])^(m + 1)*Simp[a*c*d*(m - 1) + b*(d^2 + c^2*(m + 1)) + d*(a*d*(m - 1) + b*c*(m + 2))*Sin[e + f*x], x], x], x] /; FreeQ[{a, b, c, d, e, f}, x] && NeQ[b*c - a*d, 0] && EqQ[a^2 - b^2, 0] && LtQ[m, -1]
  2761. Int[Times[Power[Plus[Pattern[a, Blank[]], Times[Optional[Pattern[b, Blank[]]], sin[Plus[Optional[Pattern[e, Blank[]]], Times[Optional[Pattern[f, Blank[]]], Pattern[x, Blank[]]]]]]], Pattern[m, Blank[]]], Power[Plus[Pattern[c, Blank[]], Times[Optional[Pattern[d, Blank[]]], sin[Plus[Optional[Pattern[e, Blank[]]], Times[Optional[Pattern[f, Blank[]]], Pattern[x, Blank[]]]]]]], 2]], Pattern[x, Blank[Symbol]]] := -Simp[(d^2*Cos[e + f*x]*(a + b*Sin[e + f*x])^(m + 1))/(b*f*(m + 2)), x] + Dist[1/(b*(m + 2)), Int[(a + b*Sin[e + f*x])^m*Simp[b*(d^2*(m + 1) + c^2*(m + 2)) - d*(a*d - 2*b*c*(m + 2))*Sin[e + f*x], x], x], x] /; FreeQ[{a, b, c, d, e, f, m}, x] && NeQ[b*c - a*d, 0] && EqQ[a^2 - b^2, 0] && !LtQ[m, -1]
  2762. Int[Times[Power[Plus[Pattern[a, Blank[]], Times[Optional[Pattern[b, Blank[]]], sin[Plus[Optional[Pattern[e, Blank[]]], Times[Optional[Pattern[f, Blank[]]], Pattern[x, Blank[]]]]]]], Pattern[m, Blank[]]], Power[Plus[Optional[Pattern[c, Blank[]]], Times[Optional[Pattern[d, Blank[]]], sin[Plus[Optional[Pattern[e, Blank[]]], Times[Optional[Pattern[f, Blank[]]], Pattern[x, Blank[]]]]]]], Pattern[n, Blank[]]]], Pattern[x, Blank[Symbol]]] := -Simp[(b^2*(b*c - a*d)*Cos[e + f*x]*(a + b*Sin[e + f*x])^(m - 2)*(c + d*Sin[e + f*x])^(n + 1))/(d*f*(n + 1)*(b*c + a*d)), x] + Dist[b^2/(d*(n + 1)*(b*c + a*d)), Int[(a + b*Sin[e + f*x])^(m - 2)*(c + d*Sin[e + f*x])^(n + 1)*Simp[a*c*(m - 2) - b*d*(m - 2*n - 4) - (b*c*(m - 1) - a*d*(m + 2*n + 1))*Sin[e + f*x], x], x], x] /; FreeQ[{a, b, c, d, e, f}, x] && NeQ[b*c - a*d, 0] && EqQ[a^2 - b^2, 0] && NeQ[c^2 - d^2, 0] && GtQ[m, 1] && LtQ[n, -1] && (IntegersQ[2*m, 2*n] || IntegerQ[m + 1/2] || (IntegerQ[m] && EqQ[c, 0]))
  2763. Int[Times[Power[Plus[Pattern[a, Blank[]], Times[Optional[Pattern[b, Blank[]]], sin[Plus[Optional[Pattern[e, Blank[]]], Times[Optional[Pattern[f, Blank[]]], Pattern[x, Blank[]]]]]]], Pattern[m, Blank[]]], Power[Plus[Optional[Pattern[c, Blank[]]], Times[Optional[Pattern[d, Blank[]]], sin[Plus[Optional[Pattern[e, Blank[]]], Times[Optional[Pattern[f, Blank[]]], Pattern[x, Blank[]]]]]]], Pattern[n, Blank[]]]], Pattern[x, Blank[Symbol]]] := -Simp[(b^2*Cos[e + f*x]*(a + b*Sin[e + f*x])^(m - 2)*(c + d*Sin[e + f*x])^(n + 1))/(d*f*(m + n)), x] + Dist[1/(d*(m + n)), Int[(a + b*Sin[e + f*x])^(m - 2)*(c + d*Sin[e + f*x])^n*Simp[a*b*c*(m - 2) + b^2*d*(n + 1) + a^2*d*(m + n) - b*(b*c*(m - 1) - a*d*(3*m + 2*n - 2))*Sin[e + f*x], x], x], x] /; FreeQ[{a, b, c, d, e, f, n}, x] && NeQ[b*c - a*d, 0] && EqQ[a^2 - b^2, 0] && NeQ[c^2 - d^2, 0] && GtQ[m, 1] && !LtQ[n, -1] && (IntegersQ[2*m, 2*n] || IntegerQ[m + 1/2] || (IntegerQ[m] && EqQ[c, 0]))
  2764. Int[Times[Power[Plus[Pattern[a, Blank[]], Times[Optional[Pattern[b, Blank[]]], sin[Plus[Optional[Pattern[e, Blank[]]], Times[Optional[Pattern[f, Blank[]]], Pattern[x, Blank[]]]]]]], Pattern[m, Blank[]]], Power[Plus[Optional[Pattern[c, Blank[]]], Times[Optional[Pattern[d, Blank[]]], sin[Plus[Optional[Pattern[e, Blank[]]], Times[Optional[Pattern[f, Blank[]]], Pattern[x, Blank[]]]]]]], Pattern[n, Blank[]]]], Pattern[x, Blank[Symbol]]] := Simp[(b*Cos[e + f*x]*(a + b*Sin[e + f*x])^m*(c + d*Sin[e + f*x])^n)/(a*f*(2*m + 1)), x] - Dist[1/(a*b*(2*m + 1)), Int[(a + b*Sin[e + f*x])^(m + 1)*(c + d*Sin[e + f*x])^(n - 1)*Simp[a*d*n - b*c*(m + 1) - b*d*(m + n + 1)*Sin[e + f*x], x], x], x] /; FreeQ[{a, b, c, d, e, f}, x] && NeQ[b*c - a*d, 0] && EqQ[a^2 - b^2, 0] && NeQ[c^2 - d^2, 0] && LtQ[m, -1] && LtQ[0, n, 1] && (IntegersQ[2*m, 2*n] || (IntegerQ[m] && EqQ[c, 0]))
  2765. Int[Times[Power[Plus[Pattern[a, Blank[]], Times[Optional[Pattern[b, Blank[]]], sin[Plus[Optional[Pattern[e, Blank[]]], Times[Optional[Pattern[f, Blank[]]], Pattern[x, Blank[]]]]]]], Pattern[m, Blank[]]], Power[Plus[Optional[Pattern[c, Blank[]]], Times[Optional[Pattern[d, Blank[]]], sin[Plus[Optional[Pattern[e, Blank[]]], Times[Optional[Pattern[f, Blank[]]], Pattern[x, Blank[]]]]]]], Pattern[n, Blank[]]]], Pattern[x, Blank[Symbol]]] := Simp[((b*c - a*d)*Cos[e + f*x]*(a + b*Sin[e + f*x])^m*(c + d*Sin[e + f*x])^(n - 1))/(a*f*(2*m + 1)), x] + Dist[1/(a*b*(2*m + 1)), Int[(a + b*Sin[e + f*x])^(m + 1)*(c + d*Sin[e + f*x])^(n - 2)*Simp[b*(c^2*(m + 1) + d^2*(n - 1)) + a*c*d*(m - n + 1) + d*(a*d*(m - n + 1) + b*c*(m + n))*Sin[e + f*x], x], x], x] /; FreeQ[{a, b, c, d, e, f}, x] && NeQ[b*c - a*d, 0] && EqQ[a^2 - b^2, 0] && NeQ[c^2 - d^2, 0] && LtQ[m, -1] && GtQ[n, 1] && (IntegersQ[2*m, 2*n] || (IntegerQ[m] && EqQ[c, 0]))
  2766. Int[Times[Power[Plus[Pattern[a, Blank[]], Times[Optional[Pattern[b, Blank[]]], sin[Plus[Optional[Pattern[e, Blank[]]], Times[Optional[Pattern[f, Blank[]]], Pattern[x, Blank[]]]]]]], Pattern[m, Blank[]]], Power[Plus[Optional[Pattern[c, Blank[]]], Times[Optional[Pattern[d, Blank[]]], sin[Plus[Optional[Pattern[e, Blank[]]], Times[Optional[Pattern[f, Blank[]]], Pattern[x, Blank[]]]]]]], Pattern[n, Blank[]]]], Pattern[x, Blank[Symbol]]] := Simp[(b^2*Cos[e + f*x]*(a + b*Sin[e + f*x])^m*(c + d*Sin[e + f*x])^(n + 1))/(a*f*(2*m + 1)*(b*c - a*d)), x] + Dist[1/(a*(2*m + 1)*(b*c - a*d)), Int[(a + b*Sin[e + f*x])^(m + 1)*(c + d*Sin[e + f*x])^n*Simp[b*c*(m + 1) - a*d*(2*m + n + 2) + b*d*(m + n + 2)*Sin[e + f*x], x], x], x] /; FreeQ[{a, b, c, d, e, f, n}, x] && NeQ[b*c - a*d, 0] && EqQ[a^2 - b^2, 0] && NeQ[c^2 - d^2, 0] && LtQ[m, -1] && !GtQ[n, 0] && (IntegersQ[2*m, 2*n] || (IntegerQ[m] && EqQ[c, 0]))
  2767. Int[Times[Power[Plus[Pattern[a, Blank[]], Times[Optional[Pattern[b, Blank[]]], sin[Plus[Optional[Pattern[e, Blank[]]], Times[Optional[Pattern[f, Blank[]]], Pattern[x, Blank[]]]]]]], -1], Power[Plus[Optional[Pattern[c, Blank[]]], Times[Optional[Pattern[d, Blank[]]], sin[Plus[Optional[Pattern[e, Blank[]]], Times[Optional[Pattern[f, Blank[]]], Pattern[x, Blank[]]]]]]], Pattern[n, Blank[]]]], Pattern[x, Blank[Symbol]]] := -Simp[((b*c - a*d)*Cos[e + f*x]*(c + d*Sin[e + f*x])^(n - 1))/(a*f*(a + b*Sin[e + f*x])), x] - Dist[d/(a*b), Int[(c + d*Sin[e + f*x])^(n - 2)*Simp[b*d*(n - 1) - a*c*n + (b*c*(n - 1) - a*d*n)*Sin[e + f*x], x], x], x] /; FreeQ[{a, b, c, d, e, f}, x] && NeQ[b*c - a*d, 0] && EqQ[a^2 - b^2, 0] && NeQ[c^2 - d^2, 0] && GtQ[n, 1] && (IntegerQ[2*n] || EqQ[c, 0])
  2768. Int[Times[Power[Plus[Pattern[a, Blank[]], Times[Optional[Pattern[b, Blank[]]], sin[Plus[Optional[Pattern[e, Blank[]]], Times[Optional[Pattern[f, Blank[]]], Pattern[x, Blank[]]]]]]], -1], Power[Plus[Optional[Pattern[c, Blank[]]], Times[Optional[Pattern[d, Blank[]]], sin[Plus[Optional[Pattern[e, Blank[]]], Times[Optional[Pattern[f, Blank[]]], Pattern[x, Blank[]]]]]]], Pattern[n, Blank[]]]], Pattern[x, Blank[Symbol]]] := -Simp[(b^2*Cos[e + f*x]*(c + d*Sin[e + f*x])^(n + 1))/(a*f*(b*c - a*d)*(a + b*Sin[e + f*x])), x] + Dist[d/(a*(b*c - a*d)), Int[(c + d*Sin[e + f*x])^n*(a*n - b*(n + 1)*Sin[e + f*x]), x], x] /; FreeQ[{a, b, c, d, e, f}, x] && NeQ[b*c - a*d, 0] && EqQ[a^2 - b^2, 0] && NeQ[c^2 - d^2, 0] && LtQ[n, 0] && (IntegerQ[2*n] || EqQ[c, 0])
  2769. Int[Times[Power[Plus[Pattern[a, Blank[]], Times[Optional[Pattern[b, Blank[]]], sin[Plus[Optional[Pattern[e, Blank[]]], Times[Optional[Pattern[f, Blank[]]], Pattern[x, Blank[]]]]]]], -1], Power[Plus[Optional[Pattern[c, Blank[]]], Times[Optional[Pattern[d, Blank[]]], sin[Plus[Optional[Pattern[e, Blank[]]], Times[Optional[Pattern[f, Blank[]]], Pattern[x, Blank[]]]]]]], Pattern[n, Blank[]]]], Pattern[x, Blank[Symbol]]] := -Simp[(b*Cos[e + f*x]*(c + d*Sin[e + f*x])^n)/(a*f*(a + b*Sin[e + f*x])), x] + Dist[(d*n)/(a*b), Int[(c + d*Sin[e + f*x])^(n - 1)*(a - b*Sin[e + f*x]), x], x] /; FreeQ[{a, b, c, d, e, f, n}, x] && NeQ[b*c - a*d, 0] && EqQ[a^2 - b^2, 0] && NeQ[c^2 - d^2, 0] && (IntegerQ[2*n] || EqQ[c, 0])
  2770. Int[Times[Power[Plus[Pattern[a, Blank[]], Times[Optional[Pattern[b, Blank[]]], sin[Plus[Optional[Pattern[e, Blank[]]], Times[Optional[Pattern[f, Blank[]]], Pattern[x, Blank[]]]]]]], Rational[1, 2]], Power[Plus[Optional[Pattern[c, Blank[]]], Times[Optional[Pattern[d, Blank[]]], sin[Plus[Optional[Pattern[e, Blank[]]], Times[Optional[Pattern[f, Blank[]]], Pattern[x, Blank[]]]]]]], Pattern[n, Blank[]]]], Pattern[x, Blank[Symbol]]] := Simp[(-2*b*Cos[e + f*x]*(c + d*Sin[e + f*x])^n)/(f*(2*n + 1)*Sqrt[a + b*Sin[e + f*x]]), x] + Dist[(2*n*(b*c + a*d))/(b*(2*n + 1)), Int[Sqrt[a + b*Sin[e + f*x]]*(c + d*Sin[e + f*x])^(n - 1), x], x] /; FreeQ[{a, b, c, d, e, f}, x] && NeQ[b*c - a*d, 0] && EqQ[a^2 - b^2, 0] && NeQ[c^2 - d^2, 0] && GtQ[n, 0] && IntegerQ[2*n]
  2771. Int[Times[Power[Plus[Pattern[a, Blank[]], Times[Optional[Pattern[b, Blank[]]], sin[Plus[Optional[Pattern[e, Blank[]]], Times[Optional[Pattern[f, Blank[]]], Pattern[x, Blank[]]]]]]], Rational[1, 2]], Power[Plus[Optional[Pattern[c, Blank[]]], Times[Optional[Pattern[d, Blank[]]], sin[Plus[Optional[Pattern[e, Blank[]]], Times[Optional[Pattern[f, Blank[]]], Pattern[x, Blank[]]]]]]], Rational[-3, 2]]], Pattern[x, Blank[Symbol]]] := Simp[(-2*b^2*Cos[e + f*x])/(f*(b*c + a*d)*Sqrt[a + b*Sin[e + f*x]]*Sqrt[c + d*Sin[e + f*x]]), x] /; FreeQ[{a, b, c, d, e, f}, x] && NeQ[b*c - a*d, 0] && EqQ[a^2 - b^2, 0] && NeQ[c^2 - d^2, 0]
  2772. Int[Times[Power[Plus[Pattern[a, Blank[]], Times[Optional[Pattern[b, Blank[]]], sin[Plus[Optional[Pattern[e, Blank[]]], Times[Optional[Pattern[f, Blank[]]], Pattern[x, Blank[]]]]]]], Rational[1, 2]], Power[Plus[Optional[Pattern[c, Blank[]]], Times[Optional[Pattern[d, Blank[]]], sin[Plus[Optional[Pattern[e, Blank[]]], Times[Optional[Pattern[f, Blank[]]], Pattern[x, Blank[]]]]]]], Pattern[n, Blank[]]]], Pattern[x, Blank[Symbol]]] := Simp[((b*c - a*d)*Cos[e + f*x]*(c + d*Sin[e + f*x])^(n + 1))/(f*(n + 1)*(c^2 - d^2)*Sqrt[a + b*Sin[e + f*x]]), x] + Dist[((2*n + 3)*(b*c - a*d))/(2*b*(n + 1)*(c^2 - d^2)), Int[Sqrt[a + b*Sin[e + f*x]]*(c + d*Sin[e + f*x])^(n + 1), x], x] /; FreeQ[{a, b, c, d, e, f}, x] && NeQ[b*c - a*d, 0] && EqQ[a^2 - b^2, 0] && NeQ[c^2 - d^2, 0] && LtQ[n, -1] && NeQ[2*n + 3, 0] && IntegerQ[2*n]
  2773. Int[Times[Power[Plus[Pattern[a, Blank[]], Times[Optional[Pattern[b, Blank[]]], sin[Plus[Optional[Pattern[e, Blank[]]], Times[Optional[Pattern[f, Blank[]]], Pattern[x, Blank[]]]]]]], Rational[1, 2]], Power[Plus[Optional[Pattern[c, Blank[]]], Times[Optional[Pattern[d, Blank[]]], sin[Plus[Optional[Pattern[e, Blank[]]], Times[Optional[Pattern[f, Blank[]]], Pattern[x, Blank[]]]]]]], -1]], Pattern[x, Blank[Symbol]]] := Dist[(-2*b)/f, Subst[Int[1/(b*c + a*d - d*x^2), x], x, (b*Cos[e + f*x])/Sqrt[a + b*Sin[e + f*x]]], x] /; FreeQ[{a, b, c, d, e, f}, x] && NeQ[b*c - a*d, 0] && EqQ[a^2 - b^2, 0] && NeQ[c^2 - d^2, 0]
  2774. Int[Times[Power[Times[Optional[Pattern[d, Blank[]]], sin[Plus[Optional[Pattern[e, Blank[]]], Times[Optional[Pattern[f, Blank[]]], Pattern[x, Blank[]]]]]], Rational[-1, 2]], Power[Plus[Pattern[a, Blank[]], Times[Optional[Pattern[b, Blank[]]], sin[Plus[Optional[Pattern[e, Blank[]]], Times[Optional[Pattern[f, Blank[]]], Pattern[x, Blank[]]]]]]], Rational[1, 2]]], Pattern[x, Blank[Symbol]]] := Dist[-2/f, Subst[Int[1/Sqrt[1 - x^2/a], x], x, (b*Cos[e + f*x])/Sqrt[a + b*Sin[e + f*x]]], x] /; FreeQ[{a, b, d, e, f}, x] && EqQ[a^2 - b^2, 0] && EqQ[d, a/b]
  2775. Int[Times[Power[Plus[Pattern[a, Blank[]], Times[Optional[Pattern[b, Blank[]]], sin[Plus[Optional[Pattern[e, Blank[]]], Times[Optional[Pattern[f, Blank[]]], Pattern[x, Blank[]]]]]]], Rational[1, 2]], Power[Plus[Optional[Pattern[c, Blank[]]], Times[Optional[Pattern[d, Blank[]]], sin[Plus[Optional[Pattern[e, Blank[]]], Times[Optional[Pattern[f, Blank[]]], Pattern[x, Blank[]]]]]]], Rational[-1, 2]]], Pattern[x, Blank[Symbol]]] := Dist[(-2*b)/f, Subst[Int[1/(b + d*x^2), x], x, (b*Cos[e + f*x])/(Sqrt[a + b*Sin[e + f*x]]*Sqrt[c + d*Sin[e + f*x]])], x] /; FreeQ[{a, b, c, d, e, f}, x] && NeQ[b*c - a*d, 0] && EqQ[a^2 - b^2, 0] && NeQ[c^2 - d^2, 0]
  2776. Int[Times[Power[Plus[Pattern[a, Blank[]], Times[Optional[Pattern[b, Blank[]]], sin[Plus[Optional[Pattern[e, Blank[]]], Times[Optional[Pattern[f, Blank[]]], Pattern[x, Blank[]]]]]]], Rational[1, 2]], Power[Plus[Optional[Pattern[c, Blank[]]], Times[Optional[Pattern[d, Blank[]]], sin[Plus[Optional[Pattern[e, Blank[]]], Times[Optional[Pattern[f, Blank[]]], Pattern[x, Blank[]]]]]]], Pattern[n, Blank[]]]], Pattern[x, Blank[Symbol]]] := Dist[(a^2*Cos[e + f*x])/(f*Sqrt[a + b*Sin[e + f*x]]*Sqrt[a - b*Sin[e + f*x]]), Subst[Int[(c + d*x)^n/Sqrt[a - b*x], x], x, Sin[e + f*x]], x] /; FreeQ[{a, b, c, d, e, f, n}, x] && NeQ[b*c - a*d, 0] && EqQ[a^2 - b^2, 0] && NeQ[c^2 - d^2, 0] && !IntegerQ[2*n]
  2777. Int[Times[Power[Plus[Pattern[a, Blank[]], Times[Optional[Pattern[b, Blank[]]], sin[Plus[Optional[Pattern[e, Blank[]]], Times[Optional[Pattern[f, Blank[]]], Pattern[x, Blank[]]]]]]], Rational[-1, 2]], Power[Plus[Optional[Pattern[c, Blank[]]], Times[Optional[Pattern[d, Blank[]]], sin[Plus[Optional[Pattern[e, Blank[]]], Times[Optional[Pattern[f, Blank[]]], Pattern[x, Blank[]]]]]]], Rational[1, 2]]], Pattern[x, Blank[Symbol]]] := Dist[d/b, Int[Sqrt[a + b*Sin[e + f*x]]/Sqrt[c + d*Sin[e + f*x]], x], x] + Dist[(b*c - a*d)/b, Int[1/(Sqrt[a + b*Sin[e + f*x]]*Sqrt[c + d*Sin[e + f*x]]), x], x] /; FreeQ[{a, b, c, d, e, f}, x] && NeQ[b*c - a*d, 0] && EqQ[a^2 - b^2, 0] && NeQ[c^2 - d^2, 0]
  2778. Int[Times[Power[Plus[Pattern[a, Blank[]], Times[Optional[Pattern[b, Blank[]]], sin[Plus[Optional[Pattern[e, Blank[]]], Times[Optional[Pattern[f, Blank[]]], Pattern[x, Blank[]]]]]]], Rational[-1, 2]], Power[Plus[Optional[Pattern[c, Blank[]]], Times[Optional[Pattern[d, Blank[]]], sin[Plus[Optional[Pattern[e, Blank[]]], Times[Optional[Pattern[f, Blank[]]], Pattern[x, Blank[]]]]]]], Pattern[n, Blank[]]]], Pattern[x, Blank[Symbol]]] := Simp[(-2*d*Cos[e + f*x]*(c + d*Sin[e + f*x])^(n - 1))/(f*(2*n - 1)*Sqrt[a + b*Sin[e + f*x]]), x] - Dist[1/(b*(2*n - 1)), Int[((c + d*Sin[e + f*x])^(n - 2)*Simp[a*c*d - b*(2*d^2*(n - 1) + c^2*(2*n - 1)) + d*(a*d - b*c*(4*n - 3))*Sin[e + f*x], x])/Sqrt[a + b*Sin[e + f*x]], x], x] /; FreeQ[{a, b, c, d, e, f}, x] && NeQ[b*c - a*d, 0] && EqQ[a^2 - b^2, 0] && NeQ[c^2 - d^2, 0] && GtQ[n, 1] && IntegerQ[2*n]
  2779. Int[Times[Power[Plus[Pattern[a, Blank[]], Times[Optional[Pattern[b, Blank[]]], sin[Plus[Optional[Pattern[e, Blank[]]], Times[Optional[Pattern[f, Blank[]]], Pattern[x, Blank[]]]]]]], Rational[-1, 2]], Power[Plus[Optional[Pattern[c, Blank[]]], Times[Optional[Pattern[d, Blank[]]], sin[Plus[Optional[Pattern[e, Blank[]]], Times[Optional[Pattern[f, Blank[]]], Pattern[x, Blank[]]]]]]], Pattern[n, Blank[]]]], Pattern[x, Blank[Symbol]]] := -Simp[(d*Cos[e + f*x]*(c + d*Sin[e + f*x])^(n + 1))/(f*(n + 1)*(c^2 - d^2)*Sqrt[a + b*Sin[e + f*x]]), x] - Dist[1/(2*b*(n + 1)*(c^2 - d^2)), Int[((c + d*Sin[e + f*x])^(n + 1)*Simp[a*d - 2*b*c*(n + 1) + b*d*(2*n + 3)*Sin[e + f*x], x])/Sqrt[a + b*Sin[e + f*x]], x], x] /; FreeQ[{a, b, c, d, e, f}, x] && NeQ[b*c - a*d, 0] && EqQ[a^2 - b^2, 0] && NeQ[c^2 - d^2, 0] && LtQ[n, -1] && IntegerQ[2*n]
  2780. Int[Times[Power[Plus[Pattern[a, Blank[]], Times[Optional[Pattern[b, Blank[]]], sin[Plus[Optional[Pattern[e, Blank[]]], Times[Optional[Pattern[f, Blank[]]], Pattern[x, Blank[]]]]]]], Rational[-1, 2]], Power[Plus[Optional[Pattern[c, Blank[]]], Times[Optional[Pattern[d, Blank[]]], sin[Plus[Optional[Pattern[e, Blank[]]], Times[Optional[Pattern[f, Blank[]]], Pattern[x, Blank[]]]]]]], -1]], Pattern[x, Blank[Symbol]]] := Dist[b/(b*c - a*d), Int[1/Sqrt[a + b*Sin[e + f*x]], x], x] - Dist[d/(b*c - a*d), Int[Sqrt[a + b*Sin[e + f*x]]/(c + d*Sin[e + f*x]), x], x] /; FreeQ[{a, b, c, d, e, f}, x] && NeQ[b*c - a*d, 0] && EqQ[a^2 - b^2, 0] && NeQ[c^2 - d^2, 0]
  2781. Int[Times[Power[Times[Optional[Pattern[d, Blank[]]], sin[Plus[Optional[Pattern[e, Blank[]]], Times[Optional[Pattern[f, Blank[]]], Pattern[x, Blank[]]]]]], Rational[-1, 2]], Power[Plus[Pattern[a, Blank[]], Times[Optional[Pattern[b, Blank[]]], sin[Plus[Optional[Pattern[e, Blank[]]], Times[Optional[Pattern[f, Blank[]]], Pattern[x, Blank[]]]]]]], Rational[-1, 2]]], Pattern[x, Blank[Symbol]]] := -Dist[Sqrt[2]/(Sqrt[a]*f), Subst[Int[1/Sqrt[1 - x^2], x], x, (b*Cos[e + f*x])/(a + b*Sin[e + f*x])], x] /; FreeQ[{a, b, d, e, f}, x] && EqQ[a^2 - b^2, 0] && EqQ[d, a/b] && GtQ[a, 0]
  2782. Int[Times[Power[Plus[Pattern[a, Blank[]], Times[Optional[Pattern[b, Blank[]]], sin[Plus[Optional[Pattern[e, Blank[]]], Times[Optional[Pattern[f, Blank[]]], Pattern[x, Blank[]]]]]]], Rational[-1, 2]], Power[Plus[Optional[Pattern[c, Blank[]]], Times[Optional[Pattern[d, Blank[]]], sin[Plus[Optional[Pattern[e, Blank[]]], Times[Optional[Pattern[f, Blank[]]], Pattern[x, Blank[]]]]]]], Rational[-1, 2]]], Pattern[x, Blank[Symbol]]] := Dist[(-2*a)/f, Subst[Int[1/(2*b^2 - (a*c - b*d)*x^2), x], x, (b*Cos[e + f*x])/(Sqrt[a + b*Sin[e + f*x]]*Sqrt[c + d*Sin[e + f*x]])], x] /; FreeQ[{a, b, c, d, e, f}, x] && NeQ[b*c - a*d, 0] && EqQ[a^2 - b^2, 0] && NeQ[c^2 - d^2, 0]
  2783. Int[Times[Power[Plus[Pattern[a, Blank[]], Times[Optional[Pattern[b, Blank[]]], sin[Plus[Optional[Pattern[e, Blank[]]], Times[Optional[Pattern[f, Blank[]]], Pattern[x, Blank[]]]]]]], Pattern[m, Blank[]]], Power[Plus[Optional[Pattern[c, Blank[]]], Times[Optional[Pattern[d, Blank[]]], sin[Plus[Optional[Pattern[e, Blank[]]], Times[Optional[Pattern[f, Blank[]]], Pattern[x, Blank[]]]]]]], Pattern[n, Blank[]]]], Pattern[x, Blank[Symbol]]] := -Simp[(d*Cos[e + f*x]*(a + b*Sin[e + f*x])^m*(c + d*Sin[e + f*x])^(n - 1))/(f*(m + n)), x] + Dist[1/(b*(m + n)), Int[(a + b*Sin[e + f*x])^m*(c + d*Sin[e + f*x])^(n - 2)*Simp[d*(a*c*m + b*d*(n - 1)) + b*c^2*(m + n) + d*(a*d*m + b*c*(m + 2*n - 1))*Sin[e + f*x], x], x], x] /; FreeQ[{a, b, c, d, e, f, m}, x] && NeQ[b*c - a*d, 0] && EqQ[a^2 - b^2, 0] && NeQ[c^2 - d^2, 0] && GtQ[n, 1] && IntegerQ[n]
  2784. Int[Times[Power[Plus[Pattern[a, Blank[]], Times[Optional[Pattern[b, Blank[]]], sin[Plus[Optional[Pattern[e, Blank[]]], Times[Optional[Pattern[f, Blank[]]], Pattern[x, Blank[]]]]]]], Pattern[m, Blank[]]], Power[Plus[Optional[Pattern[c, Blank[]]], Times[Optional[Pattern[d, Blank[]]], sin[Plus[Optional[Pattern[e, Blank[]]], Times[Optional[Pattern[f, Blank[]]], Pattern[x, Blank[]]]]]]], Optional[Pattern[n, Blank[]]]]], Pattern[x, Blank[Symbol]]] := Dist[(a^m*Cos[e + f*x])/(f*Sqrt[1 + Sin[e + f*x]]*Sqrt[1 - Sin[e + f*x]]), Subst[Int[((1 + (b*x)/a)^(m - 1/2)*(c + d*x)^n)/Sqrt[1 - (b*x)/a], x], x, Sin[e + f*x]], x] /; FreeQ[{a, b, c, d, e, f, n}, x] && NeQ[b*c - a*d, 0] && EqQ[a^2 - b^2, 0] && NeQ[c^2 - d^2, 0] && IntegerQ[m]
  2785. Int[Times[Power[Times[Optional[Pattern[d, Blank[]]], sin[Plus[Optional[Pattern[e, Blank[]]], Times[Optional[Pattern[f, Blank[]]], Pattern[x, Blank[]]]]]], Pattern[n, Blank[]]], Power[Plus[Pattern[a, Blank[]], Times[Optional[Pattern[b, Blank[]]], sin[Plus[Optional[Pattern[e, Blank[]]], Times[Optional[Pattern[f, Blank[]]], Pattern[x, Blank[]]]]]]], Pattern[m, Blank[]]]], Pattern[x, Blank[Symbol]]] := -Dist[(b*(d/b)^n*Cos[e + f*x])/(f*Sqrt[a + b*Sin[e + f*x]]*Sqrt[a - b*Sin[e + f*x]]), Subst[Int[((a - x)^n*(2*a - x)^(m - 1/2))/Sqrt[x], x], x, a - b*Sin[e + f*x]], x] /; FreeQ[{a, b, d, e, f, m, n}, x] && EqQ[a^2 - b^2, 0] && !IntegerQ[m] && GtQ[a, 0] && GtQ[d/b, 0]
  2786. Int[Times[Power[Times[Optional[Pattern[d, Blank[]]], sin[Plus[Optional[Pattern[e, Blank[]]], Times[Optional[Pattern[f, Blank[]]], Pattern[x, Blank[]]]]]], Optional[Pattern[n, Blank[]]]], Power[Plus[Pattern[a, Blank[]], Times[Optional[Pattern[b, Blank[]]], sin[Plus[Optional[Pattern[e, Blank[]]], Times[Optional[Pattern[f, Blank[]]], Pattern[x, Blank[]]]]]]], Pattern[m, Blank[]]]], Pattern[x, Blank[Symbol]]] := Dist[((d/b)^IntPart[n]*(d*Sin[e + f*x])^FracPart[n])/(b*Sin[e + f*x])^FracPart[n], Int[(a + b*Sin[e + f*x])^m*(b*Sin[e + f*x])^n, x], x] /; FreeQ[{a, b, d, e, f, m, n}, x] && EqQ[a^2 - b^2, 0] && !IntegerQ[m] && GtQ[a, 0] && !GtQ[d/b, 0]
  2787. Int[Times[Power[Times[Optional[Pattern[d, Blank[]]], sin[Plus[Optional[Pattern[e, Blank[]]], Times[Optional[Pattern[f, Blank[]]], Pattern[x, Blank[]]]]]], Optional[Pattern[n, Blank[]]]], Power[Plus[Pattern[a, Blank[]], Times[Optional[Pattern[b, Blank[]]], sin[Plus[Optional[Pattern[e, Blank[]]], Times[Optional[Pattern[f, Blank[]]], Pattern[x, Blank[]]]]]]], Pattern[m, Blank[]]]], Pattern[x, Blank[Symbol]]] := Dist[(a^IntPart[m]*(a + b*Sin[e + f*x])^FracPart[m])/(1 + (b*Sin[e + f*x])/a)^FracPart[m], Int[(1 + (b*Sin[e + f*x])/a)^m*(d*Sin[e + f*x])^n, x], x] /; FreeQ[{a, b, d, e, f, m, n}, x] && EqQ[a^2 - b^2, 0] && !IntegerQ[m] && !GtQ[a, 0]
  2788. Int[Times[Power[Plus[Pattern[a, Blank[]], Times[Optional[Pattern[b, Blank[]]], sin[Plus[Optional[Pattern[e, Blank[]]], Times[Optional[Pattern[f, Blank[]]], Pattern[x, Blank[]]]]]]], Pattern[m, Blank[]]], Power[Plus[Pattern[c, Blank[]], Times[Optional[Pattern[d, Blank[]]], sin[Plus[Optional[Pattern[e, Blank[]]], Times[Optional[Pattern[f, Blank[]]], Pattern[x, Blank[]]]]]]], Optional[Pattern[n, Blank[]]]]], Pattern[x, Blank[Symbol]]] := Dist[(a^2*Cos[e + f*x])/(f*Sqrt[a + b*Sin[e + f*x]]*Sqrt[a - b*Sin[e + f*x]]), Subst[Int[((a + b*x)^(m - 1/2)*(c + d*x)^n)/Sqrt[a - b*x], x], x, Sin[e + f*x]], x] /; FreeQ[{a, b, c, d, e, f, m, n}, x] && NeQ[b*c - a*d, 0] && EqQ[a^2 - b^2, 0] && NeQ[c^2 - d^2, 0] && !IntegerQ[m]
  2789. Int[Times[Power[Times[Optional[Pattern[b, Blank[]]], sin[Plus[Optional[Pattern[e, Blank[]]], Times[Optional[Pattern[f, Blank[]]], Pattern[x, Blank[]]]]]], Pattern[m, Blank[]]], Power[Plus[Pattern[c, Blank[]], Times[Optional[Pattern[d, Blank[]]], sin[Plus[Optional[Pattern[e, Blank[]]], Times[Optional[Pattern[f, Blank[]]], Pattern[x, Blank[]]]]]]], 2]], Pattern[x, Blank[Symbol]]] := Dist[(2*c*d)/b, Int[(b*Sin[e + f*x])^(m + 1), x], x] + Int[(b*Sin[e + f*x])^m*(c^2 + d^2*Sin[e + f*x]^2), x] /; FreeQ[{b, c, d, e, f, m}, x]
  2790. Int[Times[Power[Plus[Pattern[a, Blank[]], Times[Optional[Pattern[b, Blank[]]], sin[Plus[Optional[Pattern[e, Blank[]]], Times[Optional[Pattern[f, Blank[]]], Pattern[x, Blank[]]]]]]], Pattern[m, Blank[]]], Power[Plus[Optional[Pattern[c, Blank[]]], Times[Optional[Pattern[d, Blank[]]], sin[Plus[Optional[Pattern[e, Blank[]]], Times[Optional[Pattern[f, Blank[]]], Pattern[x, Blank[]]]]]]], 2]], Pattern[x, Blank[Symbol]]] := -Simp[((b^2*c^2 - 2*a*b*c*d + a^2*d^2)*Cos[e + f*x]*(a + b*Sin[e + f*x])^(m + 1))/(b*f*(m + 1)*(a^2 - b^2)), x] - Dist[1/(b*(m + 1)*(a^2 - b^2)), Int[(a + b*Sin[e + f*x])^(m + 1)*Simp[b*(m + 1)*(2*b*c*d - a*(c^2 + d^2)) + (a^2*d^2 - 2*a*b*c*d*(m + 2) + b^2*(d^2*(m + 1) + c^2*(m + 2)))*Sin[e + f*x], x], x], x] /; FreeQ[{a, b, c, d, e, f}, x] && NeQ[b*c - a*d, 0] && NeQ[a^2 - b^2, 0] && LtQ[m, -1]
  2791. Int[Times[Power[Plus[Pattern[a, Blank[]], Times[Optional[Pattern[b, Blank[]]], sin[Plus[Optional[Pattern[e, Blank[]]], Times[Optional[Pattern[f, Blank[]]], Pattern[x, Blank[]]]]]]], Pattern[m, Blank[]]], Power[Plus[Optional[Pattern[c, Blank[]]], Times[Optional[Pattern[d, Blank[]]], sin[Plus[Optional[Pattern[e, Blank[]]], Times[Optional[Pattern[f, Blank[]]], Pattern[x, Blank[]]]]]]], 2]], Pattern[x, Blank[Symbol]]] := -Simp[(d^2*Cos[e + f*x]*(a + b*Sin[e + f*x])^(m + 1))/(b*f*(m + 2)), x] + Dist[1/(b*(m + 2)), Int[(a + b*Sin[e + f*x])^m*Simp[b*(d^2*(m + 1) + c^2*(m + 2)) - d*(a*d - 2*b*c*(m + 2))*Sin[e + f*x], x], x], x] /; FreeQ[{a, b, c, d, e, f, m}, x] && NeQ[b*c - a*d, 0] && NeQ[a^2 - b^2, 0] && !LtQ[m, -1]
  2792. Int[Times[Power[Plus[Optional[Pattern[a, Blank[]]], Times[Optional[Pattern[b, Blank[]]], sin[Plus[Optional[Pattern[e, Blank[]]], Times[Optional[Pattern[f, Blank[]]], Pattern[x, Blank[]]]]]]], Pattern[m, Blank[]]], Power[Plus[Optional[Pattern[c, Blank[]]], Times[Optional[Pattern[d, Blank[]]], sin[Plus[Optional[Pattern[e, Blank[]]], Times[Optional[Pattern[f, Blank[]]], Pattern[x, Blank[]]]]]]], Pattern[n, Blank[]]]], Pattern[x, Blank[Symbol]]] := -Simp[((b^2*c^2 - 2*a*b*c*d + a^2*d^2)*Cos[e + f*x]*(a + b*Sin[e + f*x])^(m - 2)*(c + d*Sin[e + f*x])^(n + 1))/(d*f*(n + 1)*(c^2 - d^2)), x] + Dist[1/(d*(n + 1)*(c^2 - d^2)), Int[(a + b*Sin[e + f*x])^(m - 3)*(c + d*Sin[e + f*x])^(n + 1)*Simp[b*(m - 2)*(b*c - a*d)^2 + a*d*(n + 1)*(c*(a^2 + b^2) - 2*a*b*d) + (b*(n + 1)*(a*b*c^2 + c*d*(a^2 + b^2) - 3*a*b*d^2) - a*(n + 2)*(b*c - a*d)^2)*Sin[e + f*x] + b*(b^2*(c^2 - d^2) - m*(b*c - a*d)^2 + d*n*(2*a*b*c - d*(a^2 + b^2)))*Sin[e + f*x]^2, x], x], x] /; FreeQ[{a, b, c, d, e, f}, x] && NeQ[b*c - a*d, 0] && NeQ[a^2 - b^2, 0] && NeQ[c^2 - d^2, 0] && GtQ[m, 2] && LtQ[n, -1] && (IntegerQ[m] || IntegersQ[2*m, 2*n])
  2793. Int[Times[Power[Plus[Optional[Pattern[a, Blank[]]], Times[Optional[Pattern[b, Blank[]]], sin[Plus[Optional[Pattern[e, Blank[]]], Times[Optional[Pattern[f, Blank[]]], Pattern[x, Blank[]]]]]]], Pattern[m, Blank[]]], Power[Plus[Optional[Pattern[c, Blank[]]], Times[Optional[Pattern[d, Blank[]]], sin[Plus[Optional[Pattern[e, Blank[]]], Times[Optional[Pattern[f, Blank[]]], Pattern[x, Blank[]]]]]]], Pattern[n, Blank[]]]], Pattern[x, Blank[Symbol]]] := -Simp[(b^2*Cos[e + f*x]*(a + b*Sin[e + f*x])^(m - 2)*(c + d*Sin[e + f*x])^(n + 1))/(d*f*(m + n)), x] + Dist[1/(d*(m + n)), Int[(a + b*Sin[e + f*x])^(m - 3)*(c + d*Sin[e + f*x])^n*Simp[a^3*d*(m + n) + b^2*(b*c*(m - 2) + a*d*(n + 1)) - b*(a*b*c - b^2*d*(m + n - 1) - 3*a^2*d*(m + n))*Sin[e + f*x] - b^2*(b*c*(m - 1) - a*d*(3*m + 2*n - 2))*Sin[e + f*x]^2, x], x], x] /; FreeQ[{a, b, c, d, e, f, n}, x] && NeQ[b*c - a*d, 0] && NeQ[a^2 - b^2, 0] && NeQ[c^2 - d^2, 0] && GtQ[m, 2] && (IntegerQ[m] || IntegersQ[2*m, 2*n]) && !(IGtQ[n, 2] && ( !IntegerQ[m] || (EqQ[a, 0] && NeQ[c, 0])))
  2794. Int[Times[Power[Times[Optional[Pattern[d, Blank[]]], sin[Plus[Optional[Pattern[e, Blank[]]], Times[Optional[Pattern[f, Blank[]]], Pattern[x, Blank[]]]]]], Rational[1, 2]], Power[Plus[Pattern[a, Blank[]], Times[Optional[Pattern[b, Blank[]]], sin[Plus[Optional[Pattern[e, Blank[]]], Times[Optional[Pattern[f, Blank[]]], Pattern[x, Blank[]]]]]]], Rational[-3, 2]]], Pattern[x, Blank[Symbol]]] := Simp[(-2*a*d*Cos[e + f*x])/(f*(a^2 - b^2)*Sqrt[a + b*Sin[e + f*x]]*Sqrt[d*Sin[e + f*x]]), x] - Dist[d^2/(a^2 - b^2), Int[Sqrt[a + b*Sin[e + f*x]]/(d*Sin[e + f*x])^(3/2), x], x] /; FreeQ[{a, b, d, e, f}, x] && NeQ[a^2 - b^2, 0]
  2795. Int[Times[Power[Plus[Optional[Pattern[a, Blank[]]], Times[Optional[Pattern[b, Blank[]]], sin[Plus[Optional[Pattern[e, Blank[]]], Times[Optional[Pattern[f, Blank[]]], Pattern[x, Blank[]]]]]]], Rational[-3, 2]], Power[Plus[Pattern[c, Blank[]], Times[Optional[Pattern[d, Blank[]]], sin[Plus[Optional[Pattern[e, Blank[]]], Times[Optional[Pattern[f, Blank[]]], Pattern[x, Blank[]]]]]]], Rational[1, 2]]], Pattern[x, Blank[Symbol]]] := Dist[(c - d)/(a - b), Int[1/(Sqrt[a + b*Sin[e + f*x]]*Sqrt[c + d*Sin[e + f*x]]), x], x] - Dist[(b*c - a*d)/(a - b), Int[(1 + Sin[e + f*x])/((a + b*Sin[e + f*x])^(3/2)*Sqrt[c + d*Sin[e + f*x]]), x], x] /; FreeQ[{a, b, c, d, e, f}, x] && NeQ[b*c - a*d, 0] && NeQ[a^2 - b^2, 0] && NeQ[c^2 - d^2, 0]
  2796. Int[Times[Power[Plus[Optional[Pattern[a, Blank[]]], Times[Optional[Pattern[b, Blank[]]], sin[Plus[Optional[Pattern[e, Blank[]]], Times[Optional[Pattern[f, Blank[]]], Pattern[x, Blank[]]]]]]], Pattern[m, Blank[]]], Power[Plus[Optional[Pattern[c, Blank[]]], Times[Optional[Pattern[d, Blank[]]], sin[Plus[Optional[Pattern[e, Blank[]]], Times[Optional[Pattern[f, Blank[]]], Pattern[x, Blank[]]]]]]], Pattern[n, Blank[]]]], Pattern[x, Blank[Symbol]]] := -Simp[(b*Cos[e + f*x]*(a + b*Sin[e + f*x])^(m + 1)*(c + d*Sin[e + f*x])^n)/(f*(m + 1)*(a^2 - b^2)), x] + Dist[1/((m + 1)*(a^2 - b^2)), Int[(a + b*Sin[e + f*x])^(m + 1)*(c + d*Sin[e + f*x])^(n - 1)*Simp[a*c*(m + 1) + b*d*n + (a*d*(m + 1) - b*c*(m + 2))*Sin[e + f*x] - b*d*(m + n + 2)*Sin[e + f*x]^2, x], x], x] /; FreeQ[{a, b, c, d, e, f}, x] && NeQ[b*c - a*d, 0] && NeQ[a^2 - b^2, 0] && NeQ[c^2 - d^2, 0] && LtQ[m, -1] && LtQ[0, n, 1] && IntegersQ[2*m, 2*n]
  2797. Int[Times[Power[Times[Optional[Pattern[d, Blank[]]], sin[Plus[Optional[Pattern[e, Blank[]]], Times[Optional[Pattern[f, Blank[]]], Pattern[x, Blank[]]]]]], Rational[3, 2]], Power[Plus[Pattern[a, Blank[]], Times[Optional[Pattern[b, Blank[]]], sin[Plus[Optional[Pattern[e, Blank[]]], Times[Optional[Pattern[f, Blank[]]], Pattern[x, Blank[]]]]]]], Rational[-3, 2]]], Pattern[x, Blank[Symbol]]] := Dist[d/b, Int[Sqrt[d*Sin[e + f*x]]/Sqrt[a + b*Sin[e + f*x]], x], x] - Dist[(a*d)/b, Int[Sqrt[d*Sin[e + f*x]]/(a + b*Sin[e + f*x])^(3/2), x], x] /; FreeQ[{a, b, d, e, f}, x] && NeQ[a^2 - b^2, 0]
  2798. Int[Times[Power[Plus[Optional[Pattern[a, Blank[]]], Times[Optional[Pattern[b, Blank[]]], sin[Plus[Optional[Pattern[e, Blank[]]], Times[Optional[Pattern[f, Blank[]]], Pattern[x, Blank[]]]]]]], Rational[-3, 2]], Power[Plus[Pattern[c, Blank[]], Times[Optional[Pattern[d, Blank[]]], sin[Plus[Optional[Pattern[e, Blank[]]], Times[Optional[Pattern[f, Blank[]]], Pattern[x, Blank[]]]]]]], Rational[3, 2]]], Pattern[x, Blank[Symbol]]] := Dist[d^2/b^2, Int[Sqrt[a + b*Sin[e + f*x]]/Sqrt[c + d*Sin[e + f*x]], x], x] + Dist[(b*c - a*d)/b^2, Int[Simp[b*c + a*d + 2*b*d*Sin[e + f*x], x]/((a + b*Sin[e + f*x])^(3/2)*Sqrt[c + d*Sin[e + f*x]]), x], x] /; FreeQ[{a, b, c, d, e, f}, x] && NeQ[b*c - a*d, 0] && NeQ[a^2 - b^2, 0] && NeQ[c^2 - d^2, 0]
  2799. Int[Times[Power[Plus[Optional[Pattern[a, Blank[]]], Times[Optional[Pattern[b, Blank[]]], sin[Plus[Optional[Pattern[e, Blank[]]], Times[Optional[Pattern[f, Blank[]]], Pattern[x, Blank[]]]]]]], Pattern[m, Blank[]]], Power[Plus[Optional[Pattern[c, Blank[]]], Times[Optional[Pattern[d, Blank[]]], sin[Plus[Optional[Pattern[e, Blank[]]], Times[Optional[Pattern[f, Blank[]]], Pattern[x, Blank[]]]]]]], Pattern[n, Blank[]]]], Pattern[x, Blank[Symbol]]] := -Simp[((b*c - a*d)*Cos[e + f*x]*(a + b*Sin[e + f*x])^(m + 1)*(c + d*Sin[e + f*x])^(n - 1))/(f*(m + 1)*(a^2 - b^2)), x] + Dist[1/((m + 1)*(a^2 - b^2)), Int[(a + b*Sin[e + f*x])^(m + 1)*(c + d*Sin[e + f*x])^(n - 2)*Simp[c*(a*c - b*d)*(m + 1) + d*(b*c - a*d)*(n - 1) + (d*(a*c - b*d)*(m + 1) - c*(b*c - a*d)*(m + 2))*Sin[e + f*x] - d*(b*c - a*d)*(m + n + 1)*Sin[e + f*x]^2, x], x], x] /; FreeQ[{a, b, c, d, e, f}, x] && NeQ[b*c - a*d, 0] && NeQ[a^2 - b^2, 0] && NeQ[c^2 - d^2, 0] && LtQ[m, -1] && LtQ[1, n, 2] && IntegersQ[2*m, 2*n]
  2800. Int[Times[Power[Times[Optional[Pattern[d, Blank[]]], sin[Plus[Optional[Pattern[e, Blank[]]], Times[Optional[Pattern[f, Blank[]]], Pattern[x, Blank[]]]]]], Rational[-1, 2]], Power[Plus[Pattern[a, Blank[]], Times[Optional[Pattern[b, Blank[]]], sin[Plus[Optional[Pattern[e, Blank[]]], Times[Optional[Pattern[f, Blank[]]], Pattern[x, Blank[]]]]]]], Rational[-3, 2]]], Pattern[x, Blank[Symbol]]] := Simp[(2*b*Cos[e + f*x])/(f*(a^2 - b^2)*Sqrt[a + b*Sin[e + f*x]]*Sqrt[d*Sin[e + f*x]]), x] + Dist[d/(a^2 - b^2), Int[(b + a*Sin[e + f*x])/(Sqrt[a + b*Sin[e + f*x]]*(d*Sin[e + f*x])^(3/2)), x], x] /; FreeQ[{a, b, d, e, f}, x] && NeQ[a^2 - b^2, 0]
  2801. Int[Times[Power[Plus[Optional[Pattern[a, Blank[]]], Times[Optional[Pattern[b, Blank[]]], sin[Plus[Optional[Pattern[e, Blank[]]], Times[Optional[Pattern[f, Blank[]]], Pattern[x, Blank[]]]]]]], Rational[-3, 2]], Power[Plus[Optional[Pattern[c, Blank[]]], Times[Optional[Pattern[d, Blank[]]], sin[Plus[Optional[Pattern[e, Blank[]]], Times[Optional[Pattern[f, Blank[]]], Pattern[x, Blank[]]]]]]], Rational[-1, 2]]], Pattern[x, Blank[Symbol]]] := Dist[1/(a - b), Int[1/(Sqrt[a + b*Sin[e + f*x]]*Sqrt[c + d*Sin[e + f*x]]), x], x] - Dist[b/(a - b), Int[(1 + Sin[e + f*x])/((a + b*Sin[e + f*x])^(3/2)*Sqrt[c + d*Sin[e + f*x]]), x], x] /; FreeQ[{a, b, c, d, e, f}, x] && NeQ[b*c - a*d, 0] && NeQ[a^2 - b^2, 0] && NeQ[c^2 - d^2, 0]
  2802. Int[Times[Power[Plus[Optional[Pattern[a, Blank[]]], Times[Optional[Pattern[b, Blank[]]], sin[Plus[Optional[Pattern[e, Blank[]]], Times[Optional[Pattern[f, Blank[]]], Pattern[x, Blank[]]]]]]], Pattern[m, Blank[]]], Power[Plus[Optional[Pattern[c, Blank[]]], Times[Optional[Pattern[d, Blank[]]], sin[Plus[Optional[Pattern[e, Blank[]]], Times[Optional[Pattern[f, Blank[]]], Pattern[x, Blank[]]]]]]], Pattern[n, Blank[]]]], Pattern[x, Blank[Symbol]]] := -Simp[(b^2*Cos[e + f*x]*(a + b*Sin[e + f*x])^(m + 1)*(c + d*Sin[e + f*x])^(n + 1))/(f*(m + 1)*(b*c - a*d)*(a^2 - b^2)), x] + Dist[1/((m + 1)*(b*c - a*d)*(a^2 - b^2)), Int[(a + b*Sin[e + f*x])^(m + 1)*(c + d*Sin[e + f*x])^n*Simp[a*(b*c - a*d)*(m + 1) + b^2*d*(m + n + 2) - (b^2*c + b*(b*c - a*d)*(m + 1))*Sin[e + f*x] - b^2*d*(m + n + 3)*Sin[e + f*x]^2, x], x], x] /; FreeQ[{a, b, c, d, e, f, n}, x] && NeQ[b*c - a*d, 0] && NeQ[a^2 - b^2, 0] && NeQ[c^2 - d^2, 0] && LtQ[m, -1] && IntegersQ[2*m, 2*n] && ((EqQ[a, 0] && IntegerQ[m] && !IntegerQ[n]) || !(IntegerQ[2*n] && LtQ[n, -1] && ((IntegerQ[n] && !IntegerQ[m]) || EqQ[a, 0])))
  2803. Int[Times[Power[Plus[Optional[Pattern[a, Blank[]]], Times[Optional[Pattern[b, Blank[]]], sin[Plus[Optional[Pattern[e, Blank[]]], Times[Optional[Pattern[f, Blank[]]], Pattern[x, Blank[]]]]]]], -1], Power[Plus[Optional[Pattern[c, Blank[]]], Times[Optional[Pattern[d, Blank[]]], sin[Plus[Optional[Pattern[e, Blank[]]], Times[Optional[Pattern[f, Blank[]]], Pattern[x, Blank[]]]]]]], Rational[1, 2]]], Pattern[x, Blank[Symbol]]] := Dist[d/b, Int[1/Sqrt[c + d*Sin[e + f*x]], x], x] + Dist[(b*c - a*d)/b, Int[1/((a + b*Sin[e + f*x])*Sqrt[c + d*Sin[e + f*x]]), x], x] /; FreeQ[{a, b, c, d, e, f}, x] && NeQ[b*c - a*d, 0] && NeQ[a^2 - b^2, 0] && NeQ[c^2 - d^2, 0]
  2804. Int[Times[Power[Plus[Optional[Pattern[a, Blank[]]], Times[Optional[Pattern[b, Blank[]]], sin[Plus[Optional[Pattern[e, Blank[]]], Times[Optional[Pattern[f, Blank[]]], Pattern[x, Blank[]]]]]]], Rational[3, 2]], Power[Plus[Optional[Pattern[c, Blank[]]], Times[Optional[Pattern[d, Blank[]]], sin[Plus[Optional[Pattern[e, Blank[]]], Times[Optional[Pattern[f, Blank[]]], Pattern[x, Blank[]]]]]]], -1]], Pattern[x, Blank[Symbol]]] := Dist[b/d, Int[Sqrt[a + b*Sin[e + f*x]], x], x] - Dist[(b*c - a*d)/d, Int[Sqrt[a + b*Sin[e + f*x]]/(c + d*Sin[e + f*x]), x], x] /; FreeQ[{a, b, c, d, e, f}, x] && NeQ[b*c - a*d, 0] && NeQ[a^2 - b^2, 0] && NeQ[c^2 - d^2, 0]
  2805. Int[Times[Power[Plus[Optional[Pattern[a, Blank[]]], Times[Optional[Pattern[b, Blank[]]], sin[Plus[Optional[Pattern[e, Blank[]]], Times[Optional[Pattern[f, Blank[]]], Pattern[x, Blank[]]]]]]], -1], Power[Plus[Optional[Pattern[c, Blank[]]], Times[Optional[Pattern[d, Blank[]]], sin[Plus[Optional[Pattern[e, Blank[]]], Times[Optional[Pattern[f, Blank[]]], Pattern[x, Blank[]]]]]]], Rational[-1, 2]]], Pattern[x, Blank[Symbol]]] := Simp[(2*EllipticPi[(2*b)/(a + b), (1*(e - Pi/2 + f*x))/2, (2*d)/(c + d)])/(f*(a + b)*Sqrt[c + d]), x] /; FreeQ[{a, b, c, d, e, f}, x] && NeQ[b*c - a*d, 0] && NeQ[a^2 - b^2, 0] && NeQ[c^2 - d^2, 0] && GtQ[c + d, 0]
  2806. Int[Times[Power[Plus[Optional[Pattern[a, Blank[]]], Times[Optional[Pattern[b, Blank[]]], sin[Plus[Optional[Pattern[e, Blank[]]], Times[Optional[Pattern[f, Blank[]]], Pattern[x, Blank[]]]]]]], -1], Power[Plus[Optional[Pattern[c, Blank[]]], Times[Optional[Pattern[d, Blank[]]], sin[Plus[Optional[Pattern[e, Blank[]]], Times[Optional[Pattern[f, Blank[]]], Pattern[x, Blank[]]]]]]], Rational[-1, 2]]], Pattern[x, Blank[Symbol]]] := Simp[(2*EllipticPi[(-2*b)/(a - b), (1*(e + Pi/2 + f*x))/2, (-2*d)/(c - d)])/(f*(a - b)*Sqrt[c - d]), x] /; FreeQ[{a, b, c, d, e, f}, x] && NeQ[b*c - a*d, 0] && NeQ[a^2 - b^2, 0] && NeQ[c^2 - d^2, 0] && GtQ[c - d, 0]
  2807. Int[Times[Power[Plus[Optional[Pattern[a, Blank[]]], Times[Optional[Pattern[b, Blank[]]], sin[Plus[Optional[Pattern[e, Blank[]]], Times[Optional[Pattern[f, Blank[]]], Pattern[x, Blank[]]]]]]], -1], Power[Plus[Optional[Pattern[c, Blank[]]], Times[Optional[Pattern[d, Blank[]]], sin[Plus[Optional[Pattern[e, Blank[]]], Times[Optional[Pattern[f, Blank[]]], Pattern[x, Blank[]]]]]]], Rational[-1, 2]]], Pattern[x, Blank[Symbol]]] := Dist[Sqrt[(c + d*Sin[e + f*x])/(c + d)]/Sqrt[c + d*Sin[e + f*x]], Int[1/((a + b*Sin[e + f*x])*Sqrt[c/(c + d) + (d*Sin[e + f*x])/(c + d)]), x], x] /; FreeQ[{a, b, c, d, e, f}, x] && NeQ[b*c - a*d, 0] && NeQ[a^2 - b^2, 0] && NeQ[c^2 - d^2, 0] && !GtQ[c + d, 0]
  2808. Int[Times[Power[Times[Optional[Pattern[b, Blank[]]], sin[Plus[Optional[Pattern[e, Blank[]]], Times[Optional[Pattern[f, Blank[]]], Pattern[x, Blank[]]]]]], Rational[1, 2]], Power[Plus[Pattern[c, Blank[]], Times[Optional[Pattern[d, Blank[]]], sin[Plus[Optional[Pattern[e, Blank[]]], Times[Optional[Pattern[f, Blank[]]], Pattern[x, Blank[]]]]]]], Rational[-1, 2]]], Pattern[x, Blank[Symbol]]] := Simp[(2*c*Rt[b*(c + d), 2]*Tan[e + f*x]*Sqrt[1 + Csc[e + f*x]]*Sqrt[1 - Csc[e + f*x]]*EllipticPi[(c + d)/d, ArcSin[Sqrt[c + d*Sin[e + f*x]]/(Sqrt[b*Sin[e + f*x]]*Rt[(c + d)/b, 2])], -((c + d)/(c - d))])/(d*f*Sqrt[c^2 - d^2]), x] /; FreeQ[{b, c, d, e, f}, x] && GtQ[c^2 - d^2, 0] && PosQ[(c + d)/b] && GtQ[c^2, 0]
  2809. Int[Times[Power[Times[Optional[Pattern[b, Blank[]]], sin[Plus[Optional[Pattern[e, Blank[]]], Times[Optional[Pattern[f, Blank[]]], Pattern[x, Blank[]]]]]], Rational[1, 2]], Power[Plus[Pattern[c, Blank[]], Times[Optional[Pattern[d, Blank[]]], sin[Plus[Optional[Pattern[e, Blank[]]], Times[Optional[Pattern[f, Blank[]]], Pattern[x, Blank[]]]]]]], Rational[-1, 2]]], Pattern[x, Blank[Symbol]]] := Simp[(2*b*Tan[e + f*x]*Rt[(c + d)/b, 2]*Sqrt[(c*(1 + Csc[e + f*x]))/(c - d)]*Sqrt[(c*(1 - Csc[e + f*x]))/(c + d)]*EllipticPi[(c + d)/d, ArcSin[Sqrt[c + d*Sin[e + f*x]]/(Sqrt[b*Sin[e + f*x]]*Rt[(c + d)/b, 2])], -((c + d)/(c - d))])/(d*f), x] /; FreeQ[{b, c, d, e, f}, x] && NeQ[c^2 - d^2, 0] && PosQ[(c + d)/b]
  2810. Int[Times[Power[Times[Optional[Pattern[b, Blank[]]], sin[Plus[Optional[Pattern[e, Blank[]]], Times[Optional[Pattern[f, Blank[]]], Pattern[x, Blank[]]]]]], Rational[1, 2]], Power[Plus[Pattern[c, Blank[]], Times[Optional[Pattern[d, Blank[]]], sin[Plus[Optional[Pattern[e, Blank[]]], Times[Optional[Pattern[f, Blank[]]], Pattern[x, Blank[]]]]]]], Rational[-1, 2]]], Pattern[x, Blank[Symbol]]] := Dist[Sqrt[b*Sin[e + f*x]]/Sqrt[-(b*Sin[e + f*x])], Int[Sqrt[-(b*Sin[e + f*x])]/Sqrt[c + d*Sin[e + f*x]], x], x] /; FreeQ[{b, c, d, e, f}, x] && NeQ[c^2 - d^2, 0] && NegQ[(c + d)/b]
  2811. Int[Times[Power[Plus[Pattern[a, Blank[]], Times[Optional[Pattern[b, Blank[]]], sin[Plus[Optional[Pattern[e, Blank[]]], Times[Optional[Pattern[f, Blank[]]], Pattern[x, Blank[]]]]]]], Rational[1, 2]], Power[Plus[Optional[Pattern[c, Blank[]]], Times[Optional[Pattern[d, Blank[]]], sin[Plus[Optional[Pattern[e, Blank[]]], Times[Optional[Pattern[f, Blank[]]], Pattern[x, Blank[]]]]]]], Rational[-1, 2]]], Pattern[x, Blank[Symbol]]] := Simp[(2*(a + b*Sin[e + f*x])*Sqrt[((b*c - a*d)*(1 + Sin[e + f*x]))/((c - d)*(a + b*Sin[e + f*x]))]*Sqrt[-(((b*c - a*d)*(1 - Sin[e + f*x]))/((c + d)*(a + b*Sin[e + f*x])))]*EllipticPi[(b*(c + d))/(d*(a + b)), ArcSin[(Rt[(a + b)/(c + d), 2]*Sqrt[c + d*Sin[e + f*x]])/Sqrt[a + b*Sin[e + f*x]]], ((a - b)*(c + d))/((a + b)*(c - d))])/(d*f*Rt[(a + b)/(c + d), 2]*Cos[e + f*x]), x] /; FreeQ[{a, b, c, d, e, f}, x] && NeQ[b*c - a*d, 0] && NeQ[a^2 - b^2, 0] && NeQ[c^2 - d^2, 0] && PosQ[(a + b)/(c + d)]
  2812. Int[Times[Power[Plus[Pattern[a, Blank[]], Times[Optional[Pattern[b, Blank[]]], sin[Plus[Optional[Pattern[e, Blank[]]], Times[Optional[Pattern[f, Blank[]]], Pattern[x, Blank[]]]]]]], Rational[1, 2]], Power[Plus[Optional[Pattern[c, Blank[]]], Times[Optional[Pattern[d, Blank[]]], sin[Plus[Optional[Pattern[e, Blank[]]], Times[Optional[Pattern[f, Blank[]]], Pattern[x, Blank[]]]]]]], Rational[-1, 2]]], Pattern[x, Blank[Symbol]]] := Dist[Sqrt[-c - d*Sin[e + f*x]]/Sqrt[c + d*Sin[e + f*x]], Int[Sqrt[a + b*Sin[e + f*x]]/Sqrt[-c - d*Sin[e + f*x]], x], x] /; FreeQ[{a, b, c, d, e, f}, x] && NeQ[b*c - a*d, 0] && NeQ[a^2 - b^2, 0] && NeQ[c^2 - d^2, 0] && NegQ[(a + b)/(c + d)]
  2813. Int[Times[Power[Times[Optional[Pattern[d, Blank[]]], sin[Plus[Optional[Pattern[e, Blank[]]], Times[Optional[Pattern[f, Blank[]]], Pattern[x, Blank[]]]]]], Rational[-1, 2]], Power[Plus[Pattern[a, Blank[]], Times[Optional[Pattern[b, Blank[]]], sin[Plus[Optional[Pattern[e, Blank[]]], Times[Optional[Pattern[f, Blank[]]], Pattern[x, Blank[]]]]]]], Rational[-1, 2]]], Pattern[x, Blank[Symbol]]] := Simp[(-2*d*EllipticF[ArcSin[Cos[e + f*x]/(1 + d*Sin[e + f*x])], -((a - b*d)/(a + b*d))])/(f*Sqrt[a + b*d]), x] /; FreeQ[{a, b, d, e, f}, x] && LtQ[a^2 - b^2, 0] && EqQ[d^2, 1] && GtQ[b*d, 0]
  2814. Int[Times[Power[Times[Optional[Pattern[d, Blank[]]], sin[Plus[Optional[Pattern[e, Blank[]]], Times[Optional[Pattern[f, Blank[]]], Pattern[x, Blank[]]]]]], Rational[-1, 2]], Power[Plus[Pattern[a, Blank[]], Times[Optional[Pattern[b, Blank[]]], sin[Plus[Optional[Pattern[e, Blank[]]], Times[Optional[Pattern[f, Blank[]]], Pattern[x, Blank[]]]]]]], Rational[-1, 2]]], Pattern[x, Blank[Symbol]]] := Dist[Sqrt[Sign[b]*Sin[e + f*x]]/Sqrt[d*Sin[e + f*x]], Int[1/(Sqrt[a + b*Sin[e + f*x]]*Sqrt[Sign[b]*Sin[e + f*x]]), x], x] /; FreeQ[{a, b, d, e, f}, x] && LtQ[a^2 - b^2, 0] && GtQ[b^2, 0] && !(EqQ[d^2, 1] && GtQ[b*d, 0])
  2815. Int[Times[Power[Times[Optional[Pattern[d, Blank[]]], sin[Plus[Optional[Pattern[e, Blank[]]], Times[Optional[Pattern[f, Blank[]]], Pattern[x, Blank[]]]]]], Rational[-1, 2]], Power[Plus[Pattern[a, Blank[]], Times[Optional[Pattern[b, Blank[]]], sin[Plus[Optional[Pattern[e, Blank[]]], Times[Optional[Pattern[f, Blank[]]], Pattern[x, Blank[]]]]]]], Rational[-1, 2]]], Pattern[x, Blank[Symbol]]] := Simp[(-2*Sqrt[a^2]*Sqrt[-Cot[e + f*x]^2]*Rt[(a + b)/d, 2]*EllipticF[ArcSin[Sqrt[a + b*Sin[e + f*x]]/(Sqrt[d*Sin[e + f*x]]*Rt[(a + b)/d, 2])], -((a + b)/(a - b))])/(a*f*Sqrt[a^2 - b^2]*Cot[e + f*x]), x] /; FreeQ[{a, b, d, e, f}, x] && GtQ[a^2 - b^2, 0] && PosQ[(a + b)/d] && GtQ[a^2, 0]
  2816. Int[Times[Power[Times[Optional[Pattern[d, Blank[]]], sin[Plus[Optional[Pattern[e, Blank[]]], Times[Optional[Pattern[f, Blank[]]], Pattern[x, Blank[]]]]]], Rational[-1, 2]], Power[Plus[Pattern[a, Blank[]], Times[Optional[Pattern[b, Blank[]]], sin[Plus[Optional[Pattern[e, Blank[]]], Times[Optional[Pattern[f, Blank[]]], Pattern[x, Blank[]]]]]]], Rational[-1, 2]]], Pattern[x, Blank[Symbol]]] := Simp[(-2*Tan[e + f*x]*Rt[(a + b)/d, 2]*Sqrt[(a*(1 - Csc[e + f*x]))/(a + b)]*Sqrt[(a*(1 + Csc[e + f*x]))/(a - b)]*EllipticF[ArcSin[Sqrt[a + b*Sin[e + f*x]]/(Sqrt[d*Sin[e + f*x]]*Rt[(a + b)/d, 2])], -((a + b)/(a - b))])/(a*f), x] /; FreeQ[{a, b, d, e, f}, x] && NeQ[a^2 - b^2, 0] && PosQ[(a + b)/d]
  2817. Int[Times[Power[Times[Optional[Pattern[d, Blank[]]], sin[Plus[Optional[Pattern[e, Blank[]]], Times[Optional[Pattern[f, Blank[]]], Pattern[x, Blank[]]]]]], Rational[-1, 2]], Power[Plus[Pattern[a, Blank[]], Times[Optional[Pattern[b, Blank[]]], sin[Plus[Optional[Pattern[e, Blank[]]], Times[Optional[Pattern[f, Blank[]]], Pattern[x, Blank[]]]]]]], Rational[-1, 2]]], Pattern[x, Blank[Symbol]]] := Dist[Sqrt[-(d*Sin[e + f*x])]/Sqrt[d*Sin[e + f*x]], Int[1/(Sqrt[a + b*Sin[e + f*x]]*Sqrt[-(d*Sin[e + f*x])]), x], x] /; FreeQ[{a, b, d, e, f}, x] && NeQ[a^2 - b^2, 0] && NegQ[(a + b)/d]
  2818. Int[Times[Power[Plus[Pattern[a, Blank[]], Times[Optional[Pattern[b, Blank[]]], sin[Plus[Optional[Pattern[e, Blank[]]], Times[Optional[Pattern[f, Blank[]]], Pattern[x, Blank[]]]]]]], Rational[-1, 2]], Power[Plus[Pattern[c, Blank[]], Times[Optional[Pattern[d, Blank[]]], sin[Plus[Optional[Pattern[e, Blank[]]], Times[Optional[Pattern[f, Blank[]]], Pattern[x, Blank[]]]]]]], Rational[-1, 2]]], Pattern[x, Blank[Symbol]]] := Simp[(2*(c + d*Sin[e + f*x])*Sqrt[((b*c - a*d)*(1 - Sin[e + f*x]))/((a + b)*(c + d*Sin[e + f*x]))]*Sqrt[-(((b*c - a*d)*(1 + Sin[e + f*x]))/((a - b)*(c + d*Sin[e + f*x])))]*EllipticF[ArcSin[Rt[(c + d)/(a + b), 2]*(Sqrt[a + b*Sin[e + f*x]]/Sqrt[c + d*Sin[e + f*x]])], ((a + b)*(c - d))/((a - b)*(c + d))])/(f*(b*c - a*d)*Rt[(c + d)/(a + b), 2]*Cos[e + f*x]), x] /; FreeQ[{a, b, c, d, e, f}, x] && NeQ[b*c - a*d, 0] && NeQ[a^2 - b^2, 0] && NeQ[c^2 - d^2, 0] && PosQ[(c + d)/(a + b)]
  2819. Int[Times[Power[Plus[Optional[Pattern[a, Blank[]]], Times[Optional[Pattern[b, Blank[]]], sin[Plus[Optional[Pattern[e, Blank[]]], Times[Optional[Pattern[f, Blank[]]], Pattern[x, Blank[]]]]]]], Rational[-1, 2]], Power[Plus[Pattern[c, Blank[]], Times[Optional[Pattern[d, Blank[]]], sin[Plus[Optional[Pattern[e, Blank[]]], Times[Optional[Pattern[f, Blank[]]], Pattern[x, Blank[]]]]]]], Rational[-1, 2]]], Pattern[x, Blank[Symbol]]] := Dist[Sqrt[-a - b*Sin[e + f*x]]/Sqrt[a + b*Sin[e + f*x]], Int[1/(Sqrt[-a - b*Sin[e + f*x]]*Sqrt[c + d*Sin[e + f*x]]), x], x] /; FreeQ[{a, b, c, d, e, f}, x] && NeQ[b*c - a*d, 0] && NeQ[a^2 - b^2, 0] && NeQ[c^2 - d^2, 0] && NegQ[(c + d)/(a + b)]
  2820. Int[Times[Power[Times[Optional[Pattern[d, Blank[]]], sin[Plus[Optional[Pattern[e, Blank[]]], Times[Optional[Pattern[f, Blank[]]], Pattern[x, Blank[]]]]]], Rational[3, 2]], Power[Plus[Optional[Pattern[a, Blank[]]], Times[Optional[Pattern[b, Blank[]]], sin[Plus[Optional[Pattern[e, Blank[]]], Times[Optional[Pattern[f, Blank[]]], Pattern[x, Blank[]]]]]]], Rational[-1, 2]]], Pattern[x, Blank[Symbol]]] := -Dist[(a*d)/(2*b), Int[Sqrt[d*Sin[e + f*x]]/Sqrt[a + b*Sin[e + f*x]], x], x] + Dist[d/(2*b), Int[(Sqrt[d*Sin[e + f*x]]*(a + 2*b*Sin[e + f*x]))/Sqrt[a + b*Sin[e + f*x]], x], x] /; FreeQ[{a, b, d, e, f}, x] && NeQ[a^2 - b^2, 0]
  2821. Int[Times[Power[Plus[Optional[Pattern[a, Blank[]]], Times[Optional[Pattern[b, Blank[]]], sin[Plus[Optional[Pattern[e, Blank[]]], Times[Optional[Pattern[f, Blank[]]], Pattern[x, Blank[]]]]]]], Pattern[m, Blank[]]], Power[Plus[Optional[Pattern[c, Blank[]]], Times[Optional[Pattern[d, Blank[]]], sin[Plus[Optional[Pattern[e, Blank[]]], Times[Optional[Pattern[f, Blank[]]], Pattern[x, Blank[]]]]]]], Pattern[n, Blank[]]]], Pattern[x, Blank[Symbol]]] := -Simp[(b*Cos[e + f*x]*(a + b*Sin[e + f*x])^(m - 1)*(c + d*Sin[e + f*x])^n)/(f*(m + n)), x] + Dist[1/(d*(m + n)), Int[(a + b*Sin[e + f*x])^(m - 2)*(c + d*Sin[e + f*x])^(n - 1)*Simp[a^2*c*d*(m + n) + b*d*(b*c*(m - 1) + a*d*n) + (a*d*(2*b*c + a*d)*(m + n) - b*d*(a*c - b*d*(m + n - 1)))*Sin[e + f*x] + b*d*(b*c*n + a*d*(2*m + n - 1))*Sin[e + f*x]^2, x], x], x] /; FreeQ[{a, b, c, d, e, f}, x] && NeQ[b*c - a*d, 0] && NeQ[a^2 - b^2, 0] && NeQ[c^2 - d^2, 0] && LtQ[0, m, 2] && LtQ[-1, n, 2] && NeQ[m + n, 0] && (IntegerQ[m] || IntegersQ[2*m, 2*n])
  2822. Int[Times[Power[Plus[Optional[Pattern[a, Blank[]]], Times[Optional[Pattern[b, Blank[]]], sin[Plus[Optional[Pattern[e, Blank[]]], Times[Optional[Pattern[f, Blank[]]], Pattern[x, Blank[]]]]]]], Pattern[m, Blank[]]], Power[Plus[Optional[Pattern[c, Blank[]]], Times[Optional[Pattern[d, Blank[]]], sin[Plus[Optional[Pattern[e, Blank[]]], Times[Optional[Pattern[f, Blank[]]], Pattern[x, Blank[]]]]]]], Pattern[n, Blank[]]]], Pattern[x, Blank[Symbol]]] := Dist[b/d, Int[(a + b*Sin[e + f*x])^(m - 1)*(c + d*Sin[e + f*x])^(n + 1), x], x] - Dist[(b*c - a*d)/d, Int[(a + b*Sin[e + f*x])^(m - 1)*(c + d*Sin[e + f*x])^n, x], x] /; FreeQ[{a, b, c, d, e, f, n}, x] && NeQ[b*c - a*d, 0] && IGtQ[m, 0]
  2823. Int[Times[Power[Times[Optional[Pattern[d, Blank[]]], sin[Plus[Optional[Pattern[e, Blank[]]], Times[Optional[Pattern[f, Blank[]]], Pattern[x, Blank[]]]]]], Optional[Pattern[n, Blank[]]]], Power[Plus[Pattern[a, Blank[]], Times[Optional[Pattern[b, Blank[]]], sin[Plus[Optional[Pattern[e, Blank[]]], Times[Optional[Pattern[f, Blank[]]], Pattern[x, Blank[]]]]]]], -1]], Pattern[x, Blank[Symbol]]] := Dist[a, Int[(d*Sin[e + f*x])^n/(a^2 - b^2*Sin[e + f*x]^2), x], x] - Dist[b/d, Int[(d*Sin[e + f*x])^(n + 1)/(a^2 - b^2*Sin[e + f*x]^2), x], x] /; FreeQ[{a, b, d, e, f, n}, x] && NeQ[a^2 - b^2, 0]
  2824. Int[Times[Power[Times[Optional[Pattern[d, Blank[]]], sin[Plus[Optional[Pattern[e, Blank[]]], Times[Optional[Pattern[f, Blank[]]], Pattern[x, Blank[]]]]]], Optional[Pattern[n, Blank[]]]], Power[Plus[Pattern[a, Blank[]], Times[Optional[Pattern[b, Blank[]]], sin[Plus[Optional[Pattern[e, Blank[]]], Times[Optional[Pattern[f, Blank[]]], Pattern[x, Blank[]]]]]]], Optional[Pattern[m, Blank[]]]]], Pattern[x, Blank[Symbol]]] := Int[ExpandTrig[(d*sin[e + f*x])^n/((a - b*sin[e + f*x])^m/(a^2 - b^2*sin[e + f*x]^2)^m), x], x] /; FreeQ[{a, b, d, e, f, n}, x] && NeQ[a^2 - b^2, 0] && ILtQ[m, -1]
  2825. Int[Times[Power[Plus[Optional[Pattern[a, Blank[]]], Times[Optional[Pattern[b, Blank[]]], sin[Plus[Optional[Pattern[e, Blank[]]], Times[Optional[Pattern[f, Blank[]]], Pattern[x, Blank[]]]]]]], Pattern[m, Blank[]]], Power[Plus[Optional[Pattern[c, Blank[]]], Times[Optional[Pattern[d, Blank[]]], sin[Plus[Optional[Pattern[e, Blank[]]], Times[Optional[Pattern[f, Blank[]]], Pattern[x, Blank[]]]]]]], Pattern[n, Blank[]]]], Pattern[x, Blank[Symbol]]] := Unintegrable[(a + b*Sin[e + f*x])^m*(c + d*Sin[e + f*x])^n, x] /; FreeQ[{a, b, c, d, e, f, m, n}, x] && NeQ[b*c - a*d, 0] && NeQ[a^2 - b^2, 0] && NeQ[c^2 - d^2, 0]
  2826. Int[Times[Power[Times[Optional[Pattern[c, Blank[]]], Power[Times[Optional[Pattern[d, Blank[]]], sin[Plus[Optional[Pattern[e, Blank[]]], Times[Optional[Pattern[f, Blank[]]], Pattern[x, Blank[]]]]]], Pattern[p, Blank[]]]], Pattern[n, Blank[]]], Power[Plus[Optional[Pattern[a, Blank[]]], Times[Optional[Pattern[b, Blank[]]], sin[Plus[Optional[Pattern[e, Blank[]]], Times[Optional[Pattern[f, Blank[]]], Pattern[x, Blank[]]]]]]], Optional[Pattern[m, Blank[]]]]], Pattern[x, Blank[Symbol]]] := Dist[(c^IntPart[n]*(c*(d*Sin[e + f*x])^p)^FracPart[n])/(d*Sin[e + f*x])^(p*FracPart[n]), Int[(a + b*Sin[e + f*x])^m*(d*Sin[e + f*x])^(n*p), x], x] /; FreeQ[{a, b, c, d, e, f, m, n, p}, x] && !IntegerQ[n]
  2827. Int[Times[Power[Plus[Optional[Pattern[a, Blank[]]], Times[cos[Plus[Optional[Pattern[e, Blank[]]], Times[Optional[Pattern[f, Blank[]]], Pattern[x, Blank[]]]]], Optional[Pattern[b, Blank[]]]]], Optional[Pattern[m, Blank[]]]], Power[Times[Optional[Pattern[c, Blank[]]], Power[Times[cos[Plus[Optional[Pattern[e, Blank[]]], Times[Optional[Pattern[f, Blank[]]], Pattern[x, Blank[]]]]], Optional[Pattern[d, Blank[]]]], Pattern[p, Blank[]]]], Pattern[n, Blank[]]]], Pattern[x, Blank[Symbol]]] := Dist[(c^IntPart[n]*(c*(d*Cos[e + f*x])^p)^FracPart[n])/(d*Cos[e + f*x])^(p*FracPart[n]), Int[(a + b*Cos[e + f*x])^m*(d*Cos[e + f*x])^(n*p), x], x] /; FreeQ[{a, b, c, d, e, f, m, n, p}, x] && !IntegerQ[n]
  2828. Int[Times[Power[Plus[Times[csc[Plus[Optional[Pattern[e, Blank[]]], Times[Optional[Pattern[f, Blank[]]], Pattern[x, Blank[]]]]], Optional[Pattern[d, Blank[]]]], Pattern[c, Blank[]]], Optional[Pattern[n, Blank[]]]], Power[Plus[Pattern[a, Blank[]], Times[Optional[Pattern[b, Blank[]]], sin[Plus[Optional[Pattern[e, Blank[]]], Times[Optional[Pattern[f, Blank[]]], Pattern[x, Blank[]]]]]]], Optional[Pattern[m, Blank[]]]]], Pattern[x, Blank[Symbol]]] := Int[((a + b*Sin[e + f*x])^m*(d + c*Sin[e + f*x])^n)/Sin[e + f*x]^n, x] /; FreeQ[{a, b, c, d, e, f, m}, x] && IntegerQ[n]
  2829. Int[Times[Power[Plus[Times[csc[Plus[Optional[Pattern[e, Blank[]]], Times[Optional[Pattern[f, Blank[]]], Pattern[x, Blank[]]]]], Optional[Pattern[d, Blank[]]]], Pattern[c, Blank[]]], Pattern[n, Blank[]]], Power[Plus[Pattern[a, Blank[]], Times[Optional[Pattern[b, Blank[]]], sin[Plus[Optional[Pattern[e, Blank[]]], Times[Optional[Pattern[f, Blank[]]], Pattern[x, Blank[]]]]]]], Optional[Pattern[m, Blank[]]]]], Pattern[x, Blank[Symbol]]] := Int[((b + a*Csc[e + f*x])^m*(c + d*Csc[e + f*x])^n)/Csc[e + f*x]^m, x] /; FreeQ[{a, b, c, d, e, f, n}, x] && !IntegerQ[n] && IntegerQ[m]
  2830. Int[Times[Power[Plus[Times[cos[Plus[Optional[Pattern[e, Blank[]]], Times[Optional[Pattern[f, Blank[]]], Pattern[x, Blank[]]]]], Optional[Pattern[b, Blank[]]]], Pattern[a, Blank[]]], Optional[Pattern[m, Blank[]]]], Power[Plus[Pattern[c, Blank[]], Times[Optional[Pattern[d, Blank[]]], sec[Plus[Optional[Pattern[e, Blank[]]], Times[Optional[Pattern[f, Blank[]]], Pattern[x, Blank[]]]]]]], Pattern[n, Blank[]]]], Pattern[x, Blank[Symbol]]] := Int[((b + a*Sec[e + f*x])^m*(c + d*Sec[e + f*x])^n)/Sec[e + f*x]^m, x] /; FreeQ[{a, b, c, d, e, f, n}, x] && !IntegerQ[n] && IntegerQ[m]
  2831. Int[Times[Power[Plus[Times[csc[Plus[Optional[Pattern[e, Blank[]]], Times[Optional[Pattern[f, Blank[]]], Pattern[x, Blank[]]]]], Optional[Pattern[d, Blank[]]]], Pattern[c, Blank[]]], Pattern[n, Blank[]]], Power[Plus[Pattern[a, Blank[]], Times[Optional[Pattern[b, Blank[]]], sin[Plus[Optional[Pattern[e, Blank[]]], Times[Optional[Pattern[f, Blank[]]], Pattern[x, Blank[]]]]]]], Pattern[m, Blank[]]]], Pattern[x, Blank[Symbol]]] := Dist[(Sin[e + f*x]^n*(c + d*Csc[e + f*x])^n)/(d + c*Sin[e + f*x])^n, Int[((a + b*Sin[e + f*x])^m*(d + c*Sin[e + f*x])^n)/Sin[e + f*x]^n, x], x] /; FreeQ[{a, b, c, d, e, f, m, n}, x] && !IntegerQ[n] && !IntegerQ[m]
  2832. Int[Times[Power[Plus[Times[cos[Plus[Optional[Pattern[e, Blank[]]], Times[Optional[Pattern[f, Blank[]]], Pattern[x, Blank[]]]]], Optional[Pattern[b, Blank[]]]], Pattern[a, Blank[]]], Pattern[m, Blank[]]], Power[Plus[Pattern[c, Blank[]], Times[Optional[Pattern[d, Blank[]]], sec[Plus[Optional[Pattern[e, Blank[]]], Times[Optional[Pattern[f, Blank[]]], Pattern[x, Blank[]]]]]]], Pattern[n, Blank[]]]], Pattern[x, Blank[Symbol]]] := Dist[(Cos[e + f*x]^n*(c + d*Sec[e + f*x])^n)/(d + c*Cos[e + f*x])^n, Int[((a + b*Cos[e + f*x])^m*(d + c*Cos[e + f*x])^n)/Cos[e + f*x]^n, x], x] /; FreeQ[{a, b, c, d, e, f, m, n}, x] && !IntegerQ[n] && !IntegerQ[m]
  2833. Int[Times[cos[Plus[Optional[Pattern[e, Blank[]]], Times[Optional[Pattern[f, Blank[]]], Pattern[x, Blank[]]]]], Power[Plus[Pattern[a, Blank[]], Times[Optional[Pattern[b, Blank[]]], sin[Plus[Optional[Pattern[e, Blank[]]], Times[Optional[Pattern[f, Blank[]]], Pattern[x, Blank[]]]]]]], Optional[Pattern[m, Blank[]]]], Power[Plus[Optional[Pattern[c, Blank[]]], Times[Optional[Pattern[d, Blank[]]], sin[Plus[Optional[Pattern[e, Blank[]]], Times[Optional[Pattern[f, Blank[]]], Pattern[x, Blank[]]]]]]], Optional[Pattern[n, Blank[]]]]], Pattern[x, Blank[Symbol]]] := Dist[1/(b*f), Subst[Int[(a + x)^m*(c + (d*x)/b)^n, x], x, b*Sin[e + f*x]], x] /; FreeQ[{a, b, c, d, e, f, m, n}, x]
  2834. Int[Times[Power[cos[Plus[Optional[Pattern[e, Blank[]]], Times[Optional[Pattern[f, Blank[]]], Pattern[x, Blank[]]]]], Pattern[p, Blank[]]], Power[Times[Optional[Pattern[d, Blank[]]], sin[Plus[Optional[Pattern[e, Blank[]]], Times[Optional[Pattern[f, Blank[]]], Pattern[x, Blank[]]]]]], Optional[Pattern[n, Blank[]]]], Plus[Pattern[a, Blank[]], Times[Optional[Pattern[b, Blank[]]], sin[Plus[Optional[Pattern[e, Blank[]]], Times[Optional[Pattern[f, Blank[]]], Pattern[x, Blank[]]]]]]]], Pattern[x, Blank[Symbol]]] := Dist[a, Int[Cos[e + f*x]^p*(d*Sin[e + f*x])^n, x], x] + Dist[b/d, Int[Cos[e + f*x]^p*(d*Sin[e + f*x])^(n + 1), x], x] /; FreeQ[{a, b, d, e, f, n, p}, x] && IntegerQ[(p - 1)/2] && IntegerQ[n] && ((LtQ[p, 0] && NeQ[a^2 - b^2, 0]) || LtQ[0, n, p - 1] || LtQ[p + 1, -n, 2*p + 1])
  2835. Int[Times[Power[cos[Plus[Optional[Pattern[e, Blank[]]], Times[Optional[Pattern[f, Blank[]]], Pattern[x, Blank[]]]]], Pattern[p, Blank[]]], Power[Times[Optional[Pattern[d, Blank[]]], sin[Plus[Optional[Pattern[e, Blank[]]], Times[Optional[Pattern[f, Blank[]]], Pattern[x, Blank[]]]]]], Optional[Pattern[n, Blank[]]]], Power[Plus[Pattern[a, Blank[]], Times[Optional[Pattern[b, Blank[]]], sin[Plus[Optional[Pattern[e, Blank[]]], Times[Optional[Pattern[f, Blank[]]], Pattern[x, Blank[]]]]]]], -1]], Pattern[x, Blank[Symbol]]] := Dist[1/a, Int[Cos[e + f*x]^(p - 2)*(d*Sin[e + f*x])^n, x], x] - Dist[1/(b*d), Int[Cos[e + f*x]^(p - 2)*(d*Sin[e + f*x])^(n + 1), x], x] /; FreeQ[{a, b, d, e, f, n, p}, x] && IntegerQ[(p - 1)/2] && EqQ[a^2 - b^2, 0] && IntegerQ[n] && (LtQ[0, n, (p + 1)/2] || (LeQ[p, -n] && LtQ[-n, 2*p - 3]) || (GtQ[n, 0] && LeQ[n, -p]))
  2836. Int[Times[Power[cos[Plus[Optional[Pattern[e, Blank[]]], Times[Optional[Pattern[f, Blank[]]], Pattern[x, Blank[]]]]], Pattern[p, Blank[]]], Power[Plus[Pattern[a, Blank[]], Times[Optional[Pattern[b, Blank[]]], sin[Plus[Optional[Pattern[e, Blank[]]], Times[Optional[Pattern[f, Blank[]]], Pattern[x, Blank[]]]]]]], Optional[Pattern[m, Blank[]]]], Power[Plus[Optional[Pattern[c, Blank[]]], Times[Optional[Pattern[d, Blank[]]], sin[Plus[Optional[Pattern[e, Blank[]]], Times[Optional[Pattern[f, Blank[]]], Pattern[x, Blank[]]]]]]], Optional[Pattern[n, Blank[]]]]], Pattern[x, Blank[Symbol]]] := Dist[1/(b^p*f), Subst[Int[(a + x)^(m + (p - 1)/2)*(a - x)^((p - 1)/2)*(c + (d*x)/b)^n, x], x, b*Sin[e + f*x]], x] /; FreeQ[{a, b, e, f, c, d, m, n}, x] && IntegerQ[(p - 1)/2] && EqQ[a^2 - b^2, 0]
  2837. Int[Times[Power[cos[Plus[Optional[Pattern[e, Blank[]]], Times[Optional[Pattern[f, Blank[]]], Pattern[x, Blank[]]]]], Pattern[p, Blank[]]], Power[Plus[Pattern[a, Blank[]], Times[Optional[Pattern[b, Blank[]]], sin[Plus[Optional[Pattern[e, Blank[]]], Times[Optional[Pattern[f, Blank[]]], Pattern[x, Blank[]]]]]]], Optional[Pattern[m, Blank[]]]], Power[Plus[Optional[Pattern[c, Blank[]]], Times[Optional[Pattern[d, Blank[]]], sin[Plus[Optional[Pattern[e, Blank[]]], Times[Optional[Pattern[f, Blank[]]], Pattern[x, Blank[]]]]]]], Optional[Pattern[n, Blank[]]]]], Pattern[x, Blank[Symbol]]] := Dist[1/(b^p*f), Subst[Int[(a + x)^m*(c + (d*x)/b)^n*(b^2 - x^2)^((p - 1)/2), x], x, b*Sin[e + f*x]], x] /; FreeQ[{a, b, c, d, e, f, m, n}, x] && IntegerQ[(p - 1)/2] && NeQ[a^2 - b^2, 0]
  2838. Int[Times[Power[Times[cos[Plus[Optional[Pattern[e, Blank[]]], Times[Optional[Pattern[f, Blank[]]], Pattern[x, Blank[]]]]], Optional[Pattern[g, Blank[]]]], Pattern[p, Blank[]]], Power[Times[Optional[Pattern[d, Blank[]]], sin[Plus[Optional[Pattern[e, Blank[]]], Times[Optional[Pattern[f, Blank[]]], Pattern[x, Blank[]]]]]], Optional[Pattern[n, Blank[]]]], Plus[Pattern[a, Blank[]], Times[Optional[Pattern[b, Blank[]]], sin[Plus[Optional[Pattern[e, Blank[]]], Times[Optional[Pattern[f, Blank[]]], Pattern[x, Blank[]]]]]]]], Pattern[x, Blank[Symbol]]] := Dist[a, Int[(g*Cos[e + f*x])^p*(d*Sin[e + f*x])^n, x], x] + Dist[b/d, Int[(g*Cos[e + f*x])^p*(d*Sin[e + f*x])^(n + 1), x], x] /; FreeQ[{a, b, d, e, f, g, n, p}, x]
  2839. Int[Times[Power[Times[cos[Plus[Optional[Pattern[e, Blank[]]], Times[Optional[Pattern[f, Blank[]]], Pattern[x, Blank[]]]]], Optional[Pattern[g, Blank[]]]], Pattern[p, Blank[]]], Power[Times[Optional[Pattern[d, Blank[]]], sin[Plus[Optional[Pattern[e, Blank[]]], Times[Optional[Pattern[f, Blank[]]], Pattern[x, Blank[]]]]]], Optional[Pattern[n, Blank[]]]], Power[Plus[Pattern[a, Blank[]], Times[Optional[Pattern[b, Blank[]]], sin[Plus[Optional[Pattern[e, Blank[]]], Times[Optional[Pattern[f, Blank[]]], Pattern[x, Blank[]]]]]]], -1]], Pattern[x, Blank[Symbol]]] := Dist[g^2/a, Int[(g*Cos[e + f*x])^(p - 2)*(d*Sin[e + f*x])^n, x], x] - Dist[g^2/(b*d), Int[(g*Cos[e + f*x])^(p - 2)*(d*Sin[e + f*x])^(n + 1), x], x] /; FreeQ[{a, b, d, e, f, g, n, p}, x] && EqQ[a^2 - b^2, 0]
  2840. Int[Times[Power[Times[cos[Plus[Optional[Pattern[e, Blank[]]], Times[Optional[Pattern[f, Blank[]]], Pattern[x, Blank[]]]]], Optional[Pattern[g, Blank[]]]], Pattern[p, Blank[]]], Power[Plus[Pattern[a, Blank[]], Times[Optional[Pattern[b, Blank[]]], sin[Plus[Optional[Pattern[e, Blank[]]], Times[Optional[Pattern[f, Blank[]]], Pattern[x, Blank[]]]]]]], Optional[Pattern[m, Blank[]]]], Power[Plus[Pattern[c, Blank[]], Times[Optional[Pattern[d, Blank[]]], sin[Plus[Optional[Pattern[e, Blank[]]], Times[Optional[Pattern[f, Blank[]]], Pattern[x, Blank[]]]]]]], Optional[Pattern[n, Blank[]]]]], Pattern[x, Blank[Symbol]]] := Dist[(a^m*c^m)/g^(2*m), Int[(g*Cos[e + f*x])^(2*m + p)*(c + d*Sin[e + f*x])^(n - m), x], x] /; FreeQ[{a, b, c, d, e, f, g, n, p}, x] && EqQ[b*c + a*d, 0] && EqQ[a^2 - b^2, 0] && IntegerQ[m] && !(IntegerQ[n] && LtQ[n^2, m^2])
  2841. Int[Times[Power[cos[Plus[Optional[Pattern[e, Blank[]]], Times[Optional[Pattern[f, Blank[]]], Pattern[x, Blank[]]]]], Pattern[p, Blank[]]], Power[Plus[Pattern[a, Blank[]], Times[Optional[Pattern[b, Blank[]]], sin[Plus[Optional[Pattern[e, Blank[]]], Times[Optional[Pattern[f, Blank[]]], Pattern[x, Blank[]]]]]]], Optional[Pattern[m, Blank[]]]], Power[Plus[Pattern[c, Blank[]], Times[Optional[Pattern[d, Blank[]]], sin[Plus[Optional[Pattern[e, Blank[]]], Times[Optional[Pattern[f, Blank[]]], Pattern[x, Blank[]]]]]]], Optional[Pattern[n, Blank[]]]]], Pattern[x, Blank[Symbol]]] := Dist[1/(a^(p/2)*c^(p/2)), Int[(a + b*Sin[e + f*x])^(m + p/2)*(c + d*Sin[e + f*x])^(n + p/2), x], x] /; FreeQ[{a, b, c, d, e, f, n, p}, x] && EqQ[b*c + a*d, 0] && EqQ[a^2 - b^2, 0] && IntegerQ[p/2]
  2842. Int[Times[Power[Times[cos[Plus[Optional[Pattern[e, Blank[]]], Times[Optional[Pattern[f, Blank[]]], Pattern[x, Blank[]]]]], Optional[Pattern[g, Blank[]]]], Pattern[p, Blank[]]], Power[Plus[Pattern[a, Blank[]], Times[Optional[Pattern[b, Blank[]]], sin[Plus[Optional[Pattern[e, Blank[]]], Times[Optional[Pattern[f, Blank[]]], Pattern[x, Blank[]]]]]]], Rational[-1, 2]], Power[Plus[Pattern[c, Blank[]], Times[Optional[Pattern[d, Blank[]]], sin[Plus[Optional[Pattern[e, Blank[]]], Times[Optional[Pattern[f, Blank[]]], Pattern[x, Blank[]]]]]]], Rational[-1, 2]]], Pattern[x, Blank[Symbol]]] := Dist[(g*Cos[e + f*x])/(Sqrt[a + b*Sin[e + f*x]]*Sqrt[c + d*Sin[e + f*x]]), Int[(g*Cos[e + f*x])^(p - 1), x], x] /; FreeQ[{a, b, c, d, e, f, g, p}, x] && EqQ[b*c + a*d, 0] && EqQ[a^2 - b^2, 0]
  2843. Int[Times[Power[Times[cos[Plus[Optional[Pattern[e, Blank[]]], Times[Optional[Pattern[f, Blank[]]], Pattern[x, Blank[]]]]], Optional[Pattern[g, Blank[]]]], Pattern[p, Blank[]]], Power[Plus[Pattern[a, Blank[]], Times[Optional[Pattern[b, Blank[]]], sin[Plus[Optional[Pattern[e, Blank[]]], Times[Optional[Pattern[f, Blank[]]], Pattern[x, Blank[]]]]]]], Pattern[m, Blank[]]], Power[Plus[Pattern[c, Blank[]], Times[Optional[Pattern[d, Blank[]]], sin[Plus[Optional[Pattern[e, Blank[]]], Times[Optional[Pattern[f, Blank[]]], Pattern[x, Blank[]]]]]]], Pattern[n, Blank[]]]], Pattern[x, Blank[Symbol]]] := Dist[(a^IntPart[m]*c^IntPart[m]*(a + b*Sin[e + f*x])^FracPart[m]*(c + d*Sin[e + f*x])^FracPart[m])/(g^(2*IntPart[m])*(g*Cos[e + f*x])^(2*FracPart[m])), Int[(g*Cos[e + f*x])^(2*m + p)/(c + d*Sin[e + f*x]), x], x] /; FreeQ[{a, b, c, d, e, f, g, m, n, p}, x] && EqQ[b*c + a*d, 0] && EqQ[a^2 - b^2, 0] && EqQ[2*m + p - 1, 0] && EqQ[m - n - 1, 0]
  2844. Int[Times[Power[Times[cos[Plus[Optional[Pattern[e, Blank[]]], Times[Optional[Pattern[f, Blank[]]], Pattern[x, Blank[]]]]], Optional[Pattern[g, Blank[]]]], Pattern[p, Blank[]]], Power[Plus[Pattern[a, Blank[]], Times[Optional[Pattern[b, Blank[]]], sin[Plus[Optional[Pattern[e, Blank[]]], Times[Optional[Pattern[f, Blank[]]], Pattern[x, Blank[]]]]]]], Pattern[m, Blank[]]], Power[Plus[Pattern[c, Blank[]], Times[Optional[Pattern[d, Blank[]]], sin[Plus[Optional[Pattern[e, Blank[]]], Times[Optional[Pattern[f, Blank[]]], Pattern[x, Blank[]]]]]]], Pattern[n, Blank[]]]], Pattern[x, Blank[Symbol]]] := Simp[(b*(g*Cos[e + f*x])^(p + 1)*(a + b*Sin[e + f*x])^(m - 1)*(c + d*Sin[e + f*x])^n)/(f*g*(m - n - 1)), x] /; FreeQ[{a, b, c, d, e, f, g, m, n, p}, x] && EqQ[b*c + a*d, 0] && EqQ[a^2 - b^2, 0] && EqQ[2*m + p - 1, 0] && NeQ[m - n - 1, 0]
  2845. Int[Times[Power[Times[cos[Plus[Optional[Pattern[e, Blank[]]], Times[Optional[Pattern[f, Blank[]]], Pattern[x, Blank[]]]]], Optional[Pattern[g, Blank[]]]], Pattern[p, Blank[]]], Power[Plus[Pattern[a, Blank[]], Times[Optional[Pattern[b, Blank[]]], sin[Plus[Optional[Pattern[e, Blank[]]], Times[Optional[Pattern[f, Blank[]]], Pattern[x, Blank[]]]]]]], Pattern[m, Blank[]]], Power[Plus[Pattern[c, Blank[]], Times[Optional[Pattern[d, Blank[]]], sin[Plus[Optional[Pattern[e, Blank[]]], Times[Optional[Pattern[f, Blank[]]], Pattern[x, Blank[]]]]]]], Pattern[n, Blank[]]]], Pattern[x, Blank[Symbol]]] := Simp[(-2*b*(g*Cos[e + f*x])^(p + 1)*(a + b*Sin[e + f*x])^(m - 1)*(c + d*Sin[e + f*x])^n)/(f*g*(2*n + p + 1)), x] - Dist[(b*(2*m + p - 1))/(d*(2*n + p + 1)), Int[(g*Cos[e + f*x])^p*(a + b*Sin[e + f*x])^(m - 1)*(c + d*Sin[e + f*x])^(n + 1), x], x] /; FreeQ[{a, b, c, d, e, f, g, p}, x] && EqQ[b*c + a*d, 0] && EqQ[a^2 - b^2, 0] && IGtQ[Simplify[m + p/2 - 1/2], 0] && LtQ[n, -1] && NeQ[2*n + p + 1, 0] && !(ILtQ[Simplify[m + n + p], 0] && GtQ[Simplify[2*m + n + (3*p)/2 + 1], 0])
  2846. Int[Times[Power[Times[cos[Plus[Optional[Pattern[e, Blank[]]], Times[Optional[Pattern[f, Blank[]]], Pattern[x, Blank[]]]]], Optional[Pattern[g, Blank[]]]], Pattern[p, Blank[]]], Power[Plus[Pattern[a, Blank[]], Times[Optional[Pattern[b, Blank[]]], sin[Plus[Optional[Pattern[e, Blank[]]], Times[Optional[Pattern[f, Blank[]]], Pattern[x, Blank[]]]]]]], Pattern[m, Blank[]]], Power[Plus[Pattern[c, Blank[]], Times[Optional[Pattern[d, Blank[]]], sin[Plus[Optional[Pattern[e, Blank[]]], Times[Optional[Pattern[f, Blank[]]], Pattern[x, Blank[]]]]]]], Pattern[n, Blank[]]]], Pattern[x, Blank[Symbol]]] := -Simp[(b*(g*Cos[e + f*x])^(p + 1)*(a + b*Sin[e + f*x])^(m - 1)*(c + d*Sin[e + f*x])^n)/(f*g*(m + n + p)), x] + Dist[(a*(2*m + p - 1))/(m + n + p), Int[(g*Cos[e + f*x])^p*(a + b*Sin[e + f*x])^(m - 1)*(c + d*Sin[e + f*x])^n, x], x] /; FreeQ[{a, b, c, d, e, f, g, n, p}, x] && EqQ[b*c + a*d, 0] && EqQ[a^2 - b^2, 0] && IGtQ[Simplify[m + p/2 - 1/2], 0] && !LtQ[n, -1] && !(IGtQ[Simplify[n + p/2 - 1/2], 0] && GtQ[m - n, 0]) && !(ILtQ[Simplify[m + n + p], 0] && GtQ[Simplify[2*m + n + (3*p)/2 + 1], 0])
  2847. Int[Times[Power[Times[cos[Plus[Optional[Pattern[e, Blank[]]], Times[Optional[Pattern[f, Blank[]]], Pattern[x, Blank[]]]]], Optional[Pattern[g, Blank[]]]], Pattern[p, Blank[]]], Power[Plus[Pattern[a, Blank[]], Times[Optional[Pattern[b, Blank[]]], sin[Plus[Optional[Pattern[e, Blank[]]], Times[Optional[Pattern[f, Blank[]]], Pattern[x, Blank[]]]]]]], Pattern[m, Blank[]]], Power[Plus[Pattern[c, Blank[]], Times[Optional[Pattern[d, Blank[]]], sin[Plus[Optional[Pattern[e, Blank[]]], Times[Optional[Pattern[f, Blank[]]], Pattern[x, Blank[]]]]]]], Pattern[m, Blank[]]]], Pattern[x, Blank[Symbol]]] := Dist[(a^IntPart[m]*c^IntPart[m]*(a + b*Sin[e + f*x])^FracPart[m]*(c + d*Sin[e + f*x])^FracPart[m])/(g^(2*IntPart[m])*(g*Cos[e + f*x])^(2*FracPart[m])), Int[(g*Cos[e + f*x])^(2*m + p), x], x] /; FreeQ[{a, b, c, d, e, f, g, m, p}, x] && EqQ[b*c + a*d, 0] && EqQ[a^2 - b^2, 0] && EqQ[2*m + p + 1, 0]
  2848. Int[Times[Power[Times[cos[Plus[Optional[Pattern[e, Blank[]]], Times[Optional[Pattern[f, Blank[]]], Pattern[x, Blank[]]]]], Optional[Pattern[g, Blank[]]]], Pattern[p, Blank[]]], Power[Plus[Pattern[a, Blank[]], Times[Optional[Pattern[b, Blank[]]], sin[Plus[Optional[Pattern[e, Blank[]]], Times[Optional[Pattern[f, Blank[]]], Pattern[x, Blank[]]]]]]], Pattern[m, Blank[]]], Power[Plus[Pattern[c, Blank[]], Times[Optional[Pattern[d, Blank[]]], sin[Plus[Optional[Pattern[e, Blank[]]], Times[Optional[Pattern[f, Blank[]]], Pattern[x, Blank[]]]]]]], Pattern[n, Blank[]]]], Pattern[x, Blank[Symbol]]] := Simp[(b*(g*Cos[e + f*x])^(p + 1)*(a + b*Sin[e + f*x])^m*(c + d*Sin[e + f*x])^n)/(a*f*g*(m - n)), x] /; FreeQ[{a, b, c, d, e, f, g, m, n, p}, x] && EqQ[b*c + a*d, 0] && EqQ[a^2 - b^2, 0] && EqQ[m + n + p + 1, 0] && NeQ[m, n]
  2849. Int[Times[Power[Times[cos[Plus[Optional[Pattern[e, Blank[]]], Times[Optional[Pattern[f, Blank[]]], Pattern[x, Blank[]]]]], Optional[Pattern[g, Blank[]]]], Pattern[p, Blank[]]], Power[Plus[Pattern[a, Blank[]], Times[Optional[Pattern[b, Blank[]]], sin[Plus[Optional[Pattern[e, Blank[]]], Times[Optional[Pattern[f, Blank[]]], Pattern[x, Blank[]]]]]]], Pattern[m, Blank[]]], Power[Plus[Pattern[c, Blank[]], Times[Optional[Pattern[d, Blank[]]], sin[Plus[Optional[Pattern[e, Blank[]]], Times[Optional[Pattern[f, Blank[]]], Pattern[x, Blank[]]]]]]], Pattern[n, Blank[]]]], Pattern[x, Blank[Symbol]]] := Simp[(b*(g*Cos[e + f*x])^(p + 1)*(a + b*Sin[e + f*x])^m*(c + d*Sin[e + f*x])^n)/(a*f*g*(2*m + p + 1)), x] + Dist[(m + n + p + 1)/(a*(2*m + p + 1)), Int[(g*Cos[e + f*x])^p*(a + b*Sin[e + f*x])^(m + 1)*(c + d*Sin[e + f*x])^n, x], x] /; FreeQ[{a, b, c, d, e, f, g, m, n, p}, x] && EqQ[b*c + a*d, 0] && EqQ[a^2 - b^2, 0] && ILtQ[Simplify[m + n + p + 1], 0] && NeQ[2*m + p + 1, 0] && (SumSimplerQ[m, 1] || !SumSimplerQ[n, 1])
  2850. Int[Times[Power[Times[cos[Plus[Optional[Pattern[e, Blank[]]], Times[Optional[Pattern[f, Blank[]]], Pattern[x, Blank[]]]]], Optional[Pattern[g, Blank[]]]], Pattern[p, Blank[]]], Power[Plus[Pattern[a, Blank[]], Times[Optional[Pattern[b, Blank[]]], sin[Plus[Optional[Pattern[e, Blank[]]], Times[Optional[Pattern[f, Blank[]]], Pattern[x, Blank[]]]]]]], Pattern[m, Blank[]]], Power[Plus[Pattern[c, Blank[]], Times[Optional[Pattern[d, Blank[]]], sin[Plus[Optional[Pattern[e, Blank[]]], Times[Optional[Pattern[f, Blank[]]], Pattern[x, Blank[]]]]]]], Pattern[n, Blank[]]]], Pattern[x, Blank[Symbol]]] := Simp[(-2*b*(g*Cos[e + f*x])^(p + 1)*(a + b*Sin[e + f*x])^(m - 1)*(c + d*Sin[e + f*x])^n)/(f*g*(2*n + p + 1)), x] - Dist[(b*(2*m + p - 1))/(d*(2*n + p + 1)), Int[(g*Cos[e + f*x])^p*(a + b*Sin[e + f*x])^(m - 1)*(c + d*Sin[e + f*x])^(n + 1), x], x] /; FreeQ[{a, b, c, d, e, f, g, p}, x] && EqQ[b*c + a*d, 0] && EqQ[a^2 - b^2, 0] && GtQ[m, 0] && LtQ[n, -1] && NeQ[2*n + p + 1, 0] && IntegersQ[2*m, 2*n, 2*p]
  2851. Int[Times[Power[Times[cos[Plus[Optional[Pattern[e, Blank[]]], Times[Optional[Pattern[f, Blank[]]], Pattern[x, Blank[]]]]], Optional[Pattern[g, Blank[]]]], Pattern[p, Blank[]]], Power[Plus[Pattern[a, Blank[]], Times[Optional[Pattern[b, Blank[]]], sin[Plus[Optional[Pattern[e, Blank[]]], Times[Optional[Pattern[f, Blank[]]], Pattern[x, Blank[]]]]]]], Pattern[m, Blank[]]], Power[Plus[Pattern[c, Blank[]], Times[Optional[Pattern[d, Blank[]]], sin[Plus[Optional[Pattern[e, Blank[]]], Times[Optional[Pattern[f, Blank[]]], Pattern[x, Blank[]]]]]]], Pattern[n, Blank[]]]], Pattern[x, Blank[Symbol]]] := -Simp[(b*(g*Cos[e + f*x])^(p + 1)*(a + b*Sin[e + f*x])^(m - 1)*(c + d*Sin[e + f*x])^n)/(f*g*(m + n + p)), x] + Dist[(a*(2*m + p - 1))/(m + n + p), Int[(g*Cos[e + f*x])^p*(a + b*Sin[e + f*x])^(m - 1)*(c + d*Sin[e + f*x])^n, x], x] /; FreeQ[{a, b, c, d, e, f, g, n, p}, x] && EqQ[b*c + a*d, 0] && EqQ[a^2 - b^2, 0] && GtQ[m, 0] && NeQ[m + n + p, 0] && !LtQ[0, n, m] && IntegersQ[2*m, 2*n, 2*p]
  2852. Int[Times[Power[Times[cos[Plus[Optional[Pattern[e, Blank[]]], Times[Optional[Pattern[f, Blank[]]], Pattern[x, Blank[]]]]], Optional[Pattern[g, Blank[]]]], Pattern[p, Blank[]]], Power[Plus[Pattern[a, Blank[]], Times[Optional[Pattern[b, Blank[]]], sin[Plus[Optional[Pattern[e, Blank[]]], Times[Optional[Pattern[f, Blank[]]], Pattern[x, Blank[]]]]]]], Pattern[m, Blank[]]], Power[Plus[Pattern[c, Blank[]], Times[Optional[Pattern[d, Blank[]]], sin[Plus[Optional[Pattern[e, Blank[]]], Times[Optional[Pattern[f, Blank[]]], Pattern[x, Blank[]]]]]]], Pattern[n, Blank[]]]], Pattern[x, Blank[Symbol]]] := Simp[(b*(g*Cos[e + f*x])^(p + 1)*(a + b*Sin[e + f*x])^m*(c + d*Sin[e + f*x])^n)/(a*f*g*(2*m + p + 1)), x] + Dist[(m + n + p + 1)/(a*(2*m + p + 1)), Int[(g*Cos[e + f*x])^p*(a + b*Sin[e + f*x])^(m + 1)*(c + d*Sin[e + f*x])^n, x], x] /; FreeQ[{a, b, c, d, e, f, g, n, p}, x] && EqQ[b*c + a*d, 0] && EqQ[a^2 - b^2, 0] && LtQ[m, -1] && NeQ[2*m + p + 1, 0] && !LtQ[m, n, -1] && IntegersQ[2*m, 2*n, 2*p]
  2853. Int[Times[Power[Times[cos[Plus[Optional[Pattern[e, Blank[]]], Times[Optional[Pattern[f, Blank[]]], Pattern[x, Blank[]]]]], Optional[Pattern[g, Blank[]]]], Pattern[p, Blank[]]], Power[Plus[Pattern[a, Blank[]], Times[Optional[Pattern[b, Blank[]]], sin[Plus[Optional[Pattern[e, Blank[]]], Times[Optional[Pattern[f, Blank[]]], Pattern[x, Blank[]]]]]]], Pattern[m, Blank[]]], Power[Plus[Pattern[c, Blank[]], Times[Optional[Pattern[d, Blank[]]], sin[Plus[Optional[Pattern[e, Blank[]]], Times[Optional[Pattern[f, Blank[]]], Pattern[x, Blank[]]]]]]], Pattern[n, Blank[]]]], Pattern[x, Blank[Symbol]]] := Dist[(a^IntPart[m]*c^IntPart[m]*(a + b*Sin[e + f*x])^FracPart[m]*(c + d*Sin[e + f*x])^FracPart[m])/(g^(2*IntPart[m])*(g*Cos[e + f*x])^(2*FracPart[m])), Int[(g*Cos[e + f*x])^(2*m + p)*(c + d*Sin[e + f*x])^(n - m), x], x] /; FreeQ[{a, b, c, d, e, f, g, m, n, p}, x] && EqQ[b*c + a*d, 0] && EqQ[a^2 - b^2, 0] && (FractionQ[m] || !FractionQ[n])
  2854. Int[Times[Power[Times[cos[Plus[Optional[Pattern[e, Blank[]]], Times[Optional[Pattern[f, Blank[]]], Pattern[x, Blank[]]]]], Optional[Pattern[g, Blank[]]]], Pattern[p, Blank[]]], Power[Plus[Pattern[a, Blank[]], Times[Optional[Pattern[b, Blank[]]], sin[Plus[Optional[Pattern[e, Blank[]]], Times[Optional[Pattern[f, Blank[]]], Pattern[x, Blank[]]]]]]], Optional[Pattern[m, Blank[]]]], Plus[Optional[Pattern[c, Blank[]]], Times[Optional[Pattern[d, Blank[]]], sin[Plus[Optional[Pattern[e, Blank[]]], Times[Optional[Pattern[f, Blank[]]], Pattern[x, Blank[]]]]]]]], Pattern[x, Blank[Symbol]]] := -Simp[(d*(g*Cos[e + f*x])^(p + 1)*(a + b*Sin[e + f*x])^m)/(f*g*(m + p + 1)), x] /; FreeQ[{a, b, c, d, e, f, g, m, p}, x] && EqQ[a^2 - b^2, 0] && EqQ[a*d*m + b*c*(m + p + 1), 0]
  2855. Int[Times[Power[Times[cos[Plus[Optional[Pattern[e, Blank[]]], Times[Optional[Pattern[f, Blank[]]], Pattern[x, Blank[]]]]], Optional[Pattern[g, Blank[]]]], Pattern[p, Blank[]]], Power[Plus[Pattern[a, Blank[]], Times[Optional[Pattern[b, Blank[]]], sin[Plus[Optional[Pattern[e, Blank[]]], Times[Optional[Pattern[f, Blank[]]], Pattern[x, Blank[]]]]]]], Optional[Pattern[m, Blank[]]]], Plus[Optional[Pattern[c, Blank[]]], Times[Optional[Pattern[d, Blank[]]], sin[Plus[Optional[Pattern[e, Blank[]]], Times[Optional[Pattern[f, Blank[]]], Pattern[x, Blank[]]]]]]]], Pattern[x, Blank[Symbol]]] := -Simp[((b*c + a*d)*(g*Cos[e + f*x])^(p + 1)*(a + b*Sin[e + f*x])^m)/(a*f*g*(p + 1)), x] + Dist[(b*(a*d*m + b*c*(m + p + 1)))/(a*g^2*(p + 1)), Int[(g*Cos[e + f*x])^(p + 2)*(a + b*Sin[e + f*x])^(m - 1), x], x] /; FreeQ[{a, b, c, d, e, f, g}, x] && EqQ[a^2 - b^2, 0] && GtQ[m, -1] && LtQ[p, -1]
  2856. Int[Times[Power[Times[cos[Plus[Optional[Pattern[e, Blank[]]], Times[Optional[Pattern[f, Blank[]]], Pattern[x, Blank[]]]]], Optional[Pattern[g, Blank[]]]], Pattern[p, Blank[]]], Power[Plus[Pattern[a, Blank[]], Times[Optional[Pattern[b, Blank[]]], sin[Plus[Optional[Pattern[e, Blank[]]], Times[Optional[Pattern[f, Blank[]]], Pattern[x, Blank[]]]]]]], Optional[Pattern[m, Blank[]]]], Plus[Optional[Pattern[c, Blank[]]], Times[Optional[Pattern[d, Blank[]]], sin[Plus[Optional[Pattern[e, Blank[]]], Times[Optional[Pattern[f, Blank[]]], Pattern[x, Blank[]]]]]]]], Pattern[x, Blank[Symbol]]] := -Simp[(d*(g*Cos[e + f*x])^(p + 1)*(a + b*Sin[e + f*x])^m)/(f*g*(m + p + 1)), x] + Dist[(a*d*m + b*c*(m + p + 1))/(b*(m + p + 1)), Int[(g*Cos[e + f*x])^p*(a + b*Sin[e + f*x])^m, x], x] /; FreeQ[{a, b, c, d, e, f, g, m, p}, x] && EqQ[a^2 - b^2, 0] && IGtQ[Simplify[(2*m + p + 1)/2], 0] && NeQ[m + p + 1, 0]
  2857. Int[Times[Power[cos[Plus[Optional[Pattern[e, Blank[]]], Times[Optional[Pattern[f, Blank[]]], Pattern[x, Blank[]]]]], 2], Power[Plus[Pattern[a, Blank[]], Times[Optional[Pattern[b, Blank[]]], sin[Plus[Optional[Pattern[e, Blank[]]], Times[Optional[Pattern[f, Blank[]]], Pattern[x, Blank[]]]]]]], Pattern[m, Blank[]]], Plus[Optional[Pattern[c, Blank[]]], Times[Optional[Pattern[d, Blank[]]], sin[Plus[Optional[Pattern[e, Blank[]]], Times[Optional[Pattern[f, Blank[]]], Pattern[x, Blank[]]]]]]]], Pattern[x, Blank[Symbol]]] := Simp[(2*(b*c - a*d)*Cos[e + f*x]*(a + b*Sin[e + f*x])^(m + 1))/(b^2*f*(2*m + 3)), x] + Dist[1/(b^3*(2*m + 3)), Int[(a + b*Sin[e + f*x])^(m + 2)*(b*c + 2*a*d*(m + 1) - b*d*(2*m + 3)*Sin[e + f*x]), x], x] /; FreeQ[{a, b, c, d, e, f}, x] && EqQ[a^2 - b^2, 0] && LtQ[m, -3/2]
  2858. Int[Times[Power[cos[Plus[Optional[Pattern[e, Blank[]]], Times[Optional[Pattern[f, Blank[]]], Pattern[x, Blank[]]]]], 2], Power[Plus[Pattern[a, Blank[]], Times[Optional[Pattern[b, Blank[]]], sin[Plus[Optional[Pattern[e, Blank[]]], Times[Optional[Pattern[f, Blank[]]], Pattern[x, Blank[]]]]]]], Pattern[m, Blank[]]], Plus[Optional[Pattern[c, Blank[]]], Times[Optional[Pattern[d, Blank[]]], sin[Plus[Optional[Pattern[e, Blank[]]], Times[Optional[Pattern[f, Blank[]]], Pattern[x, Blank[]]]]]]]], Pattern[x, Blank[Symbol]]] := Simp[(d*Cos[e + f*x]*(a + b*Sin[e + f*x])^(m + 2))/(b^2*f*(m + 3)), x] - Dist[1/(b^2*(m + 3)), Int[(a + b*Sin[e + f*x])^(m + 1)*(b*d*(m + 2) - a*c*(m + 3) + (b*c*(m + 3) - a*d*(m + 4))*Sin[e + f*x]), x], x] /; FreeQ[{a, b, c, d, e, f}, x] && EqQ[a^2 - b^2, 0] && GeQ[m, -3/2] && LtQ[m, 0]
  2859. Int[Times[Power[Times[cos[Plus[Optional[Pattern[e, Blank[]]], Times[Optional[Pattern[f, Blank[]]], Pattern[x, Blank[]]]]], Optional[Pattern[g, Blank[]]]], Pattern[p, Blank[]]], Power[Plus[Pattern[a, Blank[]], Times[Optional[Pattern[b, Blank[]]], sin[Plus[Optional[Pattern[e, Blank[]]], Times[Optional[Pattern[f, Blank[]]], Pattern[x, Blank[]]]]]]], Pattern[m, Blank[]]], Plus[Optional[Pattern[c, Blank[]]], Times[Optional[Pattern[d, Blank[]]], sin[Plus[Optional[Pattern[e, Blank[]]], Times[Optional[Pattern[f, Blank[]]], Pattern[x, Blank[]]]]]]]], Pattern[x, Blank[Symbol]]] := Simp[((b*c - a*d)*(g*Cos[e + f*x])^(p + 1)*(a + b*Sin[e + f*x])^m)/(a*f*g*(2*m + p + 1)), x] + Dist[(a*d*m + b*c*(m + p + 1))/(a*b*(2*m + p + 1)), Int[(g*Cos[e + f*x])^p*(a + b*Sin[e + f*x])^(m + 1), x], x] /; FreeQ[{a, b, c, d, e, f, g, m, p}, x] && EqQ[a^2 - b^2, 0] && (LtQ[m, -1] || ILtQ[Simplify[m + p], 0]) && NeQ[2*m + p + 1, 0]
  2860. Int[Times[Power[Times[cos[Plus[Optional[Pattern[e, Blank[]]], Times[Optional[Pattern[f, Blank[]]], Pattern[x, Blank[]]]]], Optional[Pattern[g, Blank[]]]], Pattern[p, Blank[]]], Power[Plus[Pattern[a, Blank[]], Times[Optional[Pattern[b, Blank[]]], sin[Plus[Optional[Pattern[e, Blank[]]], Times[Optional[Pattern[f, Blank[]]], Pattern[x, Blank[]]]]]]], Optional[Pattern[m, Blank[]]]], Plus[Optional[Pattern[c, Blank[]]], Times[Optional[Pattern[d, Blank[]]], sin[Plus[Optional[Pattern[e, Blank[]]], Times[Optional[Pattern[f, Blank[]]], Pattern[x, Blank[]]]]]]]], Pattern[x, Blank[Symbol]]] := -Simp[(d*(g*Cos[e + f*x])^(p + 1)*(a + b*Sin[e + f*x])^m)/(f*g*(m + p + 1)), x] + Dist[(a*d*m + b*c*(m + p + 1))/(b*(m + p + 1)), Int[(g*Cos[e + f*x])^p*(a + b*Sin[e + f*x])^m, x], x] /; FreeQ[{a, b, c, d, e, f, g, m, p}, x] && EqQ[a^2 - b^2, 0] && NeQ[m + p + 1, 0]
  2861. Int[Times[Power[Times[cos[Plus[Optional[Pattern[e, Blank[]]], Times[Optional[Pattern[f, Blank[]]], Pattern[x, Blank[]]]]], Optional[Pattern[g, Blank[]]]], Pattern[p, Blank[]]], Power[Plus[Pattern[a, Blank[]], Times[Optional[Pattern[b, Blank[]]], sin[Plus[Optional[Pattern[e, Blank[]]], Times[Optional[Pattern[f, Blank[]]], Pattern[x, Blank[]]]]]]], Optional[Pattern[m, Blank[]]]], Plus[Optional[Pattern[c, Blank[]]], Times[Optional[Pattern[d, Blank[]]], sin[Plus[Optional[Pattern[e, Blank[]]], Times[Optional[Pattern[f, Blank[]]], Pattern[x, Blank[]]]]]]]], Pattern[x, Blank[Symbol]]] := -Simp[((g*Cos[e + f*x])^(p + 1)*(a + b*Sin[e + f*x])^m*(d + c*Sin[e + f*x]))/(f*g*(p + 1)), x] + Dist[1/(g^2*(p + 1)), Int[(g*Cos[e + f*x])^(p + 2)*(a + b*Sin[e + f*x])^(m - 1)*Simp[a*c*(p + 2) + b*d*m + b*c*(m + p + 2)*Sin[e + f*x], x], x], x] /; FreeQ[{a, b, c, d, e, f, g}, x] && NeQ[a^2 - b^2, 0] && GtQ[m, 0] && LtQ[p, -1] && IntegerQ[2*m] && !(EqQ[m, 1] && NeQ[c^2 - d^2, 0] && SimplerQ[c + d*x, a + b*x])
  2862. Int[Times[Power[Times[cos[Plus[Optional[Pattern[e, Blank[]]], Times[Optional[Pattern[f, Blank[]]], Pattern[x, Blank[]]]]], Optional[Pattern[g, Blank[]]]], Pattern[p, Blank[]]], Power[Plus[Pattern[a, Blank[]], Times[Optional[Pattern[b, Blank[]]], sin[Plus[Optional[Pattern[e, Blank[]]], Times[Optional[Pattern[f, Blank[]]], Pattern[x, Blank[]]]]]]], Optional[Pattern[m, Blank[]]]], Plus[Optional[Pattern[c, Blank[]]], Times[Optional[Pattern[d, Blank[]]], sin[Plus[Optional[Pattern[e, Blank[]]], Times[Optional[Pattern[f, Blank[]]], Pattern[x, Blank[]]]]]]]], Pattern[x, Blank[Symbol]]] := -Simp[(d*(g*Cos[e + f*x])^(p + 1)*(a + b*Sin[e + f*x])^m)/(f*g*(m + p + 1)), x] + Dist[1/(m + p + 1), Int[(g*Cos[e + f*x])^p*(a + b*Sin[e + f*x])^(m - 1)*Simp[a*c*(m + p + 1) + b*d*m + (a*d*m + b*c*(m + p + 1))*Sin[e + f*x], x], x], x] /; FreeQ[{a, b, c, d, e, f, g, p}, x] && NeQ[a^2 - b^2, 0] && GtQ[m, 0] && !LtQ[p, -1] && IntegerQ[2*m] && !(EqQ[m, 1] && NeQ[c^2 - d^2, 0] && SimplerQ[c + d*x, a + b*x])
  2863. Int[Times[Power[Times[cos[Plus[Optional[Pattern[e, Blank[]]], Times[Optional[Pattern[f, Blank[]]], Pattern[x, Blank[]]]]], Optional[Pattern[g, Blank[]]]], Pattern[p, Blank[]]], Power[Plus[Pattern[a, Blank[]], Times[Optional[Pattern[b, Blank[]]], sin[Plus[Optional[Pattern[e, Blank[]]], Times[Optional[Pattern[f, Blank[]]], Pattern[x, Blank[]]]]]]], Pattern[m, Blank[]]], Plus[Optional[Pattern[c, Blank[]]], Times[Optional[Pattern[d, Blank[]]], sin[Plus[Optional[Pattern[e, Blank[]]], Times[Optional[Pattern[f, Blank[]]], Pattern[x, Blank[]]]]]]]], Pattern[x, Blank[Symbol]]] := Simp[(g*(g*Cos[e + f*x])^(p - 1)*(a + b*Sin[e + f*x])^(m + 1)*(b*c*(m + p + 1) - a*d*p + b*d*(m + 1)*Sin[e + f*x]))/(b^2*f*(m + 1)*(m + p + 1)), x] + Dist[(g^2*(p - 1))/(b^2*(m + 1)*(m + p + 1)), Int[(g*Cos[e + f*x])^(p - 2)*(a + b*Sin[e + f*x])^(m + 1)*Simp[b*d*(m + 1) + (b*c*(m + p + 1) - a*d*p)*Sin[e + f*x], x], x], x] /; FreeQ[{a, b, c, d, e, f, g}, x] && NeQ[a^2 - b^2, 0] && LtQ[m, -1] && GtQ[p, 1] && NeQ[m + p + 1, 0] && IntegerQ[2*m]
  2864. Int[Times[Power[Times[cos[Plus[Optional[Pattern[e, Blank[]]], Times[Optional[Pattern[f, Blank[]]], Pattern[x, Blank[]]]]], Optional[Pattern[g, Blank[]]]], Pattern[p, Blank[]]], Power[Plus[Pattern[a, Blank[]], Times[Optional[Pattern[b, Blank[]]], sin[Plus[Optional[Pattern[e, Blank[]]], Times[Optional[Pattern[f, Blank[]]], Pattern[x, Blank[]]]]]]], Pattern[m, Blank[]]], Plus[Optional[Pattern[c, Blank[]]], Times[Optional[Pattern[d, Blank[]]], sin[Plus[Optional[Pattern[e, Blank[]]], Times[Optional[Pattern[f, Blank[]]], Pattern[x, Blank[]]]]]]]], Pattern[x, Blank[Symbol]]] := -Simp[((b*c - a*d)*(g*Cos[e + f*x])^(p + 1)*(a + b*Sin[e + f*x])^(m + 1))/(f*g*(a^2 - b^2)*(m + 1)), x] + Dist[1/((a^2 - b^2)*(m + 1)), Int[(g*Cos[e + f*x])^p*(a + b*Sin[e + f*x])^(m + 1)*Simp[(a*c - b*d)*(m + 1) - (b*c - a*d)*(m + p + 2)*Sin[e + f*x], x], x], x] /; FreeQ[{a, b, c, d, e, f, g, p}, x] && NeQ[a^2 - b^2, 0] && LtQ[m, -1] && IntegerQ[2*m]
  2865. Int[Times[Power[Times[cos[Plus[Optional[Pattern[e, Blank[]]], Times[Optional[Pattern[f, Blank[]]], Pattern[x, Blank[]]]]], Optional[Pattern[g, Blank[]]]], Pattern[p, Blank[]]], Power[Plus[Pattern[a, Blank[]], Times[Optional[Pattern[b, Blank[]]], sin[Plus[Optional[Pattern[e, Blank[]]], Times[Optional[Pattern[f, Blank[]]], Pattern[x, Blank[]]]]]]], Optional[Pattern[m, Blank[]]]], Plus[Optional[Pattern[c, Blank[]]], Times[Optional[Pattern[d, Blank[]]], sin[Plus[Optional[Pattern[e, Blank[]]], Times[Optional[Pattern[f, Blank[]]], Pattern[x, Blank[]]]]]]]], Pattern[x, Blank[Symbol]]] := Simp[(g*(g*Cos[e + f*x])^(p - 1)*(a + b*Sin[e + f*x])^(m + 1)*(b*c*(m + p + 1) - a*d*p + b*d*(m + p)*Sin[e + f*x]))/(b^2*f*(m + p)*(m + p + 1)), x] + Dist[(g^2*(p - 1))/(b^2*(m + p)*(m + p + 1)), Int[(g*Cos[e + f*x])^(p - 2)*(a + b*Sin[e + f*x])^m*Simp[b*(a*d*m + b*c*(m + p + 1)) + (a*b*c*(m + p + 1) - d*(a^2*p - b^2*(m + p)))*Sin[e + f*x], x], x], x] /; FreeQ[{a, b, c, d, e, f, g, m}, x] && NeQ[a^2 - b^2, 0] && GtQ[p, 1] && NeQ[m + p, 0] && NeQ[m + p + 1, 0] && IntegerQ[2*m]
  2866. Int[Times[Power[Times[cos[Plus[Optional[Pattern[e, Blank[]]], Times[Optional[Pattern[f, Blank[]]], Pattern[x, Blank[]]]]], Optional[Pattern[g, Blank[]]]], Pattern[p, Blank[]]], Power[Plus[Pattern[a, Blank[]], Times[Optional[Pattern[b, Blank[]]], sin[Plus[Optional[Pattern[e, Blank[]]], Times[Optional[Pattern[f, Blank[]]], Pattern[x, Blank[]]]]]]], Optional[Pattern[m, Blank[]]]], Plus[Optional[Pattern[c, Blank[]]], Times[Optional[Pattern[d, Blank[]]], sin[Plus[Optional[Pattern[e, Blank[]]], Times[Optional[Pattern[f, Blank[]]], Pattern[x, Blank[]]]]]]]], Pattern[x, Blank[Symbol]]] := Simp[((g*Cos[e + f*x])^(p + 1)*(a + b*Sin[e + f*x])^(m + 1)*(b*c - a*d - (a*c - b*d)*Sin[e + f*x]))/(f*g*(a^2 - b^2)*(p + 1)), x] + Dist[1/(g^2*(a^2 - b^2)*(p + 1)), Int[(g*Cos[e + f*x])^(p + 2)*(a + b*Sin[e + f*x])^m*Simp[c*(a^2*(p + 2) - b^2*(m + p + 2)) + a*b*d*m + b*(a*c - b*d)*(m + p + 3)*Sin[e + f*x], x], x], x] /; FreeQ[{a, b, c, d, e, f, g, m}, x] && NeQ[a^2 - b^2, 0] && LtQ[p, -1] && IntegerQ[2*m]
  2867. Int[Times[Power[Times[cos[Plus[Optional[Pattern[e, Blank[]]], Times[Optional[Pattern[f, Blank[]]], Pattern[x, Blank[]]]]], Optional[Pattern[g, Blank[]]]], Pattern[p, Blank[]]], Power[Plus[Pattern[a, Blank[]], Times[Optional[Pattern[b, Blank[]]], sin[Plus[Optional[Pattern[e, Blank[]]], Times[Optional[Pattern[f, Blank[]]], Pattern[x, Blank[]]]]]]], -1], Plus[Optional[Pattern[c, Blank[]]], Times[Optional[Pattern[d, Blank[]]], sin[Plus[Optional[Pattern[e, Blank[]]], Times[Optional[Pattern[f, Blank[]]], Pattern[x, Blank[]]]]]]]], Pattern[x, Blank[Symbol]]] := Dist[d/b, Int[(g*Cos[e + f*x])^p, x], x] + Dist[(b*c - a*d)/b, Int[(g*Cos[e + f*x])^p/(a + b*Sin[e + f*x]), x], x] /; FreeQ[{a, b, c, d, e, f, g}, x] && NeQ[a^2 - b^2, 0]
  2868. Int[Times[Power[Times[cos[Plus[Optional[Pattern[e, Blank[]]], Times[Optional[Pattern[f, Blank[]]], Pattern[x, Blank[]]]]], Optional[Pattern[g, Blank[]]]], Pattern[p, Blank[]]], Power[Plus[Pattern[a, Blank[]], Times[Optional[Pattern[b, Blank[]]], sin[Plus[Optional[Pattern[e, Blank[]]], Times[Optional[Pattern[f, Blank[]]], Pattern[x, Blank[]]]]]]], Pattern[m, Blank[]]], Plus[Pattern[c, Blank[]], Times[Optional[Pattern[d, Blank[]]], sin[Plus[Optional[Pattern[e, Blank[]]], Times[Optional[Pattern[f, Blank[]]], Pattern[x, Blank[]]]]]]]], Pattern[x, Blank[Symbol]]] := Dist[(c*g*(g*Cos[e + f*x])^(p - 1))/(f*(1 + Sin[e + f*x])^((p - 1)/2)*(1 - Sin[e + f*x])^((p - 1)/2)), Subst[Int[(1 + (d*x)/c)^((p + 1)/2)*(1 - (d*x)/c)^((p - 1)/2)*(a + b*x)^m, x], x, Sin[e + f*x]], x] /; FreeQ[{a, b, c, d, e, f, m, p}, x] && NeQ[a^2 - b^2, 0] && EqQ[c^2 - d^2, 0]
  2869. Int[Times[Power[cos[Plus[Optional[Pattern[e, Blank[]]], Times[Optional[Pattern[f, Blank[]]], Pattern[x, Blank[]]]]], Pattern[p, Blank[]]], Power[Times[Optional[Pattern[d, Blank[]]], sin[Plus[Optional[Pattern[e, Blank[]]], Times[Optional[Pattern[f, Blank[]]], Pattern[x, Blank[]]]]]], Pattern[n, Blank[]]], Power[Plus[Pattern[a, Blank[]], Times[Optional[Pattern[b, Blank[]]], sin[Plus[Optional[Pattern[e, Blank[]]], Times[Optional[Pattern[f, Blank[]]], Pattern[x, Blank[]]]]]]], Pattern[m, Blank[]]]], Pattern[x, Blank[Symbol]]] := Dist[a^(2*m), Int[(d*Sin[e + f*x])^n/(a - b*Sin[e + f*x])^m, x], x] /; FreeQ[{a, b, d, e, f, n}, x] && EqQ[a^2 - b^2, 0] && IntegersQ[m, p] && EqQ[2*m + p, 0]
  2870. Int[Times[Power[Times[cos[Plus[Optional[Pattern[e, Blank[]]], Times[Optional[Pattern[f, Blank[]]], Pattern[x, Blank[]]]]], Optional[Pattern[g, Blank[]]]], Pattern[p, Blank[]]], Power[sin[Plus[Optional[Pattern[e, Blank[]]], Times[Optional[Pattern[f, Blank[]]], Pattern[x, Blank[]]]]], 2], Power[Plus[Pattern[a, Blank[]], Times[Optional[Pattern[b, Blank[]]], sin[Plus[Optional[Pattern[e, Blank[]]], Times[Optional[Pattern[f, Blank[]]], Pattern[x, Blank[]]]]]]], Pattern[m, Blank[]]]], Pattern[x, Blank[Symbol]]] := -Simp[((g*Cos[e + f*x])^(p + 1)*(a + b*Sin[e + f*x])^(m + 1))/(2*b*f*g*(m + 1)), x] + Dist[a/(2*g^2), Int[(g*Cos[e + f*x])^(p + 2)*(a + b*Sin[e + f*x])^(m - 1), x], x] /; FreeQ[{a, b, e, f, g, m, p}, x] && EqQ[a^2 - b^2, 0] && EqQ[m - p, 0]
  2871. Int[Times[Power[Times[cos[Plus[Optional[Pattern[e, Blank[]]], Times[Optional[Pattern[f, Blank[]]], Pattern[x, Blank[]]]]], Optional[Pattern[g, Blank[]]]], Pattern[p, Blank[]]], Power[sin[Plus[Optional[Pattern[e, Blank[]]], Times[Optional[Pattern[f, Blank[]]], Pattern[x, Blank[]]]]], 2], Power[Plus[Pattern[a, Blank[]], Times[Optional[Pattern[b, Blank[]]], sin[Plus[Optional[Pattern[e, Blank[]]], Times[Optional[Pattern[f, Blank[]]], Pattern[x, Blank[]]]]]]], Pattern[m, Blank[]]]], Pattern[x, Blank[Symbol]]] := Simp[(b*(g*Cos[e + f*x])^(p + 1)*(a + b*Sin[e + f*x])^m)/(a*f*g*m), x] - Dist[1/g^2, Int[(g*Cos[e + f*x])^(p + 2)*(a + b*Sin[e + f*x])^m, x], x] /; FreeQ[{a, b, e, f, g, m, p}, x] && EqQ[a^2 - b^2, 0] && EqQ[m + p + 1, 0]
  2872. Int[Times[Power[cos[Plus[Optional[Pattern[e, Blank[]]], Times[Optional[Pattern[f, Blank[]]], Pattern[x, Blank[]]]]], Pattern[p, Blank[]]], Power[Times[Optional[Pattern[d, Blank[]]], sin[Plus[Optional[Pattern[e, Blank[]]], Times[Optional[Pattern[f, Blank[]]], Pattern[x, Blank[]]]]]], Pattern[n, Blank[]]], Power[Plus[Pattern[a, Blank[]], Times[Optional[Pattern[b, Blank[]]], sin[Plus[Optional[Pattern[e, Blank[]]], Times[Optional[Pattern[f, Blank[]]], Pattern[x, Blank[]]]]]]], Pattern[m, Blank[]]]], Pattern[x, Blank[Symbol]]] := Dist[1/a^p, Int[ExpandTrig[(d*sin[e + f*x])^n*(a - b*sin[e + f*x])^(p/2)*(a + b*sin[e + f*x])^(m + p/2), x], x], x] /; FreeQ[{a, b, d, e, f}, x] && EqQ[a^2 - b^2, 0] && IntegersQ[m, n, p/2] && ((GtQ[m, 0] && GtQ[p, 0] && LtQ[-m - p, n, -1]) || (GtQ[m, 2] && LtQ[p, 0] && GtQ[m + p/2, 0]))
  2873. Int[Times[Power[Times[cos[Plus[Optional[Pattern[e, Blank[]]], Times[Optional[Pattern[f, Blank[]]], Pattern[x, Blank[]]]]], Optional[Pattern[g, Blank[]]]], Pattern[p, Blank[]]], Power[Times[Optional[Pattern[d, Blank[]]], sin[Plus[Optional[Pattern[e, Blank[]]], Times[Optional[Pattern[f, Blank[]]], Pattern[x, Blank[]]]]]], Pattern[n, Blank[]]], Power[Plus[Pattern[a, Blank[]], Times[Optional[Pattern[b, Blank[]]], sin[Plus[Optional[Pattern[e, Blank[]]], Times[Optional[Pattern[f, Blank[]]], Pattern[x, Blank[]]]]]]], Pattern[m, Blank[]]]], Pattern[x, Blank[Symbol]]] := Int[ExpandTrig[(g*cos[e + f*x])^p, (d*sin[e + f*x])^n*(a + b*sin[e + f*x])^m, x], x] /; FreeQ[{a, b, d, e, f, g, n, p}, x] && EqQ[a^2 - b^2, 0] && IGtQ[m, 0]
  2874. Int[Times[Power[cos[Plus[Optional[Pattern[e, Blank[]]], Times[Optional[Pattern[f, Blank[]]], Pattern[x, Blank[]]]]], 2], Power[Times[Optional[Pattern[d, Blank[]]], sin[Plus[Optional[Pattern[e, Blank[]]], Times[Optional[Pattern[f, Blank[]]], Pattern[x, Blank[]]]]]], Pattern[n, Blank[]]], Power[Plus[Pattern[a, Blank[]], Times[Optional[Pattern[b, Blank[]]], sin[Plus[Optional[Pattern[e, Blank[]]], Times[Optional[Pattern[f, Blank[]]], Pattern[x, Blank[]]]]]]], Pattern[m, Blank[]]]], Pattern[x, Blank[Symbol]]] := Dist[1/b^2, Int[(d*Sin[e + f*x])^n*(a + b*Sin[e + f*x])^(m + 1)*(a - b*Sin[e + f*x]), x], x] /; FreeQ[{a, b, d, e, f, m, n}, x] && EqQ[a^2 - b^2, 0] && (ILtQ[m, 0] || !IGtQ[n, 0])
  2875. Int[Times[Power[Times[cos[Plus[Optional[Pattern[e, Blank[]]], Times[Optional[Pattern[f, Blank[]]], Pattern[x, Blank[]]]]], Optional[Pattern[g, Blank[]]]], Pattern[p, Blank[]]], Power[Times[Optional[Pattern[d, Blank[]]], sin[Plus[Optional[Pattern[e, Blank[]]], Times[Optional[Pattern[f, Blank[]]], Pattern[x, Blank[]]]]]], Pattern[n, Blank[]]], Power[Plus[Pattern[a, Blank[]], Times[Optional[Pattern[b, Blank[]]], sin[Plus[Optional[Pattern[e, Blank[]]], Times[Optional[Pattern[f, Blank[]]], Pattern[x, Blank[]]]]]]], Pattern[m, Blank[]]]], Pattern[x, Blank[Symbol]]] := Dist[(a/g)^(2*m), Int[((g*Cos[e + f*x])^(2*m + p)*(d*Sin[e + f*x])^n)/(a - b*Sin[e + f*x])^m, x], x] /; FreeQ[{a, b, d, e, f, g, n, p}, x] && EqQ[a^2 - b^2, 0] && ILtQ[m, 0]
  2876. Int[Times[Power[Times[cos[Plus[Optional[Pattern[e, Blank[]]], Times[Optional[Pattern[f, Blank[]]], Pattern[x, Blank[]]]]], Optional[Pattern[g, Blank[]]]], Pattern[p, Blank[]]], Power[Times[Optional[Pattern[d, Blank[]]], sin[Plus[Optional[Pattern[e, Blank[]]], Times[Optional[Pattern[f, Blank[]]], Pattern[x, Blank[]]]]]], Pattern[n, Blank[]]], Power[Plus[Pattern[a, Blank[]], Times[Optional[Pattern[b, Blank[]]], sin[Plus[Optional[Pattern[e, Blank[]]], Times[Optional[Pattern[f, Blank[]]], Pattern[x, Blank[]]]]]]], Pattern[m, Blank[]]]], Pattern[x, Blank[Symbol]]] := Dist[(a/g)^(2*m), Int[((g*Cos[e + f*x])^(2*m + p)*(d*Sin[e + f*x])^n)/(a - b*Sin[e + f*x])^m, x], x] /; FreeQ[{a, b, d, e, f, g, n}, x] && EqQ[a^2 - b^2, 0] && IntegerQ[m] && RationalQ[p] && (EqQ[2*m + p, 0] || (GtQ[2*m + p, 0] && LtQ[p, -1]))
  2877. Int[Times[Power[Times[cos[Plus[Optional[Pattern[e, Blank[]]], Times[Optional[Pattern[f, Blank[]]], Pattern[x, Blank[]]]]], Optional[Pattern[g, Blank[]]]], Pattern[p, Blank[]]], Power[sin[Plus[Optional[Pattern[e, Blank[]]], Times[Optional[Pattern[f, Blank[]]], Pattern[x, Blank[]]]]], 2], Power[Plus[Pattern[a, Blank[]], Times[Optional[Pattern[b, Blank[]]], sin[Plus[Optional[Pattern[e, Blank[]]], Times[Optional[Pattern[f, Blank[]]], Pattern[x, Blank[]]]]]]], Pattern[m, Blank[]]]], Pattern[x, Blank[Symbol]]] := Simp[(b*(g*Cos[e + f*x])^(p + 1)*(a + b*Sin[e + f*x])^m)/(a*f*g*(2*m + p + 1)), x] - Dist[1/(a^2*(2*m + p + 1)), Int[(g*Cos[e + f*x])^p*(a + b*Sin[e + f*x])^(m + 1)*(a*m - b*(2*m + p + 1)*Sin[e + f*x]), x], x] /; FreeQ[{a, b, e, f, g, p}, x] && EqQ[a^2 - b^2, 0] && LeQ[m, -2^(-1)] && NeQ[2*m + p + 1, 0]
  2878. Int[Times[Power[Times[cos[Plus[Optional[Pattern[e, Blank[]]], Times[Optional[Pattern[f, Blank[]]], Pattern[x, Blank[]]]]], Optional[Pattern[g, Blank[]]]], Pattern[p, Blank[]]], Power[sin[Plus[Optional[Pattern[e, Blank[]]], Times[Optional[Pattern[f, Blank[]]], Pattern[x, Blank[]]]]], 2], Power[Plus[Pattern[a, Blank[]], Times[Optional[Pattern[b, Blank[]]], sin[Plus[Optional[Pattern[e, Blank[]]], Times[Optional[Pattern[f, Blank[]]], Pattern[x, Blank[]]]]]]], Pattern[m, Blank[]]]], Pattern[x, Blank[Symbol]]] := -Simp[((g*Cos[e + f*x])^(p + 1)*(a + b*Sin[e + f*x])^(m + 1))/(b*f*g*(m + p + 2)), x] + Dist[1/(b*(m + p + 2)), Int[(g*Cos[e + f*x])^p*(a + b*Sin[e + f*x])^m*(b*(m + 1) - a*(p + 1)*Sin[e + f*x]), x], x] /; FreeQ[{a, b, e, f, g, m, p}, x] && EqQ[a^2 - b^2, 0] && NeQ[m + p + 2, 0]
  2879. Int[Times[Power[cos[Plus[Optional[Pattern[e, Blank[]]], Times[Optional[Pattern[f, Blank[]]], Pattern[x, Blank[]]]]], 2], Power[Times[Optional[Pattern[d, Blank[]]], sin[Plus[Optional[Pattern[e, Blank[]]], Times[Optional[Pattern[f, Blank[]]], Pattern[x, Blank[]]]]]], Pattern[n, Blank[]]], Power[Plus[Pattern[a, Blank[]], Times[Optional[Pattern[b, Blank[]]], sin[Plus[Optional[Pattern[e, Blank[]]], Times[Optional[Pattern[f, Blank[]]], Pattern[x, Blank[]]]]]]], Pattern[m, Blank[]]]], Pattern[x, Blank[Symbol]]] := Dist[1/b^2, Int[(d*Sin[e + f*x])^n*(a + b*Sin[e + f*x])^(m + 1)*(a - b*Sin[e + f*x]), x], x] /; FreeQ[{a, b, d, e, f, m, n}, x] && EqQ[a^2 - b^2, 0] && IntegersQ[2*m, 2*n]
  2880. Int[Times[Power[cos[Plus[Optional[Pattern[e, Blank[]]], Times[Optional[Pattern[f, Blank[]]], Pattern[x, Blank[]]]]], 4], Power[Times[Optional[Pattern[d, Blank[]]], sin[Plus[Optional[Pattern[e, Blank[]]], Times[Optional[Pattern[f, Blank[]]], Pattern[x, Blank[]]]]]], Pattern[n, Blank[]]], Power[Plus[Pattern[a, Blank[]], Times[Optional[Pattern[b, Blank[]]], sin[Plus[Optional[Pattern[e, Blank[]]], Times[Optional[Pattern[f, Blank[]]], Pattern[x, Blank[]]]]]]], Pattern[m, Blank[]]]], Pattern[x, Blank[Symbol]]] := Dist[-2/(a*b*d), Int[(d*Sin[e + f*x])^(n + 1)*(a + b*Sin[e + f*x])^(m + 2), x], x] + Dist[1/a^2, Int[(d*Sin[e + f*x])^n*(a + b*Sin[e + f*x])^(m + 2)*(1 + Sin[e + f*x]^2), x], x] /; FreeQ[{a, b, d, e, f, n}, x] && EqQ[a^2 - b^2, 0] && LtQ[m, -1]
  2881. Int[Times[Power[cos[Plus[Optional[Pattern[e, Blank[]]], Times[Optional[Pattern[f, Blank[]]], Pattern[x, Blank[]]]]], 4], Power[Times[Optional[Pattern[d, Blank[]]], sin[Plus[Optional[Pattern[e, Blank[]]], Times[Optional[Pattern[f, Blank[]]], Pattern[x, Blank[]]]]]], Pattern[n, Blank[]]], Power[Plus[Pattern[a, Blank[]], Times[Optional[Pattern[b, Blank[]]], sin[Plus[Optional[Pattern[e, Blank[]]], Times[Optional[Pattern[f, Blank[]]], Pattern[x, Blank[]]]]]]], Pattern[m, Blank[]]]], Pattern[x, Blank[Symbol]]] := Dist[1/d^4, Int[(d*Sin[e + f*x])^(n + 4)*(a + b*Sin[e + f*x])^m, x], x] + Int[(d*Sin[e + f*x])^n*(a + b*Sin[e + f*x])^m*(1 - 2*Sin[e + f*x]^2), x] /; FreeQ[{a, b, d, e, f, m, n}, x] && EqQ[a^2 - b^2, 0] && !IGtQ[m, 0]
  2882. Int[Times[Power[cos[Plus[Optional[Pattern[e, Blank[]]], Times[Optional[Pattern[f, Blank[]]], Pattern[x, Blank[]]]]], Pattern[p, Blank[]]], Power[Times[Optional[Pattern[d, Blank[]]], sin[Plus[Optional[Pattern[e, Blank[]]], Times[Optional[Pattern[f, Blank[]]], Pattern[x, Blank[]]]]]], Pattern[n, Blank[]]], Power[Plus[Pattern[a, Blank[]], Times[Optional[Pattern[b, Blank[]]], sin[Plus[Optional[Pattern[e, Blank[]]], Times[Optional[Pattern[f, Blank[]]], Pattern[x, Blank[]]]]]]], Pattern[m, Blank[]]]], Pattern[x, Blank[Symbol]]] := Dist[(a^m*Cos[e + f*x])/(f*Sqrt[1 + Sin[e + f*x]]*Sqrt[1 - Sin[e + f*x]]), Subst[Int[(d*x)^n*(1 + (b*x)/a)^(m + (p - 1)/2)*(1 - (b*x)/a)^((p - 1)/2), x], x, Sin[e + f*x]], x] /; FreeQ[{a, b, d, e, f, n}, x] && EqQ[a^2 - b^2, 0] && IntegerQ[p/2] && IntegerQ[m]
  2883. Int[Times[Power[cos[Plus[Optional[Pattern[e, Blank[]]], Times[Optional[Pattern[f, Blank[]]], Pattern[x, Blank[]]]]], Pattern[p, Blank[]]], Power[Times[Optional[Pattern[d, Blank[]]], sin[Plus[Optional[Pattern[e, Blank[]]], Times[Optional[Pattern[f, Blank[]]], Pattern[x, Blank[]]]]]], Pattern[n, Blank[]]], Power[Plus[Pattern[a, Blank[]], Times[Optional[Pattern[b, Blank[]]], sin[Plus[Optional[Pattern[e, Blank[]]], Times[Optional[Pattern[f, Blank[]]], Pattern[x, Blank[]]]]]]], Pattern[m, Blank[]]]], Pattern[x, Blank[Symbol]]] := Dist[Cos[e + f*x]/(a^(p - 2)*f*Sqrt[a + b*Sin[e + f*x]]*Sqrt[a - b*Sin[e + f*x]]), Subst[Int[(d*x)^n*(a + b*x)^(m + p/2 - 1/2)*(a - b*x)^(p/2 - 1/2), x], x, Sin[e + f*x]], x] /; FreeQ[{a, b, d, e, f, m, n}, x] && EqQ[a^2 - b^2, 0] && IntegerQ[p/2] && !IntegerQ[m]
  2884. Int[Times[Power[Times[cos[Plus[Optional[Pattern[e, Blank[]]], Times[Optional[Pattern[f, Blank[]]], Pattern[x, Blank[]]]]], Optional[Pattern[g, Blank[]]]], Pattern[p, Blank[]]], Power[Times[Optional[Pattern[d, Blank[]]], sin[Plus[Optional[Pattern[e, Blank[]]], Times[Optional[Pattern[f, Blank[]]], Pattern[x, Blank[]]]]]], Pattern[n, Blank[]]], Power[Plus[Pattern[a, Blank[]], Times[Optional[Pattern[b, Blank[]]], sin[Plus[Optional[Pattern[e, Blank[]]], Times[Optional[Pattern[f, Blank[]]], Pattern[x, Blank[]]]]]]], Pattern[m, Blank[]]]], Pattern[x, Blank[Symbol]]] := Int[ExpandTrig[(g*cos[e + f*x])^p, (d*sin[e + f*x])^n*(a + b*sin[e + f*x])^m, x], x] /; FreeQ[{a, b, d, e, f, g, n, p}, x] && EqQ[a^2 - b^2, 0] && IGtQ[m, 0] && (IntegerQ[p] || IGtQ[n, 0])
  2885. Int[Times[Power[Times[cos[Plus[Optional[Pattern[e, Blank[]]], Times[Optional[Pattern[f, Blank[]]], Pattern[x, Blank[]]]]], Optional[Pattern[g, Blank[]]]], Pattern[p, Blank[]]], Power[Times[Optional[Pattern[d, Blank[]]], sin[Plus[Optional[Pattern[e, Blank[]]], Times[Optional[Pattern[f, Blank[]]], Pattern[x, Blank[]]]]]], Pattern[n, Blank[]]], Power[Plus[Pattern[a, Blank[]], Times[Optional[Pattern[b, Blank[]]], sin[Plus[Optional[Pattern[e, Blank[]]], Times[Optional[Pattern[f, Blank[]]], Pattern[x, Blank[]]]]]]], Pattern[m, Blank[]]]], Pattern[x, Blank[Symbol]]] := Dist[(a^m*g*(g*Cos[e + f*x])^(p - 1))/(f*(1 + Sin[e + f*x])^((p - 1)/2)*(1 - Sin[e + f*x])^((p - 1)/2)), Subst[Int[(d*x)^n*(1 + (b*x)/a)^(m + (p - 1)/2)*(1 - (b*x)/a)^((p - 1)/2), x], x, Sin[e + f*x]], x] /; FreeQ[{a, b, d, e, f, n, p}, x] && EqQ[a^2 - b^2, 0] && IntegerQ[m]
  2886. Int[Times[Power[Times[cos[Plus[Optional[Pattern[e, Blank[]]], Times[Optional[Pattern[f, Blank[]]], Pattern[x, Blank[]]]]], Optional[Pattern[g, Blank[]]]], Pattern[p, Blank[]]], Power[Times[Optional[Pattern[d, Blank[]]], sin[Plus[Optional[Pattern[e, Blank[]]], Times[Optional[Pattern[f, Blank[]]], Pattern[x, Blank[]]]]]], Pattern[n, Blank[]]], Power[Plus[Pattern[a, Blank[]], Times[Optional[Pattern[b, Blank[]]], sin[Plus[Optional[Pattern[e, Blank[]]], Times[Optional[Pattern[f, Blank[]]], Pattern[x, Blank[]]]]]]], Pattern[m, Blank[]]]], Pattern[x, Blank[Symbol]]] := Dist[(g*(g*Cos[e + f*x])^(p - 1))/(f*(a + b*Sin[e + f*x])^((p - 1)/2)*(a - b*Sin[e + f*x])^((p - 1)/2)), Subst[Int[(d*x)^n*(a + b*x)^(m + (p - 1)/2)*(a - b*x)^((p - 1)/2), x], x, Sin[e + f*x]], x] /; FreeQ[{a, b, d, e, f, m, n, p}, x] && EqQ[a^2 - b^2, 0] && !IntegerQ[m]
  2887. Int[Times[Power[Times[cos[Plus[Optional[Pattern[e, Blank[]]], Times[Optional[Pattern[f, Blank[]]], Pattern[x, Blank[]]]]], Optional[Pattern[g, Blank[]]]], Pattern[p, Blank[]]], Power[Times[Optional[Pattern[d, Blank[]]], sin[Plus[Optional[Pattern[e, Blank[]]], Times[Optional[Pattern[f, Blank[]]], Pattern[x, Blank[]]]]]], Rational[-1, 2]], Power[Plus[Pattern[a, Blank[]], Times[Optional[Pattern[b, Blank[]]], sin[Plus[Optional[Pattern[e, Blank[]]], Times[Optional[Pattern[f, Blank[]]], Pattern[x, Blank[]]]]]]], Pattern[m, Blank[]]]], Pattern[x, Blank[Symbol]]] := -Simp[(g*(g*Cos[e + f*x])^(p - 1)*Sqrt[d*Sin[e + f*x]]*(a + b*Sin[e + f*x])^(m + 1))/(a*d*f*(m + 1)), x] + Dist[(g^2*(2*m + 3))/(2*a*(m + 1)), Int[((g*Cos[e + f*x])^(p - 2)*(a + b*Sin[e + f*x])^(m + 1))/Sqrt[d*Sin[e + f*x]], x], x] /; FreeQ[{a, b, d, e, f, g}, x] && NeQ[a^2 - b^2, 0] && LtQ[m, -1] && EqQ[m + p + 1/2, 0]
  2888. Int[Times[Power[Times[cos[Plus[Optional[Pattern[e, Blank[]]], Times[Optional[Pattern[f, Blank[]]], Pattern[x, Blank[]]]]], Optional[Pattern[g, Blank[]]]], Pattern[p, Blank[]]], Power[Times[Optional[Pattern[d, Blank[]]], sin[Plus[Optional[Pattern[e, Blank[]]], Times[Optional[Pattern[f, Blank[]]], Pattern[x, Blank[]]]]]], Rational[-1, 2]], Power[Plus[Pattern[a, Blank[]], Times[Optional[Pattern[b, Blank[]]], sin[Plus[Optional[Pattern[e, Blank[]]], Times[Optional[Pattern[f, Blank[]]], Pattern[x, Blank[]]]]]]], Pattern[m, Blank[]]]], Pattern[x, Blank[Symbol]]] := Simp[(2*(g*Cos[e + f*x])^(p + 1)*Sqrt[d*Sin[e + f*x]]*(a + b*Sin[e + f*x])^m)/(d*f*g*(2*m + 1)), x] + Dist[(2*a*m)/(g^2*(2*m + 1)), Int[((g*Cos[e + f*x])^(p + 2)*(a + b*Sin[e + f*x])^(m - 1))/Sqrt[d*Sin[e + f*x]], x], x] /; FreeQ[{a, b, e, f, g}, x] && NeQ[a^2 - b^2, 0] && GtQ[m, 0] && EqQ[m + p + 3/2, 0]
  2889. Int[Times[Power[cos[Plus[Optional[Pattern[e, Blank[]]], Times[Optional[Pattern[f, Blank[]]], Pattern[x, Blank[]]]]], 2], Power[Times[Optional[Pattern[d, Blank[]]], sin[Plus[Optional[Pattern[e, Blank[]]], Times[Optional[Pattern[f, Blank[]]], Pattern[x, Blank[]]]]]], Pattern[n, Blank[]]], Power[Plus[Pattern[a, Blank[]], Times[Optional[Pattern[b, Blank[]]], sin[Plus[Optional[Pattern[e, Blank[]]], Times[Optional[Pattern[f, Blank[]]], Pattern[x, Blank[]]]]]]], Pattern[m, Blank[]]]], Pattern[x, Blank[Symbol]]] := Int[(d*Sin[e + f*x])^n*(a + b*Sin[e + f*x])^m*(1 - Sin[e + f*x]^2), x] /; FreeQ[{a, b, d, e, f, m, n}, x] && NeQ[a^2 - b^2, 0] && (IGtQ[m, 0] || IntegersQ[2*m, 2*n])
  2890. Int[Times[Power[cos[Plus[Optional[Pattern[e, Blank[]]], Times[Optional[Pattern[f, Blank[]]], Pattern[x, Blank[]]]]], 4], Power[Times[Optional[Pattern[d, Blank[]]], sin[Plus[Optional[Pattern[e, Blank[]]], Times[Optional[Pattern[f, Blank[]]], Pattern[x, Blank[]]]]]], Pattern[n, Blank[]]], Power[Plus[Pattern[a, Blank[]], Times[Optional[Pattern[b, Blank[]]], sin[Plus[Optional[Pattern[e, Blank[]]], Times[Optional[Pattern[f, Blank[]]], Pattern[x, Blank[]]]]]]], Pattern[m, Blank[]]]], Pattern[x, Blank[Symbol]]] := Simp[(Cos[e + f*x]*(d*Sin[e + f*x])^(n + 1)*(a + b*Sin[e + f*x])^(m + 1))/(a*d*f*(n + 1)), x] + (Dist[1/(a^2*b*d*(n + 1)*(m + 1)), Int[(d*Sin[e + f*x])^(n + 1)*(a + b*Sin[e + f*x])^(m + 1)*Simp[a^2*(n + 1)*(n + 2) - b^2*(m + n + 2)*(m + n + 3) + a*b*(m + 1)*Sin[e + f*x] - (a^2*(n + 1)*(n + 3) - b^2*(m + n + 2)*(m + n + 4))*Sin[e + f*x]^2, x], x], x] - Simp[((a^2*(n + 1) - b^2*(m + n + 2))*Cos[e + f*x]*(d*Sin[e + f*x])^(n + 2)*(a + b*Sin[e + f*x])^(m + 1))/(a^2*b*d^2*f*(n + 1)*(m + 1)), x]) /; FreeQ[{a, b, d, e, f}, x] && NeQ[a^2 - b^2, 0] && IntegersQ[2*m, 2*n] && LtQ[m, -1] && LtQ[n, -1]
  2891. Int[Times[Power[cos[Plus[Optional[Pattern[e, Blank[]]], Times[Optional[Pattern[f, Blank[]]], Pattern[x, Blank[]]]]], 4], Power[Times[Optional[Pattern[d, Blank[]]], sin[Plus[Optional[Pattern[e, Blank[]]], Times[Optional[Pattern[f, Blank[]]], Pattern[x, Blank[]]]]]], Pattern[n, Blank[]]], Power[Plus[Pattern[a, Blank[]], Times[Optional[Pattern[b, Blank[]]], sin[Plus[Optional[Pattern[e, Blank[]]], Times[Optional[Pattern[f, Blank[]]], Pattern[x, Blank[]]]]]]], Pattern[m, Blank[]]]], Pattern[x, Blank[Symbol]]] := Simp[((a^2 - b^2)*Cos[e + f*x]*(a + b*Sin[e + f*x])^(m + 1)*(d*Sin[e + f*x])^(n + 1))/(a*b^2*d*f*(m + 1)), x] + (-Dist[1/(a^2*b^2*(m + 1)*(m + 2)), Int[(a + b*Sin[e + f*x])^(m + 2)*(d*Sin[e + f*x])^n*Simp[a^2*(n + 1)*(n + 3) - b^2*(m + n + 2)*(m + n + 3) + a*b*(m + 2)*Sin[e + f*x] - (a^2*(n + 2)*(n + 3) - b^2*(m + n + 2)*(m + n + 4))*Sin[e + f*x]^2, x], x], x] + Simp[((a^2*(n - m + 1) - b^2*(m + n + 2))*Cos[e + f*x]*(a + b*Sin[e + f*x])^(m + 2)*(d*Sin[e + f*x])^(n + 1))/(a^2*b^2*d*f*(m + 1)*(m + 2)), x]) /; FreeQ[{a, b, d, e, f, n}, x] && NeQ[a^2 - b^2, 0] && IntegersQ[2*m, 2*n] && LtQ[m, -1] && !LtQ[n, -1] && (LtQ[m, -2] || EqQ[m + n + 4, 0])
  2892. Int[Times[Power[cos[Plus[Optional[Pattern[e, Blank[]]], Times[Optional[Pattern[f, Blank[]]], Pattern[x, Blank[]]]]], 4], Power[Times[Optional[Pattern[d, Blank[]]], sin[Plus[Optional[Pattern[e, Blank[]]], Times[Optional[Pattern[f, Blank[]]], Pattern[x, Blank[]]]]]], Pattern[n, Blank[]]], Power[Plus[Pattern[a, Blank[]], Times[Optional[Pattern[b, Blank[]]], sin[Plus[Optional[Pattern[e, Blank[]]], Times[Optional[Pattern[f, Blank[]]], Pattern[x, Blank[]]]]]]], Pattern[m, Blank[]]]], Pattern[x, Blank[Symbol]]] := Simp[((a^2 - b^2)*Cos[e + f*x]*(a + b*Sin[e + f*x])^(m + 1)*(d*Sin[e + f*x])^(n + 1))/(a*b^2*d*f*(m + 1)), x] + (-Dist[1/(a*b^2*(m + 1)*(m + n + 4)), Int[(a + b*Sin[e + f*x])^(m + 1)*(d*Sin[e + f*x])^n*Simp[a^2*(n + 1)*(n + 3) - b^2*(m + n + 2)*(m + n + 4) + a*b*(m + 1)*Sin[e + f*x] - (a^2*(n + 2)*(n + 3) - b^2*(m + n + 3)*(m + n + 4))*Sin[e + f*x]^2, x], x], x] - Simp[(Cos[e + f*x]*(a + b*Sin[e + f*x])^(m + 2)*(d*Sin[e + f*x])^(n + 1))/(b^2*d*f*(m + n + 4)), x]) /; FreeQ[{a, b, d, e, f, n}, x] && NeQ[a^2 - b^2, 0] && IntegersQ[2*m, 2*n] && LtQ[m, -1] && !LtQ[n, -1] && NeQ[m + n + 4, 0]
  2893. Int[Times[Power[cos[Plus[Optional[Pattern[e, Blank[]]], Times[Optional[Pattern[f, Blank[]]], Pattern[x, Blank[]]]]], 4], Power[Times[Optional[Pattern[d, Blank[]]], sin[Plus[Optional[Pattern[e, Blank[]]], Times[Optional[Pattern[f, Blank[]]], Pattern[x, Blank[]]]]]], Pattern[n, Blank[]]], Power[Plus[Pattern[a, Blank[]], Times[Optional[Pattern[b, Blank[]]], sin[Plus[Optional[Pattern[e, Blank[]]], Times[Optional[Pattern[f, Blank[]]], Pattern[x, Blank[]]]]]]], Pattern[m, Blank[]]]], Pattern[x, Blank[Symbol]]] := Simp[(Cos[e + f*x]*(a + b*Sin[e + f*x])^(m + 1)*(d*Sin[e + f*x])^(n + 1))/(a*d*f*(n + 1)), x] + (-Dist[1/(a^2*d^2*(n + 1)*(n + 2)), Int[(a + b*Sin[e + f*x])^m*(d*Sin[e + f*x])^(n + 2)*Simp[a^2*n*(n + 2) - b^2*(m + n + 2)*(m + n + 3) + a*b*m*Sin[e + f*x] - (a^2*(n + 1)*(n + 2) - b^2*(m + n + 2)*(m + n + 4))*Sin[e + f*x]^2, x], x], x] - Simp[(b*(m + n + 2)*Cos[e + f*x]*(a + b*Sin[e + f*x])^(m + 1)*(d*Sin[e + f*x])^(n + 2))/(a^2*d^2*f*(n + 1)*(n + 2)), x]) /; FreeQ[{a, b, d, e, f, m}, x] && NeQ[a^2 - b^2, 0] && (IGtQ[m, 0] || IntegersQ[2*m, 2*n]) && !m < -1 && LtQ[n, -1] && (LtQ[n, -2] || EqQ[m + n + 4, 0])
  2894. Int[Times[Power[cos[Plus[Optional[Pattern[e, Blank[]]], Times[Optional[Pattern[f, Blank[]]], Pattern[x, Blank[]]]]], 4], Power[Times[Optional[Pattern[d, Blank[]]], sin[Plus[Optional[Pattern[e, Blank[]]], Times[Optional[Pattern[f, Blank[]]], Pattern[x, Blank[]]]]]], Pattern[n, Blank[]]], Power[Plus[Pattern[a, Blank[]], Times[Optional[Pattern[b, Blank[]]], sin[Plus[Optional[Pattern[e, Blank[]]], Times[Optional[Pattern[f, Blank[]]], Pattern[x, Blank[]]]]]]], Pattern[m, Blank[]]]], Pattern[x, Blank[Symbol]]] := Simp[(Cos[e + f*x]*(a + b*Sin[e + f*x])^(m + 1)*(d*Sin[e + f*x])^(n + 1))/(a*d*f*(n + 1)), x] + (Dist[1/(a*b*d*(n + 1)*(m + n + 4)), Int[(a + b*Sin[e + f*x])^m*(d*Sin[e + f*x])^(n + 1)*Simp[a^2*(n + 1)*(n + 2) - b^2*(m + n + 2)*(m + n + 4) + a*b*(m + 3)*Sin[e + f*x] - (a^2*(n + 1)*(n + 3) - b^2*(m + n + 3)*(m + n + 4))*Sin[e + f*x]^2, x], x], x] - Simp[(Cos[e + f*x]*(a + b*Sin[e + f*x])^(m + 1)*(d*Sin[e + f*x])^(n + 2))/(b*d^2*f*(m + n + 4)), x]) /; FreeQ[{a, b, d, e, f, m}, x] && NeQ[a^2 - b^2, 0] && (IGtQ[m, 0] || IntegersQ[2*m, 2*n]) && !m < -1 && LtQ[n, -1] && NeQ[m + n + 4, 0]
  2895. Int[Times[Power[cos[Plus[Optional[Pattern[e, Blank[]]], Times[Optional[Pattern[f, Blank[]]], Pattern[x, Blank[]]]]], 4], Power[Times[Optional[Pattern[d, Blank[]]], sin[Plus[Optional[Pattern[e, Blank[]]], Times[Optional[Pattern[f, Blank[]]], Pattern[x, Blank[]]]]]], Pattern[n, Blank[]]], Power[Plus[Pattern[a, Blank[]], Times[Optional[Pattern[b, Blank[]]], sin[Plus[Optional[Pattern[e, Blank[]]], Times[Optional[Pattern[f, Blank[]]], Pattern[x, Blank[]]]]]]], Pattern[m, Blank[]]]], Pattern[x, Blank[Symbol]]] := Simp[(a*(n + 3)*Cos[e + f*x]*(d*Sin[e + f*x])^(n + 1)*(a + b*Sin[e + f*x])^(m + 1))/(b^2*d*f*(m + n + 3)*(m + n + 4)), x] + (-Dist[1/(b^2*(m + n + 3)*(m + n + 4)), Int[(d*Sin[e + f*x])^n*(a + b*Sin[e + f*x])^m*Simp[a^2*(n + 1)*(n + 3) - b^2*(m + n + 3)*(m + n + 4) + a*b*m*Sin[e + f*x] - (a^2*(n + 2)*(n + 3) - b^2*(m + n + 3)*(m + n + 5))*Sin[e + f*x]^2, x], x], x] - Simp[(Cos[e + f*x]*(d*Sin[e + f*x])^(n + 2)*(a + b*Sin[e + f*x])^(m + 1))/(b*d^2*f*(m + n + 4)), x]) /; FreeQ[{a, b, d, e, f, m, n}, x] && NeQ[a^2 - b^2, 0] && (IGtQ[m, 0] || IntegersQ[2*m, 2*n]) && !m < -1 && !LtQ[n, -1] && NeQ[m + n + 3, 0] && NeQ[m + n + 4, 0]
  2896. Int[Times[Power[cos[Plus[Optional[Pattern[e, Blank[]]], Times[Optional[Pattern[f, Blank[]]], Pattern[x, Blank[]]]]], 6], Power[Times[Optional[Pattern[d, Blank[]]], sin[Plus[Optional[Pattern[e, Blank[]]], Times[Optional[Pattern[f, Blank[]]], Pattern[x, Blank[]]]]]], Pattern[n, Blank[]]], Power[Plus[Pattern[a, Blank[]], Times[Optional[Pattern[b, Blank[]]], sin[Plus[Optional[Pattern[e, Blank[]]], Times[Optional[Pattern[f, Blank[]]], Pattern[x, Blank[]]]]]]], Pattern[m, Blank[]]]], Pattern[x, Blank[Symbol]]] := Simp[(Cos[e + f*x]*(d*Sin[e + f*x])^(n + 1)*(a + b*Sin[e + f*x])^(m + 1))/(a*d*f*(n + 1)), x] + (Dist[1/(a^2*b^2*d^2*(n + 1)*(n + 2)*(m + n + 5)*(m + n + 6)), Int[(d*Sin[e + f*x])^(n + 2)*(a + b*Sin[e + f*x])^m*Simp[a^4*(n + 1)*(n + 2)*(n + 3)*(n + 5) - a^2*b^2*(n + 2)*(2*n + 1)*(m + n + 5)*(m + n + 6) + b^4*(m + n + 2)*(m + n + 3)*(m + n + 5)*(m + n + 6) + a*b*m*(a^2*(n + 1)*(n + 2) - b^2*(m + n + 5)*(m + n + 6))*Sin[e + f*x] - (a^4*(n + 1)*(n + 2)*(4 + n)*(n + 5) + b^4*(m + n + 2)*(m + n + 4)*(m + n + 5)*(m + n + 6) - a^2*b^2*(n + 1)*(n + 2)*(m + n + 5)*(2*n + 2*m + 13))*Sin[e + f*x]^2, x], x], x] - Simp[(b*(m + n + 2)*Cos[e + f*x]*(d*Sin[e + f*x])^(n + 2)*(a + b*Sin[e + f*x])^(m + 1))/(a^2*d^2*f*(n + 1)*(n + 2)), x] - Simp[(a*(n + 5)*Cos[e + f*x]*(d*Sin[e + f*x])^(n + 3)*(a + b*Sin[e + f*x])^(m + 1))/(b^2*d^3*f*(m + n + 5)*(m + n + 6)), x] + Simp[(Cos[e + f*x]*(d*Sin[e + f*x])^(n + 4)*(a + b*Sin[e + f*x])^(m + 1))/(b*d^4*f*(m + n + 6)), x]) /; FreeQ[{a, b, d, e, f, m, n}, x] && NeQ[a^2 - b^2, 0] && IntegersQ[2*m, 2*n] && NeQ[n, -1] && NeQ[n, -2] && NeQ[m + n + 5, 0] && NeQ[m + n + 6, 0] && !IGtQ[m, 0]
  2897. Int[Times[Power[cos[Plus[Optional[Pattern[e, Blank[]]], Times[Optional[Pattern[f, Blank[]]], Pattern[x, Blank[]]]]], Pattern[p, Blank[]]], Power[Times[Optional[Pattern[d, Blank[]]], sin[Plus[Optional[Pattern[e, Blank[]]], Times[Optional[Pattern[f, Blank[]]], Pattern[x, Blank[]]]]]], Pattern[n, Blank[]]], Power[Plus[Pattern[a, Blank[]], Times[Optional[Pattern[b, Blank[]]], sin[Plus[Optional[Pattern[e, Blank[]]], Times[Optional[Pattern[f, Blank[]]], Pattern[x, Blank[]]]]]]], Pattern[m, Blank[]]]], Pattern[x, Blank[Symbol]]] := Int[ExpandTrig[(d*sin[e + f*x])^n*(a + b*sin[e + f*x])^m*(1 - sin[e + f*x]^2)^(p/2), x], x] /; FreeQ[{a, b, d, e, f}, x] && NeQ[a^2 - b^2, 0] && IntegersQ[m, 2*n, p/2] && (LtQ[m, -1] || (EqQ[m, -1] && GtQ[p, 0]))
  2898. Int[Times[Power[Times[cos[Plus[Optional[Pattern[e, Blank[]]], Times[Optional[Pattern[f, Blank[]]], Pattern[x, Blank[]]]]], Optional[Pattern[g, Blank[]]]], Pattern[p, Blank[]]], Power[sin[Plus[Optional[Pattern[e, Blank[]]], Times[Optional[Pattern[f, Blank[]]], Pattern[x, Blank[]]]]], Pattern[n, Blank[]]], Power[Plus[Pattern[a, Blank[]], Times[Optional[Pattern[b, Blank[]]], sin[Plus[Optional[Pattern[e, Blank[]]], Times[Optional[Pattern[f, Blank[]]], Pattern[x, Blank[]]]]]]], -1]], Pattern[x, Blank[Symbol]]] := Int[ExpandTrig[(g*cos[e + f*x])^p, sin[e + f*x]^n/(a + b*sin[e + f*x]), x], x] /; FreeQ[{a, b, e, f, g, p}, x] && NeQ[a^2 - b^2, 0] && IntegerQ[n] && (LtQ[n, 0] || IGtQ[p + 1/2, 0])
  2899. Int[Times[Power[Times[cos[Plus[Optional[Pattern[e, Blank[]]], Times[Optional[Pattern[f, Blank[]]], Pattern[x, Blank[]]]]], Optional[Pattern[g, Blank[]]]], Pattern[p, Blank[]]], Power[Times[Optional[Pattern[d, Blank[]]], sin[Plus[Optional[Pattern[e, Blank[]]], Times[Optional[Pattern[f, Blank[]]], Pattern[x, Blank[]]]]]], Pattern[n, Blank[]]], Power[Plus[Pattern[a, Blank[]], Times[Optional[Pattern[b, Blank[]]], sin[Plus[Optional[Pattern[e, Blank[]]], Times[Optional[Pattern[f, Blank[]]], Pattern[x, Blank[]]]]]]], -1]], Pattern[x, Blank[Symbol]]] := Dist[g^2/a, Int[(g*Cos[e + f*x])^(p - 2)*(d*Sin[e + f*x])^n, x], x] + (-Dist[(b*g^2)/(a^2*d), Int[(g*Cos[e + f*x])^(p - 2)*(d*Sin[e + f*x])^(n + 1), x], x] - Dist[(g^2*(a^2 - b^2))/(a^2*d^2), Int[((g*Cos[e + f*x])^(p - 2)*(d*Sin[e + f*x])^(n + 2))/(a + b*Sin[e + f*x]), x], x]) /; FreeQ[{a, b, d, e, f, g}, x] && NeQ[a^2 - b^2, 0] && IntegersQ[2*n, 2*p] && GtQ[p, 1] && (LeQ[n, -2] || (EqQ[n, -3/2] && EqQ[p, 3/2]))
  2900. Int[Times[Power[Times[cos[Plus[Optional[Pattern[e, Blank[]]], Times[Optional[Pattern[f, Blank[]]], Pattern[x, Blank[]]]]], Optional[Pattern[g, Blank[]]]], Pattern[p, Blank[]]], Power[Times[Optional[Pattern[d, Blank[]]], sin[Plus[Optional[Pattern[e, Blank[]]], Times[Optional[Pattern[f, Blank[]]], Pattern[x, Blank[]]]]]], Pattern[n, Blank[]]], Power[Plus[Pattern[a, Blank[]], Times[Optional[Pattern[b, Blank[]]], sin[Plus[Optional[Pattern[e, Blank[]]], Times[Optional[Pattern[f, Blank[]]], Pattern[x, Blank[]]]]]]], -1]], Pattern[x, Blank[Symbol]]] := Dist[g^2/(a*b), Int[(g*Cos[e + f*x])^(p - 2)*(d*Sin[e + f*x])^n*(b - a*Sin[e + f*x]), x], x] + Dist[(g^2*(a^2 - b^2))/(a*b*d), Int[((g*Cos[e + f*x])^(p - 2)*(d*Sin[e + f*x])^(n + 1))/(a + b*Sin[e + f*x]), x], x] /; FreeQ[{a, b, d, e, f, g}, x] && NeQ[a^2 - b^2, 0] && IntegersQ[2*n, 2*p] && GtQ[p, 1] && (LtQ[n, -1] || (EqQ[p, 3/2] && EqQ[n, -2^(-1)]))
  2901. Int[Times[Power[Times[cos[Plus[Optional[Pattern[e, Blank[]]], Times[Optional[Pattern[f, Blank[]]], Pattern[x, Blank[]]]]], Optional[Pattern[g, Blank[]]]], Pattern[p, Blank[]]], Power[Times[Optional[Pattern[d, Blank[]]], sin[Plus[Optional[Pattern[e, Blank[]]], Times[Optional[Pattern[f, Blank[]]], Pattern[x, Blank[]]]]]], Pattern[n, Blank[]]], Power[Plus[Pattern[a, Blank[]], Times[Optional[Pattern[b, Blank[]]], sin[Plus[Optional[Pattern[e, Blank[]]], Times[Optional[Pattern[f, Blank[]]], Pattern[x, Blank[]]]]]]], -1]], Pattern[x, Blank[Symbol]]] := Dist[g^2/b^2, Int[(g*Cos[e + f*x])^(p - 2)*(d*Sin[e + f*x])^n*(a - b*Sin[e + f*x]), x], x] - Dist[(g^2*(a^2 - b^2))/b^2, Int[((g*Cos[e + f*x])^(p - 2)*(d*Sin[e + f*x])^n)/(a + b*Sin[e + f*x]), x], x] /; FreeQ[{a, b, d, e, f, g}, x] && NeQ[a^2 - b^2, 0] && IntegersQ[2*n, 2*p] && GtQ[p, 1]
  2902. Int[Times[Power[Times[cos[Plus[Optional[Pattern[e, Blank[]]], Times[Optional[Pattern[f, Blank[]]], Pattern[x, Blank[]]]]], Optional[Pattern[g, Blank[]]]], Pattern[p, Blank[]]], Power[Times[Optional[Pattern[d, Blank[]]], sin[Plus[Optional[Pattern[e, Blank[]]], Times[Optional[Pattern[f, Blank[]]], Pattern[x, Blank[]]]]]], Pattern[n, Blank[]]], Power[Plus[Pattern[a, Blank[]], Times[Optional[Pattern[b, Blank[]]], sin[Plus[Optional[Pattern[e, Blank[]]], Times[Optional[Pattern[f, Blank[]]], Pattern[x, Blank[]]]]]]], -1]], Pattern[x, Blank[Symbol]]] := Dist[(a*d^2)/(a^2 - b^2), Int[(g*Cos[e + f*x])^p*(d*Sin[e + f*x])^(n - 2), x], x] + (-Dist[(b*d)/(a^2 - b^2), Int[(g*Cos[e + f*x])^p*(d*Sin[e + f*x])^(n - 1), x], x] - Dist[(a^2*d^2)/(g^2*(a^2 - b^2)), Int[((g*Cos[e + f*x])^(p + 2)*(d*Sin[e + f*x])^(n - 2))/(a + b*Sin[e + f*x]), x], x]) /; FreeQ[{a, b, d, e, f, g}, x] && NeQ[a^2 - b^2, 0] && IntegersQ[2*n, 2*p] && LtQ[p, -1] && GtQ[n, 1]
  2903. Int[Times[Power[Times[cos[Plus[Optional[Pattern[e, Blank[]]], Times[Optional[Pattern[f, Blank[]]], Pattern[x, Blank[]]]]], Optional[Pattern[g, Blank[]]]], Pattern[p, Blank[]]], Power[Times[Optional[Pattern[d, Blank[]]], sin[Plus[Optional[Pattern[e, Blank[]]], Times[Optional[Pattern[f, Blank[]]], Pattern[x, Blank[]]]]]], Pattern[n, Blank[]]], Power[Plus[Pattern[a, Blank[]], Times[Optional[Pattern[b, Blank[]]], sin[Plus[Optional[Pattern[e, Blank[]]], Times[Optional[Pattern[f, Blank[]]], Pattern[x, Blank[]]]]]]], -1]], Pattern[x, Blank[Symbol]]] := -Dist[d/(a^2 - b^2), Int[(g*Cos[e + f*x])^p*(d*Sin[e + f*x])^(n - 1)*(b - a*Sin[e + f*x]), x], x] + Dist[(a*b*d)/(g^2*(a^2 - b^2)), Int[((g*Cos[e + f*x])^(p + 2)*(d*Sin[e + f*x])^(n - 1))/(a + b*Sin[e + f*x]), x], x] /; FreeQ[{a, b, d, e, f, g}, x] && NeQ[a^2 - b^2, 0] && IntegersQ[2*n, 2*p] && LtQ[p, -1] && GtQ[n, 0]
  2904. Int[Times[Power[Times[cos[Plus[Optional[Pattern[e, Blank[]]], Times[Optional[Pattern[f, Blank[]]], Pattern[x, Blank[]]]]], Optional[Pattern[g, Blank[]]]], Pattern[p, Blank[]]], Power[Times[Optional[Pattern[d, Blank[]]], sin[Plus[Optional[Pattern[e, Blank[]]], Times[Optional[Pattern[f, Blank[]]], Pattern[x, Blank[]]]]]], Pattern[n, Blank[]]], Power[Plus[Pattern[a, Blank[]], Times[Optional[Pattern[b, Blank[]]], sin[Plus[Optional[Pattern[e, Blank[]]], Times[Optional[Pattern[f, Blank[]]], Pattern[x, Blank[]]]]]]], -1]], Pattern[x, Blank[Symbol]]] := Dist[1/(a^2 - b^2), Int[(g*Cos[e + f*x])^p*(d*Sin[e + f*x])^n*(a - b*Sin[e + f*x]), x], x] - Dist[b^2/(g^2*(a^2 - b^2)), Int[((g*Cos[e + f*x])^(p + 2)*(d*Sin[e + f*x])^n)/(a + b*Sin[e + f*x]), x], x] /; FreeQ[{a, b, d, e, f, g}, x] && NeQ[a^2 - b^2, 0] && IntegersQ[2*n, 2*p] && LtQ[p, -1]
  2905. Int[Times[Power[Times[cos[Plus[Optional[Pattern[e, Blank[]]], Times[Optional[Pattern[f, Blank[]]], Pattern[x, Blank[]]]]], Optional[Pattern[g, Blank[]]]], Rational[1, 2]], Power[sin[Plus[Optional[Pattern[e, Blank[]]], Times[Optional[Pattern[f, Blank[]]], Pattern[x, Blank[]]]]], Rational[-1, 2]], Power[Plus[Pattern[a, Blank[]], Times[Optional[Pattern[b, Blank[]]], sin[Plus[Optional[Pattern[e, Blank[]]], Times[Optional[Pattern[f, Blank[]]], Pattern[x, Blank[]]]]]]], -1]], Pattern[x, Blank[Symbol]]] := Dist[(-4*Sqrt[2]*g)/f, Subst[Int[x^2/(((a + b)*g^2 + (a - b)*x^4)*Sqrt[1 - x^4/g^2]), x], x, Sqrt[g*Cos[e + f*x]]/Sqrt[1 + Sin[e + f*x]]], x] /; FreeQ[{a, b, e, f, g}, x] && NeQ[a^2 - b^2, 0]
  2906. Int[Times[Power[Times[cos[Plus[Optional[Pattern[e, Blank[]]], Times[Optional[Pattern[f, Blank[]]], Pattern[x, Blank[]]]]], Optional[Pattern[g, Blank[]]]], Rational[1, 2]], Power[Times[Pattern[d, Blank[]], sin[Plus[Optional[Pattern[e, Blank[]]], Times[Optional[Pattern[f, Blank[]]], Pattern[x, Blank[]]]]]], Rational[-1, 2]], Power[Plus[Pattern[a, Blank[]], Times[Optional[Pattern[b, Blank[]]], sin[Plus[Optional[Pattern[e, Blank[]]], Times[Optional[Pattern[f, Blank[]]], Pattern[x, Blank[]]]]]]], -1]], Pattern[x, Blank[Symbol]]] := Dist[Sqrt[Sin[e + f*x]]/Sqrt[d*Sin[e + f*x]], Int[Sqrt[g*Cos[e + f*x]]/(Sqrt[Sin[e + f*x]]*(a + b*Sin[e + f*x])), x], x] /; FreeQ[{a, b, d, e, f, g}, x] && NeQ[a^2 - b^2, 0]
  2907. Int[Times[Power[cos[Plus[Optional[Pattern[e, Blank[]]], Times[Optional[Pattern[f, Blank[]]], Pattern[x, Blank[]]]]], Rational[-1, 2]], Power[Times[Optional[Pattern[d, Blank[]]], sin[Plus[Optional[Pattern[e, Blank[]]], Times[Optional[Pattern[f, Blank[]]], Pattern[x, Blank[]]]]]], Rational[1, 2]], Power[Plus[Pattern[a, Blank[]], Times[Optional[Pattern[b, Blank[]]], sin[Plus[Optional[Pattern[e, Blank[]]], Times[Optional[Pattern[f, Blank[]]], Pattern[x, Blank[]]]]]]], -1]], Pattern[x, Blank[Symbol]]] := With[{q = Rt[-a^2 + b^2, 2]}, Dist[(2*Sqrt[2]*d*(b + q))/(f*q), Subst[Int[1/((d*(b + q) + a*x^2)*Sqrt[1 - x^4/d^2]), x], x, Sqrt[d*Sin[e + f*x]]/Sqrt[1 + Cos[e + f*x]]], x] - Dist[(2*Sqrt[2]*d*(b - q))/(f*q), Subst[Int[1/((d*(b - q) + a*x^2)*Sqrt[1 - x^4/d^2]), x], x, Sqrt[d*Sin[e + f*x]]/Sqrt[1 + Cos[e + f*x]]], x]] /; FreeQ[{a, b, d, e, f}, x] && NeQ[a^2 - b^2, 0]
  2908. Int[Times[Power[Times[cos[Plus[Optional[Pattern[e, Blank[]]], Times[Optional[Pattern[f, Blank[]]], Pattern[x, Blank[]]]]], Optional[Pattern[g, Blank[]]]], Rational[-1, 2]], Power[Times[Optional[Pattern[d, Blank[]]], sin[Plus[Optional[Pattern[e, Blank[]]], Times[Optional[Pattern[f, Blank[]]], Pattern[x, Blank[]]]]]], Rational[1, 2]], Power[Plus[Pattern[a, Blank[]], Times[Optional[Pattern[b, Blank[]]], sin[Plus[Optional[Pattern[e, Blank[]]], Times[Optional[Pattern[f, Blank[]]], Pattern[x, Blank[]]]]]]], -1]], Pattern[x, Blank[Symbol]]] := Dist[Sqrt[Cos[e + f*x]]/Sqrt[g*Cos[e + f*x]], Int[Sqrt[d*Sin[e + f*x]]/(Sqrt[Cos[e + f*x]]*(a + b*Sin[e + f*x])), x], x] /; FreeQ[{a, b, d, e, f, g}, x] && NeQ[a^2 - b^2, 0]
  2909. Int[Times[Power[Times[cos[Plus[Optional[Pattern[e, Blank[]]], Times[Optional[Pattern[f, Blank[]]], Pattern[x, Blank[]]]]], Optional[Pattern[g, Blank[]]]], Pattern[p, Blank[]]], Power[Times[Optional[Pattern[d, Blank[]]], sin[Plus[Optional[Pattern[e, Blank[]]], Times[Optional[Pattern[f, Blank[]]], Pattern[x, Blank[]]]]]], Pattern[n, Blank[]]], Power[Plus[Pattern[a, Blank[]], Times[Optional[Pattern[b, Blank[]]], sin[Plus[Optional[Pattern[e, Blank[]]], Times[Optional[Pattern[f, Blank[]]], Pattern[x, Blank[]]]]]]], -1]], Pattern[x, Blank[Symbol]]] := Dist[d/b, Int[(g*Cos[e + f*x])^p*(d*Sin[e + f*x])^(n - 1), x], x] - Dist[(a*d)/b, Int[((g*Cos[e + f*x])^p*(d*Sin[e + f*x])^(n - 1))/(a + b*Sin[e + f*x]), x], x] /; FreeQ[{a, b, d, e, f, g}, x] && NeQ[a^2 - b^2, 0] && IntegersQ[2*n, 2*p] && LtQ[-1, p, 1] && GtQ[n, 0]
  2910. Int[Times[Power[Times[cos[Plus[Optional[Pattern[e, Blank[]]], Times[Optional[Pattern[f, Blank[]]], Pattern[x, Blank[]]]]], Optional[Pattern[g, Blank[]]]], Pattern[p, Blank[]]], Power[Times[Optional[Pattern[d, Blank[]]], sin[Plus[Optional[Pattern[e, Blank[]]], Times[Optional[Pattern[f, Blank[]]], Pattern[x, Blank[]]]]]], Pattern[n, Blank[]]], Power[Plus[Pattern[a, Blank[]], Times[Optional[Pattern[b, Blank[]]], sin[Plus[Optional[Pattern[e, Blank[]]], Times[Optional[Pattern[f, Blank[]]], Pattern[x, Blank[]]]]]]], -1]], Pattern[x, Blank[Symbol]]] := Dist[1/a, Int[(g*Cos[e + f*x])^p*(d*Sin[e + f*x])^n, x], x] - Dist[b/(a*d), Int[((g*Cos[e + f*x])^p*(d*Sin[e + f*x])^(n + 1))/(a + b*Sin[e + f*x]), x], x] /; FreeQ[{a, b, d, e, f, g}, x] && NeQ[a^2 - b^2, 0] && IntegersQ[2*n, 2*p] && LtQ[-1, p, 1] && LtQ[n, 0]
  2911. Int[Times[Power[Times[cos[Plus[Optional[Pattern[e, Blank[]]], Times[Optional[Pattern[f, Blank[]]], Pattern[x, Blank[]]]]], Optional[Pattern[g, Blank[]]]], Pattern[p, Blank[]]], Power[Times[Optional[Pattern[d, Blank[]]], sin[Plus[Optional[Pattern[e, Blank[]]], Times[Optional[Pattern[f, Blank[]]], Pattern[x, Blank[]]]]]], Pattern[n, Blank[]]], Power[Plus[Pattern[a, Blank[]], Times[Optional[Pattern[b, Blank[]]], sin[Plus[Optional[Pattern[e, Blank[]]], Times[Optional[Pattern[f, Blank[]]], Pattern[x, Blank[]]]]]]], 2]], Pattern[x, Blank[Symbol]]] := Dist[(2*a*b)/d, Int[(g*Cos[e + f*x])^p*(d*Sin[e + f*x])^(n + 1), x], x] + Int[(g*Cos[e + f*x])^p*(d*Sin[e + f*x])^n*(a^2 + b^2*Sin[e + f*x]^2), x] /; FreeQ[{a, b, d, e, f, g, n, p}, x] && NeQ[a^2 - b^2, 0]
  2912. Int[Times[Power[Times[cos[Plus[Optional[Pattern[e, Blank[]]], Times[Optional[Pattern[f, Blank[]]], Pattern[x, Blank[]]]]], Optional[Pattern[g, Blank[]]]], Pattern[p, Blank[]]], Power[Times[Optional[Pattern[d, Blank[]]], sin[Plus[Optional[Pattern[e, Blank[]]], Times[Optional[Pattern[f, Blank[]]], Pattern[x, Blank[]]]]]], Pattern[n, Blank[]]], Power[Plus[Pattern[a, Blank[]], Times[Optional[Pattern[b, Blank[]]], sin[Plus[Optional[Pattern[e, Blank[]]], Times[Optional[Pattern[f, Blank[]]], Pattern[x, Blank[]]]]]]], Pattern[m, Blank[]]]], Pattern[x, Blank[Symbol]]] := Int[ExpandTrig[(g*cos[e + f*x])^p, (d*sin[e + f*x])^n*(a + b*sin[e + f*x])^m, x], x] /; FreeQ[{a, b, d, e, f, g, n, p}, x] && NeQ[a^2 - b^2, 0] && IntegerQ[m] && (GtQ[m, 0] || IntegerQ[n])
  2913. Int[Times[Power[Times[cos[Plus[Optional[Pattern[e, Blank[]]], Times[Optional[Pattern[f, Blank[]]], Pattern[x, Blank[]]]]], Optional[Pattern[g, Blank[]]]], Pattern[p, Blank[]]], Power[Times[Optional[Pattern[d, Blank[]]], sin[Plus[Optional[Pattern[e, Blank[]]], Times[Optional[Pattern[f, Blank[]]], Pattern[x, Blank[]]]]]], Pattern[n, Blank[]]], Power[Plus[Pattern[a, Blank[]], Times[Optional[Pattern[b, Blank[]]], sin[Plus[Optional[Pattern[e, Blank[]]], Times[Optional[Pattern[f, Blank[]]], Pattern[x, Blank[]]]]]]], Pattern[m, Blank[]]]], Pattern[x, Blank[Symbol]]] := Dist[g^2/a, Int[(g*Cos[e + f*x])^(p - 2)*(d*Sin[e + f*x])^n*(a + b*Sin[e + f*x])^(m + 1), x], x] + (-Dist[(b*g^2)/(a^2*d), Int[(g*Cos[e + f*x])^(p - 2)*(d*Sin[e + f*x])^(n + 1)*(a + b*Sin[e + f*x])^(m + 1), x], x] - Dist[(g^2*(a^2 - b^2))/(a^2*d^2), Int[(g*Cos[e + f*x])^(p - 2)*(d*Sin[e + f*x])^(n + 2)*(a + b*Sin[e + f*x])^m, x], x]) /; FreeQ[{a, b, d, e, f, g}, x] && NeQ[a^2 - b^2, 0] && IntegersQ[m, 2*n, 2*p] && LtQ[m, 0] && GtQ[p, 1] && (LeQ[n, -2] || (EqQ[m, -1] && EqQ[n, -3/2] && EqQ[p, 3/2]))
  2914. Int[Times[Power[cos[Plus[Optional[Pattern[e, Blank[]]], Times[Optional[Pattern[f, Blank[]]], Pattern[x, Blank[]]]]], Pattern[p, Blank[]]], Power[Plus[Pattern[a, Blank[]], Times[Optional[Pattern[b, Blank[]]], sin[Plus[Optional[Pattern[e, Blank[]]], Times[Optional[Pattern[f, Blank[]]], Pattern[x, Blank[]]]]]]], Pattern[m, Blank[]]], Power[Plus[Pattern[c, Blank[]], Times[Optional[Pattern[d, Blank[]]], sin[Plus[Optional[Pattern[e, Blank[]]], Times[Optional[Pattern[f, Blank[]]], Pattern[x, Blank[]]]]]]], Pattern[n, Blank[]]]], Pattern[x, Blank[Symbol]]] := Dist[a^(2*m), Int[(c + d*Sin[e + f*x])^n/(a - b*Sin[e + f*x])^m, x], x] /; FreeQ[{a, b, c, d, e, f, n}, x] && EqQ[a^2 - b^2, 0] && IntegersQ[m, p] && EqQ[2*m + p, 0]
  2915. Int[Times[Power[Times[cos[Plus[Optional[Pattern[e, Blank[]]], Times[Optional[Pattern[f, Blank[]]], Pattern[x, Blank[]]]]], Optional[Pattern[g, Blank[]]]], Pattern[p, Blank[]]], Power[Plus[Pattern[a, Blank[]], Times[Optional[Pattern[b, Blank[]]], sin[Plus[Optional[Pattern[e, Blank[]]], Times[Optional[Pattern[f, Blank[]]], Pattern[x, Blank[]]]]]]], Pattern[m, Blank[]]], Power[Plus[Pattern[c, Blank[]], Times[Optional[Pattern[d, Blank[]]], sin[Plus[Optional[Pattern[e, Blank[]]], Times[Optional[Pattern[f, Blank[]]], Pattern[x, Blank[]]]]]]], Pattern[n, Blank[]]]], Pattern[x, Blank[Symbol]]] := Dist[(a/g)^(2*m), Int[((g*Cos[e + f*x])^(2*m + p)*(c + d*Sin[e + f*x])^n)/(a - b*Sin[e + f*x])^m, x], x] /; FreeQ[{a, b, c, d, e, f, g, n}, x] && EqQ[a^2 - b^2, 0] && IntegerQ[m] && (EqQ[2*m + p, 0] || (GtQ[2*m + p, 0] && LtQ[p, -1]))
  2916. Int[Times[Power[cos[Plus[Optional[Pattern[e, Blank[]]], Times[Optional[Pattern[f, Blank[]]], Pattern[x, Blank[]]]]], 2], Power[Plus[Pattern[a, Blank[]], Times[Optional[Pattern[b, Blank[]]], sin[Plus[Optional[Pattern[e, Blank[]]], Times[Optional[Pattern[f, Blank[]]], Pattern[x, Blank[]]]]]]], Pattern[m, Blank[]]], Power[Plus[Pattern[c, Blank[]], Times[Optional[Pattern[d, Blank[]]], sin[Plus[Optional[Pattern[e, Blank[]]], Times[Optional[Pattern[f, Blank[]]], Pattern[x, Blank[]]]]]]], Pattern[n, Blank[]]]], Pattern[x, Blank[Symbol]]] := Dist[1/b^2, Int[(a + b*Sin[e + f*x])^(m + 1)*(c + d*Sin[e + f*x])^n*(a - b*Sin[e + f*x]), x], x] /; FreeQ[{a, b, c, d, e, f, m, n}, x] && EqQ[a^2 - b^2, 0] && IntegersQ[2*m, 2*n]
  2917. Int[Times[Power[cos[Plus[Optional[Pattern[e, Blank[]]], Times[Optional[Pattern[f, Blank[]]], Pattern[x, Blank[]]]]], Pattern[p, Blank[]]], Power[Plus[Pattern[a, Blank[]], Times[Optional[Pattern[b, Blank[]]], sin[Plus[Optional[Pattern[e, Blank[]]], Times[Optional[Pattern[f, Blank[]]], Pattern[x, Blank[]]]]]]], Pattern[m, Blank[]]], Power[Plus[Pattern[c, Blank[]], Times[Optional[Pattern[d, Blank[]]], sin[Plus[Optional[Pattern[e, Blank[]]], Times[Optional[Pattern[f, Blank[]]], Pattern[x, Blank[]]]]]]], Pattern[n, Blank[]]]], Pattern[x, Blank[Symbol]]] := Dist[(a^m*Cos[e + f*x])/(f*Sqrt[1 + Sin[e + f*x]]*Sqrt[1 - Sin[e + f*x]]), Subst[Int[(1 + (b*x)/a)^(m + (p - 1)/2)*(1 - (b*x)/a)^((p - 1)/2)*(c + d*x)^n, x], x, Sin[e + f*x]], x] /; FreeQ[{a, b, c, d, e, f, n}, x] && EqQ[a^2 - b^2, 0] && IntegerQ[p/2] && IntegerQ[m]
  2918. Int[Times[Power[cos[Plus[Optional[Pattern[e, Blank[]]], Times[Optional[Pattern[f, Blank[]]], Pattern[x, Blank[]]]]], Pattern[p, Blank[]]], Power[Plus[Pattern[a, Blank[]], Times[Optional[Pattern[b, Blank[]]], sin[Plus[Optional[Pattern[e, Blank[]]], Times[Optional[Pattern[f, Blank[]]], Pattern[x, Blank[]]]]]]], Pattern[m, Blank[]]], Power[Plus[Pattern[c, Blank[]], Times[Optional[Pattern[d, Blank[]]], sin[Plus[Optional[Pattern[e, Blank[]]], Times[Optional[Pattern[f, Blank[]]], Pattern[x, Blank[]]]]]]], Pattern[n, Blank[]]]], Pattern[x, Blank[Symbol]]] := Dist[Cos[e + f*x]/(a^(p - 2)*f*Sqrt[a + b*Sin[e + f*x]]*Sqrt[a - b*Sin[e + f*x]]), Subst[Int[(a + b*x)^(m + p/2 - 1/2)*(a - b*x)^(p/2 - 1/2)*(c + d*x)^n, x], x, Sin[e + f*x]], x] /; FreeQ[{a, b, c, d, e, f, m, n}, x] && EqQ[a^2 - b^2, 0] && IntegerQ[p/2] && !IntegerQ[m]
  2919. Int[Times[Power[Times[cos[Plus[Optional[Pattern[e, Blank[]]], Times[Optional[Pattern[f, Blank[]]], Pattern[x, Blank[]]]]], Optional[Pattern[g, Blank[]]]], Pattern[p, Blank[]]], Power[Plus[Pattern[a, Blank[]], Times[Optional[Pattern[b, Blank[]]], sin[Plus[Optional[Pattern[e, Blank[]]], Times[Optional[Pattern[f, Blank[]]], Pattern[x, Blank[]]]]]]], Pattern[m, Blank[]]], Power[Plus[Pattern[c, Blank[]], Times[Optional[Pattern[d, Blank[]]], sin[Plus[Optional[Pattern[e, Blank[]]], Times[Optional[Pattern[f, Blank[]]], Pattern[x, Blank[]]]]]]], Pattern[n, Blank[]]]], Pattern[x, Blank[Symbol]]] := Int[ExpandTrig[(g*cos[e + f*x])^p, (a + b*sin[e + f*x])^m*(c + d*sin[e + f*x])^n, x], x] /; FreeQ[{a, b, c, d, e, f, g, n, p}, x] && EqQ[a^2 - b^2, 0] && IGtQ[m, 0] && (IntegerQ[p] || IGtQ[n, 0])
  2920. Int[Times[Power[Times[cos[Plus[Optional[Pattern[e, Blank[]]], Times[Optional[Pattern[f, Blank[]]], Pattern[x, Blank[]]]]], Optional[Pattern[g, Blank[]]]], Pattern[p, Blank[]]], Power[Plus[Pattern[a, Blank[]], Times[Optional[Pattern[b, Blank[]]], sin[Plus[Optional[Pattern[e, Blank[]]], Times[Optional[Pattern[f, Blank[]]], Pattern[x, Blank[]]]]]]], Pattern[m, Blank[]]], Power[Plus[Pattern[c, Blank[]], Times[Optional[Pattern[d, Blank[]]], sin[Plus[Optional[Pattern[e, Blank[]]], Times[Optional[Pattern[f, Blank[]]], Pattern[x, Blank[]]]]]]], Pattern[n, Blank[]]]], Pattern[x, Blank[Symbol]]] := Dist[(a^m*g*(g*Cos[e + f*x])^(p - 1))/(f*(1 + Sin[e + f*x])^((p - 1)/2)*(1 - Sin[e + f*x])^((p - 1)/2)), Subst[Int[(1 + (b*x)/a)^(m + (p - 1)/2)*(1 - (b*x)/a)^((p - 1)/2)*(c + d*x)^n, x], x, Sin[e + f*x]], x] /; FreeQ[{a, b, c, d, e, f, n, p}, x] && EqQ[a^2 - b^2, 0] && IntegerQ[m]
  2921. Int[Times[Power[Times[cos[Plus[Optional[Pattern[e, Blank[]]], Times[Optional[Pattern[f, Blank[]]], Pattern[x, Blank[]]]]], Optional[Pattern[g, Blank[]]]], Pattern[p, Blank[]]], Power[Plus[Pattern[a, Blank[]], Times[Optional[Pattern[b, Blank[]]], sin[Plus[Optional[Pattern[e, Blank[]]], Times[Optional[Pattern[f, Blank[]]], Pattern[x, Blank[]]]]]]], Pattern[m, Blank[]]], Power[Plus[Pattern[c, Blank[]], Times[Optional[Pattern[d, Blank[]]], sin[Plus[Optional[Pattern[e, Blank[]]], Times[Optional[Pattern[f, Blank[]]], Pattern[x, Blank[]]]]]]], Pattern[n, Blank[]]]], Pattern[x, Blank[Symbol]]] := Dist[(g*(g*Cos[e + f*x])^(p - 1))/(f*(a + b*Sin[e + f*x])^((p - 1)/2)*(a - b*Sin[e + f*x])^((p - 1)/2)), Subst[Int[(a + b*x)^(m + (p - 1)/2)*(a - b*x)^((p - 1)/2)*(c + d*x)^n, x], x, Sin[e + f*x]], x] /; FreeQ[{a, b, c, d, e, f, m, n, p}, x] && EqQ[a^2 - b^2, 0] && !IntegerQ[m]
  2922. Int[Times[Power[cos[Plus[Optional[Pattern[e, Blank[]]], Times[Optional[Pattern[f, Blank[]]], Pattern[x, Blank[]]]]], 2], Power[Plus[Pattern[a, Blank[]], Times[Optional[Pattern[b, Blank[]]], sin[Plus[Optional[Pattern[e, Blank[]]], Times[Optional[Pattern[f, Blank[]]], Pattern[x, Blank[]]]]]]], Optional[Pattern[m, Blank[]]]], Power[Plus[Pattern[c, Blank[]], Times[Optional[Pattern[d, Blank[]]], sin[Plus[Optional[Pattern[e, Blank[]]], Times[Optional[Pattern[f, Blank[]]], Pattern[x, Blank[]]]]]]], Pattern[n, Blank[]]]], Pattern[x, Blank[Symbol]]] := Int[(a + b*Sin[e + f*x])^m*(c + d*Sin[e + f*x])^n*(1 - Sin[e + f*x]^2), x] /; FreeQ[{a, b, c, d, e, f, m, n}, x] && NeQ[a^2 - b^2, 0] && (IGtQ[m, 0] || IntegersQ[2*m, 2*n])
  2923. Int[Times[Power[cos[Plus[Optional[Pattern[e, Blank[]]], Times[Optional[Pattern[f, Blank[]]], Pattern[x, Blank[]]]]], Pattern[p, Blank[]]], Power[Plus[Pattern[a, Blank[]], Times[Optional[Pattern[b, Blank[]]], sin[Plus[Optional[Pattern[e, Blank[]]], Times[Optional[Pattern[f, Blank[]]], Pattern[x, Blank[]]]]]]], Optional[Pattern[m, Blank[]]]], Power[Plus[Pattern[c, Blank[]], Times[Optional[Pattern[d, Blank[]]], sin[Plus[Optional[Pattern[e, Blank[]]], Times[Optional[Pattern[f, Blank[]]], Pattern[x, Blank[]]]]]]], Pattern[n, Blank[]]]], Pattern[x, Blank[Symbol]]] := Int[ExpandTrig[(a + b*sin[e + f*x])^m*(c + d*sin[e + f*x])^n*(1 - sin[e + f*x]^2)^(p/2), x], x] /; FreeQ[{a, b, c, d, e, f, m, n}, x] && NeQ[a^2 - b^2, 0] && IGtQ[p/2, 0] && (IGtQ[m, 0] || IntegersQ[2*m, 2*n])
  2924. Int[Times[Power[Times[cos[Plus[Optional[Pattern[e, Blank[]]], Times[Optional[Pattern[f, Blank[]]], Pattern[x, Blank[]]]]], Optional[Pattern[g, Blank[]]]], Pattern[p, Blank[]]], Power[Plus[Pattern[a, Blank[]], Times[Optional[Pattern[b, Blank[]]], sin[Plus[Optional[Pattern[e, Blank[]]], Times[Optional[Pattern[f, Blank[]]], Pattern[x, Blank[]]]]]]], Pattern[m, Blank[]]], Power[Plus[Pattern[c, Blank[]], Times[Optional[Pattern[d, Blank[]]], sin[Plus[Optional[Pattern[e, Blank[]]], Times[Optional[Pattern[f, Blank[]]], Pattern[x, Blank[]]]]]]], Pattern[n, Blank[]]]], Pattern[x, Blank[Symbol]]] := Int[ExpandTrig[(g*cos[e + f*x])^p*(a + b*sin[e + f*x])^m*(c + d*sin[e + f*x])^n, x], x] /; FreeQ[{a, b, c, d, e, f, g, p}, x] && NeQ[a^2 - b^2, 0] && IntegersQ[2*m, 2*n]
  2925. Int[Times[Power[Times[cos[Plus[Optional[Pattern[e, Blank[]]], Times[Optional[Pattern[f, Blank[]]], Pattern[x, Blank[]]]]], Optional[Pattern[g, Blank[]]]], Pattern[p, Blank[]]], Power[Plus[Pattern[a, Blank[]], Times[Optional[Pattern[b, Blank[]]], sin[Plus[Optional[Pattern[e, Blank[]]], Times[Optional[Pattern[f, Blank[]]], Pattern[x, Blank[]]]]]]], Optional[Pattern[m, Blank[]]]], Power[Plus[Optional[Pattern[c, Blank[]]], Times[Optional[Pattern[d, Blank[]]], sin[Plus[Optional[Pattern[e, Blank[]]], Times[Optional[Pattern[f, Blank[]]], Pattern[x, Blank[]]]]]]], Optional[Pattern[n, Blank[]]]]], Pattern[x, Blank[Symbol]]] := Unintegrable[(g*Cos[e + f*x])^p*(a + b*Sin[e + f*x])^m*(c + d*Sin[e + f*x])^n, x] /; FreeQ[{a, b, c, d, e, f, g, m, n, p}, x] && NeQ[a^2 - b^2, 0]
  2926. Int[Times[Power[Times[Optional[Pattern[g, Blank[]]], sec[Plus[Optional[Pattern[e, Blank[]]], Times[Optional[Pattern[f, Blank[]]], Pattern[x, Blank[]]]]]], Pattern[p, Blank[]]], Power[Plus[Optional[Pattern[a, Blank[]]], Times[Optional[Pattern[b, Blank[]]], sin[Plus[Optional[Pattern[e, Blank[]]], Times[Optional[Pattern[f, Blank[]]], Pattern[x, Blank[]]]]]]], Optional[Pattern[m, Blank[]]]], Power[Plus[Optional[Pattern[c, Blank[]]], Times[Optional[Pattern[d, Blank[]]], sin[Plus[Optional[Pattern[e, Blank[]]], Times[Optional[Pattern[f, Blank[]]], Pattern[x, Blank[]]]]]]], Optional[Pattern[n, Blank[]]]]], Pattern[x, Blank[Symbol]]] := Dist[g^(2*IntPart[p])*(g*Cos[e + f*x])^FracPart[p]*(g*Sec[e + f*x])^FracPart[p], Int[((a + b*Sin[e + f*x])^m*(c + d*Sin[e + f*x])^n)/(g*Cos[e + f*x])^p, x], x] /; FreeQ[{a, b, c, d, e, f, g, m, n, p}, x] && !IntegerQ[p]
  2927. Int[Times[Power[Plus[Optional[Pattern[a, Blank[]]], Times[cos[Plus[Optional[Pattern[e, Blank[]]], Times[Optional[Pattern[f, Blank[]]], Pattern[x, Blank[]]]]], Optional[Pattern[b, Blank[]]]]], Optional[Pattern[m, Blank[]]]], Power[Plus[Optional[Pattern[c, Blank[]]], Times[cos[Plus[Optional[Pattern[e, Blank[]]], Times[Optional[Pattern[f, Blank[]]], Pattern[x, Blank[]]]]], Optional[Pattern[d, Blank[]]]]], Optional[Pattern[n, Blank[]]]], Power[Times[csc[Plus[Optional[Pattern[e, Blank[]]], Times[Optional[Pattern[f, Blank[]]], Pattern[x, Blank[]]]]], Optional[Pattern[g, Blank[]]]], Pattern[p, Blank[]]]], Pattern[x, Blank[Symbol]]] := Dist[g^(2*IntPart[p])*(g*Sin[e + f*x])^FracPart[p]*(g*Csc[e + f*x])^FracPart[p], Int[((a + b*Cos[e + f*x])^m*(c + d*Cos[e + f*x])^n)/(g*Sin[e + f*x])^p, x], x] /; FreeQ[{a, b, c, d, e, f, g, m, n, p}, x] && !IntegerQ[p]
  2928. Int[Times[Power[Times[Optional[Pattern[g, Blank[]]], sin[Plus[Optional[Pattern[e, Blank[]]], Times[Optional[Pattern[f, Blank[]]], Pattern[x, Blank[]]]]]], Rational[1, 2]], Power[Plus[Pattern[a, Blank[]], Times[Optional[Pattern[b, Blank[]]], sin[Plus[Optional[Pattern[e, Blank[]]], Times[Optional[Pattern[f, Blank[]]], Pattern[x, Blank[]]]]]]], Rational[1, 2]], Power[Plus[Pattern[c, Blank[]], Times[Optional[Pattern[d, Blank[]]], sin[Plus[Optional[Pattern[e, Blank[]]], Times[Optional[Pattern[f, Blank[]]], Pattern[x, Blank[]]]]]]], -1]], Pattern[x, Blank[Symbol]]] := Dist[g/d, Int[Sqrt[a + b*Sin[e + f*x]]/Sqrt[g*Sin[e + f*x]], x], x] - Dist[(c*g)/d, Int[Sqrt[a + b*Sin[e + f*x]]/(Sqrt[g*Sin[e + f*x]]*(c + d*Sin[e + f*x])), x], x] /; FreeQ[{a, b, c, d, e, f, g}, x] && NeQ[b*c - a*d, 0] && (EqQ[a^2 - b^2, 0] || EqQ[c^2 - d^2, 0])
  2929. Int[Times[Power[Times[Optional[Pattern[g, Blank[]]], sin[Plus[Optional[Pattern[e, Blank[]]], Times[Optional[Pattern[f, Blank[]]], Pattern[x, Blank[]]]]]], Rational[1, 2]], Power[Plus[Pattern[a, Blank[]], Times[Optional[Pattern[b, Blank[]]], sin[Plus[Optional[Pattern[e, Blank[]]], Times[Optional[Pattern[f, Blank[]]], Pattern[x, Blank[]]]]]]], Rational[1, 2]], Power[Plus[Pattern[c, Blank[]], Times[Optional[Pattern[d, Blank[]]], sin[Plus[Optional[Pattern[e, Blank[]]], Times[Optional[Pattern[f, Blank[]]], Pattern[x, Blank[]]]]]]], -1]], Pattern[x, Blank[Symbol]]] := Dist[b/d, Int[Sqrt[g*Sin[e + f*x]]/Sqrt[a + b*Sin[e + f*x]], x], x] - Dist[(b*c - a*d)/d, Int[Sqrt[g*Sin[e + f*x]]/(Sqrt[a + b*Sin[e + f*x]]*(c + d*Sin[e + f*x])), x], x] /; FreeQ[{a, b, c, d, e, f, g}, x] && NeQ[b*c - a*d, 0] && NeQ[a^2 - b^2, 0] && NeQ[c^2 - d^2, 0]
  2930. Int[Times[Power[Times[Optional[Pattern[g, Blank[]]], sin[Plus[Optional[Pattern[e, Blank[]]], Times[Optional[Pattern[f, Blank[]]], Pattern[x, Blank[]]]]]], Rational[-1, 2]], Power[Plus[Pattern[a, Blank[]], Times[Optional[Pattern[b, Blank[]]], sin[Plus[Optional[Pattern[e, Blank[]]], Times[Optional[Pattern[f, Blank[]]], Pattern[x, Blank[]]]]]]], Rational[1, 2]], Power[Plus[Pattern[c, Blank[]], Times[Optional[Pattern[d, Blank[]]], sin[Plus[Optional[Pattern[e, Blank[]]], Times[Optional[Pattern[f, Blank[]]], Pattern[x, Blank[]]]]]]], -1]], Pattern[x, Blank[Symbol]]] := Dist[(-2*b)/f, Subst[Int[1/(b*c + a*d + c*g*x^2), x], x, (b*Cos[e + f*x])/(Sqrt[g*Sin[e + f*x]]*Sqrt[a + b*Sin[e + f*x]])], x] /; FreeQ[{a, b, c, d, e, f, g}, x] && NeQ[b*c - a*d, 0] && EqQ[a^2 - b^2, 0]
  2931. Int[Times[Power[sin[Plus[Optional[Pattern[e, Blank[]]], Times[Optional[Pattern[f, Blank[]]], Pattern[x, Blank[]]]]], Rational[-1, 2]], Power[Plus[Pattern[a, Blank[]], Times[Optional[Pattern[b, Blank[]]], sin[Plus[Optional[Pattern[e, Blank[]]], Times[Optional[Pattern[f, Blank[]]], Pattern[x, Blank[]]]]]]], Rational[1, 2]], Power[Plus[Pattern[c, Blank[]], Times[Optional[Pattern[d, Blank[]]], sin[Plus[Optional[Pattern[e, Blank[]]], Times[Optional[Pattern[f, Blank[]]], Pattern[x, Blank[]]]]]]], -1]], Pattern[x, Blank[Symbol]]] := -Simp[(Sqrt[a + b]*EllipticE[ArcSin[Cos[e + f*x]/(1 + Sin[e + f*x])], -((a - b)/(a + b))])/(c*f), x] /; FreeQ[{a, b, c, d, e, f}, x] && EqQ[d, c] && GtQ[b^2 - a^2, 0] && GtQ[b, 0]
  2932. Int[Times[Power[Times[Optional[Pattern[g, Blank[]]], sin[Plus[Optional[Pattern[e, Blank[]]], Times[Optional[Pattern[f, Blank[]]], Pattern[x, Blank[]]]]]], Rational[-1, 2]], Power[Plus[Pattern[a, Blank[]], Times[Optional[Pattern[b, Blank[]]], sin[Plus[Optional[Pattern[e, Blank[]]], Times[Optional[Pattern[f, Blank[]]], Pattern[x, Blank[]]]]]]], Rational[1, 2]], Power[Plus[Pattern[c, Blank[]], Times[Optional[Pattern[d, Blank[]]], sin[Plus[Optional[Pattern[e, Blank[]]], Times[Optional[Pattern[f, Blank[]]], Pattern[x, Blank[]]]]]]], -1]], Pattern[x, Blank[Symbol]]] := -Simp[(Sqrt[a + b*Sin[e + f*x]]*Sqrt[(d*Sin[e + f*x])/(c + d*Sin[e + f*x])]*EllipticE[ArcSin[(c*Cos[e + f*x])/(c + d*Sin[e + f*x])], (b*c - a*d)/(b*c + a*d)])/(d*f*Sqrt[g*Sin[e + f*x]]*Sqrt[(c^2*(a + b*Sin[e + f*x]))/((a*c + b*d)*(c + d*Sin[e + f*x]))]), x] /; FreeQ[{a, b, c, d, e, f, g}, x] && NeQ[b*c - a*d, 0] && NeQ[a^2 - b^2, 0] && EqQ[c^2 - d^2, 0]
  2933. Int[Times[Power[Times[Optional[Pattern[g, Blank[]]], sin[Plus[Optional[Pattern[e, Blank[]]], Times[Optional[Pattern[f, Blank[]]], Pattern[x, Blank[]]]]]], Rational[-1, 2]], Power[Plus[Pattern[a, Blank[]], Times[Optional[Pattern[b, Blank[]]], sin[Plus[Optional[Pattern[e, Blank[]]], Times[Optional[Pattern[f, Blank[]]], Pattern[x, Blank[]]]]]]], Rational[1, 2]], Power[Plus[Pattern[c, Blank[]], Times[Optional[Pattern[d, Blank[]]], sin[Plus[Optional[Pattern[e, Blank[]]], Times[Optional[Pattern[f, Blank[]]], Pattern[x, Blank[]]]]]]], -1]], Pattern[x, Blank[Symbol]]] := Dist[a/c, Int[1/(Sqrt[g*Sin[e + f*x]]*Sqrt[a + b*Sin[e + f*x]]), x], x] + Dist[(b*c - a*d)/(c*g), Int[Sqrt[g*Sin[e + f*x]]/(Sqrt[a + b*Sin[e + f*x]]*(c + d*Sin[e + f*x])), x], x] /; FreeQ[{a, b, c, d, e, f, g}, x] && NeQ[b*c - a*d, 0] && NeQ[a^2 - b^2, 0] && NeQ[c^2 - d^2, 0]
  2934. Int[Times[Power[sin[Plus[Optional[Pattern[e, Blank[]]], Times[Optional[Pattern[f, Blank[]]], Pattern[x, Blank[]]]]], -1], Power[Plus[Pattern[a, Blank[]], Times[Optional[Pattern[b, Blank[]]], sin[Plus[Optional[Pattern[e, Blank[]]], Times[Optional[Pattern[f, Blank[]]], Pattern[x, Blank[]]]]]]], Rational[1, 2]], Power[Plus[Pattern[c, Blank[]], Times[Optional[Pattern[d, Blank[]]], sin[Plus[Optional[Pattern[e, Blank[]]], Times[Optional[Pattern[f, Blank[]]], Pattern[x, Blank[]]]]]]], -1]], Pattern[x, Blank[Symbol]]] := Dist[1/c, Int[Sqrt[a + b*Sin[e + f*x]]/Sin[e + f*x], x], x] - Dist[d/c, Int[Sqrt[a + b*Sin[e + f*x]]/(c + d*Sin[e + f*x]), x], x] /; FreeQ[{a, b, c, d, e, f}, x] && NeQ[b*c - a*d, 0] && EqQ[a^2 - b^2, 0]
  2935. Int[Times[Power[sin[Plus[Optional[Pattern[e, Blank[]]], Times[Optional[Pattern[f, Blank[]]], Pattern[x, Blank[]]]]], -1], Power[Plus[Pattern[a, Blank[]], Times[Optional[Pattern[b, Blank[]]], sin[Plus[Optional[Pattern[e, Blank[]]], Times[Optional[Pattern[f, Blank[]]], Pattern[x, Blank[]]]]]]], Rational[1, 2]], Power[Plus[Pattern[c, Blank[]], Times[Optional[Pattern[d, Blank[]]], sin[Plus[Optional[Pattern[e, Blank[]]], Times[Optional[Pattern[f, Blank[]]], Pattern[x, Blank[]]]]]]], -1]], Pattern[x, Blank[Symbol]]] := Dist[a/c, Int[1/(Sin[e + f*x]*Sqrt[a + b*Sin[e + f*x]]), x], x] + Dist[(b*c - a*d)/c, Int[1/(Sqrt[a + b*Sin[e + f*x]]*(c + d*Sin[e + f*x])), x], x] /; FreeQ[{a, b, c, d, e, f}, x] && NeQ[b*c - a*d, 0] && NeQ[a^2 - b^2, 0]
  2936. Int[Times[Power[Times[Optional[Pattern[g, Blank[]]], sin[Plus[Optional[Pattern[e, Blank[]]], Times[Optional[Pattern[f, Blank[]]], Pattern[x, Blank[]]]]]], Rational[1, 2]], Power[Plus[Pattern[a, Blank[]], Times[Optional[Pattern[b, Blank[]]], sin[Plus[Optional[Pattern[e, Blank[]]], Times[Optional[Pattern[f, Blank[]]], Pattern[x, Blank[]]]]]]], Rational[-1, 2]], Power[Plus[Pattern[c, Blank[]], Times[Optional[Pattern[d, Blank[]]], sin[Plus[Optional[Pattern[e, Blank[]]], Times[Optional[Pattern[f, Blank[]]], Pattern[x, Blank[]]]]]]], -1]], Pattern[x, Blank[Symbol]]] := -Dist[(a*g)/(b*c - a*d), Int[1/(Sqrt[g*Sin[e + f*x]]*Sqrt[a + b*Sin[e + f*x]]), x], x] + Dist[(c*g)/(b*c - a*d), Int[Sqrt[a + b*Sin[e + f*x]]/(Sqrt[g*Sin[e + f*x]]*(c + d*Sin[e + f*x])), x], x] /; FreeQ[{a, b, c, d, e, f, g}, x] && NeQ[b*c - a*d, 0] && (EqQ[a^2 - b^2, 0] || EqQ[c^2 - d^2, 0])
  2937. Int[Times[Power[Times[Optional[Pattern[g, Blank[]]], sin[Plus[Optional[Pattern[e, Blank[]]], Times[Optional[Pattern[f, Blank[]]], Pattern[x, Blank[]]]]]], Rational[1, 2]], Power[Plus[Pattern[a, Blank[]], Times[Optional[Pattern[b, Blank[]]], sin[Plus[Optional[Pattern[e, Blank[]]], Times[Optional[Pattern[f, Blank[]]], Pattern[x, Blank[]]]]]]], Rational[-1, 2]], Power[Plus[Pattern[c, Blank[]], Times[Optional[Pattern[d, Blank[]]], sin[Plus[Optional[Pattern[e, Blank[]]], Times[Optional[Pattern[f, Blank[]]], Pattern[x, Blank[]]]]]]], -1]], Pattern[x, Blank[Symbol]]] := Simp[(2*Sqrt[-Cot[e + f*x]^2]*Sqrt[g*Sin[e + f*x]]*Sqrt[(b + a*Csc[e + f*x])/(a + b)]*EllipticPi[(2*c)/(c + d), ArcSin[Sqrt[1 - Csc[e + f*x]]/Sqrt[2]], (2*a)/(a + b)])/(f*(c + d)*Cot[e + f*x]*Sqrt[a + b*Sin[e + f*x]]), x] /; FreeQ[{a, b, c, d, e, f, g}, x] && NeQ[b*c - a*d, 0] && NeQ[a^2 - b^2, 0] && NeQ[c^2 - d^2, 0]
  2938. Int[Times[Power[Times[Optional[Pattern[g, Blank[]]], sin[Plus[Optional[Pattern[e, Blank[]]], Times[Optional[Pattern[f, Blank[]]], Pattern[x, Blank[]]]]]], Rational[-1, 2]], Power[Plus[Pattern[a, Blank[]], Times[Optional[Pattern[b, Blank[]]], sin[Plus[Optional[Pattern[e, Blank[]]], Times[Optional[Pattern[f, Blank[]]], Pattern[x, Blank[]]]]]]], Rational[-1, 2]], Power[Plus[Pattern[c, Blank[]], Times[Optional[Pattern[d, Blank[]]], sin[Plus[Optional[Pattern[e, Blank[]]], Times[Optional[Pattern[f, Blank[]]], Pattern[x, Blank[]]]]]]], -1]], Pattern[x, Blank[Symbol]]] := Dist[b/(b*c - a*d), Int[1/(Sqrt[g*Sin[e + f*x]]*Sqrt[a + b*Sin[e + f*x]]), x], x] - Dist[d/(b*c - a*d), Int[Sqrt[a + b*Sin[e + f*x]]/(Sqrt[g*Sin[e + f*x]]*(c + d*Sin[e + f*x])), x], x] /; FreeQ[{a, b, c, d, e, f, g}, x] && NeQ[b*c - a*d, 0] && (EqQ[a^2 - b^2, 0] || EqQ[c^2 - d^2, 0])
  2939. Int[Times[Power[Times[Optional[Pattern[g, Blank[]]], sin[Plus[Optional[Pattern[e, Blank[]]], Times[Optional[Pattern[f, Blank[]]], Pattern[x, Blank[]]]]]], Rational[-1, 2]], Power[Plus[Pattern[a, Blank[]], Times[Optional[Pattern[b, Blank[]]], sin[Plus[Optional[Pattern[e, Blank[]]], Times[Optional[Pattern[f, Blank[]]], Pattern[x, Blank[]]]]]]], Rational[-1, 2]], Power[Plus[Pattern[c, Blank[]], Times[Optional[Pattern[d, Blank[]]], sin[Plus[Optional[Pattern[e, Blank[]]], Times[Optional[Pattern[f, Blank[]]], Pattern[x, Blank[]]]]]]], -1]], Pattern[x, Blank[Symbol]]] := Dist[1/c, Int[1/(Sqrt[g*Sin[e + f*x]]*Sqrt[a + b*Sin[e + f*x]]), x], x] - Dist[d/(c*g), Int[Sqrt[g*Sin[e + f*x]]/(Sqrt[a + b*Sin[e + f*x]]*(c + d*Sin[e + f*x])), x], x] /; FreeQ[{a, b, c, d, e, f, g}, x] && NeQ[b*c - a*d, 0] && NeQ[a^2 - b^2, 0] && NeQ[c^2 - d^2, 0]
  2940. Int[Times[Power[sin[Plus[Optional[Pattern[e, Blank[]]], Times[Optional[Pattern[f, Blank[]]], Pattern[x, Blank[]]]]], -1], Power[Plus[Pattern[a, Blank[]], Times[Optional[Pattern[b, Blank[]]], sin[Plus[Optional[Pattern[e, Blank[]]], Times[Optional[Pattern[f, Blank[]]], Pattern[x, Blank[]]]]]]], Rational[-1, 2]], Power[Plus[Pattern[c, Blank[]], Times[Optional[Pattern[d, Blank[]]], sin[Plus[Optional[Pattern[e, Blank[]]], Times[Optional[Pattern[f, Blank[]]], Pattern[x, Blank[]]]]]]], -1]], Pattern[x, Blank[Symbol]]] := Dist[d^2/(c*(b*c - a*d)), Int[Sqrt[a + b*Sin[e + f*x]]/(c + d*Sin[e + f*x]), x], x] + Dist[1/(c*(b*c - a*d)), Int[(b*c - a*d - b*d*Sin[e + f*x])/(Sin[e + f*x]*Sqrt[a + b*Sin[e + f*x]]), x], x] /; FreeQ[{a, b, c, d, e, f}, x] && NeQ[b*c - a*d, 0] && EqQ[a^2 - b^2, 0]
  2941. Int[Times[Power[sin[Plus[Optional[Pattern[e, Blank[]]], Times[Optional[Pattern[f, Blank[]]], Pattern[x, Blank[]]]]], -1], Power[Plus[Pattern[a, Blank[]], Times[Optional[Pattern[b, Blank[]]], sin[Plus[Optional[Pattern[e, Blank[]]], Times[Optional[Pattern[f, Blank[]]], Pattern[x, Blank[]]]]]]], Rational[-1, 2]], Power[Plus[Pattern[c, Blank[]], Times[Optional[Pattern[d, Blank[]]], sin[Plus[Optional[Pattern[e, Blank[]]], Times[Optional[Pattern[f, Blank[]]], Pattern[x, Blank[]]]]]]], -1]], Pattern[x, Blank[Symbol]]] := Dist[1/c, Int[1/(Sin[e + f*x]*Sqrt[a + b*Sin[e + f*x]]), x], x] - Dist[d/c, Int[1/(Sqrt[a + b*Sin[e + f*x]]*(c + d*Sin[e + f*x])), x], x] /; FreeQ[{a, b, c, d, e, f}, x] && NeQ[b*c - a*d, 0] && NeQ[a^2 - b^2, 0]
  2942. Int[Times[Power[sin[Plus[Optional[Pattern[e, Blank[]]], Times[Optional[Pattern[f, Blank[]]], Pattern[x, Blank[]]]]], -1], Power[Plus[Pattern[a, Blank[]], Times[Optional[Pattern[b, Blank[]]], sin[Plus[Optional[Pattern[e, Blank[]]], Times[Optional[Pattern[f, Blank[]]], Pattern[x, Blank[]]]]]]], Rational[1, 2]], Power[Plus[Pattern[c, Blank[]], Times[Optional[Pattern[d, Blank[]]], sin[Plus[Optional[Pattern[e, Blank[]]], Times[Optional[Pattern[f, Blank[]]], Pattern[x, Blank[]]]]]]], Rational[-1, 2]]], Pattern[x, Blank[Symbol]]] := -Dist[d/c, Int[Sqrt[a + b*Sin[e + f*x]]/Sqrt[c + d*Sin[e + f*x]], x], x] + Dist[1/c, Int[(Sqrt[a + b*Sin[e + f*x]]*Sqrt[c + d*Sin[e + f*x]])/Sin[e + f*x], x], x] /; FreeQ[{a, b, c, d, e, f}, x] && NeQ[b*c - a*d, 0] && EqQ[a^2 - b^2, 0] && EqQ[b*c + a*d, 0]
  2943. Int[Times[Power[sin[Plus[Optional[Pattern[e, Blank[]]], Times[Optional[Pattern[f, Blank[]]], Pattern[x, Blank[]]]]], -1], Power[Plus[Pattern[a, Blank[]], Times[Optional[Pattern[b, Blank[]]], sin[Plus[Optional[Pattern[e, Blank[]]], Times[Optional[Pattern[f, Blank[]]], Pattern[x, Blank[]]]]]]], Rational[1, 2]], Power[Plus[Pattern[c, Blank[]], Times[Optional[Pattern[d, Blank[]]], sin[Plus[Optional[Pattern[e, Blank[]]], Times[Optional[Pattern[f, Blank[]]], Pattern[x, Blank[]]]]]]], Rational[-1, 2]]], Pattern[x, Blank[Symbol]]] := Dist[(-2*a)/f, Subst[Int[1/(1 - a*c*x^2), x], x, Cos[e + f*x]/(Sqrt[a + b*Sin[e + f*x]]*Sqrt[c + d*Sin[e + f*x]])], x] /; FreeQ[{a, b, c, d, e, f}, x] && NeQ[b*c - a*d, 0] && EqQ[a^2 - b^2, 0] && NeQ[b*c + a*d, 0]
  2944. Int[Times[Power[sin[Plus[Optional[Pattern[e, Blank[]]], Times[Optional[Pattern[f, Blank[]]], Pattern[x, Blank[]]]]], -1], Power[Plus[Pattern[a, Blank[]], Times[Optional[Pattern[b, Blank[]]], sin[Plus[Optional[Pattern[e, Blank[]]], Times[Optional[Pattern[f, Blank[]]], Pattern[x, Blank[]]]]]]], Rational[1, 2]], Power[Plus[Pattern[c, Blank[]], Times[Optional[Pattern[d, Blank[]]], sin[Plus[Optional[Pattern[e, Blank[]]], Times[Optional[Pattern[f, Blank[]]], Pattern[x, Blank[]]]]]]], Rational[-1, 2]]], Pattern[x, Blank[Symbol]]] := Dist[(b*c - a*d)/c, Int[1/(Sqrt[a + b*Sin[e + f*x]]*Sqrt[c + d*Sin[e + f*x]]), x], x] + Dist[a/c, Int[Sqrt[c + d*Sin[e + f*x]]/(Sin[e + f*x]*Sqrt[a + b*Sin[e + f*x]]), x], x] /; FreeQ[{a, b, c, d, e, f}, x] && NeQ[b*c - a*d, 0] && NeQ[a^2 - b^2, 0] && EqQ[c^2 - d^2, 0]
  2945. Int[Times[Power[sin[Plus[Optional[Pattern[e, Blank[]]], Times[Optional[Pattern[f, Blank[]]], Pattern[x, Blank[]]]]], -1], Power[Plus[Pattern[a, Blank[]], Times[Optional[Pattern[b, Blank[]]], sin[Plus[Optional[Pattern[e, Blank[]]], Times[Optional[Pattern[f, Blank[]]], Pattern[x, Blank[]]]]]]], Rational[1, 2]], Power[Plus[Pattern[c, Blank[]], Times[Optional[Pattern[d, Blank[]]], sin[Plus[Optional[Pattern[e, Blank[]]], Times[Optional[Pattern[f, Blank[]]], Pattern[x, Blank[]]]]]]], Rational[-1, 2]]], Pattern[x, Blank[Symbol]]] := Simp[(-2*(a + b*Sin[e + f*x])*Sqrt[-(((b*c - a*d)*(1 - Sin[e + f*x]))/((c + d)*(a + b*Sin[e + f*x])))]*Sqrt[((b*c - a*d)*(1 + Sin[e + f*x]))/((c - d)*(a + b*Sin[e + f*x]))]*EllipticPi[(a*(c + d))/(c*(a + b)), ArcSin[(Rt[(a + b)/(c + d), 2]*Sqrt[c + d*Sin[e + f*x]])/Sqrt[a + b*Sin[e + f*x]]], ((a - b)*(c + d))/((a + b)*(c - d))])/(c*f*Rt[(a + b)/(c + d), 2]*Cos[e + f*x]), x] /; FreeQ[{a, b, c, d, e, f}, x] && NeQ[b*c - a*d, 0] && NeQ[a^2 - b^2, 0] && NeQ[c^2 - d^2, 0]
  2946. Int[Times[Power[sin[Plus[Optional[Pattern[e, Blank[]]], Times[Optional[Pattern[f, Blank[]]], Pattern[x, Blank[]]]]], -1], Power[Plus[Pattern[a, Blank[]], Times[Optional[Pattern[b, Blank[]]], sin[Plus[Optional[Pattern[e, Blank[]]], Times[Optional[Pattern[f, Blank[]]], Pattern[x, Blank[]]]]]]], Rational[-1, 2]], Power[Plus[Pattern[c, Blank[]], Times[Optional[Pattern[d, Blank[]]], sin[Plus[Optional[Pattern[e, Blank[]]], Times[Optional[Pattern[f, Blank[]]], Pattern[x, Blank[]]]]]]], Rational[-1, 2]]], Pattern[x, Blank[Symbol]]] := Dist[Cos[e + f*x]/(Sqrt[a + b*Sin[e + f*x]]*Sqrt[c + d*Sin[e + f*x]]), Int[1/(Cos[e + f*x]*Sin[e + f*x]), x], x] /; FreeQ[{a, b, c, d, e, f}, x] && NeQ[b*c - a*d, 0] && EqQ[a^2 - b^2, 0] && EqQ[c^2 - d^2, 0]
  2947. Int[Times[Power[sin[Plus[Optional[Pattern[e, Blank[]]], Times[Optional[Pattern[f, Blank[]]], Pattern[x, Blank[]]]]], -1], Power[Plus[Pattern[a, Blank[]], Times[Optional[Pattern[b, Blank[]]], sin[Plus[Optional[Pattern[e, Blank[]]], Times[Optional[Pattern[f, Blank[]]], Pattern[x, Blank[]]]]]]], Rational[-1, 2]], Power[Plus[Pattern[c, Blank[]], Times[Optional[Pattern[d, Blank[]]], sin[Plus[Optional[Pattern[e, Blank[]]], Times[Optional[Pattern[f, Blank[]]], Pattern[x, Blank[]]]]]]], Rational[-1, 2]]], Pattern[x, Blank[Symbol]]] := -Dist[b/a, Int[1/(Sqrt[a + b*Sin[e + f*x]]*Sqrt[c + d*Sin[e + f*x]]), x], x] + Dist[1/a, Int[Sqrt[a + b*Sin[e + f*x]]/(Sin[e + f*x]*Sqrt[c + d*Sin[e + f*x]]), x], x] /; FreeQ[{a, b, c, d, e, f}, x] && NeQ[b*c - a*d, 0] && (NeQ[a^2 - b^2, 0] || NeQ[c^2 - d^2, 0])
  2948. Int[Times[Power[sin[Plus[Optional[Pattern[e, Blank[]]], Times[Optional[Pattern[f, Blank[]]], Pattern[x, Blank[]]]]], -1], Power[Plus[Pattern[a, Blank[]], Times[Optional[Pattern[b, Blank[]]], sin[Plus[Optional[Pattern[e, Blank[]]], Times[Optional[Pattern[f, Blank[]]], Pattern[x, Blank[]]]]]]], Rational[1, 2]], Power[Plus[Pattern[c, Blank[]], Times[Optional[Pattern[d, Blank[]]], sin[Plus[Optional[Pattern[e, Blank[]]], Times[Optional[Pattern[f, Blank[]]], Pattern[x, Blank[]]]]]]], Rational[1, 2]]], Pattern[x, Blank[Symbol]]] := Dist[(Sqrt[a + b*Sin[e + f*x]]*Sqrt[c + d*Sin[e + f*x]])/Cos[e + f*x], Int[Cot[e + f*x], x], x] /; FreeQ[{a, b, c, d, e, f}, x] && NeQ[b*c - a*d, 0] && EqQ[a^2 - b^2, 0] && EqQ[c^2 - d^2, 0]
  2949. Int[Times[Power[sin[Plus[Optional[Pattern[e, Blank[]]], Times[Optional[Pattern[f, Blank[]]], Pattern[x, Blank[]]]]], -1], Power[Plus[Pattern[a, Blank[]], Times[Optional[Pattern[b, Blank[]]], sin[Plus[Optional[Pattern[e, Blank[]]], Times[Optional[Pattern[f, Blank[]]], Pattern[x, Blank[]]]]]]], Rational[1, 2]], Power[Plus[Pattern[c, Blank[]], Times[Optional[Pattern[d, Blank[]]], sin[Plus[Optional[Pattern[e, Blank[]]], Times[Optional[Pattern[f, Blank[]]], Pattern[x, Blank[]]]]]]], Rational[1, 2]]], Pattern[x, Blank[Symbol]]] := Dist[d, Int[Sqrt[a + b*Sin[e + f*x]]/Sqrt[c + d*Sin[e + f*x]], x], x] + Dist[c, Int[Sqrt[a + b*Sin[e + f*x]]/(Sin[e + f*x]*Sqrt[c + d*Sin[e + f*x]]), x], x] /; FreeQ[{a, b, c, d, e, f}, x] && NeQ[b*c - a*d, 0] && (NeQ[a^2 - b^2, 0] || NeQ[c^2 - d^2, 0])
  2950. Int[Times[Power[sin[Plus[Optional[Pattern[e, Blank[]]], Times[Optional[Pattern[f, Blank[]]], Pattern[x, Blank[]]]]], Pattern[p, Blank[]]], Power[Plus[Pattern[a, Blank[]], Times[Optional[Pattern[b, Blank[]]], sin[Plus[Optional[Pattern[e, Blank[]]], Times[Optional[Pattern[f, Blank[]]], Pattern[x, Blank[]]]]]]], Optional[Pattern[m, Blank[]]]], Power[Plus[Pattern[c, Blank[]], Times[Optional[Pattern[d, Blank[]]], sin[Plus[Optional[Pattern[e, Blank[]]], Times[Optional[Pattern[f, Blank[]]], Pattern[x, Blank[]]]]]]], Optional[Pattern[n, Blank[]]]]], Pattern[x, Blank[Symbol]]] := Dist[a^n*c^n, Int[Tan[e + f*x]^p*(a + b*Sin[e + f*x])^(m - n), x], x] /; FreeQ[{a, b, c, d, e, f, m}, x] && EqQ[b*c + a*d, 0] && EqQ[a^2 - b^2, 0] && EqQ[p + 2*n, 0] && IntegerQ[n]
  2951. Int[Times[Power[Times[Optional[Pattern[g, Blank[]]], sin[Plus[Optional[Pattern[e, Blank[]]], Times[Optional[Pattern[f, Blank[]]], Pattern[x, Blank[]]]]]], Pattern[p, Blank[]]], Power[Plus[Pattern[a, Blank[]], Times[Optional[Pattern[b, Blank[]]], sin[Plus[Optional[Pattern[e, Blank[]]], Times[Optional[Pattern[f, Blank[]]], Pattern[x, Blank[]]]]]]], Pattern[m, Blank[]]], Power[Plus[Pattern[c, Blank[]], Times[Optional[Pattern[d, Blank[]]], sin[Plus[Optional[Pattern[e, Blank[]]], Times[Optional[Pattern[f, Blank[]]], Pattern[x, Blank[]]]]]]], Pattern[n, Blank[]]]], Pattern[x, Blank[Symbol]]] := Dist[(Sqrt[a - b*Sin[e + f*x]]*Sqrt[a + b*Sin[e + f*x]])/(f*Cos[e + f*x]), Subst[Int[((g*x)^p*(a + b*x)^(m - 1/2)*(c + d*x)^n)/Sqrt[a - b*x], x], x, Sin[e + f*x]], x] /; FreeQ[{a, b, c, d, e, f, g, m, n, p}, x] && NeQ[b*c - a*d, 0] && EqQ[a^2 - b^2, 0] && NeQ[c^2 - d^2, 0] && IntegerQ[m - 1/2]
  2952. Int[Times[Power[Times[Optional[Pattern[g, Blank[]]], sin[Plus[Optional[Pattern[e, Blank[]]], Times[Optional[Pattern[f, Blank[]]], Pattern[x, Blank[]]]]]], Pattern[p, Blank[]]], Power[Plus[Pattern[a, Blank[]], Times[Optional[Pattern[b, Blank[]]], sin[Plus[Optional[Pattern[e, Blank[]]], Times[Optional[Pattern[f, Blank[]]], Pattern[x, Blank[]]]]]]], Pattern[m, Blank[]]], Power[Plus[Pattern[c, Blank[]], Times[Optional[Pattern[d, Blank[]]], sin[Plus[Optional[Pattern[e, Blank[]]], Times[Optional[Pattern[f, Blank[]]], Pattern[x, Blank[]]]]]]], Pattern[n, Blank[]]]], Pattern[x, Blank[Symbol]]] := Int[ExpandTrig[(g*sin[e + f*x])^p*(a + b*sin[e + f*x])^m*(c + d*sin[e + f*x])^n, x], x] /; FreeQ[{a, b, c, d, e, f, g, n, p}, x] && NeQ[b*c - a*d, 0] && (IntegersQ[m, n] || IntegersQ[m, p] || IntegersQ[n, p]) && NeQ[p, 2]
  2953. Int[Times[Power[Times[Optional[Pattern[g, Blank[]]], sin[Plus[Optional[Pattern[e, Blank[]]], Times[Optional[Pattern[f, Blank[]]], Pattern[x, Blank[]]]]]], Pattern[p, Blank[]]], Power[Plus[Pattern[a, Blank[]], Times[Optional[Pattern[b, Blank[]]], sin[Plus[Optional[Pattern[e, Blank[]]], Times[Optional[Pattern[f, Blank[]]], Pattern[x, Blank[]]]]]]], Pattern[m, Blank[]]], Power[Plus[Pattern[c, Blank[]], Times[Optional[Pattern[d, Blank[]]], sin[Plus[Optional[Pattern[e, Blank[]]], Times[Optional[Pattern[f, Blank[]]], Pattern[x, Blank[]]]]]]], Pattern[n, Blank[]]]], Pattern[x, Blank[Symbol]]] := Unintegrable[(g*Sin[e + f*x])^p*(a + b*Sin[e + f*x])^m*(c + d*Sin[e + f*x])^n, x] /; FreeQ[{a, b, c, d, e, f, g, m, n, p}, x] && NeQ[p, 2]
  2954. Int[Times[Power[Plus[Optional[Pattern[a, Blank[]]], Times[csc[Plus[Optional[Pattern[e, Blank[]]], Times[Optional[Pattern[f, Blank[]]], Pattern[x, Blank[]]]]], Optional[Pattern[b, Blank[]]]]], Optional[Pattern[m, Blank[]]]], Power[Plus[Times[csc[Plus[Optional[Pattern[e, Blank[]]], Times[Optional[Pattern[f, Blank[]]], Pattern[x, Blank[]]]]], Optional[Pattern[d, Blank[]]]], Pattern[c, Blank[]]], Optional[Pattern[n, Blank[]]]], Power[Times[Optional[Pattern[g, Blank[]]], sin[Plus[Optional[Pattern[e, Blank[]]], Times[Optional[Pattern[f, Blank[]]], Pattern[x, Blank[]]]]]], Optional[Pattern[p, Blank[]]]]], Pattern[x, Blank[Symbol]]] := Dist[g^(m + n), Int[(g*Sin[e + f*x])^(p - m - n)*(b + a*Sin[e + f*x])^m*(d + c*Sin[e + f*x])^n, x], x] /; FreeQ[{a, b, c, d, e, f, g, p}, x] && NeQ[b*c - a*d, 0] && !IntegerQ[p] && IntegerQ[m] && IntegerQ[n]
  2955. Int[Times[Power[Plus[Optional[Pattern[a, Blank[]]], Times[csc[Plus[Optional[Pattern[e, Blank[]]], Times[Optional[Pattern[f, Blank[]]], Pattern[x, Blank[]]]]], Optional[Pattern[b, Blank[]]]]], Optional[Pattern[m, Blank[]]]], Power[Plus[Times[csc[Plus[Optional[Pattern[e, Blank[]]], Times[Optional[Pattern[f, Blank[]]], Pattern[x, Blank[]]]]], Optional[Pattern[d, Blank[]]]], Pattern[c, Blank[]]], Optional[Pattern[n, Blank[]]]], Power[Times[Optional[Pattern[g, Blank[]]], sin[Plus[Optional[Pattern[e, Blank[]]], Times[Optional[Pattern[f, Blank[]]], Pattern[x, Blank[]]]]]], Optional[Pattern[p, Blank[]]]]], Pattern[x, Blank[Symbol]]] := Dist[(g*Csc[e + f*x])^p*(g*Sin[e + f*x])^p, Int[((a + b*Csc[e + f*x])^m*(c + d*Csc[e + f*x])^n)/(g*Csc[e + f*x])^p, x], x] /; FreeQ[{a, b, c, d, e, f, g, m, n, p}, x] && NeQ[b*c - a*d, 0] && !IntegerQ[p] && !(IntegerQ[m] && IntegerQ[n])
  2956. Int[Times[Power[Plus[Times[csc[Plus[Optional[Pattern[e, Blank[]]], Times[Optional[Pattern[f, Blank[]]], Pattern[x, Blank[]]]]], Optional[Pattern[d, Blank[]]]], Pattern[c, Blank[]]], Optional[Pattern[n, Blank[]]]], Power[Times[Optional[Pattern[g, Blank[]]], sin[Plus[Optional[Pattern[e, Blank[]]], Times[Optional[Pattern[f, Blank[]]], Pattern[x, Blank[]]]]]], Optional[Pattern[p, Blank[]]]], Power[Plus[Pattern[a, Blank[]], Times[Optional[Pattern[b, Blank[]]], sin[Plus[Optional[Pattern[e, Blank[]]], Times[Optional[Pattern[f, Blank[]]], Pattern[x, Blank[]]]]]]], Optional[Pattern[m, Blank[]]]]], Pattern[x, Blank[Symbol]]] := Dist[g^n, Int[(g*Sin[e + f*x])^(p - n)*(a + b*Sin[e + f*x])^m*(d + c*Sin[e + f*x])^n, x], x] /; FreeQ[{a, b, c, d, e, f, g, m, p}, x] && IntegerQ[n]
  2957. Int[Times[Power[Plus[Times[csc[Plus[Optional[Pattern[e, Blank[]]], Times[Optional[Pattern[f, Blank[]]], Pattern[x, Blank[]]]]], Optional[Pattern[d, Blank[]]]], Pattern[c, Blank[]]], Pattern[n, Blank[]]], Power[sin[Plus[Optional[Pattern[e, Blank[]]], Times[Optional[Pattern[f, Blank[]]], Pattern[x, Blank[]]]]], Optional[Pattern[p, Blank[]]]], Power[Plus[Pattern[a, Blank[]], Times[Optional[Pattern[b, Blank[]]], sin[Plus[Optional[Pattern[e, Blank[]]], Times[Optional[Pattern[f, Blank[]]], Pattern[x, Blank[]]]]]]], Optional[Pattern[m, Blank[]]]]], Pattern[x, Blank[Symbol]]] := Int[((b + a*Csc[e + f*x])^m*(c + d*Csc[e + f*x])^n)/Csc[e + f*x]^(m + p), x] /; FreeQ[{a, b, c, d, e, f, n}, x] && !IntegerQ[n] && IntegerQ[m] && IntegerQ[p]
  2958. Int[Times[Power[Plus[Times[csc[Plus[Optional[Pattern[e, Blank[]]], Times[Optional[Pattern[f, Blank[]]], Pattern[x, Blank[]]]]], Optional[Pattern[d, Blank[]]]], Pattern[c, Blank[]]], Pattern[n, Blank[]]], Power[Times[Optional[Pattern[g, Blank[]]], sin[Plus[Optional[Pattern[e, Blank[]]], Times[Optional[Pattern[f, Blank[]]], Pattern[x, Blank[]]]]]], Pattern[p, Blank[]]], Power[Plus[Pattern[a, Blank[]], Times[Optional[Pattern[b, Blank[]]], sin[Plus[Optional[Pattern[e, Blank[]]], Times[Optional[Pattern[f, Blank[]]], Pattern[x, Blank[]]]]]]], Optional[Pattern[m, Blank[]]]]], Pattern[x, Blank[Symbol]]] := Dist[Csc[e + f*x]^p*(g*Sin[e + f*x])^p, Int[((b + a*Csc[e + f*x])^m*(c + d*Csc[e + f*x])^n)/Csc[e + f*x]^(m + p), x], x] /; FreeQ[{a, b, c, d, e, f, g, n, p}, x] && !IntegerQ[n] && IntegerQ[m] && !IntegerQ[p]
  2959. Int[Times[Power[Plus[Times[csc[Plus[Optional[Pattern[e, Blank[]]], Times[Optional[Pattern[f, Blank[]]], Pattern[x, Blank[]]]]], Optional[Pattern[d, Blank[]]]], Pattern[c, Blank[]]], Pattern[n, Blank[]]], Power[Times[Optional[Pattern[g, Blank[]]], sin[Plus[Optional[Pattern[e, Blank[]]], Times[Optional[Pattern[f, Blank[]]], Pattern[x, Blank[]]]]]], Optional[Pattern[p, Blank[]]]], Power[Plus[Pattern[a, Blank[]], Times[Optional[Pattern[b, Blank[]]], sin[Plus[Optional[Pattern[e, Blank[]]], Times[Optional[Pattern[f, Blank[]]], Pattern[x, Blank[]]]]]]], Pattern[m, Blank[]]]], Pattern[x, Blank[Symbol]]] := Dist[((g*Sin[e + f*x])^n*(c + d*Csc[e + f*x])^n)/(d + c*Sin[e + f*x])^n, Int[(g*Sin[e + f*x])^(p - n)*(a + b*Sin[e + f*x])^m*(d + c*Sin[e + f*x])^n, x], x] /; FreeQ[{a, b, c, d, e, f, g, m, n, p}, x] && !IntegerQ[n] && !IntegerQ[m]
  2960. Int[Times[Power[Times[csc[Plus[Optional[Pattern[e, Blank[]]], Times[Optional[Pattern[f, Blank[]]], Pattern[x, Blank[]]]]], Optional[Pattern[g, Blank[]]]], Optional[Pattern[p, Blank[]]]], Power[Plus[Optional[Pattern[a, Blank[]]], Times[Optional[Pattern[b, Blank[]]], sin[Plus[Optional[Pattern[e, Blank[]]], Times[Optional[Pattern[f, Blank[]]], Pattern[x, Blank[]]]]]]], Optional[Pattern[m, Blank[]]]], Power[Plus[Pattern[c, Blank[]], Times[Optional[Pattern[d, Blank[]]], sin[Plus[Optional[Pattern[e, Blank[]]], Times[Optional[Pattern[f, Blank[]]], Pattern[x, Blank[]]]]]]], Optional[Pattern[n, Blank[]]]]], Pattern[x, Blank[Symbol]]] := Dist[g^(m + n), Int[(g*Csc[e + f*x])^(p - m - n)*(b + a*Csc[e + f*x])^m*(d + c*Csc[e + f*x])^n, x], x] /; FreeQ[{a, b, c, d, e, f, g, p}, x] && NeQ[b*c - a*d, 0] && !IntegerQ[p] && IntegerQ[m] && IntegerQ[n]
  2961. Int[Times[Power[Times[csc[Plus[Optional[Pattern[e, Blank[]]], Times[Optional[Pattern[f, Blank[]]], Pattern[x, Blank[]]]]], Optional[Pattern[g, Blank[]]]], Optional[Pattern[p, Blank[]]]], Power[Plus[Optional[Pattern[a, Blank[]]], Times[Optional[Pattern[b, Blank[]]], sin[Plus[Optional[Pattern[e, Blank[]]], Times[Optional[Pattern[f, Blank[]]], Pattern[x, Blank[]]]]]]], Optional[Pattern[m, Blank[]]]], Power[Plus[Pattern[c, Blank[]], Times[Optional[Pattern[d, Blank[]]], sin[Plus[Optional[Pattern[e, Blank[]]], Times[Optional[Pattern[f, Blank[]]], Pattern[x, Blank[]]]]]]], Optional[Pattern[n, Blank[]]]]], Pattern[x, Blank[Symbol]]] := Dist[(g*Csc[e + f*x])^p*(g*Sin[e + f*x])^p, Int[((a + b*Sin[e + f*x])^m*(c + d*Sin[e + f*x])^n)/(g*Sin[e + f*x])^p, x], x] /; FreeQ[{a, b, c, d, e, f, g, m, n, p}, x] && NeQ[b*c - a*d, 0] && !IntegerQ[p] && !(IntegerQ[m] && IntegerQ[n])
  2962. Int[Times[Power[Times[csc[Plus[Optional[Pattern[e, Blank[]]], Times[Optional[Pattern[f, Blank[]]], Pattern[x, Blank[]]]]], Optional[Pattern[g, Blank[]]]], Optional[Pattern[p, Blank[]]]], Power[Plus[Times[csc[Plus[Optional[Pattern[e, Blank[]]], Times[Optional[Pattern[f, Blank[]]], Pattern[x, Blank[]]]]], Optional[Pattern[d, Blank[]]]], Pattern[c, Blank[]]], Optional[Pattern[n, Blank[]]]], Power[Plus[Pattern[a, Blank[]], Times[Optional[Pattern[b, Blank[]]], sin[Plus[Optional[Pattern[e, Blank[]]], Times[Optional[Pattern[f, Blank[]]], Pattern[x, Blank[]]]]]]], Optional[Pattern[m, Blank[]]]]], Pattern[x, Blank[Symbol]]] := Dist[g^m, Int[(g*Csc[e + f*x])^(p - m)*(b + a*Csc[e + f*x])^m*(c + d*Csc[e + f*x])^n, x], x] /; FreeQ[{a, b, c, d, e, f, g, n, p}, x] && IntegerQ[m]
  2963. Int[Times[Power[csc[Plus[Optional[Pattern[e, Blank[]]], Times[Optional[Pattern[f, Blank[]]], Pattern[x, Blank[]]]]], Optional[Pattern[p, Blank[]]]], Power[Plus[Times[csc[Plus[Optional[Pattern[e, Blank[]]], Times[Optional[Pattern[f, Blank[]]], Pattern[x, Blank[]]]]], Optional[Pattern[d, Blank[]]]], Pattern[c, Blank[]]], Optional[Pattern[n, Blank[]]]], Power[Plus[Pattern[a, Blank[]], Times[Optional[Pattern[b, Blank[]]], sin[Plus[Optional[Pattern[e, Blank[]]], Times[Optional[Pattern[f, Blank[]]], Pattern[x, Blank[]]]]]]], Pattern[m, Blank[]]]], Pattern[x, Blank[Symbol]]] := Int[((a + b*Sin[e + f*x])^m*(d + c*Sin[e + f*x])^n)/Sin[e + f*x]^(n + p), x] /; FreeQ[{a, b, c, d, e, f, m}, x] && !IntegerQ[m] && IntegerQ[n] && IntegerQ[p]
  2964. Int[Times[Power[Times[csc[Plus[Optional[Pattern[e, Blank[]]], Times[Optional[Pattern[f, Blank[]]], Pattern[x, Blank[]]]]], Optional[Pattern[g, Blank[]]]], Pattern[p, Blank[]]], Power[Plus[Times[csc[Plus[Optional[Pattern[e, Blank[]]], Times[Optional[Pattern[f, Blank[]]], Pattern[x, Blank[]]]]], Optional[Pattern[d, Blank[]]]], Pattern[c, Blank[]]], Optional[Pattern[n, Blank[]]]], Power[Plus[Pattern[a, Blank[]], Times[Optional[Pattern[b, Blank[]]], sin[Plus[Optional[Pattern[e, Blank[]]], Times[Optional[Pattern[f, Blank[]]], Pattern[x, Blank[]]]]]]], Pattern[m, Blank[]]]], Pattern[x, Blank[Symbol]]] := Dist[Sin[e + f*x]^p*(g*Csc[e + f*x])^p, Int[((a + b*Sin[e + f*x])^m*(d + c*Sin[e + f*x])^n)/Sin[e + f*x]^(n + p), x], x] /; FreeQ[{a, b, c, d, e, f, g, m, p}, x] && !IntegerQ[m] && IntegerQ[n] && !IntegerQ[p]
  2965. Int[Times[Power[Times[csc[Plus[Optional[Pattern[e, Blank[]]], Times[Optional[Pattern[f, Blank[]]], Pattern[x, Blank[]]]]], Optional[Pattern[g, Blank[]]]], Optional[Pattern[p, Blank[]]]], Power[Plus[Times[csc[Plus[Optional[Pattern[e, Blank[]]], Times[Optional[Pattern[f, Blank[]]], Pattern[x, Blank[]]]]], Optional[Pattern[d, Blank[]]]], Pattern[c, Blank[]]], Pattern[n, Blank[]]], Power[Plus[Pattern[a, Blank[]], Times[Optional[Pattern[b, Blank[]]], sin[Plus[Optional[Pattern[e, Blank[]]], Times[Optional[Pattern[f, Blank[]]], Pattern[x, Blank[]]]]]]], Pattern[m, Blank[]]]], Pattern[x, Blank[Symbol]]] := Dist[((a + b*Sin[e + f*x])^m*(g*Csc[e + f*x])^m)/(b + a*Csc[e + f*x])^m, Int[(g*Csc[e + f*x])^(p - m)*(b + a*Csc[e + f*x])^m*(c + d*Csc[e + f*x])^n, x], x] /; FreeQ[{a, b, c, d, e, f, g, m, n, p}, x] && !IntegerQ[m] && !IntegerQ[n]
  2966. Int[Times[Power[sin[Plus[Optional[Pattern[e, Blank[]]], Times[Optional[Pattern[f, Blank[]]], Pattern[x, Blank[]]]]], Optional[Pattern[n, Blank[]]]], Power[Plus[Pattern[a, Blank[]], Times[Optional[Pattern[b, Blank[]]], sin[Plus[Optional[Pattern[e, Blank[]]], Times[Optional[Pattern[f, Blank[]]], Pattern[x, Blank[]]]]]]], Optional[Pattern[m, Blank[]]]], Plus[Optional[Pattern[A, Blank[]]], Times[Optional[Pattern[B, Blank[]]], sin[Plus[Optional[Pattern[e, Blank[]]], Times[Optional[Pattern[f, Blank[]]], Pattern[x, Blank[]]]]]]]], Pattern[x, Blank[Symbol]]] := Int[ExpandTrig[sin[e + f*x]^n*(a + b*sin[e + f*x])^m*(A + B*sin[e + f*x]), x], x] /; FreeQ[{a, b, e, f, A, B}, x] && EqQ[A*b + a*B, 0] && EqQ[a^2 - b^2, 0] && IntegerQ[m] && IntegerQ[n]
  2967. Int[Times[Power[Plus[Pattern[a, Blank[]], Times[Optional[Pattern[b, Blank[]]], sin[Plus[Optional[Pattern[e, Blank[]]], Times[Optional[Pattern[f, Blank[]]], Pattern[x, Blank[]]]]]]], Optional[Pattern[m, Blank[]]]], Plus[Optional[Pattern[A, Blank[]]], Times[Optional[Pattern[B, Blank[]]], sin[Plus[Optional[Pattern[e, Blank[]]], Times[Optional[Pattern[f, Blank[]]], Pattern[x, Blank[]]]]]]], Power[Plus[Pattern[c, Blank[]], Times[Optional[Pattern[d, Blank[]]], sin[Plus[Optional[Pattern[e, Blank[]]], Times[Optional[Pattern[f, Blank[]]], Pattern[x, Blank[]]]]]]], Optional[Pattern[n, Blank[]]]]], Pattern[x, Blank[Symbol]]] := Dist[a^m*c^m, Int[Cos[e + f*x]^(2*m)*(c + d*Sin[e + f*x])^(n - m)*(A + B*Sin[e + f*x]), x], x] /; FreeQ[{a, b, c, d, e, f, A, B, n}, x] && EqQ[b*c + a*d, 0] && EqQ[a^2 - b^2, 0] && IntegerQ[m] && !(IntegerQ[n] && ((LtQ[m, 0] && GtQ[n, 0]) || LtQ[0, n, m] || LtQ[m, n, 0]))
  2968. Int[Times[Power[Plus[Optional[Pattern[a, Blank[]]], Times[Optional[Pattern[b, Blank[]]], sin[Plus[Optional[Pattern[e, Blank[]]], Times[Optional[Pattern[f, Blank[]]], Pattern[x, Blank[]]]]]]], Optional[Pattern[m, Blank[]]]], Plus[Optional[Pattern[A, Blank[]]], Times[Optional[Pattern[B, Blank[]]], sin[Plus[Optional[Pattern[e, Blank[]]], Times[Optional[Pattern[f, Blank[]]], Pattern[x, Blank[]]]]]]], Plus[Optional[Pattern[c, Blank[]]], Times[Optional[Pattern[d, Blank[]]], sin[Plus[Optional[Pattern[e, Blank[]]], Times[Optional[Pattern[f, Blank[]]], Pattern[x, Blank[]]]]]]]], Pattern[x, Blank[Symbol]]] := Int[(a + b*Sin[e + f*x])^m*(A*c + (B*c + A*d)*Sin[e + f*x] + B*d*Sin[e + f*x]^2), x] /; FreeQ[{a, b, c, d, e, f, A, B, m}, x] && NeQ[b*c - a*d, 0]
  2969. Int[Times[Power[Plus[Pattern[a, Blank[]], Times[Optional[Pattern[b, Blank[]]], sin[Plus[Optional[Pattern[e, Blank[]]], Times[Optional[Pattern[f, Blank[]]], Pattern[x, Blank[]]]]]]], Rational[-1, 2]], Plus[Optional[Pattern[A, Blank[]]], Times[Optional[Pattern[B, Blank[]]], sin[Plus[Optional[Pattern[e, Blank[]]], Times[Optional[Pattern[f, Blank[]]], Pattern[x, Blank[]]]]]]], Power[Plus[Pattern[c, Blank[]], Times[Optional[Pattern[d, Blank[]]], sin[Plus[Optional[Pattern[e, Blank[]]], Times[Optional[Pattern[f, Blank[]]], Pattern[x, Blank[]]]]]]], Rational[-1, 2]]], Pattern[x, Blank[Symbol]]] := Dist[(A*b + a*B)/(2*a*b), Int[Sqrt[a + b*Sin[e + f*x]]/Sqrt[c + d*Sin[e + f*x]], x], x] + Dist[(B*c + A*d)/(2*c*d), Int[Sqrt[c + d*Sin[e + f*x]]/Sqrt[a + b*Sin[e + f*x]], x], x] /; FreeQ[{a, b, c, d, e, f, A, B}, x] && EqQ[b*c + a*d, 0] && EqQ[a^2 - b^2, 0]
  2970. Int[Times[Power[Plus[Pattern[a, Blank[]], Times[Optional[Pattern[b, Blank[]]], sin[Plus[Optional[Pattern[e, Blank[]]], Times[Optional[Pattern[f, Blank[]]], Pattern[x, Blank[]]]]]]], Pattern[m, Blank[]]], Plus[Optional[Pattern[A, Blank[]]], Times[Optional[Pattern[B, Blank[]]], sin[Plus[Optional[Pattern[e, Blank[]]], Times[Optional[Pattern[f, Blank[]]], Pattern[x, Blank[]]]]]]], Power[Plus[Pattern[c, Blank[]], Times[Optional[Pattern[d, Blank[]]], sin[Plus[Optional[Pattern[e, Blank[]]], Times[Optional[Pattern[f, Blank[]]], Pattern[x, Blank[]]]]]]], Optional[Pattern[n, Blank[]]]]], Pattern[x, Blank[Symbol]]] := -Simp[(B*Cos[e + f*x]*(a + b*Sin[e + f*x])^m*(c + d*Sin[e + f*x])^n)/(f*(m + n + 1)), x] /; FreeQ[{a, b, c, d, e, f, A, B, m, n}, x] && EqQ[b*c + a*d, 0] && EqQ[a^2 - b^2, 0] && EqQ[A*b*(m + n + 1) + a*B*(m - n), 0] && NeQ[m, -2^(-1)]
  2971. Int[Times[Power[Plus[Optional[Pattern[a, Blank[]]], Times[Optional[Pattern[b, Blank[]]], sin[Plus[Optional[Pattern[e, Blank[]]], Times[Optional[Pattern[f, Blank[]]], Pattern[x, Blank[]]]]]]], Rational[1, 2]], Plus[Optional[Pattern[A, Blank[]]], Times[Optional[Pattern[B, Blank[]]], sin[Plus[Optional[Pattern[e, Blank[]]], Times[Optional[Pattern[f, Blank[]]], Pattern[x, Blank[]]]]]]], Power[Plus[Pattern[c, Blank[]], Times[Optional[Pattern[d, Blank[]]], sin[Plus[Optional[Pattern[e, Blank[]]], Times[Optional[Pattern[f, Blank[]]], Pattern[x, Blank[]]]]]]], Pattern[n, Blank[]]]], Pattern[x, Blank[Symbol]]] := Dist[B/d, Int[Sqrt[a + b*Sin[e + f*x]]*(c + d*Sin[e + f*x])^(n + 1), x], x] - Dist[(B*c - A*d)/d, Int[Sqrt[a + b*Sin[e + f*x]]*(c + d*Sin[e + f*x])^n, x], x] /; FreeQ[{a, b, c, d, e, f, A, B, n}, x] && EqQ[b*c + a*d, 0] && EqQ[a^2 - b^2, 0]
  2972. Int[Times[Power[Plus[Pattern[a, Blank[]], Times[Optional[Pattern[b, Blank[]]], sin[Plus[Optional[Pattern[e, Blank[]]], Times[Optional[Pattern[f, Blank[]]], Pattern[x, Blank[]]]]]]], Pattern[m, Blank[]]], Plus[Optional[Pattern[A, Blank[]]], Times[Optional[Pattern[B, Blank[]]], sin[Plus[Optional[Pattern[e, Blank[]]], Times[Optional[Pattern[f, Blank[]]], Pattern[x, Blank[]]]]]]], Power[Plus[Pattern[c, Blank[]], Times[Optional[Pattern[d, Blank[]]], sin[Plus[Optional[Pattern[e, Blank[]]], Times[Optional[Pattern[f, Blank[]]], Pattern[x, Blank[]]]]]]], Optional[Pattern[n, Blank[]]]]], Pattern[x, Blank[Symbol]]] := Simp[((A*b - a*B)*Cos[e + f*x]*(a + b*Sin[e + f*x])^m*(c + d*Sin[e + f*x])^n)/(a*f*(2*m + 1)), x] + Dist[(a*B*(m - n) + A*b*(m + n + 1))/(a*b*(2*m + 1)), Int[(a + b*Sin[e + f*x])^(m + 1)*(c + d*Sin[e + f*x])^n, x], x] /; FreeQ[{a, b, c, d, e, f, A, B, m, n}, x] && EqQ[b*c + a*d, 0] && EqQ[a^2 - b^2, 0] && (LtQ[m, -2^(-1)] || (ILtQ[m + n, 0] && !SumSimplerQ[n, 1])) && NeQ[2*m + 1, 0]
  2973. Int[Times[Power[Plus[Pattern[a, Blank[]], Times[Optional[Pattern[b, Blank[]]], sin[Plus[Optional[Pattern[e, Blank[]]], Times[Optional[Pattern[f, Blank[]]], Pattern[x, Blank[]]]]]]], Optional[Pattern[m, Blank[]]]], Plus[Optional[Pattern[A, Blank[]]], Times[Optional[Pattern[B, Blank[]]], sin[Plus[Optional[Pattern[e, Blank[]]], Times[Optional[Pattern[f, Blank[]]], Pattern[x, Blank[]]]]]]], Power[Plus[Pattern[c, Blank[]], Times[Optional[Pattern[d, Blank[]]], sin[Plus[Optional[Pattern[e, Blank[]]], Times[Optional[Pattern[f, Blank[]]], Pattern[x, Blank[]]]]]]], Pattern[n, Blank[]]]], Pattern[x, Blank[Symbol]]] := -Simp[(B*Cos[e + f*x]*(a + b*Sin[e + f*x])^m*(c + d*Sin[e + f*x])^n)/(f*(m + n + 1)), x] - Dist[(B*c*(m - n) - A*d*(m + n + 1))/(d*(m + n + 1)), Int[(a + b*Sin[e + f*x])^m*(c + d*Sin[e + f*x])^n, x], x] /; FreeQ[{a, b, c, d, e, f, A, B, m, n}, x] && EqQ[b*c + a*d, 0] && EqQ[a^2 - b^2, 0] && !LtQ[m, -2^(-1)] && NeQ[m + n + 1, 0]
  2974. Int[Times[Power[Plus[Pattern[a, Blank[]], Times[Optional[Pattern[b, Blank[]]], sin[Plus[Optional[Pattern[e, Blank[]]], Times[Optional[Pattern[f, Blank[]]], Pattern[x, Blank[]]]]]]], Pattern[m, Blank[]]], Plus[Optional[Pattern[A, Blank[]]], Times[Optional[Pattern[B, Blank[]]], sin[Plus[Optional[Pattern[e, Blank[]]], Times[Optional[Pattern[f, Blank[]]], Pattern[x, Blank[]]]]]]], Power[Plus[Optional[Pattern[c, Blank[]]], Times[Optional[Pattern[d, Blank[]]], sin[Plus[Optional[Pattern[e, Blank[]]], Times[Optional[Pattern[f, Blank[]]], Pattern[x, Blank[]]]]]]], Pattern[n, Blank[]]]], Pattern[x, Blank[Symbol]]] := Simp[((B*c - A*d)*Cos[e + f*x]*(a + b*Sin[e + f*x])^m*(c + d*Sin[e + f*x])^(n + 1))/(f*(n + 1)*(c^2 - d^2)), x] /; FreeQ[{a, b, c, d, e, f, A, B, m, n}, x] && NeQ[b*c - a*d, 0] && EqQ[a^2 - b^2, 0] && NeQ[c^2 - d^2, 0] && EqQ[m + n + 2, 0] && EqQ[A*(a*d*m + b*c*(n + 1)) - B*(a*c*m + b*d*(n + 1)), 0]
  2975. Int[Times[Power[Plus[Pattern[a, Blank[]], Times[Optional[Pattern[b, Blank[]]], sin[Plus[Optional[Pattern[e, Blank[]]], Times[Optional[Pattern[f, Blank[]]], Pattern[x, Blank[]]]]]]], Pattern[m, Blank[]]], Plus[Optional[Pattern[A, Blank[]]], Times[Optional[Pattern[B, Blank[]]], sin[Plus[Optional[Pattern[e, Blank[]]], Times[Optional[Pattern[f, Blank[]]], Pattern[x, Blank[]]]]]]], Power[Plus[Optional[Pattern[c, Blank[]]], Times[Optional[Pattern[d, Blank[]]], sin[Plus[Optional[Pattern[e, Blank[]]], Times[Optional[Pattern[f, Blank[]]], Pattern[x, Blank[]]]]]]], Pattern[n, Blank[]]]], Pattern[x, Blank[Symbol]]] := -Simp[(b^2*(B*c - A*d)*Cos[e + f*x]*(a + b*Sin[e + f*x])^(m - 1)*(c + d*Sin[e + f*x])^(n + 1))/(d*f*(n + 1)*(b*c + a*d)), x] - Dist[b/(d*(n + 1)*(b*c + a*d)), Int[(a + b*Sin[e + f*x])^(m - 1)*(c + d*Sin[e + f*x])^(n + 1)*Simp[a*A*d*(m - n - 2) - B*(a*c*(m - 1) + b*d*(n + 1)) - (A*b*d*(m + n + 1) - B*(b*c*m - a*d*(n + 1)))*Sin[e + f*x], x], x], x] /; FreeQ[{a, b, c, d, e, f, A, B}, x] && NeQ[b*c - a*d, 0] && EqQ[a^2 - b^2, 0] && NeQ[c^2 - d^2, 0] && GtQ[m, 1/2] && LtQ[n, -1] && IntegerQ[2*m] && (IntegerQ[2*n] || EqQ[c, 0])
  2976. Int[Times[Power[Plus[Pattern[a, Blank[]], Times[Optional[Pattern[b, Blank[]]], sin[Plus[Optional[Pattern[e, Blank[]]], Times[Optional[Pattern[f, Blank[]]], Pattern[x, Blank[]]]]]]], Pattern[m, Blank[]]], Plus[Optional[Pattern[A, Blank[]]], Times[Optional[Pattern[B, Blank[]]], sin[Plus[Optional[Pattern[e, Blank[]]], Times[Optional[Pattern[f, Blank[]]], Pattern[x, Blank[]]]]]]], Power[Plus[Optional[Pattern[c, Blank[]]], Times[Optional[Pattern[d, Blank[]]], sin[Plus[Optional[Pattern[e, Blank[]]], Times[Optional[Pattern[f, Blank[]]], Pattern[x, Blank[]]]]]]], Pattern[n, Blank[]]]], Pattern[x, Blank[Symbol]]] := -Simp[(b*B*Cos[e + f*x]*(a + b*Sin[e + f*x])^(m - 1)*(c + d*Sin[e + f*x])^(n + 1))/(d*f*(m + n + 1)), x] + Dist[1/(d*(m + n + 1)), Int[(a + b*Sin[e + f*x])^(m - 1)*(c + d*Sin[e + f*x])^n*Simp[a*A*d*(m + n + 1) + B*(a*c*(m - 1) + b*d*(n + 1)) + (A*b*d*(m + n + 1) - B*(b*c*m - a*d*(2*m + n)))*Sin[e + f*x], x], x], x] /; FreeQ[{a, b, c, d, e, f, A, B, n}, x] && NeQ[b*c - a*d, 0] && EqQ[a^2 - b^2, 0] && NeQ[c^2 - d^2, 0] && GtQ[m, 1/2] && !LtQ[n, -1] && IntegerQ[2*m] && (IntegerQ[2*n] || EqQ[c, 0])
  2977. Int[Times[Power[Plus[Pattern[a, Blank[]], Times[Optional[Pattern[b, Blank[]]], sin[Plus[Optional[Pattern[e, Blank[]]], Times[Optional[Pattern[f, Blank[]]], Pattern[x, Blank[]]]]]]], Pattern[m, Blank[]]], Plus[Optional[Pattern[A, Blank[]]], Times[Optional[Pattern[B, Blank[]]], sin[Plus[Optional[Pattern[e, Blank[]]], Times[Optional[Pattern[f, Blank[]]], Pattern[x, Blank[]]]]]]], Power[Plus[Optional[Pattern[c, Blank[]]], Times[Optional[Pattern[d, Blank[]]], sin[Plus[Optional[Pattern[e, Blank[]]], Times[Optional[Pattern[f, Blank[]]], Pattern[x, Blank[]]]]]]], Pattern[n, Blank[]]]], Pattern[x, Blank[Symbol]]] := Simp[((A*b - a*B)*Cos[e + f*x]*(a + b*Sin[e + f*x])^m*(c + d*Sin[e + f*x])^n)/(a*f*(2*m + 1)), x] - Dist[1/(a*b*(2*m + 1)), Int[(a + b*Sin[e + f*x])^(m + 1)*(c + d*Sin[e + f*x])^(n - 1)*Simp[A*(a*d*n - b*c*(m + 1)) - B*(a*c*m + b*d*n) - d*(a*B*(m - n) + A*b*(m + n + 1))*Sin[e + f*x], x], x], x] /; FreeQ[{a, b, c, d, e, f, A, B}, x] && NeQ[b*c - a*d, 0] && EqQ[a^2 - b^2, 0] && NeQ[c^2 - d^2, 0] && LtQ[m, -2^(-1)] && GtQ[n, 0] && IntegerQ[2*m] && (IntegerQ[2*n] || EqQ[c, 0])
  2978. Int[Times[Power[Plus[Pattern[a, Blank[]], Times[Optional[Pattern[b, Blank[]]], sin[Plus[Optional[Pattern[e, Blank[]]], Times[Optional[Pattern[f, Blank[]]], Pattern[x, Blank[]]]]]]], Pattern[m, Blank[]]], Plus[Optional[Pattern[A, Blank[]]], Times[Optional[Pattern[B, Blank[]]], sin[Plus[Optional[Pattern[e, Blank[]]], Times[Optional[Pattern[f, Blank[]]], Pattern[x, Blank[]]]]]]], Power[Plus[Optional[Pattern[c, Blank[]]], Times[Optional[Pattern[d, Blank[]]], sin[Plus[Optional[Pattern[e, Blank[]]], Times[Optional[Pattern[f, Blank[]]], Pattern[x, Blank[]]]]]]], Pattern[n, Blank[]]]], Pattern[x, Blank[Symbol]]] := Simp[(b*(A*b - a*B)*Cos[e + f*x]*(a + b*Sin[e + f*x])^m*(c + d*Sin[e + f*x])^(n + 1))/(a*f*(2*m + 1)*(b*c - a*d)), x] + Dist[1/(a*(2*m + 1)*(b*c - a*d)), Int[(a + b*Sin[e + f*x])^(m + 1)*(c + d*Sin[e + f*x])^n*Simp[B*(a*c*m + b*d*(n + 1)) + A*(b*c*(m + 1) - a*d*(2*m + n + 2)) + d*(A*b - a*B)*(m + n + 2)*Sin[e + f*x], x], x], x] /; FreeQ[{a, b, c, d, e, f, A, B, n}, x] && NeQ[b*c - a*d, 0] && EqQ[a^2 - b^2, 0] && NeQ[c^2 - d^2, 0] && LtQ[m, -2^(-1)] && !GtQ[n, 0] && IntegerQ[2*m] && (IntegerQ[2*n] || EqQ[c, 0])
  2979. Int[Times[Power[Plus[Pattern[a, Blank[]], Times[Optional[Pattern[b, Blank[]]], sin[Plus[Optional[Pattern[e, Blank[]]], Times[Optional[Pattern[f, Blank[]]], Pattern[x, Blank[]]]]]]], Rational[1, 2]], Plus[Optional[Pattern[A, Blank[]]], Times[Optional[Pattern[B, Blank[]]], sin[Plus[Optional[Pattern[e, Blank[]]], Times[Optional[Pattern[f, Blank[]]], Pattern[x, Blank[]]]]]]], Power[Plus[Optional[Pattern[c, Blank[]]], Times[Optional[Pattern[d, Blank[]]], sin[Plus[Optional[Pattern[e, Blank[]]], Times[Optional[Pattern[f, Blank[]]], Pattern[x, Blank[]]]]]]], Pattern[n, Blank[]]]], Pattern[x, Blank[Symbol]]] := Simp[(-2*b*B*Cos[e + f*x]*(c + d*Sin[e + f*x])^(n + 1))/(d*f*(2*n + 3)*Sqrt[a + b*Sin[e + f*x]]), x] /; FreeQ[{a, b, c, d, e, f, A, B, n}, x] && NeQ[b*c - a*d, 0] && EqQ[a^2 - b^2, 0] && NeQ[c^2 - d^2, 0] && EqQ[A*b*d*(2*n + 3) - B*(b*c - 2*a*d*(n + 1)), 0]
  2980. Int[Times[Power[Plus[Pattern[a, Blank[]], Times[Optional[Pattern[b, Blank[]]], sin[Plus[Optional[Pattern[e, Blank[]]], Times[Optional[Pattern[f, Blank[]]], Pattern[x, Blank[]]]]]]], Rational[1, 2]], Plus[Optional[Pattern[A, Blank[]]], Times[Optional[Pattern[B, Blank[]]], sin[Plus[Optional[Pattern[e, Blank[]]], Times[Optional[Pattern[f, Blank[]]], Pattern[x, Blank[]]]]]]], Power[Plus[Optional[Pattern[c, Blank[]]], Times[Optional[Pattern[d, Blank[]]], sin[Plus[Optional[Pattern[e, Blank[]]], Times[Optional[Pattern[f, Blank[]]], Pattern[x, Blank[]]]]]]], Pattern[n, Blank[]]]], Pattern[x, Blank[Symbol]]] := -Simp[(b^2*(B*c - A*d)*Cos[e + f*x]*(c + d*Sin[e + f*x])^(n + 1))/(d*f*(n + 1)*(b*c + a*d)*Sqrt[a + b*Sin[e + f*x]]), x] + Dist[(A*b*d*(2*n + 3) - B*(b*c - 2*a*d*(n + 1)))/(2*d*(n + 1)*(b*c + a*d)), Int[Sqrt[a + b*Sin[e + f*x]]*(c + d*Sin[e + f*x])^(n + 1), x], x] /; FreeQ[{a, b, c, d, e, f, A, B}, x] && NeQ[b*c - a*d, 0] && EqQ[a^2 - b^2, 0] && NeQ[c^2 - d^2, 0] && LtQ[n, -1]
  2981. Int[Times[Power[Plus[Pattern[a, Blank[]], Times[Optional[Pattern[b, Blank[]]], sin[Plus[Optional[Pattern[e, Blank[]]], Times[Optional[Pattern[f, Blank[]]], Pattern[x, Blank[]]]]]]], Rational[1, 2]], Plus[Optional[Pattern[A, Blank[]]], Times[Optional[Pattern[B, Blank[]]], sin[Plus[Optional[Pattern[e, Blank[]]], Times[Optional[Pattern[f, Blank[]]], Pattern[x, Blank[]]]]]]], Power[Plus[Optional[Pattern[c, Blank[]]], Times[Optional[Pattern[d, Blank[]]], sin[Plus[Optional[Pattern[e, Blank[]]], Times[Optional[Pattern[f, Blank[]]], Pattern[x, Blank[]]]]]]], Pattern[n, Blank[]]]], Pattern[x, Blank[Symbol]]] := Simp[(-2*b*B*Cos[e + f*x]*(c + d*Sin[e + f*x])^(n + 1))/(d*f*(2*n + 3)*Sqrt[a + b*Sin[e + f*x]]), x] + Dist[(A*b*d*(2*n + 3) - B*(b*c - 2*a*d*(n + 1)))/(b*d*(2*n + 3)), Int[Sqrt[a + b*Sin[e + f*x]]*(c + d*Sin[e + f*x])^n, x], x] /; FreeQ[{a, b, c, d, e, f, A, B, n}, x] && NeQ[b*c - a*d, 0] && EqQ[a^2 - b^2, 0] && NeQ[c^2 - d^2, 0] && !LtQ[n, -1]
  2982. Int[Times[Power[Plus[Pattern[a, Blank[]], Times[Optional[Pattern[b, Blank[]]], sin[Plus[Optional[Pattern[e, Blank[]]], Times[Optional[Pattern[f, Blank[]]], Pattern[x, Blank[]]]]]]], Rational[-1, 2]], Plus[Optional[Pattern[A, Blank[]]], Times[Optional[Pattern[B, Blank[]]], sin[Plus[Optional[Pattern[e, Blank[]]], Times[Optional[Pattern[f, Blank[]]], Pattern[x, Blank[]]]]]]], Power[Plus[Optional[Pattern[c, Blank[]]], Times[Optional[Pattern[d, Blank[]]], sin[Plus[Optional[Pattern[e, Blank[]]], Times[Optional[Pattern[f, Blank[]]], Pattern[x, Blank[]]]]]]], Rational[-1, 2]]], Pattern[x, Blank[Symbol]]] := Dist[(A*b - a*B)/b, Int[1/(Sqrt[a + b*Sin[e + f*x]]*Sqrt[c + d*Sin[e + f*x]]), x], x] + Dist[B/b, Int[Sqrt[a + b*Sin[e + f*x]]/Sqrt[c + d*Sin[e + f*x]], x], x] /; FreeQ[{a, b, c, d, e, f, A, B}, x] && NeQ[b*c - a*d, 0] && EqQ[a^2 - b^2, 0] && NeQ[c^2 - d^2, 0]
  2983. Int[Times[Power[Plus[Pattern[a, Blank[]], Times[Optional[Pattern[b, Blank[]]], sin[Plus[Optional[Pattern[e, Blank[]]], Times[Optional[Pattern[f, Blank[]]], Pattern[x, Blank[]]]]]]], Pattern[m, Blank[]]], Plus[Optional[Pattern[A, Blank[]]], Times[Optional[Pattern[B, Blank[]]], sin[Plus[Optional[Pattern[e, Blank[]]], Times[Optional[Pattern[f, Blank[]]], Pattern[x, Blank[]]]]]]], Power[Plus[Optional[Pattern[c, Blank[]]], Times[Optional[Pattern[d, Blank[]]], sin[Plus[Optional[Pattern[e, Blank[]]], Times[Optional[Pattern[f, Blank[]]], Pattern[x, Blank[]]]]]]], Pattern[n, Blank[]]]], Pattern[x, Blank[Symbol]]] := -Simp[(B*Cos[e + f*x]*(a + b*Sin[e + f*x])^m*(c + d*Sin[e + f*x])^n)/(f*(m + n + 1)), x] + Dist[1/(b*(m + n + 1)), Int[(a + b*Sin[e + f*x])^m*(c + d*Sin[e + f*x])^(n - 1)*Simp[A*b*c*(m + n + 1) + B*(a*c*m + b*d*n) + (A*b*d*(m + n + 1) + B*(a*d*m + b*c*n))*Sin[e + f*x], x], x], x] /; FreeQ[{a, b, c, d, e, f, A, B, m}, x] && NeQ[b*c - a*d, 0] && EqQ[a^2 - b^2, 0] && NeQ[c^2 - d^2, 0] && GtQ[n, 0] && (IntegerQ[n] || EqQ[m + 1/2, 0])
  2984. Int[Times[Power[Plus[Pattern[a, Blank[]], Times[Optional[Pattern[b, Blank[]]], sin[Plus[Optional[Pattern[e, Blank[]]], Times[Optional[Pattern[f, Blank[]]], Pattern[x, Blank[]]]]]]], Pattern[m, Blank[]]], Plus[Optional[Pattern[A, Blank[]]], Times[Optional[Pattern[B, Blank[]]], sin[Plus[Optional[Pattern[e, Blank[]]], Times[Optional[Pattern[f, Blank[]]], Pattern[x, Blank[]]]]]]], Power[Plus[Optional[Pattern[c, Blank[]]], Times[Optional[Pattern[d, Blank[]]], sin[Plus[Optional[Pattern[e, Blank[]]], Times[Optional[Pattern[f, Blank[]]], Pattern[x, Blank[]]]]]]], Pattern[n, Blank[]]]], Pattern[x, Blank[Symbol]]] := Simp[((B*c - A*d)*Cos[e + f*x]*(a + b*Sin[e + f*x])^m*(c + d*Sin[e + f*x])^(n + 1))/(f*(n + 1)*(c^2 - d^2)), x] + Dist[1/(b*(n + 1)*(c^2 - d^2)), Int[(a + b*Sin[e + f*x])^m*(c + d*Sin[e + f*x])^(n + 1)*Simp[A*(a*d*m + b*c*(n + 1)) - B*(a*c*m + b*d*(n + 1)) + b*(B*c - A*d)*(m + n + 2)*Sin[e + f*x], x], x], x] /; FreeQ[{a, b, c, d, e, f, A, B, m}, x] && NeQ[b*c - a*d, 0] && EqQ[a^2 - b^2, 0] && NeQ[c^2 - d^2, 0] && LtQ[n, -1] && (IntegerQ[n] || EqQ[m + 1/2, 0])
  2985. Int[Times[Power[Plus[Pattern[a, Blank[]], Times[Optional[Pattern[b, Blank[]]], sin[Plus[Optional[Pattern[e, Blank[]]], Times[Optional[Pattern[f, Blank[]]], Pattern[x, Blank[]]]]]]], Rational[-1, 2]], Plus[Optional[Pattern[A, Blank[]]], Times[Optional[Pattern[B, Blank[]]], sin[Plus[Optional[Pattern[e, Blank[]]], Times[Optional[Pattern[f, Blank[]]], Pattern[x, Blank[]]]]]]], Power[Plus[Optional[Pattern[c, Blank[]]], Times[Optional[Pattern[d, Blank[]]], sin[Plus[Optional[Pattern[e, Blank[]]], Times[Optional[Pattern[f, Blank[]]], Pattern[x, Blank[]]]]]]], -1]], Pattern[x, Blank[Symbol]]] := Dist[(A*b - a*B)/(b*c - a*d), Int[1/Sqrt[a + b*Sin[e + f*x]], x], x] + Dist[(B*c - A*d)/(b*c - a*d), Int[Sqrt[a + b*Sin[e + f*x]]/(c + d*Sin[e + f*x]), x], x] /; FreeQ[{a, b, c, d, e, f, A, B}, x] && NeQ[b*c - a*d, 0] && EqQ[a^2 - b^2, 0] && NeQ[c^2 - d^2, 0]
  2986. Int[Times[Power[Plus[Pattern[a, Blank[]], Times[Optional[Pattern[b, Blank[]]], sin[Plus[Optional[Pattern[e, Blank[]]], Times[Optional[Pattern[f, Blank[]]], Pattern[x, Blank[]]]]]]], Pattern[m, Blank[]]], Plus[Optional[Pattern[A, Blank[]]], Times[Optional[Pattern[B, Blank[]]], sin[Plus[Optional[Pattern[e, Blank[]]], Times[Optional[Pattern[f, Blank[]]], Pattern[x, Blank[]]]]]]], Power[Plus[Optional[Pattern[c, Blank[]]], Times[Optional[Pattern[d, Blank[]]], sin[Plus[Optional[Pattern[e, Blank[]]], Times[Optional[Pattern[f, Blank[]]], Pattern[x, Blank[]]]]]]], -1]], Pattern[x, Blank[Symbol]]] := Dist[B/d, Int[(a + b*Sin[e + f*x])^m, x], x] - Dist[(B*c - A*d)/d, Int[(a + b*Sin[e + f*x])^m/(c + d*Sin[e + f*x]), x], x] /; FreeQ[{a, b, c, d, e, f, A, B, m}, x] && NeQ[b*c - a*d, 0] && EqQ[a^2 - b^2, 0] && NeQ[c^2 - d^2, 0] && NeQ[m + 1/2, 0]
  2987. Int[Times[Power[Plus[Pattern[a, Blank[]], Times[Optional[Pattern[b, Blank[]]], sin[Plus[Optional[Pattern[e, Blank[]]], Times[Optional[Pattern[f, Blank[]]], Pattern[x, Blank[]]]]]]], Optional[Pattern[m, Blank[]]]], Plus[Optional[Pattern[A, Blank[]]], Times[Optional[Pattern[B, Blank[]]], sin[Plus[Optional[Pattern[e, Blank[]]], Times[Optional[Pattern[f, Blank[]]], Pattern[x, Blank[]]]]]]], Power[Plus[Optional[Pattern[c, Blank[]]], Times[Optional[Pattern[d, Blank[]]], sin[Plus[Optional[Pattern[e, Blank[]]], Times[Optional[Pattern[f, Blank[]]], Pattern[x, Blank[]]]]]]], Pattern[n, Blank[]]]], Pattern[x, Blank[Symbol]]] := Dist[(A*b - a*B)/b, Int[(a + b*Sin[e + f*x])^m*(c + d*Sin[e + f*x])^n, x], x] + Dist[B/b, Int[(a + b*Sin[e + f*x])^(m + 1)*(c + d*Sin[e + f*x])^n, x], x] /; FreeQ[{a, b, c, d, e, f, A, B, m, n}, x] && NeQ[b*c - a*d, 0] && EqQ[a^2 - b^2, 0] && NeQ[c^2 - d^2, 0] && NeQ[A*b + a*B, 0]
  2988. Int[Times[Power[Plus[Optional[Pattern[a, Blank[]]], Times[Optional[Pattern[b, Blank[]]], sin[Plus[Optional[Pattern[e, Blank[]]], Times[Optional[Pattern[f, Blank[]]], Pattern[x, Blank[]]]]]]], 2], Plus[Optional[Pattern[A, Blank[]]], Times[Optional[Pattern[B, Blank[]]], sin[Plus[Optional[Pattern[e, Blank[]]], Times[Optional[Pattern[f, Blank[]]], Pattern[x, Blank[]]]]]]], Power[Plus[Optional[Pattern[c, Blank[]]], Times[Optional[Pattern[d, Blank[]]], sin[Plus[Optional[Pattern[e, Blank[]]], Times[Optional[Pattern[f, Blank[]]], Pattern[x, Blank[]]]]]]], Pattern[n, Blank[]]]], Pattern[x, Blank[Symbol]]] := Simp[((B*c - A*d)*(b*c - a*d)^2*Cos[e + f*x]*(c + d*Sin[e + f*x])^(n + 1))/(f*d^2*(n + 1)*(c^2 - d^2)), x] - Dist[1/(d^2*(n + 1)*(c^2 - d^2)), Int[(c + d*Sin[e + f*x])^(n + 1)*Simp[d*(n + 1)*(B*(b*c - a*d)^2 - A*d*(a^2*c + b^2*c - 2*a*b*d)) - ((B*c - A*d)*(a^2*d^2*(n + 2) + b^2*(c^2 + d^2*(n + 1))) + 2*a*b*d*(A*c*d*(n + 2) - B*(c^2 + d^2*(n + 1))))*Sin[e + f*x] - b^2*B*d*(n + 1)*(c^2 - d^2)*Sin[e + f*x]^2, x], x], x] /; FreeQ[{a, b, c, d, e, f, A, B}, x] && NeQ[b*c - a*d, 0] && NeQ[a^2 - b^2, 0] && NeQ[c^2 - d^2, 0] && LtQ[n, -1]
  2989. Int[Times[Power[Plus[Optional[Pattern[a, Blank[]]], Times[Optional[Pattern[b, Blank[]]], sin[Plus[Optional[Pattern[e, Blank[]]], Times[Optional[Pattern[f, Blank[]]], Pattern[x, Blank[]]]]]]], Pattern[m, Blank[]]], Plus[Optional[Pattern[A, Blank[]]], Times[Optional[Pattern[B, Blank[]]], sin[Plus[Optional[Pattern[e, Blank[]]], Times[Optional[Pattern[f, Blank[]]], Pattern[x, Blank[]]]]]]], Power[Plus[Optional[Pattern[c, Blank[]]], Times[Optional[Pattern[d, Blank[]]], sin[Plus[Optional[Pattern[e, Blank[]]], Times[Optional[Pattern[f, Blank[]]], Pattern[x, Blank[]]]]]]], Pattern[n, Blank[]]]], Pattern[x, Blank[Symbol]]] := -Simp[((b*c - a*d)*(B*c - A*d)*Cos[e + f*x]*(a + b*Sin[e + f*x])^(m - 1)*(c + d*Sin[e + f*x])^(n + 1))/(d*f*(n + 1)*(c^2 - d^2)), x] + Dist[1/(d*(n + 1)*(c^2 - d^2)), Int[(a + b*Sin[e + f*x])^(m - 2)*(c + d*Sin[e + f*x])^(n + 1)*Simp[b*(b*c - a*d)*(B*c - A*d)*(m - 1) + a*d*(a*A*c + b*B*c - (A*b + a*B)*d)*(n + 1) + (b*(b*d*(B*c - A*d) + a*(A*c*d + B*(c^2 - 2*d^2)))*(n + 1) - a*(b*c - a*d)*(B*c - A*d)*(n + 2))*Sin[e + f*x] + b*(d*(A*b*c + a*B*c - a*A*d)*(m + n + 1) - b*B*(c^2*m + d^2*(n + 1)))*Sin[e + f*x]^2, x], x], x] /; FreeQ[{a, b, c, d, e, f, A, B}, x] && NeQ[b*c - a*d, 0] && NeQ[a^2 - b^2, 0] && NeQ[c^2 - d^2, 0] && GtQ[m, 1] && LtQ[n, -1]
  2990. Int[Times[Power[Plus[Optional[Pattern[a, Blank[]]], Times[Optional[Pattern[b, Blank[]]], sin[Plus[Optional[Pattern[e, Blank[]]], Times[Optional[Pattern[f, Blank[]]], Pattern[x, Blank[]]]]]]], Pattern[m, Blank[]]], Plus[Optional[Pattern[A, Blank[]]], Times[Optional[Pattern[B, Blank[]]], sin[Plus[Optional[Pattern[e, Blank[]]], Times[Optional[Pattern[f, Blank[]]], Pattern[x, Blank[]]]]]]], Power[Plus[Optional[Pattern[c, Blank[]]], Times[Optional[Pattern[d, Blank[]]], sin[Plus[Optional[Pattern[e, Blank[]]], Times[Optional[Pattern[f, Blank[]]], Pattern[x, Blank[]]]]]]], Pattern[n, Blank[]]]], Pattern[x, Blank[Symbol]]] := -Simp[(b*B*Cos[e + f*x]*(a + b*Sin[e + f*x])^(m - 1)*(c + d*Sin[e + f*x])^(n + 1))/(d*f*(m + n + 1)), x] + Dist[1/(d*(m + n + 1)), Int[(a + b*Sin[e + f*x])^(m - 2)*(c + d*Sin[e + f*x])^n*Simp[a^2*A*d*(m + n + 1) + b*B*(b*c*(m - 1) + a*d*(n + 1)) + (a*d*(2*A*b + a*B)*(m + n + 1) - b*B*(a*c - b*d*(m + n)))*Sin[e + f*x] + b*(A*b*d*(m + n + 1) - B*(b*c*m - a*d*(2*m + n)))*Sin[e + f*x]^2, x], x], x] /; FreeQ[{a, b, c, d, e, f, A, B, n}, x] && NeQ[b*c - a*d, 0] && NeQ[a^2 - b^2, 0] && NeQ[c^2 - d^2, 0] && GtQ[m, 1] && !(IGtQ[n, 1] && ( !IntegerQ[m] || (EqQ[a, 0] && NeQ[c, 0])))
  2991. Int[Times[Power[Times[Optional[Pattern[b, Blank[]]], sin[Plus[Optional[Pattern[e, Blank[]]], Times[Optional[Pattern[f, Blank[]]], Pattern[x, Blank[]]]]]], Rational[-3, 2]], Plus[Optional[Pattern[A, Blank[]]], Times[Optional[Pattern[B, Blank[]]], sin[Plus[Optional[Pattern[e, Blank[]]], Times[Optional[Pattern[f, Blank[]]], Pattern[x, Blank[]]]]]]], Power[Plus[Pattern[c, Blank[]], Times[Optional[Pattern[d, Blank[]]], sin[Plus[Optional[Pattern[e, Blank[]]], Times[Optional[Pattern[f, Blank[]]], Pattern[x, Blank[]]]]]]], Rational[1, 2]]], Pattern[x, Blank[Symbol]]] := Dist[(B*d)/b^2, Int[Sqrt[b*Sin[e + f*x]]/Sqrt[c + d*Sin[e + f*x]], x], x] + Int[(A*c + (B*c + A*d)*Sin[e + f*x])/((b*Sin[e + f*x])^(3/2)*Sqrt[c + d*Sin[e + f*x]]), x] /; FreeQ[{b, c, d, e, f, A, B}, x] && NeQ[c^2 - d^2, 0]
  2992. Int[Times[Power[Plus[Pattern[a, Blank[]], Times[Optional[Pattern[b, Blank[]]], sin[Plus[Optional[Pattern[e, Blank[]]], Times[Optional[Pattern[f, Blank[]]], Pattern[x, Blank[]]]]]]], Rational[-3, 2]], Plus[Optional[Pattern[A, Blank[]]], Times[Optional[Pattern[B, Blank[]]], sin[Plus[Optional[Pattern[e, Blank[]]], Times[Optional[Pattern[f, Blank[]]], Pattern[x, Blank[]]]]]]], Power[Plus[Optional[Pattern[c, Blank[]]], Times[Optional[Pattern[d, Blank[]]], sin[Plus[Optional[Pattern[e, Blank[]]], Times[Optional[Pattern[f, Blank[]]], Pattern[x, Blank[]]]]]]], Rational[1, 2]]], Pattern[x, Blank[Symbol]]] := Dist[B/b, Int[Sqrt[c + d*Sin[e + f*x]]/Sqrt[a + b*Sin[e + f*x]], x], x] + Dist[(A*b - a*B)/b, Int[Sqrt[c + d*Sin[e + f*x]]/(a + b*Sin[e + f*x])^(3/2), x], x] /; FreeQ[{a, b, c, d, e, f, A, B}, x] && NeQ[b*c - a*d, 0] && NeQ[a^2 - b^2, 0] && NeQ[c^2 - d^2, 0]
  2993. Int[Times[Power[Times[Optional[Pattern[d, Blank[]]], sin[Plus[Optional[Pattern[e, Blank[]]], Times[Optional[Pattern[f, Blank[]]], Pattern[x, Blank[]]]]]], Rational[-1, 2]], Power[Plus[Pattern[a, Blank[]], Times[Optional[Pattern[b, Blank[]]], sin[Plus[Optional[Pattern[e, Blank[]]], Times[Optional[Pattern[f, Blank[]]], Pattern[x, Blank[]]]]]]], Rational[-3, 2]], Plus[Optional[Pattern[A, Blank[]]], Times[Optional[Pattern[B, Blank[]]], sin[Plus[Optional[Pattern[e, Blank[]]], Times[Optional[Pattern[f, Blank[]]], Pattern[x, Blank[]]]]]]]], Pattern[x, Blank[Symbol]]] := Simp[(2*(A*b - a*B)*Cos[e + f*x])/(f*(a^2 - b^2)*Sqrt[a + b*Sin[e + f*x]]*Sqrt[d*Sin[e + f*x]]), x] + Dist[d/(a^2 - b^2), Int[(A*b - a*B + (a*A - b*B)*Sin[e + f*x])/(Sqrt[a + b*Sin[e + f*x]]*(d*Sin[e + f*x])^(3/2)), x], x] /; FreeQ[{a, b, d, e, f, A, B}, x] && NeQ[a^2 - b^2, 0]
  2994. Int[Times[Power[Times[Optional[Pattern[b, Blank[]]], sin[Plus[Optional[Pattern[e, Blank[]]], Times[Optional[Pattern[f, Blank[]]], Pattern[x, Blank[]]]]]], Rational[-3, 2]], Plus[Pattern[A, Blank[]], Times[Optional[Pattern[B, Blank[]]], sin[Plus[Optional[Pattern[e, Blank[]]], Times[Optional[Pattern[f, Blank[]]], Pattern[x, Blank[]]]]]]], Power[Plus[Pattern[c, Blank[]], Times[Optional[Pattern[d, Blank[]]], sin[Plus[Optional[Pattern[e, Blank[]]], Times[Optional[Pattern[f, Blank[]]], Pattern[x, Blank[]]]]]]], Rational[-1, 2]]], Pattern[x, Blank[Symbol]]] := Simp[(-2*A*(c - d)*Tan[e + f*x]*Rt[(c + d)/b, 2]*Sqrt[(c*(1 + Csc[e + f*x]))/(c - d)]*Sqrt[(c*(1 - Csc[e + f*x]))/(c + d)]*EllipticE[ArcSin[Sqrt[c + d*Sin[e + f*x]]/(Sqrt[b*Sin[e + f*x]]*Rt[(c + d)/b, 2])], -((c + d)/(c - d))])/(f*b*c^2), x] /; FreeQ[{b, c, d, e, f, A, B}, x] && NeQ[c^2 - d^2, 0] && EqQ[A, B] && PosQ[(c + d)/b]
  2995. Int[Times[Power[Times[Optional[Pattern[b, Blank[]]], sin[Plus[Optional[Pattern[e, Blank[]]], Times[Optional[Pattern[f, Blank[]]], Pattern[x, Blank[]]]]]], Rational[-3, 2]], Plus[Pattern[A, Blank[]], Times[Optional[Pattern[B, Blank[]]], sin[Plus[Optional[Pattern[e, Blank[]]], Times[Optional[Pattern[f, Blank[]]], Pattern[x, Blank[]]]]]]], Power[Plus[Pattern[c, Blank[]], Times[Optional[Pattern[d, Blank[]]], sin[Plus[Optional[Pattern[e, Blank[]]], Times[Optional[Pattern[f, Blank[]]], Pattern[x, Blank[]]]]]]], Rational[-1, 2]]], Pattern[x, Blank[Symbol]]] := -Dist[Sqrt[-(b*Sin[e + f*x])]/Sqrt[b*Sin[e + f*x]], Int[(A + B*Sin[e + f*x])/((-(b*Sin[e + f*x]))^(3/2)*Sqrt[c + d*Sin[e + f*x]]), x], x] /; FreeQ[{b, c, d, e, f, A, B}, x] && NeQ[c^2 - d^2, 0] && EqQ[A, B] && NegQ[(c + d)/b]
  2996. Int[Times[Power[Plus[Pattern[a, Blank[]], Times[Optional[Pattern[b, Blank[]]], sin[Plus[Optional[Pattern[e, Blank[]]], Times[Optional[Pattern[f, Blank[]]], Pattern[x, Blank[]]]]]]], Rational[-3, 2]], Plus[Pattern[A, Blank[]], Times[Optional[Pattern[B, Blank[]]], sin[Plus[Optional[Pattern[e, Blank[]]], Times[Optional[Pattern[f, Blank[]]], Pattern[x, Blank[]]]]]]], Power[Plus[Pattern[c, Blank[]], Times[Optional[Pattern[d, Blank[]]], sin[Plus[Optional[Pattern[e, Blank[]]], Times[Optional[Pattern[f, Blank[]]], Pattern[x, Blank[]]]]]]], Rational[-1, 2]]], Pattern[x, Blank[Symbol]]] := Simp[(-2*A*(c - d)*(a + b*Sin[e + f*x])*Sqrt[((b*c - a*d)*(1 + Sin[e + f*x]))/((c - d)*(a + b*Sin[e + f*x]))]*Sqrt[-(((b*c - a*d)*(1 - Sin[e + f*x]))/((c + d)*(a + b*Sin[e + f*x])))]*EllipticE[ArcSin[(Rt[(a + b)/(c + d), 2]*Sqrt[c + d*Sin[e + f*x]])/Sqrt[a + b*Sin[e + f*x]]], ((a - b)*(c + d))/((a + b)*(c - d))])/(f*(b*c - a*d)^2*Rt[(a + b)/(c + d), 2]*Cos[e + f*x]), x] /; FreeQ[{a, b, c, d, e, f, A, B}, x] && NeQ[b*c - a*d, 0] && NeQ[a^2 - b^2, 0] && NeQ[c^2 - d^2, 0] && EqQ[A, B] && PosQ[(a + b)/(c + d)]
  2997. Int[Times[Power[Plus[Pattern[a, Blank[]], Times[Optional[Pattern[b, Blank[]]], sin[Plus[Optional[Pattern[e, Blank[]]], Times[Optional[Pattern[f, Blank[]]], Pattern[x, Blank[]]]]]]], Rational[-3, 2]], Plus[Pattern[A, Blank[]], Times[Optional[Pattern[B, Blank[]]], sin[Plus[Optional[Pattern[e, Blank[]]], Times[Optional[Pattern[f, Blank[]]], Pattern[x, Blank[]]]]]]], Power[Plus[Pattern[c, Blank[]], Times[Optional[Pattern[d, Blank[]]], sin[Plus[Optional[Pattern[e, Blank[]]], Times[Optional[Pattern[f, Blank[]]], Pattern[x, Blank[]]]]]]], Rational[-1, 2]]], Pattern[x, Blank[Symbol]]] := Dist[Sqrt[-c - d*Sin[e + f*x]]/Sqrt[c + d*Sin[e + f*x]], Int[(A + B*Sin[e + f*x])/((a + b*Sin[e + f*x])^(3/2)*Sqrt[-c - d*Sin[e + f*x]]), x], x] /; FreeQ[{a, b, c, d, e, f, A, B}, x] && NeQ[b*c - a*d, 0] && NeQ[a^2 - b^2, 0] && NeQ[c^2 - d^2, 0] && EqQ[A, B] && NegQ[(a + b)/(c + d)]
  2998. Int[Times[Power[Plus[Optional[Pattern[a, Blank[]]], Times[Optional[Pattern[b, Blank[]]], sin[Plus[Optional[Pattern[e, Blank[]]], Times[Optional[Pattern[f, Blank[]]], Pattern[x, Blank[]]]]]]], Rational[-3, 2]], Plus[Optional[Pattern[A, Blank[]]], Times[Optional[Pattern[B, Blank[]]], sin[Plus[Optional[Pattern[e, Blank[]]], Times[Optional[Pattern[f, Blank[]]], Pattern[x, Blank[]]]]]]], Power[Plus[Pattern[c, Blank[]], Times[Optional[Pattern[d, Blank[]]], sin[Plus[Optional[Pattern[e, Blank[]]], Times[Optional[Pattern[f, Blank[]]], Pattern[x, Blank[]]]]]]], Rational[-1, 2]]], Pattern[x, Blank[Symbol]]] := Dist[(A - B)/(a - b), Int[1/(Sqrt[a + b*Sin[e + f*x]]*Sqrt[c + d*Sin[e + f*x]]), x], x] - Dist[(A*b - a*B)/(a - b), Int[(1 + Sin[e + f*x])/((a + b*Sin[e + f*x])^(3/2)*Sqrt[c + d*Sin[e + f*x]]), x], x] /; FreeQ[{a, b, c, d, e, f, A, B}, x] && NeQ[b*c - a*d, 0] && NeQ[a^2 - b^2, 0] && NeQ[c^2 - d^2, 0] && NeQ[A, B]
  2999. Int[Times[Power[Plus[Optional[Pattern[a, Blank[]]], Times[Optional[Pattern[b, Blank[]]], sin[Plus[Optional[Pattern[e, Blank[]]], Times[Optional[Pattern[f, Blank[]]], Pattern[x, Blank[]]]]]]], Pattern[m, Blank[]]], Plus[Optional[Pattern[A, Blank[]]], Times[Optional[Pattern[B, Blank[]]], sin[Plus[Optional[Pattern[e, Blank[]]], Times[Optional[Pattern[f, Blank[]]], Pattern[x, Blank[]]]]]]], Power[Plus[Optional[Pattern[c, Blank[]]], Times[Optional[Pattern[d, Blank[]]], sin[Plus[Optional[Pattern[e, Blank[]]], Times[Optional[Pattern[f, Blank[]]], Pattern[x, Blank[]]]]]]], Pattern[n, Blank[]]]], Pattern[x, Blank[Symbol]]] := Simp[((B*a - A*b)*Cos[e + f*x]*(a + b*Sin[e + f*x])^(m + 1)*(c + d*Sin[e + f*x])^n)/(f*(m + 1)*(a^2 - b^2)), x] + Dist[1/((m + 1)*(a^2 - b^2)), Int[(a + b*Sin[e + f*x])^(m + 1)*(c + d*Sin[e + f*x])^(n - 1)*Simp[c*(a*A - b*B)*(m + 1) + d*n*(A*b - a*B) + (d*(a*A - b*B)*(m + 1) - c*(A*b - a*B)*(m + 2))*Sin[e + f*x] - d*(A*b - a*B)*(m + n + 2)*Sin[e + f*x]^2, x], x], x] /; FreeQ[{a, b, c, d, e, f, A, B}, x] && NeQ[b*c - a*d, 0] && NeQ[a^2 - b^2, 0] && NeQ[c^2 - d^2, 0] && LtQ[m, -1] && GtQ[n, 0]
  3000. Int[Times[Power[Plus[Optional[Pattern[a, Blank[]]], Times[Optional[Pattern[b, Blank[]]], sin[Plus[Optional[Pattern[e, Blank[]]], Times[Optional[Pattern[f, Blank[]]], Pattern[x, Blank[]]]]]]], Pattern[m, Blank[]]], Plus[Optional[Pattern[A, Blank[]]], Times[Optional[Pattern[B, Blank[]]], sin[Plus[Optional[Pattern[e, Blank[]]], Times[Optional[Pattern[f, Blank[]]], Pattern[x, Blank[]]]]]]], Power[Plus[Optional[Pattern[c, Blank[]]], Times[Optional[Pattern[d, Blank[]]], sin[Plus[Optional[Pattern[e, Blank[]]], Times[Optional[Pattern[f, Blank[]]], Pattern[x, Blank[]]]]]]], Pattern[n, Blank[]]]], Pattern[x, Blank[Symbol]]] := -Simp[((A*b^2 - a*b*B)*Cos[e + f*x]*(a + b*Sin[e + f*x])^(m + 1)*(c + d*Sin[e + f*x])^(1 + n))/(f*(m + 1)*(b*c - a*d)*(a^2 - b^2)), x] + Dist[1/((m + 1)*(b*c - a*d)*(a^2 - b^2)), Int[(a + b*Sin[e + f*x])^(m + 1)*(c + d*Sin[e + f*x])^n*Simp[(a*A - b*B)*(b*c - a*d)*(m + 1) + b*d*(A*b - a*B)*(m + n + 2) + (A*b - a*B)*(a*d*(m + 1) - b*c*(m + 2))*Sin[e + f*x] - b*d*(A*b - a*B)*(m + n + 3)*Sin[e + f*x]^2, x], x], x] /; FreeQ[{a, b, c, d, e, f, A, B, n}, x] && NeQ[b*c - a*d, 0] && NeQ[a^2 - b^2, 0] && NeQ[c^2 - d^2, 0] && RationalQ[m] && m < -1 && ((EqQ[a, 0] && IntegerQ[m] && !IntegerQ[n]) || !(IntegerQ[2*n] && LtQ[n, -1] && ((IntegerQ[n] && !IntegerQ[m]) || EqQ[a, 0])))
  3001. Int[Times[Power[Plus[Optional[Pattern[a, Blank[]]], Times[Optional[Pattern[b, Blank[]]], sin[Plus[Optional[Pattern[e, Blank[]]], Times[Optional[Pattern[f, Blank[]]], Pattern[x, Blank[]]]]]]], -1], Plus[Optional[Pattern[A, Blank[]]], Times[Optional[Pattern[B, Blank[]]], sin[Plus[Optional[Pattern[e, Blank[]]], Times[Optional[Pattern[f, Blank[]]], Pattern[x, Blank[]]]]]]], Power[Plus[Optional[Pattern[c, Blank[]]], Times[Optional[Pattern[d, Blank[]]], sin[Plus[Optional[Pattern[e, Blank[]]], Times[Optional[Pattern[f, Blank[]]], Pattern[x, Blank[]]]]]]], -1]], Pattern[x, Blank[Symbol]]] := Dist[(A*b - a*B)/(b*c - a*d), Int[1/(a + b*Sin[e + f*x]), x], x] + Dist[(B*c - A*d)/(b*c - a*d), Int[1/(c + d*Sin[e + f*x]), x], x] /; FreeQ[{a, b, c, d, e, f, A, B}, x] && NeQ[b*c - a*d, 0] && NeQ[a^2 - b^2, 0] && NeQ[c^2 - d^2, 0]
  3002. Int[Times[Power[Plus[Optional[Pattern[a, Blank[]]], Times[Optional[Pattern[b, Blank[]]], sin[Plus[Optional[Pattern[e, Blank[]]], Times[Optional[Pattern[f, Blank[]]], Pattern[x, Blank[]]]]]]], Pattern[m, Blank[]]], Plus[Optional[Pattern[A, Blank[]]], Times[Optional[Pattern[B, Blank[]]], sin[Plus[Optional[Pattern[e, Blank[]]], Times[Optional[Pattern[f, Blank[]]], Pattern[x, Blank[]]]]]]], Power[Plus[Optional[Pattern[c, Blank[]]], Times[Optional[Pattern[d, Blank[]]], sin[Plus[Optional[Pattern[e, Blank[]]], Times[Optional[Pattern[f, Blank[]]], Pattern[x, Blank[]]]]]]], -1]], Pattern[x, Blank[Symbol]]] := Dist[B/d, Int[(a + b*Sin[e + f*x])^m, x], x] - Dist[(B*c - A*d)/d, Int[(a + b*Sin[e + f*x])^m/(c + d*Sin[e + f*x]), x], x] /; FreeQ[{a, b, c, d, e, f, A, B, m}, x] && NeQ[b*c - a*d, 0] && NeQ[a^2 - b^2, 0] && NeQ[c^2 - d^2, 0]
  3003. Int[Times[Power[Plus[Optional[Pattern[a, Blank[]]], Times[Optional[Pattern[b, Blank[]]], sin[Plus[Optional[Pattern[e, Blank[]]], Times[Optional[Pattern[f, Blank[]]], Pattern[x, Blank[]]]]]]], Rational[1, 2]], Plus[Optional[Pattern[A, Blank[]]], Times[Optional[Pattern[B, Blank[]]], sin[Plus[Optional[Pattern[e, Blank[]]], Times[Optional[Pattern[f, Blank[]]], Pattern[x, Blank[]]]]]]], Power[Plus[Optional[Pattern[c, Blank[]]], Times[Optional[Pattern[d, Blank[]]], sin[Plus[Optional[Pattern[e, Blank[]]], Times[Optional[Pattern[f, Blank[]]], Pattern[x, Blank[]]]]]]], Pattern[n, Blank[]]]], Pattern[x, Blank[Symbol]]] := Simp[(-2*B*Cos[e + f*x]*Sqrt[a + b*Sin[e + f*x]]*(c + d*Sin[e + f*x])^n)/(f*(2*n + 3)), x] + Dist[1/(2*n + 3), Int[((c + d*Sin[e + f*x])^(n - 1)*Simp[a*A*c*(2*n + 3) + B*(b*c + 2*a*d*n) + (B*(a*c + b*d)*(2*n + 1) + A*(b*c + a*d)*(2*n + 3))*Sin[e + f*x] + (A*b*d*(2*n + 3) + B*(a*d + 2*b*c*n))*Sin[e + f*x]^2, x])/Sqrt[a + b*Sin[e + f*x]], x], x] /; FreeQ[{a, b, c, d, e, f, A, B}, x] && NeQ[b*c - a*d, 0] && NeQ[a^2 - b^2, 0] && NeQ[c^2 - d^2, 0] && EqQ[n^2, 1/4]
  3004. Int[Times[Power[sin[Plus[Optional[Pattern[e, Blank[]]], Times[Optional[Pattern[f, Blank[]]], Pattern[x, Blank[]]]]], Rational[-1, 2]], Power[Plus[Pattern[a, Blank[]], Times[Optional[Pattern[b, Blank[]]], sin[Plus[Optional[Pattern[e, Blank[]]], Times[Optional[Pattern[f, Blank[]]], Pattern[x, Blank[]]]]]]], Rational[-1, 2]], Plus[Pattern[A, Blank[]], Times[Optional[Pattern[B, Blank[]]], sin[Plus[Optional[Pattern[e, Blank[]]], Times[Optional[Pattern[f, Blank[]]], Pattern[x, Blank[]]]]]]]], Pattern[x, Blank[Symbol]]] := Simp[(4*A*EllipticPi[-1, -ArcSin[Cos[e + f*x]/(1 + Sin[e + f*x])], -((a - b)/(a + b))])/(f*Sqrt[a + b]), x] /; FreeQ[{a, b, e, f, A, B}, x] && GtQ[b, 0] && GtQ[b^2 - a^2, 0] && EqQ[A, B]
  3005. Int[Times[Power[Times[Pattern[d, Blank[]], sin[Plus[Optional[Pattern[e, Blank[]]], Times[Optional[Pattern[f, Blank[]]], Pattern[x, Blank[]]]]]], Rational[-1, 2]], Power[Plus[Pattern[a, Blank[]], Times[Optional[Pattern[b, Blank[]]], sin[Plus[Optional[Pattern[e, Blank[]]], Times[Optional[Pattern[f, Blank[]]], Pattern[x, Blank[]]]]]]], Rational[-1, 2]], Plus[Pattern[A, Blank[]], Times[Optional[Pattern[B, Blank[]]], sin[Plus[Optional[Pattern[e, Blank[]]], Times[Optional[Pattern[f, Blank[]]], Pattern[x, Blank[]]]]]]]], Pattern[x, Blank[Symbol]]] := Dist[Sqrt[Sin[e + f*x]]/Sqrt[d*Sin[e + f*x]], Int[(A + B*Sin[e + f*x])/(Sqrt[Sin[e + f*x]]*Sqrt[a + b*Sin[e + f*x]]), x], x] /; FreeQ[{a, b, e, f, d, A, B}, x] && GtQ[b, 0] && GtQ[b^2 - a^2, 0] && EqQ[A, B]
  3006. Int[Times[Power[Plus[Pattern[a, Blank[]], Times[Optional[Pattern[b, Blank[]]], sin[Plus[Optional[Pattern[e, Blank[]]], Times[Optional[Pattern[f, Blank[]]], Pattern[x, Blank[]]]]]]], Rational[-1, 2]], Plus[Optional[Pattern[A, Blank[]]], Times[Optional[Pattern[B, Blank[]]], sin[Plus[Optional[Pattern[e, Blank[]]], Times[Optional[Pattern[f, Blank[]]], Pattern[x, Blank[]]]]]]], Power[Plus[Optional[Pattern[c, Blank[]]], Times[Optional[Pattern[d, Blank[]]], sin[Plus[Optional[Pattern[e, Blank[]]], Times[Optional[Pattern[f, Blank[]]], Pattern[x, Blank[]]]]]]], Rational[-1, 2]]], Pattern[x, Blank[Symbol]]] := Dist[B/d, Int[Sqrt[c + d*Sin[e + f*x]]/Sqrt[a + b*Sin[e + f*x]], x], x] - Dist[(B*c - A*d)/d, Int[1/(Sqrt[a + b*Sin[e + f*x]]*Sqrt[c + d*Sin[e + f*x]]), x], x] /; FreeQ[{a, b, c, d, e, f, A, B}, x] && NeQ[b*c - a*d, 0] && NeQ[a^2 - b^2, 0] && NeQ[c^2 - d^2, 0]
  3007. Int[Times[Power[Plus[Optional[Pattern[a, Blank[]]], Times[Optional[Pattern[b, Blank[]]], sin[Plus[Optional[Pattern[e, Blank[]]], Times[Optional[Pattern[f, Blank[]]], Pattern[x, Blank[]]]]]]], Pattern[m, Blank[]]], Plus[Optional[Pattern[A, Blank[]]], Times[Optional[Pattern[B, Blank[]]], sin[Plus[Optional[Pattern[e, Blank[]]], Times[Optional[Pattern[f, Blank[]]], Pattern[x, Blank[]]]]]]], Power[Plus[Optional[Pattern[c, Blank[]]], Times[Optional[Pattern[d, Blank[]]], sin[Plus[Optional[Pattern[e, Blank[]]], Times[Optional[Pattern[f, Blank[]]], Pattern[x, Blank[]]]]]]], Optional[Pattern[n, Blank[]]]]], Pattern[x, Blank[Symbol]]] := Unintegrable[(a + b*Sin[e + f*x])^m*(A + B*Sin[e + f*x])*(c + d*Sin[e + f*x])^n, x] /; FreeQ[{a, b, c, d, e, f, A, B, m, n}, x] && NeQ[b*c - a*d, 0] && NeQ[a^2 - b^2, 0] && NeQ[c^2 - d^2, 0]
  3008. Int[Times[Power[Plus[Pattern[a, Blank[]], Times[Optional[Pattern[b, Blank[]]], sin[Plus[Optional[Pattern[e, Blank[]]], Times[Optional[Pattern[f, Blank[]]], Pattern[x, Blank[]]]]]]], Optional[Pattern[m, Blank[]]]], Power[Plus[Optional[Pattern[A, Blank[]]], Times[Optional[Pattern[B, Blank[]]], sin[Plus[Optional[Pattern[e, Blank[]]], Times[Optional[Pattern[f, Blank[]]], Pattern[x, Blank[]]]]]]], Pattern[p, Blank[]]], Power[Plus[Pattern[c, Blank[]], Times[Optional[Pattern[d, Blank[]]], sin[Plus[Optional[Pattern[e, Blank[]]], Times[Optional[Pattern[f, Blank[]]], Pattern[x, Blank[]]]]]]], Optional[Pattern[n, Blank[]]]]], Pattern[x, Blank[Symbol]]] := Dist[(Sqrt[a + b*Sin[e + f*x]]*Sqrt[c + d*Sin[e + f*x]])/(f*Cos[e + f*x]), Subst[Int[(a + b*x)^(m - 1/2)*(c + d*x)^(n - 1/2)*(A + B*x)^p, x], x, Sin[e + f*x]], x] /; FreeQ[{a, b, c, d, e, f, A, B, m, n, p}, x] && EqQ[b*c + a*d, 0] && EqQ[a^2 - b^2, 0]
  3009. Int[Times[Power[Plus[Optional[Pattern[A, Blank[]]], Times[cos[Plus[Optional[Pattern[e, Blank[]]], Times[Optional[Pattern[f, Blank[]]], Pattern[x, Blank[]]]]], Optional[Pattern[B, Blank[]]]]], Pattern[p, Blank[]]], Power[Plus[Times[cos[Plus[Optional[Pattern[e, Blank[]]], Times[Optional[Pattern[f, Blank[]]], Pattern[x, Blank[]]]]], Optional[Pattern[b, Blank[]]]], Pattern[a, Blank[]]], Optional[Pattern[m, Blank[]]]], Power[Plus[Times[cos[Plus[Optional[Pattern[e, Blank[]]], Times[Optional[Pattern[f, Blank[]]], Pattern[x, Blank[]]]]], Optional[Pattern[d, Blank[]]]], Pattern[c, Blank[]]], Optional[Pattern[n, Blank[]]]]], Pattern[x, Blank[Symbol]]] := -Dist[(Sqrt[a + b*Cos[e + f*x]]*Sqrt[c + d*Cos[e + f*x]])/(f*Sin[e + f*x]), Subst[Int[(a + b*x)^(m - 1/2)*(c + d*x)^(n - 1/2)*(A + B*x)^p, x], x, Cos[e + f*x]], x] /; FreeQ[{a, b, c, d, e, f, A, B, m, n, p}, x] && EqQ[b*c + a*d, 0] && EqQ[a^2 - b^2, 0]
  3010. Int[Times[Power[Times[Optional[Pattern[b, Blank[]]], sin[Plus[Optional[Pattern[e, Blank[]]], Times[Optional[Pattern[f, Blank[]]], Pattern[x, Blank[]]]]]], Optional[Pattern[m, Blank[]]]], Plus[Times[Optional[Pattern[B, Blank[]]], sin[Plus[Optional[Pattern[e, Blank[]]], Times[Optional[Pattern[f, Blank[]]], Pattern[x, Blank[]]]]]], Times[Optional[Pattern[C, Blank[]]], Power[sin[Plus[Optional[Pattern[e, Blank[]]], Times[Optional[Pattern[f, Blank[]]], Pattern[x, Blank[]]]]], 2]]]], Pattern[x, Blank[Symbol]]] := Dist[1/b, Int[(b*Sin[e + f*x])^(m + 1)*(B + C*Sin[e + f*x]), x], x] /; FreeQ[{b, e, f, B, C, m}, x]
  3011. Int[Times[Power[Times[Optional[Pattern[b, Blank[]]], sin[Plus[Optional[Pattern[e, Blank[]]], Times[Optional[Pattern[f, Blank[]]], Pattern[x, Blank[]]]]]], Optional[Pattern[m, Blank[]]]], Plus[Pattern[A, Blank[]], Times[Optional[Pattern[C, Blank[]]], Power[sin[Plus[Optional[Pattern[e, Blank[]]], Times[Optional[Pattern[f, Blank[]]], Pattern[x, Blank[]]]]], 2]]]], Pattern[x, Blank[Symbol]]] := Simp[(A*Cos[e + f*x]*(b*Sin[e + f*x])^(m + 1))/(b*f*(m + 1)), x] /; FreeQ[{b, e, f, A, C, m}, x] && EqQ[A*(m + 2) + C*(m + 1), 0]
  3012. Int[Times[Power[Times[Optional[Pattern[b, Blank[]]], sin[Plus[Optional[Pattern[e, Blank[]]], Times[Optional[Pattern[f, Blank[]]], Pattern[x, Blank[]]]]]], Pattern[m, Blank[]]], Plus[Pattern[A, Blank[]], Times[Optional[Pattern[C, Blank[]]], Power[sin[Plus[Optional[Pattern[e, Blank[]]], Times[Optional[Pattern[f, Blank[]]], Pattern[x, Blank[]]]]], 2]]]], Pattern[x, Blank[Symbol]]] := Simp[(A*Cos[e + f*x]*(b*Sin[e + f*x])^(m + 1))/(b*f*(m + 1)), x] + Dist[(A*(m + 2) + C*(m + 1))/(b^2*(m + 1)), Int[(b*Sin[e + f*x])^(m + 2), x], x] /; FreeQ[{b, e, f, A, C}, x] && LtQ[m, -1]
  3013. Int[Times[Power[sin[Plus[Optional[Pattern[e, Blank[]]], Times[Optional[Pattern[f, Blank[]]], Pattern[x, Blank[]]]]], Optional[Pattern[m, Blank[]]]], Plus[Pattern[A, Blank[]], Times[Optional[Pattern[C, Blank[]]], Power[sin[Plus[Optional[Pattern[e, Blank[]]], Times[Optional[Pattern[f, Blank[]]], Pattern[x, Blank[]]]]], 2]]]], Pattern[x, Blank[Symbol]]] := -Dist[f^(-1), Subst[Int[(1 - x^2)^((m - 1)/2)*(A + C - C*x^2), x], x, Cos[e + f*x]], x] /; FreeQ[{e, f, A, C}, x] && IGtQ[(m + 1)/2, 0]
  3014. Int[Times[Power[Times[Optional[Pattern[b, Blank[]]], sin[Plus[Optional[Pattern[e, Blank[]]], Times[Optional[Pattern[f, Blank[]]], Pattern[x, Blank[]]]]]], Optional[Pattern[m, Blank[]]]], Plus[Pattern[A, Blank[]], Times[Optional[Pattern[C, Blank[]]], Power[sin[Plus[Optional[Pattern[e, Blank[]]], Times[Optional[Pattern[f, Blank[]]], Pattern[x, Blank[]]]]], 2]]]], Pattern[x, Blank[Symbol]]] := -Simp[(C*Cos[e + f*x]*(b*Sin[e + f*x])^(m + 1))/(b*f*(m + 2)), x] + Dist[(A*(m + 2) + C*(m + 1))/(m + 2), Int[(b*Sin[e + f*x])^m, x], x] /; FreeQ[{b, e, f, A, C, m}, x] && !LtQ[m, -1]
  3015. Int[Times[Power[Plus[Pattern[a, Blank[]], Times[Optional[Pattern[b, Blank[]]], sin[Plus[Optional[Pattern[e, Blank[]]], Times[Optional[Pattern[f, Blank[]]], Pattern[x, Blank[]]]]]]], Optional[Pattern[m, Blank[]]]], Plus[Optional[Pattern[A, Blank[]]], Times[Optional[Pattern[B, Blank[]]], sin[Plus[Optional[Pattern[e, Blank[]]], Times[Optional[Pattern[f, Blank[]]], Pattern[x, Blank[]]]]]], Times[Optional[Pattern[C, Blank[]]], Power[sin[Plus[Optional[Pattern[e, Blank[]]], Times[Optional[Pattern[f, Blank[]]], Pattern[x, Blank[]]]]], 2]]]], Pattern[x, Blank[Symbol]]] := Dist[1/b^2, Int[(a + b*Sin[e + f*x])^(m + 1)*Simp[b*B - a*C + b*C*Sin[e + f*x], x], x], x] /; FreeQ[{a, b, e, f, A, B, C, m}, x] && EqQ[A*b^2 - a*b*B + a^2*C, 0]
  3016. Int[Times[Power[Plus[Pattern[a, Blank[]], Times[Optional[Pattern[b, Blank[]]], sin[Plus[Optional[Pattern[e, Blank[]]], Times[Optional[Pattern[f, Blank[]]], Pattern[x, Blank[]]]]]]], Optional[Pattern[m, Blank[]]]], Plus[Optional[Pattern[A, Blank[]]], Times[Optional[Pattern[C, Blank[]]], Power[sin[Plus[Optional[Pattern[e, Blank[]]], Times[Optional[Pattern[f, Blank[]]], Pattern[x, Blank[]]]]], 2]]]], Pattern[x, Blank[Symbol]]] := Dist[C/b^2, Int[(a + b*Sin[e + f*x])^(m + 1)*Simp[-a + b*Sin[e + f*x], x], x], x] /; FreeQ[{a, b, e, f, A, C, m}, x] && EqQ[A*b^2 + a^2*C, 0]
  3017. Int[Times[Power[Plus[Pattern[a, Blank[]], Times[Optional[Pattern[b, Blank[]]], sin[Plus[Optional[Pattern[e, Blank[]]], Times[Optional[Pattern[f, Blank[]]], Pattern[x, Blank[]]]]]]], Optional[Pattern[m, Blank[]]]], Plus[Optional[Pattern[A, Blank[]]], Times[Optional[Pattern[B, Blank[]]], sin[Plus[Optional[Pattern[e, Blank[]]], Times[Optional[Pattern[f, Blank[]]], Pattern[x, Blank[]]]]]], Times[Optional[Pattern[C, Blank[]]], Power[sin[Plus[Optional[Pattern[e, Blank[]]], Times[Optional[Pattern[f, Blank[]]], Pattern[x, Blank[]]]]], 2]]]], Pattern[x, Blank[Symbol]]] := Dist[A - C, Int[(a + b*Sin[e + f*x])^m*(1 + Sin[e + f*x]), x], x] + Dist[C, Int[(a + b*Sin[e + f*x])^m*(1 + Sin[e + f*x])^2, x], x] /; FreeQ[{a, b, e, f, A, B, C, m}, x] && EqQ[A - B + C, 0] && !IntegerQ[2*m]
  3018. Int[Times[Power[Plus[Pattern[a, Blank[]], Times[Optional[Pattern[b, Blank[]]], sin[Plus[Optional[Pattern[e, Blank[]]], Times[Optional[Pattern[f, Blank[]]], Pattern[x, Blank[]]]]]]], Optional[Pattern[m, Blank[]]]], Plus[Optional[Pattern[A, Blank[]]], Times[Optional[Pattern[C, Blank[]]], Power[sin[Plus[Optional[Pattern[e, Blank[]]], Times[Optional[Pattern[f, Blank[]]], Pattern[x, Blank[]]]]], 2]]]], Pattern[x, Blank[Symbol]]] := Dist[A - C, Int[(a + b*Sin[e + f*x])^m*(1 + Sin[e + f*x]), x], x] + Dist[C, Int[(a + b*Sin[e + f*x])^m*(1 + Sin[e + f*x])^2, x], x] /; FreeQ[{a, b, e, f, A, C, m}, x] && EqQ[A + C, 0] && !IntegerQ[2*m]
  3019. Int[Times[Power[Plus[Pattern[a, Blank[]], Times[Optional[Pattern[b, Blank[]]], sin[Plus[Optional[Pattern[e, Blank[]]], Times[Optional[Pattern[f, Blank[]]], Pattern[x, Blank[]]]]]]], Pattern[m, Blank[]]], Plus[Optional[Pattern[A, Blank[]]], Times[Optional[Pattern[B, Blank[]]], sin[Plus[Optional[Pattern[e, Blank[]]], Times[Optional[Pattern[f, Blank[]]], Pattern[x, Blank[]]]]]], Times[Optional[Pattern[C, Blank[]]], Power[sin[Plus[Optional[Pattern[e, Blank[]]], Times[Optional[Pattern[f, Blank[]]], Pattern[x, Blank[]]]]], 2]]]], Pattern[x, Blank[Symbol]]] := Simp[((A*b - a*B + b*C)*Cos[e + f*x]*(a + b*Sin[e + f*x])^m)/(a*f*(2*m + 1)), x] + Dist[1/(a^2*(2*m + 1)), Int[(a + b*Sin[e + f*x])^(m + 1)*Simp[a*A*(m + 1) + m*(b*B - a*C) + b*C*(2*m + 1)*Sin[e + f*x], x], x], x] /; FreeQ[{a, b, e, f, A, B, C}, x] && LtQ[m, -1] && EqQ[a^2 - b^2, 0]
  3020. Int[Times[Power[Plus[Pattern[a, Blank[]], Times[Optional[Pattern[b, Blank[]]], sin[Plus[Optional[Pattern[e, Blank[]]], Times[Optional[Pattern[f, Blank[]]], Pattern[x, Blank[]]]]]]], Pattern[m, Blank[]]], Plus[Optional[Pattern[A, Blank[]]], Times[Optional[Pattern[C, Blank[]]], Power[sin[Plus[Optional[Pattern[e, Blank[]]], Times[Optional[Pattern[f, Blank[]]], Pattern[x, Blank[]]]]], 2]]]], Pattern[x, Blank[Symbol]]] := Simp[(b*(A + C)*Cos[e + f*x]*(a + b*Sin[e + f*x])^m)/(a*f*(2*m + 1)), x] + Dist[1/(a^2*(2*m + 1)), Int[(a + b*Sin[e + f*x])^(m + 1)*Simp[a*A*(m + 1) - a*C*m + b*C*(2*m + 1)*Sin[e + f*x], x], x], x] /; FreeQ[{a, b, e, f, A, C}, x] && LtQ[m, -1] && EqQ[a^2 - b^2, 0]
  3021. Int[Times[Power[Plus[Optional[Pattern[a, Blank[]]], Times[Optional[Pattern[b, Blank[]]], sin[Plus[Optional[Pattern[e, Blank[]]], Times[Optional[Pattern[f, Blank[]]], Pattern[x, Blank[]]]]]]], Pattern[m, Blank[]]], Plus[Optional[Pattern[A, Blank[]]], Times[Optional[Pattern[B, Blank[]]], sin[Plus[Optional[Pattern[e, Blank[]]], Times[Optional[Pattern[f, Blank[]]], Pattern[x, Blank[]]]]]], Times[Optional[Pattern[C, Blank[]]], Power[sin[Plus[Optional[Pattern[e, Blank[]]], Times[Optional[Pattern[f, Blank[]]], Pattern[x, Blank[]]]]], 2]]]], Pattern[x, Blank[Symbol]]] := -Simp[((A*b^2 - a*b*B + a^2*C)*Cos[e + f*x]*(a + b*Sin[e + f*x])^(m + 1))/(b*f*(m + 1)*(a^2 - b^2)), x] + Dist[1/(b*(m + 1)*(a^2 - b^2)), Int[(a + b*Sin[e + f*x])^(m + 1)*Simp[b*(a*A - b*B + a*C)*(m + 1) - (A*b^2 - a*b*B + a^2*C + b*(A*b - a*B + b*C)*(m + 1))*Sin[e + f*x], x], x], x] /; FreeQ[{a, b, e, f, A, B, C}, x] && LtQ[m, -1] && NeQ[a^2 - b^2, 0]
  3022. Int[Times[Power[Plus[Pattern[a, Blank[]], Times[Optional[Pattern[b, Blank[]]], sin[Plus[Optional[Pattern[e, Blank[]]], Times[Optional[Pattern[f, Blank[]]], Pattern[x, Blank[]]]]]]], Pattern[m, Blank[]]], Plus[Optional[Pattern[A, Blank[]]], Times[Optional[Pattern[C, Blank[]]], Power[sin[Plus[Optional[Pattern[e, Blank[]]], Times[Optional[Pattern[f, Blank[]]], Pattern[x, Blank[]]]]], 2]]]], Pattern[x, Blank[Symbol]]] := -Simp[((A*b^2 + a^2*C)*Cos[e + f*x]*(a + b*Sin[e + f*x])^(m + 1))/(b*f*(m + 1)*(a^2 - b^2)), x] + Dist[1/(b*(m + 1)*(a^2 - b^2)), Int[(a + b*Sin[e + f*x])^(m + 1)*Simp[a*b*(A + C)*(m + 1) - (A*b^2 + a^2*C + b^2*(A + C)*(m + 1))*Sin[e + f*x], x], x], x] /; FreeQ[{a, b, e, f, A, C}, x] && LtQ[m, -1] && NeQ[a^2 - b^2, 0]
  3023. Int[Times[Power[Plus[Optional[Pattern[a, Blank[]]], Times[Optional[Pattern[b, Blank[]]], sin[Plus[Optional[Pattern[e, Blank[]]], Times[Optional[Pattern[f, Blank[]]], Pattern[x, Blank[]]]]]]], Optional[Pattern[m, Blank[]]]], Plus[Optional[Pattern[A, Blank[]]], Times[Optional[Pattern[B, Blank[]]], sin[Plus[Optional[Pattern[e, Blank[]]], Times[Optional[Pattern[f, Blank[]]], Pattern[x, Blank[]]]]]], Times[Optional[Pattern[C, Blank[]]], Power[sin[Plus[Optional[Pattern[e, Blank[]]], Times[Optional[Pattern[f, Blank[]]], Pattern[x, Blank[]]]]], 2]]]], Pattern[x, Blank[Symbol]]] := -Simp[(C*Cos[e + f*x]*(a + b*Sin[e + f*x])^(m + 1))/(b*f*(m + 2)), x] + Dist[1/(b*(m + 2)), Int[(a + b*Sin[e + f*x])^m*Simp[A*b*(m + 2) + b*C*(m + 1) + (b*B*(m + 2) - a*C)*Sin[e + f*x], x], x], x] /; FreeQ[{a, b, e, f, A, B, C, m}, x] && !LtQ[m, -1]
  3024. Int[Times[Power[Plus[Pattern[a, Blank[]], Times[Optional[Pattern[b, Blank[]]], sin[Plus[Optional[Pattern[e, Blank[]]], Times[Optional[Pattern[f, Blank[]]], Pattern[x, Blank[]]]]]]], Optional[Pattern[m, Blank[]]]], Plus[Optional[Pattern[A, Blank[]]], Times[Optional[Pattern[C, Blank[]]], Power[sin[Plus[Optional[Pattern[e, Blank[]]], Times[Optional[Pattern[f, Blank[]]], Pattern[x, Blank[]]]]], 2]]]], Pattern[x, Blank[Symbol]]] := -Simp[(C*Cos[e + f*x]*(a + b*Sin[e + f*x])^(m + 1))/(b*f*(m + 2)), x] + Dist[1/(b*(m + 2)), Int[(a + b*Sin[e + f*x])^m*Simp[A*b*(m + 2) + b*C*(m + 1) - a*C*Sin[e + f*x], x], x], x] /; FreeQ[{a, b, e, f, A, C, m}, x] && !LtQ[m, -1]
  3025. Int[Times[Power[Times[Optional[Pattern[b, Blank[]]], Power[sin[Plus[Optional[Pattern[e, Blank[]]], Times[Optional[Pattern[f, Blank[]]], Pattern[x, Blank[]]]]], Pattern[p, Blank[]]]], Pattern[m, Blank[]]], Plus[Optional[Pattern[A, Blank[]]], Times[Optional[Pattern[B, Blank[]]], sin[Plus[Optional[Pattern[e, Blank[]]], Times[Optional[Pattern[f, Blank[]]], Pattern[x, Blank[]]]]]], Times[Optional[Pattern[C, Blank[]]], Power[sin[Plus[Optional[Pattern[e, Blank[]]], Times[Optional[Pattern[f, Blank[]]], Pattern[x, Blank[]]]]], 2]]]], Pattern[x, Blank[Symbol]]] := Dist[(b*Sin[e + f*x]^p)^m/(b*Sin[e + f*x])^(m*p), Int[(b*Sin[e + f*x])^(m*p)*(A + B*Sin[e + f*x] + C*Sin[e + f*x]^2), x], x] /; FreeQ[{b, e, f, A, B, C, m, p}, x] && !IntegerQ[m]
  3026. Int[Times[Power[Times[Power[cos[Plus[Optional[Pattern[e, Blank[]]], Times[Optional[Pattern[f, Blank[]]], Pattern[x, Blank[]]]]], Pattern[p, Blank[]]], Optional[Pattern[b, Blank[]]]], Pattern[m, Blank[]]], Plus[Optional[Pattern[A, Blank[]]], Times[cos[Plus[Optional[Pattern[e, Blank[]]], Times[Optional[Pattern[f, Blank[]]], Pattern[x, Blank[]]]]], Optional[Pattern[B, Blank[]]]], Times[Power[cos[Plus[Optional[Pattern[e, Blank[]]], Times[Optional[Pattern[f, Blank[]]], Pattern[x, Blank[]]]]], 2], Optional[Pattern[C, Blank[]]]]]], Pattern[x, Blank[Symbol]]] := Dist[(b*Cos[e + f*x]^p)^m/(b*Cos[e + f*x])^(m*p), Int[(b*Cos[e + f*x])^(m*p)*(A + B*Cos[e + f*x] + C*Cos[e + f*x]^2), x], x] /; FreeQ[{b, e, f, A, B, C, m, p}, x] && !IntegerQ[m]
  3027. Int[Times[Power[Times[Optional[Pattern[b, Blank[]]], Power[sin[Plus[Optional[Pattern[e, Blank[]]], Times[Optional[Pattern[f, Blank[]]], Pattern[x, Blank[]]]]], Pattern[p, Blank[]]]], Pattern[m, Blank[]]], Plus[Optional[Pattern[A, Blank[]]], Times[Optional[Pattern[C, Blank[]]], Power[sin[Plus[Optional[Pattern[e, Blank[]]], Times[Optional[Pattern[f, Blank[]]], Pattern[x, Blank[]]]]], 2]]]], Pattern[x, Blank[Symbol]]] := Dist[(b*Sin[e + f*x]^p)^m/(b*Sin[e + f*x])^(m*p), Int[(b*Sin[e + f*x])^(m*p)*(A + C*Sin[e + f*x]^2), x], x] /; FreeQ[{b, e, f, A, C, m, p}, x] && !IntegerQ[m]
  3028. Int[Times[Power[Times[Power[cos[Plus[Optional[Pattern[e, Blank[]]], Times[Optional[Pattern[f, Blank[]]], Pattern[x, Blank[]]]]], Pattern[p, Blank[]]], Optional[Pattern[b, Blank[]]]], Pattern[m, Blank[]]], Plus[Optional[Pattern[A, Blank[]]], Times[Power[cos[Plus[Optional[Pattern[e, Blank[]]], Times[Optional[Pattern[f, Blank[]]], Pattern[x, Blank[]]]]], 2], Optional[Pattern[C, Blank[]]]]]], Pattern[x, Blank[Symbol]]] := Dist[(b*Cos[e + f*x]^p)^m/(b*Cos[e + f*x])^(m*p), Int[(b*Cos[e + f*x])^(m*p)*(A + C*Cos[e + f*x]^2), x], x] /; FreeQ[{b, e, f, A, C, m, p}, x] && !IntegerQ[m]
  3029. Int[Times[Power[Plus[Optional[Pattern[a, Blank[]]], Times[Optional[Pattern[b, Blank[]]], sin[Plus[Optional[Pattern[e, Blank[]]], Times[Optional[Pattern[f, Blank[]]], Pattern[x, Blank[]]]]]]], Optional[Pattern[m, Blank[]]]], Power[Plus[Optional[Pattern[c, Blank[]]], Times[Optional[Pattern[d, Blank[]]], sin[Plus[Optional[Pattern[e, Blank[]]], Times[Optional[Pattern[f, Blank[]]], Pattern[x, Blank[]]]]]]], Optional[Pattern[n, Blank[]]]], Plus[Optional[Pattern[A, Blank[]]], Times[Optional[Pattern[B, Blank[]]], sin[Plus[Optional[Pattern[e, Blank[]]], Times[Optional[Pattern[f, Blank[]]], Pattern[x, Blank[]]]]]], Times[Optional[Pattern[C, Blank[]]], Power[sin[Plus[Optional[Pattern[e, Blank[]]], Times[Optional[Pattern[f, Blank[]]], Pattern[x, Blank[]]]]], 2]]]], Pattern[x, Blank[Symbol]]] := Dist[1/b^2, Int[(a + b*Sin[e + f*x])^(m + 1)*(c + d*Sin[e + f*x])^n*(b*B - a*C + b*C*Sin[e + f*x]), x], x] /; FreeQ[{a, b, c, d, e, f, A, B, C, m, n}, x] && NeQ[b*c - a*d, 0] && EqQ[A*b^2 - a*b*B + a^2*C, 0]
  3030. Int[Times[Power[Plus[Optional[Pattern[a, Blank[]]], Times[Optional[Pattern[b, Blank[]]], sin[Plus[Optional[Pattern[e, Blank[]]], Times[Optional[Pattern[f, Blank[]]], Pattern[x, Blank[]]]]]]], Optional[Pattern[m, Blank[]]]], Power[Plus[Optional[Pattern[c, Blank[]]], Times[Optional[Pattern[d, Blank[]]], sin[Plus[Optional[Pattern[e, Blank[]]], Times[Optional[Pattern[f, Blank[]]], Pattern[x, Blank[]]]]]]], Optional[Pattern[n, Blank[]]]], Plus[Optional[Pattern[A, Blank[]]], Times[Optional[Pattern[C, Blank[]]], Power[sin[Plus[Optional[Pattern[e, Blank[]]], Times[Optional[Pattern[f, Blank[]]], Pattern[x, Blank[]]]]], 2]]]], Pattern[x, Blank[Symbol]]] := -Dist[C/b^2, Int[(a + b*Sin[e + f*x])^(m + 1)*(c + d*Sin[e + f*x])^n*(a - b*Sin[e + f*x]), x], x] /; FreeQ[{a, b, c, d, e, f, A, C, m, n}, x] && NeQ[b*c - a*d, 0] && EqQ[A*b^2 + a^2*C, 0]
  3031. Int[Times[Power[Plus[Optional[Pattern[a, Blank[]]], Times[Optional[Pattern[b, Blank[]]], sin[Plus[Optional[Pattern[e, Blank[]]], Times[Optional[Pattern[f, Blank[]]], Pattern[x, Blank[]]]]]]], Pattern[m, Blank[]]], Plus[Optional[Pattern[c, Blank[]]], Times[Optional[Pattern[d, Blank[]]], sin[Plus[Optional[Pattern[e, Blank[]]], Times[Optional[Pattern[f, Blank[]]], Pattern[x, Blank[]]]]]]], Plus[Optional[Pattern[A, Blank[]]], Times[Optional[Pattern[B, Blank[]]], sin[Plus[Optional[Pattern[e, Blank[]]], Times[Optional[Pattern[f, Blank[]]], Pattern[x, Blank[]]]]]], Times[Optional[Pattern[C, Blank[]]], Power[sin[Plus[Optional[Pattern[e, Blank[]]], Times[Optional[Pattern[f, Blank[]]], Pattern[x, Blank[]]]]], 2]]]], Pattern[x, Blank[Symbol]]] := -Simp[((b*c - a*d)*(A*b^2 - a*b*B + a^2*C)*Cos[e + f*x]*(a + b*Sin[e + f*x])^(m + 1))/(b^2*f*(m + 1)*(a^2 - b^2)), x] - Dist[1/(b^2*(m + 1)*(a^2 - b^2)), Int[(a + b*Sin[e + f*x])^(m + 1)*Simp[b*(m + 1)*((b*B - a*C)*(b*c - a*d) - A*b*(a*c - b*d)) + (b*B*(a^2*d + b^2*d*(m + 1) - a*b*c*(m + 2)) + (b*c - a*d)*(A*b^2*(m + 2) + C*(a^2 + b^2*(m + 1))))*Sin[e + f*x] - b*C*d*(m + 1)*(a^2 - b^2)*Sin[e + f*x]^2, x], x], x] /; FreeQ[{a, b, c, d, e, f, A, B, C}, x] && NeQ[b*c - a*d, 0] && NeQ[a^2 - b^2, 0] && LtQ[m, -1]
  3032. Int[Times[Power[Plus[Optional[Pattern[a, Blank[]]], Times[Optional[Pattern[b, Blank[]]], sin[Plus[Optional[Pattern[e, Blank[]]], Times[Optional[Pattern[f, Blank[]]], Pattern[x, Blank[]]]]]]], Pattern[m, Blank[]]], Plus[Optional[Pattern[c, Blank[]]], Times[Optional[Pattern[d, Blank[]]], sin[Plus[Optional[Pattern[e, Blank[]]], Times[Optional[Pattern[f, Blank[]]], Pattern[x, Blank[]]]]]]], Plus[Optional[Pattern[A, Blank[]]], Times[Optional[Pattern[C, Blank[]]], Power[sin[Plus[Optional[Pattern[e, Blank[]]], Times[Optional[Pattern[f, Blank[]]], Pattern[x, Blank[]]]]], 2]]]], Pattern[x, Blank[Symbol]]] := -Simp[((b*c - a*d)*(A*b^2 + a^2*C)*Cos[e + f*x]*(a + b*Sin[e + f*x])^(m + 1))/(b^2*f*(m + 1)*(a^2 - b^2)), x] + Dist[1/(b^2*(m + 1)*(a^2 - b^2)), Int[(a + b*Sin[e + f*x])^(m + 1)*Simp[b*(m + 1)*(a*C*(b*c - a*d) + A*b*(a*c - b*d)) - ((b*c - a*d)*(A*b^2*(m + 2) + C*(a^2 + b^2*(m + 1))))*Sin[e + f*x] + b*C*d*(m + 1)*(a^2 - b^2)*Sin[e + f*x]^2, x], x], x] /; FreeQ[{a, b, c, d, e, f, A, C}, x] && NeQ[b*c - a*d, 0] && NeQ[a^2 - b^2, 0] && LtQ[m, -1]
  3033. Int[Times[Power[Plus[Optional[Pattern[a, Blank[]]], Times[Optional[Pattern[b, Blank[]]], sin[Plus[Optional[Pattern[e, Blank[]]], Times[Optional[Pattern[f, Blank[]]], Pattern[x, Blank[]]]]]]], Optional[Pattern[m, Blank[]]]], Plus[Pattern[c, Blank[]], Times[Optional[Pattern[d, Blank[]]], sin[Plus[Optional[Pattern[e, Blank[]]], Times[Optional[Pattern[f, Blank[]]], Pattern[x, Blank[]]]]]]], Plus[Optional[Pattern[A, Blank[]]], Times[Optional[Pattern[B, Blank[]]], sin[Plus[Optional[Pattern[e, Blank[]]], Times[Optional[Pattern[f, Blank[]]], Pattern[x, Blank[]]]]]], Times[Optional[Pattern[C, Blank[]]], Power[sin[Plus[Optional[Pattern[e, Blank[]]], Times[Optional[Pattern[f, Blank[]]], Pattern[x, Blank[]]]]], 2]]]], Pattern[x, Blank[Symbol]]] := -Simp[(C*d*Cos[e + f*x]*Sin[e + f*x]*(a + b*Sin[e + f*x])^(m + 1))/(b*f*(m + 3)), x] + Dist[1/(b*(m + 3)), Int[(a + b*Sin[e + f*x])^m*Simp[a*C*d + A*b*c*(m + 3) + b*(B*c*(m + 3) + d*(C*(m + 2) + A*(m + 3)))*Sin[e + f*x] - (2*a*C*d - b*(c*C + B*d)*(m + 3))*Sin[e + f*x]^2, x], x], x] /; FreeQ[{a, b, c, d, e, f, A, B, C, m}, x] && NeQ[b*c - a*d, 0] && NeQ[a^2 - b^2, 0] && !LtQ[m, -1]
  3034. Int[Times[Power[Plus[Optional[Pattern[a, Blank[]]], Times[Optional[Pattern[b, Blank[]]], sin[Plus[Optional[Pattern[e, Blank[]]], Times[Optional[Pattern[f, Blank[]]], Pattern[x, Blank[]]]]]]], Optional[Pattern[m, Blank[]]]], Plus[Pattern[c, Blank[]], Times[Optional[Pattern[d, Blank[]]], sin[Plus[Optional[Pattern[e, Blank[]]], Times[Optional[Pattern[f, Blank[]]], Pattern[x, Blank[]]]]]]], Plus[Optional[Pattern[A, Blank[]]], Times[Optional[Pattern[C, Blank[]]], Power[sin[Plus[Optional[Pattern[e, Blank[]]], Times[Optional[Pattern[f, Blank[]]], Pattern[x, Blank[]]]]], 2]]]], Pattern[x, Blank[Symbol]]] := -Simp[(C*d*Cos[e + f*x]*Sin[e + f*x]*(a + b*Sin[e + f*x])^(m + 1))/(b*f*(m + 3)), x] + Dist[1/(b*(m + 3)), Int[(a + b*Sin[e + f*x])^m*Simp[a*C*d + A*b*c*(m + 3) + b*d*(C*(m + 2) + A*(m + 3))*Sin[e + f*x] - (2*a*C*d - b*c*C*(m + 3))*Sin[e + f*x]^2, x], x], x] /; FreeQ[{a, b, c, d, e, f, A, C, m}, x] && NeQ[b*c - a*d, 0] && NeQ[a^2 - b^2, 0] && !LtQ[m, -1]
  3035. Int[Times[Power[Plus[Pattern[a, Blank[]], Times[Optional[Pattern[b, Blank[]]], sin[Plus[Optional[Pattern[e, Blank[]]], Times[Optional[Pattern[f, Blank[]]], Pattern[x, Blank[]]]]]]], Pattern[m, Blank[]]], Power[Plus[Optional[Pattern[c, Blank[]]], Times[Optional[Pattern[d, Blank[]]], sin[Plus[Optional[Pattern[e, Blank[]]], Times[Optional[Pattern[f, Blank[]]], Pattern[x, Blank[]]]]]]], Optional[Pattern[n, Blank[]]]], Plus[Optional[Pattern[A, Blank[]]], Times[Optional[Pattern[B, Blank[]]], sin[Plus[Optional[Pattern[e, Blank[]]], Times[Optional[Pattern[f, Blank[]]], Pattern[x, Blank[]]]]]], Times[Optional[Pattern[C, Blank[]]], Power[sin[Plus[Optional[Pattern[e, Blank[]]], Times[Optional[Pattern[f, Blank[]]], Pattern[x, Blank[]]]]], 2]]]], Pattern[x, Blank[Symbol]]] := Simp[((a*A - b*B + a*C)*Cos[e + f*x]*(a + b*Sin[e + f*x])^m*(c + d*Sin[e + f*x])^(n + 1))/(2*b*c*f*(2*m + 1)), x] - Dist[1/(2*b*c*d*(2*m + 1)), Int[(a + b*Sin[e + f*x])^(m + 1)*(c + d*Sin[e + f*x])^n*Simp[A*(c^2*(m + 1) + d^2*(2*m + n + 2)) - B*c*d*(m - n - 1) - C*(c^2*m - d^2*(n + 1)) + d*((A*c + B*d)*(m + n + 2) - c*C*(3*m - n))*Sin[e + f*x], x], x], x] /; FreeQ[{a, b, c, d, e, f, A, B, C, m, n}, x] && EqQ[b*c + a*d, 0] && EqQ[a^2 - b^2, 0] && (LtQ[m, -2^(-1)] || (EqQ[m + n + 2, 0] && NeQ[2*m + 1, 0]))
  3036. Int[Times[Power[Plus[Pattern[a, Blank[]], Times[Optional[Pattern[b, Blank[]]], sin[Plus[Optional[Pattern[e, Blank[]]], Times[Optional[Pattern[f, Blank[]]], Pattern[x, Blank[]]]]]]], Pattern[m, Blank[]]], Power[Plus[Optional[Pattern[c, Blank[]]], Times[Optional[Pattern[d, Blank[]]], sin[Plus[Optional[Pattern[e, Blank[]]], Times[Optional[Pattern[f, Blank[]]], Pattern[x, Blank[]]]]]]], Optional[Pattern[n, Blank[]]]], Plus[Optional[Pattern[A, Blank[]]], Times[Optional[Pattern[C, Blank[]]], Power[sin[Plus[Optional[Pattern[e, Blank[]]], Times[Optional[Pattern[f, Blank[]]], Pattern[x, Blank[]]]]], 2]]]], Pattern[x, Blank[Symbol]]] := Simp[((a*A + a*C)*Cos[e + f*x]*(a + b*Sin[e + f*x])^m*(c + d*Sin[e + f*x])^(n + 1))/(2*b*c*f*(2*m + 1)), x] - Dist[1/(2*b*c*d*(2*m + 1)), Int[(a + b*Sin[e + f*x])^(m + 1)*(c + d*Sin[e + f*x])^n*Simp[A*(c^2*(m + 1) + d^2*(2*m + n + 2)) - C*(c^2*m - d^2*(n + 1)) + d*(A*c*(m + n + 2) - c*C*(3*m - n))*Sin[e + f*x], x], x], x] /; FreeQ[{a, b, c, d, e, f, A, C, m, n}, x] && EqQ[b*c + a*d, 0] && EqQ[a^2 - b^2, 0] && (LtQ[m, -2^(-1)] || (EqQ[m + n + 2, 0] && NeQ[2*m + 1, 0]))
  3037. Int[Times[Power[Plus[Optional[Pattern[a, Blank[]]], Times[Optional[Pattern[b, Blank[]]], sin[Plus[Optional[Pattern[e, Blank[]]], Times[Optional[Pattern[f, Blank[]]], Pattern[x, Blank[]]]]]]], Optional[Pattern[m, Blank[]]]], Power[Plus[Optional[Pattern[c, Blank[]]], Times[Optional[Pattern[d, Blank[]]], sin[Plus[Optional[Pattern[e, Blank[]]], Times[Optional[Pattern[f, Blank[]]], Pattern[x, Blank[]]]]]]], Rational[-1, 2]], Plus[Optional[Pattern[A, Blank[]]], Times[Optional[Pattern[B, Blank[]]], sin[Plus[Optional[Pattern[e, Blank[]]], Times[Optional[Pattern[f, Blank[]]], Pattern[x, Blank[]]]]]], Times[Optional[Pattern[C, Blank[]]], Power[sin[Plus[Optional[Pattern[e, Blank[]]], Times[Optional[Pattern[f, Blank[]]], Pattern[x, Blank[]]]]], 2]]]], Pattern[x, Blank[Symbol]]] := Simp[(-2*C*Cos[e + f*x]*(a + b*Sin[e + f*x])^(m + 1))/(b*f*(2*m + 3)*Sqrt[c + d*Sin[e + f*x]]), x] + Int[((a + b*Sin[e + f*x])^m*Simp[A + C + B*Sin[e + f*x], x])/Sqrt[c + d*Sin[e + f*x]], x] /; FreeQ[{a, b, c, d, e, f, A, B, C, m}, x] && EqQ[b*c + a*d, 0] && EqQ[a^2 - b^2, 0] && !LtQ[m, -2^(-1)]
  3038. Int[Times[Power[Plus[Optional[Pattern[a, Blank[]]], Times[Optional[Pattern[b, Blank[]]], sin[Plus[Optional[Pattern[e, Blank[]]], Times[Optional[Pattern[f, Blank[]]], Pattern[x, Blank[]]]]]]], Optional[Pattern[m, Blank[]]]], Power[Plus[Optional[Pattern[c, Blank[]]], Times[Optional[Pattern[d, Blank[]]], sin[Plus[Optional[Pattern[e, Blank[]]], Times[Optional[Pattern[f, Blank[]]], Pattern[x, Blank[]]]]]]], Rational[-1, 2]], Plus[Optional[Pattern[A, Blank[]]], Times[Optional[Pattern[C, Blank[]]], Power[sin[Plus[Optional[Pattern[e, Blank[]]], Times[Optional[Pattern[f, Blank[]]], Pattern[x, Blank[]]]]], 2]]]], Pattern[x, Blank[Symbol]]] := Simp[(-2*C*Cos[e + f*x]*(a + b*Sin[e + f*x])^(m + 1))/(b*f*(2*m + 3)*Sqrt[c + d*Sin[e + f*x]]), x] + Dist[A + C, Int[(a + b*Sin[e + f*x])^m/Sqrt[c + d*Sin[e + f*x]], x], x] /; FreeQ[{a, b, c, d, e, f, A, C, m}, x] && EqQ[b*c + a*d, 0] && EqQ[a^2 - b^2, 0] && !LtQ[m, -2^(-1)]
  3039. Int[Times[Power[Plus[Pattern[a, Blank[]], Times[Optional[Pattern[b, Blank[]]], sin[Plus[Optional[Pattern[e, Blank[]]], Times[Optional[Pattern[f, Blank[]]], Pattern[x, Blank[]]]]]]], Optional[Pattern[m, Blank[]]]], Power[Plus[Optional[Pattern[c, Blank[]]], Times[Optional[Pattern[d, Blank[]]], sin[Plus[Optional[Pattern[e, Blank[]]], Times[Optional[Pattern[f, Blank[]]], Pattern[x, Blank[]]]]]]], Optional[Pattern[n, Blank[]]]], Plus[Optional[Pattern[A, Blank[]]], Times[Optional[Pattern[B, Blank[]]], sin[Plus[Optional[Pattern[e, Blank[]]], Times[Optional[Pattern[f, Blank[]]], Pattern[x, Blank[]]]]]], Times[Optional[Pattern[C, Blank[]]], Power[sin[Plus[Optional[Pattern[e, Blank[]]], Times[Optional[Pattern[f, Blank[]]], Pattern[x, Blank[]]]]], 2]]]], Pattern[x, Blank[Symbol]]] := -Simp[(C*Cos[e + f*x]*(a + b*Sin[e + f*x])^m*(c + d*Sin[e + f*x])^(n + 1))/(d*f*(m + n + 2)), x] + Dist[1/(b*d*(m + n + 2)), Int[(a + b*Sin[e + f*x])^m*(c + d*Sin[e + f*x])^n*Simp[A*b*d*(m + n + 2) + C*(a*c*m + b*d*(n + 1)) + (b*B*d*(m + n + 2) - b*c*C*(2*m + 1))*Sin[e + f*x], x], x], x] /; FreeQ[{a, b, c, d, e, f, A, B, C, m, n}, x] && EqQ[b*c + a*d, 0] && EqQ[a^2 - b^2, 0] && !LtQ[m, -2^(-1)] && NeQ[m + n + 2, 0]
  3040. Int[Times[Power[Plus[Pattern[a, Blank[]], Times[Optional[Pattern[b, Blank[]]], sin[Plus[Optional[Pattern[e, Blank[]]], Times[Optional[Pattern[f, Blank[]]], Pattern[x, Blank[]]]]]]], Optional[Pattern[m, Blank[]]]], Power[Plus[Optional[Pattern[c, Blank[]]], Times[Optional[Pattern[d, Blank[]]], sin[Plus[Optional[Pattern[e, Blank[]]], Times[Optional[Pattern[f, Blank[]]], Pattern[x, Blank[]]]]]]], Optional[Pattern[n, Blank[]]]], Plus[Optional[Pattern[A, Blank[]]], Times[Optional[Pattern[C, Blank[]]], Power[sin[Plus[Optional[Pattern[e, Blank[]]], Times[Optional[Pattern[f, Blank[]]], Pattern[x, Blank[]]]]], 2]]]], Pattern[x, Blank[Symbol]]] := -Simp[(C*Cos[e + f*x]*(a + b*Sin[e + f*x])^m*(c + d*Sin[e + f*x])^(n + 1))/(d*f*(m + n + 2)), x] + Dist[1/(b*d*(m + n + 2)), Int[(a + b*Sin[e + f*x])^m*(c + d*Sin[e + f*x])^n*Simp[A*b*d*(m + n + 2) + C*(a*c*m + b*d*(n + 1)) - b*c*C*(2*m + 1)*Sin[e + f*x], x], x], x] /; FreeQ[{a, b, c, d, e, f, A, C, m, n}, x] && EqQ[b*c + a*d, 0] && EqQ[a^2 - b^2, 0] && !LtQ[m, -2^(-1)] && NeQ[m + n + 2, 0]
  3041. Int[Times[Power[Plus[Pattern[a, Blank[]], Times[Optional[Pattern[b, Blank[]]], sin[Plus[Optional[Pattern[e, Blank[]]], Times[Optional[Pattern[f, Blank[]]], Pattern[x, Blank[]]]]]]], Pattern[m, Blank[]]], Power[Plus[Optional[Pattern[c, Blank[]]], Times[Optional[Pattern[d, Blank[]]], sin[Plus[Optional[Pattern[e, Blank[]]], Times[Optional[Pattern[f, Blank[]]], Pattern[x, Blank[]]]]]]], Optional[Pattern[n, Blank[]]]], Plus[Optional[Pattern[A, Blank[]]], Times[Optional[Pattern[B, Blank[]]], sin[Plus[Optional[Pattern[e, Blank[]]], Times[Optional[Pattern[f, Blank[]]], Pattern[x, Blank[]]]]]], Times[Optional[Pattern[C, Blank[]]], Power[sin[Plus[Optional[Pattern[e, Blank[]]], Times[Optional[Pattern[f, Blank[]]], Pattern[x, Blank[]]]]], 2]]]], Pattern[x, Blank[Symbol]]] := Simp[((a*A - b*B + a*C)*Cos[e + f*x]*(a + b*Sin[e + f*x])^m*(c + d*Sin[e + f*x])^(n + 1))/(f*(b*c - a*d)*(2*m + 1)), x] + Dist[1/(b*(b*c - a*d)*(2*m + 1)), Int[(a + b*Sin[e + f*x])^(m + 1)*(c + d*Sin[e + f*x])^n*Simp[A*(a*c*(m + 1) - b*d*(2*m + n + 2)) + B*(b*c*m + a*d*(n + 1)) - C*(a*c*m + b*d*(n + 1)) + (d*(a*A - b*B)*(m + n + 2) + C*(b*c*(2*m + 1) - a*d*(m - n - 1)))*Sin[e + f*x], x], x], x] /; FreeQ[{a, b, c, d, e, f, A, B, C, n}, x] && NeQ[b*c - a*d, 0] && EqQ[a^2 - b^2, 0] && NeQ[c^2 - d^2, 0] && LtQ[m, -2^(-1)]
  3042. Int[Times[Power[Plus[Pattern[a, Blank[]], Times[Optional[Pattern[b, Blank[]]], sin[Plus[Optional[Pattern[e, Blank[]]], Times[Optional[Pattern[f, Blank[]]], Pattern[x, Blank[]]]]]]], Pattern[m, Blank[]]], Power[Plus[Optional[Pattern[c, Blank[]]], Times[Optional[Pattern[d, Blank[]]], sin[Plus[Optional[Pattern[e, Blank[]]], Times[Optional[Pattern[f, Blank[]]], Pattern[x, Blank[]]]]]]], Optional[Pattern[n, Blank[]]]], Plus[Optional[Pattern[A, Blank[]]], Times[Optional[Pattern[C, Blank[]]], Power[sin[Plus[Optional[Pattern[e, Blank[]]], Times[Optional[Pattern[f, Blank[]]], Pattern[x, Blank[]]]]], 2]]]], Pattern[x, Blank[Symbol]]] := Simp[(a*(A + C)*Cos[e + f*x]*(a + b*Sin[e + f*x])^m*(c + d*Sin[e + f*x])^(n + 1))/(f*(b*c - a*d)*(2*m + 1)), x] + Dist[1/(b*(b*c - a*d)*(2*m + 1)), Int[(a + b*Sin[e + f*x])^(m + 1)*(c + d*Sin[e + f*x])^n*Simp[A*(a*c*(m + 1) - b*d*(2*m + n + 2)) - C*(a*c*m + b*d*(n + 1)) + (a*A*d*(m + n + 2) + C*(b*c*(2*m + 1) - a*d*(m - n - 1)))*Sin[e + f*x], x], x], x] /; FreeQ[{a, b, c, d, e, f, A, C, n}, x] && NeQ[b*c - a*d, 0] && EqQ[a^2 - b^2, 0] && NeQ[c^2 - d^2, 0] && LtQ[m, -2^(-1)]
  3043. Int[Times[Power[Plus[Pattern[a, Blank[]], Times[Optional[Pattern[b, Blank[]]], sin[Plus[Optional[Pattern[e, Blank[]]], Times[Optional[Pattern[f, Blank[]]], Pattern[x, Blank[]]]]]]], Optional[Pattern[m, Blank[]]]], Power[Plus[Optional[Pattern[c, Blank[]]], Times[Optional[Pattern[d, Blank[]]], sin[Plus[Optional[Pattern[e, Blank[]]], Times[Optional[Pattern[f, Blank[]]], Pattern[x, Blank[]]]]]]], Pattern[n, Blank[]]], Plus[Optional[Pattern[A, Blank[]]], Times[Optional[Pattern[B, Blank[]]], sin[Plus[Optional[Pattern[e, Blank[]]], Times[Optional[Pattern[f, Blank[]]], Pattern[x, Blank[]]]]]], Times[Optional[Pattern[C, Blank[]]], Power[sin[Plus[Optional[Pattern[e, Blank[]]], Times[Optional[Pattern[f, Blank[]]], Pattern[x, Blank[]]]]], 2]]]], Pattern[x, Blank[Symbol]]] := -Simp[((c^2*C - B*c*d + A*d^2)*Cos[e + f*x]*(a + b*Sin[e + f*x])^m*(c + d*Sin[e + f*x])^(n + 1))/(d*f*(n + 1)*(c^2 - d^2)), x] + Dist[1/(b*d*(n + 1)*(c^2 - d^2)), Int[(a + b*Sin[e + f*x])^m*(c + d*Sin[e + f*x])^(n + 1)*Simp[A*d*(a*d*m + b*c*(n + 1)) + (c*C - B*d)*(a*c*m + b*d*(n + 1)) + b*(d*(B*c - A*d)*(m + n + 2) - C*(c^2*(m + 1) + d^2*(n + 1)))*Sin[e + f*x], x], x], x] /; FreeQ[{a, b, c, d, e, f, A, B, C, m}, x] && NeQ[b*c - a*d, 0] && EqQ[a^2 - b^2, 0] && NeQ[c^2 - d^2, 0] && !LtQ[m, -2^(-1)] && (LtQ[n, -1] || EqQ[m + n + 2, 0])
  3044. Int[Times[Power[Plus[Pattern[a, Blank[]], Times[Optional[Pattern[b, Blank[]]], sin[Plus[Optional[Pattern[e, Blank[]]], Times[Optional[Pattern[f, Blank[]]], Pattern[x, Blank[]]]]]]], Optional[Pattern[m, Blank[]]]], Power[Plus[Optional[Pattern[c, Blank[]]], Times[Optional[Pattern[d, Blank[]]], sin[Plus[Optional[Pattern[e, Blank[]]], Times[Optional[Pattern[f, Blank[]]], Pattern[x, Blank[]]]]]]], Pattern[n, Blank[]]], Plus[Optional[Pattern[A, Blank[]]], Times[Optional[Pattern[C, Blank[]]], Power[sin[Plus[Optional[Pattern[e, Blank[]]], Times[Optional[Pattern[f, Blank[]]], Pattern[x, Blank[]]]]], 2]]]], Pattern[x, Blank[Symbol]]] := -Simp[((c^2*C + A*d^2)*Cos[e + f*x]*(a + b*Sin[e + f*x])^m*(c + d*Sin[e + f*x])^(n + 1))/(d*f*(n + 1)*(c^2 - d^2)), x] + Dist[1/(b*d*(n + 1)*(c^2 - d^2)), Int[(a + b*Sin[e + f*x])^m*(c + d*Sin[e + f*x])^(n + 1)*Simp[A*d*(a*d*m + b*c*(n + 1)) + c*C*(a*c*m + b*d*(n + 1)) - b*(A*d^2*(m + n + 2) + C*(c^2*(m + 1) + d^2*(n + 1)))*Sin[e + f*x], x], x], x] /; FreeQ[{a, b, c, d, e, f, A, C, m}, x] && NeQ[b*c - a*d, 0] && EqQ[a^2 - b^2, 0] && NeQ[c^2 - d^2, 0] && !LtQ[m, -2^(-1)] && (LtQ[n, -1] || EqQ[m + n + 2, 0])
  3045. Int[Times[Power[Plus[Pattern[a, Blank[]], Times[Optional[Pattern[b, Blank[]]], sin[Plus[Optional[Pattern[e, Blank[]]], Times[Optional[Pattern[f, Blank[]]], Pattern[x, Blank[]]]]]]], Optional[Pattern[m, Blank[]]]], Power[Plus[Optional[Pattern[c, Blank[]]], Times[Optional[Pattern[d, Blank[]]], sin[Plus[Optional[Pattern[e, Blank[]]], Times[Optional[Pattern[f, Blank[]]], Pattern[x, Blank[]]]]]]], Optional[Pattern[n, Blank[]]]], Plus[Optional[Pattern[A, Blank[]]], Times[Optional[Pattern[B, Blank[]]], sin[Plus[Optional[Pattern[e, Blank[]]], Times[Optional[Pattern[f, Blank[]]], Pattern[x, Blank[]]]]]], Times[Optional[Pattern[C, Blank[]]], Power[sin[Plus[Optional[Pattern[e, Blank[]]], Times[Optional[Pattern[f, Blank[]]], Pattern[x, Blank[]]]]], 2]]]], Pattern[x, Blank[Symbol]]] := -Simp[(C*Cos[e + f*x]*(a + b*Sin[e + f*x])^m*(c + d*Sin[e + f*x])^(n + 1))/(d*f*(m + n + 2)), x] + Dist[1/(b*d*(m + n + 2)), Int[(a + b*Sin[e + f*x])^m*(c + d*Sin[e + f*x])^n*Simp[A*b*d*(m + n + 2) + C*(a*c*m + b*d*(n + 1)) + (C*(a*d*m - b*c*(m + 1)) + b*B*d*(m + n + 2))*Sin[e + f*x], x], x], x] /; FreeQ[{a, b, c, d, e, f, A, B, C, m, n}, x] && NeQ[b*c - a*d, 0] && EqQ[a^2 - b^2, 0] && NeQ[c^2 - d^2, 0] && !LtQ[m, -2^(-1)] && NeQ[m + n + 2, 0]
  3046. Int[Times[Power[Plus[Pattern[a, Blank[]], Times[Optional[Pattern[b, Blank[]]], sin[Plus[Optional[Pattern[e, Blank[]]], Times[Optional[Pattern[f, Blank[]]], Pattern[x, Blank[]]]]]]], Optional[Pattern[m, Blank[]]]], Power[Plus[Optional[Pattern[c, Blank[]]], Times[Optional[Pattern[d, Blank[]]], sin[Plus[Optional[Pattern[e, Blank[]]], Times[Optional[Pattern[f, Blank[]]], Pattern[x, Blank[]]]]]]], Optional[Pattern[n, Blank[]]]], Plus[Optional[Pattern[A, Blank[]]], Times[Optional[Pattern[C, Blank[]]], Power[sin[Plus[Optional[Pattern[e, Blank[]]], Times[Optional[Pattern[f, Blank[]]], Pattern[x, Blank[]]]]], 2]]]], Pattern[x, Blank[Symbol]]] := -Simp[(C*Cos[e + f*x]*(a + b*Sin[e + f*x])^m*(c + d*Sin[e + f*x])^(n + 1))/(d*f*(m + n + 2)), x] + Dist[1/(b*d*(m + n + 2)), Int[(a + b*Sin[e + f*x])^m*(c + d*Sin[e + f*x])^n*Simp[A*b*d*(m + n + 2) + C*(a*c*m + b*d*(n + 1)) + C*(a*d*m - b*c*(m + 1))*Sin[e + f*x], x], x], x] /; FreeQ[{a, b, c, d, e, f, A, C, m, n}, x] && NeQ[b*c - a*d, 0] && EqQ[a^2 - b^2, 0] && NeQ[c^2 - d^2, 0] && !LtQ[m, -2^(-1)] && NeQ[m + n + 2, 0]
  3047. Int[Times[Power[Plus[Optional[Pattern[a, Blank[]]], Times[Optional[Pattern[b, Blank[]]], sin[Plus[Optional[Pattern[e, Blank[]]], Times[Optional[Pattern[f, Blank[]]], Pattern[x, Blank[]]]]]]], Pattern[m, Blank[]]], Power[Plus[Optional[Pattern[c, Blank[]]], Times[Optional[Pattern[d, Blank[]]], sin[Plus[Optional[Pattern[e, Blank[]]], Times[Optional[Pattern[f, Blank[]]], Pattern[x, Blank[]]]]]]], Pattern[n, Blank[]]], Plus[Optional[Pattern[A, Blank[]]], Times[Optional[Pattern[B, Blank[]]], sin[Plus[Optional[Pattern[e, Blank[]]], Times[Optional[Pattern[f, Blank[]]], Pattern[x, Blank[]]]]]], Times[Optional[Pattern[C, Blank[]]], Power[sin[Plus[Optional[Pattern[e, Blank[]]], Times[Optional[Pattern[f, Blank[]]], Pattern[x, Blank[]]]]], 2]]]], Pattern[x, Blank[Symbol]]] := -Simp[((c^2*C - B*c*d + A*d^2)*Cos[e + f*x]*(a + b*Sin[e + f*x])^m*(c + d*Sin[e + f*x])^(n + 1))/(d*f*(n + 1)*(c^2 - d^2)), x] + Dist[1/(d*(n + 1)*(c^2 - d^2)), Int[(a + b*Sin[e + f*x])^(m - 1)*(c + d*Sin[e + f*x])^(n + 1)*Simp[A*d*(b*d*m + a*c*(n + 1)) + (c*C - B*d)*(b*c*m + a*d*(n + 1)) - (d*(A*(a*d*(n + 2) - b*c*(n + 1)) + B*(b*d*(n + 1) - a*c*(n + 2))) - C*(b*c*d*(n + 1) - a*(c^2 + d^2*(n + 1))))*Sin[e + f*x] + b*(d*(B*c - A*d)*(m + n + 2) - C*(c^2*(m + 1) + d^2*(n + 1)))*Sin[e + f*x]^2, x], x], x] /; FreeQ[{a, b, c, d, e, f, A, B, C}, x] && NeQ[b*c - a*d, 0] && NeQ[a^2 - b^2, 0] && NeQ[c^2 - d^2, 0] && GtQ[m, 0] && LtQ[n, -1]
  3048. Int[Times[Power[Plus[Optional[Pattern[a, Blank[]]], Times[Optional[Pattern[b, Blank[]]], sin[Plus[Optional[Pattern[e, Blank[]]], Times[Optional[Pattern[f, Blank[]]], Pattern[x, Blank[]]]]]]], Pattern[m, Blank[]]], Power[Plus[Optional[Pattern[c, Blank[]]], Times[Optional[Pattern[d, Blank[]]], sin[Plus[Optional[Pattern[e, Blank[]]], Times[Optional[Pattern[f, Blank[]]], Pattern[x, Blank[]]]]]]], Pattern[n, Blank[]]], Plus[Optional[Pattern[A, Blank[]]], Times[Optional[Pattern[C, Blank[]]], Power[sin[Plus[Optional[Pattern[e, Blank[]]], Times[Optional[Pattern[f, Blank[]]], Pattern[x, Blank[]]]]], 2]]]], Pattern[x, Blank[Symbol]]] := -Simp[((c^2*C + A*d^2)*Cos[e + f*x]*(a + b*Sin[e + f*x])^m*(c + d*Sin[e + f*x])^(n + 1))/(d*f*(n + 1)*(c^2 - d^2)), x] + Dist[1/(d*(n + 1)*(c^2 - d^2)), Int[(a + b*Sin[e + f*x])^(m - 1)*(c + d*Sin[e + f*x])^(n + 1)*Simp[A*d*(b*d*m + a*c*(n + 1)) + c*C*(b*c*m + a*d*(n + 1)) - (A*d*(a*d*(n + 2) - b*c*(n + 1)) - C*(b*c*d*(n + 1) - a*(c^2 + d^2*(n + 1))))*Sin[e + f*x] - b*(A*d^2*(m + n + 2) + C*(c^2*(m + 1) + d^2*(n + 1)))*Sin[e + f*x]^2, x], x], x] /; FreeQ[{a, b, c, d, e, f, A, C}, x] && NeQ[b*c - a*d, 0] && NeQ[a^2 - b^2, 0] && NeQ[c^2 - d^2, 0] && GtQ[m, 0] && LtQ[n, -1]
  3049. Int[Times[Power[Plus[Optional[Pattern[a, Blank[]]], Times[Optional[Pattern[b, Blank[]]], sin[Plus[Optional[Pattern[e, Blank[]]], Times[Optional[Pattern[f, Blank[]]], Pattern[x, Blank[]]]]]]], Optional[Pattern[m, Blank[]]]], Power[Plus[Optional[Pattern[c, Blank[]]], Times[Optional[Pattern[d, Blank[]]], sin[Plus[Optional[Pattern[e, Blank[]]], Times[Optional[Pattern[f, Blank[]]], Pattern[x, Blank[]]]]]]], Optional[Pattern[n, Blank[]]]], Plus[Optional[Pattern[A, Blank[]]], Times[Optional[Pattern[B, Blank[]]], sin[Plus[Optional[Pattern[e, Blank[]]], Times[Optional[Pattern[f, Blank[]]], Pattern[x, Blank[]]]]]], Times[Optional[Pattern[C, Blank[]]], Power[sin[Plus[Optional[Pattern[e, Blank[]]], Times[Optional[Pattern[f, Blank[]]], Pattern[x, Blank[]]]]], 2]]]], Pattern[x, Blank[Symbol]]] := -Simp[(C*Cos[e + f*x]*(a + b*Sin[e + f*x])^m*(c + d*Sin[e + f*x])^(n + 1))/(d*f*(m + n + 2)), x] + Dist[1/(d*(m + n + 2)), Int[(a + b*Sin[e + f*x])^(m - 1)*(c + d*Sin[e + f*x])^n*Simp[a*A*d*(m + n + 2) + C*(b*c*m + a*d*(n + 1)) + (d*(A*b + a*B)*(m + n + 2) - C*(a*c - b*d*(m + n + 1)))*Sin[e + f*x] + (C*(a*d*m - b*c*(m + 1)) + b*B*d*(m + n + 2))*Sin[e + f*x]^2, x], x], x] /; FreeQ[{a, b, c, d, e, f, A, B, C, n}, x] && NeQ[b*c - a*d, 0] && NeQ[a^2 - b^2, 0] && NeQ[c^2 - d^2, 0] && GtQ[m, 0] && !(IGtQ[n, 0] && ( !IntegerQ[m] || (EqQ[a, 0] && NeQ[c, 0])))
  3050. Int[Times[Power[Plus[Optional[Pattern[a, Blank[]]], Times[Optional[Pattern[b, Blank[]]], sin[Plus[Optional[Pattern[e, Blank[]]], Times[Optional[Pattern[f, Blank[]]], Pattern[x, Blank[]]]]]]], Optional[Pattern[m, Blank[]]]], Power[Plus[Optional[Pattern[c, Blank[]]], Times[Optional[Pattern[d, Blank[]]], sin[Plus[Optional[Pattern[e, Blank[]]], Times[Optional[Pattern[f, Blank[]]], Pattern[x, Blank[]]]]]]], Optional[Pattern[n, Blank[]]]], Plus[Optional[Pattern[A, Blank[]]], Times[Optional[Pattern[C, Blank[]]], Power[sin[Plus[Optional[Pattern[e, Blank[]]], Times[Optional[Pattern[f, Blank[]]], Pattern[x, Blank[]]]]], 2]]]], Pattern[x, Blank[Symbol]]] := -Simp[(C*Cos[e + f*x]*(a + b*Sin[e + f*x])^m*(c + d*Sin[e + f*x])^(n + 1))/(d*f*(m + n + 2)), x] + Dist[1/(d*(m + n + 2)), Int[(a + b*Sin[e + f*x])^(m - 1)*(c + d*Sin[e + f*x])^n*Simp[a*A*d*(m + n + 2) + C*(b*c*m + a*d*(n + 1)) + (A*b*d*(m + n + 2) - C*(a*c - b*d*(m + n + 1)))*Sin[e + f*x] + C*(a*d*m - b*c*(m + 1))*Sin[e + f*x]^2, x], x], x] /; FreeQ[{a, b, c, d, e, f, A, C, n}, x] && NeQ[b*c - a*d, 0] && NeQ[a^2 - b^2, 0] && NeQ[c^2 - d^2, 0] && GtQ[m, 0] && !(IGtQ[n, 0] && ( !IntegerQ[m] || (EqQ[a, 0] && NeQ[c, 0])))
  3051. Int[Times[Power[Times[Optional[Pattern[d, Blank[]]], sin[Plus[Optional[Pattern[e, Blank[]]], Times[Optional[Pattern[f, Blank[]]], Pattern[x, Blank[]]]]]], Rational[-1, 2]], Power[Plus[Pattern[a, Blank[]], Times[Optional[Pattern[b, Blank[]]], sin[Plus[Optional[Pattern[e, Blank[]]], Times[Optional[Pattern[f, Blank[]]], Pattern[x, Blank[]]]]]]], Rational[-3, 2]], Plus[Optional[Pattern[A, Blank[]]], Times[Optional[Pattern[B, Blank[]]], sin[Plus[Optional[Pattern[e, Blank[]]], Times[Optional[Pattern[f, Blank[]]], Pattern[x, Blank[]]]]]], Times[Optional[Pattern[C, Blank[]]], Power[sin[Plus[Optional[Pattern[e, Blank[]]], Times[Optional[Pattern[f, Blank[]]], Pattern[x, Blank[]]]]], 2]]]], Pattern[x, Blank[Symbol]]] := Dist[C/(b*d), Int[Sqrt[d*Sin[e + f*x]]/Sqrt[a + b*Sin[e + f*x]], x], x] + Dist[1/b, Int[(A*b + (b*B - a*C)*Sin[e + f*x])/((a + b*Sin[e + f*x])^(3/2)*Sqrt[d*Sin[e + f*x]]), x], x] /; FreeQ[{a, b, d, e, f, A, B, C}, x] && NeQ[a^2 - b^2, 0]
  3052. Int[Times[Power[Times[Optional[Pattern[d, Blank[]]], sin[Plus[Optional[Pattern[e, Blank[]]], Times[Optional[Pattern[f, Blank[]]], Pattern[x, Blank[]]]]]], Rational[-1, 2]], Power[Plus[Pattern[a, Blank[]], Times[Optional[Pattern[b, Blank[]]], sin[Plus[Optional[Pattern[e, Blank[]]], Times[Optional[Pattern[f, Blank[]]], Pattern[x, Blank[]]]]]]], Rational[-3, 2]], Plus[Optional[Pattern[A, Blank[]]], Times[Optional[Pattern[C, Blank[]]], Power[sin[Plus[Optional[Pattern[e, Blank[]]], Times[Optional[Pattern[f, Blank[]]], Pattern[x, Blank[]]]]], 2]]]], Pattern[x, Blank[Symbol]]] := Dist[C/(b*d), Int[Sqrt[d*Sin[e + f*x]]/Sqrt[a + b*Sin[e + f*x]], x], x] + Dist[1/b, Int[(A*b - a*C*Sin[e + f*x])/((a + b*Sin[e + f*x])^(3/2)*Sqrt[d*Sin[e + f*x]]), x], x] /; FreeQ[{a, b, d, e, f, A, C}, x] && NeQ[a^2 - b^2, 0]
  3053. Int[Times[Power[Plus[Optional[Pattern[a, Blank[]]], Times[Optional[Pattern[b, Blank[]]], sin[Plus[Optional[Pattern[e, Blank[]]], Times[Optional[Pattern[f, Blank[]]], Pattern[x, Blank[]]]]]]], Rational[-3, 2]], Power[Plus[Optional[Pattern[c, Blank[]]], Times[Optional[Pattern[d, Blank[]]], sin[Plus[Optional[Pattern[e, Blank[]]], Times[Optional[Pattern[f, Blank[]]], Pattern[x, Blank[]]]]]]], Rational[-1, 2]], Plus[Optional[Pattern[A, Blank[]]], Times[Optional[Pattern[B, Blank[]]], sin[Plus[Optional[Pattern[e, Blank[]]], Times[Optional[Pattern[f, Blank[]]], Pattern[x, Blank[]]]]]], Times[Optional[Pattern[C, Blank[]]], Power[sin[Plus[Optional[Pattern[e, Blank[]]], Times[Optional[Pattern[f, Blank[]]], Pattern[x, Blank[]]]]], 2]]]], Pattern[x, Blank[Symbol]]] := Dist[C/b^2, Int[Sqrt[a + b*Sin[e + f*x]]/Sqrt[c + d*Sin[e + f*x]], x], x] + Dist[1/b^2, Int[(A*b^2 - a^2*C + b*(b*B - 2*a*C)*Sin[e + f*x])/((a + b*Sin[e + f*x])^(3/2)*Sqrt[c + d*Sin[e + f*x]]), x], x] /; FreeQ[{a, b, c, d, e, f, A, B, C}, x] && NeQ[b*c - a*d, 0] && NeQ[a^2 - b^2, 0] && NeQ[c^2 - d^2, 0]
  3054. Int[Times[Power[Plus[Optional[Pattern[a, Blank[]]], Times[Optional[Pattern[b, Blank[]]], sin[Plus[Optional[Pattern[e, Blank[]]], Times[Optional[Pattern[f, Blank[]]], Pattern[x, Blank[]]]]]]], Rational[-3, 2]], Power[Plus[Optional[Pattern[c, Blank[]]], Times[Optional[Pattern[d, Blank[]]], sin[Plus[Optional[Pattern[e, Blank[]]], Times[Optional[Pattern[f, Blank[]]], Pattern[x, Blank[]]]]]]], Rational[-1, 2]], Plus[Optional[Pattern[A, Blank[]]], Times[Optional[Pattern[C, Blank[]]], Power[sin[Plus[Optional[Pattern[e, Blank[]]], Times[Optional[Pattern[f, Blank[]]], Pattern[x, Blank[]]]]], 2]]]], Pattern[x, Blank[Symbol]]] := Dist[C/b^2, Int[Sqrt[a + b*Sin[e + f*x]]/Sqrt[c + d*Sin[e + f*x]], x], x] + Dist[1/b^2, Int[(A*b^2 - a^2*C - 2*a*b*C*Sin[e + f*x])/((a + b*Sin[e + f*x])^(3/2)*Sqrt[c + d*Sin[e + f*x]]), x], x] /; FreeQ[{a, b, c, d, e, f, A, C}, x] && NeQ[b*c - a*d, 0] && NeQ[a^2 - b^2, 0] && NeQ[c^2 - d^2, 0]
  3055. Int[Times[Power[Plus[Optional[Pattern[a, Blank[]]], Times[Optional[Pattern[b, Blank[]]], sin[Plus[Optional[Pattern[e, Blank[]]], Times[Optional[Pattern[f, Blank[]]], Pattern[x, Blank[]]]]]]], Pattern[m, Blank[]]], Power[Plus[Optional[Pattern[c, Blank[]]], Times[Optional[Pattern[d, Blank[]]], sin[Plus[Optional[Pattern[e, Blank[]]], Times[Optional[Pattern[f, Blank[]]], Pattern[x, Blank[]]]]]]], Pattern[n, Blank[]]], Plus[Optional[Pattern[A, Blank[]]], Times[Optional[Pattern[B, Blank[]]], sin[Plus[Optional[Pattern[e, Blank[]]], Times[Optional[Pattern[f, Blank[]]], Pattern[x, Blank[]]]]]], Times[Optional[Pattern[C, Blank[]]], Power[sin[Plus[Optional[Pattern[e, Blank[]]], Times[Optional[Pattern[f, Blank[]]], Pattern[x, Blank[]]]]], 2]]]], Pattern[x, Blank[Symbol]]] := -Simp[((A*b^2 - a*b*B + a^2*C)*Cos[e + f*x]*(a + b*Sin[e + f*x])^(m + 1)*(c + d*Sin[e + f*x])^(n + 1))/(f*(m + 1)*(b*c - a*d)*(a^2 - b^2)), x] + Dist[1/((m + 1)*(b*c - a*d)*(a^2 - b^2)), Int[(a + b*Sin[e + f*x])^(m + 1)*(c + d*Sin[e + f*x])^n*Simp[(m + 1)*(b*c - a*d)*(a*A - b*B + a*C) + d*(A*b^2 - a*b*B + a^2*C)*(m + n + 2) - (c*(A*b^2 - a*b*B + a^2*C) + (m + 1)*(b*c - a*d)*(A*b - a*B + b*C))*Sin[e + f*x] - d*(A*b^2 - a*b*B + a^2*C)*(m + n + 3)*Sin[e + f*x]^2, x], x], x] /; FreeQ[{a, b, c, d, e, f, A, B, C, n}, x] && NeQ[b*c - a*d, 0] && NeQ[a^2 - b^2, 0] && NeQ[c^2 - d^2, 0] && LtQ[m, -1] && ((EqQ[a, 0] && IntegerQ[m] && !IntegerQ[n]) || !(IntegerQ[2*n] && LtQ[n, -1] && ((IntegerQ[n] && !IntegerQ[m]) || EqQ[a, 0])))
  3056. Int[Times[Power[Plus[Optional[Pattern[a, Blank[]]], Times[Optional[Pattern[b, Blank[]]], sin[Plus[Optional[Pattern[e, Blank[]]], Times[Optional[Pattern[f, Blank[]]], Pattern[x, Blank[]]]]]]], Pattern[m, Blank[]]], Power[Plus[Optional[Pattern[c, Blank[]]], Times[Optional[Pattern[d, Blank[]]], sin[Plus[Optional[Pattern[e, Blank[]]], Times[Optional[Pattern[f, Blank[]]], Pattern[x, Blank[]]]]]]], Pattern[n, Blank[]]], Plus[Optional[Pattern[A, Blank[]]], Times[Optional[Pattern[C, Blank[]]], Power[sin[Plus[Optional[Pattern[e, Blank[]]], Times[Optional[Pattern[f, Blank[]]], Pattern[x, Blank[]]]]], 2]]]], Pattern[x, Blank[Symbol]]] := -Simp[((A*b^2 + a^2*C)*Cos[e + f*x]*(a + b*Sin[e + f*x])^(m + 1)*(c + d*Sin[e + f*x])^(n + 1))/(f*(m + 1)*(b*c - a*d)*(a^2 - b^2)), x] + Dist[1/((m + 1)*(b*c - a*d)*(a^2 - b^2)), Int[(a + b*Sin[e + f*x])^(m + 1)*(c + d*Sin[e + f*x])^n*Simp[a*(m + 1)*(b*c - a*d)*(A + C) + d*(A*b^2 + a^2*C)*(m + n + 2) - (c*(A*b^2 + a^2*C) + b*(m + 1)*(b*c - a*d)*(A + C))*Sin[e + f*x] - d*(A*b^2 + a^2*C)*(m + n + 3)*Sin[e + f*x]^2, x], x], x] /; FreeQ[{a, b, c, d, e, f, A, C, n}, x] && NeQ[b*c - a*d, 0] && NeQ[a^2 - b^2, 0] && NeQ[c^2 - d^2, 0] && LtQ[m, -1] && ((EqQ[a, 0] && IntegerQ[m] && !IntegerQ[n]) || !(IntegerQ[2*n] && LtQ[n, -1] && ((IntegerQ[n] && !IntegerQ[m]) || EqQ[a, 0])))
  3057. Int[Times[Power[Plus[Pattern[a, Blank[]], Times[Optional[Pattern[b, Blank[]]], sin[Plus[Optional[Pattern[e, Blank[]]], Times[Optional[Pattern[f, Blank[]]], Pattern[x, Blank[]]]]]]], -1], Power[Plus[Optional[Pattern[c, Blank[]]], Times[Optional[Pattern[d, Blank[]]], sin[Plus[Optional[Pattern[e, Blank[]]], Times[Optional[Pattern[f, Blank[]]], Pattern[x, Blank[]]]]]]], -1], Plus[Optional[Pattern[A, Blank[]]], Times[Optional[Pattern[B, Blank[]]], sin[Plus[Optional[Pattern[e, Blank[]]], Times[Optional[Pattern[f, Blank[]]], Pattern[x, Blank[]]]]]], Times[Optional[Pattern[C, Blank[]]], Power[sin[Plus[Optional[Pattern[e, Blank[]]], Times[Optional[Pattern[f, Blank[]]], Pattern[x, Blank[]]]]], 2]]]], Pattern[x, Blank[Symbol]]] := Simp[(C*x)/(b*d), x] + (Dist[(A*b^2 - a*b*B + a^2*C)/(b*(b*c - a*d)), Int[1/(a + b*Sin[e + f*x]), x], x] - Dist[(c^2*C - B*c*d + A*d^2)/(d*(b*c - a*d)), Int[1/(c + d*Sin[e + f*x]), x], x]) /; FreeQ[{a, b, c, d, e, f, A, B, C}, x] && NeQ[b*c - a*d, 0] && NeQ[a^2 - b^2, 0] && NeQ[c^2 - d^2, 0]
  3058. Int[Times[Power[Plus[Pattern[a, Blank[]], Times[Optional[Pattern[b, Blank[]]], sin[Plus[Optional[Pattern[e, Blank[]]], Times[Optional[Pattern[f, Blank[]]], Pattern[x, Blank[]]]]]]], -1], Power[Plus[Optional[Pattern[c, Blank[]]], Times[Optional[Pattern[d, Blank[]]], sin[Plus[Optional[Pattern[e, Blank[]]], Times[Optional[Pattern[f, Blank[]]], Pattern[x, Blank[]]]]]]], -1], Plus[Optional[Pattern[A, Blank[]]], Times[Optional[Pattern[C, Blank[]]], Power[sin[Plus[Optional[Pattern[e, Blank[]]], Times[Optional[Pattern[f, Blank[]]], Pattern[x, Blank[]]]]], 2]]]], Pattern[x, Blank[Symbol]]] := Simp[(C*x)/(b*d), x] + (Dist[(A*b^2 + a^2*C)/(b*(b*c - a*d)), Int[1/(a + b*Sin[e + f*x]), x], x] - Dist[(c^2*C + A*d^2)/(d*(b*c - a*d)), Int[1/(c + d*Sin[e + f*x]), x], x]) /; FreeQ[{a, b, c, d, e, f, A, C}, x] && NeQ[b*c - a*d, 0] && NeQ[a^2 - b^2, 0] && NeQ[c^2 - d^2, 0]
  3059. Int[Times[Power[Plus[Optional[Pattern[a, Blank[]]], Times[Optional[Pattern[b, Blank[]]], sin[Plus[Optional[Pattern[e, Blank[]]], Times[Optional[Pattern[f, Blank[]]], Pattern[x, Blank[]]]]]]], Rational[-1, 2]], Power[Plus[Optional[Pattern[c, Blank[]]], Times[Optional[Pattern[d, Blank[]]], sin[Plus[Optional[Pattern[e, Blank[]]], Times[Optional[Pattern[f, Blank[]]], Pattern[x, Blank[]]]]]]], -1], Plus[Optional[Pattern[A, Blank[]]], Times[Optional[Pattern[B, Blank[]]], sin[Plus[Optional[Pattern[e, Blank[]]], Times[Optional[Pattern[f, Blank[]]], Pattern[x, Blank[]]]]]], Times[Optional[Pattern[C, Blank[]]], Power[sin[Plus[Optional[Pattern[e, Blank[]]], Times[Optional[Pattern[f, Blank[]]], Pattern[x, Blank[]]]]], 2]]]], Pattern[x, Blank[Symbol]]] := Dist[C/(b*d), Int[Sqrt[a + b*Sin[e + f*x]], x], x] - Dist[1/(b*d), Int[Simp[a*c*C - A*b*d + (b*c*C - b*B*d + a*C*d)*Sin[e + f*x], x]/(Sqrt[a + b*Sin[e + f*x]]*(c + d*Sin[e + f*x])), x], x] /; FreeQ[{a, b, c, d, e, f, A, B, C}, x] && NeQ[b*c - a*d, 0] && NeQ[a^2 - b^2, 0] && NeQ[c^2 - d^2, 0]
  3060. Int[Times[Power[Plus[Optional[Pattern[a, Blank[]]], Times[Optional[Pattern[b, Blank[]]], sin[Plus[Optional[Pattern[e, Blank[]]], Times[Optional[Pattern[f, Blank[]]], Pattern[x, Blank[]]]]]]], Rational[-1, 2]], Power[Plus[Optional[Pattern[c, Blank[]]], Times[Optional[Pattern[d, Blank[]]], sin[Plus[Optional[Pattern[e, Blank[]]], Times[Optional[Pattern[f, Blank[]]], Pattern[x, Blank[]]]]]]], -1], Plus[Optional[Pattern[A, Blank[]]], Times[Optional[Pattern[C, Blank[]]], Power[sin[Plus[Optional[Pattern[e, Blank[]]], Times[Optional[Pattern[f, Blank[]]], Pattern[x, Blank[]]]]], 2]]]], Pattern[x, Blank[Symbol]]] := Dist[C/(b*d), Int[Sqrt[a + b*Sin[e + f*x]], x], x] - Dist[1/(b*d), Int[Simp[a*c*C - A*b*d + (b*c*C + a*C*d)*Sin[e + f*x], x]/(Sqrt[a + b*Sin[e + f*x]]*(c + d*Sin[e + f*x])), x], x] /; FreeQ[{a, b, c, d, e, f, A, C}, x] && NeQ[b*c - a*d, 0] && NeQ[a^2 - b^2, 0] && NeQ[c^2 - d^2, 0]
  3061. Int[Times[Power[Plus[Optional[Pattern[a, Blank[]]], Times[Optional[Pattern[b, Blank[]]], sin[Plus[Optional[Pattern[e, Blank[]]], Times[Optional[Pattern[f, Blank[]]], Pattern[x, Blank[]]]]]]], Rational[-1, 2]], Power[Plus[Pattern[c, Blank[]], Times[Optional[Pattern[d, Blank[]]], sin[Plus[Optional[Pattern[e, Blank[]]], Times[Optional[Pattern[f, Blank[]]], Pattern[x, Blank[]]]]]]], Rational[-1, 2]], Plus[Optional[Pattern[A, Blank[]]], Times[Optional[Pattern[B, Blank[]]], sin[Plus[Optional[Pattern[e, Blank[]]], Times[Optional[Pattern[f, Blank[]]], Pattern[x, Blank[]]]]]], Times[Optional[Pattern[C, Blank[]]], Power[sin[Plus[Optional[Pattern[e, Blank[]]], Times[Optional[Pattern[f, Blank[]]], Pattern[x, Blank[]]]]], 2]]]], Pattern[x, Blank[Symbol]]] := -Simp[(C*Cos[e + f*x]*Sqrt[c + d*Sin[e + f*x]])/(d*f*Sqrt[a + b*Sin[e + f*x]]), x] + Dist[1/(2*d), Int[(1*Simp[2*a*A*d - C*(b*c - a*d) - 2*(a*c*C - d*(A*b + a*B))*Sin[e + f*x] + (2*b*B*d - C*(b*c + a*d))*Sin[e + f*x]^2, x])/((a + b*Sin[e + f*x])^(3/2)*Sqrt[c + d*Sin[e + f*x]]), x], x] /; FreeQ[{a, b, c, d, e, f, A, B, C}, x] && NeQ[b*c - a*d, 0] && NeQ[a^2 - b^2, 0] && NeQ[c^2 - d^2, 0]
  3062. Int[Times[Power[Plus[Optional[Pattern[a, Blank[]]], Times[Optional[Pattern[b, Blank[]]], sin[Plus[Optional[Pattern[e, Blank[]]], Times[Optional[Pattern[f, Blank[]]], Pattern[x, Blank[]]]]]]], Rational[-1, 2]], Power[Plus[Pattern[c, Blank[]], Times[Optional[Pattern[d, Blank[]]], sin[Plus[Optional[Pattern[e, Blank[]]], Times[Optional[Pattern[f, Blank[]]], Pattern[x, Blank[]]]]]]], Rational[-1, 2]], Plus[Optional[Pattern[A, Blank[]]], Times[Optional[Pattern[C, Blank[]]], Power[sin[Plus[Optional[Pattern[e, Blank[]]], Times[Optional[Pattern[f, Blank[]]], Pattern[x, Blank[]]]]], 2]]]], Pattern[x, Blank[Symbol]]] := -Simp[(C*Cos[e + f*x]*Sqrt[c + d*Sin[e + f*x]])/(d*f*Sqrt[a + b*Sin[e + f*x]]), x] + Dist[1/(2*d), Int[(1*Simp[2*a*A*d - C*(b*c - a*d) - 2*(a*c*C - A*b*d)*Sin[e + f*x] - C*(b*c + a*d)*Sin[e + f*x]^2, x])/((a + b*Sin[e + f*x])^(3/2)*Sqrt[c + d*Sin[e + f*x]]), x], x] /; FreeQ[{a, b, c, d, e, f, A, C}, x] && NeQ[b*c - a*d, 0] && NeQ[a^2 - b^2, 0] && NeQ[c^2 - d^2, 0]
  3063. Int[Times[Power[Times[Optional[Pattern[d, Blank[]]], sin[Plus[Optional[Pattern[e, Blank[]]], Times[Optional[Pattern[f, Blank[]]], Pattern[x, Blank[]]]]]], Optional[Pattern[n, Blank[]]]], Power[Plus[Pattern[a, Blank[]], Times[Optional[Pattern[b, Blank[]]], sin[Plus[Optional[Pattern[e, Blank[]]], Times[Optional[Pattern[f, Blank[]]], Pattern[x, Blank[]]]]]]], -1], Plus[Optional[Pattern[A, Blank[]]], Times[Optional[Pattern[B, Blank[]]], sin[Plus[Optional[Pattern[e, Blank[]]], Times[Optional[Pattern[f, Blank[]]], Pattern[x, Blank[]]]]]], Times[Optional[Pattern[C, Blank[]]], Power[sin[Plus[Optional[Pattern[e, Blank[]]], Times[Optional[Pattern[f, Blank[]]], Pattern[x, Blank[]]]]], 2]]]], Pattern[x, Blank[Symbol]]] := Dist[(b*B - a*C)/b^2, Int[(d*Sin[e + f*x])^n, x], x] + (Dist[(A*b^2 - a*b*B + a^2*C)/b^2, Int[(d*Sin[e + f*x])^n/(a + b*Sin[e + f*x]), x], x] + Dist[C/(b*d), Int[(d*Sin[e + f*x])^(n + 1), x], x]) /; FreeQ[{a, b, d, e, f, A, B, C, n}, x] && NeQ[a^2 - b^2, 0]
  3064. Int[Times[Power[Times[Optional[Pattern[d, Blank[]]], sin[Plus[Optional[Pattern[e, Blank[]]], Times[Optional[Pattern[f, Blank[]]], Pattern[x, Blank[]]]]]], Optional[Pattern[n, Blank[]]]], Power[Plus[Pattern[a, Blank[]], Times[Optional[Pattern[b, Blank[]]], sin[Plus[Optional[Pattern[e, Blank[]]], Times[Optional[Pattern[f, Blank[]]], Pattern[x, Blank[]]]]]]], -1], Plus[Optional[Pattern[A, Blank[]]], Times[Optional[Pattern[C, Blank[]]], Power[sin[Plus[Optional[Pattern[e, Blank[]]], Times[Optional[Pattern[f, Blank[]]], Pattern[x, Blank[]]]]], 2]]]], Pattern[x, Blank[Symbol]]] := -Dist[(a*C)/b^2, Int[(d*Sin[e + f*x])^n, x], x] + (Dist[(A*b^2 + a^2*C)/b^2, Int[(d*Sin[e + f*x])^n/(a + b*Sin[e + f*x]), x], x] + Dist[C/(b*d), Int[(d*Sin[e + f*x])^(n + 1), x], x]) /; FreeQ[{a, b, d, e, f, A, C, n}, x] && NeQ[a^2 - b^2, 0]
  3065. Int[Times[Power[Plus[Optional[Pattern[a, Blank[]]], Times[Optional[Pattern[b, Blank[]]], sin[Plus[Optional[Pattern[e, Blank[]]], Times[Optional[Pattern[f, Blank[]]], Pattern[x, Blank[]]]]]]], Pattern[m, Blank[]]], Power[Plus[Optional[Pattern[c, Blank[]]], Times[Optional[Pattern[d, Blank[]]], sin[Plus[Optional[Pattern[e, Blank[]]], Times[Optional[Pattern[f, Blank[]]], Pattern[x, Blank[]]]]]]], Pattern[n, Blank[]]], Plus[Optional[Pattern[A, Blank[]]], Times[Optional[Pattern[B, Blank[]]], sin[Plus[Optional[Pattern[e, Blank[]]], Times[Optional[Pattern[f, Blank[]]], Pattern[x, Blank[]]]]]], Times[Optional[Pattern[C, Blank[]]], Power[sin[Plus[Optional[Pattern[e, Blank[]]], Times[Optional[Pattern[f, Blank[]]], Pattern[x, Blank[]]]]], 2]]]], Pattern[x, Blank[Symbol]]] := Unintegrable[(a + b*Sin[e + f*x])^m*(c + d*Sin[e + f*x])^n*(A + B*Sin[e + f*x] + C*Sin[e + f*x]^2), x] /; FreeQ[{a, b, c, d, e, f, A, B, C, m, n}, x] && NeQ[b*c - a*d, 0] && NeQ[a^2 - b^2, 0] && NeQ[c^2 - d^2, 0]
  3066. Int[Times[Power[Plus[Optional[Pattern[a, Blank[]]], Times[Optional[Pattern[b, Blank[]]], sin[Plus[Optional[Pattern[e, Blank[]]], Times[Optional[Pattern[f, Blank[]]], Pattern[x, Blank[]]]]]]], Pattern[m, Blank[]]], Power[Plus[Optional[Pattern[c, Blank[]]], Times[Optional[Pattern[d, Blank[]]], sin[Plus[Optional[Pattern[e, Blank[]]], Times[Optional[Pattern[f, Blank[]]], Pattern[x, Blank[]]]]]]], Pattern[n, Blank[]]], Plus[Optional[Pattern[A, Blank[]]], Times[Optional[Pattern[C, Blank[]]], Power[sin[Plus[Optional[Pattern[e, Blank[]]], Times[Optional[Pattern[f, Blank[]]], Pattern[x, Blank[]]]]], 2]]]], Pattern[x, Blank[Symbol]]] := Unintegrable[(a + b*Sin[e + f*x])^m*(c + d*Sin[e + f*x])^n*(A + C*Sin[e + f*x]^2), x] /; FreeQ[{a, b, c, d, e, f, A, C, m, n}, x] && NeQ[b*c - a*d, 0] && NeQ[a^2 - b^2, 0] && NeQ[c^2 - d^2, 0]
  3067. Int[Times[Power[Times[Optional[Pattern[b, Blank[]]], Power[sin[Plus[Optional[Pattern[e, Blank[]]], Times[Optional[Pattern[f, Blank[]]], Pattern[x, Blank[]]]]], Pattern[p, Blank[]]]], Pattern[m, Blank[]]], Power[Plus[Optional[Pattern[c, Blank[]]], Times[Optional[Pattern[d, Blank[]]], sin[Plus[Optional[Pattern[e, Blank[]]], Times[Optional[Pattern[f, Blank[]]], Pattern[x, Blank[]]]]]]], Optional[Pattern[n, Blank[]]]], Plus[Optional[Pattern[A, Blank[]]], Times[Optional[Pattern[B, Blank[]]], sin[Plus[Optional[Pattern[e, Blank[]]], Times[Optional[Pattern[f, Blank[]]], Pattern[x, Blank[]]]]]], Times[Optional[Pattern[C, Blank[]]], Power[sin[Plus[Optional[Pattern[e, Blank[]]], Times[Optional[Pattern[f, Blank[]]], Pattern[x, Blank[]]]]], 2]]]], Pattern[x, Blank[Symbol]]] := Dist[(b*Sin[e + f*x]^p)^m/(b*Sin[e + f*x])^(m*p), Int[(b*Sin[e + f*x])^(m*p)*(c + d*Sin[e + f*x])^n*(A + B*Sin[e + f*x] + C*Sin[e + f*x]^2), x], x] /; FreeQ[{b, c, d, e, f, A, B, C, m, n, p}, x] && !IntegerQ[m]
  3068. Int[Times[Power[Times[Power[cos[Plus[Optional[Pattern[e, Blank[]]], Times[Optional[Pattern[f, Blank[]]], Pattern[x, Blank[]]]]], Pattern[p, Blank[]]], Optional[Pattern[b, Blank[]]]], Pattern[m, Blank[]]], Plus[Optional[Pattern[A, Blank[]]], Times[cos[Plus[Optional[Pattern[e, Blank[]]], Times[Optional[Pattern[f, Blank[]]], Pattern[x, Blank[]]]]], Optional[Pattern[B, Blank[]]]], Times[Power[cos[Plus[Optional[Pattern[e, Blank[]]], Times[Optional[Pattern[f, Blank[]]], Pattern[x, Blank[]]]]], 2], Optional[Pattern[C, Blank[]]]]], Power[Plus[Optional[Pattern[c, Blank[]]], Times[cos[Plus[Optional[Pattern[e, Blank[]]], Times[Optional[Pattern[f, Blank[]]], Pattern[x, Blank[]]]]], Optional[Pattern[d, Blank[]]]]], Optional[Pattern[n, Blank[]]]]], Pattern[x, Blank[Symbol]]] := Dist[(b*Cos[e + f*x]^p)^m/(b*Cos[e + f*x])^(m*p), Int[(b*Cos[e + f*x])^(m*p)*(c + d*Cos[e + f*x])^n*(A + B*Cos[e + f*x] + C*Cos[e + f*x]^2), x], x] /; FreeQ[{b, c, d, e, f, A, B, C, m, n, p}, x] && !IntegerQ[m]
  3069. Int[Times[Power[Times[Optional[Pattern[b, Blank[]]], Power[sin[Plus[Optional[Pattern[e, Blank[]]], Times[Optional[Pattern[f, Blank[]]], Pattern[x, Blank[]]]]], Pattern[p, Blank[]]]], Pattern[m, Blank[]]], Power[Plus[Optional[Pattern[c, Blank[]]], Times[Optional[Pattern[d, Blank[]]], sin[Plus[Optional[Pattern[e, Blank[]]], Times[Optional[Pattern[f, Blank[]]], Pattern[x, Blank[]]]]]]], Optional[Pattern[n, Blank[]]]], Plus[Optional[Pattern[A, Blank[]]], Times[Optional[Pattern[C, Blank[]]], Power[sin[Plus[Optional[Pattern[e, Blank[]]], Times[Optional[Pattern[f, Blank[]]], Pattern[x, Blank[]]]]], 2]]]], Pattern[x, Blank[Symbol]]] := Dist[(b*Sin[e + f*x]^p)^m/(b*Sin[e + f*x])^(m*p), Int[(b*Sin[e + f*x])^(m*p)*(c + d*Sin[e + f*x])^n*(A + C*Sin[e + f*x]^2), x], x] /; FreeQ[{b, c, d, e, f, A, C, m, n, p}, x] && !IntegerQ[m]
  3070. Int[Times[Power[Times[Power[cos[Plus[Optional[Pattern[e, Blank[]]], Times[Optional[Pattern[f, Blank[]]], Pattern[x, Blank[]]]]], Pattern[p, Blank[]]], Optional[Pattern[b, Blank[]]]], Pattern[m, Blank[]]], Plus[Optional[Pattern[A, Blank[]]], Times[Power[cos[Plus[Optional[Pattern[e, Blank[]]], Times[Optional[Pattern[f, Blank[]]], Pattern[x, Blank[]]]]], 2], Optional[Pattern[C, Blank[]]]]], Power[Plus[Optional[Pattern[c, Blank[]]], Times[cos[Plus[Optional[Pattern[e, Blank[]]], Times[Optional[Pattern[f, Blank[]]], Pattern[x, Blank[]]]]], Optional[Pattern[d, Blank[]]]]], Optional[Pattern[n, Blank[]]]]], Pattern[x, Blank[Symbol]]] := Dist[(b*Cos[e + f*x]^p)^m/(b*Cos[e + f*x])^(m*p), Int[(b*Cos[e + f*x])^(m*p)*(c + d*Cos[e + f*x])^n*(A + C*Cos[e + f*x]^2), x], x] /; FreeQ[{b, c, d, e, f, A, C, m, n, p}, x] && !IntegerQ[m]
  3071. Int[Power[Plus[Times[cos[Plus[Optional[Pattern[c, Blank[]]], Times[Optional[Pattern[d, Blank[]]], Pattern[x, Blank[]]]]], Optional[Pattern[a, Blank[]]]], Times[Optional[Pattern[b, Blank[]]], sin[Plus[Optional[Pattern[c, Blank[]]], Times[Optional[Pattern[d, Blank[]]], Pattern[x, Blank[]]]]]]], Pattern[n, Blank[]]], Pattern[x, Blank[Symbol]]] := Simp[(a*(a*Cos[c + d*x] + b*Sin[c + d*x])^n)/(b*d*n), x] /; FreeQ[{a, b, c, d, n}, x] && EqQ[a^2 + b^2, 0]
  3072. Int[Power[Plus[Times[cos[Plus[Optional[Pattern[c, Blank[]]], Times[Optional[Pattern[d, Blank[]]], Pattern[x, Blank[]]]]], Optional[Pattern[a, Blank[]]]], Times[Optional[Pattern[b, Blank[]]], sin[Plus[Optional[Pattern[c, Blank[]]], Times[Optional[Pattern[d, Blank[]]], Pattern[x, Blank[]]]]]]], Pattern[n, Blank[]]], Pattern[x, Blank[Symbol]]] := -Dist[d^(-1), Subst[Int[(a^2 + b^2 - x^2)^((n - 1)/2), x], x, b*Cos[c + d*x] - a*Sin[c + d*x]], x] /; FreeQ[{a, b, c, d}, x] && NeQ[a^2 + b^2, 0] && IGtQ[(n - 1)/2, 0]
  3073. Int[Power[Plus[Times[cos[Plus[Optional[Pattern[c, Blank[]]], Times[Optional[Pattern[d, Blank[]]], Pattern[x, Blank[]]]]], Optional[Pattern[a, Blank[]]]], Times[Optional[Pattern[b, Blank[]]], sin[Plus[Optional[Pattern[c, Blank[]]], Times[Optional[Pattern[d, Blank[]]], Pattern[x, Blank[]]]]]]], Pattern[n, Blank[]]], Pattern[x, Blank[Symbol]]] := -Simp[((b*Cos[c + d*x] - a*Sin[c + d*x])*(a*Cos[c + d*x] + b*Sin[c + d*x])^(n - 1))/(d*n), x] + Dist[((n - 1)*(a^2 + b^2))/n, Int[(a*Cos[c + d*x] + b*Sin[c + d*x])^(n - 2), x], x] /; FreeQ[{a, b, c, d}, x] && NeQ[a^2 + b^2, 0] && !IntegerQ[(n - 1)/2] && GtQ[n, 1]
  3074. Int[Power[Plus[Times[cos[Plus[Optional[Pattern[c, Blank[]]], Times[Optional[Pattern[d, Blank[]]], Pattern[x, Blank[]]]]], Optional[Pattern[a, Blank[]]]], Times[Optional[Pattern[b, Blank[]]], sin[Plus[Optional[Pattern[c, Blank[]]], Times[Optional[Pattern[d, Blank[]]], Pattern[x, Blank[]]]]]]], -1], Pattern[x, Blank[Symbol]]] := -Dist[d^(-1), Subst[Int[1/(a^2 + b^2 - x^2), x], x, b*Cos[c + d*x] - a*Sin[c + d*x]], x] /; FreeQ[{a, b, c, d}, x] && NeQ[a^2 + b^2, 0]
  3075. Int[Power[Plus[Times[cos[Plus[Optional[Pattern[c, Blank[]]], Times[Optional[Pattern[d, Blank[]]], Pattern[x, Blank[]]]]], Optional[Pattern[a, Blank[]]]], Times[Optional[Pattern[b, Blank[]]], sin[Plus[Optional[Pattern[c, Blank[]]], Times[Optional[Pattern[d, Blank[]]], Pattern[x, Blank[]]]]]]], -2], Pattern[x, Blank[Symbol]]] := Simp[Sin[c + d*x]/(a*d*(a*Cos[c + d*x] + b*Sin[c + d*x])), x] /; FreeQ[{a, b, c, d}, x] && NeQ[a^2 + b^2, 0]
  3076. Int[Power[Plus[Times[cos[Plus[Optional[Pattern[c, Blank[]]], Times[Optional[Pattern[d, Blank[]]], Pattern[x, Blank[]]]]], Optional[Pattern[a, Blank[]]]], Times[Optional[Pattern[b, Blank[]]], sin[Plus[Optional[Pattern[c, Blank[]]], Times[Optional[Pattern[d, Blank[]]], Pattern[x, Blank[]]]]]]], Pattern[n, Blank[]]], Pattern[x, Blank[Symbol]]] := Simp[((b*Cos[c + d*x] - a*Sin[c + d*x])*(a*Cos[c + d*x] + b*Sin[c + d*x])^(n + 1))/(d*(n + 1)*(a^2 + b^2)), x] + Dist[(n + 2)/((n + 1)*(a^2 + b^2)), Int[(a*Cos[c + d*x] + b*Sin[c + d*x])^(n + 2), x], x] /; FreeQ[{a, b, c, d}, x] && NeQ[a^2 + b^2, 0] && LtQ[n, -1] && NeQ[n, -2]
  3077. Int[Power[Plus[Times[cos[Plus[Optional[Pattern[c, Blank[]]], Times[Optional[Pattern[d, Blank[]]], Pattern[x, Blank[]]]]], Optional[Pattern[a, Blank[]]]], Times[Optional[Pattern[b, Blank[]]], sin[Plus[Optional[Pattern[c, Blank[]]], Times[Optional[Pattern[d, Blank[]]], Pattern[x, Blank[]]]]]]], Pattern[n, Blank[]]], Pattern[x, Blank[Symbol]]] := Dist[(a^2 + b^2)^(n/2), Int[Cos[c + d*x - ArcTan[a, b]]^n, x], x] /; FreeQ[{a, b, c, d, n}, x] && !(GeQ[n, 1] || LeQ[n, -1]) && GtQ[a^2 + b^2, 0]
  3078. Int[Power[Plus[Times[cos[Plus[Optional[Pattern[c, Blank[]]], Times[Optional[Pattern[d, Blank[]]], Pattern[x, Blank[]]]]], Optional[Pattern[a, Blank[]]]], Times[Optional[Pattern[b, Blank[]]], sin[Plus[Optional[Pattern[c, Blank[]]], Times[Optional[Pattern[d, Blank[]]], Pattern[x, Blank[]]]]]]], Pattern[n, Blank[]]], Pattern[x, Blank[Symbol]]] := Dist[(a*Cos[c + d*x] + b*Sin[c + d*x])^n/((a*Cos[c + d*x] + b*Sin[c + d*x])/Sqrt[a^2 + b^2])^n, Int[Cos[c + d*x - ArcTan[a, b]]^n, x], x] /; FreeQ[{a, b, c, d, n}, x] && !(GeQ[n, 1] || LeQ[n, -1]) && !(GtQ[a^2 + b^2, 0] || EqQ[a^2 + b^2, 0])
  3079. Int[Times[Power[sin[Plus[Optional[Pattern[c, Blank[]]], Times[Optional[Pattern[d, Blank[]]], Pattern[x, Blank[]]]]], Pattern[m, Blank[]]], Power[Plus[Times[cos[Plus[Optional[Pattern[c, Blank[]]], Times[Optional[Pattern[d, Blank[]]], Pattern[x, Blank[]]]]], Optional[Pattern[a, Blank[]]]], Times[Optional[Pattern[b, Blank[]]], sin[Plus[Optional[Pattern[c, Blank[]]], Times[Optional[Pattern[d, Blank[]]], Pattern[x, Blank[]]]]]]], Pattern[n, Blank[]]]], Pattern[x, Blank[Symbol]]] := -Simp[(a*(a*Cos[c + d*x] + b*Sin[c + d*x])^(n - 1))/(d*(n - 1)*Sin[c + d*x]^(n - 1)), x] + Dist[2*b, Int[(a*Cos[c + d*x] + b*Sin[c + d*x])^(n - 1)/Sin[c + d*x]^(n - 1), x], x] /; FreeQ[{a, b, c, d}, x] && EqQ[m + n, 0] && EqQ[a^2 + b^2, 0] && GtQ[n, 1]
  3080. Int[Times[Power[cos[Plus[Optional[Pattern[c, Blank[]]], Times[Optional[Pattern[d, Blank[]]], Pattern[x, Blank[]]]]], Pattern[m, Blank[]]], Power[Plus[Times[cos[Plus[Optional[Pattern[c, Blank[]]], Times[Optional[Pattern[d, Blank[]]], Pattern[x, Blank[]]]]], Optional[Pattern[a, Blank[]]]], Times[Optional[Pattern[b, Blank[]]], sin[Plus[Optional[Pattern[c, Blank[]]], Times[Optional[Pattern[d, Blank[]]], Pattern[x, Blank[]]]]]]], Pattern[n, Blank[]]]], Pattern[x, Blank[Symbol]]] := Simp[(b*(a*Cos[c + d*x] + b*Sin[c + d*x])^(n - 1))/(d*(n - 1)*Cos[c + d*x]^(n - 1)), x] + Dist[2*a, Int[(a*Cos[c + d*x] + b*Sin[c + d*x])^(n - 1)/Cos[c + d*x]^(n - 1), x], x] /; FreeQ[{a, b, c, d}, x] && EqQ[m + n, 0] && EqQ[a^2 + b^2, 0] && GtQ[n, 1]
  3081. Int[Times[Power[sin[Plus[Optional[Pattern[c, Blank[]]], Times[Optional[Pattern[d, Blank[]]], Pattern[x, Blank[]]]]], Optional[Pattern[m, Blank[]]]], Power[Plus[Times[cos[Plus[Optional[Pattern[c, Blank[]]], Times[Optional[Pattern[d, Blank[]]], Pattern[x, Blank[]]]]], Optional[Pattern[a, Blank[]]]], Times[Optional[Pattern[b, Blank[]]], sin[Plus[Optional[Pattern[c, Blank[]]], Times[Optional[Pattern[d, Blank[]]], Pattern[x, Blank[]]]]]]], Pattern[n, Blank[]]]], Pattern[x, Blank[Symbol]]] := Simp[(a*(a*Cos[c + d*x] + b*Sin[c + d*x])^n)/(2*b*d*n*Sin[c + d*x]^n), x] + Dist[1/(2*b), Int[(a*Cos[c + d*x] + b*Sin[c + d*x])^(n + 1)/Sin[c + d*x]^(n + 1), x], x] /; FreeQ[{a, b, c, d}, x] && EqQ[m + n, 0] && EqQ[a^2 + b^2, 0] && LtQ[n, 0]
  3082. Int[Times[Power[cos[Plus[Optional[Pattern[c, Blank[]]], Times[Optional[Pattern[d, Blank[]]], Pattern[x, Blank[]]]]], Optional[Pattern[m, Blank[]]]], Power[Plus[Times[cos[Plus[Optional[Pattern[c, Blank[]]], Times[Optional[Pattern[d, Blank[]]], Pattern[x, Blank[]]]]], Optional[Pattern[a, Blank[]]]], Times[Optional[Pattern[b, Blank[]]], sin[Plus[Optional[Pattern[c, Blank[]]], Times[Optional[Pattern[d, Blank[]]], Pattern[x, Blank[]]]]]]], Pattern[n, Blank[]]]], Pattern[x, Blank[Symbol]]] := -Simp[(b*(a*Cos[c + d*x] + b*Sin[c + d*x])^n)/(2*a*d*n*Cos[c + d*x]^n), x] + Dist[1/(2*a), Int[(a*Cos[c + d*x] + b*Sin[c + d*x])^(n + 1)/Cos[c + d*x]^(n + 1), x], x] /; FreeQ[{a, b, c, d}, x] && EqQ[m + n, 0] && EqQ[a^2 + b^2, 0] && LtQ[n, 0]
  3083. Int[Times[Power[sin[Plus[Optional[Pattern[c, Blank[]]], Times[Optional[Pattern[d, Blank[]]], Pattern[x, Blank[]]]]], Optional[Pattern[m, Blank[]]]], Power[Plus[Times[cos[Plus[Optional[Pattern[c, Blank[]]], Times[Optional[Pattern[d, Blank[]]], Pattern[x, Blank[]]]]], Optional[Pattern[a, Blank[]]]], Times[Optional[Pattern[b, Blank[]]], sin[Plus[Optional[Pattern[c, Blank[]]], Times[Optional[Pattern[d, Blank[]]], Pattern[x, Blank[]]]]]]], Pattern[n, Blank[]]]], Pattern[x, Blank[Symbol]]] := Simp[(a*(a*Cos[c + d*x] + b*Sin[c + d*x])^n*Hypergeometric2F1[1, n, n + 1, (b + a*Cot[c + d*x])/(2*b)])/(2*b*d*n*Sin[c + d*x]^n), x] /; FreeQ[{a, b, c, d, n}, x] && EqQ[m + n, 0] && EqQ[a^2 + b^2, 0] && !IntegerQ[n]
  3084. Int[Times[Power[cos[Plus[Optional[Pattern[c, Blank[]]], Times[Optional[Pattern[d, Blank[]]], Pattern[x, Blank[]]]]], Optional[Pattern[m, Blank[]]]], Power[Plus[Times[cos[Plus[Optional[Pattern[c, Blank[]]], Times[Optional[Pattern[d, Blank[]]], Pattern[x, Blank[]]]]], Optional[Pattern[a, Blank[]]]], Times[Optional[Pattern[b, Blank[]]], sin[Plus[Optional[Pattern[c, Blank[]]], Times[Optional[Pattern[d, Blank[]]], Pattern[x, Blank[]]]]]]], Pattern[n, Blank[]]]], Pattern[x, Blank[Symbol]]] := -Simp[(b*(a*Cos[c + d*x] + b*Sin[c + d*x])^n*Hypergeometric2F1[1, n, n + 1, (a + b*Tan[c + d*x])/(2*a)])/(2*a*d*n*Cos[c + d*x]^n), x] /; FreeQ[{a, b, c, d, n}, x] && EqQ[m + n, 0] && EqQ[a^2 + b^2, 0] && !IntegerQ[n]
  3085. Int[Times[Power[sin[Plus[Optional[Pattern[c, Blank[]]], Times[Optional[Pattern[d, Blank[]]], Pattern[x, Blank[]]]]], Pattern[m, Blank[]]], Power[Plus[Times[cos[Plus[Optional[Pattern[c, Blank[]]], Times[Optional[Pattern[d, Blank[]]], Pattern[x, Blank[]]]]], Optional[Pattern[a, Blank[]]]], Times[Optional[Pattern[b, Blank[]]], sin[Plus[Optional[Pattern[c, Blank[]]], Times[Optional[Pattern[d, Blank[]]], Pattern[x, Blank[]]]]]]], Optional[Pattern[n, Blank[]]]]], Pattern[x, Blank[Symbol]]] := Int[(b + a*Cot[c + d*x])^n, x] /; FreeQ[{a, b, c, d}, x] && EqQ[m + n, 0] && IntegerQ[n] && NeQ[a^2 + b^2, 0]
  3086. Int[Times[Power[cos[Plus[Optional[Pattern[c, Blank[]]], Times[Optional[Pattern[d, Blank[]]], Pattern[x, Blank[]]]]], Pattern[m, Blank[]]], Power[Plus[Times[cos[Plus[Optional[Pattern[c, Blank[]]], Times[Optional[Pattern[d, Blank[]]], Pattern[x, Blank[]]]]], Optional[Pattern[a, Blank[]]]], Times[Optional[Pattern[b, Blank[]]], sin[Plus[Optional[Pattern[c, Blank[]]], Times[Optional[Pattern[d, Blank[]]], Pattern[x, Blank[]]]]]]], Optional[Pattern[n, Blank[]]]]], Pattern[x, Blank[Symbol]]] := Int[(a + b*Tan[c + d*x])^n, x] /; FreeQ[{a, b, c, d}, x] && EqQ[m + n, 0] && IntegerQ[n] && NeQ[a^2 + b^2, 0]
  3087. Int[Times[Power[sin[Plus[Optional[Pattern[c, Blank[]]], Times[Optional[Pattern[d, Blank[]]], Pattern[x, Blank[]]]]], Optional[Pattern[m, Blank[]]]], Power[Plus[Times[cos[Plus[Optional[Pattern[c, Blank[]]], Times[Optional[Pattern[d, Blank[]]], Pattern[x, Blank[]]]]], Optional[Pattern[a, Blank[]]]], Times[Optional[Pattern[b, Blank[]]], sin[Plus[Optional[Pattern[c, Blank[]]], Times[Optional[Pattern[d, Blank[]]], Pattern[x, Blank[]]]]]]], Pattern[n, Blank[]]]], Pattern[x, Blank[Symbol]]] := Dist[1/d, Subst[Int[(x^m*(a + b*x)^n)/(1 + x^2)^((m + n + 2)/2), x], x, Tan[c + d*x]], x] /; FreeQ[{a, b, c, d}, x] && IntegerQ[n] && IntegerQ[(m + n)/2] && NeQ[n, -1] && !(GtQ[n, 0] && GtQ[m, 1])
  3088. Int[Times[Power[cos[Plus[Optional[Pattern[c, Blank[]]], Times[Optional[Pattern[d, Blank[]]], Pattern[x, Blank[]]]]], Optional[Pattern[m, Blank[]]]], Power[Plus[Times[cos[Plus[Optional[Pattern[c, Blank[]]], Times[Optional[Pattern[d, Blank[]]], Pattern[x, Blank[]]]]], Optional[Pattern[a, Blank[]]]], Times[Optional[Pattern[b, Blank[]]], sin[Plus[Optional[Pattern[c, Blank[]]], Times[Optional[Pattern[d, Blank[]]], Pattern[x, Blank[]]]]]]], Pattern[n, Blank[]]]], Pattern[x, Blank[Symbol]]] := -Dist[d^(-1), Subst[Int[(x^m*(b + a*x)^n)/(1 + x^2)^((m + n + 2)/2), x], x, Cot[c + d*x]], x] /; FreeQ[{a, b, c, d}, x] && IntegerQ[n] && IntegerQ[(m + n)/2] && NeQ[n, -1] && !(GtQ[n, 0] && GtQ[m, 1])
  3089. Int[Times[Power[sin[Plus[Optional[Pattern[c, Blank[]]], Times[Optional[Pattern[d, Blank[]]], Pattern[x, Blank[]]]]], Optional[Pattern[m, Blank[]]]], Power[Plus[Times[cos[Plus[Optional[Pattern[c, Blank[]]], Times[Optional[Pattern[d, Blank[]]], Pattern[x, Blank[]]]]], Optional[Pattern[a, Blank[]]]], Times[Optional[Pattern[b, Blank[]]], sin[Plus[Optional[Pattern[c, Blank[]]], Times[Optional[Pattern[d, Blank[]]], Pattern[x, Blank[]]]]]]], Optional[Pattern[n, Blank[]]]]], Pattern[x, Blank[Symbol]]] := Int[ExpandTrig[sin[c + d*x]^m*(a*cos[c + d*x] + b*sin[c + d*x])^n, x], x] /; FreeQ[{a, b, c, d}, x] && IntegerQ[m] && IGtQ[n, 0]
  3090. Int[Times[Power[cos[Plus[Optional[Pattern[c, Blank[]]], Times[Optional[Pattern[d, Blank[]]], Pattern[x, Blank[]]]]], Optional[Pattern[m, Blank[]]]], Power[Plus[Times[cos[Plus[Optional[Pattern[c, Blank[]]], Times[Optional[Pattern[d, Blank[]]], Pattern[x, Blank[]]]]], Optional[Pattern[a, Blank[]]]], Times[Optional[Pattern[b, Blank[]]], sin[Plus[Optional[Pattern[c, Blank[]]], Times[Optional[Pattern[d, Blank[]]], Pattern[x, Blank[]]]]]]], Optional[Pattern[n, Blank[]]]]], Pattern[x, Blank[Symbol]]] := Int[ExpandTrig[cos[c + d*x]^m*(a*cos[c + d*x] + b*sin[c + d*x])^n, x], x] /; FreeQ[{a, b, c, d}, x] && IntegerQ[m] && IGtQ[n, 0]
  3091. Int[Times[Power[sin[Plus[Optional[Pattern[c, Blank[]]], Times[Optional[Pattern[d, Blank[]]], Pattern[x, Blank[]]]]], Optional[Pattern[m, Blank[]]]], Power[Plus[Times[cos[Plus[Optional[Pattern[c, Blank[]]], Times[Optional[Pattern[d, Blank[]]], Pattern[x, Blank[]]]]], Optional[Pattern[a, Blank[]]]], Times[Optional[Pattern[b, Blank[]]], sin[Plus[Optional[Pattern[c, Blank[]]], Times[Optional[Pattern[d, Blank[]]], Pattern[x, Blank[]]]]]]], Pattern[n, Blank[]]]], Pattern[x, Blank[Symbol]]] := Dist[a^n*b^n, Int[Sin[c + d*x]^m/(b*Cos[c + d*x] + a*Sin[c + d*x])^n, x], x] /; FreeQ[{a, b, c, d, m}, x] && EqQ[a^2 + b^2, 0] && ILtQ[n, 0]
  3092. Int[Times[Power[cos[Plus[Optional[Pattern[c, Blank[]]], Times[Optional[Pattern[d, Blank[]]], Pattern[x, Blank[]]]]], Optional[Pattern[m, Blank[]]]], Power[Plus[Times[cos[Plus[Optional[Pattern[c, Blank[]]], Times[Optional[Pattern[d, Blank[]]], Pattern[x, Blank[]]]]], Optional[Pattern[a, Blank[]]]], Times[Optional[Pattern[b, Blank[]]], sin[Plus[Optional[Pattern[c, Blank[]]], Times[Optional[Pattern[d, Blank[]]], Pattern[x, Blank[]]]]]]], Pattern[n, Blank[]]]], Pattern[x, Blank[Symbol]]] := Dist[a^n*b^n, Int[Cos[c + d*x]^m/(b*Cos[c + d*x] + a*Sin[c + d*x])^n, x], x] /; FreeQ[{a, b, c, d, m}, x] && EqQ[a^2 + b^2, 0] && ILtQ[n, 0]
  3093. Int[Times[Power[sin[Plus[Optional[Pattern[c, Blank[]]], Times[Optional[Pattern[d, Blank[]]], Pattern[x, Blank[]]]]], -1], Power[Plus[Times[cos[Plus[Optional[Pattern[c, Blank[]]], Times[Optional[Pattern[d, Blank[]]], Pattern[x, Blank[]]]]], Optional[Pattern[a, Blank[]]]], Times[Optional[Pattern[b, Blank[]]], sin[Plus[Optional[Pattern[c, Blank[]]], Times[Optional[Pattern[d, Blank[]]], Pattern[x, Blank[]]]]]]], Pattern[n, Blank[]]]], Pattern[x, Blank[Symbol]]] := -Simp[(a*Cos[c + d*x] + b*Sin[c + d*x])^(n + 1)/(a*d*(n + 1)), x] + (Dist[1/a^2, Int[(a*Cos[c + d*x] + b*Sin[c + d*x])^(n + 2)/Sin[c + d*x], x], x] - Dist[b/a^2, Int[(a*Cos[c + d*x] + b*Sin[c + d*x])^(n + 1), x], x]) /; FreeQ[{a, b, c, d}, x] && NeQ[a^2 + b^2, 0] && LtQ[n, -1]
  3094. Int[Times[Power[cos[Plus[Optional[Pattern[c, Blank[]]], Times[Optional[Pattern[d, Blank[]]], Pattern[x, Blank[]]]]], -1], Power[Plus[Times[cos[Plus[Optional[Pattern[c, Blank[]]], Times[Optional[Pattern[d, Blank[]]], Pattern[x, Blank[]]]]], Optional[Pattern[a, Blank[]]]], Times[Optional[Pattern[b, Blank[]]], sin[Plus[Optional[Pattern[c, Blank[]]], Times[Optional[Pattern[d, Blank[]]], Pattern[x, Blank[]]]]]]], Pattern[n, Blank[]]]], Pattern[x, Blank[Symbol]]] := Simp[(a*Cos[c + d*x] + b*Sin[c + d*x])^(n + 1)/(b*d*(n + 1)), x] + (Dist[1/b^2, Int[(a*Cos[c + d*x] + b*Sin[c + d*x])^(n + 2)/Cos[c + d*x], x], x] - Dist[a/b^2, Int[(a*Cos[c + d*x] + b*Sin[c + d*x])^(n + 1), x], x]) /; FreeQ[{a, b, c, d}, x] && NeQ[a^2 + b^2, 0] && LtQ[n, -1]
  3095. Int[Times[Power[sin[Plus[Optional[Pattern[c, Blank[]]], Times[Optional[Pattern[d, Blank[]]], Pattern[x, Blank[]]]]], Pattern[m, Blank[]]], Power[Plus[Times[cos[Plus[Optional[Pattern[c, Blank[]]], Times[Optional[Pattern[d, Blank[]]], Pattern[x, Blank[]]]]], Optional[Pattern[a, Blank[]]]], Times[Optional[Pattern[b, Blank[]]], sin[Plus[Optional[Pattern[c, Blank[]]], Times[Optional[Pattern[d, Blank[]]], Pattern[x, Blank[]]]]]]], Pattern[n, Blank[]]]], Pattern[x, Blank[Symbol]]] := -Dist[a^2 + b^2, Int[Sin[c + d*x]^(m + 2)*(a*Cos[c + d*x] + b*Sin[c + d*x])^(n - 2), x], x] + (Dist[a^2, Int[Sin[c + d*x]^m*(a*Cos[c + d*x] + b*Sin[c + d*x])^(n - 2), x], x] + Dist[2*b, Int[Sin[c + d*x]^(m + 1)*(a*Cos[c + d*x] + b*Sin[c + d*x])^(n - 1), x], x]) /; FreeQ[{a, b, c, d}, x] && NeQ[a^2 + b^2, 0] && GtQ[n, 1] && LtQ[m, -1]
  3096. Int[Times[Power[cos[Plus[Optional[Pattern[c, Blank[]]], Times[Optional[Pattern[d, Blank[]]], Pattern[x, Blank[]]]]], Pattern[m, Blank[]]], Power[Plus[Times[cos[Plus[Optional[Pattern[c, Blank[]]], Times[Optional[Pattern[d, Blank[]]], Pattern[x, Blank[]]]]], Optional[Pattern[a, Blank[]]]], Times[Optional[Pattern[b, Blank[]]], sin[Plus[Optional[Pattern[c, Blank[]]], Times[Optional[Pattern[d, Blank[]]], Pattern[x, Blank[]]]]]]], Pattern[n, Blank[]]]], Pattern[x, Blank[Symbol]]] := -Dist[a^2 + b^2, Int[Cos[c + d*x]^(m + 2)*(a*Cos[c + d*x] + b*Sin[c + d*x])^(n - 2), x], x] + (Dist[2*a, Int[Cos[c + d*x]^(m + 1)*(a*Cos[c + d*x] + b*Sin[c + d*x])^(n - 1), x], x] + Dist[b^2, Int[Cos[c + d*x]^m*(a*Cos[c + d*x] + b*Sin[c + d*x])^(n - 2), x], x]) /; FreeQ[{a, b, c, d}, x] && NeQ[a^2 + b^2, 0] && GtQ[n, 1] && LtQ[m, -1]
  3097. Int[Times[sin[Plus[Optional[Pattern[c, Blank[]]], Times[Optional[Pattern[d, Blank[]]], Pattern[x, Blank[]]]]], Power[Plus[Times[cos[Plus[Optional[Pattern[c, Blank[]]], Times[Optional[Pattern[d, Blank[]]], Pattern[x, Blank[]]]]], Optional[Pattern[a, Blank[]]]], Times[Optional[Pattern[b, Blank[]]], sin[Plus[Optional[Pattern[c, Blank[]]], Times[Optional[Pattern[d, Blank[]]], Pattern[x, Blank[]]]]]]], -1]], Pattern[x, Blank[Symbol]]] := Simp[(b*x)/(a^2 + b^2), x] - Dist[a/(a^2 + b^2), Int[(b*Cos[c + d*x] - a*Sin[c + d*x])/(a*Cos[c + d*x] + b*Sin[c + d*x]), x], x] /; FreeQ[{a, b, c, d}, x] && NeQ[a^2 + b^2, 0]
  3098. Int[Times[cos[Plus[Optional[Pattern[c, Blank[]]], Times[Optional[Pattern[d, Blank[]]], Pattern[x, Blank[]]]]], Power[Plus[Times[cos[Plus[Optional[Pattern[c, Blank[]]], Times[Optional[Pattern[d, Blank[]]], Pattern[x, Blank[]]]]], Optional[Pattern[a, Blank[]]]], Times[Optional[Pattern[b, Blank[]]], sin[Plus[Optional[Pattern[c, Blank[]]], Times[Optional[Pattern[d, Blank[]]], Pattern[x, Blank[]]]]]]], -1]], Pattern[x, Blank[Symbol]]] := Simp[(a*x)/(a^2 + b^2), x] + Dist[b/(a^2 + b^2), Int[(b*Cos[c + d*x] - a*Sin[c + d*x])/(a*Cos[c + d*x] + b*Sin[c + d*x]), x], x] /; FreeQ[{a, b, c, d}, x] && NeQ[a^2 + b^2, 0]
  3099. Int[Times[Power[sin[Plus[Optional[Pattern[c, Blank[]]], Times[Optional[Pattern[d, Blank[]]], Pattern[x, Blank[]]]]], Pattern[m, Blank[]]], Power[Plus[Times[cos[Plus[Optional[Pattern[c, Blank[]]], Times[Optional[Pattern[d, Blank[]]], Pattern[x, Blank[]]]]], Optional[Pattern[a, Blank[]]]], Times[Optional[Pattern[b, Blank[]]], sin[Plus[Optional[Pattern[c, Blank[]]], Times[Optional[Pattern[d, Blank[]]], Pattern[x, Blank[]]]]]]], -1]], Pattern[x, Blank[Symbol]]] := -Simp[(a*Sin[c + d*x]^(m - 1))/(d*(a^2 + b^2)*(m - 1)), x] + (Dist[a^2/(a^2 + b^2), Int[Sin[c + d*x]^(m - 2)/(a*Cos[c + d*x] + b*Sin[c + d*x]), x], x] + Dist[b/(a^2 + b^2), Int[Sin[c + d*x]^(m - 1), x], x]) /; FreeQ[{a, b, c, d}, x] && NeQ[a^2 + b^2, 0] && GtQ[m, 1]
  3100. Int[Times[Power[cos[Plus[Optional[Pattern[c, Blank[]]], Times[Optional[Pattern[d, Blank[]]], Pattern[x, Blank[]]]]], Pattern[m, Blank[]]], Power[Plus[Times[cos[Plus[Optional[Pattern[c, Blank[]]], Times[Optional[Pattern[d, Blank[]]], Pattern[x, Blank[]]]]], Optional[Pattern[a, Blank[]]]], Times[Optional[Pattern[b, Blank[]]], sin[Plus[Optional[Pattern[c, Blank[]]], Times[Optional[Pattern[d, Blank[]]], Pattern[x, Blank[]]]]]]], -1]], Pattern[x, Blank[Symbol]]] := Simp[(b*Cos[c + d*x]^(m - 1))/(d*(a^2 + b^2)*(m - 1)), x] + (Dist[a/(a^2 + b^2), Int[Cos[c + d*x]^(m - 1), x], x] + Dist[b^2/(a^2 + b^2), Int[Cos[c + d*x]^(m - 2)/(a*Cos[c + d*x] + b*Sin[c + d*x]), x], x]) /; FreeQ[{a, b, c, d}, x] && NeQ[a^2 + b^2, 0] && GtQ[m, 1]
  3101. Int[Times[Power[sin[Plus[Optional[Pattern[c, Blank[]]], Times[Optional[Pattern[d, Blank[]]], Pattern[x, Blank[]]]]], -1], Power[Plus[Times[cos[Plus[Optional[Pattern[c, Blank[]]], Times[Optional[Pattern[d, Blank[]]], Pattern[x, Blank[]]]]], Optional[Pattern[a, Blank[]]]], Times[Optional[Pattern[b, Blank[]]], sin[Plus[Optional[Pattern[c, Blank[]]], Times[Optional[Pattern[d, Blank[]]], Pattern[x, Blank[]]]]]]], -1]], Pattern[x, Blank[Symbol]]] := Dist[1/a, Int[Cot[c + d*x], x], x] - Dist[1/a, Int[(b*Cos[c + d*x] - a*Sin[c + d*x])/(a*Cos[c + d*x] + b*Sin[c + d*x]), x], x] /; FreeQ[{a, b, c, d}, x] && NeQ[a^2 + b^2, 0]
  3102. Int[Times[Power[cos[Plus[Optional[Pattern[c, Blank[]]], Times[Optional[Pattern[d, Blank[]]], Pattern[x, Blank[]]]]], -1], Power[Plus[Times[cos[Plus[Optional[Pattern[c, Blank[]]], Times[Optional[Pattern[d, Blank[]]], Pattern[x, Blank[]]]]], Optional[Pattern[a, Blank[]]]], Times[Optional[Pattern[b, Blank[]]], sin[Plus[Optional[Pattern[c, Blank[]]], Times[Optional[Pattern[d, Blank[]]], Pattern[x, Blank[]]]]]]], -1]], Pattern[x, Blank[Symbol]]] := Dist[1/b, Int[Tan[c + d*x], x], x] + Dist[1/b, Int[(b*Cos[c + d*x] - a*Sin[c + d*x])/(a*Cos[c + d*x] + b*Sin[c + d*x]), x], x] /; FreeQ[{a, b, c, d}, x] && NeQ[a^2 + b^2, 0]
  3103. Int[Times[Power[sin[Plus[Optional[Pattern[c, Blank[]]], Times[Optional[Pattern[d, Blank[]]], Pattern[x, Blank[]]]]], Pattern[m, Blank[]]], Power[Plus[Times[cos[Plus[Optional[Pattern[c, Blank[]]], Times[Optional[Pattern[d, Blank[]]], Pattern[x, Blank[]]]]], Optional[Pattern[a, Blank[]]]], Times[Optional[Pattern[b, Blank[]]], sin[Plus[Optional[Pattern[c, Blank[]]], Times[Optional[Pattern[d, Blank[]]], Pattern[x, Blank[]]]]]]], -1]], Pattern[x, Blank[Symbol]]] := Simp[Sin[c + d*x]^(m + 1)/(a*d*(m + 1)), x] + (-Dist[b/a^2, Int[Sin[c + d*x]^(m + 1), x], x] + Dist[(a^2 + b^2)/a^2, Int[Sin[c + d*x]^(m + 2)/(a*Cos[c + d*x] + b*Sin[c + d*x]), x], x]) /; FreeQ[{a, b, c, d}, x] && NeQ[a^2 + b^2, 0] && LtQ[m, -1]
  3104. Int[Times[Power[cos[Plus[Optional[Pattern[c, Blank[]]], Times[Optional[Pattern[d, Blank[]]], Pattern[x, Blank[]]]]], Pattern[m, Blank[]]], Power[Plus[Times[cos[Plus[Optional[Pattern[c, Blank[]]], Times[Optional[Pattern[d, Blank[]]], Pattern[x, Blank[]]]]], Optional[Pattern[a, Blank[]]]], Times[Optional[Pattern[b, Blank[]]], sin[Plus[Optional[Pattern[c, Blank[]]], Times[Optional[Pattern[d, Blank[]]], Pattern[x, Blank[]]]]]]], -1]], Pattern[x, Blank[Symbol]]] := -Simp[Cos[c + d*x]^(m + 1)/(b*d*(m + 1)), x] + (-Dist[a/b^2, Int[Cos[c + d*x]^(m + 1), x], x] + Dist[(a^2 + b^2)/b^2, Int[Cos[c + d*x]^(m + 2)/(a*Cos[c + d*x] + b*Sin[c + d*x]), x], x]) /; FreeQ[{a, b, c, d}, x] && NeQ[a^2 + b^2, 0] && LtQ[m, -1]
  3105. Int[Times[Power[sin[Plus[Optional[Pattern[c, Blank[]]], Times[Optional[Pattern[d, Blank[]]], Pattern[x, Blank[]]]]], Pattern[m, Blank[]]], Power[Plus[Times[cos[Plus[Optional[Pattern[c, Blank[]]], Times[Optional[Pattern[d, Blank[]]], Pattern[x, Blank[]]]]], Optional[Pattern[a, Blank[]]]], Times[Optional[Pattern[b, Blank[]]], sin[Plus[Optional[Pattern[c, Blank[]]], Times[Optional[Pattern[d, Blank[]]], Pattern[x, Blank[]]]]]]], Pattern[n, Blank[]]]], Pattern[x, Blank[Symbol]]] := Dist[(a^2 + b^2)/a^2, Int[Sin[c + d*x]^(m + 2)*(a*Cos[c + d*x] + b*Sin[c + d*x])^n, x], x] + (Dist[1/a^2, Int[Sin[c + d*x]^m*(a*Cos[c + d*x] + b*Sin[c + d*x])^(n + 2), x], x] - Dist[(2*b)/a^2, Int[Sin[c + d*x]^(m + 1)*(a*Cos[c + d*x] + b*Sin[c + d*x])^(n + 1), x], x]) /; FreeQ[{a, b, c, d}, x] && NeQ[a^2 + b^2, 0] && LtQ[n, -1] && LtQ[m, -1]
  3106. Int[Times[Power[cos[Plus[Optional[Pattern[c, Blank[]]], Times[Optional[Pattern[d, Blank[]]], Pattern[x, Blank[]]]]], Pattern[m, Blank[]]], Power[Plus[Times[cos[Plus[Optional[Pattern[c, Blank[]]], Times[Optional[Pattern[d, Blank[]]], Pattern[x, Blank[]]]]], Optional[Pattern[a, Blank[]]]], Times[Optional[Pattern[b, Blank[]]], sin[Plus[Optional[Pattern[c, Blank[]]], Times[Optional[Pattern[d, Blank[]]], Pattern[x, Blank[]]]]]]], Pattern[n, Blank[]]]], Pattern[x, Blank[Symbol]]] := Dist[(a^2 + b^2)/b^2, Int[Cos[c + d*x]^(m + 2)*(a*Cos[c + d*x] + b*Sin[c + d*x])^n, x], x] + (Dist[1/b^2, Int[Cos[c + d*x]^m*(a*Cos[c + d*x] + b*Sin[c + d*x])^(n + 2), x], x] - Dist[(2*a)/b^2, Int[Cos[c + d*x]^(m + 1)*(a*Cos[c + d*x] + b*Sin[c + d*x])^(n + 1), x], x]) /; FreeQ[{a, b, c, d}, x] && NeQ[a^2 + b^2, 0] && LtQ[n, -1] && LtQ[m, -1]
  3107. Int[Times[Power[cos[Plus[Optional[Pattern[c, Blank[]]], Times[Optional[Pattern[d, Blank[]]], Pattern[x, Blank[]]]]], Optional[Pattern[m, Blank[]]]], Power[sin[Plus[Optional[Pattern[c, Blank[]]], Times[Optional[Pattern[d, Blank[]]], Pattern[x, Blank[]]]]], Optional[Pattern[n, Blank[]]]], Power[Plus[Times[cos[Plus[Optional[Pattern[c, Blank[]]], Times[Optional[Pattern[d, Blank[]]], Pattern[x, Blank[]]]]], Optional[Pattern[a, Blank[]]]], Times[Optional[Pattern[b, Blank[]]], sin[Plus[Optional[Pattern[c, Blank[]]], Times[Optional[Pattern[d, Blank[]]], Pattern[x, Blank[]]]]]]], Optional[Pattern[p, Blank[]]]]], Pattern[x, Blank[Symbol]]] := Int[ExpandTrig[cos[c + d*x]^m*sin[c + d*x]^n*(a*cos[c + d*x] + b*sin[c + d*x])^p, x], x] /; FreeQ[{a, b, c, d, m, n}, x] && IGtQ[p, 0]
  3108. Int[Times[Power[cos[Plus[Optional[Pattern[c, Blank[]]], Times[Optional[Pattern[d, Blank[]]], Pattern[x, Blank[]]]]], Optional[Pattern[m, Blank[]]]], Power[sin[Plus[Optional[Pattern[c, Blank[]]], Times[Optional[Pattern[d, Blank[]]], Pattern[x, Blank[]]]]], Optional[Pattern[n, Blank[]]]], Power[Plus[Times[cos[Plus[Optional[Pattern[c, Blank[]]], Times[Optional[Pattern[d, Blank[]]], Pattern[x, Blank[]]]]], Optional[Pattern[a, Blank[]]]], Times[Optional[Pattern[b, Blank[]]], sin[Plus[Optional[Pattern[c, Blank[]]], Times[Optional[Pattern[d, Blank[]]], Pattern[x, Blank[]]]]]]], Pattern[p, Blank[]]]], Pattern[x, Blank[Symbol]]] := Dist[a^p*b^p, Int[(Cos[c + d*x]^m*Sin[c + d*x]^n)/(b*Cos[c + d*x] + a*Sin[c + d*x])^p, x], x] /; FreeQ[{a, b, c, d, m, n}, x] && EqQ[a^2 + b^2, 0] && ILtQ[p, 0]
  3109. Int[Times[Power[cos[Plus[Optional[Pattern[c, Blank[]]], Times[Optional[Pattern[d, Blank[]]], Pattern[x, Blank[]]]]], Optional[Pattern[m, Blank[]]]], Power[sin[Plus[Optional[Pattern[c, Blank[]]], Times[Optional[Pattern[d, Blank[]]], Pattern[x, Blank[]]]]], Optional[Pattern[n, Blank[]]]], Power[Plus[Times[cos[Plus[Optional[Pattern[c, Blank[]]], Times[Optional[Pattern[d, Blank[]]], Pattern[x, Blank[]]]]], Optional[Pattern[a, Blank[]]]], Times[Optional[Pattern[b, Blank[]]], sin[Plus[Optional[Pattern[c, Blank[]]], Times[Optional[Pattern[d, Blank[]]], Pattern[x, Blank[]]]]]]], -1]], Pattern[x, Blank[Symbol]]] := Dist[b/(a^2 + b^2), Int[Cos[c + d*x]^m*Sin[c + d*x]^(n - 1), x], x] + (Dist[a/(a^2 + b^2), Int[Cos[c + d*x]^(m - 1)*Sin[c + d*x]^n, x], x] - Dist[(a*b)/(a^2 + b^2), Int[(Cos[c + d*x]^(m - 1)*Sin[c + d*x]^(n - 1))/(a*Cos[c + d*x] + b*Sin[c + d*x]), x], x]) /; FreeQ[{a, b, c, d}, x] && NeQ[a^2 + b^2, 0] && IGtQ[m, 0] && IGtQ[n, 0]
  3110. Int[Times[Power[cos[Plus[Optional[Pattern[c, Blank[]]], Times[Optional[Pattern[d, Blank[]]], Pattern[x, Blank[]]]]], Optional[Pattern[m, Blank[]]]], Power[sin[Plus[Optional[Pattern[c, Blank[]]], Times[Optional[Pattern[d, Blank[]]], Pattern[x, Blank[]]]]], Optional[Pattern[n, Blank[]]]], Power[Plus[Times[cos[Plus[Optional[Pattern[c, Blank[]]], Times[Optional[Pattern[d, Blank[]]], Pattern[x, Blank[]]]]], Optional[Pattern[a, Blank[]]]], Times[Optional[Pattern[b, Blank[]]], sin[Plus[Optional[Pattern[c, Blank[]]], Times[Optional[Pattern[d, Blank[]]], Pattern[x, Blank[]]]]]]], -1]], Pattern[x, Blank[Symbol]]] := Int[ExpandTrig[(cos[c + d*x]^m*sin[c + d*x]^n)/(a*cos[c + d*x] + b*sin[c + d*x]), x], x] /; FreeQ[{a, b, c, d, m, n}, x] && IntegersQ[m, n]
  3111. Int[Times[Power[cos[Plus[Optional[Pattern[c, Blank[]]], Times[Optional[Pattern[d, Blank[]]], Pattern[x, Blank[]]]]], Optional[Pattern[m, Blank[]]]], Power[sin[Plus[Optional[Pattern[c, Blank[]]], Times[Optional[Pattern[d, Blank[]]], Pattern[x, Blank[]]]]], Optional[Pattern[n, Blank[]]]], Power[Plus[Times[cos[Plus[Optional[Pattern[c, Blank[]]], Times[Optional[Pattern[d, Blank[]]], Pattern[x, Blank[]]]]], Optional[Pattern[a, Blank[]]]], Times[Optional[Pattern[b, Blank[]]], sin[Plus[Optional[Pattern[c, Blank[]]], Times[Optional[Pattern[d, Blank[]]], Pattern[x, Blank[]]]]]]], Pattern[p, Blank[]]]], Pattern[x, Blank[Symbol]]] := Dist[b/(a^2 + b^2), Int[Cos[c + d*x]^m*Sin[c + d*x]^(n - 1)*(a*Cos[c + d*x] + b*Sin[c + d*x])^(p + 1), x], x] + (Dist[a/(a^2 + b^2), Int[Cos[c + d*x]^(m - 1)*Sin[c + d*x]^n*(a*Cos[c + d*x] + b*Sin[c + d*x])^(p + 1), x], x] - Dist[(a*b)/(a^2 + b^2), Int[Cos[c + d*x]^(m - 1)*Sin[c + d*x]^(n - 1)*(a*Cos[c + d*x] + b*Sin[c + d*x])^p, x], x]) /; FreeQ[{a, b, c, d}, x] && NeQ[a^2 + b^2, 0] && IGtQ[m, 0] && IGtQ[n, 0] && ILtQ[p, 0]
  3112. Int[Power[Plus[Times[cos[Plus[Optional[Pattern[d, Blank[]]], Times[Optional[Pattern[e, Blank[]]], Pattern[x, Blank[]]]]], Optional[Pattern[b, Blank[]]]], Pattern[a, Blank[]], Times[Optional[Pattern[c, Blank[]]], sin[Plus[Optional[Pattern[d, Blank[]]], Times[Optional[Pattern[e, Blank[]]], Pattern[x, Blank[]]]]]]], Rational[1, 2]], Pattern[x, Blank[Symbol]]] := Simp[(-2*(c*Cos[d + e*x] - b*Sin[d + e*x]))/(e*Sqrt[a + b*Cos[d + e*x] + c*Sin[d + e*x]]), x] /; FreeQ[{a, b, c, d, e}, x] && EqQ[a^2 - b^2 - c^2, 0]
  3113. Int[Power[Plus[Times[cos[Plus[Optional[Pattern[d, Blank[]]], Times[Optional[Pattern[e, Blank[]]], Pattern[x, Blank[]]]]], Optional[Pattern[b, Blank[]]]], Pattern[a, Blank[]], Times[Optional[Pattern[c, Blank[]]], sin[Plus[Optional[Pattern[d, Blank[]]], Times[Optional[Pattern[e, Blank[]]], Pattern[x, Blank[]]]]]]], Pattern[n, Blank[]]], Pattern[x, Blank[Symbol]]] := -Simp[((c*Cos[d + e*x] - b*Sin[d + e*x])*(a + b*Cos[d + e*x] + c*Sin[d + e*x])^(n - 1))/(e*n), x] + Dist[(a*(2*n - 1))/n, Int[(a + b*Cos[d + e*x] + c*Sin[d + e*x])^(n - 1), x], x] /; FreeQ[{a, b, c, d, e}, x] && EqQ[a^2 - b^2 - c^2, 0] && GtQ[n, 0]
  3114. Int[Power[Plus[Times[cos[Plus[Optional[Pattern[d, Blank[]]], Times[Optional[Pattern[e, Blank[]]], Pattern[x, Blank[]]]]], Optional[Pattern[b, Blank[]]]], Pattern[a, Blank[]], Times[Optional[Pattern[c, Blank[]]], sin[Plus[Optional[Pattern[d, Blank[]]], Times[Optional[Pattern[e, Blank[]]], Pattern[x, Blank[]]]]]]], -1], Pattern[x, Blank[Symbol]]] := -Simp[(c - a*Sin[d + e*x])/(c*e*(c*Cos[d + e*x] - b*Sin[d + e*x])), x] /; FreeQ[{a, b, c, d, e}, x] && EqQ[a^2 - b^2 - c^2, 0]
  3115. Int[Power[Plus[Times[cos[Plus[Optional[Pattern[d, Blank[]]], Times[Optional[Pattern[e, Blank[]]], Pattern[x, Blank[]]]]], Optional[Pattern[b, Blank[]]]], Pattern[a, Blank[]], Times[Optional[Pattern[c, Blank[]]], sin[Plus[Optional[Pattern[d, Blank[]]], Times[Optional[Pattern[e, Blank[]]], Pattern[x, Blank[]]]]]]], Rational[-1, 2]], Pattern[x, Blank[Symbol]]] := Int[1/Sqrt[a + Sqrt[b^2 + c^2]*Cos[d + e*x - ArcTan[b, c]]], x] /; FreeQ[{a, b, c, d, e}, x] && EqQ[a^2 - b^2 - c^2, 0]
  3116. Int[Power[Plus[Times[cos[Plus[Optional[Pattern[d, Blank[]]], Times[Optional[Pattern[e, Blank[]]], Pattern[x, Blank[]]]]], Optional[Pattern[b, Blank[]]]], Pattern[a, Blank[]], Times[Optional[Pattern[c, Blank[]]], sin[Plus[Optional[Pattern[d, Blank[]]], Times[Optional[Pattern[e, Blank[]]], Pattern[x, Blank[]]]]]]], Pattern[n, Blank[]]], Pattern[x, Blank[Symbol]]] := Simp[((c*Cos[d + e*x] - b*Sin[d + e*x])*(a + b*Cos[d + e*x] + c*Sin[d + e*x])^n)/(a*e*(2*n + 1)), x] + Dist[(n + 1)/(a*(2*n + 1)), Int[(a + b*Cos[d + e*x] + c*Sin[d + e*x])^(n + 1), x], x] /; FreeQ[{a, b, c, d, e}, x] && EqQ[a^2 - b^2 - c^2, 0] && LtQ[n, -1]
  3117. Int[Power[Plus[Times[cos[Plus[Optional[Pattern[d, Blank[]]], Times[Optional[Pattern[e, Blank[]]], Pattern[x, Blank[]]]]], Optional[Pattern[b, Blank[]]]], Pattern[a, Blank[]], Times[Optional[Pattern[c, Blank[]]], sin[Plus[Optional[Pattern[d, Blank[]]], Times[Optional[Pattern[e, Blank[]]], Pattern[x, Blank[]]]]]]], Rational[1, 2]], Pattern[x, Blank[Symbol]]] := Dist[b/(c*e), Subst[Int[Sqrt[a + x]/x, x], x, b*Cos[d + e*x] + c*Sin[d + e*x]], x] /; FreeQ[{a, b, c, d, e}, x] && EqQ[b^2 + c^2, 0]
  3118. Int[Power[Plus[Times[cos[Plus[Optional[Pattern[d, Blank[]]], Times[Optional[Pattern[e, Blank[]]], Pattern[x, Blank[]]]]], Optional[Pattern[b, Blank[]]]], Pattern[a, Blank[]], Times[Optional[Pattern[c, Blank[]]], sin[Plus[Optional[Pattern[d, Blank[]]], Times[Optional[Pattern[e, Blank[]]], Pattern[x, Blank[]]]]]]], Rational[1, 2]], Pattern[x, Blank[Symbol]]] := Int[Sqrt[a + Sqrt[b^2 + c^2]*Cos[d + e*x - ArcTan[b, c]]], x] /; FreeQ[{a, b, c, d, e}, x] && NeQ[b^2 + c^2, 0] && GtQ[a + Sqrt[b^2 + c^2], 0]
  3119. Int[Power[Plus[Times[cos[Plus[Optional[Pattern[d, Blank[]]], Times[Optional[Pattern[e, Blank[]]], Pattern[x, Blank[]]]]], Optional[Pattern[b, Blank[]]]], Pattern[a, Blank[]], Times[Optional[Pattern[c, Blank[]]], sin[Plus[Optional[Pattern[d, Blank[]]], Times[Optional[Pattern[e, Blank[]]], Pattern[x, Blank[]]]]]]], Rational[1, 2]], Pattern[x, Blank[Symbol]]] := Dist[Sqrt[a + b*Cos[d + e*x] + c*Sin[d + e*x]]/Sqrt[(a + b*Cos[d + e*x] + c*Sin[d + e*x])/(a + Sqrt[b^2 + c^2])], Int[Sqrt[a/(a + Sqrt[b^2 + c^2]) + (Sqrt[b^2 + c^2]*Cos[d + e*x - ArcTan[b, c]])/(a + Sqrt[b^2 + c^2])], x], x] /; FreeQ[{a, b, c, d, e}, x] && NeQ[a^2 - b^2 - c^2, 0] && NeQ[b^2 + c^2, 0] && !GtQ[a + Sqrt[b^2 + c^2], 0]
  3120. Int[Power[Plus[Times[cos[Plus[Optional[Pattern[d, Blank[]]], Times[Optional[Pattern[e, Blank[]]], Pattern[x, Blank[]]]]], Optional[Pattern[b, Blank[]]]], Pattern[a, Blank[]], Times[Optional[Pattern[c, Blank[]]], sin[Plus[Optional[Pattern[d, Blank[]]], Times[Optional[Pattern[e, Blank[]]], Pattern[x, Blank[]]]]]]], Pattern[n, Blank[]]], Pattern[x, Blank[Symbol]]] := -Simp[((c*Cos[d + e*x] - b*Sin[d + e*x])*(a + b*Cos[d + e*x] + c*Sin[d + e*x])^(n - 1))/(e*n), x] + Dist[1/n, Int[Simp[n*a^2 + (n - 1)*(b^2 + c^2) + a*b*(2*n - 1)*Cos[d + e*x] + a*c*(2*n - 1)*Sin[d + e*x], x]*(a + b*Cos[d + e*x] + c*Sin[d + e*x])^(n - 2), x], x] /; FreeQ[{a, b, c, d, e}, x] && NeQ[a^2 - b^2 - c^2, 0] && GtQ[n, 1]
  3121. Int[Power[Plus[Times[cos[Plus[Optional[Pattern[d, Blank[]]], Times[Optional[Pattern[e, Blank[]]], Pattern[x, Blank[]]]]], Optional[Pattern[b, Blank[]]]], Pattern[a, Blank[]], Times[Optional[Pattern[c, Blank[]]], sin[Plus[Optional[Pattern[d, Blank[]]], Times[Optional[Pattern[e, Blank[]]], Pattern[x, Blank[]]]]]]], -1], Pattern[x, Blank[Symbol]]] := Module[{f = FreeFactors[Cot[(d + e*x)/2], x]}, -Dist[f/e, Subst[Int[1/(a + c*f*x), x], x, Cot[(d + e*x)/2]/f], x]] /; FreeQ[{a, b, c, d, e}, x] && EqQ[a + b, 0]
  3122. Int[Power[Plus[Times[cos[Plus[Optional[Pattern[d, Blank[]]], Times[Optional[Pattern[e, Blank[]]], Pattern[x, Blank[]]]]], Optional[Pattern[b, Blank[]]]], Pattern[a, Blank[]], Times[Optional[Pattern[c, Blank[]]], sin[Plus[Optional[Pattern[d, Blank[]]], Times[Optional[Pattern[e, Blank[]]], Pattern[x, Blank[]]]]]]], -1], Pattern[x, Blank[Symbol]]] := Module[{f = FreeFactors[Tan[(d + e*x)/2 + Pi/4], x]}, Dist[f/e, Subst[Int[1/(a + b*f*x), x], x, Tan[(d + e*x)/2 + Pi/4]/f], x]] /; FreeQ[{a, b, c, d, e}, x] && EqQ[a + c, 0]
  3123. Int[Power[Plus[Times[cos[Plus[Optional[Pattern[d, Blank[]]], Times[Optional[Pattern[e, Blank[]]], Pattern[x, Blank[]]]]], Optional[Pattern[b, Blank[]]]], Pattern[a, Blank[]], Times[Optional[Pattern[c, Blank[]]], sin[Plus[Optional[Pattern[d, Blank[]]], Times[Optional[Pattern[e, Blank[]]], Pattern[x, Blank[]]]]]]], -1], Pattern[x, Blank[Symbol]]] := Module[{f = FreeFactors[Cot[(d + e*x)/2 + Pi/4], x]}, -Dist[f/e, Subst[Int[1/(a + b*f*x), x], x, Cot[(d + e*x)/2 + Pi/4]/f], x]] /; FreeQ[{a, b, c, d, e}, x] && EqQ[a - c, 0] && NeQ[a - b, 0]
  3124. Int[Power[Plus[Times[cos[Plus[Optional[Pattern[d, Blank[]]], Times[Optional[Pattern[e, Blank[]]], Pattern[x, Blank[]]]]], Optional[Pattern[b, Blank[]]]], Pattern[a, Blank[]], Times[Optional[Pattern[c, Blank[]]], sin[Plus[Optional[Pattern[d, Blank[]]], Times[Optional[Pattern[e, Blank[]]], Pattern[x, Blank[]]]]]]], -1], Pattern[x, Blank[Symbol]]] := Module[{f = FreeFactors[Tan[(d + e*x)/2], x]}, Dist[(2*f)/e, Subst[Int[1/(a + b + 2*c*f*x + (a - b)*f^2*x^2), x], x, Tan[(d + e*x)/2]/f], x]] /; FreeQ[{a, b, c, d, e}, x] && NeQ[a^2 - b^2 - c^2, 0]
  3125. Int[Power[Plus[Times[cos[Plus[Optional[Pattern[d, Blank[]]], Times[Optional[Pattern[e, Blank[]]], Pattern[x, Blank[]]]]], Optional[Pattern[b, Blank[]]]], Pattern[a, Blank[]], Times[Optional[Pattern[c, Blank[]]], sin[Plus[Optional[Pattern[d, Blank[]]], Times[Optional[Pattern[e, Blank[]]], Pattern[x, Blank[]]]]]]], Rational[-1, 2]], Pattern[x, Blank[Symbol]]] := Dist[b/(c*e), Subst[Int[1/(x*Sqrt[a + x]), x], x, b*Cos[d + e*x] + c*Sin[d + e*x]], x] /; FreeQ[{a, b, c, d, e}, x] && EqQ[b^2 + c^2, 0]
  3126. Int[Power[Plus[Times[cos[Plus[Optional[Pattern[d, Blank[]]], Times[Optional[Pattern[e, Blank[]]], Pattern[x, Blank[]]]]], Optional[Pattern[b, Blank[]]]], Pattern[a, Blank[]], Times[Optional[Pattern[c, Blank[]]], sin[Plus[Optional[Pattern[d, Blank[]]], Times[Optional[Pattern[e, Blank[]]], Pattern[x, Blank[]]]]]]], Rational[-1, 2]], Pattern[x, Blank[Symbol]]] := Int[1/Sqrt[a + Sqrt[b^2 + c^2]*Cos[d + e*x - ArcTan[b, c]]], x] /; FreeQ[{a, b, c, d, e}, x] && NeQ[b^2 + c^2, 0] && GtQ[a + Sqrt[b^2 + c^2], 0]
  3127. Int[Power[Plus[Times[cos[Plus[Optional[Pattern[d, Blank[]]], Times[Optional[Pattern[e, Blank[]]], Pattern[x, Blank[]]]]], Optional[Pattern[b, Blank[]]]], Pattern[a, Blank[]], Times[Optional[Pattern[c, Blank[]]], sin[Plus[Optional[Pattern[d, Blank[]]], Times[Optional[Pattern[e, Blank[]]], Pattern[x, Blank[]]]]]]], Rational[-1, 2]], Pattern[x, Blank[Symbol]]] := Dist[Sqrt[(a + b*Cos[d + e*x] + c*Sin[d + e*x])/(a + Sqrt[b^2 + c^2])]/Sqrt[a + b*Cos[d + e*x] + c*Sin[d + e*x]], Int[1/Sqrt[a/(a + Sqrt[b^2 + c^2]) + (Sqrt[b^2 + c^2]*Cos[d + e*x - ArcTan[b, c]])/(a + Sqrt[b^2 + c^2])], x], x] /; FreeQ[{a, b, c, d, e}, x] && NeQ[a^2 - b^2 - c^2, 0] && NeQ[b^2 + c^2, 0] && !GtQ[a + Sqrt[b^2 + c^2], 0]
  3128. Int[Power[Plus[Times[cos[Plus[Optional[Pattern[d, Blank[]]], Times[Optional[Pattern[e, Blank[]]], Pattern[x, Blank[]]]]], Optional[Pattern[b, Blank[]]]], Pattern[a, Blank[]], Times[Optional[Pattern[c, Blank[]]], sin[Plus[Optional[Pattern[d, Blank[]]], Times[Optional[Pattern[e, Blank[]]], Pattern[x, Blank[]]]]]]], Rational[-3, 2]], Pattern[x, Blank[Symbol]]] := Simp[(2*(c*Cos[d + e*x] - b*Sin[d + e*x]))/(e*(a^2 - b^2 - c^2)*Sqrt[a + b*Cos[d + e*x] + c*Sin[d + e*x]]), x] + Dist[1/(a^2 - b^2 - c^2), Int[Sqrt[a + b*Cos[d + e*x] + c*Sin[d + e*x]], x], x] /; FreeQ[{a, b, c, d, e}, x] && NeQ[a^2 - b^2 - c^2, 0]
  3129. Int[Power[Plus[Times[cos[Plus[Optional[Pattern[d, Blank[]]], Times[Optional[Pattern[e, Blank[]]], Pattern[x, Blank[]]]]], Optional[Pattern[b, Blank[]]]], Pattern[a, Blank[]], Times[Optional[Pattern[c, Blank[]]], sin[Plus[Optional[Pattern[d, Blank[]]], Times[Optional[Pattern[e, Blank[]]], Pattern[x, Blank[]]]]]]], Pattern[n, Blank[]]], Pattern[x, Blank[Symbol]]] := Simp[((-(c*Cos[d + e*x]) + b*Sin[d + e*x])*(a + b*Cos[d + e*x] + c*Sin[d + e*x])^(n + 1))/(e*(n + 1)*(a^2 - b^2 - c^2)), x] + Dist[1/((n + 1)*(a^2 - b^2 - c^2)), Int[(a*(n + 1) - b*(n + 2)*Cos[d + e*x] - c*(n + 2)*Sin[d + e*x])*(a + b*Cos[d + e*x] + c*Sin[d + e*x])^(n + 1), x], x] /; FreeQ[{a, b, c, d, e}, x] && NeQ[a^2 - b^2 - c^2, 0] && LtQ[n, -1] && NeQ[n, -3/2]
  3130. Int[Times[Power[Plus[Times[cos[Plus[Optional[Pattern[d, Blank[]]], Times[Optional[Pattern[e, Blank[]]], Pattern[x, Blank[]]]]], Optional[Pattern[b, Blank[]]]], Pattern[a, Blank[]], Times[Optional[Pattern[c, Blank[]]], sin[Plus[Optional[Pattern[d, Blank[]]], Times[Optional[Pattern[e, Blank[]]], Pattern[x, Blank[]]]]]]], -1], Plus[Optional[Pattern[A, Blank[]]], Times[cos[Plus[Optional[Pattern[d, Blank[]]], Times[Optional[Pattern[e, Blank[]]], Pattern[x, Blank[]]]]], Optional[Pattern[B, Blank[]]]], Times[Optional[Pattern[C, Blank[]]], sin[Plus[Optional[Pattern[d, Blank[]]], Times[Optional[Pattern[e, Blank[]]], Pattern[x, Blank[]]]]]]]], Pattern[x, Blank[Symbol]]] := Simp[((2*a*A - b*B - c*C)*x)/(2*a^2), x] + (-Simp[((b*B + c*C)*(b*Cos[d + e*x] - c*Sin[d + e*x]))/(2*a*b*c*e), x] + Simp[((a^2*(b*B - c*C) - 2*a*A*b^2 + b^2*(b*B + c*C))*Log[RemoveContent[a + b*Cos[d + e*x] + c*Sin[d + e*x], x]])/(2*a^2*b*c*e), x]) /; FreeQ[{a, b, c, d, e, A, B, C}, x] && EqQ[b^2 + c^2, 0]
  3131. Int[Times[Power[Plus[Times[cos[Plus[Optional[Pattern[d, Blank[]]], Times[Optional[Pattern[e, Blank[]]], Pattern[x, Blank[]]]]], Optional[Pattern[b, Blank[]]]], Pattern[a, Blank[]], Times[Optional[Pattern[c, Blank[]]], sin[Plus[Optional[Pattern[d, Blank[]]], Times[Optional[Pattern[e, Blank[]]], Pattern[x, Blank[]]]]]]], -1], Plus[Optional[Pattern[A, Blank[]]], Times[Optional[Pattern[C, Blank[]]], sin[Plus[Optional[Pattern[d, Blank[]]], Times[Optional[Pattern[e, Blank[]]], Pattern[x, Blank[]]]]]]]], Pattern[x, Blank[Symbol]]] := Simp[((2*a*A - c*C)*x)/(2*a^2), x] + (-Simp[(C*Cos[d + e*x])/(2*a*e), x] + Simp[(c*C*Sin[d + e*x])/(2*a*b*e), x] + Simp[((-(a^2*C) + 2*a*c*A + b^2*C)*Log[RemoveContent[a + b*Cos[d + e*x] + c*Sin[d + e*x], x]])/(2*a^2*b*e), x]) /; FreeQ[{a, b, c, d, e, A, C}, x] && EqQ[b^2 + c^2, 0]
  3132. Int[Times[Plus[Optional[Pattern[A, Blank[]]], Times[cos[Plus[Optional[Pattern[d, Blank[]]], Times[Optional[Pattern[e, Blank[]]], Pattern[x, Blank[]]]]], Optional[Pattern[B, Blank[]]]]], Power[Plus[Times[cos[Plus[Optional[Pattern[d, Blank[]]], Times[Optional[Pattern[e, Blank[]]], Pattern[x, Blank[]]]]], Optional[Pattern[b, Blank[]]]], Pattern[a, Blank[]], Times[Optional[Pattern[c, Blank[]]], sin[Plus[Optional[Pattern[d, Blank[]]], Times[Optional[Pattern[e, Blank[]]], Pattern[x, Blank[]]]]]]], -1]], Pattern[x, Blank[Symbol]]] := Simp[((2*a*A - b*B)*x)/(2*a^2), x] + (Simp[(B*Sin[d + e*x])/(2*a*e), x] - Simp[(b*B*Cos[d + e*x])/(2*a*c*e), x] + Simp[((a^2*B - 2*a*b*A + b^2*B)*Log[RemoveContent[a + b*Cos[d + e*x] + c*Sin[d + e*x], x]])/(2*a^2*c*e), x]) /; FreeQ[{a, b, c, d, e, A, B}, x] && EqQ[b^2 + c^2, 0]
  3133. Int[Times[Power[Plus[Optional[Pattern[a, Blank[]]], Times[cos[Plus[Optional[Pattern[d, Blank[]]], Times[Optional[Pattern[e, Blank[]]], Pattern[x, Blank[]]]]], Optional[Pattern[b, Blank[]]]], Times[Optional[Pattern[c, Blank[]]], sin[Plus[Optional[Pattern[d, Blank[]]], Times[Optional[Pattern[e, Blank[]]], Pattern[x, Blank[]]]]]]], -1], Plus[Optional[Pattern[A, Blank[]]], Times[cos[Plus[Optional[Pattern[d, Blank[]]], Times[Optional[Pattern[e, Blank[]]], Pattern[x, Blank[]]]]], Optional[Pattern[B, Blank[]]]], Times[Optional[Pattern[C, Blank[]]], sin[Plus[Optional[Pattern[d, Blank[]]], Times[Optional[Pattern[e, Blank[]]], Pattern[x, Blank[]]]]]]]], Pattern[x, Blank[Symbol]]] := Simp[((b*B + c*C)*x)/(b^2 + c^2), x] + Simp[((c*B - b*C)*Log[a + b*Cos[d + e*x] + c*Sin[d + e*x]])/(e*(b^2 + c^2)), x] /; FreeQ[{a, b, c, d, e, A, B, C}, x] && NeQ[b^2 + c^2, 0] && EqQ[A*(b^2 + c^2) - a*(b*B + c*C), 0]
  3134. Int[Times[Power[Plus[Optional[Pattern[a, Blank[]]], Times[cos[Plus[Optional[Pattern[d, Blank[]]], Times[Optional[Pattern[e, Blank[]]], Pattern[x, Blank[]]]]], Optional[Pattern[b, Blank[]]]], Times[Optional[Pattern[c, Blank[]]], sin[Plus[Optional[Pattern[d, Blank[]]], Times[Optional[Pattern[e, Blank[]]], Pattern[x, Blank[]]]]]]], -1], Plus[Optional[Pattern[A, Blank[]]], Times[Optional[Pattern[C, Blank[]]], sin[Plus[Optional[Pattern[d, Blank[]]], Times[Optional[Pattern[e, Blank[]]], Pattern[x, Blank[]]]]]]]], Pattern[x, Blank[Symbol]]] := Simp[(c*C*x)/(b^2 + c^2), x] - Simp[(b*C*Log[a + b*Cos[d + e*x] + c*Sin[d + e*x]])/(e*(b^2 + c^2)), x] /; FreeQ[{a, b, c, d, e, A, C}, x] && NeQ[b^2 + c^2, 0] && EqQ[A*(b^2 + c^2) - a*c*C, 0]
  3135. Int[Times[Plus[Optional[Pattern[A, Blank[]]], Times[cos[Plus[Optional[Pattern[d, Blank[]]], Times[Optional[Pattern[e, Blank[]]], Pattern[x, Blank[]]]]], Optional[Pattern[B, Blank[]]]]], Power[Plus[Optional[Pattern[a, Blank[]]], Times[cos[Plus[Optional[Pattern[d, Blank[]]], Times[Optional[Pattern[e, Blank[]]], Pattern[x, Blank[]]]]], Optional[Pattern[b, Blank[]]]], Times[Optional[Pattern[c, Blank[]]], sin[Plus[Optional[Pattern[d, Blank[]]], Times[Optional[Pattern[e, Blank[]]], Pattern[x, Blank[]]]]]]], -1]], Pattern[x, Blank[Symbol]]] := Simp[(b*B*x)/(b^2 + c^2), x] + Simp[(c*B*Log[a + b*Cos[d + e*x] + c*Sin[d + e*x]])/(e*(b^2 + c^2)), x] /; FreeQ[{a, b, c, d, e, A, B}, x] && NeQ[b^2 + c^2, 0] && EqQ[A*(b^2 + c^2) - a*b*B, 0]
  3136. Int[Times[Power[Plus[Optional[Pattern[a, Blank[]]], Times[cos[Plus[Optional[Pattern[d, Blank[]]], Times[Optional[Pattern[e, Blank[]]], Pattern[x, Blank[]]]]], Optional[Pattern[b, Blank[]]]], Times[Optional[Pattern[c, Blank[]]], sin[Plus[Optional[Pattern[d, Blank[]]], Times[Optional[Pattern[e, Blank[]]], Pattern[x, Blank[]]]]]]], -1], Plus[Optional[Pattern[A, Blank[]]], Times[cos[Plus[Optional[Pattern[d, Blank[]]], Times[Optional[Pattern[e, Blank[]]], Pattern[x, Blank[]]]]], Optional[Pattern[B, Blank[]]]], Times[Optional[Pattern[C, Blank[]]], sin[Plus[Optional[Pattern[d, Blank[]]], Times[Optional[Pattern[e, Blank[]]], Pattern[x, Blank[]]]]]]]], Pattern[x, Blank[Symbol]]] := Simp[((b*B + c*C)*x)/(b^2 + c^2), x] + (Dist[(A*(b^2 + c^2) - a*(b*B + c*C))/(b^2 + c^2), Int[1/(a + b*Cos[d + e*x] + c*Sin[d + e*x]), x], x] + Simp[((c*B - b*C)*Log[a + b*Cos[d + e*x] + c*Sin[d + e*x]])/(e*(b^2 + c^2)), x]) /; FreeQ[{a, b, c, d, e, A, B, C}, x] && NeQ[b^2 + c^2, 0] && NeQ[A*(b^2 + c^2) - a*(b*B + c*C), 0]
  3137. Int[Times[Power[Plus[Optional[Pattern[a, Blank[]]], Times[cos[Plus[Optional[Pattern[d, Blank[]]], Times[Optional[Pattern[e, Blank[]]], Pattern[x, Blank[]]]]], Optional[Pattern[b, Blank[]]]], Times[Optional[Pattern[c, Blank[]]], sin[Plus[Optional[Pattern[d, Blank[]]], Times[Optional[Pattern[e, Blank[]]], Pattern[x, Blank[]]]]]]], -1], Plus[Optional[Pattern[A, Blank[]]], Times[Optional[Pattern[C, Blank[]]], sin[Plus[Optional[Pattern[d, Blank[]]], Times[Optional[Pattern[e, Blank[]]], Pattern[x, Blank[]]]]]]]], Pattern[x, Blank[Symbol]]] := Simp[(c*C*(d + e*x))/(e*(b^2 + c^2)), x] + (Dist[(A*(b^2 + c^2) - a*c*C)/(b^2 + c^2), Int[1/(a + b*Cos[d + e*x] + c*Sin[d + e*x]), x], x] - Simp[(b*C*Log[a + b*Cos[d + e*x] + c*Sin[d + e*x]])/(e*(b^2 + c^2)), x]) /; FreeQ[{a, b, c, d, e, A, C}, x] && NeQ[b^2 + c^2, 0] && NeQ[A*(b^2 + c^2) - a*c*C, 0]
  3138. Int[Times[Plus[Optional[Pattern[A, Blank[]]], Times[cos[Plus[Optional[Pattern[d, Blank[]]], Times[Optional[Pattern[e, Blank[]]], Pattern[x, Blank[]]]]], Optional[Pattern[B, Blank[]]]]], Power[Plus[Optional[Pattern[a, Blank[]]], Times[cos[Plus[Optional[Pattern[d, Blank[]]], Times[Optional[Pattern[e, Blank[]]], Pattern[x, Blank[]]]]], Optional[Pattern[b, Blank[]]]], Times[Optional[Pattern[c, Blank[]]], sin[Plus[Optional[Pattern[d, Blank[]]], Times[Optional[Pattern[e, Blank[]]], Pattern[x, Blank[]]]]]]], -1]], Pattern[x, Blank[Symbol]]] := Simp[(b*B*(d + e*x))/(e*(b^2 + c^2)), x] + (Dist[(A*(b^2 + c^2) - a*b*B)/(b^2 + c^2), Int[1/(a + b*Cos[d + e*x] + c*Sin[d + e*x]), x], x] + Simp[(c*B*Log[a + b*Cos[d + e*x] + c*Sin[d + e*x]])/(e*(b^2 + c^2)), x]) /; FreeQ[{a, b, c, d, e, A, B}, x] && NeQ[b^2 + c^2, 0] && NeQ[A*(b^2 + c^2) - a*b*B, 0]
  3139. Int[Times[Power[Plus[Times[cos[Plus[Optional[Pattern[d, Blank[]]], Times[Optional[Pattern[e, Blank[]]], Pattern[x, Blank[]]]]], Optional[Pattern[b, Blank[]]]], Pattern[a, Blank[]], Times[Optional[Pattern[c, Blank[]]], sin[Plus[Optional[Pattern[d, Blank[]]], Times[Optional[Pattern[e, Blank[]]], Pattern[x, Blank[]]]]]]], Optional[Pattern[n, Blank[]]]], Plus[Optional[Pattern[A, Blank[]]], Times[cos[Plus[Optional[Pattern[d, Blank[]]], Times[Optional[Pattern[e, Blank[]]], Pattern[x, Blank[]]]]], Optional[Pattern[B, Blank[]]]], Times[Optional[Pattern[C, Blank[]]], sin[Plus[Optional[Pattern[d, Blank[]]], Times[Optional[Pattern[e, Blank[]]], Pattern[x, Blank[]]]]]]]], Pattern[x, Blank[Symbol]]] := Simp[((B*c - b*C - a*C*Cos[d + e*x] + a*B*Sin[d + e*x])*(a + b*Cos[d + e*x] + c*Sin[d + e*x])^n)/(a*e*(n + 1)), x] /; FreeQ[{a, b, c, d, e, A, B, C, n}, x] && NeQ[n, -1] && EqQ[a^2 - b^2 - c^2, 0] && EqQ[(b*B + c*C)*n + a*A*(n + 1), 0]
  3140. Int[Times[Power[Plus[Times[cos[Plus[Optional[Pattern[d, Blank[]]], Times[Optional[Pattern[e, Blank[]]], Pattern[x, Blank[]]]]], Optional[Pattern[b, Blank[]]]], Pattern[a, Blank[]], Times[Optional[Pattern[c, Blank[]]], sin[Plus[Optional[Pattern[d, Blank[]]], Times[Optional[Pattern[e, Blank[]]], Pattern[x, Blank[]]]]]]], Optional[Pattern[n, Blank[]]]], Plus[Optional[Pattern[A, Blank[]]], Times[Optional[Pattern[C, Blank[]]], sin[Plus[Optional[Pattern[d, Blank[]]], Times[Optional[Pattern[e, Blank[]]], Pattern[x, Blank[]]]]]]]], Pattern[x, Blank[Symbol]]] := -Simp[((b*C + a*C*Cos[d + e*x])*(a + b*Cos[d + e*x] + c*Sin[d + e*x])^n)/(a*e*(n + 1)), x] /; FreeQ[{a, b, c, d, e, A, C, n}, x] && NeQ[n, -1] && EqQ[a^2 - b^2 - c^2, 0] && EqQ[c*C*n + a*A*(n + 1), 0]
  3141. Int[Times[Plus[Optional[Pattern[A, Blank[]]], Times[cos[Plus[Optional[Pattern[d, Blank[]]], Times[Optional[Pattern[e, Blank[]]], Pattern[x, Blank[]]]]], Optional[Pattern[B, Blank[]]]]], Power[Plus[Times[cos[Plus[Optional[Pattern[d, Blank[]]], Times[Optional[Pattern[e, Blank[]]], Pattern[x, Blank[]]]]], Optional[Pattern[b, Blank[]]]], Pattern[a, Blank[]], Times[Optional[Pattern[c, Blank[]]], sin[Plus[Optional[Pattern[d, Blank[]]], Times[Optional[Pattern[e, Blank[]]], Pattern[x, Blank[]]]]]]], Optional[Pattern[n, Blank[]]]]], Pattern[x, Blank[Symbol]]] := Simp[((B*c + a*B*Sin[d + e*x])*(a + b*Cos[d + e*x] + c*Sin[d + e*x])^n)/(a*e*(n + 1)), x] /; FreeQ[{a, b, c, d, e, A, B, n}, x] && NeQ[n, -1] && EqQ[a^2 - b^2 - c^2, 0] && EqQ[b*B*n + a*A*(n + 1), 0]
  3142. Int[Times[Power[Plus[Times[cos[Plus[Optional[Pattern[d, Blank[]]], Times[Optional[Pattern[e, Blank[]]], Pattern[x, Blank[]]]]], Optional[Pattern[b, Blank[]]]], Pattern[a, Blank[]], Times[Optional[Pattern[c, Blank[]]], sin[Plus[Optional[Pattern[d, Blank[]]], Times[Optional[Pattern[e, Blank[]]], Pattern[x, Blank[]]]]]]], Optional[Pattern[n, Blank[]]]], Plus[Optional[Pattern[A, Blank[]]], Times[cos[Plus[Optional[Pattern[d, Blank[]]], Times[Optional[Pattern[e, Blank[]]], Pattern[x, Blank[]]]]], Optional[Pattern[B, Blank[]]]], Times[Optional[Pattern[C, Blank[]]], sin[Plus[Optional[Pattern[d, Blank[]]], Times[Optional[Pattern[e, Blank[]]], Pattern[x, Blank[]]]]]]]], Pattern[x, Blank[Symbol]]] := Simp[((B*c - b*C - a*C*Cos[d + e*x] + a*B*Sin[d + e*x])*(a + b*Cos[d + e*x] + c*Sin[d + e*x])^n)/(a*e*(n + 1)), x] + Dist[((b*B + c*C)*n + a*A*(n + 1))/(a*(n + 1)), Int[(a + b*Cos[d + e*x] + c*Sin[d + e*x])^n, x], x] /; FreeQ[{a, b, c, d, e, A, B, C, n}, x] && NeQ[n, -1] && EqQ[a^2 - b^2 - c^2, 0] && NeQ[(b*B + c*C)*n + a*A*(n + 1), 0]
  3143. Int[Times[Power[Plus[Times[cos[Plus[Optional[Pattern[d, Blank[]]], Times[Optional[Pattern[e, Blank[]]], Pattern[x, Blank[]]]]], Optional[Pattern[b, Blank[]]]], Pattern[a, Blank[]], Times[Optional[Pattern[c, Blank[]]], sin[Plus[Optional[Pattern[d, Blank[]]], Times[Optional[Pattern[e, Blank[]]], Pattern[x, Blank[]]]]]]], Optional[Pattern[n, Blank[]]]], Plus[Optional[Pattern[A, Blank[]]], Times[Optional[Pattern[C, Blank[]]], sin[Plus[Optional[Pattern[d, Blank[]]], Times[Optional[Pattern[e, Blank[]]], Pattern[x, Blank[]]]]]]]], Pattern[x, Blank[Symbol]]] := -Simp[((b*C + a*C*Cos[d + e*x])*(a + b*Cos[d + e*x] + c*Sin[d + e*x])^n)/(a*e*(n + 1)), x] + Dist[(c*C*n + a*A*(n + 1))/(a*(n + 1)), Int[(a + b*Cos[d + e*x] + c*Sin[d + e*x])^n, x], x] /; FreeQ[{a, b, c, d, e, A, C, n}, x] && NeQ[n, -1] && EqQ[a^2 - b^2 - c^2, 0] && NeQ[c*C*n + a*A*(n + 1), 0]
  3144. Int[Times[Plus[Optional[Pattern[A, Blank[]]], Times[cos[Plus[Optional[Pattern[d, Blank[]]], Times[Optional[Pattern[e, Blank[]]], Pattern[x, Blank[]]]]], Optional[Pattern[B, Blank[]]]]], Power[Plus[Times[cos[Plus[Optional[Pattern[d, Blank[]]], Times[Optional[Pattern[e, Blank[]]], Pattern[x, Blank[]]]]], Optional[Pattern[b, Blank[]]]], Pattern[a, Blank[]], Times[Optional[Pattern[c, Blank[]]], sin[Plus[Optional[Pattern[d, Blank[]]], Times[Optional[Pattern[e, Blank[]]], Pattern[x, Blank[]]]]]]], Optional[Pattern[n, Blank[]]]]], Pattern[x, Blank[Symbol]]] := Simp[((B*c + a*B*Sin[d + e*x])*(a + b*Cos[d + e*x] + c*Sin[d + e*x])^n)/(a*e*(n + 1)), x] + Dist[(b*B*n + a*A*(n + 1))/(a*(n + 1)), Int[(a + b*Cos[d + e*x] + c*Sin[d + e*x])^n, x], x] /; FreeQ[{a, b, c, d, e, A, B, n}, x] && NeQ[n, -1] && EqQ[a^2 - b^2 - c^2, 0] && NeQ[b*B*n + a*A*(n + 1), 0]
  3145. Int[Times[Power[Plus[Times[cos[Plus[Optional[Pattern[d, Blank[]]], Times[Optional[Pattern[e, Blank[]]], Pattern[x, Blank[]]]]], Optional[Pattern[b, Blank[]]]], Times[Optional[Pattern[c, Blank[]]], sin[Plus[Optional[Pattern[d, Blank[]]], Times[Optional[Pattern[e, Blank[]]], Pattern[x, Blank[]]]]]]], Optional[Pattern[n, Blank[]]]], Plus[Times[cos[Plus[Optional[Pattern[d, Blank[]]], Times[Optional[Pattern[e, Blank[]]], Pattern[x, Blank[]]]]], Optional[Pattern[B, Blank[]]]], Times[Optional[Pattern[C, Blank[]]], sin[Plus[Optional[Pattern[d, Blank[]]], Times[Optional[Pattern[e, Blank[]]], Pattern[x, Blank[]]]]]]]], Pattern[x, Blank[Symbol]]] := Simp[((c*B - b*C)*(b*Cos[d + e*x] + c*Sin[d + e*x])^(n + 1))/(e*(n + 1)*(b^2 + c^2)), x] /; FreeQ[{b, c, d, e, B, C}, x] && NeQ[n, -1] && NeQ[b^2 + c^2, 0] && EqQ[b*B + c*C, 0]
  3146. Int[Times[Power[Plus[Times[cos[Plus[Optional[Pattern[d, Blank[]]], Times[Optional[Pattern[e, Blank[]]], Pattern[x, Blank[]]]]], Optional[Pattern[b, Blank[]]]], Pattern[a, Blank[]], Times[Optional[Pattern[c, Blank[]]], sin[Plus[Optional[Pattern[d, Blank[]]], Times[Optional[Pattern[e, Blank[]]], Pattern[x, Blank[]]]]]]], Optional[Pattern[n, Blank[]]]], Plus[Optional[Pattern[A, Blank[]]], Times[cos[Plus[Optional[Pattern[d, Blank[]]], Times[Optional[Pattern[e, Blank[]]], Pattern[x, Blank[]]]]], Optional[Pattern[B, Blank[]]]], Times[Optional[Pattern[C, Blank[]]], sin[Plus[Optional[Pattern[d, Blank[]]], Times[Optional[Pattern[e, Blank[]]], Pattern[x, Blank[]]]]]]]], Pattern[x, Blank[Symbol]]] := Simp[((B*c - b*C - a*C*Cos[d + e*x] + a*B*Sin[d + e*x])*(a + b*Cos[d + e*x] + c*Sin[d + e*x])^n)/(a*e*(n + 1)), x] + Dist[1/(a*(n + 1)), Int[(a + b*Cos[d + e*x] + c*Sin[d + e*x])^(n - 1)*Simp[a*(b*B + c*C)*n + a^2*A*(n + 1) + (n*(a^2*B - B*c^2 + b*c*C) + a*b*A*(n + 1))*Cos[d + e*x] + (n*(b*B*c + a^2*C - b^2*C) + a*c*A*(n + 1))*Sin[d + e*x], x], x], x] /; FreeQ[{a, b, c, d, e, A, B, C}, x] && GtQ[n, 0] && NeQ[a^2 - b^2 - c^2, 0]
  3147. Int[Times[Power[Plus[Times[cos[Plus[Optional[Pattern[d, Blank[]]], Times[Optional[Pattern[e, Blank[]]], Pattern[x, Blank[]]]]], Optional[Pattern[b, Blank[]]]], Pattern[a, Blank[]], Times[Optional[Pattern[c, Blank[]]], sin[Plus[Optional[Pattern[d, Blank[]]], Times[Optional[Pattern[e, Blank[]]], Pattern[x, Blank[]]]]]]], Optional[Pattern[n, Blank[]]]], Plus[Optional[Pattern[A, Blank[]]], Times[Optional[Pattern[C, Blank[]]], sin[Plus[Optional[Pattern[d, Blank[]]], Times[Optional[Pattern[e, Blank[]]], Pattern[x, Blank[]]]]]]]], Pattern[x, Blank[Symbol]]] := -Simp[((b*C + a*C*Cos[d + e*x])*(a + b*Cos[d + e*x] + c*Sin[d + e*x])^n)/(a*e*(n + 1)), x] + Dist[1/(a*(n + 1)), Int[(a + b*Cos[d + e*x] + c*Sin[d + e*x])^(n - 1)*Simp[a*c*C*n + a^2*A*(n + 1) + (c*b*C*n + a*b*A*(n + 1))*Cos[d + e*x] + (a^2*C*n - b^2*C*n + a*c*A*(n + 1))*Sin[d + e*x], x], x], x] /; FreeQ[{a, b, c, d, e, A, C}, x] && GtQ[n, 0] && NeQ[a^2 - b^2 - c^2, 0]
  3148. Int[Times[Plus[Optional[Pattern[A, Blank[]]], Times[cos[Plus[Optional[Pattern[d, Blank[]]], Times[Optional[Pattern[e, Blank[]]], Pattern[x, Blank[]]]]], Optional[Pattern[B, Blank[]]]]], Power[Plus[Times[cos[Plus[Optional[Pattern[d, Blank[]]], Times[Optional[Pattern[e, Blank[]]], Pattern[x, Blank[]]]]], Optional[Pattern[b, Blank[]]]], Pattern[a, Blank[]], Times[Optional[Pattern[c, Blank[]]], sin[Plus[Optional[Pattern[d, Blank[]]], Times[Optional[Pattern[e, Blank[]]], Pattern[x, Blank[]]]]]]], Optional[Pattern[n, Blank[]]]]], Pattern[x, Blank[Symbol]]] := Simp[((B*c + a*B*Sin[d + e*x])*(a + b*Cos[d + e*x] + c*Sin[d + e*x])^n)/(a*e*(n + 1)), x] + Dist[1/(a*(n + 1)), Int[(a + b*Cos[d + e*x] + c*Sin[d + e*x])^(n - 1)*Simp[a*b*B*n + a^2*A*(n + 1) + (a^2*B*n - c^2*B*n + a*b*A*(n + 1))*Cos[d + e*x] + (b*c*B*n + a*c*A*(n + 1))*Sin[d + e*x], x], x], x] /; FreeQ[{a, b, c, d, e, A, B}, x] && GtQ[n, 0] && NeQ[a^2 - b^2 - c^2, 0]
  3149. Int[Times[Power[Plus[Times[cos[Plus[Optional[Pattern[d, Blank[]]], Times[Optional[Pattern[e, Blank[]]], Pattern[x, Blank[]]]]], Optional[Pattern[b, Blank[]]]], Pattern[a, Blank[]], Times[Optional[Pattern[c, Blank[]]], sin[Plus[Optional[Pattern[d, Blank[]]], Times[Optional[Pattern[e, Blank[]]], Pattern[x, Blank[]]]]]]], Rational[-1, 2]], Plus[Optional[Pattern[A, Blank[]]], Times[cos[Plus[Optional[Pattern[d, Blank[]]], Times[Optional[Pattern[e, Blank[]]], Pattern[x, Blank[]]]]], Optional[Pattern[B, Blank[]]]], Times[Optional[Pattern[C, Blank[]]], sin[Plus[Optional[Pattern[d, Blank[]]], Times[Optional[Pattern[e, Blank[]]], Pattern[x, Blank[]]]]]]]], Pattern[x, Blank[Symbol]]] := Dist[B/b, Int[Sqrt[a + b*Cos[d + e*x] + c*Sin[d + e*x]], x], x] + Dist[(A*b - a*B)/b, Int[1/Sqrt[a + b*Cos[d + e*x] + c*Sin[d + e*x]], x], x] /; FreeQ[{a, b, c, d, e, A, B, C}, x] && EqQ[B*c - b*C, 0] && NeQ[A*b - a*B, 0]
  3150. Int[Times[Power[Plus[Optional[Pattern[a, Blank[]]], Times[cos[Plus[Optional[Pattern[d, Blank[]]], Times[Optional[Pattern[e, Blank[]]], Pattern[x, Blank[]]]]], Optional[Pattern[b, Blank[]]]], Times[Optional[Pattern[c, Blank[]]], sin[Plus[Optional[Pattern[d, Blank[]]], Times[Optional[Pattern[e, Blank[]]], Pattern[x, Blank[]]]]]]], -2], Plus[Optional[Pattern[A, Blank[]]], Times[cos[Plus[Optional[Pattern[d, Blank[]]], Times[Optional[Pattern[e, Blank[]]], Pattern[x, Blank[]]]]], Optional[Pattern[B, Blank[]]]], Times[Optional[Pattern[C, Blank[]]], sin[Plus[Optional[Pattern[d, Blank[]]], Times[Optional[Pattern[e, Blank[]]], Pattern[x, Blank[]]]]]]]], Pattern[x, Blank[Symbol]]] := Simp[(c*B - b*C - (a*C - c*A)*Cos[d + e*x] + (a*B - b*A)*Sin[d + e*x])/(e*(a^2 - b^2 - c^2)*(a + b*Cos[d + e*x] + c*Sin[d + e*x])), x] /; FreeQ[{a, b, c, d, e, A, B, C}, x] && NeQ[a^2 - b^2 - c^2, 0] && EqQ[a*A - b*B - c*C, 0]
  3151. Int[Times[Power[Plus[Optional[Pattern[a, Blank[]]], Times[cos[Plus[Optional[Pattern[d, Blank[]]], Times[Optional[Pattern[e, Blank[]]], Pattern[x, Blank[]]]]], Optional[Pattern[b, Blank[]]]], Times[Optional[Pattern[c, Blank[]]], sin[Plus[Optional[Pattern[d, Blank[]]], Times[Optional[Pattern[e, Blank[]]], Pattern[x, Blank[]]]]]]], -2], Plus[Optional[Pattern[A, Blank[]]], Times[Optional[Pattern[C, Blank[]]], sin[Plus[Optional[Pattern[d, Blank[]]], Times[Optional[Pattern[e, Blank[]]], Pattern[x, Blank[]]]]]]]], Pattern[x, Blank[Symbol]]] := -Simp[(b*C + (a*C - c*A)*Cos[d + e*x] + b*A*Sin[d + e*x])/(e*(a^2 - b^2 - c^2)*(a + b*Cos[d + e*x] + c*Sin[d + e*x])), x] /; FreeQ[{a, b, c, d, e, A, C}, x] && NeQ[a^2 - b^2 - c^2, 0] && EqQ[a*A - c*C, 0]
  3152. Int[Times[Plus[Optional[Pattern[A, Blank[]]], Times[cos[Plus[Optional[Pattern[d, Blank[]]], Times[Optional[Pattern[e, Blank[]]], Pattern[x, Blank[]]]]], Optional[Pattern[B, Blank[]]]]], Power[Plus[Optional[Pattern[a, Blank[]]], Times[cos[Plus[Optional[Pattern[d, Blank[]]], Times[Optional[Pattern[e, Blank[]]], Pattern[x, Blank[]]]]], Optional[Pattern[b, Blank[]]]], Times[Optional[Pattern[c, Blank[]]], sin[Plus[Optional[Pattern[d, Blank[]]], Times[Optional[Pattern[e, Blank[]]], Pattern[x, Blank[]]]]]]], -2]], Pattern[x, Blank[Symbol]]] := Simp[(c*B + c*A*Cos[d + e*x] + (a*B - b*A)*Sin[d + e*x])/(e*(a^2 - b^2 - c^2)*(a + b*Cos[d + e*x] + c*Sin[d + e*x])), x] /; FreeQ[{a, b, c, d, e, A, B}, x] && NeQ[a^2 - b^2 - c^2, 0] && EqQ[a*A - b*B, 0]
  3153. Int[Times[Power[Plus[Optional[Pattern[a, Blank[]]], Times[cos[Plus[Optional[Pattern[d, Blank[]]], Times[Optional[Pattern[e, Blank[]]], Pattern[x, Blank[]]]]], Optional[Pattern[b, Blank[]]]], Times[Optional[Pattern[c, Blank[]]], sin[Plus[Optional[Pattern[d, Blank[]]], Times[Optional[Pattern[e, Blank[]]], Pattern[x, Blank[]]]]]]], -2], Plus[Optional[Pattern[A, Blank[]]], Times[cos[Plus[Optional[Pattern[d, Blank[]]], Times[Optional[Pattern[e, Blank[]]], Pattern[x, Blank[]]]]], Optional[Pattern[B, Blank[]]]], Times[Optional[Pattern[C, Blank[]]], sin[Plus[Optional[Pattern[d, Blank[]]], Times[Optional[Pattern[e, Blank[]]], Pattern[x, Blank[]]]]]]]], Pattern[x, Blank[Symbol]]] := Simp[(c*B - b*C - (a*C - c*A)*Cos[d + e*x] + (a*B - b*A)*Sin[d + e*x])/(e*(a^2 - b^2 - c^2)*(a + b*Cos[d + e*x] + c*Sin[d + e*x])), x] + Dist[(a*A - b*B - c*C)/(a^2 - b^2 - c^2), Int[1/(a + b*Cos[d + e*x] + c*Sin[d + e*x]), x], x] /; FreeQ[{a, b, c, d, e, A, B, C}, x] && NeQ[a^2 - b^2 - c^2, 0] && NeQ[a*A - b*B - c*C, 0]
  3154. Int[Times[Power[Plus[Optional[Pattern[a, Blank[]]], Times[cos[Plus[Optional[Pattern[d, Blank[]]], Times[Optional[Pattern[e, Blank[]]], Pattern[x, Blank[]]]]], Optional[Pattern[b, Blank[]]]], Times[Optional[Pattern[c, Blank[]]], sin[Plus[Optional[Pattern[d, Blank[]]], Times[Optional[Pattern[e, Blank[]]], Pattern[x, Blank[]]]]]]], -2], Plus[Optional[Pattern[A, Blank[]]], Times[Optional[Pattern[C, Blank[]]], sin[Plus[Optional[Pattern[d, Blank[]]], Times[Optional[Pattern[e, Blank[]]], Pattern[x, Blank[]]]]]]]], Pattern[x, Blank[Symbol]]] := -Simp[(b*C + (a*C - c*A)*Cos[d + e*x] + b*A*Sin[d + e*x])/(e*(a^2 - b^2 - c^2)*(a + b*Cos[d + e*x] + c*Sin[d + e*x])), x] + Dist[(a*A - c*C)/(a^2 - b^2 - c^2), Int[1/(a + b*Cos[d + e*x] + c*Sin[d + e*x]), x], x] /; FreeQ[{a, b, c, d, e, A, C}, x] && NeQ[a^2 - b^2 - c^2, 0] && NeQ[a*A - c*C, 0]
  3155. Int[Times[Plus[Optional[Pattern[A, Blank[]]], Times[cos[Plus[Optional[Pattern[d, Blank[]]], Times[Optional[Pattern[e, Blank[]]], Pattern[x, Blank[]]]]], Optional[Pattern[B, Blank[]]]]], Power[Plus[Optional[Pattern[a, Blank[]]], Times[cos[Plus[Optional[Pattern[d, Blank[]]], Times[Optional[Pattern[e, Blank[]]], Pattern[x, Blank[]]]]], Optional[Pattern[b, Blank[]]]], Times[Optional[Pattern[c, Blank[]]], sin[Plus[Optional[Pattern[d, Blank[]]], Times[Optional[Pattern[e, Blank[]]], Pattern[x, Blank[]]]]]]], -2]], Pattern[x, Blank[Symbol]]] := Simp[(c*B + c*A*Cos[d + e*x] + (a*B - b*A)*Sin[d + e*x])/(e*(a^2 - b^2 - c^2)*(a + b*Cos[d + e*x] + c*Sin[d + e*x])), x] + Dist[(a*A - b*B)/(a^2 - b^2 - c^2), Int[1/(a + b*Cos[d + e*x] + c*Sin[d + e*x]), x], x] /; FreeQ[{a, b, c, d, e, A, B}, x] && NeQ[a^2 - b^2 - c^2, 0] && NeQ[a*A - b*B, 0]
  3156. Int[Times[Power[Plus[Optional[Pattern[a, Blank[]]], Times[cos[Plus[Optional[Pattern[d, Blank[]]], Times[Optional[Pattern[e, Blank[]]], Pattern[x, Blank[]]]]], Optional[Pattern[b, Blank[]]]], Times[Optional[Pattern[c, Blank[]]], sin[Plus[Optional[Pattern[d, Blank[]]], Times[Optional[Pattern[e, Blank[]]], Pattern[x, Blank[]]]]]]], Pattern[n, Blank[]]], Plus[Optional[Pattern[A, Blank[]]], Times[cos[Plus[Optional[Pattern[d, Blank[]]], Times[Optional[Pattern[e, Blank[]]], Pattern[x, Blank[]]]]], Optional[Pattern[B, Blank[]]]], Times[Optional[Pattern[C, Blank[]]], sin[Plus[Optional[Pattern[d, Blank[]]], Times[Optional[Pattern[e, Blank[]]], Pattern[x, Blank[]]]]]]]], Pattern[x, Blank[Symbol]]] := -Simp[((c*B - b*C - (a*C - c*A)*Cos[d + e*x] + (a*B - b*A)*Sin[d + e*x])*(a + b*Cos[d + e*x] + c*Sin[d + e*x])^(n + 1))/(e*(n + 1)*(a^2 - b^2 - c^2)), x] + Dist[1/((n + 1)*(a^2 - b^2 - c^2)), Int[(a + b*Cos[d + e*x] + c*Sin[d + e*x])^(n + 1)*Simp[(n + 1)*(a*A - b*B - c*C) + (n + 2)*(a*B - b*A)*Cos[d + e*x] + (n + 2)*(a*C - c*A)*Sin[d + e*x], x], x], x] /; FreeQ[{a, b, c, d, e, A, B, C}, x] && LtQ[n, -1] && NeQ[a^2 - b^2 - c^2, 0] && NeQ[n, -2]
  3157. Int[Times[Power[Plus[Optional[Pattern[a, Blank[]]], Times[cos[Plus[Optional[Pattern[d, Blank[]]], Times[Optional[Pattern[e, Blank[]]], Pattern[x, Blank[]]]]], Optional[Pattern[b, Blank[]]]], Times[Optional[Pattern[c, Blank[]]], sin[Plus[Optional[Pattern[d, Blank[]]], Times[Optional[Pattern[e, Blank[]]], Pattern[x, Blank[]]]]]]], Pattern[n, Blank[]]], Plus[Optional[Pattern[A, Blank[]]], Times[Optional[Pattern[C, Blank[]]], sin[Plus[Optional[Pattern[d, Blank[]]], Times[Optional[Pattern[e, Blank[]]], Pattern[x, Blank[]]]]]]]], Pattern[x, Blank[Symbol]]] := Simp[((b*C + (a*C - c*A)*Cos[d + e*x] + b*A*Sin[d + e*x])*(a + b*Cos[d + e*x] + c*Sin[d + e*x])^(n + 1))/(e*(n + 1)*(a^2 - b^2 - c^2)), x] + Dist[1/((n + 1)*(a^2 - b^2 - c^2)), Int[(a + b*Cos[d + e*x] + c*Sin[d + e*x])^(n + 1)*Simp[(n + 1)*(a*A - c*C) - (n + 2)*b*A*Cos[d + e*x] + (n + 2)*(a*C - c*A)*Sin[d + e*x], x], x], x] /; FreeQ[{a, b, c, d, e, A, C}, x] && LtQ[n, -1] && NeQ[a^2 - b^2 - c^2, 0] && NeQ[n, -2]
  3158. Int[Times[Plus[Optional[Pattern[A, Blank[]]], Times[cos[Plus[Optional[Pattern[d, Blank[]]], Times[Optional[Pattern[e, Blank[]]], Pattern[x, Blank[]]]]], Optional[Pattern[B, Blank[]]]]], Power[Plus[Optional[Pattern[a, Blank[]]], Times[cos[Plus[Optional[Pattern[d, Blank[]]], Times[Optional[Pattern[e, Blank[]]], Pattern[x, Blank[]]]]], Optional[Pattern[b, Blank[]]]], Times[Optional[Pattern[c, Blank[]]], sin[Plus[Optional[Pattern[d, Blank[]]], Times[Optional[Pattern[e, Blank[]]], Pattern[x, Blank[]]]]]]], Pattern[n, Blank[]]]], Pattern[x, Blank[Symbol]]] := -Simp[((c*B + c*A*Cos[d + e*x] + (a*B - b*A)*Sin[d + e*x])*(a + b*Cos[d + e*x] + c*Sin[d + e*x])^(n + 1))/(e*(n + 1)*(a^2 - b^2 - c^2)), x] + Dist[1/((n + 1)*(a^2 - b^2 - c^2)), Int[(a + b*Cos[d + e*x] + c*Sin[d + e*x])^(n + 1)*Simp[(n + 1)*(a*A - b*B) + (n + 2)*(a*B - b*A)*Cos[d + e*x] - (n + 2)*c*A*Sin[d + e*x], x], x], x] /; FreeQ[{a, b, c, d, e, A, B}, x] && LtQ[n, -1] && NeQ[a^2 - b^2 - c^2, 0] && NeQ[n, -2]
  3159. Int[Power[Plus[Optional[Pattern[a, Blank[]]], Times[Optional[Pattern[b, Blank[]]], sec[Plus[Optional[Pattern[d, Blank[]]], Times[Optional[Pattern[e, Blank[]]], Pattern[x, Blank[]]]]]], Times[Optional[Pattern[c, Blank[]]], tan[Plus[Optional[Pattern[d, Blank[]]], Times[Optional[Pattern[e, Blank[]]], Pattern[x, Blank[]]]]]]], -1], Pattern[x, Blank[Symbol]]] := Int[Cos[d + e*x]/(b + a*Cos[d + e*x] + c*Sin[d + e*x]), x] /; FreeQ[{a, b, c, d, e}, x]
  3160. Int[Power[Plus[Optional[Pattern[a, Blank[]]], Times[csc[Plus[Optional[Pattern[d, Blank[]]], Times[Optional[Pattern[e, Blank[]]], Pattern[x, Blank[]]]]], Optional[Pattern[b, Blank[]]]], Times[cot[Plus[Optional[Pattern[d, Blank[]]], Times[Optional[Pattern[e, Blank[]]], Pattern[x, Blank[]]]]], Optional[Pattern[c, Blank[]]]]], -1], Pattern[x, Blank[Symbol]]] := Int[Sin[d + e*x]/(b + a*Sin[d + e*x] + c*Cos[d + e*x]), x] /; FreeQ[{a, b, c, d, e}, x]
  3161. Int[Times[Power[cos[Plus[Optional[Pattern[d, Blank[]]], Times[Optional[Pattern[e, Blank[]]], Pattern[x, Blank[]]]]], Optional[Pattern[n, Blank[]]]], Power[Plus[Optional[Pattern[a, Blank[]]], Times[Optional[Pattern[b, Blank[]]], sec[Plus[Optional[Pattern[d, Blank[]]], Times[Optional[Pattern[e, Blank[]]], Pattern[x, Blank[]]]]]], Times[Optional[Pattern[c, Blank[]]], tan[Plus[Optional[Pattern[d, Blank[]]], Times[Optional[Pattern[e, Blank[]]], Pattern[x, Blank[]]]]]]], Optional[Pattern[n, Blank[]]]]], Pattern[x, Blank[Symbol]]] := Int[(b + a*Cos[d + e*x] + c*Sin[d + e*x])^n, x] /; FreeQ[{a, b, c, d, e}, x] && IntegerQ[n]
  3162. Int[Times[Power[Plus[Optional[Pattern[a, Blank[]]], Times[csc[Plus[Optional[Pattern[d, Blank[]]], Times[Optional[Pattern[e, Blank[]]], Pattern[x, Blank[]]]]], Optional[Pattern[b, Blank[]]]], Times[cot[Plus[Optional[Pattern[d, Blank[]]], Times[Optional[Pattern[e, Blank[]]], Pattern[x, Blank[]]]]], Optional[Pattern[c, Blank[]]]]], Optional[Pattern[n, Blank[]]]], Power[sin[Plus[Optional[Pattern[d, Blank[]]], Times[Optional[Pattern[e, Blank[]]], Pattern[x, Blank[]]]]], Optional[Pattern[n, Blank[]]]]], Pattern[x, Blank[Symbol]]] := Int[(b + a*Sin[d + e*x] + c*Cos[d + e*x])^n, x] /; FreeQ[{a, b, c, d, e}, x] && IntegerQ[n]
  3163. Int[Times[Power[cos[Plus[Optional[Pattern[d, Blank[]]], Times[Optional[Pattern[e, Blank[]]], Pattern[x, Blank[]]]]], Pattern[n, Blank[]]], Power[Plus[Optional[Pattern[a, Blank[]]], Times[Optional[Pattern[b, Blank[]]], sec[Plus[Optional[Pattern[d, Blank[]]], Times[Optional[Pattern[e, Blank[]]], Pattern[x, Blank[]]]]]], Times[Optional[Pattern[c, Blank[]]], tan[Plus[Optional[Pattern[d, Blank[]]], Times[Optional[Pattern[e, Blank[]]], Pattern[x, Blank[]]]]]]], Pattern[n, Blank[]]]], Pattern[x, Blank[Symbol]]] := Dist[(Cos[d + e*x]^n*(a + b*Sec[d + e*x] + c*Tan[d + e*x])^n)/(b + a*Cos[d + e*x] + c*Sin[d + e*x])^n, Int[(b + a*Cos[d + e*x] + c*Sin[d + e*x])^n, x], x] /; FreeQ[{a, b, c, d, e}, x] && !IntegerQ[n]
  3164. Int[Times[Power[Plus[Optional[Pattern[a, Blank[]]], Times[csc[Plus[Optional[Pattern[d, Blank[]]], Times[Optional[Pattern[e, Blank[]]], Pattern[x, Blank[]]]]], Optional[Pattern[b, Blank[]]]], Times[cot[Plus[Optional[Pattern[d, Blank[]]], Times[Optional[Pattern[e, Blank[]]], Pattern[x, Blank[]]]]], Optional[Pattern[c, Blank[]]]]], Pattern[n, Blank[]]], Power[sin[Plus[Optional[Pattern[d, Blank[]]], Times[Optional[Pattern[e, Blank[]]], Pattern[x, Blank[]]]]], Pattern[n, Blank[]]]], Pattern[x, Blank[Symbol]]] := Dist[(Sin[d + e*x]^n*(a + b*Csc[d + e*x] + c*Cot[d + e*x])^n)/(b + a*Sin[d + e*x] + c*Cos[d + e*x])^n, Int[(b + a*Sin[d + e*x] + c*Cos[d + e*x])^n, x], x] /; FreeQ[{a, b, c, d, e}, x] && !IntegerQ[n]
  3165. Int[Times[Power[sec[Plus[Optional[Pattern[d, Blank[]]], Times[Optional[Pattern[e, Blank[]]], Pattern[x, Blank[]]]]], Optional[Pattern[n, Blank[]]]], Power[Plus[Optional[Pattern[a, Blank[]]], Times[Optional[Pattern[b, Blank[]]], sec[Plus[Optional[Pattern[d, Blank[]]], Times[Optional[Pattern[e, Blank[]]], Pattern[x, Blank[]]]]]], Times[Optional[Pattern[c, Blank[]]], tan[Plus[Optional[Pattern[d, Blank[]]], Times[Optional[Pattern[e, Blank[]]], Pattern[x, Blank[]]]]]]], Pattern[m, Blank[]]]], Pattern[x, Blank[Symbol]]] := Int[1/(b + a*Cos[d + e*x] + c*Sin[d + e*x])^n, x] /; FreeQ[{a, b, c, d, e}, x] && EqQ[m + n, 0] && IntegerQ[n]
  3166. Int[Times[Power[csc[Plus[Optional[Pattern[d, Blank[]]], Times[Optional[Pattern[e, Blank[]]], Pattern[x, Blank[]]]]], Optional[Pattern[n, Blank[]]]], Power[Plus[Optional[Pattern[a, Blank[]]], Times[csc[Plus[Optional[Pattern[d, Blank[]]], Times[Optional[Pattern[e, Blank[]]], Pattern[x, Blank[]]]]], Optional[Pattern[b, Blank[]]]], Times[cot[Plus[Optional[Pattern[d, Blank[]]], Times[Optional[Pattern[e, Blank[]]], Pattern[x, Blank[]]]]], Optional[Pattern[c, Blank[]]]]], Pattern[m, Blank[]]]], Pattern[x, Blank[Symbol]]] := Int[1/(b + a*Sin[d + e*x] + c*Cos[d + e*x])^n, x] /; FreeQ[{a, b, c, d, e}, x] && EqQ[m + n, 0] && IntegerQ[n]
  3167. Int[Times[Power[sec[Plus[Optional[Pattern[d, Blank[]]], Times[Optional[Pattern[e, Blank[]]], Pattern[x, Blank[]]]]], Optional[Pattern[n, Blank[]]]], Power[Plus[Optional[Pattern[a, Blank[]]], Times[Optional[Pattern[b, Blank[]]], sec[Plus[Optional[Pattern[d, Blank[]]], Times[Optional[Pattern[e, Blank[]]], Pattern[x, Blank[]]]]]], Times[Optional[Pattern[c, Blank[]]], tan[Plus[Optional[Pattern[d, Blank[]]], Times[Optional[Pattern[e, Blank[]]], Pattern[x, Blank[]]]]]]], Pattern[m, Blank[]]]], Pattern[x, Blank[Symbol]]] := Dist[(Sec[d + e*x]^n*(b + a*Cos[d + e*x] + c*Sin[d + e*x])^n)/(a + b*Sec[d + e*x] + c*Tan[d + e*x])^n, Int[1/(b + a*Cos[d + e*x] + c*Sin[d + e*x])^n, x], x] /; FreeQ[{a, b, c, d, e}, x] && EqQ[m + n, 0] && !IntegerQ[n]
  3168. Int[Times[Power[csc[Plus[Optional[Pattern[d, Blank[]]], Times[Optional[Pattern[e, Blank[]]], Pattern[x, Blank[]]]]], Optional[Pattern[n, Blank[]]]], Power[Plus[Optional[Pattern[a, Blank[]]], Times[csc[Plus[Optional[Pattern[d, Blank[]]], Times[Optional[Pattern[e, Blank[]]], Pattern[x, Blank[]]]]], Optional[Pattern[b, Blank[]]]], Times[cot[Plus[Optional[Pattern[d, Blank[]]], Times[Optional[Pattern[e, Blank[]]], Pattern[x, Blank[]]]]], Optional[Pattern[c, Blank[]]]]], Pattern[m, Blank[]]]], Pattern[x, Blank[Symbol]]] := Dist[(Csc[d + e*x]^n*(b + a*Sin[d + e*x] + c*Cos[d + e*x])^n)/(a + b*Csc[d + e*x] + c*Cot[d + e*x])^n, Int[1/(b + a*Sin[d + e*x] + c*Cos[d + e*x])^n, x], x] /; FreeQ[{a, b, c, d, e}, x] && EqQ[m + n, 0] && !IntegerQ[n]
  3169. Int[Times[Plus[Pattern[a, Blank[]], Times[Optional[Pattern[b, Blank[]]], Power[sin[Plus[Optional[Pattern[e, Blank[]]], Times[Optional[Pattern[f, Blank[]]], Pattern[x, Blank[]]]]], 2]]], Plus[Optional[Pattern[A, Blank[]]], Times[Optional[Pattern[B, Blank[]]], Power[sin[Plus[Optional[Pattern[e, Blank[]]], Times[Optional[Pattern[f, Blank[]]], Pattern[x, Blank[]]]]], 2]]]], Pattern[x, Blank[Symbol]]] := Simp[((4*A*(2*a + b) + B*(4*a + 3*b))*x)/8, x] + (-Simp[(b*B*Cos[e + f*x]*Sin[e + f*x]^3)/(4*f), x] - Simp[((4*A*b + B*(4*a + 3*b))*Cos[e + f*x]*Sin[e + f*x])/(8*f), x]) /; FreeQ[{a, b, e, f, A, B}, x]
  3170. Int[Times[Power[Plus[Pattern[a, Blank[]], Times[Optional[Pattern[b, Blank[]]], Power[sin[Plus[Optional[Pattern[e, Blank[]]], Times[Optional[Pattern[f, Blank[]]], Pattern[x, Blank[]]]]], 2]]], Pattern[p, Blank[]]], Plus[Optional[Pattern[A, Blank[]]], Times[Optional[Pattern[B, Blank[]]], Power[sin[Plus[Optional[Pattern[e, Blank[]]], Times[Optional[Pattern[f, Blank[]]], Pattern[x, Blank[]]]]], 2]]]], Pattern[x, Blank[Symbol]]] := -Simp[(B*Cos[e + f*x]*Sin[e + f*x]*(a + b*Sin[e + f*x]^2)^p)/(2*f*(p + 1)), x] + Dist[1/(2*(p + 1)), Int[(a + b*Sin[e + f*x]^2)^(p - 1)*Simp[a*B + 2*a*A*(p + 1) + (2*A*b*(p + 1) + B*(b + 2*a*p + 2*b*p))*Sin[e + f*x]^2, x], x], x] /; FreeQ[{a, b, e, f, A, B}, x] && GtQ[p, 0]
  3171. Int[Times[Power[Plus[Pattern[a, Blank[]], Times[Optional[Pattern[b, Blank[]]], Power[sin[Plus[Optional[Pattern[e, Blank[]]], Times[Optional[Pattern[f, Blank[]]], Pattern[x, Blank[]]]]], 2]]], -1], Plus[Optional[Pattern[A, Blank[]]], Times[Optional[Pattern[B, Blank[]]], Power[sin[Plus[Optional[Pattern[e, Blank[]]], Times[Optional[Pattern[f, Blank[]]], Pattern[x, Blank[]]]]], 2]]]], Pattern[x, Blank[Symbol]]] := Simp[(B*x)/b, x] + Dist[(A*b - a*B)/b, Int[1/(a + b*Sin[e + f*x]^2), x], x] /; FreeQ[{a, b, e, f, A, B}, x]
  3172. Int[Times[Power[Plus[Pattern[a, Blank[]], Times[Optional[Pattern[b, Blank[]]], Power[sin[Plus[Optional[Pattern[e, Blank[]]], Times[Optional[Pattern[f, Blank[]]], Pattern[x, Blank[]]]]], 2]]], Rational[-1, 2]], Plus[Optional[Pattern[A, Blank[]]], Times[Optional[Pattern[B, Blank[]]], Power[sin[Plus[Optional[Pattern[e, Blank[]]], Times[Optional[Pattern[f, Blank[]]], Pattern[x, Blank[]]]]], 2]]]], Pattern[x, Blank[Symbol]]] := Dist[B/b, Int[Sqrt[a + b*Sin[e + f*x]^2], x], x] + Dist[(A*b - a*B)/b, Int[1/Sqrt[a + b*Sin[e + f*x]^2], x], x] /; FreeQ[{a, b, e, f, A, B}, x]
  3173. Int[Times[Power[Plus[Pattern[a, Blank[]], Times[Optional[Pattern[b, Blank[]]], Power[sin[Plus[Optional[Pattern[e, Blank[]]], Times[Optional[Pattern[f, Blank[]]], Pattern[x, Blank[]]]]], 2]]], Pattern[p, Blank[]]], Plus[Optional[Pattern[A, Blank[]]], Times[Optional[Pattern[B, Blank[]]], Power[sin[Plus[Optional[Pattern[e, Blank[]]], Times[Optional[Pattern[f, Blank[]]], Pattern[x, Blank[]]]]], 2]]]], Pattern[x, Blank[Symbol]]] := -Simp[((A*b - a*B)*Cos[e + f*x]*Sin[e + f*x]*(a + b*Sin[e + f*x]^2)^(p + 1))/(2*a*f*(a + b)*(p + 1)), x] - Dist[1/(2*a*(a + b)*(p + 1)), Int[(a + b*Sin[e + f*x]^2)^(p + 1)*Simp[a*B - A*(2*a*(p + 1) + b*(2*p + 3)) + 2*(A*b - a*B)*(p + 2)*Sin[e + f*x]^2, x], x], x] /; FreeQ[{a, b, e, f, A, B}, x] && LtQ[p, -1] && NeQ[a + b, 0]
  3174. Int[Times[Power[Plus[Optional[Pattern[a, Blank[]]], Times[Optional[Pattern[b, Blank[]]], Power[sin[Plus[Optional[Pattern[e, Blank[]]], Times[Optional[Pattern[f, Blank[]]], Pattern[x, Blank[]]]]], 2]]], Pattern[p, Blank[]]], Plus[Optional[Pattern[A, Blank[]]], Times[Optional[Pattern[B, Blank[]]], Power[sin[Plus[Optional[Pattern[e, Blank[]]], Times[Optional[Pattern[f, Blank[]]], Pattern[x, Blank[]]]]], 2]]]], Pattern[x, Blank[Symbol]]] := With[{ff = FreeFactors[Tan[e + f*x], x]}, Dist[(ff*(a + b*Sin[e + f*x]^2)^p*(Sec[e + f*x]^2)^p)/(f*(a + (a + b)*Tan[e + f*x]^2)^p), Subst[Int[((a + (a + b)*ff^2*x^2)^p*(A + (A + B)*ff^2*x^2))/(1 + ff^2*x^2)^(p + 2), x], x, Tan[e + f*x]/ff], x]] /; FreeQ[{a, b, e, f, A, B}, x] && !IntegerQ[p]
  3175. Int[Times[Optional[Pattern[u, Blank[]]], Power[Plus[Pattern[a, Blank[]], Times[Optional[Pattern[b, Blank[]]], Power[sin[Plus[Optional[Pattern[e, Blank[]]], Times[Optional[Pattern[f, Blank[]]], Pattern[x, Blank[]]]]], 2]]], Pattern[p, Blank[]]]], Pattern[x, Blank[Symbol]]] := Dist[a^p, Int[ActivateTrig[u*cos[e + f*x]^(2*p)], x], x] /; FreeQ[{a, b, e, f, p}, x] && EqQ[a + b, 0] && IntegerQ[p]
  3176. Int[Times[Optional[Pattern[u, Blank[]]], Power[Plus[Pattern[a, Blank[]], Times[Optional[Pattern[b, Blank[]]], Power[sin[Plus[Optional[Pattern[e, Blank[]]], Times[Optional[Pattern[f, Blank[]]], Pattern[x, Blank[]]]]], 2]]], Pattern[p, Blank[]]]], Pattern[x, Blank[Symbol]]] := Int[ActivateTrig[u*(a*cos[e + f*x]^2)^p], x] /; FreeQ[{a, b, e, f, p}, x] && EqQ[a + b, 0]
  3177. Int[Power[Plus[Pattern[a, Blank[]], Times[Optional[Pattern[b, Blank[]]], Power[sin[Plus[Optional[Pattern[e, Blank[]]], Times[Optional[Pattern[f, Blank[]]], Pattern[x, Blank[]]]]], 2]]], Rational[1, 2]], Pattern[x, Blank[Symbol]]] := Simp[(Sqrt[a]*EllipticE[e + f*x, -(b/a)])/f, x] /; FreeQ[{a, b, e, f}, x] && GtQ[a, 0]
  3178. Int[Power[Plus[Pattern[a, Blank[]], Times[Optional[Pattern[b, Blank[]]], Power[sin[Plus[Optional[Pattern[e, Blank[]]], Times[Optional[Pattern[f, Blank[]]], Pattern[x, Blank[]]]]], 2]]], Rational[1, 2]], Pattern[x, Blank[Symbol]]] := Dist[Sqrt[a + b*Sin[e + f*x]^2]/Sqrt[1 + (b*Sin[e + f*x]^2)/a], Int[Sqrt[1 + (b*Sin[e + f*x]^2)/a], x], x] /; FreeQ[{a, b, e, f}, x] && !GtQ[a, 0]
  3179. Int[Power[Plus[Pattern[a, Blank[]], Times[Optional[Pattern[b, Blank[]]], Power[sin[Plus[Optional[Pattern[e, Blank[]]], Times[Optional[Pattern[f, Blank[]]], Pattern[x, Blank[]]]]], 2]]], 2], Pattern[x, Blank[Symbol]]] := Simp[((8*a^2 + 8*a*b + 3*b^2)*x)/8, x] + (-Simp[(b^2*Cos[e + f*x]*Sin[e + f*x]^3)/(4*f), x] - Simp[(b*(8*a + 3*b)*Cos[e + f*x]*Sin[e + f*x])/(8*f), x]) /; FreeQ[{a, b, e, f}, x]
  3180. Int[Power[Plus[Pattern[a, Blank[]], Times[Optional[Pattern[b, Blank[]]], Power[sin[Plus[Optional[Pattern[e, Blank[]]], Times[Optional[Pattern[f, Blank[]]], Pattern[x, Blank[]]]]], 2]]], Pattern[p, Blank[]]], Pattern[x, Blank[Symbol]]] := -Simp[(b*Cos[e + f*x]*Sin[e + f*x]*(a + b*Sin[e + f*x]^2)^(p - 1))/(2*f*p), x] + Dist[1/(2*p), Int[(a + b*Sin[e + f*x]^2)^(p - 2)*Simp[a*(b + 2*a*p) + b*(2*a + b)*(2*p - 1)*Sin[e + f*x]^2, x], x], x] /; FreeQ[{a, b, e, f}, x] && NeQ[a + b, 0] && GtQ[p, 1]
  3181. Int[Power[Plus[Pattern[a, Blank[]], Times[Optional[Pattern[b, Blank[]]], Power[sin[Plus[Optional[Pattern[e, Blank[]]], Times[Optional[Pattern[f, Blank[]]], Pattern[x, Blank[]]]]], 2]]], -1], Pattern[x, Blank[Symbol]]] := With[{ff = FreeFactors[Tan[e + f*x], x]}, Dist[ff/f, Subst[Int[1/(a + (a + b)*ff^2*x^2), x], x, Tan[e + f*x]/ff], x]] /; FreeQ[{a, b, e, f}, x]
  3182. Int[Power[Plus[Pattern[a, Blank[]], Times[Optional[Pattern[b, Blank[]]], Power[sin[Plus[Optional[Pattern[e, Blank[]]], Times[Optional[Pattern[f, Blank[]]], Pattern[x, Blank[]]]]], 2]]], Rational[-1, 2]], Pattern[x, Blank[Symbol]]] := Simp[(1*EllipticF[e + f*x, -(b/a)])/(Sqrt[a]*f), x] /; FreeQ[{a, b, e, f}, x] && GtQ[a, 0]
  3183. Int[Power[Plus[Pattern[a, Blank[]], Times[Optional[Pattern[b, Blank[]]], Power[sin[Plus[Optional[Pattern[e, Blank[]]], Times[Optional[Pattern[f, Blank[]]], Pattern[x, Blank[]]]]], 2]]], Rational[-1, 2]], Pattern[x, Blank[Symbol]]] := Dist[Sqrt[1 + (b*Sin[e + f*x]^2)/a]/Sqrt[a + b*Sin[e + f*x]^2], Int[1/Sqrt[1 + (b*Sin[e + f*x]^2)/a], x], x] /; FreeQ[{a, b, e, f}, x] && !GtQ[a, 0]
  3184. Int[Power[Plus[Pattern[a, Blank[]], Times[Optional[Pattern[b, Blank[]]], Power[sin[Plus[Optional[Pattern[e, Blank[]]], Times[Optional[Pattern[f, Blank[]]], Pattern[x, Blank[]]]]], 2]]], Pattern[p, Blank[]]], Pattern[x, Blank[Symbol]]] := -Simp[(b*Cos[e + f*x]*Sin[e + f*x]*(a + b*Sin[e + f*x]^2)^(p + 1))/(2*a*f*(p + 1)*(a + b)), x] + Dist[1/(2*a*(p + 1)*(a + b)), Int[(a + b*Sin[e + f*x]^2)^(p + 1)*Simp[2*a*(p + 1) + b*(2*p + 3) - 2*b*(p + 2)*Sin[e + f*x]^2, x], x], x] /; FreeQ[{a, b, e, f}, x] && NeQ[a + b, 0] && LtQ[p, -1]
  3185. Int[Power[Plus[Pattern[a, Blank[]], Times[Optional[Pattern[b, Blank[]]], Power[sin[Plus[Optional[Pattern[e, Blank[]]], Times[Optional[Pattern[f, Blank[]]], Pattern[x, Blank[]]]]], 2]]], Optional[Pattern[p, Blank[]]]], Pattern[x, Blank[Symbol]]] := With[{ff = FreeFactors[Sin[e + f*x], x]}, Dist[(ff*Sqrt[Cos[e + f*x]^2])/(f*Cos[e + f*x]), Subst[Int[(a + b*ff^2*x^2)^p/Sqrt[1 - ff^2*x^2], x], x, Sin[e + f*x]/ff], x]] /; FreeQ[{a, b, e, f, p}, x] && !IntegerQ[p]
  3186. Int[Times[Power[sin[Plus[Optional[Pattern[e, Blank[]]], Times[Optional[Pattern[f, Blank[]]], Pattern[x, Blank[]]]]], Optional[Pattern[m, Blank[]]]], Power[Plus[Pattern[a, Blank[]], Times[Optional[Pattern[b, Blank[]]], Power[sin[Plus[Optional[Pattern[e, Blank[]]], Times[Optional[Pattern[f, Blank[]]], Pattern[x, Blank[]]]]], 2]]], Optional[Pattern[p, Blank[]]]]], Pattern[x, Blank[Symbol]]] := With[{ff = FreeFactors[Cos[e + f*x], x]}, -Dist[ff/f, Subst[Int[(1 - ff^2*x^2)^((m - 1)/2)*(a + b - b*ff^2*x^2)^p, x], x, Cos[e + f*x]/ff], x]] /; FreeQ[{a, b, e, f, p}, x] && IntegerQ[(m - 1)/2]
  3187. Int[Times[Power[sin[Plus[Optional[Pattern[e, Blank[]]], Times[Optional[Pattern[f, Blank[]]], Pattern[x, Blank[]]]]], Pattern[m, Blank[]]], Power[Plus[Pattern[a, Blank[]], Times[Optional[Pattern[b, Blank[]]], Power[sin[Plus[Optional[Pattern[e, Blank[]]], Times[Optional[Pattern[f, Blank[]]], Pattern[x, Blank[]]]]], 2]]], Optional[Pattern[p, Blank[]]]]], Pattern[x, Blank[Symbol]]] := With[{ff = FreeFactors[Tan[e + f*x], x]}, Dist[ff^(m + 1)/f, Subst[Int[(x^m*(a + (a + b)*ff^2*x^2)^p)/(1 + ff^2*x^2)^(m/2 + p + 1), x], x, Tan[e + f*x]/ff], x]] /; FreeQ[{a, b, e, f}, x] && IntegerQ[m/2] && IntegerQ[p]
  3188. Int[Times[Power[sin[Plus[Optional[Pattern[e, Blank[]]], Times[Optional[Pattern[f, Blank[]]], Pattern[x, Blank[]]]]], Pattern[m, Blank[]]], Power[Plus[Pattern[a, Blank[]], Times[Optional[Pattern[b, Blank[]]], Power[sin[Plus[Optional[Pattern[e, Blank[]]], Times[Optional[Pattern[f, Blank[]]], Pattern[x, Blank[]]]]], 2]]], Optional[Pattern[p, Blank[]]]]], Pattern[x, Blank[Symbol]]] := With[{ff = FreeFactors[Sin[e + f*x], x]}, Dist[(ff^(m + 1)*Sqrt[Cos[e + f*x]^2])/(f*Cos[e + f*x]), Subst[Int[(x^m*(a + b*ff^2*x^2)^p)/Sqrt[1 - ff^2*x^2], x], x, Sin[e + f*x]/ff], x]] /; FreeQ[{a, b, e, f, p}, x] && IntegerQ[m/2] && !IntegerQ[p]
  3189. Int[Times[Power[Times[Optional[Pattern[d, Blank[]]], sin[Plus[Optional[Pattern[e, Blank[]]], Times[Optional[Pattern[f, Blank[]]], Pattern[x, Blank[]]]]]], Pattern[m, Blank[]]], Power[Plus[Pattern[a, Blank[]], Times[Optional[Pattern[b, Blank[]]], Power[sin[Plus[Optional[Pattern[e, Blank[]]], Times[Optional[Pattern[f, Blank[]]], Pattern[x, Blank[]]]]], 2]]], Optional[Pattern[p, Blank[]]]]], Pattern[x, Blank[Symbol]]] := With[{ff = FreeFactors[Cos[e + f*x], x]}, -Dist[(ff*d^(2*IntPart[(m - 1)/2] + 1)*(d*Sin[e + f*x])^(2*FracPart[(m - 1)/2]))/(f*(Sin[e + f*x]^2)^FracPart[(m - 1)/2]), Subst[Int[(1 - ff^2*x^2)^((m - 1)/2)*(a + b - b*ff^2*x^2)^p, x], x, Cos[e + f*x]/ff], x]] /; FreeQ[{a, b, d, e, f, m, p}, x] && !IntegerQ[m]
  3190. Int[Times[Power[cos[Plus[Optional[Pattern[e, Blank[]]], Times[Optional[Pattern[f, Blank[]]], Pattern[x, Blank[]]]]], Optional[Pattern[m, Blank[]]]], Power[Plus[Pattern[a, Blank[]], Times[Optional[Pattern[b, Blank[]]], Power[sin[Plus[Optional[Pattern[e, Blank[]]], Times[Optional[Pattern[f, Blank[]]], Pattern[x, Blank[]]]]], 2]]], Optional[Pattern[p, Blank[]]]]], Pattern[x, Blank[Symbol]]] := With[{ff = FreeFactors[Sin[e + f*x], x]}, Dist[ff/f, Subst[Int[(1 - ff^2*x^2)^((m - 1)/2)*(a + b*ff^2*x^2)^p, x], x, Sin[e + f*x]/ff], x]] /; FreeQ[{a, b, e, f, p}, x] && IntegerQ[(m - 1)/2]
  3191. Int[Times[Power[cos[Plus[Optional[Pattern[e, Blank[]]], Times[Optional[Pattern[f, Blank[]]], Pattern[x, Blank[]]]]], Pattern[m, Blank[]]], Power[Plus[Pattern[a, Blank[]], Times[Optional[Pattern[b, Blank[]]], Power[sin[Plus[Optional[Pattern[e, Blank[]]], Times[Optional[Pattern[f, Blank[]]], Pattern[x, Blank[]]]]], 2]]], Optional[Pattern[p, Blank[]]]]], Pattern[x, Blank[Symbol]]] := With[{ff = FreeFactors[Tan[e + f*x], x]}, Dist[ff/f, Subst[Int[(a + (a + b)*ff^2*x^2)^p/(1 + ff^2*x^2)^(m/2 + p + 1), x], x, Tan[e + f*x]/ff], x]] /; FreeQ[{a, b, e, f}, x] && IntegerQ[m/2] && IntegerQ[p]
  3192. Int[Times[Power[cos[Plus[Optional[Pattern[e, Blank[]]], Times[Optional[Pattern[f, Blank[]]], Pattern[x, Blank[]]]]], Pattern[m, Blank[]]], Power[Plus[Pattern[a, Blank[]], Times[Optional[Pattern[b, Blank[]]], Power[sin[Plus[Optional[Pattern[e, Blank[]]], Times[Optional[Pattern[f, Blank[]]], Pattern[x, Blank[]]]]], 2]]], Optional[Pattern[p, Blank[]]]]], Pattern[x, Blank[Symbol]]] := With[{ff = FreeFactors[Sin[e + f*x], x]}, Dist[(ff*Sqrt[Cos[e + f*x]^2])/(f*Cos[e + f*x]), Subst[Int[(1 - ff^2*x^2)^((m - 1)/2)*(a + b*ff^2*x^2)^p, x], x, Sin[e + f*x]/ff], x]] /; FreeQ[{a, b, e, f, p}, x] && IntegerQ[m/2] && !IntegerQ[p]
  3193. Int[Times[Power[Times[cos[Plus[Optional[Pattern[e, Blank[]]], Times[Optional[Pattern[f, Blank[]]], Pattern[x, Blank[]]]]], Optional[Pattern[d, Blank[]]]], Pattern[m, Blank[]]], Power[Plus[Pattern[a, Blank[]], Times[Optional[Pattern[b, Blank[]]], Power[sin[Plus[Optional[Pattern[e, Blank[]]], Times[Optional[Pattern[f, Blank[]]], Pattern[x, Blank[]]]]], 2]]], Optional[Pattern[p, Blank[]]]]], Pattern[x, Blank[Symbol]]] := With[{ff = FreeFactors[Sin[e + f*x], x]}, Dist[(ff*d^(2*IntPart[(m - 1)/2] + 1)*(d*Cos[e + f*x])^(2*FracPart[(m - 1)/2]))/(f*(Cos[e + f*x]^2)^FracPart[(m - 1)/2]), Subst[Int[(1 - ff^2*x^2)^((m - 1)/2)*(a + b*ff^2*x^2)^p, x], x, Sin[e + f*x]/ff], x]] /; FreeQ[{a, b, d, e, f, m, p}, x] && !IntegerQ[m]
  3194. Int[Times[Power[Plus[Pattern[a, Blank[]], Times[Optional[Pattern[b, Blank[]]], Power[sin[Plus[Optional[Pattern[e, Blank[]]], Times[Optional[Pattern[f, Blank[]]], Pattern[x, Blank[]]]]], 2]]], Optional[Pattern[p, Blank[]]]], Power[tan[Plus[Optional[Pattern[e, Blank[]]], Times[Optional[Pattern[f, Blank[]]], Pattern[x, Blank[]]]]], Optional[Pattern[m, Blank[]]]]], Pattern[x, Blank[Symbol]]] := With[{ff = FreeFactors[Sin[e + f*x]^2, x]}, Dist[ff^((m + 1)/2)/(2*f), Subst[Int[(x^((m - 1)/2)*(a + b*ff*x)^p)/(1 - ff*x)^((m + 1)/2), x], x, Sin[e + f*x]^2/ff], x]] /; FreeQ[{a, b, e, f, p}, x] && IntegerQ[(m - 1)/2]
  3195. Int[Times[Power[Plus[Pattern[a, Blank[]], Times[Optional[Pattern[b, Blank[]]], Power[sin[Plus[Optional[Pattern[e, Blank[]]], Times[Optional[Pattern[f, Blank[]]], Pattern[x, Blank[]]]]], 2]]], Optional[Pattern[p, Blank[]]]], Power[Times[Optional[Pattern[d, Blank[]]], tan[Plus[Optional[Pattern[e, Blank[]]], Times[Optional[Pattern[f, Blank[]]], Pattern[x, Blank[]]]]]], Pattern[m, Blank[]]]], Pattern[x, Blank[Symbol]]] := With[{ff = FreeFactors[Tan[e + f*x], x]}, Dist[ff/f, Subst[Int[((d*ff*x)^m*(a + (a + b)*ff^2*x^2)^p)/(1 + ff^2*x^2)^(p + 1), x], x, Tan[e + f*x]/ff], x]] /; FreeQ[{a, b, d, e, f, m}, x] && IntegerQ[p]
  3196. Int[Times[Power[Plus[Pattern[a, Blank[]], Times[Optional[Pattern[b, Blank[]]], Power[sin[Plus[Optional[Pattern[e, Blank[]]], Times[Optional[Pattern[f, Blank[]]], Pattern[x, Blank[]]]]], 2]]], Optional[Pattern[p, Blank[]]]], Power[tan[Plus[Optional[Pattern[e, Blank[]]], Times[Optional[Pattern[f, Blank[]]], Pattern[x, Blank[]]]]], Pattern[m, Blank[]]]], Pattern[x, Blank[Symbol]]] := With[{ff = FreeFactors[Sin[e + f*x], x]}, Dist[(ff^(m + 1)*Sqrt[Cos[e + f*x]^2])/(f*Cos[e + f*x]), Subst[Int[(x^m*(a + b*ff^2*x^2)^p)/(1 - ff^2*x^2)^((m + 1)/2), x], x, Sin[e + f*x]/ff], x]] /; FreeQ[{a, b, e, f, p}, x] && IntegerQ[m/2] && !IntegerQ[p]
  3197. Int[Times[Power[Plus[Pattern[a, Blank[]], Times[Optional[Pattern[b, Blank[]]], Power[sin[Plus[Optional[Pattern[e, Blank[]]], Times[Optional[Pattern[f, Blank[]]], Pattern[x, Blank[]]]]], 2]]], Optional[Pattern[p, Blank[]]]], Power[Times[Optional[Pattern[d, Blank[]]], tan[Plus[Optional[Pattern[e, Blank[]]], Times[Optional[Pattern[f, Blank[]]], Pattern[x, Blank[]]]]]], Pattern[m, Blank[]]]], Pattern[x, Blank[Symbol]]] := With[{ff = FreeFactors[Sin[e + f*x], x]}, Dist[(ff*(d*Tan[e + f*x])^(m + 1)*(Cos[e + f*x]^2)^((m + 1)/2))/(d*f*Sin[e + f*x]^(m + 1)), Subst[Int[((ff*x)^m*(a + b*ff^2*x^2)^p)/(1 - ff^2*x^2)^((m + 1)/2), x], x, Sin[e + f*x]/ff], x]] /; FreeQ[{a, b, d, e, f, m, p}, x] && !IntegerQ[m]
  3198. Int[Times[Power[cos[Plus[Optional[Pattern[e, Blank[]]], Times[Optional[Pattern[f, Blank[]]], Pattern[x, Blank[]]]]], Optional[Pattern[m, Blank[]]]], Power[Times[Optional[Pattern[d, Blank[]]], sin[Plus[Optional[Pattern[e, Blank[]]], Times[Optional[Pattern[f, Blank[]]], Pattern[x, Blank[]]]]]], Optional[Pattern[n, Blank[]]]], Power[Plus[Pattern[a, Blank[]], Times[Optional[Pattern[b, Blank[]]], Power[sin[Plus[Optional[Pattern[e, Blank[]]], Times[Optional[Pattern[f, Blank[]]], Pattern[x, Blank[]]]]], 2]]], Optional[Pattern[p, Blank[]]]]], Pattern[x, Blank[Symbol]]] := With[{ff = FreeFactors[Sin[e + f*x], x]}, Dist[ff/f, Subst[Int[(d*ff*x)^n*(1 - ff^2*x^2)^((m - 1)/2)*(a + b*ff^2*x^2)^p, x], x, Sin[e + f*x]/ff], x]] /; FreeQ[{a, b, d, e, f, n, p}, x] && IntegerQ[(m - 1)/2]
  3199. Int[Times[Power[sin[Plus[Optional[Pattern[e, Blank[]]], Times[Optional[Pattern[f, Blank[]]], Pattern[x, Blank[]]]]], Optional[Pattern[n, Blank[]]]], Power[Times[Optional[Pattern[c, Blank[]]], sin[Plus[Optional[Pattern[e, Blank[]]], Times[Optional[Pattern[f, Blank[]]], Pattern[x, Blank[]]]]]], Pattern[m, Blank[]]], Power[Plus[Pattern[a, Blank[]], Times[Optional[Pattern[b, Blank[]]], Power[sin[Plus[Optional[Pattern[e, Blank[]]], Times[Optional[Pattern[f, Blank[]]], Pattern[x, Blank[]]]]], 2]]], Optional[Pattern[p, Blank[]]]]], Pattern[x, Blank[Symbol]]] := With[{ff = FreeFactors[Cos[e + f*x], x]}, -Dist[ff/f, Subst[Int[(c*ff*x)^m*(1 - ff^2*x^2)^((n - 1)/2)*(a + b - b*ff^2*x^2)^p, x], x, Cos[e + f*x]/ff], x]] /; FreeQ[{a, b, c, e, f, m, p}, x] && IntegerQ[(n - 1)/2]
  3200. Int[Times[Power[cos[Plus[Optional[Pattern[e, Blank[]]], Times[Optional[Pattern[f, Blank[]]], Pattern[x, Blank[]]]]], Pattern[m, Blank[]]], Power[sin[Plus[Optional[Pattern[e, Blank[]]], Times[Optional[Pattern[f, Blank[]]], Pattern[x, Blank[]]]]], Pattern[n, Blank[]]], Power[Plus[Pattern[a, Blank[]], Times[Optional[Pattern[b, Blank[]]], Power[sin[Plus[Optional[Pattern[e, Blank[]]], Times[Optional[Pattern[f, Blank[]]], Pattern[x, Blank[]]]]], 2]]], Optional[Pattern[p, Blank[]]]]], Pattern[x, Blank[Symbol]]] := With[{ff = FreeFactors[Tan[e + f*x], x]}, Dist[ff^(n + 1)/f, Subst[Int[(x^n*(a + (a + b)*ff^2*x^2)^p)/(1 + ff^2*x^2)^((m + n)/2 + p + 1), x], x, Tan[e + f*x]/ff], x]] /; FreeQ[{a, b, e, f}, x] && IntegerQ[m/2] && IntegerQ[n/2] && IntegerQ[p]
  3201. Int[Times[Power[cos[Plus[Optional[Pattern[e, Blank[]]], Times[Optional[Pattern[f, Blank[]]], Pattern[x, Blank[]]]]], Pattern[m, Blank[]]], Power[Times[Optional[Pattern[d, Blank[]]], sin[Plus[Optional[Pattern[e, Blank[]]], Times[Optional[Pattern[f, Blank[]]], Pattern[x, Blank[]]]]]], Optional[Pattern[n, Blank[]]]], Power[Plus[Pattern[a, Blank[]], Times[Optional[Pattern[b, Blank[]]], Power[sin[Plus[Optional[Pattern[e, Blank[]]], Times[Optional[Pattern[f, Blank[]]], Pattern[x, Blank[]]]]], 2]]], Optional[Pattern[p, Blank[]]]]], Pattern[x, Blank[Symbol]]] := With[{ff = FreeFactors[Sin[e + f*x], x]}, Dist[(ff*Sqrt[Cos[e + f*x]^2])/(f*Cos[e + f*x]), Subst[Int[(d*ff*x)^n*(1 - ff^2*x^2)^((m - 1)/2)*(a + b*ff^2*x^2)^p, x], x, Sin[e + f*x]/ff], x]] /; FreeQ[{a, b, d, e, f, n, p}, x] && IntegerQ[m/2]
  3202. Int[Times[Power[Times[cos[Plus[Optional[Pattern[e, Blank[]]], Times[Optional[Pattern[f, Blank[]]], Pattern[x, Blank[]]]]], Optional[Pattern[c, Blank[]]]], Pattern[m, Blank[]]], Power[Times[Optional[Pattern[d, Blank[]]], sin[Plus[Optional[Pattern[e, Blank[]]], Times[Optional[Pattern[f, Blank[]]], Pattern[x, Blank[]]]]]], Optional[Pattern[n, Blank[]]]], Power[Plus[Pattern[a, Blank[]], Times[Optional[Pattern[b, Blank[]]], Power[sin[Plus[Optional[Pattern[e, Blank[]]], Times[Optional[Pattern[f, Blank[]]], Pattern[x, Blank[]]]]], 2]]], Optional[Pattern[p, Blank[]]]]], Pattern[x, Blank[Symbol]]] := With[{ff = FreeFactors[Sin[e + f*x], x]}, Dist[(ff*c^(2*IntPart[(m - 1)/2] + 1)*(c*Cos[e + f*x])^(2*FracPart[(m - 1)/2]))/(f*(Cos[e + f*x]^2)^FracPart[(m - 1)/2]), Subst[Int[(d*ff*x)^n*(1 - ff^2*x^2)^((m - 1)/2)*(a + b*ff^2*x^2)^p, x], x, Sin[e + f*x]/ff], x]] /; FreeQ[{a, b, c, d, e, f, m, n, p}, x] && !IntegerQ[m]
  3203. Int[Power[Times[Optional[Pattern[b, Blank[]]], Power[sin[Plus[Optional[Pattern[e, Blank[]]], Times[Optional[Pattern[f, Blank[]]], Pattern[x, Blank[]]]]], 2]], Pattern[p, Blank[]]], Pattern[x, Blank[Symbol]]] := -Simp[(Cot[e + f*x]*(b*Sin[e + f*x]^2)^p)/(2*f*p), x] + Dist[(b*(2*p - 1))/(2*p), Int[(b*Sin[e + f*x]^2)^(p - 1), x], x] /; FreeQ[{b, e, f}, x] && !IntegerQ[p] && GtQ[p, 1]
  3204. Int[Power[Times[Optional[Pattern[b, Blank[]]], Power[sin[Plus[Optional[Pattern[e, Blank[]]], Times[Optional[Pattern[f, Blank[]]], Pattern[x, Blank[]]]]], 2]], Pattern[p, Blank[]]], Pattern[x, Blank[Symbol]]] := Simp[(Cot[e + f*x]*(b*Sin[e + f*x]^2)^(p + 1))/(b*f*(2*p + 1)), x] + Dist[(2*(p + 1))/(b*(2*p + 1)), Int[(b*Sin[e + f*x]^2)^(p + 1), x], x] /; FreeQ[{b, e, f}, x] && !IntegerQ[p] && LtQ[p, -1]
  3205. Int[Times[Power[Times[Optional[Pattern[b, Blank[]]], Power[sin[Plus[Optional[Pattern[e, Blank[]]], Times[Optional[Pattern[f, Blank[]]], Pattern[x, Blank[]]]]], Pattern[n, Blank[]]]], Optional[Pattern[p, Blank[]]]], Power[tan[Plus[Optional[Pattern[e, Blank[]]], Times[Optional[Pattern[f, Blank[]]], Pattern[x, Blank[]]]]], Optional[Pattern[m, Blank[]]]]], Pattern[x, Blank[Symbol]]] := With[{ff = FreeFactors[Sin[e + f*x]^2, x]}, Dist[ff^((m + 1)/2)/(2*f), Subst[Int[(x^((m - 1)/2)*(b*ff^(n/2)*x^(n/2))^p)/(1 - ff*x)^((m + 1)/2), x], x, Sin[e + f*x]^2/ff], x]] /; FreeQ[{b, e, f, p}, x] && IntegerQ[(m - 1)/2] && IntegerQ[n/2]
  3206. Int[Times[Power[Times[Optional[Pattern[b, Blank[]]], Power[Times[Optional[Pattern[c, Blank[]]], sin[Plus[Optional[Pattern[e, Blank[]]], Times[Optional[Pattern[f, Blank[]]], Pattern[x, Blank[]]]]]], Pattern[n, Blank[]]]], Optional[Pattern[p, Blank[]]]], Power[tan[Plus[Optional[Pattern[e, Blank[]]], Times[Optional[Pattern[f, Blank[]]], Pattern[x, Blank[]]]]], Optional[Pattern[m, Blank[]]]]], Pattern[x, Blank[Symbol]]] := With[{ff = FreeFactors[Sin[e + f*x], x]}, Dist[ff^(m + 1)/f, Subst[Int[(x^m*(b*(c*ff*x)^n)^p)/(1 - ff^2*x^2)^((m + 1)/2), x], x, Sin[e + f*x]/ff], x]] /; FreeQ[{b, c, e, f, n, p}, x] && ILtQ[(m - 1)/2, 0]
  3207. Int[Times[Optional[Pattern[u, Blank[]]], Power[Times[Optional[Pattern[b, Blank[]]], Power[sin[Plus[Optional[Pattern[e, Blank[]]], Times[Optional[Pattern[f, Blank[]]], Pattern[x, Blank[]]]]], Pattern[n, Blank[]]]], Pattern[p, Blank[]]]], Pattern[x, Blank[Symbol]]] := With[{ff = FreeFactors[Sin[e + f*x], x]}, Dist[((b*ff^n)^IntPart[p]*(b*Sin[e + f*x]^n)^FracPart[p])/(Sin[e + f*x]/ff)^(n*FracPart[p]), Int[ActivateTrig[u]*(Sin[e + f*x]/ff)^(n*p), x], x]] /; FreeQ[{b, e, f, n, p}, x] && !IntegerQ[p] && IntegerQ[n] && (EqQ[u, 1] || MatchQ[u, ((d_.)*(trig_)[e + f*x])^(m_.) /; FreeQ[{d, m}, x] && MemberQ[{sin, cos, tan, cot, sec, csc}, trig]])
  3208. Int[Times[Optional[Pattern[u, Blank[]]], Power[Times[Optional[Pattern[b, Blank[]]], Power[Times[Optional[Pattern[c, Blank[]]], sin[Plus[Optional[Pattern[e, Blank[]]], Times[Optional[Pattern[f, Blank[]]], Pattern[x, Blank[]]]]]], Pattern[n, Blank[]]]], Pattern[p, Blank[]]]], Pattern[x, Blank[Symbol]]] := Dist[(b^IntPart[p]*(b*(c*Sin[e + f*x])^n)^FracPart[p])/(c*Sin[e + f*x])^(n*FracPart[p]), Int[ActivateTrig[u]*(c*Sin[e + f*x])^(n*p), x], x] /; FreeQ[{b, c, e, f, n, p}, x] && !IntegerQ[p] && !IntegerQ[n] && (EqQ[u, 1] || MatchQ[u, ((d_.)*(trig_)[e + f*x])^(m_.) /; FreeQ[{d, m}, x] && MemberQ[{sin, cos, tan, cot, sec, csc}, trig]])
  3209. Int[Power[Plus[Pattern[a, Blank[]], Times[Optional[Pattern[b, Blank[]]], Power[sin[Plus[Optional[Pattern[e, Blank[]]], Times[Optional[Pattern[f, Blank[]]], Pattern[x, Blank[]]]]], 4]]], Optional[Pattern[p, Blank[]]]], Pattern[x, Blank[Symbol]]] := With[{ff = FreeFactors[Tan[e + f*x], x]}, Dist[ff/f, Subst[Int[(a + 2*a*ff^2*x^2 + (a + b)*ff^4*x^4)^p/(1 + ff^2*x^2)^(2*p + 1), x], x, Tan[e + f*x]/ff], x]] /; FreeQ[{a, b, e, f}, x] && IntegerQ[p]
  3210. Int[Power[Plus[Pattern[a, Blank[]], Times[Optional[Pattern[b, Blank[]]], Power[sin[Plus[Optional[Pattern[e, Blank[]]], Times[Optional[Pattern[f, Blank[]]], Pattern[x, Blank[]]]]], 4]]], Pattern[p, Blank[]]], Pattern[x, Blank[Symbol]]] := With[{ff = FreeFactors[Tan[e + f*x], x]}, Dist[(ff*(a + b*Sin[e + f*x]^4)^p*(Sec[e + f*x]^2)^(2*p))/(f*(a + 2*a*Tan[e + f*x]^2 + (a + b)*Tan[e + f*x]^4)^p), Subst[Int[(a + 2*a*ff^2*x^2 + (a + b)*ff^4*x^4)^p/(1 + ff^2*x^2)^(2*p + 1), x], x, Tan[e + f*x]/ff], x]] /; FreeQ[{a, b, e, f, p}, x] && IntegerQ[p - 1/2]
  3211. Int[Power[Plus[Pattern[a, Blank[]], Times[Optional[Pattern[b, Blank[]]], Power[sin[Plus[Optional[Pattern[e, Blank[]]], Times[Optional[Pattern[f, Blank[]]], Pattern[x, Blank[]]]]], Pattern[n, Blank[]]]]], -1], Pattern[x, Blank[Symbol]]] := Module[{k}, Dist[2/(a*n), Sum[Int[1/(1 - Sin[e + f*x]^2/((-1)^((4*k)/n)*Rt[-(a/b), n/2])), x], {k, 1, n/2}], x]] /; FreeQ[{a, b, e, f}, x] && IntegerQ[n/2]
  3212. Int[Power[Plus[Pattern[a, Blank[]], Times[Optional[Pattern[b, Blank[]]], Power[sin[Plus[Optional[Pattern[e, Blank[]]], Times[Optional[Pattern[f, Blank[]]], Pattern[x, Blank[]]]]], Pattern[n, Blank[]]]]], Pattern[p, Blank[]]], Pattern[x, Blank[Symbol]]] := With[{ff = FreeFactors[Tan[e + f*x], x]}, Dist[ff/f, Subst[Int[(b*ff^n*x^n + a*(1 + ff^2*x^2)^(n/2))^p/(1 + ff^2*x^2)^((n*p)/2 + 1), x], x, Tan[e + f*x]/ff], x]] /; FreeQ[{a, b, e, f}, x] && IntegerQ[n/2] && IGtQ[p, 0]
  3213. Int[Power[Plus[Pattern[a, Blank[]], Times[Optional[Pattern[b, Blank[]]], Power[Times[Optional[Pattern[c, Blank[]]], sin[Plus[Optional[Pattern[e, Blank[]]], Times[Optional[Pattern[f, Blank[]]], Pattern[x, Blank[]]]]]], Pattern[n, Blank[]]]]], Pattern[p, Blank[]]], Pattern[x, Blank[Symbol]]] := Int[ExpandTrig[(a + b*(c*sin[e + f*x])^n)^p, x], x] /; FreeQ[{a, b, c, e, f, n}, x] && (IGtQ[p, 0] || (EqQ[p, -1] && IntegerQ[n]))
  3214. Int[Power[Plus[Pattern[a, Blank[]], Times[Optional[Pattern[b, Blank[]]], Power[Times[Optional[Pattern[c, Blank[]]], sin[Plus[Optional[Pattern[e, Blank[]]], Times[Optional[Pattern[f, Blank[]]], Pattern[x, Blank[]]]]]], Pattern[n, Blank[]]]]], Pattern[p, Blank[]]], Pattern[x, Blank[Symbol]]] := Unintegrable[(a + b*(c*Sin[e + f*x])^n)^p, x] /; FreeQ[{a, b, c, e, f, n, p}, x]
  3215. Int[Times[Power[sin[Plus[Optional[Pattern[e, Blank[]]], Times[Optional[Pattern[f, Blank[]]], Pattern[x, Blank[]]]]], Optional[Pattern[m, Blank[]]]], Power[Plus[Pattern[a, Blank[]], Times[Optional[Pattern[b, Blank[]]], Power[sin[Plus[Optional[Pattern[e, Blank[]]], Times[Optional[Pattern[f, Blank[]]], Pattern[x, Blank[]]]]], 4]]], Optional[Pattern[p, Blank[]]]]], Pattern[x, Blank[Symbol]]] := With[{ff = FreeFactors[Cos[e + f*x], x]}, -Dist[ff/f, Subst[Int[(1 - ff^2*x^2)^((m - 1)/2)*(a + b - 2*b*ff^2*x^2 + b*ff^4*x^4)^p, x], x, Cos[e + f*x]/ff], x]] /; FreeQ[{a, b, e, f, p}, x] && IntegerQ[(m - 1)/2]
  3216. Int[Times[Power[sin[Plus[Optional[Pattern[e, Blank[]]], Times[Optional[Pattern[f, Blank[]]], Pattern[x, Blank[]]]]], Optional[Pattern[m, Blank[]]]], Power[Plus[Pattern[a, Blank[]], Times[Optional[Pattern[b, Blank[]]], Power[sin[Plus[Optional[Pattern[e, Blank[]]], Times[Optional[Pattern[f, Blank[]]], Pattern[x, Blank[]]]]], Pattern[n, Blank[]]]]], Optional[Pattern[p, Blank[]]]]], Pattern[x, Blank[Symbol]]] := With[{ff = FreeFactors[Cos[e + f*x], x]}, -Dist[ff/f, Subst[Int[(1 - ff^2*x^2)^((m - 1)/2)*(a + b*(1 - ff^2*x^2)^(n/2))^p, x], x, Cos[e + f*x]/ff], x]] /; FreeQ[{a, b, e, f, p}, x] && IntegerQ[(m - 1)/2] && IntegerQ[n/2]
  3217. Int[Times[Power[sin[Plus[Optional[Pattern[e, Blank[]]], Times[Optional[Pattern[f, Blank[]]], Pattern[x, Blank[]]]]], Pattern[m, Blank[]]], Power[Plus[Pattern[a, Blank[]], Times[Optional[Pattern[b, Blank[]]], Power[sin[Plus[Optional[Pattern[e, Blank[]]], Times[Optional[Pattern[f, Blank[]]], Pattern[x, Blank[]]]]], 4]]], Optional[Pattern[p, Blank[]]]]], Pattern[x, Blank[Symbol]]] := With[{ff = FreeFactors[Tan[e + f*x], x]}, Dist[ff^(m + 1)/f, Subst[Int[(x^m*(a + 2*a*ff^2*x^2 + (a + b)*ff^4*x^4)^p)/(1 + ff^2*x^2)^(m/2 + 2*p + 1), x], x, Tan[e + f*x]/ff], x]] /; FreeQ[{a, b, e, f}, x] && IntegerQ[m/2] && IntegerQ[p]
  3218. Int[Times[Power[sin[Plus[Optional[Pattern[e, Blank[]]], Times[Optional[Pattern[f, Blank[]]], Pattern[x, Blank[]]]]], Pattern[m, Blank[]]], Power[Plus[Pattern[a, Blank[]], Times[Optional[Pattern[b, Blank[]]], Power[sin[Plus[Optional[Pattern[e, Blank[]]], Times[Optional[Pattern[f, Blank[]]], Pattern[x, Blank[]]]]], Pattern[n, Blank[]]]]], Optional[Pattern[p, Blank[]]]]], Pattern[x, Blank[Symbol]]] := With[{ff = FreeFactors[Tan[e + f*x], x]}, Dist[ff^(m + 1)/f, Subst[Int[(x^m*(a*(1 + ff^2*x^2)^(n/2) + b*ff^n*x^n)^p)/(1 + ff^2*x^2)^(m/2 + (n*p)/2 + 1), x], x, Tan[e + f*x]/ff], x]] /; FreeQ[{a, b, e, f}, x] && IntegerQ[m/2] && IntegerQ[n/2] && IntegerQ[p]
  3219. Int[Times[Power[sin[Plus[Optional[Pattern[e, Blank[]]], Times[Optional[Pattern[f, Blank[]]], Pattern[x, Blank[]]]]], Pattern[m, Blank[]]], Power[Plus[Pattern[a, Blank[]], Times[Optional[Pattern[b, Blank[]]], Power[sin[Plus[Optional[Pattern[e, Blank[]]], Times[Optional[Pattern[f, Blank[]]], Pattern[x, Blank[]]]]], 4]]], Pattern[p, Blank[]]]], Pattern[x, Blank[Symbol]]] := With[{ff = FreeFactors[Tan[e + f*x], x]}, Dist[(ff^(m + 1)*(a + b*Sin[e + f*x]^4)^p*(Sec[e + f*x]^2)^(2*p))/(f*Apart[a*(1 + Tan[e + f*x]^2)^2 + b*Tan[e + f*x]^4]^p), Subst[Int[(x^m*ExpandToSum[a*(1 + ff^2*x^2)^2 + b*ff^4*x^4, x]^p)/(1 + ff^2*x^2)^(m/2 + 2*p + 1), x], x, Tan[e + f*x]/ff], x]] /; FreeQ[{a, b, e, f, p}, x] && IntegerQ[m/2] && IntegerQ[p - 1/2]
  3220. Int[Times[Power[sin[Plus[Optional[Pattern[e, Blank[]]], Times[Optional[Pattern[f, Blank[]]], Pattern[x, Blank[]]]]], Optional[Pattern[m, Blank[]]]], Power[Plus[Pattern[a, Blank[]], Times[Optional[Pattern[b, Blank[]]], Power[sin[Plus[Optional[Pattern[e, Blank[]]], Times[Optional[Pattern[f, Blank[]]], Pattern[x, Blank[]]]]], Pattern[n, Blank[]]]]], Optional[Pattern[p, Blank[]]]]], Pattern[x, Blank[Symbol]]] := Int[ExpandTrig[sin[e + f*x]^m*(a + b*sin[e + f*x]^n)^p, x], x] /; FreeQ[{a, b, e, f}, x] && IntegersQ[m, p] && (EqQ[n, 4] || GtQ[p, 0] || (EqQ[p, -1] && IntegerQ[n]))
  3221. Int[Times[Power[Times[Optional[Pattern[d, Blank[]]], sin[Plus[Optional[Pattern[e, Blank[]]], Times[Optional[Pattern[f, Blank[]]], Pattern[x, Blank[]]]]]], Optional[Pattern[m, Blank[]]]], Power[Plus[Pattern[a, Blank[]], Times[Optional[Pattern[b, Blank[]]], Power[Times[Optional[Pattern[c, Blank[]]], sin[Plus[Optional[Pattern[e, Blank[]]], Times[Optional[Pattern[f, Blank[]]], Pattern[x, Blank[]]]]]], Pattern[n, Blank[]]]]], Optional[Pattern[p, Blank[]]]]], Pattern[x, Blank[Symbol]]] := Int[ExpandTrig[(d*sin[e + f*x])^m*(a + b*(c*sin[e + f*x])^n)^p, x], x] /; FreeQ[{a, b, c, d, e, f, m, n}, x] && IGtQ[p, 0]
  3222. Int[Times[Power[Times[Optional[Pattern[d, Blank[]]], sin[Plus[Optional[Pattern[e, Blank[]]], Times[Optional[Pattern[f, Blank[]]], Pattern[x, Blank[]]]]]], Optional[Pattern[m, Blank[]]]], Power[Plus[Pattern[a, Blank[]], Times[Optional[Pattern[b, Blank[]]], Power[Times[Optional[Pattern[c, Blank[]]], sin[Plus[Optional[Pattern[e, Blank[]]], Times[Optional[Pattern[f, Blank[]]], Pattern[x, Blank[]]]]]], Pattern[n, Blank[]]]]], Optional[Pattern[p, Blank[]]]]], Pattern[x, Blank[Symbol]]] := Unintegrable[(d*Sin[e + f*x])^m*(a + b*(c*Sin[e + f*x])^n)^p, x] /; FreeQ[{a, b, c, d, e, f, m, n, p}, x]
  3223. Int[Times[Power[cos[Plus[Optional[Pattern[e, Blank[]]], Times[Optional[Pattern[f, Blank[]]], Pattern[x, Blank[]]]]], Optional[Pattern[m, Blank[]]]], Power[Plus[Pattern[a, Blank[]], Times[Optional[Pattern[b, Blank[]]], Power[Times[Optional[Pattern[c, Blank[]]], sin[Plus[Optional[Pattern[e, Blank[]]], Times[Optional[Pattern[f, Blank[]]], Pattern[x, Blank[]]]]]], Pattern[n, Blank[]]]]], Optional[Pattern[p, Blank[]]]]], Pattern[x, Blank[Symbol]]] := With[{ff = FreeFactors[Sin[e + f*x], x]}, Dist[ff/f, Subst[Int[(1 - ff^2*x^2)^((m - 1)/2)*(a + b*(c*ff*x)^n)^p, x], x, Sin[e + f*x]/ff], x]] /; FreeQ[{a, b, c, e, f, n, p}, x] && IntegerQ[(m - 1)/2] && (EqQ[n, 4] || GtQ[m, 0] || IGtQ[p, 0] || IntegersQ[m, p])
  3224. Int[Times[Power[cos[Plus[Optional[Pattern[e, Blank[]]], Times[Optional[Pattern[f, Blank[]]], Pattern[x, Blank[]]]]], Pattern[m, Blank[]]], Power[Plus[Pattern[a, Blank[]], Times[Optional[Pattern[b, Blank[]]], Power[sin[Plus[Optional[Pattern[e, Blank[]]], Times[Optional[Pattern[f, Blank[]]], Pattern[x, Blank[]]]]], 4]]], Optional[Pattern[p, Blank[]]]]], Pattern[x, Blank[Symbol]]] := With[{ff = FreeFactors[Tan[e + f*x], x]}, Dist[ff/f, Subst[Int[(a + 2*a*ff^2*x^2 + (a + b)*ff^4*x^4)^p/(1 + ff^2*x^2)^(m/2 + 2*p + 1), x], x, Tan[e + f*x]/ff], x]] /; FreeQ[{a, b, e, f}, x] && IntegerQ[m/2] && IntegerQ[p]
  3225. Int[Times[Power[cos[Plus[Optional[Pattern[e, Blank[]]], Times[Optional[Pattern[f, Blank[]]], Pattern[x, Blank[]]]]], Pattern[m, Blank[]]], Power[Plus[Pattern[a, Blank[]], Times[Optional[Pattern[b, Blank[]]], Power[sin[Plus[Optional[Pattern[e, Blank[]]], Times[Optional[Pattern[f, Blank[]]], Pattern[x, Blank[]]]]], Pattern[n, Blank[]]]]], Optional[Pattern[p, Blank[]]]]], Pattern[x, Blank[Symbol]]] := With[{ff = FreeFactors[Tan[e + f*x], x]}, Dist[ff/f, Subst[Int[(b*ff^n*x^n + a*(1 + ff^2*x^2)^(n/2))^p/(1 + ff^2*x^2)^(m/2 + (n*p)/2 + 1), x], x, Tan[e + f*x]/ff], x]] /; FreeQ[{a, b, e, f}, x] && IntegerQ[m/2] && IntegerQ[n/2] && IntegerQ[p]
  3226. Int[Times[Power[cos[Plus[Optional[Pattern[e, Blank[]]], Times[Optional[Pattern[f, Blank[]]], Pattern[x, Blank[]]]]], Pattern[m, Blank[]]], Power[Plus[Pattern[a, Blank[]], Times[Optional[Pattern[b, Blank[]]], Power[sin[Plus[Optional[Pattern[e, Blank[]]], Times[Optional[Pattern[f, Blank[]]], Pattern[x, Blank[]]]]], Pattern[n, Blank[]]]]], -1]], Pattern[x, Blank[Symbol]]] := Int[Expand[(1 - Sin[e + f*x]^2)^(m/2)/(a + b*Sin[e + f*x]^n), x], x] /; FreeQ[{a, b, e, f}, x] && IGtQ[m/2, 0] && IntegerQ[(n - 1)/2]
  3227. Int[Times[Power[Times[cos[Plus[Optional[Pattern[e, Blank[]]], Times[Optional[Pattern[f, Blank[]]], Pattern[x, Blank[]]]]], Optional[Pattern[d, Blank[]]]], Optional[Pattern[m, Blank[]]]], Power[Plus[Pattern[a, Blank[]], Times[Optional[Pattern[b, Blank[]]], Power[Times[Optional[Pattern[c, Blank[]]], sin[Plus[Optional[Pattern[e, Blank[]]], Times[Optional[Pattern[f, Blank[]]], Pattern[x, Blank[]]]]]], Pattern[n, Blank[]]]]], Optional[Pattern[p, Blank[]]]]], Pattern[x, Blank[Symbol]]] := Int[ExpandTrig[(d*cos[e + f*x])^m*(a + b*(c*sin[e + f*x])^n)^p, x], x] /; FreeQ[{a, b, c, d, e, f, m, n}, x] && IGtQ[p, 0]
  3228. Int[Times[Power[Times[cos[Plus[Optional[Pattern[e, Blank[]]], Times[Optional[Pattern[f, Blank[]]], Pattern[x, Blank[]]]]], Optional[Pattern[d, Blank[]]]], Optional[Pattern[m, Blank[]]]], Power[Plus[Pattern[a, Blank[]], Times[Optional[Pattern[b, Blank[]]], Power[Times[Optional[Pattern[c, Blank[]]], sin[Plus[Optional[Pattern[e, Blank[]]], Times[Optional[Pattern[f, Blank[]]], Pattern[x, Blank[]]]]]], Pattern[n, Blank[]]]]], Optional[Pattern[p, Blank[]]]]], Pattern[x, Blank[Symbol]]] := Unintegrable[(d*Cos[e + f*x])^m*(a + b*(c*Sin[e + f*x])^n)^p, x] /; FreeQ[{a, b, c, d, e, f, m, n, p}, x]
  3229. Int[Times[Power[Plus[Pattern[a, Blank[]], Times[Optional[Pattern[b, Blank[]]], Power[sin[Plus[Optional[Pattern[e, Blank[]]], Times[Optional[Pattern[f, Blank[]]], Pattern[x, Blank[]]]]], Pattern[n, Blank[]]]]], Optional[Pattern[p, Blank[]]]], Power[tan[Plus[Optional[Pattern[e, Blank[]]], Times[Optional[Pattern[f, Blank[]]], Pattern[x, Blank[]]]]], Optional[Pattern[m, Blank[]]]]], Pattern[x, Blank[Symbol]]] := With[{ff = FreeFactors[Sin[e + f*x]^2, x]}, Dist[ff^((m + 1)/2)/(2*f), Subst[Int[(x^((m - 1)/2)*(a + b*ff^(n/2)*x^(n/2))^p)/(1 - ff*x)^((m + 1)/2), x], x, Sin[e + f*x]^2/ff], x]] /; FreeQ[{a, b, e, f, p}, x] && IntegerQ[(m - 1)/2] && IntegerQ[n/2]
  3230. Int[Times[Power[Plus[Pattern[a, Blank[]], Times[Optional[Pattern[b, Blank[]]], Power[Times[Optional[Pattern[c, Blank[]]], sin[Plus[Optional[Pattern[e, Blank[]]], Times[Optional[Pattern[f, Blank[]]], Pattern[x, Blank[]]]]]], Pattern[n, Blank[]]]]], Optional[Pattern[p, Blank[]]]], Power[tan[Plus[Optional[Pattern[e, Blank[]]], Times[Optional[Pattern[f, Blank[]]], Pattern[x, Blank[]]]]], Optional[Pattern[m, Blank[]]]]], Pattern[x, Blank[Symbol]]] := With[{ff = FreeFactors[Sin[e + f*x], x]}, Dist[ff^(m + 1)/f, Subst[Int[(x^m*(a + b*(c*ff*x)^n)^p)/(1 - ff^2*x^2)^((m + 1)/2), x], x, Sin[e + f*x]/ff], x]] /; FreeQ[{a, b, c, e, f, n, p}, x] && ILtQ[(m - 1)/2, 0]
  3231. Int[Times[Power[Plus[Pattern[a, Blank[]], Times[Optional[Pattern[b, Blank[]]], Power[sin[Plus[Optional[Pattern[e, Blank[]]], Times[Optional[Pattern[f, Blank[]]], Pattern[x, Blank[]]]]], 4]]], Optional[Pattern[p, Blank[]]]], Power[Times[Optional[Pattern[d, Blank[]]], tan[Plus[Optional[Pattern[e, Blank[]]], Times[Optional[Pattern[f, Blank[]]], Pattern[x, Blank[]]]]]], Pattern[m, Blank[]]]], Pattern[x, Blank[Symbol]]] := With[{ff = FreeFactors[Tan[e + f*x], x]}, Dist[ff/f, Subst[Int[((d*ff*x)^m*ExpandToSum[a*(1 + ff^2*x^2)^2 + b*ff^4*x^4, x]^p)/(1 + ff^2*x^2)^(2*p + 1), x], x, Tan[e + f*x]/ff], x]] /; FreeQ[{a, b, d, e, f, m}, x] && IntegerQ[p]
  3232. Int[Times[Power[Plus[Pattern[a, Blank[]], Times[Optional[Pattern[b, Blank[]]], Power[sin[Plus[Optional[Pattern[e, Blank[]]], Times[Optional[Pattern[f, Blank[]]], Pattern[x, Blank[]]]]], 4]]], Pattern[p, Blank[]]], Power[Times[Optional[Pattern[d, Blank[]]], tan[Plus[Optional[Pattern[e, Blank[]]], Times[Optional[Pattern[f, Blank[]]], Pattern[x, Blank[]]]]]], Pattern[m, Blank[]]]], Pattern[x, Blank[Symbol]]] := With[{ff = FreeFactors[Tan[e + f*x], x]}, Dist[(ff*(a + b*Sin[e + f*x]^4)^p*(Sec[e + f*x]^2)^(2*p))/(f*Apart[a*(1 + Tan[e + f*x]^2)^2 + b*Tan[e + f*x]^4]^p), Subst[Int[((d*ff*x)^m*ExpandToSum[a*(1 + ff^2*x^2)^2 + b*ff^4*x^4, x]^p)/(1 + ff^2*x^2)^(2*p + 1), x], x, Tan[e + f*x]/ff], x]] /; FreeQ[{a, b, d, e, f, m}, x] && IntegerQ[p - 1/2]
  3233. Int[Times[Power[Plus[Pattern[a, Blank[]], Times[Optional[Pattern[b, Blank[]]], Power[sin[Plus[Optional[Pattern[e, Blank[]]], Times[Optional[Pattern[f, Blank[]]], Pattern[x, Blank[]]]]], Pattern[n, Blank[]]]]], Optional[Pattern[p, Blank[]]]], Power[Times[Optional[Pattern[d, Blank[]]], tan[Plus[Optional[Pattern[e, Blank[]]], Times[Optional[Pattern[f, Blank[]]], Pattern[x, Blank[]]]]]], Pattern[m, Blank[]]]], Pattern[x, Blank[Symbol]]] := With[{ff = FreeFactors[Tan[e + f*x], x]}, Dist[ff^(m + 1)/f, Subst[Int[((d*x)^m*(b*ff^n*x^n + a*(1 + ff^2*x^2)^(n/2))^p)/(1 + ff^2*x^2)^((n*p)/2 + 1), x], x, Tan[e + f*x]/ff], x]] /; FreeQ[{a, b, d, e, f, m}, x] && IntegerQ[n/2] && IGtQ[p, 0]
  3234. Int[Times[Power[Plus[Pattern[a, Blank[]], Times[Optional[Pattern[b, Blank[]]], Power[Times[Optional[Pattern[c, Blank[]]], sin[Plus[Optional[Pattern[e, Blank[]]], Times[Optional[Pattern[f, Blank[]]], Pattern[x, Blank[]]]]]], Pattern[n, Blank[]]]]], Optional[Pattern[p, Blank[]]]], Power[Times[Optional[Pattern[d, Blank[]]], tan[Plus[Optional[Pattern[e, Blank[]]], Times[Optional[Pattern[f, Blank[]]], Pattern[x, Blank[]]]]]], Optional[Pattern[m, Blank[]]]]], Pattern[x, Blank[Symbol]]] := Int[ExpandTrig[(d*tan[e + f*x])^m*(a + b*(c*sin[e + f*x])^n)^p, x], x] /; FreeQ[{a, b, c, d, e, f, m, n}, x] && IGtQ[p, 0]
  3235. Int[Times[Power[Plus[Pattern[a, Blank[]], Times[Optional[Pattern[b, Blank[]]], Power[Times[Optional[Pattern[c, Blank[]]], sin[Plus[Optional[Pattern[e, Blank[]]], Times[Optional[Pattern[f, Blank[]]], Pattern[x, Blank[]]]]]], Pattern[n, Blank[]]]]], Optional[Pattern[p, Blank[]]]], Power[Times[Optional[Pattern[d, Blank[]]], tan[Plus[Optional[Pattern[e, Blank[]]], Times[Optional[Pattern[f, Blank[]]], Pattern[x, Blank[]]]]]], Optional[Pattern[m, Blank[]]]]], Pattern[x, Blank[Symbol]]] := Unintegrable[(a + b*(c*Sin[e + f*x])^n)^p*(d*Tan[e + f*x])^m, x] /; FreeQ[{a, b, c, d, e, f, m, n, p}, x]
  3236. Int[Times[Power[Times[cot[Plus[Optional[Pattern[e, Blank[]]], Times[Optional[Pattern[f, Blank[]]], Pattern[x, Blank[]]]]], Optional[Pattern[d, Blank[]]]], Pattern[m, Blank[]]], Power[Plus[Pattern[a, Blank[]], Times[Optional[Pattern[b, Blank[]]], Power[Times[Optional[Pattern[c, Blank[]]], sin[Plus[Optional[Pattern[e, Blank[]]], Times[Optional[Pattern[f, Blank[]]], Pattern[x, Blank[]]]]]], Pattern[n, Blank[]]]]], Pattern[p, Blank[]]]], Pattern[x, Blank[Symbol]]] := Dist[(d*Cot[e + f*x])^FracPart[m]*(Tan[e + f*x]/d)^FracPart[m], Int[(a + b*(c*Sin[e + f*x])^n)^p/(Tan[e + f*x]/d)^m, x], x] /; FreeQ[{a, b, c, d, e, f, m, n, p}, x] && !IntegerQ[m]
  3237. Int[Times[Power[Times[Optional[Pattern[d, Blank[]]], sec[Plus[Optional[Pattern[e, Blank[]]], Times[Optional[Pattern[f, Blank[]]], Pattern[x, Blank[]]]]]], Pattern[m, Blank[]]], Power[Plus[Pattern[a, Blank[]], Times[Optional[Pattern[b, Blank[]]], Power[Times[Optional[Pattern[c, Blank[]]], sin[Plus[Optional[Pattern[e, Blank[]]], Times[Optional[Pattern[f, Blank[]]], Pattern[x, Blank[]]]]]], Pattern[n, Blank[]]]]], Pattern[p, Blank[]]]], Pattern[x, Blank[Symbol]]] := Dist[(d*Sec[e + f*x])^FracPart[m]*(Cos[e + f*x]/d)^FracPart[m], Int[(a + b*(c*Sin[e + f*x])^n)^p/(Cos[e + f*x]/d)^m, x], x] /; FreeQ[{a, b, c, d, e, f, m, n, p}, x] && !IntegerQ[m]
  3238. Int[Times[Power[Times[csc[Plus[Optional[Pattern[e, Blank[]]], Times[Optional[Pattern[f, Blank[]]], Pattern[x, Blank[]]]]], Optional[Pattern[d, Blank[]]]], Pattern[m, Blank[]]], Power[Plus[Pattern[a, Blank[]], Times[Optional[Pattern[b, Blank[]]], Power[sin[Plus[Optional[Pattern[e, Blank[]]], Times[Optional[Pattern[f, Blank[]]], Pattern[x, Blank[]]]]], Optional[Pattern[n, Blank[]]]]]], Optional[Pattern[p, Blank[]]]]], Pattern[x, Blank[Symbol]]] := Dist[d^(n*p), Int[(d*Csc[e + f*x])^(m - n*p)*(b + a*Csc[e + f*x]^n)^p, x], x] /; FreeQ[{a, b, d, e, f, m, n, p}, x] && !IntegerQ[m] && IntegersQ[n, p]
  3239. Int[Times[Power[Times[csc[Plus[Optional[Pattern[e, Blank[]]], Times[Optional[Pattern[f, Blank[]]], Pattern[x, Blank[]]]]], Optional[Pattern[d, Blank[]]]], Pattern[m, Blank[]]], Power[Plus[Pattern[a, Blank[]], Times[Optional[Pattern[b, Blank[]]], Power[Times[Optional[Pattern[c, Blank[]]], sin[Plus[Optional[Pattern[e, Blank[]]], Times[Optional[Pattern[f, Blank[]]], Pattern[x, Blank[]]]]]], Pattern[n, Blank[]]]]], Pattern[p, Blank[]]]], Pattern[x, Blank[Symbol]]] := Dist[(d*Csc[e + f*x])^FracPart[m]*(Sin[e + f*x]/d)^FracPart[m], Int[(a + b*(c*Sin[e + f*x])^n)^p/(Sin[e + f*x]/d)^m, x], x] /; FreeQ[{a, b, c, d, e, f, m, n, p}, x] && !IntegerQ[m]
  3240. Int[Times[Power[sin[Plus[Optional[Pattern[d, Blank[]]], Times[Optional[Pattern[e, Blank[]]], Pattern[x, Blank[]]]]], Pattern[m, Blank[]]], Power[Plus[Times[Power[cos[Plus[Optional[Pattern[d, Blank[]]], Times[Optional[Pattern[e, Blank[]]], Pattern[x, Blank[]]]]], Pattern[p, Blank[]]], Optional[Pattern[b, Blank[]]]], Pattern[a, Blank[]], Times[Optional[Pattern[c, Blank[]]], Power[sin[Plus[Optional[Pattern[d, Blank[]]], Times[Optional[Pattern[e, Blank[]]], Pattern[x, Blank[]]]]], Pattern[q, Blank[]]]]], Pattern[n, Blank[]]]], Pattern[x, Blank[Symbol]]] := Module[{f = FreeFactors[Cot[d + e*x], x]}, -Dist[f/e, Subst[Int[ExpandToSum[c + b*(1 + f^2*x^2)^(q/2 - p/2) + a*(1 + f^2*x^2)^(q/2), x]^n/(1 + f^2*x^2)^(m/2 + (n*q)/2 + 1), x], x, Cot[d + e*x]/f], x]] /; FreeQ[{a, b, c, d, e}, x] && IntegerQ[m/2] && IntegerQ[p/2] && IntegerQ[q/2] && IntegerQ[n] && GtQ[p, 0] && LeQ[p, q]
  3241. Int[Times[Power[cos[Plus[Optional[Pattern[d, Blank[]]], Times[Optional[Pattern[e, Blank[]]], Pattern[x, Blank[]]]]], Pattern[m, Blank[]]], Power[Plus[Times[Power[cos[Plus[Optional[Pattern[d, Blank[]]], Times[Optional[Pattern[e, Blank[]]], Pattern[x, Blank[]]]]], Pattern[q, Blank[]]], Optional[Pattern[c, Blank[]]]], Pattern[a, Blank[]], Times[Optional[Pattern[b, Blank[]]], Power[sin[Plus[Optional[Pattern[d, Blank[]]], Times[Optional[Pattern[e, Blank[]]], Pattern[x, Blank[]]]]], Pattern[p, Blank[]]]]], Pattern[n, Blank[]]]], Pattern[x, Blank[Symbol]]] := Module[{f = FreeFactors[Tan[d + e*x], x]}, Dist[f/e, Subst[Int[ExpandToSum[c + b*(1 + f^2*x^2)^(q/2 - p/2) + a*(1 + f^2*x^2)^(q/2), x]^n/(1 + f^2*x^2)^(m/2 + (n*q)/2 + 1), x], x, Tan[d + e*x]/f], x]] /; FreeQ[{a, b, c, d, e}, x] && IntegerQ[m/2] && IntegerQ[p/2] && IntegerQ[q/2] && IntegerQ[n] && GtQ[p, 0] && LeQ[p, q]
  3242. Int[Times[Power[sin[Plus[Optional[Pattern[d, Blank[]]], Times[Optional[Pattern[e, Blank[]]], Pattern[x, Blank[]]]]], Pattern[m, Blank[]]], Power[Plus[Times[Power[cos[Plus[Optional[Pattern[d, Blank[]]], Times[Optional[Pattern[e, Blank[]]], Pattern[x, Blank[]]]]], Pattern[p, Blank[]]], Optional[Pattern[b, Blank[]]]], Pattern[a, Blank[]], Times[Optional[Pattern[c, Blank[]]], Power[sin[Plus[Optional[Pattern[d, Blank[]]], Times[Optional[Pattern[e, Blank[]]], Pattern[x, Blank[]]]]], Pattern[q, Blank[]]]]], Pattern[n, Blank[]]]], Pattern[x, Blank[Symbol]]] := Module[{f = FreeFactors[Cot[d + e*x], x]}, -Dist[f/e, Subst[Int[ExpandToSum[a*(1 + f^2*x^2)^(p/2) + b*f^p*x^p + c*(1 + f^2*x^2)^(p/2 - q/2), x]^n/(1 + f^2*x^2)^(m/2 + (n*p)/2 + 1), x], x, Cot[d + e*x]/f], x]] /; FreeQ[{a, b, c, d, e}, x] && IntegerQ[m/2] && IntegerQ[p/2] && IntegerQ[q/2] && IntegerQ[n] && LtQ[0, q, p]
  3243. Int[Times[Power[cos[Plus[Optional[Pattern[d, Blank[]]], Times[Optional[Pattern[e, Blank[]]], Pattern[x, Blank[]]]]], Pattern[m, Blank[]]], Power[Plus[Times[Power[cos[Plus[Optional[Pattern[d, Blank[]]], Times[Optional[Pattern[e, Blank[]]], Pattern[x, Blank[]]]]], Pattern[q, Blank[]]], Optional[Pattern[c, Blank[]]]], Pattern[a, Blank[]], Times[Optional[Pattern[b, Blank[]]], Power[sin[Plus[Optional[Pattern[d, Blank[]]], Times[Optional[Pattern[e, Blank[]]], Pattern[x, Blank[]]]]], Pattern[p, Blank[]]]]], Pattern[n, Blank[]]]], Pattern[x, Blank[Symbol]]] := Module[{f = FreeFactors[Tan[d + e*x], x]}, Dist[f/e, Subst[Int[ExpandToSum[a*(1 + f^2*x^2)^(p/2) + b*f^p*x^p + c*(1 + f^2*x^2)^(p/2 - q/2), x]^n/(1 + f^2*x^2)^(m/2 + (n*p)/2 + 1), x], x, Tan[d + e*x]/f], x]] /; FreeQ[{a, b, c, d, e}, x] && IntegerQ[m/2] && IntegerQ[p/2] && IntegerQ[q/2] && IntegerQ[n] && LtQ[0, q, p]
  3244. Int[Power[Plus[Optional[Pattern[a, Blank[]]], Times[Optional[Pattern[b, Blank[]]], Power[sin[Plus[Optional[Pattern[d, Blank[]]], Times[Optional[Pattern[e, Blank[]]], Pattern[x, Blank[]]]]], Optional[Pattern[n, Blank[]]]]], Times[Optional[Pattern[c, Blank[]]], Power[sin[Plus[Optional[Pattern[d, Blank[]]], Times[Optional[Pattern[e, Blank[]]], Pattern[x, Blank[]]]]], Optional[Pattern[n2, Blank[]]]]]], Optional[Pattern[p, Blank[]]]], Pattern[x, Blank[Symbol]]] := Dist[1/(4^p*c^p), Int[(b + 2*c*Sin[d + e*x]^n)^(2*p), x], x] /; FreeQ[{a, b, c, d, e, n}, x] && EqQ[n2, 2*n] && EqQ[b^2 - 4*a*c, 0] && IntegerQ[p]
  3245. Int[Power[Plus[Optional[Pattern[a, Blank[]]], Times[Power[cos[Plus[Optional[Pattern[d, Blank[]]], Times[Optional[Pattern[e, Blank[]]], Pattern[x, Blank[]]]]], Optional[Pattern[n, Blank[]]]], Optional[Pattern[b, Blank[]]]], Times[Power[cos[Plus[Optional[Pattern[d, Blank[]]], Times[Optional[Pattern[e, Blank[]]], Pattern[x, Blank[]]]]], Optional[Pattern[n2, Blank[]]]], Optional[Pattern[c, Blank[]]]]], Optional[Pattern[p, Blank[]]]], Pattern[x, Blank[Symbol]]] := Dist[1/(4^p*c^p), Int[(b + 2*c*Cos[d + e*x]^n)^(2*p), x], x] /; FreeQ[{a, b, c, d, e, n}, x] && EqQ[n2, 2*n] && EqQ[b^2 - 4*a*c, 0] && IntegerQ[p]
  3246. Int[Power[Plus[Optional[Pattern[a, Blank[]]], Times[Optional[Pattern[b, Blank[]]], Power[sin[Plus[Optional[Pattern[d, Blank[]]], Times[Optional[Pattern[e, Blank[]]], Pattern[x, Blank[]]]]], Optional[Pattern[n, Blank[]]]]], Times[Optional[Pattern[c, Blank[]]], Power[sin[Plus[Optional[Pattern[d, Blank[]]], Times[Optional[Pattern[e, Blank[]]], Pattern[x, Blank[]]]]], Optional[Pattern[n2, Blank[]]]]]], Pattern[p, Blank[]]], Pattern[x, Blank[Symbol]]] := Dist[(a + b*Sin[d + e*x]^n + c*Sin[d + e*x]^(2*n))^p/(b + 2*c*Sin[d + e*x]^n)^(2*p), Int[u*(b + 2*c*Sin[d + e*x]^n)^(2*p), x], x] /; FreeQ[{a, b, c, d, e, n, p}, x] && EqQ[n2, 2*n] && EqQ[b^2 - 4*a*c, 0] && !IntegerQ[p]
  3247. Int[Power[Plus[Optional[Pattern[a, Blank[]]], Times[Power[cos[Plus[Optional[Pattern[d, Blank[]]], Times[Optional[Pattern[e, Blank[]]], Pattern[x, Blank[]]]]], Optional[Pattern[n, Blank[]]]], Optional[Pattern[b, Blank[]]]], Times[Power[cos[Plus[Optional[Pattern[d, Blank[]]], Times[Optional[Pattern[e, Blank[]]], Pattern[x, Blank[]]]]], Optional[Pattern[n2, Blank[]]]], Optional[Pattern[c, Blank[]]]]], Pattern[p, Blank[]]], Pattern[x, Blank[Symbol]]] := Dist[(a + b*Cos[d + e*x]^n + c*Cos[d + e*x]^(2*n))^p/(b + 2*c*Cos[d + e*x]^n)^(2*p), Int[u*(b + 2*c*Cos[d + e*x]^n)^(2*p), x], x] /; FreeQ[{a, b, c, d, e, n, p}, x] && EqQ[n2, 2*n] && EqQ[b^2 - 4*a*c, 0] && !IntegerQ[p]
  3248. Int[Power[Plus[Optional[Pattern[a, Blank[]]], Times[Optional[Pattern[b, Blank[]]], Power[sin[Plus[Optional[Pattern[d, Blank[]]], Times[Optional[Pattern[e, Blank[]]], Pattern[x, Blank[]]]]], Optional[Pattern[n, Blank[]]]]], Times[Optional[Pattern[c, Blank[]]], Power[sin[Plus[Optional[Pattern[d, Blank[]]], Times[Optional[Pattern[e, Blank[]]], Pattern[x, Blank[]]]]], Optional[Pattern[n2, Blank[]]]]]], -1], Pattern[x, Blank[Symbol]]] := Module[{q = Rt[b^2 - 4*a*c, 2]}, Dist[(2*c)/q, Int[1/(b - q + 2*c*Sin[d + e*x]^n), x], x] - Dist[(2*c)/q, Int[1/(b + q + 2*c*Sin[d + e*x]^n), x], x]] /; FreeQ[{a, b, c, d, e, n}, x] && EqQ[n2, 2*n] && NeQ[b^2 - 4*a*c, 0]
  3249. Int[Power[Plus[Optional[Pattern[a, Blank[]]], Times[Power[cos[Plus[Optional[Pattern[d, Blank[]]], Times[Optional[Pattern[e, Blank[]]], Pattern[x, Blank[]]]]], Optional[Pattern[n, Blank[]]]], Optional[Pattern[b, Blank[]]]], Times[Power[cos[Plus[Optional[Pattern[d, Blank[]]], Times[Optional[Pattern[e, Blank[]]], Pattern[x, Blank[]]]]], Optional[Pattern[n2, Blank[]]]], Optional[Pattern[c, Blank[]]]]], -1], Pattern[x, Blank[Symbol]]] := Module[{q = Rt[b^2 - 4*a*c, 2]}, Dist[(2*c)/q, Int[1/(b - q + 2*c*Cos[d + e*x]^n), x], x] - Dist[(2*c)/q, Int[1/(b + q + 2*c*Cos[d + e*x]^n), x], x]] /; FreeQ[{a, b, c, d, e, n}, x] && EqQ[n2, 2*n] && NeQ[b^2 - 4*a*c, 0]
  3250. Int[Times[Power[sin[Plus[Optional[Pattern[d, Blank[]]], Times[Optional[Pattern[e, Blank[]]], Pattern[x, Blank[]]]]], Optional[Pattern[m, Blank[]]]], Power[Plus[Optional[Pattern[a, Blank[]]], Times[Optional[Pattern[b, Blank[]]], Power[sin[Plus[Optional[Pattern[d, Blank[]]], Times[Optional[Pattern[e, Blank[]]], Pattern[x, Blank[]]]]], Optional[Pattern[n, Blank[]]]]], Times[Optional[Pattern[c, Blank[]]], Power[sin[Plus[Optional[Pattern[d, Blank[]]], Times[Optional[Pattern[e, Blank[]]], Pattern[x, Blank[]]]]], Optional[Pattern[n2, Blank[]]]]]], Pattern[p, Blank[]]]], Pattern[x, Blank[Symbol]]] := Dist[1/(4^p*c^p), Int[Sin[d + e*x]^m*(b + 2*c*Sin[d + e*x]^n)^(2*p), x], x] /; FreeQ[{a, b, c, d, e, m, n}, x] && EqQ[n2, 2*n] && EqQ[b^2 - 4*a*c, 0] && IntegerQ[p]
  3251. Int[Times[Power[cos[Plus[Optional[Pattern[d, Blank[]]], Times[Optional[Pattern[e, Blank[]]], Pattern[x, Blank[]]]]], Optional[Pattern[m, Blank[]]]], Power[Plus[Optional[Pattern[a, Blank[]]], Times[Power[cos[Plus[Optional[Pattern[d, Blank[]]], Times[Optional[Pattern[e, Blank[]]], Pattern[x, Blank[]]]]], Optional[Pattern[n, Blank[]]]], Optional[Pattern[b, Blank[]]]], Times[Power[cos[Plus[Optional[Pattern[d, Blank[]]], Times[Optional[Pattern[e, Blank[]]], Pattern[x, Blank[]]]]], Optional[Pattern[n2, Blank[]]]], Optional[Pattern[c, Blank[]]]]], Pattern[p, Blank[]]]], Pattern[x, Blank[Symbol]]] := Dist[1/(4^p*c^p), Int[Cos[d + e*x]^m*(b + 2*c*Cos[d + e*x]^n)^(2*p), x], x] /; FreeQ[{a, b, c, d, e, m, n}, x] && EqQ[n2, 2*n] && EqQ[b^2 - 4*a*c, 0] && IntegerQ[p]
  3252. Int[Times[Power[sin[Plus[Optional[Pattern[d, Blank[]]], Times[Optional[Pattern[e, Blank[]]], Pattern[x, Blank[]]]]], Optional[Pattern[m, Blank[]]]], Power[Plus[Optional[Pattern[a, Blank[]]], Times[Optional[Pattern[b, Blank[]]], Power[sin[Plus[Optional[Pattern[d, Blank[]]], Times[Optional[Pattern[e, Blank[]]], Pattern[x, Blank[]]]]], Optional[Pattern[n, Blank[]]]]], Times[Optional[Pattern[c, Blank[]]], Power[sin[Plus[Optional[Pattern[d, Blank[]]], Times[Optional[Pattern[e, Blank[]]], Pattern[x, Blank[]]]]], Optional[Pattern[n2, Blank[]]]]]], Pattern[p, Blank[]]]], Pattern[x, Blank[Symbol]]] := Dist[(a + b*Sin[d + e*x]^n + c*Sin[d + e*x]^(2*n))^p/(b + 2*c*Sin[d + e*x]^n)^(2*p), Int[Sin[d + e*x]^m*(b + 2*c*Sin[d + e*x]^n)^(2*p), x], x] /; FreeQ[{a, b, c, d, e, m, n, p}, x] && EqQ[n2, 2*n] && EqQ[b^2 - 4*a*c, 0] && !IntegerQ[p]
  3253. Int[Times[Power[cos[Plus[Optional[Pattern[d, Blank[]]], Times[Optional[Pattern[e, Blank[]]], Pattern[x, Blank[]]]]], Optional[Pattern[m, Blank[]]]], Power[Plus[Optional[Pattern[a, Blank[]]], Times[Power[cos[Plus[Optional[Pattern[d, Blank[]]], Times[Optional[Pattern[e, Blank[]]], Pattern[x, Blank[]]]]], Optional[Pattern[n, Blank[]]]], Optional[Pattern[b, Blank[]]]], Times[Power[cos[Plus[Optional[Pattern[d, Blank[]]], Times[Optional[Pattern[e, Blank[]]], Pattern[x, Blank[]]]]], Optional[Pattern[n2, Blank[]]]], Optional[Pattern[c, Blank[]]]]], Pattern[p, Blank[]]]], Pattern[x, Blank[Symbol]]] := Dist[(a + b*Cos[d + e*x]^n + c*Cos[d + e*x]^(2*n))^p/(b + 2*c*Cos[d + e*x]^n)^(2*p), Int[Cos[d + e*x]^m*(b + 2*c*Cos[d + e*x]^n)^(2*p), x], x] /; FreeQ[{a, b, c, d, e, m, n, p}, x] && EqQ[n2, 2*n] && EqQ[b^2 - 4*a*c, 0] && !IntegerQ[p]
  3254. Int[Times[Power[sin[Plus[Optional[Pattern[d, Blank[]]], Times[Optional[Pattern[e, Blank[]]], Pattern[x, Blank[]]]]], Pattern[m, Blank[]]], Power[Plus[Optional[Pattern[a, Blank[]]], Times[Optional[Pattern[b, Blank[]]], Power[sin[Plus[Optional[Pattern[d, Blank[]]], Times[Optional[Pattern[e, Blank[]]], Pattern[x, Blank[]]]]], Pattern[n, Blank[]]]], Times[Optional[Pattern[c, Blank[]]], Power[sin[Plus[Optional[Pattern[d, Blank[]]], Times[Optional[Pattern[e, Blank[]]], Pattern[x, Blank[]]]]], Pattern[n2, Blank[]]]]], Pattern[p, Blank[]]]], Pattern[x, Blank[Symbol]]] := Module[{f = FreeFactors[Cot[d + e*x], x]}, -Dist[f/e, Subst[Int[ExpandToSum[c + b*(1 + x^2)^(n/2) + a*(1 + x^2)^n, x]^p/(1 + f^2*x^2)^(m/2 + n*p + 1), x], x, Cot[d + e*x]/f], x]] /; FreeQ[{a, b, c, d, e}, x] && EqQ[n2, 2*n] && IntegerQ[m/2] && NeQ[b^2 - 4*a*c, 0] && IntegerQ[n/2] && IntegerQ[p]
  3255. Int[Times[Power[cos[Plus[Optional[Pattern[d, Blank[]]], Times[Optional[Pattern[e, Blank[]]], Pattern[x, Blank[]]]]], Optional[Pattern[m, Blank[]]]], Power[Plus[Optional[Pattern[a, Blank[]]], Times[Power[cos[Plus[Optional[Pattern[d, Blank[]]], Times[Optional[Pattern[e, Blank[]]], Pattern[x, Blank[]]]]], Pattern[n, Blank[]]], Optional[Pattern[b, Blank[]]]], Times[Power[cos[Plus[Optional[Pattern[d, Blank[]]], Times[Optional[Pattern[e, Blank[]]], Pattern[x, Blank[]]]]], Pattern[n2, Blank[]]], Optional[Pattern[c, Blank[]]]]], Pattern[p, Blank[]]]], Pattern[x, Blank[Symbol]]] := Module[{f = FreeFactors[Tan[d + e*x], x]}, Dist[f/e, Subst[Int[ExpandToSum[c + b*(1 + x^2)^(n/2) + a*(1 + x^2)^n, x]^p/(1 + f^2*x^2)^(m/2 + n*p + 1), x], x, Tan[d + e*x]/f], x]] /; FreeQ[{a, b, c, d, e}, x] && EqQ[n2, 2*n] && IntegerQ[m/2] && NeQ[b^2 - 4*a*c, 0] && IntegerQ[n/2] && IntegerQ[p]
  3256. Int[Times[Power[sin[Plus[Optional[Pattern[d, Blank[]]], Times[Optional[Pattern[e, Blank[]]], Pattern[x, Blank[]]]]], Optional[Pattern[m, Blank[]]]], Power[Plus[Optional[Pattern[a, Blank[]]], Times[Optional[Pattern[b, Blank[]]], Power[sin[Plus[Optional[Pattern[d, Blank[]]], Times[Optional[Pattern[e, Blank[]]], Pattern[x, Blank[]]]]], Optional[Pattern[n, Blank[]]]]], Times[Optional[Pattern[c, Blank[]]], Power[sin[Plus[Optional[Pattern[d, Blank[]]], Times[Optional[Pattern[e, Blank[]]], Pattern[x, Blank[]]]]], Optional[Pattern[n2, Blank[]]]]]], Pattern[p, Blank[]]]], Pattern[x, Blank[Symbol]]] := Int[ExpandTrig[sin[d + e*x]^m*(a + b*sin[d + e*x]^n + c*sin[d + e*x]^(2*n))^p, x], x] /; FreeQ[{a, b, c, d, e}, x] && EqQ[n2, 2*n] && NeQ[b^2 - 4*a*c, 0] && IntegersQ[m, n, p]
  3257. Int[Times[Power[cos[Plus[Optional[Pattern[d, Blank[]]], Times[Optional[Pattern[e, Blank[]]], Pattern[x, Blank[]]]]], Optional[Pattern[m, Blank[]]]], Power[Plus[Optional[Pattern[a, Blank[]]], Times[Power[cos[Plus[Optional[Pattern[d, Blank[]]], Times[Optional[Pattern[e, Blank[]]], Pattern[x, Blank[]]]]], Optional[Pattern[n, Blank[]]]], Optional[Pattern[b, Blank[]]]], Times[Power[cos[Plus[Optional[Pattern[d, Blank[]]], Times[Optional[Pattern[e, Blank[]]], Pattern[x, Blank[]]]]], Optional[Pattern[n2, Blank[]]]], Optional[Pattern[c, Blank[]]]]], Pattern[p, Blank[]]]], Pattern[x, Blank[Symbol]]] := Int[ExpandTrig[cos[d + e*x]^m*(a + b*cos[d + e*x]^n + c*cos[d + e*x]^(2*n))^p, x], x] /; FreeQ[{a, b, c, d, e}, x] && EqQ[n2, 2*n] && NeQ[b^2 - 4*a*c, 0] && IntegersQ[m, n, p]
  3258. Int[Times[Power[cos[Plus[Optional[Pattern[d, Blank[]]], Times[Optional[Pattern[e, Blank[]]], Pattern[x, Blank[]]]]], Optional[Pattern[m, Blank[]]]], Power[Plus[Optional[Pattern[a, Blank[]]], Times[Optional[Pattern[b, Blank[]]], Power[Times[Optional[Pattern[f, Blank[]]], sin[Plus[Optional[Pattern[d, Blank[]]], Times[Optional[Pattern[e, Blank[]]], Pattern[x, Blank[]]]]]], Optional[Pattern[n, Blank[]]]]], Times[Optional[Pattern[c, Blank[]]], Power[Times[Optional[Pattern[f, Blank[]]], sin[Plus[Optional[Pattern[d, Blank[]]], Times[Optional[Pattern[e, Blank[]]], Pattern[x, Blank[]]]]]], Optional[Pattern[n2, Blank[]]]]]], Optional[Pattern[p, Blank[]]]]], Pattern[x, Blank[Symbol]]] := Module[{g = FreeFactors[Sin[d + e*x], x]}, Dist[g/e, Subst[Int[(1 - g^2*x^2)^((m - 1)/2)*(a + b*(f*g*x)^n + c*(f*g*x)^(2*n))^p, x], x, Sin[d + e*x]/g], x]] /; FreeQ[{a, b, c, d, e, f, n, p}, x] && EqQ[n2, 2*n] && IntegerQ[(m - 1)/2]
  3259. Int[Times[Power[Plus[Optional[Pattern[a, Blank[]]], Times[Optional[Pattern[b, Blank[]]], Power[Times[cos[Plus[Optional[Pattern[d, Blank[]]], Times[Optional[Pattern[e, Blank[]]], Pattern[x, Blank[]]]]], Optional[Pattern[f, Blank[]]]], Optional[Pattern[n, Blank[]]]]], Times[Optional[Pattern[c, Blank[]]], Power[Times[cos[Plus[Optional[Pattern[d, Blank[]]], Times[Optional[Pattern[e, Blank[]]], Pattern[x, Blank[]]]]], Optional[Pattern[f, Blank[]]]], Optional[Pattern[n2, Blank[]]]]]], Optional[Pattern[p, Blank[]]]], Power[sin[Plus[Optional[Pattern[d, Blank[]]], Times[Optional[Pattern[e, Blank[]]], Pattern[x, Blank[]]]]], Optional[Pattern[m, Blank[]]]]], Pattern[x, Blank[Symbol]]] := Module[{g = FreeFactors[Cos[d + e*x], x]}, -Dist[g/e, Subst[Int[(1 - g^2*x^2)^((m - 1)/2)*(a + b*(f*g*x)^n + c*(f*g*x)^(2*n))^p, x], x, Cos[d + e*x]/g], x]] /; FreeQ[{a, b, c, d, e, f, n, p}, x] && EqQ[n2, 2*n] && IntegerQ[(m - 1)/2]
  3260. Int[Times[Power[cos[Plus[Optional[Pattern[d, Blank[]]], Times[Optional[Pattern[e, Blank[]]], Pattern[x, Blank[]]]]], Pattern[m, Blank[]]], Power[Plus[Optional[Pattern[a, Blank[]]], Times[Optional[Pattern[b, Blank[]]], Power[sin[Plus[Optional[Pattern[d, Blank[]]], Times[Optional[Pattern[e, Blank[]]], Pattern[x, Blank[]]]]], Optional[Pattern[n, Blank[]]]]], Times[Optional[Pattern[c, Blank[]]], Power[sin[Plus[Optional[Pattern[d, Blank[]]], Times[Optional[Pattern[e, Blank[]]], Pattern[x, Blank[]]]]], Optional[Pattern[n2, Blank[]]]]]], Optional[Pattern[p, Blank[]]]]], Pattern[x, Blank[Symbol]]] := Dist[1/(4^p*c^p), Int[Cos[d + e*x]^m*(b + 2*c*Sin[d + e*x]^n)^(2*p), x], x] /; FreeQ[{a, b, c, d, e, m, n}, x] && EqQ[n2, 2*n] && !IntegerQ[(m - 1)/2] && EqQ[b^2 - 4*a*c, 0] && IntegerQ[p]
  3261. Int[Times[Power[Plus[Optional[Pattern[a, Blank[]]], Times[Power[cos[Plus[Optional[Pattern[d, Blank[]]], Times[Optional[Pattern[e, Blank[]]], Pattern[x, Blank[]]]]], Optional[Pattern[n, Blank[]]]], Optional[Pattern[b, Blank[]]]], Times[Power[cos[Plus[Optional[Pattern[d, Blank[]]], Times[Optional[Pattern[e, Blank[]]], Pattern[x, Blank[]]]]], Optional[Pattern[n2, Blank[]]]], Optional[Pattern[c, Blank[]]]]], Optional[Pattern[p, Blank[]]]], Power[sin[Plus[Optional[Pattern[d, Blank[]]], Times[Optional[Pattern[e, Blank[]]], Pattern[x, Blank[]]]]], Pattern[m, Blank[]]]], Pattern[x, Blank[Symbol]]] := Dist[1/(4^p*c^p), Int[Sin[d + e*x]^m*(b + 2*c*Cos[d + e*x]^n)^(2*p), x], x] /; FreeQ[{a, b, c, d, e, m, n}, x] && EqQ[n2, 2*n] && !IntegerQ[(m - 1)/2] && EqQ[b^2 - 4*a*c, 0] && IntegerQ[p]
  3262. Int[Times[Power[cos[Plus[Optional[Pattern[d, Blank[]]], Times[Optional[Pattern[e, Blank[]]], Pattern[x, Blank[]]]]], Pattern[m, Blank[]]], Power[Plus[Optional[Pattern[a, Blank[]]], Times[Optional[Pattern[b, Blank[]]], Power[sin[Plus[Optional[Pattern[d, Blank[]]], Times[Optional[Pattern[e, Blank[]]], Pattern[x, Blank[]]]]], Optional[Pattern[n, Blank[]]]]], Times[Optional[Pattern[c, Blank[]]], Power[sin[Plus[Optional[Pattern[d, Blank[]]], Times[Optional[Pattern[e, Blank[]]], Pattern[x, Blank[]]]]], Optional[Pattern[n2, Blank[]]]]]], Pattern[p, Blank[]]]], Pattern[x, Blank[Symbol]]] := Dist[(a + b*Sin[d + e*x]^n + c*Sin[d + e*x]^(2*n))^p/(b + 2*c*Sin[d + e*x]^n)^(2*p), Int[Cos[d + e*x]^m*(b + 2*c*Sin[d + e*x]^n)^(2*p), x], x] /; FreeQ[{a, b, c, d, e, m, n, p}, x] && EqQ[n2, 2*n] && !IntegerQ[(m - 1)/2] && EqQ[b^2 - 4*a*c, 0] && !IntegerQ[p]
  3263. Int[Times[Power[Plus[Optional[Pattern[a, Blank[]]], Times[Power[cos[Plus[Optional[Pattern[d, Blank[]]], Times[Optional[Pattern[e, Blank[]]], Pattern[x, Blank[]]]]], Optional[Pattern[n, Blank[]]]], Optional[Pattern[b, Blank[]]]], Times[Power[cos[Plus[Optional[Pattern[d, Blank[]]], Times[Optional[Pattern[e, Blank[]]], Pattern[x, Blank[]]]]], Optional[Pattern[n2, Blank[]]]], Optional[Pattern[c, Blank[]]]]], Pattern[p, Blank[]]], Power[sin[Plus[Optional[Pattern[d, Blank[]]], Times[Optional[Pattern[e, Blank[]]], Pattern[x, Blank[]]]]], Pattern[m, Blank[]]]], Pattern[x, Blank[Symbol]]] := Dist[(a + b*Cos[d + e*x]^n + c*Cos[d + e*x]^(2*n))^p/(b + 2*c*Cos[d + e*x]^n)^(2*p), Int[Sin[d + e*x]^m*(b + 2*c*Cos[d + e*x]^n)^(2*p), x], x] /; FreeQ[{a, b, c, d, e, m, n, p}, x] && EqQ[n2, 2*n] && !IntegerQ[(m - 1)/2] && EqQ[b^2 - 4*a*c, 0] && !IntegerQ[p]
  3264. Int[Times[Power[cos[Plus[Optional[Pattern[d, Blank[]]], Times[Optional[Pattern[e, Blank[]]], Pattern[x, Blank[]]]]], Pattern[m, Blank[]]], Power[Plus[Optional[Pattern[a, Blank[]]], Times[Optional[Pattern[b, Blank[]]], Power[sin[Plus[Optional[Pattern[d, Blank[]]], Times[Optional[Pattern[e, Blank[]]], Pattern[x, Blank[]]]]], Pattern[n, Blank[]]]], Times[Optional[Pattern[c, Blank[]]], Power[sin[Plus[Optional[Pattern[d, Blank[]]], Times[Optional[Pattern[e, Blank[]]], Pattern[x, Blank[]]]]], Pattern[n2, Blank[]]]]], Optional[Pattern[p, Blank[]]]]], Pattern[x, Blank[Symbol]]] := Module[{f = FreeFactors[Cot[d + e*x], x]}, -Dist[f^(m + 1)/e, Subst[Int[(x^m*ExpandToSum[c + b*(1 + x^2)^(n/2) + a*(1 + x^2)^n, x]^p)/(1 + f^2*x^2)^(m/2 + n*p + 1), x], x, Cot[d + e*x]/f], x]] /; FreeQ[{a, b, c, d, e}, x] && EqQ[n2, 2*n] && IntegerQ[m/2] && NeQ[b^2 - 4*a*c, 0] && IntegerQ[n/2] && IntegerQ[p]
  3265. Int[Times[Power[Plus[Optional[Pattern[a, Blank[]]], Times[Power[cos[Plus[Optional[Pattern[d, Blank[]]], Times[Optional[Pattern[e, Blank[]]], Pattern[x, Blank[]]]]], Pattern[n, Blank[]]], Optional[Pattern[b, Blank[]]]], Times[Power[cos[Plus[Optional[Pattern[d, Blank[]]], Times[Optional[Pattern[e, Blank[]]], Pattern[x, Blank[]]]]], Pattern[n2, Blank[]]], Optional[Pattern[c, Blank[]]]]], Optional[Pattern[p, Blank[]]]], Power[sin[Plus[Optional[Pattern[d, Blank[]]], Times[Optional[Pattern[e, Blank[]]], Pattern[x, Blank[]]]]], Optional[Pattern[m, Blank[]]]]], Pattern[x, Blank[Symbol]]] := Module[{f = FreeFactors[Tan[d + e*x], x]}, Dist[f^(m + 1)/e, Subst[Int[(x^m*ExpandToSum[c + b*(1 + x^2)^(n/2) + a*(1 + x^2)^n, x]^p)/(1 + f^2*x^2)^(m/2 + n*p + 1), x], x, Tan[d + e*x]/f], x]] /; FreeQ[{a, b, c, d, e}, x] && EqQ[n2, 2*n] && IntegerQ[m/2] && NeQ[b^2 - 4*a*c, 0] && IntegerQ[n/2] && IntegerQ[p]
  3266. Int[Times[Power[cos[Plus[Optional[Pattern[d, Blank[]]], Times[Optional[Pattern[e, Blank[]]], Pattern[x, Blank[]]]]], Optional[Pattern[m, Blank[]]]], Power[Plus[Optional[Pattern[a, Blank[]]], Times[Optional[Pattern[b, Blank[]]], Power[sin[Plus[Optional[Pattern[d, Blank[]]], Times[Optional[Pattern[e, Blank[]]], Pattern[x, Blank[]]]]], Optional[Pattern[n, Blank[]]]]], Times[Optional[Pattern[c, Blank[]]], Power[sin[Plus[Optional[Pattern[d, Blank[]]], Times[Optional[Pattern[e, Blank[]]], Pattern[x, Blank[]]]]], Optional[Pattern[n2, Blank[]]]]]], Optional[Pattern[p, Blank[]]]]], Pattern[x, Blank[Symbol]]] := Int[ExpandTrig[(1 - sin[d + e*x]^2)^(m/2)*(a + b*sin[d + e*x]^n + c*sin[d + e*x]^(2*n))^p, x], x] /; FreeQ[{a, b, c, d, e}, x] && EqQ[n2, 2*n] && IntegerQ[m/2] && NeQ[b^2 - 4*a*c, 0] && IntegersQ[n, p]
  3267. Int[Times[Power[Plus[Optional[Pattern[a, Blank[]]], Times[Power[cos[Plus[Optional[Pattern[d, Blank[]]], Times[Optional[Pattern[e, Blank[]]], Pattern[x, Blank[]]]]], Optional[Pattern[n, Blank[]]]], Optional[Pattern[b, Blank[]]]], Times[Power[cos[Plus[Optional[Pattern[d, Blank[]]], Times[Optional[Pattern[e, Blank[]]], Pattern[x, Blank[]]]]], Optional[Pattern[n2, Blank[]]]], Optional[Pattern[c, Blank[]]]]], Optional[Pattern[p, Blank[]]]], Power[sin[Plus[Optional[Pattern[d, Blank[]]], Times[Optional[Pattern[e, Blank[]]], Pattern[x, Blank[]]]]], Optional[Pattern[m, Blank[]]]]], Pattern[x, Blank[Symbol]]] := Int[ExpandTrig[(1 - cos[d + e*x]^2)^(m/2)*(a + b*cos[d + e*x]^n + c*cos[d + e*x]^(2*n))^p, x], x] /; FreeQ[{a, b, c, d, e}, x] && EqQ[n2, 2*n] && IntegerQ[m/2] && NeQ[b^2 - 4*a*c, 0] && IntegersQ[n, p]
  3268. Int[Times[Power[Plus[Pattern[a, Blank[]], Times[Optional[Pattern[c, Blank[]]], Power[Times[Optional[Pattern[f, Blank[]]], sin[Plus[Optional[Pattern[d, Blank[]]], Times[Optional[Pattern[e, Blank[]]], Pattern[x, Blank[]]]]]], Optional[Pattern[n2, Blank[]]]]], Times[Optional[Pattern[b, Blank[]]], Power[Times[Optional[Pattern[f, Blank[]]], sin[Plus[Optional[Pattern[d, Blank[]]], Times[Optional[Pattern[e, Blank[]]], Pattern[x, Blank[]]]]]], Pattern[n, Blank[]]]]], Optional[Pattern[p, Blank[]]]], Power[tan[Plus[Optional[Pattern[d, Blank[]]], Times[Optional[Pattern[e, Blank[]]], Pattern[x, Blank[]]]]], Optional[Pattern[m, Blank[]]]]], Pattern[x, Blank[Symbol]]] := Module[{g = FreeFactors[Sin[d + e*x], x]}, Dist[g^(m + 1)/e, Subst[Int[(x^m*(a + b*(f*g*x)^n + c*(f*g*x)^(2*n))^p)/(1 - g^2*x^2)^((m + 1)/2), x], x, Sin[d + e*x]/g], x]] /; FreeQ[{a, b, c, d, e, f, n}, x] && IntegerQ[(m - 1)/2] && IntegerQ[2*p]
  3269. Int[Times[Power[cot[Plus[Optional[Pattern[d, Blank[]]], Times[Optional[Pattern[e, Blank[]]], Pattern[x, Blank[]]]]], Optional[Pattern[m, Blank[]]]], Power[Plus[Times[Optional[Pattern[c, Blank[]]], Power[Times[cos[Plus[Optional[Pattern[d, Blank[]]], Times[Optional[Pattern[e, Blank[]]], Pattern[x, Blank[]]]]], Optional[Pattern[f, Blank[]]]], Optional[Pattern[n2, Blank[]]]]], Times[Optional[Pattern[b, Blank[]]], Power[Times[cos[Plus[Optional[Pattern[d, Blank[]]], Times[Optional[Pattern[e, Blank[]]], Pattern[x, Blank[]]]]], Optional[Pattern[f, Blank[]]]], Pattern[n, Blank[]]]], Pattern[a, Blank[]]], Optional[Pattern[p, Blank[]]]]], Pattern[x, Blank[Symbol]]] := Module[{g = FreeFactors[Cos[d + e*x], x]}, -Dist[g^(m + 1)/e, Subst[Int[(x^m*(a + b*(f*g*x)^n + c*(f*g*x)^(2*n))^p)/(1 - g^2*x^2)^((m + 1)/2), x], x, Cos[d + e*x]/g], x]] /; FreeQ[{a, b, c, d, e, f, n}, x] && IntegerQ[(m - 1)/2] && IntegerQ[2*p]
  3270. Int[Times[Power[Plus[Optional[Pattern[a, Blank[]]], Times[Optional[Pattern[b, Blank[]]], Power[sin[Plus[Optional[Pattern[d, Blank[]]], Times[Optional[Pattern[e, Blank[]]], Pattern[x, Blank[]]]]], Optional[Pattern[n, Blank[]]]]], Times[Optional[Pattern[c, Blank[]]], Power[sin[Plus[Optional[Pattern[d, Blank[]]], Times[Optional[Pattern[e, Blank[]]], Pattern[x, Blank[]]]]], Optional[Pattern[n2, Blank[]]]]]], Optional[Pattern[p, Blank[]]]], Power[tan[Plus[Optional[Pattern[d, Blank[]]], Times[Optional[Pattern[e, Blank[]]], Pattern[x, Blank[]]]]], Pattern[m, Blank[]]]], Pattern[x, Blank[Symbol]]] := Dist[1/(4^p*c^p), Int[Tan[d + e*x]^m*(b + 2*c*Sin[d + e*x]^n)^(2*p), x], x] /; FreeQ[{a, b, c, d, e, m, n}, x] && EqQ[n2, 2*n] && !IntegerQ[(m - 1)/2] && EqQ[b^2 - 4*a*c, 0] && IntegerQ[p]
  3271. Int[Times[Power[cot[Plus[Optional[Pattern[d, Blank[]]], Times[Optional[Pattern[e, Blank[]]], Pattern[x, Blank[]]]]], Pattern[m, Blank[]]], Power[Plus[Optional[Pattern[a, Blank[]]], Times[Power[cos[Plus[Optional[Pattern[d, Blank[]]], Times[Optional[Pattern[e, Blank[]]], Pattern[x, Blank[]]]]], Optional[Pattern[n, Blank[]]]], Optional[Pattern[b, Blank[]]]], Times[Power[cos[Plus[Optional[Pattern[d, Blank[]]], Times[Optional[Pattern[e, Blank[]]], Pattern[x, Blank[]]]]], Optional[Pattern[n2, Blank[]]]], Optional[Pattern[c, Blank[]]]]], Optional[Pattern[p, Blank[]]]]], Pattern[x, Blank[Symbol]]] := Dist[1/(4^p*c^p), Int[Cot[d + e*x]^m*(b + 2*c*Cos[d + e*x]^n)^(2*p), x], x] /; FreeQ[{a, b, c, d, e, m, n}, x] && EqQ[n2, 2*n] && !IntegerQ[(m - 1)/2] && EqQ[b^2 - 4*a*c, 0] && IntegerQ[p]
  3272. Int[Times[Power[Plus[Optional[Pattern[a, Blank[]]], Times[Optional[Pattern[b, Blank[]]], Power[sin[Plus[Optional[Pattern[d, Blank[]]], Times[Optional[Pattern[e, Blank[]]], Pattern[x, Blank[]]]]], Optional[Pattern[n, Blank[]]]]], Times[Optional[Pattern[c, Blank[]]], Power[sin[Plus[Optional[Pattern[d, Blank[]]], Times[Optional[Pattern[e, Blank[]]], Pattern[x, Blank[]]]]], Optional[Pattern[n2, Blank[]]]]]], Pattern[p, Blank[]]], Power[tan[Plus[Optional[Pattern[d, Blank[]]], Times[Optional[Pattern[e, Blank[]]], Pattern[x, Blank[]]]]], Pattern[m, Blank[]]]], Pattern[x, Blank[Symbol]]] := Dist[(a + b*Sin[d + e*x]^n + c*Sin[d + e*x]^(2*n))^p/(b + 2*c*Sin[d + e*x]^n)^(2*p), Int[Tan[d + e*x]^m*(b + 2*c*Sin[d + e*x]^n)^(2*p), x], x] /; FreeQ[{a, b, c, d, e, m, n, p}, x] && EqQ[n2, 2*n] && !IntegerQ[(m - 1)/2] && EqQ[b^2 - 4*a*c, 0] && !IntegerQ[p]
  3273. Int[Times[Power[cot[Plus[Optional[Pattern[d, Blank[]]], Times[Optional[Pattern[e, Blank[]]], Pattern[x, Blank[]]]]], Pattern[m, Blank[]]], Power[Plus[Optional[Pattern[a, Blank[]]], Times[Power[cos[Plus[Optional[Pattern[d, Blank[]]], Times[Optional[Pattern[e, Blank[]]], Pattern[x, Blank[]]]]], Optional[Pattern[n, Blank[]]]], Optional[Pattern[b, Blank[]]]], Times[Power[cos[Plus[Optional[Pattern[d, Blank[]]], Times[Optional[Pattern[e, Blank[]]], Pattern[x, Blank[]]]]], Optional[Pattern[n2, Blank[]]]], Optional[Pattern[c, Blank[]]]]], Pattern[p, Blank[]]]], Pattern[x, Blank[Symbol]]] := Dist[(a + b*Cos[d + e*x]^n + c*Cos[d + e*x]^(2*n))^p/(b + 2*c*Cos[d + e*x]^n)^(2*p), Int[Cot[d + e*x]^m*(b + 2*c*Cos[d + e*x]^n)^(2*p), x], x] /; FreeQ[{a, b, c, d, e, m, n, p}, x] && EqQ[n2, 2*n] && !IntegerQ[(m - 1)/2] && EqQ[b^2 - 4*a*c, 0] && !IntegerQ[p]
  3274. Int[Times[Power[Plus[Optional[Pattern[a, Blank[]]], Times[Optional[Pattern[b, Blank[]]], Power[sin[Plus[Optional[Pattern[d, Blank[]]], Times[Optional[Pattern[e, Blank[]]], Pattern[x, Blank[]]]]], Pattern[n, Blank[]]]], Times[Optional[Pattern[c, Blank[]]], Power[sin[Plus[Optional[Pattern[d, Blank[]]], Times[Optional[Pattern[e, Blank[]]], Pattern[x, Blank[]]]]], Pattern[n2, Blank[]]]]], Optional[Pattern[p, Blank[]]]], Power[tan[Plus[Optional[Pattern[d, Blank[]]], Times[Optional[Pattern[e, Blank[]]], Pattern[x, Blank[]]]]], Optional[Pattern[m, Blank[]]]]], Pattern[x, Blank[Symbol]]] := Module[{f = FreeFactors[Tan[d + e*x], x]}, Dist[f^(m + 1)/e, Subst[Int[(x^m*ExpandToSum[c*x^(2*n) + b*x^n*(1 + x^2)^(n/2) + a*(1 + x^2)^n, x]^p)/(1 + f^2*x^2)^(n*p + 1), x], x, Tan[d + e*x]/f], x]] /; FreeQ[{a, b, c, d, e, m}, x] && EqQ[n2, 2*n] && !IntegerQ[(m - 1)/2] && NeQ[b^2 - 4*a*c, 0] && IntegerQ[n/2] && IntegerQ[p]
  3275. Int[Times[Power[cot[Plus[Optional[Pattern[d, Blank[]]], Times[Optional[Pattern[e, Blank[]]], Pattern[x, Blank[]]]]], Optional[Pattern[m, Blank[]]]], Power[Plus[Optional[Pattern[a, Blank[]]], Times[Power[cos[Plus[Optional[Pattern[d, Blank[]]], Times[Optional[Pattern[e, Blank[]]], Pattern[x, Blank[]]]]], Pattern[n, Blank[]]], Optional[Pattern[b, Blank[]]]], Times[Power[cos[Plus[Optional[Pattern[d, Blank[]]], Times[Optional[Pattern[e, Blank[]]], Pattern[x, Blank[]]]]], Pattern[n2, Blank[]]], Optional[Pattern[c, Blank[]]]]], Optional[Pattern[p, Blank[]]]]], Pattern[x, Blank[Symbol]]] := Module[{f = FreeFactors[Cot[d + e*x], x]}, -Dist[f^(m + 1)/e, Subst[Int[(x^m*ExpandToSum[c*x^(2*n) + b*x^n*(1 + x^2)^(n/2) + a*(1 + x^2)^n, x]^p)/(1 + f^2*x^2)^(n*p + 1), x], x, Cot[d + e*x]/f], x]] /; FreeQ[{a, b, c, d, e, m}, x] && EqQ[n2, 2*n] && !IntegerQ[(m - 1)/2] && NeQ[b^2 - 4*a*c, 0] && IntegerQ[n/2] && IntegerQ[p]
  3276. Int[Times[Power[Plus[Optional[Pattern[a, Blank[]]], Times[Optional[Pattern[b, Blank[]]], Power[sin[Plus[Optional[Pattern[d, Blank[]]], Times[Optional[Pattern[e, Blank[]]], Pattern[x, Blank[]]]]], Optional[Pattern[n, Blank[]]]]], Times[Optional[Pattern[c, Blank[]]], Power[sin[Plus[Optional[Pattern[d, Blank[]]], Times[Optional[Pattern[e, Blank[]]], Pattern[x, Blank[]]]]], Optional[Pattern[n2, Blank[]]]]]], Optional[Pattern[p, Blank[]]]], Power[tan[Plus[Optional[Pattern[d, Blank[]]], Times[Optional[Pattern[e, Blank[]]], Pattern[x, Blank[]]]]], Optional[Pattern[m, Blank[]]]]], Pattern[x, Blank[Symbol]]] := Int[ExpandTrig[(sin[d + e*x]^m*(a + b*sin[d + e*x]^n + c*sin[d + e*x]^(2*n))^p)/(1 - sin[d + e*x]^2)^(m/2), x], x] /; FreeQ[{a, b, c, d, e}, x] && EqQ[n2, 2*n] && IntegerQ[m/2] && NeQ[b^2 - 4*a*c, 0] && IntegersQ[n, p]
  3277. Int[Times[Power[cot[Plus[Optional[Pattern[d, Blank[]]], Times[Optional[Pattern[e, Blank[]]], Pattern[x, Blank[]]]]], Optional[Pattern[m, Blank[]]]], Power[Plus[Optional[Pattern[a, Blank[]]], Times[Power[cos[Plus[Optional[Pattern[d, Blank[]]], Times[Optional[Pattern[e, Blank[]]], Pattern[x, Blank[]]]]], Optional[Pattern[n, Blank[]]]], Optional[Pattern[b, Blank[]]]], Times[Power[cos[Plus[Optional[Pattern[d, Blank[]]], Times[Optional[Pattern[e, Blank[]]], Pattern[x, Blank[]]]]], Optional[Pattern[n2, Blank[]]]], Optional[Pattern[c, Blank[]]]]], Optional[Pattern[p, Blank[]]]]], Pattern[x, Blank[Symbol]]] := Int[ExpandTrig[(cos[d + e*x]^m*(a + b*cos[d + e*x]^n + c*cos[d + e*x]^(2*n))^p)/(1 - cos[d + e*x]^2)^(m/2), x], x] /; FreeQ[{a, b, c, d, e}, x] && EqQ[n2, 2*n] && IntegerQ[m/2] && NeQ[b^2 - 4*a*c, 0] && IntegersQ[n, p]
  3278. Int[Times[Power[cot[Plus[Optional[Pattern[d, Blank[]]], Times[Optional[Pattern[e, Blank[]]], Pattern[x, Blank[]]]]], Optional[Pattern[m, Blank[]]]], Power[Plus[Pattern[a, Blank[]], Times[Optional[Pattern[c, Blank[]]], Power[Times[Optional[Pattern[f, Blank[]]], sin[Plus[Optional[Pattern[d, Blank[]]], Times[Optional[Pattern[e, Blank[]]], Pattern[x, Blank[]]]]]], Optional[Pattern[n2, Blank[]]]]], Times[Optional[Pattern[b, Blank[]]], Power[Times[Optional[Pattern[f, Blank[]]], sin[Plus[Optional[Pattern[d, Blank[]]], Times[Optional[Pattern[e, Blank[]]], Pattern[x, Blank[]]]]]], Pattern[n, Blank[]]]]], Optional[Pattern[p, Blank[]]]]], Pattern[x, Blank[Symbol]]] := Module[{g = FreeFactors[Sin[d + e*x], x]}, Dist[g^(m + 1)/e, Subst[Int[((1 - g^2*x^2)^((m - 1)/2)*(a + b*(f*g*x)^n + c*(f*g*x)^(2*n))^p)/x^m, x], x, Sin[d + e*x]/g], x]] /; FreeQ[{a, b, c, d, e, f, n}, x] && IntegerQ[(m - 1)/2] && IntegerQ[2*p]
  3279. Int[Times[Power[Plus[Times[Optional[Pattern[c, Blank[]]], Power[Times[cos[Plus[Optional[Pattern[d, Blank[]]], Times[Optional[Pattern[e, Blank[]]], Pattern[x, Blank[]]]]], Optional[Pattern[f, Blank[]]]], Optional[Pattern[n2, Blank[]]]]], Times[Optional[Pattern[b, Blank[]]], Power[Times[cos[Plus[Optional[Pattern[d, Blank[]]], Times[Optional[Pattern[e, Blank[]]], Pattern[x, Blank[]]]]], Optional[Pattern[f, Blank[]]]], Pattern[n, Blank[]]]], Pattern[a, Blank[]]], Optional[Pattern[p, Blank[]]]], Power[tan[Plus[Optional[Pattern[d, Blank[]]], Times[Optional[Pattern[e, Blank[]]], Pattern[x, Blank[]]]]], Optional[Pattern[m, Blank[]]]]], Pattern[x, Blank[Symbol]]] := Module[{g = FreeFactors[Cos[d + e*x], x]}, -Dist[g^(m + 1)/e, Subst[Int[((1 - g^2*x^2)^((m - 1)/2)*(a + b*(f*g*x)^n + c*(f*g*x)^(2*n))^p)/x^m, x], x, Cos[d + e*x]/g], x]] /; FreeQ[{a, b, c, d, e, f, n}, x] && IntegerQ[(m - 1)/2] && IntegerQ[2*p]
  3280. Int[Times[Power[cot[Plus[Optional[Pattern[d, Blank[]]], Times[Optional[Pattern[e, Blank[]]], Pattern[x, Blank[]]]]], Pattern[m, Blank[]]], Power[Plus[Optional[Pattern[a, Blank[]]], Times[Optional[Pattern[b, Blank[]]], Power[sin[Plus[Optional[Pattern[d, Blank[]]], Times[Optional[Pattern[e, Blank[]]], Pattern[x, Blank[]]]]], Optional[Pattern[n, Blank[]]]]], Times[Optional[Pattern[c, Blank[]]], Power[sin[Plus[Optional[Pattern[d, Blank[]]], Times[Optional[Pattern[e, Blank[]]], Pattern[x, Blank[]]]]], Optional[Pattern[n2, Blank[]]]]]], Optional[Pattern[p, Blank[]]]]], Pattern[x, Blank[Symbol]]] := Dist[1/(4^p*c^p), Int[Cot[d + e*x]^m*(b + 2*c*Sin[d + e*x]^n)^(2*p), x], x] /; FreeQ[{a, b, c, d, e, m, n}, x] && EqQ[n2, 2*n] && !IntegerQ[(m - 1)/2] && EqQ[b^2 - 4*a*c, 0] && IntegerQ[p]
  3281. Int[Times[Power[Plus[Optional[Pattern[a, Blank[]]], Times[Power[cos[Plus[Optional[Pattern[d, Blank[]]], Times[Optional[Pattern[e, Blank[]]], Pattern[x, Blank[]]]]], Optional[Pattern[n, Blank[]]]], Optional[Pattern[b, Blank[]]]], Times[Power[cos[Plus[Optional[Pattern[d, Blank[]]], Times[Optional[Pattern[e, Blank[]]], Pattern[x, Blank[]]]]], Optional[Pattern[n2, Blank[]]]], Optional[Pattern[c, Blank[]]]]], Optional[Pattern[p, Blank[]]]], Power[tan[Plus[Optional[Pattern[d, Blank[]]], Times[Optional[Pattern[e, Blank[]]], Pattern[x, Blank[]]]]], Pattern[m, Blank[]]]], Pattern[x, Blank[Symbol]]] := Dist[1/(4^p*c^p), Int[Tan[d + e*x]^m*(b + 2*c*Cos[d + e*x]^n)^(2*p), x], x] /; FreeQ[{a, b, c, d, e, m, n}, x] && EqQ[n2, 2*n] && !IntegerQ[(m - 1)/2] && EqQ[b^2 - 4*a*c, 0] && IntegerQ[p]
  3282. Int[Times[Power[cot[Plus[Optional[Pattern[d, Blank[]]], Times[Optional[Pattern[e, Blank[]]], Pattern[x, Blank[]]]]], Pattern[m, Blank[]]], Power[Plus[Optional[Pattern[a, Blank[]]], Times[Optional[Pattern[b, Blank[]]], Power[sin[Plus[Optional[Pattern[d, Blank[]]], Times[Optional[Pattern[e, Blank[]]], Pattern[x, Blank[]]]]], Optional[Pattern[n, Blank[]]]]], Times[Optional[Pattern[c, Blank[]]], Power[sin[Plus[Optional[Pattern[d, Blank[]]], Times[Optional[Pattern[e, Blank[]]], Pattern[x, Blank[]]]]], Optional[Pattern[n2, Blank[]]]]]], Pattern[p, Blank[]]]], Pattern[x, Blank[Symbol]]] := Dist[(a + b*Sin[d + e*x]^n + c*Sin[d + e*x]^(2*n))^p/(b + 2*c*Sin[d + e*x]^n)^(2*p), Int[Cot[d + e*x]^m*(b + 2*c*Sin[d + e*x]^n)^(2*p), x], x] /; FreeQ[{a, b, c, d, e, m, n, p}, x] && EqQ[n2, 2*n] && !IntegerQ[(m - 1)/2] && EqQ[b^2 - 4*a*c, 0] && !IntegerQ[p]
  3283. Int[Times[Power[Plus[Optional[Pattern[a, Blank[]]], Times[Power[cos[Plus[Optional[Pattern[d, Blank[]]], Times[Optional[Pattern[e, Blank[]]], Pattern[x, Blank[]]]]], Optional[Pattern[n, Blank[]]]], Optional[Pattern[b, Blank[]]]], Times[Power[cos[Plus[Optional[Pattern[d, Blank[]]], Times[Optional[Pattern[e, Blank[]]], Pattern[x, Blank[]]]]], Optional[Pattern[n2, Blank[]]]], Optional[Pattern[c, Blank[]]]]], Pattern[p, Blank[]]], Power[tan[Plus[Optional[Pattern[d, Blank[]]], Times[Optional[Pattern[e, Blank[]]], Pattern[x, Blank[]]]]], Pattern[m, Blank[]]]], Pattern[x, Blank[Symbol]]] := Dist[(a + b*Cos[d + e*x]^n + c*Cos[d + e*x]^(2*n))^p/(b + 2*c*Cos[d + e*x]^n)^(2*p), Int[Tan[d + e*x]^m*(b + 2*c*Cos[d + e*x]^n)^(2*p), x], x] /; FreeQ[{a, b, c, d, e, m, n, p}, x] && EqQ[n2, 2*n] && !IntegerQ[(m - 1)/2] && EqQ[b^2 - 4*a*c, 0] && !IntegerQ[p]
  3284. Int[Times[Power[cot[Plus[Optional[Pattern[d, Blank[]]], Times[Optional[Pattern[e, Blank[]]], Pattern[x, Blank[]]]]], Optional[Pattern[m, Blank[]]]], Power[Plus[Pattern[a, Blank[]], Times[Optional[Pattern[b, Blank[]]], Power[sin[Plus[Optional[Pattern[d, Blank[]]], Times[Optional[Pattern[e, Blank[]]], Pattern[x, Blank[]]]]], Pattern[n, Blank[]]]], Times[Optional[Pattern[c, Blank[]]], Power[sin[Plus[Optional[Pattern[d, Blank[]]], Times[Optional[Pattern[e, Blank[]]], Pattern[x, Blank[]]]]], Pattern[n2, Blank[]]]]], Optional[Pattern[p, Blank[]]]]], Pattern[x, Blank[Symbol]]] := Module[{f = FreeFactors[Cot[d + e*x], x]}, -Dist[f^(m + 1)/e, Subst[Int[(x^m*ExpandToSum[c + b*(1 + f^2*x^2)^(n/2) + a*(1 + f^2*x^2)^n, x]^p)/(1 + f^2*x^2)^(n*p + 1), x], x, Cot[d + e*x]/f], x]] /; FreeQ[{a, b, c, d, e, m}, x] && EqQ[n2, 2*n] && IntegerQ[n/2] && IntegerQ[p]
  3285. Int[Times[Power[Plus[Times[Power[cos[Plus[Optional[Pattern[d, Blank[]]], Times[Optional[Pattern[e, Blank[]]], Pattern[x, Blank[]]]]], Pattern[n, Blank[]]], Optional[Pattern[b, Blank[]]]], Times[Power[cos[Plus[Optional[Pattern[d, Blank[]]], Times[Optional[Pattern[e, Blank[]]], Pattern[x, Blank[]]]]], Pattern[n2, Blank[]]], Optional[Pattern[c, Blank[]]]], Pattern[a, Blank[]]], Optional[Pattern[p, Blank[]]]], Power[tan[Plus[Optional[Pattern[d, Blank[]]], Times[Optional[Pattern[e, Blank[]]], Pattern[x, Blank[]]]]], Optional[Pattern[m, Blank[]]]]], Pattern[x, Blank[Symbol]]] := Module[{f = FreeFactors[Tan[d + e*x], x]}, Dist[f^(m + 1)/e, Subst[Int[(x^m*ExpandToSum[c + b*(1 + f^2*x^2)^(n/2) + a*(1 + f^2*x^2)^n, x]^p)/(1 + f^2*x^2)^(n*p + 1), x], x, Tan[d + e*x]/f], x]] /; FreeQ[{a, b, c, d, e, m}, x] && EqQ[n2, 2*n] && IntegerQ[n/2] && IntegerQ[p]
  3286. Int[Times[Power[cot[Plus[Optional[Pattern[d, Blank[]]], Times[Optional[Pattern[e, Blank[]]], Pattern[x, Blank[]]]]], Optional[Pattern[m, Blank[]]]], Power[Plus[Optional[Pattern[a, Blank[]]], Times[Optional[Pattern[b, Blank[]]], Power[sin[Plus[Optional[Pattern[d, Blank[]]], Times[Optional[Pattern[e, Blank[]]], Pattern[x, Blank[]]]]], Optional[Pattern[n, Blank[]]]]], Times[Optional[Pattern[c, Blank[]]], Power[sin[Plus[Optional[Pattern[d, Blank[]]], Times[Optional[Pattern[e, Blank[]]], Pattern[x, Blank[]]]]], Optional[Pattern[n2, Blank[]]]]]], Optional[Pattern[p, Blank[]]]]], Pattern[x, Blank[Symbol]]] := Int[ExpandTrig[((1 - sin[d + e*x]^2)^(m/2)*(a + b*sin[d + e*x]^n + c*sin[d + e*x]^(2*n))^p)/sin[d + e*x]^m, x], x] /; FreeQ[{a, b, c, d, e}, x] && EqQ[n2, 2*n] && IntegerQ[m/2] && NeQ[b^2 - 4*a*c, 0] && IntegersQ[n, p]
  3287. Int[Times[Power[Plus[Optional[Pattern[a, Blank[]]], Times[Power[cos[Plus[Optional[Pattern[d, Blank[]]], Times[Optional[Pattern[e, Blank[]]], Pattern[x, Blank[]]]]], Optional[Pattern[n, Blank[]]]], Optional[Pattern[b, Blank[]]]], Times[Power[cos[Plus[Optional[Pattern[d, Blank[]]], Times[Optional[Pattern[e, Blank[]]], Pattern[x, Blank[]]]]], Optional[Pattern[n2, Blank[]]]], Optional[Pattern[c, Blank[]]]]], Optional[Pattern[p, Blank[]]]], Power[tan[Plus[Optional[Pattern[d, Blank[]]], Times[Optional[Pattern[e, Blank[]]], Pattern[x, Blank[]]]]], Optional[Pattern[m, Blank[]]]]], Pattern[x, Blank[Symbol]]] := Int[ExpandTrig[((1 - cos[d + e*x]^2)^(m/2)*(a + b*cos[d + e*x]^n + c*cos[d + e*x]^(2*n))^p)/cos[d + e*x]^m, x], x] /; FreeQ[{a, b, c, d, e}, x] && EqQ[n2, 2*n] && IntegerQ[m/2] && NeQ[b^2 - 4*a*c, 0] && IntegersQ[n, p]
  3288. Int[Times[Plus[Pattern[A, Blank[]], Times[Optional[Pattern[B, Blank[]]], sin[Plus[Optional[Pattern[d, Blank[]]], Times[Optional[Pattern[e, Blank[]]], Pattern[x, Blank[]]]]]]], Power[Plus[Pattern[a, Blank[]], Times[Optional[Pattern[b, Blank[]]], sin[Plus[Optional[Pattern[d, Blank[]]], Times[Optional[Pattern[e, Blank[]]], Pattern[x, Blank[]]]]]], Times[Optional[Pattern[c, Blank[]]], Power[sin[Plus[Optional[Pattern[d, Blank[]]], Times[Optional[Pattern[e, Blank[]]], Pattern[x, Blank[]]]]], 2]]], Pattern[n, Blank[]]]], Pattern[x, Blank[Symbol]]] := Dist[1/(4^n*c^n), Int[(A + B*Sin[d + e*x])*(b + 2*c*Sin[d + e*x])^(2*n), x], x] /; FreeQ[{a, b, c, d, e, A, B}, x] && EqQ[b^2 - 4*a*c, 0] && IntegerQ[n]
  3289. Int[Times[Power[Plus[Times[cos[Plus[Optional[Pattern[d, Blank[]]], Times[Optional[Pattern[e, Blank[]]], Pattern[x, Blank[]]]]], Optional[Pattern[b, Blank[]]]], Times[Power[cos[Plus[Optional[Pattern[d, Blank[]]], Times[Optional[Pattern[e, Blank[]]], Pattern[x, Blank[]]]]], 2], Optional[Pattern[c, Blank[]]]], Pattern[a, Blank[]]], Pattern[n, Blank[]]], Plus[Times[cos[Plus[Optional[Pattern[d, Blank[]]], Times[Optional[Pattern[e, Blank[]]], Pattern[x, Blank[]]]]], Optional[Pattern[B, Blank[]]]], Pattern[A, Blank[]]]], Pattern[x, Blank[Symbol]]] := Dist[1/(4^n*c^n), Int[(A + B*Cos[d + e*x])*(b + 2*c*Cos[d + e*x])^(2*n), x], x] /; FreeQ[{a, b, c, d, e, A, B}, x] && EqQ[b^2 - 4*a*c, 0] && IntegerQ[n]
  3290. Int[Times[Plus[Pattern[A, Blank[]], Times[Optional[Pattern[B, Blank[]]], sin[Plus[Optional[Pattern[d, Blank[]]], Times[Optional[Pattern[e, Blank[]]], Pattern[x, Blank[]]]]]]], Power[Plus[Pattern[a, Blank[]], Times[Optional[Pattern[b, Blank[]]], sin[Plus[Optional[Pattern[d, Blank[]]], Times[Optional[Pattern[e, Blank[]]], Pattern[x, Blank[]]]]]], Times[Optional[Pattern[c, Blank[]]], Power[sin[Plus[Optional[Pattern[d, Blank[]]], Times[Optional[Pattern[e, Blank[]]], Pattern[x, Blank[]]]]], 2]]], Pattern[n, Blank[]]]], Pattern[x, Blank[Symbol]]] := Dist[(a + b*Sin[d + e*x] + c*Sin[d + e*x]^2)^n/(b + 2*c*Sin[d + e*x])^(2*n), Int[(A + B*Sin[d + e*x])*(b + 2*c*Sin[d + e*x])^(2*n), x], x] /; FreeQ[{a, b, c, d, e, A, B}, x] && EqQ[b^2 - 4*a*c, 0] && !IntegerQ[n]
  3291. Int[Times[Power[Plus[Times[cos[Plus[Optional[Pattern[d, Blank[]]], Times[Optional[Pattern[e, Blank[]]], Pattern[x, Blank[]]]]], Optional[Pattern[b, Blank[]]]], Times[Power[cos[Plus[Optional[Pattern[d, Blank[]]], Times[Optional[Pattern[e, Blank[]]], Pattern[x, Blank[]]]]], 2], Optional[Pattern[c, Blank[]]]], Pattern[a, Blank[]]], Pattern[n, Blank[]]], Plus[Times[cos[Plus[Optional[Pattern[d, Blank[]]], Times[Optional[Pattern[e, Blank[]]], Pattern[x, Blank[]]]]], Optional[Pattern[B, Blank[]]]], Pattern[A, Blank[]]]], Pattern[x, Blank[Symbol]]] := Dist[(a + b*Cos[d + e*x] + c*Cos[d + e*x]^2)^n/(b + 2*c*Cos[d + e*x])^(2*n), Int[(A + B*Cos[d + e*x])*(b + 2*c*Cos[d + e*x])^(2*n), x], x] /; FreeQ[{a, b, c, d, e, A, B}, x] && EqQ[b^2 - 4*a*c, 0] && !IntegerQ[n]
  3292. Int[Times[Plus[Pattern[A, Blank[]], Times[Optional[Pattern[B, Blank[]]], sin[Plus[Optional[Pattern[d, Blank[]]], Times[Optional[Pattern[e, Blank[]]], Pattern[x, Blank[]]]]]]], Power[Plus[Optional[Pattern[a, Blank[]]], Times[Optional[Pattern[b, Blank[]]], sin[Plus[Optional[Pattern[d, Blank[]]], Times[Optional[Pattern[e, Blank[]]], Pattern[x, Blank[]]]]]], Times[Optional[Pattern[c, Blank[]]], Power[sin[Plus[Optional[Pattern[d, Blank[]]], Times[Optional[Pattern[e, Blank[]]], Pattern[x, Blank[]]]]], 2]]], -1]], Pattern[x, Blank[Symbol]]] := Module[{q = Rt[b^2 - 4*a*c, 2]}, Dist[B + (b*B - 2*A*c)/q, Int[1/(b + q + 2*c*Sin[d + e*x]), x], x] + Dist[B - (b*B - 2*A*c)/q, Int[1/(b - q + 2*c*Sin[d + e*x]), x], x]] /; FreeQ[{a, b, c, d, e, A, B}, x] && NeQ[b^2 - 4*a*c, 0]
  3293. Int[Times[Power[Plus[Optional[Pattern[a, Blank[]]], Times[cos[Plus[Optional[Pattern[d, Blank[]]], Times[Optional[Pattern[e, Blank[]]], Pattern[x, Blank[]]]]], Optional[Pattern[b, Blank[]]]], Times[Power[cos[Plus[Optional[Pattern[d, Blank[]]], Times[Optional[Pattern[e, Blank[]]], Pattern[x, Blank[]]]]], 2], Optional[Pattern[c, Blank[]]]]], -1], Plus[Times[cos[Plus[Optional[Pattern[d, Blank[]]], Times[Optional[Pattern[e, Blank[]]], Pattern[x, Blank[]]]]], Optional[Pattern[B, Blank[]]]], Pattern[A, Blank[]]]], Pattern[x, Blank[Symbol]]] := Module[{q = Rt[b^2 - 4*a*c, 2]}, Dist[B + (b*B - 2*A*c)/q, Int[1/(b + q + 2*c*Cos[d + e*x]), x], x] + Dist[B - (b*B - 2*A*c)/q, Int[1/(b - q + 2*c*Cos[d + e*x]), x], x]] /; FreeQ[{a, b, c, d, e, A, B}, x] && NeQ[b^2 - 4*a*c, 0]
  3294. Int[Times[Plus[Pattern[A, Blank[]], Times[Optional[Pattern[B, Blank[]]], sin[Plus[Optional[Pattern[d, Blank[]]], Times[Optional[Pattern[e, Blank[]]], Pattern[x, Blank[]]]]]]], Power[Plus[Optional[Pattern[a, Blank[]]], Times[Optional[Pattern[b, Blank[]]], sin[Plus[Optional[Pattern[d, Blank[]]], Times[Optional[Pattern[e, Blank[]]], Pattern[x, Blank[]]]]]], Times[Optional[Pattern[c, Blank[]]], Power[sin[Plus[Optional[Pattern[d, Blank[]]], Times[Optional[Pattern[e, Blank[]]], Pattern[x, Blank[]]]]], 2]]], Pattern[n, Blank[]]]], Pattern[x, Blank[Symbol]]] := Int[ExpandTrig[(A + B*sin[d + e*x])*(a + b*sin[d + e*x] + c*sin[d + e*x]^2)^n, x], x] /; FreeQ[{a, b, c, d, e, A, B}, x] && NeQ[b^2 - 4*a*c, 0] && IntegerQ[n]
  3295. Int[Times[Power[Plus[Optional[Pattern[a, Blank[]]], Times[cos[Plus[Optional[Pattern[d, Blank[]]], Times[Optional[Pattern[e, Blank[]]], Pattern[x, Blank[]]]]], Optional[Pattern[b, Blank[]]]], Times[Power[cos[Plus[Optional[Pattern[d, Blank[]]], Times[Optional[Pattern[e, Blank[]]], Pattern[x, Blank[]]]]], 2], Optional[Pattern[c, Blank[]]]]], Pattern[n, Blank[]]], Plus[Times[cos[Plus[Optional[Pattern[d, Blank[]]], Times[Optional[Pattern[e, Blank[]]], Pattern[x, Blank[]]]]], Optional[Pattern[B, Blank[]]]], Pattern[A, Blank[]]]], Pattern[x, Blank[Symbol]]] := Int[ExpandTrig[(A + B*cos[d + e*x])*(a + b*cos[d + e*x] + c*cos[d + e*x]^2)^n, x], x] /; FreeQ[{a, b, c, d, e, A, B}, x] && NeQ[b^2 - 4*a*c, 0] && IntegerQ[n]
  3296. Int[Times[Power[Plus[Optional[Pattern[c, Blank[]]], Times[Optional[Pattern[d, Blank[]]], Pattern[x, Blank[]]]], Optional[Pattern[m, Blank[]]]], sin[Plus[Optional[Pattern[e, Blank[]]], Times[Optional[Pattern[f, Blank[]]], Pattern[x, Blank[]]]]]], Pattern[x, Blank[Symbol]]] := -Simp[((c + d*x)^m*Cos[e + f*x])/f, x] + Dist[(d*m)/f, Int[(c + d*x)^(m - 1)*Cos[e + f*x], x], x] /; FreeQ[{c, d, e, f}, x] && GtQ[m, 0]
  3297. Int[Times[Power[Plus[Optional[Pattern[c, Blank[]]], Times[Optional[Pattern[d, Blank[]]], Pattern[x, Blank[]]]], Pattern[m, Blank[]]], sin[Plus[Optional[Pattern[e, Blank[]]], Times[Optional[Pattern[f, Blank[]]], Pattern[x, Blank[]]]]]], Pattern[x, Blank[Symbol]]] := Simp[((c + d*x)^(m + 1)*Sin[e + f*x])/(d*(m + 1)), x] - Dist[f/(d*(m + 1)), Int[(c + d*x)^(m + 1)*Cos[e + f*x], x], x] /; FreeQ[{c, d, e, f}, x] && LtQ[m, -1]
  3298. Int[Times[Power[Plus[Optional[Pattern[c, Blank[]]], Times[Optional[Pattern[d, Blank[]]], Pattern[x, Blank[]]]], -1], sin[Plus[Optional[Pattern[e, Blank[]]], Times[Complex[0, Pattern[fz, Blank[]]], Optional[Pattern[f, Blank[]]], Pattern[x, Blank[]]]]]], Pattern[x, Blank[Symbol]]] := Simp[(I*SinhIntegral[(c*f*fz)/d + f*fz*x])/d, x] /; FreeQ[{c, d, e, f, fz}, x] && EqQ[d*e - c*f*fz*I, 0]
  3299. Int[Times[Power[Plus[Optional[Pattern[c, Blank[]]], Times[Optional[Pattern[d, Blank[]]], Pattern[x, Blank[]]]], -1], sin[Plus[Optional[Pattern[e, Blank[]]], Times[Optional[Pattern[f, Blank[]]], Pattern[x, Blank[]]]]]], Pattern[x, Blank[Symbol]]] := Simp[SinIntegral[e + f*x]/d, x] /; FreeQ[{c, d, e, f}, x] && EqQ[d*e - c*f, 0]
  3300. Int[Times[Power[Plus[Optional[Pattern[c, Blank[]]], Times[Optional[Pattern[d, Blank[]]], Pattern[x, Blank[]]]], -1], sin[Plus[Optional[Pattern[e, Blank[]]], Times[Complex[0, Pattern[fz, Blank[]]], Optional[Pattern[f, Blank[]]], Pattern[x, Blank[]]]]]], Pattern[x, Blank[Symbol]]] := Simp[CoshIntegral[-((c*f*fz)/d) - f*fz*x]/d, x] /; FreeQ[{c, d, e, f, fz}, x] && EqQ[d*(e - Pi/2) - c*f*fz*I, 0] && NegQ[(c*f*fz)/d, 0]
  3301. Int[Times[Power[Plus[Optional[Pattern[c, Blank[]]], Times[Optional[Pattern[d, Blank[]]], Pattern[x, Blank[]]]], -1], sin[Plus[Optional[Pattern[e, Blank[]]], Times[Complex[0, Pattern[fz, Blank[]]], Optional[Pattern[f, Blank[]]], Pattern[x, Blank[]]]]]], Pattern[x, Blank[Symbol]]] := Simp[CoshIntegral[(c*f*fz)/d + f*fz*x]/d, x] /; FreeQ[{c, d, e, f, fz}, x] && EqQ[d*(e - Pi/2) - c*f*fz*I, 0]
  3302. Int[Times[Power[Plus[Optional[Pattern[c, Blank[]]], Times[Optional[Pattern[d, Blank[]]], Pattern[x, Blank[]]]], -1], sin[Plus[Optional[Pattern[e, Blank[]]], Times[Optional[Pattern[f, Blank[]]], Pattern[x, Blank[]]]]]], Pattern[x, Blank[Symbol]]] := Simp[CosIntegral[e - Pi/2 + f*x]/d, x] /; FreeQ[{c, d, e, f}, x] && EqQ[d*(e - Pi/2) - c*f, 0]
  3303. Int[Times[Power[Plus[Optional[Pattern[c, Blank[]]], Times[Optional[Pattern[d, Blank[]]], Pattern[x, Blank[]]]], -1], sin[Plus[Optional[Pattern[e, Blank[]]], Times[Optional[Pattern[f, Blank[]]], Pattern[x, Blank[]]]]]], Pattern[x, Blank[Symbol]]] := Dist[Cos[(d*e - c*f)/d], Int[Sin[(c*f)/d + f*x]/(c + d*x), x], x] + Dist[Sin[(d*e - c*f)/d], Int[Cos[(c*f)/d + f*x]/(c + d*x), x], x] /; FreeQ[{c, d, e, f}, x] && NeQ[d*e - c*f, 0]
  3304. Int[Times[Power[Plus[Optional[Pattern[c, Blank[]]], Times[Optional[Pattern[d, Blank[]]], Pattern[x, Blank[]]]], Rational[-1, 2]], sin[Plus[Times[Rational[1, 2], Pi], Optional[Pattern[e, Blank[]]], Times[Optional[Pattern[f, Blank[]]], Pattern[x, Blank[]]]]]], Pattern[x, Blank[Symbol]]] := Dist[2/d, Subst[Int[Cos[(f*x^2)/d], x], x, Sqrt[c + d*x]], x] /; FreeQ[{c, d, e, f}, x] && ComplexFreeQ[f] && EqQ[d*e - c*f, 0]
  3305. Int[Times[Power[Plus[Optional[Pattern[c, Blank[]]], Times[Optional[Pattern[d, Blank[]]], Pattern[x, Blank[]]]], Rational[-1, 2]], sin[Plus[Optional[Pattern[e, Blank[]]], Times[Optional[Pattern[f, Blank[]]], Pattern[x, Blank[]]]]]], Pattern[x, Blank[Symbol]]] := Dist[2/d, Subst[Int[Sin[(f*x^2)/d], x], x, Sqrt[c + d*x]], x] /; FreeQ[{c, d, e, f}, x] && ComplexFreeQ[f] && EqQ[d*e - c*f, 0]
  3306. Int[Times[Power[Plus[Optional[Pattern[c, Blank[]]], Times[Optional[Pattern[d, Blank[]]], Pattern[x, Blank[]]]], Rational[-1, 2]], sin[Plus[Optional[Pattern[e, Blank[]]], Times[Optional[Pattern[f, Blank[]]], Pattern[x, Blank[]]]]]], Pattern[x, Blank[Symbol]]] := Dist[Cos[(d*e - c*f)/d], Int[Sin[(c*f)/d + f*x]/Sqrt[c + d*x], x], x] + Dist[Sin[(d*e - c*f)/d], Int[Cos[(c*f)/d + f*x]/Sqrt[c + d*x], x], x] /; FreeQ[{c, d, e, f}, x] && ComplexFreeQ[f] && NeQ[d*e - c*f, 0]
  3307. Int[Times[Power[Plus[Optional[Pattern[c, Blank[]]], Times[Optional[Pattern[d, Blank[]]], Pattern[x, Blank[]]]], Optional[Pattern[m, Blank[]]]], sin[Plus[Optional[Pattern[e, Blank[]]], Times[Pi, Optional[Pattern[k, Blank[]]]], Times[Optional[Pattern[f, Blank[]]], Pattern[x, Blank[]]]]]], Pattern[x, Blank[Symbol]]] := Dist[I/2, Int[(c + d*x)^m/(E^(I*k*Pi)*E^(I*(e + f*x))), x], x] - Dist[I/2, Int[(c + d*x)^m*E^(I*k*Pi)*E^(I*(e + f*x)), x], x] /; FreeQ[{c, d, e, f, m}, x] && IntegerQ[2*k]
  3308. Int[Times[Power[Plus[Optional[Pattern[c, Blank[]]], Times[Optional[Pattern[d, Blank[]]], Pattern[x, Blank[]]]], Optional[Pattern[m, Blank[]]]], sin[Plus[Optional[Pattern[e, Blank[]]], Times[Optional[Pattern[f, Blank[]]], Pattern[x, Blank[]]]]]], Pattern[x, Blank[Symbol]]] := Dist[I/2, Int[(c + d*x)^m/E^(I*(e + f*x)), x], x] - Dist[I/2, Int[(c + d*x)^m*E^(I*(e + f*x)), x], x] /; FreeQ[{c, d, e, f, m}, x]
  3309. Int[Times[Power[Plus[Optional[Pattern[c, Blank[]]], Times[Optional[Pattern[d, Blank[]]], Pattern[x, Blank[]]]], Optional[Pattern[m, Blank[]]]], Power[sin[Plus[Optional[Pattern[e, Blank[]]], Times[Rational[1, 2], Optional[Pattern[f, Blank[]]], Pattern[x, Blank[]]]]], 2]], Pattern[x, Blank[Symbol]]] := Dist[1/2, Int[(c + d*x)^m, x], x] - Dist[1/2, Int[(c + d*x)^m*Cos[2*e + f*x], x], x] /; FreeQ[{c, d, e, f, m}, x]
  3310. Int[Times[Plus[Optional[Pattern[c, Blank[]]], Times[Optional[Pattern[d, Blank[]]], Pattern[x, Blank[]]]], Power[Times[Optional[Pattern[b, Blank[]]], sin[Plus[Optional[Pattern[e, Blank[]]], Times[Optional[Pattern[f, Blank[]]], Pattern[x, Blank[]]]]]], Pattern[n, Blank[]]]], Pattern[x, Blank[Symbol]]] := Simp[(d*(b*Sin[e + f*x])^n)/(f^2*n^2), x] + (Dist[(b^2*(n - 1))/n, Int[(c + d*x)*(b*Sin[e + f*x])^(n - 2), x], x] - Simp[(b*(c + d*x)*Cos[e + f*x]*(b*Sin[e + f*x])^(n - 1))/(f*n), x]) /; FreeQ[{b, c, d, e, f}, x] && GtQ[n, 1]
  3311. Int[Times[Power[Plus[Optional[Pattern[c, Blank[]]], Times[Optional[Pattern[d, Blank[]]], Pattern[x, Blank[]]]], Pattern[m, Blank[]]], Power[Times[Optional[Pattern[b, Blank[]]], sin[Plus[Optional[Pattern[e, Blank[]]], Times[Optional[Pattern[f, Blank[]]], Pattern[x, Blank[]]]]]], Pattern[n, Blank[]]]], Pattern[x, Blank[Symbol]]] := Simp[(d*m*(c + d*x)^(m - 1)*(b*Sin[e + f*x])^n)/(f^2*n^2), x] + (Dist[(b^2*(n - 1))/n, Int[(c + d*x)^m*(b*Sin[e + f*x])^(n - 2), x], x] - Dist[(d^2*m*(m - 1))/(f^2*n^2), Int[(c + d*x)^(m - 2)*(b*Sin[e + f*x])^n, x], x] - Simp[(b*(c + d*x)^m*Cos[e + f*x]*(b*Sin[e + f*x])^(n - 1))/(f*n), x]) /; FreeQ[{b, c, d, e, f}, x] && GtQ[n, 1] && GtQ[m, 1]
  3312. Int[Times[Power[Plus[Optional[Pattern[c, Blank[]]], Times[Optional[Pattern[d, Blank[]]], Pattern[x, Blank[]]]], Pattern[m, Blank[]]], Power[sin[Plus[Optional[Pattern[e, Blank[]]], Times[Optional[Pattern[f, Blank[]]], Pattern[x, Blank[]]]]], Pattern[n, Blank[]]]], Pattern[x, Blank[Symbol]]] := Int[ExpandTrigReduce[(c + d*x)^m, Sin[e + f*x]^n, x], x] /; FreeQ[{c, d, e, f, m}, x] && IGtQ[n, 1] && ( !RationalQ[m] || (GeQ[m, -1] && LtQ[m, 1]))
  3313. Int[Times[Power[Plus[Optional[Pattern[c, Blank[]]], Times[Optional[Pattern[d, Blank[]]], Pattern[x, Blank[]]]], Pattern[m, Blank[]]], Power[sin[Plus[Optional[Pattern[e, Blank[]]], Times[Optional[Pattern[f, Blank[]]], Pattern[x, Blank[]]]]], Pattern[n, Blank[]]]], Pattern[x, Blank[Symbol]]] := Simp[((c + d*x)^(m + 1)*Sin[e + f*x]^n)/(d*(m + 1)), x] - Dist[(f*n)/(d*(m + 1)), Int[ExpandTrigReduce[(c + d*x)^(m + 1), Cos[e + f*x]*Sin[e + f*x]^(n - 1), x], x], x] /; FreeQ[{c, d, e, f, m}, x] && IGtQ[n, 1] && GeQ[m, -2] && LtQ[m, -1]
  3314. Int[Times[Power[Plus[Optional[Pattern[c, Blank[]]], Times[Optional[Pattern[d, Blank[]]], Pattern[x, Blank[]]]], Pattern[m, Blank[]]], Power[Times[Optional[Pattern[b, Blank[]]], sin[Plus[Optional[Pattern[e, Blank[]]], Times[Optional[Pattern[f, Blank[]]], Pattern[x, Blank[]]]]]], Pattern[n, Blank[]]]], Pattern[x, Blank[Symbol]]] := Simp[((c + d*x)^(m + 1)*(b*Sin[e + f*x])^n)/(d*(m + 1)), x] + (Dist[(b^2*f^2*n*(n - 1))/(d^2*(m + 1)*(m + 2)), Int[(c + d*x)^(m + 2)*(b*Sin[e + f*x])^(n - 2), x], x] - Dist[(f^2*n^2)/(d^2*(m + 1)*(m + 2)), Int[(c + d*x)^(m + 2)*(b*Sin[e + f*x])^n, x], x] - Simp[(b*f*n*(c + d*x)^(m + 2)*Cos[e + f*x]*(b*Sin[e + f*x])^(n - 1))/(d^2*(m + 1)*(m + 2)), x]) /; FreeQ[{b, c, d, e, f}, x] && GtQ[n, 1] && LtQ[m, -2]
  3315. Int[Times[Plus[Optional[Pattern[c, Blank[]]], Times[Optional[Pattern[d, Blank[]]], Pattern[x, Blank[]]]], Power[Times[Optional[Pattern[b, Blank[]]], sin[Plus[Optional[Pattern[e, Blank[]]], Times[Optional[Pattern[f, Blank[]]], Pattern[x, Blank[]]]]]], Pattern[n, Blank[]]]], Pattern[x, Blank[Symbol]]] := Simp[((c + d*x)*Cos[e + f*x]*(b*Sin[e + f*x])^(n + 1))/(b*f*(n + 1)), x] + (Dist[(n + 2)/(b^2*(n + 1)), Int[(c + d*x)*(b*Sin[e + f*x])^(n + 2), x], x] - Simp[(d*(b*Sin[e + f*x])^(n + 2))/(b^2*f^2*(n + 1)*(n + 2)), x]) /; FreeQ[{b, c, d, e, f}, x] && LtQ[n, -1] && NeQ[n, -2]
  3316. Int[Times[Power[Plus[Optional[Pattern[c, Blank[]]], Times[Optional[Pattern[d, Blank[]]], Pattern[x, Blank[]]]], Optional[Pattern[m, Blank[]]]], Power[Times[Optional[Pattern[b, Blank[]]], sin[Plus[Optional[Pattern[e, Blank[]]], Times[Optional[Pattern[f, Blank[]]], Pattern[x, Blank[]]]]]], Pattern[n, Blank[]]]], Pattern[x, Blank[Symbol]]] := Simp[((c + d*x)^m*Cos[e + f*x]*(b*Sin[e + f*x])^(n + 1))/(b*f*(n + 1)), x] + (Dist[(n + 2)/(b^2*(n + 1)), Int[(c + d*x)^m*(b*Sin[e + f*x])^(n + 2), x], x] + Dist[(d^2*m*(m - 1))/(b^2*f^2*(n + 1)*(n + 2)), Int[(c + d*x)^(m - 2)*(b*Sin[e + f*x])^(n + 2), x], x] - Simp[(d*m*(c + d*x)^(m - 1)*(b*Sin[e + f*x])^(n + 2))/(b^2*f^2*(n + 1)*(n + 2)), x]) /; FreeQ[{b, c, d, e, f}, x] && LtQ[n, -1] && NeQ[n, -2] && GtQ[m, 1]
  3317. Int[Times[Power[Plus[Optional[Pattern[c, Blank[]]], Times[Optional[Pattern[d, Blank[]]], Pattern[x, Blank[]]]], Optional[Pattern[m, Blank[]]]], Power[Plus[Pattern[a, Blank[]], Times[Optional[Pattern[b, Blank[]]], sin[Plus[Optional[Pattern[e, Blank[]]], Times[Optional[Pattern[f, Blank[]]], Pattern[x, Blank[]]]]]]], Optional[Pattern[n, Blank[]]]]], Pattern[x, Blank[Symbol]]] := Int[ExpandIntegrand[(c + d*x)^m, (a + b*Sin[e + f*x])^n, x], x] /; FreeQ[{a, b, c, d, e, f, m}, x] && IGtQ[n, 0] && (EqQ[n, 1] || IGtQ[m, 0] || NeQ[a^2 - b^2, 0])
  3318. Int[Times[Power[Plus[Optional[Pattern[c, Blank[]]], Times[Optional[Pattern[d, Blank[]]], Pattern[x, Blank[]]]], Optional[Pattern[m, Blank[]]]], Power[Plus[Pattern[a, Blank[]], Times[Optional[Pattern[b, Blank[]]], sin[Plus[Optional[Pattern[e, Blank[]]], Times[Optional[Pattern[f, Blank[]]], Pattern[x, Blank[]]]]]]], Optional[Pattern[n, Blank[]]]]], Pattern[x, Blank[Symbol]]] := Dist[(2*a)^n, Int[(c + d*x)^m*Sin[(1*(e + (Pi*a)/(2*b)))/2 + (f*x)/2]^(2*n), x], x] /; FreeQ[{a, b, c, d, e, f, m}, x] && EqQ[a^2 - b^2, 0] && IntegerQ[n] && (GtQ[n, 0] || IGtQ[m, 0])
  3319. Int[Times[Power[Plus[Optional[Pattern[c, Blank[]]], Times[Optional[Pattern[d, Blank[]]], Pattern[x, Blank[]]]], Optional[Pattern[m, Blank[]]]], Power[Plus[Pattern[a, Blank[]], Times[Optional[Pattern[b, Blank[]]], sin[Plus[Optional[Pattern[e, Blank[]]], Times[Optional[Pattern[f, Blank[]]], Pattern[x, Blank[]]]]]]], Pattern[n, Blank[]]]], Pattern[x, Blank[Symbol]]] := Dist[((2*a)^IntPart[n]*(a + b*Sin[e + f*x])^FracPart[n])/Sin[e/2 + (a*Pi)/(4*b) + (f*x)/2]^(2*FracPart[n]), Int[(c + d*x)^m*Sin[e/2 + (a*Pi)/(4*b) + (f*x)/2]^(2*n), x], x] /; FreeQ[{a, b, c, d, e, f, m}, x] && EqQ[a^2 - b^2, 0] && IntegerQ[n + 1/2] && (GtQ[n, 0] || IGtQ[m, 0])
  3320. Int[Times[Power[Plus[Optional[Pattern[c, Blank[]]], Times[Optional[Pattern[d, Blank[]]], Pattern[x, Blank[]]]], Optional[Pattern[m, Blank[]]]], Power[Plus[Pattern[a, Blank[]], Times[Optional[Pattern[b, Blank[]]], sin[Plus[Optional[Pattern[e, Blank[]]], Times[Pi, Optional[Pattern[k, Blank[]]]], Times[Complex[0, Pattern[fz, Blank[]]], Optional[Pattern[f, Blank[]]], Pattern[x, Blank[]]]]]]], -1]], Pattern[x, Blank[Symbol]]] := Dist[2, Int[((c + d*x)^m*E^(-(I*e) + f*fz*x))/(E^(I*Pi*(k - 1/2))*(b + (2*a*E^(-(I*e) + f*fz*x))/E^(I*Pi*(k - 1/2)) - (b*E^(2*(-(I*e) + f*fz*x)))/E^(2*I*k*Pi))), x], x] /; FreeQ[{a, b, c, d, e, f, fz}, x] && IntegerQ[2*k] && NeQ[a^2 - b^2, 0] && IGtQ[m, 0]
  3321. Int[Times[Power[Plus[Optional[Pattern[c, Blank[]]], Times[Optional[Pattern[d, Blank[]]], Pattern[x, Blank[]]]], Optional[Pattern[m, Blank[]]]], Power[Plus[Pattern[a, Blank[]], Times[Optional[Pattern[b, Blank[]]], sin[Plus[Optional[Pattern[e, Blank[]]], Times[Pi, Optional[Pattern[k, Blank[]]]], Times[Optional[Pattern[f, Blank[]]], Pattern[x, Blank[]]]]]]], -1]], Pattern[x, Blank[Symbol]]] := Dist[2, Int[((c + d*x)^m*E^(I*Pi*(k - 1/2))*E^(I*(e + f*x)))/(b + 2*a*E^(I*Pi*(k - 1/2))*E^(I*(e + f*x)) - b*E^(2*I*k*Pi)*E^(2*I*(e + f*x))), x], x] /; FreeQ[{a, b, c, d, e, f}, x] && IntegerQ[2*k] && NeQ[a^2 - b^2, 0] && IGtQ[m, 0]
  3322. Int[Times[Power[Plus[Optional[Pattern[c, Blank[]]], Times[Optional[Pattern[d, Blank[]]], Pattern[x, Blank[]]]], Optional[Pattern[m, Blank[]]]], Power[Plus[Pattern[a, Blank[]], Times[Optional[Pattern[b, Blank[]]], sin[Plus[Optional[Pattern[e, Blank[]]], Times[Complex[0, Pattern[fz, Blank[]]], Optional[Pattern[f, Blank[]]], Pattern[x, Blank[]]]]]]], -1]], Pattern[x, Blank[Symbol]]] := Dist[2, Int[((c + d*x)^m*E^(-(I*e) + f*fz*x))/(-(I*b) + 2*a*E^(-(I*e) + f*fz*x) + I*b*E^(2*(-(I*e) + f*fz*x))), x], x] /; FreeQ[{a, b, c, d, e, f, fz}, x] && NeQ[a^2 - b^2, 0] && IGtQ[m, 0]
  3323. Int[Times[Power[Plus[Optional[Pattern[c, Blank[]]], Times[Optional[Pattern[d, Blank[]]], Pattern[x, Blank[]]]], Optional[Pattern[m, Blank[]]]], Power[Plus[Pattern[a, Blank[]], Times[Optional[Pattern[b, Blank[]]], sin[Plus[Optional[Pattern[e, Blank[]]], Times[Optional[Pattern[f, Blank[]]], Pattern[x, Blank[]]]]]]], -1]], Pattern[x, Blank[Symbol]]] := Dist[2, Int[((c + d*x)^m*E^(I*(e + f*x)))/(I*b + 2*a*E^(I*(e + f*x)) - I*b*E^(2*I*(e + f*x))), x], x] /; FreeQ[{a, b, c, d, e, f}, x] && NeQ[a^2 - b^2, 0] && IGtQ[m, 0]
  3324. Int[Times[Power[Plus[Optional[Pattern[c, Blank[]]], Times[Optional[Pattern[d, Blank[]]], Pattern[x, Blank[]]]], Optional[Pattern[m, Blank[]]]], Power[Plus[Pattern[a, Blank[]], Times[Optional[Pattern[b, Blank[]]], sin[Plus[Optional[Pattern[e, Blank[]]], Times[Optional[Pattern[f, Blank[]]], Pattern[x, Blank[]]]]]]], -2]], Pattern[x, Blank[Symbol]]] := Simp[(b*(c + d*x)^m*Cos[e + f*x])/(f*(a^2 - b^2)*(a + b*Sin[e + f*x])), x] + (Dist[a/(a^2 - b^2), Int[(c + d*x)^m/(a + b*Sin[e + f*x]), x], x] - Dist[(b*d*m)/(f*(a^2 - b^2)), Int[((c + d*x)^(m - 1)*Cos[e + f*x])/(a + b*Sin[e + f*x]), x], x]) /; FreeQ[{a, b, c, d, e, f}, x] && NeQ[a^2 - b^2, 0] && IGtQ[m, 0]
  3325. Int[Times[Power[Plus[Optional[Pattern[c, Blank[]]], Times[Optional[Pattern[d, Blank[]]], Pattern[x, Blank[]]]], Optional[Pattern[m, Blank[]]]], Power[Plus[Pattern[a, Blank[]], Times[Optional[Pattern[b, Blank[]]], sin[Plus[Optional[Pattern[e, Blank[]]], Times[Optional[Pattern[f, Blank[]]], Pattern[x, Blank[]]]]]]], Pattern[n, Blank[]]]], Pattern[x, Blank[Symbol]]] := -Simp[(b*(c + d*x)^m*Cos[e + f*x]*(a + b*Sin[e + f*x])^(n + 1))/(f*(n + 1)*(a^2 - b^2)), x] + (Dist[a/(a^2 - b^2), Int[(c + d*x)^m*(a + b*Sin[e + f*x])^(n + 1), x], x] - Dist[(b*(n + 2))/((n + 1)*(a^2 - b^2)), Int[(c + d*x)^m*Sin[e + f*x]*(a + b*Sin[e + f*x])^(n + 1), x], x] + Dist[(b*d*m)/(f*(n + 1)*(a^2 - b^2)), Int[(c + d*x)^(m - 1)*Cos[e + f*x]*(a + b*Sin[e + f*x])^(n + 1), x], x]) /; FreeQ[{a, b, c, d, e, f}, x] && NeQ[a^2 - b^2, 0] && ILtQ[n, -2] && IGtQ[m, 0]
  3326. Int[Times[Power[Plus[Optional[Pattern[c, Blank[]]], Times[Optional[Pattern[d, Blank[]]], Pattern[x, Blank[]]]], Optional[Pattern[m, Blank[]]]], Power[Plus[Optional[Pattern[a, Blank[]]], Times[Optional[Pattern[b, Blank[]]], sin[Plus[Optional[Pattern[e, Blank[]]], Times[Optional[Pattern[f, Blank[]]], Pattern[x, Blank[]]]]]]], Optional[Pattern[n, Blank[]]]]], Pattern[x, Blank[Symbol]]] := Unintegrable[(c + d*x)^m*(a + b*Sin[e + f*x])^n, x] /; FreeQ[{a, b, c, d, e, f, m, n}, x]
  3327. Int[Times[Power[Pattern[u, Blank[]], Optional[Pattern[m, Blank[]]]], Power[Plus[Optional[Pattern[a, Blank[]]], Times[Optional[Pattern[b, Blank[]]], Sin[Pattern[v, Blank[]]]]], Optional[Pattern[n, Blank[]]]]], Pattern[x, Blank[Symbol]]] := Int[ExpandToSum[u, x]^m*(a + b*Sin[ExpandToSum[v, x]])^n, x] /; FreeQ[{a, b, m, n}, x] && LinearQ[{u, v}, x] && !LinearMatchQ[{u, v}, x]
  3328. Int[Times[Power[Plus[Optional[Pattern[a, Blank[]]], Times[Cos[Pattern[v, Blank[]]], Optional[Pattern[b, Blank[]]]]], Optional[Pattern[n, Blank[]]]], Power[Pattern[u, Blank[]], Optional[Pattern[m, Blank[]]]]], Pattern[x, Blank[Symbol]]] := Int[ExpandToSum[u, x]^m*(a + b*Cos[ExpandToSum[v, x]])^n, x] /; FreeQ[{a, b, m, n}, x] && LinearQ[{u, v}, x] && !LinearMatchQ[{u, v}, x]
  3329. Int[Times[Power[Plus[Pattern[a, Blank[]], Times[Optional[Pattern[b, Blank[]]], Power[Pattern[x, Blank[]], Pattern[n, Blank[]]]]], Optional[Pattern[p, Blank[]]]], Sin[Plus[Optional[Pattern[c, Blank[]]], Times[Optional[Pattern[d, Blank[]]], Pattern[x, Blank[]]]]]], Pattern[x, Blank[Symbol]]] := Int[ExpandIntegrand[Sin[c + d*x], (a + b*x^n)^p, x], x] /; FreeQ[{a, b, c, d, n}, x] && IGtQ[p, 0]
  3330. Int[Times[Cos[Plus[Optional[Pattern[c, Blank[]]], Times[Optional[Pattern[d, Blank[]]], Pattern[x, Blank[]]]]], Power[Plus[Pattern[a, Blank[]], Times[Optional[Pattern[b, Blank[]]], Power[Pattern[x, Blank[]], Pattern[n, Blank[]]]]], Optional[Pattern[p, Blank[]]]]], Pattern[x, Blank[Symbol]]] := Int[ExpandIntegrand[Cos[c + d*x], (a + b*x^n)^p, x], x] /; FreeQ[{a, b, c, d, n}, x] && IGtQ[p, 0]
  3331. Int[Times[Power[Plus[Pattern[a, Blank[]], Times[Optional[Pattern[b, Blank[]]], Power[Pattern[x, Blank[]], Pattern[n, Blank[]]]]], Pattern[p, Blank[]]], Sin[Plus[Optional[Pattern[c, Blank[]]], Times[Optional[Pattern[d, Blank[]]], Pattern[x, Blank[]]]]]], Pattern[x, Blank[Symbol]]] := Simp[(x^(-n + 1)*(a + b*x^n)^(p + 1)*Sin[c + d*x])/(b*n*(p + 1)), x] + (-Dist[(-n + 1)/(b*n*(p + 1)), Int[((a + b*x^n)^(p + 1)*Sin[c + d*x])/x^n, x], x] - Dist[d/(b*n*(p + 1)), Int[x^(-n + 1)*(a + b*x^n)^(p + 1)*Cos[c + d*x], x], x]) /; FreeQ[{a, b, c, d}, x] && ILtQ[p, -1] && IGtQ[n, 2]
  3332. Int[Times[Cos[Plus[Optional[Pattern[c, Blank[]]], Times[Optional[Pattern[d, Blank[]]], Pattern[x, Blank[]]]]], Power[Plus[Pattern[a, Blank[]], Times[Optional[Pattern[b, Blank[]]], Power[Pattern[x, Blank[]], Pattern[n, Blank[]]]]], Pattern[p, Blank[]]]], Pattern[x, Blank[Symbol]]] := Simp[(x^(-n + 1)*(a + b*x^n)^(p + 1)*Cos[c + d*x])/(b*n*(p + 1)), x] + (-Dist[(-n + 1)/(b*n*(p + 1)), Int[((a + b*x^n)^(p + 1)*Cos[c + d*x])/x^n, x], x] + Dist[d/(b*n*(p + 1)), Int[x^(-n + 1)*(a + b*x^n)^(p + 1)*Sin[c + d*x], x], x]) /; FreeQ[{a, b, c, d}, x] && ILtQ[p, -1] && IGtQ[n, 2]
  3333. Int[Times[Power[Plus[Pattern[a, Blank[]], Times[Optional[Pattern[b, Blank[]]], Power[Pattern[x, Blank[]], Pattern[n, Blank[]]]]], Pattern[p, Blank[]]], Sin[Plus[Optional[Pattern[c, Blank[]]], Times[Optional[Pattern[d, Blank[]]], Pattern[x, Blank[]]]]]], Pattern[x, Blank[Symbol]]] := Int[ExpandIntegrand[Sin[c + d*x], (a + b*x^n)^p, x], x] /; FreeQ[{a, b, c, d}, x] && ILtQ[p, 0] && IGtQ[n, 0] && (EqQ[n, 2] || EqQ[p, -1])
  3334. Int[Times[Cos[Plus[Optional[Pattern[c, Blank[]]], Times[Optional[Pattern[d, Blank[]]], Pattern[x, Blank[]]]]], Power[Plus[Pattern[a, Blank[]], Times[Optional[Pattern[b, Blank[]]], Power[Pattern[x, Blank[]], Pattern[n, Blank[]]]]], Pattern[p, Blank[]]]], Pattern[x, Blank[Symbol]]] := Int[ExpandIntegrand[Cos[c + d*x], (a + b*x^n)^p, x], x] /; FreeQ[{a, b, c, d}, x] && ILtQ[p, 0] && IGtQ[n, 0] && (EqQ[n, 2] || EqQ[p, -1])
  3335. Int[Times[Power[Plus[Pattern[a, Blank[]], Times[Optional[Pattern[b, Blank[]]], Power[Pattern[x, Blank[]], Pattern[n, Blank[]]]]], Pattern[p, Blank[]]], Sin[Plus[Optional[Pattern[c, Blank[]]], Times[Optional[Pattern[d, Blank[]]], Pattern[x, Blank[]]]]]], Pattern[x, Blank[Symbol]]] := Int[x^(n*p)*(b + a/x^n)^p*Sin[c + d*x], x] /; FreeQ[{a, b, c, d}, x] && ILtQ[p, 0] && ILtQ[n, 0]
  3336. Int[Times[Cos[Plus[Optional[Pattern[c, Blank[]]], Times[Optional[Pattern[d, Blank[]]], Pattern[x, Blank[]]]]], Power[Plus[Pattern[a, Blank[]], Times[Optional[Pattern[b, Blank[]]], Power[Pattern[x, Blank[]], Pattern[n, Blank[]]]]], Pattern[p, Blank[]]]], Pattern[x, Blank[Symbol]]] := Int[x^(n*p)*(b + a/x^n)^p*Cos[c + d*x], x] /; FreeQ[{a, b, c, d}, x] && ILtQ[p, 0] && ILtQ[n, 0]
  3337. Int[Times[Power[Plus[Pattern[a, Blank[]], Times[Optional[Pattern[b, Blank[]]], Power[Pattern[x, Blank[]], Pattern[n, Blank[]]]]], Pattern[p, Blank[]]], Sin[Plus[Optional[Pattern[c, Blank[]]], Times[Optional[Pattern[d, Blank[]]], Pattern[x, Blank[]]]]]], Pattern[x, Blank[Symbol]]] := Unintegrable[(a + b*x^n)^p*Sin[c + d*x], x] /; FreeQ[{a, b, c, d, n, p}, x]
  3338. Int[Times[Cos[Plus[Optional[Pattern[c, Blank[]]], Times[Optional[Pattern[d, Blank[]]], Pattern[x, Blank[]]]]], Power[Plus[Pattern[a, Blank[]], Times[Optional[Pattern[b, Blank[]]], Power[Pattern[x, Blank[]], Pattern[n, Blank[]]]]], Pattern[p, Blank[]]]], Pattern[x, Blank[Symbol]]] := Unintegrable[(a + b*x^n)^p*Cos[c + d*x], x] /; FreeQ[{a, b, c, d, n, p}, x]
  3339. Int[Times[Power[Times[Optional[Pattern[e, Blank[]]], Pattern[x, Blank[]]], Optional[Pattern[m, Blank[]]]], Power[Plus[Pattern[a, Blank[]], Times[Optional[Pattern[b, Blank[]]], Power[Pattern[x, Blank[]], Pattern[n, Blank[]]]]], Optional[Pattern[p, Blank[]]]], Sin[Plus[Optional[Pattern[c, Blank[]]], Times[Optional[Pattern[d, Blank[]]], Pattern[x, Blank[]]]]]], Pattern[x, Blank[Symbol]]] := Int[ExpandIntegrand[Sin[c + d*x], (e*x)^m*(a + b*x^n)^p, x], x] /; FreeQ[{a, b, c, d, e, m, n}, x] && IGtQ[p, 0]
  3340. Int[Times[Cos[Plus[Optional[Pattern[c, Blank[]]], Times[Optional[Pattern[d, Blank[]]], Pattern[x, Blank[]]]]], Power[Times[Optional[Pattern[e, Blank[]]], Pattern[x, Blank[]]], Optional[Pattern[m, Blank[]]]], Power[Plus[Pattern[a, Blank[]], Times[Optional[Pattern[b, Blank[]]], Power[Pattern[x, Blank[]], Pattern[n, Blank[]]]]], Optional[Pattern[p, Blank[]]]]], Pattern[x, Blank[Symbol]]] := Int[ExpandIntegrand[Cos[c + d*x], (e*x)^m*(a + b*x^n)^p, x], x] /; FreeQ[{a, b, c, d, e, m, n}, x] && IGtQ[p, 0]
  3341. Int[Times[Power[Times[Optional[Pattern[e, Blank[]]], Pattern[x, Blank[]]], Optional[Pattern[m, Blank[]]]], Power[Plus[Pattern[a, Blank[]], Times[Optional[Pattern[b, Blank[]]], Power[Pattern[x, Blank[]], Pattern[n, Blank[]]]]], Pattern[p, Blank[]]], Sin[Plus[Optional[Pattern[c, Blank[]]], Times[Optional[Pattern[d, Blank[]]], Pattern[x, Blank[]]]]]], Pattern[x, Blank[Symbol]]] := Simp[(e^m*(a + b*x^n)^(p + 1)*Sin[c + d*x])/(b*n*(p + 1)), x] - Dist[(d*e^m)/(b*n*(p + 1)), Int[(a + b*x^n)^(p + 1)*Cos[c + d*x], x], x] /; FreeQ[{a, b, c, d, e, m, n}, x] && ILtQ[p, -1] && EqQ[m, n - 1] && (IntegerQ[n] || GtQ[e, 0])
  3342. Int[Times[Cos[Plus[Optional[Pattern[c, Blank[]]], Times[Optional[Pattern[d, Blank[]]], Pattern[x, Blank[]]]]], Power[Times[Optional[Pattern[e, Blank[]]], Pattern[x, Blank[]]], Optional[Pattern[m, Blank[]]]], Power[Plus[Pattern[a, Blank[]], Times[Optional[Pattern[b, Blank[]]], Power[Pattern[x, Blank[]], Pattern[n, Blank[]]]]], Pattern[p, Blank[]]]], Pattern[x, Blank[Symbol]]] := Simp[(e^m*(a + b*x^n)^(p + 1)*Cos[c + d*x])/(b*n*(p + 1)), x] + Dist[(d*e^m)/(b*n*(p + 1)), Int[(a + b*x^n)^(p + 1)*Sin[c + d*x], x], x] /; FreeQ[{a, b, c, d, e, m, n}, x] && ILtQ[p, -1] && EqQ[m, n - 1] && (IntegerQ[n] || GtQ[e, 0])
  3343. Int[Times[Power[Pattern[x, Blank[]], Optional[Pattern[m, Blank[]]]], Power[Plus[Pattern[a, Blank[]], Times[Optional[Pattern[b, Blank[]]], Power[Pattern[x, Blank[]], Pattern[n, Blank[]]]]], Pattern[p, Blank[]]], Sin[Plus[Optional[Pattern[c, Blank[]]], Times[Optional[Pattern[d, Blank[]]], Pattern[x, Blank[]]]]]], Pattern[x, Blank[Symbol]]] := Simp[(x^(m - n + 1)*(a + b*x^n)^(p + 1)*Sin[c + d*x])/(b*n*(p + 1)), x] + (-Dist[(m - n + 1)/(b*n*(p + 1)), Int[x^(m - n)*(a + b*x^n)^(p + 1)*Sin[c + d*x], x], x] - Dist[d/(b*n*(p + 1)), Int[x^(m - n + 1)*(a + b*x^n)^(p + 1)*Cos[c + d*x], x], x]) /; FreeQ[{a, b, c, d, m}, x] && ILtQ[p, -1] && IGtQ[n, 0] && (GtQ[m - n + 1, 0] || GtQ[n, 2]) && RationalQ[m]
  3344. Int[Times[Cos[Plus[Optional[Pattern[c, Blank[]]], Times[Optional[Pattern[d, Blank[]]], Pattern[x, Blank[]]]]], Power[Pattern[x, Blank[]], Optional[Pattern[m, Blank[]]]], Power[Plus[Pattern[a, Blank[]], Times[Optional[Pattern[b, Blank[]]], Power[Pattern[x, Blank[]], Pattern[n, Blank[]]]]], Pattern[p, Blank[]]]], Pattern[x, Blank[Symbol]]] := Simp[(x^(m - n + 1)*(a + b*x^n)^(p + 1)*Cos[c + d*x])/(b*n*(p + 1)), x] + (-Dist[(m - n + 1)/(b*n*(p + 1)), Int[x^(m - n)*(a + b*x^n)^(p + 1)*Cos[c + d*x], x], x] + Dist[d/(b*n*(p + 1)), Int[x^(m - n + 1)*(a + b*x^n)^(p + 1)*Sin[c + d*x], x], x]) /; FreeQ[{a, b, c, d, m}, x] && ILtQ[p, -1] && IGtQ[n, 0] && (GtQ[m - n + 1, 0] || GtQ[n, 2]) && RationalQ[m]
  3345. Int[Times[Power[Pattern[x, Blank[]], Optional[Pattern[m, Blank[]]]], Power[Plus[Pattern[a, Blank[]], Times[Optional[Pattern[b, Blank[]]], Power[Pattern[x, Blank[]], Pattern[n, Blank[]]]]], Pattern[p, Blank[]]], Sin[Plus[Optional[Pattern[c, Blank[]]], Times[Optional[Pattern[d, Blank[]]], Pattern[x, Blank[]]]]]], Pattern[x, Blank[Symbol]]] := Int[ExpandIntegrand[Sin[c + d*x], x^m*(a + b*x^n)^p, x], x] /; FreeQ[{a, b, c, d, m}, x] && ILtQ[p, 0] && IGtQ[n, 0] && (EqQ[n, 2] || EqQ[p, -1]) && IntegerQ[m]
  3346. Int[Times[Cos[Plus[Optional[Pattern[c, Blank[]]], Times[Optional[Pattern[d, Blank[]]], Pattern[x, Blank[]]]]], Power[Pattern[x, Blank[]], Optional[Pattern[m, Blank[]]]], Power[Plus[Pattern[a, Blank[]], Times[Optional[Pattern[b, Blank[]]], Power[Pattern[x, Blank[]], Pattern[n, Blank[]]]]], Pattern[p, Blank[]]]], Pattern[x, Blank[Symbol]]] := Int[ExpandIntegrand[Cos[c + d*x], x^m*(a + b*x^n)^p, x], x] /; FreeQ[{a, b, c, d, m}, x] && ILtQ[p, 0] && IGtQ[n, 0] && (EqQ[n, 2] || EqQ[p, -1]) && IntegerQ[m]
  3347. Int[Times[Power[Pattern[x, Blank[]], Optional[Pattern[m, Blank[]]]], Power[Plus[Pattern[a, Blank[]], Times[Optional[Pattern[b, Blank[]]], Power[Pattern[x, Blank[]], Pattern[n, Blank[]]]]], Pattern[p, Blank[]]], Sin[Plus[Optional[Pattern[c, Blank[]]], Times[Optional[Pattern[d, Blank[]]], Pattern[x, Blank[]]]]]], Pattern[x, Blank[Symbol]]] := Int[x^(m + n*p)*(b + a/x^n)^p*Sin[c + d*x], x] /; FreeQ[{a, b, c, d, m}, x] && ILtQ[p, 0] && ILtQ[n, 0]
  3348. Int[Times[Cos[Plus[Optional[Pattern[c, Blank[]]], Times[Optional[Pattern[d, Blank[]]], Pattern[x, Blank[]]]]], Power[Pattern[x, Blank[]], Optional[Pattern[m, Blank[]]]], Power[Plus[Pattern[a, Blank[]], Times[Optional[Pattern[b, Blank[]]], Power[Pattern[x, Blank[]], Pattern[n, Blank[]]]]], Pattern[p, Blank[]]]], Pattern[x, Blank[Symbol]]] := Int[x^(m + n*p)*(b + a/x^n)^p*Cos[c + d*x], x] /; FreeQ[{a, b, c, d, m}, x] && ILtQ[p, 0] && ILtQ[n, 0]
  3349. Int[Times[Power[Times[Optional[Pattern[e, Blank[]]], Pattern[x, Blank[]]], Optional[Pattern[m, Blank[]]]], Power[Plus[Pattern[a, Blank[]], Times[Optional[Pattern[b, Blank[]]], Power[Pattern[x, Blank[]], Pattern[n, Blank[]]]]], Optional[Pattern[p, Blank[]]]], Sin[Plus[Optional[Pattern[c, Blank[]]], Times[Optional[Pattern[d, Blank[]]], Pattern[x, Blank[]]]]]], Pattern[x, Blank[Symbol]]] := Unintegrable[(e*x)^m*(a + b*x^n)^p*Sin[c + d*x], x] /; FreeQ[{a, b, c, d, e, m, n, p}, x]
  3350. Int[Times[Cos[Plus[Optional[Pattern[c, Blank[]]], Times[Optional[Pattern[d, Blank[]]], Pattern[x, Blank[]]]]], Power[Times[Optional[Pattern[e, Blank[]]], Pattern[x, Blank[]]], Optional[Pattern[m, Blank[]]]], Power[Plus[Pattern[a, Blank[]], Times[Optional[Pattern[b, Blank[]]], Power[Pattern[x, Blank[]], Pattern[n, Blank[]]]]], Optional[Pattern[p, Blank[]]]]], Pattern[x, Blank[Symbol]]] := Unintegrable[(e*x)^m*(a + b*x^n)^p*Cos[c + d*x], x] /; FreeQ[{a, b, c, d, e, m, n, p}, x]
  3351. Int[Sin[Times[Optional[Pattern[d, Blank[]]], Power[Plus[Optional[Pattern[e, Blank[]]], Times[Optional[Pattern[f, Blank[]]], Pattern[x, Blank[]]]], 2]]], Pattern[x, Blank[Symbol]]] := Simp[(Sqrt[Pi/2]*FresnelS[Sqrt[2/Pi]*Rt[d, 2]*(e + f*x)])/(f*Rt[d, 2]), x] /; FreeQ[{d, e, f}, x]
  3352. Int[Cos[Times[Optional[Pattern[d, Blank[]]], Power[Plus[Optional[Pattern[e, Blank[]]], Times[Optional[Pattern[f, Blank[]]], Pattern[x, Blank[]]]], 2]]], Pattern[x, Blank[Symbol]]] := Simp[(Sqrt[Pi/2]*FresnelC[Sqrt[2/Pi]*Rt[d, 2]*(e + f*x)])/(f*Rt[d, 2]), x] /; FreeQ[{d, e, f}, x]
  3353. Int[Sin[Plus[Pattern[c, Blank[]], Times[Optional[Pattern[d, Blank[]]], Power[Plus[Optional[Pattern[e, Blank[]]], Times[Optional[Pattern[f, Blank[]]], Pattern[x, Blank[]]]], 2]]]], Pattern[x, Blank[Symbol]]] := Dist[Sin[c], Int[Cos[d*(e + f*x)^2], x], x] + Dist[Cos[c], Int[Sin[d*(e + f*x)^2], x], x] /; FreeQ[{c, d, e, f}, x]
  3354. Int[Cos[Plus[Pattern[c, Blank[]], Times[Optional[Pattern[d, Blank[]]], Power[Plus[Optional[Pattern[e, Blank[]]], Times[Optional[Pattern[f, Blank[]]], Pattern[x, Blank[]]]], 2]]]], Pattern[x, Blank[Symbol]]] := Dist[Cos[c], Int[Cos[d*(e + f*x)^2], x], x] - Dist[Sin[c], Int[Sin[d*(e + f*x)^2], x], x] /; FreeQ[{c, d, e, f}, x]
  3355. Int[Sin[Plus[Optional[Pattern[c, Blank[]]], Times[Optional[Pattern[d, Blank[]]], Power[Plus[Optional[Pattern[e, Blank[]]], Times[Optional[Pattern[f, Blank[]]], Pattern[x, Blank[]]]], Pattern[n, Blank[]]]]]], Pattern[x, Blank[Symbol]]] := Dist[I/2, Int[E^(-(c*I) - d*I*(e + f*x)^n), x], x] - Dist[I/2, Int[E^(c*I + d*I*(e + f*x)^n), x], x] /; FreeQ[{c, d, e, f}, x] && IGtQ[n, 2]
  3356. Int[Cos[Plus[Optional[Pattern[c, Blank[]]], Times[Optional[Pattern[d, Blank[]]], Power[Plus[Optional[Pattern[e, Blank[]]], Times[Optional[Pattern[f, Blank[]]], Pattern[x, Blank[]]]], Pattern[n, Blank[]]]]]], Pattern[x, Blank[Symbol]]] := Dist[1/2, Int[E^(-(c*I) - d*I*(e + f*x)^n), x], x] + Dist[1/2, Int[E^(c*I + d*I*(e + f*x)^n), x], x] /; FreeQ[{c, d, e, f}, x] && IGtQ[n, 2]
  3357. Int[Power[Plus[Optional[Pattern[a, Blank[]]], Times[Optional[Pattern[b, Blank[]]], Sin[Plus[Optional[Pattern[c, Blank[]]], Times[Optional[Pattern[d, Blank[]]], Power[Plus[Optional[Pattern[e, Blank[]]], Times[Optional[Pattern[f, Blank[]]], Pattern[x, Blank[]]]], Pattern[n, Blank[]]]]]]]], Pattern[p, Blank[]]], Pattern[x, Blank[Symbol]]] := Int[ExpandTrigReduce[(a + b*Sin[c + d*(e + f*x)^n])^p, x], x] /; FreeQ[{a, b, c, d, e, f}, x] && IGtQ[p, 1] && IGtQ[n, 1]
  3358. Int[Power[Plus[Optional[Pattern[a, Blank[]]], Times[Cos[Plus[Optional[Pattern[c, Blank[]]], Times[Optional[Pattern[d, Blank[]]], Power[Plus[Optional[Pattern[e, Blank[]]], Times[Optional[Pattern[f, Blank[]]], Pattern[x, Blank[]]]], Pattern[n, Blank[]]]]]], Optional[Pattern[b, Blank[]]]]], Pattern[p, Blank[]]], Pattern[x, Blank[Symbol]]] := Int[ExpandTrigReduce[(a + b*Cos[c + d*(e + f*x)^n])^p, x], x] /; FreeQ[{a, b, c, d, e, f}, x] && IGtQ[p, 1] && IGtQ[n, 1]
  3359. Int[Power[Plus[Optional[Pattern[a, Blank[]]], Times[Optional[Pattern[b, Blank[]]], Sin[Plus[Optional[Pattern[c, Blank[]]], Times[Optional[Pattern[d, Blank[]]], Power[Plus[Optional[Pattern[e, Blank[]]], Times[Optional[Pattern[f, Blank[]]], Pattern[x, Blank[]]]], Pattern[n, Blank[]]]]]]]], Optional[Pattern[p, Blank[]]]], Pattern[x, Blank[Symbol]]] := -Dist[f^(-1), Subst[Int[(a + b*Sin[c + d/x^n])^p/x^2, x], x, 1/(e + f*x)], x] /; FreeQ[{a, b, c, d, e, f}, x] && IGtQ[p, 0] && ILtQ[n, 0] && EqQ[n, -2]
  3360. Int[Power[Plus[Optional[Pattern[a, Blank[]]], Times[Cos[Plus[Optional[Pattern[c, Blank[]]], Times[Optional[Pattern[d, Blank[]]], Power[Plus[Optional[Pattern[e, Blank[]]], Times[Optional[Pattern[f, Blank[]]], Pattern[x, Blank[]]]], Pattern[n, Blank[]]]]]], Optional[Pattern[b, Blank[]]]]], Optional[Pattern[p, Blank[]]]], Pattern[x, Blank[Symbol]]] := -Dist[f^(-1), Subst[Int[(a + b*Cos[c + d/x^n])^p/x^2, x], x, 1/(e + f*x)], x] /; FreeQ[{a, b, c, d, e, f}, x] && IGtQ[p, 0] && ILtQ[n, 0] && EqQ[n, -2]
  3361. Int[Power[Plus[Optional[Pattern[a, Blank[]]], Times[Optional[Pattern[b, Blank[]]], Sin[Plus[Optional[Pattern[c, Blank[]]], Times[Optional[Pattern[d, Blank[]]], Power[Plus[Optional[Pattern[e, Blank[]]], Times[Optional[Pattern[f, Blank[]]], Pattern[x, Blank[]]]], Pattern[n, Blank[]]]]]]]], Optional[Pattern[p, Blank[]]]], Pattern[x, Blank[Symbol]]] := Dist[1/(n*f), Subst[Int[x^(1/n - 1)*(a + b*Sin[c + d*x])^p, x], x, (e + f*x)^n], x] /; FreeQ[{a, b, c, d, e, f}, x] && IGtQ[p, 0] && IntegerQ[1/n]
  3362. Int[Power[Plus[Optional[Pattern[a, Blank[]]], Times[Cos[Plus[Optional[Pattern[c, Blank[]]], Times[Optional[Pattern[d, Blank[]]], Power[Plus[Optional[Pattern[e, Blank[]]], Times[Optional[Pattern[f, Blank[]]], Pattern[x, Blank[]]]], Pattern[n, Blank[]]]]]], Optional[Pattern[b, Blank[]]]]], Optional[Pattern[p, Blank[]]]], Pattern[x, Blank[Symbol]]] := Dist[1/(n*f), Subst[Int[x^(1/n - 1)*(a + b*Cos[c + d*x])^p, x], x, (e + f*x)^n], x] /; FreeQ[{a, b, c, d, e, f}, x] && IGtQ[p, 0] && IntegerQ[1/n]
  3363. Int[Power[Plus[Optional[Pattern[a, Blank[]]], Times[Optional[Pattern[b, Blank[]]], Sin[Plus[Optional[Pattern[c, Blank[]]], Times[Optional[Pattern[d, Blank[]]], Power[Plus[Optional[Pattern[e, Blank[]]], Times[Optional[Pattern[f, Blank[]]], Pattern[x, Blank[]]]], Pattern[n, Blank[]]]]]]]], Optional[Pattern[p, Blank[]]]], Pattern[x, Blank[Symbol]]] := Module[{k = Denominator[n]}, Dist[k/f, Subst[Int[x^(k - 1)*(a + b*Sin[c + d*x^(k*n)])^p, x], x, (e + f*x)^(1/k)], x]] /; FreeQ[{a, b, c, d, e, f}, x] && IGtQ[p, 0] && FractionQ[n]
  3364. Int[Power[Plus[Optional[Pattern[a, Blank[]]], Times[Cos[Plus[Optional[Pattern[c, Blank[]]], Times[Optional[Pattern[d, Blank[]]], Power[Plus[Optional[Pattern[e, Blank[]]], Times[Optional[Pattern[f, Blank[]]], Pattern[x, Blank[]]]], Pattern[n, Blank[]]]]]], Optional[Pattern[b, Blank[]]]]], Optional[Pattern[p, Blank[]]]], Pattern[x, Blank[Symbol]]] := Module[{k = Denominator[n]}, Dist[k/f, Subst[Int[x^(k - 1)*(a + b*Cos[c + d*x^(k*n)])^p, x], x, (e + f*x)^(1/k)], x]] /; FreeQ[{a, b, c, d, e, f}, x] && IGtQ[p, 0] && FractionQ[n]
  3365. Int[Sin[Plus[Optional[Pattern[c, Blank[]]], Times[Optional[Pattern[d, Blank[]]], Power[Plus[Optional[Pattern[e, Blank[]]], Times[Optional[Pattern[f, Blank[]]], Pattern[x, Blank[]]]], Pattern[n, Blank[]]]]]], Pattern[x, Blank[Symbol]]] := Dist[I/2, Int[E^(-(c*I) - d*I*(e + f*x)^n), x], x] - Dist[I/2, Int[E^(c*I + d*I*(e + f*x)^n), x], x] /; FreeQ[{c, d, e, f, n}, x]
  3366. Int[Cos[Plus[Optional[Pattern[c, Blank[]]], Times[Optional[Pattern[d, Blank[]]], Power[Plus[Optional[Pattern[e, Blank[]]], Times[Optional[Pattern[f, Blank[]]], Pattern[x, Blank[]]]], Pattern[n, Blank[]]]]]], Pattern[x, Blank[Symbol]]] := Dist[1/2, Int[E^(-(c*I) - d*I*(e + f*x)^n), x], x] + Dist[1/2, Int[E^(c*I + d*I*(e + f*x)^n), x], x] /; FreeQ[{c, d, e, f, n}, x]
  3367. Int[Power[Plus[Optional[Pattern[a, Blank[]]], Times[Optional[Pattern[b, Blank[]]], Sin[Plus[Optional[Pattern[c, Blank[]]], Times[Optional[Pattern[d, Blank[]]], Power[Plus[Optional[Pattern[e, Blank[]]], Times[Optional[Pattern[f, Blank[]]], Pattern[x, Blank[]]]], Pattern[n, Blank[]]]]]]]], Pattern[p, Blank[]]], Pattern[x, Blank[Symbol]]] := Int[ExpandTrigReduce[(a + b*Sin[c + d*(e + f*x)^n])^p, x], x] /; FreeQ[{a, b, c, d, e, f, n}, x] && IGtQ[p, 1]
  3368. Int[Power[Plus[Optional[Pattern[a, Blank[]]], Times[Cos[Plus[Optional[Pattern[c, Blank[]]], Times[Optional[Pattern[d, Blank[]]], Power[Plus[Optional[Pattern[e, Blank[]]], Times[Optional[Pattern[f, Blank[]]], Pattern[x, Blank[]]]], Pattern[n, Blank[]]]]]], Optional[Pattern[b, Blank[]]]]], Pattern[p, Blank[]]], Pattern[x, Blank[Symbol]]] := Int[ExpandTrigReduce[(a + b*Cos[c + d*(e + f*x)^n])^p, x], x] /; FreeQ[{a, b, c, d, e, f, n}, x] && IGtQ[p, 1]
  3369. Int[Power[Plus[Optional[Pattern[a, Blank[]]], Times[Optional[Pattern[b, Blank[]]], Sin[Plus[Optional[Pattern[c, Blank[]]], Times[Optional[Pattern[d, Blank[]]], Power[Plus[Optional[Pattern[e, Blank[]]], Times[Optional[Pattern[f, Blank[]]], Pattern[x, Blank[]]]], Pattern[n, Blank[]]]]]]]], Pattern[p, Blank[]]], Pattern[x, Blank[Symbol]]] := Unintegrable[(a + b*Sin[c + d*(e + f*x)^n])^p, x] /; FreeQ[{a, b, c, d, e, f, n, p}, x]
  3370. Int[Power[Plus[Optional[Pattern[a, Blank[]]], Times[Cos[Plus[Optional[Pattern[c, Blank[]]], Times[Optional[Pattern[d, Blank[]]], Power[Plus[Optional[Pattern[e, Blank[]]], Times[Optional[Pattern[f, Blank[]]], Pattern[x, Blank[]]]], Pattern[n, Blank[]]]]]], Optional[Pattern[b, Blank[]]]]], Pattern[p, Blank[]]], Pattern[x, Blank[Symbol]]] := Unintegrable[(a + b*Cos[c + d*(e + f*x)^n])^p, x] /; FreeQ[{a, b, c, d, e, f, n, p}, x]
  3371. Int[Power[Plus[Optional[Pattern[a, Blank[]]], Times[Optional[Pattern[b, Blank[]]], Sin[Plus[Optional[Pattern[c, Blank[]]], Times[Optional[Pattern[d, Blank[]]], Power[Pattern[u, Blank[]], Pattern[n, Blank[]]]]]]]], Optional[Pattern[p, Blank[]]]], Pattern[x, Blank[Symbol]]] := Int[(a + b*Sin[c + d*ExpandToSum[u, x]^n])^p, x] /; FreeQ[{a, b, c, d, n, p}, x] && LinearQ[u, x] && !LinearMatchQ[u, x]
  3372. Int[Power[Plus[Optional[Pattern[a, Blank[]]], Times[Cos[Plus[Optional[Pattern[c, Blank[]]], Times[Optional[Pattern[d, Blank[]]], Power[Pattern[u, Blank[]], Pattern[n, Blank[]]]]]], Optional[Pattern[b, Blank[]]]]], Optional[Pattern[p, Blank[]]]], Pattern[x, Blank[Symbol]]] := Int[(a + b*Cos[c + d*ExpandToSum[u, x]^n])^p, x] /; FreeQ[{a, b, c, d, n, p}, x] && LinearQ[u, x] && !LinearMatchQ[u, x]
  3373. Int[Power[Plus[Optional[Pattern[a, Blank[]]], Times[Optional[Pattern[b, Blank[]]], Sin[Pattern[u, Blank[]]]]], Optional[Pattern[p, Blank[]]]], Pattern[x, Blank[Symbol]]] := Int[(a + b*Sin[ExpandToSum[u, x]])^p, x] /; FreeQ[{a, b, p}, x] && BinomialQ[u, x] && !BinomialMatchQ[u, x]
  3374. Int[Power[Plus[Optional[Pattern[a, Blank[]]], Times[Cos[Pattern[u, Blank[]]], Optional[Pattern[b, Blank[]]]]], Optional[Pattern[p, Blank[]]]], Pattern[x, Blank[Symbol]]] := Int[(a + b*Cos[ExpandToSum[u, x]])^p, x] /; FreeQ[{a, b, p}, x] && BinomialQ[u, x] && !BinomialMatchQ[u, x]
  3375. Int[Times[Power[Pattern[x, Blank[]], -1], Sin[Times[Optional[Pattern[d, Blank[]]], Power[Pattern[x, Blank[]], Pattern[n, Blank[]]]]]], Pattern[x, Blank[Symbol]]] := Simp[SinIntegral[d*x^n]/n, x] /; FreeQ[{d, n}, x]
  3376. Int[Times[Cos[Times[Optional[Pattern[d, Blank[]]], Power[Pattern[x, Blank[]], Pattern[n, Blank[]]]]], Power[Pattern[x, Blank[]], -1]], Pattern[x, Blank[Symbol]]] := Simp[CosIntegral[d*x^n]/n, x] /; FreeQ[{d, n}, x]
  3377. Int[Times[Power[Pattern[x, Blank[]], -1], Sin[Plus[Pattern[c, Blank[]], Times[Optional[Pattern[d, Blank[]]], Power[Pattern[x, Blank[]], Pattern[n, Blank[]]]]]]], Pattern[x, Blank[Symbol]]] := Dist[Sin[c], Int[Cos[d*x^n]/x, x], x] + Dist[Cos[c], Int[Sin[d*x^n]/x, x], x] /; FreeQ[{c, d, n}, x]
  3378. Int[Times[Cos[Plus[Pattern[c, Blank[]], Times[Optional[Pattern[d, Blank[]]], Power[Pattern[x, Blank[]], Pattern[n, Blank[]]]]]], Power[Pattern[x, Blank[]], -1]], Pattern[x, Blank[Symbol]]] := Dist[Cos[c], Int[Cos[d*x^n]/x, x], x] - Dist[Sin[c], Int[Sin[d*x^n]/x, x], x] /; FreeQ[{c, d, n}, x]
  3379. Int[Times[Power[Pattern[x, Blank[]], Optional[Pattern[m, Blank[]]]], Power[Plus[Optional[Pattern[a, Blank[]]], Times[Optional[Pattern[b, Blank[]]], Sin[Plus[Optional[Pattern[c, Blank[]]], Times[Optional[Pattern[d, Blank[]]], Power[Pattern[x, Blank[]], Pattern[n, Blank[]]]]]]]], Optional[Pattern[p, Blank[]]]]], Pattern[x, Blank[Symbol]]] := Dist[1/n, Subst[Int[x^(Simplify[(m + 1)/n] - 1)*(a + b*Sin[c + d*x])^p, x], x, x^n], x] /; FreeQ[{a, b, c, d, m, n, p}, x] && IntegerQ[Simplify[(m + 1)/n]] && (EqQ[p, 1] || EqQ[m, n - 1] || (IntegerQ[p] && GtQ[Simplify[(m + 1)/n], 0]))
  3380. Int[Times[Power[Plus[Optional[Pattern[a, Blank[]]], Times[Cos[Plus[Optional[Pattern[c, Blank[]]], Times[Optional[Pattern[d, Blank[]]], Power[Pattern[x, Blank[]], Pattern[n, Blank[]]]]]], Optional[Pattern[b, Blank[]]]]], Optional[Pattern[p, Blank[]]]], Power[Pattern[x, Blank[]], Optional[Pattern[m, Blank[]]]]], Pattern[x, Blank[Symbol]]] := Dist[1/n, Subst[Int[x^(Simplify[(m + 1)/n] - 1)*(a + b*Cos[c + d*x])^p, x], x, x^n], x] /; FreeQ[{a, b, c, d, m, n, p}, x] && IntegerQ[Simplify[(m + 1)/n]] && (EqQ[p, 1] || EqQ[m, n - 1] || (IntegerQ[p] && GtQ[Simplify[(m + 1)/n], 0]))
  3381. Int[Times[Power[Times[Pattern[e, Blank[]], Pattern[x, Blank[]]], Pattern[m, Blank[]]], Power[Plus[Optional[Pattern[a, Blank[]]], Times[Optional[Pattern[b, Blank[]]], Sin[Plus[Optional[Pattern[c, Blank[]]], Times[Optional[Pattern[d, Blank[]]], Power[Pattern[x, Blank[]], Pattern[n, Blank[]]]]]]]], Optional[Pattern[p, Blank[]]]]], Pattern[x, Blank[Symbol]]] := Dist[(e^IntPart[m]*(e*x)^FracPart[m])/x^FracPart[m], Int[x^m*(a + b*Sin[c + d*x^n])^p, x], x] /; FreeQ[{a, b, c, d, e, m, n, p}, x] && IntegerQ[Simplify[(m + 1)/n]]
  3382. Int[Times[Power[Plus[Optional[Pattern[a, Blank[]]], Times[Cos[Plus[Optional[Pattern[c, Blank[]]], Times[Optional[Pattern[d, Blank[]]], Power[Pattern[x, Blank[]], Pattern[n, Blank[]]]]]], Optional[Pattern[b, Blank[]]]]], Optional[Pattern[p, Blank[]]]], Power[Times[Pattern[e, Blank[]], Pattern[x, Blank[]]], Pattern[m, Blank[]]]], Pattern[x, Blank[Symbol]]] := Dist[(e^IntPart[m]*(e*x)^FracPart[m])/x^FracPart[m], Int[x^m*(a + b*Cos[c + d*x^n])^p, x], x] /; FreeQ[{a, b, c, d, e, m, n, p}, x] && IntegerQ[Simplify[(m + 1)/n]]
  3383. Int[Times[Power[Pattern[x, Blank[]], Optional[Pattern[m, Blank[]]]], Sin[Plus[Optional[Pattern[a, Blank[]]], Times[Optional[Pattern[b, Blank[]]], Power[Pattern[x, Blank[]], Pattern[n, Blank[]]]]]]], Pattern[x, Blank[Symbol]]] := Dist[2/n, Subst[Int[Sin[a + b*x^2], x], x, x^(n/2)], x] /; FreeQ[{a, b, m, n}, x] && EqQ[m, n/2 - 1]
  3384. Int[Times[Cos[Plus[Optional[Pattern[a, Blank[]]], Times[Optional[Pattern[b, Blank[]]], Power[Pattern[x, Blank[]], Pattern[n, Blank[]]]]]], Power[Pattern[x, Blank[]], Optional[Pattern[m, Blank[]]]]], Pattern[x, Blank[Symbol]]] := Dist[2/n, Subst[Int[Cos[a + b*x^2], x], x, x^(n/2)], x] /; FreeQ[{a, b, m, n}, x] && EqQ[m, n/2 - 1]
  3385. Int[Times[Power[Times[Optional[Pattern[e, Blank[]]], Pattern[x, Blank[]]], Optional[Pattern[m, Blank[]]]], Sin[Plus[Optional[Pattern[c, Blank[]]], Times[Optional[Pattern[d, Blank[]]], Power[Pattern[x, Blank[]], Pattern[n, Blank[]]]]]]], Pattern[x, Blank[Symbol]]] := -Simp[(e^(n - 1)*(e*x)^(m - n + 1)*Cos[c + d*x^n])/(d*n), x] + Dist[(e^n*(m - n + 1))/(d*n), Int[(e*x)^(m - n)*Cos[c + d*x^n], x], x] /; FreeQ[{c, d, e}, x] && IGtQ[n, 0] && LtQ[n, m + 1]
  3386. Int[Times[Cos[Plus[Optional[Pattern[c, Blank[]]], Times[Optional[Pattern[d, Blank[]]], Power[Pattern[x, Blank[]], Pattern[n, Blank[]]]]]], Power[Times[Optional[Pattern[e, Blank[]]], Pattern[x, Blank[]]], Optional[Pattern[m, Blank[]]]]], Pattern[x, Blank[Symbol]]] := Simp[(e^(n - 1)*(e*x)^(m - n + 1)*Sin[c + d*x^n])/(d*n), x] - Dist[(e^n*(m - n + 1))/(d*n), Int[(e*x)^(m - n)*Sin[c + d*x^n], x], x] /; FreeQ[{c, d, e}, x] && IGtQ[n, 0] && LtQ[n, m + 1]
  3387. Int[Times[Power[Times[Optional[Pattern[e, Blank[]]], Pattern[x, Blank[]]], Pattern[m, Blank[]]], Sin[Plus[Optional[Pattern[c, Blank[]]], Times[Optional[Pattern[d, Blank[]]], Power[Pattern[x, Blank[]], Pattern[n, Blank[]]]]]]], Pattern[x, Blank[Symbol]]] := Simp[((e*x)^(m + 1)*Sin[c + d*x^n])/(e*(m + 1)), x] - Dist[(d*n)/(e^n*(m + 1)), Int[(e*x)^(m + n)*Cos[c + d*x^n], x], x] /; FreeQ[{c, d, e}, x] && IGtQ[n, 0] && LtQ[m, -1]
  3388. Int[Times[Cos[Plus[Optional[Pattern[c, Blank[]]], Times[Optional[Pattern[d, Blank[]]], Power[Pattern[x, Blank[]], Pattern[n, Blank[]]]]]], Power[Times[Optional[Pattern[e, Blank[]]], Pattern[x, Blank[]]], Pattern[m, Blank[]]]], Pattern[x, Blank[Symbol]]] := Simp[((e*x)^(m + 1)*Cos[c + d*x^n])/(e*(m + 1)), x] + Dist[(d*n)/(e^n*(m + 1)), Int[(e*x)^(m + n)*Sin[c + d*x^n], x], x] /; FreeQ[{c, d, e}, x] && IGtQ[n, 0] && LtQ[m, -1]
  3389. Int[Times[Power[Times[Optional[Pattern[e, Blank[]]], Pattern[x, Blank[]]], Optional[Pattern[m, Blank[]]]], Sin[Plus[Optional[Pattern[c, Blank[]]], Times[Optional[Pattern[d, Blank[]]], Power[Pattern[x, Blank[]], Pattern[n, Blank[]]]]]]], Pattern[x, Blank[Symbol]]] := Dist[I/2, Int[(e*x)^m*E^(-(c*I) - d*I*x^n), x], x] - Dist[I/2, Int[(e*x)^m*E^(c*I + d*I*x^n), x], x] /; FreeQ[{c, d, e, m}, x] && IGtQ[n, 0]
  3390. Int[Times[Cos[Plus[Optional[Pattern[c, Blank[]]], Times[Optional[Pattern[d, Blank[]]], Power[Pattern[x, Blank[]], Pattern[n, Blank[]]]]]], Power[Times[Optional[Pattern[e, Blank[]]], Pattern[x, Blank[]]], Optional[Pattern[m, Blank[]]]]], Pattern[x, Blank[Symbol]]] := Dist[1/2, Int[(e*x)^m*E^(-(c*I) - d*I*x^n), x], x] + Dist[1/2, Int[(e*x)^m*E^(c*I + d*I*x^n), x], x] /; FreeQ[{c, d, e, m}, x] && IGtQ[n, 0]
  3391. Int[Times[Power[Pattern[x, Blank[]], Optional[Pattern[m, Blank[]]]], Power[Sin[Plus[Optional[Pattern[a, Blank[]]], Times[Rational[1, 2], Optional[Pattern[b, Blank[]]], Power[Pattern[x, Blank[]], Pattern[n, Blank[]]]]]], 2]], Pattern[x, Blank[Symbol]]] := Dist[1/2, Int[x^m, x], x] - Dist[1/2, Int[x^m*Cos[2*a + b*x^n], x], x] /; FreeQ[{a, b, m, n}, x]
  3392. Int[Times[Power[Cos[Plus[Optional[Pattern[a, Blank[]]], Times[Rational[1, 2], Optional[Pattern[b, Blank[]]], Power[Pattern[x, Blank[]], Pattern[n, Blank[]]]]]], 2], Power[Pattern[x, Blank[]], Optional[Pattern[m, Blank[]]]]], Pattern[x, Blank[Symbol]]] := Dist[1/2, Int[x^m, x], x] + Dist[1/2, Int[x^m*Cos[2*a + b*x^n], x], x] /; FreeQ[{a, b, m, n}, x]
  3393. Int[Times[Power[Pattern[x, Blank[]], Optional[Pattern[m, Blank[]]]], Power[Sin[Plus[Optional[Pattern[a, Blank[]]], Times[Optional[Pattern[b, Blank[]]], Power[Pattern[x, Blank[]], Pattern[n, Blank[]]]]]], Pattern[p, Blank[]]]], Pattern[x, Blank[Symbol]]] := Simp[(x^(m + 1)*Sin[a + b*x^n]^p)/(m + 1), x] - Dist[(b*n*p)/(m + 1), Int[Sin[a + b*x^n]^(p - 1)*Cos[a + b*x^n], x], x] /; FreeQ[{a, b}, x] && IGtQ[p, 1] && EqQ[m + n, 0] && NeQ[n, 1] && IntegerQ[n]
  3394. Int[Times[Power[Cos[Plus[Optional[Pattern[a, Blank[]]], Times[Optional[Pattern[b, Blank[]]], Power[Pattern[x, Blank[]], Pattern[n, Blank[]]]]]], Pattern[p, Blank[]]], Power[Pattern[x, Blank[]], Optional[Pattern[m, Blank[]]]]], Pattern[x, Blank[Symbol]]] := Simp[(x^(m + 1)*Cos[a + b*x^n]^p)/(m + 1), x] + Dist[(b*n*p)/(m + 1), Int[Cos[a + b*x^n]^(p - 1)*Sin[a + b*x^n], x], x] /; FreeQ[{a, b}, x] && IGtQ[p, 1] && EqQ[m + n, 0] && NeQ[n, 1] && IntegerQ[n]
  3395. Int[Times[Power[Pattern[x, Blank[]], Optional[Pattern[m, Blank[]]]], Power[Sin[Plus[Optional[Pattern[a, Blank[]]], Times[Optional[Pattern[b, Blank[]]], Power[Pattern[x, Blank[]], Pattern[n, Blank[]]]]]], Pattern[p, Blank[]]]], Pattern[x, Blank[Symbol]]] := Simp[(n*Sin[a + b*x^n]^p)/(b^2*n^2*p^2), x] + (Dist[(p - 1)/p, Int[x^m*Sin[a + b*x^n]^(p - 2), x], x] - Simp[(x^n*Cos[a + b*x^n]*Sin[a + b*x^n]^(p - 1))/(b*n*p), x]) /; FreeQ[{a, b, m, n}, x] && EqQ[m - 2*n + 1, 0] && GtQ[p, 1]
  3396. Int[Times[Power[Cos[Plus[Optional[Pattern[a, Blank[]]], Times[Optional[Pattern[b, Blank[]]], Power[Pattern[x, Blank[]], Pattern[n, Blank[]]]]]], Pattern[p, Blank[]]], Power[Pattern[x, Blank[]], Optional[Pattern[m, Blank[]]]]], Pattern[x, Blank[Symbol]]] := Simp[(n*Cos[a + b*x^n]^p)/(b^2*n^2*p^2), x] + (Dist[(p - 1)/p, Int[x^m*Cos[a + b*x^n]^(p - 2), x], x] + Simp[(x^n*Sin[a + b*x^n]*Cos[a + b*x^n]^(p - 1))/(b*n*p), x]) /; FreeQ[{a, b, m, n}, x] && EqQ[m - 2*n + 1, 0] && GtQ[p, 1]
  3397. Int[Times[Power[Pattern[x, Blank[]], Optional[Pattern[m, Blank[]]]], Power[Sin[Plus[Optional[Pattern[a, Blank[]]], Times[Optional[Pattern[b, Blank[]]], Power[Pattern[x, Blank[]], Pattern[n, Blank[]]]]]], Pattern[p, Blank[]]]], Pattern[x, Blank[Symbol]]] := Simp[((m - n + 1)*x^(m - 2*n + 1)*Sin[a + b*x^n]^p)/(b^2*n^2*p^2), x] + (Dist[(p - 1)/p, Int[x^m*Sin[a + b*x^n]^(p - 2), x], x] - Dist[((m - n + 1)*(m - 2*n + 1))/(b^2*n^2*p^2), Int[x^(m - 2*n)*Sin[a + b*x^n]^p, x], x] - Simp[(x^(m - n + 1)*Cos[a + b*x^n]*Sin[a + b*x^n]^(p - 1))/(b*n*p), x]) /; FreeQ[{a, b}, x] && GtQ[p, 1] && IGtQ[n, 0] && IGtQ[m, 2*n - 1]
  3398. Int[Times[Power[Cos[Plus[Optional[Pattern[a, Blank[]]], Times[Optional[Pattern[b, Blank[]]], Power[Pattern[x, Blank[]], Pattern[n, Blank[]]]]]], Pattern[p, Blank[]]], Power[Pattern[x, Blank[]], Optional[Pattern[m, Blank[]]]]], Pattern[x, Blank[Symbol]]] := Simp[((m - n + 1)*x^(m - 2*n + 1)*Cos[a + b*x^n]^p)/(b^2*n^2*p^2), x] + (Dist[(p - 1)/p, Int[x^m*Cos[a + b*x^n]^(p - 2), x], x] - Dist[((m - n + 1)*(m - 2*n + 1))/(b^2*n^2*p^2), Int[x^(m - 2*n)*Cos[a + b*x^n]^p, x], x] + Simp[(x^(m - n + 1)*Sin[a + b*x^n]*Cos[a + b*x^n]^(p - 1))/(b*n*p), x]) /; FreeQ[{a, b}, x] && GtQ[p, 1] && IGtQ[n, 0] && IGtQ[m, 2*n - 1]
  3399. Int[Times[Power[Pattern[x, Blank[]], Optional[Pattern[m, Blank[]]]], Power[Sin[Plus[Optional[Pattern[a, Blank[]]], Times[Optional[Pattern[b, Blank[]]], Power[Pattern[x, Blank[]], Pattern[n, Blank[]]]]]], Pattern[p, Blank[]]]], Pattern[x, Blank[Symbol]]] := Simp[(x^(m + 1)*Sin[a + b*x^n]^p)/(m + 1), x] + (Dist[(b^2*n^2*p*(p - 1))/((m + 1)*(m + n + 1)), Int[x^(m + 2*n)*Sin[a + b*x^n]^(p - 2), x], x] - Dist[(b^2*n^2*p^2)/((m + 1)*(m + n + 1)), Int[x^(m + 2*n)*Sin[a + b*x^n]^p, x], x] - Simp[(b*n*p*x^(m + n + 1)*Cos[a + b*x^n]*Sin[a + b*x^n]^(p - 1))/((m + 1)*(m + n + 1)), x]) /; FreeQ[{a, b}, x] && GtQ[p, 1] && IGtQ[n, 0] && ILtQ[m, -2*n + 1] && NeQ[m + n + 1, 0]
  3400. Int[Times[Power[Cos[Plus[Optional[Pattern[a, Blank[]]], Times[Optional[Pattern[b, Blank[]]], Power[Pattern[x, Blank[]], Pattern[n, Blank[]]]]]], Pattern[p, Blank[]]], Power[Pattern[x, Blank[]], Optional[Pattern[m, Blank[]]]]], Pattern[x, Blank[Symbol]]] := Simp[(x^(m + 1)*Cos[a + b*x^n]^p)/(m + 1), x] + (Dist[(b^2*n^2*p*(p - 1))/((m + 1)*(m + n + 1)), Int[x^(m + 2*n)*Cos[a + b*x^n]^(p - 2), x], x] - Dist[(b^2*n^2*p^2)/((m + 1)*(m + n + 1)), Int[x^(m + 2*n)*Cos[a + b*x^n]^p, x], x] + Simp[(b*n*p*x^(m + n + 1)*Sin[a + b*x^n]*Cos[a + b*x^n]^(p - 1))/((m + 1)*(m + n + 1)), x]) /; FreeQ[{a, b}, x] && GtQ[p, 1] && IGtQ[n, 0] && ILtQ[m, -2*n + 1] && NeQ[m + n + 1, 0]
  3401. Int[Times[Power[Times[Optional[Pattern[e, Blank[]]], Pattern[x, Blank[]]], Pattern[m, Blank[]]], Power[Plus[Optional[Pattern[a, Blank[]]], Times[Optional[Pattern[b, Blank[]]], Sin[Plus[Optional[Pattern[c, Blank[]]], Times[Optional[Pattern[d, Blank[]]], Power[Pattern[x, Blank[]], Pattern[n, Blank[]]]]]]]], Optional[Pattern[p, Blank[]]]]], Pattern[x, Blank[Symbol]]] := With[{k = Denominator[m]}, Dist[k/e, Subst[Int[x^(k*(m + 1) - 1)*(a + b*Sin[c + (d*x^(k*n))/e^n])^p, x], x, (e*x)^(1/k)], x]] /; FreeQ[{a, b, c, d, e}, x] && IntegerQ[p] && IGtQ[n, 0] && FractionQ[m]
  3402. Int[Times[Power[Plus[Optional[Pattern[a, Blank[]]], Times[Cos[Plus[Optional[Pattern[c, Blank[]]], Times[Optional[Pattern[d, Blank[]]], Power[Pattern[x, Blank[]], Pattern[n, Blank[]]]]]], Optional[Pattern[b, Blank[]]]]], Optional[Pattern[p, Blank[]]]], Power[Times[Optional[Pattern[e, Blank[]]], Pattern[x, Blank[]]], Pattern[m, Blank[]]]], Pattern[x, Blank[Symbol]]] := With[{k = Denominator[m]}, Dist[k/e, Subst[Int[x^(k*(m + 1) - 1)*(a + b*Cos[c + (d*x^(k*n))/e^n])^p, x], x, (e*x)^(1/k)], x]] /; FreeQ[{a, b, c, d, e}, x] && IntegerQ[p] && IGtQ[n, 0] && FractionQ[m]
  3403. Int[Times[Power[Times[Optional[Pattern[e, Blank[]]], Pattern[x, Blank[]]], Optional[Pattern[m, Blank[]]]], Power[Plus[Optional[Pattern[a, Blank[]]], Times[Optional[Pattern[b, Blank[]]], Sin[Plus[Optional[Pattern[c, Blank[]]], Times[Optional[Pattern[d, Blank[]]], Power[Pattern[x, Blank[]], Pattern[n, Blank[]]]]]]]], Pattern[p, Blank[]]]], Pattern[x, Blank[Symbol]]] := Int[ExpandTrigReduce[(e*x)^m, (a + b*Sin[c + d*x^n])^p, x], x] /; FreeQ[{a, b, c, d, e, m}, x] && IGtQ[p, 1] && IGtQ[n, 0]
  3404. Int[Times[Power[Plus[Optional[Pattern[a, Blank[]]], Times[Cos[Plus[Optional[Pattern[c, Blank[]]], Times[Optional[Pattern[d, Blank[]]], Power[Pattern[x, Blank[]], Pattern[n, Blank[]]]]]], Optional[Pattern[b, Blank[]]]]], Pattern[p, Blank[]]], Power[Times[Optional[Pattern[e, Blank[]]], Pattern[x, Blank[]]], Optional[Pattern[m, Blank[]]]]], Pattern[x, Blank[Symbol]]] := Int[ExpandTrigReduce[(e*x)^m, (a + b*Cos[c + d*x^n])^p, x], x] /; FreeQ[{a, b, c, d, e, m}, x] && IGtQ[p, 1] && IGtQ[n, 0]
  3405. Int[Times[Power[Pattern[x, Blank[]], Optional[Pattern[m, Blank[]]]], Power[Sin[Plus[Optional[Pattern[a, Blank[]]], Times[Optional[Pattern[b, Blank[]]], Power[Pattern[x, Blank[]], Pattern[n, Blank[]]]]]], Pattern[p, Blank[]]]], Pattern[x, Blank[Symbol]]] := Simp[(x^n*Cos[a + b*x^n]*Sin[a + b*x^n]^(p + 1))/(b*n*(p + 1)), x] + (Dist[(p + 2)/(p + 1), Int[x^m*Sin[a + b*x^n]^(p + 2), x], x] - Simp[(n*Sin[a + b*x^n]^(p + 2))/(b^2*n^2*(p + 1)*(p + 2)), x]) /; FreeQ[{a, b, m, n}, x] && EqQ[m - 2*n + 1, 0] && LtQ[p, -1] && NeQ[p, -2]
  3406. Int[Times[Power[Cos[Plus[Optional[Pattern[a, Blank[]]], Times[Optional[Pattern[b, Blank[]]], Power[Pattern[x, Blank[]], Pattern[n, Blank[]]]]]], Pattern[p, Blank[]]], Power[Pattern[x, Blank[]], Optional[Pattern[m, Blank[]]]]], Pattern[x, Blank[Symbol]]] := -Simp[(x^n*Sin[a + b*x^n]*Cos[a + b*x^n]^(p + 1))/(b*n*(p + 1)), x] + (Dist[(p + 2)/(p + 1), Int[x^m*Cos[a + b*x^n]^(p + 2), x], x] - Simp[(n*Cos[a + b*x^n]^(p + 2))/(b^2*n^2*(p + 1)*(p + 2)), x]) /; FreeQ[{a, b, m, n}, x] && EqQ[m - 2*n + 1, 0] && LtQ[p, -1] && NeQ[p, -2]
  3407. Int[Times[Power[Pattern[x, Blank[]], Optional[Pattern[m, Blank[]]]], Power[Sin[Plus[Optional[Pattern[a, Blank[]]], Times[Optional[Pattern[b, Blank[]]], Power[Pattern[x, Blank[]], Pattern[n, Blank[]]]]]], Pattern[p, Blank[]]]], Pattern[x, Blank[Symbol]]] := Simp[(x^(m - n + 1)*Cos[a + b*x^n]*Sin[a + b*x^n]^(p + 1))/(b*n*(p + 1)), x] + (Dist[(p + 2)/(p + 1), Int[x^m*Sin[a + b*x^n]^(p + 2), x], x] + Dist[((m - n + 1)*(m - 2*n + 1))/(b^2*n^2*(p + 1)*(p + 2)), Int[x^(m - 2*n)*Sin[a + b*x^n]^(p + 2), x], x] - Simp[((m - n + 1)*x^(m - 2*n + 1)*Sin[a + b*x^n]^(p + 2))/(b^2*n^2*(p + 1)*(p + 2)), x]) /; FreeQ[{a, b}, x] && LtQ[p, -1] && NeQ[p, -2] && IGtQ[n, 0] && IGtQ[m, 2*n - 1]
  3408. Int[Times[Power[Cos[Plus[Optional[Pattern[a, Blank[]]], Times[Optional[Pattern[b, Blank[]]], Power[Pattern[x, Blank[]], Pattern[n, Blank[]]]]]], Pattern[p, Blank[]]], Power[Pattern[x, Blank[]], Optional[Pattern[m, Blank[]]]]], Pattern[x, Blank[Symbol]]] := -Simp[(x^(m - n + 1)*Sin[a + b*x^n]*Cos[a + b*x^n]^(p + 1))/(b*n*(p + 1)), x] + (Dist[(p + 2)/(p + 1), Int[x^m*Cos[a + b*x^n]^(p + 2), x], x] + Dist[((m - n + 1)*(m - 2*n + 1))/(b^2*n^2*(p + 1)*(p + 2)), Int[x^(m - 2*n)*Cos[a + b*x^n]^(p + 2), x], x] - Simp[((m - n + 1)*x^(m - 2*n + 1)*Cos[a + b*x^n]^(p + 2))/(b^2*n^2*(p + 1)*(p + 2)), x]) /; FreeQ[{a, b}, x] && LtQ[p, -1] && NeQ[p, -2] && IGtQ[n, 0] && IGtQ[m, 2*n - 1]
  3409. Int[Times[Power[Pattern[x, Blank[]], Optional[Pattern[m, Blank[]]]], Power[Plus[Optional[Pattern[a, Blank[]]], Times[Optional[Pattern[b, Blank[]]], Sin[Plus[Optional[Pattern[c, Blank[]]], Times[Optional[Pattern[d, Blank[]]], Power[Pattern[x, Blank[]], Pattern[n, Blank[]]]]]]]], Optional[Pattern[p, Blank[]]]]], Pattern[x, Blank[Symbol]]] := -Subst[Int[(a + b*Sin[c + d/x^n])^p/x^(m + 2), x], x, 1/x] /; FreeQ[{a, b, c, d}, x] && IGtQ[p, 0] && ILtQ[n, 0] && IntegerQ[m] && EqQ[n, -2]
  3410. Int[Times[Power[Plus[Optional[Pattern[a, Blank[]]], Times[Cos[Plus[Optional[Pattern[c, Blank[]]], Times[Optional[Pattern[d, Blank[]]], Power[Pattern[x, Blank[]], Pattern[n, Blank[]]]]]], Optional[Pattern[b, Blank[]]]]], Optional[Pattern[p, Blank[]]]], Power[Pattern[x, Blank[]], Optional[Pattern[m, Blank[]]]]], Pattern[x, Blank[Symbol]]] := -Subst[Int[(a + b*Cos[c + d/x^n])^p/x^(m + 2), x], x, 1/x] /; FreeQ[{a, b, c, d}, x] && IGtQ[p, 0] && ILtQ[n, 0] && IntegerQ[m] && EqQ[n, -2]
  3411. Int[Times[Power[Times[Optional[Pattern[e, Blank[]]], Pattern[x, Blank[]]], Pattern[m, Blank[]]], Power[Plus[Optional[Pattern[a, Blank[]]], Times[Optional[Pattern[b, Blank[]]], Sin[Plus[Optional[Pattern[c, Blank[]]], Times[Optional[Pattern[d, Blank[]]], Power[Pattern[x, Blank[]], Pattern[n, Blank[]]]]]]]], Optional[Pattern[p, Blank[]]]]], Pattern[x, Blank[Symbol]]] := With[{k = Denominator[m]}, -Dist[k/e, Subst[Int[(a + b*Sin[c + d/(e^n*x^(k*n))])^p/x^(k*(m + 1) + 1), x], x, 1/(e*x)^(1/k)], x]] /; FreeQ[{a, b, c, d, e}, x] && IGtQ[p, 0] && ILtQ[n, 0] && FractionQ[m]
  3412. Int[Times[Power[Plus[Optional[Pattern[a, Blank[]]], Times[Cos[Plus[Optional[Pattern[c, Blank[]]], Times[Optional[Pattern[d, Blank[]]], Power[Pattern[x, Blank[]], Pattern[n, Blank[]]]]]], Optional[Pattern[b, Blank[]]]]], Optional[Pattern[p, Blank[]]]], Power[Times[Optional[Pattern[e, Blank[]]], Pattern[x, Blank[]]], Pattern[m, Blank[]]]], Pattern[x, Blank[Symbol]]] := With[{k = Denominator[m]}, -Dist[k/e, Subst[Int[(a + b*Cos[c + d/(e^n*x^(k*n))])^p/x^(k*(m + 1) + 1), x], x, 1/(e*x)^(1/k)], x]] /; FreeQ[{a, b, c, d, e}, x] && IGtQ[p, 0] && ILtQ[n, 0] && FractionQ[m]
  3413. Int[Times[Power[Times[Optional[Pattern[e, Blank[]]], Pattern[x, Blank[]]], Pattern[m, Blank[]]], Power[Plus[Optional[Pattern[a, Blank[]]], Times[Optional[Pattern[b, Blank[]]], Sin[Plus[Optional[Pattern[c, Blank[]]], Times[Optional[Pattern[d, Blank[]]], Power[Pattern[x, Blank[]], Pattern[n, Blank[]]]]]]]], Optional[Pattern[p, Blank[]]]]], Pattern[x, Blank[Symbol]]] := -Dist[(e*x)^m*(x^(-1))^m, Subst[Int[(a + b*Sin[c + d/x^n])^p/x^(m + 2), x], x, 1/x], x] /; FreeQ[{a, b, c, d, e, m}, x] && IGtQ[p, 0] && ILtQ[n, 0] && !RationalQ[m]
  3414. Int[Times[Power[Plus[Optional[Pattern[a, Blank[]]], Times[Cos[Plus[Optional[Pattern[c, Blank[]]], Times[Optional[Pattern[d, Blank[]]], Power[Pattern[x, Blank[]], Pattern[n, Blank[]]]]]], Optional[Pattern[b, Blank[]]]]], Optional[Pattern[p, Blank[]]]], Power[Times[Optional[Pattern[e, Blank[]]], Pattern[x, Blank[]]], Pattern[m, Blank[]]]], Pattern[x, Blank[Symbol]]] := -Dist[(e*x)^m*(x^(-1))^m, Subst[Int[(a + b*Cos[c + d/x^n])^p/x^(m + 2), x], x, 1/x], x] /; FreeQ[{a, b, c, d, e, m}, x] && IGtQ[p, 0] && ILtQ[n, 0] && !RationalQ[m]
  3415. Int[Times[Power[Pattern[x, Blank[]], Optional[Pattern[m, Blank[]]]], Power[Plus[Optional[Pattern[a, Blank[]]], Times[Optional[Pattern[b, Blank[]]], Sin[Plus[Optional[Pattern[c, Blank[]]], Times[Optional[Pattern[d, Blank[]]], Power[Pattern[x, Blank[]], Pattern[n, Blank[]]]]]]]], Optional[Pattern[p, Blank[]]]]], Pattern[x, Blank[Symbol]]] := Module[{k = Denominator[n]}, Dist[k, Subst[Int[x^(k*(m + 1) - 1)*(a + b*Sin[c + d*x^(k*n)])^p, x], x, x^(1/k)], x]] /; FreeQ[{a, b, c, d, m}, x] && IntegerQ[p] && FractionQ[n]
  3416. Int[Times[Power[Plus[Optional[Pattern[a, Blank[]]], Times[Cos[Plus[Optional[Pattern[c, Blank[]]], Times[Optional[Pattern[d, Blank[]]], Power[Pattern[x, Blank[]], Pattern[n, Blank[]]]]]], Optional[Pattern[b, Blank[]]]]], Optional[Pattern[p, Blank[]]]], Power[Pattern[x, Blank[]], Optional[Pattern[m, Blank[]]]]], Pattern[x, Blank[Symbol]]] := Module[{k = Denominator[n]}, Dist[k, Subst[Int[x^(k*(m + 1) - 1)*(a + b*Cos[c + d*x^(k*n)])^p, x], x, x^(1/k)], x]] /; FreeQ[{a, b, c, d, m}, x] && IntegerQ[p] && FractionQ[n]
  3417. Int[Times[Power[Times[Pattern[e, Blank[]], Pattern[x, Blank[]]], Pattern[m, Blank[]]], Power[Plus[Optional[Pattern[a, Blank[]]], Times[Optional[Pattern[b, Blank[]]], Sin[Plus[Optional[Pattern[c, Blank[]]], Times[Optional[Pattern[d, Blank[]]], Power[Pattern[x, Blank[]], Pattern[n, Blank[]]]]]]]], Optional[Pattern[p, Blank[]]]]], Pattern[x, Blank[Symbol]]] := Dist[(e^IntPart[m]*(e*x)^FracPart[m])/x^FracPart[m], Int[x^m*(a + b*Sin[c + d*x^n])^p, x], x] /; FreeQ[{a, b, c, d, e, m}, x] && IntegerQ[p] && FractionQ[n]
  3418. Int[Times[Power[Plus[Optional[Pattern[a, Blank[]]], Times[Cos[Plus[Optional[Pattern[c, Blank[]]], Times[Optional[Pattern[d, Blank[]]], Power[Pattern[x, Blank[]], Pattern[n, Blank[]]]]]], Optional[Pattern[b, Blank[]]]]], Optional[Pattern[p, Blank[]]]], Power[Times[Pattern[e, Blank[]], Pattern[x, Blank[]]], Pattern[m, Blank[]]]], Pattern[x, Blank[Symbol]]] := Dist[(e^IntPart[m]*(e*x)^FracPart[m])/x^FracPart[m], Int[x^m*(a + b*Cos[c + d*x^n])^p, x], x] /; FreeQ[{a, b, c, d, e, m}, x] && IntegerQ[p] && FractionQ[n]
  3419. Int[Times[Power[Pattern[x, Blank[]], Optional[Pattern[m, Blank[]]]], Power[Plus[Optional[Pattern[a, Blank[]]], Times[Optional[Pattern[b, Blank[]]], Sin[Plus[Optional[Pattern[c, Blank[]]], Times[Optional[Pattern[d, Blank[]]], Power[Pattern[x, Blank[]], Pattern[n, Blank[]]]]]]]], Optional[Pattern[p, Blank[]]]]], Pattern[x, Blank[Symbol]]] := Dist[1/(m + 1), Subst[Int[(a + b*Sin[c + d*x^Simplify[n/(m + 1)]])^p, x], x, x^(m + 1)], x] /; FreeQ[{a, b, c, d, m, n}, x] && IntegerQ[p] && NeQ[m, -1] && IGtQ[Simplify[n/(m + 1)], 0] && !IntegerQ[n]
  3420. Int[Times[Power[Plus[Optional[Pattern[a, Blank[]]], Times[Cos[Plus[Optional[Pattern[c, Blank[]]], Times[Optional[Pattern[d, Blank[]]], Power[Pattern[x, Blank[]], Pattern[n, Blank[]]]]]], Optional[Pattern[b, Blank[]]]]], Optional[Pattern[p, Blank[]]]], Power[Pattern[x, Blank[]], Optional[Pattern[m, Blank[]]]]], Pattern[x, Blank[Symbol]]] := Dist[1/(m + 1), Subst[Int[(a + b*Cos[c + d*x^Simplify[n/(m + 1)]])^p, x], x, x^(m + 1)], x] /; FreeQ[{a, b, c, d, m, n}, x] && IntegerQ[p] && NeQ[m, -1] && IGtQ[Simplify[n/(m + 1)], 0] && !IntegerQ[n]
  3421. Int[Times[Power[Times[Pattern[e, Blank[]], Pattern[x, Blank[]]], Pattern[m, Blank[]]], Power[Plus[Optional[Pattern[a, Blank[]]], Times[Optional[Pattern[b, Blank[]]], Sin[Plus[Optional[Pattern[c, Blank[]]], Times[Optional[Pattern[d, Blank[]]], Power[Pattern[x, Blank[]], Pattern[n, Blank[]]]]]]]], Optional[Pattern[p, Blank[]]]]], Pattern[x, Blank[Symbol]]] := Dist[(e^IntPart[m]*(e*x)^FracPart[m])/x^FracPart[m], Int[x^m*(a + b*Sin[c + d*x^n])^p, x], x] /; FreeQ[{a, b, c, d, e, m, n}, x] && IntegerQ[p] && NeQ[m, -1] && IGtQ[Simplify[n/(m + 1)], 0] && !IntegerQ[n]
  3422. Int[Times[Power[Plus[Optional[Pattern[a, Blank[]]], Times[Cos[Plus[Optional[Pattern[c, Blank[]]], Times[Optional[Pattern[d, Blank[]]], Power[Pattern[x, Blank[]], Pattern[n, Blank[]]]]]], Optional[Pattern[b, Blank[]]]]], Optional[Pattern[p, Blank[]]]], Power[Times[Pattern[e, Blank[]], Pattern[x, Blank[]]], Pattern[m, Blank[]]]], Pattern[x, Blank[Symbol]]] := Dist[(e^IntPart[m]*(e*x)^FracPart[m])/x^FracPart[m], Int[x^m*(a + b*Cos[c + d*x^n])^p, x], x] /; FreeQ[{a, b, c, d, e, m, n}, x] && IntegerQ[p] && NeQ[m, -1] && IGtQ[Simplify[n/(m + 1)], 0] && !IntegerQ[n]
  3423. Int[Times[Power[Times[Optional[Pattern[e, Blank[]]], Pattern[x, Blank[]]], Optional[Pattern[m, Blank[]]]], Sin[Plus[Optional[Pattern[c, Blank[]]], Times[Optional[Pattern[d, Blank[]]], Power[Pattern[x, Blank[]], Pattern[n, Blank[]]]]]]], Pattern[x, Blank[Symbol]]] := Dist[I/2, Int[(e*x)^m*E^(-(c*I) - d*I*x^n), x], x] - Dist[I/2, Int[(e*x)^m*E^(c*I + d*I*x^n), x], x] /; FreeQ[{c, d, e, m, n}, x]
  3424. Int[Times[Cos[Plus[Optional[Pattern[c, Blank[]]], Times[Optional[Pattern[d, Blank[]]], Power[Pattern[x, Blank[]], Pattern[n, Blank[]]]]]], Power[Times[Optional[Pattern[e, Blank[]]], Pattern[x, Blank[]]], Optional[Pattern[m, Blank[]]]]], Pattern[x, Blank[Symbol]]] := Dist[1/2, Int[(e*x)^m*E^(-(c*I) - d*I*x^n), x], x] + Dist[1/2, Int[(e*x)^m*E^(c*I + d*I*x^n), x], x] /; FreeQ[{c, d, e, m, n}, x]
  3425. Int[Times[Power[Times[Optional[Pattern[e, Blank[]]], Pattern[x, Blank[]]], Optional[Pattern[m, Blank[]]]], Power[Plus[Optional[Pattern[a, Blank[]]], Times[Optional[Pattern[b, Blank[]]], Sin[Plus[Optional[Pattern[c, Blank[]]], Times[Optional[Pattern[d, Blank[]]], Power[Pattern[x, Blank[]], Pattern[n, Blank[]]]]]]]], Pattern[p, Blank[]]]], Pattern[x, Blank[Symbol]]] := Int[ExpandTrigReduce[(e*x)^m, (a + b*Sin[c + d*x^n])^p, x], x] /; FreeQ[{a, b, c, d, e, m, n}, x] && IGtQ[p, 0]
  3426. Int[Times[Power[Plus[Optional[Pattern[a, Blank[]]], Times[Cos[Plus[Optional[Pattern[c, Blank[]]], Times[Optional[Pattern[d, Blank[]]], Power[Pattern[x, Blank[]], Pattern[n, Blank[]]]]]], Optional[Pattern[b, Blank[]]]]], Pattern[p, Blank[]]], Power[Times[Optional[Pattern[e, Blank[]]], Pattern[x, Blank[]]], Optional[Pattern[m, Blank[]]]]], Pattern[x, Blank[Symbol]]] := Int[ExpandTrigReduce[(e*x)^m, (a + b*Cos[c + d*x^n])^p, x], x] /; FreeQ[{a, b, c, d, e, m, n}, x] && IGtQ[p, 0]
  3427. Int[Times[Power[Times[Optional[Pattern[e, Blank[]]], Pattern[x, Blank[]]], Optional[Pattern[m, Blank[]]]], Power[Plus[Optional[Pattern[a, Blank[]]], Times[Optional[Pattern[b, Blank[]]], Sin[Plus[Optional[Pattern[c, Blank[]]], Times[Optional[Pattern[d, Blank[]]], Power[Pattern[x, Blank[]], Pattern[n, Blank[]]]]]]]], Optional[Pattern[p, Blank[]]]]], Pattern[x, Blank[Symbol]]] := Unintegrable[(e*x)^m*(a + b*Sin[c + d*x^n])^p, x] /; FreeQ[{a, b, c, d, e, m, n, p}, x]
  3428. Int[Times[Power[Plus[Optional[Pattern[a, Blank[]]], Times[Cos[Plus[Optional[Pattern[c, Blank[]]], Times[Optional[Pattern[d, Blank[]]], Power[Pattern[x, Blank[]], Pattern[n, Blank[]]]]]], Optional[Pattern[b, Blank[]]]]], Optional[Pattern[p, Blank[]]]], Power[Times[Optional[Pattern[e, Blank[]]], Pattern[x, Blank[]]], Optional[Pattern[m, Blank[]]]]], Pattern[x, Blank[Symbol]]] := Unintegrable[(e*x)^m*(a + b*Cos[c + d*x^n])^p, x] /; FreeQ[{a, b, c, d, e, m, n, p}, x]
  3429. Int[Times[Power[Times[Pattern[e, Blank[]], Pattern[x, Blank[]]], Optional[Pattern[m, Blank[]]]], Power[Plus[Optional[Pattern[a, Blank[]]], Times[Optional[Pattern[b, Blank[]]], Sin[Pattern[u, Blank[]]]]], Optional[Pattern[p, Blank[]]]]], Pattern[x, Blank[Symbol]]] := Int[(e*x)^m*(a + b*Sin[ExpandToSum[u, x]])^p, x] /; FreeQ[{a, b, e, m, p}, x] && BinomialQ[u, x] && !BinomialMatchQ[u, x]
  3430. Int[Times[Power[Plus[Optional[Pattern[a, Blank[]]], Times[Cos[Pattern[u, Blank[]]], Optional[Pattern[b, Blank[]]]]], Optional[Pattern[p, Blank[]]]], Power[Times[Pattern[e, Blank[]], Pattern[x, Blank[]]], Optional[Pattern[m, Blank[]]]]], Pattern[x, Blank[Symbol]]] := Int[(e*x)^m*(a + b*Cos[ExpandToSum[u, x]])^p, x] /; FreeQ[{a, b, e, m, p}, x] && BinomialQ[u, x] && !BinomialMatchQ[u, x]
  3431. Int[Times[Power[Plus[Optional[Pattern[g, Blank[]]], Times[Optional[Pattern[h, Blank[]]], Pattern[x, Blank[]]]], Optional[Pattern[m, Blank[]]]], Power[Plus[Optional[Pattern[a, Blank[]]], Times[Optional[Pattern[b, Blank[]]], Sin[Plus[Optional[Pattern[c, Blank[]]], Times[Optional[Pattern[d, Blank[]]], Power[Plus[Optional[Pattern[e, Blank[]]], Times[Optional[Pattern[f, Blank[]]], Pattern[x, Blank[]]]], Pattern[n, Blank[]]]]]]]], Optional[Pattern[p, Blank[]]]]], Pattern[x, Blank[Symbol]]] := Dist[1/(n*f), Subst[Int[ExpandIntegrand[(a + b*Sin[c + d*x])^p, x^(1/n - 1)*(g - (e*h)/f + (h*x^(1/n))/f)^m, x], x], x, (e + f*x)^n], x] /; FreeQ[{a, b, c, d, e, f, g, h, m}, x] && IGtQ[p, 0] && IntegerQ[1/n]
  3432. Int[Times[Power[Plus[Optional[Pattern[a, Blank[]]], Times[Cos[Plus[Optional[Pattern[c, Blank[]]], Times[Optional[Pattern[d, Blank[]]], Power[Plus[Optional[Pattern[e, Blank[]]], Times[Optional[Pattern[f, Blank[]]], Pattern[x, Blank[]]]], Pattern[n, Blank[]]]]]], Optional[Pattern[b, Blank[]]]]], Optional[Pattern[p, Blank[]]]], Power[Plus[Optional[Pattern[g, Blank[]]], Times[Optional[Pattern[h, Blank[]]], Pattern[x, Blank[]]]], Optional[Pattern[m, Blank[]]]]], Pattern[x, Blank[Symbol]]] := Dist[1/(n*f), Subst[Int[ExpandIntegrand[(a + b*Cos[c + d*x])^p, x^(1/n - 1)*(g - (e*h)/f + (h*x^(1/n))/f)^m, x], x], x, (e + f*x)^n], x] /; FreeQ[{a, b, c, d, e, f, g, h, m}, x] && IGtQ[p, 0] && IntegerQ[1/n]
  3433. Int[Times[Power[Plus[Optional[Pattern[g, Blank[]]], Times[Optional[Pattern[h, Blank[]]], Pattern[x, Blank[]]]], Optional[Pattern[m, Blank[]]]], Power[Plus[Optional[Pattern[a, Blank[]]], Times[Optional[Pattern[b, Blank[]]], Sin[Plus[Optional[Pattern[c, Blank[]]], Times[Optional[Pattern[d, Blank[]]], Power[Plus[Optional[Pattern[e, Blank[]]], Times[Optional[Pattern[f, Blank[]]], Pattern[x, Blank[]]]], Pattern[n, Blank[]]]]]]]], Optional[Pattern[p, Blank[]]]]], Pattern[x, Blank[Symbol]]] := Module[{k = If[FractionQ[n], Denominator[n], 1]}, Dist[k/f^(m + 1), Subst[Int[ExpandIntegrand[(a + b*Sin[c + d*x^(k*n)])^p, x^(k - 1)*(f*g - e*h + h*x^k)^m, x], x], x, (e + f*x)^(1/k)], x]] /; FreeQ[{a, b, c, d, e, f, g, h}, x] && IGtQ[p, 0] && IGtQ[m, 0]
  3434. Int[Times[Power[Plus[Optional[Pattern[a, Blank[]]], Times[Cos[Plus[Optional[Pattern[c, Blank[]]], Times[Optional[Pattern[d, Blank[]]], Power[Plus[Optional[Pattern[e, Blank[]]], Times[Optional[Pattern[f, Blank[]]], Pattern[x, Blank[]]]], Pattern[n, Blank[]]]]]], Optional[Pattern[b, Blank[]]]]], Optional[Pattern[p, Blank[]]]], Power[Plus[Optional[Pattern[g, Blank[]]], Times[Optional[Pattern[h, Blank[]]], Pattern[x, Blank[]]]], Optional[Pattern[m, Blank[]]]]], Pattern[x, Blank[Symbol]]] := Module[{k = If[FractionQ[n], Denominator[n], 1]}, Dist[k/f^(m + 1), Subst[Int[ExpandIntegrand[(a + b*Cos[c + d*x^(k*n)])^p, x^(k - 1)*(f*g - e*h + h*x^k)^m, x], x], x, (e + f*x)^(1/k)], x]] /; FreeQ[{a, b, c, d, e, f, g, h}, x] && IGtQ[p, 0] && IGtQ[m, 0]
  3435. Int[Times[Power[Plus[Optional[Pattern[g, Blank[]]], Times[Optional[Pattern[h, Blank[]]], Pattern[x, Blank[]]]], Optional[Pattern[m, Blank[]]]], Power[Plus[Optional[Pattern[a, Blank[]]], Times[Optional[Pattern[b, Blank[]]], Sin[Plus[Optional[Pattern[c, Blank[]]], Times[Optional[Pattern[d, Blank[]]], Power[Plus[Optional[Pattern[e, Blank[]]], Times[Optional[Pattern[f, Blank[]]], Pattern[x, Blank[]]]], Pattern[n, Blank[]]]]]]]], Optional[Pattern[p, Blank[]]]]], Pattern[x, Blank[Symbol]]] := Dist[1/f, Subst[Int[((h*x)/f)^m*(a + b*Sin[c + d*x^n])^p, x], x, e + f*x], x] /; FreeQ[{a, b, c, d, e, f, g, h, m}, x] && IGtQ[p, 0] && EqQ[f*g - e*h, 0]
  3436. Int[Times[Power[Plus[Optional[Pattern[a, Blank[]]], Times[Cos[Plus[Optional[Pattern[c, Blank[]]], Times[Optional[Pattern[d, Blank[]]], Power[Plus[Optional[Pattern[e, Blank[]]], Times[Optional[Pattern[f, Blank[]]], Pattern[x, Blank[]]]], Pattern[n, Blank[]]]]]], Optional[Pattern[b, Blank[]]]]], Optional[Pattern[p, Blank[]]]], Power[Plus[Optional[Pattern[g, Blank[]]], Times[Optional[Pattern[h, Blank[]]], Pattern[x, Blank[]]]], Optional[Pattern[m, Blank[]]]]], Pattern[x, Blank[Symbol]]] := Dist[1/f, Subst[Int[((h*x)/f)^m*(a + b*Cos[c + d*x^n])^p, x], x, e + f*x], x] /; FreeQ[{a, b, c, d, e, f, g, h, m}, x] && IGtQ[p, 0] && EqQ[f*g - e*h, 0]
  3437. Int[Times[Power[Plus[Optional[Pattern[g, Blank[]]], Times[Optional[Pattern[h, Blank[]]], Pattern[x, Blank[]]]], Optional[Pattern[m, Blank[]]]], Power[Plus[Optional[Pattern[a, Blank[]]], Times[Optional[Pattern[b, Blank[]]], Sin[Plus[Optional[Pattern[c, Blank[]]], Times[Optional[Pattern[d, Blank[]]], Power[Plus[Optional[Pattern[e, Blank[]]], Times[Optional[Pattern[f, Blank[]]], Pattern[x, Blank[]]]], Pattern[n, Blank[]]]]]]]], Optional[Pattern[p, Blank[]]]]], Pattern[x, Blank[Symbol]]] := Unintegrable[(g + h*x)^m*(a + b*Sin[c + d*(e + f*x)^n])^p, x] /; FreeQ[{a, b, c, d, e, f, g, h, m, n, p}, x]
  3438. Int[Times[Power[Plus[Optional[Pattern[a, Blank[]]], Times[Cos[Plus[Optional[Pattern[c, Blank[]]], Times[Optional[Pattern[d, Blank[]]], Power[Plus[Optional[Pattern[e, Blank[]]], Times[Optional[Pattern[f, Blank[]]], Pattern[x, Blank[]]]], Pattern[n, Blank[]]]]]], Optional[Pattern[b, Blank[]]]]], Optional[Pattern[p, Blank[]]]], Power[Plus[Optional[Pattern[g, Blank[]]], Times[Optional[Pattern[h, Blank[]]], Pattern[x, Blank[]]]], Optional[Pattern[m, Blank[]]]]], Pattern[x, Blank[Symbol]]] := Unintegrable[(g + h*x)^m*(a + b*Cos[c + d*(e + f*x)^n])^p, x] /; FreeQ[{a, b, c, d, e, f, g, h, m, n, p}, x]
  3439. Int[Times[Power[Pattern[v, Blank[]], Optional[Pattern[m, Blank[]]]], Power[Plus[Optional[Pattern[a, Blank[]]], Times[Optional[Pattern[b, Blank[]]], Sin[Plus[Optional[Pattern[c, Blank[]]], Times[Optional[Pattern[d, Blank[]]], Power[Pattern[u, Blank[]], Pattern[n, Blank[]]]]]]]], Optional[Pattern[p, Blank[]]]]], Pattern[x, Blank[Symbol]]] := Int[ExpandToSum[v, x]^m*(a + b*Sin[c + d*ExpandToSum[u, x]^n])^p, x] /; FreeQ[{a, b, c, d, m, n, p}, x] && LinearQ[u, x] && LinearQ[v, x] && !(LinearMatchQ[u, x] && LinearMatchQ[v, x])
  3440. Int[Times[Power[Plus[Optional[Pattern[a, Blank[]]], Times[Cos[Plus[Optional[Pattern[c, Blank[]]], Times[Optional[Pattern[d, Blank[]]], Power[Pattern[u, Blank[]], Pattern[n, Blank[]]]]]], Optional[Pattern[b, Blank[]]]]], Optional[Pattern[p, Blank[]]]], Power[Pattern[v, Blank[]], Optional[Pattern[m, Blank[]]]]], Pattern[x, Blank[Symbol]]] := Int[ExpandToSum[v, x]^m*(a + b*Cos[c + d*ExpandToSum[u, x]^n])^p, x] /; FreeQ[{a, b, c, d, m, n, p}, x] && LinearQ[u, x] && LinearQ[v, x] && !(LinearMatchQ[u, x] && LinearMatchQ[v, x])
  3441. Int[Times[Cos[Plus[Optional[Pattern[a, Blank[]]], Times[Optional[Pattern[b, Blank[]]], Power[Pattern[x, Blank[]], Optional[Pattern[n, Blank[]]]]]]], Power[Pattern[x, Blank[]], Optional[Pattern[m, Blank[]]]], Power[Sin[Plus[Optional[Pattern[a, Blank[]]], Times[Optional[Pattern[b, Blank[]]], Power[Pattern[x, Blank[]], Optional[Pattern[n, Blank[]]]]]]], Optional[Pattern[p, Blank[]]]]], Pattern[x, Blank[Symbol]]] := Simp[Sin[a + b*x^n]^(p + 1)/(b*n*(p + 1)), x] /; FreeQ[{a, b, m, n, p}, x] && EqQ[m, n - 1] && NeQ[p, -1]
  3442. Int[Times[Power[Cos[Plus[Optional[Pattern[a, Blank[]]], Times[Optional[Pattern[b, Blank[]]], Power[Pattern[x, Blank[]], Optional[Pattern[n, Blank[]]]]]]], Optional[Pattern[p, Blank[]]]], Power[Pattern[x, Blank[]], Optional[Pattern[m, Blank[]]]], Sin[Plus[Optional[Pattern[a, Blank[]]], Times[Optional[Pattern[b, Blank[]]], Power[Pattern[x, Blank[]], Optional[Pattern[n, Blank[]]]]]]]], Pattern[x, Blank[Symbol]]] := -Simp[Cos[a + b*x^n]^(p + 1)/(b*n*(p + 1)), x] /; FreeQ[{a, b, m, n, p}, x] && EqQ[m, n - 1] && NeQ[p, -1]
  3443. Int[Times[Cos[Plus[Optional[Pattern[a, Blank[]]], Times[Optional[Pattern[b, Blank[]]], Power[Pattern[x, Blank[]], Optional[Pattern[n, Blank[]]]]]]], Power[Pattern[x, Blank[]], Optional[Pattern[m, Blank[]]]], Power[Sin[Plus[Optional[Pattern[a, Blank[]]], Times[Optional[Pattern[b, Blank[]]], Power[Pattern[x, Blank[]], Optional[Pattern[n, Blank[]]]]]]], Optional[Pattern[p, Blank[]]]]], Pattern[x, Blank[Symbol]]] := Simp[(x^(m - n + 1)*Sin[a + b*x^n]^(p + 1))/(b*n*(p + 1)), x] - Dist[(m - n + 1)/(b*n*(p + 1)), Int[x^(m - n)*Sin[a + b*x^n]^(p + 1), x], x] /; FreeQ[{a, b, p}, x] && LtQ[0, n, m + 1] && NeQ[p, -1]
  3444. Int[Times[Power[Cos[Plus[Optional[Pattern[a, Blank[]]], Times[Optional[Pattern[b, Blank[]]], Power[Pattern[x, Blank[]], Optional[Pattern[n, Blank[]]]]]]], Optional[Pattern[p, Blank[]]]], Power[Pattern[x, Blank[]], Optional[Pattern[m, Blank[]]]], Sin[Plus[Optional[Pattern[a, Blank[]]], Times[Optional[Pattern[b, Blank[]]], Power[Pattern[x, Blank[]], Optional[Pattern[n, Blank[]]]]]]]], Pattern[x, Blank[Symbol]]] := -Simp[(x^(m - n + 1)*Cos[a + b*x^n]^(p + 1))/(b*n*(p + 1)), x] + Dist[(m - n + 1)/(b*n*(p + 1)), Int[x^(m - n)*Cos[a + b*x^n]^(p + 1), x], x] /; FreeQ[{a, b, p}, x] && LtQ[0, n, m + 1] && NeQ[p, -1]
  3445. Int[Sin[Plus[Optional[Pattern[a, Blank[]]], Times[Optional[Pattern[b, Blank[]]], Pattern[x, Blank[]]], Times[Optional[Pattern[c, Blank[]]], Power[Pattern[x, Blank[]], 2]]]], Pattern[x, Blank[Symbol]]] := Int[Sin[(b + 2*c*x)^2/(4*c)], x] /; FreeQ[{a, b, c}, x] && EqQ[b^2 - 4*a*c, 0]
  3446. Int[Cos[Plus[Optional[Pattern[a, Blank[]]], Times[Optional[Pattern[b, Blank[]]], Pattern[x, Blank[]]], Times[Optional[Pattern[c, Blank[]]], Power[Pattern[x, Blank[]], 2]]]], Pattern[x, Blank[Symbol]]] := Int[Cos[(b + 2*c*x)^2/(4*c)], x] /; FreeQ[{a, b, c}, x] && EqQ[b^2 - 4*a*c, 0]
  3447. Int[Sin[Plus[Optional[Pattern[a, Blank[]]], Times[Optional[Pattern[b, Blank[]]], Pattern[x, Blank[]]], Times[Optional[Pattern[c, Blank[]]], Power[Pattern[x, Blank[]], 2]]]], Pattern[x, Blank[Symbol]]] := Dist[Cos[(b^2 - 4*a*c)/(4*c)], Int[Sin[(b + 2*c*x)^2/(4*c)], x], x] - Dist[Sin[(b^2 - 4*a*c)/(4*c)], Int[Cos[(b + 2*c*x)^2/(4*c)], x], x] /; FreeQ[{a, b, c}, x] && NeQ[b^2 - 4*a*c, 0]
  3448. Int[Cos[Plus[Optional[Pattern[a, Blank[]]], Times[Optional[Pattern[b, Blank[]]], Pattern[x, Blank[]]], Times[Optional[Pattern[c, Blank[]]], Power[Pattern[x, Blank[]], 2]]]], Pattern[x, Blank[Symbol]]] := Dist[Cos[(b^2 - 4*a*c)/(4*c)], Int[Cos[(b + 2*c*x)^2/(4*c)], x], x] + Dist[Sin[(b^2 - 4*a*c)/(4*c)], Int[Sin[(b + 2*c*x)^2/(4*c)], x], x] /; FreeQ[{a, b, c}, x] && NeQ[b^2 - 4*a*c, 0]
  3449. Int[Power[Sin[Plus[Optional[Pattern[a, Blank[]]], Times[Optional[Pattern[b, Blank[]]], Pattern[x, Blank[]]], Times[Optional[Pattern[c, Blank[]]], Power[Pattern[x, Blank[]], 2]]]], Pattern[n, Blank[]]], Pattern[x, Blank[Symbol]]] := Int[ExpandTrigReduce[Sin[a + b*x + c*x^2]^n, x], x] /; FreeQ[{a, b, c}, x] && IGtQ[n, 1]
  3450. Int[Power[Cos[Plus[Optional[Pattern[a, Blank[]]], Times[Optional[Pattern[b, Blank[]]], Pattern[x, Blank[]]], Times[Optional[Pattern[c, Blank[]]], Power[Pattern[x, Blank[]], 2]]]], Pattern[n, Blank[]]], Pattern[x, Blank[Symbol]]] := Int[ExpandTrigReduce[Cos[a + b*x + c*x^2]^n, x], x] /; FreeQ[{a, b, c}, x] && IGtQ[n, 1]
  3451. Int[Power[Sin[Plus[Optional[Pattern[a, Blank[]]], Times[Optional[Pattern[b, Blank[]]], Pattern[x, Blank[]]], Times[Optional[Pattern[c, Blank[]]], Power[Pattern[x, Blank[]], 2]]]], Optional[Pattern[n, Blank[]]]], Pattern[x, Blank[Symbol]]] := Unintegrable[Sin[a + b*x + c*x^2]^n, x] /; FreeQ[{a, b, c, n}, x]
  3452. Int[Power[Cos[Plus[Optional[Pattern[a, Blank[]]], Times[Optional[Pattern[b, Blank[]]], Pattern[x, Blank[]]], Times[Optional[Pattern[c, Blank[]]], Power[Pattern[x, Blank[]], 2]]]], Optional[Pattern[n, Blank[]]]], Pattern[x, Blank[Symbol]]] := Unintegrable[Cos[a + b*x + c*x^2]^n, x] /; FreeQ[{a, b, c, n}, x]
  3453. Int[Power[Sin[Pattern[v, Blank[]]], Optional[Pattern[n, Blank[]]]], Pattern[x, Blank[Symbol]]] := Int[Sin[ExpandToSum[v, x]]^n, x] /; IGtQ[n, 0] && QuadraticQ[v, x] && !QuadraticMatchQ[v, x]
  3454. Int[Power[Cos[Pattern[v, Blank[]]], Optional[Pattern[n, Blank[]]]], Pattern[x, Blank[Symbol]]] := Int[Cos[ExpandToSum[v, x]]^n, x] /; IGtQ[n, 0] && QuadraticQ[v, x] && !QuadraticMatchQ[v, x]
  3455. Int[Times[Plus[Pattern[d, Blank[]], Times[Optional[Pattern[e, Blank[]]], Pattern[x, Blank[]]]], Sin[Plus[Optional[Pattern[a, Blank[]]], Times[Optional[Pattern[b, Blank[]]], Pattern[x, Blank[]]], Times[Optional[Pattern[c, Blank[]]], Power[Pattern[x, Blank[]], 2]]]]], Pattern[x, Blank[Symbol]]] := -Simp[(e*Cos[a + b*x + c*x^2])/(2*c), x] /; FreeQ[{a, b, c, d, e}, x] && EqQ[2*c*d - b*e, 0]
  3456. Int[Times[Cos[Plus[Optional[Pattern[a, Blank[]]], Times[Optional[Pattern[b, Blank[]]], Pattern[x, Blank[]]], Times[Optional[Pattern[c, Blank[]]], Power[Pattern[x, Blank[]], 2]]]], Plus[Pattern[d, Blank[]], Times[Optional[Pattern[e, Blank[]]], Pattern[x, Blank[]]]]], Pattern[x, Blank[Symbol]]] := Simp[(e*Sin[a + b*x + c*x^2])/(2*c), x] /; FreeQ[{a, b, c, d, e}, x] && EqQ[2*c*d - b*e, 0]
  3457. Int[Times[Power[Plus[Optional[Pattern[d, Blank[]]], Times[Optional[Pattern[e, Blank[]]], Pattern[x, Blank[]]]], Pattern[m, Blank[]]], Sin[Plus[Optional[Pattern[a, Blank[]]], Times[Optional[Pattern[b, Blank[]]], Pattern[x, Blank[]]], Times[Optional[Pattern[c, Blank[]]], Power[Pattern[x, Blank[]], 2]]]]], Pattern[x, Blank[Symbol]]] := -Simp[(e*(d + e*x)^(m - 1)*Cos[a + b*x + c*x^2])/(2*c), x] + Dist[(e^2*(m - 1))/(2*c), Int[(d + e*x)^(m - 2)*Cos[a + b*x + c*x^2], x], x] /; FreeQ[{a, b, c, d, e}, x] && EqQ[2*c*d - b*e, 0] && GtQ[m, 1]
  3458. Int[Times[Cos[Plus[Optional[Pattern[a, Blank[]]], Times[Optional[Pattern[b, Blank[]]], Pattern[x, Blank[]]], Times[Optional[Pattern[c, Blank[]]], Power[Pattern[x, Blank[]], 2]]]], Power[Plus[Optional[Pattern[d, Blank[]]], Times[Optional[Pattern[e, Blank[]]], Pattern[x, Blank[]]]], Pattern[m, Blank[]]]], Pattern[x, Blank[Symbol]]] := Simp[(e*(d + e*x)^(m - 1)*Sin[a + b*x + c*x^2])/(2*c), x] - Dist[(e^2*(m - 1))/(2*c), Int[(d + e*x)^(m - 2)*Sin[a + b*x + c*x^2], x], x] /; FreeQ[{a, b, c, d, e}, x] && EqQ[2*c*d - b*e, 0] && GtQ[m, 1]
  3459. Int[Times[Power[Plus[Optional[Pattern[d, Blank[]]], Times[Optional[Pattern[e, Blank[]]], Pattern[x, Blank[]]]], Pattern[m, Blank[]]], Sin[Plus[Optional[Pattern[a, Blank[]]], Times[Optional[Pattern[b, Blank[]]], Pattern[x, Blank[]]], Times[Optional[Pattern[c, Blank[]]], Power[Pattern[x, Blank[]], 2]]]]], Pattern[x, Blank[Symbol]]] := Simp[((d + e*x)^(m + 1)*Sin[a + b*x + c*x^2])/(e*(m + 1)), x] - Dist[(2*c)/(e^2*(m + 1)), Int[(d + e*x)^(m + 2)*Cos[a + b*x + c*x^2], x], x] /; FreeQ[{a, b, c, d, e}, x] && EqQ[2*c*d - b*e, 0] && LtQ[m, -1]
  3460. Int[Times[Cos[Plus[Optional[Pattern[a, Blank[]]], Times[Optional[Pattern[b, Blank[]]], Pattern[x, Blank[]]], Times[Optional[Pattern[c, Blank[]]], Power[Pattern[x, Blank[]], 2]]]], Power[Plus[Optional[Pattern[d, Blank[]]], Times[Optional[Pattern[e, Blank[]]], Pattern[x, Blank[]]]], Pattern[m, Blank[]]]], Pattern[x, Blank[Symbol]]] := Simp[((d + e*x)^(m + 1)*Cos[a + b*x + c*x^2])/(e*(m + 1)), x] + Dist[(2*c)/(e^2*(m + 1)), Int[(d + e*x)^(m + 2)*Sin[a + b*x + c*x^2], x], x] /; FreeQ[{a, b, c, d, e}, x] && EqQ[2*c*d - b*e, 0] && LtQ[m, -1]
  3461. Int[Times[Plus[Optional[Pattern[d, Blank[]]], Times[Optional[Pattern[e, Blank[]]], Pattern[x, Blank[]]]], Sin[Plus[Optional[Pattern[a, Blank[]]], Times[Optional[Pattern[b, Blank[]]], Pattern[x, Blank[]]], Times[Optional[Pattern[c, Blank[]]], Power[Pattern[x, Blank[]], 2]]]]], Pattern[x, Blank[Symbol]]] := -Simp[(e*Cos[a + b*x + c*x^2])/(2*c), x] + Dist[(2*c*d - b*e)/(2*c), Int[Sin[a + b*x + c*x^2], x], x] /; FreeQ[{a, b, c, d, e}, x] && NeQ[2*c*d - b*e, 0]
  3462. Int[Times[Cos[Plus[Optional[Pattern[a, Blank[]]], Times[Optional[Pattern[b, Blank[]]], Pattern[x, Blank[]]], Times[Optional[Pattern[c, Blank[]]], Power[Pattern[x, Blank[]], 2]]]], Plus[Optional[Pattern[d, Blank[]]], Times[Optional[Pattern[e, Blank[]]], Pattern[x, Blank[]]]]], Pattern[x, Blank[Symbol]]] := Simp[(e*Sin[a + b*x + c*x^2])/(2*c), x] + Dist[(2*c*d - b*e)/(2*c), Int[Cos[a + b*x + c*x^2], x], x] /; FreeQ[{a, b, c, d, e}, x] && NeQ[2*c*d - b*e, 0]
  3463. Int[Times[Power[Plus[Optional[Pattern[d, Blank[]]], Times[Optional[Pattern[e, Blank[]]], Pattern[x, Blank[]]]], Pattern[m, Blank[]]], Sin[Plus[Optional[Pattern[a, Blank[]]], Times[Optional[Pattern[b, Blank[]]], Pattern[x, Blank[]]], Times[Optional[Pattern[c, Blank[]]], Power[Pattern[x, Blank[]], 2]]]]], Pattern[x, Blank[Symbol]]] := -Simp[(e*(d + e*x)^(m - 1)*Cos[a + b*x + c*x^2])/(2*c), x] + (Dist[(e^2*(m - 1))/(2*c), Int[(d + e*x)^(m - 2)*Cos[a + b*x + c*x^2], x], x] - Dist[(b*e - 2*c*d)/(2*c), Int[(d + e*x)^(m - 1)*Sin[a + b*x + c*x^2], x], x]) /; FreeQ[{a, b, c, d, e}, x] && NeQ[b*e - 2*c*d, 0] && GtQ[m, 1]
  3464. Int[Times[Cos[Plus[Optional[Pattern[a, Blank[]]], Times[Optional[Pattern[b, Blank[]]], Pattern[x, Blank[]]], Times[Optional[Pattern[c, Blank[]]], Power[Pattern[x, Blank[]], 2]]]], Power[Plus[Optional[Pattern[d, Blank[]]], Times[Optional[Pattern[e, Blank[]]], Pattern[x, Blank[]]]], Pattern[m, Blank[]]]], Pattern[x, Blank[Symbol]]] := Simp[(e*(d + e*x)^(m - 1)*Sin[a + b*x + c*x^2])/(2*c), x] + (-Dist[(e^2*(m - 1))/(2*c), Int[(d + e*x)^(m - 2)*Sin[a + b*x + c*x^2], x], x] - Dist[(b*e - 2*c*d)/(2*c), Int[(d + e*x)^(m - 1)*Cos[a + b*x + c*x^2], x], x]) /; FreeQ[{a, b, c, d, e}, x] && NeQ[b*e - 2*c*d, 0] && GtQ[m, 1]
  3465. Int[Times[Power[Plus[Optional[Pattern[d, Blank[]]], Times[Optional[Pattern[e, Blank[]]], Pattern[x, Blank[]]]], Pattern[m, Blank[]]], Sin[Plus[Optional[Pattern[a, Blank[]]], Times[Optional[Pattern[b, Blank[]]], Pattern[x, Blank[]]], Times[Optional[Pattern[c, Blank[]]], Power[Pattern[x, Blank[]], 2]]]]], Pattern[x, Blank[Symbol]]] := Simp[((d + e*x)^(m + 1)*Sin[a + b*x + c*x^2])/(e*(m + 1)), x] + (-Dist[(2*c)/(e^2*(m + 1)), Int[(d + e*x)^(m + 2)*Cos[a + b*x + c*x^2], x], x] - Dist[(b*e - 2*c*d)/(e^2*(m + 1)), Int[(d + e*x)^(m + 1)*Cos[a + b*x + c*x^2], x], x]) /; FreeQ[{a, b, c, d, e}, x] && NeQ[b*e - 2*c*d, 0] && LtQ[m, -1]
  3466. Int[Times[Cos[Plus[Optional[Pattern[a, Blank[]]], Times[Optional[Pattern[b, Blank[]]], Pattern[x, Blank[]]], Times[Optional[Pattern[c, Blank[]]], Power[Pattern[x, Blank[]], 2]]]], Power[Plus[Optional[Pattern[d, Blank[]]], Times[Optional[Pattern[e, Blank[]]], Pattern[x, Blank[]]]], Pattern[m, Blank[]]]], Pattern[x, Blank[Symbol]]] := Simp[((d + e*x)^(m + 1)*Cos[a + b*x + c*x^2])/(e*(m + 1)), x] + (Dist[(2*c)/(e^2*(m + 1)), Int[(d + e*x)^(m + 2)*Sin[a + b*x + c*x^2], x], x] + Dist[(b*e - 2*c*d)/(e^2*(m + 1)), Int[(d + e*x)^(m + 1)*Sin[a + b*x + c*x^2], x], x]) /; FreeQ[{a, b, c, d, e}, x] && NeQ[b*e - 2*c*d, 0] && LtQ[m, -1]
  3467. Int[Times[Power[Plus[Optional[Pattern[d, Blank[]]], Times[Optional[Pattern[e, Blank[]]], Pattern[x, Blank[]]]], Optional[Pattern[m, Blank[]]]], Power[Sin[Plus[Optional[Pattern[a, Blank[]]], Times[Optional[Pattern[b, Blank[]]], Pattern[x, Blank[]]], Times[Optional[Pattern[c, Blank[]]], Power[Pattern[x, Blank[]], 2]]]], Pattern[n, Blank[]]]], Pattern[x, Blank[Symbol]]] := Int[ExpandTrigReduce[(d + e*x)^m, Sin[a + b*x + c*x^2]^n, x], x] /; FreeQ[{a, b, c, d, e, m}, x] && IGtQ[n, 1]
  3468. Int[Times[Power[Cos[Plus[Optional[Pattern[a, Blank[]]], Times[Optional[Pattern[b, Blank[]]], Pattern[x, Blank[]]], Times[Optional[Pattern[c, Blank[]]], Power[Pattern[x, Blank[]], 2]]]], Pattern[n, Blank[]]], Power[Plus[Optional[Pattern[d, Blank[]]], Times[Optional[Pattern[e, Blank[]]], Pattern[x, Blank[]]]], Optional[Pattern[m, Blank[]]]]], Pattern[x, Blank[Symbol]]] := Int[ExpandTrigReduce[(d + e*x)^m, Cos[a + b*x + c*x^2]^n, x], x] /; FreeQ[{a, b, c, d, e, m}, x] && IGtQ[n, 1]
  3469. Int[Times[Power[Plus[Optional[Pattern[d, Blank[]]], Times[Optional[Pattern[e, Blank[]]], Pattern[x, Blank[]]]], Optional[Pattern[m, Blank[]]]], Power[Sin[Plus[Optional[Pattern[a, Blank[]]], Times[Optional[Pattern[b, Blank[]]], Pattern[x, Blank[]]], Times[Optional[Pattern[c, Blank[]]], Power[Pattern[x, Blank[]], 2]]]], Optional[Pattern[n, Blank[]]]]], Pattern[x, Blank[Symbol]]] := Unintegrable[(d + e*x)^m*Sin[a + b*x + c*x^2]^n, x] /; FreeQ[{a, b, c, d, e, m, n}, x]
  3470. Int[Times[Power[Cos[Plus[Optional[Pattern[a, Blank[]]], Times[Optional[Pattern[b, Blank[]]], Pattern[x, Blank[]]], Times[Optional[Pattern[c, Blank[]]], Power[Pattern[x, Blank[]], 2]]]], Optional[Pattern[n, Blank[]]]], Power[Plus[Optional[Pattern[d, Blank[]]], Times[Optional[Pattern[e, Blank[]]], Pattern[x, Blank[]]]], Optional[Pattern[m, Blank[]]]]], Pattern[x, Blank[Symbol]]] := Unintegrable[(d + e*x)^m*Cos[a + b*x + c*x^2]^n, x] /; FreeQ[{a, b, c, d, e, m, n}, x]
  3471. Int[Times[Power[Pattern[u, Blank[]], Optional[Pattern[m, Blank[]]]], Power[Sin[Pattern[v, Blank[]]], Optional[Pattern[n, Blank[]]]]], Pattern[x, Blank[Symbol]]] := Int[ExpandToSum[u, x]^m*Sin[ExpandToSum[v, x]]^n, x] /; FreeQ[m, x] && IGtQ[n, 0] && LinearQ[u, x] && QuadraticQ[v, x] && !(LinearMatchQ[u, x] && QuadraticMatchQ[v, x])
  3472. Int[Times[Power[Cos[Pattern[v, Blank[]]], Optional[Pattern[n, Blank[]]]], Power[Pattern[u, Blank[]], Optional[Pattern[m, Blank[]]]]], Pattern[x, Blank[Symbol]]] := Int[ExpandToSum[u, x]^m*Cos[ExpandToSum[v, x]]^n, x] /; FreeQ[m, x] && IGtQ[n, 0] && LinearQ[u, x] && QuadraticQ[v, x] && !(LinearMatchQ[u, x] && QuadraticMatchQ[v, x])
  3473. Int[Power[Times[Optional[Pattern[b, Blank[]]], tan[Plus[Optional[Pattern[c, Blank[]]], Times[Optional[Pattern[d, Blank[]]], Pattern[x, Blank[]]]]]], Pattern[n, Blank[]]], Pattern[x, Blank[Symbol]]] := Simp[(b*(b*Tan[c + d*x])^(n - 1))/(d*(n - 1)), x] - Dist[b^2, Int[(b*Tan[c + d*x])^(n - 2), x], x] /; FreeQ[{b, c, d}, x] && GtQ[n, 1]
  3474. Int[Power[Times[Optional[Pattern[b, Blank[]]], tan[Plus[Optional[Pattern[c, Blank[]]], Times[Optional[Pattern[d, Blank[]]], Pattern[x, Blank[]]]]]], Pattern[n, Blank[]]], Pattern[x, Blank[Symbol]]] := Simp[(b*Tan[c + d*x])^(n + 1)/(b*d*(n + 1)), x] - Dist[1/b^2, Int[(b*Tan[c + d*x])^(n + 2), x], x] /; FreeQ[{b, c, d}, x] && LtQ[n, -1]
  3475. Int[tan[Plus[Optional[Pattern[c, Blank[]]], Times[Optional[Pattern[d, Blank[]]], Pattern[x, Blank[]]]]], Pattern[x, Blank[Symbol]]] := -Simp[Log[RemoveContent[Cos[c + d*x], x]]/d, x] /; FreeQ[{c, d}, x]
  3476. Int[Power[Times[Optional[Pattern[b, Blank[]]], tan[Plus[Optional[Pattern[c, Blank[]]], Times[Optional[Pattern[d, Blank[]]], Pattern[x, Blank[]]]]]], Pattern[n, Blank[]]], Pattern[x, Blank[Symbol]]] := Dist[b/d, Subst[Int[x^n/(b^2 + x^2), x], x, b*Tan[c + d*x]], x] /; FreeQ[{b, c, d, n}, x] && !IntegerQ[n]
  3477. Int[Power[Plus[Pattern[a, Blank[]], Times[Optional[Pattern[b, Blank[]]], tan[Plus[Optional[Pattern[c, Blank[]]], Times[Optional[Pattern[d, Blank[]]], Pattern[x, Blank[]]]]]]], 2], Pattern[x, Blank[Symbol]]] := Simp[(a^2 - b^2)*x, x] + (Dist[2*a*b, Int[Tan[c + d*x], x], x] + Simp[(b^2*Tan[c + d*x])/d, x]) /; FreeQ[{a, b, c, d}, x]
  3478. Int[Power[Plus[Pattern[a, Blank[]], Times[Optional[Pattern[b, Blank[]]], tan[Plus[Optional[Pattern[c, Blank[]]], Times[Optional[Pattern[d, Blank[]]], Pattern[x, Blank[]]]]]]], Pattern[n, Blank[]]], Pattern[x, Blank[Symbol]]] := Simp[(b*(a + b*Tan[c + d*x])^(n - 1))/(d*(n - 1)), x] + Dist[2*a, Int[(a + b*Tan[c + d*x])^(n - 1), x], x] /; FreeQ[{a, b, c, d}, x] && EqQ[a^2 + b^2, 0] && GtQ[n, 1]
  3479. Int[Power[Plus[Pattern[a, Blank[]], Times[Optional[Pattern[b, Blank[]]], tan[Plus[Optional[Pattern[c, Blank[]]], Times[Optional[Pattern[d, Blank[]]], Pattern[x, Blank[]]]]]]], Pattern[n, Blank[]]], Pattern[x, Blank[Symbol]]] := Simp[(a*(a + b*Tan[c + d*x])^n)/(2*b*d*n), x] + Dist[1/(2*a), Int[(a + b*Tan[c + d*x])^(n + 1), x], x] /; FreeQ[{a, b, c, d}, x] && EqQ[a^2 + b^2, 0] && LtQ[n, 0]
  3480. Int[Power[Plus[Pattern[a, Blank[]], Times[Optional[Pattern[b, Blank[]]], tan[Plus[Optional[Pattern[c, Blank[]]], Times[Optional[Pattern[d, Blank[]]], Pattern[x, Blank[]]]]]]], Rational[1, 2]], Pattern[x, Blank[Symbol]]] := Dist[(-2*b)/d, Subst[Int[1/(2*a - x^2), x], x, Sqrt[a + b*Tan[c + d*x]]], x] /; FreeQ[{a, b, c, d}, x] && EqQ[a^2 + b^2, 0]
  3481. Int[Power[Plus[Pattern[a, Blank[]], Times[Optional[Pattern[b, Blank[]]], tan[Plus[Optional[Pattern[c, Blank[]]], Times[Optional[Pattern[d, Blank[]]], Pattern[x, Blank[]]]]]]], Pattern[n, Blank[]]], Pattern[x, Blank[Symbol]]] := -Dist[b/d, Subst[Int[(a + x)^(n - 1)/(a - x), x], x, b*Tan[c + d*x]], x] /; FreeQ[{a, b, c, d, n}, x] && EqQ[a^2 + b^2, 0]
  3482. Int[Power[Plus[Pattern[a, Blank[]], Times[Optional[Pattern[b, Blank[]]], tan[Plus[Optional[Pattern[c, Blank[]]], Times[Optional[Pattern[d, Blank[]]], Pattern[x, Blank[]]]]]]], Pattern[n, Blank[]]], Pattern[x, Blank[Symbol]]] := Simp[(b*(a + b*Tan[c + d*x])^(n - 1))/(d*(n - 1)), x] + Int[(a^2 - b^2 + 2*a*b*Tan[c + d*x])*(a + b*Tan[c + d*x])^(n - 2), x] /; FreeQ[{a, b, c, d}, x] && NeQ[a^2 + b^2, 0] && GtQ[n, 1]
  3483. Int[Power[Plus[Pattern[a, Blank[]], Times[Optional[Pattern[b, Blank[]]], tan[Plus[Optional[Pattern[c, Blank[]]], Times[Optional[Pattern[d, Blank[]]], Pattern[x, Blank[]]]]]]], Pattern[n, Blank[]]], Pattern[x, Blank[Symbol]]] := Simp[(b*(a + b*Tan[c + d*x])^(n + 1))/(d*(n + 1)*(a^2 + b^2)), x] + Dist[1/(a^2 + b^2), Int[(a - b*Tan[c + d*x])*(a + b*Tan[c + d*x])^(n + 1), x], x] /; FreeQ[{a, b, c, d}, x] && NeQ[a^2 + b^2, 0] && LtQ[n, -1]
  3484. Int[Power[Plus[Pattern[a, Blank[]], Times[Optional[Pattern[b, Blank[]]], tan[Plus[Optional[Pattern[c, Blank[]]], Times[Optional[Pattern[d, Blank[]]], Pattern[x, Blank[]]]]]]], -1], Pattern[x, Blank[Symbol]]] := Simp[(a*x)/(a^2 + b^2), x] + Dist[b/(a^2 + b^2), Int[(b - a*Tan[c + d*x])/(a + b*Tan[c + d*x]), x], x] /; FreeQ[{a, b, c, d}, x] && NeQ[a^2 + b^2, 0]
  3485. Int[Power[Plus[Pattern[a, Blank[]], Times[Optional[Pattern[b, Blank[]]], tan[Plus[Optional[Pattern[c, Blank[]]], Times[Optional[Pattern[d, Blank[]]], Pattern[x, Blank[]]]]]]], Pattern[n, Blank[]]], Pattern[x, Blank[Symbol]]] := Dist[b/d, Subst[Int[(a + x)^n/(b^2 + x^2), x], x, b*Tan[c + d*x]], x] /; FreeQ[{a, b, c, d, n}, x] && NeQ[a^2 + b^2, 0]
  3486. Int[Times[Power[Times[Optional[Pattern[d, Blank[]]], sec[Plus[Optional[Pattern[e, Blank[]]], Times[Optional[Pattern[f, Blank[]]], Pattern[x, Blank[]]]]]], Optional[Pattern[m, Blank[]]]], Plus[Pattern[a, Blank[]], Times[Optional[Pattern[b, Blank[]]], tan[Plus[Optional[Pattern[e, Blank[]]], Times[Optional[Pattern[f, Blank[]]], Pattern[x, Blank[]]]]]]]], Pattern[x, Blank[Symbol]]] := Simp[(b*(d*Sec[e + f*x])^m)/(f*m), x] + Dist[a, Int[(d*Sec[e + f*x])^m, x], x] /; FreeQ[{a, b, d, e, f, m}, x] && (IntegerQ[2*m] || NeQ[a^2 + b^2, 0])
  3487. Int[Times[Power[sec[Plus[Optional[Pattern[e, Blank[]]], Times[Optional[Pattern[f, Blank[]]], Pattern[x, Blank[]]]]], Pattern[m, Blank[]]], Power[Plus[Pattern[a, Blank[]], Times[Optional[Pattern[b, Blank[]]], tan[Plus[Optional[Pattern[e, Blank[]]], Times[Optional[Pattern[f, Blank[]]], Pattern[x, Blank[]]]]]]], Pattern[n, Blank[]]]], Pattern[x, Blank[Symbol]]] := Dist[1/(a^(m - 2)*b*f), Subst[Int[(a - x)^(m/2 - 1)*(a + x)^(n + m/2 - 1), x], x, b*Tan[e + f*x]], x] /; FreeQ[{a, b, e, f, n}, x] && EqQ[a^2 + b^2, 0] && IntegerQ[m/2]
  3488. Int[Times[Power[Times[Optional[Pattern[d, Blank[]]], sec[Plus[Optional[Pattern[e, Blank[]]], Times[Optional[Pattern[f, Blank[]]], Pattern[x, Blank[]]]]]], Optional[Pattern[m, Blank[]]]], Power[Plus[Pattern[a, Blank[]], Times[Optional[Pattern[b, Blank[]]], tan[Plus[Optional[Pattern[e, Blank[]]], Times[Optional[Pattern[f, Blank[]]], Pattern[x, Blank[]]]]]]], Pattern[n, Blank[]]]], Pattern[x, Blank[Symbol]]] := Simp[(b*(d*Sec[e + f*x])^m*(a + b*Tan[e + f*x])^n)/(a*f*m), x] /; FreeQ[{a, b, d, e, f, m, n}, x] && EqQ[a^2 + b^2, 0] && EqQ[Simplify[m + n], 0]
  3489. Int[Times[sec[Plus[Optional[Pattern[e, Blank[]]], Times[Optional[Pattern[f, Blank[]]], Pattern[x, Blank[]]]]], Power[Plus[Pattern[a, Blank[]], Times[Optional[Pattern[b, Blank[]]], tan[Plus[Optional[Pattern[e, Blank[]]], Times[Optional[Pattern[f, Blank[]]], Pattern[x, Blank[]]]]]]], Rational[-1, 2]]], Pattern[x, Blank[Symbol]]] := Dist[(-2*a)/(b*f), Subst[Int[1/(2 - a*x^2), x], x, Sec[e + f*x]/Sqrt[a + b*Tan[e + f*x]]], x] /; FreeQ[{a, b, e, f}, x] && EqQ[a^2 + b^2, 0]
  3490. Int[Times[Power[Times[Optional[Pattern[d, Blank[]]], sec[Plus[Optional[Pattern[e, Blank[]]], Times[Optional[Pattern[f, Blank[]]], Pattern[x, Blank[]]]]]], Optional[Pattern[m, Blank[]]]], Power[Plus[Pattern[a, Blank[]], Times[Optional[Pattern[b, Blank[]]], tan[Plus[Optional[Pattern[e, Blank[]]], Times[Optional[Pattern[f, Blank[]]], Pattern[x, Blank[]]]]]]], Pattern[n, Blank[]]]], Pattern[x, Blank[Symbol]]] := Simp[(b*(d*Sec[e + f*x])^m*(a + b*Tan[e + f*x])^n)/(a*f*m), x] + Dist[a/(2*d^2), Int[(d*Sec[e + f*x])^(m + 2)*(a + b*Tan[e + f*x])^(n - 1), x], x] /; FreeQ[{a, b, d, e, f}, x] && EqQ[a^2 + b^2, 0] && EqQ[m/2 + n, 0] && GtQ[n, 0]
  3491. Int[Times[Power[Times[Optional[Pattern[d, Blank[]]], sec[Plus[Optional[Pattern[e, Blank[]]], Times[Optional[Pattern[f, Blank[]]], Pattern[x, Blank[]]]]]], Optional[Pattern[m, Blank[]]]], Power[Plus[Pattern[a, Blank[]], Times[Optional[Pattern[b, Blank[]]], tan[Plus[Optional[Pattern[e, Blank[]]], Times[Optional[Pattern[f, Blank[]]], Pattern[x, Blank[]]]]]]], Pattern[n, Blank[]]]], Pattern[x, Blank[Symbol]]] := Simp[(2*d^2*(d*Sec[e + f*x])^(m - 2)*(a + b*Tan[e + f*x])^(n + 1))/(b*f*(m - 2)), x] + Dist[(2*d^2)/a, Int[(d*Sec[e + f*x])^(m - 2)*(a + b*Tan[e + f*x])^(n + 1), x], x] /; FreeQ[{a, b, d, e, f}, x] && EqQ[a^2 + b^2, 0] && EqQ[m/2 + n, 0] && LtQ[n, -1]
  3492. Int[Times[Power[Times[Optional[Pattern[d, Blank[]]], sec[Plus[Optional[Pattern[e, Blank[]]], Times[Optional[Pattern[f, Blank[]]], Pattern[x, Blank[]]]]]], Pattern[m, Blank[]]], Power[Plus[Pattern[a, Blank[]], Times[Optional[Pattern[b, Blank[]]], tan[Plus[Optional[Pattern[e, Blank[]]], Times[Optional[Pattern[f, Blank[]]], Pattern[x, Blank[]]]]]]], Pattern[n, Blank[]]]], Pattern[x, Blank[Symbol]]] := Dist[((a/d)^(2*IntPart[n])*(a + b*Tan[e + f*x])^FracPart[n]*(a - b*Tan[e + f*x])^FracPart[n])/(d*Sec[e + f*x])^(2*FracPart[n]), Int[1/(a - b*Tan[e + f*x])^n, x], x] /; FreeQ[{a, b, d, e, f, m, n}, x] && EqQ[a^2 + b^2, 0] && EqQ[Simplify[m/2 + n], 0]
  3493. Int[Times[Power[Times[Optional[Pattern[d, Blank[]]], sec[Plus[Optional[Pattern[e, Blank[]]], Times[Optional[Pattern[f, Blank[]]], Pattern[x, Blank[]]]]]], Optional[Pattern[m, Blank[]]]], Power[Plus[Pattern[a, Blank[]], Times[Optional[Pattern[b, Blank[]]], tan[Plus[Optional[Pattern[e, Blank[]]], Times[Optional[Pattern[f, Blank[]]], Pattern[x, Blank[]]]]]]], Pattern[n, Blank[]]]], Pattern[x, Blank[Symbol]]] := Simp[(2*b*(d*Sec[e + f*x])^m*(a + b*Tan[e + f*x])^(n - 1))/(f*m), x] /; FreeQ[{a, b, d, e, f, m, n}, x] && EqQ[a^2 + b^2, 0] && EqQ[Simplify[m/2 + n - 1], 0]
  3494. Int[Times[Power[Times[Optional[Pattern[d, Blank[]]], sec[Plus[Optional[Pattern[e, Blank[]]], Times[Optional[Pattern[f, Blank[]]], Pattern[x, Blank[]]]]]], Optional[Pattern[m, Blank[]]]], Power[Plus[Pattern[a, Blank[]], Times[Optional[Pattern[b, Blank[]]], tan[Plus[Optional[Pattern[e, Blank[]]], Times[Optional[Pattern[f, Blank[]]], Pattern[x, Blank[]]]]]]], Pattern[n, Blank[]]]], Pattern[x, Blank[Symbol]]] := Simp[(b*(d*Sec[e + f*x])^m*(a + b*Tan[e + f*x])^(n - 1))/(f*(m + n - 1)), x] + Dist[(a*(m + 2*n - 2))/(m + n - 1), Int[(d*Sec[e + f*x])^m*(a + b*Tan[e + f*x])^(n - 1), x], x] /; FreeQ[{a, b, d, e, f, m, n}, x] && EqQ[a^2 + b^2, 0] && IGtQ[Simplify[m/2 + n - 1], 0] && !IntegerQ[n]
  3495. Int[Times[Power[Times[Optional[Pattern[d, Blank[]]], sec[Plus[Optional[Pattern[e, Blank[]]], Times[Optional[Pattern[f, Blank[]]], Pattern[x, Blank[]]]]]], Rational[1, 2]], Power[Plus[Pattern[a, Blank[]], Times[Optional[Pattern[b, Blank[]]], tan[Plus[Optional[Pattern[e, Blank[]]], Times[Optional[Pattern[f, Blank[]]], Pattern[x, Blank[]]]]]]], Rational[1, 2]]], Pattern[x, Blank[Symbol]]] := Dist[(-4*b*d^2)/f, Subst[Int[x^2/(a^2 + d^2*x^4), x], x, Sqrt[a + b*Tan[e + f*x]]/Sqrt[d*Sec[e + f*x]]], x] /; FreeQ[{a, b, d, e, f}, x] && EqQ[a^2 + b^2, 0]
  3496. Int[Times[Power[Times[Optional[Pattern[d, Blank[]]], sec[Plus[Optional[Pattern[e, Blank[]]], Times[Optional[Pattern[f, Blank[]]], Pattern[x, Blank[]]]]]], Pattern[m, Blank[]]], Power[Plus[Pattern[a, Blank[]], Times[Optional[Pattern[b, Blank[]]], tan[Plus[Optional[Pattern[e, Blank[]]], Times[Optional[Pattern[f, Blank[]]], Pattern[x, Blank[]]]]]]], Pattern[n, Blank[]]]], Pattern[x, Blank[Symbol]]] := Simp[(2*b*(d*Sec[e + f*x])^m*(a + b*Tan[e + f*x])^(n - 1))/(f*m), x] - Dist[(b^2*(m + 2*n - 2))/(d^2*m), Int[(d*Sec[e + f*x])^(m + 2)*(a + b*Tan[e + f*x])^(n - 2), x], x] /; FreeQ[{a, b, d, e, f}, x] && EqQ[a^2 + b^2, 0] && GtQ[n, 1] && ((IGtQ[n/2, 0] && ILtQ[m - 1/2, 0]) || (EqQ[n, 2] && LtQ[m, 0]) || (LeQ[m, -1] && GtQ[m + n, 0]) || (ILtQ[m, 0] && LtQ[m/2 + n - 1, 0] && IntegerQ[n]) || (EqQ[n, 3/2] && EqQ[m, -2^(-1)])) && IntegerQ[2*m]
  3497. Int[Times[Power[Times[Optional[Pattern[d, Blank[]]], sec[Plus[Optional[Pattern[e, Blank[]]], Times[Optional[Pattern[f, Blank[]]], Pattern[x, Blank[]]]]]], Pattern[m, Blank[]]], Power[Plus[Pattern[a, Blank[]], Times[Optional[Pattern[b, Blank[]]], tan[Plus[Optional[Pattern[e, Blank[]]], Times[Optional[Pattern[f, Blank[]]], Pattern[x, Blank[]]]]]]], Pattern[n, Blank[]]]], Pattern[x, Blank[Symbol]]] := Simp[(b*(d*Sec[e + f*x])^m*(a + b*Tan[e + f*x])^n)/(a*f*m), x] + Dist[(a*(m + n))/(m*d^2), Int[(d*Sec[e + f*x])^(m + 2)*(a + b*Tan[e + f*x])^(n - 1), x], x] /; FreeQ[{a, b, d, e, f}, x] && EqQ[a^2 + b^2, 0] && GtQ[n, 0] && LtQ[m, -1] && IntegersQ[2*m, 2*n]
  3498. Int[Times[Power[Times[Optional[Pattern[d, Blank[]]], sec[Plus[Optional[Pattern[e, Blank[]]], Times[Optional[Pattern[f, Blank[]]], Pattern[x, Blank[]]]]]], Optional[Pattern[m, Blank[]]]], Power[Plus[Pattern[a, Blank[]], Times[Optional[Pattern[b, Blank[]]], tan[Plus[Optional[Pattern[e, Blank[]]], Times[Optional[Pattern[f, Blank[]]], Pattern[x, Blank[]]]]]]], Pattern[n, Blank[]]]], Pattern[x, Blank[Symbol]]] := Simp[(b*(d*Sec[e + f*x])^m*(a + b*Tan[e + f*x])^(n - 1))/(f*(m + n - 1)), x] + Dist[(a*(m + 2*n - 2))/(m + n - 1), Int[(d*Sec[e + f*x])^m*(a + b*Tan[e + f*x])^(n - 1), x], x] /; FreeQ[{a, b, d, e, f, m}, x] && EqQ[a^2 + b^2, 0] && GtQ[n, 0] && NeQ[m + n - 1, 0] && IntegersQ[2*m, 2*n]
  3499. Int[Times[Power[Times[Optional[Pattern[d, Blank[]]], sec[Plus[Optional[Pattern[e, Blank[]]], Times[Optional[Pattern[f, Blank[]]], Pattern[x, Blank[]]]]]], Rational[3, 2]], Power[Plus[Pattern[a, Blank[]], Times[Optional[Pattern[b, Blank[]]], tan[Plus[Optional[Pattern[e, Blank[]]], Times[Optional[Pattern[f, Blank[]]], Pattern[x, Blank[]]]]]]], Rational[-1, 2]]], Pattern[x, Blank[Symbol]]] := Dist[(d*Sec[e + f*x])/(Sqrt[a - b*Tan[e + f*x]]*Sqrt[a + b*Tan[e + f*x]]), Int[Sqrt[d*Sec[e + f*x]]*Sqrt[a - b*Tan[e + f*x]], x], x] /; FreeQ[{a, b, d, e, f}, x] && EqQ[a^2 + b^2, 0]
  3500. Int[Times[Power[Times[Optional[Pattern[d, Blank[]]], sec[Plus[Optional[Pattern[e, Blank[]]], Times[Optional[Pattern[f, Blank[]]], Pattern[x, Blank[]]]]]], Pattern[m, Blank[]]], Power[Plus[Pattern[a, Blank[]], Times[Optional[Pattern[b, Blank[]]], tan[Plus[Optional[Pattern[e, Blank[]]], Times[Optional[Pattern[f, Blank[]]], Pattern[x, Blank[]]]]]]], Pattern[n, Blank[]]]], Pattern[x, Blank[Symbol]]] := Simp[(2*d^2*(d*Sec[e + f*x])^(m - 2)*(a + b*Tan[e + f*x])^(n + 1))/(b*f*(m + 2*n)), x] - Dist[(d^2*(m - 2))/(b^2*(m + 2*n)), Int[(d*Sec[e + f*x])^(m - 2)*(a + b*Tan[e + f*x])^(n + 2), x], x] /; FreeQ[{a, b, d, e, f, m}, x] && EqQ[a^2 + b^2, 0] && LtQ[n, -1] && ((ILtQ[n/2, 0] && IGtQ[m - 1/2, 0]) || EqQ[n, -2] || IGtQ[m + n, 0] || (IntegersQ[n, m + 1/2] && GtQ[2*m + n + 1, 0])) && IntegerQ[2*m]
  3501. Int[Times[Power[Times[Optional[Pattern[d, Blank[]]], sec[Plus[Optional[Pattern[e, Blank[]]], Times[Optional[Pattern[f, Blank[]]], Pattern[x, Blank[]]]]]], Optional[Pattern[m, Blank[]]]], Power[Plus[Pattern[a, Blank[]], Times[Optional[Pattern[b, Blank[]]], tan[Plus[Optional[Pattern[e, Blank[]]], Times[Optional[Pattern[f, Blank[]]], Pattern[x, Blank[]]]]]]], Pattern[n, Blank[]]]], Pattern[x, Blank[Symbol]]] := Simp[(d^2*(d*Sec[e + f*x])^(m - 2)*(a + b*Tan[e + f*x])^(n + 1))/(b*f*(m + n - 1)), x] + Dist[(d^2*(m - 2))/(a*(m + n - 1)), Int[(d*Sec[e + f*x])^(m - 2)*(a + b*Tan[e + f*x])^(n + 1), x], x] /; FreeQ[{a, b, d, e, f}, x] && EqQ[a^2 + b^2, 0] && LtQ[n, 0] && GtQ[m, 1] && !ILtQ[m + n, 0] && NeQ[m + n - 1, 0] && IntegersQ[2*m, 2*n]
  3502. Int[Times[Power[Times[Optional[Pattern[d, Blank[]]], sec[Plus[Optional[Pattern[e, Blank[]]], Times[Optional[Pattern[f, Blank[]]], Pattern[x, Blank[]]]]]], Optional[Pattern[m, Blank[]]]], Power[Plus[Pattern[a, Blank[]], Times[Optional[Pattern[b, Blank[]]], tan[Plus[Optional[Pattern[e, Blank[]]], Times[Optional[Pattern[f, Blank[]]], Pattern[x, Blank[]]]]]]], Pattern[n, Blank[]]]], Pattern[x, Blank[Symbol]]] := Simp[(a*(d*Sec[e + f*x])^m*(a + b*Tan[e + f*x])^n)/(b*f*(m + 2*n)), x] + Dist[Simplify[m + n]/(a*(m + 2*n)), Int[(d*Sec[e + f*x])^m*(a + b*Tan[e + f*x])^(n + 1), x], x] /; FreeQ[{a, b, d, e, f, m}, x] && EqQ[a^2 + b^2, 0] && LtQ[n, 0] && NeQ[m + 2*n, 0] && IntegersQ[2*m, 2*n]
  3503. Int[Times[Power[Times[Optional[Pattern[d, Blank[]]], sec[Plus[Optional[Pattern[e, Blank[]]], Times[Optional[Pattern[f, Blank[]]], Pattern[x, Blank[]]]]]], Optional[Pattern[m, Blank[]]]], Power[Plus[Pattern[a, Blank[]], Times[Optional[Pattern[b, Blank[]]], tan[Plus[Optional[Pattern[e, Blank[]]], Times[Optional[Pattern[f, Blank[]]], Pattern[x, Blank[]]]]]]], Pattern[n, Blank[]]]], Pattern[x, Blank[Symbol]]] := Simp[(b*(d*Sec[e + f*x])^m*(a + b*Tan[e + f*x])^(n - 1))/(f*Simplify[m + n - 1]), x] + Dist[(a*(m + 2*n - 2))/Simplify[m + n - 1], Int[(d*Sec[e + f*x])^m*(a + b*Tan[e + f*x])^(n - 1), x], x] /; FreeQ[{a, b, d, e, f, m, n}, x] && EqQ[a^2 + b^2, 0] && IGtQ[Simplify[m + n - 1], 0] && RationalQ[n]
  3504. Int[Times[Power[Times[Optional[Pattern[d, Blank[]]], sec[Plus[Optional[Pattern[e, Blank[]]], Times[Optional[Pattern[f, Blank[]]], Pattern[x, Blank[]]]]]], Optional[Pattern[m, Blank[]]]], Power[Plus[Pattern[a, Blank[]], Times[Optional[Pattern[b, Blank[]]], tan[Plus[Optional[Pattern[e, Blank[]]], Times[Optional[Pattern[f, Blank[]]], Pattern[x, Blank[]]]]]]], Pattern[n, Blank[]]]], Pattern[x, Blank[Symbol]]] := Simp[(a*(d*Sec[e + f*x])^m*(a + b*Tan[e + f*x])^n)/(b*f*(m + 2*n)), x] + Dist[Simplify[m + n]/(a*(m + 2*n)), Int[(d*Sec[e + f*x])^m*(a + b*Tan[e + f*x])^(n + 1), x], x] /; FreeQ[{a, b, d, e, f, m, n}, x] && EqQ[a^2 + b^2, 0] && ILtQ[Simplify[m + n], 0] && NeQ[m + 2*n, 0]
  3505. Int[Times[Power[Times[Optional[Pattern[d, Blank[]]], sec[Plus[Optional[Pattern[e, Blank[]]], Times[Optional[Pattern[f, Blank[]]], Pattern[x, Blank[]]]]]], Optional[Pattern[m, Blank[]]]], Power[Plus[Pattern[a, Blank[]], Times[Optional[Pattern[b, Blank[]]], tan[Plus[Optional[Pattern[e, Blank[]]], Times[Optional[Pattern[f, Blank[]]], Pattern[x, Blank[]]]]]]], Optional[Pattern[n, Blank[]]]]], Pattern[x, Blank[Symbol]]] := Dist[(d*Sec[e + f*x])^m/((a + b*Tan[e + f*x])^(m/2)*(a - b*Tan[e + f*x])^(m/2)), Int[(a + b*Tan[e + f*x])^(m/2 + n)*(a - b*Tan[e + f*x])^(m/2), x], x] /; FreeQ[{a, b, d, e, f, m, n}, x] && EqQ[a^2 + b^2, 0]
  3506. Int[Times[Power[sec[Plus[Optional[Pattern[e, Blank[]]], Times[Optional[Pattern[f, Blank[]]], Pattern[x, Blank[]]]]], Pattern[m, Blank[]]], Power[Plus[Pattern[a, Blank[]], Times[Optional[Pattern[b, Blank[]]], tan[Plus[Optional[Pattern[e, Blank[]]], Times[Optional[Pattern[f, Blank[]]], Pattern[x, Blank[]]]]]]], Pattern[n, Blank[]]]], Pattern[x, Blank[Symbol]]] := Dist[1/(b*f), Subst[Int[(a + x)^n*(1 + x^2/b^2)^(m/2 - 1), x], x, b*Tan[e + f*x]], x] /; FreeQ[{a, b, e, f, n}, x] && NeQ[a^2 + b^2, 0] && IntegerQ[m/2]
  3507. Int[Times[Power[sec[Plus[Optional[Pattern[e, Blank[]]], Times[Optional[Pattern[f, Blank[]]], Pattern[x, Blank[]]]]], -1], Power[Plus[Pattern[a, Blank[]], Times[Optional[Pattern[b, Blank[]]], tan[Plus[Optional[Pattern[e, Blank[]]], Times[Optional[Pattern[f, Blank[]]], Pattern[x, Blank[]]]]]]], 2]], Pattern[x, Blank[Symbol]]] := Simp[(b^2*ArcTanh[Sin[e + f*x]])/f, x] + (-Simp[(2*a*b*Cos[e + f*x])/f, x] + Simp[((a^2 - b^2)*Sin[e + f*x])/f, x]) /; FreeQ[{a, b, e, f}, x] && NeQ[a^2 + b^2, 0]
  3508. Int[Times[Power[Times[Optional[Pattern[d, Blank[]]], sec[Plus[Optional[Pattern[e, Blank[]]], Times[Optional[Pattern[f, Blank[]]], Pattern[x, Blank[]]]]]], Optional[Pattern[m, Blank[]]]], Power[Plus[Pattern[a, Blank[]], Times[Optional[Pattern[b, Blank[]]], tan[Plus[Optional[Pattern[e, Blank[]]], Times[Optional[Pattern[f, Blank[]]], Pattern[x, Blank[]]]]]]], 2]], Pattern[x, Blank[Symbol]]] := Simp[(b*(d*Sec[e + f*x])^m*(a + b*Tan[e + f*x]))/(f*(m + 1)), x] + Dist[1/(m + 1), Int[(d*Sec[e + f*x])^m*(a^2*(m + 1) - b^2 + a*b*(m + 2)*Tan[e + f*x]), x], x] /; FreeQ[{a, b, d, e, f, m}, x] && NeQ[a^2 + b^2, 0] && NeQ[m, -1]
  3509. Int[Times[sec[Plus[Optional[Pattern[e, Blank[]]], Times[Optional[Pattern[f, Blank[]]], Pattern[x, Blank[]]]]], Power[Plus[Pattern[a, Blank[]], Times[Optional[Pattern[b, Blank[]]], tan[Plus[Optional[Pattern[e, Blank[]]], Times[Optional[Pattern[f, Blank[]]], Pattern[x, Blank[]]]]]]], -1]], Pattern[x, Blank[Symbol]]] := -Dist[f^(-1), Subst[Int[1/(a^2 + b^2 - x^2), x], x, (b - a*Tan[e + f*x])/Sec[e + f*x]], x] /; FreeQ[{a, b, e, f}, x] && NeQ[a^2 + b^2, 0]
  3510. Int[Times[Power[Times[Optional[Pattern[d, Blank[]]], sec[Plus[Optional[Pattern[e, Blank[]]], Times[Optional[Pattern[f, Blank[]]], Pattern[x, Blank[]]]]]], Pattern[m, Blank[]]], Power[Plus[Pattern[a, Blank[]], Times[Optional[Pattern[b, Blank[]]], tan[Plus[Optional[Pattern[e, Blank[]]], Times[Optional[Pattern[f, Blank[]]], Pattern[x, Blank[]]]]]]], -1]], Pattern[x, Blank[Symbol]]] := -Dist[d^2/b^2, Int[(d*Sec[e + f*x])^(m - 2)*(a - b*Tan[e + f*x]), x], x] + Dist[(d^2*(a^2 + b^2))/b^2, Int[(d*Sec[e + f*x])^(m - 2)/(a + b*Tan[e + f*x]), x], x] /; FreeQ[{a, b, d, e, f}, x] && NeQ[a^2 + b^2, 0] && IGtQ[m, 1]
  3511. Int[Times[Power[Times[Optional[Pattern[d, Blank[]]], sec[Plus[Optional[Pattern[e, Blank[]]], Times[Optional[Pattern[f, Blank[]]], Pattern[x, Blank[]]]]]], Pattern[m, Blank[]]], Power[Plus[Pattern[a, Blank[]], Times[Optional[Pattern[b, Blank[]]], tan[Plus[Optional[Pattern[e, Blank[]]], Times[Optional[Pattern[f, Blank[]]], Pattern[x, Blank[]]]]]]], -1]], Pattern[x, Blank[Symbol]]] := Dist[1/(a^2 + b^2), Int[(d*Sec[e + f*x])^m*(a - b*Tan[e + f*x]), x], x] + Dist[b^2/(d^2*(a^2 + b^2)), Int[(d*Sec[e + f*x])^(m + 2)/(a + b*Tan[e + f*x]), x], x] /; FreeQ[{a, b, d, e, f}, x] && NeQ[a^2 + b^2, 0] && ILtQ[m, 0]
  3512. Int[Times[Power[Times[Optional[Pattern[d, Blank[]]], sec[Plus[Optional[Pattern[e, Blank[]]], Times[Optional[Pattern[f, Blank[]]], Pattern[x, Blank[]]]]]], Optional[Pattern[m, Blank[]]]], Power[Plus[Pattern[a, Blank[]], Times[Optional[Pattern[b, Blank[]]], tan[Plus[Optional[Pattern[e, Blank[]]], Times[Optional[Pattern[f, Blank[]]], Pattern[x, Blank[]]]]]]], Pattern[n, Blank[]]]], Pattern[x, Blank[Symbol]]] := Dist[(d^(2*IntPart[m/2])*(d*Sec[e + f*x])^(2*FracPart[m/2]))/(b*f*(Sec[e + f*x]^2)^FracPart[m/2]), Subst[Int[(a + x)^n*(1 + x^2/b^2)^(m/2 - 1), x], x, b*Tan[e + f*x]], x] /; FreeQ[{a, b, d, e, f, m, n}, x] && NeQ[a^2 + b^2, 0] && !IntegerQ[m/2]
  3513. Int[Times[Power[Times[cos[Plus[Optional[Pattern[e, Blank[]]], Times[Optional[Pattern[f, Blank[]]], Pattern[x, Blank[]]]]], Optional[Pattern[d, Blank[]]]], Rational[-1, 2]], Power[Plus[Pattern[a, Blank[]], Times[Optional[Pattern[b, Blank[]]], tan[Plus[Optional[Pattern[e, Blank[]]], Times[Optional[Pattern[f, Blank[]]], Pattern[x, Blank[]]]]]]], Rational[1, 2]]], Pattern[x, Blank[Symbol]]] := Dist[(-4*b)/f, Subst[Int[x^2/(a^2*d^2 + x^4), x], x, Sqrt[d*Cos[e + f*x]]*Sqrt[a + b*Tan[e + f*x]]], x] /; FreeQ[{a, b, d, e, f}, x] && EqQ[a^2 + b^2, 0]
  3514. Int[Times[Power[Times[cos[Plus[Optional[Pattern[e, Blank[]]], Times[Optional[Pattern[f, Blank[]]], Pattern[x, Blank[]]]]], Optional[Pattern[d, Blank[]]]], Rational[-3, 2]], Power[Plus[Pattern[a, Blank[]], Times[Optional[Pattern[b, Blank[]]], tan[Plus[Optional[Pattern[e, Blank[]]], Times[Optional[Pattern[f, Blank[]]], Pattern[x, Blank[]]]]]]], Rational[-1, 2]]], Pattern[x, Blank[Symbol]]] := Dist[1/(d*Cos[e + f*x]*Sqrt[a - b*Tan[e + f*x]]*Sqrt[a + b*Tan[e + f*x]]), Int[Sqrt[a - b*Tan[e + f*x]]/Sqrt[d*Cos[e + f*x]], x], x] /; FreeQ[{a, b, d, e, f}, x] && EqQ[a^2 + b^2, 0]
  3515. Int[Times[Power[Times[cos[Plus[Optional[Pattern[e, Blank[]]], Times[Optional[Pattern[f, Blank[]]], Pattern[x, Blank[]]]]], Optional[Pattern[d, Blank[]]]], Pattern[m, Blank[]]], Power[Plus[Pattern[a, Blank[]], Times[Optional[Pattern[b, Blank[]]], tan[Plus[Optional[Pattern[e, Blank[]]], Times[Optional[Pattern[f, Blank[]]], Pattern[x, Blank[]]]]]]], Optional[Pattern[n, Blank[]]]]], Pattern[x, Blank[Symbol]]] := Dist[(d*Cos[e + f*x])^m*(d*Sec[e + f*x])^m, Int[(a + b*Tan[e + f*x])^n/(d*Sec[e + f*x])^m, x], x] /; FreeQ[{a, b, d, e, f, m, n}, x] && !IntegerQ[m]
  3516. Int[Times[Power[sin[Plus[Optional[Pattern[e, Blank[]]], Times[Optional[Pattern[f, Blank[]]], Pattern[x, Blank[]]]]], Pattern[m, Blank[]]], Power[Plus[Pattern[a, Blank[]], Times[Optional[Pattern[b, Blank[]]], tan[Plus[Optional[Pattern[e, Blank[]]], Times[Optional[Pattern[f, Blank[]]], Pattern[x, Blank[]]]]]]], Pattern[n, Blank[]]]], Pattern[x, Blank[Symbol]]] := Dist[b/f, Subst[Int[(x^m*(a + x)^n)/(b^2 + x^2)^(m/2 + 1), x], x, b*Tan[e + f*x]], x] /; FreeQ[{a, b, e, f, n}, x] && IntegerQ[m/2]
  3517. Int[Times[Power[sin[Plus[Optional[Pattern[e, Blank[]]], Times[Optional[Pattern[f, Blank[]]], Pattern[x, Blank[]]]]], Optional[Pattern[m, Blank[]]]], Power[Plus[Pattern[a, Blank[]], Times[Optional[Pattern[b, Blank[]]], tan[Plus[Optional[Pattern[e, Blank[]]], Times[Optional[Pattern[f, Blank[]]], Pattern[x, Blank[]]]]]]], Optional[Pattern[n, Blank[]]]]], Pattern[x, Blank[Symbol]]] := Int[Expand[Sin[e + f*x]^m*(a + b*Tan[e + f*x])^n, x], x] /; FreeQ[{a, b, e, f}, x] && IntegerQ[(m - 1)/2] && IGtQ[n, 0]
  3518. Int[Times[Power[sin[Plus[Optional[Pattern[e, Blank[]]], Times[Optional[Pattern[f, Blank[]]], Pattern[x, Blank[]]]]], Optional[Pattern[m, Blank[]]]], Power[Plus[Pattern[a, Blank[]], Times[Optional[Pattern[b, Blank[]]], tan[Plus[Optional[Pattern[e, Blank[]]], Times[Optional[Pattern[f, Blank[]]], Pattern[x, Blank[]]]]]]], Optional[Pattern[n, Blank[]]]]], Pattern[x, Blank[Symbol]]] := Int[(Sin[e + f*x]^m*(a*Cos[e + f*x] + b*Sin[e + f*x])^n)/Cos[e + f*x]^n, x] /; FreeQ[{a, b, e, f}, x] && IntegerQ[(m - 1)/2] && ILtQ[n, 0] && ((LtQ[m, 5] && GtQ[n, -4]) || (EqQ[m, 5] && EqQ[n, -1]))
  3519. Int[Times[Power[Times[csc[Plus[Optional[Pattern[e, Blank[]]], Times[Optional[Pattern[f, Blank[]]], Pattern[x, Blank[]]]]], Optional[Pattern[d, Blank[]]]], Pattern[m, Blank[]]], Power[Plus[Optional[Pattern[a, Blank[]]], Times[Optional[Pattern[b, Blank[]]], tan[Plus[Optional[Pattern[e, Blank[]]], Times[Optional[Pattern[f, Blank[]]], Pattern[x, Blank[]]]]]]], Optional[Pattern[n, Blank[]]]]], Pattern[x, Blank[Symbol]]] := Dist[(d*Csc[e + f*x])^FracPart[m]*(Sin[e + f*x]/d)^FracPart[m], Int[(a + b*Tan[e + f*x])^n/(Sin[e + f*x]/d)^m, x], x] /; FreeQ[{a, b, d, e, f, m, n}, x] && !IntegerQ[m]
  3520. Int[Times[Power[cos[Plus[Optional[Pattern[e, Blank[]]], Times[Optional[Pattern[f, Blank[]]], Pattern[x, Blank[]]]]], Optional[Pattern[m, Blank[]]]], Power[sin[Plus[Optional[Pattern[e, Blank[]]], Times[Optional[Pattern[f, Blank[]]], Pattern[x, Blank[]]]]], Optional[Pattern[p, Blank[]]]], Power[Plus[Pattern[a, Blank[]], Times[Optional[Pattern[b, Blank[]]], tan[Plus[Optional[Pattern[e, Blank[]]], Times[Optional[Pattern[f, Blank[]]], Pattern[x, Blank[]]]]]]], Optional[Pattern[n, Blank[]]]]], Pattern[x, Blank[Symbol]]] := Int[Cos[e + f*x]^(m - n)*Sin[e + f*x]^p*(a*Cos[e + f*x] + b*Sin[e + f*x])^n, x] /; FreeQ[{a, b, e, f, m, p}, x] && IntegerQ[n]
  3521. Int[Times[Power[cos[Plus[Optional[Pattern[e, Blank[]]], Times[Optional[Pattern[f, Blank[]]], Pattern[x, Blank[]]]]], Optional[Pattern[p, Blank[]]]], Power[Plus[Times[cot[Plus[Optional[Pattern[e, Blank[]]], Times[Optional[Pattern[f, Blank[]]], Pattern[x, Blank[]]]]], Optional[Pattern[b, Blank[]]]], Pattern[a, Blank[]]], Optional[Pattern[n, Blank[]]]], Power[sin[Plus[Optional[Pattern[e, Blank[]]], Times[Optional[Pattern[f, Blank[]]], Pattern[x, Blank[]]]]], Optional[Pattern[m, Blank[]]]]], Pattern[x, Blank[Symbol]]] := Int[Sin[e + f*x]^(m - n)*Cos[e + f*x]^p*(a*Sin[e + f*x] + b*Cos[e + f*x])^n, x] /; FreeQ[{a, b, e, f, m, p}, x] && IntegerQ[n]
  3522. Int[Times[Power[Plus[Pattern[a, Blank[]], Times[Optional[Pattern[b, Blank[]]], tan[Plus[Optional[Pattern[e, Blank[]]], Times[Optional[Pattern[f, Blank[]]], Pattern[x, Blank[]]]]]]], Optional[Pattern[m, Blank[]]]], Power[Plus[Pattern[c, Blank[]], Times[Optional[Pattern[d, Blank[]]], tan[Plus[Optional[Pattern[e, Blank[]]], Times[Optional[Pattern[f, Blank[]]], Pattern[x, Blank[]]]]]]], Optional[Pattern[n, Blank[]]]]], Pattern[x, Blank[Symbol]]] := Dist[a^m*c^m, Int[Sec[e + f*x]^(2*m)*(c + d*Tan[e + f*x])^(n - m), x], x] /; FreeQ[{a, b, c, d, e, f, n}, x] && EqQ[b*c + a*d, 0] && EqQ[a^2 + b^2, 0] && IntegerQ[m] && !(IGtQ[n, 0] && (LtQ[m, 0] || GtQ[m, n]))
  3523. Int[Times[Power[Plus[Pattern[a, Blank[]], Times[Optional[Pattern[b, Blank[]]], tan[Plus[Optional[Pattern[e, Blank[]]], Times[Optional[Pattern[f, Blank[]]], Pattern[x, Blank[]]]]]]], Pattern[m, Blank[]]], Power[Plus[Pattern[c, Blank[]], Times[Optional[Pattern[d, Blank[]]], tan[Plus[Optional[Pattern[e, Blank[]]], Times[Optional[Pattern[f, Blank[]]], Pattern[x, Blank[]]]]]]], Pattern[n, Blank[]]]], Pattern[x, Blank[Symbol]]] := Dist[(a*c)/f, Subst[Int[(a + b*x)^(m - 1)*(c + d*x)^(n - 1), x], x, Tan[e + f*x]], x] /; FreeQ[{a, b, c, d, e, f, m, n}, x] && EqQ[b*c + a*d, 0] && EqQ[a^2 + b^2, 0]
  3524. Int[Times[Plus[Pattern[a, Blank[]], Times[Optional[Pattern[b, Blank[]]], tan[Plus[Optional[Pattern[e, Blank[]]], Times[Optional[Pattern[f, Blank[]]], Pattern[x, Blank[]]]]]]], Plus[Pattern[c, Blank[]], Times[Optional[Pattern[d, Blank[]]], tan[Plus[Optional[Pattern[e, Blank[]]], Times[Optional[Pattern[f, Blank[]]], Pattern[x, Blank[]]]]]]]], Pattern[x, Blank[Symbol]]] := Simp[(a*c - b*d)*x, x] + Simp[(b*d*Tan[e + f*x])/f, x] /; FreeQ[{a, b, c, d, e, f}, x] && NeQ[b*c - a*d, 0] && EqQ[b*c + a*d, 0]
  3525. Int[Times[Plus[Pattern[a, Blank[]], Times[Optional[Pattern[b, Blank[]]], tan[Plus[Optional[Pattern[e, Blank[]]], Times[Optional[Pattern[f, Blank[]]], Pattern[x, Blank[]]]]]]], Plus[Optional[Pattern[c, Blank[]]], Times[Optional[Pattern[d, Blank[]]], tan[Plus[Optional[Pattern[e, Blank[]]], Times[Optional[Pattern[f, Blank[]]], Pattern[x, Blank[]]]]]]]], Pattern[x, Blank[Symbol]]] := Simp[(a*c - b*d)*x, x] + (Dist[b*c + a*d, Int[Tan[e + f*x], x], x] + Simp[(b*d*Tan[e + f*x])/f, x]) /; FreeQ[{a, b, c, d, e, f}, x] && NeQ[b*c - a*d, 0] && NeQ[b*c + a*d, 0]
  3526. Int[Times[Power[Plus[Pattern[a, Blank[]], Times[Optional[Pattern[b, Blank[]]], tan[Plus[Optional[Pattern[e, Blank[]]], Times[Optional[Pattern[f, Blank[]]], Pattern[x, Blank[]]]]]]], Pattern[m, Blank[]]], Plus[Optional[Pattern[c, Blank[]]], Times[Optional[Pattern[d, Blank[]]], tan[Plus[Optional[Pattern[e, Blank[]]], Times[Optional[Pattern[f, Blank[]]], Pattern[x, Blank[]]]]]]]], Pattern[x, Blank[Symbol]]] := -Simp[((b*c - a*d)*(a + b*Tan[e + f*x])^m)/(2*a*f*m), x] + Dist[(b*c + a*d)/(2*a*b), Int[(a + b*Tan[e + f*x])^(m + 1), x], x] /; FreeQ[{a, b, c, d, e, f}, x] && NeQ[b*c - a*d, 0] && EqQ[a^2 + b^2, 0] && LtQ[m, 0]
  3527. Int[Times[Power[Plus[Pattern[a, Blank[]], Times[Optional[Pattern[b, Blank[]]], tan[Plus[Optional[Pattern[e, Blank[]]], Times[Optional[Pattern[f, Blank[]]], Pattern[x, Blank[]]]]]]], Pattern[m, Blank[]]], Plus[Optional[Pattern[c, Blank[]]], Times[Optional[Pattern[d, Blank[]]], tan[Plus[Optional[Pattern[e, Blank[]]], Times[Optional[Pattern[f, Blank[]]], Pattern[x, Blank[]]]]]]]], Pattern[x, Blank[Symbol]]] := Simp[(d*(a + b*Tan[e + f*x])^m)/(f*m), x] + Dist[(b*c + a*d)/b, Int[(a + b*Tan[e + f*x])^m, x], x] /; FreeQ[{a, b, c, d, e, f, m}, x] && NeQ[b*c - a*d, 0] && EqQ[a^2 + b^2, 0] && !LtQ[m, 0]
  3528. Int[Times[Power[Plus[Optional[Pattern[a, Blank[]]], Times[Optional[Pattern[b, Blank[]]], tan[Plus[Optional[Pattern[e, Blank[]]], Times[Optional[Pattern[f, Blank[]]], Pattern[x, Blank[]]]]]]], Pattern[m, Blank[]]], Plus[Optional[Pattern[c, Blank[]]], Times[Optional[Pattern[d, Blank[]]], tan[Plus[Optional[Pattern[e, Blank[]]], Times[Optional[Pattern[f, Blank[]]], Pattern[x, Blank[]]]]]]]], Pattern[x, Blank[Symbol]]] := Simp[(d*(a + b*Tan[e + f*x])^m)/(f*m), x] + Int[(a + b*Tan[e + f*x])^(m - 1)*Simp[a*c - b*d + (b*c + a*d)*Tan[e + f*x], x], x] /; FreeQ[{a, b, c, d, e, f}, x] && NeQ[b*c - a*d, 0] && NeQ[a^2 + b^2, 0] && GtQ[m, 0]
  3529. Int[Times[Power[Plus[Optional[Pattern[a, Blank[]]], Times[Optional[Pattern[b, Blank[]]], tan[Plus[Optional[Pattern[e, Blank[]]], Times[Optional[Pattern[f, Blank[]]], Pattern[x, Blank[]]]]]]], Pattern[m, Blank[]]], Plus[Optional[Pattern[c, Blank[]]], Times[Optional[Pattern[d, Blank[]]], tan[Plus[Optional[Pattern[e, Blank[]]], Times[Optional[Pattern[f, Blank[]]], Pattern[x, Blank[]]]]]]]], Pattern[x, Blank[Symbol]]] := Simp[((b*c - a*d)*(a + b*Tan[e + f*x])^(m + 1))/(f*(m + 1)*(a^2 + b^2)), x] + Dist[1/(a^2 + b^2), Int[(a + b*Tan[e + f*x])^(m + 1)*Simp[a*c + b*d - (b*c - a*d)*Tan[e + f*x], x], x], x] /; FreeQ[{a, b, c, d, e, f}, x] && NeQ[b*c - a*d, 0] && NeQ[a^2 + b^2, 0] && LtQ[m, -1]
  3530. Int[Times[Power[Plus[Pattern[a, Blank[]], Times[Optional[Pattern[b, Blank[]]], tan[Plus[Optional[Pattern[e, Blank[]]], Times[Optional[Pattern[f, Blank[]]], Pattern[x, Blank[]]]]]]], -1], Plus[Pattern[c, Blank[]], Times[Optional[Pattern[d, Blank[]]], tan[Plus[Optional[Pattern[e, Blank[]]], Times[Optional[Pattern[f, Blank[]]], Pattern[x, Blank[]]]]]]]], Pattern[x, Blank[Symbol]]] := Simp[(c*Log[RemoveContent[a*Cos[e + f*x] + b*Sin[e + f*x], x]])/(b*f), x] /; FreeQ[{a, b, c, d, e, f}, x] && NeQ[b*c - a*d, 0] && NeQ[a^2 + b^2, 0] && EqQ[a*c + b*d, 0]
  3531. Int[Times[Power[Plus[Optional[Pattern[a, Blank[]]], Times[Optional[Pattern[b, Blank[]]], tan[Plus[Optional[Pattern[e, Blank[]]], Times[Optional[Pattern[f, Blank[]]], Pattern[x, Blank[]]]]]]], -1], Plus[Optional[Pattern[c, Blank[]]], Times[Optional[Pattern[d, Blank[]]], tan[Plus[Optional[Pattern[e, Blank[]]], Times[Optional[Pattern[f, Blank[]]], Pattern[x, Blank[]]]]]]]], Pattern[x, Blank[Symbol]]] := Simp[((a*c + b*d)*x)/(a^2 + b^2), x] + Dist[(b*c - a*d)/(a^2 + b^2), Int[(b - a*Tan[e + f*x])/(a + b*Tan[e + f*x]), x], x] /; FreeQ[{a, b, c, d, e, f}, x] && NeQ[b*c - a*d, 0] && NeQ[a^2 + b^2, 0] && NeQ[a*c + b*d, 0]
  3532. Int[Times[Power[Times[Optional[Pattern[b, Blank[]]], tan[Plus[Optional[Pattern[e, Blank[]]], Times[Optional[Pattern[f, Blank[]]], Pattern[x, Blank[]]]]]], Rational[-1, 2]], Plus[Pattern[c, Blank[]], Times[Optional[Pattern[d, Blank[]]], tan[Plus[Optional[Pattern[e, Blank[]]], Times[Optional[Pattern[f, Blank[]]], Pattern[x, Blank[]]]]]]]], Pattern[x, Blank[Symbol]]] := Dist[(-2*d^2)/f, Subst[Int[1/(2*c*d + b*x^2), x], x, (c - d*Tan[e + f*x])/Sqrt[b*Tan[e + f*x]]], x] /; FreeQ[{b, c, d, e, f}, x] && EqQ[c^2 - d^2, 0]
  3533. Int[Times[Power[Times[Optional[Pattern[b, Blank[]]], tan[Plus[Optional[Pattern[e, Blank[]]], Times[Optional[Pattern[f, Blank[]]], Pattern[x, Blank[]]]]]], Rational[-1, 2]], Plus[Pattern[c, Blank[]], Times[Optional[Pattern[d, Blank[]]], tan[Plus[Optional[Pattern[e, Blank[]]], Times[Optional[Pattern[f, Blank[]]], Pattern[x, Blank[]]]]]]]], Pattern[x, Blank[Symbol]]] := Dist[(2*c^2)/f, Subst[Int[1/(b*c - d*x^2), x], x, Sqrt[b*Tan[e + f*x]]], x] /; FreeQ[{b, c, d, e, f}, x] && EqQ[c^2 + d^2, 0]
  3534. Int[Times[Power[Times[Optional[Pattern[b, Blank[]]], tan[Plus[Optional[Pattern[e, Blank[]]], Times[Optional[Pattern[f, Blank[]]], Pattern[x, Blank[]]]]]], Rational[-1, 2]], Plus[Pattern[c, Blank[]], Times[Optional[Pattern[d, Blank[]]], tan[Plus[Optional[Pattern[e, Blank[]]], Times[Optional[Pattern[f, Blank[]]], Pattern[x, Blank[]]]]]]]], Pattern[x, Blank[Symbol]]] := Dist[2/f, Subst[Int[(b*c + d*x^2)/(b^2 + x^4), x], x, Sqrt[b*Tan[e + f*x]]], x] /; FreeQ[{b, c, d, e, f}, x] && NeQ[c^2 - d^2, 0] && NeQ[c^2 + d^2, 0]
  3535. Int[Times[Power[Plus[Pattern[a, Blank[]], Times[Optional[Pattern[b, Blank[]]], tan[Plus[Optional[Pattern[e, Blank[]]], Times[Optional[Pattern[f, Blank[]]], Pattern[x, Blank[]]]]]]], Rational[-1, 2]], Plus[Optional[Pattern[c, Blank[]]], Times[Optional[Pattern[d, Blank[]]], tan[Plus[Optional[Pattern[e, Blank[]]], Times[Optional[Pattern[f, Blank[]]], Pattern[x, Blank[]]]]]]]], Pattern[x, Blank[Symbol]]] := Dist[(-2*d^2)/f, Subst[Int[1/(2*b*c*d - 4*a*d^2 + x^2), x], x, (b*c - 2*a*d - b*d*Tan[e + f*x])/Sqrt[a + b*Tan[e + f*x]]], x] /; FreeQ[{a, b, c, d, e, f}, x] && NeQ[b*c - a*d, 0] && NeQ[a^2 + b^2, 0] && NeQ[c^2 + d^2, 0] && EqQ[2*a*c*d - b*(c^2 - d^2), 0]
  3536. Int[Times[Power[Plus[Pattern[a, Blank[]], Times[Optional[Pattern[b, Blank[]]], tan[Plus[Optional[Pattern[e, Blank[]]], Times[Optional[Pattern[f, Blank[]]], Pattern[x, Blank[]]]]]]], Rational[-1, 2]], Plus[Optional[Pattern[c, Blank[]]], Times[Optional[Pattern[d, Blank[]]], tan[Plus[Optional[Pattern[e, Blank[]]], Times[Optional[Pattern[f, Blank[]]], Pattern[x, Blank[]]]]]]]], Pattern[x, Blank[Symbol]]] := With[{q = Rt[a^2 + b^2, 2]}, Dist[1/(2*q), Int[(a*c + b*d + c*q + (b*c - a*d + d*q)*Tan[e + f*x])/Sqrt[a + b*Tan[e + f*x]], x], x] - Dist[1/(2*q), Int[(a*c + b*d - c*q + (b*c - a*d - d*q)*Tan[e + f*x])/Sqrt[a + b*Tan[e + f*x]], x], x]] /; FreeQ[{a, b, c, d, e, f}, x] && NeQ[b*c - a*d, 0] && NeQ[a^2 + b^2, 0] && NeQ[c^2 + d^2, 0] && NeQ[2*a*c*d - b*(c^2 - d^2), 0] && (PerfectSquareQ[a^2 + b^2] || RationalQ[a, b, c, d])
  3537. Int[Times[Power[Plus[Optional[Pattern[a, Blank[]]], Times[Optional[Pattern[b, Blank[]]], tan[Plus[Optional[Pattern[e, Blank[]]], Times[Optional[Pattern[f, Blank[]]], Pattern[x, Blank[]]]]]]], Pattern[m, Blank[]]], Plus[Pattern[c, Blank[]], Times[Optional[Pattern[d, Blank[]]], tan[Plus[Optional[Pattern[e, Blank[]]], Times[Optional[Pattern[f, Blank[]]], Pattern[x, Blank[]]]]]]]], Pattern[x, Blank[Symbol]]] := Dist[(c*d)/f, Subst[Int[(a + (b*x)/d)^m/(d^2 + c*x), x], x, d*Tan[e + f*x]], x] /; FreeQ[{a, b, c, d, e, f, m}, x] && NeQ[b*c - a*d, 0] && NeQ[a^2 + b^2, 0] && EqQ[c^2 + d^2, 0]
  3538. Int[Times[Power[Times[Optional[Pattern[b, Blank[]]], tan[Plus[Optional[Pattern[e, Blank[]]], Times[Optional[Pattern[f, Blank[]]], Pattern[x, Blank[]]]]]], Pattern[m, Blank[]]], Plus[Pattern[c, Blank[]], Times[Optional[Pattern[d, Blank[]]], tan[Plus[Optional[Pattern[e, Blank[]]], Times[Optional[Pattern[f, Blank[]]], Pattern[x, Blank[]]]]]]]], Pattern[x, Blank[Symbol]]] := Dist[c, Int[(b*Tan[e + f*x])^m, x], x] + Dist[d/b, Int[(b*Tan[e + f*x])^(m + 1), x], x] /; FreeQ[{b, c, d, e, f, m}, x] && NeQ[c^2 + d^2, 0] && !IntegerQ[2*m]
  3539. Int[Times[Power[Plus[Optional[Pattern[a, Blank[]]], Times[Optional[Pattern[b, Blank[]]], tan[Plus[Optional[Pattern[e, Blank[]]], Times[Optional[Pattern[f, Blank[]]], Pattern[x, Blank[]]]]]]], Pattern[m, Blank[]]], Plus[Optional[Pattern[c, Blank[]]], Times[Optional[Pattern[d, Blank[]]], tan[Plus[Optional[Pattern[e, Blank[]]], Times[Optional[Pattern[f, Blank[]]], Pattern[x, Blank[]]]]]]]], Pattern[x, Blank[Symbol]]] := Dist[(c + I*d)/2, Int[(a + b*Tan[e + f*x])^m*(1 - I*Tan[e + f*x]), x], x] + Dist[(c - I*d)/2, Int[(a + b*Tan[e + f*x])^m*(1 + I*Tan[e + f*x]), x], x] /; FreeQ[{a, b, c, d, e, f, m}, x] && NeQ[b*c - a*d, 0] && NeQ[a^2 + b^2, 0] && NeQ[c^2 + d^2, 0] && !IntegerQ[m]
  3540. Int[Times[Power[Plus[Pattern[a, Blank[]], Times[Optional[Pattern[b, Blank[]]], tan[Plus[Optional[Pattern[e, Blank[]]], Times[Optional[Pattern[f, Blank[]]], Pattern[x, Blank[]]]]]]], Pattern[m, Blank[]]], Power[Plus[Optional[Pattern[c, Blank[]]], Times[Optional[Pattern[d, Blank[]]], tan[Plus[Optional[Pattern[e, Blank[]]], Times[Optional[Pattern[f, Blank[]]], Pattern[x, Blank[]]]]]]], 2]], Pattern[x, Blank[Symbol]]] := -Simp[(b*(a*c + b*d)^2*(a + b*Tan[e + f*x])^m)/(2*a^3*f*m), x] + Dist[1/(2*a^2), Int[(a + b*Tan[e + f*x])^(m + 1)*Simp[a*c^2 - 2*b*c*d + a*d^2 - 2*b*d^2*Tan[e + f*x], x], x], x] /; FreeQ[{a, b, c, d, e, f}, x] && NeQ[b*c - a*d, 0] && LeQ[m, -1] && EqQ[a^2 + b^2, 0]
  3541. Int[Times[Power[Plus[Optional[Pattern[a, Blank[]]], Times[Optional[Pattern[b, Blank[]]], tan[Plus[Optional[Pattern[e, Blank[]]], Times[Optional[Pattern[f, Blank[]]], Pattern[x, Blank[]]]]]]], -1], Power[Plus[Optional[Pattern[c, Blank[]]], Times[Optional[Pattern[d, Blank[]]], tan[Plus[Optional[Pattern[e, Blank[]]], Times[Optional[Pattern[f, Blank[]]], Pattern[x, Blank[]]]]]]], 2]], Pattern[x, Blank[Symbol]]] := Simp[(d*(2*b*c - a*d)*x)/b^2, x] + (Dist[d^2/b, Int[Tan[e + f*x], x], x] + Dist[(b*c - a*d)^2/b^2, Int[1/(a + b*Tan[e + f*x]), x], x]) /; FreeQ[{a, b, c, d, e, f}, x] && NeQ[b*c - a*d, 0] && NeQ[a^2 + b^2, 0]
  3542. Int[Times[Power[Plus[Optional[Pattern[a, Blank[]]], Times[Optional[Pattern[b, Blank[]]], tan[Plus[Optional[Pattern[e, Blank[]]], Times[Optional[Pattern[f, Blank[]]], Pattern[x, Blank[]]]]]]], Pattern[m, Blank[]]], Power[Plus[Optional[Pattern[c, Blank[]]], Times[Optional[Pattern[d, Blank[]]], tan[Plus[Optional[Pattern[e, Blank[]]], Times[Optional[Pattern[f, Blank[]]], Pattern[x, Blank[]]]]]]], 2]], Pattern[x, Blank[Symbol]]] := Simp[((b*c - a*d)^2*(a + b*Tan[e + f*x])^(m + 1))/(b*f*(m + 1)*(a^2 + b^2)), x] + Dist[1/(a^2 + b^2), Int[(a + b*Tan[e + f*x])^(m + 1)*Simp[a*c^2 + 2*b*c*d - a*d^2 - (b*c^2 - 2*a*c*d - b*d^2)*Tan[e + f*x], x], x], x] /; FreeQ[{a, b, c, d, e, f}, x] && NeQ[b*c - a*d, 0] && LtQ[m, -1] && NeQ[a^2 + b^2, 0]
  3543. Int[Times[Power[Plus[Optional[Pattern[a, Blank[]]], Times[Optional[Pattern[b, Blank[]]], tan[Plus[Optional[Pattern[e, Blank[]]], Times[Optional[Pattern[f, Blank[]]], Pattern[x, Blank[]]]]]]], Pattern[m, Blank[]]], Power[Plus[Optional[Pattern[c, Blank[]]], Times[Optional[Pattern[d, Blank[]]], tan[Plus[Optional[Pattern[e, Blank[]]], Times[Optional[Pattern[f, Blank[]]], Pattern[x, Blank[]]]]]]], 2]], Pattern[x, Blank[Symbol]]] := Simp[(d^2*(a + b*Tan[e + f*x])^(m + 1))/(b*f*(m + 1)), x] + Int[(a + b*Tan[e + f*x])^m*Simp[c^2 - d^2 + 2*c*d*Tan[e + f*x], x], x] /; FreeQ[{a, b, c, d, e, f, m}, x] && NeQ[b*c - a*d, 0] && !LeQ[m, -1] && !(EqQ[m, 2] && EqQ[a, 0])
  3544. Int[Times[Power[Plus[Pattern[a, Blank[]], Times[Optional[Pattern[b, Blank[]]], tan[Plus[Optional[Pattern[e, Blank[]]], Times[Optional[Pattern[f, Blank[]]], Pattern[x, Blank[]]]]]]], Rational[1, 2]], Power[Plus[Optional[Pattern[c, Blank[]]], Times[Optional[Pattern[d, Blank[]]], tan[Plus[Optional[Pattern[e, Blank[]]], Times[Optional[Pattern[f, Blank[]]], Pattern[x, Blank[]]]]]]], Rational[-1, 2]]], Pattern[x, Blank[Symbol]]] := Dist[(-2*a*b)/f, Subst[Int[1/(a*c - b*d - 2*a^2*x^2), x], x, Sqrt[c + d*Tan[e + f*x]]/Sqrt[a + b*Tan[e + f*x]]], x] /; FreeQ[{a, b, c, d, e, f}, x] && NeQ[b*c - a*d, 0] && EqQ[a^2 + b^2, 0] && NeQ[c^2 + d^2, 0]
  3545. Int[Times[Power[Plus[Pattern[a, Blank[]], Times[Optional[Pattern[b, Blank[]]], tan[Plus[Optional[Pattern[e, Blank[]]], Times[Optional[Pattern[f, Blank[]]], Pattern[x, Blank[]]]]]]], Pattern[m, Blank[]]], Power[Plus[Optional[Pattern[c, Blank[]]], Times[Optional[Pattern[d, Blank[]]], tan[Plus[Optional[Pattern[e, Blank[]]], Times[Optional[Pattern[f, Blank[]]], Pattern[x, Blank[]]]]]]], Pattern[n, Blank[]]]], Pattern[x, Blank[Symbol]]] := Simp[(a*b*(a + b*Tan[e + f*x])^(m - 1)*(c + d*Tan[e + f*x])^(n + 1))/(f*(m - 1)*(a*c - b*d)), x] + Dist[(2*a^2)/(a*c - b*d), Int[(a + b*Tan[e + f*x])^(m - 1)*(c + d*Tan[e + f*x])^(n + 1), x], x] /; FreeQ[{a, b, c, d, e, f}, x] && NeQ[b*c - a*d, 0] && EqQ[a^2 + b^2, 0] && NeQ[c^2 + d^2, 0] && EqQ[m + n, 0] && GtQ[m, 1/2]
  3546. Int[Times[Power[Plus[Pattern[a, Blank[]], Times[Optional[Pattern[b, Blank[]]], tan[Plus[Optional[Pattern[e, Blank[]]], Times[Optional[Pattern[f, Blank[]]], Pattern[x, Blank[]]]]]]], Pattern[m, Blank[]]], Power[Plus[Optional[Pattern[c, Blank[]]], Times[Optional[Pattern[d, Blank[]]], tan[Plus[Optional[Pattern[e, Blank[]]], Times[Optional[Pattern[f, Blank[]]], Pattern[x, Blank[]]]]]]], Pattern[n, Blank[]]]], Pattern[x, Blank[Symbol]]] := Simp[(a*(a + b*Tan[e + f*x])^m*(c + d*Tan[e + f*x])^n)/(2*b*f*m), x] - Dist[(a*c - b*d)/(2*b^2), Int[(a + b*Tan[e + f*x])^(m + 1)*(c + d*Tan[e + f*x])^(n - 1), x], x] /; FreeQ[{a, b, c, d, e, f}, x] && NeQ[b*c - a*d, 0] && EqQ[a^2 + b^2, 0] && NeQ[c^2 + d^2, 0] && EqQ[m + n, 0] && LeQ[m, -2^(-1)]
  3547. Int[Times[Power[Plus[Pattern[a, Blank[]], Times[Optional[Pattern[b, Blank[]]], tan[Plus[Optional[Pattern[e, Blank[]]], Times[Optional[Pattern[f, Blank[]]], Pattern[x, Blank[]]]]]]], Pattern[m, Blank[]]], Power[Plus[Optional[Pattern[c, Blank[]]], Times[Optional[Pattern[d, Blank[]]], tan[Plus[Optional[Pattern[e, Blank[]]], Times[Optional[Pattern[f, Blank[]]], Pattern[x, Blank[]]]]]]], Pattern[n, Blank[]]]], Pattern[x, Blank[Symbol]]] := Simp[(a*(a + b*Tan[e + f*x])^m*(c + d*Tan[e + f*x])^(n + 1))/(2*f*m*(b*c - a*d)), x] + Dist[1/(2*a), Int[(a + b*Tan[e + f*x])^(m + 1)*(c + d*Tan[e + f*x])^n, x], x] /; FreeQ[{a, b, c, d, e, f}, x] && NeQ[b*c - a*d, 0] && EqQ[a^2 + b^2, 0] && NeQ[c^2 + d^2, 0] && EqQ[m + n + 1, 0] && LtQ[m, -1]
  3548. Int[Times[Power[Plus[Pattern[a, Blank[]], Times[Optional[Pattern[b, Blank[]]], tan[Plus[Optional[Pattern[e, Blank[]]], Times[Optional[Pattern[f, Blank[]]], Pattern[x, Blank[]]]]]]], Pattern[m, Blank[]]], Power[Plus[Optional[Pattern[c, Blank[]]], Times[Optional[Pattern[d, Blank[]]], tan[Plus[Optional[Pattern[e, Blank[]]], Times[Optional[Pattern[f, Blank[]]], Pattern[x, Blank[]]]]]]], Pattern[n, Blank[]]]], Pattern[x, Blank[Symbol]]] := -Simp[(d*(a + b*Tan[e + f*x])^m*(c + d*Tan[e + f*x])^(n + 1))/(f*m*(c^2 + d^2)), x] + Dist[a/(a*c - b*d), Int[(a + b*Tan[e + f*x])^m*(c + d*Tan[e + f*x])^(n + 1), x], x] /; FreeQ[{a, b, c, d, e, f, m, n}, x] && NeQ[b*c - a*d, 0] && EqQ[a^2 + b^2, 0] && NeQ[c^2 + d^2, 0] && EqQ[m + n + 1, 0] && !LtQ[m, -1]
  3549. Int[Times[Power[Plus[Pattern[a, Blank[]], Times[Optional[Pattern[b, Blank[]]], tan[Plus[Optional[Pattern[e, Blank[]]], Times[Optional[Pattern[f, Blank[]]], Pattern[x, Blank[]]]]]]], -1], Power[Plus[Optional[Pattern[c, Blank[]]], Times[Optional[Pattern[d, Blank[]]], tan[Plus[Optional[Pattern[e, Blank[]]], Times[Optional[Pattern[f, Blank[]]], Pattern[x, Blank[]]]]]]], Pattern[n, Blank[]]]], Pattern[x, Blank[Symbol]]] := -Simp[((a*c + b*d)*(c + d*Tan[e + f*x])^n)/(2*(b*c - a*d)*f*(a + b*Tan[e + f*x])), x] + Dist[1/(2*a*(b*c - a*d)), Int[(c + d*Tan[e + f*x])^(n - 1)*Simp[a*c*d*(n - 1) + b*c^2 + b*d^2*n - d*(b*c - a*d)*(n - 1)*Tan[e + f*x], x], x], x] /; FreeQ[{a, b, c, d, e, f}, x] && NeQ[b*c - a*d, 0] && EqQ[a^2 + b^2, 0] && NeQ[c^2 + d^2, 0] && LtQ[0, n, 1]
  3550. Int[Times[Power[Plus[Pattern[a, Blank[]], Times[Optional[Pattern[b, Blank[]]], tan[Plus[Optional[Pattern[e, Blank[]]], Times[Optional[Pattern[f, Blank[]]], Pattern[x, Blank[]]]]]]], -1], Power[Plus[Optional[Pattern[c, Blank[]]], Times[Optional[Pattern[d, Blank[]]], tan[Plus[Optional[Pattern[e, Blank[]]], Times[Optional[Pattern[f, Blank[]]], Pattern[x, Blank[]]]]]]], Pattern[n, Blank[]]]], Pattern[x, Blank[Symbol]]] := Simp[((b*c - a*d)*(c + d*Tan[e + f*x])^(n - 1))/(2*a*f*(a + b*Tan[e + f*x])), x] + Dist[1/(2*a^2), Int[(c + d*Tan[e + f*x])^(n - 2)*Simp[a*c^2 + a*d^2*(n - 1) - b*c*d*n - d*(a*c*(n - 2) + b*d*n)*Tan[e + f*x], x], x], x] /; FreeQ[{a, b, c, d, e, f}, x] && NeQ[b*c - a*d, 0] && EqQ[a^2 + b^2, 0] && NeQ[c^2 + d^2, 0] && GtQ[n, 1]
  3551. Int[Times[Power[Plus[Optional[Pattern[a, Blank[]]], Times[Optional[Pattern[b, Blank[]]], tan[Plus[Optional[Pattern[e, Blank[]]], Times[Optional[Pattern[f, Blank[]]], Pattern[x, Blank[]]]]]]], -1], Power[Plus[Optional[Pattern[c, Blank[]]], Times[Optional[Pattern[d, Blank[]]], tan[Plus[Optional[Pattern[e, Blank[]]], Times[Optional[Pattern[f, Blank[]]], Pattern[x, Blank[]]]]]]], -1]], Pattern[x, Blank[Symbol]]] := Dist[b/(b*c - a*d), Int[1/(a + b*Tan[e + f*x]), x], x] - Dist[d/(b*c - a*d), Int[1/(c + d*Tan[e + f*x]), x], x] /; FreeQ[{a, b, c, d, e, f}, x] && NeQ[b*c - a*d, 0] && EqQ[a^2 + b^2, 0] && NeQ[c^2 + d^2, 0]
  3552. Int[Times[Power[Plus[Pattern[a, Blank[]], Times[Optional[Pattern[b, Blank[]]], tan[Plus[Optional[Pattern[e, Blank[]]], Times[Optional[Pattern[f, Blank[]]], Pattern[x, Blank[]]]]]]], -1], Power[Plus[Optional[Pattern[c, Blank[]]], Times[Optional[Pattern[d, Blank[]]], tan[Plus[Optional[Pattern[e, Blank[]]], Times[Optional[Pattern[f, Blank[]]], Pattern[x, Blank[]]]]]]], Pattern[n, Blank[]]]], Pattern[x, Blank[Symbol]]] := -Simp[(a*(c + d*Tan[e + f*x])^(n + 1))/(2*f*(b*c - a*d)*(a + b*Tan[e + f*x])), x] + Dist[1/(2*a*(b*c - a*d)), Int[(c + d*Tan[e + f*x])^n*Simp[b*c + a*d*(n - 1) - b*d*n*Tan[e + f*x], x], x], x] /; FreeQ[{a, b, c, d, e, f, n}, x] && NeQ[b*c - a*d, 0] && EqQ[a^2 + b^2, 0] && NeQ[c^2 + d^2, 0] && !GtQ[n, 0]
  3553. Int[Times[Power[Plus[Pattern[a, Blank[]], Times[Optional[Pattern[b, Blank[]]], tan[Plus[Optional[Pattern[e, Blank[]]], Times[Optional[Pattern[f, Blank[]]], Pattern[x, Blank[]]]]]]], Pattern[m, Blank[]]], Power[Plus[Optional[Pattern[c, Blank[]]], Times[Optional[Pattern[d, Blank[]]], tan[Plus[Optional[Pattern[e, Blank[]]], Times[Optional[Pattern[f, Blank[]]], Pattern[x, Blank[]]]]]]], Pattern[n, Blank[]]]], Pattern[x, Blank[Symbol]]] := -Simp[(a^2*(b*c - a*d)*(a + b*Tan[e + f*x])^(m - 2)*(c + d*Tan[e + f*x])^(n + 1))/(d*f*(b*c + a*d)*(n + 1)), x] + Dist[a/(d*(b*c + a*d)*(n + 1)), Int[(a + b*Tan[e + f*x])^(m - 2)*(c + d*Tan[e + f*x])^(n + 1)*Simp[b*(b*c*(m - 2) - a*d*(m - 2*n - 4)) + (a*b*c*(m - 2) + b^2*d*(n + 1) - a^2*d*(m + n - 1))*Tan[e + f*x], x], x], x] /; FreeQ[{a, b, c, d, e, f}, x] && NeQ[b*c - a*d, 0] && EqQ[a^2 + b^2, 0] && NeQ[c^2 + d^2, 0] && GtQ[m, 1] && LtQ[n, -1] && (IntegerQ[m] || IntegersQ[2*m, 2*n])
  3554. Int[Times[Power[Plus[Pattern[a, Blank[]], Times[Optional[Pattern[b, Blank[]]], tan[Plus[Optional[Pattern[e, Blank[]]], Times[Optional[Pattern[f, Blank[]]], Pattern[x, Blank[]]]]]]], Rational[3, 2]], Power[Plus[Optional[Pattern[c, Blank[]]], Times[Optional[Pattern[d, Blank[]]], tan[Plus[Optional[Pattern[e, Blank[]]], Times[Optional[Pattern[f, Blank[]]], Pattern[x, Blank[]]]]]]], -1]], Pattern[x, Blank[Symbol]]] := Dist[(2*a^2)/(a*c - b*d), Int[Sqrt[a + b*Tan[e + f*x]], x], x] - Dist[(2*b*c*d + a*(c^2 - d^2))/(a*(c^2 + d^2)), Int[((a - b*Tan[e + f*x])*Sqrt[a + b*Tan[e + f*x]])/(c + d*Tan[e + f*x]), x], x] /; FreeQ[{a, b, c, d, e, f}, x] && NeQ[b*c - a*d, 0] && EqQ[a^2 + b^2, 0] && NeQ[c^2 + d^2, 0]
  3555. Int[Times[Power[Plus[Pattern[a, Blank[]], Times[Optional[Pattern[b, Blank[]]], tan[Plus[Optional[Pattern[e, Blank[]]], Times[Optional[Pattern[f, Blank[]]], Pattern[x, Blank[]]]]]]], Rational[3, 2]], Power[Plus[Optional[Pattern[c, Blank[]]], Times[Optional[Pattern[d, Blank[]]], tan[Plus[Optional[Pattern[e, Blank[]]], Times[Optional[Pattern[f, Blank[]]], Pattern[x, Blank[]]]]]]], Rational[-1, 2]]], Pattern[x, Blank[Symbol]]] := Dist[2*a, Int[Sqrt[a + b*Tan[e + f*x]]/Sqrt[c + d*Tan[e + f*x]], x], x] + Dist[b/a, Int[((b + a*Tan[e + f*x])*Sqrt[a + b*Tan[e + f*x]])/Sqrt[c + d*Tan[e + f*x]], x], x] /; FreeQ[{a, b, c, d, e, f}, x] && NeQ[b*c - a*d, 0] && EqQ[a^2 + b^2, 0] && NeQ[c^2 + d^2, 0]
  3556. Int[Times[Power[Plus[Pattern[a, Blank[]], Times[Optional[Pattern[b, Blank[]]], tan[Plus[Optional[Pattern[e, Blank[]]], Times[Optional[Pattern[f, Blank[]]], Pattern[x, Blank[]]]]]]], Pattern[m, Blank[]]], Power[Plus[Optional[Pattern[c, Blank[]]], Times[Optional[Pattern[d, Blank[]]], tan[Plus[Optional[Pattern[e, Blank[]]], Times[Optional[Pattern[f, Blank[]]], Pattern[x, Blank[]]]]]]], Pattern[n, Blank[]]]], Pattern[x, Blank[Symbol]]] := Simp[(b^2*(a + b*Tan[e + f*x])^(m - 2)*(c + d*Tan[e + f*x])^(n + 1))/(d*f*(m + n - 1)), x] + Dist[a/(d*(m + n - 1)), Int[(a + b*Tan[e + f*x])^(m - 2)*(c + d*Tan[e + f*x])^n*Simp[b*c*(m - 2) + a*d*(m + 2*n) + (a*c*(m - 2) + b*d*(3*m + 2*n - 4))*Tan[e + f*x], x], x], x] /; FreeQ[{a, b, c, d, e, f, n}, x] && NeQ[b*c - a*d, 0] && EqQ[a^2 + b^2, 0] && NeQ[c^2 + d^2, 0] && IntegerQ[2*m] && GtQ[m, 1] && NeQ[m + n - 1, 0] && (IntegerQ[m] || IntegersQ[2*m, 2*n])
  3557. Int[Times[Power[Plus[Pattern[a, Blank[]], Times[Optional[Pattern[b, Blank[]]], tan[Plus[Optional[Pattern[e, Blank[]]], Times[Optional[Pattern[f, Blank[]]], Pattern[x, Blank[]]]]]]], Pattern[m, Blank[]]], Power[Plus[Optional[Pattern[c, Blank[]]], Times[Optional[Pattern[d, Blank[]]], tan[Plus[Optional[Pattern[e, Blank[]]], Times[Optional[Pattern[f, Blank[]]], Pattern[x, Blank[]]]]]]], Rational[1, 2]]], Pattern[x, Blank[Symbol]]] := -Simp[(b*(a + b*Tan[e + f*x])^m*Sqrt[c + d*Tan[e + f*x]])/(2*a*f*m), x] + Dist[1/(4*a^2*m), Int[((a + b*Tan[e + f*x])^(m + 1)*Simp[2*a*c*m + b*d + a*d*(2*m + 1)*Tan[e + f*x], x])/Sqrt[c + d*Tan[e + f*x]], x], x] /; FreeQ[{a, b, c, d, e, f}, x] && NeQ[b*c - a*d, 0] && EqQ[a^2 + b^2, 0] && NeQ[c^2 + d^2, 0] && LtQ[m, 0] && IntegersQ[2*m]
  3558. Int[Times[Power[Plus[Pattern[a, Blank[]], Times[Optional[Pattern[b, Blank[]]], tan[Plus[Optional[Pattern[e, Blank[]]], Times[Optional[Pattern[f, Blank[]]], Pattern[x, Blank[]]]]]]], Pattern[m, Blank[]]], Power[Plus[Optional[Pattern[c, Blank[]]], Times[Optional[Pattern[d, Blank[]]], tan[Plus[Optional[Pattern[e, Blank[]]], Times[Optional[Pattern[f, Blank[]]], Pattern[x, Blank[]]]]]]], Pattern[n, Blank[]]]], Pattern[x, Blank[Symbol]]] := -Simp[((b*c - a*d)*(a + b*Tan[e + f*x])^m*(c + d*Tan[e + f*x])^(n - 1))/(2*a*f*m), x] + Dist[1/(2*a^2*m), Int[(a + b*Tan[e + f*x])^(m + 1)*(c + d*Tan[e + f*x])^(n - 2)*Simp[c*(a*c*m + b*d*(n - 1)) - d*(b*c*m + a*d*(n - 1)) - d*(b*d*(m - n + 1) - a*c*(m + n - 1))*Tan[e + f*x], x], x], x] /; FreeQ[{a, b, c, d, e, f}, x] && NeQ[b*c - a*d, 0] && EqQ[a^2 + b^2, 0] && NeQ[c^2 + d^2, 0] && LtQ[m, 0] && GtQ[n, 1] && (IntegerQ[m] || IntegersQ[2*m, 2*n])
  3559. Int[Times[Power[Plus[Pattern[a, Blank[]], Times[Optional[Pattern[b, Blank[]]], tan[Plus[Optional[Pattern[e, Blank[]]], Times[Optional[Pattern[f, Blank[]]], Pattern[x, Blank[]]]]]]], Pattern[m, Blank[]]], Power[Plus[Optional[Pattern[c, Blank[]]], Times[Optional[Pattern[d, Blank[]]], tan[Plus[Optional[Pattern[e, Blank[]]], Times[Optional[Pattern[f, Blank[]]], Pattern[x, Blank[]]]]]]], Pattern[n, Blank[]]]], Pattern[x, Blank[Symbol]]] := Simp[(a*(a + b*Tan[e + f*x])^m*(c + d*Tan[e + f*x])^(n + 1))/(2*f*m*(b*c - a*d)), x] + Dist[1/(2*a*m*(b*c - a*d)), Int[(a + b*Tan[e + f*x])^(m + 1)*(c + d*Tan[e + f*x])^n*Simp[b*c*m - a*d*(2*m + n + 1) + b*d*(m + n + 1)*Tan[e + f*x], x], x], x] /; FreeQ[{a, b, c, d, e, f, n}, x] && NeQ[b*c - a*d, 0] && EqQ[a^2 + b^2, 0] && NeQ[c^2 + d^2, 0] && LtQ[m, 0] && (IntegerQ[m] || IntegersQ[2*m, 2*n])
  3560. Int[Times[Power[Plus[Pattern[a, Blank[]], Times[Optional[Pattern[b, Blank[]]], tan[Plus[Optional[Pattern[e, Blank[]]], Times[Optional[Pattern[f, Blank[]]], Pattern[x, Blank[]]]]]]], Pattern[m, Blank[]]], Power[Plus[Optional[Pattern[c, Blank[]]], Times[Optional[Pattern[d, Blank[]]], tan[Plus[Optional[Pattern[e, Blank[]]], Times[Optional[Pattern[f, Blank[]]], Pattern[x, Blank[]]]]]]], Pattern[n, Blank[]]]], Pattern[x, Blank[Symbol]]] := Simp[(d*(a + b*Tan[e + f*x])^m*(c + d*Tan[e + f*x])^(n - 1))/(f*(m + n - 1)), x] - Dist[1/(a*(m + n - 1)), Int[(a + b*Tan[e + f*x])^m*(c + d*Tan[e + f*x])^(n - 2)*Simp[d*(b*c*m + a*d*(-1 + n)) - a*c^2*(m + n - 1) + d*(b*d*m - a*c*(m + 2*n - 2))*Tan[e + f*x], x], x], x] /; FreeQ[{a, b, c, d, e, f, m}, x] && NeQ[b*c - a*d, 0] && EqQ[a^2 + b^2, 0] && NeQ[c^2 + d^2, 0] && GtQ[n, 1] && NeQ[m + n - 1, 0] && (IntegerQ[n] || IntegersQ[2*m, 2*n])
  3561. Int[Times[Power[Plus[Pattern[a, Blank[]], Times[Optional[Pattern[b, Blank[]]], tan[Plus[Optional[Pattern[e, Blank[]]], Times[Optional[Pattern[f, Blank[]]], Pattern[x, Blank[]]]]]]], Pattern[m, Blank[]]], Power[Plus[Optional[Pattern[c, Blank[]]], Times[Optional[Pattern[d, Blank[]]], tan[Plus[Optional[Pattern[e, Blank[]]], Times[Optional[Pattern[f, Blank[]]], Pattern[x, Blank[]]]]]]], Pattern[n, Blank[]]]], Pattern[x, Blank[Symbol]]] := Simp[(d*(a + b*Tan[e + f*x])^m*(c + d*Tan[e + f*x])^(n + 1))/(f*(n + 1)*(c^2 + d^2)), x] - Dist[1/(a*(c^2 + d^2)*(n + 1)), Int[(a + b*Tan[e + f*x])^m*(c + d*Tan[e + f*x])^(n + 1)*Simp[b*d*m - a*c*(n + 1) + a*d*(m + n + 1)*Tan[e + f*x], x], x], x] /; FreeQ[{a, b, c, d, e, f, m}, x] && NeQ[b*c - a*d, 0] && EqQ[a^2 + b^2, 0] && NeQ[c^2 + d^2, 0] && LtQ[n, -1] && (IntegerQ[n] || IntegersQ[2*m, 2*n])
  3562. Int[Times[Power[Plus[Pattern[a, Blank[]], Times[Optional[Pattern[b, Blank[]]], tan[Plus[Optional[Pattern[e, Blank[]]], Times[Optional[Pattern[f, Blank[]]], Pattern[x, Blank[]]]]]]], Pattern[m, Blank[]]], Power[Plus[Optional[Pattern[c, Blank[]]], Times[Optional[Pattern[d, Blank[]]], tan[Plus[Optional[Pattern[e, Blank[]]], Times[Optional[Pattern[f, Blank[]]], Pattern[x, Blank[]]]]]]], -1]], Pattern[x, Blank[Symbol]]] := Dist[a/(a*c - b*d), Int[(a + b*Tan[e + f*x])^m, x], x] - Dist[d/(a*c - b*d), Int[((a + b*Tan[e + f*x])^m*(b + a*Tan[e + f*x]))/(c + d*Tan[e + f*x]), x], x] /; FreeQ[{a, b, c, d, e, f, m}, x] && NeQ[b*c - a*d, 0] && EqQ[a^2 + b^2, 0] && NeQ[c^2 + d^2, 0]
  3563. Int[Times[Power[Plus[Pattern[a, Blank[]], Times[Optional[Pattern[b, Blank[]]], tan[Plus[Optional[Pattern[e, Blank[]]], Times[Optional[Pattern[f, Blank[]]], Pattern[x, Blank[]]]]]]], Rational[1, 2]], Power[Plus[Optional[Pattern[c, Blank[]]], Times[Optional[Pattern[d, Blank[]]], tan[Plus[Optional[Pattern[e, Blank[]]], Times[Optional[Pattern[f, Blank[]]], Pattern[x, Blank[]]]]]]], Rational[1, 2]]], Pattern[x, Blank[Symbol]]] := Dist[(a*c - b*d)/a, Int[Sqrt[a + b*Tan[e + f*x]]/Sqrt[c + d*Tan[e + f*x]], x], x] + Dist[d/a, Int[(Sqrt[a + b*Tan[e + f*x]]*(b + a*Tan[e + f*x]))/Sqrt[c + d*Tan[e + f*x]], x], x] /; FreeQ[{a, b, c, d, e, f}, x] && NeQ[b*c - a*d, 0] && EqQ[a^2 + b^2, 0] && NeQ[c^2 + d^2, 0]
  3564. Int[Times[Power[Plus[Pattern[a, Blank[]], Times[Optional[Pattern[b, Blank[]]], tan[Plus[Optional[Pattern[e, Blank[]]], Times[Optional[Pattern[f, Blank[]]], Pattern[x, Blank[]]]]]]], Pattern[m, Blank[]]], Power[Plus[Optional[Pattern[c, Blank[]]], Times[Optional[Pattern[d, Blank[]]], tan[Plus[Optional[Pattern[e, Blank[]]], Times[Optional[Pattern[f, Blank[]]], Pattern[x, Blank[]]]]]]], Pattern[n, Blank[]]]], Pattern[x, Blank[Symbol]]] := Dist[(a*b)/f, Subst[Int[((a + x)^(m - 1)*(c + (d*x)/b)^n)/(b^2 + a*x), x], x, b*Tan[e + f*x]], x] /; FreeQ[{a, b, c, d, e, f, m, n}, x] && NeQ[b*c - a*d, 0] && EqQ[a^2 + b^2, 0] && NeQ[c^2 + d^2, 0]
  3565. Int[Times[Power[Plus[Optional[Pattern[a, Blank[]]], Times[Optional[Pattern[b, Blank[]]], tan[Plus[Optional[Pattern[e, Blank[]]], Times[Optional[Pattern[f, Blank[]]], Pattern[x, Blank[]]]]]]], Pattern[m, Blank[]]], Power[Plus[Optional[Pattern[c, Blank[]]], Times[Optional[Pattern[d, Blank[]]], tan[Plus[Optional[Pattern[e, Blank[]]], Times[Optional[Pattern[f, Blank[]]], Pattern[x, Blank[]]]]]]], Pattern[n, Blank[]]]], Pattern[x, Blank[Symbol]]] := Simp[((b*c - a*d)^2*(a + b*Tan[e + f*x])^(m - 2)*(c + d*Tan[e + f*x])^(n + 1))/(d*f*(n + 1)*(c^2 + d^2)), x] - Dist[1/(d*(n + 1)*(c^2 + d^2)), Int[(a + b*Tan[e + f*x])^(m - 3)*(c + d*Tan[e + f*x])^(n + 1)*Simp[a^2*d*(b*d*(m - 2) - a*c*(n + 1)) + b*(b*c - 2*a*d)*(b*c*(m - 2) + a*d*(n + 1)) - d*(n + 1)*(3*a^2*b*c - b^3*c - a^3*d + 3*a*b^2*d)*Tan[e + f*x] - b*(a*d*(2*b*c - a*d)*(m + n - 1) - b^2*(c^2*(m - 2) - d^2*(n + 1)))*Tan[e + f*x]^2, x], x], x] /; FreeQ[{a, b, c, d, e, f}, x] && NeQ[b*c - a*d, 0] && NeQ[a^2 + b^2, 0] && NeQ[c^2 + d^2, 0] && GtQ[m, 2] && LtQ[n, -1] && IntegerQ[2*m]
  3566. Int[Times[Power[Plus[Optional[Pattern[a, Blank[]]], Times[Optional[Pattern[b, Blank[]]], tan[Plus[Optional[Pattern[e, Blank[]]], Times[Optional[Pattern[f, Blank[]]], Pattern[x, Blank[]]]]]]], Pattern[m, Blank[]]], Power[Plus[Optional[Pattern[c, Blank[]]], Times[Optional[Pattern[d, Blank[]]], tan[Plus[Optional[Pattern[e, Blank[]]], Times[Optional[Pattern[f, Blank[]]], Pattern[x, Blank[]]]]]]], Pattern[n, Blank[]]]], Pattern[x, Blank[Symbol]]] := Simp[(b^2*(a + b*Tan[e + f*x])^(m - 2)*(c + d*Tan[e + f*x])^(n + 1))/(d*f*(m + n - 1)), x] + Dist[1/(d*(m + n - 1)), Int[(a + b*Tan[e + f*x])^(m - 3)*(c + d*Tan[e + f*x])^n*Simp[a^3*d*(m + n - 1) - b^2*(b*c*(m - 2) + a*d*(1 + n)) + b*d*(m + n - 1)*(3*a^2 - b^2)*Tan[e + f*x] - b^2*(b*c*(m - 2) - a*d*(3*m + 2*n - 4))*Tan[e + f*x]^2, x], x], x] /; FreeQ[{a, b, c, d, e, f, n}, x] && NeQ[b*c - a*d, 0] && NeQ[a^2 + b^2, 0] && NeQ[c^2 + d^2, 0] && IntegerQ[2*m] && GtQ[m, 2] && (GeQ[n, -1] || IntegerQ[m]) && !(IGtQ[n, 2] && ( !IntegerQ[m] || (EqQ[c, 0] && NeQ[a, 0])))
  3567. Int[Times[Power[Plus[Optional[Pattern[a, Blank[]]], Times[Optional[Pattern[b, Blank[]]], tan[Plus[Optional[Pattern[e, Blank[]]], Times[Optional[Pattern[f, Blank[]]], Pattern[x, Blank[]]]]]]], Pattern[m, Blank[]]], Power[Plus[Optional[Pattern[c, Blank[]]], Times[Optional[Pattern[d, Blank[]]], tan[Plus[Optional[Pattern[e, Blank[]]], Times[Optional[Pattern[f, Blank[]]], Pattern[x, Blank[]]]]]]], Pattern[n, Blank[]]]], Pattern[x, Blank[Symbol]]] := Simp[((b*c - a*d)*(a + b*Tan[e + f*x])^(m + 1)*(c + d*Tan[e + f*x])^(n - 1))/(f*(m + 1)*(a^2 + b^2)), x] + Dist[1/((m + 1)*(a^2 + b^2)), Int[(a + b*Tan[e + f*x])^(m + 1)*(c + d*Tan[e + f*x])^(n - 2)*Simp[a*c^2*(m + 1) + a*d^2*(n - 1) + b*c*d*(m - n + 2) - (b*c^2 - 2*a*c*d - b*d^2)*(m + 1)*Tan[e + f*x] - d*(b*c - a*d)*(m + n)*Tan[e + f*x]^2, x], x], x] /; FreeQ[{a, b, c, d, e, f}, x] && NeQ[b*c - a*d, 0] && NeQ[a^2 + b^2, 0] && NeQ[c^2 + d^2, 0] && LtQ[m, -1] && LtQ[1, n, 2] && IntegerQ[2*m]
  3568. Int[Times[Power[Plus[Optional[Pattern[a, Blank[]]], Times[Optional[Pattern[b, Blank[]]], tan[Plus[Optional[Pattern[e, Blank[]]], Times[Optional[Pattern[f, Blank[]]], Pattern[x, Blank[]]]]]]], Pattern[m, Blank[]]], Power[Plus[Optional[Pattern[c, Blank[]]], Times[Optional[Pattern[d, Blank[]]], tan[Plus[Optional[Pattern[e, Blank[]]], Times[Optional[Pattern[f, Blank[]]], Pattern[x, Blank[]]]]]]], Pattern[n, Blank[]]]], Pattern[x, Blank[Symbol]]] := Simp[(b*(a + b*Tan[e + f*x])^(m + 1)*(c + d*Tan[e + f*x])^n)/(f*(m + 1)*(a^2 + b^2)), x] + Dist[1/((m + 1)*(a^2 + b^2)), Int[(a + b*Tan[e + f*x])^(m + 1)*(c + d*Tan[e + f*x])^(n - 1)*Simp[a*c*(m + 1) - b*d*n - (b*c - a*d)*(m + 1)*Tan[e + f*x] - b*d*(m + n + 1)*Tan[e + f*x]^2, x], x], x] /; FreeQ[{a, b, c, d, e, f}, x] && NeQ[b*c - a*d, 0] && NeQ[a^2 + b^2, 0] && NeQ[c^2 + d^2, 0] && LtQ[m, -1] && GtQ[n, 0] && IntegerQ[2*m]
  3569. Int[Times[Power[Plus[Optional[Pattern[a, Blank[]]], Times[Optional[Pattern[b, Blank[]]], tan[Plus[Optional[Pattern[e, Blank[]]], Times[Optional[Pattern[f, Blank[]]], Pattern[x, Blank[]]]]]]], Pattern[m, Blank[]]], Power[Plus[Optional[Pattern[c, Blank[]]], Times[Optional[Pattern[d, Blank[]]], tan[Plus[Optional[Pattern[e, Blank[]]], Times[Optional[Pattern[f, Blank[]]], Pattern[x, Blank[]]]]]]], Pattern[n, Blank[]]]], Pattern[x, Blank[Symbol]]] := Simp[(b^2*(a + b*Tan[e + f*x])^(m + 1)*(c + d*Tan[e + f*x])^(n + 1))/(f*(m + 1)*(a^2 + b^2)*(b*c - a*d)), x] + Dist[1/((m + 1)*(a^2 + b^2)*(b*c - a*d)), Int[(a + b*Tan[e + f*x])^(m + 1)*(c + d*Tan[e + f*x])^n*Simp[a*(b*c - a*d)*(m + 1) - b^2*d*(m + n + 2) - b*(b*c - a*d)*(m + 1)*Tan[e + f*x] - b^2*d*(m + n + 2)*Tan[e + f*x]^2, x], x], x] /; FreeQ[{a, b, c, d, e, f, n}, x] && NeQ[b*c - a*d, 0] && NeQ[a^2 + b^2, 0] && NeQ[c^2 + d^2, 0] && IntegerQ[2*m] && LtQ[m, -1] && (LtQ[n, 0] || IntegerQ[m]) && !(ILtQ[n, -1] && ( !IntegerQ[m] || (EqQ[c, 0] && NeQ[a, 0])))
  3570. Int[Times[Power[Plus[Optional[Pattern[a, Blank[]]], Times[Optional[Pattern[b, Blank[]]], tan[Plus[Optional[Pattern[e, Blank[]]], Times[Optional[Pattern[f, Blank[]]], Pattern[x, Blank[]]]]]]], Pattern[m, Blank[]]], Power[Plus[Optional[Pattern[c, Blank[]]], Times[Optional[Pattern[d, Blank[]]], tan[Plus[Optional[Pattern[e, Blank[]]], Times[Optional[Pattern[f, Blank[]]], Pattern[x, Blank[]]]]]]], Pattern[n, Blank[]]]], Pattern[x, Blank[Symbol]]] := Simp[(b*(a + b*Tan[e + f*x])^(m - 1)*(c + d*Tan[e + f*x])^n)/(f*(m + n - 1)), x] + Dist[1/(m + n - 1), Int[(a + b*Tan[e + f*x])^(m - 2)*(c + d*Tan[e + f*x])^(n - 1)*Simp[a^2*c*(m + n - 1) - b*(b*c*(m - 1) + a*d*n) + (2*a*b*c + a^2*d - b^2*d)*(m + n - 1)*Tan[e + f*x] + b*(b*c*n + a*d*(2*m + n - 2))*Tan[e + f*x]^2, x], x], x] /; FreeQ[{a, b, c, d, e, f}, x] && NeQ[b*c - a*d, 0] && NeQ[a^2 + b^2, 0] && NeQ[c^2 + d^2, 0] && GtQ[m, 1] && GtQ[n, 0] && IntegerQ[2*n]
  3571. Int[Times[Power[Plus[Pattern[a, Blank[]], Times[Optional[Pattern[b, Blank[]]], tan[Plus[Optional[Pattern[e, Blank[]]], Times[Optional[Pattern[f, Blank[]]], Pattern[x, Blank[]]]]]]], -1], Power[Plus[Optional[Pattern[c, Blank[]]], Times[Optional[Pattern[d, Blank[]]], tan[Plus[Optional[Pattern[e, Blank[]]], Times[Optional[Pattern[f, Blank[]]], Pattern[x, Blank[]]]]]]], -1]], Pattern[x, Blank[Symbol]]] := Simp[((a*c - b*d)*x)/((a^2 + b^2)*(c^2 + d^2)), x] + (Dist[b^2/((b*c - a*d)*(a^2 + b^2)), Int[(b - a*Tan[e + f*x])/(a + b*Tan[e + f*x]), x], x] - Dist[d^2/((b*c - a*d)*(c^2 + d^2)), Int[(d - c*Tan[e + f*x])/(c + d*Tan[e + f*x]), x], x]) /; FreeQ[{a, b, c, d, e, f}, x] && NeQ[b*c - a*d, 0] && NeQ[a^2 + b^2, 0] && NeQ[c^2 + d^2, 0]
  3572. Int[Times[Power[Plus[Optional[Pattern[a, Blank[]]], Times[Optional[Pattern[b, Blank[]]], tan[Plus[Optional[Pattern[e, Blank[]]], Times[Optional[Pattern[f, Blank[]]], Pattern[x, Blank[]]]]]]], Rational[1, 2]], Power[Plus[Optional[Pattern[c, Blank[]]], Times[Optional[Pattern[d, Blank[]]], tan[Plus[Optional[Pattern[e, Blank[]]], Times[Optional[Pattern[f, Blank[]]], Pattern[x, Blank[]]]]]]], -1]], Pattern[x, Blank[Symbol]]] := Dist[1/(c^2 + d^2), Int[Simp[a*c + b*d + (b*c - a*d)*Tan[e + f*x], x]/Sqrt[a + b*Tan[e + f*x]], x], x] - Dist[(d*(b*c - a*d))/(c^2 + d^2), Int[(1 + Tan[e + f*x]^2)/(Sqrt[a + b*Tan[e + f*x]]*(c + d*Tan[e + f*x])), x], x] /; FreeQ[{a, b, c, d}, x] && NeQ[b*c - a*d, 0] && NeQ[a^2 + b^2, 0] && NeQ[c^2 + d^2, 0]
  3573. Int[Times[Power[Plus[Optional[Pattern[a, Blank[]]], Times[Optional[Pattern[b, Blank[]]], tan[Plus[Optional[Pattern[e, Blank[]]], Times[Optional[Pattern[f, Blank[]]], Pattern[x, Blank[]]]]]]], Rational[3, 2]], Power[Plus[Optional[Pattern[c, Blank[]]], Times[Optional[Pattern[d, Blank[]]], tan[Plus[Optional[Pattern[e, Blank[]]], Times[Optional[Pattern[f, Blank[]]], Pattern[x, Blank[]]]]]]], -1]], Pattern[x, Blank[Symbol]]] := Dist[1/(c^2 + d^2), Int[Simp[a^2*c - b^2*c + 2*a*b*d + (2*a*b*c - a^2*d + b^2*d)*Tan[e + f*x], x]/Sqrt[a + b*Tan[e + f*x]], x], x] + Dist[(b*c - a*d)^2/(c^2 + d^2), Int[(1 + Tan[e + f*x]^2)/(Sqrt[a + b*Tan[e + f*x]]*(c + d*Tan[e + f*x])), x], x] /; FreeQ[{a, b, c, d, e, f}, x] && NeQ[b*c - a*d, 0] && NeQ[a^2 + b^2, 0] && NeQ[c^2 + d^2, 0]
  3574. Int[Times[Power[Plus[Optional[Pattern[a, Blank[]]], Times[Optional[Pattern[b, Blank[]]], tan[Plus[Optional[Pattern[e, Blank[]]], Times[Optional[Pattern[f, Blank[]]], Pattern[x, Blank[]]]]]]], Pattern[m, Blank[]]], Power[Plus[Optional[Pattern[c, Blank[]]], Times[Optional[Pattern[d, Blank[]]], tan[Plus[Optional[Pattern[e, Blank[]]], Times[Optional[Pattern[f, Blank[]]], Pattern[x, Blank[]]]]]]], -1]], Pattern[x, Blank[Symbol]]] := Dist[1/(c^2 + d^2), Int[(a + b*Tan[e + f*x])^m*(c - d*Tan[e + f*x]), x], x] + Dist[d^2/(c^2 + d^2), Int[((a + b*Tan[e + f*x])^m*(1 + Tan[e + f*x]^2))/(c + d*Tan[e + f*x]), x], x] /; FreeQ[{a, b, c, d, e, f, m}, x] && NeQ[b*c - a*d, 0] && NeQ[a^2 + b^2, 0] && NeQ[c^2 + d^2, 0] && !IntegerQ[m]
  3575. Int[Times[Power[Plus[Optional[Pattern[a, Blank[]]], Times[Optional[Pattern[b, Blank[]]], tan[Plus[Optional[Pattern[e, Blank[]]], Times[Optional[Pattern[f, Blank[]]], Pattern[x, Blank[]]]]]]], Pattern[m, Blank[]]], Power[Plus[Pattern[c, Blank[]], Times[Optional[Pattern[d, Blank[]]], tan[Plus[Optional[Pattern[e, Blank[]]], Times[Optional[Pattern[f, Blank[]]], Pattern[x, Blank[]]]]]]], Pattern[n, Blank[]]]], Pattern[x, Blank[Symbol]]] := With[{ff = FreeFactors[Tan[e + f*x], x]}, Dist[ff/f, Subst[Int[((a + b*ff*x)^m*(c + d*ff*x)^n)/(1 + ff^2*x^2), x], x, Tan[e + f*x]/ff], x]] /; FreeQ[{a, b, c, d, e, f, m, n}, x] && NeQ[b*c - a*d, 0] && NeQ[a^2 + b^2, 0] && NeQ[c^2 + d^2, 0]
  3576. Int[Times[Power[Times[Optional[Pattern[d, Blank[]]], Power[tan[Plus[Optional[Pattern[e, Blank[]]], Times[Optional[Pattern[f, Blank[]]], Pattern[x, Blank[]]]]], -1]], Pattern[n, Blank[]]], Power[Plus[Optional[Pattern[a, Blank[]]], Times[Optional[Pattern[b, Blank[]]], tan[Plus[Optional[Pattern[e, Blank[]]], Times[Optional[Pattern[f, Blank[]]], Pattern[x, Blank[]]]]]]], Optional[Pattern[m, Blank[]]]]], Pattern[x, Blank[Symbol]]] := Dist[d^m, Int[(b + a*Cot[e + f*x])^m*(d*Cot[e + f*x])^(n - m), x], x] /; FreeQ[{a, b, d, e, f, n}, x] && !IntegerQ[n] && IntegerQ[m]
  3577. Int[Times[Power[Plus[Optional[Pattern[a, Blank[]]], Times[cot[Plus[Optional[Pattern[e, Blank[]]], Times[Optional[Pattern[f, Blank[]]], Pattern[x, Blank[]]]]], Optional[Pattern[b, Blank[]]]]], Optional[Pattern[m, Blank[]]]], Power[Times[Power[cot[Plus[Optional[Pattern[e, Blank[]]], Times[Optional[Pattern[f, Blank[]]], Pattern[x, Blank[]]]]], -1], Optional[Pattern[d, Blank[]]]], Pattern[n, Blank[]]]], Pattern[x, Blank[Symbol]]] := Dist[d^m, Int[(b + a*Tan[e + f*x])^m*(d*Tan[e + f*x])^(n - m), x], x] /; FreeQ[{a, b, d, e, f, n}, x] && !IntegerQ[n] && IntegerQ[m]
  3578. Int[Times[Power[Times[Optional[Pattern[c, Blank[]]], Power[Times[Optional[Pattern[d, Blank[]]], tan[Plus[Optional[Pattern[e, Blank[]]], Times[Optional[Pattern[f, Blank[]]], Pattern[x, Blank[]]]]]], Pattern[p, Blank[]]]], Pattern[n, Blank[]]], Power[Plus[Optional[Pattern[a, Blank[]]], Times[Optional[Pattern[b, Blank[]]], tan[Plus[Optional[Pattern[e, Blank[]]], Times[Optional[Pattern[f, Blank[]]], Pattern[x, Blank[]]]]]]], Optional[Pattern[m, Blank[]]]]], Pattern[x, Blank[Symbol]]] := Dist[(c^IntPart[n]*(c*(d*Tan[e + f*x])^p)^FracPart[n])/(d*Tan[e + f*x])^(p*FracPart[n]), Int[(a + b*Tan[e + f*x])^m*(d*Tan[e + f*x])^(n*p), x], x] /; FreeQ[{a, b, c, d, e, f, m, n, p}, x] && !IntegerQ[n] && !IntegerQ[m]
  3579. Int[Times[Power[Plus[Optional[Pattern[a, Blank[]]], Times[cot[Plus[Optional[Pattern[e, Blank[]]], Times[Optional[Pattern[f, Blank[]]], Pattern[x, Blank[]]]]], Optional[Pattern[b, Blank[]]]]], Optional[Pattern[m, Blank[]]]], Power[Times[Optional[Pattern[c, Blank[]]], Power[Times[cot[Plus[Optional[Pattern[e, Blank[]]], Times[Optional[Pattern[f, Blank[]]], Pattern[x, Blank[]]]]], Optional[Pattern[d, Blank[]]]], Pattern[p, Blank[]]]], Pattern[n, Blank[]]]], Pattern[x, Blank[Symbol]]] := Dist[(c^IntPart[n]*(c*(d*Cot[e + f*x])^p)^FracPart[n])/(d*Cot[e + f*x])^(p*FracPart[n]), Int[(a + b*Cot[e + f*x])^m*(d*Cot[e + f*x])^(n*p), x], x] /; FreeQ[{a, b, c, d, e, f, m, n, p}, x] && !IntegerQ[n] && !IntegerQ[m]
  3580. Int[Times[Power[Times[Optional[Pattern[g, Blank[]]], tan[Plus[Optional[Pattern[e, Blank[]]], Times[Optional[Pattern[f, Blank[]]], Pattern[x, Blank[]]]]]], Optional[Pattern[p, Blank[]]]], Power[Plus[Pattern[a, Blank[]], Times[Optional[Pattern[b, Blank[]]], tan[Plus[Optional[Pattern[e, Blank[]]], Times[Optional[Pattern[f, Blank[]]], Pattern[x, Blank[]]]]]]], Pattern[m, Blank[]]], Power[Plus[Pattern[c, Blank[]], Times[Optional[Pattern[d, Blank[]]], tan[Plus[Optional[Pattern[e, Blank[]]], Times[Optional[Pattern[f, Blank[]]], Pattern[x, Blank[]]]]]]], Pattern[n, Blank[]]]], Pattern[x, Blank[Symbol]]] := Unintegrable[(g*Tan[e + f*x])^p*(a + b*Tan[e + f*x])^m*(c + d*Tan[e + f*x])^n, x] /; FreeQ[{a, b, c, d, e, f, g, m, n, p}, x]
  3581. Int[Times[Power[Times[cot[Plus[Optional[Pattern[e, Blank[]]], Times[Optional[Pattern[f, Blank[]]], Pattern[x, Blank[]]]]], Optional[Pattern[g, Blank[]]]], Pattern[p, Blank[]]], Power[Plus[Optional[Pattern[a, Blank[]]], Times[Optional[Pattern[b, Blank[]]], tan[Plus[Optional[Pattern[e, Blank[]]], Times[Optional[Pattern[f, Blank[]]], Pattern[x, Blank[]]]]]]], Optional[Pattern[m, Blank[]]]], Power[Plus[Pattern[c, Blank[]], Times[Optional[Pattern[d, Blank[]]], tan[Plus[Optional[Pattern[e, Blank[]]], Times[Optional[Pattern[f, Blank[]]], Pattern[x, Blank[]]]]]]], Optional[Pattern[n, Blank[]]]]], Pattern[x, Blank[Symbol]]] := Dist[g^(m + n), Int[(g*Cot[e + f*x])^(p - m - n)*(b + a*Cot[e + f*x])^m*(d + c*Cot[e + f*x])^n, x], x] /; FreeQ[{a, b, c, d, e, f, g, p}, x] && !IntegerQ[p] && IntegerQ[m] && IntegerQ[n]
  3582. Int[Times[Power[Plus[Optional[Pattern[a, Blank[]]], Times[cot[Plus[Optional[Pattern[e, Blank[]]], Times[Optional[Pattern[f, Blank[]]], Pattern[x, Blank[]]]]], Optional[Pattern[b, Blank[]]]]], Optional[Pattern[m, Blank[]]]], Power[Plus[Times[cot[Plus[Optional[Pattern[e, Blank[]]], Times[Optional[Pattern[f, Blank[]]], Pattern[x, Blank[]]]]], Optional[Pattern[d, Blank[]]]], Pattern[c, Blank[]]], Optional[Pattern[n, Blank[]]]], Power[Times[Optional[Pattern[g, Blank[]]], tan[Plus[Optional[Pattern[e, Blank[]]], Times[Optional[Pattern[f, Blank[]]], Pattern[x, Blank[]]]]]], Pattern[p, Blank[]]]], Pattern[x, Blank[Symbol]]] := Dist[g^(m + n), Int[(g*Tan[e + f*x])^(p - m - n)*(b + a*Tan[e + f*x])^m*(d + c*Tan[e + f*x])^n, x], x] /; FreeQ[{a, b, c, d, e, f, g, p}, x] && !IntegerQ[p] && IntegerQ[m] && IntegerQ[n]
  3583. Int[Times[Power[Times[Optional[Pattern[g, Blank[]]], Power[tan[Plus[Optional[Pattern[e, Blank[]]], Times[Optional[Pattern[f, Blank[]]], Pattern[x, Blank[]]]]], Pattern[q, Blank[]]]], Pattern[p, Blank[]]], Power[Plus[Optional[Pattern[a, Blank[]]], Times[Optional[Pattern[b, Blank[]]], tan[Plus[Optional[Pattern[e, Blank[]]], Times[Optional[Pattern[f, Blank[]]], Pattern[x, Blank[]]]]]]], Optional[Pattern[m, Blank[]]]], Power[Plus[Pattern[c, Blank[]], Times[Optional[Pattern[d, Blank[]]], tan[Plus[Optional[Pattern[e, Blank[]]], Times[Optional[Pattern[f, Blank[]]], Pattern[x, Blank[]]]]]]], Optional[Pattern[n, Blank[]]]]], Pattern[x, Blank[Symbol]]] := Dist[(g*Tan[e + f*x]^q)^p/(g*Tan[e + f*x])^(p*q), Int[(g*Tan[e + f*x])^(p*q)*(a + b*Tan[e + f*x])^m*(c + d*Tan[e + f*x])^n, x], x] /; FreeQ[{a, b, c, d, e, f, g, m, n, p, q}, x] && !IntegerQ[p] && !(IntegerQ[m] && IntegerQ[n])
  3584. Int[Times[Power[Plus[Times[cot[Plus[Optional[Pattern[e, Blank[]]], Times[Optional[Pattern[f, Blank[]]], Pattern[x, Blank[]]]]], Optional[Pattern[d, Blank[]]]], Pattern[c, Blank[]]], Optional[Pattern[n, Blank[]]]], Power[Times[Optional[Pattern[g, Blank[]]], tan[Plus[Optional[Pattern[e, Blank[]]], Times[Optional[Pattern[f, Blank[]]], Pattern[x, Blank[]]]]]], Optional[Pattern[p, Blank[]]]], Power[Plus[Pattern[a, Blank[]], Times[Optional[Pattern[b, Blank[]]], tan[Plus[Optional[Pattern[e, Blank[]]], Times[Optional[Pattern[f, Blank[]]], Pattern[x, Blank[]]]]]]], Optional[Pattern[m, Blank[]]]]], Pattern[x, Blank[Symbol]]] := Dist[g^n, Int[(g*Tan[e + f*x])^(p - n)*(a + b*Tan[e + f*x])^m*(d + c*Tan[e + f*x])^n, x], x] /; FreeQ[{a, b, c, d, e, f, g, m, p}, x] && IntegerQ[n]
  3585. Int[Times[Power[Plus[Times[cot[Plus[Optional[Pattern[e, Blank[]]], Times[Optional[Pattern[f, Blank[]]], Pattern[x, Blank[]]]]], Optional[Pattern[d, Blank[]]]], Pattern[c, Blank[]]], Pattern[n, Blank[]]], Power[tan[Plus[Optional[Pattern[e, Blank[]]], Times[Optional[Pattern[f, Blank[]]], Pattern[x, Blank[]]]]], Optional[Pattern[p, Blank[]]]], Power[Plus[Pattern[a, Blank[]], Times[Optional[Pattern[b, Blank[]]], tan[Plus[Optional[Pattern[e, Blank[]]], Times[Optional[Pattern[f, Blank[]]], Pattern[x, Blank[]]]]]]], Optional[Pattern[m, Blank[]]]]], Pattern[x, Blank[Symbol]]] := Int[((b + a*Cot[e + f*x])^m*(c + d*Cot[e + f*x])^n)/Cot[e + f*x]^(m + p), x] /; FreeQ[{a, b, c, d, e, f, n}, x] && !IntegerQ[n] && IntegerQ[m] && IntegerQ[p]
  3586. Int[Times[Power[Plus[Times[cot[Plus[Optional[Pattern[e, Blank[]]], Times[Optional[Pattern[f, Blank[]]], Pattern[x, Blank[]]]]], Optional[Pattern[d, Blank[]]]], Pattern[c, Blank[]]], Pattern[n, Blank[]]], Power[Times[Optional[Pattern[g, Blank[]]], tan[Plus[Optional[Pattern[e, Blank[]]], Times[Optional[Pattern[f, Blank[]]], Pattern[x, Blank[]]]]]], Pattern[p, Blank[]]], Power[Plus[Pattern[a, Blank[]], Times[Optional[Pattern[b, Blank[]]], tan[Plus[Optional[Pattern[e, Blank[]]], Times[Optional[Pattern[f, Blank[]]], Pattern[x, Blank[]]]]]]], Optional[Pattern[m, Blank[]]]]], Pattern[x, Blank[Symbol]]] := Dist[Cot[e + f*x]^p*(g*Tan[e + f*x])^p, Int[((b + a*Cot[e + f*x])^m*(c + d*Cot[e + f*x])^n)/Cot[e + f*x]^(m + p), x], x] /; FreeQ[{a, b, c, d, e, f, g, n, p}, x] && !IntegerQ[n] && IntegerQ[m] && !IntegerQ[p]
  3587. Int[Times[Power[Plus[Times[cot[Plus[Optional[Pattern[e, Blank[]]], Times[Optional[Pattern[f, Blank[]]], Pattern[x, Blank[]]]]], Optional[Pattern[d, Blank[]]]], Pattern[c, Blank[]]], Pattern[n, Blank[]]], Power[Times[Optional[Pattern[g, Blank[]]], tan[Plus[Optional[Pattern[e, Blank[]]], Times[Optional[Pattern[f, Blank[]]], Pattern[x, Blank[]]]]]], Optional[Pattern[p, Blank[]]]], Power[Plus[Pattern[a, Blank[]], Times[Optional[Pattern[b, Blank[]]], tan[Plus[Optional[Pattern[e, Blank[]]], Times[Optional[Pattern[f, Blank[]]], Pattern[x, Blank[]]]]]]], Pattern[m, Blank[]]]], Pattern[x, Blank[Symbol]]] := Dist[((g*Tan[e + f*x])^n*(c + d*Cot[e + f*x])^n)/(d + c*Tan[e + f*x])^n, Int[(g*Tan[e + f*x])^(p - n)*(a + b*Tan[e + f*x])^m*(d + c*Tan[e + f*x])^n, x], x] /; FreeQ[{a, b, c, d, e, f, g, m, n, p}, x] && !IntegerQ[n] && !IntegerQ[m]
  3588. Int[Times[Power[Plus[Pattern[a, Blank[]], Times[Optional[Pattern[b, Blank[]]], tan[Plus[Optional[Pattern[e, Blank[]]], Times[Optional[Pattern[f, Blank[]]], Pattern[x, Blank[]]]]]]], Optional[Pattern[m, Blank[]]]], Plus[Optional[Pattern[A, Blank[]]], Times[Optional[Pattern[B, Blank[]]], tan[Plus[Optional[Pattern[e, Blank[]]], Times[Optional[Pattern[f, Blank[]]], Pattern[x, Blank[]]]]]]], Power[Plus[Pattern[c, Blank[]], Times[Optional[Pattern[d, Blank[]]], tan[Plus[Optional[Pattern[e, Blank[]]], Times[Optional[Pattern[f, Blank[]]], Pattern[x, Blank[]]]]]]], Optional[Pattern[n, Blank[]]]]], Pattern[x, Blank[Symbol]]] := Dist[(a*c)/f, Subst[Int[(a + b*x)^(m - 1)*(c + d*x)^(n - 1)*(A + B*x), x], x, Tan[e + f*x]], x] /; FreeQ[{a, b, c, d, e, f, A, B, m, n}, x] && EqQ[b*c + a*d, 0] && EqQ[a^2 + b^2, 0]
  3589. Int[Times[Power[Plus[Optional[Pattern[a, Blank[]]], Times[Optional[Pattern[b, Blank[]]], tan[Plus[Optional[Pattern[e, Blank[]]], Times[Optional[Pattern[f, Blank[]]], Pattern[x, Blank[]]]]]]], -1], Plus[Optional[Pattern[A, Blank[]]], Times[Optional[Pattern[B, Blank[]]], tan[Plus[Optional[Pattern[e, Blank[]]], Times[Optional[Pattern[f, Blank[]]], Pattern[x, Blank[]]]]]]], Plus[Optional[Pattern[c, Blank[]]], Times[Optional[Pattern[d, Blank[]]], tan[Plus[Optional[Pattern[e, Blank[]]], Times[Optional[Pattern[f, Blank[]]], Pattern[x, Blank[]]]]]]]], Pattern[x, Blank[Symbol]]] := Dist[(B*d)/b, Int[Tan[e + f*x], x], x] + Dist[1/b, Int[Simp[A*b*c + (A*b*d + B*(b*c - a*d))*Tan[e + f*x], x]/(a + b*Tan[e + f*x]), x], x] /; FreeQ[{a, b, c, d, e, f, A, B}, x] && NeQ[b*c - a*d, 0]
  3590. Int[Times[Power[Plus[Pattern[a, Blank[]], Times[Optional[Pattern[b, Blank[]]], tan[Plus[Optional[Pattern[e, Blank[]]], Times[Optional[Pattern[f, Blank[]]], Pattern[x, Blank[]]]]]]], Pattern[m, Blank[]]], Plus[Optional[Pattern[A, Blank[]]], Times[Optional[Pattern[B, Blank[]]], tan[Plus[Optional[Pattern[e, Blank[]]], Times[Optional[Pattern[f, Blank[]]], Pattern[x, Blank[]]]]]]], Plus[Optional[Pattern[c, Blank[]]], Times[Optional[Pattern[d, Blank[]]], tan[Plus[Optional[Pattern[e, Blank[]]], Times[Optional[Pattern[f, Blank[]]], Pattern[x, Blank[]]]]]]]], Pattern[x, Blank[Symbol]]] := -Simp[((A*b - a*B)*(a*c + b*d)*(a + b*Tan[e + f*x])^m)/(2*a^2*f*m), x] + Dist[1/(2*a*b), Int[(a + b*Tan[e + f*x])^(m + 1)*Simp[A*b*c + a*B*c + a*A*d + b*B*d + 2*a*B*d*Tan[e + f*x], x], x], x] /; FreeQ[{a, b, c, d, e, f, A, B}, x] && NeQ[b*c - a*d, 0] && LtQ[m, -1] && EqQ[a^2 + b^2, 0]
  3591. Int[Times[Power[Plus[Optional[Pattern[a, Blank[]]], Times[Optional[Pattern[b, Blank[]]], tan[Plus[Optional[Pattern[e, Blank[]]], Times[Optional[Pattern[f, Blank[]]], Pattern[x, Blank[]]]]]]], Pattern[m, Blank[]]], Plus[Optional[Pattern[A, Blank[]]], Times[Optional[Pattern[B, Blank[]]], tan[Plus[Optional[Pattern[e, Blank[]]], Times[Optional[Pattern[f, Blank[]]], Pattern[x, Blank[]]]]]]], Plus[Optional[Pattern[c, Blank[]]], Times[Optional[Pattern[d, Blank[]]], tan[Plus[Optional[Pattern[e, Blank[]]], Times[Optional[Pattern[f, Blank[]]], Pattern[x, Blank[]]]]]]]], Pattern[x, Blank[Symbol]]] := Simp[((b*c - a*d)*(A*b - a*B)*(a + b*Tan[e + f*x])^(m + 1))/(b*f*(m + 1)*(a^2 + b^2)), x] + Dist[1/(a^2 + b^2), Int[(a + b*Tan[e + f*x])^(m + 1)*Simp[a*A*c + b*B*c + A*b*d - a*B*d - (A*b*c - a*B*c - a*A*d - b*B*d)*Tan[e + f*x], x], x], x] /; FreeQ[{a, b, c, d, e, f, A, B}, x] && NeQ[b*c - a*d, 0] && LtQ[m, -1] && NeQ[a^2 + b^2, 0]
  3592. Int[Times[Power[Plus[Optional[Pattern[a, Blank[]]], Times[Optional[Pattern[b, Blank[]]], tan[Plus[Optional[Pattern[e, Blank[]]], Times[Optional[Pattern[f, Blank[]]], Pattern[x, Blank[]]]]]]], Optional[Pattern[m, Blank[]]]], Plus[Optional[Pattern[A, Blank[]]], Times[Optional[Pattern[B, Blank[]]], tan[Plus[Optional[Pattern[e, Blank[]]], Times[Optional[Pattern[f, Blank[]]], Pattern[x, Blank[]]]]]]], Plus[Optional[Pattern[c, Blank[]]], Times[Optional[Pattern[d, Blank[]]], tan[Plus[Optional[Pattern[e, Blank[]]], Times[Optional[Pattern[f, Blank[]]], Pattern[x, Blank[]]]]]]]], Pattern[x, Blank[Symbol]]] := Simp[(B*d*(a + b*Tan[e + f*x])^(m + 1))/(b*f*(m + 1)), x] + Int[(a + b*Tan[e + f*x])^m*Simp[A*c - B*d + (B*c + A*d)*Tan[e + f*x], x], x] /; FreeQ[{a, b, c, d, e, f, A, B, m}, x] && NeQ[b*c - a*d, 0] && !LeQ[m, -1]
  3593. Int[Times[Power[Plus[Pattern[a, Blank[]], Times[Optional[Pattern[b, Blank[]]], tan[Plus[Optional[Pattern[e, Blank[]]], Times[Optional[Pattern[f, Blank[]]], Pattern[x, Blank[]]]]]]], Pattern[m, Blank[]]], Plus[Optional[Pattern[A, Blank[]]], Times[Optional[Pattern[B, Blank[]]], tan[Plus[Optional[Pattern[e, Blank[]]], Times[Optional[Pattern[f, Blank[]]], Pattern[x, Blank[]]]]]]], Power[Plus[Optional[Pattern[c, Blank[]]], Times[Optional[Pattern[d, Blank[]]], tan[Plus[Optional[Pattern[e, Blank[]]], Times[Optional[Pattern[f, Blank[]]], Pattern[x, Blank[]]]]]]], Pattern[n, Blank[]]]], Pattern[x, Blank[Symbol]]] := -Simp[(a^2*(B*c - A*d)*(a + b*Tan[e + f*x])^(m - 1)*(c + d*Tan[e + f*x])^(n + 1))/(d*f*(b*c + a*d)*(n + 1)), x] - Dist[a/(d*(b*c + a*d)*(n + 1)), Int[(a + b*Tan[e + f*x])^(m - 1)*(c + d*Tan[e + f*x])^(n + 1)*Simp[A*b*d*(m - n - 2) - B*(b*c*(m - 1) + a*d*(n + 1)) + (a*A*d*(m + n) - B*(a*c*(m - 1) + b*d*(n + 1)))*Tan[e + f*x], x], x], x] /; FreeQ[{a, b, c, d, e, f, A, B}, x] && NeQ[b*c - a*d, 0] && EqQ[a^2 + b^2, 0] && GtQ[m, 1] && LtQ[n, -1]
  3594. Int[Times[Power[Plus[Pattern[a, Blank[]], Times[Optional[Pattern[b, Blank[]]], tan[Plus[Optional[Pattern[e, Blank[]]], Times[Optional[Pattern[f, Blank[]]], Pattern[x, Blank[]]]]]]], Pattern[m, Blank[]]], Plus[Optional[Pattern[A, Blank[]]], Times[Optional[Pattern[B, Blank[]]], tan[Plus[Optional[Pattern[e, Blank[]]], Times[Optional[Pattern[f, Blank[]]], Pattern[x, Blank[]]]]]]], Power[Plus[Optional[Pattern[c, Blank[]]], Times[Optional[Pattern[d, Blank[]]], tan[Plus[Optional[Pattern[e, Blank[]]], Times[Optional[Pattern[f, Blank[]]], Pattern[x, Blank[]]]]]]], Pattern[n, Blank[]]]], Pattern[x, Blank[Symbol]]] := Simp[(b*B*(a + b*Tan[e + f*x])^(m - 1)*(c + d*Tan[e + f*x])^(n + 1))/(d*f*(m + n)), x] + Dist[1/(d*(m + n)), Int[(a + b*Tan[e + f*x])^(m - 1)*(c + d*Tan[e + f*x])^n*Simp[a*A*d*(m + n) + B*(a*c*(m - 1) - b*d*(n + 1)) - (B*(b*c - a*d)*(m - 1) - d*(A*b + a*B)*(m + n))*Tan[e + f*x], x], x], x] /; FreeQ[{a, b, c, d, e, f, A, B, n}, x] && NeQ[b*c - a*d, 0] && EqQ[a^2 + b^2, 0] && GtQ[m, 1] && !LtQ[n, -1]
  3595. Int[Times[Power[Plus[Pattern[a, Blank[]], Times[Optional[Pattern[b, Blank[]]], tan[Plus[Optional[Pattern[e, Blank[]]], Times[Optional[Pattern[f, Blank[]]], Pattern[x, Blank[]]]]]]], Pattern[m, Blank[]]], Plus[Optional[Pattern[A, Blank[]]], Times[Optional[Pattern[B, Blank[]]], tan[Plus[Optional[Pattern[e, Blank[]]], Times[Optional[Pattern[f, Blank[]]], Pattern[x, Blank[]]]]]]], Power[Plus[Optional[Pattern[c, Blank[]]], Times[Optional[Pattern[d, Blank[]]], tan[Plus[Optional[Pattern[e, Blank[]]], Times[Optional[Pattern[f, Blank[]]], Pattern[x, Blank[]]]]]]], Pattern[n, Blank[]]]], Pattern[x, Blank[Symbol]]] := -Simp[((A*b - a*B)*(a + b*Tan[e + f*x])^m*(c + d*Tan[e + f*x])^n)/(2*a*f*m), x] + Dist[1/(2*a^2*m), Int[(a + b*Tan[e + f*x])^(m + 1)*(c + d*Tan[e + f*x])^(n - 1)*Simp[A*(a*c*m + b*d*n) - B*(b*c*m + a*d*n) - d*(b*B*(m - n) - a*A*(m + n))*Tan[e + f*x], x], x], x] /; FreeQ[{a, b, c, d, e, f, A, B}, x] && NeQ[b*c - a*d, 0] && EqQ[a^2 + b^2, 0] && LtQ[m, 0] && GtQ[n, 0]
  3596. Int[Times[Power[Plus[Pattern[a, Blank[]], Times[Optional[Pattern[b, Blank[]]], tan[Plus[Optional[Pattern[e, Blank[]]], Times[Optional[Pattern[f, Blank[]]], Pattern[x, Blank[]]]]]]], Pattern[m, Blank[]]], Plus[Optional[Pattern[A, Blank[]]], Times[Optional[Pattern[B, Blank[]]], tan[Plus[Optional[Pattern[e, Blank[]]], Times[Optional[Pattern[f, Blank[]]], Pattern[x, Blank[]]]]]]], Power[Plus[Optional[Pattern[c, Blank[]]], Times[Optional[Pattern[d, Blank[]]], tan[Plus[Optional[Pattern[e, Blank[]]], Times[Optional[Pattern[f, Blank[]]], Pattern[x, Blank[]]]]]]], Pattern[n, Blank[]]]], Pattern[x, Blank[Symbol]]] := Simp[((a*A + b*B)*(a + b*Tan[e + f*x])^m*(c + d*Tan[e + f*x])^(n + 1))/(2*f*m*(b*c - a*d)), x] + Dist[1/(2*a*m*(b*c - a*d)), Int[(a + b*Tan[e + f*x])^(m + 1)*(c + d*Tan[e + f*x])^n*Simp[A*(b*c*m - a*d*(2*m + n + 1)) + B*(a*c*m - b*d*(n + 1)) + d*(A*b - a*B)*(m + n + 1)*Tan[e + f*x], x], x], x] /; FreeQ[{a, b, c, d, e, f, A, B, n}, x] && NeQ[b*c - a*d, 0] && EqQ[a^2 + b^2, 0] && LtQ[m, 0] && !GtQ[n, 0]
  3597. Int[Times[Power[Plus[Pattern[a, Blank[]], Times[Optional[Pattern[b, Blank[]]], tan[Plus[Optional[Pattern[e, Blank[]]], Times[Optional[Pattern[f, Blank[]]], Pattern[x, Blank[]]]]]]], Pattern[m, Blank[]]], Plus[Optional[Pattern[A, Blank[]]], Times[Optional[Pattern[B, Blank[]]], tan[Plus[Optional[Pattern[e, Blank[]]], Times[Optional[Pattern[f, Blank[]]], Pattern[x, Blank[]]]]]]], Power[Plus[Optional[Pattern[c, Blank[]]], Times[Optional[Pattern[d, Blank[]]], tan[Plus[Optional[Pattern[e, Blank[]]], Times[Optional[Pattern[f, Blank[]]], Pattern[x, Blank[]]]]]]], Pattern[n, Blank[]]]], Pattern[x, Blank[Symbol]]] := Simp[(B*(a + b*Tan[e + f*x])^m*(c + d*Tan[e + f*x])^n)/(f*(m + n)), x] + Dist[1/(a*(m + n)), Int[(a + b*Tan[e + f*x])^m*(c + d*Tan[e + f*x])^(n - 1)*Simp[a*A*c*(m + n) - B*(b*c*m + a*d*n) + (a*A*d*(m + n) - B*(b*d*m - a*c*n))*Tan[e + f*x], x], x], x] /; FreeQ[{a, b, c, d, e, f, A, B, m}, x] && NeQ[b*c - a*d, 0] && EqQ[a^2 + b^2, 0] && GtQ[n, 0]
  3598. Int[Times[Power[Plus[Pattern[a, Blank[]], Times[Optional[Pattern[b, Blank[]]], tan[Plus[Optional[Pattern[e, Blank[]]], Times[Optional[Pattern[f, Blank[]]], Pattern[x, Blank[]]]]]]], Pattern[m, Blank[]]], Plus[Optional[Pattern[A, Blank[]]], Times[Optional[Pattern[B, Blank[]]], tan[Plus[Optional[Pattern[e, Blank[]]], Times[Optional[Pattern[f, Blank[]]], Pattern[x, Blank[]]]]]]], Power[Plus[Optional[Pattern[c, Blank[]]], Times[Optional[Pattern[d, Blank[]]], tan[Plus[Optional[Pattern[e, Blank[]]], Times[Optional[Pattern[f, Blank[]]], Pattern[x, Blank[]]]]]]], Pattern[n, Blank[]]]], Pattern[x, Blank[Symbol]]] := Simp[((A*d - B*c)*(a + b*Tan[e + f*x])^m*(c + d*Tan[e + f*x])^(n + 1))/(f*(n + 1)*(c^2 + d^2)), x] - Dist[1/(a*(n + 1)*(c^2 + d^2)), Int[(a + b*Tan[e + f*x])^m*(c + d*Tan[e + f*x])^(n + 1)*Simp[A*(b*d*m - a*c*(n + 1)) - B*(b*c*m + a*d*(n + 1)) - a*(B*c - A*d)*(m + n + 1)*Tan[e + f*x], x], x], x] /; FreeQ[{a, b, c, d, e, f, A, B, m}, x] && NeQ[b*c - a*d, 0] && EqQ[a^2 + b^2, 0] && LtQ[n, -1]
  3599. Int[Times[Power[Plus[Pattern[a, Blank[]], Times[Optional[Pattern[b, Blank[]]], tan[Plus[Optional[Pattern[e, Blank[]]], Times[Optional[Pattern[f, Blank[]]], Pattern[x, Blank[]]]]]]], Pattern[m, Blank[]]], Plus[Optional[Pattern[A, Blank[]]], Times[Optional[Pattern[B, Blank[]]], tan[Plus[Optional[Pattern[e, Blank[]]], Times[Optional[Pattern[f, Blank[]]], Pattern[x, Blank[]]]]]]], Power[Plus[Optional[Pattern[c, Blank[]]], Times[Optional[Pattern[d, Blank[]]], tan[Plus[Optional[Pattern[e, Blank[]]], Times[Optional[Pattern[f, Blank[]]], Pattern[x, Blank[]]]]]]], Pattern[n, Blank[]]]], Pattern[x, Blank[Symbol]]] := Dist[(b*B)/f, Subst[Int[(a + b*x)^(m - 1)*(c + d*x)^n, x], x, Tan[e + f*x]], x] /; FreeQ[{a, b, c, d, e, f, A, B, m, n}, x] && NeQ[b*c - a*d, 0] && EqQ[a^2 + b^2, 0] && EqQ[A*b + a*B, 0]
  3600. Int[Times[Power[Plus[Pattern[a, Blank[]], Times[Optional[Pattern[b, Blank[]]], tan[Plus[Optional[Pattern[e, Blank[]]], Times[Optional[Pattern[f, Blank[]]], Pattern[x, Blank[]]]]]]], Pattern[m, Blank[]]], Plus[Optional[Pattern[A, Blank[]]], Times[Optional[Pattern[B, Blank[]]], tan[Plus[Optional[Pattern[e, Blank[]]], Times[Optional[Pattern[f, Blank[]]], Pattern[x, Blank[]]]]]]], Power[Plus[Optional[Pattern[c, Blank[]]], Times[Optional[Pattern[d, Blank[]]], tan[Plus[Optional[Pattern[e, Blank[]]], Times[Optional[Pattern[f, Blank[]]], Pattern[x, Blank[]]]]]]], -1]], Pattern[x, Blank[Symbol]]] := Dist[(A*b + a*B)/(b*c + a*d), Int[(a + b*Tan[e + f*x])^m, x], x] - Dist[(B*c - A*d)/(b*c + a*d), Int[((a + b*Tan[e + f*x])^m*(a - b*Tan[e + f*x]))/(c + d*Tan[e + f*x]), x], x] /; FreeQ[{a, b, c, d, e, f, A, B, m}, x] && NeQ[b*c - a*d, 0] && EqQ[a^2 + b^2, 0] && NeQ[A*b + a*B, 0]
  3601. Int[Times[Power[Plus[Pattern[a, Blank[]], Times[Optional[Pattern[b, Blank[]]], tan[Plus[Optional[Pattern[e, Blank[]]], Times[Optional[Pattern[f, Blank[]]], Pattern[x, Blank[]]]]]]], Pattern[m, Blank[]]], Plus[Optional[Pattern[A, Blank[]]], Times[Optional[Pattern[B, Blank[]]], tan[Plus[Optional[Pattern[e, Blank[]]], Times[Optional[Pattern[f, Blank[]]], Pattern[x, Blank[]]]]]]], Power[Plus[Optional[Pattern[c, Blank[]]], Times[Optional[Pattern[d, Blank[]]], tan[Plus[Optional[Pattern[e, Blank[]]], Times[Optional[Pattern[f, Blank[]]], Pattern[x, Blank[]]]]]]], Pattern[n, Blank[]]]], Pattern[x, Blank[Symbol]]] := Dist[(A*b + a*B)/b, Int[(a + b*Tan[e + f*x])^m*(c + d*Tan[e + f*x])^n, x], x] - Dist[B/b, Int[(a + b*Tan[e + f*x])^m*(c + d*Tan[e + f*x])^n*(a - b*Tan[e + f*x]), x], x] /; FreeQ[{a, b, c, d, e, f, A, B, m, n}, x] && NeQ[b*c - a*d, 0] && EqQ[a^2 + b^2, 0] && NeQ[A*b + a*B, 0]
  3602. Int[Times[Power[Plus[Optional[Pattern[a, Blank[]]], Times[Optional[Pattern[b, Blank[]]], tan[Plus[Optional[Pattern[e, Blank[]]], Times[Optional[Pattern[f, Blank[]]], Pattern[x, Blank[]]]]]]], Pattern[m, Blank[]]], Plus[Pattern[A, Blank[]], Times[Optional[Pattern[B, Blank[]]], tan[Plus[Optional[Pattern[e, Blank[]]], Times[Optional[Pattern[f, Blank[]]], Pattern[x, Blank[]]]]]]], Power[Plus[Optional[Pattern[c, Blank[]]], Times[Optional[Pattern[d, Blank[]]], tan[Plus[Optional[Pattern[e, Blank[]]], Times[Optional[Pattern[f, Blank[]]], Pattern[x, Blank[]]]]]]], Pattern[n, Blank[]]]], Pattern[x, Blank[Symbol]]] := Dist[A^2/f, Subst[Int[((a + b*x)^m*(c + d*x)^n)/(A - B*x), x], x, Tan[e + f*x]], x] /; FreeQ[{a, b, c, d, e, f, A, B, m, n}, x] && NeQ[b*c - a*d, 0] && NeQ[a^2 + b^2, 0] && !IntegerQ[m] && !IntegerQ[n] && !IntegersQ[2*m, 2*n] && EqQ[A^2 + B^2, 0]
  3603. Int[Times[Power[Plus[Optional[Pattern[a, Blank[]]], Times[Optional[Pattern[b, Blank[]]], tan[Plus[Optional[Pattern[e, Blank[]]], Times[Optional[Pattern[f, Blank[]]], Pattern[x, Blank[]]]]]]], Pattern[m, Blank[]]], Plus[Optional[Pattern[A, Blank[]]], Times[Optional[Pattern[B, Blank[]]], tan[Plus[Optional[Pattern[e, Blank[]]], Times[Optional[Pattern[f, Blank[]]], Pattern[x, Blank[]]]]]]], Power[Plus[Optional[Pattern[c, Blank[]]], Times[Optional[Pattern[d, Blank[]]], tan[Plus[Optional[Pattern[e, Blank[]]], Times[Optional[Pattern[f, Blank[]]], Pattern[x, Blank[]]]]]]], Pattern[n, Blank[]]]], Pattern[x, Blank[Symbol]]] := Dist[(A + I*B)/2, Int[(a + b*Tan[e + f*x])^m*(c + d*Tan[e + f*x])^n*(1 - I*Tan[e + f*x]), x], x] + Dist[(A - I*B)/2, Int[(a + b*Tan[e + f*x])^m*(c + d*Tan[e + f*x])^n*(1 + I*Tan[e + f*x]), x], x] /; FreeQ[{a, b, c, d, e, f, A, B, m, n}, x] && NeQ[b*c - a*d, 0] && NeQ[a^2 + b^2, 0] && !IntegerQ[m] && !IntegerQ[n] && !IntegersQ[2*m, 2*n] && NeQ[A^2 + B^2, 0]
  3604. Int[Times[Power[Plus[Optional[Pattern[a, Blank[]]], Times[Optional[Pattern[b, Blank[]]], tan[Plus[Optional[Pattern[e, Blank[]]], Times[Optional[Pattern[f, Blank[]]], Pattern[x, Blank[]]]]]]], 2], Plus[Optional[Pattern[A, Blank[]]], Times[Optional[Pattern[B, Blank[]]], tan[Plus[Optional[Pattern[e, Blank[]]], Times[Optional[Pattern[f, Blank[]]], Pattern[x, Blank[]]]]]]], Power[Plus[Optional[Pattern[c, Blank[]]], Times[Optional[Pattern[d, Blank[]]], tan[Plus[Optional[Pattern[e, Blank[]]], Times[Optional[Pattern[f, Blank[]]], Pattern[x, Blank[]]]]]]], Pattern[n, Blank[]]]], Pattern[x, Blank[Symbol]]] := -Simp[((B*c - A*d)*(b*c - a*d)^2*(c + d*Tan[e + f*x])^(n + 1))/(f*d^2*(n + 1)*(c^2 + d^2)), x] + Dist[1/(d*(c^2 + d^2)), Int[(c + d*Tan[e + f*x])^(n + 1)*Simp[B*(b*c - a*d)^2 + A*d*(a^2*c - b^2*c + 2*a*b*d) + d*(B*(a^2*c - b^2*c + 2*a*b*d) + A*(2*a*b*c - a^2*d + b^2*d))*Tan[e + f*x] + b^2*B*(c^2 + d^2)*Tan[e + f*x]^2, x], x], x] /; FreeQ[{a, b, c, d, e, f, A, B}, x] && NeQ[b*c - a*d, 0] && NeQ[a^2 + b^2, 0] && NeQ[c^2 + d^2, 0] && LtQ[n, -1]
  3605. Int[Times[Power[Plus[Optional[Pattern[a, Blank[]]], Times[Optional[Pattern[b, Blank[]]], tan[Plus[Optional[Pattern[e, Blank[]]], Times[Optional[Pattern[f, Blank[]]], Pattern[x, Blank[]]]]]]], Pattern[m, Blank[]]], Plus[Optional[Pattern[A, Blank[]]], Times[Optional[Pattern[B, Blank[]]], tan[Plus[Optional[Pattern[e, Blank[]]], Times[Optional[Pattern[f, Blank[]]], Pattern[x, Blank[]]]]]]], Power[Plus[Optional[Pattern[c, Blank[]]], Times[Optional[Pattern[d, Blank[]]], tan[Plus[Optional[Pattern[e, Blank[]]], Times[Optional[Pattern[f, Blank[]]], Pattern[x, Blank[]]]]]]], Pattern[n, Blank[]]]], Pattern[x, Blank[Symbol]]] := Simp[((b*c - a*d)*(B*c - A*d)*(a + b*Tan[e + f*x])^(m - 1)*(c + d*Tan[e + f*x])^(n + 1))/(d*f*(n + 1)*(c^2 + d^2)), x] - Dist[1/(d*(n + 1)*(c^2 + d^2)), Int[(a + b*Tan[e + f*x])^(m - 2)*(c + d*Tan[e + f*x])^(n + 1)*Simp[a*A*d*(b*d*(m - 1) - a*c*(n + 1)) + (b*B*c - (A*b + a*B)*d)*(b*c*(m - 1) + a*d*(n + 1)) - d*((a*A - b*B)*(b*c - a*d) + (A*b + a*B)*(a*c + b*d))*(n + 1)*Tan[e + f*x] - b*(d*(A*b*c + a*B*c - a*A*d)*(m + n) - b*B*(c^2*(m - 1) - d^2*(n + 1)))*Tan[e + f*x]^2, x], x], x] /; FreeQ[{a, b, c, d, e, f, A, B}, x] && NeQ[b*c - a*d, 0] && NeQ[a^2 + b^2, 0] && NeQ[c^2 + d^2, 0] && GtQ[m, 1] && LtQ[n, -1] && (IntegerQ[m] || IntegersQ[2*m, 2*n])
  3606. Int[Times[Power[Plus[Optional[Pattern[a, Blank[]]], Times[Optional[Pattern[b, Blank[]]], tan[Plus[Optional[Pattern[e, Blank[]]], Times[Optional[Pattern[f, Blank[]]], Pattern[x, Blank[]]]]]]], 2], Plus[Optional[Pattern[A, Blank[]]], Times[Optional[Pattern[B, Blank[]]], tan[Plus[Optional[Pattern[e, Blank[]]], Times[Optional[Pattern[f, Blank[]]], Pattern[x, Blank[]]]]]]], Power[Plus[Optional[Pattern[c, Blank[]]], Times[Optional[Pattern[d, Blank[]]], tan[Plus[Optional[Pattern[e, Blank[]]], Times[Optional[Pattern[f, Blank[]]], Pattern[x, Blank[]]]]]]], -1]], Pattern[x, Blank[Symbol]]] := Simp[(b^2*B*Tan[e + f*x])/(d*f), x] + Dist[1/d, Int[(a^2*A*d - b^2*B*c + (2*a*A*b + B*(a^2 - b^2))*d*Tan[e + f*x] + (A*b^2*d - b*B*(b*c - 2*a*d))*Tan[e + f*x]^2)/(c + d*Tan[e + f*x]), x], x] /; FreeQ[{a, b, c, d, e, f, A, B}, x] && NeQ[b*c - a*d, 0] && NeQ[a^2 + b^2, 0] && NeQ[c^2 + d^2, 0]
  3607. Int[Times[Power[Plus[Optional[Pattern[a, Blank[]]], Times[Optional[Pattern[b, Blank[]]], tan[Plus[Optional[Pattern[e, Blank[]]], Times[Optional[Pattern[f, Blank[]]], Pattern[x, Blank[]]]]]]], Pattern[m, Blank[]]], Plus[Optional[Pattern[A, Blank[]]], Times[Optional[Pattern[B, Blank[]]], tan[Plus[Optional[Pattern[e, Blank[]]], Times[Optional[Pattern[f, Blank[]]], Pattern[x, Blank[]]]]]]], Power[Plus[Optional[Pattern[c, Blank[]]], Times[Optional[Pattern[d, Blank[]]], tan[Plus[Optional[Pattern[e, Blank[]]], Times[Optional[Pattern[f, Blank[]]], Pattern[x, Blank[]]]]]]], Pattern[n, Blank[]]]], Pattern[x, Blank[Symbol]]] := Simp[(b*B*(a + b*Tan[e + f*x])^(m - 1)*(c + d*Tan[e + f*x])^(n + 1))/(d*f*(m + n)), x] + Dist[1/(d*(m + n)), Int[(a + b*Tan[e + f*x])^(m - 2)*(c + d*Tan[e + f*x])^n*Simp[a^2*A*d*(m + n) - b*B*(b*c*(m - 1) + a*d*(n + 1)) + d*(m + n)*(2*a*A*b + B*(a^2 - b^2))*Tan[e + f*x] - (b*B*(b*c - a*d)*(m - 1) - b*(A*b + a*B)*d*(m + n))*Tan[e + f*x]^2, x], x], x] /; FreeQ[{a, b, c, d, e, f, A, B, n}, x] && NeQ[b*c - a*d, 0] && NeQ[a^2 + b^2, 0] && NeQ[c^2 + d^2, 0] && GtQ[m, 1] && (IntegerQ[m] || IntegersQ[2*m, 2*n]) && !(IGtQ[n, 1] && ( !IntegerQ[m] || (EqQ[c, 0] && NeQ[a, 0])))
  3608. Int[Times[Power[Plus[Optional[Pattern[a, Blank[]]], Times[Optional[Pattern[b, Blank[]]], tan[Plus[Optional[Pattern[e, Blank[]]], Times[Optional[Pattern[f, Blank[]]], Pattern[x, Blank[]]]]]]], Pattern[m, Blank[]]], Plus[Optional[Pattern[A, Blank[]]], Times[Optional[Pattern[B, Blank[]]], tan[Plus[Optional[Pattern[e, Blank[]]], Times[Optional[Pattern[f, Blank[]]], Pattern[x, Blank[]]]]]]], Power[Plus[Optional[Pattern[c, Blank[]]], Times[Optional[Pattern[d, Blank[]]], tan[Plus[Optional[Pattern[e, Blank[]]], Times[Optional[Pattern[f, Blank[]]], Pattern[x, Blank[]]]]]]], Pattern[n, Blank[]]]], Pattern[x, Blank[Symbol]]] := Simp[((A*b - a*B)*(a + b*Tan[e + f*x])^(m + 1)*(c + d*Tan[e + f*x])^n)/(f*(m + 1)*(a^2 + b^2)), x] + Dist[1/(b*(m + 1)*(a^2 + b^2)), Int[(a + b*Tan[e + f*x])^(m + 1)*(c + d*Tan[e + f*x])^(n - 1)*Simp[b*B*(b*c*(m + 1) + a*d*n) + A*b*(a*c*(m + 1) - b*d*n) - b*(A*(b*c - a*d) - B*(a*c + b*d))*(m + 1)*Tan[e + f*x] - b*d*(A*b - a*B)*(m + n + 1)*Tan[e + f*x]^2, x], x], x] /; FreeQ[{a, b, c, d, e, f, A, B}, x] && NeQ[b*c - a*d, 0] && NeQ[a^2 + b^2, 0] && NeQ[c^2 + d^2, 0] && LtQ[m, -1] && LtQ[0, n, 1] && (IntegerQ[m] || IntegersQ[2*m, 2*n])
  3609. Int[Times[Power[Plus[Optional[Pattern[a, Blank[]]], Times[Optional[Pattern[b, Blank[]]], tan[Plus[Optional[Pattern[e, Blank[]]], Times[Optional[Pattern[f, Blank[]]], Pattern[x, Blank[]]]]]]], Pattern[m, Blank[]]], Plus[Optional[Pattern[A, Blank[]]], Times[Optional[Pattern[B, Blank[]]], tan[Plus[Optional[Pattern[e, Blank[]]], Times[Optional[Pattern[f, Blank[]]], Pattern[x, Blank[]]]]]]], Power[Plus[Optional[Pattern[c, Blank[]]], Times[Optional[Pattern[d, Blank[]]], tan[Plus[Optional[Pattern[e, Blank[]]], Times[Optional[Pattern[f, Blank[]]], Pattern[x, Blank[]]]]]]], Pattern[n, Blank[]]]], Pattern[x, Blank[Symbol]]] := Simp[(b*(A*b - a*B)*(a + b*Tan[e + f*x])^(m + 1)*(c + d*Tan[e + f*x])^(n + 1))/(f*(m + 1)*(b*c - a*d)*(a^2 + b^2)), x] + Dist[1/((m + 1)*(b*c - a*d)*(a^2 + b^2)), Int[(a + b*Tan[e + f*x])^(m + 1)*(c + d*Tan[e + f*x])^n*Simp[b*B*(b*c*(m + 1) + a*d*(n + 1)) + A*(a*(b*c - a*d)*(m + 1) - b^2*d*(m + n + 2)) - (A*b - a*B)*(b*c - a*d)*(m + 1)*Tan[e + f*x] - b*d*(A*b - a*B)*(m + n + 2)*Tan[e + f*x]^2, x], x], x] /; FreeQ[{a, b, c, d, e, f, A, B, n}, x] && NeQ[b*c - a*d, 0] && NeQ[a^2 + b^2, 0] && NeQ[c^2 + d^2, 0] && LtQ[m, -1] && (IntegerQ[m] || IntegersQ[2*m, 2*n]) && !(ILtQ[n, -1] && ( !IntegerQ[m] || (EqQ[c, 0] && NeQ[a, 0])))
  3610. Int[Times[Power[Plus[Optional[Pattern[a, Blank[]]], Times[Optional[Pattern[b, Blank[]]], tan[Plus[Optional[Pattern[e, Blank[]]], Times[Optional[Pattern[f, Blank[]]], Pattern[x, Blank[]]]]]]], Pattern[m, Blank[]]], Plus[Optional[Pattern[A, Blank[]]], Times[Optional[Pattern[B, Blank[]]], tan[Plus[Optional[Pattern[e, Blank[]]], Times[Optional[Pattern[f, Blank[]]], Pattern[x, Blank[]]]]]]], Power[Plus[Optional[Pattern[c, Blank[]]], Times[Optional[Pattern[d, Blank[]]], tan[Plus[Optional[Pattern[e, Blank[]]], Times[Optional[Pattern[f, Blank[]]], Pattern[x, Blank[]]]]]]], Pattern[n, Blank[]]]], Pattern[x, Blank[Symbol]]] := Simp[(B*(a + b*Tan[e + f*x])^m*(c + d*Tan[e + f*x])^n)/(f*(m + n)), x] + Dist[1/(m + n), Int[(a + b*Tan[e + f*x])^(m - 1)*(c + d*Tan[e + f*x])^(n - 1)*Simp[a*A*c*(m + n) - B*(b*c*m + a*d*n) + (A*b*c + a*B*c + a*A*d - b*B*d)*(m + n)*Tan[e + f*x] + (A*b*d*(m + n) + B*(a*d*m + b*c*n))*Tan[e + f*x]^2, x], x], x] /; FreeQ[{a, b, c, d, e, f, A, B}, x] && NeQ[b*c - a*d, 0] && NeQ[a^2 + b^2, 0] && NeQ[c^2 + d^2, 0] && LtQ[0, m, 1] && LtQ[0, n, 1]
  3611. Int[Times[Power[Plus[Pattern[a, Blank[]], Times[Optional[Pattern[b, Blank[]]], tan[Plus[Optional[Pattern[e, Blank[]]], Times[Optional[Pattern[f, Blank[]]], Pattern[x, Blank[]]]]]]], -1], Plus[Optional[Pattern[A, Blank[]]], Times[Optional[Pattern[B, Blank[]]], tan[Plus[Optional[Pattern[e, Blank[]]], Times[Optional[Pattern[f, Blank[]]], Pattern[x, Blank[]]]]]]], Power[Plus[Optional[Pattern[c, Blank[]]], Times[Optional[Pattern[d, Blank[]]], tan[Plus[Optional[Pattern[e, Blank[]]], Times[Optional[Pattern[f, Blank[]]], Pattern[x, Blank[]]]]]]], -1]], Pattern[x, Blank[Symbol]]] := Simp[((B*(b*c + a*d) + A*(a*c - b*d))*x)/((a^2 + b^2)*(c^2 + d^2)), x] + (Dist[(b*(A*b - a*B))/((b*c - a*d)*(a^2 + b^2)), Int[(b - a*Tan[e + f*x])/(a + b*Tan[e + f*x]), x], x] + Dist[(d*(B*c - A*d))/((b*c - a*d)*(c^2 + d^2)), Int[(d - c*Tan[e + f*x])/(c + d*Tan[e + f*x]), x], x]) /; FreeQ[{a, b, c, d, e, f, A, B}, x] && NeQ[b*c - a*d, 0] && NeQ[a^2 + b^2, 0] && NeQ[c^2 + d^2, 0]
  3612. Int[Times[Power[Plus[Optional[Pattern[a, Blank[]]], Times[Optional[Pattern[b, Blank[]]], tan[Plus[Optional[Pattern[e, Blank[]]], Times[Optional[Pattern[f, Blank[]]], Pattern[x, Blank[]]]]]]], -1], Plus[Optional[Pattern[A, Blank[]]], Times[Optional[Pattern[B, Blank[]]], tan[Plus[Optional[Pattern[e, Blank[]]], Times[Optional[Pattern[f, Blank[]]], Pattern[x, Blank[]]]]]]], Power[Plus[Optional[Pattern[c, Blank[]]], Times[Optional[Pattern[d, Blank[]]], tan[Plus[Optional[Pattern[e, Blank[]]], Times[Optional[Pattern[f, Blank[]]], Pattern[x, Blank[]]]]]]], Rational[1, 2]]], Pattern[x, Blank[Symbol]]] := Dist[1/(a^2 + b^2), Int[Simp[A*(a*c + b*d) + B*(b*c - a*d) - (A*(b*c - a*d) - B*(a*c + b*d))*Tan[e + f*x], x]/Sqrt[c + d*Tan[e + f*x]], x], x] - Dist[((b*c - a*d)*(B*a - A*b))/(a^2 + b^2), Int[(1 + Tan[e + f*x]^2)/((a + b*Tan[e + f*x])*Sqrt[c + d*Tan[e + f*x]]), x], x] /; FreeQ[{a, b, c, d, e, f, A, B}, x] && NeQ[b*c - a*d, 0] && NeQ[a^2 + b^2, 0] && NeQ[c^2 + d^2, 0]
  3613. Int[Times[Power[Plus[Optional[Pattern[a, Blank[]]], Times[Optional[Pattern[b, Blank[]]], tan[Plus[Optional[Pattern[e, Blank[]]], Times[Optional[Pattern[f, Blank[]]], Pattern[x, Blank[]]]]]]], -1], Plus[Optional[Pattern[A, Blank[]]], Times[Optional[Pattern[B, Blank[]]], tan[Plus[Optional[Pattern[e, Blank[]]], Times[Optional[Pattern[f, Blank[]]], Pattern[x, Blank[]]]]]]], Power[Plus[Optional[Pattern[c, Blank[]]], Times[Optional[Pattern[d, Blank[]]], tan[Plus[Optional[Pattern[e, Blank[]]], Times[Optional[Pattern[f, Blank[]]], Pattern[x, Blank[]]]]]]], Pattern[n, Blank[]]]], Pattern[x, Blank[Symbol]]] := Dist[1/(a^2 + b^2), Int[(c + d*Tan[e + f*x])^n*Simp[a*A + b*B - (A*b - a*B)*Tan[e + f*x], x], x], x] + Dist[(b*(A*b - a*B))/(a^2 + b^2), Int[((c + d*Tan[e + f*x])^n*(1 + Tan[e + f*x]^2))/(a + b*Tan[e + f*x]), x], x] /; FreeQ[{a, b, c, d, e, f, A, B, n}, x] && NeQ[b*c - a*d, 0] && NeQ[a^2 + b^2, 0] && NeQ[c^2 + d^2, 0]
  3614. Int[Times[Power[Plus[Optional[Pattern[a, Blank[]]], Times[Optional[Pattern[b, Blank[]]], tan[Plus[Optional[Pattern[e, Blank[]]], Times[Optional[Pattern[f, Blank[]]], Pattern[x, Blank[]]]]]]], Rational[1, 2]], Plus[Optional[Pattern[A, Blank[]]], Times[Optional[Pattern[B, Blank[]]], tan[Plus[Optional[Pattern[e, Blank[]]], Times[Optional[Pattern[f, Blank[]]], Pattern[x, Blank[]]]]]]], Power[Plus[Optional[Pattern[c, Blank[]]], Times[Optional[Pattern[d, Blank[]]], tan[Plus[Optional[Pattern[e, Blank[]]], Times[Optional[Pattern[f, Blank[]]], Pattern[x, Blank[]]]]]]], Rational[-1, 2]]], Pattern[x, Blank[Symbol]]] := Int[Simp[a*A - b*B + (A*b + a*B)*Tan[e + f*x], x]/(Sqrt[a + b*Tan[e + f*x]]*Sqrt[c + d*Tan[e + f*x]]), x] + Dist[b*B, Int[(1 + Tan[e + f*x]^2)/(Sqrt[a + b*Tan[e + f*x]]*Sqrt[c + d*Tan[e + f*x]]), x], x] /; FreeQ[{a, b, c, d, e, f, A, B}, x] && NeQ[b*c - a*d, 0] && NeQ[a^2 + b^2, 0] && NeQ[c^2 + d^2, 0]
  3615. Int[Times[Power[Plus[Optional[Pattern[a, Blank[]]], Times[Optional[Pattern[b, Blank[]]], tan[Plus[Optional[Pattern[e, Blank[]]], Times[Optional[Pattern[f, Blank[]]], Pattern[x, Blank[]]]]]]], Pattern[m, Blank[]]], Plus[Optional[Pattern[A, Blank[]]], Times[Optional[Pattern[B, Blank[]]], tan[Plus[Optional[Pattern[e, Blank[]]], Times[Optional[Pattern[f, Blank[]]], Pattern[x, Blank[]]]]]]], Power[Plus[Optional[Pattern[c, Blank[]]], Times[Optional[Pattern[d, Blank[]]], tan[Plus[Optional[Pattern[e, Blank[]]], Times[Optional[Pattern[f, Blank[]]], Pattern[x, Blank[]]]]]]], Pattern[n, Blank[]]]], Pattern[x, Blank[Symbol]]] := Dist[A^2/f, Subst[Int[((a + b*x)^m*(c + d*x)^n)/(A - B*x), x], x, Tan[e + f*x]], x] /; FreeQ[{a, b, c, d, e, f, A, B, m, n}, x] && NeQ[b*c - a*d, 0] && NeQ[a^2 + b^2, 0] && EqQ[A^2 + B^2, 0]
  3616. Int[Times[Power[Plus[Optional[Pattern[a, Blank[]]], Times[Optional[Pattern[b, Blank[]]], tan[Plus[Optional[Pattern[e, Blank[]]], Times[Optional[Pattern[f, Blank[]]], Pattern[x, Blank[]]]]]]], Pattern[m, Blank[]]], Plus[Optional[Pattern[A, Blank[]]], Times[Optional[Pattern[B, Blank[]]], tan[Plus[Optional[Pattern[e, Blank[]]], Times[Optional[Pattern[f, Blank[]]], Pattern[x, Blank[]]]]]]], Power[Plus[Optional[Pattern[c, Blank[]]], Times[Optional[Pattern[d, Blank[]]], tan[Plus[Optional[Pattern[e, Blank[]]], Times[Optional[Pattern[f, Blank[]]], Pattern[x, Blank[]]]]]]], Pattern[n, Blank[]]]], Pattern[x, Blank[Symbol]]] := Dist[(A + I*B)/2, Int[(a + b*Tan[e + f*x])^m*(c + d*Tan[e + f*x])^n*(1 - I*Tan[e + f*x]), x], x] + Dist[(A - I*B)/2, Int[(a + b*Tan[e + f*x])^m*(c + d*Tan[e + f*x])^n*(1 + I*Tan[e + f*x]), x], x] /; FreeQ[{a, b, c, d, e, f, A, B, m, n}, x] && NeQ[b*c - a*d, 0] && NeQ[a^2 + b^2, 0] && NeQ[A^2 + B^2, 0]
  3617. Int[Times[Power[Plus[Optional[Pattern[a, Blank[]]], Times[Optional[Pattern[b, Blank[]]], tan[Plus[Optional[Pattern[e, Blank[]]], Times[Optional[Pattern[f, Blank[]]], Pattern[x, Blank[]]]]]]], Optional[Pattern[m, Blank[]]]], Plus[Pattern[A, Blank[]], Times[Optional[Pattern[C, Blank[]]], Power[tan[Plus[Optional[Pattern[e, Blank[]]], Times[Optional[Pattern[f, Blank[]]], Pattern[x, Blank[]]]]], 2]]]], Pattern[x, Blank[Symbol]]] := Dist[A/(b*f), Subst[Int[(a + x)^m, x], x, b*Tan[e + f*x]], x] /; FreeQ[{a, b, e, f, A, C, m}, x] && EqQ[A, C]
  3618. Int[Times[Power[Plus[Optional[Pattern[a, Blank[]]], Times[cot[Plus[Optional[Pattern[e, Blank[]]], Times[Optional[Pattern[f, Blank[]]], Pattern[x, Blank[]]]]], Optional[Pattern[b, Blank[]]]]], Optional[Pattern[m, Blank[]]]], Plus[Times[Power[cot[Plus[Optional[Pattern[e, Blank[]]], Times[Optional[Pattern[f, Blank[]]], Pattern[x, Blank[]]]]], 2], Optional[Pattern[C, Blank[]]]], Pattern[A, Blank[]]]], Pattern[x, Blank[Symbol]]] := -Dist[A/(b*f), Subst[Int[(a + x)^m, x], x, b*Cot[e + f*x]], x] /; FreeQ[{a, b, e, f, A, C, m}, x] && EqQ[A, C]
  3619. Int[Times[Power[Plus[Optional[Pattern[a, Blank[]]], Times[Optional[Pattern[b, Blank[]]], tan[Plus[Optional[Pattern[e, Blank[]]], Times[Optional[Pattern[f, Blank[]]], Pattern[x, Blank[]]]]]]], Optional[Pattern[m, Blank[]]]], Plus[Optional[Pattern[A, Blank[]]], Times[Optional[Pattern[B, Blank[]]], tan[Plus[Optional[Pattern[e, Blank[]]], Times[Optional[Pattern[f, Blank[]]], Pattern[x, Blank[]]]]]], Times[Optional[Pattern[C, Blank[]]], Power[tan[Plus[Optional[Pattern[e, Blank[]]], Times[Optional[Pattern[f, Blank[]]], Pattern[x, Blank[]]]]], 2]]]], Pattern[x, Blank[Symbol]]] := Dist[1/b^2, Int[(a + b*Tan[e + f*x])^(m + 1)*Simp[b*B - a*C + b*C*Tan[e + f*x], x], x], x] /; FreeQ[{a, b, e, f, A, B, C, m}, x] && EqQ[A*b^2 - a*b*B + a^2*C, 0]
  3620. Int[Times[Power[Plus[Optional[Pattern[a, Blank[]]], Times[Optional[Pattern[b, Blank[]]], tan[Plus[Optional[Pattern[e, Blank[]]], Times[Optional[Pattern[f, Blank[]]], Pattern[x, Blank[]]]]]]], Optional[Pattern[m, Blank[]]]], Plus[Optional[Pattern[A, Blank[]]], Times[Optional[Pattern[C, Blank[]]], Power[tan[Plus[Optional[Pattern[e, Blank[]]], Times[Optional[Pattern[f, Blank[]]], Pattern[x, Blank[]]]]], 2]]]], Pattern[x, Blank[Symbol]]] := -Dist[C/b^2, Int[(a + b*Tan[e + f*x])^(m + 1)*(a - b*Tan[e + f*x]), x], x] /; FreeQ[{a, b, e, f, A, C, m}, x] && EqQ[A*b^2 + a^2*C, 0]
  3621. Int[Times[Power[Plus[Optional[Pattern[a, Blank[]]], Times[Optional[Pattern[b, Blank[]]], tan[Plus[Optional[Pattern[e, Blank[]]], Times[Optional[Pattern[f, Blank[]]], Pattern[x, Blank[]]]]]]], Optional[Pattern[m, Blank[]]]], Plus[Optional[Pattern[A, Blank[]]], Times[Optional[Pattern[B, Blank[]]], tan[Plus[Optional[Pattern[e, Blank[]]], Times[Optional[Pattern[f, Blank[]]], Pattern[x, Blank[]]]]]], Times[Optional[Pattern[C, Blank[]]], Power[tan[Plus[Optional[Pattern[e, Blank[]]], Times[Optional[Pattern[f, Blank[]]], Pattern[x, Blank[]]]]], 2]]]], Pattern[x, Blank[Symbol]]] := -Simp[((a*A + b*B - a*C)*Tan[e + f*x]*(a + b*Tan[e + f*x])^m)/(2*a*f*m), x] + Dist[1/(2*a^2*m), Int[(a + b*Tan[e + f*x])^(m + 1)*Simp[(b*B - a*C) + a*A*(2*m + 1) - (b*C*(m - 1) + (A*b - a*B)*(m + 1))*Tan[e + f*x], x], x], x] /; FreeQ[{a, b, e, f, A, B, C}, x] && NeQ[A*b^2 - a*b*B + a^2*C, 0] && LeQ[m, -1] && EqQ[a^2 + b^2, 0]
  3622. Int[Times[Power[Plus[Optional[Pattern[a, Blank[]]], Times[Optional[Pattern[b, Blank[]]], tan[Plus[Optional[Pattern[e, Blank[]]], Times[Optional[Pattern[f, Blank[]]], Pattern[x, Blank[]]]]]]], Optional[Pattern[m, Blank[]]]], Plus[Optional[Pattern[A, Blank[]]], Times[Optional[Pattern[C, Blank[]]], Power[tan[Plus[Optional[Pattern[e, Blank[]]], Times[Optional[Pattern[f, Blank[]]], Pattern[x, Blank[]]]]], 2]]]], Pattern[x, Blank[Symbol]]] := -Simp[((a*A - a*C)*Tan[e + f*x]*(a + b*Tan[e + f*x])^m)/(2*a*f*m), x] + Dist[1/(2*a^2*m), Int[(a + b*Tan[e + f*x])^(m + 1)*Simp[-(a*C) + a*A*(2*m + 1) - (b*C*(m - 1) + A*b*(m + 1))*Tan[e + f*x], x], x], x] /; FreeQ[{a, b, e, f, A, C}, x] && NeQ[A*b^2 + a^2*C, 0] && LeQ[m, -1] && EqQ[a^2 + b^2, 0]
  3623. Int[Times[Power[Plus[Optional[Pattern[a, Blank[]]], Times[Optional[Pattern[b, Blank[]]], tan[Plus[Optional[Pattern[e, Blank[]]], Times[Optional[Pattern[f, Blank[]]], Pattern[x, Blank[]]]]]]], -1], Plus[Pattern[A, Blank[]], Times[Optional[Pattern[B, Blank[]]], tan[Plus[Optional[Pattern[e, Blank[]]], Times[Optional[Pattern[f, Blank[]]], Pattern[x, Blank[]]]]]], Times[Optional[Pattern[C, Blank[]]], Power[tan[Plus[Optional[Pattern[e, Blank[]]], Times[Optional[Pattern[f, Blank[]]], Pattern[x, Blank[]]]]], 2]]]], Pattern[x, Blank[Symbol]]] := Simp[((a*A + b*B - a*C)*x)/(a^2 + b^2), x] + Dist[(A*b^2 - a*b*B + a^2*C)/(a^2 + b^2), Int[(1 + Tan[e + f*x]^2)/(a + b*Tan[e + f*x]), x], x] /; FreeQ[{a, b, e, f, A, B, C}, x] && NeQ[a^2 + b^2, 0] && EqQ[A*b - a*B - b*C, 0]
  3624. Int[Times[Power[tan[Plus[Optional[Pattern[e, Blank[]]], Times[Optional[Pattern[f, Blank[]]], Pattern[x, Blank[]]]]], -1], Plus[Pattern[A, Blank[]], Times[Optional[Pattern[B, Blank[]]], tan[Plus[Optional[Pattern[e, Blank[]]], Times[Optional[Pattern[f, Blank[]]], Pattern[x, Blank[]]]]]], Times[Optional[Pattern[C, Blank[]]], Power[tan[Plus[Optional[Pattern[e, Blank[]]], Times[Optional[Pattern[f, Blank[]]], Pattern[x, Blank[]]]]], 2]]]], Pattern[x, Blank[Symbol]]] := Simp[B*x, x] + (Dist[A, Int[1/Tan[e + f*x], x], x] + Dist[C, Int[Tan[e + f*x], x], x]) /; FreeQ[{e, f, A, B, C}, x] && NeQ[A, C]
  3625. Int[Times[Power[tan[Plus[Optional[Pattern[e, Blank[]]], Times[Optional[Pattern[f, Blank[]]], Pattern[x, Blank[]]]]], -1], Plus[Pattern[A, Blank[]], Times[Optional[Pattern[C, Blank[]]], Power[tan[Plus[Optional[Pattern[e, Blank[]]], Times[Optional[Pattern[f, Blank[]]], Pattern[x, Blank[]]]]], 2]]]], Pattern[x, Blank[Symbol]]] := Dist[A, Int[1/Tan[e + f*x], x], x] + Dist[C, Int[Tan[e + f*x], x], x] /; FreeQ[{e, f, A, C}, x] && NeQ[A, C]
  3626. Int[Times[Power[Plus[Optional[Pattern[a, Blank[]]], Times[Optional[Pattern[b, Blank[]]], tan[Plus[Optional[Pattern[e, Blank[]]], Times[Optional[Pattern[f, Blank[]]], Pattern[x, Blank[]]]]]]], -1], Plus[Pattern[A, Blank[]], Times[Optional[Pattern[B, Blank[]]], tan[Plus[Optional[Pattern[e, Blank[]]], Times[Optional[Pattern[f, Blank[]]], Pattern[x, Blank[]]]]]], Times[Optional[Pattern[C, Blank[]]], Power[tan[Plus[Optional[Pattern[e, Blank[]]], Times[Optional[Pattern[f, Blank[]]], Pattern[x, Blank[]]]]], 2]]]], Pattern[x, Blank[Symbol]]] := Simp[((a*A + b*B - a*C)*x)/(a^2 + b^2), x] + (Dist[(A*b^2 - a*b*B + a^2*C)/(a^2 + b^2), Int[(1 + Tan[e + f*x]^2)/(a + b*Tan[e + f*x]), x], x] - Dist[(A*b - a*B - b*C)/(a^2 + b^2), Int[Tan[e + f*x], x], x]) /; FreeQ[{a, b, e, f, A, B, C}, x] && NeQ[A*b^2 - a*b*B + a^2*C, 0] && NeQ[a^2 + b^2, 0] && NeQ[A*b - a*B - b*C, 0]
  3627. Int[Times[Power[Plus[Pattern[a, Blank[]], Times[Optional[Pattern[b, Blank[]]], tan[Plus[Optional[Pattern[e, Blank[]]], Times[Optional[Pattern[f, Blank[]]], Pattern[x, Blank[]]]]]]], -1], Plus[Pattern[A, Blank[]], Times[Optional[Pattern[C, Blank[]]], Power[tan[Plus[Optional[Pattern[e, Blank[]]], Times[Optional[Pattern[f, Blank[]]], Pattern[x, Blank[]]]]], 2]]]], Pattern[x, Blank[Symbol]]] := Simp[(a*(A - C)*x)/(a^2 + b^2), x] + (Dist[(a^2*C + A*b^2)/(a^2 + b^2), Int[(1 + Tan[e + f*x]^2)/(a + b*Tan[e + f*x]), x], x] - Dist[(b*(A - C))/(a^2 + b^2), Int[Tan[e + f*x], x], x]) /; FreeQ[{a, b, e, f, A, C}, x] && NeQ[a^2*C + A*b^2, 0] && NeQ[a^2 + b^2, 0] && NeQ[A, C]
  3628. Int[Times[Power[Plus[Optional[Pattern[a, Blank[]]], Times[Optional[Pattern[b, Blank[]]], tan[Plus[Optional[Pattern[e, Blank[]]], Times[Optional[Pattern[f, Blank[]]], Pattern[x, Blank[]]]]]]], Pattern[m, Blank[]]], Plus[Optional[Pattern[A, Blank[]]], Times[Optional[Pattern[B, Blank[]]], tan[Plus[Optional[Pattern[e, Blank[]]], Times[Optional[Pattern[f, Blank[]]], Pattern[x, Blank[]]]]]], Times[Optional[Pattern[C, Blank[]]], Power[tan[Plus[Optional[Pattern[e, Blank[]]], Times[Optional[Pattern[f, Blank[]]], Pattern[x, Blank[]]]]], 2]]]], Pattern[x, Blank[Symbol]]] := Simp[((A*b^2 - a*b*B + a^2*C)*(a + b*Tan[e + f*x])^(m + 1))/(b*f*(m + 1)*(a^2 + b^2)), x] + Dist[1/(a^2 + b^2), Int[(a + b*Tan[e + f*x])^(m + 1)*Simp[b*B + a*(A - C) - (A*b - a*B - b*C)*Tan[e + f*x], x], x], x] /; FreeQ[{a, b, e, f, A, B, C}, x] && NeQ[A*b^2 - a*b*B + a^2*C, 0] && LtQ[m, -1] && NeQ[a^2 + b^2, 0]
  3629. Int[Times[Power[Plus[Optional[Pattern[a, Blank[]]], Times[Optional[Pattern[b, Blank[]]], tan[Plus[Optional[Pattern[e, Blank[]]], Times[Optional[Pattern[f, Blank[]]], Pattern[x, Blank[]]]]]]], Pattern[m, Blank[]]], Plus[Optional[Pattern[A, Blank[]]], Times[Optional[Pattern[C, Blank[]]], Power[tan[Plus[Optional[Pattern[e, Blank[]]], Times[Optional[Pattern[f, Blank[]]], Pattern[x, Blank[]]]]], 2]]]], Pattern[x, Blank[Symbol]]] := Simp[((A*b^2 + a^2*C)*(a + b*Tan[e + f*x])^(m + 1))/(b*f*(m + 1)*(a^2 + b^2)), x] + Dist[1/(a^2 + b^2), Int[(a + b*Tan[e + f*x])^(m + 1)*Simp[a*(A - C) - (A*b - b*C)*Tan[e + f*x], x], x], x] /; FreeQ[{a, b, e, f, A, C}, x] && NeQ[A*b^2 + a^2*C, 0] && LtQ[m, -1] && NeQ[a^2 + b^2, 0]
  3630. Int[Times[Power[Plus[Optional[Pattern[a, Blank[]]], Times[Optional[Pattern[b, Blank[]]], tan[Plus[Optional[Pattern[e, Blank[]]], Times[Optional[Pattern[f, Blank[]]], Pattern[x, Blank[]]]]]]], Optional[Pattern[m, Blank[]]]], Plus[Optional[Pattern[A, Blank[]]], Times[Optional[Pattern[B, Blank[]]], tan[Plus[Optional[Pattern[e, Blank[]]], Times[Optional[Pattern[f, Blank[]]], Pattern[x, Blank[]]]]]], Times[Optional[Pattern[C, Blank[]]], Power[tan[Plus[Optional[Pattern[e, Blank[]]], Times[Optional[Pattern[f, Blank[]]], Pattern[x, Blank[]]]]], 2]]]], Pattern[x, Blank[Symbol]]] := Simp[(C*(a + b*Tan[e + f*x])^(m + 1))/(b*f*(m + 1)), x] + Int[(a + b*Tan[e + f*x])^m*Simp[A - C + B*Tan[e + f*x], x], x] /; FreeQ[{a, b, e, f, A, B, C, m}, x] && NeQ[A*b^2 - a*b*B + a^2*C, 0] && !LeQ[m, -1]
  3631. Int[Times[Power[Plus[Optional[Pattern[a, Blank[]]], Times[Optional[Pattern[b, Blank[]]], tan[Plus[Optional[Pattern[e, Blank[]]], Times[Optional[Pattern[f, Blank[]]], Pattern[x, Blank[]]]]]]], Optional[Pattern[m, Blank[]]]], Plus[Optional[Pattern[A, Blank[]]], Times[Optional[Pattern[C, Blank[]]], Power[tan[Plus[Optional[Pattern[e, Blank[]]], Times[Optional[Pattern[f, Blank[]]], Pattern[x, Blank[]]]]], 2]]]], Pattern[x, Blank[Symbol]]] := Simp[(C*(a + b*Tan[e + f*x])^(m + 1))/(b*f*(m + 1)), x] + Dist[A - C, Int[(a + b*Tan[e + f*x])^m, x], x] /; FreeQ[{a, b, e, f, A, C, m}, x] && NeQ[A*b^2 + a^2*C, 0] && !LeQ[m, -1]
  3632. Int[Times[Power[Plus[Optional[Pattern[a, Blank[]]], Times[Optional[Pattern[b, Blank[]]], tan[Plus[Optional[Pattern[e, Blank[]]], Times[Optional[Pattern[f, Blank[]]], Pattern[x, Blank[]]]]]]], Optional[Pattern[m, Blank[]]]], Power[Plus[Optional[Pattern[c, Blank[]]], Times[Optional[Pattern[d, Blank[]]], tan[Plus[Optional[Pattern[e, Blank[]]], Times[Optional[Pattern[f, Blank[]]], Pattern[x, Blank[]]]]]]], Optional[Pattern[n, Blank[]]]], Plus[Optional[Pattern[A, Blank[]]], Times[Optional[Pattern[B, Blank[]]], tan[Plus[Optional[Pattern[e, Blank[]]], Times[Optional[Pattern[f, Blank[]]], Pattern[x, Blank[]]]]]], Times[Optional[Pattern[C, Blank[]]], Power[tan[Plus[Optional[Pattern[e, Blank[]]], Times[Optional[Pattern[f, Blank[]]], Pattern[x, Blank[]]]]], 2]]]], Pattern[x, Blank[Symbol]]] := Dist[1/b^2, Int[(a + b*Tan[e + f*x])^(m + 1)*(c + d*Tan[e + f*x])^n*(b*B - a*C + b*C*Tan[e + f*x]), x], x] /; FreeQ[{a, b, c, d, e, f, A, B, C, m, n}, x] && NeQ[b*c - a*d, 0] && EqQ[A*b^2 - a*b*B + a^2*C, 0]
  3633. Int[Times[Power[Plus[Optional[Pattern[a, Blank[]]], Times[Optional[Pattern[b, Blank[]]], tan[Plus[Optional[Pattern[e, Blank[]]], Times[Optional[Pattern[f, Blank[]]], Pattern[x, Blank[]]]]]]], Optional[Pattern[m, Blank[]]]], Power[Plus[Optional[Pattern[c, Blank[]]], Times[Optional[Pattern[d, Blank[]]], tan[Plus[Optional[Pattern[e, Blank[]]], Times[Optional[Pattern[f, Blank[]]], Pattern[x, Blank[]]]]]]], Optional[Pattern[n, Blank[]]]], Plus[Optional[Pattern[A, Blank[]]], Times[Optional[Pattern[C, Blank[]]], Power[tan[Plus[Optional[Pattern[e, Blank[]]], Times[Optional[Pattern[f, Blank[]]], Pattern[x, Blank[]]]]], 2]]]], Pattern[x, Blank[Symbol]]] := -Dist[C/b^2, Int[(a + b*Tan[e + f*x])^(m + 1)*(c + d*Tan[e + f*x])^n*(a - b*Tan[e + f*x]), x], x] /; FreeQ[{a, b, c, d, e, f, A, C, m, n}, x] && NeQ[b*c - a*d, 0] && EqQ[A*b^2 + a^2*C, 0]
  3634. Int[Times[Power[Plus[Optional[Pattern[a, Blank[]]], Times[Optional[Pattern[b, Blank[]]], tan[Plus[Optional[Pattern[e, Blank[]]], Times[Optional[Pattern[f, Blank[]]], Pattern[x, Blank[]]]]]]], Optional[Pattern[m, Blank[]]]], Power[Plus[Optional[Pattern[c, Blank[]]], Times[Optional[Pattern[d, Blank[]]], tan[Plus[Optional[Pattern[e, Blank[]]], Times[Optional[Pattern[f, Blank[]]], Pattern[x, Blank[]]]]]]], Optional[Pattern[n, Blank[]]]], Plus[Pattern[A, Blank[]], Times[Optional[Pattern[C, Blank[]]], Power[tan[Plus[Optional[Pattern[e, Blank[]]], Times[Optional[Pattern[f, Blank[]]], Pattern[x, Blank[]]]]], 2]]]], Pattern[x, Blank[Symbol]]] := Dist[A/f, Subst[Int[(a + b*x)^m*(c + d*x)^n, x], x, Tan[e + f*x]], x] /; FreeQ[{a, b, c, d, e, f, A, C, m, n}, x] && EqQ[A, C]
  3635. Int[Times[Plus[Optional[Pattern[a, Blank[]]], Times[Optional[Pattern[b, Blank[]]], tan[Plus[Optional[Pattern[e, Blank[]]], Times[Optional[Pattern[f, Blank[]]], Pattern[x, Blank[]]]]]]], Power[Plus[Optional[Pattern[c, Blank[]]], Times[Optional[Pattern[d, Blank[]]], tan[Plus[Optional[Pattern[e, Blank[]]], Times[Optional[Pattern[f, Blank[]]], Pattern[x, Blank[]]]]]]], Pattern[n, Blank[]]], Plus[Optional[Pattern[A, Blank[]]], Times[Optional[Pattern[B, Blank[]]], tan[Plus[Optional[Pattern[e, Blank[]]], Times[Optional[Pattern[f, Blank[]]], Pattern[x, Blank[]]]]]], Times[Optional[Pattern[C, Blank[]]], Power[tan[Plus[Optional[Pattern[e, Blank[]]], Times[Optional[Pattern[f, Blank[]]], Pattern[x, Blank[]]]]], 2]]]], Pattern[x, Blank[Symbol]]] := -Simp[((b*c - a*d)*(c^2*C - B*c*d + A*d^2)*(c + d*Tan[e + f*x])^(n + 1))/(d^2*f*(n + 1)*(c^2 + d^2)), x] + Dist[1/(d*(c^2 + d^2)), Int[(c + d*Tan[e + f*x])^(n + 1)*Simp[a*d*(A*c - c*C + B*d) + b*(c^2*C - B*c*d + A*d^2) + d*(A*b*c + a*B*c - b*c*C - a*A*d + b*B*d + a*C*d)*Tan[e + f*x] + b*C*(c^2 + d^2)*Tan[e + f*x]^2, x], x], x] /; FreeQ[{a, b, c, d, e, f, A, B, C}, x] && NeQ[b*c - a*d, 0] && NeQ[c^2 + d^2, 0] && LtQ[n, -1]
  3636. Int[Times[Plus[Optional[Pattern[a, Blank[]]], Times[Optional[Pattern[b, Blank[]]], tan[Plus[Optional[Pattern[e, Blank[]]], Times[Optional[Pattern[f, Blank[]]], Pattern[x, Blank[]]]]]]], Power[Plus[Optional[Pattern[c, Blank[]]], Times[Optional[Pattern[d, Blank[]]], tan[Plus[Optional[Pattern[e, Blank[]]], Times[Optional[Pattern[f, Blank[]]], Pattern[x, Blank[]]]]]]], Pattern[n, Blank[]]], Plus[Optional[Pattern[A, Blank[]]], Times[Optional[Pattern[C, Blank[]]], Power[tan[Plus[Optional[Pattern[e, Blank[]]], Times[Optional[Pattern[f, Blank[]]], Pattern[x, Blank[]]]]], 2]]]], Pattern[x, Blank[Symbol]]] := -Simp[((b*c - a*d)*(c^2*C + A*d^2)*(c + d*Tan[e + f*x])^(n + 1))/(d^2*f*(n + 1)*(c^2 + d^2)), x] + Dist[1/(d*(c^2 + d^2)), Int[(c + d*Tan[e + f*x])^(n + 1)*Simp[a*d*(A*c - c*C) + b*(c^2*C + A*d^2) + d*(A*b*c - b*c*C - a*A*d + a*C*d)*Tan[e + f*x] + b*C*(c^2 + d^2)*Tan[e + f*x]^2, x], x], x] /; FreeQ[{a, b, c, d, e, f, A, C}, x] && NeQ[b*c - a*d, 0] && NeQ[c^2 + d^2, 0] && LtQ[n, -1]
  3637. Int[Times[Plus[Pattern[a, Blank[]], Times[Optional[Pattern[b, Blank[]]], tan[Plus[Optional[Pattern[e, Blank[]]], Times[Optional[Pattern[f, Blank[]]], Pattern[x, Blank[]]]]]]], Power[Plus[Optional[Pattern[c, Blank[]]], Times[Optional[Pattern[d, Blank[]]], tan[Plus[Optional[Pattern[e, Blank[]]], Times[Optional[Pattern[f, Blank[]]], Pattern[x, Blank[]]]]]]], Optional[Pattern[n, Blank[]]]], Plus[Optional[Pattern[A, Blank[]]], Times[Optional[Pattern[B, Blank[]]], tan[Plus[Optional[Pattern[e, Blank[]]], Times[Optional[Pattern[f, Blank[]]], Pattern[x, Blank[]]]]]], Times[Optional[Pattern[C, Blank[]]], Power[tan[Plus[Optional[Pattern[e, Blank[]]], Times[Optional[Pattern[f, Blank[]]], Pattern[x, Blank[]]]]], 2]]]], Pattern[x, Blank[Symbol]]] := Simp[(b*C*Tan[e + f*x]*(c + d*Tan[e + f*x])^(n + 1))/(d*f*(n + 2)), x] - Dist[1/(d*(n + 2)), Int[(c + d*Tan[e + f*x])^n*Simp[b*c*C - a*A*d*(n + 2) - (A*b + a*B - b*C)*d*(n + 2)*Tan[e + f*x] - (a*C*d*(n + 2) - b*(c*C - B*d*(n + 2)))*Tan[e + f*x]^2, x], x], x] /; FreeQ[{a, b, c, d, e, f, A, B, C, n}, x] && NeQ[b*c - a*d, 0] && NeQ[c^2 + d^2, 0] && !LtQ[n, -1]
  3638. Int[Times[Plus[Pattern[a, Blank[]], Times[Optional[Pattern[b, Blank[]]], tan[Plus[Optional[Pattern[e, Blank[]]], Times[Optional[Pattern[f, Blank[]]], Pattern[x, Blank[]]]]]]], Power[Plus[Optional[Pattern[c, Blank[]]], Times[Optional[Pattern[d, Blank[]]], tan[Plus[Optional[Pattern[e, Blank[]]], Times[Optional[Pattern[f, Blank[]]], Pattern[x, Blank[]]]]]]], Optional[Pattern[n, Blank[]]]], Plus[Optional[Pattern[A, Blank[]]], Times[Optional[Pattern[C, Blank[]]], Power[tan[Plus[Optional[Pattern[e, Blank[]]], Times[Optional[Pattern[f, Blank[]]], Pattern[x, Blank[]]]]], 2]]]], Pattern[x, Blank[Symbol]]] := Simp[(b*C*Tan[e + f*x]*(c + d*Tan[e + f*x])^(n + 1))/(d*f*(n + 2)), x] - Dist[1/(d*(n + 2)), Int[(c + d*Tan[e + f*x])^n*Simp[b*c*C - a*A*d*(n + 2) - (A*b - b*C)*d*(n + 2)*Tan[e + f*x] - (a*C*d*(n + 2) - b*c*C)*Tan[e + f*x]^2, x], x], x] /; FreeQ[{a, b, c, d, e, f, A, C, n}, x] && NeQ[b*c - a*d, 0] && NeQ[c^2 + d^2, 0] && !LtQ[n, -1]
  3639. Int[Times[Power[Plus[Pattern[a, Blank[]], Times[Optional[Pattern[b, Blank[]]], tan[Plus[Optional[Pattern[e, Blank[]]], Times[Optional[Pattern[f, Blank[]]], Pattern[x, Blank[]]]]]]], Pattern[m, Blank[]]], Power[Plus[Optional[Pattern[c, Blank[]]], Times[Optional[Pattern[d, Blank[]]], tan[Plus[Optional[Pattern[e, Blank[]]], Times[Optional[Pattern[f, Blank[]]], Pattern[x, Blank[]]]]]]], Optional[Pattern[n, Blank[]]]], Plus[Optional[Pattern[A, Blank[]]], Times[Optional[Pattern[B, Blank[]]], tan[Plus[Optional[Pattern[e, Blank[]]], Times[Optional[Pattern[f, Blank[]]], Pattern[x, Blank[]]]]]], Times[Optional[Pattern[C, Blank[]]], Power[tan[Plus[Optional[Pattern[e, Blank[]]], Times[Optional[Pattern[f, Blank[]]], Pattern[x, Blank[]]]]], 2]]]], Pattern[x, Blank[Symbol]]] := Simp[((a*A + b*B - a*C)*(a + b*Tan[e + f*x])^m*(c + d*Tan[e + f*x])^(n + 1))/(2*f*m*(b*c - a*d)), x] + Dist[1/(2*a*m*(b*c - a*d)), Int[(a + b*Tan[e + f*x])^(m + 1)*(c + d*Tan[e + f*x])^n*Simp[b*(c*(A + C)*m - B*d*(n + 1)) + a*(B*c*m + C*d*(n + 1) - A*d*(2*m + n + 1)) + (b*C*d*(m - n - 1) + A*b*d*(m + n + 1) + a*(2*c*C*m - B*d*(m + n + 1)))*Tan[e + f*x], x], x], x] /; FreeQ[{a, b, c, d, e, f, A, B, C, n}, x] && NeQ[b*c - a*d, 0] && EqQ[a^2 + b^2, 0] && (LtQ[m, 0] || EqQ[m + n + 1, 0])
  3640. Int[Times[Power[Plus[Pattern[a, Blank[]], Times[Optional[Pattern[b, Blank[]]], tan[Plus[Optional[Pattern[e, Blank[]]], Times[Optional[Pattern[f, Blank[]]], Pattern[x, Blank[]]]]]]], Pattern[m, Blank[]]], Power[Plus[Optional[Pattern[c, Blank[]]], Times[Optional[Pattern[d, Blank[]]], tan[Plus[Optional[Pattern[e, Blank[]]], Times[Optional[Pattern[f, Blank[]]], Pattern[x, Blank[]]]]]]], Optional[Pattern[n, Blank[]]]], Plus[Optional[Pattern[A, Blank[]]], Times[Optional[Pattern[C, Blank[]]], Power[tan[Plus[Optional[Pattern[e, Blank[]]], Times[Optional[Pattern[f, Blank[]]], Pattern[x, Blank[]]]]], 2]]]], Pattern[x, Blank[Symbol]]] := Simp[(a*(A - C)*(a + b*Tan[e + f*x])^m*(c + d*Tan[e + f*x])^(n + 1))/(2*f*m*(b*c - a*d)), x] + Dist[1/(2*a*m*(b*c - a*d)), Int[(a + b*Tan[e + f*x])^(m + 1)*(c + d*Tan[e + f*x])^n*Simp[b*c*(A + C)*m + a*(C*d*(n + 1) - A*d*(2*m + n + 1)) + (b*C*d*(m - n - 1) + A*b*d*(m + n + 1) + 2*a*c*C*m)*Tan[e + f*x], x], x], x] /; FreeQ[{a, b, c, d, e, f, A, C, n}, x] && NeQ[b*c - a*d, 0] && EqQ[a^2 + b^2, 0] && (LtQ[m, 0] || EqQ[m + n + 1, 0])
  3641. Int[Times[Power[Plus[Pattern[a, Blank[]], Times[Optional[Pattern[b, Blank[]]], tan[Plus[Optional[Pattern[e, Blank[]]], Times[Optional[Pattern[f, Blank[]]], Pattern[x, Blank[]]]]]]], Optional[Pattern[m, Blank[]]]], Power[Plus[Optional[Pattern[c, Blank[]]], Times[Optional[Pattern[d, Blank[]]], tan[Plus[Optional[Pattern[e, Blank[]]], Times[Optional[Pattern[f, Blank[]]], Pattern[x, Blank[]]]]]]], Pattern[n, Blank[]]], Plus[Optional[Pattern[A, Blank[]]], Times[Optional[Pattern[B, Blank[]]], tan[Plus[Optional[Pattern[e, Blank[]]], Times[Optional[Pattern[f, Blank[]]], Pattern[x, Blank[]]]]]], Times[Optional[Pattern[C, Blank[]]], Power[tan[Plus[Optional[Pattern[e, Blank[]]], Times[Optional[Pattern[f, Blank[]]], Pattern[x, Blank[]]]]], 2]]]], Pattern[x, Blank[Symbol]]] := Simp[((c^2*C - B*c*d + A*d^2)*(a + b*Tan[e + f*x])^m*(c + d*Tan[e + f*x])^(n + 1))/(d*f*(n + 1)*(c^2 + d^2)), x] - Dist[1/(a*d*(n + 1)*(c^2 + d^2)), Int[(a + b*Tan[e + f*x])^m*(c + d*Tan[e + f*x])^(n + 1)*Simp[b*(c^2*C - B*c*d + A*d^2)*m - a*d*(n + 1)*(A*c - c*C + B*d) - a*(d*(B*c - A*d)*(m + n + 1) - C*(c^2*m - d^2*(n + 1)))*Tan[e + f*x], x], x], x] /; FreeQ[{a, b, c, d, e, f, A, B, C, m}, x] && NeQ[b*c - a*d, 0] && EqQ[a^2 + b^2, 0] && !LtQ[m, 0] && LtQ[n, -1] && NeQ[c^2 + d^2, 0]
  3642. Int[Times[Power[Plus[Pattern[a, Blank[]], Times[Optional[Pattern[b, Blank[]]], tan[Plus[Optional[Pattern[e, Blank[]]], Times[Optional[Pattern[f, Blank[]]], Pattern[x, Blank[]]]]]]], Optional[Pattern[m, Blank[]]]], Power[Plus[Optional[Pattern[c, Blank[]]], Times[Optional[Pattern[d, Blank[]]], tan[Plus[Optional[Pattern[e, Blank[]]], Times[Optional[Pattern[f, Blank[]]], Pattern[x, Blank[]]]]]]], Pattern[n, Blank[]]], Plus[Optional[Pattern[A, Blank[]]], Times[Optional[Pattern[C, Blank[]]], Power[tan[Plus[Optional[Pattern[e, Blank[]]], Times[Optional[Pattern[f, Blank[]]], Pattern[x, Blank[]]]]], 2]]]], Pattern[x, Blank[Symbol]]] := Simp[((c^2*C + A*d^2)*(a + b*Tan[e + f*x])^m*(c + d*Tan[e + f*x])^(n + 1))/(d*f*(n + 1)*(c^2 + d^2)), x] - Dist[1/(a*d*(n + 1)*(c^2 + d^2)), Int[(a + b*Tan[e + f*x])^m*(c + d*Tan[e + f*x])^(n + 1)*Simp[b*(c^2*C + A*d^2)*m - a*d*(n + 1)*(A*c - c*C) - a*(-(A*d^2*(m + n + 1)) - C*(c^2*m - d^2*(n + 1)))*Tan[e + f*x], x], x], x] /; FreeQ[{a, b, c, d, e, f, A, C, m}, x] && NeQ[b*c - a*d, 0] && EqQ[a^2 + b^2, 0] && !LtQ[m, 0] && LtQ[n, -1] && NeQ[c^2 + d^2, 0]
  3643. Int[Times[Power[Plus[Pattern[a, Blank[]], Times[Optional[Pattern[b, Blank[]]], tan[Plus[Optional[Pattern[e, Blank[]]], Times[Optional[Pattern[f, Blank[]]], Pattern[x, Blank[]]]]]]], Optional[Pattern[m, Blank[]]]], Power[Plus[Optional[Pattern[c, Blank[]]], Times[Optional[Pattern[d, Blank[]]], tan[Plus[Optional[Pattern[e, Blank[]]], Times[Optional[Pattern[f, Blank[]]], Pattern[x, Blank[]]]]]]], Optional[Pattern[n, Blank[]]]], Plus[Optional[Pattern[A, Blank[]]], Times[Optional[Pattern[B, Blank[]]], tan[Plus[Optional[Pattern[e, Blank[]]], Times[Optional[Pattern[f, Blank[]]], Pattern[x, Blank[]]]]]], Times[Optional[Pattern[C, Blank[]]], Power[tan[Plus[Optional[Pattern[e, Blank[]]], Times[Optional[Pattern[f, Blank[]]], Pattern[x, Blank[]]]]], 2]]]], Pattern[x, Blank[Symbol]]] := Simp[(C*(a + b*Tan[e + f*x])^m*(c + d*Tan[e + f*x])^(n + 1))/(d*f*(m + n + 1)), x] + Dist[1/(b*d*(m + n + 1)), Int[(a + b*Tan[e + f*x])^m*(c + d*Tan[e + f*x])^n*Simp[A*b*d*(m + n + 1) + C*(a*c*m - b*d*(n + 1)) - (C*m*(b*c - a*d) - b*B*d*(m + n + 1))*Tan[e + f*x], x], x], x] /; FreeQ[{a, b, c, d, e, f, A, B, C, m, n}, x] && NeQ[b*c - a*d, 0] && EqQ[a^2 + b^2, 0] && !LtQ[m, 0] && NeQ[m + n + 1, 0]
  3644. Int[Times[Power[Plus[Pattern[a, Blank[]], Times[Optional[Pattern[b, Blank[]]], tan[Plus[Optional[Pattern[e, Blank[]]], Times[Optional[Pattern[f, Blank[]]], Pattern[x, Blank[]]]]]]], Optional[Pattern[m, Blank[]]]], Power[Plus[Optional[Pattern[c, Blank[]]], Times[Optional[Pattern[d, Blank[]]], tan[Plus[Optional[Pattern[e, Blank[]]], Times[Optional[Pattern[f, Blank[]]], Pattern[x, Blank[]]]]]]], Optional[Pattern[n, Blank[]]]], Plus[Optional[Pattern[A, Blank[]]], Times[Optional[Pattern[C, Blank[]]], Power[tan[Plus[Optional[Pattern[e, Blank[]]], Times[Optional[Pattern[f, Blank[]]], Pattern[x, Blank[]]]]], 2]]]], Pattern[x, Blank[Symbol]]] := Simp[(C*(a + b*Tan[e + f*x])^m*(c + d*Tan[e + f*x])^(n + 1))/(d*f*(m + n + 1)), x] + Dist[1/(b*d*(m + n + 1)), Int[(a + b*Tan[e + f*x])^m*(c + d*Tan[e + f*x])^n*Simp[A*b*d*(m + n + 1) + C*(a*c*m - b*d*(n + 1)) - C*m*(b*c - a*d)*Tan[e + f*x], x], x], x] /; FreeQ[{a, b, c, d, e, f, A, C, m, n}, x] && NeQ[b*c - a*d, 0] && EqQ[a^2 + b^2, 0] && !LtQ[m, 0] && NeQ[m + n + 1, 0]
  3645. Int[Times[Power[Plus[Optional[Pattern[a, Blank[]]], Times[Optional[Pattern[b, Blank[]]], tan[Plus[Optional[Pattern[e, Blank[]]], Times[Optional[Pattern[f, Blank[]]], Pattern[x, Blank[]]]]]]], Pattern[m, Blank[]]], Power[Plus[Optional[Pattern[c, Blank[]]], Times[Optional[Pattern[d, Blank[]]], tan[Plus[Optional[Pattern[e, Blank[]]], Times[Optional[Pattern[f, Blank[]]], Pattern[x, Blank[]]]]]]], Pattern[n, Blank[]]], Plus[Optional[Pattern[A, Blank[]]], Times[Optional[Pattern[B, Blank[]]], tan[Plus[Optional[Pattern[e, Blank[]]], Times[Optional[Pattern[f, Blank[]]], Pattern[x, Blank[]]]]]], Times[Optional[Pattern[C, Blank[]]], Power[tan[Plus[Optional[Pattern[e, Blank[]]], Times[Optional[Pattern[f, Blank[]]], Pattern[x, Blank[]]]]], 2]]]], Pattern[x, Blank[Symbol]]] := Simp[((A*d^2 + c*(c*C - B*d))*(a + b*Tan[e + f*x])^m*(c + d*Tan[e + f*x])^(n + 1))/(d*f*(n + 1)*(c^2 + d^2)), x] - Dist[1/(d*(n + 1)*(c^2 + d^2)), Int[(a + b*Tan[e + f*x])^(m - 1)*(c + d*Tan[e + f*x])^(n + 1)*Simp[A*d*(b*d*m - a*c*(n + 1)) + (c*C - B*d)*(b*c*m + a*d*(n + 1)) - d*(n + 1)*((A - C)*(b*c - a*d) + B*(a*c + b*d))*Tan[e + f*x] - b*(d*(B*c - A*d)*(m + n + 1) - C*(c^2*m - d^2*(n + 1)))*Tan[e + f*x]^2, x], x], x] /; FreeQ[{a, b, c, d, e, f, A, B, C}, x] && NeQ[b*c - a*d, 0] && NeQ[a^2 + b^2, 0] && NeQ[c^2 + d^2, 0] && GtQ[m, 0] && LtQ[n, -1]
  3646. Int[Times[Power[Plus[Optional[Pattern[a, Blank[]]], Times[Optional[Pattern[b, Blank[]]], tan[Plus[Optional[Pattern[e, Blank[]]], Times[Optional[Pattern[f, Blank[]]], Pattern[x, Blank[]]]]]]], Pattern[m, Blank[]]], Power[Plus[Optional[Pattern[c, Blank[]]], Times[Optional[Pattern[d, Blank[]]], tan[Plus[Optional[Pattern[e, Blank[]]], Times[Optional[Pattern[f, Blank[]]], Pattern[x, Blank[]]]]]]], Pattern[n, Blank[]]], Plus[Optional[Pattern[A, Blank[]]], Times[Optional[Pattern[C, Blank[]]], Power[tan[Plus[Optional[Pattern[e, Blank[]]], Times[Optional[Pattern[f, Blank[]]], Pattern[x, Blank[]]]]], 2]]]], Pattern[x, Blank[Symbol]]] := Simp[((A*d^2 + c^2*C)*(a + b*Tan[e + f*x])^m*(c + d*Tan[e + f*x])^(n + 1))/(d*f*(n + 1)*(c^2 + d^2)), x] - Dist[1/(d*(n + 1)*(c^2 + d^2)), Int[(a + b*Tan[e + f*x])^(m - 1)*(c + d*Tan[e + f*x])^(n + 1)*Simp[A*d*(b*d*m - a*c*(n + 1)) + c*C*(b*c*m + a*d*(n + 1)) - d*(n + 1)*((A - C)*(b*c - a*d))*Tan[e + f*x] + b*(A*d^2*(m + n + 1) + C*(c^2*m - d^2*(n + 1)))*Tan[e + f*x]^2, x], x], x] /; FreeQ[{a, b, c, d, e, f, A, C}, x] && NeQ[b*c - a*d, 0] && NeQ[a^2 + b^2, 0] && NeQ[c^2 + d^2, 0] && GtQ[m, 0] && LtQ[n, -1]
  3647. Int[Times[Power[Plus[Optional[Pattern[a, Blank[]]], Times[Optional[Pattern[b, Blank[]]], tan[Plus[Optional[Pattern[e, Blank[]]], Times[Optional[Pattern[f, Blank[]]], Pattern[x, Blank[]]]]]]], Optional[Pattern[m, Blank[]]]], Power[Plus[Optional[Pattern[c, Blank[]]], Times[Optional[Pattern[d, Blank[]]], tan[Plus[Optional[Pattern[e, Blank[]]], Times[Optional[Pattern[f, Blank[]]], Pattern[x, Blank[]]]]]]], Pattern[n, Blank[]]], Plus[Optional[Pattern[A, Blank[]]], Times[Optional[Pattern[B, Blank[]]], tan[Plus[Optional[Pattern[e, Blank[]]], Times[Optional[Pattern[f, Blank[]]], Pattern[x, Blank[]]]]]], Times[Optional[Pattern[C, Blank[]]], Power[tan[Plus[Optional[Pattern[e, Blank[]]], Times[Optional[Pattern[f, Blank[]]], Pattern[x, Blank[]]]]], 2]]]], Pattern[x, Blank[Symbol]]] := Simp[(C*(a + b*Tan[e + f*x])^m*(c + d*Tan[e + f*x])^(n + 1))/(d*f*(m + n + 1)), x] + Dist[1/(d*(m + n + 1)), Int[(a + b*Tan[e + f*x])^(m - 1)*(c + d*Tan[e + f*x])^n*Simp[a*A*d*(m + n + 1) - C*(b*c*m + a*d*(n + 1)) + d*(A*b + a*B - b*C)*(m + n + 1)*Tan[e + f*x] - (C*m*(b*c - a*d) - b*B*d*(m + n + 1))*Tan[e + f*x]^2, x], x], x] /; FreeQ[{a, b, c, d, e, f, A, B, C, n}, x] && NeQ[b*c - a*d, 0] && NeQ[a^2 + b^2, 0] && NeQ[c^2 + d^2, 0] && GtQ[m, 0] && !(IGtQ[n, 0] && ( !IntegerQ[m] || (EqQ[c, 0] && NeQ[a, 0])))
  3648. Int[Times[Power[Plus[Optional[Pattern[a, Blank[]]], Times[Optional[Pattern[b, Blank[]]], tan[Plus[Optional[Pattern[e, Blank[]]], Times[Optional[Pattern[f, Blank[]]], Pattern[x, Blank[]]]]]]], Optional[Pattern[m, Blank[]]]], Power[Plus[Optional[Pattern[c, Blank[]]], Times[Optional[Pattern[d, Blank[]]], tan[Plus[Optional[Pattern[e, Blank[]]], Times[Optional[Pattern[f, Blank[]]], Pattern[x, Blank[]]]]]]], Pattern[n, Blank[]]], Plus[Optional[Pattern[A, Blank[]]], Times[Optional[Pattern[C, Blank[]]], Power[tan[Plus[Optional[Pattern[e, Blank[]]], Times[Optional[Pattern[f, Blank[]]], Pattern[x, Blank[]]]]], 2]]]], Pattern[x, Blank[Symbol]]] := Simp[(C*(a + b*Tan[e + f*x])^m*(c + d*Tan[e + f*x])^(n + 1))/(d*f*(m + n + 1)), x] + Dist[1/(d*(m + n + 1)), Int[(a + b*Tan[e + f*x])^(m - 1)*(c + d*Tan[e + f*x])^n*Simp[a*A*d*(m + n + 1) - C*(b*c*m + a*d*(n + 1)) + d*(A*b - b*C)*(m + n + 1)*Tan[e + f*x] - C*m*(b*c - a*d)*Tan[e + f*x]^2, x], x], x] /; FreeQ[{a, b, c, d, e, f, A, C, n}, x] && NeQ[b*c - a*d, 0] && NeQ[a^2 + b^2, 0] && NeQ[c^2 + d^2, 0] && GtQ[m, 0] && !(IGtQ[n, 0] && ( !IntegerQ[m] || (EqQ[c, 0] && NeQ[a, 0])))
  3649. Int[Times[Power[Plus[Optional[Pattern[a, Blank[]]], Times[Optional[Pattern[b, Blank[]]], tan[Plus[Optional[Pattern[e, Blank[]]], Times[Optional[Pattern[f, Blank[]]], Pattern[x, Blank[]]]]]]], Pattern[m, Blank[]]], Power[Plus[Optional[Pattern[c, Blank[]]], Times[Optional[Pattern[d, Blank[]]], tan[Plus[Optional[Pattern[e, Blank[]]], Times[Optional[Pattern[f, Blank[]]], Pattern[x, Blank[]]]]]]], Pattern[n, Blank[]]], Plus[Optional[Pattern[A, Blank[]]], Times[Optional[Pattern[B, Blank[]]], tan[Plus[Optional[Pattern[e, Blank[]]], Times[Optional[Pattern[f, Blank[]]], Pattern[x, Blank[]]]]]], Times[Optional[Pattern[C, Blank[]]], Power[tan[Plus[Optional[Pattern[e, Blank[]]], Times[Optional[Pattern[f, Blank[]]], Pattern[x, Blank[]]]]], 2]]]], Pattern[x, Blank[Symbol]]] := Simp[((A*b^2 - a*(b*B - a*C))*(a + b*Tan[e + f*x])^(m + 1)*(c + d*Tan[e + f*x])^(n + 1))/(f*(m + 1)*(b*c - a*d)*(a^2 + b^2)), x] + Dist[1/((m + 1)*(b*c - a*d)*(a^2 + b^2)), Int[(a + b*Tan[e + f*x])^(m + 1)*(c + d*Tan[e + f*x])^n*Simp[A*(a*(b*c - a*d)*(m + 1) - b^2*d*(m + n + 2)) + (b*B - a*C)*(b*c*(m + 1) + a*d*(n + 1)) - (m + 1)*(b*c - a*d)*(A*b - a*B - b*C)*Tan[e + f*x] - d*(A*b^2 - a*(b*B - a*C))*(m + n + 2)*Tan[e + f*x]^2, x], x], x] /; FreeQ[{a, b, c, d, e, f, A, B, C, n}, x] && NeQ[b*c - a*d, 0] && NeQ[a^2 + b^2, 0] && NeQ[c^2 + d^2, 0] && LtQ[m, -1] && !(ILtQ[n, -1] && ( !IntegerQ[m] || (EqQ[c, 0] && NeQ[a, 0])))
  3650. Int[Times[Power[Plus[Optional[Pattern[a, Blank[]]], Times[Optional[Pattern[b, Blank[]]], tan[Plus[Optional[Pattern[e, Blank[]]], Times[Optional[Pattern[f, Blank[]]], Pattern[x, Blank[]]]]]]], Pattern[m, Blank[]]], Power[Plus[Optional[Pattern[c, Blank[]]], Times[Optional[Pattern[d, Blank[]]], tan[Plus[Optional[Pattern[e, Blank[]]], Times[Optional[Pattern[f, Blank[]]], Pattern[x, Blank[]]]]]]], Pattern[n, Blank[]]], Plus[Optional[Pattern[A, Blank[]]], Times[Optional[Pattern[C, Blank[]]], Power[tan[Plus[Optional[Pattern[e, Blank[]]], Times[Optional[Pattern[f, Blank[]]], Pattern[x, Blank[]]]]], 2]]]], Pattern[x, Blank[Symbol]]] := Simp[((A*b^2 + a^2*C)*(a + b*Tan[e + f*x])^(m + 1)*(c + d*Tan[e + f*x])^(n + 1))/(f*(m + 1)*(b*c - a*d)*(a^2 + b^2)), x] + Dist[1/((m + 1)*(b*c - a*d)*(a^2 + b^2)), Int[(a + b*Tan[e + f*x])^(m + 1)*(c + d*Tan[e + f*x])^n*Simp[A*(a*(b*c - a*d)*(m + 1) - b^2*d*(m + n + 2)) - a*C*(b*c*(m + 1) + a*d*(n + 1)) - (m + 1)*(b*c - a*d)*(A*b - b*C)*Tan[e + f*x] - d*(A*b^2 + a^2*C)*(m + n + 2)*Tan[e + f*x]^2, x], x], x] /; FreeQ[{a, b, c, d, e, f, A, C, n}, x] && NeQ[b*c - a*d, 0] && NeQ[a^2 + b^2, 0] && NeQ[c^2 + d^2, 0] && LtQ[m, -1] && !(ILtQ[n, -1] && ( !IntegerQ[m] || (EqQ[c, 0] && NeQ[a, 0])))
  3651. Int[Times[Power[Plus[Pattern[a, Blank[]], Times[Optional[Pattern[b, Blank[]]], tan[Plus[Optional[Pattern[e, Blank[]]], Times[Optional[Pattern[f, Blank[]]], Pattern[x, Blank[]]]]]]], -1], Power[Plus[Optional[Pattern[c, Blank[]]], Times[Optional[Pattern[d, Blank[]]], tan[Plus[Optional[Pattern[e, Blank[]]], Times[Optional[Pattern[f, Blank[]]], Pattern[x, Blank[]]]]]]], -1], Plus[Optional[Pattern[A, Blank[]]], Times[Optional[Pattern[B, Blank[]]], tan[Plus[Optional[Pattern[e, Blank[]]], Times[Optional[Pattern[f, Blank[]]], Pattern[x, Blank[]]]]]], Times[Optional[Pattern[C, Blank[]]], Power[tan[Plus[Optional[Pattern[e, Blank[]]], Times[Optional[Pattern[f, Blank[]]], Pattern[x, Blank[]]]]], 2]]]], Pattern[x, Blank[Symbol]]] := Simp[((a*(A*c - c*C + B*d) + b*(B*c - A*d + C*d))*x)/((a^2 + b^2)*(c^2 + d^2)), x] + (Dist[(A*b^2 - a*b*B + a^2*C)/((b*c - a*d)*(a^2 + b^2)), Int[(b - a*Tan[e + f*x])/(a + b*Tan[e + f*x]), x], x] - Dist[(c^2*C - B*c*d + A*d^2)/((b*c - a*d)*(c^2 + d^2)), Int[(d - c*Tan[e + f*x])/(c + d*Tan[e + f*x]), x], x]) /; FreeQ[{a, b, c, d, e, f, A, B, C}, x] && NeQ[b*c - a*d, 0] && NeQ[a^2 + b^2, 0] && NeQ[c^2 + d^2, 0]
  3652. Int[Times[Power[Plus[Pattern[a, Blank[]], Times[Optional[Pattern[b, Blank[]]], tan[Plus[Optional[Pattern[e, Blank[]]], Times[Optional[Pattern[f, Blank[]]], Pattern[x, Blank[]]]]]]], -1], Power[Plus[Optional[Pattern[c, Blank[]]], Times[Optional[Pattern[d, Blank[]]], tan[Plus[Optional[Pattern[e, Blank[]]], Times[Optional[Pattern[f, Blank[]]], Pattern[x, Blank[]]]]]]], -1], Plus[Optional[Pattern[A, Blank[]]], Times[Optional[Pattern[C, Blank[]]], Power[tan[Plus[Optional[Pattern[e, Blank[]]], Times[Optional[Pattern[f, Blank[]]], Pattern[x, Blank[]]]]], 2]]]], Pattern[x, Blank[Symbol]]] := Simp[((a*(A*c - c*C) - b*(A*d - C*d))*x)/((a^2 + b^2)*(c^2 + d^2)), x] + (Dist[(A*b^2 + a^2*C)/((b*c - a*d)*(a^2 + b^2)), Int[(b - a*Tan[e + f*x])/(a + b*Tan[e + f*x]), x], x] - Dist[(c^2*C + A*d^2)/((b*c - a*d)*(c^2 + d^2)), Int[(d - c*Tan[e + f*x])/(c + d*Tan[e + f*x]), x], x]) /; FreeQ[{a, b, c, d, e, f, A, C}, x] && NeQ[b*c - a*d, 0] && NeQ[a^2 + b^2, 0] && NeQ[c^2 + d^2, 0]
  3653. Int[Times[Power[Plus[Optional[Pattern[a, Blank[]]], Times[Optional[Pattern[b, Blank[]]], tan[Plus[Optional[Pattern[e, Blank[]]], Times[Optional[Pattern[f, Blank[]]], Pattern[x, Blank[]]]]]]], -1], Power[Plus[Optional[Pattern[c, Blank[]]], Times[Optional[Pattern[d, Blank[]]], tan[Plus[Optional[Pattern[e, Blank[]]], Times[Optional[Pattern[f, Blank[]]], Pattern[x, Blank[]]]]]]], Pattern[n, Blank[]]], Plus[Optional[Pattern[A, Blank[]]], Times[Optional[Pattern[B, Blank[]]], tan[Plus[Optional[Pattern[e, Blank[]]], Times[Optional[Pattern[f, Blank[]]], Pattern[x, Blank[]]]]]], Times[Optional[Pattern[C, Blank[]]], Power[tan[Plus[Optional[Pattern[e, Blank[]]], Times[Optional[Pattern[f, Blank[]]], Pattern[x, Blank[]]]]], 2]]]], Pattern[x, Blank[Symbol]]] := Dist[1/(a^2 + b^2), Int[(c + d*Tan[e + f*x])^n*Simp[b*B + a*(A - C) + (a*B - b*(A - C))*Tan[e + f*x], x], x], x] + Dist[(A*b^2 - a*b*B + a^2*C)/(a^2 + b^2), Int[((c + d*Tan[e + f*x])^n*(1 + Tan[e + f*x]^2))/(a + b*Tan[e + f*x]), x], x] /; FreeQ[{a, b, c, d, e, f, A, B, C, n}, x] && NeQ[b*c - a*d, 0] && NeQ[a^2 + b^2, 0] && NeQ[c^2 + d^2, 0] && !GtQ[n, 0] && !LeQ[n, -1]
  3654. Int[Times[Power[Plus[Optional[Pattern[a, Blank[]]], Times[Optional[Pattern[b, Blank[]]], tan[Plus[Optional[Pattern[e, Blank[]]], Times[Optional[Pattern[f, Blank[]]], Pattern[x, Blank[]]]]]]], -1], Power[Plus[Optional[Pattern[c, Blank[]]], Times[Optional[Pattern[d, Blank[]]], tan[Plus[Optional[Pattern[e, Blank[]]], Times[Optional[Pattern[f, Blank[]]], Pattern[x, Blank[]]]]]]], Pattern[n, Blank[]]], Plus[Optional[Pattern[A, Blank[]]], Times[Optional[Pattern[C, Blank[]]], Power[tan[Plus[Optional[Pattern[e, Blank[]]], Times[Optional[Pattern[f, Blank[]]], Pattern[x, Blank[]]]]], 2]]]], Pattern[x, Blank[Symbol]]] := Dist[1/(a^2 + b^2), Int[(c + d*Tan[e + f*x])^n*Simp[a*(A - C) - (A*b - b*C)*Tan[e + f*x], x], x], x] + Dist[(A*b^2 + a^2*C)/(a^2 + b^2), Int[((c + d*Tan[e + f*x])^n*(1 + Tan[e + f*x]^2))/(a + b*Tan[e + f*x]), x], x] /; FreeQ[{a, b, c, d, e, f, A, C, n}, x] && NeQ[b*c - a*d, 0] && NeQ[a^2 + b^2, 0] && NeQ[c^2 + d^2, 0] && !GtQ[n, 0] && !LeQ[n, -1]
  3655. Int[Times[Power[Plus[Optional[Pattern[a, Blank[]]], Times[Optional[Pattern[b, Blank[]]], tan[Plus[Optional[Pattern[e, Blank[]]], Times[Optional[Pattern[f, Blank[]]], Pattern[x, Blank[]]]]]]], Pattern[m, Blank[]]], Power[Plus[Optional[Pattern[c, Blank[]]], Times[Optional[Pattern[d, Blank[]]], tan[Plus[Optional[Pattern[e, Blank[]]], Times[Optional[Pattern[f, Blank[]]], Pattern[x, Blank[]]]]]]], Pattern[n, Blank[]]], Plus[Optional[Pattern[A, Blank[]]], Times[Optional[Pattern[B, Blank[]]], tan[Plus[Optional[Pattern[e, Blank[]]], Times[Optional[Pattern[f, Blank[]]], Pattern[x, Blank[]]]]]], Times[Optional[Pattern[C, Blank[]]], Power[tan[Plus[Optional[Pattern[e, Blank[]]], Times[Optional[Pattern[f, Blank[]]], Pattern[x, Blank[]]]]], 2]]]], Pattern[x, Blank[Symbol]]] := With[{ff = FreeFactors[Tan[e + f*x], x]}, Dist[ff/f, Subst[Int[((a + b*ff*x)^m*(c + d*ff*x)^n*(A + B*ff*x + C*ff^2*x^2))/(1 + ff^2*x^2), x], x, Tan[e + f*x]/ff], x]] /; FreeQ[{a, b, c, d, e, f, A, B, C, m, n}, x] && NeQ[b*c - a*d, 0] && NeQ[a^2 + b^2, 0] && NeQ[c^2 + d^2, 0]
  3656. Int[Times[Power[Plus[Optional[Pattern[a, Blank[]]], Times[Optional[Pattern[b, Blank[]]], tan[Plus[Optional[Pattern[e, Blank[]]], Times[Optional[Pattern[f, Blank[]]], Pattern[x, Blank[]]]]]]], Pattern[m, Blank[]]], Power[Plus[Optional[Pattern[c, Blank[]]], Times[Optional[Pattern[d, Blank[]]], tan[Plus[Optional[Pattern[e, Blank[]]], Times[Optional[Pattern[f, Blank[]]], Pattern[x, Blank[]]]]]]], Pattern[n, Blank[]]], Plus[Optional[Pattern[A, Blank[]]], Times[Optional[Pattern[C, Blank[]]], Power[tan[Plus[Optional[Pattern[e, Blank[]]], Times[Optional[Pattern[f, Blank[]]], Pattern[x, Blank[]]]]], 2]]]], Pattern[x, Blank[Symbol]]] := With[{ff = FreeFactors[Tan[e + f*x], x]}, Dist[ff/f, Subst[Int[((a + b*ff*x)^m*(c + d*ff*x)^n*(A + C*ff^2*x^2))/(1 + ff^2*x^2), x], x, Tan[e + f*x]/ff], x]] /; FreeQ[{a, b, c, d, e, f, A, C, m, n}, x] && NeQ[b*c - a*d, 0] && NeQ[a^2 + b^2, 0] && NeQ[c^2 + d^2, 0]
  3657. Int[Times[Optional[Pattern[u, Blank[]]], Power[Plus[Pattern[a, Blank[]], Times[Optional[Pattern[b, Blank[]]], Power[tan[Plus[Optional[Pattern[e, Blank[]]], Times[Optional[Pattern[f, Blank[]]], Pattern[x, Blank[]]]]], 2]]], Pattern[p, Blank[]]]], Pattern[x, Blank[Symbol]]] := Int[ActivateTrig[u*(a*sec[e + f*x]^2)^p], x] /; FreeQ[{a, b, e, f, p}, x] && EqQ[a, b]
  3658. Int[Times[Optional[Pattern[u, Blank[]]], Power[Times[Optional[Pattern[b, Blank[]]], Power[tan[Plus[Optional[Pattern[e, Blank[]]], Times[Optional[Pattern[f, Blank[]]], Pattern[x, Blank[]]]]], Pattern[n, Blank[]]]], Pattern[p, Blank[]]]], Pattern[x, Blank[Symbol]]] := With[{ff = FreeFactors[Tan[e + f*x], x]}, Dist[((b*ff^n)^IntPart[p]*(b*Tan[e + f*x]^n)^FracPart[p])/(Tan[e + f*x]/ff)^(n*FracPart[p]), Int[ActivateTrig[u]*(Tan[e + f*x]/ff)^(n*p), x], x]] /; FreeQ[{b, e, f, n, p}, x] && !IntegerQ[p] && IntegerQ[n] && (EqQ[u, 1] || MatchQ[u, ((d_.)*(trig_)[e + f*x])^(m_.) /; FreeQ[{d, m}, x] && MemberQ[{sin, cos, tan, cot, sec, csc}, trig]])
  3659. Int[Times[Optional[Pattern[u, Blank[]]], Power[Times[Optional[Pattern[b, Blank[]]], Power[Times[Optional[Pattern[c, Blank[]]], tan[Plus[Optional[Pattern[e, Blank[]]], Times[Optional[Pattern[f, Blank[]]], Pattern[x, Blank[]]]]]], Pattern[n, Blank[]]]], Pattern[p, Blank[]]]], Pattern[x, Blank[Symbol]]] := Dist[(b^IntPart[p]*(b*(c*Tan[e + f*x])^n)^FracPart[p])/(c*Tan[e + f*x])^(n*FracPart[p]), Int[ActivateTrig[u]*(c*Tan[e + f*x])^(n*p), x], x] /; FreeQ[{b, c, e, f, n, p}, x] && !IntegerQ[p] && !IntegerQ[n] && (EqQ[u, 1] || MatchQ[u, ((d_.)*(trig_)[e + f*x])^(m_.) /; FreeQ[{d, m}, x] && MemberQ[{sin, cos, tan, cot, sec, csc}, trig]])
  3660. Int[Power[Plus[Pattern[a, Blank[]], Times[Optional[Pattern[b, Blank[]]], Power[tan[Plus[Optional[Pattern[e, Blank[]]], Times[Optional[Pattern[f, Blank[]]], Pattern[x, Blank[]]]]], 2]]], -1], Pattern[x, Blank[Symbol]]] := Simp[x/(a - b), x] - Dist[b/(a - b), Int[Sec[e + f*x]^2/(a + b*Tan[e + f*x]^2), x], x] /; FreeQ[{a, b, e, f}, x] && NeQ[a, b]
  3661. Int[Power[Plus[Pattern[a, Blank[]], Times[Optional[Pattern[b, Blank[]]], Power[Times[Optional[Pattern[c, Blank[]]], tan[Plus[Optional[Pattern[e, Blank[]]], Times[Optional[Pattern[f, Blank[]]], Pattern[x, Blank[]]]]]], Pattern[n, Blank[]]]]], Pattern[p, Blank[]]], Pattern[x, Blank[Symbol]]] := With[{ff = FreeFactors[Tan[e + f*x], x]}, Dist[(c*ff)/f, Subst[Int[(a + b*(ff*x)^n)^p/(c^2 + ff^2*x^2), x], x, (c*Tan[e + f*x])/ff], x]] /; FreeQ[{a, b, c, e, f, n, p}, x] && (IntegersQ[n, p] || IGtQ[p, 0] || EqQ[n^2, 4] || EqQ[n^2, 16])
  3662. Int[Power[Plus[Pattern[a, Blank[]], Times[Optional[Pattern[b, Blank[]]], Power[Times[Optional[Pattern[c, Blank[]]], tan[Plus[Optional[Pattern[e, Blank[]]], Times[Optional[Pattern[f, Blank[]]], Pattern[x, Blank[]]]]]], Pattern[n, Blank[]]]]], Optional[Pattern[p, Blank[]]]], Pattern[x, Blank[Symbol]]] := Unintegrable[(a + b*(c*Tan[e + f*x])^n)^p, x] /; FreeQ[{a, b, c, e, f, n, p}, x]
  3663. Int[Times[Power[sin[Plus[Optional[Pattern[e, Blank[]]], Times[Optional[Pattern[f, Blank[]]], Pattern[x, Blank[]]]]], Pattern[m, Blank[]]], Power[Plus[Pattern[a, Blank[]], Times[Optional[Pattern[b, Blank[]]], Power[Times[Optional[Pattern[c, Blank[]]], tan[Plus[Optional[Pattern[e, Blank[]]], Times[Optional[Pattern[f, Blank[]]], Pattern[x, Blank[]]]]]], Pattern[n, Blank[]]]]], Optional[Pattern[p, Blank[]]]]], Pattern[x, Blank[Symbol]]] := With[{ff = FreeFactors[Tan[e + f*x], x]}, Dist[(c*ff^(m + 1))/f, Subst[Int[(x^m*(a + b*(ff*x)^n)^p)/(c^2 + ff^2*x^2)^(m/2 + 1), x], x, (c*Tan[e + f*x])/ff], x]] /; FreeQ[{a, b, c, e, f, n, p}, x] && IntegerQ[m/2]
  3664. Int[Times[Power[sin[Plus[Optional[Pattern[e, Blank[]]], Times[Optional[Pattern[f, Blank[]]], Pattern[x, Blank[]]]]], Optional[Pattern[m, Blank[]]]], Power[Plus[Pattern[a, Blank[]], Times[Optional[Pattern[b, Blank[]]], Power[tan[Plus[Optional[Pattern[e, Blank[]]], Times[Optional[Pattern[f, Blank[]]], Pattern[x, Blank[]]]]], 2]]], Optional[Pattern[p, Blank[]]]]], Pattern[x, Blank[Symbol]]] := With[{ff = FreeFactors[Sec[e + f*x], x]}, Dist[1/(f*ff^m), Subst[Int[((-1 + ff^2*x^2)^((m - 1)/2)*(a - b + b*ff^2*x^2)^p)/x^(m + 1), x], x, Sec[e + f*x]/ff], x]] /; FreeQ[{a, b, e, f, p}, x] && IntegerQ[(m - 1)/2]
  3665. Int[Times[Power[sin[Plus[Optional[Pattern[e, Blank[]]], Times[Optional[Pattern[f, Blank[]]], Pattern[x, Blank[]]]]], Optional[Pattern[m, Blank[]]]], Power[Plus[Pattern[a, Blank[]], Times[Optional[Pattern[b, Blank[]]], Power[tan[Plus[Optional[Pattern[e, Blank[]]], Times[Optional[Pattern[f, Blank[]]], Pattern[x, Blank[]]]]], Pattern[n, Blank[]]]]], Optional[Pattern[p, Blank[]]]]], Pattern[x, Blank[Symbol]]] := With[{ff = FreeFactors[Sec[e + f*x], x]}, Dist[1/(f*ff^m), Subst[Int[((-1 + ff^2*x^2)^((m - 1)/2)*(a + b*(-1 + ff^2*x^2)^(n/2))^p)/x^(m + 1), x], x, Sec[e + f*x]/ff], x]] /; FreeQ[{a, b, e, f, p}, x] && IntegerQ[(m - 1)/2] && IntegerQ[n/2]
  3666. Int[Times[Power[Times[Optional[Pattern[d, Blank[]]], sin[Plus[Optional[Pattern[e, Blank[]]], Times[Optional[Pattern[f, Blank[]]], Pattern[x, Blank[]]]]]], Optional[Pattern[m, Blank[]]]], Power[Plus[Pattern[a, Blank[]], Times[Optional[Pattern[b, Blank[]]], Power[Times[Optional[Pattern[c, Blank[]]], tan[Plus[Optional[Pattern[e, Blank[]]], Times[Optional[Pattern[f, Blank[]]], Pattern[x, Blank[]]]]]], Pattern[n, Blank[]]]]], Optional[Pattern[p, Blank[]]]]], Pattern[x, Blank[Symbol]]] := Int[ExpandTrig[(d*sin[e + f*x])^m*(a + b*(c*tan[e + f*x])^n)^p, x], x] /; FreeQ[{a, b, c, d, e, f, m, n}, x] && IGtQ[p, 0]
  3667. Int[Times[Power[Times[Optional[Pattern[d, Blank[]]], sin[Plus[Optional[Pattern[e, Blank[]]], Times[Optional[Pattern[f, Blank[]]], Pattern[x, Blank[]]]]]], Pattern[m, Blank[]]], Power[Plus[Pattern[a, Blank[]], Times[Optional[Pattern[b, Blank[]]], Power[tan[Plus[Optional[Pattern[e, Blank[]]], Times[Optional[Pattern[f, Blank[]]], Pattern[x, Blank[]]]]], 2]]], Pattern[p, Blank[]]]], Pattern[x, Blank[Symbol]]] := With[{ff = FreeFactors[Tan[e + f*x], x]}, Dist[(ff*(d*Sin[e + f*x])^m*(Sec[e + f*x]^2)^(m/2))/(f*Tan[e + f*x]^m), Subst[Int[((ff*x)^m*(a + b*ff^2*x^2)^p)/(1 + ff^2*x^2)^(m/2 + 1), x], x, Tan[e + f*x]/ff], x]] /; FreeQ[{a, b, d, e, f, m, p}, x] && !IntegerQ[m]
  3668. Int[Times[Power[Times[Optional[Pattern[d, Blank[]]], sin[Plus[Optional[Pattern[e, Blank[]]], Times[Optional[Pattern[f, Blank[]]], Pattern[x, Blank[]]]]]], Optional[Pattern[m, Blank[]]]], Power[Plus[Pattern[a, Blank[]], Times[Optional[Pattern[b, Blank[]]], Power[Times[Optional[Pattern[c, Blank[]]], tan[Plus[Optional[Pattern[e, Blank[]]], Times[Optional[Pattern[f, Blank[]]], Pattern[x, Blank[]]]]]], Pattern[n, Blank[]]]]], Optional[Pattern[p, Blank[]]]]], Pattern[x, Blank[Symbol]]] := Unintegrable[(d*Sin[e + f*x])^m*(a + b*(c*Tan[e + f*x])^n)^p, x] /; FreeQ[{a, b, c, d, e, f, m, n, p}, x]
  3669. Int[Times[Power[Times[cos[Plus[Optional[Pattern[e, Blank[]]], Times[Optional[Pattern[f, Blank[]]], Pattern[x, Blank[]]]]], Optional[Pattern[d, Blank[]]]], Pattern[m, Blank[]]], Power[Plus[Pattern[a, Blank[]], Times[Optional[Pattern[b, Blank[]]], Power[Times[Optional[Pattern[c, Blank[]]], tan[Plus[Optional[Pattern[e, Blank[]]], Times[Optional[Pattern[f, Blank[]]], Pattern[x, Blank[]]]]]], Pattern[n, Blank[]]]]], Pattern[p, Blank[]]]], Pattern[x, Blank[Symbol]]] := Dist[(d*Cos[e + f*x])^FracPart[m]*(Sec[e + f*x]/d)^FracPart[m], Int[(a + b*(c*Tan[e + f*x])^n)^p/(Sec[e + f*x]/d)^m, x], x] /; FreeQ[{a, b, c, d, e, f, m, n, p}, x] && !IntegerQ[m]
  3670. Int[Times[Power[Times[Optional[Pattern[d, Blank[]]], tan[Plus[Optional[Pattern[e, Blank[]]], Times[Optional[Pattern[f, Blank[]]], Pattern[x, Blank[]]]]]], Optional[Pattern[m, Blank[]]]], Power[Plus[Pattern[a, Blank[]], Times[Optional[Pattern[b, Blank[]]], Power[Times[Optional[Pattern[c, Blank[]]], tan[Plus[Optional[Pattern[e, Blank[]]], Times[Optional[Pattern[f, Blank[]]], Pattern[x, Blank[]]]]]], Pattern[n, Blank[]]]]], Optional[Pattern[p, Blank[]]]]], Pattern[x, Blank[Symbol]]] := With[{ff = FreeFactors[Tan[e + f*x], x]}, Dist[(c*ff)/f, Subst[Int[(((d*ff*x)/c)^m*(a + b*(ff*x)^n)^p)/(c^2 + ff^2*x^2), x], x, (c*Tan[e + f*x])/ff], x]] /; FreeQ[{a, b, c, d, e, f, m, n, p}, x] && (IGtQ[p, 0] || EqQ[n, 2] || EqQ[n, 4] || (IntegerQ[p] && RationalQ[n]))
  3671. Int[Times[Power[Times[Optional[Pattern[d, Blank[]]], tan[Plus[Optional[Pattern[e, Blank[]]], Times[Optional[Pattern[f, Blank[]]], Pattern[x, Blank[]]]]]], Optional[Pattern[m, Blank[]]]], Power[Plus[Pattern[a, Blank[]], Times[Optional[Pattern[b, Blank[]]], Power[Times[Optional[Pattern[c, Blank[]]], tan[Plus[Optional[Pattern[e, Blank[]]], Times[Optional[Pattern[f, Blank[]]], Pattern[x, Blank[]]]]]], Pattern[n, Blank[]]]]], Optional[Pattern[p, Blank[]]]]], Pattern[x, Blank[Symbol]]] := Int[ExpandTrig[(d*tan[e + f*x])^m*(a + b*(c*tan[e + f*x])^n)^p, x], x] /; FreeQ[{a, b, c, d, e, f, m, n}, x] && IGtQ[p, 0]
  3672. Int[Times[Power[Times[Optional[Pattern[d, Blank[]]], tan[Plus[Optional[Pattern[e, Blank[]]], Times[Optional[Pattern[f, Blank[]]], Pattern[x, Blank[]]]]]], Optional[Pattern[m, Blank[]]]], Power[Plus[Pattern[a, Blank[]], Times[Optional[Pattern[b, Blank[]]], Power[Times[Optional[Pattern[c, Blank[]]], tan[Plus[Optional[Pattern[e, Blank[]]], Times[Optional[Pattern[f, Blank[]]], Pattern[x, Blank[]]]]]], Pattern[n, Blank[]]]]], Optional[Pattern[p, Blank[]]]]], Pattern[x, Blank[Symbol]]] := Unintegrable[(d*Tan[e + f*x])^m*(a + b*(c*Tan[e + f*x])^n)^p, x] /; FreeQ[{a, b, c, d, e, f, m, n, p}, x]
  3673. Int[Times[Power[Times[cot[Plus[Optional[Pattern[e, Blank[]]], Times[Optional[Pattern[f, Blank[]]], Pattern[x, Blank[]]]]], Optional[Pattern[d, Blank[]]]], Pattern[m, Blank[]]], Power[Plus[Pattern[a, Blank[]], Times[Optional[Pattern[b, Blank[]]], Power[tan[Plus[Optional[Pattern[e, Blank[]]], Times[Optional[Pattern[f, Blank[]]], Pattern[x, Blank[]]]]], Optional[Pattern[n, Blank[]]]]]], Optional[Pattern[p, Blank[]]]]], Pattern[x, Blank[Symbol]]] := Dist[d^(n*p), Int[(d*Cot[e + f*x])^(m - n*p)*(b + a*Cot[e + f*x]^n)^p, x], x] /; FreeQ[{a, b, d, e, f, m, n, p}, x] && !IntegerQ[m] && IntegersQ[n, p]
  3674. Int[Times[Power[Times[cot[Plus[Optional[Pattern[e, Blank[]]], Times[Optional[Pattern[f, Blank[]]], Pattern[x, Blank[]]]]], Optional[Pattern[d, Blank[]]]], Pattern[m, Blank[]]], Power[Plus[Pattern[a, Blank[]], Times[Optional[Pattern[b, Blank[]]], Power[Times[Optional[Pattern[c, Blank[]]], tan[Plus[Optional[Pattern[e, Blank[]]], Times[Optional[Pattern[f, Blank[]]], Pattern[x, Blank[]]]]]], Pattern[n, Blank[]]]]], Pattern[p, Blank[]]]], Pattern[x, Blank[Symbol]]] := Dist[(d*Cot[e + f*x])^FracPart[m]*(Tan[e + f*x]/d)^FracPart[m], Int[(a + b*(c*Tan[e + f*x])^n)^p/(Tan[e + f*x]/d)^m, x], x] /; FreeQ[{a, b, c, d, e, f, m, n, p}, x] && !IntegerQ[m]
  3675. Int[Times[Power[sec[Plus[Optional[Pattern[e, Blank[]]], Times[Optional[Pattern[f, Blank[]]], Pattern[x, Blank[]]]]], Pattern[m, Blank[]]], Power[Plus[Pattern[a, Blank[]], Times[Optional[Pattern[b, Blank[]]], Power[Times[Optional[Pattern[c, Blank[]]], tan[Plus[Optional[Pattern[e, Blank[]]], Times[Optional[Pattern[f, Blank[]]], Pattern[x, Blank[]]]]]], Pattern[n, Blank[]]]]], Optional[Pattern[p, Blank[]]]]], Pattern[x, Blank[Symbol]]] := With[{ff = FreeFactors[Tan[e + f*x], x]}, Dist[ff/(c^(m - 1)*f), Subst[Int[(c^2 + ff^2*x^2)^(m/2 - 1)*(a + b*(ff*x)^n)^p, x], x, (c*Tan[e + f*x])/ff], x]] /; FreeQ[{a, b, c, e, f, n, p}, x] && IntegerQ[m/2] && (IntegersQ[n, p] || IGtQ[m, 0] || IGtQ[p, 0] || EqQ[n^2, 4] || EqQ[n^2, 16])
  3676. Int[Times[Power[sec[Plus[Optional[Pattern[e, Blank[]]], Times[Optional[Pattern[f, Blank[]]], Pattern[x, Blank[]]]]], Optional[Pattern[m, Blank[]]]], Power[Plus[Pattern[a, Blank[]], Times[Optional[Pattern[b, Blank[]]], Power[tan[Plus[Optional[Pattern[e, Blank[]]], Times[Optional[Pattern[f, Blank[]]], Pattern[x, Blank[]]]]], Pattern[n, Blank[]]]]], Optional[Pattern[p, Blank[]]]]], Pattern[x, Blank[Symbol]]] := With[{ff = FreeFactors[Sin[e + f*x], x]}, Dist[ff/f, Subst[Int[ExpandToSum[b*(ff*x)^n + a*(1 - ff^2*x^2)^(n/2), x]^p/(1 - ff^2*x^2)^((m + n*p + 1)/2), x], x, Sin[e + f*x]/ff], x]] /; FreeQ[{a, b, e, f}, x] && IntegerQ[(m - 1)/2] && IntegerQ[n/2] && IntegerQ[p]
  3677. Int[Times[Power[sec[Plus[Optional[Pattern[e, Blank[]]], Times[Optional[Pattern[f, Blank[]]], Pattern[x, Blank[]]]]], Optional[Pattern[m, Blank[]]]], Power[Plus[Pattern[a, Blank[]], Times[Optional[Pattern[b, Blank[]]], Power[tan[Plus[Optional[Pattern[e, Blank[]]], Times[Optional[Pattern[f, Blank[]]], Pattern[x, Blank[]]]]], Pattern[n, Blank[]]]]], Optional[Pattern[p, Blank[]]]]], Pattern[x, Blank[Symbol]]] := With[{ff = FreeFactors[Sin[e + f*x], x]}, Dist[ff/f, Subst[Int[(1*((b*(ff*x)^n + a*(1 - ff^2*x^2)^(n/2))/(1 - ff^2*x^2)^(n/2))^p)/(1 - ff^2*x^2)^((m + 1)/2), x], x, Sin[e + f*x]/ff], x]] /; FreeQ[{a, b, e, f, p}, x] && IntegerQ[(m - 1)/2] && IntegerQ[n/2] && !IntegerQ[p]
  3678. Int[Times[Power[Times[Optional[Pattern[d, Blank[]]], sec[Plus[Optional[Pattern[e, Blank[]]], Times[Optional[Pattern[f, Blank[]]], Pattern[x, Blank[]]]]]], Optional[Pattern[m, Blank[]]]], Power[Plus[Pattern[a, Blank[]], Times[Optional[Pattern[b, Blank[]]], Power[Times[Optional[Pattern[c, Blank[]]], tan[Plus[Optional[Pattern[e, Blank[]]], Times[Optional[Pattern[f, Blank[]]], Pattern[x, Blank[]]]]]], Pattern[n, Blank[]]]]], Optional[Pattern[p, Blank[]]]]], Pattern[x, Blank[Symbol]]] := Int[ExpandTrig[(d*sec[e + f*x])^m*(a + b*(c*tan[e + f*x])^n)^p, x], x] /; FreeQ[{a, b, c, d, e, f, m, n}, x] && IGtQ[p, 0]
  3679. Int[Times[Power[Times[Optional[Pattern[d, Blank[]]], sec[Plus[Optional[Pattern[e, Blank[]]], Times[Optional[Pattern[f, Blank[]]], Pattern[x, Blank[]]]]]], Pattern[m, Blank[]]], Power[Plus[Pattern[a, Blank[]], Times[Optional[Pattern[b, Blank[]]], Power[tan[Plus[Optional[Pattern[e, Blank[]]], Times[Optional[Pattern[f, Blank[]]], Pattern[x, Blank[]]]]], 2]]], Pattern[p, Blank[]]]], Pattern[x, Blank[Symbol]]] := With[{ff = FreeFactors[Tan[e + f*x], x]}, Dist[(ff*(d*Sec[e + f*x])^m)/(f*(Sec[e + f*x]^2)^(m/2)), Subst[Int[(1 + ff^2*x^2)^(m/2 - 1)*(a + b*ff^2*x^2)^p, x], x, Tan[e + f*x]/ff], x]] /; FreeQ[{a, b, d, e, f, m, p}, x] && !IntegerQ[m]
  3680. Int[Times[Power[Times[Optional[Pattern[d, Blank[]]], sec[Plus[Optional[Pattern[e, Blank[]]], Times[Optional[Pattern[f, Blank[]]], Pattern[x, Blank[]]]]]], Optional[Pattern[m, Blank[]]]], Power[Plus[Pattern[a, Blank[]], Times[Optional[Pattern[b, Blank[]]], Power[Times[Optional[Pattern[c, Blank[]]], tan[Plus[Optional[Pattern[e, Blank[]]], Times[Optional[Pattern[f, Blank[]]], Pattern[x, Blank[]]]]]], Pattern[n, Blank[]]]]], Optional[Pattern[p, Blank[]]]]], Pattern[x, Blank[Symbol]]] := Unintegrable[(d*Sec[e + f*x])^m*(a + b*(c*Tan[e + f*x])^n)^p, x] /; FreeQ[{a, b, c, d, e, f, m, n, p}, x]
  3681. Int[Times[Power[Times[csc[Plus[Optional[Pattern[e, Blank[]]], Times[Optional[Pattern[f, Blank[]]], Pattern[x, Blank[]]]]], Optional[Pattern[d, Blank[]]]], Pattern[m, Blank[]]], Power[Plus[Pattern[a, Blank[]], Times[Optional[Pattern[b, Blank[]]], Power[Times[Optional[Pattern[c, Blank[]]], tan[Plus[Optional[Pattern[e, Blank[]]], Times[Optional[Pattern[f, Blank[]]], Pattern[x, Blank[]]]]]], Pattern[n, Blank[]]]]], Pattern[p, Blank[]]]], Pattern[x, Blank[Symbol]]] := Dist[(d*Csc[e + f*x])^FracPart[m]*(Sin[e + f*x]/d)^FracPart[m], Int[(a + b*(c*Tan[e + f*x])^n)^p/(Sin[e + f*x]/d)^m, x], x] /; FreeQ[{a, b, c, d, e, f, m, n, p}, x] && !IntegerQ[m]
  3682. Int[Power[Plus[Pattern[a, Blank[]], Times[Optional[Pattern[b, Blank[]]], Power[tan[Plus[Optional[Pattern[d, Blank[]]], Times[Optional[Pattern[e, Blank[]]], Pattern[x, Blank[]]]]], Optional[Pattern[n, Blank[]]]]], Times[Optional[Pattern[c, Blank[]]], Power[tan[Plus[Optional[Pattern[d, Blank[]]], Times[Optional[Pattern[e, Blank[]]], Pattern[x, Blank[]]]]], Optional[Pattern[n2, Blank[]]]]]], Optional[Pattern[p, Blank[]]]], Pattern[x, Blank[Symbol]]] := Dist[1/(4^p*c^p), Int[(b + 2*c*Tan[d + e*x]^n)^(2*p), x], x] /; FreeQ[{a, b, c, d, e, n}, x] && EqQ[n2, 2*n] && EqQ[b^2 - 4*a*c, 0] && IntegerQ[p]
  3683. Int[Power[Plus[Times[Power[cot[Plus[Optional[Pattern[d, Blank[]]], Times[Optional[Pattern[e, Blank[]]], Pattern[x, Blank[]]]]], Optional[Pattern[n, Blank[]]]], Optional[Pattern[b, Blank[]]]], Times[Power[cot[Plus[Optional[Pattern[d, Blank[]]], Times[Optional[Pattern[e, Blank[]]], Pattern[x, Blank[]]]]], Optional[Pattern[n2, Blank[]]]], Optional[Pattern[c, Blank[]]]], Pattern[a, Blank[]]], Optional[Pattern[p, Blank[]]]], Pattern[x, Blank[Symbol]]] := Dist[1/(4^p*c^p), Int[(b + 2*c*Cot[d + e*x]^n)^(2*p), x], x] /; FreeQ[{a, b, c, d, e, n}, x] && EqQ[n2, 2*n] && EqQ[b^2 - 4*a*c, 0] && IntegerQ[p]
  3684. Int[Power[Plus[Pattern[a, Blank[]], Times[Optional[Pattern[b, Blank[]]], Power[tan[Plus[Optional[Pattern[d, Blank[]]], Times[Optional[Pattern[e, Blank[]]], Pattern[x, Blank[]]]]], Optional[Pattern[n, Blank[]]]]], Times[Optional[Pattern[c, Blank[]]], Power[tan[Plus[Optional[Pattern[d, Blank[]]], Times[Optional[Pattern[e, Blank[]]], Pattern[x, Blank[]]]]], Optional[Pattern[n2, Blank[]]]]]], Pattern[p, Blank[]]], Pattern[x, Blank[Symbol]]] := Dist[(a + b*Tan[d + e*x]^n + c*Tan[d + e*x]^(2*n))^p/(b + 2*c*Tan[d + e*x]^n)^(2*p), Int[(b + 2*c*Tan[d + e*x]^n)^(2*p), x], x] /; FreeQ[{a, b, c, d, e, n}, x] && EqQ[n2, 2*n] && EqQ[b^2 - 4*a*c, 0] && !IntegerQ[p]
  3685. Int[Power[Plus[Times[Power[cot[Plus[Optional[Pattern[d, Blank[]]], Times[Optional[Pattern[e, Blank[]]], Pattern[x, Blank[]]]]], Optional[Pattern[n, Blank[]]]], Optional[Pattern[b, Blank[]]]], Times[Power[cot[Plus[Optional[Pattern[d, Blank[]]], Times[Optional[Pattern[e, Blank[]]], Pattern[x, Blank[]]]]], Optional[Pattern[n2, Blank[]]]], Optional[Pattern[c, Blank[]]]], Pattern[a, Blank[]]], Pattern[p, Blank[]]], Pattern[x, Blank[Symbol]]] := Dist[(a + b*Cot[d + e*x]^n + c*Cot[d + e*x]^(2*n))^p/(b + 2*c*Cot[d + e*x]^n)^(2*p), Int[(b + 2*c*Cot[d + e*x]^n)^(2*p), x], x] /; FreeQ[{a, b, c, d, e, n}, x] && EqQ[n2, 2*n] && EqQ[b^2 - 4*a*c, 0] && !IntegerQ[p]
  3686. Int[Power[Plus[Optional[Pattern[a, Blank[]]], Times[Optional[Pattern[b, Blank[]]], Power[tan[Plus[Optional[Pattern[d, Blank[]]], Times[Optional[Pattern[e, Blank[]]], Pattern[x, Blank[]]]]], Optional[Pattern[n, Blank[]]]]], Times[Optional[Pattern[c, Blank[]]], Power[tan[Plus[Optional[Pattern[d, Blank[]]], Times[Optional[Pattern[e, Blank[]]], Pattern[x, Blank[]]]]], Optional[Pattern[n2, Blank[]]]]]], -1], Pattern[x, Blank[Symbol]]] := Module[{q = Rt[b^2 - 4*a*c, 2]}, Dist[(2*c)/q, Int[1/(b - q + 2*c*Tan[d + e*x]^n), x], x] - Dist[(2*c)/q, Int[1/(b + q + 2*c*Tan[d + e*x]^n), x], x]] /; FreeQ[{a, b, c, d, e, n}, x] && EqQ[n2, 2*n] && NeQ[b^2 - 4*a*c, 0]
  3687. Int[Power[Plus[Optional[Pattern[a, Blank[]]], Times[Power[cot[Plus[Optional[Pattern[d, Blank[]]], Times[Optional[Pattern[e, Blank[]]], Pattern[x, Blank[]]]]], Optional[Pattern[n, Blank[]]]], Optional[Pattern[b, Blank[]]]], Times[Power[cot[Plus[Optional[Pattern[d, Blank[]]], Times[Optional[Pattern[e, Blank[]]], Pattern[x, Blank[]]]]], Optional[Pattern[n2, Blank[]]]], Optional[Pattern[c, Blank[]]]]], -1], Pattern[x, Blank[Symbol]]] := Module[{q = Rt[b^2 - 4*a*c, 2]}, Dist[(2*c)/q, Int[1/(b - q + 2*c*Cot[d + e*x]^n), x], x] - Dist[(2*c)/q, Int[1/(b + q + 2*c*Cot[d + e*x]^n), x], x]] /; FreeQ[{a, b, c, d, e, n}, x] && EqQ[n2, 2*n] && NeQ[b^2 - 4*a*c, 0]
  3688. Int[Times[Power[sin[Plus[Optional[Pattern[d, Blank[]]], Times[Optional[Pattern[e, Blank[]]], Pattern[x, Blank[]]]]], Pattern[m, Blank[]]], Power[Plus[Optional[Pattern[a, Blank[]]], Times[Optional[Pattern[b, Blank[]]], Power[Times[Optional[Pattern[f, Blank[]]], tan[Plus[Optional[Pattern[d, Blank[]]], Times[Optional[Pattern[e, Blank[]]], Pattern[x, Blank[]]]]]], Optional[Pattern[n, Blank[]]]]], Times[Optional[Pattern[c, Blank[]]], Power[Times[Optional[Pattern[f, Blank[]]], tan[Plus[Optional[Pattern[d, Blank[]]], Times[Optional[Pattern[e, Blank[]]], Pattern[x, Blank[]]]]]], Optional[Pattern[n2, Blank[]]]]]], Pattern[p, Blank[]]]], Pattern[x, Blank[Symbol]]] := Dist[f/e, Subst[Int[(x^m*(a + b*x^n + c*x^(2*n))^p)/(f^2 + x^2)^(m/2 + 1), x], x, f*Tan[d + e*x]], x] /; FreeQ[{a, b, c, d, e, f, n, p}, x] && EqQ[n2, 2*n] && IntegerQ[m/2]
  3689. Int[Times[Power[cos[Plus[Optional[Pattern[d, Blank[]]], Times[Optional[Pattern[e, Blank[]]], Pattern[x, Blank[]]]]], Pattern[m, Blank[]]], Power[Plus[Optional[Pattern[a, Blank[]]], Times[Optional[Pattern[b, Blank[]]], Power[Times[cot[Plus[Optional[Pattern[d, Blank[]]], Times[Optional[Pattern[e, Blank[]]], Pattern[x, Blank[]]]]], Optional[Pattern[f, Blank[]]]], Optional[Pattern[n, Blank[]]]]], Times[Optional[Pattern[c, Blank[]]], Power[Times[cot[Plus[Optional[Pattern[d, Blank[]]], Times[Optional[Pattern[e, Blank[]]], Pattern[x, Blank[]]]]], Optional[Pattern[f, Blank[]]]], Optional[Pattern[n2, Blank[]]]]]], Pattern[p, Blank[]]]], Pattern[x, Blank[Symbol]]] := -Dist[f/e, Subst[Int[(x^m*(a + b*x^n + c*x^(2*n))^p)/(f^2 + x^2)^(m/2 + 1), x], x, f*Cot[d + e*x]], x] /; FreeQ[{a, b, c, d, e, f, n, p}, x] && EqQ[n2, 2*n] && IntegerQ[m/2]
  3690. Int[Times[Power[sin[Plus[Optional[Pattern[d, Blank[]]], Times[Optional[Pattern[e, Blank[]]], Pattern[x, Blank[]]]]], Optional[Pattern[m, Blank[]]]], Power[Plus[Optional[Pattern[a, Blank[]]], Times[Optional[Pattern[b, Blank[]]], Power[tan[Plus[Optional[Pattern[d, Blank[]]], Times[Optional[Pattern[e, Blank[]]], Pattern[x, Blank[]]]]], Optional[Pattern[n, Blank[]]]]], Times[Optional[Pattern[c, Blank[]]], Power[tan[Plus[Optional[Pattern[d, Blank[]]], Times[Optional[Pattern[e, Blank[]]], Pattern[x, Blank[]]]]], Optional[Pattern[n2, Blank[]]]]]], Optional[Pattern[p, Blank[]]]]], Pattern[x, Blank[Symbol]]] := Module[{g = FreeFactors[Cos[d + e*x], x]}, -Dist[g/e, Subst[Int[((1 - g^2*x^2)^((m - 1)/2)*ExpandToSum[a*(g*x)^(2*n) + b*(g*x)^n*(1 - g^2*x^2)^(n/2) + c*(1 - g^2*x^2)^n, x]^p)/(g*x)^(2*n*p), x], x, Cos[d + e*x]/g], x]] /; FreeQ[{a, b, c, d, e}, x] && EqQ[n2, 2*n] && IntegerQ[(m - 1)/2] && IntegerQ[n/2] && IntegerQ[p]
  3691. Int[Times[Power[cos[Plus[Optional[Pattern[d, Blank[]]], Times[Optional[Pattern[e, Blank[]]], Pattern[x, Blank[]]]]], Optional[Pattern[m, Blank[]]]], Power[Plus[Optional[Pattern[a, Blank[]]], Times[Power[cot[Plus[Optional[Pattern[d, Blank[]]], Times[Optional[Pattern[e, Blank[]]], Pattern[x, Blank[]]]]], Optional[Pattern[n, Blank[]]]], Optional[Pattern[b, Blank[]]]], Times[Optional[Pattern[c, Blank[]]], Power[tan[Plus[Optional[Pattern[d, Blank[]]], Times[Optional[Pattern[e, Blank[]]], Pattern[x, Blank[]]]]], Optional[Pattern[n2, Blank[]]]]]], Optional[Pattern[p, Blank[]]]]], Pattern[x, Blank[Symbol]]] := Module[{g = FreeFactors[Sin[d + e*x], x]}, Dist[g/e, Subst[Int[((1 - g^2*x^2)^((m - 1)/2)*ExpandToSum[a*(g*x)^(2*n) + b*(g*x)^n*(1 - g^2*x^2)^(n/2) + c*(1 - g^2*x^2)^n, x]^p)/(g*x)^(2*n*p), x], x, Sin[d + e*x]/g], x]] /; FreeQ[{a, b, c, d, e}, x] && EqQ[n2, 2*n] && IntegerQ[(m - 1)/2] && IntegerQ[n/2] && IntegerQ[p]
  3692. Int[Times[Power[cos[Plus[Optional[Pattern[d, Blank[]]], Times[Optional[Pattern[e, Blank[]]], Pattern[x, Blank[]]]]], Pattern[m, Blank[]]], Power[Plus[Optional[Pattern[a, Blank[]]], Times[Optional[Pattern[b, Blank[]]], Power[Times[Optional[Pattern[f, Blank[]]], tan[Plus[Optional[Pattern[d, Blank[]]], Times[Optional[Pattern[e, Blank[]]], Pattern[x, Blank[]]]]]], Optional[Pattern[n, Blank[]]]]], Times[Optional[Pattern[c, Blank[]]], Power[Times[Optional[Pattern[f, Blank[]]], tan[Plus[Optional[Pattern[d, Blank[]]], Times[Optional[Pattern[e, Blank[]]], Pattern[x, Blank[]]]]]], Optional[Pattern[n2, Blank[]]]]]], Optional[Pattern[p, Blank[]]]]], Pattern[x, Blank[Symbol]]] := Dist[f^(m + 1)/e, Subst[Int[(a + b*x^n + c*x^(2*n))^p/(f^2 + x^2)^(m/2 + 1), x], x, f*Tan[d + e*x]], x] /; FreeQ[{a, b, c, d, e, f, n, p}, x] && EqQ[n2, 2*n] && IntegerQ[m/2]
  3693. Int[Times[Power[Plus[Optional[Pattern[a, Blank[]]], Times[Optional[Pattern[b, Blank[]]], Power[Times[cot[Plus[Optional[Pattern[d, Blank[]]], Times[Optional[Pattern[e, Blank[]]], Pattern[x, Blank[]]]]], Optional[Pattern[f, Blank[]]]], Optional[Pattern[n, Blank[]]]]], Times[Optional[Pattern[c, Blank[]]], Power[Times[cot[Plus[Optional[Pattern[d, Blank[]]], Times[Optional[Pattern[e, Blank[]]], Pattern[x, Blank[]]]]], Optional[Pattern[f, Blank[]]]], Optional[Pattern[n2, Blank[]]]]]], Optional[Pattern[p, Blank[]]]], Power[sin[Plus[Optional[Pattern[d, Blank[]]], Times[Optional[Pattern[e, Blank[]]], Pattern[x, Blank[]]]]], Pattern[m, Blank[]]]], Pattern[x, Blank[Symbol]]] := -Dist[f^(m + 1)/e, Subst[Int[(a + b*x^n + c*x^(2*n))^p/(f^2 + x^2)^(m/2 + 1), x], x, f*Cot[d + e*x]], x] /; FreeQ[{a, b, c, d, e, f, n, p}, x] && EqQ[n2, 2*n] && IntegerQ[m/2]
  3694. Int[Times[Power[cos[Plus[Optional[Pattern[d, Blank[]]], Times[Optional[Pattern[e, Blank[]]], Pattern[x, Blank[]]]]], Pattern[m, Blank[]]], Power[Plus[Optional[Pattern[a, Blank[]]], Times[Optional[Pattern[b, Blank[]]], Power[tan[Plus[Optional[Pattern[d, Blank[]]], Times[Optional[Pattern[e, Blank[]]], Pattern[x, Blank[]]]]], Optional[Pattern[n, Blank[]]]]], Times[Optional[Pattern[c, Blank[]]], Power[tan[Plus[Optional[Pattern[d, Blank[]]], Times[Optional[Pattern[e, Blank[]]], Pattern[x, Blank[]]]]], Optional[Pattern[n2, Blank[]]]]]], Optional[Pattern[p, Blank[]]]]], Pattern[x, Blank[Symbol]]] := Module[{g = FreeFactors[Sin[d + e*x], x]}, Dist[g/e, Subst[Int[(1 - g^2*x^2)^((m - 2*n*p - 1)/2)*ExpandToSum[c*x^(2*n) + b*x^n*(1 - x^2)^(n/2) + a*(1 - x^2)^n, x]^p, x], x, Sin[d + e*x]/g], x]] /; FreeQ[{a, b, c, d, e}, x] && EqQ[n2, 2*n] && IntegerQ[(m - 1)/2] && IntegerQ[n/2] && IntegerQ[p]
  3695. Int[Times[Power[Plus[Optional[Pattern[a, Blank[]]], Times[Power[cot[Plus[Optional[Pattern[d, Blank[]]], Times[Optional[Pattern[e, Blank[]]], Pattern[x, Blank[]]]]], Optional[Pattern[n, Blank[]]]], Optional[Pattern[b, Blank[]]]], Times[Power[cot[Plus[Optional[Pattern[d, Blank[]]], Times[Optional[Pattern[e, Blank[]]], Pattern[x, Blank[]]]]], Optional[Pattern[n2, Blank[]]]], Optional[Pattern[c, Blank[]]]]], Optional[Pattern[p, Blank[]]]], Power[sin[Plus[Optional[Pattern[d, Blank[]]], Times[Optional[Pattern[e, Blank[]]], Pattern[x, Blank[]]]]], Pattern[m, Blank[]]]], Pattern[x, Blank[Symbol]]] := Module[{g = FreeFactors[Cos[d + e*x], x]}, -Dist[g/e, Subst[Int[(1 - g^2*x^2)^((m - 2*n*p - 1)/2)*ExpandToSum[c*x^(2*n) + b*x^n*(1 - x^2)^(n/2) + a*(1 - x^2)^n, x]^p, x], x, Cos[d + e*x]/g], x]] /; FreeQ[{a, b, c, d, e}, x] && EqQ[n2, 2*n] && IntegerQ[(m - 1)/2] && IntegerQ[n/2] && IntegerQ[p]
  3696. Int[Times[Power[tan[Plus[Optional[Pattern[d, Blank[]]], Times[Optional[Pattern[e, Blank[]]], Pattern[x, Blank[]]]]], Optional[Pattern[m, Blank[]]]], Power[Plus[Optional[Pattern[a, Blank[]]], Times[Optional[Pattern[b, Blank[]]], Power[tan[Plus[Optional[Pattern[d, Blank[]]], Times[Optional[Pattern[e, Blank[]]], Pattern[x, Blank[]]]]], Optional[Pattern[n, Blank[]]]]], Times[Optional[Pattern[c, Blank[]]], Power[tan[Plus[Optional[Pattern[d, Blank[]]], Times[Optional[Pattern[e, Blank[]]], Pattern[x, Blank[]]]]], Optional[Pattern[n2, Blank[]]]]]], Pattern[p, Blank[]]]], Pattern[x, Blank[Symbol]]] := Dist[1/(4^p*c^p), Int[Tan[d + e*x]^m*(b + 2*c*Tan[d + e*x]^n)^(2*p), x], x] /; FreeQ[{a, b, c, d, e, m, n}, x] && EqQ[n2, 2*n] && EqQ[b^2 - 4*a*c, 0] && IntegerQ[p]
  3697. Int[Times[Power[cot[Plus[Optional[Pattern[d, Blank[]]], Times[Optional[Pattern[e, Blank[]]], Pattern[x, Blank[]]]]], Optional[Pattern[m, Blank[]]]], Power[Plus[Optional[Pattern[a, Blank[]]], Times[Power[cot[Plus[Optional[Pattern[d, Blank[]]], Times[Optional[Pattern[e, Blank[]]], Pattern[x, Blank[]]]]], Optional[Pattern[n, Blank[]]]], Optional[Pattern[b, Blank[]]]], Times[Power[cot[Plus[Optional[Pattern[d, Blank[]]], Times[Optional[Pattern[e, Blank[]]], Pattern[x, Blank[]]]]], Optional[Pattern[n2, Blank[]]]], Optional[Pattern[c, Blank[]]]]], Pattern[p, Blank[]]]], Pattern[x, Blank[Symbol]]] := Dist[1/(4^p*c^p), Int[Cot[d + e*x]^m*(b + 2*c*Cot[d + e*x]^n)^(2*p), x], x] /; FreeQ[{a, b, c, d, e, m, n}, x] && EqQ[n2, 2*n] && EqQ[b^2 - 4*a*c, 0] && IntegerQ[p]
  3698. Int[Times[Power[tan[Plus[Optional[Pattern[d, Blank[]]], Times[Optional[Pattern[e, Blank[]]], Pattern[x, Blank[]]]]], Optional[Pattern[m, Blank[]]]], Power[Plus[Optional[Pattern[a, Blank[]]], Times[Optional[Pattern[b, Blank[]]], Power[tan[Plus[Optional[Pattern[d, Blank[]]], Times[Optional[Pattern[e, Blank[]]], Pattern[x, Blank[]]]]], Optional[Pattern[n, Blank[]]]]], Times[Optional[Pattern[c, Blank[]]], Power[tan[Plus[Optional[Pattern[d, Blank[]]], Times[Optional[Pattern[e, Blank[]]], Pattern[x, Blank[]]]]], Optional[Pattern[n2, Blank[]]]]]], Pattern[p, Blank[]]]], Pattern[x, Blank[Symbol]]] := Dist[(a + b*Tan[d + e*x]^n + c*Tan[d + e*x]^(2*n))^p/(b + 2*c*Tan[d + e*x]^n)^(2*p), Int[Tan[d + e*x]^m*(b + 2*c*Tan[d + e*x]^n)^(2*p), x], x] /; FreeQ[{a, b, c, d, e, m, n, p}, x] && EqQ[n2, 2*n] && EqQ[b^2 - 4*a*c, 0] && !IntegerQ[p]
  3699. Int[Times[Power[cot[Plus[Optional[Pattern[d, Blank[]]], Times[Optional[Pattern[e, Blank[]]], Pattern[x, Blank[]]]]], Optional[Pattern[m, Blank[]]]], Power[Plus[Optional[Pattern[a, Blank[]]], Times[Power[cot[Plus[Optional[Pattern[d, Blank[]]], Times[Optional[Pattern[e, Blank[]]], Pattern[x, Blank[]]]]], Optional[Pattern[n, Blank[]]]], Optional[Pattern[b, Blank[]]]], Times[Power[cot[Plus[Optional[Pattern[d, Blank[]]], Times[Optional[Pattern[e, Blank[]]], Pattern[x, Blank[]]]]], Optional[Pattern[n2, Blank[]]]], Optional[Pattern[c, Blank[]]]]], Pattern[p, Blank[]]]], Pattern[x, Blank[Symbol]]] := Dist[(a + b*Cot[d + e*x]^n + c*Cot[d + e*x]^(2*n))^p/(b + 2*c*Cot[d + e*x]^n)^(2*p), Int[Cot[d + e*x]^m*(b + 2*c*Cot[d + e*x]^n)^(2*p), x], x] /; FreeQ[{a, b, c, d, e, m, n, p}, x] && EqQ[n2, 2*n] && EqQ[b^2 - 4*a*c, 0] && !IntegerQ[p]
  3700. Int[Times[Power[tan[Plus[Optional[Pattern[d, Blank[]]], Times[Optional[Pattern[e, Blank[]]], Pattern[x, Blank[]]]]], Optional[Pattern[m, Blank[]]]], Power[Plus[Optional[Pattern[a, Blank[]]], Times[Optional[Pattern[b, Blank[]]], Power[Times[Optional[Pattern[f, Blank[]]], tan[Plus[Optional[Pattern[d, Blank[]]], Times[Optional[Pattern[e, Blank[]]], Pattern[x, Blank[]]]]]], Optional[Pattern[n, Blank[]]]]], Times[Optional[Pattern[c, Blank[]]], Power[Times[Optional[Pattern[f, Blank[]]], tan[Plus[Optional[Pattern[d, Blank[]]], Times[Optional[Pattern[e, Blank[]]], Pattern[x, Blank[]]]]]], Optional[Pattern[n2, Blank[]]]]]], Pattern[p, Blank[]]]], Pattern[x, Blank[Symbol]]] := Dist[f/e, Subst[Int[((x/f)^m*(a + b*x^n + c*x^(2*n))^p)/(f^2 + x^2), x], x, f*Tan[d + e*x]], x] /; FreeQ[{a, b, c, d, e, f, m, n, p}, x] && EqQ[n2, 2*n] && NeQ[b^2 - 4*a*c, 0]
  3701. Int[Times[Power[cot[Plus[Optional[Pattern[d, Blank[]]], Times[Optional[Pattern[e, Blank[]]], Pattern[x, Blank[]]]]], Optional[Pattern[m, Blank[]]]], Power[Plus[Optional[Pattern[a, Blank[]]], Times[Optional[Pattern[b, Blank[]]], Power[Times[cot[Plus[Optional[Pattern[d, Blank[]]], Times[Optional[Pattern[e, Blank[]]], Pattern[x, Blank[]]]]], Optional[Pattern[f, Blank[]]]], Optional[Pattern[n, Blank[]]]]], Times[Optional[Pattern[c, Blank[]]], Power[Times[cot[Plus[Optional[Pattern[d, Blank[]]], Times[Optional[Pattern[e, Blank[]]], Pattern[x, Blank[]]]]], Optional[Pattern[f, Blank[]]]], Optional[Pattern[n2, Blank[]]]]]], Pattern[p, Blank[]]]], Pattern[x, Blank[Symbol]]] := -Dist[f/e, Subst[Int[((x/f)^m*(a + b*x^n + c*x^(2*n))^p)/(f^2 + x^2), x], x, f*Cot[d + e*x]], x] /; FreeQ[{a, b, c, d, e, f, m, n, p}, x] && EqQ[n2, 2*n] && NeQ[b^2 - 4*a*c, 0]
  3702. Int[Times[Power[cot[Plus[Optional[Pattern[d, Blank[]]], Times[Optional[Pattern[e, Blank[]]], Pattern[x, Blank[]]]]], Optional[Pattern[m, Blank[]]]], Power[Plus[Optional[Pattern[a, Blank[]]], Times[Optional[Pattern[b, Blank[]]], Power[tan[Plus[Optional[Pattern[d, Blank[]]], Times[Optional[Pattern[e, Blank[]]], Pattern[x, Blank[]]]]], Optional[Pattern[n, Blank[]]]]], Times[Optional[Pattern[c, Blank[]]], Power[tan[Plus[Optional[Pattern[d, Blank[]]], Times[Optional[Pattern[e, Blank[]]], Pattern[x, Blank[]]]]], Optional[Pattern[n2, Blank[]]]]]], Optional[Pattern[p, Blank[]]]]], Pattern[x, Blank[Symbol]]] := Dist[1/(4^p*c^p), Int[Cot[d + e*x]^m*(b + 2*c*Tan[d + e*x]^n)^(2*p), x], x] /; FreeQ[{a, b, c, d, e, m, n}, x] && EqQ[n2, 2*n] && EqQ[b^2 - 4*a*c, 0] && IntegerQ[p]
  3703. Int[Times[Power[Plus[Optional[Pattern[a, Blank[]]], Times[Power[cot[Plus[Optional[Pattern[d, Blank[]]], Times[Optional[Pattern[e, Blank[]]], Pattern[x, Blank[]]]]], Optional[Pattern[n, Blank[]]]], Optional[Pattern[b, Blank[]]]], Times[Power[cot[Plus[Optional[Pattern[d, Blank[]]], Times[Optional[Pattern[e, Blank[]]], Pattern[x, Blank[]]]]], Optional[Pattern[n2, Blank[]]]], Optional[Pattern[c, Blank[]]]]], Optional[Pattern[p, Blank[]]]], Power[tan[Plus[Optional[Pattern[d, Blank[]]], Times[Optional[Pattern[e, Blank[]]], Pattern[x, Blank[]]]]], Optional[Pattern[m, Blank[]]]]], Pattern[x, Blank[Symbol]]] := Dist[1/(4^p*c^p), Int[Tan[d + e*x]^m*(b + 2*c*Cot[d + e*x]^n)^(2*p), x], x] /; FreeQ[{a, b, c, d, e, m, n}, x] && EqQ[n2, 2*n] && EqQ[b^2 - 4*a*c, 0] && IntegerQ[p]
  3704. Int[Times[Power[cot[Plus[Optional[Pattern[d, Blank[]]], Times[Optional[Pattern[e, Blank[]]], Pattern[x, Blank[]]]]], Optional[Pattern[m, Blank[]]]], Power[Plus[Optional[Pattern[a, Blank[]]], Times[Optional[Pattern[b, Blank[]]], Power[tan[Plus[Optional[Pattern[d, Blank[]]], Times[Optional[Pattern[e, Blank[]]], Pattern[x, Blank[]]]]], Optional[Pattern[n, Blank[]]]]], Times[Optional[Pattern[c, Blank[]]], Power[tan[Plus[Optional[Pattern[d, Blank[]]], Times[Optional[Pattern[e, Blank[]]], Pattern[x, Blank[]]]]], Optional[Pattern[n2, Blank[]]]]]], Pattern[p, Blank[]]]], Pattern[x, Blank[Symbol]]] := Dist[(a + b*Tan[d + e*x]^n + c*Tan[d + e*x]^(2*n))^p/(b + 2*c*Tan[d + e*x]^n)^(2*p), Int[Cot[d + e*x]^m*(b + 2*c*Tan[d + e*x]^n)^(2*p), x], x] /; FreeQ[{a, b, c, d, e, m, n, p}, x] && EqQ[n2, 2*n] && EqQ[b^2 - 4*a*c, 0] && !IntegerQ[p]
  3705. Int[Times[Power[Plus[Optional[Pattern[a, Blank[]]], Times[Power[cot[Plus[Optional[Pattern[d, Blank[]]], Times[Optional[Pattern[e, Blank[]]], Pattern[x, Blank[]]]]], Optional[Pattern[n, Blank[]]]], Optional[Pattern[b, Blank[]]]], Times[Power[cot[Plus[Optional[Pattern[d, Blank[]]], Times[Optional[Pattern[e, Blank[]]], Pattern[x, Blank[]]]]], Optional[Pattern[n2, Blank[]]]], Optional[Pattern[c, Blank[]]]]], Pattern[p, Blank[]]], Power[tan[Plus[Optional[Pattern[d, Blank[]]], Times[Optional[Pattern[e, Blank[]]], Pattern[x, Blank[]]]]], Optional[Pattern[m, Blank[]]]]], Pattern[x, Blank[Symbol]]] := Dist[(a + b*Cot[d + e*x]^n + c*Cot[d + e*x]^(2*n))^p/(b + 2*c*Cot[d + e*x]^n)^(2*p), Int[Tan[d + e*x]^m*(b + 2*c*Cot[d + e*x]^n)^(2*p), x], x] /; FreeQ[{a, b, c, d, e, m, n, p}, x] && EqQ[n2, 2*n] && EqQ[b^2 - 4*a*c, 0] && !IntegerQ[p]
  3706. Int[Times[Power[cot[Plus[Optional[Pattern[d, Blank[]]], Times[Optional[Pattern[e, Blank[]]], Pattern[x, Blank[]]]]], Optional[Pattern[m, Blank[]]]], Power[Plus[Optional[Pattern[a, Blank[]]], Times[Optional[Pattern[b, Blank[]]], Power[tan[Plus[Optional[Pattern[d, Blank[]]], Times[Optional[Pattern[e, Blank[]]], Pattern[x, Blank[]]]]], Pattern[n, Blank[]]]], Times[Optional[Pattern[c, Blank[]]], Power[tan[Plus[Optional[Pattern[d, Blank[]]], Times[Optional[Pattern[e, Blank[]]], Pattern[x, Blank[]]]]], Pattern[n2, Blank[]]]]], Optional[Pattern[p, Blank[]]]]], Pattern[x, Blank[Symbol]]] := Module[{g = FreeFactors[Cot[d + e*x], x]}, Dist[g/e, Subst[Int[((g*x)^(m - 2*n*p)*(c + b*(g*x)^n + a*(g*x)^(2*n))^p)/(1 + g^2*x^2), x], x, Cot[d + e*x]/g], x]] /; FreeQ[{a, b, c, d, e, m, p}, x] && EqQ[n2, 2*n] && NeQ[b^2 - 4*a*c, 0] && IntegerQ[n/2]
  3707. Int[Times[Power[Plus[Optional[Pattern[a, Blank[]]], Times[Power[cot[Plus[Optional[Pattern[d, Blank[]]], Times[Optional[Pattern[e, Blank[]]], Pattern[x, Blank[]]]]], Pattern[n, Blank[]]], Optional[Pattern[b, Blank[]]]], Times[Power[cot[Plus[Optional[Pattern[d, Blank[]]], Times[Optional[Pattern[e, Blank[]]], Pattern[x, Blank[]]]]], Pattern[n2, Blank[]]], Optional[Pattern[c, Blank[]]]]], Optional[Pattern[p, Blank[]]]], Power[tan[Plus[Optional[Pattern[d, Blank[]]], Times[Optional[Pattern[e, Blank[]]], Pattern[x, Blank[]]]]], Optional[Pattern[m, Blank[]]]]], Pattern[x, Blank[Symbol]]] := Module[{g = FreeFactors[Tan[d + e*x], x]}, -Dist[g/e, Subst[Int[((g*x)^(m - 2*n*p)*(c + b*(g*x)^n + a*(g*x)^(2*n))^p)/(1 + g^2*x^2), x], x, Tan[d + e*x]/g], x]] /; FreeQ[{a, b, c, d, e, m, p}, x] && EqQ[n2, 2*n] && NeQ[b^2 - 4*a*c, 0] && IntegerQ[n/2]
  3708. Int[Times[Plus[Pattern[A, Blank[]], Times[Optional[Pattern[B, Blank[]]], tan[Plus[Optional[Pattern[d, Blank[]]], Times[Optional[Pattern[e, Blank[]]], Pattern[x, Blank[]]]]]]], Power[Plus[Pattern[a, Blank[]], Times[Optional[Pattern[b, Blank[]]], tan[Plus[Optional[Pattern[d, Blank[]]], Times[Optional[Pattern[e, Blank[]]], Pattern[x, Blank[]]]]]], Times[Optional[Pattern[c, Blank[]]], Power[tan[Plus[Optional[Pattern[d, Blank[]]], Times[Optional[Pattern[e, Blank[]]], Pattern[x, Blank[]]]]], 2]]], Pattern[n, Blank[]]]], Pattern[x, Blank[Symbol]]] := Dist[1/(4^n*c^n), Int[(A + B*Tan[d + e*x])*(b + 2*c*Tan[d + e*x])^(2*n), x], x] /; FreeQ[{a, b, c, d, e, A, B}, x] && EqQ[b^2 - 4*a*c, 0] && IntegerQ[n]
  3709. Int[Times[Power[Plus[Times[cot[Plus[Optional[Pattern[d, Blank[]]], Times[Optional[Pattern[e, Blank[]]], Pattern[x, Blank[]]]]], Optional[Pattern[b, Blank[]]]], Times[Power[cot[Plus[Optional[Pattern[d, Blank[]]], Times[Optional[Pattern[e, Blank[]]], Pattern[x, Blank[]]]]], 2], Optional[Pattern[c, Blank[]]]], Pattern[a, Blank[]]], Pattern[n, Blank[]]], Plus[Times[cot[Plus[Optional[Pattern[d, Blank[]]], Times[Optional[Pattern[e, Blank[]]], Pattern[x, Blank[]]]]], Optional[Pattern[B, Blank[]]]], Pattern[A, Blank[]]]], Pattern[x, Blank[Symbol]]] := Dist[1/(4^n*c^n), Int[(A + B*Cot[d + e*x])*(b + 2*c*Cot[d + e*x])^(2*n), x], x] /; FreeQ[{a, b, c, d, e, A, B}, x] && EqQ[b^2 - 4*a*c, 0] && IntegerQ[n]
  3710. Int[Times[Plus[Pattern[A, Blank[]], Times[Optional[Pattern[B, Blank[]]], tan[Plus[Optional[Pattern[d, Blank[]]], Times[Optional[Pattern[e, Blank[]]], Pattern[x, Blank[]]]]]]], Power[Plus[Pattern[a, Blank[]], Times[Optional[Pattern[b, Blank[]]], tan[Plus[Optional[Pattern[d, Blank[]]], Times[Optional[Pattern[e, Blank[]]], Pattern[x, Blank[]]]]]], Times[Optional[Pattern[c, Blank[]]], Power[tan[Plus[Optional[Pattern[d, Blank[]]], Times[Optional[Pattern[e, Blank[]]], Pattern[x, Blank[]]]]], 2]]], Pattern[n, Blank[]]]], Pattern[x, Blank[Symbol]]] := Dist[(a + b*Tan[d + e*x] + c*Tan[d + e*x]^2)^n/(b + 2*c*Tan[d + e*x])^(2*n), Int[(A + B*Tan[d + e*x])*(b + 2*c*Tan[d + e*x])^(2*n), x], x] /; FreeQ[{a, b, c, d, e, A, B}, x] && EqQ[b^2 - 4*a*c, 0] && !IntegerQ[n]
  3711. Int[Times[Power[Plus[Times[cot[Plus[Optional[Pattern[d, Blank[]]], Times[Optional[Pattern[e, Blank[]]], Pattern[x, Blank[]]]]], Optional[Pattern[b, Blank[]]]], Times[Power[cot[Plus[Optional[Pattern[d, Blank[]]], Times[Optional[Pattern[e, Blank[]]], Pattern[x, Blank[]]]]], 2], Optional[Pattern[c, Blank[]]]], Pattern[a, Blank[]]], Pattern[n, Blank[]]], Plus[Times[cot[Plus[Optional[Pattern[d, Blank[]]], Times[Optional[Pattern[e, Blank[]]], Pattern[x, Blank[]]]]], Optional[Pattern[B, Blank[]]]], Pattern[A, Blank[]]]], Pattern[x, Blank[Symbol]]] := Dist[(a + b*Cot[d + e*x] + c*Cot[d + e*x]^2)^n/(b + 2*c*Cot[d + e*x])^(2*n), Int[(A + B*Cot[d + e*x])*(b + 2*c*Cot[d + e*x])^(2*n), x], x] /; FreeQ[{a, b, c, d, e, A, B}, x] && EqQ[b^2 - 4*a*c, 0] && !IntegerQ[n]
  3712. Int[Times[Plus[Pattern[A, Blank[]], Times[Optional[Pattern[B, Blank[]]], tan[Plus[Optional[Pattern[d, Blank[]]], Times[Optional[Pattern[e, Blank[]]], Pattern[x, Blank[]]]]]]], Power[Plus[Optional[Pattern[a, Blank[]]], Times[Optional[Pattern[b, Blank[]]], tan[Plus[Optional[Pattern[d, Blank[]]], Times[Optional[Pattern[e, Blank[]]], Pattern[x, Blank[]]]]]], Times[Optional[Pattern[c, Blank[]]], Power[tan[Plus[Optional[Pattern[d, Blank[]]], Times[Optional[Pattern[e, Blank[]]], Pattern[x, Blank[]]]]], 2]]], -1]], Pattern[x, Blank[Symbol]]] := Module[{q = Rt[b^2 - 4*a*c, 2]}, Dist[B + (b*B - 2*A*c)/q, Int[1/Simp[b + q + 2*c*Tan[d + e*x], x], x], x] + Dist[B - (b*B - 2*A*c)/q, Int[1/Simp[b - q + 2*c*Tan[d + e*x], x], x], x]] /; FreeQ[{a, b, c, d, e, A, B}, x] && NeQ[b^2 - 4*a*c, 0]
  3713. Int[Times[Power[Plus[Optional[Pattern[a, Blank[]]], Times[cot[Plus[Optional[Pattern[d, Blank[]]], Times[Optional[Pattern[e, Blank[]]], Pattern[x, Blank[]]]]], Optional[Pattern[b, Blank[]]]], Times[Power[cot[Plus[Optional[Pattern[d, Blank[]]], Times[Optional[Pattern[e, Blank[]]], Pattern[x, Blank[]]]]], 2], Optional[Pattern[c, Blank[]]]]], -1], Plus[Times[cot[Plus[Optional[Pattern[d, Blank[]]], Times[Optional[Pattern[e, Blank[]]], Pattern[x, Blank[]]]]], Optional[Pattern[B, Blank[]]]], Pattern[A, Blank[]]]], Pattern[x, Blank[Symbol]]] := Module[{q = Rt[b^2 - 4*a*c, 2]}, Dist[B + (b*B - 2*A*c)/q, Int[1/Simp[b + q + 2*c*Cot[d + e*x], x], x], x] + Dist[B - (b*B - 2*A*c)/q, Int[1/Simp[b - q + 2*c*Cot[d + e*x], x], x], x]] /; FreeQ[{a, b, c, d, e, A, B}, x] && NeQ[b^2 - 4*a*c, 0]
  3714. Int[Times[Plus[Pattern[A, Blank[]], Times[Optional[Pattern[B, Blank[]]], tan[Plus[Optional[Pattern[d, Blank[]]], Times[Optional[Pattern[e, Blank[]]], Pattern[x, Blank[]]]]]]], Power[Plus[Optional[Pattern[a, Blank[]]], Times[Optional[Pattern[b, Blank[]]], tan[Plus[Optional[Pattern[d, Blank[]]], Times[Optional[Pattern[e, Blank[]]], Pattern[x, Blank[]]]]]], Times[Optional[Pattern[c, Blank[]]], Power[tan[Plus[Optional[Pattern[d, Blank[]]], Times[Optional[Pattern[e, Blank[]]], Pattern[x, Blank[]]]]], 2]]], Pattern[n, Blank[]]]], Pattern[x, Blank[Symbol]]] := Int[ExpandTrig[(A + B*tan[d + e*x])*(a + b*tan[d + e*x] + c*tan[d + e*x]^2)^n, x], x] /; FreeQ[{a, b, c, d, e, A, B}, x] && NeQ[b^2 - 4*a*c, 0] && IntegerQ[n]
  3715. Int[Times[Power[Plus[Optional[Pattern[a, Blank[]]], Times[cot[Plus[Optional[Pattern[d, Blank[]]], Times[Optional[Pattern[e, Blank[]]], Pattern[x, Blank[]]]]], Optional[Pattern[b, Blank[]]]], Times[Power[cot[Plus[Optional[Pattern[d, Blank[]]], Times[Optional[Pattern[e, Blank[]]], Pattern[x, Blank[]]]]], 2], Optional[Pattern[c, Blank[]]]]], Pattern[n, Blank[]]], Plus[Times[cot[Plus[Optional[Pattern[d, Blank[]]], Times[Optional[Pattern[e, Blank[]]], Pattern[x, Blank[]]]]], Optional[Pattern[B, Blank[]]]], Pattern[A, Blank[]]]], Pattern[x, Blank[Symbol]]] := Int[ExpandTrig[(A + B*cot[d + e*x])*(a + b*cot[d + e*x] + c*cot[d + e*x]^2)^n, x], x] /; FreeQ[{a, b, c, d, e, A, B}, x] && NeQ[b^2 - 4*a*c, 0] && IntegerQ[n]
  3716. Int[Times[Power[Plus[Optional[Pattern[c, Blank[]]], Times[Optional[Pattern[d, Blank[]]], Pattern[x, Blank[]]]], Optional[Pattern[m, Blank[]]]], tan[Plus[Optional[Pattern[e, Blank[]]], Times[Pi, Optional[Pattern[k, Blank[]]]], Times[Complex[0, Pattern[fz, Blank[]]], Optional[Pattern[f, Blank[]]], Pattern[x, Blank[]]]]]], Pattern[x, Blank[Symbol]]] := -Simp[(I*(c + d*x)^(m + 1))/(d*(m + 1)), x] + Dist[2*I, Int[((c + d*x)^m*E^(2*(-(I*e) + f*fz*x)))/(E^(2*I*k*Pi)*(1 + E^(2*(-(I*e) + f*fz*x))/E^(2*I*k*Pi))), x], x] /; FreeQ[{c, d, e, f, fz}, x] && IntegerQ[4*k] && IGtQ[m, 0]
  3717. Int[Times[Power[Plus[Optional[Pattern[c, Blank[]]], Times[Optional[Pattern[d, Blank[]]], Pattern[x, Blank[]]]], Optional[Pattern[m, Blank[]]]], tan[Plus[Optional[Pattern[e, Blank[]]], Times[Pi, Optional[Pattern[k, Blank[]]]], Times[Optional[Pattern[f, Blank[]]], Pattern[x, Blank[]]]]]], Pattern[x, Blank[Symbol]]] := Simp[(I*(c + d*x)^(m + 1))/(d*(m + 1)), x] - Dist[2*I, Int[((c + d*x)^m*E^(2*I*k*Pi)*E^(2*I*(e + f*x)))/(1 + E^(2*I*k*Pi)*E^(2*I*(e + f*x))), x], x] /; FreeQ[{c, d, e, f}, x] && IntegerQ[4*k] && IGtQ[m, 0]
  3718. Int[Times[Power[Plus[Optional[Pattern[c, Blank[]]], Times[Optional[Pattern[d, Blank[]]], Pattern[x, Blank[]]]], Optional[Pattern[m, Blank[]]]], tan[Plus[Optional[Pattern[e, Blank[]]], Times[Complex[0, Pattern[fz, Blank[]]], Optional[Pattern[f, Blank[]]], Pattern[x, Blank[]]]]]], Pattern[x, Blank[Symbol]]] := -Simp[(I*(c + d*x)^(m + 1))/(d*(m + 1)), x] + Dist[2*I, Int[((c + d*x)^m*E^(2*(-(I*e) + f*fz*x)))/(1 + E^(2*(-(I*e) + f*fz*x))), x], x] /; FreeQ[{c, d, e, f, fz}, x] && IGtQ[m, 0]
  3719. Int[Times[Power[Plus[Optional[Pattern[c, Blank[]]], Times[Optional[Pattern[d, Blank[]]], Pattern[x, Blank[]]]], Optional[Pattern[m, Blank[]]]], tan[Plus[Optional[Pattern[e, Blank[]]], Times[Optional[Pattern[f, Blank[]]], Pattern[x, Blank[]]]]]], Pattern[x, Blank[Symbol]]] := Simp[(I*(c + d*x)^(m + 1))/(d*(m + 1)), x] - Dist[2*I, Int[((c + d*x)^m*E^(2*I*(e + f*x)))/(1 + E^(2*I*(e + f*x))), x], x] /; FreeQ[{c, d, e, f}, x] && IGtQ[m, 0]
  3720. Int[Times[Power[Plus[Optional[Pattern[c, Blank[]]], Times[Optional[Pattern[d, Blank[]]], Pattern[x, Blank[]]]], Optional[Pattern[m, Blank[]]]], Power[Times[Optional[Pattern[b, Blank[]]], tan[Plus[Optional[Pattern[e, Blank[]]], Times[Optional[Pattern[f, Blank[]]], Pattern[x, Blank[]]]]]], Pattern[n, Blank[]]]], Pattern[x, Blank[Symbol]]] := Simp[(b*(c + d*x)^m*(b*Tan[e + f*x])^(n - 1))/(f*(n - 1)), x] + (-Dist[(b*d*m)/(f*(n - 1)), Int[(c + d*x)^(m - 1)*(b*Tan[e + f*x])^(n - 1), x], x] - Dist[b^2, Int[(c + d*x)^m*(b*Tan[e + f*x])^(n - 2), x], x]) /; FreeQ[{b, c, d, e, f}, x] && GtQ[n, 1] && GtQ[m, 0]
  3721. Int[Times[Power[Plus[Optional[Pattern[c, Blank[]]], Times[Optional[Pattern[d, Blank[]]], Pattern[x, Blank[]]]], Optional[Pattern[m, Blank[]]]], Power[Times[Optional[Pattern[b, Blank[]]], tan[Plus[Optional[Pattern[e, Blank[]]], Times[Optional[Pattern[f, Blank[]]], Pattern[x, Blank[]]]]]], Pattern[n, Blank[]]]], Pattern[x, Blank[Symbol]]] := Simp[((c + d*x)^m*(b*Tan[e + f*x])^(n + 1))/(b*f*(n + 1)), x] + (-Dist[(d*m)/(b*f*(n + 1)), Int[(c + d*x)^(m - 1)*(b*Tan[e + f*x])^(n + 1), x], x] - Dist[1/b^2, Int[(c + d*x)^m*(b*Tan[e + f*x])^(n + 2), x], x]) /; FreeQ[{b, c, d, e, f}, x] && LtQ[n, -1] && GtQ[m, 0]
  3722. Int[Times[Power[Plus[Optional[Pattern[c, Blank[]]], Times[Optional[Pattern[d, Blank[]]], Pattern[x, Blank[]]]], Optional[Pattern[m, Blank[]]]], Power[Plus[Pattern[a, Blank[]], Times[Optional[Pattern[b, Blank[]]], tan[Plus[Optional[Pattern[e, Blank[]]], Times[Optional[Pattern[f, Blank[]]], Pattern[x, Blank[]]]]]]], Optional[Pattern[n, Blank[]]]]], Pattern[x, Blank[Symbol]]] := Int[ExpandIntegrand[(c + d*x)^m, (a + b*Tan[e + f*x])^n, x], x] /; FreeQ[{a, b, c, d, e, f, m}, x] && IGtQ[m, 0] && IGtQ[n, 0]
  3723. Int[Times[Power[Plus[Optional[Pattern[c, Blank[]]], Times[Optional[Pattern[d, Blank[]]], Pattern[x, Blank[]]]], Optional[Pattern[m, Blank[]]]], Power[Plus[Pattern[a, Blank[]], Times[Optional[Pattern[b, Blank[]]], tan[Plus[Optional[Pattern[e, Blank[]]], Times[Optional[Pattern[f, Blank[]]], Pattern[x, Blank[]]]]]]], -1]], Pattern[x, Blank[Symbol]]] := Simp[(c + d*x)^(m + 1)/(2*a*d*(m + 1)), x] + (Dist[(a*d*m)/(2*b*f), Int[(c + d*x)^(m - 1)/(a + b*Tan[e + f*x]), x], x] - Simp[(a*(c + d*x)^m)/(2*b*f*(a + b*Tan[e + f*x])), x]) /; FreeQ[{a, b, c, d, e, f}, x] && EqQ[a^2 + b^2, 0] && GtQ[m, 0]
  3724. Int[Times[Power[Plus[Optional[Pattern[c, Blank[]]], Times[Optional[Pattern[d, Blank[]]], Pattern[x, Blank[]]]], -2], Power[Plus[Pattern[a, Blank[]], Times[Optional[Pattern[b, Blank[]]], tan[Plus[Optional[Pattern[e, Blank[]]], Times[Optional[Pattern[f, Blank[]]], Pattern[x, Blank[]]]]]]], -1]], Pattern[x, Blank[Symbol]]] := -Simp[(d*(c + d*x)*(a + b*Tan[e + f*x]))^(-1), x] + (-Dist[f/(a*d), Int[Sin[2*e + 2*f*x]/(c + d*x), x], x] + Dist[f/(b*d), Int[Cos[2*e + 2*f*x]/(c + d*x), x], x]) /; FreeQ[{a, b, c, d, e, f}, x] && EqQ[a^2 + b^2, 0]
  3725. Int[Times[Power[Plus[Optional[Pattern[c, Blank[]]], Times[Optional[Pattern[d, Blank[]]], Pattern[x, Blank[]]]], Pattern[m, Blank[]]], Power[Plus[Pattern[a, Blank[]], Times[Optional[Pattern[b, Blank[]]], tan[Plus[Optional[Pattern[e, Blank[]]], Times[Optional[Pattern[f, Blank[]]], Pattern[x, Blank[]]]]]]], -1]], Pattern[x, Blank[Symbol]]] := Simp[(f*(c + d*x)^(m + 2))/(b*d^2*(m + 1)*(m + 2)), x] + (Dist[(2*b*f)/(a*d*(m + 1)), Int[(c + d*x)^(m + 1)/(a + b*Tan[e + f*x]), x], x] + Simp[(c + d*x)^(m + 1)/(d*(m + 1)*(a + b*Tan[e + f*x])), x]) /; FreeQ[{a, b, c, d, e, f}, x] && EqQ[a^2 + b^2, 0] && LtQ[m, -1] && NeQ[m, -2]
  3726. Int[Times[Power[Plus[Optional[Pattern[c, Blank[]]], Times[Optional[Pattern[d, Blank[]]], Pattern[x, Blank[]]]], -1], Power[Plus[Pattern[a, Blank[]], Times[Optional[Pattern[b, Blank[]]], tan[Plus[Optional[Pattern[e, Blank[]]], Times[Optional[Pattern[f, Blank[]]], Pattern[x, Blank[]]]]]]], -1]], Pattern[x, Blank[Symbol]]] := Simp[Log[c + d*x]/(2*a*d), x] + (Dist[1/(2*a), Int[Cos[2*e + 2*f*x]/(c + d*x), x], x] + Dist[1/(2*b), Int[Sin[2*e + 2*f*x]/(c + d*x), x], x]) /; FreeQ[{a, b, c, d, e, f}, x] && EqQ[a^2 + b^2, 0]
  3727. Int[Times[Power[Plus[Optional[Pattern[c, Blank[]]], Times[Optional[Pattern[d, Blank[]]], Pattern[x, Blank[]]]], Pattern[m, Blank[]]], Power[Plus[Pattern[a, Blank[]], Times[Optional[Pattern[b, Blank[]]], tan[Plus[Optional[Pattern[e, Blank[]]], Times[Optional[Pattern[f, Blank[]]], Pattern[x, Blank[]]]]]]], -1]], Pattern[x, Blank[Symbol]]] := Simp[(c + d*x)^(m + 1)/(2*a*d*(m + 1)), x] + Dist[1/(2*a), Int[(c + d*x)^m*E^((2*a*(e + f*x))/b), x], x] /; FreeQ[{a, b, c, d, e, f, m}, x] && EqQ[a^2 + b^2, 0] && !IntegerQ[m]
  3728. Int[Times[Power[Plus[Optional[Pattern[c, Blank[]]], Times[Optional[Pattern[d, Blank[]]], Pattern[x, Blank[]]]], Pattern[m, Blank[]]], Power[Plus[Pattern[a, Blank[]], Times[Optional[Pattern[b, Blank[]]], tan[Plus[Optional[Pattern[e, Blank[]]], Times[Optional[Pattern[f, Blank[]]], Pattern[x, Blank[]]]]]]], Pattern[n, Blank[]]]], Pattern[x, Blank[Symbol]]] := Int[ExpandIntegrand[(c + d*x)^m, (1/(2*a) + Cos[2*e + 2*f*x]/(2*a) + Sin[2*e + 2*f*x]/(2*b))^(-n), x], x] /; FreeQ[{a, b, c, d, e, f}, x] && EqQ[a^2 + b^2, 0] && ILtQ[m, 0] && ILtQ[n, 0]
  3729. Int[Times[Power[Plus[Optional[Pattern[c, Blank[]]], Times[Optional[Pattern[d, Blank[]]], Pattern[x, Blank[]]]], Pattern[m, Blank[]]], Power[Plus[Pattern[a, Blank[]], Times[Optional[Pattern[b, Blank[]]], tan[Plus[Optional[Pattern[e, Blank[]]], Times[Optional[Pattern[f, Blank[]]], Pattern[x, Blank[]]]]]]], Pattern[n, Blank[]]]], Pattern[x, Blank[Symbol]]] := Int[ExpandIntegrand[(c + d*x)^m, (1/(2*a) + E^((2*a*(e + f*x))/b)/(2*a))^(-n), x], x] /; FreeQ[{a, b, c, d, e, f, m}, x] && EqQ[a^2 + b^2, 0] && ILtQ[n, 0]
  3730. Int[Times[Power[Plus[Optional[Pattern[c, Blank[]]], Times[Optional[Pattern[d, Blank[]]], Pattern[x, Blank[]]]], Optional[Pattern[m, Blank[]]]], Power[Plus[Pattern[a, Blank[]], Times[Optional[Pattern[b, Blank[]]], tan[Plus[Optional[Pattern[e, Blank[]]], Times[Optional[Pattern[f, Blank[]]], Pattern[x, Blank[]]]]]]], Pattern[n, Blank[]]]], Pattern[x, Blank[Symbol]]] := With[{u = IntHide[(a + b*Tan[e + f*x])^n, x]}, Dist[(c + d*x)^m, u, x] - Dist[d*m, Int[Dist[(c + d*x)^(m - 1), u, x], x], x]] /; FreeQ[{a, b, c, d, e, f}, x] && EqQ[a^2 + b^2, 0] && ILtQ[n, -1] && GtQ[m, 0]
  3731. Int[Times[Power[Plus[Optional[Pattern[c, Blank[]]], Times[Optional[Pattern[d, Blank[]]], Pattern[x, Blank[]]]], Optional[Pattern[m, Blank[]]]], Power[Plus[Pattern[a, Blank[]], Times[Optional[Pattern[b, Blank[]]], tan[Plus[Optional[Pattern[e, Blank[]]], Times[Pi, Optional[Pattern[k, Blank[]]]], Times[Optional[Pattern[f, Blank[]]], Pattern[x, Blank[]]]]]]], -1]], Pattern[x, Blank[Symbol]]] := Simp[(c + d*x)^(m + 1)/(d*(m + 1)*(a + I*b)), x] + Dist[2*I*b, Int[((c + d*x)^m*E^(2*I*k*Pi)*E^Simp[2*I*(e + f*x), x])/((a + I*b)^2 + (a^2 + b^2)*E^(2*I*k*Pi)*E^Simp[2*I*(e + f*x), x]), x], x] /; FreeQ[{a, b, c, d, e, f}, x] && IntegerQ[4*k] && NeQ[a^2 + b^2, 0] && IGtQ[m, 0]
  3732. Int[Times[Power[Plus[Optional[Pattern[c, Blank[]]], Times[Optional[Pattern[d, Blank[]]], Pattern[x, Blank[]]]], Optional[Pattern[m, Blank[]]]], Power[Plus[Pattern[a, Blank[]], Times[Optional[Pattern[b, Blank[]]], tan[Plus[Optional[Pattern[e, Blank[]]], Times[Optional[Pattern[f, Blank[]]], Pattern[x, Blank[]]]]]]], -1]], Pattern[x, Blank[Symbol]]] := Simp[(c + d*x)^(m + 1)/(d*(m + 1)*(a + I*b)), x] + Dist[2*I*b, Int[((c + d*x)^m*E^Simp[2*I*(e + f*x), x])/((a + I*b)^2 + (a^2 + b^2)*E^Simp[2*I*(e + f*x), x]), x], x] /; FreeQ[{a, b, c, d, e, f}, x] && NeQ[a^2 + b^2, 0] && IGtQ[m, 0]
  3733. Int[Times[Plus[Optional[Pattern[c, Blank[]]], Times[Optional[Pattern[d, Blank[]]], Pattern[x, Blank[]]]], Power[Plus[Pattern[a, Blank[]], Times[Optional[Pattern[b, Blank[]]], tan[Plus[Optional[Pattern[e, Blank[]]], Times[Optional[Pattern[f, Blank[]]], Pattern[x, Blank[]]]]]]], -2]], Pattern[x, Blank[Symbol]]] := -Simp[(c + d*x)^2/(2*d*(a^2 + b^2)), x] + (Dist[1/(f*(a^2 + b^2)), Int[(b*d + 2*a*c*f + 2*a*d*f*x)/(a + b*Tan[e + f*x]), x], x] - Simp[(b*(c + d*x))/(f*(a^2 + b^2)*(a + b*Tan[e + f*x])), x]) /; FreeQ[{a, b, c, d, e, f}, x] && NeQ[a^2 + b^2, 0]
  3734. Int[Times[Power[Plus[Optional[Pattern[c, Blank[]]], Times[Optional[Pattern[d, Blank[]]], Pattern[x, Blank[]]]], Optional[Pattern[m, Blank[]]]], Power[Plus[Pattern[a, Blank[]], Times[Optional[Pattern[b, Blank[]]], tan[Plus[Optional[Pattern[e, Blank[]]], Times[Optional[Pattern[f, Blank[]]], Pattern[x, Blank[]]]]]]], Pattern[n, Blank[]]]], Pattern[x, Blank[Symbol]]] := Int[ExpandIntegrand[(c + d*x)^m, (1/(a - I*b) - (2*I*b)/(a^2 + b^2 + (a - I*b)^2*E^(2*I*(e + f*x))))^(-n), x], x] /; FreeQ[{a, b, c, d, e, f}, x] && NeQ[a^2 + b^2, 0] && ILtQ[n, 0] && IGtQ[m, 0]
  3735. Int[Times[Power[Plus[Optional[Pattern[c, Blank[]]], Times[Optional[Pattern[d, Blank[]]], Pattern[x, Blank[]]]], Optional[Pattern[m, Blank[]]]], Power[tan[Plus[Optional[Pattern[e, Blank[]]], Times[Optional[Pattern[f, Blank[]]], Pattern[x, Blank[]]]]], Optional[Pattern[n, Blank[]]]]], Pattern[x, Blank[Symbol]]] := Simp[If[MatchQ[f, (f1_.)*(Complex[0, j_])], If[MatchQ[e, (e1_.) + Pi/2], I^n*Unintegrable[(c + d*x)^m*Coth[-(I*(e - Pi/2)) - I*f*x]^n, x], I^n*Unintegrable[(c + d*x)^m*Tanh[-(I*e) - I*f*x]^n, x]], If[MatchQ[e, (e1_.) + Pi/2], (-1)^n*Unintegrable[(c + d*x)^m*Cot[e - Pi/2 + f*x]^n, x], Unintegrable[(c + d*x)^m*Tan[e + f*x]^n, x]]], x] /; FreeQ[{c, d, e, f, m, n}, x] && IntegerQ[n]
  3736. Int[Times[Power[Plus[Optional[Pattern[c, Blank[]]], Times[Optional[Pattern[d, Blank[]]], Pattern[x, Blank[]]]], Optional[Pattern[m, Blank[]]]], Power[Plus[Optional[Pattern[a, Blank[]]], Times[Optional[Pattern[b, Blank[]]], tan[Plus[Optional[Pattern[e, Blank[]]], Times[Optional[Pattern[f, Blank[]]], Pattern[x, Blank[]]]]]]], Optional[Pattern[n, Blank[]]]]], Pattern[x, Blank[Symbol]]] := Unintegrable[(c + d*x)^m*(a + b*Tan[e + f*x])^n, x] /; FreeQ[{a, b, c, d, e, f, m, n}, x]
  3737. Int[Times[Power[Pattern[u, Blank[]], Optional[Pattern[m, Blank[]]]], Power[Plus[Optional[Pattern[a, Blank[]]], Times[Optional[Pattern[b, Blank[]]], Tan[Pattern[v, Blank[]]]]], Optional[Pattern[n, Blank[]]]]], Pattern[x, Blank[Symbol]]] := Int[ExpandToSum[u, x]^m*(a + b*Tan[ExpandToSum[v, x]])^n, x] /; FreeQ[{a, b, m, n}, x] && LinearQ[{u, v}, x] && !LinearMatchQ[{u, v}, x]
  3738. Int[Times[Power[Plus[Optional[Pattern[a, Blank[]]], Times[Cot[Pattern[v, Blank[]]], Optional[Pattern[b, Blank[]]]]], Optional[Pattern[n, Blank[]]]], Power[Pattern[u, Blank[]], Optional[Pattern[m, Blank[]]]]], Pattern[x, Blank[Symbol]]] := Int[ExpandToSum[u, x]^m*(a + b*Cot[ExpandToSum[v, x]])^n, x] /; FreeQ[{a, b, m, n}, x] && LinearQ[{u, v}, x] && !LinearMatchQ[{u, v}, x]
  3739. Int[Power[Plus[Optional[Pattern[a, Blank[]]], Times[Optional[Pattern[b, Blank[]]], Tan[Plus[Optional[Pattern[c, Blank[]]], Times[Optional[Pattern[d, Blank[]]], Power[Pattern[x, Blank[]], Pattern[n, Blank[]]]]]]]], Optional[Pattern[p, Blank[]]]], Pattern[x, Blank[Symbol]]] := Dist[1/n, Subst[Int[x^(1/n - 1)*(a + b*Tan[c + d*x])^p, x], x, x^n], x] /; FreeQ[{a, b, c, d, p}, x] && IGtQ[1/n, 0] && IntegerQ[p]
  3740. Int[Power[Plus[Optional[Pattern[a, Blank[]]], Times[Cot[Plus[Optional[Pattern[c, Blank[]]], Times[Optional[Pattern[d, Blank[]]], Power[Pattern[x, Blank[]], Pattern[n, Blank[]]]]]], Optional[Pattern[b, Blank[]]]]], Optional[Pattern[p, Blank[]]]], Pattern[x, Blank[Symbol]]] := Dist[1/n, Subst[Int[x^(1/n - 1)*(a + b*Cot[c + d*x])^p, x], x, x^n], x] /; FreeQ[{a, b, c, d, p}, x] && IGtQ[1/n, 0] && IntegerQ[p]
  3741. Int[Power[Plus[Optional[Pattern[a, Blank[]]], Times[Optional[Pattern[b, Blank[]]], Tan[Plus[Optional[Pattern[c, Blank[]]], Times[Optional[Pattern[d, Blank[]]], Power[Pattern[x, Blank[]], Pattern[n, Blank[]]]]]]]], Optional[Pattern[p, Blank[]]]], Pattern[x, Blank[Symbol]]] := Unintegrable[(a + b*Tan[c + d*x^n])^p, x] /; FreeQ[{a, b, c, d, n, p}, x]
  3742. Int[Power[Plus[Optional[Pattern[a, Blank[]]], Times[Cot[Plus[Optional[Pattern[c, Blank[]]], Times[Optional[Pattern[d, Blank[]]], Power[Pattern[x, Blank[]], Pattern[n, Blank[]]]]]], Optional[Pattern[b, Blank[]]]]], Optional[Pattern[p, Blank[]]]], Pattern[x, Blank[Symbol]]] := Unintegrable[(a + b*Cot[c + d*x^n])^p, x] /; FreeQ[{a, b, c, d, n, p}, x]
  3743. Int[Power[Plus[Optional[Pattern[a, Blank[]]], Times[Optional[Pattern[b, Blank[]]], Tan[Plus[Optional[Pattern[c, Blank[]]], Times[Optional[Pattern[d, Blank[]]], Power[Pattern[u, Blank[]], Pattern[n, Blank[]]]]]]]], Optional[Pattern[p, Blank[]]]], Pattern[x, Blank[Symbol]]] := Dist[1/Coefficient[u, x, 1], Subst[Int[(a + b*Tan[c + d*x^n])^p, x], x, u], x] /; FreeQ[{a, b, c, d, n, p}, x] && LinearQ[u, x] && NeQ[u, x]
  3744. Int[Power[Plus[Optional[Pattern[a, Blank[]]], Times[Cot[Plus[Optional[Pattern[c, Blank[]]], Times[Optional[Pattern[d, Blank[]]], Power[Pattern[u, Blank[]], Pattern[n, Blank[]]]]]], Optional[Pattern[b, Blank[]]]]], Optional[Pattern[p, Blank[]]]], Pattern[x, Blank[Symbol]]] := Dist[1/Coefficient[u, x, 1], Subst[Int[(a + b*Cot[c + d*x^n])^p, x], x, u], x] /; FreeQ[{a, b, c, d, n, p}, x] && LinearQ[u, x] && NeQ[u, x]
  3745. Int[Power[Plus[Optional[Pattern[a, Blank[]]], Times[Optional[Pattern[b, Blank[]]], Tan[Pattern[u, Blank[]]]]], Optional[Pattern[p, Blank[]]]], Pattern[x, Blank[Symbol]]] := Int[(a + b*Tan[ExpandToSum[u, x]])^p, x] /; FreeQ[{a, b, p}, x] && BinomialQ[u, x] && !BinomialMatchQ[u, x]
  3746. Int[Power[Plus[Optional[Pattern[a, Blank[]]], Times[Cot[Pattern[u, Blank[]]], Optional[Pattern[b, Blank[]]]]], Optional[Pattern[p, Blank[]]]], Pattern[x, Blank[Symbol]]] := Int[(a + b*Cot[ExpandToSum[u, x]])^p, x] /; FreeQ[{a, b, p}, x] && BinomialQ[u, x] && !BinomialMatchQ[u, x]
  3747. Int[Times[Power[Pattern[x, Blank[]], Optional[Pattern[m, Blank[]]]], Power[Plus[Optional[Pattern[a, Blank[]]], Times[Optional[Pattern[b, Blank[]]], Tan[Plus[Optional[Pattern[c, Blank[]]], Times[Optional[Pattern[d, Blank[]]], Power[Pattern[x, Blank[]], Pattern[n, Blank[]]]]]]]], Optional[Pattern[p, Blank[]]]]], Pattern[x, Blank[Symbol]]] := Dist[1/n, Subst[Int[x^(Simplify[(m + 1)/n] - 1)*(a + b*Tan[c + d*x])^p, x], x, x^n], x] /; FreeQ[{a, b, c, d, m, n, p}, x] && IGtQ[Simplify[(m + 1)/n], 0] && IntegerQ[p]
  3748. Int[Times[Power[Plus[Optional[Pattern[a, Blank[]]], Times[Cot[Plus[Optional[Pattern[c, Blank[]]], Times[Optional[Pattern[d, Blank[]]], Power[Pattern[x, Blank[]], Pattern[n, Blank[]]]]]], Optional[Pattern[b, Blank[]]]]], Optional[Pattern[p, Blank[]]]], Power[Pattern[x, Blank[]], Optional[Pattern[m, Blank[]]]]], Pattern[x, Blank[Symbol]]] := Dist[1/n, Subst[Int[x^(Simplify[(m + 1)/n] - 1)*(a + b*Cot[c + d*x])^p, x], x, x^n], x] /; FreeQ[{a, b, c, d, m, n, p}, x] && IGtQ[Simplify[(m + 1)/n], 0] && IntegerQ[p]
  3749. Int[Times[Power[Pattern[x, Blank[]], Optional[Pattern[m, Blank[]]]], Power[Tan[Plus[Optional[Pattern[c, Blank[]]], Times[Optional[Pattern[d, Blank[]]], Power[Pattern[x, Blank[]], Pattern[n, Blank[]]]]]], 2]], Pattern[x, Blank[Symbol]]] := Simp[(x^(m - n + 1)*Tan[c + d*x^n])/(d*n), x] + (-Dist[(m - n + 1)/(d*n), Int[x^(m - n)*Tan[c + d*x^n], x], x] - Int[x^m, x]) /; FreeQ[{c, d, m, n}, x]
  3750. Int[Times[Power[Cot[Plus[Optional[Pattern[c, Blank[]]], Times[Optional[Pattern[d, Blank[]]], Power[Pattern[x, Blank[]], Pattern[n, Blank[]]]]]], 2], Power[Pattern[x, Blank[]], Optional[Pattern[m, Blank[]]]]], Pattern[x, Blank[Symbol]]] := -Simp[(x^(m - n + 1)*Cot[c + d*x^n])/(d*n), x] + (Dist[(m - n + 1)/(d*n), Int[x^(m - n)*Cot[c + d*x^n], x], x] - Int[x^m, x]) /; FreeQ[{c, d, m, n}, x]
  3751. Int[Times[Power[Pattern[x, Blank[]], Optional[Pattern[m, Blank[]]]], Power[Plus[Optional[Pattern[a, Blank[]]], Times[Optional[Pattern[b, Blank[]]], Tan[Plus[Optional[Pattern[c, Blank[]]], Times[Optional[Pattern[d, Blank[]]], Power[Pattern[x, Blank[]], Pattern[n, Blank[]]]]]]]], Optional[Pattern[p, Blank[]]]]], Pattern[x, Blank[Symbol]]] := Unintegrable[x^m*(a + b*Tan[c + d*x^n])^p, x] /; FreeQ[{a, b, c, d, m, n, p}, x]
  3752. Int[Times[Power[Plus[Optional[Pattern[a, Blank[]]], Times[Cot[Plus[Optional[Pattern[c, Blank[]]], Times[Optional[Pattern[d, Blank[]]], Power[Pattern[x, Blank[]], Pattern[n, Blank[]]]]]], Optional[Pattern[b, Blank[]]]]], Optional[Pattern[p, Blank[]]]], Power[Pattern[x, Blank[]], Optional[Pattern[m, Blank[]]]]], Pattern[x, Blank[Symbol]]] := Unintegrable[x^m*(a + b*Cot[c + d*x^n])^p, x] /; FreeQ[{a, b, c, d, m, n, p}, x]
  3753. Int[Times[Power[Times[Pattern[e, Blank[]], Pattern[x, Blank[]]], Optional[Pattern[m, Blank[]]]], Power[Plus[Optional[Pattern[a, Blank[]]], Times[Optional[Pattern[b, Blank[]]], Tan[Plus[Optional[Pattern[c, Blank[]]], Times[Optional[Pattern[d, Blank[]]], Power[Pattern[x, Blank[]], Pattern[n, Blank[]]]]]]]], Optional[Pattern[p, Blank[]]]]], Pattern[x, Blank[Symbol]]] := Dist[(e^IntPart[m]*(e*x)^FracPart[m])/x^FracPart[m], Int[x^m*(a + b*Tan[c + d*x^n])^p, x], x] /; FreeQ[{a, b, c, d, e, m, n, p}, x]
  3754. Int[Times[Power[Plus[Optional[Pattern[a, Blank[]]], Times[Cot[Plus[Optional[Pattern[c, Blank[]]], Times[Optional[Pattern[d, Blank[]]], Power[Pattern[x, Blank[]], Pattern[n, Blank[]]]]]], Optional[Pattern[b, Blank[]]]]], Optional[Pattern[p, Blank[]]]], Power[Times[Pattern[e, Blank[]], Pattern[x, Blank[]]], Optional[Pattern[m, Blank[]]]]], Pattern[x, Blank[Symbol]]] := Dist[(e^IntPart[m]*(e*x)^FracPart[m])/x^FracPart[m], Int[x^m*(a + b*Cot[c + d*x^n])^p, x], x] /; FreeQ[{a, b, c, d, e, m, n, p}, x]
  3755. Int[Times[Power[Times[Pattern[e, Blank[]], Pattern[x, Blank[]]], Optional[Pattern[m, Blank[]]]], Power[Plus[Optional[Pattern[a, Blank[]]], Times[Optional[Pattern[b, Blank[]]], Tan[Pattern[u, Blank[]]]]], Optional[Pattern[p, Blank[]]]]], Pattern[x, Blank[Symbol]]] := Int[(e*x)^m*(a + b*Tan[ExpandToSum[u, x]])^p, x] /; FreeQ[{a, b, e, m, p}, x] && BinomialQ[u, x] && !BinomialMatchQ[u, x]
  3756. Int[Times[Power[Plus[Optional[Pattern[a, Blank[]]], Times[Cot[Pattern[u, Blank[]]], Optional[Pattern[b, Blank[]]]]], Optional[Pattern[p, Blank[]]]], Power[Times[Pattern[e, Blank[]], Pattern[x, Blank[]]], Optional[Pattern[m, Blank[]]]]], Pattern[x, Blank[Symbol]]] := Int[(e*x)^m*(a + b*Cot[ExpandToSum[u, x]])^p, x] /; FreeQ[{a, b, e, m, p}, x] && BinomialQ[u, x] && !BinomialMatchQ[u, x]
  3757. Int[Times[Power[Pattern[x, Blank[]], Optional[Pattern[m, Blank[]]]], Power[Sec[Plus[Optional[Pattern[a, Blank[]]], Times[Optional[Pattern[b, Blank[]]], Power[Pattern[x, Blank[]], Optional[Pattern[n, Blank[]]]]]]], Optional[Pattern[p, Blank[]]]], Power[Tan[Plus[Optional[Pattern[a, Blank[]]], Times[Optional[Pattern[b, Blank[]]], Power[Pattern[x, Blank[]], Optional[Pattern[n, Blank[]]]]]]], Optional[Pattern[q, Blank[]]]]], Pattern[x, Blank[Symbol]]] := Simp[(x^(m - n + 1)*Sec[a + b*x^n]^p)/(b*n*p), x] - Dist[(m - n + 1)/(b*n*p), Int[x^(m - n)*Sec[a + b*x^n]^p, x], x] /; FreeQ[{a, b, p}, x] && IntegerQ[n] && GeQ[m, n] && EqQ[q, 1]
  3758. Int[Times[Power[Cot[Plus[Optional[Pattern[a, Blank[]]], Times[Optional[Pattern[b, Blank[]]], Power[Pattern[x, Blank[]], Optional[Pattern[n, Blank[]]]]]]], Optional[Pattern[q, Blank[]]]], Power[Csc[Plus[Optional[Pattern[a, Blank[]]], Times[Optional[Pattern[b, Blank[]]], Power[Pattern[x, Blank[]], Optional[Pattern[n, Blank[]]]]]]], Optional[Pattern[p, Blank[]]]], Power[Pattern[x, Blank[]], Optional[Pattern[m, Blank[]]]]], Pattern[x, Blank[Symbol]]] := -Simp[(x^(m - n + 1)*Csc[a + b*x^n]^p)/(b*n*p), x] + Dist[(m - n + 1)/(b*n*p), Int[x^(m - n)*Csc[a + b*x^n]^p, x], x] /; FreeQ[{a, b, p}, x] && IntegerQ[n] && GeQ[m, n] && EqQ[q, 1]
  3759. Int[Power[Tan[Plus[Optional[Pattern[a, Blank[]]], Times[Optional[Pattern[b, Blank[]]], Pattern[x, Blank[]]], Times[Optional[Pattern[c, Blank[]]], Power[Pattern[x, Blank[]], 2]]]], Optional[Pattern[n, Blank[]]]], Pattern[x, Blank[Symbol]]] := Unintegrable[Tan[a + b*x + c*x^2]^n, x] /; FreeQ[{a, b, c, n}, x]
  3760. Int[Power[Cot[Plus[Optional[Pattern[a, Blank[]]], Times[Optional[Pattern[b, Blank[]]], Pattern[x, Blank[]]], Times[Optional[Pattern[c, Blank[]]], Power[Pattern[x, Blank[]], 2]]]], Optional[Pattern[n, Blank[]]]], Pattern[x, Blank[Symbol]]] := Unintegrable[Cot[a + b*x + c*x^2]^n, x] /; FreeQ[{a, b, c, n}, x]
  3761. Int[Times[Plus[Pattern[d, Blank[]], Times[Optional[Pattern[e, Blank[]]], Pattern[x, Blank[]]]], Tan[Plus[Optional[Pattern[a, Blank[]]], Times[Optional[Pattern[b, Blank[]]], Pattern[x, Blank[]]], Times[Optional[Pattern[c, Blank[]]], Power[Pattern[x, Blank[]], 2]]]]], Pattern[x, Blank[Symbol]]] := -Simp[(e*Log[Cos[a + b*x + c*x^2]])/(2*c), x] /; FreeQ[{a, b, c, d, e}, x] && EqQ[2*c*d - b*e, 0]
  3762. Int[Times[Cot[Plus[Optional[Pattern[a, Blank[]]], Times[Optional[Pattern[b, Blank[]]], Pattern[x, Blank[]]], Times[Optional[Pattern[c, Blank[]]], Power[Pattern[x, Blank[]], 2]]]], Plus[Pattern[d, Blank[]], Times[Optional[Pattern[e, Blank[]]], Pattern[x, Blank[]]]]], Pattern[x, Blank[Symbol]]] := Simp[(e*Log[Sin[a + b*x + c*x^2]])/(2*c), x] /; FreeQ[{a, b, c, d, e}, x] && EqQ[2*c*d - b*e, 0]
  3763. Int[Times[Plus[Optional[Pattern[d, Blank[]]], Times[Optional[Pattern[e, Blank[]]], Pattern[x, Blank[]]]], Tan[Plus[Optional[Pattern[a, Blank[]]], Times[Optional[Pattern[b, Blank[]]], Pattern[x, Blank[]]], Times[Optional[Pattern[c, Blank[]]], Power[Pattern[x, Blank[]], 2]]]]], Pattern[x, Blank[Symbol]]] := -Simp[(e*Log[Cos[a + b*x + c*x^2]])/(2*c), x] + Dist[(2*c*d - b*e)/(2*c), Int[Tan[a + b*x + c*x^2], x], x] /; FreeQ[{a, b, c, d, e}, x] && NeQ[2*c*d - b*e, 0]
  3764. Int[Times[Cot[Plus[Optional[Pattern[a, Blank[]]], Times[Optional[Pattern[b, Blank[]]], Pattern[x, Blank[]]], Times[Optional[Pattern[c, Blank[]]], Power[Pattern[x, Blank[]], 2]]]], Plus[Optional[Pattern[d, Blank[]]], Times[Optional[Pattern[e, Blank[]]], Pattern[x, Blank[]]]]], Pattern[x, Blank[Symbol]]] := Simp[(e*Log[Sin[a + b*x + c*x^2]])/(2*c), x] + Dist[(2*c*d - b*e)/(2*c), Int[Cot[a + b*x + c*x^2], x], x] /; FreeQ[{a, b, c, d, e}, x] && NeQ[2*c*d - b*e, 0]
  3765. Int[Times[Power[Plus[Optional[Pattern[d, Blank[]]], Times[Optional[Pattern[e, Blank[]]], Pattern[x, Blank[]]]], Optional[Pattern[m, Blank[]]]], Power[Tan[Plus[Optional[Pattern[a, Blank[]]], Times[Optional[Pattern[b, Blank[]]], Pattern[x, Blank[]]], Times[Optional[Pattern[c, Blank[]]], Power[Pattern[x, Blank[]], 2]]]], Optional[Pattern[n, Blank[]]]]], Pattern[x, Blank[Symbol]]] := Unintegrable[(d + e*x)^m*Tan[a + b*x + c*x^2]^n, x] /; FreeQ[{a, b, c, d, e, m, n}, x]
  3766. Int[Times[Power[Cot[Plus[Optional[Pattern[a, Blank[]]], Times[Optional[Pattern[b, Blank[]]], Pattern[x, Blank[]]], Times[Optional[Pattern[c, Blank[]]], Power[Pattern[x, Blank[]], 2]]]], Optional[Pattern[n, Blank[]]]], Power[Plus[Optional[Pattern[d, Blank[]]], Times[Optional[Pattern[e, Blank[]]], Pattern[x, Blank[]]]], Optional[Pattern[m, Blank[]]]]], Pattern[x, Blank[Symbol]]] := Unintegrable[(d + e*x)^m*Cot[a + b*x + c*x^2]^n, x] /; FreeQ[{a, b, c, d, e, m, n}, x]
  3767. Int[Power[csc[Plus[Optional[Pattern[c, Blank[]]], Times[Optional[Pattern[d, Blank[]]], Pattern[x, Blank[]]]]], Pattern[n, Blank[]]], Pattern[x, Blank[Symbol]]] := -Dist[d^(-1), Subst[Int[ExpandIntegrand[(1 + x^2)^(n/2 - 1), x], x], x, Cot[c + d*x]], x] /; FreeQ[{c, d}, x] && IGtQ[n/2, 0]
  3768. Int[Power[Times[csc[Plus[Optional[Pattern[c, Blank[]]], Times[Optional[Pattern[d, Blank[]]], Pattern[x, Blank[]]]]], Optional[Pattern[b, Blank[]]]], Pattern[n, Blank[]]], Pattern[x, Blank[Symbol]]] := -Simp[(b*Cos[c + d*x]*(b*Csc[c + d*x])^(n - 1))/(d*(n - 1)), x] + Dist[(b^2*(n - 2))/(n - 1), Int[(b*Csc[c + d*x])^(n - 2), x], x] /; FreeQ[{b, c, d}, x] && GtQ[n, 1] && IntegerQ[2*n]
  3769. Int[Power[Times[csc[Plus[Optional[Pattern[c, Blank[]]], Times[Optional[Pattern[d, Blank[]]], Pattern[x, Blank[]]]]], Optional[Pattern[b, Blank[]]]], Pattern[n, Blank[]]], Pattern[x, Blank[Symbol]]] := Simp[(Cos[c + d*x]*(b*Csc[c + d*x])^(n + 1))/(b*d*n), x] + Dist[(n + 1)/(b^2*n), Int[(b*Csc[c + d*x])^(n + 2), x], x] /; FreeQ[{b, c, d}, x] && LtQ[n, -1] && IntegerQ[2*n]
  3770. Int[csc[Plus[Optional[Pattern[c, Blank[]]], Times[Optional[Pattern[d, Blank[]]], Pattern[x, Blank[]]]]], Pattern[x, Blank[Symbol]]] := -Simp[ArcTanh[Cos[c + d*x]]/d, x] /; FreeQ[{c, d}, x]
  3771. Int[Power[Times[csc[Plus[Optional[Pattern[c, Blank[]]], Times[Optional[Pattern[d, Blank[]]], Pattern[x, Blank[]]]]], Optional[Pattern[b, Blank[]]]], Pattern[n, Blank[]]], Pattern[x, Blank[Symbol]]] := Dist[(b*Csc[c + d*x])^n*Sin[c + d*x]^n, Int[1/Sin[c + d*x]^n, x], x] /; FreeQ[{b, c, d}, x] && EqQ[n^2, 1/4]
  3772. Int[Power[Times[csc[Plus[Optional[Pattern[c, Blank[]]], Times[Optional[Pattern[d, Blank[]]], Pattern[x, Blank[]]]]], Optional[Pattern[b, Blank[]]]], Pattern[n, Blank[]]], Pattern[x, Blank[Symbol]]] := Simp[(b*Csc[c + d*x])^(n - 1)*((Sin[c + d*x]/b)^(n - 1)*Int[1/(Sin[c + d*x]/b)^n, x]), x] /; FreeQ[{b, c, d, n}, x] && !IntegerQ[n]
  3773. Int[Power[Plus[Times[csc[Plus[Optional[Pattern[c, Blank[]]], Times[Optional[Pattern[d, Blank[]]], Pattern[x, Blank[]]]]], Optional[Pattern[b, Blank[]]]], Pattern[a, Blank[]]], 2], Pattern[x, Blank[Symbol]]] := Simp[a^2*x, x] + (Dist[2*a*b, Int[Csc[c + d*x], x], x] + Dist[b^2, Int[Csc[c + d*x]^2, x], x]) /; FreeQ[{a, b, c, d}, x]
  3774. Int[Power[Plus[Times[csc[Plus[Optional[Pattern[c, Blank[]]], Times[Optional[Pattern[d, Blank[]]], Pattern[x, Blank[]]]]], Optional[Pattern[b, Blank[]]]], Pattern[a, Blank[]]], Rational[1, 2]], Pattern[x, Blank[Symbol]]] := Dist[(-2*b)/d, Subst[Int[1/(a + x^2), x], x, (b*Cot[c + d*x])/Sqrt[a + b*Csc[c + d*x]]], x] /; FreeQ[{a, b, c, d}, x] && EqQ[a^2 - b^2, 0]
  3775. Int[Power[Plus[Times[csc[Plus[Optional[Pattern[c, Blank[]]], Times[Optional[Pattern[d, Blank[]]], Pattern[x, Blank[]]]]], Optional[Pattern[b, Blank[]]]], Pattern[a, Blank[]]], Pattern[n, Blank[]]], Pattern[x, Blank[Symbol]]] := -Simp[(b^2*Cot[c + d*x]*(a + b*Csc[c + d*x])^(n - 2))/(d*(n - 1)), x] + Dist[a/(n - 1), Int[(a + b*Csc[c + d*x])^(n - 2)*(a*(n - 1) + b*(3*n - 4)*Csc[c + d*x]), x], x] /; FreeQ[{a, b, c, d}, x] && EqQ[a^2 - b^2, 0] && GtQ[n, 1] && IntegerQ[2*n]
  3776. Int[Power[Plus[Times[csc[Plus[Optional[Pattern[c, Blank[]]], Times[Optional[Pattern[d, Blank[]]], Pattern[x, Blank[]]]]], Optional[Pattern[b, Blank[]]]], Pattern[a, Blank[]]], Rational[-1, 2]], Pattern[x, Blank[Symbol]]] := Dist[1/a, Int[Sqrt[a + b*Csc[c + d*x]], x], x] - Dist[b/a, Int[Csc[c + d*x]/Sqrt[a + b*Csc[c + d*x]], x], x] /; FreeQ[{a, b, c, d}, x] && EqQ[a^2 - b^2, 0]
  3777. Int[Power[Plus[Times[csc[Plus[Optional[Pattern[c, Blank[]]], Times[Optional[Pattern[d, Blank[]]], Pattern[x, Blank[]]]]], Optional[Pattern[b, Blank[]]]], Pattern[a, Blank[]]], Pattern[n, Blank[]]], Pattern[x, Blank[Symbol]]] := -Simp[(Cot[c + d*x]*(a + b*Csc[c + d*x])^n)/(d*(2*n + 1)), x] + Dist[1/(a^2*(2*n + 1)), Int[(a + b*Csc[c + d*x])^(n + 1)*(a*(2*n + 1) - b*(n + 1)*Csc[c + d*x]), x], x] /; FreeQ[{a, b, c, d}, x] && EqQ[a^2 - b^2, 0] && LeQ[n, -1] && IntegerQ[2*n]
  3778. Int[Power[Plus[Times[csc[Plus[Optional[Pattern[c, Blank[]]], Times[Optional[Pattern[d, Blank[]]], Pattern[x, Blank[]]]]], Optional[Pattern[b, Blank[]]]], Pattern[a, Blank[]]], Pattern[n, Blank[]]], Pattern[x, Blank[Symbol]]] := Dist[(a^n*Cot[c + d*x])/(d*Sqrt[1 + Csc[c + d*x]]*Sqrt[1 - Csc[c + d*x]]), Subst[Int[(1 + (b*x)/a)^(n - 1/2)/(x*Sqrt[1 - (b*x)/a]), x], x, Csc[c + d*x]], x] /; FreeQ[{a, b, c, d, n}, x] && EqQ[a^2 - b^2, 0] && !IntegerQ[2*n] && GtQ[a, 0]
  3779. Int[Power[Plus[Times[csc[Plus[Optional[Pattern[c, Blank[]]], Times[Optional[Pattern[d, Blank[]]], Pattern[x, Blank[]]]]], Optional[Pattern[b, Blank[]]]], Pattern[a, Blank[]]], Pattern[n, Blank[]]], Pattern[x, Blank[Symbol]]] := Dist[(a^IntPart[n]*(a + b*Csc[c + d*x])^FracPart[n])/(1 + (b*Csc[c + d*x])/a)^FracPart[n], Int[(1 + (b*Csc[c + d*x])/a)^n, x], x] /; FreeQ[{a, b, c, d, n}, x] && EqQ[a^2 - b^2, 0] && !IntegerQ[2*n] && !GtQ[a, 0]
  3780. Int[Power[Plus[Times[csc[Plus[Optional[Pattern[c, Blank[]]], Times[Optional[Pattern[d, Blank[]]], Pattern[x, Blank[]]]]], Optional[Pattern[b, Blank[]]]], Pattern[a, Blank[]]], Rational[1, 2]], Pattern[x, Blank[Symbol]]] := Simp[(2*(a + b*Csc[c + d*x])*Sqrt[(b*(1 + Csc[c + d*x]))/(a + b*Csc[c + d*x])]*Sqrt[-((b*(1 - Csc[c + d*x]))/(a + b*Csc[c + d*x]))]*EllipticPi[a/(a + b), ArcSin[Rt[a + b, 2]/Sqrt[a + b*Csc[c + d*x]]], (a - b)/(a + b)])/(d*Rt[a + b, 2]*Cot[c + d*x]), x] /; FreeQ[{a, b, c, d}, x] && NeQ[a^2 - b^2, 0]
  3781. Int[Power[Plus[Times[csc[Plus[Optional[Pattern[c, Blank[]]], Times[Optional[Pattern[d, Blank[]]], Pattern[x, Blank[]]]]], Optional[Pattern[b, Blank[]]]], Pattern[a, Blank[]]], Rational[3, 2]], Pattern[x, Blank[Symbol]]] := Int[(a^2 + b*(2*a - b)*Csc[c + d*x])/Sqrt[a + b*Csc[c + d*x]], x] + Dist[b^2, Int[(Csc[c + d*x]*(1 + Csc[c + d*x]))/Sqrt[a + b*Csc[c + d*x]], x], x] /; FreeQ[{a, b, c, d}, x] && NeQ[a^2 - b^2, 0]
  3782. Int[Power[Plus[Times[csc[Plus[Optional[Pattern[c, Blank[]]], Times[Optional[Pattern[d, Blank[]]], Pattern[x, Blank[]]]]], Optional[Pattern[b, Blank[]]]], Pattern[a, Blank[]]], Pattern[n, Blank[]]], Pattern[x, Blank[Symbol]]] := -Simp[(b^2*Cot[c + d*x]*(a + b*Csc[c + d*x])^(n - 2))/(d*(n - 1)), x] + Dist[1/(n - 1), Int[(a + b*Csc[c + d*x])^(n - 3)*Simp[a^3*(n - 1) + (b*(b^2*(n - 2) + 3*a^2*(n - 1)))*Csc[c + d*x] + (a*b^2*(3*n - 4))*Csc[c + d*x]^2, x], x], x] /; FreeQ[{a, b, c, d}, x] && NeQ[a^2 - b^2, 0] && GtQ[n, 2] && IntegerQ[2*n]
  3783. Int[Power[Plus[Times[csc[Plus[Optional[Pattern[c, Blank[]]], Times[Optional[Pattern[d, Blank[]]], Pattern[x, Blank[]]]]], Optional[Pattern[b, Blank[]]]], Pattern[a, Blank[]]], -1], Pattern[x, Blank[Symbol]]] := Simp[x/a, x] - Dist[1/a, Int[1/(1 + (a*Sin[c + d*x])/b), x], x] /; FreeQ[{a, b, c, d}, x] && NeQ[a^2 - b^2, 0]
  3784. Int[Power[Plus[Times[csc[Plus[Optional[Pattern[c, Blank[]]], Times[Optional[Pattern[d, Blank[]]], Pattern[x, Blank[]]]]], Optional[Pattern[b, Blank[]]]], Pattern[a, Blank[]]], Rational[-1, 2]], Pattern[x, Blank[Symbol]]] := Simp[(2*Rt[a + b, 2]*Sqrt[(b*(1 - Csc[c + d*x]))/(a + b)]*Sqrt[-((b*(1 + Csc[c + d*x]))/(a - b))]*EllipticPi[(a + b)/a, ArcSin[Sqrt[a + b*Csc[c + d*x]]/Rt[a + b, 2]], (a + b)/(a - b)])/(a*d*Cot[c + d*x]), x] /; FreeQ[{a, b, c, d}, x] && NeQ[a^2 - b^2, 0]
  3785. Int[Power[Plus[Times[csc[Plus[Optional[Pattern[c, Blank[]]], Times[Optional[Pattern[d, Blank[]]], Pattern[x, Blank[]]]]], Optional[Pattern[b, Blank[]]]], Pattern[a, Blank[]]], Pattern[n, Blank[]]], Pattern[x, Blank[Symbol]]] := Simp[(b^2*Cot[c + d*x]*(a + b*Csc[c + d*x])^(n + 1))/(a*d*(n + 1)*(a^2 - b^2)), x] + Dist[1/(a*(n + 1)*(a^2 - b^2)), Int[(a + b*Csc[c + d*x])^(n + 1)*Simp[(a^2 - b^2)*(n + 1) - a*b*(n + 1)*Csc[c + d*x] + b^2*(n + 2)*Csc[c + d*x]^2, x], x], x] /; FreeQ[{a, b, c, d}, x] && NeQ[a^2 - b^2, 0] && LtQ[n, -1] && IntegerQ[2*n]
  3786. Int[Power[Plus[Times[csc[Plus[Optional[Pattern[c, Blank[]]], Times[Optional[Pattern[d, Blank[]]], Pattern[x, Blank[]]]]], Optional[Pattern[b, Blank[]]]], Pattern[a, Blank[]]], Pattern[n, Blank[]]], Pattern[x, Blank[Symbol]]] := Unintegrable[(a + b*Csc[c + d*x])^n, x] /; FreeQ[{a, b, c, d, n}, x] && NeQ[a^2 - b^2, 0] && !IntegerQ[2*n]
  3787. Int[Times[Power[Times[csc[Plus[Optional[Pattern[e, Blank[]]], Times[Optional[Pattern[f, Blank[]]], Pattern[x, Blank[]]]]], Optional[Pattern[d, Blank[]]]], Optional[Pattern[n, Blank[]]]], Plus[Times[csc[Plus[Optional[Pattern[e, Blank[]]], Times[Optional[Pattern[f, Blank[]]], Pattern[x, Blank[]]]]], Optional[Pattern[b, Blank[]]]], Pattern[a, Blank[]]]], Pattern[x, Blank[Symbol]]] := Dist[a, Int[(d*Csc[e + f*x])^n, x], x] + Dist[b/d, Int[(d*Csc[e + f*x])^(n + 1), x], x] /; FreeQ[{a, b, d, e, f, n}, x]
  3788. Int[Times[Power[Times[csc[Plus[Optional[Pattern[e, Blank[]]], Times[Optional[Pattern[f, Blank[]]], Pattern[x, Blank[]]]]], Optional[Pattern[d, Blank[]]]], Optional[Pattern[n, Blank[]]]], Power[Plus[Times[csc[Plus[Optional[Pattern[e, Blank[]]], Times[Optional[Pattern[f, Blank[]]], Pattern[x, Blank[]]]]], Optional[Pattern[b, Blank[]]]], Pattern[a, Blank[]]], 2]], Pattern[x, Blank[Symbol]]] := Dist[(2*a*b)/d, Int[(d*Csc[e + f*x])^(n + 1), x], x] + Int[(d*Csc[e + f*x])^n*(a^2 + b^2*Csc[e + f*x]^2), x] /; FreeQ[{a, b, d, e, f, n}, x]
  3789. Int[Times[Power[csc[Plus[Optional[Pattern[e, Blank[]]], Times[Optional[Pattern[f, Blank[]]], Pattern[x, Blank[]]]]], 2], Power[Plus[Times[csc[Plus[Optional[Pattern[e, Blank[]]], Times[Optional[Pattern[f, Blank[]]], Pattern[x, Blank[]]]]], Optional[Pattern[b, Blank[]]]], Pattern[a, Blank[]]], -1]], Pattern[x, Blank[Symbol]]] := Dist[1/b, Int[Csc[e + f*x], x], x] - Dist[a/b, Int[Csc[e + f*x]/(a + b*Csc[e + f*x]), x], x] /; FreeQ[{a, b, e, f}, x]
  3790. Int[Times[Power[csc[Plus[Optional[Pattern[e, Blank[]]], Times[Optional[Pattern[f, Blank[]]], Pattern[x, Blank[]]]]], 3], Power[Plus[Times[csc[Plus[Optional[Pattern[e, Blank[]]], Times[Optional[Pattern[f, Blank[]]], Pattern[x, Blank[]]]]], Optional[Pattern[b, Blank[]]]], Pattern[a, Blank[]]], -1]], Pattern[x, Blank[Symbol]]] := -Simp[Cot[e + f*x]/(b*f), x] - Dist[a/b, Int[Csc[e + f*x]^2/(a + b*Csc[e + f*x]), x], x] /; FreeQ[{a, b, e, f}, x]
  3791. Int[Times[Power[Times[csc[Plus[Optional[Pattern[e, Blank[]]], Times[Optional[Pattern[f, Blank[]]], Pattern[x, Blank[]]]]], Optional[Pattern[d, Blank[]]]], Optional[Pattern[n, Blank[]]]], Power[Plus[Times[csc[Plus[Optional[Pattern[e, Blank[]]], Times[Optional[Pattern[f, Blank[]]], Pattern[x, Blank[]]]]], Optional[Pattern[b, Blank[]]]], Pattern[a, Blank[]]], Pattern[m, Blank[]]]], Pattern[x, Blank[Symbol]]] := Int[ExpandTrig[(a + b*csc[e + f*x])^m*(d*csc[e + f*x])^n, x], x] /; FreeQ[{a, b, d, e, f, m, n}, x] && EqQ[a^2 - b^2, 0] && IGtQ[m, 0] && RationalQ[n]
  3792. Int[Times[csc[Plus[Optional[Pattern[e, Blank[]]], Times[Optional[Pattern[f, Blank[]]], Pattern[x, Blank[]]]]], Power[Plus[Times[csc[Plus[Optional[Pattern[e, Blank[]]], Times[Optional[Pattern[f, Blank[]]], Pattern[x, Blank[]]]]], Optional[Pattern[b, Blank[]]]], Pattern[a, Blank[]]], Rational[1, 2]]], Pattern[x, Blank[Symbol]]] := Simp[(-2*b*Cot[e + f*x])/(f*Sqrt[a + b*Csc[e + f*x]]), x] /; FreeQ[{a, b, e, f}, x] && EqQ[a^2 - b^2, 0]
  3793. Int[Times[csc[Plus[Optional[Pattern[e, Blank[]]], Times[Optional[Pattern[f, Blank[]]], Pattern[x, Blank[]]]]], Power[Plus[Times[csc[Plus[Optional[Pattern[e, Blank[]]], Times[Optional[Pattern[f, Blank[]]], Pattern[x, Blank[]]]]], Optional[Pattern[b, Blank[]]]], Pattern[a, Blank[]]], Pattern[m, Blank[]]]], Pattern[x, Blank[Symbol]]] := -Simp[(b*Cot[e + f*x]*(a + b*Csc[e + f*x])^(m - 1))/(f*m), x] + Dist[(a*(2*m - 1))/m, Int[Csc[e + f*x]*(a + b*Csc[e + f*x])^(m - 1), x], x] /; FreeQ[{a, b, e, f}, x] && EqQ[a^2 - b^2, 0] && GtQ[m, 1/2] && IntegerQ[2*m]
  3794. Int[Times[csc[Plus[Optional[Pattern[e, Blank[]]], Times[Optional[Pattern[f, Blank[]]], Pattern[x, Blank[]]]]], Power[Plus[Times[csc[Plus[Optional[Pattern[e, Blank[]]], Times[Optional[Pattern[f, Blank[]]], Pattern[x, Blank[]]]]], Optional[Pattern[b, Blank[]]]], Pattern[a, Blank[]]], -1]], Pattern[x, Blank[Symbol]]] := -Simp[Cot[e + f*x]/(f*(b + a*Csc[e + f*x])), x] /; FreeQ[{a, b, e, f}, x] && EqQ[a^2 - b^2, 0]
  3795. Int[Times[csc[Plus[Optional[Pattern[e, Blank[]]], Times[Optional[Pattern[f, Blank[]]], Pattern[x, Blank[]]]]], Power[Plus[Times[csc[Plus[Optional[Pattern[e, Blank[]]], Times[Optional[Pattern[f, Blank[]]], Pattern[x, Blank[]]]]], Optional[Pattern[b, Blank[]]]], Pattern[a, Blank[]]], Rational[-1, 2]]], Pattern[x, Blank[Symbol]]] := Dist[-2/f, Subst[Int[1/(2*a + x^2), x], x, (b*Cot[e + f*x])/Sqrt[a + b*Csc[e + f*x]]], x] /; FreeQ[{a, b, e, f}, x] && EqQ[a^2 - b^2, 0]
  3796. Int[Times[csc[Plus[Optional[Pattern[e, Blank[]]], Times[Optional[Pattern[f, Blank[]]], Pattern[x, Blank[]]]]], Power[Plus[Times[csc[Plus[Optional[Pattern[e, Blank[]]], Times[Optional[Pattern[f, Blank[]]], Pattern[x, Blank[]]]]], Optional[Pattern[b, Blank[]]]], Pattern[a, Blank[]]], Pattern[m, Blank[]]]], Pattern[x, Blank[Symbol]]] := Simp[(b*Cot[e + f*x]*(a + b*Csc[e + f*x])^m)/(a*f*(2*m + 1)), x] + Dist[(m + 1)/(a*(2*m + 1)), Int[Csc[e + f*x]*(a + b*Csc[e + f*x])^(m + 1), x], x] /; FreeQ[{a, b, e, f}, x] && EqQ[a^2 - b^2, 0] && LtQ[m, -2^(-1)] && IntegerQ[2*m]
  3797. Int[Times[Power[csc[Plus[Optional[Pattern[e, Blank[]]], Times[Optional[Pattern[f, Blank[]]], Pattern[x, Blank[]]]]], 2], Power[Plus[Times[csc[Plus[Optional[Pattern[e, Blank[]]], Times[Optional[Pattern[f, Blank[]]], Pattern[x, Blank[]]]]], Optional[Pattern[b, Blank[]]]], Pattern[a, Blank[]]], Pattern[m, Blank[]]]], Pattern[x, Blank[Symbol]]] := -Simp[(Cot[e + f*x]*(a + b*Csc[e + f*x])^m)/(f*(2*m + 1)), x] + Dist[m/(b*(2*m + 1)), Int[Csc[e + f*x]*(a + b*Csc[e + f*x])^(m + 1), x], x] /; FreeQ[{a, b, e, f}, x] && EqQ[a^2 - b^2, 0] && LtQ[m, -2^(-1)]
  3798. Int[Times[Power[csc[Plus[Optional[Pattern[e, Blank[]]], Times[Optional[Pattern[f, Blank[]]], Pattern[x, Blank[]]]]], 2], Power[Plus[Times[csc[Plus[Optional[Pattern[e, Blank[]]], Times[Optional[Pattern[f, Blank[]]], Pattern[x, Blank[]]]]], Optional[Pattern[b, Blank[]]]], Pattern[a, Blank[]]], Pattern[m, Blank[]]]], Pattern[x, Blank[Symbol]]] := -Simp[(Cot[e + f*x]*(a + b*Csc[e + f*x])^m)/(f*(m + 1)), x] + Dist[(a*m)/(b*(m + 1)), Int[Csc[e + f*x]*(a + b*Csc[e + f*x])^m, x], x] /; FreeQ[{a, b, e, f, m}, x] && EqQ[a^2 - b^2, 0] && !LtQ[m, -2^(-1)]
  3799. Int[Times[Power[csc[Plus[Optional[Pattern[e, Blank[]]], Times[Optional[Pattern[f, Blank[]]], Pattern[x, Blank[]]]]], 3], Power[Plus[Times[csc[Plus[Optional[Pattern[e, Blank[]]], Times[Optional[Pattern[f, Blank[]]], Pattern[x, Blank[]]]]], Optional[Pattern[b, Blank[]]]], Pattern[a, Blank[]]], Pattern[m, Blank[]]]], Pattern[x, Blank[Symbol]]] := Simp[(b*Cot[e + f*x]*(a + b*Csc[e + f*x])^m)/(a*f*(2*m + 1)), x] - Dist[1/(a^2*(2*m + 1)), Int[Csc[e + f*x]*(a + b*Csc[e + f*x])^(m + 1)*(a*m - b*(2*m + 1)*Csc[e + f*x]), x], x] /; FreeQ[{a, b, e, f}, x] && EqQ[a^2 - b^2, 0] && LtQ[m, -2^(-1)]
  3800. Int[Times[Power[csc[Plus[Optional[Pattern[e, Blank[]]], Times[Optional[Pattern[f, Blank[]]], Pattern[x, Blank[]]]]], 3], Power[Plus[Times[csc[Plus[Optional[Pattern[e, Blank[]]], Times[Optional[Pattern[f, Blank[]]], Pattern[x, Blank[]]]]], Optional[Pattern[b, Blank[]]]], Pattern[a, Blank[]]], Pattern[m, Blank[]]]], Pattern[x, Blank[Symbol]]] := -Simp[(Cot[e + f*x]*(a + b*Csc[e + f*x])^(m + 1))/(b*f*(m + 2)), x] + Dist[1/(b*(m + 2)), Int[Csc[e + f*x]*(a + b*Csc[e + f*x])^m*(b*(m + 1) - a*Csc[e + f*x]), x], x] /; FreeQ[{a, b, e, f, m}, x] && EqQ[a^2 - b^2, 0] && !LtQ[m, -2^(-1)]
  3801. Int[Times[Power[Times[csc[Plus[Optional[Pattern[e, Blank[]]], Times[Optional[Pattern[f, Blank[]]], Pattern[x, Blank[]]]]], Optional[Pattern[d, Blank[]]]], Rational[1, 2]], Power[Plus[Times[csc[Plus[Optional[Pattern[e, Blank[]]], Times[Optional[Pattern[f, Blank[]]], Pattern[x, Blank[]]]]], Optional[Pattern[b, Blank[]]]], Pattern[a, Blank[]]], Rational[1, 2]]], Pattern[x, Blank[Symbol]]] := Dist[(-2*a*Sqrt[(a*d)/b])/(b*f), Subst[Int[1/Sqrt[1 + x^2/a], x], x, (b*Cot[e + f*x])/Sqrt[a + b*Csc[e + f*x]]], x] /; FreeQ[{a, b, d, e, f}, x] && EqQ[a^2 - b^2, 0] && GtQ[(a*d)/b, 0]
  3802. Int[Times[Power[Times[csc[Plus[Optional[Pattern[e, Blank[]]], Times[Optional[Pattern[f, Blank[]]], Pattern[x, Blank[]]]]], Optional[Pattern[d, Blank[]]]], Rational[1, 2]], Power[Plus[Times[csc[Plus[Optional[Pattern[e, Blank[]]], Times[Optional[Pattern[f, Blank[]]], Pattern[x, Blank[]]]]], Optional[Pattern[b, Blank[]]]], Pattern[a, Blank[]]], Rational[1, 2]]], Pattern[x, Blank[Symbol]]] := Dist[(-2*b*d)/f, Subst[Int[1/(b - d*x^2), x], x, (b*Cot[e + f*x])/(Sqrt[a + b*Csc[e + f*x]]*Sqrt[d*Csc[e + f*x]])], x] /; FreeQ[{a, b, d, e, f}, x] && EqQ[a^2 - b^2, 0] && !GtQ[(a*d)/b, 0]
  3803. Int[Times[Power[Times[csc[Plus[Optional[Pattern[e, Blank[]]], Times[Optional[Pattern[f, Blank[]]], Pattern[x, Blank[]]]]], Optional[Pattern[d, Blank[]]]], Pattern[n, Blank[]]], Power[Plus[Times[csc[Plus[Optional[Pattern[e, Blank[]]], Times[Optional[Pattern[f, Blank[]]], Pattern[x, Blank[]]]]], Optional[Pattern[b, Blank[]]]], Pattern[a, Blank[]]], Rational[1, 2]]], Pattern[x, Blank[Symbol]]] := Simp[(-2*b*d*Cot[e + f*x]*(d*Csc[e + f*x])^(n - 1))/(f*(2*n - 1)*Sqrt[a + b*Csc[e + f*x]]), x] + Dist[(2*a*d*(n - 1))/(b*(2*n - 1)), Int[Sqrt[a + b*Csc[e + f*x]]*(d*Csc[e + f*x])^(n - 1), x], x] /; FreeQ[{a, b, d, e, f}, x] && EqQ[a^2 - b^2, 0] && GtQ[n, 1] && IntegerQ[2*n]
  3804. Int[Times[Power[Times[csc[Plus[Optional[Pattern[e, Blank[]]], Times[Optional[Pattern[f, Blank[]]], Pattern[x, Blank[]]]]], Optional[Pattern[d, Blank[]]]], Rational[-1, 2]], Power[Plus[Times[csc[Plus[Optional[Pattern[e, Blank[]]], Times[Optional[Pattern[f, Blank[]]], Pattern[x, Blank[]]]]], Optional[Pattern[b, Blank[]]]], Pattern[a, Blank[]]], Rational[1, 2]]], Pattern[x, Blank[Symbol]]] := Simp[(-2*a*Cot[e + f*x])/(f*Sqrt[a + b*Csc[e + f*x]]*Sqrt[d*Csc[e + f*x]]), x] /; FreeQ[{a, b, d, e, f}, x] && EqQ[a^2 - b^2, 0]
  3805. Int[Times[Power[Times[csc[Plus[Optional[Pattern[e, Blank[]]], Times[Optional[Pattern[f, Blank[]]], Pattern[x, Blank[]]]]], Optional[Pattern[d, Blank[]]]], Pattern[n, Blank[]]], Power[Plus[Times[csc[Plus[Optional[Pattern[e, Blank[]]], Times[Optional[Pattern[f, Blank[]]], Pattern[x, Blank[]]]]], Optional[Pattern[b, Blank[]]]], Pattern[a, Blank[]]], Rational[1, 2]]], Pattern[x, Blank[Symbol]]] := Simp[(a*Cot[e + f*x]*(d*Csc[e + f*x])^n)/(f*n*Sqrt[a + b*Csc[e + f*x]]), x] + Dist[(a*(2*n + 1))/(2*b*d*n), Int[Sqrt[a + b*Csc[e + f*x]]*(d*Csc[e + f*x])^(n + 1), x], x] /; FreeQ[{a, b, d, e, f}, x] && EqQ[a^2 - b^2, 0] && LtQ[n, -2^(-1)] && IntegerQ[2*n]
  3806. Int[Times[Power[Times[csc[Plus[Optional[Pattern[e, Blank[]]], Times[Optional[Pattern[f, Blank[]]], Pattern[x, Blank[]]]]], Optional[Pattern[d, Blank[]]]], Pattern[n, Blank[]]], Power[Plus[Times[csc[Plus[Optional[Pattern[e, Blank[]]], Times[Optional[Pattern[f, Blank[]]], Pattern[x, Blank[]]]]], Optional[Pattern[b, Blank[]]]], Pattern[a, Blank[]]], Rational[1, 2]]], Pattern[x, Blank[Symbol]]] := Dist[(a^2*d*Cot[e + f*x])/(f*Sqrt[a + b*Csc[e + f*x]]*Sqrt[a - b*Csc[e + f*x]]), Subst[Int[(d*x)^(n - 1)/Sqrt[a - b*x], x], x, Csc[e + f*x]], x] /; FreeQ[{a, b, d, e, f, n}, x] && EqQ[a^2 - b^2, 0]
  3807. Int[Times[Power[Times[csc[Plus[Optional[Pattern[e, Blank[]]], Times[Optional[Pattern[f, Blank[]]], Pattern[x, Blank[]]]]], Optional[Pattern[d, Blank[]]]], Rational[1, 2]], Power[Plus[Times[csc[Plus[Optional[Pattern[e, Blank[]]], Times[Optional[Pattern[f, Blank[]]], Pattern[x, Blank[]]]]], Optional[Pattern[b, Blank[]]]], Pattern[a, Blank[]]], Rational[-1, 2]]], Pattern[x, Blank[Symbol]]] := -Dist[(Sqrt[2]*Sqrt[a])/(b*f), Subst[Int[1/Sqrt[1 + x^2], x], x, (b*Cot[e + f*x])/(a + b*Csc[e + f*x])], x] /; FreeQ[{a, b, d, e, f}, x] && EqQ[a^2 - b^2, 0] && EqQ[d - a/b, 0] && GtQ[a, 0]
  3808. Int[Times[Power[Times[csc[Plus[Optional[Pattern[e, Blank[]]], Times[Optional[Pattern[f, Blank[]]], Pattern[x, Blank[]]]]], Optional[Pattern[d, Blank[]]]], Rational[1, 2]], Power[Plus[Times[csc[Plus[Optional[Pattern[e, Blank[]]], Times[Optional[Pattern[f, Blank[]]], Pattern[x, Blank[]]]]], Optional[Pattern[b, Blank[]]]], Pattern[a, Blank[]]], Rational[-1, 2]]], Pattern[x, Blank[Symbol]]] := Dist[(-2*b*d)/(a*f), Subst[Int[1/(2*b - d*x^2), x], x, (b*Cot[e + f*x])/(Sqrt[a + b*Csc[e + f*x]]*Sqrt[d*Csc[e + f*x]])], x] /; FreeQ[{a, b, d, e, f}, x] && EqQ[a^2 - b^2, 0]
  3809. Int[Times[Power[Times[csc[Plus[Optional[Pattern[e, Blank[]]], Times[Optional[Pattern[f, Blank[]]], Pattern[x, Blank[]]]]], Optional[Pattern[d, Blank[]]]], Pattern[n, Blank[]]], Power[Plus[Times[csc[Plus[Optional[Pattern[e, Blank[]]], Times[Optional[Pattern[f, Blank[]]], Pattern[x, Blank[]]]]], Optional[Pattern[b, Blank[]]]], Pattern[a, Blank[]]], Pattern[m, Blank[]]]], Pattern[x, Blank[Symbol]]] := -Simp[(a*Cot[e + f*x]*(a + b*Csc[e + f*x])^(m - 1)*(d*Csc[e + f*x])^n)/(f*m), x] + Dist[(b*(2*m - 1))/(d*m), Int[(a + b*Csc[e + f*x])^(m - 1)*(d*Csc[e + f*x])^(n + 1), x], x] /; FreeQ[{a, b, d, e, f, m, n}, x] && EqQ[a^2 - b^2, 0] && EqQ[m + n, 0] && GtQ[m, 1/2] && IntegerQ[2*m]
  3810. Int[Times[Power[Times[csc[Plus[Optional[Pattern[e, Blank[]]], Times[Optional[Pattern[f, Blank[]]], Pattern[x, Blank[]]]]], Optional[Pattern[d, Blank[]]]], Pattern[n, Blank[]]], Power[Plus[Times[csc[Plus[Optional[Pattern[e, Blank[]]], Times[Optional[Pattern[f, Blank[]]], Pattern[x, Blank[]]]]], Optional[Pattern[b, Blank[]]]], Pattern[a, Blank[]]], Pattern[m, Blank[]]]], Pattern[x, Blank[Symbol]]] := Simp[(b*d*Cot[e + f*x]*(a + b*Csc[e + f*x])^m*(d*Csc[e + f*x])^(n - 1))/(a*f*(2*m + 1)), x] + Dist[(d*(m + 1))/(b*(2*m + 1)), Int[(a + b*Csc[e + f*x])^(m + 1)*(d*Csc[e + f*x])^(n - 1), x], x] /; FreeQ[{a, b, d, e, f, m, n}, x] && EqQ[a^2 - b^2, 0] && EqQ[m + n, 0] && LtQ[m, -2^(-1)] && IntegerQ[2*m]
  3811. Int[Times[Power[Times[csc[Plus[Optional[Pattern[e, Blank[]]], Times[Optional[Pattern[f, Blank[]]], Pattern[x, Blank[]]]]], Optional[Pattern[d, Blank[]]]], Pattern[n, Blank[]]], Power[Plus[Times[csc[Plus[Optional[Pattern[e, Blank[]]], Times[Optional[Pattern[f, Blank[]]], Pattern[x, Blank[]]]]], Optional[Pattern[b, Blank[]]]], Pattern[a, Blank[]]], Pattern[m, Blank[]]]], Pattern[x, Blank[Symbol]]] := -Simp[(Cot[e + f*x]*(a + b*Csc[e + f*x])^m*(d*Csc[e + f*x])^n)/(f*(2*m + 1)), x] + Dist[m/(a*(2*m + 1)), Int[(a + b*Csc[e + f*x])^(m + 1)*(d*Csc[e + f*x])^n, x], x] /; FreeQ[{a, b, d, e, f}, x] && EqQ[a^2 - b^2, 0] && EqQ[m + n + 1, 0] && LtQ[m, -2^(-1)]
  3812. Int[Times[Power[Times[csc[Plus[Optional[Pattern[e, Blank[]]], Times[Optional[Pattern[f, Blank[]]], Pattern[x, Blank[]]]]], Optional[Pattern[d, Blank[]]]], Pattern[n, Blank[]]], Power[Plus[Times[csc[Plus[Optional[Pattern[e, Blank[]]], Times[Optional[Pattern[f, Blank[]]], Pattern[x, Blank[]]]]], Optional[Pattern[b, Blank[]]]], Pattern[a, Blank[]]], Pattern[m, Blank[]]]], Pattern[x, Blank[Symbol]]] := -Simp[(Cot[e + f*x]*(a + b*Csc[e + f*x])^m*(d*Csc[e + f*x])^n)/(f*(m + 1)), x] + Dist[(a*m)/(b*d*(m + 1)), Int[(a + b*Csc[e + f*x])^m*(d*Csc[e + f*x])^(n + 1), x], x] /; FreeQ[{a, b, d, e, f, m, n}, x] && EqQ[a^2 - b^2, 0] && EqQ[m + n + 1, 0] && !LtQ[m, -2^(-1)]
  3813. Int[Times[Power[Times[csc[Plus[Optional[Pattern[e, Blank[]]], Times[Optional[Pattern[f, Blank[]]], Pattern[x, Blank[]]]]], Optional[Pattern[d, Blank[]]]], Pattern[n, Blank[]]], Power[Plus[Times[csc[Plus[Optional[Pattern[e, Blank[]]], Times[Optional[Pattern[f, Blank[]]], Pattern[x, Blank[]]]]], Optional[Pattern[b, Blank[]]]], Pattern[a, Blank[]]], Pattern[m, Blank[]]]], Pattern[x, Blank[Symbol]]] := Simp[(b^2*Cot[e + f*x]*(a + b*Csc[e + f*x])^(m - 2)*(d*Csc[e + f*x])^n)/(f*n), x] - Dist[a/(d*n), Int[(a + b*Csc[e + f*x])^(m - 2)*(d*Csc[e + f*x])^(n + 1)*(b*(m - 2*n - 2) - a*(m + 2*n - 1)*Csc[e + f*x]), x], x] /; FreeQ[{a, b, d, e, f}, x] && EqQ[a^2 - b^2, 0] && GtQ[m, 1] && (LtQ[n, -1] || (EqQ[m, 3/2] && EqQ[n, -2^(-1)])) && IntegerQ[2*m]
  3814. Int[Times[Power[Times[csc[Plus[Optional[Pattern[e, Blank[]]], Times[Optional[Pattern[f, Blank[]]], Pattern[x, Blank[]]]]], Optional[Pattern[d, Blank[]]]], Pattern[n, Blank[]]], Power[Plus[Times[csc[Plus[Optional[Pattern[e, Blank[]]], Times[Optional[Pattern[f, Blank[]]], Pattern[x, Blank[]]]]], Optional[Pattern[b, Blank[]]]], Pattern[a, Blank[]]], Pattern[m, Blank[]]]], Pattern[x, Blank[Symbol]]] := -Simp[(b^2*Cot[e + f*x]*(a + b*Csc[e + f*x])^(m - 2)*(d*Csc[e + f*x])^n)/(f*(m + n - 1)), x] + Dist[b/(m + n - 1), Int[(a + b*Csc[e + f*x])^(m - 2)*(d*Csc[e + f*x])^n*(b*(m + 2*n - 1) + a*(3*m + 2*n - 4)*Csc[e + f*x]), x], x] /; FreeQ[{a, b, d, e, f, n}, x] && EqQ[a^2 - b^2, 0] && GtQ[m, 1] && NeQ[m + n - 1, 0] && IntegerQ[2*m]
  3815. Int[Times[Power[Times[csc[Plus[Optional[Pattern[e, Blank[]]], Times[Optional[Pattern[f, Blank[]]], Pattern[x, Blank[]]]]], Optional[Pattern[d, Blank[]]]], Pattern[n, Blank[]]], Power[Plus[Times[csc[Plus[Optional[Pattern[e, Blank[]]], Times[Optional[Pattern[f, Blank[]]], Pattern[x, Blank[]]]]], Optional[Pattern[b, Blank[]]]], Pattern[a, Blank[]]], Pattern[m, Blank[]]]], Pattern[x, Blank[Symbol]]] := Simp[(b*d*Cot[e + f*x]*(a + b*Csc[e + f*x])^m*(d*Csc[e + f*x])^(n - 1))/(a*f*(2*m + 1)), x] - Dist[d/(a*b*(2*m + 1)), Int[(a + b*Csc[e + f*x])^(m + 1)*(d*Csc[e + f*x])^(n - 1)*(a*(n - 1) - b*(m + n)*Csc[e + f*x]), x], x] /; FreeQ[{a, b, d, e, f}, x] && EqQ[a^2 - b^2, 0] && LtQ[m, -1] && LtQ[1, n, 2] && (IntegersQ[2*m, 2*n] || IntegerQ[m])
  3816. Int[Times[Power[Times[csc[Plus[Optional[Pattern[e, Blank[]]], Times[Optional[Pattern[f, Blank[]]], Pattern[x, Blank[]]]]], Optional[Pattern[d, Blank[]]]], Pattern[n, Blank[]]], Power[Plus[Times[csc[Plus[Optional[Pattern[e, Blank[]]], Times[Optional[Pattern[f, Blank[]]], Pattern[x, Blank[]]]]], Optional[Pattern[b, Blank[]]]], Pattern[a, Blank[]]], Pattern[m, Blank[]]]], Pattern[x, Blank[Symbol]]] := -Simp[(d^2*Cot[e + f*x]*(a + b*Csc[e + f*x])^m*(d*Csc[e + f*x])^(n - 2))/(f*(2*m + 1)), x] + Dist[d^2/(a*b*(2*m + 1)), Int[(a + b*Csc[e + f*x])^(m + 1)*(d*Csc[e + f*x])^(n - 2)*(b*(n - 2) + a*(m - n + 2)*Csc[e + f*x]), x], x] /; FreeQ[{a, b, d, e, f}, x] && EqQ[a^2 - b^2, 0] && LtQ[m, -1] && GtQ[n, 2] && (IntegersQ[2*m, 2*n] || IntegerQ[m])
  3817. Int[Times[Power[Times[csc[Plus[Optional[Pattern[e, Blank[]]], Times[Optional[Pattern[f, Blank[]]], Pattern[x, Blank[]]]]], Optional[Pattern[d, Blank[]]]], Pattern[n, Blank[]]], Power[Plus[Times[csc[Plus[Optional[Pattern[e, Blank[]]], Times[Optional[Pattern[f, Blank[]]], Pattern[x, Blank[]]]]], Optional[Pattern[b, Blank[]]]], Pattern[a, Blank[]]], Pattern[m, Blank[]]]], Pattern[x, Blank[Symbol]]] := -Simp[(Cot[e + f*x]*(a + b*Csc[e + f*x])^m*(d*Csc[e + f*x])^n)/(f*(2*m + 1)), x] + Dist[1/(a^2*(2*m + 1)), Int[(a + b*Csc[e + f*x])^(m + 1)*(d*Csc[e + f*x])^n*(a*(2*m + n + 1) - b*(m + n + 1)*Csc[e + f*x]), x], x] /; FreeQ[{a, b, d, e, f, n}, x] && EqQ[a^2 - b^2, 0] && LtQ[m, -1] && (IntegersQ[2*m, 2*n] || IntegerQ[m])
  3818. Int[Times[Power[Times[csc[Plus[Optional[Pattern[e, Blank[]]], Times[Optional[Pattern[f, Blank[]]], Pattern[x, Blank[]]]]], Optional[Pattern[d, Blank[]]]], Pattern[n, Blank[]]], Power[Plus[Times[csc[Plus[Optional[Pattern[e, Blank[]]], Times[Optional[Pattern[f, Blank[]]], Pattern[x, Blank[]]]]], Optional[Pattern[b, Blank[]]]], Pattern[a, Blank[]]], -1]], Pattern[x, Blank[Symbol]]] := Simp[(d^2*Cot[e + f*x]*(d*Csc[e + f*x])^(n - 2))/(f*(a + b*Csc[e + f*x])), x] - Dist[d^2/(a*b), Int[(d*Csc[e + f*x])^(n - 2)*(b*(n - 2) - a*(n - 1)*Csc[e + f*x]), x], x] /; FreeQ[{a, b, d, e, f}, x] && EqQ[a^2 - b^2, 0] && GtQ[n, 1]
  3819. Int[Times[Power[Times[csc[Plus[Optional[Pattern[e, Blank[]]], Times[Optional[Pattern[f, Blank[]]], Pattern[x, Blank[]]]]], Optional[Pattern[d, Blank[]]]], Pattern[n, Blank[]]], Power[Plus[Times[csc[Plus[Optional[Pattern[e, Blank[]]], Times[Optional[Pattern[f, Blank[]]], Pattern[x, Blank[]]]]], Optional[Pattern[b, Blank[]]]], Pattern[a, Blank[]]], -1]], Pattern[x, Blank[Symbol]]] := Simp[(Cot[e + f*x]*(d*Csc[e + f*x])^n)/(f*(a + b*Csc[e + f*x])), x] - Dist[1/a^2, Int[(d*Csc[e + f*x])^n*(a*(n - 1) - b*n*Csc[e + f*x]), x], x] /; FreeQ[{a, b, d, e, f}, x] && EqQ[a^2 - b^2, 0] && LtQ[n, 0]
  3820. Int[Times[Power[Times[csc[Plus[Optional[Pattern[e, Blank[]]], Times[Optional[Pattern[f, Blank[]]], Pattern[x, Blank[]]]]], Optional[Pattern[d, Blank[]]]], Pattern[n, Blank[]]], Power[Plus[Times[csc[Plus[Optional[Pattern[e, Blank[]]], Times[Optional[Pattern[f, Blank[]]], Pattern[x, Blank[]]]]], Optional[Pattern[b, Blank[]]]], Pattern[a, Blank[]]], -1]], Pattern[x, Blank[Symbol]]] := -Simp[(b*d*Cot[e + f*x]*(d*Csc[e + f*x])^(n - 1))/(a*f*(a + b*Csc[e + f*x])), x] + Dist[(d*(n - 1))/(a*b), Int[(d*Csc[e + f*x])^(n - 1)*(a - b*Csc[e + f*x]), x], x] /; FreeQ[{a, b, d, e, f, n}, x] && EqQ[a^2 - b^2, 0]
  3821. Int[Times[Power[Times[csc[Plus[Optional[Pattern[e, Blank[]]], Times[Optional[Pattern[f, Blank[]]], Pattern[x, Blank[]]]]], Optional[Pattern[d, Blank[]]]], Rational[3, 2]], Power[Plus[Times[csc[Plus[Optional[Pattern[e, Blank[]]], Times[Optional[Pattern[f, Blank[]]], Pattern[x, Blank[]]]]], Optional[Pattern[b, Blank[]]]], Pattern[a, Blank[]]], Rational[-1, 2]]], Pattern[x, Blank[Symbol]]] := Dist[d/b, Int[Sqrt[a + b*Csc[e + f*x]]*Sqrt[d*Csc[e + f*x]], x], x] - Dist[(a*d)/b, Int[Sqrt[d*Csc[e + f*x]]/Sqrt[a + b*Csc[e + f*x]], x], x] /; FreeQ[{a, b, d, e, f}, x] && EqQ[a^2 - b^2, 0]
  3822. Int[Times[Power[Times[csc[Plus[Optional[Pattern[e, Blank[]]], Times[Optional[Pattern[f, Blank[]]], Pattern[x, Blank[]]]]], Optional[Pattern[d, Blank[]]]], Pattern[n, Blank[]]], Power[Plus[Times[csc[Plus[Optional[Pattern[e, Blank[]]], Times[Optional[Pattern[f, Blank[]]], Pattern[x, Blank[]]]]], Optional[Pattern[b, Blank[]]]], Pattern[a, Blank[]]], Rational[-1, 2]]], Pattern[x, Blank[Symbol]]] := Simp[(-2*d^2*Cot[e + f*x]*(d*Csc[e + f*x])^(n - 2))/(f*(2*n - 3)*Sqrt[a + b*Csc[e + f*x]]), x] + Dist[d^2/(b*(2*n - 3)), Int[((d*Csc[e + f*x])^(n - 2)*(2*b*(n - 2) - a*Csc[e + f*x]))/Sqrt[a + b*Csc[e + f*x]], x], x] /; FreeQ[{a, b, d, e, f}, x] && EqQ[a^2 - b^2, 0] && GtQ[n, 2] && IntegerQ[2*n]
  3823. Int[Times[Power[Times[csc[Plus[Optional[Pattern[e, Blank[]]], Times[Optional[Pattern[f, Blank[]]], Pattern[x, Blank[]]]]], Optional[Pattern[d, Blank[]]]], Pattern[n, Blank[]]], Power[Plus[Times[csc[Plus[Optional[Pattern[e, Blank[]]], Times[Optional[Pattern[f, Blank[]]], Pattern[x, Blank[]]]]], Optional[Pattern[b, Blank[]]]], Pattern[a, Blank[]]], Rational[-1, 2]]], Pattern[x, Blank[Symbol]]] := Simp[(Cot[e + f*x]*(d*Csc[e + f*x])^n)/(f*n*Sqrt[a + b*Csc[e + f*x]]), x] + Dist[1/(2*b*d*n), Int[((d*Csc[e + f*x])^(n + 1)*(a + b*(2*n + 1)*Csc[e + f*x]))/Sqrt[a + b*Csc[e + f*x]], x], x] /; FreeQ[{a, b, d, e, f}, x] && EqQ[a^2 - b^2, 0] && LtQ[n, 0] && IntegerQ[2*n]
  3824. Int[Times[Power[Times[csc[Plus[Optional[Pattern[e, Blank[]]], Times[Optional[Pattern[f, Blank[]]], Pattern[x, Blank[]]]]], Optional[Pattern[d, Blank[]]]], Pattern[n, Blank[]]], Power[Plus[Times[csc[Plus[Optional[Pattern[e, Blank[]]], Times[Optional[Pattern[f, Blank[]]], Pattern[x, Blank[]]]]], Optional[Pattern[b, Blank[]]]], Pattern[a, Blank[]]], Pattern[m, Blank[]]]], Pattern[x, Blank[Symbol]]] := -Simp[(d^2*Cot[e + f*x]*(a + b*Csc[e + f*x])^m*(d*Csc[e + f*x])^(n - 2))/(f*(m + n - 1)), x] + Dist[d^2/(b*(m + n - 1)), Int[(a + b*Csc[e + f*x])^m*(d*Csc[e + f*x])^(n - 2)*(b*(n - 2) + a*m*Csc[e + f*x]), x], x] /; FreeQ[{a, b, d, e, f, m}, x] && EqQ[a^2 - b^2, 0] && GtQ[n, 2] && NeQ[m + n - 1, 0] && IntegerQ[n]
  3825. Int[Times[Power[Times[csc[Plus[Optional[Pattern[e, Blank[]]], Times[Optional[Pattern[f, Blank[]]], Pattern[x, Blank[]]]]], Optional[Pattern[d, Blank[]]]], Pattern[n, Blank[]]], Power[Plus[Times[csc[Plus[Optional[Pattern[e, Blank[]]], Times[Optional[Pattern[f, Blank[]]], Pattern[x, Blank[]]]]], Optional[Pattern[b, Blank[]]]], Pattern[a, Blank[]]], Pattern[m, Blank[]]]], Pattern[x, Blank[Symbol]]] := -Dist[(((a*d)/b)^n*Cot[e + f*x])/(a^(n - 2)*f*Sqrt[a + b*Csc[e + f*x]]*Sqrt[a - b*Csc[e + f*x]]), Subst[Int[((a - x)^(n - 1)*(2*a - x)^(m - 1/2))/Sqrt[x], x], x, a - b*Csc[e + f*x]], x] /; FreeQ[{a, b, d, e, f, m, n}, x] && EqQ[a^2 - b^2, 0] && !IntegerQ[m] && GtQ[a, 0] && !IntegerQ[n] && GtQ[(a*d)/b, 0]
  3826. Int[Times[Power[Times[csc[Plus[Optional[Pattern[e, Blank[]]], Times[Optional[Pattern[f, Blank[]]], Pattern[x, Blank[]]]]], Optional[Pattern[d, Blank[]]]], Pattern[n, Blank[]]], Power[Plus[Times[csc[Plus[Optional[Pattern[e, Blank[]]], Times[Optional[Pattern[f, Blank[]]], Pattern[x, Blank[]]]]], Optional[Pattern[b, Blank[]]]], Pattern[a, Blank[]]], Pattern[m, Blank[]]]], Pattern[x, Blank[Symbol]]] := -Dist[((-((a*d)/b))^n*Cot[e + f*x])/(a^(n - 1)*f*Sqrt[a + b*Csc[e + f*x]]*Sqrt[a - b*Csc[e + f*x]]), Subst[Int[(x^(m - 1/2)*(a - x)^(n - 1))/Sqrt[2*a - x], x], x, a + b*Csc[e + f*x]], x] /; FreeQ[{a, b, d, e, f, m, n}, x] && EqQ[a^2 - b^2, 0] && !IntegerQ[m] && GtQ[a, 0] && !IntegerQ[n] && LtQ[(a*d)/b, 0]
  3827. Int[Times[Power[Times[csc[Plus[Optional[Pattern[e, Blank[]]], Times[Optional[Pattern[f, Blank[]]], Pattern[x, Blank[]]]]], Optional[Pattern[d, Blank[]]]], Optional[Pattern[n, Blank[]]]], Power[Plus[Times[csc[Plus[Optional[Pattern[e, Blank[]]], Times[Optional[Pattern[f, Blank[]]], Pattern[x, Blank[]]]]], Optional[Pattern[b, Blank[]]]], Pattern[a, Blank[]]], Pattern[m, Blank[]]]], Pattern[x, Blank[Symbol]]] := Dist[(a^2*d*Cot[e + f*x])/(f*Sqrt[a + b*Csc[e + f*x]]*Sqrt[a - b*Csc[e + f*x]]), Subst[Int[((d*x)^(n - 1)*(a + b*x)^(m - 1/2))/Sqrt[a - b*x], x], x, Csc[e + f*x]], x] /; FreeQ[{a, b, d, e, f, m, n}, x] && EqQ[a^2 - b^2, 0] && !IntegerQ[m] && GtQ[a, 0]
  3828. Int[Times[Power[Times[csc[Plus[Optional[Pattern[e, Blank[]]], Times[Optional[Pattern[f, Blank[]]], Pattern[x, Blank[]]]]], Optional[Pattern[d, Blank[]]]], Optional[Pattern[n, Blank[]]]], Power[Plus[Times[csc[Plus[Optional[Pattern[e, Blank[]]], Times[Optional[Pattern[f, Blank[]]], Pattern[x, Blank[]]]]], Optional[Pattern[b, Blank[]]]], Pattern[a, Blank[]]], Pattern[m, Blank[]]]], Pattern[x, Blank[Symbol]]] := Dist[(a^IntPart[m]*(a + b*Csc[e + f*x])^FracPart[m])/(1 + (b*Csc[e + f*x])/a)^FracPart[m], Int[(1 + (b*Csc[e + f*x])/a)^m*(d*Csc[e + f*x])^n, x], x] /; FreeQ[{a, b, d, e, f, m, n}, x] && EqQ[a^2 - b^2, 0] && !IntegerQ[m] && !GtQ[a, 0]
  3829. Int[Times[csc[Plus[Optional[Pattern[e, Blank[]]], Times[Optional[Pattern[f, Blank[]]], Pattern[x, Blank[]]]]], Power[Plus[Times[csc[Plus[Optional[Pattern[e, Blank[]]], Times[Optional[Pattern[f, Blank[]]], Pattern[x, Blank[]]]]], Optional[Pattern[b, Blank[]]]], Pattern[a, Blank[]]], Rational[1, 2]]], Pattern[x, Blank[Symbol]]] := Dist[a - b, Int[Csc[e + f*x]/Sqrt[a + b*Csc[e + f*x]], x], x] + Dist[b, Int[(Csc[e + f*x]*(1 + Csc[e + f*x]))/Sqrt[a + b*Csc[e + f*x]], x], x] /; FreeQ[{a, b, e, f}, x] && NeQ[a^2 - b^2, 0]
  3830. Int[Times[csc[Plus[Optional[Pattern[e, Blank[]]], Times[Optional[Pattern[f, Blank[]]], Pattern[x, Blank[]]]]], Power[Plus[Times[csc[Plus[Optional[Pattern[e, Blank[]]], Times[Optional[Pattern[f, Blank[]]], Pattern[x, Blank[]]]]], Optional[Pattern[b, Blank[]]]], Pattern[a, Blank[]]], Pattern[m, Blank[]]]], Pattern[x, Blank[Symbol]]] := -Simp[(b*Cot[e + f*x]*(a + b*Csc[e + f*x])^(m - 1))/(f*m), x] + Dist[1/m, Int[Csc[e + f*x]*(a + b*Csc[e + f*x])^(m - 2)*(b^2*(m - 1) + a^2*m + a*b*(2*m - 1)*Csc[e + f*x]), x], x] /; FreeQ[{a, b, e, f}, x] && NeQ[a^2 - b^2, 0] && GtQ[m, 1] && IntegerQ[2*m]
  3831. Int[Times[csc[Plus[Optional[Pattern[e, Blank[]]], Times[Optional[Pattern[f, Blank[]]], Pattern[x, Blank[]]]]], Power[Plus[Times[csc[Plus[Optional[Pattern[e, Blank[]]], Times[Optional[Pattern[f, Blank[]]], Pattern[x, Blank[]]]]], Optional[Pattern[b, Blank[]]]], Pattern[a, Blank[]]], -1]], Pattern[x, Blank[Symbol]]] := Dist[1/b, Int[1/(1 + (a*Sin[e + f*x])/b), x], x] /; FreeQ[{a, b, e, f}, x] && NeQ[a^2 - b^2, 0]
  3832. Int[Times[csc[Plus[Optional[Pattern[e, Blank[]]], Times[Optional[Pattern[f, Blank[]]], Pattern[x, Blank[]]]]], Power[Plus[Times[csc[Plus[Optional[Pattern[e, Blank[]]], Times[Optional[Pattern[f, Blank[]]], Pattern[x, Blank[]]]]], Optional[Pattern[b, Blank[]]]], Pattern[a, Blank[]]], Rational[-1, 2]]], Pattern[x, Blank[Symbol]]] := Simp[(-2*Rt[a + b, 2]*Sqrt[(b*(1 - Csc[e + f*x]))/(a + b)]*Sqrt[-((b*(1 + Csc[e + f*x]))/(a - b))]*EllipticF[ArcSin[Sqrt[a + b*Csc[e + f*x]]/Rt[a + b, 2]], (a + b)/(a - b)])/(b*f*Cot[e + f*x]), x] /; FreeQ[{a, b, e, f}, x] && NeQ[a^2 - b^2, 0]
  3833. Int[Times[csc[Plus[Optional[Pattern[e, Blank[]]], Times[Optional[Pattern[f, Blank[]]], Pattern[x, Blank[]]]]], Power[Plus[Times[csc[Plus[Optional[Pattern[e, Blank[]]], Times[Optional[Pattern[f, Blank[]]], Pattern[x, Blank[]]]]], Optional[Pattern[b, Blank[]]]], Pattern[a, Blank[]]], Pattern[m, Blank[]]]], Pattern[x, Blank[Symbol]]] := -Simp[(b*Cot[e + f*x]*(a + b*Csc[e + f*x])^(m + 1))/(f*(m + 1)*(a^2 - b^2)), x] + Dist[1/((m + 1)*(a^2 - b^2)), Int[Csc[e + f*x]*(a + b*Csc[e + f*x])^(m + 1)*(a*(m + 1) - b*(m + 2)*Csc[e + f*x]), x], x] /; FreeQ[{a, b, e, f}, x] && NeQ[a^2 - b^2, 0] && LtQ[m, -1] && IntegerQ[2*m]
  3834. Int[Times[csc[Plus[Optional[Pattern[e, Blank[]]], Times[Optional[Pattern[f, Blank[]]], Pattern[x, Blank[]]]]], Power[Plus[Times[csc[Plus[Optional[Pattern[e, Blank[]]], Times[Optional[Pattern[f, Blank[]]], Pattern[x, Blank[]]]]], Optional[Pattern[b, Blank[]]]], Pattern[a, Blank[]]], Pattern[m, Blank[]]]], Pattern[x, Blank[Symbol]]] := Dist[Cot[e + f*x]/(f*Sqrt[1 + Csc[e + f*x]]*Sqrt[1 - Csc[e + f*x]]), Subst[Int[(a + b*x)^m/(Sqrt[1 + x]*Sqrt[1 - x]), x], x, Csc[e + f*x]], x] /; FreeQ[{a, b, e, f, m}, x] && NeQ[a^2 - b^2, 0] && !IntegerQ[2*m]
  3835. Int[Times[Power[csc[Plus[Optional[Pattern[e, Blank[]]], Times[Optional[Pattern[f, Blank[]]], Pattern[x, Blank[]]]]], 2], Power[Plus[Times[csc[Plus[Optional[Pattern[e, Blank[]]], Times[Optional[Pattern[f, Blank[]]], Pattern[x, Blank[]]]]], Optional[Pattern[b, Blank[]]]], Pattern[a, Blank[]]], Pattern[m, Blank[]]]], Pattern[x, Blank[Symbol]]] := -Simp[(Cot[e + f*x]*(a + b*Csc[e + f*x])^m)/(f*(m + 1)), x] + Dist[m/(m + 1), Int[Csc[e + f*x]*(a + b*Csc[e + f*x])^(m - 1)*(b + a*Csc[e + f*x]), x], x] /; FreeQ[{a, b, e, f}, x] && NeQ[a^2 - b^2, 0] && GtQ[m, 0]
  3836. Int[Times[Power[csc[Plus[Optional[Pattern[e, Blank[]]], Times[Optional[Pattern[f, Blank[]]], Pattern[x, Blank[]]]]], 2], Power[Plus[Times[csc[Plus[Optional[Pattern[e, Blank[]]], Times[Optional[Pattern[f, Blank[]]], Pattern[x, Blank[]]]]], Optional[Pattern[b, Blank[]]]], Pattern[a, Blank[]]], Pattern[m, Blank[]]]], Pattern[x, Blank[Symbol]]] := Simp[(a*Cot[e + f*x]*(a + b*Csc[e + f*x])^(m + 1))/(f*(m + 1)*(a^2 - b^2)), x] - Dist[1/((m + 1)*(a^2 - b^2)), Int[Csc[e + f*x]*(a + b*Csc[e + f*x])^(m + 1)*(b*(m + 1) - a*(m + 2)*Csc[e + f*x]), x], x] /; FreeQ[{a, b, e, f}, x] && NeQ[a^2 - b^2, 0] && LtQ[m, -1]
  3837. Int[Times[Power[csc[Plus[Optional[Pattern[e, Blank[]]], Times[Optional[Pattern[f, Blank[]]], Pattern[x, Blank[]]]]], 2], Power[Plus[Times[csc[Plus[Optional[Pattern[e, Blank[]]], Times[Optional[Pattern[f, Blank[]]], Pattern[x, Blank[]]]]], Optional[Pattern[b, Blank[]]]], Pattern[a, Blank[]]], Rational[-1, 2]]], Pattern[x, Blank[Symbol]]] := -Int[Csc[e + f*x]/Sqrt[a + b*Csc[e + f*x]], x] + Int[(Csc[e + f*x]*(1 + Csc[e + f*x]))/Sqrt[a + b*Csc[e + f*x]], x] /; FreeQ[{a, b, e, f}, x] && NeQ[a^2 - b^2, 0]
  3838. Int[Times[Power[csc[Plus[Optional[Pattern[e, Blank[]]], Times[Optional[Pattern[f, Blank[]]], Pattern[x, Blank[]]]]], 2], Power[Plus[Times[csc[Plus[Optional[Pattern[e, Blank[]]], Times[Optional[Pattern[f, Blank[]]], Pattern[x, Blank[]]]]], Optional[Pattern[b, Blank[]]]], Pattern[a, Blank[]]], Pattern[m, Blank[]]]], Pattern[x, Blank[Symbol]]] := -Dist[a/b, Int[Csc[e + f*x]*(a + b*Csc[e + f*x])^m, x], x] + Dist[1/b, Int[Csc[e + f*x]*(a + b*Csc[e + f*x])^(m + 1), x], x] /; FreeQ[{a, b, e, f, m}, x] && NeQ[a^2 - b^2, 0]
  3839. Int[Times[Power[csc[Plus[Optional[Pattern[e, Blank[]]], Times[Optional[Pattern[f, Blank[]]], Pattern[x, Blank[]]]]], 3], Power[Plus[Times[csc[Plus[Optional[Pattern[e, Blank[]]], Times[Optional[Pattern[f, Blank[]]], Pattern[x, Blank[]]]]], Optional[Pattern[b, Blank[]]]], Pattern[a, Blank[]]], Pattern[m, Blank[]]]], Pattern[x, Blank[Symbol]]] := -Simp[(a^2*Cot[e + f*x]*(a + b*Csc[e + f*x])^(m + 1))/(b*f*(m + 1)*(a^2 - b^2)), x] + Dist[1/(b*(m + 1)*(a^2 - b^2)), Int[Csc[e + f*x]*(a + b*Csc[e + f*x])^(m + 1)*Simp[a*b*(m + 1) - (a^2 + b^2*(m + 1))*Csc[e + f*x], x], x], x] /; FreeQ[{a, b, e, f}, x] && NeQ[a^2 - b^2, 0] && LtQ[m, -1]
  3840. Int[Times[Power[csc[Plus[Optional[Pattern[e, Blank[]]], Times[Optional[Pattern[f, Blank[]]], Pattern[x, Blank[]]]]], 3], Power[Plus[Times[csc[Plus[Optional[Pattern[e, Blank[]]], Times[Optional[Pattern[f, Blank[]]], Pattern[x, Blank[]]]]], Optional[Pattern[b, Blank[]]]], Pattern[a, Blank[]]], Pattern[m, Blank[]]]], Pattern[x, Blank[Symbol]]] := -Simp[(Cot[e + f*x]*(a + b*Csc[e + f*x])^(m + 1))/(b*f*(m + 2)), x] + Dist[1/(b*(m + 2)), Int[Csc[e + f*x]*(a + b*Csc[e + f*x])^m*(b*(m + 1) - a*Csc[e + f*x]), x], x] /; FreeQ[{a, b, e, f, m}, x] && NeQ[a^2 - b^2, 0] && !LtQ[m, -1]
  3841. Int[Times[Power[Times[csc[Plus[Optional[Pattern[e, Blank[]]], Times[Optional[Pattern[f, Blank[]]], Pattern[x, Blank[]]]]], Optional[Pattern[d, Blank[]]]], Pattern[n, Blank[]]], Power[Plus[Times[csc[Plus[Optional[Pattern[e, Blank[]]], Times[Optional[Pattern[f, Blank[]]], Pattern[x, Blank[]]]]], Optional[Pattern[b, Blank[]]]], Pattern[a, Blank[]]], Pattern[m, Blank[]]]], Pattern[x, Blank[Symbol]]] := Simp[(a^2*Cot[e + f*x]*(a + b*Csc[e + f*x])^(m - 2)*(d*Csc[e + f*x])^n)/(f*n), x] - Dist[1/(d*n), Int[(a + b*Csc[e + f*x])^(m - 3)*(d*Csc[e + f*x])^(n + 1)*Simp[a^2*b*(m - 2*n - 2) - a*(3*b^2*n + a^2*(n + 1))*Csc[e + f*x] - b*(b^2*n + a^2*(m + n - 1))*Csc[e + f*x]^2, x], x], x] /; FreeQ[{a, b, d, e, f}, x] && NeQ[a^2 - b^2, 0] && GtQ[m, 2] && ((IntegerQ[m] && LtQ[n, -1]) || (IntegersQ[m + 1/2, 2*n] && LeQ[n, -1]))
  3842. Int[Times[Power[Times[csc[Plus[Optional[Pattern[e, Blank[]]], Times[Optional[Pattern[f, Blank[]]], Pattern[x, Blank[]]]]], Optional[Pattern[d, Blank[]]]], Pattern[n, Blank[]]], Power[Plus[Times[csc[Plus[Optional[Pattern[e, Blank[]]], Times[Optional[Pattern[f, Blank[]]], Pattern[x, Blank[]]]]], Optional[Pattern[b, Blank[]]]], Pattern[a, Blank[]]], Pattern[m, Blank[]]]], Pattern[x, Blank[Symbol]]] := -Simp[(b^2*Cot[e + f*x]*(a + b*Csc[e + f*x])^(m - 2)*(d*Csc[e + f*x])^n)/(f*(m + n - 1)), x] + Dist[1/(d*(m + n - 1)), Int[(a + b*Csc[e + f*x])^(m - 3)*(d*Csc[e + f*x])^n*Simp[a^3*d*(m + n - 1) + a*b^2*d*n + b*(b^2*d*(m + n - 2) + 3*a^2*d*(m + n - 1))*Csc[e + f*x] + a*b^2*d*(3*m + 2*n - 4)*Csc[e + f*x]^2, x], x], x] /; FreeQ[{a, b, d, e, f, n}, x] && NeQ[a^2 - b^2, 0] && GtQ[m, 2] && (IntegerQ[m] || IntegersQ[2*m, 2*n]) && !(IGtQ[n, 2] && !IntegerQ[m])
  3843. Int[Times[Power[Times[csc[Plus[Optional[Pattern[e, Blank[]]], Times[Optional[Pattern[f, Blank[]]], Pattern[x, Blank[]]]]], Optional[Pattern[d, Blank[]]]], Pattern[n, Blank[]]], Power[Plus[Times[csc[Plus[Optional[Pattern[e, Blank[]]], Times[Optional[Pattern[f, Blank[]]], Pattern[x, Blank[]]]]], Optional[Pattern[b, Blank[]]]], Pattern[a, Blank[]]], Pattern[m, Blank[]]]], Pattern[x, Blank[Symbol]]] := -Simp[(b*d*Cot[e + f*x]*(a + b*Csc[e + f*x])^(m + 1)*(d*Csc[e + f*x])^(n - 1))/(f*(m + 1)*(a^2 - b^2)), x] + Dist[1/((m + 1)*(a^2 - b^2)), Int[(a + b*Csc[e + f*x])^(m + 1)*(d*Csc[e + f*x])^(n - 1)*Simp[b*d*(n - 1) + a*d*(m + 1)*Csc[e + f*x] - b*d*(m + n + 1)*Csc[e + f*x]^2, x], x], x] /; FreeQ[{a, b, d, e, f}, x] && NeQ[a^2 - b^2, 0] && LtQ[m, -1] && LtQ[0, n, 1] && IntegersQ[2*m, 2*n]
  3844. Int[Times[Power[Times[csc[Plus[Optional[Pattern[e, Blank[]]], Times[Optional[Pattern[f, Blank[]]], Pattern[x, Blank[]]]]], Optional[Pattern[d, Blank[]]]], Pattern[n, Blank[]]], Power[Plus[Times[csc[Plus[Optional[Pattern[e, Blank[]]], Times[Optional[Pattern[f, Blank[]]], Pattern[x, Blank[]]]]], Optional[Pattern[b, Blank[]]]], Pattern[a, Blank[]]], Pattern[m, Blank[]]]], Pattern[x, Blank[Symbol]]] := Simp[(a*d^2*Cot[e + f*x]*(a + b*Csc[e + f*x])^(m + 1)*(d*Csc[e + f*x])^(n - 2))/(f*(m + 1)*(a^2 - b^2)), x] - Dist[d^2/((m + 1)*(a^2 - b^2)), Int[(a + b*Csc[e + f*x])^(m + 1)*(d*Csc[e + f*x])^(n - 2)*(a*(n - 2) + b*(m + 1)*Csc[e + f*x] - a*(m + n)*Csc[e + f*x]^2), x], x] /; FreeQ[{a, b, d, e, f}, x] && NeQ[a^2 - b^2, 0] && LtQ[m, -1] && LtQ[1, n, 2] && IntegersQ[2*m, 2*n]
  3845. Int[Times[Power[Times[csc[Plus[Optional[Pattern[e, Blank[]]], Times[Optional[Pattern[f, Blank[]]], Pattern[x, Blank[]]]]], Optional[Pattern[d, Blank[]]]], Pattern[n, Blank[]]], Power[Plus[Times[csc[Plus[Optional[Pattern[e, Blank[]]], Times[Optional[Pattern[f, Blank[]]], Pattern[x, Blank[]]]]], Optional[Pattern[b, Blank[]]]], Pattern[a, Blank[]]], Pattern[m, Blank[]]]], Pattern[x, Blank[Symbol]]] := -Simp[(a^2*d^3*Cot[e + f*x]*(a + b*Csc[e + f*x])^(m + 1)*(d*Csc[e + f*x])^(n - 3))/(b*f*(m + 1)*(a^2 - b^2)), x] + Dist[d^3/(b*(m + 1)*(a^2 - b^2)), Int[(a + b*Csc[e + f*x])^(m + 1)*(d*Csc[e + f*x])^(n - 3)*Simp[a^2*(n - 3) + a*b*(m + 1)*Csc[e + f*x] - (a^2*(n - 2) + b^2*(m + 1))*Csc[e + f*x]^2, x], x], x] /; FreeQ[{a, b, d, e, f}, x] && NeQ[a^2 - b^2, 0] && LtQ[m, -1] && (IGtQ[n, 3] || (IntegersQ[n + 1/2, 2*m] && GtQ[n, 2]))
  3846. Int[Times[Power[Times[csc[Plus[Optional[Pattern[e, Blank[]]], Times[Optional[Pattern[f, Blank[]]], Pattern[x, Blank[]]]]], Optional[Pattern[d, Blank[]]]], Pattern[n, Blank[]]], Power[Plus[Times[csc[Plus[Optional[Pattern[e, Blank[]]], Times[Optional[Pattern[f, Blank[]]], Pattern[x, Blank[]]]]], Optional[Pattern[b, Blank[]]]], Pattern[a, Blank[]]], Pattern[m, Blank[]]]], Pattern[x, Blank[Symbol]]] := Simp[(Cot[e + f*x]*(a + b*Csc[e + f*x])^(m + 1)*(d*Csc[e + f*x])^n)/(a*f*n), x] - Dist[1/(a*d*n), Int[(a + b*Csc[e + f*x])^m*(d*Csc[e + f*x])^(n + 1)*Simp[b*(m + n + 1) - a*(n + 1)*Csc[e + f*x] - b*(m + n + 2)*Csc[e + f*x]^2, x], x], x] /; FreeQ[{a, b, d, e, f}, x] && NeQ[a^2 - b^2, 0] && ILtQ[m + 1/2, 0] && ILtQ[n, 0]
  3847. Int[Times[Power[Times[csc[Plus[Optional[Pattern[e, Blank[]]], Times[Optional[Pattern[f, Blank[]]], Pattern[x, Blank[]]]]], Optional[Pattern[d, Blank[]]]], Pattern[n, Blank[]]], Power[Plus[Times[csc[Plus[Optional[Pattern[e, Blank[]]], Times[Optional[Pattern[f, Blank[]]], Pattern[x, Blank[]]]]], Optional[Pattern[b, Blank[]]]], Pattern[a, Blank[]]], Pattern[m, Blank[]]]], Pattern[x, Blank[Symbol]]] := Simp[(b^2*Cot[e + f*x]*(a + b*Csc[e + f*x])^(m + 1)*(d*Csc[e + f*x])^n)/(a*f*(m + 1)*(a^2 - b^2)), x] + Dist[1/(a*(m + 1)*(a^2 - b^2)), Int[(a + b*Csc[e + f*x])^(m + 1)*(d*Csc[e + f*x])^n*(a^2*(m + 1) - b^2*(m + n + 1) - a*b*(m + 1)*Csc[e + f*x] + b^2*(m + n + 2)*Csc[e + f*x]^2), x], x] /; FreeQ[{a, b, d, e, f, n}, x] && NeQ[a^2 - b^2, 0] && LtQ[m, -1] && IntegersQ[2*m, 2*n]
  3848. Int[Times[Power[Times[csc[Plus[Optional[Pattern[e, Blank[]]], Times[Optional[Pattern[f, Blank[]]], Pattern[x, Blank[]]]]], Optional[Pattern[d, Blank[]]]], Rational[1, 2]], Power[Plus[Times[csc[Plus[Optional[Pattern[e, Blank[]]], Times[Optional[Pattern[f, Blank[]]], Pattern[x, Blank[]]]]], Optional[Pattern[b, Blank[]]]], Pattern[a, Blank[]]], -1]], Pattern[x, Blank[Symbol]]] := Dist[(Sqrt[d*Sin[e + f*x]]*Sqrt[d*Csc[e + f*x]])/d, Int[Sqrt[d*Sin[e + f*x]]/(b + a*Sin[e + f*x]), x], x] /; FreeQ[{a, b, d, e, f}, x] && NeQ[a^2 - b^2, 0]
  3849. Int[Times[Power[Times[csc[Plus[Optional[Pattern[e, Blank[]]], Times[Optional[Pattern[f, Blank[]]], Pattern[x, Blank[]]]]], Optional[Pattern[d, Blank[]]]], Rational[3, 2]], Power[Plus[Times[csc[Plus[Optional[Pattern[e, Blank[]]], Times[Optional[Pattern[f, Blank[]]], Pattern[x, Blank[]]]]], Optional[Pattern[b, Blank[]]]], Pattern[a, Blank[]]], -1]], Pattern[x, Blank[Symbol]]] := Dist[d*Sqrt[d*Sin[e + f*x]]*Sqrt[d*Csc[e + f*x]], Int[1/(Sqrt[d*Sin[e + f*x]]*(b + a*Sin[e + f*x])), x], x] /; FreeQ[{a, b, d, e, f}, x] && NeQ[a^2 - b^2, 0]
  3850. Int[Times[Power[Times[csc[Plus[Optional[Pattern[e, Blank[]]], Times[Optional[Pattern[f, Blank[]]], Pattern[x, Blank[]]]]], Optional[Pattern[d, Blank[]]]], Rational[5, 2]], Power[Plus[Times[csc[Plus[Optional[Pattern[e, Blank[]]], Times[Optional[Pattern[f, Blank[]]], Pattern[x, Blank[]]]]], Optional[Pattern[b, Blank[]]]], Pattern[a, Blank[]]], -1]], Pattern[x, Blank[Symbol]]] := Dist[d/b, Int[(d*Csc[e + f*x])^(3/2), x], x] - Dist[(a*d)/b, Int[(d*Csc[e + f*x])^(3/2)/(a + b*Csc[e + f*x]), x], x] /; FreeQ[{a, b, d, e, f}, x] && NeQ[a^2 - b^2, 0]
  3851. Int[Times[Power[Times[csc[Plus[Optional[Pattern[e, Blank[]]], Times[Optional[Pattern[f, Blank[]]], Pattern[x, Blank[]]]]], Optional[Pattern[d, Blank[]]]], Pattern[n, Blank[]]], Power[Plus[Times[csc[Plus[Optional[Pattern[e, Blank[]]], Times[Optional[Pattern[f, Blank[]]], Pattern[x, Blank[]]]]], Optional[Pattern[b, Blank[]]]], Pattern[a, Blank[]]], -1]], Pattern[x, Blank[Symbol]]] := -Simp[(d^3*Cot[e + f*x]*(d*Csc[e + f*x])^(n - 3))/(b*f*(n - 2)), x] + Dist[d^3/(b*(n - 2)), Int[((d*Csc[e + f*x])^(n - 3)*Simp[a*(n - 3) + b*(n - 3)*Csc[e + f*x] - a*(n - 2)*Csc[e + f*x]^2, x])/(a + b*Csc[e + f*x]), x], x] /; FreeQ[{a, b, d, e, f}, x] && NeQ[a^2 - b^2, 0] && GtQ[n, 3]
  3852. Int[Times[Power[Times[csc[Plus[Optional[Pattern[e, Blank[]]], Times[Optional[Pattern[f, Blank[]]], Pattern[x, Blank[]]]]], Optional[Pattern[d, Blank[]]]], Rational[-1, 2]], Power[Plus[Times[csc[Plus[Optional[Pattern[e, Blank[]]], Times[Optional[Pattern[f, Blank[]]], Pattern[x, Blank[]]]]], Optional[Pattern[b, Blank[]]]], Pattern[a, Blank[]]], -1]], Pattern[x, Blank[Symbol]]] := Dist[b^2/(a^2*d^2), Int[(d*Csc[e + f*x])^(3/2)/(a + b*Csc[e + f*x]), x], x] + Dist[1/a^2, Int[(a - b*Csc[e + f*x])/Sqrt[d*Csc[e + f*x]], x], x] /; FreeQ[{a, b, d, e, f}, x] && NeQ[a^2 - b^2, 0]
  3853. Int[Times[Power[Times[csc[Plus[Optional[Pattern[e, Blank[]]], Times[Optional[Pattern[f, Blank[]]], Pattern[x, Blank[]]]]], Optional[Pattern[d, Blank[]]]], Pattern[n, Blank[]]], Power[Plus[Times[csc[Plus[Optional[Pattern[e, Blank[]]], Times[Optional[Pattern[f, Blank[]]], Pattern[x, Blank[]]]]], Optional[Pattern[b, Blank[]]]], Pattern[a, Blank[]]], -1]], Pattern[x, Blank[Symbol]]] := Simp[(Cot[e + f*x]*(d*Csc[e + f*x])^n)/(a*f*n), x] - Dist[1/(a*d*n), Int[((d*Csc[e + f*x])^(n + 1)*Simp[b*n - a*(n + 1)*Csc[e + f*x] - b*(n + 1)*Csc[e + f*x]^2, x])/(a + b*Csc[e + f*x]), x], x] /; FreeQ[{a, b, d, e, f}, x] && NeQ[a^2 - b^2, 0] && LeQ[n, -1] && IntegerQ[2*n]
  3854. Int[Times[Power[Times[csc[Plus[Optional[Pattern[e, Blank[]]], Times[Optional[Pattern[f, Blank[]]], Pattern[x, Blank[]]]]], Optional[Pattern[d, Blank[]]]], Rational[1, 2]], Power[Plus[Times[csc[Plus[Optional[Pattern[e, Blank[]]], Times[Optional[Pattern[f, Blank[]]], Pattern[x, Blank[]]]]], Optional[Pattern[b, Blank[]]]], Pattern[a, Blank[]]], Rational[1, 2]]], Pattern[x, Blank[Symbol]]] := Dist[a, Int[Sqrt[d*Csc[e + f*x]]/Sqrt[a + b*Csc[e + f*x]], x], x] + Dist[b/d, Int[(d*Csc[e + f*x])^(3/2)/Sqrt[a + b*Csc[e + f*x]], x], x] /; FreeQ[{a, b, d, e, f}, x] && NeQ[a^2 - b^2, 0]
  3855. Int[Times[Power[Times[csc[Plus[Optional[Pattern[e, Blank[]]], Times[Optional[Pattern[f, Blank[]]], Pattern[x, Blank[]]]]], Optional[Pattern[d, Blank[]]]], Pattern[n, Blank[]]], Power[Plus[Times[csc[Plus[Optional[Pattern[e, Blank[]]], Times[Optional[Pattern[f, Blank[]]], Pattern[x, Blank[]]]]], Optional[Pattern[b, Blank[]]]], Pattern[a, Blank[]]], Rational[1, 2]]], Pattern[x, Blank[Symbol]]] := Simp[(-2*d*Cos[e + f*x]*Sqrt[a + b*Csc[e + f*x]]*(d*Csc[e + f*x])^(n - 1))/(f*(2*n - 1)), x] + Dist[d^2/(2*n - 1), Int[((d*Csc[e + f*x])^(n - 2)*Simp[2*a*(n - 2) + b*(2*n - 3)*Csc[e + f*x] + a*Csc[e + f*x]^2, x])/Sqrt[a + b*Csc[e + f*x]], x], x] /; FreeQ[{a, b, d, e, f}, x] && NeQ[a^2 - b^2, 0] && GtQ[n, 1] && IntegerQ[2*n]
  3856. Int[Times[Power[Times[csc[Plus[Optional[Pattern[e, Blank[]]], Times[Optional[Pattern[f, Blank[]]], Pattern[x, Blank[]]]]], Optional[Pattern[d, Blank[]]]], Rational[-1, 2]], Power[Plus[Times[csc[Plus[Optional[Pattern[e, Blank[]]], Times[Optional[Pattern[f, Blank[]]], Pattern[x, Blank[]]]]], Optional[Pattern[b, Blank[]]]], Pattern[a, Blank[]]], Rational[1, 2]]], Pattern[x, Blank[Symbol]]] := Dist[Sqrt[a + b*Csc[e + f*x]]/(Sqrt[d*Csc[e + f*x]]*Sqrt[b + a*Sin[e + f*x]]), Int[Sqrt[b + a*Sin[e + f*x]], x], x] /; FreeQ[{a, b, d, e, f}, x] && NeQ[a^2 - b^2, 0]
  3857. Int[Times[Power[Times[csc[Plus[Optional[Pattern[e, Blank[]]], Times[Optional[Pattern[f, Blank[]]], Pattern[x, Blank[]]]]], Optional[Pattern[d, Blank[]]]], Pattern[n, Blank[]]], Power[Plus[Times[csc[Plus[Optional[Pattern[e, Blank[]]], Times[Optional[Pattern[f, Blank[]]], Pattern[x, Blank[]]]]], Optional[Pattern[b, Blank[]]]], Pattern[a, Blank[]]], Rational[1, 2]]], Pattern[x, Blank[Symbol]]] := Simp[(Cot[e + f*x]*Sqrt[a + b*Csc[e + f*x]]*(d*Csc[e + f*x])^n)/(f*n), x] - Dist[1/(2*d*n), Int[((d*Csc[e + f*x])^(n + 1)*Simp[b - 2*a*(n + 1)*Csc[e + f*x] - b*(2*n + 3)*Csc[e + f*x]^2, x])/Sqrt[a + b*Csc[e + f*x]], x], x] /; FreeQ[{a, b, d, e, f}, x] && NeQ[a^2 - b^2, 0] && LeQ[n, -1] && IntegerQ[2*n]
  3858. Int[Times[Power[Times[csc[Plus[Optional[Pattern[e, Blank[]]], Times[Optional[Pattern[f, Blank[]]], Pattern[x, Blank[]]]]], Optional[Pattern[d, Blank[]]]], Rational[1, 2]], Power[Plus[Times[csc[Plus[Optional[Pattern[e, Blank[]]], Times[Optional[Pattern[f, Blank[]]], Pattern[x, Blank[]]]]], Optional[Pattern[b, Blank[]]]], Pattern[a, Blank[]]], Rational[-1, 2]]], Pattern[x, Blank[Symbol]]] := Dist[(Sqrt[d*Csc[e + f*x]]*Sqrt[b + a*Sin[e + f*x]])/Sqrt[a + b*Csc[e + f*x]], Int[1/Sqrt[b + a*Sin[e + f*x]], x], x] /; FreeQ[{a, b, d, e, f}, x] && NeQ[a^2 - b^2, 0]
  3859. Int[Times[Power[Times[csc[Plus[Optional[Pattern[e, Blank[]]], Times[Optional[Pattern[f, Blank[]]], Pattern[x, Blank[]]]]], Optional[Pattern[d, Blank[]]]], Rational[3, 2]], Power[Plus[Times[csc[Plus[Optional[Pattern[e, Blank[]]], Times[Optional[Pattern[f, Blank[]]], Pattern[x, Blank[]]]]], Optional[Pattern[b, Blank[]]]], Pattern[a, Blank[]]], Rational[-1, 2]]], Pattern[x, Blank[Symbol]]] := Dist[(d*Sqrt[d*Csc[e + f*x]]*Sqrt[b + a*Sin[e + f*x]])/Sqrt[a + b*Csc[e + f*x]], Int[1/(Sin[e + f*x]*Sqrt[b + a*Sin[e + f*x]]), x], x] /; FreeQ[{a, b, d, e, f}, x] && NeQ[a^2 - b^2, 0]
  3860. Int[Times[Power[Times[csc[Plus[Optional[Pattern[e, Blank[]]], Times[Optional[Pattern[f, Blank[]]], Pattern[x, Blank[]]]]], Optional[Pattern[d, Blank[]]]], Pattern[n, Blank[]]], Power[Plus[Times[csc[Plus[Optional[Pattern[e, Blank[]]], Times[Optional[Pattern[f, Blank[]]], Pattern[x, Blank[]]]]], Optional[Pattern[b, Blank[]]]], Pattern[a, Blank[]]], Rational[-1, 2]]], Pattern[x, Blank[Symbol]]] := Simp[(-2*d^2*Cos[e + f*x]*(d*Csc[e + f*x])^(n - 2)*Sqrt[a + b*Csc[e + f*x]])/(b*f*(2*n - 3)), x] + Dist[d^3/(b*(2*n - 3)), Int[((d*Csc[e + f*x])^(n - 3)*Simp[2*a*(n - 3) + b*(2*n - 5)*Csc[e + f*x] - 2*a*(n - 2)*Csc[e + f*x]^2, x])/Sqrt[a + b*Csc[e + f*x]], x], x] /; FreeQ[{a, b, d, e, f}, x] && NeQ[a^2 - b^2, 0] && GtQ[n, 2] && IntegerQ[2*n]
  3861. Int[Times[Power[csc[Plus[Optional[Pattern[e, Blank[]]], Times[Optional[Pattern[f, Blank[]]], Pattern[x, Blank[]]]]], -1], Power[Plus[Times[csc[Plus[Optional[Pattern[e, Blank[]]], Times[Optional[Pattern[f, Blank[]]], Pattern[x, Blank[]]]]], Optional[Pattern[b, Blank[]]]], Pattern[a, Blank[]]], Rational[-1, 2]]], Pattern[x, Blank[Symbol]]] := -Simp[(Cos[e + f*x]*Sqrt[a + b*Csc[e + f*x]])/(a*f), x] - Dist[b/(2*a), Int[(1 + Csc[e + f*x]^2)/Sqrt[a + b*Csc[e + f*x]], x], x] /; FreeQ[{a, b, e, f}, x] && NeQ[a^2 - b^2, 0]
  3862. Int[Times[Power[Times[csc[Plus[Optional[Pattern[e, Blank[]]], Times[Optional[Pattern[f, Blank[]]], Pattern[x, Blank[]]]]], Optional[Pattern[d, Blank[]]]], Rational[-1, 2]], Power[Plus[Times[csc[Plus[Optional[Pattern[e, Blank[]]], Times[Optional[Pattern[f, Blank[]]], Pattern[x, Blank[]]]]], Optional[Pattern[b, Blank[]]]], Pattern[a, Blank[]]], Rational[-1, 2]]], Pattern[x, Blank[Symbol]]] := Dist[1/a, Int[Sqrt[a + b*Csc[e + f*x]]/Sqrt[d*Csc[e + f*x]], x], x] - Dist[b/(a*d), Int[Sqrt[d*Csc[e + f*x]]/Sqrt[a + b*Csc[e + f*x]], x], x] /; FreeQ[{a, b, d, e, f}, x] && NeQ[a^2 - b^2, 0]
  3863. Int[Times[Power[Times[csc[Plus[Optional[Pattern[e, Blank[]]], Times[Optional[Pattern[f, Blank[]]], Pattern[x, Blank[]]]]], Optional[Pattern[d, Blank[]]]], Pattern[n, Blank[]]], Power[Plus[Times[csc[Plus[Optional[Pattern[e, Blank[]]], Times[Optional[Pattern[f, Blank[]]], Pattern[x, Blank[]]]]], Optional[Pattern[b, Blank[]]]], Pattern[a, Blank[]]], Rational[-1, 2]]], Pattern[x, Blank[Symbol]]] := Simp[(Cos[e + f*x]*(d*Csc[e + f*x])^(n + 1)*Sqrt[a + b*Csc[e + f*x]])/(a*d*f*n), x] + Dist[1/(2*a*d*n), Int[((d*Csc[e + f*x])^(n + 1)*Simp[-(b*(2*n + 1)) + 2*a*(n + 1)*Csc[e + f*x] + b*(2*n + 3)*Csc[e + f*x]^2, x])/Sqrt[a + b*Csc[e + f*x]], x], x] /; FreeQ[{a, b, d, e, f}, x] && NeQ[a^2 - b^2, 0] && LtQ[n, -1] && IntegerQ[2*n]
  3864. Int[Times[Power[Times[csc[Plus[Optional[Pattern[e, Blank[]]], Times[Optional[Pattern[f, Blank[]]], Pattern[x, Blank[]]]]], Optional[Pattern[d, Blank[]]]], Pattern[n, Blank[]]], Power[Plus[Times[csc[Plus[Optional[Pattern[e, Blank[]]], Times[Optional[Pattern[f, Blank[]]], Pattern[x, Blank[]]]]], Optional[Pattern[b, Blank[]]]], Pattern[a, Blank[]]], Rational[3, 2]]], Pattern[x, Blank[Symbol]]] := Simp[(a*Cot[e + f*x]*Sqrt[a + b*Csc[e + f*x]]*(d*Csc[e + f*x])^n)/(f*n), x] + Dist[1/(2*d*n), Int[((d*Csc[e + f*x])^(n + 1)*Simp[a*b*(2*n - 1) + 2*(b^2*n + a^2*(n + 1))*Csc[e + f*x] + a*b*(2*n + 3)*Csc[e + f*x]^2, x])/Sqrt[a + b*Csc[e + f*x]], x], x] /; FreeQ[{a, b, d, e, f}, x] && NeQ[a^2 - b^2, 0] && LeQ[n, -1] && IntegersQ[2*n]
  3865. Int[Times[Power[Times[csc[Plus[Optional[Pattern[e, Blank[]]], Times[Optional[Pattern[f, Blank[]]], Pattern[x, Blank[]]]]], Optional[Pattern[d, Blank[]]]], Pattern[n, Blank[]]], Power[Plus[Times[csc[Plus[Optional[Pattern[e, Blank[]]], Times[Optional[Pattern[f, Blank[]]], Pattern[x, Blank[]]]]], Optional[Pattern[b, Blank[]]]], Pattern[a, Blank[]]], Pattern[m, Blank[]]]], Pattern[x, Blank[Symbol]]] := -Simp[(d^3*Cot[e + f*x]*(a + b*Csc[e + f*x])^(m + 1)*(d*Csc[e + f*x])^(n - 3))/(b*f*(m + n - 1)), x] + Dist[d^3/(b*(m + n - 1)), Int[(a + b*Csc[e + f*x])^m*(d*Csc[e + f*x])^(n - 3)*Simp[a*(n - 3) + b*(m + n - 2)*Csc[e + f*x] - a*(n - 2)*Csc[e + f*x]^2, x], x], x] /; FreeQ[{a, b, d, e, f, m}, x] && NeQ[a^2 - b^2, 0] && GtQ[n, 3] && (IntegerQ[n] || IntegersQ[2*m, 2*n]) && !IGtQ[m, 2]
  3866. Int[Times[Power[Times[csc[Plus[Optional[Pattern[e, Blank[]]], Times[Optional[Pattern[f, Blank[]]], Pattern[x, Blank[]]]]], Optional[Pattern[d, Blank[]]]], Pattern[n, Blank[]]], Power[Plus[Times[csc[Plus[Optional[Pattern[e, Blank[]]], Times[Optional[Pattern[f, Blank[]]], Pattern[x, Blank[]]]]], Optional[Pattern[b, Blank[]]]], Pattern[a, Blank[]]], Pattern[m, Blank[]]]], Pattern[x, Blank[Symbol]]] := -Simp[(b*d*Cot[e + f*x]*(a + b*Csc[e + f*x])^(m - 1)*(d*Csc[e + f*x])^(n - 1))/(f*(m + n - 1)), x] + Dist[d/(m + n - 1), Int[(a + b*Csc[e + f*x])^(m - 2)*(d*Csc[e + f*x])^(n - 1)*Simp[a*b*(n - 1) + (b^2*(m + n - 2) + a^2*(m + n - 1))*Csc[e + f*x] + a*b*(2*m + n - 2)*Csc[e + f*x]^2, x], x], x] /; FreeQ[{a, b, d, e, f}, x] && NeQ[a^2 - b^2, 0] && LtQ[0, m, 2] && LtQ[0, n, 3] && NeQ[m + n - 1, 0] && (IntegerQ[m] || IntegersQ[2*m, 2*n])
  3867. Int[Times[Power[Times[csc[Plus[Optional[Pattern[e, Blank[]]], Times[Optional[Pattern[f, Blank[]]], Pattern[x, Blank[]]]]], Optional[Pattern[d, Blank[]]]], Pattern[n, Blank[]]], Power[Plus[Times[csc[Plus[Optional[Pattern[e, Blank[]]], Times[Optional[Pattern[f, Blank[]]], Pattern[x, Blank[]]]]], Optional[Pattern[b, Blank[]]]], Pattern[a, Blank[]]], Pattern[m, Blank[]]]], Pattern[x, Blank[Symbol]]] := -Simp[(d^2*Cot[e + f*x]*(a + b*Csc[e + f*x])^m*(d*Csc[e + f*x])^(n - 2))/(f*(m + n - 1)), x] + Dist[d^2/(b*(m + n - 1)), Int[(a + b*Csc[e + f*x])^(m - 1)*(d*Csc[e + f*x])^(n - 2)*Simp[a*b*(n - 2) + b^2*(m + n - 2)*Csc[e + f*x] + a*b*m*Csc[e + f*x]^2, x], x], x] /; FreeQ[{a, b, d, e, f}, x] && NeQ[a^2 - b^2, 0] && LtQ[-1, m, 2] && LtQ[1, n, 3] && NeQ[m + n - 1, 0] && (IntegerQ[n] || IntegersQ[2*m, 2*n])
  3868. Int[Times[Power[Times[csc[Plus[Optional[Pattern[e, Blank[]]], Times[Optional[Pattern[f, Blank[]]], Pattern[x, Blank[]]]]], Optional[Pattern[d, Blank[]]]], Rational[-1, 2]], Power[Plus[Times[csc[Plus[Optional[Pattern[e, Blank[]]], Times[Optional[Pattern[f, Blank[]]], Pattern[x, Blank[]]]]], Optional[Pattern[b, Blank[]]]], Pattern[a, Blank[]]], Rational[3, 2]]], Pattern[x, Blank[Symbol]]] := Dist[a, Int[Sqrt[a + b*Csc[e + f*x]]/Sqrt[d*Csc[e + f*x]], x], x] + Dist[b/d, Int[Sqrt[a + b*Csc[e + f*x]]*Sqrt[d*Csc[e + f*x]], x], x] /; FreeQ[{a, b, d, e, f}, x] && NeQ[a^2 - b^2, 0]
  3869. Int[Times[Power[Times[csc[Plus[Optional[Pattern[e, Blank[]]], Times[Optional[Pattern[f, Blank[]]], Pattern[x, Blank[]]]]], Optional[Pattern[d, Blank[]]]], Optional[Pattern[n, Blank[]]]], Power[Plus[Times[csc[Plus[Optional[Pattern[e, Blank[]]], Times[Optional[Pattern[f, Blank[]]], Pattern[x, Blank[]]]]], Optional[Pattern[b, Blank[]]]], Pattern[a, Blank[]]], Optional[Pattern[m, Blank[]]]]], Pattern[x, Blank[Symbol]]] := Dist[Sin[e + f*x]^n*(d*Csc[e + f*x])^n, Int[(b + a*Sin[e + f*x])^m/Sin[e + f*x]^(m + n), x], x] /; FreeQ[{a, b, d, e, f, n}, x] && NeQ[a^2 - b^2, 0] && IntegerQ[m]
  3870. Int[Times[Power[Times[csc[Plus[Optional[Pattern[e, Blank[]]], Times[Optional[Pattern[f, Blank[]]], Pattern[x, Blank[]]]]], Optional[Pattern[d, Blank[]]]], Optional[Pattern[n, Blank[]]]], Power[Plus[Times[csc[Plus[Optional[Pattern[e, Blank[]]], Times[Optional[Pattern[f, Blank[]]], Pattern[x, Blank[]]]]], Optional[Pattern[b, Blank[]]]], Pattern[a, Blank[]]], Optional[Pattern[m, Blank[]]]]], Pattern[x, Blank[Symbol]]] := Unintegrable[(d*Csc[e + f*x])^n*(a + b*Csc[e + f*x])^m, x] /; FreeQ[{a, b, d, e, f, m, n}, x]
  3871. Int[Times[Power[Times[cos[Plus[Optional[Pattern[e, Blank[]]], Times[Optional[Pattern[f, Blank[]]], Pattern[x, Blank[]]]]], Optional[Pattern[d, Blank[]]]], Pattern[m, Blank[]]], Power[Plus[Optional[Pattern[a, Blank[]]], Times[Optional[Pattern[b, Blank[]]], sec[Plus[Optional[Pattern[e, Blank[]]], Times[Optional[Pattern[f, Blank[]]], Pattern[x, Blank[]]]]]]], Pattern[p, Blank[]]]], Pattern[x, Blank[Symbol]]] := Dist[(d*Cos[e + f*x])^FracPart[m]*(Sec[e + f*x]/d)^FracPart[m], Int[(a + b*Sec[e + f*x])^p/(Sec[e + f*x]/d)^m, x], x] /; FreeQ[{a, b, d, e, f, m, p}, x] && !IntegerQ[m] && !IntegerQ[p]
  3872. Int[Times[Power[Times[cos[Plus[Optional[Pattern[e, Blank[]]], Times[Optional[Pattern[f, Blank[]]], Pattern[x, Blank[]]]]], Optional[Pattern[g, Blank[]]]], Optional[Pattern[p, Blank[]]]], Power[Plus[Times[csc[Plus[Optional[Pattern[e, Blank[]]], Times[Optional[Pattern[f, Blank[]]], Pattern[x, Blank[]]]]], Optional[Pattern[b, Blank[]]]], Pattern[a, Blank[]]], Optional[Pattern[m, Blank[]]]]], Pattern[x, Blank[Symbol]]] := Int[((g*Cos[e + f*x])^p*(b + a*Sin[e + f*x])^m)/Sin[e + f*x]^m, x] /; FreeQ[{a, b, e, f, g, p}, x] && IntegerQ[m]
  3873. Int[Times[Power[cos[Plus[Optional[Pattern[e, Blank[]]], Times[Optional[Pattern[f, Blank[]]], Pattern[x, Blank[]]]]], Optional[Pattern[p, Blank[]]]], Power[Plus[Times[csc[Plus[Optional[Pattern[e, Blank[]]], Times[Optional[Pattern[f, Blank[]]], Pattern[x, Blank[]]]]], Optional[Pattern[b, Blank[]]]], Pattern[a, Blank[]]], Pattern[m, Blank[]]]], Pattern[x, Blank[Symbol]]] := -Dist[(f*b^(p - 1))^(-1), Subst[Int[((-a + b*x)^((p - 1)/2)*(a + b*x)^(m + (p - 1)/2))/x^(p + 1), x], x, Csc[e + f*x]], x] /; FreeQ[{a, b, e, f, m}, x] && IntegerQ[(p - 1)/2] && EqQ[a^2 - b^2, 0]
  3874. Int[Times[Power[cos[Plus[Optional[Pattern[e, Blank[]]], Times[Optional[Pattern[f, Blank[]]], Pattern[x, Blank[]]]]], Optional[Pattern[p, Blank[]]]], Power[Plus[Times[csc[Plus[Optional[Pattern[e, Blank[]]], Times[Optional[Pattern[f, Blank[]]], Pattern[x, Blank[]]]]], Optional[Pattern[b, Blank[]]]], Pattern[a, Blank[]]], Pattern[m, Blank[]]]], Pattern[x, Blank[Symbol]]] := -Dist[f^(-1), Subst[Int[((-1 + x)^((p - 1)/2)*(1 + x)^((p - 1)/2)*(a + b*x)^m)/x^(p + 1), x], x, Csc[e + f*x]], x] /; FreeQ[{a, b, e, f, m}, x] && IntegerQ[(p - 1)/2] && NeQ[a^2 - b^2, 0]
  3875. Int[Times[Power[cos[Plus[Optional[Pattern[e, Blank[]]], Times[Optional[Pattern[f, Blank[]]], Pattern[x, Blank[]]]]], -2], Power[Plus[Times[csc[Plus[Optional[Pattern[e, Blank[]]], Times[Optional[Pattern[f, Blank[]]], Pattern[x, Blank[]]]]], Optional[Pattern[b, Blank[]]]], Pattern[a, Blank[]]], Pattern[m, Blank[]]]], Pattern[x, Blank[Symbol]]] := Simp[(Tan[e + f*x]*(a + b*Csc[e + f*x])^m)/f, x] + Dist[b*m, Int[Csc[e + f*x]*(a + b*Csc[e + f*x])^(m - 1), x], x] /; FreeQ[{a, b, e, f, m}, x]
  3876. Int[Times[Power[Times[cos[Plus[Optional[Pattern[e, Blank[]]], Times[Optional[Pattern[f, Blank[]]], Pattern[x, Blank[]]]]], Optional[Pattern[g, Blank[]]]], Optional[Pattern[p, Blank[]]]], Power[Plus[Times[csc[Plus[Optional[Pattern[e, Blank[]]], Times[Optional[Pattern[f, Blank[]]], Pattern[x, Blank[]]]]], Optional[Pattern[b, Blank[]]]], Pattern[a, Blank[]]], Pattern[m, Blank[]]]], Pattern[x, Blank[Symbol]]] := Dist[(Sin[e + f*x]^FracPart[m]*(a + b*Csc[e + f*x])^FracPart[m])/(b + a*Sin[e + f*x])^FracPart[m], Int[((g*Cos[e + f*x])^p*(b + a*Sin[e + f*x])^m)/Sin[e + f*x]^m, x], x] /; FreeQ[{a, b, e, f, g, m, p}, x] && (EqQ[a^2 - b^2, 0] || IntegersQ[2*m, p])
  3877. Int[Times[Power[Times[cos[Plus[Optional[Pattern[e, Blank[]]], Times[Optional[Pattern[f, Blank[]]], Pattern[x, Blank[]]]]], Optional[Pattern[g, Blank[]]]], Optional[Pattern[p, Blank[]]]], Power[Plus[Times[csc[Plus[Optional[Pattern[e, Blank[]]], Times[Optional[Pattern[f, Blank[]]], Pattern[x, Blank[]]]]], Optional[Pattern[b, Blank[]]]], Pattern[a, Blank[]]], Optional[Pattern[m, Blank[]]]]], Pattern[x, Blank[Symbol]]] := Unintegrable[(g*Cos[e + f*x])^p*(a + b*Csc[e + f*x])^m, x] /; FreeQ[{a, b, e, f, g, m, p}, x]
  3878. Int[Times[Power[Plus[Times[csc[Plus[Optional[Pattern[e, Blank[]]], Times[Optional[Pattern[f, Blank[]]], Pattern[x, Blank[]]]]], Optional[Pattern[b, Blank[]]]], Pattern[a, Blank[]]], Optional[Pattern[m, Blank[]]]], Power[Times[Optional[Pattern[g, Blank[]]], sec[Plus[Optional[Pattern[e, Blank[]]], Times[Optional[Pattern[f, Blank[]]], Pattern[x, Blank[]]]]]], Pattern[p, Blank[]]]], Pattern[x, Blank[Symbol]]] := Dist[g^IntPart[p]*(g*Sec[e + f*x])^FracPart[p]*Cos[e + f*x]^FracPart[p], Int[(a + b*Csc[e + f*x])^m/Cos[e + f*x]^p, x], x] /; FreeQ[{a, b, e, f, g, m, p}, x] && !IntegerQ[p]
  3879. Int[Times[Power[cot[Plus[Optional[Pattern[c, Blank[]]], Times[Optional[Pattern[d, Blank[]]], Pattern[x, Blank[]]]]], Optional[Pattern[m, Blank[]]]], Power[Plus[Times[csc[Plus[Optional[Pattern[c, Blank[]]], Times[Optional[Pattern[d, Blank[]]], Pattern[x, Blank[]]]]], Optional[Pattern[b, Blank[]]]], Pattern[a, Blank[]]], Optional[Pattern[n, Blank[]]]]], Pattern[x, Blank[Symbol]]] := Dist[1/(a^(m - n - 1)*b^n*d), Subst[Int[((a - b*x)^((m - 1)/2)*(a + b*x)^((m - 1)/2 + n))/x^(m + n), x], x, Sin[c + d*x]], x] /; FreeQ[{a, b, c, d}, x] && IntegerQ[(m - 1)/2] && EqQ[a^2 - b^2, 0] && IntegerQ[n]
  3880. Int[Times[Power[cot[Plus[Optional[Pattern[c, Blank[]]], Times[Optional[Pattern[d, Blank[]]], Pattern[x, Blank[]]]]], Optional[Pattern[m, Blank[]]]], Power[Plus[Times[csc[Plus[Optional[Pattern[c, Blank[]]], Times[Optional[Pattern[d, Blank[]]], Pattern[x, Blank[]]]]], Optional[Pattern[b, Blank[]]]], Pattern[a, Blank[]]], Pattern[n, Blank[]]]], Pattern[x, Blank[Symbol]]] := -Dist[(d*b^(m - 1))^(-1), Subst[Int[((-a + b*x)^((m - 1)/2)*(a + b*x)^((m - 1)/2 + n))/x, x], x, Csc[c + d*x]], x] /; FreeQ[{a, b, c, d, n}, x] && IntegerQ[(m - 1)/2] && EqQ[a^2 - b^2, 0] && !IntegerQ[n]
  3881. Int[Times[Power[Times[cot[Plus[Optional[Pattern[c, Blank[]]], Times[Optional[Pattern[d, Blank[]]], Pattern[x, Blank[]]]]], Optional[Pattern[e, Blank[]]]], Pattern[m, Blank[]]], Plus[Times[csc[Plus[Optional[Pattern[c, Blank[]]], Times[Optional[Pattern[d, Blank[]]], Pattern[x, Blank[]]]]], Optional[Pattern[b, Blank[]]]], Pattern[a, Blank[]]]], Pattern[x, Blank[Symbol]]] := -Simp[(e*(e*Cot[c + d*x])^(m - 1)*(a*m + b*(m - 1)*Csc[c + d*x]))/(d*m*(m - 1)), x] - Dist[e^2/m, Int[(e*Cot[c + d*x])^(m - 2)*(a*m + b*(m - 1)*Csc[c + d*x]), x], x] /; FreeQ[{a, b, c, d, e}, x] && GtQ[m, 1]
  3882. Int[Times[Power[Times[cot[Plus[Optional[Pattern[c, Blank[]]], Times[Optional[Pattern[d, Blank[]]], Pattern[x, Blank[]]]]], Optional[Pattern[e, Blank[]]]], Pattern[m, Blank[]]], Plus[Times[csc[Plus[Optional[Pattern[c, Blank[]]], Times[Optional[Pattern[d, Blank[]]], Pattern[x, Blank[]]]]], Optional[Pattern[b, Blank[]]]], Pattern[a, Blank[]]]], Pattern[x, Blank[Symbol]]] := -Simp[((e*Cot[c + d*x])^(m + 1)*(a + b*Csc[c + d*x]))/(d*e*(m + 1)), x] - Dist[1/(e^2*(m + 1)), Int[(e*Cot[c + d*x])^(m + 2)*(a*(m + 1) + b*(m + 2)*Csc[c + d*x]), x], x] /; FreeQ[{a, b, c, d, e}, x] && LtQ[m, -1]
  3883. Int[Times[Power[cot[Plus[Optional[Pattern[c, Blank[]]], Times[Optional[Pattern[d, Blank[]]], Pattern[x, Blank[]]]]], -1], Plus[Times[csc[Plus[Optional[Pattern[c, Blank[]]], Times[Optional[Pattern[d, Blank[]]], Pattern[x, Blank[]]]]], Optional[Pattern[b, Blank[]]]], Pattern[a, Blank[]]]], Pattern[x, Blank[Symbol]]] := Int[(b + a*Sin[c + d*x])/Cos[c + d*x], x] /; FreeQ[{a, b, c, d}, x]
  3884. Int[Times[Power[Times[cot[Plus[Optional[Pattern[c, Blank[]]], Times[Optional[Pattern[d, Blank[]]], Pattern[x, Blank[]]]]], Optional[Pattern[e, Blank[]]]], Optional[Pattern[m, Blank[]]]], Plus[Times[csc[Plus[Optional[Pattern[c, Blank[]]], Times[Optional[Pattern[d, Blank[]]], Pattern[x, Blank[]]]]], Optional[Pattern[b, Blank[]]]], Pattern[a, Blank[]]]], Pattern[x, Blank[Symbol]]] := Dist[a, Int[(e*Cot[c + d*x])^m, x], x] + Dist[b, Int[(e*Cot[c + d*x])^m*Csc[c + d*x], x], x] /; FreeQ[{a, b, c, d, e, m}, x]
  3885. Int[Times[Power[cot[Plus[Optional[Pattern[c, Blank[]]], Times[Optional[Pattern[d, Blank[]]], Pattern[x, Blank[]]]]], Optional[Pattern[m, Blank[]]]], Power[Plus[Times[csc[Plus[Optional[Pattern[c, Blank[]]], Times[Optional[Pattern[d, Blank[]]], Pattern[x, Blank[]]]]], Optional[Pattern[b, Blank[]]]], Pattern[a, Blank[]]], Pattern[n, Blank[]]]], Pattern[x, Blank[Symbol]]] := -Dist[(-1)^((m - 1)/2)/(d*b^(m - 1)), Subst[Int[((b^2 - x^2)^((m - 1)/2)*(a + x)^n)/x, x], x, b*Csc[c + d*x]], x] /; FreeQ[{a, b, c, d, n}, x] && IntegerQ[(m - 1)/2] && NeQ[a^2 - b^2, 0]
  3886. Int[Times[Power[Times[cot[Plus[Optional[Pattern[c, Blank[]]], Times[Optional[Pattern[d, Blank[]]], Pattern[x, Blank[]]]]], Optional[Pattern[e, Blank[]]]], Pattern[m, Blank[]]], Power[Plus[Times[csc[Plus[Optional[Pattern[c, Blank[]]], Times[Optional[Pattern[d, Blank[]]], Pattern[x, Blank[]]]]], Optional[Pattern[b, Blank[]]]], Pattern[a, Blank[]]], Pattern[n, Blank[]]]], Pattern[x, Blank[Symbol]]] := Int[ExpandIntegrand[(e*Cot[c + d*x])^m, (a + b*Csc[c + d*x])^n, x], x] /; FreeQ[{a, b, c, d, e, m}, x] && IGtQ[n, 0]
  3887. Int[Times[Power[cot[Plus[Optional[Pattern[c, Blank[]]], Times[Optional[Pattern[d, Blank[]]], Pattern[x, Blank[]]]]], Optional[Pattern[m, Blank[]]]], Power[Plus[Times[csc[Plus[Optional[Pattern[c, Blank[]]], Times[Optional[Pattern[d, Blank[]]], Pattern[x, Blank[]]]]], Optional[Pattern[b, Blank[]]]], Pattern[a, Blank[]]], Optional[Pattern[n, Blank[]]]]], Pattern[x, Blank[Symbol]]] := Dist[(-2*a^(m/2 + n + 1/2))/d, Subst[Int[(x^m*(2 + a*x^2)^(m/2 + n - 1/2))/(1 + a*x^2), x], x, Cot[c + d*x]/Sqrt[a + b*Csc[c + d*x]]], x] /; FreeQ[{a, b, c, d}, x] && EqQ[a^2 - b^2, 0] && IntegerQ[m/2] && IntegerQ[n - 1/2]
  3888. Int[Times[Power[Times[cot[Plus[Optional[Pattern[c, Blank[]]], Times[Optional[Pattern[d, Blank[]]], Pattern[x, Blank[]]]]], Optional[Pattern[e, Blank[]]]], Pattern[m, Blank[]]], Power[Plus[Times[csc[Plus[Optional[Pattern[c, Blank[]]], Times[Optional[Pattern[d, Blank[]]], Pattern[x, Blank[]]]]], Optional[Pattern[b, Blank[]]]], Pattern[a, Blank[]]], Pattern[n, Blank[]]]], Pattern[x, Blank[Symbol]]] := Dist[a^(2*n)/e^(2*n), Int[(e*Cot[c + d*x])^(m + 2*n)/(-a + b*Csc[c + d*x])^n, x], x] /; FreeQ[{a, b, c, d, e, m}, x] && EqQ[a^2 - b^2, 0] && ILtQ[n, 0]
  3889. Int[Times[Power[Times[cot[Plus[Optional[Pattern[c, Blank[]]], Times[Optional[Pattern[d, Blank[]]], Pattern[x, Blank[]]]]], Optional[Pattern[e, Blank[]]]], Pattern[m, Blank[]]], Power[Plus[Times[csc[Plus[Optional[Pattern[c, Blank[]]], Times[Optional[Pattern[d, Blank[]]], Pattern[x, Blank[]]]]], Optional[Pattern[b, Blank[]]]], Pattern[a, Blank[]]], Pattern[n, Blank[]]]], Pattern[x, Blank[Symbol]]] := -Simp[(2^(m + n + 1)*(e*Cot[c + d*x])^(m + 1)*(a + b*Csc[c + d*x])^n*(a/(a + b*Csc[c + d*x]))^(m + n + 1)*AppellF1[(m + 1)/2, m + n, 1, (m + 3)/2, -((a - b*Csc[c + d*x])/(a + b*Csc[c + d*x])), (a - b*Csc[c + d*x])/(a + b*Csc[c + d*x])])/(d*e*(m + 1)), x] /; FreeQ[{a, b, c, d, e, m, n}, x] && EqQ[a^2 - b^2, 0] && !IntegerQ[n]
  3890. Int[Times[Power[Times[cot[Plus[Optional[Pattern[c, Blank[]]], Times[Optional[Pattern[d, Blank[]]], Pattern[x, Blank[]]]]], Optional[Pattern[e, Blank[]]]], Rational[1, 2]], Power[Plus[Times[csc[Plus[Optional[Pattern[c, Blank[]]], Times[Optional[Pattern[d, Blank[]]], Pattern[x, Blank[]]]]], Optional[Pattern[b, Blank[]]]], Pattern[a, Blank[]]], -1]], Pattern[x, Blank[Symbol]]] := Dist[1/a, Int[Sqrt[e*Cot[c + d*x]], x], x] - Dist[b/a, Int[Sqrt[e*Cot[c + d*x]]/(b + a*Sin[c + d*x]), x], x] /; FreeQ[{a, b, c, d, e}, x] && NeQ[a^2 - b^2, 0]
  3891. Int[Times[Power[Times[cot[Plus[Optional[Pattern[c, Blank[]]], Times[Optional[Pattern[d, Blank[]]], Pattern[x, Blank[]]]]], Optional[Pattern[e, Blank[]]]], Pattern[m, Blank[]]], Power[Plus[Times[csc[Plus[Optional[Pattern[c, Blank[]]], Times[Optional[Pattern[d, Blank[]]], Pattern[x, Blank[]]]]], Optional[Pattern[b, Blank[]]]], Pattern[a, Blank[]]], -1]], Pattern[x, Blank[Symbol]]] := -Dist[e^2/b^2, Int[(e*Cot[c + d*x])^(m - 2)*(a - b*Csc[c + d*x]), x], x] + Dist[(e^2*(a^2 - b^2))/b^2, Int[(e*Cot[c + d*x])^(m - 2)/(a + b*Csc[c + d*x]), x], x] /; FreeQ[{a, b, c, d, e}, x] && NeQ[a^2 - b^2, 0] && IGtQ[m - 1/2, 0]
  3892. Int[Times[Power[Times[cot[Plus[Optional[Pattern[c, Blank[]]], Times[Optional[Pattern[d, Blank[]]], Pattern[x, Blank[]]]]], Optional[Pattern[e, Blank[]]]], Rational[-1, 2]], Power[Plus[Times[csc[Plus[Optional[Pattern[c, Blank[]]], Times[Optional[Pattern[d, Blank[]]], Pattern[x, Blank[]]]]], Optional[Pattern[b, Blank[]]]], Pattern[a, Blank[]]], -1]], Pattern[x, Blank[Symbol]]] := Dist[1/a, Int[1/Sqrt[e*Cot[c + d*x]], x], x] - Dist[b/a, Int[1/(Sqrt[e*Cot[c + d*x]]*(b + a*Sin[c + d*x])), x], x] /; FreeQ[{a, b, c, d, e}, x] && NeQ[a^2 - b^2, 0]
  3893. Int[Times[Power[Times[cot[Plus[Optional[Pattern[c, Blank[]]], Times[Optional[Pattern[d, Blank[]]], Pattern[x, Blank[]]]]], Optional[Pattern[e, Blank[]]]], Pattern[m, Blank[]]], Power[Plus[Times[csc[Plus[Optional[Pattern[c, Blank[]]], Times[Optional[Pattern[d, Blank[]]], Pattern[x, Blank[]]]]], Optional[Pattern[b, Blank[]]]], Pattern[a, Blank[]]], -1]], Pattern[x, Blank[Symbol]]] := Dist[1/(a^2 - b^2), Int[(e*Cot[c + d*x])^m*(a - b*Csc[c + d*x]), x], x] + Dist[b^2/(e^2*(a^2 - b^2)), Int[(e*Cot[c + d*x])^(m + 2)/(a + b*Csc[c + d*x]), x], x] /; FreeQ[{a, b, c, d, e}, x] && NeQ[a^2 - b^2, 0] && ILtQ[m + 1/2, 0]
  3894. Int[Times[Power[cot[Plus[Optional[Pattern[c, Blank[]]], Times[Optional[Pattern[d, Blank[]]], Pattern[x, Blank[]]]]], 2], Power[Plus[Times[csc[Plus[Optional[Pattern[c, Blank[]]], Times[Optional[Pattern[d, Blank[]]], Pattern[x, Blank[]]]]], Optional[Pattern[b, Blank[]]]], Pattern[a, Blank[]]], Pattern[n, Blank[]]]], Pattern[x, Blank[Symbol]]] := Int[(-1 + Csc[c + d*x]^2)*(a + b*Csc[c + d*x])^n, x] /; FreeQ[{a, b, c, d, n}, x] && NeQ[a^2 - b^2, 0]
  3895. Int[Times[Power[cot[Plus[Optional[Pattern[c, Blank[]]], Times[Optional[Pattern[d, Blank[]]], Pattern[x, Blank[]]]]], Pattern[m, Blank[]]], Power[Plus[Times[csc[Plus[Optional[Pattern[c, Blank[]]], Times[Optional[Pattern[d, Blank[]]], Pattern[x, Blank[]]]]], Optional[Pattern[b, Blank[]]]], Pattern[a, Blank[]]], Pattern[n, Blank[]]]], Pattern[x, Blank[Symbol]]] := Int[ExpandIntegrand[(a + b*Csc[c + d*x])^n, (-1 + Csc[c + d*x]^2)^(m/2), x], x] /; FreeQ[{a, b, c, d, n}, x] && NeQ[a^2 - b^2, 0] && IGtQ[m/2, 0] && IntegerQ[n - 1/2]
  3896. Int[Times[Power[cot[Plus[Optional[Pattern[c, Blank[]]], Times[Optional[Pattern[d, Blank[]]], Pattern[x, Blank[]]]]], Pattern[m, Blank[]]], Power[Plus[Times[csc[Plus[Optional[Pattern[c, Blank[]]], Times[Optional[Pattern[d, Blank[]]], Pattern[x, Blank[]]]]], Optional[Pattern[b, Blank[]]]], Pattern[a, Blank[]]], Pattern[n, Blank[]]]], Pattern[x, Blank[Symbol]]] := Int[ExpandIntegrand[(a + b*Csc[c + d*x])^n, (-1 + Sec[c + d*x]^2)^(-(m/2)), x], x] /; FreeQ[{a, b, c, d, n}, x] && NeQ[a^2 - b^2, 0] && ILtQ[m/2, 0] && IntegerQ[n - 1/2] && EqQ[m, -2]
  3897. Int[Times[Power[Times[cot[Plus[Optional[Pattern[c, Blank[]]], Times[Optional[Pattern[d, Blank[]]], Pattern[x, Blank[]]]]], Optional[Pattern[e, Blank[]]]], Pattern[m, Blank[]]], Power[Plus[Times[csc[Plus[Optional[Pattern[c, Blank[]]], Times[Optional[Pattern[d, Blank[]]], Pattern[x, Blank[]]]]], Optional[Pattern[b, Blank[]]]], Pattern[a, Blank[]]], Pattern[n, Blank[]]]], Pattern[x, Blank[Symbol]]] := Int[ExpandIntegrand[(e*Cot[c + d*x])^m, (a + b*Csc[c + d*x])^n, x], x] /; FreeQ[{a, b, c, d, e, m}, x] && NeQ[a^2 - b^2, 0] && IGtQ[n, 0]
  3898. Int[Times[Power[cot[Plus[Optional[Pattern[c, Blank[]]], Times[Optional[Pattern[d, Blank[]]], Pattern[x, Blank[]]]]], Optional[Pattern[m, Blank[]]]], Power[Plus[Times[csc[Plus[Optional[Pattern[c, Blank[]]], Times[Optional[Pattern[d, Blank[]]], Pattern[x, Blank[]]]]], Optional[Pattern[b, Blank[]]]], Pattern[a, Blank[]]], Pattern[n, Blank[]]]], Pattern[x, Blank[Symbol]]] := Int[(Cos[c + d*x]^m*(b + a*Sin[c + d*x])^n)/Sin[c + d*x]^(m + n), x] /; FreeQ[{a, b, c, d}, x] && NeQ[a^2 - b^2, 0] && IntegerQ[n] && IntegerQ[m] && (IntegerQ[m/2] || LeQ[m, 1])
  3899. Int[Times[Power[Plus[Optional[Pattern[a, Blank[]]], Times[csc[Plus[Optional[Pattern[c, Blank[]]], Times[Optional[Pattern[d, Blank[]]], Pattern[x, Blank[]]]]], Optional[Pattern[b, Blank[]]]]], Optional[Pattern[n, Blank[]]]], Power[Times[cot[Plus[Optional[Pattern[c, Blank[]]], Times[Optional[Pattern[d, Blank[]]], Pattern[x, Blank[]]]]], Optional[Pattern[e, Blank[]]]], Optional[Pattern[m, Blank[]]]]], Pattern[x, Blank[Symbol]]] := Unintegrable[(e*Cot[c + d*x])^m*(a + b*Csc[c + d*x])^n, x] /; FreeQ[{a, b, c, d, e, m, n}, x]
  3900. Int[Times[Power[Times[cot[Plus[Optional[Pattern[c, Blank[]]], Times[Optional[Pattern[d, Blank[]]], Pattern[x, Blank[]]]]], Optional[Pattern[e, Blank[]]]], Pattern[m, Blank[]]], Power[Plus[Pattern[a, Blank[]], Times[Optional[Pattern[b, Blank[]]], sec[Plus[Optional[Pattern[c, Blank[]]], Times[Optional[Pattern[d, Blank[]]], Pattern[x, Blank[]]]]]]], Optional[Pattern[n, Blank[]]]]], Pattern[x, Blank[Symbol]]] := Dist[(e*Cot[c + d*x])^m*Tan[c + d*x]^m, Int[(a + b*Sec[c + d*x])^n/Tan[c + d*x]^m, x], x] /; FreeQ[{a, b, c, d, e, m, n}, x] && !IntegerQ[m]
  3901. Int[Times[Power[Plus[Pattern[a, Blank[]], Times[Optional[Pattern[b, Blank[]]], sec[Plus[Optional[Pattern[c, Blank[]]], Times[Optional[Pattern[d, Blank[]]], Pattern[x, Blank[]]]]]]], Optional[Pattern[n, Blank[]]]], Power[Times[Optional[Pattern[e, Blank[]]], Power[tan[Plus[Optional[Pattern[c, Blank[]]], Times[Optional[Pattern[d, Blank[]]], Pattern[x, Blank[]]]]], Pattern[p, Blank[]]]], Pattern[m, Blank[]]]], Pattern[x, Blank[Symbol]]] := Dist[(e*Tan[c + d*x]^p)^m/(e*Tan[c + d*x])^(m*p), Int[(e*Tan[c + d*x])^(m*p)*(a + b*Sec[c + d*x])^n, x], x] /; FreeQ[{a, b, c, d, e, m, n, p}, x] && !IntegerQ[m]
  3902. Int[Times[Power[Times[Power[cot[Plus[Optional[Pattern[c, Blank[]]], Times[Optional[Pattern[d, Blank[]]], Pattern[x, Blank[]]]]], Pattern[p, Blank[]]], Optional[Pattern[e, Blank[]]]], Pattern[m, Blank[]]], Power[Plus[Times[csc[Plus[Optional[Pattern[c, Blank[]]], Times[Optional[Pattern[d, Blank[]]], Pattern[x, Blank[]]]]], Optional[Pattern[b, Blank[]]]], Pattern[a, Blank[]]], Optional[Pattern[n, Blank[]]]]], Pattern[x, Blank[Symbol]]] := Dist[(e*Cot[c + d*x]^p)^m/(e*Cot[c + d*x])^(m*p), Int[(e*Cot[c + d*x])^(m*p)*(a + b*Csc[c + d*x])^n, x], x] /; FreeQ[{a, b, c, d, e, m, n, p}, x] && !IntegerQ[m]
  3903. Int[Times[Power[Plus[Times[csc[Plus[Optional[Pattern[e, Blank[]]], Times[Optional[Pattern[f, Blank[]]], Pattern[x, Blank[]]]]], Optional[Pattern[b, Blank[]]]], Pattern[a, Blank[]]], Optional[Pattern[m, Blank[]]]], Power[Plus[Times[csc[Plus[Optional[Pattern[e, Blank[]]], Times[Optional[Pattern[f, Blank[]]], Pattern[x, Blank[]]]]], Optional[Pattern[d, Blank[]]]], Pattern[c, Blank[]]], Pattern[n, Blank[]]]], Pattern[x, Blank[Symbol]]] := Dist[c^n, Int[ExpandTrig[(1 + (d*csc[e + f*x])/c)^n, (a + b*csc[e + f*x])^m, x], x], x] /; FreeQ[{a, b, c, d, e, f, n}, x] && EqQ[b*c + a*d, 0] && EqQ[a^2 - b^2, 0] && IGtQ[m, 0] && ILtQ[n, 0] && LtQ[m + n, 2]
  3904. Int[Times[Power[Plus[Times[csc[Plus[Optional[Pattern[e, Blank[]]], Times[Optional[Pattern[f, Blank[]]], Pattern[x, Blank[]]]]], Optional[Pattern[b, Blank[]]]], Pattern[a, Blank[]]], Optional[Pattern[m, Blank[]]]], Power[Plus[Times[csc[Plus[Optional[Pattern[e, Blank[]]], Times[Optional[Pattern[f, Blank[]]], Pattern[x, Blank[]]]]], Optional[Pattern[d, Blank[]]]], Pattern[c, Blank[]]], Optional[Pattern[n, Blank[]]]]], Pattern[x, Blank[Symbol]]] := Dist[(-(a*c))^m, Int[Cot[e + f*x]^(2*m)*(c + d*Csc[e + f*x])^(n - m), x], x] /; FreeQ[{a, b, c, d, e, f, n}, x] && EqQ[b*c + a*d, 0] && EqQ[a^2 - b^2, 0] && IntegerQ[m] && RationalQ[n] && !(IntegerQ[n] && GtQ[m - n, 0])
  3905. Int[Times[Power[Plus[Times[csc[Plus[Optional[Pattern[e, Blank[]]], Times[Optional[Pattern[f, Blank[]]], Pattern[x, Blank[]]]]], Optional[Pattern[b, Blank[]]]], Pattern[a, Blank[]]], Pattern[m, Blank[]]], Power[Plus[Times[csc[Plus[Optional[Pattern[e, Blank[]]], Times[Optional[Pattern[f, Blank[]]], Pattern[x, Blank[]]]]], Optional[Pattern[d, Blank[]]]], Pattern[c, Blank[]]], Pattern[m, Blank[]]]], Pattern[x, Blank[Symbol]]] := Dist[((-(a*c))^(m + 1/2)*Cot[e + f*x])/(Sqrt[a + b*Csc[e + f*x]]*Sqrt[c + d*Csc[e + f*x]]), Int[Cot[e + f*x]^(2*m), x], x] /; FreeQ[{a, b, c, d, e, f, m}, x] && EqQ[b*c + a*d, 0] && EqQ[a^2 - b^2, 0] && IntegerQ[m + 1/2]
  3906. Int[Times[Power[Plus[Times[csc[Plus[Optional[Pattern[e, Blank[]]], Times[Optional[Pattern[f, Blank[]]], Pattern[x, Blank[]]]]], Optional[Pattern[b, Blank[]]]], Pattern[a, Blank[]]], Rational[1, 2]], Power[Plus[Times[csc[Plus[Optional[Pattern[e, Blank[]]], Times[Optional[Pattern[f, Blank[]]], Pattern[x, Blank[]]]]], Optional[Pattern[d, Blank[]]]], Pattern[c, Blank[]]], Optional[Pattern[n, Blank[]]]]], Pattern[x, Blank[Symbol]]] := Simp[(2*a*c*Cot[e + f*x]*(c + d*Csc[e + f*x])^(n - 1))/(f*(2*n - 1)*Sqrt[a + b*Csc[e + f*x]]), x] + Dist[c, Int[Sqrt[a + b*Csc[e + f*x]]*(c + d*Csc[e + f*x])^(n - 1), x], x] /; FreeQ[{a, b, c, d, e, f}, x] && EqQ[b*c + a*d, 0] && EqQ[a^2 - b^2, 0] && GtQ[n, 1/2]
  3907. Int[Times[Power[Plus[Times[csc[Plus[Optional[Pattern[e, Blank[]]], Times[Optional[Pattern[f, Blank[]]], Pattern[x, Blank[]]]]], Optional[Pattern[b, Blank[]]]], Pattern[a, Blank[]]], Rational[1, 2]], Power[Plus[Times[csc[Plus[Optional[Pattern[e, Blank[]]], Times[Optional[Pattern[f, Blank[]]], Pattern[x, Blank[]]]]], Optional[Pattern[d, Blank[]]]], Pattern[c, Blank[]]], Pattern[n, Blank[]]]], Pattern[x, Blank[Symbol]]] := Simp[(-2*a*Cot[e + f*x]*(c + d*Csc[e + f*x])^n)/(f*(2*n + 1)*Sqrt[a + b*Csc[e + f*x]]), x] + Dist[1/c, Int[Sqrt[a + b*Csc[e + f*x]]*(c + d*Csc[e + f*x])^(n + 1), x], x] /; FreeQ[{a, b, c, d, e, f}, x] && EqQ[b*c + a*d, 0] && EqQ[a^2 - b^2, 0] && LtQ[n, -2^(-1)]
  3908. Int[Times[Power[Plus[Times[csc[Plus[Optional[Pattern[e, Blank[]]], Times[Optional[Pattern[f, Blank[]]], Pattern[x, Blank[]]]]], Optional[Pattern[b, Blank[]]]], Pattern[a, Blank[]]], Rational[3, 2]], Power[Plus[Times[csc[Plus[Optional[Pattern[e, Blank[]]], Times[Optional[Pattern[f, Blank[]]], Pattern[x, Blank[]]]]], Optional[Pattern[d, Blank[]]]], Pattern[c, Blank[]]], Optional[Pattern[n, Blank[]]]]], Pattern[x, Blank[Symbol]]] := Simp[(-4*a^2*Cot[e + f*x]*(c + d*Csc[e + f*x])^n)/(f*(2*n + 1)*Sqrt[a + b*Csc[e + f*x]]), x] + Dist[a/c, Int[Sqrt[a + b*Csc[e + f*x]]*(c + d*Csc[e + f*x])^(n + 1), x], x] /; FreeQ[{a, b, c, d, e, f}, x] && EqQ[b*c + a*d, 0] && EqQ[a^2 - b^2, 0] && LtQ[n, -2^(-1)]
  3909. Int[Times[Power[Plus[Times[csc[Plus[Optional[Pattern[e, Blank[]]], Times[Optional[Pattern[f, Blank[]]], Pattern[x, Blank[]]]]], Optional[Pattern[b, Blank[]]]], Pattern[a, Blank[]]], Rational[3, 2]], Power[Plus[Times[csc[Plus[Optional[Pattern[e, Blank[]]], Times[Optional[Pattern[f, Blank[]]], Pattern[x, Blank[]]]]], Optional[Pattern[d, Blank[]]]], Pattern[c, Blank[]]], Optional[Pattern[n, Blank[]]]]], Pattern[x, Blank[Symbol]]] := Simp[(-2*a^2*Cot[e + f*x]*(c + d*Csc[e + f*x])^n)/(f*(2*n + 1)*Sqrt[a + b*Csc[e + f*x]]), x] + Dist[a, Int[Sqrt[a + b*Csc[e + f*x]]*(c + d*Csc[e + f*x])^n, x], x] /; FreeQ[{a, b, c, d, e, f, n}, x] && EqQ[b*c + a*d, 0] && EqQ[a^2 - b^2, 0] && !LeQ[n, -2^(-1)]
  3910. Int[Times[Power[Plus[Times[csc[Plus[Optional[Pattern[e, Blank[]]], Times[Optional[Pattern[f, Blank[]]], Pattern[x, Blank[]]]]], Optional[Pattern[b, Blank[]]]], Pattern[a, Blank[]]], Rational[5, 2]], Power[Plus[Times[csc[Plus[Optional[Pattern[e, Blank[]]], Times[Optional[Pattern[f, Blank[]]], Pattern[x, Blank[]]]]], Optional[Pattern[d, Blank[]]]], Pattern[c, Blank[]]], Optional[Pattern[n, Blank[]]]]], Pattern[x, Blank[Symbol]]] := Simp[(-8*a^3*Cot[e + f*x]*(c + d*Csc[e + f*x])^n)/(f*(2*n + 1)*Sqrt[a + b*Csc[e + f*x]]), x] + Dist[a^2/c^2, Int[Sqrt[a + b*Csc[e + f*x]]*(c + d*Csc[e + f*x])^(n + 2), x], x] /; FreeQ[{a, b, c, d, e, f}, x] && EqQ[b*c + a*d, 0] && EqQ[a^2 - b^2, 0] && LtQ[n, -2^(-1)]
  3911. Int[Times[Power[Plus[Times[csc[Plus[Optional[Pattern[e, Blank[]]], Times[Optional[Pattern[f, Blank[]]], Pattern[x, Blank[]]]]], Optional[Pattern[b, Blank[]]]], Pattern[a, Blank[]]], Pattern[m, Blank[]]], Power[Plus[Times[csc[Plus[Optional[Pattern[e, Blank[]]], Times[Optional[Pattern[f, Blank[]]], Pattern[x, Blank[]]]]], Optional[Pattern[d, Blank[]]]], Pattern[c, Blank[]]], Pattern[n, Blank[]]]], Pattern[x, Blank[Symbol]]] := -Dist[(a*c*Cot[e + f*x])/(f*Sqrt[a + b*Csc[e + f*x]]*Sqrt[c + d*Csc[e + f*x]]), Subst[Int[((b + a*x)^(m - 1/2)*(d + c*x)^(n - 1/2))/x^(m + n), x], x, Sin[e + f*x]], x] /; FreeQ[{a, b, c, d, e, f}, x] && EqQ[b*c + a*d, 0] && EqQ[a^2 - b^2, 0] && IntegerQ[m - 1/2] && EqQ[m + n, 0]
  3912. Int[Times[Power[Plus[Times[csc[Plus[Optional[Pattern[e, Blank[]]], Times[Optional[Pattern[f, Blank[]]], Pattern[x, Blank[]]]]], Optional[Pattern[b, Blank[]]]], Pattern[a, Blank[]]], Optional[Pattern[m, Blank[]]]], Power[Plus[Times[csc[Plus[Optional[Pattern[e, Blank[]]], Times[Optional[Pattern[f, Blank[]]], Pattern[x, Blank[]]]]], Optional[Pattern[d, Blank[]]]], Pattern[c, Blank[]]], Optional[Pattern[n, Blank[]]]]], Pattern[x, Blank[Symbol]]] := Dist[(a*c*Cot[e + f*x])/(f*Sqrt[a + b*Csc[e + f*x]]*Sqrt[c + d*Csc[e + f*x]]), Subst[Int[((a + b*x)^(m - 1/2)*(c + d*x)^(n - 1/2))/x, x], x, Csc[e + f*x]], x] /; FreeQ[{a, b, c, d, e, f, m, n}, x] && EqQ[b*c + a*d, 0] && EqQ[a^2 - b^2, 0]
  3913. Int[Times[Plus[Times[csc[Plus[Optional[Pattern[e, Blank[]]], Times[Optional[Pattern[f, Blank[]]], Pattern[x, Blank[]]]]], Optional[Pattern[b, Blank[]]]], Pattern[a, Blank[]]], Plus[Times[csc[Plus[Optional[Pattern[e, Blank[]]], Times[Optional[Pattern[f, Blank[]]], Pattern[x, Blank[]]]]], Optional[Pattern[d, Blank[]]]], Pattern[c, Blank[]]]], Pattern[x, Blank[Symbol]]] := Simp[a*c*x, x] + Dist[b*d, Int[Csc[e + f*x]^2, x], x] /; FreeQ[{a, b, c, d, e, f}, x] && EqQ[b*c + a*d, 0]
  3914. Int[Times[Plus[Times[csc[Plus[Optional[Pattern[e, Blank[]]], Times[Optional[Pattern[f, Blank[]]], Pattern[x, Blank[]]]]], Optional[Pattern[b, Blank[]]]], Pattern[a, Blank[]]], Plus[Times[csc[Plus[Optional[Pattern[e, Blank[]]], Times[Optional[Pattern[f, Blank[]]], Pattern[x, Blank[]]]]], Optional[Pattern[d, Blank[]]]], Pattern[c, Blank[]]]], Pattern[x, Blank[Symbol]]] := Simp[a*c*x, x] + (Dist[b*d, Int[Csc[e + f*x]^2, x], x] + Dist[b*c + a*d, Int[Csc[e + f*x], x], x]) /; FreeQ[{a, b, c, d, e, f}, x] && NeQ[b*c - a*d, 0] && NeQ[b*c + a*d, 0]
  3915. Int[Times[Power[Plus[Times[csc[Plus[Optional[Pattern[e, Blank[]]], Times[Optional[Pattern[f, Blank[]]], Pattern[x, Blank[]]]]], Optional[Pattern[b, Blank[]]]], Pattern[a, Blank[]]], Rational[1, 2]], Plus[Times[csc[Plus[Optional[Pattern[e, Blank[]]], Times[Optional[Pattern[f, Blank[]]], Pattern[x, Blank[]]]]], Optional[Pattern[d, Blank[]]]], Pattern[c, Blank[]]]], Pattern[x, Blank[Symbol]]] := Dist[c, Int[Sqrt[a + b*Csc[e + f*x]], x], x] + Dist[d, Int[Sqrt[a + b*Csc[e + f*x]]*Csc[e + f*x], x], x] /; FreeQ[{a, b, c, d, e, f}, x] && NeQ[b*c - a*d, 0] && EqQ[a^2 - b^2, 0]
  3916. Int[Times[Power[Plus[Times[csc[Plus[Optional[Pattern[e, Blank[]]], Times[Optional[Pattern[f, Blank[]]], Pattern[x, Blank[]]]]], Optional[Pattern[b, Blank[]]]], Pattern[a, Blank[]]], Rational[1, 2]], Plus[Times[csc[Plus[Optional[Pattern[e, Blank[]]], Times[Optional[Pattern[f, Blank[]]], Pattern[x, Blank[]]]]], Optional[Pattern[d, Blank[]]]], Pattern[c, Blank[]]]], Pattern[x, Blank[Symbol]]] := Dist[a*c, Int[1/Sqrt[a + b*Csc[e + f*x]], x], x] + Int[(Csc[e + f*x]*(b*c + a*d + b*d*Csc[e + f*x]))/Sqrt[a + b*Csc[e + f*x]], x] /; FreeQ[{a, b, c, d, e, f}, x] && NeQ[b*c - a*d, 0] && NeQ[a^2 - b^2, 0]
  3917. Int[Times[Power[Plus[Times[csc[Plus[Optional[Pattern[e, Blank[]]], Times[Optional[Pattern[f, Blank[]]], Pattern[x, Blank[]]]]], Optional[Pattern[b, Blank[]]]], Pattern[a, Blank[]]], Pattern[m, Blank[]]], Plus[Times[csc[Plus[Optional[Pattern[e, Blank[]]], Times[Optional[Pattern[f, Blank[]]], Pattern[x, Blank[]]]]], Optional[Pattern[d, Blank[]]]], Pattern[c, Blank[]]]], Pattern[x, Blank[Symbol]]] := -Simp[(b*d*Cot[e + f*x]*(a + b*Csc[e + f*x])^(m - 1))/(f*m), x] + Dist[1/m, Int[(a + b*Csc[e + f*x])^(m - 1)*Simp[a*c*m + (b*c*m + a*d*(2*m - 1))*Csc[e + f*x], x], x], x] /; FreeQ[{a, b, c, d, e, f}, x] && NeQ[b*c - a*d, 0] && GtQ[m, 1] && EqQ[a^2 - b^2, 0] && IntegerQ[2*m]
  3918. Int[Times[Power[Plus[Times[csc[Plus[Optional[Pattern[e, Blank[]]], Times[Optional[Pattern[f, Blank[]]], Pattern[x, Blank[]]]]], Optional[Pattern[b, Blank[]]]], Pattern[a, Blank[]]], Pattern[m, Blank[]]], Plus[Times[csc[Plus[Optional[Pattern[e, Blank[]]], Times[Optional[Pattern[f, Blank[]]], Pattern[x, Blank[]]]]], Optional[Pattern[d, Blank[]]]], Pattern[c, Blank[]]]], Pattern[x, Blank[Symbol]]] := -Simp[(b*d*Cot[e + f*x]*(a + b*Csc[e + f*x])^(m - 1))/(f*m), x] + Dist[1/m, Int[(a + b*Csc[e + f*x])^(m - 2)*Simp[a^2*c*m + (b^2*d*(m - 1) + 2*a*b*c*m + a^2*d*m)*Csc[e + f*x] + b*(b*c*m + a*d*(2*m - 1))*Csc[e + f*x]^2, x], x], x] /; FreeQ[{a, b, c, d, e, f}, x] && NeQ[b*c - a*d, 0] && GtQ[m, 1] && NeQ[a^2 - b^2, 0] && IntegerQ[2*m]
  3919. Int[Times[Power[Plus[Times[csc[Plus[Optional[Pattern[e, Blank[]]], Times[Optional[Pattern[f, Blank[]]], Pattern[x, Blank[]]]]], Optional[Pattern[b, Blank[]]]], Pattern[a, Blank[]]], -1], Plus[Times[csc[Plus[Optional[Pattern[e, Blank[]]], Times[Optional[Pattern[f, Blank[]]], Pattern[x, Blank[]]]]], Optional[Pattern[d, Blank[]]]], Pattern[c, Blank[]]]], Pattern[x, Blank[Symbol]]] := Simp[(c*x)/a, x] - Dist[(b*c - a*d)/a, Int[Csc[e + f*x]/(a + b*Csc[e + f*x]), x], x] /; FreeQ[{a, b, c, d, e, f}, x] && NeQ[b*c - a*d, 0]
  3920. Int[Times[Power[Plus[Times[csc[Plus[Optional[Pattern[e, Blank[]]], Times[Optional[Pattern[f, Blank[]]], Pattern[x, Blank[]]]]], Optional[Pattern[b, Blank[]]]], Pattern[a, Blank[]]], Rational[-1, 2]], Plus[Times[csc[Plus[Optional[Pattern[e, Blank[]]], Times[Optional[Pattern[f, Blank[]]], Pattern[x, Blank[]]]]], Optional[Pattern[d, Blank[]]]], Pattern[c, Blank[]]]], Pattern[x, Blank[Symbol]]] := Dist[c/a, Int[Sqrt[a + b*Csc[e + f*x]], x], x] - Dist[(b*c - a*d)/a, Int[Csc[e + f*x]/Sqrt[a + b*Csc[e + f*x]], x], x] /; FreeQ[{a, b, c, d, e, f}, x] && NeQ[b*c - a*d, 0] && EqQ[a^2 - b^2, 0]
  3921. Int[Times[Power[Plus[Times[csc[Plus[Optional[Pattern[e, Blank[]]], Times[Optional[Pattern[f, Blank[]]], Pattern[x, Blank[]]]]], Optional[Pattern[b, Blank[]]]], Pattern[a, Blank[]]], Rational[-1, 2]], Plus[Times[csc[Plus[Optional[Pattern[e, Blank[]]], Times[Optional[Pattern[f, Blank[]]], Pattern[x, Blank[]]]]], Optional[Pattern[d, Blank[]]]], Pattern[c, Blank[]]]], Pattern[x, Blank[Symbol]]] := Dist[c, Int[1/Sqrt[a + b*Csc[e + f*x]], x], x] + Dist[d, Int[Csc[e + f*x]/Sqrt[a + b*Csc[e + f*x]], x], x] /; FreeQ[{a, b, c, d, e, f}, x] && NeQ[b*c - a*d, 0] && NeQ[a^2 - b^2, 0]
  3922. Int[Times[Power[Plus[Times[csc[Plus[Optional[Pattern[e, Blank[]]], Times[Optional[Pattern[f, Blank[]]], Pattern[x, Blank[]]]]], Optional[Pattern[b, Blank[]]]], Pattern[a, Blank[]]], Pattern[m, Blank[]]], Plus[Times[csc[Plus[Optional[Pattern[e, Blank[]]], Times[Optional[Pattern[f, Blank[]]], Pattern[x, Blank[]]]]], Optional[Pattern[d, Blank[]]]], Pattern[c, Blank[]]]], Pattern[x, Blank[Symbol]]] := -Simp[((b*c - a*d)*Cot[e + f*x]*(a + b*Csc[e + f*x])^m)/(b*f*(2*m + 1)), x] + Dist[1/(a^2*(2*m + 1)), Int[(a + b*Csc[e + f*x])^(m + 1)*Simp[a*c*(2*m + 1) - (b*c - a*d)*(m + 1)*Csc[e + f*x], x], x], x] /; FreeQ[{a, b, c, d, e, f}, x] && NeQ[b*c - a*d, 0] && LtQ[m, -1] && EqQ[a^2 - b^2, 0] && IntegerQ[2*m]
  3923. Int[Times[Power[Plus[Times[csc[Plus[Optional[Pattern[e, Blank[]]], Times[Optional[Pattern[f, Blank[]]], Pattern[x, Blank[]]]]], Optional[Pattern[b, Blank[]]]], Pattern[a, Blank[]]], Pattern[m, Blank[]]], Plus[Times[csc[Plus[Optional[Pattern[e, Blank[]]], Times[Optional[Pattern[f, Blank[]]], Pattern[x, Blank[]]]]], Optional[Pattern[d, Blank[]]]], Pattern[c, Blank[]]]], Pattern[x, Blank[Symbol]]] := Simp[(b*(b*c - a*d)*Cot[e + f*x]*(a + b*Csc[e + f*x])^(m + 1))/(a*f*(m + 1)*(a^2 - b^2)), x] + Dist[1/(a*(m + 1)*(a^2 - b^2)), Int[(a + b*Csc[e + f*x])^(m + 1)*Simp[c*(a^2 - b^2)*(m + 1) - (a*(b*c - a*d)*(m + 1))*Csc[e + f*x] + b*(b*c - a*d)*(m + 2)*Csc[e + f*x]^2, x], x], x] /; FreeQ[{a, b, c, d, e, f}, x] && NeQ[b*c - a*d, 0] && LtQ[m, -1] && NeQ[a^2 - b^2, 0] && IntegerQ[2*m]
  3924. Int[Times[Power[Plus[Times[csc[Plus[Optional[Pattern[e, Blank[]]], Times[Optional[Pattern[f, Blank[]]], Pattern[x, Blank[]]]]], Optional[Pattern[b, Blank[]]]], Pattern[a, Blank[]]], Pattern[m, Blank[]]], Plus[Times[csc[Plus[Optional[Pattern[e, Blank[]]], Times[Optional[Pattern[f, Blank[]]], Pattern[x, Blank[]]]]], Optional[Pattern[d, Blank[]]]], Pattern[c, Blank[]]]], Pattern[x, Blank[Symbol]]] := Dist[c, Int[(a + b*Csc[e + f*x])^m, x], x] + Dist[d, Int[(a + b*Csc[e + f*x])^m*Csc[e + f*x], x], x] /; FreeQ[{a, b, c, d, e, f, m}, x] && NeQ[b*c - a*d, 0] && !IntegerQ[2*m]
  3925. Int[Times[Power[Plus[Times[csc[Plus[Optional[Pattern[e, Blank[]]], Times[Optional[Pattern[f, Blank[]]], Pattern[x, Blank[]]]]], Optional[Pattern[b, Blank[]]]], Pattern[a, Blank[]]], Rational[1, 2]], Power[Plus[Times[csc[Plus[Optional[Pattern[e, Blank[]]], Times[Optional[Pattern[f, Blank[]]], Pattern[x, Blank[]]]]], Optional[Pattern[d, Blank[]]]], Pattern[c, Blank[]]], -1]], Pattern[x, Blank[Symbol]]] := Dist[1/c, Int[Sqrt[a + b*Csc[e + f*x]], x], x] - Dist[d/c, Int[(Csc[e + f*x]*Sqrt[a + b*Csc[e + f*x]])/(c + d*Csc[e + f*x]), x], x] /; FreeQ[{a, b, c, d, e, f}, x] && NeQ[b*c - a*d, 0] && (EqQ[a^2 - b^2, 0] || EqQ[c^2 - d^2, 0])
  3926. Int[Times[Power[Plus[Times[csc[Plus[Optional[Pattern[e, Blank[]]], Times[Optional[Pattern[f, Blank[]]], Pattern[x, Blank[]]]]], Optional[Pattern[b, Blank[]]]], Pattern[a, Blank[]]], Rational[1, 2]], Power[Plus[Times[csc[Plus[Optional[Pattern[e, Blank[]]], Times[Optional[Pattern[f, Blank[]]], Pattern[x, Blank[]]]]], Optional[Pattern[d, Blank[]]]], Pattern[c, Blank[]]], -1]], Pattern[x, Blank[Symbol]]] := Dist[a/c, Int[1/Sqrt[a + b*Csc[e + f*x]], x], x] + Dist[(b*c - a*d)/c, Int[Csc[e + f*x]/(Sqrt[a + b*Csc[e + f*x]]*(c + d*Csc[e + f*x])), x], x] /; FreeQ[{a, b, c, d, e, f}, x] && NeQ[b*c - a*d, 0] && NeQ[a^2 - b^2, 0] && NeQ[c^2 - d^2, 0]
  3927. Int[Times[Power[Plus[Times[csc[Plus[Optional[Pattern[e, Blank[]]], Times[Optional[Pattern[f, Blank[]]], Pattern[x, Blank[]]]]], Optional[Pattern[b, Blank[]]]], Pattern[a, Blank[]]], Rational[3, 2]], Power[Plus[Times[csc[Plus[Optional[Pattern[e, Blank[]]], Times[Optional[Pattern[f, Blank[]]], Pattern[x, Blank[]]]]], Optional[Pattern[d, Blank[]]]], Pattern[c, Blank[]]], -1]], Pattern[x, Blank[Symbol]]] := Dist[a/c, Int[Sqrt[a + b*Csc[e + f*x]], x], x] + Dist[(b*c - a*d)/c, Int[(Csc[e + f*x]*Sqrt[a + b*Csc[e + f*x]])/(c + d*Csc[e + f*x]), x], x] /; FreeQ[{a, b, c, d, e, f}, x] && NeQ[b*c - a*d, 0] && (EqQ[a^2 - b^2, 0] || EqQ[c^2 - d^2, 0])
  3928. Int[Times[Power[Plus[Times[csc[Plus[Optional[Pattern[e, Blank[]]], Times[Optional[Pattern[f, Blank[]]], Pattern[x, Blank[]]]]], Optional[Pattern[b, Blank[]]]], Pattern[a, Blank[]]], Rational[3, 2]], Power[Plus[Times[csc[Plus[Optional[Pattern[e, Blank[]]], Times[Optional[Pattern[f, Blank[]]], Pattern[x, Blank[]]]]], Optional[Pattern[d, Blank[]]]], Pattern[c, Blank[]]], -1]], Pattern[x, Blank[Symbol]]] := Dist[1/(c*d), Int[(a^2*d + b^2*c*Csc[e + f*x])/Sqrt[a + b*Csc[e + f*x]], x], x] - Dist[(b*c - a*d)^2/(c*d), Int[Csc[e + f*x]/(Sqrt[a + b*Csc[e + f*x]]*(c + d*Csc[e + f*x])), x], x] /; FreeQ[{a, b, c, d, e, f}, x] && NeQ[b*c - a*d, 0] && NeQ[a^2 - b^2, 0] && NeQ[c^2 - d^2, 0]
  3929. Int[Times[Power[Plus[Times[csc[Plus[Optional[Pattern[e, Blank[]]], Times[Optional[Pattern[f, Blank[]]], Pattern[x, Blank[]]]]], Optional[Pattern[b, Blank[]]]], Pattern[a, Blank[]]], Rational[-1, 2]], Power[Plus[Times[csc[Plus[Optional[Pattern[e, Blank[]]], Times[Optional[Pattern[f, Blank[]]], Pattern[x, Blank[]]]]], Optional[Pattern[d, Blank[]]]], Pattern[c, Blank[]]], -1]], Pattern[x, Blank[Symbol]]] := Dist[1/(c*(b*c - a*d)), Int[(b*c - a*d - b*d*Csc[e + f*x])/Sqrt[a + b*Csc[e + f*x]], x], x] + Dist[d^2/(c*(b*c - a*d)), Int[(Csc[e + f*x]*Sqrt[a + b*Csc[e + f*x]])/(c + d*Csc[e + f*x]), x], x] /; FreeQ[{a, b, c, d, e, f}, x] && NeQ[b*c - a*d, 0] && (EqQ[a^2 - b^2, 0] || EqQ[c^2 - d^2, 0])
  3930. Int[Times[Power[Plus[Times[csc[Plus[Optional[Pattern[e, Blank[]]], Times[Optional[Pattern[f, Blank[]]], Pattern[x, Blank[]]]]], Optional[Pattern[b, Blank[]]]], Pattern[a, Blank[]]], Rational[-1, 2]], Power[Plus[Times[csc[Plus[Optional[Pattern[e, Blank[]]], Times[Optional[Pattern[f, Blank[]]], Pattern[x, Blank[]]]]], Optional[Pattern[d, Blank[]]]], Pattern[c, Blank[]]], -1]], Pattern[x, Blank[Symbol]]] := Dist[1/c, Int[1/Sqrt[a + b*Csc[e + f*x]], x], x] - Dist[d/c, Int[Csc[e + f*x]/(Sqrt[a + b*Csc[e + f*x]]*(c + d*Csc[e + f*x])), x], x] /; FreeQ[{a, b, c, d, e, f}, x] && NeQ[b*c - a*d, 0] && NeQ[a^2 - b^2, 0]
  3931. Int[Times[Power[Plus[Times[csc[Plus[Optional[Pattern[e, Blank[]]], Times[Optional[Pattern[f, Blank[]]], Pattern[x, Blank[]]]]], Optional[Pattern[b, Blank[]]]], Pattern[a, Blank[]]], Rational[1, 2]], Power[Plus[Times[csc[Plus[Optional[Pattern[e, Blank[]]], Times[Optional[Pattern[f, Blank[]]], Pattern[x, Blank[]]]]], Optional[Pattern[d, Blank[]]]], Pattern[c, Blank[]]], Rational[1, 2]]], Pattern[x, Blank[Symbol]]] := Dist[(Sqrt[a + b*Csc[e + f*x]]*Sqrt[c + d*Csc[e + f*x]])/Cot[e + f*x], Int[Cot[e + f*x], x], x] /; FreeQ[{a, b, c, d, e, f}, x] && NeQ[b*c - a*d, 0] && EqQ[a^2 - b^2, 0] && EqQ[c^2 - d^2, 0]
  3932. Int[Times[Power[Plus[Times[csc[Plus[Optional[Pattern[e, Blank[]]], Times[Optional[Pattern[f, Blank[]]], Pattern[x, Blank[]]]]], Optional[Pattern[b, Blank[]]]], Pattern[a, Blank[]]], Rational[1, 2]], Power[Plus[Times[csc[Plus[Optional[Pattern[e, Blank[]]], Times[Optional[Pattern[f, Blank[]]], Pattern[x, Blank[]]]]], Optional[Pattern[d, Blank[]]]], Pattern[c, Blank[]]], Rational[1, 2]]], Pattern[x, Blank[Symbol]]] := Dist[c, Int[Sqrt[a + b*Csc[e + f*x]]/Sqrt[c + d*Csc[e + f*x]], x], x] + Dist[d, Int[(Csc[e + f*x]*Sqrt[a + b*Csc[e + f*x]])/Sqrt[c + d*Csc[e + f*x]], x], x] /; FreeQ[{a, b, c, d, e, f}, x] && NeQ[b*c - a*d, 0]
  3933. Int[Times[Power[Plus[Times[csc[Plus[Optional[Pattern[e, Blank[]]], Times[Optional[Pattern[f, Blank[]]], Pattern[x, Blank[]]]]], Optional[Pattern[b, Blank[]]]], Pattern[a, Blank[]]], Rational[1, 2]], Power[Plus[Times[csc[Plus[Optional[Pattern[e, Blank[]]], Times[Optional[Pattern[f, Blank[]]], Pattern[x, Blank[]]]]], Optional[Pattern[d, Blank[]]]], Pattern[c, Blank[]]], Rational[-1, 2]]], Pattern[x, Blank[Symbol]]] := Dist[1/c, Int[Sqrt[a + b*Csc[e + f*x]]*Sqrt[c + d*Csc[e + f*x]], x], x] - Dist[d/c, Int[(Csc[e + f*x]*Sqrt[a + b*Csc[e + f*x]])/Sqrt[c + d*Csc[e + f*x]], x], x] /; FreeQ[{a, b, c, d, e, f}, x] && NeQ[b*c - a*d, 0] && EqQ[a^2 - b^2, 0] && EqQ[c^2 - d^2, 0]
  3934. Int[Times[Power[Plus[Times[csc[Plus[Optional[Pattern[e, Blank[]]], Times[Optional[Pattern[f, Blank[]]], Pattern[x, Blank[]]]]], Optional[Pattern[b, Blank[]]]], Pattern[a, Blank[]]], Rational[1, 2]], Power[Plus[Times[csc[Plus[Optional[Pattern[e, Blank[]]], Times[Optional[Pattern[f, Blank[]]], Pattern[x, Blank[]]]]], Optional[Pattern[d, Blank[]]]], Pattern[c, Blank[]]], Rational[-1, 2]]], Pattern[x, Blank[Symbol]]] := Dist[(-2*a)/f, Subst[Int[1/(1 + a*c*x^2), x], x, Cot[e + f*x]/(Sqrt[a + b*Csc[e + f*x]]*Sqrt[c + d*Csc[e + f*x]])], x] /; FreeQ[{a, b, c, d, e, f}, x] && NeQ[b*c - a*d, 0] && EqQ[a^2 - b^2, 0] && NeQ[c^2 - d^2, 0]
  3935. Int[Times[Power[Plus[Times[csc[Plus[Optional[Pattern[e, Blank[]]], Times[Optional[Pattern[f, Blank[]]], Pattern[x, Blank[]]]]], Optional[Pattern[b, Blank[]]]], Pattern[a, Blank[]]], Rational[1, 2]], Power[Plus[Times[csc[Plus[Optional[Pattern[e, Blank[]]], Times[Optional[Pattern[f, Blank[]]], Pattern[x, Blank[]]]]], Optional[Pattern[d, Blank[]]]], Pattern[c, Blank[]]], Rational[-1, 2]]], Pattern[x, Blank[Symbol]]] := Dist[a/c, Int[Sqrt[c + d*Csc[e + f*x]]/Sqrt[a + b*Csc[e + f*x]], x], x] + Dist[(b*c - a*d)/c, Int[Csc[e + f*x]/(Sqrt[a + b*Csc[e + f*x]]*Sqrt[c + d*Csc[e + f*x]]), x], x] /; FreeQ[{a, b, c, d, e, f}, x] && NeQ[b*c - a*d, 0] && NeQ[a^2 - b^2, 0] && EqQ[c^2 - d^2, 0]
  3936. Int[Times[Power[Plus[Times[csc[Plus[Optional[Pattern[e, Blank[]]], Times[Optional[Pattern[f, Blank[]]], Pattern[x, Blank[]]]]], Optional[Pattern[b, Blank[]]]], Pattern[a, Blank[]]], Rational[1, 2]], Power[Plus[Times[csc[Plus[Optional[Pattern[e, Blank[]]], Times[Optional[Pattern[f, Blank[]]], Pattern[x, Blank[]]]]], Optional[Pattern[d, Blank[]]]], Pattern[c, Blank[]]], Rational[-1, 2]]], Pattern[x, Blank[Symbol]]] := Simp[(2*(a + b*Csc[e + f*x])*Sqrt[((b*c - a*d)*(1 + Csc[e + f*x]))/((c - d)*(a + b*Csc[e + f*x]))]*Sqrt[-(((b*c - a*d)*(1 - Csc[e + f*x]))/((c + d)*(a + b*Csc[e + f*x])))]*EllipticPi[(a*(c + d))/(c*(a + b)), ArcSin[(Rt[(a + b)/(c + d), 2]*Sqrt[c + d*Csc[e + f*x]])/Sqrt[a + b*Csc[e + f*x]]], ((a - b)*(c + d))/((a + b)*(c - d))])/(c*f*Rt[(a + b)/(c + d), 2]*Cot[e + f*x]), x] /; FreeQ[{a, b, c, d, e, f}, x] && NeQ[b*c - a*d, 0] && NeQ[a^2 - b^2, 0] && NeQ[c^2 - d^2, 0]
  3937. Int[Times[Power[Plus[Times[csc[Plus[Optional[Pattern[e, Blank[]]], Times[Optional[Pattern[f, Blank[]]], Pattern[x, Blank[]]]]], Optional[Pattern[b, Blank[]]]], Pattern[a, Blank[]]], Rational[-1, 2]], Power[Plus[Times[csc[Plus[Optional[Pattern[e, Blank[]]], Times[Optional[Pattern[f, Blank[]]], Pattern[x, Blank[]]]]], Optional[Pattern[d, Blank[]]]], Pattern[c, Blank[]]], Rational[-1, 2]]], Pattern[x, Blank[Symbol]]] := Dist[Cot[e + f*x]/(Sqrt[a + b*Csc[e + f*x]]*Sqrt[c + d*Csc[e + f*x]]), Int[1/Cot[e + f*x], x], x] /; FreeQ[{a, b, c, d, e, f}, x] && NeQ[b*c - a*d, 0] && EqQ[a^2 - b^2, 0] && EqQ[c^2 - d^2, 0]
  3938. Int[Times[Power[Plus[Times[csc[Plus[Optional[Pattern[e, Blank[]]], Times[Optional[Pattern[f, Blank[]]], Pattern[x, Blank[]]]]], Optional[Pattern[b, Blank[]]]], Pattern[a, Blank[]]], Rational[-1, 2]], Power[Plus[Times[csc[Plus[Optional[Pattern[e, Blank[]]], Times[Optional[Pattern[f, Blank[]]], Pattern[x, Blank[]]]]], Optional[Pattern[d, Blank[]]]], Pattern[c, Blank[]]], Rational[-1, 2]]], Pattern[x, Blank[Symbol]]] := Dist[1/a, Int[Sqrt[a + b*Csc[e + f*x]]/Sqrt[c + d*Csc[e + f*x]], x], x] - Dist[b/a, Int[Csc[e + f*x]/(Sqrt[a + b*Csc[e + f*x]]*Sqrt[c + d*Csc[e + f*x]]), x], x] /; FreeQ[{a, b, c, d, e, f}, x] && NeQ[b*c - a*d, 0]
  3939. Int[Times[Power[Plus[Times[csc[Plus[Optional[Pattern[e, Blank[]]], Times[Optional[Pattern[f, Blank[]]], Pattern[x, Blank[]]]]], Optional[Pattern[b, Blank[]]]], Pattern[a, Blank[]]], Rational[1, 2]], Power[Plus[Times[csc[Plus[Optional[Pattern[e, Blank[]]], Times[Optional[Pattern[f, Blank[]]], Pattern[x, Blank[]]]]], Optional[Pattern[d, Blank[]]]], Pattern[c, Blank[]]], Rational[-3, 2]]], Pattern[x, Blank[Symbol]]] := Dist[1/c, Int[Sqrt[a + b*Csc[e + f*x]]/Sqrt[c + d*Csc[e + f*x]], x], x] - Dist[d/c, Int[(Csc[e + f*x]*Sqrt[a + b*Csc[e + f*x]])/(c + d*Csc[e + f*x])^(3/2), x], x] /; FreeQ[{a, b, c, d, e, f}, x] && NeQ[b*c - a*d, 0] && NeQ[c^2 - d^2, 0]
  3940. Int[Times[Power[Plus[Times[csc[Plus[Optional[Pattern[e, Blank[]]], Times[Optional[Pattern[f, Blank[]]], Pattern[x, Blank[]]]]], Optional[Pattern[b, Blank[]]]], Pattern[a, Blank[]]], Optional[Pattern[m, Blank[]]]], Power[Plus[Times[csc[Plus[Optional[Pattern[e, Blank[]]], Times[Optional[Pattern[f, Blank[]]], Pattern[x, Blank[]]]]], Optional[Pattern[d, Blank[]]]], Pattern[c, Blank[]]], Optional[Pattern[n, Blank[]]]]], Pattern[x, Blank[Symbol]]] := Dist[(a^2*Cot[e + f*x])/(f*Sqrt[a + b*Csc[e + f*x]]*Sqrt[a - b*Csc[e + f*x]]), Subst[Int[((a + b*x)^(m - 1/2)*(c + d*x)^n)/(x*Sqrt[a - b*x]), x], x, Csc[e + f*x]], x] /; FreeQ[{a, b, c, d, e, f, m, n}, x] && NeQ[b*c - a*d, 0] && EqQ[a^2 - b^2, 0] && NeQ[c^2 - d^2, 0] && IntegerQ[m - 1/2]
  3941. Int[Times[Power[Plus[Times[csc[Plus[Optional[Pattern[e, Blank[]]], Times[Optional[Pattern[f, Blank[]]], Pattern[x, Blank[]]]]], Optional[Pattern[b, Blank[]]]], Pattern[a, Blank[]]], Pattern[m, Blank[]]], Power[Plus[Times[csc[Plus[Optional[Pattern[e, Blank[]]], Times[Optional[Pattern[f, Blank[]]], Pattern[x, Blank[]]]]], Optional[Pattern[d, Blank[]]]], Pattern[c, Blank[]]], Pattern[n, Blank[]]]], Pattern[x, Blank[Symbol]]] := Int[((b + a*Sin[e + f*x])^m*(d + c*Sin[e + f*x])^n)/Sin[e + f*x]^(m + n), x] /; FreeQ[{a, b, c, d, e, f, m, n}, x] && NeQ[b*c - a*d, 0] && IntegerQ[m] && IntegerQ[n] && LeQ[-2, m + n, 0]
  3942. Int[Times[Power[Plus[Times[csc[Plus[Optional[Pattern[e, Blank[]]], Times[Optional[Pattern[f, Blank[]]], Pattern[x, Blank[]]]]], Optional[Pattern[b, Blank[]]]], Pattern[a, Blank[]]], Pattern[m, Blank[]]], Power[Plus[Times[csc[Plus[Optional[Pattern[e, Blank[]]], Times[Optional[Pattern[f, Blank[]]], Pattern[x, Blank[]]]]], Optional[Pattern[d, Blank[]]]], Pattern[c, Blank[]]], Pattern[n, Blank[]]]], Pattern[x, Blank[Symbol]]] := Dist[(Sqrt[d + c*Sin[e + f*x]]*Sqrt[a + b*Csc[e + f*x]])/(Sqrt[b + a*Sin[e + f*x]]*Sqrt[c + d*Csc[e + f*x]]), Int[((b + a*Sin[e + f*x])^m*(d + c*Sin[e + f*x])^n)/Sin[e + f*x]^(m + n), x], x] /; FreeQ[{a, b, c, d, e, f, m, n}, x] && NeQ[b*c - a*d, 0] && IntegerQ[m + 1/2] && IntegerQ[n + 1/2] && LeQ[-2, m + n, 0]
  3943. Int[Times[Power[Plus[Times[csc[Plus[Optional[Pattern[e, Blank[]]], Times[Optional[Pattern[f, Blank[]]], Pattern[x, Blank[]]]]], Optional[Pattern[b, Blank[]]]], Pattern[a, Blank[]]], Pattern[m, Blank[]]], Power[Plus[Times[csc[Plus[Optional[Pattern[e, Blank[]]], Times[Optional[Pattern[f, Blank[]]], Pattern[x, Blank[]]]]], Optional[Pattern[d, Blank[]]]], Pattern[c, Blank[]]], Pattern[n, Blank[]]]], Pattern[x, Blank[Symbol]]] := Dist[(Sin[e + f*x]^(m + n)*(a + b*Csc[e + f*x])^m*(c + d*Csc[e + f*x])^n)/((b + a*Sin[e + f*x])^m*(d + c*Sin[e + f*x])^n), Int[((b + a*Sin[e + f*x])^m*(d + c*Sin[e + f*x])^n)/Sin[e + f*x]^Simplify[m + n], x], x] /; FreeQ[{a, b, c, d, e, f, m, n}, x] && NeQ[b*c - a*d, 0] && EqQ[m + n, 0] && !IntegerQ[2*m]
  3944. Int[Times[Power[Plus[Times[csc[Plus[Optional[Pattern[e, Blank[]]], Times[Optional[Pattern[f, Blank[]]], Pattern[x, Blank[]]]]], Optional[Pattern[b, Blank[]]]], Pattern[a, Blank[]]], Pattern[m, Blank[]]], Power[Plus[Times[csc[Plus[Optional[Pattern[e, Blank[]]], Times[Optional[Pattern[f, Blank[]]], Pattern[x, Blank[]]]]], Optional[Pattern[d, Blank[]]]], Pattern[c, Blank[]]], Pattern[n, Blank[]]]], Pattern[x, Blank[Symbol]]] := Int[ExpandTrig[(a + b*csc[e + f*x])^m, (c + d*csc[e + f*x])^n, x], x] /; FreeQ[{a, b, c, d, e, f, m, n}, x] && IGtQ[n, 0]
  3945. Int[Times[Power[Plus[Times[csc[Plus[Optional[Pattern[e, Blank[]]], Times[Optional[Pattern[f, Blank[]]], Pattern[x, Blank[]]]]], Optional[Pattern[b, Blank[]]]], Pattern[a, Blank[]]], Optional[Pattern[m, Blank[]]]], Power[Plus[Times[csc[Plus[Optional[Pattern[e, Blank[]]], Times[Optional[Pattern[f, Blank[]]], Pattern[x, Blank[]]]]], Optional[Pattern[d, Blank[]]]], Pattern[c, Blank[]]], Optional[Pattern[n, Blank[]]]]], Pattern[x, Blank[Symbol]]] := Unintegrable[(a + b*Csc[e + f*x])^m*(c + d*Csc[e + f*x])^n, x] /; FreeQ[{a, b, c, d, e, f, m, n}, x]
  3946. Int[Times[Power[Times[Optional[Pattern[d, Blank[]]], Power[sec[Plus[Optional[Pattern[e, Blank[]]], Times[Optional[Pattern[f, Blank[]]], Pattern[x, Blank[]]]]], -1]], Pattern[n, Blank[]]], Power[Plus[Optional[Pattern[a, Blank[]]], Times[Optional[Pattern[b, Blank[]]], sec[Plus[Optional[Pattern[e, Blank[]]], Times[Optional[Pattern[f, Blank[]]], Pattern[x, Blank[]]]]]]], Optional[Pattern[m, Blank[]]]]], Pattern[x, Blank[Symbol]]] := Dist[d^m, Int[(b + a*Cos[e + f*x])^m*(d*Cos[e + f*x])^(n - m), x], x] /; FreeQ[{a, b, d, e, f, n}, x] && !IntegerQ[n] && IntegerQ[m]
  3947. Int[Times[Power[Plus[Optional[Pattern[a, Blank[]]], Times[csc[Plus[Optional[Pattern[e, Blank[]]], Times[Optional[Pattern[f, Blank[]]], Pattern[x, Blank[]]]]], Optional[Pattern[b, Blank[]]]]], Optional[Pattern[m, Blank[]]]], Power[Times[Power[csc[Plus[Optional[Pattern[e, Blank[]]], Times[Optional[Pattern[f, Blank[]]], Pattern[x, Blank[]]]]], -1], Optional[Pattern[d, Blank[]]]], Pattern[n, Blank[]]]], Pattern[x, Blank[Symbol]]] := Dist[d^m, Int[(b + a*Sin[e + f*x])^m*(d*Sin[e + f*x])^(n - m), x], x] /; FreeQ[{a, b, d, e, f, n}, x] && !IntegerQ[n] && IntegerQ[m]
  3948. Int[Times[Power[Times[Optional[Pattern[c, Blank[]]], Power[Times[Optional[Pattern[d, Blank[]]], sec[Plus[Optional[Pattern[e, Blank[]]], Times[Optional[Pattern[f, Blank[]]], Pattern[x, Blank[]]]]]], Pattern[p, Blank[]]]], Pattern[n, Blank[]]], Power[Plus[Optional[Pattern[a, Blank[]]], Times[Optional[Pattern[b, Blank[]]], sec[Plus[Optional[Pattern[e, Blank[]]], Times[Optional[Pattern[f, Blank[]]], Pattern[x, Blank[]]]]]]], Optional[Pattern[m, Blank[]]]]], Pattern[x, Blank[Symbol]]] := Dist[(c^IntPart[n]*(c*(d*Sec[e + f*x])^p)^FracPart[n])/(d*Sec[e + f*x])^(p*FracPart[n]), Int[(a + b*Sec[e + f*x])^m*(d*Sec[e + f*x])^(n*p), x], x] /; FreeQ[{a, b, c, d, e, f, m, n, p}, x] && !IntegerQ[n]
  3949. Int[Times[Power[Plus[Optional[Pattern[a, Blank[]]], Times[csc[Plus[Optional[Pattern[e, Blank[]]], Times[Optional[Pattern[f, Blank[]]], Pattern[x, Blank[]]]]], Optional[Pattern[b, Blank[]]]]], Optional[Pattern[m, Blank[]]]], Power[Times[Optional[Pattern[c, Blank[]]], Power[Times[csc[Plus[Optional[Pattern[e, Blank[]]], Times[Optional[Pattern[f, Blank[]]], Pattern[x, Blank[]]]]], Optional[Pattern[d, Blank[]]]], Pattern[p, Blank[]]]], Pattern[n, Blank[]]]], Pattern[x, Blank[Symbol]]] := Dist[(c^IntPart[n]*(c*(d*Csc[e + f*x])^p)^FracPart[n])/(d*Csc[e + f*x])^(p*FracPart[n]), Int[(a + b*Cos[e + f*x])^m*(d*Cos[e + f*x])^(n*p), x], x] /; FreeQ[{a, b, c, d, e, f, m, n, p}, x] && !IntegerQ[n]
  3950. Int[Times[csc[Plus[Optional[Pattern[e, Blank[]]], Times[Optional[Pattern[f, Blank[]]], Pattern[x, Blank[]]]]], Power[Plus[Times[csc[Plus[Optional[Pattern[e, Blank[]]], Times[Optional[Pattern[f, Blank[]]], Pattern[x, Blank[]]]]], Optional[Pattern[b, Blank[]]]], Pattern[a, Blank[]]], Pattern[m, Blank[]]], Power[Plus[Times[csc[Plus[Optional[Pattern[e, Blank[]]], Times[Optional[Pattern[f, Blank[]]], Pattern[x, Blank[]]]]], Optional[Pattern[d, Blank[]]]], Pattern[c, Blank[]]], Optional[Pattern[n, Blank[]]]]], Pattern[x, Blank[Symbol]]] := Simp[(b*Cot[e + f*x]*(a + b*Csc[e + f*x])^m*(c + d*Csc[e + f*x])^n)/(a*f*(2*m + 1)), x] /; FreeQ[{a, b, c, d, e, f, m, n}, x] && EqQ[b*c + a*d, 0] && EqQ[a^2 - b^2, 0] && EqQ[m + n + 1, 0] && NeQ[2*m + 1, 0]
  3951. Int[Times[csc[Plus[Optional[Pattern[e, Blank[]]], Times[Optional[Pattern[f, Blank[]]], Pattern[x, Blank[]]]]], Power[Plus[Times[csc[Plus[Optional[Pattern[e, Blank[]]], Times[Optional[Pattern[f, Blank[]]], Pattern[x, Blank[]]]]], Optional[Pattern[b, Blank[]]]], Pattern[a, Blank[]]], Pattern[m, Blank[]]], Power[Plus[Times[csc[Plus[Optional[Pattern[e, Blank[]]], Times[Optional[Pattern[f, Blank[]]], Pattern[x, Blank[]]]]], Optional[Pattern[d, Blank[]]]], Pattern[c, Blank[]]], Optional[Pattern[n, Blank[]]]]], Pattern[x, Blank[Symbol]]] := Simp[(b*Cot[e + f*x]*(a + b*Csc[e + f*x])^m*(c + d*Csc[e + f*x])^n)/(a*f*(2*m + 1)), x] + Dist[(m + n + 1)/(a*(2*m + 1)), Int[Csc[e + f*x]*(a + b*Csc[e + f*x])^(m + 1)*(c + d*Csc[e + f*x])^n, x], x] /; FreeQ[{a, b, c, d, e, f, m, n}, x] && EqQ[b*c + a*d, 0] && EqQ[a^2 - b^2, 0] && ILtQ[m + n + 1, 0] && NeQ[2*m + 1, 0] && !LtQ[n, 0] && !(IGtQ[n + 1/2, 0] && LtQ[n + 1/2, -(m + n)])
  3952. Int[Times[csc[Plus[Optional[Pattern[e, Blank[]]], Times[Optional[Pattern[f, Blank[]]], Pattern[x, Blank[]]]]], Power[Plus[Times[csc[Plus[Optional[Pattern[e, Blank[]]], Times[Optional[Pattern[f, Blank[]]], Pattern[x, Blank[]]]]], Optional[Pattern[b, Blank[]]]], Pattern[a, Blank[]]], Rational[-1, 2]], Power[Plus[Times[csc[Plus[Optional[Pattern[e, Blank[]]], Times[Optional[Pattern[f, Blank[]]], Pattern[x, Blank[]]]]], Optional[Pattern[d, Blank[]]]], Pattern[c, Blank[]]], Rational[1, 2]]], Pattern[x, Blank[Symbol]]] := Simp[(a*c*Log[1 + (b*Csc[e + f*x])/a]*Cot[e + f*x])/(b*f*Sqrt[a + b*Csc[e + f*x]]*Sqrt[c + d*Csc[e + f*x]]), x] /; FreeQ[{a, b, c, d, e, f}, x] && EqQ[b*c + a*d, 0] && EqQ[a^2 - b^2, 0]
  3953. Int[Times[csc[Plus[Optional[Pattern[e, Blank[]]], Times[Optional[Pattern[f, Blank[]]], Pattern[x, Blank[]]]]], Power[Plus[Times[csc[Plus[Optional[Pattern[e, Blank[]]], Times[Optional[Pattern[f, Blank[]]], Pattern[x, Blank[]]]]], Optional[Pattern[b, Blank[]]]], Pattern[a, Blank[]]], Optional[Pattern[m, Blank[]]]], Power[Plus[Times[csc[Plus[Optional[Pattern[e, Blank[]]], Times[Optional[Pattern[f, Blank[]]], Pattern[x, Blank[]]]]], Optional[Pattern[d, Blank[]]]], Pattern[c, Blank[]]], Rational[1, 2]]], Pattern[x, Blank[Symbol]]] := Simp[(2*a*c*Cot[e + f*x]*(a + b*Csc[e + f*x])^m)/(b*f*(2*m + 1)*Sqrt[c + d*Csc[e + f*x]]), x] /; FreeQ[{a, b, c, d, e, f, m}, x] && EqQ[b*c + a*d, 0] && EqQ[a^2 - b^2, 0] && NeQ[m, -2^(-1)]
  3954. Int[Times[csc[Plus[Optional[Pattern[e, Blank[]]], Times[Optional[Pattern[f, Blank[]]], Pattern[x, Blank[]]]]], Power[Plus[Times[csc[Plus[Optional[Pattern[e, Blank[]]], Times[Optional[Pattern[f, Blank[]]], Pattern[x, Blank[]]]]], Optional[Pattern[b, Blank[]]]], Pattern[a, Blank[]]], Pattern[m, Blank[]]], Power[Plus[Times[csc[Plus[Optional[Pattern[e, Blank[]]], Times[Optional[Pattern[f, Blank[]]], Pattern[x, Blank[]]]]], Optional[Pattern[d, Blank[]]]], Pattern[c, Blank[]]], Optional[Pattern[n, Blank[]]]]], Pattern[x, Blank[Symbol]]] := Simp[(2*a*c*Cot[e + f*x]*(a + b*Csc[e + f*x])^m*(c + d*Csc[e + f*x])^(n - 1))/(b*f*(2*m + 1)), x] - Dist[(d*(2*n - 1))/(b*(2*m + 1)), Int[Csc[e + f*x]*(a + b*Csc[e + f*x])^(m + 1)*(c + d*Csc[e + f*x])^(n - 1), x], x] /; FreeQ[{a, b, c, d, e, f}, x] && EqQ[b*c + a*d, 0] && EqQ[a^2 - b^2, 0] && IGtQ[n - 1/2, 0] && LtQ[m, -2^(-1)]
  3955. Int[Times[csc[Plus[Optional[Pattern[e, Blank[]]], Times[Optional[Pattern[f, Blank[]]], Pattern[x, Blank[]]]]], Power[Plus[Times[csc[Plus[Optional[Pattern[e, Blank[]]], Times[Optional[Pattern[f, Blank[]]], Pattern[x, Blank[]]]]], Optional[Pattern[b, Blank[]]]], Pattern[a, Blank[]]], Optional[Pattern[m, Blank[]]]], Power[Plus[Times[csc[Plus[Optional[Pattern[e, Blank[]]], Times[Optional[Pattern[f, Blank[]]], Pattern[x, Blank[]]]]], Optional[Pattern[d, Blank[]]]], Pattern[c, Blank[]]], Pattern[n, Blank[]]]], Pattern[x, Blank[Symbol]]] := -Simp[(d*Cot[e + f*x]*(a + b*Csc[e + f*x])^m*(c + d*Csc[e + f*x])^(n - 1))/(f*(m + n)), x] + Dist[(c*(2*n - 1))/(m + n), Int[Csc[e + f*x]*(a + b*Csc[e + f*x])^m*(c + d*Csc[e + f*x])^(n - 1), x], x] /; FreeQ[{a, b, c, d, e, f, m}, x] && EqQ[b*c + a*d, 0] && EqQ[a^2 - b^2, 0] && IGtQ[n - 1/2, 0] && !LtQ[m, -2^(-1)] && !(IGtQ[m - 1/2, 0] && LtQ[m, n])
  3956. Int[Times[csc[Plus[Optional[Pattern[e, Blank[]]], Times[Optional[Pattern[f, Blank[]]], Pattern[x, Blank[]]]]], Power[Plus[Times[csc[Plus[Optional[Pattern[e, Blank[]]], Times[Optional[Pattern[f, Blank[]]], Pattern[x, Blank[]]]]], Optional[Pattern[b, Blank[]]]], Pattern[a, Blank[]]], Rational[-1, 2]], Power[Plus[Times[csc[Plus[Optional[Pattern[e, Blank[]]], Times[Optional[Pattern[f, Blank[]]], Pattern[x, Blank[]]]]], Optional[Pattern[d, Blank[]]]], Pattern[c, Blank[]]], Optional[Pattern[n, Blank[]]]]], Pattern[x, Blank[Symbol]]] := Simp[(-2*d*Cot[e + f*x]*(c + d*Csc[e + f*x])^(n - 1))/(f*(2*n - 1)*Sqrt[a + b*Csc[e + f*x]]), x] + Dist[(2*c*(2*n - 1))/(2*n - 1), Int[(Csc[e + f*x]*(c + d*Csc[e + f*x])^(n - 1))/Sqrt[a + b*Csc[e + f*x]], x], x] /; FreeQ[{a, b, c, d, e, f}, x] && EqQ[b*c + a*d, 0] && EqQ[a^2 - b^2, 0] && IGtQ[n, 0]
  3957. Int[Times[csc[Plus[Optional[Pattern[e, Blank[]]], Times[Optional[Pattern[f, Blank[]]], Pattern[x, Blank[]]]]], Power[Plus[Times[csc[Plus[Optional[Pattern[e, Blank[]]], Times[Optional[Pattern[f, Blank[]]], Pattern[x, Blank[]]]]], Optional[Pattern[b, Blank[]]]], Pattern[a, Blank[]]], Pattern[m, Blank[]]], Power[Plus[Times[csc[Plus[Optional[Pattern[e, Blank[]]], Times[Optional[Pattern[f, Blank[]]], Pattern[x, Blank[]]]]], Optional[Pattern[d, Blank[]]]], Pattern[c, Blank[]]], Optional[Pattern[n, Blank[]]]]], Pattern[x, Blank[Symbol]]] := Simp[(2*a*c*Cot[e + f*x]*(a + b*Csc[e + f*x])^m*(c + d*Csc[e + f*x])^(n - 1))/(b*f*(2*m + 1)), x] - Dist[(d*(2*n - 1))/(b*(2*m + 1)), Int[Csc[e + f*x]*(a + b*Csc[e + f*x])^(m + 1)*(c + d*Csc[e + f*x])^(n - 1), x], x] /; FreeQ[{a, b, c, d, e, f}, x] && EqQ[b*c + a*d, 0] && EqQ[a^2 - b^2, 0] && IGtQ[n, 0] && LtQ[m, -2^(-1)] && IntegerQ[2*m]
  3958. Int[Times[csc[Plus[Optional[Pattern[e, Blank[]]], Times[Optional[Pattern[f, Blank[]]], Pattern[x, Blank[]]]]], Power[Plus[Times[csc[Plus[Optional[Pattern[e, Blank[]]], Times[Optional[Pattern[f, Blank[]]], Pattern[x, Blank[]]]]], Optional[Pattern[b, Blank[]]]], Pattern[a, Blank[]]], Optional[Pattern[m, Blank[]]]], Power[Plus[Times[csc[Plus[Optional[Pattern[e, Blank[]]], Times[Optional[Pattern[f, Blank[]]], Pattern[x, Blank[]]]]], Optional[Pattern[d, Blank[]]]], Pattern[c, Blank[]]], Optional[Pattern[n, Blank[]]]]], Pattern[x, Blank[Symbol]]] := Dist[(-(a*c))^m, Int[ExpandTrig[csc[e + f*x]*cot[e + f*x]^(2*m), (c + d*csc[e + f*x])^(n - m), x], x], x] /; FreeQ[{a, b, c, d, e, f, n}, x] && EqQ[b*c + a*d, 0] && EqQ[a^2 - b^2, 0] && IntegersQ[m, n] && GeQ[n - m, 0] && GtQ[m*n, 0]
  3959. Int[Times[csc[Plus[Optional[Pattern[e, Blank[]]], Times[Optional[Pattern[f, Blank[]]], Pattern[x, Blank[]]]]], Power[Plus[Times[csc[Plus[Optional[Pattern[e, Blank[]]], Times[Optional[Pattern[f, Blank[]]], Pattern[x, Blank[]]]]], Optional[Pattern[b, Blank[]]]], Pattern[a, Blank[]]], Pattern[m, Blank[]]], Power[Plus[Times[csc[Plus[Optional[Pattern[e, Blank[]]], Times[Optional[Pattern[f, Blank[]]], Pattern[x, Blank[]]]]], Optional[Pattern[d, Blank[]]]], Pattern[c, Blank[]]], Pattern[m, Blank[]]]], Pattern[x, Blank[Symbol]]] := Dist[((-(a*c))^(m + 1/2)*Cot[e + f*x])/(Sqrt[a + b*Csc[e + f*x]]*Sqrt[c + d*Csc[e + f*x]]), Int[Csc[e + f*x]*Cot[e + f*x]^(2*m), x], x] /; FreeQ[{a, b, c, d, e, f}, x] && EqQ[b*c + a*d, 0] && EqQ[a^2 - b^2, 0] && IntegerQ[m + 1/2]
  3960. Int[Times[csc[Plus[Optional[Pattern[e, Blank[]]], Times[Optional[Pattern[f, Blank[]]], Pattern[x, Blank[]]]]], Power[Plus[Times[csc[Plus[Optional[Pattern[e, Blank[]]], Times[Optional[Pattern[f, Blank[]]], Pattern[x, Blank[]]]]], Optional[Pattern[b, Blank[]]]], Pattern[a, Blank[]]], Pattern[m, Blank[]]], Power[Plus[Times[csc[Plus[Optional[Pattern[e, Blank[]]], Times[Optional[Pattern[f, Blank[]]], Pattern[x, Blank[]]]]], Optional[Pattern[d, Blank[]]]], Pattern[c, Blank[]]], Pattern[n, Blank[]]]], Pattern[x, Blank[Symbol]]] := Simp[(b*Cot[e + f*x]*(a + b*Csc[e + f*x])^m*(c + d*Csc[e + f*x])^n)/(a*f*(2*m + 1)), x] + Dist[(m + n + 1)/(a*(2*m + 1)), Int[Csc[e + f*x]*(a + b*Csc[e + f*x])^(m + 1)*(c + d*Csc[e + f*x])^n, x], x] /; FreeQ[{a, b, c, d, e, f, n}, x] && EqQ[b*c + a*d, 0] && EqQ[a^2 - b^2, 0] && ((ILtQ[m, 0] && ILtQ[n - 1/2, 0]) || (ILtQ[m - 1/2, 0] && ILtQ[n - 1/2, 0] && LtQ[m, n]))
  3961. Int[Times[csc[Plus[Optional[Pattern[e, Blank[]]], Times[Optional[Pattern[f, Blank[]]], Pattern[x, Blank[]]]]], Power[Plus[Times[csc[Plus[Optional[Pattern[e, Blank[]]], Times[Optional[Pattern[f, Blank[]]], Pattern[x, Blank[]]]]], Optional[Pattern[b, Blank[]]]], Pattern[a, Blank[]]], Optional[Pattern[m, Blank[]]]], Power[Plus[Times[csc[Plus[Optional[Pattern[e, Blank[]]], Times[Optional[Pattern[f, Blank[]]], Pattern[x, Blank[]]]]], Optional[Pattern[d, Blank[]]]], Pattern[c, Blank[]]], Pattern[n, Blank[]]]], Pattern[x, Blank[Symbol]]] := Dist[(a*c*Cot[e + f*x])/(f*Sqrt[a + b*Csc[e + f*x]]*Sqrt[c + d*Csc[e + f*x]]), Subst[Int[(a + b*x)^(m - 1/2)*(c + d*x)^(n - 1/2), x], x, Csc[e + f*x]], x] /; FreeQ[{a, b, c, d, e, f, m, n}, x] && EqQ[b*c + a*d, 0] && EqQ[a^2 - b^2, 0]
  3962. Int[Times[Power[Times[csc[Plus[Optional[Pattern[e, Blank[]]], Times[Optional[Pattern[f, Blank[]]], Pattern[x, Blank[]]]]], Optional[Pattern[g, Blank[]]]], Optional[Pattern[p, Blank[]]]], Power[Plus[Times[csc[Plus[Optional[Pattern[e, Blank[]]], Times[Optional[Pattern[f, Blank[]]], Pattern[x, Blank[]]]]], Optional[Pattern[b, Blank[]]]], Pattern[a, Blank[]]], Optional[Pattern[m, Blank[]]]], Power[Plus[Times[csc[Plus[Optional[Pattern[e, Blank[]]], Times[Optional[Pattern[f, Blank[]]], Pattern[x, Blank[]]]]], Optional[Pattern[d, Blank[]]]], Pattern[c, Blank[]]], Optional[Pattern[n, Blank[]]]]], Pattern[x, Blank[Symbol]]] := Dist[(-(a*c))^m, Int[ExpandTrig[(g*csc[e + f*x])^p*cot[e + f*x]^(2*m), (c + d*csc[e + f*x])^(n - m), x], x], x] /; FreeQ[{a, b, c, d, e, f, g, n, p}, x] && EqQ[b*c + a*d, 0] && EqQ[a^2 - b^2, 0] && IntegersQ[m, n] && GeQ[n - m, 0] && GtQ[m*n, 0]
  3963. Int[Times[Power[Times[csc[Plus[Optional[Pattern[e, Blank[]]], Times[Optional[Pattern[f, Blank[]]], Pattern[x, Blank[]]]]], Optional[Pattern[g, Blank[]]]], Optional[Pattern[p, Blank[]]]], Power[Plus[Times[csc[Plus[Optional[Pattern[e, Blank[]]], Times[Optional[Pattern[f, Blank[]]], Pattern[x, Blank[]]]]], Optional[Pattern[b, Blank[]]]], Pattern[a, Blank[]]], Pattern[m, Blank[]]], Power[Plus[Times[csc[Plus[Optional[Pattern[e, Blank[]]], Times[Optional[Pattern[f, Blank[]]], Pattern[x, Blank[]]]]], Optional[Pattern[d, Blank[]]]], Pattern[c, Blank[]]], Pattern[m, Blank[]]]], Pattern[x, Blank[Symbol]]] := Dist[((-(a*c))^(m + 1/2)*Cot[e + f*x])/(Sqrt[a + b*Csc[e + f*x]]*Sqrt[c + d*Csc[e + f*x]]), Int[(g*Csc[e + f*x])^p*Cot[e + f*x]^(2*m), x], x] /; FreeQ[{a, b, c, d, e, f, g, p}, x] && EqQ[b*c + a*d, 0] && EqQ[a^2 - b^2, 0] && IntegerQ[m + 1/2]
  3964. Int[Times[Power[Times[csc[Plus[Optional[Pattern[e, Blank[]]], Times[Optional[Pattern[f, Blank[]]], Pattern[x, Blank[]]]]], Optional[Pattern[g, Blank[]]]], Optional[Pattern[p, Blank[]]]], Power[Plus[Times[csc[Plus[Optional[Pattern[e, Blank[]]], Times[Optional[Pattern[f, Blank[]]], Pattern[x, Blank[]]]]], Optional[Pattern[b, Blank[]]]], Pattern[a, Blank[]]], Pattern[m, Blank[]]], Power[Plus[Times[csc[Plus[Optional[Pattern[e, Blank[]]], Times[Optional[Pattern[f, Blank[]]], Pattern[x, Blank[]]]]], Optional[Pattern[d, Blank[]]]], Pattern[c, Blank[]]], Pattern[n, Blank[]]]], Pattern[x, Blank[Symbol]]] := Dist[(a*c*g*Cot[e + f*x])/(f*Sqrt[a + b*Csc[e + f*x]]*Sqrt[c + d*Csc[e + f*x]]), Subst[Int[(g*x)^(p - 1)*(a + b*x)^(m - 1/2)*(c + d*x)^(n - 1/2), x], x, Csc[e + f*x]], x] /; FreeQ[{a, b, c, d, e, f, g, m, n, p}, x] && EqQ[b*c + a*d, 0] && EqQ[a^2 - b^2, 0]
  3965. Int[Times[Power[Times[csc[Plus[Optional[Pattern[e, Blank[]]], Times[Optional[Pattern[f, Blank[]]], Pattern[x, Blank[]]]]], Optional[Pattern[g, Blank[]]]], Rational[1, 2]], Power[Plus[Times[csc[Plus[Optional[Pattern[e, Blank[]]], Times[Optional[Pattern[f, Blank[]]], Pattern[x, Blank[]]]]], Optional[Pattern[b, Blank[]]]], Pattern[a, Blank[]]], Rational[1, 2]], Power[Plus[Times[csc[Plus[Optional[Pattern[e, Blank[]]], Times[Optional[Pattern[f, Blank[]]], Pattern[x, Blank[]]]]], Optional[Pattern[d, Blank[]]]], Pattern[c, Blank[]]], -1]], Pattern[x, Blank[Symbol]]] := Dist[(-2*b*g)/f, Subst[Int[1/(b*c + a*d - c*g*x^2), x], x, (b*Cot[e + f*x])/(Sqrt[g*Csc[e + f*x]]*Sqrt[a + b*Csc[e + f*x]])], x] /; FreeQ[{a, b, c, d, e, f, g}, x] && NeQ[b*c - a*d, 0] && EqQ[a^2 - b^2, 0]
  3966. Int[Times[Power[Times[csc[Plus[Optional[Pattern[e, Blank[]]], Times[Optional[Pattern[f, Blank[]]], Pattern[x, Blank[]]]]], Optional[Pattern[g, Blank[]]]], Rational[1, 2]], Power[Plus[Times[csc[Plus[Optional[Pattern[e, Blank[]]], Times[Optional[Pattern[f, Blank[]]], Pattern[x, Blank[]]]]], Optional[Pattern[b, Blank[]]]], Pattern[a, Blank[]]], Rational[1, 2]], Power[Plus[Times[csc[Plus[Optional[Pattern[e, Blank[]]], Times[Optional[Pattern[f, Blank[]]], Pattern[x, Blank[]]]]], Optional[Pattern[d, Blank[]]]], Pattern[c, Blank[]]], -1]], Pattern[x, Blank[Symbol]]] := Dist[a/c, Int[Sqrt[g*Csc[e + f*x]]/Sqrt[a + b*Csc[e + f*x]], x], x] + Dist[(b*c - a*d)/(c*g), Int[(g*Csc[e + f*x])^(3/2)/(Sqrt[a + b*Csc[e + f*x]]*(c + d*Csc[e + f*x])), x], x] /; FreeQ[{a, b, c, d, e, f, g}, x] && NeQ[b*c - a*d, 0] && NeQ[a^2 - b^2, 0]
  3967. Int[Times[csc[Plus[Optional[Pattern[e, Blank[]]], Times[Optional[Pattern[f, Blank[]]], Pattern[x, Blank[]]]]], Power[Plus[Times[csc[Plus[Optional[Pattern[e, Blank[]]], Times[Optional[Pattern[f, Blank[]]], Pattern[x, Blank[]]]]], Optional[Pattern[b, Blank[]]]], Pattern[a, Blank[]]], Rational[1, 2]], Power[Plus[Times[csc[Plus[Optional[Pattern[e, Blank[]]], Times[Optional[Pattern[f, Blank[]]], Pattern[x, Blank[]]]]], Optional[Pattern[d, Blank[]]]], Pattern[c, Blank[]]], -1]], Pattern[x, Blank[Symbol]]] := Dist[(-2*b)/f, Subst[Int[1/(b*c + a*d + d*x^2), x], x, (b*Cot[e + f*x])/Sqrt[a + b*Csc[e + f*x]]], x] /; FreeQ[{a, b, c, d, e, f}, x] && NeQ[b*c - a*d, 0] && EqQ[a^2 - b^2, 0]
  3968. Int[Times[csc[Plus[Optional[Pattern[e, Blank[]]], Times[Optional[Pattern[f, Blank[]]], Pattern[x, Blank[]]]]], Power[Plus[Times[csc[Plus[Optional[Pattern[e, Blank[]]], Times[Optional[Pattern[f, Blank[]]], Pattern[x, Blank[]]]]], Optional[Pattern[b, Blank[]]]], Pattern[a, Blank[]]], Rational[1, 2]], Power[Plus[Times[csc[Plus[Optional[Pattern[e, Blank[]]], Times[Optional[Pattern[f, Blank[]]], Pattern[x, Blank[]]]]], Optional[Pattern[d, Blank[]]]], Pattern[c, Blank[]]], -1]], Pattern[x, Blank[Symbol]]] := -Simp[(Sqrt[a + b*Csc[e + f*x]]*Sqrt[c/(c + d*Csc[e + f*x])]*EllipticE[ArcSin[(c*Cot[e + f*x])/(c + d*Csc[e + f*x])], -((b*c - a*d)/(b*c + a*d))])/(d*f*Sqrt[(c*d*(a + b*Csc[e + f*x]))/((b*c + a*d)*(c + d*Csc[e + f*x]))]), x] /; FreeQ[{a, b, c, d, e, f}, x] && NeQ[b*c - a*d, 0] && NeQ[a^2 - b^2, 0] && EqQ[c^2 - d^2, 0]
  3969. Int[Times[csc[Plus[Optional[Pattern[e, Blank[]]], Times[Optional[Pattern[f, Blank[]]], Pattern[x, Blank[]]]]], Power[Plus[Times[csc[Plus[Optional[Pattern[e, Blank[]]], Times[Optional[Pattern[f, Blank[]]], Pattern[x, Blank[]]]]], Optional[Pattern[b, Blank[]]]], Pattern[a, Blank[]]], Rational[1, 2]], Power[Plus[Times[csc[Plus[Optional[Pattern[e, Blank[]]], Times[Optional[Pattern[f, Blank[]]], Pattern[x, Blank[]]]]], Optional[Pattern[d, Blank[]]]], Pattern[c, Blank[]]], -1]], Pattern[x, Blank[Symbol]]] := Dist[b/d, Int[Csc[e + f*x]/Sqrt[a + b*Csc[e + f*x]], x], x] - Dist[(b*c - a*d)/d, Int[Csc[e + f*x]/(Sqrt[a + b*Csc[e + f*x]]*(c + d*Csc[e + f*x])), x], x] /; FreeQ[{a, b, c, d, e, f}, x] && NeQ[b*c - a*d, 0] && NeQ[a^2 - b^2, 0] && NeQ[c^2 - d^2, 0]
  3970. Int[Times[Power[Times[csc[Plus[Optional[Pattern[e, Blank[]]], Times[Optional[Pattern[f, Blank[]]], Pattern[x, Blank[]]]]], Optional[Pattern[g, Blank[]]]], Rational[3, 2]], Power[Plus[Times[csc[Plus[Optional[Pattern[e, Blank[]]], Times[Optional[Pattern[f, Blank[]]], Pattern[x, Blank[]]]]], Optional[Pattern[b, Blank[]]]], Pattern[a, Blank[]]], Rational[1, 2]], Power[Plus[Times[csc[Plus[Optional[Pattern[e, Blank[]]], Times[Optional[Pattern[f, Blank[]]], Pattern[x, Blank[]]]]], Optional[Pattern[d, Blank[]]]], Pattern[c, Blank[]]], -1]], Pattern[x, Blank[Symbol]]] := Dist[g/d, Int[Sqrt[g*Csc[e + f*x]]*Sqrt[a + b*Csc[e + f*x]], x], x] - Dist[(c*g)/d, Int[(Sqrt[g*Csc[e + f*x]]*Sqrt[a + b*Csc[e + f*x]])/(c + d*Csc[e + f*x]), x], x] /; FreeQ[{a, b, c, d, e, f, g}, x] && NeQ[b*c - a*d, 0] && EqQ[a^2 - b^2, 0]
  3971. Int[Times[Power[Times[csc[Plus[Optional[Pattern[e, Blank[]]], Times[Optional[Pattern[f, Blank[]]], Pattern[x, Blank[]]]]], Optional[Pattern[g, Blank[]]]], Rational[3, 2]], Power[Plus[Times[csc[Plus[Optional[Pattern[e, Blank[]]], Times[Optional[Pattern[f, Blank[]]], Pattern[x, Blank[]]]]], Optional[Pattern[b, Blank[]]]], Pattern[a, Blank[]]], Rational[1, 2]], Power[Plus[Times[csc[Plus[Optional[Pattern[e, Blank[]]], Times[Optional[Pattern[f, Blank[]]], Pattern[x, Blank[]]]]], Optional[Pattern[d, Blank[]]]], Pattern[c, Blank[]]], -1]], Pattern[x, Blank[Symbol]]] := Dist[b/d, Int[(g*Csc[e + f*x])^(3/2)/Sqrt[a + b*Csc[e + f*x]], x], x] - Dist[(b*c - a*d)/d, Int[(g*Csc[e + f*x])^(3/2)/(Sqrt[a + b*Csc[e + f*x]]*(c + d*Csc[e + f*x])), x], x] /; FreeQ[{a, b, c, d, e, f, g}, x] && NeQ[b*c - a*d, 0] && NeQ[a^2 - b^2, 0]
  3972. Int[Times[csc[Plus[Optional[Pattern[e, Blank[]]], Times[Optional[Pattern[f, Blank[]]], Pattern[x, Blank[]]]]], Power[Plus[Times[csc[Plus[Optional[Pattern[e, Blank[]]], Times[Optional[Pattern[f, Blank[]]], Pattern[x, Blank[]]]]], Optional[Pattern[b, Blank[]]]], Pattern[a, Blank[]]], Rational[-1, 2]], Power[Plus[Times[csc[Plus[Optional[Pattern[e, Blank[]]], Times[Optional[Pattern[f, Blank[]]], Pattern[x, Blank[]]]]], Optional[Pattern[d, Blank[]]]], Pattern[c, Blank[]]], -1]], Pattern[x, Blank[Symbol]]] := Dist[b/(b*c - a*d), Int[Csc[e + f*x]/Sqrt[a + b*Csc[e + f*x]], x], x] - Dist[d/(b*c - a*d), Int[(Csc[e + f*x]*Sqrt[a + b*Csc[e + f*x]])/(c + d*Csc[e + f*x]), x], x] /; FreeQ[{a, b, c, d, e, f}, x] && NeQ[b*c - a*d, 0] && (EqQ[a^2 - b^2, 0] || EqQ[c^2 - d^2, 0])
  3973. Int[Times[csc[Plus[Optional[Pattern[e, Blank[]]], Times[Optional[Pattern[f, Blank[]]], Pattern[x, Blank[]]]]], Power[Plus[Times[csc[Plus[Optional[Pattern[e, Blank[]]], Times[Optional[Pattern[f, Blank[]]], Pattern[x, Blank[]]]]], Optional[Pattern[b, Blank[]]]], Pattern[a, Blank[]]], Rational[-1, 2]], Power[Plus[Times[csc[Plus[Optional[Pattern[e, Blank[]]], Times[Optional[Pattern[f, Blank[]]], Pattern[x, Blank[]]]]], Optional[Pattern[d, Blank[]]]], Pattern[c, Blank[]]], -1]], Pattern[x, Blank[Symbol]]] := Simp[(-2*Cot[e + f*x]*Sqrt[(a + b*Csc[e + f*x])/(a + b)]*EllipticPi[(2*d)/(c + d), ArcSin[Sqrt[1 - Csc[e + f*x]]/Sqrt[2]], (2*b)/(a + b)])/(f*(c + d)*Sqrt[a + b*Csc[e + f*x]]*Sqrt[-Cot[e + f*x]^2]), x] /; FreeQ[{a, b, c, d, e, f}, x] && NeQ[b*c - a*d, 0] && NeQ[a^2 - b^2, 0] && NeQ[c^2 - d^2, 0]
  3974. Int[Times[Power[Times[csc[Plus[Optional[Pattern[e, Blank[]]], Times[Optional[Pattern[f, Blank[]]], Pattern[x, Blank[]]]]], Optional[Pattern[g, Blank[]]]], Rational[3, 2]], Power[Plus[Times[csc[Plus[Optional[Pattern[e, Blank[]]], Times[Optional[Pattern[f, Blank[]]], Pattern[x, Blank[]]]]], Optional[Pattern[b, Blank[]]]], Pattern[a, Blank[]]], Rational[-1, 2]], Power[Plus[Times[csc[Plus[Optional[Pattern[e, Blank[]]], Times[Optional[Pattern[f, Blank[]]], Pattern[x, Blank[]]]]], Optional[Pattern[d, Blank[]]]], Pattern[c, Blank[]]], -1]], Pattern[x, Blank[Symbol]]] := -Dist[(a*g)/(b*c - a*d), Int[Sqrt[g*Csc[e + f*x]]/Sqrt[a + b*Csc[e + f*x]], x], x] + Dist[(c*g)/(b*c - a*d), Int[(Sqrt[g*Csc[e + f*x]]*Sqrt[a + b*Csc[e + f*x]])/(c + d*Csc[e + f*x]), x], x] /; FreeQ[{a, b, c, d, e, f, g}, x] && NeQ[b*c - a*d, 0] && EqQ[a^2 - b^2, 0]
  3975. Int[Times[Power[Times[csc[Plus[Optional[Pattern[e, Blank[]]], Times[Optional[Pattern[f, Blank[]]], Pattern[x, Blank[]]]]], Optional[Pattern[g, Blank[]]]], Rational[3, 2]], Power[Plus[Times[csc[Plus[Optional[Pattern[e, Blank[]]], Times[Optional[Pattern[f, Blank[]]], Pattern[x, Blank[]]]]], Optional[Pattern[b, Blank[]]]], Pattern[a, Blank[]]], Rational[-1, 2]], Power[Plus[Times[csc[Plus[Optional[Pattern[e, Blank[]]], Times[Optional[Pattern[f, Blank[]]], Pattern[x, Blank[]]]]], Optional[Pattern[d, Blank[]]]], Pattern[c, Blank[]]], -1]], Pattern[x, Blank[Symbol]]] := Dist[(g*Sqrt[g*Csc[e + f*x]]*Sqrt[b + a*Sin[e + f*x]])/Sqrt[a + b*Csc[e + f*x]], Int[1/(Sqrt[b + a*Sin[e + f*x]]*(d + c*Sin[e + f*x])), x], x] /; FreeQ[{a, b, c, d, e, f, g}, x] && NeQ[b*c - a*d, 0] && NeQ[a^2 - b^2, 0]
  3976. Int[Times[Power[csc[Plus[Optional[Pattern[e, Blank[]]], Times[Optional[Pattern[f, Blank[]]], Pattern[x, Blank[]]]]], 2], Power[Plus[Times[csc[Plus[Optional[Pattern[e, Blank[]]], Times[Optional[Pattern[f, Blank[]]], Pattern[x, Blank[]]]]], Optional[Pattern[b, Blank[]]]], Pattern[a, Blank[]]], Rational[-1, 2]], Power[Plus[Times[csc[Plus[Optional[Pattern[e, Blank[]]], Times[Optional[Pattern[f, Blank[]]], Pattern[x, Blank[]]]]], Optional[Pattern[d, Blank[]]]], Pattern[c, Blank[]]], -1]], Pattern[x, Blank[Symbol]]] := -Dist[a/(b*c - a*d), Int[Csc[e + f*x]/Sqrt[a + b*Csc[e + f*x]], x], x] + Dist[c/(b*c - a*d), Int[(Csc[e + f*x]*Sqrt[a + b*Csc[e + f*x]])/(c + d*Csc[e + f*x]), x], x] /; FreeQ[{a, b, c, d, e, f}, x] && NeQ[b*c - a*d, 0] && (EqQ[a^2 - b^2, 0] || EqQ[c^2 - d^2, 0])
  3977. Int[Times[Power[csc[Plus[Optional[Pattern[e, Blank[]]], Times[Optional[Pattern[f, Blank[]]], Pattern[x, Blank[]]]]], 2], Power[Plus[Times[csc[Plus[Optional[Pattern[e, Blank[]]], Times[Optional[Pattern[f, Blank[]]], Pattern[x, Blank[]]]]], Optional[Pattern[b, Blank[]]]], Pattern[a, Blank[]]], Rational[-1, 2]], Power[Plus[Times[csc[Plus[Optional[Pattern[e, Blank[]]], Times[Optional[Pattern[f, Blank[]]], Pattern[x, Blank[]]]]], Optional[Pattern[d, Blank[]]]], Pattern[c, Blank[]]], -1]], Pattern[x, Blank[Symbol]]] := Dist[1/d, Int[Csc[e + f*x]/Sqrt[a + b*Csc[e + f*x]], x], x] - Dist[c/d, Int[Csc[e + f*x]/(Sqrt[a + b*Csc[e + f*x]]*(c + d*Csc[e + f*x])), x], x] /; FreeQ[{a, b, c, d, e, f}, x] && NeQ[b*c - a*d, 0] && NeQ[a^2 - b^2, 0] && NeQ[c^2 - d^2, 0]
  3978. Int[Times[Power[Times[csc[Plus[Optional[Pattern[e, Blank[]]], Times[Optional[Pattern[f, Blank[]]], Pattern[x, Blank[]]]]], Optional[Pattern[g, Blank[]]]], Rational[5, 2]], Power[Plus[Times[csc[Plus[Optional[Pattern[e, Blank[]]], Times[Optional[Pattern[f, Blank[]]], Pattern[x, Blank[]]]]], Optional[Pattern[b, Blank[]]]], Pattern[a, Blank[]]], Rational[-1, 2]], Power[Plus[Times[csc[Plus[Optional[Pattern[e, Blank[]]], Times[Optional[Pattern[f, Blank[]]], Pattern[x, Blank[]]]]], Optional[Pattern[d, Blank[]]]], Pattern[c, Blank[]]], -1]], Pattern[x, Blank[Symbol]]] := -Dist[(c^2*g^2)/(d*(b*c - a*d)), Int[(Sqrt[g*Csc[e + f*x]]*Sqrt[a + b*Csc[e + f*x]])/(c + d*Csc[e + f*x]), x], x] + Dist[g^2/(d*(b*c - a*d)), Int[(Sqrt[g*Csc[e + f*x]]*(a*c + (b*c - a*d)*Csc[e + f*x]))/Sqrt[a + b*Csc[e + f*x]], x], x] /; FreeQ[{a, b, c, d, e, f, g}, x] && NeQ[b*c - a*d, 0] && EqQ[a^2 - b^2, 0]
  3979. Int[Times[Power[Times[csc[Plus[Optional[Pattern[e, Blank[]]], Times[Optional[Pattern[f, Blank[]]], Pattern[x, Blank[]]]]], Optional[Pattern[g, Blank[]]]], Rational[5, 2]], Power[Plus[Times[csc[Plus[Optional[Pattern[e, Blank[]]], Times[Optional[Pattern[f, Blank[]]], Pattern[x, Blank[]]]]], Optional[Pattern[b, Blank[]]]], Pattern[a, Blank[]]], Rational[-1, 2]], Power[Plus[Times[csc[Plus[Optional[Pattern[e, Blank[]]], Times[Optional[Pattern[f, Blank[]]], Pattern[x, Blank[]]]]], Optional[Pattern[d, Blank[]]]], Pattern[c, Blank[]]], -1]], Pattern[x, Blank[Symbol]]] := Dist[g/d, Int[(g*Csc[e + f*x])^(3/2)/Sqrt[a + b*Csc[e + f*x]], x], x] - Dist[(c*g)/d, Int[(g*Csc[e + f*x])^(3/2)/(Sqrt[a + b*Csc[e + f*x]]*(c + d*Csc[e + f*x])), x], x] /; FreeQ[{a, b, c, d, e, f, g}, x] && NeQ[b*c - a*d, 0] && NeQ[a^2 - b^2, 0]
  3980. Int[Times[csc[Plus[Optional[Pattern[e, Blank[]]], Times[Optional[Pattern[f, Blank[]]], Pattern[x, Blank[]]]]], Power[Plus[Times[csc[Plus[Optional[Pattern[e, Blank[]]], Times[Optional[Pattern[f, Blank[]]], Pattern[x, Blank[]]]]], Optional[Pattern[b, Blank[]]]], Pattern[a, Blank[]]], Rational[1, 2]], Power[Plus[Times[csc[Plus[Optional[Pattern[e, Blank[]]], Times[Optional[Pattern[f, Blank[]]], Pattern[x, Blank[]]]]], Optional[Pattern[d, Blank[]]]], Pattern[c, Blank[]]], Rational[-1, 2]]], Pattern[x, Blank[Symbol]]] := Dist[(-2*b)/f, Subst[Int[1/(1 - b*d*x^2), x], x, Cot[e + f*x]/(Sqrt[a + b*Csc[e + f*x]]*Sqrt[c + d*Csc[e + f*x]])], x] /; FreeQ[{a, b, c, d, e, f}, x] && NeQ[b*c - a*d, 0] && EqQ[a^2 - b^2, 0] && NeQ[c^2 - d^2, 0]
  3981. Int[Times[csc[Plus[Optional[Pattern[e, Blank[]]], Times[Optional[Pattern[f, Blank[]]], Pattern[x, Blank[]]]]], Power[Plus[Times[csc[Plus[Optional[Pattern[e, Blank[]]], Times[Optional[Pattern[f, Blank[]]], Pattern[x, Blank[]]]]], Optional[Pattern[b, Blank[]]]], Pattern[a, Blank[]]], Rational[1, 2]], Power[Plus[Times[csc[Plus[Optional[Pattern[e, Blank[]]], Times[Optional[Pattern[f, Blank[]]], Pattern[x, Blank[]]]]], Optional[Pattern[d, Blank[]]]], Pattern[c, Blank[]]], Rational[-1, 2]]], Pattern[x, Blank[Symbol]]] := -Dist[(b*c - a*d)/d, Int[Csc[e + f*x]/(Sqrt[a + b*Csc[e + f*x]]*Sqrt[c + d*Csc[e + f*x]]), x], x] + Dist[b/d, Int[(Csc[e + f*x]*Sqrt[c + d*Csc[e + f*x]])/Sqrt[a + b*Csc[e + f*x]], x], x] /; FreeQ[{a, b, c, d, e, f}, x] && NeQ[b*c - a*d, 0] && NeQ[a^2 - b^2, 0] && EqQ[c^2 - d^2, 0]
  3982. Int[Times[csc[Plus[Optional[Pattern[e, Blank[]]], Times[Optional[Pattern[f, Blank[]]], Pattern[x, Blank[]]]]], Power[Plus[Times[csc[Plus[Optional[Pattern[e, Blank[]]], Times[Optional[Pattern[f, Blank[]]], Pattern[x, Blank[]]]]], Optional[Pattern[b, Blank[]]]], Pattern[a, Blank[]]], Rational[1, 2]], Power[Plus[Times[csc[Plus[Optional[Pattern[e, Blank[]]], Times[Optional[Pattern[f, Blank[]]], Pattern[x, Blank[]]]]], Optional[Pattern[d, Blank[]]]], Pattern[c, Blank[]]], Rational[-1, 2]]], Pattern[x, Blank[Symbol]]] := Simp[(-2*(a + b*Csc[e + f*x])*Sqrt[-(((b*c - a*d)*(1 - Csc[e + f*x]))/((c + d)*(a + b*Csc[e + f*x])))]*Sqrt[((b*c - a*d)*(1 + Csc[e + f*x]))/((c - d)*(a + b*Csc[e + f*x]))]*EllipticPi[(b*(c + d))/(d*(a + b)), ArcSin[(Sqrt[(a + b)/(c + d)]*Sqrt[c + d*Csc[e + f*x]])/Sqrt[a + b*Csc[e + f*x]]], ((a - b)*(c + d))/((a + b)*(c - d))])/(d*f*Sqrt[(a + b)/(c + d)]*Cot[e + f*x]), x] /; FreeQ[{a, b, c, d, e, f}, x] && NeQ[b*c - a*d, 0] && NeQ[a^2 - b^2, 0] && NeQ[c^2 - d^2, 0]
  3983. Int[Times[csc[Plus[Optional[Pattern[e, Blank[]]], Times[Optional[Pattern[f, Blank[]]], Pattern[x, Blank[]]]]], Power[Plus[Times[csc[Plus[Optional[Pattern[e, Blank[]]], Times[Optional[Pattern[f, Blank[]]], Pattern[x, Blank[]]]]], Optional[Pattern[b, Blank[]]]], Pattern[a, Blank[]]], Rational[-1, 2]], Power[Plus[Times[csc[Plus[Optional[Pattern[e, Blank[]]], Times[Optional[Pattern[f, Blank[]]], Pattern[x, Blank[]]]]], Optional[Pattern[d, Blank[]]]], Pattern[c, Blank[]]], Rational[-1, 2]]], Pattern[x, Blank[Symbol]]] := Dist[(-2*a)/(b*f), Subst[Int[1/(2 + (a*c - b*d)*x^2), x], x, Cot[e + f*x]/(Sqrt[a + b*Csc[e + f*x]]*Sqrt[c + d*Csc[e + f*x]])], x] /; FreeQ[{a, b, c, d, e, f}, x] && NeQ[b*c - a*d, 0] && EqQ[a^2 - b^2, 0] && NeQ[c^2 - d^2, 0]
  3984. Int[Times[csc[Plus[Optional[Pattern[e, Blank[]]], Times[Optional[Pattern[f, Blank[]]], Pattern[x, Blank[]]]]], Power[Plus[Times[csc[Plus[Optional[Pattern[e, Blank[]]], Times[Optional[Pattern[f, Blank[]]], Pattern[x, Blank[]]]]], Optional[Pattern[b, Blank[]]]], Pattern[a, Blank[]]], Rational[-1, 2]], Power[Plus[Times[csc[Plus[Optional[Pattern[e, Blank[]]], Times[Optional[Pattern[f, Blank[]]], Pattern[x, Blank[]]]]], Optional[Pattern[d, Blank[]]]], Pattern[c, Blank[]]], Rational[-1, 2]]], Pattern[x, Blank[Symbol]]] := Simp[(-2*(c + d*Csc[e + f*x])*Sqrt[((b*c - a*d)*(1 - Csc[e + f*x]))/((a + b)*(c + d*Csc[e + f*x]))]*Sqrt[-(((b*c - a*d)*(1 + Csc[e + f*x]))/((a - b)*(c + d*Csc[e + f*x])))]*EllipticF[ArcSin[Rt[(c + d)/(a + b), 2]*(Sqrt[a + b*Csc[e + f*x]]/Sqrt[c + d*Csc[e + f*x]])], ((a + b)*(c - d))/((a - b)*(c + d))])/(f*(b*c - a*d)*Rt[(c + d)/(a + b), 2]*Cot[e + f*x]), x] /; FreeQ[{a, b, c, d, e, f}, x] && NeQ[b*c - a*d, 0] && NeQ[a^2 - b^2, 0] && NeQ[c^2 - d^2, 0]
  3985. Int[Times[Power[csc[Plus[Optional[Pattern[e, Blank[]]], Times[Optional[Pattern[f, Blank[]]], Pattern[x, Blank[]]]]], 2], Power[Plus[Times[csc[Plus[Optional[Pattern[e, Blank[]]], Times[Optional[Pattern[f, Blank[]]], Pattern[x, Blank[]]]]], Optional[Pattern[b, Blank[]]]], Pattern[a, Blank[]]], Rational[-1, 2]], Power[Plus[Times[csc[Plus[Optional[Pattern[e, Blank[]]], Times[Optional[Pattern[f, Blank[]]], Pattern[x, Blank[]]]]], Optional[Pattern[d, Blank[]]]], Pattern[c, Blank[]]], Rational[-1, 2]]], Pattern[x, Blank[Symbol]]] := -Dist[a/b, Int[Csc[e + f*x]/(Sqrt[a + b*Csc[e + f*x]]*Sqrt[c + d*Csc[e + f*x]]), x], x] + Dist[1/b, Int[(Csc[e + f*x]*Sqrt[a + b*Csc[e + f*x]])/Sqrt[c + d*Csc[e + f*x]], x], x] /; FreeQ[{a, b, c, d, e, f}, x] && NeQ[b*c - a*d, 0]
  3986. Int[Times[csc[Plus[Optional[Pattern[e, Blank[]]], Times[Optional[Pattern[f, Blank[]]], Pattern[x, Blank[]]]]], Power[Plus[Times[csc[Plus[Optional[Pattern[e, Blank[]]], Times[Optional[Pattern[f, Blank[]]], Pattern[x, Blank[]]]]], Optional[Pattern[b, Blank[]]]], Pattern[a, Blank[]]], Rational[1, 2]], Power[Plus[Times[csc[Plus[Optional[Pattern[e, Blank[]]], Times[Optional[Pattern[f, Blank[]]], Pattern[x, Blank[]]]]], Optional[Pattern[d, Blank[]]]], Pattern[c, Blank[]]], Rational[-3, 2]]], Pattern[x, Blank[Symbol]]] := Dist[(a - b)/(c - d), Int[Csc[e + f*x]/(Sqrt[a + b*Csc[e + f*x]]*Sqrt[c + d*Csc[e + f*x]]), x], x] + Dist[(b*c - a*d)/(c - d), Int[(Csc[e + f*x]*(1 + Csc[e + f*x]))/(Sqrt[a + b*Csc[e + f*x]]*(c + d*Csc[e + f*x])^(3/2)), x], x] /; FreeQ[{a, b, c, d, e, f}, x] && NeQ[b*c - a*d, 0] && NeQ[a^2 - b^2, 0] && NeQ[c^2 - d^2, 0]
  3987. Int[Times[Power[Times[csc[Plus[Optional[Pattern[e, Blank[]]], Times[Optional[Pattern[f, Blank[]]], Pattern[x, Blank[]]]]], Optional[Pattern[g, Blank[]]]], Optional[Pattern[p, Blank[]]]], Power[Plus[Times[csc[Plus[Optional[Pattern[e, Blank[]]], Times[Optional[Pattern[f, Blank[]]], Pattern[x, Blank[]]]]], Optional[Pattern[b, Blank[]]]], Pattern[a, Blank[]]], Pattern[m, Blank[]]], Power[Plus[Times[csc[Plus[Optional[Pattern[e, Blank[]]], Times[Optional[Pattern[f, Blank[]]], Pattern[x, Blank[]]]]], Optional[Pattern[d, Blank[]]]], Pattern[c, Blank[]]], Pattern[n, Blank[]]]], Pattern[x, Blank[Symbol]]] := Dist[(a^2*g*Cot[e + f*x])/(f*Sqrt[a + b*Csc[e + f*x]]*Sqrt[a - b*Csc[e + f*x]]), Subst[Int[((g*x)^(p - 1)*(a + b*x)^(m - 1/2)*(c + d*x)^n)/Sqrt[a - b*x], x], x, Csc[e + f*x]], x] /; FreeQ[{a, b, c, d, e, f, g, m, n, p}, x] && NeQ[b*c - a*d, 0] && EqQ[a^2 - b^2, 0] && NeQ[c^2 - d^2, 0] && (EqQ[p, 1] || IntegerQ[m - 1/2])
  3988. Int[Times[Power[Times[csc[Plus[Optional[Pattern[e, Blank[]]], Times[Optional[Pattern[f, Blank[]]], Pattern[x, Blank[]]]]], Optional[Pattern[g, Blank[]]]], Optional[Pattern[p, Blank[]]]], Power[Plus[Times[csc[Plus[Optional[Pattern[e, Blank[]]], Times[Optional[Pattern[f, Blank[]]], Pattern[x, Blank[]]]]], Optional[Pattern[b, Blank[]]]], Pattern[a, Blank[]]], Pattern[m, Blank[]]], Power[Plus[Times[csc[Plus[Optional[Pattern[e, Blank[]]], Times[Optional[Pattern[f, Blank[]]], Pattern[x, Blank[]]]]], Optional[Pattern[d, Blank[]]]], Pattern[c, Blank[]]], Pattern[n, Blank[]]]], Pattern[x, Blank[Symbol]]] := Dist[1/g^(m + n), Int[(g*Csc[e + f*x])^(m + n + p)*(b + a*Sin[e + f*x])^m*(d + c*Sin[e + f*x])^n, x], x] /; FreeQ[{a, b, c, d, e, f, g, p}, x] && NeQ[b*c - a*d, 0] && IntegerQ[m] && IntegerQ[n]
  3989. Int[Times[Power[Times[csc[Plus[Optional[Pattern[e, Blank[]]], Times[Optional[Pattern[f, Blank[]]], Pattern[x, Blank[]]]]], Optional[Pattern[g, Blank[]]]], Optional[Pattern[p, Blank[]]]], Power[Plus[Times[csc[Plus[Optional[Pattern[e, Blank[]]], Times[Optional[Pattern[f, Blank[]]], Pattern[x, Blank[]]]]], Optional[Pattern[b, Blank[]]]], Pattern[a, Blank[]]], Pattern[m, Blank[]]], Power[Plus[Times[csc[Plus[Optional[Pattern[e, Blank[]]], Times[Optional[Pattern[f, Blank[]]], Pattern[x, Blank[]]]]], Optional[Pattern[d, Blank[]]]], Pattern[c, Blank[]]], Pattern[n, Blank[]]]], Pattern[x, Blank[Symbol]]] := Dist[((g*Csc[e + f*x])^(m + p)*(c + d*Csc[e + f*x])^n)/(g^m*(d + c*Sin[e + f*x])^n), Int[(b + a*Sin[e + f*x])^m*(d + c*Sin[e + f*x])^n, x], x] /; FreeQ[{a, b, c, d, e, f, g, n, p}, x] && NeQ[b*c - a*d, 0] && EqQ[m + n + p, 0] && IntegerQ[m]
  3990. Int[Times[Power[Times[csc[Plus[Optional[Pattern[e, Blank[]]], Times[Optional[Pattern[f, Blank[]]], Pattern[x, Blank[]]]]], Optional[Pattern[g, Blank[]]]], Optional[Pattern[p, Blank[]]]], Power[Plus[Times[csc[Plus[Optional[Pattern[e, Blank[]]], Times[Optional[Pattern[f, Blank[]]], Pattern[x, Blank[]]]]], Optional[Pattern[b, Blank[]]]], Pattern[a, Blank[]]], Pattern[m, Blank[]]], Power[Plus[Times[csc[Plus[Optional[Pattern[e, Blank[]]], Times[Optional[Pattern[f, Blank[]]], Pattern[x, Blank[]]]]], Optional[Pattern[d, Blank[]]]], Pattern[c, Blank[]]], Pattern[n, Blank[]]]], Pattern[x, Blank[Symbol]]] := Dist[((g*Csc[e + f*x])^p*(a + b*Csc[e + f*x])^m*(c + d*Csc[e + f*x])^n)/((b + a*Sin[e + f*x])^m*(d + c*Sin[e + f*x])^n), Int[(b + a*Sin[e + f*x])^m*(d + c*Sin[e + f*x])^n, x], x] /; FreeQ[{a, b, c, d, e, f, g, m, n, p}, x] && NeQ[b*c - a*d, 0] && EqQ[m + n + p, 0] && !IntegerQ[m]
  3991. Int[Times[Power[csc[Plus[Optional[Pattern[e, Blank[]]], Times[Optional[Pattern[f, Blank[]]], Pattern[x, Blank[]]]]], Optional[Pattern[p, Blank[]]]], Power[Plus[Times[csc[Plus[Optional[Pattern[e, Blank[]]], Times[Optional[Pattern[f, Blank[]]], Pattern[x, Blank[]]]]], Optional[Pattern[b, Blank[]]]], Pattern[a, Blank[]]], Pattern[m, Blank[]]], Power[Plus[Times[csc[Plus[Optional[Pattern[e, Blank[]]], Times[Optional[Pattern[f, Blank[]]], Pattern[x, Blank[]]]]], Optional[Pattern[d, Blank[]]]], Pattern[c, Blank[]]], Pattern[n, Blank[]]]], Pattern[x, Blank[Symbol]]] := Dist[(Sqrt[d + c*Sin[e + f*x]]*Sqrt[a + b*Csc[e + f*x]])/(Sqrt[b + a*Sin[e + f*x]]*Sqrt[c + d*Csc[e + f*x]]), Int[((b + a*Sin[e + f*x])^m*(d + c*Sin[e + f*x])^n)/Sin[e + f*x]^(m + n + p), x], x] /; FreeQ[{a, b, c, d, e, f, m, n}, x] && NeQ[b*c - a*d, 0] && IntegerQ[m - 1/2] && IntegerQ[n - 1/2] && IntegerQ[p] && LeQ[-2, m + n + p, -1]
  3992. Int[Times[Power[Times[csc[Plus[Optional[Pattern[e, Blank[]]], Times[Optional[Pattern[f, Blank[]]], Pattern[x, Blank[]]]]], Optional[Pattern[g, Blank[]]]], Optional[Pattern[p, Blank[]]]], Power[Plus[Times[csc[Plus[Optional[Pattern[e, Blank[]]], Times[Optional[Pattern[f, Blank[]]], Pattern[x, Blank[]]]]], Optional[Pattern[b, Blank[]]]], Pattern[a, Blank[]]], Pattern[m, Blank[]]], Power[Plus[Times[csc[Plus[Optional[Pattern[e, Blank[]]], Times[Optional[Pattern[f, Blank[]]], Pattern[x, Blank[]]]]], Optional[Pattern[d, Blank[]]]], Pattern[c, Blank[]]], Pattern[n, Blank[]]]], Pattern[x, Blank[Symbol]]] := Int[ExpandTrig[(g*csc[e + f*x])^p*(a + b*csc[e + f*x])^m*(c + d*csc[e + f*x])^n, x], x] /; FreeQ[{a, b, c, d, e, f, g, m, n, p}, x] && NeQ[b*c - a*d, 0] && (IntegersQ[m, n] || IntegersQ[m, p] || IntegersQ[n, p])
  3993. Int[Times[Power[Plus[Optional[Pattern[a, Blank[]]], Times[csc[Plus[Optional[Pattern[e, Blank[]]], Times[Optional[Pattern[f, Blank[]]], Pattern[x, Blank[]]]]], Optional[Pattern[b, Blank[]]]]], Pattern[m, Blank[]]], Power[Plus[Optional[Pattern[c, Blank[]]], Times[csc[Plus[Optional[Pattern[e, Blank[]]], Times[Optional[Pattern[f, Blank[]]], Pattern[x, Blank[]]]]], Optional[Pattern[d, Blank[]]]]], Pattern[n, Blank[]]], Power[Times[csc[Plus[Optional[Pattern[e, Blank[]]], Times[Optional[Pattern[f, Blank[]]], Pattern[x, Blank[]]]]], Optional[Pattern[g, Blank[]]]], Optional[Pattern[p, Blank[]]]]], Pattern[x, Blank[Symbol]]] := Unintegrable[(g*Csc[e + f*x])^p*(a + b*Csc[e + f*x])^m*(c + d*Csc[e + f*x])^n, x] /; FreeQ[{a, b, c, d, e, f, g, m, n, p}, x]
  3994. Int[Times[sec[Plus[Optional[Pattern[e, Blank[]]], Times[Optional[Pattern[f, Blank[]]], Pattern[x, Blank[]]]]], Power[Plus[Pattern[a, Blank[]], Times[Optional[Pattern[b, Blank[]]], sec[Plus[Optional[Pattern[e, Blank[]]], Times[Optional[Pattern[f, Blank[]]], Pattern[x, Blank[]]]]]]], Rational[-1, 2]], Plus[Pattern[A, Blank[]], Times[Optional[Pattern[B, Blank[]]], sec[Plus[Optional[Pattern[e, Blank[]]], Times[Optional[Pattern[f, Blank[]]], Pattern[x, Blank[]]]]]]], Power[Plus[Pattern[c, Blank[]], Times[Optional[Pattern[d, Blank[]]], sec[Plus[Optional[Pattern[e, Blank[]]], Times[Optional[Pattern[f, Blank[]]], Pattern[x, Blank[]]]]]]], Rational[-3, 2]]], Pattern[x, Blank[Symbol]]] := Simp[(2*A*(1 + Sec[e + f*x])*Sqrt[((b*c - a*d)*(1 - Sec[e + f*x]))/((a + b)*(c + d*Sec[e + f*x]))]*EllipticE[ArcSin[(Rt[(c + d)/(a + b), 2]*Sqrt[a + b*Sec[e + f*x]])/Sqrt[c + d*Sec[e + f*x]]], ((a + b)*(c - d))/((a - b)*(c + d))])/(f*(b*c - a*d)*Rt[(c + d)/(a + b), 2]*Tan[e + f*x]*Sqrt[-(((b*c - a*d)*(1 + Sec[e + f*x]))/((a - b)*(c + d*Sec[e + f*x])))]), x] /; FreeQ[{a, b, c, d, e, f, A, B}, x] && NeQ[b*c - a*d, 0] && NeQ[a^2 - b^2, 0] && NeQ[c^2 - d^2, 0] && EqQ[A, B]
  3995. Int[Times[csc[Plus[Optional[Pattern[e, Blank[]]], Times[Optional[Pattern[f, Blank[]]], Pattern[x, Blank[]]]]], Power[Plus[Times[csc[Plus[Optional[Pattern[e, Blank[]]], Times[Optional[Pattern[f, Blank[]]], Pattern[x, Blank[]]]]], Optional[Pattern[b, Blank[]]]], Pattern[a, Blank[]]], Rational[-1, 2]], Plus[Times[csc[Plus[Optional[Pattern[e, Blank[]]], Times[Optional[Pattern[f, Blank[]]], Pattern[x, Blank[]]]]], Optional[Pattern[B, Blank[]]]], Pattern[A, Blank[]]], Power[Plus[Times[csc[Plus[Optional[Pattern[e, Blank[]]], Times[Optional[Pattern[f, Blank[]]], Pattern[x, Blank[]]]]], Optional[Pattern[d, Blank[]]]], Pattern[c, Blank[]]], Rational[-3, 2]]], Pattern[x, Blank[Symbol]]] := Simp[(-2*A*(1 + Csc[e + f*x])*Sqrt[((b*c - a*d)*(1 - Csc[e + f*x]))/((a + b)*(c + d*Csc[e + f*x]))]*EllipticE[ArcSin[(Rt[(c + d)/(a + b), 2]*Sqrt[a + b*Csc[e + f*x]])/Sqrt[c + d*Csc[e + f*x]]], ((a + b)*(c - d))/((a - b)*(c + d))])/(f*(b*c - a*d)*Rt[(c + d)/(a + b), 2]*Cot[e + f*x]*Sqrt[-(((b*c - a*d)*(1 + Csc[e + f*x]))/((a - b)*(c + d*Csc[e + f*x])))]), x] /; FreeQ[{a, b, c, d, e, f, A, B}, x] && NeQ[b*c - a*d, 0] && NeQ[a^2 - b^2, 0] && NeQ[c^2 - d^2, 0] && EqQ[A, B]
  3996. Int[Times[Power[Times[csc[Plus[Optional[Pattern[e, Blank[]]], Times[Optional[Pattern[f, Blank[]]], Pattern[x, Blank[]]]]], Optional[Pattern[d, Blank[]]]], Pattern[n, Blank[]]], Plus[Times[csc[Plus[Optional[Pattern[e, Blank[]]], Times[Optional[Pattern[f, Blank[]]], Pattern[x, Blank[]]]]], Optional[Pattern[b, Blank[]]]], Pattern[a, Blank[]]], Plus[Times[csc[Plus[Optional[Pattern[e, Blank[]]], Times[Optional[Pattern[f, Blank[]]], Pattern[x, Blank[]]]]], Optional[Pattern[B, Blank[]]]], Pattern[A, Blank[]]]], Pattern[x, Blank[Symbol]]] := Simp[(A*a*Cot[e + f*x]*(d*Csc[e + f*x])^n)/(f*n), x] + Dist[1/(d*n), Int[(d*Csc[e + f*x])^(n + 1)*Simp[n*(B*a + A*b) + (B*b*n + A*a*(n + 1))*Csc[e + f*x], x], x], x] /; FreeQ[{a, b, d, e, f, A, B}, x] && NeQ[A*b - a*B, 0] && LeQ[n, -1]
  3997. Int[Times[Power[Times[csc[Plus[Optional[Pattern[e, Blank[]]], Times[Optional[Pattern[f, Blank[]]], Pattern[x, Blank[]]]]], Optional[Pattern[d, Blank[]]]], Optional[Pattern[n, Blank[]]]], Plus[Times[csc[Plus[Optional[Pattern[e, Blank[]]], Times[Optional[Pattern[f, Blank[]]], Pattern[x, Blank[]]]]], Optional[Pattern[b, Blank[]]]], Pattern[a, Blank[]]], Plus[Times[csc[Plus[Optional[Pattern[e, Blank[]]], Times[Optional[Pattern[f, Blank[]]], Pattern[x, Blank[]]]]], Optional[Pattern[B, Blank[]]]], Pattern[A, Blank[]]]], Pattern[x, Blank[Symbol]]] := -Simp[(b*B*Cot[e + f*x]*(d*Csc[e + f*x])^n)/(f*(n + 1)), x] + Dist[1/(n + 1), Int[(d*Csc[e + f*x])^n*Simp[A*a*(n + 1) + B*b*n + (A*b + B*a)*(n + 1)*Csc[e + f*x], x], x], x] /; FreeQ[{a, b, d, e, f, A, B}, x] && NeQ[A*b - a*B, 0] && !LeQ[n, -1]
  3998. Int[Times[csc[Plus[Optional[Pattern[e, Blank[]]], Times[Optional[Pattern[f, Blank[]]], Pattern[x, Blank[]]]]], Power[Plus[Times[csc[Plus[Optional[Pattern[e, Blank[]]], Times[Optional[Pattern[f, Blank[]]], Pattern[x, Blank[]]]]], Optional[Pattern[b, Blank[]]]], Pattern[a, Blank[]]], -1], Plus[Times[csc[Plus[Optional[Pattern[e, Blank[]]], Times[Optional[Pattern[f, Blank[]]], Pattern[x, Blank[]]]]], Optional[Pattern[B, Blank[]]]], Pattern[A, Blank[]]]], Pattern[x, Blank[Symbol]]] := Dist[B/b, Int[Csc[e + f*x], x], x] + Dist[(A*b - a*B)/b, Int[Csc[e + f*x]/(a + b*Csc[e + f*x]), x], x] /; FreeQ[{a, b, e, f, A, B}, x] && NeQ[A*b - a*B, 0]
  3999. Int[Times[csc[Plus[Optional[Pattern[e, Blank[]]], Times[Optional[Pattern[f, Blank[]]], Pattern[x, Blank[]]]]], Power[Plus[Times[csc[Plus[Optional[Pattern[e, Blank[]]], Times[Optional[Pattern[f, Blank[]]], Pattern[x, Blank[]]]]], Optional[Pattern[b, Blank[]]]], Pattern[a, Blank[]]], Pattern[m, Blank[]]], Plus[Times[csc[Plus[Optional[Pattern[e, Blank[]]], Times[Optional[Pattern[f, Blank[]]], Pattern[x, Blank[]]]]], Optional[Pattern[B, Blank[]]]], Pattern[A, Blank[]]]], Pattern[x, Blank[Symbol]]] := -Simp[(B*Cot[e + f*x]*(a + b*Csc[e + f*x])^m)/(f*(m + 1)), x] /; FreeQ[{a, b, A, B, e, f, m}, x] && NeQ[A*b - a*B, 0] && EqQ[a^2 - b^2, 0] && EqQ[a*B*m + A*b*(m + 1), 0]
  4000. Int[Times[csc[Plus[Optional[Pattern[e, Blank[]]], Times[Optional[Pattern[f, Blank[]]], Pattern[x, Blank[]]]]], Power[Plus[Times[csc[Plus[Optional[Pattern[e, Blank[]]], Times[Optional[Pattern[f, Blank[]]], Pattern[x, Blank[]]]]], Optional[Pattern[b, Blank[]]]], Pattern[a, Blank[]]], Pattern[m, Blank[]]], Plus[Times[csc[Plus[Optional[Pattern[e, Blank[]]], Times[Optional[Pattern[f, Blank[]]], Pattern[x, Blank[]]]]], Optional[Pattern[B, Blank[]]]], Pattern[A, Blank[]]]], Pattern[x, Blank[Symbol]]] := Simp[((A*b - a*B)*Cot[e + f*x]*(a + b*Csc[e + f*x])^m)/(a*f*(2*m + 1)), x] + Dist[(a*B*m + A*b*(m + 1))/(a*b*(2*m + 1)), Int[Csc[e + f*x]*(a + b*Csc[e + f*x])^(m + 1), x], x] /; FreeQ[{a, b, A, B, e, f}, x] && NeQ[A*b - a*B, 0] && EqQ[a^2 - b^2, 0] && NeQ[a*B*m + A*b*(m + 1), 0] && LtQ[m, -2^(-1)]
  4001. Int[Times[csc[Plus[Optional[Pattern[e, Blank[]]], Times[Optional[Pattern[f, Blank[]]], Pattern[x, Blank[]]]]], Power[Plus[Times[csc[Plus[Optional[Pattern[e, Blank[]]], Times[Optional[Pattern[f, Blank[]]], Pattern[x, Blank[]]]]], Optional[Pattern[b, Blank[]]]], Pattern[a, Blank[]]], Pattern[m, Blank[]]], Plus[Times[csc[Plus[Optional[Pattern[e, Blank[]]], Times[Optional[Pattern[f, Blank[]]], Pattern[x, Blank[]]]]], Optional[Pattern[B, Blank[]]]], Pattern[A, Blank[]]]], Pattern[x, Blank[Symbol]]] := -Simp[(B*Cot[e + f*x]*(a + b*Csc[e + f*x])^m)/(f*(m + 1)), x] + Dist[(a*B*m + A*b*(m + 1))/(b*(m + 1)), Int[Csc[e + f*x]*(a + b*Csc[e + f*x])^m, x], x] /; FreeQ[{a, b, A, B, e, f, m}, x] && NeQ[A*b - a*B, 0] && EqQ[a^2 - b^2, 0] && NeQ[a*B*m + A*b*(m + 1), 0] && !LtQ[m, -2^(-1)]
  4002. Int[Times[csc[Plus[Optional[Pattern[e, Blank[]]], Times[Optional[Pattern[f, Blank[]]], Pattern[x, Blank[]]]]], Power[Plus[Times[csc[Plus[Optional[Pattern[e, Blank[]]], Times[Optional[Pattern[f, Blank[]]], Pattern[x, Blank[]]]]], Optional[Pattern[b, Blank[]]]], Pattern[a, Blank[]]], Pattern[m, Blank[]]], Plus[Times[csc[Plus[Optional[Pattern[e, Blank[]]], Times[Optional[Pattern[f, Blank[]]], Pattern[x, Blank[]]]]], Optional[Pattern[B, Blank[]]]], Pattern[A, Blank[]]]], Pattern[x, Blank[Symbol]]] := -Simp[(B*Cot[e + f*x]*(a + b*Csc[e + f*x])^m)/(f*(m + 1)), x] + Dist[1/(m + 1), Int[Csc[e + f*x]*(a + b*Csc[e + f*x])^(m - 1)*Simp[b*B*m + a*A*(m + 1) + (a*B*m + A*b*(m + 1))*Csc[e + f*x], x], x], x] /; FreeQ[{a, b, A, B, e, f}, x] && NeQ[A*b - a*B, 0] && NeQ[a^2 - b^2, 0] && GtQ[m, 0]
  4003. Int[Times[csc[Plus[Optional[Pattern[e, Blank[]]], Times[Optional[Pattern[f, Blank[]]], Pattern[x, Blank[]]]]], Power[Plus[Times[csc[Plus[Optional[Pattern[e, Blank[]]], Times[Optional[Pattern[f, Blank[]]], Pattern[x, Blank[]]]]], Optional[Pattern[b, Blank[]]]], Pattern[a, Blank[]]], Pattern[m, Blank[]]], Plus[Times[csc[Plus[Optional[Pattern[e, Blank[]]], Times[Optional[Pattern[f, Blank[]]], Pattern[x, Blank[]]]]], Optional[Pattern[B, Blank[]]]], Pattern[A, Blank[]]]], Pattern[x, Blank[Symbol]]] := -Simp[((A*b - a*B)*Cot[e + f*x]*(a + b*Csc[e + f*x])^(m + 1))/(f*(m + 1)*(a^2 - b^2)), x] + Dist[1/((m + 1)*(a^2 - b^2)), Int[Csc[e + f*x]*(a + b*Csc[e + f*x])^(m + 1)*Simp[(a*A - b*B)*(m + 1) - (A*b - a*B)*(m + 2)*Csc[e + f*x], x], x], x] /; FreeQ[{a, b, A, B, e, f}, x] && NeQ[A*b - a*B, 0] && NeQ[a^2 - b^2, 0] && LtQ[m, -1]
  4004. Int[Times[csc[Plus[Optional[Pattern[e, Blank[]]], Times[Optional[Pattern[f, Blank[]]], Pattern[x, Blank[]]]]], Power[Plus[Times[csc[Plus[Optional[Pattern[e, Blank[]]], Times[Optional[Pattern[f, Blank[]]], Pattern[x, Blank[]]]]], Optional[Pattern[b, Blank[]]]], Pattern[a, Blank[]]], Rational[-1, 2]], Plus[Times[csc[Plus[Optional[Pattern[e, Blank[]]], Times[Optional[Pattern[f, Blank[]]], Pattern[x, Blank[]]]]], Optional[Pattern[B, Blank[]]]], Pattern[A, Blank[]]]], Pattern[x, Blank[Symbol]]] := Simp[(-2*(A*b - a*B)*Rt[a + (b*B)/A, 2]*Sqrt[(b*(1 - Csc[e + f*x]))/(a + b)]*Sqrt[-((b*(1 + Csc[e + f*x]))/(a - b))]*EllipticE[ArcSin[Sqrt[a + b*Csc[e + f*x]]/Rt[a + (b*B)/A, 2]], (a*A + b*B)/(a*A - b*B)])/(b^2*f*Cot[e + f*x]), x] /; FreeQ[{a, b, e, f, A, B}, x] && NeQ[a^2 - b^2, 0] && EqQ[A^2 - B^2, 0]
  4005. Int[Times[csc[Plus[Optional[Pattern[e, Blank[]]], Times[Optional[Pattern[f, Blank[]]], Pattern[x, Blank[]]]]], Power[Plus[Times[csc[Plus[Optional[Pattern[e, Blank[]]], Times[Optional[Pattern[f, Blank[]]], Pattern[x, Blank[]]]]], Optional[Pattern[b, Blank[]]]], Pattern[a, Blank[]]], Rational[-1, 2]], Plus[Times[csc[Plus[Optional[Pattern[e, Blank[]]], Times[Optional[Pattern[f, Blank[]]], Pattern[x, Blank[]]]]], Optional[Pattern[B, Blank[]]]], Pattern[A, Blank[]]]], Pattern[x, Blank[Symbol]]] := Dist[A - B, Int[Csc[e + f*x]/Sqrt[a + b*Csc[e + f*x]], x], x] + Dist[B, Int[(Csc[e + f*x]*(1 + Csc[e + f*x]))/Sqrt[a + b*Csc[e + f*x]], x], x] /; FreeQ[{a, b, e, f, A, B}, x] && NeQ[a^2 - b^2, 0] && NeQ[A^2 - B^2, 0]
  4006. Int[Times[csc[Plus[Optional[Pattern[e, Blank[]]], Times[Optional[Pattern[f, Blank[]]], Pattern[x, Blank[]]]]], Power[Plus[Times[csc[Plus[Optional[Pattern[e, Blank[]]], Times[Optional[Pattern[f, Blank[]]], Pattern[x, Blank[]]]]], Optional[Pattern[b, Blank[]]]], Pattern[a, Blank[]]], Pattern[m, Blank[]]], Plus[Times[csc[Plus[Optional[Pattern[e, Blank[]]], Times[Optional[Pattern[f, Blank[]]], Pattern[x, Blank[]]]]], Optional[Pattern[B, Blank[]]]], Pattern[A, Blank[]]]], Pattern[x, Blank[Symbol]]] := Simp[(2*Sqrt[2]*A*(a + b*Csc[e + f*x])^m*(A - B*Csc[e + f*x])*Sqrt[(A + B*Csc[e + f*x])/A]*AppellF1[1/2, -(1/2), -m, 3/2, (A - B*Csc[e + f*x])/(2*A), (b*(A - B*Csc[e + f*x]))/(A*b + a*B)])/(B*f*Cot[e + f*x]*((A*(a + b*Csc[e + f*x]))/(a*A + b*B))^m), x] /; FreeQ[{a, b, A, B, e, f}, x] && NeQ[A*b - a*B, 0] && NeQ[a^2 - b^2, 0] && EqQ[A^2 - B^2, 0] && !IntegerQ[2*m]
  4007. Int[Times[csc[Plus[Optional[Pattern[e, Blank[]]], Times[Optional[Pattern[f, Blank[]]], Pattern[x, Blank[]]]]], Power[Plus[Times[csc[Plus[Optional[Pattern[e, Blank[]]], Times[Optional[Pattern[f, Blank[]]], Pattern[x, Blank[]]]]], Optional[Pattern[b, Blank[]]]], Pattern[a, Blank[]]], Pattern[m, Blank[]]], Plus[Times[csc[Plus[Optional[Pattern[e, Blank[]]], Times[Optional[Pattern[f, Blank[]]], Pattern[x, Blank[]]]]], Optional[Pattern[B, Blank[]]]], Pattern[A, Blank[]]]], Pattern[x, Blank[Symbol]]] := Dist[(A*b - a*B)/b, Int[Csc[e + f*x]*(a + b*Csc[e + f*x])^m, x], x] + Dist[B/b, Int[Csc[e + f*x]*(a + b*Csc[e + f*x])^(m + 1), x], x] /; FreeQ[{a, b, A, B, e, f, m}, x] && NeQ[A*b - a*B, 0] && NeQ[a^2 - b^2, 0]
  4008. Int[Times[Power[csc[Plus[Optional[Pattern[e, Blank[]]], Times[Optional[Pattern[f, Blank[]]], Pattern[x, Blank[]]]]], 2], Power[Plus[Times[csc[Plus[Optional[Pattern[e, Blank[]]], Times[Optional[Pattern[f, Blank[]]], Pattern[x, Blank[]]]]], Optional[Pattern[b, Blank[]]]], Pattern[a, Blank[]]], Pattern[m, Blank[]]], Plus[Times[csc[Plus[Optional[Pattern[e, Blank[]]], Times[Optional[Pattern[f, Blank[]]], Pattern[x, Blank[]]]]], Optional[Pattern[B, Blank[]]]], Pattern[A, Blank[]]]], Pattern[x, Blank[Symbol]]] := -Simp[((A*b - a*B)*Cot[e + f*x]*(a + b*Csc[e + f*x])^m)/(b*f*(2*m + 1)), x] + Dist[1/(b^2*(2*m + 1)), Int[Csc[e + f*x]*(a + b*Csc[e + f*x])^(m + 1)*Simp[A*b*m - a*B*m + b*B*(2*m + 1)*Csc[e + f*x], x], x], x] /; FreeQ[{a, b, e, f, A, B}, x] && NeQ[A*b - a*B, 0] && EqQ[a^2 - b^2, 0] && LtQ[m, -2^(-1)]
  4009. Int[Times[Power[csc[Plus[Optional[Pattern[e, Blank[]]], Times[Optional[Pattern[f, Blank[]]], Pattern[x, Blank[]]]]], 2], Power[Plus[Times[csc[Plus[Optional[Pattern[e, Blank[]]], Times[Optional[Pattern[f, Blank[]]], Pattern[x, Blank[]]]]], Optional[Pattern[b, Blank[]]]], Pattern[a, Blank[]]], Pattern[m, Blank[]]], Plus[Times[csc[Plus[Optional[Pattern[e, Blank[]]], Times[Optional[Pattern[f, Blank[]]], Pattern[x, Blank[]]]]], Optional[Pattern[B, Blank[]]]], Pattern[A, Blank[]]]], Pattern[x, Blank[Symbol]]] := Simp[(a*(A*b - a*B)*Cot[e + f*x]*(a + b*Csc[e + f*x])^(m + 1))/(b*f*(m + 1)*(a^2 - b^2)), x] - Dist[1/(b*(m + 1)*(a^2 - b^2)), Int[Csc[e + f*x]*(a + b*Csc[e + f*x])^(m + 1)*Simp[b*(A*b - a*B)*(m + 1) - (a*A*b*(m + 2) - B*(a^2 + b^2*(m + 1)))*Csc[e + f*x], x], x], x] /; FreeQ[{a, b, e, f, A, B}, x] && NeQ[A*b - a*B, 0] && NeQ[a^2 - b^2, 0] && LtQ[m, -1]
  4010. Int[Times[Power[csc[Plus[Optional[Pattern[e, Blank[]]], Times[Optional[Pattern[f, Blank[]]], Pattern[x, Blank[]]]]], 2], Power[Plus[Times[csc[Plus[Optional[Pattern[e, Blank[]]], Times[Optional[Pattern[f, Blank[]]], Pattern[x, Blank[]]]]], Optional[Pattern[b, Blank[]]]], Pattern[a, Blank[]]], Pattern[m, Blank[]]], Plus[Times[csc[Plus[Optional[Pattern[e, Blank[]]], Times[Optional[Pattern[f, Blank[]]], Pattern[x, Blank[]]]]], Optional[Pattern[B, Blank[]]]], Pattern[A, Blank[]]]], Pattern[x, Blank[Symbol]]] := -Simp[(B*Cot[e + f*x]*(a + b*Csc[e + f*x])^(m + 1))/(b*f*(m + 2)), x] + Dist[1/(b*(m + 2)), Int[Csc[e + f*x]*(a + b*Csc[e + f*x])^m*Simp[b*B*(m + 1) + (A*b*(m + 2) - a*B)*Csc[e + f*x], x], x], x] /; FreeQ[{a, b, e, f, A, B, m}, x] && NeQ[A*b - a*B, 0] && !LtQ[m, -1]
  4011. Int[Times[Power[Times[csc[Plus[Optional[Pattern[e, Blank[]]], Times[Optional[Pattern[f, Blank[]]], Pattern[x, Blank[]]]]], Optional[Pattern[d, Blank[]]]], Pattern[n, Blank[]]], Power[Plus[Times[csc[Plus[Optional[Pattern[e, Blank[]]], Times[Optional[Pattern[f, Blank[]]], Pattern[x, Blank[]]]]], Optional[Pattern[b, Blank[]]]], Pattern[a, Blank[]]], Pattern[m, Blank[]]], Plus[Times[csc[Plus[Optional[Pattern[e, Blank[]]], Times[Optional[Pattern[f, Blank[]]], Pattern[x, Blank[]]]]], Optional[Pattern[B, Blank[]]]], Pattern[A, Blank[]]]], Pattern[x, Blank[Symbol]]] := Simp[(A*Cot[e + f*x]*(a + b*Csc[e + f*x])^m*(d*Csc[e + f*x])^n)/(f*n), x] /; FreeQ[{a, b, d, e, f, A, B, m, n}, x] && NeQ[A*b - a*B, 0] && EqQ[a^2 - b^2, 0] && EqQ[m + n + 1, 0] && EqQ[a*A*m - b*B*n, 0]
  4012. Int[Times[Power[Times[csc[Plus[Optional[Pattern[e, Blank[]]], Times[Optional[Pattern[f, Blank[]]], Pattern[x, Blank[]]]]], Optional[Pattern[d, Blank[]]]], Pattern[n, Blank[]]], Power[Plus[Times[csc[Plus[Optional[Pattern[e, Blank[]]], Times[Optional[Pattern[f, Blank[]]], Pattern[x, Blank[]]]]], Optional[Pattern[b, Blank[]]]], Pattern[a, Blank[]]], Pattern[m, Blank[]]], Plus[Times[csc[Plus[Optional[Pattern[e, Blank[]]], Times[Optional[Pattern[f, Blank[]]], Pattern[x, Blank[]]]]], Optional[Pattern[B, Blank[]]]], Pattern[A, Blank[]]]], Pattern[x, Blank[Symbol]]] := -Simp[((A*b - a*B)*Cot[e + f*x]*(a + b*Csc[e + f*x])^m*(d*Csc[e + f*x])^n)/(b*f*(2*m + 1)), x] + Dist[(a*A*m + b*B*(m + 1))/(a^2*(2*m + 1)), Int[(a + b*Csc[e + f*x])^(m + 1)*(d*Csc[e + f*x])^n, x], x] /; FreeQ[{a, b, d, e, f, A, B, n}, x] && NeQ[A*b - a*B, 0] && EqQ[a^2 - b^2, 0] && EqQ[m + n + 1, 0] && LeQ[m, -1]
  4013. Int[Times[Power[Times[csc[Plus[Optional[Pattern[e, Blank[]]], Times[Optional[Pattern[f, Blank[]]], Pattern[x, Blank[]]]]], Optional[Pattern[d, Blank[]]]], Pattern[n, Blank[]]], Power[Plus[Times[csc[Plus[Optional[Pattern[e, Blank[]]], Times[Optional[Pattern[f, Blank[]]], Pattern[x, Blank[]]]]], Optional[Pattern[b, Blank[]]]], Pattern[a, Blank[]]], Pattern[m, Blank[]]], Plus[Times[csc[Plus[Optional[Pattern[e, Blank[]]], Times[Optional[Pattern[f, Blank[]]], Pattern[x, Blank[]]]]], Optional[Pattern[B, Blank[]]]], Pattern[A, Blank[]]]], Pattern[x, Blank[Symbol]]] := Simp[(A*Cot[e + f*x]*(a + b*Csc[e + f*x])^m*(d*Csc[e + f*x])^n)/(f*n), x] - Dist[(a*A*m - b*B*n)/(b*d*n), Int[(a + b*Csc[e + f*x])^m*(d*Csc[e + f*x])^(n + 1), x], x] /; FreeQ[{a, b, d, e, f, A, B, m, n}, x] && NeQ[A*b - a*B, 0] && EqQ[a^2 - b^2, 0] && EqQ[m + n + 1, 0] && !LeQ[m, -1]
  4014. Int[Times[Power[Times[csc[Plus[Optional[Pattern[e, Blank[]]], Times[Optional[Pattern[f, Blank[]]], Pattern[x, Blank[]]]]], Optional[Pattern[d, Blank[]]]], Pattern[n, Blank[]]], Power[Plus[Times[csc[Plus[Optional[Pattern[e, Blank[]]], Times[Optional[Pattern[f, Blank[]]], Pattern[x, Blank[]]]]], Optional[Pattern[b, Blank[]]]], Pattern[a, Blank[]]], Rational[1, 2]], Plus[Times[csc[Plus[Optional[Pattern[e, Blank[]]], Times[Optional[Pattern[f, Blank[]]], Pattern[x, Blank[]]]]], Optional[Pattern[B, Blank[]]]], Pattern[A, Blank[]]]], Pattern[x, Blank[Symbol]]] := Simp[(-2*b*B*Cot[e + f*x]*(d*Csc[e + f*x])^n)/(f*(2*n + 1)*Sqrt[a + b*Csc[e + f*x]]), x] /; FreeQ[{a, b, d, e, f, A, B, n}, x] && NeQ[A*b - a*B, 0] && EqQ[a^2 - b^2, 0] && EqQ[A*b*(2*n + 1) + 2*a*B*n, 0]
  4015. Int[Times[Power[Times[csc[Plus[Optional[Pattern[e, Blank[]]], Times[Optional[Pattern[f, Blank[]]], Pattern[x, Blank[]]]]], Optional[Pattern[d, Blank[]]]], Pattern[n, Blank[]]], Power[Plus[Times[csc[Plus[Optional[Pattern[e, Blank[]]], Times[Optional[Pattern[f, Blank[]]], Pattern[x, Blank[]]]]], Optional[Pattern[b, Blank[]]]], Pattern[a, Blank[]]], Rational[1, 2]], Plus[Times[csc[Plus[Optional[Pattern[e, Blank[]]], Times[Optional[Pattern[f, Blank[]]], Pattern[x, Blank[]]]]], Optional[Pattern[B, Blank[]]]], Pattern[A, Blank[]]]], Pattern[x, Blank[Symbol]]] := Simp[(A*b^2*Cot[e + f*x]*(d*Csc[e + f*x])^n)/(a*f*n*Sqrt[a + b*Csc[e + f*x]]), x] + Dist[(A*b*(2*n + 1) + 2*a*B*n)/(2*a*d*n), Int[Sqrt[a + b*Csc[e + f*x]]*(d*Csc[e + f*x])^(n + 1), x], x] /; FreeQ[{a, b, d, e, f, A, B}, x] && NeQ[A*b - a*B, 0] && EqQ[a^2 - b^2, 0] && NeQ[A*b*(2*n + 1) + 2*a*B*n, 0] && LtQ[n, 0]
  4016. Int[Times[Power[Times[csc[Plus[Optional[Pattern[e, Blank[]]], Times[Optional[Pattern[f, Blank[]]], Pattern[x, Blank[]]]]], Optional[Pattern[d, Blank[]]]], Pattern[n, Blank[]]], Power[Plus[Times[csc[Plus[Optional[Pattern[e, Blank[]]], Times[Optional[Pattern[f, Blank[]]], Pattern[x, Blank[]]]]], Optional[Pattern[b, Blank[]]]], Pattern[a, Blank[]]], Rational[1, 2]], Plus[Times[csc[Plus[Optional[Pattern[e, Blank[]]], Times[Optional[Pattern[f, Blank[]]], Pattern[x, Blank[]]]]], Optional[Pattern[B, Blank[]]]], Pattern[A, Blank[]]]], Pattern[x, Blank[Symbol]]] := Simp[(-2*b*B*Cot[e + f*x]*(d*Csc[e + f*x])^n)/(f*(2*n + 1)*Sqrt[a + b*Csc[e + f*x]]), x] + Dist[(A*b*(2*n + 1) + 2*a*B*n)/(b*(2*n + 1)), Int[Sqrt[a + b*Csc[e + f*x]]*(d*Csc[e + f*x])^n, x], x] /; FreeQ[{a, b, d, e, f, A, B, n}, x] && NeQ[A*b - a*B, 0] && EqQ[a^2 - b^2, 0] && NeQ[A*b*(2*n + 1) + 2*a*B*n, 0] && !LtQ[n, 0]
  4017. Int[Times[Power[Times[csc[Plus[Optional[Pattern[e, Blank[]]], Times[Optional[Pattern[f, Blank[]]], Pattern[x, Blank[]]]]], Optional[Pattern[d, Blank[]]]], Pattern[n, Blank[]]], Power[Plus[Times[csc[Plus[Optional[Pattern[e, Blank[]]], Times[Optional[Pattern[f, Blank[]]], Pattern[x, Blank[]]]]], Optional[Pattern[b, Blank[]]]], Pattern[a, Blank[]]], Pattern[m, Blank[]]], Plus[Times[csc[Plus[Optional[Pattern[e, Blank[]]], Times[Optional[Pattern[f, Blank[]]], Pattern[x, Blank[]]]]], Optional[Pattern[B, Blank[]]]], Pattern[A, Blank[]]]], Pattern[x, Blank[Symbol]]] := Simp[(a*A*Cot[e + f*x]*(a + b*Csc[e + f*x])^(m - 1)*(d*Csc[e + f*x])^n)/(f*n), x] - Dist[b/(a*d*n), Int[(a + b*Csc[e + f*x])^(m - 1)*(d*Csc[e + f*x])^(n + 1)*Simp[a*A*(m - n - 1) - b*B*n - (a*B*n + A*b*(m + n))*Csc[e + f*x], x], x], x] /; FreeQ[{a, b, d, e, f, A, B}, x] && NeQ[A*b - a*B, 0] && EqQ[a^2 - b^2, 0] && GtQ[m, 1/2] && LtQ[n, -1]
  4018. Int[Times[Power[Times[csc[Plus[Optional[Pattern[e, Blank[]]], Times[Optional[Pattern[f, Blank[]]], Pattern[x, Blank[]]]]], Optional[Pattern[d, Blank[]]]], Pattern[n, Blank[]]], Power[Plus[Times[csc[Plus[Optional[Pattern[e, Blank[]]], Times[Optional[Pattern[f, Blank[]]], Pattern[x, Blank[]]]]], Optional[Pattern[b, Blank[]]]], Pattern[a, Blank[]]], Pattern[m, Blank[]]], Plus[Times[csc[Plus[Optional[Pattern[e, Blank[]]], Times[Optional[Pattern[f, Blank[]]], Pattern[x, Blank[]]]]], Optional[Pattern[B, Blank[]]]], Pattern[A, Blank[]]]], Pattern[x, Blank[Symbol]]] := -Simp[(b*B*Cot[e + f*x]*(a + b*Csc[e + f*x])^(m - 1)*(d*Csc[e + f*x])^n)/(f*(m + n)), x] + Dist[1/(d*(m + n)), Int[(a + b*Csc[e + f*x])^(m - 1)*(d*Csc[e + f*x])^n*Simp[a*A*d*(m + n) + B*(b*d*n) + (A*b*d*(m + n) + a*B*d*(2*m + n - 1))*Csc[e + f*x], x], x], x] /; FreeQ[{a, b, d, e, f, A, B, n}, x] && NeQ[A*b - a*B, 0] && EqQ[a^2 - b^2, 0] && GtQ[m, 1/2] && !LtQ[n, -1]
  4019. Int[Times[Power[Times[csc[Plus[Optional[Pattern[e, Blank[]]], Times[Optional[Pattern[f, Blank[]]], Pattern[x, Blank[]]]]], Optional[Pattern[d, Blank[]]]], Pattern[n, Blank[]]], Power[Plus[Times[csc[Plus[Optional[Pattern[e, Blank[]]], Times[Optional[Pattern[f, Blank[]]], Pattern[x, Blank[]]]]], Optional[Pattern[b, Blank[]]]], Pattern[a, Blank[]]], Pattern[m, Blank[]]], Plus[Times[csc[Plus[Optional[Pattern[e, Blank[]]], Times[Optional[Pattern[f, Blank[]]], Pattern[x, Blank[]]]]], Optional[Pattern[B, Blank[]]]], Pattern[A, Blank[]]]], Pattern[x, Blank[Symbol]]] := Simp[(d*(A*b - a*B)*Cot[e + f*x]*(a + b*Csc[e + f*x])^m*(d*Csc[e + f*x])^(n - 1))/(a*f*(2*m + 1)), x] - Dist[1/(a*b*(2*m + 1)), Int[(a + b*Csc[e + f*x])^(m + 1)*(d*Csc[e + f*x])^(n - 1)*Simp[A*(a*d*(n - 1)) - B*(b*d*(n - 1)) - d*(a*B*(m - n + 1) + A*b*(m + n))*Csc[e + f*x], x], x], x] /; FreeQ[{a, b, d, e, f, A, B}, x] && NeQ[A*b - a*B, 0] && EqQ[a^2 - b^2, 0] && LtQ[m, -2^(-1)] && GtQ[n, 0]
  4020. Int[Times[Power[Times[csc[Plus[Optional[Pattern[e, Blank[]]], Times[Optional[Pattern[f, Blank[]]], Pattern[x, Blank[]]]]], Optional[Pattern[d, Blank[]]]], Pattern[n, Blank[]]], Power[Plus[Times[csc[Plus[Optional[Pattern[e, Blank[]]], Times[Optional[Pattern[f, Blank[]]], Pattern[x, Blank[]]]]], Optional[Pattern[b, Blank[]]]], Pattern[a, Blank[]]], Pattern[m, Blank[]]], Plus[Times[csc[Plus[Optional[Pattern[e, Blank[]]], Times[Optional[Pattern[f, Blank[]]], Pattern[x, Blank[]]]]], Optional[Pattern[B, Blank[]]]], Pattern[A, Blank[]]]], Pattern[x, Blank[Symbol]]] := -Simp[((A*b - a*B)*Cot[e + f*x]*(a + b*Csc[e + f*x])^m*(d*Csc[e + f*x])^n)/(b*f*(2*m + 1)), x] - Dist[1/(a^2*(2*m + 1)), Int[(a + b*Csc[e + f*x])^(m + 1)*(d*Csc[e + f*x])^n*Simp[b*B*n - a*A*(2*m + n + 1) + (A*b - a*B)*(m + n + 1)*Csc[e + f*x], x], x], x] /; FreeQ[{a, b, d, e, f, A, B, n}, x] && NeQ[A*b - a*B, 0] && EqQ[a^2 - b^2, 0] && LtQ[m, -2^(-1)] && !GtQ[n, 0]
  4021. Int[Times[Power[Times[csc[Plus[Optional[Pattern[e, Blank[]]], Times[Optional[Pattern[f, Blank[]]], Pattern[x, Blank[]]]]], Optional[Pattern[d, Blank[]]]], Pattern[n, Blank[]]], Power[Plus[Times[csc[Plus[Optional[Pattern[e, Blank[]]], Times[Optional[Pattern[f, Blank[]]], Pattern[x, Blank[]]]]], Optional[Pattern[b, Blank[]]]], Pattern[a, Blank[]]], Pattern[m, Blank[]]], Plus[Times[csc[Plus[Optional[Pattern[e, Blank[]]], Times[Optional[Pattern[f, Blank[]]], Pattern[x, Blank[]]]]], Optional[Pattern[B, Blank[]]]], Pattern[A, Blank[]]]], Pattern[x, Blank[Symbol]]] := -Simp[(B*d*Cot[e + f*x]*(a + b*Csc[e + f*x])^m*(d*Csc[e + f*x])^(n - 1))/(f*(m + n)), x] + Dist[d/(b*(m + n)), Int[(a + b*Csc[e + f*x])^m*(d*Csc[e + f*x])^(n - 1)*Simp[b*B*(n - 1) + (A*b*(m + n) + a*B*m)*Csc[e + f*x], x], x], x] /; FreeQ[{a, b, d, e, f, A, B, m}, x] && NeQ[A*b - a*B, 0] && EqQ[a^2 - b^2, 0] && GtQ[n, 1]
  4022. Int[Times[Power[Times[csc[Plus[Optional[Pattern[e, Blank[]]], Times[Optional[Pattern[f, Blank[]]], Pattern[x, Blank[]]]]], Optional[Pattern[d, Blank[]]]], Pattern[n, Blank[]]], Power[Plus[Times[csc[Plus[Optional[Pattern[e, Blank[]]], Times[Optional[Pattern[f, Blank[]]], Pattern[x, Blank[]]]]], Optional[Pattern[b, Blank[]]]], Pattern[a, Blank[]]], Pattern[m, Blank[]]], Plus[Times[csc[Plus[Optional[Pattern[e, Blank[]]], Times[Optional[Pattern[f, Blank[]]], Pattern[x, Blank[]]]]], Optional[Pattern[B, Blank[]]]], Pattern[A, Blank[]]]], Pattern[x, Blank[Symbol]]] := Simp[(A*Cot[e + f*x]*(a + b*Csc[e + f*x])^m*(d*Csc[e + f*x])^n)/(f*n), x] - Dist[1/(b*d*n), Int[(a + b*Csc[e + f*x])^m*(d*Csc[e + f*x])^(n + 1)*Simp[a*A*m - b*B*n - A*b*(m + n + 1)*Csc[e + f*x], x], x], x] /; FreeQ[{a, b, d, e, f, A, B, m}, x] && NeQ[A*b - a*B, 0] && EqQ[a^2 - b^2, 0] && LtQ[n, 0]
  4023. Int[Times[Power[Times[csc[Plus[Optional[Pattern[e, Blank[]]], Times[Optional[Pattern[f, Blank[]]], Pattern[x, Blank[]]]]], Optional[Pattern[d, Blank[]]]], Pattern[n, Blank[]]], Power[Plus[Times[csc[Plus[Optional[Pattern[e, Blank[]]], Times[Optional[Pattern[f, Blank[]]], Pattern[x, Blank[]]]]], Optional[Pattern[b, Blank[]]]], Pattern[a, Blank[]]], Pattern[m, Blank[]]], Plus[Times[csc[Plus[Optional[Pattern[e, Blank[]]], Times[Optional[Pattern[f, Blank[]]], Pattern[x, Blank[]]]]], Optional[Pattern[B, Blank[]]]], Pattern[A, Blank[]]]], Pattern[x, Blank[Symbol]]] := Dist[(A*b - a*B)/b, Int[(a + b*Csc[e + f*x])^m*(d*Csc[e + f*x])^n, x], x] + Dist[B/b, Int[(a + b*Csc[e + f*x])^(m + 1)*(d*Csc[e + f*x])^n, x], x] /; FreeQ[{a, b, d, e, f, A, B, m}, x] && NeQ[A*b - a*B, 0] && EqQ[a^2 - b^2, 0]
  4024. Int[Times[Power[Times[csc[Plus[Optional[Pattern[e, Blank[]]], Times[Optional[Pattern[f, Blank[]]], Pattern[x, Blank[]]]]], Optional[Pattern[d, Blank[]]]], Pattern[n, Blank[]]], Power[Plus[Times[csc[Plus[Optional[Pattern[e, Blank[]]], Times[Optional[Pattern[f, Blank[]]], Pattern[x, Blank[]]]]], Optional[Pattern[b, Blank[]]]], Pattern[a, Blank[]]], 2], Plus[Times[csc[Plus[Optional[Pattern[e, Blank[]]], Times[Optional[Pattern[f, Blank[]]], Pattern[x, Blank[]]]]], Optional[Pattern[B, Blank[]]]], Pattern[A, Blank[]]]], Pattern[x, Blank[Symbol]]] := Simp[(a^2*A*Cos[e + f*x]*(d*Csc[e + f*x])^(n + 1))/(d*f*n), x] + Dist[1/(d*n), Int[(d*Csc[e + f*x])^(n + 1)*(a*(2*A*b + a*B)*n + (2*a*b*B*n + A*(b^2*n + a^2*(n + 1)))*Csc[e + f*x] + b^2*B*n*Csc[e + f*x]^2), x], x] /; FreeQ[{a, b, d, e, f, A, B}, x] && NeQ[A*b - a*B, 0] && NeQ[a^2 - b^2, 0] && LeQ[n, -1]
  4025. Int[Times[Power[Times[csc[Plus[Optional[Pattern[e, Blank[]]], Times[Optional[Pattern[f, Blank[]]], Pattern[x, Blank[]]]]], Optional[Pattern[d, Blank[]]]], Pattern[n, Blank[]]], Power[Plus[Times[csc[Plus[Optional[Pattern[e, Blank[]]], Times[Optional[Pattern[f, Blank[]]], Pattern[x, Blank[]]]]], Optional[Pattern[b, Blank[]]]], Pattern[a, Blank[]]], Pattern[m, Blank[]]], Plus[Times[csc[Plus[Optional[Pattern[e, Blank[]]], Times[Optional[Pattern[f, Blank[]]], Pattern[x, Blank[]]]]], Optional[Pattern[B, Blank[]]]], Pattern[A, Blank[]]]], Pattern[x, Blank[Symbol]]] := Simp[(a*A*Cot[e + f*x]*(a + b*Csc[e + f*x])^(m - 1)*(d*Csc[e + f*x])^n)/(f*n), x] + Dist[1/(d*n), Int[(a + b*Csc[e + f*x])^(m - 2)*(d*Csc[e + f*x])^(n + 1)*Simp[a*(a*B*n - A*b*(m - n - 1)) + (2*a*b*B*n + A*(b^2*n + a^2*(1 + n)))*Csc[e + f*x] + b*(b*B*n + a*A*(m + n))*Csc[e + f*x]^2, x], x], x] /; FreeQ[{a, b, d, e, f, A, B}, x] && NeQ[A*b - a*B, 0] && NeQ[a^2 - b^2, 0] && GtQ[m, 1] && LeQ[n, -1]
  4026. Int[Times[Power[Times[csc[Plus[Optional[Pattern[e, Blank[]]], Times[Optional[Pattern[f, Blank[]]], Pattern[x, Blank[]]]]], Optional[Pattern[d, Blank[]]]], Pattern[n, Blank[]]], Power[Plus[Times[csc[Plus[Optional[Pattern[e, Blank[]]], Times[Optional[Pattern[f, Blank[]]], Pattern[x, Blank[]]]]], Optional[Pattern[b, Blank[]]]], Pattern[a, Blank[]]], Pattern[m, Blank[]]], Plus[Times[csc[Plus[Optional[Pattern[e, Blank[]]], Times[Optional[Pattern[f, Blank[]]], Pattern[x, Blank[]]]]], Optional[Pattern[B, Blank[]]]], Pattern[A, Blank[]]]], Pattern[x, Blank[Symbol]]] := -Simp[(b*B*Cot[e + f*x]*(a + b*Csc[e + f*x])^(m - 1)*(d*Csc[e + f*x])^n)/(f*(m + n)), x] + Dist[1/(m + n), Int[(a + b*Csc[e + f*x])^(m - 2)*(d*Csc[e + f*x])^n*Simp[a^2*A*(m + n) + a*b*B*n + (a*(2*A*b + a*B)*(m + n) + b^2*B*(m + n - 1))*Csc[e + f*x] + b*(A*b*(m + n) + a*B*(2*m + n - 1))*Csc[e + f*x]^2, x], x], x] /; FreeQ[{a, b, d, e, f, A, B, n}, x] && NeQ[A*b - a*B, 0] && NeQ[a^2 - b^2, 0] && GtQ[m, 1] && !(IGtQ[n, 1] && !IntegerQ[m])
  4027. Int[Times[Power[Times[csc[Plus[Optional[Pattern[e, Blank[]]], Times[Optional[Pattern[f, Blank[]]], Pattern[x, Blank[]]]]], Optional[Pattern[d, Blank[]]]], Pattern[n, Blank[]]], Power[Plus[Times[csc[Plus[Optional[Pattern[e, Blank[]]], Times[Optional[Pattern[f, Blank[]]], Pattern[x, Blank[]]]]], Optional[Pattern[b, Blank[]]]], Pattern[a, Blank[]]], Pattern[m, Blank[]]], Plus[Times[csc[Plus[Optional[Pattern[e, Blank[]]], Times[Optional[Pattern[f, Blank[]]], Pattern[x, Blank[]]]]], Optional[Pattern[B, Blank[]]]], Pattern[A, Blank[]]]], Pattern[x, Blank[Symbol]]] := -Simp[(d*(A*b - a*B)*Cot[e + f*x]*(a + b*Csc[e + f*x])^(m + 1)*(d*Csc[e + f*x])^(n - 1))/(f*(m + 1)*(a^2 - b^2)), x] + Dist[1/((m + 1)*(a^2 - b^2)), Int[(a + b*Csc[e + f*x])^(m + 1)*(d*Csc[e + f*x])^(n - 1)*Simp[d*(n - 1)*(A*b - a*B) + d*(a*A - b*B)*(m + 1)*Csc[e + f*x] - d*(A*b - a*B)*(m + n + 1)*Csc[e + f*x]^2, x], x], x] /; FreeQ[{a, b, d, e, f, A, B}, x] && NeQ[A*b - a*B, 0] && NeQ[a^2 - b^2, 0] && LtQ[m, -1] && LtQ[0, n, 1]
  4028. Int[Times[Power[csc[Plus[Optional[Pattern[e, Blank[]]], Times[Optional[Pattern[f, Blank[]]], Pattern[x, Blank[]]]]], 3], Power[Plus[Times[csc[Plus[Optional[Pattern[e, Blank[]]], Times[Optional[Pattern[f, Blank[]]], Pattern[x, Blank[]]]]], Optional[Pattern[b, Blank[]]]], Pattern[a, Blank[]]], Pattern[m, Blank[]]], Plus[Times[csc[Plus[Optional[Pattern[e, Blank[]]], Times[Optional[Pattern[f, Blank[]]], Pattern[x, Blank[]]]]], Optional[Pattern[B, Blank[]]]], Pattern[A, Blank[]]]], Pattern[x, Blank[Symbol]]] := -Simp[(a^2*(A*b - a*B)*Cot[e + f*x]*(a + b*Csc[e + f*x])^(m + 1))/(b^2*f*(m + 1)*(a^2 - b^2)), x] + Dist[1/(b^2*(m + 1)*(a^2 - b^2)), Int[Csc[e + f*x]*(a + b*Csc[e + f*x])^(m + 1)*Simp[a*b*(A*b - a*B)*(m + 1) - (A*b - a*B)*(a^2 + b^2*(m + 1))*Csc[e + f*x] + b*B*(m + 1)*(a^2 - b^2)*Csc[e + f*x]^2, x], x], x] /; FreeQ[{a, b, e, f, A, B}, x] && NeQ[A*b - a*B, 0] && NeQ[a^2 - b^2, 0] && LtQ[m, -1]
  4029. Int[Times[Power[Times[csc[Plus[Optional[Pattern[e, Blank[]]], Times[Optional[Pattern[f, Blank[]]], Pattern[x, Blank[]]]]], Optional[Pattern[d, Blank[]]]], Pattern[n, Blank[]]], Power[Plus[Times[csc[Plus[Optional[Pattern[e, Blank[]]], Times[Optional[Pattern[f, Blank[]]], Pattern[x, Blank[]]]]], Optional[Pattern[b, Blank[]]]], Pattern[a, Blank[]]], Pattern[m, Blank[]]], Plus[Times[csc[Plus[Optional[Pattern[e, Blank[]]], Times[Optional[Pattern[f, Blank[]]], Pattern[x, Blank[]]]]], Optional[Pattern[B, Blank[]]]], Pattern[A, Blank[]]]], Pattern[x, Blank[Symbol]]] := Simp[(a*d^2*(A*b - a*B)*Cot[e + f*x]*(a + b*Csc[e + f*x])^(m + 1)*(d*Csc[e + f*x])^(n - 2))/(b*f*(m + 1)*(a^2 - b^2)), x] - Dist[d/(b*(m + 1)*(a^2 - b^2)), Int[(a + b*Csc[e + f*x])^(m + 1)*(d*Csc[e + f*x])^(n - 2)*Simp[a*d*(A*b - a*B)*(n - 2) + b*d*(A*b - a*B)*(m + 1)*Csc[e + f*x] - (a*A*b*d*(m + n) - d*B*(a^2*(n - 1) + b^2*(m + 1)))*Csc[e + f*x]^2, x], x], x] /; FreeQ[{a, b, d, e, f, A, B}, x] && NeQ[A*b - a*B, 0] && NeQ[a^2 - b^2, 0] && LtQ[m, -1] && GtQ[n, 1]
  4030. Int[Times[Power[Times[csc[Plus[Optional[Pattern[e, Blank[]]], Times[Optional[Pattern[f, Blank[]]], Pattern[x, Blank[]]]]], Optional[Pattern[d, Blank[]]]], Pattern[n, Blank[]]], Power[Plus[Times[csc[Plus[Optional[Pattern[e, Blank[]]], Times[Optional[Pattern[f, Blank[]]], Pattern[x, Blank[]]]]], Optional[Pattern[b, Blank[]]]], Pattern[a, Blank[]]], Pattern[m, Blank[]]], Plus[Times[csc[Plus[Optional[Pattern[e, Blank[]]], Times[Optional[Pattern[f, Blank[]]], Pattern[x, Blank[]]]]], Optional[Pattern[B, Blank[]]]], Pattern[A, Blank[]]]], Pattern[x, Blank[Symbol]]] := Simp[(b*(A*b - a*B)*Cot[e + f*x]*(a + b*Csc[e + f*x])^(m + 1)*(d*Csc[e + f*x])^n)/(a*f*(m + 1)*(a^2 - b^2)), x] + Dist[1/(a*(m + 1)*(a^2 - b^2)), Int[(a + b*Csc[e + f*x])^(m + 1)*(d*Csc[e + f*x])^n*Simp[A*(a^2*(m + 1) - b^2*(m + n + 1)) + a*b*B*n - a*(A*b - a*B)*(m + 1)*Csc[e + f*x] + b*(A*b - a*B)*(m + n + 2)*Csc[e + f*x]^2, x], x], x] /; FreeQ[{a, b, d, e, f, A, B, n}, x] && NeQ[A*b - a*B, 0] && NeQ[a^2 - b^2, 0] && LtQ[m, -1] && !(ILtQ[m + 1/2, 0] && ILtQ[n, 0])
  4031. Int[Times[Power[Times[csc[Plus[Optional[Pattern[e, Blank[]]], Times[Optional[Pattern[f, Blank[]]], Pattern[x, Blank[]]]]], Optional[Pattern[d, Blank[]]]], Pattern[n, Blank[]]], Power[Plus[Times[csc[Plus[Optional[Pattern[e, Blank[]]], Times[Optional[Pattern[f, Blank[]]], Pattern[x, Blank[]]]]], Optional[Pattern[b, Blank[]]]], Pattern[a, Blank[]]], Pattern[m, Blank[]]], Plus[Times[csc[Plus[Optional[Pattern[e, Blank[]]], Times[Optional[Pattern[f, Blank[]]], Pattern[x, Blank[]]]]], Optional[Pattern[B, Blank[]]]], Pattern[A, Blank[]]]], Pattern[x, Blank[Symbol]]] := -Simp[(B*d*Cot[e + f*x]*(a + b*Csc[e + f*x])^m*(d*Csc[e + f*x])^(n - 1))/(f*(m + n)), x] + Dist[d/(m + n), Int[(a + b*Csc[e + f*x])^(m - 1)*(d*Csc[e + f*x])^(n - 1)*Simp[a*B*(n - 1) + (b*B*(m + n - 1) + a*A*(m + n))*Csc[e + f*x] + (a*B*m + A*b*(m + n))*Csc[e + f*x]^2, x], x], x] /; FreeQ[{a, b, d, e, f, A, B}, x] && NeQ[A*b - a*B, 0] && NeQ[a^2 - b^2, 0] && LtQ[0, m, 1] && GtQ[n, 0]
  4032. Int[Times[Power[Times[csc[Plus[Optional[Pattern[e, Blank[]]], Times[Optional[Pattern[f, Blank[]]], Pattern[x, Blank[]]]]], Optional[Pattern[d, Blank[]]]], Pattern[n, Blank[]]], Power[Plus[Times[csc[Plus[Optional[Pattern[e, Blank[]]], Times[Optional[Pattern[f, Blank[]]], Pattern[x, Blank[]]]]], Optional[Pattern[b, Blank[]]]], Pattern[a, Blank[]]], Pattern[m, Blank[]]], Plus[Times[csc[Plus[Optional[Pattern[e, Blank[]]], Times[Optional[Pattern[f, Blank[]]], Pattern[x, Blank[]]]]], Optional[Pattern[B, Blank[]]]], Pattern[A, Blank[]]]], Pattern[x, Blank[Symbol]]] := Simp[(A*Cot[e + f*x]*(a + b*Csc[e + f*x])^m*(d*Csc[e + f*x])^n)/(f*n), x] - Dist[1/(d*n), Int[(a + b*Csc[e + f*x])^(m - 1)*(d*Csc[e + f*x])^(n + 1)*Simp[A*b*m - a*B*n - (b*B*n + a*A*(n + 1))*Csc[e + f*x] - A*b*(m + n + 1)*Csc[e + f*x]^2, x], x], x] /; FreeQ[{a, b, d, e, f, A, B}, x] && NeQ[A*b - a*B, 0] && NeQ[a^2 - b^2, 0] && LtQ[0, m, 1] && LeQ[n, -1]
  4033. Int[Times[Power[Times[csc[Plus[Optional[Pattern[e, Blank[]]], Times[Optional[Pattern[f, Blank[]]], Pattern[x, Blank[]]]]], Optional[Pattern[d, Blank[]]]], Pattern[n, Blank[]]], Power[Plus[Times[csc[Plus[Optional[Pattern[e, Blank[]]], Times[Optional[Pattern[f, Blank[]]], Pattern[x, Blank[]]]]], Optional[Pattern[b, Blank[]]]], Pattern[a, Blank[]]], Pattern[m, Blank[]]], Plus[Times[csc[Plus[Optional[Pattern[e, Blank[]]], Times[Optional[Pattern[f, Blank[]]], Pattern[x, Blank[]]]]], Optional[Pattern[B, Blank[]]]], Pattern[A, Blank[]]]], Pattern[x, Blank[Symbol]]] := -Simp[(B*d^2*Cot[e + f*x]*(a + b*Csc[e + f*x])^(m + 1)*(d*Csc[e + f*x])^(n - 2))/(b*f*(m + n)), x] + Dist[d^2/(b*(m + n)), Int[(a + b*Csc[e + f*x])^m*(d*Csc[e + f*x])^(n - 2)*Simp[a*B*(n - 2) + B*b*(m + n - 1)*Csc[e + f*x] + (A*b*(m + n) - a*B*(n - 1))*Csc[e + f*x]^2, x], x], x] /; FreeQ[{a, b, d, e, f, A, B, m}, x] && NeQ[A*b - a*B, 0] && NeQ[a^2 - b^2, 0] && GtQ[n, 1] && NeQ[m + n, 0] && !IGtQ[m, 1]
  4034. Int[Times[Power[Times[csc[Plus[Optional[Pattern[e, Blank[]]], Times[Optional[Pattern[f, Blank[]]], Pattern[x, Blank[]]]]], Optional[Pattern[d, Blank[]]]], Pattern[n, Blank[]]], Power[Plus[Times[csc[Plus[Optional[Pattern[e, Blank[]]], Times[Optional[Pattern[f, Blank[]]], Pattern[x, Blank[]]]]], Optional[Pattern[b, Blank[]]]], Pattern[a, Blank[]]], Pattern[m, Blank[]]], Plus[Times[csc[Plus[Optional[Pattern[e, Blank[]]], Times[Optional[Pattern[f, Blank[]]], Pattern[x, Blank[]]]]], Optional[Pattern[B, Blank[]]]], Pattern[A, Blank[]]]], Pattern[x, Blank[Symbol]]] := Simp[(A*Cot[e + f*x]*(a + b*Csc[e + f*x])^(m + 1)*(d*Csc[e + f*x])^n)/(a*f*n), x] + Dist[1/(a*d*n), Int[(a + b*Csc[e + f*x])^m*(d*Csc[e + f*x])^(n + 1)*Simp[a*B*n - A*b*(m + n + 1) + A*a*(n + 1)*Csc[e + f*x] + A*b*(m + n + 2)*Csc[e + f*x]^2, x], x], x] /; FreeQ[{a, b, d, e, f, A, B, m}, x] && NeQ[A*b - a*B, 0] && NeQ[a^2 - b^2, 0] && LeQ[n, -1]
  4035. Int[Times[Power[Times[csc[Plus[Optional[Pattern[e, Blank[]]], Times[Optional[Pattern[f, Blank[]]], Pattern[x, Blank[]]]]], Optional[Pattern[d, Blank[]]]], Rational[-1, 2]], Power[Plus[Times[csc[Plus[Optional[Pattern[e, Blank[]]], Times[Optional[Pattern[f, Blank[]]], Pattern[x, Blank[]]]]], Optional[Pattern[b, Blank[]]]], Pattern[a, Blank[]]], Rational[-1, 2]], Plus[Times[csc[Plus[Optional[Pattern[e, Blank[]]], Times[Optional[Pattern[f, Blank[]]], Pattern[x, Blank[]]]]], Optional[Pattern[B, Blank[]]]], Pattern[A, Blank[]]]], Pattern[x, Blank[Symbol]]] := Dist[A/a, Int[Sqrt[a + b*Csc[e + f*x]]/Sqrt[d*Csc[e + f*x]], x], x] - Dist[(A*b - a*B)/(a*d), Int[Sqrt[d*Csc[e + f*x]]/Sqrt[a + b*Csc[e + f*x]], x], x] /; FreeQ[{a, b, d, e, f, A, B}, x] && NeQ[A*b - a*B, 0] && NeQ[a^2 - b^2, 0]
  4036. Int[Times[Power[Times[csc[Plus[Optional[Pattern[e, Blank[]]], Times[Optional[Pattern[f, Blank[]]], Pattern[x, Blank[]]]]], Optional[Pattern[d, Blank[]]]], Rational[1, 2]], Power[Plus[Times[csc[Plus[Optional[Pattern[e, Blank[]]], Times[Optional[Pattern[f, Blank[]]], Pattern[x, Blank[]]]]], Optional[Pattern[b, Blank[]]]], Pattern[a, Blank[]]], Rational[-1, 2]], Plus[Times[csc[Plus[Optional[Pattern[e, Blank[]]], Times[Optional[Pattern[f, Blank[]]], Pattern[x, Blank[]]]]], Optional[Pattern[B, Blank[]]]], Pattern[A, Blank[]]]], Pattern[x, Blank[Symbol]]] := Dist[A, Int[Sqrt[d*Csc[e + f*x]]/Sqrt[a + b*Csc[e + f*x]], x], x] + Dist[B/d, Int[(d*Csc[e + f*x])^(3/2)/Sqrt[a + b*Csc[e + f*x]], x], x] /; FreeQ[{a, b, d, e, f, A, B}, x] && NeQ[A*b - a*B, 0] && NeQ[a^2 - b^2, 0]
  4037. Int[Times[Power[Times[csc[Plus[Optional[Pattern[e, Blank[]]], Times[Optional[Pattern[f, Blank[]]], Pattern[x, Blank[]]]]], Optional[Pattern[d, Blank[]]]], Rational[-1, 2]], Power[Plus[Times[csc[Plus[Optional[Pattern[e, Blank[]]], Times[Optional[Pattern[f, Blank[]]], Pattern[x, Blank[]]]]], Optional[Pattern[b, Blank[]]]], Pattern[a, Blank[]]], Rational[1, 2]], Plus[Times[csc[Plus[Optional[Pattern[e, Blank[]]], Times[Optional[Pattern[f, Blank[]]], Pattern[x, Blank[]]]]], Optional[Pattern[B, Blank[]]]], Pattern[A, Blank[]]]], Pattern[x, Blank[Symbol]]] := Dist[B/d, Int[Sqrt[a + b*Csc[e + f*x]]*Sqrt[d*Csc[e + f*x]], x], x] + Dist[A, Int[Sqrt[a + b*Csc[e + f*x]]/Sqrt[d*Csc[e + f*x]], x], x] /; FreeQ[{a, b, d, e, f, A, B}, x] && NeQ[A*b - a*B, 0] && NeQ[a^2 - b^2, 0]
  4038. Int[Times[Power[Times[csc[Plus[Optional[Pattern[e, Blank[]]], Times[Optional[Pattern[f, Blank[]]], Pattern[x, Blank[]]]]], Optional[Pattern[d, Blank[]]]], Pattern[n, Blank[]]], Power[Plus[Times[csc[Plus[Optional[Pattern[e, Blank[]]], Times[Optional[Pattern[f, Blank[]]], Pattern[x, Blank[]]]]], Optional[Pattern[b, Blank[]]]], Pattern[a, Blank[]]], -1], Plus[Times[csc[Plus[Optional[Pattern[e, Blank[]]], Times[Optional[Pattern[f, Blank[]]], Pattern[x, Blank[]]]]], Optional[Pattern[B, Blank[]]]], Pattern[A, Blank[]]]], Pattern[x, Blank[Symbol]]] := Dist[A/a, Int[(d*Csc[e + f*x])^n, x], x] - Dist[(A*b - a*B)/(a*d), Int[(d*Csc[e + f*x])^(n + 1)/(a + b*Csc[e + f*x]), x], x] /; FreeQ[{a, b, d, e, f, A, B, n}, x] && NeQ[A*b - a*B, 0] && NeQ[a^2 - b^2, 0]
  4039. Int[Times[Power[Times[csc[Plus[Optional[Pattern[e, Blank[]]], Times[Optional[Pattern[f, Blank[]]], Pattern[x, Blank[]]]]], Optional[Pattern[d, Blank[]]]], Optional[Pattern[n, Blank[]]]], Power[Plus[Times[csc[Plus[Optional[Pattern[e, Blank[]]], Times[Optional[Pattern[f, Blank[]]], Pattern[x, Blank[]]]]], Optional[Pattern[b, Blank[]]]], Pattern[a, Blank[]]], Pattern[m, Blank[]]], Plus[Times[csc[Plus[Optional[Pattern[e, Blank[]]], Times[Optional[Pattern[f, Blank[]]], Pattern[x, Blank[]]]]], Optional[Pattern[B, Blank[]]]], Pattern[A, Blank[]]]], Pattern[x, Blank[Symbol]]] := Unintegrable[(d*Csc[e + f*x])^n*(a + b*Csc[e + f*x])^m*(A + B*Csc[e + f*x]), x] /; FreeQ[{a, b, d, e, f, A, B, m, n}, x] && NeQ[A*b - a*B, 0] && NeQ[a^2 - b^2, 0]
  4040. Int[Times[Power[Plus[Optional[Pattern[A, Blank[]]], Times[csc[Plus[Optional[Pattern[e, Blank[]]], Times[Optional[Pattern[f, Blank[]]], Pattern[x, Blank[]]]]], Optional[Pattern[B, Blank[]]]]], Optional[Pattern[p, Blank[]]]], Power[Plus[Times[csc[Plus[Optional[Pattern[e, Blank[]]], Times[Optional[Pattern[f, Blank[]]], Pattern[x, Blank[]]]]], Optional[Pattern[b, Blank[]]]], Pattern[a, Blank[]]], Optional[Pattern[m, Blank[]]]], Power[Plus[Times[csc[Plus[Optional[Pattern[e, Blank[]]], Times[Optional[Pattern[f, Blank[]]], Pattern[x, Blank[]]]]], Optional[Pattern[d, Blank[]]]], Pattern[c, Blank[]]], Optional[Pattern[n, Blank[]]]]], Pattern[x, Blank[Symbol]]] := Dist[(-(a*c))^m, Int[(Cos[e + f*x]^(2*m)*(d + c*Sin[e + f*x])^(n - m)*(B + A*Sin[e + f*x])^p)/Sin[e + f*x]^(m + n + p), x], x] /; FreeQ[{a, b, c, d, e, f, A, B, n, p}, x] && EqQ[b*c + a*d, 0] && EqQ[a^2 - b^2, 0] && IntegersQ[m, n, p]
  4041. Int[Times[Plus[Optional[Pattern[A, Blank[]]], Times[csc[Plus[Optional[Pattern[e, Blank[]]], Times[Optional[Pattern[f, Blank[]]], Pattern[x, Blank[]]]]], Optional[Pattern[B, Blank[]]]], Times[Power[csc[Plus[Optional[Pattern[e, Blank[]]], Times[Optional[Pattern[f, Blank[]]], Pattern[x, Blank[]]]]], 2], Optional[Pattern[C, Blank[]]]]], Power[Plus[Times[csc[Plus[Optional[Pattern[e, Blank[]]], Times[Optional[Pattern[f, Blank[]]], Pattern[x, Blank[]]]]], Optional[Pattern[b, Blank[]]]], Pattern[a, Blank[]]], Optional[Pattern[m, Blank[]]]]], Pattern[x, Blank[Symbol]]] := Dist[1/b^2, Int[(a + b*Csc[e + f*x])^(m + 1)*Simp[b*B - a*C + b*C*Csc[e + f*x], x], x], x] /; FreeQ[{a, b, e, f, A, B, C, m}, x] && EqQ[A*b^2 - a*b*B + a^2*C, 0]
  4042. Int[Times[Plus[Optional[Pattern[A, Blank[]]], Times[Power[csc[Plus[Optional[Pattern[e, Blank[]]], Times[Optional[Pattern[f, Blank[]]], Pattern[x, Blank[]]]]], 2], Optional[Pattern[C, Blank[]]]]], Power[Plus[Times[csc[Plus[Optional[Pattern[e, Blank[]]], Times[Optional[Pattern[f, Blank[]]], Pattern[x, Blank[]]]]], Optional[Pattern[b, Blank[]]]], Pattern[a, Blank[]]], Optional[Pattern[m, Blank[]]]]], Pattern[x, Blank[Symbol]]] := Dist[C/b^2, Int[(a + b*Csc[e + f*x])^(m + 1)*Simp[-a + b*Csc[e + f*x], x], x], x] /; FreeQ[{a, b, e, f, A, C, m}, x] && EqQ[A*b^2 + a^2*C, 0]
  4043. Int[Times[Power[Times[csc[Plus[Optional[Pattern[e, Blank[]]], Times[Optional[Pattern[f, Blank[]]], Pattern[x, Blank[]]]]], Optional[Pattern[b, Blank[]]]], Optional[Pattern[m, Blank[]]]], Plus[Times[Power[csc[Plus[Optional[Pattern[e, Blank[]]], Times[Optional[Pattern[f, Blank[]]], Pattern[x, Blank[]]]]], 2], Optional[Pattern[C, Blank[]]]], Pattern[A, Blank[]]]], Pattern[x, Blank[Symbol]]] := Simp[(A*Cot[e + f*x]*(b*Csc[e + f*x])^m)/(f*m), x] /; FreeQ[{b, e, f, A, C, m}, x] && EqQ[C*m + A*(m + 1), 0]
  4044. Int[Times[Power[csc[Plus[Optional[Pattern[e, Blank[]]], Times[Optional[Pattern[f, Blank[]]], Pattern[x, Blank[]]]]], Optional[Pattern[m, Blank[]]]], Plus[Times[Power[csc[Plus[Optional[Pattern[e, Blank[]]], Times[Optional[Pattern[f, Blank[]]], Pattern[x, Blank[]]]]], 2], Optional[Pattern[C, Blank[]]]], Pattern[A, Blank[]]]], Pattern[x, Blank[Symbol]]] := Int[(C + A*Sin[e + f*x]^2)/Sin[e + f*x]^(m + 2), x] /; FreeQ[{e, f, A, C}, x] && NeQ[C*m + A*(m + 1), 0] && ILtQ[(m + 1)/2, 0]
  4045. Int[Times[Power[Times[csc[Plus[Optional[Pattern[e, Blank[]]], Times[Optional[Pattern[f, Blank[]]], Pattern[x, Blank[]]]]], Optional[Pattern[b, Blank[]]]], Optional[Pattern[m, Blank[]]]], Plus[Times[Power[csc[Plus[Optional[Pattern[e, Blank[]]], Times[Optional[Pattern[f, Blank[]]], Pattern[x, Blank[]]]]], 2], Optional[Pattern[C, Blank[]]]], Pattern[A, Blank[]]]], Pattern[x, Blank[Symbol]]] := Simp[(A*Cot[e + f*x]*(b*Csc[e + f*x])^m)/(f*m), x] + Dist[(C*m + A*(m + 1))/(b^2*m), Int[(b*Csc[e + f*x])^(m + 2), x], x] /; FreeQ[{b, e, f, A, C}, x] && NeQ[C*m + A*(m + 1), 0] && LeQ[m, -1]
  4046. Int[Times[Power[Times[csc[Plus[Optional[Pattern[e, Blank[]]], Times[Optional[Pattern[f, Blank[]]], Pattern[x, Blank[]]]]], Optional[Pattern[b, Blank[]]]], Optional[Pattern[m, Blank[]]]], Plus[Times[Power[csc[Plus[Optional[Pattern[e, Blank[]]], Times[Optional[Pattern[f, Blank[]]], Pattern[x, Blank[]]]]], 2], Optional[Pattern[C, Blank[]]]], Pattern[A, Blank[]]]], Pattern[x, Blank[Symbol]]] := -Simp[(C*Cot[e + f*x]*(b*Csc[e + f*x])^m)/(f*(m + 1)), x] + Dist[(C*m + A*(m + 1))/(m + 1), Int[(b*Csc[e + f*x])^m, x], x] /; FreeQ[{b, e, f, A, C, m}, x] && NeQ[C*m + A*(m + 1), 0] && !LeQ[m, -1]
  4047. Int[Times[Power[Times[csc[Plus[Optional[Pattern[e, Blank[]]], Times[Optional[Pattern[f, Blank[]]], Pattern[x, Blank[]]]]], Optional[Pattern[b, Blank[]]]], Optional[Pattern[m, Blank[]]]], Plus[Optional[Pattern[A, Blank[]]], Times[csc[Plus[Optional[Pattern[e, Blank[]]], Times[Optional[Pattern[f, Blank[]]], Pattern[x, Blank[]]]]], Optional[Pattern[B, Blank[]]]], Times[Power[csc[Plus[Optional[Pattern[e, Blank[]]], Times[Optional[Pattern[f, Blank[]]], Pattern[x, Blank[]]]]], 2], Optional[Pattern[C, Blank[]]]]]], Pattern[x, Blank[Symbol]]] := Dist[B/b, Int[(b*Csc[e + f*x])^(m + 1), x], x] + Int[(b*Csc[e + f*x])^m*(A + C*Csc[e + f*x]^2), x] /; FreeQ[{b, e, f, A, B, C, m}, x]
  4048. Int[Times[Plus[Optional[Pattern[A, Blank[]]], Times[csc[Plus[Optional[Pattern[e, Blank[]]], Times[Optional[Pattern[f, Blank[]]], Pattern[x, Blank[]]]]], Optional[Pattern[B, Blank[]]]], Times[Power[csc[Plus[Optional[Pattern[e, Blank[]]], Times[Optional[Pattern[f, Blank[]]], Pattern[x, Blank[]]]]], 2], Optional[Pattern[C, Blank[]]]]], Plus[Times[csc[Plus[Optional[Pattern[e, Blank[]]], Times[Optional[Pattern[f, Blank[]]], Pattern[x, Blank[]]]]], Optional[Pattern[b, Blank[]]]], Pattern[a, Blank[]]]], Pattern[x, Blank[Symbol]]] := -Simp[(b*C*Csc[e + f*x]*Cot[e + f*x])/(2*f), x] + Dist[1/2, Int[Simp[2*A*a + (2*B*a + b*(2*A + C))*Csc[e + f*x] + 2*(a*C + B*b)*Csc[e + f*x]^2, x], x], x] /; FreeQ[{a, b, e, f, A, B, C}, x]
  4049. Int[Times[Plus[Optional[Pattern[A, Blank[]]], Times[Power[csc[Plus[Optional[Pattern[e, Blank[]]], Times[Optional[Pattern[f, Blank[]]], Pattern[x, Blank[]]]]], 2], Optional[Pattern[C, Blank[]]]]], Plus[Times[csc[Plus[Optional[Pattern[e, Blank[]]], Times[Optional[Pattern[f, Blank[]]], Pattern[x, Blank[]]]]], Optional[Pattern[b, Blank[]]]], Pattern[a, Blank[]]]], Pattern[x, Blank[Symbol]]] := -Simp[(b*C*Csc[e + f*x]*Cot[e + f*x])/(2*f), x] + Dist[1/2, Int[Simp[2*A*a + b*(2*A + C)*Csc[e + f*x] + 2*a*C*Csc[e + f*x]^2, x], x], x] /; FreeQ[{a, b, e, f, A, C}, x]
  4050. Int[Times[Plus[Optional[Pattern[A, Blank[]]], Times[csc[Plus[Optional[Pattern[e, Blank[]]], Times[Optional[Pattern[f, Blank[]]], Pattern[x, Blank[]]]]], Optional[Pattern[B, Blank[]]]], Times[Power[csc[Plus[Optional[Pattern[e, Blank[]]], Times[Optional[Pattern[f, Blank[]]], Pattern[x, Blank[]]]]], 2], Optional[Pattern[C, Blank[]]]]], Power[Plus[Times[csc[Plus[Optional[Pattern[e, Blank[]]], Times[Optional[Pattern[f, Blank[]]], Pattern[x, Blank[]]]]], Optional[Pattern[b, Blank[]]]], Pattern[a, Blank[]]], -1]], Pattern[x, Blank[Symbol]]] := Dist[C/b, Int[Csc[e + f*x], x], x] + Dist[1/b, Int[(A*b + (b*B - a*C)*Csc[e + f*x])/(a + b*Csc[e + f*x]), x], x] /; FreeQ[{a, b, e, f, A, B, C}, x]
  4051. Int[Times[Plus[Optional[Pattern[A, Blank[]]], Times[Power[csc[Plus[Optional[Pattern[e, Blank[]]], Times[Optional[Pattern[f, Blank[]]], Pattern[x, Blank[]]]]], 2], Optional[Pattern[C, Blank[]]]]], Power[Plus[Times[csc[Plus[Optional[Pattern[e, Blank[]]], Times[Optional[Pattern[f, Blank[]]], Pattern[x, Blank[]]]]], Optional[Pattern[b, Blank[]]]], Pattern[a, Blank[]]], -1]], Pattern[x, Blank[Symbol]]] := Dist[C/b, Int[Csc[e + f*x], x], x] + Dist[1/b, Int[(A*b - a*C*Csc[e + f*x])/(a + b*Csc[e + f*x]), x], x] /; FreeQ[{a, b, e, f, A, C}, x]
  4052. Int[Times[Plus[Optional[Pattern[A, Blank[]]], Times[csc[Plus[Optional[Pattern[e, Blank[]]], Times[Optional[Pattern[f, Blank[]]], Pattern[x, Blank[]]]]], Optional[Pattern[B, Blank[]]]], Times[Power[csc[Plus[Optional[Pattern[e, Blank[]]], Times[Optional[Pattern[f, Blank[]]], Pattern[x, Blank[]]]]], 2], Optional[Pattern[C, Blank[]]]]], Power[Plus[Times[csc[Plus[Optional[Pattern[e, Blank[]]], Times[Optional[Pattern[f, Blank[]]], Pattern[x, Blank[]]]]], Optional[Pattern[b, Blank[]]]], Pattern[a, Blank[]]], Pattern[m, Blank[]]]], Pattern[x, Blank[Symbol]]] := -Simp[((a*A - b*B + a*C)*Cot[e + f*x]*(a + b*Csc[e + f*x])^m)/(a*f*(2*m + 1)), x] + Dist[1/(a*b*(2*m + 1)), Int[(a + b*Csc[e + f*x])^(m + 1)*Simp[A*b*(2*m + 1) + (b*B*(m + 1) - a*(A*(m + 1) - C*m))*Csc[e + f*x], x], x], x] /; FreeQ[{a, b, e, f, A, B, C}, x] && EqQ[a^2 - b^2, 0] && LtQ[m, -2^(-1)]
  4053. Int[Times[Plus[Optional[Pattern[A, Blank[]]], Times[Power[csc[Plus[Optional[Pattern[e, Blank[]]], Times[Optional[Pattern[f, Blank[]]], Pattern[x, Blank[]]]]], 2], Optional[Pattern[C, Blank[]]]]], Power[Plus[Times[csc[Plus[Optional[Pattern[e, Blank[]]], Times[Optional[Pattern[f, Blank[]]], Pattern[x, Blank[]]]]], Optional[Pattern[b, Blank[]]]], Pattern[a, Blank[]]], Pattern[m, Blank[]]]], Pattern[x, Blank[Symbol]]] := -Simp[(a*(A + C)*Cot[e + f*x]*(a + b*Csc[e + f*x])^m)/(a*f*(2*m + 1)), x] + Dist[1/(a*b*(2*m + 1)), Int[(a + b*Csc[e + f*x])^(m + 1)*Simp[A*b*(2*m + 1) - a*(A*(m + 1) - C*m)*Csc[e + f*x], x], x], x] /; FreeQ[{a, b, e, f, A, C}, x] && EqQ[a^2 - b^2, 0] && LtQ[m, -2^(-1)]
  4054. Int[Times[Plus[Optional[Pattern[A, Blank[]]], Times[csc[Plus[Optional[Pattern[e, Blank[]]], Times[Optional[Pattern[f, Blank[]]], Pattern[x, Blank[]]]]], Optional[Pattern[B, Blank[]]]], Times[Power[csc[Plus[Optional[Pattern[e, Blank[]]], Times[Optional[Pattern[f, Blank[]]], Pattern[x, Blank[]]]]], 2], Optional[Pattern[C, Blank[]]]]], Power[Plus[Times[csc[Plus[Optional[Pattern[e, Blank[]]], Times[Optional[Pattern[f, Blank[]]], Pattern[x, Blank[]]]]], Optional[Pattern[b, Blank[]]]], Pattern[a, Blank[]]], Optional[Pattern[m, Blank[]]]]], Pattern[x, Blank[Symbol]]] := -Simp[(C*Cot[e + f*x]*(a + b*Csc[e + f*x])^m)/(f*(m + 1)), x] + Dist[1/(b*(m + 1)), Int[(a + b*Csc[e + f*x])^m*Simp[A*b*(m + 1) + (a*C*m + b*B*(m + 1))*Csc[e + f*x], x], x], x] /; FreeQ[{a, b, e, f, A, B, C, m}, x] && EqQ[a^2 - b^2, 0] && !LtQ[m, -2^(-1)]
  4055. Int[Times[Plus[Optional[Pattern[A, Blank[]]], Times[Power[csc[Plus[Optional[Pattern[e, Blank[]]], Times[Optional[Pattern[f, Blank[]]], Pattern[x, Blank[]]]]], 2], Optional[Pattern[C, Blank[]]]]], Power[Plus[Times[csc[Plus[Optional[Pattern[e, Blank[]]], Times[Optional[Pattern[f, Blank[]]], Pattern[x, Blank[]]]]], Optional[Pattern[b, Blank[]]]], Pattern[a, Blank[]]], Optional[Pattern[m, Blank[]]]]], Pattern[x, Blank[Symbol]]] := -Simp[(C*Cot[e + f*x]*(a + b*Csc[e + f*x])^m)/(f*(m + 1)), x] + Dist[1/(b*(m + 1)), Int[(a + b*Csc[e + f*x])^m*Simp[A*b*(m + 1) + a*C*m*Csc[e + f*x], x], x], x] /; FreeQ[{a, b, e, f, A, C, m}, x] && EqQ[a^2 - b^2, 0] && !LtQ[m, -2^(-1)]
  4056. Int[Times[Plus[Optional[Pattern[A, Blank[]]], Times[csc[Plus[Optional[Pattern[e, Blank[]]], Times[Optional[Pattern[f, Blank[]]], Pattern[x, Blank[]]]]], Optional[Pattern[B, Blank[]]]], Times[Power[csc[Plus[Optional[Pattern[e, Blank[]]], Times[Optional[Pattern[f, Blank[]]], Pattern[x, Blank[]]]]], 2], Optional[Pattern[C, Blank[]]]]], Power[Plus[Times[csc[Plus[Optional[Pattern[e, Blank[]]], Times[Optional[Pattern[f, Blank[]]], Pattern[x, Blank[]]]]], Optional[Pattern[b, Blank[]]]], Pattern[a, Blank[]]], Optional[Pattern[m, Blank[]]]]], Pattern[x, Blank[Symbol]]] := -Simp[(C*Cot[e + f*x]*(a + b*Csc[e + f*x])^m)/(f*(m + 1)), x] + Dist[1/(m + 1), Int[(a + b*Csc[e + f*x])^(m - 1)*Simp[a*A*(m + 1) + ((A*b + a*B)*(m + 1) + b*C*m)*Csc[e + f*x] + (b*B*(m + 1) + a*C*m)*Csc[e + f*x]^2, x], x], x] /; FreeQ[{a, b, e, f, A, B, C}, x] && NeQ[a^2 - b^2, 0] && IGtQ[2*m, 0]
  4057. Int[Times[Plus[Optional[Pattern[A, Blank[]]], Times[Power[csc[Plus[Optional[Pattern[e, Blank[]]], Times[Optional[Pattern[f, Blank[]]], Pattern[x, Blank[]]]]], 2], Optional[Pattern[C, Blank[]]]]], Power[Plus[Times[csc[Plus[Optional[Pattern[e, Blank[]]], Times[Optional[Pattern[f, Blank[]]], Pattern[x, Blank[]]]]], Optional[Pattern[b, Blank[]]]], Pattern[a, Blank[]]], Optional[Pattern[m, Blank[]]]]], Pattern[x, Blank[Symbol]]] := -Simp[(C*Cot[e + f*x]*(a + b*Csc[e + f*x])^m)/(f*(m + 1)), x] + Dist[1/(m + 1), Int[(a + b*Csc[e + f*x])^(m - 1)*Simp[a*A*(m + 1) + (A*b*(m + 1) + b*C*m)*Csc[e + f*x] + a*C*m*Csc[e + f*x]^2, x], x], x] /; FreeQ[{a, b, e, f, A, C}, x] && NeQ[a^2 - b^2, 0] && IGtQ[2*m, 0]
  4058. Int[Times[Plus[Optional[Pattern[A, Blank[]]], Times[csc[Plus[Optional[Pattern[e, Blank[]]], Times[Optional[Pattern[f, Blank[]]], Pattern[x, Blank[]]]]], Optional[Pattern[B, Blank[]]]], Times[Power[csc[Plus[Optional[Pattern[e, Blank[]]], Times[Optional[Pattern[f, Blank[]]], Pattern[x, Blank[]]]]], 2], Optional[Pattern[C, Blank[]]]]], Power[Plus[Times[csc[Plus[Optional[Pattern[e, Blank[]]], Times[Optional[Pattern[f, Blank[]]], Pattern[x, Blank[]]]]], Optional[Pattern[b, Blank[]]]], Pattern[a, Blank[]]], Rational[-1, 2]]], Pattern[x, Blank[Symbol]]] := Int[(A + (B - C)*Csc[e + f*x])/Sqrt[a + b*Csc[e + f*x]], x] + Dist[C, Int[(Csc[e + f*x]*(1 + Csc[e + f*x]))/Sqrt[a + b*Csc[e + f*x]], x], x] /; FreeQ[{a, b, e, f, A, B, C}, x] && NeQ[a^2 - b^2, 0]
  4059. Int[Times[Plus[Optional[Pattern[A, Blank[]]], Times[Power[csc[Plus[Optional[Pattern[e, Blank[]]], Times[Optional[Pattern[f, Blank[]]], Pattern[x, Blank[]]]]], 2], Optional[Pattern[C, Blank[]]]]], Power[Plus[Times[csc[Plus[Optional[Pattern[e, Blank[]]], Times[Optional[Pattern[f, Blank[]]], Pattern[x, Blank[]]]]], Optional[Pattern[b, Blank[]]]], Pattern[a, Blank[]]], Rational[-1, 2]]], Pattern[x, Blank[Symbol]]] := Int[(A - C*Csc[e + f*x])/Sqrt[a + b*Csc[e + f*x]], x] + Dist[C, Int[(Csc[e + f*x]*(1 + Csc[e + f*x]))/Sqrt[a + b*Csc[e + f*x]], x], x] /; FreeQ[{a, b, e, f, A, C}, x] && NeQ[a^2 - b^2, 0]
  4060. Int[Times[Plus[Optional[Pattern[A, Blank[]]], Times[csc[Plus[Optional[Pattern[e, Blank[]]], Times[Optional[Pattern[f, Blank[]]], Pattern[x, Blank[]]]]], Optional[Pattern[B, Blank[]]]], Times[Power[csc[Plus[Optional[Pattern[e, Blank[]]], Times[Optional[Pattern[f, Blank[]]], Pattern[x, Blank[]]]]], 2], Optional[Pattern[C, Blank[]]]]], Power[Plus[Times[csc[Plus[Optional[Pattern[e, Blank[]]], Times[Optional[Pattern[f, Blank[]]], Pattern[x, Blank[]]]]], Optional[Pattern[b, Blank[]]]], Pattern[a, Blank[]]], Pattern[m, Blank[]]]], Pattern[x, Blank[Symbol]]] := Simp[((A*b^2 - a*b*B + a^2*C)*Cot[e + f*x]*(a + b*Csc[e + f*x])^(m + 1))/(a*f*(m + 1)*(a^2 - b^2)), x] + Dist[1/(a*(m + 1)*(a^2 - b^2)), Int[(a + b*Csc[e + f*x])^(m + 1)*Simp[A*(a^2 - b^2)*(m + 1) - a*(A*b - a*B + b*C)*(m + 1)*Csc[e + f*x] + (A*b^2 - a*b*B + a^2*C)*(m + 2)*Csc[e + f*x]^2, x], x], x] /; FreeQ[{a, b, e, f, A, B, C}, x] && NeQ[a^2 - b^2, 0] && LtQ[m, -1]
  4061. Int[Times[Plus[Optional[Pattern[A, Blank[]]], Times[Power[csc[Plus[Optional[Pattern[e, Blank[]]], Times[Optional[Pattern[f, Blank[]]], Pattern[x, Blank[]]]]], 2], Optional[Pattern[C, Blank[]]]]], Power[Plus[Times[csc[Plus[Optional[Pattern[e, Blank[]]], Times[Optional[Pattern[f, Blank[]]], Pattern[x, Blank[]]]]], Optional[Pattern[b, Blank[]]]], Pattern[a, Blank[]]], Pattern[m, Blank[]]]], Pattern[x, Blank[Symbol]]] := Simp[((A*b^2 + a^2*C)*Cot[e + f*x]*(a + b*Csc[e + f*x])^(m + 1))/(a*f*(m + 1)*(a^2 - b^2)), x] + Dist[1/(a*(m + 1)*(a^2 - b^2)), Int[(a + b*Csc[e + f*x])^(m + 1)*Simp[A*(a^2 - b^2)*(m + 1) - a*b*(A + C)*(m + 1)*Csc[e + f*x] + (A*b^2 + a^2*C)*(m + 2)*Csc[e + f*x]^2, x], x], x] /; FreeQ[{a, b, e, f, A, C}, x] && NeQ[a^2 - b^2, 0] && IntegerQ[2*m] && LtQ[m, -1]
  4062. Int[Times[Plus[Optional[Pattern[A, Blank[]]], Times[csc[Plus[Optional[Pattern[e, Blank[]]], Times[Optional[Pattern[f, Blank[]]], Pattern[x, Blank[]]]]], Optional[Pattern[B, Blank[]]]], Times[Power[csc[Plus[Optional[Pattern[e, Blank[]]], Times[Optional[Pattern[f, Blank[]]], Pattern[x, Blank[]]]]], 2], Optional[Pattern[C, Blank[]]]]], Power[Plus[Times[csc[Plus[Optional[Pattern[e, Blank[]]], Times[Optional[Pattern[f, Blank[]]], Pattern[x, Blank[]]]]], Optional[Pattern[b, Blank[]]]], Pattern[a, Blank[]]], Pattern[m, Blank[]]]], Pattern[x, Blank[Symbol]]] := Dist[1/b, Int[(a + b*Csc[e + f*x])^m*(A*b + (b*B - a*C)*Csc[e + f*x]), x], x] + Dist[C/b, Int[Csc[e + f*x]*(a + b*Csc[e + f*x])^(m + 1), x], x] /; FreeQ[{a, b, e, f, A, B, C, m}, x] && NeQ[a^2 - b^2, 0] && !IntegerQ[2*m]
  4063. Int[Times[Plus[Optional[Pattern[A, Blank[]]], Times[Power[csc[Plus[Optional[Pattern[e, Blank[]]], Times[Optional[Pattern[f, Blank[]]], Pattern[x, Blank[]]]]], 2], Optional[Pattern[C, Blank[]]]]], Power[Plus[Times[csc[Plus[Optional[Pattern[e, Blank[]]], Times[Optional[Pattern[f, Blank[]]], Pattern[x, Blank[]]]]], Optional[Pattern[b, Blank[]]]], Pattern[a, Blank[]]], Pattern[m, Blank[]]]], Pattern[x, Blank[Symbol]]] := Dist[1/b, Int[(a + b*Csc[e + f*x])^m*(A*b - a*C*Csc[e + f*x]), x], x] + Dist[C/b, Int[Csc[e + f*x]*(a + b*Csc[e + f*x])^(m + 1), x], x] /; FreeQ[{a, b, e, f, A, C, m}, x] && NeQ[a^2 - b^2, 0] && !IntegerQ[2*m]
  4064. Int[Times[Power[Times[cos[Plus[Optional[Pattern[e, Blank[]]], Times[Optional[Pattern[f, Blank[]]], Pattern[x, Blank[]]]]], Optional[Pattern[b, Blank[]]]], Pattern[m, Blank[]]], Plus[Optional[Pattern[A, Blank[]]], Times[Optional[Pattern[B, Blank[]]], sec[Plus[Optional[Pattern[e, Blank[]]], Times[Optional[Pattern[f, Blank[]]], Pattern[x, Blank[]]]]]], Times[Optional[Pattern[C, Blank[]]], Power[sec[Plus[Optional[Pattern[e, Blank[]]], Times[Optional[Pattern[f, Blank[]]], Pattern[x, Blank[]]]]], 2]]]], Pattern[x, Blank[Symbol]]] := Dist[b^2, Int[(b*Cos[e + f*x])^(m - 2)*(C + B*Cos[e + f*x] + A*Cos[e + f*x]^2), x], x] /; FreeQ[{b, e, f, A, B, C, m}, x] && !IntegerQ[m]
  4065. Int[Times[Plus[Optional[Pattern[A, Blank[]]], Times[csc[Plus[Optional[Pattern[e, Blank[]]], Times[Optional[Pattern[f, Blank[]]], Pattern[x, Blank[]]]]], Optional[Pattern[B, Blank[]]]], Times[Power[csc[Plus[Optional[Pattern[e, Blank[]]], Times[Optional[Pattern[f, Blank[]]], Pattern[x, Blank[]]]]], 2], Optional[Pattern[C, Blank[]]]]], Power[Times[Optional[Pattern[b, Blank[]]], sin[Plus[Optional[Pattern[e, Blank[]]], Times[Optional[Pattern[f, Blank[]]], Pattern[x, Blank[]]]]]], Pattern[m, Blank[]]]], Pattern[x, Blank[Symbol]]] := Dist[b^2, Int[(b*Sin[e + f*x])^(m - 2)*(C + B*Sin[e + f*x] + A*Sin[e + f*x]^2), x], x] /; FreeQ[{b, e, f, A, B, C, m}, x] && !IntegerQ[m]
  4066. Int[Times[Power[Times[cos[Plus[Optional[Pattern[e, Blank[]]], Times[Optional[Pattern[f, Blank[]]], Pattern[x, Blank[]]]]], Optional[Pattern[b, Blank[]]]], Pattern[m, Blank[]]], Plus[Optional[Pattern[A, Blank[]]], Times[Optional[Pattern[C, Blank[]]], Power[sec[Plus[Optional[Pattern[e, Blank[]]], Times[Optional[Pattern[f, Blank[]]], Pattern[x, Blank[]]]]], 2]]]], Pattern[x, Blank[Symbol]]] := Dist[b^2, Int[(b*Cos[e + f*x])^(m - 2)*(C + A*Cos[e + f*x]^2), x], x] /; FreeQ[{b, e, f, A, C, m}, x] && !IntegerQ[m]
  4067. Int[Times[Plus[Optional[Pattern[A, Blank[]]], Times[Power[csc[Plus[Optional[Pattern[e, Blank[]]], Times[Optional[Pattern[f, Blank[]]], Pattern[x, Blank[]]]]], 2], Optional[Pattern[C, Blank[]]]]], Power[Times[Optional[Pattern[b, Blank[]]], sin[Plus[Optional[Pattern[e, Blank[]]], Times[Optional[Pattern[f, Blank[]]], Pattern[x, Blank[]]]]]], Pattern[m, Blank[]]]], Pattern[x, Blank[Symbol]]] := Dist[b^2, Int[(b*Sin[e + f*x])^(m - 2)*(C + A*Sin[e + f*x]^2), x], x] /; FreeQ[{b, e, f, A, C, m}, x] && !IntegerQ[m]
  4068. Int[Times[Power[Times[Optional[Pattern[a, Blank[]]], Power[Times[Optional[Pattern[b, Blank[]]], sec[Plus[Optional[Pattern[e, Blank[]]], Times[Optional[Pattern[f, Blank[]]], Pattern[x, Blank[]]]]]], Pattern[p, Blank[]]]], Pattern[m, Blank[]]], Plus[Optional[Pattern[A, Blank[]]], Times[Optional[Pattern[B, Blank[]]], sec[Plus[Optional[Pattern[e, Blank[]]], Times[Optional[Pattern[f, Blank[]]], Pattern[x, Blank[]]]]]], Times[Optional[Pattern[C, Blank[]]], Power[sec[Plus[Optional[Pattern[e, Blank[]]], Times[Optional[Pattern[f, Blank[]]], Pattern[x, Blank[]]]]], 2]]]], Pattern[x, Blank[Symbol]]] := Dist[(a^IntPart[m]*(a*(b*Sec[e + f*x])^p)^FracPart[m])/(b*Sec[e + f*x])^(p*FracPart[m]), Int[(b*Sec[e + f*x])^(m*p)*(A + B*Sec[e + f*x] + C*Sec[e + f*x]^2), x], x] /; FreeQ[{a, b, e, f, A, B, C, m, p}, x] && !IntegerQ[m]
  4069. Int[Times[Power[Times[Optional[Pattern[a, Blank[]]], Power[Times[csc[Plus[Optional[Pattern[e, Blank[]]], Times[Optional[Pattern[f, Blank[]]], Pattern[x, Blank[]]]]], Optional[Pattern[b, Blank[]]]], Pattern[p, Blank[]]]], Pattern[m, Blank[]]], Plus[Optional[Pattern[A, Blank[]]], Times[csc[Plus[Optional[Pattern[e, Blank[]]], Times[Optional[Pattern[f, Blank[]]], Pattern[x, Blank[]]]]], Optional[Pattern[B, Blank[]]]], Times[Power[csc[Plus[Optional[Pattern[e, Blank[]]], Times[Optional[Pattern[f, Blank[]]], Pattern[x, Blank[]]]]], 2], Optional[Pattern[C, Blank[]]]]]], Pattern[x, Blank[Symbol]]] := Dist[(a^IntPart[m]*(a*(b*Csc[e + f*x])^p)^FracPart[m])/(b*Csc[e + f*x])^(p*FracPart[m]), Int[(b*Csc[e + f*x])^(m*p)*(A + B*Csc[e + f*x] + C*Csc[e + f*x]^2), x], x] /; FreeQ[{a, b, e, f, A, B, C, m, p}, x] && !IntegerQ[m]
  4070. Int[Times[Power[Times[Optional[Pattern[a, Blank[]]], Power[Times[Optional[Pattern[b, Blank[]]], sec[Plus[Optional[Pattern[e, Blank[]]], Times[Optional[Pattern[f, Blank[]]], Pattern[x, Blank[]]]]]], Pattern[p, Blank[]]]], Pattern[m, Blank[]]], Plus[Optional[Pattern[A, Blank[]]], Times[Optional[Pattern[C, Blank[]]], Power[sec[Plus[Optional[Pattern[e, Blank[]]], Times[Optional[Pattern[f, Blank[]]], Pattern[x, Blank[]]]]], 2]]]], Pattern[x, Blank[Symbol]]] := Dist[(a^IntPart[m]*(a*(b*Sec[e + f*x])^p)^FracPart[m])/(b*Sec[e + f*x])^(p*FracPart[m]), Int[(b*Sec[e + f*x])^(m*p)*(A + C*Sec[e + f*x]^2), x], x] /; FreeQ[{a, b, e, f, A, C, m, p}, x] && !IntegerQ[m]
  4071. Int[Times[Power[Times[Optional[Pattern[a, Blank[]]], Power[Times[csc[Plus[Optional[Pattern[e, Blank[]]], Times[Optional[Pattern[f, Blank[]]], Pattern[x, Blank[]]]]], Optional[Pattern[b, Blank[]]]], Pattern[p, Blank[]]]], Pattern[m, Blank[]]], Plus[Optional[Pattern[A, Blank[]]], Times[Power[csc[Plus[Optional[Pattern[e, Blank[]]], Times[Optional[Pattern[f, Blank[]]], Pattern[x, Blank[]]]]], 2], Optional[Pattern[C, Blank[]]]]]], Pattern[x, Blank[Symbol]]] := Dist[(a^IntPart[m]*(a*(b*Csc[e + f*x])^p)^FracPart[m])/(b*Csc[e + f*x])^(p*FracPart[m]), Int[(b*Csc[e + f*x])^(m*p)*(A + C*Csc[e + f*x]^2), x], x] /; FreeQ[{a, b, e, f, A, C, m, p}, x] && !IntegerQ[m]
  4072. Int[Times[Power[Plus[Optional[Pattern[a, Blank[]]], Times[csc[Plus[Optional[Pattern[e, Blank[]]], Times[Optional[Pattern[f, Blank[]]], Pattern[x, Blank[]]]]], Optional[Pattern[b, Blank[]]]]], Optional[Pattern[m, Blank[]]]], Plus[Optional[Pattern[A, Blank[]]], Times[csc[Plus[Optional[Pattern[e, Blank[]]], Times[Optional[Pattern[f, Blank[]]], Pattern[x, Blank[]]]]], Optional[Pattern[B, Blank[]]]], Times[Power[csc[Plus[Optional[Pattern[e, Blank[]]], Times[Optional[Pattern[f, Blank[]]], Pattern[x, Blank[]]]]], 2], Optional[Pattern[C, Blank[]]]]], Power[Plus[Optional[Pattern[c, Blank[]]], Times[csc[Plus[Optional[Pattern[e, Blank[]]], Times[Optional[Pattern[f, Blank[]]], Pattern[x, Blank[]]]]], Optional[Pattern[d, Blank[]]]]], Optional[Pattern[n, Blank[]]]]], Pattern[x, Blank[Symbol]]] := Dist[1/b^2, Int[(a + b*Csc[e + f*x])^(m + 1)*(c + d*Csc[e + f*x])^n*(b*B - a*C + b*C*Csc[e + f*x]), x], x] /; FreeQ[{a, b, c, d, e, f, A, B, C, m, n}, x] && EqQ[A*b^2 - a*b*B + a^2*C, 0]
  4073. Int[Times[Power[Plus[Optional[Pattern[a, Blank[]]], Times[csc[Plus[Optional[Pattern[e, Blank[]]], Times[Optional[Pattern[f, Blank[]]], Pattern[x, Blank[]]]]], Optional[Pattern[b, Blank[]]]]], Optional[Pattern[m, Blank[]]]], Plus[Optional[Pattern[A, Blank[]]], Times[Power[csc[Plus[Optional[Pattern[e, Blank[]]], Times[Optional[Pattern[f, Blank[]]], Pattern[x, Blank[]]]]], 2], Optional[Pattern[C, Blank[]]]]], Power[Plus[Optional[Pattern[c, Blank[]]], Times[csc[Plus[Optional[Pattern[e, Blank[]]], Times[Optional[Pattern[f, Blank[]]], Pattern[x, Blank[]]]]], Optional[Pattern[d, Blank[]]]]], Optional[Pattern[n, Blank[]]]]], Pattern[x, Blank[Symbol]]] := -Dist[C/b^2, Int[(a + b*Csc[e + f*x])^(m + 1)*(c + d*Csc[e + f*x])^n*(a - b*Csc[e + f*x]), x], x] /; FreeQ[{a, b, c, d, e, f, A, C, m, n}, x] && EqQ[A*b^2 + a^2*C, 0]
  4074. Int[Times[Plus[Optional[Pattern[A, Blank[]]], Times[csc[Plus[Optional[Pattern[e, Blank[]]], Times[Optional[Pattern[f, Blank[]]], Pattern[x, Blank[]]]]], Optional[Pattern[B, Blank[]]]], Times[Power[csc[Plus[Optional[Pattern[e, Blank[]]], Times[Optional[Pattern[f, Blank[]]], Pattern[x, Blank[]]]]], 2], Optional[Pattern[C, Blank[]]]]], Power[Times[csc[Plus[Optional[Pattern[e, Blank[]]], Times[Optional[Pattern[f, Blank[]]], Pattern[x, Blank[]]]]], Optional[Pattern[d, Blank[]]]], Pattern[n, Blank[]]], Plus[Times[csc[Plus[Optional[Pattern[e, Blank[]]], Times[Optional[Pattern[f, Blank[]]], Pattern[x, Blank[]]]]], Optional[Pattern[b, Blank[]]]], Pattern[a, Blank[]]]], Pattern[x, Blank[Symbol]]] := Simp[(A*a*Cot[e + f*x]*(d*Csc[e + f*x])^n)/(f*n), x] + Dist[1/(d*n), Int[(d*Csc[e + f*x])^(n + 1)*Simp[n*(B*a + A*b) + (n*(a*C + B*b) + A*a*(n + 1))*Csc[e + f*x] + b*C*n*Csc[e + f*x]^2, x], x], x] /; FreeQ[{a, b, d, e, f, A, B, C}, x] && LtQ[n, -1]
  4075. Int[Times[Plus[Optional[Pattern[A, Blank[]]], Times[Power[csc[Plus[Optional[Pattern[e, Blank[]]], Times[Optional[Pattern[f, Blank[]]], Pattern[x, Blank[]]]]], 2], Optional[Pattern[C, Blank[]]]]], Power[Times[csc[Plus[Optional[Pattern[e, Blank[]]], Times[Optional[Pattern[f, Blank[]]], Pattern[x, Blank[]]]]], Optional[Pattern[d, Blank[]]]], Pattern[n, Blank[]]], Plus[Times[csc[Plus[Optional[Pattern[e, Blank[]]], Times[Optional[Pattern[f, Blank[]]], Pattern[x, Blank[]]]]], Optional[Pattern[b, Blank[]]]], Pattern[a, Blank[]]]], Pattern[x, Blank[Symbol]]] := Simp[(A*a*Cot[e + f*x]*(d*Csc[e + f*x])^n)/(f*n), x] + Dist[1/(d*n), Int[(d*Csc[e + f*x])^(n + 1)*Simp[A*b*n + a*(C*n + A*(n + 1))*Csc[e + f*x] + b*C*n*Csc[e + f*x]^2, x], x], x] /; FreeQ[{a, b, d, e, f, A, C}, x] && LtQ[n, -1]
  4076. Int[Times[Plus[Optional[Pattern[A, Blank[]]], Times[csc[Plus[Optional[Pattern[e, Blank[]]], Times[Optional[Pattern[f, Blank[]]], Pattern[x, Blank[]]]]], Optional[Pattern[B, Blank[]]]], Times[Power[csc[Plus[Optional[Pattern[e, Blank[]]], Times[Optional[Pattern[f, Blank[]]], Pattern[x, Blank[]]]]], 2], Optional[Pattern[C, Blank[]]]]], Power[Times[csc[Plus[Optional[Pattern[e, Blank[]]], Times[Optional[Pattern[f, Blank[]]], Pattern[x, Blank[]]]]], Optional[Pattern[d, Blank[]]]], Optional[Pattern[n, Blank[]]]], Plus[Times[csc[Plus[Optional[Pattern[e, Blank[]]], Times[Optional[Pattern[f, Blank[]]], Pattern[x, Blank[]]]]], Optional[Pattern[b, Blank[]]]], Pattern[a, Blank[]]]], Pattern[x, Blank[Symbol]]] := -Simp[(b*C*Csc[e + f*x]*Cot[e + f*x]*(d*Csc[e + f*x])^n)/(f*(n + 2)), x] + Dist[1/(n + 2), Int[(d*Csc[e + f*x])^n*Simp[A*a*(n + 2) + (B*a*(n + 2) + b*(C*(n + 1) + A*(n + 2)))*Csc[e + f*x] + (a*C + B*b)*(n + 2)*Csc[e + f*x]^2, x], x], x] /; FreeQ[{a, b, d, e, f, A, B, C, n}, x] && !LtQ[n, -1]
  4077. Int[Times[Plus[Optional[Pattern[A, Blank[]]], Times[Power[csc[Plus[Optional[Pattern[e, Blank[]]], Times[Optional[Pattern[f, Blank[]]], Pattern[x, Blank[]]]]], 2], Optional[Pattern[C, Blank[]]]]], Power[Times[csc[Plus[Optional[Pattern[e, Blank[]]], Times[Optional[Pattern[f, Blank[]]], Pattern[x, Blank[]]]]], Optional[Pattern[d, Blank[]]]], Optional[Pattern[n, Blank[]]]], Plus[Times[csc[Plus[Optional[Pattern[e, Blank[]]], Times[Optional[Pattern[f, Blank[]]], Pattern[x, Blank[]]]]], Optional[Pattern[b, Blank[]]]], Pattern[a, Blank[]]]], Pattern[x, Blank[Symbol]]] := -Simp[(b*C*Csc[e + f*x]*Cot[e + f*x]*(d*Csc[e + f*x])^n)/(f*(n + 2)), x] + Dist[1/(n + 2), Int[(d*Csc[e + f*x])^n*Simp[A*a*(n + 2) + b*(C*(n + 1) + A*(n + 2))*Csc[e + f*x] + a*C*(n + 2)*Csc[e + f*x]^2, x], x], x] /; FreeQ[{a, b, d, e, f, A, C, n}, x] && !LtQ[n, -1]
  4078. Int[Times[csc[Plus[Optional[Pattern[e, Blank[]]], Times[Optional[Pattern[f, Blank[]]], Pattern[x, Blank[]]]]], Plus[Optional[Pattern[A, Blank[]]], Times[csc[Plus[Optional[Pattern[e, Blank[]]], Times[Optional[Pattern[f, Blank[]]], Pattern[x, Blank[]]]]], Optional[Pattern[B, Blank[]]]], Times[Power[csc[Plus[Optional[Pattern[e, Blank[]]], Times[Optional[Pattern[f, Blank[]]], Pattern[x, Blank[]]]]], 2], Optional[Pattern[C, Blank[]]]]], Power[Plus[Times[csc[Plus[Optional[Pattern[e, Blank[]]], Times[Optional[Pattern[f, Blank[]]], Pattern[x, Blank[]]]]], Optional[Pattern[b, Blank[]]]], Pattern[a, Blank[]]], Pattern[m, Blank[]]]], Pattern[x, Blank[Symbol]]] := -Simp[((a*A - b*B + a*C)*Cot[e + f*x]*Csc[e + f*x]*(a + b*Csc[e + f*x])^m)/(a*f*(2*m + 1)), x] - Dist[1/(a*b*(2*m + 1)), Int[Csc[e + f*x]*(a + b*Csc[e + f*x])^(m + 1)*Simp[a*B - b*C - 2*A*b*(m + 1) - (b*B*(m + 2) - a*(A*(m + 2) - C*(m - 1)))*Csc[e + f*x], x], x], x] /; FreeQ[{a, b, e, f, A, B, C}, x] && LtQ[m, -1] && EqQ[a^2 - b^2, 0]
  4079. Int[Times[csc[Plus[Optional[Pattern[e, Blank[]]], Times[Optional[Pattern[f, Blank[]]], Pattern[x, Blank[]]]]], Plus[Optional[Pattern[A, Blank[]]], Times[Power[csc[Plus[Optional[Pattern[e, Blank[]]], Times[Optional[Pattern[f, Blank[]]], Pattern[x, Blank[]]]]], 2], Optional[Pattern[C, Blank[]]]]], Power[Plus[Times[csc[Plus[Optional[Pattern[e, Blank[]]], Times[Optional[Pattern[f, Blank[]]], Pattern[x, Blank[]]]]], Optional[Pattern[b, Blank[]]]], Pattern[a, Blank[]]], Pattern[m, Blank[]]]], Pattern[x, Blank[Symbol]]] := -Simp[((A + C)*Cot[e + f*x]*Csc[e + f*x]*(a + b*Csc[e + f*x])^m)/(f*(2*m + 1)), x] - Dist[1/(a*b*(2*m + 1)), Int[Csc[e + f*x]*(a + b*Csc[e + f*x])^(m + 1)*Simp[-(b*C) - 2*A*b*(m + 1) + a*(A*(m + 2) - C*(m - 1))*Csc[e + f*x], x], x], x] /; FreeQ[{a, b, e, f, A, C}, x] && LtQ[m, -1] && EqQ[a^2 - b^2, 0]
  4080. Int[Times[csc[Plus[Optional[Pattern[e, Blank[]]], Times[Optional[Pattern[f, Blank[]]], Pattern[x, Blank[]]]]], Plus[Optional[Pattern[A, Blank[]]], Times[csc[Plus[Optional[Pattern[e, Blank[]]], Times[Optional[Pattern[f, Blank[]]], Pattern[x, Blank[]]]]], Optional[Pattern[B, Blank[]]]], Times[Power[csc[Plus[Optional[Pattern[e, Blank[]]], Times[Optional[Pattern[f, Blank[]]], Pattern[x, Blank[]]]]], 2], Optional[Pattern[C, Blank[]]]]], Power[Plus[Times[csc[Plus[Optional[Pattern[e, Blank[]]], Times[Optional[Pattern[f, Blank[]]], Pattern[x, Blank[]]]]], Optional[Pattern[b, Blank[]]]], Pattern[a, Blank[]]], Pattern[m, Blank[]]]], Pattern[x, Blank[Symbol]]] := -Simp[((A*b^2 - a*b*B + a^2*C)*Cot[e + f*x]*(a + b*Csc[e + f*x])^(m + 1))/(b*f*(m + 1)*(a^2 - b^2)), x] + Dist[1/(b*(m + 1)*(a^2 - b^2)), Int[Csc[e + f*x]*(a + b*Csc[e + f*x])^(m + 1)*Simp[b*(a*A - b*B + a*C)*(m + 1) - (A*b^2 - a*b*B + a^2*C + b*(A*b - a*B + b*C)*(m + 1))*Csc[e + f*x], x], x], x] /; FreeQ[{a, b, e, f, A, B, C}, x] && LtQ[m, -1] && NeQ[a^2 - b^2, 0]
  4081. Int[Times[csc[Plus[Optional[Pattern[e, Blank[]]], Times[Optional[Pattern[f, Blank[]]], Pattern[x, Blank[]]]]], Plus[Optional[Pattern[A, Blank[]]], Times[Power[csc[Plus[Optional[Pattern[e, Blank[]]], Times[Optional[Pattern[f, Blank[]]], Pattern[x, Blank[]]]]], 2], Optional[Pattern[C, Blank[]]]]], Power[Plus[Times[csc[Plus[Optional[Pattern[e, Blank[]]], Times[Optional[Pattern[f, Blank[]]], Pattern[x, Blank[]]]]], Optional[Pattern[b, Blank[]]]], Pattern[a, Blank[]]], Pattern[m, Blank[]]]], Pattern[x, Blank[Symbol]]] := -Simp[((A*b^2 + a^2*C)*Cot[e + f*x]*(a + b*Csc[e + f*x])^(m + 1))/(b*f*(m + 1)*(a^2 - b^2)), x] + Dist[1/(b*(m + 1)*(a^2 - b^2)), Int[Csc[e + f*x]*(a + b*Csc[e + f*x])^(m + 1)*Simp[a*b*(A + C)*(m + 1) - (A*b^2 + a^2*C + b*(A*b + b*C)*(m + 1))*Csc[e + f*x], x], x], x] /; FreeQ[{a, b, e, f, A, C}, x] && LtQ[m, -1] && NeQ[a^2 - b^2, 0]
  4082. Int[Times[csc[Plus[Optional[Pattern[e, Blank[]]], Times[Optional[Pattern[f, Blank[]]], Pattern[x, Blank[]]]]], Plus[Optional[Pattern[A, Blank[]]], Times[csc[Plus[Optional[Pattern[e, Blank[]]], Times[Optional[Pattern[f, Blank[]]], Pattern[x, Blank[]]]]], Optional[Pattern[B, Blank[]]]], Times[Power[csc[Plus[Optional[Pattern[e, Blank[]]], Times[Optional[Pattern[f, Blank[]]], Pattern[x, Blank[]]]]], 2], Optional[Pattern[C, Blank[]]]]], Power[Plus[Times[csc[Plus[Optional[Pattern[e, Blank[]]], Times[Optional[Pattern[f, Blank[]]], Pattern[x, Blank[]]]]], Optional[Pattern[b, Blank[]]]], Pattern[a, Blank[]]], Pattern[m, Blank[]]]], Pattern[x, Blank[Symbol]]] := -Simp[(C*Cot[e + f*x]*(a + b*Csc[e + f*x])^(m + 1))/(b*f*(m + 2)), x] + Dist[1/(b*(m + 2)), Int[Csc[e + f*x]*(a + b*Csc[e + f*x])^m*Simp[b*A*(m + 2) + b*C*(m + 1) + (b*B*(m + 2) - a*C)*Csc[e + f*x], x], x], x] /; FreeQ[{a, b, e, f, A, B, C, m}, x] && !LtQ[m, -1]
  4083. Int[Times[csc[Plus[Optional[Pattern[e, Blank[]]], Times[Optional[Pattern[f, Blank[]]], Pattern[x, Blank[]]]]], Plus[Optional[Pattern[A, Blank[]]], Times[Power[csc[Plus[Optional[Pattern[e, Blank[]]], Times[Optional[Pattern[f, Blank[]]], Pattern[x, Blank[]]]]], 2], Optional[Pattern[C, Blank[]]]]], Power[Plus[Times[csc[Plus[Optional[Pattern[e, Blank[]]], Times[Optional[Pattern[f, Blank[]]], Pattern[x, Blank[]]]]], Optional[Pattern[b, Blank[]]]], Pattern[a, Blank[]]], Pattern[m, Blank[]]]], Pattern[x, Blank[Symbol]]] := -Simp[(C*Cot[e + f*x]*(a + b*Csc[e + f*x])^(m + 1))/(b*f*(m + 2)), x] + Dist[1/(b*(m + 2)), Int[Csc[e + f*x]*(a + b*Csc[e + f*x])^m*Simp[b*A*(m + 2) + b*C*(m + 1) - a*C*Csc[e + f*x], x], x], x] /; FreeQ[{a, b, e, f, A, C, m}, x] && !LtQ[m, -1]
  4084. Int[Times[Plus[Optional[Pattern[A, Blank[]]], Times[csc[Plus[Optional[Pattern[e, Blank[]]], Times[Optional[Pattern[f, Blank[]]], Pattern[x, Blank[]]]]], Optional[Pattern[B, Blank[]]]], Times[Power[csc[Plus[Optional[Pattern[e, Blank[]]], Times[Optional[Pattern[f, Blank[]]], Pattern[x, Blank[]]]]], 2], Optional[Pattern[C, Blank[]]]]], Power[Times[csc[Plus[Optional[Pattern[e, Blank[]]], Times[Optional[Pattern[f, Blank[]]], Pattern[x, Blank[]]]]], Optional[Pattern[d, Blank[]]]], Pattern[n, Blank[]]], Power[Plus[Times[csc[Plus[Optional[Pattern[e, Blank[]]], Times[Optional[Pattern[f, Blank[]]], Pattern[x, Blank[]]]]], Optional[Pattern[b, Blank[]]]], Pattern[a, Blank[]]], Pattern[m, Blank[]]]], Pattern[x, Blank[Symbol]]] := -Simp[((a*A - b*B + a*C)*Cot[e + f*x]*(a + b*Csc[e + f*x])^m*(d*Csc[e + f*x])^n)/(a*f*(2*m + 1)), x] - Dist[1/(a*b*(2*m + 1)), Int[(a + b*Csc[e + f*x])^(m + 1)*(d*Csc[e + f*x])^n*Simp[a*B*n - b*C*n - A*b*(2*m + n + 1) - (b*B*(m + n + 1) - a*(A*(m + n + 1) - C*(m - n)))*Csc[e + f*x], x], x], x] /; FreeQ[{a, b, d, e, f, A, B, C, n}, x] && EqQ[a^2 - b^2, 0] && LtQ[m, -2^(-1)]
  4085. Int[Times[Plus[Optional[Pattern[A, Blank[]]], Times[Power[csc[Plus[Optional[Pattern[e, Blank[]]], Times[Optional[Pattern[f, Blank[]]], Pattern[x, Blank[]]]]], 2], Optional[Pattern[C, Blank[]]]]], Power[Times[csc[Plus[Optional[Pattern[e, Blank[]]], Times[Optional[Pattern[f, Blank[]]], Pattern[x, Blank[]]]]], Optional[Pattern[d, Blank[]]]], Pattern[n, Blank[]]], Power[Plus[Times[csc[Plus[Optional[Pattern[e, Blank[]]], Times[Optional[Pattern[f, Blank[]]], Pattern[x, Blank[]]]]], Optional[Pattern[b, Blank[]]]], Pattern[a, Blank[]]], Pattern[m, Blank[]]]], Pattern[x, Blank[Symbol]]] := -Simp[(a*(A + C)*Cot[e + f*x]*(a + b*Csc[e + f*x])^m*(d*Csc[e + f*x])^n)/(a*f*(2*m + 1)), x] + Dist[1/(a*b*(2*m + 1)), Int[(a + b*Csc[e + f*x])^(m + 1)*(d*Csc[e + f*x])^n*Simp[b*C*n + A*b*(2*m + n + 1) - (a*(A*(m + n + 1) - C*(m - n)))*Csc[e + f*x], x], x], x] /; FreeQ[{a, b, d, e, f, A, C, n}, x] && EqQ[a^2 - b^2, 0] && LtQ[m, -2^(-1)]
  4086. Int[Times[Plus[Optional[Pattern[A, Blank[]]], Times[csc[Plus[Optional[Pattern[e, Blank[]]], Times[Optional[Pattern[f, Blank[]]], Pattern[x, Blank[]]]]], Optional[Pattern[B, Blank[]]]], Times[Power[csc[Plus[Optional[Pattern[e, Blank[]]], Times[Optional[Pattern[f, Blank[]]], Pattern[x, Blank[]]]]], 2], Optional[Pattern[C, Blank[]]]]], Power[Times[csc[Plus[Optional[Pattern[e, Blank[]]], Times[Optional[Pattern[f, Blank[]]], Pattern[x, Blank[]]]]], Optional[Pattern[d, Blank[]]]], Pattern[n, Blank[]]], Power[Plus[Times[csc[Plus[Optional[Pattern[e, Blank[]]], Times[Optional[Pattern[f, Blank[]]], Pattern[x, Blank[]]]]], Optional[Pattern[b, Blank[]]]], Pattern[a, Blank[]]], Pattern[m, Blank[]]]], Pattern[x, Blank[Symbol]]] := Simp[(A*Cot[e + f*x]*(a + b*Csc[e + f*x])^m*(d*Csc[e + f*x])^n)/(f*n), x] - Dist[1/(b*d*n), Int[(a + b*Csc[e + f*x])^m*(d*Csc[e + f*x])^(n + 1)*Simp[a*A*m - b*B*n - b*(A*(m + n + 1) + C*n)*Csc[e + f*x], x], x], x] /; FreeQ[{a, b, d, e, f, A, B, C, m}, x] && EqQ[a^2 - b^2, 0] && !LtQ[m, -2^(-1)] && (LtQ[n, -2^(-1)] || EqQ[m + n + 1, 0])
  4087. Int[Times[Plus[Optional[Pattern[A, Blank[]]], Times[Power[csc[Plus[Optional[Pattern[e, Blank[]]], Times[Optional[Pattern[f, Blank[]]], Pattern[x, Blank[]]]]], 2], Optional[Pattern[C, Blank[]]]]], Power[Times[csc[Plus[Optional[Pattern[e, Blank[]]], Times[Optional[Pattern[f, Blank[]]], Pattern[x, Blank[]]]]], Optional[Pattern[d, Blank[]]]], Pattern[n, Blank[]]], Power[Plus[Times[csc[Plus[Optional[Pattern[e, Blank[]]], Times[Optional[Pattern[f, Blank[]]], Pattern[x, Blank[]]]]], Optional[Pattern[b, Blank[]]]], Pattern[a, Blank[]]], Pattern[m, Blank[]]]], Pattern[x, Blank[Symbol]]] := Simp[(A*Cot[e + f*x]*(a + b*Csc[e + f*x])^m*(d*Csc[e + f*x])^n)/(f*n), x] - Dist[1/(b*d*n), Int[(a + b*Csc[e + f*x])^m*(d*Csc[e + f*x])^(n + 1)*Simp[a*A*m - b*(A*(m + n + 1) + C*n)*Csc[e + f*x], x], x], x] /; FreeQ[{a, b, d, e, f, A, C, m}, x] && EqQ[a^2 - b^2, 0] && !LtQ[m, -2^(-1)] && (LtQ[n, -2^(-1)] || EqQ[m + n + 1, 0])
  4088. Int[Times[Plus[Optional[Pattern[A, Blank[]]], Times[csc[Plus[Optional[Pattern[e, Blank[]]], Times[Optional[Pattern[f, Blank[]]], Pattern[x, Blank[]]]]], Optional[Pattern[B, Blank[]]]], Times[Power[csc[Plus[Optional[Pattern[e, Blank[]]], Times[Optional[Pattern[f, Blank[]]], Pattern[x, Blank[]]]]], 2], Optional[Pattern[C, Blank[]]]]], Power[Times[csc[Plus[Optional[Pattern[e, Blank[]]], Times[Optional[Pattern[f, Blank[]]], Pattern[x, Blank[]]]]], Optional[Pattern[d, Blank[]]]], Pattern[n, Blank[]]], Power[Plus[Times[csc[Plus[Optional[Pattern[e, Blank[]]], Times[Optional[Pattern[f, Blank[]]], Pattern[x, Blank[]]]]], Optional[Pattern[b, Blank[]]]], Pattern[a, Blank[]]], Pattern[m, Blank[]]]], Pattern[x, Blank[Symbol]]] := -Simp[(C*Cot[e + f*x]*(a + b*Csc[e + f*x])^m*(d*Csc[e + f*x])^n)/(f*(m + n + 1)), x] + Dist[1/(b*(m + n + 1)), Int[(a + b*Csc[e + f*x])^m*(d*Csc[e + f*x])^n*Simp[A*b*(m + n + 1) + b*C*n + (a*C*m + b*B*(m + n + 1))*Csc[e + f*x], x], x], x] /; FreeQ[{a, b, d, e, f, A, B, C, m, n}, x] && EqQ[a^2 - b^2, 0] && !LtQ[m, -2^(-1)] && !LtQ[n, -2^(-1)] && NeQ[m + n + 1, 0]
  4089. Int[Times[Plus[Optional[Pattern[A, Blank[]]], Times[Power[csc[Plus[Optional[Pattern[e, Blank[]]], Times[Optional[Pattern[f, Blank[]]], Pattern[x, Blank[]]]]], 2], Optional[Pattern[C, Blank[]]]]], Power[Times[csc[Plus[Optional[Pattern[e, Blank[]]], Times[Optional[Pattern[f, Blank[]]], Pattern[x, Blank[]]]]], Optional[Pattern[d, Blank[]]]], Pattern[n, Blank[]]], Power[Plus[Times[csc[Plus[Optional[Pattern[e, Blank[]]], Times[Optional[Pattern[f, Blank[]]], Pattern[x, Blank[]]]]], Optional[Pattern[b, Blank[]]]], Pattern[a, Blank[]]], Pattern[m, Blank[]]]], Pattern[x, Blank[Symbol]]] := -Simp[(C*Cot[e + f*x]*(a + b*Csc[e + f*x])^m*(d*Csc[e + f*x])^n)/(f*(m + n + 1)), x] + Dist[1/(b*(m + n + 1)), Int[(a + b*Csc[e + f*x])^m*(d*Csc[e + f*x])^n*Simp[A*b*(m + n + 1) + b*C*n + a*C*m*Csc[e + f*x], x], x], x] /; FreeQ[{a, b, d, e, f, A, C, m, n}, x] && EqQ[a^2 - b^2, 0] && !LtQ[m, -2^(-1)] && !LtQ[n, -2^(-1)] && NeQ[m + n + 1, 0]
  4090. Int[Times[Power[csc[Plus[Optional[Pattern[e, Blank[]]], Times[Optional[Pattern[f, Blank[]]], Pattern[x, Blank[]]]]], 2], Plus[Optional[Pattern[A, Blank[]]], Times[csc[Plus[Optional[Pattern[e, Blank[]]], Times[Optional[Pattern[f, Blank[]]], Pattern[x, Blank[]]]]], Optional[Pattern[B, Blank[]]]], Times[Power[csc[Plus[Optional[Pattern[e, Blank[]]], Times[Optional[Pattern[f, Blank[]]], Pattern[x, Blank[]]]]], 2], Optional[Pattern[C, Blank[]]]]], Power[Plus[Times[csc[Plus[Optional[Pattern[e, Blank[]]], Times[Optional[Pattern[f, Blank[]]], Pattern[x, Blank[]]]]], Optional[Pattern[b, Blank[]]]], Pattern[a, Blank[]]], Pattern[m, Blank[]]]], Pattern[x, Blank[Symbol]]] := Simp[(a*(A*b^2 - a*b*B + a^2*C)*Cot[e + f*x]*(a + b*Csc[e + f*x])^(m + 1))/(b^2*f*(m + 1)*(a^2 - b^2)), x] - Dist[1/(b^2*(m + 1)*(a^2 - b^2)), Int[Csc[e + f*x]*(a + b*Csc[e + f*x])^(m + 1)*Simp[b*(m + 1)*(-(a*(b*B - a*C)) + A*b^2) + (b*B*(a^2 + b^2*(m + 1)) - a*(A*b^2*(m + 2) + C*(a^2 + b^2*(m + 1))))*Csc[e + f*x] - b*C*(m + 1)*(a^2 - b^2)*Csc[e + f*x]^2, x], x], x] /; FreeQ[{a, b, e, f, A, B, C}, x] && NeQ[a^2 - b^2, 0] && LtQ[m, -1]
  4091. Int[Times[Power[csc[Plus[Optional[Pattern[e, Blank[]]], Times[Optional[Pattern[f, Blank[]]], Pattern[x, Blank[]]]]], 2], Plus[Optional[Pattern[A, Blank[]]], Times[Power[csc[Plus[Optional[Pattern[e, Blank[]]], Times[Optional[Pattern[f, Blank[]]], Pattern[x, Blank[]]]]], 2], Optional[Pattern[C, Blank[]]]]], Power[Plus[Times[csc[Plus[Optional[Pattern[e, Blank[]]], Times[Optional[Pattern[f, Blank[]]], Pattern[x, Blank[]]]]], Optional[Pattern[b, Blank[]]]], Pattern[a, Blank[]]], Pattern[m, Blank[]]]], Pattern[x, Blank[Symbol]]] := Simp[(a*(A*b^2 + a^2*C)*Cot[e + f*x]*(a + b*Csc[e + f*x])^(m + 1))/(b^2*f*(m + 1)*(a^2 - b^2)), x] - Dist[1/(b^2*(m + 1)*(a^2 - b^2)), Int[Csc[e + f*x]*(a + b*Csc[e + f*x])^(m + 1)*Simp[b*(m + 1)*(a^2*C + A*b^2) - a*(A*b^2*(m + 2) + C*(a^2 + b^2*(m + 1)))*Csc[e + f*x] - b*C*(m + 1)*(a^2 - b^2)*Csc[e + f*x]^2, x], x], x] /; FreeQ[{a, b, e, f, A, C}, x] && NeQ[a^2 - b^2, 0] && LtQ[m, -1]
  4092. Int[Times[Power[csc[Plus[Optional[Pattern[e, Blank[]]], Times[Optional[Pattern[f, Blank[]]], Pattern[x, Blank[]]]]], 2], Plus[Optional[Pattern[A, Blank[]]], Times[csc[Plus[Optional[Pattern[e, Blank[]]], Times[Optional[Pattern[f, Blank[]]], Pattern[x, Blank[]]]]], Optional[Pattern[B, Blank[]]]], Times[Power[csc[Plus[Optional[Pattern[e, Blank[]]], Times[Optional[Pattern[f, Blank[]]], Pattern[x, Blank[]]]]], 2], Optional[Pattern[C, Blank[]]]]], Power[Plus[Times[csc[Plus[Optional[Pattern[e, Blank[]]], Times[Optional[Pattern[f, Blank[]]], Pattern[x, Blank[]]]]], Optional[Pattern[b, Blank[]]]], Pattern[a, Blank[]]], Pattern[m, Blank[]]]], Pattern[x, Blank[Symbol]]] := -Simp[(C*Csc[e + f*x]*Cot[e + f*x]*(a + b*Csc[e + f*x])^(m + 1))/(b*f*(m + 3)), x] + Dist[1/(b*(m + 3)), Int[Csc[e + f*x]*(a + b*Csc[e + f*x])^m*Simp[a*C + b*(C*(m + 2) + A*(m + 3))*Csc[e + f*x] - (2*a*C - b*B*(m + 3))*Csc[e + f*x]^2, x], x], x] /; FreeQ[{a, b, e, f, A, B, C, m}, x] && NeQ[a^2 - b^2, 0] && !LtQ[m, -1]
  4093. Int[Times[Power[csc[Plus[Optional[Pattern[e, Blank[]]], Times[Optional[Pattern[f, Blank[]]], Pattern[x, Blank[]]]]], 2], Plus[Optional[Pattern[A, Blank[]]], Times[Power[csc[Plus[Optional[Pattern[e, Blank[]]], Times[Optional[Pattern[f, Blank[]]], Pattern[x, Blank[]]]]], 2], Optional[Pattern[C, Blank[]]]]], Power[Plus[Times[csc[Plus[Optional[Pattern[e, Blank[]]], Times[Optional[Pattern[f, Blank[]]], Pattern[x, Blank[]]]]], Optional[Pattern[b, Blank[]]]], Pattern[a, Blank[]]], Pattern[m, Blank[]]]], Pattern[x, Blank[Symbol]]] := -Simp[(C*Csc[e + f*x]*Cot[e + f*x]*(a + b*Csc[e + f*x])^(m + 1))/(b*f*(m + 3)), x] + Dist[1/(b*(m + 3)), Int[Csc[e + f*x]*(a + b*Csc[e + f*x])^m*Simp[a*C + b*(C*(m + 2) + A*(m + 3))*Csc[e + f*x] - 2*a*C*Csc[e + f*x]^2, x], x], x] /; FreeQ[{a, b, e, f, A, C, m}, x] && NeQ[a^2 - b^2, 0] && !LtQ[m, -1]
  4094. Int[Times[Plus[Optional[Pattern[A, Blank[]]], Times[csc[Plus[Optional[Pattern[e, Blank[]]], Times[Optional[Pattern[f, Blank[]]], Pattern[x, Blank[]]]]], Optional[Pattern[B, Blank[]]]], Times[Power[csc[Plus[Optional[Pattern[e, Blank[]]], Times[Optional[Pattern[f, Blank[]]], Pattern[x, Blank[]]]]], 2], Optional[Pattern[C, Blank[]]]]], Power[Times[csc[Plus[Optional[Pattern[e, Blank[]]], Times[Optional[Pattern[f, Blank[]]], Pattern[x, Blank[]]]]], Optional[Pattern[d, Blank[]]]], Pattern[n, Blank[]]], Power[Plus[Times[csc[Plus[Optional[Pattern[e, Blank[]]], Times[Optional[Pattern[f, Blank[]]], Pattern[x, Blank[]]]]], Optional[Pattern[b, Blank[]]]], Pattern[a, Blank[]]], Pattern[m, Blank[]]]], Pattern[x, Blank[Symbol]]] := Simp[(A*Cot[e + f*x]*(a + b*Csc[e + f*x])^m*(d*Csc[e + f*x])^n)/(f*n), x] - Dist[1/(d*n), Int[(a + b*Csc[e + f*x])^(m - 1)*(d*Csc[e + f*x])^(n + 1)*Simp[A*b*m - a*B*n - (b*B*n + a*(C*n + A*(n + 1)))*Csc[e + f*x] - b*(C*n + A*(m + n + 1))*Csc[e + f*x]^2, x], x], x] /; FreeQ[{a, b, d, e, f, A, B, C}, x] && NeQ[a^2 - b^2, 0] && GtQ[m, 0] && LeQ[n, -1]
  4095. Int[Times[Plus[Optional[Pattern[A, Blank[]]], Times[Power[csc[Plus[Optional[Pattern[e, Blank[]]], Times[Optional[Pattern[f, Blank[]]], Pattern[x, Blank[]]]]], 2], Optional[Pattern[C, Blank[]]]]], Power[Times[csc[Plus[Optional[Pattern[e, Blank[]]], Times[Optional[Pattern[f, Blank[]]], Pattern[x, Blank[]]]]], Optional[Pattern[d, Blank[]]]], Pattern[n, Blank[]]], Power[Plus[Times[csc[Plus[Optional[Pattern[e, Blank[]]], Times[Optional[Pattern[f, Blank[]]], Pattern[x, Blank[]]]]], Optional[Pattern[b, Blank[]]]], Pattern[a, Blank[]]], Pattern[m, Blank[]]]], Pattern[x, Blank[Symbol]]] := Simp[(A*Cot[e + f*x]*(a + b*Csc[e + f*x])^m*(d*Csc[e + f*x])^n)/(f*n), x] - Dist[1/(d*n), Int[(a + b*Csc[e + f*x])^(m - 1)*(d*Csc[e + f*x])^(n + 1)*Simp[A*b*m - a*(C*n + A*(n + 1))*Csc[e + f*x] - b*(C*n + A*(m + n + 1))*Csc[e + f*x]^2, x], x], x] /; FreeQ[{a, b, d, e, f, A, C}, x] && NeQ[a^2 - b^2, 0] && GtQ[m, 0] && LeQ[n, -1]
  4096. Int[Times[Plus[Optional[Pattern[A, Blank[]]], Times[csc[Plus[Optional[Pattern[e, Blank[]]], Times[Optional[Pattern[f, Blank[]]], Pattern[x, Blank[]]]]], Optional[Pattern[B, Blank[]]]], Times[Power[csc[Plus[Optional[Pattern[e, Blank[]]], Times[Optional[Pattern[f, Blank[]]], Pattern[x, Blank[]]]]], 2], Optional[Pattern[C, Blank[]]]]], Power[Times[csc[Plus[Optional[Pattern[e, Blank[]]], Times[Optional[Pattern[f, Blank[]]], Pattern[x, Blank[]]]]], Optional[Pattern[d, Blank[]]]], Pattern[n, Blank[]]], Power[Plus[Times[csc[Plus[Optional[Pattern[e, Blank[]]], Times[Optional[Pattern[f, Blank[]]], Pattern[x, Blank[]]]]], Optional[Pattern[b, Blank[]]]], Pattern[a, Blank[]]], Pattern[m, Blank[]]]], Pattern[x, Blank[Symbol]]] := -Simp[(C*Cot[e + f*x]*(a + b*Csc[e + f*x])^m*(d*Csc[e + f*x])^n)/(f*(m + n + 1)), x] + Dist[1/(m + n + 1), Int[(a + b*Csc[e + f*x])^(m - 1)*(d*Csc[e + f*x])^n*Simp[a*A*(m + n + 1) + a*C*n + ((A*b + a*B)*(m + n + 1) + b*C*(m + n))*Csc[e + f*x] + (b*B*(m + n + 1) + a*C*m)*Csc[e + f*x]^2, x], x], x] /; FreeQ[{a, b, d, e, f, A, B, C, n}, x] && NeQ[a^2 - b^2, 0] && GtQ[m, 0] && !LeQ[n, -1]
  4097. Int[Times[Plus[Optional[Pattern[A, Blank[]]], Times[Power[csc[Plus[Optional[Pattern[e, Blank[]]], Times[Optional[Pattern[f, Blank[]]], Pattern[x, Blank[]]]]], 2], Optional[Pattern[C, Blank[]]]]], Power[Times[csc[Plus[Optional[Pattern[e, Blank[]]], Times[Optional[Pattern[f, Blank[]]], Pattern[x, Blank[]]]]], Optional[Pattern[d, Blank[]]]], Pattern[n, Blank[]]], Power[Plus[Times[csc[Plus[Optional[Pattern[e, Blank[]]], Times[Optional[Pattern[f, Blank[]]], Pattern[x, Blank[]]]]], Optional[Pattern[b, Blank[]]]], Pattern[a, Blank[]]], Pattern[m, Blank[]]]], Pattern[x, Blank[Symbol]]] := -Simp[(C*Cot[e + f*x]*(a + b*Csc[e + f*x])^m*(d*Csc[e + f*x])^n)/(f*(m + n + 1)), x] + Dist[1/(m + n + 1), Int[(a + b*Csc[e + f*x])^(m - 1)*(d*Csc[e + f*x])^n*Simp[a*A*(m + n + 1) + a*C*n + b*(A*(m + n + 1) + C*(m + n))*Csc[e + f*x] + a*C*m*Csc[e + f*x]^2, x], x], x] /; FreeQ[{a, b, d, e, f, A, C, n}, x] && NeQ[a^2 - b^2, 0] && GtQ[m, 0] && !LeQ[n, -1]
  4098. Int[Times[Plus[Optional[Pattern[A, Blank[]]], Times[csc[Plus[Optional[Pattern[e, Blank[]]], Times[Optional[Pattern[f, Blank[]]], Pattern[x, Blank[]]]]], Optional[Pattern[B, Blank[]]]], Times[Power[csc[Plus[Optional[Pattern[e, Blank[]]], Times[Optional[Pattern[f, Blank[]]], Pattern[x, Blank[]]]]], 2], Optional[Pattern[C, Blank[]]]]], Power[Times[csc[Plus[Optional[Pattern[e, Blank[]]], Times[Optional[Pattern[f, Blank[]]], Pattern[x, Blank[]]]]], Optional[Pattern[d, Blank[]]]], Pattern[n, Blank[]]], Power[Plus[Times[csc[Plus[Optional[Pattern[e, Blank[]]], Times[Optional[Pattern[f, Blank[]]], Pattern[x, Blank[]]]]], Optional[Pattern[b, Blank[]]]], Pattern[a, Blank[]]], Pattern[m, Blank[]]]], Pattern[x, Blank[Symbol]]] := -Simp[(d*(A*b^2 - a*b*B + a^2*C)*Cot[e + f*x]*(a + b*Csc[e + f*x])^(m + 1)*(d*Csc[e + f*x])^(n - 1))/(b*f*(a^2 - b^2)*(m + 1)), x] + Dist[d/(b*(a^2 - b^2)*(m + 1)), Int[(a + b*Csc[e + f*x])^(m + 1)*(d*Csc[e + f*x])^(n - 1)*Simp[A*b^2*(n - 1) - a*(b*B - a*C)*(n - 1) + b*(a*A - b*B + a*C)*(m + 1)*Csc[e + f*x] - (b*(A*b - a*B)*(m + n + 1) + C*(a^2*n + b^2*(m + 1)))*Csc[e + f*x]^2, x], x], x] /; FreeQ[{a, b, d, e, f, A, B, C}, x] && NeQ[a^2 - b^2, 0] && LtQ[m, -1] && GtQ[n, 0]
  4099. Int[Times[Plus[Optional[Pattern[A, Blank[]]], Times[Power[csc[Plus[Optional[Pattern[e, Blank[]]], Times[Optional[Pattern[f, Blank[]]], Pattern[x, Blank[]]]]], 2], Optional[Pattern[C, Blank[]]]]], Power[Times[csc[Plus[Optional[Pattern[e, Blank[]]], Times[Optional[Pattern[f, Blank[]]], Pattern[x, Blank[]]]]], Optional[Pattern[d, Blank[]]]], Pattern[n, Blank[]]], Power[Plus[Times[csc[Plus[Optional[Pattern[e, Blank[]]], Times[Optional[Pattern[f, Blank[]]], Pattern[x, Blank[]]]]], Optional[Pattern[b, Blank[]]]], Pattern[a, Blank[]]], Pattern[m, Blank[]]]], Pattern[x, Blank[Symbol]]] := -Simp[(d*(A*b^2 + a^2*C)*Cot[e + f*x]*(a + b*Csc[e + f*x])^(m + 1)*(d*Csc[e + f*x])^(n - 1))/(b*f*(a^2 - b^2)*(m + 1)), x] + Dist[d/(b*(a^2 - b^2)*(m + 1)), Int[(a + b*Csc[e + f*x])^(m + 1)*(d*Csc[e + f*x])^(n - 1)*Simp[A*b^2*(n - 1) + a^2*C*(n - 1) + a*b*(A + C)*(m + 1)*Csc[e + f*x] - (A*b^2*(m + n + 1) + C*(a^2*n + b^2*(m + 1)))*Csc[e + f*x]^2, x], x], x] /; FreeQ[{a, b, d, e, f, A, C}, x] && NeQ[a^2 - b^2, 0] && LtQ[m, -1] && GtQ[n, 0]
  4100. Int[Times[Plus[Optional[Pattern[A, Blank[]]], Times[csc[Plus[Optional[Pattern[e, Blank[]]], Times[Optional[Pattern[f, Blank[]]], Pattern[x, Blank[]]]]], Optional[Pattern[B, Blank[]]]], Times[Power[csc[Plus[Optional[Pattern[e, Blank[]]], Times[Optional[Pattern[f, Blank[]]], Pattern[x, Blank[]]]]], 2], Optional[Pattern[C, Blank[]]]]], Power[Times[csc[Plus[Optional[Pattern[e, Blank[]]], Times[Optional[Pattern[f, Blank[]]], Pattern[x, Blank[]]]]], Optional[Pattern[d, Blank[]]]], Pattern[n, Blank[]]], Power[Plus[Times[csc[Plus[Optional[Pattern[e, Blank[]]], Times[Optional[Pattern[f, Blank[]]], Pattern[x, Blank[]]]]], Optional[Pattern[b, Blank[]]]], Pattern[a, Blank[]]], Pattern[m, Blank[]]]], Pattern[x, Blank[Symbol]]] := Simp[((A*b^2 - a*b*B + a^2*C)*Cot[e + f*x]*(a + b*Csc[e + f*x])^(m + 1)*(d*Csc[e + f*x])^n)/(a*f*(m + 1)*(a^2 - b^2)), x] + Dist[1/(a*(m + 1)*(a^2 - b^2)), Int[(a + b*Csc[e + f*x])^(m + 1)*(d*Csc[e + f*x])^n*Simp[a*(a*A - b*B + a*C)*(m + 1) - (A*b^2 - a*b*B + a^2*C)*(m + n + 1) - a*(A*b - a*B + b*C)*(m + 1)*Csc[e + f*x] + (A*b^2 - a*b*B + a^2*C)*(m + n + 2)*Csc[e + f*x]^2, x], x], x] /; FreeQ[{a, b, d, e, f, A, B, C, n}, x] && NeQ[a^2 - b^2, 0] && LtQ[m, -1] && !(ILtQ[m + 1/2, 0] && ILtQ[n, 0])
  4101. Int[Times[Plus[Optional[Pattern[A, Blank[]]], Times[Power[csc[Plus[Optional[Pattern[e, Blank[]]], Times[Optional[Pattern[f, Blank[]]], Pattern[x, Blank[]]]]], 2], Optional[Pattern[C, Blank[]]]]], Power[Times[csc[Plus[Optional[Pattern[e, Blank[]]], Times[Optional[Pattern[f, Blank[]]], Pattern[x, Blank[]]]]], Optional[Pattern[d, Blank[]]]], Pattern[n, Blank[]]], Power[Plus[Times[csc[Plus[Optional[Pattern[e, Blank[]]], Times[Optional[Pattern[f, Blank[]]], Pattern[x, Blank[]]]]], Optional[Pattern[b, Blank[]]]], Pattern[a, Blank[]]], Pattern[m, Blank[]]]], Pattern[x, Blank[Symbol]]] := Simp[((A*b^2 + a^2*C)*Cot[e + f*x]*(a + b*Csc[e + f*x])^(m + 1)*(d*Csc[e + f*x])^n)/(a*f*(m + 1)*(a^2 - b^2)), x] + Dist[1/(a*(m + 1)*(a^2 - b^2)), Int[(a + b*Csc[e + f*x])^(m + 1)*(d*Csc[e + f*x])^n*Simp[a^2*(A + C)*(m + 1) - (A*b^2 + a^2*C)*(m + n + 1) - a*b*(A + C)*(m + 1)*Csc[e + f*x] + (A*b^2 + a^2*C)*(m + n + 2)*Csc[e + f*x]^2, x], x], x] /; FreeQ[{a, b, d, e, f, A, C, n}, x] && NeQ[a^2 - b^2, 0] && LtQ[m, -1] && !(ILtQ[m + 1/2, 0] && ILtQ[n, 0])
  4102. Int[Times[Plus[Optional[Pattern[A, Blank[]]], Times[csc[Plus[Optional[Pattern[e, Blank[]]], Times[Optional[Pattern[f, Blank[]]], Pattern[x, Blank[]]]]], Optional[Pattern[B, Blank[]]]], Times[Power[csc[Plus[Optional[Pattern[e, Blank[]]], Times[Optional[Pattern[f, Blank[]]], Pattern[x, Blank[]]]]], 2], Optional[Pattern[C, Blank[]]]]], Power[Times[csc[Plus[Optional[Pattern[e, Blank[]]], Times[Optional[Pattern[f, Blank[]]], Pattern[x, Blank[]]]]], Optional[Pattern[d, Blank[]]]], Pattern[n, Blank[]]], Power[Plus[Times[csc[Plus[Optional[Pattern[e, Blank[]]], Times[Optional[Pattern[f, Blank[]]], Pattern[x, Blank[]]]]], Optional[Pattern[b, Blank[]]]], Pattern[a, Blank[]]], Pattern[m, Blank[]]]], Pattern[x, Blank[Symbol]]] := -Simp[(C*d*Cot[e + f*x]*(a + b*Csc[e + f*x])^(m + 1)*(d*Csc[e + f*x])^(n - 1))/(b*f*(m + n + 1)), x] + Dist[d/(b*(m + n + 1)), Int[(a + b*Csc[e + f*x])^m*(d*Csc[e + f*x])^(n - 1)*Simp[a*C*(n - 1) + (A*b*(m + n + 1) + b*C*(m + n))*Csc[e + f*x] + (b*B*(m + n + 1) - a*C*n)*Csc[e + f*x]^2, x], x], x] /; FreeQ[{a, b, d, e, f, A, B, C, m}, x] && NeQ[a^2 - b^2, 0] && GtQ[n, 0]
  4103. Int[Times[Plus[Optional[Pattern[A, Blank[]]], Times[Power[csc[Plus[Optional[Pattern[e, Blank[]]], Times[Optional[Pattern[f, Blank[]]], Pattern[x, Blank[]]]]], 2], Optional[Pattern[C, Blank[]]]]], Power[Times[csc[Plus[Optional[Pattern[e, Blank[]]], Times[Optional[Pattern[f, Blank[]]], Pattern[x, Blank[]]]]], Optional[Pattern[d, Blank[]]]], Pattern[n, Blank[]]], Power[Plus[Times[csc[Plus[Optional[Pattern[e, Blank[]]], Times[Optional[Pattern[f, Blank[]]], Pattern[x, Blank[]]]]], Optional[Pattern[b, Blank[]]]], Pattern[a, Blank[]]], Pattern[m, Blank[]]]], Pattern[x, Blank[Symbol]]] := -Simp[(C*d*Cot[e + f*x]*(a + b*Csc[e + f*x])^(m + 1)*(d*Csc[e + f*x])^(n - 1))/(b*f*(m + n + 1)), x] + Dist[d/(b*(m + n + 1)), Int[(a + b*Csc[e + f*x])^m*(d*Csc[e + f*x])^(n - 1)*Simp[a*C*(n - 1) + (A*b*(m + n + 1) + b*C*(m + n))*Csc[e + f*x] - a*C*n*Csc[e + f*x]^2, x], x], x] /; FreeQ[{a, b, d, e, f, A, C, m}, x] && NeQ[a^2 - b^2, 0] && GtQ[n, 0]
  4104. Int[Times[Plus[Optional[Pattern[A, Blank[]]], Times[csc[Plus[Optional[Pattern[e, Blank[]]], Times[Optional[Pattern[f, Blank[]]], Pattern[x, Blank[]]]]], Optional[Pattern[B, Blank[]]]], Times[Power[csc[Plus[Optional[Pattern[e, Blank[]]], Times[Optional[Pattern[f, Blank[]]], Pattern[x, Blank[]]]]], 2], Optional[Pattern[C, Blank[]]]]], Power[Times[csc[Plus[Optional[Pattern[e, Blank[]]], Times[Optional[Pattern[f, Blank[]]], Pattern[x, Blank[]]]]], Optional[Pattern[d, Blank[]]]], Pattern[n, Blank[]]], Power[Plus[Times[csc[Plus[Optional[Pattern[e, Blank[]]], Times[Optional[Pattern[f, Blank[]]], Pattern[x, Blank[]]]]], Optional[Pattern[b, Blank[]]]], Pattern[a, Blank[]]], Pattern[m, Blank[]]]], Pattern[x, Blank[Symbol]]] := Simp[(A*Cot[e + f*x]*(a + b*Csc[e + f*x])^(m + 1)*(d*Csc[e + f*x])^n)/(a*f*n), x] + Dist[1/(a*d*n), Int[(a + b*Csc[e + f*x])^m*(d*Csc[e + f*x])^(n + 1)*Simp[a*B*n - A*b*(m + n + 1) + a*(A + A*n + C*n)*Csc[e + f*x] + A*b*(m + n + 2)*Csc[e + f*x]^2, x], x], x] /; FreeQ[{a, b, d, e, f, A, B, C, m}, x] && NeQ[a^2 - b^2, 0] && LeQ[n, -1]
  4105. Int[Times[Plus[Optional[Pattern[A, Blank[]]], Times[Power[csc[Plus[Optional[Pattern[e, Blank[]]], Times[Optional[Pattern[f, Blank[]]], Pattern[x, Blank[]]]]], 2], Optional[Pattern[C, Blank[]]]]], Power[Times[csc[Plus[Optional[Pattern[e, Blank[]]], Times[Optional[Pattern[f, Blank[]]], Pattern[x, Blank[]]]]], Optional[Pattern[d, Blank[]]]], Pattern[n, Blank[]]], Power[Plus[Times[csc[Plus[Optional[Pattern[e, Blank[]]], Times[Optional[Pattern[f, Blank[]]], Pattern[x, Blank[]]]]], Optional[Pattern[b, Blank[]]]], Pattern[a, Blank[]]], Pattern[m, Blank[]]]], Pattern[x, Blank[Symbol]]] := Simp[(A*Cot[e + f*x]*(a + b*Csc[e + f*x])^(m + 1)*(d*Csc[e + f*x])^n)/(a*f*n), x] + Dist[1/(a*d*n), Int[(a + b*Csc[e + f*x])^m*(d*Csc[e + f*x])^(n + 1)*Simp[-(A*b*(m + n + 1)) + a*(A + A*n + C*n)*Csc[e + f*x] + A*b*(m + n + 2)*Csc[e + f*x]^2, x], x], x] /; FreeQ[{a, b, d, e, f, A, C, m}, x] && NeQ[a^2 - b^2, 0] && LeQ[n, -1]
  4106. Int[Times[Plus[Optional[Pattern[A, Blank[]]], Times[csc[Plus[Optional[Pattern[e, Blank[]]], Times[Optional[Pattern[f, Blank[]]], Pattern[x, Blank[]]]]], Optional[Pattern[B, Blank[]]]], Times[Power[csc[Plus[Optional[Pattern[e, Blank[]]], Times[Optional[Pattern[f, Blank[]]], Pattern[x, Blank[]]]]], 2], Optional[Pattern[C, Blank[]]]]], Power[Times[csc[Plus[Optional[Pattern[e, Blank[]]], Times[Optional[Pattern[f, Blank[]]], Pattern[x, Blank[]]]]], Optional[Pattern[d, Blank[]]]], Rational[-1, 2]], Power[Plus[Times[csc[Plus[Optional[Pattern[e, Blank[]]], Times[Optional[Pattern[f, Blank[]]], Pattern[x, Blank[]]]]], Optional[Pattern[b, Blank[]]]], Pattern[a, Blank[]]], -1]], Pattern[x, Blank[Symbol]]] := Dist[(A*b^2 - a*b*B + a^2*C)/(a^2*d^2), Int[(d*Csc[e + f*x])^(3/2)/(a + b*Csc[e + f*x]), x], x] + Dist[1/a^2, Int[(a*A - (A*b - a*B)*Csc[e + f*x])/Sqrt[d*Csc[e + f*x]], x], x] /; FreeQ[{a, b, d, e, f, A, B, C}, x] && NeQ[a^2 - b^2, 0]
  4107. Int[Times[Plus[Optional[Pattern[A, Blank[]]], Times[Power[csc[Plus[Optional[Pattern[e, Blank[]]], Times[Optional[Pattern[f, Blank[]]], Pattern[x, Blank[]]]]], 2], Optional[Pattern[C, Blank[]]]]], Power[Times[csc[Plus[Optional[Pattern[e, Blank[]]], Times[Optional[Pattern[f, Blank[]]], Pattern[x, Blank[]]]]], Optional[Pattern[d, Blank[]]]], Rational[-1, 2]], Power[Plus[Times[csc[Plus[Optional[Pattern[e, Blank[]]], Times[Optional[Pattern[f, Blank[]]], Pattern[x, Blank[]]]]], Optional[Pattern[b, Blank[]]]], Pattern[a, Blank[]]], -1]], Pattern[x, Blank[Symbol]]] := Dist[(A*b^2 + a^2*C)/(a^2*d^2), Int[(d*Csc[e + f*x])^(3/2)/(a + b*Csc[e + f*x]), x], x] + Dist[1/a^2, Int[(a*A - A*b*Csc[e + f*x])/Sqrt[d*Csc[e + f*x]], x], x] /; FreeQ[{a, b, d, e, f, A, C}, x] && NeQ[a^2 - b^2, 0]
  4108. Int[Times[Plus[Optional[Pattern[A, Blank[]]], Times[csc[Plus[Optional[Pattern[e, Blank[]]], Times[Optional[Pattern[f, Blank[]]], Pattern[x, Blank[]]]]], Optional[Pattern[B, Blank[]]]], Times[Power[csc[Plus[Optional[Pattern[e, Blank[]]], Times[Optional[Pattern[f, Blank[]]], Pattern[x, Blank[]]]]], 2], Optional[Pattern[C, Blank[]]]]], Power[Times[csc[Plus[Optional[Pattern[e, Blank[]]], Times[Optional[Pattern[f, Blank[]]], Pattern[x, Blank[]]]]], Optional[Pattern[d, Blank[]]]], Rational[-1, 2]], Power[Plus[Times[csc[Plus[Optional[Pattern[e, Blank[]]], Times[Optional[Pattern[f, Blank[]]], Pattern[x, Blank[]]]]], Optional[Pattern[b, Blank[]]]], Pattern[a, Blank[]]], Rational[-1, 2]]], Pattern[x, Blank[Symbol]]] := Dist[C/d^2, Int[(d*Csc[e + f*x])^(3/2)/Sqrt[a + b*Csc[e + f*x]], x], x] + Int[(A + B*Csc[e + f*x])/(Sqrt[d*Csc[e + f*x]]*Sqrt[a + b*Csc[e + f*x]]), x] /; FreeQ[{a, b, d, e, f, A, B, C}, x] && NeQ[a^2 - b^2, 0]
  4109. Int[Times[Plus[Optional[Pattern[A, Blank[]]], Times[Power[csc[Plus[Optional[Pattern[e, Blank[]]], Times[Optional[Pattern[f, Blank[]]], Pattern[x, Blank[]]]]], 2], Optional[Pattern[C, Blank[]]]]], Power[Times[csc[Plus[Optional[Pattern[e, Blank[]]], Times[Optional[Pattern[f, Blank[]]], Pattern[x, Blank[]]]]], Optional[Pattern[d, Blank[]]]], Rational[-1, 2]], Power[Plus[Times[csc[Plus[Optional[Pattern[e, Blank[]]], Times[Optional[Pattern[f, Blank[]]], Pattern[x, Blank[]]]]], Optional[Pattern[b, Blank[]]]], Pattern[a, Blank[]]], Rational[-1, 2]]], Pattern[x, Blank[Symbol]]] := Dist[C/d^2, Int[(d*Csc[e + f*x])^(3/2)/Sqrt[a + b*Csc[e + f*x]], x], x] + Dist[A, Int[1/(Sqrt[d*Csc[e + f*x]]*Sqrt[a + b*Csc[e + f*x]]), x], x] /; FreeQ[{a, b, d, e, f, A, C}, x] && NeQ[a^2 - b^2, 0]
  4110. Int[Times[Plus[Optional[Pattern[A, Blank[]]], Times[csc[Plus[Optional[Pattern[e, Blank[]]], Times[Optional[Pattern[f, Blank[]]], Pattern[x, Blank[]]]]], Optional[Pattern[B, Blank[]]]], Times[Power[csc[Plus[Optional[Pattern[e, Blank[]]], Times[Optional[Pattern[f, Blank[]]], Pattern[x, Blank[]]]]], 2], Optional[Pattern[C, Blank[]]]]], Power[Times[csc[Plus[Optional[Pattern[e, Blank[]]], Times[Optional[Pattern[f, Blank[]]], Pattern[x, Blank[]]]]], Optional[Pattern[d, Blank[]]]], Optional[Pattern[n, Blank[]]]], Power[Plus[Times[csc[Plus[Optional[Pattern[e, Blank[]]], Times[Optional[Pattern[f, Blank[]]], Pattern[x, Blank[]]]]], Optional[Pattern[b, Blank[]]]], Pattern[a, Blank[]]], Optional[Pattern[m, Blank[]]]]], Pattern[x, Blank[Symbol]]] := Unintegrable[(d*Csc[e + f*x])^n*(a + b*Csc[e + f*x])^m*(A + B*Csc[e + f*x] + C*Csc[e + f*x]^2), x] /; FreeQ[{a, b, d, e, f, A, B, C, m, n}, x]
  4111. Int[Times[Plus[Optional[Pattern[A, Blank[]]], Times[Power[csc[Plus[Optional[Pattern[e, Blank[]]], Times[Optional[Pattern[f, Blank[]]], Pattern[x, Blank[]]]]], 2], Optional[Pattern[C, Blank[]]]]], Power[Times[csc[Plus[Optional[Pattern[e, Blank[]]], Times[Optional[Pattern[f, Blank[]]], Pattern[x, Blank[]]]]], Optional[Pattern[d, Blank[]]]], Optional[Pattern[n, Blank[]]]], Power[Plus[Times[csc[Plus[Optional[Pattern[e, Blank[]]], Times[Optional[Pattern[f, Blank[]]], Pattern[x, Blank[]]]]], Optional[Pattern[b, Blank[]]]], Pattern[a, Blank[]]], Optional[Pattern[m, Blank[]]]]], Pattern[x, Blank[Symbol]]] := Unintegrable[(d*Csc[e + f*x])^n*(a + b*Csc[e + f*x])^m*(A + C*Csc[e + f*x]^2), x] /; FreeQ[{a, b, d, e, f, A, C, m, n}, x]
  4112. Int[Times[Power[Times[cos[Plus[Optional[Pattern[e, Blank[]]], Times[Optional[Pattern[f, Blank[]]], Pattern[x, Blank[]]]]], Optional[Pattern[d, Blank[]]]], Pattern[n, Blank[]]], Power[Plus[Pattern[a, Blank[]], Times[Optional[Pattern[b, Blank[]]], sec[Plus[Optional[Pattern[e, Blank[]]], Times[Optional[Pattern[f, Blank[]]], Pattern[x, Blank[]]]]]]], Optional[Pattern[m, Blank[]]]], Plus[Optional[Pattern[A, Blank[]]], Times[Optional[Pattern[B, Blank[]]], sec[Plus[Optional[Pattern[e, Blank[]]], Times[Optional[Pattern[f, Blank[]]], Pattern[x, Blank[]]]]]], Times[Optional[Pattern[C, Blank[]]], Power[sec[Plus[Optional[Pattern[e, Blank[]]], Times[Optional[Pattern[f, Blank[]]], Pattern[x, Blank[]]]]], 2]]]], Pattern[x, Blank[Symbol]]] := Dist[d^(m + 2), Int[(b + a*Cos[e + f*x])^m*(d*Cos[e + f*x])^(n - m - 2)*(C + B*Cos[e + f*x] + A*Cos[e + f*x]^2), x], x] /; FreeQ[{a, b, d, e, f, A, B, C, n}, x] && !IntegerQ[n] && IntegerQ[m]
  4113. Int[Times[Plus[Optional[Pattern[A, Blank[]]], Times[csc[Plus[Optional[Pattern[e, Blank[]]], Times[Optional[Pattern[f, Blank[]]], Pattern[x, Blank[]]]]], Optional[Pattern[B, Blank[]]]], Times[Power[csc[Plus[Optional[Pattern[e, Blank[]]], Times[Optional[Pattern[f, Blank[]]], Pattern[x, Blank[]]]]], 2], Optional[Pattern[C, Blank[]]]]], Power[Plus[Times[csc[Plus[Optional[Pattern[e, Blank[]]], Times[Optional[Pattern[f, Blank[]]], Pattern[x, Blank[]]]]], Optional[Pattern[b, Blank[]]]], Pattern[a, Blank[]]], Optional[Pattern[m, Blank[]]]], Power[Times[Optional[Pattern[d, Blank[]]], sin[Plus[Optional[Pattern[e, Blank[]]], Times[Optional[Pattern[f, Blank[]]], Pattern[x, Blank[]]]]]], Pattern[n, Blank[]]]], Pattern[x, Blank[Symbol]]] := Dist[d^(m + 2), Int[(b + a*Sin[e + f*x])^m*(d*Sin[e + f*x])^(n - m - 2)*(C + B*Sin[e + f*x] + A*Sin[e + f*x]^2), x], x] /; FreeQ[{a, b, d, e, f, A, B, C, n}, x] && !IntegerQ[n] && IntegerQ[m]
  4114. Int[Times[Power[Times[cos[Plus[Optional[Pattern[e, Blank[]]], Times[Optional[Pattern[f, Blank[]]], Pattern[x, Blank[]]]]], Optional[Pattern[d, Blank[]]]], Pattern[n, Blank[]]], Power[Plus[Pattern[a, Blank[]], Times[Optional[Pattern[b, Blank[]]], sec[Plus[Optional[Pattern[e, Blank[]]], Times[Optional[Pattern[f, Blank[]]], Pattern[x, Blank[]]]]]]], Optional[Pattern[m, Blank[]]]], Plus[Optional[Pattern[A, Blank[]]], Times[Optional[Pattern[C, Blank[]]], Power[sec[Plus[Optional[Pattern[e, Blank[]]], Times[Optional[Pattern[f, Blank[]]], Pattern[x, Blank[]]]]], 2]]]], Pattern[x, Blank[Symbol]]] := Dist[d^(m + 2), Int[(b + a*Cos[e + f*x])^m*(d*Cos[e + f*x])^(n - m - 2)*(C + A*Cos[e + f*x]^2), x], x] /; FreeQ[{a, b, d, e, f, A, C, n}, x] && !IntegerQ[n] && IntegerQ[m]
  4115. Int[Times[Plus[Optional[Pattern[A, Blank[]]], Times[Power[csc[Plus[Optional[Pattern[e, Blank[]]], Times[Optional[Pattern[f, Blank[]]], Pattern[x, Blank[]]]]], 2], Optional[Pattern[C, Blank[]]]]], Power[Plus[Times[csc[Plus[Optional[Pattern[e, Blank[]]], Times[Optional[Pattern[f, Blank[]]], Pattern[x, Blank[]]]]], Optional[Pattern[b, Blank[]]]], Pattern[a, Blank[]]], Optional[Pattern[m, Blank[]]]], Power[Times[Optional[Pattern[d, Blank[]]], sin[Plus[Optional[Pattern[e, Blank[]]], Times[Optional[Pattern[f, Blank[]]], Pattern[x, Blank[]]]]]], Pattern[n, Blank[]]]], Pattern[x, Blank[Symbol]]] := Dist[d^(m + 2), Int[(b + a*Sin[e + f*x])^m*(d*Sin[e + f*x])^(n - m - 2)*(C + A*Sin[e + f*x]^2), x], x] /; FreeQ[{a, b, d, e, f, A, C, n}, x] && !IntegerQ[n] && IntegerQ[m]
  4116. Int[Times[Power[Times[Optional[Pattern[c, Blank[]]], Power[Times[Optional[Pattern[d, Blank[]]], sec[Plus[Optional[Pattern[e, Blank[]]], Times[Optional[Pattern[f, Blank[]]], Pattern[x, Blank[]]]]]], Pattern[p, Blank[]]]], Pattern[n, Blank[]]], Power[Plus[Pattern[a, Blank[]], Times[Optional[Pattern[b, Blank[]]], sec[Plus[Optional[Pattern[e, Blank[]]], Times[Optional[Pattern[f, Blank[]]], Pattern[x, Blank[]]]]]]], Optional[Pattern[m, Blank[]]]], Plus[Optional[Pattern[A, Blank[]]], Times[Optional[Pattern[B, Blank[]]], sec[Plus[Optional[Pattern[e, Blank[]]], Times[Optional[Pattern[f, Blank[]]], Pattern[x, Blank[]]]]]], Times[Optional[Pattern[C, Blank[]]], Power[sec[Plus[Optional[Pattern[e, Blank[]]], Times[Optional[Pattern[f, Blank[]]], Pattern[x, Blank[]]]]], 2]]]], Pattern[x, Blank[Symbol]]] := Dist[(c^IntPart[n]*(c*(d*Sec[e + f*x])^p)^FracPart[n])/(d*Sec[e + f*x])^(p*FracPart[n]), Int[(a + b*Sec[e + f*x])^m*(d*Sec[e + f*x])^(n*p)*(A + B*Sec[e + f*x] + C*Sec[e + f*x]^2), x], x] /; FreeQ[{a, b, c, d, e, f, A, B, C, m, n, p}, x] && !IntegerQ[n]
  4117. Int[Times[Plus[Optional[Pattern[A, Blank[]]], Times[csc[Plus[Optional[Pattern[e, Blank[]]], Times[Optional[Pattern[f, Blank[]]], Pattern[x, Blank[]]]]], Optional[Pattern[B, Blank[]]]], Times[Power[csc[Plus[Optional[Pattern[e, Blank[]]], Times[Optional[Pattern[f, Blank[]]], Pattern[x, Blank[]]]]], 2], Optional[Pattern[C, Blank[]]]]], Power[Times[Optional[Pattern[c, Blank[]]], Power[Times[csc[Plus[Optional[Pattern[e, Blank[]]], Times[Optional[Pattern[f, Blank[]]], Pattern[x, Blank[]]]]], Optional[Pattern[d, Blank[]]]], Pattern[p, Blank[]]]], Pattern[n, Blank[]]], Power[Plus[Times[csc[Plus[Optional[Pattern[e, Blank[]]], Times[Optional[Pattern[f, Blank[]]], Pattern[x, Blank[]]]]], Optional[Pattern[b, Blank[]]]], Pattern[a, Blank[]]], Optional[Pattern[m, Blank[]]]]], Pattern[x, Blank[Symbol]]] := Dist[(c^IntPart[n]*(c*(d*Csc[e + f*x])^p)^FracPart[n])/(d*Csc[e + f*x])^(p*FracPart[n]), Int[(a + b*Csc[e + f*x])^m*(d*Csc[e + f*x])^(n*p)*(A + B*Csc[e + f*x] + C*Csc[e + f*x]^2), x], x] /; FreeQ[{a, b, c, d, e, f, A, B, C, m, n, p}, x] && !IntegerQ[n]
  4118. Int[Times[Power[Times[Optional[Pattern[c, Blank[]]], Power[Times[Optional[Pattern[d, Blank[]]], sec[Plus[Optional[Pattern[e, Blank[]]], Times[Optional[Pattern[f, Blank[]]], Pattern[x, Blank[]]]]]], Pattern[p, Blank[]]]], Pattern[n, Blank[]]], Power[Plus[Pattern[a, Blank[]], Times[Optional[Pattern[b, Blank[]]], sec[Plus[Optional[Pattern[e, Blank[]]], Times[Optional[Pattern[f, Blank[]]], Pattern[x, Blank[]]]]]]], Optional[Pattern[m, Blank[]]]], Plus[Optional[Pattern[A, Blank[]]], Times[Optional[Pattern[C, Blank[]]], Power[sec[Plus[Optional[Pattern[e, Blank[]]], Times[Optional[Pattern[f, Blank[]]], Pattern[x, Blank[]]]]], 2]]]], Pattern[x, Blank[Symbol]]] := Dist[(c^IntPart[n]*(c*(d*Sec[e + f*x])^p)^FracPart[n])/(d*Sec[e + f*x])^(p*FracPart[n]), Int[(a + b*Sec[e + f*x])^m*(d*Sec[e + f*x])^(n*p)*(A + C*Sec[e + f*x]^2), x], x] /; FreeQ[{a, b, c, d, e, f, A, C, m, n, p}, x] && !IntegerQ[n]
  4119. Int[Times[Plus[Optional[Pattern[A, Blank[]]], Times[Power[csc[Plus[Optional[Pattern[e, Blank[]]], Times[Optional[Pattern[f, Blank[]]], Pattern[x, Blank[]]]]], 2], Optional[Pattern[C, Blank[]]]]], Power[Times[Optional[Pattern[c, Blank[]]], Power[Times[csc[Plus[Optional[Pattern[e, Blank[]]], Times[Optional[Pattern[f, Blank[]]], Pattern[x, Blank[]]]]], Optional[Pattern[d, Blank[]]]], Pattern[p, Blank[]]]], Pattern[n, Blank[]]], Power[Plus[Times[csc[Plus[Optional[Pattern[e, Blank[]]], Times[Optional[Pattern[f, Blank[]]], Pattern[x, Blank[]]]]], Optional[Pattern[b, Blank[]]]], Pattern[a, Blank[]]], Optional[Pattern[m, Blank[]]]]], Pattern[x, Blank[Symbol]]] := Dist[(c^IntPart[n]*(c*(d*Csc[e + f*x])^p)^FracPart[n])/(d*Csc[e + f*x])^(p*FracPart[n]), Int[(a + b*Csc[e + f*x])^m*(d*Csc[e + f*x])^(n*p)*(A + C*Csc[e + f*x]^2), x], x] /; FreeQ[{a, b, c, d, e, f, A, C, m, n, p}, x] && !IntegerQ[n]
  4120. Int[Times[Optional[Pattern[u, Blank[]]], Power[Plus[Pattern[a, Blank[]], Times[Optional[Pattern[b, Blank[]]], Power[sec[Plus[Optional[Pattern[e, Blank[]]], Times[Optional[Pattern[f, Blank[]]], Pattern[x, Blank[]]]]], 2]]], Pattern[p, Blank[]]]], Pattern[x, Blank[Symbol]]] := Dist[b^p, Int[ActivateTrig[u*tan[e + f*x]^(2*p)], x], x] /; FreeQ[{a, b, e, f, p}, x] && EqQ[a + b, 0] && IntegerQ[p]
  4121. Int[Times[Optional[Pattern[u, Blank[]]], Power[Plus[Pattern[a, Blank[]], Times[Optional[Pattern[b, Blank[]]], Power[sec[Plus[Optional[Pattern[e, Blank[]]], Times[Optional[Pattern[f, Blank[]]], Pattern[x, Blank[]]]]], 2]]], Pattern[p, Blank[]]]], Pattern[x, Blank[Symbol]]] := Int[ActivateTrig[u*(b*tan[e + f*x]^2)^p], x] /; FreeQ[{a, b, e, f, p}, x] && EqQ[a + b, 0]
  4122. Int[Power[Times[Optional[Pattern[b, Blank[]]], Power[sec[Plus[Optional[Pattern[e, Blank[]]], Times[Optional[Pattern[f, Blank[]]], Pattern[x, Blank[]]]]], 2]], Pattern[p, Blank[]]], Pattern[x, Blank[Symbol]]] := With[{ff = FreeFactors[Tan[e + f*x], x]}, Dist[(b*ff)/f, Subst[Int[(b + b*ff^2*x^2)^(p - 1), x], x, Tan[e + f*x]/ff], x]] /; FreeQ[{b, e, f, p}, x] && !IntegerQ[p]
  4123. Int[Power[Times[Optional[Pattern[b, Blank[]]], Power[Times[Optional[Pattern[c, Blank[]]], sec[Plus[Optional[Pattern[e, Blank[]]], Times[Optional[Pattern[f, Blank[]]], Pattern[x, Blank[]]]]]], Pattern[n, Blank[]]]], Pattern[p, Blank[]]], Pattern[x, Blank[Symbol]]] := Dist[(b^IntPart[p]*(b*(c*Sec[e + f*x])^n)^FracPart[p])/(c*Sec[e + f*x])^(n*FracPart[p]), Int[(c*Sec[e + f*x])^(n*p), x], x] /; FreeQ[{b, c, e, f, n, p}, x] && !IntegerQ[p]
  4124. Int[Times[Power[Times[Optional[Pattern[b, Blank[]]], Power[sec[Plus[Optional[Pattern[e, Blank[]]], Times[Optional[Pattern[f, Blank[]]], Pattern[x, Blank[]]]]], 2]], Optional[Pattern[p, Blank[]]]], Power[tan[Plus[Optional[Pattern[e, Blank[]]], Times[Optional[Pattern[f, Blank[]]], Pattern[x, Blank[]]]]], Optional[Pattern[m, Blank[]]]]], Pattern[x, Blank[Symbol]]] := Dist[b/(2*f), Subst[Int[(-1 + x)^((m - 1)/2)*(b*x)^(p - 1), x], x, Sec[e + f*x]^2], x] /; FreeQ[{b, e, f, p}, x] && !IntegerQ[p] && IntegerQ[(m - 1)/2]
  4125. Int[Times[Optional[Pattern[u, Blank[]]], Power[Times[Optional[Pattern[b, Blank[]]], Power[sec[Plus[Optional[Pattern[e, Blank[]]], Times[Optional[Pattern[f, Blank[]]], Pattern[x, Blank[]]]]], Pattern[n, Blank[]]]], Pattern[p, Blank[]]]], Pattern[x, Blank[Symbol]]] := With[{ff = FreeFactors[Sec[e + f*x], x]}, Dist[((b*ff^n)^IntPart[p]*(b*Sec[e + f*x]^n)^FracPart[p])/(Sec[e + f*x]/ff)^(n*FracPart[p]), Int[ActivateTrig[u]*(Sec[e + f*x]/ff)^(n*p), x], x]] /; FreeQ[{b, e, f, n, p}, x] && !IntegerQ[p] && IntegerQ[n] && (EqQ[u, 1] || MatchQ[u, ((d_.)*(trig_)[e + f*x])^(m_.) /; FreeQ[{d, m}, x] && MemberQ[{sin, cos, tan, cot, sec, csc}, trig]])
  4126. Int[Times[Optional[Pattern[u, Blank[]]], Power[Times[Optional[Pattern[b, Blank[]]], Power[Times[Optional[Pattern[c, Blank[]]], sec[Plus[Optional[Pattern[e, Blank[]]], Times[Optional[Pattern[f, Blank[]]], Pattern[x, Blank[]]]]]], Pattern[n, Blank[]]]], Pattern[p, Blank[]]]], Pattern[x, Blank[Symbol]]] := Dist[(b^IntPart[p]*(b*(c*Sec[e + f*x])^n)^FracPart[p])/(c*Sec[e + f*x])^(n*FracPart[p]), Int[ActivateTrig[u]*(c*Sec[e + f*x])^(n*p), x], x] /; FreeQ[{b, c, e, f, n, p}, x] && !IntegerQ[p] && !IntegerQ[n] && (EqQ[u, 1] || MatchQ[u, ((d_.)*(trig_)[e + f*x])^(m_.) /; FreeQ[{d, m}, x] && MemberQ[{sin, cos, tan, cot, sec, csc}, trig]])
  4127. Int[Power[Plus[Pattern[a, Blank[]], Times[Optional[Pattern[b, Blank[]]], Power[sec[Plus[Optional[Pattern[e, Blank[]]], Times[Optional[Pattern[f, Blank[]]], Pattern[x, Blank[]]]]], 2]]], -1], Pattern[x, Blank[Symbol]]] := Simp[x/a, x] - Dist[b/a, Int[1/(b + a*Cos[e + f*x]^2), x], x] /; FreeQ[{a, b, e, f}, x] && NeQ[a + b, 0]
  4128. Int[Power[Plus[Pattern[a, Blank[]], Times[Optional[Pattern[b, Blank[]]], Power[sec[Plus[Optional[Pattern[e, Blank[]]], Times[Optional[Pattern[f, Blank[]]], Pattern[x, Blank[]]]]], 2]]], Pattern[p, Blank[]]], Pattern[x, Blank[Symbol]]] := With[{ff = FreeFactors[Tan[e + f*x], x]}, Dist[ff/f, Subst[Int[(a + b + b*ff^2*x^2)^p/(1 + ff^2*x^2), x], x, Tan[e + f*x]/ff], x]] /; FreeQ[{a, b, e, f, p}, x] && NeQ[a + b, 0] && NeQ[p, -1]
  4129. Int[Power[Plus[Pattern[a, Blank[]], Times[Optional[Pattern[b, Blank[]]], Power[sec[Plus[Optional[Pattern[e, Blank[]]], Times[Optional[Pattern[f, Blank[]]], Pattern[x, Blank[]]]]], 4]]], Pattern[p, Blank[]]], Pattern[x, Blank[Symbol]]] := With[{ff = FreeFactors[Tan[e + f*x], x]}, Dist[ff/f, Subst[Int[(a + b + 2*b*ff^2*x^2 + b*ff^4*x^4)^p/(1 + ff^2*x^2), x], x, Tan[e + f*x]/ff], x]] /; FreeQ[{a, b, e, f, p}, x] && IntegerQ[2*p]
  4130. Int[Power[Plus[Pattern[a, Blank[]], Times[Optional[Pattern[b, Blank[]]], Power[sec[Plus[Optional[Pattern[e, Blank[]]], Times[Optional[Pattern[f, Blank[]]], Pattern[x, Blank[]]]]], Pattern[n, Blank[]]]]], Pattern[p, Blank[]]], Pattern[x, Blank[Symbol]]] := With[{ff = FreeFactors[Tan[e + f*x], x]}, Dist[ff/f, Subst[Int[(a + b*(1 + ff^2*x^2)^(n/2))^p/(1 + ff^2*x^2), x], x, Tan[e + f*x]/ff], x]] /; FreeQ[{a, b, e, f, p}, x] && IntegerQ[n/2] && IGtQ[p, -2]
  4131. Int[Power[Plus[Pattern[a, Blank[]], Times[Optional[Pattern[b, Blank[]]], Power[Times[Optional[Pattern[c, Blank[]]], sec[Plus[Optional[Pattern[e, Blank[]]], Times[Optional[Pattern[f, Blank[]]], Pattern[x, Blank[]]]]]], Pattern[n, Blank[]]]]], Pattern[p, Blank[]]], Pattern[x, Blank[Symbol]]] := Unintegrable[(a + b*(c*Sec[e + f*x])^n)^p, x] /; FreeQ[{a, b, c, e, f, n, p}, x]
  4132. Int[Times[Power[Plus[Pattern[a, Blank[]], Times[Optional[Pattern[b, Blank[]]], Power[sec[Plus[Optional[Pattern[e, Blank[]]], Times[Optional[Pattern[f, Blank[]]], Pattern[x, Blank[]]]]], Pattern[n, Blank[]]]]], Optional[Pattern[p, Blank[]]]], Power[sin[Plus[Optional[Pattern[e, Blank[]]], Times[Optional[Pattern[f, Blank[]]], Pattern[x, Blank[]]]]], Pattern[m, Blank[]]]], Pattern[x, Blank[Symbol]]] := With[{ff = FreeFactors[Tan[e + f*x], x]}, Dist[ff^(m + 1)/f, Subst[Int[(x^m*ExpandToSum[a + b*(1 + ff^2*x^2)^(n/2), x]^p)/(1 + ff^2*x^2)^(m/2 + 1), x], x, Tan[e + f*x]/ff], x]] /; FreeQ[{a, b, e, f, p}, x] && IntegerQ[m/2] && IntegerQ[n/2]
  4133. Int[Times[Power[Plus[Pattern[a, Blank[]], Times[Optional[Pattern[b, Blank[]]], Power[sec[Plus[Optional[Pattern[e, Blank[]]], Times[Optional[Pattern[f, Blank[]]], Pattern[x, Blank[]]]]], Pattern[n, Blank[]]]]], Optional[Pattern[p, Blank[]]]], Power[sin[Plus[Optional[Pattern[e, Blank[]]], Times[Optional[Pattern[f, Blank[]]], Pattern[x, Blank[]]]]], Optional[Pattern[m, Blank[]]]]], Pattern[x, Blank[Symbol]]] := With[{ff = FreeFactors[Cos[e + f*x], x]}, -Dist[ff/f, Subst[Int[((1 - ff^2*x^2)^((m - 1)/2)*(b + a*(ff*x)^n)^p)/(ff*x)^(n*p), x], x, Cos[e + f*x]/ff], x]] /; FreeQ[{a, b, e, f}, x] && IntegerQ[(m - 1)/2] && IntegerQ[n] && IntegerQ[p]
  4134. Int[Times[Power[Plus[Pattern[a, Blank[]], Times[Optional[Pattern[b, Blank[]]], Power[Times[Optional[Pattern[c, Blank[]]], sec[Plus[Optional[Pattern[e, Blank[]]], Times[Optional[Pattern[f, Blank[]]], Pattern[x, Blank[]]]]]], Pattern[n, Blank[]]]]], Optional[Pattern[p, Blank[]]]], Power[sin[Plus[Optional[Pattern[e, Blank[]]], Times[Optional[Pattern[f, Blank[]]], Pattern[x, Blank[]]]]], Optional[Pattern[m, Blank[]]]]], Pattern[x, Blank[Symbol]]] := With[{ff = FreeFactors[Cos[e + f*x], x]}, Dist[1/(f*ff^m), Subst[Int[((-1 + ff^2*x^2)^((m - 1)/2)*(a + b*(c*ff*x)^n)^p)/x^(m + 1), x], x, Sec[e + f*x]/ff], x]] /; FreeQ[{a, b, c, e, f, n, p}, x] && IntegerQ[(m - 1)/2] && (GtQ[m, 0] || EqQ[n, 2] || EqQ[n, 4])
  4135. Int[Times[Power[Plus[Pattern[a, Blank[]], Times[Optional[Pattern[b, Blank[]]], Power[Times[Optional[Pattern[c, Blank[]]], sec[Plus[Optional[Pattern[e, Blank[]]], Times[Optional[Pattern[f, Blank[]]], Pattern[x, Blank[]]]]]], Pattern[n, Blank[]]]]], Optional[Pattern[p, Blank[]]]], Power[Times[Optional[Pattern[d, Blank[]]], sin[Plus[Optional[Pattern[e, Blank[]]], Times[Optional[Pattern[f, Blank[]]], Pattern[x, Blank[]]]]]], Optional[Pattern[m, Blank[]]]]], Pattern[x, Blank[Symbol]]] := Unintegrable[(a + b*(c*Sec[e + f*x])^n)^p*(d*Sin[e + f*x])^m, x] /; FreeQ[{a, b, c, d, e, f, m, n, p}, x]
  4136. Int[Times[Power[Times[cos[Plus[Optional[Pattern[e, Blank[]]], Times[Optional[Pattern[f, Blank[]]], Pattern[x, Blank[]]]]], Optional[Pattern[d, Blank[]]]], Pattern[m, Blank[]]], Power[Plus[Pattern[a, Blank[]], Times[Optional[Pattern[b, Blank[]]], Power[sec[Plus[Optional[Pattern[e, Blank[]]], Times[Optional[Pattern[f, Blank[]]], Pattern[x, Blank[]]]]], Optional[Pattern[n, Blank[]]]]]], Optional[Pattern[p, Blank[]]]]], Pattern[x, Blank[Symbol]]] := Dist[d^(n*p), Int[(d*Cos[e + f*x])^(m - n*p)*(b + a*Cos[e + f*x]^n)^p, x], x] /; FreeQ[{a, b, d, e, f, m, n, p}, x] && !IntegerQ[m] && IntegersQ[n, p]
  4137. Int[Times[Power[Times[cos[Plus[Optional[Pattern[e, Blank[]]], Times[Optional[Pattern[f, Blank[]]], Pattern[x, Blank[]]]]], Optional[Pattern[d, Blank[]]]], Pattern[m, Blank[]]], Power[Plus[Pattern[a, Blank[]], Times[Optional[Pattern[b, Blank[]]], Power[Times[Optional[Pattern[c, Blank[]]], sec[Plus[Optional[Pattern[e, Blank[]]], Times[Optional[Pattern[f, Blank[]]], Pattern[x, Blank[]]]]]], Pattern[n, Blank[]]]]], Pattern[p, Blank[]]]], Pattern[x, Blank[Symbol]]] := Dist[(d*Cos[e + f*x])^FracPart[m]*(Sec[e + f*x]/d)^FracPart[m], Int[(a + b*(c*Sec[e + f*x])^n)^p/(Sec[e + f*x]/d)^m, x], x] /; FreeQ[{a, b, c, d, e, f, m, n, p}, x] && !IntegerQ[m]
  4138. Int[Times[Power[Plus[Pattern[a, Blank[]], Times[Optional[Pattern[b, Blank[]]], Power[sec[Plus[Optional[Pattern[e, Blank[]]], Times[Optional[Pattern[f, Blank[]]], Pattern[x, Blank[]]]]], Pattern[n, Blank[]]]]], Optional[Pattern[p, Blank[]]]], Power[tan[Plus[Optional[Pattern[e, Blank[]]], Times[Optional[Pattern[f, Blank[]]], Pattern[x, Blank[]]]]], Optional[Pattern[m, Blank[]]]]], Pattern[x, Blank[Symbol]]] := Module[{ff = FreeFactors[Cos[e + f*x], x]}, -Dist[(f*ff^(m + n*p - 1))^(-1), Subst[Int[((1 - ff^2*x^2)^((m - 1)/2)*(b + a*(ff*x)^n)^p)/x^(m + n*p), x], x, Cos[e + f*x]/ff], x]] /; FreeQ[{a, b, e, f, n}, x] && IntegerQ[(m - 1)/2] && IntegerQ[n] && IntegerQ[p]
  4139. Int[Times[Power[Plus[Pattern[a, Blank[]], Times[Optional[Pattern[b, Blank[]]], Power[Times[Optional[Pattern[c, Blank[]]], sec[Plus[Optional[Pattern[e, Blank[]]], Times[Optional[Pattern[f, Blank[]]], Pattern[x, Blank[]]]]]], Pattern[n, Blank[]]]]], Optional[Pattern[p, Blank[]]]], Power[tan[Plus[Optional[Pattern[e, Blank[]]], Times[Optional[Pattern[f, Blank[]]], Pattern[x, Blank[]]]]], Optional[Pattern[m, Blank[]]]]], Pattern[x, Blank[Symbol]]] := With[{ff = FreeFactors[Sec[e + f*x], x]}, Dist[1/f, Subst[Int[((-1 + ff^2*x^2)^((m - 1)/2)*(a + b*(c*ff*x)^n)^p)/x, x], x, Sec[e + f*x]/ff], x]] /; FreeQ[{a, b, c, e, f, n, p}, x] && IntegerQ[(m - 1)/2] && (GtQ[m, 0] || EqQ[n, 2] || EqQ[n, 4] || IGtQ[p, 0] || IntegersQ[2*n, p])
  4140. Int[Times[Power[Times[Optional[Pattern[b, Blank[]]], Power[sec[Plus[Optional[Pattern[e, Blank[]]], Times[Optional[Pattern[f, Blank[]]], Pattern[x, Blank[]]]]], 2]], Optional[Pattern[p, Blank[]]]], Power[Times[Optional[Pattern[d, Blank[]]], tan[Plus[Optional[Pattern[e, Blank[]]], Times[Optional[Pattern[f, Blank[]]], Pattern[x, Blank[]]]]]], Pattern[m, Blank[]]]], Pattern[x, Blank[Symbol]]] := With[{ff = FreeFactors[Tan[e + f*x], x]}, Dist[(b*ff)/f, Subst[Int[(d*ff*x)^m*(b + b*ff^2*x^2)^(p - 1), x], x, Tan[e + f*x]/ff], x]] /; FreeQ[{b, d, e, f, m, p}, x]
  4141. Int[Times[Power[Plus[Pattern[a, Blank[]], Times[Optional[Pattern[b, Blank[]]], Power[sec[Plus[Optional[Pattern[e, Blank[]]], Times[Optional[Pattern[f, Blank[]]], Pattern[x, Blank[]]]]], Pattern[n, Blank[]]]]], Optional[Pattern[p, Blank[]]]], Power[Times[Optional[Pattern[d, Blank[]]], tan[Plus[Optional[Pattern[e, Blank[]]], Times[Optional[Pattern[f, Blank[]]], Pattern[x, Blank[]]]]]], Pattern[m, Blank[]]]], Pattern[x, Blank[Symbol]]] := With[{ff = FreeFactors[Tan[e + f*x], x]}, Dist[ff/f, Subst[Int[((d*ff*x)^m*(a + b*(1 + ff^2*x^2)^(n/2))^p)/(1 + ff^2*x^2), x], x, Tan[e + f*x]/ff], x]] /; FreeQ[{a, b, d, e, f, m, p}, x] && IntegerQ[n/2] && (IntegerQ[m/2] || EqQ[n, 2])
  4142. Int[Times[Power[Times[Optional[Pattern[b, Blank[]]], Power[Times[Optional[Pattern[c, Blank[]]], sec[Plus[Optional[Pattern[e, Blank[]]], Times[Optional[Pattern[f, Blank[]]], Pattern[x, Blank[]]]]]], Pattern[n, Blank[]]]], Optional[Pattern[p, Blank[]]]], Power[Times[Optional[Pattern[d, Blank[]]], tan[Plus[Optional[Pattern[e, Blank[]]], Times[Optional[Pattern[f, Blank[]]], Pattern[x, Blank[]]]]]], Pattern[m, Blank[]]]], Pattern[x, Blank[Symbol]]] := Simp[(d*(d*Tan[e + f*x])^(m - 1)*(b*(c*Sec[e + f*x])^n)^p)/(f*(p*n + m - 1)), x] - Dist[(d^2*(m - 1))/(p*n + m - 1), Int[(d*Tan[e + f*x])^(m - 2)*(b*(c*Sec[e + f*x])^n)^p, x], x] /; FreeQ[{b, c, d, e, f, p, n}, x] && GtQ[m, 1] && NeQ[p*n + m - 1, 0] && IntegersQ[2*p*n, 2*m]
  4143. Int[Times[Power[Times[Optional[Pattern[b, Blank[]]], Power[Times[Optional[Pattern[c, Blank[]]], sec[Plus[Optional[Pattern[e, Blank[]]], Times[Optional[Pattern[f, Blank[]]], Pattern[x, Blank[]]]]]], Pattern[n, Blank[]]]], Optional[Pattern[p, Blank[]]]], Power[Times[Optional[Pattern[d, Blank[]]], tan[Plus[Optional[Pattern[e, Blank[]]], Times[Optional[Pattern[f, Blank[]]], Pattern[x, Blank[]]]]]], Pattern[m, Blank[]]]], Pattern[x, Blank[Symbol]]] := Simp[((d*Tan[e + f*x])^(m + 1)*(b*(c*Sec[e + f*x])^n)^p)/(d*f*(m + 1)), x] - Dist[(p*n + m + 1)/(d^2*(m + 1)), Int[(d*Tan[e + f*x])^(m + 2)*(b*(c*Sec[e + f*x])^n)^p, x], x] /; FreeQ[{b, c, d, e, f, p, n}, x] && LtQ[m, -1] && NeQ[p*n + m + 1, 0] && IntegersQ[2*p*n, 2*m]
  4144. Int[Times[Power[Plus[Pattern[a, Blank[]], Times[Optional[Pattern[b, Blank[]]], Power[Times[Optional[Pattern[c, Blank[]]], sec[Plus[Optional[Pattern[e, Blank[]]], Times[Optional[Pattern[f, Blank[]]], Pattern[x, Blank[]]]]]], Pattern[n, Blank[]]]]], Optional[Pattern[p, Blank[]]]], Power[Times[Optional[Pattern[d, Blank[]]], tan[Plus[Optional[Pattern[e, Blank[]]], Times[Optional[Pattern[f, Blank[]]], Pattern[x, Blank[]]]]]], Optional[Pattern[m, Blank[]]]]], Pattern[x, Blank[Symbol]]] := Unintegrable[(a + b*(c*Sec[e + f*x])^n)^p*(d*Tan[e + f*x])^m, x] /; FreeQ[{a, b, c, d, e, f, m, n, p}, x]
  4145. Int[Times[Power[Times[cot[Plus[Optional[Pattern[e, Blank[]]], Times[Optional[Pattern[f, Blank[]]], Pattern[x, Blank[]]]]], Optional[Pattern[d, Blank[]]]], Pattern[m, Blank[]]], Power[Plus[Pattern[a, Blank[]], Times[Optional[Pattern[b, Blank[]]], Power[Times[Optional[Pattern[c, Blank[]]], sec[Plus[Optional[Pattern[e, Blank[]]], Times[Optional[Pattern[f, Blank[]]], Pattern[x, Blank[]]]]]], Pattern[n, Blank[]]]]], Pattern[p, Blank[]]]], Pattern[x, Blank[Symbol]]] := Dist[(d*Cot[e + f*x])^FracPart[m]*(Tan[e + f*x]/d)^FracPart[m], Int[(a + b*(c*Sec[e + f*x])^n)^p/(Tan[e + f*x]/d)^m, x], x] /; FreeQ[{a, b, c, d, e, f, m, n, p}, x] && !IntegerQ[m]
  4146. Int[Times[Power[sec[Plus[Optional[Pattern[e, Blank[]]], Times[Optional[Pattern[f, Blank[]]], Pattern[x, Blank[]]]]], Pattern[m, Blank[]]], Power[Plus[Pattern[a, Blank[]], Times[Optional[Pattern[b, Blank[]]], Power[sec[Plus[Optional[Pattern[e, Blank[]]], Times[Optional[Pattern[f, Blank[]]], Pattern[x, Blank[]]]]], Pattern[n, Blank[]]]]], Pattern[p, Blank[]]]], Pattern[x, Blank[Symbol]]] := With[{ff = FreeFactors[Tan[e + f*x], x]}, Dist[ff/f, Subst[Int[(1 + ff^2*x^2)^(m/2 - 1)*ExpandToSum[a + b*(1 + ff^2*x^2)^(n/2), x]^p, x], x, Tan[e + f*x]/ff], x]] /; FreeQ[{a, b, e, f, p}, x] && IntegerQ[m/2] && IntegerQ[n/2]
  4147. Int[Times[Power[sec[Plus[Optional[Pattern[e, Blank[]]], Times[Optional[Pattern[f, Blank[]]], Pattern[x, Blank[]]]]], Optional[Pattern[m, Blank[]]]], Power[Plus[Pattern[a, Blank[]], Times[Optional[Pattern[b, Blank[]]], Power[sec[Plus[Optional[Pattern[e, Blank[]]], Times[Optional[Pattern[f, Blank[]]], Pattern[x, Blank[]]]]], Pattern[n, Blank[]]]]], Pattern[p, Blank[]]]], Pattern[x, Blank[Symbol]]] := With[{ff = FreeFactors[Sin[e + f*x], x]}, Dist[ff/f, Subst[Int[ExpandToSum[b + a*(1 - ff^2*x^2)^(n/2), x]^p/(1 - ff^2*x^2)^((m + n*p + 1)/2), x], x, Sin[e + f*x]/ff], x]] /; FreeQ[{a, b, e, f}, x] && IntegerQ[(m - 1)/2] && IntegerQ[n/2] && IntegerQ[p]
  4148. Int[Times[Power[sec[Plus[Optional[Pattern[e, Blank[]]], Times[Optional[Pattern[f, Blank[]]], Pattern[x, Blank[]]]]], Optional[Pattern[m, Blank[]]]], Power[Plus[Pattern[a, Blank[]], Times[Optional[Pattern[b, Blank[]]], Power[sec[Plus[Optional[Pattern[e, Blank[]]], Times[Optional[Pattern[f, Blank[]]], Pattern[x, Blank[]]]]], Pattern[n, Blank[]]]]], Pattern[p, Blank[]]]], Pattern[x, Blank[Symbol]]] := With[{ff = FreeFactors[Sin[e + f*x], x]}, Dist[ff/f, Subst[Int[(a + b/(1 - ff^2*x^2)^(n/2))^p/(1 - ff^2*x^2)^((m + 1)/2), x], x, Sin[e + f*x]/ff], x]] /; FreeQ[{a, b, e, f, p}, x] && IntegerQ[(m - 1)/2] && IntegerQ[n/2] && !IntegerQ[p]
  4149. Int[Times[Power[sec[Plus[Optional[Pattern[e, Blank[]]], Times[Optional[Pattern[f, Blank[]]], Pattern[x, Blank[]]]]], Optional[Pattern[m, Blank[]]]], Power[Plus[Pattern[a, Blank[]], Times[Optional[Pattern[b, Blank[]]], Power[sec[Plus[Optional[Pattern[e, Blank[]]], Times[Optional[Pattern[f, Blank[]]], Pattern[x, Blank[]]]]], Pattern[n, Blank[]]]]], Pattern[p, Blank[]]]], Pattern[x, Blank[Symbol]]] := Int[ExpandTrig[sec[e + f*x]^m*(a + b*sec[e + f*x]^n)^p, x], x] /; FreeQ[{a, b, e, f}, x] && IntegersQ[m, n, p]
  4150. Int[Times[Power[Times[Optional[Pattern[d, Blank[]]], sec[Plus[Optional[Pattern[e, Blank[]]], Times[Optional[Pattern[f, Blank[]]], Pattern[x, Blank[]]]]]], Optional[Pattern[m, Blank[]]]], Power[Plus[Pattern[a, Blank[]], Times[Optional[Pattern[b, Blank[]]], Power[Times[Optional[Pattern[c, Blank[]]], sec[Plus[Optional[Pattern[e, Blank[]]], Times[Optional[Pattern[f, Blank[]]], Pattern[x, Blank[]]]]]], Pattern[n, Blank[]]]]], Optional[Pattern[p, Blank[]]]]], Pattern[x, Blank[Symbol]]] := Unintegrable[(d*Sec[e + f*x])^m*(a + b*(c*Sec[e + f*x])^n)^p, x] /; FreeQ[{a, b, c, d, e, f, m, n, p}, x]
  4151. Int[Times[Power[Times[csc[Plus[Optional[Pattern[e, Blank[]]], Times[Optional[Pattern[f, Blank[]]], Pattern[x, Blank[]]]]], Optional[Pattern[d, Blank[]]]], Pattern[m, Blank[]]], Power[Plus[Pattern[a, Blank[]], Times[Optional[Pattern[b, Blank[]]], Power[Times[Optional[Pattern[c, Blank[]]], sec[Plus[Optional[Pattern[e, Blank[]]], Times[Optional[Pattern[f, Blank[]]], Pattern[x, Blank[]]]]]], Pattern[n, Blank[]]]]], Pattern[p, Blank[]]]], Pattern[x, Blank[Symbol]]] := Dist[(d*Csc[e + f*x])^FracPart[m]*(Sin[e + f*x]/d)^FracPart[m], Int[(a + b*(c*Sec[e + f*x])^n)^p/(Sin[e + f*x]/d)^m, x], x] /; FreeQ[{a, b, c, d, e, f, m, n, p}, x] && !IntegerQ[m]
  4152. Int[Power[Plus[Optional[Pattern[a, Blank[]]], Times[Optional[Pattern[b, Blank[]]], Power[sec[Plus[Optional[Pattern[d, Blank[]]], Times[Optional[Pattern[e, Blank[]]], Pattern[x, Blank[]]]]], Optional[Pattern[n, Blank[]]]]], Times[Optional[Pattern[c, Blank[]]], Power[sec[Plus[Optional[Pattern[d, Blank[]]], Times[Optional[Pattern[e, Blank[]]], Pattern[x, Blank[]]]]], Optional[Pattern[n2, Blank[]]]]]], Optional[Pattern[p, Blank[]]]], Pattern[x, Blank[Symbol]]] := Dist[1/(4^p*c^p), Int[(b + 2*c*Sec[d + e*x]^n)^(2*p), x], x] /; FreeQ[{a, b, c, d, e, n}, x] && EqQ[n2, 2*n] && EqQ[b^2 - 4*a*c, 0] && IntegerQ[p]
  4153. Int[Power[Plus[Optional[Pattern[a, Blank[]]], Times[Power[csc[Plus[Optional[Pattern[d, Blank[]]], Times[Optional[Pattern[e, Blank[]]], Pattern[x, Blank[]]]]], Optional[Pattern[n, Blank[]]]], Optional[Pattern[b, Blank[]]]], Times[Power[csc[Plus[Optional[Pattern[d, Blank[]]], Times[Optional[Pattern[e, Blank[]]], Pattern[x, Blank[]]]]], Optional[Pattern[n2, Blank[]]]], Optional[Pattern[c, Blank[]]]]], Optional[Pattern[p, Blank[]]]], Pattern[x, Blank[Symbol]]] := Dist[1/(4^p*c^p), Int[(b + 2*c*Csc[d + e*x]^n)^(2*p), x], x] /; FreeQ[{a, b, c, d, e, n}, x] && EqQ[n2, 2*n] && EqQ[b^2 - 4*a*c, 0] && IntegerQ[p]
  4154. Int[Power[Plus[Optional[Pattern[a, Blank[]]], Times[Optional[Pattern[b, Blank[]]], Power[sec[Plus[Optional[Pattern[d, Blank[]]], Times[Optional[Pattern[e, Blank[]]], Pattern[x, Blank[]]]]], Optional[Pattern[n, Blank[]]]]], Times[Optional[Pattern[c, Blank[]]], Power[sec[Plus[Optional[Pattern[d, Blank[]]], Times[Optional[Pattern[e, Blank[]]], Pattern[x, Blank[]]]]], Optional[Pattern[n2, Blank[]]]]]], Pattern[p, Blank[]]], Pattern[x, Blank[Symbol]]] := Dist[(a + b*Sec[d + e*x]^n + c*Sec[d + e*x]^(2*n))^p/(b + 2*c*Sec[d + e*x]^n)^(2*p), Int[u*(b + 2*c*Sec[d + e*x]^n)^(2*p), x], x] /; FreeQ[{a, b, c, d, e, n, p}, x] && EqQ[n2, 2*n] && EqQ[b^2 - 4*a*c, 0] && !IntegerQ[p]
  4155. Int[Power[Plus[Optional[Pattern[a, Blank[]]], Times[Power[csc[Plus[Optional[Pattern[d, Blank[]]], Times[Optional[Pattern[e, Blank[]]], Pattern[x, Blank[]]]]], Optional[Pattern[n, Blank[]]]], Optional[Pattern[b, Blank[]]]], Times[Power[csc[Plus[Optional[Pattern[d, Blank[]]], Times[Optional[Pattern[e, Blank[]]], Pattern[x, Blank[]]]]], Optional[Pattern[n2, Blank[]]]], Optional[Pattern[c, Blank[]]]]], Pattern[p, Blank[]]], Pattern[x, Blank[Symbol]]] := Dist[(a + b*Csc[d + e*x]^n + c*Csc[d + e*x]^(2*n))^p/(b + 2*c*Csc[d + e*x]^n)^(2*p), Int[u*(b + 2*c*Csc[d + e*x]^n)^(2*p), x], x] /; FreeQ[{a, b, c, d, e, n, p}, x] && EqQ[n2, 2*n] && EqQ[b^2 - 4*a*c, 0] && !IntegerQ[p]
  4156. Int[Power[Plus[Optional[Pattern[a, Blank[]]], Times[Optional[Pattern[b, Blank[]]], Power[sec[Plus[Optional[Pattern[d, Blank[]]], Times[Optional[Pattern[e, Blank[]]], Pattern[x, Blank[]]]]], Optional[Pattern[n, Blank[]]]]], Times[Optional[Pattern[c, Blank[]]], Power[sec[Plus[Optional[Pattern[d, Blank[]]], Times[Optional[Pattern[e, Blank[]]], Pattern[x, Blank[]]]]], Optional[Pattern[n2, Blank[]]]]]], -1], Pattern[x, Blank[Symbol]]] := Module[{q = Rt[b^2 - 4*a*c, 2]}, Dist[(2*c)/q, Int[1/(b - q + 2*c*Sec[d + e*x]^n), x], x] - Dist[(2*c)/q, Int[1/(b + q + 2*c*Sec[d + e*x]^n), x], x]] /; FreeQ[{a, b, c, d, e, n}, x] && EqQ[n2, 2*n] && NeQ[b^2 - 4*a*c, 0]
  4157. Int[Power[Plus[Optional[Pattern[a, Blank[]]], Times[Power[csc[Plus[Optional[Pattern[d, Blank[]]], Times[Optional[Pattern[e, Blank[]]], Pattern[x, Blank[]]]]], Optional[Pattern[n, Blank[]]]], Optional[Pattern[b, Blank[]]]], Times[Power[csc[Plus[Optional[Pattern[d, Blank[]]], Times[Optional[Pattern[e, Blank[]]], Pattern[x, Blank[]]]]], Optional[Pattern[n2, Blank[]]]], Optional[Pattern[c, Blank[]]]]], -1], Pattern[x, Blank[Symbol]]] := Module[{q = Rt[b^2 - 4*a*c, 2]}, Dist[(2*c)/q, Int[1/(b - q + 2*c*Csc[d + e*x]^n), x], x] - Dist[(2*c)/q, Int[1/(b + q + 2*c*Csc[d + e*x]^n), x], x]] /; FreeQ[{a, b, c, d, e, n}, x] && EqQ[n2, 2*n] && NeQ[b^2 - 4*a*c, 0]
  4158. Int[Times[Power[Plus[Optional[Pattern[a, Blank[]]], Times[Optional[Pattern[b, Blank[]]], Power[sec[Plus[Optional[Pattern[d, Blank[]]], Times[Optional[Pattern[e, Blank[]]], Pattern[x, Blank[]]]]], Optional[Pattern[n, Blank[]]]]], Times[Optional[Pattern[c, Blank[]]], Power[sec[Plus[Optional[Pattern[d, Blank[]]], Times[Optional[Pattern[e, Blank[]]], Pattern[x, Blank[]]]]], Pattern[n2, Blank[]]]]], Optional[Pattern[p, Blank[]]]], Power[sin[Plus[Optional[Pattern[d, Blank[]]], Times[Optional[Pattern[e, Blank[]]], Pattern[x, Blank[]]]]], Optional[Pattern[m, Blank[]]]]], Pattern[x, Blank[Symbol]]] := Module[{f = FreeFactors[Cos[d + e*x], x]}, -Dist[f/e, Subst[Int[((1 - f^2*x^2)^((m - 1)/2)*(b + a*(f*x)^n)^p)/(f*x)^(n*p), x], x, Cos[d + e*x]/f], x]] /; FreeQ[{a, b, c, d, e}, x] && EqQ[n2, 2*n] && IntegerQ[(m - 1)/2] && IntegersQ[n, p]
  4159. Int[Times[Power[cos[Plus[Optional[Pattern[d, Blank[]]], Times[Optional[Pattern[e, Blank[]]], Pattern[x, Blank[]]]]], Optional[Pattern[m, Blank[]]]], Power[Plus[Optional[Pattern[a, Blank[]]], Times[Power[csc[Plus[Optional[Pattern[d, Blank[]]], Times[Optional[Pattern[e, Blank[]]], Pattern[x, Blank[]]]]], Optional[Pattern[n, Blank[]]]], Optional[Pattern[b, Blank[]]]], Times[Power[csc[Plus[Optional[Pattern[d, Blank[]]], Times[Optional[Pattern[e, Blank[]]], Pattern[x, Blank[]]]]], Pattern[n2, Blank[]]], Optional[Pattern[c, Blank[]]]]], Optional[Pattern[p, Blank[]]]]], Pattern[x, Blank[Symbol]]] := Module[{f = FreeFactors[Sin[d + e*x], x]}, Dist[f/e, Subst[Int[((1 - f^2*x^2)^((m - 1)/2)*(b + a*(f*x)^n)^p)/(f*x)^(n*p), x], x, Sin[d + e*x]/f], x]] /; FreeQ[{a, b, c, d, e}, x] && EqQ[n2, 2*n] && IntegerQ[(m - 1)/2] && IntegersQ[n, p]
  4160. Int[Times[Power[Plus[Optional[Pattern[a, Blank[]]], Times[Optional[Pattern[b, Blank[]]], Power[sec[Plus[Optional[Pattern[d, Blank[]]], Times[Optional[Pattern[e, Blank[]]], Pattern[x, Blank[]]]]], Pattern[n, Blank[]]]], Times[Optional[Pattern[c, Blank[]]], Power[sec[Plus[Optional[Pattern[d, Blank[]]], Times[Optional[Pattern[e, Blank[]]], Pattern[x, Blank[]]]]], Pattern[n2, Blank[]]]]], Optional[Pattern[p, Blank[]]]], Power[sin[Plus[Optional[Pattern[d, Blank[]]], Times[Optional[Pattern[e, Blank[]]], Pattern[x, Blank[]]]]], Pattern[m, Blank[]]]], Pattern[x, Blank[Symbol]]] := Module[{f = FreeFactors[Tan[d + e*x], x]}, Dist[f^(m + 1)/e, Subst[Int[(x^m*ExpandToSum[a + b*(1 + f^2*x^2)^(n/2) + c*(1 + f^2*x^2)^n, x]^p)/(1 + f^2*x^2)^(m/2 + 1), x], x, Tan[d + e*x]/f], x]] /; FreeQ[{a, b, c, d, e, p}, x] && EqQ[n2, 2*n] && IntegerQ[m/2] && IntegerQ[n/2]
  4161. Int[Times[Power[cos[Plus[Optional[Pattern[d, Blank[]]], Times[Optional[Pattern[e, Blank[]]], Pattern[x, Blank[]]]]], Pattern[m, Blank[]]], Power[Plus[Optional[Pattern[a, Blank[]]], Times[Power[csc[Plus[Optional[Pattern[d, Blank[]]], Times[Optional[Pattern[e, Blank[]]], Pattern[x, Blank[]]]]], Pattern[n, Blank[]]], Optional[Pattern[b, Blank[]]]], Times[Power[csc[Plus[Optional[Pattern[d, Blank[]]], Times[Optional[Pattern[e, Blank[]]], Pattern[x, Blank[]]]]], Pattern[n2, Blank[]]], Optional[Pattern[c, Blank[]]]]], Optional[Pattern[p, Blank[]]]]], Pattern[x, Blank[Symbol]]] := Module[{f = FreeFactors[Cot[d + e*x], x]}, -Dist[f^(m + 1)/e, Subst[Int[(x^m*ExpandToSum[a + b*(1 + f^2*x^2)^(n/2) + c*(1 + f^2*x^2)^n, x]^p)/(1 + f^2*x^2)^(m/2 + 1), x], x, Cot[d + e*x]/f], x]] /; FreeQ[{a, b, c, d, e, p}, x] && EqQ[n2, 2*n] && IntegerQ[m/2] && IntegerQ[n/2]
  4162. Int[Times[Power[sec[Plus[Optional[Pattern[d, Blank[]]], Times[Optional[Pattern[e, Blank[]]], Pattern[x, Blank[]]]]], Optional[Pattern[m, Blank[]]]], Power[Plus[Optional[Pattern[a, Blank[]]], Times[Optional[Pattern[b, Blank[]]], Power[sec[Plus[Optional[Pattern[d, Blank[]]], Times[Optional[Pattern[e, Blank[]]], Pattern[x, Blank[]]]]], Optional[Pattern[n, Blank[]]]]], Times[Optional[Pattern[c, Blank[]]], Power[sec[Plus[Optional[Pattern[d, Blank[]]], Times[Optional[Pattern[e, Blank[]]], Pattern[x, Blank[]]]]], Optional[Pattern[n2, Blank[]]]]]], Pattern[p, Blank[]]]], Pattern[x, Blank[Symbol]]] := Dist[1/(4^p*c^p), Int[Sec[d + e*x]^m*(b + 2*c*Sec[d + e*x]^n)^(2*p), x], x] /; FreeQ[{a, b, c, d, e, m, n}, x] && EqQ[n2, 2*n] && EqQ[b^2 - 4*a*c, 0] && IntegerQ[p]
  4163. Int[Times[Power[csc[Plus[Optional[Pattern[d, Blank[]]], Times[Optional[Pattern[e, Blank[]]], Pattern[x, Blank[]]]]], Optional[Pattern[m, Blank[]]]], Power[Plus[Optional[Pattern[a, Blank[]]], Times[Power[csc[Plus[Optional[Pattern[d, Blank[]]], Times[Optional[Pattern[e, Blank[]]], Pattern[x, Blank[]]]]], Optional[Pattern[n, Blank[]]]], Optional[Pattern[b, Blank[]]]], Times[Power[csc[Plus[Optional[Pattern[d, Blank[]]], Times[Optional[Pattern[e, Blank[]]], Pattern[x, Blank[]]]]], Optional[Pattern[n2, Blank[]]]], Optional[Pattern[c, Blank[]]]]], Pattern[p, Blank[]]]], Pattern[x, Blank[Symbol]]] := Dist[1/(4^p*c^p), Int[Csc[d + e*x]^m*(b + 2*c*Csc[d + e*x]^n)^(2*p), x], x] /; FreeQ[{a, b, c, d, e, m, n}, x] && EqQ[n2, 2*n] && EqQ[b^2 - 4*a*c, 0] && IntegerQ[p]
  4164. Int[Times[Power[sec[Plus[Optional[Pattern[d, Blank[]]], Times[Optional[Pattern[e, Blank[]]], Pattern[x, Blank[]]]]], Optional[Pattern[m, Blank[]]]], Power[Plus[Optional[Pattern[a, Blank[]]], Times[Optional[Pattern[b, Blank[]]], Power[sec[Plus[Optional[Pattern[d, Blank[]]], Times[Optional[Pattern[e, Blank[]]], Pattern[x, Blank[]]]]], Optional[Pattern[n, Blank[]]]]], Times[Optional[Pattern[c, Blank[]]], Power[sec[Plus[Optional[Pattern[d, Blank[]]], Times[Optional[Pattern[e, Blank[]]], Pattern[x, Blank[]]]]], Optional[Pattern[n2, Blank[]]]]]], Pattern[p, Blank[]]]], Pattern[x, Blank[Symbol]]] := Dist[(a + b*Sec[d + e*x]^n + c*Sec[d + e*x]^(2*n))^p/(b + 2*c*Sec[d + e*x]^n)^(2*p), Int[Sec[d + e*x]^m*(b + 2*c*Sec[d + e*x]^n)^(2*p), x], x] /; FreeQ[{a, b, c, d, e, m, n, p}, x] && EqQ[n2, 2*n] && EqQ[b^2 - 4*a*c, 0] && !IntegerQ[p]
  4165. Int[Times[Power[csc[Plus[Optional[Pattern[d, Blank[]]], Times[Optional[Pattern[e, Blank[]]], Pattern[x, Blank[]]]]], Optional[Pattern[m, Blank[]]]], Power[Plus[Optional[Pattern[a, Blank[]]], Times[Power[csc[Plus[Optional[Pattern[d, Blank[]]], Times[Optional[Pattern[e, Blank[]]], Pattern[x, Blank[]]]]], Optional[Pattern[n, Blank[]]]], Optional[Pattern[b, Blank[]]]], Times[Power[csc[Plus[Optional[Pattern[d, Blank[]]], Times[Optional[Pattern[e, Blank[]]], Pattern[x, Blank[]]]]], Optional[Pattern[n2, Blank[]]]], Optional[Pattern[c, Blank[]]]]], Pattern[p, Blank[]]]], Pattern[x, Blank[Symbol]]] := Dist[(a + b*Csc[d + e*x]^n + c*Csc[d + e*x]^(2*n))^p/(b + 2*c*Csc[d + e*x]^n)^(2*p), Int[Csc[d + e*x]^m*(b + 2*c*Csc[d + e*x]^n)^(2*p), x], x] /; FreeQ[{a, b, c, d, e, m, n, p}, x] && EqQ[n2, 2*n] && EqQ[b^2 - 4*a*c, 0] && !IntegerQ[p]
  4166. Int[Times[Power[sec[Plus[Optional[Pattern[d, Blank[]]], Times[Optional[Pattern[e, Blank[]]], Pattern[x, Blank[]]]]], Optional[Pattern[m, Blank[]]]], Power[Plus[Optional[Pattern[a, Blank[]]], Times[Optional[Pattern[b, Blank[]]], Power[sec[Plus[Optional[Pattern[d, Blank[]]], Times[Optional[Pattern[e, Blank[]]], Pattern[x, Blank[]]]]], Optional[Pattern[n, Blank[]]]]], Times[Optional[Pattern[c, Blank[]]], Power[sec[Plus[Optional[Pattern[d, Blank[]]], Times[Optional[Pattern[e, Blank[]]], Pattern[x, Blank[]]]]], Optional[Pattern[n2, Blank[]]]]]], Pattern[p, Blank[]]]], Pattern[x, Blank[Symbol]]] := Int[ExpandTrig[sec[d + e*x]^m*(a + b*sec[d + e*x]^n + c*sec[d + e*x]^(2*n))^p, x], x] /; FreeQ[{a, b, c, d, e}, x] && EqQ[n2, 2*n] && IntegersQ[m, n, p]
  4167. Int[Times[Power[csc[Plus[Optional[Pattern[d, Blank[]]], Times[Optional[Pattern[e, Blank[]]], Pattern[x, Blank[]]]]], Optional[Pattern[m, Blank[]]]], Power[Plus[Optional[Pattern[a, Blank[]]], Times[Power[csc[Plus[Optional[Pattern[d, Blank[]]], Times[Optional[Pattern[e, Blank[]]], Pattern[x, Blank[]]]]], Optional[Pattern[n, Blank[]]]], Optional[Pattern[b, Blank[]]]], Times[Power[csc[Plus[Optional[Pattern[d, Blank[]]], Times[Optional[Pattern[e, Blank[]]], Pattern[x, Blank[]]]]], Optional[Pattern[n2, Blank[]]]], Optional[Pattern[c, Blank[]]]]], Pattern[p, Blank[]]]], Pattern[x, Blank[Symbol]]] := Int[ExpandTrig[csc[d + e*x]^m*(a + b*csc[d + e*x]^n + c*csc[d + e*x]^(2*n))^p, x], x] /; FreeQ[{a, b, c, d, e}, x] && EqQ[n2, 2*n] && IntegersQ[m, n, p]
  4168. Int[Times[Power[Plus[Pattern[a, Blank[]], Times[Optional[Pattern[b, Blank[]]], Power[sec[Plus[Optional[Pattern[d, Blank[]]], Times[Optional[Pattern[e, Blank[]]], Pattern[x, Blank[]]]]], Optional[Pattern[n, Blank[]]]]], Times[Optional[Pattern[c, Blank[]]], Power[sec[Plus[Optional[Pattern[d, Blank[]]], Times[Optional[Pattern[e, Blank[]]], Pattern[x, Blank[]]]]], Optional[Pattern[n2, Blank[]]]]]], Optional[Pattern[p, Blank[]]]], Power[tan[Plus[Optional[Pattern[d, Blank[]]], Times[Optional[Pattern[e, Blank[]]], Pattern[x, Blank[]]]]], Optional[Pattern[m, Blank[]]]]], Pattern[x, Blank[Symbol]]] := Module[{f = FreeFactors[Cos[d + e*x], x]}, -Dist[(e*f^(m + n*p - 1))^(-1), Subst[Int[((1 - f^2*x^2)^((m - 1)/2)*(c + b*(f*x)^n + c*(f*x)^(2*n))^p)/x^(m + 2*n*p), x], x, Cos[d + e*x]/f], x]] /; FreeQ[{a, b, c, d, e, n}, x] && EqQ[n2, 2*n] && IntegerQ[(m - 1)/2] && IntegerQ[n] && IntegerQ[p]
  4169. Int[Times[Power[cot[Plus[Optional[Pattern[d, Blank[]]], Times[Optional[Pattern[e, Blank[]]], Pattern[x, Blank[]]]]], Optional[Pattern[m, Blank[]]]], Power[Plus[Times[Power[csc[Plus[Optional[Pattern[d, Blank[]]], Times[Optional[Pattern[e, Blank[]]], Pattern[x, Blank[]]]]], Optional[Pattern[n, Blank[]]]], Optional[Pattern[b, Blank[]]]], Pattern[a, Blank[]], Times[Optional[Pattern[c, Blank[]]], Power[sec[Plus[Optional[Pattern[d, Blank[]]], Times[Optional[Pattern[e, Blank[]]], Pattern[x, Blank[]]]]], Optional[Pattern[n2, Blank[]]]]]], Optional[Pattern[p, Blank[]]]]], Pattern[x, Blank[Symbol]]] := Module[{f = FreeFactors[Sin[d + e*x], x]}, Dist[1/(e*f^(m + n*p - 1)), Subst[Int[((1 - f^2*x^2)^((m - 1)/2)*(c + b*(f*x)^n + c*(f*x)^(2*n))^p)/x^(m + 2*n*p), x], x, Sin[d + e*x]/f], x]] /; FreeQ[{a, b, c, d, e, n}, x] && EqQ[n2, 2*n] && IntegerQ[(m - 1)/2] && IntegerQ[n] && IntegerQ[p]
  4170. Int[Times[Power[Plus[Pattern[a, Blank[]], Times[Optional[Pattern[c, Blank[]]], Power[sec[Plus[Optional[Pattern[d, Blank[]]], Times[Optional[Pattern[e, Blank[]]], Pattern[x, Blank[]]]]], Optional[Pattern[n2, Blank[]]]]], Times[Optional[Pattern[b, Blank[]]], Power[sec[Plus[Optional[Pattern[d, Blank[]]], Times[Optional[Pattern[e, Blank[]]], Pattern[x, Blank[]]]]], Pattern[n, Blank[]]]]], Optional[Pattern[p, Blank[]]]], Power[tan[Plus[Optional[Pattern[d, Blank[]]], Times[Optional[Pattern[e, Blank[]]], Pattern[x, Blank[]]]]], Optional[Pattern[m, Blank[]]]]], Pattern[x, Blank[Symbol]]] := Module[{f = FreeFactors[Tan[d + e*x], x]}, Dist[f^(m + 1)/e, Subst[Int[(x^m*ExpandToSum[a + b*(1 + f^2*x^2)^(n/2) + c*(1 + f^2*x^2)^n, x]^p)/(1 + f^2*x^2), x], x, Tan[d + e*x]/f], x]] /; FreeQ[{a, b, c, d, e}, x] && EqQ[n2, 2*n] && IntegerQ[m/2] && IntegerQ[n/2]
  4171. Int[Times[Power[cot[Plus[Optional[Pattern[d, Blank[]]], Times[Optional[Pattern[e, Blank[]]], Pattern[x, Blank[]]]]], Optional[Pattern[m, Blank[]]]], Power[Plus[Times[Power[csc[Plus[Optional[Pattern[d, Blank[]]], Times[Optional[Pattern[e, Blank[]]], Pattern[x, Blank[]]]]], Pattern[n, Blank[]]], Optional[Pattern[b, Blank[]]]], Pattern[a, Blank[]], Times[Optional[Pattern[c, Blank[]]], Power[sec[Plus[Optional[Pattern[d, Blank[]]], Times[Optional[Pattern[e, Blank[]]], Pattern[x, Blank[]]]]], Optional[Pattern[n2, Blank[]]]]]], Optional[Pattern[p, Blank[]]]]], Pattern[x, Blank[Symbol]]] := Module[{f = FreeFactors[Cot[d + e*x], x]}, -Dist[f^(m + 1)/e, Subst[Int[(x^m*ExpandToSum[a + b*(1 + f^2*x^2)^(n/2) + c*(1 + f^2*x^2)^n, x]^p)/(1 + f^2*x^2), x], x, Cot[d + e*x]/f], x]] /; FreeQ[{a, b, c, d, e}, x] && EqQ[n2, 2*n] && IntegerQ[m/2] && IntegerQ[n/2]
  4172. Int[Times[Plus[Pattern[A, Blank[]], Times[Optional[Pattern[B, Blank[]]], sec[Plus[Optional[Pattern[d, Blank[]]], Times[Optional[Pattern[e, Blank[]]], Pattern[x, Blank[]]]]]]], Power[Plus[Pattern[a, Blank[]], Times[Optional[Pattern[b, Blank[]]], sec[Plus[Optional[Pattern[d, Blank[]]], Times[Optional[Pattern[e, Blank[]]], Pattern[x, Blank[]]]]]], Times[Optional[Pattern[c, Blank[]]], Power[sec[Plus[Optional[Pattern[d, Blank[]]], Times[Optional[Pattern[e, Blank[]]], Pattern[x, Blank[]]]]], 2]]], Pattern[n, Blank[]]]], Pattern[x, Blank[Symbol]]] := Dist[1/(4^n*c^n), Int[(A + B*Sec[d + e*x])*(b + 2*c*Sec[d + e*x])^(2*n), x], x] /; FreeQ[{a, b, c, d, e, A, B}, x] && EqQ[b^2 - 4*a*c, 0] && IntegerQ[n]
  4173. Int[Times[Power[Plus[Times[csc[Plus[Optional[Pattern[d, Blank[]]], Times[Optional[Pattern[e, Blank[]]], Pattern[x, Blank[]]]]], Optional[Pattern[b, Blank[]]]], Times[Power[csc[Plus[Optional[Pattern[d, Blank[]]], Times[Optional[Pattern[e, Blank[]]], Pattern[x, Blank[]]]]], 2], Optional[Pattern[c, Blank[]]]], Pattern[a, Blank[]]], Pattern[n, Blank[]]], Plus[Times[csc[Plus[Optional[Pattern[d, Blank[]]], Times[Optional[Pattern[e, Blank[]]], Pattern[x, Blank[]]]]], Optional[Pattern[B, Blank[]]]], Pattern[A, Blank[]]]], Pattern[x, Blank[Symbol]]] := Dist[1/(4^n*c^n), Int[(A + B*Csc[d + e*x])*(b + 2*c*Csc[d + e*x])^(2*n), x], x] /; FreeQ[{a, b, c, d, e, A, B}, x] && EqQ[b^2 - 4*a*c, 0] && IntegerQ[n]
  4174. Int[Times[Plus[Pattern[A, Blank[]], Times[Optional[Pattern[B, Blank[]]], sec[Plus[Optional[Pattern[d, Blank[]]], Times[Optional[Pattern[e, Blank[]]], Pattern[x, Blank[]]]]]]], Power[Plus[Pattern[a, Blank[]], Times[Optional[Pattern[b, Blank[]]], sec[Plus[Optional[Pattern[d, Blank[]]], Times[Optional[Pattern[e, Blank[]]], Pattern[x, Blank[]]]]]], Times[Optional[Pattern[c, Blank[]]], Power[sec[Plus[Optional[Pattern[d, Blank[]]], Times[Optional[Pattern[e, Blank[]]], Pattern[x, Blank[]]]]], 2]]], Pattern[n, Blank[]]]], Pattern[x, Blank[Symbol]]] := Dist[(a + b*Sec[d + e*x] + c*Sec[d + e*x]^2)^n/(b + 2*c*Sec[d + e*x])^(2*n), Int[(A + B*Sec[d + e*x])*(b + 2*c*Sec[d + e*x])^(2*n), x], x] /; FreeQ[{a, b, c, d, e, A, B}, x] && EqQ[b^2 - 4*a*c, 0] && !IntegerQ[n]
  4175. Int[Times[Power[Plus[Times[csc[Plus[Optional[Pattern[d, Blank[]]], Times[Optional[Pattern[e, Blank[]]], Pattern[x, Blank[]]]]], Optional[Pattern[b, Blank[]]]], Times[Power[csc[Plus[Optional[Pattern[d, Blank[]]], Times[Optional[Pattern[e, Blank[]]], Pattern[x, Blank[]]]]], 2], Optional[Pattern[c, Blank[]]]], Pattern[a, Blank[]]], Pattern[n, Blank[]]], Plus[Times[csc[Plus[Optional[Pattern[d, Blank[]]], Times[Optional[Pattern[e, Blank[]]], Pattern[x, Blank[]]]]], Optional[Pattern[B, Blank[]]]], Pattern[A, Blank[]]]], Pattern[x, Blank[Symbol]]] := Dist[(a + b*Csc[d + e*x] + c*Csc[d + e*x]^2)^n/(b + 2*c*Csc[d + e*x])^(2*n), Int[(A + B*Csc[d + e*x])*(b + 2*c*Csc[d + e*x])^(2*n), x], x] /; FreeQ[{a, b, c, d, e, A, B}, x] && EqQ[b^2 - 4*a*c, 0] && !IntegerQ[n]
  4176. Int[Times[Plus[Pattern[A, Blank[]], Times[Optional[Pattern[B, Blank[]]], sec[Plus[Optional[Pattern[d, Blank[]]], Times[Optional[Pattern[e, Blank[]]], Pattern[x, Blank[]]]]]]], Power[Plus[Optional[Pattern[a, Blank[]]], Times[Optional[Pattern[b, Blank[]]], sec[Plus[Optional[Pattern[d, Blank[]]], Times[Optional[Pattern[e, Blank[]]], Pattern[x, Blank[]]]]]], Times[Optional[Pattern[c, Blank[]]], Power[sec[Plus[Optional[Pattern[d, Blank[]]], Times[Optional[Pattern[e, Blank[]]], Pattern[x, Blank[]]]]], 2]]], -1]], Pattern[x, Blank[Symbol]]] := Module[{q = Rt[b^2 - 4*a*c, 2]}, Dist[B + (b*B - 2*A*c)/q, Int[1/(b + q + 2*c*Sec[d + e*x]), x], x] + Dist[B - (b*B - 2*A*c)/q, Int[1/(b - q + 2*c*Sec[d + e*x]), x], x]] /; FreeQ[{a, b, c, d, e, A, B}, x] && NeQ[b^2 - 4*a*c, 0]
  4177. Int[Times[Power[Plus[Optional[Pattern[a, Blank[]]], Times[csc[Plus[Optional[Pattern[d, Blank[]]], Times[Optional[Pattern[e, Blank[]]], Pattern[x, Blank[]]]]], Optional[Pattern[b, Blank[]]]], Times[Power[csc[Plus[Optional[Pattern[d, Blank[]]], Times[Optional[Pattern[e, Blank[]]], Pattern[x, Blank[]]]]], 2], Optional[Pattern[c, Blank[]]]]], -1], Plus[Times[csc[Plus[Optional[Pattern[d, Blank[]]], Times[Optional[Pattern[e, Blank[]]], Pattern[x, Blank[]]]]], Optional[Pattern[B, Blank[]]]], Pattern[A, Blank[]]]], Pattern[x, Blank[Symbol]]] := Module[{q = Rt[b^2 - 4*a*c, 2]}, Dist[B + (b*B - 2*A*c)/q, Int[1/(b + q + 2*c*Csc[d + e*x]), x], x] + Dist[B - (b*B - 2*A*c)/q, Int[1/(b - q + 2*c*Csc[d + e*x]), x], x]] /; FreeQ[{a, b, c, d, e, A, B}, x] && NeQ[b^2 - 4*a*c, 0]
  4178. Int[Times[Plus[Pattern[A, Blank[]], Times[Optional[Pattern[B, Blank[]]], sec[Plus[Optional[Pattern[d, Blank[]]], Times[Optional[Pattern[e, Blank[]]], Pattern[x, Blank[]]]]]]], Power[Plus[Optional[Pattern[a, Blank[]]], Times[Optional[Pattern[b, Blank[]]], sec[Plus[Optional[Pattern[d, Blank[]]], Times[Optional[Pattern[e, Blank[]]], Pattern[x, Blank[]]]]]], Times[Optional[Pattern[c, Blank[]]], Power[sec[Plus[Optional[Pattern[d, Blank[]]], Times[Optional[Pattern[e, Blank[]]], Pattern[x, Blank[]]]]], 2]]], Pattern[n, Blank[]]]], Pattern[x, Blank[Symbol]]] := Int[ExpandTrig[(A + B*sec[d + e*x])*(a + b*sec[d + e*x] + c*sec[d + e*x]^2)^n, x], x] /; FreeQ[{a, b, c, d, e, A, B}, x] && NeQ[b^2 - 4*a*c, 0] && IntegerQ[n]
  4179. Int[Times[Power[Plus[Optional[Pattern[a, Blank[]]], Times[csc[Plus[Optional[Pattern[d, Blank[]]], Times[Optional[Pattern[e, Blank[]]], Pattern[x, Blank[]]]]], Optional[Pattern[b, Blank[]]]], Times[Power[csc[Plus[Optional[Pattern[d, Blank[]]], Times[Optional[Pattern[e, Blank[]]], Pattern[x, Blank[]]]]], 2], Optional[Pattern[c, Blank[]]]]], Pattern[n, Blank[]]], Plus[Times[csc[Plus[Optional[Pattern[d, Blank[]]], Times[Optional[Pattern[e, Blank[]]], Pattern[x, Blank[]]]]], Optional[Pattern[B, Blank[]]]], Pattern[A, Blank[]]]], Pattern[x, Blank[Symbol]]] := Int[ExpandTrig[(A + B*csc[d + e*x])*(a + b*csc[d + e*x] + c*csc[d + e*x]^2)^n, x], x] /; FreeQ[{a, b, c, d, e, A, B}, x] && NeQ[b^2 - 4*a*c, 0] && IntegerQ[n]
  4180. Int[Times[csc[Plus[Optional[Pattern[e, Blank[]]], Times[Pi, Optional[Pattern[k, Blank[]]]], Times[Complex[0, Pattern[fz, Blank[]]], Optional[Pattern[f, Blank[]]], Pattern[x, Blank[]]]]], Power[Plus[Optional[Pattern[c, Blank[]]], Times[Optional[Pattern[d, Blank[]]], Pattern[x, Blank[]]]], Optional[Pattern[m, Blank[]]]]], Pattern[x, Blank[Symbol]]] := Simp[(-2*(c + d*x)^m*ArcTanh[E^(-(I*e) + f*fz*x)/E^(I*k*Pi)])/(f*fz*I), x] + (-Dist[(d*m)/(f*fz*I), Int[(c + d*x)^(m - 1)*Log[1 - E^(-(I*e) + f*fz*x)/E^(I*k*Pi)], x], x] + Dist[(d*m)/(f*fz*I), Int[(c + d*x)^(m - 1)*Log[1 + E^(-(I*e) + f*fz*x)/E^(I*k*Pi)], x], x]) /; FreeQ[{c, d, e, f, fz}, x] && IntegerQ[2*k] && IGtQ[m, 0]
  4181. Int[Times[csc[Plus[Optional[Pattern[e, Blank[]]], Times[Pi, Optional[Pattern[k, Blank[]]]], Times[Optional[Pattern[f, Blank[]]], Pattern[x, Blank[]]]]], Power[Plus[Optional[Pattern[c, Blank[]]], Times[Optional[Pattern[d, Blank[]]], Pattern[x, Blank[]]]], Optional[Pattern[m, Blank[]]]]], Pattern[x, Blank[Symbol]]] := Simp[(-2*(c + d*x)^m*ArcTanh[E^(I*k*Pi)*E^(I*(e + f*x))])/f, x] + (-Dist[(d*m)/f, Int[(c + d*x)^(m - 1)*Log[1 - E^(I*k*Pi)*E^(I*(e + f*x))], x], x] + Dist[(d*m)/f, Int[(c + d*x)^(m - 1)*Log[1 + E^(I*k*Pi)*E^(I*(e + f*x))], x], x]) /; FreeQ[{c, d, e, f}, x] && IntegerQ[2*k] && IGtQ[m, 0]
  4182. Int[Times[csc[Plus[Optional[Pattern[e, Blank[]]], Times[Complex[0, Pattern[fz, Blank[]]], Optional[Pattern[f, Blank[]]], Pattern[x, Blank[]]]]], Power[Plus[Optional[Pattern[c, Blank[]]], Times[Optional[Pattern[d, Blank[]]], Pattern[x, Blank[]]]], Optional[Pattern[m, Blank[]]]]], Pattern[x, Blank[Symbol]]] := Simp[(-2*(c + d*x)^m*ArcTanh[E^(-(I*e) + f*fz*x)])/(f*fz*I), x] + (-Dist[(d*m)/(f*fz*I), Int[(c + d*x)^(m - 1)*Log[1 - E^(-(I*e) + f*fz*x)], x], x] + Dist[(d*m)/(f*fz*I), Int[(c + d*x)^(m - 1)*Log[1 + E^(-(I*e) + f*fz*x)], x], x]) /; FreeQ[{c, d, e, f, fz}, x] && IGtQ[m, 0]
  4183. Int[Times[csc[Plus[Optional[Pattern[e, Blank[]]], Times[Optional[Pattern[f, Blank[]]], Pattern[x, Blank[]]]]], Power[Plus[Optional[Pattern[c, Blank[]]], Times[Optional[Pattern[d, Blank[]]], Pattern[x, Blank[]]]], Optional[Pattern[m, Blank[]]]]], Pattern[x, Blank[Symbol]]] := Simp[(-2*(c + d*x)^m*ArcTanh[E^(I*(e + f*x))])/f, x] + (-Dist[(d*m)/f, Int[(c + d*x)^(m - 1)*Log[1 - E^(I*(e + f*x))], x], x] + Dist[(d*m)/f, Int[(c + d*x)^(m - 1)*Log[1 + E^(I*(e + f*x))], x], x]) /; FreeQ[{c, d, e, f}, x] && IGtQ[m, 0]
  4184. Int[Times[Power[csc[Plus[Optional[Pattern[e, Blank[]]], Times[Optional[Pattern[f, Blank[]]], Pattern[x, Blank[]]]]], 2], Power[Plus[Optional[Pattern[c, Blank[]]], Times[Optional[Pattern[d, Blank[]]], Pattern[x, Blank[]]]], Optional[Pattern[m, Blank[]]]]], Pattern[x, Blank[Symbol]]] := -Simp[((c + d*x)^m*Cot[e + f*x])/f, x] + Dist[(d*m)/f, Int[(c + d*x)^(m - 1)*Cot[e + f*x], x], x] /; FreeQ[{c, d, e, f}, x] && GtQ[m, 0]
  4185. Int[Times[Power[Times[csc[Plus[Optional[Pattern[e, Blank[]]], Times[Optional[Pattern[f, Blank[]]], Pattern[x, Blank[]]]]], Optional[Pattern[b, Blank[]]]], Pattern[n, Blank[]]], Plus[Optional[Pattern[c, Blank[]]], Times[Optional[Pattern[d, Blank[]]], Pattern[x, Blank[]]]]], Pattern[x, Blank[Symbol]]] := -Simp[(b^2*(c + d*x)*Cot[e + f*x]*(b*Csc[e + f*x])^(n - 2))/(f*(n - 1)), x] + (Dist[(b^2*(n - 2))/(n - 1), Int[(c + d*x)*(b*Csc[e + f*x])^(n - 2), x], x] - Simp[(b^2*d*(b*Csc[e + f*x])^(n - 2))/(f^2*(n - 1)*(n - 2)), x]) /; FreeQ[{b, c, d, e, f}, x] && GtQ[n, 1] && NeQ[n, 2]
  4186. Int[Times[Power[Times[csc[Plus[Optional[Pattern[e, Blank[]]], Times[Optional[Pattern[f, Blank[]]], Pattern[x, Blank[]]]]], Optional[Pattern[b, Blank[]]]], Pattern[n, Blank[]]], Power[Plus[Optional[Pattern[c, Blank[]]], Times[Optional[Pattern[d, Blank[]]], Pattern[x, Blank[]]]], Pattern[m, Blank[]]]], Pattern[x, Blank[Symbol]]] := -Simp[(b^2*(c + d*x)^m*Cot[e + f*x]*(b*Csc[e + f*x])^(n - 2))/(f*(n - 1)), x] + (Dist[(b^2*d^2*m*(m - 1))/(f^2*(n - 1)*(n - 2)), Int[(c + d*x)^(m - 2)*(b*Csc[e + f*x])^(n - 2), x], x] + Dist[(b^2*(n - 2))/(n - 1), Int[(c + d*x)^m*(b*Csc[e + f*x])^(n - 2), x], x] - Simp[(b^2*d*m*(c + d*x)^(m - 1)*(b*Csc[e + f*x])^(n - 2))/(f^2*(n - 1)*(n - 2)), x]) /; FreeQ[{b, c, d, e, f}, x] && GtQ[n, 1] && NeQ[n, 2] && GtQ[m, 1]
  4187. Int[Times[Power[Times[csc[Plus[Optional[Pattern[e, Blank[]]], Times[Optional[Pattern[f, Blank[]]], Pattern[x, Blank[]]]]], Optional[Pattern[b, Blank[]]]], Pattern[n, Blank[]]], Plus[Optional[Pattern[c, Blank[]]], Times[Optional[Pattern[d, Blank[]]], Pattern[x, Blank[]]]]], Pattern[x, Blank[Symbol]]] := Simp[(d*(b*Csc[e + f*x])^n)/(f^2*n^2), x] + (Dist[(n + 1)/(b^2*n), Int[(c + d*x)*(b*Csc[e + f*x])^(n + 2), x], x] + Simp[((c + d*x)*Cos[e + f*x]*(b*Csc[e + f*x])^(n + 1))/(b*f*n), x]) /; FreeQ[{b, c, d, e, f}, x] && LtQ[n, -1]
  4188. Int[Times[Power[Times[csc[Plus[Optional[Pattern[e, Blank[]]], Times[Optional[Pattern[f, Blank[]]], Pattern[x, Blank[]]]]], Optional[Pattern[b, Blank[]]]], Pattern[n, Blank[]]], Power[Plus[Optional[Pattern[c, Blank[]]], Times[Optional[Pattern[d, Blank[]]], Pattern[x, Blank[]]]], Pattern[m, Blank[]]]], Pattern[x, Blank[Symbol]]] := Simp[(d*m*(c + d*x)^(m - 1)*(b*Csc[e + f*x])^n)/(f^2*n^2), x] + (Dist[(n + 1)/(b^2*n), Int[(c + d*x)^m*(b*Csc[e + f*x])^(n + 2), x], x] - Dist[(d^2*m*(m - 1))/(f^2*n^2), Int[(c + d*x)^(m - 2)*(b*Csc[e + f*x])^n, x], x] + Simp[((c + d*x)^m*Cos[e + f*x]*(b*Csc[e + f*x])^(n + 1))/(b*f*n), x]) /; FreeQ[{b, c, d, e, f}, x] && LtQ[n, -1] && GtQ[m, 1]
  4189. Int[Times[Power[Times[csc[Plus[Optional[Pattern[e, Blank[]]], Times[Optional[Pattern[f, Blank[]]], Pattern[x, Blank[]]]]], Optional[Pattern[b, Blank[]]]], Pattern[n, Blank[]]], Power[Plus[Optional[Pattern[c, Blank[]]], Times[Optional[Pattern[d, Blank[]]], Pattern[x, Blank[]]]], Optional[Pattern[m, Blank[]]]]], Pattern[x, Blank[Symbol]]] := Dist[(b*Sin[e + f*x])^n*(b*Csc[e + f*x])^n, Int[(c + d*x)^m/(b*Sin[e + f*x])^n, x], x] /; FreeQ[{b, c, d, e, f, m, n}, x] && !IntegerQ[n]
  4190. Int[Times[Power[Plus[Times[csc[Plus[Optional[Pattern[e, Blank[]]], Times[Optional[Pattern[f, Blank[]]], Pattern[x, Blank[]]]]], Optional[Pattern[b, Blank[]]]], Pattern[a, Blank[]]], Optional[Pattern[n, Blank[]]]], Power[Plus[Optional[Pattern[c, Blank[]]], Times[Optional[Pattern[d, Blank[]]], Pattern[x, Blank[]]]], Optional[Pattern[m, Blank[]]]]], Pattern[x, Blank[Symbol]]] := Int[ExpandIntegrand[(c + d*x)^m, (a + b*Csc[e + f*x])^n, x], x] /; FreeQ[{a, b, c, d, e, f, m}, x] && IGtQ[m, 0] && IGtQ[n, 0]
  4191. Int[Times[Power[Plus[Times[csc[Plus[Optional[Pattern[e, Blank[]]], Times[Optional[Pattern[f, Blank[]]], Pattern[x, Blank[]]]]], Optional[Pattern[b, Blank[]]]], Pattern[a, Blank[]]], Optional[Pattern[n, Blank[]]]], Power[Plus[Optional[Pattern[c, Blank[]]], Times[Optional[Pattern[d, Blank[]]], Pattern[x, Blank[]]]], Optional[Pattern[m, Blank[]]]]], Pattern[x, Blank[Symbol]]] := Int[ExpandIntegrand[(c + d*x)^m, 1/(Sin[e + f*x]^n/(b + a*Sin[e + f*x])^n), x], x] /; FreeQ[{a, b, c, d, e, f}, x] && ILtQ[n, 0] && IGtQ[m, 0]
  4192. Int[Times[Power[csc[Plus[Optional[Pattern[e, Blank[]]], Times[Optional[Pattern[f, Blank[]]], Pattern[x, Blank[]]]]], Optional[Pattern[n, Blank[]]]], Power[Plus[Optional[Pattern[c, Blank[]]], Times[Optional[Pattern[d, Blank[]]], Pattern[x, Blank[]]]], Optional[Pattern[m, Blank[]]]]], Pattern[x, Blank[Symbol]]] := Simp[If[MatchQ[f, (f1_.)*(Complex[0, j_])], If[MatchQ[e, (e1_.) + Pi/2], Unintegrable[(c + d*x)^m*Sech[I*(e - Pi/2) + I*f*x]^n, x], (-I)^n*Unintegrable[(c + d*x)^m*Csch[-(I*e) - I*f*x]^n, x]], If[MatchQ[e, (e1_.) + Pi/2], Unintegrable[(c + d*x)^m*Sec[e - Pi/2 + f*x]^n, x], Unintegrable[(c + d*x)^m*Csc[e + f*x]^n, x]]], x] /; FreeQ[{c, d, e, f, m, n}, x] && IntegerQ[n]
  4193. Int[Times[Power[Plus[Optional[Pattern[a, Blank[]]], Times[csc[Plus[Optional[Pattern[e, Blank[]]], Times[Optional[Pattern[f, Blank[]]], Pattern[x, Blank[]]]]], Optional[Pattern[b, Blank[]]]]], Optional[Pattern[n, Blank[]]]], Power[Plus[Optional[Pattern[c, Blank[]]], Times[Optional[Pattern[d, Blank[]]], Pattern[x, Blank[]]]], Optional[Pattern[m, Blank[]]]]], Pattern[x, Blank[Symbol]]] := Unintegrable[(c + d*x)^m*(a + b*Csc[e + f*x])^n, x] /; FreeQ[{a, b, c, d, e, f, m, n}, x]
  4194. Int[Times[Power[Pattern[u, Blank[]], Optional[Pattern[m, Blank[]]]], Power[Plus[Optional[Pattern[a, Blank[]]], Times[Optional[Pattern[b, Blank[]]], Sec[Pattern[v, Blank[]]]]], Optional[Pattern[n, Blank[]]]]], Pattern[x, Blank[Symbol]]] := Int[ExpandToSum[u, x]^m*(a + b*Sec[ExpandToSum[v, x]])^n, x] /; FreeQ[{a, b, m, n}, x] && LinearQ[{u, v}, x] && !LinearMatchQ[{u, v}, x]
  4195. Int[Times[Power[Plus[Optional[Pattern[a, Blank[]]], Times[Csc[Pattern[v, Blank[]]], Optional[Pattern[b, Blank[]]]]], Optional[Pattern[n, Blank[]]]], Power[Pattern[u, Blank[]], Optional[Pattern[m, Blank[]]]]], Pattern[x, Blank[Symbol]]] := Int[ExpandToSum[u, x]^m*(a + b*Csc[ExpandToSum[v, x]])^n, x] /; FreeQ[{a, b, m, n}, x] && LinearQ[{u, v}, x] && !LinearMatchQ[{u, v}, x]
  4196. Int[Power[Plus[Optional[Pattern[a, Blank[]]], Times[Optional[Pattern[b, Blank[]]], Sec[Plus[Optional[Pattern[c, Blank[]]], Times[Optional[Pattern[d, Blank[]]], Power[Pattern[x, Blank[]], Pattern[n, Blank[]]]]]]]], Optional[Pattern[p, Blank[]]]], Pattern[x, Blank[Symbol]]] := Dist[1/n, Subst[Int[x^(1/n - 1)*(a + b*Sec[c + d*x])^p, x], x, x^n], x] /; FreeQ[{a, b, c, d, p}, x] && IGtQ[1/n, 0] && IntegerQ[p]
  4197. Int[Power[Plus[Optional[Pattern[a, Blank[]]], Times[Csc[Plus[Optional[Pattern[c, Blank[]]], Times[Optional[Pattern[d, Blank[]]], Power[Pattern[x, Blank[]], Pattern[n, Blank[]]]]]], Optional[Pattern[b, Blank[]]]]], Optional[Pattern[p, Blank[]]]], Pattern[x, Blank[Symbol]]] := Dist[1/n, Subst[Int[x^(1/n - 1)*(a + b*Csc[c + d*x])^p, x], x, x^n], x] /; FreeQ[{a, b, c, d, p}, x] && IGtQ[1/n, 0] && IntegerQ[p]
  4198. Int[Power[Plus[Optional[Pattern[a, Blank[]]], Times[Optional[Pattern[b, Blank[]]], Sec[Plus[Optional[Pattern[c, Blank[]]], Times[Optional[Pattern[d, Blank[]]], Power[Pattern[x, Blank[]], Pattern[n, Blank[]]]]]]]], Optional[Pattern[p, Blank[]]]], Pattern[x, Blank[Symbol]]] := Unintegrable[(a + b*Sec[c + d*x^n])^p, x] /; FreeQ[{a, b, c, d, n, p}, x]
  4199. Int[Power[Plus[Optional[Pattern[a, Blank[]]], Times[Csc[Plus[Optional[Pattern[c, Blank[]]], Times[Optional[Pattern[d, Blank[]]], Power[Pattern[x, Blank[]], Pattern[n, Blank[]]]]]], Optional[Pattern[b, Blank[]]]]], Optional[Pattern[p, Blank[]]]], Pattern[x, Blank[Symbol]]] := Unintegrable[(a + b*Csc[c + d*x^n])^p, x] /; FreeQ[{a, b, c, d, n, p}, x]
  4200. Int[Power[Plus[Optional[Pattern[a, Blank[]]], Times[Optional[Pattern[b, Blank[]]], Sec[Plus[Optional[Pattern[c, Blank[]]], Times[Optional[Pattern[d, Blank[]]], Power[Pattern[u, Blank[]], Pattern[n, Blank[]]]]]]]], Optional[Pattern[p, Blank[]]]], Pattern[x, Blank[Symbol]]] := Dist[1/Coefficient[u, x, 1], Subst[Int[(a + b*Sec[c + d*x^n])^p, x], x, u], x] /; FreeQ[{a, b, c, d, n, p}, x] && LinearQ[u, x] && NeQ[u, x]
  4201. Int[Power[Plus[Optional[Pattern[a, Blank[]]], Times[Csc[Plus[Optional[Pattern[c, Blank[]]], Times[Optional[Pattern[d, Blank[]]], Power[Pattern[u, Blank[]], Pattern[n, Blank[]]]]]], Optional[Pattern[b, Blank[]]]]], Optional[Pattern[p, Blank[]]]], Pattern[x, Blank[Symbol]]] := Dist[1/Coefficient[u, x, 1], Subst[Int[(a + b*Csc[c + d*x^n])^p, x], x, u], x] /; FreeQ[{a, b, c, d, n, p}, x] && LinearQ[u, x] && NeQ[u, x]
  4202. Int[Power[Plus[Optional[Pattern[a, Blank[]]], Times[Optional[Pattern[b, Blank[]]], Sec[Pattern[u, Blank[]]]]], Optional[Pattern[p, Blank[]]]], Pattern[x, Blank[Symbol]]] := Int[(a + b*Sec[ExpandToSum[u, x]])^p, x] /; FreeQ[{a, b, p}, x] && BinomialQ[u, x] && !BinomialMatchQ[u, x]
  4203. Int[Power[Plus[Optional[Pattern[a, Blank[]]], Times[Csc[Pattern[u, Blank[]]], Optional[Pattern[b, Blank[]]]]], Optional[Pattern[p, Blank[]]]], Pattern[x, Blank[Symbol]]] := Int[(a + b*Csc[ExpandToSum[u, x]])^p, x] /; FreeQ[{a, b, p}, x] && BinomialQ[u, x] && !BinomialMatchQ[u, x]
  4204. Int[Times[Power[Pattern[x, Blank[]], Optional[Pattern[m, Blank[]]]], Power[Plus[Optional[Pattern[a, Blank[]]], Times[Optional[Pattern[b, Blank[]]], Sec[Plus[Optional[Pattern[c, Blank[]]], Times[Optional[Pattern[d, Blank[]]], Power[Pattern[x, Blank[]], Pattern[n, Blank[]]]]]]]], Optional[Pattern[p, Blank[]]]]], Pattern[x, Blank[Symbol]]] := Dist[1/n, Subst[Int[x^(Simplify[(m + 1)/n] - 1)*(a + b*Sec[c + d*x])^p, x], x, x^n], x] /; FreeQ[{a, b, c, d, m, n, p}, x] && IGtQ[Simplify[(m + 1)/n], 0] && IntegerQ[p]
  4205. Int[Times[Power[Plus[Optional[Pattern[a, Blank[]]], Times[Csc[Plus[Optional[Pattern[c, Blank[]]], Times[Optional[Pattern[d, Blank[]]], Power[Pattern[x, Blank[]], Pattern[n, Blank[]]]]]], Optional[Pattern[b, Blank[]]]]], Optional[Pattern[p, Blank[]]]], Power[Pattern[x, Blank[]], Optional[Pattern[m, Blank[]]]]], Pattern[x, Blank[Symbol]]] := Dist[1/n, Subst[Int[x^(Simplify[(m + 1)/n] - 1)*(a + b*Csc[c + d*x])^p, x], x, x^n], x] /; FreeQ[{a, b, c, d, m, n, p}, x] && IGtQ[Simplify[(m + 1)/n], 0] && IntegerQ[p]
  4206. Int[Times[Power[Pattern[x, Blank[]], Optional[Pattern[m, Blank[]]]], Power[Plus[Optional[Pattern[a, Blank[]]], Times[Optional[Pattern[b, Blank[]]], Sec[Plus[Optional[Pattern[c, Blank[]]], Times[Optional[Pattern[d, Blank[]]], Power[Pattern[x, Blank[]], Pattern[n, Blank[]]]]]]]], Optional[Pattern[p, Blank[]]]]], Pattern[x, Blank[Symbol]]] := Unintegrable[x^m*(a + b*Sec[c + d*x^n])^p, x] /; FreeQ[{a, b, c, d, m, n, p}, x]
  4207. Int[Times[Power[Plus[Optional[Pattern[a, Blank[]]], Times[Csc[Plus[Optional[Pattern[c, Blank[]]], Times[Optional[Pattern[d, Blank[]]], Power[Pattern[x, Blank[]], Pattern[n, Blank[]]]]]], Optional[Pattern[b, Blank[]]]]], Optional[Pattern[p, Blank[]]]], Power[Pattern[x, Blank[]], Optional[Pattern[m, Blank[]]]]], Pattern[x, Blank[Symbol]]] := Unintegrable[x^m*(a + b*Csc[c + d*x^n])^p, x] /; FreeQ[{a, b, c, d, m, n, p}, x]
  4208. Int[Times[Power[Times[Pattern[e, Blank[]], Pattern[x, Blank[]]], Optional[Pattern[m, Blank[]]]], Power[Plus[Optional[Pattern[a, Blank[]]], Times[Optional[Pattern[b, Blank[]]], Sec[Plus[Optional[Pattern[c, Blank[]]], Times[Optional[Pattern[d, Blank[]]], Power[Pattern[x, Blank[]], Pattern[n, Blank[]]]]]]]], Optional[Pattern[p, Blank[]]]]], Pattern[x, Blank[Symbol]]] := Dist[(e^IntPart[m]*(e*x)^FracPart[m])/x^FracPart[m], Int[x^m*(a + b*Sec[c + d*x^n])^p, x], x] /; FreeQ[{a, b, c, d, e, m, n, p}, x]
  4209. Int[Times[Power[Plus[Optional[Pattern[a, Blank[]]], Times[Csc[Plus[Optional[Pattern[c, Blank[]]], Times[Optional[Pattern[d, Blank[]]], Power[Pattern[x, Blank[]], Pattern[n, Blank[]]]]]], Optional[Pattern[b, Blank[]]]]], Optional[Pattern[p, Blank[]]]], Power[Times[Pattern[e, Blank[]], Pattern[x, Blank[]]], Optional[Pattern[m, Blank[]]]]], Pattern[x, Blank[Symbol]]] := Dist[(e^IntPart[m]*(e*x)^FracPart[m])/x^FracPart[m], Int[x^m*(a + b*Csc[c + d*x^n])^p, x], x] /; FreeQ[{a, b, c, d, e, m, n, p}, x]
  4210. Int[Times[Power[Times[Pattern[e, Blank[]], Pattern[x, Blank[]]], Optional[Pattern[m, Blank[]]]], Power[Plus[Optional[Pattern[a, Blank[]]], Times[Optional[Pattern[b, Blank[]]], Sec[Pattern[u, Blank[]]]]], Optional[Pattern[p, Blank[]]]]], Pattern[x, Blank[Symbol]]] := Int[(e*x)^m*(a + b*Sec[ExpandToSum[u, x]])^p, x] /; FreeQ[{a, b, e, m, p}, x] && BinomialQ[u, x] && !BinomialMatchQ[u, x]
  4211. Int[Times[Power[Plus[Optional[Pattern[a, Blank[]]], Times[Csc[Pattern[u, Blank[]]], Optional[Pattern[b, Blank[]]]]], Optional[Pattern[p, Blank[]]]], Power[Times[Pattern[e, Blank[]], Pattern[x, Blank[]]], Optional[Pattern[m, Blank[]]]]], Pattern[x, Blank[Symbol]]] := Int[(e*x)^m*(a + b*Csc[ExpandToSum[u, x]])^p, x] /; FreeQ[{a, b, e, m, p}, x] && BinomialQ[u, x] && !BinomialMatchQ[u, x]
  4212. Int[Times[Power[Pattern[x, Blank[]], Optional[Pattern[m, Blank[]]]], Power[Sec[Plus[Optional[Pattern[a, Blank[]]], Times[Optional[Pattern[b, Blank[]]], Power[Pattern[x, Blank[]], Optional[Pattern[n, Blank[]]]]]]], Pattern[p, Blank[]]], Sin[Plus[Optional[Pattern[a, Blank[]]], Times[Optional[Pattern[b, Blank[]]], Power[Pattern[x, Blank[]], Optional[Pattern[n, Blank[]]]]]]]], Pattern[x, Blank[Symbol]]] := Simp[(x^(m - n + 1)*Sec[a + b*x^n]^(p - 1))/(b*n*(p - 1)), x] - Dist[(m - n + 1)/(b*n*(p - 1)), Int[x^(m - n)*Sec[a + b*x^n]^(p - 1), x], x] /; FreeQ[{a, b, p}, x] && IntegerQ[n] && GeQ[m - n, 0] && NeQ[p, 1]
  4213. Int[Times[Cos[Plus[Optional[Pattern[a, Blank[]]], Times[Optional[Pattern[b, Blank[]]], Power[Pattern[x, Blank[]], Optional[Pattern[n, Blank[]]]]]]], Power[Csc[Plus[Optional[Pattern[a, Blank[]]], Times[Optional[Pattern[b, Blank[]]], Power[Pattern[x, Blank[]], Optional[Pattern[n, Blank[]]]]]]], Pattern[p, Blank[]]], Power[Pattern[x, Blank[]], Optional[Pattern[m, Blank[]]]]], Pattern[x, Blank[Symbol]]] := -Simp[(x^(m - n + 1)*Csc[a + b*x^n]^(p - 1))/(b*n*(p - 1)), x] + Dist[(m - n + 1)/(b*n*(p - 1)), Int[x^(m - n)*Csc[a + b*x^n]^(p - 1), x], x] /; FreeQ[{a, b, p}, x] && IntegerQ[n] && GeQ[m - n, 0] && NeQ[p, 1]
  4214. Int[Times[Pattern[u, Blank[]], Power[Times[Optional[Pattern[d, Blank[]]], sin[Plus[Optional[Pattern[a, Blank[]]], Times[Optional[Pattern[b, Blank[]]], Pattern[x, Blank[]]]]]], Optional[Pattern[n, Blank[]]]], Power[Times[Optional[Pattern[c, Blank[]]], tan[Plus[Optional[Pattern[a, Blank[]]], Times[Optional[Pattern[b, Blank[]]], Pattern[x, Blank[]]]]]], Optional[Pattern[m, Blank[]]]]], Pattern[x, Blank[Symbol]]] := Dist[((c*Tan[a + b*x])^m*(d*Cos[a + b*x])^m)/(d*Sin[a + b*x])^m, Int[(ActivateTrig[u]*(d*Sin[a + b*x])^(m + n))/(d*Cos[a + b*x])^m, x], x] /; FreeQ[{a, b, c, d, m, n}, x] && KnownSineIntegrandQ[u, x] && !IntegerQ[m]
  4215. Int[Times[Power[Times[cos[Plus[Optional[Pattern[a, Blank[]]], Times[Optional[Pattern[b, Blank[]]], Pattern[x, Blank[]]]]], Optional[Pattern[d, Blank[]]]], Optional[Pattern[n, Blank[]]]], Pattern[u, Blank[]], Power[Times[Optional[Pattern[c, Blank[]]], tan[Plus[Optional[Pattern[a, Blank[]]], Times[Optional[Pattern[b, Blank[]]], Pattern[x, Blank[]]]]]], Optional[Pattern[m, Blank[]]]]], Pattern[x, Blank[Symbol]]] := Dist[((c*Tan[a + b*x])^m*(d*Cos[a + b*x])^m)/(d*Sin[a + b*x])^m, Int[(ActivateTrig[u]*(d*Sin[a + b*x])^m)/(d*Cos[a + b*x])^(m - n), x], x] /; FreeQ[{a, b, c, d, m, n}, x] && KnownSineIntegrandQ[u, x] && !IntegerQ[m]
  4216. Int[Times[Power[Times[cot[Plus[Optional[Pattern[a, Blank[]]], Times[Optional[Pattern[b, Blank[]]], Pattern[x, Blank[]]]]], Optional[Pattern[c, Blank[]]]], Optional[Pattern[m, Blank[]]]], Pattern[u, Blank[]], Power[Times[Optional[Pattern[d, Blank[]]], sin[Plus[Optional[Pattern[a, Blank[]]], Times[Optional[Pattern[b, Blank[]]], Pattern[x, Blank[]]]]]], Optional[Pattern[n, Blank[]]]]], Pattern[x, Blank[Symbol]]] := Dist[((c*Cot[a + b*x])^m*(d*Sin[a + b*x])^m)/(d*Cos[a + b*x])^m, Int[(ActivateTrig[u]*(d*Cos[a + b*x])^m)/(d*Sin[a + b*x])^(m - n), x], x] /; FreeQ[{a, b, c, d, m, n}, x] && KnownSineIntegrandQ[u, x] && !IntegerQ[m]
  4217. Int[Times[Power[Times[cot[Plus[Optional[Pattern[a, Blank[]]], Times[Optional[Pattern[b, Blank[]]], Pattern[x, Blank[]]]]], Optional[Pattern[c, Blank[]]]], Optional[Pattern[m, Blank[]]]], Power[Times[cos[Plus[Optional[Pattern[a, Blank[]]], Times[Optional[Pattern[b, Blank[]]], Pattern[x, Blank[]]]]], Optional[Pattern[d, Blank[]]]], Optional[Pattern[n, Blank[]]]], Pattern[u, Blank[]]], Pattern[x, Blank[Symbol]]] := Dist[((c*Cot[a + b*x])^m*(d*Sin[a + b*x])^m)/(d*Cos[a + b*x])^m, Int[(ActivateTrig[u]*(d*Cos[a + b*x])^(m + n))/(d*Sin[a + b*x])^m, x], x] /; FreeQ[{a, b, c, d, m, n}, x] && KnownSineIntegrandQ[u, x] && !IntegerQ[m]
  4218. Int[Times[Power[Times[cos[Plus[Optional[Pattern[a, Blank[]]], Times[Optional[Pattern[b, Blank[]]], Pattern[x, Blank[]]]]], Optional[Pattern[d, Blank[]]]], Optional[Pattern[n, Blank[]]]], Pattern[u, Blank[]], Power[Times[Optional[Pattern[c, Blank[]]], sec[Plus[Optional[Pattern[a, Blank[]]], Times[Optional[Pattern[b, Blank[]]], Pattern[x, Blank[]]]]]], Optional[Pattern[m, Blank[]]]]], Pattern[x, Blank[Symbol]]] := Dist[(c*Csc[a + b*x])^m*(d*Sin[a + b*x])^m, Int[ActivateTrig[u]*(d*Sin[a + b*x])^(n - m), x], x] /; FreeQ[{a, b, c, d, m, n}, x] && KnownSineIntegrandQ[u, x]
  4219. Int[Times[Pattern[u, Blank[]], Power[Times[Optional[Pattern[c, Blank[]]], tan[Plus[Optional[Pattern[a, Blank[]]], Times[Optional[Pattern[b, Blank[]]], Pattern[x, Blank[]]]]]], Optional[Pattern[m, Blank[]]]]], Pattern[x, Blank[Symbol]]] := Dist[((c*Tan[a + b*x])^m*(c*Cos[a + b*x])^m)/(c*Sin[a + b*x])^m, Int[(ActivateTrig[u]*(c*Sin[a + b*x])^m)/(c*Cos[a + b*x])^m, x], x] /; FreeQ[{a, b, c, m}, x] && !IntegerQ[m] && KnownSineIntegrandQ[u, x]
  4220. Int[Times[Power[Times[cot[Plus[Optional[Pattern[a, Blank[]]], Times[Optional[Pattern[b, Blank[]]], Pattern[x, Blank[]]]]], Optional[Pattern[c, Blank[]]]], Optional[Pattern[m, Blank[]]]], Pattern[u, Blank[]]], Pattern[x, Blank[Symbol]]] := Dist[((c*Cot[a + b*x])^m*(c*Sin[a + b*x])^m)/(c*Cos[a + b*x])^m, Int[(ActivateTrig[u]*(c*Cos[a + b*x])^m)/(c*Sin[a + b*x])^m, x], x] /; FreeQ[{a, b, c, m}, x] && !IntegerQ[m] && KnownSineIntegrandQ[u, x]
  4221. Int[Times[Pattern[u, Blank[]], Power[Times[Optional[Pattern[c, Blank[]]], sec[Plus[Optional[Pattern[a, Blank[]]], Times[Optional[Pattern[b, Blank[]]], Pattern[x, Blank[]]]]]], Optional[Pattern[m, Blank[]]]]], Pattern[x, Blank[Symbol]]] := Dist[(c*Sec[a + b*x])^m*(c*Cos[a + b*x])^m, Int[ActivateTrig[u]/(c*Cos[a + b*x])^m, x], x] /; FreeQ[{a, b, c, m}, x] && !IntegerQ[m] && KnownSineIntegrandQ[u, x]
  4222. Int[Times[Power[Times[csc[Plus[Optional[Pattern[a, Blank[]]], Times[Optional[Pattern[b, Blank[]]], Pattern[x, Blank[]]]]], Optional[Pattern[c, Blank[]]]], Optional[Pattern[m, Blank[]]]], Pattern[u, Blank[]]], Pattern[x, Blank[Symbol]]] := Dist[(c*Csc[a + b*x])^m*(c*Sin[a + b*x])^m, Int[ActivateTrig[u]/(c*Sin[a + b*x])^m, x], x] /; FreeQ[{a, b, c, m}, x] && !IntegerQ[m] && KnownSineIntegrandQ[u, x]
  4223. Int[Times[Plus[Times[csc[Plus[Optional[Pattern[a, Blank[]]], Times[Optional[Pattern[b, Blank[]]], Pattern[x, Blank[]]]]], Optional[Pattern[B, Blank[]]]], Pattern[A, Blank[]]], Pattern[u, Blank[]], Power[Times[Optional[Pattern[c, Blank[]]], sin[Plus[Optional[Pattern[a, Blank[]]], Times[Optional[Pattern[b, Blank[]]], Pattern[x, Blank[]]]]]], Optional[Pattern[n, Blank[]]]]], Pattern[x, Blank[Symbol]]] := Dist[c, Int[ActivateTrig[u]*(c*Sin[a + b*x])^(n - 1)*(B + A*Sin[a + b*x]), x], x] /; FreeQ[{a, b, c, A, B, n}, x] && KnownSineIntegrandQ[u, x]
  4224. Int[Times[Power[Times[cos[Plus[Optional[Pattern[a, Blank[]]], Times[Optional[Pattern[b, Blank[]]], Pattern[x, Blank[]]]]], Optional[Pattern[c, Blank[]]]], Optional[Pattern[n, Blank[]]]], Pattern[u, Blank[]], Plus[Pattern[A, Blank[]], Times[Optional[Pattern[B, Blank[]]], sec[Plus[Optional[Pattern[a, Blank[]]], Times[Optional[Pattern[b, Blank[]]], Pattern[x, Blank[]]]]]]]], Pattern[x, Blank[Symbol]]] := Dist[c, Int[ActivateTrig[u]*(c*Cos[a + b*x])^(n - 1)*(B + A*Cos[a + b*x]), x], x] /; FreeQ[{a, b, c, A, B, n}, x] && KnownSineIntegrandQ[u, x]
  4225. Int[Times[Plus[Times[csc[Plus[Optional[Pattern[a, Blank[]]], Times[Optional[Pattern[b, Blank[]]], Pattern[x, Blank[]]]]], Optional[Pattern[B, Blank[]]]], Pattern[A, Blank[]]], Pattern[u, Blank[]]], Pattern[x, Blank[Symbol]]] := Int[(ActivateTrig[u]*(B + A*Sin[a + b*x]))/Sin[a + b*x], x] /; FreeQ[{a, b, A, B}, x] && KnownSineIntegrandQ[u, x]
  4226. Int[Times[Pattern[u, Blank[]], Plus[Pattern[A, Blank[]], Times[Optional[Pattern[B, Blank[]]], sec[Plus[Optional[Pattern[a, Blank[]]], Times[Optional[Pattern[b, Blank[]]], Pattern[x, Blank[]]]]]]]], Pattern[x, Blank[Symbol]]] := Int[(ActivateTrig[u]*(B + A*Cos[a + b*x]))/Cos[a + b*x], x] /; FreeQ[{a, b, A, B}, x] && KnownSineIntegrandQ[u, x]
  4227. Int[Times[Plus[Optional[Pattern[A, Blank[]]], Times[csc[Plus[Optional[Pattern[a, Blank[]]], Times[Optional[Pattern[b, Blank[]]], Pattern[x, Blank[]]]]], Optional[Pattern[B, Blank[]]]], Times[Power[csc[Plus[Optional[Pattern[a, Blank[]]], Times[Optional[Pattern[b, Blank[]]], Pattern[x, Blank[]]]]], 2], Optional[Pattern[C, Blank[]]]]], Optional[Pattern[u, Blank[]]], Power[Times[Optional[Pattern[c, Blank[]]], sin[Plus[Optional[Pattern[a, Blank[]]], Times[Optional[Pattern[b, Blank[]]], Pattern[x, Blank[]]]]]], Optional[Pattern[n, Blank[]]]]], Pattern[x, Blank[Symbol]]] := Dist[c^2, Int[ActivateTrig[u]*(c*Sin[a + b*x])^(n - 2)*(C + B*Sin[a + b*x] + A*Sin[a + b*x]^2), x], x] /; FreeQ[{a, b, c, A, B, C, n}, x] && KnownSineIntegrandQ[u, x]
  4228. Int[Times[Power[Times[cos[Plus[Optional[Pattern[a, Blank[]]], Times[Optional[Pattern[b, Blank[]]], Pattern[x, Blank[]]]]], Optional[Pattern[c, Blank[]]]], Optional[Pattern[n, Blank[]]]], Optional[Pattern[u, Blank[]]], Plus[Optional[Pattern[A, Blank[]]], Times[Optional[Pattern[B, Blank[]]], sec[Plus[Optional[Pattern[a, Blank[]]], Times[Optional[Pattern[b, Blank[]]], Pattern[x, Blank[]]]]]], Times[Optional[Pattern[C, Blank[]]], Power[sec[Plus[Optional[Pattern[a, Blank[]]], Times[Optional[Pattern[b, Blank[]]], Pattern[x, Blank[]]]]], 2]]]], Pattern[x, Blank[Symbol]]] := Dist[c^2, Int[ActivateTrig[u]*(c*Cos[a + b*x])^(n - 2)*(C + B*Cos[a + b*x] + A*Cos[a + b*x]^2), x], x] /; FreeQ[{a, b, c, A, B, C, n}, x] && KnownSineIntegrandQ[u, x]
  4229. Int[Times[Optional[Pattern[u, Blank[]]], Plus[Times[Power[csc[Plus[Optional[Pattern[a, Blank[]]], Times[Optional[Pattern[b, Blank[]]], Pattern[x, Blank[]]]]], 2], Optional[Pattern[C, Blank[]]]], Pattern[A, Blank[]]], Power[Times[Optional[Pattern[c, Blank[]]], sin[Plus[Optional[Pattern[a, Blank[]]], Times[Optional[Pattern[b, Blank[]]], Pattern[x, Blank[]]]]]], Optional[Pattern[n, Blank[]]]]], Pattern[x, Blank[Symbol]]] := Dist[c^2, Int[ActivateTrig[u]*(c*Sin[a + b*x])^(n - 2)*(C + A*Sin[a + b*x]^2), x], x] /; FreeQ[{a, b, c, A, C, n}, x] && KnownSineIntegrandQ[u, x]
  4230. Int[Times[Power[Times[cos[Plus[Optional[Pattern[a, Blank[]]], Times[Optional[Pattern[b, Blank[]]], Pattern[x, Blank[]]]]], Optional[Pattern[c, Blank[]]]], Optional[Pattern[n, Blank[]]]], Optional[Pattern[u, Blank[]]], Plus[Pattern[A, Blank[]], Times[Optional[Pattern[C, Blank[]]], Power[sec[Plus[Optional[Pattern[a, Blank[]]], Times[Optional[Pattern[b, Blank[]]], Pattern[x, Blank[]]]]], 2]]]], Pattern[x, Blank[Symbol]]] := Dist[c^2, Int[ActivateTrig[u]*(c*Cos[a + b*x])^(n - 2)*(C + A*Cos[a + b*x]^2), x], x] /; FreeQ[{a, b, c, A, C, n}, x] && KnownSineIntegrandQ[u, x]
  4231. Int[Times[Plus[Optional[Pattern[A, Blank[]]], Times[csc[Plus[Optional[Pattern[a, Blank[]]], Times[Optional[Pattern[b, Blank[]]], Pattern[x, Blank[]]]]], Optional[Pattern[B, Blank[]]]], Times[Power[csc[Plus[Optional[Pattern[a, Blank[]]], Times[Optional[Pattern[b, Blank[]]], Pattern[x, Blank[]]]]], 2], Optional[Pattern[C, Blank[]]]]], Pattern[u, Blank[]]], Pattern[x, Blank[Symbol]]] := Int[(ActivateTrig[u]*(C + B*Sin[a + b*x] + A*Sin[a + b*x]^2))/Sin[a + b*x]^2, x] /; FreeQ[{a, b, A, B, C}, x] && KnownSineIntegrandQ[u, x]
  4232. Int[Times[Pattern[u, Blank[]], Plus[Optional[Pattern[A, Blank[]]], Times[Optional[Pattern[B, Blank[]]], sec[Plus[Optional[Pattern[a, Blank[]]], Times[Optional[Pattern[b, Blank[]]], Pattern[x, Blank[]]]]]], Times[Optional[Pattern[C, Blank[]]], Power[sec[Plus[Optional[Pattern[a, Blank[]]], Times[Optional[Pattern[b, Blank[]]], Pattern[x, Blank[]]]]], 2]]]], Pattern[x, Blank[Symbol]]] := Int[(ActivateTrig[u]*(C + B*Cos[a + b*x] + A*Cos[a + b*x]^2))/Cos[a + b*x]^2, x] /; FreeQ[{a, b, A, B, C}, x] && KnownSineIntegrandQ[u, x]
  4233. Int[Times[Plus[Times[Power[csc[Plus[Optional[Pattern[a, Blank[]]], Times[Optional[Pattern[b, Blank[]]], Pattern[x, Blank[]]]]], 2], Optional[Pattern[C, Blank[]]]], Pattern[A, Blank[]]], Pattern[u, Blank[]]], Pattern[x, Blank[Symbol]]] := Int[(ActivateTrig[u]*(C + A*Sin[a + b*x]^2))/Sin[a + b*x]^2, x] /; FreeQ[{a, b, A, C}, x] && KnownSineIntegrandQ[u, x]
  4234. Int[Times[Pattern[u, Blank[]], Plus[Pattern[A, Blank[]], Times[Optional[Pattern[C, Blank[]]], Power[sec[Plus[Optional[Pattern[a, Blank[]]], Times[Optional[Pattern[b, Blank[]]], Pattern[x, Blank[]]]]], 2]]]], Pattern[x, Blank[Symbol]]] := Int[(ActivateTrig[u]*(C + A*Cos[a + b*x]^2))/Cos[a + b*x]^2, x] /; FreeQ[{a, b, A, C}, x] && KnownSineIntegrandQ[u, x]
  4235. Int[Times[Pattern[u, Blank[]], Plus[Optional[Pattern[A, Blank[]]], Times[csc[Plus[Optional[Pattern[a, Blank[]]], Times[Optional[Pattern[b, Blank[]]], Pattern[x, Blank[]]]]], Optional[Pattern[C, Blank[]]]], Times[Optional[Pattern[B, Blank[]]], sin[Plus[Optional[Pattern[a, Blank[]]], Times[Optional[Pattern[b, Blank[]]], Pattern[x, Blank[]]]]]]]], Pattern[x, Blank[Symbol]]] := Int[(ActivateTrig[u]*(C + A*Sin[a + b*x] + B*Sin[a + b*x]^2))/Sin[a + b*x], x] /; FreeQ[{a, b, A, B, C}, x]
  4236. Int[Times[Pattern[u, Blank[]], Plus[Optional[Pattern[A, Blank[]]], Times[cos[Plus[Optional[Pattern[a, Blank[]]], Times[Optional[Pattern[b, Blank[]]], Pattern[x, Blank[]]]]], Optional[Pattern[B, Blank[]]]], Times[Optional[Pattern[C, Blank[]]], sec[Plus[Optional[Pattern[a, Blank[]]], Times[Optional[Pattern[b, Blank[]]], Pattern[x, Blank[]]]]]]]], Pattern[x, Blank[Symbol]]] := Int[(ActivateTrig[u]*(C + A*Cos[a + b*x] + B*Cos[a + b*x]^2))/Cos[a + b*x], x] /; FreeQ[{a, b, A, B, C}, x]
  4237. Int[Times[Pattern[u, Blank[]], Plus[Times[Optional[Pattern[A, Blank[]]], Power[sin[Plus[Optional[Pattern[a, Blank[]]], Times[Optional[Pattern[b, Blank[]]], Pattern[x, Blank[]]]]], Optional[Pattern[n, Blank[]]]]], Times[Optional[Pattern[B, Blank[]]], Power[sin[Plus[Optional[Pattern[a, Blank[]]], Times[Optional[Pattern[b, Blank[]]], Pattern[x, Blank[]]]]], Pattern[n1, Blank[]]]], Times[Optional[Pattern[C, Blank[]]], Power[sin[Plus[Optional[Pattern[a, Blank[]]], Times[Optional[Pattern[b, Blank[]]], Pattern[x, Blank[]]]]], Pattern[n2, Blank[]]]]]], Pattern[x, Blank[Symbol]]] := Int[ActivateTrig[u]*Sin[a + b*x]^n*(A + B*Sin[a + b*x] + C*Sin[a + b*x]^2), x] /; FreeQ[{a, b, A, B, C, n}, x] && EqQ[n1, n + 1] && EqQ[n2, n + 2]
  4238. Int[Times[Plus[Times[Power[cos[Plus[Optional[Pattern[a, Blank[]]], Times[Optional[Pattern[b, Blank[]]], Pattern[x, Blank[]]]]], Optional[Pattern[n, Blank[]]]], Optional[Pattern[A, Blank[]]]], Times[Power[cos[Plus[Optional[Pattern[a, Blank[]]], Times[Optional[Pattern[b, Blank[]]], Pattern[x, Blank[]]]]], Pattern[n1, Blank[]]], Optional[Pattern[B, Blank[]]]], Times[Power[cos[Plus[Optional[Pattern[a, Blank[]]], Times[Optional[Pattern[b, Blank[]]], Pattern[x, Blank[]]]]], Pattern[n2, Blank[]]], Optional[Pattern[C, Blank[]]]]], Pattern[u, Blank[]]], Pattern[x, Blank[Symbol]]] := Int[ActivateTrig[u]*Cos[a + b*x]^n*(A + B*Cos[a + b*x] + C*Cos[a + b*x]^2), x] /; FreeQ[{a, b, A, B, C, n}, x] && EqQ[n1, n + 1] && EqQ[n2, n + 2]
  4239. Int[Times[Power[Times[cot[Plus[Optional[Pattern[a, Blank[]]], Times[Optional[Pattern[b, Blank[]]], Pattern[x, Blank[]]]]], Optional[Pattern[c, Blank[]]]], Optional[Pattern[m, Blank[]]]], Pattern[u, Blank[]], Power[Times[Optional[Pattern[d, Blank[]]], tan[Plus[Optional[Pattern[a, Blank[]]], Times[Optional[Pattern[b, Blank[]]], Pattern[x, Blank[]]]]]], Optional[Pattern[n, Blank[]]]]], Pattern[x, Blank[Symbol]]] := Dist[(c*Cot[a + b*x])^m*(d*Tan[a + b*x])^m, Int[ActivateTrig[u]*(d*Tan[a + b*x])^(n - m), x], x] /; FreeQ[{a, b, c, d, m, n}, x] && KnownTangentIntegrandQ[u, x]
  4240. Int[Times[Power[Times[cos[Plus[Optional[Pattern[a, Blank[]]], Times[Optional[Pattern[b, Blank[]]], Pattern[x, Blank[]]]]], Optional[Pattern[d, Blank[]]]], Optional[Pattern[n, Blank[]]]], Pattern[u, Blank[]], Power[Times[Optional[Pattern[c, Blank[]]], tan[Plus[Optional[Pattern[a, Blank[]]], Times[Optional[Pattern[b, Blank[]]], Pattern[x, Blank[]]]]]], Optional[Pattern[m, Blank[]]]]], Pattern[x, Blank[Symbol]]] := Dist[((c*Tan[a + b*x])^m*(d*Cos[a + b*x])^m)/(d*Sin[a + b*x])^m, Int[(ActivateTrig[u]*(d*Sin[a + b*x])^m)/(d*Cos[a + b*x])^(m - n), x], x] /; FreeQ[{a, b, c, d, m, n}, x] && KnownCotangentIntegrandQ[u, x]
  4241. Int[Times[Power[Times[cot[Plus[Optional[Pattern[a, Blank[]]], Times[Optional[Pattern[b, Blank[]]], Pattern[x, Blank[]]]]], Optional[Pattern[c, Blank[]]]], Optional[Pattern[m, Blank[]]]], Pattern[u, Blank[]]], Pattern[x, Blank[Symbol]]] := Dist[(c*Cot[a + b*x])^m*(c*Tan[a + b*x])^m, Int[ActivateTrig[u]/(c*Tan[a + b*x])^m, x], x] /; FreeQ[{a, b, c, m}, x] && !IntegerQ[m] && KnownTangentIntegrandQ[u, x]
  4242. Int[Times[Pattern[u, Blank[]], Power[Times[Optional[Pattern[c, Blank[]]], tan[Plus[Optional[Pattern[a, Blank[]]], Times[Optional[Pattern[b, Blank[]]], Pattern[x, Blank[]]]]]], Optional[Pattern[m, Blank[]]]]], Pattern[x, Blank[Symbol]]] := Dist[(c*Cot[a + b*x])^m*(c*Tan[a + b*x])^m, Int[ActivateTrig[u]/(c*Cot[a + b*x])^m, x], x] /; FreeQ[{a, b, c, m}, x] && !IntegerQ[m] && KnownCotangentIntegrandQ[u, x]
  4243. Int[Times[Plus[Times[cot[Plus[Optional[Pattern[a, Blank[]]], Times[Optional[Pattern[b, Blank[]]], Pattern[x, Blank[]]]]], Optional[Pattern[B, Blank[]]]], Pattern[A, Blank[]]], Pattern[u, Blank[]], Power[Times[Optional[Pattern[c, Blank[]]], tan[Plus[Optional[Pattern[a, Blank[]]], Times[Optional[Pattern[b, Blank[]]], Pattern[x, Blank[]]]]]], Optional[Pattern[n, Blank[]]]]], Pattern[x, Blank[Symbol]]] := Dist[c, Int[ActivateTrig[u]*(c*Tan[a + b*x])^(n - 1)*(B + A*Tan[a + b*x]), x], x] /; FreeQ[{a, b, c, A, B, n}, x] && KnownTangentIntegrandQ[u, x]
  4244. Int[Times[Power[Times[cot[Plus[Optional[Pattern[a, Blank[]]], Times[Optional[Pattern[b, Blank[]]], Pattern[x, Blank[]]]]], Optional[Pattern[c, Blank[]]]], Optional[Pattern[n, Blank[]]]], Pattern[u, Blank[]], Plus[Pattern[A, Blank[]], Times[Optional[Pattern[B, Blank[]]], tan[Plus[Optional[Pattern[a, Blank[]]], Times[Optional[Pattern[b, Blank[]]], Pattern[x, Blank[]]]]]]]], Pattern[x, Blank[Symbol]]] := Dist[c, Int[ActivateTrig[u]*(c*Cot[a + b*x])^(n - 1)*(B + A*Cot[a + b*x]), x], x] /; FreeQ[{a, b, c, A, B, n}, x] && KnownCotangentIntegrandQ[u, x]
  4245. Int[Times[Plus[Times[cot[Plus[Optional[Pattern[a, Blank[]]], Times[Optional[Pattern[b, Blank[]]], Pattern[x, Blank[]]]]], Optional[Pattern[B, Blank[]]]], Pattern[A, Blank[]]], Pattern[u, Blank[]]], Pattern[x, Blank[Symbol]]] := Int[(ActivateTrig[u]*(B + A*Tan[a + b*x]))/Tan[a + b*x], x] /; FreeQ[{a, b, A, B}, x] && KnownTangentIntegrandQ[u, x]
  4246. Int[Times[Pattern[u, Blank[]], Plus[Pattern[A, Blank[]], Times[Optional[Pattern[B, Blank[]]], tan[Plus[Optional[Pattern[a, Blank[]]], Times[Optional[Pattern[b, Blank[]]], Pattern[x, Blank[]]]]]]]], Pattern[x, Blank[Symbol]]] := Int[(ActivateTrig[u]*(B + A*Cot[a + b*x]))/Cot[a + b*x], x] /; FreeQ[{a, b, A, B}, x] && KnownCotangentIntegrandQ[u, x]
  4247. Int[Times[Plus[Optional[Pattern[A, Blank[]]], Times[cot[Plus[Optional[Pattern[a, Blank[]]], Times[Optional[Pattern[b, Blank[]]], Pattern[x, Blank[]]]]], Optional[Pattern[B, Blank[]]]], Times[Power[cot[Plus[Optional[Pattern[a, Blank[]]], Times[Optional[Pattern[b, Blank[]]], Pattern[x, Blank[]]]]], 2], Optional[Pattern[C, Blank[]]]]], Optional[Pattern[u, Blank[]]], Power[Times[Optional[Pattern[c, Blank[]]], tan[Plus[Optional[Pattern[a, Blank[]]], Times[Optional[Pattern[b, Blank[]]], Pattern[x, Blank[]]]]]], Optional[Pattern[n, Blank[]]]]], Pattern[x, Blank[Symbol]]] := Dist[c^2, Int[ActivateTrig[u]*(c*Tan[a + b*x])^(n - 2)*(C + B*Tan[a + b*x] + A*Tan[a + b*x]^2), x], x] /; FreeQ[{a, b, c, A, B, C, n}, x] && KnownTangentIntegrandQ[u, x]
  4248. Int[Times[Power[Times[cot[Plus[Optional[Pattern[a, Blank[]]], Times[Optional[Pattern[b, Blank[]]], Pattern[x, Blank[]]]]], Optional[Pattern[c, Blank[]]]], Optional[Pattern[n, Blank[]]]], Optional[Pattern[u, Blank[]]], Plus[Optional[Pattern[A, Blank[]]], Times[Optional[Pattern[B, Blank[]]], tan[Plus[Optional[Pattern[a, Blank[]]], Times[Optional[Pattern[b, Blank[]]], Pattern[x, Blank[]]]]]], Times[Optional[Pattern[C, Blank[]]], Power[tan[Plus[Optional[Pattern[a, Blank[]]], Times[Optional[Pattern[b, Blank[]]], Pattern[x, Blank[]]]]], 2]]]], Pattern[x, Blank[Symbol]]] := Dist[c^2, Int[ActivateTrig[u]*(c*Cot[a + b*x])^(n - 2)*(C + B*Cot[a + b*x] + A*Cot[a + b*x]^2), x], x] /; FreeQ[{a, b, c, A, B, C, n}, x] && KnownCotangentIntegrandQ[u, x]
  4249. Int[Times[Optional[Pattern[u, Blank[]]], Plus[Times[Power[cot[Plus[Optional[Pattern[a, Blank[]]], Times[Optional[Pattern[b, Blank[]]], Pattern[x, Blank[]]]]], 2], Optional[Pattern[C, Blank[]]]], Pattern[A, Blank[]]], Power[Times[Optional[Pattern[c, Blank[]]], tan[Plus[Optional[Pattern[a, Blank[]]], Times[Optional[Pattern[b, Blank[]]], Pattern[x, Blank[]]]]]], Optional[Pattern[n, Blank[]]]]], Pattern[x, Blank[Symbol]]] := Dist[c^2, Int[ActivateTrig[u]*(c*Tan[a + b*x])^(n - 2)*(C + A*Tan[a + b*x]^2), x], x] /; FreeQ[{a, b, c, A, C, n}, x] && KnownTangentIntegrandQ[u, x]
  4250. Int[Times[Power[Times[cot[Plus[Optional[Pattern[a, Blank[]]], Times[Optional[Pattern[b, Blank[]]], Pattern[x, Blank[]]]]], Optional[Pattern[c, Blank[]]]], Optional[Pattern[n, Blank[]]]], Optional[Pattern[u, Blank[]]], Plus[Pattern[A, Blank[]], Times[Optional[Pattern[C, Blank[]]], Power[tan[Plus[Optional[Pattern[a, Blank[]]], Times[Optional[Pattern[b, Blank[]]], Pattern[x, Blank[]]]]], 2]]]], Pattern[x, Blank[Symbol]]] := Dist[c^2, Int[ActivateTrig[u]*(c*Cot[a + b*x])^(n - 2)*(C + A*Cot[a + b*x]^2), x], x] /; FreeQ[{a, b, c, A, C, n}, x] && KnownCotangentIntegrandQ[u, x]
  4251. Int[Times[Plus[Optional[Pattern[A, Blank[]]], Times[cot[Plus[Optional[Pattern[a, Blank[]]], Times[Optional[Pattern[b, Blank[]]], Pattern[x, Blank[]]]]], Optional[Pattern[B, Blank[]]]], Times[Power[cot[Plus[Optional[Pattern[a, Blank[]]], Times[Optional[Pattern[b, Blank[]]], Pattern[x, Blank[]]]]], 2], Optional[Pattern[C, Blank[]]]]], Pattern[u, Blank[]]], Pattern[x, Blank[Symbol]]] := Int[(ActivateTrig[u]*(C + B*Tan[a + b*x] + A*Tan[a + b*x]^2))/Tan[a + b*x]^2, x] /; FreeQ[{a, b, A, B, C}, x] && KnownTangentIntegrandQ[u, x]
  4252. Int[Times[Pattern[u, Blank[]], Plus[Optional[Pattern[A, Blank[]]], Times[Optional[Pattern[B, Blank[]]], tan[Plus[Optional[Pattern[a, Blank[]]], Times[Optional[Pattern[b, Blank[]]], Pattern[x, Blank[]]]]]], Times[Optional[Pattern[C, Blank[]]], Power[tan[Plus[Optional[Pattern[a, Blank[]]], Times[Optional[Pattern[b, Blank[]]], Pattern[x, Blank[]]]]], 2]]]], Pattern[x, Blank[Symbol]]] := Int[(ActivateTrig[u]*(C + B*Cot[a + b*x] + A*Cot[a + b*x]^2))/Cot[a + b*x]^2, x] /; FreeQ[{a, b, A, B, C}, x] && KnownCotangentIntegrandQ[u, x]
  4253. Int[Times[Plus[Times[Power[cot[Plus[Optional[Pattern[a, Blank[]]], Times[Optional[Pattern[b, Blank[]]], Pattern[x, Blank[]]]]], 2], Optional[Pattern[C, Blank[]]]], Pattern[A, Blank[]]], Pattern[u, Blank[]]], Pattern[x, Blank[Symbol]]] := Int[(ActivateTrig[u]*(C + A*Tan[a + b*x]^2))/Tan[a + b*x]^2, x] /; FreeQ[{a, b, A, C}, x] && KnownTangentIntegrandQ[u, x]
  4254. Int[Times[Pattern[u, Blank[]], Plus[Pattern[A, Blank[]], Times[Optional[Pattern[C, Blank[]]], Power[tan[Plus[Optional[Pattern[a, Blank[]]], Times[Optional[Pattern[b, Blank[]]], Pattern[x, Blank[]]]]], 2]]]], Pattern[x, Blank[Symbol]]] := Int[(ActivateTrig[u]*(C + A*Cot[a + b*x]^2))/Cot[a + b*x]^2, x] /; FreeQ[{a, b, A, C}, x] && KnownCotangentIntegrandQ[u, x]
  4255. Int[Times[Pattern[u, Blank[]], Plus[Optional[Pattern[A, Blank[]]], Times[cot[Plus[Optional[Pattern[a, Blank[]]], Times[Optional[Pattern[b, Blank[]]], Pattern[x, Blank[]]]]], Optional[Pattern[C, Blank[]]]], Times[Optional[Pattern[B, Blank[]]], tan[Plus[Optional[Pattern[a, Blank[]]], Times[Optional[Pattern[b, Blank[]]], Pattern[x, Blank[]]]]]]]], Pattern[x, Blank[Symbol]]] := Int[(ActivateTrig[u]*(C + A*Tan[a + b*x] + B*Tan[a + b*x]^2))/Tan[a + b*x], x] /; FreeQ[{a, b, A, B, C}, x]
  4256. Int[Times[Pattern[u, Blank[]], Plus[Times[Optional[Pattern[A, Blank[]]], Power[tan[Plus[Optional[Pattern[a, Blank[]]], Times[Optional[Pattern[b, Blank[]]], Pattern[x, Blank[]]]]], Optional[Pattern[n, Blank[]]]]], Times[Optional[Pattern[B, Blank[]]], Power[tan[Plus[Optional[Pattern[a, Blank[]]], Times[Optional[Pattern[b, Blank[]]], Pattern[x, Blank[]]]]], Pattern[n1, Blank[]]]], Times[Optional[Pattern[C, Blank[]]], Power[tan[Plus[Optional[Pattern[a, Blank[]]], Times[Optional[Pattern[b, Blank[]]], Pattern[x, Blank[]]]]], Pattern[n2, Blank[]]]]]], Pattern[x, Blank[Symbol]]] := Int[ActivateTrig[u]*Tan[a + b*x]^n*(A + B*Tan[a + b*x] + C*Tan[a + b*x]^2), x] /; FreeQ[{a, b, A, B, C, n}, x] && EqQ[n1, n + 1] && EqQ[n2, n + 2]
  4257. Int[Times[Plus[Times[Power[cot[Plus[Optional[Pattern[a, Blank[]]], Times[Optional[Pattern[b, Blank[]]], Pattern[x, Blank[]]]]], Optional[Pattern[n, Blank[]]]], Optional[Pattern[A, Blank[]]]], Times[Power[cot[Plus[Optional[Pattern[a, Blank[]]], Times[Optional[Pattern[b, Blank[]]], Pattern[x, Blank[]]]]], Pattern[n1, Blank[]]], Optional[Pattern[B, Blank[]]]], Times[Power[cot[Plus[Optional[Pattern[a, Blank[]]], Times[Optional[Pattern[b, Blank[]]], Pattern[x, Blank[]]]]], Pattern[n2, Blank[]]], Optional[Pattern[C, Blank[]]]]], Pattern[u, Blank[]]], Pattern[x, Blank[Symbol]]] := Int[ActivateTrig[u]*Cot[a + b*x]^n*(A + B*Cot[a + b*x] + C*Cot[a + b*x]^2), x] /; FreeQ[{a, b, A, B, C, n}, x] && EqQ[n1, n + 1] && EqQ[n2, n + 2]
  4258. Int[Times[Power[Times[csc[Plus[Optional[Pattern[a, Blank[]]], Times[Optional[Pattern[b, Blank[]]], Pattern[x, Blank[]]]]], Optional[Pattern[d, Blank[]]]], Optional[Pattern[n, Blank[]]]], Pattern[u, Blank[]], Power[Times[Optional[Pattern[c, Blank[]]], sin[Plus[Optional[Pattern[a, Blank[]]], Times[Optional[Pattern[b, Blank[]]], Pattern[x, Blank[]]]]]], Optional[Pattern[m, Blank[]]]]], Pattern[x, Blank[Symbol]]] := Dist[(c*Sin[a + b*x])^m*(d*Csc[a + b*x])^m, Int[ActivateTrig[u]*(d*Csc[a + b*x])^(n - m), x], x] /; FreeQ[{a, b, c, d, m, n}, x] && KnownSecantIntegrandQ[u, x]
  4259. Int[Times[Power[Times[cos[Plus[Optional[Pattern[a, Blank[]]], Times[Optional[Pattern[b, Blank[]]], Pattern[x, Blank[]]]]], Optional[Pattern[c, Blank[]]]], Optional[Pattern[m, Blank[]]]], Pattern[u, Blank[]], Power[Times[Optional[Pattern[d, Blank[]]], sec[Plus[Optional[Pattern[a, Blank[]]], Times[Optional[Pattern[b, Blank[]]], Pattern[x, Blank[]]]]]], Optional[Pattern[n, Blank[]]]]], Pattern[x, Blank[Symbol]]] := Dist[(c*Cos[a + b*x])^m*(d*Sec[a + b*x])^m, Int[ActivateTrig[u]*(d*Sec[a + b*x])^(n - m), x], x] /; FreeQ[{a, b, c, d, m, n}, x] && KnownSecantIntegrandQ[u, x]
  4260. Int[Times[Pattern[u, Blank[]], Power[Times[Optional[Pattern[d, Blank[]]], sec[Plus[Optional[Pattern[a, Blank[]]], Times[Optional[Pattern[b, Blank[]]], Pattern[x, Blank[]]]]]], Optional[Pattern[n, Blank[]]]], Power[Times[Optional[Pattern[c, Blank[]]], tan[Plus[Optional[Pattern[a, Blank[]]], Times[Optional[Pattern[b, Blank[]]], Pattern[x, Blank[]]]]]], Optional[Pattern[m, Blank[]]]]], Pattern[x, Blank[Symbol]]] := Dist[((c*Tan[a + b*x])^m*(d*Csc[a + b*x])^m)/(d*Sec[a + b*x])^m, Int[(ActivateTrig[u]*(d*Sec[a + b*x])^(m + n))/(d*Csc[a + b*x])^m, x], x] /; FreeQ[{a, b, c, d, m, n}, x] && KnownSecantIntegrandQ[u, x] && !IntegerQ[m]
  4261. Int[Times[Power[Times[csc[Plus[Optional[Pattern[a, Blank[]]], Times[Optional[Pattern[b, Blank[]]], Pattern[x, Blank[]]]]], Optional[Pattern[d, Blank[]]]], Optional[Pattern[n, Blank[]]]], Pattern[u, Blank[]], Power[Times[Optional[Pattern[c, Blank[]]], tan[Plus[Optional[Pattern[a, Blank[]]], Times[Optional[Pattern[b, Blank[]]], Pattern[x, Blank[]]]]]], Optional[Pattern[m, Blank[]]]]], Pattern[x, Blank[Symbol]]] := Dist[((c*Tan[a + b*x])^m*(d*Csc[a + b*x])^m)/(d*Sec[a + b*x])^m, Int[(ActivateTrig[u]*(d*Sec[a + b*x])^m)/(d*Csc[a + b*x])^(m - n), x], x] /; FreeQ[{a, b, c, d, m, n}, x] && KnownSecantIntegrandQ[u, x] && !IntegerQ[m]
  4262. Int[Times[Power[Times[cot[Plus[Optional[Pattern[a, Blank[]]], Times[Optional[Pattern[b, Blank[]]], Pattern[x, Blank[]]]]], Optional[Pattern[c, Blank[]]]], Optional[Pattern[m, Blank[]]]], Pattern[u, Blank[]], Power[Times[Optional[Pattern[d, Blank[]]], sec[Plus[Optional[Pattern[a, Blank[]]], Times[Optional[Pattern[b, Blank[]]], Pattern[x, Blank[]]]]]], Optional[Pattern[n, Blank[]]]]], Pattern[x, Blank[Symbol]]] := Dist[((c*Cot[a + b*x])^m*(d*Sec[a + b*x])^m)/(d*Csc[a + b*x])^m, Int[(ActivateTrig[u]*(d*Csc[a + b*x])^m)/(d*Sec[a + b*x])^(m - n), x], x] /; FreeQ[{a, b, c, d, m, n}, x] && KnownSecantIntegrandQ[u, x] && !IntegerQ[m]
  4263. Int[Times[Power[Times[cot[Plus[Optional[Pattern[a, Blank[]]], Times[Optional[Pattern[b, Blank[]]], Pattern[x, Blank[]]]]], Optional[Pattern[c, Blank[]]]], Optional[Pattern[m, Blank[]]]], Power[Times[csc[Plus[Optional[Pattern[a, Blank[]]], Times[Optional[Pattern[b, Blank[]]], Pattern[x, Blank[]]]]], Optional[Pattern[d, Blank[]]]], Optional[Pattern[n, Blank[]]]], Pattern[u, Blank[]]], Pattern[x, Blank[Symbol]]] := Dist[((c*Cot[a + b*x])^m*(d*Sec[a + b*x])^m)/(d*Csc[a + b*x])^m, Int[(ActivateTrig[u]*(d*Csc[a + b*x])^(m + n))/(d*Sec[a + b*x])^m, x], x] /; FreeQ[{a, b, c, d, m, n}, x] && KnownSecantIntegrandQ[u, x] && !IntegerQ[m]
  4264. Int[Times[Pattern[u, Blank[]], Power[Times[Optional[Pattern[c, Blank[]]], sin[Plus[Optional[Pattern[a, Blank[]]], Times[Optional[Pattern[b, Blank[]]], Pattern[x, Blank[]]]]]], Optional[Pattern[m, Blank[]]]]], Pattern[x, Blank[Symbol]]] := Dist[(c*Csc[a + b*x])^m*(c*Sin[a + b*x])^m, Int[ActivateTrig[u]/(c*Csc[a + b*x])^m, x], x] /; FreeQ[{a, b, c, m}, x] && !IntegerQ[m] && KnownSecantIntegrandQ[u, x]
  4265. Int[Times[Power[Times[cos[Plus[Optional[Pattern[a, Blank[]]], Times[Optional[Pattern[b, Blank[]]], Pattern[x, Blank[]]]]], Optional[Pattern[c, Blank[]]]], Optional[Pattern[m, Blank[]]]], Pattern[u, Blank[]]], Pattern[x, Blank[Symbol]]] := Dist[(c*Cos[a + b*x])^m*(c*Sec[a + b*x])^m, Int[ActivateTrig[u]/(c*Sec[a + b*x])^m, x], x] /; FreeQ[{a, b, c, m}, x] && !IntegerQ[m] && KnownSecantIntegrandQ[u, x]
  4266. Int[Times[Pattern[u, Blank[]], Power[Times[Optional[Pattern[c, Blank[]]], tan[Plus[Optional[Pattern[a, Blank[]]], Times[Optional[Pattern[b, Blank[]]], Pattern[x, Blank[]]]]]], Optional[Pattern[m, Blank[]]]]], Pattern[x, Blank[Symbol]]] := Dist[((c*Tan[a + b*x])^m*(c*Csc[a + b*x])^m)/(c*Sec[a + b*x])^m, Int[(ActivateTrig[u]*(c*Sec[a + b*x])^m)/(c*Csc[a + b*x])^m, x], x] /; FreeQ[{a, b, c, m}, x] && !IntegerQ[m] && KnownSecantIntegrandQ[u, x]
  4267. Int[Times[Power[Times[cot[Plus[Optional[Pattern[a, Blank[]]], Times[Optional[Pattern[b, Blank[]]], Pattern[x, Blank[]]]]], Optional[Pattern[c, Blank[]]]], Optional[Pattern[m, Blank[]]]], Pattern[u, Blank[]]], Pattern[x, Blank[Symbol]]] := Dist[((c*Cot[a + b*x])^m*(c*Sec[a + b*x])^m)/(c*Csc[a + b*x])^m, Int[(ActivateTrig[u]*(c*Csc[a + b*x])^m)/(c*Sec[a + b*x])^m, x], x] /; FreeQ[{a, b, c, m}, x] && !IntegerQ[m] && KnownSecantIntegrandQ[u, x]
  4268. Int[Times[Plus[Times[cos[Plus[Optional[Pattern[a, Blank[]]], Times[Optional[Pattern[b, Blank[]]], Pattern[x, Blank[]]]]], Optional[Pattern[B, Blank[]]]], Pattern[A, Blank[]]], Pattern[u, Blank[]], Power[Times[Optional[Pattern[c, Blank[]]], sec[Plus[Optional[Pattern[a, Blank[]]], Times[Optional[Pattern[b, Blank[]]], Pattern[x, Blank[]]]]]], Optional[Pattern[n, Blank[]]]]], Pattern[x, Blank[Symbol]]] := Dist[c, Int[ActivateTrig[u]*(c*Sec[a + b*x])^(n - 1)*(B + A*Sec[a + b*x]), x], x] /; FreeQ[{a, b, c, A, B, n}, x] && KnownSecantIntegrandQ[u, x]
  4269. Int[Times[Power[Times[csc[Plus[Optional[Pattern[a, Blank[]]], Times[Optional[Pattern[b, Blank[]]], Pattern[x, Blank[]]]]], Optional[Pattern[c, Blank[]]]], Optional[Pattern[n, Blank[]]]], Pattern[u, Blank[]], Plus[Pattern[A, Blank[]], Times[Optional[Pattern[B, Blank[]]], sin[Plus[Optional[Pattern[a, Blank[]]], Times[Optional[Pattern[b, Blank[]]], Pattern[x, Blank[]]]]]]]], Pattern[x, Blank[Symbol]]] := Dist[c, Int[ActivateTrig[u]*(c*Csc[a + b*x])^(n - 1)*(B + A*Csc[a + b*x]), x], x] /; FreeQ[{a, b, c, A, B, n}, x] && KnownSecantIntegrandQ[u, x]
  4270. Int[Times[Plus[Times[cos[Plus[Optional[Pattern[a, Blank[]]], Times[Optional[Pattern[b, Blank[]]], Pattern[x, Blank[]]]]], Optional[Pattern[B, Blank[]]]], Pattern[A, Blank[]]], Pattern[u, Blank[]]], Pattern[x, Blank[Symbol]]] := Int[(ActivateTrig[u]*(B + A*Sec[a + b*x]))/Sec[a + b*x], x] /; FreeQ[{a, b, A, B}, x] && KnownSecantIntegrandQ[u, x]
  4271. Int[Times[Pattern[u, Blank[]], Plus[Pattern[A, Blank[]], Times[Optional[Pattern[B, Blank[]]], sin[Plus[Optional[Pattern[a, Blank[]]], Times[Optional[Pattern[b, Blank[]]], Pattern[x, Blank[]]]]]]]], Pattern[x, Blank[Symbol]]] := Int[(ActivateTrig[u]*(B + A*Csc[a + b*x]))/Csc[a + b*x], x] /; FreeQ[{a, b, A, B}, x] && KnownSecantIntegrandQ[u, x]
  4272. Int[Times[Plus[Optional[Pattern[A, Blank[]]], Times[cos[Plus[Optional[Pattern[a, Blank[]]], Times[Optional[Pattern[b, Blank[]]], Pattern[x, Blank[]]]]], Optional[Pattern[B, Blank[]]]], Times[Power[cos[Plus[Optional[Pattern[a, Blank[]]], Times[Optional[Pattern[b, Blank[]]], Pattern[x, Blank[]]]]], 2], Optional[Pattern[C, Blank[]]]]], Optional[Pattern[u, Blank[]]], Power[Times[Optional[Pattern[c, Blank[]]], sec[Plus[Optional[Pattern[a, Blank[]]], Times[Optional[Pattern[b, Blank[]]], Pattern[x, Blank[]]]]]], Optional[Pattern[n, Blank[]]]]], Pattern[x, Blank[Symbol]]] := Dist[c^2, Int[ActivateTrig[u]*(c*Sec[a + b*x])^(n - 2)*(C + B*Sec[a + b*x] + A*Sec[a + b*x]^2), x], x] /; FreeQ[{a, b, c, A, B, C, n}, x] && KnownSecantIntegrandQ[u, x]
  4273. Int[Times[Power[Times[csc[Plus[Optional[Pattern[a, Blank[]]], Times[Optional[Pattern[b, Blank[]]], Pattern[x, Blank[]]]]], Optional[Pattern[c, Blank[]]]], Optional[Pattern[n, Blank[]]]], Optional[Pattern[u, Blank[]]], Plus[Optional[Pattern[A, Blank[]]], Times[Optional[Pattern[B, Blank[]]], sin[Plus[Optional[Pattern[a, Blank[]]], Times[Optional[Pattern[b, Blank[]]], Pattern[x, Blank[]]]]]], Times[Optional[Pattern[C, Blank[]]], Power[sin[Plus[Optional[Pattern[a, Blank[]]], Times[Optional[Pattern[b, Blank[]]], Pattern[x, Blank[]]]]], 2]]]], Pattern[x, Blank[Symbol]]] := Dist[c^2, Int[ActivateTrig[u]*(c*Csc[a + b*x])^(n - 2)*(C + B*Csc[a + b*x] + A*Csc[a + b*x]^2), x], x] /; FreeQ[{a, b, c, A, B, C, n}, x] && KnownSecantIntegrandQ[u, x]
  4274. Int[Times[Optional[Pattern[u, Blank[]]], Plus[Times[Power[cos[Plus[Optional[Pattern[a, Blank[]]], Times[Optional[Pattern[b, Blank[]]], Pattern[x, Blank[]]]]], 2], Optional[Pattern[C, Blank[]]]], Pattern[A, Blank[]]], Power[Times[Optional[Pattern[c, Blank[]]], sec[Plus[Optional[Pattern[a, Blank[]]], Times[Optional[Pattern[b, Blank[]]], Pattern[x, Blank[]]]]]], Optional[Pattern[n, Blank[]]]]], Pattern[x, Blank[Symbol]]] := Dist[c^2, Int[ActivateTrig[u]*(c*Sec[a + b*x])^(n - 2)*(C + A*Sec[a + b*x]^2), x], x] /; FreeQ[{a, b, c, A, C, n}, x] && KnownSecantIntegrandQ[u, x]
  4275. Int[Times[Power[Times[csc[Plus[Optional[Pattern[a, Blank[]]], Times[Optional[Pattern[b, Blank[]]], Pattern[x, Blank[]]]]], Optional[Pattern[c, Blank[]]]], Optional[Pattern[n, Blank[]]]], Optional[Pattern[u, Blank[]]], Plus[Pattern[A, Blank[]], Times[Optional[Pattern[C, Blank[]]], Power[sin[Plus[Optional[Pattern[a, Blank[]]], Times[Optional[Pattern[b, Blank[]]], Pattern[x, Blank[]]]]], 2]]]], Pattern[x, Blank[Symbol]]] := Dist[c^2, Int[ActivateTrig[u]*(c*Csc[a + b*x])^(n - 2)*(C + A*Csc[a + b*x]^2), x], x] /; FreeQ[{a, b, c, A, C, n}, x] && KnownSecantIntegrandQ[u, x]
  4276. Int[Times[Plus[Optional[Pattern[A, Blank[]]], Times[cos[Plus[Optional[Pattern[a, Blank[]]], Times[Optional[Pattern[b, Blank[]]], Pattern[x, Blank[]]]]], Optional[Pattern[B, Blank[]]]], Times[Power[cos[Plus[Optional[Pattern[a, Blank[]]], Times[Optional[Pattern[b, Blank[]]], Pattern[x, Blank[]]]]], 2], Optional[Pattern[C, Blank[]]]]], Pattern[u, Blank[]]], Pattern[x, Blank[Symbol]]] := Int[(ActivateTrig[u]*(C + B*Sec[a + b*x] + A*Sec[a + b*x]^2))/Sec[a + b*x]^2, x] /; FreeQ[{a, b, A, B, C}, x] && KnownSecantIntegrandQ[u, x]
  4277. Int[Times[Pattern[u, Blank[]], Plus[Optional[Pattern[A, Blank[]]], Times[Optional[Pattern[B, Blank[]]], sin[Plus[Optional[Pattern[a, Blank[]]], Times[Optional[Pattern[b, Blank[]]], Pattern[x, Blank[]]]]]], Times[Optional[Pattern[C, Blank[]]], Power[sin[Plus[Optional[Pattern[a, Blank[]]], Times[Optional[Pattern[b, Blank[]]], Pattern[x, Blank[]]]]], 2]]]], Pattern[x, Blank[Symbol]]] := Int[(ActivateTrig[u]*(C + B*Csc[a + b*x] + A*Csc[a + b*x]^2))/Csc[a + b*x]^2, x] /; FreeQ[{a, b, A, B, C}, x] && KnownSecantIntegrandQ[u, x]
  4278. Int[Times[Plus[Times[Power[cos[Plus[Optional[Pattern[a, Blank[]]], Times[Optional[Pattern[b, Blank[]]], Pattern[x, Blank[]]]]], 2], Optional[Pattern[C, Blank[]]]], Pattern[A, Blank[]]], Pattern[u, Blank[]]], Pattern[x, Blank[Symbol]]] := Int[(ActivateTrig[u]*(C + A*Sec[a + b*x]^2))/Sec[a + b*x]^2, x] /; FreeQ[{a, b, A, C}, x] && KnownSecantIntegrandQ[u, x]
  4279. Int[Times[Pattern[u, Blank[]], Plus[Pattern[A, Blank[]], Times[Optional[Pattern[C, Blank[]]], Power[sin[Plus[Optional[Pattern[a, Blank[]]], Times[Optional[Pattern[b, Blank[]]], Pattern[x, Blank[]]]]], 2]]]], Pattern[x, Blank[Symbol]]] := Int[(ActivateTrig[u]*(C + A*Csc[a + b*x]^2))/Csc[a + b*x]^2, x] /; FreeQ[{a, b, A, C}, x] && KnownSecantIntegrandQ[u, x]
  4280. Int[Times[Pattern[u, Blank[]], Plus[Times[Optional[Pattern[A, Blank[]]], Power[sec[Plus[Optional[Pattern[a, Blank[]]], Times[Optional[Pattern[b, Blank[]]], Pattern[x, Blank[]]]]], Optional[Pattern[n, Blank[]]]]], Times[Optional[Pattern[B, Blank[]]], Power[sec[Plus[Optional[Pattern[a, Blank[]]], Times[Optional[Pattern[b, Blank[]]], Pattern[x, Blank[]]]]], Pattern[n1, Blank[]]]], Times[Optional[Pattern[C, Blank[]]], Power[sec[Plus[Optional[Pattern[a, Blank[]]], Times[Optional[Pattern[b, Blank[]]], Pattern[x, Blank[]]]]], Pattern[n2, Blank[]]]]]], Pattern[x, Blank[Symbol]]] := Int[ActivateTrig[u]*Sec[a + b*x]^n*(A + B*Sec[a + b*x] + C*Sec[a + b*x]^2), x] /; FreeQ[{a, b, A, B, C, n}, x] && EqQ[n1, n + 1] && EqQ[n2, n + 2]
  4281. Int[Times[Plus[Times[Power[csc[Plus[Optional[Pattern[a, Blank[]]], Times[Optional[Pattern[b, Blank[]]], Pattern[x, Blank[]]]]], Optional[Pattern[n, Blank[]]]], Optional[Pattern[A, Blank[]]]], Times[Power[csc[Plus[Optional[Pattern[a, Blank[]]], Times[Optional[Pattern[b, Blank[]]], Pattern[x, Blank[]]]]], Pattern[n1, Blank[]]], Optional[Pattern[B, Blank[]]]], Times[Power[csc[Plus[Optional[Pattern[a, Blank[]]], Times[Optional[Pattern[b, Blank[]]], Pattern[x, Blank[]]]]], Pattern[n2, Blank[]]], Optional[Pattern[C, Blank[]]]]], Pattern[u, Blank[]]], Pattern[x, Blank[Symbol]]] := Int[ActivateTrig[u]*Csc[a + b*x]^n*(A + B*Csc[a + b*x] + C*Csc[a + b*x]^2), x] /; FreeQ[{a, b, A, B, C, n}, x] && EqQ[n1, n + 1] && EqQ[n2, n + 2]
  4282. Int[Times[sin[Plus[Optional[Pattern[a, Blank[]]], Times[Optional[Pattern[b, Blank[]]], Pattern[x, Blank[]]]]], sin[Plus[Optional[Pattern[c, Blank[]]], Times[Optional[Pattern[d, Blank[]]], Pattern[x, Blank[]]]]]], Pattern[x, Blank[Symbol]]] := Simp[Sin[a - c + (b - d)*x]/(2*(b - d)), x] - Simp[Sin[a + c + (b + d)*x]/(2*(b + d)), x] /; FreeQ[{a, b, c, d}, x] && NeQ[b^2 - d^2, 0]
  4283. Int[Times[cos[Plus[Optional[Pattern[a, Blank[]]], Times[Optional[Pattern[b, Blank[]]], Pattern[x, Blank[]]]]], cos[Plus[Optional[Pattern[c, Blank[]]], Times[Optional[Pattern[d, Blank[]]], Pattern[x, Blank[]]]]]], Pattern[x, Blank[Symbol]]] := Simp[Sin[a - c + (b - d)*x]/(2*(b - d)), x] + Simp[Sin[a + c + (b + d)*x]/(2*(b + d)), x] /; FreeQ[{a, b, c, d}, x] && NeQ[b^2 - d^2, 0]
  4284. Int[Times[cos[Plus[Optional[Pattern[c, Blank[]]], Times[Optional[Pattern[d, Blank[]]], Pattern[x, Blank[]]]]], sin[Plus[Optional[Pattern[a, Blank[]]], Times[Optional[Pattern[b, Blank[]]], Pattern[x, Blank[]]]]]], Pattern[x, Blank[Symbol]]] := -Simp[Cos[a - c + (b - d)*x]/(2*(b - d)), x] - Simp[Cos[a + c + (b + d)*x]/(2*(b + d)), x] /; FreeQ[{a, b, c, d}, x] && NeQ[b^2 - d^2, 0]
  4285. Int[Times[Power[cos[Plus[Optional[Pattern[a, Blank[]]], Times[Optional[Pattern[b, Blank[]]], Pattern[x, Blank[]]]]], 2], Power[Times[Optional[Pattern[g, Blank[]]], sin[Plus[Optional[Pattern[c, Blank[]]], Times[Optional[Pattern[d, Blank[]]], Pattern[x, Blank[]]]]]], Pattern[p, Blank[]]]], Pattern[x, Blank[Symbol]]] := Dist[1/2, Int[(g*Sin[c + d*x])^p, x], x] + Dist[1/2, Int[Cos[c + d*x]*(g*Sin[c + d*x])^p, x], x] /; FreeQ[{a, b, c, d, g}, x] && EqQ[b*c - a*d, 0] && EqQ[d/b, 2] && IGtQ[p/2, 0]
  4286. Int[Times[Power[sin[Plus[Optional[Pattern[a, Blank[]]], Times[Optional[Pattern[b, Blank[]]], Pattern[x, Blank[]]]]], 2], Power[Times[Optional[Pattern[g, Blank[]]], sin[Plus[Optional[Pattern[c, Blank[]]], Times[Optional[Pattern[d, Blank[]]], Pattern[x, Blank[]]]]]], Pattern[p, Blank[]]]], Pattern[x, Blank[Symbol]]] := Dist[1/2, Int[(g*Sin[c + d*x])^p, x], x] - Dist[1/2, Int[Cos[c + d*x]*(g*Sin[c + d*x])^p, x], x] /; FreeQ[{a, b, c, d, g}, x] && EqQ[b*c - a*d, 0] && EqQ[d/b, 2] && IGtQ[p/2, 0]
  4287. Int[Times[Power[Times[cos[Plus[Optional[Pattern[a, Blank[]]], Times[Optional[Pattern[b, Blank[]]], Pattern[x, Blank[]]]]], Optional[Pattern[e, Blank[]]]], Optional[Pattern[m, Blank[]]]], Power[sin[Plus[Optional[Pattern[c, Blank[]]], Times[Optional[Pattern[d, Blank[]]], Pattern[x, Blank[]]]]], Optional[Pattern[p, Blank[]]]]], Pattern[x, Blank[Symbol]]] := Dist[2^p/e^p, Int[(e*Cos[a + b*x])^(m + p)*Sin[a + b*x]^p, x], x] /; FreeQ[{a, b, c, d, e, m}, x] && EqQ[b*c - a*d, 0] && EqQ[d/b, 2] && IntegerQ[p]
  4288. Int[Times[Power[Times[Optional[Pattern[f, Blank[]]], sin[Plus[Optional[Pattern[a, Blank[]]], Times[Optional[Pattern[b, Blank[]]], Pattern[x, Blank[]]]]]], Optional[Pattern[n, Blank[]]]], Power[sin[Plus[Optional[Pattern[c, Blank[]]], Times[Optional[Pattern[d, Blank[]]], Pattern[x, Blank[]]]]], Optional[Pattern[p, Blank[]]]]], Pattern[x, Blank[Symbol]]] := Dist[2^p/f^p, Int[Cos[a + b*x]^p*(f*Sin[a + b*x])^(n + p), x], x] /; FreeQ[{a, b, c, d, f, n}, x] && EqQ[b*c - a*d, 0] && EqQ[d/b, 2] && IntegerQ[p]
  4289. Int[Times[Power[Times[cos[Plus[Optional[Pattern[a, Blank[]]], Times[Optional[Pattern[b, Blank[]]], Pattern[x, Blank[]]]]], Optional[Pattern[e, Blank[]]]], Pattern[m, Blank[]]], Power[Times[Optional[Pattern[g, Blank[]]], sin[Plus[Optional[Pattern[c, Blank[]]], Times[Optional[Pattern[d, Blank[]]], Pattern[x, Blank[]]]]]], Pattern[p, Blank[]]]], Pattern[x, Blank[Symbol]]] := Simp[(e^2*(e*Cos[a + b*x])^(m - 2)*(g*Sin[c + d*x])^(p + 1))/(2*b*g*(p + 1)), x] /; FreeQ[{a, b, c, d, e, g, m, p}, x] && EqQ[b*c - a*d, 0] && EqQ[d/b, 2] && !IntegerQ[p] && EqQ[m + p - 1, 0]
  4290. Int[Times[Power[Times[Optional[Pattern[e, Blank[]]], sin[Plus[Optional[Pattern[a, Blank[]]], Times[Optional[Pattern[b, Blank[]]], Pattern[x, Blank[]]]]]], Pattern[m, Blank[]]], Power[Times[Optional[Pattern[g, Blank[]]], sin[Plus[Optional[Pattern[c, Blank[]]], Times[Optional[Pattern[d, Blank[]]], Pattern[x, Blank[]]]]]], Pattern[p, Blank[]]]], Pattern[x, Blank[Symbol]]] := -Simp[(e^2*(e*Sin[a + b*x])^(m - 2)*(g*Sin[c + d*x])^(p + 1))/(2*b*g*(p + 1)), x] /; FreeQ[{a, b, c, d, e, g, m, p}, x] && EqQ[b*c - a*d, 0] && EqQ[d/b, 2] && !IntegerQ[p] && EqQ[m + p - 1, 0]
  4291. Int[Times[Power[Times[cos[Plus[Optional[Pattern[a, Blank[]]], Times[Optional[Pattern[b, Blank[]]], Pattern[x, Blank[]]]]], Optional[Pattern[e, Blank[]]]], Optional[Pattern[m, Blank[]]]], Power[Times[Optional[Pattern[g, Blank[]]], sin[Plus[Optional[Pattern[c, Blank[]]], Times[Optional[Pattern[d, Blank[]]], Pattern[x, Blank[]]]]]], Pattern[p, Blank[]]]], Pattern[x, Blank[Symbol]]] := -Simp[((e*Cos[a + b*x])^m*(g*Sin[c + d*x])^(p + 1))/(b*g*m), x] /; FreeQ[{a, b, c, d, e, g, m, p}, x] && EqQ[b*c - a*d, 0] && EqQ[d/b, 2] && !IntegerQ[p] && EqQ[m + 2*p + 2, 0]
  4292. Int[Times[Power[Times[Optional[Pattern[e, Blank[]]], sin[Plus[Optional[Pattern[a, Blank[]]], Times[Optional[Pattern[b, Blank[]]], Pattern[x, Blank[]]]]]], Optional[Pattern[m, Blank[]]]], Power[Times[Optional[Pattern[g, Blank[]]], sin[Plus[Optional[Pattern[c, Blank[]]], Times[Optional[Pattern[d, Blank[]]], Pattern[x, Blank[]]]]]], Pattern[p, Blank[]]]], Pattern[x, Blank[Symbol]]] := Simp[((e*Sin[a + b*x])^m*(g*Sin[c + d*x])^(p + 1))/(b*g*m), x] /; FreeQ[{a, b, c, d, e, g, m, p}, x] && EqQ[b*c - a*d, 0] && EqQ[d/b, 2] && !IntegerQ[p] && EqQ[m + 2*p + 2, 0]
  4293. Int[Times[Power[Times[cos[Plus[Optional[Pattern[a, Blank[]]], Times[Optional[Pattern[b, Blank[]]], Pattern[x, Blank[]]]]], Optional[Pattern[e, Blank[]]]], Pattern[m, Blank[]]], Power[Times[Optional[Pattern[g, Blank[]]], sin[Plus[Optional[Pattern[c, Blank[]]], Times[Optional[Pattern[d, Blank[]]], Pattern[x, Blank[]]]]]], Pattern[p, Blank[]]]], Pattern[x, Blank[Symbol]]] := Simp[(e^2*(e*Cos[a + b*x])^(m - 2)*(g*Sin[c + d*x])^(p + 1))/(2*b*g*(p + 1)), x] + Dist[(e^4*(m + p - 1))/(4*g^2*(p + 1)), Int[(e*Cos[a + b*x])^(m - 4)*(g*Sin[c + d*x])^(p + 2), x], x] /; FreeQ[{a, b, c, d, e, g}, x] && EqQ[b*c - a*d, 0] && EqQ[d/b, 2] && !IntegerQ[p] && GtQ[m, 2] && LtQ[p, -1] && (GtQ[m, 3] || EqQ[p, -3/2]) && IntegersQ[2*m, 2*p]
  4294. Int[Times[Power[Times[Optional[Pattern[e, Blank[]]], sin[Plus[Optional[Pattern[a, Blank[]]], Times[Optional[Pattern[b, Blank[]]], Pattern[x, Blank[]]]]]], Pattern[m, Blank[]]], Power[Times[Optional[Pattern[g, Blank[]]], sin[Plus[Optional[Pattern[c, Blank[]]], Times[Optional[Pattern[d, Blank[]]], Pattern[x, Blank[]]]]]], Pattern[p, Blank[]]]], Pattern[x, Blank[Symbol]]] := -Simp[(e^2*(e*Sin[a + b*x])^(m - 2)*(g*Sin[c + d*x])^(p + 1))/(2*b*g*(p + 1)), x] + Dist[(e^4*(m + p - 1))/(4*g^2*(p + 1)), Int[(e*Sin[a + b*x])^(m - 4)*(g*Sin[c + d*x])^(p + 2), x], x] /; FreeQ[{a, b, c, d, e, g}, x] && EqQ[b*c - a*d, 0] && EqQ[d/b, 2] && !IntegerQ[p] && GtQ[m, 2] && LtQ[p, -1] && (GtQ[m, 3] || EqQ[p, -3/2]) && IntegersQ[2*m, 2*p]
  4295. Int[Times[Power[Times[cos[Plus[Optional[Pattern[a, Blank[]]], Times[Optional[Pattern[b, Blank[]]], Pattern[x, Blank[]]]]], Optional[Pattern[e, Blank[]]]], Pattern[m, Blank[]]], Power[Times[Optional[Pattern[g, Blank[]]], sin[Plus[Optional[Pattern[c, Blank[]]], Times[Optional[Pattern[d, Blank[]]], Pattern[x, Blank[]]]]]], Pattern[p, Blank[]]]], Pattern[x, Blank[Symbol]]] := Simp[((e*Cos[a + b*x])^m*(g*Sin[c + d*x])^(p + 1))/(2*b*g*(p + 1)), x] + Dist[(e^2*(m + 2*p + 2))/(4*g^2*(p + 1)), Int[(e*Cos[a + b*x])^(m - 2)*(g*Sin[c + d*x])^(p + 2), x], x] /; FreeQ[{a, b, c, d, e, g}, x] && EqQ[b*c - a*d, 0] && EqQ[d/b, 2] && !IntegerQ[p] && GtQ[m, 1] && LtQ[p, -1] && NeQ[m + 2*p + 2, 0] && (LtQ[p, -2] || EqQ[m, 2]) && IntegersQ[2*m, 2*p]
  4296. Int[Times[Power[Times[Optional[Pattern[e, Blank[]]], sin[Plus[Optional[Pattern[a, Blank[]]], Times[Optional[Pattern[b, Blank[]]], Pattern[x, Blank[]]]]]], Pattern[m, Blank[]]], Power[Times[Optional[Pattern[g, Blank[]]], sin[Plus[Optional[Pattern[c, Blank[]]], Times[Optional[Pattern[d, Blank[]]], Pattern[x, Blank[]]]]]], Pattern[p, Blank[]]]], Pattern[x, Blank[Symbol]]] := -Simp[((e*Sin[a + b*x])^m*(g*Sin[c + d*x])^(p + 1))/(2*b*g*(p + 1)), x] + Dist[(e^2*(m + 2*p + 2))/(4*g^2*(p + 1)), Int[(e*Sin[a + b*x])^(m - 2)*(g*Sin[c + d*x])^(p + 2), x], x] /; FreeQ[{a, b, c, d, e, g}, x] && EqQ[b*c - a*d, 0] && EqQ[d/b, 2] && !IntegerQ[p] && GtQ[m, 1] && LtQ[p, -1] && NeQ[m + 2*p + 2, 0] && (LtQ[p, -2] || EqQ[m, 2]) && IntegersQ[2*m, 2*p]
  4297. Int[Times[Power[Times[cos[Plus[Optional[Pattern[a, Blank[]]], Times[Optional[Pattern[b, Blank[]]], Pattern[x, Blank[]]]]], Optional[Pattern[e, Blank[]]]], Pattern[m, Blank[]]], Power[Times[Optional[Pattern[g, Blank[]]], sin[Plus[Optional[Pattern[c, Blank[]]], Times[Optional[Pattern[d, Blank[]]], Pattern[x, Blank[]]]]]], Pattern[p, Blank[]]]], Pattern[x, Blank[Symbol]]] := Simp[(e^2*(e*Cos[a + b*x])^(m - 2)*(g*Sin[c + d*x])^(p + 1))/(2*b*g*(m + 2*p)), x] + Dist[(e^2*(m + p - 1))/(m + 2*p), Int[(e*Cos[a + b*x])^(m - 2)*(g*Sin[c + d*x])^p, x], x] /; FreeQ[{a, b, c, d, e, g, p}, x] && EqQ[b*c - a*d, 0] && EqQ[d/b, 2] && !IntegerQ[p] && GtQ[m, 1] && NeQ[m + 2*p, 0] && IntegersQ[2*m, 2*p]
  4298. Int[Times[Power[Times[Optional[Pattern[e, Blank[]]], sin[Plus[Optional[Pattern[a, Blank[]]], Times[Optional[Pattern[b, Blank[]]], Pattern[x, Blank[]]]]]], Pattern[m, Blank[]]], Power[Times[Optional[Pattern[g, Blank[]]], sin[Plus[Optional[Pattern[c, Blank[]]], Times[Optional[Pattern[d, Blank[]]], Pattern[x, Blank[]]]]]], Pattern[p, Blank[]]]], Pattern[x, Blank[Symbol]]] := -Simp[(e^2*(e*Sin[a + b*x])^(m - 2)*(g*Sin[c + d*x])^(p + 1))/(2*b*g*(m + 2*p)), x] + Dist[(e^2*(m + p - 1))/(m + 2*p), Int[(e*Sin[a + b*x])^(m - 2)*(g*Sin[c + d*x])^p, x], x] /; FreeQ[{a, b, c, d, e, g, p}, x] && EqQ[b*c - a*d, 0] && EqQ[d/b, 2] && !IntegerQ[p] && GtQ[m, 1] && NeQ[m + 2*p, 0] && IntegersQ[2*m, 2*p]
  4299. Int[Times[Power[Times[cos[Plus[Optional[Pattern[a, Blank[]]], Times[Optional[Pattern[b, Blank[]]], Pattern[x, Blank[]]]]], Optional[Pattern[e, Blank[]]]], Pattern[m, Blank[]]], Power[Times[Optional[Pattern[g, Blank[]]], sin[Plus[Optional[Pattern[c, Blank[]]], Times[Optional[Pattern[d, Blank[]]], Pattern[x, Blank[]]]]]], Pattern[p, Blank[]]]], Pattern[x, Blank[Symbol]]] := -Simp[((e*Cos[a + b*x])^m*(g*Sin[c + d*x])^(p + 1))/(2*b*g*(m + p + 1)), x] + Dist[(m + 2*p + 2)/(e^2*(m + p + 1)), Int[(e*Cos[a + b*x])^(m + 2)*(g*Sin[c + d*x])^p, x], x] /; FreeQ[{a, b, c, d, e, g, p}, x] && EqQ[b*c - a*d, 0] && EqQ[d/b, 2] && !IntegerQ[p] && LtQ[m, -1] && NeQ[m + 2*p + 2, 0] && NeQ[m + p + 1, 0] && IntegersQ[2*m, 2*p]
  4300. Int[Times[Power[Times[Optional[Pattern[e, Blank[]]], sin[Plus[Optional[Pattern[a, Blank[]]], Times[Optional[Pattern[b, Blank[]]], Pattern[x, Blank[]]]]]], Pattern[m, Blank[]]], Power[Times[Optional[Pattern[g, Blank[]]], sin[Plus[Optional[Pattern[c, Blank[]]], Times[Optional[Pattern[d, Blank[]]], Pattern[x, Blank[]]]]]], Pattern[p, Blank[]]]], Pattern[x, Blank[Symbol]]] := Simp[((e*Sin[a + b*x])^m*(g*Sin[c + d*x])^(p + 1))/(2*b*g*(m + p + 1)), x] + Dist[(m + 2*p + 2)/(e^2*(m + p + 1)), Int[(e*Sin[a + b*x])^(m + 2)*(g*Sin[c + d*x])^p, x], x] /; FreeQ[{a, b, c, d, e, g, p}, x] && EqQ[b*c - a*d, 0] && EqQ[d/b, 2] && !IntegerQ[p] && LtQ[m, -1] && NeQ[m + 2*p + 2, 0] && NeQ[m + p + 1, 0] && IntegersQ[2*m, 2*p]
  4301. Int[Times[cos[Plus[Optional[Pattern[a, Blank[]]], Times[Optional[Pattern[b, Blank[]]], Pattern[x, Blank[]]]]], Power[Times[Optional[Pattern[g, Blank[]]], sin[Plus[Optional[Pattern[c, Blank[]]], Times[Optional[Pattern[d, Blank[]]], Pattern[x, Blank[]]]]]], Pattern[p, Blank[]]]], Pattern[x, Blank[Symbol]]] := Simp[(2*Sin[a + b*x]*(g*Sin[c + d*x])^p)/(d*(2*p + 1)), x] + Dist[(2*p*g)/(2*p + 1), Int[Sin[a + b*x]*(g*Sin[c + d*x])^(p - 1), x], x] /; FreeQ[{a, b, c, d, g}, x] && EqQ[b*c - a*d, 0] && EqQ[d/b, 2] && !IntegerQ[p] && GtQ[p, 0] && IntegerQ[2*p]
  4302. Int[Times[sin[Plus[Optional[Pattern[a, Blank[]]], Times[Optional[Pattern[b, Blank[]]], Pattern[x, Blank[]]]]], Power[Times[Optional[Pattern[g, Blank[]]], sin[Plus[Optional[Pattern[c, Blank[]]], Times[Optional[Pattern[d, Blank[]]], Pattern[x, Blank[]]]]]], Pattern[p, Blank[]]]], Pattern[x, Blank[Symbol]]] := Simp[(-2*Cos[a + b*x]*(g*Sin[c + d*x])^p)/(d*(2*p + 1)), x] + Dist[(2*p*g)/(2*p + 1), Int[Cos[a + b*x]*(g*Sin[c + d*x])^(p - 1), x], x] /; FreeQ[{a, b, c, d, g}, x] && EqQ[b*c - a*d, 0] && EqQ[d/b, 2] && !IntegerQ[p] && GtQ[p, 0] && IntegerQ[2*p]
  4303. Int[Times[cos[Plus[Optional[Pattern[a, Blank[]]], Times[Optional[Pattern[b, Blank[]]], Pattern[x, Blank[]]]]], Power[Times[Optional[Pattern[g, Blank[]]], sin[Plus[Optional[Pattern[c, Blank[]]], Times[Optional[Pattern[d, Blank[]]], Pattern[x, Blank[]]]]]], Pattern[p, Blank[]]]], Pattern[x, Blank[Symbol]]] := Simp[(Cos[a + b*x]*(g*Sin[c + d*x])^(p + 1))/(2*b*g*(p + 1)), x] + Dist[(2*p + 3)/(2*g*(p + 1)), Int[Sin[a + b*x]*(g*Sin[c + d*x])^(p + 1), x], x] /; FreeQ[{a, b, c, d, g}, x] && EqQ[b*c - a*d, 0] && EqQ[d/b, 2] && !IntegerQ[p] && LtQ[p, -1] && IntegerQ[2*p]
  4304. Int[Times[sin[Plus[Optional[Pattern[a, Blank[]]], Times[Optional[Pattern[b, Blank[]]], Pattern[x, Blank[]]]]], Power[Times[Optional[Pattern[g, Blank[]]], sin[Plus[Optional[Pattern[c, Blank[]]], Times[Optional[Pattern[d, Blank[]]], Pattern[x, Blank[]]]]]], Pattern[p, Blank[]]]], Pattern[x, Blank[Symbol]]] := -Simp[(Sin[a + b*x]*(g*Sin[c + d*x])^(p + 1))/(2*b*g*(p + 1)), x] + Dist[(2*p + 3)/(2*g*(p + 1)), Int[Cos[a + b*x]*(g*Sin[c + d*x])^(p + 1), x], x] /; FreeQ[{a, b, c, d, g}, x] && EqQ[b*c - a*d, 0] && EqQ[d/b, 2] && !IntegerQ[p] && LtQ[p, -1] && IntegerQ[2*p]
  4305. Int[Times[cos[Plus[Optional[Pattern[a, Blank[]]], Times[Optional[Pattern[b, Blank[]]], Pattern[x, Blank[]]]]], Power[sin[Plus[Optional[Pattern[c, Blank[]]], Times[Optional[Pattern[d, Blank[]]], Pattern[x, Blank[]]]]], Rational[-1, 2]]], Pattern[x, Blank[Symbol]]] := -Simp[ArcSin[Cos[a + b*x] - Sin[a + b*x]]/d, x] + Simp[Log[Cos[a + b*x] + Sin[a + b*x] + Sqrt[Sin[c + d*x]]]/d, x] /; FreeQ[{a, b, c, d}, x] && EqQ[b*c - a*d, 0] && EqQ[d/b, 2]
  4306. Int[Times[sin[Plus[Optional[Pattern[a, Blank[]]], Times[Optional[Pattern[b, Blank[]]], Pattern[x, Blank[]]]]], Power[sin[Plus[Optional[Pattern[c, Blank[]]], Times[Optional[Pattern[d, Blank[]]], Pattern[x, Blank[]]]]], Rational[-1, 2]]], Pattern[x, Blank[Symbol]]] := -Simp[ArcSin[Cos[a + b*x] - Sin[a + b*x]]/d, x] - Simp[Log[Cos[a + b*x] + Sin[a + b*x] + Sqrt[Sin[c + d*x]]]/d, x] /; FreeQ[{a, b, c, d}, x] && EqQ[b*c - a*d, 0] && EqQ[d/b, 2]
  4307. Int[Times[Power[cos[Plus[Optional[Pattern[a, Blank[]]], Times[Optional[Pattern[b, Blank[]]], Pattern[x, Blank[]]]]], -1], Power[Times[Optional[Pattern[g, Blank[]]], sin[Plus[Optional[Pattern[c, Blank[]]], Times[Optional[Pattern[d, Blank[]]], Pattern[x, Blank[]]]]]], Pattern[p, Blank[]]]], Pattern[x, Blank[Symbol]]] := Dist[2*g, Int[Sin[a + b*x]*(g*Sin[c + d*x])^(p - 1), x], x] /; FreeQ[{a, b, c, d, g, p}, x] && EqQ[b*c - a*d, 0] && EqQ[d/b, 2] && !IntegerQ[p] && IntegerQ[2*p]
  4308. Int[Times[Power[sin[Plus[Optional[Pattern[a, Blank[]]], Times[Optional[Pattern[b, Blank[]]], Pattern[x, Blank[]]]]], -1], Power[Times[Optional[Pattern[g, Blank[]]], sin[Plus[Optional[Pattern[c, Blank[]]], Times[Optional[Pattern[d, Blank[]]], Pattern[x, Blank[]]]]]], Pattern[p, Blank[]]]], Pattern[x, Blank[Symbol]]] := Dist[2*g, Int[Cos[a + b*x]*(g*Sin[c + d*x])^(p - 1), x], x] /; FreeQ[{a, b, c, d, g, p}, x] && EqQ[b*c - a*d, 0] && EqQ[d/b, 2] && !IntegerQ[p] && IntegerQ[2*p]
  4309. Int[Times[Power[Times[cos[Plus[Optional[Pattern[a, Blank[]]], Times[Optional[Pattern[b, Blank[]]], Pattern[x, Blank[]]]]], Optional[Pattern[e, Blank[]]]], Optional[Pattern[m, Blank[]]]], Power[Times[Optional[Pattern[g, Blank[]]], sin[Plus[Optional[Pattern[c, Blank[]]], Times[Optional[Pattern[d, Blank[]]], Pattern[x, Blank[]]]]]], Pattern[p, Blank[]]]], Pattern[x, Blank[Symbol]]] := Dist[(g*Sin[c + d*x])^p/((e*Cos[a + b*x])^p*Sin[a + b*x]^p), Int[(e*Cos[a + b*x])^(m + p)*Sin[a + b*x]^p, x], x] /; FreeQ[{a, b, c, d, e, g, m, p}, x] && EqQ[b*c - a*d, 0] && EqQ[d/b, 2] && !IntegerQ[p]
  4310. Int[Times[Power[Times[Optional[Pattern[f, Blank[]]], sin[Plus[Optional[Pattern[a, Blank[]]], Times[Optional[Pattern[b, Blank[]]], Pattern[x, Blank[]]]]]], Optional[Pattern[n, Blank[]]]], Power[Times[Optional[Pattern[g, Blank[]]], sin[Plus[Optional[Pattern[c, Blank[]]], Times[Optional[Pattern[d, Blank[]]], Pattern[x, Blank[]]]]]], Pattern[p, Blank[]]]], Pattern[x, Blank[Symbol]]] := Dist[(g*Sin[c + d*x])^p/(Cos[a + b*x]^p*(f*Sin[a + b*x])^p), Int[Cos[a + b*x]^p*(f*Sin[a + b*x])^(n + p), x], x] /; FreeQ[{a, b, c, d, f, g, n, p}, x] && EqQ[b*c - a*d, 0] && EqQ[d/b, 2] && !IntegerQ[p]
  4311. Int[Times[Power[cos[Plus[Optional[Pattern[a, Blank[]]], Times[Optional[Pattern[b, Blank[]]], Pattern[x, Blank[]]]]], 2], Power[sin[Plus[Optional[Pattern[a, Blank[]]], Times[Optional[Pattern[b, Blank[]]], Pattern[x, Blank[]]]]], 2], Power[Times[Optional[Pattern[g, Blank[]]], sin[Plus[Optional[Pattern[c, Blank[]]], Times[Optional[Pattern[d, Blank[]]], Pattern[x, Blank[]]]]]], Pattern[p, Blank[]]]], Pattern[x, Blank[Symbol]]] := Dist[1/4, Int[(g*Sin[c + d*x])^p, x], x] - Dist[1/4, Int[Cos[c + d*x]^2*(g*Sin[c + d*x])^p, x], x] /; FreeQ[{a, b, c, d, g}, x] && EqQ[b*c - a*d, 0] && EqQ[d/b, 2] && IGtQ[p/2, 0]
  4312. Int[Times[Power[Times[cos[Plus[Optional[Pattern[a, Blank[]]], Times[Optional[Pattern[b, Blank[]]], Pattern[x, Blank[]]]]], Optional[Pattern[e, Blank[]]]], Optional[Pattern[m, Blank[]]]], Power[Times[Optional[Pattern[f, Blank[]]], sin[Plus[Optional[Pattern[a, Blank[]]], Times[Optional[Pattern[b, Blank[]]], Pattern[x, Blank[]]]]]], Optional[Pattern[n, Blank[]]]], Power[sin[Plus[Optional[Pattern[c, Blank[]]], Times[Optional[Pattern[d, Blank[]]], Pattern[x, Blank[]]]]], Optional[Pattern[p, Blank[]]]]], Pattern[x, Blank[Symbol]]] := Dist[2^p/(e^p*f^p), Int[(e*Cos[a + b*x])^(m + p)*(f*Sin[a + b*x])^(n + p), x], x] /; FreeQ[{a, b, c, d, e, f, m, n}, x] && EqQ[b*c - a*d, 0] && EqQ[d/b, 2] && IntegerQ[p]
  4313. Int[Times[Power[Times[cos[Plus[Optional[Pattern[a, Blank[]]], Times[Optional[Pattern[b, Blank[]]], Pattern[x, Blank[]]]]], Optional[Pattern[e, Blank[]]]], Optional[Pattern[m, Blank[]]]], Power[Times[Optional[Pattern[f, Blank[]]], sin[Plus[Optional[Pattern[a, Blank[]]], Times[Optional[Pattern[b, Blank[]]], Pattern[x, Blank[]]]]]], Optional[Pattern[n, Blank[]]]], Power[Times[Optional[Pattern[g, Blank[]]], sin[Plus[Optional[Pattern[c, Blank[]]], Times[Optional[Pattern[d, Blank[]]], Pattern[x, Blank[]]]]]], Pattern[p, Blank[]]]], Pattern[x, Blank[Symbol]]] := Simp[(e*(e*Cos[a + b*x])^(m - 1)*(f*Sin[a + b*x])^(n + 1)*(g*Sin[c + d*x])^p)/(b*f*(n + p + 1)), x] /; FreeQ[{a, b, c, d, e, f, g, m, n, p}, x] && EqQ[b*c - a*d, 0] && EqQ[d/b, 2] && !IntegerQ[p] && EqQ[m + p + 1, 0]
  4314. Int[Times[Power[Times[cos[Plus[Optional[Pattern[a, Blank[]]], Times[Optional[Pattern[b, Blank[]]], Pattern[x, Blank[]]]]], Optional[Pattern[f, Blank[]]]], Pattern[n, Blank[]]], Power[Times[Optional[Pattern[e, Blank[]]], sin[Plus[Optional[Pattern[a, Blank[]]], Times[Optional[Pattern[b, Blank[]]], Pattern[x, Blank[]]]]]], Pattern[m, Blank[]]], Power[Times[Optional[Pattern[g, Blank[]]], sin[Plus[Optional[Pattern[c, Blank[]]], Times[Optional[Pattern[d, Blank[]]], Pattern[x, Blank[]]]]]], Pattern[p, Blank[]]]], Pattern[x, Blank[Symbol]]] := -Simp[(e*(e*Sin[a + b*x])^(m - 1)*(f*Cos[a + b*x])^(n + 1)*(g*Sin[c + d*x])^p)/(b*f*(n + p + 1)), x] /; FreeQ[{a, b, c, d, e, f, g, m, n, p}, x] && EqQ[b*c - a*d, 0] && EqQ[d/b, 2] && !IntegerQ[p] && EqQ[m + p + 1, 0]
  4315. Int[Times[Power[Times[cos[Plus[Optional[Pattern[a, Blank[]]], Times[Optional[Pattern[b, Blank[]]], Pattern[x, Blank[]]]]], Optional[Pattern[e, Blank[]]]], Optional[Pattern[m, Blank[]]]], Power[Times[Optional[Pattern[f, Blank[]]], sin[Plus[Optional[Pattern[a, Blank[]]], Times[Optional[Pattern[b, Blank[]]], Pattern[x, Blank[]]]]]], Optional[Pattern[n, Blank[]]]], Power[Times[Optional[Pattern[g, Blank[]]], sin[Plus[Optional[Pattern[c, Blank[]]], Times[Optional[Pattern[d, Blank[]]], Pattern[x, Blank[]]]]]], Pattern[p, Blank[]]]], Pattern[x, Blank[Symbol]]] := -Simp[((e*Cos[a + b*x])^(m + 1)*(f*Sin[a + b*x])^(n + 1)*(g*Sin[c + d*x])^p)/(b*e*f*(m + p + 1)), x] /; FreeQ[{a, b, c, d, e, f, g, m, n, p}, x] && EqQ[b*c - a*d, 0] && EqQ[d/b, 2] && !IntegerQ[p] && EqQ[m + n + 2*p + 2, 0] && NeQ[m + p + 1, 0]
  4316. Int[Times[Power[Times[cos[Plus[Optional[Pattern[a, Blank[]]], Times[Optional[Pattern[b, Blank[]]], Pattern[x, Blank[]]]]], Optional[Pattern[e, Blank[]]]], Pattern[m, Blank[]]], Power[Times[Optional[Pattern[f, Blank[]]], sin[Plus[Optional[Pattern[a, Blank[]]], Times[Optional[Pattern[b, Blank[]]], Pattern[x, Blank[]]]]]], Pattern[n, Blank[]]], Power[Times[Optional[Pattern[g, Blank[]]], sin[Plus[Optional[Pattern[c, Blank[]]], Times[Optional[Pattern[d, Blank[]]], Pattern[x, Blank[]]]]]], Pattern[p, Blank[]]]], Pattern[x, Blank[Symbol]]] := Simp[(e^2*(e*Cos[a + b*x])^(m - 2)*(f*Sin[a + b*x])^n*(g*Sin[c + d*x])^(p + 1))/(2*b*g*(n + p + 1)), x] + Dist[(e^4*(m + p - 1))/(4*g^2*(n + p + 1)), Int[(e*Cos[a + b*x])^(m - 4)*(f*Sin[a + b*x])^n*(g*Sin[c + d*x])^(p + 2), x], x] /; FreeQ[{a, b, c, d, e, f, g, n}, x] && EqQ[b*c - a*d, 0] && EqQ[d/b, 2] && !IntegerQ[p] && GtQ[m, 3] && LtQ[p, -1] && NeQ[n + p + 1, 0] && IntegersQ[2*m, 2*n, 2*p]
  4317. Int[Times[Power[Times[cos[Plus[Optional[Pattern[a, Blank[]]], Times[Optional[Pattern[b, Blank[]]], Pattern[x, Blank[]]]]], Optional[Pattern[f, Blank[]]]], Pattern[n, Blank[]]], Power[Times[Optional[Pattern[e, Blank[]]], sin[Plus[Optional[Pattern[a, Blank[]]], Times[Optional[Pattern[b, Blank[]]], Pattern[x, Blank[]]]]]], Pattern[m, Blank[]]], Power[Times[Optional[Pattern[g, Blank[]]], sin[Plus[Optional[Pattern[c, Blank[]]], Times[Optional[Pattern[d, Blank[]]], Pattern[x, Blank[]]]]]], Pattern[p, Blank[]]]], Pattern[x, Blank[Symbol]]] := -Simp[(e^2*(e*Sin[a + b*x])^(m - 2)*(f*Cos[a + b*x])^n*(g*Sin[c + d*x])^(p + 1))/(2*b*g*(n + p + 1)), x] + Dist[(e^4*(m + p - 1))/(4*g^2*(n + p + 1)), Int[(e*Sin[a + b*x])^(m - 4)*(f*Cos[a + b*x])^n*(g*Sin[c + d*x])^(p + 2), x], x] /; FreeQ[{a, b, c, d, e, f, g, n}, x] && EqQ[b*c - a*d, 0] && EqQ[d/b, 2] && !IntegerQ[p] && GtQ[m, 3] && LtQ[p, -1] && NeQ[n + p + 1, 0] && IntegersQ[2*m, 2*n, 2*p]
  4318. Int[Times[Power[Times[cos[Plus[Optional[Pattern[a, Blank[]]], Times[Optional[Pattern[b, Blank[]]], Pattern[x, Blank[]]]]], Optional[Pattern[e, Blank[]]]], Pattern[m, Blank[]]], Power[Times[Optional[Pattern[f, Blank[]]], sin[Plus[Optional[Pattern[a, Blank[]]], Times[Optional[Pattern[b, Blank[]]], Pattern[x, Blank[]]]]]], Optional[Pattern[n, Blank[]]]], Power[Times[Optional[Pattern[g, Blank[]]], sin[Plus[Optional[Pattern[c, Blank[]]], Times[Optional[Pattern[d, Blank[]]], Pattern[x, Blank[]]]]]], Pattern[p, Blank[]]]], Pattern[x, Blank[Symbol]]] := Simp[((e*Cos[a + b*x])^m*(f*Sin[a + b*x])^n*(g*Sin[c + d*x])^(p + 1))/(2*b*g*(n + p + 1)), x] + Dist[(e^2*(m + n + 2*p + 2))/(4*g^2*(n + p + 1)), Int[(e*Cos[a + b*x])^(m - 2)*(f*Sin[a + b*x])^n*(g*Sin[c + d*x])^(p + 2), x], x] /; FreeQ[{a, b, c, d, e, f, g, n}, x] && EqQ[b*c - a*d, 0] && EqQ[d/b, 2] && !IntegerQ[p] && GtQ[m, 1] && LtQ[p, -1] && NeQ[m + n + 2*p + 2, 0] && NeQ[n + p + 1, 0] && IntegersQ[2*m, 2*n, 2*p] && (LtQ[p, -2] || EqQ[m, 2] || EqQ[m, 3])
  4319. Int[Times[Power[Times[cos[Plus[Optional[Pattern[a, Blank[]]], Times[Optional[Pattern[b, Blank[]]], Pattern[x, Blank[]]]]], Optional[Pattern[f, Blank[]]]], Optional[Pattern[n, Blank[]]]], Power[Times[Optional[Pattern[e, Blank[]]], sin[Plus[Optional[Pattern[a, Blank[]]], Times[Optional[Pattern[b, Blank[]]], Pattern[x, Blank[]]]]]], Pattern[m, Blank[]]], Power[Times[Optional[Pattern[g, Blank[]]], sin[Plus[Optional[Pattern[c, Blank[]]], Times[Optional[Pattern[d, Blank[]]], Pattern[x, Blank[]]]]]], Pattern[p, Blank[]]]], Pattern[x, Blank[Symbol]]] := -Simp[((e*Sin[a + b*x])^m*(f*Cos[a + b*x])^n*(g*Sin[c + d*x])^(p + 1))/(2*b*g*(n + p + 1)), x] + Dist[(e^2*(m + n + 2*p + 2))/(4*g^2*(n + p + 1)), Int[(e*Sin[a + b*x])^(m - 2)*(f*Cos[a + b*x])^n*(g*Sin[c + d*x])^(p + 2), x], x] /; FreeQ[{a, b, c, d, e, f, g, n}, x] && EqQ[b*c - a*d, 0] && EqQ[d/b, 2] && !IntegerQ[p] && GtQ[m, 1] && LtQ[p, -1] && NeQ[m + n + 2*p + 2, 0] && NeQ[n + p + 1, 0] && IntegersQ[2*m, 2*n, 2*p] && (LtQ[p, -2] || EqQ[m, 2] || EqQ[m, 3])
  4320. Int[Times[Power[Times[cos[Plus[Optional[Pattern[a, Blank[]]], Times[Optional[Pattern[b, Blank[]]], Pattern[x, Blank[]]]]], Optional[Pattern[e, Blank[]]]], Pattern[m, Blank[]]], Power[Times[Optional[Pattern[f, Blank[]]], sin[Plus[Optional[Pattern[a, Blank[]]], Times[Optional[Pattern[b, Blank[]]], Pattern[x, Blank[]]]]]], Pattern[n, Blank[]]], Power[Times[Optional[Pattern[g, Blank[]]], sin[Plus[Optional[Pattern[c, Blank[]]], Times[Optional[Pattern[d, Blank[]]], Pattern[x, Blank[]]]]]], Pattern[p, Blank[]]]], Pattern[x, Blank[Symbol]]] := Simp[(e*(e*Cos[a + b*x])^(m - 1)*(f*Sin[a + b*x])^(n + 1)*(g*Sin[c + d*x])^p)/(b*f*(n + p + 1)), x] + Dist[(e^2*(m + p - 1))/(f^2*(n + p + 1)), Int[(e*Cos[a + b*x])^(m - 2)*(f*Sin[a + b*x])^(n + 2)*(g*Sin[c + d*x])^p, x], x] /; FreeQ[{a, b, c, d, e, f, g, p}, x] && EqQ[b*c - a*d, 0] && EqQ[d/b, 2] && !IntegerQ[p] && GtQ[m, 1] && LtQ[n, -1] && NeQ[n + p + 1, 0] && IntegersQ[2*m, 2*n, 2*p]
  4321. Int[Times[Power[Times[cos[Plus[Optional[Pattern[a, Blank[]]], Times[Optional[Pattern[b, Blank[]]], Pattern[x, Blank[]]]]], Optional[Pattern[f, Blank[]]]], Pattern[n, Blank[]]], Power[Times[Optional[Pattern[e, Blank[]]], sin[Plus[Optional[Pattern[a, Blank[]]], Times[Optional[Pattern[b, Blank[]]], Pattern[x, Blank[]]]]]], Pattern[m, Blank[]]], Power[Times[Optional[Pattern[g, Blank[]]], sin[Plus[Optional[Pattern[c, Blank[]]], Times[Optional[Pattern[d, Blank[]]], Pattern[x, Blank[]]]]]], Pattern[p, Blank[]]]], Pattern[x, Blank[Symbol]]] := -Simp[(e*(e*Sin[a + b*x])^(m - 1)*(f*Cos[a + b*x])^(n + 1)*(g*Sin[c + d*x])^p)/(b*f*(n + p + 1)), x] + Dist[(e^2*(m + p - 1))/(f^2*(n + p + 1)), Int[(e*Sin[a + b*x])^(m - 2)*(f*Cos[a + b*x])^(n + 2)*(g*Sin[c + d*x])^p, x], x] /; FreeQ[{a, b, c, d, e, f, g, p}, x] && EqQ[b*c - a*d, 0] && EqQ[d/b, 2] && !IntegerQ[p] && GtQ[m, 1] && LtQ[n, -1] && NeQ[n + p + 1, 0] && IntegersQ[2*m, 2*n, 2*p]
  4322. Int[Times[Power[Times[cos[Plus[Optional[Pattern[a, Blank[]]], Times[Optional[Pattern[b, Blank[]]], Pattern[x, Blank[]]]]], Optional[Pattern[e, Blank[]]]], Pattern[m, Blank[]]], Power[Times[Optional[Pattern[f, Blank[]]], sin[Plus[Optional[Pattern[a, Blank[]]], Times[Optional[Pattern[b, Blank[]]], Pattern[x, Blank[]]]]]], Optional[Pattern[n, Blank[]]]], Power[Times[Optional[Pattern[g, Blank[]]], sin[Plus[Optional[Pattern[c, Blank[]]], Times[Optional[Pattern[d, Blank[]]], Pattern[x, Blank[]]]]]], Pattern[p, Blank[]]]], Pattern[x, Blank[Symbol]]] := Simp[(e*(e*Cos[a + b*x])^(m - 1)*(f*Sin[a + b*x])^(n + 1)*(g*Sin[c + d*x])^p)/(b*f*(m + n + 2*p)), x] + Dist[(e^2*(m + p - 1))/(m + n + 2*p), Int[(e*Cos[a + b*x])^(m - 2)*(f*Sin[a + b*x])^n*(g*Sin[c + d*x])^p, x], x] /; FreeQ[{a, b, c, d, e, f, g, n, p}, x] && EqQ[b*c - a*d, 0] && EqQ[d/b, 2] && !IntegerQ[p] && GtQ[m, 1] && NeQ[m + n + 2*p, 0] && IntegersQ[2*m, 2*n, 2*p]
  4323. Int[Times[Power[Times[cos[Plus[Optional[Pattern[a, Blank[]]], Times[Optional[Pattern[b, Blank[]]], Pattern[x, Blank[]]]]], Optional[Pattern[f, Blank[]]]], Optional[Pattern[n, Blank[]]]], Power[Times[Optional[Pattern[e, Blank[]]], sin[Plus[Optional[Pattern[a, Blank[]]], Times[Optional[Pattern[b, Blank[]]], Pattern[x, Blank[]]]]]], Pattern[m, Blank[]]], Power[Times[Optional[Pattern[g, Blank[]]], sin[Plus[Optional[Pattern[c, Blank[]]], Times[Optional[Pattern[d, Blank[]]], Pattern[x, Blank[]]]]]], Pattern[p, Blank[]]]], Pattern[x, Blank[Symbol]]] := -Simp[(e*(e*Sin[a + b*x])^(m - 1)*(f*Cos[a + b*x])^(n + 1)*(g*Sin[c + d*x])^p)/(b*f*(m + n + 2*p)), x] + Dist[(e^2*(m + p - 1))/(m + n + 2*p), Int[(e*Sin[a + b*x])^(m - 2)*(f*Cos[a + b*x])^n*(g*Sin[c + d*x])^p, x], x] /; FreeQ[{a, b, c, d, e, f, g, n, p}, x] && EqQ[b*c - a*d, 0] && EqQ[d/b, 2] && !IntegerQ[p] && GtQ[m, 1] && NeQ[m + n + 2*p, 0] && IntegersQ[2*m, 2*n, 2*p]
  4324. Int[Times[Power[Times[cos[Plus[Optional[Pattern[a, Blank[]]], Times[Optional[Pattern[b, Blank[]]], Pattern[x, Blank[]]]]], Optional[Pattern[e, Blank[]]]], Pattern[m, Blank[]]], Power[Times[Optional[Pattern[f, Blank[]]], sin[Plus[Optional[Pattern[a, Blank[]]], Times[Optional[Pattern[b, Blank[]]], Pattern[x, Blank[]]]]]], Optional[Pattern[n, Blank[]]]], Power[Times[Optional[Pattern[g, Blank[]]], sin[Plus[Optional[Pattern[c, Blank[]]], Times[Optional[Pattern[d, Blank[]]], Pattern[x, Blank[]]]]]], Pattern[p, Blank[]]]], Pattern[x, Blank[Symbol]]] := -Simp[(f*(e*Cos[a + b*x])^(m + 1)*(f*Sin[a + b*x])^(n - 1)*(g*Sin[c + d*x])^p)/(b*e*(m + n + 2*p)), x] + Dist[(2*f*g*(n + p - 1))/(e*(m + n + 2*p)), Int[(e*Cos[a + b*x])^(m + 1)*(f*Sin[a + b*x])^(n - 1)*(g*Sin[c + d*x])^(p - 1), x], x] /; FreeQ[{a, b, c, d, e, f, g}, x] && EqQ[b*c - a*d, 0] && EqQ[d/b, 2] && !IntegerQ[p] && LtQ[m, -1] && GtQ[n, 0] && GtQ[p, 0] && NeQ[m + n + 2*p, 0] && IntegersQ[2*m, 2*n, 2*p]
  4325. Int[Times[Power[Times[cos[Plus[Optional[Pattern[a, Blank[]]], Times[Optional[Pattern[b, Blank[]]], Pattern[x, Blank[]]]]], Optional[Pattern[f, Blank[]]]], Optional[Pattern[n, Blank[]]]], Power[Times[Optional[Pattern[e, Blank[]]], sin[Plus[Optional[Pattern[a, Blank[]]], Times[Optional[Pattern[b, Blank[]]], Pattern[x, Blank[]]]]]], Pattern[m, Blank[]]], Power[Times[Optional[Pattern[g, Blank[]]], sin[Plus[Optional[Pattern[c, Blank[]]], Times[Optional[Pattern[d, Blank[]]], Pattern[x, Blank[]]]]]], Pattern[p, Blank[]]]], Pattern[x, Blank[Symbol]]] := Simp[(f*(e*Sin[a + b*x])^(m + 1)*(f*Cos[a + b*x])^(n - 1)*(g*Sin[c + d*x])^p)/(b*e*(m + n + 2*p)), x] + Dist[(2*f*g*(n + p - 1))/(e*(m + n + 2*p)), Int[(e*Sin[a + b*x])^(m + 1)*(f*Cos[a + b*x])^(n - 1)*(g*Sin[c + d*x])^(p - 1), x], x] /; FreeQ[{a, b, c, d, e, f, g}, x] && EqQ[b*c - a*d, 0] && EqQ[d/b, 2] && !IntegerQ[p] && LtQ[m, -1] && GtQ[n, 0] && GtQ[p, 0] && NeQ[m + n + 2*p, 0] && IntegersQ[2*m, 2*n, 2*p]
  4326. Int[Times[Power[Times[cos[Plus[Optional[Pattern[a, Blank[]]], Times[Optional[Pattern[b, Blank[]]], Pattern[x, Blank[]]]]], Optional[Pattern[e, Blank[]]]], Pattern[m, Blank[]]], Power[Times[Optional[Pattern[f, Blank[]]], sin[Plus[Optional[Pattern[a, Blank[]]], Times[Optional[Pattern[b, Blank[]]], Pattern[x, Blank[]]]]]], Optional[Pattern[n, Blank[]]]], Power[Times[Optional[Pattern[g, Blank[]]], sin[Plus[Optional[Pattern[c, Blank[]]], Times[Optional[Pattern[d, Blank[]]], Pattern[x, Blank[]]]]]], Pattern[p, Blank[]]]], Pattern[x, Blank[Symbol]]] := -Simp[((e*Cos[a + b*x])^(m + 1)*(f*Sin[a + b*x])^(n + 1)*(g*Sin[c + d*x])^p)/(b*e*f*(m + p + 1)), x] + Dist[(f*(m + n + 2*p + 2))/(2*e*g*(m + p + 1)), Int[(e*Cos[a + b*x])^(m + 1)*(f*Sin[a + b*x])^(n - 1)*(g*Sin[c + d*x])^(p + 1), x], x] /; FreeQ[{a, b, c, d, e, f, g}, x] && EqQ[b*c - a*d, 0] && EqQ[d/b, 2] && !IntegerQ[p] && LtQ[m, -1] && GtQ[n, 0] && LtQ[p, -1] && NeQ[m + n + 2*p + 2, 0] && NeQ[m + p + 1, 0] && IntegersQ[2*m, 2*n, 2*p]
  4327. Int[Times[Power[Times[cos[Plus[Optional[Pattern[a, Blank[]]], Times[Optional[Pattern[b, Blank[]]], Pattern[x, Blank[]]]]], Optional[Pattern[f, Blank[]]]], Optional[Pattern[n, Blank[]]]], Power[Times[Optional[Pattern[e, Blank[]]], sin[Plus[Optional[Pattern[a, Blank[]]], Times[Optional[Pattern[b, Blank[]]], Pattern[x, Blank[]]]]]], Pattern[m, Blank[]]], Power[Times[Optional[Pattern[g, Blank[]]], sin[Plus[Optional[Pattern[c, Blank[]]], Times[Optional[Pattern[d, Blank[]]], Pattern[x, Blank[]]]]]], Pattern[p, Blank[]]]], Pattern[x, Blank[Symbol]]] := Simp[((e*Sin[a + b*x])^(m + 1)*(f*Cos[a + b*x])^(n + 1)*(g*Sin[c + d*x])^p)/(b*e*f*(m + p + 1)), x] + Dist[(f*(m + n + 2*p + 2))/(2*e*g*(m + p + 1)), Int[(e*Sin[a + b*x])^(m + 1)*(f*Cos[a + b*x])^(n - 1)*(g*Sin[c + d*x])^(p + 1), x], x] /; FreeQ[{a, b, c, d, e, f, g}, x] && EqQ[b*c - a*d, 0] && EqQ[d/b, 2] && !IntegerQ[p] && LtQ[m, -1] && GtQ[n, 0] && LtQ[p, -1] && NeQ[m + n + 2*p + 2, 0] && NeQ[m + p + 1, 0] && IntegersQ[2*m, 2*n, 2*p]
  4328. Int[Times[Power[Times[cos[Plus[Optional[Pattern[a, Blank[]]], Times[Optional[Pattern[b, Blank[]]], Pattern[x, Blank[]]]]], Optional[Pattern[e, Blank[]]]], Pattern[m, Blank[]]], Power[Times[Optional[Pattern[f, Blank[]]], sin[Plus[Optional[Pattern[a, Blank[]]], Times[Optional[Pattern[b, Blank[]]], Pattern[x, Blank[]]]]]], Optional[Pattern[n, Blank[]]]], Power[Times[Optional[Pattern[g, Blank[]]], sin[Plus[Optional[Pattern[c, Blank[]]], Times[Optional[Pattern[d, Blank[]]], Pattern[x, Blank[]]]]]], Pattern[p, Blank[]]]], Pattern[x, Blank[Symbol]]] := -Simp[((e*Cos[a + b*x])^(m + 1)*(f*Sin[a + b*x])^(n + 1)*(g*Sin[c + d*x])^p)/(b*e*f*(m + p + 1)), x] + Dist[(m + n + 2*p + 2)/(e^2*(m + p + 1)), Int[(e*Cos[a + b*x])^(m + 2)*(f*Sin[a + b*x])^n*(g*Sin[c + d*x])^p, x], x] /; FreeQ[{a, b, c, d, e, f, g, n, p}, x] && EqQ[b*c - a*d, 0] && EqQ[d/b, 2] && !IntegerQ[p] && LtQ[m, -1] && NeQ[m + n + 2*p + 2, 0] && NeQ[m + p + 1, 0] && IntegersQ[2*m, 2*n, 2*p]
  4329. Int[Times[Power[Times[cos[Plus[Optional[Pattern[a, Blank[]]], Times[Optional[Pattern[b, Blank[]]], Pattern[x, Blank[]]]]], Optional[Pattern[f, Blank[]]]], Optional[Pattern[n, Blank[]]]], Power[Times[Optional[Pattern[e, Blank[]]], sin[Plus[Optional[Pattern[a, Blank[]]], Times[Optional[Pattern[b, Blank[]]], Pattern[x, Blank[]]]]]], Pattern[m, Blank[]]], Power[Times[Optional[Pattern[g, Blank[]]], sin[Plus[Optional[Pattern[c, Blank[]]], Times[Optional[Pattern[d, Blank[]]], Pattern[x, Blank[]]]]]], Pattern[p, Blank[]]]], Pattern[x, Blank[Symbol]]] := Simp[((e*Sin[a + b*x])^(m + 1)*(f*Cos[a + b*x])^(n + 1)*(g*Sin[c + d*x])^p)/(b*e*f*(m + p + 1)), x] + Dist[(m + n + 2*p + 2)/(e^2*(m + p + 1)), Int[(e*Sin[a + b*x])^(m + 2)*(f*Cos[a + b*x])^n*(g*Sin[c + d*x])^p, x], x] /; FreeQ[{a, b, c, d, e, f, g, n, p}, x] && EqQ[b*c - a*d, 0] && EqQ[d/b, 2] && !IntegerQ[p] && LtQ[m, -1] && NeQ[m + n + 2*p + 2, 0] && NeQ[m + p + 1, 0] && IntegersQ[2*m, 2*n, 2*p]
  4330. Int[Times[Power[Times[cos[Plus[Optional[Pattern[a, Blank[]]], Times[Optional[Pattern[b, Blank[]]], Pattern[x, Blank[]]]]], Optional[Pattern[e, Blank[]]]], Optional[Pattern[m, Blank[]]]], Power[Times[Optional[Pattern[f, Blank[]]], sin[Plus[Optional[Pattern[a, Blank[]]], Times[Optional[Pattern[b, Blank[]]], Pattern[x, Blank[]]]]]], Optional[Pattern[n, Blank[]]]], Power[Times[Optional[Pattern[g, Blank[]]], sin[Plus[Optional[Pattern[c, Blank[]]], Times[Optional[Pattern[d, Blank[]]], Pattern[x, Blank[]]]]]], Pattern[p, Blank[]]]], Pattern[x, Blank[Symbol]]] := Dist[(g*Sin[c + d*x])^p/((e*Cos[a + b*x])^p*(f*Sin[a + b*x])^p), Int[(e*Cos[a + b*x])^(m + p)*(f*Sin[a + b*x])^(n + p), x], x] /; FreeQ[{a, b, c, d, e, f, g, m, n, p}, x] && EqQ[b*c - a*d, 0] && EqQ[d/b, 2] && !IntegerQ[p]
  4331. Int[Times[Power[Times[cos[Plus[Optional[Pattern[a, Blank[]]], Times[Optional[Pattern[b, Blank[]]], Pattern[x, Blank[]]]]], Optional[Pattern[e, Blank[]]]], Optional[Pattern[m, Blank[]]]], sin[Plus[Optional[Pattern[c, Blank[]]], Times[Optional[Pattern[d, Blank[]]], Pattern[x, Blank[]]]]]], Pattern[x, Blank[Symbol]]] := -Simp[((m + 2)*(e*Cos[a + b*x])^(m + 1)*Cos[(m + 1)*(a + b*x)])/(d*e*(m + 1)), x] /; FreeQ[{a, b, c, d, e, m}, x] && EqQ[b*c - a*d, 0] && EqQ[d/b, Abs[m + 2]]
  4332. Int[Power[Times[Optional[Pattern[a, Blank[]]], Power[Pattern[F, Blank[]][Plus[Optional[Pattern[c, Blank[]]], Times[Optional[Pattern[d, Blank[]]], Pattern[x, Blank[]]]]], Pattern[p, Blank[]]]], Pattern[n, Blank[]]], Pattern[x, Blank[Symbol]]] := With[{v = ActivateTrig[F[c + d*x]]}, Dist[(a^IntPart[n]*(v/NonfreeFactors[v, x])^(p*IntPart[n])*(a*v^p)^FracPart[n])/NonfreeFactors[v, x]^(p*FracPart[n]), Int[NonfreeFactors[v, x]^(n*p), x], x]] /; FreeQ[{a, c, d, n, p}, x] && InertTrigQ[F] && !IntegerQ[n] && IntegerQ[p]
  4333. Int[Power[Times[Optional[Pattern[a, Blank[]]], Power[Times[Optional[Pattern[b, Blank[]]], Pattern[F, Blank[]][Plus[Optional[Pattern[c, Blank[]]], Times[Optional[Pattern[d, Blank[]]], Pattern[x, Blank[]]]]]], Pattern[p, Blank[]]]], Optional[Pattern[n, Blank[]]]], Pattern[x, Blank[Symbol]]] := With[{v = ActivateTrig[F[c + d*x]]}, Dist[(a^IntPart[n]*(a*(b*v)^p)^FracPart[n])/(b*v)^(p*FracPart[n]), Int[(b*v)^(n*p), x], x]] /; FreeQ[{a, b, c, d, n, p}, x] && InertTrigQ[F] && !IntegerQ[n] && !IntegerQ[p]
  4334. Int[Times[Pattern[u, Blank[]], Pattern[F, Blank[]][Times[Optional[Pattern[c, Blank[]]], Plus[Optional[Pattern[a, Blank[]]], Times[Optional[Pattern[b, Blank[]]], Pattern[x, Blank[]]]]]]], Pattern[x, Blank[Symbol]]] := With[{d = FreeFactors[Sin[c*(a + b*x)], x]}, Dist[d/(b*c), Subst[Int[SubstFor[1, Sin[c*(a + b*x)]/d, u, x], x], x, Sin[c*(a + b*x)]/d], x] /; FunctionOfQ[Sin[c*(a + b*x)]/d, u, x, True]] /; FreeQ[{a, b, c}, x] && (EqQ[F, Cos] || EqQ[F, cos])
  4335. Int[Times[Pattern[u, Blank[]], Pattern[F, Blank[]][Times[Optional[Pattern[c, Blank[]]], Plus[Optional[Pattern[a, Blank[]]], Times[Optional[Pattern[b, Blank[]]], Pattern[x, Blank[]]]]]]], Pattern[x, Blank[Symbol]]] := With[{d = FreeFactors[Cos[c*(a + b*x)], x]}, -Dist[d/(b*c), Subst[Int[SubstFor[1, Cos[c*(a + b*x)]/d, u, x], x], x, Cos[c*(a + b*x)]/d], x] /; FunctionOfQ[Cos[c*(a + b*x)]/d, u, x, True]] /; FreeQ[{a, b, c}, x] && (EqQ[F, Sin] || EqQ[F, sin])
  4336. Int[Times[Cosh[Times[Optional[Pattern[c, Blank[]]], Plus[Optional[Pattern[a, Blank[]]], Times[Optional[Pattern[b, Blank[]]], Pattern[x, Blank[]]]]]], Pattern[u, Blank[]]], Pattern[x, Blank[Symbol]]] := With[{d = FreeFactors[Sinh[c*(a + b*x)], x]}, Dist[d/(b*c), Subst[Int[SubstFor[1, Sinh[c*(a + b*x)]/d, u, x], x], x, Sinh[c*(a + b*x)]/d], x] /; FunctionOfQ[Sinh[c*(a + b*x)]/d, u, x, True]] /; FreeQ[{a, b, c}, x]
  4337. Int[Times[Pattern[u, Blank[]], Sinh[Times[Optional[Pattern[c, Blank[]]], Plus[Optional[Pattern[a, Blank[]]], Times[Optional[Pattern[b, Blank[]]], Pattern[x, Blank[]]]]]]], Pattern[x, Blank[Symbol]]] := With[{d = FreeFactors[Cosh[c*(a + b*x)], x]}, Dist[d/(b*c), Subst[Int[SubstFor[1, Cosh[c*(a + b*x)]/d, u, x], x], x, Cosh[c*(a + b*x)]/d], x] /; FunctionOfQ[Cosh[c*(a + b*x)]/d, u, x, True]] /; FreeQ[{a, b, c}, x]
  4338. Int[Times[Pattern[u, Blank[]], Pattern[F, Blank[]][Times[Optional[Pattern[c, Blank[]]], Plus[Optional[Pattern[a, Blank[]]], Times[Optional[Pattern[b, Blank[]]], Pattern[x, Blank[]]]]]]], Pattern[x, Blank[Symbol]]] := With[{d = FreeFactors[Sin[c*(a + b*x)], x]}, Dist[1/(b*c), Subst[Int[SubstFor[1/x, Sin[c*(a + b*x)]/d, u, x], x], x, Sin[c*(a + b*x)]/d], x] /; FunctionOfQ[Sin[c*(a + b*x)]/d, u, x, True]] /; FreeQ[{a, b, c}, x] && (EqQ[F, Cot] || EqQ[F, cot])
  4339. Int[Times[Pattern[u, Blank[]], Pattern[F, Blank[]][Times[Optional[Pattern[c, Blank[]]], Plus[Optional[Pattern[a, Blank[]]], Times[Optional[Pattern[b, Blank[]]], Pattern[x, Blank[]]]]]]], Pattern[x, Blank[Symbol]]] := With[{d = FreeFactors[Cos[c*(a + b*x)], x]}, -Dist[(b*c)^(-1), Subst[Int[SubstFor[1/x, Cos[c*(a + b*x)]/d, u, x], x], x, Cos[c*(a + b*x)]/d], x] /; FunctionOfQ[Cos[c*(a + b*x)]/d, u, x, True]] /; FreeQ[{a, b, c}, x] && (EqQ[F, Tan] || EqQ[F, tan])
  4340. Int[Times[Coth[Times[Optional[Pattern[c, Blank[]]], Plus[Optional[Pattern[a, Blank[]]], Times[Optional[Pattern[b, Blank[]]], Pattern[x, Blank[]]]]]], Pattern[u, Blank[]]], Pattern[x, Blank[Symbol]]] := With[{d = FreeFactors[Sinh[c*(a + b*x)], x]}, Dist[1/(b*c), Subst[Int[SubstFor[1/x, Sinh[c*(a + b*x)]/d, u, x], x], x, Sinh[c*(a + b*x)]/d], x] /; FunctionOfQ[Sinh[c*(a + b*x)]/d, u, x, True]] /; FreeQ[{a, b, c}, x]
  4341. Int[Times[Pattern[u, Blank[]], Tanh[Times[Optional[Pattern[c, Blank[]]], Plus[Optional[Pattern[a, Blank[]]], Times[Optional[Pattern[b, Blank[]]], Pattern[x, Blank[]]]]]]], Pattern[x, Blank[Symbol]]] := With[{d = FreeFactors[Cosh[c*(a + b*x)], x]}, Dist[1/(b*c), Subst[Int[SubstFor[1/x, Cosh[c*(a + b*x)]/d, u, x], x], x, Cosh[c*(a + b*x)]/d], x] /; FunctionOfQ[Cosh[c*(a + b*x)]/d, u, x, True]] /; FreeQ[{a, b, c}, x]
  4342. Int[Times[Pattern[u, Blank[]], Power[Pattern[F, Blank[]][Times[Optional[Pattern[c, Blank[]]], Plus[Optional[Pattern[a, Blank[]]], Times[Optional[Pattern[b, Blank[]]], Pattern[x, Blank[]]]]]], 2]], Pattern[x, Blank[Symbol]]] := With[{d = FreeFactors[Tan[c*(a + b*x)], x]}, Dist[d/(b*c), Subst[Int[SubstFor[1, Tan[c*(a + b*x)]/d, u, x], x], x, Tan[c*(a + b*x)]/d], x] /; FunctionOfQ[Tan[c*(a + b*x)]/d, u, x, True]] /; FreeQ[{a, b, c}, x] && NonsumQ[u] && (EqQ[F, Sec] || EqQ[F, sec])
  4343. Int[Times[Power[cos[Times[Optional[Pattern[c, Blank[]]], Plus[Optional[Pattern[a, Blank[]]], Times[Optional[Pattern[b, Blank[]]], Pattern[x, Blank[]]]]]], -2], Pattern[u, Blank[]]], Pattern[x, Blank[Symbol]]] := With[{d = FreeFactors[Tan[c*(a + b*x)], x]}, Dist[d/(b*c), Subst[Int[SubstFor[1, Tan[c*(a + b*x)]/d, u, x], x], x, Tan[c*(a + b*x)]/d], x] /; FunctionOfQ[Tan[c*(a + b*x)]/d, u, x, True]] /; FreeQ[{a, b, c}, x] && NonsumQ[u]
  4344. Int[Times[Pattern[u, Blank[]], Power[Pattern[F, Blank[]][Times[Optional[Pattern[c, Blank[]]], Plus[Optional[Pattern[a, Blank[]]], Times[Optional[Pattern[b, Blank[]]], Pattern[x, Blank[]]]]]], 2]], Pattern[x, Blank[Symbol]]] := With[{d = FreeFactors[Cot[c*(a + b*x)], x]}, -Dist[d/(b*c), Subst[Int[SubstFor[1, Cot[c*(a + b*x)]/d, u, x], x], x, Cot[c*(a + b*x)]/d], x] /; FunctionOfQ[Cot[c*(a + b*x)]/d, u, x, True]] /; FreeQ[{a, b, c}, x] && NonsumQ[u] && (EqQ[F, Csc] || EqQ[F, csc])
  4345. Int[Times[Pattern[u, Blank[]], Power[sin[Times[Optional[Pattern[c, Blank[]]], Plus[Optional[Pattern[a, Blank[]]], Times[Optional[Pattern[b, Blank[]]], Pattern[x, Blank[]]]]]], -2]], Pattern[x, Blank[Symbol]]] := With[{d = FreeFactors[Cot[c*(a + b*x)], x]}, -Dist[d/(b*c), Subst[Int[SubstFor[1, Cot[c*(a + b*x)]/d, u, x], x], x, Cot[c*(a + b*x)]/d], x] /; FunctionOfQ[Cot[c*(a + b*x)]/d, u, x, True]] /; FreeQ[{a, b, c}, x] && NonsumQ[u]
  4346. Int[Times[Pattern[u, Blank[]], Power[Sech[Times[Optional[Pattern[c, Blank[]]], Plus[Optional[Pattern[a, Blank[]]], Times[Optional[Pattern[b, Blank[]]], Pattern[x, Blank[]]]]]], 2]], Pattern[x, Blank[Symbol]]] := With[{d = FreeFactors[Tanh[c*(a + b*x)], x]}, Dist[d/(b*c), Subst[Int[SubstFor[1, Tanh[c*(a + b*x)]/d, u, x], x], x, Tanh[c*(a + b*x)]/d], x] /; FunctionOfQ[Tanh[c*(a + b*x)]/d, u, x, True]] /; FreeQ[{a, b, c}, x] && NonsumQ[u]
  4347. Int[Times[Power[Csch[Times[Optional[Pattern[c, Blank[]]], Plus[Optional[Pattern[a, Blank[]]], Times[Optional[Pattern[b, Blank[]]], Pattern[x, Blank[]]]]]], 2], Pattern[u, Blank[]]], Pattern[x, Blank[Symbol]]] := With[{d = FreeFactors[Coth[c*(a + b*x)], x]}, -Dist[d/(b*c), Subst[Int[SubstFor[1, Coth[c*(a + b*x)]/d, u, x], x], x, Coth[c*(a + b*x)]/d], x] /; FunctionOfQ[Coth[c*(a + b*x)]/d, u, x, True]] /; FreeQ[{a, b, c}, x] && NonsumQ[u]
  4348. Int[Times[Pattern[u, Blank[]], Power[Pattern[F, Blank[]][Times[Optional[Pattern[c, Blank[]]], Plus[Optional[Pattern[a, Blank[]]], Times[Optional[Pattern[b, Blank[]]], Pattern[x, Blank[]]]]]], Optional[Pattern[n, Blank[]]]]], Pattern[x, Blank[Symbol]]] := With[{d = FreeFactors[Tan[c*(a + b*x)], x]}, Dist[1/(b*c*d^(n - 1)), Subst[Int[SubstFor[1/(x^n*(1 + d^2*x^2)), Tan[c*(a + b*x)]/d, u, x], x], x, Tan[c*(a + b*x)]/d], x] /; FunctionOfQ[Tan[c*(a + b*x)]/d, u, x, True] && TryPureTanSubst[ActivateTrig[u]*Cot[c*(a + b*x)]^n, x]] /; FreeQ[{a, b, c}, x] && IntegerQ[n] && (EqQ[F, Cot] || EqQ[F, cot])
  4349. Int[Times[Pattern[u, Blank[]], Power[Pattern[F, Blank[]][Times[Optional[Pattern[c, Blank[]]], Plus[Optional[Pattern[a, Blank[]]], Times[Optional[Pattern[b, Blank[]]], Pattern[x, Blank[]]]]]], Optional[Pattern[n, Blank[]]]]], Pattern[x, Blank[Symbol]]] := With[{d = FreeFactors[Cot[c*(a + b*x)], x]}, -Dist[(b*c*d^(n - 1))^(-1), Subst[Int[SubstFor[1/(x^n*(1 + d^2*x^2)), Cot[c*(a + b*x)]/d, u, x], x], x, Cot[c*(a + b*x)]/d], x] /; FunctionOfQ[Cot[c*(a + b*x)]/d, u, x, True] && TryPureTanSubst[ActivateTrig[u]*Tan[c*(a + b*x)]^n, x]] /; FreeQ[{a, b, c}, x] && IntegerQ[n] && (EqQ[F, Tan] || EqQ[F, tan])
  4350. Int[Times[Power[Coth[Times[Optional[Pattern[c, Blank[]]], Plus[Optional[Pattern[a, Blank[]]], Times[Optional[Pattern[b, Blank[]]], Pattern[x, Blank[]]]]]], Optional[Pattern[n, Blank[]]]], Pattern[u, Blank[]]], Pattern[x, Blank[Symbol]]] := With[{d = FreeFactors[Tanh[c*(a + b*x)], x]}, Dist[1/(b*c*d^(n - 1)), Subst[Int[SubstFor[1/(x^n*(1 - d^2*x^2)), Tanh[c*(a + b*x)]/d, u, x], x], x, Tanh[c*(a + b*x)]/d], x] /; FunctionOfQ[Tanh[c*(a + b*x)]/d, u, x, True] && TryPureTanSubst[ActivateTrig[u]*Coth[c*(a + b*x)]^n, x]] /; FreeQ[{a, b, c}, x] && IntegerQ[n]
  4351. Int[Times[Pattern[u, Blank[]], Power[Tanh[Times[Optional[Pattern[c, Blank[]]], Plus[Optional[Pattern[a, Blank[]]], Times[Optional[Pattern[b, Blank[]]], Pattern[x, Blank[]]]]]], Optional[Pattern[n, Blank[]]]]], Pattern[x, Blank[Symbol]]] := With[{d = FreeFactors[Coth[c*(a + b*x)], x]}, Dist[1/(b*c*d^(n - 1)), Subst[Int[SubstFor[1/(x^n*(1 - d^2*x^2)), Coth[c*(a + b*x)]/d, u, x], x], x, Coth[c*(a + b*x)]/d], x] /; FunctionOfQ[Coth[c*(a + b*x)]/d, u, x, True] && TryPureTanSubst[ActivateTrig[u]*Tanh[c*(a + b*x)]^n, x]] /; FreeQ[{a, b, c}, x] && IntegerQ[n]
  4352. Int[Pattern[u, Blank[]], Pattern[x, Blank[Symbol]]] := With[{v = FunctionOfTrig[u, x]}, Simp[With[{d = FreeFactors[Cot[v], x]}, Dist[-(d/Coefficient[v, x, 1]), Subst[Int[SubstFor[1/(1 + d^2*x^2), Cot[v]/d, u, x], x], x, Cot[v]/d], x]], x] /; !FalseQ[v] && FunctionOfQ[NonfreeFactors[Cot[v], x], u, x, True] && TryPureTanSubst[ActivateTrig[u], x]]
  4353. Int[Pattern[u, Blank[]], Pattern[x, Blank[Symbol]]] := With[{v = FunctionOfTrig[u, x]}, Simp[With[{d = FreeFactors[Tan[v], x]}, Dist[d/Coefficient[v, x, 1], Subst[Int[SubstFor[1/(1 + d^2*x^2), Tan[v]/d, u, x], x], x, Tan[v]/d], x]], x] /; !FalseQ[v] && FunctionOfQ[NonfreeFactors[Tan[v], x], u, x, True] && TryPureTanSubst[ActivateTrig[u], x]]
  4354. Int[Times[Power[Pattern[F, Blank[]][Plus[Optional[Pattern[a, Blank[]]], Times[Optional[Pattern[b, Blank[]]], Pattern[x, Blank[]]]]], Optional[Pattern[p, Blank[]]]], Power[Pattern[G, Blank[]][Plus[Optional[Pattern[c, Blank[]]], Times[Optional[Pattern[d, Blank[]]], Pattern[x, Blank[]]]]], Optional[Pattern[q, Blank[]]]]], Pattern[x, Blank[Symbol]]] := Int[ExpandTrigReduce[ActivateTrig[F[a + b*x]^p*G[c + d*x]^q], x], x] /; FreeQ[{a, b, c, d}, x] && (EqQ[F, sin] || EqQ[F, cos]) && (EqQ[G, sin] || EqQ[G, cos]) && IGtQ[p, 0] && IGtQ[q, 0]
  4355. Int[Times[Power[Pattern[F, Blank[]][Plus[Optional[Pattern[a, Blank[]]], Times[Optional[Pattern[b, Blank[]]], Pattern[x, Blank[]]]]], Optional[Pattern[p, Blank[]]]], Power[Pattern[G, Blank[]][Plus[Optional[Pattern[c, Blank[]]], Times[Optional[Pattern[d, Blank[]]], Pattern[x, Blank[]]]]], Optional[Pattern[q, Blank[]]]], Power[Pattern[H, Blank[]][Plus[Optional[Pattern[e, Blank[]]], Times[Optional[Pattern[f, Blank[]]], Pattern[x, Blank[]]]]], Optional[Pattern[r, Blank[]]]]], Pattern[x, Blank[Symbol]]] := Int[ExpandTrigReduce[ActivateTrig[F[a + b*x]^p*G[c + d*x]^q*H[e + f*x]^r], x], x] /; FreeQ[{a, b, c, d, e, f}, x] && (EqQ[F, sin] || EqQ[F, cos]) && (EqQ[G, sin] || EqQ[G, cos]) && (EqQ[H, sin] || EqQ[H, cos]) && IGtQ[p, 0] && IGtQ[q, 0] && IGtQ[r, 0]
  4356. Int[Times[Pattern[u, Blank[]], Pattern[F, Blank[]][Times[Optional[Pattern[c, Blank[]]], Plus[Optional[Pattern[a, Blank[]]], Times[Optional[Pattern[b, Blank[]]], Pattern[x, Blank[]]]]]]], Pattern[x, Blank[Symbol]]] := With[{d = FreeFactors[Sin[c*(a + b*x)], x]}, Dist[d/(b*c), Subst[Int[SubstFor[1, Sin[c*(a + b*x)]/d, u, x], x], x, Sin[c*(a + b*x)]/d], x] /; FunctionOfQ[Sin[c*(a + b*x)]/d, u, x]] /; FreeQ[{a, b, c}, x] && (EqQ[F, Cos] || EqQ[F, cos])
  4357. Int[Times[Pattern[u, Blank[]], Pattern[F, Blank[]][Times[Optional[Pattern[c, Blank[]]], Plus[Optional[Pattern[a, Blank[]]], Times[Optional[Pattern[b, Blank[]]], Pattern[x, Blank[]]]]]]], Pattern[x, Blank[Symbol]]] := With[{d = FreeFactors[Cos[c*(a + b*x)], x]}, -Dist[d/(b*c), Subst[Int[SubstFor[1, Cos[c*(a + b*x)]/d, u, x], x], x, Cos[c*(a + b*x)]/d], x] /; FunctionOfQ[Cos[c*(a + b*x)]/d, u, x]] /; FreeQ[{a, b, c}, x] && (EqQ[F, Sin] || EqQ[F, sin])
  4358. Int[Times[Cosh[Times[Optional[Pattern[c, Blank[]]], Plus[Optional[Pattern[a, Blank[]]], Times[Optional[Pattern[b, Blank[]]], Pattern[x, Blank[]]]]]], Pattern[u, Blank[]]], Pattern[x, Blank[Symbol]]] := With[{d = FreeFactors[Sinh[c*(a + b*x)], x]}, Dist[d/(b*c), Subst[Int[SubstFor[1, Sinh[c*(a + b*x)]/d, u, x], x], x, Sinh[c*(a + b*x)]/d], x] /; FunctionOfQ[Sinh[c*(a + b*x)]/d, u, x]] /; FreeQ[{a, b, c}, x]
  4359. Int[Times[Pattern[u, Blank[]], Sinh[Times[Optional[Pattern[c, Blank[]]], Plus[Optional[Pattern[a, Blank[]]], Times[Optional[Pattern[b, Blank[]]], Pattern[x, Blank[]]]]]]], Pattern[x, Blank[Symbol]]] := With[{d = FreeFactors[Cosh[c*(a + b*x)], x]}, Dist[d/(b*c), Subst[Int[SubstFor[1, Cosh[c*(a + b*x)]/d, u, x], x], x, Cosh[c*(a + b*x)]/d], x] /; FunctionOfQ[Cosh[c*(a + b*x)]/d, u, x]] /; FreeQ[{a, b, c}, x]
  4360. Int[Times[Pattern[u, Blank[]], Pattern[F, Blank[]][Times[Optional[Pattern[c, Blank[]]], Plus[Optional[Pattern[a, Blank[]]], Times[Optional[Pattern[b, Blank[]]], Pattern[x, Blank[]]]]]]], Pattern[x, Blank[Symbol]]] := With[{d = FreeFactors[Sin[c*(a + b*x)], x]}, Dist[1/(b*c), Subst[Int[SubstFor[1/x, Sin[c*(a + b*x)]/d, u, x], x], x, Sin[c*(a + b*x)]/d], x] /; FunctionOfQ[Sin[c*(a + b*x)]/d, u, x]] /; FreeQ[{a, b, c}, x] && (EqQ[F, Cot] || EqQ[F, cot])
  4361. Int[Times[Pattern[u, Blank[]], Pattern[F, Blank[]][Times[Optional[Pattern[c, Blank[]]], Plus[Optional[Pattern[a, Blank[]]], Times[Optional[Pattern[b, Blank[]]], Pattern[x, Blank[]]]]]]], Pattern[x, Blank[Symbol]]] := With[{d = FreeFactors[Cos[c*(a + b*x)], x]}, -Dist[(b*c)^(-1), Subst[Int[SubstFor[1/x, Cos[c*(a + b*x)]/d, u, x], x], x, Cos[c*(a + b*x)]/d], x] /; FunctionOfQ[Cos[c*(a + b*x)]/d, u, x]] /; FreeQ[{a, b, c}, x] && (EqQ[F, Tan] || EqQ[F, tan])
  4362. Int[Times[Coth[Times[Optional[Pattern[c, Blank[]]], Plus[Optional[Pattern[a, Blank[]]], Times[Optional[Pattern[b, Blank[]]], Pattern[x, Blank[]]]]]], Pattern[u, Blank[]]], Pattern[x, Blank[Symbol]]] := With[{d = FreeFactors[Sinh[c*(a + b*x)], x]}, Dist[1/(b*c), Subst[Int[SubstFor[1/x, Sinh[c*(a + b*x)]/d, u, x], x], x, Sinh[c*(a + b*x)]/d], x] /; FunctionOfQ[Sinh[c*(a + b*x)]/d, u, x]] /; FreeQ[{a, b, c}, x]
  4363. Int[Times[Pattern[u, Blank[]], Tanh[Times[Optional[Pattern[c, Blank[]]], Plus[Optional[Pattern[a, Blank[]]], Times[Optional[Pattern[b, Blank[]]], Pattern[x, Blank[]]]]]]], Pattern[x, Blank[Symbol]]] := With[{d = FreeFactors[Cosh[c*(a + b*x)], x]}, Dist[1/(b*c), Subst[Int[SubstFor[1/x, Cosh[c*(a + b*x)]/d, u, x], x], x, Cosh[c*(a + b*x)]/d], x] /; FunctionOfQ[Cosh[c*(a + b*x)]/d, u, x]] /; FreeQ[{a, b, c}, x]
  4364. Int[Times[Pattern[u, Blank[]], Power[Pattern[F, Blank[]][Times[Optional[Pattern[c, Blank[]]], Plus[Optional[Pattern[a, Blank[]]], Times[Optional[Pattern[b, Blank[]]], Pattern[x, Blank[]]]]]], Pattern[n, Blank[]]]], Pattern[x, Blank[Symbol]]] := With[{d = FreeFactors[Sin[c*(a + b*x)], x]}, Dist[d/(b*c), Subst[Int[SubstFor[(1 - d^2*x^2)^((n - 1)/2), Sin[c*(a + b*x)]/d, u, x], x], x, Sin[c*(a + b*x)]/d], x] /; FunctionOfQ[Sin[c*(a + b*x)]/d, u, x]] /; FreeQ[{a, b, c}, x] && IntegerQ[(n - 1)/2] && NonsumQ[u] && (EqQ[F, Cos] || EqQ[F, cos])
  4365. Int[Times[Pattern[u, Blank[]], Power[Pattern[F, Blank[]][Times[Optional[Pattern[c, Blank[]]], Plus[Optional[Pattern[a, Blank[]]], Times[Optional[Pattern[b, Blank[]]], Pattern[x, Blank[]]]]]], Pattern[n, Blank[]]]], Pattern[x, Blank[Symbol]]] := With[{d = FreeFactors[Sin[c*(a + b*x)], x]}, Dist[d/(b*c), Subst[Int[SubstFor[(1 - d^2*x^2)^((-n - 1)/2), Sin[c*(a + b*x)]/d, u, x], x], x, Sin[c*(a + b*x)]/d], x] /; FunctionOfQ[Sin[c*(a + b*x)]/d, u, x]] /; FreeQ[{a, b, c}, x] && IntegerQ[(n - 1)/2] && NonsumQ[u] && (EqQ[F, Sec] || EqQ[F, sec])
  4366. Int[Times[Pattern[u, Blank[]], Power[Pattern[F, Blank[]][Times[Optional[Pattern[c, Blank[]]], Plus[Optional[Pattern[a, Blank[]]], Times[Optional[Pattern[b, Blank[]]], Pattern[x, Blank[]]]]]], Pattern[n, Blank[]]]], Pattern[x, Blank[Symbol]]] := With[{d = FreeFactors[Cos[c*(a + b*x)], x]}, -Dist[d/(b*c), Subst[Int[SubstFor[(1 - d^2*x^2)^((n - 1)/2), Cos[c*(a + b*x)]/d, u, x], x], x, Cos[c*(a + b*x)]/d], x] /; FunctionOfQ[Cos[c*(a + b*x)]/d, u, x]] /; FreeQ[{a, b, c}, x] && IntegerQ[(n - 1)/2] && NonsumQ[u] && (EqQ[F, Sin] || EqQ[F, sin])
  4367. Int[Times[Pattern[u, Blank[]], Power[Pattern[F, Blank[]][Times[Optional[Pattern[c, Blank[]]], Plus[Optional[Pattern[a, Blank[]]], Times[Optional[Pattern[b, Blank[]]], Pattern[x, Blank[]]]]]], Pattern[n, Blank[]]]], Pattern[x, Blank[Symbol]]] := With[{d = FreeFactors[Cos[c*(a + b*x)], x]}, -Dist[d/(b*c), Subst[Int[SubstFor[(1 - d^2*x^2)^((-n - 1)/2), Cos[c*(a + b*x)]/d, u, x], x], x, Cos[c*(a + b*x)]/d], x] /; FunctionOfQ[Cos[c*(a + b*x)]/d, u, x]] /; FreeQ[{a, b, c}, x] && IntegerQ[(n - 1)/2] && NonsumQ[u] && (EqQ[F, Csc] || EqQ[F, csc])
  4368. Int[Times[Power[Cosh[Times[Optional[Pattern[c, Blank[]]], Plus[Optional[Pattern[a, Blank[]]], Times[Optional[Pattern[b, Blank[]]], Pattern[x, Blank[]]]]]], Pattern[n, Blank[]]], Pattern[u, Blank[]]], Pattern[x, Blank[Symbol]]] := With[{d = FreeFactors[Sinh[c*(a + b*x)], x]}, Dist[d/(b*c), Subst[Int[SubstFor[(1 + d^2*x^2)^((n - 1)/2), Sinh[c*(a + b*x)]/d, u, x], x], x, Sinh[c*(a + b*x)]/d], x] /; FunctionOfQ[Sinh[c*(a + b*x)]/d, u, x]] /; FreeQ[{a, b, c}, x] && IntegerQ[(n - 1)/2] && NonsumQ[u]
  4369. Int[Times[Pattern[u, Blank[]], Power[Sech[Times[Optional[Pattern[c, Blank[]]], Plus[Optional[Pattern[a, Blank[]]], Times[Optional[Pattern[b, Blank[]]], Pattern[x, Blank[]]]]]], Pattern[n, Blank[]]]], Pattern[x, Blank[Symbol]]] := With[{d = FreeFactors[Sinh[c*(a + b*x)], x]}, Dist[d/(b*c), Subst[Int[SubstFor[(1 + d^2*x^2)^((-n - 1)/2), Sinh[c*(a + b*x)]/d, u, x], x], x, Sinh[c*(a + b*x)]/d], x] /; FunctionOfQ[Sinh[c*(a + b*x)]/d, u, x]] /; FreeQ[{a, b, c}, x] && IntegerQ[(n - 1)/2] && NonsumQ[u]
  4370. Int[Times[Pattern[u, Blank[]], Power[Sinh[Times[Optional[Pattern[c, Blank[]]], Plus[Optional[Pattern[a, Blank[]]], Times[Optional[Pattern[b, Blank[]]], Pattern[x, Blank[]]]]]], Pattern[n, Blank[]]]], Pattern[x, Blank[Symbol]]] := With[{d = FreeFactors[Cosh[c*(a + b*x)], x]}, Dist[d/(b*c), Subst[Int[SubstFor[(-1 + d^2*x^2)^((n - 1)/2), Cosh[c*(a + b*x)]/d, u, x], x], x, Cosh[c*(a + b*x)]/d], x] /; FunctionOfQ[Cosh[c*(a + b*x)]/d, u, x]] /; FreeQ[{a, b, c}, x] && IntegerQ[(n - 1)/2] && NonsumQ[u]
  4371. Int[Times[Power[Csch[Times[Optional[Pattern[c, Blank[]]], Plus[Optional[Pattern[a, Blank[]]], Times[Optional[Pattern[b, Blank[]]], Pattern[x, Blank[]]]]]], Pattern[n, Blank[]]], Pattern[u, Blank[]]], Pattern[x, Blank[Symbol]]] := With[{d = FreeFactors[Cosh[c*(a + b*x)], x]}, Dist[d/(b*c), Subst[Int[SubstFor[(-1 + d^2*x^2)^((-n - 1)/2), Cosh[c*(a + b*x)]/d, u, x], x], x, Cosh[c*(a + b*x)]/d], x] /; FunctionOfQ[Cosh[c*(a + b*x)]/d, u, x]] /; FreeQ[{a, b, c}, x] && IntegerQ[(n - 1)/2] && NonsumQ[u]
  4372. Int[Times[Pattern[u, Blank[]], Power[Pattern[F, Blank[]][Times[Optional[Pattern[c, Blank[]]], Plus[Optional[Pattern[a, Blank[]]], Times[Optional[Pattern[b, Blank[]]], Pattern[x, Blank[]]]]]], Pattern[n, Blank[]]]], Pattern[x, Blank[Symbol]]] := With[{d = FreeFactors[Sin[c*(a + b*x)], x]}, Dist[1/(b*c*d^(n - 1)), Subst[Int[SubstFor[(1 - d^2*x^2)^((n - 1)/2)/x^n, Sin[c*(a + b*x)]/d, u, x], x], x, Sin[c*(a + b*x)]/d], x] /; FunctionOfQ[Sin[c*(a + b*x)]/d, u, x]] /; FreeQ[{a, b, c}, x] && IntegerQ[(n - 1)/2] && NonsumQ[u] && (EqQ[F, Cot] || EqQ[F, cot])
  4373. Int[Times[Pattern[u, Blank[]], Power[Pattern[F, Blank[]][Times[Optional[Pattern[c, Blank[]]], Plus[Optional[Pattern[a, Blank[]]], Times[Optional[Pattern[b, Blank[]]], Pattern[x, Blank[]]]]]], Pattern[n, Blank[]]]], Pattern[x, Blank[Symbol]]] := With[{d = FreeFactors[Cos[c*(a + b*x)], x]}, -Dist[(b*c*d^(n - 1))^(-1), Subst[Int[SubstFor[(1 - d^2*x^2)^((n - 1)/2)/x^n, Cos[c*(a + b*x)]/d, u, x], x], x, Cos[c*(a + b*x)]/d], x] /; FunctionOfQ[Cos[c*(a + b*x)]/d, u, x]] /; FreeQ[{a, b, c}, x] && IntegerQ[(n - 1)/2] && NonsumQ[u] && (EqQ[F, Tan] || EqQ[F, tan])
  4374. Int[Times[Power[Coth[Times[Optional[Pattern[c, Blank[]]], Plus[Optional[Pattern[a, Blank[]]], Times[Optional[Pattern[b, Blank[]]], Pattern[x, Blank[]]]]]], Pattern[n, Blank[]]], Pattern[u, Blank[]]], Pattern[x, Blank[Symbol]]] := With[{d = FreeFactors[Sinh[c*(a + b*x)], x]}, Dist[1/(b*c*d^(n - 1)), Subst[Int[SubstFor[(1 + d^2*x^2)^((n - 1)/2)/x^n, Sinh[c*(a + b*x)]/d, u, x], x], x, Sinh[c*(a + b*x)]/d], x] /; FunctionOfQ[Sinh[c*(a + b*x)]/d, u, x]] /; FreeQ[{a, b, c}, x] && IntegerQ[(n - 1)/2] && NonsumQ[u]
  4375. Int[Times[Pattern[u, Blank[]], Power[Tanh[Times[Optional[Pattern[c, Blank[]]], Plus[Optional[Pattern[a, Blank[]]], Times[Optional[Pattern[b, Blank[]]], Pattern[x, Blank[]]]]]], Pattern[n, Blank[]]]], Pattern[x, Blank[Symbol]]] := With[{d = FreeFactors[Cosh[c*(a + b*x)], x]}, Dist[1/(b*c*d^(n - 1)), Subst[Int[SubstFor[(-1 + d^2*x^2)^((n - 1)/2)/x^n, Cosh[c*(a + b*x)]/d, u, x], x], x, Cosh[c*(a + b*x)]/d], x] /; FunctionOfQ[Cosh[c*(a + b*x)]/d, u, x]] /; FreeQ[{a, b, c}, x] && IntegerQ[(n - 1)/2] && NonsumQ[u]
  4376. Int[Times[Pattern[u, Blank[]], Plus[Pattern[v, Blank[]], Times[Optional[Pattern[d, Blank[]]], Power[Pattern[F, Blank[]][Times[Optional[Pattern[c, Blank[]]], Plus[Optional[Pattern[a, Blank[]]], Times[Optional[Pattern[b, Blank[]]], Pattern[x, Blank[]]]]]], Optional[Pattern[n, Blank[]]]]]]], Pattern[x, Blank[Symbol]]] := With[{e = FreeFactors[Sin[c*(a + b*x)], x]}, Int[ActivateTrig[u*v], x] + Dist[d, Int[ActivateTrig[u]*Cos[c*(a + b*x)]^n, x], x] /; FunctionOfQ[Sin[c*(a + b*x)]/e, u, x]] /; FreeQ[{a, b, c, d}, x] && !FreeQ[v, x] && IntegerQ[(n - 1)/2] && NonsumQ[u] && (EqQ[F, Cos] || EqQ[F, cos])
  4377. Int[Times[Pattern[u, Blank[]], Plus[Pattern[v, Blank[]], Times[Optional[Pattern[d, Blank[]]], Power[Pattern[F, Blank[]][Times[Optional[Pattern[c, Blank[]]], Plus[Optional[Pattern[a, Blank[]]], Times[Optional[Pattern[b, Blank[]]], Pattern[x, Blank[]]]]]], Optional[Pattern[n, Blank[]]]]]]], Pattern[x, Blank[Symbol]]] := With[{e = FreeFactors[Cos[c*(a + b*x)], x]}, Int[ActivateTrig[u*v], x] + Dist[d, Int[ActivateTrig[u]*Sin[c*(a + b*x)]^n, x], x] /; FunctionOfQ[Cos[c*(a + b*x)]/e, u, x]] /; FreeQ[{a, b, c, d}, x] && !FreeQ[v, x] && IntegerQ[(n - 1)/2] && NonsumQ[u] && (EqQ[F, Sin] || EqQ[F, sin])
  4378. Int[Pattern[u, Blank[]], Pattern[x, Blank[Symbol]]] := With[{v = FunctionOfTrig[u, x]}, Simp[With[{d = FreeFactors[Sin[v], x]}, Dist[d/Coefficient[v, x, 1], Subst[Int[SubstFor[1, Sin[v]/d, u/Cos[v], x], x], x, Sin[v]/d], x]], x] /; !FalseQ[v] && FunctionOfQ[NonfreeFactors[Sin[v], x], u/Cos[v], x]]
  4379. Int[Pattern[u, Blank[]], Pattern[x, Blank[Symbol]]] := With[{v = FunctionOfTrig[u, x]}, Simp[With[{d = FreeFactors[Cos[v], x]}, Dist[-(d/Coefficient[v, x, 1]), Subst[Int[SubstFor[1, Cos[v]/d, u/Sin[v], x], x], x, Cos[v]/d], x]], x] /; !FalseQ[v] && FunctionOfQ[NonfreeFactors[Cos[v], x], u/Sin[v], x]]
  4380. Int[Times[Optional[Pattern[u, Blank[]]], Power[Plus[Optional[Pattern[a, Blank[]]], Times[Power[cos[Plus[Optional[Pattern[d, Blank[]]], Times[Optional[Pattern[e, Blank[]]], Pattern[x, Blank[]]]]], 2], Optional[Pattern[b, Blank[]]]], Times[Optional[Pattern[c, Blank[]]], Power[sin[Plus[Optional[Pattern[d, Blank[]]], Times[Optional[Pattern[e, Blank[]]], Pattern[x, Blank[]]]]], 2]]], Optional[Pattern[p, Blank[]]]]], Pattern[x, Blank[Symbol]]] := Dist[(a + c)^p, Int[ActivateTrig[u], x], x] /; FreeQ[{a, b, c, d, e, p}, x] && EqQ[b - c, 0]
  4381. Int[Times[Optional[Pattern[u, Blank[]]], Power[Plus[Optional[Pattern[a, Blank[]]], Times[Optional[Pattern[c, Blank[]]], Power[sec[Plus[Optional[Pattern[d, Blank[]]], Times[Optional[Pattern[e, Blank[]]], Pattern[x, Blank[]]]]], 2]], Times[Optional[Pattern[b, Blank[]]], Power[tan[Plus[Optional[Pattern[d, Blank[]]], Times[Optional[Pattern[e, Blank[]]], Pattern[x, Blank[]]]]], 2]]], Optional[Pattern[p, Blank[]]]]], Pattern[x, Blank[Symbol]]] := Dist[(a + c)^p, Int[ActivateTrig[u], x], x] /; FreeQ[{a, b, c, d, e, p}, x] && EqQ[b + c, 0]
  4382. Int[Times[Power[Plus[Optional[Pattern[a, Blank[]]], Times[Power[cot[Plus[Optional[Pattern[d, Blank[]]], Times[Optional[Pattern[e, Blank[]]], Pattern[x, Blank[]]]]], 2], Optional[Pattern[b, Blank[]]]], Times[Power[csc[Plus[Optional[Pattern[d, Blank[]]], Times[Optional[Pattern[e, Blank[]]], Pattern[x, Blank[]]]]], 2], Optional[Pattern[c, Blank[]]]]], Optional[Pattern[p, Blank[]]]], Optional[Pattern[u, Blank[]]]], Pattern[x, Blank[Symbol]]] := Dist[(a + c)^p, Int[ActivateTrig[u], x], x] /; FreeQ[{a, b, c, d, e, p}, x] && EqQ[b + c, 0]
  4383. Int[Times[Pattern[u, Blank[]], Power[Pattern[y, Blank[]], -1]], Pattern[x, Blank[Symbol]]] := With[{q = DerivativeDivides[ActivateTrig[y], ActivateTrig[u], x]}, Simp[q*Log[RemoveContent[ActivateTrig[y], x]], x] /; !FalseQ[q]] /; !InertTrigFreeQ[u]
  4384. Int[Times[Pattern[u, Blank[]], Power[Pattern[w, Blank[]], -1], Power[Pattern[y, Blank[]], -1]], Pattern[x, Blank[Symbol]]] := With[{q = DerivativeDivides[ActivateTrig[y*w], ActivateTrig[u], x]}, Simp[q*Log[RemoveContent[ActivateTrig[y*w], x]], x] /; !FalseQ[q]] /; !InertTrigFreeQ[u]
  4385. Int[Times[Pattern[u, Blank[]], Power[Pattern[y, Blank[]], Optional[Pattern[m, Blank[]]]]], Pattern[x, Blank[Symbol]]] := With[{q = DerivativeDivides[ActivateTrig[y], ActivateTrig[u], x]}, Simp[(q*ActivateTrig[y^(m + 1)])/(m + 1), x] /; !FalseQ[q]] /; FreeQ[m, x] && NeQ[m, -1] && !InertTrigFreeQ[u]
  4386. Int[Times[Pattern[u, Blank[]], Power[Pattern[y, Blank[]], Optional[Pattern[m, Blank[]]]], Power[Pattern[z, Blank[]], Optional[Pattern[n, Blank[]]]]], Pattern[x, Blank[Symbol]]] := With[{q = DerivativeDivides[ActivateTrig[y*z], ActivateTrig[u*z^(n - m)], x]}, Simp[(q*ActivateTrig[y^(m + 1)*z^(m + 1)])/(m + 1), x] /; !FalseQ[q]] /; FreeQ[{m, n}, x] && NeQ[m, -1] && !InertTrigFreeQ[u]
  4387. Int[Times[Optional[Pattern[u, Blank[]]], Power[Times[Optional[Pattern[a, Blank[]]], Power[Pattern[F, Blank[]][Plus[Optional[Pattern[c, Blank[]]], Times[Optional[Pattern[d, Blank[]]], Pattern[x, Blank[]]]]], Pattern[p, Blank[]]]], Pattern[n, Blank[]]]], Pattern[x, Blank[Symbol]]] := With[{v = ActivateTrig[F[c + d*x]]}, Dist[(a^IntPart[n]*(v/NonfreeFactors[v, x])^(p*IntPart[n])*(a*v^p)^FracPart[n])/NonfreeFactors[v, x]^(p*FracPart[n]), Int[ActivateTrig[u]*NonfreeFactors[v, x]^(n*p), x], x]] /; FreeQ[{a, c, d, n, p}, x] && InertTrigQ[F] && !IntegerQ[n] && IntegerQ[p]
  4388. Int[Times[Optional[Pattern[u, Blank[]]], Power[Times[Optional[Pattern[a, Blank[]]], Power[Times[Optional[Pattern[b, Blank[]]], Pattern[F, Blank[]][Plus[Optional[Pattern[c, Blank[]]], Times[Optional[Pattern[d, Blank[]]], Pattern[x, Blank[]]]]]], Pattern[p, Blank[]]]], Optional[Pattern[n, Blank[]]]]], Pattern[x, Blank[Symbol]]] := With[{v = ActivateTrig[F[c + d*x]]}, Dist[(a^IntPart[n]*(a*(b*v)^p)^FracPart[n])/(b*v)^(p*FracPart[n]), Int[ActivateTrig[u]*(b*v)^(n*p), x], x]] /; FreeQ[{a, b, c, d, n, p}, x] && InertTrigQ[F] && !IntegerQ[n] && !IntegerQ[p]
  4389. Int[Pattern[u, Blank[]], Pattern[x, Blank[Symbol]]] := With[{v = FunctionOfTrig[u, x]}, With[{d = FreeFactors[Tan[v], x]}, Dist[d/Coefficient[v, x, 1], Subst[Int[SubstFor[1/(1 + d^2*x^2), Tan[v]/d, u, x], x], x, Tan[v]/d], x]] /; !FalseQ[v] && FunctionOfQ[NonfreeFactors[Tan[v], x], u, x]] /; InverseFunctionFreeQ[u, x] && !MatchQ[u, (v_.)*((c_.)*tan[w_]^(n_.)*tan[z_]^(n_.))^(p_.) /; FreeQ[{c, p}, x] && IntegerQ[n] && LinearQ[w, x] && EqQ[z, 2*w]]
  4390. Int[Times[Pattern[u, Blank[]], Power[Times[Optional[Pattern[c, Blank[]]], sin[Pattern[v, Blank[]]]], Pattern[m, Blank[]]]], Pattern[x, Blank[Symbol]]] := With[{w = FunctionOfTrig[(u*Sin[v/2]^(2*m))/(c*Tan[v/2])^m, x]}, Dist[((c*Sin[v])^m*(c*Tan[v/2])^m)/Sin[v/2]^(2*m), Int[(u*Sin[v/2]^(2*m))/(c*Tan[v/2])^m, x], x] /; !FalseQ[w] && FunctionOfQ[NonfreeFactors[Tan[w], x], (u*Sin[v/2]^(2*m))/(c*Tan[v/2])^m, x]] /; FreeQ[c, x] && LinearQ[v, x] && IntegerQ[m + 1/2] && !SumQ[u] && InverseFunctionFreeQ[u, x]
  4391. Int[Times[Optional[Pattern[u, Blank[]]], Power[Plus[Times[Optional[Pattern[b, Blank[]]], Power[sec[Plus[Optional[Pattern[c, Blank[]]], Times[Optional[Pattern[d, Blank[]]], Pattern[x, Blank[]]]]], Optional[Pattern[n, Blank[]]]]], Times[Optional[Pattern[a, Blank[]]], Power[tan[Plus[Optional[Pattern[c, Blank[]]], Times[Optional[Pattern[d, Blank[]]], Pattern[x, Blank[]]]]], Optional[Pattern[n, Blank[]]]]]], Pattern[p, Blank[]]]], Pattern[x, Blank[Symbol]]] := Int[ActivateTrig[u]*Sec[c + d*x]^(n*p)*(b + a*Sin[c + d*x]^n)^p, x] /; FreeQ[{a, b, c, d}, x] && IntegersQ[n, p]
  4392. Int[Times[Power[Plus[Times[Power[cot[Plus[Optional[Pattern[c, Blank[]]], Times[Optional[Pattern[d, Blank[]]], Pattern[x, Blank[]]]]], Optional[Pattern[n, Blank[]]]], Optional[Pattern[a, Blank[]]]], Times[Power[csc[Plus[Optional[Pattern[c, Blank[]]], Times[Optional[Pattern[d, Blank[]]], Pattern[x, Blank[]]]]], Optional[Pattern[n, Blank[]]]], Optional[Pattern[b, Blank[]]]]], Pattern[p, Blank[]]], Optional[Pattern[u, Blank[]]]], Pattern[x, Blank[Symbol]]] := Int[ActivateTrig[u]*Csc[c + d*x]^(n*p)*(b + a*Cos[c + d*x]^n)^p, x] /; FreeQ[{a, b, c, d}, x] && IntegersQ[n, p]
  4393. Int[Times[Pattern[u, Blank[]], Power[Plus[Times[Pattern[a, Blank[]], Power[Pattern[F, Blank[]][Plus[Optional[Pattern[c, Blank[]]], Times[Optional[Pattern[d, Blank[]]], Pattern[x, Blank[]]]]], Optional[Pattern[p, Blank[]]]]], Times[Optional[Pattern[b, Blank[]]], Power[Pattern[F, Blank[]][Plus[Optional[Pattern[c, Blank[]]], Times[Optional[Pattern[d, Blank[]]], Pattern[x, Blank[]]]]], Optional[Pattern[q, Blank[]]]]]], Optional[Pattern[n, Blank[]]]]], Pattern[x, Blank[Symbol]]] := Int[ActivateTrig[u*F[c + d*x]^(n*p)*(a + b*F[c + d*x]^(q - p))^n], x] /; FreeQ[{a, b, c, d, p, q}, x] && InertTrigQ[F] && IntegerQ[n] && PosQ[q - p]
  4394. Int[Times[Pattern[u, Blank[]], Power[Plus[Times[Pattern[a, Blank[]], Power[Pattern[F, Blank[]][Plus[Optional[Pattern[d, Blank[]]], Times[Optional[Pattern[e, Blank[]]], Pattern[x, Blank[]]]]], Optional[Pattern[p, Blank[]]]]], Times[Optional[Pattern[b, Blank[]]], Power[Pattern[F, Blank[]][Plus[Optional[Pattern[d, Blank[]]], Times[Optional[Pattern[e, Blank[]]], Pattern[x, Blank[]]]]], Optional[Pattern[q, Blank[]]]]], Times[Optional[Pattern[c, Blank[]]], Power[Pattern[F, Blank[]][Plus[Optional[Pattern[d, Blank[]]], Times[Optional[Pattern[e, Blank[]]], Pattern[x, Blank[]]]]], Optional[Pattern[r, Blank[]]]]]], Optional[Pattern[n, Blank[]]]]], Pattern[x, Blank[Symbol]]] := Int[ActivateTrig[u*F[d + e*x]^(n*p)*(a + b*F[d + e*x]^(q - p) + c*F[d + e*x]^(r - p))^n], x] /; FreeQ[{a, b, c, d, e, p, q, r}, x] && InertTrigQ[F] && IntegerQ[n] && PosQ[q - p] && PosQ[r - p]
  4395. Int[Times[Pattern[u, Blank[]], Power[Plus[Pattern[a, Blank[]], Times[Optional[Pattern[b, Blank[]]], Power[Pattern[F, Blank[]][Plus[Optional[Pattern[d, Blank[]]], Times[Optional[Pattern[e, Blank[]]], Pattern[x, Blank[]]]]], Optional[Pattern[p, Blank[]]]]], Times[Optional[Pattern[c, Blank[]]], Power[Pattern[F, Blank[]][Plus[Optional[Pattern[d, Blank[]]], Times[Optional[Pattern[e, Blank[]]], Pattern[x, Blank[]]]]], Optional[Pattern[q, Blank[]]]]]], Optional[Pattern[n, Blank[]]]]], Pattern[x, Blank[Symbol]]] := Int[ActivateTrig[u*F[d + e*x]^(n*p)*(b + a/F[d + e*x]^p + c*F[d + e*x]^(q - p))^n], x] /; FreeQ[{a, b, c, d, e, p, q}, x] && InertTrigQ[F] && IntegerQ[n] && NegQ[p]
  4396. Int[Times[Optional[Pattern[u, Blank[]]], Power[Plus[Times[cos[Plus[Optional[Pattern[c, Blank[]]], Times[Optional[Pattern[d, Blank[]]], Pattern[x, Blank[]]]]], Optional[Pattern[a, Blank[]]]], Times[Optional[Pattern[b, Blank[]]], sin[Plus[Optional[Pattern[c, Blank[]]], Times[Optional[Pattern[d, Blank[]]], Pattern[x, Blank[]]]]]]], Optional[Pattern[n, Blank[]]]]], Pattern[x, Blank[Symbol]]] := Int[ActivateTrig[u]*(a/E^((a*(c + d*x))/b))^n, x] /; FreeQ[{a, b, c, d, n}, x] && EqQ[a^2 + b^2, 0]
  4397. Int[Pattern[u, Blank[]], Pattern[x, Blank[Symbol]]] := Int[TrigSimplify[u], x] /; TrigSimplifyQ[u]
  4398. Int[Times[Optional[Pattern[u, Blank[]]], Power[Times[Pattern[a, Blank[]], Pattern[v, Blank[]]], Pattern[p, Blank[]]]], Pattern[x, Blank[Symbol]]] := With[{uu = ActivateTrig[u], vv = ActivateTrig[v]}, Dist[(a^IntPart[p]*(a*vv)^FracPart[p])/vv^FracPart[p], Int[uu*vv^p, x], x]] /; FreeQ[{a, p}, x] && !IntegerQ[p] && !InertTrigFreeQ[v]
  4399. Int[Times[Optional[Pattern[u, Blank[]]], Power[Power[Pattern[v, Blank[]], Pattern[m, Blank[]]], Pattern[p, Blank[]]]], Pattern[x, Blank[Symbol]]] := With[{uu = ActivateTrig[u], vv = ActivateTrig[v]}, Dist[(vv^m)^FracPart[p]/vv^(m*FracPart[p]), Int[uu*vv^(m*p), x], x]] /; FreeQ[{m, p}, x] && !IntegerQ[p] && !InertTrigFreeQ[v]
  4400. Int[Times[Optional[Pattern[u, Blank[]]], Power[Times[Power[Pattern[v, Blank[]], Optional[Pattern[m, Blank[]]]], Power[Pattern[w, Blank[]], Optional[Pattern[n, Blank[]]]]], Pattern[p, Blank[]]]], Pattern[x, Blank[Symbol]]] := With[{uu = ActivateTrig[u], vv = ActivateTrig[v], ww = ActivateTrig[w]}, Dist[(vv^m*ww^n)^FracPart[p]/(vv^(m*FracPart[p])*ww^(n*FracPart[p])), Int[uu*vv^(m*p)*ww^(n*p), x], x]] /; FreeQ[{m, n, p}, x] && !IntegerQ[p] && ( !InertTrigFreeQ[v] || !InertTrigFreeQ[w])
  4401. Int[Pattern[u, Blank[]], Pattern[x, Blank[Symbol]]] := With[{v = ExpandTrig[u, x]}, Int[v, x] /; SumQ[v]] /; !InertTrigFreeQ[u]
  4402. Int[Pattern[u, Blank[]], Pattern[x, Blank[Symbol]]] := With[{w = Block[{$ShowSteps = False, $StepCounter = Null}, Int[SubstFor[1/(1 + FreeFactors[Tan[FunctionOfTrig[u, x]/2], x]^2*x^2), Tan[FunctionOfTrig[u, x]/2]/FreeFactors[Tan[FunctionOfTrig[u, x]/2], x], u, x], x]]}, Module[{v = FunctionOfTrig[u, x], d}, Simp[d = FreeFactors[Tan[v/2], x]; Dist[(2*d)/Coefficient[v, x, 1], Subst[Int[SubstFor[1/(1 + d^2*x^2), Tan[v/2]/d, u, x], x], x, Tan[v/2]/d], x], x]] /; CalculusFreeQ[w, x]] /; InverseFunctionFreeQ[u, x] && !FalseQ[FunctionOfTrig[u, x]]
  4403. Int[Pattern[u, Blank[]], Pattern[x, Blank[Symbol]]] := With[{v = ActivateTrig[u]}, CannotIntegrate[v, x]] /; !InertTrigFreeQ[u]
  4404. Int[Times[Cos[Plus[Optional[Pattern[a, Blank[]]], Times[Optional[Pattern[b, Blank[]]], Pattern[x, Blank[]]]]], Power[Plus[Optional[Pattern[c, Blank[]]], Times[Optional[Pattern[d, Blank[]]], Pattern[x, Blank[]]]], Optional[Pattern[m, Blank[]]]], Power[Sin[Plus[Optional[Pattern[a, Blank[]]], Times[Optional[Pattern[b, Blank[]]], Pattern[x, Blank[]]]]], Optional[Pattern[n, Blank[]]]]], Pattern[x, Blank[Symbol]]] := Simp[((c + d*x)^m*Sin[a + b*x]^(n + 1))/(b*(n + 1)), x] - Dist[(d*m)/(b*(n + 1)), Int[(c + d*x)^(m - 1)*Sin[a + b*x]^(n + 1), x], x] /; FreeQ[{a, b, c, d, n}, x] && IGtQ[m, 0] && NeQ[n, -1]
  4405. Int[Times[Power[Cos[Plus[Optional[Pattern[a, Blank[]]], Times[Optional[Pattern[b, Blank[]]], Pattern[x, Blank[]]]]], Optional[Pattern[n, Blank[]]]], Power[Plus[Optional[Pattern[c, Blank[]]], Times[Optional[Pattern[d, Blank[]]], Pattern[x, Blank[]]]], Optional[Pattern[m, Blank[]]]], Sin[Plus[Optional[Pattern[a, Blank[]]], Times[Optional[Pattern[b, Blank[]]], Pattern[x, Blank[]]]]]], Pattern[x, Blank[Symbol]]] := -Simp[((c + d*x)^m*Cos[a + b*x]^(n + 1))/(b*(n + 1)), x] + Dist[(d*m)/(b*(n + 1)), Int[(c + d*x)^(m - 1)*Cos[a + b*x]^(n + 1), x], x] /; FreeQ[{a, b, c, d, n}, x] && IGtQ[m, 0] && NeQ[n, -1]
  4406. Int[Times[Power[Cos[Plus[Optional[Pattern[a, Blank[]]], Times[Optional[Pattern[b, Blank[]]], Pattern[x, Blank[]]]]], Optional[Pattern[p, Blank[]]]], Power[Plus[Optional[Pattern[c, Blank[]]], Times[Optional[Pattern[d, Blank[]]], Pattern[x, Blank[]]]], Optional[Pattern[m, Blank[]]]], Power[Sin[Plus[Optional[Pattern[a, Blank[]]], Times[Optional[Pattern[b, Blank[]]], Pattern[x, Blank[]]]]], Optional[Pattern[n, Blank[]]]]], Pattern[x, Blank[Symbol]]] := Int[ExpandTrigReduce[(c + d*x)^m, Sin[a + b*x]^n*Cos[a + b*x]^p, x], x] /; FreeQ[{a, b, c, d, m}, x] && IGtQ[n, 0] && IGtQ[p, 0]
  4407. Int[Times[Power[Plus[Optional[Pattern[c, Blank[]]], Times[Optional[Pattern[d, Blank[]]], Pattern[x, Blank[]]]], Optional[Pattern[m, Blank[]]]], Power[Sin[Plus[Optional[Pattern[a, Blank[]]], Times[Optional[Pattern[b, Blank[]]], Pattern[x, Blank[]]]]], Optional[Pattern[n, Blank[]]]], Power[Tan[Plus[Optional[Pattern[a, Blank[]]], Times[Optional[Pattern[b, Blank[]]], Pattern[x, Blank[]]]]], Optional[Pattern[p, Blank[]]]]], Pattern[x, Blank[Symbol]]] := -Int[(c + d*x)^m*Sin[a + b*x]^n*Tan[a + b*x]^(p - 2), x] + Int[(c + d*x)^m*Sin[a + b*x]^(n - 2)*Tan[a + b*x]^p, x] /; FreeQ[{a, b, c, d, m}, x] && IGtQ[n, 0] && IGtQ[p, 0]
  4408. Int[Times[Power[Cos[Plus[Optional[Pattern[a, Blank[]]], Times[Optional[Pattern[b, Blank[]]], Pattern[x, Blank[]]]]], Optional[Pattern[n, Blank[]]]], Power[Cot[Plus[Optional[Pattern[a, Blank[]]], Times[Optional[Pattern[b, Blank[]]], Pattern[x, Blank[]]]]], Optional[Pattern[p, Blank[]]]], Power[Plus[Optional[Pattern[c, Blank[]]], Times[Optional[Pattern[d, Blank[]]], Pattern[x, Blank[]]]], Optional[Pattern[m, Blank[]]]]], Pattern[x, Blank[Symbol]]] := -Int[(c + d*x)^m*Cos[a + b*x]^n*Cot[a + b*x]^(p - 2), x] + Int[(c + d*x)^m*Cos[a + b*x]^(n - 2)*Cot[a + b*x]^p, x] /; FreeQ[{a, b, c, d, m}, x] && IGtQ[n, 0] && IGtQ[p, 0]
  4409. Int[Times[Power[Plus[Optional[Pattern[c, Blank[]]], Times[Optional[Pattern[d, Blank[]]], Pattern[x, Blank[]]]], Optional[Pattern[m, Blank[]]]], Power[Sec[Plus[Optional[Pattern[a, Blank[]]], Times[Optional[Pattern[b, Blank[]]], Pattern[x, Blank[]]]]], Optional[Pattern[n, Blank[]]]], Power[Tan[Plus[Optional[Pattern[a, Blank[]]], Times[Optional[Pattern[b, Blank[]]], Pattern[x, Blank[]]]]], Optional[Pattern[p, Blank[]]]]], Pattern[x, Blank[Symbol]]] := Simp[((c + d*x)^m*Sec[a + b*x]^n)/(b*n), x] - Dist[(d*m)/(b*n), Int[(c + d*x)^(m - 1)*Sec[a + b*x]^n, x], x] /; FreeQ[{a, b, c, d, n}, x] && EqQ[p, 1] && GtQ[m, 0]
  4410. Int[Times[Power[Cot[Plus[Optional[Pattern[a, Blank[]]], Times[Optional[Pattern[b, Blank[]]], Pattern[x, Blank[]]]]], Optional[Pattern[p, Blank[]]]], Power[Csc[Plus[Optional[Pattern[a, Blank[]]], Times[Optional[Pattern[b, Blank[]]], Pattern[x, Blank[]]]]], Optional[Pattern[n, Blank[]]]], Power[Plus[Optional[Pattern[c, Blank[]]], Times[Optional[Pattern[d, Blank[]]], Pattern[x, Blank[]]]], Optional[Pattern[m, Blank[]]]]], Pattern[x, Blank[Symbol]]] := -Simp[((c + d*x)^m*Csc[a + b*x]^n)/(b*n), x] + Dist[(d*m)/(b*n), Int[(c + d*x)^(m - 1)*Csc[a + b*x]^n, x], x] /; FreeQ[{a, b, c, d, n}, x] && EqQ[p, 1] && GtQ[m, 0]
  4411. Int[Times[Power[Plus[Optional[Pattern[c, Blank[]]], Times[Optional[Pattern[d, Blank[]]], Pattern[x, Blank[]]]], Optional[Pattern[m, Blank[]]]], Power[Sec[Plus[Optional[Pattern[a, Blank[]]], Times[Optional[Pattern[b, Blank[]]], Pattern[x, Blank[]]]]], 2], Power[Tan[Plus[Optional[Pattern[a, Blank[]]], Times[Optional[Pattern[b, Blank[]]], Pattern[x, Blank[]]]]], Optional[Pattern[n, Blank[]]]]], Pattern[x, Blank[Symbol]]] := Simp[((c + d*x)^m*Tan[a + b*x]^(n + 1))/(b*(n + 1)), x] - Dist[(d*m)/(b*(n + 1)), Int[(c + d*x)^(m - 1)*Tan[a + b*x]^(n + 1), x], x] /; FreeQ[{a, b, c, d, n}, x] && IGtQ[m, 0] && NeQ[n, -1]
  4412. Int[Times[Power[Cot[Plus[Optional[Pattern[a, Blank[]]], Times[Optional[Pattern[b, Blank[]]], Pattern[x, Blank[]]]]], Optional[Pattern[n, Blank[]]]], Power[Csc[Plus[Optional[Pattern[a, Blank[]]], Times[Optional[Pattern[b, Blank[]]], Pattern[x, Blank[]]]]], 2], Power[Plus[Optional[Pattern[c, Blank[]]], Times[Optional[Pattern[d, Blank[]]], Pattern[x, Blank[]]]], Optional[Pattern[m, Blank[]]]]], Pattern[x, Blank[Symbol]]] := -Simp[((c + d*x)^m*Cot[a + b*x]^(n + 1))/(b*(n + 1)), x] + Dist[(d*m)/(b*(n + 1)), Int[(c + d*x)^(m - 1)*Cot[a + b*x]^(n + 1), x], x] /; FreeQ[{a, b, c, d, n}, x] && IGtQ[m, 0] && NeQ[n, -1]
  4413. Int[Times[Power[Plus[Optional[Pattern[c, Blank[]]], Times[Optional[Pattern[d, Blank[]]], Pattern[x, Blank[]]]], Optional[Pattern[m, Blank[]]]], Sec[Plus[Optional[Pattern[a, Blank[]]], Times[Optional[Pattern[b, Blank[]]], Pattern[x, Blank[]]]]], Power[Tan[Plus[Optional[Pattern[a, Blank[]]], Times[Optional[Pattern[b, Blank[]]], Pattern[x, Blank[]]]]], Pattern[p, Blank[]]]], Pattern[x, Blank[Symbol]]] := -Int[(c + d*x)^m*Sec[a + b*x]*Tan[a + b*x]^(p - 2), x] + Int[(c + d*x)^m*Sec[a + b*x]^3*Tan[a + b*x]^(p - 2), x] /; FreeQ[{a, b, c, d, m}, x] && IGtQ[p/2, 0]
  4414. Int[Times[Power[Plus[Optional[Pattern[c, Blank[]]], Times[Optional[Pattern[d, Blank[]]], Pattern[x, Blank[]]]], Optional[Pattern[m, Blank[]]]], Power[Sec[Plus[Optional[Pattern[a, Blank[]]], Times[Optional[Pattern[b, Blank[]]], Pattern[x, Blank[]]]]], Optional[Pattern[n, Blank[]]]], Power[Tan[Plus[Optional[Pattern[a, Blank[]]], Times[Optional[Pattern[b, Blank[]]], Pattern[x, Blank[]]]]], Pattern[p, Blank[]]]], Pattern[x, Blank[Symbol]]] := -Int[(c + d*x)^m*Sec[a + b*x]^n*Tan[a + b*x]^(p - 2), x] + Int[(c + d*x)^m*Sec[a + b*x]^(n + 2)*Tan[a + b*x]^(p - 2), x] /; FreeQ[{a, b, c, d, m, n}, x] && IGtQ[p/2, 0]
  4415. Int[Times[Power[Cot[Plus[Optional[Pattern[a, Blank[]]], Times[Optional[Pattern[b, Blank[]]], Pattern[x, Blank[]]]]], Pattern[p, Blank[]]], Csc[Plus[Optional[Pattern[a, Blank[]]], Times[Optional[Pattern[b, Blank[]]], Pattern[x, Blank[]]]]], Power[Plus[Optional[Pattern[c, Blank[]]], Times[Optional[Pattern[d, Blank[]]], Pattern[x, Blank[]]]], Optional[Pattern[m, Blank[]]]]], Pattern[x, Blank[Symbol]]] := -Int[(c + d*x)^m*Csc[a + b*x]*Cot[a + b*x]^(p - 2), x] + Int[(c + d*x)^m*Csc[a + b*x]^3*Cot[a + b*x]^(p - 2), x] /; FreeQ[{a, b, c, d, m}, x] && IGtQ[p/2, 0]
  4416. Int[Times[Power[Cot[Plus[Optional[Pattern[a, Blank[]]], Times[Optional[Pattern[b, Blank[]]], Pattern[x, Blank[]]]]], Pattern[p, Blank[]]], Power[Csc[Plus[Optional[Pattern[a, Blank[]]], Times[Optional[Pattern[b, Blank[]]], Pattern[x, Blank[]]]]], Optional[Pattern[n, Blank[]]]], Power[Plus[Optional[Pattern[c, Blank[]]], Times[Optional[Pattern[d, Blank[]]], Pattern[x, Blank[]]]], Optional[Pattern[m, Blank[]]]]], Pattern[x, Blank[Symbol]]] := -Int[(c + d*x)^m*Csc[a + b*x]^n*Cot[a + b*x]^(p - 2), x] + Int[(c + d*x)^m*Csc[a + b*x]^(n + 2)*Cot[a + b*x]^(p - 2), x] /; FreeQ[{a, b, c, d, m, n}, x] && IGtQ[p/2, 0]
  4417. Int[Times[Power[Plus[Optional[Pattern[c, Blank[]]], Times[Optional[Pattern[d, Blank[]]], Pattern[x, Blank[]]]], Optional[Pattern[m, Blank[]]]], Power[Sec[Plus[Optional[Pattern[a, Blank[]]], Times[Optional[Pattern[b, Blank[]]], Pattern[x, Blank[]]]]], Optional[Pattern[n, Blank[]]]], Power[Tan[Plus[Optional[Pattern[a, Blank[]]], Times[Optional[Pattern[b, Blank[]]], Pattern[x, Blank[]]]]], Optional[Pattern[p, Blank[]]]]], Pattern[x, Blank[Symbol]]] := Module[{u = IntHide[Sec[a + b*x]^n*Tan[a + b*x]^p, x]}, Dist[(c + d*x)^m, u, x] - Dist[d*m, Int[(c + d*x)^(m - 1)*u, x], x]] /; FreeQ[{a, b, c, d, n, p}, x] && IGtQ[m, 0] && (IntegerQ[n/2] || IntegerQ[(p - 1)/2])
  4418. Int[Times[Power[Cot[Plus[Optional[Pattern[a, Blank[]]], Times[Optional[Pattern[b, Blank[]]], Pattern[x, Blank[]]]]], Optional[Pattern[p, Blank[]]]], Power[Csc[Plus[Optional[Pattern[a, Blank[]]], Times[Optional[Pattern[b, Blank[]]], Pattern[x, Blank[]]]]], Optional[Pattern[n, Blank[]]]], Power[Plus[Optional[Pattern[c, Blank[]]], Times[Optional[Pattern[d, Blank[]]], Pattern[x, Blank[]]]], Optional[Pattern[m, Blank[]]]]], Pattern[x, Blank[Symbol]]] := Module[{u = IntHide[Csc[a + b*x]^n*Cot[a + b*x]^p, x]}, Dist[(c + d*x)^m, u, x] - Dist[d*m, Int[(c + d*x)^(m - 1)*u, x], x]] /; FreeQ[{a, b, c, d, n, p}, x] && IGtQ[m, 0] && (IntegerQ[n/2] || IntegerQ[(p - 1)/2])
  4419. Int[Times[Power[Csc[Plus[Optional[Pattern[a, Blank[]]], Times[Optional[Pattern[b, Blank[]]], Pattern[x, Blank[]]]]], Optional[Pattern[n, Blank[]]]], Power[Plus[Optional[Pattern[c, Blank[]]], Times[Optional[Pattern[d, Blank[]]], Pattern[x, Blank[]]]], Optional[Pattern[m, Blank[]]]], Power[Sec[Plus[Optional[Pattern[a, Blank[]]], Times[Optional[Pattern[b, Blank[]]], Pattern[x, Blank[]]]]], Optional[Pattern[n, Blank[]]]]], Pattern[x, Blank[Symbol]]] := Dist[2^n, Int[(c + d*x)^m*Csc[2*a + 2*b*x]^n, x], x] /; FreeQ[{a, b, c, d, m}, x] && IntegerQ[n] && RationalQ[m]
  4420. Int[Times[Power[Csc[Plus[Optional[Pattern[a, Blank[]]], Times[Optional[Pattern[b, Blank[]]], Pattern[x, Blank[]]]]], Optional[Pattern[n, Blank[]]]], Power[Plus[Optional[Pattern[c, Blank[]]], Times[Optional[Pattern[d, Blank[]]], Pattern[x, Blank[]]]], Optional[Pattern[m, Blank[]]]], Power[Sec[Plus[Optional[Pattern[a, Blank[]]], Times[Optional[Pattern[b, Blank[]]], Pattern[x, Blank[]]]]], Optional[Pattern[p, Blank[]]]]], Pattern[x, Blank[Symbol]]] := Module[{u = IntHide[Csc[a + b*x]^n*Sec[a + b*x]^p, x]}, Dist[(c + d*x)^m, u, x] - Dist[d*m, Int[(c + d*x)^(m - 1)*u, x], x]] /; FreeQ[{a, b, c, d}, x] && IntegersQ[n, p] && GtQ[m, 0] && NeQ[n, p]
  4421. Int[Times[Power[Pattern[u, Blank[]], Optional[Pattern[m, Blank[]]]], Power[Pattern[F, Blank[]][Pattern[v, Blank[]]], Optional[Pattern[n, Blank[]]]], Power[Pattern[G, Blank[]][Pattern[w, Blank[]]], Optional[Pattern[p, Blank[]]]]], Pattern[x, Blank[Symbol]]] := Int[ExpandToSum[u, x]^m*F[ExpandToSum[v, x]]^n*G[ExpandToSum[v, x]]^p, x] /; FreeQ[{m, n, p}, x] && TrigQ[F] && TrigQ[G] && EqQ[v, w] && LinearQ[{u, v, w}, x] && !LinearMatchQ[{u, v, w}, x]
  4422. Int[Times[Cos[Plus[Optional[Pattern[c, Blank[]]], Times[Optional[Pattern[d, Blank[]]], Pattern[x, Blank[]]]]], Power[Plus[Optional[Pattern[e, Blank[]]], Times[Optional[Pattern[f, Blank[]]], Pattern[x, Blank[]]]], Optional[Pattern[m, Blank[]]]], Power[Plus[Pattern[a, Blank[]], Times[Optional[Pattern[b, Blank[]]], Sin[Plus[Optional[Pattern[c, Blank[]]], Times[Optional[Pattern[d, Blank[]]], Pattern[x, Blank[]]]]]]], Optional[Pattern[n, Blank[]]]]], Pattern[x, Blank[Symbol]]] := Simp[((e + f*x)^m*(a + b*Sin[c + d*x])^(n + 1))/(b*d*(n + 1)), x] - Dist[(f*m)/(b*d*(n + 1)), Int[(e + f*x)^(m - 1)*(a + b*Sin[c + d*x])^(n + 1), x], x] /; FreeQ[{a, b, c, d, e, f, n}, x] && IGtQ[m, 0] && NeQ[n, -1]
  4423. Int[Times[Power[Plus[Times[Cos[Plus[Optional[Pattern[c, Blank[]]], Times[Optional[Pattern[d, Blank[]]], Pattern[x, Blank[]]]]], Optional[Pattern[b, Blank[]]]], Pattern[a, Blank[]]], Optional[Pattern[n, Blank[]]]], Power[Plus[Optional[Pattern[e, Blank[]]], Times[Optional[Pattern[f, Blank[]]], Pattern[x, Blank[]]]], Optional[Pattern[m, Blank[]]]], Sin[Plus[Optional[Pattern[c, Blank[]]], Times[Optional[Pattern[d, Blank[]]], Pattern[x, Blank[]]]]]], Pattern[x, Blank[Symbol]]] := -Simp[((e + f*x)^m*(a + b*Cos[c + d*x])^(n + 1))/(b*d*(n + 1)), x] + Dist[(f*m)/(b*d*(n + 1)), Int[(e + f*x)^(m - 1)*(a + b*Cos[c + d*x])^(n + 1), x], x] /; FreeQ[{a, b, c, d, e, f, n}, x] && IGtQ[m, 0] && NeQ[n, -1]
  4424. Int[Times[Power[Plus[Optional[Pattern[e, Blank[]]], Times[Optional[Pattern[f, Blank[]]], Pattern[x, Blank[]]]], Optional[Pattern[m, Blank[]]]], Power[Sec[Plus[Optional[Pattern[c, Blank[]]], Times[Optional[Pattern[d, Blank[]]], Pattern[x, Blank[]]]]], 2], Power[Plus[Pattern[a, Blank[]], Times[Optional[Pattern[b, Blank[]]], Tan[Plus[Optional[Pattern[c, Blank[]]], Times[Optional[Pattern[d, Blank[]]], Pattern[x, Blank[]]]]]]], Optional[Pattern[n, Blank[]]]]], Pattern[x, Blank[Symbol]]] := Simp[((e + f*x)^m*(a + b*Tan[c + d*x])^(n + 1))/(b*d*(n + 1)), x] - Dist[(f*m)/(b*d*(n + 1)), Int[(e + f*x)^(m - 1)*(a + b*Tan[c + d*x])^(n + 1), x], x] /; FreeQ[{a, b, c, d, e, f, n}, x] && IGtQ[m, 0] && NeQ[n, -1]
  4425. Int[Times[Power[Csc[Plus[Optional[Pattern[c, Blank[]]], Times[Optional[Pattern[d, Blank[]]], Pattern[x, Blank[]]]]], 2], Power[Plus[Times[Cot[Plus[Optional[Pattern[c, Blank[]]], Times[Optional[Pattern[d, Blank[]]], Pattern[x, Blank[]]]]], Optional[Pattern[b, Blank[]]]], Pattern[a, Blank[]]], Optional[Pattern[n, Blank[]]]], Power[Plus[Optional[Pattern[e, Blank[]]], Times[Optional[Pattern[f, Blank[]]], Pattern[x, Blank[]]]], Optional[Pattern[m, Blank[]]]]], Pattern[x, Blank[Symbol]]] := -Simp[((e + f*x)^m*(a + b*Cot[c + d*x])^(n + 1))/(b*d*(n + 1)), x] + Dist[(f*m)/(b*d*(n + 1)), Int[(e + f*x)^(m - 1)*(a + b*Cot[c + d*x])^(n + 1), x], x] /; FreeQ[{a, b, c, d, e, f, n}, x] && IGtQ[m, 0] && NeQ[n, -1]
  4426. Int[Times[Power[Plus[Optional[Pattern[e, Blank[]]], Times[Optional[Pattern[f, Blank[]]], Pattern[x, Blank[]]]], Optional[Pattern[m, Blank[]]]], Sec[Plus[Optional[Pattern[c, Blank[]]], Times[Optional[Pattern[d, Blank[]]], Pattern[x, Blank[]]]]], Power[Plus[Pattern[a, Blank[]], Times[Optional[Pattern[b, Blank[]]], Sec[Plus[Optional[Pattern[c, Blank[]]], Times[Optional[Pattern[d, Blank[]]], Pattern[x, Blank[]]]]]]], Optional[Pattern[n, Blank[]]]], Tan[Plus[Optional[Pattern[c, Blank[]]], Times[Optional[Pattern[d, Blank[]]], Pattern[x, Blank[]]]]]], Pattern[x, Blank[Symbol]]] := Simp[((e + f*x)^m*(a + b*Sec[c + d*x])^(n + 1))/(b*d*(n + 1)), x] - Dist[(f*m)/(b*d*(n + 1)), Int[(e + f*x)^(m - 1)*(a + b*Sec[c + d*x])^(n + 1), x], x] /; FreeQ[{a, b, c, d, e, f, n}, x] && IGtQ[m, 0] && NeQ[n, -1]
  4427. Int[Times[Cot[Plus[Optional[Pattern[c, Blank[]]], Times[Optional[Pattern[d, Blank[]]], Pattern[x, Blank[]]]]], Csc[Plus[Optional[Pattern[c, Blank[]]], Times[Optional[Pattern[d, Blank[]]], Pattern[x, Blank[]]]]], Power[Plus[Times[Csc[Plus[Optional[Pattern[c, Blank[]]], Times[Optional[Pattern[d, Blank[]]], Pattern[x, Blank[]]]]], Optional[Pattern[b, Blank[]]]], Pattern[a, Blank[]]], Optional[Pattern[n, Blank[]]]], Power[Plus[Optional[Pattern[e, Blank[]]], Times[Optional[Pattern[f, Blank[]]], Pattern[x, Blank[]]]], Optional[Pattern[m, Blank[]]]]], Pattern[x, Blank[Symbol]]] := -Simp[((e + f*x)^m*(a + b*Csc[c + d*x])^(n + 1))/(b*d*(n + 1)), x] + Dist[(f*m)/(b*d*(n + 1)), Int[(e + f*x)^(m - 1)*(a + b*Csc[c + d*x])^(n + 1), x], x] /; FreeQ[{a, b, c, d, e, f, n}, x] && IGtQ[m, 0] && NeQ[n, -1]
  4428. Int[Times[Power[Plus[Optional[Pattern[e, Blank[]]], Times[Optional[Pattern[f, Blank[]]], Pattern[x, Blank[]]]], Optional[Pattern[m, Blank[]]]], Power[Sin[Plus[Optional[Pattern[a, Blank[]]], Times[Optional[Pattern[b, Blank[]]], Pattern[x, Blank[]]]]], Optional[Pattern[p, Blank[]]]], Power[Sin[Plus[Optional[Pattern[c, Blank[]]], Times[Optional[Pattern[d, Blank[]]], Pattern[x, Blank[]]]]], Optional[Pattern[q, Blank[]]]]], Pattern[x, Blank[Symbol]]] := Int[ExpandTrigReduce[(e + f*x)^m, Sin[a + b*x]^p*Sin[c + d*x]^q, x], x] /; FreeQ[{a, b, c, d, e, f}, x] && IGtQ[p, 0] && IGtQ[q, 0] && IntegerQ[m]
  4429. Int[Times[Power[Cos[Plus[Optional[Pattern[a, Blank[]]], Times[Optional[Pattern[b, Blank[]]], Pattern[x, Blank[]]]]], Optional[Pattern[p, Blank[]]]], Power[Cos[Plus[Optional[Pattern[c, Blank[]]], Times[Optional[Pattern[d, Blank[]]], Pattern[x, Blank[]]]]], Optional[Pattern[q, Blank[]]]], Power[Plus[Optional[Pattern[e, Blank[]]], Times[Optional[Pattern[f, Blank[]]], Pattern[x, Blank[]]]], Optional[Pattern[m, Blank[]]]]], Pattern[x, Blank[Symbol]]] := Int[ExpandTrigReduce[(e + f*x)^m, Cos[a + b*x]^p*Cos[c + d*x]^q, x], x] /; FreeQ[{a, b, c, d, e, f}, x] && IGtQ[p, 0] && IGtQ[q, 0] && IntegerQ[m]
  4430. Int[Times[Power[Cos[Plus[Optional[Pattern[c, Blank[]]], Times[Optional[Pattern[d, Blank[]]], Pattern[x, Blank[]]]]], Optional[Pattern[q, Blank[]]]], Power[Plus[Optional[Pattern[e, Blank[]]], Times[Optional[Pattern[f, Blank[]]], Pattern[x, Blank[]]]], Optional[Pattern[m, Blank[]]]], Power[Sin[Plus[Optional[Pattern[a, Blank[]]], Times[Optional[Pattern[b, Blank[]]], Pattern[x, Blank[]]]]], Optional[Pattern[p, Blank[]]]]], Pattern[x, Blank[Symbol]]] := Int[ExpandTrigReduce[(e + f*x)^m, Sin[a + b*x]^p*Cos[c + d*x]^q, x], x] /; FreeQ[{a, b, c, d, e, f, m}, x] && IGtQ[p, 0] && IGtQ[q, 0]
  4431. Int[Times[Power[Plus[Optional[Pattern[e, Blank[]]], Times[Optional[Pattern[f, Blank[]]], Pattern[x, Blank[]]]], Optional[Pattern[m, Blank[]]]], Power[Pattern[F, Blank[]][Plus[Optional[Pattern[a, Blank[]]], Times[Optional[Pattern[b, Blank[]]], Pattern[x, Blank[]]]]], Optional[Pattern[p, Blank[]]]], Power[Pattern[G, Blank[]][Plus[Optional[Pattern[c, Blank[]]], Times[Optional[Pattern[d, Blank[]]], Pattern[x, Blank[]]]]], Optional[Pattern[q, Blank[]]]]], Pattern[x, Blank[Symbol]]] := Int[ExpandTrigExpand[(e + f*x)^m*G[c + d*x]^q, F, c + d*x, p, b/d, x], x] /; FreeQ[{a, b, c, d, e, f, m}, x] && MemberQ[{Sin, Cos}, F] && MemberQ[{Sec, Csc}, G] && IGtQ[p, 0] && IGtQ[q, 0] && EqQ[b*c - a*d, 0] && IGtQ[b/d, 1]
  4432. Int[Times[Power[Pattern[F, Blank[]], Times[Optional[Pattern[c, Blank[]]], Plus[Optional[Pattern[a, Blank[]]], Times[Optional[Pattern[b, Blank[]]], Pattern[x, Blank[]]]]]], Sin[Plus[Optional[Pattern[d, Blank[]]], Times[Optional[Pattern[e, Blank[]]], Pattern[x, Blank[]]]]]], Pattern[x, Blank[Symbol]]] := Simp[(b*c*Log[F]*F^(c*(a + b*x))*Sin[d + e*x])/(e^2 + b^2*c^2*Log[F]^2), x] - Simp[(e*F^(c*(a + b*x))*Cos[d + e*x])/(e^2 + b^2*c^2*Log[F]^2), x] /; FreeQ[{F, a, b, c, d, e}, x] && NeQ[e^2 + b^2*c^2*Log[F]^2, 0]
  4433. Int[Times[Cos[Plus[Optional[Pattern[d, Blank[]]], Times[Optional[Pattern[e, Blank[]]], Pattern[x, Blank[]]]]], Power[Pattern[F, Blank[]], Times[Optional[Pattern[c, Blank[]]], Plus[Optional[Pattern[a, Blank[]]], Times[Optional[Pattern[b, Blank[]]], Pattern[x, Blank[]]]]]]], Pattern[x, Blank[Symbol]]] := Simp[(b*c*Log[F]*F^(c*(a + b*x))*Cos[d + e*x])/(e^2 + b^2*c^2*Log[F]^2), x] + Simp[(e*F^(c*(a + b*x))*Sin[d + e*x])/(e^2 + b^2*c^2*Log[F]^2), x] /; FreeQ[{F, a, b, c, d, e}, x] && NeQ[e^2 + b^2*c^2*Log[F]^2, 0]
  4434. Int[Times[Power[Pattern[F, Blank[]], Times[Optional[Pattern[c, Blank[]]], Plus[Optional[Pattern[a, Blank[]]], Times[Optional[Pattern[b, Blank[]]], Pattern[x, Blank[]]]]]], Power[Sin[Plus[Optional[Pattern[d, Blank[]]], Times[Optional[Pattern[e, Blank[]]], Pattern[x, Blank[]]]]], Pattern[n, Blank[]]]], Pattern[x, Blank[Symbol]]] := Simp[(b*c*Log[F]*F^(c*(a + b*x))*Sin[d + e*x]^n)/(e^2*n^2 + b^2*c^2*Log[F]^2), x] + (Dist[(n*(n - 1)*e^2)/(e^2*n^2 + b^2*c^2*Log[F]^2), Int[F^(c*(a + b*x))*Sin[d + e*x]^(n - 2), x], x] - Simp[(e*n*F^(c*(a + b*x))*Cos[d + e*x]*Sin[d + e*x]^(n - 1))/(e^2*n^2 + b^2*c^2*Log[F]^2), x]) /; FreeQ[{F, a, b, c, d, e}, x] && NeQ[e^2*n^2 + b^2*c^2*Log[F]^2, 0] && GtQ[n, 1]
  4435. Int[Times[Power[Cos[Plus[Optional[Pattern[d, Blank[]]], Times[Optional[Pattern[e, Blank[]]], Pattern[x, Blank[]]]]], Pattern[m, Blank[]]], Power[Pattern[F, Blank[]], Times[Optional[Pattern[c, Blank[]]], Plus[Optional[Pattern[a, Blank[]]], Times[Optional[Pattern[b, Blank[]]], Pattern[x, Blank[]]]]]]], Pattern[x, Blank[Symbol]]] := Simp[(b*c*Log[F]*F^(c*(a + b*x))*Cos[d + e*x]^m)/(e^2*m^2 + b^2*c^2*Log[F]^2), x] + (Dist[(m*(m - 1)*e^2)/(e^2*m^2 + b^2*c^2*Log[F]^2), Int[F^(c*(a + b*x))*Cos[d + e*x]^(m - 2), x], x] + Simp[(e*m*F^(c*(a + b*x))*Sin[d + e*x]*Cos[d + e*x]^(m - 1))/(e^2*m^2 + b^2*c^2*Log[F]^2), x]) /; FreeQ[{F, a, b, c, d, e}, x] && NeQ[e^2*m^2 + b^2*c^2*Log[F]^2, 0] && GtQ[m, 1]
  4436. Int[Times[Power[Pattern[F, Blank[]], Times[Optional[Pattern[c, Blank[]]], Plus[Optional[Pattern[a, Blank[]]], Times[Optional[Pattern[b, Blank[]]], Pattern[x, Blank[]]]]]], Power[Sin[Plus[Optional[Pattern[d, Blank[]]], Times[Optional[Pattern[e, Blank[]]], Pattern[x, Blank[]]]]], Pattern[n, Blank[]]]], Pattern[x, Blank[Symbol]]] := -Simp[(b*c*Log[F]*F^(c*(a + b*x))*Sin[d + e*x]^(n + 2))/(e^2*(n + 1)*(n + 2)), x] + Simp[(F^(c*(a + b*x))*Cos[d + e*x]*Sin[d + e*x]^(n + 1))/(e*(n + 1)), x] /; FreeQ[{F, a, b, c, d, e, n}, x] && EqQ[e^2*(n + 2)^2 + b^2*c^2*Log[F]^2, 0] && NeQ[n, -1] && NeQ[n, -2]
  4437. Int[Times[Power[Cos[Plus[Optional[Pattern[d, Blank[]]], Times[Optional[Pattern[e, Blank[]]], Pattern[x, Blank[]]]]], Pattern[n, Blank[]]], Power[Pattern[F, Blank[]], Times[Optional[Pattern[c, Blank[]]], Plus[Optional[Pattern[a, Blank[]]], Times[Optional[Pattern[b, Blank[]]], Pattern[x, Blank[]]]]]]], Pattern[x, Blank[Symbol]]] := -Simp[(b*c*Log[F]*F^(c*(a + b*x))*Cos[d + e*x]^(n + 2))/(e^2*(n + 1)*(n + 2)), x] - Simp[(F^(c*(a + b*x))*Sin[d + e*x]*Cos[d + e*x]^(n + 1))/(e*(n + 1)), x] /; FreeQ[{F, a, b, c, d, e, n}, x] && EqQ[e^2*(n + 2)^2 + b^2*c^2*Log[F]^2, 0] && NeQ[n, -1] && NeQ[n, -2]
  4438. Int[Times[Power[Pattern[F, Blank[]], Times[Optional[Pattern[c, Blank[]]], Plus[Optional[Pattern[a, Blank[]]], Times[Optional[Pattern[b, Blank[]]], Pattern[x, Blank[]]]]]], Power[Sin[Plus[Optional[Pattern[d, Blank[]]], Times[Optional[Pattern[e, Blank[]]], Pattern[x, Blank[]]]]], Pattern[n, Blank[]]]], Pattern[x, Blank[Symbol]]] := -Simp[(b*c*Log[F]*F^(c*(a + b*x))*Sin[d + e*x]^(n + 2))/(e^2*(n + 1)*(n + 2)), x] + (Dist[(e^2*(n + 2)^2 + b^2*c^2*Log[F]^2)/(e^2*(n + 1)*(n + 2)), Int[F^(c*(a + b*x))*Sin[d + e*x]^(n + 2), x], x] + Simp[(F^(c*(a + b*x))*Cos[d + e*x]*Sin[d + e*x]^(n + 1))/(e*(n + 1)), x]) /; FreeQ[{F, a, b, c, d, e}, x] && NeQ[e^2*(n + 2)^2 + b^2*c^2*Log[F]^2, 0] && LtQ[n, -1] && NeQ[n, -2]
  4439. Int[Times[Power[Cos[Plus[Optional[Pattern[d, Blank[]]], Times[Optional[Pattern[e, Blank[]]], Pattern[x, Blank[]]]]], Pattern[n, Blank[]]], Power[Pattern[F, Blank[]], Times[Optional[Pattern[c, Blank[]]], Plus[Optional[Pattern[a, Blank[]]], Times[Optional[Pattern[b, Blank[]]], Pattern[x, Blank[]]]]]]], Pattern[x, Blank[Symbol]]] := -Simp[(b*c*Log[F]*F^(c*(a + b*x))*Cos[d + e*x]^(n + 2))/(e^2*(n + 1)*(n + 2)), x] + (Dist[(e^2*(n + 2)^2 + b^2*c^2*Log[F]^2)/(e^2*(n + 1)*(n + 2)), Int[F^(c*(a + b*x))*Cos[d + e*x]^(n + 2), x], x] - Simp[(F^(c*(a + b*x))*Sin[d + e*x]*Cos[d + e*x]^(n + 1))/(e*(n + 1)), x]) /; FreeQ[{F, a, b, c, d, e}, x] && NeQ[e^2*(n + 2)^2 + b^2*c^2*Log[F]^2, 0] && LtQ[n, -1] && NeQ[n, -2]
  4440. Int[Times[Power[Pattern[F, Blank[]], Times[Optional[Pattern[c, Blank[]]], Plus[Optional[Pattern[a, Blank[]]], Times[Optional[Pattern[b, Blank[]]], Pattern[x, Blank[]]]]]], Power[Sin[Plus[Optional[Pattern[d, Blank[]]], Times[Optional[Pattern[e, Blank[]]], Pattern[x, Blank[]]]]], Pattern[n, Blank[]]]], Pattern[x, Blank[Symbol]]] := Dist[(E^(I*n*(d + e*x))*Sin[d + e*x]^n)/(-1 + E^(2*I*(d + e*x)))^n, Int[(F^(c*(a + b*x))*(-1 + E^(2*I*(d + e*x)))^n)/E^(I*n*(d + e*x)), x], x] /; FreeQ[{F, a, b, c, d, e, n}, x] && !IntegerQ[n]
  4441. Int[Times[Power[Cos[Plus[Optional[Pattern[d, Blank[]]], Times[Optional[Pattern[e, Blank[]]], Pattern[x, Blank[]]]]], Pattern[n, Blank[]]], Power[Pattern[F, Blank[]], Times[Optional[Pattern[c, Blank[]]], Plus[Optional[Pattern[a, Blank[]]], Times[Optional[Pattern[b, Blank[]]], Pattern[x, Blank[]]]]]]], Pattern[x, Blank[Symbol]]] := Dist[(E^(I*n*(d + e*x))*Cos[d + e*x]^n)/(1 + E^(2*I*(d + e*x)))^n, Int[(F^(c*(a + b*x))*(1 + E^(2*I*(d + e*x)))^n)/E^(I*n*(d + e*x)), x], x] /; FreeQ[{F, a, b, c, d, e, n}, x] && !IntegerQ[n]
  4442. Int[Times[Power[Pattern[F, Blank[]], Times[Optional[Pattern[c, Blank[]]], Plus[Optional[Pattern[a, Blank[]]], Times[Optional[Pattern[b, Blank[]]], Pattern[x, Blank[]]]]]], Power[Tan[Plus[Optional[Pattern[d, Blank[]]], Times[Optional[Pattern[e, Blank[]]], Pattern[x, Blank[]]]]], Optional[Pattern[n, Blank[]]]]], Pattern[x, Blank[Symbol]]] := Dist[I^n, Int[ExpandIntegrand[(F^(c*(a + b*x))*(1 - E^(2*I*(d + e*x)))^n)/(1 + E^(2*I*(d + e*x)))^n, x], x], x] /; FreeQ[{F, a, b, c, d, e}, x] && IntegerQ[n]
  4443. Int[Times[Power[Cot[Plus[Optional[Pattern[d, Blank[]]], Times[Optional[Pattern[e, Blank[]]], Pattern[x, Blank[]]]]], Optional[Pattern[n, Blank[]]]], Power[Pattern[F, Blank[]], Times[Optional[Pattern[c, Blank[]]], Plus[Optional[Pattern[a, Blank[]]], Times[Optional[Pattern[b, Blank[]]], Pattern[x, Blank[]]]]]]], Pattern[x, Blank[Symbol]]] := Dist[(-I)^n, Int[ExpandIntegrand[(F^(c*(a + b*x))*(1 + E^(2*I*(d + e*x)))^n)/(1 - E^(2*I*(d + e*x)))^n, x], x], x] /; FreeQ[{F, a, b, c, d, e}, x] && IntegerQ[n]
  4444. Int[Times[Power[Pattern[F, Blank[]], Times[Optional[Pattern[c, Blank[]]], Plus[Optional[Pattern[a, Blank[]]], Times[Optional[Pattern[b, Blank[]]], Pattern[x, Blank[]]]]]], Power[Sec[Plus[Optional[Pattern[d, Blank[]]], Times[Optional[Pattern[e, Blank[]]], Pattern[x, Blank[]]]]], Pattern[n, Blank[]]]], Pattern[x, Blank[Symbol]]] := Simp[b*c*Log[F]*F^(c*(a + b*x))*(Sec[d + e*x]^n/(e^2*n^2 + b^2*c^2*Log[F]^2)), x] + (Dist[e^2*n*((n + 1)/(e^2*n^2 + b^2*c^2*Log[F]^2)), Int[F^(c*(a + b*x))*Sec[d + e*x]^(n + 2), x], x] - Simp[e*n*F^(c*(a + b*x))*Sec[d + e*x]^(n + 1)*(Sin[d + e*x]/(e^2*n^2 + b^2*c^2*Log[F]^2)), x]) /; FreeQ[{F, a, b, c, d, e}, x] && NeQ[e^2*n^2 + b^2*c^2*Log[F]^2, 0] && LtQ[n, -1]
  4445. Int[Times[Power[Csc[Plus[Optional[Pattern[d, Blank[]]], Times[Optional[Pattern[e, Blank[]]], Pattern[x, Blank[]]]]], Pattern[n, Blank[]]], Power[Pattern[F, Blank[]], Times[Optional[Pattern[c, Blank[]]], Plus[Optional[Pattern[a, Blank[]]], Times[Optional[Pattern[b, Blank[]]], Pattern[x, Blank[]]]]]]], Pattern[x, Blank[Symbol]]] := Simp[b*c*Log[F]*F^(c*(a + b*x))*(Csc[d + e*x]^n/(e^2*n^2 + b^2*c^2*Log[F]^2)), x] + (Dist[e^2*n*((n + 1)/(e^2*n^2 + b^2*c^2*Log[F]^2)), Int[F^(c*(a + b*x))*Csc[d + e*x]^(n + 2), x], x] + Simp[e*n*F^(c*(a + b*x))*Csc[d + e*x]^(n + 1)*(Cos[d + e*x]/(e^2*n^2 + b^2*c^2*Log[F]^2)), x]) /; FreeQ[{F, a, b, c, d, e}, x] && NeQ[e^2*n^2 + b^2*c^2*Log[F]^2, 0] && LtQ[n, -1]
  4446. Int[Times[Power[Pattern[F, Blank[]], Times[Optional[Pattern[c, Blank[]]], Plus[Optional[Pattern[a, Blank[]]], Times[Optional[Pattern[b, Blank[]]], Pattern[x, Blank[]]]]]], Power[Sec[Plus[Optional[Pattern[d, Blank[]]], Times[Optional[Pattern[e, Blank[]]], Pattern[x, Blank[]]]]], Pattern[n, Blank[]]]], Pattern[x, Blank[Symbol]]] := -Simp[(b*c*Log[F]*F^(c*(a + b*x))*Sec[d + e*x]^(n - 2))/(e^2*(n - 1)*(n - 2)), x] + Simp[(F^(c*(a + b*x))*Sec[d + e*x]^(n - 1)*Sin[d + e*x])/(e*(n - 1)), x] /; FreeQ[{F, a, b, c, d, e, n}, x] && EqQ[b^2*c^2*Log[F]^2 + e^2*(n - 2)^2, 0] && NeQ[n, 1] && NeQ[n, 2]
  4447. Int[Times[Power[Csc[Plus[Optional[Pattern[d, Blank[]]], Times[Optional[Pattern[e, Blank[]]], Pattern[x, Blank[]]]]], Pattern[n, Blank[]]], Power[Pattern[F, Blank[]], Times[Optional[Pattern[c, Blank[]]], Plus[Optional[Pattern[a, Blank[]]], Times[Optional[Pattern[b, Blank[]]], Pattern[x, Blank[]]]]]]], Pattern[x, Blank[Symbol]]] := -Simp[(b*c*Log[F]*F^(c*(a + b*x))*Csc[d + e*x]^(n - 2))/(e^2*(n - 1)*(n - 2)), x] + Simp[(F^(c*(a + b*x))*Csc[d + e*x]^(n - 1)*Cos[d + e*x])/(e*(n - 1)), x] /; FreeQ[{F, a, b, c, d, e, n}, x] && EqQ[b^2*c^2*Log[F]^2 + e^2*(n - 2)^2, 0] && NeQ[n, 1] && NeQ[n, 2]
  4448. Int[Times[Power[Pattern[F, Blank[]], Times[Optional[Pattern[c, Blank[]]], Plus[Optional[Pattern[a, Blank[]]], Times[Optional[Pattern[b, Blank[]]], Pattern[x, Blank[]]]]]], Power[Sec[Plus[Optional[Pattern[d, Blank[]]], Times[Optional[Pattern[e, Blank[]]], Pattern[x, Blank[]]]]], Pattern[n, Blank[]]]], Pattern[x, Blank[Symbol]]] := -Simp[(b*c*Log[F]*F^(c*(a + b*x))*Sec[d + e*x]^(n - 2))/(e^2*(n - 1)*(n - 2)), x] + (Dist[(e^2*(n - 2)^2 + b^2*c^2*Log[F]^2)/(e^2*(n - 1)*(n - 2)), Int[F^(c*(a + b*x))*Sec[d + e*x]^(n - 2), x], x] + Simp[(F^(c*(a + b*x))*Sec[d + e*x]^(n - 1)*Sin[d + e*x])/(e*(n - 1)), x]) /; FreeQ[{F, a, b, c, d, e}, x] && NeQ[b^2*c^2*Log[F]^2 + e^2*(n - 2)^2, 0] && GtQ[n, 1] && NeQ[n, 2]
  4449. Int[Times[Power[Csc[Plus[Optional[Pattern[d, Blank[]]], Times[Optional[Pattern[e, Blank[]]], Pattern[x, Blank[]]]]], Pattern[n, Blank[]]], Power[Pattern[F, Blank[]], Times[Optional[Pattern[c, Blank[]]], Plus[Optional[Pattern[a, Blank[]]], Times[Optional[Pattern[b, Blank[]]], Pattern[x, Blank[]]]]]]], Pattern[x, Blank[Symbol]]] := -Simp[(b*c*Log[F]*F^(c*(a + b*x))*Csc[d + e*x]^(n - 2))/(e^2*(n - 1)*(n - 2)), x] + (Dist[(e^2*(n - 2)^2 + b^2*c^2*Log[F]^2)/(e^2*(n - 1)*(n - 2)), Int[F^(c*(a + b*x))*Csc[d + e*x]^(n - 2), x], x] - Simp[(F^(c*(a + b*x))*Csc[d + e*x]^(n - 1)*Cos[d + e*x])/(e*(n - 1)), x]) /; FreeQ[{F, a, b, c, d, e}, x] && NeQ[b^2*c^2*Log[F]^2 + e^2*(n - 2)^2, 0] && GtQ[n, 1] && NeQ[n, 2]
  4450. Int[Times[Power[Pattern[F, Blank[]], Times[Optional[Pattern[c, Blank[]]], Plus[Optional[Pattern[a, Blank[]]], Times[Optional[Pattern[b, Blank[]]], Pattern[x, Blank[]]]]]], Power[Sec[Plus[Optional[Pattern[d, Blank[]]], Times[Pi, Optional[Pattern[k, Blank[]]]], Times[Optional[Pattern[e, Blank[]]], Pattern[x, Blank[]]]]], Optional[Pattern[n, Blank[]]]]], Pattern[x, Blank[Symbol]]] := Simp[(2^n*E^(I*k*n*Pi)*E^(I*n*(d + e*x))*F^(c*(a + b*x))*Hypergeometric2F1[n, n/2 - (I*b*c*Log[F])/(2*e), 1 + n/2 - (I*b*c*Log[F])/(2*e), -(E^(2*I*k*Pi)*E^(2*I*(d + e*x)))])/(I*e*n + b*c*Log[F]), x] /; FreeQ[{F, a, b, c, d, e}, x] && IntegerQ[4*k] && IntegerQ[n]
  4451. Int[Times[Power[Pattern[F, Blank[]], Times[Optional[Pattern[c, Blank[]]], Plus[Optional[Pattern[a, Blank[]]], Times[Optional[Pattern[b, Blank[]]], Pattern[x, Blank[]]]]]], Power[Sec[Plus[Optional[Pattern[d, Blank[]]], Times[Optional[Pattern[e, Blank[]]], Pattern[x, Blank[]]]]], Optional[Pattern[n, Blank[]]]]], Pattern[x, Blank[Symbol]]] := Simp[(2^n*E^(I*n*(d + e*x))*F^(c*(a + b*x))*Hypergeometric2F1[n, n/2 - (I*b*c*Log[F])/(2*e), 1 + n/2 - (I*b*c*Log[F])/(2*e), -E^(2*I*(d + e*x))])/(I*e*n + b*c*Log[F]), x] /; FreeQ[{F, a, b, c, d, e}, x] && IntegerQ[n]
  4452. Int[Times[Power[Csc[Plus[Optional[Pattern[d, Blank[]]], Times[Pi, Optional[Pattern[k, Blank[]]]], Times[Optional[Pattern[e, Blank[]]], Pattern[x, Blank[]]]]], Optional[Pattern[n, Blank[]]]], Power[Pattern[F, Blank[]], Times[Optional[Pattern[c, Blank[]]], Plus[Optional[Pattern[a, Blank[]]], Times[Optional[Pattern[b, Blank[]]], Pattern[x, Blank[]]]]]]], Pattern[x, Blank[Symbol]]] := Simp[(-2*I)^n*E^(I*k*n*Pi)*E^(I*n*(d + e*x))*(F^(c*(a + b*x))/(I*e*n + b*c*Log[F]))*Hypergeometric2F1[n, n/2 - (I*b*c*Log[F])/(2*e), 1 + n/2 - (I*b*c*Log[F])/(2*e), E^(2*I*k*Pi)*E^(2*I*(d + e*x))], x] /; FreeQ[{F, a, b, c, d, e}, x] && IntegerQ[4*k] && IntegerQ[n]
  4453. Int[Times[Power[Csc[Plus[Optional[Pattern[d, Blank[]]], Times[Optional[Pattern[e, Blank[]]], Pattern[x, Blank[]]]]], Optional[Pattern[n, Blank[]]]], Power[Pattern[F, Blank[]], Times[Optional[Pattern[c, Blank[]]], Plus[Optional[Pattern[a, Blank[]]], Times[Optional[Pattern[b, Blank[]]], Pattern[x, Blank[]]]]]]], Pattern[x, Blank[Symbol]]] := Simp[(-2*I)^n*E^(I*n*(d + e*x))*(F^(c*(a + b*x))/(I*e*n + b*c*Log[F]))*Hypergeometric2F1[n, n/2 - (I*b*c*Log[F])/(2*e), 1 + n/2 - (I*b*c*Log[F])/(2*e), E^(2*I*(d + e*x))], x] /; FreeQ[{F, a, b, c, d, e}, x] && IntegerQ[n]
  4454. Int[Times[Power[Pattern[F, Blank[]], Times[Optional[Pattern[c, Blank[]]], Plus[Optional[Pattern[a, Blank[]]], Times[Optional[Pattern[b, Blank[]]], Pattern[x, Blank[]]]]]], Power[Sec[Plus[Optional[Pattern[d, Blank[]]], Times[Optional[Pattern[e, Blank[]]], Pattern[x, Blank[]]]]], Optional[Pattern[n, Blank[]]]]], Pattern[x, Blank[Symbol]]] := Dist[((1 + E^(2*I*(d + e*x)))^n*Sec[d + e*x]^n)/E^(I*n*(d + e*x)), Int[SimplifyIntegrand[(F^(c*(a + b*x))*E^(I*n*(d + e*x)))/(1 + E^(2*I*(d + e*x)))^n, x], x], x] /; FreeQ[{F, a, b, c, d, e}, x] && !IntegerQ[n]
  4455. Int[Times[Power[Csc[Plus[Optional[Pattern[d, Blank[]]], Times[Optional[Pattern[e, Blank[]]], Pattern[x, Blank[]]]]], Optional[Pattern[n, Blank[]]]], Power[Pattern[F, Blank[]], Times[Optional[Pattern[c, Blank[]]], Plus[Optional[Pattern[a, Blank[]]], Times[Optional[Pattern[b, Blank[]]], Pattern[x, Blank[]]]]]]], Pattern[x, Blank[Symbol]]] := Dist[((1 - E^(-2*I*(d + e*x)))^n*Csc[d + e*x]^n)/E^(-(I*n*(d + e*x))), Int[SimplifyIntegrand[F^(c*(a + b*x))/(E^(I*n*(d + e*x))*(1 - E^(-2*I*(d + e*x)))^n), x], x], x] /; FreeQ[{F, a, b, c, d, e}, x] && !IntegerQ[n]
  4456. Int[Times[Power[Pattern[F, Blank[]], Times[Optional[Pattern[c, Blank[]]], Plus[Optional[Pattern[a, Blank[]]], Times[Optional[Pattern[b, Blank[]]], Pattern[x, Blank[]]]]]], Power[Plus[Pattern[f, Blank[]], Times[Optional[Pattern[g, Blank[]]], Sin[Plus[Optional[Pattern[d, Blank[]]], Times[Optional[Pattern[e, Blank[]]], Pattern[x, Blank[]]]]]]], Optional[Pattern[n, Blank[]]]]], Pattern[x, Blank[Symbol]]] := Dist[2^n*f^n, Int[F^(c*(a + b*x))*Cos[d/2 + (e*x)/2 - (f*Pi)/(4*g)]^(2*n), x], x] /; FreeQ[{F, a, b, c, d, e, f, g}, x] && EqQ[f^2 - g^2, 0] && ILtQ[n, 0]
  4457. Int[Times[Power[Plus[Times[Cos[Plus[Optional[Pattern[d, Blank[]]], Times[Optional[Pattern[e, Blank[]]], Pattern[x, Blank[]]]]], Optional[Pattern[g, Blank[]]]], Pattern[f, Blank[]]], Optional[Pattern[n, Blank[]]]], Power[Pattern[F, Blank[]], Times[Optional[Pattern[c, Blank[]]], Plus[Optional[Pattern[a, Blank[]]], Times[Optional[Pattern[b, Blank[]]], Pattern[x, Blank[]]]]]]], Pattern[x, Blank[Symbol]]] := Dist[2^n*f^n, Int[F^(c*(a + b*x))*Cos[d/2 + (e*x)/2]^(2*n), x], x] /; FreeQ[{F, a, b, c, d, e, f, g}, x] && EqQ[f - g, 0] && ILtQ[n, 0]
  4458. Int[Times[Power[Plus[Times[Cos[Plus[Optional[Pattern[d, Blank[]]], Times[Optional[Pattern[e, Blank[]]], Pattern[x, Blank[]]]]], Optional[Pattern[g, Blank[]]]], Pattern[f, Blank[]]], Optional[Pattern[n, Blank[]]]], Power[Pattern[F, Blank[]], Times[Optional[Pattern[c, Blank[]]], Plus[Optional[Pattern[a, Blank[]]], Times[Optional[Pattern[b, Blank[]]], Pattern[x, Blank[]]]]]]], Pattern[x, Blank[Symbol]]] := Dist[2^n*f^n, Int[F^(c*(a + b*x))*Sin[d/2 + (e*x)/2]^(2*n), x], x] /; FreeQ[{F, a, b, c, d, e, f, g}, x] && EqQ[f + g, 0] && ILtQ[n, 0]
  4459. Int[Times[Power[Cos[Plus[Optional[Pattern[d, Blank[]]], Times[Optional[Pattern[e, Blank[]]], Pattern[x, Blank[]]]]], Optional[Pattern[m, Blank[]]]], Power[Pattern[F, Blank[]], Times[Optional[Pattern[c, Blank[]]], Plus[Optional[Pattern[a, Blank[]]], Times[Optional[Pattern[b, Blank[]]], Pattern[x, Blank[]]]]]], Power[Plus[Pattern[f, Blank[]], Times[Optional[Pattern[g, Blank[]]], Sin[Plus[Optional[Pattern[d, Blank[]]], Times[Optional[Pattern[e, Blank[]]], Pattern[x, Blank[]]]]]]], Optional[Pattern[n, Blank[]]]]], Pattern[x, Blank[Symbol]]] := Dist[g^n, Int[F^(c*(a + b*x))*Tan[(f*Pi)/(4*g) - d/2 - (e*x)/2]^m, x], x] /; FreeQ[{F, a, b, c, d, e, f, g}, x] && EqQ[f^2 - g^2, 0] && IntegersQ[m, n] && EqQ[m + n, 0]
  4460. Int[Times[Power[Plus[Times[Cos[Plus[Optional[Pattern[d, Blank[]]], Times[Optional[Pattern[e, Blank[]]], Pattern[x, Blank[]]]]], Optional[Pattern[g, Blank[]]]], Pattern[f, Blank[]]], Optional[Pattern[n, Blank[]]]], Power[Pattern[F, Blank[]], Times[Optional[Pattern[c, Blank[]]], Plus[Optional[Pattern[a, Blank[]]], Times[Optional[Pattern[b, Blank[]]], Pattern[x, Blank[]]]]]], Power[Sin[Plus[Optional[Pattern[d, Blank[]]], Times[Optional[Pattern[e, Blank[]]], Pattern[x, Blank[]]]]], Optional[Pattern[m, Blank[]]]]], Pattern[x, Blank[Symbol]]] := Dist[f^n, Int[F^(c*(a + b*x))*Tan[d/2 + (e*x)/2]^m, x], x] /; FreeQ[{F, a, b, c, d, e, f, g}, x] && EqQ[f - g, 0] && IntegersQ[m, n] && EqQ[m + n, 0]
  4461. Int[Times[Power[Plus[Times[Cos[Plus[Optional[Pattern[d, Blank[]]], Times[Optional[Pattern[e, Blank[]]], Pattern[x, Blank[]]]]], Optional[Pattern[g, Blank[]]]], Pattern[f, Blank[]]], Optional[Pattern[n, Blank[]]]], Power[Pattern[F, Blank[]], Times[Optional[Pattern[c, Blank[]]], Plus[Optional[Pattern[a, Blank[]]], Times[Optional[Pattern[b, Blank[]]], Pattern[x, Blank[]]]]]], Power[Sin[Plus[Optional[Pattern[d, Blank[]]], Times[Optional[Pattern[e, Blank[]]], Pattern[x, Blank[]]]]], Optional[Pattern[m, Blank[]]]]], Pattern[x, Blank[Symbol]]] := Dist[f^n, Int[F^(c*(a + b*x))*Cot[d/2 + (e*x)/2]^m, x], x] /; FreeQ[{F, a, b, c, d, e, f, g}, x] && EqQ[f + g, 0] && IntegersQ[m, n] && EqQ[m + n, 0]
  4462. Int[Times[Power[Pattern[F, Blank[]], Times[Optional[Pattern[c, Blank[]]], Plus[Optional[Pattern[a, Blank[]]], Times[Optional[Pattern[b, Blank[]]], Pattern[x, Blank[]]]]]], Plus[Times[Cos[Plus[Optional[Pattern[d, Blank[]]], Times[Optional[Pattern[e, Blank[]]], Pattern[x, Blank[]]]]], Optional[Pattern[i, Blank[]]]], Pattern[h, Blank[]]], Power[Plus[Pattern[f, Blank[]], Times[Optional[Pattern[g, Blank[]]], Sin[Plus[Optional[Pattern[d, Blank[]]], Times[Optional[Pattern[e, Blank[]]], Pattern[x, Blank[]]]]]]], -1]], Pattern[x, Blank[Symbol]]] := Dist[2*i, Int[F^(c*(a + b*x))*(Cos[d + e*x]/(f + g*Sin[d + e*x])), x], x] + Int[F^(c*(a + b*x))*((h - i*Cos[d + e*x])/(f + g*Sin[d + e*x])), x] /; FreeQ[{F, a, b, c, d, e, f, g, h, i}, x] && EqQ[f^2 - g^2, 0] && EqQ[h^2 - i^2, 0] && EqQ[g*h - f*i, 0]
  4463. Int[Times[Power[Plus[Times[Cos[Plus[Optional[Pattern[d, Blank[]]], Times[Optional[Pattern[e, Blank[]]], Pattern[x, Blank[]]]]], Optional[Pattern[g, Blank[]]]], Pattern[f, Blank[]]], -1], Power[Pattern[F, Blank[]], Times[Optional[Pattern[c, Blank[]]], Plus[Optional[Pattern[a, Blank[]]], Times[Optional[Pattern[b, Blank[]]], Pattern[x, Blank[]]]]]], Plus[Pattern[h, Blank[]], Times[Optional[Pattern[i, Blank[]]], Sin[Plus[Optional[Pattern[d, Blank[]]], Times[Optional[Pattern[e, Blank[]]], Pattern[x, Blank[]]]]]]]], Pattern[x, Blank[Symbol]]] := Dist[2*i, Int[F^(c*(a + b*x))*(Sin[d + e*x]/(f + g*Cos[d + e*x])), x], x] + Int[F^(c*(a + b*x))*((h - i*Sin[d + e*x])/(f + g*Cos[d + e*x])), x] /; FreeQ[{F, a, b, c, d, e, f, g, h, i}, x] && EqQ[f^2 - g^2, 0] && EqQ[h^2 - i^2, 0] && EqQ[g*h + f*i, 0]
  4464. Int[Times[Power[Pattern[F, Blank[]], Times[Optional[Pattern[c, Blank[]]], Pattern[u, Blank[]]]], Power[Pattern[G, Blank[]][Pattern[v, Blank[]]], Optional[Pattern[n, Blank[]]]]], Pattern[x, Blank[Symbol]]] := Int[F^(c*ExpandToSum[u, x])*G[ExpandToSum[v, x]]^n, x] /; FreeQ[{F, c, n}, x] && TrigQ[G] && LinearQ[{u, v}, x] && !LinearMatchQ[{u, v}, x]
  4465. Int[Times[Power[Pattern[F, Blank[]], Times[Optional[Pattern[c, Blank[]]], Plus[Optional[Pattern[a, Blank[]]], Times[Optional[Pattern[b, Blank[]]], Pattern[x, Blank[]]]]]], Power[Times[Optional[Pattern[f, Blank[]]], Pattern[x, Blank[]]], Optional[Pattern[m, Blank[]]]], Power[Sin[Plus[Optional[Pattern[d, Blank[]]], Times[Optional[Pattern[e, Blank[]]], Pattern[x, Blank[]]]]], Optional[Pattern[n, Blank[]]]]], Pattern[x, Blank[Symbol]]] := Module[{u = IntHide[F^(c*(a + b*x))*Sin[d + e*x]^n, x]}, Dist[(f*x)^m, u, x] - Dist[f*m, Int[(f*x)^(m - 1)*u, x], x]] /; FreeQ[{F, a, b, c, d, e, f}, x] && IGtQ[n, 0] && GtQ[m, 0]
  4466. Int[Times[Power[Cos[Plus[Optional[Pattern[d, Blank[]]], Times[Optional[Pattern[e, Blank[]]], Pattern[x, Blank[]]]]], Optional[Pattern[n, Blank[]]]], Power[Pattern[F, Blank[]], Times[Optional[Pattern[c, Blank[]]], Plus[Optional[Pattern[a, Blank[]]], Times[Optional[Pattern[b, Blank[]]], Pattern[x, Blank[]]]]]], Power[Times[Optional[Pattern[f, Blank[]]], Pattern[x, Blank[]]], Optional[Pattern[m, Blank[]]]]], Pattern[x, Blank[Symbol]]] := Module[{u = IntHide[F^(c*(a + b*x))*Cos[d + e*x]^n, x]}, Dist[(f*x)^m, u, x] - Dist[f*m, Int[(f*x)^(m - 1)*u, x], x]] /; FreeQ[{F, a, b, c, d, e, f}, x] && IGtQ[n, 0] && GtQ[m, 0]
  4467. Int[Times[Power[Pattern[F, Blank[]], Times[Optional[Pattern[c, Blank[]]], Plus[Optional[Pattern[a, Blank[]]], Times[Optional[Pattern[b, Blank[]]], Pattern[x, Blank[]]]]]], Power[Times[Optional[Pattern[f, Blank[]]], Pattern[x, Blank[]]], Pattern[m, Blank[]]], Sin[Plus[Optional[Pattern[d, Blank[]]], Times[Optional[Pattern[e, Blank[]]], Pattern[x, Blank[]]]]]], Pattern[x, Blank[Symbol]]] := Simp[((f*x)^(m + 1)*F^(c*(a + b*x))*Sin[d + e*x])/(f*(m + 1)), x] + (-Dist[e/(f*(m + 1)), Int[(f*x)^(m + 1)*F^(c*(a + b*x))*Cos[d + e*x], x], x] - Dist[(b*c*Log[F])/(f*(m + 1)), Int[(f*x)^(m + 1)*F^(c*(a + b*x))*Sin[d + e*x], x], x]) /; FreeQ[{F, a, b, c, d, e, f, m}, x] && (LtQ[m, -1] || SumSimplerQ[m, 1])
  4468. Int[Times[Cos[Plus[Optional[Pattern[d, Blank[]]], Times[Optional[Pattern[e, Blank[]]], Pattern[x, Blank[]]]]], Power[Pattern[F, Blank[]], Times[Optional[Pattern[c, Blank[]]], Plus[Optional[Pattern[a, Blank[]]], Times[Optional[Pattern[b, Blank[]]], Pattern[x, Blank[]]]]]], Power[Times[Optional[Pattern[f, Blank[]]], Pattern[x, Blank[]]], Pattern[m, Blank[]]]], Pattern[x, Blank[Symbol]]] := Simp[((f*x)^(m + 1)*F^(c*(a + b*x))*Cos[d + e*x])/(f*(m + 1)), x] + (Dist[e/(f*(m + 1)), Int[(f*x)^(m + 1)*F^(c*(a + b*x))*Sin[d + e*x], x], x] - Dist[(b*c*Log[F])/(f*(m + 1)), Int[(f*x)^(m + 1)*F^(c*(a + b*x))*Cos[d + e*x], x], x]) /; FreeQ[{F, a, b, c, d, e, f, m}, x] && (LtQ[m, -1] || SumSimplerQ[m, 1])
  4469. Int[Times[Power[Cos[Plus[Optional[Pattern[f, Blank[]]], Times[Optional[Pattern[g, Blank[]]], Pattern[x, Blank[]]]]], Optional[Pattern[n, Blank[]]]], Power[Pattern[F, Blank[]], Times[Optional[Pattern[c, Blank[]]], Plus[Optional[Pattern[a, Blank[]]], Times[Optional[Pattern[b, Blank[]]], Pattern[x, Blank[]]]]]], Power[Sin[Plus[Optional[Pattern[d, Blank[]]], Times[Optional[Pattern[e, Blank[]]], Pattern[x, Blank[]]]]], Optional[Pattern[m, Blank[]]]]], Pattern[x, Blank[Symbol]]] := Int[ExpandTrigReduce[F^(c*(a + b*x)), Sin[d + e*x]^m*Cos[f + g*x]^n, x], x] /; FreeQ[{F, a, b, c, d, e, f, g}, x] && IGtQ[m, 0] && IGtQ[n, 0]
  4470. Int[Times[Power[Cos[Plus[Optional[Pattern[f, Blank[]]], Times[Optional[Pattern[g, Blank[]]], Pattern[x, Blank[]]]]], Optional[Pattern[n, Blank[]]]], Power[Pattern[F, Blank[]], Times[Optional[Pattern[c, Blank[]]], Plus[Optional[Pattern[a, Blank[]]], Times[Optional[Pattern[b, Blank[]]], Pattern[x, Blank[]]]]]], Power[Pattern[x, Blank[]], Optional[Pattern[p, Blank[]]]], Power[Sin[Plus[Optional[Pattern[d, Blank[]]], Times[Optional[Pattern[e, Blank[]]], Pattern[x, Blank[]]]]], Optional[Pattern[m, Blank[]]]]], Pattern[x, Blank[Symbol]]] := Int[ExpandTrigReduce[x^p*F^(c*(a + b*x)), Sin[d + e*x]^m*Cos[f + g*x]^n, x], x] /; FreeQ[{F, a, b, c, d, e, f, g}, x] && IGtQ[m, 0] && IGtQ[n, 0] && IGtQ[p, 0]
  4471. Int[Times[Power[Pattern[F, Blank[]], Times[Optional[Pattern[c, Blank[]]], Plus[Optional[Pattern[a, Blank[]]], Times[Optional[Pattern[b, Blank[]]], Pattern[x, Blank[]]]]]], Power[Pattern[G, Blank[]][Plus[Optional[Pattern[d, Blank[]]], Times[Optional[Pattern[e, Blank[]]], Pattern[x, Blank[]]]]], Optional[Pattern[m, Blank[]]]], Power[Pattern[H, Blank[]][Plus[Optional[Pattern[d, Blank[]]], Times[Optional[Pattern[e, Blank[]]], Pattern[x, Blank[]]]]], Optional[Pattern[n, Blank[]]]]], Pattern[x, Blank[Symbol]]] := Int[ExpandTrigToExp[F^(c*(a + b*x)), G[d + e*x]^m*H[d + e*x]^n, x], x] /; FreeQ[{F, a, b, c, d, e}, x] && IGtQ[m, 0] && IGtQ[n, 0] && TrigQ[G] && TrigQ[H]
  4472. Int[Times[Power[Pattern[F, Blank[]], Pattern[u, Blank[]]], Power[Sin[Pattern[v, Blank[]]], Optional[Pattern[n, Blank[]]]]], Pattern[x, Blank[Symbol]]] := Int[ExpandTrigToExp[F^u, Sin[v]^n, x], x] /; FreeQ[F, x] && (LinearQ[u, x] || PolyQ[u, x, 2]) && (LinearQ[v, x] || PolyQ[v, x, 2]) && IGtQ[n, 0]
  4473. Int[Times[Power[Cos[Pattern[v, Blank[]]], Optional[Pattern[n, Blank[]]]], Power[Pattern[F, Blank[]], Pattern[u, Blank[]]]], Pattern[x, Blank[Symbol]]] := Int[ExpandTrigToExp[F^u, Cos[v]^n, x], x] /; FreeQ[F, x] && (LinearQ[u, x] || PolyQ[u, x, 2]) && (LinearQ[v, x] || PolyQ[v, x, 2]) && IGtQ[n, 0]
  4474. Int[Times[Power[Cos[Pattern[v, Blank[]]], Optional[Pattern[n, Blank[]]]], Power[Pattern[F, Blank[]], Pattern[u, Blank[]]], Power[Sin[Pattern[v, Blank[]]], Optional[Pattern[m, Blank[]]]]], Pattern[x, Blank[Symbol]]] := Int[ExpandTrigToExp[F^u, Sin[v]^m*Cos[v]^n, x], x] /; FreeQ[F, x] && (LinearQ[u, x] || PolyQ[u, x, 2]) && (LinearQ[v, x] || PolyQ[v, x, 2]) && IGtQ[m, 0] && IGtQ[n, 0]
  4475. Int[Sin[Times[Plus[Optional[Pattern[a, Blank[]]], Times[Log[Times[Optional[Pattern[c, Blank[]]], Power[Pattern[x, Blank[]], Optional[Pattern[n, Blank[]]]]]], Optional[Pattern[b, Blank[]]]]], Optional[Pattern[d, Blank[]]]]], Pattern[x, Blank[Symbol]]] := Simp[(x*Sin[d*(a + b*Log[c*x^n])])/(b^2*d^2*n^2 + 1), x] - Simp[(b*d*n*x*Cos[d*(a + b*Log[c*x^n])])/(b^2*d^2*n^2 + 1), x] /; FreeQ[{a, b, c, d, n}, x] && NeQ[b^2*d^2*n^2 + 1, 0]
  4476. Int[Cos[Times[Plus[Optional[Pattern[a, Blank[]]], Times[Log[Times[Optional[Pattern[c, Blank[]]], Power[Pattern[x, Blank[]], Optional[Pattern[n, Blank[]]]]]], Optional[Pattern[b, Blank[]]]]], Optional[Pattern[d, Blank[]]]]], Pattern[x, Blank[Symbol]]] := Simp[(x*Cos[d*(a + b*Log[c*x^n])])/(b^2*d^2*n^2 + 1), x] + Simp[(b*d*n*x*Sin[d*(a + b*Log[c*x^n])])/(b^2*d^2*n^2 + 1), x] /; FreeQ[{a, b, c, d, n}, x] && NeQ[b^2*d^2*n^2 + 1, 0]
  4477. Int[Power[Sin[Times[Plus[Optional[Pattern[a, Blank[]]], Times[Log[Times[Optional[Pattern[c, Blank[]]], Power[Pattern[x, Blank[]], Optional[Pattern[n, Blank[]]]]]], Optional[Pattern[b, Blank[]]]]], Optional[Pattern[d, Blank[]]]]], Pattern[p, Blank[]]], Pattern[x, Blank[Symbol]]] := Simp[(x*Sin[d*(a + b*Log[c*x^n])]^p)/(b^2*d^2*n^2*p^2 + 1), x] + (Dist[(b^2*d^2*n^2*p*(p - 1))/(b^2*d^2*n^2*p^2 + 1), Int[Sin[d*(a + b*Log[c*x^n])]^(p - 2), x], x] - Simp[(b*d*n*p*x*Cos[d*(a + b*Log[c*x^n])]*Sin[d*(a + b*Log[c*x^n])]^(p - 1))/(b^2*d^2*n^2*p^2 + 1), x]) /; FreeQ[{a, b, c, d, n}, x] && IGtQ[p, 1] && NeQ[b^2*d^2*n^2*p^2 + 1, 0]
  4478. Int[Power[Cos[Times[Plus[Optional[Pattern[a, Blank[]]], Times[Log[Times[Optional[Pattern[c, Blank[]]], Power[Pattern[x, Blank[]], Optional[Pattern[n, Blank[]]]]]], Optional[Pattern[b, Blank[]]]]], Optional[Pattern[d, Blank[]]]]], Pattern[p, Blank[]]], Pattern[x, Blank[Symbol]]] := Simp[(x*Cos[d*(a + b*Log[c*x^n])]^p)/(b^2*d^2*n^2*p^2 + 1), x] + (Dist[(b^2*d^2*n^2*p*(p - 1))/(b^2*d^2*n^2*p^2 + 1), Int[Cos[d*(a + b*Log[c*x^n])]^(p - 2), x], x] + Simp[(b*d*n*p*x*Cos[d*(a + b*Log[c*x^n])]^(p - 1)*Sin[d*(a + b*Log[c*x^n])])/(b^2*d^2*n^2*p^2 + 1), x]) /; FreeQ[{a, b, c, d, n}, x] && IGtQ[p, 1] && NeQ[b^2*d^2*n^2*p^2 + 1, 0]
  4479. Int[Power[Sin[Times[Plus[Optional[Pattern[a, Blank[]]], Times[Log[Pattern[x, Blank[]]], Optional[Pattern[b, Blank[]]]]], Optional[Pattern[d, Blank[]]]]], Optional[Pattern[p, Blank[]]]], Pattern[x, Blank[Symbol]]] := Dist[1/(2^p*b^p*d^p*p^p), Int[ExpandIntegrand[(E^(a*b*d^2*p)/x^p^(-1) - x^(1/p)/E^(a*b*d^2*p))^p, x], x], x] /; FreeQ[{a, b, d}, x] && IGtQ[p, 0] && EqQ[b^2*d^2*p^2 + 1, 0]
  4480. Int[Power[Cos[Times[Plus[Optional[Pattern[a, Blank[]]], Times[Log[Pattern[x, Blank[]]], Optional[Pattern[b, Blank[]]]]], Optional[Pattern[d, Blank[]]]]], Optional[Pattern[p, Blank[]]]], Pattern[x, Blank[Symbol]]] := Dist[1/2^p, Int[ExpandIntegrand[(E^(a*b*d^2*p)/x^p^(-1) + x^(1/p)/E^(a*b*d^2*p))^p, x], x], x] /; FreeQ[{a, b, d}, x] && IGtQ[p, 0] && EqQ[b^2*d^2*p^2 + 1, 0]
  4481. Int[Power[Sin[Times[Plus[Optional[Pattern[a, Blank[]]], Times[Log[Pattern[x, Blank[]]], Optional[Pattern[b, Blank[]]]]], Optional[Pattern[d, Blank[]]]]], Pattern[p, Blank[]]], Pattern[x, Blank[Symbol]]] := Dist[(Sin[d*(a + b*Log[x])]^p*x^(I*b*d*p))/(1 - E^(2*I*a*d)*x^(2*I*b*d))^p, Int[(1 - E^(2*I*a*d)*x^(2*I*b*d))^p/x^(I*b*d*p), x], x] /; FreeQ[{a, b, d, p}, x] && !IntegerQ[p]
  4482. Int[Power[Cos[Times[Plus[Optional[Pattern[a, Blank[]]], Times[Log[Pattern[x, Blank[]]], Optional[Pattern[b, Blank[]]]]], Optional[Pattern[d, Blank[]]]]], Pattern[p, Blank[]]], Pattern[x, Blank[Symbol]]] := Dist[(Cos[d*(a + b*Log[x])]^p*x^(I*b*d*p))/(1 + E^(2*I*a*d)*x^(2*I*b*d))^p, Int[(1 + E^(2*I*a*d)*x^(2*I*b*d))^p/x^(I*b*d*p), x], x] /; FreeQ[{a, b, d, p}, x] && !IntegerQ[p]
  4483. Int[Power[Sin[Times[Plus[Optional[Pattern[a, Blank[]]], Times[Log[Times[Optional[Pattern[c, Blank[]]], Power[Pattern[x, Blank[]], Optional[Pattern[n, Blank[]]]]]], Optional[Pattern[b, Blank[]]]]], Optional[Pattern[d, Blank[]]]]], Optional[Pattern[p, Blank[]]]], Pattern[x, Blank[Symbol]]] := Dist[x/(n*(c*x^n)^(1/n)), Subst[Int[x^(1/n - 1)*Sin[d*(a + b*Log[x])]^p, x], x, c*x^n], x] /; FreeQ[{a, b, c, d, n, p}, x] && (NeQ[c, 1] || NeQ[n, 1])
  4484. Int[Power[Cos[Times[Plus[Optional[Pattern[a, Blank[]]], Times[Log[Times[Optional[Pattern[c, Blank[]]], Power[Pattern[x, Blank[]], Optional[Pattern[n, Blank[]]]]]], Optional[Pattern[b, Blank[]]]]], Optional[Pattern[d, Blank[]]]]], Optional[Pattern[p, Blank[]]]], Pattern[x, Blank[Symbol]]] := Dist[x/(n*(c*x^n)^(1/n)), Subst[Int[x^(1/n - 1)*Cos[d*(a + b*Log[x])]^p, x], x, c*x^n], x] /; FreeQ[{a, b, c, d, n, p}, x] && (NeQ[c, 1] || NeQ[n, 1])
  4485. Int[Times[Power[Times[Optional[Pattern[e, Blank[]]], Pattern[x, Blank[]]], Optional[Pattern[m, Blank[]]]], Sin[Times[Plus[Optional[Pattern[a, Blank[]]], Times[Log[Times[Optional[Pattern[c, Blank[]]], Power[Pattern[x, Blank[]], Optional[Pattern[n, Blank[]]]]]], Optional[Pattern[b, Blank[]]]]], Optional[Pattern[d, Blank[]]]]]], Pattern[x, Blank[Symbol]]] := Simp[((m + 1)*(e*x)^(m + 1)*Sin[d*(a + b*Log[c*x^n])])/(b^2*d^2*e*n^2 + e*(m + 1)^2), x] - Simp[(b*d*n*(e*x)^(m + 1)*Cos[d*(a + b*Log[c*x^n])])/(b^2*d^2*e*n^2 + e*(m + 1)^2), x] /; FreeQ[{a, b, c, d, e, m, n}, x] && NeQ[b^2*d^2*n^2 + (m + 1)^2, 0]
  4486. Int[Times[Cos[Times[Plus[Optional[Pattern[a, Blank[]]], Times[Log[Times[Optional[Pattern[c, Blank[]]], Power[Pattern[x, Blank[]], Optional[Pattern[n, Blank[]]]]]], Optional[Pattern[b, Blank[]]]]], Optional[Pattern[d, Blank[]]]]], Power[Times[Optional[Pattern[e, Blank[]]], Pattern[x, Blank[]]], Optional[Pattern[m, Blank[]]]]], Pattern[x, Blank[Symbol]]] := Simp[((m + 1)*(e*x)^(m + 1)*Cos[d*(a + b*Log[c*x^n])])/(b^2*d^2*e*n^2 + e*(m + 1)^2), x] + Simp[(b*d*n*(e*x)^(m + 1)*Sin[d*(a + b*Log[c*x^n])])/(b^2*d^2*e*n^2 + e*(m + 1)^2), x] /; FreeQ[{a, b, c, d, e, m, n}, x] && NeQ[b^2*d^2*n^2 + (m + 1)^2, 0]
  4487. Int[Times[Power[Times[Optional[Pattern[e, Blank[]]], Pattern[x, Blank[]]], Optional[Pattern[m, Blank[]]]], Power[Sin[Times[Plus[Optional[Pattern[a, Blank[]]], Times[Log[Times[Optional[Pattern[c, Blank[]]], Power[Pattern[x, Blank[]], Optional[Pattern[n, Blank[]]]]]], Optional[Pattern[b, Blank[]]]]], Optional[Pattern[d, Blank[]]]]], Pattern[p, Blank[]]]], Pattern[x, Blank[Symbol]]] := Simp[((m + 1)*(e*x)^(m + 1)*Sin[d*(a + b*Log[c*x^n])]^p)/(b^2*d^2*e*n^2*p^2 + e*(m + 1)^2), x] + (Dist[(b^2*d^2*n^2*p*(p - 1))/(b^2*d^2*n^2*p^2 + (m + 1)^2), Int[(e*x)^m*Sin[d*(a + b*Log[c*x^n])]^(p - 2), x], x] - Simp[(b*d*n*p*(e*x)^(m + 1)*Cos[d*(a + b*Log[c*x^n])]*Sin[d*(a + b*Log[c*x^n])]^(p - 1))/(b^2*d^2*e*n^2*p^2 + e*(m + 1)^2), x]) /; FreeQ[{a, b, c, d, e, m, n}, x] && IGtQ[p, 1] && NeQ[b^2*d^2*n^2*p^2 + (m + 1)^2, 0]
  4488. Int[Times[Power[Cos[Times[Plus[Optional[Pattern[a, Blank[]]], Times[Log[Times[Optional[Pattern[c, Blank[]]], Power[Pattern[x, Blank[]], Optional[Pattern[n, Blank[]]]]]], Optional[Pattern[b, Blank[]]]]], Optional[Pattern[d, Blank[]]]]], Pattern[p, Blank[]]], Power[Times[Optional[Pattern[e, Blank[]]], Pattern[x, Blank[]]], Optional[Pattern[m, Blank[]]]]], Pattern[x, Blank[Symbol]]] := Simp[((m + 1)*(e*x)^(m + 1)*Cos[d*(a + b*Log[c*x^n])]^p)/(b^2*d^2*e*n^2*p^2 + e*(m + 1)^2), x] + (Dist[(b^2*d^2*n^2*p*(p - 1))/(b^2*d^2*n^2*p^2 + (m + 1)^2), Int[(e*x)^m*Cos[d*(a + b*Log[c*x^n])]^(p - 2), x], x] + Simp[(b*d*n*p*(e*x)^(m + 1)*Sin[d*(a + b*Log[c*x^n])]*Cos[d*(a + b*Log[c*x^n])]^(p - 1))/(b^2*d^2*e*n^2*p^2 + e*(m + 1)^2), x]) /; FreeQ[{a, b, c, d, e, m, n}, x] && IGtQ[p, 1] && NeQ[b^2*d^2*n^2*p^2 + (m + 1)^2, 0]
  4489. Int[Times[Power[Times[Optional[Pattern[e, Blank[]]], Pattern[x, Blank[]]], Optional[Pattern[m, Blank[]]]], Power[Sin[Times[Plus[Optional[Pattern[a, Blank[]]], Times[Log[Pattern[x, Blank[]]], Optional[Pattern[b, Blank[]]]]], Optional[Pattern[d, Blank[]]]]], Optional[Pattern[p, Blank[]]]]], Pattern[x, Blank[Symbol]]] := Dist[(m + 1)^p/(2^p*b^p*d^p*p^p), Int[ExpandIntegrand[(e*x)^m*(E^((a*b*d^2*p)/(m + 1))/x^((m + 1)/p) - x^((m + 1)/p)/E^((a*b*d^2*p)/(m + 1)))^p, x], x], x] /; FreeQ[{a, b, d, e, m}, x] && IGtQ[p, 0] && EqQ[b^2*d^2*p^2 + (m + 1)^2, 0]
  4490. Int[Times[Power[Cos[Times[Plus[Optional[Pattern[a, Blank[]]], Times[Log[Pattern[x, Blank[]]], Optional[Pattern[b, Blank[]]]]], Optional[Pattern[d, Blank[]]]]], Optional[Pattern[p, Blank[]]]], Power[Times[Optional[Pattern[e, Blank[]]], Pattern[x, Blank[]]], Optional[Pattern[m, Blank[]]]]], Pattern[x, Blank[Symbol]]] := Dist[1/2^p, Int[ExpandIntegrand[(e*x)^m*(E^((a*b*d^2*p)/(m + 1))/x^((m + 1)/p) + x^((m + 1)/p)/E^((a*b*d^2*p)/(m + 1)))^p, x], x], x] /; FreeQ[{a, b, d, e, m}, x] && IGtQ[p, 0] && EqQ[b^2*d^2*p^2 + (m + 1)^2, 0]
  4491. Int[Times[Power[Times[Optional[Pattern[e, Blank[]]], Pattern[x, Blank[]]], Optional[Pattern[m, Blank[]]]], Power[Sin[Times[Plus[Optional[Pattern[a, Blank[]]], Times[Log[Pattern[x, Blank[]]], Optional[Pattern[b, Blank[]]]]], Optional[Pattern[d, Blank[]]]]], Pattern[p, Blank[]]]], Pattern[x, Blank[Symbol]]] := Dist[(Sin[d*(a + b*Log[x])]^p*x^(I*b*d*p))/(1 - E^(2*I*a*d)*x^(2*I*b*d))^p, Int[((e*x)^m*(1 - E^(2*I*a*d)*x^(2*I*b*d))^p)/x^(I*b*d*p), x], x] /; FreeQ[{a, b, d, e, m, p}, x] && !IntegerQ[p]
  4492. Int[Times[Power[Cos[Times[Plus[Optional[Pattern[a, Blank[]]], Times[Log[Pattern[x, Blank[]]], Optional[Pattern[b, Blank[]]]]], Optional[Pattern[d, Blank[]]]]], Pattern[p, Blank[]]], Power[Times[Optional[Pattern[e, Blank[]]], Pattern[x, Blank[]]], Optional[Pattern[m, Blank[]]]]], Pattern[x, Blank[Symbol]]] := Dist[(Cos[d*(a + b*Log[x])]^p*x^(I*b*d*p))/(1 + E^(2*I*a*d)*x^(2*I*b*d))^p, Int[((e*x)^m*(1 + E^(2*I*a*d)*x^(2*I*b*d))^p)/x^(I*b*d*p), x], x] /; FreeQ[{a, b, d, e, m, p}, x] && !IntegerQ[p]
  4493. Int[Times[Power[Times[Optional[Pattern[e, Blank[]]], Pattern[x, Blank[]]], Optional[Pattern[m, Blank[]]]], Power[Sin[Times[Plus[Optional[Pattern[a, Blank[]]], Times[Log[Times[Optional[Pattern[c, Blank[]]], Power[Pattern[x, Blank[]], Optional[Pattern[n, Blank[]]]]]], Optional[Pattern[b, Blank[]]]]], Optional[Pattern[d, Blank[]]]]], Optional[Pattern[p, Blank[]]]]], Pattern[x, Blank[Symbol]]] := Dist[(e*x)^(m + 1)/(e*n*(c*x^n)^((m + 1)/n)), Subst[Int[x^((m + 1)/n - 1)*Sin[d*(a + b*Log[x])]^p, x], x, c*x^n], x] /; FreeQ[{a, b, c, d, e, m, n, p}, x] && (NeQ[c, 1] || NeQ[n, 1])
  4494. Int[Times[Power[Cos[Times[Plus[Optional[Pattern[a, Blank[]]], Times[Log[Times[Optional[Pattern[c, Blank[]]], Power[Pattern[x, Blank[]], Optional[Pattern[n, Blank[]]]]]], Optional[Pattern[b, Blank[]]]]], Optional[Pattern[d, Blank[]]]]], Optional[Pattern[p, Blank[]]]], Power[Times[Optional[Pattern[e, Blank[]]], Pattern[x, Blank[]]], Optional[Pattern[m, Blank[]]]]], Pattern[x, Blank[Symbol]]] := Dist[(e*x)^(m + 1)/(e*n*(c*x^n)^((m + 1)/n)), Subst[Int[x^((m + 1)/n - 1)*Cos[d*(a + b*Log[x])]^p, x], x, c*x^n], x] /; FreeQ[{a, b, c, d, e, m, n, p}, x] && (NeQ[c, 1] || NeQ[n, 1])
  4495. Int[Times[Power[Times[Plus[Optional[Pattern[e, Blank[]]], Times[Log[Times[Optional[Pattern[g, Blank[]]], Power[Pattern[x, Blank[]], Optional[Pattern[m, Blank[]]]]]], Optional[Pattern[f, Blank[]]]]], Optional[Pattern[h, Blank[]]]], Optional[Pattern[q, Blank[]]]], Sin[Times[Plus[Optional[Pattern[a, Blank[]]], Times[Log[Times[Optional[Pattern[c, Blank[]]], Power[Pattern[x, Blank[]], Optional[Pattern[n, Blank[]]]]]], Optional[Pattern[b, Blank[]]]]], Optional[Pattern[d, Blank[]]]]]], Pattern[x, Blank[Symbol]]] := Dist[I/(E^(I*a*d)*(c*x^n)^(I*b*d)*(2/x^(I*b*d*n))), Int[(h*(e + f*Log[g*x^m]))^q/x^(I*b*d*n), x], x] - Dist[(I*E^(I*a*d)*(c*x^n)^(I*b*d))/(2*x^(I*b*d*n)), Int[x^(I*b*d*n)*(h*(e + f*Log[g*x^m]))^q, x], x] /; FreeQ[{a, b, c, d, e, f, g, h, m, n, q}, x]
  4496. Int[Times[Cos[Times[Plus[Optional[Pattern[a, Blank[]]], Times[Log[Times[Optional[Pattern[c, Blank[]]], Power[Pattern[x, Blank[]], Optional[Pattern[n, Blank[]]]]]], Optional[Pattern[b, Blank[]]]]], Optional[Pattern[d, Blank[]]]]], Power[Times[Plus[Optional[Pattern[e, Blank[]]], Times[Log[Times[Optional[Pattern[g, Blank[]]], Power[Pattern[x, Blank[]], Optional[Pattern[m, Blank[]]]]]], Optional[Pattern[f, Blank[]]]]], Optional[Pattern[h, Blank[]]]], Optional[Pattern[q, Blank[]]]]], Pattern[x, Blank[Symbol]]] := Dist[1/(E^(I*a*d)*(c*x^n)^(I*b*d)*(2/x^(I*b*d*n))), Int[(h*(e + f*Log[g*x^m]))^q/x^(I*b*d*n), x], x] + Dist[(E^(I*a*d)*(c*x^n)^(I*b*d))/(2*x^(I*b*d*n)), Int[x^(I*b*d*n)*(h*(e + f*Log[g*x^m]))^q, x], x] /; FreeQ[{a, b, c, d, e, f, g, h, m, n, q}, x]
  4497. Int[Times[Power[Times[Plus[Optional[Pattern[e, Blank[]]], Times[Log[Times[Optional[Pattern[g, Blank[]]], Power[Pattern[x, Blank[]], Optional[Pattern[m, Blank[]]]]]], Optional[Pattern[f, Blank[]]]]], Optional[Pattern[h, Blank[]]]], Optional[Pattern[q, Blank[]]]], Power[Times[Optional[Pattern[i, Blank[]]], Pattern[x, Blank[]]], Optional[Pattern[r, Blank[]]]], Sin[Times[Plus[Optional[Pattern[a, Blank[]]], Times[Log[Times[Optional[Pattern[c, Blank[]]], Power[Pattern[x, Blank[]], Optional[Pattern[n, Blank[]]]]]], Optional[Pattern[b, Blank[]]]]], Optional[Pattern[d, Blank[]]]]]], Pattern[x, Blank[Symbol]]] := Dist[(I*(i*x)^r)/(E^(I*a*d)*(c*x^n)^(I*b*d)*(2*x^(r - I*b*d*n))), Int[x^(r - I*b*d*n)*(h*(e + f*Log[g*x^m]))^q, x], x] - Dist[(I*E^(I*a*d)*(i*x)^r*(c*x^n)^(I*b*d))/(2*x^(r + I*b*d*n)), Int[x^(r + I*b*d*n)*(h*(e + f*Log[g*x^m]))^q, x], x] /; FreeQ[{a, b, c, d, e, f, g, h, i, m, n, q, r}, x]
  4498. Int[Times[Cos[Times[Plus[Optional[Pattern[a, Blank[]]], Times[Log[Times[Optional[Pattern[c, Blank[]]], Power[Pattern[x, Blank[]], Optional[Pattern[n, Blank[]]]]]], Optional[Pattern[b, Blank[]]]]], Optional[Pattern[d, Blank[]]]]], Power[Times[Plus[Optional[Pattern[e, Blank[]]], Times[Log[Times[Optional[Pattern[g, Blank[]]], Power[Pattern[x, Blank[]], Optional[Pattern[m, Blank[]]]]]], Optional[Pattern[f, Blank[]]]]], Optional[Pattern[h, Blank[]]]], Optional[Pattern[q, Blank[]]]], Power[Times[Optional[Pattern[i, Blank[]]], Pattern[x, Blank[]]], Optional[Pattern[r, Blank[]]]]], Pattern[x, Blank[Symbol]]] := Dist[(i*x)^r/(E^(I*a*d)*(c*x^n)^(I*b*d)*(2*x^(r - I*b*d*n))), Int[x^(r - I*b*d*n)*(h*(e + f*Log[g*x^m]))^q, x], x] + Dist[(E^(I*a*d)*(i*x)^r*(c*x^n)^(I*b*d))/(2*x^(r + I*b*d*n)), Int[x^(r + I*b*d*n)*(h*(e + f*Log[g*x^m]))^q, x], x] /; FreeQ[{a, b, c, d, e, f, g, h, i, m, n, q, r}, x]
  4499. Int[Power[Sec[Times[Plus[Optional[Pattern[a, Blank[]]], Times[Log[Pattern[x, Blank[]]], Optional[Pattern[b, Blank[]]]]], Optional[Pattern[d, Blank[]]]]], Optional[Pattern[p, Blank[]]]], Pattern[x, Blank[Symbol]]] := Dist[2^p*E^(I*a*d*p), Int[x^(I*b*d*p)/(1 + E^(2*I*a*d)*x^(2*I*b*d))^p, x], x] /; FreeQ[{a, b, d}, x] && IntegerQ[p]
  4500. Int[Power[Csc[Times[Plus[Optional[Pattern[a, Blank[]]], Times[Log[Pattern[x, Blank[]]], Optional[Pattern[b, Blank[]]]]], Optional[Pattern[d, Blank[]]]]], Optional[Pattern[p, Blank[]]]], Pattern[x, Blank[Symbol]]] := Dist[(-2*I)^p*E^(I*a*d*p), Int[x^(I*b*d*p)/(1 - E^(2*I*a*d)*x^(2*I*b*d))^p, x], x] /; FreeQ[{a, b, d}, x] && IntegerQ[p]
  4501. Int[Power[Sec[Times[Plus[Optional[Pattern[a, Blank[]]], Times[Log[Pattern[x, Blank[]]], Optional[Pattern[b, Blank[]]]]], Optional[Pattern[d, Blank[]]]]], Optional[Pattern[p, Blank[]]]], Pattern[x, Blank[Symbol]]] := Dist[(Sec[d*(a + b*Log[x])]^p*(1 + E^(2*I*a*d)*x^(2*I*b*d))^p)/x^(I*b*d*p), Int[x^(I*b*d*p)/(1 + E^(2*I*a*d)*x^(2*I*b*d))^p, x], x] /; FreeQ[{a, b, d, p}, x] && !IntegerQ[p]
  4502. Int[Power[Csc[Times[Plus[Optional[Pattern[a, Blank[]]], Times[Log[Pattern[x, Blank[]]], Optional[Pattern[b, Blank[]]]]], Optional[Pattern[d, Blank[]]]]], Optional[Pattern[p, Blank[]]]], Pattern[x, Blank[Symbol]]] := Dist[(Csc[d*(a + b*Log[x])]^p*(1 - E^(2*I*a*d)*x^(2*I*b*d))^p)/x^(I*b*d*p), Int[x^(I*b*d*p)/(1 - E^(2*I*a*d)*x^(2*I*b*d))^p, x], x] /; FreeQ[{a, b, d, p}, x] && !IntegerQ[p]
  4503. Int[Power[Sec[Times[Plus[Optional[Pattern[a, Blank[]]], Times[Log[Times[Optional[Pattern[c, Blank[]]], Power[Pattern[x, Blank[]], Optional[Pattern[n, Blank[]]]]]], Optional[Pattern[b, Blank[]]]]], Optional[Pattern[d, Blank[]]]]], Optional[Pattern[p, Blank[]]]], Pattern[x, Blank[Symbol]]] := Dist[x/(n*(c*x^n)^(1/n)), Subst[Int[x^(1/n - 1)*Sec[d*(a + b*Log[x])]^p, x], x, c*x^n], x] /; FreeQ[{a, b, c, d, n, p}, x] && (NeQ[c, 1] || NeQ[n, 1])
  4504. Int[Power[Csc[Times[Plus[Optional[Pattern[a, Blank[]]], Times[Log[Times[Optional[Pattern[c, Blank[]]], Power[Pattern[x, Blank[]], Optional[Pattern[n, Blank[]]]]]], Optional[Pattern[b, Blank[]]]]], Optional[Pattern[d, Blank[]]]]], Optional[Pattern[p, Blank[]]]], Pattern[x, Blank[Symbol]]] := Dist[x/(n*(c*x^n)^(1/n)), Subst[Int[x^(1/n - 1)*Csc[d*(a + b*Log[x])]^p, x], x, c*x^n], x] /; FreeQ[{a, b, c, d, n, p}, x] && (NeQ[c, 1] || NeQ[n, 1])
  4505. Int[Times[Power[Times[Optional[Pattern[e, Blank[]]], Pattern[x, Blank[]]], Optional[Pattern[m, Blank[]]]], Power[Sec[Times[Plus[Optional[Pattern[a, Blank[]]], Times[Log[Pattern[x, Blank[]]], Optional[Pattern[b, Blank[]]]]], Optional[Pattern[d, Blank[]]]]], Optional[Pattern[p, Blank[]]]]], Pattern[x, Blank[Symbol]]] := Dist[2^p*E^(I*a*d*p), Int[((e*x)^m*x^(I*b*d*p))/(1 + E^(2*I*a*d)*x^(2*I*b*d))^p, x], x] /; FreeQ[{a, b, d, e, m}, x] && IntegerQ[p]
  4506. Int[Times[Power[Csc[Times[Plus[Optional[Pattern[a, Blank[]]], Times[Log[Pattern[x, Blank[]]], Optional[Pattern[b, Blank[]]]]], Optional[Pattern[d, Blank[]]]]], Optional[Pattern[p, Blank[]]]], Power[Times[Optional[Pattern[e, Blank[]]], Pattern[x, Blank[]]], Optional[Pattern[m, Blank[]]]]], Pattern[x, Blank[Symbol]]] := Dist[(-2*I)^p*E^(I*a*d*p), Int[((e*x)^m*x^(I*b*d*p))/(1 - E^(2*I*a*d)*x^(2*I*b*d))^p, x], x] /; FreeQ[{a, b, d, e, m}, x] && IntegerQ[p]
  4507. Int[Times[Power[Times[Optional[Pattern[e, Blank[]]], Pattern[x, Blank[]]], Optional[Pattern[m, Blank[]]]], Power[Sec[Times[Plus[Optional[Pattern[a, Blank[]]], Times[Log[Pattern[x, Blank[]]], Optional[Pattern[b, Blank[]]]]], Optional[Pattern[d, Blank[]]]]], Optional[Pattern[p, Blank[]]]]], Pattern[x, Blank[Symbol]]] := Dist[(Sec[d*(a + b*Log[x])]^p*(1 + E^(2*I*a*d)*x^(2*I*b*d))^p)/x^(I*b*d*p), Int[((e*x)^m*x^(I*b*d*p))/(1 + E^(2*I*a*d)*x^(2*I*b*d))^p, x], x] /; FreeQ[{a, b, d, e, m, p}, x] && !IntegerQ[p]
  4508. Int[Times[Power[Csc[Times[Plus[Optional[Pattern[a, Blank[]]], Times[Log[Pattern[x, Blank[]]], Optional[Pattern[b, Blank[]]]]], Optional[Pattern[d, Blank[]]]]], Optional[Pattern[p, Blank[]]]], Power[Times[Optional[Pattern[e, Blank[]]], Pattern[x, Blank[]]], Optional[Pattern[m, Blank[]]]]], Pattern[x, Blank[Symbol]]] := Dist[(Csc[d*(a + b*Log[x])]^p*(1 - E^(2*I*a*d)*x^(2*I*b*d))^p)/x^(I*b*d*p), Int[((e*x)^m*x^(I*b*d*p))/(1 - E^(2*I*a*d)*x^(2*I*b*d))^p, x], x] /; FreeQ[{a, b, d, e, m, p}, x] && !IntegerQ[p]
  4509. Int[Times[Power[Times[Optional[Pattern[e, Blank[]]], Pattern[x, Blank[]]], Optional[Pattern[m, Blank[]]]], Power[Sec[Times[Plus[Optional[Pattern[a, Blank[]]], Times[Log[Times[Optional[Pattern[c, Blank[]]], Power[Pattern[x, Blank[]], Optional[Pattern[n, Blank[]]]]]], Optional[Pattern[b, Blank[]]]]], Optional[Pattern[d, Blank[]]]]], Optional[Pattern[p, Blank[]]]]], Pattern[x, Blank[Symbol]]] := Dist[(e*x)^(m + 1)/(e*n*(c*x^n)^((m + 1)/n)), Subst[Int[x^((m + 1)/n - 1)*Sec[d*(a + b*Log[x])]^p, x], x, c*x^n], x] /; FreeQ[{a, b, c, d, e, m, n, p}, x] && (NeQ[c, 1] || NeQ[n, 1])
  4510. Int[Times[Power[Csc[Times[Plus[Optional[Pattern[a, Blank[]]], Times[Log[Times[Optional[Pattern[c, Blank[]]], Power[Pattern[x, Blank[]], Optional[Pattern[n, Blank[]]]]]], Optional[Pattern[b, Blank[]]]]], Optional[Pattern[d, Blank[]]]]], Optional[Pattern[p, Blank[]]]], Power[Times[Optional[Pattern[e, Blank[]]], Pattern[x, Blank[]]], Optional[Pattern[m, Blank[]]]]], Pattern[x, Blank[Symbol]]] := Dist[(e*x)^(m + 1)/(e*n*(c*x^n)^((m + 1)/n)), Subst[Int[x^((m + 1)/n - 1)*Csc[d*(a + b*Log[x])]^p, x], x, c*x^n], x] /; FreeQ[{a, b, c, d, e, m, n, p}, x] && (NeQ[c, 1] || NeQ[n, 1])
  4511. Int[Times[Log[Times[Optional[Pattern[b, Blank[]]], Pattern[x, Blank[]]]], Sin[Times[Log[Times[Optional[Pattern[b, Blank[]]], Pattern[x, Blank[]]]], Optional[Pattern[a, Blank[]]], Pattern[x, Blank[]]]]], Pattern[x, Blank[Symbol]]] := -Simp[Cos[a*x*Log[b*x]]/a, x] - Int[Sin[a*x*Log[b*x]], x] /; FreeQ[{a, b}, x]
  4512. Int[Times[Cos[Times[Log[Times[Optional[Pattern[b, Blank[]]], Pattern[x, Blank[]]]], Optional[Pattern[a, Blank[]]], Pattern[x, Blank[]]]], Log[Times[Optional[Pattern[b, Blank[]]], Pattern[x, Blank[]]]]], Pattern[x, Blank[Symbol]]] := Simp[Sin[a*x*Log[b*x]]/a, x] - Int[Cos[a*x*Log[b*x]], x] /; FreeQ[{a, b}, x]
  4513. Int[Times[Log[Times[Optional[Pattern[b, Blank[]]], Pattern[x, Blank[]]]], Power[Pattern[x, Blank[]], Optional[Pattern[m, Blank[]]]], Sin[Times[Log[Times[Optional[Pattern[b, Blank[]]], Pattern[x, Blank[]]]], Optional[Pattern[a, Blank[]]], Power[Pattern[x, Blank[]], Optional[Pattern[n, Blank[]]]]]]], Pattern[x, Blank[Symbol]]] := -Simp[Cos[a*x^n*Log[b*x]]/(a*n), x] - Dist[1/n, Int[x^m*Sin[a*x^n*Log[b*x]], x], x] /; FreeQ[{a, b, m, n}, x] && EqQ[m, n - 1]
  4514. Int[Times[Cos[Times[Log[Times[Optional[Pattern[b, Blank[]]], Pattern[x, Blank[]]]], Optional[Pattern[a, Blank[]]], Power[Pattern[x, Blank[]], Optional[Pattern[n, Blank[]]]]]], Log[Times[Optional[Pattern[b, Blank[]]], Pattern[x, Blank[]]]], Power[Pattern[x, Blank[]], Optional[Pattern[m, Blank[]]]]], Pattern[x, Blank[Symbol]]] := Simp[Sin[a*x^n*Log[b*x]]/(a*n), x] - Dist[1/n, Int[x^m*Cos[a*x^n*Log[b*x]], x], x] /; FreeQ[{a, b, m, n}, x] && EqQ[m, n - 1]
  4515. Int[Times[Power[Plus[Optional[Pattern[e, Blank[]]], Times[Optional[Pattern[f, Blank[]]], Pattern[x, Blank[]]]], Optional[Pattern[m, Blank[]]]], Power[Sin[Plus[Optional[Pattern[c, Blank[]]], Times[Optional[Pattern[d, Blank[]]], Pattern[x, Blank[]]]]], Optional[Pattern[n, Blank[]]]], Power[Plus[Pattern[a, Blank[]], Times[Optional[Pattern[b, Blank[]]], Sin[Plus[Optional[Pattern[c, Blank[]]], Times[Optional[Pattern[d, Blank[]]], Pattern[x, Blank[]]]]]]], -1]], Pattern[x, Blank[Symbol]]] := Dist[1/b, Int[(e + f*x)^m*Sin[c + d*x]^(n - 1), x], x] - Dist[a/b, Int[((e + f*x)^m*Sin[c + d*x]^(n - 1))/(a + b*Sin[c + d*x]), x], x] /; FreeQ[{a, b, c, d, e, f}, x] && IGtQ[m, 0] && IGtQ[n, 0]
  4516. Int[Times[Power[Cos[Plus[Optional[Pattern[c, Blank[]]], Times[Optional[Pattern[d, Blank[]]], Pattern[x, Blank[]]]]], Optional[Pattern[n, Blank[]]]], Power[Plus[Times[Cos[Plus[Optional[Pattern[c, Blank[]]], Times[Optional[Pattern[d, Blank[]]], Pattern[x, Blank[]]]]], Optional[Pattern[b, Blank[]]]], Pattern[a, Blank[]]], -1], Power[Plus[Optional[Pattern[e, Blank[]]], Times[Optional[Pattern[f, Blank[]]], Pattern[x, Blank[]]]], Optional[Pattern[m, Blank[]]]]], Pattern[x, Blank[Symbol]]] := Dist[1/b, Int[(e + f*x)^m*Cos[c + d*x]^(n - 1), x], x] - Dist[a/b, Int[((e + f*x)^m*Cos[c + d*x]^(n - 1))/(a + b*Cos[c + d*x]), x], x] /; FreeQ[{a, b, c, d, e, f}, x] && IGtQ[m, 0] && IGtQ[n, 0]
  4517. Int[Times[Cos[Plus[Optional[Pattern[c, Blank[]]], Times[Optional[Pattern[d, Blank[]]], Pattern[x, Blank[]]]]], Power[Plus[Optional[Pattern[e, Blank[]]], Times[Optional[Pattern[f, Blank[]]], Pattern[x, Blank[]]]], Optional[Pattern[m, Blank[]]]], Power[Plus[Pattern[a, Blank[]], Times[Optional[Pattern[b, Blank[]]], Sin[Plus[Optional[Pattern[c, Blank[]]], Times[Optional[Pattern[d, Blank[]]], Pattern[x, Blank[]]]]]]], -1]], Pattern[x, Blank[Symbol]]] := -Simp[(I*(e + f*x)^(m + 1))/(b*f*(m + 1)), x] + Dist[2, Int[((e + f*x)^m*E^(I*(c + d*x)))/(a - I*b*E^(I*(c + d*x))), x], x] /; FreeQ[{a, b, c, d, e, f}, x] && IGtQ[m, 0] && EqQ[a^2 - b^2, 0]
  4518. Int[Times[Power[Plus[Times[Cos[Plus[Optional[Pattern[c, Blank[]]], Times[Optional[Pattern[d, Blank[]]], Pattern[x, Blank[]]]]], Optional[Pattern[b, Blank[]]]], Pattern[a, Blank[]]], -1], Power[Plus[Optional[Pattern[e, Blank[]]], Times[Optional[Pattern[f, Blank[]]], Pattern[x, Blank[]]]], Optional[Pattern[m, Blank[]]]], Sin[Plus[Optional[Pattern[c, Blank[]]], Times[Optional[Pattern[d, Blank[]]], Pattern[x, Blank[]]]]]], Pattern[x, Blank[Symbol]]] := Simp[(I*(e + f*x)^(m + 1))/(b*f*(m + 1)), x] - Dist[2*I, Int[((e + f*x)^m*E^(I*(c + d*x)))/(a + b*E^(I*(c + d*x))), x], x] /; FreeQ[{a, b, c, d, e, f}, x] && IGtQ[m, 0] && EqQ[a^2 - b^2, 0]
  4519. Int[Times[Cos[Plus[Optional[Pattern[c, Blank[]]], Times[Optional[Pattern[d, Blank[]]], Pattern[x, Blank[]]]]], Power[Plus[Optional[Pattern[e, Blank[]]], Times[Optional[Pattern[f, Blank[]]], Pattern[x, Blank[]]]], Optional[Pattern[m, Blank[]]]], Power[Plus[Pattern[a, Blank[]], Times[Optional[Pattern[b, Blank[]]], Sin[Plus[Optional[Pattern[c, Blank[]]], Times[Optional[Pattern[d, Blank[]]], Pattern[x, Blank[]]]]]]], -1]], Pattern[x, Blank[Symbol]]] := -Simp[(I*(e + f*x)^(m + 1))/(b*f*(m + 1)), x] + (Int[((e + f*x)^m*E^(I*(c + d*x)))/(a - Rt[a^2 - b^2, 2] - I*b*E^(I*(c + d*x))), x] + Int[((e + f*x)^m*E^(I*(c + d*x)))/(a + Rt[a^2 - b^2, 2] - I*b*E^(I*(c + d*x))), x]) /; FreeQ[{a, b, c, d, e, f}, x] && IGtQ[m, 0] && PosQ[a^2 - b^2]
  4520. Int[Times[Power[Plus[Times[Cos[Plus[Optional[Pattern[c, Blank[]]], Times[Optional[Pattern[d, Blank[]]], Pattern[x, Blank[]]]]], Optional[Pattern[b, Blank[]]]], Pattern[a, Blank[]]], -1], Power[Plus[Optional[Pattern[e, Blank[]]], Times[Optional[Pattern[f, Blank[]]], Pattern[x, Blank[]]]], Optional[Pattern[m, Blank[]]]], Sin[Plus[Optional[Pattern[c, Blank[]]], Times[Optional[Pattern[d, Blank[]]], Pattern[x, Blank[]]]]]], Pattern[x, Blank[Symbol]]] := Simp[(I*(e + f*x)^(m + 1))/(b*f*(m + 1)), x] + (-Dist[I, Int[((e + f*x)^m*E^(I*(c + d*x)))/(a - Rt[a^2 - b^2, 2] + b*E^(I*(c + d*x))), x], x] - Dist[I, Int[((e + f*x)^m*E^(I*(c + d*x)))/(a + Rt[a^2 - b^2, 2] + b*E^(I*(c + d*x))), x], x]) /; FreeQ[{a, b, c, d, e, f}, x] && IGtQ[m, 0] && PosQ[a^2 - b^2]
  4521. Int[Times[Cos[Plus[Optional[Pattern[c, Blank[]]], Times[Optional[Pattern[d, Blank[]]], Pattern[x, Blank[]]]]], Power[Plus[Optional[Pattern[e, Blank[]]], Times[Optional[Pattern[f, Blank[]]], Pattern[x, Blank[]]]], Optional[Pattern[m, Blank[]]]], Power[Plus[Pattern[a, Blank[]], Times[Optional[Pattern[b, Blank[]]], Sin[Plus[Optional[Pattern[c, Blank[]]], Times[Optional[Pattern[d, Blank[]]], Pattern[x, Blank[]]]]]]], -1]], Pattern[x, Blank[Symbol]]] := -Simp[(I*(e + f*x)^(m + 1))/(b*f*(m + 1)), x] + (Dist[I, Int[((e + f*x)^m*E^(I*(c + d*x)))/(I*a - Rt[-a^2 + b^2, 2] + b*E^(I*(c + d*x))), x], x] + Dist[I, Int[((e + f*x)^m*E^(I*(c + d*x)))/(I*a + Rt[-a^2 + b^2, 2] + b*E^(I*(c + d*x))), x], x]) /; FreeQ[{a, b, c, d, e, f}, x] && IGtQ[m, 0] && NegQ[a^2 - b^2]
  4522. Int[Times[Power[Plus[Times[Cos[Plus[Optional[Pattern[c, Blank[]]], Times[Optional[Pattern[d, Blank[]]], Pattern[x, Blank[]]]]], Optional[Pattern[b, Blank[]]]], Pattern[a, Blank[]]], -1], Power[Plus[Optional[Pattern[e, Blank[]]], Times[Optional[Pattern[f, Blank[]]], Pattern[x, Blank[]]]], Optional[Pattern[m, Blank[]]]], Sin[Plus[Optional[Pattern[c, Blank[]]], Times[Optional[Pattern[d, Blank[]]], Pattern[x, Blank[]]]]]], Pattern[x, Blank[Symbol]]] := Simp[(I*(e + f*x)^(m + 1))/(b*f*(m + 1)), x] + (Int[((e + f*x)^m*E^(I*(c + d*x)))/(I*a - Rt[-a^2 + b^2, 2] + I*b*E^(I*(c + d*x))), x] + Int[((e + f*x)^m*E^(I*(c + d*x)))/(I*a + Rt[-a^2 + b^2, 2] + I*b*E^(I*(c + d*x))), x]) /; FreeQ[{a, b, c, d, e, f}, x] && IGtQ[m, 0] && NegQ[a^2 - b^2]
  4523. Int[Times[Power[Cos[Plus[Optional[Pattern[c, Blank[]]], Times[Optional[Pattern[d, Blank[]]], Pattern[x, Blank[]]]]], Pattern[n, Blank[]]], Power[Plus[Optional[Pattern[e, Blank[]]], Times[Optional[Pattern[f, Blank[]]], Pattern[x, Blank[]]]], Optional[Pattern[m, Blank[]]]], Power[Plus[Pattern[a, Blank[]], Times[Optional[Pattern[b, Blank[]]], Sin[Plus[Optional[Pattern[c, Blank[]]], Times[Optional[Pattern[d, Blank[]]], Pattern[x, Blank[]]]]]]], -1]], Pattern[x, Blank[Symbol]]] := Dist[1/a, Int[(e + f*x)^m*Cos[c + d*x]^(n - 2), x], x] - Dist[1/b, Int[(e + f*x)^m*Cos[c + d*x]^(n - 2)*Sin[c + d*x], x], x] /; FreeQ[{a, b, c, d, e, f, m}, x] && IGtQ[n, 1] && EqQ[a^2 - b^2, 0]
  4524. Int[Times[Power[Plus[Times[Cos[Plus[Optional[Pattern[c, Blank[]]], Times[Optional[Pattern[d, Blank[]]], Pattern[x, Blank[]]]]], Optional[Pattern[b, Blank[]]]], Pattern[a, Blank[]]], -1], Power[Plus[Optional[Pattern[e, Blank[]]], Times[Optional[Pattern[f, Blank[]]], Pattern[x, Blank[]]]], Optional[Pattern[m, Blank[]]]], Power[Sin[Plus[Optional[Pattern[c, Blank[]]], Times[Optional[Pattern[d, Blank[]]], Pattern[x, Blank[]]]]], Pattern[n, Blank[]]]], Pattern[x, Blank[Symbol]]] := Dist[1/a, Int[(e + f*x)^m*Sin[c + d*x]^(n - 2), x], x] - Dist[1/b, Int[(e + f*x)^m*Sin[c + d*x]^(n - 2)*Cos[c + d*x], x], x] /; FreeQ[{a, b, c, d, e, f, m}, x] && IGtQ[n, 1] && EqQ[a^2 - b^2, 0]
  4525. Int[Times[Power[Cos[Plus[Optional[Pattern[c, Blank[]]], Times[Optional[Pattern[d, Blank[]]], Pattern[x, Blank[]]]]], Pattern[n, Blank[]]], Power[Plus[Optional[Pattern[e, Blank[]]], Times[Optional[Pattern[f, Blank[]]], Pattern[x, Blank[]]]], Optional[Pattern[m, Blank[]]]], Power[Plus[Pattern[a, Blank[]], Times[Optional[Pattern[b, Blank[]]], Sin[Plus[Optional[Pattern[c, Blank[]]], Times[Optional[Pattern[d, Blank[]]], Pattern[x, Blank[]]]]]]], -1]], Pattern[x, Blank[Symbol]]] := Dist[a/b^2, Int[(e + f*x)^m*Cos[c + d*x]^(n - 2), x], x] + (-Dist[1/b, Int[(e + f*x)^m*Cos[c + d*x]^(n - 2)*Sin[c + d*x], x], x] - Dist[(a^2 - b^2)/b^2, Int[((e + f*x)^m*Cos[c + d*x]^(n - 2))/(a + b*Sin[c + d*x]), x], x]) /; FreeQ[{a, b, c, d, e, f}, x] && IGtQ[n, 1] && NeQ[a^2 - b^2, 0] && IGtQ[m, 0]
  4526. Int[Times[Power[Plus[Times[Cos[Plus[Optional[Pattern[c, Blank[]]], Times[Optional[Pattern[d, Blank[]]], Pattern[x, Blank[]]]]], Optional[Pattern[b, Blank[]]]], Pattern[a, Blank[]]], -1], Power[Plus[Optional[Pattern[e, Blank[]]], Times[Optional[Pattern[f, Blank[]]], Pattern[x, Blank[]]]], Optional[Pattern[m, Blank[]]]], Power[Sin[Plus[Optional[Pattern[c, Blank[]]], Times[Optional[Pattern[d, Blank[]]], Pattern[x, Blank[]]]]], Pattern[n, Blank[]]]], Pattern[x, Blank[Symbol]]] := Dist[a/b^2, Int[(e + f*x)^m*Sin[c + d*x]^(n - 2), x], x] + (-Dist[1/b, Int[(e + f*x)^m*Sin[c + d*x]^(n - 2)*Cos[c + d*x], x], x] - Dist[(a^2 - b^2)/b^2, Int[((e + f*x)^m*Sin[c + d*x]^(n - 2))/(a + b*Cos[c + d*x]), x], x]) /; FreeQ[{a, b, c, d, e, f}, x] && IGtQ[n, 1] && NeQ[a^2 - b^2, 0] && IGtQ[m, 0]
  4527. Int[Times[Power[Plus[Optional[Pattern[e, Blank[]]], Times[Optional[Pattern[f, Blank[]]], Pattern[x, Blank[]]]], Optional[Pattern[m, Blank[]]]], Power[Plus[Pattern[a, Blank[]], Times[Optional[Pattern[b, Blank[]]], Sin[Plus[Optional[Pattern[c, Blank[]]], Times[Optional[Pattern[d, Blank[]]], Pattern[x, Blank[]]]]]]], -1], Power[Tan[Plus[Optional[Pattern[c, Blank[]]], Times[Optional[Pattern[d, Blank[]]], Pattern[x, Blank[]]]]], Optional[Pattern[n, Blank[]]]]], Pattern[x, Blank[Symbol]]] := Dist[1/b, Int[(e + f*x)^m*Sec[c + d*x]*Tan[c + d*x]^(n - 1), x], x] - Dist[a/b, Int[((e + f*x)^m*Sec[c + d*x]*Tan[c + d*x]^(n - 1))/(a + b*Sin[c + d*x]), x], x] /; FreeQ[{a, b, c, d, e, f}, x] && IGtQ[m, 0] && IGtQ[n, 0]
  4528. Int[Times[Power[Cot[Plus[Optional[Pattern[c, Blank[]]], Times[Optional[Pattern[d, Blank[]]], Pattern[x, Blank[]]]]], Optional[Pattern[n, Blank[]]]], Power[Plus[Times[Cos[Plus[Optional[Pattern[c, Blank[]]], Times[Optional[Pattern[d, Blank[]]], Pattern[x, Blank[]]]]], Optional[Pattern[b, Blank[]]]], Pattern[a, Blank[]]], -1], Power[Plus[Optional[Pattern[e, Blank[]]], Times[Optional[Pattern[f, Blank[]]], Pattern[x, Blank[]]]], Optional[Pattern[m, Blank[]]]]], Pattern[x, Blank[Symbol]]] := Dist[1/b, Int[(e + f*x)^m*Csc[c + d*x]*Cot[c + d*x]^(n - 1), x], x] - Dist[a/b, Int[((e + f*x)^m*Csc[c + d*x]*Cot[c + d*x]^(n - 1))/(a + b*Cos[c + d*x]), x], x] /; FreeQ[{a, b, c, d, e, f}, x] && IGtQ[m, 0] && IGtQ[n, 0]
  4529. Int[Times[Power[Cot[Plus[Optional[Pattern[c, Blank[]]], Times[Optional[Pattern[d, Blank[]]], Pattern[x, Blank[]]]]], Optional[Pattern[n, Blank[]]]], Power[Plus[Optional[Pattern[e, Blank[]]], Times[Optional[Pattern[f, Blank[]]], Pattern[x, Blank[]]]], Optional[Pattern[m, Blank[]]]], Power[Plus[Pattern[a, Blank[]], Times[Optional[Pattern[b, Blank[]]], Sin[Plus[Optional[Pattern[c, Blank[]]], Times[Optional[Pattern[d, Blank[]]], Pattern[x, Blank[]]]]]]], -1]], Pattern[x, Blank[Symbol]]] := Dist[1/a, Int[(e + f*x)^m*Cot[c + d*x]^n, x], x] - Dist[b/a, Int[((e + f*x)^m*Cos[c + d*x]*Cot[c + d*x]^(n - 1))/(a + b*Sin[c + d*x]), x], x] /; FreeQ[{a, b, c, d, e, f}, x] && IGtQ[m, 0] && IGtQ[n, 0]
  4530. Int[Times[Power[Plus[Times[Cos[Plus[Optional[Pattern[c, Blank[]]], Times[Optional[Pattern[d, Blank[]]], Pattern[x, Blank[]]]]], Optional[Pattern[b, Blank[]]]], Pattern[a, Blank[]]], -1], Power[Plus[Optional[Pattern[e, Blank[]]], Times[Optional[Pattern[f, Blank[]]], Pattern[x, Blank[]]]], Optional[Pattern[m, Blank[]]]], Power[Tan[Plus[Optional[Pattern[c, Blank[]]], Times[Optional[Pattern[d, Blank[]]], Pattern[x, Blank[]]]]], Optional[Pattern[n, Blank[]]]]], Pattern[x, Blank[Symbol]]] := Dist[1/a, Int[(e + f*x)^m*Tan[c + d*x]^n, x], x] - Dist[b/a, Int[((e + f*x)^m*Sin[c + d*x]*Tan[c + d*x]^(n - 1))/(a + b*Cos[c + d*x]), x], x] /; FreeQ[{a, b, c, d, e, f}, x] && IGtQ[m, 0] && IGtQ[n, 0]
  4531. Int[Times[Power[Plus[Optional[Pattern[e, Blank[]]], Times[Optional[Pattern[f, Blank[]]], Pattern[x, Blank[]]]], Optional[Pattern[m, Blank[]]]], Power[Sec[Plus[Optional[Pattern[c, Blank[]]], Times[Optional[Pattern[d, Blank[]]], Pattern[x, Blank[]]]]], Optional[Pattern[n, Blank[]]]], Power[Plus[Pattern[a, Blank[]], Times[Optional[Pattern[b, Blank[]]], Sin[Plus[Optional[Pattern[c, Blank[]]], Times[Optional[Pattern[d, Blank[]]], Pattern[x, Blank[]]]]]]], -1]], Pattern[x, Blank[Symbol]]] := Dist[1/a, Int[(e + f*x)^m*Sec[c + d*x]^(n + 2), x], x] - Dist[1/b, Int[(e + f*x)^m*Sec[c + d*x]^(n + 1)*Tan[c + d*x], x], x] /; FreeQ[{a, b, c, d, e, f, n}, x] && IGtQ[m, 0] && EqQ[a^2 - b^2, 0]
  4532. Int[Times[Power[Csc[Plus[Optional[Pattern[c, Blank[]]], Times[Optional[Pattern[d, Blank[]]], Pattern[x, Blank[]]]]], Optional[Pattern[n, Blank[]]]], Power[Plus[Times[Cos[Plus[Optional[Pattern[c, Blank[]]], Times[Optional[Pattern[d, Blank[]]], Pattern[x, Blank[]]]]], Optional[Pattern[b, Blank[]]]], Pattern[a, Blank[]]], -1], Power[Plus[Optional[Pattern[e, Blank[]]], Times[Optional[Pattern[f, Blank[]]], Pattern[x, Blank[]]]], Optional[Pattern[m, Blank[]]]]], Pattern[x, Blank[Symbol]]] := Dist[1/a, Int[(e + f*x)^m*Csc[c + d*x]^(n + 2), x], x] - Dist[1/b, Int[(e + f*x)^m*Csc[c + d*x]^(n + 1)*Cot[c + d*x], x], x] /; FreeQ[{a, b, c, d, e, f, n}, x] && IGtQ[m, 0] && EqQ[a^2 - b^2, 0]
  4533. Int[Times[Power[Plus[Optional[Pattern[e, Blank[]]], Times[Optional[Pattern[f, Blank[]]], Pattern[x, Blank[]]]], Optional[Pattern[m, Blank[]]]], Power[Sec[Plus[Optional[Pattern[c, Blank[]]], Times[Optional[Pattern[d, Blank[]]], Pattern[x, Blank[]]]]], Optional[Pattern[n, Blank[]]]], Power[Plus[Pattern[a, Blank[]], Times[Optional[Pattern[b, Blank[]]], Sin[Plus[Optional[Pattern[c, Blank[]]], Times[Optional[Pattern[d, Blank[]]], Pattern[x, Blank[]]]]]]], -1]], Pattern[x, Blank[Symbol]]] := -Dist[b^2/(a^2 - b^2), Int[((e + f*x)^m*Sec[c + d*x]^(n - 2))/(a + b*Sin[c + d*x]), x], x] + Dist[1/(a^2 - b^2), Int[(e + f*x)^m*Sec[c + d*x]^n*(a - b*Sin[c + d*x]), x], x] /; FreeQ[{a, b, c, d, e, f}, x] && IGtQ[m, 0] && NeQ[a^2 - b^2, 0] && IGtQ[n, 0]
  4534. Int[Times[Power[Csc[Plus[Optional[Pattern[c, Blank[]]], Times[Optional[Pattern[d, Blank[]]], Pattern[x, Blank[]]]]], Optional[Pattern[n, Blank[]]]], Power[Plus[Times[Cos[Plus[Optional[Pattern[c, Blank[]]], Times[Optional[Pattern[d, Blank[]]], Pattern[x, Blank[]]]]], Optional[Pattern[b, Blank[]]]], Pattern[a, Blank[]]], -1], Power[Plus[Optional[Pattern[e, Blank[]]], Times[Optional[Pattern[f, Blank[]]], Pattern[x, Blank[]]]], Optional[Pattern[m, Blank[]]]]], Pattern[x, Blank[Symbol]]] := -Dist[b^2/(a^2 - b^2), Int[((e + f*x)^m*Csc[c + d*x]^(n - 2))/(a + b*Cos[c + d*x]), x], x] + Dist[1/(a^2 - b^2), Int[(e + f*x)^m*Csc[c + d*x]^n*(a - b*Cos[c + d*x]), x], x] /; FreeQ[{a, b, c, d, e, f}, x] && IGtQ[m, 0] && NeQ[a^2 - b^2, 0] && IGtQ[n, 0]
  4535. Int[Times[Power[Csc[Plus[Optional[Pattern[c, Blank[]]], Times[Optional[Pattern[d, Blank[]]], Pattern[x, Blank[]]]]], Optional[Pattern[n, Blank[]]]], Power[Plus[Optional[Pattern[e, Blank[]]], Times[Optional[Pattern[f, Blank[]]], Pattern[x, Blank[]]]], Optional[Pattern[m, Blank[]]]], Power[Plus[Pattern[a, Blank[]], Times[Optional[Pattern[b, Blank[]]], Sin[Plus[Optional[Pattern[c, Blank[]]], Times[Optional[Pattern[d, Blank[]]], Pattern[x, Blank[]]]]]]], -1]], Pattern[x, Blank[Symbol]]] := Dist[1/a, Int[(e + f*x)^m*Csc[c + d*x]^n, x], x] - Dist[b/a, Int[((e + f*x)^m*Csc[c + d*x]^(n - 1))/(a + b*Sin[c + d*x]), x], x] /; FreeQ[{a, b, c, d, e, f}, x] && IGtQ[m, 0] && IGtQ[n, 0]
  4536. Int[Times[Power[Plus[Times[Cos[Plus[Optional[Pattern[c, Blank[]]], Times[Optional[Pattern[d, Blank[]]], Pattern[x, Blank[]]]]], Optional[Pattern[b, Blank[]]]], Pattern[a, Blank[]]], -1], Power[Plus[Optional[Pattern[e, Blank[]]], Times[Optional[Pattern[f, Blank[]]], Pattern[x, Blank[]]]], Optional[Pattern[m, Blank[]]]], Power[Sec[Plus[Optional[Pattern[c, Blank[]]], Times[Optional[Pattern[d, Blank[]]], Pattern[x, Blank[]]]]], Optional[Pattern[n, Blank[]]]]], Pattern[x, Blank[Symbol]]] := Dist[1/a, Int[(e + f*x)^m*Sec[c + d*x]^n, x], x] - Dist[b/a, Int[((e + f*x)^m*Sec[c + d*x]^(n - 1))/(a + b*Cos[c + d*x]), x], x] /; FreeQ[{a, b, c, d, e, f}, x] && IGtQ[m, 0] && IGtQ[n, 0]
  4537. Int[Times[Power[Plus[Optional[Pattern[e, Blank[]]], Times[Optional[Pattern[f, Blank[]]], Pattern[x, Blank[]]]], Optional[Pattern[m, Blank[]]]], Power[Plus[Pattern[a, Blank[]], Times[Optional[Pattern[b, Blank[]]], Sin[Plus[Optional[Pattern[c, Blank[]]], Times[Optional[Pattern[d, Blank[]]], Pattern[x, Blank[]]]]]]], -1], Power[Pattern[F, Blank[]][Plus[Optional[Pattern[c, Blank[]]], Times[Optional[Pattern[d, Blank[]]], Pattern[x, Blank[]]]]], Optional[Pattern[n, Blank[]]]]], Pattern[x, Blank[Symbol]]] := Unintegrable[((e + f*x)^m*F[c + d*x]^n)/(a + b*Sin[c + d*x]), x] /; FreeQ[{a, b, c, d, e, f, m, n}, x] && TrigQ[F]
  4538. Int[Times[Power[Plus[Times[Cos[Plus[Optional[Pattern[c, Blank[]]], Times[Optional[Pattern[d, Blank[]]], Pattern[x, Blank[]]]]], Optional[Pattern[b, Blank[]]]], Pattern[a, Blank[]]], -1], Power[Plus[Optional[Pattern[e, Blank[]]], Times[Optional[Pattern[f, Blank[]]], Pattern[x, Blank[]]]], Optional[Pattern[m, Blank[]]]], Power[Pattern[F, Blank[]][Plus[Optional[Pattern[c, Blank[]]], Times[Optional[Pattern[d, Blank[]]], Pattern[x, Blank[]]]]], Optional[Pattern[n, Blank[]]]]], Pattern[x, Blank[Symbol]]] := Unintegrable[((e + f*x)^m*F[c + d*x]^n)/(a + b*Cos[c + d*x]), x] /; FreeQ[{a, b, c, d, e, f, m, n}, x] && TrigQ[F]
  4539. Int[Times[Power[Cos[Plus[Optional[Pattern[c, Blank[]]], Times[Optional[Pattern[d, Blank[]]], Pattern[x, Blank[]]]]], Optional[Pattern[p, Blank[]]]], Power[Plus[Optional[Pattern[e, Blank[]]], Times[Optional[Pattern[f, Blank[]]], Pattern[x, Blank[]]]], Optional[Pattern[m, Blank[]]]], Power[Sin[Plus[Optional[Pattern[c, Blank[]]], Times[Optional[Pattern[d, Blank[]]], Pattern[x, Blank[]]]]], Optional[Pattern[n, Blank[]]]], Power[Plus[Pattern[a, Blank[]], Times[Optional[Pattern[b, Blank[]]], Sin[Plus[Optional[Pattern[c, Blank[]]], Times[Optional[Pattern[d, Blank[]]], Pattern[x, Blank[]]]]]]], -1]], Pattern[x, Blank[Symbol]]] := Dist[1/b, Int[(e + f*x)^m*Cos[c + d*x]^p*Sin[c + d*x]^(n - 1), x], x] - Dist[a/b, Int[((e + f*x)^m*Cos[c + d*x]^p*Sin[c + d*x]^(n - 1))/(a + b*Sin[c + d*x]), x], x] /; FreeQ[{a, b, c, d, e, f}, x] && IGtQ[m, 0] && IGtQ[n, 0] && IGtQ[p, 0]
  4540. Int[Times[Power[Cos[Plus[Optional[Pattern[c, Blank[]]], Times[Optional[Pattern[d, Blank[]]], Pattern[x, Blank[]]]]], Optional[Pattern[n, Blank[]]]], Power[Plus[Times[Cos[Plus[Optional[Pattern[c, Blank[]]], Times[Optional[Pattern[d, Blank[]]], Pattern[x, Blank[]]]]], Optional[Pattern[b, Blank[]]]], Pattern[a, Blank[]]], -1], Power[Plus[Optional[Pattern[e, Blank[]]], Times[Optional[Pattern[f, Blank[]]], Pattern[x, Blank[]]]], Optional[Pattern[m, Blank[]]]], Power[Sin[Plus[Optional[Pattern[c, Blank[]]], Times[Optional[Pattern[d, Blank[]]], Pattern[x, Blank[]]]]], Optional[Pattern[p, Blank[]]]]], Pattern[x, Blank[Symbol]]] := Dist[1/b, Int[(e + f*x)^m*Sin[c + d*x]^p*Cos[c + d*x]^(n - 1), x], x] - Dist[a/b, Int[((e + f*x)^m*Sin[c + d*x]^p*Cos[c + d*x]^(n - 1))/(a + b*Cos[c + d*x]), x], x] /; FreeQ[{a, b, c, d, e, f}, x] && IGtQ[m, 0] && IGtQ[n, 0] && IGtQ[p, 0]
  4541. Int[Times[Power[Cos[Plus[Optional[Pattern[c, Blank[]]], Times[Optional[Pattern[d, Blank[]]], Pattern[x, Blank[]]]]], Optional[Pattern[p, Blank[]]]], Power[Plus[Optional[Pattern[e, Blank[]]], Times[Optional[Pattern[f, Blank[]]], Pattern[x, Blank[]]]], Optional[Pattern[m, Blank[]]]], Power[Plus[Pattern[a, Blank[]], Times[Optional[Pattern[b, Blank[]]], Sin[Plus[Optional[Pattern[c, Blank[]]], Times[Optional[Pattern[d, Blank[]]], Pattern[x, Blank[]]]]]]], -1], Power[Tan[Plus[Optional[Pattern[c, Blank[]]], Times[Optional[Pattern[d, Blank[]]], Pattern[x, Blank[]]]]], Optional[Pattern[n, Blank[]]]]], Pattern[x, Blank[Symbol]]] := Dist[1/b, Int[(e + f*x)^m*Cos[c + d*x]^(p - 1)*Tan[c + d*x]^(n - 1), x], x] - Dist[a/b, Int[((e + f*x)^m*Cos[c + d*x]^(p - 1)*Tan[c + d*x]^(n - 1))/(a + b*Sin[c + d*x]), x], x] /; FreeQ[{a, b, c, d, e, f}, x] && IGtQ[m, 0] && IGtQ[n, 0] && IGtQ[p, 0]
  4542. Int[Times[Power[Cot[Plus[Optional[Pattern[c, Blank[]]], Times[Optional[Pattern[d, Blank[]]], Pattern[x, Blank[]]]]], Optional[Pattern[n, Blank[]]]], Power[Plus[Times[Cos[Plus[Optional[Pattern[c, Blank[]]], Times[Optional[Pattern[d, Blank[]]], Pattern[x, Blank[]]]]], Optional[Pattern[b, Blank[]]]], Pattern[a, Blank[]]], -1], Power[Plus[Optional[Pattern[e, Blank[]]], Times[Optional[Pattern[f, Blank[]]], Pattern[x, Blank[]]]], Optional[Pattern[m, Blank[]]]], Power[Sin[Plus[Optional[Pattern[c, Blank[]]], Times[Optional[Pattern[d, Blank[]]], Pattern[x, Blank[]]]]], Optional[Pattern[p, Blank[]]]]], Pattern[x, Blank[Symbol]]] := Dist[1/b, Int[(e + f*x)^m*Sin[c + d*x]^(p - 1)*Cot[c + d*x]^(n - 1), x], x] - Dist[a/b, Int[((e + f*x)^m*Sin[c + d*x]^(p - 1)*Cot[c + d*x]^(n - 1))/(a + b*Cos[c + d*x]), x], x] /; FreeQ[{a, b, c, d, e, f}, x] && IGtQ[m, 0] && IGtQ[n, 0] && IGtQ[p, 0]
  4543. Int[Times[Power[Cos[Plus[Optional[Pattern[c, Blank[]]], Times[Optional[Pattern[d, Blank[]]], Pattern[x, Blank[]]]]], Optional[Pattern[p, Blank[]]]], Power[Cot[Plus[Optional[Pattern[c, Blank[]]], Times[Optional[Pattern[d, Blank[]]], Pattern[x, Blank[]]]]], Optional[Pattern[n, Blank[]]]], Power[Plus[Optional[Pattern[e, Blank[]]], Times[Optional[Pattern[f, Blank[]]], Pattern[x, Blank[]]]], Optional[Pattern[m, Blank[]]]], Power[Plus[Pattern[a, Blank[]], Times[Optional[Pattern[b, Blank[]]], Sin[Plus[Optional[Pattern[c, Blank[]]], Times[Optional[Pattern[d, Blank[]]], Pattern[x, Blank[]]]]]]], -1]], Pattern[x, Blank[Symbol]]] := Dist[1/a, Int[(e + f*x)^m*Cos[c + d*x]^p*Cot[c + d*x]^n, x], x] - Dist[b/a, Int[((e + f*x)^m*Cos[c + d*x]^(p + 1)*Cot[c + d*x]^(n - 1))/(a + b*Sin[c + d*x]), x], x] /; FreeQ[{a, b, c, d, e, f}, x] && IGtQ[m, 0] && IGtQ[n, 0] && IGtQ[p, 0]
  4544. Int[Times[Power[Plus[Times[Cos[Plus[Optional[Pattern[c, Blank[]]], Times[Optional[Pattern[d, Blank[]]], Pattern[x, Blank[]]]]], Optional[Pattern[b, Blank[]]]], Pattern[a, Blank[]]], -1], Power[Plus[Optional[Pattern[e, Blank[]]], Times[Optional[Pattern[f, Blank[]]], Pattern[x, Blank[]]]], Optional[Pattern[m, Blank[]]]], Power[Sin[Plus[Optional[Pattern[c, Blank[]]], Times[Optional[Pattern[d, Blank[]]], Pattern[x, Blank[]]]]], Optional[Pattern[p, Blank[]]]], Power[Tan[Plus[Optional[Pattern[c, Blank[]]], Times[Optional[Pattern[d, Blank[]]], Pattern[x, Blank[]]]]], Optional[Pattern[n, Blank[]]]]], Pattern[x, Blank[Symbol]]] := Dist[1/a, Int[(e + f*x)^m*Sin[c + d*x]^p*Tan[c + d*x]^n, x], x] - Dist[b/a, Int[((e + f*x)^m*Sin[c + d*x]^(p + 1)*Tan[c + d*x]^(n - 1))/(a + b*Cos[c + d*x]), x], x] /; FreeQ[{a, b, c, d, e, f}, x] && IGtQ[m, 0] && IGtQ[n, 0] && IGtQ[p, 0]
  4545. Int[Times[Power[Cos[Plus[Optional[Pattern[c, Blank[]]], Times[Optional[Pattern[d, Blank[]]], Pattern[x, Blank[]]]]], Optional[Pattern[p, Blank[]]]], Power[Csc[Plus[Optional[Pattern[c, Blank[]]], Times[Optional[Pattern[d, Blank[]]], Pattern[x, Blank[]]]]], Optional[Pattern[n, Blank[]]]], Power[Plus[Optional[Pattern[e, Blank[]]], Times[Optional[Pattern[f, Blank[]]], Pattern[x, Blank[]]]], Optional[Pattern[m, Blank[]]]], Power[Plus[Pattern[a, Blank[]], Times[Optional[Pattern[b, Blank[]]], Sin[Plus[Optional[Pattern[c, Blank[]]], Times[Optional[Pattern[d, Blank[]]], Pattern[x, Blank[]]]]]]], -1]], Pattern[x, Blank[Symbol]]] := Dist[1/a, Int[(e + f*x)^m*Cos[c + d*x]^p*Csc[c + d*x]^n, x], x] - Dist[b/a, Int[((e + f*x)^m*Cos[c + d*x]^p*Csc[c + d*x]^(n - 1))/(a + b*Sin[c + d*x]), x], x] /; FreeQ[{a, b, c, d, e, f}, x] && IGtQ[m, 0] && IGtQ[n, 0] && IGtQ[p, 0]
  4546. Int[Times[Power[Plus[Times[Cos[Plus[Optional[Pattern[c, Blank[]]], Times[Optional[Pattern[d, Blank[]]], Pattern[x, Blank[]]]]], Optional[Pattern[b, Blank[]]]], Pattern[a, Blank[]]], -1], Power[Plus[Optional[Pattern[e, Blank[]]], Times[Optional[Pattern[f, Blank[]]], Pattern[x, Blank[]]]], Optional[Pattern[m, Blank[]]]], Power[Sec[Plus[Optional[Pattern[c, Blank[]]], Times[Optional[Pattern[d, Blank[]]], Pattern[x, Blank[]]]]], Optional[Pattern[n, Blank[]]]], Power[Sin[Plus[Optional[Pattern[c, Blank[]]], Times[Optional[Pattern[d, Blank[]]], Pattern[x, Blank[]]]]], Optional[Pattern[p, Blank[]]]]], Pattern[x, Blank[Symbol]]] := Dist[1/a, Int[(e + f*x)^m*Sin[c + d*x]^p*Sec[c + d*x]^n, x], x] - Dist[b/a, Int[((e + f*x)^m*Sin[c + d*x]^p*Sec[c + d*x]^(n - 1))/(a + b*Cos[c + d*x]), x], x] /; FreeQ[{a, b, c, d, e, f}, x] && IGtQ[m, 0] && IGtQ[n, 0] && IGtQ[p, 0]
  4547. Int[Times[Power[Cos[Plus[Optional[Pattern[c, Blank[]]], Times[Optional[Pattern[d, Blank[]]], Pattern[x, Blank[]]]]], Optional[Pattern[p, Blank[]]]], Power[Plus[Optional[Pattern[e, Blank[]]], Times[Optional[Pattern[f, Blank[]]], Pattern[x, Blank[]]]], Optional[Pattern[m, Blank[]]]], Power[Plus[Pattern[a, Blank[]], Times[Optional[Pattern[b, Blank[]]], Sin[Plus[Optional[Pattern[c, Blank[]]], Times[Optional[Pattern[d, Blank[]]], Pattern[x, Blank[]]]]]]], -1], Power[Pattern[F, Blank[]][Plus[Optional[Pattern[c, Blank[]]], Times[Optional[Pattern[d, Blank[]]], Pattern[x, Blank[]]]]], Optional[Pattern[n, Blank[]]]]], Pattern[x, Blank[Symbol]]] := Unintegrable[((e + f*x)^m*Cos[c + d*x]^p*F[c + d*x]^n)/(a + b*Sin[c + d*x]), x] /; FreeQ[{a, b, c, d, e, f, m, n, p}, x] && TrigQ[F]
  4548. Int[Times[Power[Plus[Times[Cos[Plus[Optional[Pattern[c, Blank[]]], Times[Optional[Pattern[d, Blank[]]], Pattern[x, Blank[]]]]], Optional[Pattern[b, Blank[]]]], Pattern[a, Blank[]]], -1], Power[Plus[Optional[Pattern[e, Blank[]]], Times[Optional[Pattern[f, Blank[]]], Pattern[x, Blank[]]]], Optional[Pattern[m, Blank[]]]], Power[Sin[Plus[Optional[Pattern[c, Blank[]]], Times[Optional[Pattern[d, Blank[]]], Pattern[x, Blank[]]]]], Optional[Pattern[p, Blank[]]]], Power[Pattern[F, Blank[]][Plus[Optional[Pattern[c, Blank[]]], Times[Optional[Pattern[d, Blank[]]], Pattern[x, Blank[]]]]], Optional[Pattern[n, Blank[]]]]], Pattern[x, Blank[Symbol]]] := Unintegrable[((e + f*x)^m*F[c + d*x]^n*Sin[c + d*x]^p)/(a + b*Cos[c + d*x]), x] /; FreeQ[{a, b, c, d, e, f, m, n}, x] && TrigQ[F]
  4549. Int[Times[Power[Plus[Optional[Pattern[e, Blank[]]], Times[Optional[Pattern[f, Blank[]]], Pattern[x, Blank[]]]], Optional[Pattern[m, Blank[]]]], Power[Plus[Pattern[a, Blank[]], Times[Optional[Pattern[b, Blank[]]], Sec[Plus[Optional[Pattern[c, Blank[]]], Times[Optional[Pattern[d, Blank[]]], Pattern[x, Blank[]]]]]]], -1], Power[Pattern[F, Blank[]][Plus[Optional[Pattern[c, Blank[]]], Times[Optional[Pattern[d, Blank[]]], Pattern[x, Blank[]]]]], Optional[Pattern[n, Blank[]]]]], Pattern[x, Blank[Symbol]]] := Int[((e + f*x)^m*Cos[c + d*x]*F[c + d*x]^n)/(b + a*Cos[c + d*x]), x] /; FreeQ[{a, b, c, d, e, f}, x] && TrigQ[F] && IntegersQ[m, n]
  4550. Int[Times[Power[Plus[Times[Csc[Plus[Optional[Pattern[c, Blank[]]], Times[Optional[Pattern[d, Blank[]]], Pattern[x, Blank[]]]]], Optional[Pattern[b, Blank[]]]], Pattern[a, Blank[]]], -1], Power[Plus[Optional[Pattern[e, Blank[]]], Times[Optional[Pattern[f, Blank[]]], Pattern[x, Blank[]]]], Optional[Pattern[m, Blank[]]]], Power[Pattern[F, Blank[]][Plus[Optional[Pattern[c, Blank[]]], Times[Optional[Pattern[d, Blank[]]], Pattern[x, Blank[]]]]], Optional[Pattern[n, Blank[]]]]], Pattern[x, Blank[Symbol]]] := Int[((e + f*x)^m*Sin[c + d*x]*F[c + d*x]^n)/(b + a*Sin[c + d*x]), x] /; FreeQ[{a, b, c, d, e, f}, x] && TrigQ[F] && IntegersQ[m, n]
  4551. Int[Times[Power[Plus[Optional[Pattern[e, Blank[]]], Times[Optional[Pattern[f, Blank[]]], Pattern[x, Blank[]]]], Optional[Pattern[m, Blank[]]]], Power[Plus[Pattern[a, Blank[]], Times[Optional[Pattern[b, Blank[]]], Sec[Plus[Optional[Pattern[c, Blank[]]], Times[Optional[Pattern[d, Blank[]]], Pattern[x, Blank[]]]]]]], -1], Power[Pattern[F, Blank[]][Plus[Optional[Pattern[c, Blank[]]], Times[Optional[Pattern[d, Blank[]]], Pattern[x, Blank[]]]]], Optional[Pattern[n, Blank[]]]], Power[Pattern[G, Blank[]][Plus[Optional[Pattern[c, Blank[]]], Times[Optional[Pattern[d, Blank[]]], Pattern[x, Blank[]]]]], Optional[Pattern[p, Blank[]]]]], Pattern[x, Blank[Symbol]]] := Int[((e + f*x)^m*Cos[c + d*x]*F[c + d*x]^n*G[c + d*x]^p)/(b + a*Cos[c + d*x]), x] /; FreeQ[{a, b, c, d, e, f}, x] && TrigQ[F] && TrigQ[G] && IntegersQ[m, n, p]
  4552. Int[Times[Power[Plus[Times[Csc[Plus[Optional[Pattern[c, Blank[]]], Times[Optional[Pattern[d, Blank[]]], Pattern[x, Blank[]]]]], Optional[Pattern[b, Blank[]]]], Pattern[a, Blank[]]], -1], Power[Plus[Optional[Pattern[e, Blank[]]], Times[Optional[Pattern[f, Blank[]]], Pattern[x, Blank[]]]], Optional[Pattern[m, Blank[]]]], Power[Pattern[F, Blank[]][Plus[Optional[Pattern[c, Blank[]]], Times[Optional[Pattern[d, Blank[]]], Pattern[x, Blank[]]]]], Optional[Pattern[n, Blank[]]]], Power[Pattern[G, Blank[]][Plus[Optional[Pattern[c, Blank[]]], Times[Optional[Pattern[d, Blank[]]], Pattern[x, Blank[]]]]], Optional[Pattern[p, Blank[]]]]], Pattern[x, Blank[Symbol]]] := Int[((e + f*x)^m*Sin[c + d*x]*F[c + d*x]^n*G[c + d*x]^p)/(b + a*Sin[c + d*x]), x] /; FreeQ[{a, b, c, d, e, f}, x] && TrigQ[F] && TrigQ[G] && IntegersQ[m, n, p]
  4553. Int[Times[Power[Sin[Plus[Optional[Pattern[a, Blank[]]], Times[Optional[Pattern[b, Blank[]]], Pattern[x, Blank[]]]]], Optional[Pattern[p, Blank[]]]], Power[Sin[Plus[Optional[Pattern[c, Blank[]]], Times[Optional[Pattern[d, Blank[]]], Pattern[x, Blank[]]]]], Optional[Pattern[q, Blank[]]]]], Pattern[x, Blank[Symbol]]] := Dist[1/2^(p + q), Int[ExpandIntegrand[(I/E^(I*(c + d*x)) - I*E^(I*(c + d*x)))^q, (I/E^(I*(a + b*x)) - I*E^(I*(a + b*x)))^p, x], x], x] /; FreeQ[{a, b, c, d, q}, x] && IGtQ[p, 0] && !IntegerQ[q]
  4554. Int[Times[Power[Cos[Plus[Optional[Pattern[a, Blank[]]], Times[Optional[Pattern[b, Blank[]]], Pattern[x, Blank[]]]]], Optional[Pattern[p, Blank[]]]], Power[Cos[Plus[Optional[Pattern[c, Blank[]]], Times[Optional[Pattern[d, Blank[]]], Pattern[x, Blank[]]]]], Optional[Pattern[q, Blank[]]]]], Pattern[x, Blank[Symbol]]] := Dist[1/2^(p + q), Int[ExpandIntegrand[(E^(-(I*(c + d*x))) + E^(I*(c + d*x)))^q, (E^(-(I*(a + b*x))) + E^(I*(a + b*x)))^p, x], x], x] /; FreeQ[{a, b, c, d, q}, x] && IGtQ[p, 0] && !IntegerQ[q]
  4555. Int[Times[Power[Cos[Plus[Optional[Pattern[c, Blank[]]], Times[Optional[Pattern[d, Blank[]]], Pattern[x, Blank[]]]]], Optional[Pattern[q, Blank[]]]], Power[Sin[Plus[Optional[Pattern[a, Blank[]]], Times[Optional[Pattern[b, Blank[]]], Pattern[x, Blank[]]]]], Optional[Pattern[p, Blank[]]]]], Pattern[x, Blank[Symbol]]] := Dist[1/2^(p + q), Int[ExpandIntegrand[(E^(-(I*(c + d*x))) + E^(I*(c + d*x)))^q, (I/E^(I*(a + b*x)) - I*E^(I*(a + b*x)))^p, x], x], x] /; FreeQ[{a, b, c, d, q}, x] && IGtQ[p, 0] && !IntegerQ[q]
  4556. Int[Times[Power[Cos[Plus[Optional[Pattern[a, Blank[]]], Times[Optional[Pattern[b, Blank[]]], Pattern[x, Blank[]]]]], Optional[Pattern[p, Blank[]]]], Power[Sin[Plus[Optional[Pattern[c, Blank[]]], Times[Optional[Pattern[d, Blank[]]], Pattern[x, Blank[]]]]], Optional[Pattern[q, Blank[]]]]], Pattern[x, Blank[Symbol]]] := Dist[1/2^(p + q), Int[ExpandIntegrand[(I/E^(I*(c + d*x)) - I*E^(I*(c + d*x)))^q, (E^(-(I*(a + b*x))) + E^(I*(a + b*x)))^p, x], x], x] /; FreeQ[{a, b, c, d, q}, x] && IGtQ[p, 0] && !IntegerQ[q]
  4557. Int[Times[Sin[Plus[Optional[Pattern[a, Blank[]]], Times[Optional[Pattern[b, Blank[]]], Pattern[x, Blank[]]]]], Tan[Plus[Optional[Pattern[c, Blank[]]], Times[Optional[Pattern[d, Blank[]]], Pattern[x, Blank[]]]]]], Pattern[x, Blank[Symbol]]] := Int[1/(E^(I*(a + b*x))*2) - E^(I*(a + b*x))/2 - 1/(E^(I*(a + b*x))*(1 + E^(2*I*(c + d*x)))) + E^(I*(a + b*x))/(1 + E^(2*I*(c + d*x))), x] /; FreeQ[{a, b, c, d}, x] && NeQ[b^2 - d^2, 0]
  4558. Int[Times[Cos[Plus[Optional[Pattern[a, Blank[]]], Times[Optional[Pattern[b, Blank[]]], Pattern[x, Blank[]]]]], Cot[Plus[Optional[Pattern[c, Blank[]]], Times[Optional[Pattern[d, Blank[]]], Pattern[x, Blank[]]]]]], Pattern[x, Blank[Symbol]]] := Int[I/(E^(I*(a + b*x))*2) + (I*E^(I*(a + b*x)))/2 - I/(E^(I*(a + b*x))*(1 - E^(2*I*(c + d*x)))) - (I*E^(I*(a + b*x)))/(1 - E^(2*I*(c + d*x))), x] /; FreeQ[{a, b, c, d}, x] && NeQ[b^2 - d^2, 0]
  4559. Int[Times[Cot[Plus[Optional[Pattern[c, Blank[]]], Times[Optional[Pattern[d, Blank[]]], Pattern[x, Blank[]]]]], Sin[Plus[Optional[Pattern[a, Blank[]]], Times[Optional[Pattern[b, Blank[]]], Pattern[x, Blank[]]]]]], Pattern[x, Blank[Symbol]]] := Int[-(1/(E^(I*(a + b*x))*2)) + E^(I*(a + b*x))/2 + 1/(E^(I*(a + b*x))*(1 - E^(2*I*(c + d*x)))) - E^(I*(a + b*x))/(1 - E^(2*I*(c + d*x))), x] /; FreeQ[{a, b, c, d}, x] && NeQ[b^2 - d^2, 0]
  4560. Int[Times[Cos[Plus[Optional[Pattern[a, Blank[]]], Times[Optional[Pattern[b, Blank[]]], Pattern[x, Blank[]]]]], Tan[Plus[Optional[Pattern[c, Blank[]]], Times[Optional[Pattern[d, Blank[]]], Pattern[x, Blank[]]]]]], Pattern[x, Blank[Symbol]]] := Int[-(I/(E^(I*(a + b*x))*2)) - (I*E^(I*(a + b*x)))/2 + I/(E^(I*(a + b*x))*(1 + E^(2*I*(c + d*x)))) + (I*E^(I*(a + b*x)))/(1 + E^(2*I*(c + d*x))), x] /; FreeQ[{a, b, c, d}, x] && NeQ[b^2 - d^2, 0]
  4561. Int[Power[Sin[Times[Optional[Pattern[a, Blank[]]], Power[Plus[Optional[Pattern[c, Blank[]]], Times[Optional[Pattern[d, Blank[]]], Pattern[x, Blank[]]]], -1]]], Optional[Pattern[n, Blank[]]]], Pattern[x, Blank[Symbol]]] := -Dist[d^(-1), Subst[Int[Sin[a*x]^n/x^2, x], x, 1/(c + d*x)], x] /; FreeQ[{a, c, d}, x] && IGtQ[n, 0]
  4562. Int[Power[Cos[Times[Optional[Pattern[a, Blank[]]], Power[Plus[Optional[Pattern[c, Blank[]]], Times[Optional[Pattern[d, Blank[]]], Pattern[x, Blank[]]]], -1]]], Optional[Pattern[n, Blank[]]]], Pattern[x, Blank[Symbol]]] := -Dist[d^(-1), Subst[Int[Cos[a*x]^n/x^2, x], x, 1/(c + d*x)], x] /; FreeQ[{a, c, d}, x] && IGtQ[n, 0]
  4563. Int[Power[Sin[Times[Optional[Pattern[e, Blank[]]], Plus[Optional[Pattern[a, Blank[]]], Times[Optional[Pattern[b, Blank[]]], Pattern[x, Blank[]]]], Power[Plus[Optional[Pattern[c, Blank[]]], Times[Optional[Pattern[d, Blank[]]], Pattern[x, Blank[]]]], -1]]], Optional[Pattern[n, Blank[]]]], Pattern[x, Blank[Symbol]]] := -Dist[d^(-1), Subst[Int[Sin[(b*e)/d - (e*(b*c - a*d)*x)/d]^n/x^2, x], x, 1/(c + d*x)], x] /; FreeQ[{a, b, c, d}, x] && IGtQ[n, 0] && NeQ[b*c - a*d, 0]
  4564. Int[Power[Cos[Times[Optional[Pattern[e, Blank[]]], Plus[Optional[Pattern[a, Blank[]]], Times[Optional[Pattern[b, Blank[]]], Pattern[x, Blank[]]]], Power[Plus[Optional[Pattern[c, Blank[]]], Times[Optional[Pattern[d, Blank[]]], Pattern[x, Blank[]]]], -1]]], Optional[Pattern[n, Blank[]]]], Pattern[x, Blank[Symbol]]] := -Dist[d^(-1), Subst[Int[Cos[(b*e)/d - (e*(b*c - a*d)*x)/d]^n/x^2, x], x, 1/(c + d*x)], x] /; FreeQ[{a, b, c, d}, x] && IGtQ[n, 0] && NeQ[b*c - a*d, 0]
  4565. Int[Power[Sin[Pattern[u, Blank[]]], Optional[Pattern[n, Blank[]]]], Pattern[x, Blank[Symbol]]] := Module[{lst = QuotientOfLinearsParts[u, x]}, Int[Sin[(lst[[1]] + lst[[2]]*x)/(lst[[3]] + lst[[4]]*x)]^n, x]] /; IGtQ[n, 0] && QuotientOfLinearsQ[u, x]
  4566. Int[Power[Cos[Pattern[u, Blank[]]], Optional[Pattern[n, Blank[]]]], Pattern[x, Blank[Symbol]]] := Module[{lst = QuotientOfLinearsParts[u, x]}, Int[Cos[(lst[[1]] + lst[[2]]*x)/(lst[[3]] + lst[[4]]*x)]^n, x]] /; IGtQ[n, 0] && QuotientOfLinearsQ[u, x]
  4567. Int[Times[Optional[Pattern[u, Blank[]]], Power[Sin[Pattern[v, Blank[]]], Optional[Pattern[p, Blank[]]]], Power[Sin[Pattern[w, Blank[]]], Optional[Pattern[q, Blank[]]]]], Pattern[x, Blank[Symbol]]] := Int[u*Sin[v]^(p + q), x] /; EqQ[w, v]
  4568. Int[Times[Power[Cos[Pattern[v, Blank[]]], Optional[Pattern[p, Blank[]]]], Power[Cos[Pattern[w, Blank[]]], Optional[Pattern[q, Blank[]]]], Optional[Pattern[u, Blank[]]]], Pattern[x, Blank[Symbol]]] := Int[u*Cos[v]^(p + q), x] /; EqQ[w, v]
  4569. Int[Times[Power[Sin[Pattern[v, Blank[]]], Optional[Pattern[p, Blank[]]]], Power[Sin[Pattern[w, Blank[]]], Optional[Pattern[q, Blank[]]]]], Pattern[x, Blank[Symbol]]] := Int[ExpandTrigReduce[Sin[v]^p*Sin[w]^q, x], x] /; ((PolynomialQ[v, x] && PolynomialQ[w, x]) || (BinomialQ[{v, w}, x] && IndependentQ[Cancel[v/w], x])) && IGtQ[p, 0] && IGtQ[q, 0]
  4570. Int[Times[Power[Cos[Pattern[v, Blank[]]], Optional[Pattern[p, Blank[]]]], Power[Cos[Pattern[w, Blank[]]], Optional[Pattern[q, Blank[]]]]], Pattern[x, Blank[Symbol]]] := Int[ExpandTrigReduce[Cos[v]^p*Cos[w]^q, x], x] /; ((PolynomialQ[v, x] && PolynomialQ[w, x]) || (BinomialQ[{v, w}, x] && IndependentQ[Cancel[v/w], x])) && IGtQ[p, 0] && IGtQ[q, 0]
  4571. Int[Times[Power[Pattern[x, Blank[]], Optional[Pattern[m, Blank[]]]], Power[Sin[Pattern[v, Blank[]]], Optional[Pattern[p, Blank[]]]], Power[Sin[Pattern[w, Blank[]]], Optional[Pattern[q, Blank[]]]]], Pattern[x, Blank[Symbol]]] := Int[ExpandTrigReduce[x^m, Sin[v]^p*Sin[w]^q, x], x] /; IGtQ[m, 0] && IGtQ[p, 0] && IGtQ[q, 0] && ((PolynomialQ[v, x] && PolynomialQ[w, x]) || (BinomialQ[{v, w}, x] && IndependentQ[Cancel[v/w], x]))
  4572. Int[Times[Power[Cos[Pattern[v, Blank[]]], Optional[Pattern[p, Blank[]]]], Power[Cos[Pattern[w, Blank[]]], Optional[Pattern[q, Blank[]]]], Power[Pattern[x, Blank[]], Optional[Pattern[m, Blank[]]]]], Pattern[x, Blank[Symbol]]] := Int[ExpandTrigReduce[x^m, Cos[v]^p*Cos[w]^q, x], x] /; IGtQ[m, 0] && IGtQ[p, 0] && IGtQ[q, 0] && ((PolynomialQ[v, x] && PolynomialQ[w, x]) || (BinomialQ[{v, w}, x] && IndependentQ[Cancel[v/w], x]))
  4573. Int[Times[Power[Cos[Pattern[w, Blank[]]], Optional[Pattern[p, Blank[]]]], Optional[Pattern[u, Blank[]]], Power[Sin[Pattern[v, Blank[]]], Optional[Pattern[p, Blank[]]]]], Pattern[x, Blank[Symbol]]] := Dist[1/2^p, Int[u*Sin[2*v]^p, x], x] /; EqQ[w, v] && IntegerQ[p]
  4574. Int[Times[Power[Cos[Pattern[w, Blank[]]], Optional[Pattern[q, Blank[]]]], Power[Sin[Pattern[v, Blank[]]], Optional[Pattern[p, Blank[]]]]], Pattern[x, Blank[Symbol]]] := Int[ExpandTrigReduce[Sin[v]^p*Cos[w]^q, x], x] /; IGtQ[p, 0] && IGtQ[q, 0] && ((PolynomialQ[v, x] && PolynomialQ[w, x]) || (BinomialQ[{v, w}, x] && IndependentQ[Cancel[v/w], x]))
  4575. Int[Times[Power[Cos[Pattern[w, Blank[]]], Optional[Pattern[q, Blank[]]]], Power[Pattern[x, Blank[]], Optional[Pattern[m, Blank[]]]], Power[Sin[Pattern[v, Blank[]]], Optional[Pattern[p, Blank[]]]]], Pattern[x, Blank[Symbol]]] := Int[ExpandTrigReduce[x^m, Sin[v]^p*Cos[w]^q, x], x] /; IGtQ[m, 0] && IGtQ[p, 0] && IGtQ[q, 0] && ((PolynomialQ[v, x] && PolynomialQ[w, x]) || (BinomialQ[{v, w}, x] && IndependentQ[Cancel[v/w], x]))
  4576. Int[Times[Sin[Pattern[v, Blank[]]], Power[Tan[Pattern[w, Blank[]]], Optional[Pattern[n, Blank[]]]]], Pattern[x, Blank[Symbol]]] := -Int[Cos[v]*Tan[w]^(n - 1), x] + Dist[Cos[v - w], Int[Sec[w]*Tan[w]^(n - 1), x], x] /; GtQ[n, 0] && FreeQ[v - w, x] && NeQ[w, v]
  4577. Int[Times[Cos[Pattern[v, Blank[]]], Power[Cot[Pattern[w, Blank[]]], Optional[Pattern[n, Blank[]]]]], Pattern[x, Blank[Symbol]]] := -Int[Sin[v]*Cot[w]^(n - 1), x] + Dist[Cos[v - w], Int[Csc[w]*Cot[w]^(n - 1), x], x] /; GtQ[n, 0] && FreeQ[v - w, x] && NeQ[w, v]
  4578. Int[Times[Power[Cot[Pattern[w, Blank[]]], Optional[Pattern[n, Blank[]]]], Sin[Pattern[v, Blank[]]]], Pattern[x, Blank[Symbol]]] := Int[Cos[v]*Cot[w]^(n - 1), x] + Dist[Sin[v - w], Int[Csc[w]*Cot[w]^(n - 1), x], x] /; GtQ[n, 0] && FreeQ[v - w, x] && NeQ[w, v]
  4579. Int[Times[Cos[Pattern[v, Blank[]]], Power[Tan[Pattern[w, Blank[]]], Optional[Pattern[n, Blank[]]]]], Pattern[x, Blank[Symbol]]] := Int[Sin[v]*Tan[w]^(n - 1), x] - Dist[Sin[v - w], Int[Sec[w]*Tan[w]^(n - 1), x], x] /; GtQ[n, 0] && FreeQ[v - w, x] && NeQ[w, v]
  4580. Int[Times[Power[Sec[Pattern[w, Blank[]]], Optional[Pattern[n, Blank[]]]], Sin[Pattern[v, Blank[]]]], Pattern[x, Blank[Symbol]]] := Dist[Cos[v - w], Int[Tan[w]*Sec[w]^(n - 1), x], x] + Dist[Sin[v - w], Int[Sec[w]^(n - 1), x], x] /; GtQ[n, 0] && FreeQ[v - w, x] && NeQ[w, v]
  4581. Int[Times[Cos[Pattern[v, Blank[]]], Power[Csc[Pattern[w, Blank[]]], Optional[Pattern[n, Blank[]]]]], Pattern[x, Blank[Symbol]]] := Dist[Cos[v - w], Int[Cot[w]*Csc[w]^(n - 1), x], x] - Dist[Sin[v - w], Int[Csc[w]^(n - 1), x], x] /; GtQ[n, 0] && FreeQ[v - w, x] && NeQ[w, v]
  4582. Int[Times[Power[Csc[Pattern[w, Blank[]]], Optional[Pattern[n, Blank[]]]], Sin[Pattern[v, Blank[]]]], Pattern[x, Blank[Symbol]]] := Dist[Sin[v - w], Int[Cot[w]*Csc[w]^(n - 1), x], x] + Dist[Cos[v - w], Int[Csc[w]^(n - 1), x], x] /; GtQ[n, 0] && FreeQ[v - w, x] && NeQ[w, v]
  4583. Int[Times[Cos[Pattern[v, Blank[]]], Power[Sec[Pattern[w, Blank[]]], Optional[Pattern[n, Blank[]]]]], Pattern[x, Blank[Symbol]]] := -Dist[Sin[v - w], Int[Tan[w]*Sec[w]^(n - 1), x], x] + Dist[Cos[v - w], Int[Sec[w]^(n - 1), x], x] /; GtQ[n, 0] && FreeQ[v - w, x] && NeQ[w, v]
  4584. Int[Times[Power[Plus[Optional[Pattern[e, Blank[]]], Times[Optional[Pattern[f, Blank[]]], Pattern[x, Blank[]]]], Optional[Pattern[m, Blank[]]]], Power[Plus[Pattern[a, Blank[]], Times[Cos[Plus[Optional[Pattern[c, Blank[]]], Times[Optional[Pattern[d, Blank[]]], Pattern[x, Blank[]]]]], Optional[Pattern[b, Blank[]]], Sin[Plus[Optional[Pattern[c, Blank[]]], Times[Optional[Pattern[d, Blank[]]], Pattern[x, Blank[]]]]]]], Optional[Pattern[n, Blank[]]]]], Pattern[x, Blank[Symbol]]] := Int[(e + f*x)^m*(a + (b*Sin[2*c + 2*d*x])/2)^n, x] /; FreeQ[{a, b, c, d, e, f, m, n}, x]
  4585. Int[Times[Power[Pattern[x, Blank[]], Optional[Pattern[m, Blank[]]]], Power[Plus[Pattern[a, Blank[]], Times[Optional[Pattern[b, Blank[]]], Power[Sin[Plus[Optional[Pattern[c, Blank[]]], Times[Optional[Pattern[d, Blank[]]], Pattern[x, Blank[]]]]], 2]]], Pattern[n, Blank[]]]], Pattern[x, Blank[Symbol]]] := Dist[1/2^n, Int[x^m*(2*a + b - b*Cos[2*c + 2*d*x])^n, x], x] /; FreeQ[{a, b, c, d}, x] && NeQ[a + b, 0] && IGtQ[m, 0] && ILtQ[n, 0] && (EqQ[n, -1] || (EqQ[m, 1] && EqQ[n, -2]))
  4586. Int[Times[Power[Plus[Times[Power[Cos[Plus[Optional[Pattern[c, Blank[]]], Times[Optional[Pattern[d, Blank[]]], Pattern[x, Blank[]]]]], 2], Optional[Pattern[b, Blank[]]]], Pattern[a, Blank[]]], Pattern[n, Blank[]]], Power[Pattern[x, Blank[]], Optional[Pattern[m, Blank[]]]]], Pattern[x, Blank[Symbol]]] := Dist[1/2^n, Int[x^m*(2*a + b + b*Cos[2*c + 2*d*x])^n, x], x] /; FreeQ[{a, b, c, d}, x] && NeQ[a + b, 0] && IGtQ[m, 0] && ILtQ[n, 0] && (EqQ[n, -1] || (EqQ[m, 1] && EqQ[n, -2]))
  4587. Int[Times[Power[Plus[Optional[Pattern[f, Blank[]]], Times[Optional[Pattern[g, Blank[]]], Pattern[x, Blank[]]]], Optional[Pattern[m, Blank[]]]], Power[Plus[Optional[Pattern[a, Blank[]]], Times[Power[Cos[Plus[Optional[Pattern[d, Blank[]]], Times[Optional[Pattern[e, Blank[]]], Pattern[x, Blank[]]]]], 2], Optional[Pattern[b, Blank[]]]], Times[Optional[Pattern[c, Blank[]]], Power[Sin[Plus[Optional[Pattern[d, Blank[]]], Times[Optional[Pattern[e, Blank[]]], Pattern[x, Blank[]]]]], 2]]], -1]], Pattern[x, Blank[Symbol]]] := Dist[2, Int[(f + g*x)^m/(2*a + b + c + (b - c)*Cos[2*d + 2*e*x]), x], x] /; FreeQ[{a, b, c, d, e, f, g}, x] && IGtQ[m, 0] && NeQ[a + b, 0] && NeQ[a + c, 0]
  4588. Int[Times[Power[Plus[Optional[Pattern[f, Blank[]]], Times[Optional[Pattern[g, Blank[]]], Pattern[x, Blank[]]]], Optional[Pattern[m, Blank[]]]], Power[Sec[Plus[Optional[Pattern[d, Blank[]]], Times[Optional[Pattern[e, Blank[]]], Pattern[x, Blank[]]]]], 2], Power[Plus[Pattern[b, Blank[]], Times[Optional[Pattern[c, Blank[]]], Power[Tan[Plus[Optional[Pattern[d, Blank[]]], Times[Optional[Pattern[e, Blank[]]], Pattern[x, Blank[]]]]], 2]]], -1]], Pattern[x, Blank[Symbol]]] := Dist[2, Int[(f + g*x)^m/(b + c + (b - c)*Cos[2*d + 2*e*x]), x], x] /; FreeQ[{b, c, d, e, f, g}, x] && IGtQ[m, 0]
  4589. Int[Times[Power[Plus[Optional[Pattern[f, Blank[]]], Times[Optional[Pattern[g, Blank[]]], Pattern[x, Blank[]]]], Optional[Pattern[m, Blank[]]]], Power[Sec[Plus[Optional[Pattern[d, Blank[]]], Times[Optional[Pattern[e, Blank[]]], Pattern[x, Blank[]]]]], 2], Power[Plus[Optional[Pattern[b, Blank[]]], Times[Optional[Pattern[a, Blank[]]], Power[Sec[Plus[Optional[Pattern[d, Blank[]]], Times[Optional[Pattern[e, Blank[]]], Pattern[x, Blank[]]]]], 2]], Times[Optional[Pattern[c, Blank[]]], Power[Tan[Plus[Optional[Pattern[d, Blank[]]], Times[Optional[Pattern[e, Blank[]]], Pattern[x, Blank[]]]]], 2]]], -1]], Pattern[x, Blank[Symbol]]] := Dist[2, Int[(f + g*x)^m/(2*a + b + c + (b - c)*Cos[2*d + 2*e*x]), x], x] /; FreeQ[{a, b, c, d, e, f, g}, x] && IGtQ[m, 0] && NeQ[a + b, 0] && NeQ[a + c, 0]
  4590. Int[Times[Power[Csc[Plus[Optional[Pattern[d, Blank[]]], Times[Optional[Pattern[e, Blank[]]], Pattern[x, Blank[]]]]], 2], Power[Plus[Times[Power[Cot[Plus[Optional[Pattern[d, Blank[]]], Times[Optional[Pattern[e, Blank[]]], Pattern[x, Blank[]]]]], 2], Optional[Pattern[b, Blank[]]]], Pattern[c, Blank[]]], -1], Power[Plus[Optional[Pattern[f, Blank[]]], Times[Optional[Pattern[g, Blank[]]], Pattern[x, Blank[]]]], Optional[Pattern[m, Blank[]]]]], Pattern[x, Blank[Symbol]]] := Dist[2, Int[(f + g*x)^m/(b + c + (b - c)*Cos[2*d + 2*e*x]), x], x] /; FreeQ[{b, c, d, e, f, g}, x] && IGtQ[m, 0]
  4591. Int[Times[Power[Csc[Plus[Optional[Pattern[d, Blank[]]], Times[Optional[Pattern[e, Blank[]]], Pattern[x, Blank[]]]]], 2], Power[Plus[Times[Power[Csc[Plus[Optional[Pattern[d, Blank[]]], Times[Optional[Pattern[e, Blank[]]], Pattern[x, Blank[]]]]], 2], Optional[Pattern[a, Blank[]]]], Times[Power[Cot[Plus[Optional[Pattern[d, Blank[]]], Times[Optional[Pattern[e, Blank[]]], Pattern[x, Blank[]]]]], 2], Optional[Pattern[b, Blank[]]]], Optional[Pattern[c, Blank[]]]], -1], Power[Plus[Optional[Pattern[f, Blank[]]], Times[Optional[Pattern[g, Blank[]]], Pattern[x, Blank[]]]], Optional[Pattern[m, Blank[]]]]], Pattern[x, Blank[Symbol]]] := Dist[2, Int[(f + g*x)^m/(2*a + b + c + (b - c)*Cos[2*d + 2*e*x]), x], x] /; FreeQ[{a, b, c, d, e, f, g}, x] && IGtQ[m, 0] && NeQ[a + b, 0] && NeQ[a + c, 0]
  4592. Int[Times[Plus[Optional[Pattern[e, Blank[]]], Times[Optional[Pattern[f, Blank[]]], Pattern[x, Blank[]]]], Power[Plus[Pattern[a, Blank[]], Times[Optional[Pattern[b, Blank[]]], Sin[Plus[Optional[Pattern[c, Blank[]]], Times[Optional[Pattern[d, Blank[]]], Pattern[x, Blank[]]]]]]], -2], Plus[Pattern[A, Blank[]], Times[Optional[Pattern[B, Blank[]]], Sin[Plus[Optional[Pattern[c, Blank[]]], Times[Optional[Pattern[d, Blank[]]], Pattern[x, Blank[]]]]]]]], Pattern[x, Blank[Symbol]]] := -Simp[(B*(e + f*x)*Cos[c + d*x])/(a*d*(a + b*Sin[c + d*x])), x] + Dist[(B*f)/(a*d), Int[Cos[c + d*x]/(a + b*Sin[c + d*x]), x], x] /; FreeQ[{a, b, c, d, e, f, A, B}, x] && EqQ[a*A - b*B, 0]
  4593. Int[Times[Power[Plus[Times[Cos[Plus[Optional[Pattern[c, Blank[]]], Times[Optional[Pattern[d, Blank[]]], Pattern[x, Blank[]]]]], Optional[Pattern[b, Blank[]]]], Pattern[a, Blank[]]], -2], Plus[Times[Cos[Plus[Optional[Pattern[c, Blank[]]], Times[Optional[Pattern[d, Blank[]]], Pattern[x, Blank[]]]]], Optional[Pattern[B, Blank[]]]], Pattern[A, Blank[]]], Plus[Optional[Pattern[e, Blank[]]], Times[Optional[Pattern[f, Blank[]]], Pattern[x, Blank[]]]]], Pattern[x, Blank[Symbol]]] := Simp[(B*(e + f*x)*Sin[c + d*x])/(a*d*(a + b*Cos[c + d*x])), x] - Dist[(B*f)/(a*d), Int[Sin[c + d*x]/(a + b*Cos[c + d*x]), x], x] /; FreeQ[{a, b, c, d, e, f, A, B}, x] && EqQ[a*A - b*B, 0]
  4594. Int[Times[Power[Pattern[x, Blank[]], 2], Power[Plus[Times[Cos[Times[Optional[Pattern[a, Blank[]]], Pattern[x, Blank[]]]], Optional[Pattern[d, Blank[]]], Pattern[x, Blank[]]], Times[Optional[Pattern[c, Blank[]]], Sin[Times[Optional[Pattern[a, Blank[]]], Pattern[x, Blank[]]]]]], -2]], Pattern[x, Blank[Symbol]]] := Simp[x/(a*d*Sin[a*x]*(c*Sin[a*x] + d*x*Cos[a*x])), x] + Dist[1/d^2, Int[1/Sin[a*x]^2, x], x] /; FreeQ[{a, c, d}, x] && EqQ[a*c + d, 0]
  4595. Int[Times[Power[Pattern[x, Blank[]], 2], Power[Plus[Times[Cos[Times[Optional[Pattern[a, Blank[]]], Pattern[x, Blank[]]]], Optional[Pattern[c, Blank[]]]], Times[Optional[Pattern[d, Blank[]]], Pattern[x, Blank[]], Sin[Times[Optional[Pattern[a, Blank[]]], Pattern[x, Blank[]]]]]], -2]], Pattern[x, Blank[Symbol]]] := -Simp[x/(a*d*Cos[a*x]*(c*Cos[a*x] + d*x*Sin[a*x])), x] + Dist[1/d^2, Int[1/Cos[a*x]^2, x], x] /; FreeQ[{a, c, d}, x] && EqQ[a*c - d, 0]
  4596. Int[Times[Power[Sin[Times[Optional[Pattern[a, Blank[]]], Pattern[x, Blank[]]]], 2], Power[Plus[Times[Cos[Times[Optional[Pattern[a, Blank[]]], Pattern[x, Blank[]]]], Optional[Pattern[d, Blank[]]], Pattern[x, Blank[]]], Times[Optional[Pattern[c, Blank[]]], Sin[Times[Optional[Pattern[a, Blank[]]], Pattern[x, Blank[]]]]]], -2]], Pattern[x, Blank[Symbol]]] := Simp[1/(d^2*x), x] + Simp[Sin[a*x]/(a*d*x*(d*x*Cos[a*x] + c*Sin[a*x])), x] /; FreeQ[{a, c, d}, x] && EqQ[a*c + d, 0]
  4597. Int[Times[Power[Cos[Times[Optional[Pattern[a, Blank[]]], Pattern[x, Blank[]]]], 2], Power[Plus[Times[Cos[Times[Optional[Pattern[a, Blank[]]], Pattern[x, Blank[]]]], Optional[Pattern[c, Blank[]]]], Times[Optional[Pattern[d, Blank[]]], Pattern[x, Blank[]], Sin[Times[Optional[Pattern[a, Blank[]]], Pattern[x, Blank[]]]]]], -2]], Pattern[x, Blank[Symbol]]] := Simp[1/(d^2*x), x] - Simp[Cos[a*x]/(a*d*x*(d*x*Sin[a*x] + c*Cos[a*x])), x] /; FreeQ[{a, c, d}, x] && EqQ[a*c - d, 0]
  4598. Int[Times[Power[Times[Optional[Pattern[b, Blank[]]], Pattern[x, Blank[]]], Pattern[m, Blank[]]], Power[Sin[Times[Optional[Pattern[a, Blank[]]], Pattern[x, Blank[]]]], Pattern[n, Blank[]]], Power[Plus[Times[Cos[Times[Optional[Pattern[a, Blank[]]], Pattern[x, Blank[]]]], Optional[Pattern[d, Blank[]]], Pattern[x, Blank[]]], Times[Optional[Pattern[c, Blank[]]], Sin[Times[Optional[Pattern[a, Blank[]]], Pattern[x, Blank[]]]]]], -2]], Pattern[x, Blank[Symbol]]] := Simp[(b*(b*x)^(m - 1)*Sin[a*x]^(n - 1))/(a*d*(c*Sin[a*x] + d*x*Cos[a*x])), x] - Dist[(b^2*(n - 1))/d^2, Int[(b*x)^(m - 2)*Sin[a*x]^(n - 2), x], x] /; FreeQ[{a, b, c, d, m, n}, x] && EqQ[a*c + d, 0] && EqQ[m, 2 - n]
  4599. Int[Times[Power[Cos[Times[Optional[Pattern[a, Blank[]]], Pattern[x, Blank[]]]], Pattern[n, Blank[]]], Power[Times[Optional[Pattern[b, Blank[]]], Pattern[x, Blank[]]], Pattern[m, Blank[]]], Power[Plus[Times[Cos[Times[Optional[Pattern[a, Blank[]]], Pattern[x, Blank[]]]], Optional[Pattern[c, Blank[]]]], Times[Optional[Pattern[d, Blank[]]], Pattern[x, Blank[]], Sin[Times[Optional[Pattern[a, Blank[]]], Pattern[x, Blank[]]]]]], -2]], Pattern[x, Blank[Symbol]]] := -Simp[(b*(b*x)^(m - 1)*Cos[a*x]^(n - 1))/(a*d*(c*Cos[a*x] + d*x*Sin[a*x])), x] - Dist[(b^2*(n - 1))/d^2, Int[(b*x)^(m - 2)*Cos[a*x]^(n - 2), x], x] /; FreeQ[{a, b, c, d, m, n}, x] && EqQ[a*c - d, 0] && EqQ[m, 2 - n]
  4600. Int[Times[Power[Csc[Times[Optional[Pattern[a, Blank[]]], Pattern[x, Blank[]]]], Optional[Pattern[n, Blank[]]]], Power[Times[Optional[Pattern[b, Blank[]]], Pattern[x, Blank[]]], Optional[Pattern[m, Blank[]]]], Power[Plus[Times[Cos[Times[Optional[Pattern[a, Blank[]]], Pattern[x, Blank[]]]], Optional[Pattern[d, Blank[]]], Pattern[x, Blank[]]], Times[Optional[Pattern[c, Blank[]]], Sin[Times[Optional[Pattern[a, Blank[]]], Pattern[x, Blank[]]]]]], -2]], Pattern[x, Blank[Symbol]]] := Simp[(b*(b*x)^(m - 1)*Csc[a*x]^(n + 1))/(a*d*(c*Sin[a*x] + d*x*Cos[a*x])), x] + Dist[(b^2*(n + 1))/d^2, Int[(b*x)^(m - 2)*Csc[a*x]^(n + 2), x], x] /; FreeQ[{a, b, c, d, m, n}, x] && EqQ[a*c + d, 0] && EqQ[m, n + 2]
  4601. Int[Times[Power[Times[Optional[Pattern[b, Blank[]]], Pattern[x, Blank[]]], Optional[Pattern[m, Blank[]]]], Power[Sec[Times[Optional[Pattern[a, Blank[]]], Pattern[x, Blank[]]]], Optional[Pattern[n, Blank[]]]], Power[Plus[Times[Cos[Times[Optional[Pattern[a, Blank[]]], Pattern[x, Blank[]]]], Optional[Pattern[c, Blank[]]]], Times[Optional[Pattern[d, Blank[]]], Pattern[x, Blank[]], Sin[Times[Optional[Pattern[a, Blank[]]], Pattern[x, Blank[]]]]]], -2]], Pattern[x, Blank[Symbol]]] := -Simp[(b*(b*x)^(m - 1)*Sec[a*x]^(n + 1))/(a*d*(c*Cos[a*x] + d*x*Sin[a*x])), x] + Dist[(b^2*(n + 1))/d^2, Int[(b*x)^(m - 2)*Sec[a*x]^(n + 2), x], x] /; FreeQ[{a, b, c, d, m, n}, x] && EqQ[a*c - d, 0] && EqQ[m, n + 2]
  4602. Int[Times[Power[Plus[Optional[Pattern[g, Blank[]]], Times[Optional[Pattern[h, Blank[]]], Pattern[x, Blank[]]]], Optional[Pattern[p, Blank[]]]], Power[Plus[Pattern[a, Blank[]], Times[Optional[Pattern[b, Blank[]]], Sin[Plus[Optional[Pattern[e, Blank[]]], Times[Optional[Pattern[f, Blank[]]], Pattern[x, Blank[]]]]]]], Optional[Pattern[m, Blank[]]]], Power[Plus[Pattern[c, Blank[]], Times[Optional[Pattern[d, Blank[]]], Sin[Plus[Optional[Pattern[e, Blank[]]], Times[Optional[Pattern[f, Blank[]]], Pattern[x, Blank[]]]]]]], Optional[Pattern[n, Blank[]]]]], Pattern[x, Blank[Symbol]]] := Dist[a^m*c^m, Int[(g + h*x)^p*Cos[e + f*x]^(2*m)*(c + d*Sin[e + f*x])^(n - m), x], x] /; FreeQ[{a, b, c, d, e, f, g, h}, x] && EqQ[b*c + a*d, 0] && EqQ[a^2 - b^2, 0] && IntegerQ[m] && IGtQ[n - m, 0]
  4603. Int[Times[Power[Plus[Times[Cos[Plus[Optional[Pattern[e, Blank[]]], Times[Optional[Pattern[f, Blank[]]], Pattern[x, Blank[]]]]], Optional[Pattern[b, Blank[]]]], Pattern[a, Blank[]]], Optional[Pattern[m, Blank[]]]], Power[Plus[Times[Cos[Plus[Optional[Pattern[e, Blank[]]], Times[Optional[Pattern[f, Blank[]]], Pattern[x, Blank[]]]]], Optional[Pattern[d, Blank[]]]], Pattern[c, Blank[]]], Optional[Pattern[n, Blank[]]]], Power[Plus[Optional[Pattern[g, Blank[]]], Times[Optional[Pattern[h, Blank[]]], Pattern[x, Blank[]]]], Optional[Pattern[p, Blank[]]]]], Pattern[x, Blank[Symbol]]] := Dist[a^m*c^m, Int[(g + h*x)^p*Sin[e + f*x]^(2*m)*(c + d*Cos[e + f*x])^(n - m), x], x] /; FreeQ[{a, b, c, d, e, f, g, h}, x] && EqQ[b*c + a*d, 0] && EqQ[a^2 - b^2, 0] && IntegerQ[m] && IGtQ[n - m, 0]
  4604. Int[Times[Power[Plus[Optional[Pattern[g, Blank[]]], Times[Optional[Pattern[h, Blank[]]], Pattern[x, Blank[]]]], Optional[Pattern[p, Blank[]]]], Power[Plus[Pattern[a, Blank[]], Times[Optional[Pattern[b, Blank[]]], Sin[Plus[Optional[Pattern[e, Blank[]]], Times[Optional[Pattern[f, Blank[]]], Pattern[x, Blank[]]]]]]], Pattern[m, Blank[]]], Power[Plus[Pattern[c, Blank[]], Times[Optional[Pattern[d, Blank[]]], Sin[Plus[Optional[Pattern[e, Blank[]]], Times[Optional[Pattern[f, Blank[]]], Pattern[x, Blank[]]]]]]], Pattern[n, Blank[]]]], Pattern[x, Blank[Symbol]]] := Dist[(a^IntPart[m]*c^IntPart[m]*(a + b*Sin[e + f*x])^FracPart[m]*(c + d*Sin[e + f*x])^FracPart[m])/Cos[e + f*x]^(2*FracPart[m]), Int[(g + h*x)^p*Cos[e + f*x]^(2*m)*(c + d*Sin[e + f*x])^(n - m), x], x] /; FreeQ[{a, b, c, d, e, f, g, h}, x] && EqQ[b*c + a*d, 0] && EqQ[a^2 - b^2, 0] && IntegerQ[p] && IntegerQ[2*m] && IGeQ[n - m, 0]
  4605. Int[Times[Power[Plus[Times[Cos[Plus[Optional[Pattern[e, Blank[]]], Times[Optional[Pattern[f, Blank[]]], Pattern[x, Blank[]]]]], Optional[Pattern[b, Blank[]]]], Pattern[a, Blank[]]], Pattern[m, Blank[]]], Power[Plus[Times[Cos[Plus[Optional[Pattern[e, Blank[]]], Times[Optional[Pattern[f, Blank[]]], Pattern[x, Blank[]]]]], Optional[Pattern[d, Blank[]]]], Pattern[c, Blank[]]], Pattern[n, Blank[]]], Power[Plus[Optional[Pattern[g, Blank[]]], Times[Optional[Pattern[h, Blank[]]], Pattern[x, Blank[]]]], Optional[Pattern[p, Blank[]]]]], Pattern[x, Blank[Symbol]]] := Dist[(a^IntPart[m]*c^IntPart[m]*(a + b*Cos[e + f*x])^FracPart[m]*(c + d*Cos[e + f*x])^FracPart[m])/Sin[e + f*x]^(2*FracPart[m]), Int[(g + h*x)^p*Sin[e + f*x]^(2*m)*(c + d*Cos[e + f*x])^(n - m), x], x] /; FreeQ[{a, b, c, d, e, f, g, h}, x] && EqQ[b*c + a*d, 0] && EqQ[a^2 - b^2, 0] && IntegerQ[p] && IntegerQ[2*m] && IGeQ[n - m, 0]
  4606. Int[Times[Power[Sec[Pattern[v, Blank[]]], Optional[Pattern[m, Blank[]]]], Power[Plus[Pattern[a, Blank[]], Times[Optional[Pattern[b, Blank[]]], Tan[Pattern[v, Blank[]]]]], Optional[Pattern[n, Blank[]]]]], Pattern[x, Blank[Symbol]]] := Int[(a*Cos[v] + b*Sin[v])^n, x] /; FreeQ[{a, b}, x] && IntegerQ[(m - 1)/2] && EqQ[m + n, 0]
  4607. Int[Times[Power[Csc[Pattern[v, Blank[]]], Optional[Pattern[m, Blank[]]]], Power[Plus[Times[Cot[Pattern[v, Blank[]]], Optional[Pattern[b, Blank[]]]], Pattern[a, Blank[]]], Optional[Pattern[n, Blank[]]]]], Pattern[x, Blank[Symbol]]] := Int[(b*Cos[v] + a*Sin[v])^n, x] /; FreeQ[{a, b}, x] && IntegerQ[(m - 1)/2] && EqQ[m + n, 0]
  4608. Int[Times[Optional[Pattern[u, Blank[]]], Power[Sin[Plus[Optional[Pattern[a, Blank[]]], Times[Optional[Pattern[b, Blank[]]], Pattern[x, Blank[]]]]], Optional[Pattern[m, Blank[]]]], Power[Sin[Plus[Optional[Pattern[c, Blank[]]], Times[Optional[Pattern[d, Blank[]]], Pattern[x, Blank[]]]]], Optional[Pattern[n, Blank[]]]]], Pattern[x, Blank[Symbol]]] := Int[ExpandTrigReduce[u, Sin[a + b*x]^m*Sin[c + d*x]^n, x], x] /; FreeQ[{a, b, c, d}, x] && IGtQ[m, 0] && IGtQ[n, 0]
  4609. Int[Times[Power[Cos[Plus[Optional[Pattern[a, Blank[]]], Times[Optional[Pattern[b, Blank[]]], Pattern[x, Blank[]]]]], Optional[Pattern[m, Blank[]]]], Power[Cos[Plus[Optional[Pattern[c, Blank[]]], Times[Optional[Pattern[d, Blank[]]], Pattern[x, Blank[]]]]], Optional[Pattern[n, Blank[]]]], Optional[Pattern[u, Blank[]]]], Pattern[x, Blank[Symbol]]] := Int[ExpandTrigReduce[u, Cos[a + b*x]^m*Cos[c + d*x]^n, x], x] /; FreeQ[{a, b, c, d}, x] && IGtQ[m, 0] && IGtQ[n, 0]
  4610. Int[Times[Sec[Plus[Optional[Pattern[a, Blank[]]], Times[Optional[Pattern[b, Blank[]]], Pattern[x, Blank[]]]]], Sec[Plus[Pattern[c, Blank[]], Times[Optional[Pattern[d, Blank[]]], Pattern[x, Blank[]]]]]], Pattern[x, Blank[Symbol]]] := -Dist[Csc[(b*c - a*d)/d], Int[Tan[a + b*x], x], x] + Dist[Csc[(b*c - a*d)/b], Int[Tan[c + d*x], x], x] /; FreeQ[{a, b, c, d}, x] && EqQ[b^2 - d^2, 0] && NeQ[b*c - a*d, 0]
  4611. Int[Times[Csc[Plus[Optional[Pattern[a, Blank[]]], Times[Optional[Pattern[b, Blank[]]], Pattern[x, Blank[]]]]], Csc[Plus[Pattern[c, Blank[]], Times[Optional[Pattern[d, Blank[]]], Pattern[x, Blank[]]]]]], Pattern[x, Blank[Symbol]]] := Dist[Csc[(b*c - a*d)/b], Int[Cot[a + b*x], x], x] - Dist[Csc[(b*c - a*d)/d], Int[Cot[c + d*x], x], x] /; FreeQ[{a, b, c, d}, x] && EqQ[b^2 - d^2, 0] && NeQ[b*c - a*d, 0]
  4612. Int[Times[Tan[Plus[Optional[Pattern[a, Blank[]]], Times[Optional[Pattern[b, Blank[]]], Pattern[x, Blank[]]]]], Tan[Plus[Pattern[c, Blank[]], Times[Optional[Pattern[d, Blank[]]], Pattern[x, Blank[]]]]]], Pattern[x, Blank[Symbol]]] := -Simp[(b*x)/d, x] + Dist[(b*Cos[(b*c - a*d)/d])/d, Int[Sec[a + b*x]*Sec[c + d*x], x], x] /; FreeQ[{a, b, c, d}, x] && EqQ[b^2 - d^2, 0] && NeQ[b*c - a*d, 0]
  4613. Int[Times[Cot[Plus[Optional[Pattern[a, Blank[]]], Times[Optional[Pattern[b, Blank[]]], Pattern[x, Blank[]]]]], Cot[Plus[Pattern[c, Blank[]], Times[Optional[Pattern[d, Blank[]]], Pattern[x, Blank[]]]]]], Pattern[x, Blank[Symbol]]] := -Simp[(b*x)/d, x] + Dist[Cos[(b*c - a*d)/d], Int[Csc[a + b*x]*Csc[c + d*x], x], x] /; FreeQ[{a, b, c, d}, x] && EqQ[b^2 - d^2, 0] && NeQ[b*c - a*d, 0]
  4614. Int[Times[Optional[Pattern[u, Blank[]]], Power[Plus[Times[Cos[Pattern[v, Blank[]]], Optional[Pattern[a, Blank[]]]], Times[Optional[Pattern[b, Blank[]]], Sin[Pattern[v, Blank[]]]]], Optional[Pattern[n, Blank[]]]]], Pattern[x, Blank[Symbol]]] := Int[u*(a/E^((a*v)/b))^n, x] /; FreeQ[{a, b, n}, x] && EqQ[a^2 + b^2, 0]
  4615. Int[Sin[Times[Power[Plus[Optional[Pattern[a, Blank[]]], Times[Log[Times[Optional[Pattern[c, Blank[]]], Power[Pattern[x, Blank[]], Optional[Pattern[n, Blank[]]]]]], Optional[Pattern[b, Blank[]]]]], 2], Optional[Pattern[d, Blank[]]]]], Pattern[x, Blank[Symbol]]] := Dist[I/2, Int[E^(-(I*d*(a + b*Log[c*x^n])^2)), x], x] - Dist[I/2, Int[E^(I*d*(a + b*Log[c*x^n])^2), x], x] /; FreeQ[{a, b, c, d, n}, x]
  4616. Int[Cos[Times[Power[Plus[Optional[Pattern[a, Blank[]]], Times[Log[Times[Optional[Pattern[c, Blank[]]], Power[Pattern[x, Blank[]], Optional[Pattern[n, Blank[]]]]]], Optional[Pattern[b, Blank[]]]]], 2], Optional[Pattern[d, Blank[]]]]], Pattern[x, Blank[Symbol]]] := Dist[1/2, Int[E^(-(I*d*(a + b*Log[c*x^n])^2)), x], x] + Dist[1/2, Int[E^(I*d*(a + b*Log[c*x^n])^2), x], x] /; FreeQ[{a, b, c, d, n}, x]
  4617. Int[Times[Power[Times[Optional[Pattern[e, Blank[]]], Pattern[x, Blank[]]], Optional[Pattern[m, Blank[]]]], Sin[Times[Power[Plus[Optional[Pattern[a, Blank[]]], Times[Log[Times[Optional[Pattern[c, Blank[]]], Power[Pattern[x, Blank[]], Optional[Pattern[n, Blank[]]]]]], Optional[Pattern[b, Blank[]]]]], 2], Optional[Pattern[d, Blank[]]]]]], Pattern[x, Blank[Symbol]]] := Dist[I/2, Int[(e*x)^m/E^(I*d*(a + b*Log[c*x^n])^2), x], x] - Dist[I/2, Int[(e*x)^m*E^(I*d*(a + b*Log[c*x^n])^2), x], x] /; FreeQ[{a, b, c, d, e, m, n}, x]
  4618. Int[Times[Cos[Times[Power[Plus[Optional[Pattern[a, Blank[]]], Times[Log[Times[Optional[Pattern[c, Blank[]]], Power[Pattern[x, Blank[]], Optional[Pattern[n, Blank[]]]]]], Optional[Pattern[b, Blank[]]]]], 2], Optional[Pattern[d, Blank[]]]]], Power[Times[Optional[Pattern[e, Blank[]]], Pattern[x, Blank[]]], Optional[Pattern[m, Blank[]]]]], Pattern[x, Blank[Symbol]]] := Dist[1/2, Int[(e*x)^m/E^(I*d*(a + b*Log[c*x^n])^2), x], x] + Dist[1/2, Int[(e*x)^m*E^(I*d*(a + b*Log[c*x^n])^2), x], x] /; FreeQ[{a, b, c, d, e, m, n}, x]
  4619. Int[Power[Plus[Optional[Pattern[a, Blank[]]], Times[ArcSin[Times[Optional[Pattern[c, Blank[]]], Pattern[x, Blank[]]]], Optional[Pattern[b, Blank[]]]]], Optional[Pattern[n, Blank[]]]], Pattern[x, Blank[Symbol]]] := Simp[x*(a + b*ArcSin[c*x])^n, x] - Dist[b*c*n, Int[(x*(a + b*ArcSin[c*x])^(n - 1))/Sqrt[1 - c^2*x^2], x], x] /; FreeQ[{a, b, c}, x] && GtQ[n, 0]
  4620. Int[Power[Plus[Optional[Pattern[a, Blank[]]], Times[ArcCos[Times[Optional[Pattern[c, Blank[]]], Pattern[x, Blank[]]]], Optional[Pattern[b, Blank[]]]]], Optional[Pattern[n, Blank[]]]], Pattern[x, Blank[Symbol]]] := Simp[x*(a + b*ArcCos[c*x])^n, x] + Dist[b*c*n, Int[(x*(a + b*ArcCos[c*x])^(n - 1))/Sqrt[1 - c^2*x^2], x], x] /; FreeQ[{a, b, c}, x] && GtQ[n, 0]
  4621. Int[Power[Plus[Optional[Pattern[a, Blank[]]], Times[ArcSin[Times[Optional[Pattern[c, Blank[]]], Pattern[x, Blank[]]]], Optional[Pattern[b, Blank[]]]]], Pattern[n, Blank[]]], Pattern[x, Blank[Symbol]]] := Simp[(Sqrt[1 - c^2*x^2]*(a + b*ArcSin[c*x])^(n + 1))/(b*c*(n + 1)), x] + Dist[c/(b*(n + 1)), Int[(x*(a + b*ArcSin[c*x])^(n + 1))/Sqrt[1 - c^2*x^2], x], x] /; FreeQ[{a, b, c}, x] && LtQ[n, -1]
  4622. Int[Power[Plus[Optional[Pattern[a, Blank[]]], Times[ArcCos[Times[Optional[Pattern[c, Blank[]]], Pattern[x, Blank[]]]], Optional[Pattern[b, Blank[]]]]], Pattern[n, Blank[]]], Pattern[x, Blank[Symbol]]] := -Simp[(Sqrt[1 - c^2*x^2]*(a + b*ArcCos[c*x])^(n + 1))/(b*c*(n + 1)), x] - Dist[c/(b*(n + 1)), Int[(x*(a + b*ArcCos[c*x])^(n + 1))/Sqrt[1 - c^2*x^2], x], x] /; FreeQ[{a, b, c}, x] && LtQ[n, -1]
  4623. Int[Power[Plus[Optional[Pattern[a, Blank[]]], Times[ArcSin[Times[Optional[Pattern[c, Blank[]]], Pattern[x, Blank[]]]], Optional[Pattern[b, Blank[]]]]], Pattern[n, Blank[]]], Pattern[x, Blank[Symbol]]] := Dist[1/(b*c), Subst[Int[x^n*Cos[a/b - x/b], x], x, a + b*ArcSin[c*x]], x] /; FreeQ[{a, b, c, n}, x]
  4624. Int[Power[Plus[Optional[Pattern[a, Blank[]]], Times[ArcCos[Times[Optional[Pattern[c, Blank[]]], Pattern[x, Blank[]]]], Optional[Pattern[b, Blank[]]]]], Pattern[n, Blank[]]], Pattern[x, Blank[Symbol]]] := Dist[1/(b*c), Subst[Int[x^n*Sin[a/b - x/b], x], x, a + b*ArcCos[c*x]], x] /; FreeQ[{a, b, c, n}, x]
  4625. Int[Times[Power[Plus[Optional[Pattern[a, Blank[]]], Times[ArcSin[Times[Optional[Pattern[c, Blank[]]], Pattern[x, Blank[]]]], Optional[Pattern[b, Blank[]]]]], Optional[Pattern[n, Blank[]]]], Power[Pattern[x, Blank[]], -1]], Pattern[x, Blank[Symbol]]] := Subst[Int[(a + b*x)^n/Tan[x], x], x, ArcSin[c*x]] /; FreeQ[{a, b, c}, x] && IGtQ[n, 0]
  4626. Int[Times[Power[Plus[Optional[Pattern[a, Blank[]]], Times[ArcCos[Times[Optional[Pattern[c, Blank[]]], Pattern[x, Blank[]]]], Optional[Pattern[b, Blank[]]]]], Optional[Pattern[n, Blank[]]]], Power[Pattern[x, Blank[]], -1]], Pattern[x, Blank[Symbol]]] := -Subst[Int[(a + b*x)^n/Cot[x], x], x, ArcCos[c*x]] /; FreeQ[{a, b, c}, x] && IGtQ[n, 0]
  4627. Int[Times[Power[Plus[Optional[Pattern[a, Blank[]]], Times[ArcSin[Times[Optional[Pattern[c, Blank[]]], Pattern[x, Blank[]]]], Optional[Pattern[b, Blank[]]]]], Optional[Pattern[n, Blank[]]]], Power[Times[Optional[Pattern[d, Blank[]]], Pattern[x, Blank[]]], Optional[Pattern[m, Blank[]]]]], Pattern[x, Blank[Symbol]]] := Simp[((d*x)^(m + 1)*(a + b*ArcSin[c*x])^n)/(d*(m + 1)), x] - Dist[(b*c*n)/(d*(m + 1)), Int[((d*x)^(m + 1)*(a + b*ArcSin[c*x])^(n - 1))/Sqrt[1 - c^2*x^2], x], x] /; FreeQ[{a, b, c, d, m}, x] && IGtQ[n, 0] && NeQ[m, -1]
  4628. Int[Times[Power[Plus[Optional[Pattern[a, Blank[]]], Times[ArcCos[Times[Optional[Pattern[c, Blank[]]], Pattern[x, Blank[]]]], Optional[Pattern[b, Blank[]]]]], Optional[Pattern[n, Blank[]]]], Power[Times[Optional[Pattern[d, Blank[]]], Pattern[x, Blank[]]], Optional[Pattern[m, Blank[]]]]], Pattern[x, Blank[Symbol]]] := Simp[((d*x)^(m + 1)*(a + b*ArcCos[c*x])^n)/(d*(m + 1)), x] + Dist[(b*c*n)/(d*(m + 1)), Int[((d*x)^(m + 1)*(a + b*ArcCos[c*x])^(n - 1))/Sqrt[1 - c^2*x^2], x], x] /; FreeQ[{a, b, c, d, m}, x] && IGtQ[n, 0] && NeQ[m, -1]
  4629. Int[Times[Power[Plus[Optional[Pattern[a, Blank[]]], Times[ArcSin[Times[Optional[Pattern[c, Blank[]]], Pattern[x, Blank[]]]], Optional[Pattern[b, Blank[]]]]], Pattern[n, Blank[]]], Power[Pattern[x, Blank[]], Optional[Pattern[m, Blank[]]]]], Pattern[x, Blank[Symbol]]] := Simp[(x^(m + 1)*(a + b*ArcSin[c*x])^n)/(m + 1), x] - Dist[(b*c*n)/(m + 1), Int[(x^(m + 1)*(a + b*ArcSin[c*x])^(n - 1))/Sqrt[1 - c^2*x^2], x], x] /; FreeQ[{a, b, c}, x] && IGtQ[m, 0] && GtQ[n, 0]
  4630. Int[Times[Power[Plus[Optional[Pattern[a, Blank[]]], Times[ArcCos[Times[Optional[Pattern[c, Blank[]]], Pattern[x, Blank[]]]], Optional[Pattern[b, Blank[]]]]], Pattern[n, Blank[]]], Power[Pattern[x, Blank[]], Optional[Pattern[m, Blank[]]]]], Pattern[x, Blank[Symbol]]] := Simp[(x^(m + 1)*(a + b*ArcCos[c*x])^n)/(m + 1), x] + Dist[(b*c*n)/(m + 1), Int[(x^(m + 1)*(a + b*ArcCos[c*x])^(n - 1))/Sqrt[1 - c^2*x^2], x], x] /; FreeQ[{a, b, c}, x] && IGtQ[m, 0] && GtQ[n, 0]
  4631. Int[Times[Power[Plus[Optional[Pattern[a, Blank[]]], Times[ArcSin[Times[Optional[Pattern[c, Blank[]]], Pattern[x, Blank[]]]], Optional[Pattern[b, Blank[]]]]], Pattern[n, Blank[]]], Power[Pattern[x, Blank[]], Optional[Pattern[m, Blank[]]]]], Pattern[x, Blank[Symbol]]] := Simp[(x^m*Sqrt[1 - c^2*x^2]*(a + b*ArcSin[c*x])^(n + 1))/(b*c*(n + 1)), x] - Dist[1/(b*c^(m + 1)*(n + 1)), Subst[Int[ExpandTrigReduce[(a + b*x)^(n + 1), Sin[x]^(m - 1)*(m - (m + 1)*Sin[x]^2), x], x], x, ArcSin[c*x]], x] /; FreeQ[{a, b, c}, x] && IGtQ[m, 0] && GeQ[n, -2] && LtQ[n, -1]
  4632. Int[Times[Power[Plus[Optional[Pattern[a, Blank[]]], Times[ArcCos[Times[Optional[Pattern[c, Blank[]]], Pattern[x, Blank[]]]], Optional[Pattern[b, Blank[]]]]], Pattern[n, Blank[]]], Power[Pattern[x, Blank[]], Optional[Pattern[m, Blank[]]]]], Pattern[x, Blank[Symbol]]] := -Simp[(x^m*Sqrt[1 - c^2*x^2]*(a + b*ArcCos[c*x])^(n + 1))/(b*c*(n + 1)), x] - Dist[1/(b*c^(m + 1)*(n + 1)), Subst[Int[ExpandTrigReduce[(a + b*x)^(n + 1), Cos[x]^(m - 1)*(m - (m + 1)*Cos[x]^2), x], x], x, ArcCos[c*x]], x] /; FreeQ[{a, b, c}, x] && IGtQ[m, 0] && GeQ[n, -2] && LtQ[n, -1]
  4633. Int[Times[Power[Plus[Optional[Pattern[a, Blank[]]], Times[ArcSin[Times[Optional[Pattern[c, Blank[]]], Pattern[x, Blank[]]]], Optional[Pattern[b, Blank[]]]]], Pattern[n, Blank[]]], Power[Pattern[x, Blank[]], Optional[Pattern[m, Blank[]]]]], Pattern[x, Blank[Symbol]]] := Simp[(x^m*Sqrt[1 - c^2*x^2]*(a + b*ArcSin[c*x])^(n + 1))/(b*c*(n + 1)), x] + (Dist[(c*(m + 1))/(b*(n + 1)), Int[(x^(m + 1)*(a + b*ArcSin[c*x])^(n + 1))/Sqrt[1 - c^2*x^2], x], x] - Dist[m/(b*c*(n + 1)), Int[(x^(m - 1)*(a + b*ArcSin[c*x])^(n + 1))/Sqrt[1 - c^2*x^2], x], x]) /; FreeQ[{a, b, c}, x] && IGtQ[m, 0] && LtQ[n, -2]
  4634. Int[Times[Power[Plus[Optional[Pattern[a, Blank[]]], Times[ArcCos[Times[Optional[Pattern[c, Blank[]]], Pattern[x, Blank[]]]], Optional[Pattern[b, Blank[]]]]], Pattern[n, Blank[]]], Power[Pattern[x, Blank[]], Optional[Pattern[m, Blank[]]]]], Pattern[x, Blank[Symbol]]] := -Simp[(x^m*Sqrt[1 - c^2*x^2]*(a + b*ArcCos[c*x])^(n + 1))/(b*c*(n + 1)), x] + (-Dist[(c*(m + 1))/(b*(n + 1)), Int[(x^(m + 1)*(a + b*ArcCos[c*x])^(n + 1))/Sqrt[1 - c^2*x^2], x], x] + Dist[m/(b*c*(n + 1)), Int[(x^(m - 1)*(a + b*ArcCos[c*x])^(n + 1))/Sqrt[1 - c^2*x^2], x], x]) /; FreeQ[{a, b, c}, x] && IGtQ[m, 0] && LtQ[n, -2]
  4635. Int[Times[Power[Plus[Optional[Pattern[a, Blank[]]], Times[ArcSin[Times[Optional[Pattern[c, Blank[]]], Pattern[x, Blank[]]]], Optional[Pattern[b, Blank[]]]]], Pattern[n, Blank[]]], Power[Pattern[x, Blank[]], Optional[Pattern[m, Blank[]]]]], Pattern[x, Blank[Symbol]]] := Dist[1/c^(m + 1), Subst[Int[(a + b*x)^n*Sin[x]^m*Cos[x], x], x, ArcSin[c*x]], x] /; FreeQ[{a, b, c, n}, x] && IGtQ[m, 0]
  4636. Int[Times[Power[Plus[Optional[Pattern[a, Blank[]]], Times[ArcCos[Times[Optional[Pattern[c, Blank[]]], Pattern[x, Blank[]]]], Optional[Pattern[b, Blank[]]]]], Pattern[n, Blank[]]], Power[Pattern[x, Blank[]], Optional[Pattern[m, Blank[]]]]], Pattern[x, Blank[Symbol]]] := -Dist[(c^(m + 1))^(-1), Subst[Int[(a + b*x)^n*Cos[x]^m*Sin[x], x], x, ArcCos[c*x]], x] /; FreeQ[{a, b, c, n}, x] && IGtQ[m, 0]
  4637. Int[Times[Power[Plus[Optional[Pattern[a, Blank[]]], Times[ArcSin[Times[Optional[Pattern[c, Blank[]]], Pattern[x, Blank[]]]], Optional[Pattern[b, Blank[]]]]], Optional[Pattern[n, Blank[]]]], Power[Times[Optional[Pattern[d, Blank[]]], Pattern[x, Blank[]]], Optional[Pattern[m, Blank[]]]]], Pattern[x, Blank[Symbol]]] := Unintegrable[(d*x)^m*(a + b*ArcSin[c*x])^n, x] /; FreeQ[{a, b, c, d, m, n}, x]
  4638. Int[Times[Power[Plus[Optional[Pattern[a, Blank[]]], Times[ArcCos[Times[Optional[Pattern[c, Blank[]]], Pattern[x, Blank[]]]], Optional[Pattern[b, Blank[]]]]], Optional[Pattern[n, Blank[]]]], Power[Times[Optional[Pattern[d, Blank[]]], Pattern[x, Blank[]]], Optional[Pattern[m, Blank[]]]]], Pattern[x, Blank[Symbol]]] := Unintegrable[(d*x)^m*(a + b*ArcCos[c*x])^n, x] /; FreeQ[{a, b, c, d, m, n}, x]
  4639. Int[Times[Power[Plus[Optional[Pattern[a, Blank[]]], Times[ArcSin[Times[Optional[Pattern[c, Blank[]]], Pattern[x, Blank[]]]], Optional[Pattern[b, Blank[]]]]], -1], Power[Plus[Pattern[d, Blank[]], Times[Optional[Pattern[e, Blank[]]], Power[Pattern[x, Blank[]], 2]]], Rational[-1, 2]]], Pattern[x, Blank[Symbol]]] := Simp[Log[a + b*ArcSin[c*x]]/(b*c*Sqrt[d]), x] /; FreeQ[{a, b, c, d, e}, x] && EqQ[c^2*d + e, 0] && GtQ[d, 0]
  4640. Int[Times[Power[Plus[Optional[Pattern[a, Blank[]]], Times[ArcCos[Times[Optional[Pattern[c, Blank[]]], Pattern[x, Blank[]]]], Optional[Pattern[b, Blank[]]]]], -1], Power[Plus[Pattern[d, Blank[]], Times[Optional[Pattern[e, Blank[]]], Power[Pattern[x, Blank[]], 2]]], Rational[-1, 2]]], Pattern[x, Blank[Symbol]]] := -Simp[Log[a + b*ArcCos[c*x]]/(b*c*Sqrt[d]), x] /; FreeQ[{a, b, c, d, e}, x] && EqQ[c^2*d + e, 0] && GtQ[d, 0]
  4641. Int[Times[Power[Plus[Optional[Pattern[a, Blank[]]], Times[ArcSin[Times[Optional[Pattern[c, Blank[]]], Pattern[x, Blank[]]]], Optional[Pattern[b, Blank[]]]]], Optional[Pattern[n, Blank[]]]], Power[Plus[Pattern[d, Blank[]], Times[Optional[Pattern[e, Blank[]]], Power[Pattern[x, Blank[]], 2]]], Rational[-1, 2]]], Pattern[x, Blank[Symbol]]] := Simp[(a + b*ArcSin[c*x])^(n + 1)/(b*c*Sqrt[d]*(n + 1)), x] /; FreeQ[{a, b, c, d, e, n}, x] && EqQ[c^2*d + e, 0] && GtQ[d, 0] && NeQ[n, -1]
  4642. Int[Times[Power[Plus[Optional[Pattern[a, Blank[]]], Times[ArcCos[Times[Optional[Pattern[c, Blank[]]], Pattern[x, Blank[]]]], Optional[Pattern[b, Blank[]]]]], Optional[Pattern[n, Blank[]]]], Power[Plus[Pattern[d, Blank[]], Times[Optional[Pattern[e, Blank[]]], Power[Pattern[x, Blank[]], 2]]], Rational[-1, 2]]], Pattern[x, Blank[Symbol]]] := -Simp[(a + b*ArcCos[c*x])^(n + 1)/(b*c*Sqrt[d]*(n + 1)), x] /; FreeQ[{a, b, c, d, e, n}, x] && EqQ[c^2*d + e, 0] && GtQ[d, 0] && NeQ[n, -1]
  4643. Int[Times[Power[Plus[Optional[Pattern[a, Blank[]]], Times[ArcSin[Times[Optional[Pattern[c, Blank[]]], Pattern[x, Blank[]]]], Optional[Pattern[b, Blank[]]]]], Optional[Pattern[n, Blank[]]]], Power[Plus[Pattern[d, Blank[]], Times[Optional[Pattern[e, Blank[]]], Power[Pattern[x, Blank[]], 2]]], Rational[-1, 2]]], Pattern[x, Blank[Symbol]]] := Dist[Sqrt[1 - c^2*x^2]/Sqrt[d + e*x^2], Int[(a + b*ArcSin[c*x])^n/Sqrt[1 - c^2*x^2], x], x] /; FreeQ[{a, b, c, d, e, n}, x] && EqQ[c^2*d + e, 0] && !GtQ[d, 0]
  4644. Int[Times[Power[Plus[Optional[Pattern[a, Blank[]]], Times[ArcCos[Times[Optional[Pattern[c, Blank[]]], Pattern[x, Blank[]]]], Optional[Pattern[b, Blank[]]]]], Optional[Pattern[n, Blank[]]]], Power[Plus[Pattern[d, Blank[]], Times[Optional[Pattern[e, Blank[]]], Power[Pattern[x, Blank[]], 2]]], Rational[-1, 2]]], Pattern[x, Blank[Symbol]]] := Dist[Sqrt[1 - c^2*x^2]/Sqrt[d + e*x^2], Int[(a + b*ArcCos[c*x])^n/Sqrt[1 - c^2*x^2], x], x] /; FreeQ[{a, b, c, d, e, n}, x] && EqQ[c^2*d + e, 0] && !GtQ[d, 0]
  4645. Int[Times[Plus[Optional[Pattern[a, Blank[]]], Times[ArcSin[Times[Optional[Pattern[c, Blank[]]], Pattern[x, Blank[]]]], Optional[Pattern[b, Blank[]]]]], Power[Plus[Pattern[d, Blank[]], Times[Optional[Pattern[e, Blank[]]], Power[Pattern[x, Blank[]], 2]]], Optional[Pattern[p, Blank[]]]]], Pattern[x, Blank[Symbol]]] := With[{u = IntHide[(d + e*x^2)^p, x]}, Dist[a + b*ArcSin[c*x], u, x] - Dist[b*c, Int[SimplifyIntegrand[u/Sqrt[1 - c^2*x^2], x], x], x]] /; FreeQ[{a, b, c, d, e}, x] && EqQ[c^2*d + e, 0] && IGtQ[p, 0]
  4646. Int[Times[Plus[Optional[Pattern[a, Blank[]]], Times[ArcCos[Times[Optional[Pattern[c, Blank[]]], Pattern[x, Blank[]]]], Optional[Pattern[b, Blank[]]]]], Power[Plus[Pattern[d, Blank[]], Times[Optional[Pattern[e, Blank[]]], Power[Pattern[x, Blank[]], 2]]], Optional[Pattern[p, Blank[]]]]], Pattern[x, Blank[Symbol]]] := With[{u = IntHide[(d + e*x^2)^p, x]}, Dist[a + b*ArcCos[c*x], u, x] + Dist[b*c, Int[SimplifyIntegrand[u/Sqrt[1 - c^2*x^2], x], x], x]] /; FreeQ[{a, b, c, d, e}, x] && EqQ[c^2*d + e, 0] && IGtQ[p, 0]
  4647. Int[Times[Power[Plus[Optional[Pattern[a, Blank[]]], Times[ArcSin[Times[Optional[Pattern[c, Blank[]]], Pattern[x, Blank[]]]], Optional[Pattern[b, Blank[]]]]], Optional[Pattern[n, Blank[]]]], Power[Plus[Pattern[d, Blank[]], Times[Optional[Pattern[e, Blank[]]], Power[Pattern[x, Blank[]], 2]]], Rational[1, 2]]], Pattern[x, Blank[Symbol]]] := Simp[(x*Sqrt[d + e*x^2]*(a + b*ArcSin[c*x])^n)/2, x] + (Dist[Sqrt[d + e*x^2]/(2*Sqrt[1 - c^2*x^2]), Int[(a + b*ArcSin[c*x])^n/Sqrt[1 - c^2*x^2], x], x] - Dist[(b*c*n*Sqrt[d + e*x^2])/(2*Sqrt[1 - c^2*x^2]), Int[x*(a + b*ArcSin[c*x])^(n - 1), x], x]) /; FreeQ[{a, b, c, d, e}, x] && EqQ[c^2*d + e, 0] && GtQ[n, 0]
  4648. Int[Times[Power[Plus[Optional[Pattern[a, Blank[]]], Times[ArcCos[Times[Optional[Pattern[c, Blank[]]], Pattern[x, Blank[]]]], Optional[Pattern[b, Blank[]]]]], Optional[Pattern[n, Blank[]]]], Power[Plus[Pattern[d, Blank[]], Times[Optional[Pattern[e, Blank[]]], Power[Pattern[x, Blank[]], 2]]], Rational[1, 2]]], Pattern[x, Blank[Symbol]]] := Simp[(x*Sqrt[d + e*x^2]*(a + b*ArcCos[c*x])^n)/2, x] + (Dist[Sqrt[d + e*x^2]/(2*Sqrt[1 - c^2*x^2]), Int[(a + b*ArcCos[c*x])^n/Sqrt[1 - c^2*x^2], x], x] + Dist[(b*c*n*Sqrt[d + e*x^2])/(2*Sqrt[1 - c^2*x^2]), Int[x*(a + b*ArcCos[c*x])^(n - 1), x], x]) /; FreeQ[{a, b, c, d, e}, x] && EqQ[c^2*d + e, 0] && GtQ[n, 0]
  4649. Int[Times[Power[Plus[Optional[Pattern[a, Blank[]]], Times[ArcSin[Times[Optional[Pattern[c, Blank[]]], Pattern[x, Blank[]]]], Optional[Pattern[b, Blank[]]]]], Optional[Pattern[n, Blank[]]]], Power[Plus[Pattern[d, Blank[]], Times[Optional[Pattern[e, Blank[]]], Power[Pattern[x, Blank[]], 2]]], Optional[Pattern[p, Blank[]]]]], Pattern[x, Blank[Symbol]]] := Simp[(x*(d + e*x^2)^p*(a + b*ArcSin[c*x])^n)/(2*p + 1), x] + (Dist[(2*d*p)/(2*p + 1), Int[(d + e*x^2)^(p - 1)*(a + b*ArcSin[c*x])^n, x], x] - Dist[(b*c*n*d^IntPart[p]*(d + e*x^2)^FracPart[p])/((2*p + 1)*(1 - c^2*x^2)^FracPart[p]), Int[x*(1 - c^2*x^2)^(p - 1/2)*(a + b*ArcSin[c*x])^(n - 1), x], x]) /; FreeQ[{a, b, c, d, e}, x] && EqQ[c^2*d + e, 0] && GtQ[n, 0] && GtQ[p, 0]
  4650. Int[Times[Power[Plus[Optional[Pattern[a, Blank[]]], Times[ArcCos[Times[Optional[Pattern[c, Blank[]]], Pattern[x, Blank[]]]], Optional[Pattern[b, Blank[]]]]], Optional[Pattern[n, Blank[]]]], Power[Plus[Pattern[d, Blank[]], Times[Optional[Pattern[e, Blank[]]], Power[Pattern[x, Blank[]], 2]]], Optional[Pattern[p, Blank[]]]]], Pattern[x, Blank[Symbol]]] := Simp[(x*(d + e*x^2)^p*(a + b*ArcCos[c*x])^n)/(2*p + 1), x] + (Dist[(2*d*p)/(2*p + 1), Int[(d + e*x^2)^(p - 1)*(a + b*ArcCos[c*x])^n, x], x] + Dist[(b*c*n*d^IntPart[p]*(d + e*x^2)^FracPart[p])/((2*p + 1)*(1 - c^2*x^2)^FracPart[p]), Int[x*(1 - c^2*x^2)^(p - 1/2)*(a + b*ArcCos[c*x])^(n - 1), x], x]) /; FreeQ[{a, b, c, d, e}, x] && EqQ[c^2*d + e, 0] && GtQ[n, 0] && GtQ[p, 0]
  4651. Int[Times[Power[Plus[Optional[Pattern[a, Blank[]]], Times[ArcSin[Times[Optional[Pattern[c, Blank[]]], Pattern[x, Blank[]]]], Optional[Pattern[b, Blank[]]]]], Optional[Pattern[n, Blank[]]]], Power[Plus[Pattern[d, Blank[]], Times[Optional[Pattern[e, Blank[]]], Power[Pattern[x, Blank[]], 2]]], Rational[-3, 2]]], Pattern[x, Blank[Symbol]]] := Simp[(x*(a + b*ArcSin[c*x])^n)/(d*Sqrt[d + e*x^2]), x] - Dist[(b*c*n)/Sqrt[d], Int[(x*(a + b*ArcSin[c*x])^(n - 1))/(d + e*x^2), x], x] /; FreeQ[{a, b, c, d, e}, x] && EqQ[c^2*d + e, 0] && GtQ[n, 0] && GtQ[d, 0]
  4652. Int[Times[Power[Plus[Optional[Pattern[a, Blank[]]], Times[ArcCos[Times[Optional[Pattern[c, Blank[]]], Pattern[x, Blank[]]]], Optional[Pattern[b, Blank[]]]]], Optional[Pattern[n, Blank[]]]], Power[Plus[Pattern[d, Blank[]], Times[Optional[Pattern[e, Blank[]]], Power[Pattern[x, Blank[]], 2]]], Rational[-3, 2]]], Pattern[x, Blank[Symbol]]] := Simp[(x*(a + b*ArcCos[c*x])^n)/(d*Sqrt[d + e*x^2]), x] + Dist[(b*c*n)/Sqrt[d], Int[(x*(a + b*ArcCos[c*x])^(n - 1))/(d + e*x^2), x], x] /; FreeQ[{a, b, c, d, e}, x] && EqQ[c^2*d + e, 0] && GtQ[n, 0] && GtQ[d, 0]
  4653. Int[Times[Power[Plus[Optional[Pattern[a, Blank[]]], Times[ArcSin[Times[Optional[Pattern[c, Blank[]]], Pattern[x, Blank[]]]], Optional[Pattern[b, Blank[]]]]], Optional[Pattern[n, Blank[]]]], Power[Plus[Pattern[d, Blank[]], Times[Optional[Pattern[e, Blank[]]], Power[Pattern[x, Blank[]], 2]]], Rational[-3, 2]]], Pattern[x, Blank[Symbol]]] := Simp[(x*(a + b*ArcSin[c*x])^n)/(d*Sqrt[d + e*x^2]), x] - Dist[(b*c*n*Sqrt[1 - c^2*x^2])/(d*Sqrt[d + e*x^2]), Int[(x*(a + b*ArcSin[c*x])^(n - 1))/(1 - c^2*x^2), x], x] /; FreeQ[{a, b, c, d, e}, x] && EqQ[c^2*d + e, 0] && GtQ[n, 0]
  4654. Int[Times[Power[Plus[Optional[Pattern[a, Blank[]]], Times[ArcCos[Times[Optional[Pattern[c, Blank[]]], Pattern[x, Blank[]]]], Optional[Pattern[b, Blank[]]]]], Optional[Pattern[n, Blank[]]]], Power[Plus[Pattern[d, Blank[]], Times[Optional[Pattern[e, Blank[]]], Power[Pattern[x, Blank[]], 2]]], Rational[-3, 2]]], Pattern[x, Blank[Symbol]]] := Simp[(x*(a + b*ArcCos[c*x])^n)/(d*Sqrt[d + e*x^2]), x] + Dist[(b*c*n*Sqrt[1 - c^2*x^2])/(d*Sqrt[d + e*x^2]), Int[(x*(a + b*ArcCos[c*x])^(n - 1))/(1 - c^2*x^2), x], x] /; FreeQ[{a, b, c, d, e}, x] && EqQ[c^2*d + e, 0] && GtQ[n, 0]
  4655. Int[Times[Power[Plus[Optional[Pattern[a, Blank[]]], Times[ArcSin[Times[Optional[Pattern[c, Blank[]]], Pattern[x, Blank[]]]], Optional[Pattern[b, Blank[]]]]], Optional[Pattern[n, Blank[]]]], Power[Plus[Pattern[d, Blank[]], Times[Optional[Pattern[e, Blank[]]], Power[Pattern[x, Blank[]], 2]]], Pattern[p, Blank[]]]], Pattern[x, Blank[Symbol]]] := -Simp[(x*(d + e*x^2)^(p + 1)*(a + b*ArcSin[c*x])^n)/(2*d*(p + 1)), x] + (Dist[(2*p + 3)/(2*d*(p + 1)), Int[(d + e*x^2)^(p + 1)*(a + b*ArcSin[c*x])^n, x], x] + Dist[(b*c*n*d^IntPart[p]*(d + e*x^2)^FracPart[p])/(2*(p + 1)*(1 - c^2*x^2)^FracPart[p]), Int[x*(1 - c^2*x^2)^(p + 1/2)*(a + b*ArcSin[c*x])^(n - 1), x], x]) /; FreeQ[{a, b, c, d, e}, x] && EqQ[c^2*d + e, 0] && GtQ[n, 0] && LtQ[p, -1] && NeQ[p, -3/2]
  4656. Int[Times[Power[Plus[Optional[Pattern[a, Blank[]]], Times[ArcCos[Times[Optional[Pattern[c, Blank[]]], Pattern[x, Blank[]]]], Optional[Pattern[b, Blank[]]]]], Optional[Pattern[n, Blank[]]]], Power[Plus[Pattern[d, Blank[]], Times[Optional[Pattern[e, Blank[]]], Power[Pattern[x, Blank[]], 2]]], Pattern[p, Blank[]]]], Pattern[x, Blank[Symbol]]] := -Simp[(x*(d + e*x^2)^(p + 1)*(a + b*ArcCos[c*x])^n)/(2*d*(p + 1)), x] + (Dist[(2*p + 3)/(2*d*(p + 1)), Int[(d + e*x^2)^(p + 1)*(a + b*ArcCos[c*x])^n, x], x] - Dist[(b*c*n*d^IntPart[p]*(d + e*x^2)^FracPart[p])/(2*(p + 1)*(1 - c^2*x^2)^FracPart[p]), Int[x*(1 - c^2*x^2)^(p + 1/2)*(a + b*ArcCos[c*x])^(n - 1), x], x]) /; FreeQ[{a, b, c, d, e}, x] && EqQ[c^2*d + e, 0] && GtQ[n, 0] && LtQ[p, -1] && NeQ[p, -3/2]
  4657. Int[Times[Power[Plus[Optional[Pattern[a, Blank[]]], Times[ArcSin[Times[Optional[Pattern[c, Blank[]]], Pattern[x, Blank[]]]], Optional[Pattern[b, Blank[]]]]], Optional[Pattern[n, Blank[]]]], Power[Plus[Pattern[d, Blank[]], Times[Optional[Pattern[e, Blank[]]], Power[Pattern[x, Blank[]], 2]]], -1]], Pattern[x, Blank[Symbol]]] := Dist[1/(c*d), Subst[Int[(a + b*x)^n*Sec[x], x], x, ArcSin[c*x]], x] /; FreeQ[{a, b, c, d, e}, x] && EqQ[c^2*d + e, 0] && IGtQ[n, 0]
  4658. Int[Times[Power[Plus[Optional[Pattern[a, Blank[]]], Times[ArcCos[Times[Optional[Pattern[c, Blank[]]], Pattern[x, Blank[]]]], Optional[Pattern[b, Blank[]]]]], Optional[Pattern[n, Blank[]]]], Power[Plus[Pattern[d, Blank[]], Times[Optional[Pattern[e, Blank[]]], Power[Pattern[x, Blank[]], 2]]], -1]], Pattern[x, Blank[Symbol]]] := -Dist[(c*d)^(-1), Subst[Int[(a + b*x)^n*Csc[x], x], x, ArcCos[c*x]], x] /; FreeQ[{a, b, c, d, e}, x] && EqQ[c^2*d + e, 0] && IGtQ[n, 0]
  4659. Int[Times[Power[Plus[Optional[Pattern[a, Blank[]]], Times[ArcSin[Times[Optional[Pattern[c, Blank[]]], Pattern[x, Blank[]]]], Optional[Pattern[b, Blank[]]]]], Pattern[n, Blank[]]], Power[Plus[Pattern[d, Blank[]], Times[Optional[Pattern[e, Blank[]]], Power[Pattern[x, Blank[]], 2]]], Optional[Pattern[p, Blank[]]]]], Pattern[x, Blank[Symbol]]] := Simp[(Sqrt[1 - c^2*x^2]*(d + e*x^2)^p*(a + b*ArcSin[c*x])^(n + 1))/(b*c*(n + 1)), x] + Dist[(c*(2*p + 1)*d^IntPart[p]*(d + e*x^2)^FracPart[p])/(b*(n + 1)*(1 - c^2*x^2)^FracPart[p]), Int[x*(1 - c^2*x^2)^(p - 1/2)*(a + b*ArcSin[c*x])^(n + 1), x], x] /; FreeQ[{a, b, c, d, e, p}, x] && EqQ[c^2*d + e, 0] && LtQ[n, -1]
  4660. Int[Times[Power[Plus[Optional[Pattern[a, Blank[]]], Times[ArcCos[Times[Optional[Pattern[c, Blank[]]], Pattern[x, Blank[]]]], Optional[Pattern[b, Blank[]]]]], Pattern[n, Blank[]]], Power[Plus[Pattern[d, Blank[]], Times[Optional[Pattern[e, Blank[]]], Power[Pattern[x, Blank[]], 2]]], Optional[Pattern[p, Blank[]]]]], Pattern[x, Blank[Symbol]]] := -Simp[(Sqrt[1 - c^2*x^2]*(d + e*x^2)^p*(a + b*ArcCos[c*x])^(n + 1))/(b*c*(n + 1)), x] - Dist[(c*(2*p + 1)*d^IntPart[p]*(d + e*x^2)^FracPart[p])/(b*(n + 1)*(1 - c^2*x^2)^FracPart[p]), Int[x*(1 - c^2*x^2)^(p - 1/2)*(a + b*ArcCos[c*x])^(n + 1), x], x] /; FreeQ[{a, b, c, d, e, p}, x] && EqQ[c^2*d + e, 0] && LtQ[n, -1]
  4661. Int[Times[Power[Plus[Optional[Pattern[a, Blank[]]], Times[ArcSin[Times[Optional[Pattern[c, Blank[]]], Pattern[x, Blank[]]]], Optional[Pattern[b, Blank[]]]]], Optional[Pattern[n, Blank[]]]], Power[Plus[Pattern[d, Blank[]], Times[Optional[Pattern[e, Blank[]]], Power[Pattern[x, Blank[]], 2]]], Optional[Pattern[p, Blank[]]]]], Pattern[x, Blank[Symbol]]] := Dist[d^p/c, Subst[Int[(a + b*x)^n*Cos[x]^(2*p + 1), x], x, ArcSin[c*x]], x] /; FreeQ[{a, b, c, d, e, n}, x] && EqQ[c^2*d + e, 0] && IGtQ[2*p, 0] && (IntegerQ[p] || GtQ[d, 0])
  4662. Int[Times[Power[Plus[Optional[Pattern[a, Blank[]]], Times[ArcCos[Times[Optional[Pattern[c, Blank[]]], Pattern[x, Blank[]]]], Optional[Pattern[b, Blank[]]]]], Optional[Pattern[n, Blank[]]]], Power[Plus[Pattern[d, Blank[]], Times[Optional[Pattern[e, Blank[]]], Power[Pattern[x, Blank[]], 2]]], Optional[Pattern[p, Blank[]]]]], Pattern[x, Blank[Symbol]]] := -Dist[d^p/c, Subst[Int[(a + b*x)^n*Sin[x]^(2*p + 1), x], x, ArcCos[c*x]], x] /; FreeQ[{a, b, c, d, e, n}, x] && EqQ[c^2*d + e, 0] && IGtQ[2*p, 0] && (IntegerQ[p] || GtQ[d, 0])
  4663. Int[Times[Power[Plus[Optional[Pattern[a, Blank[]]], Times[ArcSin[Times[Optional[Pattern[c, Blank[]]], Pattern[x, Blank[]]]], Optional[Pattern[b, Blank[]]]]], Optional[Pattern[n, Blank[]]]], Power[Plus[Pattern[d, Blank[]], Times[Optional[Pattern[e, Blank[]]], Power[Pattern[x, Blank[]], 2]]], Optional[Pattern[p, Blank[]]]]], Pattern[x, Blank[Symbol]]] := Dist[(d^(p - 1/2)*Sqrt[d + e*x^2])/Sqrt[1 - c^2*x^2], Int[(1 - c^2*x^2)^p*(a + b*ArcSin[c*x])^n, x], x] /; FreeQ[{a, b, c, d, e, n}, x] && EqQ[c^2*d + e, 0] && IGtQ[2*p, 0] && !(IntegerQ[p] || GtQ[d, 0])
  4664. Int[Times[Power[Plus[Optional[Pattern[a, Blank[]]], Times[ArcCos[Times[Optional[Pattern[c, Blank[]]], Pattern[x, Blank[]]]], Optional[Pattern[b, Blank[]]]]], Optional[Pattern[n, Blank[]]]], Power[Plus[Pattern[d, Blank[]], Times[Optional[Pattern[e, Blank[]]], Power[Pattern[x, Blank[]], 2]]], Optional[Pattern[p, Blank[]]]]], Pattern[x, Blank[Symbol]]] := Dist[(d^(p - 1/2)*Sqrt[d + e*x^2])/Sqrt[1 - c^2*x^2], Int[(1 - c^2*x^2)^p*(a + b*ArcCos[c*x])^n, x], x] /; FreeQ[{a, b, c, d, e, n}, x] && EqQ[c^2*d + e, 0] && IGtQ[2*p, 0] && !(IntegerQ[p] || GtQ[d, 0])
  4665. Int[Times[Plus[Optional[Pattern[a, Blank[]]], Times[ArcSin[Times[Optional[Pattern[c, Blank[]]], Pattern[x, Blank[]]]], Optional[Pattern[b, Blank[]]]]], Power[Plus[Pattern[d, Blank[]], Times[Optional[Pattern[e, Blank[]]], Power[Pattern[x, Blank[]], 2]]], Optional[Pattern[p, Blank[]]]]], Pattern[x, Blank[Symbol]]] := With[{u = IntHide[(d + e*x^2)^p, x]}, Dist[a + b*ArcSin[c*x], u, x] - Dist[b*c, Int[SimplifyIntegrand[u/Sqrt[1 - c^2*x^2], x], x], x]] /; FreeQ[{a, b, c, d, e}, x] && NeQ[c^2*d + e, 0] && (IGtQ[p, 0] || ILtQ[p + 1/2, 0])
  4666. Int[Times[Plus[Optional[Pattern[a, Blank[]]], Times[ArcCos[Times[Optional[Pattern[c, Blank[]]], Pattern[x, Blank[]]]], Optional[Pattern[b, Blank[]]]]], Power[Plus[Pattern[d, Blank[]], Times[Optional[Pattern[e, Blank[]]], Power[Pattern[x, Blank[]], 2]]], Optional[Pattern[p, Blank[]]]]], Pattern[x, Blank[Symbol]]] := With[{u = IntHide[(d + e*x^2)^p, x]}, Dist[a + b*ArcCos[c*x], u, x] + Dist[b*c, Int[SimplifyIntegrand[u/Sqrt[1 - c^2*x^2], x], x], x]] /; FreeQ[{a, b, c, d, e}, x] && NeQ[c^2*d + e, 0] && (IGtQ[p, 0] || ILtQ[p + 1/2, 0])
  4667. Int[Times[Power[Plus[Optional[Pattern[a, Blank[]]], Times[ArcSin[Times[Optional[Pattern[c, Blank[]]], Pattern[x, Blank[]]]], Optional[Pattern[b, Blank[]]]]], Optional[Pattern[n, Blank[]]]], Power[Plus[Pattern[d, Blank[]], Times[Optional[Pattern[e, Blank[]]], Power[Pattern[x, Blank[]], 2]]], Optional[Pattern[p, Blank[]]]]], Pattern[x, Blank[Symbol]]] := Int[ExpandIntegrand[(a + b*ArcSin[c*x])^n, (d + e*x^2)^p, x], x] /; FreeQ[{a, b, c, d, e, n}, x] && NeQ[c^2*d + e, 0] && IntegerQ[p] && (GtQ[p, 0] || IGtQ[n, 0])
  4668. Int[Times[Power[Plus[Optional[Pattern[a, Blank[]]], Times[ArcCos[Times[Optional[Pattern[c, Blank[]]], Pattern[x, Blank[]]]], Optional[Pattern[b, Blank[]]]]], Optional[Pattern[n, Blank[]]]], Power[Plus[Pattern[d, Blank[]], Times[Optional[Pattern[e, Blank[]]], Power[Pattern[x, Blank[]], 2]]], Optional[Pattern[p, Blank[]]]]], Pattern[x, Blank[Symbol]]] := Int[ExpandIntegrand[(a + b*ArcCos[c*x])^n, (d + e*x^2)^p, x], x] /; FreeQ[{a, b, c, d, e, n}, x] && NeQ[c^2*d + e, 0] && IntegerQ[p] && (GtQ[p, 0] || IGtQ[n, 0])
  4669. Int[Times[Power[Plus[Optional[Pattern[a, Blank[]]], Times[ArcSin[Times[Optional[Pattern[c, Blank[]]], Pattern[x, Blank[]]]], Optional[Pattern[b, Blank[]]]]], Optional[Pattern[n, Blank[]]]], Power[Plus[Pattern[d, Blank[]], Times[Optional[Pattern[e, Blank[]]], Power[Pattern[x, Blank[]], 2]]], Optional[Pattern[p, Blank[]]]]], Pattern[x, Blank[Symbol]]] := Unintegrable[(d + e*x^2)^p*(a + b*ArcSin[c*x])^n, x] /; FreeQ[{a, b, c, d, e, n, p}, x]
  4670. Int[Times[Power[Plus[Optional[Pattern[a, Blank[]]], Times[ArcCos[Times[Optional[Pattern[c, Blank[]]], Pattern[x, Blank[]]]], Optional[Pattern[b, Blank[]]]]], Optional[Pattern[n, Blank[]]]], Power[Plus[Pattern[d, Blank[]], Times[Optional[Pattern[e, Blank[]]], Power[Pattern[x, Blank[]], 2]]], Optional[Pattern[p, Blank[]]]]], Pattern[x, Blank[Symbol]]] := Unintegrable[(d + e*x^2)^p*(a + b*ArcCos[c*x])^n, x] /; FreeQ[{a, b, c, d, e, n, p}, x]
  4671. Int[Times[Power[Plus[Optional[Pattern[a, Blank[]]], Times[ArcSin[Times[Optional[Pattern[c, Blank[]]], Pattern[x, Blank[]]]], Optional[Pattern[b, Blank[]]]]], Optional[Pattern[n, Blank[]]]], Power[Plus[Pattern[d, Blank[]], Times[Optional[Pattern[e, Blank[]]], Pattern[x, Blank[]]]], Pattern[p, Blank[]]], Power[Plus[Pattern[f, Blank[]], Times[Optional[Pattern[g, Blank[]]], Pattern[x, Blank[]]]], Pattern[q, Blank[]]]], Pattern[x, Blank[Symbol]]] := Dist[(-((d^2*g)/e))^q, Int[(d + e*x)^(p - q)*(1 - c^2*x^2)^q*(a + b*ArcSin[c*x])^n, x], x] /; FreeQ[{a, b, c, d, e, f, g, n}, x] && EqQ[e*f + d*g, 0] && EqQ[c^2*d^2 - e^2, 0] && HalfIntegerQ[p, q] && GeQ[p - q, 0] && GtQ[d, 0] && LtQ[g/e, 0]
  4672. Int[Times[Power[Plus[Optional[Pattern[a, Blank[]]], Times[ArcCos[Times[Optional[Pattern[c, Blank[]]], Pattern[x, Blank[]]]], Optional[Pattern[b, Blank[]]]]], Optional[Pattern[n, Blank[]]]], Power[Plus[Pattern[d, Blank[]], Times[Optional[Pattern[e, Blank[]]], Pattern[x, Blank[]]]], Pattern[p, Blank[]]], Power[Plus[Pattern[f, Blank[]], Times[Optional[Pattern[g, Blank[]]], Pattern[x, Blank[]]]], Pattern[q, Blank[]]]], Pattern[x, Blank[Symbol]]] := Dist[(-((d^2*g)/e))^q, Int[(d + e*x)^(p - q)*(1 - c^2*x^2)^q*(a + b*ArcCos[c*x])^n, x], x] /; FreeQ[{a, b, c, d, e, f, g, n}, x] && EqQ[e*f + d*g, 0] && EqQ[c^2*d^2 - e^2, 0] && HalfIntegerQ[p, q] && GeQ[p - q, 0] && GtQ[d, 0] && LtQ[g/e, 0]
  4673. Int[Times[Power[Plus[Optional[Pattern[a, Blank[]]], Times[ArcSin[Times[Optional[Pattern[c, Blank[]]], Pattern[x, Blank[]]]], Optional[Pattern[b, Blank[]]]]], Optional[Pattern[n, Blank[]]]], Power[Plus[Pattern[d, Blank[]], Times[Optional[Pattern[e, Blank[]]], Pattern[x, Blank[]]]], Pattern[p, Blank[]]], Power[Plus[Pattern[f, Blank[]], Times[Optional[Pattern[g, Blank[]]], Pattern[x, Blank[]]]], Pattern[q, Blank[]]]], Pattern[x, Blank[Symbol]]] := Dist[((d + e*x)^q*(f + g*x)^q)/(1 - c^2*x^2)^q, Int[(d + e*x)^(p - q)*(1 - c^2*x^2)^q*(a + b*ArcSin[c*x])^n, x], x] /; FreeQ[{a, b, c, d, e, f, g, n}, x] && EqQ[e*f + d*g, 0] && EqQ[c^2*d^2 - e^2, 0] && HalfIntegerQ[p, q] && GeQ[p - q, 0]
  4674. Int[Times[Power[Plus[Optional[Pattern[a, Blank[]]], Times[ArcCos[Times[Optional[Pattern[c, Blank[]]], Pattern[x, Blank[]]]], Optional[Pattern[b, Blank[]]]]], Optional[Pattern[n, Blank[]]]], Power[Plus[Pattern[d, Blank[]], Times[Optional[Pattern[e, Blank[]]], Pattern[x, Blank[]]]], Pattern[p, Blank[]]], Power[Plus[Pattern[f, Blank[]], Times[Optional[Pattern[g, Blank[]]], Pattern[x, Blank[]]]], Pattern[q, Blank[]]]], Pattern[x, Blank[Symbol]]] := Dist[((d + e*x)^q*(f + g*x)^q)/(1 - c^2*x^2)^q, Int[(d + e*x)^(p - q)*(1 - c^2*x^2)^q*(a + b*ArcCos[c*x])^n, x], x] /; FreeQ[{a, b, c, d, e, f, g, n}, x] && EqQ[e*f + d*g, 0] && EqQ[c^2*d^2 - e^2, 0] && HalfIntegerQ[p, q] && GeQ[p - q, 0]
  4675. Int[Times[Power[Plus[Optional[Pattern[a, Blank[]]], Times[ArcSin[Times[Optional[Pattern[c, Blank[]]], Pattern[x, Blank[]]]], Optional[Pattern[b, Blank[]]]]], Optional[Pattern[n, Blank[]]]], Pattern[x, Blank[]], Power[Plus[Pattern[d, Blank[]], Times[Optional[Pattern[e, Blank[]]], Power[Pattern[x, Blank[]], 2]]], -1]], Pattern[x, Blank[Symbol]]] := -Dist[e^(-1), Subst[Int[(a + b*x)^n*Tan[x], x], x, ArcSin[c*x]], x] /; FreeQ[{a, b, c, d, e}, x] && EqQ[c^2*d + e, 0] && IGtQ[n, 0]
  4676. Int[Times[Power[Plus[Optional[Pattern[a, Blank[]]], Times[ArcCos[Times[Optional[Pattern[c, Blank[]]], Pattern[x, Blank[]]]], Optional[Pattern[b, Blank[]]]]], Optional[Pattern[n, Blank[]]]], Pattern[x, Blank[]], Power[Plus[Pattern[d, Blank[]], Times[Optional[Pattern[e, Blank[]]], Power[Pattern[x, Blank[]], 2]]], -1]], Pattern[x, Blank[Symbol]]] := Dist[1/e, Subst[Int[(a + b*x)^n*Cot[x], x], x, ArcCos[c*x]], x] /; FreeQ[{a, b, c, d, e}, x] && EqQ[c^2*d + e, 0] && IGtQ[n, 0]
  4677. Int[Times[Power[Plus[Optional[Pattern[a, Blank[]]], Times[ArcSin[Times[Optional[Pattern[c, Blank[]]], Pattern[x, Blank[]]]], Optional[Pattern[b, Blank[]]]]], Optional[Pattern[n, Blank[]]]], Pattern[x, Blank[]], Power[Plus[Pattern[d, Blank[]], Times[Optional[Pattern[e, Blank[]]], Power[Pattern[x, Blank[]], 2]]], Optional[Pattern[p, Blank[]]]]], Pattern[x, Blank[Symbol]]] := Simp[((d + e*x^2)^(p + 1)*(a + b*ArcSin[c*x])^n)/(2*e*(p + 1)), x] + Dist[(b*n*d^IntPart[p]*(d + e*x^2)^FracPart[p])/(2*c*(p + 1)*(1 - c^2*x^2)^FracPart[p]), Int[(1 - c^2*x^2)^(p + 1/2)*(a + b*ArcSin[c*x])^(n - 1), x], x] /; FreeQ[{a, b, c, d, e, p}, x] && EqQ[c^2*d + e, 0] && GtQ[n, 0] && NeQ[p, -1]
  4678. Int[Times[Power[Plus[Optional[Pattern[a, Blank[]]], Times[ArcCos[Times[Optional[Pattern[c, Blank[]]], Pattern[x, Blank[]]]], Optional[Pattern[b, Blank[]]]]], Optional[Pattern[n, Blank[]]]], Pattern[x, Blank[]], Power[Plus[Pattern[d, Blank[]], Times[Optional[Pattern[e, Blank[]]], Power[Pattern[x, Blank[]], 2]]], Optional[Pattern[p, Blank[]]]]], Pattern[x, Blank[Symbol]]] := Simp[((d + e*x^2)^(p + 1)*(a + b*ArcCos[c*x])^n)/(2*e*(p + 1)), x] - Dist[(b*n*d^IntPart[p]*(d + e*x^2)^FracPart[p])/(2*c*(p + 1)*(1 - c^2*x^2)^FracPart[p]), Int[(1 - c^2*x^2)^(p + 1/2)*(a + b*ArcCos[c*x])^(n - 1), x], x] /; FreeQ[{a, b, c, d, e, p}, x] && EqQ[c^2*d + e, 0] && GtQ[n, 0] && NeQ[p, -1]
  4679. Int[Times[Power[Plus[Optional[Pattern[a, Blank[]]], Times[ArcSin[Times[Optional[Pattern[c, Blank[]]], Pattern[x, Blank[]]]], Optional[Pattern[b, Blank[]]]]], Optional[Pattern[n, Blank[]]]], Power[Pattern[x, Blank[]], -1], Power[Plus[Pattern[d, Blank[]], Times[Optional[Pattern[e, Blank[]]], Power[Pattern[x, Blank[]], 2]]], -1]], Pattern[x, Blank[Symbol]]] := Dist[1/d, Subst[Int[(a + b*x)^n/(Cos[x]*Sin[x]), x], x, ArcSin[c*x]], x] /; FreeQ[{a, b, c, d, e}, x] && EqQ[c^2*d + e, 0] && IGtQ[n, 0]
  4680. Int[Times[Power[Plus[Optional[Pattern[a, Blank[]]], Times[ArcCos[Times[Optional[Pattern[c, Blank[]]], Pattern[x, Blank[]]]], Optional[Pattern[b, Blank[]]]]], Optional[Pattern[n, Blank[]]]], Power[Pattern[x, Blank[]], -1], Power[Plus[Pattern[d, Blank[]], Times[Optional[Pattern[e, Blank[]]], Power[Pattern[x, Blank[]], 2]]], -1]], Pattern[x, Blank[Symbol]]] := -Dist[d^(-1), Subst[Int[(a + b*x)^n/(Cos[x]*Sin[x]), x], x, ArcCos[c*x]], x] /; FreeQ[{a, b, c, d, e}, x] && EqQ[c^2*d + e, 0] && IGtQ[n, 0]
  4681. Int[Times[Power[Plus[Optional[Pattern[a, Blank[]]], Times[ArcSin[Times[Optional[Pattern[c, Blank[]]], Pattern[x, Blank[]]]], Optional[Pattern[b, Blank[]]]]], Optional[Pattern[n, Blank[]]]], Power[Times[Optional[Pattern[f, Blank[]]], Pattern[x, Blank[]]], Pattern[m, Blank[]]], Power[Plus[Pattern[d, Blank[]], Times[Optional[Pattern[e, Blank[]]], Power[Pattern[x, Blank[]], 2]]], Pattern[p, Blank[]]]], Pattern[x, Blank[Symbol]]] := Simp[((f*x)^(m + 1)*(d + e*x^2)^(p + 1)*(a + b*ArcSin[c*x])^n)/(d*f*(m + 1)), x] - Dist[(b*c*n*d^IntPart[p]*(d + e*x^2)^FracPart[p])/(f*(m + 1)*(1 - c^2*x^2)^FracPart[p]), Int[(f*x)^(m + 1)*(1 - c^2*x^2)^(p + 1/2)*(a + b*ArcSin[c*x])^(n - 1), x], x] /; FreeQ[{a, b, c, d, e, f, m, p}, x] && EqQ[c^2*d + e, 0] && GtQ[n, 0] && EqQ[m + 2*p + 3, 0] && NeQ[m, -1]
  4682. Int[Times[Power[Plus[Optional[Pattern[a, Blank[]]], Times[ArcCos[Times[Optional[Pattern[c, Blank[]]], Pattern[x, Blank[]]]], Optional[Pattern[b, Blank[]]]]], Optional[Pattern[n, Blank[]]]], Power[Times[Optional[Pattern[f, Blank[]]], Pattern[x, Blank[]]], Pattern[m, Blank[]]], Power[Plus[Pattern[d, Blank[]], Times[Optional[Pattern[e, Blank[]]], Power[Pattern[x, Blank[]], 2]]], Pattern[p, Blank[]]]], Pattern[x, Blank[Symbol]]] := Simp[((f*x)^(m + 1)*(d + e*x^2)^(p + 1)*(a + b*ArcCos[c*x])^n)/(d*f*(m + 1)), x] + Dist[(b*c*n*d^IntPart[p]*(d + e*x^2)^FracPart[p])/(f*(m + 1)*(1 - c^2*x^2)^FracPart[p]), Int[(f*x)^(m + 1)*(1 - c^2*x^2)^(p + 1/2)*(a + b*ArcCos[c*x])^(n - 1), x], x] /; FreeQ[{a, b, c, d, e, f, m, p}, x] && EqQ[c^2*d + e, 0] && GtQ[n, 0] && EqQ[m + 2*p + 3, 0] && NeQ[m, -1]
  4683. Int[Times[Plus[Optional[Pattern[a, Blank[]]], Times[ArcSin[Times[Optional[Pattern[c, Blank[]]], Pattern[x, Blank[]]]], Optional[Pattern[b, Blank[]]]]], Power[Pattern[x, Blank[]], -1], Power[Plus[Pattern[d, Blank[]], Times[Optional[Pattern[e, Blank[]]], Power[Pattern[x, Blank[]], 2]]], Optional[Pattern[p, Blank[]]]]], Pattern[x, Blank[Symbol]]] := Simp[((d + e*x^2)^p*(a + b*ArcSin[c*x]))/(2*p), x] + (Dist[d, Int[((d + e*x^2)^(p - 1)*(a + b*ArcSin[c*x]))/x, x], x] - Dist[(b*c*d^p)/(2*p), Int[(1 - c^2*x^2)^(p - 1/2), x], x]) /; FreeQ[{a, b, c, d, e}, x] && EqQ[c^2*d + e, 0] && IGtQ[p, 0]
  4684. Int[Times[Plus[Optional[Pattern[a, Blank[]]], Times[ArcCos[Times[Optional[Pattern[c, Blank[]]], Pattern[x, Blank[]]]], Optional[Pattern[b, Blank[]]]]], Power[Pattern[x, Blank[]], -1], Power[Plus[Pattern[d, Blank[]], Times[Optional[Pattern[e, Blank[]]], Power[Pattern[x, Blank[]], 2]]], Optional[Pattern[p, Blank[]]]]], Pattern[x, Blank[Symbol]]] := Simp[((d + e*x^2)^p*(a + b*ArcCos[c*x]))/(2*p), x] + (Dist[d, Int[((d + e*x^2)^(p - 1)*(a + b*ArcCos[c*x]))/x, x], x] + Dist[(b*c*d^p)/(2*p), Int[(1 - c^2*x^2)^(p - 1/2), x], x]) /; FreeQ[{a, b, c, d, e}, x] && EqQ[c^2*d + e, 0] && IGtQ[p, 0]
  4685. Int[Times[Plus[Optional[Pattern[a, Blank[]]], Times[ArcSin[Times[Optional[Pattern[c, Blank[]]], Pattern[x, Blank[]]]], Optional[Pattern[b, Blank[]]]]], Power[Times[Optional[Pattern[f, Blank[]]], Pattern[x, Blank[]]], Pattern[m, Blank[]]], Power[Plus[Pattern[d, Blank[]], Times[Optional[Pattern[e, Blank[]]], Power[Pattern[x, Blank[]], 2]]], Optional[Pattern[p, Blank[]]]]], Pattern[x, Blank[Symbol]]] := Simp[((f*x)^(m + 1)*(d + e*x^2)^p*(a + b*ArcSin[c*x]))/(f*(m + 1)), x] + (-Dist[(b*c*d^p)/(f*(m + 1)), Int[(f*x)^(m + 1)*(1 - c^2*x^2)^(p - 1/2), x], x] - Dist[(2*e*p)/(f^2*(m + 1)), Int[(f*x)^(m + 2)*(d + e*x^2)^(p - 1)*(a + b*ArcSin[c*x]), x], x]) /; FreeQ[{a, b, c, d, e, f}, x] && EqQ[c^2*d + e, 0] && IGtQ[p, 0] && ILtQ[(m + 1)/2, 0]
  4686. Int[Times[Plus[Optional[Pattern[a, Blank[]]], Times[ArcCos[Times[Optional[Pattern[c, Blank[]]], Pattern[x, Blank[]]]], Optional[Pattern[b, Blank[]]]]], Power[Times[Optional[Pattern[f, Blank[]]], Pattern[x, Blank[]]], Pattern[m, Blank[]]], Power[Plus[Pattern[d, Blank[]], Times[Optional[Pattern[e, Blank[]]], Power[Pattern[x, Blank[]], 2]]], Optional[Pattern[p, Blank[]]]]], Pattern[x, Blank[Symbol]]] := Simp[((f*x)^(m + 1)*(d + e*x^2)^p*(a + b*ArcCos[c*x]))/(f*(m + 1)), x] + (Dist[(b*c*d^p)/(f*(m + 1)), Int[(f*x)^(m + 1)*(1 - c^2*x^2)^(p - 1/2), x], x] - Dist[(2*e*p)/(f^2*(m + 1)), Int[(f*x)^(m + 2)*(d + e*x^2)^(p - 1)*(a + b*ArcCos[c*x]), x], x]) /; FreeQ[{a, b, c, d, e, f}, x] && EqQ[c^2*d + e, 0] && IGtQ[p, 0] && ILtQ[(m + 1)/2, 0]
  4687. Int[Times[Plus[Optional[Pattern[a, Blank[]]], Times[ArcSin[Times[Optional[Pattern[c, Blank[]]], Pattern[x, Blank[]]]], Optional[Pattern[b, Blank[]]]]], Power[Times[Optional[Pattern[f, Blank[]]], Pattern[x, Blank[]]], Pattern[m, Blank[]]], Power[Plus[Pattern[d, Blank[]], Times[Optional[Pattern[e, Blank[]]], Power[Pattern[x, Blank[]], 2]]], Optional[Pattern[p, Blank[]]]]], Pattern[x, Blank[Symbol]]] := With[{u = IntHide[(f*x)^m*(d + e*x^2)^p, x]}, Dist[a + b*ArcSin[c*x], u, x] - Dist[b*c, Int[SimplifyIntegrand[u/Sqrt[1 - c^2*x^2], x], x], x]] /; FreeQ[{a, b, c, d, e, f, m}, x] && EqQ[c^2*d + e, 0] && IGtQ[p, 0]
  4688. Int[Times[Plus[Optional[Pattern[a, Blank[]]], Times[ArcCos[Times[Optional[Pattern[c, Blank[]]], Pattern[x, Blank[]]]], Optional[Pattern[b, Blank[]]]]], Power[Times[Optional[Pattern[f, Blank[]]], Pattern[x, Blank[]]], Pattern[m, Blank[]]], Power[Plus[Pattern[d, Blank[]], Times[Optional[Pattern[e, Blank[]]], Power[Pattern[x, Blank[]], 2]]], Optional[Pattern[p, Blank[]]]]], Pattern[x, Blank[Symbol]]] := With[{u = IntHide[(f*x)^m*(d + e*x^2)^p, x]}, Dist[a + b*ArcCos[c*x], u, x] + Dist[b*c, Int[SimplifyIntegrand[u/Sqrt[1 - c^2*x^2], x], x], x]] /; FreeQ[{a, b, c, d, e, f, m}, x] && EqQ[c^2*d + e, 0] && IGtQ[p, 0]
  4689. Int[Times[Plus[Optional[Pattern[a, Blank[]]], Times[ArcSin[Times[Optional[Pattern[c, Blank[]]], Pattern[x, Blank[]]]], Optional[Pattern[b, Blank[]]]]], Power[Pattern[x, Blank[]], Pattern[m, Blank[]]], Power[Plus[Pattern[d, Blank[]], Times[Optional[Pattern[e, Blank[]]], Power[Pattern[x, Blank[]], 2]]], Pattern[p, Blank[]]]], Pattern[x, Blank[Symbol]]] := With[{u = IntHide[x^m*(1 - c^2*x^2)^p, x]}, Dist[d^p*(a + b*ArcSin[c*x]), u, x] - Dist[b*c*d^p, Int[SimplifyIntegrand[u/Sqrt[1 - c^2*x^2], x], x], x]] /; FreeQ[{a, b, c, d, e}, x] && EqQ[c^2*d + e, 0] && IntegerQ[p - 1/2] && (IGtQ[(m + 1)/2, 0] || ILtQ[(m + 2*p + 3)/2, 0]) && NeQ[p, -2^(-1)] && GtQ[d, 0]
  4690. Int[Times[Plus[Optional[Pattern[a, Blank[]]], Times[ArcCos[Times[Optional[Pattern[c, Blank[]]], Pattern[x, Blank[]]]], Optional[Pattern[b, Blank[]]]]], Power[Pattern[x, Blank[]], Pattern[m, Blank[]]], Power[Plus[Pattern[d, Blank[]], Times[Optional[Pattern[e, Blank[]]], Power[Pattern[x, Blank[]], 2]]], Pattern[p, Blank[]]]], Pattern[x, Blank[Symbol]]] := With[{u = IntHide[x^m*(1 - c^2*x^2)^p, x]}, Dist[d^p*(a + b*ArcCos[c*x]), u, x] + Dist[b*c*d^p, Int[SimplifyIntegrand[u/Sqrt[1 - c^2*x^2], x], x], x]] /; FreeQ[{a, b, c, d, e}, x] && EqQ[c^2*d + e, 0] && IntegerQ[p - 1/2] && (IGtQ[(m + 1)/2, 0] || ILtQ[(m + 2*p + 3)/2, 0]) && NeQ[p, -2^(-1)] && GtQ[d, 0]
  4691. Int[Times[Plus[Optional[Pattern[a, Blank[]]], Times[ArcSin[Times[Optional[Pattern[c, Blank[]]], Pattern[x, Blank[]]]], Optional[Pattern[b, Blank[]]]]], Power[Pattern[x, Blank[]], Pattern[m, Blank[]]], Power[Plus[Pattern[d, Blank[]], Times[Optional[Pattern[e, Blank[]]], Power[Pattern[x, Blank[]], 2]]], Pattern[p, Blank[]]]], Pattern[x, Blank[Symbol]]] := With[{u = IntHide[x^m*(1 - c^2*x^2)^p, x]}, Dist[a + b*ArcSin[c*x], Int[x^m*(d + e*x^2)^p, x], x] - Dist[(b*c*d^(p - 1/2)*Sqrt[d + e*x^2])/Sqrt[1 - c^2*x^2], Int[SimplifyIntegrand[u/Sqrt[1 - c^2*x^2], x], x], x]] /; FreeQ[{a, b, c, d, e}, x] && EqQ[c^2*d + e, 0] && IGtQ[p + 1/2, 0] && (IGtQ[(m + 1)/2, 0] || ILtQ[(m + 2*p + 3)/2, 0])
  4692. Int[Times[Plus[Optional[Pattern[a, Blank[]]], Times[ArcCos[Times[Optional[Pattern[c, Blank[]]], Pattern[x, Blank[]]]], Optional[Pattern[b, Blank[]]]]], Power[Pattern[x, Blank[]], Pattern[m, Blank[]]], Power[Plus[Pattern[d, Blank[]], Times[Optional[Pattern[e, Blank[]]], Power[Pattern[x, Blank[]], 2]]], Pattern[p, Blank[]]]], Pattern[x, Blank[Symbol]]] := With[{u = IntHide[x^m*(1 - c^2*x^2)^p, x]}, Dist[a + b*ArcCos[c*x], Int[x^m*(d + e*x^2)^p, x], x] + Dist[(b*c*d^(p - 1/2)*Sqrt[d + e*x^2])/Sqrt[1 - c^2*x^2], Int[SimplifyIntegrand[u/Sqrt[1 - c^2*x^2], x], x], x]] /; FreeQ[{a, b, c, d, e}, x] && EqQ[c^2*d + e, 0] && IGtQ[p + 1/2, 0] && (IGtQ[(m + 1)/2, 0] || ILtQ[(m + 2*p + 3)/2, 0])
  4693. Int[Times[Power[Plus[Optional[Pattern[a, Blank[]]], Times[ArcSin[Times[Optional[Pattern[c, Blank[]]], Pattern[x, Blank[]]]], Optional[Pattern[b, Blank[]]]]], Optional[Pattern[n, Blank[]]]], Power[Times[Optional[Pattern[f, Blank[]]], Pattern[x, Blank[]]], Pattern[m, Blank[]]], Power[Plus[Pattern[d, Blank[]], Times[Optional[Pattern[e, Blank[]]], Power[Pattern[x, Blank[]], 2]]], Rational[1, 2]]], Pattern[x, Blank[Symbol]]] := Simp[((f*x)^(m + 1)*Sqrt[d + e*x^2]*(a + b*ArcSin[c*x])^n)/(f*(m + 1)), x] + (-Dist[(b*c*n*Sqrt[d + e*x^2])/(f*(m + 1)*Sqrt[1 - c^2*x^2]), Int[(f*x)^(m + 1)*(a + b*ArcSin[c*x])^(n - 1), x], x] + Dist[(c^2*Sqrt[d + e*x^2])/(f^2*(m + 1)*Sqrt[1 - c^2*x^2]), Int[((f*x)^(m + 2)*(a + b*ArcSin[c*x])^n)/Sqrt[1 - c^2*x^2], x], x]) /; FreeQ[{a, b, c, d, e, f}, x] && EqQ[c^2*d + e, 0] && GtQ[n, 0] && LtQ[m, -1]
  4694. Int[Times[Power[Plus[Optional[Pattern[a, Blank[]]], Times[ArcCos[Times[Optional[Pattern[c, Blank[]]], Pattern[x, Blank[]]]], Optional[Pattern[b, Blank[]]]]], Optional[Pattern[n, Blank[]]]], Power[Times[Optional[Pattern[f, Blank[]]], Pattern[x, Blank[]]], Pattern[m, Blank[]]], Power[Plus[Pattern[d, Blank[]], Times[Optional[Pattern[e, Blank[]]], Power[Pattern[x, Blank[]], 2]]], Rational[1, 2]]], Pattern[x, Blank[Symbol]]] := Simp[((f*x)^(m + 1)*Sqrt[d + e*x^2]*(a + b*ArcCos[c*x])^n)/(f*(m + 1)), x] + (Dist[(b*c*n*Sqrt[d + e*x^2])/(f*(m + 1)*Sqrt[1 - c^2*x^2]), Int[(f*x)^(m + 1)*(a + b*ArcCos[c*x])^(n - 1), x], x] + Dist[(c^2*Sqrt[d + e*x^2])/(f^2*(m + 1)*Sqrt[1 - c^2*x^2]), Int[((f*x)^(m + 2)*(a + b*ArcCos[c*x])^n)/Sqrt[1 - c^2*x^2], x], x]) /; FreeQ[{a, b, c, d, e, f}, x] && EqQ[c^2*d + e, 0] && GtQ[n, 0] && LtQ[m, -1]
  4695. Int[Times[Power[Plus[Optional[Pattern[a, Blank[]]], Times[ArcSin[Times[Optional[Pattern[c, Blank[]]], Pattern[x, Blank[]]]], Optional[Pattern[b, Blank[]]]]], Optional[Pattern[n, Blank[]]]], Power[Times[Optional[Pattern[f, Blank[]]], Pattern[x, Blank[]]], Pattern[m, Blank[]]], Power[Plus[Pattern[d, Blank[]], Times[Optional[Pattern[e, Blank[]]], Power[Pattern[x, Blank[]], 2]]], Optional[Pattern[p, Blank[]]]]], Pattern[x, Blank[Symbol]]] := Simp[((f*x)^(m + 1)*(d + e*x^2)^p*(a + b*ArcSin[c*x])^n)/(f*(m + 1)), x] + (-Dist[(2*e*p)/(f^2*(m + 1)), Int[(f*x)^(m + 2)*(d + e*x^2)^(p - 1)*(a + b*ArcSin[c*x])^n, x], x] - Dist[(b*c*n*d^IntPart[p]*(d + e*x^2)^FracPart[p])/(f*(m + 1)*(1 - c^2*x^2)^FracPart[p]), Int[(f*x)^(m + 1)*(1 - c^2*x^2)^(p - 1/2)*(a + b*ArcSin[c*x])^(n - 1), x], x]) /; FreeQ[{a, b, c, d, e, f}, x] && EqQ[c^2*d + e, 0] && GtQ[n, 0] && GtQ[p, 0] && LtQ[m, -1]
  4696. Int[Times[Power[Plus[Optional[Pattern[a, Blank[]]], Times[ArcCos[Times[Optional[Pattern[c, Blank[]]], Pattern[x, Blank[]]]], Optional[Pattern[b, Blank[]]]]], Optional[Pattern[n, Blank[]]]], Power[Times[Optional[Pattern[f, Blank[]]], Pattern[x, Blank[]]], Pattern[m, Blank[]]], Power[Plus[Pattern[d, Blank[]], Times[Optional[Pattern[e, Blank[]]], Power[Pattern[x, Blank[]], 2]]], Optional[Pattern[p, Blank[]]]]], Pattern[x, Blank[Symbol]]] := Simp[((f*x)^(m + 1)*(d + e*x^2)^p*(a + b*ArcCos[c*x])^n)/(f*(m + 1)), x] + (-Dist[(2*e*p)/(f^2*(m + 1)), Int[(f*x)^(m + 2)*(d + e*x^2)^(p - 1)*(a + b*ArcCos[c*x])^n, x], x] + Dist[(b*c*n*d^IntPart[p]*(d + e*x^2)^FracPart[p])/(f*(m + 1)*(1 - c^2*x^2)^FracPart[p]), Int[(f*x)^(m + 1)*(1 - c^2*x^2)^(p - 1/2)*(a + b*ArcCos[c*x])^(n - 1), x], x]) /; FreeQ[{a, b, c, d, e, f}, x] && EqQ[c^2*d + e, 0] && GtQ[n, 0] && GtQ[p, 0] && LtQ[m, -1]
  4697. Int[Times[Power[Plus[Optional[Pattern[a, Blank[]]], Times[ArcSin[Times[Optional[Pattern[c, Blank[]]], Pattern[x, Blank[]]]], Optional[Pattern[b, Blank[]]]]], Optional[Pattern[n, Blank[]]]], Power[Times[Optional[Pattern[f, Blank[]]], Pattern[x, Blank[]]], Pattern[m, Blank[]]], Power[Plus[Pattern[d, Blank[]], Times[Optional[Pattern[e, Blank[]]], Power[Pattern[x, Blank[]], 2]]], Rational[1, 2]]], Pattern[x, Blank[Symbol]]] := Simp[((f*x)^(m + 1)*Sqrt[d + e*x^2]*(a + b*ArcSin[c*x])^n)/(f*(m + 2)), x] + (Dist[Sqrt[d + e*x^2]/((m + 2)*Sqrt[1 - c^2*x^2]), Int[((f*x)^m*(a + b*ArcSin[c*x])^n)/Sqrt[1 - c^2*x^2], x], x] - Dist[(b*c*n*Sqrt[d + e*x^2])/(f*(m + 2)*Sqrt[1 - c^2*x^2]), Int[(f*x)^(m + 1)*(a + b*ArcSin[c*x])^(n - 1), x], x]) /; FreeQ[{a, b, c, d, e, f, m}, x] && EqQ[c^2*d + e, 0] && GtQ[n, 0] && !LtQ[m, -1] && (RationalQ[m] || EqQ[n, 1])
  4698. Int[Times[Power[Plus[Optional[Pattern[a, Blank[]]], Times[ArcCos[Times[Optional[Pattern[c, Blank[]]], Pattern[x, Blank[]]]], Optional[Pattern[b, Blank[]]]]], Optional[Pattern[n, Blank[]]]], Power[Times[Optional[Pattern[f, Blank[]]], Pattern[x, Blank[]]], Pattern[m, Blank[]]], Power[Plus[Pattern[d, Blank[]], Times[Optional[Pattern[e, Blank[]]], Power[Pattern[x, Blank[]], 2]]], Rational[1, 2]]], Pattern[x, Blank[Symbol]]] := Simp[((f*x)^(m + 1)*Sqrt[d + e*x^2]*(a + b*ArcCos[c*x])^n)/(f*(m + 2)), x] + (Dist[Sqrt[d + e*x^2]/((m + 2)*Sqrt[1 - c^2*x^2]), Int[((f*x)^m*(a + b*ArcCos[c*x])^n)/Sqrt[1 - c^2*x^2], x], x] + Dist[(b*c*n*Sqrt[d + e*x^2])/(f*(m + 2)*Sqrt[1 - c^2*x^2]), Int[(f*x)^(m + 1)*(a + b*ArcCos[c*x])^(n - 1), x], x]) /; FreeQ[{a, b, c, d, e, f, m}, x] && EqQ[c^2*d + e, 0] && GtQ[n, 0] && !LtQ[m, -1] && (RationalQ[m] || EqQ[n, 1])
  4699. Int[Times[Power[Plus[Optional[Pattern[a, Blank[]]], Times[ArcSin[Times[Optional[Pattern[c, Blank[]]], Pattern[x, Blank[]]]], Optional[Pattern[b, Blank[]]]]], Optional[Pattern[n, Blank[]]]], Power[Times[Optional[Pattern[f, Blank[]]], Pattern[x, Blank[]]], Pattern[m, Blank[]]], Power[Plus[Pattern[d, Blank[]], Times[Optional[Pattern[e, Blank[]]], Power[Pattern[x, Blank[]], 2]]], Optional[Pattern[p, Blank[]]]]], Pattern[x, Blank[Symbol]]] := Simp[((f*x)^(m + 1)*(d + e*x^2)^p*(a + b*ArcSin[c*x])^n)/(f*(m + 2*p + 1)), x] + (Dist[(2*d*p)/(m + 2*p + 1), Int[(f*x)^m*(d + e*x^2)^(p - 1)*(a + b*ArcSin[c*x])^n, x], x] - Dist[(b*c*n*d^IntPart[p]*(d + e*x^2)^FracPart[p])/(f*(m + 2*p + 1)*(1 - c^2*x^2)^FracPart[p]), Int[(f*x)^(m + 1)*(1 - c^2*x^2)^(p - 1/2)*(a + b*ArcSin[c*x])^(n - 1), x], x]) /; FreeQ[{a, b, c, d, e, f, m}, x] && EqQ[c^2*d + e, 0] && GtQ[n, 0] && GtQ[p, 0] && !LtQ[m, -1] && (RationalQ[m] || EqQ[n, 1])
  4700. Int[Times[Power[Plus[Optional[Pattern[a, Blank[]]], Times[ArcCos[Times[Optional[Pattern[c, Blank[]]], Pattern[x, Blank[]]]], Optional[Pattern[b, Blank[]]]]], Optional[Pattern[n, Blank[]]]], Power[Times[Optional[Pattern[f, Blank[]]], Pattern[x, Blank[]]], Pattern[m, Blank[]]], Power[Plus[Pattern[d, Blank[]], Times[Optional[Pattern[e, Blank[]]], Power[Pattern[x, Blank[]], 2]]], Optional[Pattern[p, Blank[]]]]], Pattern[x, Blank[Symbol]]] := Simp[((f*x)^(m + 1)*(d + e*x^2)^p*(a + b*ArcCos[c*x])^n)/(f*(m + 2*p + 1)), x] + (Dist[(2*d*p)/(m + 2*p + 1), Int[(f*x)^m*(d + e*x^2)^(p - 1)*(a + b*ArcCos[c*x])^n, x], x] + Dist[(b*c*n*d^IntPart[p]*(d + e*x^2)^FracPart[p])/(f*(m + 2*p + 1)*(1 - c^2*x^2)^FracPart[p]), Int[(f*x)^(m + 1)*(1 - c^2*x^2)^(p - 1/2)*(a + b*ArcCos[c*x])^(n - 1), x], x]) /; FreeQ[{a, b, c, d, e, f, m}, x] && EqQ[c^2*d + e, 0] && GtQ[n, 0] && GtQ[p, 0] && !LtQ[m, -1] && (RationalQ[m] || EqQ[n, 1])
  4701. Int[Times[Power[Plus[Optional[Pattern[a, Blank[]]], Times[ArcSin[Times[Optional[Pattern[c, Blank[]]], Pattern[x, Blank[]]]], Optional[Pattern[b, Blank[]]]]], Optional[Pattern[n, Blank[]]]], Power[Times[Optional[Pattern[f, Blank[]]], Pattern[x, Blank[]]], Pattern[m, Blank[]]], Power[Plus[Pattern[d, Blank[]], Times[Optional[Pattern[e, Blank[]]], Power[Pattern[x, Blank[]], 2]]], Pattern[p, Blank[]]]], Pattern[x, Blank[Symbol]]] := Simp[((f*x)^(m + 1)*(d + e*x^2)^(p + 1)*(a + b*ArcSin[c*x])^n)/(d*f*(m + 1)), x] + (Dist[(c^2*(m + 2*p + 3))/(f^2*(m + 1)), Int[(f*x)^(m + 2)*(d + e*x^2)^p*(a + b*ArcSin[c*x])^n, x], x] - Dist[(b*c*n*d^IntPart[p]*(d + e*x^2)^FracPart[p])/(f*(m + 1)*(1 - c^2*x^2)^FracPart[p]), Int[(f*x)^(m + 1)*(1 - c^2*x^2)^(p + 1/2)*(a + b*ArcSin[c*x])^(n - 1), x], x]) /; FreeQ[{a, b, c, d, e, f, p}, x] && EqQ[c^2*d + e, 0] && GtQ[n, 0] && LtQ[m, -1] && IntegerQ[m]
  4702. Int[Times[Power[Plus[Optional[Pattern[a, Blank[]]], Times[ArcCos[Times[Optional[Pattern[c, Blank[]]], Pattern[x, Blank[]]]], Optional[Pattern[b, Blank[]]]]], Optional[Pattern[n, Blank[]]]], Power[Times[Optional[Pattern[f, Blank[]]], Pattern[x, Blank[]]], Pattern[m, Blank[]]], Power[Plus[Pattern[d, Blank[]], Times[Optional[Pattern[e, Blank[]]], Power[Pattern[x, Blank[]], 2]]], Pattern[p, Blank[]]]], Pattern[x, Blank[Symbol]]] := Simp[((f*x)^(m + 1)*(d + e*x^2)^(p + 1)*(a + b*ArcCos[c*x])^n)/(d*f*(m + 1)), x] + (Dist[(c^2*(m + 2*p + 3))/(f^2*(m + 1)), Int[(f*x)^(m + 2)*(d + e*x^2)^p*(a + b*ArcCos[c*x])^n, x], x] + Dist[(b*c*n*d^IntPart[p]*(d + e*x^2)^FracPart[p])/(f*(m + 1)*(1 - c^2*x^2)^FracPart[p]), Int[(f*x)^(m + 1)*(1 - c^2*x^2)^(p + 1/2)*(a + b*ArcCos[c*x])^(n - 1), x], x]) /; FreeQ[{a, b, c, d, e, f, p}, x] && EqQ[c^2*d + e, 0] && GtQ[n, 0] && LtQ[m, -1] && IntegerQ[m]
  4703. Int[Times[Power[Plus[Optional[Pattern[a, Blank[]]], Times[ArcSin[Times[Optional[Pattern[c, Blank[]]], Pattern[x, Blank[]]]], Optional[Pattern[b, Blank[]]]]], Optional[Pattern[n, Blank[]]]], Power[Times[Optional[Pattern[f, Blank[]]], Pattern[x, Blank[]]], Pattern[m, Blank[]]], Power[Plus[Pattern[d, Blank[]], Times[Optional[Pattern[e, Blank[]]], Power[Pattern[x, Blank[]], 2]]], Pattern[p, Blank[]]]], Pattern[x, Blank[Symbol]]] := Simp[(f*(f*x)^(m - 1)*(d + e*x^2)^(p + 1)*(a + b*ArcSin[c*x])^n)/(2*e*(p + 1)), x] + (-Dist[(f^2*(m - 1))/(2*e*(p + 1)), Int[(f*x)^(m - 2)*(d + e*x^2)^(p + 1)*(a + b*ArcSin[c*x])^n, x], x] + Dist[(b*f*n*d^IntPart[p]*(d + e*x^2)^FracPart[p])/(2*c*(p + 1)*(1 - c^2*x^2)^FracPart[p]), Int[(f*x)^(m - 1)*(1 - c^2*x^2)^(p + 1/2)*(a + b*ArcSin[c*x])^(n - 1), x], x]) /; FreeQ[{a, b, c, d, e, f}, x] && EqQ[c^2*d + e, 0] && GtQ[n, 0] && LtQ[p, -1] && GtQ[m, 1]
  4704. Int[Times[Power[Plus[Optional[Pattern[a, Blank[]]], Times[ArcCos[Times[Optional[Pattern[c, Blank[]]], Pattern[x, Blank[]]]], Optional[Pattern[b, Blank[]]]]], Optional[Pattern[n, Blank[]]]], Power[Times[Optional[Pattern[f, Blank[]]], Pattern[x, Blank[]]], Pattern[m, Blank[]]], Power[Plus[Pattern[d, Blank[]], Times[Optional[Pattern[e, Blank[]]], Power[Pattern[x, Blank[]], 2]]], Pattern[p, Blank[]]]], Pattern[x, Blank[Symbol]]] := Simp[(f*(f*x)^(m - 1)*(d + e*x^2)^(p + 1)*(a + b*ArcCos[c*x])^n)/(2*e*(p + 1)), x] + (-Dist[(f^2*(m - 1))/(2*e*(p + 1)), Int[(f*x)^(m - 2)*(d + e*x^2)^(p + 1)*(a + b*ArcCos[c*x])^n, x], x] - Dist[(b*f*n*d^IntPart[p]*(d + e*x^2)^FracPart[p])/(2*c*(p + 1)*(1 - c^2*x^2)^FracPart[p]), Int[(f*x)^(m - 1)*(1 - c^2*x^2)^(p + 1/2)*(a + b*ArcCos[c*x])^(n - 1), x], x]) /; FreeQ[{a, b, c, d, e, f}, x] && EqQ[c^2*d + e, 0] && GtQ[n, 0] && LtQ[p, -1] && GtQ[m, 1]
  4705. Int[Times[Power[Plus[Optional[Pattern[a, Blank[]]], Times[ArcSin[Times[Optional[Pattern[c, Blank[]]], Pattern[x, Blank[]]]], Optional[Pattern[b, Blank[]]]]], Optional[Pattern[n, Blank[]]]], Power[Times[Optional[Pattern[f, Blank[]]], Pattern[x, Blank[]]], Pattern[m, Blank[]]], Power[Plus[Pattern[d, Blank[]], Times[Optional[Pattern[e, Blank[]]], Power[Pattern[x, Blank[]], 2]]], Pattern[p, Blank[]]]], Pattern[x, Blank[Symbol]]] := -Simp[((f*x)^(m + 1)*(d + e*x^2)^(p + 1)*(a + b*ArcSin[c*x])^n)/(2*d*f*(p + 1)), x] + (Dist[(m + 2*p + 3)/(2*d*(p + 1)), Int[(f*x)^m*(d + e*x^2)^(p + 1)*(a + b*ArcSin[c*x])^n, x], x] + Dist[(b*c*n*d^IntPart[p]*(d + e*x^2)^FracPart[p])/(2*f*(p + 1)*(1 - c^2*x^2)^FracPart[p]), Int[(f*x)^(m + 1)*(1 - c^2*x^2)^(p + 1/2)*(a + b*ArcSin[c*x])^(n - 1), x], x]) /; FreeQ[{a, b, c, d, e, f, m}, x] && EqQ[c^2*d + e, 0] && GtQ[n, 0] && LtQ[p, -1] && !GtQ[m, 1] && (IntegerQ[m] || IntegerQ[p] || EqQ[n, 1])
  4706. Int[Times[Power[Plus[Optional[Pattern[a, Blank[]]], Times[ArcCos[Times[Optional[Pattern[c, Blank[]]], Pattern[x, Blank[]]]], Optional[Pattern[b, Blank[]]]]], Optional[Pattern[n, Blank[]]]], Power[Times[Optional[Pattern[f, Blank[]]], Pattern[x, Blank[]]], Pattern[m, Blank[]]], Power[Plus[Pattern[d, Blank[]], Times[Optional[Pattern[e, Blank[]]], Power[Pattern[x, Blank[]], 2]]], Pattern[p, Blank[]]]], Pattern[x, Blank[Symbol]]] := -Simp[((f*x)^(m + 1)*(d + e*x^2)^(p + 1)*(a + b*ArcCos[c*x])^n)/(2*d*f*(p + 1)), x] + (Dist[(m + 2*p + 3)/(2*d*(p + 1)), Int[(f*x)^m*(d + e*x^2)^(p + 1)*(a + b*ArcCos[c*x])^n, x], x] - Dist[(b*c*n*d^IntPart[p]*(d + e*x^2)^FracPart[p])/(2*f*(p + 1)*(1 - c^2*x^2)^FracPart[p]), Int[(f*x)^(m + 1)*(1 - c^2*x^2)^(p + 1/2)*(a + b*ArcCos[c*x])^(n - 1), x], x]) /; FreeQ[{a, b, c, d, e, f, m}, x] && EqQ[c^2*d + e, 0] && GtQ[n, 0] && LtQ[p, -1] && !GtQ[m, 1] && (IntegerQ[m] || IntegerQ[p] || EqQ[n, 1])
  4707. Int[Times[Power[Plus[Optional[Pattern[a, Blank[]]], Times[ArcSin[Times[Optional[Pattern[c, Blank[]]], Pattern[x, Blank[]]]], Optional[Pattern[b, Blank[]]]]], Optional[Pattern[n, Blank[]]]], Power[Times[Optional[Pattern[f, Blank[]]], Pattern[x, Blank[]]], Pattern[m, Blank[]]], Power[Plus[Pattern[d, Blank[]], Times[Optional[Pattern[e, Blank[]]], Power[Pattern[x, Blank[]], 2]]], Rational[-1, 2]]], Pattern[x, Blank[Symbol]]] := Simp[(f*(f*x)^(m - 1)*Sqrt[d + e*x^2]*(a + b*ArcSin[c*x])^n)/(e*m), x] + (Dist[(f^2*(m - 1))/(c^2*m), Int[((f*x)^(m - 2)*(a + b*ArcSin[c*x])^n)/Sqrt[d + e*x^2], x], x] + Dist[(b*f*n*Sqrt[1 - c^2*x^2])/(c*m*Sqrt[d + e*x^2]), Int[(f*x)^(m - 1)*(a + b*ArcSin[c*x])^(n - 1), x], x]) /; FreeQ[{a, b, c, d, e, f}, x] && EqQ[c^2*d + e, 0] && GtQ[n, 0] && GtQ[m, 1] && IntegerQ[m]
  4708. Int[Times[Power[Plus[Optional[Pattern[a, Blank[]]], Times[ArcCos[Times[Optional[Pattern[c, Blank[]]], Pattern[x, Blank[]]]], Optional[Pattern[b, Blank[]]]]], Optional[Pattern[n, Blank[]]]], Power[Times[Optional[Pattern[f, Blank[]]], Pattern[x, Blank[]]], Pattern[m, Blank[]]], Power[Plus[Pattern[d, Blank[]], Times[Optional[Pattern[e, Blank[]]], Power[Pattern[x, Blank[]], 2]]], Rational[-1, 2]]], Pattern[x, Blank[Symbol]]] := Simp[(f*(f*x)^(m - 1)*Sqrt[d + e*x^2]*(a + b*ArcCos[c*x])^n)/(e*m), x] + (Dist[(f^2*(m - 1))/(c^2*m), Int[((f*x)^(m - 2)*(a + b*ArcCos[c*x])^n)/Sqrt[d + e*x^2], x], x] - Dist[(b*f*n*Sqrt[1 - c^2*x^2])/(c*m*Sqrt[d + e*x^2]), Int[(f*x)^(m - 1)*(a + b*ArcCos[c*x])^(n - 1), x], x]) /; FreeQ[{a, b, c, d, e, f}, x] && EqQ[c^2*d + e, 0] && GtQ[n, 0] && GtQ[m, 1] && IntegerQ[m]
  4709. Int[Times[Power[Plus[Optional[Pattern[a, Blank[]]], Times[ArcSin[Times[Optional[Pattern[c, Blank[]]], Pattern[x, Blank[]]]], Optional[Pattern[b, Blank[]]]]], Optional[Pattern[n, Blank[]]]], Power[Pattern[x, Blank[]], Pattern[m, Blank[]]], Power[Plus[Pattern[d, Blank[]], Times[Optional[Pattern[e, Blank[]]], Power[Pattern[x, Blank[]], 2]]], Rational[-1, 2]]], Pattern[x, Blank[Symbol]]] := Dist[1/(c^(m + 1)*Sqrt[d]), Subst[Int[(a + b*x)^n*Sin[x]^m, x], x, ArcSin[c*x]], x] /; FreeQ[{a, b, c, d, e}, x] && EqQ[c^2*d + e, 0] && GtQ[d, 0] && IGtQ[n, 0] && IntegerQ[m]
  4710. Int[Times[Power[Plus[Optional[Pattern[a, Blank[]]], Times[ArcCos[Times[Optional[Pattern[c, Blank[]]], Pattern[x, Blank[]]]], Optional[Pattern[b, Blank[]]]]], Optional[Pattern[n, Blank[]]]], Power[Pattern[x, Blank[]], Pattern[m, Blank[]]], Power[Plus[Pattern[d, Blank[]], Times[Optional[Pattern[e, Blank[]]], Power[Pattern[x, Blank[]], 2]]], Rational[-1, 2]]], Pattern[x, Blank[Symbol]]] := -Dist[(c^(m + 1)*Sqrt[d])^(-1), Subst[Int[(a + b*x)^n*Cos[x]^m, x], x, ArcCos[c*x]], x] /; FreeQ[{a, b, c, d, e}, x] && EqQ[c^2*d + e, 0] && GtQ[d, 0] && IGtQ[n, 0] && IntegerQ[m]
  4711. Int[Times[Plus[Optional[Pattern[a, Blank[]]], Times[ArcSin[Times[Optional[Pattern[c, Blank[]]], Pattern[x, Blank[]]]], Optional[Pattern[b, Blank[]]]]], Power[Times[Optional[Pattern[f, Blank[]]], Pattern[x, Blank[]]], Pattern[m, Blank[]]], Power[Plus[Pattern[d, Blank[]], Times[Optional[Pattern[e, Blank[]]], Power[Pattern[x, Blank[]], 2]]], Rational[-1, 2]]], Pattern[x, Blank[Symbol]]] := Simp[((f*x)^(m + 1)*(a + b*ArcSin[c*x])*Hypergeometric2F1[1/2, (1 + m)/2, (3 + m)/2, c^2*x^2])/(Sqrt[d]*f*(m + 1)), x] - Simp[(b*c*(f*x)^(m + 2)*HypergeometricPFQ[{1, 1 + m/2, 1 + m/2}, {3/2 + m/2, 2 + m/2}, c^2*x^2])/(Sqrt[d]*f^2*(m + 1)*(m + 2)), x] /; FreeQ[{a, b, c, d, e, f, m}, x] && EqQ[c^2*d + e, 0] && GtQ[d, 0] && !IntegerQ[m]
  4712. Int[Times[Plus[Optional[Pattern[a, Blank[]]], Times[ArcCos[Times[Optional[Pattern[c, Blank[]]], Pattern[x, Blank[]]]], Optional[Pattern[b, Blank[]]]]], Power[Times[Optional[Pattern[f, Blank[]]], Pattern[x, Blank[]]], Pattern[m, Blank[]]], Power[Plus[Pattern[d, Blank[]], Times[Optional[Pattern[e, Blank[]]], Power[Pattern[x, Blank[]], 2]]], Rational[-1, 2]]], Pattern[x, Blank[Symbol]]] := Simp[((f*x)^(m + 1)*(a + b*ArcCos[c*x])*Hypergeometric2F1[1/2, (1 + m)/2, (3 + m)/2, c^2*x^2])/(Sqrt[d]*f*(m + 1)), x] + Simp[(b*c*(f*x)^(m + 2)*HypergeometricPFQ[{1, 1 + m/2, 1 + m/2}, {3/2 + m/2, 2 + m/2}, c^2*x^2])/(Sqrt[d]*f^2*(m + 1)*(m + 2)), x] /; FreeQ[{a, b, c, d, e, f, m}, x] && EqQ[c^2*d + e, 0] && GtQ[d, 0] && !IntegerQ[m]
  4713. Int[Times[Power[Plus[Optional[Pattern[a, Blank[]]], Times[ArcSin[Times[Optional[Pattern[c, Blank[]]], Pattern[x, Blank[]]]], Optional[Pattern[b, Blank[]]]]], Optional[Pattern[n, Blank[]]]], Power[Times[Optional[Pattern[f, Blank[]]], Pattern[x, Blank[]]], Pattern[m, Blank[]]], Power[Plus[Pattern[d, Blank[]], Times[Optional[Pattern[e, Blank[]]], Power[Pattern[x, Blank[]], 2]]], Rational[-1, 2]]], Pattern[x, Blank[Symbol]]] := Dist[Sqrt[1 - c^2*x^2]/Sqrt[d + e*x^2], Int[((f*x)^m*(a + b*ArcSin[c*x])^n)/Sqrt[1 - c^2*x^2], x], x] /; FreeQ[{a, b, c, d, e, f, m}, x] && EqQ[c^2*d + e, 0] && GtQ[n, 0] && !GtQ[d, 0] && (IntegerQ[m] || EqQ[n, 1])
  4714. Int[Times[Power[Plus[Optional[Pattern[a, Blank[]]], Times[ArcCos[Times[Optional[Pattern[c, Blank[]]], Pattern[x, Blank[]]]], Optional[Pattern[b, Blank[]]]]], Optional[Pattern[n, Blank[]]]], Power[Times[Optional[Pattern[f, Blank[]]], Pattern[x, Blank[]]], Pattern[m, Blank[]]], Power[Plus[Pattern[d, Blank[]], Times[Optional[Pattern[e, Blank[]]], Power[Pattern[x, Blank[]], 2]]], Rational[-1, 2]]], Pattern[x, Blank[Symbol]]] := Dist[Sqrt[1 - c^2*x^2]/Sqrt[d + e*x^2], Int[((f*x)^m*(a + b*ArcCos[c*x])^n)/Sqrt[1 - c^2*x^2], x], x] /; FreeQ[{a, b, c, d, e, f, m}, x] && EqQ[c^2*d + e, 0] && GtQ[n, 0] && !GtQ[d, 0] && (IntegerQ[m] || EqQ[n, 1])
  4715. Int[Times[Power[Plus[Optional[Pattern[a, Blank[]]], Times[ArcSin[Times[Optional[Pattern[c, Blank[]]], Pattern[x, Blank[]]]], Optional[Pattern[b, Blank[]]]]], Optional[Pattern[n, Blank[]]]], Power[Times[Optional[Pattern[f, Blank[]]], Pattern[x, Blank[]]], Pattern[m, Blank[]]], Power[Plus[Pattern[d, Blank[]], Times[Optional[Pattern[e, Blank[]]], Power[Pattern[x, Blank[]], 2]]], Pattern[p, Blank[]]]], Pattern[x, Blank[Symbol]]] := Simp[(f*(f*x)^(m - 1)*(d + e*x^2)^(p + 1)*(a + b*ArcSin[c*x])^n)/(e*(m + 2*p + 1)), x] + (Dist[(f^2*(m - 1))/(c^2*(m + 2*p + 1)), Int[(f*x)^(m - 2)*(d + e*x^2)^p*(a + b*ArcSin[c*x])^n, x], x] + Dist[(b*f*n*d^IntPart[p]*(d + e*x^2)^FracPart[p])/(c*(m + 2*p + 1)*(1 - c^2*x^2)^FracPart[p]), Int[(f*x)^(m - 1)*(1 - c^2*x^2)^(p + 1/2)*(a + b*ArcSin[c*x])^(n - 1), x], x]) /; FreeQ[{a, b, c, d, e, f, p}, x] && EqQ[c^2*d + e, 0] && GtQ[n, 0] && GtQ[m, 1] && NeQ[m + 2*p + 1, 0] && IntegerQ[m]
  4716. Int[Times[Power[Plus[Optional[Pattern[a, Blank[]]], Times[ArcCos[Times[Optional[Pattern[c, Blank[]]], Pattern[x, Blank[]]]], Optional[Pattern[b, Blank[]]]]], Optional[Pattern[n, Blank[]]]], Power[Times[Optional[Pattern[f, Blank[]]], Pattern[x, Blank[]]], Pattern[m, Blank[]]], Power[Plus[Pattern[d, Blank[]], Times[Optional[Pattern[e, Blank[]]], Power[Pattern[x, Blank[]], 2]]], Pattern[p, Blank[]]]], Pattern[x, Blank[Symbol]]] := Simp[(f*(f*x)^(m - 1)*(d + e*x^2)^(p + 1)*(a + b*ArcCos[c*x])^n)/(e*(m + 2*p + 1)), x] + (Dist[(f^2*(m - 1))/(c^2*(m + 2*p + 1)), Int[(f*x)^(m - 2)*(d + e*x^2)^p*(a + b*ArcCos[c*x])^n, x], x] - Dist[(b*f*n*d^IntPart[p]*(d + e*x^2)^FracPart[p])/(c*(m + 2*p + 1)*(1 - c^2*x^2)^FracPart[p]), Int[(f*x)^(m - 1)*(1 - c^2*x^2)^(p + 1/2)*(a + b*ArcCos[c*x])^(n - 1), x], x]) /; FreeQ[{a, b, c, d, e, f, p}, x] && EqQ[c^2*d + e, 0] && GtQ[n, 0] && GtQ[m, 1] && NeQ[m + 2*p + 1, 0] && IntegerQ[m]
  4717. Int[Times[Power[Plus[Optional[Pattern[a, Blank[]]], Times[ArcSin[Times[Optional[Pattern[c, Blank[]]], Pattern[x, Blank[]]]], Optional[Pattern[b, Blank[]]]]], Pattern[n, Blank[]]], Power[Times[Optional[Pattern[f, Blank[]]], Pattern[x, Blank[]]], Optional[Pattern[m, Blank[]]]], Power[Plus[Pattern[d, Blank[]], Times[Optional[Pattern[e, Blank[]]], Power[Pattern[x, Blank[]], 2]]], Optional[Pattern[p, Blank[]]]]], Pattern[x, Blank[Symbol]]] := Simp[((f*x)^m*Sqrt[1 - c^2*x^2]*(d + e*x^2)^p*(a + b*ArcSin[c*x])^(n + 1))/(b*c*(n + 1)), x] - Dist[(f*m*d^IntPart[p]*(d + e*x^2)^FracPart[p])/(b*c*(n + 1)*(1 - c^2*x^2)^FracPart[p]), Int[(f*x)^(m - 1)*(1 - c^2*x^2)^(p - 1/2)*(a + b*ArcSin[c*x])^(n + 1), x], x] /; FreeQ[{a, b, c, d, e, f, m, p}, x] && EqQ[c^2*d + e, 0] && LtQ[n, -1] && EqQ[m + 2*p + 1, 0]
  4718. Int[Times[Power[Plus[Optional[Pattern[a, Blank[]]], Times[ArcCos[Times[Optional[Pattern[c, Blank[]]], Pattern[x, Blank[]]]], Optional[Pattern[b, Blank[]]]]], Pattern[n, Blank[]]], Power[Times[Optional[Pattern[f, Blank[]]], Pattern[x, Blank[]]], Optional[Pattern[m, Blank[]]]], Power[Plus[Pattern[d, Blank[]], Times[Optional[Pattern[e, Blank[]]], Power[Pattern[x, Blank[]], 2]]], Optional[Pattern[p, Blank[]]]]], Pattern[x, Blank[Symbol]]] := -Simp[((f*x)^m*Sqrt[1 - c^2*x^2]*(d + e*x^2)^p*(a + b*ArcCos[c*x])^(n + 1))/(b*c*(n + 1)), x] + Dist[(f*m*d^IntPart[p]*(d + e*x^2)^FracPart[p])/(b*c*(n + 1)*(1 - c^2*x^2)^FracPart[p]), Int[(f*x)^(m - 1)*(1 - c^2*x^2)^(p - 1/2)*(a + b*ArcCos[c*x])^(n + 1), x], x] /; FreeQ[{a, b, c, d, e, f, m, p}, x] && EqQ[c^2*d + e, 0] && LtQ[n, -1] && EqQ[m + 2*p + 1, 0]
  4719. Int[Times[Power[Plus[Optional[Pattern[a, Blank[]]], Times[ArcSin[Times[Optional[Pattern[c, Blank[]]], Pattern[x, Blank[]]]], Optional[Pattern[b, Blank[]]]]], Pattern[n, Blank[]]], Power[Times[Optional[Pattern[f, Blank[]]], Pattern[x, Blank[]]], Optional[Pattern[m, Blank[]]]], Power[Plus[Pattern[d, Blank[]], Times[Optional[Pattern[e, Blank[]]], Power[Pattern[x, Blank[]], 2]]], Rational[-1, 2]]], Pattern[x, Blank[Symbol]]] := Simp[((f*x)^m*(a + b*ArcSin[c*x])^(n + 1))/(b*c*Sqrt[d]*(n + 1)), x] - Dist[(f*m)/(b*c*Sqrt[d]*(n + 1)), Int[(f*x)^(m - 1)*(a + b*ArcSin[c*x])^(n + 1), x], x] /; FreeQ[{a, b, c, d, e, f, m}, x] && EqQ[c^2*d + e, 0] && LtQ[n, -1] && GtQ[d, 0]
  4720. Int[Times[Power[Plus[Optional[Pattern[a, Blank[]]], Times[ArcCos[Times[Optional[Pattern[c, Blank[]]], Pattern[x, Blank[]]]], Optional[Pattern[b, Blank[]]]]], Pattern[n, Blank[]]], Power[Times[Optional[Pattern[f, Blank[]]], Pattern[x, Blank[]]], Optional[Pattern[m, Blank[]]]], Power[Plus[Pattern[d, Blank[]], Times[Optional[Pattern[e, Blank[]]], Power[Pattern[x, Blank[]], 2]]], Rational[-1, 2]]], Pattern[x, Blank[Symbol]]] := -Simp[((f*x)^m*(a + b*ArcCos[c*x])^(n + 1))/(b*c*Sqrt[d]*(n + 1)), x] + Dist[(f*m)/(b*c*Sqrt[d]*(n + 1)), Int[(f*x)^(m - 1)*(a + b*ArcCos[c*x])^(n + 1), x], x] /; FreeQ[{a, b, c, d, e, f, m}, x] && EqQ[c^2*d + e, 0] && LtQ[n, -1] && GtQ[d, 0]
  4721. Int[Times[Power[Plus[Optional[Pattern[a, Blank[]]], Times[ArcSin[Times[Optional[Pattern[c, Blank[]]], Pattern[x, Blank[]]]], Optional[Pattern[b, Blank[]]]]], Pattern[n, Blank[]]], Power[Times[Optional[Pattern[f, Blank[]]], Pattern[x, Blank[]]], Optional[Pattern[m, Blank[]]]], Power[Plus[Pattern[d, Blank[]], Times[Optional[Pattern[e, Blank[]]], Power[Pattern[x, Blank[]], 2]]], Optional[Pattern[p, Blank[]]]]], Pattern[x, Blank[Symbol]]] := Simp[((f*x)^m*Sqrt[1 - c^2*x^2]*(d + e*x^2)^p*(a + b*ArcSin[c*x])^(n + 1))/(b*c*(n + 1)), x] + (-Dist[(f*m*d^IntPart[p]*(d + e*x^2)^FracPart[p])/(b*c*(n + 1)*(1 - c^2*x^2)^FracPart[p]), Int[(f*x)^(m - 1)*(1 - c^2*x^2)^(p - 1/2)*(a + b*ArcSin[c*x])^(n + 1), x], x] + Dist[(c*(m + 2*p + 1)*d^IntPart[p]*(d + e*x^2)^FracPart[p])/(b*f*(n + 1)*(1 - c^2*x^2)^FracPart[p]), Int[(f*x)^(m + 1)*(1 - c^2*x^2)^(p - 1/2)*(a + b*ArcSin[c*x])^(n + 1), x], x]) /; FreeQ[{a, b, c, d, e, f}, x] && EqQ[c^2*d + e, 0] && LtQ[n, -1] && IGtQ[m, -3] && IGtQ[2*p, 0]
  4722. Int[Times[Power[Plus[Optional[Pattern[a, Blank[]]], Times[ArcCos[Times[Optional[Pattern[c, Blank[]]], Pattern[x, Blank[]]]], Optional[Pattern[b, Blank[]]]]], Pattern[n, Blank[]]], Power[Times[Optional[Pattern[f, Blank[]]], Pattern[x, Blank[]]], Optional[Pattern[m, Blank[]]]], Power[Plus[Pattern[d, Blank[]], Times[Optional[Pattern[e, Blank[]]], Power[Pattern[x, Blank[]], 2]]], Optional[Pattern[p, Blank[]]]]], Pattern[x, Blank[Symbol]]] := -Simp[((f*x)^m*Sqrt[1 - c^2*x^2]*(d + e*x^2)^p*(a + b*ArcCos[c*x])^(n + 1))/(b*c*(n + 1)), x] + (Dist[(f*m*d^IntPart[p]*(d + e*x^2)^FracPart[p])/(b*c*(n + 1)*(1 - c^2*x^2)^FracPart[p]), Int[(f*x)^(m - 1)*(1 - c^2*x^2)^(p - 1/2)*(a + b*ArcCos[c*x])^(n + 1), x], x] - Dist[(c*(m + 2*p + 1)*d^IntPart[p]*(d + e*x^2)^FracPart[p])/(b*f*(n + 1)*(1 - c^2*x^2)^FracPart[p]), Int[(f*x)^(m + 1)*(1 - c^2*x^2)^(p - 1/2)*(a + b*ArcCos[c*x])^(n + 1), x], x]) /; FreeQ[{a, b, c, d, e, f}, x] && EqQ[c^2*d + e, 0] && LtQ[n, -1] && IGtQ[m, -3] && IGtQ[2*p, 0]
  4723. Int[Times[Power[Plus[Optional[Pattern[a, Blank[]]], Times[ArcSin[Times[Optional[Pattern[c, Blank[]]], Pattern[x, Blank[]]]], Optional[Pattern[b, Blank[]]]]], Optional[Pattern[n, Blank[]]]], Power[Pattern[x, Blank[]], Optional[Pattern[m, Blank[]]]], Power[Plus[Pattern[d, Blank[]], Times[Optional[Pattern[e, Blank[]]], Power[Pattern[x, Blank[]], 2]]], Optional[Pattern[p, Blank[]]]]], Pattern[x, Blank[Symbol]]] := Dist[d^p/c^(m + 1), Subst[Int[(a + b*x)^n*Sin[x]^m*Cos[x]^(2*p + 1), x], x, ArcSin[c*x]], x] /; FreeQ[{a, b, c, d, e, n}, x] && EqQ[c^2*d + e, 0] && IntegerQ[2*p] && GtQ[p, -1] && IGtQ[m, 0] && (IntegerQ[p] || GtQ[d, 0])
  4724. Int[Times[Power[Plus[Optional[Pattern[a, Blank[]]], Times[ArcCos[Times[Optional[Pattern[c, Blank[]]], Pattern[x, Blank[]]]], Optional[Pattern[b, Blank[]]]]], Optional[Pattern[n, Blank[]]]], Power[Pattern[x, Blank[]], Optional[Pattern[m, Blank[]]]], Power[Plus[Pattern[d, Blank[]], Times[Optional[Pattern[e, Blank[]]], Power[Pattern[x, Blank[]], 2]]], Optional[Pattern[p, Blank[]]]]], Pattern[x, Blank[Symbol]]] := -Dist[d^p/c^(m + 1), Subst[Int[(a + b*x)^n*Cos[x]^m*Sin[x]^(2*p + 1), x], x, ArcCos[c*x]], x] /; FreeQ[{a, b, c, d, e, n}, x] && EqQ[c^2*d + e, 0] && IntegerQ[2*p] && GtQ[p, -1] && IGtQ[m, 0] && (IntegerQ[p] || GtQ[d, 0])
  4725. Int[Times[Power[Plus[Optional[Pattern[a, Blank[]]], Times[ArcSin[Times[Optional[Pattern[c, Blank[]]], Pattern[x, Blank[]]]], Optional[Pattern[b, Blank[]]]]], Pattern[n, Blank[]]], Power[Pattern[x, Blank[]], Optional[Pattern[m, Blank[]]]], Power[Plus[Pattern[d, Blank[]], Times[Optional[Pattern[e, Blank[]]], Power[Pattern[x, Blank[]], 2]]], Pattern[p, Blank[]]]], Pattern[x, Blank[Symbol]]] := Dist[(d^IntPart[p]*(d + e*x^2)^FracPart[p])/(1 - c^2*x^2)^FracPart[p], Int[x^m*(1 - c^2*x^2)^p*(a + b*ArcSin[c*x])^n, x], x] /; FreeQ[{a, b, c, d, e, n}, x] && EqQ[c^2*d + e, 0] && IntegerQ[2*p] && GtQ[p, -1] && IGtQ[m, 0] && !(IntegerQ[p] || GtQ[d, 0])
  4726. Int[Times[Power[Plus[Optional[Pattern[a, Blank[]]], Times[ArcCos[Times[Optional[Pattern[c, Blank[]]], Pattern[x, Blank[]]]], Optional[Pattern[b, Blank[]]]]], Pattern[n, Blank[]]], Power[Pattern[x, Blank[]], Optional[Pattern[m, Blank[]]]], Power[Plus[Pattern[d, Blank[]], Times[Optional[Pattern[e, Blank[]]], Power[Pattern[x, Blank[]], 2]]], Pattern[p, Blank[]]]], Pattern[x, Blank[Symbol]]] := Dist[(d^IntPart[p]*(d + e*x^2)^FracPart[p])/(1 - c^2*x^2)^FracPart[p], Int[x^m*(1 - c^2*x^2)^p*(a + b*ArcCos[c*x])^n, x], x] /; FreeQ[{a, b, c, d, e, n}, x] && EqQ[c^2*d + e, 0] && IntegerQ[2*p] && GtQ[p, -1] && IGtQ[m, 0] && !(IntegerQ[p] || GtQ[d, 0])
  4727. Int[Times[Power[Plus[Optional[Pattern[a, Blank[]]], Times[ArcSin[Times[Optional[Pattern[c, Blank[]]], Pattern[x, Blank[]]]], Optional[Pattern[b, Blank[]]]]], Optional[Pattern[n, Blank[]]]], Power[Times[Optional[Pattern[f, Blank[]]], Pattern[x, Blank[]]], Pattern[m, Blank[]]], Power[Plus[Pattern[d, Blank[]], Times[Optional[Pattern[e, Blank[]]], Power[Pattern[x, Blank[]], 2]]], Pattern[p, Blank[]]]], Pattern[x, Blank[Symbol]]] := Int[ExpandIntegrand[(a + b*ArcSin[c*x])^n/Sqrt[d + e*x^2], (f*x)^m*(d + e*x^2)^(p + 1/2), x], x] /; FreeQ[{a, b, c, d, e, f, m, n}, x] && EqQ[c^2*d + e, 0] && GtQ[d, 0] && IGtQ[p + 1/2, 0] && !IGtQ[(m + 1)/2, 0] && (EqQ[m, -1] || EqQ[m, -2])
  4728. Int[Times[Power[Plus[Optional[Pattern[a, Blank[]]], Times[ArcCos[Times[Optional[Pattern[c, Blank[]]], Pattern[x, Blank[]]]], Optional[Pattern[b, Blank[]]]]], Optional[Pattern[n, Blank[]]]], Power[Times[Optional[Pattern[f, Blank[]]], Pattern[x, Blank[]]], Pattern[m, Blank[]]], Power[Plus[Pattern[d, Blank[]], Times[Optional[Pattern[e, Blank[]]], Power[Pattern[x, Blank[]], 2]]], Pattern[p, Blank[]]]], Pattern[x, Blank[Symbol]]] := Int[ExpandIntegrand[(a + b*ArcCos[c*x])^n/Sqrt[d + e*x^2], (f*x)^m*(d + e*x^2)^(p + 1/2), x], x] /; FreeQ[{a, b, c, d, e, f, m, n}, x] && EqQ[c^2*d + e, 0] && GtQ[d, 0] && IGtQ[p + 1/2, 0] && !IGtQ[(m + 1)/2, 0] && (EqQ[m, -1] || EqQ[m, -2])
  4729. Int[Times[Plus[Optional[Pattern[a, Blank[]]], Times[ArcSin[Times[Optional[Pattern[c, Blank[]]], Pattern[x, Blank[]]]], Optional[Pattern[b, Blank[]]]]], Pattern[x, Blank[]], Power[Plus[Pattern[d, Blank[]], Times[Optional[Pattern[e, Blank[]]], Power[Pattern[x, Blank[]], 2]]], Optional[Pattern[p, Blank[]]]]], Pattern[x, Blank[Symbol]]] := Simp[((d + e*x^2)^(p + 1)*(a + b*ArcSin[c*x]))/(2*e*(p + 1)), x] - Dist[(b*c)/(2*e*(p + 1)), Int[(d + e*x^2)^(p + 1)/Sqrt[1 - c^2*x^2], x], x] /; FreeQ[{a, b, c, d, e, p}, x] && NeQ[c^2*d + e, 0] && NeQ[p, -1]
  4730. Int[Times[Plus[Optional[Pattern[a, Blank[]]], Times[ArcCos[Times[Optional[Pattern[c, Blank[]]], Pattern[x, Blank[]]]], Optional[Pattern[b, Blank[]]]]], Pattern[x, Blank[]], Power[Plus[Pattern[d, Blank[]], Times[Optional[Pattern[e, Blank[]]], Power[Pattern[x, Blank[]], 2]]], Optional[Pattern[p, Blank[]]]]], Pattern[x, Blank[Symbol]]] := Simp[((d + e*x^2)^(p + 1)*(a + b*ArcCos[c*x]))/(2*e*(p + 1)), x] + Dist[(b*c)/(2*e*(p + 1)), Int[(d + e*x^2)^(p + 1)/Sqrt[1 - c^2*x^2], x], x] /; FreeQ[{a, b, c, d, e, p}, x] && NeQ[c^2*d + e, 0] && NeQ[p, -1]
  4731. Int[Times[Plus[Optional[Pattern[a, Blank[]]], Times[ArcSin[Times[Optional[Pattern[c, Blank[]]], Pattern[x, Blank[]]]], Optional[Pattern[b, Blank[]]]]], Power[Times[Optional[Pattern[f, Blank[]]], Pattern[x, Blank[]]], Optional[Pattern[m, Blank[]]]], Power[Plus[Pattern[d, Blank[]], Times[Optional[Pattern[e, Blank[]]], Power[Pattern[x, Blank[]], 2]]], Optional[Pattern[p, Blank[]]]]], Pattern[x, Blank[Symbol]]] := With[{u = IntHide[(f*x)^m*(d + e*x^2)^p, x]}, Dist[a + b*ArcSin[c*x], u, x] - Dist[b*c, Int[SimplifyIntegrand[u/Sqrt[1 - c^2*x^2], x], x], x]] /; FreeQ[{a, b, c, d, e, f, m}, x] && NeQ[c^2*d + e, 0] && IntegerQ[p] && (GtQ[p, 0] || (IGtQ[(m - 1)/2, 0] && LeQ[m + p, 0]))
  4732. Int[Times[Plus[Optional[Pattern[a, Blank[]]], Times[ArcCos[Times[Optional[Pattern[c, Blank[]]], Pattern[x, Blank[]]]], Optional[Pattern[b, Blank[]]]]], Power[Times[Optional[Pattern[f, Blank[]]], Pattern[x, Blank[]]], Optional[Pattern[m, Blank[]]]], Power[Plus[Pattern[d, Blank[]], Times[Optional[Pattern[e, Blank[]]], Power[Pattern[x, Blank[]], 2]]], Optional[Pattern[p, Blank[]]]]], Pattern[x, Blank[Symbol]]] := With[{u = IntHide[(f*x)^m*(d + e*x^2)^p, x]}, Dist[a + b*ArcCos[c*x], u, x] + Dist[b*c, Int[SimplifyIntegrand[u/Sqrt[1 - c^2*x^2], x], x], x]] /; FreeQ[{a, b, c, d, e, f, m}, x] && NeQ[c^2*d + e, 0] && IntegerQ[p] && (GtQ[p, 0] || (IGtQ[(m - 1)/2, 0] && LeQ[m + p, 0]))
  4733. Int[Times[Power[Plus[Optional[Pattern[a, Blank[]]], Times[ArcSin[Times[Optional[Pattern[c, Blank[]]], Pattern[x, Blank[]]]], Optional[Pattern[b, Blank[]]]]], Optional[Pattern[n, Blank[]]]], Power[Times[Optional[Pattern[f, Blank[]]], Pattern[x, Blank[]]], Optional[Pattern[m, Blank[]]]], Power[Plus[Pattern[d, Blank[]], Times[Optional[Pattern[e, Blank[]]], Power[Pattern[x, Blank[]], 2]]], Optional[Pattern[p, Blank[]]]]], Pattern[x, Blank[Symbol]]] := Int[ExpandIntegrand[(a + b*ArcSin[c*x])^n, (f*x)^m*(d + e*x^2)^p, x], x] /; FreeQ[{a, b, c, d, e, f}, x] && NeQ[c^2*d + e, 0] && IGtQ[n, 0] && IntegerQ[p] && IntegerQ[m]
  4734. Int[Times[Power[Plus[Optional[Pattern[a, Blank[]]], Times[ArcCos[Times[Optional[Pattern[c, Blank[]]], Pattern[x, Blank[]]]], Optional[Pattern[b, Blank[]]]]], Optional[Pattern[n, Blank[]]]], Power[Times[Optional[Pattern[f, Blank[]]], Pattern[x, Blank[]]], Optional[Pattern[m, Blank[]]]], Power[Plus[Pattern[d, Blank[]], Times[Optional[Pattern[e, Blank[]]], Power[Pattern[x, Blank[]], 2]]], Optional[Pattern[p, Blank[]]]]], Pattern[x, Blank[Symbol]]] := Int[ExpandIntegrand[(a + b*ArcCos[c*x])^n, (f*x)^m*(d + e*x^2)^p, x], x] /; FreeQ[{a, b, c, d, e, f}, x] && NeQ[c^2*d + e, 0] && IGtQ[n, 0] && IntegerQ[p] && IntegerQ[m]
  4735. Int[Times[Power[Plus[Optional[Pattern[a, Blank[]]], Times[ArcSin[Times[Optional[Pattern[c, Blank[]]], Pattern[x, Blank[]]]], Optional[Pattern[b, Blank[]]]]], Optional[Pattern[n, Blank[]]]], Power[Times[Optional[Pattern[f, Blank[]]], Pattern[x, Blank[]]], Optional[Pattern[m, Blank[]]]], Power[Plus[Pattern[d, Blank[]], Times[Optional[Pattern[e, Blank[]]], Power[Pattern[x, Blank[]], 2]]], Optional[Pattern[p, Blank[]]]]], Pattern[x, Blank[Symbol]]] := Unintegrable[(f*x)^m*(d + e*x^2)^p*(a + b*ArcSin[c*x])^n, x] /; FreeQ[{a, b, c, d, e, f, m, n, p}, x]
  4736. Int[Times[Power[Plus[Optional[Pattern[a, Blank[]]], Times[ArcCos[Times[Optional[Pattern[c, Blank[]]], Pattern[x, Blank[]]]], Optional[Pattern[b, Blank[]]]]], Optional[Pattern[n, Blank[]]]], Power[Times[Optional[Pattern[f, Blank[]]], Pattern[x, Blank[]]], Optional[Pattern[m, Blank[]]]], Power[Plus[Pattern[d, Blank[]], Times[Optional[Pattern[e, Blank[]]], Power[Pattern[x, Blank[]], 2]]], Optional[Pattern[p, Blank[]]]]], Pattern[x, Blank[Symbol]]] := Unintegrable[(f*x)^m*(d + e*x^2)^p*(a + b*ArcCos[c*x])^n, x] /; FreeQ[{a, b, c, d, e, f, m, n, p}, x]
  4737. Int[Times[Power[Plus[Optional[Pattern[a, Blank[]]], Times[ArcSin[Times[Optional[Pattern[c, Blank[]]], Pattern[x, Blank[]]]], Optional[Pattern[b, Blank[]]]]], Optional[Pattern[n, Blank[]]]], Power[Times[Optional[Pattern[h, Blank[]]], Pattern[x, Blank[]]], Optional[Pattern[m, Blank[]]]], Power[Plus[Pattern[d, Blank[]], Times[Optional[Pattern[e, Blank[]]], Pattern[x, Blank[]]]], Pattern[p, Blank[]]], Power[Plus[Pattern[f, Blank[]], Times[Optional[Pattern[g, Blank[]]], Pattern[x, Blank[]]]], Pattern[q, Blank[]]]], Pattern[x, Blank[Symbol]]] := Dist[(-((d^2*g)/e))^q, Int[(h*x)^m*(d + e*x)^(p - q)*(1 - c^2*x^2)^q*(a + b*ArcSin[c*x])^n, x], x] /; FreeQ[{a, b, c, d, e, f, g, h, m, n}, x] && EqQ[e*f + d*g, 0] && EqQ[c^2*d^2 - e^2, 0] && HalfIntegerQ[p, q] && GeQ[p - q, 0] && GtQ[d, 0] && LtQ[g/e, 0]
  4738. Int[Times[Power[Plus[Optional[Pattern[a, Blank[]]], Times[ArcCos[Times[Optional[Pattern[c, Blank[]]], Pattern[x, Blank[]]]], Optional[Pattern[b, Blank[]]]]], Optional[Pattern[n, Blank[]]]], Power[Times[Optional[Pattern[h, Blank[]]], Pattern[x, Blank[]]], Optional[Pattern[m, Blank[]]]], Power[Plus[Pattern[d, Blank[]], Times[Optional[Pattern[e, Blank[]]], Pattern[x, Blank[]]]], Pattern[p, Blank[]]], Power[Plus[Pattern[f, Blank[]], Times[Optional[Pattern[g, Blank[]]], Pattern[x, Blank[]]]], Pattern[q, Blank[]]]], Pattern[x, Blank[Symbol]]] := Dist[(-((d^2*g)/e))^q, Int[(h*x)^m*(d + e*x)^(p - q)*(1 - c^2*x^2)^q*(a + b*ArcCos[c*x])^n, x], x] /; FreeQ[{a, b, c, d, e, f, g, h, m, n}, x] && EqQ[e*f + d*g, 0] && EqQ[c^2*d^2 - e^2, 0] && HalfIntegerQ[p, q] && GeQ[p - q, 0] && GtQ[d, 0] && LtQ[g/e, 0]
  4739. Int[Times[Power[Plus[Optional[Pattern[a, Blank[]]], Times[ArcSin[Times[Optional[Pattern[c, Blank[]]], Pattern[x, Blank[]]]], Optional[Pattern[b, Blank[]]]]], Optional[Pattern[n, Blank[]]]], Power[Times[Optional[Pattern[h, Blank[]]], Pattern[x, Blank[]]], Optional[Pattern[m, Blank[]]]], Power[Plus[Pattern[d, Blank[]], Times[Optional[Pattern[e, Blank[]]], Pattern[x, Blank[]]]], Pattern[p, Blank[]]], Power[Plus[Pattern[f, Blank[]], Times[Optional[Pattern[g, Blank[]]], Pattern[x, Blank[]]]], Pattern[q, Blank[]]]], Pattern[x, Blank[Symbol]]] := Dist[((-((d^2*g)/e))^IntPart[q]*(d + e*x)^FracPart[q]*(f + g*x)^FracPart[q])/(1 - c^2*x^2)^FracPart[q], Int[(h*x)^m*(d + e*x)^(p - q)*(1 - c^2*x^2)^q*(a + b*ArcSin[c*x])^n, x], x] /; FreeQ[{a, b, c, d, e, f, g, h, m, n}, x] && EqQ[e*f + d*g, 0] && EqQ[c^2*d^2 - e^2, 0] && HalfIntegerQ[p, q] && GeQ[p - q, 0]
  4740. Int[Times[Power[Plus[Optional[Pattern[a, Blank[]]], Times[ArcCos[Times[Optional[Pattern[c, Blank[]]], Pattern[x, Blank[]]]], Optional[Pattern[b, Blank[]]]]], Optional[Pattern[n, Blank[]]]], Power[Times[Optional[Pattern[h, Blank[]]], Pattern[x, Blank[]]], Optional[Pattern[m, Blank[]]]], Power[Plus[Pattern[d, Blank[]], Times[Optional[Pattern[e, Blank[]]], Pattern[x, Blank[]]]], Pattern[p, Blank[]]], Power[Plus[Pattern[f, Blank[]], Times[Optional[Pattern[g, Blank[]]], Pattern[x, Blank[]]]], Pattern[q, Blank[]]]], Pattern[x, Blank[Symbol]]] := Dist[((-((d^2*g)/e))^IntPart[q]*(d + e*x)^FracPart[q]*(f + g*x)^FracPart[q])/(1 - c^2*x^2)^FracPart[q], Int[(h*x)^m*(d + e*x)^(p - q)*(1 - c^2*x^2)^q*(a + b*ArcCos[c*x])^n, x], x] /; FreeQ[{a, b, c, d, e, f, g, h, m, n}, x] && EqQ[e*f + d*g, 0] && EqQ[c^2*d^2 - e^2, 0] && HalfIntegerQ[p, q] && GeQ[p - q, 0]
  4741. Int[Times[Power[Plus[Optional[Pattern[a, Blank[]]], Times[ArcSin[Times[Optional[Pattern[c, Blank[]]], Pattern[x, Blank[]]]], Optional[Pattern[b, Blank[]]]]], Optional[Pattern[n, Blank[]]]], Power[Plus[Pattern[d, Blank[]], Times[Optional[Pattern[e, Blank[]]], Pattern[x, Blank[]]]], -1]], Pattern[x, Blank[Symbol]]] := Subst[Int[((a + b*x)^n*Cos[x])/(c*d + e*Sin[x]), x], x, ArcSin[c*x]] /; FreeQ[{a, b, c, d, e}, x] && IGtQ[n, 0]
  4742. Int[Times[Power[Plus[Optional[Pattern[a, Blank[]]], Times[ArcCos[Times[Optional[Pattern[c, Blank[]]], Pattern[x, Blank[]]]], Optional[Pattern[b, Blank[]]]]], Optional[Pattern[n, Blank[]]]], Power[Plus[Pattern[d, Blank[]], Times[Optional[Pattern[e, Blank[]]], Pattern[x, Blank[]]]], -1]], Pattern[x, Blank[Symbol]]] := -Subst[Int[((a + b*x)^n*Sin[x])/(c*d + e*Cos[x]), x], x, ArcCos[c*x]] /; FreeQ[{a, b, c, d, e}, x] && IGtQ[n, 0]
  4743. Int[Times[Power[Plus[Optional[Pattern[a, Blank[]]], Times[ArcSin[Times[Optional[Pattern[c, Blank[]]], Pattern[x, Blank[]]]], Optional[Pattern[b, Blank[]]]]], Optional[Pattern[n, Blank[]]]], Power[Plus[Pattern[d, Blank[]], Times[Optional[Pattern[e, Blank[]]], Pattern[x, Blank[]]]], Optional[Pattern[m, Blank[]]]]], Pattern[x, Blank[Symbol]]] := Simp[((d + e*x)^(m + 1)*(a + b*ArcSin[c*x])^n)/(e*(m + 1)), x] - Dist[(b*c*n)/(e*(m + 1)), Int[((d + e*x)^(m + 1)*(a + b*ArcSin[c*x])^(n - 1))/Sqrt[1 - c^2*x^2], x], x] /; FreeQ[{a, b, c, d, e, m}, x] && IGtQ[n, 0] && NeQ[m, -1]
  4744. Int[Times[Power[Plus[Optional[Pattern[a, Blank[]]], Times[ArcCos[Times[Optional[Pattern[c, Blank[]]], Pattern[x, Blank[]]]], Optional[Pattern[b, Blank[]]]]], Optional[Pattern[n, Blank[]]]], Power[Plus[Pattern[d, Blank[]], Times[Optional[Pattern[e, Blank[]]], Pattern[x, Blank[]]]], Optional[Pattern[m, Blank[]]]]], Pattern[x, Blank[Symbol]]] := Simp[((d + e*x)^(m + 1)*(a + b*ArcCos[c*x])^n)/(e*(m + 1)), x] + Dist[(b*c*n)/(e*(m + 1)), Int[((d + e*x)^(m + 1)*(a + b*ArcCos[c*x])^(n - 1))/Sqrt[1 - c^2*x^2], x], x] /; FreeQ[{a, b, c, d, e, m}, x] && IGtQ[n, 0] && NeQ[m, -1]
  4745. Int[Times[Power[Plus[Optional[Pattern[a, Blank[]]], Times[ArcSin[Times[Optional[Pattern[c, Blank[]]], Pattern[x, Blank[]]]], Optional[Pattern[b, Blank[]]]]], Pattern[n, Blank[]]], Power[Plus[Pattern[d, Blank[]], Times[Optional[Pattern[e, Blank[]]], Pattern[x, Blank[]]]], Optional[Pattern[m, Blank[]]]]], Pattern[x, Blank[Symbol]]] := Int[ExpandIntegrand[(d + e*x)^m*(a + b*ArcSin[c*x])^n, x], x] /; FreeQ[{a, b, c, d, e}, x] && IGtQ[m, 0] && LtQ[n, -1]
  4746. Int[Times[Power[Plus[Optional[Pattern[a, Blank[]]], Times[ArcCos[Times[Optional[Pattern[c, Blank[]]], Pattern[x, Blank[]]]], Optional[Pattern[b, Blank[]]]]], Pattern[n, Blank[]]], Power[Plus[Pattern[d, Blank[]], Times[Optional[Pattern[e, Blank[]]], Pattern[x, Blank[]]]], Optional[Pattern[m, Blank[]]]]], Pattern[x, Blank[Symbol]]] := Int[ExpandIntegrand[(d + e*x)^m*(a + b*ArcCos[c*x])^n, x], x] /; FreeQ[{a, b, c, d, e}, x] && IGtQ[m, 0] && LtQ[n, -1]
  4747. Int[Times[Power[Plus[Optional[Pattern[a, Blank[]]], Times[ArcSin[Times[Optional[Pattern[c, Blank[]]], Pattern[x, Blank[]]]], Optional[Pattern[b, Blank[]]]]], Pattern[n, Blank[]]], Power[Plus[Optional[Pattern[d, Blank[]]], Times[Optional[Pattern[e, Blank[]]], Pattern[x, Blank[]]]], Optional[Pattern[m, Blank[]]]]], Pattern[x, Blank[Symbol]]] := Dist[1/c^(m + 1), Subst[Int[(a + b*x)^n*Cos[x]*(c*d + e*Sin[x])^m, x], x, ArcSin[c*x]], x] /; FreeQ[{a, b, c, d, e, n}, x] && IGtQ[m, 0]
  4748. Int[Times[Power[Plus[Optional[Pattern[a, Blank[]]], Times[ArcCos[Times[Optional[Pattern[c, Blank[]]], Pattern[x, Blank[]]]], Optional[Pattern[b, Blank[]]]]], Pattern[n, Blank[]]], Power[Plus[Optional[Pattern[d, Blank[]]], Times[Optional[Pattern[e, Blank[]]], Pattern[x, Blank[]]]], Optional[Pattern[m, Blank[]]]]], Pattern[x, Blank[Symbol]]] := -Dist[(c^(m + 1))^(-1), Subst[Int[(a + b*x)^n*Sin[x]*(c*d + e*Cos[x])^m, x], x, ArcCos[c*x]], x] /; FreeQ[{a, b, c, d, e, n}, x] && IGtQ[m, 0]
  4749. Int[Times[Plus[Optional[Pattern[a, Blank[]]], Times[ArcSin[Times[Optional[Pattern[c, Blank[]]], Pattern[x, Blank[]]]], Optional[Pattern[b, Blank[]]]]], Pattern[Px, Blank[]]], Pattern[x, Blank[Symbol]]] := With[{u = IntHide[ExpandExpression[Px, x], x]}, Dist[a + b*ArcSin[c*x], u, x] - Dist[b*c, Int[SimplifyIntegrand[u/Sqrt[1 - c^2*x^2], x], x], x]] /; FreeQ[{a, b, c}, x] && PolynomialQ[Px, x]
  4750. Int[Times[Plus[Optional[Pattern[a, Blank[]]], Times[ArcCos[Times[Optional[Pattern[c, Blank[]]], Pattern[x, Blank[]]]], Optional[Pattern[b, Blank[]]]]], Pattern[Px, Blank[]]], Pattern[x, Blank[Symbol]]] := With[{u = IntHide[ExpandExpression[Px, x], x]}, Dist[a + b*ArcCos[c*x], u, x] + Dist[b*c, Int[SimplifyIntegrand[u/Sqrt[1 - c^2*x^2], x], x], x]] /; FreeQ[{a, b, c}, x] && PolynomialQ[Px, x]
  4751. Int[Times[Power[Plus[Optional[Pattern[a, Blank[]]], Times[ArcSin[Times[Optional[Pattern[c, Blank[]]], Pattern[x, Blank[]]]], Optional[Pattern[b, Blank[]]]]], Pattern[n, Blank[]]], Pattern[Px, Blank[]]], Pattern[x, Blank[Symbol]]] := Int[ExpandIntegrand[Px*(a + b*ArcSin[c*x])^n, x], x] /; FreeQ[{a, b, c, n}, x] && PolynomialQ[Px, x]
  4752. Int[Times[Power[Plus[Optional[Pattern[a, Blank[]]], Times[ArcCos[Times[Optional[Pattern[c, Blank[]]], Pattern[x, Blank[]]]], Optional[Pattern[b, Blank[]]]]], Pattern[n, Blank[]]], Pattern[Px, Blank[]]], Pattern[x, Blank[Symbol]]] := Int[ExpandIntegrand[Px*(a + b*ArcCos[c*x])^n, x], x] /; FreeQ[{a, b, c, n}, x] && PolynomialQ[Px, x]
  4753. Int[Times[Plus[Optional[Pattern[a, Blank[]]], Times[ArcSin[Times[Optional[Pattern[c, Blank[]]], Pattern[x, Blank[]]]], Optional[Pattern[b, Blank[]]]]], Pattern[Px, Blank[]], Power[Plus[Optional[Pattern[d, Blank[]]], Times[Optional[Pattern[e, Blank[]]], Pattern[x, Blank[]]]], Optional[Pattern[m, Blank[]]]]], Pattern[x, Blank[Symbol]]] := With[{u = IntHide[Px*(d + e*x)^m, x]}, Dist[a + b*ArcSin[c*x], u, x] - Dist[b*c, Int[SimplifyIntegrand[u/Sqrt[1 - c^2*x^2], x], x], x]] /; FreeQ[{a, b, c, d, e, m}, x] && PolynomialQ[Px, x]
  4754. Int[Times[Plus[Optional[Pattern[a, Blank[]]], Times[ArcCos[Times[Optional[Pattern[c, Blank[]]], Pattern[x, Blank[]]]], Optional[Pattern[b, Blank[]]]]], Pattern[Px, Blank[]], Power[Plus[Optional[Pattern[d, Blank[]]], Times[Optional[Pattern[e, Blank[]]], Pattern[x, Blank[]]]], Optional[Pattern[m, Blank[]]]]], Pattern[x, Blank[Symbol]]] := With[{u = IntHide[Px*(d + e*x)^m, x]}, Dist[a + b*ArcCos[c*x], u, x] + Dist[b*c, Int[SimplifyIntegrand[u/Sqrt[1 - c^2*x^2], x], x], x]] /; FreeQ[{a, b, c, d, e, m}, x] && PolynomialQ[Px, x]
  4755. Int[Times[Power[Plus[Optional[Pattern[a, Blank[]]], Times[ArcSin[Times[Optional[Pattern[c, Blank[]]], Pattern[x, Blank[]]]], Optional[Pattern[b, Blank[]]]]], Pattern[n, Blank[]]], Power[Plus[Pattern[d, Blank[]], Times[Optional[Pattern[e, Blank[]]], Pattern[x, Blank[]]]], Pattern[m, Blank[]]], Power[Plus[Optional[Pattern[f, Blank[]]], Times[Optional[Pattern[g, Blank[]]], Pattern[x, Blank[]]]], Optional[Pattern[p, Blank[]]]]], Pattern[x, Blank[Symbol]]] := With[{u = IntHide[(f + g*x)^p*(d + e*x)^m, x]}, Dist[(a + b*ArcSin[c*x])^n, u, x] - Dist[b*c*n, Int[SimplifyIntegrand[(u*(a + b*ArcSin[c*x])^(n - 1))/Sqrt[1 - c^2*x^2], x], x], x]] /; FreeQ[{a, b, c, d, e, f, g}, x] && IGtQ[n, 0] && IGtQ[p, 0] && ILtQ[m, 0] && LtQ[m + p + 1, 0]
  4756. Int[Times[Power[Plus[Optional[Pattern[a, Blank[]]], Times[ArcCos[Times[Optional[Pattern[c, Blank[]]], Pattern[x, Blank[]]]], Optional[Pattern[b, Blank[]]]]], Pattern[n, Blank[]]], Power[Plus[Pattern[d, Blank[]], Times[Optional[Pattern[e, Blank[]]], Pattern[x, Blank[]]]], Pattern[m, Blank[]]], Power[Plus[Optional[Pattern[f, Blank[]]], Times[Optional[Pattern[g, Blank[]]], Pattern[x, Blank[]]]], Optional[Pattern[p, Blank[]]]]], Pattern[x, Blank[Symbol]]] := With[{u = IntHide[(f + g*x)^p*(d + e*x)^m, x]}, Dist[(a + b*ArcCos[c*x])^n, u, x] + Dist[b*c*n, Int[SimplifyIntegrand[(u*(a + b*ArcCos[c*x])^(n - 1))/Sqrt[1 - c^2*x^2], x], x], x]] /; FreeQ[{a, b, c, d, e, f, g}, x] && IGtQ[n, 0] && IGtQ[p, 0] && ILtQ[m, 0] && LtQ[m + p + 1, 0]
  4757. Int[Times[Power[Plus[Optional[Pattern[a, Blank[]]], Times[ArcSin[Times[Optional[Pattern[c, Blank[]]], Pattern[x, Blank[]]]], Optional[Pattern[b, Blank[]]]]], Pattern[n, Blank[]]], Power[Plus[Pattern[d, Blank[]], Times[Optional[Pattern[e, Blank[]]], Pattern[x, Blank[]]]], -2], Power[Plus[Optional[Pattern[f, Blank[]]], Times[Optional[Pattern[g, Blank[]]], Pattern[x, Blank[]]], Times[Optional[Pattern[h, Blank[]]], Power[Pattern[x, Blank[]], 2]]], Optional[Pattern[p, Blank[]]]]], Pattern[x, Blank[Symbol]]] := With[{u = IntHide[(f + g*x + h*x^2)^p/(d + e*x)^2, x]}, Dist[(a + b*ArcSin[c*x])^n, u, x] - Dist[b*c*n, Int[SimplifyIntegrand[(u*(a + b*ArcSin[c*x])^(n - 1))/Sqrt[1 - c^2*x^2], x], x], x]] /; FreeQ[{a, b, c, d, e, f, g, h}, x] && IGtQ[n, 0] && IGtQ[p, 0] && EqQ[e*g - 2*d*h, 0]
  4758. Int[Times[Power[Plus[Optional[Pattern[a, Blank[]]], Times[ArcCos[Times[Optional[Pattern[c, Blank[]]], Pattern[x, Blank[]]]], Optional[Pattern[b, Blank[]]]]], Pattern[n, Blank[]]], Power[Plus[Pattern[d, Blank[]], Times[Optional[Pattern[e, Blank[]]], Pattern[x, Blank[]]]], -2], Power[Plus[Optional[Pattern[f, Blank[]]], Times[Optional[Pattern[g, Blank[]]], Pattern[x, Blank[]]], Times[Optional[Pattern[h, Blank[]]], Power[Pattern[x, Blank[]], 2]]], Optional[Pattern[p, Blank[]]]]], Pattern[x, Blank[Symbol]]] := With[{u = IntHide[(f + g*x + h*x^2)^p/(d + e*x)^2, x]}, Dist[(a + b*ArcCos[c*x])^n, u, x] + Dist[b*c*n, Int[SimplifyIntegrand[(u*(a + b*ArcCos[c*x])^(n - 1))/Sqrt[1 - c^2*x^2], x], x], x]] /; FreeQ[{a, b, c, d, e, f, g, h}, x] && IGtQ[n, 0] && IGtQ[p, 0] && EqQ[e*g - 2*d*h, 0]
  4759. Int[Times[Power[Plus[Optional[Pattern[a, Blank[]]], Times[ArcSin[Times[Optional[Pattern[c, Blank[]]], Pattern[x, Blank[]]]], Optional[Pattern[b, Blank[]]]]], Pattern[n, Blank[]]], Pattern[Px, Blank[]], Power[Plus[Pattern[d, Blank[]], Times[Optional[Pattern[e, Blank[]]], Pattern[x, Blank[]]]], Optional[Pattern[m, Blank[]]]]], Pattern[x, Blank[Symbol]]] := Int[ExpandIntegrand[Px*(d + e*x)^m*(a + b*ArcSin[c*x])^n, x], x] /; FreeQ[{a, b, c, d, e}, x] && PolynomialQ[Px, x] && IGtQ[n, 0] && IntegerQ[m]
  4760. Int[Times[Power[Plus[Optional[Pattern[a, Blank[]]], Times[ArcCos[Times[Optional[Pattern[c, Blank[]]], Pattern[x, Blank[]]]], Optional[Pattern[b, Blank[]]]]], Pattern[n, Blank[]]], Pattern[Px, Blank[]], Power[Plus[Pattern[d, Blank[]], Times[Optional[Pattern[e, Blank[]]], Pattern[x, Blank[]]]], Optional[Pattern[m, Blank[]]]]], Pattern[x, Blank[Symbol]]] := Int[ExpandIntegrand[Px*(d + e*x)^m*(a + b*ArcCos[c*x])^n, x], x] /; FreeQ[{a, b, c, d, e}, x] && PolynomialQ[Px, x] && IGtQ[n, 0] && IntegerQ[m]
  4761. Int[Times[Plus[Optional[Pattern[a, Blank[]]], Times[ArcSin[Times[Optional[Pattern[c, Blank[]]], Pattern[x, Blank[]]]], Optional[Pattern[b, Blank[]]]]], Power[Plus[Pattern[f, Blank[]], Times[Optional[Pattern[g, Blank[]]], Pattern[x, Blank[]]]], Optional[Pattern[m, Blank[]]]], Power[Plus[Pattern[d, Blank[]], Times[Optional[Pattern[e, Blank[]]], Power[Pattern[x, Blank[]], 2]]], Pattern[p, Blank[]]]], Pattern[x, Blank[Symbol]]] := With[{u = IntHide[(f + g*x)^m*(d + e*x^2)^p, x]}, Dist[a + b*ArcSin[c*x], u, x] - Dist[b*c, Int[Dist[1/Sqrt[1 - c^2*x^2], u, x], x], x]] /; FreeQ[{a, b, c, d, e, f, g}, x] && EqQ[c^2*d + e, 0] && IGtQ[m, 0] && ILtQ[p + 1/2, 0] && GtQ[d, 0] && (LtQ[m, -2*p - 1] || GtQ[m, 3])
  4762. Int[Times[Plus[Optional[Pattern[a, Blank[]]], Times[ArcCos[Times[Optional[Pattern[c, Blank[]]], Pattern[x, Blank[]]]], Optional[Pattern[b, Blank[]]]]], Power[Plus[Pattern[f, Blank[]], Times[Optional[Pattern[g, Blank[]]], Pattern[x, Blank[]]]], Optional[Pattern[m, Blank[]]]], Power[Plus[Pattern[d, Blank[]], Times[Optional[Pattern[e, Blank[]]], Power[Pattern[x, Blank[]], 2]]], Pattern[p, Blank[]]]], Pattern[x, Blank[Symbol]]] := With[{u = IntHide[(f + g*x)^m*(d + e*x^2)^p, x]}, Dist[a + b*ArcCos[c*x], u, x] + Dist[b*c, Int[Dist[1/Sqrt[1 - c^2*x^2], u, x], x], x]] /; FreeQ[{a, b, c, d, e, f, g}, x] && EqQ[c^2*d + e, 0] && IGtQ[m, 0] && ILtQ[p + 1/2, 0] && GtQ[d, 0] && (LtQ[m, -2*p - 1] || GtQ[m, 3])
  4763. Int[Times[Power[Plus[Optional[Pattern[a, Blank[]]], Times[ArcSin[Times[Optional[Pattern[c, Blank[]]], Pattern[x, Blank[]]]], Optional[Pattern[b, Blank[]]]]], Optional[Pattern[n, Blank[]]]], Power[Plus[Pattern[f, Blank[]], Times[Optional[Pattern[g, Blank[]]], Pattern[x, Blank[]]]], Optional[Pattern[m, Blank[]]]], Power[Plus[Pattern[d, Blank[]], Times[Optional[Pattern[e, Blank[]]], Power[Pattern[x, Blank[]], 2]]], Pattern[p, Blank[]]]], Pattern[x, Blank[Symbol]]] := Int[ExpandIntegrand[(d + e*x^2)^p*(a + b*ArcSin[c*x])^n, (f + g*x)^m, x], x] /; FreeQ[{a, b, c, d, e, f, g}, x] && EqQ[c^2*d + e, 0] && IGtQ[m, 0] && IntegerQ[p + 1/2] && GtQ[d, 0] && IGtQ[n, 0] && (m == 1 || p > 0 || (n == 1 && p > -1) || (m == 2 && p < -2))
  4764. Int[Times[Power[Plus[Optional[Pattern[a, Blank[]]], Times[ArcCos[Times[Optional[Pattern[c, Blank[]]], Pattern[x, Blank[]]]], Optional[Pattern[b, Blank[]]]]], Optional[Pattern[n, Blank[]]]], Power[Plus[Pattern[f, Blank[]], Times[Optional[Pattern[g, Blank[]]], Pattern[x, Blank[]]]], Optional[Pattern[m, Blank[]]]], Power[Plus[Pattern[d, Blank[]], Times[Optional[Pattern[e, Blank[]]], Power[Pattern[x, Blank[]], 2]]], Pattern[p, Blank[]]]], Pattern[x, Blank[Symbol]]] := Int[ExpandIntegrand[(d + e*x^2)^p*(a + b*ArcCos[c*x])^n, (f + g*x)^m, x], x] /; FreeQ[{a, b, c, d, e, f, g}, x] && EqQ[c^2*d + e, 0] && IGtQ[m, 0] && IntegerQ[p + 1/2] && GtQ[d, 0] && IGtQ[n, 0] && (m == 1 || p > 0 || (n == 1 && p > -1) || (m == 2 && p < -2))
  4765. Int[Times[Power[Plus[Optional[Pattern[a, Blank[]]], Times[ArcSin[Times[Optional[Pattern[c, Blank[]]], Pattern[x, Blank[]]]], Optional[Pattern[b, Blank[]]]]], Optional[Pattern[n, Blank[]]]], Power[Plus[Pattern[f, Blank[]], Times[Optional[Pattern[g, Blank[]]], Pattern[x, Blank[]]]], Pattern[m, Blank[]]], Power[Plus[Pattern[d, Blank[]], Times[Optional[Pattern[e, Blank[]]], Power[Pattern[x, Blank[]], 2]]], Rational[1, 2]]], Pattern[x, Blank[Symbol]]] := Simp[((f + g*x)^m*(d + e*x^2)*(a + b*ArcSin[c*x])^(n + 1))/(b*c*Sqrt[d]*(n + 1)), x] - Dist[1/(b*c*Sqrt[d]*(n + 1)), Int[(d*g*m + 2*e*f*x + e*g*(m + 2)*x^2)*(f + g*x)^(m - 1)*(a + b*ArcSin[c*x])^(n + 1), x], x] /; FreeQ[{a, b, c, d, e, f, g}, x] && EqQ[c^2*d + e, 0] && ILtQ[m, 0] && GtQ[d, 0] && IGtQ[n, 0]
  4766. Int[Times[Power[Plus[Optional[Pattern[a, Blank[]]], Times[ArcCos[Times[Optional[Pattern[c, Blank[]]], Pattern[x, Blank[]]]], Optional[Pattern[b, Blank[]]]]], Optional[Pattern[n, Blank[]]]], Power[Plus[Pattern[f, Blank[]], Times[Optional[Pattern[g, Blank[]]], Pattern[x, Blank[]]]], Pattern[m, Blank[]]], Power[Plus[Pattern[d, Blank[]], Times[Optional[Pattern[e, Blank[]]], Power[Pattern[x, Blank[]], 2]]], Rational[1, 2]]], Pattern[x, Blank[Symbol]]] := -Simp[((f + g*x)^m*(d + e*x^2)*(a + b*ArcCos[c*x])^(n + 1))/(b*c*Sqrt[d]*(n + 1)), x] + Dist[1/(b*c*Sqrt[d]*(n + 1)), Int[(d*g*m + 2*e*f*x + e*g*(m + 2)*x^2)*(f + g*x)^(m - 1)*(a + b*ArcCos[c*x])^(n + 1), x], x] /; FreeQ[{a, b, c, d, e, f, g}, x] && EqQ[c^2*d + e, 0] && ILtQ[m, 0] && GtQ[d, 0] && IGtQ[n, 0]
  4767. Int[Times[Power[Plus[Optional[Pattern[a, Blank[]]], Times[ArcSin[Times[Optional[Pattern[c, Blank[]]], Pattern[x, Blank[]]]], Optional[Pattern[b, Blank[]]]]], Optional[Pattern[n, Blank[]]]], Power[Plus[Pattern[f, Blank[]], Times[Optional[Pattern[g, Blank[]]], Pattern[x, Blank[]]]], Optional[Pattern[m, Blank[]]]], Power[Plus[Pattern[d, Blank[]], Times[Optional[Pattern[e, Blank[]]], Power[Pattern[x, Blank[]], 2]]], Pattern[p, Blank[]]]], Pattern[x, Blank[Symbol]]] := Int[ExpandIntegrand[Sqrt[d + e*x^2]*(a + b*ArcSin[c*x])^n, (f + g*x)^m*(d + e*x^2)^(p - 1/2), x], x] /; FreeQ[{a, b, c, d, e, f, g}, x] && EqQ[c^2*d + e, 0] && IntegerQ[m] && IGtQ[p + 1/2, 0] && GtQ[d, 0] && IGtQ[n, 0]
  4768. Int[Times[Power[Plus[Optional[Pattern[a, Blank[]]], Times[ArcCos[Times[Optional[Pattern[c, Blank[]]], Pattern[x, Blank[]]]], Optional[Pattern[b, Blank[]]]]], Optional[Pattern[n, Blank[]]]], Power[Plus[Pattern[f, Blank[]], Times[Optional[Pattern[g, Blank[]]], Pattern[x, Blank[]]]], Optional[Pattern[m, Blank[]]]], Power[Plus[Pattern[d, Blank[]], Times[Optional[Pattern[e, Blank[]]], Power[Pattern[x, Blank[]], 2]]], Pattern[p, Blank[]]]], Pattern[x, Blank[Symbol]]] := Int[ExpandIntegrand[Sqrt[d + e*x^2]*(a + b*ArcCos[c*x])^n, (f + g*x)^m*(d + e*x^2)^(p - 1/2), x], x] /; FreeQ[{a, b, c, d, e, f, g}, x] && EqQ[c^2*d + e, 0] && IntegerQ[m] && IGtQ[p + 1/2, 0] && GtQ[d, 0] && IGtQ[n, 0]
  4769. Int[Times[Power[Plus[Optional[Pattern[a, Blank[]]], Times[ArcSin[Times[Optional[Pattern[c, Blank[]]], Pattern[x, Blank[]]]], Optional[Pattern[b, Blank[]]]]], Optional[Pattern[n, Blank[]]]], Power[Plus[Pattern[f, Blank[]], Times[Optional[Pattern[g, Blank[]]], Pattern[x, Blank[]]]], Optional[Pattern[m, Blank[]]]], Power[Plus[Pattern[d, Blank[]], Times[Optional[Pattern[e, Blank[]]], Power[Pattern[x, Blank[]], 2]]], Pattern[p, Blank[]]]], Pattern[x, Blank[Symbol]]] := Simp[((f + g*x)^m*(d + e*x^2)^(p + 1/2)*(a + b*ArcSin[c*x])^(n + 1))/(b*c*Sqrt[d]*(n + 1)), x] - Dist[1/(b*c*Sqrt[d]*(n + 1)), Int[ExpandIntegrand[(f + g*x)^(m - 1)*(a + b*ArcSin[c*x])^(n + 1), (d*g*m + e*f*(2*p + 1)*x + e*g*(m + 2*p + 1)*x^2)*(d + e*x^2)^(p - 1/2), x], x], x] /; FreeQ[{a, b, c, d, e, f, g}, x] && EqQ[c^2*d + e, 0] && ILtQ[m, 0] && IGtQ[p - 1/2, 0] && GtQ[d, 0] && IGtQ[n, 0]
  4770. Int[Times[Power[Plus[Optional[Pattern[a, Blank[]]], Times[ArcCos[Times[Optional[Pattern[c, Blank[]]], Pattern[x, Blank[]]]], Optional[Pattern[b, Blank[]]]]], Optional[Pattern[n, Blank[]]]], Power[Plus[Pattern[f, Blank[]], Times[Optional[Pattern[g, Blank[]]], Pattern[x, Blank[]]]], Optional[Pattern[m, Blank[]]]], Power[Plus[Pattern[d, Blank[]], Times[Optional[Pattern[e, Blank[]]], Power[Pattern[x, Blank[]], 2]]], Pattern[p, Blank[]]]], Pattern[x, Blank[Symbol]]] := -Simp[((f + g*x)^m*(d + e*x^2)^(p + 1/2)*(a + b*ArcCos[c*x])^(n + 1))/(b*c*Sqrt[d]*(n + 1)), x] + Dist[1/(b*c*Sqrt[d]*(n + 1)), Int[ExpandIntegrand[(f + g*x)^(m - 1)*(a + b*ArcCos[c*x])^(n + 1), (d*g*m + e*f*(2*p + 1)*x + e*g*(m + 2*p + 1)*x^2)*(d + e*x^2)^(p - 1/2), x], x], x] /; FreeQ[{a, b, c, d, e, f, g}, x] && EqQ[c^2*d + e, 0] && ILtQ[m, 0] && IGtQ[p - 1/2, 0] && GtQ[d, 0] && IGtQ[n, 0]
  4771. Int[Times[Power[Plus[Optional[Pattern[a, Blank[]]], Times[ArcSin[Times[Optional[Pattern[c, Blank[]]], Pattern[x, Blank[]]]], Optional[Pattern[b, Blank[]]]]], Pattern[n, Blank[]]], Power[Plus[Pattern[f, Blank[]], Times[Optional[Pattern[g, Blank[]]], Pattern[x, Blank[]]]], Optional[Pattern[m, Blank[]]]], Power[Plus[Pattern[d, Blank[]], Times[Optional[Pattern[e, Blank[]]], Power[Pattern[x, Blank[]], 2]]], Rational[-1, 2]]], Pattern[x, Blank[Symbol]]] := Simp[((f + g*x)^m*(a + b*ArcSin[c*x])^(n + 1))/(b*c*Sqrt[d]*(n + 1)), x] - Dist[(g*m)/(b*c*Sqrt[d]*(n + 1)), Int[(f + g*x)^(m - 1)*(a + b*ArcSin[c*x])^(n + 1), x], x] /; FreeQ[{a, b, c, d, e, f, g}, x] && EqQ[c^2*d + e, 0] && IGtQ[m, 0] && GtQ[d, 0] && LtQ[n, -1]
  4772. Int[Times[Power[Plus[Optional[Pattern[a, Blank[]]], Times[ArcCos[Times[Optional[Pattern[c, Blank[]]], Pattern[x, Blank[]]]], Optional[Pattern[b, Blank[]]]]], Pattern[n, Blank[]]], Power[Plus[Pattern[f, Blank[]], Times[Optional[Pattern[g, Blank[]]], Pattern[x, Blank[]]]], Optional[Pattern[m, Blank[]]]], Power[Plus[Pattern[d, Blank[]], Times[Optional[Pattern[e, Blank[]]], Power[Pattern[x, Blank[]], 2]]], Rational[-1, 2]]], Pattern[x, Blank[Symbol]]] := -Simp[((f + g*x)^m*(a + b*ArcCos[c*x])^(n + 1))/(b*c*Sqrt[d]*(n + 1)), x] + Dist[(g*m)/(b*c*Sqrt[d]*(n + 1)), Int[(f + g*x)^(m - 1)*(a + b*ArcCos[c*x])^(n + 1), x], x] /; FreeQ[{a, b, c, d, e, f, g}, x] && EqQ[c^2*d + e, 0] && IGtQ[m, 0] && GtQ[d, 0] && LtQ[n, -1]
  4773. Int[Times[Power[Plus[Optional[Pattern[a, Blank[]]], Times[ArcSin[Times[Optional[Pattern[c, Blank[]]], Pattern[x, Blank[]]]], Optional[Pattern[b, Blank[]]]]], Optional[Pattern[n, Blank[]]]], Power[Plus[Pattern[f, Blank[]], Times[Optional[Pattern[g, Blank[]]], Pattern[x, Blank[]]]], Optional[Pattern[m, Blank[]]]], Power[Plus[Pattern[d, Blank[]], Times[Optional[Pattern[e, Blank[]]], Power[Pattern[x, Blank[]], 2]]], Rational[-1, 2]]], Pattern[x, Blank[Symbol]]] := Dist[1/(c^(m + 1)*Sqrt[d]), Subst[Int[(a + b*x)^n*(c*f + g*Sin[x])^m, x], x, ArcSin[c*x]], x] /; FreeQ[{a, b, c, d, e, f, g, n}, x] && EqQ[c^2*d + e, 0] && IntegerQ[m] && GtQ[d, 0] && (GtQ[m, 0] || IGtQ[n, 0])
  4774. Int[Times[Power[Plus[Optional[Pattern[a, Blank[]]], Times[ArcCos[Times[Optional[Pattern[c, Blank[]]], Pattern[x, Blank[]]]], Optional[Pattern[b, Blank[]]]]], Optional[Pattern[n, Blank[]]]], Power[Plus[Pattern[f, Blank[]], Times[Optional[Pattern[g, Blank[]]], Pattern[x, Blank[]]]], Optional[Pattern[m, Blank[]]]], Power[Plus[Pattern[d, Blank[]], Times[Optional[Pattern[e, Blank[]]], Power[Pattern[x, Blank[]], 2]]], Rational[-1, 2]]], Pattern[x, Blank[Symbol]]] := -Dist[(c^(m + 1)*Sqrt[d])^(-1), Subst[Int[(a + b*x)^n*(c*f + g*Cos[x])^m, x], x, ArcCos[c*x]], x] /; FreeQ[{a, b, c, d, e, f, g, n}, x] && EqQ[c^2*d + e, 0] && IntegerQ[m] && GtQ[d, 0] && (GtQ[m, 0] || IGtQ[n, 0])
  4775. Int[Times[Power[Plus[Optional[Pattern[a, Blank[]]], Times[ArcSin[Times[Optional[Pattern[c, Blank[]]], Pattern[x, Blank[]]]], Optional[Pattern[b, Blank[]]]]], Optional[Pattern[n, Blank[]]]], Power[Plus[Pattern[f, Blank[]], Times[Optional[Pattern[g, Blank[]]], Pattern[x, Blank[]]]], Optional[Pattern[m, Blank[]]]], Power[Plus[Pattern[d, Blank[]], Times[Optional[Pattern[e, Blank[]]], Power[Pattern[x, Blank[]], 2]]], Pattern[p, Blank[]]]], Pattern[x, Blank[Symbol]]] := Int[ExpandIntegrand[(a + b*ArcSin[c*x])^n/Sqrt[d + e*x^2], (f + g*x)^m*(d + e*x^2)^(p + 1/2), x], x] /; FreeQ[{a, b, c, d, e, f, g}, x] && EqQ[c^2*d + e, 0] && IntegerQ[m] && ILtQ[p + 1/2, 0] && GtQ[d, 0] && IGtQ[n, 0]
  4776. Int[Times[Power[Plus[Optional[Pattern[a, Blank[]]], Times[ArcCos[Times[Optional[Pattern[c, Blank[]]], Pattern[x, Blank[]]]], Optional[Pattern[b, Blank[]]]]], Optional[Pattern[n, Blank[]]]], Power[Plus[Pattern[f, Blank[]], Times[Optional[Pattern[g, Blank[]]], Pattern[x, Blank[]]]], Optional[Pattern[m, Blank[]]]], Power[Plus[Pattern[d, Blank[]], Times[Optional[Pattern[e, Blank[]]], Power[Pattern[x, Blank[]], 2]]], Pattern[p, Blank[]]]], Pattern[x, Blank[Symbol]]] := Int[ExpandIntegrand[(a + b*ArcCos[c*x])^n/Sqrt[d + e*x^2], (f + g*x)^m*(d + e*x^2)^(p + 1/2), x], x] /; FreeQ[{a, b, c, d, e, f, g}, x] && EqQ[c^2*d + e, 0] && IntegerQ[m] && ILtQ[p + 1/2, 0] && GtQ[d, 0] && IGtQ[n, 0]
  4777. Int[Times[Power[Plus[Optional[Pattern[a, Blank[]]], Times[ArcSin[Times[Optional[Pattern[c, Blank[]]], Pattern[x, Blank[]]]], Optional[Pattern[b, Blank[]]]]], Optional[Pattern[n, Blank[]]]], Power[Plus[Pattern[f, Blank[]], Times[Optional[Pattern[g, Blank[]]], Pattern[x, Blank[]]]], Optional[Pattern[m, Blank[]]]], Power[Plus[Pattern[d, Blank[]], Times[Optional[Pattern[e, Blank[]]], Power[Pattern[x, Blank[]], 2]]], Pattern[p, Blank[]]]], Pattern[x, Blank[Symbol]]] := Dist[(d^IntPart[p]*(d + e*x^2)^FracPart[p])/(1 - c^2*x^2)^FracPart[p], Int[(f + g*x)^m*(1 - c^2*x^2)^p*(a + b*ArcSin[c*x])^n, x], x] /; FreeQ[{a, b, c, d, e, f, g, n}, x] && EqQ[c^2*d + e, 0] && IntegerQ[m] && IntegerQ[p - 1/2] && !GtQ[d, 0]
  4778. Int[Times[Power[Plus[Optional[Pattern[a, Blank[]]], Times[ArcCos[Times[Optional[Pattern[c, Blank[]]], Pattern[x, Blank[]]]], Optional[Pattern[b, Blank[]]]]], Optional[Pattern[n, Blank[]]]], Power[Plus[Pattern[f, Blank[]], Times[Optional[Pattern[g, Blank[]]], Pattern[x, Blank[]]]], Optional[Pattern[m, Blank[]]]], Power[Plus[Pattern[d, Blank[]], Times[Optional[Pattern[e, Blank[]]], Power[Pattern[x, Blank[]], 2]]], Pattern[p, Blank[]]]], Pattern[x, Blank[Symbol]]] := Dist[(d^IntPart[p]*(d + e*x^2)^FracPart[p])/(1 - c^2*x^2)^FracPart[p], Int[(f + g*x)^m*(1 - c^2*x^2)^p*(a + b*ArcCos[c*x])^n, x], x] /; FreeQ[{a, b, c, d, e, f, g, n}, x] && EqQ[c^2*d + e, 0] && IntegerQ[m] && IntegerQ[p - 1/2] && !GtQ[d, 0]
  4779. Int[Times[Log[Times[Optional[Pattern[h, Blank[]]], Power[Plus[Optional[Pattern[f, Blank[]]], Times[Optional[Pattern[g, Blank[]]], Pattern[x, Blank[]]]], Optional[Pattern[m, Blank[]]]]]], Power[Plus[Optional[Pattern[a, Blank[]]], Times[ArcSin[Times[Optional[Pattern[c, Blank[]]], Pattern[x, Blank[]]]], Optional[Pattern[b, Blank[]]]]], Optional[Pattern[n, Blank[]]]], Power[Plus[Pattern[d, Blank[]], Times[Optional[Pattern[e, Blank[]]], Power[Pattern[x, Blank[]], 2]]], Rational[-1, 2]]], Pattern[x, Blank[Symbol]]] := Simp[(Log[h*(f + g*x)^m]*(a + b*ArcSin[c*x])^(n + 1))/(b*c*Sqrt[d]*(n + 1)), x] - Dist[(g*m)/(b*c*Sqrt[d]*(n + 1)), Int[(a + b*ArcSin[c*x])^(n + 1)/(f + g*x), x], x] /; FreeQ[{a, b, c, d, e, f, g, h, m}, x] && EqQ[c^2*d + e, 0] && GtQ[d, 0] && IGtQ[n, 0]
  4780. Int[Times[Log[Times[Optional[Pattern[h, Blank[]]], Power[Plus[Optional[Pattern[f, Blank[]]], Times[Optional[Pattern[g, Blank[]]], Pattern[x, Blank[]]]], Optional[Pattern[m, Blank[]]]]]], Power[Plus[Optional[Pattern[a, Blank[]]], Times[ArcCos[Times[Optional[Pattern[c, Blank[]]], Pattern[x, Blank[]]]], Optional[Pattern[b, Blank[]]]]], Optional[Pattern[n, Blank[]]]], Power[Plus[Pattern[d, Blank[]], Times[Optional[Pattern[e, Blank[]]], Power[Pattern[x, Blank[]], 2]]], Rational[-1, 2]]], Pattern[x, Blank[Symbol]]] := -Simp[(Log[h*(f + g*x)^m]*(a + b*ArcCos[c*x])^(n + 1))/(b*c*Sqrt[d]*(n + 1)), x] + Dist[(g*m)/(b*c*Sqrt[d]*(n + 1)), Int[(a + b*ArcCos[c*x])^(n + 1)/(f + g*x), x], x] /; FreeQ[{a, b, c, d, e, f, g, h, m}, x] && EqQ[c^2*d + e, 0] && GtQ[d, 0] && IGtQ[n, 0]
  4781. Int[Times[Log[Times[Optional[Pattern[h, Blank[]]], Power[Plus[Optional[Pattern[f, Blank[]]], Times[Optional[Pattern[g, Blank[]]], Pattern[x, Blank[]]]], Optional[Pattern[m, Blank[]]]]]], Power[Plus[Optional[Pattern[a, Blank[]]], Times[ArcSin[Times[Optional[Pattern[c, Blank[]]], Pattern[x, Blank[]]]], Optional[Pattern[b, Blank[]]]]], Optional[Pattern[n, Blank[]]]], Power[Plus[Pattern[d, Blank[]], Times[Optional[Pattern[e, Blank[]]], Power[Pattern[x, Blank[]], 2]]], Pattern[p, Blank[]]]], Pattern[x, Blank[Symbol]]] := Dist[(d^IntPart[p]*(d + e*x^2)^FracPart[p])/(1 - c^2*x^2)^FracPart[p], Int[Log[h*(f + g*x)^m]*(1 - c^2*x^2)^p*(a + b*ArcSin[c*x])^n, x], x] /; FreeQ[{a, b, c, d, e, f, g, h, m, n}, x] && EqQ[c^2*d + e, 0] && IntegerQ[p - 1/2] && !GtQ[d, 0]
  4782. Int[Times[Log[Times[Optional[Pattern[h, Blank[]]], Power[Plus[Optional[Pattern[f, Blank[]]], Times[Optional[Pattern[g, Blank[]]], Pattern[x, Blank[]]]], Optional[Pattern[m, Blank[]]]]]], Power[Plus[Optional[Pattern[a, Blank[]]], Times[ArcCos[Times[Optional[Pattern[c, Blank[]]], Pattern[x, Blank[]]]], Optional[Pattern[b, Blank[]]]]], Optional[Pattern[n, Blank[]]]], Power[Plus[Pattern[d, Blank[]], Times[Optional[Pattern[e, Blank[]]], Power[Pattern[x, Blank[]], 2]]], Pattern[p, Blank[]]]], Pattern[x, Blank[Symbol]]] := Dist[(d^IntPart[p]*(d + e*x^2)^FracPart[p])/(1 - c^2*x^2)^FracPart[p], Int[Log[h*(f + g*x)^m]*(1 - c^2*x^2)^p*(a + b*ArcCos[c*x])^n, x], x] /; FreeQ[{a, b, c, d, e, f, g, h, m, n}, x] && EqQ[c^2*d + e, 0] && IntegerQ[p - 1/2] && !GtQ[d, 0]
  4783. Int[Times[Plus[Optional[Pattern[a, Blank[]]], Times[ArcSin[Times[Optional[Pattern[c, Blank[]]], Pattern[x, Blank[]]]], Optional[Pattern[b, Blank[]]]]], Power[Plus[Pattern[d, Blank[]], Times[Optional[Pattern[e, Blank[]]], Pattern[x, Blank[]]]], Pattern[m, Blank[]]], Power[Plus[Pattern[f, Blank[]], Times[Optional[Pattern[g, Blank[]]], Pattern[x, Blank[]]]], Pattern[m, Blank[]]]], Pattern[x, Blank[Symbol]]] := With[{u = IntHide[(d + e*x)^m*(f + g*x)^m, x]}, Dist[a + b*ArcSin[c*x], u, x] - Dist[b*c, Int[Dist[1/Sqrt[1 - c^2*x^2], u, x], x], x]] /; FreeQ[{a, b, c, d, e, f, g}, x] && ILtQ[m + 1/2, 0]
  4784. Int[Times[Plus[Optional[Pattern[a, Blank[]]], Times[ArcCos[Times[Optional[Pattern[c, Blank[]]], Pattern[x, Blank[]]]], Optional[Pattern[b, Blank[]]]]], Power[Plus[Pattern[d, Blank[]], Times[Optional[Pattern[e, Blank[]]], Pattern[x, Blank[]]]], Pattern[m, Blank[]]], Power[Plus[Pattern[f, Blank[]], Times[Optional[Pattern[g, Blank[]]], Pattern[x, Blank[]]]], Pattern[m, Blank[]]]], Pattern[x, Blank[Symbol]]] := With[{u = IntHide[(d + e*x)^m*(f + g*x)^m, x]}, Dist[a + b*ArcCos[c*x], u, x] + Dist[b*c, Int[Dist[1/Sqrt[1 - c^2*x^2], u, x], x], x]] /; FreeQ[{a, b, c, d, e, f, g}, x] && ILtQ[m + 1/2, 0]
  4785. Int[Times[Power[Plus[Optional[Pattern[a, Blank[]]], Times[ArcSin[Times[Optional[Pattern[c, Blank[]]], Pattern[x, Blank[]]]], Optional[Pattern[b, Blank[]]]]], Optional[Pattern[n, Blank[]]]], Power[Plus[Pattern[d, Blank[]], Times[Optional[Pattern[e, Blank[]]], Pattern[x, Blank[]]]], Optional[Pattern[m, Blank[]]]], Power[Plus[Pattern[f, Blank[]], Times[Optional[Pattern[g, Blank[]]], Pattern[x, Blank[]]]], Optional[Pattern[m, Blank[]]]]], Pattern[x, Blank[Symbol]]] := Int[ExpandIntegrand[(d + e*x)^m*(f + g*x)^m*(a + b*ArcSin[c*x])^n, x], x] /; FreeQ[{a, b, c, d, e, f, g, n}, x] && IntegerQ[m]
  4786. Int[Times[Power[Plus[Optional[Pattern[a, Blank[]]], Times[ArcCos[Times[Optional[Pattern[c, Blank[]]], Pattern[x, Blank[]]]], Optional[Pattern[b, Blank[]]]]], Optional[Pattern[n, Blank[]]]], Power[Plus[Pattern[d, Blank[]], Times[Optional[Pattern[e, Blank[]]], Pattern[x, Blank[]]]], Optional[Pattern[m, Blank[]]]], Power[Plus[Pattern[f, Blank[]], Times[Optional[Pattern[g, Blank[]]], Pattern[x, Blank[]]]], Optional[Pattern[m, Blank[]]]]], Pattern[x, Blank[Symbol]]] := Int[ExpandIntegrand[(d + e*x)^m*(f + g*x)^m*(a + b*ArcCos[c*x])^n, x], x] /; FreeQ[{a, b, c, d, e, f, g, n}, x] && IntegerQ[m]
  4787. Int[Times[Plus[Optional[Pattern[a, Blank[]]], Times[ArcSin[Times[Optional[Pattern[c, Blank[]]], Pattern[x, Blank[]]]], Optional[Pattern[b, Blank[]]]]], Pattern[u, Blank[]]], Pattern[x, Blank[Symbol]]] := With[{v = IntHide[u, x]}, Dist[a + b*ArcSin[c*x], v, x] - Dist[b*c, Int[SimplifyIntegrand[v/Sqrt[1 - c^2*x^2], x], x], x] /; InverseFunctionFreeQ[v, x]] /; FreeQ[{a, b, c}, x]
  4788. Int[Times[Plus[Optional[Pattern[a, Blank[]]], Times[ArcCos[Times[Optional[Pattern[c, Blank[]]], Pattern[x, Blank[]]]], Optional[Pattern[b, Blank[]]]]], Pattern[u, Blank[]]], Pattern[x, Blank[Symbol]]] := With[{v = IntHide[u, x]}, Dist[a + b*ArcCos[c*x], v, x] + Dist[b*c, Int[SimplifyIntegrand[v/Sqrt[1 - c^2*x^2], x], x], x] /; InverseFunctionFreeQ[v, x]] /; FreeQ[{a, b, c}, x]
  4789. Int[Times[Power[Plus[Optional[Pattern[a, Blank[]]], Times[ArcSin[Times[Optional[Pattern[c, Blank[]]], Pattern[x, Blank[]]]], Optional[Pattern[b, Blank[]]]]], Optional[Pattern[n, Blank[]]]], Pattern[Px, Blank[]], Power[Plus[Pattern[d, Blank[]], Times[Optional[Pattern[e, Blank[]]], Power[Pattern[x, Blank[]], 2]]], Pattern[p, Blank[]]]], Pattern[x, Blank[Symbol]]] := With[{u = ExpandIntegrand[Px*(d + e*x^2)^p*(a + b*ArcSin[c*x])^n, x]}, Int[u, x] /; SumQ[u]] /; FreeQ[{a, b, c, d, e, n}, x] && PolynomialQ[Px, x] && EqQ[c^2*d + e, 0] && IntegerQ[p - 1/2]
  4790. Int[Times[Power[Plus[Optional[Pattern[a, Blank[]]], Times[ArcCos[Times[Optional[Pattern[c, Blank[]]], Pattern[x, Blank[]]]], Optional[Pattern[b, Blank[]]]]], Optional[Pattern[n, Blank[]]]], Pattern[Px, Blank[]], Power[Plus[Pattern[d, Blank[]], Times[Optional[Pattern[e, Blank[]]], Power[Pattern[x, Blank[]], 2]]], Pattern[p, Blank[]]]], Pattern[x, Blank[Symbol]]] := With[{u = ExpandIntegrand[Px*(d + e*x^2)^p*(a + b*ArcCos[c*x])^n, x]}, Int[u, x] /; SumQ[u]] /; FreeQ[{a, b, c, d, e, n}, x] && PolynomialQ[Px, x] && EqQ[c^2*d + e, 0] && IntegerQ[p - 1/2]
  4791. Int[Times[Power[Plus[Optional[Pattern[a, Blank[]]], Times[ArcSin[Times[Optional[Pattern[c, Blank[]]], Pattern[x, Blank[]]]], Optional[Pattern[b, Blank[]]]]], Optional[Pattern[n, Blank[]]]], Optional[Pattern[Px, Blank[]]], Power[Plus[Pattern[f, Blank[]], Times[Optional[Pattern[g, Blank[]]], Power[Plus[Pattern[d, Blank[]], Times[Optional[Pattern[e, Blank[]]], Power[Pattern[x, Blank[]], 2]]], Pattern[p, Blank[]]]]], Optional[Pattern[m, Blank[]]]]], Pattern[x, Blank[Symbol]]] := With[{u = ExpandIntegrand[Px*(f + g*(d + e*x^2)^p)^m*(a + b*ArcSin[c*x])^n, x]}, Int[u, x] /; SumQ[u]] /; FreeQ[{a, b, c, d, e, f, g}, x] && PolynomialQ[Px, x] && EqQ[c^2*d + e, 0] && IGtQ[p + 1/2, 0] && IntegersQ[m, n]
  4792. Int[Times[Power[Plus[Optional[Pattern[a, Blank[]]], Times[ArcCos[Times[Optional[Pattern[c, Blank[]]], Pattern[x, Blank[]]]], Optional[Pattern[b, Blank[]]]]], Optional[Pattern[n, Blank[]]]], Optional[Pattern[Px, Blank[]]], Power[Plus[Pattern[f, Blank[]], Times[Optional[Pattern[g, Blank[]]], Power[Plus[Pattern[d, Blank[]], Times[Optional[Pattern[e, Blank[]]], Power[Pattern[x, Blank[]], 2]]], Pattern[p, Blank[]]]]], Optional[Pattern[m, Blank[]]]]], Pattern[x, Blank[Symbol]]] := With[{u = ExpandIntegrand[Px*(f + g*(d + e*x^2)^p)^m*(a + b*ArcCos[c*x])^n, x]}, Int[u, x] /; SumQ[u]] /; FreeQ[{a, b, c, d, e, f, g}, x] && PolynomialQ[Px, x] && EqQ[c^2*d + e, 0] && IGtQ[p + 1/2, 0] && IntegersQ[m, n]
  4793. Int[Times[Power[ArcSin[Times[Optional[Pattern[c, Blank[]]], Pattern[x, Blank[]]]], Optional[Pattern[n, Blank[]]]], Pattern[RFx, Blank[]]], Pattern[x, Blank[Symbol]]] := With[{u = ExpandIntegrand[ArcSin[c*x]^n, RFx, x]}, Int[u, x] /; SumQ[u]] /; FreeQ[c, x] && RationalFunctionQ[RFx, x] && IGtQ[n, 0]
  4794. Int[Times[Power[ArcCos[Times[Optional[Pattern[c, Blank[]]], Pattern[x, Blank[]]]], Optional[Pattern[n, Blank[]]]], Pattern[RFx, Blank[]]], Pattern[x, Blank[Symbol]]] := With[{u = ExpandIntegrand[ArcCos[c*x]^n, RFx, x]}, Int[u, x] /; SumQ[u]] /; FreeQ[c, x] && RationalFunctionQ[RFx, x] && IGtQ[n, 0]
  4795. Int[Times[Power[Plus[Times[ArcSin[Times[Optional[Pattern[c, Blank[]]], Pattern[x, Blank[]]]], Optional[Pattern[b, Blank[]]]], Pattern[a, Blank[]]], Optional[Pattern[n, Blank[]]]], Pattern[RFx, Blank[]]], Pattern[x, Blank[Symbol]]] := Int[ExpandIntegrand[RFx*(a + b*ArcSin[c*x])^n, x], x] /; FreeQ[{a, b, c}, x] && RationalFunctionQ[RFx, x] && IGtQ[n, 0]
  4796. Int[Times[Power[Plus[Times[ArcCos[Times[Optional[Pattern[c, Blank[]]], Pattern[x, Blank[]]]], Optional[Pattern[b, Blank[]]]], Pattern[a, Blank[]]], Optional[Pattern[n, Blank[]]]], Pattern[RFx, Blank[]]], Pattern[x, Blank[Symbol]]] := Int[ExpandIntegrand[RFx*(a + b*ArcCos[c*x])^n, x], x] /; FreeQ[{a, b, c}, x] && RationalFunctionQ[RFx, x] && IGtQ[n, 0]
  4797. Int[Times[Power[ArcSin[Times[Optional[Pattern[c, Blank[]]], Pattern[x, Blank[]]]], Optional[Pattern[n, Blank[]]]], Pattern[RFx, Blank[]], Power[Plus[Pattern[d, Blank[]], Times[Optional[Pattern[e, Blank[]]], Power[Pattern[x, Blank[]], 2]]], Pattern[p, Blank[]]]], Pattern[x, Blank[Symbol]]] := With[{u = ExpandIntegrand[(d + e*x^2)^p*ArcSin[c*x]^n, RFx, x]}, Int[u, x] /; SumQ[u]] /; FreeQ[{c, d, e}, x] && RationalFunctionQ[RFx, x] && IGtQ[n, 0] && EqQ[c^2*d + e, 0] && IntegerQ[p - 1/2]
  4798. Int[Times[Power[ArcCos[Times[Optional[Pattern[c, Blank[]]], Pattern[x, Blank[]]]], Optional[Pattern[n, Blank[]]]], Pattern[RFx, Blank[]], Power[Plus[Pattern[d, Blank[]], Times[Optional[Pattern[e, Blank[]]], Power[Pattern[x, Blank[]], 2]]], Pattern[p, Blank[]]]], Pattern[x, Blank[Symbol]]] := With[{u = ExpandIntegrand[(d + e*x^2)^p*ArcCos[c*x]^n, RFx, x]}, Int[u, x] /; SumQ[u]] /; FreeQ[{c, d, e}, x] && RationalFunctionQ[RFx, x] && IGtQ[n, 0] && EqQ[c^2*d + e, 0] && IntegerQ[p - 1/2]
  4799. Int[Times[Power[Plus[Times[ArcSin[Times[Optional[Pattern[c, Blank[]]], Pattern[x, Blank[]]]], Optional[Pattern[b, Blank[]]]], Pattern[a, Blank[]]], Optional[Pattern[n, Blank[]]]], Pattern[RFx, Blank[]], Power[Plus[Pattern[d, Blank[]], Times[Optional[Pattern[e, Blank[]]], Power[Pattern[x, Blank[]], 2]]], Pattern[p, Blank[]]]], Pattern[x, Blank[Symbol]]] := Int[ExpandIntegrand[(d + e*x^2)^p, RFx*(a + b*ArcSin[c*x])^n, x], x] /; FreeQ[{a, b, c, d, e}, x] && RationalFunctionQ[RFx, x] && IGtQ[n, 0] && EqQ[c^2*d + e, 0] && IntegerQ[p - 1/2]
  4800. Int[Times[Power[Plus[Times[ArcCos[Times[Optional[Pattern[c, Blank[]]], Pattern[x, Blank[]]]], Optional[Pattern[b, Blank[]]]], Pattern[a, Blank[]]], Optional[Pattern[n, Blank[]]]], Pattern[RFx, Blank[]], Power[Plus[Pattern[d, Blank[]], Times[Optional[Pattern[e, Blank[]]], Power[Pattern[x, Blank[]], 2]]], Pattern[p, Blank[]]]], Pattern[x, Blank[Symbol]]] := Int[ExpandIntegrand[(d + e*x^2)^p, RFx*(a + b*ArcCos[c*x])^n, x], x] /; FreeQ[{a, b, c, d, e}, x] && RationalFunctionQ[RFx, x] && IGtQ[n, 0] && EqQ[c^2*d + e, 0] && IntegerQ[p - 1/2]
  4801. Int[Times[Power[Plus[Optional[Pattern[a, Blank[]]], Times[ArcSin[Times[Optional[Pattern[c, Blank[]]], Pattern[x, Blank[]]]], Optional[Pattern[b, Blank[]]]]], Optional[Pattern[n, Blank[]]]], Optional[Pattern[u, Blank[]]]], Pattern[x, Blank[Symbol]]] := Unintegrable[u*(a + b*ArcSin[c*x])^n, x] /; FreeQ[{a, b, c, n}, x]
  4802. Int[Times[Power[Plus[Optional[Pattern[a, Blank[]]], Times[ArcCos[Times[Optional[Pattern[c, Blank[]]], Pattern[x, Blank[]]]], Optional[Pattern[b, Blank[]]]]], Optional[Pattern[n, Blank[]]]], Optional[Pattern[u, Blank[]]]], Pattern[x, Blank[Symbol]]] := Unintegrable[u*(a + b*ArcCos[c*x])^n, x] /; FreeQ[{a, b, c, n}, x]
  4803. Int[Power[Plus[Optional[Pattern[a, Blank[]]], Times[ArcSin[Plus[Pattern[c, Blank[]], Times[Optional[Pattern[d, Blank[]]], Pattern[x, Blank[]]]]], Optional[Pattern[b, Blank[]]]]], Optional[Pattern[n, Blank[]]]], Pattern[x, Blank[Symbol]]] := Dist[1/d, Subst[Int[(a + b*ArcSin[x])^n, x], x, c + d*x], x] /; FreeQ[{a, b, c, d, n}, x]
  4804. Int[Power[Plus[Optional[Pattern[a, Blank[]]], Times[ArcCos[Plus[Pattern[c, Blank[]], Times[Optional[Pattern[d, Blank[]]], Pattern[x, Blank[]]]]], Optional[Pattern[b, Blank[]]]]], Optional[Pattern[n, Blank[]]]], Pattern[x, Blank[Symbol]]] := Dist[1/d, Subst[Int[(a + b*ArcCos[x])^n, x], x, c + d*x], x] /; FreeQ[{a, b, c, d, n}, x]
  4805. Int[Times[Power[Plus[Optional[Pattern[a, Blank[]]], Times[ArcSin[Plus[Pattern[c, Blank[]], Times[Optional[Pattern[d, Blank[]]], Pattern[x, Blank[]]]]], Optional[Pattern[b, Blank[]]]]], Optional[Pattern[n, Blank[]]]], Power[Plus[Optional[Pattern[e, Blank[]]], Times[Optional[Pattern[f, Blank[]]], Pattern[x, Blank[]]]], Optional[Pattern[m, Blank[]]]]], Pattern[x, Blank[Symbol]]] := Dist[1/d, Subst[Int[((d*e - c*f)/d + (f*x)/d)^m*(a + b*ArcSin[x])^n, x], x, c + d*x], x] /; FreeQ[{a, b, c, d, e, f, m, n}, x]
  4806. Int[Times[Power[Plus[Optional[Pattern[a, Blank[]]], Times[ArcCos[Plus[Pattern[c, Blank[]], Times[Optional[Pattern[d, Blank[]]], Pattern[x, Blank[]]]]], Optional[Pattern[b, Blank[]]]]], Optional[Pattern[n, Blank[]]]], Power[Plus[Optional[Pattern[e, Blank[]]], Times[Optional[Pattern[f, Blank[]]], Pattern[x, Blank[]]]], Optional[Pattern[m, Blank[]]]]], Pattern[x, Blank[Symbol]]] := Dist[1/d, Subst[Int[((d*e - c*f)/d + (f*x)/d)^m*(a + b*ArcCos[x])^n, x], x, c + d*x], x] /; FreeQ[{a, b, c, d, e, f, m, n}, x]
  4807. Int[Times[Power[Plus[Optional[Pattern[a, Blank[]]], Times[ArcSin[Plus[Pattern[c, Blank[]], Times[Optional[Pattern[d, Blank[]]], Pattern[x, Blank[]]]]], Optional[Pattern[b, Blank[]]]]], Optional[Pattern[n, Blank[]]]], Power[Plus[Optional[Pattern[A, Blank[]]], Times[Optional[Pattern[B, Blank[]]], Pattern[x, Blank[]]], Times[Optional[Pattern[C, Blank[]]], Power[Pattern[x, Blank[]], 2]]], Optional[Pattern[p, Blank[]]]]], Pattern[x, Blank[Symbol]]] := Dist[1/d, Subst[Int[(-(C/d^2) + (C*x^2)/d^2)^p*(a + b*ArcSin[x])^n, x], x, c + d*x], x] /; FreeQ[{a, b, c, d, A, B, C, n, p}, x] && EqQ[B*(1 - c^2) + 2*A*c*d, 0] && EqQ[2*c*C - B*d, 0]
  4808. Int[Times[Power[Plus[Optional[Pattern[a, Blank[]]], Times[ArcCos[Plus[Pattern[c, Blank[]], Times[Optional[Pattern[d, Blank[]]], Pattern[x, Blank[]]]]], Optional[Pattern[b, Blank[]]]]], Optional[Pattern[n, Blank[]]]], Power[Plus[Optional[Pattern[A, Blank[]]], Times[Optional[Pattern[B, Blank[]]], Pattern[x, Blank[]]], Times[Optional[Pattern[C, Blank[]]], Power[Pattern[x, Blank[]], 2]]], Optional[Pattern[p, Blank[]]]]], Pattern[x, Blank[Symbol]]] := Dist[1/d, Subst[Int[(-(C/d^2) + (C*x^2)/d^2)^p*(a + b*ArcCos[x])^n, x], x, c + d*x], x] /; FreeQ[{a, b, c, d, A, B, C, n, p}, x] && EqQ[B*(1 - c^2) + 2*A*c*d, 0] && EqQ[2*c*C - B*d, 0]
  4809. Int[Times[Power[Plus[Optional[Pattern[a, Blank[]]], Times[ArcSin[Plus[Pattern[c, Blank[]], Times[Optional[Pattern[d, Blank[]]], Pattern[x, Blank[]]]]], Optional[Pattern[b, Blank[]]]]], Optional[Pattern[n, Blank[]]]], Power[Plus[Optional[Pattern[e, Blank[]]], Times[Optional[Pattern[f, Blank[]]], Pattern[x, Blank[]]]], Optional[Pattern[m, Blank[]]]], Power[Plus[Optional[Pattern[A, Blank[]]], Times[Optional[Pattern[B, Blank[]]], Pattern[x, Blank[]]], Times[Optional[Pattern[C, Blank[]]], Power[Pattern[x, Blank[]], 2]]], Optional[Pattern[p, Blank[]]]]], Pattern[x, Blank[Symbol]]] := Dist[1/d, Subst[Int[((d*e - c*f)/d + (f*x)/d)^m*(-(C/d^2) + (C*x^2)/d^2)^p*(a + b*ArcSin[x])^n, x], x, c + d*x], x] /; FreeQ[{a, b, c, d, e, f, A, B, C, m, n, p}, x] && EqQ[B*(1 - c^2) + 2*A*c*d, 0] && EqQ[2*c*C - B*d, 0]
  4810. Int[Times[Power[Plus[Optional[Pattern[a, Blank[]]], Times[ArcCos[Plus[Pattern[c, Blank[]], Times[Optional[Pattern[d, Blank[]]], Pattern[x, Blank[]]]]], Optional[Pattern[b, Blank[]]]]], Optional[Pattern[n, Blank[]]]], Power[Plus[Optional[Pattern[e, Blank[]]], Times[Optional[Pattern[f, Blank[]]], Pattern[x, Blank[]]]], Optional[Pattern[m, Blank[]]]], Power[Plus[Optional[Pattern[A, Blank[]]], Times[Optional[Pattern[B, Blank[]]], Pattern[x, Blank[]]], Times[Optional[Pattern[C, Blank[]]], Power[Pattern[x, Blank[]], 2]]], Optional[Pattern[p, Blank[]]]]], Pattern[x, Blank[Symbol]]] := Dist[1/d, Subst[Int[((d*e - c*f)/d + (f*x)/d)^m*(-(C/d^2) + (C*x^2)/d^2)^p*(a + b*ArcCos[x])^n, x], x, c + d*x], x] /; FreeQ[{a, b, c, d, e, f, A, B, C, m, n, p}, x] && EqQ[B*(1 - c^2) + 2*A*c*d, 0] && EqQ[2*c*C - B*d, 0]
  4811. Int[Power[Plus[Optional[Pattern[a, Blank[]]], Times[ArcSin[Plus[Pattern[c, Blank[]], Times[Optional[Pattern[d, Blank[]]], Power[Pattern[x, Blank[]], 2]]]], Optional[Pattern[b, Blank[]]]]], Rational[1, 2]], Pattern[x, Blank[Symbol]]] := Simp[x*Sqrt[a + b*ArcSin[c + d*x^2]], x] + (-Simp[(Sqrt[Pi]*x*(Cos[a/(2*b)] + c*Sin[a/(2*b)])*FresnelC[Sqrt[c/(Pi*b)]*Sqrt[a + b*ArcSin[c + d*x^2]]])/(Sqrt[c/b]*(Cos[ArcSin[c + d*x^2]/2] - c*Sin[ArcSin[c + d*x^2]/2])), x] + Simp[(Sqrt[Pi]*x*(Cos[a/(2*b)] - c*Sin[a/(2*b)])*FresnelS[Sqrt[c/(Pi*b)]*Sqrt[a + b*ArcSin[c + d*x^2]]])/(Sqrt[c/b]*(Cos[ArcSin[c + d*x^2]/2] - c*Sin[ArcSin[c + d*x^2]/2])), x]) /; FreeQ[{a, b, c, d}, x] && EqQ[c^2, 1]
  4812. Int[Power[Plus[Optional[Pattern[a, Blank[]]], Times[ArcCos[Plus[1, Times[Optional[Pattern[d, Blank[]]], Power[Pattern[x, Blank[]], 2]]]], Optional[Pattern[b, Blank[]]]]], Rational[1, 2]], Pattern[x, Blank[Symbol]]] := Simp[(-2*Sqrt[a + b*ArcCos[1 + d*x^2]]*Sin[ArcCos[1 + d*x^2]/2]^2)/(d*x), x] + (-Simp[(2*Sqrt[Pi]*Sin[a/(2*b)]*Sin[ArcCos[1 + d*x^2]/2]*FresnelC[Sqrt[1/(Pi*b)]*Sqrt[a + b*ArcCos[1 + d*x^2]]])/(Sqrt[1/b]*d*x), x] + Simp[(2*Sqrt[Pi]*Cos[a/(2*b)]*Sin[ArcCos[1 + d*x^2]/2]*FresnelS[Sqrt[1/(Pi*b)]*Sqrt[a + b*ArcCos[1 + d*x^2]]])/(Sqrt[1/b]*d*x), x]) /; FreeQ[{a, b, d}, x]
  4813. Int[Power[Plus[Optional[Pattern[a, Blank[]]], Times[ArcCos[Plus[-1, Times[Optional[Pattern[d, Blank[]]], Power[Pattern[x, Blank[]], 2]]]], Optional[Pattern[b, Blank[]]]]], Rational[1, 2]], Pattern[x, Blank[Symbol]]] := Simp[(2*Sqrt[a + b*ArcCos[-1 + d*x^2]]*Cos[(1/2)*ArcCos[-1 + d*x^2]]^2)/(d*x), x] + (-Simp[(2*Sqrt[Pi]*Cos[a/(2*b)]*Cos[ArcCos[-1 + d*x^2]/2]*FresnelC[Sqrt[1/(Pi*b)]*Sqrt[a + b*ArcCos[-1 + d*x^2]]])/(Sqrt[1/b]*d*x), x] - Simp[(2*Sqrt[Pi]*Sin[a/(2*b)]*Cos[ArcCos[-1 + d*x^2]/2]*FresnelS[Sqrt[1/(Pi*b)]*Sqrt[a + b*ArcCos[-1 + d*x^2]]])/(Sqrt[1/b]*d*x), x]) /; FreeQ[{a, b, d}, x]
  4814. Int[Power[Plus[Optional[Pattern[a, Blank[]]], Times[ArcSin[Plus[Pattern[c, Blank[]], Times[Optional[Pattern[d, Blank[]]], Power[Pattern[x, Blank[]], 2]]]], Optional[Pattern[b, Blank[]]]]], Pattern[n, Blank[]]], Pattern[x, Blank[Symbol]]] := Simp[x*(a + b*ArcSin[c + d*x^2])^n, x] + (-Dist[4*b^2*n*(n - 1), Int[(a + b*ArcSin[c + d*x^2])^(n - 2), x], x] + Simp[(2*b*n*Sqrt[-2*c*d*x^2 - d^2*x^4]*(a + b*ArcSin[c + d*x^2])^(n - 1))/(d*x), x]) /; FreeQ[{a, b, c, d}, x] && EqQ[c^2, 1] && GtQ[n, 1]
  4815. Int[Power[Plus[Optional[Pattern[a, Blank[]]], Times[ArcCos[Plus[Pattern[c, Blank[]], Times[Optional[Pattern[d, Blank[]]], Power[Pattern[x, Blank[]], 2]]]], Optional[Pattern[b, Blank[]]]]], Pattern[n, Blank[]]], Pattern[x, Blank[Symbol]]] := Simp[x*(a + b*ArcCos[c + d*x^2])^n, x] + (-Dist[4*b^2*n*(n - 1), Int[(a + b*ArcCos[c + d*x^2])^(n - 2), x], x] - Simp[(2*b*n*Sqrt[-2*c*d*x^2 - d^2*x^4]*(a + b*ArcCos[c + d*x^2])^(n - 1))/(d*x), x]) /; FreeQ[{a, b, c, d}, x] && EqQ[c^2, 1] && GtQ[n, 1]
  4816. Int[Power[Plus[Optional[Pattern[a, Blank[]]], Times[ArcSin[Plus[Pattern[c, Blank[]], Times[Optional[Pattern[d, Blank[]]], Power[Pattern[x, Blank[]], 2]]]], Optional[Pattern[b, Blank[]]]]], -1], Pattern[x, Blank[Symbol]]] := -Simp[(x*(c*Cos[a/(2*b)] - Sin[a/(2*b)])*CosIntegral[(c/(2*b))*(a + b*ArcSin[c + d*x^2])])/(2*b*(Cos[ArcSin[c + d*x^2]/2] - c*Sin[ArcSin[c + d*x^2]/2])), x] - Simp[(x*(c*Cos[a/(2*b)] + Sin[a/(2*b)])*SinIntegral[(c/(2*b))*(a + b*ArcSin[c + d*x^2])])/(2*b*(Cos[ArcSin[c + d*x^2]/2] - c*Sin[ArcSin[c + d*x^2]/2])), x] /; FreeQ[{a, b, c, d}, x] && EqQ[c^2, 1]
  4817. Int[Power[Plus[Optional[Pattern[a, Blank[]]], Times[ArcCos[Plus[1, Times[Optional[Pattern[d, Blank[]]], Power[Pattern[x, Blank[]], 2]]]], Optional[Pattern[b, Blank[]]]]], -1], Pattern[x, Blank[Symbol]]] := Simp[(x*Cos[a/(2*b)]*CosIntegral[(a + b*ArcCos[1 + d*x^2])/(2*b)])/(Sqrt[2]*b*Sqrt[-(d*x^2)]), x] + Simp[(x*Sin[a/(2*b)]*SinIntegral[(a + b*ArcCos[1 + d*x^2])/(2*b)])/(Sqrt[2]*b*Sqrt[-(d*x^2)]), x] /; FreeQ[{a, b, d}, x]
  4818. Int[Power[Plus[Optional[Pattern[a, Blank[]]], Times[ArcCos[Plus[-1, Times[Optional[Pattern[d, Blank[]]], Power[Pattern[x, Blank[]], 2]]]], Optional[Pattern[b, Blank[]]]]], -1], Pattern[x, Blank[Symbol]]] := Simp[(x*Sin[a/(2*b)]*CosIntegral[(a + b*ArcCos[-1 + d*x^2])/(2*b)])/(Sqrt[2]*b*Sqrt[d*x^2]), x] - Simp[(x*Cos[a/(2*b)]*SinIntegral[(a + b*ArcCos[-1 + d*x^2])/(2*b)])/(Sqrt[2]*b*Sqrt[d*x^2]), x] /; FreeQ[{a, b, d}, x]
  4819. Int[Power[Plus[Optional[Pattern[a, Blank[]]], Times[ArcSin[Plus[Pattern[c, Blank[]], Times[Optional[Pattern[d, Blank[]]], Power[Pattern[x, Blank[]], 2]]]], Optional[Pattern[b, Blank[]]]]], Rational[-1, 2]], Pattern[x, Blank[Symbol]]] := -Simp[(Sqrt[Pi]*x*(Cos[a/(2*b)] - c*Sin[a/(2*b)])*FresnelC[(1*Sqrt[a + b*ArcSin[c + d*x^2]])/(Sqrt[b*c]*Sqrt[Pi])])/(Sqrt[b*c]*(Cos[ArcSin[c + d*x^2]/2] - c*Sin[ArcSin[c + d*x^2]/2])), x] - Simp[(Sqrt[Pi]*x*(Cos[a/(2*b)] + c*Sin[a/(2*b)])*FresnelS[(1/(Sqrt[b*c]*Sqrt[Pi]))*Sqrt[a + b*ArcSin[c + d*x^2]]])/(Sqrt[b*c]*(Cos[ArcSin[c + d*x^2]/2] - c*Sin[ArcSin[c + d*x^2]/2])), x] /; FreeQ[{a, b, c, d}, x] && EqQ[c^2, 1]
  4820. Int[Power[Plus[Optional[Pattern[a, Blank[]]], Times[ArcCos[Plus[1, Times[Optional[Pattern[d, Blank[]]], Power[Pattern[x, Blank[]], 2]]]], Optional[Pattern[b, Blank[]]]]], Rational[-1, 2]], Pattern[x, Blank[Symbol]]] := Simp[(-2*Sqrt[Pi/b]*Cos[a/(2*b)]*Sin[ArcCos[1 + d*x^2]/2]*FresnelC[Sqrt[1/(Pi*b)]*Sqrt[a + b*ArcCos[1 + d*x^2]]])/(d*x), x] - Simp[(2*Sqrt[Pi/b]*Sin[a/(2*b)]*Sin[ArcCos[1 + d*x^2]/2]*FresnelS[Sqrt[1/(Pi*b)]*Sqrt[a + b*ArcCos[1 + d*x^2]]])/(d*x), x] /; FreeQ[{a, b, d}, x]
  4821. Int[Power[Plus[Optional[Pattern[a, Blank[]]], Times[ArcCos[Plus[-1, Times[Optional[Pattern[d, Blank[]]], Power[Pattern[x, Blank[]], 2]]]], Optional[Pattern[b, Blank[]]]]], Rational[-1, 2]], Pattern[x, Blank[Symbol]]] := Simp[(2*Sqrt[Pi/b]*Sin[a/(2*b)]*Cos[ArcCos[-1 + d*x^2]/2]*FresnelC[Sqrt[1/(Pi*b)]*Sqrt[a + b*ArcCos[-1 + d*x^2]]])/(d*x), x] - Simp[(2*Sqrt[Pi/b]*Cos[a/(2*b)]*Cos[ArcCos[-1 + d*x^2]/2]*FresnelS[Sqrt[1/(Pi*b)]*Sqrt[a + b*ArcCos[-1 + d*x^2]]])/(d*x), x] /; FreeQ[{a, b, d}, x]
  4822. Int[Power[Plus[Optional[Pattern[a, Blank[]]], Times[ArcSin[Plus[Pattern[c, Blank[]], Times[Optional[Pattern[d, Blank[]]], Power[Pattern[x, Blank[]], 2]]]], Optional[Pattern[b, Blank[]]]]], Rational[-3, 2]], Pattern[x, Blank[Symbol]]] := -Simp[Sqrt[-2*c*d*x^2 - d^2*x^4]/(b*d*x*Sqrt[a + b*ArcSin[c + d*x^2]]), x] + (-Simp[((c/b)^(3/2)*Sqrt[Pi]*x*(Cos[a/(2*b)] + c*Sin[a/(2*b)])*FresnelC[Sqrt[c/(Pi*b)]*Sqrt[a + b*ArcSin[c + d*x^2]]])/(Cos[(1/2)*ArcSin[c + d*x^2]] - c*Sin[ArcSin[c + d*x^2]/2]), x] + Simp[((c/b)^(3/2)*Sqrt[Pi]*x*(Cos[a/(2*b)] - c*Sin[a/(2*b)])*FresnelS[Sqrt[c/(Pi*b)]*Sqrt[a + b*ArcSin[c + d*x^2]]])/(Cos[(1/2)*ArcSin[c + d*x^2]] - c*Sin[ArcSin[c + d*x^2]/2]), x]) /; FreeQ[{a, b, c, d}, x] && EqQ[c^2, 1]
  4823. Int[Power[Plus[Optional[Pattern[a, Blank[]]], Times[ArcCos[Plus[1, Times[Optional[Pattern[d, Blank[]]], Power[Pattern[x, Blank[]], 2]]]], Optional[Pattern[b, Blank[]]]]], Rational[-3, 2]], Pattern[x, Blank[Symbol]]] := Simp[Sqrt[-2*d*x^2 - d^2*x^4]/(b*d*x*Sqrt[a + b*ArcCos[1 + d*x^2]]), x] + (-Simp[(2*(1/b)^(3/2)*Sqrt[Pi]*Sin[a/(2*b)]*Sin[ArcCos[1 + d*x^2]/2]*FresnelC[Sqrt[1/(Pi*b)]*Sqrt[a + b*ArcCos[1 + d*x^2]]])/(d*x), x] + Simp[(2*(1/b)^(3/2)*Sqrt[Pi]*Cos[a/(2*b)]*Sin[ArcCos[1 + d*x^2]/2]*FresnelS[Sqrt[1/(Pi*b)]*Sqrt[a + b*ArcCos[1 + d*x^2]]])/(d*x), x]) /; FreeQ[{a, b, d}, x]
  4824. Int[Power[Plus[Optional[Pattern[a, Blank[]]], Times[ArcCos[Plus[-1, Times[Optional[Pattern[d, Blank[]]], Power[Pattern[x, Blank[]], 2]]]], Optional[Pattern[b, Blank[]]]]], Rational[-3, 2]], Pattern[x, Blank[Symbol]]] := Simp[Sqrt[2*d*x^2 - d^2*x^4]/(b*d*x*Sqrt[a + b*ArcCos[-1 + d*x^2]]), x] + (-Simp[(2*(1/b)^(3/2)*Sqrt[Pi]*Cos[a/(2*b)]*Cos[ArcCos[-1 + d*x^2]/2]*FresnelC[Sqrt[1/(Pi*b)]*Sqrt[a + b*ArcCos[-1 + d*x^2]]])/(d*x), x] - Simp[(2*(1/b)^(3/2)*Sqrt[Pi]*Sin[a/(2*b)]*Cos[ArcCos[-1 + d*x^2]/2]*FresnelS[Sqrt[1/(Pi*b)]*Sqrt[a + b*ArcCos[-1 + d*x^2]]])/(d*x), x]) /; FreeQ[{a, b, d}, x]
  4825. Int[Power[Plus[Optional[Pattern[a, Blank[]]], Times[ArcSin[Plus[Pattern[c, Blank[]], Times[Optional[Pattern[d, Blank[]]], Power[Pattern[x, Blank[]], 2]]]], Optional[Pattern[b, Blank[]]]]], -2], Pattern[x, Blank[Symbol]]] := -Simp[Sqrt[-2*c*d*x^2 - d^2*x^4]/(2*b*d*x*(a + b*ArcSin[c + d*x^2])), x] + (-Simp[(x*(Cos[a/(2*b)] + c*Sin[a/(2*b)])*CosIntegral[(c/(2*b))*(a + b*ArcSin[c + d*x^2])])/(4*b^2*(Cos[ArcSin[c + d*x^2]/2] - c*Sin[ArcSin[c + d*x^2]/2])), x] + Simp[(x*(Cos[a/(2*b)] - c*Sin[a/(2*b)])*SinIntegral[(c/(2*b))*(a + b*ArcSin[c + d*x^2])])/(4*b^2*(Cos[ArcSin[c + d*x^2]/2] - c*Sin[ArcSin[c + d*x^2]/2])), x]) /; FreeQ[{a, b, c, d}, x] && EqQ[c^2, 1]
  4826. Int[Power[Plus[Optional[Pattern[a, Blank[]]], Times[ArcCos[Plus[1, Times[Optional[Pattern[d, Blank[]]], Power[Pattern[x, Blank[]], 2]]]], Optional[Pattern[b, Blank[]]]]], -2], Pattern[x, Blank[Symbol]]] := Simp[Sqrt[-2*d*x^2 - d^2*x^4]/(2*b*d*x*(a + b*ArcCos[1 + d*x^2])), x] + (Simp[(x*Sin[a/(2*b)]*CosIntegral[(a + b*ArcCos[1 + d*x^2])/(2*b)])/(2*Sqrt[2]*b^2*Sqrt[(-d)*x^2]), x] - Simp[(x*Cos[a/(2*b)]*SinIntegral[(a + b*ArcCos[1 + d*x^2])/(2*b)])/(2*Sqrt[2]*b^2*Sqrt[(-d)*x^2]), x]) /; FreeQ[{a, b, d}, x]
  4827. Int[Power[Plus[Optional[Pattern[a, Blank[]]], Times[ArcCos[Plus[-1, Times[Optional[Pattern[d, Blank[]]], Power[Pattern[x, Blank[]], 2]]]], Optional[Pattern[b, Blank[]]]]], -2], Pattern[x, Blank[Symbol]]] := Simp[Sqrt[2*d*x^2 - d^2*x^4]/(2*b*d*x*(a + b*ArcCos[-1 + d*x^2])), x] + (-Simp[(x*Cos[a/(2*b)]*CosIntegral[(a + b*ArcCos[-1 + d*x^2])/(2*b)])/(2*Sqrt[2]*b^2*Sqrt[d*x^2]), x] - Simp[(x*Sin[a/(2*b)]*SinIntegral[(a + b*ArcCos[-1 + d*x^2])/(2*b)])/(2*Sqrt[2]*b^2*Sqrt[d*x^2]), x]) /; FreeQ[{a, b, d}, x]
  4828. Int[Power[Plus[Optional[Pattern[a, Blank[]]], Times[ArcSin[Plus[Pattern[c, Blank[]], Times[Optional[Pattern[d, Blank[]]], Power[Pattern[x, Blank[]], 2]]]], Optional[Pattern[b, Blank[]]]]], Pattern[n, Blank[]]], Pattern[x, Blank[Symbol]]] := Simp[(x*(a + b*ArcSin[c + d*x^2])^(n + 2))/(4*b^2*(n + 1)*(n + 2)), x] + (-Dist[1/(4*b^2*(n + 1)*(n + 2)), Int[(a + b*ArcSin[c + d*x^2])^(n + 2), x], x] + Simp[(Sqrt[-2*c*d*x^2 - d^2*x^4]*(a + b*ArcSin[c + d*x^2])^(n + 1))/(2*b*d*(n + 1)*x), x]) /; FreeQ[{a, b, c, d}, x] && EqQ[c^2, 1] && LtQ[n, -1] && NeQ[n, -2]
  4829. Int[Power[Plus[Optional[Pattern[a, Blank[]]], Times[ArcCos[Plus[Pattern[c, Blank[]], Times[Optional[Pattern[d, Blank[]]], Power[Pattern[x, Blank[]], 2]]]], Optional[Pattern[b, Blank[]]]]], Pattern[n, Blank[]]], Pattern[x, Blank[Symbol]]] := Simp[(x*(a + b*ArcCos[c + d*x^2])^(n + 2))/(4*b^2*(n + 1)*(n + 2)), x] + (-Dist[1/(4*b^2*(n + 1)*(n + 2)), Int[(a + b*ArcCos[c + d*x^2])^(n + 2), x], x] - Simp[(Sqrt[-2*c*d*x^2 - d^2*x^4]*(a + b*ArcCos[c + d*x^2])^(n + 1))/(2*b*d*(n + 1)*x), x]) /; FreeQ[{a, b, c, d}, x] && EqQ[c^2, 1] && LtQ[n, -1] && NeQ[n, -2]
  4830. Int[Times[Power[ArcSin[Times[Optional[Pattern[a, Blank[]]], Power[Pattern[x, Blank[]], Pattern[p, Blank[]]]]], Optional[Pattern[n, Blank[]]]], Power[Pattern[x, Blank[]], -1]], Pattern[x, Blank[Symbol]]] := Dist[1/p, Subst[Int[x^n*Cot[x], x], x, ArcSin[a*x^p]], x] /; FreeQ[{a, p}, x] && IGtQ[n, 0]
  4831. Int[Times[Power[ArcCos[Times[Optional[Pattern[a, Blank[]]], Power[Pattern[x, Blank[]], Pattern[p, Blank[]]]]], Optional[Pattern[n, Blank[]]]], Power[Pattern[x, Blank[]], -1]], Pattern[x, Blank[Symbol]]] := -Dist[p^(-1), Subst[Int[x^n*Tan[x], x], x, ArcCos[a*x^p]], x] /; FreeQ[{a, p}, x] && IGtQ[n, 0]
  4832. Int[Times[Power[ArcSin[Times[Optional[Pattern[c, Blank[]]], Power[Plus[Optional[Pattern[a, Blank[]]], Times[Optional[Pattern[b, Blank[]]], Power[Pattern[x, Blank[]], Optional[Pattern[n, Blank[]]]]]], -1]]], Optional[Pattern[m, Blank[]]]], Optional[Pattern[u, Blank[]]]], Pattern[x, Blank[Symbol]]] := Int[u*ArcCsc[a/c + (b*x^n)/c]^m, x] /; FreeQ[{a, b, c, n, m}, x]
  4833. Int[Times[Power[ArcCos[Times[Optional[Pattern[c, Blank[]]], Power[Plus[Optional[Pattern[a, Blank[]]], Times[Optional[Pattern[b, Blank[]]], Power[Pattern[x, Blank[]], Optional[Pattern[n, Blank[]]]]]], -1]]], Optional[Pattern[m, Blank[]]]], Optional[Pattern[u, Blank[]]]], Pattern[x, Blank[Symbol]]] := Int[u*ArcSec[a/c + (b*x^n)/c]^m, x] /; FreeQ[{a, b, c, n, m}, x]
  4834. Int[Times[Power[ArcSin[Power[Plus[1, Times[Optional[Pattern[b, Blank[]]], Power[Pattern[x, Blank[]], 2]]], Rational[1, 2]]], Optional[Pattern[n, Blank[]]]], Power[Plus[1, Times[Optional[Pattern[b, Blank[]]], Power[Pattern[x, Blank[]], 2]]], Rational[-1, 2]]], Pattern[x, Blank[Symbol]]] := Dist[Sqrt[-(b*x^2)]/(b*x), Subst[Int[ArcSin[x]^n/Sqrt[1 - x^2], x], x, Sqrt[1 + b*x^2]], x] /; FreeQ[{b, n}, x]
  4835. Int[Times[Power[ArcCos[Power[Plus[1, Times[Optional[Pattern[b, Blank[]]], Power[Pattern[x, Blank[]], 2]]], Rational[1, 2]]], Optional[Pattern[n, Blank[]]]], Power[Plus[1, Times[Optional[Pattern[b, Blank[]]], Power[Pattern[x, Blank[]], 2]]], Rational[-1, 2]]], Pattern[x, Blank[Symbol]]] := Dist[Sqrt[-(b*x^2)]/(b*x), Subst[Int[ArcCos[x]^n/Sqrt[1 - x^2], x], x, Sqrt[1 + b*x^2]], x] /; FreeQ[{b, n}, x]
  4836. Int[Times[Optional[Pattern[u, Blank[]]], Power[Pattern[f, Blank[]], Times[Power[ArcSin[Plus[Optional[Pattern[a, Blank[]]], Times[Optional[Pattern[b, Blank[]]], Pattern[x, Blank[]]]]], Optional[Pattern[n, Blank[]]]], Optional[Pattern[c, Blank[]]]]]], Pattern[x, Blank[Symbol]]] := Dist[1/b, Subst[Int[(u /. x -> -(a/b) + Sin[x]/b)*f^(c*x^n)*Cos[x], x], x, ArcSin[a + b*x]], x] /; FreeQ[{a, b, c, f}, x] && IGtQ[n, 0]
  4837. Int[Times[Optional[Pattern[u, Blank[]]], Power[Pattern[f, Blank[]], Times[Power[ArcCos[Plus[Optional[Pattern[a, Blank[]]], Times[Optional[Pattern[b, Blank[]]], Pattern[x, Blank[]]]]], Optional[Pattern[n, Blank[]]]], Optional[Pattern[c, Blank[]]]]]], Pattern[x, Blank[Symbol]]] := -Dist[b^(-1), Subst[Int[(u /. x -> -(a/b) + Cos[x]/b)*f^(c*x^n)*Sin[x], x], x, ArcCos[a + b*x]], x] /; FreeQ[{a, b, c, f}, x] && IGtQ[n, 0]
  4838. Int[ArcSin[Plus[Times[Optional[Pattern[a, Blank[]]], Power[Pattern[x, Blank[]], 2]], Times[Optional[Pattern[b, Blank[]]], Power[Plus[Pattern[c, Blank[]], Times[Optional[Pattern[d, Blank[]]], Power[Pattern[x, Blank[]], 2]]], Rational[1, 2]]]]], Pattern[x, Blank[Symbol]]] := Simp[x*ArcSin[a*x^2 + b*Sqrt[c + d*x^2]], x] - Dist[(x*Sqrt[b^2*d + a^2*x^2 + 2*a*b*Sqrt[c + d*x^2]])/Sqrt[(-x^2)*(b^2*d + a^2*x^2 + 2*a*b*Sqrt[c + d*x^2])], Int[(x*(b*d + 2*a*Sqrt[c + d*x^2]))/(Sqrt[c + d*x^2]*Sqrt[b^2*d + a^2*x^2 + 2*a*b*Sqrt[c + d*x^2]]), x], x] /; FreeQ[{a, b, c, d}, x] && EqQ[b^2*c, 1]
  4839. Int[ArcCos[Plus[Times[Optional[Pattern[a, Blank[]]], Power[Pattern[x, Blank[]], 2]], Times[Optional[Pattern[b, Blank[]]], Power[Plus[Pattern[c, Blank[]], Times[Optional[Pattern[d, Blank[]]], Power[Pattern[x, Blank[]], 2]]], Rational[1, 2]]]]], Pattern[x, Blank[Symbol]]] := Simp[x*ArcCos[a*x^2 + b*Sqrt[c + d*x^2]], x] + Dist[(x*Sqrt[b^2*d + a^2*x^2 + 2*a*b*Sqrt[c + d*x^2]])/Sqrt[(-x^2)*(b^2*d + a^2*x^2 + 2*a*b*Sqrt[c + d*x^2])], Int[(x*(b*d + 2*a*Sqrt[c + d*x^2]))/(Sqrt[c + d*x^2]*Sqrt[b^2*d + a^2*x^2 + 2*a*b*Sqrt[c + d*x^2]]), x], x] /; FreeQ[{a, b, c, d}, x] && EqQ[b^2*c, 1]
  4840. Int[ArcSin[Pattern[u, Blank[]]], Pattern[x, Blank[Symbol]]] := Simp[x*ArcSin[u], x] - Int[SimplifyIntegrand[(x*D[u, x])/Sqrt[1 - u^2], x], x] /; InverseFunctionFreeQ[u, x] && !FunctionOfExponentialQ[u, x]
  4841. Int[ArcCos[Pattern[u, Blank[]]], Pattern[x, Blank[Symbol]]] := Simp[x*ArcCos[u], x] + Int[SimplifyIntegrand[(x*D[u, x])/Sqrt[1 - u^2], x], x] /; InverseFunctionFreeQ[u, x] && !FunctionOfExponentialQ[u, x]
  4842. Int[Times[Plus[Optional[Pattern[a, Blank[]]], Times[ArcSin[Pattern[u, Blank[]]], Optional[Pattern[b, Blank[]]]]], Power[Plus[Optional[Pattern[c, Blank[]]], Times[Optional[Pattern[d, Blank[]]], Pattern[x, Blank[]]]], Optional[Pattern[m, Blank[]]]]], Pattern[x, Blank[Symbol]]] := Simp[((c + d*x)^(m + 1)*(a + b*ArcSin[u]))/(d*(m + 1)), x] - Dist[b/(d*(m + 1)), Int[SimplifyIntegrand[((c + d*x)^(m + 1)*D[u, x])/Sqrt[1 - u^2], x], x], x] /; FreeQ[{a, b, c, d, m}, x] && NeQ[m, -1] && InverseFunctionFreeQ[u, x] && !FunctionOfQ[(c + d*x)^(m + 1), u, x] && !FunctionOfExponentialQ[u, x]
  4843. Int[Times[Plus[Optional[Pattern[a, Blank[]]], Times[ArcCos[Pattern[u, Blank[]]], Optional[Pattern[b, Blank[]]]]], Power[Plus[Optional[Pattern[c, Blank[]]], Times[Optional[Pattern[d, Blank[]]], Pattern[x, Blank[]]]], Optional[Pattern[m, Blank[]]]]], Pattern[x, Blank[Symbol]]] := Simp[((c + d*x)^(m + 1)*(a + b*ArcCos[u]))/(d*(m + 1)), x] + Dist[b/(d*(m + 1)), Int[SimplifyIntegrand[((c + d*x)^(m + 1)*D[u, x])/Sqrt[1 - u^2], x], x], x] /; FreeQ[{a, b, c, d, m}, x] && NeQ[m, -1] && InverseFunctionFreeQ[u, x] && !FunctionOfQ[(c + d*x)^(m + 1), u, x] && !FunctionOfExponentialQ[u, x]
  4844. Int[Times[Plus[Optional[Pattern[a, Blank[]]], Times[ArcSin[Pattern[u, Blank[]]], Optional[Pattern[b, Blank[]]]]], Pattern[v, Blank[]]], Pattern[x, Blank[Symbol]]] := With[{w = IntHide[v, x]}, Dist[a + b*ArcSin[u], w, x] - Dist[b, Int[SimplifyIntegrand[(w*D[u, x])/Sqrt[1 - u^2], x], x], x] /; InverseFunctionFreeQ[w, x]] /; FreeQ[{a, b}, x] && InverseFunctionFreeQ[u, x] && !MatchQ[v, ((c_.) + (d_.)*x)^(m_.) /; FreeQ[{c, d, m}, x]]
  4845. Int[Times[Plus[Optional[Pattern[a, Blank[]]], Times[ArcCos[Pattern[u, Blank[]]], Optional[Pattern[b, Blank[]]]]], Pattern[v, Blank[]]], Pattern[x, Blank[Symbol]]] := With[{w = IntHide[v, x]}, Dist[a + b*ArcCos[u], w, x] + Dist[b, Int[SimplifyIntegrand[(w*D[u, x])/Sqrt[1 - u^2], x], x], x] /; InverseFunctionFreeQ[w, x]] /; FreeQ[{a, b}, x] && InverseFunctionFreeQ[u, x] && !MatchQ[v, ((c_.) + (d_.)*x)^(m_.) /; FreeQ[{c, d, m}, x]]
  4846. Int[Power[Plus[Optional[Pattern[a, Blank[]]], Times[ArcTan[Times[Optional[Pattern[c, Blank[]]], Pattern[x, Blank[]]]], Optional[Pattern[b, Blank[]]]]], Optional[Pattern[p, Blank[]]]], Pattern[x, Blank[Symbol]]] := Simp[x*(a + b*ArcTan[c*x])^p, x] - Dist[b*c*p, Int[(x*(a + b*ArcTan[c*x])^(p - 1))/(1 + c^2*x^2), x], x] /; FreeQ[{a, b, c}, x] && IGtQ[p, 0]
  4847. Int[Power[Plus[Optional[Pattern[a, Blank[]]], Times[ArcCot[Times[Optional[Pattern[c, Blank[]]], Pattern[x, Blank[]]]], Optional[Pattern[b, Blank[]]]]], Optional[Pattern[p, Blank[]]]], Pattern[x, Blank[Symbol]]] := Simp[x*(a + b*ArcCot[c*x])^p, x] + Dist[b*c*p, Int[(x*(a + b*ArcCot[c*x])^(p - 1))/(1 + c^2*x^2), x], x] /; FreeQ[{a, b, c}, x] && IGtQ[p, 0]
  4848. Int[Times[Plus[Optional[Pattern[a, Blank[]]], Times[ArcTan[Times[Optional[Pattern[c, Blank[]]], Pattern[x, Blank[]]]], Optional[Pattern[b, Blank[]]]]], Power[Pattern[x, Blank[]], -1]], Pattern[x, Blank[Symbol]]] := Simp[a*Log[x], x] + (Dist[(I*b)/2, Int[Log[1 - I*c*x]/x, x], x] - Dist[(I*b)/2, Int[Log[1 + I*c*x]/x, x], x]) /; FreeQ[{a, b, c}, x]
  4849. Int[Times[Plus[Optional[Pattern[a, Blank[]]], Times[ArcCot[Times[Optional[Pattern[c, Blank[]]], Pattern[x, Blank[]]]], Optional[Pattern[b, Blank[]]]]], Power[Pattern[x, Blank[]], -1]], Pattern[x, Blank[Symbol]]] := Simp[a*Log[x], x] + (Dist[(I*b)/2, Int[Log[1 - I/(c*x)]/x, x], x] - Dist[(I*b)/2, Int[Log[1 + I/(c*x)]/x, x], x]) /; FreeQ[{a, b, c}, x]
  4850. Int[Times[Power[Plus[Optional[Pattern[a, Blank[]]], Times[ArcTan[Times[Optional[Pattern[c, Blank[]]], Pattern[x, Blank[]]]], Optional[Pattern[b, Blank[]]]]], Pattern[p, Blank[]]], Power[Pattern[x, Blank[]], -1]], Pattern[x, Blank[Symbol]]] := Simp[2*(a + b*ArcTan[c*x])^p*ArcTanh[1 - 2/(1 + I*c*x)], x] - Dist[2*b*c*p, Int[((a + b*ArcTan[c*x])^(p - 1)*ArcTanh[1 - 2/(1 + I*c*x)])/(1 + c^2*x^2), x], x] /; FreeQ[{a, b, c}, x] && IGtQ[p, 1]
  4851. Int[Times[Power[Plus[Optional[Pattern[a, Blank[]]], Times[ArcCot[Times[Optional[Pattern[c, Blank[]]], Pattern[x, Blank[]]]], Optional[Pattern[b, Blank[]]]]], Pattern[p, Blank[]]], Power[Pattern[x, Blank[]], -1]], Pattern[x, Blank[Symbol]]] := Simp[2*(a + b*ArcCot[c*x])^p*ArcCoth[1 - 2/(1 + I*c*x)], x] + Dist[2*b*c*p, Int[((a + b*ArcCot[c*x])^(p - 1)*ArcCoth[1 - 2/(1 + I*c*x)])/(1 + c^2*x^2), x], x] /; FreeQ[{a, b, c}, x] && IGtQ[p, 1]
  4852. Int[Times[Power[Plus[Optional[Pattern[a, Blank[]]], Times[ArcTan[Times[Optional[Pattern[c, Blank[]]], Pattern[x, Blank[]]]], Optional[Pattern[b, Blank[]]]]], Optional[Pattern[p, Blank[]]]], Power[Times[Optional[Pattern[d, Blank[]]], Pattern[x, Blank[]]], Optional[Pattern[m, Blank[]]]]], Pattern[x, Blank[Symbol]]] := Simp[((d*x)^(m + 1)*(a + b*ArcTan[c*x])^p)/(d*(m + 1)), x] - Dist[(b*c*p)/(d*(m + 1)), Int[((d*x)^(m + 1)*(a + b*ArcTan[c*x])^(p - 1))/(1 + c^2*x^2), x], x] /; FreeQ[{a, b, c, d, m}, x] && IGtQ[p, 0] && (EqQ[p, 1] || IntegerQ[m]) && NeQ[m, -1]
  4853. Int[Times[Power[Plus[Optional[Pattern[a, Blank[]]], Times[ArcCot[Times[Optional[Pattern[c, Blank[]]], Pattern[x, Blank[]]]], Optional[Pattern[b, Blank[]]]]], Optional[Pattern[p, Blank[]]]], Power[Times[Optional[Pattern[d, Blank[]]], Pattern[x, Blank[]]], Optional[Pattern[m, Blank[]]]]], Pattern[x, Blank[Symbol]]] := Simp[((d*x)^(m + 1)*(a + b*ArcCot[c*x])^p)/(d*(m + 1)), x] + Dist[(b*c*p)/(d*(m + 1)), Int[((d*x)^(m + 1)*(a + b*ArcCot[c*x])^(p - 1))/(1 + c^2*x^2), x], x] /; FreeQ[{a, b, c, d, m}, x] && IGtQ[p, 0] && (EqQ[p, 1] || IntegerQ[m]) && NeQ[m, -1]
  4854. Int[Times[Power[Plus[Optional[Pattern[a, Blank[]]], Times[ArcTan[Times[Optional[Pattern[c, Blank[]]], Pattern[x, Blank[]]]], Optional[Pattern[b, Blank[]]]]], Optional[Pattern[p, Blank[]]]], Power[Plus[Pattern[d, Blank[]], Times[Optional[Pattern[e, Blank[]]], Pattern[x, Blank[]]]], -1]], Pattern[x, Blank[Symbol]]] := -Simp[((a + b*ArcTan[c*x])^p*Log[2/(1 + (e*x)/d)])/e, x] + Dist[(b*c*p)/e, Int[((a + b*ArcTan[c*x])^(p - 1)*Log[2/(1 + (e*x)/d)])/(1 + c^2*x^2), x], x] /; FreeQ[{a, b, c, d, e}, x] && IGtQ[p, 0] && EqQ[c^2*d^2 + e^2, 0]
  4855. Int[Times[Power[Plus[Optional[Pattern[a, Blank[]]], Times[ArcCot[Times[Optional[Pattern[c, Blank[]]], Pattern[x, Blank[]]]], Optional[Pattern[b, Blank[]]]]], Optional[Pattern[p, Blank[]]]], Power[Plus[Pattern[d, Blank[]], Times[Optional[Pattern[e, Blank[]]], Pattern[x, Blank[]]]], -1]], Pattern[x, Blank[Symbol]]] := -Simp[((a + b*ArcCot[c*x])^p*Log[2/(1 + (e*x)/d)])/e, x] - Dist[(b*c*p)/e, Int[((a + b*ArcCot[c*x])^(p - 1)*Log[2/(1 + (e*x)/d)])/(1 + c^2*x^2), x], x] /; FreeQ[{a, b, c, d, e}, x] && IGtQ[p, 0] && EqQ[c^2*d^2 + e^2, 0]
  4856. Int[Times[Plus[Optional[Pattern[a, Blank[]]], Times[ArcTan[Times[Optional[Pattern[c, Blank[]]], Pattern[x, Blank[]]]], Optional[Pattern[b, Blank[]]]]], Power[Plus[Pattern[d, Blank[]], Times[Optional[Pattern[e, Blank[]]], Pattern[x, Blank[]]]], -1]], Pattern[x, Blank[Symbol]]] := -Simp[((a + b*ArcTan[c*x])*Log[2/(1 - I*c*x)])/e, x] + (Dist[(b*c)/e, Int[Log[2/(1 - I*c*x)]/(1 + c^2*x^2), x], x] - Dist[(b*c)/e, Int[Log[(2*c*(d + e*x))/((c*d + I*e)*(1 - I*c*x))]/(1 + c^2*x^2), x], x] + Simp[((a + b*ArcTan[c*x])*Log[(2*c*(d + e*x))/((c*d + I*e)*(1 - I*c*x))])/e, x]) /; FreeQ[{a, b, c, d, e}, x] && NeQ[c^2*d^2 + e^2, 0]
  4857. Int[Times[Plus[Optional[Pattern[a, Blank[]]], Times[ArcCot[Times[Optional[Pattern[c, Blank[]]], Pattern[x, Blank[]]]], Optional[Pattern[b, Blank[]]]]], Power[Plus[Pattern[d, Blank[]], Times[Optional[Pattern[e, Blank[]]], Pattern[x, Blank[]]]], -1]], Pattern[x, Blank[Symbol]]] := -Simp[((a + b*ArcCot[c*x])*Log[2/(1 - I*c*x)])/e, x] + (-Dist[(b*c)/e, Int[Log[2/(1 - I*c*x)]/(1 + c^2*x^2), x], x] + Dist[(b*c)/e, Int[Log[(2*c*(d + e*x))/((c*d + I*e)*(1 - I*c*x))]/(1 + c^2*x^2), x], x] + Simp[((a + b*ArcCot[c*x])*Log[(2*c*(d + e*x))/((c*d + I*e)*(1 - I*c*x))])/e, x]) /; FreeQ[{a, b, c, d, e}, x] && NeQ[c^2*d^2 + e^2, 0]
  4858. Int[Times[Power[Plus[Optional[Pattern[a, Blank[]]], Times[ArcTan[Times[Optional[Pattern[c, Blank[]]], Pattern[x, Blank[]]]], Optional[Pattern[b, Blank[]]]]], 2], Power[Plus[Pattern[d, Blank[]], Times[Optional[Pattern[e, Blank[]]], Pattern[x, Blank[]]]], -1]], Pattern[x, Blank[Symbol]]] := -Simp[((a + b*ArcTan[c*x])^2*Log[2/(1 - I*c*x)])/e, x] + (Simp[((a + b*ArcTan[c*x])^2*Log[(2*c*(d + e*x))/((c*d + I*e)*(1 - I*c*x))])/e, x] + Simp[(I*b*(a + b*ArcTan[c*x])*PolyLog[2, 1 - 2/(1 - I*c*x)])/e, x] - Simp[(I*b*(a + b*ArcTan[c*x])*PolyLog[2, 1 - (2*c*(d + e*x))/((c*d + I*e)*(1 - I*c*x))])/e, x] - Simp[(b^2*PolyLog[3, 1 - 2/(1 - I*c*x)])/(2*e), x] + Simp[(b^2*PolyLog[3, 1 - (2*c*(d + e*x))/((c*d + I*e)*(1 - I*c*x))])/(2*e), x]) /; FreeQ[{a, b, c, d, e}, x] && NeQ[c^2*d^2 + e^2, 0]
  4859. Int[Times[Power[Plus[Optional[Pattern[a, Blank[]]], Times[ArcCot[Times[Optional[Pattern[c, Blank[]]], Pattern[x, Blank[]]]], Optional[Pattern[b, Blank[]]]]], 2], Power[Plus[Pattern[d, Blank[]], Times[Optional[Pattern[e, Blank[]]], Pattern[x, Blank[]]]], -1]], Pattern[x, Blank[Symbol]]] := -Simp[((a + b*ArcCot[c*x])^2*Log[2/(1 - I*c*x)])/e, x] + (Simp[((a + b*ArcCot[c*x])^2*Log[(2*c*(d + e*x))/((c*d + I*e)*(1 - I*c*x))])/e, x] - Simp[(I*b*(a + b*ArcCot[c*x])*PolyLog[2, 1 - 2/(1 - I*c*x)])/e, x] + Simp[(I*b*(a + b*ArcCot[c*x])*PolyLog[2, 1 - (2*c*(d + e*x))/((c*d + I*e)*(1 - I*c*x))])/e, x] - Simp[(b^2*PolyLog[3, 1 - 2/(1 - I*c*x)])/(2*e), x] + Simp[(b^2*PolyLog[3, 1 - (2*c*(d + e*x))/((c*d + I*e)*(1 - I*c*x))])/(2*e), x]) /; FreeQ[{a, b, c, d, e}, x] && NeQ[c^2*d^2 + e^2, 0]
  4860. Int[Times[Power[Plus[Optional[Pattern[a, Blank[]]], Times[ArcTan[Times[Optional[Pattern[c, Blank[]]], Pattern[x, Blank[]]]], Optional[Pattern[b, Blank[]]]]], 3], Power[Plus[Pattern[d, Blank[]], Times[Optional[Pattern[e, Blank[]]], Pattern[x, Blank[]]]], -1]], Pattern[x, Blank[Symbol]]] := -Simp[((a + b*ArcTan[c*x])^3*Log[2/(1 - I*c*x)])/e, x] + (Simp[((a + b*ArcTan[c*x])^3*Log[(2*c*(d + e*x))/((c*d + I*e)*(1 - I*c*x))])/e, x] + Simp[(3*I*b*(a + b*ArcTan[c*x])^2*PolyLog[2, 1 - 2/(1 - I*c*x)])/(2*e), x] - Simp[(3*I*b*(a + b*ArcTan[c*x])^2*PolyLog[2, 1 - (2*c*(d + e*x))/((c*d + I*e)*(1 - I*c*x))])/(2*e), x] - Simp[(3*b^2*(a + b*ArcTan[c*x])*PolyLog[3, 1 - 2/(1 - I*c*x)])/(2*e), x] + Simp[(3*b^2*(a + b*ArcTan[c*x])*PolyLog[3, 1 - (2*c*(d + e*x))/((c*d + I*e)*(1 - I*c*x))])/(2*e), x] - Simp[(3*I*b^3*PolyLog[4, 1 - 2/(1 - I*c*x)])/(4*e), x] + Simp[(3*I*b^3*PolyLog[4, 1 - (2*c*(d + e*x))/((c*d + I*e)*(1 - I*c*x))])/(4*e), x]) /; FreeQ[{a, b, c, d, e}, x] && NeQ[c^2*d^2 + e^2, 0]
  4861. Int[Times[Power[Plus[Optional[Pattern[a, Blank[]]], Times[ArcCot[Times[Optional[Pattern[c, Blank[]]], Pattern[x, Blank[]]]], Optional[Pattern[b, Blank[]]]]], 3], Power[Plus[Pattern[d, Blank[]], Times[Optional[Pattern[e, Blank[]]], Pattern[x, Blank[]]]], -1]], Pattern[x, Blank[Symbol]]] := -Simp[((a + b*ArcCot[c*x])^3*Log[2/(1 - I*c*x)])/e, x] + (Simp[((a + b*ArcCot[c*x])^3*Log[(2*c*(d + e*x))/((c*d + I*e)*(1 - I*c*x))])/e, x] - Simp[(3*I*b*(a + b*ArcCot[c*x])^2*PolyLog[2, 1 - 2/(1 - I*c*x)])/(2*e), x] + Simp[(3*I*b*(a + b*ArcCot[c*x])^2*PolyLog[2, 1 - (2*c*(d + e*x))/((c*d + I*e)*(1 - I*c*x))])/(2*e), x] - Simp[(3*b^2*(a + b*ArcCot[c*x])*PolyLog[3, 1 - 2/(1 - I*c*x)])/(2*e), x] + Simp[(3*b^2*(a + b*ArcCot[c*x])*PolyLog[3, 1 - (2*c*(d + e*x))/((c*d + I*e)*(1 - I*c*x))])/(2*e), x] + Simp[(3*I*b^3*PolyLog[4, 1 - 2/(1 - I*c*x)])/(4*e), x] - Simp[(3*I*b^3*PolyLog[4, 1 - (2*c*(d + e*x))/((c*d + I*e)*(1 - I*c*x))])/(4*e), x]) /; FreeQ[{a, b, c, d, e}, x] && NeQ[c^2*d^2 + e^2, 0]
  4862. Int[Times[Plus[Optional[Pattern[a, Blank[]]], Times[ArcTan[Times[Optional[Pattern[c, Blank[]]], Pattern[x, Blank[]]]], Optional[Pattern[b, Blank[]]]]], Power[Plus[Pattern[d, Blank[]], Times[Optional[Pattern[e, Blank[]]], Pattern[x, Blank[]]]], Optional[Pattern[q, Blank[]]]]], Pattern[x, Blank[Symbol]]] := Simp[((d + e*x)^(q + 1)*(a + b*ArcTan[c*x]))/(e*(q + 1)), x] - Dist[(b*c)/(e*(q + 1)), Int[(d + e*x)^(q + 1)/(1 + c^2*x^2), x], x] /; FreeQ[{a, b, c, d, e, q}, x] && NeQ[q, -1]
  4863. Int[Times[Plus[Optional[Pattern[a, Blank[]]], Times[ArcCot[Times[Optional[Pattern[c, Blank[]]], Pattern[x, Blank[]]]], Optional[Pattern[b, Blank[]]]]], Power[Plus[Pattern[d, Blank[]], Times[Optional[Pattern[e, Blank[]]], Pattern[x, Blank[]]]], Optional[Pattern[q, Blank[]]]]], Pattern[x, Blank[Symbol]]] := Simp[((d + e*x)^(q + 1)*(a + b*ArcCot[c*x]))/(e*(q + 1)), x] + Dist[(b*c)/(e*(q + 1)), Int[(d + e*x)^(q + 1)/(1 + c^2*x^2), x], x] /; FreeQ[{a, b, c, d, e, q}, x] && NeQ[q, -1]
  4864. Int[Times[Power[Plus[Optional[Pattern[a, Blank[]]], Times[ArcTan[Times[Optional[Pattern[c, Blank[]]], Pattern[x, Blank[]]]], Optional[Pattern[b, Blank[]]]]], Pattern[p, Blank[]]], Power[Plus[Pattern[d, Blank[]], Times[Optional[Pattern[e, Blank[]]], Pattern[x, Blank[]]]], Optional[Pattern[q, Blank[]]]]], Pattern[x, Blank[Symbol]]] := Simp[((d + e*x)^(q + 1)*(a + b*ArcTan[c*x])^p)/(e*(q + 1)), x] - Dist[(b*c*p)/(e*(q + 1)), Int[ExpandIntegrand[(a + b*ArcTan[c*x])^(p - 1), (d + e*x)^(q + 1)/(1 + c^2*x^2), x], x], x] /; FreeQ[{a, b, c, d, e}, x] && IGtQ[p, 1] && IntegerQ[q] && NeQ[q, -1]
  4865. Int[Times[Power[Plus[Optional[Pattern[a, Blank[]]], Times[ArcCot[Times[Optional[Pattern[c, Blank[]]], Pattern[x, Blank[]]]], Optional[Pattern[b, Blank[]]]]], Pattern[p, Blank[]]], Power[Plus[Pattern[d, Blank[]], Times[Optional[Pattern[e, Blank[]]], Pattern[x, Blank[]]]], Optional[Pattern[q, Blank[]]]]], Pattern[x, Blank[Symbol]]] := Simp[((d + e*x)^(q + 1)*(a + b*ArcCot[c*x])^p)/(e*(q + 1)), x] + Dist[(b*c*p)/(e*(q + 1)), Int[ExpandIntegrand[(a + b*ArcCot[c*x])^(p - 1), (d + e*x)^(q + 1)/(1 + c^2*x^2), x], x], x] /; FreeQ[{a, b, c, d, e}, x] && IGtQ[p, 1] && IntegerQ[q] && NeQ[q, -1]
  4866. Int[Times[Power[Plus[Optional[Pattern[a, Blank[]]], Times[ArcTan[Times[Optional[Pattern[c, Blank[]]], Pattern[x, Blank[]]]], Optional[Pattern[b, Blank[]]]]], Optional[Pattern[p, Blank[]]]], Power[Times[Optional[Pattern[f, Blank[]]], Pattern[x, Blank[]]], Optional[Pattern[m, Blank[]]]], Power[Plus[Pattern[d, Blank[]], Times[Optional[Pattern[e, Blank[]]], Pattern[x, Blank[]]]], -1]], Pattern[x, Blank[Symbol]]] := Dist[f/e, Int[(f*x)^(m - 1)*(a + b*ArcTan[c*x])^p, x], x] - Dist[(d*f)/e, Int[((f*x)^(m - 1)*(a + b*ArcTan[c*x])^p)/(d + e*x), x], x] /; FreeQ[{a, b, c, d, e, f}, x] && IGtQ[p, 0] && EqQ[c^2*d^2 + e^2, 0] && GtQ[m, 0]
  4867. Int[Times[Power[Plus[Optional[Pattern[a, Blank[]]], Times[ArcCot[Times[Optional[Pattern[c, Blank[]]], Pattern[x, Blank[]]]], Optional[Pattern[b, Blank[]]]]], Optional[Pattern[p, Blank[]]]], Power[Times[Optional[Pattern[f, Blank[]]], Pattern[x, Blank[]]], Optional[Pattern[m, Blank[]]]], Power[Plus[Pattern[d, Blank[]], Times[Optional[Pattern[e, Blank[]]], Pattern[x, Blank[]]]], -1]], Pattern[x, Blank[Symbol]]] := Dist[f/e, Int[(f*x)^(m - 1)*(a + b*ArcCot[c*x])^p, x], x] - Dist[(d*f)/e, Int[((f*x)^(m - 1)*(a + b*ArcCot[c*x])^p)/(d + e*x), x], x] /; FreeQ[{a, b, c, d, e, f}, x] && IGtQ[p, 0] && EqQ[c^2*d^2 + e^2, 0] && GtQ[m, 0]
  4868. Int[Times[Power[Plus[Optional[Pattern[a, Blank[]]], Times[ArcTan[Times[Optional[Pattern[c, Blank[]]], Pattern[x, Blank[]]]], Optional[Pattern[b, Blank[]]]]], Optional[Pattern[p, Blank[]]]], Power[Pattern[x, Blank[]], -1], Power[Plus[Pattern[d, Blank[]], Times[Optional[Pattern[e, Blank[]]], Pattern[x, Blank[]]]], -1]], Pattern[x, Blank[Symbol]]] := Simp[((a + b*ArcTan[c*x])^p*Log[2 - 2/(1 + (e*x)/d)])/d, x] - Dist[(b*c*p)/d, Int[((a + b*ArcTan[c*x])^(p - 1)*Log[2 - 2/(1 + (e*x)/d)])/(1 + c^2*x^2), x], x] /; FreeQ[{a, b, c, d, e}, x] && IGtQ[p, 0] && EqQ[c^2*d^2 + e^2, 0]
  4869. Int[Times[Power[Plus[Optional[Pattern[a, Blank[]]], Times[ArcCot[Times[Optional[Pattern[c, Blank[]]], Pattern[x, Blank[]]]], Optional[Pattern[b, Blank[]]]]], Optional[Pattern[p, Blank[]]]], Power[Pattern[x, Blank[]], -1], Power[Plus[Pattern[d, Blank[]], Times[Optional[Pattern[e, Blank[]]], Pattern[x, Blank[]]]], -1]], Pattern[x, Blank[Symbol]]] := Simp[((a + b*ArcCot[c*x])^p*Log[2 - 2/(1 + (e*x)/d)])/d, x] + Dist[(b*c*p)/d, Int[((a + b*ArcCot[c*x])^(p - 1)*Log[2 - 2/(1 + (e*x)/d)])/(1 + c^2*x^2), x], x] /; FreeQ[{a, b, c, d, e}, x] && IGtQ[p, 0] && EqQ[c^2*d^2 + e^2, 0]
  4870. Int[Times[Power[Plus[Optional[Pattern[a, Blank[]]], Times[ArcTan[Times[Optional[Pattern[c, Blank[]]], Pattern[x, Blank[]]]], Optional[Pattern[b, Blank[]]]]], Optional[Pattern[p, Blank[]]]], Power[Times[Optional[Pattern[f, Blank[]]], Pattern[x, Blank[]]], Pattern[m, Blank[]]], Power[Plus[Pattern[d, Blank[]], Times[Optional[Pattern[e, Blank[]]], Pattern[x, Blank[]]]], -1]], Pattern[x, Blank[Symbol]]] := Dist[1/d, Int[(f*x)^m*(a + b*ArcTan[c*x])^p, x], x] - Dist[e/(d*f), Int[((f*x)^(m + 1)*(a + b*ArcTan[c*x])^p)/(d + e*x), x], x] /; FreeQ[{a, b, c, d, e, f}, x] && IGtQ[p, 0] && EqQ[c^2*d^2 + e^2, 0] && LtQ[m, -1]
  4871. Int[Times[Power[Plus[Optional[Pattern[a, Blank[]]], Times[ArcCot[Times[Optional[Pattern[c, Blank[]]], Pattern[x, Blank[]]]], Optional[Pattern[b, Blank[]]]]], Optional[Pattern[p, Blank[]]]], Power[Times[Optional[Pattern[f, Blank[]]], Pattern[x, Blank[]]], Pattern[m, Blank[]]], Power[Plus[Pattern[d, Blank[]], Times[Optional[Pattern[e, Blank[]]], Pattern[x, Blank[]]]], -1]], Pattern[x, Blank[Symbol]]] := Dist[1/d, Int[(f*x)^m*(a + b*ArcCot[c*x])^p, x], x] - Dist[e/(d*f), Int[((f*x)^(m + 1)*(a + b*ArcCot[c*x])^p)/(d + e*x), x], x] /; FreeQ[{a, b, c, d, e, f}, x] && IGtQ[p, 0] && EqQ[c^2*d^2 + e^2, 0] && LtQ[m, -1]
  4872. Int[Times[Plus[Optional[Pattern[a, Blank[]]], Times[ArcTan[Times[Optional[Pattern[c, Blank[]]], Pattern[x, Blank[]]]], Optional[Pattern[b, Blank[]]]]], Power[Times[Optional[Pattern[f, Blank[]]], Pattern[x, Blank[]]], Optional[Pattern[m, Blank[]]]], Power[Plus[Optional[Pattern[d, Blank[]]], Times[Optional[Pattern[e, Blank[]]], Pattern[x, Blank[]]]], Optional[Pattern[q, Blank[]]]]], Pattern[x, Blank[Symbol]]] := With[{u = IntHide[(f*x)^m*(d + e*x)^q, x]}, Dist[a + b*ArcTan[c*x], u, x] - Dist[b*c, Int[SimplifyIntegrand[u/(1 + c^2*x^2), x], x], x]] /; FreeQ[{a, b, c, d, e, f, q}, x] && NeQ[q, -1] && IntegerQ[2*m] && ((IGtQ[m, 0] && IGtQ[q, 0]) || (ILtQ[m + q + 1, 0] && LtQ[m*q, 0]))
  4873. Int[Times[Plus[Optional[Pattern[a, Blank[]]], Times[ArcCot[Times[Optional[Pattern[c, Blank[]]], Pattern[x, Blank[]]]], Optional[Pattern[b, Blank[]]]]], Power[Times[Optional[Pattern[f, Blank[]]], Pattern[x, Blank[]]], Optional[Pattern[m, Blank[]]]], Power[Plus[Optional[Pattern[d, Blank[]]], Times[Optional[Pattern[e, Blank[]]], Pattern[x, Blank[]]]], Optional[Pattern[q, Blank[]]]]], Pattern[x, Blank[Symbol]]] := With[{u = IntHide[(f*x)^m*(d + e*x)^q, x]}, Dist[a + b*ArcCot[c*x], u, x] + Dist[b*c, Int[SimplifyIntegrand[u/(1 + c^2*x^2), x], x], x]] /; FreeQ[{a, b, c, d, e, f, q}, x] && NeQ[q, -1] && IntegerQ[2*m] && ((IGtQ[m, 0] && IGtQ[q, 0]) || (ILtQ[m + q + 1, 0] && LtQ[m*q, 0]))
  4874. Int[Times[Power[Plus[Optional[Pattern[a, Blank[]]], Times[ArcTan[Times[Optional[Pattern[c, Blank[]]], Pattern[x, Blank[]]]], Optional[Pattern[b, Blank[]]]]], Pattern[p, Blank[]]], Power[Times[Optional[Pattern[f, Blank[]]], Pattern[x, Blank[]]], Optional[Pattern[m, Blank[]]]], Power[Plus[Optional[Pattern[d, Blank[]]], Times[Optional[Pattern[e, Blank[]]], Pattern[x, Blank[]]]], Pattern[q, Blank[]]]], Pattern[x, Blank[Symbol]]] := With[{u = IntHide[(f*x)^m*(d + e*x)^q, x]}, Dist[(a + b*ArcTan[c*x])^p, u, x] - Dist[b*c*p, Int[ExpandIntegrand[(a + b*ArcTan[c*x])^(p - 1), u/(1 + c^2*x^2), x], x], x]] /; FreeQ[{a, b, c, d, e, f, q}, x] && IGtQ[p, 1] && EqQ[c^2*d^2 + e^2, 0] && IntegersQ[m, q] && NeQ[m, -1] && NeQ[q, -1] && ILtQ[m + q + 1, 0] && LtQ[m*q, 0]
  4875. Int[Times[Power[Plus[Optional[Pattern[a, Blank[]]], Times[ArcCot[Times[Optional[Pattern[c, Blank[]]], Pattern[x, Blank[]]]], Optional[Pattern[b, Blank[]]]]], Pattern[p, Blank[]]], Power[Times[Optional[Pattern[f, Blank[]]], Pattern[x, Blank[]]], Optional[Pattern[m, Blank[]]]], Power[Plus[Optional[Pattern[d, Blank[]]], Times[Optional[Pattern[e, Blank[]]], Pattern[x, Blank[]]]], Pattern[q, Blank[]]]], Pattern[x, Blank[Symbol]]] := With[{u = IntHide[(f*x)^m*(d + e*x)^q, x]}, Dist[(a + b*ArcCot[c*x])^p, u, x] + Dist[b*c*p, Int[ExpandIntegrand[(a + b*ArcCot[c*x])^(p - 1), u/(1 + c^2*x^2), x], x], x]] /; FreeQ[{a, b, c, d, e, f, q}, x] && IGtQ[p, 1] && EqQ[c^2*d^2 + e^2, 0] && IntegersQ[m, q] && NeQ[m, -1] && NeQ[q, -1] && ILtQ[m + q + 1, 0] && LtQ[m*q, 0]
  4876. Int[Times[Power[Plus[Optional[Pattern[a, Blank[]]], Times[ArcTan[Times[Optional[Pattern[c, Blank[]]], Pattern[x, Blank[]]]], Optional[Pattern[b, Blank[]]]]], Optional[Pattern[p, Blank[]]]], Power[Times[Optional[Pattern[f, Blank[]]], Pattern[x, Blank[]]], Optional[Pattern[m, Blank[]]]], Power[Plus[Pattern[d, Blank[]], Times[Optional[Pattern[e, Blank[]]], Pattern[x, Blank[]]]], Optional[Pattern[q, Blank[]]]]], Pattern[x, Blank[Symbol]]] := Int[ExpandIntegrand[(a + b*ArcTan[c*x])^p, (f*x)^m*(d + e*x)^q, x], x] /; FreeQ[{a, b, c, d, e, f, m}, x] && IGtQ[p, 0] && IntegerQ[q] && (GtQ[q, 0] || NeQ[a, 0] || IntegerQ[m])
  4877. Int[Times[Power[Plus[Optional[Pattern[a, Blank[]]], Times[ArcCot[Times[Optional[Pattern[c, Blank[]]], Pattern[x, Blank[]]]], Optional[Pattern[b, Blank[]]]]], Optional[Pattern[p, Blank[]]]], Power[Times[Optional[Pattern[f, Blank[]]], Pattern[x, Blank[]]], Optional[Pattern[m, Blank[]]]], Power[Plus[Pattern[d, Blank[]], Times[Optional[Pattern[e, Blank[]]], Pattern[x, Blank[]]]], Optional[Pattern[q, Blank[]]]]], Pattern[x, Blank[Symbol]]] := Int[ExpandIntegrand[(a + b*ArcCot[c*x])^p, (f*x)^m*(d + e*x)^q, x], x] /; FreeQ[{a, b, c, d, e, f, m}, x] && IGtQ[p, 0] && IntegerQ[q] && (GtQ[q, 0] || NeQ[a, 0] || IntegerQ[m])
  4878. Int[Times[Plus[Optional[Pattern[a, Blank[]]], Times[ArcTan[Times[Optional[Pattern[c, Blank[]]], Pattern[x, Blank[]]]], Optional[Pattern[b, Blank[]]]]], Power[Plus[Pattern[d, Blank[]], Times[Optional[Pattern[e, Blank[]]], Power[Pattern[x, Blank[]], 2]]], Optional[Pattern[q, Blank[]]]]], Pattern[x, Blank[Symbol]]] := -Simp[(b*(d + e*x^2)^q)/(2*c*q*(2*q + 1)), x] + (Dist[(2*d*q)/(2*q + 1), Int[(d + e*x^2)^(q - 1)*(a + b*ArcTan[c*x]), x], x] + Simp[(x*(d + e*x^2)^q*(a + b*ArcTan[c*x]))/(2*q + 1), x]) /; FreeQ[{a, b, c, d, e}, x] && EqQ[e, c^2*d] && GtQ[q, 0]
  4879. Int[Times[Plus[Optional[Pattern[a, Blank[]]], Times[ArcCot[Times[Optional[Pattern[c, Blank[]]], Pattern[x, Blank[]]]], Optional[Pattern[b, Blank[]]]]], Power[Plus[Pattern[d, Blank[]], Times[Optional[Pattern[e, Blank[]]], Power[Pattern[x, Blank[]], 2]]], Optional[Pattern[q, Blank[]]]]], Pattern[x, Blank[Symbol]]] := Simp[(b*(d + e*x^2)^q)/(2*c*q*(2*q + 1)), x] + (Dist[(2*d*q)/(2*q + 1), Int[(d + e*x^2)^(q - 1)*(a + b*ArcCot[c*x]), x], x] + Simp[(x*(d + e*x^2)^q*(a + b*ArcCot[c*x]))/(2*q + 1), x]) /; FreeQ[{a, b, c, d, e}, x] && EqQ[e, c^2*d] && GtQ[q, 0]
  4880. Int[Times[Power[Plus[Optional[Pattern[a, Blank[]]], Times[ArcTan[Times[Optional[Pattern[c, Blank[]]], Pattern[x, Blank[]]]], Optional[Pattern[b, Blank[]]]]], Pattern[p, Blank[]]], Power[Plus[Pattern[d, Blank[]], Times[Optional[Pattern[e, Blank[]]], Power[Pattern[x, Blank[]], 2]]], Optional[Pattern[q, Blank[]]]]], Pattern[x, Blank[Symbol]]] := -Simp[(b*p*(d + e*x^2)^q*(a + b*ArcTan[c*x])^(p - 1))/(2*c*q*(2*q + 1)), x] + (Dist[(2*d*q)/(2*q + 1), Int[(d + e*x^2)^(q - 1)*(a + b*ArcTan[c*x])^p, x], x] + Dist[(b^2*d*p*(p - 1))/(2*q*(2*q + 1)), Int[(d + e*x^2)^(q - 1)*(a + b*ArcTan[c*x])^(p - 2), x], x] + Simp[(x*(d + e*x^2)^q*(a + b*ArcTan[c*x])^p)/(2*q + 1), x]) /; FreeQ[{a, b, c, d, e}, x] && EqQ[e, c^2*d] && GtQ[q, 0] && GtQ[p, 1]
  4881. Int[Times[Power[Plus[Optional[Pattern[a, Blank[]]], Times[ArcCot[Times[Optional[Pattern[c, Blank[]]], Pattern[x, Blank[]]]], Optional[Pattern[b, Blank[]]]]], Pattern[p, Blank[]]], Power[Plus[Pattern[d, Blank[]], Times[Optional[Pattern[e, Blank[]]], Power[Pattern[x, Blank[]], 2]]], Optional[Pattern[q, Blank[]]]]], Pattern[x, Blank[Symbol]]] := Simp[(b*p*(d + e*x^2)^q*(a + b*ArcCot[c*x])^(p - 1))/(2*c*q*(2*q + 1)), x] + (Dist[(2*d*q)/(2*q + 1), Int[(d + e*x^2)^(q - 1)*(a + b*ArcCot[c*x])^p, x], x] + Dist[(b^2*d*p*(p - 1))/(2*q*(2*q + 1)), Int[(d + e*x^2)^(q - 1)*(a + b*ArcCot[c*x])^(p - 2), x], x] + Simp[(x*(d + e*x^2)^q*(a + b*ArcCot[c*x])^p)/(2*q + 1), x]) /; FreeQ[{a, b, c, d, e}, x] && EqQ[e, c^2*d] && GtQ[q, 0] && GtQ[p, 1]
  4882. Int[Times[Power[Plus[Optional[Pattern[a, Blank[]]], Times[ArcTan[Times[Optional[Pattern[c, Blank[]]], Pattern[x, Blank[]]]], Optional[Pattern[b, Blank[]]]]], -1], Power[Plus[Pattern[d, Blank[]], Times[Optional[Pattern[e, Blank[]]], Power[Pattern[x, Blank[]], 2]]], -1]], Pattern[x, Blank[Symbol]]] := Simp[Log[RemoveContent[a + b*ArcTan[c*x], x]]/(b*c*d), x] /; FreeQ[{a, b, c, d, e}, x] && EqQ[e, c^2*d]
  4883. Int[Times[Power[Plus[Optional[Pattern[a, Blank[]]], Times[ArcCot[Times[Optional[Pattern[c, Blank[]]], Pattern[x, Blank[]]]], Optional[Pattern[b, Blank[]]]]], -1], Power[Plus[Pattern[d, Blank[]], Times[Optional[Pattern[e, Blank[]]], Power[Pattern[x, Blank[]], 2]]], -1]], Pattern[x, Blank[Symbol]]] := -Simp[Log[RemoveContent[a + b*ArcCot[c*x], x]]/(b*c*d), x] /; FreeQ[{a, b, c, d, e}, x] && EqQ[e, c^2*d]
  4884. Int[Times[Power[Plus[Optional[Pattern[a, Blank[]]], Times[ArcTan[Times[Optional[Pattern[c, Blank[]]], Pattern[x, Blank[]]]], Optional[Pattern[b, Blank[]]]]], Optional[Pattern[p, Blank[]]]], Power[Plus[Pattern[d, Blank[]], Times[Optional[Pattern[e, Blank[]]], Power[Pattern[x, Blank[]], 2]]], -1]], Pattern[x, Blank[Symbol]]] := Simp[(a + b*ArcTan[c*x])^(p + 1)/(b*c*d*(p + 1)), x] /; FreeQ[{a, b, c, d, e, p}, x] && EqQ[e, c^2*d] && NeQ[p, -1]
  4885. Int[Times[Power[Plus[Optional[Pattern[a, Blank[]]], Times[ArcCot[Times[Optional[Pattern[c, Blank[]]], Pattern[x, Blank[]]]], Optional[Pattern[b, Blank[]]]]], Optional[Pattern[p, Blank[]]]], Power[Plus[Pattern[d, Blank[]], Times[Optional[Pattern[e, Blank[]]], Power[Pattern[x, Blank[]], 2]]], -1]], Pattern[x, Blank[Symbol]]] := -Simp[(a + b*ArcCot[c*x])^(p + 1)/(b*c*d*(p + 1)), x] /; FreeQ[{a, b, c, d, e, p}, x] && EqQ[e, c^2*d] && NeQ[p, -1]
  4886. Int[Times[Plus[Optional[Pattern[a, Blank[]]], Times[ArcTan[Times[Optional[Pattern[c, Blank[]]], Pattern[x, Blank[]]]], Optional[Pattern[b, Blank[]]]]], Power[Plus[Pattern[d, Blank[]], Times[Optional[Pattern[e, Blank[]]], Power[Pattern[x, Blank[]], 2]]], Rational[-1, 2]]], Pattern[x, Blank[Symbol]]] := Simp[(-2*I*(a + b*ArcTan[c*x])*ArcTan[Sqrt[1 + I*c*x]/Sqrt[1 - I*c*x]])/(c*Sqrt[d]), x] + (Simp[(I*b*PolyLog[2, -((I*Sqrt[1 + I*c*x])/Sqrt[1 - I*c*x])])/(c*Sqrt[d]), x] - Simp[(I*b*PolyLog[2, (I*Sqrt[1 + I*c*x])/Sqrt[1 - I*c*x]])/(c*Sqrt[d]), x]) /; FreeQ[{a, b, c, d, e}, x] && EqQ[e, c^2*d] && GtQ[d, 0]
  4887. Int[Times[Plus[Optional[Pattern[a, Blank[]]], Times[ArcCot[Times[Optional[Pattern[c, Blank[]]], Pattern[x, Blank[]]]], Optional[Pattern[b, Blank[]]]]], Power[Plus[Pattern[d, Blank[]], Times[Optional[Pattern[e, Blank[]]], Power[Pattern[x, Blank[]], 2]]], Rational[-1, 2]]], Pattern[x, Blank[Symbol]]] := Simp[(-2*I*(a + b*ArcCot[c*x])*ArcTan[Sqrt[1 + I*c*x]/Sqrt[1 - I*c*x]])/(c*Sqrt[d]), x] + (-Simp[(I*b*PolyLog[2, -((I*Sqrt[1 + I*c*x])/Sqrt[1 - I*c*x])])/(c*Sqrt[d]), x] + Simp[(I*b*PolyLog[2, (I*Sqrt[1 + I*c*x])/Sqrt[1 - I*c*x]])/(c*Sqrt[d]), x]) /; FreeQ[{a, b, c, d, e}, x] && EqQ[e, c^2*d] && GtQ[d, 0]
  4888. Int[Times[Power[Plus[Optional[Pattern[a, Blank[]]], Times[ArcTan[Times[Optional[Pattern[c, Blank[]]], Pattern[x, Blank[]]]], Optional[Pattern[b, Blank[]]]]], Optional[Pattern[p, Blank[]]]], Power[Plus[Pattern[d, Blank[]], Times[Optional[Pattern[e, Blank[]]], Power[Pattern[x, Blank[]], 2]]], Rational[-1, 2]]], Pattern[x, Blank[Symbol]]] := Dist[1/(c*Sqrt[d]), Subst[Int[(a + b*x)^p*Sec[x], x], x, ArcTan[c*x]], x] /; FreeQ[{a, b, c, d, e}, x] && EqQ[e, c^2*d] && IGtQ[p, 0] && GtQ[d, 0]
  4889. Int[Times[Power[Plus[Optional[Pattern[a, Blank[]]], Times[ArcCot[Times[Optional[Pattern[c, Blank[]]], Pattern[x, Blank[]]]], Optional[Pattern[b, Blank[]]]]], Optional[Pattern[p, Blank[]]]], Power[Plus[Pattern[d, Blank[]], Times[Optional[Pattern[e, Blank[]]], Power[Pattern[x, Blank[]], 2]]], Rational[-1, 2]]], Pattern[x, Blank[Symbol]]] := -Dist[(x*Sqrt[1 + 1/(c^2*x^2)])/Sqrt[d + e*x^2], Subst[Int[(a + b*x)^p*Csc[x], x], x, ArcCot[c*x]], x] /; FreeQ[{a, b, c, d, e}, x] && EqQ[e, c^2*d] && IGtQ[p, 0] && GtQ[d, 0]
  4890. Int[Times[Power[Plus[Optional[Pattern[a, Blank[]]], Times[ArcTan[Times[Optional[Pattern[c, Blank[]]], Pattern[x, Blank[]]]], Optional[Pattern[b, Blank[]]]]], Optional[Pattern[p, Blank[]]]], Power[Plus[Pattern[d, Blank[]], Times[Optional[Pattern[e, Blank[]]], Power[Pattern[x, Blank[]], 2]]], Rational[-1, 2]]], Pattern[x, Blank[Symbol]]] := Dist[Sqrt[1 + c^2*x^2]/Sqrt[d + e*x^2], Int[(a + b*ArcTan[c*x])^p/Sqrt[1 + c^2*x^2], x], x] /; FreeQ[{a, b, c, d, e}, x] && EqQ[e, c^2*d] && IGtQ[p, 0] && !GtQ[d, 0]
  4891. Int[Times[Power[Plus[Optional[Pattern[a, Blank[]]], Times[ArcCot[Times[Optional[Pattern[c, Blank[]]], Pattern[x, Blank[]]]], Optional[Pattern[b, Blank[]]]]], Optional[Pattern[p, Blank[]]]], Power[Plus[Pattern[d, Blank[]], Times[Optional[Pattern[e, Blank[]]], Power[Pattern[x, Blank[]], 2]]], Rational[-1, 2]]], Pattern[x, Blank[Symbol]]] := Dist[Sqrt[1 + c^2*x^2]/Sqrt[d + e*x^2], Int[(a + b*ArcCot[c*x])^p/Sqrt[1 + c^2*x^2], x], x] /; FreeQ[{a, b, c, d, e}, x] && EqQ[e, c^2*d] && IGtQ[p, 0] && !GtQ[d, 0]
  4892. Int[Times[Power[Plus[Optional[Pattern[a, Blank[]]], Times[ArcTan[Times[Optional[Pattern[c, Blank[]]], Pattern[x, Blank[]]]], Optional[Pattern[b, Blank[]]]]], Optional[Pattern[p, Blank[]]]], Power[Plus[Pattern[d, Blank[]], Times[Optional[Pattern[e, Blank[]]], Power[Pattern[x, Blank[]], 2]]], -2]], Pattern[x, Blank[Symbol]]] := Simp[(x*(a + b*ArcTan[c*x])^p)/(2*d*(d + e*x^2)), x] + (-Dist[(b*c*p)/2, Int[(x*(a + b*ArcTan[c*x])^(p - 1))/(d + e*x^2)^2, x], x] + Simp[(a + b*ArcTan[c*x])^(p + 1)/(2*b*c*d^2*(p + 1)), x]) /; FreeQ[{a, b, c, d, e}, x] && EqQ[e, c^2*d] && GtQ[p, 0]
  4893. Int[Times[Power[Plus[Optional[Pattern[a, Blank[]]], Times[ArcCot[Times[Optional[Pattern[c, Blank[]]], Pattern[x, Blank[]]]], Optional[Pattern[b, Blank[]]]]], Optional[Pattern[p, Blank[]]]], Power[Plus[Pattern[d, Blank[]], Times[Optional[Pattern[e, Blank[]]], Power[Pattern[x, Blank[]], 2]]], -2]], Pattern[x, Blank[Symbol]]] := Simp[(x*(a + b*ArcCot[c*x])^p)/(2*d*(d + e*x^2)), x] + (Dist[(b*c*p)/2, Int[(x*(a + b*ArcCot[c*x])^(p - 1))/(d + e*x^2)^2, x], x] - Simp[(a + b*ArcCot[c*x])^(p + 1)/(2*b*c*d^2*(p + 1)), x]) /; FreeQ[{a, b, c, d, e}, x] && EqQ[e, c^2*d] && GtQ[p, 0]
  4894. Int[Times[Plus[Optional[Pattern[a, Blank[]]], Times[ArcTan[Times[Optional[Pattern[c, Blank[]]], Pattern[x, Blank[]]]], Optional[Pattern[b, Blank[]]]]], Power[Plus[Pattern[d, Blank[]], Times[Optional[Pattern[e, Blank[]]], Power[Pattern[x, Blank[]], 2]]], Rational[-3, 2]]], Pattern[x, Blank[Symbol]]] := Simp[b/(c*d*Sqrt[d + e*x^2]), x] + Simp[(x*(a + b*ArcTan[c*x]))/(d*Sqrt[d + e*x^2]), x] /; FreeQ[{a, b, c, d, e}, x] && EqQ[e, c^2*d]
  4895. Int[Times[Plus[Optional[Pattern[a, Blank[]]], Times[ArcCot[Times[Optional[Pattern[c, Blank[]]], Pattern[x, Blank[]]]], Optional[Pattern[b, Blank[]]]]], Power[Plus[Pattern[d, Blank[]], Times[Optional[Pattern[e, Blank[]]], Power[Pattern[x, Blank[]], 2]]], Rational[-3, 2]]], Pattern[x, Blank[Symbol]]] := -Simp[b/(c*d*Sqrt[d + e*x^2]), x] + Simp[(x*(a + b*ArcCot[c*x]))/(d*Sqrt[d + e*x^2]), x] /; FreeQ[{a, b, c, d, e}, x] && EqQ[e, c^2*d]
  4896. Int[Times[Plus[Optional[Pattern[a, Blank[]]], Times[ArcTan[Times[Optional[Pattern[c, Blank[]]], Pattern[x, Blank[]]]], Optional[Pattern[b, Blank[]]]]], Power[Plus[Pattern[d, Blank[]], Times[Optional[Pattern[e, Blank[]]], Power[Pattern[x, Blank[]], 2]]], Pattern[q, Blank[]]]], Pattern[x, Blank[Symbol]]] := Simp[(b*(d + e*x^2)^(q + 1))/(4*c*d*(q + 1)^2), x] + (Dist[(2*q + 3)/(2*d*(q + 1)), Int[(d + e*x^2)^(q + 1)*(a + b*ArcTan[c*x]), x], x] - Simp[(x*(d + e*x^2)^(q + 1)*(a + b*ArcTan[c*x]))/(2*d*(q + 1)), x]) /; FreeQ[{a, b, c, d, e}, x] && EqQ[e, c^2*d] && LtQ[q, -1] && NeQ[q, -3/2]
  4897. Int[Times[Plus[Optional[Pattern[a, Blank[]]], Times[ArcCot[Times[Optional[Pattern[c, Blank[]]], Pattern[x, Blank[]]]], Optional[Pattern[b, Blank[]]]]], Power[Plus[Pattern[d, Blank[]], Times[Optional[Pattern[e, Blank[]]], Power[Pattern[x, Blank[]], 2]]], Pattern[q, Blank[]]]], Pattern[x, Blank[Symbol]]] := -Simp[(b*(d + e*x^2)^(q + 1))/(4*c*d*(q + 1)^2), x] + (Dist[(2*q + 3)/(2*d*(q + 1)), Int[(d + e*x^2)^(q + 1)*(a + b*ArcCot[c*x]), x], x] - Simp[(x*(d + e*x^2)^(q + 1)*(a + b*ArcCot[c*x]))/(2*d*(q + 1)), x]) /; FreeQ[{a, b, c, d, e}, x] && EqQ[e, c^2*d] && LtQ[q, -1] && NeQ[q, -3/2]
  4898. Int[Times[Power[Plus[Optional[Pattern[a, Blank[]]], Times[ArcTan[Times[Optional[Pattern[c, Blank[]]], Pattern[x, Blank[]]]], Optional[Pattern[b, Blank[]]]]], Pattern[p, Blank[]]], Power[Plus[Pattern[d, Blank[]], Times[Optional[Pattern[e, Blank[]]], Power[Pattern[x, Blank[]], 2]]], Rational[-3, 2]]], Pattern[x, Blank[Symbol]]] := Simp[(b*p*(a + b*ArcTan[c*x])^(p - 1))/(c*d*Sqrt[d + e*x^2]), x] + (-Dist[b^2*p*(p - 1), Int[(a + b*ArcTan[c*x])^(p - 2)/(d + e*x^2)^(3/2), x], x] + Simp[(x*(a + b*ArcTan[c*x])^p)/(d*Sqrt[d + e*x^2]), x]) /; FreeQ[{a, b, c, d, e}, x] && EqQ[e, c^2*d] && GtQ[p, 1]
  4899. Int[Times[Power[Plus[Optional[Pattern[a, Blank[]]], Times[ArcCot[Times[Optional[Pattern[c, Blank[]]], Pattern[x, Blank[]]]], Optional[Pattern[b, Blank[]]]]], Pattern[p, Blank[]]], Power[Plus[Pattern[d, Blank[]], Times[Optional[Pattern[e, Blank[]]], Power[Pattern[x, Blank[]], 2]]], Rational[-3, 2]]], Pattern[x, Blank[Symbol]]] := -Simp[(b*p*(a + b*ArcCot[c*x])^(p - 1))/(c*d*Sqrt[d + e*x^2]), x] + (-Dist[b^2*p*(p - 1), Int[(a + b*ArcCot[c*x])^(p - 2)/(d + e*x^2)^(3/2), x], x] + Simp[(x*(a + b*ArcCot[c*x])^p)/(d*Sqrt[d + e*x^2]), x]) /; FreeQ[{a, b, c, d, e}, x] && EqQ[e, c^2*d] && GtQ[p, 1]
  4900. Int[Times[Power[Plus[Optional[Pattern[a, Blank[]]], Times[ArcTan[Times[Optional[Pattern[c, Blank[]]], Pattern[x, Blank[]]]], Optional[Pattern[b, Blank[]]]]], Pattern[p, Blank[]]], Power[Plus[Pattern[d, Blank[]], Times[Optional[Pattern[e, Blank[]]], Power[Pattern[x, Blank[]], 2]]], Pattern[q, Blank[]]]], Pattern[x, Blank[Symbol]]] := Simp[(b*p*(d + e*x^2)^(q + 1)*(a + b*ArcTan[c*x])^(p - 1))/(4*c*d*(q + 1)^2), x] + (Dist[(2*q + 3)/(2*d*(q + 1)), Int[(d + e*x^2)^(q + 1)*(a + b*ArcTan[c*x])^p, x], x] - Dist[(b^2*p*(p - 1))/(4*(q + 1)^2), Int[(d + e*x^2)^q*(a + b*ArcTan[c*x])^(p - 2), x], x] - Simp[(x*(d + e*x^2)^(q + 1)*(a + b*ArcTan[c*x])^p)/(2*d*(q + 1)), x]) /; FreeQ[{a, b, c, d, e}, x] && EqQ[e, c^2*d] && LtQ[q, -1] && GtQ[p, 1] && NeQ[q, -3/2]
  4901. Int[Times[Power[Plus[Optional[Pattern[a, Blank[]]], Times[ArcCot[Times[Optional[Pattern[c, Blank[]]], Pattern[x, Blank[]]]], Optional[Pattern[b, Blank[]]]]], Pattern[p, Blank[]]], Power[Plus[Pattern[d, Blank[]], Times[Optional[Pattern[e, Blank[]]], Power[Pattern[x, Blank[]], 2]]], Pattern[q, Blank[]]]], Pattern[x, Blank[Symbol]]] := -Simp[(b*p*(d + e*x^2)^(q + 1)*(a + b*ArcCot[c*x])^(p - 1))/(4*c*d*(q + 1)^2), x] + (Dist[(2*q + 3)/(2*d*(q + 1)), Int[(d + e*x^2)^(q + 1)*(a + b*ArcCot[c*x])^p, x], x] - Dist[(b^2*p*(p - 1))/(4*(q + 1)^2), Int[(d + e*x^2)^q*(a + b*ArcCot[c*x])^(p - 2), x], x] - Simp[(x*(d + e*x^2)^(q + 1)*(a + b*ArcCot[c*x])^p)/(2*d*(q + 1)), x]) /; FreeQ[{a, b, c, d, e}, x] && EqQ[e, c^2*d] && LtQ[q, -1] && GtQ[p, 1] && NeQ[q, -3/2]
  4902. Int[Times[Power[Plus[Optional[Pattern[a, Blank[]]], Times[ArcTan[Times[Optional[Pattern[c, Blank[]]], Pattern[x, Blank[]]]], Optional[Pattern[b, Blank[]]]]], Pattern[p, Blank[]]], Power[Plus[Pattern[d, Blank[]], Times[Optional[Pattern[e, Blank[]]], Power[Pattern[x, Blank[]], 2]]], Pattern[q, Blank[]]]], Pattern[x, Blank[Symbol]]] := Simp[((d + e*x^2)^(q + 1)*(a + b*ArcTan[c*x])^(p + 1))/(b*c*d*(p + 1)), x] - Dist[(2*c*(q + 1))/(b*(p + 1)), Int[x*(d + e*x^2)^q*(a + b*ArcTan[c*x])^(p + 1), x], x] /; FreeQ[{a, b, c, d, e}, x] && EqQ[e, c^2*d] && LtQ[q, -1] && LtQ[p, -1]
  4903. Int[Times[Power[Plus[Optional[Pattern[a, Blank[]]], Times[ArcCot[Times[Optional[Pattern[c, Blank[]]], Pattern[x, Blank[]]]], Optional[Pattern[b, Blank[]]]]], Pattern[p, Blank[]]], Power[Plus[Pattern[d, Blank[]], Times[Optional[Pattern[e, Blank[]]], Power[Pattern[x, Blank[]], 2]]], Pattern[q, Blank[]]]], Pattern[x, Blank[Symbol]]] := -Simp[((d + e*x^2)^(q + 1)*(a + b*ArcCot[c*x])^(p + 1))/(b*c*d*(p + 1)), x] + Dist[(2*c*(q + 1))/(b*(p + 1)), Int[x*(d + e*x^2)^q*(a + b*ArcCot[c*x])^(p + 1), x], x] /; FreeQ[{a, b, c, d, e}, x] && EqQ[e, c^2*d] && LtQ[q, -1] && LtQ[p, -1]
  4904. Int[Times[Power[Plus[Optional[Pattern[a, Blank[]]], Times[ArcTan[Times[Optional[Pattern[c, Blank[]]], Pattern[x, Blank[]]]], Optional[Pattern[b, Blank[]]]]], Optional[Pattern[p, Blank[]]]], Power[Plus[Pattern[d, Blank[]], Times[Optional[Pattern[e, Blank[]]], Power[Pattern[x, Blank[]], 2]]], Pattern[q, Blank[]]]], Pattern[x, Blank[Symbol]]] := Dist[d^q/c, Subst[Int[(a + b*x)^p/Cos[x]^(2*(q + 1)), x], x, ArcTan[c*x]], x] /; FreeQ[{a, b, c, d, e, p}, x] && EqQ[e, c^2*d] && ILtQ[2*(q + 1), 0] && (IntegerQ[q] || GtQ[d, 0])
  4905. Int[Times[Power[Plus[Optional[Pattern[a, Blank[]]], Times[ArcTan[Times[Optional[Pattern[c, Blank[]]], Pattern[x, Blank[]]]], Optional[Pattern[b, Blank[]]]]], Optional[Pattern[p, Blank[]]]], Power[Plus[Pattern[d, Blank[]], Times[Optional[Pattern[e, Blank[]]], Power[Pattern[x, Blank[]], 2]]], Pattern[q, Blank[]]]], Pattern[x, Blank[Symbol]]] := Dist[(d^(q + 1/2)*Sqrt[1 + c^2*x^2])/Sqrt[d + e*x^2], Int[(1 + c^2*x^2)^q*(a + b*ArcTan[c*x])^p, x], x] /; FreeQ[{a, b, c, d, e, p}, x] && EqQ[e, c^2*d] && ILtQ[2*(q + 1), 0] && !(IntegerQ[q] || GtQ[d, 0])
  4906. Int[Times[Power[Plus[Optional[Pattern[a, Blank[]]], Times[ArcCot[Times[Optional[Pattern[c, Blank[]]], Pattern[x, Blank[]]]], Optional[Pattern[b, Blank[]]]]], Optional[Pattern[p, Blank[]]]], Power[Plus[Pattern[d, Blank[]], Times[Optional[Pattern[e, Blank[]]], Power[Pattern[x, Blank[]], 2]]], Pattern[q, Blank[]]]], Pattern[x, Blank[Symbol]]] := -Dist[d^q/c, Subst[Int[(a + b*x)^p/Sin[x]^(2*(q + 1)), x], x, ArcCot[c*x]], x] /; FreeQ[{a, b, c, d, e, p}, x] && EqQ[e, c^2*d] && ILtQ[2*(q + 1), 0] && IntegerQ[q]
  4907. Int[Times[Power[Plus[Optional[Pattern[a, Blank[]]], Times[ArcCot[Times[Optional[Pattern[c, Blank[]]], Pattern[x, Blank[]]]], Optional[Pattern[b, Blank[]]]]], Optional[Pattern[p, Blank[]]]], Power[Plus[Pattern[d, Blank[]], Times[Optional[Pattern[e, Blank[]]], Power[Pattern[x, Blank[]], 2]]], Pattern[q, Blank[]]]], Pattern[x, Blank[Symbol]]] := -Dist[(d^(q + 1/2)*x*Sqrt[(1 + c^2*x^2)/(c^2*x^2)])/Sqrt[d + e*x^2], Subst[Int[(a + b*x)^p/Sin[x]^(2*(q + 1)), x], x, ArcCot[c*x]], x] /; FreeQ[{a, b, c, d, e, p}, x] && EqQ[e, c^2*d] && ILtQ[2*(q + 1), 0] && !IntegerQ[q]
  4908. Int[Times[ArcTan[Times[Optional[Pattern[c, Blank[]]], Pattern[x, Blank[]]]], Power[Plus[Optional[Pattern[d, Blank[]]], Times[Optional[Pattern[e, Blank[]]], Power[Pattern[x, Blank[]], 2]]], -1]], Pattern[x, Blank[Symbol]]] := Dist[I/2, Int[Log[1 - I*c*x]/(d + e*x^2), x], x] - Dist[I/2, Int[Log[1 + I*c*x]/(d + e*x^2), x], x] /; FreeQ[{c, d, e}, x]
  4909. Int[Times[ArcCot[Times[Optional[Pattern[c, Blank[]]], Pattern[x, Blank[]]]], Power[Plus[Optional[Pattern[d, Blank[]]], Times[Optional[Pattern[e, Blank[]]], Power[Pattern[x, Blank[]], 2]]], -1]], Pattern[x, Blank[Symbol]]] := Dist[I/2, Int[Log[1 - I/(c*x)]/(d + e*x^2), x], x] - Dist[I/2, Int[Log[1 + I/(c*x)]/(d + e*x^2), x], x] /; FreeQ[{c, d, e}, x]
  4910. Int[Times[Plus[Times[ArcTan[Times[Optional[Pattern[c, Blank[]]], Pattern[x, Blank[]]]], Optional[Pattern[b, Blank[]]]], Pattern[a, Blank[]]], Power[Plus[Optional[Pattern[d, Blank[]]], Times[Optional[Pattern[e, Blank[]]], Power[Pattern[x, Blank[]], 2]]], -1]], Pattern[x, Blank[Symbol]]] := Dist[a, Int[1/(d + e*x^2), x], x] + Dist[b, Int[ArcTan[c*x]/(d + e*x^2), x], x] /; FreeQ[{a, b, c, d, e}, x]
  4911. Int[Times[Plus[Times[ArcCot[Times[Optional[Pattern[c, Blank[]]], Pattern[x, Blank[]]]], Optional[Pattern[b, Blank[]]]], Pattern[a, Blank[]]], Power[Plus[Optional[Pattern[d, Blank[]]], Times[Optional[Pattern[e, Blank[]]], Power[Pattern[x, Blank[]], 2]]], -1]], Pattern[x, Blank[Symbol]]] := Dist[a, Int[1/(d + e*x^2), x], x] + Dist[b, Int[ArcCot[c*x]/(d + e*x^2), x], x] /; FreeQ[{a, b, c, d, e}, x]
  4912. Int[Times[Plus[Optional[Pattern[a, Blank[]]], Times[ArcTan[Times[Optional[Pattern[c, Blank[]]], Pattern[x, Blank[]]]], Optional[Pattern[b, Blank[]]]]], Power[Plus[Optional[Pattern[d, Blank[]]], Times[Optional[Pattern[e, Blank[]]], Power[Pattern[x, Blank[]], 2]]], Optional[Pattern[q, Blank[]]]]], Pattern[x, Blank[Symbol]]] := With[{u = IntHide[(d + e*x^2)^q, x]}, Dist[a + b*ArcTan[c*x], u, x] - Dist[b*c, Int[u/(1 + c^2*x^2), x], x]] /; FreeQ[{a, b, c, d, e}, x] && (IntegerQ[q] || ILtQ[q + 1/2, 0])
  4913. Int[Times[Plus[Optional[Pattern[a, Blank[]]], Times[ArcCot[Times[Optional[Pattern[c, Blank[]]], Pattern[x, Blank[]]]], Optional[Pattern[b, Blank[]]]]], Power[Plus[Optional[Pattern[d, Blank[]]], Times[Optional[Pattern[e, Blank[]]], Power[Pattern[x, Blank[]], 2]]], Optional[Pattern[q, Blank[]]]]], Pattern[x, Blank[Symbol]]] := With[{u = IntHide[(d + e*x^2)^q, x]}, Dist[a + b*ArcCot[c*x], u, x] + Dist[b*c, Int[u/(1 + c^2*x^2), x], x]] /; FreeQ[{a, b, c, d, e}, x] && (IntegerQ[q] || ILtQ[q + 1/2, 0])
  4914. Int[Times[Power[Plus[Optional[Pattern[a, Blank[]]], Times[ArcTan[Times[Optional[Pattern[c, Blank[]]], Pattern[x, Blank[]]]], Optional[Pattern[b, Blank[]]]]], Optional[Pattern[p, Blank[]]]], Power[Plus[Pattern[d, Blank[]], Times[Optional[Pattern[e, Blank[]]], Power[Pattern[x, Blank[]], 2]]], Optional[Pattern[q, Blank[]]]]], Pattern[x, Blank[Symbol]]] := Int[ExpandIntegrand[(a + b*ArcTan[c*x])^p, (d + e*x^2)^q, x], x] /; FreeQ[{a, b, c, d, e}, x] && IntegerQ[q] && IGtQ[p, 0]
  4915. Int[Times[Power[Plus[Optional[Pattern[a, Blank[]]], Times[ArcCot[Times[Optional[Pattern[c, Blank[]]], Pattern[x, Blank[]]]], Optional[Pattern[b, Blank[]]]]], Optional[Pattern[p, Blank[]]]], Power[Plus[Pattern[d, Blank[]], Times[Optional[Pattern[e, Blank[]]], Power[Pattern[x, Blank[]], 2]]], Optional[Pattern[q, Blank[]]]]], Pattern[x, Blank[Symbol]]] := Int[ExpandIntegrand[(a + b*ArcCot[c*x])^p, (d + e*x^2)^q, x], x] /; FreeQ[{a, b, c, d, e}, x] && IntegerQ[q] && IGtQ[p, 0]
  4916. Int[Times[Power[Plus[Optional[Pattern[a, Blank[]]], Times[ArcTan[Times[Optional[Pattern[c, Blank[]]], Pattern[x, Blank[]]]], Optional[Pattern[b, Blank[]]]]], Optional[Pattern[p, Blank[]]]], Power[Times[Optional[Pattern[f, Blank[]]], Pattern[x, Blank[]]], Pattern[m, Blank[]]], Power[Plus[Pattern[d, Blank[]], Times[Optional[Pattern[e, Blank[]]], Power[Pattern[x, Blank[]], 2]]], -1]], Pattern[x, Blank[Symbol]]] := Dist[f^2/e, Int[(f*x)^(m - 2)*(a + b*ArcTan[c*x])^p, x], x] - Dist[(d*f^2)/e, Int[((f*x)^(m - 2)*(a + b*ArcTan[c*x])^p)/(d + e*x^2), x], x] /; FreeQ[{a, b, c, d, e, f}, x] && GtQ[p, 0] && GtQ[m, 1]
  4917. Int[Times[Power[Plus[Optional[Pattern[a, Blank[]]], Times[ArcCot[Times[Optional[Pattern[c, Blank[]]], Pattern[x, Blank[]]]], Optional[Pattern[b, Blank[]]]]], Optional[Pattern[p, Blank[]]]], Power[Times[Optional[Pattern[f, Blank[]]], Pattern[x, Blank[]]], Pattern[m, Blank[]]], Power[Plus[Pattern[d, Blank[]], Times[Optional[Pattern[e, Blank[]]], Power[Pattern[x, Blank[]], 2]]], -1]], Pattern[x, Blank[Symbol]]] := Dist[f^2/e, Int[(f*x)^(m - 2)*(a + b*ArcCot[c*x])^p, x], x] - Dist[(d*f^2)/e, Int[((f*x)^(m - 2)*(a + b*ArcCot[c*x])^p)/(d + e*x^2), x], x] /; FreeQ[{a, b, c, d, e, f}, x] && GtQ[p, 0] && GtQ[m, 1]
  4918. Int[Times[Power[Plus[Optional[Pattern[a, Blank[]]], Times[ArcTan[Times[Optional[Pattern[c, Blank[]]], Pattern[x, Blank[]]]], Optional[Pattern[b, Blank[]]]]], Optional[Pattern[p, Blank[]]]], Power[Times[Optional[Pattern[f, Blank[]]], Pattern[x, Blank[]]], Pattern[m, Blank[]]], Power[Plus[Pattern[d, Blank[]], Times[Optional[Pattern[e, Blank[]]], Power[Pattern[x, Blank[]], 2]]], -1]], Pattern[x, Blank[Symbol]]] := Dist[1/d, Int[(f*x)^m*(a + b*ArcTan[c*x])^p, x], x] - Dist[e/(d*f^2), Int[((f*x)^(m + 2)*(a + b*ArcTan[c*x])^p)/(d + e*x^2), x], x] /; FreeQ[{a, b, c, d, e, f}, x] && GtQ[p, 0] && LtQ[m, -1]
  4919. Int[Times[Power[Plus[Optional[Pattern[a, Blank[]]], Times[ArcCot[Times[Optional[Pattern[c, Blank[]]], Pattern[x, Blank[]]]], Optional[Pattern[b, Blank[]]]]], Optional[Pattern[p, Blank[]]]], Power[Times[Optional[Pattern[f, Blank[]]], Pattern[x, Blank[]]], Pattern[m, Blank[]]], Power[Plus[Pattern[d, Blank[]], Times[Optional[Pattern[e, Blank[]]], Power[Pattern[x, Blank[]], 2]]], -1]], Pattern[x, Blank[Symbol]]] := Dist[1/d, Int[(f*x)^m*(a + b*ArcCot[c*x])^p, x], x] - Dist[e/(d*f^2), Int[((f*x)^(m + 2)*(a + b*ArcCot[c*x])^p)/(d + e*x^2), x], x] /; FreeQ[{a, b, c, d, e, f}, x] && GtQ[p, 0] && LtQ[m, -1]
  4920. Int[Times[Power[Plus[Optional[Pattern[a, Blank[]]], Times[ArcTan[Times[Optional[Pattern[c, Blank[]]], Pattern[x, Blank[]]]], Optional[Pattern[b, Blank[]]]]], Optional[Pattern[p, Blank[]]]], Pattern[x, Blank[]], Power[Plus[Pattern[d, Blank[]], Times[Optional[Pattern[e, Blank[]]], Power[Pattern[x, Blank[]], 2]]], -1]], Pattern[x, Blank[Symbol]]] := -Simp[(I*(a + b*ArcTan[c*x])^(p + 1))/(b*e*(p + 1)), x] - Dist[1/(c*d), Int[(a + b*ArcTan[c*x])^p/(I - c*x), x], x] /; FreeQ[{a, b, c, d, e}, x] && EqQ[e, c^2*d] && IGtQ[p, 0]
  4921. Int[Times[Power[Plus[Optional[Pattern[a, Blank[]]], Times[ArcCot[Times[Optional[Pattern[c, Blank[]]], Pattern[x, Blank[]]]], Optional[Pattern[b, Blank[]]]]], Optional[Pattern[p, Blank[]]]], Pattern[x, Blank[]], Power[Plus[Pattern[d, Blank[]], Times[Optional[Pattern[e, Blank[]]], Power[Pattern[x, Blank[]], 2]]], -1]], Pattern[x, Blank[Symbol]]] := Simp[(I*(a + b*ArcCot[c*x])^(p + 1))/(b*e*(p + 1)), x] - Dist[1/(c*d), Int[(a + b*ArcCot[c*x])^p/(I - c*x), x], x] /; FreeQ[{a, b, c, d, e}, x] && EqQ[e, c^2*d] && IGtQ[p, 0]
  4922. Int[Times[Power[Plus[Optional[Pattern[a, Blank[]]], Times[ArcTan[Times[Optional[Pattern[c, Blank[]]], Pattern[x, Blank[]]]], Optional[Pattern[b, Blank[]]]]], Pattern[p, Blank[]]], Pattern[x, Blank[]], Power[Plus[Pattern[d, Blank[]], Times[Optional[Pattern[e, Blank[]]], Power[Pattern[x, Blank[]], 2]]], -1]], Pattern[x, Blank[Symbol]]] := Simp[(x*(a + b*ArcTan[c*x])^(p + 1))/(b*c*d*(p + 1)), x] - Dist[1/(b*c*d*(p + 1)), Int[(a + b*ArcTan[c*x])^(p + 1), x], x] /; FreeQ[{a, b, c, d, e}, x] && EqQ[e, c^2*d] && !IGtQ[p, 0] && NeQ[p, -1]
  4923. Int[Times[Power[Plus[Optional[Pattern[a, Blank[]]], Times[ArcCot[Times[Optional[Pattern[c, Blank[]]], Pattern[x, Blank[]]]], Optional[Pattern[b, Blank[]]]]], Pattern[p, Blank[]]], Pattern[x, Blank[]], Power[Plus[Pattern[d, Blank[]], Times[Optional[Pattern[e, Blank[]]], Power[Pattern[x, Blank[]], 2]]], -1]], Pattern[x, Blank[Symbol]]] := -Simp[(x*(a + b*ArcCot[c*x])^(p + 1))/(b*c*d*(p + 1)), x] + Dist[1/(b*c*d*(p + 1)), Int[(a + b*ArcCot[c*x])^(p + 1), x], x] /; FreeQ[{a, b, c, d, e}, x] && EqQ[e, c^2*d] && !IGtQ[p, 0] && NeQ[p, -1]
  4924. Int[Times[Power[Plus[Optional[Pattern[a, Blank[]]], Times[ArcTan[Times[Optional[Pattern[c, Blank[]]], Pattern[x, Blank[]]]], Optional[Pattern[b, Blank[]]]]], Optional[Pattern[p, Blank[]]]], Power[Pattern[x, Blank[]], -1], Power[Plus[Pattern[d, Blank[]], Times[Optional[Pattern[e, Blank[]]], Power[Pattern[x, Blank[]], 2]]], -1]], Pattern[x, Blank[Symbol]]] := -Simp[(I*(a + b*ArcTan[c*x])^(p + 1))/(b*d*(p + 1)), x] + Dist[I/d, Int[(a + b*ArcTan[c*x])^p/(x*(I + c*x)), x], x] /; FreeQ[{a, b, c, d, e}, x] && EqQ[e, c^2*d] && GtQ[p, 0]
  4925. Int[Times[Power[Plus[Optional[Pattern[a, Blank[]]], Times[ArcCot[Times[Optional[Pattern[c, Blank[]]], Pattern[x, Blank[]]]], Optional[Pattern[b, Blank[]]]]], Optional[Pattern[p, Blank[]]]], Power[Pattern[x, Blank[]], -1], Power[Plus[Pattern[d, Blank[]], Times[Optional[Pattern[e, Blank[]]], Power[Pattern[x, Blank[]], 2]]], -1]], Pattern[x, Blank[Symbol]]] := Simp[(I*(a + b*ArcCot[c*x])^(p + 1))/(b*d*(p + 1)), x] + Dist[I/d, Int[(a + b*ArcCot[c*x])^p/(x*(I + c*x)), x], x] /; FreeQ[{a, b, c, d, e}, x] && EqQ[e, c^2*d] && GtQ[p, 0]
  4926. Int[Times[Power[Plus[Optional[Pattern[a, Blank[]]], Times[ArcTan[Times[Optional[Pattern[c, Blank[]]], Pattern[x, Blank[]]]], Optional[Pattern[b, Blank[]]]]], Pattern[p, Blank[]]], Power[Times[Optional[Pattern[f, Blank[]]], Pattern[x, Blank[]]], Pattern[m, Blank[]]], Power[Plus[Pattern[d, Blank[]], Times[Optional[Pattern[e, Blank[]]], Power[Pattern[x, Blank[]], 2]]], -1]], Pattern[x, Blank[Symbol]]] := Simp[((f*x)^m*(a + b*ArcTan[c*x])^(p + 1))/(b*c*d*(p + 1)), x] - Dist[(f*m)/(b*c*d*(p + 1)), Int[(f*x)^(m - 1)*(a + b*ArcTan[c*x])^(p + 1), x], x] /; FreeQ[{a, b, c, d, e, f, m}, x] && EqQ[e, c^2*d] && LtQ[p, -1]
  4927. Int[Times[Power[Plus[Optional[Pattern[a, Blank[]]], Times[ArcCot[Times[Optional[Pattern[c, Blank[]]], Pattern[x, Blank[]]]], Optional[Pattern[b, Blank[]]]]], Pattern[p, Blank[]]], Power[Times[Optional[Pattern[f, Blank[]]], Pattern[x, Blank[]]], Pattern[m, Blank[]]], Power[Plus[Pattern[d, Blank[]], Times[Optional[Pattern[e, Blank[]]], Power[Pattern[x, Blank[]], 2]]], -1]], Pattern[x, Blank[Symbol]]] := -Simp[((f*x)^m*(a + b*ArcCot[c*x])^(p + 1))/(b*c*d*(p + 1)), x] + Dist[(f*m)/(b*c*d*(p + 1)), Int[(f*x)^(m - 1)*(a + b*ArcCot[c*x])^(p + 1), x], x] /; FreeQ[{a, b, c, d, e, f, m}, x] && EqQ[e, c^2*d] && LtQ[p, -1]
  4928. Int[Times[Plus[Optional[Pattern[a, Blank[]]], Times[ArcTan[Times[Optional[Pattern[c, Blank[]]], Pattern[x, Blank[]]]], Optional[Pattern[b, Blank[]]]]], Power[Pattern[x, Blank[]], Optional[Pattern[m, Blank[]]]], Power[Plus[Pattern[d, Blank[]], Times[Optional[Pattern[e, Blank[]]], Power[Pattern[x, Blank[]], 2]]], -1]], Pattern[x, Blank[Symbol]]] := Int[ExpandIntegrand[a + b*ArcTan[c*x], x^m/(d + e*x^2), x], x] /; FreeQ[{a, b, c, d, e}, x] && IntegerQ[m] && !(EqQ[m, 1] && NeQ[a, 0])
  4929. Int[Times[Plus[Optional[Pattern[a, Blank[]]], Times[ArcCot[Times[Optional[Pattern[c, Blank[]]], Pattern[x, Blank[]]]], Optional[Pattern[b, Blank[]]]]], Power[Pattern[x, Blank[]], Optional[Pattern[m, Blank[]]]], Power[Plus[Pattern[d, Blank[]], Times[Optional[Pattern[e, Blank[]]], Power[Pattern[x, Blank[]], 2]]], -1]], Pattern[x, Blank[Symbol]]] := Int[ExpandIntegrand[a + b*ArcCot[c*x], x^m/(d + e*x^2), x], x] /; FreeQ[{a, b, c, d, e}, x] && IntegerQ[m] && !(EqQ[m, 1] && NeQ[a, 0])
  4930. Int[Times[Power[Plus[Optional[Pattern[a, Blank[]]], Times[ArcTan[Times[Optional[Pattern[c, Blank[]]], Pattern[x, Blank[]]]], Optional[Pattern[b, Blank[]]]]], Optional[Pattern[p, Blank[]]]], Pattern[x, Blank[]], Power[Plus[Pattern[d, Blank[]], Times[Optional[Pattern[e, Blank[]]], Power[Pattern[x, Blank[]], 2]]], Optional[Pattern[q, Blank[]]]]], Pattern[x, Blank[Symbol]]] := Simp[((d + e*x^2)^(q + 1)*(a + b*ArcTan[c*x])^p)/(2*e*(q + 1)), x] - Dist[(b*p)/(2*c*(q + 1)), Int[(d + e*x^2)^q*(a + b*ArcTan[c*x])^(p - 1), x], x] /; FreeQ[{a, b, c, d, e, q}, x] && EqQ[e, c^2*d] && GtQ[p, 0] && NeQ[q, -1]
  4931. Int[Times[Power[Plus[Optional[Pattern[a, Blank[]]], Times[ArcCot[Times[Optional[Pattern[c, Blank[]]], Pattern[x, Blank[]]]], Optional[Pattern[b, Blank[]]]]], Optional[Pattern[p, Blank[]]]], Pattern[x, Blank[]], Power[Plus[Pattern[d, Blank[]], Times[Optional[Pattern[e, Blank[]]], Power[Pattern[x, Blank[]], 2]]], Optional[Pattern[q, Blank[]]]]], Pattern[x, Blank[Symbol]]] := Simp[((d + e*x^2)^(q + 1)*(a + b*ArcCot[c*x])^p)/(2*e*(q + 1)), x] + Dist[(b*p)/(2*c*(q + 1)), Int[(d + e*x^2)^q*(a + b*ArcCot[c*x])^(p - 1), x], x] /; FreeQ[{a, b, c, d, e, q}, x] && EqQ[e, c^2*d] && GtQ[p, 0] && NeQ[q, -1]
  4932. Int[Times[Power[Plus[Optional[Pattern[a, Blank[]]], Times[ArcTan[Times[Optional[Pattern[c, Blank[]]], Pattern[x, Blank[]]]], Optional[Pattern[b, Blank[]]]]], Pattern[p, Blank[]]], Pattern[x, Blank[]], Power[Plus[Pattern[d, Blank[]], Times[Optional[Pattern[e, Blank[]]], Power[Pattern[x, Blank[]], 2]]], -2]], Pattern[x, Blank[Symbol]]] := Simp[(x*(a + b*ArcTan[c*x])^(p + 1))/(b*c*d*(p + 1)*(d + e*x^2)), x] + (-Dist[4/(b^2*(p + 1)*(p + 2)), Int[(x*(a + b*ArcTan[c*x])^(p + 2))/(d + e*x^2)^2, x], x] - Simp[((1 - c^2*x^2)*(a + b*ArcTan[c*x])^(p + 2))/(b^2*e*(p + 1)*(p + 2)*(d + e*x^2)), x]) /; FreeQ[{a, b, c, d, e}, x] && EqQ[e, c^2*d] && LtQ[p, -1] && NeQ[p, -2]
  4933. Int[Times[Power[Plus[Optional[Pattern[a, Blank[]]], Times[ArcCot[Times[Optional[Pattern[c, Blank[]]], Pattern[x, Blank[]]]], Optional[Pattern[b, Blank[]]]]], Pattern[p, Blank[]]], Pattern[x, Blank[]], Power[Plus[Pattern[d, Blank[]], Times[Optional[Pattern[e, Blank[]]], Power[Pattern[x, Blank[]], 2]]], -2]], Pattern[x, Blank[Symbol]]] := -Simp[(x*(a + b*ArcCot[c*x])^(p + 1))/(b*c*d*(p + 1)*(d + e*x^2)), x] + (-Dist[4/(b^2*(p + 1)*(p + 2)), Int[(x*(a + b*ArcCot[c*x])^(p + 2))/(d + e*x^2)^2, x], x] - Simp[((1 - c^2*x^2)*(a + b*ArcCot[c*x])^(p + 2))/(b^2*e*(p + 1)*(p + 2)*(d + e*x^2)), x]) /; FreeQ[{a, b, c, d, e}, x] && EqQ[e, c^2*d] && LtQ[p, -1] && NeQ[p, -2]
  4934. Int[Times[Plus[Optional[Pattern[a, Blank[]]], Times[ArcTan[Times[Optional[Pattern[c, Blank[]]], Pattern[x, Blank[]]]], Optional[Pattern[b, Blank[]]]]], Power[Pattern[x, Blank[]], 2], Power[Plus[Pattern[d, Blank[]], Times[Optional[Pattern[e, Blank[]]], Power[Pattern[x, Blank[]], 2]]], Pattern[q, Blank[]]]], Pattern[x, Blank[Symbol]]] := -Simp[(b*(d + e*x^2)^(q + 1))/(4*c^3*d*(q + 1)^2), x] + (-Dist[1/(2*c^2*d*(q + 1)), Int[(d + e*x^2)^(q + 1)*(a + b*ArcTan[c*x]), x], x] + Simp[(x*(d + e*x^2)^(q + 1)*(a + b*ArcTan[c*x]))/(2*c^2*d*(q + 1)), x]) /; FreeQ[{a, b, c, d, e}, x] && EqQ[e, c^2*d] && LtQ[q, -1] && NeQ[q, -5/2]
  4935. Int[Times[Plus[Optional[Pattern[a, Blank[]]], Times[ArcCot[Times[Optional[Pattern[c, Blank[]]], Pattern[x, Blank[]]]], Optional[Pattern[b, Blank[]]]]], Power[Pattern[x, Blank[]], 2], Power[Plus[Pattern[d, Blank[]], Times[Optional[Pattern[e, Blank[]]], Power[Pattern[x, Blank[]], 2]]], Pattern[q, Blank[]]]], Pattern[x, Blank[Symbol]]] := Simp[(b*(d + e*x^2)^(q + 1))/(4*c^3*d*(q + 1)^2), x] + (-Dist[1/(2*c^2*d*(q + 1)), Int[(d + e*x^2)^(q + 1)*(a + b*ArcCot[c*x]), x], x] + Simp[(x*(d + e*x^2)^(q + 1)*(a + b*ArcCot[c*x]))/(2*c^2*d*(q + 1)), x]) /; FreeQ[{a, b, c, d, e}, x] && EqQ[e, c^2*d] && LtQ[q, -1] && NeQ[q, -5/2]
  4936. Int[Times[Power[Plus[Optional[Pattern[a, Blank[]]], Times[ArcTan[Times[Optional[Pattern[c, Blank[]]], Pattern[x, Blank[]]]], Optional[Pattern[b, Blank[]]]]], Optional[Pattern[p, Blank[]]]], Power[Pattern[x, Blank[]], 2], Power[Plus[Pattern[d, Blank[]], Times[Optional[Pattern[e, Blank[]]], Power[Pattern[x, Blank[]], 2]]], -2]], Pattern[x, Blank[Symbol]]] := Simp[(a + b*ArcTan[c*x])^(p + 1)/(2*b*c^3*d^2*(p + 1)), x] + (Dist[(b*p)/(2*c), Int[(x*(a + b*ArcTan[c*x])^(p - 1))/(d + e*x^2)^2, x], x] - Simp[(x*(a + b*ArcTan[c*x])^p)/(2*c^2*d*(d + e*x^2)), x]) /; FreeQ[{a, b, c, d, e}, x] && EqQ[e, c^2*d] && GtQ[p, 0]
  4937. Int[Times[Power[Plus[Optional[Pattern[a, Blank[]]], Times[ArcCot[Times[Optional[Pattern[c, Blank[]]], Pattern[x, Blank[]]]], Optional[Pattern[b, Blank[]]]]], Optional[Pattern[p, Blank[]]]], Power[Pattern[x, Blank[]], 2], Power[Plus[Pattern[d, Blank[]], Times[Optional[Pattern[e, Blank[]]], Power[Pattern[x, Blank[]], 2]]], -2]], Pattern[x, Blank[Symbol]]] := -Simp[(a + b*ArcCot[c*x])^(p + 1)/(2*b*c^3*d^2*(p + 1)), x] + (-Dist[(b*p)/(2*c), Int[(x*(a + b*ArcCot[c*x])^(p - 1))/(d + e*x^2)^2, x], x] - Simp[(x*(a + b*ArcCot[c*x])^p)/(2*c^2*d*(d + e*x^2)), x]) /; FreeQ[{a, b, c, d, e}, x] && EqQ[e, c^2*d] && GtQ[p, 0]
  4938. Int[Times[Plus[Optional[Pattern[a, Blank[]]], Times[ArcTan[Times[Optional[Pattern[c, Blank[]]], Pattern[x, Blank[]]]], Optional[Pattern[b, Blank[]]]]], Power[Times[Optional[Pattern[f, Blank[]]], Pattern[x, Blank[]]], Pattern[m, Blank[]]], Power[Plus[Pattern[d, Blank[]], Times[Optional[Pattern[e, Blank[]]], Power[Pattern[x, Blank[]], 2]]], Pattern[q, Blank[]]]], Pattern[x, Blank[Symbol]]] := Simp[(b*(f*x)^m*(d + e*x^2)^(q + 1))/(c*d*m^2), x] + (Dist[(f^2*(m - 1))/(c^2*d*m), Int[(f*x)^(m - 2)*(d + e*x^2)^(q + 1)*(a + b*ArcTan[c*x]), x], x] - Simp[(f*(f*x)^(m - 1)*(d + e*x^2)^(q + 1)*(a + b*ArcTan[c*x]))/(c^2*d*m), x]) /; FreeQ[{a, b, c, d, e, f}, x] && EqQ[e, c^2*d] && EqQ[m + 2*q + 2, 0] && LtQ[q, -1]
  4939. Int[Times[Plus[Optional[Pattern[a, Blank[]]], Times[ArcCot[Times[Optional[Pattern[c, Blank[]]], Pattern[x, Blank[]]]], Optional[Pattern[b, Blank[]]]]], Power[Times[Optional[Pattern[f, Blank[]]], Pattern[x, Blank[]]], Pattern[m, Blank[]]], Power[Plus[Pattern[d, Blank[]], Times[Optional[Pattern[e, Blank[]]], Power[Pattern[x, Blank[]], 2]]], Pattern[q, Blank[]]]], Pattern[x, Blank[Symbol]]] := -Simp[(b*(f*x)^m*(d + e*x^2)^(q + 1))/(c*d*m^2), x] + (Dist[(f^2*(m - 1))/(c^2*d*m), Int[(f*x)^(m - 2)*(d + e*x^2)^(q + 1)*(a + b*ArcCot[c*x]), x], x] - Simp[(f*(f*x)^(m - 1)*(d + e*x^2)^(q + 1)*(a + b*ArcCot[c*x]))/(c^2*d*m), x]) /; FreeQ[{a, b, c, d, e, f}, x] && EqQ[e, c^2*d] && EqQ[m + 2*q + 2, 0] && LtQ[q, -1]
  4940. Int[Times[Power[Plus[Optional[Pattern[a, Blank[]]], Times[ArcTan[Times[Optional[Pattern[c, Blank[]]], Pattern[x, Blank[]]]], Optional[Pattern[b, Blank[]]]]], Pattern[p, Blank[]]], Power[Times[Optional[Pattern[f, Blank[]]], Pattern[x, Blank[]]], Pattern[m, Blank[]]], Power[Plus[Pattern[d, Blank[]], Times[Optional[Pattern[e, Blank[]]], Power[Pattern[x, Blank[]], 2]]], Pattern[q, Blank[]]]], Pattern[x, Blank[Symbol]]] := Simp[(b*p*(f*x)^m*(d + e*x^2)^(q + 1)*(a + b*ArcTan[c*x])^(p - 1))/(c*d*m^2), x] + (Dist[(f^2*(m - 1))/(c^2*d*m), Int[(f*x)^(m - 2)*(d + e*x^2)^(q + 1)*(a + b*ArcTan[c*x])^p, x], x] - Dist[(b^2*p*(p - 1))/m^2, Int[(f*x)^m*(d + e*x^2)^q*(a + b*ArcTan[c*x])^(p - 2), x], x] - Simp[(f*(f*x)^(m - 1)*(d + e*x^2)^(q + 1)*(a + b*ArcTan[c*x])^p)/(c^2*d*m), x]) /; FreeQ[{a, b, c, d, e, f, m}, x] && EqQ[e, c^2*d] && EqQ[m + 2*q + 2, 0] && LtQ[q, -1] && GtQ[p, 1]
  4941. Int[Times[Power[Plus[Optional[Pattern[a, Blank[]]], Times[ArcCot[Times[Optional[Pattern[c, Blank[]]], Pattern[x, Blank[]]]], Optional[Pattern[b, Blank[]]]]], Pattern[p, Blank[]]], Power[Times[Optional[Pattern[f, Blank[]]], Pattern[x, Blank[]]], Pattern[m, Blank[]]], Power[Plus[Pattern[d, Blank[]], Times[Optional[Pattern[e, Blank[]]], Power[Pattern[x, Blank[]], 2]]], Pattern[q, Blank[]]]], Pattern[x, Blank[Symbol]]] := -Simp[(b*p*(f*x)^m*(d + e*x^2)^(q + 1)*(a + b*ArcCot[c*x])^(p - 1))/(c*d*m^2), x] + (Dist[(f^2*(m - 1))/(c^2*d*m), Int[(f*x)^(m - 2)*(d + e*x^2)^(q + 1)*(a + b*ArcCot[c*x])^p, x], x] - Dist[(b^2*p*(p - 1))/m^2, Int[(f*x)^m*(d + e*x^2)^q*(a + b*ArcCot[c*x])^(p - 2), x], x] - Simp[(f*(f*x)^(m - 1)*(d + e*x^2)^(q + 1)*(a + b*ArcCot[c*x])^p)/(c^2*d*m), x]) /; FreeQ[{a, b, c, d, e, f, m}, x] && EqQ[e, c^2*d] && EqQ[m + 2*q + 2, 0] && LtQ[q, -1] && GtQ[p, 1]
  4942. Int[Times[Power[Plus[Optional[Pattern[a, Blank[]]], Times[ArcTan[Times[Optional[Pattern[c, Blank[]]], Pattern[x, Blank[]]]], Optional[Pattern[b, Blank[]]]]], Pattern[p, Blank[]]], Power[Times[Optional[Pattern[f, Blank[]]], Pattern[x, Blank[]]], Optional[Pattern[m, Blank[]]]], Power[Plus[Pattern[d, Blank[]], Times[Optional[Pattern[e, Blank[]]], Power[Pattern[x, Blank[]], 2]]], Optional[Pattern[q, Blank[]]]]], Pattern[x, Blank[Symbol]]] := Simp[((f*x)^m*(d + e*x^2)^(q + 1)*(a + b*ArcTan[c*x])^(p + 1))/(b*c*d*(p + 1)), x] - Dist[(f*m)/(b*c*(p + 1)), Int[(f*x)^(m - 1)*(d + e*x^2)^q*(a + b*ArcTan[c*x])^(p + 1), x], x] /; FreeQ[{a, b, c, d, e, f, m, q}, x] && EqQ[e, c^2*d] && EqQ[m + 2*q + 2, 0] && LtQ[p, -1]
  4943. Int[Times[Power[Plus[Optional[Pattern[a, Blank[]]], Times[ArcCot[Times[Optional[Pattern[c, Blank[]]], Pattern[x, Blank[]]]], Optional[Pattern[b, Blank[]]]]], Pattern[p, Blank[]]], Power[Times[Optional[Pattern[f, Blank[]]], Pattern[x, Blank[]]], Optional[Pattern[m, Blank[]]]], Power[Plus[Pattern[d, Blank[]], Times[Optional[Pattern[e, Blank[]]], Power[Pattern[x, Blank[]], 2]]], Optional[Pattern[q, Blank[]]]]], Pattern[x, Blank[Symbol]]] := -Simp[((f*x)^m*(d + e*x^2)^(q + 1)*(a + b*ArcCot[c*x])^(p + 1))/(b*c*d*(p + 1)), x] + Dist[(f*m)/(b*c*(p + 1)), Int[(f*x)^(m - 1)*(d + e*x^2)^q*(a + b*ArcCot[c*x])^(p + 1), x], x] /; FreeQ[{a, b, c, d, e, f, m, q}, x] && EqQ[e, c^2*d] && EqQ[m + 2*q + 2, 0] && LtQ[p, -1]
  4944. Int[Times[Power[Plus[Optional[Pattern[a, Blank[]]], Times[ArcTan[Times[Optional[Pattern[c, Blank[]]], Pattern[x, Blank[]]]], Optional[Pattern[b, Blank[]]]]], Optional[Pattern[p, Blank[]]]], Power[Times[Optional[Pattern[f, Blank[]]], Pattern[x, Blank[]]], Optional[Pattern[m, Blank[]]]], Power[Plus[Pattern[d, Blank[]], Times[Optional[Pattern[e, Blank[]]], Power[Pattern[x, Blank[]], 2]]], Optional[Pattern[q, Blank[]]]]], Pattern[x, Blank[Symbol]]] := Simp[((f*x)^(m + 1)*(d + e*x^2)^(q + 1)*(a + b*ArcTan[c*x])^p)/(d*f*(m + 1)), x] - Dist[(b*c*p)/(f*(m + 1)), Int[(f*x)^(m + 1)*(d + e*x^2)^q*(a + b*ArcTan[c*x])^(p - 1), x], x] /; FreeQ[{a, b, c, d, e, f, m, q}, x] && EqQ[e, c^2*d] && EqQ[m + 2*q + 3, 0] && GtQ[p, 0] && NeQ[m, -1]
  4945. Int[Times[Power[Plus[Optional[Pattern[a, Blank[]]], Times[ArcCot[Times[Optional[Pattern[c, Blank[]]], Pattern[x, Blank[]]]], Optional[Pattern[b, Blank[]]]]], Optional[Pattern[p, Blank[]]]], Power[Times[Optional[Pattern[f, Blank[]]], Pattern[x, Blank[]]], Optional[Pattern[m, Blank[]]]], Power[Plus[Pattern[d, Blank[]], Times[Optional[Pattern[e, Blank[]]], Power[Pattern[x, Blank[]], 2]]], Optional[Pattern[q, Blank[]]]]], Pattern[x, Blank[Symbol]]] := Simp[((f*x)^(m + 1)*(d + e*x^2)^(q + 1)*(a + b*ArcCot[c*x])^p)/(d*f*(m + 1)), x] + Dist[(b*c*p)/(f*(m + 1)), Int[(f*x)^(m + 1)*(d + e*x^2)^q*(a + b*ArcCot[c*x])^(p - 1), x], x] /; FreeQ[{a, b, c, d, e, f, m, q}, x] && EqQ[e, c^2*d] && EqQ[m + 2*q + 3, 0] && GtQ[p, 0] && NeQ[m, -1]
  4946. Int[Times[Plus[Optional[Pattern[a, Blank[]]], Times[ArcTan[Times[Optional[Pattern[c, Blank[]]], Pattern[x, Blank[]]]], Optional[Pattern[b, Blank[]]]]], Power[Times[Optional[Pattern[f, Blank[]]], Pattern[x, Blank[]]], Pattern[m, Blank[]]], Power[Plus[Pattern[d, Blank[]], Times[Optional[Pattern[e, Blank[]]], Power[Pattern[x, Blank[]], 2]]], Rational[1, 2]]], Pattern[x, Blank[Symbol]]] := Simp[((f*x)^(m + 1)*Sqrt[d + e*x^2]*(a + b*ArcTan[c*x]))/(f*(m + 2)), x] + (Dist[d/(m + 2), Int[((f*x)^m*(a + b*ArcTan[c*x]))/Sqrt[d + e*x^2], x], x] - Dist[(b*c*d)/(f*(m + 2)), Int[(f*x)^(m + 1)/Sqrt[d + e*x^2], x], x]) /; FreeQ[{a, b, c, d, e, f, m}, x] && EqQ[e, c^2*d] && NeQ[m, -2]
  4947. Int[Times[Plus[Optional[Pattern[a, Blank[]]], Times[ArcCot[Times[Optional[Pattern[c, Blank[]]], Pattern[x, Blank[]]]], Optional[Pattern[b, Blank[]]]]], Power[Times[Optional[Pattern[f, Blank[]]], Pattern[x, Blank[]]], Pattern[m, Blank[]]], Power[Plus[Pattern[d, Blank[]], Times[Optional[Pattern[e, Blank[]]], Power[Pattern[x, Blank[]], 2]]], Rational[1, 2]]], Pattern[x, Blank[Symbol]]] := Simp[((f*x)^(m + 1)*Sqrt[d + e*x^2]*(a + b*ArcCot[c*x]))/(f*(m + 2)), x] + (Dist[d/(m + 2), Int[((f*x)^m*(a + b*ArcCot[c*x]))/Sqrt[d + e*x^2], x], x] + Dist[(b*c*d)/(f*(m + 2)), Int[(f*x)^(m + 1)/Sqrt[d + e*x^2], x], x]) /; FreeQ[{a, b, c, d, e, f, m}, x] && EqQ[e, c^2*d] && NeQ[m, -2]
  4948. Int[Times[Power[Plus[Optional[Pattern[a, Blank[]]], Times[ArcTan[Times[Optional[Pattern[c, Blank[]]], Pattern[x, Blank[]]]], Optional[Pattern[b, Blank[]]]]], Optional[Pattern[p, Blank[]]]], Power[Times[Optional[Pattern[f, Blank[]]], Pattern[x, Blank[]]], Pattern[m, Blank[]]], Power[Plus[Pattern[d, Blank[]], Times[Optional[Pattern[e, Blank[]]], Power[Pattern[x, Blank[]], 2]]], Pattern[q, Blank[]]]], Pattern[x, Blank[Symbol]]] := Int[ExpandIntegrand[(f*x)^m*(d + e*x^2)^q*(a + b*ArcTan[c*x])^p, x], x] /; FreeQ[{a, b, c, d, e, f, m}, x] && EqQ[e, c^2*d] && IGtQ[p, 0] && IGtQ[q, 1] && (EqQ[p, 1] || IntegerQ[m])
  4949. Int[Times[Power[Plus[Optional[Pattern[a, Blank[]]], Times[ArcCot[Times[Optional[Pattern[c, Blank[]]], Pattern[x, Blank[]]]], Optional[Pattern[b, Blank[]]]]], Optional[Pattern[p, Blank[]]]], Power[Times[Optional[Pattern[f, Blank[]]], Pattern[x, Blank[]]], Pattern[m, Blank[]]], Power[Plus[Pattern[d, Blank[]], Times[Optional[Pattern[e, Blank[]]], Power[Pattern[x, Blank[]], 2]]], Pattern[q, Blank[]]]], Pattern[x, Blank[Symbol]]] := Int[ExpandIntegrand[(f*x)^m*(d + e*x^2)^q*(a + b*ArcCot[c*x])^p, x], x] /; FreeQ[{a, b, c, d, e, f, m}, x] && EqQ[e, c^2*d] && IGtQ[p, 0] && IGtQ[q, 1] && (EqQ[p, 1] || IntegerQ[m])
  4950. Int[Times[Power[Plus[Optional[Pattern[a, Blank[]]], Times[ArcTan[Times[Optional[Pattern[c, Blank[]]], Pattern[x, Blank[]]]], Optional[Pattern[b, Blank[]]]]], Optional[Pattern[p, Blank[]]]], Power[Times[Optional[Pattern[f, Blank[]]], Pattern[x, Blank[]]], Pattern[m, Blank[]]], Power[Plus[Pattern[d, Blank[]], Times[Optional[Pattern[e, Blank[]]], Power[Pattern[x, Blank[]], 2]]], Optional[Pattern[q, Blank[]]]]], Pattern[x, Blank[Symbol]]] := Dist[d, Int[(f*x)^m*(d + e*x^2)^(q - 1)*(a + b*ArcTan[c*x])^p, x], x] + Dist[(c^2*d)/f^2, Int[(f*x)^(m + 2)*(d + e*x^2)^(q - 1)*(a + b*ArcTan[c*x])^p, x], x] /; FreeQ[{a, b, c, d, e, f, m}, x] && EqQ[e, c^2*d] && GtQ[q, 0] && IGtQ[p, 0] && (RationalQ[m] || (EqQ[p, 1] && IntegerQ[q]))
  4951. Int[Times[Power[Plus[Optional[Pattern[a, Blank[]]], Times[ArcCot[Times[Optional[Pattern[c, Blank[]]], Pattern[x, Blank[]]]], Optional[Pattern[b, Blank[]]]]], Optional[Pattern[p, Blank[]]]], Power[Times[Optional[Pattern[f, Blank[]]], Pattern[x, Blank[]]], Pattern[m, Blank[]]], Power[Plus[Pattern[d, Blank[]], Times[Optional[Pattern[e, Blank[]]], Power[Pattern[x, Blank[]], 2]]], Optional[Pattern[q, Blank[]]]]], Pattern[x, Blank[Symbol]]] := Dist[d, Int[(f*x)^m*(d + e*x^2)^(q - 1)*(a + b*ArcCot[c*x])^p, x], x] + Dist[(c^2*d)/f^2, Int[(f*x)^(m + 2)*(d + e*x^2)^(q - 1)*(a + b*ArcCot[c*x])^p, x], x] /; FreeQ[{a, b, c, d, e, f, m}, x] && EqQ[e, c^2*d] && GtQ[q, 0] && IGtQ[p, 0] && (RationalQ[m] || (EqQ[p, 1] && IntegerQ[q]))
  4952. Int[Times[Power[Plus[Optional[Pattern[a, Blank[]]], Times[ArcTan[Times[Optional[Pattern[c, Blank[]]], Pattern[x, Blank[]]]], Optional[Pattern[b, Blank[]]]]], Optional[Pattern[p, Blank[]]]], Power[Times[Optional[Pattern[f, Blank[]]], Pattern[x, Blank[]]], Pattern[m, Blank[]]], Power[Plus[Pattern[d, Blank[]], Times[Optional[Pattern[e, Blank[]]], Power[Pattern[x, Blank[]], 2]]], Rational[-1, 2]]], Pattern[x, Blank[Symbol]]] := Simp[(f*(f*x)^(m - 1)*Sqrt[d + e*x^2]*(a + b*ArcTan[c*x])^p)/(c^2*d*m), x] + (-Dist[(b*f*p)/(c*m), Int[((f*x)^(m - 1)*(a + b*ArcTan[c*x])^(p - 1))/Sqrt[d + e*x^2], x], x] - Dist[(f^2*(m - 1))/(c^2*m), Int[((f*x)^(m - 2)*(a + b*ArcTan[c*x])^p)/Sqrt[d + e*x^2], x], x]) /; FreeQ[{a, b, c, d, e, f}, x] && EqQ[e, c^2*d] && GtQ[p, 0] && GtQ[m, 1]
  4953. Int[Times[Power[Plus[Optional[Pattern[a, Blank[]]], Times[ArcCot[Times[Optional[Pattern[c, Blank[]]], Pattern[x, Blank[]]]], Optional[Pattern[b, Blank[]]]]], Optional[Pattern[p, Blank[]]]], Power[Times[Optional[Pattern[f, Blank[]]], Pattern[x, Blank[]]], Pattern[m, Blank[]]], Power[Plus[Pattern[d, Blank[]], Times[Optional[Pattern[e, Blank[]]], Power[Pattern[x, Blank[]], 2]]], Rational[-1, 2]]], Pattern[x, Blank[Symbol]]] := Simp[(f*(f*x)^(m - 1)*Sqrt[d + e*x^2]*(a + b*ArcCot[c*x])^p)/(c^2*d*m), x] + (Dist[(b*f*p)/(c*m), Int[((f*x)^(m - 1)*(a + b*ArcCot[c*x])^(p - 1))/Sqrt[d + e*x^2], x], x] - Dist[(f^2*(m - 1))/(c^2*m), Int[((f*x)^(m - 2)*(a + b*ArcCot[c*x])^p)/Sqrt[d + e*x^2], x], x]) /; FreeQ[{a, b, c, d, e, f}, x] && EqQ[e, c^2*d] && GtQ[p, 0] && GtQ[m, 1]
  4954. Int[Times[Plus[Optional[Pattern[a, Blank[]]], Times[ArcTan[Times[Optional[Pattern[c, Blank[]]], Pattern[x, Blank[]]]], Optional[Pattern[b, Blank[]]]]], Power[Pattern[x, Blank[]], -1], Power[Plus[Pattern[d, Blank[]], Times[Optional[Pattern[e, Blank[]]], Power[Pattern[x, Blank[]], 2]]], Rational[-1, 2]]], Pattern[x, Blank[Symbol]]] := Simp[(-2*(a + b*ArcTan[c*x])*ArcTanh[Sqrt[1 + I*c*x]/Sqrt[1 - I*c*x]])/Sqrt[d], x] + (Simp[(I*b*PolyLog[2, -(Sqrt[1 + I*c*x]/Sqrt[1 - I*c*x])])/Sqrt[d], x] - Simp[(I*b*PolyLog[2, Sqrt[1 + I*c*x]/Sqrt[1 - I*c*x]])/Sqrt[d], x]) /; FreeQ[{a, b, c, d, e}, x] && EqQ[e, c^2*d] && GtQ[d, 0]
  4955. Int[Times[Plus[Optional[Pattern[a, Blank[]]], Times[ArcCot[Times[Optional[Pattern[c, Blank[]]], Pattern[x, Blank[]]]], Optional[Pattern[b, Blank[]]]]], Power[Pattern[x, Blank[]], -1], Power[Plus[Pattern[d, Blank[]], Times[Optional[Pattern[e, Blank[]]], Power[Pattern[x, Blank[]], 2]]], Rational[-1, 2]]], Pattern[x, Blank[Symbol]]] := Simp[(-2*(a + b*ArcCot[c*x])*ArcTanh[Sqrt[1 + I*c*x]/Sqrt[1 - I*c*x]])/Sqrt[d], x] + (-Simp[(I*b*PolyLog[2, -(Sqrt[1 + I*c*x]/Sqrt[1 - I*c*x])])/Sqrt[d], x] + Simp[(I*b*PolyLog[2, Sqrt[1 + I*c*x]/Sqrt[1 - I*c*x]])/Sqrt[d], x]) /; FreeQ[{a, b, c, d, e}, x] && EqQ[e, c^2*d] && GtQ[d, 0]
  4956. Int[Times[Power[Plus[Optional[Pattern[a, Blank[]]], Times[ArcTan[Times[Optional[Pattern[c, Blank[]]], Pattern[x, Blank[]]]], Optional[Pattern[b, Blank[]]]]], Pattern[p, Blank[]]], Power[Pattern[x, Blank[]], -1], Power[Plus[Pattern[d, Blank[]], Times[Optional[Pattern[e, Blank[]]], Power[Pattern[x, Blank[]], 2]]], Rational[-1, 2]]], Pattern[x, Blank[Symbol]]] := Dist[1/Sqrt[d], Subst[Int[(a + b*x)^p*Csc[x], x], x, ArcTan[c*x]], x] /; FreeQ[{a, b, c, d, e}, x] && EqQ[e, c^2*d] && IGtQ[p, 0] && GtQ[d, 0]
  4957. Int[Times[Power[Plus[Optional[Pattern[a, Blank[]]], Times[ArcCot[Times[Optional[Pattern[c, Blank[]]], Pattern[x, Blank[]]]], Optional[Pattern[b, Blank[]]]]], Pattern[p, Blank[]]], Power[Pattern[x, Blank[]], -1], Power[Plus[Pattern[d, Blank[]], Times[Optional[Pattern[e, Blank[]]], Power[Pattern[x, Blank[]], 2]]], Rational[-1, 2]]], Pattern[x, Blank[Symbol]]] := -Dist[(c*x*Sqrt[1 + 1/(c^2*x^2)])/Sqrt[d + e*x^2], Subst[Int[(a + b*x)^p*Sec[x], x], x, ArcCot[c*x]], x] /; FreeQ[{a, b, c, d, e}, x] && EqQ[e, c^2*d] && IGtQ[p, 0] && GtQ[d, 0]
  4958. Int[Times[Power[Plus[Optional[Pattern[a, Blank[]]], Times[ArcTan[Times[Optional[Pattern[c, Blank[]]], Pattern[x, Blank[]]]], Optional[Pattern[b, Blank[]]]]], Optional[Pattern[p, Blank[]]]], Power[Pattern[x, Blank[]], -1], Power[Plus[Pattern[d, Blank[]], Times[Optional[Pattern[e, Blank[]]], Power[Pattern[x, Blank[]], 2]]], Rational[-1, 2]]], Pattern[x, Blank[Symbol]]] := Dist[Sqrt[1 + c^2*x^2]/Sqrt[d + e*x^2], Int[(a + b*ArcTan[c*x])^p/(x*Sqrt[1 + c^2*x^2]), x], x] /; FreeQ[{a, b, c, d, e}, x] && EqQ[e, c^2*d] && IGtQ[p, 0] && !GtQ[d, 0]
  4959. Int[Times[Power[Plus[Optional[Pattern[a, Blank[]]], Times[ArcCot[Times[Optional[Pattern[c, Blank[]]], Pattern[x, Blank[]]]], Optional[Pattern[b, Blank[]]]]], Optional[Pattern[p, Blank[]]]], Power[Pattern[x, Blank[]], -1], Power[Plus[Pattern[d, Blank[]], Times[Optional[Pattern[e, Blank[]]], Power[Pattern[x, Blank[]], 2]]], Rational[-1, 2]]], Pattern[x, Blank[Symbol]]] := Dist[Sqrt[1 + c^2*x^2]/Sqrt[d + e*x^2], Int[(a + b*ArcCot[c*x])^p/(x*Sqrt[1 + c^2*x^2]), x], x] /; FreeQ[{a, b, c, d, e}, x] && EqQ[e, c^2*d] && IGtQ[p, 0] && !GtQ[d, 0]
  4960. Int[Times[Power[Plus[Optional[Pattern[a, Blank[]]], Times[ArcTan[Times[Optional[Pattern[c, Blank[]]], Pattern[x, Blank[]]]], Optional[Pattern[b, Blank[]]]]], Optional[Pattern[p, Blank[]]]], Power[Pattern[x, Blank[]], -2], Power[Plus[Pattern[d, Blank[]], Times[Optional[Pattern[e, Blank[]]], Power[Pattern[x, Blank[]], 2]]], Rational[-1, 2]]], Pattern[x, Blank[Symbol]]] := -Simp[(Sqrt[d + e*x^2]*(a + b*ArcTan[c*x])^p)/(d*x), x] + Dist[b*c*p, Int[(a + b*ArcTan[c*x])^(p - 1)/(x*Sqrt[d + e*x^2]), x], x] /; FreeQ[{a, b, c, d, e}, x] && EqQ[e, c^2*d] && GtQ[p, 0]
  4961. Int[Times[Power[Plus[Optional[Pattern[a, Blank[]]], Times[ArcCot[Times[Optional[Pattern[c, Blank[]]], Pattern[x, Blank[]]]], Optional[Pattern[b, Blank[]]]]], Optional[Pattern[p, Blank[]]]], Power[Pattern[x, Blank[]], -2], Power[Plus[Pattern[d, Blank[]], Times[Optional[Pattern[e, Blank[]]], Power[Pattern[x, Blank[]], 2]]], Rational[-1, 2]]], Pattern[x, Blank[Symbol]]] := -Simp[(Sqrt[d + e*x^2]*(a + b*ArcCot[c*x])^p)/(d*x), x] - Dist[b*c*p, Int[(a + b*ArcCot[c*x])^(p - 1)/(x*Sqrt[d + e*x^2]), x], x] /; FreeQ[{a, b, c, d, e}, x] && EqQ[e, c^2*d] && GtQ[p, 0]
  4962. Int[Times[Power[Plus[Optional[Pattern[a, Blank[]]], Times[ArcTan[Times[Optional[Pattern[c, Blank[]]], Pattern[x, Blank[]]]], Optional[Pattern[b, Blank[]]]]], Optional[Pattern[p, Blank[]]]], Power[Times[Optional[Pattern[f, Blank[]]], Pattern[x, Blank[]]], Pattern[m, Blank[]]], Power[Plus[Pattern[d, Blank[]], Times[Optional[Pattern[e, Blank[]]], Power[Pattern[x, Blank[]], 2]]], Rational[-1, 2]]], Pattern[x, Blank[Symbol]]] := Simp[((f*x)^(m + 1)*Sqrt[d + e*x^2]*(a + b*ArcTan[c*x])^p)/(d*f*(m + 1)), x] + (-Dist[(b*c*p)/(f*(m + 1)), Int[((f*x)^(m + 1)*(a + b*ArcTan[c*x])^(p - 1))/Sqrt[d + e*x^2], x], x] - Dist[(c^2*(m + 2))/(f^2*(m + 1)), Int[((f*x)^(m + 2)*(a + b*ArcTan[c*x])^p)/Sqrt[d + e*x^2], x], x]) /; FreeQ[{a, b, c, d, e, f}, x] && EqQ[e, c^2*d] && GtQ[p, 0] && LtQ[m, -1] && NeQ[m, -2]
  4963. Int[Times[Power[Plus[Optional[Pattern[a, Blank[]]], Times[ArcCot[Times[Optional[Pattern[c, Blank[]]], Pattern[x, Blank[]]]], Optional[Pattern[b, Blank[]]]]], Optional[Pattern[p, Blank[]]]], Power[Times[Optional[Pattern[f, Blank[]]], Pattern[x, Blank[]]], Pattern[m, Blank[]]], Power[Plus[Pattern[d, Blank[]], Times[Optional[Pattern[e, Blank[]]], Power[Pattern[x, Blank[]], 2]]], Rational[-1, 2]]], Pattern[x, Blank[Symbol]]] := Simp[((f*x)^(m + 1)*Sqrt[d + e*x^2]*(a + b*ArcCot[c*x])^p)/(d*f*(m + 1)), x] + (Dist[(b*c*p)/(f*(m + 1)), Int[((f*x)^(m + 1)*(a + b*ArcCot[c*x])^(p - 1))/Sqrt[d + e*x^2], x], x] - Dist[(c^2*(m + 2))/(f^2*(m + 1)), Int[((f*x)^(m + 2)*(a + b*ArcCot[c*x])^p)/Sqrt[d + e*x^2], x], x]) /; FreeQ[{a, b, c, d, e, f}, x] && EqQ[e, c^2*d] && GtQ[p, 0] && LtQ[m, -1] && NeQ[m, -2]
  4964. Int[Times[Power[Plus[Optional[Pattern[a, Blank[]]], Times[ArcTan[Times[Optional[Pattern[c, Blank[]]], Pattern[x, Blank[]]]], Optional[Pattern[b, Blank[]]]]], Optional[Pattern[p, Blank[]]]], Power[Pattern[x, Blank[]], Pattern[m, Blank[]]], Power[Plus[Pattern[d, Blank[]], Times[Optional[Pattern[e, Blank[]]], Power[Pattern[x, Blank[]], 2]]], Pattern[q, Blank[]]]], Pattern[x, Blank[Symbol]]] := Dist[1/e, Int[x^(m - 2)*(d + e*x^2)^(q + 1)*(a + b*ArcTan[c*x])^p, x], x] - Dist[d/e, Int[x^(m - 2)*(d + e*x^2)^q*(a + b*ArcTan[c*x])^p, x], x] /; FreeQ[{a, b, c, d, e}, x] && EqQ[e, c^2*d] && IntegersQ[p, 2*q] && LtQ[q, -1] && IGtQ[m, 1] && NeQ[p, -1]
  4965. Int[Times[Power[Plus[Optional[Pattern[a, Blank[]]], Times[ArcCot[Times[Optional[Pattern[c, Blank[]]], Pattern[x, Blank[]]]], Optional[Pattern[b, Blank[]]]]], Optional[Pattern[p, Blank[]]]], Power[Pattern[x, Blank[]], Pattern[m, Blank[]]], Power[Plus[Pattern[d, Blank[]], Times[Optional[Pattern[e, Blank[]]], Power[Pattern[x, Blank[]], 2]]], Pattern[q, Blank[]]]], Pattern[x, Blank[Symbol]]] := Dist[1/e, Int[x^(m - 2)*(d + e*x^2)^(q + 1)*(a + b*ArcCot[c*x])^p, x], x] - Dist[d/e, Int[x^(m - 2)*(d + e*x^2)^q*(a + b*ArcCot[c*x])^p, x], x] /; FreeQ[{a, b, c, d, e}, x] && EqQ[e, c^2*d] && IntegersQ[p, 2*q] && LtQ[q, -1] && IGtQ[m, 1] && NeQ[p, -1]
  4966. Int[Times[Power[Plus[Optional[Pattern[a, Blank[]]], Times[ArcTan[Times[Optional[Pattern[c, Blank[]]], Pattern[x, Blank[]]]], Optional[Pattern[b, Blank[]]]]], Optional[Pattern[p, Blank[]]]], Power[Pattern[x, Blank[]], Pattern[m, Blank[]]], Power[Plus[Pattern[d, Blank[]], Times[Optional[Pattern[e, Blank[]]], Power[Pattern[x, Blank[]], 2]]], Pattern[q, Blank[]]]], Pattern[x, Blank[Symbol]]] := Dist[1/d, Int[x^m*(d + e*x^2)^(q + 1)*(a + b*ArcTan[c*x])^p, x], x] - Dist[e/d, Int[x^(m + 2)*(d + e*x^2)^q*(a + b*ArcTan[c*x])^p, x], x] /; FreeQ[{a, b, c, d, e}, x] && EqQ[e, c^2*d] && IntegersQ[p, 2*q] && LtQ[q, -1] && ILtQ[m, 0] && NeQ[p, -1]
  4967. Int[Times[Power[Plus[Optional[Pattern[a, Blank[]]], Times[ArcCot[Times[Optional[Pattern[c, Blank[]]], Pattern[x, Blank[]]]], Optional[Pattern[b, Blank[]]]]], Optional[Pattern[p, Blank[]]]], Power[Pattern[x, Blank[]], Pattern[m, Blank[]]], Power[Plus[Pattern[d, Blank[]], Times[Optional[Pattern[e, Blank[]]], Power[Pattern[x, Blank[]], 2]]], Pattern[q, Blank[]]]], Pattern[x, Blank[Symbol]]] := Dist[1/d, Int[x^m*(d + e*x^2)^(q + 1)*(a + b*ArcCot[c*x])^p, x], x] - Dist[e/d, Int[x^(m + 2)*(d + e*x^2)^q*(a + b*ArcCot[c*x])^p, x], x] /; FreeQ[{a, b, c, d, e}, x] && EqQ[e, c^2*d] && IntegersQ[p, 2*q] && LtQ[q, -1] && ILtQ[m, 0] && NeQ[p, -1]
  4968. Int[Times[Power[Plus[Optional[Pattern[a, Blank[]]], Times[ArcTan[Times[Optional[Pattern[c, Blank[]]], Pattern[x, Blank[]]]], Optional[Pattern[b, Blank[]]]]], Optional[Pattern[p, Blank[]]]], Power[Pattern[x, Blank[]], Optional[Pattern[m, Blank[]]]], Power[Plus[Pattern[d, Blank[]], Times[Optional[Pattern[e, Blank[]]], Power[Pattern[x, Blank[]], 2]]], Pattern[q, Blank[]]]], Pattern[x, Blank[Symbol]]] := Simp[(x^m*(d + e*x^2)^(q + 1)*(a + b*ArcTan[c*x])^(p + 1))/(b*c*d*(p + 1)), x] + (-Dist[(c*(m + 2*q + 2))/(b*(p + 1)), Int[x^(m + 1)*(d + e*x^2)^q*(a + b*ArcTan[c*x])^(p + 1), x], x] - Dist[m/(b*c*(p + 1)), Int[x^(m - 1)*(d + e*x^2)^q*(a + b*ArcTan[c*x])^(p + 1), x], x]) /; FreeQ[{a, b, c, d, e, m}, x] && EqQ[e, c^2*d] && IntegerQ[m] && LtQ[q, -1] && LtQ[p, -1] && NeQ[m + 2*q + 2, 0]
  4969. Int[Times[Power[Plus[Optional[Pattern[a, Blank[]]], Times[ArcCot[Times[Optional[Pattern[c, Blank[]]], Pattern[x, Blank[]]]], Optional[Pattern[b, Blank[]]]]], Optional[Pattern[p, Blank[]]]], Power[Pattern[x, Blank[]], Optional[Pattern[m, Blank[]]]], Power[Plus[Pattern[d, Blank[]], Times[Optional[Pattern[e, Blank[]]], Power[Pattern[x, Blank[]], 2]]], Pattern[q, Blank[]]]], Pattern[x, Blank[Symbol]]] := -Simp[(x^m*(d + e*x^2)^(q + 1)*(a + b*ArcCot[c*x])^(p + 1))/(b*c*d*(p + 1)), x] + (Dist[(c*(m + 2*q + 2))/(b*(p + 1)), Int[x^(m + 1)*(d + e*x^2)^q*(a + b*ArcCot[c*x])^(p + 1), x], x] + Dist[m/(b*c*(p + 1)), Int[x^(m - 1)*(d + e*x^2)^q*(a + b*ArcCot[c*x])^(p + 1), x], x]) /; FreeQ[{a, b, c, d, e, m}, x] && EqQ[e, c^2*d] && IntegerQ[m] && LtQ[q, -1] && LtQ[p, -1] && NeQ[m + 2*q + 2, 0]
  4970. Int[Times[Power[Plus[Optional[Pattern[a, Blank[]]], Times[ArcTan[Times[Optional[Pattern[c, Blank[]]], Pattern[x, Blank[]]]], Optional[Pattern[b, Blank[]]]]], Optional[Pattern[p, Blank[]]]], Power[Pattern[x, Blank[]], Optional[Pattern[m, Blank[]]]], Power[Plus[Pattern[d, Blank[]], Times[Optional[Pattern[e, Blank[]]], Power[Pattern[x, Blank[]], 2]]], Pattern[q, Blank[]]]], Pattern[x, Blank[Symbol]]] := Dist[d^q/c^(m + 1), Subst[Int[((a + b*x)^p*Sin[x]^m)/Cos[x]^(m + 2*(q + 1)), x], x, ArcTan[c*x]], x] /; FreeQ[{a, b, c, d, e, p}, x] && EqQ[e, c^2*d] && IGtQ[m, 0] && ILtQ[m + 2*q + 1, 0] && (IntegerQ[q] || GtQ[d, 0])
  4971. Int[Times[Power[Plus[Optional[Pattern[a, Blank[]]], Times[ArcTan[Times[Optional[Pattern[c, Blank[]]], Pattern[x, Blank[]]]], Optional[Pattern[b, Blank[]]]]], Optional[Pattern[p, Blank[]]]], Power[Pattern[x, Blank[]], Optional[Pattern[m, Blank[]]]], Power[Plus[Pattern[d, Blank[]], Times[Optional[Pattern[e, Blank[]]], Power[Pattern[x, Blank[]], 2]]], Pattern[q, Blank[]]]], Pattern[x, Blank[Symbol]]] := Dist[(d^(q + 1/2)*Sqrt[1 + c^2*x^2])/Sqrt[d + e*x^2], Int[x^m*(1 + c^2*x^2)^q*(a + b*ArcTan[c*x])^p, x], x] /; FreeQ[{a, b, c, d, e, p}, x] && EqQ[e, c^2*d] && IGtQ[m, 0] && ILtQ[m + 2*q + 1, 0] && !(IntegerQ[q] || GtQ[d, 0])
  4972. Int[Times[Power[Plus[Optional[Pattern[a, Blank[]]], Times[ArcCot[Times[Optional[Pattern[c, Blank[]]], Pattern[x, Blank[]]]], Optional[Pattern[b, Blank[]]]]], Optional[Pattern[p, Blank[]]]], Power[Pattern[x, Blank[]], Optional[Pattern[m, Blank[]]]], Power[Plus[Pattern[d, Blank[]], Times[Optional[Pattern[e, Blank[]]], Power[Pattern[x, Blank[]], 2]]], Pattern[q, Blank[]]]], Pattern[x, Blank[Symbol]]] := -Dist[d^q/c^(m + 1), Subst[Int[((a + b*x)^p*Cos[x]^m)/Sin[x]^(m + 2*(q + 1)), x], x, ArcCot[c*x]], x] /; FreeQ[{a, b, c, d, e, p}, x] && EqQ[e, c^2*d] && IGtQ[m, 0] && ILtQ[m + 2*q + 1, 0] && IntegerQ[q]
  4973. Int[Times[Power[Plus[Optional[Pattern[a, Blank[]]], Times[ArcCot[Times[Optional[Pattern[c, Blank[]]], Pattern[x, Blank[]]]], Optional[Pattern[b, Blank[]]]]], Optional[Pattern[p, Blank[]]]], Power[Pattern[x, Blank[]], Optional[Pattern[m, Blank[]]]], Power[Plus[Pattern[d, Blank[]], Times[Optional[Pattern[e, Blank[]]], Power[Pattern[x, Blank[]], 2]]], Pattern[q, Blank[]]]], Pattern[x, Blank[Symbol]]] := -Dist[(d^(q + 1/2)*x*Sqrt[(1 + c^2*x^2)/(c^2*x^2)])/(c^m*Sqrt[d + e*x^2]), Subst[Int[((a + b*x)^p*Cos[x]^m)/Sin[x]^(m + 2*(q + 1)), x], x, ArcCot[c*x]], x] /; FreeQ[{a, b, c, d, e, p}, x] && EqQ[e, c^2*d] && IGtQ[m, 0] && ILtQ[m + 2*q + 1, 0] && !IntegerQ[q]
  4974. Int[Times[Plus[Optional[Pattern[a, Blank[]]], Times[ArcTan[Times[Optional[Pattern[c, Blank[]]], Pattern[x, Blank[]]]], Optional[Pattern[b, Blank[]]]]], Pattern[x, Blank[]], Power[Plus[Optional[Pattern[d, Blank[]]], Times[Optional[Pattern[e, Blank[]]], Power[Pattern[x, Blank[]], 2]]], Optional[Pattern[q, Blank[]]]]], Pattern[x, Blank[Symbol]]] := Simp[((d + e*x^2)^(q + 1)*(a + b*ArcTan[c*x]))/(2*e*(q + 1)), x] - Dist[(b*c)/(2*e*(q + 1)), Int[(d + e*x^2)^(q + 1)/(1 + c^2*x^2), x], x] /; FreeQ[{a, b, c, d, e, q}, x] && NeQ[q, -1]
  4975. Int[Times[Plus[Optional[Pattern[a, Blank[]]], Times[ArcCot[Times[Optional[Pattern[c, Blank[]]], Pattern[x, Blank[]]]], Optional[Pattern[b, Blank[]]]]], Pattern[x, Blank[]], Power[Plus[Optional[Pattern[d, Blank[]]], Times[Optional[Pattern[e, Blank[]]], Power[Pattern[x, Blank[]], 2]]], Optional[Pattern[q, Blank[]]]]], Pattern[x, Blank[Symbol]]] := Simp[((d + e*x^2)^(q + 1)*(a + b*ArcCot[c*x]))/(2*e*(q + 1)), x] + Dist[(b*c)/(2*e*(q + 1)), Int[(d + e*x^2)^(q + 1)/(1 + c^2*x^2), x], x] /; FreeQ[{a, b, c, d, e, q}, x] && NeQ[q, -1]
  4976. Int[Times[Plus[Optional[Pattern[a, Blank[]]], Times[ArcTan[Times[Optional[Pattern[c, Blank[]]], Pattern[x, Blank[]]]], Optional[Pattern[b, Blank[]]]]], Power[Times[Optional[Pattern[f, Blank[]]], Pattern[x, Blank[]]], Optional[Pattern[m, Blank[]]]], Power[Plus[Optional[Pattern[d, Blank[]]], Times[Optional[Pattern[e, Blank[]]], Power[Pattern[x, Blank[]], 2]]], Optional[Pattern[q, Blank[]]]]], Pattern[x, Blank[Symbol]]] := With[{u = IntHide[(f*x)^m*(d + e*x^2)^q, x]}, Dist[a + b*ArcTan[c*x], u, x] - Dist[b*c, Int[SimplifyIntegrand[u/(1 + c^2*x^2), x], x], x]] /; FreeQ[{a, b, c, d, e, f, m, q}, x] && ((IGtQ[q, 0] && !(ILtQ[(m - 1)/2, 0] && GtQ[m + 2*q + 3, 0])) || (IGtQ[(m + 1)/2, 0] && !(ILtQ[q, 0] && GtQ[m + 2*q + 3, 0])) || (ILtQ[(m + 2*q + 1)/2, 0] && !ILtQ[(m - 1)/2, 0]))
  4977. Int[Times[Plus[Optional[Pattern[a, Blank[]]], Times[ArcCot[Times[Optional[Pattern[c, Blank[]]], Pattern[x, Blank[]]]], Optional[Pattern[b, Blank[]]]]], Power[Times[Optional[Pattern[f, Blank[]]], Pattern[x, Blank[]]], Optional[Pattern[m, Blank[]]]], Power[Plus[Optional[Pattern[d, Blank[]]], Times[Optional[Pattern[e, Blank[]]], Power[Pattern[x, Blank[]], 2]]], Optional[Pattern[q, Blank[]]]]], Pattern[x, Blank[Symbol]]] := With[{u = IntHide[(f*x)^m*(d + e*x^2)^q, x]}, Dist[a + b*ArcCot[c*x], u, x] + Dist[b*c, Int[SimplifyIntegrand[u/(1 + c^2*x^2), x], x], x]] /; FreeQ[{a, b, c, d, e, f, m, q}, x] && ((IGtQ[q, 0] && !(ILtQ[(m - 1)/2, 0] && GtQ[m + 2*q + 3, 0])) || (IGtQ[(m + 1)/2, 0] && !(ILtQ[q, 0] && GtQ[m + 2*q + 3, 0])) || (ILtQ[(m + 2*q + 1)/2, 0] && !ILtQ[(m - 1)/2, 0]))
  4978. Int[Times[Power[Plus[Optional[Pattern[a, Blank[]]], Times[ArcTan[Times[Optional[Pattern[c, Blank[]]], Pattern[x, Blank[]]]], Optional[Pattern[b, Blank[]]]]], Optional[Pattern[p, Blank[]]]], Pattern[x, Blank[]], Power[Plus[Pattern[d, Blank[]], Times[Optional[Pattern[e, Blank[]]], Power[Pattern[x, Blank[]], 2]]], -2]], Pattern[x, Blank[Symbol]]] := Dist[1/(4*d^2*Rt[-(e/d), 2]), Int[(a + b*ArcTan[c*x])^p/(1 - Rt[-(e/d), 2]*x)^2, x], x] - Dist[1/(4*d^2*Rt[-(e/d), 2]), Int[(a + b*ArcTan[c*x])^p/(1 + Rt[-(e/d), 2]*x)^2, x], x] /; FreeQ[{a, b, c, d, e}, x] && IGtQ[p, 0]
  4979. Int[Times[Power[Plus[Optional[Pattern[a, Blank[]]], Times[ArcCot[Times[Optional[Pattern[c, Blank[]]], Pattern[x, Blank[]]]], Optional[Pattern[b, Blank[]]]]], Optional[Pattern[p, Blank[]]]], Pattern[x, Blank[]], Power[Plus[Pattern[d, Blank[]], Times[Optional[Pattern[e, Blank[]]], Power[Pattern[x, Blank[]], 2]]], -2]], Pattern[x, Blank[Symbol]]] := Dist[1/(4*d^2*Rt[-(e/d), 2]), Int[(a + b*ArcCot[c*x])^p/(1 - Rt[-(e/d), 2]*x)^2, x], x] - Dist[1/(4*d^2*Rt[-(e/d), 2]), Int[(a + b*ArcCot[c*x])^p/(1 + Rt[-(e/d), 2]*x)^2, x], x] /; FreeQ[{a, b, c, d, e}, x] && IGtQ[p, 0]
  4980. Int[Times[Power[Plus[Optional[Pattern[a, Blank[]]], Times[ArcTan[Times[Optional[Pattern[c, Blank[]]], Pattern[x, Blank[]]]], Optional[Pattern[b, Blank[]]]]], Optional[Pattern[p, Blank[]]]], Power[Times[Optional[Pattern[f, Blank[]]], Pattern[x, Blank[]]], Optional[Pattern[m, Blank[]]]], Power[Plus[Pattern[d, Blank[]], Times[Optional[Pattern[e, Blank[]]], Power[Pattern[x, Blank[]], 2]]], Optional[Pattern[q, Blank[]]]]], Pattern[x, Blank[Symbol]]] := With[{u = ExpandIntegrand[(a + b*ArcTan[c*x])^p, (f*x)^m*(d + e*x^2)^q, x]}, Int[u, x] /; SumQ[u]] /; FreeQ[{a, b, c, d, e, f, m}, x] && IntegerQ[q] && IGtQ[p, 0] && ((EqQ[p, 1] && GtQ[q, 0]) || IntegerQ[m])
  4981. Int[Times[Power[Plus[Optional[Pattern[a, Blank[]]], Times[ArcCot[Times[Optional[Pattern[c, Blank[]]], Pattern[x, Blank[]]]], Optional[Pattern[b, Blank[]]]]], Optional[Pattern[p, Blank[]]]], Power[Times[Optional[Pattern[f, Blank[]]], Pattern[x, Blank[]]], Optional[Pattern[m, Blank[]]]], Power[Plus[Pattern[d, Blank[]], Times[Optional[Pattern[e, Blank[]]], Power[Pattern[x, Blank[]], 2]]], Optional[Pattern[q, Blank[]]]]], Pattern[x, Blank[Symbol]]] := With[{u = ExpandIntegrand[(a + b*ArcCot[c*x])^p, (f*x)^m*(d + e*x^2)^q, x]}, Int[u, x] /; SumQ[u]] /; FreeQ[{a, b, c, d, e, f, m}, x] && IntegerQ[q] && IGtQ[p, 0] && ((EqQ[p, 1] && GtQ[q, 0]) || IntegerQ[m])
  4982. Int[Times[Plus[Times[ArcTan[Times[Optional[Pattern[c, Blank[]]], Pattern[x, Blank[]]]], Optional[Pattern[b, Blank[]]]], Pattern[a, Blank[]]], Power[Times[Optional[Pattern[f, Blank[]]], Pattern[x, Blank[]]], Optional[Pattern[m, Blank[]]]], Power[Plus[Pattern[d, Blank[]], Times[Optional[Pattern[e, Blank[]]], Power[Pattern[x, Blank[]], 2]]], Optional[Pattern[q, Blank[]]]]], Pattern[x, Blank[Symbol]]] := Dist[a, Int[(f*x)^m*(d + e*x^2)^q, x], x] + Dist[b, Int[(f*x)^m*(d + e*x^2)^q*ArcTan[c*x], x], x] /; FreeQ[{a, b, c, d, e, f, m, q}, x]
  4983. Int[Times[Plus[Times[ArcCot[Times[Optional[Pattern[c, Blank[]]], Pattern[x, Blank[]]]], Optional[Pattern[b, Blank[]]]], Pattern[a, Blank[]]], Power[Times[Optional[Pattern[f, Blank[]]], Pattern[x, Blank[]]], Optional[Pattern[m, Blank[]]]], Power[Plus[Pattern[d, Blank[]], Times[Optional[Pattern[e, Blank[]]], Power[Pattern[x, Blank[]], 2]]], Optional[Pattern[q, Blank[]]]]], Pattern[x, Blank[Symbol]]] := Dist[a, Int[(f*x)^m*(d + e*x^2)^q, x], x] + Dist[b, Int[(f*x)^m*(d + e*x^2)^q*ArcCot[c*x], x], x] /; FreeQ[{a, b, c, d, e, f, m, q}, x]
  4984. Int[Times[Power[Plus[Optional[Pattern[a, Blank[]]], Times[ArcTan[Times[Optional[Pattern[c, Blank[]]], Pattern[x, Blank[]]]], Optional[Pattern[b, Blank[]]]]], Optional[Pattern[p, Blank[]]]], Power[Plus[Pattern[f, Blank[]], Times[Optional[Pattern[g, Blank[]]], Pattern[x, Blank[]]]], Optional[Pattern[m, Blank[]]]], Power[Plus[Pattern[d, Blank[]], Times[Optional[Pattern[e, Blank[]]], Power[Pattern[x, Blank[]], 2]]], -1]], Pattern[x, Blank[Symbol]]] := Int[ExpandIntegrand[(a + b*ArcTan[c*x])^p/(d + e*x^2), (f + g*x)^m, x], x] /; FreeQ[{a, b, c, d, e, f, g}, x] && IGtQ[p, 0] && EqQ[e, c^2*d] && IGtQ[m, 0]
  4985. Int[Times[Power[Plus[Optional[Pattern[a, Blank[]]], Times[ArcCot[Times[Optional[Pattern[c, Blank[]]], Pattern[x, Blank[]]]], Optional[Pattern[b, Blank[]]]]], Optional[Pattern[p, Blank[]]]], Power[Plus[Pattern[f, Blank[]], Times[Optional[Pattern[g, Blank[]]], Pattern[x, Blank[]]]], Optional[Pattern[m, Blank[]]]], Power[Plus[Pattern[d, Blank[]], Times[Optional[Pattern[e, Blank[]]], Power[Pattern[x, Blank[]], 2]]], -1]], Pattern[x, Blank[Symbol]]] := Int[ExpandIntegrand[(a + b*ArcCot[c*x])^p/(d + e*x^2), (f + g*x)^m, x], x] /; FreeQ[{a, b, c, d, e, f, g}, x] && IGtQ[p, 0] && EqQ[e, c^2*d] && IGtQ[m, 0]
  4986. Int[Times[ArcTanh[Pattern[u, Blank[]]], Power[Plus[Optional[Pattern[a, Blank[]]], Times[ArcTan[Times[Optional[Pattern[c, Blank[]]], Pattern[x, Blank[]]]], Optional[Pattern[b, Blank[]]]]], Optional[Pattern[p, Blank[]]]], Power[Plus[Pattern[d, Blank[]], Times[Optional[Pattern[e, Blank[]]], Power[Pattern[x, Blank[]], 2]]], -1]], Pattern[x, Blank[Symbol]]] := Dist[1/2, Int[(Log[1 + u]*(a + b*ArcTan[c*x])^p)/(d + e*x^2), x], x] - Dist[1/2, Int[(Log[1 - u]*(a + b*ArcTan[c*x])^p)/(d + e*x^2), x], x] /; FreeQ[{a, b, c, d, e}, x] && IGtQ[p, 0] && EqQ[e, c^2*d] && EqQ[u^2 - (1 - (2*I)/(I + c*x))^2, 0]
  4987. Int[Times[ArcCoth[Pattern[u, Blank[]]], Power[Plus[Optional[Pattern[a, Blank[]]], Times[ArcCot[Times[Optional[Pattern[c, Blank[]]], Pattern[x, Blank[]]]], Optional[Pattern[b, Blank[]]]]], Optional[Pattern[p, Blank[]]]], Power[Plus[Pattern[d, Blank[]], Times[Optional[Pattern[e, Blank[]]], Power[Pattern[x, Blank[]], 2]]], -1]], Pattern[x, Blank[Symbol]]] := Dist[1/2, Int[(Log[SimplifyIntegrand[1 + 1/u, x]]*(a + b*ArcCot[c*x])^p)/(d + e*x^2), x], x] - Dist[1/2, Int[(Log[SimplifyIntegrand[1 - 1/u, x]]*(a + b*ArcCot[c*x])^p)/(d + e*x^2), x], x] /; FreeQ[{a, b, c, d, e}, x] && IGtQ[p, 0] && EqQ[e, c^2*d] && EqQ[u^2 - (1 - (2*I)/(I + c*x))^2, 0]
  4988. Int[Times[ArcTanh[Pattern[u, Blank[]]], Power[Plus[Optional[Pattern[a, Blank[]]], Times[ArcTan[Times[Optional[Pattern[c, Blank[]]], Pattern[x, Blank[]]]], Optional[Pattern[b, Blank[]]]]], Optional[Pattern[p, Blank[]]]], Power[Plus[Pattern[d, Blank[]], Times[Optional[Pattern[e, Blank[]]], Power[Pattern[x, Blank[]], 2]]], -1]], Pattern[x, Blank[Symbol]]] := Dist[1/2, Int[(Log[1 + u]*(a + b*ArcTan[c*x])^p)/(d + e*x^2), x], x] - Dist[1/2, Int[(Log[1 - u]*(a + b*ArcTan[c*x])^p)/(d + e*x^2), x], x] /; FreeQ[{a, b, c, d, e}, x] && IGtQ[p, 0] && EqQ[e, c^2*d] && EqQ[u^2 - (1 - (2*I)/(I - c*x))^2, 0]
  4989. Int[Times[ArcCoth[Pattern[u, Blank[]]], Power[Plus[Optional[Pattern[a, Blank[]]], Times[ArcCot[Times[Optional[Pattern[c, Blank[]]], Pattern[x, Blank[]]]], Optional[Pattern[b, Blank[]]]]], Optional[Pattern[p, Blank[]]]], Power[Plus[Pattern[d, Blank[]], Times[Optional[Pattern[e, Blank[]]], Power[Pattern[x, Blank[]], 2]]], -1]], Pattern[x, Blank[Symbol]]] := Dist[1/2, Int[(Log[SimplifyIntegrand[1 + 1/u, x]]*(a + b*ArcCot[c*x])^p)/(d + e*x^2), x], x] - Dist[1/2, Int[(Log[SimplifyIntegrand[1 - 1/u, x]]*(a + b*ArcCot[c*x])^p)/(d + e*x^2), x], x] /; FreeQ[{a, b, c, d, e}, x] && IGtQ[p, 0] && EqQ[e, c^2*d] && EqQ[u^2 - (1 - (2*I)/(I - c*x))^2, 0]
  4990. Int[Times[Log[Plus[Pattern[f, Blank[]], Times[Optional[Pattern[g, Blank[]]], Pattern[x, Blank[]]]]], Power[Plus[Optional[Pattern[a, Blank[]]], Times[ArcTan[Times[Optional[Pattern[c, Blank[]]], Pattern[x, Blank[]]]], Optional[Pattern[b, Blank[]]]]], Optional[Pattern[p, Blank[]]]], Power[Plus[Pattern[d, Blank[]], Times[Optional[Pattern[e, Blank[]]], Power[Pattern[x, Blank[]], 2]]], -1]], Pattern[x, Blank[Symbol]]] := Simp[((a + b*ArcTan[c*x])^(p + 1)*Log[f + g*x])/(b*c*d*(p + 1)), x] - Dist[g/(b*c*d*(p + 1)), Int[(a + b*ArcTan[c*x])^(p + 1)/(f + g*x), x], x] /; FreeQ[{a, b, c, d, e, f, g}, x] && IGtQ[p, 0] && EqQ[e, c^2*d] && EqQ[c^2*f^2 + g^2, 0]
  4991. Int[Times[Log[Plus[Pattern[f, Blank[]], Times[Optional[Pattern[g, Blank[]]], Pattern[x, Blank[]]]]], Power[Plus[Optional[Pattern[a, Blank[]]], Times[ArcCot[Times[Optional[Pattern[c, Blank[]]], Pattern[x, Blank[]]]], Optional[Pattern[b, Blank[]]]]], Optional[Pattern[p, Blank[]]]], Power[Plus[Pattern[d, Blank[]], Times[Optional[Pattern[e, Blank[]]], Power[Pattern[x, Blank[]], 2]]], -1]], Pattern[x, Blank[Symbol]]] := Simp[((a + b*ArcCot[c*x])^(p + 1)*Log[f + g*x])/(b*c*d*(p + 1)), x] - Dist[g/(b*c*d*(p + 1)), Int[(a + b*ArcCot[c*x])^(p + 1)/(f + g*x), x], x] /; FreeQ[{a, b, c, d, e, f, g}, x] && IGtQ[p, 0] && EqQ[e, c^2*d] && EqQ[c^2*f^2 + g^2, 0]
  4992. Int[Times[Log[Pattern[u, Blank[]]], Power[Plus[Optional[Pattern[a, Blank[]]], Times[ArcTan[Times[Optional[Pattern[c, Blank[]]], Pattern[x, Blank[]]]], Optional[Pattern[b, Blank[]]]]], Optional[Pattern[p, Blank[]]]], Power[Plus[Pattern[d, Blank[]], Times[Optional[Pattern[e, Blank[]]], Power[Pattern[x, Blank[]], 2]]], -1]], Pattern[x, Blank[Symbol]]] := Simp[(I*(a + b*ArcTan[c*x])^p*PolyLog[2, 1 - u])/(2*c*d), x] - Dist[(b*p*I)/2, Int[((a + b*ArcTan[c*x])^(p - 1)*PolyLog[2, 1 - u])/(d + e*x^2), x], x] /; FreeQ[{a, b, c, d, e}, x] && IGtQ[p, 0] && EqQ[e, c^2*d] && EqQ[(1 - u)^2 - (1 - (2*I)/(I + c*x))^2, 0]
  4993. Int[Times[Log[Pattern[u, Blank[]]], Power[Plus[Optional[Pattern[a, Blank[]]], Times[ArcCot[Times[Optional[Pattern[c, Blank[]]], Pattern[x, Blank[]]]], Optional[Pattern[b, Blank[]]]]], Optional[Pattern[p, Blank[]]]], Power[Plus[Pattern[d, Blank[]], Times[Optional[Pattern[e, Blank[]]], Power[Pattern[x, Blank[]], 2]]], -1]], Pattern[x, Blank[Symbol]]] := Simp[(I*(a + b*ArcCot[c*x])^p*PolyLog[2, 1 - u])/(2*c*d), x] + Dist[(b*p*I)/2, Int[((a + b*ArcCot[c*x])^(p - 1)*PolyLog[2, 1 - u])/(d + e*x^2), x], x] /; FreeQ[{a, b, c, d, e}, x] && IGtQ[p, 0] && EqQ[e, c^2*d] && EqQ[(1 - u)^2 - (1 - (2*I)/(I + c*x))^2, 0]
  4994. Int[Times[Log[Pattern[u, Blank[]]], Power[Plus[Optional[Pattern[a, Blank[]]], Times[ArcTan[Times[Optional[Pattern[c, Blank[]]], Pattern[x, Blank[]]]], Optional[Pattern[b, Blank[]]]]], Optional[Pattern[p, Blank[]]]], Power[Plus[Pattern[d, Blank[]], Times[Optional[Pattern[e, Blank[]]], Power[Pattern[x, Blank[]], 2]]], -1]], Pattern[x, Blank[Symbol]]] := -Simp[(I*(a + b*ArcTan[c*x])^p*PolyLog[2, 1 - u])/(2*c*d), x] + Dist[(b*p*I)/2, Int[((a + b*ArcTan[c*x])^(p - 1)*PolyLog[2, 1 - u])/(d + e*x^2), x], x] /; FreeQ[{a, b, c, d, e}, x] && IGtQ[p, 0] && EqQ[e, c^2*d] && EqQ[(1 - u)^2 - (1 - (2*I)/(I - c*x))^2, 0]
  4995. Int[Times[Log[Pattern[u, Blank[]]], Power[Plus[Optional[Pattern[a, Blank[]]], Times[ArcCot[Times[Optional[Pattern[c, Blank[]]], Pattern[x, Blank[]]]], Optional[Pattern[b, Blank[]]]]], Optional[Pattern[p, Blank[]]]], Power[Plus[Pattern[d, Blank[]], Times[Optional[Pattern[e, Blank[]]], Power[Pattern[x, Blank[]], 2]]], -1]], Pattern[x, Blank[Symbol]]] := -Simp[(I*(a + b*ArcCot[c*x])^p*PolyLog[2, 1 - u])/(2*c*d), x] - Dist[(b*p*I)/2, Int[((a + b*ArcCot[c*x])^(p - 1)*PolyLog[2, 1 - u])/(d + e*x^2), x], x] /; FreeQ[{a, b, c, d, e}, x] && IGtQ[p, 0] && EqQ[e, c^2*d] && EqQ[(1 - u)^2 - (1 - (2*I)/(I - c*x))^2, 0]
  4996. Int[Times[Power[Plus[Optional[Pattern[a, Blank[]]], Times[ArcTan[Times[Optional[Pattern[c, Blank[]]], Pattern[x, Blank[]]]], Optional[Pattern[b, Blank[]]]]], Optional[Pattern[p, Blank[]]]], Power[Plus[Pattern[d, Blank[]], Times[Optional[Pattern[e, Blank[]]], Power[Pattern[x, Blank[]], 2]]], -1], PolyLog[Pattern[k, Blank[]], Pattern[u, Blank[]]]], Pattern[x, Blank[Symbol]]] := -Simp[(I*(a + b*ArcTan[c*x])^p*PolyLog[k + 1, u])/(2*c*d), x] + Dist[(b*p*I)/2, Int[((a + b*ArcTan[c*x])^(p - 1)*PolyLog[k + 1, u])/(d + e*x^2), x], x] /; FreeQ[{a, b, c, d, e, k}, x] && IGtQ[p, 0] && EqQ[e, c^2*d] && EqQ[u^2 - (1 - (2*I)/(I + c*x))^2, 0]
  4997. Int[Times[Power[Plus[Optional[Pattern[a, Blank[]]], Times[ArcCot[Times[Optional[Pattern[c, Blank[]]], Pattern[x, Blank[]]]], Optional[Pattern[b, Blank[]]]]], Optional[Pattern[p, Blank[]]]], Power[Plus[Pattern[d, Blank[]], Times[Optional[Pattern[e, Blank[]]], Power[Pattern[x, Blank[]], 2]]], -1], PolyLog[Pattern[k, Blank[]], Pattern[u, Blank[]]]], Pattern[x, Blank[Symbol]]] := -Simp[(I*(a + b*ArcCot[c*x])^p*PolyLog[k + 1, u])/(2*c*d), x] - Dist[(b*p*I)/2, Int[((a + b*ArcCot[c*x])^(p - 1)*PolyLog[k + 1, u])/(d + e*x^2), x], x] /; FreeQ[{a, b, c, d, e, k}, x] && IGtQ[p, 0] && EqQ[e, c^2*d] && EqQ[u^2 - (1 - (2*I)/(I + c*x))^2, 0]
  4998. Int[Times[Power[Plus[Optional[Pattern[a, Blank[]]], Times[ArcTan[Times[Optional[Pattern[c, Blank[]]], Pattern[x, Blank[]]]], Optional[Pattern[b, Blank[]]]]], Optional[Pattern[p, Blank[]]]], Power[Plus[Pattern[d, Blank[]], Times[Optional[Pattern[e, Blank[]]], Power[Pattern[x, Blank[]], 2]]], -1], PolyLog[Pattern[k, Blank[]], Pattern[u, Blank[]]]], Pattern[x, Blank[Symbol]]] := Simp[(I*(a + b*ArcTan[c*x])^p*PolyLog[k + 1, u])/(2*c*d), x] - Dist[(b*p*I)/2, Int[((a + b*ArcTan[c*x])^(p - 1)*PolyLog[k + 1, u])/(d + e*x^2), x], x] /; FreeQ[{a, b, c, d, e, k}, x] && IGtQ[p, 0] && EqQ[e, c^2*d] && EqQ[u^2 - (1 - (2*I)/(I - c*x))^2, 0]
  4999. Int[Times[Power[Plus[Optional[Pattern[a, Blank[]]], Times[ArcCot[Times[Optional[Pattern[c, Blank[]]], Pattern[x, Blank[]]]], Optional[Pattern[b, Blank[]]]]], Optional[Pattern[p, Blank[]]]], Power[Plus[Pattern[d, Blank[]], Times[Optional[Pattern[e, Blank[]]], Power[Pattern[x, Blank[]], 2]]], -1], PolyLog[Pattern[k, Blank[]], Pattern[u, Blank[]]]], Pattern[x, Blank[Symbol]]] := Simp[(I*(a + b*ArcCot[c*x])^p*PolyLog[k + 1, u])/(2*c*d), x] + Dist[(b*p*I)/2, Int[((a + b*ArcCot[c*x])^(p - 1)*PolyLog[k + 1, u])/(d + e*x^2), x], x] /; FreeQ[{a, b, c, d, e, k}, x] && IGtQ[p, 0] && EqQ[e, c^2*d] && EqQ[u^2 - (1 - (2*I)/(I - c*x))^2, 0]
  5000. Int[Times[Power[Plus[Optional[Pattern[a, Blank[]]], Times[ArcCot[Times[Optional[Pattern[c, Blank[]]], Pattern[x, Blank[]]]], Optional[Pattern[b, Blank[]]]]], -1], Power[Plus[Optional[Pattern[a, Blank[]]], Times[ArcTan[Times[Optional[Pattern[c, Blank[]]], Pattern[x, Blank[]]]], Optional[Pattern[b, Blank[]]]]], -1], Power[Plus[Pattern[d, Blank[]], Times[Optional[Pattern[e, Blank[]]], Power[Pattern[x, Blank[]], 2]]], -1]], Pattern[x, Blank[Symbol]]] := Simp[(-Log[a + b*ArcCot[c*x]] + Log[a + b*ArcTan[c*x]])/(b*c*d*(2*a + b*ArcCot[c*x] + b*ArcTan[c*x])), x] /; FreeQ[{a, b, c, d, e}, x] && EqQ[e, c^2*d]
  5001. Int[Times[Power[Plus[Optional[Pattern[a, Blank[]]], Times[ArcCot[Times[Optional[Pattern[c, Blank[]]], Pattern[x, Blank[]]]], Optional[Pattern[b, Blank[]]]]], Optional[Pattern[q, Blank[]]]], Power[Plus[Optional[Pattern[a, Blank[]]], Times[ArcTan[Times[Optional[Pattern[c, Blank[]]], Pattern[x, Blank[]]]], Optional[Pattern[b, Blank[]]]]], Optional[Pattern[p, Blank[]]]], Power[Plus[Pattern[d, Blank[]], Times[Optional[Pattern[e, Blank[]]], Power[Pattern[x, Blank[]], 2]]], -1]], Pattern[x, Blank[Symbol]]] := -Simp[((a + b*ArcCot[c*x])^(q + 1)*(a + b*ArcTan[c*x])^p)/(b*c*d*(q + 1)), x] + Dist[p/(q + 1), Int[((a + b*ArcCot[c*x])^(q + 1)*(a + b*ArcTan[c*x])^(p - 1))/(d + e*x^2), x], x] /; FreeQ[{a, b, c, d, e}, x] && EqQ[e, c^2*d] && IGtQ[p, 0] && IGeQ[q, p]
  5002. Int[Times[Power[Plus[Optional[Pattern[a, Blank[]]], Times[ArcCot[Times[Optional[Pattern[c, Blank[]]], Pattern[x, Blank[]]]], Optional[Pattern[b, Blank[]]]]], Optional[Pattern[p, Blank[]]]], Power[Plus[Optional[Pattern[a, Blank[]]], Times[ArcTan[Times[Optional[Pattern[c, Blank[]]], Pattern[x, Blank[]]]], Optional[Pattern[b, Blank[]]]]], Optional[Pattern[q, Blank[]]]], Power[Plus[Pattern[d, Blank[]], Times[Optional[Pattern[e, Blank[]]], Power[Pattern[x, Blank[]], 2]]], -1]], Pattern[x, Blank[Symbol]]] := Simp[((a + b*ArcTan[c*x])^(q + 1)*(a + b*ArcCot[c*x])^p)/(b*c*d*(q + 1)), x] + Dist[p/(q + 1), Int[((a + b*ArcTan[c*x])^(q + 1)*(a + b*ArcCot[c*x])^(p - 1))/(d + e*x^2), x], x] /; FreeQ[{a, b, c, d, e}, x] && EqQ[e, c^2*d] && IGtQ[p, 0] && IGeQ[q, p]
  5003. Int[Times[ArcTan[Times[Optional[Pattern[a, Blank[]]], Pattern[x, Blank[]]]], Power[Plus[Pattern[c, Blank[]], Times[Optional[Pattern[d, Blank[]]], Power[Pattern[x, Blank[]], Optional[Pattern[n, Blank[]]]]]], -1]], Pattern[x, Blank[Symbol]]] := Dist[I/2, Int[Log[1 - I*a*x]/(c + d*x^n), x], x] - Dist[I/2, Int[Log[1 + I*a*x]/(c + d*x^n), x], x] /; FreeQ[{a, c, d}, x] && IntegerQ[n] && !(EqQ[n, 2] && EqQ[d, a^2*c])
  5004. Int[Times[ArcCot[Times[Optional[Pattern[a, Blank[]]], Pattern[x, Blank[]]]], Power[Plus[Pattern[c, Blank[]], Times[Optional[Pattern[d, Blank[]]], Power[Pattern[x, Blank[]], Optional[Pattern[n, Blank[]]]]]], -1]], Pattern[x, Blank[Symbol]]] := Dist[I/2, Int[Log[1 - I/(a*x)]/(c + d*x^n), x], x] - Dist[I/2, Int[Log[1 + I/(a*x)]/(c + d*x^n), x], x] /; FreeQ[{a, c, d}, x] && IntegerQ[n] && !(EqQ[n, 2] && EqQ[d, a^2*c])
  5005. Int[Times[ArcTan[Times[Optional[Pattern[c, Blank[]]], Power[Pattern[x, Blank[]], Optional[Pattern[n, Blank[]]]]]], Log[Times[Optional[Pattern[d, Blank[]]], Power[Pattern[x, Blank[]], Optional[Pattern[m, Blank[]]]]]], Power[Pattern[x, Blank[]], -1]], Pattern[x, Blank[Symbol]]] := Dist[I/2, Int[(Log[d*x^m]*Log[1 - I*c*x^n])/x, x], x] - Dist[I/2, Int[(Log[d*x^m]*Log[1 + I*c*x^n])/x, x], x] /; FreeQ[{c, d, m, n}, x]
  5006. Int[Times[ArcCot[Times[Optional[Pattern[c, Blank[]]], Power[Pattern[x, Blank[]], Optional[Pattern[n, Blank[]]]]]], Log[Times[Optional[Pattern[d, Blank[]]], Power[Pattern[x, Blank[]], Optional[Pattern[m, Blank[]]]]]], Power[Pattern[x, Blank[]], -1]], Pattern[x, Blank[Symbol]]] := Dist[I/2, Int[(Log[d*x^m]*Log[1 - I/(c*x^n)])/x, x], x] - Dist[I/2, Int[(Log[d*x^m]*Log[1 + I/(c*x^n)])/x, x], x] /; FreeQ[{c, d, m, n}, x]
  5007. Int[Times[Log[Times[Optional[Pattern[d, Blank[]]], Power[Pattern[x, Blank[]], Optional[Pattern[m, Blank[]]]]]], Plus[Times[ArcTan[Times[Optional[Pattern[c, Blank[]]], Power[Pattern[x, Blank[]], Optional[Pattern[n, Blank[]]]]]], Optional[Pattern[b, Blank[]]]], Pattern[a, Blank[]]], Power[Pattern[x, Blank[]], -1]], Pattern[x, Blank[Symbol]]] := Dist[a, Int[Log[d*x^m]/x, x], x] + Dist[b, Int[(Log[d*x^m]*ArcTan[c*x^n])/x, x], x] /; FreeQ[{a, b, c, d, m, n}, x]
  5008. Int[Times[Log[Times[Optional[Pattern[d, Blank[]]], Power[Pattern[x, Blank[]], Optional[Pattern[m, Blank[]]]]]], Plus[Times[ArcCot[Times[Optional[Pattern[c, Blank[]]], Power[Pattern[x, Blank[]], Optional[Pattern[n, Blank[]]]]]], Optional[Pattern[b, Blank[]]]], Pattern[a, Blank[]]], Power[Pattern[x, Blank[]], -1]], Pattern[x, Blank[Symbol]]] := Dist[a, Int[Log[d*x^m]/x, x], x] + Dist[b, Int[(Log[d*x^m]*ArcCot[c*x^n])/x, x], x] /; FreeQ[{a, b, c, d, m, n}, x]
  5009. Int[Times[Plus[Optional[Pattern[a, Blank[]]], Times[ArcTan[Times[Optional[Pattern[c, Blank[]]], Pattern[x, Blank[]]]], Optional[Pattern[b, Blank[]]]]], Plus[Optional[Pattern[d, Blank[]]], Times[Log[Plus[Optional[Pattern[f, Blank[]]], Times[Optional[Pattern[g, Blank[]]], Power[Pattern[x, Blank[]], 2]]]], Optional[Pattern[e, Blank[]]]]]], Pattern[x, Blank[Symbol]]] := Simp[x*(d + e*Log[f + g*x^2])*(a + b*ArcTan[c*x]), x] + (-Dist[b*c, Int[(x*(d + e*Log[f + g*x^2]))/(1 + c^2*x^2), x], x] - Dist[2*e*g, Int[(x^2*(a + b*ArcTan[c*x]))/(f + g*x^2), x], x]) /; FreeQ[{a, b, c, d, e, f, g}, x]
  5010. Int[Times[Plus[Optional[Pattern[a, Blank[]]], Times[ArcCot[Times[Optional[Pattern[c, Blank[]]], Pattern[x, Blank[]]]], Optional[Pattern[b, Blank[]]]]], Plus[Optional[Pattern[d, Blank[]]], Times[Log[Plus[Optional[Pattern[f, Blank[]]], Times[Optional[Pattern[g, Blank[]]], Power[Pattern[x, Blank[]], 2]]]], Optional[Pattern[e, Blank[]]]]]], Pattern[x, Blank[Symbol]]] := Simp[x*(d + e*Log[f + g*x^2])*(a + b*ArcCot[c*x]), x] + (Dist[b*c, Int[(x*(d + e*Log[f + g*x^2]))/(1 + c^2*x^2), x], x] - Dist[2*e*g, Int[(x^2*(a + b*ArcCot[c*x]))/(f + g*x^2), x], x]) /; FreeQ[{a, b, c, d, e, f, g}, x]
  5011. Int[Times[ArcTan[Times[Optional[Pattern[c, Blank[]]], Pattern[x, Blank[]]]], Log[Plus[Optional[Pattern[f, Blank[]]], Times[Optional[Pattern[g, Blank[]]], Power[Pattern[x, Blank[]], 2]]]], Power[Pattern[x, Blank[]], -1]], Pattern[x, Blank[Symbol]]] := Dist[Log[f + g*x^2] - Log[1 - I*c*x] - Log[1 + I*c*x], Int[ArcTan[c*x]/x, x], x] + (Dist[I/2, Int[Log[1 - I*c*x]^2/x, x], x] - Dist[I/2, Int[Log[1 + I*c*x]^2/x, x], x]) /; FreeQ[{c, f, g}, x] && EqQ[g, c^2*f]
  5012. Int[Times[ArcCot[Times[Optional[Pattern[c, Blank[]]], Pattern[x, Blank[]]]], Log[Plus[Optional[Pattern[f, Blank[]]], Times[Optional[Pattern[g, Blank[]]], Power[Pattern[x, Blank[]], 2]]]], Power[Pattern[x, Blank[]], -1]], Pattern[x, Blank[Symbol]]] := Dist[Log[f + g*x^2] - Log[c^2*x^2] - Log[1 - I/(c*x)] - Log[1 + I/(c*x)], Int[ArcCot[c*x]/x, x], x] + (Dist[I/2, Int[Log[1 - I/(c*x)]^2/x, x], x] - Dist[I/2, Int[Log[1 + I/(c*x)]^2/x, x], x] + Int[(Log[c^2*x^2]*ArcCot[c*x])/x, x]) /; FreeQ[{c, f, g}, x] && EqQ[g, c^2*f]
  5013. Int[Times[Log[Plus[Optional[Pattern[f, Blank[]]], Times[Optional[Pattern[g, Blank[]]], Power[Pattern[x, Blank[]], 2]]]], Plus[Times[ArcTan[Times[Optional[Pattern[c, Blank[]]], Pattern[x, Blank[]]]], Optional[Pattern[b, Blank[]]]], Pattern[a, Blank[]]], Power[Pattern[x, Blank[]], -1]], Pattern[x, Blank[Symbol]]] := Dist[a, Int[Log[f + g*x^2]/x, x], x] + Dist[b, Int[(Log[f + g*x^2]*ArcTan[c*x])/x, x], x] /; FreeQ[{a, b, c, f, g}, x]
  5014. Int[Times[Log[Plus[Optional[Pattern[f, Blank[]]], Times[Optional[Pattern[g, Blank[]]], Power[Pattern[x, Blank[]], 2]]]], Plus[Times[ArcCot[Times[Optional[Pattern[c, Blank[]]], Pattern[x, Blank[]]]], Optional[Pattern[b, Blank[]]]], Pattern[a, Blank[]]], Power[Pattern[x, Blank[]], -1]], Pattern[x, Blank[Symbol]]] := Dist[a, Int[Log[f + g*x^2]/x, x], x] + Dist[b, Int[(Log[f + g*x^2]*ArcCot[c*x])/x, x], x] /; FreeQ[{a, b, c, f, g}, x]
  5015. Int[Times[Plus[Optional[Pattern[a, Blank[]]], Times[ArcTan[Times[Optional[Pattern[c, Blank[]]], Pattern[x, Blank[]]]], Optional[Pattern[b, Blank[]]]]], Plus[Times[Log[Plus[Optional[Pattern[f, Blank[]]], Times[Optional[Pattern[g, Blank[]]], Power[Pattern[x, Blank[]], 2]]]], Optional[Pattern[e, Blank[]]]], Pattern[d, Blank[]]], Power[Pattern[x, Blank[]], -1]], Pattern[x, Blank[Symbol]]] := Dist[d, Int[(a + b*ArcTan[c*x])/x, x], x] + Dist[e, Int[(Log[f + g*x^2]*(a + b*ArcTan[c*x]))/x, x], x] /; FreeQ[{a, b, c, d, e, f, g}, x]
  5016. Int[Times[Plus[Optional[Pattern[a, Blank[]]], Times[ArcCot[Times[Optional[Pattern[c, Blank[]]], Pattern[x, Blank[]]]], Optional[Pattern[b, Blank[]]]]], Plus[Times[Log[Plus[Optional[Pattern[f, Blank[]]], Times[Optional[Pattern[g, Blank[]]], Power[Pattern[x, Blank[]], 2]]]], Optional[Pattern[e, Blank[]]]], Pattern[d, Blank[]]], Power[Pattern[x, Blank[]], -1]], Pattern[x, Blank[Symbol]]] := Dist[d, Int[(a + b*ArcCot[c*x])/x, x], x] + Dist[e, Int[(Log[f + g*x^2]*(a + b*ArcCot[c*x]))/x, x], x] /; FreeQ[{a, b, c, d, e, f, g}, x]
  5017. Int[Times[Plus[Optional[Pattern[a, Blank[]]], Times[ArcTan[Times[Optional[Pattern[c, Blank[]]], Pattern[x, Blank[]]]], Optional[Pattern[b, Blank[]]]]], Plus[Optional[Pattern[d, Blank[]]], Times[Log[Plus[Optional[Pattern[f, Blank[]]], Times[Optional[Pattern[g, Blank[]]], Power[Pattern[x, Blank[]], 2]]]], Optional[Pattern[e, Blank[]]]]], Power[Pattern[x, Blank[]], Optional[Pattern[m, Blank[]]]]], Pattern[x, Blank[Symbol]]] := Simp[(x^(m + 1)*(d + e*Log[f + g*x^2])*(a + b*ArcTan[c*x]))/(m + 1), x] + (-Dist[(b*c)/(m + 1), Int[(x^(m + 1)*(d + e*Log[f + g*x^2]))/(1 + c^2*x^2), x], x] - Dist[(2*e*g)/(m + 1), Int[(x^(m + 2)*(a + b*ArcTan[c*x]))/(f + g*x^2), x], x]) /; FreeQ[{a, b, c, d, e, f, g}, x] && ILtQ[m/2, 0]
  5018. Int[Times[Plus[Optional[Pattern[a, Blank[]]], Times[ArcCot[Times[Optional[Pattern[c, Blank[]]], Pattern[x, Blank[]]]], Optional[Pattern[b, Blank[]]]]], Plus[Optional[Pattern[d, Blank[]]], Times[Log[Plus[Optional[Pattern[f, Blank[]]], Times[Optional[Pattern[g, Blank[]]], Power[Pattern[x, Blank[]], 2]]]], Optional[Pattern[e, Blank[]]]]], Power[Pattern[x, Blank[]], Optional[Pattern[m, Blank[]]]]], Pattern[x, Blank[Symbol]]] := Simp[(x^(m + 1)*(d + e*Log[f + g*x^2])*(a + b*ArcCot[c*x]))/(m + 1), x] + (Dist[(b*c)/(m + 1), Int[(x^(m + 1)*(d + e*Log[f + g*x^2]))/(1 + c^2*x^2), x], x] - Dist[(2*e*g)/(m + 1), Int[(x^(m + 2)*(a + b*ArcCot[c*x]))/(f + g*x^2), x], x]) /; FreeQ[{a, b, c, d, e, f, g}, x] && ILtQ[m/2, 0]
  5019. Int[Times[Plus[Optional[Pattern[a, Blank[]]], Times[ArcTan[Times[Optional[Pattern[c, Blank[]]], Pattern[x, Blank[]]]], Optional[Pattern[b, Blank[]]]]], Plus[Optional[Pattern[d, Blank[]]], Times[Log[Plus[Optional[Pattern[f, Blank[]]], Times[Optional[Pattern[g, Blank[]]], Power[Pattern[x, Blank[]], 2]]]], Optional[Pattern[e, Blank[]]]]], Power[Pattern[x, Blank[]], Optional[Pattern[m, Blank[]]]]], Pattern[x, Blank[Symbol]]] := With[{u = IntHide[x^m*(d + e*Log[f + g*x^2]), x]}, Dist[a + b*ArcTan[c*x], u, x] - Dist[b*c, Int[ExpandIntegrand[u/(1 + c^2*x^2), x], x], x]] /; FreeQ[{a, b, c, d, e, f, g}, x] && IGtQ[(m + 1)/2, 0]
  5020. Int[Times[Plus[Optional[Pattern[a, Blank[]]], Times[ArcCot[Times[Optional[Pattern[c, Blank[]]], Pattern[x, Blank[]]]], Optional[Pattern[b, Blank[]]]]], Plus[Optional[Pattern[d, Blank[]]], Times[Log[Plus[Optional[Pattern[f, Blank[]]], Times[Optional[Pattern[g, Blank[]]], Power[Pattern[x, Blank[]], 2]]]], Optional[Pattern[e, Blank[]]]]], Power[Pattern[x, Blank[]], Optional[Pattern[m, Blank[]]]]], Pattern[x, Blank[Symbol]]] := With[{u = IntHide[x^m*(d + e*Log[f + g*x^2]), x]}, Dist[a + b*ArcCot[c*x], u, x] + Dist[b*c, Int[ExpandIntegrand[u/(1 + c^2*x^2), x], x], x]] /; FreeQ[{a, b, c, d, e, f, g}, x] && IGtQ[(m + 1)/2, 0]
  5021. Int[Times[Plus[Optional[Pattern[a, Blank[]]], Times[ArcTan[Times[Optional[Pattern[c, Blank[]]], Pattern[x, Blank[]]]], Optional[Pattern[b, Blank[]]]]], Plus[Optional[Pattern[d, Blank[]]], Times[Log[Plus[Optional[Pattern[f, Blank[]]], Times[Optional[Pattern[g, Blank[]]], Power[Pattern[x, Blank[]], 2]]]], Optional[Pattern[e, Blank[]]]]], Power[Pattern[x, Blank[]], Optional[Pattern[m, Blank[]]]]], Pattern[x, Blank[Symbol]]] := With[{u = IntHide[x^m*(a + b*ArcTan[c*x]), x]}, Dist[d + e*Log[f + g*x^2], u, x] - Dist[2*e*g, Int[ExpandIntegrand[(x*u)/(f + g*x^2), x], x], x]] /; FreeQ[{a, b, c, d, e, f, g}, x] && IntegerQ[m] && NeQ[m, -1]
  5022. Int[Times[Plus[Optional[Pattern[a, Blank[]]], Times[ArcCot[Times[Optional[Pattern[c, Blank[]]], Pattern[x, Blank[]]]], Optional[Pattern[b, Blank[]]]]], Plus[Optional[Pattern[d, Blank[]]], Times[Log[Plus[Optional[Pattern[f, Blank[]]], Times[Optional[Pattern[g, Blank[]]], Power[Pattern[x, Blank[]], 2]]]], Optional[Pattern[e, Blank[]]]]], Power[Pattern[x, Blank[]], Optional[Pattern[m, Blank[]]]]], Pattern[x, Blank[Symbol]]] := With[{u = IntHide[x^m*(a + b*ArcCot[c*x]), x]}, Dist[d + e*Log[f + g*x^2], u, x] - Dist[2*e*g, Int[ExpandIntegrand[(x*u)/(f + g*x^2), x], x], x]] /; FreeQ[{a, b, c, d, e, f, g}, x] && IntegerQ[m] && NeQ[m, -1]
  5023. Int[Times[Power[Plus[Optional[Pattern[a, Blank[]]], Times[ArcTan[Times[Optional[Pattern[c, Blank[]]], Pattern[x, Blank[]]]], Optional[Pattern[b, Blank[]]]]], 2], Plus[Optional[Pattern[d, Blank[]]], Times[Log[Plus[Pattern[f, Blank[]], Times[Optional[Pattern[g, Blank[]]], Power[Pattern[x, Blank[]], 2]]]], Optional[Pattern[e, Blank[]]]]], Pattern[x, Blank[]]], Pattern[x, Blank[Symbol]]] := Simp[((f + g*x^2)*(d + e*Log[f + g*x^2])*(a + b*ArcTan[c*x])^2)/(2*g), x] + (-Dist[b/c, Int[(d + e*Log[f + g*x^2])*(a + b*ArcTan[c*x]), x], x] + Dist[b*c*e, Int[(x^2*(a + b*ArcTan[c*x]))/(1 + c^2*x^2), x], x] - Simp[(e*x^2*(a + b*ArcTan[c*x])^2)/2, x]) /; FreeQ[{a, b, c, d, e, f, g}, x] && EqQ[g, c^2*f]
  5024. Int[Times[Power[Plus[Optional[Pattern[a, Blank[]]], Times[ArcCot[Times[Optional[Pattern[c, Blank[]]], Pattern[x, Blank[]]]], Optional[Pattern[b, Blank[]]]]], 2], Plus[Optional[Pattern[d, Blank[]]], Times[Log[Plus[Pattern[f, Blank[]], Times[Optional[Pattern[g, Blank[]]], Power[Pattern[x, Blank[]], 2]]]], Optional[Pattern[e, Blank[]]]]], Pattern[x, Blank[]]], Pattern[x, Blank[Symbol]]] := Simp[((f + g*x^2)*(d + e*Log[f + g*x^2])*(a + b*ArcCot[c*x])^2)/(2*g), x] + (Dist[b/c, Int[(d + e*Log[f + g*x^2])*(a + b*ArcCot[c*x]), x], x] - Dist[b*c*e, Int[(x^2*(a + b*ArcCot[c*x]))/(1 + c^2*x^2), x], x] - Simp[(e*x^2*(a + b*ArcCot[c*x])^2)/2, x]) /; FreeQ[{a, b, c, d, e, f, g}, x] && EqQ[g, c^2*f]
  5025. Int[Times[Power[Plus[Optional[Pattern[a, Blank[]]], Times[ArcTan[Times[Optional[Pattern[c, Blank[]]], Pattern[x, Blank[]]]], Optional[Pattern[b, Blank[]]]]], Optional[Pattern[p, Blank[]]]], Optional[Pattern[u, Blank[]]]], Pattern[x, Blank[Symbol]]] := Unintegrable[u*(a + b*ArcTan[c*x])^p, x] /; FreeQ[{a, b, c, p}, x] && (EqQ[u, 1] || MatchQ[u, ((d_.) + (e_.)*x)^(q_.) /; FreeQ[{d, e, q}, x]] || MatchQ[u, ((f_.)*x)^(m_.)*((d_.) + (e_.)*x)^(q_.) /; FreeQ[{d, e, f, m, q}, x]] || MatchQ[u, ((d_.) + (e_.)*x^2)^(q_.) /; FreeQ[{d, e, q}, x]] || MatchQ[u, ((f_.)*x)^(m_.)*((d_.) + (e_.)*x^2)^(q_.) /; FreeQ[{d, e, f, m, q}, x]])
  5026. Int[Times[Power[Plus[Optional[Pattern[a, Blank[]]], Times[ArcCot[Times[Optional[Pattern[c, Blank[]]], Pattern[x, Blank[]]]], Optional[Pattern[b, Blank[]]]]], Optional[Pattern[p, Blank[]]]], Optional[Pattern[u, Blank[]]]], Pattern[x, Blank[Symbol]]] := Unintegrable[u*(a + b*ArcCot[c*x])^p, x] /; FreeQ[{a, b, c, p}, x] && (EqQ[u, 1] || MatchQ[u, ((d_.) + (e_.)*x)^(q_.) /; FreeQ[{d, e, q}, x]] || MatchQ[u, ((f_.)*x)^(m_.)*((d_.) + (e_.)*x)^(q_.) /; FreeQ[{d, e, f, m, q}, x]] || MatchQ[u, ((d_.) + (e_.)*x^2)^(q_.) /; FreeQ[{d, e, q}, x]] || MatchQ[u, ((f_.)*x)^(m_.)*((d_.) + (e_.)*x^2)^(q_.) /; FreeQ[{d, e, f, m, q}, x]])
  5027. Int[ArcTan[Times[Optional[Pattern[c, Blank[]]], Power[Pattern[x, Blank[]], Pattern[n, Blank[]]]]], Pattern[x, Blank[Symbol]]] := Simp[x*ArcTan[c*x^n], x] - Dist[c*n, Int[x^n/(1 + c^2*x^(2*n)), x], x] /; FreeQ[{c, n}, x]
  5028. Int[ArcCot[Times[Optional[Pattern[c, Blank[]]], Power[Pattern[x, Blank[]], Pattern[n, Blank[]]]]], Pattern[x, Blank[Symbol]]] := Simp[x*ArcCot[c*x^n], x] + Dist[c*n, Int[x^n/(1 + c^2*x^(2*n)), x], x] /; FreeQ[{c, n}, x]
  5029. Int[Power[Plus[Optional[Pattern[a, Blank[]]], Times[ArcTan[Times[Optional[Pattern[c, Blank[]]], Power[Pattern[x, Blank[]], Optional[Pattern[n, Blank[]]]]]], Optional[Pattern[b, Blank[]]]]], Optional[Pattern[p, Blank[]]]], Pattern[x, Blank[Symbol]]] := Int[ExpandIntegrand[(a + (I*b*Log[1 - I*c*x^n])/2 - (I*b*Log[1 + I*c*x^n])/2)^p, x], x] /; FreeQ[{a, b, c, n}, x] && IGtQ[p, 0] && IntegerQ[n]
  5030. Int[Power[Plus[Optional[Pattern[a, Blank[]]], Times[ArcCot[Times[Optional[Pattern[c, Blank[]]], Power[Pattern[x, Blank[]], Optional[Pattern[n, Blank[]]]]]], Optional[Pattern[b, Blank[]]]]], Optional[Pattern[p, Blank[]]]], Pattern[x, Blank[Symbol]]] := Int[ExpandIntegrand[(a + (I*b*Log[1 - I/(x^n*c)])/2 - (I*b*Log[1 + I/(x^n*c)])/2)^p, x], x] /; FreeQ[{a, b, c, n}, x] && IGtQ[p, 0] && IntegerQ[n]
  5031. Int[Times[Power[Plus[Optional[Pattern[a, Blank[]]], Times[ArcTan[Times[Optional[Pattern[c, Blank[]]], Power[Pattern[x, Blank[]], Pattern[n, Blank[]]]]], Optional[Pattern[b, Blank[]]]]], Optional[Pattern[p, Blank[]]]], Power[Pattern[x, Blank[]], -1]], Pattern[x, Blank[Symbol]]] := Dist[1/n, Subst[Int[(a + b*ArcTan[c*x])^p/x, x], x, x^n], x] /; FreeQ[{a, b, c, n}, x] && IGtQ[p, 0]
  5032. Int[Times[Power[Plus[Optional[Pattern[a, Blank[]]], Times[ArcCot[Times[Optional[Pattern[c, Blank[]]], Power[Pattern[x, Blank[]], Pattern[n, Blank[]]]]], Optional[Pattern[b, Blank[]]]]], Optional[Pattern[p, Blank[]]]], Power[Pattern[x, Blank[]], -1]], Pattern[x, Blank[Symbol]]] := Dist[1/n, Subst[Int[(a + b*ArcCot[c*x])^p/x, x], x, x^n], x] /; FreeQ[{a, b, c, n}, x] && IGtQ[p, 0]
  5033. Int[Times[Plus[Optional[Pattern[a, Blank[]]], Times[ArcTan[Times[Optional[Pattern[c, Blank[]]], Power[Pattern[x, Blank[]], Pattern[n, Blank[]]]]], Optional[Pattern[b, Blank[]]]]], Power[Times[Optional[Pattern[d, Blank[]]], Pattern[x, Blank[]]], Optional[Pattern[m, Blank[]]]]], Pattern[x, Blank[Symbol]]] := Simp[((d*x)^(m + 1)*(a + b*ArcTan[c*x^n]))/(d*(m + 1)), x] - Dist[(b*c*n)/(d*(m + 1)), Int[(x^(n - 1)*(d*x)^(m + 1))/(1 + c^2*x^(2*n)), x], x] /; FreeQ[{a, b, c, d, m, n}, x] && NeQ[m, -1]
  5034. Int[Times[Plus[Optional[Pattern[a, Blank[]]], Times[ArcCot[Times[Optional[Pattern[c, Blank[]]], Power[Pattern[x, Blank[]], Pattern[n, Blank[]]]]], Optional[Pattern[b, Blank[]]]]], Power[Times[Optional[Pattern[d, Blank[]]], Pattern[x, Blank[]]], Optional[Pattern[m, Blank[]]]]], Pattern[x, Blank[Symbol]]] := Simp[((d*x)^(m + 1)*(a + b*ArcCot[c*x^n]))/(d*(m + 1)), x] + Dist[(b*c*n)/(d*(m + 1)), Int[(x^(n - 1)*(d*x)^(m + 1))/(1 + c^2*x^(2*n)), x], x] /; FreeQ[{a, b, c, d, m, n}, x] && NeQ[m, -1]
  5035. Int[Times[Power[Plus[Optional[Pattern[a, Blank[]]], Times[ArcTan[Times[Optional[Pattern[c, Blank[]]], Power[Pattern[x, Blank[]], Optional[Pattern[n, Blank[]]]]]], Optional[Pattern[b, Blank[]]]]], Optional[Pattern[p, Blank[]]]], Power[Times[Optional[Pattern[d, Blank[]]], Pattern[x, Blank[]]], Optional[Pattern[m, Blank[]]]]], Pattern[x, Blank[Symbol]]] := Int[ExpandIntegrand[(d*x)^m*(a + (I*b*Log[1 - I*c*x^n])/2 - (I*b*Log[1 + I*c*x^n])/2)^p, x], x] /; FreeQ[{a, b, c, d, m, n}, x] && IGtQ[p, 0] && IntegerQ[m] && IntegerQ[n]
  5036. Int[Times[Power[Plus[Optional[Pattern[a, Blank[]]], Times[ArcCot[Times[Optional[Pattern[c, Blank[]]], Power[Pattern[x, Blank[]], Optional[Pattern[n, Blank[]]]]]], Optional[Pattern[b, Blank[]]]]], Optional[Pattern[p, Blank[]]]], Power[Times[Optional[Pattern[d, Blank[]]], Pattern[x, Blank[]]], Optional[Pattern[m, Blank[]]]]], Pattern[x, Blank[Symbol]]] := Int[ExpandIntegrand[(d*x)^m*(a + (I*b*Log[1 - I/(x^n*c)])/2 - (I*b*Log[1 + I/(x^n*c)])/2)^p, x], x] /; FreeQ[{a, b, c, d, m, n}, x] && IGtQ[p, 0] && IntegerQ[m] && IntegerQ[n]
  5037. Int[Times[Power[Plus[Optional[Pattern[a, Blank[]]], Times[ArcTan[Times[Optional[Pattern[c, Blank[]]], Power[Pattern[x, Blank[]], Pattern[n, Blank[]]]]], Optional[Pattern[b, Blank[]]]]], Optional[Pattern[p, Blank[]]]], Optional[Pattern[u, Blank[]]]], Pattern[x, Blank[Symbol]]] := Unintegrable[u*(a + b*ArcTan[c*x^n])^p, x] /; FreeQ[{a, b, c, n, p}, x] && (EqQ[u, 1] || MatchQ[u, ((d_.)*x)^(m_.) /; FreeQ[{d, m}, x]])
  5038. Int[Times[Power[Plus[Optional[Pattern[a, Blank[]]], Times[ArcCot[Times[Optional[Pattern[c, Blank[]]], Power[Pattern[x, Blank[]], Pattern[n, Blank[]]]]], Optional[Pattern[b, Blank[]]]]], Optional[Pattern[p, Blank[]]]], Optional[Pattern[u, Blank[]]]], Pattern[x, Blank[Symbol]]] := Unintegrable[u*(a + b*ArcCot[c*x^n])^p, x] /; FreeQ[{a, b, c, n, p}, x] && (EqQ[u, 1] || MatchQ[u, ((d_.)*x)^(m_.) /; FreeQ[{d, m}, x]])
  5039. Int[Power[Plus[Optional[Pattern[a, Blank[]]], Times[ArcTan[Plus[Pattern[c, Blank[]], Times[Optional[Pattern[d, Blank[]]], Pattern[x, Blank[]]]]], Optional[Pattern[b, Blank[]]]]], Optional[Pattern[p, Blank[]]]], Pattern[x, Blank[Symbol]]] := Dist[1/d, Subst[Int[(a + b*ArcTan[x])^p, x], x, c + d*x], x] /; FreeQ[{a, b, c, d}, x] && IGtQ[p, 0]
  5040. Int[Power[Plus[Optional[Pattern[a, Blank[]]], Times[ArcCot[Plus[Pattern[c, Blank[]], Times[Optional[Pattern[d, Blank[]]], Pattern[x, Blank[]]]]], Optional[Pattern[b, Blank[]]]]], Optional[Pattern[p, Blank[]]]], Pattern[x, Blank[Symbol]]] := Dist[1/d, Subst[Int[(a + b*ArcCot[x])^p, x], x, c + d*x], x] /; FreeQ[{a, b, c, d}, x] && IGtQ[p, 0]
  5041. Int[Power[Plus[Optional[Pattern[a, Blank[]]], Times[ArcTan[Plus[Pattern[c, Blank[]], Times[Optional[Pattern[d, Blank[]]], Pattern[x, Blank[]]]]], Optional[Pattern[b, Blank[]]]]], Pattern[p, Blank[]]], Pattern[x, Blank[Symbol]]] := Unintegrable[(a + b*ArcTan[c + d*x])^p, x] /; FreeQ[{a, b, c, d, p}, x] && !IGtQ[p, 0]
  5042. Int[Power[Plus[Optional[Pattern[a, Blank[]]], Times[ArcCot[Plus[Pattern[c, Blank[]], Times[Optional[Pattern[d, Blank[]]], Pattern[x, Blank[]]]]], Optional[Pattern[b, Blank[]]]]], Pattern[p, Blank[]]], Pattern[x, Blank[Symbol]]] := Unintegrable[(a + b*ArcCot[c + d*x])^p, x] /; FreeQ[{a, b, c, d, p}, x] && !IGtQ[p, 0]
  5043. Int[Times[Power[Plus[Optional[Pattern[a, Blank[]]], Times[ArcTan[Plus[Pattern[c, Blank[]], Times[Optional[Pattern[d, Blank[]]], Pattern[x, Blank[]]]]], Optional[Pattern[b, Blank[]]]]], Optional[Pattern[p, Blank[]]]], Power[Plus[Optional[Pattern[e, Blank[]]], Times[Optional[Pattern[f, Blank[]]], Pattern[x, Blank[]]]], Optional[Pattern[m, Blank[]]]]], Pattern[x, Blank[Symbol]]] := Dist[1/d, Subst[Int[((f*x)/d)^m*(a + b*ArcTan[x])^p, x], x, c + d*x], x] /; FreeQ[{a, b, c, d, e, f, m}, x] && EqQ[d*e - c*f, 0] && IGtQ[p, 0]
  5044. Int[Times[Power[Plus[Optional[Pattern[a, Blank[]]], Times[ArcCot[Plus[Pattern[c, Blank[]], Times[Optional[Pattern[d, Blank[]]], Pattern[x, Blank[]]]]], Optional[Pattern[b, Blank[]]]]], Optional[Pattern[p, Blank[]]]], Power[Plus[Optional[Pattern[e, Blank[]]], Times[Optional[Pattern[f, Blank[]]], Pattern[x, Blank[]]]], Optional[Pattern[m, Blank[]]]]], Pattern[x, Blank[Symbol]]] := Dist[1/d, Subst[Int[((f*x)/d)^m*(a + b*ArcCot[x])^p, x], x, c + d*x], x] /; FreeQ[{a, b, c, d, e, f, m}, x] && EqQ[d*e - c*f, 0] && IGtQ[p, 0]
  5045. Int[Times[Power[Plus[Optional[Pattern[a, Blank[]]], Times[ArcTan[Plus[Pattern[c, Blank[]], Times[Optional[Pattern[d, Blank[]]], Pattern[x, Blank[]]]]], Optional[Pattern[b, Blank[]]]]], Optional[Pattern[p, Blank[]]]], Power[Plus[Optional[Pattern[e, Blank[]]], Times[Optional[Pattern[f, Blank[]]], Pattern[x, Blank[]]]], Pattern[m, Blank[]]]], Pattern[x, Blank[Symbol]]] := Simp[((e + f*x)^(m + 1)*(a + b*ArcTan[c + d*x])^p)/(f*(m + 1)), x] - Dist[(b*d*p)/(f*(m + 1)), Int[((e + f*x)^(m + 1)*(a + b*ArcTan[c + d*x])^(p - 1))/(1 + (c + d*x)^2), x], x] /; FreeQ[{a, b, c, d, e, f}, x] && IGtQ[p, 0] && ILtQ[m, -1]
  5046. Int[Times[Power[Plus[Optional[Pattern[a, Blank[]]], Times[ArcCot[Plus[Pattern[c, Blank[]], Times[Optional[Pattern[d, Blank[]]], Pattern[x, Blank[]]]]], Optional[Pattern[b, Blank[]]]]], Optional[Pattern[p, Blank[]]]], Power[Plus[Optional[Pattern[e, Blank[]]], Times[Optional[Pattern[f, Blank[]]], Pattern[x, Blank[]]]], Pattern[m, Blank[]]]], Pattern[x, Blank[Symbol]]] := Simp[((e + f*x)^(m + 1)*(a + b*ArcCot[c + d*x])^p)/(f*(m + 1)), x] + Dist[(b*d*p)/(f*(m + 1)), Int[((e + f*x)^(m + 1)*(a + b*ArcCot[c + d*x])^(p - 1))/(1 + (c + d*x)^2), x], x] /; FreeQ[{a, b, c, d, e, f}, x] && IGtQ[p, 0] && ILtQ[m, -1]
  5047. Int[Times[Power[Plus[Optional[Pattern[a, Blank[]]], Times[ArcTan[Plus[Pattern[c, Blank[]], Times[Optional[Pattern[d, Blank[]]], Pattern[x, Blank[]]]]], Optional[Pattern[b, Blank[]]]]], Optional[Pattern[p, Blank[]]]], Power[Plus[Optional[Pattern[e, Blank[]]], Times[Optional[Pattern[f, Blank[]]], Pattern[x, Blank[]]]], Optional[Pattern[m, Blank[]]]]], Pattern[x, Blank[Symbol]]] := Dist[1/d, Subst[Int[((d*e - c*f)/d + (f*x)/d)^m*(a + b*ArcTan[x])^p, x], x, c + d*x], x] /; FreeQ[{a, b, c, d, e, f, m, p}, x] && IGtQ[p, 0]
  5048. Int[Times[Power[Plus[Optional[Pattern[a, Blank[]]], Times[ArcCot[Plus[Pattern[c, Blank[]], Times[Optional[Pattern[d, Blank[]]], Pattern[x, Blank[]]]]], Optional[Pattern[b, Blank[]]]]], Optional[Pattern[p, Blank[]]]], Power[Plus[Optional[Pattern[e, Blank[]]], Times[Optional[Pattern[f, Blank[]]], Pattern[x, Blank[]]]], Optional[Pattern[m, Blank[]]]]], Pattern[x, Blank[Symbol]]] := Dist[1/d, Subst[Int[((d*e - c*f)/d + (f*x)/d)^m*(a + b*ArcCot[x])^p, x], x, c + d*x], x] /; FreeQ[{a, b, c, d, e, f, m, p}, x] && IGtQ[p, 0]
  5049. Int[Times[Power[Plus[Optional[Pattern[a, Blank[]]], Times[ArcTan[Plus[Pattern[c, Blank[]], Times[Optional[Pattern[d, Blank[]]], Pattern[x, Blank[]]]]], Optional[Pattern[b, Blank[]]]]], Pattern[p, Blank[]]], Power[Plus[Optional[Pattern[e, Blank[]]], Times[Optional[Pattern[f, Blank[]]], Pattern[x, Blank[]]]], Optional[Pattern[m, Blank[]]]]], Pattern[x, Blank[Symbol]]] := Unintegrable[(e + f*x)^m*(a + b*ArcTan[c + d*x])^p, x] /; FreeQ[{a, b, c, d, e, f, m, p}, x] && !IGtQ[p, 0]
  5050. Int[Times[Power[Plus[Optional[Pattern[a, Blank[]]], Times[ArcCot[Plus[Pattern[c, Blank[]], Times[Optional[Pattern[d, Blank[]]], Pattern[x, Blank[]]]]], Optional[Pattern[b, Blank[]]]]], Pattern[p, Blank[]]], Power[Plus[Optional[Pattern[e, Blank[]]], Times[Optional[Pattern[f, Blank[]]], Pattern[x, Blank[]]]], Optional[Pattern[m, Blank[]]]]], Pattern[x, Blank[Symbol]]] := Unintegrable[(e + f*x)^m*(a + b*ArcCot[c + d*x])^p, x] /; FreeQ[{a, b, c, d, e, f, m, p}, x] && !IGtQ[p, 0]
  5051. Int[Times[ArcTan[Plus[Pattern[a, Blank[]], Times[Optional[Pattern[b, Blank[]]], Pattern[x, Blank[]]]]], Power[Plus[Pattern[c, Blank[]], Times[Optional[Pattern[d, Blank[]]], Power[Pattern[x, Blank[]], Optional[Pattern[n, Blank[]]]]]], -1]], Pattern[x, Blank[Symbol]]] := Dist[I/2, Int[Log[1 - I*a - I*b*x]/(c + d*x^n), x], x] - Dist[I/2, Int[Log[1 + I*a + I*b*x]/(c + d*x^n), x], x] /; FreeQ[{a, b, c, d}, x] && RationalQ[n]
  5052. Int[Times[ArcCot[Plus[Pattern[a, Blank[]], Times[Optional[Pattern[b, Blank[]]], Pattern[x, Blank[]]]]], Power[Plus[Pattern[c, Blank[]], Times[Optional[Pattern[d, Blank[]]], Power[Pattern[x, Blank[]], Optional[Pattern[n, Blank[]]]]]], -1]], Pattern[x, Blank[Symbol]]] := Dist[I/2, Int[Log[(-I + a + b*x)/(a + b*x)]/(c + d*x^n), x], x] - Dist[I/2, Int[Log[(I + a + b*x)/(a + b*x)]/(c + d*x^n), x], x] /; FreeQ[{a, b, c, d}, x] && RationalQ[n]
  5053. Int[Times[ArcTan[Plus[Pattern[a, Blank[]], Times[Optional[Pattern[b, Blank[]]], Pattern[x, Blank[]]]]], Power[Plus[Pattern[c, Blank[]], Times[Optional[Pattern[d, Blank[]]], Power[Pattern[x, Blank[]], Pattern[n, Blank[]]]]], -1]], Pattern[x, Blank[Symbol]]] := Unintegrable[ArcTan[a + b*x]/(c + d*x^n), x] /; FreeQ[{a, b, c, d, n}, x] && !RationalQ[n]
  5054. Int[Times[ArcCot[Plus[Pattern[a, Blank[]], Times[Optional[Pattern[b, Blank[]]], Pattern[x, Blank[]]]]], Power[Plus[Pattern[c, Blank[]], Times[Optional[Pattern[d, Blank[]]], Power[Pattern[x, Blank[]], Pattern[n, Blank[]]]]], -1]], Pattern[x, Blank[Symbol]]] := Unintegrable[ArcCot[a + b*x]/(c + d*x^n), x] /; FreeQ[{a, b, c, d, n}, x] && !RationalQ[n]
  5055. Int[Times[Power[Plus[Optional[Pattern[a, Blank[]]], Times[ArcTan[Plus[Pattern[c, Blank[]], Times[Optional[Pattern[d, Blank[]]], Pattern[x, Blank[]]]]], Optional[Pattern[b, Blank[]]]]], Optional[Pattern[p, Blank[]]]], Power[Plus[Optional[Pattern[A, Blank[]]], Times[Optional[Pattern[B, Blank[]]], Pattern[x, Blank[]]], Times[Optional[Pattern[C, Blank[]]], Power[Pattern[x, Blank[]], 2]]], Optional[Pattern[q, Blank[]]]]], Pattern[x, Blank[Symbol]]] := Dist[1/d, Subst[Int[(C/d^2 + (C*x^2)/d^2)^q*(a + b*ArcTan[x])^p, x], x, c + d*x], x] /; FreeQ[{a, b, c, d, A, B, C, p, q}, x] && EqQ[B*(1 + c^2) - 2*A*c*d, 0] && EqQ[2*c*C - B*d, 0]
  5056. Int[Times[Power[Plus[Optional[Pattern[a, Blank[]]], Times[ArcCot[Plus[Pattern[c, Blank[]], Times[Optional[Pattern[d, Blank[]]], Pattern[x, Blank[]]]]], Optional[Pattern[b, Blank[]]]]], Optional[Pattern[p, Blank[]]]], Power[Plus[Optional[Pattern[A, Blank[]]], Times[Optional[Pattern[B, Blank[]]], Pattern[x, Blank[]]], Times[Optional[Pattern[C, Blank[]]], Power[Pattern[x, Blank[]], 2]]], Optional[Pattern[q, Blank[]]]]], Pattern[x, Blank[Symbol]]] := Dist[1/d, Subst[Int[(C/d^2 + (C*x^2)/d^2)^q*(a + b*ArcCot[x])^p, x], x, c + d*x], x] /; FreeQ[{a, b, c, d, A, B, C, p, q}, x] && EqQ[B*(1 + c^2) - 2*A*c*d, 0] && EqQ[2*c*C - B*d, 0]
  5057. Int[Times[Power[Plus[Optional[Pattern[a, Blank[]]], Times[ArcTan[Plus[Pattern[c, Blank[]], Times[Optional[Pattern[d, Blank[]]], Pattern[x, Blank[]]]]], Optional[Pattern[b, Blank[]]]]], Optional[Pattern[p, Blank[]]]], Power[Plus[Optional[Pattern[e, Blank[]]], Times[Optional[Pattern[f, Blank[]]], Pattern[x, Blank[]]]], Optional[Pattern[m, Blank[]]]], Power[Plus[Optional[Pattern[A, Blank[]]], Times[Optional[Pattern[B, Blank[]]], Pattern[x, Blank[]]], Times[Optional[Pattern[C, Blank[]]], Power[Pattern[x, Blank[]], 2]]], Optional[Pattern[q, Blank[]]]]], Pattern[x, Blank[Symbol]]] := Dist[1/d, Subst[Int[((d*e - c*f)/d + (f*x)/d)^m*(C/d^2 + (C*x^2)/d^2)^q*(a + b*ArcTan[x])^p, x], x, c + d*x], x] /; FreeQ[{a, b, c, d, e, f, A, B, C, m, p, q}, x] && EqQ[B*(1 + c^2) - 2*A*c*d, 0] && EqQ[2*c*C - B*d, 0]
  5058. Int[Times[Power[Plus[Optional[Pattern[a, Blank[]]], Times[ArcCot[Plus[Pattern[c, Blank[]], Times[Optional[Pattern[d, Blank[]]], Pattern[x, Blank[]]]]], Optional[Pattern[b, Blank[]]]]], Optional[Pattern[p, Blank[]]]], Power[Plus[Optional[Pattern[e, Blank[]]], Times[Optional[Pattern[f, Blank[]]], Pattern[x, Blank[]]]], Optional[Pattern[m, Blank[]]]], Power[Plus[Optional[Pattern[A, Blank[]]], Times[Optional[Pattern[B, Blank[]]], Pattern[x, Blank[]]], Times[Optional[Pattern[C, Blank[]]], Power[Pattern[x, Blank[]], 2]]], Optional[Pattern[q, Blank[]]]]], Pattern[x, Blank[Symbol]]] := Dist[1/d, Subst[Int[((d*e - c*f)/d + (f*x)/d)^m*(C/d^2 + (C*x^2)/d^2)^q*(a + b*ArcCot[x])^p, x], x, c + d*x], x] /; FreeQ[{a, b, c, d, e, f, A, B, C, m, p, q}, x] && EqQ[B*(1 + c^2) - 2*A*c*d, 0] && EqQ[2*c*C - B*d, 0]
  5059. Int[Power[E, Times[ArcTan[Times[Optional[Pattern[a, Blank[]]], Pattern[x, Blank[]]]], Pattern[n, Blank[]]]], Pattern[x, Blank[Symbol]]] := Int[(1 - I*a*x)^((I*n + 1)/2)/((1 + I*a*x)^((I*n - 1)/2)*Sqrt[1 + a^2*x^2]), x] /; FreeQ[a, x] && IntegerQ[(I*n - 1)/2]
  5060. Int[Times[Power[E, Times[ArcTan[Times[Optional[Pattern[a, Blank[]]], Pattern[x, Blank[]]]], Pattern[n, Blank[]]]], Power[Pattern[x, Blank[]], Optional[Pattern[m, Blank[]]]]], Pattern[x, Blank[Symbol]]] := Int[x^m*((1 - I*a*x)^((I*n + 1)/2)/((1 + I*a*x)^((I*n - 1)/2)*Sqrt[1 + a^2*x^2])), x] /; FreeQ[{a, m}, x] && IntegerQ[(I*n - 1)/2]
  5061. Int[Power[E, Times[ArcTan[Times[Optional[Pattern[a, Blank[]]], Pattern[x, Blank[]]]], Optional[Pattern[n, Blank[]]]]], Pattern[x, Blank[Symbol]]] := Int[(1 - I*a*x)^((I*n)/2)/(1 + I*a*x)^((I*n)/2), x] /; FreeQ[{a, n}, x] && !IntegerQ[(I*n - 1)/2]
  5062. Int[Times[Power[E, Times[ArcTan[Times[Optional[Pattern[a, Blank[]]], Pattern[x, Blank[]]]], Optional[Pattern[n, Blank[]]]]], Power[Pattern[x, Blank[]], Optional[Pattern[m, Blank[]]]]], Pattern[x, Blank[Symbol]]] := Int[(x^m*(1 - I*a*x)^((I*n)/2))/(1 + I*a*x)^((I*n)/2), x] /; FreeQ[{a, m, n}, x] && !IntegerQ[(I*n - 1)/2]
  5063. Int[Times[Power[E, Times[ArcTan[Times[Optional[Pattern[a, Blank[]]], Pattern[x, Blank[]]]], Optional[Pattern[n, Blank[]]]]], Optional[Pattern[u, Blank[]]], Power[Plus[Pattern[c, Blank[]], Times[Optional[Pattern[d, Blank[]]], Pattern[x, Blank[]]]], Optional[Pattern[p, Blank[]]]]], Pattern[x, Blank[Symbol]]] := Dist[c^p, Int[(u*(1 + (d*x)/c)^p*(1 - I*a*x)^((I*n)/2))/(1 + I*a*x)^((I*n)/2), x], x] /; FreeQ[{a, c, d, n, p}, x] && EqQ[a^2*c^2 + d^2, 0] && (IntegerQ[p] || GtQ[c, 0])
  5064. Int[Times[Power[E, Times[ArcTan[Times[Optional[Pattern[a, Blank[]]], Pattern[x, Blank[]]]], Optional[Pattern[n, Blank[]]]]], Optional[Pattern[u, Blank[]]], Power[Plus[Pattern[c, Blank[]], Times[Optional[Pattern[d, Blank[]]], Pattern[x, Blank[]]]], Optional[Pattern[p, Blank[]]]]], Pattern[x, Blank[Symbol]]] := Int[(u*(c + d*x)^p*(1 - I*a*x)^((I*n)/2))/(1 + I*a*x)^((I*n)/2), x] /; FreeQ[{a, c, d, n, p}, x] && EqQ[a^2*c^2 + d^2, 0] && !(IntegerQ[p] || GtQ[c, 0])
  5065. Int[Times[Power[E, Times[ArcTan[Times[Optional[Pattern[a, Blank[]]], Pattern[x, Blank[]]]], Optional[Pattern[n, Blank[]]]]], Optional[Pattern[u, Blank[]]], Power[Plus[Pattern[c, Blank[]], Times[Optional[Pattern[d, Blank[]]], Power[Pattern[x, Blank[]], -1]]], Optional[Pattern[p, Blank[]]]]], Pattern[x, Blank[Symbol]]] := Dist[d^p, Int[(u*(1 + (c*x)/d)^p*E^(n*ArcTan[a*x]))/x^p, x], x] /; FreeQ[{a, c, d, n}, x] && EqQ[c^2 + a^2*d^2, 0] && IntegerQ[p]
  5066. Int[Times[Power[E, Times[ArcTanh[Times[Optional[Pattern[a, Blank[]]], Pattern[x, Blank[]]]], Pattern[n, Blank[]]]], Optional[Pattern[u, Blank[]]], Power[Plus[Pattern[c, Blank[]], Times[Optional[Pattern[d, Blank[]]], Power[Pattern[x, Blank[]], -1]]], Pattern[p, Blank[]]]], Pattern[x, Blank[Symbol]]] := Dist[(-1)^(n/2)*c^p, Int[(u*(1 + d/(c*x))^p*(1 - 1/(I*a*x))^((I*n)/2))/(1 + 1/(I*a*x))^((I*n)/2), x], x] /; FreeQ[{a, c, d, p}, x] && EqQ[c^2 + a^2*d^2, 0] && !IntegerQ[p] && IntegerQ[(I*n)/2] && GtQ[c, 0]
  5067. Int[Times[Power[E, Times[ArcTan[Times[Optional[Pattern[a, Blank[]]], Pattern[x, Blank[]]]], Pattern[n, Blank[]]]], Optional[Pattern[u, Blank[]]], Power[Plus[Pattern[c, Blank[]], Times[Optional[Pattern[d, Blank[]]], Power[Pattern[x, Blank[]], -1]]], Pattern[p, Blank[]]]], Pattern[x, Blank[Symbol]]] := Int[(u*(c + d/x)^p*(1 - I*a*x)^((I*n)/2))/(1 + I*a*x)^((I*n)/2), x] /; FreeQ[{a, c, d, p}, x] && EqQ[c^2 + a^2*d^2, 0] && !IntegerQ[p] && IntegerQ[(I*n)/2] && !GtQ[c, 0]
  5068. Int[Times[Power[E, Times[ArcTan[Times[Optional[Pattern[a, Blank[]]], Pattern[x, Blank[]]]], Optional[Pattern[n, Blank[]]]]], Optional[Pattern[u, Blank[]]], Power[Plus[Pattern[c, Blank[]], Times[Optional[Pattern[d, Blank[]]], Power[Pattern[x, Blank[]], -1]]], Pattern[p, Blank[]]]], Pattern[x, Blank[Symbol]]] := Dist[(x^p*(c + d/x)^p)/(1 + (c*x)/d)^p, Int[(u*(1 + (c*x)/d)^p*E^(n*ArcTan[a*x]))/x^p, x], x] /; FreeQ[{a, c, d, n, p}, x] && EqQ[c^2 + a^2*d^2, 0] && !IntegerQ[p]
  5069. Int[Times[Power[E, Times[ArcTan[Times[Optional[Pattern[a, Blank[]]], Pattern[x, Blank[]]]], Optional[Pattern[n, Blank[]]]]], Power[Plus[Pattern[c, Blank[]], Times[Optional[Pattern[d, Blank[]]], Power[Pattern[x, Blank[]], 2]]], Rational[-3, 2]]], Pattern[x, Blank[Symbol]]] := Simp[((n + a*x)*E^(n*ArcTan[a*x]))/(a*c*(n^2 + 1)*Sqrt[c + d*x^2]), x] /; FreeQ[{a, c, d, n}, x] && EqQ[d, a^2*c] && !IntegerQ[I*n]
  5070. Int[Times[Power[E, Times[ArcTan[Times[Optional[Pattern[a, Blank[]]], Pattern[x, Blank[]]]], Optional[Pattern[n, Blank[]]]]], Power[Plus[Pattern[c, Blank[]], Times[Optional[Pattern[d, Blank[]]], Power[Pattern[x, Blank[]], 2]]], Pattern[p, Blank[]]]], Pattern[x, Blank[Symbol]]] := Simp[((n - 2*a*(p + 1)*x)*(c + d*x^2)^(p + 1)*E^(n*ArcTan[a*x]))/(a*c*(n^2 + 4*(p + 1)^2)), x] + Dist[(2*(p + 1)*(2*p + 3))/(c*(n^2 + 4*(p + 1)^2)), Int[(c + d*x^2)^(p + 1)*E^(n*ArcTan[a*x]), x], x] /; FreeQ[{a, c, d, n}, x] && EqQ[d, a^2*c] && LtQ[p, -1] && !IntegerQ[I*n] && NeQ[n^2 + 4*(p + 1)^2, 0] && IntegerQ[2*p]
  5071. Int[Times[Power[E, Times[ArcTan[Times[Optional[Pattern[a, Blank[]]], Pattern[x, Blank[]]]], Optional[Pattern[n, Blank[]]]]], Power[Plus[Pattern[c, Blank[]], Times[Optional[Pattern[d, Blank[]]], Power[Pattern[x, Blank[]], 2]]], -1]], Pattern[x, Blank[Symbol]]] := Simp[E^(n*ArcTan[a*x])/(a*c*n), x] /; FreeQ[{a, c, d, n}, x] && EqQ[d, a^2*c]
  5072. Int[Times[Power[E, Times[ArcTan[Times[Optional[Pattern[a, Blank[]]], Pattern[x, Blank[]]]], Pattern[n, Blank[]]]], Power[Plus[Pattern[c, Blank[]], Times[Optional[Pattern[d, Blank[]]], Power[Pattern[x, Blank[]], 2]]], Optional[Pattern[p, Blank[]]]]], Pattern[x, Blank[Symbol]]] := Dist[c^p, Int[(1 + a^2*x^2)^(p - (I*n)/2)*(1 - I*a*x)^(I*n), x], x] /; FreeQ[{a, c, d, p}, x] && EqQ[d, a^2*c] && IntegerQ[p] && IntegerQ[(I*n + 1)/2] && !IntegerQ[p - (I*n)/2]
  5073. Int[Times[Power[E, Times[ArcTan[Times[Optional[Pattern[a, Blank[]]], Pattern[x, Blank[]]]], Optional[Pattern[n, Blank[]]]]], Power[Plus[Pattern[c, Blank[]], Times[Optional[Pattern[d, Blank[]]], Power[Pattern[x, Blank[]], 2]]], Optional[Pattern[p, Blank[]]]]], Pattern[x, Blank[Symbol]]] := Dist[c^p, Int[(1 - I*a*x)^(p + (I*n)/2)*(1 + I*a*x)^(p - (I*n)/2), x], x] /; FreeQ[{a, c, d, n, p}, x] && EqQ[d, a^2*c] && (IntegerQ[p] || GtQ[c, 0])
  5074. Int[Times[Power[E, Times[ArcTan[Times[Optional[Pattern[a, Blank[]]], Pattern[x, Blank[]]]], Pattern[n, Blank[]]]], Power[Plus[Pattern[c, Blank[]], Times[Optional[Pattern[d, Blank[]]], Power[Pattern[x, Blank[]], 2]]], Pattern[p, Blank[]]]], Pattern[x, Blank[Symbol]]] := Dist[c^((I*n)/2), Int[(c + d*x^2)^(p - (I*n)/2)*(1 - I*a*x)^(I*n), x], x] /; FreeQ[{a, c, d, p}, x] && EqQ[d, a^2*c] && !(IntegerQ[p] || GtQ[c, 0]) && IGtQ[(I*n)/2, 0]
  5075. Int[Times[Power[E, Times[ArcTan[Times[Optional[Pattern[a, Blank[]]], Pattern[x, Blank[]]]], Pattern[n, Blank[]]]], Power[Plus[Pattern[c, Blank[]], Times[Optional[Pattern[d, Blank[]]], Power[Pattern[x, Blank[]], 2]]], Pattern[p, Blank[]]]], Pattern[x, Blank[Symbol]]] := Dist[1/c^((I*n)/2), Int[(c + d*x^2)^(p + (I*n)/2)/(1 + I*a*x)^(I*n), x], x] /; FreeQ[{a, c, d, p}, x] && EqQ[d, a^2*c] && !(IntegerQ[p] || GtQ[c, 0]) && ILtQ[(I*n)/2, 0]
  5076. Int[Times[Power[E, Times[ArcTan[Times[Optional[Pattern[a, Blank[]]], Pattern[x, Blank[]]]], Optional[Pattern[n, Blank[]]]]], Power[Plus[Pattern[c, Blank[]], Times[Optional[Pattern[d, Blank[]]], Power[Pattern[x, Blank[]], 2]]], Pattern[p, Blank[]]]], Pattern[x, Blank[Symbol]]] := Dist[(c^IntPart[p]*(c + d*x^2)^FracPart[p])/(1 + a^2*x^2)^FracPart[p], Int[(1 + a^2*x^2)^p*E^(n*ArcTan[a*x]), x], x] /; FreeQ[{a, c, d, n, p}, x] && EqQ[d, a^2*c] && !(IntegerQ[p] || GtQ[c, 0])
  5077. Int[Times[Power[E, Times[ArcTan[Times[Optional[Pattern[a, Blank[]]], Pattern[x, Blank[]]]], Optional[Pattern[n, Blank[]]]]], Pattern[x, Blank[]], Power[Plus[Pattern[c, Blank[]], Times[Optional[Pattern[d, Blank[]]], Power[Pattern[x, Blank[]], 2]]], Rational[-3, 2]]], Pattern[x, Blank[Symbol]]] := -Simp[((1 - a*n*x)*E^(n*ArcTan[a*x]))/(d*(n^2 + 1)*Sqrt[c + d*x^2]), x] /; FreeQ[{a, c, d, n}, x] && EqQ[d, a^2*c] && !IntegerQ[I*n]
  5078. Int[Times[Power[E, Times[ArcTan[Times[Optional[Pattern[a, Blank[]]], Pattern[x, Blank[]]]], Optional[Pattern[n, Blank[]]]]], Pattern[x, Blank[]], Power[Plus[Pattern[c, Blank[]], Times[Optional[Pattern[d, Blank[]]], Power[Pattern[x, Blank[]], 2]]], Pattern[p, Blank[]]]], Pattern[x, Blank[Symbol]]] := Simp[((c + d*x^2)^(p + 1)*E^(n*ArcTan[a*x]))/(2*d*(p + 1)), x] - Dist[(a*c*n)/(2*d*(p + 1)), Int[(c + d*x^2)^p*E^(n*ArcTan[a*x]), x], x] /; FreeQ[{a, c, d, n}, x] && EqQ[d, a^2*c] && LtQ[p, -1] && !IntegerQ[I*n] && IntegerQ[2*p]
  5079. Int[Times[Power[E, Times[ArcTan[Times[Optional[Pattern[a, Blank[]]], Pattern[x, Blank[]]]], Optional[Pattern[n, Blank[]]]]], Power[Pattern[x, Blank[]], 2], Power[Plus[Pattern[c, Blank[]], Times[Optional[Pattern[d, Blank[]]], Power[Pattern[x, Blank[]], 2]]], Optional[Pattern[p, Blank[]]]]], Pattern[x, Blank[Symbol]]] := -Simp[((1 - a*n*x)*(c + d*x^2)^(p + 1)*E^(n*ArcTan[a*x]))/(a*d*n*(n^2 + 1)), x] /; FreeQ[{a, c, d, n}, x] && EqQ[d, a^2*c] && EqQ[n^2 - 2*(p + 1), 0] && !IntegerQ[I*n]
  5080. Int[Times[Power[E, Times[ArcTan[Times[Optional[Pattern[a, Blank[]]], Pattern[x, Blank[]]]], Optional[Pattern[n, Blank[]]]]], Power[Pattern[x, Blank[]], 2], Power[Plus[Pattern[c, Blank[]], Times[Optional[Pattern[d, Blank[]]], Power[Pattern[x, Blank[]], 2]]], Pattern[p, Blank[]]]], Pattern[x, Blank[Symbol]]] := -Simp[((n - 2*(p + 1)*a*x)*(c + d*x^2)^(p + 1)*E^(n*ArcTan[a*x]))/(a*d*(n^2 + 4*(p + 1)^2)), x] + Dist[(n^2 - 2*(p + 1))/(d*(n^2 + 4*(p + 1)^2)), Int[(c + d*x^2)^(p + 1)*E^(n*ArcTan[a*x]), x], x] /; FreeQ[{a, c, d, n}, x] && EqQ[d, a^2*c] && LtQ[p, -1] && !IntegerQ[I*n] && NeQ[n^2 + 4*(p + 1)^2, 0] && IntegerQ[2*p]
  5081. Int[Times[Power[E, Times[ArcTan[Times[Optional[Pattern[a, Blank[]]], Pattern[x, Blank[]]]], Pattern[n, Blank[]]]], Power[Pattern[x, Blank[]], Optional[Pattern[m, Blank[]]]], Power[Plus[Pattern[c, Blank[]], Times[Optional[Pattern[d, Blank[]]], Power[Pattern[x, Blank[]], 2]]], Optional[Pattern[p, Blank[]]]]], Pattern[x, Blank[Symbol]]] := Dist[c^p, Int[x^m*(1 + a^2*x^2)^(p - (I*n)/2)*(1 - I*a*x)^(I*n), x], x] /; FreeQ[{a, c, d, m, p}, x] && EqQ[d, a^2*c] && (IntegerQ[p] || GtQ[c, 0]) && IntegerQ[(I*n + 1)/2] && !IntegerQ[p - (I*n)/2]
  5082. Int[Times[Power[E, Times[ArcTan[Times[Optional[Pattern[a, Blank[]]], Pattern[x, Blank[]]]], Optional[Pattern[n, Blank[]]]]], Power[Pattern[x, Blank[]], Optional[Pattern[m, Blank[]]]], Power[Plus[Pattern[c, Blank[]], Times[Optional[Pattern[d, Blank[]]], Power[Pattern[x, Blank[]], 2]]], Optional[Pattern[p, Blank[]]]]], Pattern[x, Blank[Symbol]]] := Dist[c^p, Int[x^m*(1 - I*a*x)^(p + (I*n)/2)*(1 + I*a*x)^(p - (I*n)/2), x], x] /; FreeQ[{a, c, d, m, n, p}, x] && EqQ[d, a^2*c] && (IntegerQ[p] || GtQ[c, 0])
  5083. Int[Times[Power[E, Times[ArcTan[Times[Optional[Pattern[a, Blank[]]], Pattern[x, Blank[]]]], Pattern[n, Blank[]]]], Power[Pattern[x, Blank[]], Optional[Pattern[m, Blank[]]]], Power[Plus[Pattern[c, Blank[]], Times[Optional[Pattern[d, Blank[]]], Power[Pattern[x, Blank[]], 2]]], Pattern[p, Blank[]]]], Pattern[x, Blank[Symbol]]] := Dist[c^((I*n)/2), Int[x^m*(c + d*x^2)^(p - (I*n)/2)*(1 - I*a*x)^(I*n), x], x] /; FreeQ[{a, c, d, m, p}, x] && EqQ[d, a^2*c] && !(IntegerQ[p] || GtQ[c, 0]) && IGtQ[(I*n)/2, 0]
  5084. Int[Times[Power[E, Times[ArcTan[Times[Optional[Pattern[a, Blank[]]], Pattern[x, Blank[]]]], Pattern[n, Blank[]]]], Power[Pattern[x, Blank[]], Optional[Pattern[m, Blank[]]]], Power[Plus[Pattern[c, Blank[]], Times[Optional[Pattern[d, Blank[]]], Power[Pattern[x, Blank[]], 2]]], Pattern[p, Blank[]]]], Pattern[x, Blank[Symbol]]] := Dist[1/c^((I*n)/2), Int[(x^m*(c + d*x^2)^(p + (I*n)/2))/(1 + I*a*x)^(I*n), x], x] /; FreeQ[{a, c, d, m, p}, x] && EqQ[d, a^2*c] && !(IntegerQ[p] || GtQ[c, 0]) && ILtQ[(I*n)/2, 0]
  5085. Int[Times[Power[E, Times[ArcTan[Times[Optional[Pattern[a, Blank[]]], Pattern[x, Blank[]]]], Optional[Pattern[n, Blank[]]]]], Power[Pattern[x, Blank[]], Optional[Pattern[m, Blank[]]]], Power[Plus[Pattern[c, Blank[]], Times[Optional[Pattern[d, Blank[]]], Power[Pattern[x, Blank[]], 2]]], Pattern[p, Blank[]]]], Pattern[x, Blank[Symbol]]] := Dist[(c^IntPart[p]*(c + d*x^2)^FracPart[p])/(1 + a^2*x^2)^FracPart[p], Int[x^m*(1 + a^2*x^2)^p*E^(n*ArcTan[a*x]), x], x] /; FreeQ[{a, c, d, m, n, p}, x] && EqQ[d, a^2*c] && !(IntegerQ[p] || GtQ[c, 0])
  5086. Int[Times[Power[E, Times[ArcTan[Times[Optional[Pattern[a, Blank[]]], Pattern[x, Blank[]]]], Optional[Pattern[n, Blank[]]]]], Pattern[u, Blank[]], Power[Plus[Pattern[c, Blank[]], Times[Optional[Pattern[d, Blank[]]], Power[Pattern[x, Blank[]], 2]]], Optional[Pattern[p, Blank[]]]]], Pattern[x, Blank[Symbol]]] := Dist[c^p, Int[u*(1 - I*a*x)^(p + (I*n)/2)*(1 + I*a*x)^(p - (I*n)/2), x], x] /; FreeQ[{a, c, d, n, p}, x] && EqQ[d, a^2*c] && (IntegerQ[p] || GtQ[c, 0])
  5087. Int[Times[Power[E, Times[ArcTan[Times[Optional[Pattern[a, Blank[]]], Pattern[x, Blank[]]]], Pattern[n, Blank[]]]], Pattern[u, Blank[]], Power[Plus[Pattern[c, Blank[]], Times[Optional[Pattern[d, Blank[]]], Power[Pattern[x, Blank[]], 2]]], Optional[Pattern[p, Blank[]]]]], Pattern[x, Blank[Symbol]]] := Dist[(c^IntPart[p]*(c + d*x^2)^FracPart[p])/((1 - I*a*x)^FracPart[p]*(1 + I*a*x)^FracPart[p]), Int[u*(1 - I*a*x)^(p + (I*n)/2)*(1 + I*a*x)^(p - (I*n)/2), x], x] /; FreeQ[{a, c, d, n, p}, x] && EqQ[d, a^2*c] && (IntegerQ[p] || GtQ[c, 0]) && IntegerQ[(I*n)/2]
  5088. Int[Times[Power[E, Times[ArcTan[Times[Optional[Pattern[a, Blank[]]], Pattern[x, Blank[]]]], Optional[Pattern[n, Blank[]]]]], Pattern[u, Blank[]], Power[Plus[Pattern[c, Blank[]], Times[Optional[Pattern[d, Blank[]]], Power[Pattern[x, Blank[]], 2]]], Pattern[p, Blank[]]]], Pattern[x, Blank[Symbol]]] := Dist[(c^IntPart[p]*(c + d*x^2)^FracPart[p])/(1 + a^2*x^2)^FracPart[p], Int[u*(1 + a^2*x^2)^p*E^(n*ArcTan[a*x]), x], x] /; FreeQ[{a, c, d, n, p}, x] && EqQ[d, a^2*c] && !(IntegerQ[p] || GtQ[c, 0]) && !IntegerQ[(I*n)/2]
  5089. Int[Times[Power[E, Times[ArcTan[Times[Optional[Pattern[a, Blank[]]], Pattern[x, Blank[]]]], Optional[Pattern[n, Blank[]]]]], Optional[Pattern[u, Blank[]]], Power[Plus[Pattern[c, Blank[]], Times[Optional[Pattern[d, Blank[]]], Power[Pattern[x, Blank[]], -2]]], Optional[Pattern[p, Blank[]]]]], Pattern[x, Blank[Symbol]]] := Dist[d^p, Int[(u*(1 + a^2*x^2)^p*E^(n*ArcTan[a*x]))/x^(2*p), x], x] /; FreeQ[{a, c, d, n}, x] && EqQ[c - a^2*d, 0] && IntegerQ[p]
  5090. Int[Times[Power[E, Times[ArcTan[Times[Optional[Pattern[a, Blank[]]], Pattern[x, Blank[]]]], Pattern[n, Blank[]]]], Optional[Pattern[u, Blank[]]], Power[Plus[Pattern[c, Blank[]], Times[Optional[Pattern[d, Blank[]]], Power[Pattern[x, Blank[]], -2]]], Pattern[p, Blank[]]]], Pattern[x, Blank[Symbol]]] := Dist[c^p, Int[u*(1 - I/(a*x))^p*(1 + I/(a*x))^p*E^(n*ArcTan[a*x]), x], x] /; FreeQ[{a, c, d, p}, x] && EqQ[c - a^2*d, 0] && !IntegerQ[p] && IntegerQ[(I*n)/2] && GtQ[c, 0]
  5091. Int[Times[Power[E, Times[ArcTan[Times[Optional[Pattern[a, Blank[]]], Pattern[x, Blank[]]]], Pattern[n, Blank[]]]], Optional[Pattern[u, Blank[]]], Power[Plus[Pattern[c, Blank[]], Times[Optional[Pattern[d, Blank[]]], Power[Pattern[x, Blank[]], -2]]], Pattern[p, Blank[]]]], Pattern[x, Blank[Symbol]]] := Dist[(x^(2*p)*(c + d/x^2)^p)/((1 - I*a*x)^p*(1 + I*a*x)^p), Int[(u*(1 - I*a*x)^p*(1 + I*a*x)^p*E^(n*ArcTan[a*x]))/x^(2*p), x], x] /; FreeQ[{a, c, d, n, p}, x] && EqQ[c - a^2*d, 0] && !IntegerQ[p] && IntegerQ[(I*n)/2] && !GtQ[c, 0]
  5092. Int[Times[Power[E, Times[ArcTan[Times[Optional[Pattern[a, Blank[]]], Pattern[x, Blank[]]]], Optional[Pattern[n, Blank[]]]]], Optional[Pattern[u, Blank[]]], Power[Plus[Pattern[c, Blank[]], Times[Optional[Pattern[d, Blank[]]], Power[Pattern[x, Blank[]], -2]]], Pattern[p, Blank[]]]], Pattern[x, Blank[Symbol]]] := Dist[(x^(2*p)*(c + d/x^2)^p)/(1 + a^2*x^2)^p, Int[(u*(1 + a^2*x^2)^p*E^(n*ArcTan[a*x]))/x^(2*p), x], x] /; FreeQ[{a, c, d, n, p}, x] && EqQ[c - a^2*d, 0] && !IntegerQ[p] && !IntegerQ[(I*n)/2]
  5093. Int[Power[E, Times[ArcTan[Times[Optional[Pattern[c, Blank[]]], Plus[Pattern[a, Blank[]], Times[Optional[Pattern[b, Blank[]]], Pattern[x, Blank[]]]]]], Optional[Pattern[n, Blank[]]]]], Pattern[x, Blank[Symbol]]] := Int[(1 - I*a*c - I*b*c*x)^((I*n)/2)/(1 + I*a*c + I*b*c*x)^((I*n)/2), x] /; FreeQ[{a, b, c, n}, x]
  5094. Int[Times[Power[E, Times[ArcTan[Times[Optional[Pattern[c, Blank[]]], Plus[Pattern[a, Blank[]], Times[Optional[Pattern[b, Blank[]]], Pattern[x, Blank[]]]]]], Pattern[n, Blank[]]]], Power[Pattern[x, Blank[]], Pattern[m, Blank[]]]], Pattern[x, Blank[Symbol]]] := Dist[4/(I^m*n*b^(m + 1)*c^(m + 1)), Subst[Int[(x^(2/(I*n))*(1 - I*a*c - (1 + I*a*c)*x^(2/(I*n)))^m)/(1 + x^(2/(I*n)))^(m + 2), x], x, (1 - I*c*(a + b*x))^((I*n)/2)/(1 + I*c*(a + b*x))^((I*n)/2)], x] /; FreeQ[{a, b, c}, x] && ILtQ[m, 0] && LtQ[-1, I*n, 1]
  5095. Int[Times[Power[E, Times[ArcTan[Times[Optional[Pattern[c, Blank[]]], Plus[Pattern[a, Blank[]], Times[Optional[Pattern[b, Blank[]]], Pattern[x, Blank[]]]]]], Optional[Pattern[n, Blank[]]]]], Power[Plus[Optional[Pattern[d, Blank[]]], Times[Optional[Pattern[e, Blank[]]], Pattern[x, Blank[]]]], Optional[Pattern[m, Blank[]]]]], Pattern[x, Blank[Symbol]]] := Int[((d + e*x)^m*(1 - I*a*c - I*b*c*x)^((I*n)/2))/(1 + I*a*c + I*b*c*x)^((I*n)/2), x] /; FreeQ[{a, b, c, d, e, m, n}, x]
  5096. Int[Times[Power[E, Times[ArcTan[Plus[Pattern[a, Blank[]], Times[Optional[Pattern[b, Blank[]]], Pattern[x, Blank[]]]]], Optional[Pattern[n, Blank[]]]]], Optional[Pattern[u, Blank[]]], Power[Plus[Pattern[c, Blank[]], Times[Optional[Pattern[d, Blank[]]], Pattern[x, Blank[]]], Times[Optional[Pattern[e, Blank[]]], Power[Pattern[x, Blank[]], 2]]], Optional[Pattern[p, Blank[]]]]], Pattern[x, Blank[Symbol]]] := Dist[(c/(1 + a^2))^p, Int[u*(1 - I*a - I*b*x)^(p + (I*n)/2)*(1 + I*a + I*b*x)^(p - (I*n)/2), x], x] /; FreeQ[{a, b, c, d, e, n, p}, x] && EqQ[b*d, 2*a*e] && EqQ[b^2*c - e*(1 + a^2), 0] && (IntegerQ[p] || GtQ[c/(1 + a^2), 0])
  5097. Int[Times[Power[E, Times[ArcTan[Plus[Pattern[a, Blank[]], Times[Optional[Pattern[b, Blank[]]], Pattern[x, Blank[]]]]], Optional[Pattern[n, Blank[]]]]], Optional[Pattern[u, Blank[]]], Power[Plus[Pattern[c, Blank[]], Times[Optional[Pattern[d, Blank[]]], Pattern[x, Blank[]]], Times[Optional[Pattern[e, Blank[]]], Power[Pattern[x, Blank[]], 2]]], Optional[Pattern[p, Blank[]]]]], Pattern[x, Blank[Symbol]]] := Dist[(c + d*x + e*x^2)^p/(1 + a^2 + 2*a*b*x + b^2*x^2)^p, Int[u*(1 + a^2 + 2*a*b*x + b^2*x^2)^p*E^(n*ArcTan[a*x]), x], x] /; FreeQ[{a, b, c, d, e, n, p}, x] && EqQ[b*d, 2*a*e] && EqQ[b^2*c - e*(1 + a^2), 0] && !(IntegerQ[p] || GtQ[c/(1 + a^2), 0])
  5098. Int[Times[Power[E, Times[ArcTan[Times[Optional[Pattern[c, Blank[]]], Power[Plus[Optional[Pattern[a, Blank[]]], Times[Optional[Pattern[b, Blank[]]], Pattern[x, Blank[]]]], -1]]], Optional[Pattern[n, Blank[]]]]], Optional[Pattern[u, Blank[]]]], Pattern[x, Blank[Symbol]]] := Int[u*E^(n*ArcCot[a/c + (b*x)/c]), x] /; FreeQ[{a, b, c, n}, x]
  5099. Int[Times[Power[E, Times[ArcCot[Times[Optional[Pattern[a, Blank[]]], Pattern[x, Blank[]]]], Pattern[n, Blank[]]]], Optional[Pattern[u, Blank[]]]], Pattern[x, Blank[Symbol]]] := Dist[(-1)^((I*n)/2), Int[u/E^(n*ArcTan[a*x]), x], x] /; FreeQ[a, x] && IntegerQ[(I*n)/2]
  5100. Int[Power[E, Times[ArcCot[Times[Optional[Pattern[a, Blank[]]], Pattern[x, Blank[]]]], Pattern[n, Blank[]]]], Pattern[x, Blank[Symbol]]] := -Subst[Int[(1 - (I*x)/a)^((I*n + 1)/2)/(x^2*(1 + (I*x)/a)^((I*n - 1)/2)*Sqrt[1 + x^2/a^2]), x], x, 1/x] /; FreeQ[a, x] && IntegerQ[(I*n - 1)/2]
  5101. Int[Times[Power[E, Times[ArcCot[Times[Optional[Pattern[a, Blank[]]], Pattern[x, Blank[]]]], Pattern[n, Blank[]]]], Power[Pattern[x, Blank[]], Optional[Pattern[m, Blank[]]]]], Pattern[x, Blank[Symbol]]] := -Subst[Int[(1 - (I*x)/a)^((I*n + 1)/2)/(x^(m + 2)*(1 + (I*x)/a)^((I*n - 1)/2)*Sqrt[1 + x^2/a^2]), x], x, 1/x] /; FreeQ[a, x] && IntegerQ[(I*n - 1)/2] && IntegerQ[m]
  5102. Int[Power[E, Times[ArcCot[Times[Optional[Pattern[a, Blank[]]], Pattern[x, Blank[]]]], Optional[Pattern[n, Blank[]]]]], Pattern[x, Blank[Symbol]]] := -Subst[Int[(1 - (I*x)/a)^((I*n)/2)/(x^2*(1 + (I*x)/a)^((I*n)/2)), x], x, 1/x] /; FreeQ[{a, n}, x] && !IntegerQ[I*n]
  5103. Int[Times[Power[E, Times[ArcCot[Times[Optional[Pattern[a, Blank[]]], Pattern[x, Blank[]]]], Optional[Pattern[n, Blank[]]]]], Power[Pattern[x, Blank[]], Optional[Pattern[m, Blank[]]]]], Pattern[x, Blank[Symbol]]] := -Subst[Int[(1 - (I*x)/a)^(n/2)/(x^(m + 2)*(1 + (I*x)/a)^(n/2)), x], x, 1/x] /; FreeQ[{a, n}, x] && !IntegerQ[I*n] && IntegerQ[m]
  5104. Int[Times[Power[E, Times[ArcCot[Times[Optional[Pattern[a, Blank[]]], Pattern[x, Blank[]]]], Pattern[n, Blank[]]]], Power[Pattern[x, Blank[]], Pattern[m, Blank[]]]], Pattern[x, Blank[Symbol]]] := -Dist[x^m*(1/x)^m, Subst[Int[(1 - (I*x)/a)^((I*n + 1)/2)/(x^(m + 2)*(1 + (I*x)/a)^((I*n - 1)/2)*Sqrt[1 + x^2/a^2]), x], x, 1/x], x] /; FreeQ[{a, m}, x] && IntegerQ[(I*n - 1)/2] && !IntegerQ[m]
  5105. Int[Times[Power[E, Times[ArcCot[Times[Optional[Pattern[a, Blank[]]], Pattern[x, Blank[]]]], Optional[Pattern[n, Blank[]]]]], Power[Pattern[x, Blank[]], Pattern[m, Blank[]]]], Pattern[x, Blank[Symbol]]] := -Subst[Int[(1 - (I*x)/a)^(n/2)/(x^(m + 2)*(1 + (I*x)/a)^(n/2)), x], x, 1/x] /; FreeQ[{a, m, n}, x] && !IntegerQ[(I*n)/2] && !IntegerQ[m]
  5106. Int[Times[Power[E, Times[ArcCot[Times[Optional[Pattern[a, Blank[]]], Pattern[x, Blank[]]]], Optional[Pattern[n, Blank[]]]]], Optional[Pattern[u, Blank[]]], Power[Plus[Pattern[c, Blank[]], Times[Optional[Pattern[d, Blank[]]], Pattern[x, Blank[]]]], Optional[Pattern[p, Blank[]]]]], Pattern[x, Blank[Symbol]]] := Dist[d^p, Int[u*x^p*(1 + c/(d*x))^p*E^(n*ArcCot[a*x]), x], x] /; FreeQ[{a, c, d, n}, x] && EqQ[a^2*c^2 + d^2, 0] && !IntegerQ[(I*n)/2] && IntegerQ[p]
  5107. Int[Times[Power[E, Times[ArcCot[Times[Optional[Pattern[a, Blank[]]], Pattern[x, Blank[]]]], Optional[Pattern[n, Blank[]]]]], Optional[Pattern[u, Blank[]]], Power[Plus[Pattern[c, Blank[]], Times[Optional[Pattern[d, Blank[]]], Pattern[x, Blank[]]]], Pattern[p, Blank[]]]], Pattern[x, Blank[Symbol]]] := Dist[(c + d*x)^p/(x^p*(1 + c/(d*x))^p), Int[u*x^p*(1 + c/(d*x))^p*E^(n*ArcCot[a*x]), x], x] /; FreeQ[{a, c, d, n, p}, x] && EqQ[a^2*c^2 + d^2, 0] && !IntegerQ[(I*n)/2] && !IntegerQ[p]
  5108. Int[Times[Power[E, Times[ArcCot[Times[Optional[Pattern[a, Blank[]]], Pattern[x, Blank[]]]], Optional[Pattern[n, Blank[]]]]], Power[Plus[Pattern[c, Blank[]], Times[Optional[Pattern[d, Blank[]]], Power[Pattern[x, Blank[]], -1]]], Optional[Pattern[p, Blank[]]]]], Pattern[x, Blank[Symbol]]] := -Dist[c^p, Subst[Int[((1 + (d*x)/c)^p*(1 - (I*x)/a)^((I*n)/2))/(x^2*(1 + (I*x)/a)^((I*n)/2)), x], x, 1/x], x] /; FreeQ[{a, c, d, n, p}, x] && EqQ[c^2 + a^2*d^2, 0] && !IntegerQ[(I*n)/2] && (IntegerQ[p] || GtQ[c, 0])
  5109. Int[Times[Power[E, Times[ArcCot[Times[Optional[Pattern[a, Blank[]]], Pattern[x, Blank[]]]], Optional[Pattern[n, Blank[]]]]], Power[Plus[Pattern[c, Blank[]], Times[Optional[Pattern[d, Blank[]]], Power[Pattern[x, Blank[]], -1]]], Optional[Pattern[p, Blank[]]]], Power[Pattern[x, Blank[]], Optional[Pattern[m, Blank[]]]]], Pattern[x, Blank[Symbol]]] := -Dist[c^p, Subst[Int[((1 + (d*x)/c)^p*(1 - (I*x)/a)^((I*n)/2))/(x^(m + 2)*(1 + (I*x)/a)^((I*n)/2)), x], x, 1/x], x] /; FreeQ[{a, c, d, m, n, p}, x] && EqQ[c^2 + a^2*d^2, 0] && !IntegerQ[(I*n)/2] && (IntegerQ[p] || GtQ[c, 0]) && IntegerQ[m]
  5110. Int[Times[Power[E, Times[ArcCot[Times[Optional[Pattern[a, Blank[]]], Pattern[x, Blank[]]]], Optional[Pattern[n, Blank[]]]]], Power[Plus[Pattern[c, Blank[]], Times[Optional[Pattern[d, Blank[]]], Power[Pattern[x, Blank[]], -1]]], Pattern[p, Blank[]]]], Pattern[x, Blank[Symbol]]] := Dist[(c + d/x)^p/(1 + d/(c*x))^p, Int[(1 + d/(c*x))^p*E^(n*ArcCot[a*x]), x], x] /; FreeQ[{a, c, d, n, p}, x] && EqQ[c^2 + a^2*d^2, 0] && !IntegerQ[(I*n)/2] && !(IntegerQ[p] || GtQ[c, 0])
  5111. Int[Times[Power[E, Times[ArcCot[Times[Optional[Pattern[a, Blank[]]], Pattern[x, Blank[]]]], Optional[Pattern[n, Blank[]]]]], Power[Plus[Pattern[c, Blank[]], Times[Optional[Pattern[d, Blank[]]], Power[Pattern[x, Blank[]], -1]]], Optional[Pattern[p, Blank[]]]], Power[Pattern[x, Blank[]], Pattern[m, Blank[]]]], Pattern[x, Blank[Symbol]]] := -Dist[c^p*x^m*(1/x)^m, Subst[Int[((1 + (d*x)/c)^p*(1 - (I*x)/a)^((I*n)/2))/(x^(m + 2)*(1 + (I*x)/a)^((I*n)/2)), x], x, 1/x], x] /; FreeQ[{a, c, d, m, n, p}, x] && EqQ[c^2 + a^2*d^2, 0] && !IntegerQ[(I*n)/2] && (IntegerQ[p] || GtQ[c, 0]) && !IntegerQ[m]
  5112. Int[Times[Power[E, Times[ArcCot[Times[Optional[Pattern[a, Blank[]]], Pattern[x, Blank[]]]], Optional[Pattern[n, Blank[]]]]], Optional[Pattern[u, Blank[]]], Power[Plus[Pattern[c, Blank[]], Times[Optional[Pattern[d, Blank[]]], Power[Pattern[x, Blank[]], -1]]], Pattern[p, Blank[]]]], Pattern[x, Blank[Symbol]]] := Dist[(c + d/x)^p/(1 + d/(c*x))^p, Int[u*(1 + d/(c*x))^p*E^(n*ArcCot[a*x]), x], x] /; FreeQ[{a, c, d, n, p}, x] && EqQ[c^2 + a^2*d^2, 0] && !IntegerQ[(I*n)/2] && !(IntegerQ[p] || GtQ[c, 0])
  5113. Int[Times[Power[E, Times[ArcCot[Times[Optional[Pattern[a, Blank[]]], Pattern[x, Blank[]]]], Optional[Pattern[n, Blank[]]]]], Power[Plus[Pattern[c, Blank[]], Times[Optional[Pattern[d, Blank[]]], Power[Pattern[x, Blank[]], 2]]], -1]], Pattern[x, Blank[Symbol]]] := -Simp[E^(n*ArcCot[a*x])/(a*c*n), x] /; FreeQ[{a, c, d, n}, x] && EqQ[d, a^2*c]
  5114. Int[Times[Power[E, Times[ArcCot[Times[Optional[Pattern[a, Blank[]]], Pattern[x, Blank[]]]], Optional[Pattern[n, Blank[]]]]], Power[Plus[Pattern[c, Blank[]], Times[Optional[Pattern[d, Blank[]]], Power[Pattern[x, Blank[]], 2]]], Rational[-3, 2]]], Pattern[x, Blank[Symbol]]] := -Simp[((n - a*x)*E^(n*ArcCot[a*x]))/(a*c*(n^2 + 1)*Sqrt[c + d*x^2]), x] /; FreeQ[{a, c, d, n}, x] && EqQ[d, a^2*c] && !IntegerQ[(I*n - 1)/2]
  5115. Int[Times[Power[E, Times[ArcCot[Times[Optional[Pattern[a, Blank[]]], Pattern[x, Blank[]]]], Optional[Pattern[n, Blank[]]]]], Power[Plus[Pattern[c, Blank[]], Times[Optional[Pattern[d, Blank[]]], Power[Pattern[x, Blank[]], 2]]], Pattern[p, Blank[]]]], Pattern[x, Blank[Symbol]]] := -Simp[((n + 2*a*(p + 1)*x)*(c + d*x^2)^(p + 1)*E^(n*ArcCot[a*x]))/(a*c*(n^2 + 4*(p + 1)^2)), x] + Dist[(2*(p + 1)*(2*p + 3))/(c*(n^2 + 4*(p + 1)^2)), Int[(c + d*x^2)^(p + 1)*E^(n*ArcCot[a*x]), x], x] /; FreeQ[{a, c, d, n}, x] && EqQ[d, a^2*c] && LtQ[p, -1] && NeQ[p, -3/2] && NeQ[n^2 + 4*(p + 1)^2, 0] && !(IntegerQ[p] && IntegerQ[(I*n)/2]) && !( !IntegerQ[p] && IntegerQ[(I*n - 1)/2])
  5116. Int[Times[Power[E, Times[ArcCot[Times[Optional[Pattern[a, Blank[]]], Pattern[x, Blank[]]]], Optional[Pattern[n, Blank[]]]]], Pattern[x, Blank[]], Power[Plus[Pattern[c, Blank[]], Times[Optional[Pattern[d, Blank[]]], Power[Pattern[x, Blank[]], 2]]], Rational[-3, 2]]], Pattern[x, Blank[Symbol]]] := -Simp[((1 + a*n*x)*E^(n*ArcCot[a*x]))/(a^2*c*(n^2 + 1)*Sqrt[c + d*x^2]), x] /; FreeQ[{a, c, d, n}, x] && EqQ[d, a^2*c] && !IntegerQ[(I*n - 1)/2]
  5117. Int[Times[Power[E, Times[ArcCot[Times[Optional[Pattern[a, Blank[]]], Pattern[x, Blank[]]]], Optional[Pattern[n, Blank[]]]]], Pattern[x, Blank[]], Power[Plus[Pattern[c, Blank[]], Times[Optional[Pattern[d, Blank[]]], Power[Pattern[x, Blank[]], 2]]], Pattern[p, Blank[]]]], Pattern[x, Blank[Symbol]]] := Simp[((2*(p + 1) - a*n*x)*(c + d*x^2)^(p + 1)*E^(n*ArcCot[a*x]))/(a^2*c*(n^2 + 4*(p + 1)^2)), x] + Dist[(n*(2*p + 3))/(a*c*(n^2 + 4*(p + 1)^2)), Int[(c + d*x^2)^(p + 1)*E^(n*ArcCot[a*x]), x], x] /; FreeQ[{a, c, d, n}, x] && EqQ[d, a^2*c] && LeQ[p, -1] && NeQ[p, -3/2] && NeQ[n^2 + 4*(p + 1)^2, 0] && !(IntegerQ[p] && IntegerQ[(I*n)/2]) && !( !IntegerQ[p] && IntegerQ[(I*n - 1)/2])
  5118. Int[Times[Power[E, Times[ArcCot[Times[Optional[Pattern[a, Blank[]]], Pattern[x, Blank[]]]], Optional[Pattern[n, Blank[]]]]], Power[Pattern[x, Blank[]], 2], Power[Plus[Pattern[c, Blank[]], Times[Optional[Pattern[d, Blank[]]], Power[Pattern[x, Blank[]], 2]]], Optional[Pattern[p, Blank[]]]]], Pattern[x, Blank[Symbol]]] := Simp[((n + 2*(p + 1)*a*x)*(c + d*x^2)^(p + 1)*E^(n*ArcCot[a*x]))/(a^3*c*n^2*(n^2 + 1)), x] /; FreeQ[{a, c, d, n}, x] && EqQ[d, a^2*c] && EqQ[n^2 - 2*(p + 1), 0] && NeQ[n^2 + 1, 0]
  5119. Int[Times[Power[E, Times[ArcCot[Times[Optional[Pattern[a, Blank[]]], Pattern[x, Blank[]]]], Optional[Pattern[n, Blank[]]]]], Power[Pattern[x, Blank[]], 2], Power[Plus[Pattern[c, Blank[]], Times[Optional[Pattern[d, Blank[]]], Power[Pattern[x, Blank[]], 2]]], Pattern[p, Blank[]]]], Pattern[x, Blank[Symbol]]] := Simp[((n + 2*(p + 1)*a*x)*(c + d*x^2)^(p + 1)*E^(n*ArcCot[a*x]))/(a^3*c*(n^2 + 4*(p + 1)^2)), x] + Dist[(n^2 - 2*(p + 1))/(a^2*c*(n^2 + 4*(p + 1)^2)), Int[(c + d*x^2)^(p + 1)*E^(n*ArcCot[a*x]), x], x] /; FreeQ[{a, c, d, n}, x] && EqQ[d, a^2*c] && LeQ[p, -1] && NeQ[n^2 - 2*(p + 1), 0] && NeQ[n^2 + 4*(p + 1)^2, 0] && !(IntegerQ[p] && IntegerQ[(I*n)/2]) && !( !IntegerQ[p] && IntegerQ[(I*n - 1)/2])
  5120. Int[Times[Power[E, Times[ArcCot[Times[Optional[Pattern[a, Blank[]]], Pattern[x, Blank[]]]], Optional[Pattern[n, Blank[]]]]], Power[Pattern[x, Blank[]], Optional[Pattern[m, Blank[]]]], Power[Plus[Pattern[c, Blank[]], Times[Optional[Pattern[d, Blank[]]], Power[Pattern[x, Blank[]], 2]]], Pattern[p, Blank[]]]], Pattern[x, Blank[Symbol]]] := -Dist[c^p/a^(m + 1), Subst[Int[(E^(n*x)*Cot[x]^(m + 2*(p + 1)))/Cos[x]^(2*(p + 1)), x], x, ArcCot[a*x]], x] /; FreeQ[{a, c, d, n}, x] && EqQ[d, a^2*c] && IntegerQ[m] && LeQ[3, m, -2*(p + 1)] && IntegerQ[p]
  5121. Int[Times[Power[E, Times[ArcCot[Times[Optional[Pattern[a, Blank[]]], Pattern[x, Blank[]]]], Optional[Pattern[n, Blank[]]]]], Optional[Pattern[u, Blank[]]], Power[Plus[Pattern[c, Blank[]], Times[Optional[Pattern[d, Blank[]]], Power[Pattern[x, Blank[]], 2]]], Optional[Pattern[p, Blank[]]]]], Pattern[x, Blank[Symbol]]] := Dist[d^p, Int[u*x^(2*p)*(1 + 1/(a^2*x^2))^p*E^(n*ArcCot[a*x]), x], x] /; FreeQ[{a, c, d, n}, x] && EqQ[d, a^2*c] && !IntegerQ[(I*n)/2] && IntegerQ[p]
  5122. Int[Times[Power[E, Times[ArcCot[Times[Optional[Pattern[a, Blank[]]], Pattern[x, Blank[]]]], Optional[Pattern[n, Blank[]]]]], Optional[Pattern[u, Blank[]]], Power[Plus[Pattern[c, Blank[]], Times[Optional[Pattern[d, Blank[]]], Power[Pattern[x, Blank[]], 2]]], Pattern[p, Blank[]]]], Pattern[x, Blank[Symbol]]] := Dist[(c + d*x^2)^p/(x^(2*p)*(1 + 1/(a^2*x^2))^p), Int[u*x^(2*p)*(1 + 1/(a^2*x^2))^p*E^(n*ArcCot[a*x]), x], x] /; FreeQ[{a, c, d, n, p}, x] && EqQ[d, a^2*c] && !IntegerQ[(I*n)/2] && !IntegerQ[p]
  5123. Int[Times[Power[E, Times[ArcCot[Times[Optional[Pattern[a, Blank[]]], Pattern[x, Blank[]]]], Optional[Pattern[n, Blank[]]]]], Optional[Pattern[u, Blank[]]], Power[Plus[Pattern[c, Blank[]], Times[Optional[Pattern[d, Blank[]]], Power[Pattern[x, Blank[]], -2]]], Optional[Pattern[p, Blank[]]]]], Pattern[x, Blank[Symbol]]] := Dist[c^p/(I*a)^(2*p), Int[(u*(-1 + I*a*x)^(p - (I*n)/2)*(1 + I*a*x)^(p + (I*n)/2))/x^(2*p), x], x] /; FreeQ[{a, c, d, n, p}, x] && EqQ[c, a^2*d] && !IntegerQ[(I*n)/2] && (IntegerQ[p] || GtQ[c, 0]) && IntegersQ[2*p, p + (I*n)/2]
  5124. Int[Times[Power[E, Times[ArcCot[Times[Optional[Pattern[a, Blank[]]], Pattern[x, Blank[]]]], Optional[Pattern[n, Blank[]]]]], Power[Plus[Pattern[c, Blank[]], Times[Optional[Pattern[d, Blank[]]], Power[Pattern[x, Blank[]], -2]]], Optional[Pattern[p, Blank[]]]]], Pattern[x, Blank[Symbol]]] := -Dist[c^p, Subst[Int[((1 - (I*x)/a)^(p + (I*n)/2)*(1 + (I*x)/a)^(p - (I*n)/2))/x^2, x], x, 1/x], x] /; FreeQ[{a, c, d, n, p}, x] && EqQ[c, a^2*d] && !IntegerQ[(I*n)/2] && (IntegerQ[p] || GtQ[c, 0]) && !(IntegerQ[2*p] && IntegerQ[p + (I*n)/2])
  5125. Int[Times[Power[E, Times[ArcCot[Times[Optional[Pattern[a, Blank[]]], Pattern[x, Blank[]]]], Optional[Pattern[n, Blank[]]]]], Power[Plus[Pattern[c, Blank[]], Times[Optional[Pattern[d, Blank[]]], Power[Pattern[x, Blank[]], -2]]], Optional[Pattern[p, Blank[]]]], Power[Pattern[x, Blank[]], Optional[Pattern[m, Blank[]]]]], Pattern[x, Blank[Symbol]]] := -Dist[c^p, Subst[Int[((1 - (I*x)/a)^(p + (I*n)/2)*(1 + (I*x)/a)^(p - (I*n)/2))/x^(m + 2), x], x, 1/x], x] /; FreeQ[{a, c, d, n, p}, x] && EqQ[c, a^2*d] && !IntegerQ[(I*n)/2] && (IntegerQ[p] || GtQ[c, 0]) && !(IntegerQ[2*p] && IntegerQ[p + (I*n)/2]) && IntegerQ[m]
  5126. Int[Times[Power[E, Times[ArcCot[Times[Optional[Pattern[a, Blank[]]], Pattern[x, Blank[]]]], Optional[Pattern[n, Blank[]]]]], Power[Plus[Pattern[c, Blank[]], Times[Optional[Pattern[d, Blank[]]], Power[Pattern[x, Blank[]], -2]]], Optional[Pattern[p, Blank[]]]], Power[Pattern[x, Blank[]], Pattern[m, Blank[]]]], Pattern[x, Blank[Symbol]]] := -Dist[c^p*x^m*(1/x)^m, Subst[Int[((1 - (I*x)/a)^(p + (I*n)/2)*(1 + (I*x)/a)^(p - (I*n)/2))/x^(m + 2), x], x, 1/x], x] /; FreeQ[{a, c, d, m, n, p}, x] && EqQ[c, a^2*d] && !IntegerQ[(I*n)/2] && (IntegerQ[p] || GtQ[c, 0]) && !(IntegerQ[2*p] && IntegerQ[p + (I*n)/2]) && !IntegerQ[m]
  5127. Int[Times[Power[E, Times[ArcCot[Times[Optional[Pattern[a, Blank[]]], Pattern[x, Blank[]]]], Optional[Pattern[n, Blank[]]]]], Optional[Pattern[u, Blank[]]], Power[Plus[Pattern[c, Blank[]], Times[Optional[Pattern[d, Blank[]]], Power[Pattern[x, Blank[]], -2]]], Pattern[p, Blank[]]]], Pattern[x, Blank[Symbol]]] := Dist[(c + d/x^2)^p/(1 + 1/(a^2*x^2))^p, Int[u*(1 + 1/(a^2*x^2))^p*E^(n*ArcCot[a*x]), x], x] /; FreeQ[{a, c, d, n, p}, x] && EqQ[c, a^2*d] && !IntegerQ[(I*n)/2] && !(IntegerQ[p] || GtQ[c, 0])
  5128. Int[Times[Power[E, Times[ArcCot[Times[Optional[Pattern[c, Blank[]]], Plus[Pattern[a, Blank[]], Times[Optional[Pattern[b, Blank[]]], Pattern[x, Blank[]]]]]], Pattern[n, Blank[]]]], Optional[Pattern[u, Blank[]]]], Pattern[x, Blank[Symbol]]] := Dist[(-1)^((I*n)/2), Int[u/E^(n*ArcTan[c*(a + b*x)]), x], x] /; FreeQ[{a, b, c}, x] && IntegerQ[(I*n)/2]
  5129. Int[Power[E, Times[ArcCot[Times[Optional[Pattern[c, Blank[]]], Plus[Pattern[a, Blank[]], Times[Optional[Pattern[b, Blank[]]], Pattern[x, Blank[]]]]]], Optional[Pattern[n, Blank[]]]]], Pattern[x, Blank[Symbol]]] := Dist[((I*c*(a + b*x))^((I*n)/2)*(1 + 1/(I*c*(a + b*x)))^((I*n)/2))/(1 + I*a*c + I*b*c*x)^((I*n)/2), Int[(1 + I*a*c + I*b*c*x)^((I*n)/2)/(-1 + I*a*c + I*b*c*x)^((I*n)/2), x], x] /; FreeQ[{a, b, c, n}, x] && !IntegerQ[(I*n)/2]
  5130. Int[Times[Power[E, Times[ArcCoth[Times[Optional[Pattern[c, Blank[]]], Plus[Pattern[a, Blank[]], Times[Optional[Pattern[b, Blank[]]], Pattern[x, Blank[]]]]]], Pattern[n, Blank[]]]], Power[Pattern[x, Blank[]], Pattern[m, Blank[]]]], Pattern[x, Blank[Symbol]]] := Dist[4/(I^m*n*b^(m + 1)*c^(m + 1)), Subst[Int[(x^(2/(I*n))*(1 + I*a*c + (1 - I*a*c)*x^(2/(I*n)))^m)/(-1 + x^(2/(I*n)))^(m + 2), x], x, (1 + 1/(I*c*(a + b*x)))^((I*n)/2)/(1 - 1/(I*c*(a + b*x)))^((I*n)/2)], x] /; FreeQ[{a, b, c}, x] && ILtQ[m, 0] && LtQ[-1, I*n, 1]
  5131. Int[Times[Power[E, Times[ArcCoth[Times[Optional[Pattern[c, Blank[]]], Plus[Pattern[a, Blank[]], Times[Optional[Pattern[b, Blank[]]], Pattern[x, Blank[]]]]]], Optional[Pattern[n, Blank[]]]]], Power[Plus[Optional[Pattern[d, Blank[]]], Times[Optional[Pattern[e, Blank[]]], Pattern[x, Blank[]]]], Optional[Pattern[m, Blank[]]]]], Pattern[x, Blank[Symbol]]] := Dist[((I*c*(a + b*x))^((I*n)/2)*(1 + 1/(I*c*(a + b*x)))^((I*n)/2))/(1 + I*a*c + I*b*c*x)^((I*n)/2), Int[((d + e*x)^m*(1 + I*a*c + I*b*c*x)^((I*n)/2))/(-1 + I*a*c + I*b*c*x)^((I*n)/2), x], x] /; FreeQ[{a, b, c, d, e, m, n}, x] && !IntegerQ[(I*n)/2]
  5132. Int[Times[Power[E, Times[ArcCot[Plus[Pattern[a, Blank[]], Times[Optional[Pattern[b, Blank[]]], Pattern[x, Blank[]]]]], Optional[Pattern[n, Blank[]]]]], Optional[Pattern[u, Blank[]]], Power[Plus[Pattern[c, Blank[]], Times[Optional[Pattern[d, Blank[]]], Pattern[x, Blank[]]], Times[Optional[Pattern[e, Blank[]]], Power[Pattern[x, Blank[]], 2]]], Optional[Pattern[p, Blank[]]]]], Pattern[x, Blank[Symbol]]] := Dist[(c/(1 + a^2))^p*((I*a + I*b*x)/(1 + I*a + I*b*x))^((I*n)/2)*((1 + I*a + I*b*x)/(I*a + I*b*x))^((I*n)/2)*((1 - I*a - I*b*x)^((I*n)/2)/(-1 + I*a + I*b*x)^((I*n)/2)), Int[u*(1 - I*a - I*b*x)^(p - (I*n)/2)*(1 + I*a + I*b*x)^(p + (I*n)/2), x], x] /; FreeQ[{a, b, c, d, e, n, p}, x] && !IntegerQ[(I*n)/2] && EqQ[b*d - 2*a*e, 0] && EqQ[b^2*c - e*(1 + a^2), 0] && (IntegerQ[p] || GtQ[c/(1 + a^2), 0])
  5133. Int[Times[Power[E, Times[ArcCot[Plus[Pattern[a, Blank[]], Times[Optional[Pattern[b, Blank[]]], Pattern[x, Blank[]]]]], Optional[Pattern[n, Blank[]]]]], Optional[Pattern[u, Blank[]]], Power[Plus[Pattern[c, Blank[]], Times[Optional[Pattern[d, Blank[]]], Pattern[x, Blank[]]], Times[Optional[Pattern[e, Blank[]]], Power[Pattern[x, Blank[]], 2]]], Optional[Pattern[p, Blank[]]]]], Pattern[x, Blank[Symbol]]] := Dist[(c + d*x + e*x^2)^p/(1 + a^2 + 2*a*b*x + b^2*x^2)^p, Int[u*(1 + a^2 + 2*a*b*x + b^2*x^2)^p*E^(n*ArcCot[a*x]), x], x] /; FreeQ[{a, b, c, d, e, n, p}, x] && !IntegerQ[(I*n)/2] && EqQ[b*d - 2*a*e, 0] && EqQ[b^2*c - e*(1 + a^2), 0] && !(IntegerQ[p] || GtQ[c/(1 + a^2), 0])
  5134. Int[Times[Power[E, Times[ArcCot[Times[Optional[Pattern[c, Blank[]]], Power[Plus[Optional[Pattern[a, Blank[]]], Times[Optional[Pattern[b, Blank[]]], Pattern[x, Blank[]]]], -1]]], Optional[Pattern[n, Blank[]]]]], Optional[Pattern[u, Blank[]]]], Pattern[x, Blank[Symbol]]] := Int[u*E^(n*ArcTan[a/c + (b*x)/c]), x] /; FreeQ[{a, b, c, n}, x]
  5135. Int[ArcTan[Plus[Pattern[a, Blank[]], Times[Optional[Pattern[b, Blank[]]], Power[Pattern[x, Blank[]], Pattern[n, Blank[]]]]]], Pattern[x, Blank[Symbol]]] := Simp[x*ArcTan[a + b*x^n], x] - Dist[b*n, Int[x^n/(1 + a^2 + 2*a*b*x^n + b^2*x^(2*n)), x], x] /; FreeQ[{a, b, n}, x]
  5136. Int[ArcCot[Plus[Pattern[a, Blank[]], Times[Optional[Pattern[b, Blank[]]], Power[Pattern[x, Blank[]], Pattern[n, Blank[]]]]]], Pattern[x, Blank[Symbol]]] := Simp[x*ArcCot[a + b*x^n], x] + Dist[b*n, Int[x^n/(1 + a^2 + 2*a*b*x^n + b^2*x^(2*n)), x], x] /; FreeQ[{a, b, n}, x]
  5137. Int[Times[ArcTan[Plus[Optional[Pattern[a, Blank[]]], Times[Optional[Pattern[b, Blank[]]], Power[Pattern[x, Blank[]], Pattern[n, Blank[]]]]]], Power[Pattern[x, Blank[]], -1]], Pattern[x, Blank[Symbol]]] := Dist[I/2, Int[Log[1 - I*a - I*b*x^n]/x, x], x] - Dist[I/2, Int[Log[1 + I*a + I*b*x^n]/x, x], x] /; FreeQ[{a, b, n}, x]
  5138. Int[Times[ArcCot[Plus[Optional[Pattern[a, Blank[]]], Times[Optional[Pattern[b, Blank[]]], Power[Pattern[x, Blank[]], Pattern[n, Blank[]]]]]], Power[Pattern[x, Blank[]], -1]], Pattern[x, Blank[Symbol]]] := Dist[I/2, Int[Log[1 - I/(a + b*x^n)]/x, x], x] - Dist[I/2, Int[Log[1 + I/(a + b*x^n)]/x, x], x] /; FreeQ[{a, b, n}, x]
  5139. Int[Times[ArcTan[Plus[Pattern[a, Blank[]], Times[Optional[Pattern[b, Blank[]]], Power[Pattern[x, Blank[]], Pattern[n, Blank[]]]]]], Power[Pattern[x, Blank[]], Optional[Pattern[m, Blank[]]]]], Pattern[x, Blank[Symbol]]] := Simp[(x^(m + 1)*ArcTan[a + b*x^n])/(m + 1), x] - Dist[(b*n)/(m + 1), Int[x^(m + n)/(1 + a^2 + 2*a*b*x^n + b^2*x^(2*n)), x], x] /; FreeQ[{a, b}, x] && RationalQ[m, n] && m + 1 != 0 && m + 1 != n
  5140. Int[Times[ArcCot[Plus[Pattern[a, Blank[]], Times[Optional[Pattern[b, Blank[]]], Power[Pattern[x, Blank[]], Pattern[n, Blank[]]]]]], Power[Pattern[x, Blank[]], Optional[Pattern[m, Blank[]]]]], Pattern[x, Blank[Symbol]]] := Simp[(x^(m + 1)*ArcCot[a + b*x^n])/(m + 1), x] + Dist[(b*n)/(m + 1), Int[x^(m + n)/(1 + a^2 + 2*a*b*x^n + b^2*x^(2*n)), x], x] /; FreeQ[{a, b}, x] && RationalQ[m, n] && m + 1 != 0 && m + 1 != n
  5141. Int[ArcTan[Plus[Optional[Pattern[a, Blank[]]], Times[Optional[Pattern[b, Blank[]]], Power[Pattern[f, Blank[]], Plus[Optional[Pattern[c, Blank[]]], Times[Optional[Pattern[d, Blank[]]], Pattern[x, Blank[]]]]]]]], Pattern[x, Blank[Symbol]]] := Dist[I/2, Int[Log[1 - I*a - I*b*f^(c + d*x)], x], x] - Dist[I/2, Int[Log[1 + I*a + I*b*f^(c + d*x)], x], x] /; FreeQ[{a, b, c, d, f}, x]
  5142. Int[ArcCot[Plus[Optional[Pattern[a, Blank[]]], Times[Optional[Pattern[b, Blank[]]], Power[Pattern[f, Blank[]], Plus[Optional[Pattern[c, Blank[]]], Times[Optional[Pattern[d, Blank[]]], Pattern[x, Blank[]]]]]]]], Pattern[x, Blank[Symbol]]] := Dist[I/2, Int[Log[1 - I/(a + b*f^(c + d*x))], x], x] - Dist[I/2, Int[Log[1 + I/(a + b*f^(c + d*x))], x], x] /; FreeQ[{a, b, c, d, f}, x]
  5143. Int[Times[ArcTan[Plus[Optional[Pattern[a, Blank[]]], Times[Optional[Pattern[b, Blank[]]], Power[Pattern[f, Blank[]], Plus[Optional[Pattern[c, Blank[]]], Times[Optional[Pattern[d, Blank[]]], Pattern[x, Blank[]]]]]]]], Power[Pattern[x, Blank[]], Optional[Pattern[m, Blank[]]]]], Pattern[x, Blank[Symbol]]] := Dist[I/2, Int[x^m*Log[1 - I*a - I*b*f^(c + d*x)], x], x] - Dist[I/2, Int[x^m*Log[1 + I*a + I*b*f^(c + d*x)], x], x] /; FreeQ[{a, b, c, d, f}, x] && IntegerQ[m] && m > 0
  5144. Int[Times[ArcCot[Plus[Optional[Pattern[a, Blank[]]], Times[Optional[Pattern[b, Blank[]]], Power[Pattern[f, Blank[]], Plus[Optional[Pattern[c, Blank[]]], Times[Optional[Pattern[d, Blank[]]], Pattern[x, Blank[]]]]]]]], Power[Pattern[x, Blank[]], Optional[Pattern[m, Blank[]]]]], Pattern[x, Blank[Symbol]]] := Dist[I/2, Int[x^m*Log[1 - I/(a + b*f^(c + d*x))], x], x] - Dist[I/2, Int[x^m*Log[1 + I/(a + b*f^(c + d*x))], x], x] /; FreeQ[{a, b, c, d, f}, x] && IntegerQ[m] && m > 0
  5145. Int[Times[Power[ArcTan[Times[Optional[Pattern[c, Blank[]]], Power[Plus[Optional[Pattern[a, Blank[]]], Times[Optional[Pattern[b, Blank[]]], Power[Pattern[x, Blank[]], Optional[Pattern[n, Blank[]]]]]], -1]]], Optional[Pattern[m, Blank[]]]], Optional[Pattern[u, Blank[]]]], Pattern[x, Blank[Symbol]]] := Int[u*ArcCot[a/c + (b*x^n)/c]^m, x] /; FreeQ[{a, b, c, n, m}, x]
  5146. Int[Times[Power[ArcCot[Times[Optional[Pattern[c, Blank[]]], Power[Plus[Optional[Pattern[a, Blank[]]], Times[Optional[Pattern[b, Blank[]]], Power[Pattern[x, Blank[]], Optional[Pattern[n, Blank[]]]]]], -1]]], Optional[Pattern[m, Blank[]]]], Optional[Pattern[u, Blank[]]]], Pattern[x, Blank[Symbol]]] := Int[u*ArcTan[a/c + (b*x^n)/c]^m, x] /; FreeQ[{a, b, c, n, m}, x]
  5147. Int[ArcTan[Times[Optional[Pattern[c, Blank[]]], Pattern[x, Blank[]], Power[Plus[Optional[Pattern[a, Blank[]]], Times[Optional[Pattern[b, Blank[]]], Power[Pattern[x, Blank[]], 2]]], Rational[-1, 2]]]], Pattern[x, Blank[Symbol]]] := Simp[x*ArcTan[(c*x)/Sqrt[a + b*x^2]], x] - Dist[c, Int[x/Sqrt[a + b*x^2], x], x] /; FreeQ[{a, b, c}, x] && EqQ[b + c^2, 0]
  5148. Int[ArcCot[Times[Optional[Pattern[c, Blank[]]], Pattern[x, Blank[]], Power[Plus[Optional[Pattern[a, Blank[]]], Times[Optional[Pattern[b, Blank[]]], Power[Pattern[x, Blank[]], 2]]], Rational[-1, 2]]]], Pattern[x, Blank[Symbol]]] := Simp[x*ArcCot[(c*x)/Sqrt[a + b*x^2]], x] + Dist[c, Int[x/Sqrt[a + b*x^2], x], x] /; FreeQ[{a, b, c}, x] && EqQ[b + c^2, 0]
  5149. Int[Times[ArcTan[Times[Optional[Pattern[c, Blank[]]], Pattern[x, Blank[]], Power[Plus[Optional[Pattern[a, Blank[]]], Times[Optional[Pattern[b, Blank[]]], Power[Pattern[x, Blank[]], 2]]], Rational[-1, 2]]]], Power[Pattern[x, Blank[]], -1]], Pattern[x, Blank[Symbol]]] := Simp[ArcTan[(c*x)/Sqrt[a + b*x^2]]*Log[x], x] - Dist[c, Int[Log[x]/Sqrt[a + b*x^2], x], x] /; FreeQ[{a, b, c}, x] && EqQ[b + c^2, 0]
  5150. Int[Times[ArcCot[Times[Optional[Pattern[c, Blank[]]], Pattern[x, Blank[]], Power[Plus[Optional[Pattern[a, Blank[]]], Times[Optional[Pattern[b, Blank[]]], Power[Pattern[x, Blank[]], 2]]], Rational[-1, 2]]]], Power[Pattern[x, Blank[]], -1]], Pattern[x, Blank[Symbol]]] := Simp[ArcCot[(c*x)/Sqrt[a + b*x^2]]*Log[x], x] + Dist[c, Int[Log[x]/Sqrt[a + b*x^2], x], x] /; FreeQ[{a, b, c}, x] && EqQ[b + c^2, 0]
  5151. Int[Times[ArcTan[Times[Optional[Pattern[c, Blank[]]], Pattern[x, Blank[]], Power[Plus[Optional[Pattern[a, Blank[]]], Times[Optional[Pattern[b, Blank[]]], Power[Pattern[x, Blank[]], 2]]], Rational[-1, 2]]]], Power[Times[Optional[Pattern[d, Blank[]]], Pattern[x, Blank[]]], Optional[Pattern[m, Blank[]]]]], Pattern[x, Blank[Symbol]]] := Simp[((d*x)^(m + 1)*ArcTan[(c*x)/Sqrt[a + b*x^2]])/(d*(m + 1)), x] - Dist[c/(d*(m + 1)), Int[(d*x)^(m + 1)/Sqrt[a + b*x^2], x], x] /; FreeQ[{a, b, c, d, m}, x] && EqQ[b + c^2, 0] && NeQ[m, -1]
  5152. Int[Times[ArcCot[Times[Optional[Pattern[c, Blank[]]], Pattern[x, Blank[]], Power[Plus[Optional[Pattern[a, Blank[]]], Times[Optional[Pattern[b, Blank[]]], Power[Pattern[x, Blank[]], 2]]], Rational[-1, 2]]]], Power[Times[Optional[Pattern[d, Blank[]]], Pattern[x, Blank[]]], Optional[Pattern[m, Blank[]]]]], Pattern[x, Blank[Symbol]]] := Simp[((d*x)^(m + 1)*ArcCot[(c*x)/Sqrt[a + b*x^2]])/(d*(m + 1)), x] + Dist[c/(d*(m + 1)), Int[(d*x)^(m + 1)/Sqrt[a + b*x^2], x], x] /; FreeQ[{a, b, c, d, m}, x] && EqQ[b + c^2, 0] && NeQ[m, -1]
  5153. Int[Times[Power[ArcTan[Times[Optional[Pattern[c, Blank[]]], Pattern[x, Blank[]], Power[Plus[Optional[Pattern[a, Blank[]]], Times[Optional[Pattern[b, Blank[]]], Power[Pattern[x, Blank[]], 2]]], Rational[-1, 2]]]], -1], Power[Plus[Optional[Pattern[a, Blank[]]], Times[Optional[Pattern[b, Blank[]]], Power[Pattern[x, Blank[]], 2]]], Rational[-1, 2]]], Pattern[x, Blank[Symbol]]] := Simp[(1*Log[ArcTan[(c*x)/Sqrt[a + b*x^2]]])/c, x] /; FreeQ[{a, b, c}, x] && EqQ[b + c^2, 0]
  5154. Int[Times[Power[ArcCot[Times[Optional[Pattern[c, Blank[]]], Pattern[x, Blank[]], Power[Plus[Optional[Pattern[a, Blank[]]], Times[Optional[Pattern[b, Blank[]]], Power[Pattern[x, Blank[]], 2]]], Rational[-1, 2]]]], -1], Power[Plus[Optional[Pattern[a, Blank[]]], Times[Optional[Pattern[b, Blank[]]], Power[Pattern[x, Blank[]], 2]]], Rational[-1, 2]]], Pattern[x, Blank[Symbol]]] := -Simp[Log[ArcCot[(c*x)/Sqrt[a + b*x^2]]]/c, x] /; FreeQ[{a, b, c}, x] && EqQ[b + c^2, 0]
  5155. Int[Times[Power[ArcTan[Times[Optional[Pattern[c, Blank[]]], Pattern[x, Blank[]], Power[Plus[Optional[Pattern[a, Blank[]]], Times[Optional[Pattern[b, Blank[]]], Power[Pattern[x, Blank[]], 2]]], Rational[-1, 2]]]], Optional[Pattern[m, Blank[]]]], Power[Plus[Optional[Pattern[a, Blank[]]], Times[Optional[Pattern[b, Blank[]]], Power[Pattern[x, Blank[]], 2]]], Rational[-1, 2]]], Pattern[x, Blank[Symbol]]] := Simp[ArcTan[(c*x)/Sqrt[a + b*x^2]]^(m + 1)/(c*(m + 1)), x] /; FreeQ[{a, b, c, m}, x] && EqQ[b + c^2, 0] && NeQ[m, -1]
  5156. Int[Times[Power[ArcCot[Times[Optional[Pattern[c, Blank[]]], Pattern[x, Blank[]], Power[Plus[Optional[Pattern[a, Blank[]]], Times[Optional[Pattern[b, Blank[]]], Power[Pattern[x, Blank[]], 2]]], Rational[-1, 2]]]], Optional[Pattern[m, Blank[]]]], Power[Plus[Optional[Pattern[a, Blank[]]], Times[Optional[Pattern[b, Blank[]]], Power[Pattern[x, Blank[]], 2]]], Rational[-1, 2]]], Pattern[x, Blank[Symbol]]] := -Simp[ArcCot[(c*x)/Sqrt[a + b*x^2]]^(m + 1)/(c*(m + 1)), x] /; FreeQ[{a, b, c, m}, x] && EqQ[b + c^2, 0] && NeQ[m, -1]
  5157. Int[Times[Power[ArcTan[Times[Optional[Pattern[c, Blank[]]], Pattern[x, Blank[]], Power[Plus[Optional[Pattern[a, Blank[]]], Times[Optional[Pattern[b, Blank[]]], Power[Pattern[x, Blank[]], 2]]], Rational[-1, 2]]]], Optional[Pattern[m, Blank[]]]], Power[Plus[Optional[Pattern[d, Blank[]]], Times[Optional[Pattern[e, Blank[]]], Power[Pattern[x, Blank[]], 2]]], Rational[-1, 2]]], Pattern[x, Blank[Symbol]]] := Dist[Sqrt[a + b*x^2]/Sqrt[d + e*x^2], Int[ArcTan[(c*x)/Sqrt[a + b*x^2]]^m/Sqrt[a + b*x^2], x], x] /; FreeQ[{a, b, c, d, e, m}, x] && EqQ[b + c^2, 0] && EqQ[b*d - a*e, 0]
  5158. Int[Times[Power[ArcCot[Times[Optional[Pattern[c, Blank[]]], Pattern[x, Blank[]], Power[Plus[Optional[Pattern[a, Blank[]]], Times[Optional[Pattern[b, Blank[]]], Power[Pattern[x, Blank[]], 2]]], Rational[-1, 2]]]], Optional[Pattern[m, Blank[]]]], Power[Plus[Optional[Pattern[d, Blank[]]], Times[Optional[Pattern[e, Blank[]]], Power[Pattern[x, Blank[]], 2]]], Rational[-1, 2]]], Pattern[x, Blank[Symbol]]] := Dist[Sqrt[a + b*x^2]/Sqrt[d + e*x^2], Int[ArcCot[(c*x)/Sqrt[a + b*x^2]]^m/Sqrt[a + b*x^2], x], x] /; FreeQ[{a, b, c, d, e, m}, x] && EqQ[b + c^2, 0] && EqQ[b*d - a*e, 0]
  5159. Int[Times[ArcTan[Plus[Pattern[v, Blank[]], Times[Optional[Pattern[s, Blank[]]], Power[Pattern[w, Blank[]], Rational[1, 2]]]]], Optional[Pattern[u, Blank[]]]], Pattern[x, Blank[Symbol]]] := Dist[(Pi*s)/4, Int[u, x], x] + Dist[1/2, Int[u*ArcTan[v], x], x] /; EqQ[s^2, 1] && EqQ[w, v^2 + 1]
  5160. Int[Times[ArcCot[Plus[Pattern[v, Blank[]], Times[Optional[Pattern[s, Blank[]]], Power[Pattern[w, Blank[]], Rational[1, 2]]]]], Optional[Pattern[u, Blank[]]]], Pattern[x, Blank[Symbol]]] := Dist[(Pi*s)/4, Int[u, x], x] - Dist[1/2, Int[u*ArcTan[v], x], x] /; EqQ[s^2, 1] && EqQ[w, v^2 + 1]
  5161. Int[Times[Pattern[u, Blank[]], Power[Pattern[v, Blank[]], Optional[Pattern[n, Blank[]]]]], Pattern[x, Blank[Symbol]]] := With[{tmp = InverseFunctionOfLinear[u, x]}, Dist[(-(Discriminant[v, x]/(4*Coefficient[v, x, 2])))^n/Coefficient[tmp[[1]], x, 1], Subst[Int[SimplifyIntegrand[SubstForInverseFunction[u, tmp, x]*Sec[x]^(2*(n + 1)), x], x], x, tmp], x] /; !FalseQ[tmp] && EqQ[Head[tmp], ArcTan] && EqQ[Discriminant[v, x]*tmp[[1]]^2 + D[v, x]^2, 0]] /; QuadraticQ[v, x] && ILtQ[n, 0] && NegQ[Discriminant[v, x]] && MatchQ[u, (r_.)*(f_)^(w_) /; FreeQ[f, x]]
  5162. Int[Times[Pattern[u, Blank[]], Power[Pattern[v, Blank[]], Optional[Pattern[n, Blank[]]]]], Pattern[x, Blank[Symbol]]] := With[{tmp = InverseFunctionOfLinear[u, x]}, -Dist[(-(Discriminant[v, x]/(4*Coefficient[v, x, 2])))^n/Coefficient[tmp[[1]], x, 1], Subst[Int[SimplifyIntegrand[SubstForInverseFunction[u, tmp, x]*Csc[x]^(2*(n + 1)), x], x], x, tmp], x] /; !FalseQ[tmp] && EqQ[Head[tmp], ArcCot] && EqQ[Discriminant[v, x]*tmp[[1]]^2 + D[v, x]^2, 0]] /; QuadraticQ[v, x] && ILtQ[n, 0] && NegQ[Discriminant[v, x]] && MatchQ[u, (r_.)*(f_)^(w_) /; FreeQ[f, x]]
  5163. Int[ArcTan[Plus[Optional[Pattern[c, Blank[]]], Times[Optional[Pattern[d, Blank[]]], Tan[Plus[Optional[Pattern[a, Blank[]]], Times[Optional[Pattern[b, Blank[]]], Pattern[x, Blank[]]]]]]]], Pattern[x, Blank[Symbol]]] := Simp[x*ArcTan[c + d*Tan[a + b*x]], x] - Dist[I*b, Int[x/(c + I*d + c*E^(2*I*a + 2*I*b*x)), x], x] /; FreeQ[{a, b, c, d}, x] && EqQ[(c + I*d)^2, -1]
  5164. Int[ArcCot[Plus[Optional[Pattern[c, Blank[]]], Times[Optional[Pattern[d, Blank[]]], Tan[Plus[Optional[Pattern[a, Blank[]]], Times[Optional[Pattern[b, Blank[]]], Pattern[x, Blank[]]]]]]]], Pattern[x, Blank[Symbol]]] := Simp[x*ArcCot[c + d*Tan[a + b*x]], x] + Dist[I*b, Int[x/(c + I*d + c*E^(2*I*a + 2*I*b*x)), x], x] /; FreeQ[{a, b, c, d}, x] && EqQ[(c + I*d)^2, -1]
  5165. Int[ArcTan[Plus[Optional[Pattern[c, Blank[]]], Times[Cot[Plus[Optional[Pattern[a, Blank[]]], Times[Optional[Pattern[b, Blank[]]], Pattern[x, Blank[]]]]], Optional[Pattern[d, Blank[]]]]]], Pattern[x, Blank[Symbol]]] := Simp[x*ArcTan[c + d*Cot[a + b*x]], x] - Dist[I*b, Int[x/(c - I*d - c*E^(2*I*a + 2*I*b*x)), x], x] /; FreeQ[{a, b, c, d}, x] && EqQ[(c - I*d)^2, -1]
  5166. Int[ArcCot[Plus[Optional[Pattern[c, Blank[]]], Times[Cot[Plus[Optional[Pattern[a, Blank[]]], Times[Optional[Pattern[b, Blank[]]], Pattern[x, Blank[]]]]], Optional[Pattern[d, Blank[]]]]]], Pattern[x, Blank[Symbol]]] := Simp[x*ArcCot[c + d*Cot[a + b*x]], x] + Dist[I*b, Int[x/(c - I*d - c*E^(2*I*a + 2*I*b*x)), x], x] /; FreeQ[{a, b, c, d}, x] && EqQ[(c - I*d)^2, -1]
  5167. Int[ArcTan[Plus[Optional[Pattern[c, Blank[]]], Times[Optional[Pattern[d, Blank[]]], Tan[Plus[Optional[Pattern[a, Blank[]]], Times[Optional[Pattern[b, Blank[]]], Pattern[x, Blank[]]]]]]]], Pattern[x, Blank[Symbol]]] := Simp[x*ArcTan[c + d*Tan[a + b*x]], x] + (Dist[b*(1 - I*c - d), Int[(x*E^(2*I*a + 2*I*b*x))/(1 - I*c + d + (1 - I*c - d)*E^(2*I*a + 2*I*b*x)), x], x] - Dist[b*(1 + I*c + d), Int[(x*E^(2*I*a + 2*I*b*x))/(1 + I*c - d + (1 + I*c + d)*E^(2*I*a + 2*I*b*x)), x], x]) /; FreeQ[{a, b, c, d}, x] && NeQ[(c + I*d)^2, -1]
  5168. Int[ArcCot[Plus[Optional[Pattern[c, Blank[]]], Times[Optional[Pattern[d, Blank[]]], Tan[Plus[Optional[Pattern[a, Blank[]]], Times[Optional[Pattern[b, Blank[]]], Pattern[x, Blank[]]]]]]]], Pattern[x, Blank[Symbol]]] := Simp[x*ArcCot[c + d*Tan[a + b*x]], x] + (-Dist[b*(1 - I*c - d), Int[(x*E^(2*I*a + 2*I*b*x))/(1 - I*c + d + (1 - I*c - d)*E^(2*I*a + 2*I*b*x)), x], x] + Dist[b*(1 + I*c + d), Int[(x*E^(2*I*a + 2*I*b*x))/(1 + I*c - d + (1 + I*c + d)*E^(2*I*a + 2*I*b*x)), x], x]) /; FreeQ[{a, b, c, d}, x] && NeQ[(c + I*d)^2, -1]
  5169. Int[ArcTan[Plus[Optional[Pattern[c, Blank[]]], Times[Cot[Plus[Optional[Pattern[a, Blank[]]], Times[Optional[Pattern[b, Blank[]]], Pattern[x, Blank[]]]]], Optional[Pattern[d, Blank[]]]]]], Pattern[x, Blank[Symbol]]] := Simp[x*ArcTan[c + d*Cot[a + b*x]], x] + (Dist[b*(1 + I*c - d), Int[(x*E^(2*I*a + 2*I*b*x))/(1 + I*c + d - (1 + I*c - d)*E^(2*I*a + 2*I*b*x)), x], x] - Dist[b*(1 - I*c + d), Int[(x*E^(2*I*a + 2*I*b*x))/(1 - I*c - d - (1 - I*c + d)*E^(2*I*a + 2*I*b*x)), x], x]) /; FreeQ[{a, b, c, d}, x] && NeQ[(c + I*d)^2, -1]
  5170. Int[ArcCot[Plus[Optional[Pattern[c, Blank[]]], Times[Cot[Plus[Optional[Pattern[a, Blank[]]], Times[Optional[Pattern[b, Blank[]]], Pattern[x, Blank[]]]]], Optional[Pattern[d, Blank[]]]]]], Pattern[x, Blank[Symbol]]] := Simp[x*ArcCot[c + d*Cot[a + b*x]], x] + (-Dist[b*(1 + I*c - d), Int[(x*E^(2*I*a + 2*I*b*x))/(1 + I*c + d - (1 + I*c - d)*E^(2*I*a + 2*I*b*x)), x], x] + Dist[b*(1 - I*c + d), Int[(x*E^(2*I*a + 2*I*b*x))/(1 - I*c - d - (1 - I*c + d)*E^(2*I*a + 2*I*b*x)), x], x]) /; FreeQ[{a, b, c, d}, x] && NeQ[(c - I*d)^2, -1]
  5171. Int[Times[ArcTan[Plus[Optional[Pattern[c, Blank[]]], Times[Optional[Pattern[d, Blank[]]], Tan[Plus[Optional[Pattern[a, Blank[]]], Times[Optional[Pattern[b, Blank[]]], Pattern[x, Blank[]]]]]]]], Power[Plus[Optional[Pattern[e, Blank[]]], Times[Optional[Pattern[f, Blank[]]], Pattern[x, Blank[]]]], Optional[Pattern[m, Blank[]]]]], Pattern[x, Blank[Symbol]]] := Simp[((e + f*x)^(m + 1)*ArcTan[c + d*Tan[a + b*x]])/(f*(m + 1)), x] - Dist[(I*b)/(f*(m + 1)), Int[(e + f*x)^(m + 1)/(c + I*d + c*E^(2*I*a + 2*I*b*x)), x], x] /; FreeQ[{a, b, c, d, e, f}, x] && IGtQ[m, 0] && EqQ[(c + I*d)^2, -1]
  5172. Int[Times[ArcCot[Plus[Optional[Pattern[c, Blank[]]], Times[Optional[Pattern[d, Blank[]]], Tan[Plus[Optional[Pattern[a, Blank[]]], Times[Optional[Pattern[b, Blank[]]], Pattern[x, Blank[]]]]]]]], Power[Plus[Optional[Pattern[e, Blank[]]], Times[Optional[Pattern[f, Blank[]]], Pattern[x, Blank[]]]], Optional[Pattern[m, Blank[]]]]], Pattern[x, Blank[Symbol]]] := Simp[((e + f*x)^(m + 1)*ArcCot[c + d*Tan[a + b*x]])/(f*(m + 1)), x] + Dist[(I*b)/(f*(m + 1)), Int[(e + f*x)^(m + 1)/(c + I*d + c*E^(2*I*a + 2*I*b*x)), x], x] /; FreeQ[{a, b, c, d, e, f}, x] && IGtQ[m, 0] && EqQ[(c + I*d)^2, -1]
  5173. Int[Times[ArcTan[Plus[Optional[Pattern[c, Blank[]]], Times[Cot[Plus[Optional[Pattern[a, Blank[]]], Times[Optional[Pattern[b, Blank[]]], Pattern[x, Blank[]]]]], Optional[Pattern[d, Blank[]]]]]], Power[Plus[Optional[Pattern[e, Blank[]]], Times[Optional[Pattern[f, Blank[]]], Pattern[x, Blank[]]]], Optional[Pattern[m, Blank[]]]]], Pattern[x, Blank[Symbol]]] := Simp[((e + f*x)^(m + 1)*ArcTan[c + d*Cot[a + b*x]])/(f*(m + 1)), x] - Dist[(I*b)/(f*(m + 1)), Int[(e + f*x)^(m + 1)/(c - I*d - c*E^(2*I*a + 2*I*b*x)), x], x] /; FreeQ[{a, b, c, d, e, f}, x] && IGtQ[m, 0] && EqQ[(c - I*d)^2, -1]
  5174. Int[Times[ArcCot[Plus[Optional[Pattern[c, Blank[]]], Times[Cot[Plus[Optional[Pattern[a, Blank[]]], Times[Optional[Pattern[b, Blank[]]], Pattern[x, Blank[]]]]], Optional[Pattern[d, Blank[]]]]]], Power[Plus[Optional[Pattern[e, Blank[]]], Times[Optional[Pattern[f, Blank[]]], Pattern[x, Blank[]]]], Optional[Pattern[m, Blank[]]]]], Pattern[x, Blank[Symbol]]] := Simp[((e + f*x)^(m + 1)*ArcCot[c + d*Cot[a + b*x]])/(f*(m + 1)), x] + Dist[(I*b)/(f*(m + 1)), Int[(e + f*x)^(m + 1)/(c - I*d - c*E^(2*I*a + 2*I*b*x)), x], x] /; FreeQ[{a, b, c, d, e, f}, x] && IGtQ[m, 0] && EqQ[(c - I*d)^2, -1]
  5175. Int[Times[ArcTan[Plus[Optional[Pattern[c, Blank[]]], Times[Optional[Pattern[d, Blank[]]], Tan[Plus[Optional[Pattern[a, Blank[]]], Times[Optional[Pattern[b, Blank[]]], Pattern[x, Blank[]]]]]]]], Power[Plus[Optional[Pattern[e, Blank[]]], Times[Optional[Pattern[f, Blank[]]], Pattern[x, Blank[]]]], Optional[Pattern[m, Blank[]]]]], Pattern[x, Blank[Symbol]]] := Simp[((e + f*x)^(m + 1)*ArcTan[c + d*Tan[a + b*x]])/(f*(m + 1)), x] + (Dist[(b*(1 - I*c - d))/(f*(m + 1)), Int[((e + f*x)^(m + 1)*E^(2*I*a + 2*I*b*x))/(1 - I*c + d + (1 - I*c - d)*E^(2*I*a + 2*I*b*x)), x], x] - Dist[(b*(1 + I*c + d))/(f*(m + 1)), Int[((e + f*x)^(m + 1)*E^(2*I*a + 2*I*b*x))/(1 + I*c - d + (1 + I*c + d)*E^(2*I*a + 2*I*b*x)), x], x]) /; FreeQ[{a, b, c, d, e, f}, x] && IGtQ[m, 0] && NeQ[(c + I*d)^2, -1]
  5176. Int[Times[ArcCot[Plus[Optional[Pattern[c, Blank[]]], Times[Optional[Pattern[d, Blank[]]], Tan[Plus[Optional[Pattern[a, Blank[]]], Times[Optional[Pattern[b, Blank[]]], Pattern[x, Blank[]]]]]]]], Power[Plus[Optional[Pattern[e, Blank[]]], Times[Optional[Pattern[f, Blank[]]], Pattern[x, Blank[]]]], Optional[Pattern[m, Blank[]]]]], Pattern[x, Blank[Symbol]]] := Simp[((e + f*x)^(m + 1)*ArcCot[c + d*Tan[a + b*x]])/(f*(m + 1)), x] + (-Dist[(b*(1 - I*c - d))/(f*(m + 1)), Int[((e + f*x)^(m + 1)*E^(2*I*a + 2*I*b*x))/(1 - I*c + d + (1 - I*c - d)*E^(2*I*a + 2*I*b*x)), x], x] + Dist[(b*(1 + I*c + d))/(f*(m + 1)), Int[((e + f*x)^(m + 1)*E^(2*I*a + 2*I*b*x))/(1 + I*c - d + (1 + I*c + d)*E^(2*I*a + 2*I*b*x)), x], x]) /; FreeQ[{a, b, c, d, e, f}, x] && IGtQ[m, 0] && NeQ[(c + I*d)^2, -1]
  5177. Int[Times[ArcTan[Plus[Optional[Pattern[c, Blank[]]], Times[Cot[Plus[Optional[Pattern[a, Blank[]]], Times[Optional[Pattern[b, Blank[]]], Pattern[x, Blank[]]]]], Optional[Pattern[d, Blank[]]]]]], Power[Plus[Optional[Pattern[e, Blank[]]], Times[Optional[Pattern[f, Blank[]]], Pattern[x, Blank[]]]], Optional[Pattern[m, Blank[]]]]], Pattern[x, Blank[Symbol]]] := Simp[((e + f*x)^(m + 1)*ArcTan[c + d*Cot[a + b*x]])/(f*(m + 1)), x] + (Dist[(b*(1 + I*c - d))/(f*(m + 1)), Int[((e + f*x)^(m + 1)*E^(2*I*a + 2*I*b*x))/(1 + I*c + d - (1 + I*c - d)*E^(2*I*a + 2*I*b*x)), x], x] - Dist[(b*(1 - I*c + d))/(f*(m + 1)), Int[((e + f*x)^(m + 1)*E^(2*I*a + 2*I*b*x))/(1 - I*c - d - (1 - I*c + d)*E^(2*I*a + 2*I*b*x)), x], x]) /; FreeQ[{a, b, c, d, e, f}, x] && IGtQ[m, 0] && NeQ[(c - I*d)^2, -1]
  5178. Int[Times[ArcCot[Plus[Optional[Pattern[c, Blank[]]], Times[Cot[Plus[Optional[Pattern[a, Blank[]]], Times[Optional[Pattern[b, Blank[]]], Pattern[x, Blank[]]]]], Optional[Pattern[d, Blank[]]]]]], Power[Plus[Optional[Pattern[e, Blank[]]], Times[Optional[Pattern[f, Blank[]]], Pattern[x, Blank[]]]], Optional[Pattern[m, Blank[]]]]], Pattern[x, Blank[Symbol]]] := Simp[((e + f*x)^(m + 1)*ArcCot[c + d*Cot[a + b*x]])/(f*(m + 1)), x] + (-Dist[(b*(1 + I*c - d))/(f*(m + 1)), Int[((e + f*x)^(m + 1)*E^(2*I*a + 2*I*b*x))/(1 + I*c + d - (1 + I*c - d)*E^(2*I*a + 2*I*b*x)), x], x] + Dist[(b*(1 - I*c + d))/(f*(m + 1)), Int[((e + f*x)^(m + 1)*E^(2*I*a + 2*I*b*x))/(1 - I*c - d - (1 - I*c + d)*E^(2*I*a + 2*I*b*x)), x], x]) /; FreeQ[{a, b, c, d, e, f}, x] && IGtQ[m, 0] && NeQ[(c - I*d)^2, -1]
  5179. Int[ArcTan[Tanh[Plus[Optional[Pattern[a, Blank[]]], Times[Optional[Pattern[b, Blank[]]], Pattern[x, Blank[]]]]]], Pattern[x, Blank[Symbol]]] := Simp[x*ArcTan[Tanh[a + b*x]], x] - Dist[b, Int[x*Sech[2*a + 2*b*x], x], x] /; FreeQ[{a, b}, x]
  5180. Int[ArcCot[Tanh[Plus[Optional[Pattern[a, Blank[]]], Times[Optional[Pattern[b, Blank[]]], Pattern[x, Blank[]]]]]], Pattern[x, Blank[Symbol]]] := Simp[x*ArcCot[Tanh[a + b*x]], x] + Dist[b, Int[x*Sech[2*a + 2*b*x], x], x] /; FreeQ[{a, b}, x]
  5181. Int[ArcTan[Coth[Plus[Optional[Pattern[a, Blank[]]], Times[Optional[Pattern[b, Blank[]]], Pattern[x, Blank[]]]]]], Pattern[x, Blank[Symbol]]] := Simp[x*ArcTan[Coth[a + b*x]], x] + Dist[b, Int[x*Sech[2*a + 2*b*x], x], x] /; FreeQ[{a, b}, x]
  5182. Int[ArcCot[Coth[Plus[Optional[Pattern[a, Blank[]]], Times[Optional[Pattern[b, Blank[]]], Pattern[x, Blank[]]]]]], Pattern[x, Blank[Symbol]]] := Simp[x*ArcCot[Coth[a + b*x]], x] - Dist[b, Int[x*Sech[2*a + 2*b*x], x], x] /; FreeQ[{a, b}, x]
  5183. Int[Times[ArcTan[Tanh[Plus[Optional[Pattern[a, Blank[]]], Times[Optional[Pattern[b, Blank[]]], Pattern[x, Blank[]]]]]], Power[Plus[Optional[Pattern[e, Blank[]]], Times[Optional[Pattern[f, Blank[]]], Pattern[x, Blank[]]]], Optional[Pattern[m, Blank[]]]]], Pattern[x, Blank[Symbol]]] := Simp[((e + f*x)^(m + 1)*ArcTan[Tanh[a + b*x]])/(f*(m + 1)), x] - Dist[b/(f*(m + 1)), Int[(e + f*x)^(m + 1)*Sech[2*a + 2*b*x], x], x] /; FreeQ[{a, b, e, f}, x] && IGtQ[m, 0]
  5184. Int[Times[ArcCot[Tanh[Plus[Optional[Pattern[a, Blank[]]], Times[Optional[Pattern[b, Blank[]]], Pattern[x, Blank[]]]]]], Power[Plus[Optional[Pattern[e, Blank[]]], Times[Optional[Pattern[f, Blank[]]], Pattern[x, Blank[]]]], Optional[Pattern[m, Blank[]]]]], Pattern[x, Blank[Symbol]]] := Simp[((e + f*x)^(m + 1)*ArcCot[Tanh[a + b*x]])/(f*(m + 1)), x] + Dist[b/(f*(m + 1)), Int[(e + f*x)^(m + 1)*Sech[2*a + 2*b*x], x], x] /; FreeQ[{a, b, e, f}, x] && IGtQ[m, 0]
  5185. Int[Times[ArcTan[Coth[Plus[Optional[Pattern[a, Blank[]]], Times[Optional[Pattern[b, Blank[]]], Pattern[x, Blank[]]]]]], Power[Plus[Optional[Pattern[e, Blank[]]], Times[Optional[Pattern[f, Blank[]]], Pattern[x, Blank[]]]], Optional[Pattern[m, Blank[]]]]], Pattern[x, Blank[Symbol]]] := Simp[((e + f*x)^(m + 1)*ArcTan[Coth[a + b*x]])/(f*(m + 1)), x] + Dist[b/(f*(m + 1)), Int[(e + f*x)^(m + 1)*Sech[2*a + 2*b*x], x], x] /; FreeQ[{a, b, e, f}, x] && IGtQ[m, 0]
  5186. Int[Times[ArcCot[Coth[Plus[Optional[Pattern[a, Blank[]]], Times[Optional[Pattern[b, Blank[]]], Pattern[x, Blank[]]]]]], Power[Plus[Optional[Pattern[e, Blank[]]], Times[Optional[Pattern[f, Blank[]]], Pattern[x, Blank[]]]], Optional[Pattern[m, Blank[]]]]], Pattern[x, Blank[Symbol]]] := Simp[((e + f*x)^(m + 1)*ArcCot[Coth[a + b*x]])/(f*(m + 1)), x] - Dist[b/(f*(m + 1)), Int[(e + f*x)^(m + 1)*Sech[2*a + 2*b*x], x], x] /; FreeQ[{a, b, e, f}, x] && IGtQ[m, 0]
  5187. Int[ArcTan[Plus[Optional[Pattern[c, Blank[]]], Times[Optional[Pattern[d, Blank[]]], Tanh[Plus[Optional[Pattern[a, Blank[]]], Times[Optional[Pattern[b, Blank[]]], Pattern[x, Blank[]]]]]]]], Pattern[x, Blank[Symbol]]] := Simp[x*ArcTan[c + d*Tanh[a + b*x]], x] - Dist[b, Int[x/(c - d + c*E^(2*a + 2*b*x)), x], x] /; FreeQ[{a, b, c, d}, x] && EqQ[(c - d)^2, -1]
  5188. Int[ArcCot[Plus[Optional[Pattern[c, Blank[]]], Times[Optional[Pattern[d, Blank[]]], Tanh[Plus[Optional[Pattern[a, Blank[]]], Times[Optional[Pattern[b, Blank[]]], Pattern[x, Blank[]]]]]]]], Pattern[x, Blank[Symbol]]] := Simp[x*ArcCot[c + d*Tanh[a + b*x]], x] + Dist[b, Int[x/(c - d + c*E^(2*a + 2*b*x)), x], x] /; FreeQ[{a, b, c, d}, x] && EqQ[(c - d)^2, -1]
  5189. Int[ArcTan[Plus[Optional[Pattern[c, Blank[]]], Times[Coth[Plus[Optional[Pattern[a, Blank[]]], Times[Optional[Pattern[b, Blank[]]], Pattern[x, Blank[]]]]], Optional[Pattern[d, Blank[]]]]]], Pattern[x, Blank[Symbol]]] := Simp[x*ArcTan[c + d*Coth[a + b*x]], x] - Dist[b, Int[x/(c - d - c*E^(2*a + 2*b*x)), x], x] /; FreeQ[{a, b, c, d}, x] && EqQ[(c - d)^2, -1]
  5190. Int[ArcCot[Plus[Optional[Pattern[c, Blank[]]], Times[Coth[Plus[Optional[Pattern[a, Blank[]]], Times[Optional[Pattern[b, Blank[]]], Pattern[x, Blank[]]]]], Optional[Pattern[d, Blank[]]]]]], Pattern[x, Blank[Symbol]]] := Simp[x*ArcCot[c + d*Coth[a + b*x]], x] + Dist[b, Int[x/(c - d - c*E^(2*a + 2*b*x)), x], x] /; FreeQ[{a, b, c, d}, x] && EqQ[(c - d)^2, -1]
  5191. Int[ArcTan[Plus[Optional[Pattern[c, Blank[]]], Times[Optional[Pattern[d, Blank[]]], Tanh[Plus[Optional[Pattern[a, Blank[]]], Times[Optional[Pattern[b, Blank[]]], Pattern[x, Blank[]]]]]]]], Pattern[x, Blank[Symbol]]] := Simp[x*ArcTan[c + d*Tanh[a + b*x]], x] + (Dist[I*b*(I - c - d), Int[(x*E^(2*a + 2*b*x))/(I - c + d + (I - c - d)*E^(2*a + 2*b*x)), x], x] - Dist[I*b*(I + c + d), Int[(x*E^(2*a + 2*b*x))/(I + c - d + (I + c + d)*E^(2*a + 2*b*x)), x], x]) /; FreeQ[{a, b, c, d}, x] && NeQ[(c - d)^2, -1]
  5192. Int[ArcCot[Plus[Optional[Pattern[c, Blank[]]], Times[Optional[Pattern[d, Blank[]]], Tanh[Plus[Optional[Pattern[a, Blank[]]], Times[Optional[Pattern[b, Blank[]]], Pattern[x, Blank[]]]]]]]], Pattern[x, Blank[Symbol]]] := Simp[x*ArcCot[c + d*Tanh[a + b*x]], x] + (-Dist[I*b*(I - c - d), Int[(x*E^(2*a + 2*b*x))/(I - c + d + (I - c - d)*E^(2*a + 2*b*x)), x], x] + Dist[I*b*(I + c + d), Int[(x*E^(2*a + 2*b*x))/(I + c - d + (I + c + d)*E^(2*a + 2*b*x)), x], x]) /; FreeQ[{a, b, c, d}, x] && NeQ[(c - d)^2, -1]
  5193. Int[ArcTan[Plus[Optional[Pattern[c, Blank[]]], Times[Coth[Plus[Optional[Pattern[a, Blank[]]], Times[Optional[Pattern[b, Blank[]]], Pattern[x, Blank[]]]]], Optional[Pattern[d, Blank[]]]]]], Pattern[x, Blank[Symbol]]] := Simp[x*ArcTan[c + d*Coth[a + b*x]], x] + (-Dist[I*b*(I - c - d), Int[(x*E^(2*a + 2*b*x))/(I - c + d - (I - c - d)*E^(2*a + 2*b*x)), x], x] + Dist[I*b*(I + c + d), Int[(x*E^(2*a + 2*b*x))/(I + c - d - (I + c + d)*E^(2*a + 2*b*x)), x], x]) /; FreeQ[{a, b, c, d}, x] && NeQ[(c - d)^2, -1]
  5194. Int[ArcCot[Plus[Optional[Pattern[c, Blank[]]], Times[Coth[Plus[Optional[Pattern[a, Blank[]]], Times[Optional[Pattern[b, Blank[]]], Pattern[x, Blank[]]]]], Optional[Pattern[d, Blank[]]]]]], Pattern[x, Blank[Symbol]]] := Simp[x*ArcCot[c + d*Coth[a + b*x]], x] + (Dist[I*b*(I - c - d), Int[(x*E^(2*a + 2*b*x))/(I - c + d - (I - c - d)*E^(2*a + 2*b*x)), x], x] - Dist[I*b*(I + c + d), Int[(x*E^(2*a + 2*b*x))/(I + c - d - (I + c + d)*E^(2*a + 2*b*x)), x], x]) /; FreeQ[{a, b, c, d}, x] && NeQ[(c - d)^2, -1]
  5195. Int[Times[ArcTan[Plus[Optional[Pattern[c, Blank[]]], Times[Optional[Pattern[d, Blank[]]], Tanh[Plus[Optional[Pattern[a, Blank[]]], Times[Optional[Pattern[b, Blank[]]], Pattern[x, Blank[]]]]]]]], Power[Plus[Optional[Pattern[e, Blank[]]], Times[Optional[Pattern[f, Blank[]]], Pattern[x, Blank[]]]], Optional[Pattern[m, Blank[]]]]], Pattern[x, Blank[Symbol]]] := Simp[((e + f*x)^(m + 1)*ArcTan[c + d*Tanh[a + b*x]])/(f*(m + 1)), x] - Dist[b/(f*(m + 1)), Int[(e + f*x)^(m + 1)/(c - d + c*E^(2*a + 2*b*x)), x], x] /; FreeQ[{a, b, c, d, e, f}, x] && IGtQ[m, 0] && EqQ[(c - d)^2, -1]
  5196. Int[Times[ArcCot[Plus[Optional[Pattern[c, Blank[]]], Times[Optional[Pattern[d, Blank[]]], Tanh[Plus[Optional[Pattern[a, Blank[]]], Times[Optional[Pattern[b, Blank[]]], Pattern[x, Blank[]]]]]]]], Power[Plus[Optional[Pattern[e, Blank[]]], Times[Optional[Pattern[f, Blank[]]], Pattern[x, Blank[]]]], Optional[Pattern[m, Blank[]]]]], Pattern[x, Blank[Symbol]]] := Simp[((e + f*x)^(m + 1)*ArcCot[c + d*Tanh[a + b*x]])/(f*(m + 1)), x] + Dist[b/(f*(m + 1)), Int[(e + f*x)^(m + 1)/(c - d + c*E^(2*a + 2*b*x)), x], x] /; FreeQ[{a, b, c, d, e, f}, x] && IGtQ[m, 0] && EqQ[(c - d)^2, -1]
  5197. Int[Times[ArcTan[Plus[Optional[Pattern[c, Blank[]]], Times[Coth[Plus[Optional[Pattern[a, Blank[]]], Times[Optional[Pattern[b, Blank[]]], Pattern[x, Blank[]]]]], Optional[Pattern[d, Blank[]]]]]], Power[Plus[Optional[Pattern[e, Blank[]]], Times[Optional[Pattern[f, Blank[]]], Pattern[x, Blank[]]]], Optional[Pattern[m, Blank[]]]]], Pattern[x, Blank[Symbol]]] := Simp[((e + f*x)^(m + 1)*ArcTan[c + d*Coth[a + b*x]])/(f*(m + 1)), x] - Dist[b/(f*(m + 1)), Int[(e + f*x)^(m + 1)/(c - d - c*E^(2*a + 2*b*x)), x], x] /; FreeQ[{a, b, c, d, e, f}, x] && IGtQ[m, 0] && EqQ[(c - d)^2, -1]
  5198. Int[Times[ArcCot[Plus[Optional[Pattern[c, Blank[]]], Times[Coth[Plus[Optional[Pattern[a, Blank[]]], Times[Optional[Pattern[b, Blank[]]], Pattern[x, Blank[]]]]], Optional[Pattern[d, Blank[]]]]]], Power[Plus[Optional[Pattern[e, Blank[]]], Times[Optional[Pattern[f, Blank[]]], Pattern[x, Blank[]]]], Optional[Pattern[m, Blank[]]]]], Pattern[x, Blank[Symbol]]] := Simp[((e + f*x)^(m + 1)*ArcCot[c + d*Coth[a + b*x]])/(f*(m + 1)), x] + Dist[b/(f*(m + 1)), Int[(e + f*x)^(m + 1)/(c - d - c*E^(2*a + 2*b*x)), x], x] /; FreeQ[{a, b, c, d, e, f}, x] && IGtQ[m, 0] && EqQ[(c - d)^2, -1]
  5199. Int[Times[ArcTan[Plus[Optional[Pattern[c, Blank[]]], Times[Optional[Pattern[d, Blank[]]], Tanh[Plus[Optional[Pattern[a, Blank[]]], Times[Optional[Pattern[b, Blank[]]], Pattern[x, Blank[]]]]]]]], Power[Plus[Optional[Pattern[e, Blank[]]], Times[Optional[Pattern[f, Blank[]]], Pattern[x, Blank[]]]], Optional[Pattern[m, Blank[]]]]], Pattern[x, Blank[Symbol]]] := Simp[((e + f*x)^(m + 1)*ArcTan[c + d*Tanh[a + b*x]])/(f*(m + 1)), x] + (Dist[(I*b*(I - c - d))/(f*(m + 1)), Int[((e + f*x)^(m + 1)*E^(2*a + 2*b*x))/(I - c + d + (I - c - d)*E^(2*a + 2*b*x)), x], x] - Dist[(I*b*(I + c + d))/(f*(m + 1)), Int[((e + f*x)^(m + 1)*E^(2*a + 2*b*x))/(I + c - d + (I + c + d)*E^(2*a + 2*b*x)), x], x]) /; FreeQ[{a, b, c, d, e, f}, x] && IGtQ[m, 0] && NeQ[(c - d)^2, -1]
  5200. Int[Times[ArcCot[Plus[Optional[Pattern[c, Blank[]]], Times[Optional[Pattern[d, Blank[]]], Tanh[Plus[Optional[Pattern[a, Blank[]]], Times[Optional[Pattern[b, Blank[]]], Pattern[x, Blank[]]]]]]]], Power[Plus[Optional[Pattern[e, Blank[]]], Times[Optional[Pattern[f, Blank[]]], Pattern[x, Blank[]]]], Optional[Pattern[m, Blank[]]]]], Pattern[x, Blank[Symbol]]] := Simp[((e + f*x)^(m + 1)*ArcCot[c + d*Tanh[a + b*x]])/(f*(m + 1)), x] + (-Dist[(I*b*(I - c - d))/(f*(m + 1)), Int[((e + f*x)^(m + 1)*E^(2*a + 2*b*x))/(I - c + d + (I - c - d)*E^(2*a + 2*b*x)), x], x] + Dist[(I*b*(I + c + d))/(f*(m + 1)), Int[((e + f*x)^(m + 1)*E^(2*a + 2*b*x))/(I + c - d + (I + c + d)*E^(2*a + 2*b*x)), x], x]) /; FreeQ[{a, b, c, d, e, f}, x] && IGtQ[m, 0] && NeQ[(c - d)^2, -1]
  5201. Int[Times[ArcTan[Plus[Optional[Pattern[c, Blank[]]], Times[Coth[Plus[Optional[Pattern[a, Blank[]]], Times[Optional[Pattern[b, Blank[]]], Pattern[x, Blank[]]]]], Optional[Pattern[d, Blank[]]]]]], Power[Plus[Optional[Pattern[e, Blank[]]], Times[Optional[Pattern[f, Blank[]]], Pattern[x, Blank[]]]], Optional[Pattern[m, Blank[]]]]], Pattern[x, Blank[Symbol]]] := Simp[((e + f*x)^(m + 1)*ArcTan[c + d*Coth[a + b*x]])/(f*(m + 1)), x] + (-Dist[(I*b*(I - c - d))/(f*(m + 1)), Int[((e + f*x)^(m + 1)*E^(2*a + 2*b*x))/(I - c + d - (I - c - d)*E^(2*a + 2*b*x)), x], x] + Dist[(I*b*(I + c + d))/(f*(m + 1)), Int[((e + f*x)^(m + 1)*E^(2*a + 2*b*x))/(I + c - d - (I + c + d)*E^(2*a + 2*b*x)), x], x]) /; FreeQ[{a, b, c, d, e, f}, x] && IGtQ[m, 0] && NeQ[(c - d)^2, -1]
  5202. Int[Times[ArcCot[Plus[Optional[Pattern[c, Blank[]]], Times[Coth[Plus[Optional[Pattern[a, Blank[]]], Times[Optional[Pattern[b, Blank[]]], Pattern[x, Blank[]]]]], Optional[Pattern[d, Blank[]]]]]], Power[Plus[Optional[Pattern[e, Blank[]]], Times[Optional[Pattern[f, Blank[]]], Pattern[x, Blank[]]]], Optional[Pattern[m, Blank[]]]]], Pattern[x, Blank[Symbol]]] := Simp[((e + f*x)^(m + 1)*ArcCot[c + d*Coth[a + b*x]])/(f*(m + 1)), x] + (Dist[(I*b*(I - c - d))/(f*(m + 1)), Int[((e + f*x)^(m + 1)*E^(2*a + 2*b*x))/(I - c + d - (I - c - d)*E^(2*a + 2*b*x)), x], x] - Dist[(I*b*(I + c + d))/(f*(m + 1)), Int[((e + f*x)^(m + 1)*E^(2*a + 2*b*x))/(I + c - d - (I + c + d)*E^(2*a + 2*b*x)), x], x]) /; FreeQ[{a, b, c, d, e, f}, x] && IGtQ[m, 0] && NeQ[(c - d)^2, -1]
  5203. Int[ArcTan[Pattern[u, Blank[]]], Pattern[x, Blank[Symbol]]] := Simp[x*ArcTan[u], x] - Int[SimplifyIntegrand[(x*D[u, x])/(1 + u^2), x], x] /; InverseFunctionFreeQ[u, x]
  5204. Int[ArcCot[Pattern[u, Blank[]]], Pattern[x, Blank[Symbol]]] := Simp[x*ArcCot[u], x] + Int[SimplifyIntegrand[(x*D[u, x])/(1 + u^2), x], x] /; InverseFunctionFreeQ[u, x]
  5205. Int[Times[Plus[Optional[Pattern[a, Blank[]]], Times[ArcTan[Pattern[u, Blank[]]], Optional[Pattern[b, Blank[]]]]], Power[Plus[Optional[Pattern[c, Blank[]]], Times[Optional[Pattern[d, Blank[]]], Pattern[x, Blank[]]]], Optional[Pattern[m, Blank[]]]]], Pattern[x, Blank[Symbol]]] := Simp[((c + d*x)^(m + 1)*(a + b*ArcTan[u]))/(d*(m + 1)), x] - Dist[b/(d*(m + 1)), Int[SimplifyIntegrand[((c + d*x)^(m + 1)*D[u, x])/(1 + u^2), x], x], x] /; FreeQ[{a, b, c, d, m}, x] && NeQ[m, -1] && InverseFunctionFreeQ[u, x] && !FunctionOfQ[(c + d*x)^(m + 1), u, x] && FalseQ[PowerVariableExpn[u, m + 1, x]]
  5206. Int[Times[Plus[Optional[Pattern[a, Blank[]]], Times[ArcCot[Pattern[u, Blank[]]], Optional[Pattern[b, Blank[]]]]], Power[Plus[Optional[Pattern[c, Blank[]]], Times[Optional[Pattern[d, Blank[]]], Pattern[x, Blank[]]]], Optional[Pattern[m, Blank[]]]]], Pattern[x, Blank[Symbol]]] := Simp[((c + d*x)^(m + 1)*(a + b*ArcCot[u]))/(d*(m + 1)), x] + Dist[b/(d*(m + 1)), Int[SimplifyIntegrand[((c + d*x)^(m + 1)*D[u, x])/(1 + u^2), x], x], x] /; FreeQ[{a, b, c, d, m}, x] && NeQ[m, -1] && InverseFunctionFreeQ[u, x] && !FunctionOfQ[(c + d*x)^(m + 1), u, x] && FalseQ[PowerVariableExpn[u, m + 1, x]]
  5207. Int[Times[Plus[Optional[Pattern[a, Blank[]]], Times[ArcTan[Pattern[u, Blank[]]], Optional[Pattern[b, Blank[]]]]], Pattern[v, Blank[]]], Pattern[x, Blank[Symbol]]] := With[{w = IntHide[v, x]}, Dist[a + b*ArcTan[u], w, x] - Dist[b, Int[SimplifyIntegrand[(w*D[u, x])/(1 + u^2), x], x], x] /; InverseFunctionFreeQ[w, x]] /; FreeQ[{a, b}, x] && InverseFunctionFreeQ[u, x] && !MatchQ[v, ((c_.) + (d_.)*x)^(m_.) /; FreeQ[{c, d, m}, x]] && FalseQ[FunctionOfLinear[v*(a + b*ArcTan[u]), x]]
  5208. Int[Times[Plus[Optional[Pattern[a, Blank[]]], Times[ArcCot[Pattern[u, Blank[]]], Optional[Pattern[b, Blank[]]]]], Pattern[v, Blank[]]], Pattern[x, Blank[Symbol]]] := With[{w = IntHide[v, x]}, Dist[a + b*ArcCot[u], w, x] + Dist[b, Int[SimplifyIntegrand[(w*D[u, x])/(1 + u^2), x], x], x] /; InverseFunctionFreeQ[w, x]] /; FreeQ[{a, b}, x] && InverseFunctionFreeQ[u, x] && !MatchQ[v, ((c_.) + (d_.)*x)^(m_.) /; FreeQ[{c, d, m}, x]] && FalseQ[FunctionOfLinear[v*(a + b*ArcCot[u]), x]]
  5209. Int[Times[ArcTan[Pattern[v, Blank[]]], Log[Pattern[w, Blank[]]], Power[Plus[Optional[Pattern[a, Blank[]]], Times[Optional[Pattern[b, Blank[]]], Pattern[x, Blank[]]]], -1]], Pattern[x, Blank[Symbol]]] := Dist[I/2, Int[(Log[1 - I*v]*Log[w])/(a + b*x), x], x] - Dist[I/2, Int[(Log[1 + I*v]*Log[w])/(a + b*x), x], x] /; FreeQ[{a, b}, x] && LinearQ[v, x] && LinearQ[w, x] && EqQ[Simplify[D[v/(a + b*x), x]], 0] && EqQ[Simplify[D[w/(a + b*x), x]], 0]
  5210. Int[Times[ArcTan[Pattern[v, Blank[]]], Log[Pattern[w, Blank[]]]], Pattern[x, Blank[Symbol]]] := Simp[x*ArcTan[v]*Log[w], x] + (-Int[SimplifyIntegrand[(x*Log[w]*D[v, x])/(1 + v^2), x], x] - Int[SimplifyIntegrand[(x*ArcTan[v]*D[w, x])/w, x], x]) /; InverseFunctionFreeQ[v, x] && InverseFunctionFreeQ[w, x]
  5211. Int[Times[ArcCot[Pattern[v, Blank[]]], Log[Pattern[w, Blank[]]]], Pattern[x, Blank[Symbol]]] := Simp[x*ArcCot[v]*Log[w], x] + (Int[SimplifyIntegrand[(x*Log[w]*D[v, x])/(1 + v^2), x], x] - Int[SimplifyIntegrand[(x*ArcCot[v]*D[w, x])/w, x], x]) /; InverseFunctionFreeQ[v, x] && InverseFunctionFreeQ[w, x]
  5212. Int[Times[ArcTan[Pattern[v, Blank[]]], Log[Pattern[w, Blank[]]], Pattern[u, Blank[]]], Pattern[x, Blank[Symbol]]] := With[{z = IntHide[u, x]}, Dist[ArcTan[v]*Log[w], z, x] + (-Int[SimplifyIntegrand[(z*Log[w]*D[v, x])/(1 + v^2), x], x] - Int[SimplifyIntegrand[(z*ArcTan[v]*D[w, x])/w, x], x]) /; InverseFunctionFreeQ[z, x]] /; InverseFunctionFreeQ[v, x] && InverseFunctionFreeQ[w, x]
  5213. Int[Times[ArcCot[Pattern[v, Blank[]]], Log[Pattern[w, Blank[]]], Pattern[u, Blank[]]], Pattern[x, Blank[Symbol]]] := With[{z = IntHide[u, x]}, Dist[ArcCot[v]*Log[w], z, x] + (Int[SimplifyIntegrand[(z*Log[w]*D[v, x])/(1 + v^2), x], x] - Int[SimplifyIntegrand[(z*ArcCot[v]*D[w, x])/w, x], x]) /; InverseFunctionFreeQ[z, x]] /; InverseFunctionFreeQ[v, x] && InverseFunctionFreeQ[w, x]
  5214. Int[ArcSec[Times[Optional[Pattern[c, Blank[]]], Pattern[x, Blank[]]]], Pattern[x, Blank[Symbol]]] := Simp[x*ArcSec[c*x], x] - Dist[1/c, Int[1/(x*Sqrt[1 - 1/(c^2*x^2)]), x], x] /; FreeQ[c, x]
  5215. Int[ArcCsc[Times[Optional[Pattern[c, Blank[]]], Pattern[x, Blank[]]]], Pattern[x, Blank[Symbol]]] := Simp[x*ArcCsc[c*x], x] + Dist[1/c, Int[1/(x*Sqrt[1 - 1/(c^2*x^2)]), x], x] /; FreeQ[c, x]
  5216. Int[Power[Plus[Optional[Pattern[a, Blank[]]], Times[ArcSec[Times[Optional[Pattern[c, Blank[]]], Pattern[x, Blank[]]]], Optional[Pattern[b, Blank[]]]]], Pattern[n, Blank[]]], Pattern[x, Blank[Symbol]]] := Dist[1/c, Subst[Int[(a + b*x)^n*Sec[x]*Tan[x], x], x, ArcSec[c*x]], x] /; FreeQ[{a, b, c, n}, x] && IGtQ[n, 0]
  5217. Int[Power[Plus[Optional[Pattern[a, Blank[]]], Times[ArcCsc[Times[Optional[Pattern[c, Blank[]]], Pattern[x, Blank[]]]], Optional[Pattern[b, Blank[]]]]], Pattern[n, Blank[]]], Pattern[x, Blank[Symbol]]] := -Dist[c^(-1), Subst[Int[(a + b*x)^n*Csc[x]*Cot[x], x], x, ArcCsc[c*x]], x] /; FreeQ[{a, b, c, n}, x] && IGtQ[n, 0]
  5218. Int[Times[Plus[Optional[Pattern[a, Blank[]]], Times[ArcSec[Times[Optional[Pattern[c, Blank[]]], Pattern[x, Blank[]]]], Optional[Pattern[b, Blank[]]]]], Power[Pattern[x, Blank[]], -1]], Pattern[x, Blank[Symbol]]] := -Subst[Int[(a + b*ArcCos[x/c])/x, x], x, 1/x] /; FreeQ[{a, b, c}, x]
  5219. Int[Times[Plus[Optional[Pattern[a, Blank[]]], Times[ArcCsc[Times[Optional[Pattern[c, Blank[]]], Pattern[x, Blank[]]]], Optional[Pattern[b, Blank[]]]]], Power[Pattern[x, Blank[]], -1]], Pattern[x, Blank[Symbol]]] := -Subst[Int[(a + b*ArcSin[x/c])/x, x], x, 1/x] /; FreeQ[{a, b, c}, x]
  5220. Int[Times[Plus[Optional[Pattern[a, Blank[]]], Times[ArcSec[Times[Optional[Pattern[c, Blank[]]], Pattern[x, Blank[]]]], Optional[Pattern[b, Blank[]]]]], Power[Times[Optional[Pattern[d, Blank[]]], Pattern[x, Blank[]]], Optional[Pattern[m, Blank[]]]]], Pattern[x, Blank[Symbol]]] := Simp[((d*x)^(m + 1)*(a + b*ArcSec[c*x]))/(d*(m + 1)), x] - Dist[(b*d)/(c*(m + 1)), Int[(d*x)^(m - 1)/Sqrt[1 - 1/(c^2*x^2)], x], x] /; FreeQ[{a, b, c, d, m}, x] && NeQ[m, -1]
  5221. Int[Times[Plus[Optional[Pattern[a, Blank[]]], Times[ArcCsc[Times[Optional[Pattern[c, Blank[]]], Pattern[x, Blank[]]]], Optional[Pattern[b, Blank[]]]]], Power[Times[Optional[Pattern[d, Blank[]]], Pattern[x, Blank[]]], Optional[Pattern[m, Blank[]]]]], Pattern[x, Blank[Symbol]]] := Simp[((d*x)^(m + 1)*(a + b*ArcCsc[c*x]))/(d*(m + 1)), x] + Dist[(b*d)/(c*(m + 1)), Int[(d*x)^(m - 1)/Sqrt[1 - 1/(c^2*x^2)], x], x] /; FreeQ[{a, b, c, d, m}, x] && NeQ[m, -1]
  5222. Int[Times[Power[Plus[Optional[Pattern[a, Blank[]]], Times[ArcSec[Times[Optional[Pattern[c, Blank[]]], Pattern[x, Blank[]]]], Optional[Pattern[b, Blank[]]]]], Pattern[n, Blank[]]], Power[Pattern[x, Blank[]], Optional[Pattern[m, Blank[]]]]], Pattern[x, Blank[Symbol]]] := Dist[1/c^(m + 1), Subst[Int[(a + b*x)^n*Sec[x]^(m + 1)*Tan[x], x], x, ArcSec[c*x]], x] /; FreeQ[{a, b, c}, x] && IntegerQ[n] && IntegerQ[m] && (GtQ[n, 0] || LtQ[m, -1])
  5223. Int[Times[Power[Plus[Optional[Pattern[a, Blank[]]], Times[ArcCsc[Times[Optional[Pattern[c, Blank[]]], Pattern[x, Blank[]]]], Optional[Pattern[b, Blank[]]]]], Pattern[n, Blank[]]], Power[Pattern[x, Blank[]], Optional[Pattern[m, Blank[]]]]], Pattern[x, Blank[Symbol]]] := -Dist[(c^(m + 1))^(-1), Subst[Int[(a + b*x)^n*Csc[x]^(m + 1)*Cot[x], x], x, ArcCsc[c*x]], x] /; FreeQ[{a, b, c}, x] && IntegerQ[n] && IntegerQ[m] && (GtQ[n, 0] || LtQ[m, -1])
  5224. Int[Times[Plus[Optional[Pattern[a, Blank[]]], Times[ArcSec[Times[Optional[Pattern[c, Blank[]]], Pattern[x, Blank[]]]], Optional[Pattern[b, Blank[]]]]], Power[Plus[Optional[Pattern[d, Blank[]]], Times[Optional[Pattern[e, Blank[]]], Pattern[x, Blank[]]]], -1]], Pattern[x, Blank[Symbol]]] := Simp[((a + b*ArcSec[c*x])*Log[1 + ((e - Sqrt[-(c^2*d^2) + e^2])*E^(I*ArcSec[c*x]))/(c*d)])/e, x] + (-Dist[b/(c*e), Int[Log[1 + ((e - Sqrt[-(c^2*d^2) + e^2])*E^(I*ArcSec[c*x]))/(c*d)]/(x^2*Sqrt[1 - 1/(c^2*x^2)]), x], x] - Dist[b/(c*e), Int[Log[1 + ((e + Sqrt[-(c^2*d^2) + e^2])*E^(I*ArcSec[c*x]))/(c*d)]/(x^2*Sqrt[1 - 1/(c^2*x^2)]), x], x] + Dist[b/(c*e), Int[Log[1 + E^(2*I*ArcSec[c*x])]/(x^2*Sqrt[1 - 1/(c^2*x^2)]), x], x] + Simp[((a + b*ArcSec[c*x])*Log[1 + ((e + Sqrt[-(c^2*d^2) + e^2])*E^(I*ArcSec[c*x]))/(c*d)])/e, x] - Simp[((a + b*ArcSec[c*x])*Log[1 + E^(2*I*ArcSec[c*x])])/e, x]) /; FreeQ[{a, b, c, d, e}, x]
  5225. Int[Times[Plus[Optional[Pattern[a, Blank[]]], Times[ArcCsc[Times[Optional[Pattern[c, Blank[]]], Pattern[x, Blank[]]]], Optional[Pattern[b, Blank[]]]]], Power[Plus[Optional[Pattern[d, Blank[]]], Times[Optional[Pattern[e, Blank[]]], Pattern[x, Blank[]]]], -1]], Pattern[x, Blank[Symbol]]] := Simp[((a + b*ArcCsc[c*x])*Log[1 - (I*(e - Sqrt[-(c^2*d^2) + e^2])*E^(I*ArcCsc[c*x]))/(c*d)])/e, x] + (Dist[b/(c*e), Int[Log[1 - (I*(e - Sqrt[-(c^2*d^2) + e^2])*E^(I*ArcCsc[c*x]))/(c*d)]/(x^2*Sqrt[1 - 1/(c^2*x^2)]), x], x] + Dist[b/(c*e), Int[Log[1 - (I*(e + Sqrt[-(c^2*d^2) + e^2])*E^(I*ArcCsc[c*x]))/(c*d)]/(x^2*Sqrt[1 - 1/(c^2*x^2)]), x], x] - Dist[b/(c*e), Int[Log[1 - E^(2*I*ArcCsc[c*x])]/(x^2*Sqrt[1 - 1/(c^2*x^2)]), x], x] + Simp[((a + b*ArcCsc[c*x])*Log[1 - (I*(e + Sqrt[-(c^2*d^2) + e^2])*E^(I*ArcCsc[c*x]))/(c*d)])/e, x] - Simp[((a + b*ArcCsc[c*x])*Log[1 - E^(2*I*ArcCsc[c*x])])/e, x]) /; FreeQ[{a, b, c, d, e}, x]
  5226. Int[Times[Plus[Optional[Pattern[a, Blank[]]], Times[ArcSec[Times[Optional[Pattern[c, Blank[]]], Pattern[x, Blank[]]]], Optional[Pattern[b, Blank[]]]]], Power[Plus[Optional[Pattern[d, Blank[]]], Times[Optional[Pattern[e, Blank[]]], Pattern[x, Blank[]]]], Optional[Pattern[m, Blank[]]]]], Pattern[x, Blank[Symbol]]] := Simp[((d + e*x)^(m + 1)*(a + b*ArcSec[c*x]))/(e*(m + 1)), x] - Dist[b/(c*e*(m + 1)), Int[(d + e*x)^(m + 1)/(x^2*Sqrt[1 - 1/(c^2*x^2)]), x], x] /; FreeQ[{a, b, c, d, e, m}, x] && NeQ[m, -1]
  5227. Int[Times[Plus[Optional[Pattern[a, Blank[]]], Times[ArcCsc[Times[Optional[Pattern[c, Blank[]]], Pattern[x, Blank[]]]], Optional[Pattern[b, Blank[]]]]], Power[Plus[Optional[Pattern[d, Blank[]]], Times[Optional[Pattern[e, Blank[]]], Pattern[x, Blank[]]]], Optional[Pattern[m, Blank[]]]]], Pattern[x, Blank[Symbol]]] := Simp[((d + e*x)^(m + 1)*(a + b*ArcCsc[c*x]))/(e*(m + 1)), x] + Dist[b/(c*e*(m + 1)), Int[(d + e*x)^(m + 1)/(x^2*Sqrt[1 - 1/(c^2*x^2)]), x], x] /; FreeQ[{a, b, c, d, e, m}, x] && NeQ[m, -1]
  5228. Int[Times[Plus[Optional[Pattern[a, Blank[]]], Times[ArcSec[Times[Optional[Pattern[c, Blank[]]], Pattern[x, Blank[]]]], Optional[Pattern[b, Blank[]]]]], Power[Plus[Optional[Pattern[d, Blank[]]], Times[Optional[Pattern[e, Blank[]]], Power[Pattern[x, Blank[]], 2]]], Optional[Pattern[p, Blank[]]]]], Pattern[x, Blank[Symbol]]] := With[{u = IntHide[(d + e*x^2)^p, x]}, Dist[a + b*ArcSec[c*x], u, x] - Dist[(b*c*x)/Sqrt[c^2*x^2], Int[SimplifyIntegrand[u/(x*Sqrt[c^2*x^2 - 1]), x], x], x]] /; FreeQ[{a, b, c, d, e}, x] && (IGtQ[p, 0] || ILtQ[p + 1/2, 0])
  5229. Int[Times[Plus[Optional[Pattern[a, Blank[]]], Times[ArcCsc[Times[Optional[Pattern[c, Blank[]]], Pattern[x, Blank[]]]], Optional[Pattern[b, Blank[]]]]], Power[Plus[Optional[Pattern[d, Blank[]]], Times[Optional[Pattern[e, Blank[]]], Power[Pattern[x, Blank[]], 2]]], Optional[Pattern[p, Blank[]]]]], Pattern[x, Blank[Symbol]]] := With[{u = IntHide[(d + e*x^2)^p, x]}, Dist[a + b*ArcCsc[c*x], u, x] + Dist[(b*c*x)/Sqrt[c^2*x^2], Int[SimplifyIntegrand[u/(x*Sqrt[c^2*x^2 - 1]), x], x], x]] /; FreeQ[{a, b, c, d, e}, x] && (IGtQ[p, 0] || ILtQ[p + 1/2, 0])
  5230. Int[Times[Power[Plus[Optional[Pattern[a, Blank[]]], Times[ArcSec[Times[Optional[Pattern[c, Blank[]]], Pattern[x, Blank[]]]], Optional[Pattern[b, Blank[]]]]], Optional[Pattern[n, Blank[]]]], Power[Plus[Optional[Pattern[d, Blank[]]], Times[Optional[Pattern[e, Blank[]]], Power[Pattern[x, Blank[]], 2]]], Optional[Pattern[p, Blank[]]]]], Pattern[x, Blank[Symbol]]] := -Subst[Int[((e + d*x^2)^p*(a + b*ArcCos[x/c])^n)/x^(2*(p + 1)), x], x, 1/x] /; FreeQ[{a, b, c, d, e, n}, x] && IGtQ[n, 0] && IntegerQ[p]
  5231. Int[Times[Power[Plus[Optional[Pattern[a, Blank[]]], Times[ArcCsc[Times[Optional[Pattern[c, Blank[]]], Pattern[x, Blank[]]]], Optional[Pattern[b, Blank[]]]]], Optional[Pattern[n, Blank[]]]], Power[Plus[Optional[Pattern[d, Blank[]]], Times[Optional[Pattern[e, Blank[]]], Power[Pattern[x, Blank[]], 2]]], Optional[Pattern[p, Blank[]]]]], Pattern[x, Blank[Symbol]]] := -Subst[Int[((e + d*x^2)^p*(a + b*ArcSin[x/c])^n)/x^(2*(p + 1)), x], x, 1/x] /; FreeQ[{a, b, c, d, e, n}, x] && IGtQ[n, 0] && IntegerQ[p]
  5232. Int[Times[Power[Plus[Optional[Pattern[a, Blank[]]], Times[ArcSec[Times[Optional[Pattern[c, Blank[]]], Pattern[x, Blank[]]]], Optional[Pattern[b, Blank[]]]]], Optional[Pattern[n, Blank[]]]], Power[Plus[Optional[Pattern[d, Blank[]]], Times[Optional[Pattern[e, Blank[]]], Power[Pattern[x, Blank[]], 2]]], Pattern[p, Blank[]]]], Pattern[x, Blank[Symbol]]] := -Dist[Sqrt[x^2]/x, Subst[Int[((e + d*x^2)^p*(a + b*ArcCos[x/c])^n)/x^(2*(p + 1)), x], x, 1/x], x] /; FreeQ[{a, b, c, d, e, n}, x] && IGtQ[n, 0] && EqQ[c^2*d + e, 0] && IntegerQ[p + 1/2] && GtQ[e, 0] && LtQ[d, 0]
  5233. Int[Times[Power[Plus[Optional[Pattern[a, Blank[]]], Times[ArcCsc[Times[Optional[Pattern[c, Blank[]]], Pattern[x, Blank[]]]], Optional[Pattern[b, Blank[]]]]], Optional[Pattern[n, Blank[]]]], Power[Plus[Optional[Pattern[d, Blank[]]], Times[Optional[Pattern[e, Blank[]]], Power[Pattern[x, Blank[]], 2]]], Pattern[p, Blank[]]]], Pattern[x, Blank[Symbol]]] := -Dist[Sqrt[x^2]/x, Subst[Int[((e + d*x^2)^p*(a + b*ArcSin[x/c])^n)/x^(2*(p + 1)), x], x, 1/x], x] /; FreeQ[{a, b, c, d, e, n}, x] && IGtQ[n, 0] && EqQ[c^2*d + e, 0] && IntegerQ[p + 1/2] && GtQ[e, 0] && LtQ[d, 0]
  5234. Int[Times[Power[Plus[Optional[Pattern[a, Blank[]]], Times[ArcSec[Times[Optional[Pattern[c, Blank[]]], Pattern[x, Blank[]]]], Optional[Pattern[b, Blank[]]]]], Optional[Pattern[n, Blank[]]]], Power[Plus[Optional[Pattern[d, Blank[]]], Times[Optional[Pattern[e, Blank[]]], Power[Pattern[x, Blank[]], 2]]], Pattern[p, Blank[]]]], Pattern[x, Blank[Symbol]]] := -Dist[Sqrt[d + e*x^2]/(x*Sqrt[e + d/x^2]), Subst[Int[((e + d*x^2)^p*(a + b*ArcCos[x/c])^n)/x^(2*(p + 1)), x], x, 1/x], x] /; FreeQ[{a, b, c, d, e, n}, x] && IGtQ[n, 0] && EqQ[c^2*d + e, 0] && IntegerQ[p + 1/2] && !(GtQ[e, 0] && LtQ[d, 0])
  5235. Int[Times[Power[Plus[Optional[Pattern[a, Blank[]]], Times[ArcCsc[Times[Optional[Pattern[c, Blank[]]], Pattern[x, Blank[]]]], Optional[Pattern[b, Blank[]]]]], Optional[Pattern[n, Blank[]]]], Power[Plus[Optional[Pattern[d, Blank[]]], Times[Optional[Pattern[e, Blank[]]], Power[Pattern[x, Blank[]], 2]]], Pattern[p, Blank[]]]], Pattern[x, Blank[Symbol]]] := -Dist[Sqrt[d + e*x^2]/(x*Sqrt[e + d/x^2]), Subst[Int[((e + d*x^2)^p*(a + b*ArcSin[x/c])^n)/x^(2*(p + 1)), x], x, 1/x], x] /; FreeQ[{a, b, c, d, e, n}, x] && IGtQ[n, 0] && EqQ[c^2*d + e, 0] && IntegerQ[p + 1/2] && !(GtQ[e, 0] && LtQ[d, 0])
  5236. Int[Times[Plus[Optional[Pattern[a, Blank[]]], Times[ArcSec[Times[Optional[Pattern[c, Blank[]]], Pattern[x, Blank[]]]], Optional[Pattern[b, Blank[]]]]], Pattern[x, Blank[]], Power[Plus[Optional[Pattern[d, Blank[]]], Times[Optional[Pattern[e, Blank[]]], Power[Pattern[x, Blank[]], 2]]], Optional[Pattern[p, Blank[]]]]], Pattern[x, Blank[Symbol]]] := Simp[((d + e*x^2)^(p + 1)*(a + b*ArcSec[c*x]))/(2*e*(p + 1)), x] - Dist[(b*c*x)/(2*e*(p + 1)*Sqrt[c^2*x^2]), Int[(d + e*x^2)^(p + 1)/(x*Sqrt[c^2*x^2 - 1]), x], x] /; FreeQ[{a, b, c, d, e, p}, x] && NeQ[p, -1]
  5237. Int[Times[Plus[Optional[Pattern[a, Blank[]]], Times[ArcCsc[Times[Optional[Pattern[c, Blank[]]], Pattern[x, Blank[]]]], Optional[Pattern[b, Blank[]]]]], Pattern[x, Blank[]], Power[Plus[Optional[Pattern[d, Blank[]]], Times[Optional[Pattern[e, Blank[]]], Power[Pattern[x, Blank[]], 2]]], Optional[Pattern[p, Blank[]]]]], Pattern[x, Blank[Symbol]]] := Simp[((d + e*x^2)^(p + 1)*(a + b*ArcCsc[c*x]))/(2*e*(p + 1)), x] + Dist[(b*c*x)/(2*e*(p + 1)*Sqrt[c^2*x^2]), Int[(d + e*x^2)^(p + 1)/(x*Sqrt[c^2*x^2 - 1]), x], x] /; FreeQ[{a, b, c, d, e, p}, x] && NeQ[p, -1]
  5238. Int[Times[Plus[Optional[Pattern[a, Blank[]]], Times[ArcSec[Times[Optional[Pattern[c, Blank[]]], Pattern[x, Blank[]]]], Optional[Pattern[b, Blank[]]]]], Power[Times[Optional[Pattern[f, Blank[]]], Pattern[x, Blank[]]], Optional[Pattern[m, Blank[]]]], Power[Plus[Optional[Pattern[d, Blank[]]], Times[Optional[Pattern[e, Blank[]]], Power[Pattern[x, Blank[]], 2]]], Optional[Pattern[p, Blank[]]]]], Pattern[x, Blank[Symbol]]] := With[{u = IntHide[(f*x)^m*(d + e*x^2)^p, x]}, Dist[a + b*ArcSec[c*x], u, x] - Dist[(b*c*x)/Sqrt[c^2*x^2], Int[SimplifyIntegrand[u/(x*Sqrt[c^2*x^2 - 1]), x], x], x]] /; FreeQ[{a, b, c, d, e, f, m, p}, x] && ((IGtQ[p, 0] && !(ILtQ[(m - 1)/2, 0] && GtQ[m + 2*p + 3, 0])) || (IGtQ[(m + 1)/2, 0] && !(ILtQ[p, 0] && GtQ[m + 2*p + 3, 0])) || (ILtQ[(m + 2*p + 1)/2, 0] && !ILtQ[(m - 1)/2, 0]))
  5239. Int[Times[Plus[Optional[Pattern[a, Blank[]]], Times[ArcCsc[Times[Optional[Pattern[c, Blank[]]], Pattern[x, Blank[]]]], Optional[Pattern[b, Blank[]]]]], Power[Times[Optional[Pattern[f, Blank[]]], Pattern[x, Blank[]]], Optional[Pattern[m, Blank[]]]], Power[Plus[Optional[Pattern[d, Blank[]]], Times[Optional[Pattern[e, Blank[]]], Power[Pattern[x, Blank[]], 2]]], Optional[Pattern[p, Blank[]]]]], Pattern[x, Blank[Symbol]]] := With[{u = IntHide[(f*x)^m*(d + e*x^2)^p, x]}, Dist[a + b*ArcCsc[c*x], u, x] + Dist[(b*c*x)/Sqrt[c^2*x^2], Int[SimplifyIntegrand[u/(x*Sqrt[c^2*x^2 - 1]), x], x], x]] /; FreeQ[{a, b, c, d, e, f, m, p}, x] && ((IGtQ[p, 0] && !(ILtQ[(m - 1)/2, 0] && GtQ[m + 2*p + 3, 0])) || (IGtQ[(m + 1)/2, 0] && !(ILtQ[p, 0] && GtQ[m + 2*p + 3, 0])) || (ILtQ[(m + 2*p + 1)/2, 0] && !ILtQ[(m - 1)/2, 0]))
  5240. Int[Times[Power[Plus[Optional[Pattern[a, Blank[]]], Times[ArcSec[Times[Optional[Pattern[c, Blank[]]], Pattern[x, Blank[]]]], Optional[Pattern[b, Blank[]]]]], Optional[Pattern[n, Blank[]]]], Power[Pattern[x, Blank[]], Optional[Pattern[m, Blank[]]]], Power[Plus[Optional[Pattern[d, Blank[]]], Times[Optional[Pattern[e, Blank[]]], Power[Pattern[x, Blank[]], 2]]], Optional[Pattern[p, Blank[]]]]], Pattern[x, Blank[Symbol]]] := -Subst[Int[((e + d*x^2)^p*(a + b*ArcCos[x/c])^n)/x^(m + 2*(p + 1)), x], x, 1/x] /; FreeQ[{a, b, c, d, e, n}, x] && IGtQ[n, 0] && IntegerQ[m] && IntegerQ[p]
  5241. Int[Times[Power[Plus[Optional[Pattern[a, Blank[]]], Times[ArcCsc[Times[Optional[Pattern[c, Blank[]]], Pattern[x, Blank[]]]], Optional[Pattern[b, Blank[]]]]], Optional[Pattern[n, Blank[]]]], Power[Pattern[x, Blank[]], Optional[Pattern[m, Blank[]]]], Power[Plus[Optional[Pattern[d, Blank[]]], Times[Optional[Pattern[e, Blank[]]], Power[Pattern[x, Blank[]], 2]]], Optional[Pattern[p, Blank[]]]]], Pattern[x, Blank[Symbol]]] := -Subst[Int[((e + d*x^2)^p*(a + b*ArcSin[x/c])^n)/x^(m + 2*(p + 1)), x], x, 1/x] /; FreeQ[{a, b, c, d, e, n}, x] && IGtQ[n, 0] && IntegerQ[m] && IntegerQ[p]
  5242. Int[Times[Power[Plus[Optional[Pattern[a, Blank[]]], Times[ArcSec[Times[Optional[Pattern[c, Blank[]]], Pattern[x, Blank[]]]], Optional[Pattern[b, Blank[]]]]], Optional[Pattern[n, Blank[]]]], Power[Pattern[x, Blank[]], Optional[Pattern[m, Blank[]]]], Power[Plus[Optional[Pattern[d, Blank[]]], Times[Optional[Pattern[e, Blank[]]], Power[Pattern[x, Blank[]], 2]]], Pattern[p, Blank[]]]], Pattern[x, Blank[Symbol]]] := -Dist[Sqrt[x^2]/x, Subst[Int[((e + d*x^2)^p*(a + b*ArcCos[x/c])^n)/x^(m + 2*(p + 1)), x], x, 1/x], x] /; FreeQ[{a, b, c, d, e, n}, x] && IGtQ[n, 0] && EqQ[c^2*d + e, 0] && IntegerQ[m] && IntegerQ[p + 1/2] && GtQ[e, 0] && LtQ[d, 0]
  5243. Int[Times[Power[Plus[Optional[Pattern[a, Blank[]]], Times[ArcCsc[Times[Optional[Pattern[c, Blank[]]], Pattern[x, Blank[]]]], Optional[Pattern[b, Blank[]]]]], Optional[Pattern[n, Blank[]]]], Power[Pattern[x, Blank[]], Optional[Pattern[m, Blank[]]]], Power[Plus[Optional[Pattern[d, Blank[]]], Times[Optional[Pattern[e, Blank[]]], Power[Pattern[x, Blank[]], 2]]], Pattern[p, Blank[]]]], Pattern[x, Blank[Symbol]]] := -Dist[Sqrt[x^2]/x, Subst[Int[((e + d*x^2)^p*(a + b*ArcSin[x/c])^n)/x^(m + 2*(p + 1)), x], x, 1/x], x] /; FreeQ[{a, b, c, d, e, n}, x] && IGtQ[n, 0] && EqQ[c^2*d + e, 0] && IntegerQ[m] && IntegerQ[p + 1/2] && GtQ[e, 0] && LtQ[d, 0]
  5244. Int[Times[Power[Plus[Optional[Pattern[a, Blank[]]], Times[ArcSec[Times[Optional[Pattern[c, Blank[]]], Pattern[x, Blank[]]]], Optional[Pattern[b, Blank[]]]]], Optional[Pattern[n, Blank[]]]], Power[Pattern[x, Blank[]], Optional[Pattern[m, Blank[]]]], Power[Plus[Optional[Pattern[d, Blank[]]], Times[Optional[Pattern[e, Blank[]]], Power[Pattern[x, Blank[]], 2]]], Pattern[p, Blank[]]]], Pattern[x, Blank[Symbol]]] := -Dist[Sqrt[d + e*x^2]/(x*Sqrt[e + d/x^2]), Subst[Int[((e + d*x^2)^p*(a + b*ArcCos[x/c])^n)/x^(m + 2*(p + 1)), x], x, 1/x], x] /; FreeQ[{a, b, c, d, e, n}, x] && IGtQ[n, 0] && EqQ[c^2*d + e, 0] && IntegerQ[m] && IntegerQ[p + 1/2] && !(GtQ[e, 0] && LtQ[d, 0])
  5245. Int[Times[Power[Plus[Optional[Pattern[a, Blank[]]], Times[ArcCsc[Times[Optional[Pattern[c, Blank[]]], Pattern[x, Blank[]]]], Optional[Pattern[b, Blank[]]]]], Optional[Pattern[n, Blank[]]]], Power[Pattern[x, Blank[]], Optional[Pattern[m, Blank[]]]], Power[Plus[Optional[Pattern[d, Blank[]]], Times[Optional[Pattern[e, Blank[]]], Power[Pattern[x, Blank[]], 2]]], Pattern[p, Blank[]]]], Pattern[x, Blank[Symbol]]] := -Dist[Sqrt[d + e*x^2]/(x*Sqrt[e + d/x^2]), Subst[Int[((e + d*x^2)^p*(a + b*ArcSin[x/c])^n)/x^(m + 2*(p + 1)), x], x, 1/x], x] /; FreeQ[{a, b, c, d, e, n}, x] && IGtQ[n, 0] && EqQ[c^2*d + e, 0] && IntegerQ[m] && IntegerQ[p + 1/2] && !(GtQ[e, 0] && LtQ[d, 0])
  5246. Int[Times[Plus[Optional[Pattern[a, Blank[]]], Times[ArcSec[Times[Optional[Pattern[c, Blank[]]], Pattern[x, Blank[]]]], Optional[Pattern[b, Blank[]]]]], Pattern[u, Blank[]]], Pattern[x, Blank[Symbol]]] := With[{v = IntHide[u, x]}, Dist[a + b*ArcSec[c*x], v, x] - Dist[b/c, Int[SimplifyIntegrand[v/(x^2*Sqrt[1 - 1/(c^2*x^2)]), x], x], x] /; InverseFunctionFreeQ[v, x]] /; FreeQ[{a, b, c}, x]
  5247. Int[Times[Plus[Optional[Pattern[a, Blank[]]], Times[ArcCsc[Times[Optional[Pattern[c, Blank[]]], Pattern[x, Blank[]]]], Optional[Pattern[b, Blank[]]]]], Pattern[u, Blank[]]], Pattern[x, Blank[Symbol]]] := With[{v = IntHide[u, x]}, Dist[a + b*ArcCsc[c*x], v, x] + Dist[b/c, Int[SimplifyIntegrand[v/(x^2*Sqrt[1 - 1/(c^2*x^2)]), x], x], x] /; InverseFunctionFreeQ[v, x]] /; FreeQ[{a, b, c}, x]
  5248. Int[Times[Power[Plus[Optional[Pattern[a, Blank[]]], Times[ArcSec[Times[Optional[Pattern[c, Blank[]]], Pattern[x, Blank[]]]], Optional[Pattern[b, Blank[]]]]], Optional[Pattern[n, Blank[]]]], Optional[Pattern[u, Blank[]]]], Pattern[x, Blank[Symbol]]] := Unintegrable[u*(a + b*ArcSec[c*x])^n, x] /; FreeQ[{a, b, c, n}, x]
  5249. Int[Times[Power[Plus[Optional[Pattern[a, Blank[]]], Times[ArcCsc[Times[Optional[Pattern[c, Blank[]]], Pattern[x, Blank[]]]], Optional[Pattern[b, Blank[]]]]], Optional[Pattern[n, Blank[]]]], Optional[Pattern[u, Blank[]]]], Pattern[x, Blank[Symbol]]] := Unintegrable[u*(a + b*ArcCsc[c*x])^n, x] /; FreeQ[{a, b, c, n}, x]
  5250. Int[ArcSec[Plus[Pattern[c, Blank[]], Times[Optional[Pattern[d, Blank[]]], Pattern[x, Blank[]]]]], Pattern[x, Blank[Symbol]]] := Simp[((c + d*x)*ArcSec[c + d*x])/d, x] - Int[1/((c + d*x)*Sqrt[1 - 1/(c + d*x)^2]), x] /; FreeQ[{c, d}, x]
  5251. Int[ArcCsc[Plus[Pattern[c, Blank[]], Times[Optional[Pattern[d, Blank[]]], Pattern[x, Blank[]]]]], Pattern[x, Blank[Symbol]]] := Simp[((c + d*x)*ArcCsc[c + d*x])/d, x] + Int[1/((c + d*x)*Sqrt[1 - 1/(c + d*x)^2]), x] /; FreeQ[{c, d}, x]
  5252. Int[Power[Plus[Optional[Pattern[a, Blank[]]], Times[ArcSec[Plus[Pattern[c, Blank[]], Times[Optional[Pattern[d, Blank[]]], Pattern[x, Blank[]]]]], Optional[Pattern[b, Blank[]]]]], Optional[Pattern[p, Blank[]]]], Pattern[x, Blank[Symbol]]] := Dist[1/d, Subst[Int[(a + b*ArcSec[x])^p, x], x, c + d*x], x] /; FreeQ[{a, b, c, d}, x] && IGtQ[p, 0]
  5253. Int[Power[Plus[Optional[Pattern[a, Blank[]]], Times[ArcCsc[Plus[Pattern[c, Blank[]], Times[Optional[Pattern[d, Blank[]]], Pattern[x, Blank[]]]]], Optional[Pattern[b, Blank[]]]]], Optional[Pattern[p, Blank[]]]], Pattern[x, Blank[Symbol]]] := Dist[1/d, Subst[Int[(a + b*ArcCsc[x])^p, x], x, c + d*x], x] /; FreeQ[{a, b, c, d}, x] && IGtQ[p, 0]
  5254. Int[Power[Plus[Optional[Pattern[a, Blank[]]], Times[ArcSec[Plus[Pattern[c, Blank[]], Times[Optional[Pattern[d, Blank[]]], Pattern[x, Blank[]]]]], Optional[Pattern[b, Blank[]]]]], Pattern[p, Blank[]]], Pattern[x, Blank[Symbol]]] := Unintegrable[(a + b*ArcSec[c + d*x])^p, x] /; FreeQ[{a, b, c, d, p}, x] && !IGtQ[p, 0]
  5255. Int[Power[Plus[Optional[Pattern[a, Blank[]]], Times[ArcCsc[Plus[Pattern[c, Blank[]], Times[Optional[Pattern[d, Blank[]]], Pattern[x, Blank[]]]]], Optional[Pattern[b, Blank[]]]]], Pattern[p, Blank[]]], Pattern[x, Blank[Symbol]]] := Unintegrable[(a + b*ArcCsc[c + d*x])^p, x] /; FreeQ[{a, b, c, d, p}, x] && !IGtQ[p, 0]
  5256. Int[Times[Power[Plus[Optional[Pattern[a, Blank[]]], Times[ArcSec[Plus[Pattern[c, Blank[]], Times[Optional[Pattern[d, Blank[]]], Pattern[x, Blank[]]]]], Optional[Pattern[b, Blank[]]]]], Optional[Pattern[p, Blank[]]]], Power[Plus[Optional[Pattern[e, Blank[]]], Times[Optional[Pattern[f, Blank[]]], Pattern[x, Blank[]]]], Optional[Pattern[m, Blank[]]]]], Pattern[x, Blank[Symbol]]] := Dist[1/d, Subst[Int[((f*x)/d)^m*(a + b*ArcSec[x])^p, x], x, c + d*x], x] /; FreeQ[{a, b, c, d, e, f, m}, x] && EqQ[d*e - c*f, 0] && IGtQ[p, 0]
  5257. Int[Times[Power[Plus[Optional[Pattern[a, Blank[]]], Times[ArcCsc[Plus[Pattern[c, Blank[]], Times[Optional[Pattern[d, Blank[]]], Pattern[x, Blank[]]]]], Optional[Pattern[b, Blank[]]]]], Optional[Pattern[p, Blank[]]]], Power[Plus[Optional[Pattern[e, Blank[]]], Times[Optional[Pattern[f, Blank[]]], Pattern[x, Blank[]]]], Optional[Pattern[m, Blank[]]]]], Pattern[x, Blank[Symbol]]] := Dist[1/d, Subst[Int[((f*x)/d)^m*(a + b*ArcCsc[x])^p, x], x, c + d*x], x] /; FreeQ[{a, b, c, d, e, f, m}, x] && EqQ[d*e - c*f, 0] && IGtQ[p, 0]
  5258. Int[Times[Power[Plus[Optional[Pattern[a, Blank[]]], Times[ArcSec[Plus[Pattern[c, Blank[]], Times[Optional[Pattern[d, Blank[]]], Pattern[x, Blank[]]]]], Optional[Pattern[b, Blank[]]]]], Optional[Pattern[p, Blank[]]]], Power[Plus[Optional[Pattern[e, Blank[]]], Times[Optional[Pattern[f, Blank[]]], Pattern[x, Blank[]]]], Optional[Pattern[m, Blank[]]]]], Pattern[x, Blank[Symbol]]] := Dist[1/d^(m + 1), Subst[Int[(a + b*x)^p*Sec[x]*Tan[x]*(d*e - c*f + f*Sec[x])^m, x], x, ArcSec[c + d*x]], x] /; FreeQ[{a, b, c, d, e, f}, x] && IGtQ[p, 0] && IntegerQ[m]
  5259. Int[Times[Power[Plus[Optional[Pattern[a, Blank[]]], Times[ArcCsc[Plus[Pattern[c, Blank[]], Times[Optional[Pattern[d, Blank[]]], Pattern[x, Blank[]]]]], Optional[Pattern[b, Blank[]]]]], Optional[Pattern[p, Blank[]]]], Power[Plus[Optional[Pattern[e, Blank[]]], Times[Optional[Pattern[f, Blank[]]], Pattern[x, Blank[]]]], Optional[Pattern[m, Blank[]]]]], Pattern[x, Blank[Symbol]]] := -Dist[(d^(m + 1))^(-1), Subst[Int[(a + b*x)^p*Csc[x]*Cot[x]*(d*e - c*f + f*Csc[x])^m, x], x, ArcCsc[c + d*x]], x] /; FreeQ[{a, b, c, d, e, f}, x] && IGtQ[p, 0] && IntegerQ[m]
  5260. Int[Times[Power[Plus[Optional[Pattern[a, Blank[]]], Times[ArcSec[Plus[Pattern[c, Blank[]], Times[Optional[Pattern[d, Blank[]]], Pattern[x, Blank[]]]]], Optional[Pattern[b, Blank[]]]]], Optional[Pattern[p, Blank[]]]], Power[Plus[Optional[Pattern[e, Blank[]]], Times[Optional[Pattern[f, Blank[]]], Pattern[x, Blank[]]]], Optional[Pattern[m, Blank[]]]]], Pattern[x, Blank[Symbol]]] := Dist[1/d, Subst[Int[((d*e - c*f)/d + (f*x)/d)^m*(a + b*ArcSec[x])^p, x], x, c + d*x], x] /; FreeQ[{a, b, c, d, e, f, m}, x] && IGtQ[p, 0]
  5261. Int[Times[Power[Plus[Optional[Pattern[a, Blank[]]], Times[ArcCsc[Plus[Pattern[c, Blank[]], Times[Optional[Pattern[d, Blank[]]], Pattern[x, Blank[]]]]], Optional[Pattern[b, Blank[]]]]], Optional[Pattern[p, Blank[]]]], Power[Plus[Optional[Pattern[e, Blank[]]], Times[Optional[Pattern[f, Blank[]]], Pattern[x, Blank[]]]], Optional[Pattern[m, Blank[]]]]], Pattern[x, Blank[Symbol]]] := Dist[1/d, Subst[Int[((d*e - c*f)/d + (f*x)/d)^m*(a + b*ArcCsc[x])^p, x], x, c + d*x], x] /; FreeQ[{a, b, c, d, e, f, m}, x] && IGtQ[p, 0]
  5262. Int[Times[Power[Plus[Optional[Pattern[a, Blank[]]], Times[ArcSec[Plus[Pattern[c, Blank[]], Times[Optional[Pattern[d, Blank[]]], Pattern[x, Blank[]]]]], Optional[Pattern[b, Blank[]]]]], Pattern[p, Blank[]]], Power[Plus[Optional[Pattern[e, Blank[]]], Times[Optional[Pattern[f, Blank[]]], Pattern[x, Blank[]]]], Optional[Pattern[m, Blank[]]]]], Pattern[x, Blank[Symbol]]] := Unintegrable[(e + f*x)^m*(a + b*ArcSec[c + d*x])^p, x] /; FreeQ[{a, b, c, d, e, f, m, p}, x] && !IGtQ[p, 0]
  5263. Int[Times[Power[Plus[Optional[Pattern[a, Blank[]]], Times[ArcCsc[Plus[Pattern[c, Blank[]], Times[Optional[Pattern[d, Blank[]]], Pattern[x, Blank[]]]]], Optional[Pattern[b, Blank[]]]]], Pattern[p, Blank[]]], Power[Plus[Optional[Pattern[e, Blank[]]], Times[Optional[Pattern[f, Blank[]]], Pattern[x, Blank[]]]], Optional[Pattern[m, Blank[]]]]], Pattern[x, Blank[Symbol]]] := Unintegrable[(e + f*x)^m*(a + b*ArcCsc[c + d*x])^p, x] /; FreeQ[{a, b, c, d, e, f, m, p}, x] && !IGtQ[p, 0]
  5264. Int[Times[Power[ArcSec[Times[Optional[Pattern[c, Blank[]]], Power[Plus[Optional[Pattern[a, Blank[]]], Times[Optional[Pattern[b, Blank[]]], Power[Pattern[x, Blank[]], Optional[Pattern[n, Blank[]]]]]], -1]]], Optional[Pattern[m, Blank[]]]], Optional[Pattern[u, Blank[]]]], Pattern[x, Blank[Symbol]]] := Int[u*ArcCos[a/c + (b*x^n)/c]^m, x] /; FreeQ[{a, b, c, n, m}, x]
  5265. Int[Times[Power[ArcCsc[Times[Optional[Pattern[c, Blank[]]], Power[Plus[Optional[Pattern[a, Blank[]]], Times[Optional[Pattern[b, Blank[]]], Power[Pattern[x, Blank[]], Optional[Pattern[n, Blank[]]]]]], -1]]], Optional[Pattern[m, Blank[]]]], Optional[Pattern[u, Blank[]]]], Pattern[x, Blank[Symbol]]] := Int[u*ArcSin[a/c + (b*x^n)/c]^m, x] /; FreeQ[{a, b, c, n, m}, x]
  5266. Int[Times[Optional[Pattern[u, Blank[]]], Power[Pattern[f, Blank[]], Times[Power[ArcSec[Plus[Optional[Pattern[a, Blank[]]], Times[Optional[Pattern[b, Blank[]]], Pattern[x, Blank[]]]]], Optional[Pattern[n, Blank[]]]], Optional[Pattern[c, Blank[]]]]]], Pattern[x, Blank[Symbol]]] := Dist[1/b, Subst[Int[(u /. x -> -(a/b) + Sec[x]/b)*f^(c*x^n)*Sec[x]*Tan[x], x], x, ArcSec[a + b*x]], x] /; FreeQ[{a, b, c, f}, x] && IGtQ[n, 0]
  5267. Int[Times[Optional[Pattern[u, Blank[]]], Power[Pattern[f, Blank[]], Times[Power[ArcCsc[Plus[Optional[Pattern[a, Blank[]]], Times[Optional[Pattern[b, Blank[]]], Pattern[x, Blank[]]]]], Optional[Pattern[n, Blank[]]]], Optional[Pattern[c, Blank[]]]]]], Pattern[x, Blank[Symbol]]] := -Dist[b^(-1), Subst[Int[(u /. x -> -(a/b) + Csc[x]/b)*f^(c*x^n)*Csc[x]*Cot[x], x], x, ArcCsc[a + b*x]], x] /; FreeQ[{a, b, c, f}, x] && IGtQ[n, 0]
  5268. Int[ArcSec[Pattern[u, Blank[]]], Pattern[x, Blank[Symbol]]] := Simp[x*ArcSec[u], x] - Dist[u/Sqrt[u^2], Int[SimplifyIntegrand[(x*D[u, x])/(u*Sqrt[u^2 - 1]), x], x], x] /; InverseFunctionFreeQ[u, x] && !FunctionOfExponentialQ[u, x]
  5269. Int[ArcCsc[Pattern[u, Blank[]]], Pattern[x, Blank[Symbol]]] := Simp[x*ArcCsc[u], x] + Dist[u/Sqrt[u^2], Int[SimplifyIntegrand[(x*D[u, x])/(u*Sqrt[u^2 - 1]), x], x], x] /; InverseFunctionFreeQ[u, x] && !FunctionOfExponentialQ[u, x]
  5270. Int[Times[Plus[Optional[Pattern[a, Blank[]]], Times[ArcSec[Pattern[u, Blank[]]], Optional[Pattern[b, Blank[]]]]], Power[Plus[Optional[Pattern[c, Blank[]]], Times[Optional[Pattern[d, Blank[]]], Pattern[x, Blank[]]]], Optional[Pattern[m, Blank[]]]]], Pattern[x, Blank[Symbol]]] := Simp[((c + d*x)^(m + 1)*(a + b*ArcSec[u]))/(d*(m + 1)), x] - Dist[(b*u)/(d*(m + 1)*Sqrt[u^2]), Int[SimplifyIntegrand[((c + d*x)^(m + 1)*D[u, x])/(u*Sqrt[u^2 - 1]), x], x], x] /; FreeQ[{a, b, c, d, m}, x] && NeQ[m, -1] && InverseFunctionFreeQ[u, x] && !FunctionOfQ[(c + d*x)^(m + 1), u, x] && !FunctionOfExponentialQ[u, x]
  5271. Int[Times[Plus[Optional[Pattern[a, Blank[]]], Times[ArcCsc[Pattern[u, Blank[]]], Optional[Pattern[b, Blank[]]]]], Power[Plus[Optional[Pattern[c, Blank[]]], Times[Optional[Pattern[d, Blank[]]], Pattern[x, Blank[]]]], Optional[Pattern[m, Blank[]]]]], Pattern[x, Blank[Symbol]]] := Simp[((c + d*x)^(m + 1)*(a + b*ArcCsc[u]))/(d*(m + 1)), x] + Dist[(b*u)/(d*(m + 1)*Sqrt[u^2]), Int[SimplifyIntegrand[((c + d*x)^(m + 1)*D[u, x])/(u*Sqrt[u^2 - 1]), x], x], x] /; FreeQ[{a, b, c, d, m}, x] && NeQ[m, -1] && InverseFunctionFreeQ[u, x] && !FunctionOfQ[(c + d*x)^(m + 1), u, x] && !FunctionOfExponentialQ[u, x]
  5272. Int[Times[Plus[Optional[Pattern[a, Blank[]]], Times[ArcSec[Pattern[u, Blank[]]], Optional[Pattern[b, Blank[]]]]], Pattern[v, Blank[]]], Pattern[x, Blank[Symbol]]] := With[{w = IntHide[v, x]}, Dist[a + b*ArcSec[u], w, x] - Dist[(b*u)/Sqrt[u^2], Int[SimplifyIntegrand[(w*D[u, x])/(u*Sqrt[u^2 - 1]), x], x], x] /; InverseFunctionFreeQ[w, x]] /; FreeQ[{a, b}, x] && InverseFunctionFreeQ[u, x] && !MatchQ[v, ((c_.) + (d_.)*x)^(m_.) /; FreeQ[{c, d, m}, x]]
  5273. Int[Times[Plus[Optional[Pattern[a, Blank[]]], Times[ArcCsc[Pattern[u, Blank[]]], Optional[Pattern[b, Blank[]]]]], Pattern[v, Blank[]]], Pattern[x, Blank[Symbol]]] := With[{w = IntHide[v, x]}, Dist[a + b*ArcCsc[u], w, x] + Dist[(b*u)/Sqrt[u^2], Int[SimplifyIntegrand[(w*D[u, x])/(u*Sqrt[u^2 - 1]), x], x], x] /; InverseFunctionFreeQ[w, x]] /; FreeQ[{a, b}, x] && InverseFunctionFreeQ[u, x] && !MatchQ[v, ((c_.) + (d_.)*x)^(m_.) /; FreeQ[{c, d, m}, x]]
  5274. Int[Times[Power[Pattern[u, Blank[]], Optional[Pattern[m, Blank[]]]], Power[Plus[Optional[Pattern[a, Blank[]]], Times[Optional[Pattern[b, Blank[]]], Sinh[Pattern[v, Blank[]]]]], Optional[Pattern[n, Blank[]]]]], Pattern[x, Blank[Symbol]]] := Int[ExpandToSum[u, x]^m*(a + b*Sinh[ExpandToSum[v, x]])^n, x] /; FreeQ[{a, b, m, n}, x] && LinearQ[{u, v}, x] && !LinearMatchQ[{u, v}, x]
  5275. Int[Times[Power[Plus[Optional[Pattern[a, Blank[]]], Times[Cosh[Pattern[v, Blank[]]], Optional[Pattern[b, Blank[]]]]], Optional[Pattern[n, Blank[]]]], Power[Pattern[u, Blank[]], Optional[Pattern[m, Blank[]]]]], Pattern[x, Blank[Symbol]]] := Int[ExpandToSum[u, x]^m*(a + b*Cosh[ExpandToSum[v, x]])^n, x] /; FreeQ[{a, b, m, n}, x] && LinearQ[{u, v}, x] && !LinearMatchQ[{u, v}, x]
  5276. Int[Times[Power[Plus[Pattern[a, Blank[]], Times[Optional[Pattern[b, Blank[]]], Power[Pattern[x, Blank[]], Pattern[n, Blank[]]]]], Optional[Pattern[p, Blank[]]]], Sinh[Plus[Optional[Pattern[c, Blank[]]], Times[Optional[Pattern[d, Blank[]]], Pattern[x, Blank[]]]]]], Pattern[x, Blank[Symbol]]] := Int[ExpandIntegrand[Sinh[c + d*x], (a + b*x^n)^p, x], x] /; FreeQ[{a, b, c, d, n}, x] && IGtQ[p, 0]
  5277. Int[Times[Cosh[Plus[Optional[Pattern[c, Blank[]]], Times[Optional[Pattern[d, Blank[]]], Pattern[x, Blank[]]]]], Power[Plus[Pattern[a, Blank[]], Times[Optional[Pattern[b, Blank[]]], Power[Pattern[x, Blank[]], Pattern[n, Blank[]]]]], Optional[Pattern[p, Blank[]]]]], Pattern[x, Blank[Symbol]]] := Int[ExpandIntegrand[Cosh[c + d*x], (a + b*x^n)^p, x], x] /; FreeQ[{a, b, c, d, n}, x] && IGtQ[p, 0]
  5278. Int[Times[Power[Plus[Pattern[a, Blank[]], Times[Optional[Pattern[b, Blank[]]], Power[Pattern[x, Blank[]], Pattern[n, Blank[]]]]], Pattern[p, Blank[]]], Sinh[Plus[Optional[Pattern[c, Blank[]]], Times[Optional[Pattern[d, Blank[]]], Pattern[x, Blank[]]]]]], Pattern[x, Blank[Symbol]]] := Simp[(x^(-n + 1)*(a + b*x^n)^(p + 1)*Sinh[c + d*x])/(b*n*(p + 1)), x] + (-Dist[(-n + 1)/(b*n*(p + 1)), Int[((a + b*x^n)^(p + 1)*Sinh[c + d*x])/x^n, x], x] - Dist[d/(b*n*(p + 1)), Int[x^(-n + 1)*(a + b*x^n)^(p + 1)*Cosh[c + d*x], x], x]) /; FreeQ[{a, b, c, d}, x] && IntegerQ[p] && IGtQ[n, 0] && LtQ[p, -1] && GtQ[n, 2]
  5279. Int[Times[Cosh[Plus[Optional[Pattern[c, Blank[]]], Times[Optional[Pattern[d, Blank[]]], Pattern[x, Blank[]]]]], Power[Plus[Pattern[a, Blank[]], Times[Optional[Pattern[b, Blank[]]], Power[Pattern[x, Blank[]], Pattern[n, Blank[]]]]], Pattern[p, Blank[]]]], Pattern[x, Blank[Symbol]]] := Simp[(x^(-n + 1)*(a + b*x^n)^(p + 1)*Cosh[c + d*x])/(b*n*(p + 1)), x] + (-Dist[(-n + 1)/(b*n*(p + 1)), Int[((a + b*x^n)^(p + 1)*Cosh[c + d*x])/x^n, x], x] - Dist[d/(b*n*(p + 1)), Int[x^(-n + 1)*(a + b*x^n)^(p + 1)*Sinh[c + d*x], x], x]) /; FreeQ[{a, b, c, d}, x] && IntegerQ[p] && IGtQ[n, 0] && LtQ[p, -1] && GtQ[n, 2]
  5280. Int[Times[Power[Plus[Pattern[a, Blank[]], Times[Optional[Pattern[b, Blank[]]], Power[Pattern[x, Blank[]], Pattern[n, Blank[]]]]], Pattern[p, Blank[]]], Sinh[Plus[Optional[Pattern[c, Blank[]]], Times[Optional[Pattern[d, Blank[]]], Pattern[x, Blank[]]]]]], Pattern[x, Blank[Symbol]]] := Int[ExpandIntegrand[Sinh[c + d*x], (a + b*x^n)^p, x], x] /; FreeQ[{a, b, c, d}, x] && ILtQ[p, 0] && IGtQ[n, 0] && (EqQ[n, 2] || EqQ[p, -1])
  5281. Int[Times[Cosh[Plus[Optional[Pattern[c, Blank[]]], Times[Optional[Pattern[d, Blank[]]], Pattern[x, Blank[]]]]], Power[Plus[Pattern[a, Blank[]], Times[Optional[Pattern[b, Blank[]]], Power[Pattern[x, Blank[]], Pattern[n, Blank[]]]]], Pattern[p, Blank[]]]], Pattern[x, Blank[Symbol]]] := Int[ExpandIntegrand[Cosh[c + d*x], (a + b*x^n)^p, x], x] /; FreeQ[{a, b, c, d}, x] && ILtQ[p, 0] && IGtQ[n, 0] && (EqQ[n, 2] || EqQ[p, -1])
  5282. Int[Times[Power[Plus[Pattern[a, Blank[]], Times[Optional[Pattern[b, Blank[]]], Power[Pattern[x, Blank[]], Pattern[n, Blank[]]]]], Pattern[p, Blank[]]], Sinh[Plus[Optional[Pattern[c, Blank[]]], Times[Optional[Pattern[d, Blank[]]], Pattern[x, Blank[]]]]]], Pattern[x, Blank[Symbol]]] := Int[x^(n*p)*(b + a/x^n)^p*Sinh[c + d*x], x] /; FreeQ[{a, b, c, d}, x] && ILtQ[p, 0] && ILtQ[n, 0]
  5283. Int[Times[Cosh[Plus[Optional[Pattern[c, Blank[]]], Times[Optional[Pattern[d, Blank[]]], Pattern[x, Blank[]]]]], Power[Plus[Pattern[a, Blank[]], Times[Optional[Pattern[b, Blank[]]], Power[Pattern[x, Blank[]], Pattern[n, Blank[]]]]], Pattern[p, Blank[]]]], Pattern[x, Blank[Symbol]]] := Int[x^(n*p)*(b + a/x^n)^p*Cosh[c + d*x], x] /; FreeQ[{a, b, c, d}, x] && ILtQ[p, 0] && ILtQ[n, 0]
  5284. Int[Times[Power[Plus[Pattern[a, Blank[]], Times[Optional[Pattern[b, Blank[]]], Power[Pattern[x, Blank[]], Pattern[n, Blank[]]]]], Pattern[p, Blank[]]], Sinh[Plus[Optional[Pattern[c, Blank[]]], Times[Optional[Pattern[d, Blank[]]], Pattern[x, Blank[]]]]]], Pattern[x, Blank[Symbol]]] := Unintegrable[(a + b*x^n)^p*Sinh[c + d*x], x] /; FreeQ[{a, b, c, d, n, p}, x]
  5285. Int[Times[Cosh[Plus[Optional[Pattern[c, Blank[]]], Times[Optional[Pattern[d, Blank[]]], Pattern[x, Blank[]]]]], Power[Plus[Pattern[a, Blank[]], Times[Optional[Pattern[b, Blank[]]], Power[Pattern[x, Blank[]], Pattern[n, Blank[]]]]], Pattern[p, Blank[]]]], Pattern[x, Blank[Symbol]]] := Unintegrable[(a + b*x^n)^p*Cosh[c + d*x], x] /; FreeQ[{a, b, c, d, n, p}, x]
  5286. Int[Times[Power[Times[Optional[Pattern[e, Blank[]]], Pattern[x, Blank[]]], Optional[Pattern[m, Blank[]]]], Power[Plus[Pattern[a, Blank[]], Times[Optional[Pattern[b, Blank[]]], Power[Pattern[x, Blank[]], Pattern[n, Blank[]]]]], Optional[Pattern[p, Blank[]]]], Sinh[Plus[Optional[Pattern[c, Blank[]]], Times[Optional[Pattern[d, Blank[]]], Pattern[x, Blank[]]]]]], Pattern[x, Blank[Symbol]]] := Int[ExpandIntegrand[Sinh[c + d*x], (e*x)^m*(a + b*x^n)^p, x], x] /; FreeQ[{a, b, c, d, e, m, n}, x] && IGtQ[p, 0]
  5287. Int[Times[Cosh[Plus[Optional[Pattern[c, Blank[]]], Times[Optional[Pattern[d, Blank[]]], Pattern[x, Blank[]]]]], Power[Times[Optional[Pattern[e, Blank[]]], Pattern[x, Blank[]]], Optional[Pattern[m, Blank[]]]], Power[Plus[Pattern[a, Blank[]], Times[Optional[Pattern[b, Blank[]]], Power[Pattern[x, Blank[]], Pattern[n, Blank[]]]]], Optional[Pattern[p, Blank[]]]]], Pattern[x, Blank[Symbol]]] := Int[ExpandIntegrand[Cosh[c + d*x], (e*x)^m*(a + b*x^n)^p, x], x] /; FreeQ[{a, b, c, d, e, m, n}, x] && IGtQ[p, 0]
  5288. Int[Times[Power[Times[Optional[Pattern[e, Blank[]]], Pattern[x, Blank[]]], Optional[Pattern[m, Blank[]]]], Power[Plus[Pattern[a, Blank[]], Times[Optional[Pattern[b, Blank[]]], Power[Pattern[x, Blank[]], Pattern[n, Blank[]]]]], Pattern[p, Blank[]]], Sinh[Plus[Optional[Pattern[c, Blank[]]], Times[Optional[Pattern[d, Blank[]]], Pattern[x, Blank[]]]]]], Pattern[x, Blank[Symbol]]] := Simp[(e^m*(a + b*x^n)^(p + 1)*Sinh[c + d*x])/(b*n*(p + 1)), x] - Dist[(d*e^m)/(b*n*(p + 1)), Int[(a + b*x^n)^(p + 1)*Cosh[c + d*x], x], x] /; FreeQ[{a, b, c, d, e, m, n}, x] && IntegerQ[p] && EqQ[m - n + 1, 0] && LtQ[p, -1] && (IntegerQ[n] || GtQ[e, 0])
  5289. Int[Times[Cosh[Plus[Optional[Pattern[c, Blank[]]], Times[Optional[Pattern[d, Blank[]]], Pattern[x, Blank[]]]]], Power[Times[Optional[Pattern[e, Blank[]]], Pattern[x, Blank[]]], Optional[Pattern[m, Blank[]]]], Power[Plus[Pattern[a, Blank[]], Times[Optional[Pattern[b, Blank[]]], Power[Pattern[x, Blank[]], Pattern[n, Blank[]]]]], Pattern[p, Blank[]]]], Pattern[x, Blank[Symbol]]] := Simp[(e^m*(a + b*x^n)^(p + 1)*Cosh[c + d*x])/(b*n*(p + 1)), x] - Dist[(d*e^m)/(b*n*(p + 1)), Int[(a + b*x^n)^(p + 1)*Sinh[c + d*x], x], x] /; FreeQ[{a, b, c, d, e, m, n}, x] && IntegerQ[p] && EqQ[m - n + 1, 0] && LtQ[p, -1] && (IntegerQ[n] || GtQ[e, 0])
  5290. Int[Times[Power[Pattern[x, Blank[]], Optional[Pattern[m, Blank[]]]], Power[Plus[Pattern[a, Blank[]], Times[Optional[Pattern[b, Blank[]]], Power[Pattern[x, Blank[]], Pattern[n, Blank[]]]]], Pattern[p, Blank[]]], Sinh[Plus[Optional[Pattern[c, Blank[]]], Times[Optional[Pattern[d, Blank[]]], Pattern[x, Blank[]]]]]], Pattern[x, Blank[Symbol]]] := Simp[(x^(m - n + 1)*(a + b*x^n)^(p + 1)*Sinh[c + d*x])/(b*n*(p + 1)), x] + (-Dist[(m - n + 1)/(b*n*(p + 1)), Int[x^(m - n)*(a + b*x^n)^(p + 1)*Sinh[c + d*x], x], x] - Dist[d/(b*n*(p + 1)), Int[x^(m - n + 1)*(a + b*x^n)^(p + 1)*Cosh[c + d*x], x], x]) /; FreeQ[{a, b, c, d}, x] && ILtQ[p, -1] && IGtQ[n, 0] && RationalQ[m] && (GtQ[m - n + 1, 0] || GtQ[n, 2])
  5291. Int[Times[Cosh[Plus[Optional[Pattern[c, Blank[]]], Times[Optional[Pattern[d, Blank[]]], Pattern[x, Blank[]]]]], Power[Pattern[x, Blank[]], Optional[Pattern[m, Blank[]]]], Power[Plus[Pattern[a, Blank[]], Times[Optional[Pattern[b, Blank[]]], Power[Pattern[x, Blank[]], Pattern[n, Blank[]]]]], Pattern[p, Blank[]]]], Pattern[x, Blank[Symbol]]] := Simp[(x^(m - n + 1)*(a + b*x^n)^(p + 1)*Cosh[c + d*x])/(b*n*(p + 1)), x] + (-Dist[(m - n + 1)/(b*n*(p + 1)), Int[x^(m - n)*(a + b*x^n)^(p + 1)*Cosh[c + d*x], x], x] - Dist[d/(b*n*(p + 1)), Int[x^(m - n + 1)*(a + b*x^n)^(p + 1)*Sinh[c + d*x], x], x]) /; FreeQ[{a, b, c, d}, x] && ILtQ[p, -1] && IGtQ[n, 0] && RationalQ[m] && (GtQ[m - n + 1, 0] || GtQ[n, 2])
  5292. Int[Times[Power[Pattern[x, Blank[]], Optional[Pattern[m, Blank[]]]], Power[Plus[Pattern[a, Blank[]], Times[Optional[Pattern[b, Blank[]]], Power[Pattern[x, Blank[]], Pattern[n, Blank[]]]]], Pattern[p, Blank[]]], Sinh[Plus[Optional[Pattern[c, Blank[]]], Times[Optional[Pattern[d, Blank[]]], Pattern[x, Blank[]]]]]], Pattern[x, Blank[Symbol]]] := Int[ExpandIntegrand[Sinh[c + d*x], x^m*(a + b*x^n)^p, x], x] /; FreeQ[{a, b, c, d}, x] && ILtQ[p, 0] && IntegerQ[m] && IGtQ[n, 0] && (EqQ[n, 2] || EqQ[p, -1])
  5293. Int[Times[Cosh[Plus[Optional[Pattern[c, Blank[]]], Times[Optional[Pattern[d, Blank[]]], Pattern[x, Blank[]]]]], Power[Pattern[x, Blank[]], Optional[Pattern[m, Blank[]]]], Power[Plus[Pattern[a, Blank[]], Times[Optional[Pattern[b, Blank[]]], Power[Pattern[x, Blank[]], Pattern[n, Blank[]]]]], Pattern[p, Blank[]]]], Pattern[x, Blank[Symbol]]] := Int[ExpandIntegrand[Cosh[c + d*x], x^m*(a + b*x^n)^p, x], x] /; FreeQ[{a, b, c, d}, x] && ILtQ[p, 0] && IntegerQ[m] && IGtQ[n, 0] && (EqQ[n, 2] || EqQ[p, -1])
  5294. Int[Times[Power[Pattern[x, Blank[]], Optional[Pattern[m, Blank[]]]], Power[Plus[Pattern[a, Blank[]], Times[Optional[Pattern[b, Blank[]]], Power[Pattern[x, Blank[]], Pattern[n, Blank[]]]]], Pattern[p, Blank[]]], Sinh[Plus[Optional[Pattern[c, Blank[]]], Times[Optional[Pattern[d, Blank[]]], Pattern[x, Blank[]]]]]], Pattern[x, Blank[Symbol]]] := Int[x^(m + n*p)*(b + a/x^n)^p*Sinh[c + d*x], x] /; FreeQ[{a, b, c, d, m}, x] && ILtQ[p, 0] && ILtQ[n, 0]
  5295. Int[Times[Cosh[Plus[Optional[Pattern[c, Blank[]]], Times[Optional[Pattern[d, Blank[]]], Pattern[x, Blank[]]]]], Power[Pattern[x, Blank[]], Optional[Pattern[m, Blank[]]]], Power[Plus[Pattern[a, Blank[]], Times[Optional[Pattern[b, Blank[]]], Power[Pattern[x, Blank[]], Pattern[n, Blank[]]]]], Pattern[p, Blank[]]]], Pattern[x, Blank[Symbol]]] := Int[x^(m + n*p)*(b + a/x^n)^p*Cosh[c + d*x], x] /; FreeQ[{a, b, c, d, m}, x] && ILtQ[p, 0] && ILtQ[n, 0]
  5296. Int[Times[Power[Times[Optional[Pattern[e, Blank[]]], Pattern[x, Blank[]]], Optional[Pattern[m, Blank[]]]], Power[Plus[Pattern[a, Blank[]], Times[Optional[Pattern[b, Blank[]]], Power[Pattern[x, Blank[]], Pattern[n, Blank[]]]]], Optional[Pattern[p, Blank[]]]], Sinh[Plus[Optional[Pattern[c, Blank[]]], Times[Optional[Pattern[d, Blank[]]], Pattern[x, Blank[]]]]]], Pattern[x, Blank[Symbol]]] := Unintegrable[(e*x)^m*(a + b*x^n)^p*Sinh[c + d*x], x] /; FreeQ[{a, b, c, d, e, m, n, p}, x]
  5297. Int[Times[Cosh[Plus[Optional[Pattern[c, Blank[]]], Times[Optional[Pattern[d, Blank[]]], Pattern[x, Blank[]]]]], Power[Times[Optional[Pattern[e, Blank[]]], Pattern[x, Blank[]]], Optional[Pattern[m, Blank[]]]], Power[Plus[Pattern[a, Blank[]], Times[Optional[Pattern[b, Blank[]]], Power[Pattern[x, Blank[]], Pattern[n, Blank[]]]]], Optional[Pattern[p, Blank[]]]]], Pattern[x, Blank[Symbol]]] := Unintegrable[(e*x)^m*(a + b*x^n)^p*Cosh[c + d*x], x] /; FreeQ[{a, b, c, d, e, m, n, p}, x]
  5298. Int[Sinh[Plus[Optional[Pattern[c, Blank[]]], Times[Optional[Pattern[d, Blank[]]], Power[Pattern[x, Blank[]], Pattern[n, Blank[]]]]]], Pattern[x, Blank[Symbol]]] := Dist[1/2, Int[E^(c + d*x^n), x], x] - Dist[1/2, Int[E^(-c - d*x^n), x], x] /; FreeQ[{c, d}, x] && IGtQ[n, 1]
  5299. Int[Cosh[Plus[Optional[Pattern[c, Blank[]]], Times[Optional[Pattern[d, Blank[]]], Power[Pattern[x, Blank[]], Pattern[n, Blank[]]]]]], Pattern[x, Blank[Symbol]]] := Dist[1/2, Int[E^(c + d*x^n), x], x] + Dist[1/2, Int[E^(-c - d*x^n), x], x] /; FreeQ[{c, d}, x] && IGtQ[n, 1]
  5300. Int[Power[Plus[Optional[Pattern[a, Blank[]]], Times[Optional[Pattern[b, Blank[]]], Sinh[Plus[Optional[Pattern[c, Blank[]]], Times[Optional[Pattern[d, Blank[]]], Power[Pattern[x, Blank[]], Pattern[n, Blank[]]]]]]]], Pattern[p, Blank[]]], Pattern[x, Blank[Symbol]]] := Int[ExpandTrigReduce[(a + b*Sinh[c + d*x^n])^p, x], x] /; FreeQ[{a, b, c, d}, x] && IGtQ[n, 1] && IGtQ[p, 1]
  5301. Int[Power[Plus[Optional[Pattern[a, Blank[]]], Times[Cosh[Plus[Optional[Pattern[c, Blank[]]], Times[Optional[Pattern[d, Blank[]]], Power[Pattern[x, Blank[]], Pattern[n, Blank[]]]]]], Optional[Pattern[b, Blank[]]]]], Pattern[p, Blank[]]], Pattern[x, Blank[Symbol]]] := Int[ExpandTrigReduce[(a + b*Cosh[c + d*x^n])^p, x], x] /; FreeQ[{a, b, c, d}, x] && IGtQ[n, 1] && IGtQ[p, 1]
  5302. Int[Power[Plus[Optional[Pattern[a, Blank[]]], Times[Optional[Pattern[b, Blank[]]], Sinh[Plus[Optional[Pattern[c, Blank[]]], Times[Optional[Pattern[d, Blank[]]], Power[Pattern[x, Blank[]], Pattern[n, Blank[]]]]]]]], Optional[Pattern[p, Blank[]]]], Pattern[x, Blank[Symbol]]] := -Subst[Int[(a + b*Sinh[c + d/x^n])^p/x^2, x], x, 1/x] /; FreeQ[{a, b, c, d}, x] && ILtQ[n, 0] && IntegerQ[p]
  5303. Int[Power[Plus[Optional[Pattern[a, Blank[]]], Times[Cosh[Plus[Optional[Pattern[c, Blank[]]], Times[Optional[Pattern[d, Blank[]]], Power[Pattern[x, Blank[]], Pattern[n, Blank[]]]]]], Optional[Pattern[b, Blank[]]]]], Optional[Pattern[p, Blank[]]]], Pattern[x, Blank[Symbol]]] := -Subst[Int[(a + b*Cosh[c + d/x^n])^p/x^2, x], x, 1/x] /; FreeQ[{a, b, c, d}, x] && ILtQ[n, 0] && IntegerQ[p]
  5304. Int[Power[Plus[Optional[Pattern[a, Blank[]]], Times[Optional[Pattern[b, Blank[]]], Sinh[Plus[Optional[Pattern[c, Blank[]]], Times[Optional[Pattern[d, Blank[]]], Power[Pattern[x, Blank[]], Pattern[n, Blank[]]]]]]]], Optional[Pattern[p, Blank[]]]], Pattern[x, Blank[Symbol]]] := Module[{k = Denominator[n]}, Dist[k, Subst[Int[x^(k - 1)*(a + b*Sinh[c + d*x^(k*n)])^p, x], x, x^(1/k)], x]] /; FreeQ[{a, b, c, d}, x] && FractionQ[n] && IntegerQ[p]
  5305. Int[Power[Plus[Optional[Pattern[a, Blank[]]], Times[Cosh[Plus[Optional[Pattern[c, Blank[]]], Times[Optional[Pattern[d, Blank[]]], Power[Pattern[x, Blank[]], Pattern[n, Blank[]]]]]], Optional[Pattern[b, Blank[]]]]], Optional[Pattern[p, Blank[]]]], Pattern[x, Blank[Symbol]]] := Module[{k = Denominator[n]}, Dist[k, Subst[Int[x^(k - 1)*(a + b*Cosh[c + d*x^(k*n)])^p, x], x, x^(1/k)], x]] /; FreeQ[{a, b, c, d}, x] && FractionQ[n] && IntegerQ[p]
  5306. Int[Sinh[Plus[Optional[Pattern[c, Blank[]]], Times[Optional[Pattern[d, Blank[]]], Power[Pattern[x, Blank[]], Pattern[n, Blank[]]]]]], Pattern[x, Blank[Symbol]]] := Dist[1/2, Int[E^(c + d*x^n), x], x] - Dist[1/2, Int[E^(-c - d*x^n), x], x] /; FreeQ[{c, d, n}, x]
  5307. Int[Cosh[Plus[Optional[Pattern[c, Blank[]]], Times[Optional[Pattern[d, Blank[]]], Power[Pattern[x, Blank[]], Pattern[n, Blank[]]]]]], Pattern[x, Blank[Symbol]]] := Dist[1/2, Int[E^(c + d*x^n), x], x] + Dist[1/2, Int[E^(-c - d*x^n), x], x] /; FreeQ[{c, d, n}, x]
  5308. Int[Power[Plus[Optional[Pattern[a, Blank[]]], Times[Optional[Pattern[b, Blank[]]], Sinh[Plus[Optional[Pattern[c, Blank[]]], Times[Optional[Pattern[d, Blank[]]], Power[Pattern[x, Blank[]], Pattern[n, Blank[]]]]]]]], Pattern[p, Blank[]]], Pattern[x, Blank[Symbol]]] := Int[ExpandTrigReduce[(a + b*Sinh[c + d*x^n])^p, x], x] /; FreeQ[{a, b, c, d, n}, x] && IGtQ[p, 0]
  5309. Int[Power[Plus[Optional[Pattern[a, Blank[]]], Times[Cosh[Plus[Optional[Pattern[c, Blank[]]], Times[Optional[Pattern[d, Blank[]]], Power[Pattern[x, Blank[]], Pattern[n, Blank[]]]]]], Optional[Pattern[b, Blank[]]]]], Pattern[p, Blank[]]], Pattern[x, Blank[Symbol]]] := Int[ExpandTrigReduce[(a + b*Cosh[c + d*x^n])^p, x], x] /; FreeQ[{a, b, c, d, n}, x] && IGtQ[p, 0]
  5310. Int[Power[Plus[Optional[Pattern[a, Blank[]]], Times[Optional[Pattern[b, Blank[]]], Sinh[Plus[Optional[Pattern[c, Blank[]]], Times[Optional[Pattern[d, Blank[]]], Power[Pattern[u, Blank[]], Pattern[n, Blank[]]]]]]]], Optional[Pattern[p, Blank[]]]], Pattern[x, Blank[Symbol]]] := Dist[1/Coefficient[u, x, 1], Subst[Int[(a + b*Sinh[c + d*x^n])^p, x], x, u], x] /; FreeQ[{a, b, c, d, n}, x] && IntegerQ[p] && LinearQ[u, x] && NeQ[u, x]
  5311. Int[Power[Plus[Optional[Pattern[a, Blank[]]], Times[Cosh[Plus[Optional[Pattern[c, Blank[]]], Times[Optional[Pattern[d, Blank[]]], Power[Pattern[u, Blank[]], Pattern[n, Blank[]]]]]], Optional[Pattern[b, Blank[]]]]], Optional[Pattern[p, Blank[]]]], Pattern[x, Blank[Symbol]]] := Dist[1/Coefficient[u, x, 1], Subst[Int[(a + b*Cosh[c + d*x^n])^p, x], x, u], x] /; FreeQ[{a, b, c, d, n}, x] && IntegerQ[p] && LinearQ[u, x] && NeQ[u, x]
  5312. Int[Power[Plus[Optional[Pattern[a, Blank[]]], Times[Optional[Pattern[b, Blank[]]], Sinh[Plus[Optional[Pattern[c, Blank[]]], Times[Optional[Pattern[d, Blank[]]], Power[Pattern[u, Blank[]], Pattern[n, Blank[]]]]]]]], Pattern[p, Blank[]]], Pattern[x, Blank[Symbol]]] := Unintegrable[(a + b*Sinh[c + d*u^n])^p, x] /; FreeQ[{a, b, c, d, n, p}, x] && LinearQ[u, x]
  5313. Int[Power[Plus[Optional[Pattern[a, Blank[]]], Times[Cosh[Plus[Optional[Pattern[c, Blank[]]], Times[Optional[Pattern[d, Blank[]]], Power[Pattern[u, Blank[]], Pattern[n, Blank[]]]]]], Optional[Pattern[b, Blank[]]]]], Pattern[p, Blank[]]], Pattern[x, Blank[Symbol]]] := Unintegrable[(a + b*Cosh[c + d*u^n])^p, x] /; FreeQ[{a, b, c, d, n, p}, x] && LinearQ[u, x]
  5314. Int[Power[Plus[Optional[Pattern[a, Blank[]]], Times[Optional[Pattern[b, Blank[]]], Sinh[Pattern[u, Blank[]]]]], Optional[Pattern[p, Blank[]]]], Pattern[x, Blank[Symbol]]] := Int[(a + b*Sinh[ExpandToSum[u, x]])^p, x] /; FreeQ[{a, b, p}, x] && BinomialQ[u, x] && !BinomialMatchQ[u, x]
  5315. Int[Power[Plus[Optional[Pattern[a, Blank[]]], Times[Cosh[Pattern[u, Blank[]]], Optional[Pattern[b, Blank[]]]]], Optional[Pattern[p, Blank[]]]], Pattern[x, Blank[Symbol]]] := Int[(a + b*Cosh[ExpandToSum[u, x]])^p, x] /; FreeQ[{a, b, p}, x] && BinomialQ[u, x] && !BinomialMatchQ[u, x]
  5316. Int[Times[Power[Pattern[x, Blank[]], -1], Sinh[Times[Optional[Pattern[d, Blank[]]], Power[Pattern[x, Blank[]], Pattern[n, Blank[]]]]]], Pattern[x, Blank[Symbol]]] := Simp[SinhIntegral[d*x^n]/n, x] /; FreeQ[{d, n}, x]
  5317. Int[Times[Cosh[Times[Optional[Pattern[d, Blank[]]], Power[Pattern[x, Blank[]], Pattern[n, Blank[]]]]], Power[Pattern[x, Blank[]], -1]], Pattern[x, Blank[Symbol]]] := Simp[CoshIntegral[d*x^n]/n, x] /; FreeQ[{d, n}, x]
  5318. Int[Times[Power[Pattern[x, Blank[]], -1], Sinh[Plus[Pattern[c, Blank[]], Times[Optional[Pattern[d, Blank[]]], Power[Pattern[x, Blank[]], Pattern[n, Blank[]]]]]]], Pattern[x, Blank[Symbol]]] := Dist[Sinh[c], Int[Cosh[d*x^n]/x, x], x] + Dist[Cosh[c], Int[Sinh[d*x^n]/x, x], x] /; FreeQ[{c, d, n}, x]
  5319. Int[Times[Cosh[Plus[Pattern[c, Blank[]], Times[Optional[Pattern[d, Blank[]]], Power[Pattern[x, Blank[]], Pattern[n, Blank[]]]]]], Power[Pattern[x, Blank[]], -1]], Pattern[x, Blank[Symbol]]] := Dist[Cosh[c], Int[Cosh[d*x^n]/x, x], x] + Dist[Sinh[c], Int[Sinh[d*x^n]/x, x], x] /; FreeQ[{c, d, n}, x]
  5320. Int[Times[Power[Pattern[x, Blank[]], Optional[Pattern[m, Blank[]]]], Power[Plus[Optional[Pattern[a, Blank[]]], Times[Optional[Pattern[b, Blank[]]], Sinh[Plus[Optional[Pattern[c, Blank[]]], Times[Optional[Pattern[d, Blank[]]], Power[Pattern[x, Blank[]], Pattern[n, Blank[]]]]]]]], Optional[Pattern[p, Blank[]]]]], Pattern[x, Blank[Symbol]]] := Dist[1/n, Subst[Int[x^(Simplify[(m + 1)/n] - 1)*(a + b*Sinh[c + d*x])^p, x], x, x^n], x] /; FreeQ[{a, b, c, d, m, n, p}, x] && IntegerQ[Simplify[(m + 1)/n]] && (EqQ[p, 1] || EqQ[m, n - 1] || (IntegerQ[p] && GtQ[Simplify[(m + 1)/n], 0]))
  5321. Int[Times[Power[Plus[Optional[Pattern[a, Blank[]]], Times[Cosh[Plus[Optional[Pattern[c, Blank[]]], Times[Optional[Pattern[d, Blank[]]], Power[Pattern[x, Blank[]], Pattern[n, Blank[]]]]]], Optional[Pattern[b, Blank[]]]]], Optional[Pattern[p, Blank[]]]], Power[Pattern[x, Blank[]], Optional[Pattern[m, Blank[]]]]], Pattern[x, Blank[Symbol]]] := Dist[1/n, Subst[Int[x^(Simplify[(m + 1)/n] - 1)*(a + b*Cosh[c + d*x])^p, x], x, x^n], x] /; FreeQ[{a, b, c, d, m, n, p}, x] && IntegerQ[Simplify[(m + 1)/n]] && (EqQ[p, 1] || EqQ[m, n - 1] || (IntegerQ[p] && GtQ[Simplify[(m + 1)/n], 0]))
  5322. Int[Times[Power[Times[Pattern[e, Blank[]], Pattern[x, Blank[]]], Pattern[m, Blank[]]], Power[Plus[Optional[Pattern[a, Blank[]]], Times[Optional[Pattern[b, Blank[]]], Sinh[Plus[Optional[Pattern[c, Blank[]]], Times[Optional[Pattern[d, Blank[]]], Power[Pattern[x, Blank[]], Pattern[n, Blank[]]]]]]]], Optional[Pattern[p, Blank[]]]]], Pattern[x, Blank[Symbol]]] := Dist[(e^IntPart[m]*(e*x)^FracPart[m])/x^FracPart[m], Int[x^m*(a + b*Sinh[c + d*x^n])^p, x], x] /; FreeQ[{a, b, c, d, e, m, n, p}, x] && IntegerQ[Simplify[(m + 1)/n]]
  5323. Int[Times[Power[Plus[Optional[Pattern[a, Blank[]]], Times[Cosh[Plus[Optional[Pattern[c, Blank[]]], Times[Optional[Pattern[d, Blank[]]], Power[Pattern[x, Blank[]], Pattern[n, Blank[]]]]]], Optional[Pattern[b, Blank[]]]]], Optional[Pattern[p, Blank[]]]], Power[Times[Pattern[e, Blank[]], Pattern[x, Blank[]]], Pattern[m, Blank[]]]], Pattern[x, Blank[Symbol]]] := Dist[(e^IntPart[m]*(e*x)^FracPart[m])/x^FracPart[m], Int[x^m*(a + b*Cosh[c + d*x^n])^p, x], x] /; FreeQ[{a, b, c, d, e, m, n, p}, x] && IntegerQ[Simplify[(m + 1)/n]]
  5324. Int[Times[Power[Times[Optional[Pattern[e, Blank[]]], Pattern[x, Blank[]]], Optional[Pattern[m, Blank[]]]], Sinh[Plus[Optional[Pattern[c, Blank[]]], Times[Optional[Pattern[d, Blank[]]], Power[Pattern[x, Blank[]], Pattern[n, Blank[]]]]]]], Pattern[x, Blank[Symbol]]] := Simp[(e^(n - 1)*(e*x)^(m - n + 1)*Cosh[c + d*x^n])/(d*n), x] - Dist[(e^n*(m - n + 1))/(d*n), Int[(e*x)^(m - n)*Cosh[c + d*x^n], x], x] /; FreeQ[{c, d, e}, x] && IGtQ[n, 0] && LtQ[0, n, m + 1]
  5325. Int[Times[Cosh[Plus[Optional[Pattern[c, Blank[]]], Times[Optional[Pattern[d, Blank[]]], Power[Pattern[x, Blank[]], Pattern[n, Blank[]]]]]], Power[Times[Optional[Pattern[e, Blank[]]], Pattern[x, Blank[]]], Optional[Pattern[m, Blank[]]]]], Pattern[x, Blank[Symbol]]] := Simp[(e^(n - 1)*(e*x)^(m - n + 1)*Sinh[c + d*x^n])/(d*n), x] - Dist[(e^n*(m - n + 1))/(d*n), Int[(e*x)^(m - n)*Sinh[c + d*x^n], x], x] /; FreeQ[{c, d, e}, x] && IGtQ[n, 0] && LtQ[0, n, m + 1]
  5326. Int[Times[Power[Times[Optional[Pattern[e, Blank[]]], Pattern[x, Blank[]]], Pattern[m, Blank[]]], Sinh[Plus[Optional[Pattern[c, Blank[]]], Times[Optional[Pattern[d, Blank[]]], Power[Pattern[x, Blank[]], Pattern[n, Blank[]]]]]]], Pattern[x, Blank[Symbol]]] := Simp[((e*x)^(m + 1)*Sinh[c + d*x^n])/(e*(m + 1)), x] - Dist[(d*n)/(e^n*(m + 1)), Int[(e*x)^(m + n)*Cosh[c + d*x^n], x], x] /; FreeQ[{c, d, e}, x] && IGtQ[n, 0] && LtQ[m, -1]
  5327. Int[Times[Cosh[Plus[Optional[Pattern[c, Blank[]]], Times[Optional[Pattern[d, Blank[]]], Power[Pattern[x, Blank[]], Pattern[n, Blank[]]]]]], Power[Times[Optional[Pattern[e, Blank[]]], Pattern[x, Blank[]]], Pattern[m, Blank[]]]], Pattern[x, Blank[Symbol]]] := Simp[((e*x)^(m + 1)*Cosh[c + d*x^n])/(e*(m + 1)), x] - Dist[(d*n)/(e^n*(m + 1)), Int[(e*x)^(m + n)*Sinh[c + d*x^n], x], x] /; FreeQ[{c, d, e}, x] && IGtQ[n, 0] && LtQ[m, -1]
  5328. Int[Times[Power[Times[Optional[Pattern[e, Blank[]]], Pattern[x, Blank[]]], Optional[Pattern[m, Blank[]]]], Sinh[Plus[Optional[Pattern[c, Blank[]]], Times[Optional[Pattern[d, Blank[]]], Power[Pattern[x, Blank[]], Pattern[n, Blank[]]]]]]], Pattern[x, Blank[Symbol]]] := Dist[1/2, Int[(e*x)^m*E^(c + d*x^n), x], x] - Dist[1/2, Int[(e*x)^m*E^(-c - d*x^n), x], x] /; FreeQ[{c, d, e, m}, x] && IGtQ[n, 0]
  5329. Int[Times[Cosh[Plus[Optional[Pattern[c, Blank[]]], Times[Optional[Pattern[d, Blank[]]], Power[Pattern[x, Blank[]], Pattern[n, Blank[]]]]]], Power[Times[Optional[Pattern[e, Blank[]]], Pattern[x, Blank[]]], Optional[Pattern[m, Blank[]]]]], Pattern[x, Blank[Symbol]]] := Dist[1/2, Int[(e*x)^m*E^(c + d*x^n), x], x] + Dist[1/2, Int[(e*x)^m*E^(-c - d*x^n), x], x] /; FreeQ[{c, d, e, m}, x] && IGtQ[n, 0]
  5330. Int[Times[Power[Pattern[x, Blank[]], Optional[Pattern[m, Blank[]]]], Power[Sinh[Plus[Optional[Pattern[a, Blank[]]], Times[Optional[Pattern[b, Blank[]]], Power[Pattern[x, Blank[]], Pattern[n, Blank[]]]]]], Pattern[p, Blank[]]]], Pattern[x, Blank[Symbol]]] := -Simp[Sinh[a + b*x^n]^p/((n - 1)*x^(n - 1)), x] + Dist[(b*n*p)/(n - 1), Int[Sinh[a + b*x^n]^(p - 1)*Cosh[a + b*x^n], x], x] /; FreeQ[{a, b}, x] && IntegersQ[n, p] && EqQ[m + n, 0] && GtQ[p, 1] && NeQ[n, 1]
  5331. Int[Times[Power[Cosh[Plus[Optional[Pattern[a, Blank[]]], Times[Optional[Pattern[b, Blank[]]], Power[Pattern[x, Blank[]], Pattern[n, Blank[]]]]]], Pattern[p, Blank[]]], Power[Pattern[x, Blank[]], Optional[Pattern[m, Blank[]]]]], Pattern[x, Blank[Symbol]]] := -Simp[Cosh[a + b*x^n]^p/((n - 1)*x^(n - 1)), x] + Dist[(b*n*p)/(n - 1), Int[Cosh[a + b*x^n]^(p - 1)*Sinh[a + b*x^n], x], x] /; FreeQ[{a, b}, x] && IntegersQ[n, p] && EqQ[m + n, 0] && GtQ[p, 1] && NeQ[n, 1]
  5332. Int[Times[Power[Pattern[x, Blank[]], Optional[Pattern[m, Blank[]]]], Power[Sinh[Plus[Optional[Pattern[a, Blank[]]], Times[Optional[Pattern[b, Blank[]]], Power[Pattern[x, Blank[]], Pattern[n, Blank[]]]]]], Pattern[p, Blank[]]]], Pattern[x, Blank[Symbol]]] := -Simp[(n*Sinh[a + b*x^n]^p)/(b^2*n^2*p^2), x] + (-Dist[(p - 1)/p, Int[x^m*Sinh[a + b*x^n]^(p - 2), x], x] + Simp[(x^n*Cosh[a + b*x^n]*Sinh[a + b*x^n]^(p - 1))/(b*n*p), x]) /; FreeQ[{a, b, m, n}, x] && EqQ[m - 2*n + 1] && GtQ[p, 1]
  5333. Int[Times[Power[Cosh[Plus[Optional[Pattern[a, Blank[]]], Times[Optional[Pattern[b, Blank[]]], Power[Pattern[x, Blank[]], Pattern[n, Blank[]]]]]], Pattern[p, Blank[]]], Power[Pattern[x, Blank[]], Optional[Pattern[m, Blank[]]]]], Pattern[x, Blank[Symbol]]] := -Simp[(n*Cosh[a + b*x^n]^p)/(b^2*n^2*p^2), x] + (Dist[(p - 1)/p, Int[x^m*Cosh[a + b*x^n]^(p - 2), x], x] + Simp[(x^n*Sinh[a + b*x^n]*Cosh[a + b*x^n]^(p - 1))/(b*n*p), x]) /; FreeQ[{a, b, m, n}, x] && EqQ[m - 2*n + 1] && GtQ[p, 1]
  5334. Int[Times[Power[Pattern[x, Blank[]], Optional[Pattern[m, Blank[]]]], Power[Sinh[Plus[Optional[Pattern[a, Blank[]]], Times[Optional[Pattern[b, Blank[]]], Power[Pattern[x, Blank[]], Pattern[n, Blank[]]]]]], Pattern[p, Blank[]]]], Pattern[x, Blank[Symbol]]] := -Simp[((m - n + 1)*x^(m - 2*n + 1)*Sinh[a + b*x^n]^p)/(b^2*n^2*p^2), x] + (-Dist[(p - 1)/p, Int[x^m*Sinh[a + b*x^n]^(p - 2), x], x] + Dist[((m - n + 1)*(m - 2*n + 1))/(b^2*n^2*p^2), Int[x^(m - 2*n)*Sinh[a + b*x^n]^p, x], x] + Simp[(x^(m - n + 1)*Cosh[a + b*x^n]*Sinh[a + b*x^n]^(p - 1))/(b*n*p), x]) /; FreeQ[{a, b}, x] && IntegersQ[m, n] && GtQ[p, 1] && LtQ[0, 2*n, m + 1]
  5335. Int[Times[Power[Cosh[Plus[Optional[Pattern[a, Blank[]]], Times[Optional[Pattern[b, Blank[]]], Power[Pattern[x, Blank[]], Pattern[n, Blank[]]]]]], Pattern[p, Blank[]]], Power[Pattern[x, Blank[]], Optional[Pattern[m, Blank[]]]]], Pattern[x, Blank[Symbol]]] := -Simp[((m - n + 1)*x^(m - 2*n + 1)*Cosh[a + b*x^n]^p)/(b^2*n^2*p^2), x] + (Dist[(p - 1)/p, Int[x^m*Cosh[a + b*x^n]^(p - 2), x], x] + Dist[((m - n + 1)*(m - 2*n + 1))/(b^2*n^2*p^2), Int[x^(m - 2*n)*Cosh[a + b*x^n]^p, x], x] + Simp[(x^(m - n + 1)*Sinh[a + b*x^n]*Cosh[a + b*x^n]^(p - 1))/(b*n*p), x]) /; FreeQ[{a, b}, x] && IntegersQ[m, n] && GtQ[p, 1] && LtQ[0, 2*n, m + 1]
  5336. Int[Times[Power[Pattern[x, Blank[]], Optional[Pattern[m, Blank[]]]], Power[Sinh[Plus[Optional[Pattern[a, Blank[]]], Times[Optional[Pattern[b, Blank[]]], Power[Pattern[x, Blank[]], Pattern[n, Blank[]]]]]], Pattern[p, Blank[]]]], Pattern[x, Blank[Symbol]]] := Simp[(x^(m + 1)*Sinh[a + b*x^n]^p)/(m + 1), x] + (Dist[(b^2*n^2*p*(p - 1))/((m + 1)*(m + n + 1)), Int[x^(m + 2*n)*Sinh[a + b*x^n]^(p - 2), x], x] + Dist[(b^2*n^2*p^2)/((m + 1)*(m + n + 1)), Int[x^(m + 2*n)*Sinh[a + b*x^n]^p, x], x] - Simp[(b*n*p*x^(m + n + 1)*Cosh[a + b*x^n]*Sinh[a + b*x^n]^(p - 1))/((m + 1)*(m + n + 1)), x]) /; FreeQ[{a, b}, x] && IntegersQ[m, n] && GtQ[p, 1] && LtQ[0, 2*n, 1 - m] && NeQ[m + n + 1, 0]
  5337. Int[Times[Power[Cosh[Plus[Optional[Pattern[a, Blank[]]], Times[Optional[Pattern[b, Blank[]]], Power[Pattern[x, Blank[]], Pattern[n, Blank[]]]]]], Pattern[p, Blank[]]], Power[Pattern[x, Blank[]], Optional[Pattern[m, Blank[]]]]], Pattern[x, Blank[Symbol]]] := Simp[(x^(m + 1)*Cosh[a + b*x^n]^p)/(m + 1), x] + (-Dist[(b^2*n^2*p*(p - 1))/((m + 1)*(m + n + 1)), Int[x^(m + 2*n)*Cosh[a + b*x^n]^(p - 2), x], x] + Dist[(b^2*n^2*p^2)/((m + 1)*(m + n + 1)), Int[x^(m + 2*n)*Cosh[a + b*x^n]^p, x], x] - Simp[(b*n*p*x^(m + n + 1)*Sinh[a + b*x^n]*Cosh[a + b*x^n]^(p - 1))/((m + 1)*(m + n + 1)), x]) /; FreeQ[{a, b}, x] && IntegersQ[m, n] && GtQ[p, 1] && LtQ[0, 2*n, 1 - m] && NeQ[m + n + 1, 0]
  5338. Int[Times[Power[Times[Optional[Pattern[e, Blank[]]], Pattern[x, Blank[]]], Pattern[m, Blank[]]], Power[Plus[Optional[Pattern[a, Blank[]]], Times[Optional[Pattern[b, Blank[]]], Sinh[Plus[Optional[Pattern[c, Blank[]]], Times[Optional[Pattern[d, Blank[]]], Power[Pattern[x, Blank[]], Pattern[n, Blank[]]]]]]]], Optional[Pattern[p, Blank[]]]]], Pattern[x, Blank[Symbol]]] := With[{k = Denominator[m]}, Dist[k/e, Subst[Int[x^(k*(m + 1) - 1)*(a + b*Sinh[c + (d*x^(k*n))/e^n])^p, x], x, (e*x)^(1/k)], x]] /; FreeQ[{a, b, c, d, e}, x] && IntegerQ[p] && IGtQ[n, 0] && FractionQ[m]
  5339. Int[Times[Power[Plus[Optional[Pattern[a, Blank[]]], Times[Cosh[Plus[Optional[Pattern[c, Blank[]]], Times[Optional[Pattern[d, Blank[]]], Power[Pattern[x, Blank[]], Pattern[n, Blank[]]]]]], Optional[Pattern[b, Blank[]]]]], Optional[Pattern[p, Blank[]]]], Power[Times[Optional[Pattern[e, Blank[]]], Pattern[x, Blank[]]], Pattern[m, Blank[]]]], Pattern[x, Blank[Symbol]]] := With[{k = Denominator[m]}, Dist[k/e, Subst[Int[x^(k*(m + 1) - 1)*(a + b*Cosh[c + (d*x^(k*n))/e^n])^p, x], x, (e*x)^(1/k)], x]] /; FreeQ[{a, b, c, d, e}, x] && IntegerQ[p] && IGtQ[n, 0] && FractionQ[m]
  5340. Int[Times[Power[Times[Optional[Pattern[e, Blank[]]], Pattern[x, Blank[]]], Optional[Pattern[m, Blank[]]]], Power[Plus[Optional[Pattern[a, Blank[]]], Times[Optional[Pattern[b, Blank[]]], Sinh[Plus[Optional[Pattern[c, Blank[]]], Times[Optional[Pattern[d, Blank[]]], Power[Pattern[x, Blank[]], Pattern[n, Blank[]]]]]]]], Pattern[p, Blank[]]]], Pattern[x, Blank[Symbol]]] := Int[ExpandTrigReduce[(e*x)^m, (a + b*Sinh[c + d*x^n])^p, x], x] /; FreeQ[{a, b, c, d, e, m}, x] && IGtQ[p, 1] && IGtQ[n, 0]
  5341. Int[Times[Power[Plus[Optional[Pattern[a, Blank[]]], Times[Cosh[Plus[Optional[Pattern[c, Blank[]]], Times[Optional[Pattern[d, Blank[]]], Power[Pattern[x, Blank[]], Pattern[n, Blank[]]]]]], Optional[Pattern[b, Blank[]]]]], Pattern[p, Blank[]]], Power[Times[Optional[Pattern[e, Blank[]]], Pattern[x, Blank[]]], Optional[Pattern[m, Blank[]]]]], Pattern[x, Blank[Symbol]]] := Int[ExpandTrigReduce[(e*x)^m, (a + b*Cosh[c + d*x^n])^p, x], x] /; FreeQ[{a, b, c, d, e, m}, x] && IGtQ[p, 1] && IGtQ[n, 0]
  5342. Int[Times[Power[Pattern[x, Blank[]], Optional[Pattern[m, Blank[]]]], Power[Sinh[Plus[Optional[Pattern[a, Blank[]]], Times[Optional[Pattern[b, Blank[]]], Power[Pattern[x, Blank[]], Pattern[n, Blank[]]]]]], Pattern[p, Blank[]]]], Pattern[x, Blank[Symbol]]] := Simp[(x^n*Cosh[a + b*x^n]*Sinh[a + b*x^n]^(p + 1))/(b*n*(p + 1)), x] + (-Dist[(p + 2)/(p + 1), Int[x^m*Sinh[a + b*x^n]^(p + 2), x], x] - Simp[(n*Sinh[a + b*x^n]^(p + 2))/(b^2*n^2*(p + 1)*(p + 2)), x]) /; FreeQ[{a, b, m, n}, x] && EqQ[m - 2*n + 1, 0] && LtQ[p, -1] && NeQ[p, -2]
  5343. Int[Times[Power[Cosh[Plus[Optional[Pattern[a, Blank[]]], Times[Optional[Pattern[b, Blank[]]], Power[Pattern[x, Blank[]], Pattern[n, Blank[]]]]]], Pattern[p, Blank[]]], Power[Pattern[x, Blank[]], Optional[Pattern[m, Blank[]]]]], Pattern[x, Blank[Symbol]]] := -Simp[(x^n*Sinh[a + b*x^n]*Cosh[a + b*x^n]^(p + 1))/(b*n*(p + 1)), x] + (Dist[(p + 2)/(p + 1), Int[x^m*Cosh[a + b*x^n]^(p + 2), x], x] + Simp[(n*Cosh[a + b*x^n]^(p + 2))/(b^2*n^2*(p + 1)*(p + 2)), x]) /; FreeQ[{a, b, m, n}, x] && EqQ[m - 2*n + 1, 0] && LtQ[p, -1] && NeQ[p, -2]
  5344. Int[Times[Power[Pattern[x, Blank[]], Optional[Pattern[m, Blank[]]]], Power[Sinh[Plus[Optional[Pattern[a, Blank[]]], Times[Optional[Pattern[b, Blank[]]], Power[Pattern[x, Blank[]], Pattern[n, Blank[]]]]]], Pattern[p, Blank[]]]], Pattern[x, Blank[Symbol]]] := Simp[(x^(m - n + 1)*Cosh[a + b*x^n]*Sinh[a + b*x^n]^(p + 1))/(b*n*(p + 1)), x] + (-Dist[(p + 2)/(p + 1), Int[x^m*Sinh[a + b*x^n]^(p + 2), x], x] + Dist[((m - n + 1)*(m - 2*n + 1))/(b^2*n^2*(p + 1)*(p + 2)), Int[x^(m - 2*n)*Sinh[a + b*x^n]^(p + 2), x], x] - Simp[((m - n + 1)*x^(m - 2*n + 1)*Sinh[a + b*x^n]^(p + 2))/(b^2*n^2*(p + 1)*(p + 2)), x]) /; FreeQ[{a, b}, x] && IntegersQ[m, n] && LtQ[p, -1] && NeQ[p, -2] && LtQ[0, 2*n, m + 1]
  5345. Int[Times[Power[Cosh[Plus[Optional[Pattern[a, Blank[]]], Times[Optional[Pattern[b, Blank[]]], Power[Pattern[x, Blank[]], Pattern[n, Blank[]]]]]], Pattern[p, Blank[]]], Power[Pattern[x, Blank[]], Optional[Pattern[m, Blank[]]]]], Pattern[x, Blank[Symbol]]] := -Simp[(x^(m - n + 1)*Sinh[a + b*x^n]*Cosh[a + b*x^n]^(p + 1))/(b*n*(p + 1)), x] + (Dist[(p + 2)/(p + 1), Int[x^m*Cosh[a + b*x^n]^(p + 2), x], x] - Dist[((m - n + 1)*(m - 2*n + 1))/(b^2*n^2*(p + 1)*(p + 2)), Int[x^(m - 2*n)*Cosh[a + b*x^n]^(p + 2), x], x] + Simp[((m - n + 1)*x^(m - 2*n + 1)*Cosh[a + b*x^n]^(p + 2))/(b^2*n^2*(p + 1)*(p + 2)), x]) /; FreeQ[{a, b}, x] && IntegersQ[m, n] && LtQ[p, -1] && NeQ[p, -2] && LtQ[0, 2*n, m + 1]
  5346. Int[Times[Power[Pattern[x, Blank[]], Optional[Pattern[m, Blank[]]]], Power[Plus[Optional[Pattern[a, Blank[]]], Times[Optional[Pattern[b, Blank[]]], Sinh[Plus[Optional[Pattern[c, Blank[]]], Times[Optional[Pattern[d, Blank[]]], Power[Pattern[x, Blank[]], Pattern[n, Blank[]]]]]]]], Optional[Pattern[p, Blank[]]]]], Pattern[x, Blank[Symbol]]] := -Subst[Int[(a + b*Sinh[c + d/x^n])^p/x^(m + 2), x], x, 1/x] /; FreeQ[{a, b, c, d}, x] && IntegerQ[p] && ILtQ[n, 0] && IntegerQ[m]
  5347. Int[Times[Power[Plus[Optional[Pattern[a, Blank[]]], Times[Cosh[Plus[Optional[Pattern[c, Blank[]]], Times[Optional[Pattern[d, Blank[]]], Power[Pattern[x, Blank[]], Pattern[n, Blank[]]]]]], Optional[Pattern[b, Blank[]]]]], Optional[Pattern[p, Blank[]]]], Power[Pattern[x, Blank[]], Optional[Pattern[m, Blank[]]]]], Pattern[x, Blank[Symbol]]] := -Subst[Int[(a + b*Cosh[c + d/x^n])^p/x^(m + 2), x], x, 1/x] /; FreeQ[{a, b, c, d}, x] && IntegerQ[p] && ILtQ[n, 0] && IntegerQ[m]
  5348. Int[Times[Power[Times[Optional[Pattern[e, Blank[]]], Pattern[x, Blank[]]], Pattern[m, Blank[]]], Power[Plus[Optional[Pattern[a, Blank[]]], Times[Optional[Pattern[b, Blank[]]], Sinh[Plus[Optional[Pattern[c, Blank[]]], Times[Optional[Pattern[d, Blank[]]], Power[Pattern[x, Blank[]], Pattern[n, Blank[]]]]]]]], Optional[Pattern[p, Blank[]]]]], Pattern[x, Blank[Symbol]]] := With[{k = Denominator[m]}, -Dist[k/e, Subst[Int[(a + b*Sinh[c + d/(e^n*x^(k*n))])^p/x^(k*(m + 1) + 1), x], x, 1/(e*x)^(1/k)], x]] /; FreeQ[{a, b, c, d, e}, x] && IntegerQ[p] && ILtQ[n, 0] && FractionQ[m]
  5349. Int[Times[Power[Plus[Optional[Pattern[a, Blank[]]], Times[Cosh[Plus[Optional[Pattern[c, Blank[]]], Times[Optional[Pattern[d, Blank[]]], Power[Pattern[x, Blank[]], Pattern[n, Blank[]]]]]], Optional[Pattern[b, Blank[]]]]], Optional[Pattern[p, Blank[]]]], Power[Times[Optional[Pattern[e, Blank[]]], Pattern[x, Blank[]]], Pattern[m, Blank[]]]], Pattern[x, Blank[Symbol]]] := With[{k = Denominator[m]}, -Dist[k/e, Subst[Int[(a + b*Cosh[c + d/(e^n*x^(k*n))])^p/x^(k*(m + 1) + 1), x], x, 1/(e*x)^(1/k)], x]] /; FreeQ[{a, b, c, d, e}, x] && IntegerQ[p] && ILtQ[n, 0] && FractionQ[m]
  5350. Int[Times[Power[Times[Optional[Pattern[e, Blank[]]], Pattern[x, Blank[]]], Pattern[m, Blank[]]], Power[Plus[Optional[Pattern[a, Blank[]]], Times[Optional[Pattern[b, Blank[]]], Sinh[Plus[Optional[Pattern[c, Blank[]]], Times[Optional[Pattern[d, Blank[]]], Power[Pattern[x, Blank[]], Pattern[n, Blank[]]]]]]]], Optional[Pattern[p, Blank[]]]]], Pattern[x, Blank[Symbol]]] := -Dist[(e*x)^m*(x^(-1))^m, Subst[Int[(a + b*Sinh[c + d/x^n])^p/x^(m + 2), x], x, 1/x], x] /; FreeQ[{a, b, c, d, e, m}, x] && IntegerQ[p] && ILtQ[n, 0] && !RationalQ[m]
  5351. Int[Times[Power[Plus[Optional[Pattern[a, Blank[]]], Times[Cosh[Plus[Optional[Pattern[c, Blank[]]], Times[Optional[Pattern[d, Blank[]]], Power[Pattern[x, Blank[]], Pattern[n, Blank[]]]]]], Optional[Pattern[b, Blank[]]]]], Optional[Pattern[p, Blank[]]]], Power[Times[Optional[Pattern[e, Blank[]]], Pattern[x, Blank[]]], Pattern[m, Blank[]]]], Pattern[x, Blank[Symbol]]] := -Dist[(e*x)^m*(x^(-1))^m, Subst[Int[(a + b*Cosh[c + d/x^n])^p/x^(m + 2), x], x, 1/x], x] /; FreeQ[{a, b, c, d, e, m}, x] && IntegerQ[p] && ILtQ[n, 0] && !RationalQ[m]
  5352. Int[Times[Power[Pattern[x, Blank[]], Optional[Pattern[m, Blank[]]]], Power[Plus[Optional[Pattern[a, Blank[]]], Times[Optional[Pattern[b, Blank[]]], Sinh[Plus[Optional[Pattern[c, Blank[]]], Times[Optional[Pattern[d, Blank[]]], Power[Pattern[x, Blank[]], Pattern[n, Blank[]]]]]]]], Optional[Pattern[p, Blank[]]]]], Pattern[x, Blank[Symbol]]] := Module[{k = Denominator[n]}, Dist[k, Subst[Int[x^(k*(m + 1) - 1)*(a + b*Sinh[c + d*x^(k*n)])^p, x], x, x^(1/k)], x]] /; FreeQ[{a, b, c, d, m}, x] && IntegerQ[p] && FractionQ[n]
  5353. Int[Times[Power[Plus[Optional[Pattern[a, Blank[]]], Times[Cosh[Plus[Optional[Pattern[c, Blank[]]], Times[Optional[Pattern[d, Blank[]]], Power[Pattern[x, Blank[]], Pattern[n, Blank[]]]]]], Optional[Pattern[b, Blank[]]]]], Optional[Pattern[p, Blank[]]]], Power[Pattern[x, Blank[]], Optional[Pattern[m, Blank[]]]]], Pattern[x, Blank[Symbol]]] := Module[{k = Denominator[n]}, Dist[k, Subst[Int[x^(k*(m + 1) - 1)*(a + b*Cosh[c + d*x^(k*n)])^p, x], x, x^(1/k)], x]] /; FreeQ[{a, b, c, d, m}, x] && IntegerQ[p] && FractionQ[n]
  5354. Int[Times[Power[Times[Pattern[e, Blank[]], Pattern[x, Blank[]]], Pattern[m, Blank[]]], Power[Plus[Optional[Pattern[a, Blank[]]], Times[Optional[Pattern[b, Blank[]]], Sinh[Plus[Optional[Pattern[c, Blank[]]], Times[Optional[Pattern[d, Blank[]]], Power[Pattern[x, Blank[]], Pattern[n, Blank[]]]]]]]], Optional[Pattern[p, Blank[]]]]], Pattern[x, Blank[Symbol]]] := Dist[(e^IntPart[m]*(e*x)^FracPart[m])/x^FracPart[m], Int[x^m*(a + b*Sinh[c + d*x^n])^p, x], x] /; FreeQ[{a, b, c, d, e, m}, x] && IntegerQ[p] && FractionQ[n]
  5355. Int[Times[Power[Plus[Optional[Pattern[a, Blank[]]], Times[Cosh[Plus[Optional[Pattern[c, Blank[]]], Times[Optional[Pattern[d, Blank[]]], Power[Pattern[x, Blank[]], Pattern[n, Blank[]]]]]], Optional[Pattern[b, Blank[]]]]], Optional[Pattern[p, Blank[]]]], Power[Times[Pattern[e, Blank[]], Pattern[x, Blank[]]], Pattern[m, Blank[]]]], Pattern[x, Blank[Symbol]]] := Dist[(e^IntPart[m]*(e*x)^FracPart[m])/x^FracPart[m], Int[x^m*(a + b*Cosh[c + d*x^n])^p, x], x] /; FreeQ[{a, b, c, d, e, m}, x] && IntegerQ[p] && FractionQ[n]
  5356. Int[Times[Power[Pattern[x, Blank[]], Optional[Pattern[m, Blank[]]]], Power[Plus[Optional[Pattern[a, Blank[]]], Times[Optional[Pattern[b, Blank[]]], Sinh[Plus[Optional[Pattern[c, Blank[]]], Times[Optional[Pattern[d, Blank[]]], Power[Pattern[x, Blank[]], Pattern[n, Blank[]]]]]]]], Optional[Pattern[p, Blank[]]]]], Pattern[x, Blank[Symbol]]] := Dist[1/(m + 1), Subst[Int[(a + b*Sinh[c + d*x^Simplify[n/(m + 1)]])^p, x], x, x^(m + 1)], x] /; FreeQ[{a, b, c, d, m, n}, x] && IntegerQ[p] && NeQ[m, -1] && IGtQ[Simplify[n/(m + 1)], 0] && !IntegerQ[n]
  5357. Int[Times[Power[Plus[Optional[Pattern[a, Blank[]]], Times[Cosh[Plus[Optional[Pattern[c, Blank[]]], Times[Optional[Pattern[d, Blank[]]], Power[Pattern[x, Blank[]], Pattern[n, Blank[]]]]]], Optional[Pattern[b, Blank[]]]]], Optional[Pattern[p, Blank[]]]], Power[Pattern[x, Blank[]], Optional[Pattern[m, Blank[]]]]], Pattern[x, Blank[Symbol]]] := Dist[1/(m + 1), Subst[Int[(a + b*Cosh[c + d*x^Simplify[n/(m + 1)]])^p, x], x, x^(m + 1)], x] /; FreeQ[{a, b, c, d, m, n}, x] && IntegerQ[p] && NeQ[m, -1] && IGtQ[Simplify[n/(m + 1)], 0] && !IntegerQ[n]
  5358. Int[Times[Power[Times[Pattern[e, Blank[]], Pattern[x, Blank[]]], Pattern[m, Blank[]]], Power[Plus[Optional[Pattern[a, Blank[]]], Times[Optional[Pattern[b, Blank[]]], Sinh[Plus[Optional[Pattern[c, Blank[]]], Times[Optional[Pattern[d, Blank[]]], Power[Pattern[x, Blank[]], Pattern[n, Blank[]]]]]]]], Optional[Pattern[p, Blank[]]]]], Pattern[x, Blank[Symbol]]] := Dist[(e^IntPart[m]*(e*x)^FracPart[m])/x^FracPart[m], Int[x^m*(a + b*Sinh[c + d*x^n])^p, x], x] /; FreeQ[{a, b, c, d, e, m, n}, x] && IntegerQ[p] && NeQ[m, -1] && IGtQ[Simplify[n/(m + 1)], 0] && !IntegerQ[n]
  5359. Int[Times[Power[Plus[Optional[Pattern[a, Blank[]]], Times[Cosh[Plus[Optional[Pattern[c, Blank[]]], Times[Optional[Pattern[d, Blank[]]], Power[Pattern[x, Blank[]], Pattern[n, Blank[]]]]]], Optional[Pattern[b, Blank[]]]]], Optional[Pattern[p, Blank[]]]], Power[Times[Pattern[e, Blank[]], Pattern[x, Blank[]]], Pattern[m, Blank[]]]], Pattern[x, Blank[Symbol]]] := Dist[(e^IntPart[m]*(e*x)^FracPart[m])/x^FracPart[m], Int[x^m*(a + b*Cosh[c + d*x^n])^p, x], x] /; FreeQ[{a, b, c, d, e, m, n}, x] && IntegerQ[p] && NeQ[m, -1] && IGtQ[Simplify[n/(m + 1)], 0] && !IntegerQ[n]
  5360. Int[Times[Power[Times[Optional[Pattern[e, Blank[]]], Pattern[x, Blank[]]], Optional[Pattern[m, Blank[]]]], Sinh[Plus[Optional[Pattern[c, Blank[]]], Times[Optional[Pattern[d, Blank[]]], Power[Pattern[x, Blank[]], Pattern[n, Blank[]]]]]]], Pattern[x, Blank[Symbol]]] := Dist[1/2, Int[(e*x)^m*E^(c + d*x^n), x], x] - Dist[1/2, Int[(e*x)^m*E^(-c - d*x^n), x], x] /; FreeQ[{c, d, e, m, n}, x]
  5361. Int[Times[Cosh[Plus[Optional[Pattern[c, Blank[]]], Times[Optional[Pattern[d, Blank[]]], Power[Pattern[x, Blank[]], Pattern[n, Blank[]]]]]], Power[Times[Optional[Pattern[e, Blank[]]], Pattern[x, Blank[]]], Optional[Pattern[m, Blank[]]]]], Pattern[x, Blank[Symbol]]] := Dist[1/2, Int[(e*x)^m*E^(c + d*x^n), x], x] + Dist[1/2, Int[(e*x)^m*E^(-c - d*x^n), x], x] /; FreeQ[{c, d, e, m, n}, x]
  5362. Int[Times[Power[Times[Optional[Pattern[e, Blank[]]], Pattern[x, Blank[]]], Optional[Pattern[m, Blank[]]]], Power[Plus[Optional[Pattern[a, Blank[]]], Times[Optional[Pattern[b, Blank[]]], Sinh[Plus[Optional[Pattern[c, Blank[]]], Times[Optional[Pattern[d, Blank[]]], Power[Pattern[x, Blank[]], Pattern[n, Blank[]]]]]]]], Pattern[p, Blank[]]]], Pattern[x, Blank[Symbol]]] := Int[ExpandTrigReduce[(e*x)^m, (a + b*Sinh[c + d*x^n])^p, x], x] /; FreeQ[{a, b, c, d, e, m, n}, x] && IGtQ[p, 0]
  5363. Int[Times[Power[Plus[Optional[Pattern[a, Blank[]]], Times[Cosh[Plus[Optional[Pattern[c, Blank[]]], Times[Optional[Pattern[d, Blank[]]], Power[Pattern[x, Blank[]], Pattern[n, Blank[]]]]]], Optional[Pattern[b, Blank[]]]]], Pattern[p, Blank[]]], Power[Times[Optional[Pattern[e, Blank[]]], Pattern[x, Blank[]]], Optional[Pattern[m, Blank[]]]]], Pattern[x, Blank[Symbol]]] := Int[ExpandTrigReduce[(e*x)^m, (a + b*Cosh[c + d*x^n])^p, x], x] /; FreeQ[{a, b, c, d, e, m, n}, x] && IGtQ[p, 0]
  5364. Int[Times[Power[Pattern[x, Blank[]], Optional[Pattern[m, Blank[]]]], Power[Plus[Optional[Pattern[a, Blank[]]], Times[Optional[Pattern[b, Blank[]]], Sinh[Plus[Optional[Pattern[c, Blank[]]], Times[Optional[Pattern[d, Blank[]]], Power[Pattern[u, Blank[]], Pattern[n, Blank[]]]]]]]], Optional[Pattern[p, Blank[]]]]], Pattern[x, Blank[Symbol]]] := Dist[1/Coefficient[u, x, 1]^(m + 1), Subst[Int[(x - Coefficient[u, x, 0])^m*(a + b*Sinh[c + d*x^n])^p, x], x, u], x] /; FreeQ[{a, b, c, d, n, p}, x] && LinearQ[u, x] && NeQ[u, x] && IntegerQ[m]
  5365. Int[Times[Power[Plus[Optional[Pattern[a, Blank[]]], Times[Cosh[Plus[Optional[Pattern[c, Blank[]]], Times[Optional[Pattern[d, Blank[]]], Power[Pattern[u, Blank[]], Pattern[n, Blank[]]]]]], Optional[Pattern[b, Blank[]]]]], Optional[Pattern[p, Blank[]]]], Power[Pattern[x, Blank[]], Optional[Pattern[m, Blank[]]]]], Pattern[x, Blank[Symbol]]] := Dist[1/Coefficient[u, x, 1]^(m + 1), Subst[Int[(x - Coefficient[u, x, 0])^m*(a + b*Cosh[c + d*x^n])^p, x], x, u], x] /; FreeQ[{a, b, c, d, n, p}, x] && LinearQ[u, x] && NeQ[u, x] && IntegerQ[m]
  5366. Int[Times[Power[Times[Optional[Pattern[e, Blank[]]], Pattern[x, Blank[]]], Optional[Pattern[m, Blank[]]]], Power[Plus[Optional[Pattern[a, Blank[]]], Times[Optional[Pattern[b, Blank[]]], Sinh[Plus[Optional[Pattern[c, Blank[]]], Times[Optional[Pattern[d, Blank[]]], Power[Pattern[u, Blank[]], Pattern[n, Blank[]]]]]]]], Optional[Pattern[p, Blank[]]]]], Pattern[x, Blank[Symbol]]] := Unintegrable[(e*x)^m*(a + b*Sinh[c + d*u^n])^p, x] /; FreeQ[{a, b, c, d, e, m, n, p}, x] && LinearQ[u, x]
  5367. Int[Times[Power[Plus[Optional[Pattern[a, Blank[]]], Times[Cosh[Plus[Optional[Pattern[c, Blank[]]], Times[Optional[Pattern[d, Blank[]]], Power[Pattern[u, Blank[]], Pattern[n, Blank[]]]]]], Optional[Pattern[b, Blank[]]]]], Optional[Pattern[p, Blank[]]]], Power[Times[Optional[Pattern[e, Blank[]]], Pattern[x, Blank[]]], Optional[Pattern[m, Blank[]]]]], Pattern[x, Blank[Symbol]]] := Unintegrable[(e*x)^m*(a + b*Cosh[c + d*u^n])^p, x] /; FreeQ[{a, b, c, d, e, m, n, p}, x] && LinearQ[u, x]
  5368. Int[Times[Power[Times[Pattern[e, Blank[]], Pattern[x, Blank[]]], Optional[Pattern[m, Blank[]]]], Power[Plus[Optional[Pattern[a, Blank[]]], Times[Optional[Pattern[b, Blank[]]], Sinh[Pattern[u, Blank[]]]]], Optional[Pattern[p, Blank[]]]]], Pattern[x, Blank[Symbol]]] := Int[(e*x)^m*(a + b*Sinh[ExpandToSum[u, x]])^p, x] /; FreeQ[{a, b, e, m, p}, x] && BinomialQ[u, x] && !BinomialMatchQ[u, x]
  5369. Int[Times[Power[Plus[Optional[Pattern[a, Blank[]]], Times[Cosh[Pattern[u, Blank[]]], Optional[Pattern[b, Blank[]]]]], Optional[Pattern[p, Blank[]]]], Power[Times[Pattern[e, Blank[]], Pattern[x, Blank[]]], Optional[Pattern[m, Blank[]]]]], Pattern[x, Blank[Symbol]]] := Int[(e*x)^m*(a + b*Cosh[ExpandToSum[u, x]])^p, x] /; FreeQ[{a, b, e, m, p}, x] && BinomialQ[u, x] && !BinomialMatchQ[u, x]
  5370. Int[Times[Cosh[Plus[Optional[Pattern[a, Blank[]]], Times[Optional[Pattern[b, Blank[]]], Power[Pattern[x, Blank[]], Optional[Pattern[n, Blank[]]]]]]], Power[Pattern[x, Blank[]], Optional[Pattern[m, Blank[]]]], Power[Sinh[Plus[Optional[Pattern[a, Blank[]]], Times[Optional[Pattern[b, Blank[]]], Power[Pattern[x, Blank[]], Pattern[n, Blank[]]]]]], Optional[Pattern[p, Blank[]]]]], Pattern[x, Blank[Symbol]]] := Simp[Sinh[a + b*x^n]^(p + 1)/(b*n*(p + 1)), x] /; FreeQ[{a, b, m, n, p}, x] && EqQ[m, n - 1] && NeQ[p, -1]
  5371. Int[Times[Power[Cosh[Plus[Optional[Pattern[a, Blank[]]], Times[Optional[Pattern[b, Blank[]]], Power[Pattern[x, Blank[]], Pattern[n, Blank[]]]]]], Optional[Pattern[p, Blank[]]]], Power[Pattern[x, Blank[]], Optional[Pattern[m, Blank[]]]], Sinh[Plus[Optional[Pattern[a, Blank[]]], Times[Optional[Pattern[b, Blank[]]], Power[Pattern[x, Blank[]], Optional[Pattern[n, Blank[]]]]]]]], Pattern[x, Blank[Symbol]]] := Simp[Cosh[a + b*x^n]^(p + 1)/(b*n*(p + 1)), x] /; FreeQ[{a, b, m, n, p}, x] && EqQ[m, n - 1] && NeQ[p, -1]
  5372. Int[Times[Cosh[Plus[Optional[Pattern[a, Blank[]]], Times[Optional[Pattern[b, Blank[]]], Power[Pattern[x, Blank[]], Optional[Pattern[n, Blank[]]]]]]], Power[Pattern[x, Blank[]], Optional[Pattern[m, Blank[]]]], Power[Sinh[Plus[Optional[Pattern[a, Blank[]]], Times[Optional[Pattern[b, Blank[]]], Power[Pattern[x, Blank[]], Optional[Pattern[n, Blank[]]]]]]], Optional[Pattern[p, Blank[]]]]], Pattern[x, Blank[Symbol]]] := Simp[(x^(m - n + 1)*Sinh[a + b*x^n]^(p + 1))/(b*n*(p + 1)), x] - Dist[(m - n + 1)/(b*n*(p + 1)), Int[x^(m - n)*Sinh[a + b*x^n]^(p + 1), x], x] /; FreeQ[{a, b, p}, x] && LtQ[0, n, m + 1] && NeQ[p, -1]
  5373. Int[Times[Power[Cosh[Plus[Optional[Pattern[a, Blank[]]], Times[Optional[Pattern[b, Blank[]]], Power[Pattern[x, Blank[]], Optional[Pattern[n, Blank[]]]]]]], Optional[Pattern[p, Blank[]]]], Power[Pattern[x, Blank[]], Optional[Pattern[m, Blank[]]]], Sinh[Plus[Optional[Pattern[a, Blank[]]], Times[Optional[Pattern[b, Blank[]]], Power[Pattern[x, Blank[]], Optional[Pattern[n, Blank[]]]]]]]], Pattern[x, Blank[Symbol]]] := Simp[(x^(m - n + 1)*Cosh[a + b*x^n]^(p + 1))/(b*n*(p + 1)), x] - Dist[(m - n + 1)/(b*n*(p + 1)), Int[x^(m - n)*Cosh[a + b*x^n]^(p + 1), x], x] /; FreeQ[{a, b, p}, x] && LtQ[0, n, m + 1] && NeQ[p, -1]
  5374. Int[Sinh[Plus[Optional[Pattern[a, Blank[]]], Times[Optional[Pattern[b, Blank[]]], Pattern[x, Blank[]]], Times[Optional[Pattern[c, Blank[]]], Power[Pattern[x, Blank[]], 2]]]], Pattern[x, Blank[Symbol]]] := Dist[1/2, Int[E^(a + b*x + c*x^2), x], x] - Dist[1/2, Int[E^(-a - b*x - c*x^2), x], x] /; FreeQ[{a, b, c}, x]
  5375. Int[Cosh[Plus[Optional[Pattern[a, Blank[]]], Times[Optional[Pattern[b, Blank[]]], Pattern[x, Blank[]]], Times[Optional[Pattern[c, Blank[]]], Power[Pattern[x, Blank[]], 2]]]], Pattern[x, Blank[Symbol]]] := Dist[1/2, Int[E^(a + b*x + c*x^2), x], x] + Dist[1/2, Int[E^(-a - b*x - c*x^2), x], x] /; FreeQ[{a, b, c}, x]
  5376. Int[Power[Sinh[Plus[Optional[Pattern[a, Blank[]]], Times[Optional[Pattern[b, Blank[]]], Pattern[x, Blank[]]], Times[Optional[Pattern[c, Blank[]]], Power[Pattern[x, Blank[]], 2]]]], Pattern[n, Blank[]]], Pattern[x, Blank[Symbol]]] := Int[ExpandTrigReduce[Sinh[a + b*x + c*x^2]^n, x], x] /; FreeQ[{a, b, c}, x] && IGtQ[n, 1]
  5377. Int[Power[Cosh[Plus[Optional[Pattern[a, Blank[]]], Times[Optional[Pattern[b, Blank[]]], Pattern[x, Blank[]]], Times[Optional[Pattern[c, Blank[]]], Power[Pattern[x, Blank[]], 2]]]], Pattern[n, Blank[]]], Pattern[x, Blank[Symbol]]] := Int[ExpandTrigReduce[Cosh[a + b*x + c*x^2]^n, x], x] /; FreeQ[{a, b, c}, x] && IGtQ[n, 1]
  5378. Int[Power[Sinh[Pattern[v, Blank[]]], Optional[Pattern[n, Blank[]]]], Pattern[x, Blank[Symbol]]] := Int[Sinh[ExpandToSum[v, x]]^n, x] /; IGtQ[n, 0] && QuadraticQ[v, x] && !QuadraticMatchQ[v, x]
  5379. Int[Power[Cosh[Pattern[v, Blank[]]], Optional[Pattern[n, Blank[]]]], Pattern[x, Blank[Symbol]]] := Int[Cosh[ExpandToSum[v, x]]^n, x] /; IGtQ[n, 0] && QuadraticQ[v, x] && !QuadraticMatchQ[v, x]
  5380. Int[Times[Plus[Optional[Pattern[d, Blank[]]], Times[Optional[Pattern[e, Blank[]]], Pattern[x, Blank[]]]], Sinh[Plus[Optional[Pattern[a, Blank[]]], Times[Optional[Pattern[b, Blank[]]], Pattern[x, Blank[]]], Times[Optional[Pattern[c, Blank[]]], Power[Pattern[x, Blank[]], 2]]]]], Pattern[x, Blank[Symbol]]] := Simp[(e*Cosh[a + b*x + c*x^2])/(2*c), x] /; FreeQ[{a, b, c, d, e}, x] && EqQ[b*e - 2*c*d, 0]
  5381. Int[Times[Cosh[Plus[Optional[Pattern[a, Blank[]]], Times[Optional[Pattern[b, Blank[]]], Pattern[x, Blank[]]], Times[Optional[Pattern[c, Blank[]]], Power[Pattern[x, Blank[]], 2]]]], Plus[Optional[Pattern[d, Blank[]]], Times[Optional[Pattern[e, Blank[]]], Pattern[x, Blank[]]]]], Pattern[x, Blank[Symbol]]] := Simp[(e*Sinh[a + b*x + c*x^2])/(2*c), x] /; FreeQ[{a, b, c, d, e}, x] && EqQ[b*e - 2*c*d, 0]
  5382. Int[Times[Plus[Optional[Pattern[d, Blank[]]], Times[Optional[Pattern[e, Blank[]]], Pattern[x, Blank[]]]], Sinh[Plus[Optional[Pattern[a, Blank[]]], Times[Optional[Pattern[b, Blank[]]], Pattern[x, Blank[]]], Times[Optional[Pattern[c, Blank[]]], Power[Pattern[x, Blank[]], 2]]]]], Pattern[x, Blank[Symbol]]] := Simp[(e*Cosh[a + b*x + c*x^2])/(2*c), x] - Dist[(b*e - 2*c*d)/(2*c), Int[Sinh[a + b*x + c*x^2], x], x] /; FreeQ[{a, b, c, d, e}, x] && NeQ[b*e - 2*c*d, 0]
  5383. Int[Times[Cosh[Plus[Optional[Pattern[a, Blank[]]], Times[Optional[Pattern[b, Blank[]]], Pattern[x, Blank[]]], Times[Optional[Pattern[c, Blank[]]], Power[Pattern[x, Blank[]], 2]]]], Plus[Optional[Pattern[d, Blank[]]], Times[Optional[Pattern[e, Blank[]]], Pattern[x, Blank[]]]]], Pattern[x, Blank[Symbol]]] := Simp[(e*Sinh[a + b*x + c*x^2])/(2*c), x] - Dist[(b*e - 2*c*d)/(2*c), Int[Cosh[a + b*x + c*x^2], x], x] /; FreeQ[{a, b, c, d, e}, x] && NeQ[b*e - 2*c*d, 0]
  5384. Int[Times[Power[Plus[Optional[Pattern[d, Blank[]]], Times[Optional[Pattern[e, Blank[]]], Pattern[x, Blank[]]]], Pattern[m, Blank[]]], Sinh[Plus[Optional[Pattern[a, Blank[]]], Times[Optional[Pattern[b, Blank[]]], Pattern[x, Blank[]]], Times[Optional[Pattern[c, Blank[]]], Power[Pattern[x, Blank[]], 2]]]]], Pattern[x, Blank[Symbol]]] := Simp[(e*(d + e*x)^(m - 1)*Cosh[a + b*x + c*x^2])/(2*c), x] - Dist[(e^2*(m - 1))/(2*c), Int[(d + e*x)^(m - 2)*Cosh[a + b*x + c*x^2], x], x] /; FreeQ[{a, b, c, d, e}, x] && GtQ[m, 1] && EqQ[b*e - 2*c*d, 0]
  5385. Int[Times[Cosh[Plus[Optional[Pattern[a, Blank[]]], Times[Optional[Pattern[b, Blank[]]], Pattern[x, Blank[]]], Times[Optional[Pattern[c, Blank[]]], Power[Pattern[x, Blank[]], 2]]]], Power[Plus[Optional[Pattern[d, Blank[]]], Times[Optional[Pattern[e, Blank[]]], Pattern[x, Blank[]]]], Pattern[m, Blank[]]]], Pattern[x, Blank[Symbol]]] := Simp[(e*(d + e*x)^(m - 1)*Sinh[a + b*x + c*x^2])/(2*c), x] - Dist[(e^2*(m - 1))/(2*c), Int[(d + e*x)^(m - 2)*Sinh[a + b*x + c*x^2], x], x] /; FreeQ[{a, b, c, d, e}, x] && GtQ[m, 1] && EqQ[b*e - 2*c*d, 0]
  5386. Int[Times[Power[Plus[Optional[Pattern[d, Blank[]]], Times[Optional[Pattern[e, Blank[]]], Pattern[x, Blank[]]]], Pattern[m, Blank[]]], Sinh[Plus[Optional[Pattern[a, Blank[]]], Times[Optional[Pattern[b, Blank[]]], Pattern[x, Blank[]]], Times[Optional[Pattern[c, Blank[]]], Power[Pattern[x, Blank[]], 2]]]]], Pattern[x, Blank[Symbol]]] := Simp[(e*(d + e*x)^(m - 1)*Cosh[a + b*x + c*x^2])/(2*c), x] + (-Dist[(e^2*(m - 1))/(2*c), Int[(d + e*x)^(m - 2)*Cosh[a + b*x + c*x^2], x], x] - Dist[(b*e - 2*c*d)/(2*c), Int[(d + e*x)^(m - 1)*Sinh[a + b*x + c*x^2], x], x]) /; FreeQ[{a, b, c, d, e}, x] && GtQ[m, 1] && NeQ[b*e - 2*c*d, 0]
  5387. Int[Times[Cosh[Plus[Optional[Pattern[a, Blank[]]], Times[Optional[Pattern[b, Blank[]]], Pattern[x, Blank[]]], Times[Optional[Pattern[c, Blank[]]], Power[Pattern[x, Blank[]], 2]]]], Power[Plus[Optional[Pattern[d, Blank[]]], Times[Optional[Pattern[e, Blank[]]], Pattern[x, Blank[]]]], Pattern[m, Blank[]]]], Pattern[x, Blank[Symbol]]] := Simp[(e*(d + e*x)^(m - 1)*Sinh[a + b*x + c*x^2])/(2*c), x] + (-Dist[(e^2*(m - 1))/(2*c), Int[(d + e*x)^(m - 2)*Sinh[a + b*x + c*x^2], x], x] - Dist[(b*e - 2*c*d)/(2*c), Int[(d + e*x)^(m - 1)*Cosh[a + b*x + c*x^2], x], x]) /; FreeQ[{a, b, c, d, e}, x] && GtQ[m, 1] && NeQ[b*e - 2*c*d, 0]
  5388. Int[Times[Power[Plus[Optional[Pattern[d, Blank[]]], Times[Optional[Pattern[e, Blank[]]], Pattern[x, Blank[]]]], Pattern[m, Blank[]]], Sinh[Plus[Optional[Pattern[a, Blank[]]], Times[Optional[Pattern[b, Blank[]]], Pattern[x, Blank[]]], Times[Optional[Pattern[c, Blank[]]], Power[Pattern[x, Blank[]], 2]]]]], Pattern[x, Blank[Symbol]]] := Simp[((d + e*x)^(m + 1)*Sinh[a + b*x + c*x^2])/(e*(m + 1)), x] - Dist[(2*c)/(e^2*(m + 1)), Int[(d + e*x)^(m + 2)*Cosh[a + b*x + c*x^2], x], x] /; FreeQ[{a, b, c, d, e}, x] && LtQ[m, -1] && EqQ[b*e - 2*c*d, 0]
  5389. Int[Times[Cosh[Plus[Optional[Pattern[a, Blank[]]], Times[Optional[Pattern[b, Blank[]]], Pattern[x, Blank[]]], Times[Optional[Pattern[c, Blank[]]], Power[Pattern[x, Blank[]], 2]]]], Power[Plus[Optional[Pattern[d, Blank[]]], Times[Optional[Pattern[e, Blank[]]], Pattern[x, Blank[]]]], Pattern[m, Blank[]]]], Pattern[x, Blank[Symbol]]] := Simp[((d + e*x)^(m + 1)*Cosh[a + b*x + c*x^2])/(e*(m + 1)), x] - Dist[(2*c)/(e^2*(m + 1)), Int[(d + e*x)^(m + 2)*Sinh[a + b*x + c*x^2], x], x] /; FreeQ[{a, b, c, d, e}, x] && LtQ[m, -1] && EqQ[b*e - 2*c*d, 0]
  5390. Int[Times[Power[Plus[Optional[Pattern[d, Blank[]]], Times[Optional[Pattern[e, Blank[]]], Pattern[x, Blank[]]]], Pattern[m, Blank[]]], Sinh[Plus[Optional[Pattern[a, Blank[]]], Times[Optional[Pattern[b, Blank[]]], Pattern[x, Blank[]]], Times[Optional[Pattern[c, Blank[]]], Power[Pattern[x, Blank[]], 2]]]]], Pattern[x, Blank[Symbol]]] := Simp[((d + e*x)^(m + 1)*Sinh[a + b*x + c*x^2])/(e*(m + 1)), x] + (-Dist[(2*c)/(e^2*(m + 1)), Int[(d + e*x)^(m + 2)*Cosh[a + b*x + c*x^2], x], x] - Dist[(b*e - 2*c*d)/(e^2*(m + 1)), Int[(d + e*x)^(m + 1)*Cosh[a + b*x + c*x^2], x], x]) /; FreeQ[{a, b, c, d, e}, x] && LtQ[m, -1] && NeQ[b*e - 2*c*d, 0]
  5391. Int[Times[Cosh[Plus[Optional[Pattern[a, Blank[]]], Times[Optional[Pattern[b, Blank[]]], Pattern[x, Blank[]]], Times[Optional[Pattern[c, Blank[]]], Power[Pattern[x, Blank[]], 2]]]], Power[Plus[Optional[Pattern[d, Blank[]]], Times[Optional[Pattern[e, Blank[]]], Pattern[x, Blank[]]]], Pattern[m, Blank[]]]], Pattern[x, Blank[Symbol]]] := Simp[((d + e*x)^(m + 1)*Cosh[a + b*x + c*x^2])/(e*(m + 1)), x] + (-Dist[(2*c)/(e^2*(m + 1)), Int[(d + e*x)^(m + 2)*Sinh[a + b*x + c*x^2], x], x] - Dist[(b*e - 2*c*d)/(e^2*(m + 1)), Int[(d + e*x)^(m + 1)*Sinh[a + b*x + c*x^2], x], x]) /; FreeQ[{a, b, c, d, e}, x] && LtQ[m, -1] && NeQ[b*e - 2*c*d, 0]
  5392. Int[Times[Power[Plus[Optional[Pattern[d, Blank[]]], Times[Optional[Pattern[e, Blank[]]], Pattern[x, Blank[]]]], Optional[Pattern[m, Blank[]]]], Sinh[Plus[Optional[Pattern[a, Blank[]]], Times[Optional[Pattern[b, Blank[]]], Pattern[x, Blank[]]], Times[Optional[Pattern[c, Blank[]]], Power[Pattern[x, Blank[]], 2]]]]], Pattern[x, Blank[Symbol]]] := Unintegrable[(d + e*x)^m*Sinh[a + b*x + c*x^2], x] /; FreeQ[{a, b, c, d, e, m}, x]
  5393. Int[Times[Cosh[Plus[Optional[Pattern[a, Blank[]]], Times[Optional[Pattern[b, Blank[]]], Pattern[x, Blank[]]], Times[Optional[Pattern[c, Blank[]]], Power[Pattern[x, Blank[]], 2]]]], Power[Plus[Optional[Pattern[d, Blank[]]], Times[Optional[Pattern[e, Blank[]]], Pattern[x, Blank[]]]], Optional[Pattern[m, Blank[]]]]], Pattern[x, Blank[Symbol]]] := Unintegrable[(d + e*x)^m*Cosh[a + b*x + c*x^2], x] /; FreeQ[{a, b, c, d, e, m}, x]
  5394. Int[Times[Power[Plus[Optional[Pattern[d, Blank[]]], Times[Optional[Pattern[e, Blank[]]], Pattern[x, Blank[]]]], Optional[Pattern[m, Blank[]]]], Power[Sinh[Plus[Optional[Pattern[a, Blank[]]], Times[Optional[Pattern[b, Blank[]]], Pattern[x, Blank[]]], Times[Optional[Pattern[c, Blank[]]], Power[Pattern[x, Blank[]], 2]]]], Pattern[n, Blank[]]]], Pattern[x, Blank[Symbol]]] := Int[ExpandTrigReduce[(d + e*x)^m, Sinh[a + b*x + c*x^2]^n, x], x] /; FreeQ[{a, b, c, d, e, m}, x] && IGtQ[n, 1]
  5395. Int[Times[Power[Cosh[Plus[Optional[Pattern[a, Blank[]]], Times[Optional[Pattern[b, Blank[]]], Pattern[x, Blank[]]], Times[Optional[Pattern[c, Blank[]]], Power[Pattern[x, Blank[]], 2]]]], Pattern[n, Blank[]]], Power[Plus[Optional[Pattern[d, Blank[]]], Times[Optional[Pattern[e, Blank[]]], Pattern[x, Blank[]]]], Optional[Pattern[m, Blank[]]]]], Pattern[x, Blank[Symbol]]] := Int[ExpandTrigReduce[(d + e*x)^m, Cosh[a + b*x + c*x^2]^n, x], x] /; FreeQ[{a, b, c, d, e, m}, x] && IGtQ[n, 1]
  5396. Int[Times[Power[Pattern[u, Blank[]], Optional[Pattern[m, Blank[]]]], Power[Sinh[Pattern[v, Blank[]]], Optional[Pattern[n, Blank[]]]]], Pattern[x, Blank[Symbol]]] := Int[ExpandToSum[u, x]^m*Sinh[ExpandToSum[v, x]]^n, x] /; FreeQ[m, x] && IGtQ[n, 0] && LinearQ[u, x] && QuadraticQ[v, x] && !(LinearMatchQ[u, x] && QuadraticMatchQ[v, x])
  5397. Int[Times[Power[Cosh[Pattern[v, Blank[]]], Optional[Pattern[n, Blank[]]]], Power[Pattern[u, Blank[]], Optional[Pattern[m, Blank[]]]]], Pattern[x, Blank[Symbol]]] := Int[ExpandToSum[u, x]^m*Cosh[ExpandToSum[v, x]]^n, x] /; FreeQ[m, x] && IGtQ[n, 0] && LinearQ[u, x] && QuadraticQ[v, x] && !(LinearMatchQ[u, x] && QuadraticMatchQ[v, x])
  5398. Int[Times[Power[Pattern[u, Blank[]], Optional[Pattern[m, Blank[]]]], Power[Plus[Optional[Pattern[a, Blank[]]], Times[Optional[Pattern[b, Blank[]]], Tanh[Pattern[v, Blank[]]]]], Optional[Pattern[n, Blank[]]]]], Pattern[x, Blank[Symbol]]] := Int[ExpandToSum[u, x]^m*(a + b*Tanh[ExpandToSum[v, x]])^n, x] /; FreeQ[{a, b, m, n}, x] && LinearQ[{u, v}, x] && !LinearMatchQ[{u, v}, x]
  5399. Int[Times[Power[Plus[Optional[Pattern[a, Blank[]]], Times[Coth[Pattern[v, Blank[]]], Optional[Pattern[b, Blank[]]]]], Optional[Pattern[n, Blank[]]]], Power[Pattern[u, Blank[]], Optional[Pattern[m, Blank[]]]]], Pattern[x, Blank[Symbol]]] := Int[ExpandToSum[u, x]^m*(a + b*Coth[ExpandToSum[v, x]])^n, x] /; FreeQ[{a, b, m, n}, x] && LinearQ[{u, v}, x] && !LinearMatchQ[{u, v}, x]
  5400. Int[Power[Plus[Optional[Pattern[a, Blank[]]], Times[Optional[Pattern[b, Blank[]]], Tanh[Plus[Optional[Pattern[c, Blank[]]], Times[Optional[Pattern[d, Blank[]]], Power[Pattern[x, Blank[]], Pattern[n, Blank[]]]]]]]], Optional[Pattern[p, Blank[]]]], Pattern[x, Blank[Symbol]]] := Dist[1/n, Subst[Int[x^(1/n - 1)*(a + b*Tanh[c + d*x])^p, x], x, x^n], x] /; FreeQ[{a, b, c, d, p}, x] && IGtQ[1/n, 0] && IntegerQ[p]
  5401. Int[Power[Plus[Optional[Pattern[a, Blank[]]], Times[Coth[Plus[Optional[Pattern[c, Blank[]]], Times[Optional[Pattern[d, Blank[]]], Power[Pattern[x, Blank[]], Pattern[n, Blank[]]]]]], Optional[Pattern[b, Blank[]]]]], Optional[Pattern[p, Blank[]]]], Pattern[x, Blank[Symbol]]] := Dist[1/n, Subst[Int[x^(1/n - 1)*(a + b*Coth[c + d*x])^p, x], x, x^n], x] /; FreeQ[{a, b, c, d, p}, x] && IGtQ[1/n, 0] && IntegerQ[p]
  5402. Int[Power[Plus[Optional[Pattern[a, Blank[]]], Times[Optional[Pattern[b, Blank[]]], Tanh[Plus[Optional[Pattern[c, Blank[]]], Times[Optional[Pattern[d, Blank[]]], Power[Pattern[x, Blank[]], Pattern[n, Blank[]]]]]]]], Optional[Pattern[p, Blank[]]]], Pattern[x, Blank[Symbol]]] := Simp[Integral[(a + b*Tanh[c + d*x^n])^p, x], x] /; FreeQ[{a, b, c, d, n, p}, x]
  5403. Int[Power[Plus[Optional[Pattern[a, Blank[]]], Times[Coth[Plus[Optional[Pattern[c, Blank[]]], Times[Optional[Pattern[d, Blank[]]], Power[Pattern[x, Blank[]], Pattern[n, Blank[]]]]]], Optional[Pattern[b, Blank[]]]]], Optional[Pattern[p, Blank[]]]], Pattern[x, Blank[Symbol]]] := Simp[Integral[(a + b*Coth[c + d*x^n])^p, x], x] /; FreeQ[{a, b, c, d, n, p}, x]
  5404. Int[Power[Plus[Optional[Pattern[a, Blank[]]], Times[Optional[Pattern[b, Blank[]]], Tanh[Plus[Optional[Pattern[c, Blank[]]], Times[Optional[Pattern[d, Blank[]]], Power[Pattern[u, Blank[]], Pattern[n, Blank[]]]]]]]], Optional[Pattern[p, Blank[]]]], Pattern[x, Blank[Symbol]]] := Dist[1/Coefficient[u, x, 1], Subst[Int[(a + b*Tanh[c + d*x^n])^p, x], x, u], x] /; FreeQ[{a, b, c, d, n, p}, x] && LinearQ[u, x] && NeQ[u, x]
  5405. Int[Power[Plus[Optional[Pattern[a, Blank[]]], Times[Coth[Plus[Optional[Pattern[c, Blank[]]], Times[Optional[Pattern[d, Blank[]]], Power[Pattern[u, Blank[]], Pattern[n, Blank[]]]]]], Optional[Pattern[b, Blank[]]]]], Optional[Pattern[p, Blank[]]]], Pattern[x, Blank[Symbol]]] := Dist[1/Coefficient[u, x, 1], Subst[Int[(a + b*Coth[c + d*x^n])^p, x], x, u], x] /; FreeQ[{a, b, c, d, n, p}, x] && LinearQ[u, x] && NeQ[u, x]
  5406. Int[Power[Plus[Optional[Pattern[a, Blank[]]], Times[Optional[Pattern[b, Blank[]]], Tanh[Pattern[u, Blank[]]]]], Optional[Pattern[p, Blank[]]]], Pattern[x, Blank[Symbol]]] := Int[(a + b*Tanh[ExpandToSum[u, x]])^p, x] /; FreeQ[{a, b, p}, x] && BinomialQ[u, x] && !BinomialMatchQ[u, x]
  5407. Int[Power[Plus[Optional[Pattern[a, Blank[]]], Times[Coth[Pattern[u, Blank[]]], Optional[Pattern[b, Blank[]]]]], Optional[Pattern[p, Blank[]]]], Pattern[x, Blank[Symbol]]] := Int[(a + b*Coth[ExpandToSum[u, x]])^p, x] /; FreeQ[{a, b, p}, x] && BinomialQ[u, x] && !BinomialMatchQ[u, x]
  5408. Int[Times[Power[Pattern[x, Blank[]], Optional[Pattern[m, Blank[]]]], Power[Plus[Optional[Pattern[a, Blank[]]], Times[Optional[Pattern[b, Blank[]]], Tanh[Plus[Optional[Pattern[c, Blank[]]], Times[Optional[Pattern[d, Blank[]]], Power[Pattern[x, Blank[]], Pattern[n, Blank[]]]]]]]], Optional[Pattern[p, Blank[]]]]], Pattern[x, Blank[Symbol]]] := Dist[1/n, Subst[Int[x^(Simplify[(m + 1)/n] - 1)*(a + b*Tanh[c + d*x])^p, x], x, x^n], x] /; FreeQ[{a, b, c, d, m, n, p}, x] && IGtQ[Simplify[(m + 1)/n], 0] && IntegerQ[p]
  5409. Int[Times[Power[Plus[Optional[Pattern[a, Blank[]]], Times[Coth[Plus[Optional[Pattern[c, Blank[]]], Times[Optional[Pattern[d, Blank[]]], Power[Pattern[x, Blank[]], Pattern[n, Blank[]]]]]], Optional[Pattern[b, Blank[]]]]], Optional[Pattern[p, Blank[]]]], Power[Pattern[x, Blank[]], Optional[Pattern[m, Blank[]]]]], Pattern[x, Blank[Symbol]]] := Dist[1/n, Subst[Int[x^(Simplify[(m + 1)/n] - 1)*(a + b*Coth[c + d*x])^p, x], x, x^n], x] /; FreeQ[{a, b, c, d, m, n, p}, x] && IGtQ[Simplify[(m + 1)/n], 0] && IntegerQ[p]
  5410. Int[Times[Power[Pattern[x, Blank[]], Optional[Pattern[m, Blank[]]]], Power[Tanh[Plus[Optional[Pattern[c, Blank[]]], Times[Optional[Pattern[d, Blank[]]], Power[Pattern[x, Blank[]], Pattern[n, Blank[]]]]]], 2]], Pattern[x, Blank[Symbol]]] := -Simp[(x^(m - n + 1)*Tanh[c + d*x^n])/(d*n), x] + (Dist[(m - n + 1)/(d*n), Int[x^(m - n)*Tanh[c + d*x^n], x], x] + Int[x^m, x]) /; FreeQ[{c, d, m, n}, x]
  5411. Int[Times[Power[Coth[Plus[Optional[Pattern[c, Blank[]]], Times[Optional[Pattern[d, Blank[]]], Power[Pattern[x, Blank[]], Pattern[n, Blank[]]]]]], 2], Power[Pattern[x, Blank[]], Optional[Pattern[m, Blank[]]]]], Pattern[x, Blank[Symbol]]] := -Simp[(x^(m - n + 1)*Coth[c + d*x^n])/(d*n), x] + (Dist[(m - n + 1)/(d*n), Int[x^(m - n)*Coth[c + d*x^n], x], x] + Int[x^m, x]) /; FreeQ[{c, d, m, n}, x]
  5412. Int[Times[Power[Pattern[x, Blank[]], Optional[Pattern[m, Blank[]]]], Power[Plus[Optional[Pattern[a, Blank[]]], Times[Optional[Pattern[b, Blank[]]], Tanh[Plus[Optional[Pattern[c, Blank[]]], Times[Optional[Pattern[d, Blank[]]], Power[Pattern[x, Blank[]], Pattern[n, Blank[]]]]]]]], Optional[Pattern[p, Blank[]]]]], Pattern[x, Blank[Symbol]]] := Simp[Integral[x^m*(a + b*Tanh[c + d*x^n])^p, x], x] /; FreeQ[{a, b, c, d, m, n, p}, x]
  5413. Int[Times[Power[Plus[Optional[Pattern[a, Blank[]]], Times[Coth[Plus[Optional[Pattern[c, Blank[]]], Times[Optional[Pattern[d, Blank[]]], Power[Pattern[x, Blank[]], Pattern[n, Blank[]]]]]], Optional[Pattern[b, Blank[]]]]], Optional[Pattern[p, Blank[]]]], Power[Pattern[x, Blank[]], Optional[Pattern[m, Blank[]]]]], Pattern[x, Blank[Symbol]]] := Simp[Integral[x^m*(a + b*Coth[c + d*x^n])^p, x], x] /; FreeQ[{a, b, c, d, m, n, p}, x]
  5414. Int[Times[Power[Times[Pattern[e, Blank[]], Pattern[x, Blank[]]], Optional[Pattern[m, Blank[]]]], Power[Plus[Optional[Pattern[a, Blank[]]], Times[Optional[Pattern[b, Blank[]]], Tanh[Plus[Optional[Pattern[c, Blank[]]], Times[Optional[Pattern[d, Blank[]]], Power[Pattern[x, Blank[]], Pattern[n, Blank[]]]]]]]], Optional[Pattern[p, Blank[]]]]], Pattern[x, Blank[Symbol]]] := Dist[(e^IntPart[m]*(e*x)^FracPart[m])/x^FracPart[m], Int[x^m*(a + b*Tanh[c + d*x^n])^p, x], x] /; FreeQ[{a, b, c, d, e, m, n, p}, x]
  5415. Int[Times[Power[Plus[Optional[Pattern[a, Blank[]]], Times[Coth[Plus[Optional[Pattern[c, Blank[]]], Times[Optional[Pattern[d, Blank[]]], Power[Pattern[x, Blank[]], Pattern[n, Blank[]]]]]], Optional[Pattern[b, Blank[]]]]], Optional[Pattern[p, Blank[]]]], Power[Times[Pattern[e, Blank[]], Pattern[x, Blank[]]], Optional[Pattern[m, Blank[]]]]], Pattern[x, Blank[Symbol]]] := Dist[(e^IntPart[m]*(e*x)^FracPart[m])/x^FracPart[m], Int[x^m*(a + b*Coth[c + d*x^n])^p, x], x] /; FreeQ[{a, b, c, d, e, m, n, p}, x]
  5416. Int[Times[Power[Times[Pattern[e, Blank[]], Pattern[x, Blank[]]], Optional[Pattern[m, Blank[]]]], Power[Plus[Optional[Pattern[a, Blank[]]], Times[Optional[Pattern[b, Blank[]]], Tanh[Pattern[u, Blank[]]]]], Optional[Pattern[p, Blank[]]]]], Pattern[x, Blank[Symbol]]] := Int[(e*x)^m*(a + b*Tanh[ExpandToSum[u, x]])^p, x] /; FreeQ[{a, b, e, m, p}, x] && BinomialQ[u, x] && !BinomialMatchQ[u, x]
  5417. Int[Times[Power[Plus[Optional[Pattern[a, Blank[]]], Times[Coth[Pattern[u, Blank[]]], Optional[Pattern[b, Blank[]]]]], Optional[Pattern[p, Blank[]]]], Power[Times[Pattern[e, Blank[]], Pattern[x, Blank[]]], Optional[Pattern[m, Blank[]]]]], Pattern[x, Blank[Symbol]]] := Int[(e*x)^m*(a + b*Coth[ExpandToSum[u, x]])^p, x] /; FreeQ[{a, b, e, m, p}, x] && BinomialQ[u, x] && !BinomialMatchQ[u, x]
  5418. Int[Times[Power[Pattern[x, Blank[]], Optional[Pattern[m, Blank[]]]], Power[Sech[Plus[Optional[Pattern[a, Blank[]]], Times[Optional[Pattern[b, Blank[]]], Power[Pattern[x, Blank[]], Optional[Pattern[n, Blank[]]]]]]], Optional[Pattern[p, Blank[]]]], Power[Tanh[Plus[Optional[Pattern[a, Blank[]]], Times[Optional[Pattern[b, Blank[]]], Power[Pattern[x, Blank[]], Optional[Pattern[n, Blank[]]]]]]], Optional[Pattern[q, Blank[]]]]], Pattern[x, Blank[Symbol]]] := -Simp[(x^(m - n + 1)*Sech[a + b*x^n]^p)/(b*n*p), x] + Dist[(m - n + 1)/(b*n*p), Int[x^(m - n)*Sech[a + b*x^n]^p, x], x] /; FreeQ[{a, b, p}, x] && RationalQ[m] && IntegerQ[n] && GeQ[m - n, 0] && EqQ[q, 1]
  5419. Int[Times[Power[Coth[Plus[Optional[Pattern[a, Blank[]]], Times[Optional[Pattern[b, Blank[]]], Power[Pattern[x, Blank[]], Optional[Pattern[n, Blank[]]]]]]], Optional[Pattern[q, Blank[]]]], Power[Csch[Plus[Optional[Pattern[a, Blank[]]], Times[Optional[Pattern[b, Blank[]]], Power[Pattern[x, Blank[]], Optional[Pattern[n, Blank[]]]]]]], Optional[Pattern[p, Blank[]]]], Power[Pattern[x, Blank[]], Optional[Pattern[m, Blank[]]]]], Pattern[x, Blank[Symbol]]] := -Simp[(x^(m - n + 1)*Csch[a + b*x^n]^p)/(b*n*p), x] + Dist[(m - n + 1)/(b*n*p), Int[x^(m - n)*Csch[a + b*x^n]^p, x], x] /; FreeQ[{a, b, p}, x] && RationalQ[m] && IntegerQ[n] && GeQ[m - n, 0] && EqQ[q, 1]
  5420. Int[Power[Tanh[Plus[Optional[Pattern[a, Blank[]]], Times[Optional[Pattern[b, Blank[]]], Pattern[x, Blank[]]], Times[Optional[Pattern[c, Blank[]]], Power[Pattern[x, Blank[]], 2]]]], Optional[Pattern[n, Blank[]]]], Pattern[x, Blank[Symbol]]] := Simp[Integral[Tanh[a + b*x + c*x^2]^n, x], x] /; FreeQ[{a, b, c, n}, x]
  5421. Int[Power[Coth[Plus[Optional[Pattern[a, Blank[]]], Times[Optional[Pattern[b, Blank[]]], Pattern[x, Blank[]]], Times[Optional[Pattern[c, Blank[]]], Power[Pattern[x, Blank[]], 2]]]], Optional[Pattern[n, Blank[]]]], Pattern[x, Blank[Symbol]]] := Simp[Integral[Coth[a + b*x + c*x^2]^n, x], x] /; FreeQ[{a, b, c, n}, x]
  5422. Int[Times[Plus[Optional[Pattern[d, Blank[]]], Times[Optional[Pattern[e, Blank[]]], Pattern[x, Blank[]]]], Tanh[Plus[Optional[Pattern[a, Blank[]]], Times[Optional[Pattern[b, Blank[]]], Pattern[x, Blank[]]], Times[Optional[Pattern[c, Blank[]]], Power[Pattern[x, Blank[]], 2]]]]], Pattern[x, Blank[Symbol]]] := Simp[(e*Log[Cosh[a + b*x + c*x^2]])/(2*c), x] + Dist[(2*c*d - b*e)/(2*c), Int[Tanh[a + b*x + c*x^2], x], x] /; FreeQ[{a, b, c, d, e}, x]
  5423. Int[Times[Coth[Plus[Optional[Pattern[a, Blank[]]], Times[Optional[Pattern[b, Blank[]]], Pattern[x, Blank[]]], Times[Optional[Pattern[c, Blank[]]], Power[Pattern[x, Blank[]], 2]]]], Plus[Optional[Pattern[d, Blank[]]], Times[Optional[Pattern[e, Blank[]]], Pattern[x, Blank[]]]]], Pattern[x, Blank[Symbol]]] := Simp[(e*Log[Sinh[a + b*x + c*x^2]])/(2*c), x] + Dist[(2*c*d - b*e)/(2*c), Int[Coth[a + b*x + c*x^2], x], x] /; FreeQ[{a, b, c, d, e}, x]
  5424. Int[Times[Power[Plus[Optional[Pattern[d, Blank[]]], Times[Optional[Pattern[e, Blank[]]], Pattern[x, Blank[]]]], Optional[Pattern[m, Blank[]]]], Power[Tanh[Plus[Optional[Pattern[a, Blank[]]], Times[Optional[Pattern[b, Blank[]]], Pattern[x, Blank[]]], Times[Optional[Pattern[c, Blank[]]], Power[Pattern[x, Blank[]], 2]]]], Optional[Pattern[n, Blank[]]]]], Pattern[x, Blank[Symbol]]] := Simp[Integral[(d + e*x)^m*Tanh[a + b*x + c*x^2]^n, x], x] /; FreeQ[{a, b, c, d, e, m, n}, x]
  5425. Int[Times[Power[Coth[Plus[Optional[Pattern[a, Blank[]]], Times[Optional[Pattern[b, Blank[]]], Pattern[x, Blank[]]], Times[Optional[Pattern[c, Blank[]]], Power[Pattern[x, Blank[]], 2]]]], Optional[Pattern[n, Blank[]]]], Power[Plus[Optional[Pattern[d, Blank[]]], Times[Optional[Pattern[e, Blank[]]], Pattern[x, Blank[]]]], Optional[Pattern[m, Blank[]]]]], Pattern[x, Blank[Symbol]]] := Simp[Integral[(d + e*x)^m*Coth[a + b*x + c*x^2]^n, x], x] /; FreeQ[{a, b, c, d, e, m, n}, x]
  5426. Int[Times[Power[Pattern[u, Blank[]], Optional[Pattern[m, Blank[]]]], Power[Sech[Pattern[v, Blank[]]], Optional[Pattern[n, Blank[]]]]], Pattern[x, Blank[Symbol]]] := Int[ExpandToSum[u, x]^m*Sech[ExpandToSum[v, x]]^n, x] /; FreeQ[{m, n}, x] && LinearQ[{u, v}, x] && !LinearMatchQ[{u, v}, x]
  5427. Int[Times[Power[Csch[Pattern[v, Blank[]]], Optional[Pattern[n, Blank[]]]], Power[Pattern[u, Blank[]], Optional[Pattern[m, Blank[]]]]], Pattern[x, Blank[Symbol]]] := Int[ExpandToSum[u, x]^m*Csch[ExpandToSum[v, x]]^n, x] /; FreeQ[{m, n}, x] && LinearQ[{u, v}, x] && !LinearMatchQ[{u, v}, x]
  5428. Int[Power[Plus[Optional[Pattern[a, Blank[]]], Times[Optional[Pattern[b, Blank[]]], Sech[Plus[Optional[Pattern[c, Blank[]]], Times[Optional[Pattern[d, Blank[]]], Power[Pattern[x, Blank[]], Pattern[n, Blank[]]]]]]]], Optional[Pattern[p, Blank[]]]], Pattern[x, Blank[Symbol]]] := Dist[1/n, Subst[Int[x^(1/n - 1)*(a + b*Sech[c + d*x])^p, x], x, x^n], x] /; FreeQ[{a, b, c, d, p}, x] && IGtQ[1/n, 0] && IntegerQ[p]
  5429. Int[Power[Plus[Optional[Pattern[a, Blank[]]], Times[Csch[Plus[Optional[Pattern[c, Blank[]]], Times[Optional[Pattern[d, Blank[]]], Power[Pattern[x, Blank[]], Pattern[n, Blank[]]]]]], Optional[Pattern[b, Blank[]]]]], Optional[Pattern[p, Blank[]]]], Pattern[x, Blank[Symbol]]] := Dist[1/n, Subst[Int[x^(1/n - 1)*(a + b*Csch[c + d*x])^p, x], x, x^n], x] /; FreeQ[{a, b, c, d, p}, x] && IGtQ[1/n, 0] && IntegerQ[p]
  5430. Int[Power[Plus[Optional[Pattern[a, Blank[]]], Times[Optional[Pattern[b, Blank[]]], Sech[Plus[Optional[Pattern[c, Blank[]]], Times[Optional[Pattern[d, Blank[]]], Power[Pattern[x, Blank[]], Pattern[n, Blank[]]]]]]]], Optional[Pattern[p, Blank[]]]], Pattern[x, Blank[Symbol]]] := Unintegrable[(a + b*Sech[c + d*x^n])^p, x] /; FreeQ[{a, b, c, d, n, p}, x]
  5431. Int[Power[Plus[Optional[Pattern[a, Blank[]]], Times[Csch[Plus[Optional[Pattern[c, Blank[]]], Times[Optional[Pattern[d, Blank[]]], Power[Pattern[x, Blank[]], Pattern[n, Blank[]]]]]], Optional[Pattern[b, Blank[]]]]], Optional[Pattern[p, Blank[]]]], Pattern[x, Blank[Symbol]]] := Unintegrable[(a + b*Csch[c + d*x^n])^p, x] /; FreeQ[{a, b, c, d, n, p}, x]
  5432. Int[Power[Plus[Optional[Pattern[a, Blank[]]], Times[Optional[Pattern[b, Blank[]]], Sech[Plus[Optional[Pattern[c, Blank[]]], Times[Optional[Pattern[d, Blank[]]], Power[Pattern[u, Blank[]], Pattern[n, Blank[]]]]]]]], Optional[Pattern[p, Blank[]]]], Pattern[x, Blank[Symbol]]] := Dist[1/Coefficient[u, x, 1], Subst[Int[(a + b*Sech[c + d*x^n])^p, x], x, u], x] /; FreeQ[{a, b, c, d, n, p}, x] && LinearQ[u, x] && NeQ[u, x]
  5433. Int[Power[Plus[Optional[Pattern[a, Blank[]]], Times[Csch[Plus[Optional[Pattern[c, Blank[]]], Times[Optional[Pattern[d, Blank[]]], Power[Pattern[u, Blank[]], Pattern[n, Blank[]]]]]], Optional[Pattern[b, Blank[]]]]], Optional[Pattern[p, Blank[]]]], Pattern[x, Blank[Symbol]]] := Dist[1/Coefficient[u, x, 1], Subst[Int[(a + b*Csch[c + d*x^n])^p, x], x, u], x] /; FreeQ[{a, b, c, d, n, p}, x] && LinearQ[u, x] && NeQ[u, x]
  5434. Int[Power[Plus[Optional[Pattern[a, Blank[]]], Times[Optional[Pattern[b, Blank[]]], Sech[Pattern[u, Blank[]]]]], Optional[Pattern[p, Blank[]]]], Pattern[x, Blank[Symbol]]] := Int[(a + b*Sech[ExpandToSum[u, x]])^p, x] /; FreeQ[{a, b, p}, x] && BinomialQ[u, x] && !BinomialMatchQ[u, x]
  5435. Int[Power[Plus[Optional[Pattern[a, Blank[]]], Times[Csch[Pattern[u, Blank[]]], Optional[Pattern[b, Blank[]]]]], Optional[Pattern[p, Blank[]]]], Pattern[x, Blank[Symbol]]] := Int[(a + b*Csch[ExpandToSum[u, x]])^p, x] /; FreeQ[{a, b, p}, x] && BinomialQ[u, x] && !BinomialMatchQ[u, x]
  5436. Int[Times[Power[Pattern[x, Blank[]], Optional[Pattern[m, Blank[]]]], Power[Plus[Optional[Pattern[a, Blank[]]], Times[Optional[Pattern[b, Blank[]]], Sech[Plus[Optional[Pattern[c, Blank[]]], Times[Optional[Pattern[d, Blank[]]], Power[Pattern[x, Blank[]], Pattern[n, Blank[]]]]]]]], Optional[Pattern[p, Blank[]]]]], Pattern[x, Blank[Symbol]]] := Dist[1/n, Subst[Int[x^(Simplify[(m + 1)/n] - 1)*(a + b*Sech[c + d*x])^p, x], x, x^n], x] /; FreeQ[{a, b, c, d, m, n, p}, x] && IGtQ[Simplify[(m + 1)/n], 0] && IntegerQ[p]
  5437. Int[Times[Power[Plus[Optional[Pattern[a, Blank[]]], Times[Csch[Plus[Optional[Pattern[c, Blank[]]], Times[Optional[Pattern[d, Blank[]]], Power[Pattern[x, Blank[]], Pattern[n, Blank[]]]]]], Optional[Pattern[b, Blank[]]]]], Optional[Pattern[p, Blank[]]]], Power[Pattern[x, Blank[]], Optional[Pattern[m, Blank[]]]]], Pattern[x, Blank[Symbol]]] := Dist[1/n, Subst[Int[x^(Simplify[(m + 1)/n] - 1)*(a + b*Csch[c + d*x])^p, x], x, x^n], x] /; FreeQ[{a, b, c, d, m, n, p}, x] && IGtQ[Simplify[(m + 1)/n], 0] && IntegerQ[p]
  5438. Int[Times[Power[Pattern[x, Blank[]], Optional[Pattern[m, Blank[]]]], Power[Plus[Optional[Pattern[a, Blank[]]], Times[Optional[Pattern[b, Blank[]]], Sech[Plus[Optional[Pattern[c, Blank[]]], Times[Optional[Pattern[d, Blank[]]], Power[Pattern[x, Blank[]], Pattern[n, Blank[]]]]]]]], Optional[Pattern[p, Blank[]]]]], Pattern[x, Blank[Symbol]]] := Unintegrable[x^m*(a + b*Sech[c + d*x^n])^p, x] /; FreeQ[{a, b, c, d, m, n, p}, x]
  5439. Int[Times[Power[Plus[Optional[Pattern[a, Blank[]]], Times[Csch[Plus[Optional[Pattern[c, Blank[]]], Times[Optional[Pattern[d, Blank[]]], Power[Pattern[x, Blank[]], Pattern[n, Blank[]]]]]], Optional[Pattern[b, Blank[]]]]], Optional[Pattern[p, Blank[]]]], Power[Pattern[x, Blank[]], Optional[Pattern[m, Blank[]]]]], Pattern[x, Blank[Symbol]]] := Unintegrable[x^m*(a + b*Csch[c + d*x^n])^p, x] /; FreeQ[{a, b, c, d, m, n, p}, x]
  5440. Int[Times[Power[Times[Pattern[e, Blank[]], Pattern[x, Blank[]]], Optional[Pattern[m, Blank[]]]], Power[Plus[Optional[Pattern[a, Blank[]]], Times[Optional[Pattern[b, Blank[]]], Sech[Plus[Optional[Pattern[c, Blank[]]], Times[Optional[Pattern[d, Blank[]]], Power[Pattern[x, Blank[]], Pattern[n, Blank[]]]]]]]], Optional[Pattern[p, Blank[]]]]], Pattern[x, Blank[Symbol]]] := Dist[(e^IntPart[m]*(e*x)^FracPart[m])/x^FracPart[m], Int[x^m*(a + b*Sech[c + d*x^n])^p, x], x] /; FreeQ[{a, b, c, d, e, m, n, p}, x]
  5441. Int[Times[Power[Plus[Optional[Pattern[a, Blank[]]], Times[Csch[Plus[Optional[Pattern[c, Blank[]]], Times[Optional[Pattern[d, Blank[]]], Power[Pattern[x, Blank[]], Pattern[n, Blank[]]]]]], Optional[Pattern[b, Blank[]]]]], Optional[Pattern[p, Blank[]]]], Power[Times[Pattern[e, Blank[]], Pattern[x, Blank[]]], Optional[Pattern[m, Blank[]]]]], Pattern[x, Blank[Symbol]]] := Dist[(e^IntPart[m]*(e*x)^FracPart[m])/x^FracPart[m], Int[x^m*(a + b*Csch[c + d*x^n])^p, x], x] /; FreeQ[{a, b, c, d, e, m, n, p}, x]
  5442. Int[Times[Power[Times[Pattern[e, Blank[]], Pattern[x, Blank[]]], Optional[Pattern[m, Blank[]]]], Power[Plus[Optional[Pattern[a, Blank[]]], Times[Optional[Pattern[b, Blank[]]], Sech[Pattern[u, Blank[]]]]], Optional[Pattern[p, Blank[]]]]], Pattern[x, Blank[Symbol]]] := Int[(e*x)^m*(a + b*Sech[ExpandToSum[u, x]])^p, x] /; FreeQ[{a, b, e, m, p}, x] && BinomialQ[u, x] && !BinomialMatchQ[u, x]
  5443. Int[Times[Power[Plus[Optional[Pattern[a, Blank[]]], Times[Csch[Pattern[u, Blank[]]], Optional[Pattern[b, Blank[]]]]], Optional[Pattern[p, Blank[]]]], Power[Times[Pattern[e, Blank[]], Pattern[x, Blank[]]], Optional[Pattern[m, Blank[]]]]], Pattern[x, Blank[Symbol]]] := Int[(e*x)^m*(a + b*Csch[ExpandToSum[u, x]])^p, x] /; FreeQ[{a, b, e, m, p}, x] && BinomialQ[u, x] && !BinomialMatchQ[u, x]
  5444. Int[Times[Power[Pattern[x, Blank[]], Optional[Pattern[m, Blank[]]]], Power[Sech[Plus[Optional[Pattern[a, Blank[]]], Times[Optional[Pattern[b, Blank[]]], Power[Pattern[x, Blank[]], Optional[Pattern[n, Blank[]]]]]]], Pattern[p, Blank[]]], Sinh[Plus[Optional[Pattern[a, Blank[]]], Times[Optional[Pattern[b, Blank[]]], Power[Pattern[x, Blank[]], Optional[Pattern[n, Blank[]]]]]]]], Pattern[x, Blank[Symbol]]] := -Simp[(x^(m - n + 1)*Sech[a + b*x^n]^(p - 1))/(b*n*(p - 1)), x] + Dist[(m - n + 1)/(b*n*(p - 1)), Int[x^(m - n)*Sech[a + b*x^n]^(p - 1), x], x] /; FreeQ[{a, b, p}, x] && IntegerQ[n] && GeQ[m - n, 0] && NeQ[p, 1]
  5445. Int[Times[Cosh[Plus[Optional[Pattern[a, Blank[]]], Times[Optional[Pattern[b, Blank[]]], Power[Pattern[x, Blank[]], Optional[Pattern[n, Blank[]]]]]]], Power[Csch[Plus[Optional[Pattern[a, Blank[]]], Times[Optional[Pattern[b, Blank[]]], Power[Pattern[x, Blank[]], Optional[Pattern[n, Blank[]]]]]]], Pattern[p, Blank[]]], Power[Pattern[x, Blank[]], Optional[Pattern[m, Blank[]]]]], Pattern[x, Blank[Symbol]]] := -Simp[(x^(m - n + 1)*Csch[a + b*x^n]^(p - 1))/(b*n*(p - 1)), x] + Dist[(m - n + 1)/(b*n*(p - 1)), Int[x^(m - n)*Csch[a + b*x^n]^(p - 1), x], x] /; FreeQ[{a, b, p}, x] && IntegerQ[n] && GeQ[m - n, 0] && NeQ[p, 1]
  5446. Int[Times[Cosh[Plus[Optional[Pattern[a, Blank[]]], Times[Optional[Pattern[b, Blank[]]], Pattern[x, Blank[]]]]], Power[Plus[Optional[Pattern[c, Blank[]]], Times[Optional[Pattern[d, Blank[]]], Pattern[x, Blank[]]]], Optional[Pattern[m, Blank[]]]], Power[Sinh[Plus[Optional[Pattern[a, Blank[]]], Times[Optional[Pattern[b, Blank[]]], Pattern[x, Blank[]]]]], Optional[Pattern[n, Blank[]]]]], Pattern[x, Blank[Symbol]]] := Simp[((c + d*x)^m*Sinh[a + b*x]^(n + 1))/(b*(n + 1)), x] - Dist[(d*m)/(b*(n + 1)), Int[(c + d*x)^(m - 1)*Sinh[a + b*x]^(n + 1), x], x] /; FreeQ[{a, b, c, d, n}, x] && IGtQ[m, 0] && NeQ[n, -1]
  5447. Int[Times[Power[Cosh[Plus[Optional[Pattern[a, Blank[]]], Times[Optional[Pattern[b, Blank[]]], Pattern[x, Blank[]]]]], Optional[Pattern[n, Blank[]]]], Power[Plus[Optional[Pattern[c, Blank[]]], Times[Optional[Pattern[d, Blank[]]], Pattern[x, Blank[]]]], Optional[Pattern[m, Blank[]]]], Sinh[Plus[Optional[Pattern[a, Blank[]]], Times[Optional[Pattern[b, Blank[]]], Pattern[x, Blank[]]]]]], Pattern[x, Blank[Symbol]]] := Simp[((c + d*x)^m*Cosh[a + b*x]^(n + 1))/(b*(n + 1)), x] - Dist[(d*m)/(b*(n + 1)), Int[(c + d*x)^(m - 1)*Cosh[a + b*x]^(n + 1), x], x] /; FreeQ[{a, b, c, d, n}, x] && IGtQ[m, 0] && NeQ[n, -1]
  5448. Int[Times[Power[Cosh[Plus[Optional[Pattern[a, Blank[]]], Times[Optional[Pattern[b, Blank[]]], Pattern[x, Blank[]]]]], Optional[Pattern[p, Blank[]]]], Power[Plus[Optional[Pattern[c, Blank[]]], Times[Optional[Pattern[d, Blank[]]], Pattern[x, Blank[]]]], Optional[Pattern[m, Blank[]]]], Power[Sinh[Plus[Optional[Pattern[a, Blank[]]], Times[Optional[Pattern[b, Blank[]]], Pattern[x, Blank[]]]]], Optional[Pattern[n, Blank[]]]]], Pattern[x, Blank[Symbol]]] := Int[ExpandTrigReduce[(c + d*x)^m, Sinh[a + b*x]^n*Cosh[a + b*x]^p, x], x] /; FreeQ[{a, b, c, d, m}, x] && IGtQ[n, 0] && IGtQ[p, 0]
  5449. Int[Times[Power[Plus[Optional[Pattern[c, Blank[]]], Times[Optional[Pattern[d, Blank[]]], Pattern[x, Blank[]]]], Optional[Pattern[m, Blank[]]]], Power[Sinh[Plus[Optional[Pattern[a, Blank[]]], Times[Optional[Pattern[b, Blank[]]], Pattern[x, Blank[]]]]], Optional[Pattern[n, Blank[]]]], Power[Tanh[Plus[Optional[Pattern[a, Blank[]]], Times[Optional[Pattern[b, Blank[]]], Pattern[x, Blank[]]]]], Optional[Pattern[p, Blank[]]]]], Pattern[x, Blank[Symbol]]] := Int[(c + d*x)^m*Sinh[a + b*x]^n*Tanh[a + b*x]^(p - 2), x] - Int[(c + d*x)^m*Sinh[a + b*x]^(n - 2)*Tanh[a + b*x]^p, x] /; FreeQ[{a, b, c, d, m}, x] && IGtQ[n, 0] && IGtQ[p, 0]
  5450. Int[Times[Power[Cosh[Plus[Optional[Pattern[a, Blank[]]], Times[Optional[Pattern[b, Blank[]]], Pattern[x, Blank[]]]]], Optional[Pattern[n, Blank[]]]], Power[Coth[Plus[Optional[Pattern[a, Blank[]]], Times[Optional[Pattern[b, Blank[]]], Pattern[x, Blank[]]]]], Optional[Pattern[p, Blank[]]]], Power[Plus[Optional[Pattern[c, Blank[]]], Times[Optional[Pattern[d, Blank[]]], Pattern[x, Blank[]]]], Optional[Pattern[m, Blank[]]]]], Pattern[x, Blank[Symbol]]] := Int[(c + d*x)^m*Cosh[a + b*x]^n*Coth[a + b*x]^(p - 2), x] + Int[(c + d*x)^m*Cosh[a + b*x]^(n - 2)*Coth[a + b*x]^p, x] /; FreeQ[{a, b, c, d, m}, x] && IGtQ[n, 0] && IGtQ[p, 0]
  5451. Int[Times[Power[Plus[Optional[Pattern[c, Blank[]]], Times[Optional[Pattern[d, Blank[]]], Pattern[x, Blank[]]]], Optional[Pattern[m, Blank[]]]], Power[Sech[Plus[Optional[Pattern[a, Blank[]]], Times[Optional[Pattern[b, Blank[]]], Pattern[x, Blank[]]]]], Optional[Pattern[n, Blank[]]]], Power[Tanh[Plus[Optional[Pattern[a, Blank[]]], Times[Optional[Pattern[b, Blank[]]], Pattern[x, Blank[]]]]], Optional[Pattern[p, Blank[]]]]], Pattern[x, Blank[Symbol]]] := -Simp[((c + d*x)^m*Sech[a + b*x]^n)/(b*n), x] + Dist[(d*m)/(b*n), Int[(c + d*x)^(m - 1)*Sech[a + b*x]^n, x], x] /; FreeQ[{a, b, c, d, n}, x] && EqQ[p, 1] && GtQ[m, 0]
  5452. Int[Times[Power[Coth[Plus[Optional[Pattern[a, Blank[]]], Times[Optional[Pattern[b, Blank[]]], Pattern[x, Blank[]]]]], Optional[Pattern[p, Blank[]]]], Power[Csch[Plus[Optional[Pattern[a, Blank[]]], Times[Optional[Pattern[b, Blank[]]], Pattern[x, Blank[]]]]], Optional[Pattern[n, Blank[]]]], Power[Plus[Optional[Pattern[c, Blank[]]], Times[Optional[Pattern[d, Blank[]]], Pattern[x, Blank[]]]], Optional[Pattern[m, Blank[]]]]], Pattern[x, Blank[Symbol]]] := -Simp[((c + d*x)^m*Csch[a + b*x]^n)/(b*n), x] + Dist[(d*m)/(b*n), Int[(c + d*x)^(m - 1)*Csch[a + b*x]^n, x], x] /; FreeQ[{a, b, c, d, n}, x] && EqQ[p, 1] && GtQ[m, 0]
  5453. Int[Times[Power[Plus[Optional[Pattern[c, Blank[]]], Times[Optional[Pattern[d, Blank[]]], Pattern[x, Blank[]]]], Optional[Pattern[m, Blank[]]]], Power[Sech[Plus[Optional[Pattern[a, Blank[]]], Times[Optional[Pattern[b, Blank[]]], Pattern[x, Blank[]]]]], 2], Power[Tanh[Plus[Optional[Pattern[a, Blank[]]], Times[Optional[Pattern[b, Blank[]]], Pattern[x, Blank[]]]]], Optional[Pattern[n, Blank[]]]]], Pattern[x, Blank[Symbol]]] := Simp[((c + d*x)^m*Tanh[a + b*x]^(n + 1))/(b*(n + 1)), x] - Dist[(d*m)/(b*(n + 1)), Int[(c + d*x)^(m - 1)*Tanh[a + b*x]^(n + 1), x], x] /; FreeQ[{a, b, c, d, n}, x] && IGtQ[m, 0] && NeQ[n, -1]
  5454. Int[Times[Power[Coth[Plus[Optional[Pattern[a, Blank[]]], Times[Optional[Pattern[b, Blank[]]], Pattern[x, Blank[]]]]], Optional[Pattern[n, Blank[]]]], Power[Csch[Plus[Optional[Pattern[a, Blank[]]], Times[Optional[Pattern[b, Blank[]]], Pattern[x, Blank[]]]]], 2], Power[Plus[Optional[Pattern[c, Blank[]]], Times[Optional[Pattern[d, Blank[]]], Pattern[x, Blank[]]]], Optional[Pattern[m, Blank[]]]]], Pattern[x, Blank[Symbol]]] := -Simp[((c + d*x)^m*Coth[a + b*x]^(n + 1))/(b*(n + 1)), x] + Dist[(d*m)/(b*(n + 1)), Int[(c + d*x)^(m - 1)*Coth[a + b*x]^(n + 1), x], x] /; FreeQ[{a, b, c, d, n}, x] && IGtQ[m, 0] && NeQ[n, -1]
  5455. Int[Times[Power[Plus[Optional[Pattern[c, Blank[]]], Times[Optional[Pattern[d, Blank[]]], Pattern[x, Blank[]]]], Optional[Pattern[m, Blank[]]]], Sech[Plus[Optional[Pattern[a, Blank[]]], Times[Optional[Pattern[b, Blank[]]], Pattern[x, Blank[]]]]], Power[Tanh[Plus[Optional[Pattern[a, Blank[]]], Times[Optional[Pattern[b, Blank[]]], Pattern[x, Blank[]]]]], Pattern[p, Blank[]]]], Pattern[x, Blank[Symbol]]] := Int[(c + d*x)^m*Sech[a + b*x]*Tanh[a + b*x]^(p - 2), x] - Int[(c + d*x)^m*Sech[a + b*x]^3*Tanh[a + b*x]^(p - 2), x] /; FreeQ[{a, b, c, d, m}, x] && IGtQ[p/2, 0]
  5456. Int[Times[Power[Plus[Optional[Pattern[c, Blank[]]], Times[Optional[Pattern[d, Blank[]]], Pattern[x, Blank[]]]], Optional[Pattern[m, Blank[]]]], Power[Sech[Plus[Optional[Pattern[a, Blank[]]], Times[Optional[Pattern[b, Blank[]]], Pattern[x, Blank[]]]]], Optional[Pattern[n, Blank[]]]], Power[Tanh[Plus[Optional[Pattern[a, Blank[]]], Times[Optional[Pattern[b, Blank[]]], Pattern[x, Blank[]]]]], Pattern[p, Blank[]]]], Pattern[x, Blank[Symbol]]] := Int[(c + d*x)^m*Sech[a + b*x]^n*Tanh[a + b*x]^(p - 2), x] - Int[(c + d*x)^m*Sech[a + b*x]^(n + 2)*Tanh[a + b*x]^(p - 2), x] /; FreeQ[{a, b, c, d, m, n}, x] && IGtQ[p/2, 0]
  5457. Int[Times[Power[Coth[Plus[Optional[Pattern[a, Blank[]]], Times[Optional[Pattern[b, Blank[]]], Pattern[x, Blank[]]]]], Pattern[p, Blank[]]], Csch[Plus[Optional[Pattern[a, Blank[]]], Times[Optional[Pattern[b, Blank[]]], Pattern[x, Blank[]]]]], Power[Plus[Optional[Pattern[c, Blank[]]], Times[Optional[Pattern[d, Blank[]]], Pattern[x, Blank[]]]], Optional[Pattern[m, Blank[]]]]], Pattern[x, Blank[Symbol]]] := Int[(c + d*x)^m*Csch[a + b*x]*Coth[a + b*x]^(p - 2), x] + Int[(c + d*x)^m*Csch[a + b*x]^3*Coth[a + b*x]^(p - 2), x] /; FreeQ[{a, b, c, d, m}, x] && IGtQ[p/2, 0]
  5458. Int[Times[Power[Coth[Plus[Optional[Pattern[a, Blank[]]], Times[Optional[Pattern[b, Blank[]]], Pattern[x, Blank[]]]]], Pattern[p, Blank[]]], Power[Csch[Plus[Optional[Pattern[a, Blank[]]], Times[Optional[Pattern[b, Blank[]]], Pattern[x, Blank[]]]]], Optional[Pattern[n, Blank[]]]], Power[Plus[Optional[Pattern[c, Blank[]]], Times[Optional[Pattern[d, Blank[]]], Pattern[x, Blank[]]]], Optional[Pattern[m, Blank[]]]]], Pattern[x, Blank[Symbol]]] := Int[(c + d*x)^m*Csch[a + b*x]^n*Coth[a + b*x]^(p - 2), x] + Int[(c + d*x)^m*Csch[a + b*x]^(n + 2)*Coth[a + b*x]^(p - 2), x] /; FreeQ[{a, b, c, d, m, n}, x] && IGtQ[p/2, 0]
  5459. Int[Times[Power[Plus[Optional[Pattern[c, Blank[]]], Times[Optional[Pattern[d, Blank[]]], Pattern[x, Blank[]]]], Optional[Pattern[m, Blank[]]]], Power[Sech[Plus[Optional[Pattern[a, Blank[]]], Times[Optional[Pattern[b, Blank[]]], Pattern[x, Blank[]]]]], Optional[Pattern[n, Blank[]]]], Power[Tanh[Plus[Optional[Pattern[a, Blank[]]], Times[Optional[Pattern[b, Blank[]]], Pattern[x, Blank[]]]]], Optional[Pattern[p, Blank[]]]]], Pattern[x, Blank[Symbol]]] := With[{u = IntHide[Sech[a + b*x]^n*Tanh[a + b*x]^p, x]}, Dist[(c + d*x)^m, u, x] - Dist[d*m, Int[(c + d*x)^(m - 1)*u, x], x]] /; FreeQ[{a, b, c, d, n, p}, x] && IGtQ[m, 0] && (IntegerQ[n/2] || IntegerQ[(p - 1)/2])
  5460. Int[Times[Power[Coth[Plus[Optional[Pattern[a, Blank[]]], Times[Optional[Pattern[b, Blank[]]], Pattern[x, Blank[]]]]], Optional[Pattern[p, Blank[]]]], Power[Csch[Plus[Optional[Pattern[a, Blank[]]], Times[Optional[Pattern[b, Blank[]]], Pattern[x, Blank[]]]]], Optional[Pattern[n, Blank[]]]], Power[Plus[Optional[Pattern[c, Blank[]]], Times[Optional[Pattern[d, Blank[]]], Pattern[x, Blank[]]]], Optional[Pattern[m, Blank[]]]]], Pattern[x, Blank[Symbol]]] := With[{u = IntHide[Csch[a + b*x]^n*Coth[a + b*x]^p, x]}, Dist[(c + d*x)^m, u, x] - Dist[d*m, Int[(c + d*x)^(m - 1)*u, x], x]] /; FreeQ[{a, b, c, d, n, p}, x] && IGtQ[m, 0] && (IntegerQ[n/2] || IntegerQ[(p - 1)/2])
  5461. Int[Times[Power[Csch[Plus[Optional[Pattern[a, Blank[]]], Times[Optional[Pattern[b, Blank[]]], Pattern[x, Blank[]]]]], Optional[Pattern[n, Blank[]]]], Power[Plus[Optional[Pattern[c, Blank[]]], Times[Optional[Pattern[d, Blank[]]], Pattern[x, Blank[]]]], Optional[Pattern[m, Blank[]]]], Power[Sech[Plus[Optional[Pattern[a, Blank[]]], Times[Optional[Pattern[b, Blank[]]], Pattern[x, Blank[]]]]], Optional[Pattern[n, Blank[]]]]], Pattern[x, Blank[Symbol]]] := Dist[2^n, Int[(c + d*x)^m*Csch[2*a + 2*b*x]^n, x], x] /; FreeQ[{a, b, c, d}, x] && RationalQ[m] && IntegerQ[n]
  5462. Int[Times[Power[Csch[Plus[Optional[Pattern[a, Blank[]]], Times[Optional[Pattern[b, Blank[]]], Pattern[x, Blank[]]]]], Optional[Pattern[n, Blank[]]]], Power[Plus[Optional[Pattern[c, Blank[]]], Times[Optional[Pattern[d, Blank[]]], Pattern[x, Blank[]]]], Optional[Pattern[m, Blank[]]]], Power[Sech[Plus[Optional[Pattern[a, Blank[]]], Times[Optional[Pattern[b, Blank[]]], Pattern[x, Blank[]]]]], Optional[Pattern[p, Blank[]]]]], Pattern[x, Blank[Symbol]]] := With[{u = IntHide[Csch[a + b*x]^n*Sech[a + b*x]^p, x]}, Dist[(c + d*x)^m, u, x] - Dist[d*m, Int[(c + d*x)^(m - 1)*u, x], x]] /; FreeQ[{a, b, c, d}, x] && IntegersQ[n, p] && GtQ[m, 0] && NeQ[n, p]
  5463. Int[Times[Power[Pattern[u, Blank[]], Optional[Pattern[m, Blank[]]]], Power[Pattern[F, Blank[]][Pattern[v, Blank[]]], Optional[Pattern[n, Blank[]]]], Power[Pattern[G, Blank[]][Pattern[w, Blank[]]], Optional[Pattern[p, Blank[]]]]], Pattern[x, Blank[Symbol]]] := Int[ExpandToSum[u, x]^m*F[ExpandToSum[v, x]]^n*G[ExpandToSum[v, x]]^p, x] /; FreeQ[{m, n, p}, x] && HyperbolicQ[F] && HyperbolicQ[G] && EqQ[v, w] && LinearQ[{u, v, w}, x] && !LinearMatchQ[{u, v, w}, x]
  5464. Int[Times[Cosh[Plus[Optional[Pattern[c, Blank[]]], Times[Optional[Pattern[d, Blank[]]], Pattern[x, Blank[]]]]], Power[Plus[Optional[Pattern[e, Blank[]]], Times[Optional[Pattern[f, Blank[]]], Pattern[x, Blank[]]]], Optional[Pattern[m, Blank[]]]], Power[Plus[Pattern[a, Blank[]], Times[Optional[Pattern[b, Blank[]]], Sinh[Plus[Optional[Pattern[c, Blank[]]], Times[Optional[Pattern[d, Blank[]]], Pattern[x, Blank[]]]]]]], Optional[Pattern[n, Blank[]]]]], Pattern[x, Blank[Symbol]]] := Simp[((e + f*x)^m*(a + b*Sinh[c + d*x])^(n + 1))/(b*d*(n + 1)), x] - Dist[(f*m)/(b*d*(n + 1)), Int[(e + f*x)^(m - 1)*(a + b*Sinh[c + d*x])^(n + 1), x], x] /; FreeQ[{a, b, c, d, e, f, n}, x] && IGtQ[m, 0] && NeQ[n, -1]
  5465. Int[Times[Power[Plus[Times[Cosh[Plus[Optional[Pattern[c, Blank[]]], Times[Optional[Pattern[d, Blank[]]], Pattern[x, Blank[]]]]], Optional[Pattern[b, Blank[]]]], Pattern[a, Blank[]]], Optional[Pattern[n, Blank[]]]], Power[Plus[Optional[Pattern[e, Blank[]]], Times[Optional[Pattern[f, Blank[]]], Pattern[x, Blank[]]]], Optional[Pattern[m, Blank[]]]], Sinh[Plus[Optional[Pattern[c, Blank[]]], Times[Optional[Pattern[d, Blank[]]], Pattern[x, Blank[]]]]]], Pattern[x, Blank[Symbol]]] := Simp[((e + f*x)^m*(a + b*Cosh[c + d*x])^(n + 1))/(b*d*(n + 1)), x] - Dist[(f*m)/(b*d*(n + 1)), Int[(e + f*x)^(m - 1)*(a + b*Cosh[c + d*x])^(n + 1), x], x] /; FreeQ[{a, b, c, d, e, f, n}, x] && IGtQ[m, 0] && NeQ[n, -1]
  5466. Int[Times[Power[Plus[Optional[Pattern[e, Blank[]]], Times[Optional[Pattern[f, Blank[]]], Pattern[x, Blank[]]]], Optional[Pattern[m, Blank[]]]], Power[Sech[Plus[Optional[Pattern[c, Blank[]]], Times[Optional[Pattern[d, Blank[]]], Pattern[x, Blank[]]]]], 2], Power[Plus[Pattern[a, Blank[]], Times[Optional[Pattern[b, Blank[]]], Tanh[Plus[Optional[Pattern[c, Blank[]]], Times[Optional[Pattern[d, Blank[]]], Pattern[x, Blank[]]]]]]], Optional[Pattern[n, Blank[]]]]], Pattern[x, Blank[Symbol]]] := Simp[((e + f*x)^m*(a + b*Tanh[c + d*x])^(n + 1))/(b*d*(n + 1)), x] - Dist[(f*m)/(b*d*(n + 1)), Int[(e + f*x)^(m - 1)*(a + b*Tanh[c + d*x])^(n + 1), x], x] /; FreeQ[{a, b, c, d, e, f, n}, x] && IGtQ[m, 0] && NeQ[n, -1]
  5467. Int[Times[Power[Csch[Plus[Optional[Pattern[c, Blank[]]], Times[Optional[Pattern[d, Blank[]]], Pattern[x, Blank[]]]]], 2], Power[Plus[Times[Coth[Plus[Optional[Pattern[c, Blank[]]], Times[Optional[Pattern[d, Blank[]]], Pattern[x, Blank[]]]]], Optional[Pattern[b, Blank[]]]], Pattern[a, Blank[]]], Optional[Pattern[n, Blank[]]]], Power[Plus[Optional[Pattern[e, Blank[]]], Times[Optional[Pattern[f, Blank[]]], Pattern[x, Blank[]]]], Optional[Pattern[m, Blank[]]]]], Pattern[x, Blank[Symbol]]] := -Simp[((e + f*x)^m*(a + b*Coth[c + d*x])^(n + 1))/(b*d*(n + 1)), x] + Dist[(f*m)/(b*d*(n + 1)), Int[(e + f*x)^(m - 1)*(a + b*Coth[c + d*x])^(n + 1), x], x] /; FreeQ[{a, b, c, d, e, f, n}, x] && IGtQ[m, 0] && NeQ[n, -1]
  5468. Int[Times[Power[Plus[Optional[Pattern[e, Blank[]]], Times[Optional[Pattern[f, Blank[]]], Pattern[x, Blank[]]]], Optional[Pattern[m, Blank[]]]], Sech[Plus[Optional[Pattern[c, Blank[]]], Times[Optional[Pattern[d, Blank[]]], Pattern[x, Blank[]]]]], Power[Plus[Pattern[a, Blank[]], Times[Optional[Pattern[b, Blank[]]], Sech[Plus[Optional[Pattern[c, Blank[]]], Times[Optional[Pattern[d, Blank[]]], Pattern[x, Blank[]]]]]]], Optional[Pattern[n, Blank[]]]], Tanh[Plus[Optional[Pattern[c, Blank[]]], Times[Optional[Pattern[d, Blank[]]], Pattern[x, Blank[]]]]]], Pattern[x, Blank[Symbol]]] := -Simp[((e + f*x)^m*(a + b*Sech[c + d*x])^(n + 1))/(b*d*(n + 1)), x] + Dist[(f*m)/(b*d*(n + 1)), Int[(e + f*x)^(m - 1)*(a + b*Sech[c + d*x])^(n + 1), x], x] /; FreeQ[{a, b, c, d, e, f, n}, x] && IGtQ[m, 0] && NeQ[n, -1]
  5469. Int[Times[Coth[Plus[Optional[Pattern[c, Blank[]]], Times[Optional[Pattern[d, Blank[]]], Pattern[x, Blank[]]]]], Csch[Plus[Optional[Pattern[c, Blank[]]], Times[Optional[Pattern[d, Blank[]]], Pattern[x, Blank[]]]]], Power[Plus[Times[Csch[Plus[Optional[Pattern[c, Blank[]]], Times[Optional[Pattern[d, Blank[]]], Pattern[x, Blank[]]]]], Optional[Pattern[b, Blank[]]]], Pattern[a, Blank[]]], Optional[Pattern[n, Blank[]]]], Power[Plus[Optional[Pattern[e, Blank[]]], Times[Optional[Pattern[f, Blank[]]], Pattern[x, Blank[]]]], Optional[Pattern[m, Blank[]]]]], Pattern[x, Blank[Symbol]]] := -Simp[((e + f*x)^m*(a + b*Csch[c + d*x])^(n + 1))/(b*d*(n + 1)), x] + Dist[(f*m)/(b*d*(n + 1)), Int[(e + f*x)^(m - 1)*(a + b*Csch[c + d*x])^(n + 1), x], x] /; FreeQ[{a, b, c, d, e, f, n}, x] && IGtQ[m, 0] && NeQ[n, -1]
  5470. Int[Times[Power[Plus[Optional[Pattern[e, Blank[]]], Times[Optional[Pattern[f, Blank[]]], Pattern[x, Blank[]]]], Optional[Pattern[m, Blank[]]]], Power[Sinh[Plus[Optional[Pattern[a, Blank[]]], Times[Optional[Pattern[b, Blank[]]], Pattern[x, Blank[]]]]], Optional[Pattern[p, Blank[]]]], Power[Sinh[Plus[Optional[Pattern[c, Blank[]]], Times[Optional[Pattern[d, Blank[]]], Pattern[x, Blank[]]]]], Optional[Pattern[q, Blank[]]]]], Pattern[x, Blank[Symbol]]] := Int[ExpandTrigReduce[(e + f*x)^m, Sinh[a + b*x]^p*Sinh[c + d*x]^q, x], x] /; FreeQ[{a, b, c, d, e, f}, x] && IGtQ[p, 0] && IGtQ[q, 0] && IntegerQ[m]
  5471. Int[Times[Power[Cosh[Plus[Optional[Pattern[a, Blank[]]], Times[Optional[Pattern[b, Blank[]]], Pattern[x, Blank[]]]]], Optional[Pattern[p, Blank[]]]], Power[Cosh[Plus[Optional[Pattern[c, Blank[]]], Times[Optional[Pattern[d, Blank[]]], Pattern[x, Blank[]]]]], Optional[Pattern[q, Blank[]]]], Power[Plus[Optional[Pattern[e, Blank[]]], Times[Optional[Pattern[f, Blank[]]], Pattern[x, Blank[]]]], Optional[Pattern[m, Blank[]]]]], Pattern[x, Blank[Symbol]]] := Int[ExpandTrigReduce[(e + f*x)^m, Cosh[a + b*x]^p*Cosh[c + d*x]^q, x], x] /; FreeQ[{a, b, c, d, e, f}, x] && IGtQ[p, 0] && IGtQ[q, 0] && IntegerQ[m]
  5472. Int[Times[Power[Cosh[Plus[Optional[Pattern[c, Blank[]]], Times[Optional[Pattern[d, Blank[]]], Pattern[x, Blank[]]]]], Optional[Pattern[q, Blank[]]]], Power[Plus[Optional[Pattern[e, Blank[]]], Times[Optional[Pattern[f, Blank[]]], Pattern[x, Blank[]]]], Optional[Pattern[m, Blank[]]]], Power[Sinh[Plus[Optional[Pattern[a, Blank[]]], Times[Optional[Pattern[b, Blank[]]], Pattern[x, Blank[]]]]], Optional[Pattern[p, Blank[]]]]], Pattern[x, Blank[Symbol]]] := Int[ExpandTrigReduce[(e + f*x)^m, Sinh[a + b*x]^p*Cosh[c + d*x]^q, x], x] /; FreeQ[{a, b, c, d, e, f, m}, x] && IGtQ[p, 0] && IGtQ[q, 0]
  5473. Int[Times[Power[Plus[Optional[Pattern[e, Blank[]]], Times[Optional[Pattern[f, Blank[]]], Pattern[x, Blank[]]]], Optional[Pattern[m, Blank[]]]], Power[Pattern[F, Blank[]][Plus[Optional[Pattern[a, Blank[]]], Times[Optional[Pattern[b, Blank[]]], Pattern[x, Blank[]]]]], Optional[Pattern[p, Blank[]]]], Power[Pattern[G, Blank[]][Plus[Optional[Pattern[c, Blank[]]], Times[Optional[Pattern[d, Blank[]]], Pattern[x, Blank[]]]]], Optional[Pattern[q, Blank[]]]]], Pattern[x, Blank[Symbol]]] := Int[ExpandTrigExpand[(e + f*x)^m*G[c + d*x]^q, F, c + d*x, p, b/d, x], x] /; FreeQ[{a, b, c, d, e, f, m}, x] && MemberQ[{Sinh, Cosh}, F] && MemberQ[{Sech, Csch}, G] && IGtQ[p, 0] && IGtQ[q, 0] && EqQ[b*c - a*d, 0] && IGtQ[b/d, 1]
  5474. Int[Times[Power[Pattern[F, Blank[]], Times[Optional[Pattern[c, Blank[]]], Plus[Optional[Pattern[a, Blank[]]], Times[Optional[Pattern[b, Blank[]]], Pattern[x, Blank[]]]]]], Sinh[Plus[Optional[Pattern[d, Blank[]]], Times[Optional[Pattern[e, Blank[]]], Pattern[x, Blank[]]]]]], Pattern[x, Blank[Symbol]]] := -Simp[(b*c*Log[F]*F^(c*(a + b*x))*Sinh[d + e*x])/(e^2 - b^2*c^2*Log[F]^2), x] + Simp[(e*F^(c*(a + b*x))*Cosh[d + e*x])/(e^2 - b^2*c^2*Log[F]^2), x] /; FreeQ[{F, a, b, c, d, e}, x] && NeQ[e^2 - b^2*c^2*Log[F]^2, 0]
  5475. Int[Times[Cosh[Plus[Optional[Pattern[d, Blank[]]], Times[Optional[Pattern[e, Blank[]]], Pattern[x, Blank[]]]]], Power[Pattern[F, Blank[]], Times[Optional[Pattern[c, Blank[]]], Plus[Optional[Pattern[a, Blank[]]], Times[Optional[Pattern[b, Blank[]]], Pattern[x, Blank[]]]]]]], Pattern[x, Blank[Symbol]]] := -Simp[(b*c*Log[F]*F^(c*(a + b*x))*Cosh[d + e*x])/(e^2 - b^2*c^2*Log[F]^2), x] + Simp[(e*F^(c*(a + b*x))*Sinh[d + e*x])/(e^2 - b^2*c^2*Log[F]^2), x] /; FreeQ[{F, a, b, c, d, e}, x] && NeQ[e^2 - b^2*c^2*Log[F]^2, 0]
  5476. Int[Times[Power[Pattern[F, Blank[]], Times[Optional[Pattern[c, Blank[]]], Plus[Optional[Pattern[a, Blank[]]], Times[Optional[Pattern[b, Blank[]]], Pattern[x, Blank[]]]]]], Power[Sinh[Plus[Optional[Pattern[d, Blank[]]], Times[Optional[Pattern[e, Blank[]]], Pattern[x, Blank[]]]]], Pattern[n, Blank[]]]], Pattern[x, Blank[Symbol]]] := -Simp[(b*c*Log[F]*F^(c*(a + b*x))*Sinh[d + e*x]^n)/(e^2*n^2 - b^2*c^2*Log[F]^2), x] + (-Dist[(n*(n - 1)*e^2)/(e^2*n^2 - b^2*c^2*Log[F]^2), Int[F^(c*(a + b*x))*Sinh[d + e*x]^(n - 2), x], x] + Simp[(e*n*F^(c*(a + b*x))*Cosh[d + e*x]*Sinh[d + e*x]^(n - 1))/(e^2*n^2 - b^2*c^2*Log[F]^2), x]) /; FreeQ[{F, a, b, c, d, e}, x] && NeQ[e^2*n^2 - b^2*c^2*Log[F]^2, 0] && GtQ[n, 1]
  5477. Int[Times[Power[Cosh[Plus[Optional[Pattern[d, Blank[]]], Times[Optional[Pattern[e, Blank[]]], Pattern[x, Blank[]]]]], Pattern[n, Blank[]]], Power[Pattern[F, Blank[]], Times[Optional[Pattern[c, Blank[]]], Plus[Optional[Pattern[a, Blank[]]], Times[Optional[Pattern[b, Blank[]]], Pattern[x, Blank[]]]]]]], Pattern[x, Blank[Symbol]]] := -Simp[(b*c*Log[F]*F^(c*(a + b*x))*Cosh[d + e*x]^n)/(e^2*n^2 - b^2*c^2*Log[F]^2), x] + (Dist[(n*(n - 1)*e^2)/(e^2*n^2 - b^2*c^2*Log[F]^2), Int[F^(c*(a + b*x))*Cosh[d + e*x]^(n - 2), x], x] + Simp[(e*n*F^(c*(a + b*x))*Sinh[d + e*x]*Cosh[d + e*x]^(n - 1))/(e^2*n^2 - b^2*c^2*Log[F]^2), x]) /; FreeQ[{F, a, b, c, d, e}, x] && NeQ[e^2*n^2 - b^2*c^2*Log[F]^2, 0] && GtQ[n, 1]
  5478. Int[Times[Power[Pattern[F, Blank[]], Times[Optional[Pattern[c, Blank[]]], Plus[Optional[Pattern[a, Blank[]]], Times[Optional[Pattern[b, Blank[]]], Pattern[x, Blank[]]]]]], Power[Sinh[Plus[Optional[Pattern[d, Blank[]]], Times[Optional[Pattern[e, Blank[]]], Pattern[x, Blank[]]]]], Pattern[n, Blank[]]]], Pattern[x, Blank[Symbol]]] := -Simp[(b*c*Log[F]*F^(c*(a + b*x))*Sinh[d + e*x]^(n + 2))/(e^2*(n + 1)*(n + 2)), x] + Simp[(F^(c*(a + b*x))*Cosh[d + e*x]*Sinh[d + e*x]^(n + 1))/(e*(n + 1)), x] /; FreeQ[{F, a, b, c, d, e, n}, x] && EqQ[e^2*(n + 2)^2 - b^2*c^2*Log[F]^2, 0] && NeQ[n, -1] && NeQ[n, -2]
  5479. Int[Times[Power[Cosh[Plus[Optional[Pattern[d, Blank[]]], Times[Optional[Pattern[e, Blank[]]], Pattern[x, Blank[]]]]], Pattern[n, Blank[]]], Power[Pattern[F, Blank[]], Times[Optional[Pattern[c, Blank[]]], Plus[Optional[Pattern[a, Blank[]]], Times[Optional[Pattern[b, Blank[]]], Pattern[x, Blank[]]]]]]], Pattern[x, Blank[Symbol]]] := Simp[(b*c*Log[F]*F^(c*(a + b*x))*Cosh[d + e*x]^(n + 2))/(e^2*(n + 1)*(n + 2)), x] - Simp[(F^(c*(a + b*x))*Sinh[d + e*x]*Cosh[d + e*x]^(n + 1))/(e*(n + 1)), x] /; FreeQ[{F, a, b, c, d, e, n}, x] && EqQ[e^2*(n + 2)^2 - b^2*c^2*Log[F]^2, 0] && NeQ[n, -1] && NeQ[n, -2]
  5480. Int[Times[Power[Pattern[F, Blank[]], Times[Optional[Pattern[c, Blank[]]], Plus[Optional[Pattern[a, Blank[]]], Times[Optional[Pattern[b, Blank[]]], Pattern[x, Blank[]]]]]], Power[Sinh[Plus[Optional[Pattern[d, Blank[]]], Times[Optional[Pattern[e, Blank[]]], Pattern[x, Blank[]]]]], Pattern[n, Blank[]]]], Pattern[x, Blank[Symbol]]] := -Simp[(b*c*Log[F]*F^(c*(a + b*x))*Sinh[d + e*x]^(n + 2))/(e^2*(n + 1)*(n + 2)), x] + (-Dist[(e^2*(n + 2)^2 - b^2*c^2*Log[F]^2)/(e^2*(n + 1)*(n + 2)), Int[F^(c*(a + b*x))*Sinh[d + e*x]^(n + 2), x], x] + Simp[(F^(c*(a + b*x))*Cosh[d + e*x]*Sinh[d + e*x]^(n + 1))/(e*(n + 1)), x]) /; FreeQ[{F, a, b, c, d, e}, x] && NeQ[e^2*(n + 2)^2 - b^2*c^2*Log[F]^2, 0] && LtQ[n, -1] && NeQ[n, -2]
  5481. Int[Times[Power[Cosh[Plus[Optional[Pattern[d, Blank[]]], Times[Optional[Pattern[e, Blank[]]], Pattern[x, Blank[]]]]], Pattern[n, Blank[]]], Power[Pattern[F, Blank[]], Times[Optional[Pattern[c, Blank[]]], Plus[Optional[Pattern[a, Blank[]]], Times[Optional[Pattern[b, Blank[]]], Pattern[x, Blank[]]]]]]], Pattern[x, Blank[Symbol]]] := Simp[(b*c*Log[F]*F^(c*(a + b*x))*Cosh[d + e*x]^(n + 2))/(e^2*(n + 1)*(n + 2)), x] + (Dist[(e^2*(n + 2)^2 - b^2*c^2*Log[F]^2)/(e^2*(n + 1)*(n + 2)), Int[F^(c*(a + b*x))*Cosh[d + e*x]^(n + 2), x], x] - Simp[(F^(c*(a + b*x))*Sinh[d + e*x]*Cosh[d + e*x]^(n + 1))/(e*(n + 1)), x]) /; FreeQ[{F, a, b, c, d, e}, x] && NeQ[e^2*(n + 2)^2 - b^2*c^2*Log[F]^2, 0] && LtQ[n, -1] && NeQ[n, -2]
  5482. Int[Times[Power[Pattern[F, Blank[]], Times[Optional[Pattern[c, Blank[]]], Plus[Optional[Pattern[a, Blank[]]], Times[Optional[Pattern[b, Blank[]]], Pattern[x, Blank[]]]]]], Power[Sinh[Plus[Optional[Pattern[d, Blank[]]], Times[Optional[Pattern[e, Blank[]]], Pattern[x, Blank[]]]]], Pattern[n, Blank[]]]], Pattern[x, Blank[Symbol]]] := Dist[(E^(n*(d + e*x))*Sinh[d + e*x]^n)/(-1 + E^(2*(d + e*x)))^n, Int[(F^(c*(a + b*x))*(-1 + E^(2*(d + e*x)))^n)/E^(n*(d + e*x)), x], x] /; FreeQ[{F, a, b, c, d, e, n}, x] && !IntegerQ[n]
  5483. Int[Times[Power[Cosh[Plus[Optional[Pattern[d, Blank[]]], Times[Optional[Pattern[e, Blank[]]], Pattern[x, Blank[]]]]], Pattern[n, Blank[]]], Power[Pattern[F, Blank[]], Times[Optional[Pattern[c, Blank[]]], Plus[Optional[Pattern[a, Blank[]]], Times[Optional[Pattern[b, Blank[]]], Pattern[x, Blank[]]]]]]], Pattern[x, Blank[Symbol]]] := Dist[(E^(n*(d + e*x))*Cosh[d + e*x]^n)/(1 + E^(2*(d + e*x)))^n, Int[(F^(c*(a + b*x))*(1 + E^(2*(d + e*x)))^n)/E^(n*(d + e*x)), x], x] /; FreeQ[{F, a, b, c, d, e, n}, x] && !IntegerQ[n]
  5484. Int[Times[Power[Pattern[F, Blank[]], Times[Optional[Pattern[c, Blank[]]], Plus[Optional[Pattern[a, Blank[]]], Times[Optional[Pattern[b, Blank[]]], Pattern[x, Blank[]]]]]], Power[Tanh[Plus[Optional[Pattern[d, Blank[]]], Times[Optional[Pattern[e, Blank[]]], Pattern[x, Blank[]]]]], Optional[Pattern[n, Blank[]]]]], Pattern[x, Blank[Symbol]]] := Int[ExpandIntegrand[(F^(c*(a + b*x))*(-1 + E^(2*(d + e*x)))^n)/(1 + E^(2*(d + e*x)))^n, x], x] /; FreeQ[{F, a, b, c, d, e}, x] && IntegerQ[n]
  5485. Int[Times[Power[Coth[Plus[Optional[Pattern[d, Blank[]]], Times[Optional[Pattern[e, Blank[]]], Pattern[x, Blank[]]]]], Optional[Pattern[n, Blank[]]]], Power[Pattern[F, Blank[]], Times[Optional[Pattern[c, Blank[]]], Plus[Optional[Pattern[a, Blank[]]], Times[Optional[Pattern[b, Blank[]]], Pattern[x, Blank[]]]]]]], Pattern[x, Blank[Symbol]]] := Int[ExpandIntegrand[(F^(c*(a + b*x))*(1 + E^(2*(d + e*x)))^n)/(-1 + E^(2*(d + e*x)))^n, x], x] /; FreeQ[{F, a, b, c, d, e}, x] && IntegerQ[n]
  5486. Int[Times[Power[Pattern[F, Blank[]], Times[Optional[Pattern[c, Blank[]]], Plus[Optional[Pattern[a, Blank[]]], Times[Optional[Pattern[b, Blank[]]], Pattern[x, Blank[]]]]]], Power[Sech[Plus[Optional[Pattern[d, Blank[]]], Times[Optional[Pattern[e, Blank[]]], Pattern[x, Blank[]]]]], Pattern[n, Blank[]]]], Pattern[x, Blank[Symbol]]] := -Simp[b*c*Log[F]*F^(c*(a + b*x))*(Sech[d + e*x]^n/(e^2*n^2 - b^2*c^2*Log[F]^2)), x] + (Dist[e^2*n*((n + 1)/(e^2*n^2 - b^2*c^2*Log[F]^2)), Int[F^(c*(a + b*x))*Sech[d + e*x]^(n + 2), x], x] - Simp[e*n*F^(c*(a + b*x))*Sech[d + e*x]^(n + 1)*(Sinh[d + e*x]/(e^2*n^2 - b^2*c^2*Log[F]^2)), x]) /; FreeQ[{F, a, b, c, d, e}, x] && NeQ[e^2*n^2 + b^2*c^2*Log[F]^2, 0] && LtQ[n, -1]
  5487. Int[Times[Power[Csch[Plus[Optional[Pattern[d, Blank[]]], Times[Optional[Pattern[e, Blank[]]], Pattern[x, Blank[]]]]], Pattern[n, Blank[]]], Power[Pattern[F, Blank[]], Times[Optional[Pattern[c, Blank[]]], Plus[Optional[Pattern[a, Blank[]]], Times[Optional[Pattern[b, Blank[]]], Pattern[x, Blank[]]]]]]], Pattern[x, Blank[Symbol]]] := -Simp[b*c*Log[F]*F^(c*(a + b*x))*(Csch[d + e*x]^n/(e^2*n^2 - b^2*c^2*Log[F]^2)), x] + (-Dist[e^2*n*((n + 1)/(e^2*n^2 - b^2*c^2*Log[F]^2)), Int[F^(c*(a + b*x))*Csch[d + e*x]^(n + 2), x], x] - Simp[e*n*F^(c*(a + b*x))*Csch[d + e*x]^(n + 1)*(Cosh[d + e*x]/(e^2*n^2 - b^2*c^2*Log[F]^2)), x]) /; FreeQ[{F, a, b, c, d, e}, x] && NeQ[e^2*n^2 + b^2*c^2*Log[F]^2, 0] && LtQ[n, -1]
  5488. Int[Times[Power[Pattern[F, Blank[]], Times[Optional[Pattern[c, Blank[]]], Plus[Optional[Pattern[a, Blank[]]], Times[Optional[Pattern[b, Blank[]]], Pattern[x, Blank[]]]]]], Power[Sech[Plus[Optional[Pattern[d, Blank[]]], Times[Optional[Pattern[e, Blank[]]], Pattern[x, Blank[]]]]], Pattern[n, Blank[]]]], Pattern[x, Blank[Symbol]]] := Simp[(b*c*Log[F]*F^(c*(a + b*x))*Sech[d + e*x]^(n - 2))/(e^2*(n - 1)*(n - 2)), x] + Simp[(F^(c*(a + b*x))*Sech[d + e*x]^(n - 1)*Sinh[d + e*x])/(e*(n - 1)), x] /; FreeQ[{F, a, b, c, d, e, n}, x] && EqQ[e^2*(n - 2)^2 - b^2*c^2*Log[F]^2, 0] && NeQ[n, 1] && NeQ[n, 2]
  5489. Int[Times[Power[Csch[Plus[Optional[Pattern[d, Blank[]]], Times[Optional[Pattern[e, Blank[]]], Pattern[x, Blank[]]]]], Pattern[n, Blank[]]], Power[Pattern[F, Blank[]], Times[Optional[Pattern[c, Blank[]]], Plus[Optional[Pattern[a, Blank[]]], Times[Optional[Pattern[b, Blank[]]], Pattern[x, Blank[]]]]]]], Pattern[x, Blank[Symbol]]] := -Simp[(b*c*Log[F]*F^(c*(a + b*x))*Csch[d + e*x]^(n - 2))/(e^2*(n - 1)*(n - 2)), x] - Simp[(F^(c*(a + b*x))*Csch[d + e*x]^(n - 1)*Cosh[d + e*x])/(e*(n - 1)), x] /; FreeQ[{F, a, b, c, d, e, n}, x] && EqQ[e^2*(n - 2)^2 - b^2*c^2*Log[F]^2, 0] && NeQ[n, 1] && NeQ[n, 2]
  5490. Int[Times[Power[Pattern[F, Blank[]], Times[Optional[Pattern[c, Blank[]]], Plus[Optional[Pattern[a, Blank[]]], Times[Optional[Pattern[b, Blank[]]], Pattern[x, Blank[]]]]]], Power[Sech[Plus[Optional[Pattern[d, Blank[]]], Times[Optional[Pattern[e, Blank[]]], Pattern[x, Blank[]]]]], Pattern[n, Blank[]]]], Pattern[x, Blank[Symbol]]] := Simp[(b*c*Log[F]*F^(c*(a + b*x))*Sech[d + e*x]^(n - 2))/(e^2*(n - 1)*(n - 2)), x] + (Dist[(e^2*(n - 2)^2 - b^2*c^2*Log[F]^2)/(e^2*(n - 1)*(n - 2)), Int[F^(c*(a + b*x))*Sech[d + e*x]^(n - 2), x], x] + Simp[(F^(c*(a + b*x))*Sech[d + e*x]^(n - 1)*Sinh[d + e*x])/(e*(n - 1)), x]) /; FreeQ[{F, a, b, c, d, e}, x] && NeQ[e^2*(n - 2)^2 - b^2*c^2*Log[F]^2, 0] && GtQ[n, 1] && NeQ[n, 2]
  5491. Int[Times[Power[Csch[Plus[Optional[Pattern[d, Blank[]]], Times[Optional[Pattern[e, Blank[]]], Pattern[x, Blank[]]]]], Pattern[n, Blank[]]], Power[Pattern[F, Blank[]], Times[Optional[Pattern[c, Blank[]]], Plus[Optional[Pattern[a, Blank[]]], Times[Optional[Pattern[b, Blank[]]], Pattern[x, Blank[]]]]]]], Pattern[x, Blank[Symbol]]] := -Simp[(b*c*Log[F]*F^(c*(a + b*x))*Csch[d + e*x]^(n - 2))/(e^2*(n - 1)*(n - 2)), x] + (-Dist[(e^2*(n - 2)^2 - b^2*c^2*Log[F]^2)/(e^2*(n - 1)*(n - 2)), Int[F^(c*(a + b*x))*Csch[d + e*x]^(n - 2), x], x] - Simp[(F^(c*(a + b*x))*Csch[d + e*x]^(n - 1)*Cosh[d + e*x])/(e*(n - 1)), x]) /; FreeQ[{F, a, b, c, d, e}, x] && NeQ[e^2*(n - 2)^2 - b^2*c^2*Log[F]^2, 0] && GtQ[n, 1] && NeQ[n, 2]
  5492. Int[Times[Power[Pattern[F, Blank[]], Times[Optional[Pattern[c, Blank[]]], Plus[Optional[Pattern[a, Blank[]]], Times[Optional[Pattern[b, Blank[]]], Pattern[x, Blank[]]]]]], Power[Sech[Plus[Optional[Pattern[d, Blank[]]], Times[Optional[Pattern[e, Blank[]]], Pattern[x, Blank[]]]]], Optional[Pattern[n, Blank[]]]]], Pattern[x, Blank[Symbol]]] := Simp[(2^n*E^(n*(d + e*x))*F^(c*(a + b*x))*Hypergeometric2F1[n, n/2 + (b*c*Log[F])/(2*e), 1 + n/2 + (b*c*Log[F])/(2*e), -E^(2*(d + e*x))])/(e*n + b*c*Log[F]), x] /; FreeQ[{F, a, b, c, d, e}, x] && IntegerQ[n]
  5493. Int[Times[Power[Csch[Plus[Optional[Pattern[d, Blank[]]], Times[Optional[Pattern[e, Blank[]]], Pattern[x, Blank[]]]]], Optional[Pattern[n, Blank[]]]], Power[Pattern[F, Blank[]], Times[Optional[Pattern[c, Blank[]]], Plus[Optional[Pattern[a, Blank[]]], Times[Optional[Pattern[b, Blank[]]], Pattern[x, Blank[]]]]]]], Pattern[x, Blank[Symbol]]] := Simp[((-2)^n*E^(n*(d + e*x))*F^(c*(a + b*x))*Hypergeometric2F1[n, n/2 + (b*c*Log[F])/(2*e), 1 + n/2 + (b*c*Log[F])/(2*e), E^(2*(d + e*x))])/(e*n + b*c*Log[F]), x] /; FreeQ[{F, a, b, c, d, e}, x] && IntegerQ[n]
  5494. Int[Times[Power[Pattern[F, Blank[]], Times[Optional[Pattern[c, Blank[]]], Plus[Optional[Pattern[a, Blank[]]], Times[Optional[Pattern[b, Blank[]]], Pattern[x, Blank[]]]]]], Power[Sech[Plus[Optional[Pattern[d, Blank[]]], Times[Optional[Pattern[e, Blank[]]], Pattern[x, Blank[]]]]], Optional[Pattern[n, Blank[]]]]], Pattern[x, Blank[Symbol]]] := Dist[((1 + E^(2*(d + e*x)))^n*Sech[d + e*x]^n)/E^(n*(d + e*x)), Int[SimplifyIntegrand[(F^(c*(a + b*x))*E^(n*(d + e*x)))/(1 + E^(2*(d + e*x)))^n, x], x], x] /; FreeQ[{F, a, b, c, d, e}, x] && !IntegerQ[n]
  5495. Int[Times[Power[Csch[Plus[Optional[Pattern[d, Blank[]]], Times[Optional[Pattern[e, Blank[]]], Pattern[x, Blank[]]]]], Optional[Pattern[n, Blank[]]]], Power[Pattern[F, Blank[]], Times[Optional[Pattern[c, Blank[]]], Plus[Optional[Pattern[a, Blank[]]], Times[Optional[Pattern[b, Blank[]]], Pattern[x, Blank[]]]]]]], Pattern[x, Blank[Symbol]]] := Dist[((1 - E^(-2*(d + e*x)))^n*Csch[d + e*x]^n)/E^(-(n*(d + e*x))), Int[SimplifyIntegrand[F^(c*(a + b*x))/(E^(n*(d + e*x))*(1 - E^(-2*(d + e*x)))^n), x], x], x] /; FreeQ[{F, a, b, c, d, e}, x] && !IntegerQ[n]
  5496. Int[Times[Power[Pattern[F, Blank[]], Times[Optional[Pattern[c, Blank[]]], Plus[Optional[Pattern[a, Blank[]]], Times[Optional[Pattern[b, Blank[]]], Pattern[x, Blank[]]]]]], Power[Plus[Pattern[f, Blank[]], Times[Optional[Pattern[g, Blank[]]], Sinh[Plus[Optional[Pattern[d, Blank[]]], Times[Optional[Pattern[e, Blank[]]], Pattern[x, Blank[]]]]]]], Optional[Pattern[n, Blank[]]]]], Pattern[x, Blank[Symbol]]] := Dist[2^n*f^n, Int[F^(c*(a + b*x))*Cosh[d/2 + (e*x)/2 - (f*Pi)/(4*g)]^(2*n), x], x] /; FreeQ[{F, a, b, c, d, e, f, g}, x] && EqQ[f^2 + g^2, 0] && ILtQ[n, 0]
  5497. Int[Times[Power[Plus[Times[Cosh[Plus[Optional[Pattern[d, Blank[]]], Times[Optional[Pattern[e, Blank[]]], Pattern[x, Blank[]]]]], Optional[Pattern[g, Blank[]]]], Pattern[f, Blank[]]], Optional[Pattern[n, Blank[]]]], Power[Pattern[F, Blank[]], Times[Optional[Pattern[c, Blank[]]], Plus[Optional[Pattern[a, Blank[]]], Times[Optional[Pattern[b, Blank[]]], Pattern[x, Blank[]]]]]]], Pattern[x, Blank[Symbol]]] := Dist[2^n*g^n, Int[F^(c*(a + b*x))*Cosh[d/2 + (e*x)/2]^(2*n), x], x] /; FreeQ[{F, a, b, c, d, e, f, g}, x] && EqQ[f - g, 0] && ILtQ[n, 0]
  5498. Int[Times[Power[Plus[Times[Cosh[Plus[Optional[Pattern[d, Blank[]]], Times[Optional[Pattern[e, Blank[]]], Pattern[x, Blank[]]]]], Optional[Pattern[g, Blank[]]]], Pattern[f, Blank[]]], Optional[Pattern[n, Blank[]]]], Power[Pattern[F, Blank[]], Times[Optional[Pattern[c, Blank[]]], Plus[Optional[Pattern[a, Blank[]]], Times[Optional[Pattern[b, Blank[]]], Pattern[x, Blank[]]]]]]], Pattern[x, Blank[Symbol]]] := Dist[2^n*g^n, Int[F^(c*(a + b*x))*Sinh[d/2 + (e*x)/2]^(2*n), x], x] /; FreeQ[{F, a, b, c, d, e, f, g}, x] && EqQ[f + g, 0] && ILtQ[n, 0]
  5499. Int[Times[Power[Cosh[Plus[Optional[Pattern[d, Blank[]]], Times[Optional[Pattern[e, Blank[]]], Pattern[x, Blank[]]]]], Optional[Pattern[m, Blank[]]]], Power[Pattern[F, Blank[]], Times[Optional[Pattern[c, Blank[]]], Plus[Optional[Pattern[a, Blank[]]], Times[Optional[Pattern[b, Blank[]]], Pattern[x, Blank[]]]]]], Power[Plus[Pattern[f, Blank[]], Times[Optional[Pattern[g, Blank[]]], Sinh[Plus[Optional[Pattern[d, Blank[]]], Times[Optional[Pattern[e, Blank[]]], Pattern[x, Blank[]]]]]]], Optional[Pattern[n, Blank[]]]]], Pattern[x, Blank[Symbol]]] := Dist[g^n, Int[F^(c*(a + b*x))*Tanh[d/2 + (e*x)/2 - (f*Pi)/(4*g)]^m, x], x] /; FreeQ[{F, a, b, c, d, e, f, g}, x] && EqQ[f^2 + g^2, 0] && IntegersQ[m, n] && EqQ[m + n, 0]
  5500. Int[Times[Power[Plus[Times[Cosh[Plus[Optional[Pattern[d, Blank[]]], Times[Optional[Pattern[e, Blank[]]], Pattern[x, Blank[]]]]], Optional[Pattern[g, Blank[]]]], Pattern[f, Blank[]]], Optional[Pattern[n, Blank[]]]], Power[Pattern[F, Blank[]], Times[Optional[Pattern[c, Blank[]]], Plus[Optional[Pattern[a, Blank[]]], Times[Optional[Pattern[b, Blank[]]], Pattern[x, Blank[]]]]]], Power[Sinh[Plus[Optional[Pattern[d, Blank[]]], Times[Optional[Pattern[e, Blank[]]], Pattern[x, Blank[]]]]], Optional[Pattern[m, Blank[]]]]], Pattern[x, Blank[Symbol]]] := Dist[g^n, Int[F^(c*(a + b*x))*Tanh[d/2 + (e*x)/2]^m, x], x] /; FreeQ[{F, a, b, c, d, e, f, g}, x] && EqQ[f - g, 0] && IntegersQ[m, n] && EqQ[m + n, 0]
  5501. Int[Times[Power[Plus[Times[Cosh[Plus[Optional[Pattern[d, Blank[]]], Times[Optional[Pattern[e, Blank[]]], Pattern[x, Blank[]]]]], Optional[Pattern[g, Blank[]]]], Pattern[f, Blank[]]], Optional[Pattern[n, Blank[]]]], Power[Pattern[F, Blank[]], Times[Optional[Pattern[c, Blank[]]], Plus[Optional[Pattern[a, Blank[]]], Times[Optional[Pattern[b, Blank[]]], Pattern[x, Blank[]]]]]], Power[Sinh[Plus[Optional[Pattern[d, Blank[]]], Times[Optional[Pattern[e, Blank[]]], Pattern[x, Blank[]]]]], Optional[Pattern[m, Blank[]]]]], Pattern[x, Blank[Symbol]]] := Dist[g^n, Int[F^(c*(a + b*x))*Coth[d/2 + (e*x)/2]^m, x], x] /; FreeQ[{F, a, b, c, d, e, f, g}, x] && EqQ[f + g, 0] && IntegersQ[m, n] && EqQ[m + n, 0]
  5502. Int[Times[Power[Pattern[F, Blank[]], Times[Optional[Pattern[c, Blank[]]], Plus[Optional[Pattern[a, Blank[]]], Times[Optional[Pattern[b, Blank[]]], Pattern[x, Blank[]]]]]], Plus[Times[Cosh[Plus[Optional[Pattern[d, Blank[]]], Times[Optional[Pattern[e, Blank[]]], Pattern[x, Blank[]]]]], Optional[Pattern[i, Blank[]]]], Pattern[h, Blank[]]], Power[Plus[Pattern[f, Blank[]], Times[Optional[Pattern[g, Blank[]]], Sinh[Plus[Optional[Pattern[d, Blank[]]], Times[Optional[Pattern[e, Blank[]]], Pattern[x, Blank[]]]]]]], -1]], Pattern[x, Blank[Symbol]]] := Dist[2*i, Int[F^(c*(a + b*x))*(Cosh[d + e*x]/(f + g*Sinh[d + e*x])), x], x] + Int[F^(c*(a + b*x))*((h - i*Cosh[d + e*x])/(f + g*Sinh[d + e*x])), x] /; FreeQ[{F, a, b, c, d, e, f, g, h, i}, x] && EqQ[f^2 + g^2, 0] && EqQ[h^2 - i^2, 0] && EqQ[g*h - f*i, 0]
  5503. Int[Times[Power[Plus[Times[Cosh[Plus[Optional[Pattern[d, Blank[]]], Times[Optional[Pattern[e, Blank[]]], Pattern[x, Blank[]]]]], Optional[Pattern[g, Blank[]]]], Pattern[f, Blank[]]], -1], Power[Pattern[F, Blank[]], Times[Optional[Pattern[c, Blank[]]], Plus[Optional[Pattern[a, Blank[]]], Times[Optional[Pattern[b, Blank[]]], Pattern[x, Blank[]]]]]], Plus[Pattern[h, Blank[]], Times[Optional[Pattern[i, Blank[]]], Sinh[Plus[Optional[Pattern[d, Blank[]]], Times[Optional[Pattern[e, Blank[]]], Pattern[x, Blank[]]]]]]]], Pattern[x, Blank[Symbol]]] := Dist[2*i, Int[F^(c*(a + b*x))*(Sinh[d + e*x]/(f + g*Cosh[d + e*x])), x], x] + Int[F^(c*(a + b*x))*((h - i*Sinh[d + e*x])/(f + g*Cosh[d + e*x])), x] /; FreeQ[{F, a, b, c, d, e, f, g, h, i}, x] && EqQ[f^2 - g^2, 0] && EqQ[h^2 + i^2, 0] && EqQ[g*h + f*i, 0]
  5504. Int[Times[Power[Pattern[F, Blank[]], Times[Optional[Pattern[c, Blank[]]], Pattern[u, Blank[]]]], Power[Pattern[G, Blank[]][Pattern[v, Blank[]]], Optional[Pattern[n, Blank[]]]]], Pattern[x, Blank[Symbol]]] := Int[F^(c*ExpandToSum[u, x])*G[ExpandToSum[v, x]]^n, x] /; FreeQ[{F, c, n}, x] && HyperbolicQ[G] && LinearQ[{u, v}, x] && !LinearMatchQ[{u, v}, x]
  5505. Int[Times[Power[Pattern[F, Blank[]], Times[Optional[Pattern[c, Blank[]]], Plus[Optional[Pattern[a, Blank[]]], Times[Optional[Pattern[b, Blank[]]], Pattern[x, Blank[]]]]]], Power[Times[Optional[Pattern[f, Blank[]]], Pattern[x, Blank[]]], Optional[Pattern[m, Blank[]]]], Power[Sinh[Plus[Optional[Pattern[d, Blank[]]], Times[Optional[Pattern[e, Blank[]]], Pattern[x, Blank[]]]]], Optional[Pattern[n, Blank[]]]]], Pattern[x, Blank[Symbol]]] := Module[{u = IntHide[F^(c*(a + b*x))*Sinh[d + e*x]^n, x]}, Dist[(f*x)^m, u, x] - Dist[f*m, Int[(f*x)^(m - 1)*u, x], x]] /; FreeQ[{F, a, b, c, d, e, f}, x] && IGtQ[n, 0] && GtQ[m, 0]
  5506. Int[Times[Power[Cosh[Plus[Optional[Pattern[d, Blank[]]], Times[Optional[Pattern[e, Blank[]]], Pattern[x, Blank[]]]]], Optional[Pattern[n, Blank[]]]], Power[Pattern[F, Blank[]], Times[Optional[Pattern[c, Blank[]]], Plus[Optional[Pattern[a, Blank[]]], Times[Optional[Pattern[b, Blank[]]], Pattern[x, Blank[]]]]]], Power[Times[Optional[Pattern[f, Blank[]]], Pattern[x, Blank[]]], Optional[Pattern[m, Blank[]]]]], Pattern[x, Blank[Symbol]]] := Module[{u = IntHide[F^(c*(a + b*x))*Cosh[d + e*x]^n, x]}, Dist[(f*x)^m, u, x] - Dist[f*m, Int[(f*x)^(m - 1)*u, x], x]] /; FreeQ[{F, a, b, c, d, e, f}, x] && IGtQ[n, 0] && GtQ[m, 0]
  5507. Int[Times[Power[Pattern[F, Blank[]], Times[Optional[Pattern[c, Blank[]]], Plus[Optional[Pattern[a, Blank[]]], Times[Optional[Pattern[b, Blank[]]], Pattern[x, Blank[]]]]]], Power[Times[Optional[Pattern[f, Blank[]]], Pattern[x, Blank[]]], Pattern[m, Blank[]]], Sinh[Plus[Optional[Pattern[d, Blank[]]], Times[Optional[Pattern[e, Blank[]]], Pattern[x, Blank[]]]]]], Pattern[x, Blank[Symbol]]] := Simp[((f*x)^(m + 1)*F^(c*(a + b*x))*Sinh[d + e*x])/(f*(m + 1)), x] + (-Dist[e/(f*(m + 1)), Int[(f*x)^(m + 1)*F^(c*(a + b*x))*Cosh[d + e*x], x], x] - Dist[(b*c*Log[F])/(f*(m + 1)), Int[(f*x)^(m + 1)*F^(c*(a + b*x))*Sinh[d + e*x], x], x]) /; FreeQ[{F, a, b, c, d, e, f, m}, x] && (LtQ[m, -1] || SumSimplerQ[m, 1])
  5508. Int[Times[Cosh[Plus[Optional[Pattern[d, Blank[]]], Times[Optional[Pattern[e, Blank[]]], Pattern[x, Blank[]]]]], Power[Pattern[F, Blank[]], Times[Optional[Pattern[c, Blank[]]], Plus[Optional[Pattern[a, Blank[]]], Times[Optional[Pattern[b, Blank[]]], Pattern[x, Blank[]]]]]], Power[Times[Optional[Pattern[f, Blank[]]], Pattern[x, Blank[]]], Pattern[m, Blank[]]]], Pattern[x, Blank[Symbol]]] := Simp[((f*x)^(m + 1)*F^(c*(a + b*x))*Cosh[d + e*x])/(f*(m + 1)), x] + (-Dist[e/(f*(m + 1)), Int[(f*x)^(m + 1)*F^(c*(a + b*x))*Sinh[d + e*x], x], x] - Dist[(b*c*Log[F])/(f*(m + 1)), Int[(f*x)^(m + 1)*F^(c*(a + b*x))*Cosh[d + e*x], x], x]) /; FreeQ[{F, a, b, c, d, e, f, m}, x] && (LtQ[m, -1] || SumSimplerQ[m, 1])
  5509. Int[Times[Power[Cosh[Plus[Optional[Pattern[f, Blank[]]], Times[Optional[Pattern[g, Blank[]]], Pattern[x, Blank[]]]]], Optional[Pattern[n, Blank[]]]], Power[Pattern[F, Blank[]], Times[Optional[Pattern[c, Blank[]]], Plus[Optional[Pattern[a, Blank[]]], Times[Optional[Pattern[b, Blank[]]], Pattern[x, Blank[]]]]]], Power[Sinh[Plus[Optional[Pattern[d, Blank[]]], Times[Optional[Pattern[e, Blank[]]], Pattern[x, Blank[]]]]], Optional[Pattern[m, Blank[]]]]], Pattern[x, Blank[Symbol]]] := Int[ExpandTrigReduce[F^(c*(a + b*x)), Sinh[d + e*x]^m*Cosh[f + g*x]^n, x], x] /; FreeQ[{F, a, b, c, d, e, f, g}, x] && IGtQ[m, 0] && IGtQ[n, 0]
  5510. Int[Times[Power[Cosh[Plus[Optional[Pattern[f, Blank[]]], Times[Optional[Pattern[g, Blank[]]], Pattern[x, Blank[]]]]], Optional[Pattern[n, Blank[]]]], Power[Pattern[F, Blank[]], Times[Optional[Pattern[c, Blank[]]], Plus[Optional[Pattern[a, Blank[]]], Times[Optional[Pattern[b, Blank[]]], Pattern[x, Blank[]]]]]], Power[Pattern[x, Blank[]], Optional[Pattern[p, Blank[]]]], Power[Sinh[Plus[Optional[Pattern[d, Blank[]]], Times[Optional[Pattern[e, Blank[]]], Pattern[x, Blank[]]]]], Optional[Pattern[m, Blank[]]]]], Pattern[x, Blank[Symbol]]] := Int[ExpandTrigReduce[x^p*F^(c*(a + b*x)), Sinh[d + e*x]^m*Cosh[f + g*x]^n, x], x] /; FreeQ[{F, a, b, c, d, e, f, g}, x] && IGtQ[m, 0] && IGtQ[n, 0] && IGtQ[p, 0]
  5511. Int[Times[Power[Pattern[F, Blank[]], Times[Optional[Pattern[c, Blank[]]], Plus[Optional[Pattern[a, Blank[]]], Times[Optional[Pattern[b, Blank[]]], Pattern[x, Blank[]]]]]], Power[Pattern[G, Blank[]][Plus[Optional[Pattern[d, Blank[]]], Times[Optional[Pattern[e, Blank[]]], Pattern[x, Blank[]]]]], Optional[Pattern[m, Blank[]]]], Power[Pattern[H, Blank[]][Plus[Optional[Pattern[d, Blank[]]], Times[Optional[Pattern[e, Blank[]]], Pattern[x, Blank[]]]]], Optional[Pattern[n, Blank[]]]]], Pattern[x, Blank[Symbol]]] := Int[ExpandTrigToExp[F^(c*(a + b*x)), G[d + e*x]^m*H[d + e*x]^n, x], x] /; FreeQ[{F, a, b, c, d, e}, x] && IGtQ[m, 0] && IGtQ[n, 0] && HyperbolicQ[G] && HyperbolicQ[H]
  5512. Int[Times[Power[Pattern[F, Blank[]], Pattern[u, Blank[]]], Power[Sinh[Pattern[v, Blank[]]], Optional[Pattern[n, Blank[]]]]], Pattern[x, Blank[Symbol]]] := Int[ExpandTrigToExp[F^u, Sinh[v]^n, x], x] /; FreeQ[F, x] && (LinearQ[u, x] || PolyQ[u, x, 2]) && (LinearQ[v, x] || PolyQ[v, x, 2]) && IGtQ[n, 0]
  5513. Int[Times[Power[Cosh[Pattern[v, Blank[]]], Optional[Pattern[n, Blank[]]]], Power[Pattern[F, Blank[]], Pattern[u, Blank[]]]], Pattern[x, Blank[Symbol]]] := Int[ExpandTrigToExp[F^u, Cosh[v]^n, x], x] /; FreeQ[F, x] && (LinearQ[u, x] || PolyQ[u, x, 2]) && (LinearQ[v, x] || PolyQ[v, x, 2]) && IGtQ[n, 0]
  5514. Int[Times[Power[Cosh[Pattern[v, Blank[]]], Optional[Pattern[n, Blank[]]]], Power[Pattern[F, Blank[]], Pattern[u, Blank[]]], Power[Sinh[Pattern[v, Blank[]]], Optional[Pattern[m, Blank[]]]]], Pattern[x, Blank[Symbol]]] := Int[ExpandTrigToExp[F^u, Sinh[v]^m*Cosh[v]^n, x], x] /; FreeQ[F, x] && (LinearQ[u, x] || PolyQ[u, x, 2]) && (LinearQ[v, x] || PolyQ[v, x, 2]) && IGtQ[m, 0] && IGtQ[n, 0]
  5515. Int[Power[Sinh[Times[Log[Times[Optional[Pattern[c, Blank[]]], Power[Pattern[x, Blank[]], Optional[Pattern[n, Blank[]]]]]], Optional[Pattern[b, Blank[]]]]], Optional[Pattern[p, Blank[]]]], Pattern[x, Blank[Symbol]]] := Int[((c*x^n)^b/2 - 1/(2*(c*x^n)^b))^p, x] /; FreeQ[c, x] && RationalQ[b, n, p]
  5516. Int[Power[Cosh[Times[Log[Times[Optional[Pattern[c, Blank[]]], Power[Pattern[x, Blank[]], Optional[Pattern[n, Blank[]]]]]], Optional[Pattern[b, Blank[]]]]], Optional[Pattern[p, Blank[]]]], Pattern[x, Blank[Symbol]]] := Int[((c*x^n)^b/2 + 1/(2*(c*x^n)^b))^p, x] /; FreeQ[c, x] && RationalQ[b, n, p]
  5517. Int[Sinh[Times[Plus[Optional[Pattern[a, Blank[]]], Times[Log[Times[Optional[Pattern[c, Blank[]]], Power[Pattern[x, Blank[]], Optional[Pattern[n, Blank[]]]]]], Optional[Pattern[b, Blank[]]]]], Optional[Pattern[d, Blank[]]]]], Pattern[x, Blank[Symbol]]] := -Simp[(x*Sinh[d*(a + b*Log[c*x^n])])/(b^2*d^2*n^2 - 1), x] + Simp[(b*d*n*x*Cosh[d*(a + b*Log[c*x^n])])/(b^2*d^2*n^2 - 1), x] /; FreeQ[{a, b, c, d, n}, x] && NeQ[b^2*d^2*n^2 - 1, 0]
  5518. Int[Cosh[Times[Plus[Optional[Pattern[a, Blank[]]], Times[Log[Times[Optional[Pattern[c, Blank[]]], Power[Pattern[x, Blank[]], Optional[Pattern[n, Blank[]]]]]], Optional[Pattern[b, Blank[]]]]], Optional[Pattern[d, Blank[]]]]], Pattern[x, Blank[Symbol]]] := -Simp[(x*Cosh[d*(a + b*Log[c*x^n])])/(b^2*d^2*n^2 - 1), x] + Simp[(b*d*n*x*Sinh[d*(a + b*Log[c*x^n])])/(b^2*d^2*n^2 - 1), x] /; FreeQ[{a, b, c, d, n}, x] && NeQ[b^2*d^2*n^2 - 1, 0]
  5519. Int[Power[Sinh[Times[Plus[Optional[Pattern[a, Blank[]]], Times[Log[Times[Optional[Pattern[c, Blank[]]], Power[Pattern[x, Blank[]], Optional[Pattern[n, Blank[]]]]]], Optional[Pattern[b, Blank[]]]]], Optional[Pattern[d, Blank[]]]]], Pattern[p, Blank[]]], Pattern[x, Blank[Symbol]]] := -Simp[(x*Sinh[d*(a + b*Log[c*x^n])]^p)/(b^2*d^2*n^2*p^2 - 1), x] + (-Dist[(b^2*d^2*n^2*p*(p - 1))/(b^2*d^2*n^2*p^2 - 1), Int[Sinh[d*(a + b*Log[c*x^n])]^(p - 2), x], x] + Simp[(b*d*n*p*x*Cosh[d*(a + b*Log[c*x^n])]*Sinh[d*(a + b*Log[c*x^n])]^(p - 1))/(b^2*d^2*n^2*p^2 - 1), x]) /; FreeQ[{a, b, c, d, n}, x] && IGtQ[p, 1] && NeQ[b^2*d^2*n^2*p^2 - 1, 0]
  5520. Int[Power[Cosh[Times[Plus[Optional[Pattern[a, Blank[]]], Times[Log[Times[Optional[Pattern[c, Blank[]]], Power[Pattern[x, Blank[]], Optional[Pattern[n, Blank[]]]]]], Optional[Pattern[b, Blank[]]]]], Optional[Pattern[d, Blank[]]]]], Pattern[p, Blank[]]], Pattern[x, Blank[Symbol]]] := -Simp[(x*Cosh[d*(a + b*Log[c*x^n])]^p)/(b^2*d^2*n^2*p^2 - 1), x] + (Dist[(b^2*d^2*n^2*p*(p - 1))/(b^2*d^2*n^2*p^2 - 1), Int[Cosh[d*(a + b*Log[c*x^n])]^(p - 2), x], x] + Simp[(b*d*n*p*x*Cosh[d*(a + b*Log[c*x^n])]^(p - 1)*Sinh[d*(a + b*Log[c*x^n])])/(b^2*d^2*n^2*p^2 - 1), x]) /; FreeQ[{a, b, c, d, n}, x] && IGtQ[p, 1] && NeQ[b^2*d^2*n^2*p^2 - 1, 0]
  5521. Int[Power[Sinh[Times[Plus[Optional[Pattern[a, Blank[]]], Times[Log[Pattern[x, Blank[]]], Optional[Pattern[b, Blank[]]]]], Optional[Pattern[d, Blank[]]]]], Optional[Pattern[p, Blank[]]]], Pattern[x, Blank[Symbol]]] := Dist[1/(2^p*b^p*d^p*p^p), Int[ExpandIntegrand[(-(1/(E^(a*b*d^2*p)*x^p^(-1))) + E^(a*b*d^2*p)*x^(1/p))^p, x], x], x] /; FreeQ[{a, b, d}, x] && IGtQ[p, 0] && EqQ[b^2*d^2*p^2 - 1, 0]
  5522. Int[Power[Cosh[Times[Plus[Optional[Pattern[a, Blank[]]], Times[Log[Pattern[x, Blank[]]], Optional[Pattern[b, Blank[]]]]], Optional[Pattern[d, Blank[]]]]], Optional[Pattern[p, Blank[]]]], Pattern[x, Blank[Symbol]]] := Dist[1/2^p, Int[ExpandIntegrand[(1/(E^(a*b*d^2*p)*x^p^(-1)) + E^(a*b*d^2*p)*x^(1/p))^p, x], x], x] /; FreeQ[{a, b, d}, x] && IGtQ[p, 0] && EqQ[b^2*d^2*p^2 - 1, 0]
  5523. Int[Power[Sinh[Times[Plus[Optional[Pattern[a, Blank[]]], Times[Log[Pattern[x, Blank[]]], Optional[Pattern[b, Blank[]]]]], Optional[Pattern[d, Blank[]]]]], Pattern[p, Blank[]]], Pattern[x, Blank[Symbol]]] := Dist[Sinh[d*(a + b*Log[x])]^p/(x^(b*d*p)*(1 - 1/(E^(2*a*d)*x^(2*b*d)))^p), Int[x^(b*d*p)*(1 - 1/(E^(2*a*d)*x^(2*b*d)))^p, x], x] /; FreeQ[{a, b, d, p}, x] && !IntegerQ[p]
  5524. Int[Power[Cosh[Times[Plus[Optional[Pattern[a, Blank[]]], Times[Log[Pattern[x, Blank[]]], Optional[Pattern[b, Blank[]]]]], Optional[Pattern[d, Blank[]]]]], Pattern[p, Blank[]]], Pattern[x, Blank[Symbol]]] := Dist[Cosh[d*(a + b*Log[x])]^p/(x^(b*d*p)*(1 + 1/(E^(2*a*d)*x^(2*b*d)))^p), Int[x^(b*d*p)*(1 + 1/(E^(2*a*d)*x^(2*b*d)))^p, x], x] /; FreeQ[{a, b, d, p}, x] && !IntegerQ[p]
  5525. Int[Power[Sinh[Times[Plus[Optional[Pattern[a, Blank[]]], Times[Log[Times[Optional[Pattern[c, Blank[]]], Power[Pattern[x, Blank[]], Optional[Pattern[n, Blank[]]]]]], Optional[Pattern[b, Blank[]]]]], Optional[Pattern[d, Blank[]]]]], Optional[Pattern[p, Blank[]]]], Pattern[x, Blank[Symbol]]] := Dist[x/(n*(c*x^n)^(1/n)), Subst[Int[x^(1/n - 1)*Sinh[d*(a + b*Log[x])]^p, x], x, c*x^n], x] /; FreeQ[{a, b, c, d, n, p}, x] && (NeQ[c, 1] || NeQ[n, 1])
  5526. Int[Power[Cosh[Times[Plus[Optional[Pattern[a, Blank[]]], Times[Log[Times[Optional[Pattern[c, Blank[]]], Power[Pattern[x, Blank[]], Optional[Pattern[n, Blank[]]]]]], Optional[Pattern[b, Blank[]]]]], Optional[Pattern[d, Blank[]]]]], Optional[Pattern[p, Blank[]]]], Pattern[x, Blank[Symbol]]] := Dist[x/(n*(c*x^n)^(1/n)), Subst[Int[x^(1/n - 1)*Cosh[d*(a + b*Log[x])]^p, x], x, c*x^n], x] /; FreeQ[{a, b, c, d, n, p}, x] && (NeQ[c, 1] || NeQ[n, 1])
  5527. Int[Times[Power[Times[Optional[Pattern[e, Blank[]]], Pattern[x, Blank[]]], Optional[Pattern[m, Blank[]]]], Sinh[Times[Plus[Optional[Pattern[a, Blank[]]], Times[Log[Times[Optional[Pattern[c, Blank[]]], Power[Pattern[x, Blank[]], Optional[Pattern[n, Blank[]]]]]], Optional[Pattern[b, Blank[]]]]], Optional[Pattern[d, Blank[]]]]]], Pattern[x, Blank[Symbol]]] := -Simp[((m + 1)*(e*x)^(m + 1)*Sinh[d*(a + b*Log[c*x^n])])/(b^2*d^2*e*n^2 - e*(m + 1)^2), x] + Simp[(b*d*n*(e*x)^(m + 1)*Cosh[d*(a + b*Log[c*x^n])])/(b^2*d^2*e*n^2 - e*(m + 1)^2), x] /; FreeQ[{a, b, c, d, e, m, n}, x] && NeQ[b^2*d^2*n^2 - (m + 1)^2, 0]
  5528. Int[Times[Cosh[Times[Plus[Optional[Pattern[a, Blank[]]], Times[Log[Times[Optional[Pattern[c, Blank[]]], Power[Pattern[x, Blank[]], Optional[Pattern[n, Blank[]]]]]], Optional[Pattern[b, Blank[]]]]], Optional[Pattern[d, Blank[]]]]], Power[Times[Optional[Pattern[e, Blank[]]], Pattern[x, Blank[]]], Optional[Pattern[m, Blank[]]]]], Pattern[x, Blank[Symbol]]] := -Simp[((m + 1)*(e*x)^(m + 1)*Cosh[d*(a + b*Log[c*x^n])])/(b^2*d^2*e*n^2 - e*(m + 1)^2), x] + Simp[(b*d*n*(e*x)^(m + 1)*Sinh[d*(a + b*Log[c*x^n])])/(b^2*d^2*e*n^2 - e*(m + 1)^2), x] /; FreeQ[{a, b, c, d, e, m, n}, x] && NeQ[b^2*d^2*n^2 - (m + 1)^2, 0]
  5529. Int[Times[Power[Times[Optional[Pattern[e, Blank[]]], Pattern[x, Blank[]]], Optional[Pattern[m, Blank[]]]], Power[Sinh[Times[Plus[Optional[Pattern[a, Blank[]]], Times[Log[Times[Optional[Pattern[c, Blank[]]], Power[Pattern[x, Blank[]], Optional[Pattern[n, Blank[]]]]]], Optional[Pattern[b, Blank[]]]]], Optional[Pattern[d, Blank[]]]]], Pattern[p, Blank[]]]], Pattern[x, Blank[Symbol]]] := -Simp[((m + 1)*(e*x)^(m + 1)*Sinh[d*(a + b*Log[c*x^n])]^p)/(b^2*d^2*e*n^2*p^2 - e*(m + 1)^2), x] + (-Dist[(b^2*d^2*n^2*p*(p - 1))/(b^2*d^2*n^2*p^2 - (m + 1)^2), Int[(e*x)^m*Sinh[d*(a + b*Log[c*x^n])]^(p - 2), x], x] + Simp[(b*d*n*p*(e*x)^(m + 1)*Cosh[d*(a + b*Log[c*x^n])]*Sinh[d*(a + b*Log[c*x^n])]^(p - 1))/(b^2*d^2*e*n^2*p^2 - e*(m + 1)^2), x]) /; FreeQ[{a, b, c, d, e, m, n}, x] && IGtQ[p, 1] && NeQ[b^2*d^2*n^2*p^2 - (m + 1)^2, 0]
  5530. Int[Times[Power[Cosh[Times[Plus[Optional[Pattern[a, Blank[]]], Times[Log[Times[Optional[Pattern[c, Blank[]]], Power[Pattern[x, Blank[]], Optional[Pattern[n, Blank[]]]]]], Optional[Pattern[b, Blank[]]]]], Optional[Pattern[d, Blank[]]]]], Pattern[p, Blank[]]], Power[Times[Optional[Pattern[e, Blank[]]], Pattern[x, Blank[]]], Optional[Pattern[m, Blank[]]]]], Pattern[x, Blank[Symbol]]] := -Simp[((m + 1)*(e*x)^(m + 1)*Cosh[d*(a + b*Log[c*x^n])]^p)/(b^2*d^2*e*n^2*p^2 - e*(m + 1)^2), x] + (Dist[(b^2*d^2*n^2*p*(p - 1))/(b^2*d^2*n^2*p^2 - (m + 1)^2), Int[(e*x)^m*Cosh[d*(a + b*Log[c*x^n])]^(p - 2), x], x] + Simp[(b*d*n*p*(e*x)^(m + 1)*Sinh[d*(a + b*Log[c*x^n])]*Cosh[d*(a + b*Log[c*x^n])]^(p - 1))/(b^2*d^2*e*n^2*p^2 - e*(m + 1)^2), x]) /; FreeQ[{a, b, c, d, e, m, n}, x] && IGtQ[p, 1] && NeQ[b^2*d^2*n^2*p^2 - (m + 1)^2, 0]
  5531. Int[Times[Power[Times[Optional[Pattern[e, Blank[]]], Pattern[x, Blank[]]], Optional[Pattern[m, Blank[]]]], Power[Sinh[Times[Plus[Optional[Pattern[a, Blank[]]], Times[Log[Pattern[x, Blank[]]], Optional[Pattern[b, Blank[]]]]], Optional[Pattern[d, Blank[]]]]], Optional[Pattern[p, Blank[]]]]], Pattern[x, Blank[Symbol]]] := Dist[(m + 1)^p/(2^p*b^p*d^p*p^p), Int[ExpandIntegrand[(e*x)^m*(-(1/(E^((a*b*d^2*p)/(m + 1))*x^((m + 1)/p))) + E^((a*b*d^2*p)/(m + 1))*x^((m + 1)/p))^p, x], x], x] /; FreeQ[{a, b, d, e, m}, x] && IGtQ[p, 0] && EqQ[b^2*d^2*p^2 - (m + 1)^2, 0]
  5532. Int[Times[Power[Cosh[Times[Plus[Optional[Pattern[a, Blank[]]], Times[Log[Pattern[x, Blank[]]], Optional[Pattern[b, Blank[]]]]], Optional[Pattern[d, Blank[]]]]], Optional[Pattern[p, Blank[]]]], Power[Times[Optional[Pattern[e, Blank[]]], Pattern[x, Blank[]]], Optional[Pattern[m, Blank[]]]]], Pattern[x, Blank[Symbol]]] := Dist[1/2^p, Int[ExpandIntegrand[(e*x)^m*(1/(E^((a*b*d^2*p)/(m + 1))*x^((m + 1)/p)) + E^((a*b*d^2*p)/(m + 1))*x^((m + 1)/p))^p, x], x], x] /; FreeQ[{a, b, d, e, m}, x] && IGtQ[p, 0] && EqQ[b^2*d^2*p^2 - (m + 1)^2, 0]
  5533. Int[Times[Power[Times[Optional[Pattern[e, Blank[]]], Pattern[x, Blank[]]], Optional[Pattern[m, Blank[]]]], Power[Sinh[Times[Plus[Optional[Pattern[a, Blank[]]], Times[Log[Pattern[x, Blank[]]], Optional[Pattern[b, Blank[]]]]], Optional[Pattern[d, Blank[]]]]], Pattern[p, Blank[]]]], Pattern[x, Blank[Symbol]]] := Dist[Sinh[d*(a + b*Log[x])]^p/(x^(b*d*p)*(1 - 1/(E^(2*a*d)*x^(2*b*d)))^p), Int[(e*x)^m*x^(b*d*p)*(1 - 1/(E^(2*a*d)*x^(2*b*d)))^p, x], x] /; FreeQ[{a, b, d, e, m, p}, x] && !IntegerQ[p]
  5534. Int[Times[Power[Cosh[Times[Plus[Optional[Pattern[a, Blank[]]], Times[Log[Pattern[x, Blank[]]], Optional[Pattern[b, Blank[]]]]], Optional[Pattern[d, Blank[]]]]], Pattern[p, Blank[]]], Power[Times[Optional[Pattern[e, Blank[]]], Pattern[x, Blank[]]], Optional[Pattern[m, Blank[]]]]], Pattern[x, Blank[Symbol]]] := Dist[Cosh[d*(a + b*Log[x])]^p/(x^(b*d*p)*(1 + 1/(E^(2*a*d)*x^(2*b*d)))^p), Int[(e*x)^m*x^(b*d*p)*(1 + 1/(E^(2*a*d)*x^(2*b*d)))^p, x], x] /; FreeQ[{a, b, d, e, m, p}, x] && !IntegerQ[p]
  5535. Int[Times[Power[Times[Optional[Pattern[e, Blank[]]], Pattern[x, Blank[]]], Optional[Pattern[m, Blank[]]]], Power[Sinh[Times[Plus[Optional[Pattern[a, Blank[]]], Times[Log[Times[Optional[Pattern[c, Blank[]]], Power[Pattern[x, Blank[]], Optional[Pattern[n, Blank[]]]]]], Optional[Pattern[b, Blank[]]]]], Optional[Pattern[d, Blank[]]]]], Optional[Pattern[p, Blank[]]]]], Pattern[x, Blank[Symbol]]] := Dist[(e*x)^(m + 1)/(e*n*(c*x^n)^((m + 1)/n)), Subst[Int[x^((m + 1)/n - 1)*Sinh[d*(a + b*Log[x])]^p, x], x, c*x^n], x] /; FreeQ[{a, b, c, d, e, m, n, p}, x] && (NeQ[c, 1] || NeQ[n, 1])
  5536. Int[Times[Power[Cosh[Times[Plus[Optional[Pattern[a, Blank[]]], Times[Log[Times[Optional[Pattern[c, Blank[]]], Power[Pattern[x, Blank[]], Optional[Pattern[n, Blank[]]]]]], Optional[Pattern[b, Blank[]]]]], Optional[Pattern[d, Blank[]]]]], Optional[Pattern[p, Blank[]]]], Power[Times[Optional[Pattern[e, Blank[]]], Pattern[x, Blank[]]], Optional[Pattern[m, Blank[]]]]], Pattern[x, Blank[Symbol]]] := Dist[(e*x)^(m + 1)/(e*n*(c*x^n)^((m + 1)/n)), Subst[Int[x^((m + 1)/n - 1)*Cosh[d*(a + b*Log[x])]^p, x], x, c*x^n], x] /; FreeQ[{a, b, c, d, e, m, n, p}, x] && (NeQ[c, 1] || NeQ[n, 1])
  5537. Int[Times[Power[Times[Plus[Optional[Pattern[e, Blank[]]], Times[Log[Times[Optional[Pattern[g, Blank[]]], Power[Pattern[x, Blank[]], Optional[Pattern[m, Blank[]]]]]], Optional[Pattern[f, Blank[]]]]], Optional[Pattern[h, Blank[]]]], Optional[Pattern[q, Blank[]]]], Sinh[Times[Plus[Optional[Pattern[a, Blank[]]], Times[Log[Times[Optional[Pattern[c, Blank[]]], Power[Pattern[x, Blank[]], Optional[Pattern[n, Blank[]]]]]], Optional[Pattern[b, Blank[]]]]], Optional[Pattern[d, Blank[]]]]]], Pattern[x, Blank[Symbol]]] := -Dist[1/(E^(a*d)*(c*x^n)^(b*d)*(2/x^(b*d*n))), Int[(h*(e + f*Log[g*x^m]))^q/x^(b*d*n), x], x] + Dist[(E^(a*d)*(c*x^n)^(b*d))/(2*x^(b*d*n)), Int[x^(b*d*n)*(h*(e + f*Log[g*x^m]))^q, x], x] /; FreeQ[{a, b, c, d, e, f, g, h, m, n, q}, x]
  5538. Int[Times[Cosh[Times[Plus[Optional[Pattern[a, Blank[]]], Times[Log[Times[Optional[Pattern[c, Blank[]]], Power[Pattern[x, Blank[]], Optional[Pattern[n, Blank[]]]]]], Optional[Pattern[b, Blank[]]]]], Optional[Pattern[d, Blank[]]]]], Power[Times[Plus[Optional[Pattern[e, Blank[]]], Times[Log[Times[Optional[Pattern[g, Blank[]]], Power[Pattern[x, Blank[]], Optional[Pattern[m, Blank[]]]]]], Optional[Pattern[f, Blank[]]]]], Optional[Pattern[h, Blank[]]]], Optional[Pattern[q, Blank[]]]]], Pattern[x, Blank[Symbol]]] := Dist[1/(E^(a*d)*(c*x^n)^(b*d)*(2/x^(b*d*n))), Int[(h*(e + f*Log[g*x^m]))^q/x^(b*d*n), x], x] + Dist[(E^(a*d)*(c*x^n)^(b*d))/(2*x^(b*d*n)), Int[x^(b*d*n)*(h*(e + f*Log[g*x^m]))^q, x], x] /; FreeQ[{a, b, c, d, e, f, g, h, m, n, q}, x]
  5539. Int[Times[Power[Times[Plus[Optional[Pattern[e, Blank[]]], Times[Log[Times[Optional[Pattern[g, Blank[]]], Power[Pattern[x, Blank[]], Optional[Pattern[m, Blank[]]]]]], Optional[Pattern[f, Blank[]]]]], Optional[Pattern[h, Blank[]]]], Optional[Pattern[q, Blank[]]]], Power[Times[Optional[Pattern[i, Blank[]]], Pattern[x, Blank[]]], Optional[Pattern[r, Blank[]]]], Sinh[Times[Plus[Optional[Pattern[a, Blank[]]], Times[Log[Times[Optional[Pattern[c, Blank[]]], Power[Pattern[x, Blank[]], Optional[Pattern[n, Blank[]]]]]], Optional[Pattern[b, Blank[]]]]], Optional[Pattern[d, Blank[]]]]]], Pattern[x, Blank[Symbol]]] := -Dist[(i*x)^r/(E^(a*d)*(c*x^n)^(b*d)*(2*x^(r - b*d*n))), Int[x^(r - b*d*n)*(h*(e + f*Log[g*x^m]))^q, x], x] + Dist[(E^(a*d)*(i*x)^r*(c*x^n)^(b*d))/(2*x^(r + b*d*n)), Int[x^(r + b*d*n)*(h*(e + f*Log[g*x^m]))^q, x], x] /; FreeQ[{a, b, c, d, e, f, g, h, i, m, n, q, r}, x]
  5540. Int[Times[Cosh[Times[Plus[Optional[Pattern[a, Blank[]]], Times[Log[Times[Optional[Pattern[c, Blank[]]], Power[Pattern[x, Blank[]], Optional[Pattern[n, Blank[]]]]]], Optional[Pattern[b, Blank[]]]]], Optional[Pattern[d, Blank[]]]]], Power[Times[Plus[Optional[Pattern[e, Blank[]]], Times[Log[Times[Optional[Pattern[g, Blank[]]], Power[Pattern[x, Blank[]], Optional[Pattern[m, Blank[]]]]]], Optional[Pattern[f, Blank[]]]]], Optional[Pattern[h, Blank[]]]], Optional[Pattern[q, Blank[]]]], Power[Times[Optional[Pattern[i, Blank[]]], Pattern[x, Blank[]]], Optional[Pattern[r, Blank[]]]]], Pattern[x, Blank[Symbol]]] := Dist[(i*x)^r/(E^(a*d)*(c*x^n)^(b*d)*(2*x^(r - b*d*n))), Int[x^(r - b*d*n)*(h*(e + f*Log[g*x^m]))^q, x], x] + Dist[(E^(a*d)*(i*x)^r*(c*x^n)^(b*d))/(2*x^(r + b*d*n)), Int[x^(r + b*d*n)*(h*(e + f*Log[g*x^m]))^q, x], x] /; FreeQ[{a, b, c, d, e, f, g, h, i, m, n, q, r}, x]
  5541. Int[Power[Sech[Times[Plus[Optional[Pattern[a, Blank[]]], Times[Log[Pattern[x, Blank[]]], Optional[Pattern[b, Blank[]]]]], Optional[Pattern[d, Blank[]]]]], Optional[Pattern[p, Blank[]]]], Pattern[x, Blank[Symbol]]] := Dist[2^p/E^(a*d*p), Int[1/(x^(b*d*p)*(1 + 1/(E^(2*a*d)*x^(2*b*d)))^p), x], x] /; FreeQ[{a, b, d}, x] && IntegerQ[p]
  5542. Int[Power[Csch[Times[Plus[Optional[Pattern[a, Blank[]]], Times[Log[Pattern[x, Blank[]]], Optional[Pattern[b, Blank[]]]]], Optional[Pattern[d, Blank[]]]]], Optional[Pattern[p, Blank[]]]], Pattern[x, Blank[Symbol]]] := Dist[2^p/E^(a*d*p), Int[1/(x^(b*d*p)*(1 - 1/(E^(2*a*d)*x^(2*b*d)))^p), x], x] /; FreeQ[{a, b, d}, x] && IntegerQ[p]
  5543. Int[Power[Sech[Times[Plus[Optional[Pattern[a, Blank[]]], Times[Log[Pattern[x, Blank[]]], Optional[Pattern[b, Blank[]]]]], Optional[Pattern[d, Blank[]]]]], Optional[Pattern[p, Blank[]]]], Pattern[x, Blank[Symbol]]] := Dist[(Sech[d*(a + b*Log[x])]^p*(1 + 1/(E^(2*a*d)*x^(2*b*d)))^p)/x^(-(b*d*p)), Int[1/(x^(b*d*p)*(1 + 1/(E^(2*a*d)*x^(2*b*d)))^p), x], x] /; FreeQ[{a, b, d, p}, x] && !IntegerQ[p]
  5544. Int[Power[Csch[Times[Plus[Optional[Pattern[a, Blank[]]], Times[Log[Pattern[x, Blank[]]], Optional[Pattern[b, Blank[]]]]], Optional[Pattern[d, Blank[]]]]], Optional[Pattern[p, Blank[]]]], Pattern[x, Blank[Symbol]]] := Dist[(Csch[d*(a + b*Log[x])]^p*(1 - 1/(E^(2*a*d)*x^(2*b*d)))^p)/x^(-(b*d*p)), Int[1/(x^(b*d*p)*(1 - 1/(E^(2*a*d)*x^(2*b*d)))^p), x], x] /; FreeQ[{a, b, d, p}, x] && !IntegerQ[p]
  5545. Int[Power[Sech[Times[Plus[Optional[Pattern[a, Blank[]]], Times[Log[Times[Optional[Pattern[c, Blank[]]], Power[Pattern[x, Blank[]], Optional[Pattern[n, Blank[]]]]]], Optional[Pattern[b, Blank[]]]]], Optional[Pattern[d, Blank[]]]]], Optional[Pattern[p, Blank[]]]], Pattern[x, Blank[Symbol]]] := Dist[x/(n*(c*x^n)^(1/n)), Subst[Int[x^(1/n - 1)*Sech[d*(a + b*Log[x])]^p, x], x, c*x^n], x] /; FreeQ[{a, b, c, d, n, p}, x] && (NeQ[c, 1] || NeQ[n, 1])
  5546. Int[Power[Csch[Times[Plus[Optional[Pattern[a, Blank[]]], Times[Log[Times[Optional[Pattern[c, Blank[]]], Power[Pattern[x, Blank[]], Optional[Pattern[n, Blank[]]]]]], Optional[Pattern[b, Blank[]]]]], Optional[Pattern[d, Blank[]]]]], Optional[Pattern[p, Blank[]]]], Pattern[x, Blank[Symbol]]] := Dist[x/(n*(c*x^n)^(1/n)), Subst[Int[x^(1/n - 1)*Csch[d*(a + b*Log[x])]^p, x], x, c*x^n], x] /; FreeQ[{a, b, c, d, n, p}, x] && (NeQ[c, 1] || NeQ[n, 1])
  5547. Int[Times[Power[Times[Optional[Pattern[e, Blank[]]], Pattern[x, Blank[]]], Optional[Pattern[m, Blank[]]]], Power[Sech[Times[Plus[Optional[Pattern[a, Blank[]]], Times[Log[Pattern[x, Blank[]]], Optional[Pattern[b, Blank[]]]]], Optional[Pattern[d, Blank[]]]]], Optional[Pattern[p, Blank[]]]]], Pattern[x, Blank[Symbol]]] := Dist[2^p/E^(a*d*p), Int[(e*x)^m/(x^(b*d*p)*(1 + 1/(E^(2*a*d)*x^(2*b*d)))^p), x], x] /; FreeQ[{a, b, d, e, m}, x] && IntegerQ[p]
  5548. Int[Times[Power[Csch[Times[Plus[Optional[Pattern[a, Blank[]]], Times[Log[Pattern[x, Blank[]]], Optional[Pattern[b, Blank[]]]]], Optional[Pattern[d, Blank[]]]]], Optional[Pattern[p, Blank[]]]], Power[Times[Optional[Pattern[e, Blank[]]], Pattern[x, Blank[]]], Optional[Pattern[m, Blank[]]]]], Pattern[x, Blank[Symbol]]] := Dist[2^p/E^(a*d*p), Int[(e*x)^m/(x^(b*d*p)*(1 - 1/(E^(2*a*d)*x^(2*b*d)))^p), x], x] /; FreeQ[{a, b, d, e, m}, x] && IntegerQ[p]
  5549. Int[Times[Power[Times[Optional[Pattern[e, Blank[]]], Pattern[x, Blank[]]], Optional[Pattern[m, Blank[]]]], Power[Sech[Times[Plus[Optional[Pattern[a, Blank[]]], Times[Log[Pattern[x, Blank[]]], Optional[Pattern[b, Blank[]]]]], Optional[Pattern[d, Blank[]]]]], Optional[Pattern[p, Blank[]]]]], Pattern[x, Blank[Symbol]]] := Dist[(Sech[d*(a + b*Log[x])]^p*(1 + 1/(E^(2*a*d)*x^(2*b*d)))^p)/x^(-(b*d*p)), Int[(e*x)^m/(x^(b*d*p)*(1 + 1/(E^(2*a*d)*x^(2*b*d)))^p), x], x] /; FreeQ[{a, b, d, e, m, p}, x] && !IntegerQ[p]
  5550. Int[Times[Power[Csch[Times[Plus[Optional[Pattern[a, Blank[]]], Times[Log[Pattern[x, Blank[]]], Optional[Pattern[b, Blank[]]]]], Optional[Pattern[d, Blank[]]]]], Optional[Pattern[p, Blank[]]]], Power[Times[Optional[Pattern[e, Blank[]]], Pattern[x, Blank[]]], Optional[Pattern[m, Blank[]]]]], Pattern[x, Blank[Symbol]]] := Dist[(Csch[d*(a + b*Log[x])]^p*(1 - 1/(E^(2*a*d)*x^(2*b*d)))^p)/x^(-(b*d*p)), Int[(e*x)^m/(x^(b*d*p)*(1 - 1/(E^(2*a*d)*x^(2*b*d)))^p), x], x] /; FreeQ[{a, b, d, e, m, p}, x] && !IntegerQ[p]
  5551. Int[Times[Power[Times[Optional[Pattern[e, Blank[]]], Pattern[x, Blank[]]], Optional[Pattern[m, Blank[]]]], Power[Sech[Times[Plus[Optional[Pattern[a, Blank[]]], Times[Log[Times[Optional[Pattern[c, Blank[]]], Power[Pattern[x, Blank[]], Optional[Pattern[n, Blank[]]]]]], Optional[Pattern[b, Blank[]]]]], Optional[Pattern[d, Blank[]]]]], Optional[Pattern[p, Blank[]]]]], Pattern[x, Blank[Symbol]]] := Dist[(e*x)^(m + 1)/(e*n*(c*x^n)^((m + 1)/n)), Subst[Int[x^((m + 1)/n - 1)*Sech[d*(a + b*Log[x])]^p, x], x, c*x^n], x] /; FreeQ[{a, b, c, d, e, m, n, p}, x] && (NeQ[c, 1] || NeQ[n, 1])
  5552. Int[Times[Power[Csch[Times[Plus[Optional[Pattern[a, Blank[]]], Times[Log[Times[Optional[Pattern[c, Blank[]]], Power[Pattern[x, Blank[]], Optional[Pattern[n, Blank[]]]]]], Optional[Pattern[b, Blank[]]]]], Optional[Pattern[d, Blank[]]]]], Optional[Pattern[p, Blank[]]]], Power[Times[Optional[Pattern[e, Blank[]]], Pattern[x, Blank[]]], Optional[Pattern[m, Blank[]]]]], Pattern[x, Blank[Symbol]]] := Dist[(e*x)^(m + 1)/(e*n*(c*x^n)^((m + 1)/n)), Subst[Int[x^((m + 1)/n - 1)*Csch[d*(a + b*Log[x])]^p, x], x, c*x^n], x] /; FreeQ[{a, b, c, d, e, m, n, p}, x] && (NeQ[c, 1] || NeQ[n, 1])
  5553. Int[Times[Log[Times[Optional[Pattern[b, Blank[]]], Pattern[x, Blank[]]]], Sinh[Times[Log[Times[Optional[Pattern[b, Blank[]]], Pattern[x, Blank[]]]], Optional[Pattern[a, Blank[]]], Pattern[x, Blank[]]]]], Pattern[x, Blank[Symbol]]] := Simp[Cosh[a*x*Log[b*x]]/a, x] - Int[Sinh[a*x*Log[b*x]], x] /; FreeQ[{a, b}, x]
  5554. Int[Times[Cosh[Times[Log[Times[Optional[Pattern[b, Blank[]]], Pattern[x, Blank[]]]], Optional[Pattern[a, Blank[]]], Pattern[x, Blank[]]]], Log[Times[Optional[Pattern[b, Blank[]]], Pattern[x, Blank[]]]]], Pattern[x, Blank[Symbol]]] := Simp[Sinh[a*x*Log[b*x]]/a, x] - Int[Cosh[a*x*Log[b*x]], x] /; FreeQ[{a, b}, x]
  5555. Int[Times[Log[Times[Optional[Pattern[b, Blank[]]], Pattern[x, Blank[]]]], Power[Pattern[x, Blank[]], Optional[Pattern[m, Blank[]]]], Sinh[Times[Log[Times[Optional[Pattern[b, Blank[]]], Pattern[x, Blank[]]]], Optional[Pattern[a, Blank[]]], Power[Pattern[x, Blank[]], Optional[Pattern[n, Blank[]]]]]]], Pattern[x, Blank[Symbol]]] := Simp[Cosh[a*x^n*Log[b*x]]/(a*n), x] - Dist[1/n, Int[x^m*Sinh[a*x^n*Log[b*x]], x], x] /; FreeQ[{a, b, m, n}, x] && EqQ[m, n - 1]
  5556. Int[Times[Cosh[Times[Log[Times[Optional[Pattern[b, Blank[]]], Pattern[x, Blank[]]]], Optional[Pattern[a, Blank[]]], Power[Pattern[x, Blank[]], Optional[Pattern[n, Blank[]]]]]], Log[Times[Optional[Pattern[b, Blank[]]], Pattern[x, Blank[]]]], Power[Pattern[x, Blank[]], Optional[Pattern[m, Blank[]]]]], Pattern[x, Blank[Symbol]]] := Simp[Sinh[a*x^n*Log[b*x]]/(a*n), x] - Dist[1/n, Int[x^m*Cosh[a*x^n*Log[b*x]], x], x] /; FreeQ[{a, b, m, n}, x] && EqQ[m, n - 1]
  5557. Int[Times[Power[Plus[Optional[Pattern[e, Blank[]]], Times[Optional[Pattern[f, Blank[]]], Pattern[x, Blank[]]]], Optional[Pattern[m, Blank[]]]], Power[Sinh[Plus[Optional[Pattern[c, Blank[]]], Times[Optional[Pattern[d, Blank[]]], Pattern[x, Blank[]]]]], Optional[Pattern[n, Blank[]]]], Power[Plus[Pattern[a, Blank[]], Times[Optional[Pattern[b, Blank[]]], Sinh[Plus[Optional[Pattern[c, Blank[]]], Times[Optional[Pattern[d, Blank[]]], Pattern[x, Blank[]]]]]]], -1]], Pattern[x, Blank[Symbol]]] := Dist[1/b, Int[(e + f*x)^m*Sinh[c + d*x]^(n - 1), x], x] - Dist[a/b, Int[((e + f*x)^m*Sinh[c + d*x]^(n - 1))/(a + b*Sinh[c + d*x]), x], x] /; FreeQ[{a, b, c, d, e, f}, x] && IGtQ[m, 0] && IGtQ[n, 0]
  5558. Int[Times[Power[Cosh[Plus[Optional[Pattern[c, Blank[]]], Times[Optional[Pattern[d, Blank[]]], Pattern[x, Blank[]]]]], Optional[Pattern[n, Blank[]]]], Power[Plus[Times[Cosh[Plus[Optional[Pattern[c, Blank[]]], Times[Optional[Pattern[d, Blank[]]], Pattern[x, Blank[]]]]], Optional[Pattern[b, Blank[]]]], Pattern[a, Blank[]]], -1], Power[Plus[Optional[Pattern[e, Blank[]]], Times[Optional[Pattern[f, Blank[]]], Pattern[x, Blank[]]]], Optional[Pattern[m, Blank[]]]]], Pattern[x, Blank[Symbol]]] := Dist[1/b, Int[(e + f*x)^m*Cosh[c + d*x]^(n - 1), x], x] - Dist[a/b, Int[((e + f*x)^m*Cosh[c + d*x]^(n - 1))/(a + b*Cosh[c + d*x]), x], x] /; FreeQ[{a, b, c, d, e, f}, x] && IGtQ[m, 0] && IGtQ[n, 0]
  5559. Int[Times[Cosh[Plus[Optional[Pattern[c, Blank[]]], Times[Optional[Pattern[d, Blank[]]], Pattern[x, Blank[]]]]], Power[Plus[Optional[Pattern[e, Blank[]]], Times[Optional[Pattern[f, Blank[]]], Pattern[x, Blank[]]]], Optional[Pattern[m, Blank[]]]], Power[Plus[Pattern[a, Blank[]], Times[Optional[Pattern[b, Blank[]]], Sinh[Plus[Optional[Pattern[c, Blank[]]], Times[Optional[Pattern[d, Blank[]]], Pattern[x, Blank[]]]]]]], -1]], Pattern[x, Blank[Symbol]]] := -Simp[(e + f*x)^(m + 1)/(b*f*(m + 1)), x] + Dist[2, Int[((e + f*x)^m*E^(c + d*x))/(a + b*E^(c + d*x)), x], x] /; FreeQ[{a, b, c, d, e, f}, x] && IGtQ[m, 0] && EqQ[a^2 + b^2, 0]
  5560. Int[Times[Power[Plus[Times[Cosh[Plus[Optional[Pattern[c, Blank[]]], Times[Optional[Pattern[d, Blank[]]], Pattern[x, Blank[]]]]], Optional[Pattern[b, Blank[]]]], Pattern[a, Blank[]]], -1], Power[Plus[Optional[Pattern[e, Blank[]]], Times[Optional[Pattern[f, Blank[]]], Pattern[x, Blank[]]]], Optional[Pattern[m, Blank[]]]], Sinh[Plus[Optional[Pattern[c, Blank[]]], Times[Optional[Pattern[d, Blank[]]], Pattern[x, Blank[]]]]]], Pattern[x, Blank[Symbol]]] := -Simp[(e + f*x)^(m + 1)/(b*f*(m + 1)), x] + Dist[2, Int[((e + f*x)^m*E^(c + d*x))/(a + b*E^(c + d*x)), x], x] /; FreeQ[{a, b, c, d, e, f}, x] && IGtQ[m, 0] && EqQ[a^2 - b^2, 0]
  5561. Int[Times[Cosh[Plus[Optional[Pattern[c, Blank[]]], Times[Optional[Pattern[d, Blank[]]], Pattern[x, Blank[]]]]], Power[Plus[Optional[Pattern[e, Blank[]]], Times[Optional[Pattern[f, Blank[]]], Pattern[x, Blank[]]]], Optional[Pattern[m, Blank[]]]], Power[Plus[Pattern[a, Blank[]], Times[Optional[Pattern[b, Blank[]]], Sinh[Plus[Optional[Pattern[c, Blank[]]], Times[Optional[Pattern[d, Blank[]]], Pattern[x, Blank[]]]]]]], -1]], Pattern[x, Blank[Symbol]]] := -Simp[(e + f*x)^(m + 1)/(b*f*(m + 1)), x] + (Int[((e + f*x)^m*E^(c + d*x))/(a - Rt[a^2 + b^2, 2] + b*E^(c + d*x)), x] + Int[((e + f*x)^m*E^(c + d*x))/(a + Rt[a^2 + b^2, 2] + b*E^(c + d*x)), x]) /; FreeQ[{a, b, c, d, e, f}, x] && IGtQ[m, 0] && NeQ[a^2 + b^2, 0]
  5562. Int[Times[Power[Plus[Times[Cosh[Plus[Optional[Pattern[c, Blank[]]], Times[Optional[Pattern[d, Blank[]]], Pattern[x, Blank[]]]]], Optional[Pattern[b, Blank[]]]], Pattern[a, Blank[]]], -1], Power[Plus[Optional[Pattern[e, Blank[]]], Times[Optional[Pattern[f, Blank[]]], Pattern[x, Blank[]]]], Optional[Pattern[m, Blank[]]]], Sinh[Plus[Optional[Pattern[c, Blank[]]], Times[Optional[Pattern[d, Blank[]]], Pattern[x, Blank[]]]]]], Pattern[x, Blank[Symbol]]] := -Simp[(e + f*x)^(m + 1)/(b*f*(m + 1)), x] + (Int[((e + f*x)^m*E^(c + d*x))/(a - Rt[a^2 - b^2, 2] + b*E^(c + d*x)), x] + Int[((e + f*x)^m*E^(c + d*x))/(a + Rt[a^2 - b^2, 2] + b*E^(c + d*x)), x]) /; FreeQ[{a, b, c, d, e, f}, x] && IGtQ[m, 0] && NeQ[a^2 - b^2, 0]
  5563. Int[Times[Power[Cosh[Plus[Optional[Pattern[c, Blank[]]], Times[Optional[Pattern[d, Blank[]]], Pattern[x, Blank[]]]]], Pattern[n, Blank[]]], Power[Plus[Optional[Pattern[e, Blank[]]], Times[Optional[Pattern[f, Blank[]]], Pattern[x, Blank[]]]], Optional[Pattern[m, Blank[]]]], Power[Plus[Pattern[a, Blank[]], Times[Optional[Pattern[b, Blank[]]], Sinh[Plus[Optional[Pattern[c, Blank[]]], Times[Optional[Pattern[d, Blank[]]], Pattern[x, Blank[]]]]]]], -1]], Pattern[x, Blank[Symbol]]] := Dist[1/a, Int[(e + f*x)^m*Cosh[c + d*x]^(n - 2), x], x] + Dist[1/b, Int[(e + f*x)^m*Cosh[c + d*x]^(n - 2)*Sinh[c + d*x], x], x] /; FreeQ[{a, b, c, d, e, f, m}, x] && IGtQ[n, 1] && EqQ[a^2 + b^2, 0]
  5564. Int[Times[Power[Plus[Times[Cosh[Plus[Optional[Pattern[c, Blank[]]], Times[Optional[Pattern[d, Blank[]]], Pattern[x, Blank[]]]]], Optional[Pattern[b, Blank[]]]], Pattern[a, Blank[]]], -1], Power[Plus[Optional[Pattern[e, Blank[]]], Times[Optional[Pattern[f, Blank[]]], Pattern[x, Blank[]]]], Optional[Pattern[m, Blank[]]]], Power[Sinh[Plus[Optional[Pattern[c, Blank[]]], Times[Optional[Pattern[d, Blank[]]], Pattern[x, Blank[]]]]], Pattern[n, Blank[]]]], Pattern[x, Blank[Symbol]]] := -Dist[a^(-1), Int[(e + f*x)^m*Sinh[c + d*x]^(n - 2), x], x] + Dist[1/b, Int[(e + f*x)^m*Sinh[c + d*x]^(n - 2)*Cosh[c + d*x], x], x] /; FreeQ[{a, b, c, d, e, f, m}, x] && IGtQ[n, 1] && EqQ[a^2 - b^2, 0]
  5565. Int[Times[Power[Cosh[Plus[Optional[Pattern[c, Blank[]]], Times[Optional[Pattern[d, Blank[]]], Pattern[x, Blank[]]]]], Pattern[n, Blank[]]], Power[Plus[Optional[Pattern[e, Blank[]]], Times[Optional[Pattern[f, Blank[]]], Pattern[x, Blank[]]]], Optional[Pattern[m, Blank[]]]], Power[Plus[Pattern[a, Blank[]], Times[Optional[Pattern[b, Blank[]]], Sinh[Plus[Optional[Pattern[c, Blank[]]], Times[Optional[Pattern[d, Blank[]]], Pattern[x, Blank[]]]]]]], -1]], Pattern[x, Blank[Symbol]]] := -Dist[a/b^2, Int[(e + f*x)^m*Cosh[c + d*x]^(n - 2), x], x] + (Dist[1/b, Int[(e + f*x)^m*Cosh[c + d*x]^(n - 2)*Sinh[c + d*x], x], x] + Dist[(a^2 + b^2)/b^2, Int[((e + f*x)^m*Cosh[c + d*x]^(n - 2))/(a + b*Sinh[c + d*x]), x], x]) /; FreeQ[{a, b, c, d, e, f}, x] && IGtQ[n, 1] && NeQ[a^2 + b^2, 0] && IGtQ[m, 0]
  5566. Int[Times[Power[Plus[Times[Cosh[Plus[Optional[Pattern[c, Blank[]]], Times[Optional[Pattern[d, Blank[]]], Pattern[x, Blank[]]]]], Optional[Pattern[b, Blank[]]]], Pattern[a, Blank[]]], -1], Power[Plus[Optional[Pattern[e, Blank[]]], Times[Optional[Pattern[f, Blank[]]], Pattern[x, Blank[]]]], Optional[Pattern[m, Blank[]]]], Power[Sinh[Plus[Optional[Pattern[c, Blank[]]], Times[Optional[Pattern[d, Blank[]]], Pattern[x, Blank[]]]]], Pattern[n, Blank[]]]], Pattern[x, Blank[Symbol]]] := -Dist[a/b^2, Int[(e + f*x)^m*Sinh[c + d*x]^(n - 2), x], x] + (Dist[1/b, Int[(e + f*x)^m*Sinh[c + d*x]^(n - 2)*Cosh[c + d*x], x], x] + Dist[(a^2 - b^2)/b^2, Int[((e + f*x)^m*Sinh[c + d*x]^(n - 2))/(a + b*Cosh[c + d*x]), x], x]) /; FreeQ[{a, b, c, d, e, f}, x] && IGtQ[n, 1] && NeQ[a^2 - b^2, 0] && IGtQ[m, 0]
  5567. Int[Times[Power[Plus[Optional[Pattern[e, Blank[]]], Times[Optional[Pattern[f, Blank[]]], Pattern[x, Blank[]]]], Optional[Pattern[m, Blank[]]]], Power[Plus[Pattern[a, Blank[]], Times[Optional[Pattern[b, Blank[]]], Sinh[Plus[Optional[Pattern[c, Blank[]]], Times[Optional[Pattern[d, Blank[]]], Pattern[x, Blank[]]]]]]], -1], Power[Tanh[Plus[Optional[Pattern[c, Blank[]]], Times[Optional[Pattern[d, Blank[]]], Pattern[x, Blank[]]]]], Optional[Pattern[n, Blank[]]]]], Pattern[x, Blank[Symbol]]] := Dist[1/b, Int[(e + f*x)^m*Sech[c + d*x]*Tanh[c + d*x]^(n - 1), x], x] - Dist[a/b, Int[((e + f*x)^m*Sech[c + d*x]*Tanh[c + d*x]^(n - 1))/(a + b*Sinh[c + d*x]), x], x] /; FreeQ[{a, b, c, d, e, f}, x] && IGtQ[m, 0] && IGtQ[n, 0]
  5568. Int[Times[Power[Coth[Plus[Optional[Pattern[c, Blank[]]], Times[Optional[Pattern[d, Blank[]]], Pattern[x, Blank[]]]]], Optional[Pattern[n, Blank[]]]], Power[Plus[Times[Cosh[Plus[Optional[Pattern[c, Blank[]]], Times[Optional[Pattern[d, Blank[]]], Pattern[x, Blank[]]]]], Optional[Pattern[b, Blank[]]]], Pattern[a, Blank[]]], -1], Power[Plus[Optional[Pattern[e, Blank[]]], Times[Optional[Pattern[f, Blank[]]], Pattern[x, Blank[]]]], Optional[Pattern[m, Blank[]]]]], Pattern[x, Blank[Symbol]]] := Dist[1/b, Int[(e + f*x)^m*Csch[c + d*x]*Coth[c + d*x]^(n - 1), x], x] - Dist[a/b, Int[((e + f*x)^m*Csch[c + d*x]*Coth[c + d*x]^(n - 1))/(a + b*Cosh[c + d*x]), x], x] /; FreeQ[{a, b, c, d, e, f}, x] && IGtQ[m, 0] && IGtQ[n, 0]
  5569. Int[Times[Power[Coth[Plus[Optional[Pattern[c, Blank[]]], Times[Optional[Pattern[d, Blank[]]], Pattern[x, Blank[]]]]], Optional[Pattern[n, Blank[]]]], Power[Plus[Optional[Pattern[e, Blank[]]], Times[Optional[Pattern[f, Blank[]]], Pattern[x, Blank[]]]], Optional[Pattern[m, Blank[]]]], Power[Plus[Pattern[a, Blank[]], Times[Optional[Pattern[b, Blank[]]], Sinh[Plus[Optional[Pattern[c, Blank[]]], Times[Optional[Pattern[d, Blank[]]], Pattern[x, Blank[]]]]]]], -1]], Pattern[x, Blank[Symbol]]] := Dist[1/a, Int[(e + f*x)^m*Coth[c + d*x]^n, x], x] - Dist[b/a, Int[((e + f*x)^m*Cosh[c + d*x]*Coth[c + d*x]^(n - 1))/(a + b*Sinh[c + d*x]), x], x] /; FreeQ[{a, b, c, d, e, f}, x] && IGtQ[m, 0] && IGtQ[n, 0]
  5570. Int[Times[Power[Plus[Times[Cosh[Plus[Optional[Pattern[c, Blank[]]], Times[Optional[Pattern[d, Blank[]]], Pattern[x, Blank[]]]]], Optional[Pattern[b, Blank[]]]], Pattern[a, Blank[]]], -1], Power[Plus[Optional[Pattern[e, Blank[]]], Times[Optional[Pattern[f, Blank[]]], Pattern[x, Blank[]]]], Optional[Pattern[m, Blank[]]]], Power[Tanh[Plus[Optional[Pattern[c, Blank[]]], Times[Optional[Pattern[d, Blank[]]], Pattern[x, Blank[]]]]], Optional[Pattern[n, Blank[]]]]], Pattern[x, Blank[Symbol]]] := Dist[1/a, Int[(e + f*x)^m*Tanh[c + d*x]^n, x], x] - Dist[b/a, Int[((e + f*x)^m*Sinh[c + d*x]*Tanh[c + d*x]^(n - 1))/(a + b*Cosh[c + d*x]), x], x] /; FreeQ[{a, b, c, d, e, f}, x] && IGtQ[m, 0] && IGtQ[n, 0]
  5571. Int[Times[Power[Plus[Optional[Pattern[e, Blank[]]], Times[Optional[Pattern[f, Blank[]]], Pattern[x, Blank[]]]], Optional[Pattern[m, Blank[]]]], Power[Sech[Plus[Optional[Pattern[c, Blank[]]], Times[Optional[Pattern[d, Blank[]]], Pattern[x, Blank[]]]]], Optional[Pattern[n, Blank[]]]], Power[Plus[Pattern[a, Blank[]], Times[Optional[Pattern[b, Blank[]]], Sinh[Plus[Optional[Pattern[c, Blank[]]], Times[Optional[Pattern[d, Blank[]]], Pattern[x, Blank[]]]]]]], -1]], Pattern[x, Blank[Symbol]]] := Dist[1/a, Int[(e + f*x)^m*Sech[c + d*x]^(n + 2), x], x] + Dist[1/b, Int[(e + f*x)^m*Sech[c + d*x]^(n + 1)*Tanh[c + d*x], x], x] /; FreeQ[{a, b, c, d, e, f, n}, x] && IGtQ[m, 0] && EqQ[a^2 + b^2, 0]
  5572. Int[Times[Power[Csch[Plus[Optional[Pattern[c, Blank[]]], Times[Optional[Pattern[d, Blank[]]], Pattern[x, Blank[]]]]], Optional[Pattern[n, Blank[]]]], Power[Plus[Times[Cosh[Plus[Optional[Pattern[c, Blank[]]], Times[Optional[Pattern[d, Blank[]]], Pattern[x, Blank[]]]]], Optional[Pattern[b, Blank[]]]], Pattern[a, Blank[]]], -1], Power[Plus[Optional[Pattern[e, Blank[]]], Times[Optional[Pattern[f, Blank[]]], Pattern[x, Blank[]]]], Optional[Pattern[m, Blank[]]]]], Pattern[x, Blank[Symbol]]] := -Dist[a^(-1), Int[(e + f*x)^m*Csch[c + d*x]^(n + 2), x], x] + Dist[1/b, Int[(e + f*x)^m*Csch[c + d*x]^(n + 1)*Coth[c + d*x], x], x] /; FreeQ[{a, b, c, d, e, f, n}, x] && IGtQ[m, 0] && EqQ[a^2 - b^2, 0]
  5573. Int[Times[Power[Plus[Optional[Pattern[e, Blank[]]], Times[Optional[Pattern[f, Blank[]]], Pattern[x, Blank[]]]], Optional[Pattern[m, Blank[]]]], Power[Sech[Plus[Optional[Pattern[c, Blank[]]], Times[Optional[Pattern[d, Blank[]]], Pattern[x, Blank[]]]]], Optional[Pattern[n, Blank[]]]], Power[Plus[Pattern[a, Blank[]], Times[Optional[Pattern[b, Blank[]]], Sinh[Plus[Optional[Pattern[c, Blank[]]], Times[Optional[Pattern[d, Blank[]]], Pattern[x, Blank[]]]]]]], -1]], Pattern[x, Blank[Symbol]]] := Dist[b^2/(a^2 + b^2), Int[((e + f*x)^m*Sech[c + d*x]^(n - 2))/(a + b*Sinh[c + d*x]), x], x] + Dist[1/(a^2 + b^2), Int[(e + f*x)^m*Sech[c + d*x]^n*(a - b*Sinh[c + d*x]), x], x] /; FreeQ[{a, b, c, d, e, f}, x] && IGtQ[m, 0] && NeQ[a^2 + b^2, 0] && IGtQ[n, 0]
  5574. Int[Times[Power[Csch[Plus[Optional[Pattern[c, Blank[]]], Times[Optional[Pattern[d, Blank[]]], Pattern[x, Blank[]]]]], Optional[Pattern[n, Blank[]]]], Power[Plus[Times[Cosh[Plus[Optional[Pattern[c, Blank[]]], Times[Optional[Pattern[d, Blank[]]], Pattern[x, Blank[]]]]], Optional[Pattern[b, Blank[]]]], Pattern[a, Blank[]]], -1], Power[Plus[Optional[Pattern[e, Blank[]]], Times[Optional[Pattern[f, Blank[]]], Pattern[x, Blank[]]]], Optional[Pattern[m, Blank[]]]]], Pattern[x, Blank[Symbol]]] := Dist[b^2/(a^2 - b^2), Int[((e + f*x)^m*Csch[c + d*x]^(n - 2))/(a + b*Cosh[c + d*x]), x], x] + Dist[1/(a^2 - b^2), Int[(e + f*x)^m*Csch[c + d*x]^n*(a - b*Cosh[c + d*x]), x], x] /; FreeQ[{a, b, c, d, e, f}, x] && IGtQ[m, 0] && NeQ[a^2 - b^2, 0] && IGtQ[n, 0]
  5575. Int[Times[Power[Csch[Plus[Optional[Pattern[c, Blank[]]], Times[Optional[Pattern[d, Blank[]]], Pattern[x, Blank[]]]]], Optional[Pattern[n, Blank[]]]], Power[Plus[Optional[Pattern[e, Blank[]]], Times[Optional[Pattern[f, Blank[]]], Pattern[x, Blank[]]]], Optional[Pattern[m, Blank[]]]], Power[Plus[Pattern[a, Blank[]], Times[Optional[Pattern[b, Blank[]]], Sinh[Plus[Optional[Pattern[c, Blank[]]], Times[Optional[Pattern[d, Blank[]]], Pattern[x, Blank[]]]]]]], -1]], Pattern[x, Blank[Symbol]]] := Dist[1/a, Int[(e + f*x)^m*Csch[c + d*x]^n, x], x] - Dist[b/a, Int[((e + f*x)^m*Csch[c + d*x]^(n - 1))/(a + b*Sinh[c + d*x]), x], x] /; FreeQ[{a, b, c, d, e, f}, x] && IGtQ[m, 0] && IGtQ[n, 0]
  5576. Int[Times[Power[Plus[Times[Cosh[Plus[Optional[Pattern[c, Blank[]]], Times[Optional[Pattern[d, Blank[]]], Pattern[x, Blank[]]]]], Optional[Pattern[b, Blank[]]]], Pattern[a, Blank[]]], -1], Power[Plus[Optional[Pattern[e, Blank[]]], Times[Optional[Pattern[f, Blank[]]], Pattern[x, Blank[]]]], Optional[Pattern[m, Blank[]]]], Power[Sech[Plus[Optional[Pattern[c, Blank[]]], Times[Optional[Pattern[d, Blank[]]], Pattern[x, Blank[]]]]], Optional[Pattern[n, Blank[]]]]], Pattern[x, Blank[Symbol]]] := Dist[1/a, Int[(e + f*x)^m*Sech[c + d*x]^n, x], x] - Dist[b/a, Int[((e + f*x)^m*Sech[c + d*x]^(n - 1))/(a + b*Cosh[c + d*x]), x], x] /; FreeQ[{a, b, c, d, e, f}, x] && IGtQ[m, 0] && IGtQ[n, 0]
  5577. Int[Times[Power[Plus[Optional[Pattern[e, Blank[]]], Times[Optional[Pattern[f, Blank[]]], Pattern[x, Blank[]]]], Optional[Pattern[m, Blank[]]]], Power[Plus[Pattern[a, Blank[]], Times[Optional[Pattern[b, Blank[]]], Sinh[Plus[Optional[Pattern[c, Blank[]]], Times[Optional[Pattern[d, Blank[]]], Pattern[x, Blank[]]]]]]], -1], Power[Pattern[F, Blank[]][Plus[Optional[Pattern[c, Blank[]]], Times[Optional[Pattern[d, Blank[]]], Pattern[x, Blank[]]]]], Optional[Pattern[n, Blank[]]]]], Pattern[x, Blank[Symbol]]] := Unintegrable[((e + f*x)^m*F[c + d*x]^n)/(a + b*Sinh[c + d*x]), x] /; FreeQ[{a, b, c, d, e, f, m, n}, x] && HyperbolicQ[F]
  5578. Int[Times[Power[Plus[Times[Cosh[Plus[Optional[Pattern[c, Blank[]]], Times[Optional[Pattern[d, Blank[]]], Pattern[x, Blank[]]]]], Optional[Pattern[b, Blank[]]]], Pattern[a, Blank[]]], -1], Power[Plus[Optional[Pattern[e, Blank[]]], Times[Optional[Pattern[f, Blank[]]], Pattern[x, Blank[]]]], Optional[Pattern[m, Blank[]]]], Power[Pattern[F, Blank[]][Plus[Optional[Pattern[c, Blank[]]], Times[Optional[Pattern[d, Blank[]]], Pattern[x, Blank[]]]]], Optional[Pattern[n, Blank[]]]]], Pattern[x, Blank[Symbol]]] := Unintegrable[((e + f*x)^m*F[c + d*x]^n)/(a + b*Cosh[c + d*x]), x] /; FreeQ[{a, b, c, d, e, f, m, n}, x] && HyperbolicQ[F]
  5579. Int[Times[Power[Cosh[Plus[Optional[Pattern[c, Blank[]]], Times[Optional[Pattern[d, Blank[]]], Pattern[x, Blank[]]]]], Optional[Pattern[p, Blank[]]]], Power[Plus[Optional[Pattern[e, Blank[]]], Times[Optional[Pattern[f, Blank[]]], Pattern[x, Blank[]]]], Optional[Pattern[m, Blank[]]]], Power[Sinh[Plus[Optional[Pattern[c, Blank[]]], Times[Optional[Pattern[d, Blank[]]], Pattern[x, Blank[]]]]], Optional[Pattern[n, Blank[]]]], Power[Plus[Pattern[a, Blank[]], Times[Optional[Pattern[b, Blank[]]], Sinh[Plus[Optional[Pattern[c, Blank[]]], Times[Optional[Pattern[d, Blank[]]], Pattern[x, Blank[]]]]]]], -1]], Pattern[x, Blank[Symbol]]] := Dist[1/b, Int[(e + f*x)^m*Cosh[c + d*x]^p*Sinh[c + d*x]^(n - 1), x], x] - Dist[a/b, Int[((e + f*x)^m*Cosh[c + d*x]^p*Sinh[c + d*x]^(n - 1))/(a + b*Sinh[c + d*x]), x], x] /; FreeQ[{a, b, c, d, e, f}, x] && IGtQ[m, 0] && IGtQ[n, 0] && IGtQ[p, 0]
  5580. Int[Times[Power[Cosh[Plus[Optional[Pattern[c, Blank[]]], Times[Optional[Pattern[d, Blank[]]], Pattern[x, Blank[]]]]], Optional[Pattern[n, Blank[]]]], Power[Plus[Times[Cosh[Plus[Optional[Pattern[c, Blank[]]], Times[Optional[Pattern[d, Blank[]]], Pattern[x, Blank[]]]]], Optional[Pattern[b, Blank[]]]], Pattern[a, Blank[]]], -1], Power[Plus[Optional[Pattern[e, Blank[]]], Times[Optional[Pattern[f, Blank[]]], Pattern[x, Blank[]]]], Optional[Pattern[m, Blank[]]]], Power[Sinh[Plus[Optional[Pattern[c, Blank[]]], Times[Optional[Pattern[d, Blank[]]], Pattern[x, Blank[]]]]], Optional[Pattern[p, Blank[]]]]], Pattern[x, Blank[Symbol]]] := Dist[1/b, Int[(e + f*x)^m*Sinh[c + d*x]^p*Cosh[c + d*x]^(n - 1), x], x] - Dist[a/b, Int[((e + f*x)^m*Sinh[c + d*x]^p*Cosh[c + d*x]^(n - 1))/(a + b*Cosh[c + d*x]), x], x] /; FreeQ[{a, b, c, d, e, f}, x] && IGtQ[m, 0] && IGtQ[n, 0] && IGtQ[p, 0]
  5581. Int[Times[Power[Plus[Optional[Pattern[e, Blank[]]], Times[Optional[Pattern[f, Blank[]]], Pattern[x, Blank[]]]], Optional[Pattern[m, Blank[]]]], Power[Sinh[Plus[Optional[Pattern[c, Blank[]]], Times[Optional[Pattern[d, Blank[]]], Pattern[x, Blank[]]]]], Optional[Pattern[p, Blank[]]]], Power[Plus[Pattern[a, Blank[]], Times[Optional[Pattern[b, Blank[]]], Sinh[Plus[Optional[Pattern[c, Blank[]]], Times[Optional[Pattern[d, Blank[]]], Pattern[x, Blank[]]]]]]], -1], Power[Tanh[Plus[Optional[Pattern[c, Blank[]]], Times[Optional[Pattern[d, Blank[]]], Pattern[x, Blank[]]]]], Optional[Pattern[n, Blank[]]]]], Pattern[x, Blank[Symbol]]] := Dist[1/b, Int[(e + f*x)^m*Sinh[c + d*x]^(p - 1)*Tanh[c + d*x]^n, x], x] - Dist[a/b, Int[((e + f*x)^m*Sinh[c + d*x]^(p - 1)*Tanh[c + d*x]^n)/(a + b*Sinh[c + d*x]), x], x] /; FreeQ[{a, b, c, d, e, f}, x] && IGtQ[m, 0] && IGtQ[n, 0] && IGtQ[p, 0]
  5582. Int[Times[Power[Cosh[Plus[Optional[Pattern[c, Blank[]]], Times[Optional[Pattern[d, Blank[]]], Pattern[x, Blank[]]]]], Optional[Pattern[p, Blank[]]]], Power[Coth[Plus[Optional[Pattern[c, Blank[]]], Times[Optional[Pattern[d, Blank[]]], Pattern[x, Blank[]]]]], Optional[Pattern[n, Blank[]]]], Power[Plus[Times[Cosh[Plus[Optional[Pattern[c, Blank[]]], Times[Optional[Pattern[d, Blank[]]], Pattern[x, Blank[]]]]], Optional[Pattern[b, Blank[]]]], Pattern[a, Blank[]]], -1], Power[Plus[Optional[Pattern[e, Blank[]]], Times[Optional[Pattern[f, Blank[]]], Pattern[x, Blank[]]]], Optional[Pattern[m, Blank[]]]]], Pattern[x, Blank[Symbol]]] := Dist[1/b, Int[(e + f*x)^m*Cosh[c + d*x]^(p - 1)*Coth[c + d*x]^n, x], x] - Dist[a/b, Int[((e + f*x)^m*Cosh[c + d*x]^(p - 1)*Coth[c + d*x]^n)/(a + b*Cosh[c + d*x]), x], x] /; FreeQ[{a, b, c, d, e, f}, x] && IGtQ[m, 0] && IGtQ[n, 0] && IGtQ[p, 0]
  5583. Int[Times[Power[Plus[Optional[Pattern[e, Blank[]]], Times[Optional[Pattern[f, Blank[]]], Pattern[x, Blank[]]]], Optional[Pattern[m, Blank[]]]], Power[Sech[Plus[Optional[Pattern[c, Blank[]]], Times[Optional[Pattern[d, Blank[]]], Pattern[x, Blank[]]]]], Optional[Pattern[p, Blank[]]]], Power[Plus[Pattern[a, Blank[]], Times[Optional[Pattern[b, Blank[]]], Sinh[Plus[Optional[Pattern[c, Blank[]]], Times[Optional[Pattern[d, Blank[]]], Pattern[x, Blank[]]]]]]], -1], Power[Tanh[Plus[Optional[Pattern[c, Blank[]]], Times[Optional[Pattern[d, Blank[]]], Pattern[x, Blank[]]]]], Optional[Pattern[n, Blank[]]]]], Pattern[x, Blank[Symbol]]] := Dist[1/b, Int[(e + f*x)^m*Sech[c + d*x]^(p + 1)*Tanh[c + d*x]^(n - 1), x], x] - Dist[a/b, Int[((e + f*x)^m*Sech[c + d*x]^(p + 1)*Tanh[c + d*x]^(n - 1))/(a + b*Sinh[c + d*x]), x], x] /; FreeQ[{a, b, c, d, e, f}, x] && IGtQ[m, 0] && IGtQ[n, 0] && IGtQ[p, 0]
  5584. Int[Times[Power[Coth[Plus[Optional[Pattern[c, Blank[]]], Times[Optional[Pattern[d, Blank[]]], Pattern[x, Blank[]]]]], Optional[Pattern[n, Blank[]]]], Power[Csch[Plus[Optional[Pattern[c, Blank[]]], Times[Optional[Pattern[d, Blank[]]], Pattern[x, Blank[]]]]], Optional[Pattern[p, Blank[]]]], Power[Plus[Times[Cosh[Plus[Optional[Pattern[c, Blank[]]], Times[Optional[Pattern[d, Blank[]]], Pattern[x, Blank[]]]]], Optional[Pattern[b, Blank[]]]], Pattern[a, Blank[]]], -1], Power[Plus[Optional[Pattern[e, Blank[]]], Times[Optional[Pattern[f, Blank[]]], Pattern[x, Blank[]]]], Optional[Pattern[m, Blank[]]]]], Pattern[x, Blank[Symbol]]] := Dist[1/b, Int[(e + f*x)^m*Csch[c + d*x]^(p + 1)*Coth[c + d*x]^(n - 1), x], x] - Dist[a/b, Int[((e + f*x)^m*Csch[c + d*x]^(p + 1)*Coth[c + d*x]^(n - 1))/(a + b*Cosh[c + d*x]), x], x] /; FreeQ[{a, b, c, d, e, f}, x] && IGtQ[m, 0] && IGtQ[n, 0] && IGtQ[p, 0]
  5585. Int[Times[Power[Cosh[Plus[Optional[Pattern[c, Blank[]]], Times[Optional[Pattern[d, Blank[]]], Pattern[x, Blank[]]]]], Optional[Pattern[p, Blank[]]]], Power[Coth[Plus[Optional[Pattern[c, Blank[]]], Times[Optional[Pattern[d, Blank[]]], Pattern[x, Blank[]]]]], Optional[Pattern[n, Blank[]]]], Power[Plus[Optional[Pattern[e, Blank[]]], Times[Optional[Pattern[f, Blank[]]], Pattern[x, Blank[]]]], Optional[Pattern[m, Blank[]]]], Power[Plus[Pattern[a, Blank[]], Times[Optional[Pattern[b, Blank[]]], Sinh[Plus[Optional[Pattern[c, Blank[]]], Times[Optional[Pattern[d, Blank[]]], Pattern[x, Blank[]]]]]]], -1]], Pattern[x, Blank[Symbol]]] := Dist[1/a, Int[(e + f*x)^m*Cosh[c + d*x]^p*Coth[c + d*x]^n, x], x] - Dist[b/a, Int[((e + f*x)^m*Cosh[c + d*x]^(p + 1)*Coth[c + d*x]^(n - 1))/(a + b*Sinh[c + d*x]), x], x] /; FreeQ[{a, b, c, d, e, f}, x] && IGtQ[m, 0] && IGtQ[n, 0] && IGtQ[p, 0]
  5586. Int[Times[Power[Plus[Times[Cosh[Plus[Optional[Pattern[c, Blank[]]], Times[Optional[Pattern[d, Blank[]]], Pattern[x, Blank[]]]]], Optional[Pattern[b, Blank[]]]], Pattern[a, Blank[]]], -1], Power[Plus[Optional[Pattern[e, Blank[]]], Times[Optional[Pattern[f, Blank[]]], Pattern[x, Blank[]]]], Optional[Pattern[m, Blank[]]]], Power[Sinh[Plus[Optional[Pattern[c, Blank[]]], Times[Optional[Pattern[d, Blank[]]], Pattern[x, Blank[]]]]], Optional[Pattern[p, Blank[]]]], Power[Tanh[Plus[Optional[Pattern[c, Blank[]]], Times[Optional[Pattern[d, Blank[]]], Pattern[x, Blank[]]]]], Optional[Pattern[n, Blank[]]]]], Pattern[x, Blank[Symbol]]] := Dist[1/a, Int[(e + f*x)^m*Sinh[c + d*x]^p*Tanh[c + d*x]^n, x], x] - Dist[b/a, Int[((e + f*x)^m*Sinh[c + d*x]^(p + 1)*Tanh[c + d*x]^(n - 1))/(a + b*Cosh[c + d*x]), x], x] /; FreeQ[{a, b, c, d, e, f}, x] && IGtQ[m, 0] && IGtQ[n, 0] && IGtQ[p, 0]
  5587. Int[Times[Power[Coth[Plus[Optional[Pattern[c, Blank[]]], Times[Optional[Pattern[d, Blank[]]], Pattern[x, Blank[]]]]], Optional[Pattern[n, Blank[]]]], Power[Csch[Plus[Optional[Pattern[c, Blank[]]], Times[Optional[Pattern[d, Blank[]]], Pattern[x, Blank[]]]]], Optional[Pattern[p, Blank[]]]], Power[Plus[Optional[Pattern[e, Blank[]]], Times[Optional[Pattern[f, Blank[]]], Pattern[x, Blank[]]]], Optional[Pattern[m, Blank[]]]], Power[Plus[Pattern[a, Blank[]], Times[Optional[Pattern[b, Blank[]]], Sinh[Plus[Optional[Pattern[c, Blank[]]], Times[Optional[Pattern[d, Blank[]]], Pattern[x, Blank[]]]]]]], -1]], Pattern[x, Blank[Symbol]]] := Dist[1/a, Int[(e + f*x)^m*Csch[c + d*x]^p*Coth[c + d*x]^n, x], x] - Dist[b/a, Int[((e + f*x)^m*Csch[c + d*x]^(p - 1)*Coth[c + d*x]^n)/(a + b*Sinh[c + d*x]), x], x] /; FreeQ[{a, b, c, d, e, f}, x] && IGtQ[m, 0] && IGtQ[n, 0] && IGtQ[p, 0]
  5588. Int[Times[Power[Plus[Times[Cosh[Plus[Optional[Pattern[c, Blank[]]], Times[Optional[Pattern[d, Blank[]]], Pattern[x, Blank[]]]]], Optional[Pattern[b, Blank[]]]], Pattern[a, Blank[]]], -1], Power[Plus[Optional[Pattern[e, Blank[]]], Times[Optional[Pattern[f, Blank[]]], Pattern[x, Blank[]]]], Optional[Pattern[m, Blank[]]]], Power[Sech[Plus[Optional[Pattern[c, Blank[]]], Times[Optional[Pattern[d, Blank[]]], Pattern[x, Blank[]]]]], Optional[Pattern[p, Blank[]]]], Power[Tanh[Plus[Optional[Pattern[c, Blank[]]], Times[Optional[Pattern[d, Blank[]]], Pattern[x, Blank[]]]]], Optional[Pattern[n, Blank[]]]]], Pattern[x, Blank[Symbol]]] := Dist[1/a, Int[(e + f*x)^m*Sech[c + d*x]^p*Tanh[c + d*x]^n, x], x] - Dist[b/a, Int[((e + f*x)^m*Sech[c + d*x]^(p - 1)*Tanh[c + d*x]^n)/(a + b*Cosh[c + d*x]), x], x] /; FreeQ[{a, b, c, d, e, f}, x] && IGtQ[m, 0] && IGtQ[n, 0] && IGtQ[p, 0]
  5589. Int[Times[Power[Csch[Plus[Optional[Pattern[c, Blank[]]], Times[Optional[Pattern[d, Blank[]]], Pattern[x, Blank[]]]]], Optional[Pattern[n, Blank[]]]], Power[Plus[Optional[Pattern[e, Blank[]]], Times[Optional[Pattern[f, Blank[]]], Pattern[x, Blank[]]]], Optional[Pattern[m, Blank[]]]], Power[Sech[Plus[Optional[Pattern[c, Blank[]]], Times[Optional[Pattern[d, Blank[]]], Pattern[x, Blank[]]]]], Optional[Pattern[p, Blank[]]]], Power[Plus[Pattern[a, Blank[]], Times[Optional[Pattern[b, Blank[]]], Sinh[Plus[Optional[Pattern[c, Blank[]]], Times[Optional[Pattern[d, Blank[]]], Pattern[x, Blank[]]]]]]], -1]], Pattern[x, Blank[Symbol]]] := Dist[1/a, Int[(e + f*x)^m*Sech[c + d*x]^p*Csch[c + d*x]^n, x], x] - Dist[b/a, Int[((e + f*x)^m*Sech[c + d*x]^p*Csch[c + d*x]^(n - 1))/(a + b*Sinh[c + d*x]), x], x] /; FreeQ[{a, b, c, d, e, f}, x] && IGtQ[m, 0] && IGtQ[n, 0] && IGtQ[p, 0]
  5590. Int[Times[Power[Csch[Plus[Optional[Pattern[c, Blank[]]], Times[Optional[Pattern[d, Blank[]]], Pattern[x, Blank[]]]]], Optional[Pattern[p, Blank[]]]], Power[Plus[Times[Cosh[Plus[Optional[Pattern[c, Blank[]]], Times[Optional[Pattern[d, Blank[]]], Pattern[x, Blank[]]]]], Optional[Pattern[b, Blank[]]]], Pattern[a, Blank[]]], -1], Power[Plus[Optional[Pattern[e, Blank[]]], Times[Optional[Pattern[f, Blank[]]], Pattern[x, Blank[]]]], Optional[Pattern[m, Blank[]]]], Power[Sech[Plus[Optional[Pattern[c, Blank[]]], Times[Optional[Pattern[d, Blank[]]], Pattern[x, Blank[]]]]], Optional[Pattern[n, Blank[]]]]], Pattern[x, Blank[Symbol]]] := Dist[1/a, Int[(e + f*x)^m*Csch[c + d*x]^p*Sech[c + d*x]^n, x], x] - Dist[b/a, Int[((e + f*x)^m*Csch[c + d*x]^p*Sech[c + d*x]^(n - 1))/(a + b*Cosh[c + d*x]), x], x] /; FreeQ[{a, b, c, d, e, f}, x] && IGtQ[m, 0] && IGtQ[n, 0] && IGtQ[p, 0]
  5591. Int[Times[Power[Plus[Optional[Pattern[e, Blank[]]], Times[Optional[Pattern[f, Blank[]]], Pattern[x, Blank[]]]], Optional[Pattern[m, Blank[]]]], Power[Plus[Pattern[a, Blank[]], Times[Optional[Pattern[b, Blank[]]], Sinh[Plus[Optional[Pattern[c, Blank[]]], Times[Optional[Pattern[d, Blank[]]], Pattern[x, Blank[]]]]]]], -1], Power[Pattern[F, Blank[]][Plus[Optional[Pattern[c, Blank[]]], Times[Optional[Pattern[d, Blank[]]], Pattern[x, Blank[]]]]], Optional[Pattern[n, Blank[]]]], Power[Pattern[G, Blank[]][Plus[Optional[Pattern[c, Blank[]]], Times[Optional[Pattern[d, Blank[]]], Pattern[x, Blank[]]]]], Optional[Pattern[p, Blank[]]]]], Pattern[x, Blank[Symbol]]] := Unintegrable[((e + f*x)^m*F[c + d*x]^n*G[c + d*x]^p)/(a + b*Sinh[c + d*x]), x] /; FreeQ[{a, b, c, d, e, f, m, n, p}, x] && HyperbolicQ[F] && HyperbolicQ[G]
  5592. Int[Times[Power[Plus[Times[Cosh[Plus[Optional[Pattern[c, Blank[]]], Times[Optional[Pattern[d, Blank[]]], Pattern[x, Blank[]]]]], Optional[Pattern[b, Blank[]]]], Pattern[a, Blank[]]], -1], Power[Plus[Optional[Pattern[e, Blank[]]], Times[Optional[Pattern[f, Blank[]]], Pattern[x, Blank[]]]], Optional[Pattern[m, Blank[]]]], Power[Pattern[F, Blank[]][Plus[Optional[Pattern[c, Blank[]]], Times[Optional[Pattern[d, Blank[]]], Pattern[x, Blank[]]]]], Optional[Pattern[n, Blank[]]]], Power[Pattern[G, Blank[]][Plus[Optional[Pattern[c, Blank[]]], Times[Optional[Pattern[d, Blank[]]], Pattern[x, Blank[]]]]], Optional[Pattern[p, Blank[]]]]], Pattern[x, Blank[Symbol]]] := Unintegrable[((e + f*x)^m*F[c + d*x]^n*G[c + d*x]^p)/(a + b*Cosh[c + d*x]), x] /; FreeQ[{a, b, c, d, e, f, m, n, p}, x] && HyperbolicQ[F] && HyperbolicQ[G]
  5593. Int[Times[Power[Plus[Optional[Pattern[e, Blank[]]], Times[Optional[Pattern[f, Blank[]]], Pattern[x, Blank[]]]], Optional[Pattern[m, Blank[]]]], Power[Plus[Pattern[a, Blank[]], Times[Optional[Pattern[b, Blank[]]], Sech[Plus[Optional[Pattern[c, Blank[]]], Times[Optional[Pattern[d, Blank[]]], Pattern[x, Blank[]]]]]]], -1], Power[Pattern[F, Blank[]][Plus[Optional[Pattern[c, Blank[]]], Times[Optional[Pattern[d, Blank[]]], Pattern[x, Blank[]]]]], Optional[Pattern[n, Blank[]]]]], Pattern[x, Blank[Symbol]]] := Int[((e + f*x)^m*Cosh[c + d*x]*F[c + d*x]^n)/(b + a*Cosh[c + d*x]), x] /; FreeQ[{a, b, c, d, e, f}, x] && HyperbolicQ[F] && IntegersQ[m, n]
  5594. Int[Times[Power[Plus[Times[Csch[Plus[Optional[Pattern[c, Blank[]]], Times[Optional[Pattern[d, Blank[]]], Pattern[x, Blank[]]]]], Optional[Pattern[b, Blank[]]]], Pattern[a, Blank[]]], -1], Power[Plus[Optional[Pattern[e, Blank[]]], Times[Optional[Pattern[f, Blank[]]], Pattern[x, Blank[]]]], Optional[Pattern[m, Blank[]]]], Power[Pattern[F, Blank[]][Plus[Optional[Pattern[c, Blank[]]], Times[Optional[Pattern[d, Blank[]]], Pattern[x, Blank[]]]]], Optional[Pattern[n, Blank[]]]]], Pattern[x, Blank[Symbol]]] := Int[((e + f*x)^m*Sinh[c + d*x]*F[c + d*x]^n)/(b + a*Sinh[c + d*x]), x] /; FreeQ[{a, b, c, d, e, f}, x] && HyperbolicQ[F] && IntegersQ[m, n]
  5595. Int[Times[Power[Plus[Optional[Pattern[e, Blank[]]], Times[Optional[Pattern[f, Blank[]]], Pattern[x, Blank[]]]], Optional[Pattern[m, Blank[]]]], Power[Plus[Pattern[a, Blank[]], Times[Optional[Pattern[b, Blank[]]], Sech[Plus[Optional[Pattern[c, Blank[]]], Times[Optional[Pattern[d, Blank[]]], Pattern[x, Blank[]]]]]]], -1], Power[Pattern[F, Blank[]][Plus[Optional[Pattern[c, Blank[]]], Times[Optional[Pattern[d, Blank[]]], Pattern[x, Blank[]]]]], Optional[Pattern[n, Blank[]]]], Power[Pattern[G, Blank[]][Plus[Optional[Pattern[c, Blank[]]], Times[Optional[Pattern[d, Blank[]]], Pattern[x, Blank[]]]]], Optional[Pattern[p, Blank[]]]]], Pattern[x, Blank[Symbol]]] := Int[((e + f*x)^m*Cosh[c + d*x]*F[c + d*x]^n*G[c + d*x]^p)/(b + a*Cosh[c + d*x]), x] /; FreeQ[{a, b, c, d, e, f}, x] && HyperbolicQ[F] && HyperbolicQ[G] && IntegersQ[m, n, p]
  5596. Int[Times[Power[Plus[Times[Csch[Plus[Optional[Pattern[c, Blank[]]], Times[Optional[Pattern[d, Blank[]]], Pattern[x, Blank[]]]]], Optional[Pattern[b, Blank[]]]], Pattern[a, Blank[]]], -1], Power[Plus[Optional[Pattern[e, Blank[]]], Times[Optional[Pattern[f, Blank[]]], Pattern[x, Blank[]]]], Optional[Pattern[m, Blank[]]]], Power[Pattern[F, Blank[]][Plus[Optional[Pattern[c, Blank[]]], Times[Optional[Pattern[d, Blank[]]], Pattern[x, Blank[]]]]], Optional[Pattern[n, Blank[]]]], Power[Pattern[G, Blank[]][Plus[Optional[Pattern[c, Blank[]]], Times[Optional[Pattern[d, Blank[]]], Pattern[x, Blank[]]]]], Optional[Pattern[p, Blank[]]]]], Pattern[x, Blank[Symbol]]] := Int[((e + f*x)^m*Sinh[c + d*x]*F[c + d*x]^n*G[c + d*x]^p)/(b + a*Sinh[c + d*x]), x] /; FreeQ[{a, b, c, d, e, f}, x] && HyperbolicQ[F] && HyperbolicQ[G] && IntegersQ[m, n, p]
  5597. Int[Times[Power[Sinh[Plus[Optional[Pattern[a, Blank[]]], Times[Optional[Pattern[b, Blank[]]], Pattern[x, Blank[]]]]], Optional[Pattern[p, Blank[]]]], Power[Sinh[Plus[Optional[Pattern[c, Blank[]]], Times[Optional[Pattern[d, Blank[]]], Pattern[x, Blank[]]]]], Optional[Pattern[q, Blank[]]]]], Pattern[x, Blank[Symbol]]] := Dist[1/2^(p + q), Int[ExpandIntegrand[(-E^(-c - d*x) + E^(c + d*x))^q, (-E^(-a - b*x) + E^(a + b*x))^p, x], x], x] /; FreeQ[{a, b, c, d, q}, x] && IGtQ[p, 0] && !IntegerQ[q]
  5598. Int[Times[Power[Cosh[Plus[Optional[Pattern[a, Blank[]]], Times[Optional[Pattern[b, Blank[]]], Pattern[x, Blank[]]]]], Optional[Pattern[p, Blank[]]]], Power[Cosh[Plus[Optional[Pattern[c, Blank[]]], Times[Optional[Pattern[d, Blank[]]], Pattern[x, Blank[]]]]], Optional[Pattern[q, Blank[]]]]], Pattern[x, Blank[Symbol]]] := Dist[1/2^(p + q), Int[ExpandIntegrand[(E^(-c - d*x) + E^(c + d*x))^q, (E^(-a - b*x) + E^(a + b*x))^p, x], x], x] /; FreeQ[{a, b, c, d, q}, x] && IGtQ[p, 0] && !IntegerQ[q]
  5599. Int[Times[Power[Cosh[Plus[Optional[Pattern[c, Blank[]]], Times[Optional[Pattern[d, Blank[]]], Pattern[x, Blank[]]]]], Optional[Pattern[q, Blank[]]]], Power[Sinh[Plus[Optional[Pattern[a, Blank[]]], Times[Optional[Pattern[b, Blank[]]], Pattern[x, Blank[]]]]], Optional[Pattern[p, Blank[]]]]], Pattern[x, Blank[Symbol]]] := Dist[1/2^(p + q), Int[ExpandIntegrand[(E^(-c - d*x) + E^(c + d*x))^q, (-E^(-a - b*x) + E^(a + b*x))^p, x], x], x] /; FreeQ[{a, b, c, d, q}, x] && IGtQ[p, 0] && !IntegerQ[q]
  5600. Int[Times[Power[Cosh[Plus[Optional[Pattern[a, Blank[]]], Times[Optional[Pattern[b, Blank[]]], Pattern[x, Blank[]]]]], Optional[Pattern[p, Blank[]]]], Power[Sinh[Plus[Optional[Pattern[c, Blank[]]], Times[Optional[Pattern[d, Blank[]]], Pattern[x, Blank[]]]]], Optional[Pattern[q, Blank[]]]]], Pattern[x, Blank[Symbol]]] := Dist[1/2^(p + q), Int[ExpandIntegrand[(-E^(-c - d*x) + E^(c + d*x))^q, (E^(-a - b*x) + E^(a + b*x))^p, x], x], x] /; FreeQ[{a, b, c, d, q}, x] && IGtQ[p, 0] && !IntegerQ[q]
  5601. Int[Times[Sinh[Plus[Optional[Pattern[a, Blank[]]], Times[Optional[Pattern[b, Blank[]]], Pattern[x, Blank[]]]]], Tanh[Plus[Optional[Pattern[c, Blank[]]], Times[Optional[Pattern[d, Blank[]]], Pattern[x, Blank[]]]]]], Pattern[x, Blank[Symbol]]] := Int[-(1/(E^(a + b*x)*2)) + E^(a + b*x)/2 + 1/(E^(a + b*x)*(1 + E^(2*(c + d*x)))) - E^(a + b*x)/(1 + E^(2*(c + d*x))), x] /; FreeQ[{a, b, c, d}, x] && NeQ[b^2 - d^2, 0]
  5602. Int[Times[Cosh[Plus[Optional[Pattern[a, Blank[]]], Times[Optional[Pattern[b, Blank[]]], Pattern[x, Blank[]]]]], Coth[Plus[Optional[Pattern[c, Blank[]]], Times[Optional[Pattern[d, Blank[]]], Pattern[x, Blank[]]]]]], Pattern[x, Blank[Symbol]]] := Int[1/(E^(a + b*x)*2) + E^(a + b*x)/2 - 1/(E^(a + b*x)*(1 - E^(2*(c + d*x)))) - E^(a + b*x)/(1 - E^(2*(c + d*x))), x] /; FreeQ[{a, b, c, d}, x] && NeQ[b^2 - d^2, 0]
  5603. Int[Times[Coth[Plus[Optional[Pattern[c, Blank[]]], Times[Optional[Pattern[d, Blank[]]], Pattern[x, Blank[]]]]], Sinh[Plus[Optional[Pattern[a, Blank[]]], Times[Optional[Pattern[b, Blank[]]], Pattern[x, Blank[]]]]]], Pattern[x, Blank[Symbol]]] := Int[-(1/(E^(a + b*x)*2)) + E^(a + b*x)/2 + 1/(E^(a + b*x)*(1 - E^(2*(c + d*x)))) - E^(a + b*x)/(1 - E^(2*(c + d*x))), x] /; FreeQ[{a, b, c, d}, x] && NeQ[b^2 - d^2, 0]
  5604. Int[Times[Cosh[Plus[Optional[Pattern[a, Blank[]]], Times[Optional[Pattern[b, Blank[]]], Pattern[x, Blank[]]]]], Tanh[Plus[Optional[Pattern[c, Blank[]]], Times[Optional[Pattern[d, Blank[]]], Pattern[x, Blank[]]]]]], Pattern[x, Blank[Symbol]]] := Int[1/(E^(a + b*x)*2) + E^(a + b*x)/2 - 1/(E^(a + b*x)*(1 + E^(2*(c + d*x)))) - E^(a + b*x)/(1 + E^(2*(c + d*x))), x] /; FreeQ[{a, b, c, d}, x] && NeQ[b^2 - d^2, 0]
  5605. Int[Power[Sinh[Times[Optional[Pattern[a, Blank[]]], Power[Plus[Optional[Pattern[c, Blank[]]], Times[Optional[Pattern[d, Blank[]]], Pattern[x, Blank[]]]], -1]]], Optional[Pattern[n, Blank[]]]], Pattern[x, Blank[Symbol]]] := -Dist[d^(-1), Subst[Int[Sinh[a*x]^n/x^2, x], x, 1/(c + d*x)], x] /; FreeQ[{a, c, d}, x] && IGtQ[n, 0]
  5606. Int[Power[Cosh[Times[Optional[Pattern[a, Blank[]]], Power[Plus[Optional[Pattern[c, Blank[]]], Times[Optional[Pattern[d, Blank[]]], Pattern[x, Blank[]]]], -1]]], Optional[Pattern[n, Blank[]]]], Pattern[x, Blank[Symbol]]] := -Dist[d^(-1), Subst[Int[Cosh[a*x]^n/x^2, x], x, 1/(c + d*x)], x] /; FreeQ[{a, c, d}, x] && IGtQ[n, 0]
  5607. Int[Power[Sinh[Times[Optional[Pattern[e, Blank[]]], Plus[Optional[Pattern[a, Blank[]]], Times[Optional[Pattern[b, Blank[]]], Pattern[x, Blank[]]]], Power[Plus[Optional[Pattern[c, Blank[]]], Times[Optional[Pattern[d, Blank[]]], Pattern[x, Blank[]]]], -1]]], Optional[Pattern[n, Blank[]]]], Pattern[x, Blank[Symbol]]] := -Dist[d^(-1), Subst[Int[Sinh[(b*e)/d - (e*(b*c - a*d)*x)/d]^n/x^2, x], x, 1/(c + d*x)], x] /; FreeQ[{a, b, c, d}, x] && IGtQ[n, 0] && NeQ[b*c - a*d, 0]
  5608. Int[Power[Cosh[Times[Optional[Pattern[e, Blank[]]], Plus[Optional[Pattern[a, Blank[]]], Times[Optional[Pattern[b, Blank[]]], Pattern[x, Blank[]]]], Power[Plus[Optional[Pattern[c, Blank[]]], Times[Optional[Pattern[d, Blank[]]], Pattern[x, Blank[]]]], -1]]], Optional[Pattern[n, Blank[]]]], Pattern[x, Blank[Symbol]]] := -Dist[d^(-1), Subst[Int[Cosh[(b*e)/d - (e*(b*c - a*d)*x)/d]^n/x^2, x], x, 1/(c + d*x)], x] /; FreeQ[{a, b, c, d}, x] && IGtQ[n, 0] && NeQ[b*c - a*d, 0]
  5609. Int[Power[Sinh[Pattern[u, Blank[]]], Optional[Pattern[n, Blank[]]]], Pattern[x, Blank[Symbol]]] := With[{lst = QuotientOfLinearsParts[u, x]}, Int[Sinh[(lst[[1]] + lst[[2]]*x)/(lst[[3]] + lst[[4]]*x)]^n, x]] /; IGtQ[n, 0] && QuotientOfLinearsQ[u, x]
  5610. Int[Power[Cosh[Pattern[u, Blank[]]], Optional[Pattern[n, Blank[]]]], Pattern[x, Blank[Symbol]]] := With[{lst = QuotientOfLinearsParts[u, x]}, Int[Cosh[(lst[[1]] + lst[[2]]*x)/(lst[[3]] + lst[[4]]*x)]^n, x]] /; IGtQ[n, 0] && QuotientOfLinearsQ[u, x]
  5611. Int[Times[Optional[Pattern[u, Blank[]]], Power[Sinh[Pattern[v, Blank[]]], Optional[Pattern[p, Blank[]]]], Power[Sinh[Pattern[w, Blank[]]], Optional[Pattern[q, Blank[]]]]], Pattern[x, Blank[Symbol]]] := Int[u*Sinh[v]^(p + q), x] /; EqQ[w, v]
  5612. Int[Times[Power[Cosh[Pattern[v, Blank[]]], Optional[Pattern[p, Blank[]]]], Power[Cosh[Pattern[w, Blank[]]], Optional[Pattern[q, Blank[]]]], Optional[Pattern[u, Blank[]]]], Pattern[x, Blank[Symbol]]] := Int[u*Cosh[v]^(p + q), x] /; EqQ[w, v]
  5613. Int[Times[Power[Sinh[Pattern[v, Blank[]]], Optional[Pattern[p, Blank[]]]], Power[Sinh[Pattern[w, Blank[]]], Optional[Pattern[q, Blank[]]]]], Pattern[x, Blank[Symbol]]] := Int[ExpandTrigReduce[Sinh[v]^p*Sinh[w]^q, x], x] /; IGtQ[p, 0] && IGtQ[q, 0] && ((PolynomialQ[v, x] && PolynomialQ[w, x]) || (BinomialQ[{v, w}, x] && IndependentQ[Cancel[v/w], x]))
  5614. Int[Times[Power[Cosh[Pattern[v, Blank[]]], Optional[Pattern[p, Blank[]]]], Power[Cosh[Pattern[w, Blank[]]], Optional[Pattern[q, Blank[]]]]], Pattern[x, Blank[Symbol]]] := Int[ExpandTrigReduce[Cosh[v]^p*Cosh[w]^q, x], x] /; IGtQ[p, 0] && IGtQ[q, 0] && ((PolynomialQ[v, x] && PolynomialQ[w, x]) || (BinomialQ[{v, w}, x] && IndependentQ[Cancel[v/w], x]))
  5615. Int[Times[Power[Pattern[x, Blank[]], Optional[Pattern[m, Blank[]]]], Power[Sinh[Pattern[v, Blank[]]], Optional[Pattern[p, Blank[]]]], Power[Sinh[Pattern[w, Blank[]]], Optional[Pattern[q, Blank[]]]]], Pattern[x, Blank[Symbol]]] := Int[ExpandTrigReduce[x^m, Sinh[v]^p*Sinh[w]^q, x], x] /; IGtQ[m, 0] && IGtQ[p, 0] && IGtQ[q, 0] && ((PolynomialQ[v, x] && PolynomialQ[w, x]) || (BinomialQ[{v, w}, x] && IndependentQ[Cancel[v/w], x]))
  5616. Int[Times[Power[Cosh[Pattern[v, Blank[]]], Optional[Pattern[p, Blank[]]]], Power[Cosh[Pattern[w, Blank[]]], Optional[Pattern[q, Blank[]]]], Power[Pattern[x, Blank[]], Optional[Pattern[m, Blank[]]]]], Pattern[x, Blank[Symbol]]] := Int[ExpandTrigReduce[x^m, Cosh[v]^p*Cosh[w]^q, x], x] /; IGtQ[m, 0] && IGtQ[p, 0] && IGtQ[q, 0] && ((PolynomialQ[v, x] && PolynomialQ[w, x]) || (BinomialQ[{v, w}, x] && IndependentQ[Cancel[v/w], x]))
  5617. Int[Times[Power[Cosh[Pattern[w, Blank[]]], Optional[Pattern[p, Blank[]]]], Optional[Pattern[u, Blank[]]], Power[Sinh[Pattern[v, Blank[]]], Optional[Pattern[p, Blank[]]]]], Pattern[x, Blank[Symbol]]] := Dist[1/2^p, Int[u*Sinh[2*v]^p, x], x] /; EqQ[w, v] && IntegerQ[p]
  5618. Int[Times[Power[Cosh[Pattern[w, Blank[]]], Optional[Pattern[q, Blank[]]]], Power[Sinh[Pattern[v, Blank[]]], Optional[Pattern[p, Blank[]]]]], Pattern[x, Blank[Symbol]]] := Int[ExpandTrigReduce[Sinh[v]^p*Cosh[w]^q, x], x] /; IGtQ[p, 0] && IGtQ[q, 0] && ((PolynomialQ[v, x] && PolynomialQ[w, x]) || (BinomialQ[{v, w}, x] && IndependentQ[Cancel[v/w], x]))
  5619. Int[Times[Power[Cosh[Pattern[w, Blank[]]], Optional[Pattern[q, Blank[]]]], Power[Pattern[x, Blank[]], Optional[Pattern[m, Blank[]]]], Power[Sinh[Pattern[v, Blank[]]], Optional[Pattern[p, Blank[]]]]], Pattern[x, Blank[Symbol]]] := Int[ExpandTrigReduce[x^m, Sinh[v]^p*Cosh[w]^q, x], x] /; IGtQ[m, 0] && IGtQ[p, 0] && IGtQ[q, 0] && ((PolynomialQ[v, x] && PolynomialQ[w, x]) || (BinomialQ[{v, w}, x] && IndependentQ[Cancel[v/w], x]))
  5620. Int[Times[Sinh[Pattern[v, Blank[]]], Power[Tanh[Pattern[w, Blank[]]], Optional[Pattern[n, Blank[]]]]], Pattern[x, Blank[Symbol]]] := Int[Cosh[v]*Tanh[w]^(n - 1), x] - Dist[Cosh[v - w], Int[Sech[w]*Tanh[w]^(n - 1), x], x] /; GtQ[n, 0] && NeQ[w, v] && FreeQ[v - w, x]
  5621. Int[Times[Cosh[Pattern[v, Blank[]]], Power[Coth[Pattern[w, Blank[]]], Optional[Pattern[n, Blank[]]]]], Pattern[x, Blank[Symbol]]] := Int[Sinh[v]*Coth[w]^(n - 1), x] + Dist[Cosh[v - w], Int[Csch[w]*Coth[w]^(n - 1), x], x] /; GtQ[n, 0] && NeQ[w, v] && FreeQ[v - w, x]
  5622. Int[Times[Power[Coth[Pattern[w, Blank[]]], Optional[Pattern[n, Blank[]]]], Sinh[Pattern[v, Blank[]]]], Pattern[x, Blank[Symbol]]] := Int[Cosh[v]*Coth[w]^(n - 1), x] + Dist[Sinh[v - w], Int[Csch[w]*Coth[w]^(n - 1), x], x] /; GtQ[n, 0] && NeQ[w, v] && FreeQ[v - w, x]
  5623. Int[Times[Cosh[Pattern[v, Blank[]]], Power[Tanh[Pattern[w, Blank[]]], Optional[Pattern[n, Blank[]]]]], Pattern[x, Blank[Symbol]]] := Int[Sinh[v]*Tanh[w]^(n - 1), x] - Dist[Sinh[v - w], Int[Sech[w]*Tanh[w]^(n - 1), x], x] /; GtQ[n, 0] && NeQ[w, v] && FreeQ[v - w, x]
  5624. Int[Times[Power[Sech[Pattern[w, Blank[]]], Optional[Pattern[n, Blank[]]]], Sinh[Pattern[v, Blank[]]]], Pattern[x, Blank[Symbol]]] := Dist[Cosh[v - w], Int[Tanh[w]*Sech[w]^(n - 1), x], x] + Dist[Sinh[v - w], Int[Sech[w]^(n - 1), x], x] /; GtQ[n, 0] && NeQ[w, v] && FreeQ[v - w, x]
  5625. Int[Times[Cosh[Pattern[v, Blank[]]], Power[Csch[Pattern[w, Blank[]]], Optional[Pattern[n, Blank[]]]]], Pattern[x, Blank[Symbol]]] := Dist[Cosh[v - w], Int[Coth[w]*Csch[w]^(n - 1), x], x] + Dist[Sinh[v - w], Int[Csch[w]^(n - 1), x], x] /; GtQ[n, 0] && NeQ[w, v] && FreeQ[v - w, x]
  5626. Int[Times[Power[Csch[Pattern[w, Blank[]]], Optional[Pattern[n, Blank[]]]], Sinh[Pattern[v, Blank[]]]], Pattern[x, Blank[Symbol]]] := Dist[Sinh[v - w], Int[Coth[w]*Csch[w]^(n - 1), x], x] + Dist[Cosh[v - w], Int[Csch[w]^(n - 1), x], x] /; GtQ[n, 0] && NeQ[w, v] && FreeQ[v - w, x]
  5627. Int[Times[Cosh[Pattern[v, Blank[]]], Power[Sech[Pattern[w, Blank[]]], Optional[Pattern[n, Blank[]]]]], Pattern[x, Blank[Symbol]]] := Dist[Sinh[v - w], Int[Tanh[w]*Sech[w]^(n - 1), x], x] + Dist[Cosh[v - w], Int[Sech[w]^(n - 1), x], x] /; GtQ[n, 0] && NeQ[w, v] && FreeQ[v - w, x]
  5628. Int[Times[Power[Plus[Optional[Pattern[e, Blank[]]], Times[Optional[Pattern[f, Blank[]]], Pattern[x, Blank[]]]], Optional[Pattern[m, Blank[]]]], Power[Plus[Pattern[a, Blank[]], Times[Cosh[Plus[Optional[Pattern[c, Blank[]]], Times[Optional[Pattern[d, Blank[]]], Pattern[x, Blank[]]]]], Optional[Pattern[b, Blank[]]], Sinh[Plus[Optional[Pattern[c, Blank[]]], Times[Optional[Pattern[d, Blank[]]], Pattern[x, Blank[]]]]]]], Optional[Pattern[n, Blank[]]]]], Pattern[x, Blank[Symbol]]] := Int[(e + f*x)^m*(a + (b*Sinh[2*c + 2*d*x])/2)^n, x] /; FreeQ[{a, b, c, d, e, f, m, n}, x]
  5629. Int[Times[Power[Pattern[x, Blank[]], Optional[Pattern[m, Blank[]]]], Power[Plus[Pattern[a, Blank[]], Times[Optional[Pattern[b, Blank[]]], Power[Sinh[Plus[Optional[Pattern[c, Blank[]]], Times[Optional[Pattern[d, Blank[]]], Pattern[x, Blank[]]]]], 2]]], Pattern[n, Blank[]]]], Pattern[x, Blank[Symbol]]] := Dist[1/2^n, Int[x^m*(2*a - b + b*Cosh[2*c + 2*d*x])^n, x], x] /; FreeQ[{a, b, c, d}, x] && NeQ[a - b, 0] && IGtQ[m, 0] && ILtQ[n, 0] && (EqQ[n, -1] || (EqQ[m, 1] && EqQ[n, -2]))
  5630. Int[Times[Power[Plus[Times[Power[Cosh[Plus[Optional[Pattern[c, Blank[]]], Times[Optional[Pattern[d, Blank[]]], Pattern[x, Blank[]]]]], 2], Optional[Pattern[b, Blank[]]]], Pattern[a, Blank[]]], Pattern[n, Blank[]]], Power[Pattern[x, Blank[]], Optional[Pattern[m, Blank[]]]]], Pattern[x, Blank[Symbol]]] := Dist[1/2^n, Int[x^m*(2*a + b + b*Cosh[2*c + 2*d*x])^n, x], x] /; FreeQ[{a, b, c, d}, x] && NeQ[a - b, 0] && IGtQ[m, 0] && ILtQ[n, 0] && (EqQ[n, -1] || (EqQ[m, 1] && EqQ[n, -2]))
  5631. Int[Times[Power[Plus[Optional[Pattern[f, Blank[]]], Times[Optional[Pattern[g, Blank[]]], Pattern[x, Blank[]]]], Optional[Pattern[m, Blank[]]]], Power[Plus[Optional[Pattern[a, Blank[]]], Times[Power[Cosh[Plus[Optional[Pattern[d, Blank[]]], Times[Optional[Pattern[e, Blank[]]], Pattern[x, Blank[]]]]], 2], Optional[Pattern[b, Blank[]]]], Times[Optional[Pattern[c, Blank[]]], Power[Sinh[Plus[Optional[Pattern[d, Blank[]]], Times[Optional[Pattern[e, Blank[]]], Pattern[x, Blank[]]]]], 2]]], -1]], Pattern[x, Blank[Symbol]]] := Dist[2, Int[(f + g*x)^m/(2*a + b - c + (b + c)*Cosh[2*d + 2*e*x]), x], x] /; FreeQ[{a, b, c, d, e, f, g}, x] && IGtQ[m, 0] && NeQ[a + b, 0] && NeQ[a + c, 0]
  5632. Int[Times[Power[Plus[Optional[Pattern[f, Blank[]]], Times[Optional[Pattern[g, Blank[]]], Pattern[x, Blank[]]]], Optional[Pattern[m, Blank[]]]], Power[Sech[Plus[Optional[Pattern[d, Blank[]]], Times[Optional[Pattern[e, Blank[]]], Pattern[x, Blank[]]]]], 2], Power[Plus[Pattern[b, Blank[]], Times[Optional[Pattern[c, Blank[]]], Power[Tanh[Plus[Optional[Pattern[d, Blank[]]], Times[Optional[Pattern[e, Blank[]]], Pattern[x, Blank[]]]]], 2]]], -1]], Pattern[x, Blank[Symbol]]] := Dist[2, Int[(f + g*x)^m/(b - c + (b + c)*Cosh[2*d + 2*e*x]), x], x] /; FreeQ[{b, c, d, e, f, g}, x] && IGtQ[m, 0]
  5633. Int[Times[Power[Plus[Optional[Pattern[f, Blank[]]], Times[Optional[Pattern[g, Blank[]]], Pattern[x, Blank[]]]], Optional[Pattern[m, Blank[]]]], Power[Sech[Plus[Optional[Pattern[d, Blank[]]], Times[Optional[Pattern[e, Blank[]]], Pattern[x, Blank[]]]]], 2], Power[Plus[Optional[Pattern[b, Blank[]]], Times[Optional[Pattern[a, Blank[]]], Power[Sech[Plus[Optional[Pattern[d, Blank[]]], Times[Optional[Pattern[e, Blank[]]], Pattern[x, Blank[]]]]], 2]], Times[Optional[Pattern[c, Blank[]]], Power[Tanh[Plus[Optional[Pattern[d, Blank[]]], Times[Optional[Pattern[e, Blank[]]], Pattern[x, Blank[]]]]], 2]]], -1]], Pattern[x, Blank[Symbol]]] := Dist[2, Int[(f + g*x)^m/(2*a + b - c + (b + c)*Cosh[2*d + 2*e*x]), x], x] /; FreeQ[{a, b, c, d, e, f, g}, x] && IGtQ[m, 0] && NeQ[a + b, 0] && NeQ[a + c, 0]
  5634. Int[Times[Power[Csch[Plus[Optional[Pattern[d, Blank[]]], Times[Optional[Pattern[e, Blank[]]], Pattern[x, Blank[]]]]], 2], Power[Plus[Times[Power[Coth[Plus[Optional[Pattern[d, Blank[]]], Times[Optional[Pattern[e, Blank[]]], Pattern[x, Blank[]]]]], 2], Optional[Pattern[b, Blank[]]]], Pattern[c, Blank[]]], -1], Power[Plus[Optional[Pattern[f, Blank[]]], Times[Optional[Pattern[g, Blank[]]], Pattern[x, Blank[]]]], Optional[Pattern[m, Blank[]]]]], Pattern[x, Blank[Symbol]]] := Dist[2, Int[(f + g*x)^m/(b - c + (b + c)*Cosh[2*d + 2*e*x]), x], x] /; FreeQ[{b, c, d, e, f, g}, x] && IGtQ[m, 0]
  5635. Int[Times[Power[Csch[Plus[Optional[Pattern[d, Blank[]]], Times[Optional[Pattern[e, Blank[]]], Pattern[x, Blank[]]]]], 2], Power[Plus[Times[Power[Csch[Plus[Optional[Pattern[d, Blank[]]], Times[Optional[Pattern[e, Blank[]]], Pattern[x, Blank[]]]]], 2], Optional[Pattern[a, Blank[]]]], Times[Power[Coth[Plus[Optional[Pattern[d, Blank[]]], Times[Optional[Pattern[e, Blank[]]], Pattern[x, Blank[]]]]], 2], Optional[Pattern[b, Blank[]]]], Optional[Pattern[c, Blank[]]]], -1], Power[Plus[Optional[Pattern[f, Blank[]]], Times[Optional[Pattern[g, Blank[]]], Pattern[x, Blank[]]]], Optional[Pattern[m, Blank[]]]]], Pattern[x, Blank[Symbol]]] := Dist[2, Int[(f + g*x)^m/(2*a + b - c + (b + c)*Cosh[2*d + 2*e*x]), x], x] /; FreeQ[{a, b, c, d, e, f, g}, x] && IGtQ[m, 0] && NeQ[a + b, 0] && NeQ[a + c, 0]
  5636. Int[Times[Plus[Optional[Pattern[e, Blank[]]], Times[Optional[Pattern[f, Blank[]]], Pattern[x, Blank[]]]], Power[Plus[Pattern[a, Blank[]], Times[Optional[Pattern[b, Blank[]]], Sinh[Plus[Optional[Pattern[c, Blank[]]], Times[Optional[Pattern[d, Blank[]]], Pattern[x, Blank[]]]]]]], -2], Plus[Pattern[A, Blank[]], Times[Optional[Pattern[B, Blank[]]], Sinh[Plus[Optional[Pattern[c, Blank[]]], Times[Optional[Pattern[d, Blank[]]], Pattern[x, Blank[]]]]]]]], Pattern[x, Blank[Symbol]]] := Simp[(B*(e + f*x)*Cosh[c + d*x])/(a*d*(a + b*Sinh[c + d*x])), x] - Dist[(B*f)/(a*d), Int[Cosh[c + d*x]/(a + b*Sinh[c + d*x]), x], x] /; FreeQ[{a, b, c, d, e, f, A, B}, x] && EqQ[a*A + b*B, 0]
  5637. Int[Times[Power[Plus[Times[Cosh[Plus[Optional[Pattern[c, Blank[]]], Times[Optional[Pattern[d, Blank[]]], Pattern[x, Blank[]]]]], Optional[Pattern[b, Blank[]]]], Pattern[a, Blank[]]], -2], Plus[Times[Cosh[Plus[Optional[Pattern[c, Blank[]]], Times[Optional[Pattern[d, Blank[]]], Pattern[x, Blank[]]]]], Optional[Pattern[B, Blank[]]]], Pattern[A, Blank[]]], Plus[Optional[Pattern[e, Blank[]]], Times[Optional[Pattern[f, Blank[]]], Pattern[x, Blank[]]]]], Pattern[x, Blank[Symbol]]] := Simp[(B*(e + f*x)*Sinh[c + d*x])/(a*d*(a + b*Cosh[c + d*x])), x] - Dist[(B*f)/(a*d), Int[Sinh[c + d*x]/(a + b*Cosh[c + d*x]), x], x] /; FreeQ[{a, b, c, d, e, f, A, B}, x] && EqQ[a*A - b*B, 0]
  5638. Int[Times[Power[Plus[Optional[Pattern[e, Blank[]]], Times[Optional[Pattern[f, Blank[]]], Pattern[x, Blank[]]]], Optional[Pattern[m, Blank[]]]], Power[Sinh[Plus[Optional[Pattern[a, Blank[]]], Times[Optional[Pattern[b, Blank[]]], Power[Plus[Pattern[c, Blank[]], Times[Optional[Pattern[d, Blank[]]], Pattern[x, Blank[]]]], Pattern[n, Blank[]]]]]], Optional[Pattern[p, Blank[]]]]], Pattern[x, Blank[Symbol]]] := Dist[1/d^(m + 1), Subst[Int[(d*e - c*f + f*x)^m*Sinh[a + b*x^n]^p, x], x, c + d*x], x] /; FreeQ[{a, b, c, d, e, f, n}, x] && IGtQ[m, 0] && RationalQ[p]
  5639. Int[Times[Power[Cosh[Plus[Optional[Pattern[a, Blank[]]], Times[Optional[Pattern[b, Blank[]]], Power[Plus[Pattern[c, Blank[]], Times[Optional[Pattern[d, Blank[]]], Pattern[x, Blank[]]]], Pattern[n, Blank[]]]]]], Optional[Pattern[p, Blank[]]]], Power[Plus[Optional[Pattern[e, Blank[]]], Times[Optional[Pattern[f, Blank[]]], Pattern[x, Blank[]]]], Optional[Pattern[m, Blank[]]]]], Pattern[x, Blank[Symbol]]] := Dist[1/d^(m + 1), Subst[Int[(d*e - c*f + f*x)^m*Cosh[a + b*x^n]^p, x], x, c + d*x], x] /; FreeQ[{a, b, c, d, e, f, n}, x] && IGtQ[m, 0] && RationalQ[p]
  5640. Int[Times[Power[Sech[Pattern[v, Blank[]]], Optional[Pattern[m, Blank[]]]], Power[Plus[Pattern[a, Blank[]], Times[Optional[Pattern[b, Blank[]]], Tanh[Pattern[v, Blank[]]]]], Optional[Pattern[n, Blank[]]]]], Pattern[x, Blank[Symbol]]] := Int[(a*Cosh[v] + b*Sinh[v])^n, x] /; FreeQ[{a, b}, x] && IntegerQ[(m - 1)/2] && EqQ[m + n, 0]
  5641. Int[Times[Power[Csch[Pattern[v, Blank[]]], Optional[Pattern[m, Blank[]]]], Power[Plus[Times[Coth[Pattern[v, Blank[]]], Optional[Pattern[b, Blank[]]]], Pattern[a, Blank[]]], Optional[Pattern[n, Blank[]]]]], Pattern[x, Blank[Symbol]]] := Int[(b*Cosh[v] + a*Sinh[v])^n, x] /; FreeQ[{a, b}, x] && IntegerQ[(m - 1)/2] && EqQ[m + n, 0]
  5642. Int[Times[Optional[Pattern[u, Blank[]]], Power[Sinh[Plus[Optional[Pattern[a, Blank[]]], Times[Optional[Pattern[b, Blank[]]], Pattern[x, Blank[]]]]], Optional[Pattern[m, Blank[]]]], Power[Sinh[Plus[Optional[Pattern[c, Blank[]]], Times[Optional[Pattern[d, Blank[]]], Pattern[x, Blank[]]]]], Optional[Pattern[n, Blank[]]]]], Pattern[x, Blank[Symbol]]] := Int[ExpandTrigReduce[u, Sinh[a + b*x]^m*Sinh[c + d*x]^n, x], x] /; FreeQ[{a, b, c, d}, x] && IGtQ[m, 0] && IGtQ[n, 0]
  5643. Int[Times[Power[Cosh[Plus[Optional[Pattern[a, Blank[]]], Times[Optional[Pattern[b, Blank[]]], Pattern[x, Blank[]]]]], Optional[Pattern[m, Blank[]]]], Power[Cosh[Plus[Optional[Pattern[c, Blank[]]], Times[Optional[Pattern[d, Blank[]]], Pattern[x, Blank[]]]]], Optional[Pattern[n, Blank[]]]], Optional[Pattern[u, Blank[]]]], Pattern[x, Blank[Symbol]]] := Int[ExpandTrigReduce[u, Cosh[a + b*x]^m*Cosh[c + d*x]^n, x], x] /; FreeQ[{a, b, c, d}, x] && IGtQ[m, 0] && IGtQ[n, 0]
  5644. Int[Times[Sech[Plus[Optional[Pattern[a, Blank[]]], Times[Optional[Pattern[b, Blank[]]], Pattern[x, Blank[]]]]], Sech[Plus[Pattern[c, Blank[]], Times[Optional[Pattern[d, Blank[]]], Pattern[x, Blank[]]]]]], Pattern[x, Blank[Symbol]]] := -Dist[Csch[(b*c - a*d)/d], Int[Tanh[a + b*x], x], x] + Dist[Csch[(b*c - a*d)/b], Int[Tanh[c + d*x], x], x] /; FreeQ[{a, b, c, d}, x] && EqQ[b^2 - d^2, 0] && NeQ[b*c - a*d, 0]
  5645. Int[Times[Csch[Plus[Optional[Pattern[a, Blank[]]], Times[Optional[Pattern[b, Blank[]]], Pattern[x, Blank[]]]]], Csch[Plus[Pattern[c, Blank[]], Times[Optional[Pattern[d, Blank[]]], Pattern[x, Blank[]]]]]], Pattern[x, Blank[Symbol]]] := Dist[Csch[(b*c - a*d)/b], Int[Coth[a + b*x], x], x] - Dist[Csch[(b*c - a*d)/d], Int[Coth[c + d*x], x], x] /; FreeQ[{a, b, c, d}, x] && EqQ[b^2 - d^2, 0] && NeQ[b*c - a*d, 0]
  5646. Int[Times[Tanh[Plus[Optional[Pattern[a, Blank[]]], Times[Optional[Pattern[b, Blank[]]], Pattern[x, Blank[]]]]], Tanh[Plus[Pattern[c, Blank[]], Times[Optional[Pattern[d, Blank[]]], Pattern[x, Blank[]]]]]], Pattern[x, Blank[Symbol]]] := Simp[(b*x)/d, x] - Dist[(b*Cosh[(b*c - a*d)/d])/d, Int[Sech[a + b*x]*Sech[c + d*x], x], x] /; FreeQ[{a, b, c, d}, x] && EqQ[b^2 - d^2, 0] && NeQ[b*c - a*d, 0]
  5647. Int[Times[Coth[Plus[Optional[Pattern[a, Blank[]]], Times[Optional[Pattern[b, Blank[]]], Pattern[x, Blank[]]]]], Coth[Plus[Pattern[c, Blank[]], Times[Optional[Pattern[d, Blank[]]], Pattern[x, Blank[]]]]]], Pattern[x, Blank[Symbol]]] := Simp[(b*x)/d, x] + Dist[Cosh[(b*c - a*d)/d], Int[Csch[a + b*x]*Csch[c + d*x], x], x] /; FreeQ[{a, b, c, d}, x] && EqQ[b^2 - d^2, 0] && NeQ[b*c - a*d, 0]
  5648. Int[Times[Optional[Pattern[u, Blank[]]], Power[Plus[Times[Cosh[Pattern[v, Blank[]]], Optional[Pattern[a, Blank[]]]], Times[Optional[Pattern[b, Blank[]]], Sinh[Pattern[v, Blank[]]]]], Optional[Pattern[n, Blank[]]]]], Pattern[x, Blank[Symbol]]] := Int[u*(a*E^((a*v)/b))^n, x] /; FreeQ[{a, b, n}, x] && EqQ[a^2 - b^2, 0]
  5649. Int[Sinh[Times[Power[Plus[Optional[Pattern[a, Blank[]]], Times[Log[Times[Optional[Pattern[c, Blank[]]], Power[Pattern[x, Blank[]], Optional[Pattern[n, Blank[]]]]]], Optional[Pattern[b, Blank[]]]]], 2], Optional[Pattern[d, Blank[]]]]], Pattern[x, Blank[Symbol]]] := -Dist[2^(-1), Int[E^(-(d*(a + b*Log[c*x^n])^2)), x], x] + Dist[1/2, Int[E^(d*(a + b*Log[c*x^n])^2), x], x] /; FreeQ[{a, b, c, d, n}, x]
  5650. Int[Cosh[Times[Power[Plus[Optional[Pattern[a, Blank[]]], Times[Log[Times[Optional[Pattern[c, Blank[]]], Power[Pattern[x, Blank[]], Optional[Pattern[n, Blank[]]]]]], Optional[Pattern[b, Blank[]]]]], 2], Optional[Pattern[d, Blank[]]]]], Pattern[x, Blank[Symbol]]] := Dist[1/2, Int[E^(-(d*(a + b*Log[c*x^n])^2)), x], x] + Dist[1/2, Int[E^(d*(a + b*Log[c*x^n])^2), x], x] /; FreeQ[{a, b, c, d, n}, x]
  5651. Int[Times[Power[Times[Optional[Pattern[e, Blank[]]], Pattern[x, Blank[]]], Optional[Pattern[m, Blank[]]]], Sinh[Times[Power[Plus[Optional[Pattern[a, Blank[]]], Times[Log[Times[Optional[Pattern[c, Blank[]]], Power[Pattern[x, Blank[]], Optional[Pattern[n, Blank[]]]]]], Optional[Pattern[b, Blank[]]]]], 2], Optional[Pattern[d, Blank[]]]]]], Pattern[x, Blank[Symbol]]] := -Dist[2^(-1), Int[(e*x)^m/E^(d*(a + b*Log[c*x^n])^2), x], x] + Dist[1/2, Int[(e*x)^m*E^(d*(a + b*Log[c*x^n])^2), x], x] /; FreeQ[{a, b, c, d, e, m, n}, x]
  5652. Int[Times[Cosh[Times[Power[Plus[Optional[Pattern[a, Blank[]]], Times[Log[Times[Optional[Pattern[c, Blank[]]], Power[Pattern[x, Blank[]], Optional[Pattern[n, Blank[]]]]]], Optional[Pattern[b, Blank[]]]]], 2], Optional[Pattern[d, Blank[]]]]], Power[Times[Optional[Pattern[e, Blank[]]], Pattern[x, Blank[]]], Optional[Pattern[m, Blank[]]]]], Pattern[x, Blank[Symbol]]] := Dist[1/2, Int[(e*x)^m/E^(d*(a + b*Log[c*x^n])^2), x], x] + Dist[1/2, Int[(e*x)^m*E^(d*(a + b*Log[c*x^n])^2), x], x] /; FreeQ[{a, b, c, d, e, m, n}, x]
  5653. Int[Power[Plus[Optional[Pattern[a, Blank[]]], Times[ArcSinh[Times[Optional[Pattern[c, Blank[]]], Pattern[x, Blank[]]]], Optional[Pattern[b, Blank[]]]]], Optional[Pattern[n, Blank[]]]], Pattern[x, Blank[Symbol]]] := Simp[x*(a + b*ArcSinh[c*x])^n, x] - Dist[b*c*n, Int[(x*(a + b*ArcSinh[c*x])^(n - 1))/Sqrt[1 + c^2*x^2], x], x] /; FreeQ[{a, b, c}, x] && GtQ[n, 0]
  5654. Int[Power[Plus[Optional[Pattern[a, Blank[]]], Times[ArcCosh[Times[Optional[Pattern[c, Blank[]]], Pattern[x, Blank[]]]], Optional[Pattern[b, Blank[]]]]], Optional[Pattern[n, Blank[]]]], Pattern[x, Blank[Symbol]]] := Simp[x*(a + b*ArcCosh[c*x])^n, x] - Dist[b*c*n, Int[(x*(a + b*ArcCosh[c*x])^(n - 1))/(Sqrt[-1 + c*x]*Sqrt[1 + c*x]), x], x] /; FreeQ[{a, b, c}, x] && GtQ[n, 0]
  5655. Int[Power[Plus[Optional[Pattern[a, Blank[]]], Times[ArcSinh[Times[Optional[Pattern[c, Blank[]]], Pattern[x, Blank[]]]], Optional[Pattern[b, Blank[]]]]], Pattern[n, Blank[]]], Pattern[x, Blank[Symbol]]] := Simp[(Sqrt[1 + c^2*x^2]*(a + b*ArcSinh[c*x])^(n + 1))/(b*c*(n + 1)), x] - Dist[c/(b*(n + 1)), Int[(x*(a + b*ArcSinh[c*x])^(n + 1))/Sqrt[1 + c^2*x^2], x], x] /; FreeQ[{a, b, c}, x] && LtQ[n, -1]
  5656. Int[Power[Plus[Optional[Pattern[a, Blank[]]], Times[ArcCosh[Times[Optional[Pattern[c, Blank[]]], Pattern[x, Blank[]]]], Optional[Pattern[b, Blank[]]]]], Pattern[n, Blank[]]], Pattern[x, Blank[Symbol]]] := Simp[(Sqrt[-1 + c*x]*Sqrt[1 + c*x]*(a + b*ArcCosh[c*x])^(n + 1))/(b*c*(n + 1)), x] - Dist[c/(b*(n + 1)), Int[(x*(a + b*ArcCosh[c*x])^(n + 1))/(Sqrt[-1 + c*x]*Sqrt[1 + c*x]), x], x] /; FreeQ[{a, b, c}, x] && LtQ[n, -1]
  5657. Int[Power[Plus[Optional[Pattern[a, Blank[]]], Times[ArcSinh[Times[Optional[Pattern[c, Blank[]]], Pattern[x, Blank[]]]], Optional[Pattern[b, Blank[]]]]], Pattern[n, Blank[]]], Pattern[x, Blank[Symbol]]] := Dist[1/(b*c), Subst[Int[x^n*Cosh[a/b - x/b], x], x, a + b*ArcSinh[c*x]], x] /; FreeQ[{a, b, c, n}, x]
  5658. Int[Power[Plus[Optional[Pattern[a, Blank[]]], Times[ArcCosh[Times[Optional[Pattern[c, Blank[]]], Pattern[x, Blank[]]]], Optional[Pattern[b, Blank[]]]]], Pattern[n, Blank[]]], Pattern[x, Blank[Symbol]]] := -Dist[(b*c)^(-1), Subst[Int[x^n*Sinh[a/b - x/b], x], x, a + b*ArcCosh[c*x]], x] /; FreeQ[{a, b, c, n}, x]
  5659. Int[Times[Power[Plus[Optional[Pattern[a, Blank[]]], Times[ArcSinh[Times[Optional[Pattern[c, Blank[]]], Pattern[x, Blank[]]]], Optional[Pattern[b, Blank[]]]]], Optional[Pattern[n, Blank[]]]], Power[Pattern[x, Blank[]], -1]], Pattern[x, Blank[Symbol]]] := Subst[Int[(a + b*x)^n/Tanh[x], x], x, ArcSinh[c*x]] /; FreeQ[{a, b, c}, x] && IGtQ[n, 0]
  5660. Int[Times[Power[Plus[Optional[Pattern[a, Blank[]]], Times[ArcCosh[Times[Optional[Pattern[c, Blank[]]], Pattern[x, Blank[]]]], Optional[Pattern[b, Blank[]]]]], Optional[Pattern[n, Blank[]]]], Power[Pattern[x, Blank[]], -1]], Pattern[x, Blank[Symbol]]] := Subst[Int[(a + b*x)^n/Coth[x], x], x, ArcCosh[c*x]] /; FreeQ[{a, b, c}, x] && IGtQ[n, 0]
  5661. Int[Times[Power[Plus[Optional[Pattern[a, Blank[]]], Times[ArcSinh[Times[Optional[Pattern[c, Blank[]]], Pattern[x, Blank[]]]], Optional[Pattern[b, Blank[]]]]], Optional[Pattern[n, Blank[]]]], Power[Times[Optional[Pattern[d, Blank[]]], Pattern[x, Blank[]]], Optional[Pattern[m, Blank[]]]]], Pattern[x, Blank[Symbol]]] := Simp[((d*x)^(m + 1)*(a + b*ArcSinh[c*x])^n)/(d*(m + 1)), x] - Dist[(b*c*n)/(d*(m + 1)), Int[((d*x)^(m + 1)*(a + b*ArcSinh[c*x])^(n - 1))/Sqrt[1 + c^2*x^2], x], x] /; FreeQ[{a, b, c, d, m}, x] && IGtQ[n, 0] && NeQ[m, -1]
  5662. Int[Times[Power[Plus[Optional[Pattern[a, Blank[]]], Times[ArcCosh[Times[Optional[Pattern[c, Blank[]]], Pattern[x, Blank[]]]], Optional[Pattern[b, Blank[]]]]], Optional[Pattern[n, Blank[]]]], Power[Times[Optional[Pattern[d, Blank[]]], Pattern[x, Blank[]]], Optional[Pattern[m, Blank[]]]]], Pattern[x, Blank[Symbol]]] := Simp[((d*x)^(m + 1)*(a + b*ArcCosh[c*x])^n)/(d*(m + 1)), x] - Dist[(b*c*n)/(d*(m + 1)), Int[((d*x)^(m + 1)*(a + b*ArcCosh[c*x])^(n - 1))/(Sqrt[-1 + c*x]*Sqrt[1 + c*x]), x], x] /; FreeQ[{a, b, c, d, m}, x] && IGtQ[n, 0] && NeQ[m, -1]
  5663. Int[Times[Power[Plus[Optional[Pattern[a, Blank[]]], Times[ArcSinh[Times[Optional[Pattern[c, Blank[]]], Pattern[x, Blank[]]]], Optional[Pattern[b, Blank[]]]]], Pattern[n, Blank[]]], Power[Pattern[x, Blank[]], Optional[Pattern[m, Blank[]]]]], Pattern[x, Blank[Symbol]]] := Simp[(x^(m + 1)*(a + b*ArcSinh[c*x])^n)/(m + 1), x] - Dist[(b*c*n)/(m + 1), Int[(x^(m + 1)*(a + b*ArcSinh[c*x])^(n - 1))/Sqrt[1 + c^2*x^2], x], x] /; FreeQ[{a, b, c}, x] && IGtQ[m, 0] && GtQ[n, 0]
  5664. Int[Times[Power[Plus[Optional[Pattern[a, Blank[]]], Times[ArcCosh[Times[Optional[Pattern[c, Blank[]]], Pattern[x, Blank[]]]], Optional[Pattern[b, Blank[]]]]], Pattern[n, Blank[]]], Power[Pattern[x, Blank[]], Optional[Pattern[m, Blank[]]]]], Pattern[x, Blank[Symbol]]] := Simp[(x^(m + 1)*(a + b*ArcCosh[c*x])^n)/(m + 1), x] - Dist[(b*c*n)/(m + 1), Int[(x^(m + 1)*(a + b*ArcCosh[c*x])^(n - 1))/(Sqrt[-1 + c*x]*Sqrt[1 + c*x]), x], x] /; FreeQ[{a, b, c}, x] && IGtQ[m, 0] && GtQ[n, 0]
  5665. Int[Times[Power[Plus[Optional[Pattern[a, Blank[]]], Times[ArcSinh[Times[Optional[Pattern[c, Blank[]]], Pattern[x, Blank[]]]], Optional[Pattern[b, Blank[]]]]], Pattern[n, Blank[]]], Power[Pattern[x, Blank[]], Optional[Pattern[m, Blank[]]]]], Pattern[x, Blank[Symbol]]] := Simp[(x^m*Sqrt[1 + c^2*x^2]*(a + b*ArcSinh[c*x])^(n + 1))/(b*c*(n + 1)), x] - Dist[1/(b*c^(m + 1)*(n + 1)), Subst[Int[ExpandTrigReduce[(a + b*x)^(n + 1), Sinh[x]^(m - 1)*(m + (m + 1)*Sinh[x]^2), x], x], x, ArcSinh[c*x]], x] /; FreeQ[{a, b, c}, x] && IGtQ[m, 0] && GeQ[n, -2] && LtQ[n, -1]
  5666. Int[Times[Power[Plus[Optional[Pattern[a, Blank[]]], Times[ArcCosh[Times[Optional[Pattern[c, Blank[]]], Pattern[x, Blank[]]]], Optional[Pattern[b, Blank[]]]]], Pattern[n, Blank[]]], Power[Pattern[x, Blank[]], Optional[Pattern[m, Blank[]]]]], Pattern[x, Blank[Symbol]]] := Simp[(x^m*Sqrt[-1 + c*x]*Sqrt[1 + c*x]*(a + b*ArcCosh[c*x])^(n + 1))/(b*c*(n + 1)), x] + Dist[1/(b*c^(m + 1)*(n + 1)), Subst[Int[ExpandTrigReduce[(a + b*x)^(n + 1)*Cosh[x]^(m - 1)*(m - (m + 1)*Cosh[x]^2), x], x], x, ArcCosh[c*x]], x] /; FreeQ[{a, b, c}, x] && IGtQ[m, 0] && GeQ[n, -2] && LtQ[n, -1]
  5667. Int[Times[Power[Plus[Optional[Pattern[a, Blank[]]], Times[ArcSinh[Times[Optional[Pattern[c, Blank[]]], Pattern[x, Blank[]]]], Optional[Pattern[b, Blank[]]]]], Pattern[n, Blank[]]], Power[Pattern[x, Blank[]], Optional[Pattern[m, Blank[]]]]], Pattern[x, Blank[Symbol]]] := Simp[(x^m*Sqrt[1 + c^2*x^2]*(a + b*ArcSinh[c*x])^(n + 1))/(b*c*(n + 1)), x] + (-Dist[(c*(m + 1))/(b*(n + 1)), Int[(x^(m + 1)*(a + b*ArcSinh[c*x])^(n + 1))/Sqrt[1 + c^2*x^2], x], x] - Dist[m/(b*c*(n + 1)), Int[(x^(m - 1)*(a + b*ArcSinh[c*x])^(n + 1))/Sqrt[1 + c^2*x^2], x], x]) /; FreeQ[{a, b, c}, x] && IGtQ[m, 0] && LtQ[n, -2]
  5668. Int[Times[Power[Plus[Optional[Pattern[a, Blank[]]], Times[ArcCosh[Times[Optional[Pattern[c, Blank[]]], Pattern[x, Blank[]]]], Optional[Pattern[b, Blank[]]]]], Pattern[n, Blank[]]], Power[Pattern[x, Blank[]], Optional[Pattern[m, Blank[]]]]], Pattern[x, Blank[Symbol]]] := Simp[(x^m*Sqrt[-1 + c*x]*Sqrt[1 + c*x]*(a + b*ArcCosh[c*x])^(n + 1))/(b*c*(n + 1)), x] + (-Dist[(c*(m + 1))/(b*(n + 1)), Int[(x^(m + 1)*(a + b*ArcCosh[c*x])^(n + 1))/(Sqrt[-1 + c*x]*Sqrt[1 + c*x]), x], x] + Dist[m/(b*c*(n + 1)), Int[(x^(m - 1)*(a + b*ArcCosh[c*x])^(n + 1))/(Sqrt[-1 + c*x]*Sqrt[1 + c*x]), x], x]) /; FreeQ[{a, b, c}, x] && IGtQ[m, 0] && LtQ[n, -2]
  5669. Int[Times[Power[Plus[Optional[Pattern[a, Blank[]]], Times[ArcSinh[Times[Optional[Pattern[c, Blank[]]], Pattern[x, Blank[]]]], Optional[Pattern[b, Blank[]]]]], Pattern[n, Blank[]]], Power[Pattern[x, Blank[]], Optional[Pattern[m, Blank[]]]]], Pattern[x, Blank[Symbol]]] := Dist[1/c^(m + 1), Subst[Int[(a + b*x)^n*Sinh[x]^m*Cosh[x], x], x, ArcSinh[c*x]], x] /; FreeQ[{a, b, c, n}, x] && IGtQ[m, 0]
  5670. Int[Times[Power[Plus[Optional[Pattern[a, Blank[]]], Times[ArcCosh[Times[Optional[Pattern[c, Blank[]]], Pattern[x, Blank[]]]], Optional[Pattern[b, Blank[]]]]], Pattern[n, Blank[]]], Power[Pattern[x, Blank[]], Optional[Pattern[m, Blank[]]]]], Pattern[x, Blank[Symbol]]] := Dist[1/c^(m + 1), Subst[Int[(a + b*x)^n*Cosh[x]^m*Sinh[x], x], x, ArcCosh[c*x]], x] /; FreeQ[{a, b, c, n}, x] && IGtQ[m, 0]
  5671. Int[Times[Power[Plus[Optional[Pattern[a, Blank[]]], Times[ArcSinh[Times[Optional[Pattern[c, Blank[]]], Pattern[x, Blank[]]]], Optional[Pattern[b, Blank[]]]]], Optional[Pattern[n, Blank[]]]], Power[Times[Optional[Pattern[d, Blank[]]], Pattern[x, Blank[]]], Optional[Pattern[m, Blank[]]]]], Pattern[x, Blank[Symbol]]] := Unintegrable[(d*x)^m*(a + b*ArcSinh[c*x])^n, x] /; FreeQ[{a, b, c, d, m, n}, x]
  5672. Int[Times[Power[Plus[Optional[Pattern[a, Blank[]]], Times[ArcCosh[Times[Optional[Pattern[c, Blank[]]], Pattern[x, Blank[]]]], Optional[Pattern[b, Blank[]]]]], Optional[Pattern[n, Blank[]]]], Power[Times[Optional[Pattern[d, Blank[]]], Pattern[x, Blank[]]], Optional[Pattern[m, Blank[]]]]], Pattern[x, Blank[Symbol]]] := Unintegrable[(d*x)^m*(a + b*ArcCosh[c*x])^n, x] /; FreeQ[{a, b, c, d, m, n}, x]
  5673. Int[Times[Power[Plus[Optional[Pattern[a, Blank[]]], Times[ArcSinh[Times[Optional[Pattern[c, Blank[]]], Pattern[x, Blank[]]]], Optional[Pattern[b, Blank[]]]]], -1], Power[Plus[Pattern[d, Blank[]], Times[Optional[Pattern[e, Blank[]]], Power[Pattern[x, Blank[]], 2]]], Rational[-1, 2]]], Pattern[x, Blank[Symbol]]] := Simp[Log[a + b*ArcSinh[c*x]]/(b*c*Sqrt[d]), x] /; FreeQ[{a, b, c, d, e}, x] && EqQ[e, c^2*d] && GtQ[d, 0]
  5674. Int[Times[Power[Plus[Optional[Pattern[a, Blank[]]], Times[ArcCosh[Times[Optional[Pattern[c, Blank[]]], Pattern[x, Blank[]]]], Optional[Pattern[b, Blank[]]]]], -1], Power[Plus[Pattern[d1, Blank[]], Times[Optional[Pattern[e1, Blank[]]], Pattern[x, Blank[]]]], Rational[-1, 2]], Power[Plus[Pattern[d2, Blank[]], Times[Optional[Pattern[e2, Blank[]]], Pattern[x, Blank[]]]], Rational[-1, 2]]], Pattern[x, Blank[Symbol]]] := Simp[Log[a + b*ArcCosh[c*x]]/(b*c*Sqrt[-(d1*d2)]), x] /; FreeQ[{a, b, c, d1, e1, d2, e2}, x] && EqQ[e1, c*d1] && EqQ[e2, -(c*d2)] && GtQ[d1, 0] && LtQ[d2, 0]
  5675. Int[Times[Power[Plus[Optional[Pattern[a, Blank[]]], Times[ArcSinh[Times[Optional[Pattern[c, Blank[]]], Pattern[x, Blank[]]]], Optional[Pattern[b, Blank[]]]]], Optional[Pattern[n, Blank[]]]], Power[Plus[Pattern[d, Blank[]], Times[Optional[Pattern[e, Blank[]]], Power[Pattern[x, Blank[]], 2]]], Rational[-1, 2]]], Pattern[x, Blank[Symbol]]] := Simp[(a + b*ArcSinh[c*x])^(n + 1)/(b*c*Sqrt[d]*(n + 1)), x] /; FreeQ[{a, b, c, d, e, n}, x] && EqQ[e, c^2*d] && GtQ[d, 0] && NeQ[n, -1]
  5676. Int[Times[Power[Plus[Optional[Pattern[a, Blank[]]], Times[ArcCosh[Times[Optional[Pattern[c, Blank[]]], Pattern[x, Blank[]]]], Optional[Pattern[b, Blank[]]]]], Optional[Pattern[n, Blank[]]]], Power[Plus[Pattern[d1, Blank[]], Times[Optional[Pattern[e1, Blank[]]], Pattern[x, Blank[]]]], Rational[-1, 2]], Power[Plus[Pattern[d2, Blank[]], Times[Optional[Pattern[e2, Blank[]]], Pattern[x, Blank[]]]], Rational[-1, 2]]], Pattern[x, Blank[Symbol]]] := Simp[(a + b*ArcCosh[c*x])^(n + 1)/(b*c*Sqrt[-(d1*d2)]*(n + 1)), x] /; FreeQ[{a, b, c, d1, e1, d2, e2, n}, x] && EqQ[e1, c*d1] && EqQ[e2, -(c*d2)] && GtQ[d1, 0] && LtQ[d2, 0] && NeQ[n, -1]
  5677. Int[Times[Power[Plus[Optional[Pattern[a, Blank[]]], Times[ArcSinh[Times[Optional[Pattern[c, Blank[]]], Pattern[x, Blank[]]]], Optional[Pattern[b, Blank[]]]]], Optional[Pattern[n, Blank[]]]], Power[Plus[Pattern[d, Blank[]], Times[Optional[Pattern[e, Blank[]]], Power[Pattern[x, Blank[]], 2]]], Rational[-1, 2]]], Pattern[x, Blank[Symbol]]] := Dist[Sqrt[1 + c^2*x^2]/Sqrt[d + e*x^2], Int[(a + b*ArcSinh[c*x])^n/Sqrt[1 + c^2*x^2], x], x] /; FreeQ[{a, b, c, d, e, n}, x] && EqQ[e, c^2*d] && !GtQ[d, 0]
  5678. Int[Times[Power[Plus[Optional[Pattern[a, Blank[]]], Times[ArcCosh[Times[Optional[Pattern[c, Blank[]]], Pattern[x, Blank[]]]], Optional[Pattern[b, Blank[]]]]], Optional[Pattern[n, Blank[]]]], Power[Plus[Pattern[d1, Blank[]], Times[Optional[Pattern[e1, Blank[]]], Pattern[x, Blank[]]]], Rational[-1, 2]], Power[Plus[Pattern[d2, Blank[]], Times[Optional[Pattern[e2, Blank[]]], Pattern[x, Blank[]]]], Rational[-1, 2]]], Pattern[x, Blank[Symbol]]] := Dist[(Sqrt[1 + c*x]*Sqrt[-1 + c*x])/(Sqrt[d1 + e1*x]*Sqrt[d2 + e2*x]), Int[(a + b*ArcCosh[c*x])^n/(Sqrt[1 + c*x]*Sqrt[-1 + c*x]), x], x] /; FreeQ[{a, b, c, d1, e1, d2, e2, n}, x] && EqQ[e1, c*d1] && EqQ[e2, -(c*d2)] && !(GtQ[d1, 0] && LtQ[d2, 0])
  5679. Int[Times[Plus[Optional[Pattern[a, Blank[]]], Times[ArcSinh[Times[Optional[Pattern[c, Blank[]]], Pattern[x, Blank[]]]], Optional[Pattern[b, Blank[]]]]], Power[Plus[Pattern[d, Blank[]], Times[Optional[Pattern[e, Blank[]]], Power[Pattern[x, Blank[]], 2]]], Optional[Pattern[p, Blank[]]]]], Pattern[x, Blank[Symbol]]] := With[{u = IntHide[(d + e*x^2)^p, x]}, Dist[a + b*ArcSinh[c*x], u, x] - Dist[b*c, Int[SimplifyIntegrand[u/Sqrt[1 + c^2*x^2], x], x], x]] /; FreeQ[{a, b, c, d, e}, x] && EqQ[e, c^2*d] && IGtQ[p, 0]
  5680. Int[Times[Plus[Optional[Pattern[a, Blank[]]], Times[ArcCosh[Times[Optional[Pattern[c, Blank[]]], Pattern[x, Blank[]]]], Optional[Pattern[b, Blank[]]]]], Power[Plus[Pattern[d, Blank[]], Times[Optional[Pattern[e, Blank[]]], Power[Pattern[x, Blank[]], 2]]], Optional[Pattern[p, Blank[]]]]], Pattern[x, Blank[Symbol]]] := With[{u = IntHide[(d + e*x^2)^p, x]}, Dist[a + b*ArcCosh[c*x], u, x] - Dist[b*c, Int[SimplifyIntegrand[u/(Sqrt[1 + c*x]*Sqrt[-1 + c*x]), x], x], x]] /; FreeQ[{a, b, c, d, e}, x] && EqQ[c^2*d + e, 0] && IGtQ[p, 0]
  5681. Int[Times[Power[Plus[Optional[Pattern[a, Blank[]]], Times[ArcCosh[Times[Optional[Pattern[c, Blank[]]], Pattern[x, Blank[]]]], Optional[Pattern[b, Blank[]]]]], Optional[Pattern[n, Blank[]]]], Power[Plus[Pattern[d, Blank[]], Times[Optional[Pattern[e, Blank[]]], Power[Pattern[x, Blank[]], 2]]], Optional[Pattern[p, Blank[]]]]], Pattern[x, Blank[Symbol]]] := Simp[(x*(d + e*x^2)^p*(a + b*ArcCosh[c*x])^n)/(2*p + 1), x] + (-Dist[(b*c*n*(-d)^p)/(2*p + 1), Int[x*(-1 + c*x)^(p - 1/2)*(1 + c*x)^(p - 1/2)*(a + b*ArcCosh[c*x])^(n - 1), x], x] + Dist[(2*d*p)/(2*p + 1), Int[(d + e*x^2)^(p - 1)*(a + b*ArcCosh[c*x])^n, x], x]) /; FreeQ[{a, b, c, d, e}, x] && EqQ[c^2*d + e, 0] && GtQ[n, 0] && GtQ[p, 0] && IntegerQ[p]
  5682. Int[Times[Power[Plus[Optional[Pattern[a, Blank[]]], Times[ArcSinh[Times[Optional[Pattern[c, Blank[]]], Pattern[x, Blank[]]]], Optional[Pattern[b, Blank[]]]]], Optional[Pattern[n, Blank[]]]], Power[Plus[Pattern[d, Blank[]], Times[Optional[Pattern[e, Blank[]]], Power[Pattern[x, Blank[]], 2]]], Rational[1, 2]]], Pattern[x, Blank[Symbol]]] := Simp[(x*Sqrt[d + e*x^2]*(a + b*ArcSinh[c*x])^n)/2, x] + (Dist[Sqrt[d + e*x^2]/(2*Sqrt[1 + c^2*x^2]), Int[(a + b*ArcSinh[c*x])^n/Sqrt[1 + c^2*x^2], x], x] - Dist[(b*c*n*Sqrt[d + e*x^2])/(2*Sqrt[1 + c^2*x^2]), Int[x*(a + b*ArcSinh[c*x])^(n - 1), x], x]) /; FreeQ[{a, b, c, d, e}, x] && EqQ[e, c^2*d] && GtQ[n, 0]
  5683. Int[Times[Power[Plus[Optional[Pattern[a, Blank[]]], Times[ArcCosh[Times[Optional[Pattern[c, Blank[]]], Pattern[x, Blank[]]]], Optional[Pattern[b, Blank[]]]]], Optional[Pattern[n, Blank[]]]], Power[Plus[Pattern[d1, Blank[]], Times[Optional[Pattern[e1, Blank[]]], Pattern[x, Blank[]]]], Rational[1, 2]], Power[Plus[Pattern[d2, Blank[]], Times[Optional[Pattern[e2, Blank[]]], Pattern[x, Blank[]]]], Rational[1, 2]]], Pattern[x, Blank[Symbol]]] := Simp[(x*Sqrt[d1 + e1*x]*Sqrt[d2 + e2*x]*(a + b*ArcCosh[c*x])^n)/2, x] + (-Dist[(Sqrt[d1 + e1*x]*Sqrt[d2 + e2*x])/(2*Sqrt[1 + c*x]*Sqrt[-1 + c*x]), Int[(a + b*ArcCosh[c*x])^n/(Sqrt[1 + c*x]*Sqrt[-1 + c*x]), x], x] - Dist[(b*c*n*Sqrt[d1 + e1*x]*Sqrt[d2 + e2*x])/(2*Sqrt[1 + c*x]*Sqrt[-1 + c*x]), Int[x*(a + b*ArcCosh[c*x])^(n - 1), x], x]) /; FreeQ[{a, b, c, d1, e1, d2, e2}, x] && EqQ[e1, c*d1] && EqQ[e2, -(c*d2)] && GtQ[n, 0]
  5684. Int[Times[Power[Plus[Optional[Pattern[a, Blank[]]], Times[ArcSinh[Times[Optional[Pattern[c, Blank[]]], Pattern[x, Blank[]]]], Optional[Pattern[b, Blank[]]]]], Optional[Pattern[n, Blank[]]]], Power[Plus[Pattern[d, Blank[]], Times[Optional[Pattern[e, Blank[]]], Power[Pattern[x, Blank[]], 2]]], Optional[Pattern[p, Blank[]]]]], Pattern[x, Blank[Symbol]]] := Simp[(x*(d + e*x^2)^p*(a + b*ArcSinh[c*x])^n)/(2*p + 1), x] + (Dist[(2*d*p)/(2*p + 1), Int[(d + e*x^2)^(p - 1)*(a + b*ArcSinh[c*x])^n, x], x] - Dist[(b*c*n*d^IntPart[p]*(d + e*x^2)^FracPart[p])/((2*p + 1)*(1 + c^2*x^2)^FracPart[p]), Int[x*(1 + c^2*x^2)^(p - 1/2)*(a + b*ArcSinh[c*x])^(n - 1), x], x]) /; FreeQ[{a, b, c, d, e}, x] && EqQ[e, c^2*d] && GtQ[n, 0] && GtQ[p, 0]
  5685. Int[Times[Power[Plus[Optional[Pattern[a, Blank[]]], Times[ArcCosh[Times[Optional[Pattern[c, Blank[]]], Pattern[x, Blank[]]]], Optional[Pattern[b, Blank[]]]]], Optional[Pattern[n, Blank[]]]], Power[Plus[Pattern[d1, Blank[]], Times[Optional[Pattern[e1, Blank[]]], Pattern[x, Blank[]]]], Optional[Pattern[p, Blank[]]]], Power[Plus[Pattern[d2, Blank[]], Times[Optional[Pattern[e2, Blank[]]], Pattern[x, Blank[]]]], Optional[Pattern[p, Blank[]]]]], Pattern[x, Blank[Symbol]]] := Simp[(x*(d1 + e1*x)^p*(d2 + e2*x)^p*(a + b*ArcCosh[c*x])^n)/(2*p + 1), x] + (Dist[(2*d1*d2*p)/(2*p + 1), Int[(d1 + e1*x)^(p - 1)*(d2 + e2*x)^(p - 1)*(a + b*ArcCosh[c*x])^n, x], x] - Dist[(b*c*n*(-(d1*d2))^(p - 1/2)*Sqrt[d1 + e1*x]*Sqrt[d2 + e2*x])/((2*p + 1)*Sqrt[1 + c*x]*Sqrt[-1 + c*x]), Int[x*(-1 + c^2*x^2)^(p - 1/2)*(a + b*ArcCosh[c*x])^(n - 1), x], x]) /; FreeQ[{a, b, c, d1, e1, d2, e2}, x] && EqQ[e1, c*d1] && EqQ[e2, -(c*d2)] && GtQ[n, 0] && GtQ[p, 0] && IntegerQ[p - 1/2]
  5686. Int[Times[Power[Plus[Optional[Pattern[a, Blank[]]], Times[ArcCosh[Times[Optional[Pattern[c, Blank[]]], Pattern[x, Blank[]]]], Optional[Pattern[b, Blank[]]]]], Optional[Pattern[n, Blank[]]]], Power[Plus[Pattern[d1, Blank[]], Times[Optional[Pattern[e1, Blank[]]], Pattern[x, Blank[]]]], Optional[Pattern[p, Blank[]]]], Power[Plus[Pattern[d2, Blank[]], Times[Optional[Pattern[e2, Blank[]]], Pattern[x, Blank[]]]], Optional[Pattern[p, Blank[]]]]], Pattern[x, Blank[Symbol]]] := Simp[(x*(d1 + e1*x)^p*(d2 + e2*x)^p*(a + b*ArcCosh[c*x])^n)/(2*p + 1), x] + (Dist[(2*d1*d2*p)/(2*p + 1), Int[(d1 + e1*x)^(p - 1)*(d2 + e2*x)^(p - 1)*(a + b*ArcCosh[c*x])^n, x], x] - Dist[(b*c*n*(-(d1*d2))^IntPart[p]*(d1 + e1*x)^FracPart[p]*(d2 + e2*x)^FracPart[p])/((2*p + 1)*(1 + c*x)^FracPart[p]*(-1 + c*x)^FracPart[p]), Int[x*(-1 + c*x)^(p - 1/2)*(1 + c*x)^(p - 1/2)*(a + b*ArcCosh[c*x])^(n - 1), x], x]) /; FreeQ[{a, b, c, d1, e1, d2, e2}, x] && EqQ[e1, c*d1] && EqQ[e2, -(c*d2)] && GtQ[n, 0] && GtQ[p, 0]
  5687. Int[Times[Power[Plus[Optional[Pattern[a, Blank[]]], Times[ArcSinh[Times[Optional[Pattern[c, Blank[]]], Pattern[x, Blank[]]]], Optional[Pattern[b, Blank[]]]]], Optional[Pattern[n, Blank[]]]], Power[Plus[Pattern[d, Blank[]], Times[Optional[Pattern[e, Blank[]]], Power[Pattern[x, Blank[]], 2]]], Rational[-3, 2]]], Pattern[x, Blank[Symbol]]] := Simp[(x*(a + b*ArcSinh[c*x])^n)/(d*Sqrt[d + e*x^2]), x] - Dist[(b*c*n*Sqrt[1 + c^2*x^2])/(d*Sqrt[d + e*x^2]), Int[(x*(a + b*ArcSinh[c*x])^(n - 1))/(1 + c^2*x^2), x], x] /; FreeQ[{a, b, c, d, e}, x] && EqQ[e, c^2*d] && GtQ[n, 0]
  5688. Int[Times[Power[Plus[Optional[Pattern[a, Blank[]]], Times[ArcCosh[Times[Optional[Pattern[c, Blank[]]], Pattern[x, Blank[]]]], Optional[Pattern[b, Blank[]]]]], Optional[Pattern[n, Blank[]]]], Power[Plus[Pattern[d1, Blank[]], Times[Optional[Pattern[e1, Blank[]]], Pattern[x, Blank[]]]], Rational[-3, 2]], Power[Plus[Pattern[d2, Blank[]], Times[Optional[Pattern[e2, Blank[]]], Pattern[x, Blank[]]]], Rational[-3, 2]]], Pattern[x, Blank[Symbol]]] := Simp[(x*(a + b*ArcCosh[c*x])^n)/(d1*d2*Sqrt[d1 + e1*x]*Sqrt[d2 + e2*x]), x] + Dist[(b*c*n*Sqrt[1 + c*x]*Sqrt[-1 + c*x])/(d1*d2*Sqrt[d1 + e1*x]*Sqrt[d2 + e2*x]), Int[(x*(a + b*ArcCosh[c*x])^(n - 1))/(1 - c^2*x^2), x], x] /; FreeQ[{a, b, c, d1, e1, d2, e2}, x] && EqQ[e1, c*d1] && EqQ[e2, -(c*d2)] && GtQ[n, 0]
  5689. Int[Times[Power[Plus[Optional[Pattern[a, Blank[]]], Times[ArcCosh[Times[Optional[Pattern[c, Blank[]]], Pattern[x, Blank[]]]], Optional[Pattern[b, Blank[]]]]], Optional[Pattern[n, Blank[]]]], Power[Plus[Pattern[d, Blank[]], Times[Optional[Pattern[e, Blank[]]], Power[Pattern[x, Blank[]], 2]]], Pattern[p, Blank[]]]], Pattern[x, Blank[Symbol]]] := -Simp[(x*(d + e*x^2)^(p + 1)*(a + b*ArcCosh[c*x])^n)/(2*d*(p + 1)), x] + (-Dist[(b*c*n*(-d)^p)/(2*(p + 1)), Int[x*(1 + c*x)^(p + 1/2)*(-1 + c*x)^(p + 1/2)*(a + b*ArcCosh[c*x])^(n - 1), x], x] + Dist[(2*p + 3)/(2*d*(p + 1)), Int[(d + e*x^2)^(p + 1)*(a + b*ArcCosh[c*x])^n, x], x]) /; FreeQ[{a, b, c, d, e}, x] && EqQ[c^2*d + e, 0] && GtQ[n, 0] && LtQ[p, -1] && IntegerQ[p]
  5690. Int[Times[Power[Plus[Optional[Pattern[a, Blank[]]], Times[ArcSinh[Times[Optional[Pattern[c, Blank[]]], Pattern[x, Blank[]]]], Optional[Pattern[b, Blank[]]]]], Optional[Pattern[n, Blank[]]]], Power[Plus[Pattern[d, Blank[]], Times[Optional[Pattern[e, Blank[]]], Power[Pattern[x, Blank[]], 2]]], Pattern[p, Blank[]]]], Pattern[x, Blank[Symbol]]] := -Simp[(x*(d + e*x^2)^(p + 1)*(a + b*ArcSinh[c*x])^n)/(2*d*(p + 1)), x] + (Dist[(2*p + 3)/(2*d*(p + 1)), Int[(d + e*x^2)^(p + 1)*(a + b*ArcSinh[c*x])^n, x], x] + Dist[(b*c*n*d^IntPart[p]*(d + e*x^2)^FracPart[p])/(2*(p + 1)*(1 + c^2*x^2)^FracPart[p]), Int[x*(1 + c^2*x^2)^(p + 1/2)*(a + b*ArcSinh[c*x])^(n - 1), x], x]) /; FreeQ[{a, b, c, d, e}, x] && EqQ[e, c^2*d] && GtQ[n, 0] && LtQ[p, -1] && NeQ[p, -3/2]
  5691. Int[Times[Power[Plus[Optional[Pattern[a, Blank[]]], Times[ArcCosh[Times[Optional[Pattern[c, Blank[]]], Pattern[x, Blank[]]]], Optional[Pattern[b, Blank[]]]]], Optional[Pattern[n, Blank[]]]], Power[Plus[Pattern[d1, Blank[]], Times[Optional[Pattern[e1, Blank[]]], Pattern[x, Blank[]]]], Pattern[p, Blank[]]], Power[Plus[Pattern[d2, Blank[]], Times[Optional[Pattern[e2, Blank[]]], Pattern[x, Blank[]]]], Pattern[p, Blank[]]]], Pattern[x, Blank[Symbol]]] := -Simp[(x*(d1 + e1*x)^(p + 1)*(d2 + e2*x)^(p + 1)*(a + b*ArcCosh[c*x])^n)/(2*d1*d2*(p + 1)), x] + (Dist[(2*p + 3)/(2*d1*d2*(p + 1)), Int[(d1 + e1*x)^(p + 1)*(d2 + e2*x)^(p + 1)*(a + b*ArcCosh[c*x])^n, x], x] - Dist[(b*c*n*(-(d1*d2))^(p + 1/2)*Sqrt[1 + c*x]*Sqrt[-1 + c*x])/(2*(p + 1)*Sqrt[d1 + e1*x]*Sqrt[d2 + e2*x]), Int[x*(-1 + c^2*x^2)^(p + 1/2)*(a + b*ArcCosh[c*x])^(n - 1), x], x]) /; FreeQ[{a, b, c, d1, e1, d2, e2}, x] && EqQ[e1, c*d1] && EqQ[e2, -(c*d2)] && GtQ[n, 0] && LtQ[p, -1] && NeQ[p, -3/2] && IntegerQ[p + 1/2]
  5692. Int[Times[Power[Plus[Optional[Pattern[a, Blank[]]], Times[ArcCosh[Times[Optional[Pattern[c, Blank[]]], Pattern[x, Blank[]]]], Optional[Pattern[b, Blank[]]]]], Optional[Pattern[n, Blank[]]]], Power[Plus[Pattern[d1, Blank[]], Times[Optional[Pattern[e1, Blank[]]], Pattern[x, Blank[]]]], Pattern[p, Blank[]]], Power[Plus[Pattern[d2, Blank[]], Times[Optional[Pattern[e2, Blank[]]], Pattern[x, Blank[]]]], Pattern[p, Blank[]]]], Pattern[x, Blank[Symbol]]] := -Simp[(x*(d1 + e1*x)^(p + 1)*(d2 + e2*x)^(p + 1)*(a + b*ArcCosh[c*x])^n)/(2*d1*d2*(p + 1)), x] + (Dist[(2*p + 3)/(2*d1*d2*(p + 1)), Int[(d1 + e1*x)^(p + 1)*(d2 + e2*x)^(p + 1)*(a + b*ArcCosh[c*x])^n, x], x] - Dist[(b*c*n*(-(d1*d2))^IntPart[p]*(d1 + e1*x)^FracPart[p]*(d2 + e2*x)^FracPart[p])/(2*(p + 1)*(1 + c*x)^FracPart[p]*(-1 + c*x)^FracPart[p]), Int[x*(1 + c*x)^(p + 1/2)*(-1 + c*x)^(p + 1/2)*(a + b*ArcCosh[c*x])^(n - 1), x], x]) /; FreeQ[{a, b, c, d1, e1, d2, e2}, x] && EqQ[e1, c*d1] && EqQ[e2, -(c*d2)] && GtQ[n, 0] && LtQ[p, -1] && NeQ[p, -3/2]
  5693. Int[Times[Power[Plus[Optional[Pattern[a, Blank[]]], Times[ArcSinh[Times[Optional[Pattern[c, Blank[]]], Pattern[x, Blank[]]]], Optional[Pattern[b, Blank[]]]]], Optional[Pattern[n, Blank[]]]], Power[Plus[Pattern[d, Blank[]], Times[Optional[Pattern[e, Blank[]]], Power[Pattern[x, Blank[]], 2]]], -1]], Pattern[x, Blank[Symbol]]] := Dist[1/(c*d), Subst[Int[(a + b*x)^n*Sech[x], x], x, ArcSinh[c*x]], x] /; FreeQ[{a, b, c, d, e}, x] && EqQ[e, c^2*d] && IGtQ[n, 0]
  5694. Int[Times[Power[Plus[Optional[Pattern[a, Blank[]]], Times[ArcCosh[Times[Optional[Pattern[c, Blank[]]], Pattern[x, Blank[]]]], Optional[Pattern[b, Blank[]]]]], Optional[Pattern[n, Blank[]]]], Power[Plus[Pattern[d, Blank[]], Times[Optional[Pattern[e, Blank[]]], Power[Pattern[x, Blank[]], 2]]], -1]], Pattern[x, Blank[Symbol]]] := -Dist[(c*d)^(-1), Subst[Int[(a + b*x)^n*Csch[x], x], x, ArcCosh[c*x]], x] /; FreeQ[{a, b, c, d, e}, x] && EqQ[c^2*d + e, 0] && IGtQ[n, 0]
  5695. Int[Times[Power[Plus[Optional[Pattern[a, Blank[]]], Times[ArcCosh[Times[Optional[Pattern[c, Blank[]]], Pattern[x, Blank[]]]], Optional[Pattern[b, Blank[]]]]], Pattern[n, Blank[]]], Power[Plus[Pattern[d, Blank[]], Times[Optional[Pattern[e, Blank[]]], Power[Pattern[x, Blank[]], 2]]], Optional[Pattern[p, Blank[]]]]], Pattern[x, Blank[Symbol]]] := Simp[((-d)^p*(-1 + c*x)^(p + 1/2)*(1 + c*x)^(p + 1/2)*(a + b*ArcCosh[c*x])^(n + 1))/(b*c*(n + 1)), x] - Dist[(c*(-d)^p*(2*p + 1))/(b*(n + 1)), Int[x*(-1 + c*x)^(p - 1/2)*(1 + c*x)^(p - 1/2)*(a + b*ArcCosh[c*x])^(n + 1), x], x] /; FreeQ[{a, b, c, d, e, p}, x] && EqQ[c^2*d + e, 0] && LtQ[n, -1] && IntegerQ[p]
  5696. Int[Times[Power[Plus[Optional[Pattern[a, Blank[]]], Times[ArcSinh[Times[Optional[Pattern[c, Blank[]]], Pattern[x, Blank[]]]], Optional[Pattern[b, Blank[]]]]], Pattern[n, Blank[]]], Power[Plus[Pattern[d, Blank[]], Times[Optional[Pattern[e, Blank[]]], Power[Pattern[x, Blank[]], 2]]], Optional[Pattern[p, Blank[]]]]], Pattern[x, Blank[Symbol]]] := Simp[(Sqrt[1 + c^2*x^2]*(d + e*x^2)^p*(a + b*ArcSinh[c*x])^(n + 1))/(b*c*(n + 1)), x] - Dist[(c*(2*p + 1)*d^IntPart[p]*(d + e*x^2)^FracPart[p])/(b*(n + 1)*(1 + c^2*x^2)^FracPart[p]), Int[x*(1 + c^2*x^2)^(p - 1/2)*(a + b*ArcSinh[c*x])^(n + 1), x], x] /; FreeQ[{a, b, c, d, e, p}, x] && EqQ[e, c^2*d] && LtQ[n, -1]
  5697. Int[Times[Power[Plus[Optional[Pattern[a, Blank[]]], Times[ArcCosh[Times[Optional[Pattern[c, Blank[]]], Pattern[x, Blank[]]]], Optional[Pattern[b, Blank[]]]]], Pattern[n, Blank[]]], Power[Plus[Pattern[d1, Blank[]], Times[Optional[Pattern[e1, Blank[]]], Pattern[x, Blank[]]]], Optional[Pattern[p, Blank[]]]], Power[Plus[Pattern[d2, Blank[]], Times[Optional[Pattern[e2, Blank[]]], Pattern[x, Blank[]]]], Optional[Pattern[p, Blank[]]]]], Pattern[x, Blank[Symbol]]] := Simp[(Sqrt[1 + c*x]*Sqrt[-1 + c*x]*(d1 + e1*x)^p*(d2 + e2*x)^p*(a + b*ArcCosh[c*x])^(n + 1))/(b*c*(n + 1)), x] - Dist[(c*(2*p + 1)*(-(d1*d2))^(p - 1/2)*Sqrt[d1 + e1*x]*Sqrt[d2 + e2*x])/(b*(n + 1)*Sqrt[1 + c*x]*Sqrt[-1 + c*x]), Int[x*(-1 + c^2*x^2)^(p - 1/2)*(a + b*ArcCosh[c*x])^(n + 1), x], x] /; FreeQ[{a, b, c, d1, e1, d2, e2, p}, x] && EqQ[e1, c*d1] && EqQ[e2, -(c*d2)] && LtQ[n, -1] && IntegerQ[p - 1/2]
  5698. Int[Times[Power[Plus[Optional[Pattern[a, Blank[]]], Times[ArcCosh[Times[Optional[Pattern[c, Blank[]]], Pattern[x, Blank[]]]], Optional[Pattern[b, Blank[]]]]], Pattern[n, Blank[]]], Power[Plus[Pattern[d1, Blank[]], Times[Optional[Pattern[e1, Blank[]]], Pattern[x, Blank[]]]], Optional[Pattern[p, Blank[]]]], Power[Plus[Pattern[d2, Blank[]], Times[Optional[Pattern[e2, Blank[]]], Pattern[x, Blank[]]]], Optional[Pattern[p, Blank[]]]]], Pattern[x, Blank[Symbol]]] := Simp[(Sqrt[1 + c*x]*Sqrt[-1 + c*x]*(d1 + e1*x)^p*(d2 + e2*x)^p*(a + b*ArcCosh[c*x])^(n + 1))/(b*c*(n + 1)), x] - Dist[(c*(2*p + 1)*(-(d1*d2))^IntPart[p]*(d1 + e1*x)^FracPart[p]*(d2 + e2*x)^FracPart[p])/(b*(n + 1)*(1 + c*x)^FracPart[p]*(-1 + c*x)^FracPart[p]), Int[x*(1 + c*x)^(p - 1/2)*(-1 + c*x)^(p - 1/2)*(a + b*ArcCosh[c*x])^(n + 1), x], x] /; FreeQ[{a, b, c, d1, e1, d2, e2, p}, x] && EqQ[e1, c*d1] && EqQ[e2, -(c*d2)] && LtQ[n, -1]
  5699. Int[Times[Power[Plus[Optional[Pattern[a, Blank[]]], Times[ArcSinh[Times[Optional[Pattern[c, Blank[]]], Pattern[x, Blank[]]]], Optional[Pattern[b, Blank[]]]]], Optional[Pattern[n, Blank[]]]], Power[Plus[Pattern[d, Blank[]], Times[Optional[Pattern[e, Blank[]]], Power[Pattern[x, Blank[]], 2]]], Optional[Pattern[p, Blank[]]]]], Pattern[x, Blank[Symbol]]] := Dist[d^p/c, Subst[Int[(a + b*x)^n*Cosh[x]^(2*p + 1), x], x, ArcSinh[c*x]], x] /; FreeQ[{a, b, c, d, e, n}, x] && EqQ[e, c^2*d] && IGtQ[2*p, 0] && (IntegerQ[p] || GtQ[d, 0])
  5700. Int[Times[Power[Plus[Optional[Pattern[a, Blank[]]], Times[ArcCosh[Times[Optional[Pattern[c, Blank[]]], Pattern[x, Blank[]]]], Optional[Pattern[b, Blank[]]]]], Optional[Pattern[n, Blank[]]]], Power[Plus[Pattern[d, Blank[]], Times[Optional[Pattern[e, Blank[]]], Power[Pattern[x, Blank[]], 2]]], Optional[Pattern[p, Blank[]]]]], Pattern[x, Blank[Symbol]]] := Dist[(-d)^p/c, Subst[Int[(a + b*x)^n*Sinh[x]^(2*p + 1), x], x, ArcCosh[c*x]], x] /; FreeQ[{a, b, c, d, e, n}, x] && EqQ[c^2*d + e, 0] && IGtQ[p, 0]
  5701. Int[Times[Power[Plus[Optional[Pattern[a, Blank[]]], Times[ArcCosh[Times[Optional[Pattern[c, Blank[]]], Pattern[x, Blank[]]]], Optional[Pattern[b, Blank[]]]]], Optional[Pattern[n, Blank[]]]], Power[Plus[Pattern[d1, Blank[]], Times[Optional[Pattern[e1, Blank[]]], Pattern[x, Blank[]]]], Optional[Pattern[p, Blank[]]]], Power[Plus[Pattern[d2, Blank[]], Times[Optional[Pattern[e2, Blank[]]], Pattern[x, Blank[]]]], Optional[Pattern[p, Blank[]]]]], Pattern[x, Blank[Symbol]]] := Dist[(-(d1*d2))^p/c, Subst[Int[(a + b*x)^n*Sinh[x]^(2*p + 1), x], x, ArcCosh[c*x]], x] /; FreeQ[{a, b, c, d1, e1, d2, e2, n}, x] && EqQ[e1, c*d1] && EqQ[e2, -(c*d2)] && IGtQ[p + 1/2, 0] && (GtQ[d1, 0] && LtQ[d2, 0])
  5702. Int[Times[Power[Plus[Optional[Pattern[a, Blank[]]], Times[ArcSinh[Times[Optional[Pattern[c, Blank[]]], Pattern[x, Blank[]]]], Optional[Pattern[b, Blank[]]]]], Optional[Pattern[n, Blank[]]]], Power[Plus[Pattern[d, Blank[]], Times[Optional[Pattern[e, Blank[]]], Power[Pattern[x, Blank[]], 2]]], Optional[Pattern[p, Blank[]]]]], Pattern[x, Blank[Symbol]]] := Dist[(d^(p - 1/2)*Sqrt[d + e*x^2])/Sqrt[1 + c^2*x^2], Int[(1 + c^2*x^2)^p*(a + b*ArcSinh[c*x])^n, x], x] /; FreeQ[{a, b, c, d, e, n}, x] && EqQ[e, c^2*d] && IGtQ[2*p, 0] && !(IntegerQ[p] || GtQ[d, 0])
  5703. Int[Times[Power[Plus[Optional[Pattern[a, Blank[]]], Times[ArcCosh[Times[Optional[Pattern[c, Blank[]]], Pattern[x, Blank[]]]], Optional[Pattern[b, Blank[]]]]], Optional[Pattern[n, Blank[]]]], Power[Plus[Pattern[d1, Blank[]], Times[Optional[Pattern[e1, Blank[]]], Pattern[x, Blank[]]]], Optional[Pattern[p, Blank[]]]], Power[Plus[Pattern[d2, Blank[]], Times[Optional[Pattern[e2, Blank[]]], Pattern[x, Blank[]]]], Optional[Pattern[p, Blank[]]]]], Pattern[x, Blank[Symbol]]] := Dist[((-(d1*d2))^(p - 1/2)*Sqrt[d1 + e1*x]*Sqrt[d2 + e2*x])/(Sqrt[1 + c*x]*Sqrt[-1 + c*x]), Int[(1 + c*x)^p*(-1 + c*x)^p*(a + b*ArcCosh[c*x])^n, x], x] /; FreeQ[{a, b, c, d1, e1, d2, e2, n}, x] && EqQ[e1, c*d1] && EqQ[e2, -(c*d2)] && IGtQ[2*p, 0] && !(GtQ[d1, 0] && LtQ[d2, 0])
  5704. Int[Times[Plus[Optional[Pattern[a, Blank[]]], Times[ArcSinh[Times[Optional[Pattern[c, Blank[]]], Pattern[x, Blank[]]]], Optional[Pattern[b, Blank[]]]]], Power[Plus[Pattern[d, Blank[]], Times[Optional[Pattern[e, Blank[]]], Power[Pattern[x, Blank[]], 2]]], Optional[Pattern[p, Blank[]]]]], Pattern[x, Blank[Symbol]]] := With[{u = IntHide[(d + e*x^2)^p, x]}, Dist[a + b*ArcSinh[c*x], u, x] - Dist[b*c, Int[SimplifyIntegrand[u/Sqrt[1 + c^2*x^2], x], x], x]] /; FreeQ[{a, b, c, d, e}, x] && NeQ[e, c^2*d] && (IGtQ[p, 0] || ILtQ[p + 1/2, 0])
  5705. Int[Times[Plus[Optional[Pattern[a, Blank[]]], Times[ArcCosh[Times[Optional[Pattern[c, Blank[]]], Pattern[x, Blank[]]]], Optional[Pattern[b, Blank[]]]]], Power[Plus[Pattern[d, Blank[]], Times[Optional[Pattern[e, Blank[]]], Power[Pattern[x, Blank[]], 2]]], Optional[Pattern[p, Blank[]]]]], Pattern[x, Blank[Symbol]]] := With[{u = IntHide[(d + e*x^2)^p, x]}, Dist[a + b*ArcCosh[c*x], u, x] - Dist[b*c, Int[SimplifyIntegrand[u/(Sqrt[1 + c*x]*Sqrt[-1 + c*x]), x], x], x]] /; FreeQ[{a, b, c, d, e}, x] && NeQ[c^2*d + e, 0] && (IGtQ[p, 0] || ILtQ[p + 1/2, 0])
  5706. Int[Times[Power[Plus[Optional[Pattern[a, Blank[]]], Times[ArcSinh[Times[Optional[Pattern[c, Blank[]]], Pattern[x, Blank[]]]], Optional[Pattern[b, Blank[]]]]], Optional[Pattern[n, Blank[]]]], Power[Plus[Pattern[d, Blank[]], Times[Optional[Pattern[e, Blank[]]], Power[Pattern[x, Blank[]], 2]]], Optional[Pattern[p, Blank[]]]]], Pattern[x, Blank[Symbol]]] := Int[ExpandIntegrand[(a + b*ArcSinh[c*x])^n, (d + e*x^2)^p, x], x] /; FreeQ[{a, b, c, d, e, n}, x] && NeQ[e, c^2*d] && IntegerQ[p] && (p > 0 || IGtQ[n, 0])
  5707. Int[Times[Power[Plus[Optional[Pattern[a, Blank[]]], Times[ArcCosh[Times[Optional[Pattern[c, Blank[]]], Pattern[x, Blank[]]]], Optional[Pattern[b, Blank[]]]]], Optional[Pattern[n, Blank[]]]], Power[Plus[Pattern[d, Blank[]], Times[Optional[Pattern[e, Blank[]]], Power[Pattern[x, Blank[]], 2]]], Optional[Pattern[p, Blank[]]]]], Pattern[x, Blank[Symbol]]] := Int[ExpandIntegrand[(a + b*ArcCosh[c*x])^n, (d + e*x^2)^p, x], x] /; FreeQ[{a, b, c, d, e, n}, x] && NeQ[c^2*d + e, 0] && IntegerQ[p] && (p > 0 || IGtQ[n, 0])
  5708. Int[Times[Power[Plus[Optional[Pattern[a, Blank[]]], Times[ArcSinh[Times[Optional[Pattern[c, Blank[]]], Pattern[x, Blank[]]]], Optional[Pattern[b, Blank[]]]]], Optional[Pattern[n, Blank[]]]], Power[Plus[Pattern[d, Blank[]], Times[Optional[Pattern[e, Blank[]]], Power[Pattern[x, Blank[]], 2]]], Optional[Pattern[p, Blank[]]]]], Pattern[x, Blank[Symbol]]] := Unintegrable[(d + e*x^2)^p*(a + b*ArcSinh[c*x])^n, x] /; FreeQ[{a, b, c, d, e, n, p}, x]
  5709. Int[Times[Power[Plus[Optional[Pattern[a, Blank[]]], Times[ArcCosh[Times[Optional[Pattern[c, Blank[]]], Pattern[x, Blank[]]]], Optional[Pattern[b, Blank[]]]]], Optional[Pattern[n, Blank[]]]], Power[Plus[Pattern[d, Blank[]], Times[Optional[Pattern[e, Blank[]]], Power[Pattern[x, Blank[]], 2]]], Optional[Pattern[p, Blank[]]]]], Pattern[x, Blank[Symbol]]] := Unintegrable[(d + e*x^2)^p*(a + b*ArcCosh[c*x])^n, x] /; FreeQ[{a, b, c, d, e, n, p}, x] && IntegerQ[p]
  5710. Int[Times[Power[Plus[Optional[Pattern[a, Blank[]]], Times[ArcCosh[Times[Optional[Pattern[c, Blank[]]], Pattern[x, Blank[]]]], Optional[Pattern[b, Blank[]]]]], Optional[Pattern[n, Blank[]]]], Power[Plus[Pattern[d1, Blank[]], Times[Optional[Pattern[e1, Blank[]]], Pattern[x, Blank[]]]], Optional[Pattern[p, Blank[]]]], Power[Plus[Pattern[d2, Blank[]], Times[Optional[Pattern[e2, Blank[]]], Pattern[x, Blank[]]]], Optional[Pattern[p, Blank[]]]]], Pattern[x, Blank[Symbol]]] := Unintegrable[(d1 + e1*x)^p*(d2 + e2*x)^p*(a + b*ArcCosh[c*x])^n, x] /; FreeQ[{a, b, c, d1, e1, d2, e2, n, p}, x]
  5711. Int[Times[Power[Plus[Optional[Pattern[a, Blank[]]], Times[ArcSinh[Times[Optional[Pattern[c, Blank[]]], Pattern[x, Blank[]]]], Optional[Pattern[b, Blank[]]]]], Optional[Pattern[n, Blank[]]]], Power[Plus[Pattern[d, Blank[]], Times[Optional[Pattern[e, Blank[]]], Pattern[x, Blank[]]]], Pattern[p, Blank[]]], Power[Plus[Pattern[f, Blank[]], Times[Optional[Pattern[g, Blank[]]], Pattern[x, Blank[]]]], Pattern[q, Blank[]]]], Pattern[x, Blank[Symbol]]] := Dist[(-((d^2*g)/e))^q, Int[(d + e*x)^(p - q)*(1 + c^2*x^2)^q*(a + b*ArcSinh[c*x])^n, x], x] /; FreeQ[{a, b, c, d, e, f, g, n}, x] && EqQ[e*f + d*g, 0] && EqQ[c^2*d^2 + e^2, 0] && HalfIntegerQ[p, q] && GeQ[p - q, 0] && GtQ[d, 0] && LtQ[g/e, 0]
  5712. Int[Times[Power[Plus[Optional[Pattern[a, Blank[]]], Times[ArcSinh[Times[Optional[Pattern[c, Blank[]]], Pattern[x, Blank[]]]], Optional[Pattern[b, Blank[]]]]], Optional[Pattern[n, Blank[]]]], Power[Plus[Pattern[d, Blank[]], Times[Optional[Pattern[e, Blank[]]], Pattern[x, Blank[]]]], Pattern[p, Blank[]]], Power[Plus[Pattern[f, Blank[]], Times[Optional[Pattern[g, Blank[]]], Pattern[x, Blank[]]]], Pattern[q, Blank[]]]], Pattern[x, Blank[Symbol]]] := Dist[((d + e*x)^q*(f + g*x)^q)/(1 + c^2*x^2)^q, Int[(d + e*x)^(p - q)*(1 + c^2*x^2)^q*(a + b*ArcSinh[c*x])^n, x], x] /; FreeQ[{a, b, c, d, e, f, g, n}, x] && EqQ[e*f + d*g, 0] && EqQ[c^2*d^2 + e^2, 0] && HalfIntegerQ[p, q] && GeQ[p - q, 0]
  5713. Int[Times[Power[Plus[Optional[Pattern[a, Blank[]]], Times[ArcCosh[Times[Optional[Pattern[c, Blank[]]], Pattern[x, Blank[]]]], Optional[Pattern[b, Blank[]]]]], Optional[Pattern[n, Blank[]]]], Power[Plus[Pattern[d, Blank[]], Times[Optional[Pattern[e, Blank[]]], Power[Pattern[x, Blank[]], 2]]], Pattern[p, Blank[]]]], Pattern[x, Blank[Symbol]]] := Dist[((-d)^IntPart[p]*(d + e*x^2)^FracPart[p])/((1 + c*x)^FracPart[p]*(-1 + c*x)^FracPart[p]), Int[(1 + c*x)^p*(-1 + c*x)^p*(a + b*ArcCosh[c*x])^n, x], x] /; FreeQ[{a, b, c, d, e, n, p}, x] && EqQ[c^2*d + e, 0] && !IntegerQ[p]
  5714. Int[Times[Power[Plus[Optional[Pattern[a, Blank[]]], Times[ArcSinh[Times[Optional[Pattern[c, Blank[]]], Pattern[x, Blank[]]]], Optional[Pattern[b, Blank[]]]]], Optional[Pattern[n, Blank[]]]], Pattern[x, Blank[]], Power[Plus[Pattern[d, Blank[]], Times[Optional[Pattern[e, Blank[]]], Power[Pattern[x, Blank[]], 2]]], -1]], Pattern[x, Blank[Symbol]]] := Dist[1/e, Subst[Int[(a + b*x)^n*Tanh[x], x], x, ArcSinh[c*x]], x] /; FreeQ[{a, b, c, d, e}, x] && EqQ[e, c^2*d] && IGtQ[n, 0]
  5715. Int[Times[Power[Plus[Optional[Pattern[a, Blank[]]], Times[ArcCosh[Times[Optional[Pattern[c, Blank[]]], Pattern[x, Blank[]]]], Optional[Pattern[b, Blank[]]]]], Optional[Pattern[n, Blank[]]]], Pattern[x, Blank[]], Power[Plus[Pattern[d, Blank[]], Times[Optional[Pattern[e, Blank[]]], Power[Pattern[x, Blank[]], 2]]], -1]], Pattern[x, Blank[Symbol]]] := Dist[1/e, Subst[Int[(a + b*x)^n*Coth[x], x], x, ArcCosh[c*x]], x] /; FreeQ[{a, b, c, d, e}, x] && EqQ[c^2*d + e, 0] && IGtQ[n, 0]
  5716. Int[Times[Power[Plus[Optional[Pattern[a, Blank[]]], Times[ArcCosh[Times[Optional[Pattern[c, Blank[]]], Pattern[x, Blank[]]]], Optional[Pattern[b, Blank[]]]]], Optional[Pattern[n, Blank[]]]], Pattern[x, Blank[]], Power[Plus[Pattern[d, Blank[]], Times[Optional[Pattern[e, Blank[]]], Power[Pattern[x, Blank[]], 2]]], Optional[Pattern[p, Blank[]]]]], Pattern[x, Blank[Symbol]]] := Simp[((d + e*x^2)^(p + 1)*(a + b*ArcCosh[c*x])^n)/(2*e*(p + 1)), x] - Dist[(b*n*(-d)^p)/(2*c*(p + 1)), Int[(1 + c*x)^(p + 1/2)*(-1 + c*x)^(p + 1/2)*(a + b*ArcCosh[c*x])^(n - 1), x], x] /; FreeQ[{a, b, c, d, e, p}, x] && EqQ[c^2*d + e, 0] && GtQ[n, 0] && NeQ[p, -1] && IntegerQ[p]
  5717. Int[Times[Power[Plus[Optional[Pattern[a, Blank[]]], Times[ArcSinh[Times[Optional[Pattern[c, Blank[]]], Pattern[x, Blank[]]]], Optional[Pattern[b, Blank[]]]]], Optional[Pattern[n, Blank[]]]], Pattern[x, Blank[]], Power[Plus[Pattern[d, Blank[]], Times[Optional[Pattern[e, Blank[]]], Power[Pattern[x, Blank[]], 2]]], Optional[Pattern[p, Blank[]]]]], Pattern[x, Blank[Symbol]]] := Simp[((d + e*x^2)^(p + 1)*(a + b*ArcSinh[c*x])^n)/(2*e*(p + 1)), x] - Dist[(b*n*d^IntPart[p]*(d + e*x^2)^FracPart[p])/(2*c*(p + 1)*(1 + c^2*x^2)^FracPart[p]), Int[(1 + c^2*x^2)^(p + 1/2)*(a + b*ArcSinh[c*x])^(n - 1), x], x] /; FreeQ[{a, b, c, d, e, p}, x] && EqQ[e, c^2*d] && GtQ[n, 0] && NeQ[p, -1]
  5718. Int[Times[Power[Plus[Optional[Pattern[a, Blank[]]], Times[ArcCosh[Times[Optional[Pattern[c, Blank[]]], Pattern[x, Blank[]]]], Optional[Pattern[b, Blank[]]]]], Optional[Pattern[n, Blank[]]]], Pattern[x, Blank[]], Power[Plus[Pattern[d1, Blank[]], Times[Optional[Pattern[e1, Blank[]]], Pattern[x, Blank[]]]], Optional[Pattern[p, Blank[]]]], Power[Plus[Pattern[d2, Blank[]], Times[Optional[Pattern[e2, Blank[]]], Pattern[x, Blank[]]]], Optional[Pattern[p, Blank[]]]]], Pattern[x, Blank[Symbol]]] := Simp[((d1 + e1*x)^(p + 1)*(d2 + e2*x)^(p + 1)*(a + b*ArcCosh[c*x])^n)/(2*e1*e2*(p + 1)), x] - Dist[(b*n*(-(d1*d2))^IntPart[p]*(d1 + e1*x)^FracPart[p]*(d2 + e2*x)^FracPart[p])/(2*c*(p + 1)*(1 + c*x)^FracPart[p]*(-1 + c*x)^FracPart[p]), Int[(-1 + c^2*x^2)^(p + 1/2)*(a + b*ArcCosh[c*x])^(n - 1), x], x] /; FreeQ[{a, b, c, d1, e1, d2, e2, p}, x] && EqQ[e1 - c*d1, 0] && EqQ[e2 + c*d2, 0] && GtQ[n, 0] && NeQ[p, -1] && IntegerQ[p + 1/2]
  5719. Int[Times[Power[Plus[Optional[Pattern[a, Blank[]]], Times[ArcCosh[Times[Optional[Pattern[c, Blank[]]], Pattern[x, Blank[]]]], Optional[Pattern[b, Blank[]]]]], Optional[Pattern[n, Blank[]]]], Pattern[x, Blank[]], Power[Plus[Pattern[d1, Blank[]], Times[Optional[Pattern[e1, Blank[]]], Pattern[x, Blank[]]]], Optional[Pattern[p, Blank[]]]], Power[Plus[Pattern[d2, Blank[]], Times[Optional[Pattern[e2, Blank[]]], Pattern[x, Blank[]]]], Optional[Pattern[p, Blank[]]]]], Pattern[x, Blank[Symbol]]] := Simp[((d1 + e1*x)^(p + 1)*(d2 + e2*x)^(p + 1)*(a + b*ArcCosh[c*x])^n)/(2*e1*e2*(p + 1)), x] - Dist[(b*n*(-(d1*d2))^IntPart[p]*(d1 + e1*x)^FracPart[p]*(d2 + e2*x)^FracPart[p])/(2*c*(p + 1)*(1 + c*x)^FracPart[p]*(-1 + c*x)^FracPart[p]), Int[(1 + c*x)^(p + 1/2)*(-1 + c*x)^(p + 1/2)*(a + b*ArcCosh[c*x])^(n - 1), x], x] /; FreeQ[{a, b, c, d1, e1, d2, e2, p}, x] && EqQ[e1 - c*d1, 0] && EqQ[e2 + c*d2, 0] && GtQ[n, 0] && NeQ[p, -1]
  5720. Int[Times[Power[Plus[Optional[Pattern[a, Blank[]]], Times[ArcSinh[Times[Optional[Pattern[c, Blank[]]], Pattern[x, Blank[]]]], Optional[Pattern[b, Blank[]]]]], Optional[Pattern[n, Blank[]]]], Power[Pattern[x, Blank[]], -1], Power[Plus[Pattern[d, Blank[]], Times[Optional[Pattern[e, Blank[]]], Power[Pattern[x, Blank[]], 2]]], -1]], Pattern[x, Blank[Symbol]]] := Dist[1/d, Subst[Int[(a + b*x)^n/(Cosh[x]*Sinh[x]), x], x, ArcSinh[c*x]], x] /; FreeQ[{a, b, c, d, e}, x] && EqQ[e, c^2*d] && IGtQ[n, 0]
  5721. Int[Times[Power[Plus[Optional[Pattern[a, Blank[]]], Times[ArcCosh[Times[Optional[Pattern[c, Blank[]]], Pattern[x, Blank[]]]], Optional[Pattern[b, Blank[]]]]], Optional[Pattern[n, Blank[]]]], Power[Pattern[x, Blank[]], -1], Power[Plus[Pattern[d, Blank[]], Times[Optional[Pattern[e, Blank[]]], Power[Pattern[x, Blank[]], 2]]], -1]], Pattern[x, Blank[Symbol]]] := -Dist[d^(-1), Subst[Int[(a + b*x)^n/(Cosh[x]*Sinh[x]), x], x, ArcCosh[c*x]], x] /; FreeQ[{a, b, c, d, e}, x] && EqQ[c^2*d + e, 0] && IGtQ[n, 0]
  5722. Int[Times[Power[Plus[Optional[Pattern[a, Blank[]]], Times[ArcCosh[Times[Optional[Pattern[c, Blank[]]], Pattern[x, Blank[]]]], Optional[Pattern[b, Blank[]]]]], Optional[Pattern[n, Blank[]]]], Power[Times[Optional[Pattern[f, Blank[]]], Pattern[x, Blank[]]], Pattern[m, Blank[]]], Power[Plus[Pattern[d, Blank[]], Times[Optional[Pattern[e, Blank[]]], Power[Pattern[x, Blank[]], 2]]], Optional[Pattern[p, Blank[]]]]], Pattern[x, Blank[Symbol]]] := Simp[((f*x)^(m + 1)*(d + e*x^2)^(p + 1)*(a + b*ArcCosh[c*x])^n)/(d*f*(m + 1)), x] + Dist[(b*c*n*(-d)^p)/(f*(m + 1)), Int[(f*x)^(m + 1)*(1 + c*x)^(p + 1/2)*(-1 + c*x)^(p + 1/2)*(a + b*ArcCosh[c*x])^(n - 1), x], x] /; FreeQ[{a, b, c, d, e, f, m, p}, x] && EqQ[c^2*d + e, 0] && GtQ[n, 0] && EqQ[m + 2*p + 3, 0] && NeQ[m, -1] && IntegerQ[p]
  5723. Int[Times[Power[Plus[Optional[Pattern[a, Blank[]]], Times[ArcSinh[Times[Optional[Pattern[c, Blank[]]], Pattern[x, Blank[]]]], Optional[Pattern[b, Blank[]]]]], Optional[Pattern[n, Blank[]]]], Power[Times[Optional[Pattern[f, Blank[]]], Pattern[x, Blank[]]], Pattern[m, Blank[]]], Power[Plus[Pattern[d, Blank[]], Times[Optional[Pattern[e, Blank[]]], Power[Pattern[x, Blank[]], 2]]], Pattern[p, Blank[]]]], Pattern[x, Blank[Symbol]]] := Simp[((f*x)^(m + 1)*(d + e*x^2)^(p + 1)*(a + b*ArcSinh[c*x])^n)/(d*f*(m + 1)), x] - Dist[(b*c*n*d^IntPart[p]*(d + e*x^2)^FracPart[p])/(f*(m + 1)*(1 + c^2*x^2)^FracPart[p]), Int[(f*x)^(m + 1)*(1 + c^2*x^2)^(p + 1/2)*(a + b*ArcSinh[c*x])^(n - 1), x], x] /; FreeQ[{a, b, c, d, e, f, m, p}, x] && EqQ[e, c^2*d] && GtQ[n, 0] && EqQ[m + 2*p + 3, 0] && NeQ[m, -1]
  5724. Int[Times[Power[Plus[Optional[Pattern[a, Blank[]]], Times[ArcCosh[Times[Optional[Pattern[c, Blank[]]], Pattern[x, Blank[]]]], Optional[Pattern[b, Blank[]]]]], Optional[Pattern[n, Blank[]]]], Power[Times[Optional[Pattern[f, Blank[]]], Pattern[x, Blank[]]], Pattern[m, Blank[]]], Power[Plus[Pattern[d1, Blank[]], Times[Optional[Pattern[e1, Blank[]]], Pattern[x, Blank[]]]], Optional[Pattern[p, Blank[]]]], Power[Plus[Pattern[d2, Blank[]], Times[Optional[Pattern[e2, Blank[]]], Pattern[x, Blank[]]]], Optional[Pattern[p, Blank[]]]]], Pattern[x, Blank[Symbol]]] := Simp[((f*x)^(m + 1)*(d1 + e1*x)^(p + 1)*(d2 + e2*x)^(p + 1)*(a + b*ArcCosh[c*x])^n)/(d1*d2*f*(m + 1)), x] + Dist[(b*c*n*(-(d1*d2))^IntPart[p]*(d1 + e1*x)^FracPart[p]*(d2 + e2*x)^FracPart[p])/(f*(m + 1)*(1 + c*x)^FracPart[p]*(-1 + c*x)^FracPart[p]), Int[(f*x)^(m + 1)*(-1 + c^2*x^2)^(p + 1/2)*(a + b*ArcCosh[c*x])^(n - 1), x], x] /; FreeQ[{a, b, c, d1, e1, d2, e2, f, m, p}, x] && EqQ[e1 - c*d1, 0] && EqQ[e2 + c*d2, 0] && GtQ[n, 0] && EqQ[m + 2*p + 3, 0] && NeQ[m, -1] && IntegerQ[p + 1/2]
  5725. Int[Times[Power[Plus[Optional[Pattern[a, Blank[]]], Times[ArcCosh[Times[Optional[Pattern[c, Blank[]]], Pattern[x, Blank[]]]], Optional[Pattern[b, Blank[]]]]], Optional[Pattern[n, Blank[]]]], Power[Times[Optional[Pattern[f, Blank[]]], Pattern[x, Blank[]]], Pattern[m, Blank[]]], Power[Plus[Pattern[d1, Blank[]], Times[Optional[Pattern[e1, Blank[]]], Pattern[x, Blank[]]]], Optional[Pattern[p, Blank[]]]], Power[Plus[Pattern[d2, Blank[]], Times[Optional[Pattern[e2, Blank[]]], Pattern[x, Blank[]]]], Optional[Pattern[p, Blank[]]]]], Pattern[x, Blank[Symbol]]] := Simp[((f*x)^(m + 1)*(d1 + e1*x)^(p + 1)*(d2 + e2*x)^(p + 1)*(a + b*ArcCosh[c*x])^n)/(d1*d2*f*(m + 1)), x] + Dist[(b*c*n*(-(d1*d2))^IntPart[p]*(d1 + e1*x)^FracPart[p]*(d2 + e2*x)^FracPart[p])/(f*(m + 1)*(1 + c*x)^FracPart[p]*(-1 + c*x)^FracPart[p]), Int[(f*x)^(m + 1)*(1 + c*x)^(p + 1/2)*(-1 + c*x)^(p + 1/2)*(a + b*ArcCosh[c*x])^(n - 1), x], x] /; FreeQ[{a, b, c, d1, e1, d2, e2, f, m, p}, x] && EqQ[e1 - c*d1, 0] && EqQ[e2 + c*d2, 0] && GtQ[n, 0] && EqQ[m + 2*p + 3, 0] && NeQ[m, -1]
  5726. Int[Times[Plus[Optional[Pattern[a, Blank[]]], Times[ArcSinh[Times[Optional[Pattern[c, Blank[]]], Pattern[x, Blank[]]]], Optional[Pattern[b, Blank[]]]]], Power[Pattern[x, Blank[]], -1], Power[Plus[Pattern[d, Blank[]], Times[Optional[Pattern[e, Blank[]]], Power[Pattern[x, Blank[]], 2]]], Optional[Pattern[p, Blank[]]]]], Pattern[x, Blank[Symbol]]] := Simp[((d + e*x^2)^p*(a + b*ArcSinh[c*x]))/(2*p), x] + (Dist[d, Int[((d + e*x^2)^(p - 1)*(a + b*ArcSinh[c*x]))/x, x], x] - Dist[(b*c*d^p)/(2*p), Int[(1 + c^2*x^2)^(p - 1/2), x], x]) /; FreeQ[{a, b, c, d, e}, x] && EqQ[e, c^2*d] && IGtQ[p, 0]
  5727. Int[Times[Plus[Optional[Pattern[a, Blank[]]], Times[ArcCosh[Times[Optional[Pattern[c, Blank[]]], Pattern[x, Blank[]]]], Optional[Pattern[b, Blank[]]]]], Power[Pattern[x, Blank[]], -1], Power[Plus[Pattern[d, Blank[]], Times[Optional[Pattern[e, Blank[]]], Power[Pattern[x, Blank[]], 2]]], Optional[Pattern[p, Blank[]]]]], Pattern[x, Blank[Symbol]]] := Simp[((d + e*x^2)^p*(a + b*ArcCosh[c*x]))/(2*p), x] + (Dist[d, Int[((d + e*x^2)^(p - 1)*(a + b*ArcCosh[c*x]))/x, x], x] - Dist[(b*c*(-d)^p)/(2*p), Int[(1 + c*x)^(p - 1/2)*(-1 + c*x)^(p - 1/2), x], x]) /; FreeQ[{a, b, c, d, e}, x] && EqQ[c^2*d + e, 0] && IGtQ[p, 0]
  5728. Int[Times[Plus[Optional[Pattern[a, Blank[]]], Times[ArcSinh[Times[Optional[Pattern[c, Blank[]]], Pattern[x, Blank[]]]], Optional[Pattern[b, Blank[]]]]], Power[Times[Optional[Pattern[f, Blank[]]], Pattern[x, Blank[]]], Pattern[m, Blank[]]], Power[Plus[Pattern[d, Blank[]], Times[Optional[Pattern[e, Blank[]]], Power[Pattern[x, Blank[]], 2]]], Optional[Pattern[p, Blank[]]]]], Pattern[x, Blank[Symbol]]] := Simp[((f*x)^(m + 1)*(d + e*x^2)^p*(a + b*ArcSinh[c*x]))/(f*(m + 1)), x] + (-Dist[(b*c*d^p)/(f*(m + 1)), Int[(f*x)^(m + 1)*(1 + c^2*x^2)^(p - 1/2), x], x] - Dist[(2*e*p)/(f^2*(m + 1)), Int[(f*x)^(m + 2)*(d + e*x^2)^(p - 1)*(a + b*ArcSinh[c*x]), x], x]) /; FreeQ[{a, b, c, d, e, f}, x] && EqQ[e, c^2*d] && IGtQ[p, 0] && ILtQ[(m + 1)/2, 0]
  5729. Int[Times[Plus[Optional[Pattern[a, Blank[]]], Times[ArcCosh[Times[Optional[Pattern[c, Blank[]]], Pattern[x, Blank[]]]], Optional[Pattern[b, Blank[]]]]], Power[Times[Optional[Pattern[f, Blank[]]], Pattern[x, Blank[]]], Pattern[m, Blank[]]], Power[Plus[Pattern[d, Blank[]], Times[Optional[Pattern[e, Blank[]]], Power[Pattern[x, Blank[]], 2]]], Optional[Pattern[p, Blank[]]]]], Pattern[x, Blank[Symbol]]] := Simp[((f*x)^(m + 1)*(d + e*x^2)^p*(a + b*ArcCosh[c*x]))/(f*(m + 1)), x] + (-Dist[(b*c*(-d)^p)/(f*(m + 1)), Int[(f*x)^(m + 1)*(1 + c*x)^(p - 1/2)*(-1 + c*x)^(p - 1/2), x], x] - Dist[(2*e*p)/(f^2*(m + 1)), Int[(f*x)^(m + 2)*(d + e*x^2)^(p - 1)*(a + b*ArcCosh[c*x]), x], x]) /; FreeQ[{a, b, c, d, e, f}, x] && EqQ[c^2*d + e, 0] && IGtQ[p, 0] && ILtQ[(m + 1)/2, 0]
  5730. Int[Times[Plus[Optional[Pattern[a, Blank[]]], Times[ArcSinh[Times[Optional[Pattern[c, Blank[]]], Pattern[x, Blank[]]]], Optional[Pattern[b, Blank[]]]]], Power[Times[Optional[Pattern[f, Blank[]]], Pattern[x, Blank[]]], Pattern[m, Blank[]]], Power[Plus[Pattern[d, Blank[]], Times[Optional[Pattern[e, Blank[]]], Power[Pattern[x, Blank[]], 2]]], Optional[Pattern[p, Blank[]]]]], Pattern[x, Blank[Symbol]]] := With[{u = IntHide[(f*x)^m*(d + e*x^2)^p, x]}, Dist[a + b*ArcSinh[c*x], u, x] - Dist[b*c, Int[SimplifyIntegrand[u/Sqrt[1 + c^2*x^2], x], x], x]] /; FreeQ[{a, b, c, d, e, f, m}, x] && EqQ[e, c^2*d] && IGtQ[p, 0]
  5731. Int[Times[Plus[Optional[Pattern[a, Blank[]]], Times[ArcCosh[Times[Optional[Pattern[c, Blank[]]], Pattern[x, Blank[]]]], Optional[Pattern[b, Blank[]]]]], Power[Times[Optional[Pattern[f, Blank[]]], Pattern[x, Blank[]]], Pattern[m, Blank[]]], Power[Plus[Pattern[d, Blank[]], Times[Optional[Pattern[e, Blank[]]], Power[Pattern[x, Blank[]], 2]]], Optional[Pattern[p, Blank[]]]]], Pattern[x, Blank[Symbol]]] := With[{u = IntHide[(f*x)^m*(d + e*x^2)^p, x]}, Dist[a + b*ArcCosh[c*x], u, x] - Dist[b*c, Int[SimplifyIntegrand[u/(Sqrt[1 + c*x]*Sqrt[-1 + c*x]), x], x], x]] /; FreeQ[{a, b, c, d, e, f, m}, x] && EqQ[c^2*d + e, 0] && IGtQ[p, 0]
  5732. Int[Times[Plus[Optional[Pattern[a, Blank[]]], Times[ArcSinh[Times[Optional[Pattern[c, Blank[]]], Pattern[x, Blank[]]]], Optional[Pattern[b, Blank[]]]]], Power[Pattern[x, Blank[]], Pattern[m, Blank[]]], Power[Plus[Pattern[d, Blank[]], Times[Optional[Pattern[e, Blank[]]], Power[Pattern[x, Blank[]], 2]]], Pattern[p, Blank[]]]], Pattern[x, Blank[Symbol]]] := With[{u = IntHide[x^m*(1 + c^2*x^2)^p, x]}, Dist[d^p*(a + b*ArcSinh[c*x]), u, x] - Dist[b*c*d^p, Int[SimplifyIntegrand[u/Sqrt[1 + c^2*x^2], x], x], x]] /; FreeQ[{a, b, c, d, e}, x] && EqQ[e, c^2*d] && IntegerQ[p - 1/2] && (IGtQ[(m + 1)/2, 0] || ILtQ[(m + 2*p + 3)/2, 0]) && NeQ[p, -2^(-1)] && GtQ[d, 0]
  5733. Int[Times[Plus[Optional[Pattern[a, Blank[]]], Times[ArcCosh[Times[Optional[Pattern[c, Blank[]]], Pattern[x, Blank[]]]], Optional[Pattern[b, Blank[]]]]], Power[Pattern[x, Blank[]], Pattern[m, Blank[]]], Power[Plus[Pattern[d1, Blank[]], Times[Optional[Pattern[e1, Blank[]]], Pattern[x, Blank[]]]], Pattern[p, Blank[]]], Power[Plus[Pattern[d2, Blank[]], Times[Optional[Pattern[e2, Blank[]]], Pattern[x, Blank[]]]], Pattern[p, Blank[]]]], Pattern[x, Blank[Symbol]]] := With[{u = IntHide[x^m*(1 + c*x)^p*(-1 + c*x)^p, x]}, Dist[(-(d1*d2))^p*(a + b*ArcCosh[c*x]), u, x] - Dist[b*c*(-(d1*d2))^p, Int[SimplifyIntegrand[u/(Sqrt[1 + c*x]*Sqrt[-1 + c*x]), x], x], x]] /; FreeQ[{a, b, c, d1, e1, d2, e2}, x] && EqQ[e1 - c*d1, 0] && EqQ[e2 + c*d2, 0] && IntegerQ[p - 1/2] && (IGtQ[(m + 1)/2, 0] || ILtQ[(m + 2*p + 3)/2, 0]) && NeQ[p, -2^(-1)] && GtQ[d1, 0] && LtQ[d2, 0]
  5734. Int[Times[Plus[Optional[Pattern[a, Blank[]]], Times[ArcSinh[Times[Optional[Pattern[c, Blank[]]], Pattern[x, Blank[]]]], Optional[Pattern[b, Blank[]]]]], Power[Pattern[x, Blank[]], Pattern[m, Blank[]]], Power[Plus[Pattern[d, Blank[]], Times[Optional[Pattern[e, Blank[]]], Power[Pattern[x, Blank[]], 2]]], Pattern[p, Blank[]]]], Pattern[x, Blank[Symbol]]] := With[{u = IntHide[x^m*(1 + c^2*x^2)^p, x]}, Dist[a + b*ArcSinh[c*x], Int[x^m*(d + e*x^2)^p, x], x] - Dist[(b*c*d^(p - 1/2)*Sqrt[d + e*x^2])/Sqrt[1 + c^2*x^2], Int[SimplifyIntegrand[u/Sqrt[1 + c^2*x^2], x], x], x]] /; FreeQ[{a, b, c, d, e}, x] && EqQ[e, c^2*d] && IGtQ[p + 1/2, 0] && (IGtQ[(m + 1)/2, 0] || ILtQ[(m + 2*p + 3)/2, 0])
  5735. Int[Times[Plus[Optional[Pattern[a, Blank[]]], Times[ArcCosh[Times[Optional[Pattern[c, Blank[]]], Pattern[x, Blank[]]]], Optional[Pattern[b, Blank[]]]]], Power[Pattern[x, Blank[]], Pattern[m, Blank[]]], Power[Plus[Pattern[d1, Blank[]], Times[Optional[Pattern[e1, Blank[]]], Pattern[x, Blank[]]]], Pattern[p, Blank[]]], Power[Plus[Pattern[d2, Blank[]], Times[Optional[Pattern[e2, Blank[]]], Pattern[x, Blank[]]]], Pattern[p, Blank[]]]], Pattern[x, Blank[Symbol]]] := With[{u = IntHide[x^m*(1 + c*x)^p*(-1 + c*x)^p, x]}, Dist[a + b*ArcCosh[c*x], Int[x^m*(d1 + e1*x)^p*(d2 + e2*x)^p, x], x] - Dist[(b*c*(-(d1*d2))^(p - 1/2)*Sqrt[d1 + e1*x]*Sqrt[d2 + e2*x])/(Sqrt[1 + c*x]*Sqrt[-1 + c*x]), Int[SimplifyIntegrand[u/(Sqrt[1 + c*x]*Sqrt[-1 + c*x]), x], x], x]] /; FreeQ[{a, b, c, d1, e1, d2, e2}, x] && EqQ[e1 - c*d1, 0] && EqQ[e2 + c*d2, 0] && IGtQ[p + 1/2, 0] && (IGtQ[(m + 1)/2, 0] || ILtQ[(m + 2*p + 3)/2, 0])
  5736. Int[Times[Power[Plus[Optional[Pattern[a, Blank[]]], Times[ArcCosh[Times[Optional[Pattern[c, Blank[]]], Pattern[x, Blank[]]]], Optional[Pattern[b, Blank[]]]]], Optional[Pattern[n, Blank[]]]], Power[Times[Optional[Pattern[f, Blank[]]], Pattern[x, Blank[]]], Pattern[m, Blank[]]], Power[Plus[Pattern[d, Blank[]], Times[Optional[Pattern[e, Blank[]]], Power[Pattern[x, Blank[]], 2]]], Optional[Pattern[p, Blank[]]]]], Pattern[x, Blank[Symbol]]] := Simp[((f*x)^(m + 1)*(d + e*x^2)^p*(a + b*ArcCosh[c*x])^n)/(f*(m + 1)), x] + (-Dist[(b*c*n*(-d)^p)/(f*(m + 1)), Int[(f*x)^(m + 1)*(1 + c*x)^(p - 1/2)*(-1 + c*x)^(p - 1/2)*(a + b*ArcCosh[c*x])^(n - 1), x], x] - Dist[(2*e*p)/(f^2*(m + 1)), Int[(f*x)^(m + 2)*(d + e*x^2)^(p - 1)*(a + b*ArcCosh[c*x])^n, x], x]) /; FreeQ[{a, b, c, d, e, f}, x] && EqQ[c^2*d + e, 0] && GtQ[n, 0] && GtQ[p, 0] && LtQ[m, -1] && IntegerQ[p]
  5737. Int[Times[Power[Plus[Optional[Pattern[a, Blank[]]], Times[ArcSinh[Times[Optional[Pattern[c, Blank[]]], Pattern[x, Blank[]]]], Optional[Pattern[b, Blank[]]]]], Optional[Pattern[n, Blank[]]]], Power[Times[Optional[Pattern[f, Blank[]]], Pattern[x, Blank[]]], Pattern[m, Blank[]]], Power[Plus[Pattern[d, Blank[]], Times[Optional[Pattern[e, Blank[]]], Power[Pattern[x, Blank[]], 2]]], Rational[1, 2]]], Pattern[x, Blank[Symbol]]] := Simp[((f*x)^(m + 1)*Sqrt[d + e*x^2]*(a + b*ArcSinh[c*x])^n)/(f*(m + 1)), x] + (-Dist[(b*c*n*Sqrt[d + e*x^2])/(f*(m + 1)*Sqrt[1 + c^2*x^2]), Int[(f*x)^(m + 1)*(a + b*ArcSinh[c*x])^(n - 1), x], x] - Dist[(c^2*Sqrt[d + e*x^2])/(f^2*(m + 1)*Sqrt[1 + c^2*x^2]), Int[((f*x)^(m + 2)*(a + b*ArcSinh[c*x])^n)/Sqrt[1 + c^2*x^2], x], x]) /; FreeQ[{a, b, c, d, e, f}, x] && EqQ[e, c^2*d] && GtQ[n, 0] && LtQ[m, -1]
  5738. Int[Times[Power[Plus[Optional[Pattern[a, Blank[]]], Times[ArcCosh[Times[Optional[Pattern[c, Blank[]]], Pattern[x, Blank[]]]], Optional[Pattern[b, Blank[]]]]], Optional[Pattern[n, Blank[]]]], Power[Times[Optional[Pattern[f, Blank[]]], Pattern[x, Blank[]]], Pattern[m, Blank[]]], Power[Plus[Pattern[d1, Blank[]], Times[Optional[Pattern[e1, Blank[]]], Pattern[x, Blank[]]]], Rational[1, 2]], Power[Plus[Pattern[d2, Blank[]], Times[Optional[Pattern[e2, Blank[]]], Pattern[x, Blank[]]]], Rational[1, 2]]], Pattern[x, Blank[Symbol]]] := Simp[((f*x)^(m + 1)*Sqrt[d1 + e1*x]*Sqrt[d2 + e2*x]*(a + b*ArcCosh[c*x])^n)/(f*(m + 1)), x] + (-Dist[(b*c*n*Sqrt[d1 + e1*x]*Sqrt[d2 + e2*x])/(f*(m + 1)*Sqrt[1 + c*x]*Sqrt[-1 + c*x]), Int[(f*x)^(m + 1)*(a + b*ArcCosh[c*x])^(n - 1), x], x] - Dist[(c^2*Sqrt[d1 + e1*x]*Sqrt[d2 + e2*x])/(f^2*(m + 1)*Sqrt[1 + c*x]*Sqrt[-1 + c*x]), Int[((f*x)^(m + 2)*(a + b*ArcCosh[c*x])^n)/(Sqrt[1 + c*x]*Sqrt[-1 + c*x]), x], x]) /; FreeQ[{a, b, c, d1, e1, d2, e2, f}, x] && EqQ[e1 - c*d1, 0] && EqQ[e2 + c*d2, 0] && GtQ[n, 0] && LtQ[m, -1]
  5739. Int[Times[Power[Plus[Optional[Pattern[a, Blank[]]], Times[ArcSinh[Times[Optional[Pattern[c, Blank[]]], Pattern[x, Blank[]]]], Optional[Pattern[b, Blank[]]]]], Optional[Pattern[n, Blank[]]]], Power[Times[Optional[Pattern[f, Blank[]]], Pattern[x, Blank[]]], Pattern[m, Blank[]]], Power[Plus[Pattern[d, Blank[]], Times[Optional[Pattern[e, Blank[]]], Power[Pattern[x, Blank[]], 2]]], Optional[Pattern[p, Blank[]]]]], Pattern[x, Blank[Symbol]]] := Simp[((f*x)^(m + 1)*(d + e*x^2)^p*(a + b*ArcSinh[c*x])^n)/(f*(m + 1)), x] + (-Dist[(2*e*p)/(f^2*(m + 1)), Int[(f*x)^(m + 2)*(d + e*x^2)^(p - 1)*(a + b*ArcSinh[c*x])^n, x], x] - Dist[(b*c*n*d^IntPart[p]*(d + e*x^2)^FracPart[p])/(f*(m + 1)*(1 + c^2*x^2)^FracPart[p]), Int[(f*x)^(m + 1)*(1 + c^2*x^2)^(p - 1/2)*(a + b*ArcSinh[c*x])^(n - 1), x], x]) /; FreeQ[{a, b, c, d, e, f}, x] && EqQ[e, c^2*d] && GtQ[n, 0] && GtQ[p, 0] && LtQ[m, -1]
  5740. Int[Times[Power[Plus[Optional[Pattern[a, Blank[]]], Times[ArcCosh[Times[Optional[Pattern[c, Blank[]]], Pattern[x, Blank[]]]], Optional[Pattern[b, Blank[]]]]], Optional[Pattern[n, Blank[]]]], Power[Times[Optional[Pattern[f, Blank[]]], Pattern[x, Blank[]]], Pattern[m, Blank[]]], Power[Plus[Pattern[d1, Blank[]], Times[Optional[Pattern[e1, Blank[]]], Pattern[x, Blank[]]]], Pattern[p, Blank[]]], Power[Plus[Pattern[d2, Blank[]], Times[Optional[Pattern[e2, Blank[]]], Pattern[x, Blank[]]]], Pattern[p, Blank[]]]], Pattern[x, Blank[Symbol]]] := Simp[((f*x)^(m + 1)*(d1 + e1*x)^p*(d2 + e2*x)^p*(a + b*ArcCosh[c*x])^n)/(f*(m + 1)), x] + (-Dist[(2*e1*e2*p)/(f^2*(m + 1)), Int[(f*x)^(m + 2)*(d1 + e1*x)^(p - 1)*(d2 + e2*x)^(p - 1)*(a + b*ArcCosh[c*x])^n, x], x] - Dist[(b*c*n*(-(d1*d2))^(p - 1/2)*Sqrt[d1 + e1*x]*Sqrt[d2 + e2*x])/(f*(m + 1)*Sqrt[1 + c*x]*Sqrt[-1 + c*x]), Int[(f*x)^(m + 1)*(-1 + c^2*x^2)^(p - 1/2)*(a + b*ArcCosh[c*x])^(n - 1), x], x]) /; FreeQ[{a, b, c, d1, e1, d2, e2, f}, x] && EqQ[e1 - c*d1, 0] && EqQ[e2 + c*d2, 0] && GtQ[n, 0] && GtQ[p, 0] && LtQ[m, -1] && IntegerQ[p - 1/2]
  5741. Int[Times[Power[Plus[Optional[Pattern[a, Blank[]]], Times[ArcCosh[Times[Optional[Pattern[c, Blank[]]], Pattern[x, Blank[]]]], Optional[Pattern[b, Blank[]]]]], Optional[Pattern[n, Blank[]]]], Power[Times[Optional[Pattern[f, Blank[]]], Pattern[x, Blank[]]], Pattern[m, Blank[]]], Power[Plus[Pattern[d, Blank[]], Times[Optional[Pattern[e, Blank[]]], Power[Pattern[x, Blank[]], 2]]], Optional[Pattern[p, Blank[]]]]], Pattern[x, Blank[Symbol]]] := Simp[((f*x)^(m + 1)*(d + e*x^2)^p*(a + b*ArcCosh[c*x])^n)/(f*(m + 2*p + 1)), x] + (Dist[(2*d*p)/(m + 2*p + 1), Int[(f*x)^m*(d + e*x^2)^(p - 1)*(a + b*ArcCosh[c*x])^n, x], x] - Dist[(b*c*n*(-d)^p)/(f*(m + 2*p + 1)), Int[(f*x)^(m + 1)*(1 + c*x)^(p - 1/2)*(-1 + c*x)^(p - 1/2)*(a + b*ArcCosh[c*x])^(n - 1), x], x]) /; FreeQ[{a, b, c, d, e, f, m}, x] && EqQ[c^2*d + e, 0] && GtQ[n, 0] && GtQ[p, 0] && !LtQ[m, -1] && IntegerQ[p] && (RationalQ[m] || EqQ[n, 1])
  5742. Int[Times[Power[Plus[Optional[Pattern[a, Blank[]]], Times[ArcSinh[Times[Optional[Pattern[c, Blank[]]], Pattern[x, Blank[]]]], Optional[Pattern[b, Blank[]]]]], Optional[Pattern[n, Blank[]]]], Power[Times[Optional[Pattern[f, Blank[]]], Pattern[x, Blank[]]], Pattern[m, Blank[]]], Power[Plus[Pattern[d, Blank[]], Times[Optional[Pattern[e, Blank[]]], Power[Pattern[x, Blank[]], 2]]], Rational[1, 2]]], Pattern[x, Blank[Symbol]]] := Simp[((f*x)^(m + 1)*Sqrt[d + e*x^2]*(a + b*ArcSinh[c*x])^n)/(f*(m + 2)), x] + (Dist[Sqrt[d + e*x^2]/((m + 2)*Sqrt[1 + c^2*x^2]), Int[((f*x)^m*(a + b*ArcSinh[c*x])^n)/Sqrt[1 + c^2*x^2], x], x] - Dist[(b*c*n*Sqrt[d + e*x^2])/(f*(m + 2)*Sqrt[1 + c^2*x^2]), Int[(f*x)^(m + 1)*(a + b*ArcSinh[c*x])^(n - 1), x], x]) /; FreeQ[{a, b, c, d, e, f, m}, x] && EqQ[e, c^2*d] && GtQ[n, 0] && !LtQ[m, -1] && (RationalQ[m] || EqQ[n, 1])
  5743. Int[Times[Power[Plus[Optional[Pattern[a, Blank[]]], Times[ArcCosh[Times[Optional[Pattern[c, Blank[]]], Pattern[x, Blank[]]]], Optional[Pattern[b, Blank[]]]]], Optional[Pattern[n, Blank[]]]], Power[Times[Optional[Pattern[f, Blank[]]], Pattern[x, Blank[]]], Pattern[m, Blank[]]], Power[Plus[Pattern[d1, Blank[]], Times[Optional[Pattern[e1, Blank[]]], Pattern[x, Blank[]]]], Rational[1, 2]], Power[Plus[Pattern[d2, Blank[]], Times[Optional[Pattern[e2, Blank[]]], Pattern[x, Blank[]]]], Rational[1, 2]]], Pattern[x, Blank[Symbol]]] := Simp[((f*x)^(m + 1)*Sqrt[d1 + e1*x]*Sqrt[d2 + e2*x]*(a + b*ArcCosh[c*x])^n)/(f*(m + 2)), x] + (-Dist[(Sqrt[d1 + e1*x]*Sqrt[d2 + e2*x])/((m + 2)*Sqrt[1 + c*x]*Sqrt[-1 + c*x]), Int[((f*x)^m*(a + b*ArcCosh[c*x])^n)/(Sqrt[1 + c*x]*Sqrt[-1 + c*x]), x], x] - Dist[(b*c*n*Sqrt[d1 + e1*x]*Sqrt[d2 + e2*x])/(f*(m + 2)*Sqrt[1 + c*x]*Sqrt[-1 + c*x]), Int[(f*x)^(m + 1)*(a + b*ArcCosh[c*x])^(n - 1), x], x]) /; FreeQ[{a, b, c, d1, e1, d2, e2, f, m}, x] && EqQ[e1 - c*d1, 0] && EqQ[e2 + c*d2, 0] && GtQ[n, 0] && !LtQ[m, -1] && (RationalQ[m] || EqQ[n, 1])
  5744. Int[Times[Power[Plus[Optional[Pattern[a, Blank[]]], Times[ArcSinh[Times[Optional[Pattern[c, Blank[]]], Pattern[x, Blank[]]]], Optional[Pattern[b, Blank[]]]]], Optional[Pattern[n, Blank[]]]], Power[Times[Optional[Pattern[f, Blank[]]], Pattern[x, Blank[]]], Pattern[m, Blank[]]], Power[Plus[Pattern[d, Blank[]], Times[Optional[Pattern[e, Blank[]]], Power[Pattern[x, Blank[]], 2]]], Optional[Pattern[p, Blank[]]]]], Pattern[x, Blank[Symbol]]] := Simp[((f*x)^(m + 1)*(d + e*x^2)^p*(a + b*ArcSinh[c*x])^n)/(f*(m + 2*p + 1)), x] + (Dist[(2*d*p)/(m + 2*p + 1), Int[(f*x)^m*(d + e*x^2)^(p - 1)*(a + b*ArcSinh[c*x])^n, x], x] - Dist[(b*c*n*d^IntPart[p]*(d + e*x^2)^FracPart[p])/(f*(m + 2*p + 1)*(1 + c^2*x^2)^FracPart[p]), Int[(f*x)^(m + 1)*(1 + c^2*x^2)^(p - 1/2)*(a + b*ArcSinh[c*x])^(n - 1), x], x]) /; FreeQ[{a, b, c, d, e, f, m}, x] && EqQ[e, c^2*d] && GtQ[n, 0] && GtQ[p, 0] && !LtQ[m, -1] && (RationalQ[m] || EqQ[n, 1])
  5745. Int[Times[Power[Plus[Optional[Pattern[a, Blank[]]], Times[ArcCosh[Times[Optional[Pattern[c, Blank[]]], Pattern[x, Blank[]]]], Optional[Pattern[b, Blank[]]]]], Optional[Pattern[n, Blank[]]]], Power[Times[Optional[Pattern[f, Blank[]]], Pattern[x, Blank[]]], Pattern[m, Blank[]]], Power[Plus[Pattern[d1, Blank[]], Times[Optional[Pattern[e1, Blank[]]], Pattern[x, Blank[]]]], Pattern[p, Blank[]]], Power[Plus[Pattern[d2, Blank[]], Times[Optional[Pattern[e2, Blank[]]], Pattern[x, Blank[]]]], Pattern[p, Blank[]]]], Pattern[x, Blank[Symbol]]] := Simp[((f*x)^(m + 1)*(d1 + e1*x)^p*(d2 + e2*x)^p*(a + b*ArcCosh[c*x])^n)/(f*(m + 2*p + 1)), x] + (Dist[(2*d1*d2*p)/(m + 2*p + 1), Int[(f*x)^m*(d1 + e1*x)^(p - 1)*(d2 + e2*x)^(p - 1)*(a + b*ArcCosh[c*x])^n, x], x] - Dist[(b*c*n*(-(d1*d2))^(p - 1/2)*Sqrt[d1 + e1*x]*Sqrt[d2 + e2*x])/(f*(m + 2*p + 1)*Sqrt[1 + c*x]*Sqrt[-1 + c*x]), Int[(f*x)^(m + 1)*(-1 + c^2*x^2)^(p - 1/2)*(a + b*ArcCosh[c*x])^(n - 1), x], x]) /; FreeQ[{a, b, c, d1, e1, d2, e2, f, m}, x] && EqQ[e1 - c*d1, 0] && EqQ[e2 + c*d2, 0] && GtQ[n, 0] && GtQ[p, 0] && !LtQ[m, -1] && IntegerQ[p - 1/2] && (RationalQ[m] || EqQ[n, 1])
  5746. Int[Times[Power[Plus[Optional[Pattern[a, Blank[]]], Times[ArcCosh[Times[Optional[Pattern[c, Blank[]]], Pattern[x, Blank[]]]], Optional[Pattern[b, Blank[]]]]], Optional[Pattern[n, Blank[]]]], Power[Times[Optional[Pattern[f, Blank[]]], Pattern[x, Blank[]]], Pattern[m, Blank[]]], Power[Plus[Pattern[d, Blank[]], Times[Optional[Pattern[e, Blank[]]], Power[Pattern[x, Blank[]], 2]]], Pattern[p, Blank[]]]], Pattern[x, Blank[Symbol]]] := Simp[((f*x)^(m + 1)*(d + e*x^2)^(p + 1)*(a + b*ArcCosh[c*x])^n)/(d*f*(m + 1)), x] + (Dist[(b*c*n*(-d)^p)/(f*(m + 1)), Int[(f*x)^(m + 1)*(1 + c*x)^(p + 1/2)*(-1 + c*x)^(p + 1/2)*(a + b*ArcCosh[c*x])^(n - 1), x], x] + Dist[(c^2*(m + 2*p + 3))/(f^2*(m + 1)), Int[(f*x)^(m + 2)*(d + e*x^2)^p*(a + b*ArcCosh[c*x])^n, x], x]) /; FreeQ[{a, b, c, d, e, f, p}, x] && EqQ[c^2*d + e, 0] && GtQ[n, 0] && LtQ[m, -1] && IntegerQ[m] && IntegerQ[p]
  5747. Int[Times[Power[Plus[Optional[Pattern[a, Blank[]]], Times[ArcSinh[Times[Optional[Pattern[c, Blank[]]], Pattern[x, Blank[]]]], Optional[Pattern[b, Blank[]]]]], Optional[Pattern[n, Blank[]]]], Power[Times[Optional[Pattern[f, Blank[]]], Pattern[x, Blank[]]], Pattern[m, Blank[]]], Power[Plus[Pattern[d, Blank[]], Times[Optional[Pattern[e, Blank[]]], Power[Pattern[x, Blank[]], 2]]], Pattern[p, Blank[]]]], Pattern[x, Blank[Symbol]]] := Simp[((f*x)^(m + 1)*(d + e*x^2)^(p + 1)*(a + b*ArcSinh[c*x])^n)/(d*f*(m + 1)), x] + (-Dist[(c^2*(m + 2*p + 3))/(f^2*(m + 1)), Int[(f*x)^(m + 2)*(d + e*x^2)^p*(a + b*ArcSinh[c*x])^n, x], x] - Dist[(b*c*n*d^IntPart[p]*(d + e*x^2)^FracPart[p])/(f*(m + 1)*(1 + c^2*x^2)^FracPart[p]), Int[(f*x)^(m + 1)*(1 + c^2*x^2)^(p + 1/2)*(a + b*ArcSinh[c*x])^(n - 1), x], x]) /; FreeQ[{a, b, c, d, e, f, p}, x] && EqQ[e, c^2*d] && GtQ[n, 0] && LtQ[m, -1] && IntegerQ[m]
  5748. Int[Times[Power[Plus[Optional[Pattern[a, Blank[]]], Times[ArcCosh[Times[Optional[Pattern[c, Blank[]]], Pattern[x, Blank[]]]], Optional[Pattern[b, Blank[]]]]], Optional[Pattern[n, Blank[]]]], Power[Times[Optional[Pattern[f, Blank[]]], Pattern[x, Blank[]]], Pattern[m, Blank[]]], Power[Plus[Pattern[d1, Blank[]], Times[Optional[Pattern[e1, Blank[]]], Pattern[x, Blank[]]]], Pattern[p, Blank[]]], Power[Plus[Pattern[d2, Blank[]], Times[Optional[Pattern[e2, Blank[]]], Pattern[x, Blank[]]]], Pattern[p, Blank[]]]], Pattern[x, Blank[Symbol]]] := Simp[((f*x)^(m + 1)*(d1 + e1*x)^(p + 1)*(d2 + e2*x)^(p + 1)*(a + b*ArcCosh[c*x])^n)/(d1*d2*f*(m + 1)), x] + (Dist[(c^2*(m + 2*p + 3))/(f^2*(m + 1)), Int[(f*x)^(m + 2)*(d1 + e1*x)^p*(d2 + e2*x)^p*(a + b*ArcCosh[c*x])^n, x], x] + Dist[(b*c*n*(-(d1*d2))^IntPart[p]*(d1 + e1*x)^FracPart[p]*(d2 + e2*x)^FracPart[p])/(f*(m + 1)*(1 + c*x)^FracPart[p]*(-1 + c*x)^FracPart[p]), Int[(f*x)^(m + 1)*(-1 + c^2*x^2)^(p + 1/2)*(a + b*ArcCosh[c*x])^(n - 1), x], x]) /; FreeQ[{a, b, c, d1, e1, d2, e2, f, p}, x] && EqQ[e1 - c*d1, 0] && EqQ[e2 + c*d2, 0] && GtQ[n, 0] && LtQ[m, -1] && IntegerQ[m] && IntegerQ[p + 1/2]
  5749. Int[Times[Power[Plus[Optional[Pattern[a, Blank[]]], Times[ArcCosh[Times[Optional[Pattern[c, Blank[]]], Pattern[x, Blank[]]]], Optional[Pattern[b, Blank[]]]]], Optional[Pattern[n, Blank[]]]], Power[Times[Optional[Pattern[f, Blank[]]], Pattern[x, Blank[]]], Pattern[m, Blank[]]], Power[Plus[Pattern[d1, Blank[]], Times[Optional[Pattern[e1, Blank[]]], Pattern[x, Blank[]]]], Pattern[p, Blank[]]], Power[Plus[Pattern[d2, Blank[]], Times[Optional[Pattern[e2, Blank[]]], Pattern[x, Blank[]]]], Pattern[p, Blank[]]]], Pattern[x, Blank[Symbol]]] := Simp[((f*x)^(m + 1)*(d1 + e1*x)^(p + 1)*(d2 + e2*x)^(p + 1)*(a + b*ArcCosh[c*x])^n)/(d1*d2*f*(m + 1)), x] + (Dist[(c^2*(m + 2*p + 3))/(f^2*(m + 1)), Int[(f*x)^(m + 2)*(d1 + e1*x)^p*(d2 + e2*x)^p*(a + b*ArcCosh[c*x])^n, x], x] + Dist[(b*c*n*(-(d1*d2))^IntPart[p]*(d1 + e1*x)^FracPart[p]*(d2 + e2*x)^FracPart[p])/(f*(m + 1)*(1 + c*x)^FracPart[p]*(-1 + c*x)^FracPart[p]), Int[(f*x)^(m + 1)*(1 + c*x)^(p + 1/2)*(-1 + c*x)^(p + 1/2)*(a + b*ArcCosh[c*x])^(n - 1), x], x]) /; FreeQ[{a, b, c, d1, e1, d2, e2, f, p}, x] && EqQ[e1 - c*d1, 0] && EqQ[e2 + c*d2, 0] && GtQ[n, 0] && LtQ[m, -1] && IntegerQ[m]
  5750. Int[Times[Power[Plus[Optional[Pattern[a, Blank[]]], Times[ArcCosh[Times[Optional[Pattern[c, Blank[]]], Pattern[x, Blank[]]]], Optional[Pattern[b, Blank[]]]]], Optional[Pattern[n, Blank[]]]], Power[Times[Optional[Pattern[f, Blank[]]], Pattern[x, Blank[]]], Pattern[m, Blank[]]], Power[Plus[Pattern[d, Blank[]], Times[Optional[Pattern[e, Blank[]]], Power[Pattern[x, Blank[]], 2]]], Pattern[p, Blank[]]]], Pattern[x, Blank[Symbol]]] := Simp[(f*(f*x)^(m - 1)*(d + e*x^2)^(p + 1)*(a + b*ArcCosh[c*x])^n)/(2*e*(p + 1)), x] + (-Dist[(b*f*n*(-d)^p)/(2*c*(p + 1)), Int[(f*x)^(m - 1)*(1 + c*x)^(p + 1/2)*(-1 + c*x)^(p + 1/2)*(a + b*ArcCosh[c*x])^(n - 1), x], x] - Dist[(f^2*(m - 1))/(2*e*(p + 1)), Int[(f*x)^(m - 2)*(d + e*x^2)^(p + 1)*(a + b*ArcCosh[c*x])^n, x], x]) /; FreeQ[{a, b, c, d, e, f}, x] && EqQ[c^2*d + e, 0] && GtQ[n, 0] && LtQ[p, -1] && GtQ[m, 1] && IntegerQ[p]
  5751. Int[Times[Power[Plus[Optional[Pattern[a, Blank[]]], Times[ArcSinh[Times[Optional[Pattern[c, Blank[]]], Pattern[x, Blank[]]]], Optional[Pattern[b, Blank[]]]]], Optional[Pattern[n, Blank[]]]], Power[Times[Optional[Pattern[f, Blank[]]], Pattern[x, Blank[]]], Pattern[m, Blank[]]], Power[Plus[Pattern[d, Blank[]], Times[Optional[Pattern[e, Blank[]]], Power[Pattern[x, Blank[]], 2]]], Pattern[p, Blank[]]]], Pattern[x, Blank[Symbol]]] := Simp[(f*(f*x)^(m - 1)*(d + e*x^2)^(p + 1)*(a + b*ArcSinh[c*x])^n)/(2*e*(p + 1)), x] + (-Dist[(f^2*(m - 1))/(2*e*(p + 1)), Int[(f*x)^(m - 2)*(d + e*x^2)^(p + 1)*(a + b*ArcSinh[c*x])^n, x], x] - Dist[(b*f*n*d^IntPart[p]*(d + e*x^2)^FracPart[p])/(2*c*(p + 1)*(1 + c^2*x^2)^FracPart[p]), Int[(f*x)^(m - 1)*(1 + c^2*x^2)^(p + 1/2)*(a + b*ArcSinh[c*x])^(n - 1), x], x]) /; FreeQ[{a, b, c, d, e, f}, x] && EqQ[e, c^2*d] && GtQ[n, 0] && LtQ[p, -1] && GtQ[m, 1]
  5752. Int[Times[Power[Plus[Optional[Pattern[a, Blank[]]], Times[ArcCosh[Times[Optional[Pattern[c, Blank[]]], Pattern[x, Blank[]]]], Optional[Pattern[b, Blank[]]]]], Optional[Pattern[n, Blank[]]]], Power[Times[Optional[Pattern[f, Blank[]]], Pattern[x, Blank[]]], Pattern[m, Blank[]]], Power[Plus[Pattern[d1, Blank[]], Times[Optional[Pattern[e1, Blank[]]], Pattern[x, Blank[]]]], Pattern[p, Blank[]]], Power[Plus[Pattern[d2, Blank[]], Times[Optional[Pattern[e2, Blank[]]], Pattern[x, Blank[]]]], Pattern[p, Blank[]]]], Pattern[x, Blank[Symbol]]] := Simp[(f*(f*x)^(m - 1)*(d1 + e1*x)^(p + 1)*(d2 + e2*x)^(p + 1)*(a + b*ArcCosh[c*x])^n)/(2*e1*e2*(p + 1)), x] + (-Dist[(f^2*(m - 1))/(2*e1*e2*(p + 1)), Int[(f*x)^(m - 2)*(d1 + e1*x)^(p + 1)*(d2 + e2*x)^(p + 1)*(a + b*ArcCosh[c*x])^n, x], x] - Dist[(b*f*n*(-(d1*d2))^IntPart[p]*(d1 + e1*x)^FracPart[p]*(d2 + e2*x)^FracPart[p])/(2*c*(p + 1)*(1 + c*x)^FracPart[p]*(-1 + c*x)^FracPart[p]), Int[(f*x)^(m - 1)*(-1 + c^2*x^2)^(p + 1/2)*(a + b*ArcCosh[c*x])^(n - 1), x], x]) /; FreeQ[{a, b, c, d1, e1, d2, e2, f}, x] && EqQ[e1 - c*d1, 0] && EqQ[e2 + c*d2, 0] && GtQ[n, 0] && LtQ[p, -1] && GtQ[m, 1] && IntegerQ[p + 1/2]
  5753. Int[Times[Power[Plus[Optional[Pattern[a, Blank[]]], Times[ArcCosh[Times[Optional[Pattern[c, Blank[]]], Pattern[x, Blank[]]]], Optional[Pattern[b, Blank[]]]]], Optional[Pattern[n, Blank[]]]], Power[Times[Optional[Pattern[f, Blank[]]], Pattern[x, Blank[]]], Pattern[m, Blank[]]], Power[Plus[Pattern[d1, Blank[]], Times[Optional[Pattern[e1, Blank[]]], Pattern[x, Blank[]]]], Pattern[p, Blank[]]], Power[Plus[Pattern[d2, Blank[]], Times[Optional[Pattern[e2, Blank[]]], Pattern[x, Blank[]]]], Pattern[p, Blank[]]]], Pattern[x, Blank[Symbol]]] := Simp[(f*(f*x)^(m - 1)*(d1 + e1*x)^(p + 1)*(d2 + e2*x)^(p + 1)*(a + b*ArcCosh[c*x])^n)/(2*e1*e2*(p + 1)), x] + (-Dist[(f^2*(m - 1))/(2*e1*e2*(p + 1)), Int[(f*x)^(m - 2)*(d1 + e1*x)^(p + 1)*(d2 + e2*x)^(p + 1)*(a + b*ArcCosh[c*x])^n, x], x] - Dist[(b*f*n*(-(d1*d2))^IntPart[p]*(d1 + e1*x)^FracPart[p]*(d2 + e2*x)^FracPart[p])/(2*c*(p + 1)*(1 + c*x)^FracPart[p]*(-1 + c*x)^FracPart[p]), Int[(f*x)^(m - 1)*(1 + c*x)^(p + 1/2)*(-1 + c*x)^(p + 1/2)*(a + b*ArcCosh[c*x])^(n - 1), x], x]) /; FreeQ[{a, b, c, d1, e1, d2, e2, f}, x] && EqQ[e1 - c*d1, 0] && EqQ[e2 + c*d2, 0] && GtQ[n, 0] && LtQ[p, -1] && !IntegerQ[p] && GtQ[m, 1]
  5754. Int[Times[Power[Plus[Optional[Pattern[a, Blank[]]], Times[ArcCosh[Times[Optional[Pattern[c, Blank[]]], Pattern[x, Blank[]]]], Optional[Pattern[b, Blank[]]]]], Optional[Pattern[n, Blank[]]]], Power[Times[Optional[Pattern[f, Blank[]]], Pattern[x, Blank[]]], Pattern[m, Blank[]]], Power[Plus[Pattern[d, Blank[]], Times[Optional[Pattern[e, Blank[]]], Power[Pattern[x, Blank[]], 2]]], Pattern[p, Blank[]]]], Pattern[x, Blank[Symbol]]] := -Simp[((f*x)^(m + 1)*(d + e*x^2)^(p + 1)*(a + b*ArcCosh[c*x])^n)/(2*d*f*(p + 1)), x] + (Dist[(m + 2*p + 3)/(2*d*(p + 1)), Int[(f*x)^m*(d + e*x^2)^(p + 1)*(a + b*ArcCosh[c*x])^n, x], x] - Dist[(b*c*n*(-d)^p)/(2*f*(p + 1)), Int[(f*x)^(m + 1)*(1 + c*x)^(p + 1/2)*(-1 + c*x)^(p + 1/2)*(a + b*ArcCosh[c*x])^(n - 1), x], x]) /; FreeQ[{a, b, c, d, e, f, m}, x] && EqQ[c^2*d + e, 0] && GtQ[n, 0] && LtQ[p, -1] && !GtQ[m, 1] && IntegerQ[p]
  5755. Int[Times[Power[Plus[Optional[Pattern[a, Blank[]]], Times[ArcSinh[Times[Optional[Pattern[c, Blank[]]], Pattern[x, Blank[]]]], Optional[Pattern[b, Blank[]]]]], Optional[Pattern[n, Blank[]]]], Power[Times[Optional[Pattern[f, Blank[]]], Pattern[x, Blank[]]], Pattern[m, Blank[]]], Power[Plus[Pattern[d, Blank[]], Times[Optional[Pattern[e, Blank[]]], Power[Pattern[x, Blank[]], 2]]], Pattern[p, Blank[]]]], Pattern[x, Blank[Symbol]]] := -Simp[((f*x)^(m + 1)*(d + e*x^2)^(p + 1)*(a + b*ArcSinh[c*x])^n)/(2*d*f*(p + 1)), x] + (Dist[(m + 2*p + 3)/(2*d*(p + 1)), Int[(f*x)^m*(d + e*x^2)^(p + 1)*(a + b*ArcSinh[c*x])^n, x], x] + Dist[(b*c*n*d^IntPart[p]*(d + e*x^2)^FracPart[p])/(2*f*(p + 1)*(1 + c^2*x^2)^FracPart[p]), Int[(f*x)^(m + 1)*(1 + c^2*x^2)^(p + 1/2)*(a + b*ArcSinh[c*x])^(n - 1), x], x]) /; FreeQ[{a, b, c, d, e, f, m}, x] && EqQ[e, c^2*d] && GtQ[n, 0] && LtQ[p, -1] && !GtQ[m, 1] && (IntegerQ[m] || IntegerQ[p] || EqQ[n, 1])
  5756. Int[Times[Power[Plus[Optional[Pattern[a, Blank[]]], Times[ArcCosh[Times[Optional[Pattern[c, Blank[]]], Pattern[x, Blank[]]]], Optional[Pattern[b, Blank[]]]]], Optional[Pattern[n, Blank[]]]], Power[Times[Optional[Pattern[f, Blank[]]], Pattern[x, Blank[]]], Pattern[m, Blank[]]], Power[Plus[Pattern[d1, Blank[]], Times[Optional[Pattern[e1, Blank[]]], Pattern[x, Blank[]]]], Pattern[p, Blank[]]], Power[Plus[Pattern[d2, Blank[]], Times[Optional[Pattern[e2, Blank[]]], Pattern[x, Blank[]]]], Pattern[p, Blank[]]]], Pattern[x, Blank[Symbol]]] := -Simp[((f*x)^(m + 1)*(d1 + e1*x)^(p + 1)*(d2 + e2*x)^(p + 1)*(a + b*ArcCosh[c*x])^n)/(2*d1*d2*f*(p + 1)), x] + (Dist[(m + 2*p + 3)/(2*d1*d2*(p + 1)), Int[(f*x)^m*(d1 + e1*x)^(p + 1)*(d2 + e2*x)^(p + 1)*(a + b*ArcCosh[c*x])^n, x], x] - Dist[(b*c*n*(-(d1*d2))^IntPart[p]*(d1 + e1*x)^FracPart[p]*(d2 + e2*x)^FracPart[p])/(2*f*(p + 1)*(1 + c*x)^FracPart[p]*(-1 + c*x)^FracPart[p]), Int[(f*x)^(m + 1)*(-1 + c^2*x^2)^(p + 1/2)*(a + b*ArcCosh[c*x])^(n - 1), x], x]) /; FreeQ[{a, b, c, d1, e1, d2, e2, f, m}, x] && EqQ[e1 - c*d1, 0] && EqQ[e2 + c*d2, 0] && GtQ[n, 0] && LtQ[p, -1] && !GtQ[m, 1] && (IntegerQ[m] || EqQ[n, 1]) && IntegerQ[p + 1/2]
  5757. Int[Times[Power[Plus[Optional[Pattern[a, Blank[]]], Times[ArcCosh[Times[Optional[Pattern[c, Blank[]]], Pattern[x, Blank[]]]], Optional[Pattern[b, Blank[]]]]], Optional[Pattern[n, Blank[]]]], Power[Times[Optional[Pattern[f, Blank[]]], Pattern[x, Blank[]]], Pattern[m, Blank[]]], Power[Plus[Pattern[d1, Blank[]], Times[Optional[Pattern[e1, Blank[]]], Pattern[x, Blank[]]]], Pattern[p, Blank[]]], Power[Plus[Pattern[d2, Blank[]], Times[Optional[Pattern[e2, Blank[]]], Pattern[x, Blank[]]]], Pattern[p, Blank[]]]], Pattern[x, Blank[Symbol]]] := -Simp[((f*x)^(m + 1)*(d1 + e1*x)^(p + 1)*(d2 + e2*x)^(p + 1)*(a + b*ArcCosh[c*x])^n)/(2*d1*d2*f*(p + 1)), x] + (Dist[(m + 2*p + 3)/(2*d1*d2*(p + 1)), Int[(f*x)^m*(d1 + e1*x)^(p + 1)*(d2 + e2*x)^(p + 1)*(a + b*ArcCosh[c*x])^n, x], x] - Dist[(b*c*n*(-(d1*d2))^IntPart[p]*(d1 + e1*x)^FracPart[p]*(d2 + e2*x)^FracPart[p])/(2*f*(p + 1)*(1 + c*x)^FracPart[p]*(-1 + c*x)^FracPart[p]), Int[(f*x)^(m + 1)*(1 + c*x)^(p + 1/2)*(-1 + c*x)^(p + 1/2)*(a + b*ArcCosh[c*x])^(n - 1), x], x]) /; FreeQ[{a, b, c, d1, e1, d2, e2, f, m}, x] && EqQ[e1 - c*d1, 0] && EqQ[e2 + c*d2, 0] && GtQ[n, 0] && LtQ[p, -1] && !GtQ[m, 1] && (IntegerQ[m] || IntegerQ[p] || EqQ[n, 1])
  5758. Int[Times[Power[Plus[Optional[Pattern[a, Blank[]]], Times[ArcSinh[Times[Optional[Pattern[c, Blank[]]], Pattern[x, Blank[]]]], Optional[Pattern[b, Blank[]]]]], Optional[Pattern[n, Blank[]]]], Power[Times[Optional[Pattern[f, Blank[]]], Pattern[x, Blank[]]], Pattern[m, Blank[]]], Power[Plus[Pattern[d, Blank[]], Times[Optional[Pattern[e, Blank[]]], Power[Pattern[x, Blank[]], 2]]], Rational[-1, 2]]], Pattern[x, Blank[Symbol]]] := Simp[(f*(f*x)^(m - 1)*Sqrt[d + e*x^2]*(a + b*ArcSinh[c*x])^n)/(e*m), x] + (-Dist[(f^2*(m - 1))/(c^2*m), Int[((f*x)^(m - 2)*(a + b*ArcSinh[c*x])^n)/Sqrt[d + e*x^2], x], x] - Dist[(b*f*n*Sqrt[1 + c^2*x^2])/(c*m*Sqrt[d + e*x^2]), Int[(f*x)^(m - 1)*(a + b*ArcSinh[c*x])^(n - 1), x], x]) /; FreeQ[{a, b, c, d, e, f}, x] && EqQ[e, c^2*d] && GtQ[n, 0] && GtQ[m, 1] && IntegerQ[m]
  5759. Int[Times[Power[Plus[Optional[Pattern[a, Blank[]]], Times[ArcCosh[Times[Optional[Pattern[c, Blank[]]], Pattern[x, Blank[]]]], Optional[Pattern[b, Blank[]]]]], Optional[Pattern[n, Blank[]]]], Power[Times[Optional[Pattern[f, Blank[]]], Pattern[x, Blank[]]], Pattern[m, Blank[]]], Power[Plus[Pattern[d1, Blank[]], Times[Optional[Pattern[e1, Blank[]]], Pattern[x, Blank[]]]], Rational[-1, 2]], Power[Plus[Pattern[d2, Blank[]], Times[Optional[Pattern[e2, Blank[]]], Pattern[x, Blank[]]]], Rational[-1, 2]]], Pattern[x, Blank[Symbol]]] := Simp[(f*(f*x)^(m - 1)*Sqrt[d1 + e1*x]*Sqrt[d2 + e2*x]*(a + b*ArcCosh[c*x])^n)/(e1*e2*m), x] + (Dist[(f^2*(m - 1))/(c^2*m), Int[((f*x)^(m - 2)*(a + b*ArcCosh[c*x])^n)/(Sqrt[d1 + e1*x]*Sqrt[d2 + e2*x]), x], x] + Dist[(b*f*n*Sqrt[d1 + e1*x]*Sqrt[d2 + e2*x])/(c*d1*d2*m*Sqrt[1 + c*x]*Sqrt[-1 + c*x]), Int[(f*x)^(m - 1)*(a + b*ArcCosh[c*x])^(n - 1), x], x]) /; FreeQ[{a, b, c, d1, e1, d2, e2, f}, x] && EqQ[e1 - c*d1, 0] && EqQ[e2 + c*d2, 0] && GtQ[n, 0] && GtQ[m, 1] && IntegerQ[m]
  5760. Int[Times[Power[Plus[Optional[Pattern[a, Blank[]]], Times[ArcSinh[Times[Optional[Pattern[c, Blank[]]], Pattern[x, Blank[]]]], Optional[Pattern[b, Blank[]]]]], Optional[Pattern[n, Blank[]]]], Power[Pattern[x, Blank[]], Pattern[m, Blank[]]], Power[Plus[Pattern[d, Blank[]], Times[Optional[Pattern[e, Blank[]]], Power[Pattern[x, Blank[]], 2]]], Rational[-1, 2]]], Pattern[x, Blank[Symbol]]] := Dist[1/(c^(m + 1)*Sqrt[d]), Subst[Int[(a + b*x)^n*Sinh[x]^m, x], x, ArcSinh[c*x]], x] /; FreeQ[{a, b, c, d, e}, x] && EqQ[e, c^2*d] && GtQ[d, 0] && IGtQ[n, 0] && IntegerQ[m]
  5761. Int[Times[Power[Plus[Optional[Pattern[a, Blank[]]], Times[ArcCosh[Times[Optional[Pattern[c, Blank[]]], Pattern[x, Blank[]]]], Optional[Pattern[b, Blank[]]]]], Optional[Pattern[n, Blank[]]]], Power[Pattern[x, Blank[]], Pattern[m, Blank[]]], Power[Plus[Pattern[d1, Blank[]], Times[Optional[Pattern[e1, Blank[]]], Pattern[x, Blank[]]]], Rational[-1, 2]], Power[Plus[Pattern[d2, Blank[]], Times[Optional[Pattern[e2, Blank[]]], Pattern[x, Blank[]]]], Rational[-1, 2]]], Pattern[x, Blank[Symbol]]] := Dist[1/(c^(m + 1)*Sqrt[-(d1*d2)]), Subst[Int[(a + b*x)^n*Cosh[x]^m, x], x, ArcCosh[c*x]], x] /; FreeQ[{a, b, c, d1, e1, d2, e2}, x] && EqQ[e1 - c*d1, 0] && EqQ[e2 + c*d2, 0] && IGtQ[n, 0] && GtQ[d1, 0] && LtQ[d2, 0] && IntegerQ[m]
  5762. Int[Times[Plus[Optional[Pattern[a, Blank[]]], Times[ArcSinh[Times[Optional[Pattern[c, Blank[]]], Pattern[x, Blank[]]]], Optional[Pattern[b, Blank[]]]]], Power[Times[Optional[Pattern[f, Blank[]]], Pattern[x, Blank[]]], Pattern[m, Blank[]]], Power[Plus[Pattern[d, Blank[]], Times[Optional[Pattern[e, Blank[]]], Power[Pattern[x, Blank[]], 2]]], Rational[-1, 2]]], Pattern[x, Blank[Symbol]]] := Simp[((f*x)^(m + 1)*(a + b*ArcSinh[c*x])*Hypergeometric2F1[1/2, (1 + m)/2, (3 + m)/2, -(c^2*x^2)])/(Sqrt[d]*f*(m + 1)), x] - Simp[(b*c*(f*x)^(m + 2)*HypergeometricPFQ[{1, 1 + m/2, 1 + m/2}, {3/2 + m/2, 2 + m/2}, -(c^2*x^2)])/(Sqrt[d]*f^2*(m + 1)*(m + 2)), x] /; FreeQ[{a, b, c, d, e, f, m}, x] && EqQ[e, c^2*d] && GtQ[d, 0] && !IntegerQ[m]
  5763. Int[Times[Plus[Optional[Pattern[a, Blank[]]], Times[ArcCosh[Times[Optional[Pattern[c, Blank[]]], Pattern[x, Blank[]]]], Optional[Pattern[b, Blank[]]]]], Power[Times[Optional[Pattern[f, Blank[]]], Pattern[x, Blank[]]], Pattern[m, Blank[]]], Power[Plus[Pattern[d1, Blank[]], Times[Optional[Pattern[e1, Blank[]]], Pattern[x, Blank[]]]], Rational[-1, 2]], Power[Plus[Pattern[d2, Blank[]], Times[Optional[Pattern[e2, Blank[]]], Pattern[x, Blank[]]]], Rational[-1, 2]]], Pattern[x, Blank[Symbol]]] := Simp[((f*x)^(m + 1)*Sqrt[1 - c^2*x^2]*(a + b*ArcCosh[c*x])*Hypergeometric2F1[1/2, (1 + m)/2, (3 + m)/2, c^2*x^2])/(f*(m + 1)*Sqrt[d1 + e1*x]*Sqrt[d2 + e2*x]), x] + Simp[(b*c*(f*x)^(m + 2)*HypergeometricPFQ[{1, 1 + m/2, 1 + m/2}, {3/2 + m/2, 2 + m/2}, c^2*x^2])/(Sqrt[-(d1*d2)]*f^2*(m + 1)*(m + 2)), x] /; FreeQ[{a, b, c, d1, e1, d2, e2, f, m}, x] && EqQ[e1 - c*d1, 0] && EqQ[e2 + c*d2, 0] && GtQ[d1, 0] && LtQ[d2, 0] && !IntegerQ[m]
  5764. Int[Times[Power[Plus[Optional[Pattern[a, Blank[]]], Times[ArcSinh[Times[Optional[Pattern[c, Blank[]]], Pattern[x, Blank[]]]], Optional[Pattern[b, Blank[]]]]], Optional[Pattern[n, Blank[]]]], Power[Times[Optional[Pattern[f, Blank[]]], Pattern[x, Blank[]]], Pattern[m, Blank[]]], Power[Plus[Pattern[d, Blank[]], Times[Optional[Pattern[e, Blank[]]], Power[Pattern[x, Blank[]], 2]]], Rational[-1, 2]]], Pattern[x, Blank[Symbol]]] := Dist[Sqrt[1 + c^2*x^2]/Sqrt[d + e*x^2], Int[((f*x)^m*(a + b*ArcSinh[c*x])^n)/Sqrt[1 + c^2*x^2], x], x] /; FreeQ[{a, b, c, d, e, f, m}, x] && EqQ[e, c^2*d] && GtQ[n, 0] && !GtQ[d, 0] && (IntegerQ[m] || EqQ[n, 1])
  5765. Int[Times[Power[Plus[Optional[Pattern[a, Blank[]]], Times[ArcCosh[Times[Optional[Pattern[c, Blank[]]], Pattern[x, Blank[]]]], Optional[Pattern[b, Blank[]]]]], Optional[Pattern[n, Blank[]]]], Power[Times[Optional[Pattern[f, Blank[]]], Pattern[x, Blank[]]], Pattern[m, Blank[]]], Power[Plus[Pattern[d1, Blank[]], Times[Optional[Pattern[e1, Blank[]]], Pattern[x, Blank[]]]], Rational[-1, 2]], Power[Plus[Pattern[d2, Blank[]], Times[Optional[Pattern[e2, Blank[]]], Pattern[x, Blank[]]]], Rational[-1, 2]]], Pattern[x, Blank[Symbol]]] := Dist[(Sqrt[1 + c*x]*Sqrt[-1 + c*x])/(Sqrt[d1 + e1*x]*Sqrt[d2 + e2*x]), Int[((f*x)^m*(a + b*ArcCosh[c*x])^n)/(Sqrt[1 + c*x]*Sqrt[-1 + c*x]), x], x] /; FreeQ[{a, b, c, d1, e1, d2, e2, f, m}, x] && EqQ[e1 - c*d1, 0] && EqQ[e2 + c*d2, 0] && GtQ[n, 0] && !(GtQ[d1, 0] && LtQ[d2, 0]) && (IntegerQ[m] || EqQ[n, 1])
  5766. Int[Times[Power[Plus[Optional[Pattern[a, Blank[]]], Times[ArcCosh[Times[Optional[Pattern[c, Blank[]]], Pattern[x, Blank[]]]], Optional[Pattern[b, Blank[]]]]], Optional[Pattern[n, Blank[]]]], Power[Times[Optional[Pattern[f, Blank[]]], Pattern[x, Blank[]]], Pattern[m, Blank[]]], Power[Plus[Pattern[d, Blank[]], Times[Optional[Pattern[e, Blank[]]], Power[Pattern[x, Blank[]], 2]]], Pattern[p, Blank[]]]], Pattern[x, Blank[Symbol]]] := Simp[(f*(f*x)^(m - 1)*(d + e*x^2)^(p + 1)*(a + b*ArcCosh[c*x])^n)/(e*(m + 2*p + 1)), x] + (-Dist[(b*f*n*(-d)^p)/(c*(m + 2*p + 1)), Int[(f*x)^(m - 1)*(1 + c*x)^(p + 1/2)*(-1 + c*x)^(p + 1/2)*(a + b*ArcCosh[c*x])^(n - 1), x], x] + Dist[(f^2*(m - 1))/(c^2*(m + 2*p + 1)), Int[(f*x)^(m - 2)*(d + e*x^2)^p*(a + b*ArcCosh[c*x])^n, x], x]) /; FreeQ[{a, b, c, d, e, f, p}, x] && EqQ[c^2*d + e, 0] && GtQ[n, 0] && GtQ[m, 1] && NeQ[m + 2*p + 1, 0] && IntegerQ[p] && IntegerQ[m]
  5767. Int[Times[Power[Plus[Optional[Pattern[a, Blank[]]], Times[ArcSinh[Times[Optional[Pattern[c, Blank[]]], Pattern[x, Blank[]]]], Optional[Pattern[b, Blank[]]]]], Optional[Pattern[n, Blank[]]]], Power[Times[Optional[Pattern[f, Blank[]]], Pattern[x, Blank[]]], Pattern[m, Blank[]]], Power[Plus[Pattern[d, Blank[]], Times[Optional[Pattern[e, Blank[]]], Power[Pattern[x, Blank[]], 2]]], Pattern[p, Blank[]]]], Pattern[x, Blank[Symbol]]] := Simp[(f*(f*x)^(m - 1)*(d + e*x^2)^(p + 1)*(a + b*ArcSinh[c*x])^n)/(e*(m + 2*p + 1)), x] + (-Dist[(f^2*(m - 1))/(c^2*(m + 2*p + 1)), Int[(f*x)^(m - 2)*(d + e*x^2)^p*(a + b*ArcSinh[c*x])^n, x], x] - Dist[(b*f*n*d^IntPart[p]*(d + e*x^2)^FracPart[p])/(c*(m + 2*p + 1)*(1 + c^2*x^2)^FracPart[p]), Int[(f*x)^(m - 1)*(1 + c^2*x^2)^(p + 1/2)*(a + b*ArcSinh[c*x])^(n - 1), x], x]) /; FreeQ[{a, b, c, d, e, f, p}, x] && EqQ[e, c^2*d] && GtQ[n, 0] && GtQ[m, 1] && NeQ[m + 2*p + 1, 0] && IntegerQ[m]
  5768. Int[Times[Power[Plus[Optional[Pattern[a, Blank[]]], Times[ArcCosh[Times[Optional[Pattern[c, Blank[]]], Pattern[x, Blank[]]]], Optional[Pattern[b, Blank[]]]]], Optional[Pattern[n, Blank[]]]], Power[Times[Optional[Pattern[f, Blank[]]], Pattern[x, Blank[]]], Pattern[m, Blank[]]], Power[Plus[Pattern[d1, Blank[]], Times[Optional[Pattern[e1, Blank[]]], Pattern[x, Blank[]]]], Pattern[p, Blank[]]], Power[Plus[Pattern[d2, Blank[]], Times[Optional[Pattern[e2, Blank[]]], Pattern[x, Blank[]]]], Pattern[p, Blank[]]]], Pattern[x, Blank[Symbol]]] := Simp[(f*(f*x)^(m - 1)*(d1 + e1*x)^(p + 1)*(d2 + e2*x)^(p + 1)*(a + b*ArcCosh[c*x])^n)/(e1*e2*(m + 2*p + 1)), x] + (Dist[(f^2*(m - 1))/(c^2*(m + 2*p + 1)), Int[(f*x)^(m - 2)*(d1 + e1*x)^p*(d2 + e2*x)^p*(a + b*ArcCosh[c*x])^n, x], x] - Dist[(b*f*n*(-(d1*d2))^IntPart[p]*(d1 + e1*x)^FracPart[p]*(d2 + e2*x)^FracPart[p])/(c*(m + 2*p + 1)*(1 + c*x)^FracPart[p]*(-1 + c*x)^FracPart[p]), Int[(f*x)^(m - 1)*(-1 + c^2*x^2)^(p + 1/2)*(a + b*ArcCosh[c*x])^(n - 1), x], x]) /; FreeQ[{a, b, c, d1, e1, d2, e2, f, p}, x] && EqQ[e1 - c*d1, 0] && EqQ[e2 + c*d2, 0] && GtQ[n, 0] && GtQ[m, 1] && NeQ[m + 2*p + 1, 0] && IntegerQ[m] && IntegerQ[p + 1/2]
  5769. Int[Times[Power[Plus[Optional[Pattern[a, Blank[]]], Times[ArcCosh[Times[Optional[Pattern[c, Blank[]]], Pattern[x, Blank[]]]], Optional[Pattern[b, Blank[]]]]], Optional[Pattern[n, Blank[]]]], Power[Times[Optional[Pattern[f, Blank[]]], Pattern[x, Blank[]]], Pattern[m, Blank[]]], Power[Plus[Pattern[d1, Blank[]], Times[Optional[Pattern[e1, Blank[]]], Pattern[x, Blank[]]]], Pattern[p, Blank[]]], Power[Plus[Pattern[d2, Blank[]], Times[Optional[Pattern[e2, Blank[]]], Pattern[x, Blank[]]]], Pattern[p, Blank[]]]], Pattern[x, Blank[Symbol]]] := Simp[(f*(f*x)^(m - 1)*(d1 + e1*x)^(p + 1)*(d2 + e2*x)^(p + 1)*(a + b*ArcCosh[c*x])^n)/(e1*e2*(m + 2*p + 1)), x] + (Dist[(f^2*(m - 1))/(c^2*(m + 2*p + 1)), Int[(f*x)^(m - 2)*(d1 + e1*x)^p*(d2 + e2*x)^p*(a + b*ArcCosh[c*x])^n, x], x] - Dist[(b*f*n*(-(d1*d2))^IntPart[p]*(d1 + e1*x)^FracPart[p]*(d2 + e2*x)^FracPart[p])/(c*(m + 2*p + 1)*(1 + c*x)^FracPart[p]*(-1 + c*x)^FracPart[p]), Int[(f*x)^(m - 1)*(1 + c*x)^(p + 1/2)*(-1 + c*x)^(p + 1/2)*(a + b*ArcCosh[c*x])^(n - 1), x], x]) /; FreeQ[{a, b, c, d1, e1, d2, e2, f, p}, x] && EqQ[e1 - c*d1, 0] && EqQ[e2 + c*d2, 0] && GtQ[n, 0] && GtQ[m, 1] && NeQ[m + 2*p + 1, 0] && IntegerQ[m]
  5770. Int[Times[Power[Plus[Optional[Pattern[a, Blank[]]], Times[ArcCosh[Times[Optional[Pattern[c, Blank[]]], Pattern[x, Blank[]]]], Optional[Pattern[b, Blank[]]]]], Pattern[n, Blank[]]], Power[Times[Optional[Pattern[f, Blank[]]], Pattern[x, Blank[]]], Optional[Pattern[m, Blank[]]]], Power[Plus[Pattern[d, Blank[]], Times[Optional[Pattern[e, Blank[]]], Power[Pattern[x, Blank[]], 2]]], Optional[Pattern[p, Blank[]]]]], Pattern[x, Blank[Symbol]]] := Simp[((f*x)^m*Sqrt[1 + c*x]*Sqrt[-1 + c*x]*(d + e*x^2)^p*(a + b*ArcCosh[c*x])^(n + 1))/(b*c*(n + 1)), x] + Dist[(f*m*(-d)^p)/(b*c*(n + 1)), Int[(f*x)^(m - 1)*(1 + c*x)^(p - 1/2)*(-1 + c*x)^(p - 1/2)*(a + b*ArcCosh[c*x])^(n + 1), x], x] /; FreeQ[{a, b, c, d, e, f, m, p}, x] && EqQ[c^2*d + e, 0] && LtQ[n, -1] && EqQ[m + 2*p + 1, 0] && IntegerQ[p]
  5771. Int[Times[Power[Plus[Optional[Pattern[a, Blank[]]], Times[ArcSinh[Times[Optional[Pattern[c, Blank[]]], Pattern[x, Blank[]]]], Optional[Pattern[b, Blank[]]]]], Pattern[n, Blank[]]], Power[Times[Optional[Pattern[f, Blank[]]], Pattern[x, Blank[]]], Optional[Pattern[m, Blank[]]]], Power[Plus[Pattern[d, Blank[]], Times[Optional[Pattern[e, Blank[]]], Power[Pattern[x, Blank[]], 2]]], Optional[Pattern[p, Blank[]]]]], Pattern[x, Blank[Symbol]]] := Simp[((f*x)^m*Sqrt[1 + c^2*x^2]*(d + e*x^2)^p*(a + b*ArcSinh[c*x])^(n + 1))/(b*c*(n + 1)), x] - Dist[(f*m*d^IntPart[p]*(d + e*x^2)^FracPart[p])/(b*c*(n + 1)*(1 + c^2*x^2)^FracPart[p]), Int[(f*x)^(m - 1)*(1 + c^2*x^2)^(p - 1/2)*(a + b*ArcSinh[c*x])^(n + 1), x], x] /; FreeQ[{a, b, c, d, e, f, m, p}, x] && EqQ[e, c^2*d] && LtQ[n, -1] && EqQ[m + 2*p + 1, 0]
  5772. Int[Times[Power[Plus[Optional[Pattern[a, Blank[]]], Times[ArcCosh[Times[Optional[Pattern[c, Blank[]]], Pattern[x, Blank[]]]], Optional[Pattern[b, Blank[]]]]], Pattern[n, Blank[]]], Power[Times[Optional[Pattern[f, Blank[]]], Pattern[x, Blank[]]], Optional[Pattern[m, Blank[]]]], Power[Plus[Pattern[d1, Blank[]], Times[Optional[Pattern[e1, Blank[]]], Pattern[x, Blank[]]]], Optional[Pattern[p, Blank[]]]], Power[Plus[Pattern[d2, Blank[]], Times[Optional[Pattern[e2, Blank[]]], Pattern[x, Blank[]]]], Optional[Pattern[p, Blank[]]]]], Pattern[x, Blank[Symbol]]] := Simp[((f*x)^m*Sqrt[1 + c*x]*Sqrt[-1 + c*x]*(d1 + e1*x)^p*(d2 + e2*x)^p*(a + b*ArcCosh[c*x])^(n + 1))/(b*c*(n + 1)), x] + Dist[(f*m*(-(d1*d2))^IntPart[p]*(d1 + e1*x)^FracPart[p]*(d2 + e2*x)^FracPart[p])/(b*c*(n + 1)*(1 + c*x)^FracPart[p]*(-1 + c*x)^FracPart[p]), Int[(f*x)^(m - 1)*(-1 + c^2*x^2)^(p - 1/2)*(a + b*ArcCosh[c*x])^(n + 1), x], x] /; FreeQ[{a, b, c, d1, e1, d2, e2, f, m, p}, x] && EqQ[e1 - c*d1, 0] && EqQ[e2 + c*d2, 0] && LtQ[n, -1] && EqQ[m + 2*p + 1, 0] && IntegerQ[p - 1/2]
  5773. Int[Times[Power[Plus[Optional[Pattern[a, Blank[]]], Times[ArcCosh[Times[Optional[Pattern[c, Blank[]]], Pattern[x, Blank[]]]], Optional[Pattern[b, Blank[]]]]], Pattern[n, Blank[]]], Power[Times[Optional[Pattern[f, Blank[]]], Pattern[x, Blank[]]], Optional[Pattern[m, Blank[]]]], Power[Plus[Pattern[d1, Blank[]], Times[Optional[Pattern[e1, Blank[]]], Pattern[x, Blank[]]]], Optional[Pattern[p, Blank[]]]], Power[Plus[Pattern[d2, Blank[]], Times[Optional[Pattern[e2, Blank[]]], Pattern[x, Blank[]]]], Optional[Pattern[p, Blank[]]]]], Pattern[x, Blank[Symbol]]] := Simp[((f*x)^m*Sqrt[1 + c*x]*Sqrt[-1 + c*x]*(d1 + e1*x)^p*(d2 + e2*x)^p*(a + b*ArcCosh[c*x])^(n + 1))/(b*c*(n + 1)), x] + Dist[(f*m*(-(d1*d2))^IntPart[p]*(d1 + e1*x)^FracPart[p]*(d2 + e2*x)^FracPart[p])/(b*c*(n + 1)*(1 + c*x)^FracPart[p]*(-1 + c*x)^FracPart[p]), Int[(f*x)^(m - 1)*(1 + c*x)^(p - 1/2)*(-1 + c*x)^(p - 1/2)*(a + b*ArcCosh[c*x])^(n + 1), x], x] /; FreeQ[{a, b, c, d1, e1, d2, e2, f, m, p}, x] && EqQ[e1 - c*d1, 0] && EqQ[e2 + c*d2, 0] && LtQ[n, -1] && EqQ[m + 2*p + 1, 0]
  5774. Int[Times[Power[Plus[Optional[Pattern[a, Blank[]]], Times[ArcSinh[Times[Optional[Pattern[c, Blank[]]], Pattern[x, Blank[]]]], Optional[Pattern[b, Blank[]]]]], Pattern[n, Blank[]]], Power[Times[Optional[Pattern[f, Blank[]]], Pattern[x, Blank[]]], Optional[Pattern[m, Blank[]]]], Power[Plus[Pattern[d, Blank[]], Times[Optional[Pattern[e, Blank[]]], Power[Pattern[x, Blank[]], 2]]], Rational[-1, 2]]], Pattern[x, Blank[Symbol]]] := Simp[((f*x)^m*(a + b*ArcSinh[c*x])^(n + 1))/(b*c*Sqrt[d]*(n + 1)), x] - Dist[(f*m)/(b*c*Sqrt[d]*(n + 1)), Int[(f*x)^(m - 1)*(a + b*ArcSinh[c*x])^(n + 1), x], x] /; FreeQ[{a, b, c, d, e, f, m}, x] && EqQ[e, c^2*d] && LtQ[n, -1] && GtQ[d, 0]
  5775. Int[Times[Power[Plus[Optional[Pattern[a, Blank[]]], Times[ArcCosh[Times[Optional[Pattern[c, Blank[]]], Pattern[x, Blank[]]]], Optional[Pattern[b, Blank[]]]]], Pattern[n, Blank[]]], Power[Times[Optional[Pattern[f, Blank[]]], Pattern[x, Blank[]]], Optional[Pattern[m, Blank[]]]], Power[Plus[Pattern[d1, Blank[]], Times[Optional[Pattern[e1, Blank[]]], Pattern[x, Blank[]]]], Rational[-1, 2]], Power[Plus[Pattern[d2, Blank[]], Times[Optional[Pattern[e2, Blank[]]], Pattern[x, Blank[]]]], Rational[-1, 2]]], Pattern[x, Blank[Symbol]]] := Simp[((f*x)^m*(a + b*ArcCosh[c*x])^(n + 1))/(b*c*Sqrt[-(d1*d2)]*(n + 1)), x] - Dist[(f*m)/(b*c*Sqrt[-(d1*d2)]*(n + 1)), Int[(f*x)^(m - 1)*(a + b*ArcCosh[c*x])^(n + 1), x], x] /; FreeQ[{a, b, c, d1, e1, d2, e2, f, m}, x] && EqQ[e1 - c*d1, 0] && EqQ[e2 + c*d2, 0] && LtQ[n, -1] && GtQ[d1, 0] && LtQ[d2, 0]
  5776. Int[Times[Power[Plus[Optional[Pattern[a, Blank[]]], Times[ArcCosh[Times[Optional[Pattern[c, Blank[]]], Pattern[x, Blank[]]]], Optional[Pattern[b, Blank[]]]]], Pattern[n, Blank[]]], Power[Times[Optional[Pattern[f, Blank[]]], Pattern[x, Blank[]]], Optional[Pattern[m, Blank[]]]], Power[Plus[Pattern[d, Blank[]], Times[Optional[Pattern[e, Blank[]]], Power[Pattern[x, Blank[]], 2]]], Optional[Pattern[p, Blank[]]]]], Pattern[x, Blank[Symbol]]] := Simp[((f*x)^m*Sqrt[1 + c*x]*Sqrt[-1 + c*x]*(d + e*x^2)^p*(a + b*ArcCosh[c*x])^(n + 1))/(b*c*(n + 1)), x] + (Dist[(f*m*(-d)^p)/(b*c*(n + 1)), Int[(f*x)^(m - 1)*(1 + c*x)^(p - 1/2)*(-1 + c*x)^(p - 1/2)*(a + b*ArcCosh[c*x])^(n + 1), x], x] - Dist[(c*(-d)^p*(m + 2*p + 1))/(b*f*(n + 1)), Int[(f*x)^(m + 1)*(1 + c*x)^(p - 1/2)*(-1 + c*x)^(p - 1/2)*(a + b*ArcCosh[c*x])^(n + 1), x], x]) /; FreeQ[{a, b, c, d, e, f}, x] && EqQ[c^2*d + e, 0] && LtQ[n, -1] && IGtQ[m, -3] && IGtQ[p, 0]
  5777. Int[Times[Power[Plus[Optional[Pattern[a, Blank[]]], Times[ArcSinh[Times[Optional[Pattern[c, Blank[]]], Pattern[x, Blank[]]]], Optional[Pattern[b, Blank[]]]]], Pattern[n, Blank[]]], Power[Times[Optional[Pattern[f, Blank[]]], Pattern[x, Blank[]]], Optional[Pattern[m, Blank[]]]], Power[Plus[Pattern[d, Blank[]], Times[Optional[Pattern[e, Blank[]]], Power[Pattern[x, Blank[]], 2]]], Optional[Pattern[p, Blank[]]]]], Pattern[x, Blank[Symbol]]] := Simp[((f*x)^m*Sqrt[1 + c^2*x^2]*(d + e*x^2)^p*(a + b*ArcSinh[c*x])^(n + 1))/(b*c*(n + 1)), x] + (-Dist[(f*m*d^IntPart[p]*(d + e*x^2)^FracPart[p])/(b*c*(n + 1)*(1 + c^2*x^2)^FracPart[p]), Int[(f*x)^(m - 1)*(1 + c^2*x^2)^(p - 1/2)*(a + b*ArcSinh[c*x])^(n + 1), x], x] - Dist[(c*(m + 2*p + 1)*d^IntPart[p]*(d + e*x^2)^FracPart[p])/(b*f*(n + 1)*(1 + c^2*x^2)^FracPart[p]), Int[(f*x)^(m + 1)*(1 + c^2*x^2)^(p - 1/2)*(a + b*ArcSinh[c*x])^(n + 1), x], x]) /; FreeQ[{a, b, c, d, e, f}, x] && EqQ[e, c^2*d] && LtQ[n, -1] && IGtQ[m, -3] && IGtQ[2*p, 0]
  5778. Int[Times[Power[Plus[Optional[Pattern[a, Blank[]]], Times[ArcCosh[Times[Optional[Pattern[c, Blank[]]], Pattern[x, Blank[]]]], Optional[Pattern[b, Blank[]]]]], Pattern[n, Blank[]]], Power[Times[Optional[Pattern[f, Blank[]]], Pattern[x, Blank[]]], Optional[Pattern[m, Blank[]]]], Power[Plus[Pattern[d1, Blank[]], Times[Optional[Pattern[e1, Blank[]]], Pattern[x, Blank[]]]], Optional[Pattern[p, Blank[]]]], Power[Plus[Pattern[d2, Blank[]], Times[Optional[Pattern[e2, Blank[]]], Pattern[x, Blank[]]]], Optional[Pattern[p, Blank[]]]]], Pattern[x, Blank[Symbol]]] := Simp[((f*x)^m*Sqrt[1 + c*x]*Sqrt[-1 + c*x]*(d1 + e1*x)^p*(d2 + e2*x)^p*(a + b*ArcCosh[c*x])^(n + 1))/(b*c*(n + 1)), x] + (Dist[(f*m*(-(d1*d2))^IntPart[p]*(d1 + e1*x)^FracPart[p]*(d2 + e2*x)^FracPart[p])/(b*c*(n + 1)*(1 + c*x)^FracPart[p]*(-1 + c*x)^FracPart[p]), Int[(f*x)^(m - 1)*(-1 + c^2*x^2)^(p - 1/2)*(a + b*ArcCosh[c*x])^(n + 1), x], x] - Dist[(c*(m + 2*p + 1)*(-(d1*d2))^IntPart[p]*(d1 + e1*x)^FracPart[p]*(d2 + e2*x)^FracPart[p])/(b*f*(n + 1)*(1 + c*x)^FracPart[p]*(-1 + c*x)^FracPart[p]), Int[(f*x)^(m + 1)*(-1 + c^2*x^2)^(p - 1/2)*(a + b*ArcCosh[c*x])^(n + 1), x], x]) /; FreeQ[{a, b, c, d1, e1, d2, e2, f}, x] && EqQ[e1 - c*d1, 0] && EqQ[e2 + c*d2, 0] && LtQ[n, -1] && IGtQ[m, -3] && IGtQ[p + 1/2, 0]
  5779. Int[Times[Power[Plus[Optional[Pattern[a, Blank[]]], Times[ArcSinh[Times[Optional[Pattern[c, Blank[]]], Pattern[x, Blank[]]]], Optional[Pattern[b, Blank[]]]]], Optional[Pattern[n, Blank[]]]], Power[Pattern[x, Blank[]], Optional[Pattern[m, Blank[]]]], Power[Plus[Pattern[d, Blank[]], Times[Optional[Pattern[e, Blank[]]], Power[Pattern[x, Blank[]], 2]]], Optional[Pattern[p, Blank[]]]]], Pattern[x, Blank[Symbol]]] := Dist[d^p/c^(m + 1), Subst[Int[(a + b*x)^n*Sinh[x]^m*Cosh[x]^(2*p + 1), x], x, ArcSinh[c*x]], x] /; FreeQ[{a, b, c, d, e, n}, x] && EqQ[e, c^2*d] && IntegerQ[2*p] && GtQ[p, -1] && IGtQ[m, 0] && (IntegerQ[p] || GtQ[d, 0])
  5780. Int[Times[Power[Plus[Optional[Pattern[a, Blank[]]], Times[ArcCosh[Times[Optional[Pattern[c, Blank[]]], Pattern[x, Blank[]]]], Optional[Pattern[b, Blank[]]]]], Optional[Pattern[n, Blank[]]]], Power[Pattern[x, Blank[]], Optional[Pattern[m, Blank[]]]], Power[Plus[Pattern[d, Blank[]], Times[Optional[Pattern[e, Blank[]]], Power[Pattern[x, Blank[]], 2]]], Optional[Pattern[p, Blank[]]]]], Pattern[x, Blank[Symbol]]] := Dist[(-d)^p/c^(m + 1), Subst[Int[(a + b*x)^n*Cosh[x]^m*Sinh[x]^(2*p + 1), x], x, ArcCosh[c*x]], x] /; FreeQ[{a, b, c, d, e, n}, x] && EqQ[c^2*d + e, 0] && IGtQ[p, 0] && IGtQ[m, 0]
  5781. Int[Times[Power[Plus[Optional[Pattern[a, Blank[]]], Times[ArcCosh[Times[Optional[Pattern[c, Blank[]]], Pattern[x, Blank[]]]], Optional[Pattern[b, Blank[]]]]], Optional[Pattern[n, Blank[]]]], Power[Pattern[x, Blank[]], Optional[Pattern[m, Blank[]]]], Power[Plus[Pattern[d1, Blank[]], Times[Optional[Pattern[e1, Blank[]]], Pattern[x, Blank[]]]], Optional[Pattern[p, Blank[]]]], Power[Plus[Pattern[d2, Blank[]], Times[Optional[Pattern[e2, Blank[]]], Pattern[x, Blank[]]]], Optional[Pattern[p, Blank[]]]]], Pattern[x, Blank[Symbol]]] := Dist[(-(d1*d2))^p/c^(m + 1), Subst[Int[(a + b*x)^n*Cosh[x]^m*Sinh[x]^(2*p + 1), x], x, ArcCosh[c*x]], x] /; FreeQ[{a, b, c, d1, e1, d2, e2, n}, x] && EqQ[e1 - c*d1, 0] && EqQ[e2 + c*d2, 0] && IntegerQ[p + 1/2] && GtQ[p, -1] && IGtQ[m, 0] && (GtQ[d1, 0] && LtQ[d2, 0])
  5782. Int[Times[Power[Plus[Optional[Pattern[a, Blank[]]], Times[ArcSinh[Times[Optional[Pattern[c, Blank[]]], Pattern[x, Blank[]]]], Optional[Pattern[b, Blank[]]]]], Pattern[n, Blank[]]], Power[Pattern[x, Blank[]], Optional[Pattern[m, Blank[]]]], Power[Plus[Pattern[d, Blank[]], Times[Optional[Pattern[e, Blank[]]], Power[Pattern[x, Blank[]], 2]]], Pattern[p, Blank[]]]], Pattern[x, Blank[Symbol]]] := Dist[(d^IntPart[p]*(d + e*x^2)^FracPart[p])/(1 + c^2*x^2)^FracPart[p], Int[x^m*(1 + c^2*x^2)^p*(a + b*ArcSinh[c*x])^n, x], x] /; FreeQ[{a, b, c, d, e, n}, x] && EqQ[e, c^2*d] && IntegerQ[2*p] && GtQ[p, -1] && IGtQ[m, 0] && !(IntegerQ[p] || GtQ[d, 0])
  5783. Int[Times[Power[Plus[Optional[Pattern[a, Blank[]]], Times[ArcCosh[Times[Optional[Pattern[c, Blank[]]], Pattern[x, Blank[]]]], Optional[Pattern[b, Blank[]]]]], Optional[Pattern[n, Blank[]]]], Power[Pattern[x, Blank[]], Optional[Pattern[m, Blank[]]]], Power[Plus[Pattern[d1, Blank[]], Times[Optional[Pattern[e1, Blank[]]], Pattern[x, Blank[]]]], Optional[Pattern[p, Blank[]]]], Power[Plus[Pattern[d2, Blank[]], Times[Optional[Pattern[e2, Blank[]]], Pattern[x, Blank[]]]], Pattern[p, Blank[]]]], Pattern[x, Blank[Symbol]]] := Dist[((-(d1*d2))^IntPart[p]*(d1 + e1*x)^FracPart[p]*(d2 + e2*x)^FracPart[p])/((1 + c*x)^FracPart[p]*(-1 + c*x)^FracPart[p]), Int[x^m*(1 + c*x)^p*(-1 + c*x)^p*(a + b*ArcCosh[c*x])^n, x], x] /; FreeQ[{a, b, c, d1, e1, d2, e2, n}, x] && EqQ[e1 - c*d1, 0] && EqQ[e2 + c*d2, 0] && IntegerQ[2*p] && GtQ[p, -1] && IGtQ[m, 0] && !(IntegerQ[p] || (GtQ[d1, 0] && LtQ[d2, 0]))
  5784. Int[Times[Power[Plus[Optional[Pattern[a, Blank[]]], Times[ArcSinh[Times[Optional[Pattern[c, Blank[]]], Pattern[x, Blank[]]]], Optional[Pattern[b, Blank[]]]]], Optional[Pattern[n, Blank[]]]], Power[Times[Optional[Pattern[f, Blank[]]], Pattern[x, Blank[]]], Pattern[m, Blank[]]], Power[Plus[Pattern[d, Blank[]], Times[Optional[Pattern[e, Blank[]]], Power[Pattern[x, Blank[]], 2]]], Pattern[p, Blank[]]]], Pattern[x, Blank[Symbol]]] := Int[ExpandIntegrand[(a + b*ArcSinh[c*x])^n/Sqrt[d + e*x^2], (f*x)^m*(d + e*x^2)^(p + 1/2), x], x] /; FreeQ[{a, b, c, d, e, f, m, n}, x] && EqQ[e, c^2*d] && GtQ[d, 0] && IGtQ[p + 1/2, 0] && !IGtQ[(m + 1)/2, 0] && (EqQ[m, -1] || EqQ[m, -2])
  5785. Int[Times[Power[Plus[Optional[Pattern[a, Blank[]]], Times[ArcCosh[Times[Optional[Pattern[c, Blank[]]], Pattern[x, Blank[]]]], Optional[Pattern[b, Blank[]]]]], Optional[Pattern[n, Blank[]]]], Power[Times[Optional[Pattern[f, Blank[]]], Pattern[x, Blank[]]], Pattern[m, Blank[]]], Power[Plus[Pattern[d1, Blank[]], Times[Optional[Pattern[e1, Blank[]]], Pattern[x, Blank[]]]], Pattern[p, Blank[]]], Power[Plus[Pattern[d2, Blank[]], Times[Optional[Pattern[e2, Blank[]]], Pattern[x, Blank[]]]], Pattern[p, Blank[]]]], Pattern[x, Blank[Symbol]]] := Int[ExpandIntegrand[(a + b*ArcCosh[c*x])^n/(Sqrt[d1 + e1*x]*Sqrt[d2 + e2*x]), (f*x)^m*(d1 + e1*x)^(p + 1/2)*(d2 + e2*x)^(p + 1/2), x], x] /; FreeQ[{a, b, c, d1, e1, d2, e2, f, m, n}, x] && EqQ[e1 - c*d1, 0] && EqQ[e2 + c*d2, 0] && GtQ[d1, 0] && LtQ[d2, 0] && IGtQ[p + 1/2, 0] && !IGtQ[(m + 1)/2, 0] && (EqQ[m, -1] || EqQ[m, -2])
  5786. Int[Times[Plus[Optional[Pattern[a, Blank[]]], Times[ArcCosh[Times[Optional[Pattern[c, Blank[]]], Pattern[x, Blank[]]]], Optional[Pattern[b, Blank[]]]]], Power[Times[Optional[Pattern[f, Blank[]]], Pattern[x, Blank[]]], Optional[Pattern[m, Blank[]]]], Plus[Pattern[d, Blank[]], Times[Optional[Pattern[e, Blank[]]], Power[Pattern[x, Blank[]], 2]]]], Pattern[x, Blank[Symbol]]] := Simp[(d*(f*x)^(m + 1)*(a + b*ArcCosh[c*x]))/(f*(m + 1)), x] + (-Dist[(b*c)/(f*(m + 1)*(m + 3)), Int[((f*x)^(m + 1)*(d*(m + 3) + e*(m + 1)*x^2))/(Sqrt[1 + c*x]*Sqrt[-1 + c*x]), x], x] + Simp[(e*(f*x)^(m + 3)*(a + b*ArcCosh[c*x]))/(f^3*(m + 3)), x]) /; FreeQ[{a, b, c, d, e, f, m}, x] && NeQ[c^2*d + e, 0] && NeQ[m, -1] && NeQ[m, -3]
  5787. Int[Times[Plus[Optional[Pattern[a, Blank[]]], Times[ArcSinh[Times[Optional[Pattern[c, Blank[]]], Pattern[x, Blank[]]]], Optional[Pattern[b, Blank[]]]]], Pattern[x, Blank[]], Power[Plus[Pattern[d, Blank[]], Times[Optional[Pattern[e, Blank[]]], Power[Pattern[x, Blank[]], 2]]], Optional[Pattern[p, Blank[]]]]], Pattern[x, Blank[Symbol]]] := Simp[((d + e*x^2)^(p + 1)*(a + b*ArcSinh[c*x]))/(2*e*(p + 1)), x] - Dist[(b*c)/(2*e*(p + 1)), Int[(d + e*x^2)^(p + 1)/Sqrt[1 + c^2*x^2], x], x] /; FreeQ[{a, b, c, d, e, p}, x] && NeQ[e, c^2*d] && NeQ[p, -1]
  5788. Int[Times[Plus[Optional[Pattern[a, Blank[]]], Times[ArcCosh[Times[Optional[Pattern[c, Blank[]]], Pattern[x, Blank[]]]], Optional[Pattern[b, Blank[]]]]], Pattern[x, Blank[]], Power[Plus[Pattern[d, Blank[]], Times[Optional[Pattern[e, Blank[]]], Power[Pattern[x, Blank[]], 2]]], Optional[Pattern[p, Blank[]]]]], Pattern[x, Blank[Symbol]]] := Simp[((d + e*x^2)^(p + 1)*(a + b*ArcCosh[c*x]))/(2*e*(p + 1)), x] - Dist[(b*c)/(2*e*(p + 1)), Int[(d + e*x^2)^(p + 1)/(Sqrt[1 + c*x]*Sqrt[-1 + c*x]), x], x] /; FreeQ[{a, b, c, d, e, p}, x] && NeQ[c^2*d + e, 0] && NeQ[p, -1]
  5789. Int[Times[Plus[Optional[Pattern[a, Blank[]]], Times[ArcSinh[Times[Optional[Pattern[c, Blank[]]], Pattern[x, Blank[]]]], Optional[Pattern[b, Blank[]]]]], Power[Times[Optional[Pattern[f, Blank[]]], Pattern[x, Blank[]]], Optional[Pattern[m, Blank[]]]], Power[Plus[Pattern[d, Blank[]], Times[Optional[Pattern[e, Blank[]]], Power[Pattern[x, Blank[]], 2]]], Optional[Pattern[p, Blank[]]]]], Pattern[x, Blank[Symbol]]] := With[{u = IntHide[(f*x)^m*(d + e*x^2)^p, x]}, Dist[a + b*ArcSinh[c*x], u, x] - Dist[b*c, Int[SimplifyIntegrand[u/Sqrt[1 + c^2*x^2], x], x], x]] /; FreeQ[{a, b, c, d, e, f, m}, x] && NeQ[e, c^2*d] && IntegerQ[p] && (GtQ[p, 0] || (IGtQ[(m - 1)/2, 0] && LeQ[m + p, 0]))
  5790. Int[Times[Plus[Optional[Pattern[a, Blank[]]], Times[ArcCosh[Times[Optional[Pattern[c, Blank[]]], Pattern[x, Blank[]]]], Optional[Pattern[b, Blank[]]]]], Power[Times[Optional[Pattern[f, Blank[]]], Pattern[x, Blank[]]], Optional[Pattern[m, Blank[]]]], Power[Plus[Pattern[d, Blank[]], Times[Optional[Pattern[e, Blank[]]], Power[Pattern[x, Blank[]], 2]]], Optional[Pattern[p, Blank[]]]]], Pattern[x, Blank[Symbol]]] := With[{u = IntHide[(f*x)^m*(d + e*x^2)^p, x]}, Dist[a + b*ArcCosh[c*x], u, x] - Dist[b*c, Int[SimplifyIntegrand[u/(Sqrt[1 + c*x]*Sqrt[-1 + c*x]), x], x], x]] /; FreeQ[{a, b, c, d, e, f, m}, x] && NeQ[c^2*d + e, 0] && IntegerQ[p] && (GtQ[p, 0] || (IGtQ[(m - 1)/2, 0] && LeQ[m + p, 0]))
  5791. Int[Times[Power[Plus[Optional[Pattern[a, Blank[]]], Times[ArcSinh[Times[Optional[Pattern[c, Blank[]]], Pattern[x, Blank[]]]], Optional[Pattern[b, Blank[]]]]], Optional[Pattern[n, Blank[]]]], Power[Times[Optional[Pattern[f, Blank[]]], Pattern[x, Blank[]]], Optional[Pattern[m, Blank[]]]], Power[Plus[Pattern[d, Blank[]], Times[Optional[Pattern[e, Blank[]]], Power[Pattern[x, Blank[]], 2]]], Optional[Pattern[p, Blank[]]]]], Pattern[x, Blank[Symbol]]] := Int[ExpandIntegrand[(a + b*ArcSinh[c*x])^n, (f*x)^m*(d + e*x^2)^p, x], x] /; FreeQ[{a, b, c, d, e, f}, x] && NeQ[e, c^2*d] && IGtQ[n, 0] && IntegerQ[p] && IntegerQ[m]
  5792. Int[Times[Power[Plus[Optional[Pattern[a, Blank[]]], Times[ArcCosh[Times[Optional[Pattern[c, Blank[]]], Pattern[x, Blank[]]]], Optional[Pattern[b, Blank[]]]]], Optional[Pattern[n, Blank[]]]], Power[Times[Optional[Pattern[f, Blank[]]], Pattern[x, Blank[]]], Optional[Pattern[m, Blank[]]]], Power[Plus[Pattern[d, Blank[]], Times[Optional[Pattern[e, Blank[]]], Power[Pattern[x, Blank[]], 2]]], Optional[Pattern[p, Blank[]]]]], Pattern[x, Blank[Symbol]]] := Int[ExpandIntegrand[(a + b*ArcCosh[c*x])^n, (f*x)^m*(d + e*x^2)^p, x], x] /; FreeQ[{a, b, c, d, e, f}, x] && NeQ[c^2*d + e, 0] && IGtQ[n, 0] && IntegerQ[p] && IntegerQ[m]
  5793. Int[Times[Power[Plus[Optional[Pattern[a, Blank[]]], Times[ArcSinh[Times[Optional[Pattern[c, Blank[]]], Pattern[x, Blank[]]]], Optional[Pattern[b, Blank[]]]]], Optional[Pattern[n, Blank[]]]], Power[Times[Optional[Pattern[f, Blank[]]], Pattern[x, Blank[]]], Optional[Pattern[m, Blank[]]]], Power[Plus[Pattern[d, Blank[]], Times[Optional[Pattern[e, Blank[]]], Power[Pattern[x, Blank[]], 2]]], Optional[Pattern[p, Blank[]]]]], Pattern[x, Blank[Symbol]]] := Unintegrable[(f*x)^m*(d + e*x^2)^p*(a + b*ArcSinh[c*x])^n, x] /; FreeQ[{a, b, c, d, e, f, m, n, p}, x]
  5794. Int[Times[Power[Plus[Optional[Pattern[a, Blank[]]], Times[ArcCosh[Times[Optional[Pattern[c, Blank[]]], Pattern[x, Blank[]]]], Optional[Pattern[b, Blank[]]]]], Optional[Pattern[n, Blank[]]]], Power[Times[Optional[Pattern[f, Blank[]]], Pattern[x, Blank[]]], Optional[Pattern[m, Blank[]]]], Power[Plus[Pattern[d, Blank[]], Times[Optional[Pattern[e, Blank[]]], Power[Pattern[x, Blank[]], 2]]], Optional[Pattern[p, Blank[]]]]], Pattern[x, Blank[Symbol]]] := Unintegrable[(f*x)^m*(d + e*x^2)^p*(a + b*ArcCosh[c*x])^n, x] /; FreeQ[{a, b, c, d, e, f, m, n, p}, x] && IntegerQ[p]
  5795. Int[Times[Power[Plus[Optional[Pattern[a, Blank[]]], Times[ArcCosh[Times[Optional[Pattern[c, Blank[]]], Pattern[x, Blank[]]]], Optional[Pattern[b, Blank[]]]]], Optional[Pattern[n, Blank[]]]], Power[Times[Optional[Pattern[f, Blank[]]], Pattern[x, Blank[]]], Optional[Pattern[m, Blank[]]]], Power[Plus[Pattern[d1, Blank[]], Times[Optional[Pattern[e1, Blank[]]], Pattern[x, Blank[]]]], Optional[Pattern[p, Blank[]]]], Power[Plus[Pattern[d2, Blank[]], Times[Optional[Pattern[e2, Blank[]]], Pattern[x, Blank[]]]], Optional[Pattern[p, Blank[]]]]], Pattern[x, Blank[Symbol]]] := Unintegrable[(f*x)^m*(d1 + e1*x)^p*(d2 + e2*x)^p*(a + b*ArcCosh[c*x])^n, x] /; FreeQ[{a, b, c, d1, e1, d2, e2, f, m, n, p}, x]
  5796. Int[Times[Power[Plus[Optional[Pattern[a, Blank[]]], Times[ArcSinh[Times[Optional[Pattern[c, Blank[]]], Pattern[x, Blank[]]]], Optional[Pattern[b, Blank[]]]]], Optional[Pattern[n, Blank[]]]], Power[Times[Optional[Pattern[h, Blank[]]], Pattern[x, Blank[]]], Optional[Pattern[m, Blank[]]]], Power[Plus[Pattern[d, Blank[]], Times[Optional[Pattern[e, Blank[]]], Pattern[x, Blank[]]]], Pattern[p, Blank[]]], Power[Plus[Pattern[f, Blank[]], Times[Optional[Pattern[g, Blank[]]], Pattern[x, Blank[]]]], Pattern[q, Blank[]]]], Pattern[x, Blank[Symbol]]] := Dist[(-((d^2*g)/e))^q, Int[(h*x)^m*(d + e*x)^(p - q)*(1 + c^2*x^2)^q*(a + b*ArcSinh[c*x])^n, x], x] /; FreeQ[{a, b, c, d, e, f, g, h, m, n}, x] && EqQ[e*f + d*g, 0] && EqQ[c^2*d^2 + e^2, 0] && HalfIntegerQ[p, q] && GeQ[p - q, 0] && GtQ[d, 0] && LtQ[g/e, 0]
  5797. Int[Times[Power[Plus[Optional[Pattern[a, Blank[]]], Times[ArcSinh[Times[Optional[Pattern[c, Blank[]]], Pattern[x, Blank[]]]], Optional[Pattern[b, Blank[]]]]], Optional[Pattern[n, Blank[]]]], Power[Times[Optional[Pattern[h, Blank[]]], Pattern[x, Blank[]]], Optional[Pattern[m, Blank[]]]], Power[Plus[Pattern[d, Blank[]], Times[Optional[Pattern[e, Blank[]]], Pattern[x, Blank[]]]], Pattern[p, Blank[]]], Power[Plus[Pattern[f, Blank[]], Times[Optional[Pattern[g, Blank[]]], Pattern[x, Blank[]]]], Pattern[q, Blank[]]]], Pattern[x, Blank[Symbol]]] := Dist[((-((d^2*g)/e))^IntPart[q]*(d + e*x)^FracPart[q]*(f + g*x)^FracPart[q])/(1 + c^2*x^2)^FracPart[q], Int[(h*x)^m*(d + e*x)^(p - q)*(1 + c^2*x^2)^q*(a + b*ArcSinh[c*x])^n, x], x] /; FreeQ[{a, b, c, d, e, f, g, h, m, n}, x] && EqQ[e*f + d*g, 0] && EqQ[c^2*d^2 + e^2, 0] && HalfIntegerQ[p, q] && GeQ[p - q, 0]
  5798. Int[Times[Power[Plus[Optional[Pattern[a, Blank[]]], Times[ArcCosh[Times[Optional[Pattern[c, Blank[]]], Pattern[x, Blank[]]]], Optional[Pattern[b, Blank[]]]]], Optional[Pattern[n, Blank[]]]], Power[Times[Optional[Pattern[f, Blank[]]], Pattern[x, Blank[]]], Optional[Pattern[m, Blank[]]]], Power[Plus[Pattern[d, Blank[]], Times[Optional[Pattern[e, Blank[]]], Power[Pattern[x, Blank[]], 2]]], Pattern[p, Blank[]]]], Pattern[x, Blank[Symbol]]] := Dist[((-d)^IntPart[p]*(d + e*x^2)^FracPart[p])/((1 + c*x)^FracPart[p]*(-1 + c*x)^FracPart[p]), Int[(f*x)^m*(1 + c*x)^p*(-1 + c*x)^p*(a + b*ArcCosh[c*x])^n, x], x] /; FreeQ[{a, b, c, d, e, f, m, n, p}, x] && EqQ[c^2*d + e, 0] && !IntegerQ[p]
  5799. Int[Times[Power[Plus[Optional[Pattern[a, Blank[]]], Times[ArcSinh[Times[Optional[Pattern[c, Blank[]]], Pattern[x, Blank[]]]], Optional[Pattern[b, Blank[]]]]], Optional[Pattern[n, Blank[]]]], Power[Plus[Optional[Pattern[d, Blank[]]], Times[Optional[Pattern[e, Blank[]]], Pattern[x, Blank[]]]], -1]], Pattern[x, Blank[Symbol]]] := Subst[Int[((a + b*x)^n*Cosh[x])/(c*d + e*Sinh[x]), x], x, ArcSinh[c*x]] /; FreeQ[{a, b, c, d, e}, x] && IGtQ[n, 0]
  5800. Int[Times[Power[Plus[Optional[Pattern[a, Blank[]]], Times[ArcCosh[Times[Optional[Pattern[c, Blank[]]], Pattern[x, Blank[]]]], Optional[Pattern[b, Blank[]]]]], Optional[Pattern[n, Blank[]]]], Power[Plus[Optional[Pattern[d, Blank[]]], Times[Optional[Pattern[e, Blank[]]], Pattern[x, Blank[]]]], -1]], Pattern[x, Blank[Symbol]]] := Subst[Int[((a + b*x)^n*Sinh[x])/(c*d + e*Cosh[x]), x], x, ArcCosh[c*x]] /; FreeQ[{a, b, c, d, e}, x] && IGtQ[n, 0]
  5801. Int[Times[Power[Plus[Optional[Pattern[a, Blank[]]], Times[ArcSinh[Times[Optional[Pattern[c, Blank[]]], Pattern[x, Blank[]]]], Optional[Pattern[b, Blank[]]]]], Optional[Pattern[n, Blank[]]]], Power[Plus[Optional[Pattern[d, Blank[]]], Times[Optional[Pattern[e, Blank[]]], Pattern[x, Blank[]]]], Optional[Pattern[m, Blank[]]]]], Pattern[x, Blank[Symbol]]] := Simp[((d + e*x)^(m + 1)*(a + b*ArcSinh[c*x])^n)/(e*(m + 1)), x] - Dist[(b*c*n)/(e*(m + 1)), Int[((d + e*x)^(m + 1)*(a + b*ArcSinh[c*x])^(n - 1))/Sqrt[1 + c^2*x^2], x], x] /; FreeQ[{a, b, c, d, e, m}, x] && IGtQ[n, 0] && NeQ[m, -1]
  5802. Int[Times[Power[Plus[Optional[Pattern[a, Blank[]]], Times[ArcCosh[Times[Optional[Pattern[c, Blank[]]], Pattern[x, Blank[]]]], Optional[Pattern[b, Blank[]]]]], Optional[Pattern[n, Blank[]]]], Power[Plus[Optional[Pattern[d, Blank[]]], Times[Optional[Pattern[e, Blank[]]], Pattern[x, Blank[]]]], Optional[Pattern[m, Blank[]]]]], Pattern[x, Blank[Symbol]]] := Simp[((d + e*x)^(m + 1)*(a + b*ArcCosh[c*x])^n)/(e*(m + 1)), x] - Dist[(b*c*n)/(e*(m + 1)), Int[((d + e*x)^(m + 1)*(a + b*ArcCosh[c*x])^(n - 1))/(Sqrt[-1 + c*x]*Sqrt[1 + c*x]), x], x] /; FreeQ[{a, b, c, d, e, m}, x] && IGtQ[n, 0] && NeQ[m, -1]
  5803. Int[Times[Power[Plus[Optional[Pattern[a, Blank[]]], Times[ArcSinh[Times[Optional[Pattern[c, Blank[]]], Pattern[x, Blank[]]]], Optional[Pattern[b, Blank[]]]]], Pattern[n, Blank[]]], Power[Plus[Pattern[d, Blank[]], Times[Optional[Pattern[e, Blank[]]], Pattern[x, Blank[]]]], Optional[Pattern[m, Blank[]]]]], Pattern[x, Blank[Symbol]]] := Int[ExpandIntegrand[(d + e*x)^m*(a + b*ArcSinh[c*x])^n, x], x] /; FreeQ[{a, b, c, d, e}, x] && IGtQ[m, 0] && LtQ[n, -1]
  5804. Int[Times[Power[Plus[Optional[Pattern[a, Blank[]]], Times[ArcCosh[Times[Optional[Pattern[c, Blank[]]], Pattern[x, Blank[]]]], Optional[Pattern[b, Blank[]]]]], Pattern[n, Blank[]]], Power[Plus[Pattern[d, Blank[]], Times[Optional[Pattern[e, Blank[]]], Pattern[x, Blank[]]]], Optional[Pattern[m, Blank[]]]]], Pattern[x, Blank[Symbol]]] := Int[ExpandIntegrand[(d + e*x)^m*(a + b*ArcCosh[c*x])^n, x], x] /; FreeQ[{a, b, c, d, e}, x] && IGtQ[m, 0] && LtQ[n, -1]
  5805. Int[Times[Power[Plus[Optional[Pattern[a, Blank[]]], Times[ArcSinh[Times[Optional[Pattern[c, Blank[]]], Pattern[x, Blank[]]]], Optional[Pattern[b, Blank[]]]]], Optional[Pattern[n, Blank[]]]], Power[Plus[Optional[Pattern[d, Blank[]]], Times[Optional[Pattern[e, Blank[]]], Pattern[x, Blank[]]]], Optional[Pattern[m, Blank[]]]]], Pattern[x, Blank[Symbol]]] := Dist[1/c^(m + 1), Subst[Int[(a + b*x)^n*Cosh[x]*(c*d + e*Sinh[x])^m, x], x, ArcSinh[c*x]], x] /; FreeQ[{a, b, c, d, e, n}, x] && IGtQ[m, 0]
  5806. Int[Times[Power[Plus[Optional[Pattern[a, Blank[]]], Times[ArcCosh[Times[Optional[Pattern[c, Blank[]]], Pattern[x, Blank[]]]], Optional[Pattern[b, Blank[]]]]], Optional[Pattern[n, Blank[]]]], Power[Plus[Optional[Pattern[d, Blank[]]], Times[Optional[Pattern[e, Blank[]]], Pattern[x, Blank[]]]], Optional[Pattern[m, Blank[]]]]], Pattern[x, Blank[Symbol]]] := Dist[1/c^(m + 1), Subst[Int[(a + b*x)^n*(c*d + e*Cosh[x])^m*Sinh[x], x], x, ArcCosh[c*x]], x] /; FreeQ[{a, b, c, d, e, n}, x] && IGtQ[m, 0]
  5807. Int[Times[Plus[Optional[Pattern[a, Blank[]]], Times[ArcSinh[Times[Optional[Pattern[c, Blank[]]], Pattern[x, Blank[]]]], Optional[Pattern[b, Blank[]]]]], Pattern[Px, Blank[]]], Pattern[x, Blank[Symbol]]] := With[{u = IntHide[ExpandExpression[Px, x], x]}, Dist[a + b*ArcSinh[c*x], u, x] - Dist[b*c, Int[SimplifyIntegrand[u/Sqrt[1 + c^2*x^2], x], x], x]] /; FreeQ[{a, b, c}, x] && PolynomialQ[Px, x]
  5808. Int[Times[Plus[Optional[Pattern[a, Blank[]]], Times[ArcCosh[Times[Optional[Pattern[c, Blank[]]], Pattern[x, Blank[]]]], Optional[Pattern[b, Blank[]]]]], Pattern[Px, Blank[]]], Pattern[x, Blank[Symbol]]] := With[{u = IntHide[ExpandExpression[Px, x], x]}, Dist[a + b*ArcCosh[c*x], u, x] - Dist[(b*c*Sqrt[1 - c^2*x^2])/(Sqrt[-1 + c*x]*Sqrt[1 + c*x]), Int[SimplifyIntegrand[u/Sqrt[1 - c^2*x^2], x], x], x]] /; FreeQ[{a, b, c}, x] && PolynomialQ[Px, x]
  5809. Int[Times[Power[Plus[Optional[Pattern[a, Blank[]]], Times[ArcSinh[Times[Optional[Pattern[c, Blank[]]], Pattern[x, Blank[]]]], Optional[Pattern[b, Blank[]]]]], Pattern[n, Blank[]]], Pattern[Px, Blank[]]], Pattern[x, Blank[Symbol]]] := Int[ExpandIntegrand[Px*(a + b*ArcSinh[c*x])^n, x], x] /; FreeQ[{a, b, c, n}, x] && PolynomialQ[Px, x]
  5810. Int[Times[Power[Plus[Optional[Pattern[a, Blank[]]], Times[ArcCosh[Times[Optional[Pattern[c, Blank[]]], Pattern[x, Blank[]]]], Optional[Pattern[b, Blank[]]]]], Pattern[n, Blank[]]], Pattern[Px, Blank[]]], Pattern[x, Blank[Symbol]]] := Int[ExpandIntegrand[Px*(a + b*ArcCosh[c*x])^n, x], x] /; FreeQ[{a, b, c, n}, x] && PolynomialQ[Px, x]
  5811. Int[Times[Plus[Optional[Pattern[a, Blank[]]], Times[ArcSinh[Times[Optional[Pattern[c, Blank[]]], Pattern[x, Blank[]]]], Optional[Pattern[b, Blank[]]]]], Pattern[Px, Blank[]], Power[Plus[Optional[Pattern[d, Blank[]]], Times[Optional[Pattern[e, Blank[]]], Pattern[x, Blank[]]]], Optional[Pattern[m, Blank[]]]]], Pattern[x, Blank[Symbol]]] := With[{u = IntHide[Px*(d + e*x)^m, x]}, Dist[a + b*ArcSinh[c*x], u, x] - Dist[b*c, Int[SimplifyIntegrand[u/Sqrt[1 + c^2*x^2], x], x], x]] /; FreeQ[{a, b, c, d, e, m}, x] && PolynomialQ[Px, x]
  5812. Int[Times[Plus[Optional[Pattern[a, Blank[]]], Times[ArcCosh[Times[Optional[Pattern[c, Blank[]]], Pattern[x, Blank[]]]], Optional[Pattern[b, Blank[]]]]], Pattern[Px, Blank[]], Power[Plus[Optional[Pattern[d, Blank[]]], Times[Optional[Pattern[e, Blank[]]], Pattern[x, Blank[]]]], Optional[Pattern[m, Blank[]]]]], Pattern[x, Blank[Symbol]]] := With[{u = IntHide[Px*(d + e*x)^m, x]}, Dist[a + b*ArcCosh[c*x], u, x] - Dist[(b*c*Sqrt[1 - c^2*x^2])/(Sqrt[-1 + c*x]*Sqrt[1 + c*x]), Int[SimplifyIntegrand[u/Sqrt[1 - c^2*x^2], x], x], x]] /; FreeQ[{a, b, c, d, e, m}, x] && PolynomialQ[Px, x]
  5813. Int[Times[Power[Plus[Optional[Pattern[a, Blank[]]], Times[ArcSinh[Times[Optional[Pattern[c, Blank[]]], Pattern[x, Blank[]]]], Optional[Pattern[b, Blank[]]]]], Pattern[n, Blank[]]], Power[Plus[Pattern[d, Blank[]], Times[Optional[Pattern[e, Blank[]]], Pattern[x, Blank[]]]], Pattern[m, Blank[]]], Power[Plus[Optional[Pattern[f, Blank[]]], Times[Optional[Pattern[g, Blank[]]], Pattern[x, Blank[]]]], Optional[Pattern[p, Blank[]]]]], Pattern[x, Blank[Symbol]]] := With[{u = IntHide[(f + g*x)^p*(d + e*x)^m, x]}, Dist[(a + b*ArcSinh[c*x])^n, u, x] - Dist[b*c*n, Int[SimplifyIntegrand[(u*(a + b*ArcSinh[c*x])^(n - 1))/Sqrt[1 + c^2*x^2], x], x], x]] /; FreeQ[{a, b, c, d, e, f, g}, x] && IGtQ[n, 0] && IGtQ[p, 0] && ILtQ[m, 0] && LtQ[m + p + 1, 0]
  5814. Int[Times[Power[Plus[Optional[Pattern[a, Blank[]]], Times[ArcCosh[Times[Optional[Pattern[c, Blank[]]], Pattern[x, Blank[]]]], Optional[Pattern[b, Blank[]]]]], Pattern[n, Blank[]]], Power[Plus[Pattern[d, Blank[]], Times[Optional[Pattern[e, Blank[]]], Pattern[x, Blank[]]]], Pattern[m, Blank[]]], Power[Plus[Optional[Pattern[f, Blank[]]], Times[Optional[Pattern[g, Blank[]]], Pattern[x, Blank[]]]], Optional[Pattern[p, Blank[]]]]], Pattern[x, Blank[Symbol]]] := With[{u = IntHide[(f + g*x)^p*(d + e*x)^m, x]}, Dist[(a + b*ArcCosh[c*x])^n, u, x] - Dist[b*c*n, Int[SimplifyIntegrand[(u*(a + b*ArcCosh[c*x])^(n - 1))/(Sqrt[1 + c*x]*Sqrt[-1 + c*x]), x], x], x]] /; FreeQ[{a, b, c, d, e, f, g}, x] && IGtQ[n, 0] && IGtQ[p, 0] && ILtQ[m, 0] && LtQ[m + p + 1, 0]
  5815. Int[Times[Power[Plus[Optional[Pattern[a, Blank[]]], Times[ArcSinh[Times[Optional[Pattern[c, Blank[]]], Pattern[x, Blank[]]]], Optional[Pattern[b, Blank[]]]]], Pattern[n, Blank[]]], Power[Plus[Pattern[d, Blank[]], Times[Optional[Pattern[e, Blank[]]], Pattern[x, Blank[]]]], -2], Power[Plus[Optional[Pattern[f, Blank[]]], Times[Optional[Pattern[g, Blank[]]], Pattern[x, Blank[]]], Times[Optional[Pattern[h, Blank[]]], Power[Pattern[x, Blank[]], 2]]], Optional[Pattern[p, Blank[]]]]], Pattern[x, Blank[Symbol]]] := With[{u = IntHide[(f + g*x + h*x^2)^p/(d + e*x)^2, x]}, Dist[(a + b*ArcSinh[c*x])^n, u, x] - Dist[b*c*n, Int[SimplifyIntegrand[(u*(a + b*ArcSinh[c*x])^(n - 1))/Sqrt[1 + c^2*x^2], x], x], x]] /; FreeQ[{a, b, c, d, e, f, g, h}, x] && IGtQ[n, 0] && IGtQ[p, 0] && EqQ[e*g - 2*d*h, 0]
  5816. Int[Times[Power[Plus[Optional[Pattern[a, Blank[]]], Times[ArcCosh[Times[Optional[Pattern[c, Blank[]]], Pattern[x, Blank[]]]], Optional[Pattern[b, Blank[]]]]], Pattern[n, Blank[]]], Power[Plus[Pattern[d, Blank[]], Times[Optional[Pattern[e, Blank[]]], Pattern[x, Blank[]]]], -2], Power[Plus[Optional[Pattern[f, Blank[]]], Times[Optional[Pattern[g, Blank[]]], Pattern[x, Blank[]]], Times[Optional[Pattern[h, Blank[]]], Power[Pattern[x, Blank[]], 2]]], Optional[Pattern[p, Blank[]]]]], Pattern[x, Blank[Symbol]]] := With[{u = IntHide[(f + g*x + h*x^2)^p/(d + e*x)^2, x]}, Dist[(a + b*ArcCosh[c*x])^n, u, x] - Dist[b*c*n, Int[SimplifyIntegrand[(u*(a + b*ArcCosh[c*x])^(n - 1))/(Sqrt[1 + c*x]*Sqrt[-1 + c*x]), x], x], x]] /; FreeQ[{a, b, c, d, e, f, g, h}, x] && IGtQ[n, 0] && IGtQ[p, 0] && EqQ[e*g - 2*d*h, 0]
  5817. Int[Times[Power[Plus[Optional[Pattern[a, Blank[]]], Times[ArcSinh[Times[Optional[Pattern[c, Blank[]]], Pattern[x, Blank[]]]], Optional[Pattern[b, Blank[]]]]], Optional[Pattern[n, Blank[]]]], Pattern[Px, Blank[]], Power[Plus[Pattern[d, Blank[]], Times[Optional[Pattern[e, Blank[]]], Pattern[x, Blank[]]]], Optional[Pattern[m, Blank[]]]]], Pattern[x, Blank[Symbol]]] := Int[ExpandIntegrand[Px*(d + e*x)^m*(a + b*ArcSinh[c*x])^n, x], x] /; FreeQ[{a, b, c, d, e}, x] && PolynomialQ[Px, x] && IGtQ[n, 0] && IntegerQ[m]
  5818. Int[Times[Power[Plus[Optional[Pattern[a, Blank[]]], Times[ArcCosh[Times[Optional[Pattern[c, Blank[]]], Pattern[x, Blank[]]]], Optional[Pattern[b, Blank[]]]]], Optional[Pattern[n, Blank[]]]], Pattern[Px, Blank[]], Power[Plus[Pattern[d, Blank[]], Times[Optional[Pattern[e, Blank[]]], Pattern[x, Blank[]]]], Optional[Pattern[m, Blank[]]]]], Pattern[x, Blank[Symbol]]] := Int[ExpandIntegrand[Px*(d + e*x)^m*(a + b*ArcCosh[c*x])^n, x], x] /; FreeQ[{a, b, c, d, e}, x] && PolynomialQ[Px, x] && IGtQ[n, 0] && IntegerQ[m]
  5819. Int[Times[Plus[Optional[Pattern[a, Blank[]]], Times[ArcSinh[Times[Optional[Pattern[c, Blank[]]], Pattern[x, Blank[]]]], Optional[Pattern[b, Blank[]]]]], Power[Plus[Pattern[f, Blank[]], Times[Optional[Pattern[g, Blank[]]], Pattern[x, Blank[]]]], Optional[Pattern[m, Blank[]]]], Power[Plus[Pattern[d, Blank[]], Times[Optional[Pattern[e, Blank[]]], Power[Pattern[x, Blank[]], 2]]], Pattern[p, Blank[]]]], Pattern[x, Blank[Symbol]]] := With[{u = IntHide[(f + g*x)^m*(d + e*x^2)^p, x]}, Dist[a + b*ArcSinh[c*x], u, x] - Dist[b*c, Int[Dist[1/Sqrt[1 + c^2*x^2], u, x], x], x]] /; FreeQ[{a, b, c, d, e, f, g}, x] && EqQ[e, c^2*d] && IGtQ[m, 0] && ILtQ[p + 1/2, 0] && GtQ[d, 0] && (LtQ[m, -2*p - 1] || GtQ[m, 3])
  5820. Int[Times[Plus[Optional[Pattern[a, Blank[]]], Times[ArcCosh[Times[Optional[Pattern[c, Blank[]]], Pattern[x, Blank[]]]], Optional[Pattern[b, Blank[]]]]], Power[Plus[Pattern[d1, Blank[]], Times[Optional[Pattern[e1, Blank[]]], Pattern[x, Blank[]]]], Pattern[p, Blank[]]], Power[Plus[Pattern[d2, Blank[]], Times[Optional[Pattern[e2, Blank[]]], Pattern[x, Blank[]]]], Pattern[p, Blank[]]], Power[Plus[Pattern[f, Blank[]], Times[Optional[Pattern[g, Blank[]]], Pattern[x, Blank[]]]], Optional[Pattern[m, Blank[]]]]], Pattern[x, Blank[Symbol]]] := With[{u = IntHide[(f + g*x)^m*(d1 + e1*x)^p*(d2 + e2*x)^p, x]}, Dist[a + b*ArcCosh[c*x], u, x] - Dist[b*c, Int[Dist[1/(Sqrt[1 + c*x]*Sqrt[-1 + c*x]), u, x], x], x]] /; FreeQ[{a, b, c, d1, e1, d2, e2, f, g}, x] && EqQ[e1 - c*d1, 0] && EqQ[e2 + c*d2, 0] && IGtQ[m, 0] && ILtQ[p + 1/2, 0] && GtQ[d1, 0] && LtQ[d2, 0] && (LtQ[m, -2*p - 1] || GtQ[m, 3])
  5821. Int[Times[Power[Plus[Optional[Pattern[a, Blank[]]], Times[ArcSinh[Times[Optional[Pattern[c, Blank[]]], Pattern[x, Blank[]]]], Optional[Pattern[b, Blank[]]]]], Optional[Pattern[n, Blank[]]]], Power[Plus[Pattern[f, Blank[]], Times[Optional[Pattern[g, Blank[]]], Pattern[x, Blank[]]]], Optional[Pattern[m, Blank[]]]], Power[Plus[Pattern[d, Blank[]], Times[Optional[Pattern[e, Blank[]]], Power[Pattern[x, Blank[]], 2]]], Pattern[p, Blank[]]]], Pattern[x, Blank[Symbol]]] := Int[ExpandIntegrand[(d + e*x^2)^p*(a + b*ArcSinh[c*x])^n, (f + g*x)^m, x], x] /; FreeQ[{a, b, c, d, e, f, g}, x] && EqQ[e, c^2*d] && IGtQ[m, 0] && IntegerQ[p + 1/2] && GtQ[d, 0] && IGtQ[n, 0] && ((EqQ[n, 1] && GtQ[p, -1]) || GtQ[p, 0] || EqQ[m, 1] || (EqQ[m, 2] && LtQ[p, -2]))
  5822. Int[Times[Power[Plus[Optional[Pattern[a, Blank[]]], Times[ArcCosh[Times[Optional[Pattern[c, Blank[]]], Pattern[x, Blank[]]]], Optional[Pattern[b, Blank[]]]]], Optional[Pattern[n, Blank[]]]], Power[Plus[Pattern[d1, Blank[]], Times[Optional[Pattern[e1, Blank[]]], Pattern[x, Blank[]]]], Pattern[p, Blank[]]], Power[Plus[Pattern[d2, Blank[]], Times[Optional[Pattern[e2, Blank[]]], Pattern[x, Blank[]]]], Pattern[p, Blank[]]], Power[Plus[Pattern[f, Blank[]], Times[Optional[Pattern[g, Blank[]]], Pattern[x, Blank[]]]], Optional[Pattern[m, Blank[]]]]], Pattern[x, Blank[Symbol]]] := Int[ExpandIntegrand[(d1 + e1*x)^p*(d2 + e2*x)^p*(a + b*ArcCosh[c*x])^n, (f + g*x)^m, x], x] /; FreeQ[{a, b, c, d1, e1, d2, e2, f, g}, x] && EqQ[e1 - c*d1, 0] && EqQ[e2 + c*d2, 0] && IGtQ[m, 0] && IntegerQ[p + 1/2] && GtQ[d1, 0] && LtQ[d2, 0] && IGtQ[n, 0] && ((EqQ[n, 1] && GtQ[p, -1]) || GtQ[p, 0] || EqQ[m, 1] || (EqQ[m, 2] && LtQ[p, -2]))
  5823. Int[Times[Power[Plus[Optional[Pattern[a, Blank[]]], Times[ArcSinh[Times[Optional[Pattern[c, Blank[]]], Pattern[x, Blank[]]]], Optional[Pattern[b, Blank[]]]]], Optional[Pattern[n, Blank[]]]], Power[Plus[Optional[Pattern[f, Blank[]]], Times[Optional[Pattern[g, Blank[]]], Pattern[x, Blank[]]]], Pattern[m, Blank[]]], Power[Plus[Pattern[d, Blank[]], Times[Optional[Pattern[e, Blank[]]], Power[Pattern[x, Blank[]], 2]]], Rational[1, 2]]], Pattern[x, Blank[Symbol]]] := Simp[((f + g*x)^m*(d + e*x^2)*(a + b*ArcSinh[c*x])^(n + 1))/(b*c*Sqrt[d]*(n + 1)), x] - Dist[1/(b*c*Sqrt[d]*(n + 1)), Int[(d*g*m + 2*e*f*x + e*g*(m + 2)*x^2)*(f + g*x)^(m - 1)*(a + b*ArcSinh[c*x])^(n + 1), x], x] /; FreeQ[{a, b, c, d, e, f, g}, x] && EqQ[e, c^2*d] && ILtQ[m, 0] && GtQ[d, 0] && IGtQ[n, 0]
  5824. Int[Times[Power[Plus[Optional[Pattern[a, Blank[]]], Times[ArcCosh[Times[Optional[Pattern[c, Blank[]]], Pattern[x, Blank[]]]], Optional[Pattern[b, Blank[]]]]], Optional[Pattern[n, Blank[]]]], Power[Plus[Pattern[d1, Blank[]], Times[Optional[Pattern[e1, Blank[]]], Pattern[x, Blank[]]]], Rational[1, 2]], Power[Plus[Pattern[d2, Blank[]], Times[Optional[Pattern[e2, Blank[]]], Pattern[x, Blank[]]]], Rational[1, 2]], Power[Plus[Pattern[f, Blank[]], Times[Optional[Pattern[g, Blank[]]], Pattern[x, Blank[]]]], Pattern[m, Blank[]]]], Pattern[x, Blank[Symbol]]] := Simp[((f + g*x)^m*(d1*d2 + e1*e2*x^2)*(a + b*ArcCosh[c*x])^(n + 1))/(b*c*Sqrt[-(d1*d2)]*(n + 1)), x] - Dist[1/(b*c*Sqrt[-(d1*d2)]*(n + 1)), Int[(d1*d2*g*m + 2*e1*e2*f*x + e1*e2*g*(m + 2)*x^2)*(f + g*x)^(m - 1)*(a + b*ArcCosh[c*x])^(n + 1), x], x] /; FreeQ[{a, b, c, d1, e1, d2, e2, f, g}, x] && EqQ[e1 - c*d1, 0] && EqQ[e2 + c*d2, 0] && ILtQ[m, 0] && GtQ[d1, 0] && LtQ[d2, 0] && IGtQ[n, 0]
  5825. Int[Times[Power[Plus[Optional[Pattern[a, Blank[]]], Times[ArcSinh[Times[Optional[Pattern[c, Blank[]]], Pattern[x, Blank[]]]], Optional[Pattern[b, Blank[]]]]], Optional[Pattern[n, Blank[]]]], Power[Plus[Pattern[f, Blank[]], Times[Optional[Pattern[g, Blank[]]], Pattern[x, Blank[]]]], Optional[Pattern[m, Blank[]]]], Power[Plus[Pattern[d, Blank[]], Times[Optional[Pattern[e, Blank[]]], Power[Pattern[x, Blank[]], 2]]], Pattern[p, Blank[]]]], Pattern[x, Blank[Symbol]]] := Int[ExpandIntegrand[Sqrt[d + e*x^2]*(a + b*ArcSinh[c*x])^n, (f + g*x)^m*(d + e*x^2)^(p - 1/2), x], x] /; FreeQ[{a, b, c, d, e, f, g}, x] && EqQ[e, c^2*d] && IntegerQ[m] && IGtQ[p + 1/2, 0] && GtQ[d, 0] && IGtQ[n, 0]
  5826. Int[Times[Power[Plus[Optional[Pattern[a, Blank[]]], Times[ArcCosh[Times[Optional[Pattern[c, Blank[]]], Pattern[x, Blank[]]]], Optional[Pattern[b, Blank[]]]]], Optional[Pattern[n, Blank[]]]], Power[Plus[Pattern[d1, Blank[]], Times[Optional[Pattern[e1, Blank[]]], Pattern[x, Blank[]]]], Pattern[p, Blank[]]], Power[Plus[Pattern[d2, Blank[]], Times[Optional[Pattern[e2, Blank[]]], Pattern[x, Blank[]]]], Pattern[p, Blank[]]], Power[Plus[Pattern[f, Blank[]], Times[Optional[Pattern[g, Blank[]]], Pattern[x, Blank[]]]], Optional[Pattern[m, Blank[]]]]], Pattern[x, Blank[Symbol]]] := Int[ExpandIntegrand[Sqrt[d1 + e1*x]*Sqrt[d2 + e2*x]*(a + b*ArcCosh[c*x])^n, (f + g*x)^m*(d1 + e1*x)^(p - 1/2)*(d2 + e2*x)^(p - 1/2), x], x] /; FreeQ[{a, b, c, d1, e1, d2, e2, f, g}, x] && EqQ[e1 - c*d1, 0] && EqQ[e2 + c*d2, 0] && IntegerQ[m] && IGtQ[p + 1/2, 0] && GtQ[d1, 0] && LtQ[d2, 0] && IGtQ[n, 0]
  5827. Int[Times[Power[Plus[Optional[Pattern[a, Blank[]]], Times[ArcSinh[Times[Optional[Pattern[c, Blank[]]], Pattern[x, Blank[]]]], Optional[Pattern[b, Blank[]]]]], Optional[Pattern[n, Blank[]]]], Power[Plus[Pattern[f, Blank[]], Times[Optional[Pattern[g, Blank[]]], Pattern[x, Blank[]]]], Optional[Pattern[m, Blank[]]]], Power[Plus[Pattern[d, Blank[]], Times[Optional[Pattern[e, Blank[]]], Power[Pattern[x, Blank[]], 2]]], Pattern[p, Blank[]]]], Pattern[x, Blank[Symbol]]] := Simp[((f + g*x)^m*(d + e*x^2)^(p + 1/2)*(a + b*ArcSinh[c*x])^(n + 1))/(b*c*Sqrt[d]*(n + 1)), x] - Dist[1/(b*c*Sqrt[d]*(n + 1)), Int[ExpandIntegrand[(f + g*x)^(m - 1)*(a + b*ArcSinh[c*x])^(n + 1), (d*g*m + e*f*(2*p + 1)*x + e*g*(m + 2*p + 1)*x^2)*(d + e*x^2)^(p - 1/2), x], x], x] /; FreeQ[{a, b, c, d, e, f, g}, x] && EqQ[e, c^2*d] && ILtQ[m, 0] && IGtQ[p - 1/2, 0] && GtQ[d, 0] && IGtQ[n, 0]
  5828. Int[Times[Power[Plus[Optional[Pattern[a, Blank[]]], Times[ArcCosh[Times[Optional[Pattern[c, Blank[]]], Pattern[x, Blank[]]]], Optional[Pattern[b, Blank[]]]]], Optional[Pattern[n, Blank[]]]], Power[Plus[Pattern[d1, Blank[]], Times[Optional[Pattern[e1, Blank[]]], Pattern[x, Blank[]]]], Pattern[p, Blank[]]], Power[Plus[Pattern[d2, Blank[]], Times[Optional[Pattern[e2, Blank[]]], Pattern[x, Blank[]]]], Pattern[p, Blank[]]], Power[Plus[Pattern[f, Blank[]], Times[Optional[Pattern[g, Blank[]]], Pattern[x, Blank[]]]], Optional[Pattern[m, Blank[]]]]], Pattern[x, Blank[Symbol]]] := Simp[((f + g*x)^m*(d1 + e1*x)^(p + 1/2)*(d2 + e2*x)^(p + 1/2)*(a + b*ArcCosh[c*x])^(n + 1))/(b*c*Sqrt[-(d1*d2)]*(n + 1)), x] - Dist[1/(b*c*Sqrt[-(d1*d2)]*(n + 1)), Int[ExpandIntegrand[(f + g*x)^(m - 1)*(a + b*ArcCosh[c*x])^(n + 1), (d1*d2*g*m + e1*e2*f*(2*p + 1)*x + e1*e2*g*(m + 2*p + 1)*x^2)*(d1 + e1*x)^(p - 1/2)*(d2 + e2*x)^(p - 1/2), x], x], x] /; FreeQ[{a, b, c, d1, e1, d2, e2, f, g}, x] && EqQ[e1 - c*d1, 0] && EqQ[e2 + c*d2, 0] && ILtQ[m, 0] && IGtQ[p - 1/2, 0] && GtQ[d1, 0] && LtQ[d2, 0] && IGtQ[n, 0]
  5829. Int[Times[Power[Plus[Optional[Pattern[a, Blank[]]], Times[ArcSinh[Times[Optional[Pattern[c, Blank[]]], Pattern[x, Blank[]]]], Optional[Pattern[b, Blank[]]]]], Pattern[n, Blank[]]], Power[Plus[Pattern[f, Blank[]], Times[Optional[Pattern[g, Blank[]]], Pattern[x, Blank[]]]], Optional[Pattern[m, Blank[]]]], Power[Plus[Pattern[d, Blank[]], Times[Optional[Pattern[e, Blank[]]], Power[Pattern[x, Blank[]], 2]]], Rational[-1, 2]]], Pattern[x, Blank[Symbol]]] := Simp[((f + g*x)^m*(a + b*ArcSinh[c*x])^(n + 1))/(b*c*Sqrt[d]*(n + 1)), x] - Dist[(g*m)/(b*c*Sqrt[d]*(n + 1)), Int[(f + g*x)^(m - 1)*(a + b*ArcSinh[c*x])^(n + 1), x], x] /; FreeQ[{a, b, c, d, e, f, g}, x] && EqQ[e, c^2*d] && IGtQ[m, 0] && GtQ[d, 0] && LtQ[n, -1]
  5830. Int[Times[Power[Plus[Optional[Pattern[a, Blank[]]], Times[ArcCosh[Times[Optional[Pattern[c, Blank[]]], Pattern[x, Blank[]]]], Optional[Pattern[b, Blank[]]]]], Pattern[n, Blank[]]], Power[Plus[Pattern[d1, Blank[]], Times[Optional[Pattern[e1, Blank[]]], Pattern[x, Blank[]]]], Rational[-1, 2]], Power[Plus[Pattern[d2, Blank[]], Times[Optional[Pattern[e2, Blank[]]], Pattern[x, Blank[]]]], Rational[-1, 2]], Power[Plus[Pattern[f, Blank[]], Times[Optional[Pattern[g, Blank[]]], Pattern[x, Blank[]]]], Optional[Pattern[m, Blank[]]]]], Pattern[x, Blank[Symbol]]] := Simp[((f + g*x)^m*(a + b*ArcCosh[c*x])^(n + 1))/(b*c*Sqrt[-(d1*d2)]*(n + 1)), x] - Dist[(g*m)/(b*c*Sqrt[-(d1*d2)]*(n + 1)), Int[(f + g*x)^(m - 1)*(a + b*ArcCosh[c*x])^(n + 1), x], x] /; FreeQ[{a, b, c, d1, e1, d2, e2, f, g}, x] && EqQ[e1 - c*d1, 0] && EqQ[e2 + c*d2, 0] && IGtQ[m, 0] && GtQ[d1, 0] && LtQ[d2, 0] && LtQ[n, -1]
  5831. Int[Times[Power[Plus[Optional[Pattern[a, Blank[]]], Times[ArcSinh[Times[Optional[Pattern[c, Blank[]]], Pattern[x, Blank[]]]], Optional[Pattern[b, Blank[]]]]], Optional[Pattern[n, Blank[]]]], Power[Plus[Pattern[f, Blank[]], Times[Optional[Pattern[g, Blank[]]], Pattern[x, Blank[]]]], Optional[Pattern[m, Blank[]]]], Power[Plus[Pattern[d, Blank[]], Times[Optional[Pattern[e, Blank[]]], Power[Pattern[x, Blank[]], 2]]], Rational[-1, 2]]], Pattern[x, Blank[Symbol]]] := Dist[1/(c^(m + 1)*Sqrt[d]), Subst[Int[(a + b*x)^n*(c*f + g*Sinh[x])^m, x], x, ArcSinh[c*x]], x] /; FreeQ[{a, b, c, d, e, f, g, n}, x] && EqQ[e, c^2*d] && IntegerQ[m] && GtQ[d, 0] && (GtQ[m, 0] || IGtQ[n, 0])
  5832. Int[Times[Power[Plus[Optional[Pattern[a, Blank[]]], Times[ArcCosh[Times[Optional[Pattern[c, Blank[]]], Pattern[x, Blank[]]]], Optional[Pattern[b, Blank[]]]]], Optional[Pattern[n, Blank[]]]], Power[Plus[Pattern[d1, Blank[]], Times[Optional[Pattern[e1, Blank[]]], Pattern[x, Blank[]]]], Rational[-1, 2]], Power[Plus[Pattern[d2, Blank[]], Times[Optional[Pattern[e2, Blank[]]], Pattern[x, Blank[]]]], Rational[-1, 2]], Power[Plus[Pattern[f, Blank[]], Times[Optional[Pattern[g, Blank[]]], Pattern[x, Blank[]]]], Optional[Pattern[m, Blank[]]]]], Pattern[x, Blank[Symbol]]] := Dist[1/(c^(m + 1)*Sqrt[-(d1*d2)]), Subst[Int[(a + b*x)^n*(c*f + g*Cosh[x])^m, x], x, ArcCosh[c*x]], x] /; FreeQ[{a, b, c, d1, e1, d2, e2, f, g, n}, x] && EqQ[e1 - c*d1, 0] && EqQ[e2 + c*d2, 0] && IntegerQ[m] && GtQ[d1, 0] && LtQ[d2, 0] && (GtQ[m, 0] || IGtQ[n, 0])
  5833. Int[Times[Power[Plus[Optional[Pattern[a, Blank[]]], Times[ArcSinh[Times[Optional[Pattern[c, Blank[]]], Pattern[x, Blank[]]]], Optional[Pattern[b, Blank[]]]]], Optional[Pattern[n, Blank[]]]], Power[Plus[Pattern[f, Blank[]], Times[Optional[Pattern[g, Blank[]]], Pattern[x, Blank[]]]], Optional[Pattern[m, Blank[]]]], Power[Plus[Pattern[d, Blank[]], Times[Optional[Pattern[e, Blank[]]], Power[Pattern[x, Blank[]], 2]]], Pattern[p, Blank[]]]], Pattern[x, Blank[Symbol]]] := Int[ExpandIntegrand[(a + b*ArcSinh[c*x])^n/Sqrt[d + e*x^2], (f + g*x)^m*(d + e*x^2)^(p + 1/2), x], x] /; FreeQ[{a, b, c, d, e, f, g}, x] && EqQ[e, c^2*d] && IntegerQ[m] && ILtQ[p + 1/2, 0] && GtQ[d, 0] && IGtQ[n, 0]
  5834. Int[Times[Power[Plus[Optional[Pattern[a, Blank[]]], Times[ArcCosh[Times[Optional[Pattern[c, Blank[]]], Pattern[x, Blank[]]]], Optional[Pattern[b, Blank[]]]]], Optional[Pattern[n, Blank[]]]], Power[Plus[Pattern[d1, Blank[]], Times[Optional[Pattern[e1, Blank[]]], Pattern[x, Blank[]]]], Pattern[p, Blank[]]], Power[Plus[Pattern[d2, Blank[]], Times[Optional[Pattern[e2, Blank[]]], Pattern[x, Blank[]]]], Pattern[p, Blank[]]], Power[Plus[Pattern[f, Blank[]], Times[Optional[Pattern[g, Blank[]]], Pattern[x, Blank[]]]], Optional[Pattern[m, Blank[]]]]], Pattern[x, Blank[Symbol]]] := Int[ExpandIntegrand[(a + b*ArcCosh[c*x])^n/(Sqrt[d1 + e1*x]*Sqrt[d2 + e2*x]), (f + g*x)^m*(d1 + e1*x)^(p + 1/2)*(d2 + e2*x)^(p + 1/2), x], x] /; FreeQ[{a, b, c, d1, e1, d2, e2, f, g}, x] && EqQ[e1 - c*d1, 0] && EqQ[e2 + c*d2, 0] && IntegerQ[m] && ILtQ[p + 1/2, 0] && GtQ[d1, 0] && LtQ[d2, 0] && IGtQ[n, 0]
  5835. Int[Times[Power[Plus[Optional[Pattern[a, Blank[]]], Times[ArcSinh[Times[Optional[Pattern[c, Blank[]]], Pattern[x, Blank[]]]], Optional[Pattern[b, Blank[]]]]], Optional[Pattern[n, Blank[]]]], Power[Plus[Pattern[f, Blank[]], Times[Optional[Pattern[g, Blank[]]], Pattern[x, Blank[]]]], Optional[Pattern[m, Blank[]]]], Power[Plus[Pattern[d, Blank[]], Times[Optional[Pattern[e, Blank[]]], Power[Pattern[x, Blank[]], 2]]], Pattern[p, Blank[]]]], Pattern[x, Blank[Symbol]]] := Dist[(d^IntPart[p]*(d + e*x^2)^FracPart[p])/(1 + c^2*x^2)^FracPart[p], Int[(f + g*x)^m*(1 + c^2*x^2)^p*(a + b*ArcSinh[c*x])^n, x], x] /; FreeQ[{a, b, c, d, e, f, g, n}, x] && EqQ[e, c^2*d] && IntegerQ[m] && IntegerQ[p - 1/2] && !GtQ[d, 0]
  5836. Int[Times[Power[Plus[Optional[Pattern[a, Blank[]]], Times[ArcCosh[Times[Optional[Pattern[c, Blank[]]], Pattern[x, Blank[]]]], Optional[Pattern[b, Blank[]]]]], Optional[Pattern[n, Blank[]]]], Power[Plus[Pattern[f, Blank[]], Times[Optional[Pattern[g, Blank[]]], Pattern[x, Blank[]]]], Optional[Pattern[m, Blank[]]]], Power[Plus[Pattern[d, Blank[]], Times[Optional[Pattern[e, Blank[]]], Power[Pattern[x, Blank[]], 2]]], Pattern[p, Blank[]]]], Pattern[x, Blank[Symbol]]] := Dist[((-d)^IntPart[p]*(d + e*x^2)^FracPart[p])/((1 + c*x)^FracPart[p]*(-1 + c*x)^FracPart[p]), Int[(f + g*x)^m*(1 + c*x)^p*(-1 + c*x)^p*(a + b*ArcCosh[c*x])^n, x], x] /; FreeQ[{a, b, c, d, e, f, g, n}, x] && EqQ[c^2*d + e, 0] && IntegerQ[m] && IntegerQ[p - 1/2]
  5837. Int[Times[Power[Plus[Optional[Pattern[a, Blank[]]], Times[ArcCosh[Times[Optional[Pattern[c, Blank[]]], Pattern[x, Blank[]]]], Optional[Pattern[b, Blank[]]]]], Optional[Pattern[n, Blank[]]]], Power[Plus[Pattern[d1, Blank[]], Times[Optional[Pattern[e1, Blank[]]], Pattern[x, Blank[]]]], Pattern[p, Blank[]]], Power[Plus[Pattern[d2, Blank[]], Times[Optional[Pattern[e2, Blank[]]], Pattern[x, Blank[]]]], Pattern[p, Blank[]]], Power[Plus[Pattern[f, Blank[]], Times[Optional[Pattern[g, Blank[]]], Pattern[x, Blank[]]]], Optional[Pattern[m, Blank[]]]]], Pattern[x, Blank[Symbol]]] := Dist[((-(d1*d2))^IntPart[p]*(d1 + e1*x)^FracPart[p]*(d2 + e2*x)^FracPart[p])/(1 - c^2*x^2)^FracPart[p], Int[(f + g*x)^m*(1 + c*x)^p*(-1 + c*x)^p*(a + b*ArcCosh[c*x])^n, x], x] /; FreeQ[{a, b, c, d1, e1, d2, e2, f, g, n}, x] && EqQ[e1 - c*d1, 0] && EqQ[e2 + c*d2, 0] && IntegerQ[m] && IntegerQ[p - 1/2] && !(GtQ[d1, 0] && LtQ[d2, 0])
  5838. Int[Times[Log[Times[Optional[Pattern[h, Blank[]]], Power[Plus[Optional[Pattern[f, Blank[]]], Times[Optional[Pattern[g, Blank[]]], Pattern[x, Blank[]]]], Optional[Pattern[m, Blank[]]]]]], Power[Plus[Optional[Pattern[a, Blank[]]], Times[ArcSinh[Times[Optional[Pattern[c, Blank[]]], Pattern[x, Blank[]]]], Optional[Pattern[b, Blank[]]]]], Optional[Pattern[n, Blank[]]]], Power[Plus[Pattern[d, Blank[]], Times[Optional[Pattern[e, Blank[]]], Power[Pattern[x, Blank[]], 2]]], Rational[-1, 2]]], Pattern[x, Blank[Symbol]]] := Simp[(Log[h*(f + g*x)^m]*(a + b*ArcSinh[c*x])^(n + 1))/(b*c*Sqrt[d]*(n + 1)), x] - Dist[(g*m)/(b*c*Sqrt[d]*(n + 1)), Int[(a + b*ArcSinh[c*x])^(n + 1)/(f + g*x), x], x] /; FreeQ[{a, b, c, d, e, f, g, h, m}, x] && EqQ[e, c^2*d] && GtQ[d, 0] && IGtQ[n, 0]
  5839. Int[Times[Log[Times[Optional[Pattern[h, Blank[]]], Power[Plus[Optional[Pattern[f, Blank[]]], Times[Optional[Pattern[g, Blank[]]], Pattern[x, Blank[]]]], Optional[Pattern[m, Blank[]]]]]], Power[Plus[Optional[Pattern[a, Blank[]]], Times[ArcCosh[Times[Optional[Pattern[c, Blank[]]], Pattern[x, Blank[]]]], Optional[Pattern[b, Blank[]]]]], Optional[Pattern[n, Blank[]]]], Power[Plus[Pattern[d1, Blank[]], Times[Optional[Pattern[e1, Blank[]]], Pattern[x, Blank[]]]], Rational[-1, 2]], Power[Plus[Pattern[d2, Blank[]], Times[Optional[Pattern[e2, Blank[]]], Pattern[x, Blank[]]]], Rational[-1, 2]]], Pattern[x, Blank[Symbol]]] := Simp[(Log[h*(f + g*x)^m]*(a + b*ArcCosh[c*x])^(n + 1))/(b*c*Sqrt[-(d1*d2)]*(n + 1)), x] - Dist[(g*m)/(b*c*Sqrt[-(d1*d2)]*(n + 1)), Int[(a + b*ArcCosh[c*x])^(n + 1)/(f + g*x), x], x] /; FreeQ[{a, b, c, d1, e1, d2, e2, f, g, h, m}, x] && EqQ[e1 - c*d1, 0] && EqQ[e2 + c*d2, 0] && GtQ[d1, 0] && LtQ[d2, 0] && IGtQ[n, 0]
  5840. Int[Times[Log[Times[Optional[Pattern[h, Blank[]]], Power[Plus[Optional[Pattern[f, Blank[]]], Times[Optional[Pattern[g, Blank[]]], Pattern[x, Blank[]]]], Optional[Pattern[m, Blank[]]]]]], Power[Plus[Optional[Pattern[a, Blank[]]], Times[ArcSinh[Times[Optional[Pattern[c, Blank[]]], Pattern[x, Blank[]]]], Optional[Pattern[b, Blank[]]]]], Optional[Pattern[n, Blank[]]]], Power[Plus[Pattern[d, Blank[]], Times[Optional[Pattern[e, Blank[]]], Power[Pattern[x, Blank[]], 2]]], Pattern[p, Blank[]]]], Pattern[x, Blank[Symbol]]] := Dist[(d^IntPart[p]*(d + e*x^2)^FracPart[p])/(1 + c^2*x^2)^FracPart[p], Int[Log[h*(f + g*x)^m]*(1 + c^2*x^2)^p*(a + b*ArcSinh[c*x])^n, x], x] /; FreeQ[{a, b, c, d, e, f, g, h, m, n}, x] && EqQ[e, c^2*d] && IntegerQ[p - 1/2] && !GtQ[d, 0]
  5841. Int[Times[Log[Times[Optional[Pattern[h, Blank[]]], Power[Plus[Optional[Pattern[f, Blank[]]], Times[Optional[Pattern[g, Blank[]]], Pattern[x, Blank[]]]], Optional[Pattern[m, Blank[]]]]]], Power[Plus[Optional[Pattern[a, Blank[]]], Times[ArcCosh[Times[Optional[Pattern[c, Blank[]]], Pattern[x, Blank[]]]], Optional[Pattern[b, Blank[]]]]], Optional[Pattern[n, Blank[]]]], Power[Plus[Pattern[d, Blank[]], Times[Optional[Pattern[e, Blank[]]], Power[Pattern[x, Blank[]], 2]]], Pattern[p, Blank[]]]], Pattern[x, Blank[Symbol]]] := Dist[((-d)^IntPart[p]*(d + e*x^2)^FracPart[p])/((1 + c*x)^FracPart[p]*(-1 + c*x)^FracPart[p]), Int[Log[h*(f + g*x)^m]*(1 + c*x)^p*(-1 + c*x)^p*(a + b*ArcCosh[c*x])^n, x], x] /; FreeQ[{a, b, c, d, e, f, g, h, m, n}, x] && EqQ[c^2*d + e, 0] && IntegerQ[p - 1/2]
  5842. Int[Times[Log[Times[Optional[Pattern[h, Blank[]]], Power[Plus[Optional[Pattern[f, Blank[]]], Times[Optional[Pattern[g, Blank[]]], Pattern[x, Blank[]]]], Optional[Pattern[m, Blank[]]]]]], Power[Plus[Optional[Pattern[a, Blank[]]], Times[ArcCosh[Times[Optional[Pattern[c, Blank[]]], Pattern[x, Blank[]]]], Optional[Pattern[b, Blank[]]]]], Optional[Pattern[n, Blank[]]]], Power[Plus[Pattern[d1, Blank[]], Times[Optional[Pattern[e1, Blank[]]], Pattern[x, Blank[]]]], Pattern[p, Blank[]]], Power[Plus[Pattern[d2, Blank[]], Times[Optional[Pattern[e2, Blank[]]], Pattern[x, Blank[]]]], Pattern[p, Blank[]]]], Pattern[x, Blank[Symbol]]] := Dist[((-(d1*d2))^IntPart[p]*(d1 + e1*x)^FracPart[p]*(d2 + e2*x)^FracPart[p])/((1 + c*x)^FracPart[p]*(-1 + c*x)^FracPart[p]), Int[Log[h*(f + g*x)^m]*(1 + c*x)^p*(-1 + c*x)^p*(a + b*ArcCosh[c*x])^n, x], x] /; FreeQ[{a, b, c, d1, e1, d2, e2, f, g, h, m, n}, x] && EqQ[e1 - c*d1, 0] && EqQ[e2 + c*d2, 0] && IntegerQ[p - 1/2] && !(GtQ[d1, 0] && LtQ[d2, 0])
  5843. Int[Times[Plus[Optional[Pattern[a, Blank[]]], Times[ArcSinh[Times[Optional[Pattern[c, Blank[]]], Pattern[x, Blank[]]]], Optional[Pattern[b, Blank[]]]]], Power[Plus[Pattern[d, Blank[]], Times[Optional[Pattern[e, Blank[]]], Pattern[x, Blank[]]]], Pattern[m, Blank[]]], Power[Plus[Pattern[f, Blank[]], Times[Optional[Pattern[g, Blank[]]], Pattern[x, Blank[]]]], Pattern[m, Blank[]]]], Pattern[x, Blank[Symbol]]] := With[{u = IntHide[(d + e*x)^m*(f + g*x)^m, x]}, Dist[a + b*ArcSinh[c*x], u, x] - Dist[b*c, Int[Dist[1/Sqrt[1 + c^2*x^2], u, x], x], x]] /; FreeQ[{a, b, c, d, e, f, g}, x] && ILtQ[m + 1/2, 0]
  5844. Int[Times[Plus[Optional[Pattern[a, Blank[]]], Times[ArcCosh[Times[Optional[Pattern[c, Blank[]]], Pattern[x, Blank[]]]], Optional[Pattern[b, Blank[]]]]], Power[Plus[Pattern[d, Blank[]], Times[Optional[Pattern[e, Blank[]]], Pattern[x, Blank[]]]], Pattern[m, Blank[]]], Power[Plus[Pattern[f, Blank[]], Times[Optional[Pattern[g, Blank[]]], Pattern[x, Blank[]]]], Pattern[m, Blank[]]]], Pattern[x, Blank[Symbol]]] := With[{u = IntHide[(d + e*x)^m*(f + g*x)^m, x]}, Dist[a + b*ArcCosh[c*x], u, x] - Dist[b*c, Int[Dist[1/(Sqrt[1 + c*x]*Sqrt[-1 + c*x]), u, x], x], x]] /; FreeQ[{a, b, c, d, e, f, g}, x] && ILtQ[m + 1/2, 0]
  5845. Int[Times[Power[Plus[Optional[Pattern[a, Blank[]]], Times[ArcSinh[Times[Optional[Pattern[c, Blank[]]], Pattern[x, Blank[]]]], Optional[Pattern[b, Blank[]]]]], Optional[Pattern[n, Blank[]]]], Power[Plus[Pattern[d, Blank[]], Times[Optional[Pattern[e, Blank[]]], Pattern[x, Blank[]]]], Optional[Pattern[m, Blank[]]]], Power[Plus[Pattern[f, Blank[]], Times[Optional[Pattern[g, Blank[]]], Pattern[x, Blank[]]]], Optional[Pattern[m, Blank[]]]]], Pattern[x, Blank[Symbol]]] := Int[ExpandIntegrand[(a + b*ArcSinh[c*x])^n, (d + e*x)^m*(f + g*x)^m, x], x] /; FreeQ[{a, b, c, d, e, f, g, n}, x] && IntegerQ[m]
  5846. Int[Times[Power[Plus[Optional[Pattern[a, Blank[]]], Times[ArcCosh[Times[Optional[Pattern[c, Blank[]]], Pattern[x, Blank[]]]], Optional[Pattern[b, Blank[]]]]], Optional[Pattern[n, Blank[]]]], Power[Plus[Pattern[d, Blank[]], Times[Optional[Pattern[e, Blank[]]], Pattern[x, Blank[]]]], Optional[Pattern[m, Blank[]]]], Power[Plus[Pattern[f, Blank[]], Times[Optional[Pattern[g, Blank[]]], Pattern[x, Blank[]]]], Optional[Pattern[m, Blank[]]]]], Pattern[x, Blank[Symbol]]] := Int[ExpandIntegrand[(a + b*ArcCosh[c*x])^n, (d + e*x)^m*(f + g*x)^m, x], x] /; FreeQ[{a, b, c, d, e, f, g, n}, x] && IntegerQ[m]
  5847. Int[Times[Plus[Optional[Pattern[a, Blank[]]], Times[ArcSinh[Times[Optional[Pattern[c, Blank[]]], Pattern[x, Blank[]]]], Optional[Pattern[b, Blank[]]]]], Pattern[u, Blank[]]], Pattern[x, Blank[Symbol]]] := With[{v = IntHide[u, x]}, Dist[a + b*ArcSinh[c*x], v, x] - Dist[b*c, Int[SimplifyIntegrand[v/Sqrt[1 + c^2*x^2], x], x], x] /; InverseFunctionFreeQ[v, x]] /; FreeQ[{a, b, c}, x]
  5848. Int[Times[Plus[Optional[Pattern[a, Blank[]]], Times[ArcCosh[Times[Optional[Pattern[c, Blank[]]], Pattern[x, Blank[]]]], Optional[Pattern[b, Blank[]]]]], Pattern[u, Blank[]]], Pattern[x, Blank[Symbol]]] := With[{v = IntHide[u, x]}, Dist[a + b*ArcCosh[c*x], v, x] - Dist[(b*c*Sqrt[1 - c^2*x^2])/(Sqrt[-1 + c*x]*Sqrt[1 + c*x]), Int[SimplifyIntegrand[v/Sqrt[1 - c^2*x^2], x], x], x] /; InverseFunctionFreeQ[v, x]] /; FreeQ[{a, b, c}, x]
  5849. Int[Times[Power[Plus[Optional[Pattern[a, Blank[]]], Times[ArcSinh[Times[Optional[Pattern[c, Blank[]]], Pattern[x, Blank[]]]], Optional[Pattern[b, Blank[]]]]], Pattern[n, Blank[]]], Pattern[Px, Blank[]], Power[Plus[Pattern[d, Blank[]], Times[Optional[Pattern[e, Blank[]]], Power[Pattern[x, Blank[]], 2]]], Pattern[p, Blank[]]]], Pattern[x, Blank[Symbol]]] := With[{u = ExpandIntegrand[Px*(d + e*x^2)^p*(a + b*ArcSinh[c*x])^n, x]}, Int[u, x] /; SumQ[u]] /; FreeQ[{a, b, c, d, e, n}, x] && PolynomialQ[Px, x] && EqQ[e, c^2*d] && IntegerQ[p - 1/2]
  5850. Int[Times[Power[Plus[Optional[Pattern[a, Blank[]]], Times[ArcCosh[Times[Optional[Pattern[c, Blank[]]], Pattern[x, Blank[]]]], Optional[Pattern[b, Blank[]]]]], Pattern[n, Blank[]]], Pattern[Px, Blank[]], Power[Plus[Pattern[d1, Blank[]], Times[Optional[Pattern[e1, Blank[]]], Pattern[x, Blank[]]]], Pattern[p, Blank[]]], Power[Plus[Pattern[d2, Blank[]], Times[Optional[Pattern[e2, Blank[]]], Pattern[x, Blank[]]]], Pattern[p, Blank[]]]], Pattern[x, Blank[Symbol]]] := With[{u = ExpandIntegrand[Px*(d1 + e1*x)^p*(d2 + e2*x)^p*(a + b*ArcCosh[c*x])^n, x]}, Int[u, x] /; SumQ[u]] /; FreeQ[{a, b, c, d1, e1, d2, e2, n}, x] && PolynomialQ[Px, x] && EqQ[e1 - c*d1, 0] && EqQ[e2 + c*d2, 0] && IntegerQ[p - 1/2]
  5851. Int[Times[Power[Plus[Optional[Pattern[a, Blank[]]], Times[ArcSinh[Times[Optional[Pattern[c, Blank[]]], Pattern[x, Blank[]]]], Optional[Pattern[b, Blank[]]]]], Optional[Pattern[n, Blank[]]]], Optional[Pattern[Px, Blank[]]], Power[Plus[Pattern[f, Blank[]], Times[Optional[Pattern[g, Blank[]]], Power[Plus[Pattern[d, Blank[]], Times[Optional[Pattern[e, Blank[]]], Power[Pattern[x, Blank[]], 2]]], Pattern[p, Blank[]]]]], Optional[Pattern[m, Blank[]]]]], Pattern[x, Blank[Symbol]]] := With[{u = ExpandIntegrand[Px*(f + g*(d + e*x^2)^p)^m*(a + b*ArcSinh[c*x])^n, x]}, Int[u, x] /; SumQ[u]] /; FreeQ[{a, b, c, d, e, f, g}, x] && PolynomialQ[Px, x] && EqQ[e, c^2*d] && IGtQ[p + 1/2, 0] && IntegersQ[m, n]
  5852. Int[Times[Power[Plus[Optional[Pattern[a, Blank[]]], Times[ArcCosh[Times[Optional[Pattern[c, Blank[]]], Pattern[x, Blank[]]]], Optional[Pattern[b, Blank[]]]]], Optional[Pattern[n, Blank[]]]], Optional[Pattern[Px, Blank[]]], Power[Plus[Pattern[f, Blank[]], Times[Optional[Pattern[g, Blank[]]], Power[Plus[Pattern[d1, Blank[]], Times[Optional[Pattern[e1, Blank[]]], Pattern[x, Blank[]]]], Pattern[p, Blank[]]], Power[Plus[Pattern[d2, Blank[]], Times[Optional[Pattern[e2, Blank[]]], Pattern[x, Blank[]]]], Pattern[p, Blank[]]]]], Optional[Pattern[m, Blank[]]]]], Pattern[x, Blank[Symbol]]] := With[{u = ExpandIntegrand[Px*(f + g*(d1 + e1*x)^p*(d2 + e2*x)^p)^m*(a + b*ArcCosh[c*x])^n, x]}, Int[u, x] /; SumQ[u]] /; FreeQ[{a, b, c, d1, e1, d2, e2, f, g}, x] && PolynomialQ[Px, x] && EqQ[e1 - c*d1, 0] && EqQ[e2 + c*d2, 0] && IGtQ[p + 1/2, 0] && IntegersQ[m, n]
  5853. Int[Times[Power[ArcSinh[Times[Optional[Pattern[c, Blank[]]], Pattern[x, Blank[]]]], Optional[Pattern[n, Blank[]]]], Pattern[RFx, Blank[]]], Pattern[x, Blank[Symbol]]] := With[{u = ExpandIntegrand[ArcSinh[c*x]^n, RFx, x]}, Int[u, x] /; SumQ[u]] /; FreeQ[c, x] && RationalFunctionQ[RFx, x] && IGtQ[n, 0]
  5854. Int[Times[Power[ArcCosh[Times[Optional[Pattern[c, Blank[]]], Pattern[x, Blank[]]]], Optional[Pattern[n, Blank[]]]], Pattern[RFx, Blank[]]], Pattern[x, Blank[Symbol]]] := With[{u = ExpandIntegrand[ArcCosh[c*x]^n, RFx, x]}, Int[u, x] /; SumQ[u]] /; FreeQ[c, x] && RationalFunctionQ[RFx, x] && IGtQ[n, 0]
  5855. Int[Times[Power[Plus[Times[ArcSinh[Times[Optional[Pattern[c, Blank[]]], Pattern[x, Blank[]]]], Optional[Pattern[b, Blank[]]]], Pattern[a, Blank[]]], Optional[Pattern[n, Blank[]]]], Pattern[RFx, Blank[]]], Pattern[x, Blank[Symbol]]] := Int[ExpandIntegrand[RFx*(a + b*ArcSinh[c*x])^n, x], x] /; FreeQ[{a, b, c}, x] && RationalFunctionQ[RFx, x] && IGtQ[n, 0]
  5856. Int[Times[Power[Plus[Times[ArcCosh[Times[Optional[Pattern[c, Blank[]]], Pattern[x, Blank[]]]], Optional[Pattern[b, Blank[]]]], Pattern[a, Blank[]]], Optional[Pattern[n, Blank[]]]], Pattern[RFx, Blank[]]], Pattern[x, Blank[Symbol]]] := Int[ExpandIntegrand[RFx*(a + b*ArcCosh[c*x])^n, x], x] /; FreeQ[{a, b, c}, x] && RationalFunctionQ[RFx, x] && IGtQ[n, 0]
  5857. Int[Times[Power[ArcSinh[Times[Optional[Pattern[c, Blank[]]], Pattern[x, Blank[]]]], Optional[Pattern[n, Blank[]]]], Pattern[RFx, Blank[]], Power[Plus[Pattern[d, Blank[]], Times[Optional[Pattern[e, Blank[]]], Power[Pattern[x, Blank[]], 2]]], Pattern[p, Blank[]]]], Pattern[x, Blank[Symbol]]] := With[{u = ExpandIntegrand[(d + e*x^2)^p*ArcSinh[c*x]^n, RFx, x]}, Int[u, x] /; SumQ[u]] /; FreeQ[{c, d, e}, x] && RationalFunctionQ[RFx, x] && IGtQ[n, 0] && EqQ[e, c^2*d] && IntegerQ[p - 1/2]
  5858. Int[Times[Power[ArcCosh[Times[Optional[Pattern[c, Blank[]]], Pattern[x, Blank[]]]], Optional[Pattern[n, Blank[]]]], Pattern[RFx, Blank[]], Power[Plus[Pattern[d1, Blank[]], Times[Optional[Pattern[e1, Blank[]]], Pattern[x, Blank[]]]], Pattern[p, Blank[]]], Power[Plus[Pattern[d2, Blank[]], Times[Optional[Pattern[e2, Blank[]]], Pattern[x, Blank[]]]], Pattern[p, Blank[]]]], Pattern[x, Blank[Symbol]]] := With[{u = ExpandIntegrand[(d1 + e1*x)^p*(d2 + e2*x)^p*ArcCosh[c*x]^n, RFx, x]}, Int[u, x] /; SumQ[u]] /; FreeQ[{c, d1, e1, d2, e2}, x] && RationalFunctionQ[RFx, x] && IGtQ[n, 0] && EqQ[e1 - c*d1, 0] && EqQ[e2 + c*d2, 0] && IntegerQ[p - 1/2]
  5859. Int[Times[Power[Plus[Times[ArcSinh[Times[Optional[Pattern[c, Blank[]]], Pattern[x, Blank[]]]], Optional[Pattern[b, Blank[]]]], Pattern[a, Blank[]]], Optional[Pattern[n, Blank[]]]], Pattern[RFx, Blank[]], Power[Plus[Pattern[d, Blank[]], Times[Optional[Pattern[e, Blank[]]], Power[Pattern[x, Blank[]], 2]]], Pattern[p, Blank[]]]], Pattern[x, Blank[Symbol]]] := Int[ExpandIntegrand[(d + e*x^2)^p, RFx*(a + b*ArcSinh[c*x])^n, x], x] /; FreeQ[{a, b, c, d, e}, x] && RationalFunctionQ[RFx, x] && IGtQ[n, 0] && EqQ[e, c^2*d] && IntegerQ[p - 1/2]
  5860. Int[Times[Power[Plus[Times[ArcCosh[Times[Optional[Pattern[c, Blank[]]], Pattern[x, Blank[]]]], Optional[Pattern[b, Blank[]]]], Pattern[a, Blank[]]], Optional[Pattern[n, Blank[]]]], Pattern[RFx, Blank[]], Power[Plus[Pattern[d1, Blank[]], Times[Optional[Pattern[e1, Blank[]]], Pattern[x, Blank[]]]], Pattern[p, Blank[]]], Power[Plus[Pattern[d2, Blank[]], Times[Optional[Pattern[e2, Blank[]]], Pattern[x, Blank[]]]], Pattern[p, Blank[]]]], Pattern[x, Blank[Symbol]]] := Int[ExpandIntegrand[(d1 + e1*x)^p*(d2 + e2*x)^p, RFx*(a + b*ArcCosh[c*x])^n, x], x] /; FreeQ[{a, b, c, d1, e1, d2, e2}, x] && RationalFunctionQ[RFx, x] && IGtQ[n, 0] && EqQ[e1 - c*d1, 0] && EqQ[e2 + c*d2, 0] && IntegerQ[p - 1/2]
  5861. Int[Times[Power[Plus[Optional[Pattern[a, Blank[]]], Times[ArcSinh[Times[Optional[Pattern[c, Blank[]]], Pattern[x, Blank[]]]], Optional[Pattern[b, Blank[]]]]], Optional[Pattern[n, Blank[]]]], Optional[Pattern[u, Blank[]]]], Pattern[x, Blank[Symbol]]] := Unintegrable[u*(a + b*ArcSinh[c*x])^n, x] /; FreeQ[{a, b, c, n}, x]
  5862. Int[Times[Power[Plus[Optional[Pattern[a, Blank[]]], Times[ArcCosh[Times[Optional[Pattern[c, Blank[]]], Pattern[x, Blank[]]]], Optional[Pattern[b, Blank[]]]]], Optional[Pattern[n, Blank[]]]], Optional[Pattern[u, Blank[]]]], Pattern[x, Blank[Symbol]]] := Unintegrable[u*(a + b*ArcCosh[c*x])^n, x] /; FreeQ[{a, b, c, n}, x]
  5863. Int[Power[Plus[Optional[Pattern[a, Blank[]]], Times[ArcSinh[Plus[Pattern[c, Blank[]], Times[Optional[Pattern[d, Blank[]]], Pattern[x, Blank[]]]]], Optional[Pattern[b, Blank[]]]]], Optional[Pattern[n, Blank[]]]], Pattern[x, Blank[Symbol]]] := Dist[1/d, Subst[Int[(a + b*ArcSinh[x])^n, x], x, c + d*x], x] /; FreeQ[{a, b, c, d, n}, x]
  5864. Int[Power[Plus[Optional[Pattern[a, Blank[]]], Times[ArcCosh[Plus[Pattern[c, Blank[]], Times[Optional[Pattern[d, Blank[]]], Pattern[x, Blank[]]]]], Optional[Pattern[b, Blank[]]]]], Optional[Pattern[n, Blank[]]]], Pattern[x, Blank[Symbol]]] := Dist[1/d, Subst[Int[(a + b*ArcCosh[x])^n, x], x, c + d*x], x] /; FreeQ[{a, b, c, d, n}, x]
  5865. Int[Times[Power[Plus[Optional[Pattern[a, Blank[]]], Times[ArcSinh[Plus[Pattern[c, Blank[]], Times[Optional[Pattern[d, Blank[]]], Pattern[x, Blank[]]]]], Optional[Pattern[b, Blank[]]]]], Optional[Pattern[n, Blank[]]]], Power[Plus[Optional[Pattern[e, Blank[]]], Times[Optional[Pattern[f, Blank[]]], Pattern[x, Blank[]]]], Optional[Pattern[m, Blank[]]]]], Pattern[x, Blank[Symbol]]] := Dist[1/d, Subst[Int[((d*e - c*f)/d + (f*x)/d)^m*(a + b*ArcSinh[x])^n, x], x, c + d*x], x] /; FreeQ[{a, b, c, d, e, f, m, n}, x]
  5866. Int[Times[Power[Plus[Optional[Pattern[a, Blank[]]], Times[ArcCosh[Plus[Pattern[c, Blank[]], Times[Optional[Pattern[d, Blank[]]], Pattern[x, Blank[]]]]], Optional[Pattern[b, Blank[]]]]], Optional[Pattern[n, Blank[]]]], Power[Plus[Optional[Pattern[e, Blank[]]], Times[Optional[Pattern[f, Blank[]]], Pattern[x, Blank[]]]], Optional[Pattern[m, Blank[]]]]], Pattern[x, Blank[Symbol]]] := Dist[1/d, Subst[Int[((d*e - c*f)/d + (f*x)/d)^m*(a + b*ArcCosh[x])^n, x], x, c + d*x], x] /; FreeQ[{a, b, c, d, e, f, m, n}, x]
  5867. Int[Times[Power[Plus[Optional[Pattern[a, Blank[]]], Times[ArcSinh[Plus[Pattern[c, Blank[]], Times[Optional[Pattern[d, Blank[]]], Pattern[x, Blank[]]]]], Optional[Pattern[b, Blank[]]]]], Optional[Pattern[n, Blank[]]]], Power[Plus[Optional[Pattern[A, Blank[]]], Times[Optional[Pattern[B, Blank[]]], Pattern[x, Blank[]]], Times[Optional[Pattern[C, Blank[]]], Power[Pattern[x, Blank[]], 2]]], Optional[Pattern[p, Blank[]]]]], Pattern[x, Blank[Symbol]]] := Dist[1/d, Subst[Int[(C/d^2 + (C*x^2)/d^2)^p*(a + b*ArcSinh[x])^n, x], x, c + d*x], x] /; FreeQ[{a, b, c, d, A, B, C, n, p}, x] && EqQ[B*(1 + c^2) - 2*A*c*d, 0] && EqQ[2*c*C - B*d, 0]
  5868. Int[Times[Power[Plus[Optional[Pattern[a, Blank[]]], Times[ArcCosh[Plus[Pattern[c, Blank[]], Times[Optional[Pattern[d, Blank[]]], Pattern[x, Blank[]]]]], Optional[Pattern[b, Blank[]]]]], Optional[Pattern[n, Blank[]]]], Power[Plus[Optional[Pattern[A, Blank[]]], Times[Optional[Pattern[B, Blank[]]], Pattern[x, Blank[]]], Times[Optional[Pattern[C, Blank[]]], Power[Pattern[x, Blank[]], 2]]], Optional[Pattern[p, Blank[]]]]], Pattern[x, Blank[Symbol]]] := Dist[1/d, Subst[Int[(-(C/d^2) + (C*x^2)/d^2)^p*(a + b*ArcCosh[x])^n, x], x, c + d*x], x] /; FreeQ[{a, b, c, d, A, B, C, n, p}, x] && EqQ[B*(1 - c^2) + 2*A*c*d, 0] && EqQ[2*c*C - B*d, 0]
  5869. Int[Times[Power[Plus[Optional[Pattern[a, Blank[]]], Times[ArcSinh[Plus[Pattern[c, Blank[]], Times[Optional[Pattern[d, Blank[]]], Pattern[x, Blank[]]]]], Optional[Pattern[b, Blank[]]]]], Optional[Pattern[n, Blank[]]]], Power[Plus[Optional[Pattern[e, Blank[]]], Times[Optional[Pattern[f, Blank[]]], Pattern[x, Blank[]]]], Optional[Pattern[m, Blank[]]]], Power[Plus[Optional[Pattern[A, Blank[]]], Times[Optional[Pattern[B, Blank[]]], Pattern[x, Blank[]]], Times[Optional[Pattern[C, Blank[]]], Power[Pattern[x, Blank[]], 2]]], Optional[Pattern[p, Blank[]]]]], Pattern[x, Blank[Symbol]]] := Dist[1/d, Subst[Int[((d*e - c*f)/d + (f*x)/d)^m*(C/d^2 + (C*x^2)/d^2)^p*(a + b*ArcSinh[x])^n, x], x, c + d*x], x] /; FreeQ[{a, b, c, d, e, f, A, B, C, m, n, p}, x] && EqQ[B*(1 + c^2) - 2*A*c*d, 0] && EqQ[2*c*C - B*d, 0]
  5870. Int[Times[Power[Plus[Optional[Pattern[a, Blank[]]], Times[ArcCosh[Plus[Pattern[c, Blank[]], Times[Optional[Pattern[d, Blank[]]], Pattern[x, Blank[]]]]], Optional[Pattern[b, Blank[]]]]], Optional[Pattern[n, Blank[]]]], Power[Plus[Optional[Pattern[e, Blank[]]], Times[Optional[Pattern[f, Blank[]]], Pattern[x, Blank[]]]], Optional[Pattern[m, Blank[]]]], Power[Plus[Optional[Pattern[A, Blank[]]], Times[Optional[Pattern[B, Blank[]]], Pattern[x, Blank[]]], Times[Optional[Pattern[C, Blank[]]], Power[Pattern[x, Blank[]], 2]]], Optional[Pattern[p, Blank[]]]]], Pattern[x, Blank[Symbol]]] := Dist[1/d, Subst[Int[((d*e - c*f)/d + (f*x)/d)^m*(-(C/d^2) + (C*x^2)/d^2)^p*(a + b*ArcCosh[x])^n, x], x, c + d*x], x] /; FreeQ[{a, b, c, d, e, f, A, B, C, m, n, p}, x] && EqQ[B*(1 - c^2) + 2*A*c*d, 0] && EqQ[2*c*C - B*d, 0]
  5871. Int[Power[Plus[Optional[Pattern[a, Blank[]]], Times[ArcSinh[Plus[Pattern[c, Blank[]], Times[Optional[Pattern[d, Blank[]]], Power[Pattern[x, Blank[]], 2]]]], Optional[Pattern[b, Blank[]]]]], Rational[1, 2]], Pattern[x, Blank[Symbol]]] := Simp[x*Sqrt[a + b*ArcSinh[c + d*x^2]], x] + (-Simp[(Sqrt[Pi]*x*(Cosh[a/(2*b)] - c*Sinh[a/(2*b)])*FresnelC[Sqrt[-(c/(Pi*b))]*Sqrt[a + b*ArcSinh[c + d*x^2]]])/(Sqrt[-(c/b)]*(Cosh[ArcSinh[c + d*x^2]/2] + c*Sinh[ArcSinh[c + d*x^2]/2])), x] + Simp[(Sqrt[Pi]*x*(Cosh[a/(2*b)] + c*Sinh[a/(2*b)])*FresnelS[Sqrt[-(c/(Pi*b))]*Sqrt[a + b*ArcSinh[c + d*x^2]]])/(Sqrt[-(c/b)]*(Cosh[ArcSinh[c + d*x^2]/2] + c*Sinh[ArcSinh[c + d*x^2]/2])), x]) /; FreeQ[{a, b, c, d}, x] && EqQ[c^2, -1]
  5872. Int[Power[Plus[Optional[Pattern[a, Blank[]]], Times[ArcSinh[Plus[Pattern[c, Blank[]], Times[Optional[Pattern[d, Blank[]]], Power[Pattern[x, Blank[]], 2]]]], Optional[Pattern[b, Blank[]]]]], Pattern[n, Blank[]]], Pattern[x, Blank[Symbol]]] := Simp[x*(a + b*ArcSinh[c + d*x^2])^n, x] + (Dist[4*b^2*n*(n - 1), Int[(a + b*ArcSinh[c + d*x^2])^(n - 2), x], x] - Simp[(2*b*n*Sqrt[2*c*d*x^2 + d^2*x^4]*(a + b*ArcSinh[c + d*x^2])^(n - 1))/(d*x), x]) /; FreeQ[{a, b, c, d}, x] && EqQ[c^2, -1] && GtQ[n, 1]
  5873. Int[Power[Plus[Optional[Pattern[a, Blank[]]], Times[ArcSinh[Plus[Pattern[c, Blank[]], Times[Optional[Pattern[d, Blank[]]], Power[Pattern[x, Blank[]], 2]]]], Optional[Pattern[b, Blank[]]]]], -1], Pattern[x, Blank[Symbol]]] := Simp[(x*(c*Cosh[a/(2*b)] - Sinh[a/(2*b)])*CoshIntegral[(a + b*ArcSinh[c + d*x^2])/(2*b)])/(2*b*(Cosh[ArcSinh[c + d*x^2]/2] + c*Sinh[(1/2)*ArcSinh[c + d*x^2]])), x] + Simp[(x*(Cosh[a/(2*b)] - c*Sinh[a/(2*b)])*SinhIntegral[(a + b*ArcSinh[c + d*x^2])/(2*b)])/(2*b*(Cosh[ArcSinh[c + d*x^2]/2] + c*Sinh[(1/2)*ArcSinh[c + d*x^2]])), x] /; FreeQ[{a, b, c, d}, x] && EqQ[c^2, -1]
  5874. Int[Power[Plus[Optional[Pattern[a, Blank[]]], Times[ArcSinh[Plus[Pattern[c, Blank[]], Times[Optional[Pattern[d, Blank[]]], Power[Pattern[x, Blank[]], 2]]]], Optional[Pattern[b, Blank[]]]]], Rational[-1, 2]], Pattern[x, Blank[Symbol]]] := Simp[((c + 1)*Sqrt[Pi/2]*x*(Cosh[a/(2*b)] - Sinh[a/(2*b)])*Erfi[Sqrt[a + b*ArcSinh[c + d*x^2]]/Sqrt[2*b]])/(2*Sqrt[b]*(Cosh[ArcSinh[c + d*x^2]/2] + c*Sinh[ArcSinh[c + d*x^2]/2])), x] + Simp[((c - 1)*Sqrt[Pi/2]*x*(Cosh[a/(2*b)] + Sinh[a/(2*b)])*Erf[Sqrt[a + b*ArcSinh[c + d*x^2]]/Sqrt[2*b]])/(2*Sqrt[b]*(Cosh[ArcSinh[c + d*x^2]/2] + c*Sinh[ArcSinh[c + d*x^2]/2])), x] /; FreeQ[{a, b, c, d}, x] && EqQ[c^2, -1]
  5875. Int[Power[Plus[Optional[Pattern[a, Blank[]]], Times[ArcSinh[Plus[Pattern[c, Blank[]], Times[Optional[Pattern[d, Blank[]]], Power[Pattern[x, Blank[]], 2]]]], Optional[Pattern[b, Blank[]]]]], Rational[-3, 2]], Pattern[x, Blank[Symbol]]] := -Simp[Sqrt[2*c*d*x^2 + d^2*x^4]/(b*d*x*Sqrt[a + b*ArcSinh[c + d*x^2]]), x] + (-Simp[((-(c/b))^(3/2)*Sqrt[Pi]*x*(Cosh[a/(2*b)] - c*Sinh[a/(2*b)])*FresnelC[Sqrt[-(c/(Pi*b))]*Sqrt[a + b*ArcSinh[c + d*x^2]]])/(Cosh[ArcSinh[c + d*x^2]/2] + c*Sinh[ArcSinh[c + d*x^2]/2]), x] + Simp[((-(c/b))^(3/2)*Sqrt[Pi]*x*(Cosh[a/(2*b)] + c*Sinh[a/(2*b)])*FresnelS[Sqrt[-(c/(Pi*b))]*Sqrt[a + b*ArcSinh[c + d*x^2]]])/(Cosh[ArcSinh[c + d*x^2]/2] + c*Sinh[ArcSinh[c + d*x^2]/2]), x]) /; FreeQ[{a, b, c, d}, x] && EqQ[c^2, -1]
  5876. Int[Power[Plus[Optional[Pattern[a, Blank[]]], Times[ArcSinh[Plus[Pattern[c, Blank[]], Times[Optional[Pattern[d, Blank[]]], Power[Pattern[x, Blank[]], 2]]]], Optional[Pattern[b, Blank[]]]]], -2], Pattern[x, Blank[Symbol]]] := -Simp[Sqrt[2*c*d*x^2 + d^2*x^4]/(2*b*d*x*(a + b*ArcSinh[c + d*x^2])), x] + (Simp[(x*(Cosh[a/(2*b)] - c*Sinh[a/(2*b)])*CoshIntegral[(a + b*ArcSinh[c + d*x^2])/(2*b)])/(4*b^2*(Cosh[ArcSinh[c + d*x^2]/2] + c*Sinh[ArcSinh[c + d*x^2]/2])), x] + Simp[(x*(c*Cosh[a/(2*b)] - Sinh[a/(2*b)])*SinhIntegral[(a + b*ArcSinh[c + d*x^2])/(2*b)])/(4*b^2*(Cosh[ArcSinh[c + d*x^2]/2] + c*Sinh[ArcSinh[c + d*x^2]/2])), x]) /; FreeQ[{a, b, c, d}, x] && EqQ[c^2, -1]
  5877. Int[Power[Plus[Optional[Pattern[a, Blank[]]], Times[ArcSinh[Plus[Pattern[c, Blank[]], Times[Optional[Pattern[d, Blank[]]], Power[Pattern[x, Blank[]], 2]]]], Optional[Pattern[b, Blank[]]]]], Pattern[n, Blank[]]], Pattern[x, Blank[Symbol]]] := -Simp[(x*(a + b*ArcSinh[c + d*x^2])^(n + 2))/(4*b^2*(n + 1)*(n + 2)), x] + (Dist[1/(4*b^2*(n + 1)*(n + 2)), Int[(a + b*ArcSinh[c + d*x^2])^(n + 2), x], x] + Simp[(Sqrt[2*c*d*x^2 + d^2*x^4]*(a + b*ArcSinh[c + d*x^2])^(n + 1))/(2*b*d*(n + 1)*x), x]) /; FreeQ[{a, b, c, d}, x] && EqQ[c^2, -1] && LtQ[n, -1] && NeQ[n, -2]
  5878. Int[Power[Plus[Optional[Pattern[a, Blank[]]], Times[ArcCosh[Plus[1, Times[Optional[Pattern[d, Blank[]]], Power[Pattern[x, Blank[]], 2]]]], Optional[Pattern[b, Blank[]]]]], Rational[1, 2]], Pattern[x, Blank[Symbol]]] := Simp[(2*Sqrt[a + b*ArcCosh[1 + d*x^2]]*Sinh[(1/2)*ArcCosh[1 + d*x^2]]^2)/(d*x), x] + (Simp[(Sqrt[b]*Sqrt[Pi/2]*(Cosh[a/(2*b)] + Sinh[a/(2*b)])*Sinh[(1/2)*ArcCosh[1 + d*x^2]]*Erf[(1/Sqrt[2*b])*Sqrt[a + b*ArcCosh[1 + d*x^2]]])/(d*x), x] - Simp[(Sqrt[b]*Sqrt[Pi/2]*(Cosh[a/(2*b)] - Sinh[a/(2*b)])*Sinh[(1/2)*ArcCosh[1 + d*x^2]]*Erfi[(1/Sqrt[2*b])*Sqrt[a + b*ArcCosh[1 + d*x^2]]])/(d*x), x]) /; FreeQ[{a, b, d}, x]
  5879. Int[Power[Plus[Optional[Pattern[a, Blank[]]], Times[ArcCosh[Plus[-1, Times[Optional[Pattern[d, Blank[]]], Power[Pattern[x, Blank[]], 2]]]], Optional[Pattern[b, Blank[]]]]], Rational[1, 2]], Pattern[x, Blank[Symbol]]] := Simp[(2*Sqrt[a + b*ArcCosh[-1 + d*x^2]]*Cosh[(1/2)*ArcCosh[-1 + d*x^2]]^2)/(d*x), x] + (-Simp[(Sqrt[b]*Sqrt[Pi/2]*(Cosh[a/(2*b)] + Sinh[a/(2*b)])*Cosh[(1/2)*ArcCosh[-1 + d*x^2]]*Erf[(1/Sqrt[2*b])*Sqrt[a + b*ArcCosh[-1 + d*x^2]]])/(d*x), x] - Simp[(Sqrt[b]*Sqrt[Pi/2]*(Cosh[a/(2*b)] - Sinh[a/(2*b)])*Cosh[(1/2)*ArcCosh[-1 + d*x^2]]*Erfi[(1/Sqrt[2*b])*Sqrt[a + b*ArcCosh[-1 + d*x^2]]])/(d*x), x]) /; FreeQ[{a, b, d}, x]
  5880. Int[Power[Plus[Optional[Pattern[a, Blank[]]], Times[ArcCosh[Plus[Pattern[c, Blank[]], Times[Optional[Pattern[d, Blank[]]], Power[Pattern[x, Blank[]], 2]]]], Optional[Pattern[b, Blank[]]]]], Pattern[n, Blank[]]], Pattern[x, Blank[Symbol]]] := Simp[x*(a + b*ArcCosh[c + d*x^2])^n, x] + (Dist[4*b^2*n*(n - 1), Int[(a + b*ArcCosh[c + d*x^2])^(n - 2), x], x] - Simp[(2*b*n*(2*c*d*x^2 + d^2*x^4)*(a + b*ArcCosh[c + d*x^2])^(n - 1))/(d*x*Sqrt[-1 + c + d*x^2]*Sqrt[1 + c + d*x^2]), x]) /; FreeQ[{a, b, c, d}, x] && EqQ[c^2, 1] && GtQ[n, 1]
  5881. Int[Power[Plus[Optional[Pattern[a, Blank[]]], Times[ArcCosh[Plus[1, Times[Optional[Pattern[d, Blank[]]], Power[Pattern[x, Blank[]], 2]]]], Optional[Pattern[b, Blank[]]]]], -1], Pattern[x, Blank[Symbol]]] := Simp[(x*Cosh[a/(2*b)]*CoshIntegral[(a + b*ArcCosh[1 + d*x^2])/(2*b)])/(Sqrt[2]*b*Sqrt[d*x^2]), x] - Simp[(x*Sinh[a/(2*b)]*SinhIntegral[(a + b*ArcCosh[1 + d*x^2])/(2*b)])/(Sqrt[2]*b*Sqrt[d*x^2]), x] /; FreeQ[{a, b, d}, x]
  5882. Int[Power[Plus[Optional[Pattern[a, Blank[]]], Times[ArcCosh[Plus[-1, Times[Optional[Pattern[d, Blank[]]], Power[Pattern[x, Blank[]], 2]]]], Optional[Pattern[b, Blank[]]]]], -1], Pattern[x, Blank[Symbol]]] := -Simp[(x*Sinh[a/(2*b)]*CoshIntegral[(a + b*ArcCosh[-1 + d*x^2])/(2*b)])/(Sqrt[2]*b*Sqrt[d*x^2]), x] + Simp[(x*Cosh[a/(2*b)]*SinhIntegral[(a + b*ArcCosh[-1 + d*x^2])/(2*b)])/(Sqrt[2]*b*Sqrt[d*x^2]), x] /; FreeQ[{a, b, d}, x]
  5883. Int[Power[Plus[Optional[Pattern[a, Blank[]]], Times[ArcCosh[Plus[1, Times[Optional[Pattern[d, Blank[]]], Power[Pattern[x, Blank[]], 2]]]], Optional[Pattern[b, Blank[]]]]], Rational[-1, 2]], Pattern[x, Blank[Symbol]]] := Simp[(Sqrt[Pi/2]*(Cosh[a/(2*b)] - Sinh[a/(2*b)])*Sinh[ArcCosh[1 + d*x^2]/2]*Erfi[Sqrt[a + b*ArcCosh[1 + d*x^2]]/Sqrt[2*b]])/(Sqrt[b]*d*x), x] + Simp[(Sqrt[Pi/2]*(Cosh[a/(2*b)] + Sinh[a/(2*b)])*Sinh[ArcCosh[1 + d*x^2]/2]*Erf[Sqrt[a + b*ArcCosh[1 + d*x^2]]/Sqrt[2*b]])/(Sqrt[b]*d*x), x] /; FreeQ[{a, b, d}, x]
  5884. Int[Power[Plus[Optional[Pattern[a, Blank[]]], Times[ArcCosh[Plus[-1, Times[Optional[Pattern[d, Blank[]]], Power[Pattern[x, Blank[]], 2]]]], Optional[Pattern[b, Blank[]]]]], Rational[-1, 2]], Pattern[x, Blank[Symbol]]] := Simp[(Sqrt[Pi/2]*(Cosh[a/(2*b)] - Sinh[a/(2*b)])*Cosh[ArcCosh[-1 + d*x^2]/2]*Erfi[Sqrt[a + b*ArcCosh[-1 + d*x^2]]/Sqrt[2*b]])/(Sqrt[b]*d*x), x] - Simp[(Sqrt[Pi/2]*(Cosh[a/(2*b)] + Sinh[a/(2*b)])*Cosh[ArcCosh[-1 + d*x^2]/2]*Erf[Sqrt[a + b*ArcCosh[-1 + d*x^2]]/Sqrt[2*b]])/(Sqrt[b]*d*x), x] /; FreeQ[{a, b, d}, x]
  5885. Int[Power[Plus[Optional[Pattern[a, Blank[]]], Times[ArcCosh[Plus[1, Times[Optional[Pattern[d, Blank[]]], Power[Pattern[x, Blank[]], 2]]]], Optional[Pattern[b, Blank[]]]]], Rational[-3, 2]], Pattern[x, Blank[Symbol]]] := -Simp[(Sqrt[d*x^2]*Sqrt[2 + d*x^2])/(b*d*x*Sqrt[a + b*ArcCosh[1 + d*x^2]]), x] + (-Simp[(Sqrt[Pi/2]*(Cosh[a/(2*b)] + Sinh[a/(2*b)])*Sinh[ArcCosh[1 + d*x^2]/2]*Erf[Sqrt[a + b*ArcCosh[1 + d*x^2]]/Sqrt[2*b]])/(b^(3/2)*d*x), x] + Simp[(Sqrt[Pi/2]*(Cosh[a/(2*b)] - Sinh[a/(2*b)])*Sinh[ArcCosh[1 + d*x^2]/2]*Erfi[Sqrt[a + b*ArcCosh[1 + d*x^2]]/Sqrt[2*b]])/(b^(3/2)*d*x), x]) /; FreeQ[{a, b, d}, x]
  5886. Int[Power[Plus[Optional[Pattern[a, Blank[]]], Times[ArcCosh[Plus[-1, Times[Optional[Pattern[d, Blank[]]], Power[Pattern[x, Blank[]], 2]]]], Optional[Pattern[b, Blank[]]]]], Rational[-3, 2]], Pattern[x, Blank[Symbol]]] := -Simp[(Sqrt[d*x^2]*Sqrt[-2 + d*x^2])/(b*d*x*Sqrt[a + b*ArcCosh[-1 + d*x^2]]), x] + (Simp[(Sqrt[Pi/2]*(Cosh[a/(2*b)] + Sinh[a/(2*b)])*Cosh[ArcCosh[-1 + d*x^2]/2]*Erf[Sqrt[a + b*ArcCosh[-1 + d*x^2]]/Sqrt[2*b]])/(b^(3/2)*d*x), x] + Simp[(Sqrt[Pi/2]*(Cosh[a/(2*b)] - Sinh[a/(2*b)])*Cosh[ArcCosh[-1 + d*x^2]/2]*Erfi[Sqrt[a + b*ArcCosh[-1 + d*x^2]]/Sqrt[2*b]])/(b^(3/2)*d*x), x]) /; FreeQ[{a, b, d}, x]
  5887. Int[Power[Plus[Optional[Pattern[a, Blank[]]], Times[ArcCosh[Plus[1, Times[Optional[Pattern[d, Blank[]]], Power[Pattern[x, Blank[]], 2]]]], Optional[Pattern[b, Blank[]]]]], -2], Pattern[x, Blank[Symbol]]] := -Simp[(Sqrt[d*x^2]*Sqrt[2 + d*x^2])/(2*b*d*x*(a + b*ArcCosh[1 + d*x^2])), x] + (-Simp[(x*Sinh[a/(2*b)]*CoshIntegral[(a + b*ArcCosh[1 + d*x^2])/(2*b)])/(2*Sqrt[2]*b^2*Sqrt[d*x^2]), x] + Simp[(x*Cosh[a/(2*b)]*SinhIntegral[(a + b*ArcCosh[1 + d*x^2])/(2*b)])/(2*Sqrt[2]*b^2*Sqrt[d*x^2]), x]) /; FreeQ[{a, b, d}, x]
  5888. Int[Power[Plus[Optional[Pattern[a, Blank[]]], Times[ArcCosh[Plus[-1, Times[Optional[Pattern[d, Blank[]]], Power[Pattern[x, Blank[]], 2]]]], Optional[Pattern[b, Blank[]]]]], -2], Pattern[x, Blank[Symbol]]] := -Simp[(Sqrt[d*x^2]*Sqrt[-2 + d*x^2])/(2*b*d*x*(a + b*ArcCosh[-1 + d*x^2])), x] + (Simp[(x*Cosh[a/(2*b)]*CoshIntegral[(a + b*ArcCosh[-1 + d*x^2])/(2*b)])/(2*Sqrt[2]*b^2*Sqrt[d*x^2]), x] - Simp[(x*Sinh[a/(2*b)]*SinhIntegral[(a + b*ArcCosh[-1 + d*x^2])/(2*b)])/(2*Sqrt[2]*b^2*Sqrt[d*x^2]), x]) /; FreeQ[{a, b, d}, x]
  5889. Int[Power[Plus[Optional[Pattern[a, Blank[]]], Times[ArcCosh[Plus[Pattern[c, Blank[]], Times[Optional[Pattern[d, Blank[]]], Power[Pattern[x, Blank[]], 2]]]], Optional[Pattern[b, Blank[]]]]], Pattern[n, Blank[]]], Pattern[x, Blank[Symbol]]] := -Simp[(x*(a + b*ArcCosh[c + d*x^2])^(n + 2))/(4*b^2*(n + 1)*(n + 2)), x] + (Dist[1/(4*b^2*(n + 1)*(n + 2)), Int[(a + b*ArcCosh[c + d*x^2])^(n + 2), x], x] + Simp[((2*c*x^2 + d*x^4)*(a + b*ArcCosh[c + d*x^2])^(n + 1))/(2*b*(n + 1)*x*Sqrt[-1 + c + d*x^2]*Sqrt[1 + c + d*x^2]), x]) /; FreeQ[{a, b, c, d}, x] && EqQ[c^2, 1] && LtQ[n, -1] && NeQ[n, -2]
  5890. Int[Times[Power[ArcSinh[Times[Optional[Pattern[a, Blank[]]], Power[Pattern[x, Blank[]], Pattern[p, Blank[]]]]], Optional[Pattern[n, Blank[]]]], Power[Pattern[x, Blank[]], -1]], Pattern[x, Blank[Symbol]]] := Dist[1/p, Subst[Int[x^n*Coth[x], x], x, ArcSinh[a*x^p]], x] /; FreeQ[{a, p}, x] && IGtQ[n, 0]
  5891. Int[Times[Power[ArcCosh[Times[Optional[Pattern[a, Blank[]]], Power[Pattern[x, Blank[]], Pattern[p, Blank[]]]]], Optional[Pattern[n, Blank[]]]], Power[Pattern[x, Blank[]], -1]], Pattern[x, Blank[Symbol]]] := Dist[1/p, Subst[Int[x^n*Tanh[x], x], x, ArcCosh[a*x^p]], x] /; FreeQ[{a, p}, x] && IGtQ[n, 0]
  5892. Int[Times[Power[ArcSinh[Times[Optional[Pattern[c, Blank[]]], Power[Plus[Optional[Pattern[a, Blank[]]], Times[Optional[Pattern[b, Blank[]]], Power[Pattern[x, Blank[]], Optional[Pattern[n, Blank[]]]]]], -1]]], Optional[Pattern[m, Blank[]]]], Optional[Pattern[u, Blank[]]]], Pattern[x, Blank[Symbol]]] := Int[u*ArcCsch[a/c + (b*x^n)/c]^m, x] /; FreeQ[{a, b, c, n, m}, x]
  5893. Int[Times[Power[ArcCosh[Times[Optional[Pattern[c, Blank[]]], Power[Plus[Optional[Pattern[a, Blank[]]], Times[Optional[Pattern[b, Blank[]]], Power[Pattern[x, Blank[]], Optional[Pattern[n, Blank[]]]]]], -1]]], Optional[Pattern[m, Blank[]]]], Optional[Pattern[u, Blank[]]]], Pattern[x, Blank[Symbol]]] := Int[u*ArcSech[a/c + (b*x^n)/c]^m, x] /; FreeQ[{a, b, c, n, m}, x]
  5894. Int[Times[Power[ArcSinh[Power[Plus[-1, Times[Optional[Pattern[b, Blank[]]], Power[Pattern[x, Blank[]], 2]]], Rational[1, 2]]], Optional[Pattern[n, Blank[]]]], Power[Plus[-1, Times[Optional[Pattern[b, Blank[]]], Power[Pattern[x, Blank[]], 2]]], Rational[-1, 2]]], Pattern[x, Blank[Symbol]]] := Dist[Sqrt[b*x^2]/(b*x), Subst[Int[ArcSinh[x]^n/Sqrt[1 + x^2], x], x, Sqrt[-1 + b*x^2]], x] /; FreeQ[{b, n}, x]
  5895. Int[Times[Power[ArcCosh[Power[Plus[1, Times[Optional[Pattern[b, Blank[]]], Power[Pattern[x, Blank[]], 2]]], Rational[1, 2]]], Optional[Pattern[n, Blank[]]]], Power[Plus[1, Times[Optional[Pattern[b, Blank[]]], Power[Pattern[x, Blank[]], 2]]], Rational[-1, 2]]], Pattern[x, Blank[Symbol]]] := Dist[(Sqrt[-1 + Sqrt[1 + b*x^2]]*Sqrt[1 + Sqrt[1 + b*x^2]])/(b*x), Subst[Int[ArcCosh[x]^n/(Sqrt[-1 + x]*Sqrt[1 + x]), x], x, Sqrt[1 + b*x^2]], x] /; FreeQ[{b, n}, x]
  5896. Int[Power[Pattern[f, Blank[]], Times[Power[ArcSinh[Plus[Optional[Pattern[a, Blank[]]], Times[Optional[Pattern[b, Blank[]]], Pattern[x, Blank[]]]]], Optional[Pattern[n, Blank[]]]], Optional[Pattern[c, Blank[]]]]], Pattern[x, Blank[Symbol]]] := Dist[1/b, Subst[Int[f^(c*x^n)*Cosh[x], x], x, ArcSinh[a + b*x]], x] /; FreeQ[{a, b, c, f}, x] && IGtQ[n, 0]
  5897. Int[Power[Pattern[f, Blank[]], Times[Power[ArcCosh[Plus[Optional[Pattern[a, Blank[]]], Times[Optional[Pattern[b, Blank[]]], Pattern[x, Blank[]]]]], Optional[Pattern[n, Blank[]]]], Optional[Pattern[c, Blank[]]]]], Pattern[x, Blank[Symbol]]] := Dist[1/b, Subst[Int[f^(c*x^n)*Sinh[x], x], x, ArcCosh[a + b*x]], x] /; FreeQ[{a, b, c, f}, x] && IGtQ[n, 0]
  5898. Int[Times[Power[Pattern[f, Blank[]], Times[Power[ArcSinh[Plus[Optional[Pattern[a, Blank[]]], Times[Optional[Pattern[b, Blank[]]], Pattern[x, Blank[]]]]], Optional[Pattern[n, Blank[]]]], Optional[Pattern[c, Blank[]]]]], Power[Pattern[x, Blank[]], Optional[Pattern[m, Blank[]]]]], Pattern[x, Blank[Symbol]]] := Dist[1/b, Subst[Int[(-(a/b) + Sinh[x]/b)^m*f^(c*x^n)*Cosh[x], x], x, ArcSinh[a + b*x]], x] /; FreeQ[{a, b, c, f}, x] && IGtQ[m, 0] && IGtQ[n, 0]
  5899. Int[Times[Power[Pattern[f, Blank[]], Times[Power[ArcCosh[Plus[Optional[Pattern[a, Blank[]]], Times[Optional[Pattern[b, Blank[]]], Pattern[x, Blank[]]]]], Optional[Pattern[n, Blank[]]]], Optional[Pattern[c, Blank[]]]]], Power[Pattern[x, Blank[]], Optional[Pattern[m, Blank[]]]]], Pattern[x, Blank[Symbol]]] := Dist[1/b, Subst[Int[(-(a/b) + Cosh[x]/b)^m*f^(c*x^n)*Sinh[x], x], x, ArcCosh[a + b*x]], x] /; FreeQ[{a, b, c, f}, x] && IGtQ[m, 0] && IGtQ[n, 0]
  5900. Int[ArcSinh[Pattern[u, Blank[]]], Pattern[x, Blank[Symbol]]] := Simp[x*ArcSinh[u], x] - Int[SimplifyIntegrand[(x*D[u, x])/Sqrt[1 + u^2], x], x] /; InverseFunctionFreeQ[u, x] && !FunctionOfExponentialQ[u, x]
  5901. Int[ArcCosh[Pattern[u, Blank[]]], Pattern[x, Blank[Symbol]]] := Simp[x*ArcCosh[u], x] - Int[SimplifyIntegrand[(x*D[u, x])/(Sqrt[-1 + u]*Sqrt[1 + u]), x], x] /; InverseFunctionFreeQ[u, x] && !FunctionOfExponentialQ[u, x]
  5902. Int[Times[Plus[Optional[Pattern[a, Blank[]]], Times[ArcSinh[Pattern[u, Blank[]]], Optional[Pattern[b, Blank[]]]]], Power[Plus[Optional[Pattern[c, Blank[]]], Times[Optional[Pattern[d, Blank[]]], Pattern[x, Blank[]]]], Optional[Pattern[m, Blank[]]]]], Pattern[x, Blank[Symbol]]] := Simp[((c + d*x)^(m + 1)*(a + b*ArcSinh[u]))/(d*(m + 1)), x] - Dist[b/(d*(m + 1)), Int[SimplifyIntegrand[((c + d*x)^(m + 1)*D[u, x])/Sqrt[1 + u^2], x], x], x] /; FreeQ[{a, b, c, d, m}, x] && NeQ[m, -1] && InverseFunctionFreeQ[u, x] && !FunctionOfQ[(c + d*x)^(m + 1), u, x] && !FunctionOfExponentialQ[u, x]
  5903. Int[Times[Plus[Optional[Pattern[a, Blank[]]], Times[ArcCosh[Pattern[u, Blank[]]], Optional[Pattern[b, Blank[]]]]], Power[Plus[Optional[Pattern[c, Blank[]]], Times[Optional[Pattern[d, Blank[]]], Pattern[x, Blank[]]]], Optional[Pattern[m, Blank[]]]]], Pattern[x, Blank[Symbol]]] := Simp[((c + d*x)^(m + 1)*(a + b*ArcCosh[u]))/(d*(m + 1)), x] - Dist[b/(d*(m + 1)), Int[SimplifyIntegrand[((c + d*x)^(m + 1)*D[u, x])/(Sqrt[-1 + u]*Sqrt[1 + u]), x], x], x] /; FreeQ[{a, b, c, d, m}, x] && NeQ[m, -1] && InverseFunctionFreeQ[u, x] && !FunctionOfQ[(c + d*x)^(m + 1), u, x] && !FunctionOfExponentialQ[u, x]
  5904. Int[Times[Plus[Optional[Pattern[a, Blank[]]], Times[ArcSinh[Pattern[u, Blank[]]], Optional[Pattern[b, Blank[]]]]], Pattern[v, Blank[]]], Pattern[x, Blank[Symbol]]] := With[{w = IntHide[v, x]}, Dist[a + b*ArcSinh[u], w, x] - Dist[b, Int[SimplifyIntegrand[(w*D[u, x])/Sqrt[1 + u^2], x], x], x] /; InverseFunctionFreeQ[w, x]] /; FreeQ[{a, b}, x] && InverseFunctionFreeQ[u, x] && !MatchQ[v, ((c_.) + (d_.)*x)^(m_.) /; FreeQ[{c, d, m}, x]]
  5905. Int[Times[Plus[Optional[Pattern[a, Blank[]]], Times[ArcCosh[Pattern[u, Blank[]]], Optional[Pattern[b, Blank[]]]]], Pattern[v, Blank[]]], Pattern[x, Blank[Symbol]]] := With[{w = IntHide[v, x]}, Dist[a + b*ArcCosh[u], w, x] - Dist[b, Int[SimplifyIntegrand[(w*D[u, x])/(Sqrt[-1 + u]*Sqrt[1 + u]), x], x], x] /; InverseFunctionFreeQ[w, x]] /; FreeQ[{a, b}, x] && InverseFunctionFreeQ[u, x] && !MatchQ[v, ((c_.) + (d_.)*x)^(m_.) /; FreeQ[{c, d, m}, x]]
  5906. Int[Power[E, Times[ArcSinh[Pattern[u, Blank[]]], Optional[Pattern[n, Blank[]]]]], Pattern[x, Blank[Symbol]]] := Int[(u + Sqrt[1 + u^2])^n, x] /; IntegerQ[n] && PolynomialQ[u, x]
  5907. Int[Times[Power[E, Times[ArcSinh[Pattern[u, Blank[]]], Optional[Pattern[n, Blank[]]]]], Power[Pattern[x, Blank[]], Optional[Pattern[m, Blank[]]]]], Pattern[x, Blank[Symbol]]] := Int[x^m*(u + Sqrt[1 + u^2])^n, x] /; RationalQ[m] && IntegerQ[n] && PolynomialQ[u, x]
  5908. Int[Power[E, Times[ArcCosh[Pattern[u, Blank[]]], Optional[Pattern[n, Blank[]]]]], Pattern[x, Blank[Symbol]]] := Int[(u + Sqrt[-1 + u]*Sqrt[1 + u])^n, x] /; IntegerQ[n] && PolynomialQ[u, x]
  5909. Int[Times[Power[E, Times[ArcCosh[Pattern[u, Blank[]]], Optional[Pattern[n, Blank[]]]]], Power[Pattern[x, Blank[]], Optional[Pattern[m, Blank[]]]]], Pattern[x, Blank[Symbol]]] := Int[x^m*(u + Sqrt[-1 + u]*Sqrt[1 + u])^n, x] /; RationalQ[m] && IntegerQ[n] && PolynomialQ[u, x]
  5910. Int[Power[Plus[Optional[Pattern[a, Blank[]]], Times[ArcTanh[Times[Optional[Pattern[c, Blank[]]], Pattern[x, Blank[]]]], Optional[Pattern[b, Blank[]]]]], Optional[Pattern[p, Blank[]]]], Pattern[x, Blank[Symbol]]] := Simp[x*(a + b*ArcTanh[c*x])^p, x] - Dist[b*c*p, Int[(x*(a + b*ArcTanh[c*x])^(p - 1))/(1 - c^2*x^2), x], x] /; FreeQ[{a, b, c}, x] && IGtQ[p, 0]
  5911. Int[Power[Plus[Optional[Pattern[a, Blank[]]], Times[ArcCoth[Times[Optional[Pattern[c, Blank[]]], Pattern[x, Blank[]]]], Optional[Pattern[b, Blank[]]]]], Optional[Pattern[p, Blank[]]]], Pattern[x, Blank[Symbol]]] := Simp[x*(a + b*ArcCoth[c*x])^p, x] - Dist[b*c*p, Int[(x*(a + b*ArcCoth[c*x])^(p - 1))/(1 - c^2*x^2), x], x] /; FreeQ[{a, b, c}, x] && IGtQ[p, 0]
  5912. Int[Times[Plus[Optional[Pattern[a, Blank[]]], Times[ArcTanh[Times[Optional[Pattern[c, Blank[]]], Pattern[x, Blank[]]]], Optional[Pattern[b, Blank[]]]]], Power[Pattern[x, Blank[]], -1]], Pattern[x, Blank[Symbol]]] := Simp[a*Log[x], x] + (-Simp[(b*PolyLog[2, -(c*x)])/2, x] + Simp[(b*PolyLog[2, c*x])/2, x]) /; FreeQ[{a, b, c}, x]
  5913. Int[Times[Plus[Optional[Pattern[a, Blank[]]], Times[ArcCoth[Times[Optional[Pattern[c, Blank[]]], Pattern[x, Blank[]]]], Optional[Pattern[b, Blank[]]]]], Power[Pattern[x, Blank[]], -1]], Pattern[x, Blank[Symbol]]] := Simp[a*Log[x], x] + (Simp[(b*PolyLog[2, -(c*x)^(-1)])/2, x] - Simp[(b*PolyLog[2, 1/(c*x)])/2, x]) /; FreeQ[{a, b, c}, x]
  5914. Int[Times[Power[Plus[Optional[Pattern[a, Blank[]]], Times[ArcTanh[Times[Optional[Pattern[c, Blank[]]], Pattern[x, Blank[]]]], Optional[Pattern[b, Blank[]]]]], Pattern[p, Blank[]]], Power[Pattern[x, Blank[]], -1]], Pattern[x, Blank[Symbol]]] := Simp[2*(a + b*ArcTanh[c*x])^p*ArcTanh[1 - 2/(1 - c*x)], x] - Dist[2*b*c*p, Int[((a + b*ArcTanh[c*x])^(p - 1)*ArcTanh[1 - 2/(1 - c*x)])/(1 - c^2*x^2), x], x] /; FreeQ[{a, b, c}, x] && IGtQ[p, 1]
  5915. Int[Times[Power[Plus[Optional[Pattern[a, Blank[]]], Times[ArcCoth[Times[Optional[Pattern[c, Blank[]]], Pattern[x, Blank[]]]], Optional[Pattern[b, Blank[]]]]], Pattern[p, Blank[]]], Power[Pattern[x, Blank[]], -1]], Pattern[x, Blank[Symbol]]] := Simp[2*(a + b*ArcCoth[c*x])^p*ArcCoth[1 - 2/(1 - c*x)], x] - Dist[2*b*c*p, Int[((a + b*ArcCoth[c*x])^(p - 1)*ArcCoth[1 - 2/(1 - c*x)])/(1 - c^2*x^2), x], x] /; FreeQ[{a, b, c}, x] && IGtQ[p, 1]
  5916. Int[Times[Power[Plus[Optional[Pattern[a, Blank[]]], Times[ArcTanh[Times[Optional[Pattern[c, Blank[]]], Pattern[x, Blank[]]]], Optional[Pattern[b, Blank[]]]]], Optional[Pattern[p, Blank[]]]], Power[Times[Optional[Pattern[d, Blank[]]], Pattern[x, Blank[]]], Optional[Pattern[m, Blank[]]]]], Pattern[x, Blank[Symbol]]] := Simp[((d*x)^(m + 1)*(a + b*ArcTanh[c*x])^p)/(d*(m + 1)), x] - Dist[(b*c*p)/(d*(m + 1)), Int[((d*x)^(m + 1)*(a + b*ArcTanh[c*x])^(p - 1))/(1 - c^2*x^2), x], x] /; FreeQ[{a, b, c, d, m}, x] && IGtQ[p, 0] && (EqQ[p, 1] || IntegerQ[m]) && NeQ[m, -1]
  5917. Int[Times[Power[Plus[Optional[Pattern[a, Blank[]]], Times[ArcCoth[Times[Optional[Pattern[c, Blank[]]], Pattern[x, Blank[]]]], Optional[Pattern[b, Blank[]]]]], Optional[Pattern[p, Blank[]]]], Power[Times[Optional[Pattern[d, Blank[]]], Pattern[x, Blank[]]], Optional[Pattern[m, Blank[]]]]], Pattern[x, Blank[Symbol]]] := Simp[((d*x)^(m + 1)*(a + b*ArcCoth[c*x])^p)/(d*(m + 1)), x] - Dist[(b*c*p)/(d*(m + 1)), Int[((d*x)^(m + 1)*(a + b*ArcCoth[c*x])^(p - 1))/(1 - c^2*x^2), x], x] /; FreeQ[{a, b, c, d, m}, x] && IGtQ[p, 0] && (EqQ[p, 1] || IntegerQ[m]) && NeQ[m, -1]
  5918. Int[Times[Power[Plus[Optional[Pattern[a, Blank[]]], Times[ArcTanh[Times[Optional[Pattern[c, Blank[]]], Pattern[x, Blank[]]]], Optional[Pattern[b, Blank[]]]]], Optional[Pattern[p, Blank[]]]], Power[Plus[Pattern[d, Blank[]], Times[Optional[Pattern[e, Blank[]]], Pattern[x, Blank[]]]], -1]], Pattern[x, Blank[Symbol]]] := -Simp[((a + b*ArcTanh[c*x])^p*Log[2/(1 + (e*x)/d)])/e, x] + Dist[(b*c*p)/e, Int[((a + b*ArcTanh[c*x])^(p - 1)*Log[2/(1 + (e*x)/d)])/(1 - c^2*x^2), x], x] /; FreeQ[{a, b, c, d, e}, x] && IGtQ[p, 0] && EqQ[c^2*d^2 - e^2, 0]
  5919. Int[Times[Power[Plus[Optional[Pattern[a, Blank[]]], Times[ArcCoth[Times[Optional[Pattern[c, Blank[]]], Pattern[x, Blank[]]]], Optional[Pattern[b, Blank[]]]]], Optional[Pattern[p, Blank[]]]], Power[Plus[Pattern[d, Blank[]], Times[Optional[Pattern[e, Blank[]]], Pattern[x, Blank[]]]], -1]], Pattern[x, Blank[Symbol]]] := -Simp[((a + b*ArcCoth[c*x])^p*Log[2/(1 + (e*x)/d)])/e, x] + Dist[(b*c*p)/e, Int[((a + b*ArcCoth[c*x])^(p - 1)*Log[2/(1 + (e*x)/d)])/(1 - c^2*x^2), x], x] /; FreeQ[{a, b, c, d, e}, x] && IGtQ[p, 0] && EqQ[c^2*d^2 - e^2, 0]
  5920. Int[Times[Plus[Optional[Pattern[a, Blank[]]], Times[ArcTanh[Times[Optional[Pattern[c, Blank[]]], Pattern[x, Blank[]]]], Optional[Pattern[b, Blank[]]]]], Power[Plus[Pattern[d, Blank[]], Times[Optional[Pattern[e, Blank[]]], Pattern[x, Blank[]]]], -1]], Pattern[x, Blank[Symbol]]] := -Simp[((a + b*ArcTanh[c*x])*Log[2/(1 + c*x)])/e, x] + (Dist[(b*c)/e, Int[Log[2/(1 + c*x)]/(1 - c^2*x^2), x], x] - Dist[(b*c)/e, Int[Log[(2*c*(d + e*x))/((c*d + e)*(1 + c*x))]/(1 - c^2*x^2), x], x] + Simp[((a + b*ArcTanh[c*x])*Log[(2*c*(d + e*x))/((c*d + e)*(1 + c*x))])/e, x]) /; FreeQ[{a, b, c, d, e}, x] && NeQ[c^2*d^2 - e^2, 0]
  5921. Int[Times[Plus[Optional[Pattern[a, Blank[]]], Times[ArcCoth[Times[Optional[Pattern[c, Blank[]]], Pattern[x, Blank[]]]], Optional[Pattern[b, Blank[]]]]], Power[Plus[Pattern[d, Blank[]], Times[Optional[Pattern[e, Blank[]]], Pattern[x, Blank[]]]], -1]], Pattern[x, Blank[Symbol]]] := -Simp[((a + b*ArcCoth[c*x])*Log[2/(1 + c*x)])/e, x] + (Dist[(b*c)/e, Int[Log[2/(1 + c*x)]/(1 - c^2*x^2), x], x] - Dist[(b*c)/e, Int[Log[(2*c*(d + e*x))/((c*d + e)*(1 + c*x))]/(1 - c^2*x^2), x], x] + Simp[((a + b*ArcCoth[c*x])*Log[(2*c*(d + e*x))/((c*d + e)*(1 + c*x))])/e, x]) /; FreeQ[{a, b, c, d, e}, x] && NeQ[c^2*d^2 - e^2, 0]
  5922. Int[Times[Power[Plus[Optional[Pattern[a, Blank[]]], Times[ArcTanh[Times[Optional[Pattern[c, Blank[]]], Pattern[x, Blank[]]]], Optional[Pattern[b, Blank[]]]]], 2], Power[Plus[Pattern[d, Blank[]], Times[Optional[Pattern[e, Blank[]]], Pattern[x, Blank[]]]], -1]], Pattern[x, Blank[Symbol]]] := -Simp[((a + b*ArcTanh[c*x])^2*Log[2/(1 + c*x)])/e, x] + (Simp[((a + b*ArcTanh[c*x])^2*Log[(2*c*(d + e*x))/((c*d + e)*(1 + c*x))])/e, x] + Simp[(b*(a + b*ArcTanh[c*x])*PolyLog[2, 1 - 2/(1 + c*x)])/e, x] - Simp[(b*(a + b*ArcTanh[c*x])*PolyLog[2, 1 - (2*c*(d + e*x))/((c*d + e)*(1 + c*x))])/e, x] + Simp[(b^2*PolyLog[3, 1 - 2/(1 + c*x)])/(2*e), x] - Simp[(b^2*PolyLog[3, 1 - (2*c*(d + e*x))/((c*d + e)*(1 + c*x))])/(2*e), x]) /; FreeQ[{a, b, c, d, e}, x] && NeQ[c^2*d^2 - e^2, 0]
  5923. Int[Times[Power[Plus[Optional[Pattern[a, Blank[]]], Times[ArcCoth[Times[Optional[Pattern[c, Blank[]]], Pattern[x, Blank[]]]], Optional[Pattern[b, Blank[]]]]], 2], Power[Plus[Pattern[d, Blank[]], Times[Optional[Pattern[e, Blank[]]], Pattern[x, Blank[]]]], -1]], Pattern[x, Blank[Symbol]]] := -Simp[((a + b*ArcCoth[c*x])^2*Log[2/(1 + c*x)])/e, x] + (Simp[((a + b*ArcCoth[c*x])^2*Log[(2*c*(d + e*x))/((c*d + e)*(1 + c*x))])/e, x] + Simp[(b*(a + b*ArcCoth[c*x])*PolyLog[2, 1 - 2/(1 + c*x)])/e, x] - Simp[(b*(a + b*ArcCoth[c*x])*PolyLog[2, 1 - (2*c*(d + e*x))/((c*d + e)*(1 + c*x))])/e, x] + Simp[(b^2*PolyLog[3, 1 - 2/(1 + c*x)])/(2*e), x] - Simp[(b^2*PolyLog[3, 1 - (2*c*(d + e*x))/((c*d + e)*(1 + c*x))])/(2*e), x]) /; FreeQ[{a, b, c, d, e}, x] && NeQ[c^2*d^2 - e^2, 0]
  5924. Int[Times[Power[Plus[Optional[Pattern[a, Blank[]]], Times[ArcTanh[Times[Optional[Pattern[c, Blank[]]], Pattern[x, Blank[]]]], Optional[Pattern[b, Blank[]]]]], 3], Power[Plus[Pattern[d, Blank[]], Times[Optional[Pattern[e, Blank[]]], Pattern[x, Blank[]]]], -1]], Pattern[x, Blank[Symbol]]] := -Simp[((a + b*ArcTanh[c*x])^3*Log[2/(1 + c*x)])/e, x] + (Simp[((a + b*ArcTanh[c*x])^3*Log[(2*c*(d + e*x))/((c*d + e)*(1 + c*x))])/e, x] + Simp[(3*b*(a + b*ArcTanh[c*x])^2*PolyLog[2, 1 - 2/(1 + c*x)])/(2*e), x] - Simp[(3*b*(a + b*ArcTanh[c*x])^2*PolyLog[2, 1 - (2*c*(d + e*x))/((c*d + e)*(1 + c*x))])/(2*e), x] + Simp[(3*b^2*(a + b*ArcTanh[c*x])*PolyLog[3, 1 - 2/(1 + c*x)])/(2*e), x] - Simp[(3*b^2*(a + b*ArcTanh[c*x])*PolyLog[3, 1 - (2*c*(d + e*x))/((c*d + e)*(1 + c*x))])/(2*e), x] + Simp[(3*b^3*PolyLog[4, 1 - 2/(1 + c*x)])/(4*e), x] - Simp[(3*b^3*PolyLog[4, 1 - (2*c*(d + e*x))/((c*d + e)*(1 + c*x))])/(4*e), x]) /; FreeQ[{a, b, c, d, e}, x] && NeQ[c^2*d^2 - e^2, 0]
  5925. Int[Times[Power[Plus[Optional[Pattern[a, Blank[]]], Times[ArcCoth[Times[Optional[Pattern[c, Blank[]]], Pattern[x, Blank[]]]], Optional[Pattern[b, Blank[]]]]], 3], Power[Plus[Pattern[d, Blank[]], Times[Optional[Pattern[e, Blank[]]], Pattern[x, Blank[]]]], -1]], Pattern[x, Blank[Symbol]]] := -Simp[((a + b*ArcCoth[c*x])^3*Log[2/(1 + c*x)])/e, x] + (Simp[((a + b*ArcCoth[c*x])^3*Log[(2*c*(d + e*x))/((c*d + e)*(1 + c*x))])/e, x] + Simp[(3*b*(a + b*ArcCoth[c*x])^2*PolyLog[2, 1 - 2/(1 + c*x)])/(2*e), x] - Simp[(3*b*(a + b*ArcCoth[c*x])^2*PolyLog[2, 1 - (2*c*(d + e*x))/((c*d + e)*(1 + c*x))])/(2*e), x] + Simp[(3*b^2*(a + b*ArcCoth[c*x])*PolyLog[3, 1 - 2/(1 + c*x)])/(2*e), x] - Simp[(3*b^2*(a + b*ArcCoth[c*x])*PolyLog[3, 1 - (2*c*(d + e*x))/((c*d + e)*(1 + c*x))])/(2*e), x] + Simp[(3*b^3*PolyLog[4, 1 - 2/(1 + c*x)])/(4*e), x] - Simp[(3*b^3*PolyLog[4, 1 - (2*c*(d + e*x))/((c*d + e)*(1 + c*x))])/(4*e), x]) /; FreeQ[{a, b, c, d, e}, x] && NeQ[c^2*d^2 - e^2, 0]
  5926. Int[Times[Plus[Optional[Pattern[a, Blank[]]], Times[ArcTanh[Times[Optional[Pattern[c, Blank[]]], Pattern[x, Blank[]]]], Optional[Pattern[b, Blank[]]]]], Power[Plus[Pattern[d, Blank[]], Times[Optional[Pattern[e, Blank[]]], Pattern[x, Blank[]]]], Optional[Pattern[q, Blank[]]]]], Pattern[x, Blank[Symbol]]] := Simp[((d + e*x)^(q + 1)*(a + b*ArcTanh[c*x]))/(e*(q + 1)), x] - Dist[(b*c)/(e*(q + 1)), Int[(d + e*x)^(q + 1)/(1 - c^2*x^2), x], x] /; FreeQ[{a, b, c, d, e, q}, x] && NeQ[q, -1]
  5927. Int[Times[Plus[Optional[Pattern[a, Blank[]]], Times[ArcCoth[Times[Optional[Pattern[c, Blank[]]], Pattern[x, Blank[]]]], Optional[Pattern[b, Blank[]]]]], Power[Plus[Pattern[d, Blank[]], Times[Optional[Pattern[e, Blank[]]], Pattern[x, Blank[]]]], Optional[Pattern[q, Blank[]]]]], Pattern[x, Blank[Symbol]]] := Simp[((d + e*x)^(q + 1)*(a + b*ArcCoth[c*x]))/(e*(q + 1)), x] - Dist[(b*c)/(e*(q + 1)), Int[(d + e*x)^(q + 1)/(1 - c^2*x^2), x], x] /; FreeQ[{a, b, c, d, e, q}, x] && NeQ[q, -1]
  5928. Int[Times[Power[Plus[Optional[Pattern[a, Blank[]]], Times[ArcTanh[Times[Optional[Pattern[c, Blank[]]], Pattern[x, Blank[]]]], Optional[Pattern[b, Blank[]]]]], Pattern[p, Blank[]]], Power[Plus[Pattern[d, Blank[]], Times[Optional[Pattern[e, Blank[]]], Pattern[x, Blank[]]]], Optional[Pattern[q, Blank[]]]]], Pattern[x, Blank[Symbol]]] := Simp[((d + e*x)^(q + 1)*(a + b*ArcTanh[c*x])^p)/(e*(q + 1)), x] - Dist[(b*c*p)/(e*(q + 1)), Int[ExpandIntegrand[(a + b*ArcTanh[c*x])^(p - 1), (d + e*x)^(q + 1)/(1 - c^2*x^2), x], x], x] /; FreeQ[{a, b, c, d, e}, x] && IGtQ[p, 1] && IntegerQ[q] && NeQ[q, -1]
  5929. Int[Times[Power[Plus[Optional[Pattern[a, Blank[]]], Times[ArcCoth[Times[Optional[Pattern[c, Blank[]]], Pattern[x, Blank[]]]], Optional[Pattern[b, Blank[]]]]], Pattern[p, Blank[]]], Power[Plus[Pattern[d, Blank[]], Times[Optional[Pattern[e, Blank[]]], Pattern[x, Blank[]]]], Optional[Pattern[q, Blank[]]]]], Pattern[x, Blank[Symbol]]] := Simp[((d + e*x)^(q + 1)*(a + b*ArcCoth[c*x])^p)/(e*(q + 1)), x] - Dist[(b*c*p)/(e*(q + 1)), Int[ExpandIntegrand[(a + b*ArcCoth[c*x])^(p - 1), (d + e*x)^(q + 1)/(1 - c^2*x^2), x], x], x] /; FreeQ[{a, b, c, d, e}, x] && IGtQ[p, 1] && IntegerQ[q] && NeQ[q, -1]
  5930. Int[Times[Power[Plus[Optional[Pattern[a, Blank[]]], Times[ArcTanh[Times[Optional[Pattern[c, Blank[]]], Pattern[x, Blank[]]]], Optional[Pattern[b, Blank[]]]]], Optional[Pattern[p, Blank[]]]], Power[Times[Optional[Pattern[f, Blank[]]], Pattern[x, Blank[]]], Optional[Pattern[m, Blank[]]]], Power[Plus[Pattern[d, Blank[]], Times[Optional[Pattern[e, Blank[]]], Pattern[x, Blank[]]]], -1]], Pattern[x, Blank[Symbol]]] := Dist[f/e, Int[(f*x)^(m - 1)*(a + b*ArcTanh[c*x])^p, x], x] - Dist[(d*f)/e, Int[((f*x)^(m - 1)*(a + b*ArcTanh[c*x])^p)/(d + e*x), x], x] /; FreeQ[{a, b, c, d, e, f}, x] && IGtQ[p, 0] && EqQ[c^2*d^2 - e^2, 0] && GtQ[m, 0]
  5931. Int[Times[Power[Plus[Optional[Pattern[a, Blank[]]], Times[ArcCoth[Times[Optional[Pattern[c, Blank[]]], Pattern[x, Blank[]]]], Optional[Pattern[b, Blank[]]]]], Optional[Pattern[p, Blank[]]]], Power[Times[Optional[Pattern[f, Blank[]]], Pattern[x, Blank[]]], Optional[Pattern[m, Blank[]]]], Power[Plus[Pattern[d, Blank[]], Times[Optional[Pattern[e, Blank[]]], Pattern[x, Blank[]]]], -1]], Pattern[x, Blank[Symbol]]] := Dist[f/e, Int[(f*x)^(m - 1)*(a + b*ArcCoth[c*x])^p, x], x] - Dist[(d*f)/e, Int[((f*x)^(m - 1)*(a + b*ArcCoth[c*x])^p)/(d + e*x), x], x] /; FreeQ[{a, b, c, d, e, f}, x] && IGtQ[p, 0] && EqQ[c^2*d^2 - e^2, 0] && GtQ[m, 0]
  5932. Int[Times[Power[Plus[Optional[Pattern[a, Blank[]]], Times[ArcTanh[Times[Optional[Pattern[c, Blank[]]], Pattern[x, Blank[]]]], Optional[Pattern[b, Blank[]]]]], Optional[Pattern[p, Blank[]]]], Power[Pattern[x, Blank[]], -1], Power[Plus[Pattern[d, Blank[]], Times[Optional[Pattern[e, Blank[]]], Pattern[x, Blank[]]]], -1]], Pattern[x, Blank[Symbol]]] := Simp[((a + b*ArcTanh[c*x])^p*Log[2 - 2/(1 + (e*x)/d)])/d, x] - Dist[(b*c*p)/d, Int[((a + b*ArcTanh[c*x])^(p - 1)*Log[2 - 2/(1 + (e*x)/d)])/(1 - c^2*x^2), x], x] /; FreeQ[{a, b, c, d, e}, x] && IGtQ[p, 0] && EqQ[c^2*d^2 - e^2, 0]
  5933. Int[Times[Power[Plus[Optional[Pattern[a, Blank[]]], Times[ArcCoth[Times[Optional[Pattern[c, Blank[]]], Pattern[x, Blank[]]]], Optional[Pattern[b, Blank[]]]]], Optional[Pattern[p, Blank[]]]], Power[Pattern[x, Blank[]], -1], Power[Plus[Pattern[d, Blank[]], Times[Optional[Pattern[e, Blank[]]], Pattern[x, Blank[]]]], -1]], Pattern[x, Blank[Symbol]]] := Simp[((a + b*ArcCoth[c*x])^p*Log[2 - 2/(1 + (e*x)/d)])/d, x] - Dist[(b*c*p)/d, Int[((a + b*ArcCoth[c*x])^(p - 1)*Log[2 - 2/(1 + (e*x)/d)])/(1 - c^2*x^2), x], x] /; FreeQ[{a, b, c, d, e}, x] && IGtQ[p, 0] && EqQ[c^2*d^2 - e^2, 0]
  5934. Int[Times[Power[Plus[Optional[Pattern[a, Blank[]]], Times[ArcTanh[Times[Optional[Pattern[c, Blank[]]], Pattern[x, Blank[]]]], Optional[Pattern[b, Blank[]]]]], Optional[Pattern[p, Blank[]]]], Power[Times[Optional[Pattern[f, Blank[]]], Pattern[x, Blank[]]], Pattern[m, Blank[]]], Power[Plus[Pattern[d, Blank[]], Times[Optional[Pattern[e, Blank[]]], Pattern[x, Blank[]]]], -1]], Pattern[x, Blank[Symbol]]] := Dist[1/d, Int[(f*x)^m*(a + b*ArcTanh[c*x])^p, x], x] - Dist[e/(d*f), Int[((f*x)^(m + 1)*(a + b*ArcTanh[c*x])^p)/(d + e*x), x], x] /; FreeQ[{a, b, c, d, e, f}, x] && IGtQ[p, 0] && EqQ[c^2*d^2 - e^2, 0] && LtQ[m, -1]
  5935. Int[Times[Power[Plus[Optional[Pattern[a, Blank[]]], Times[ArcCoth[Times[Optional[Pattern[c, Blank[]]], Pattern[x, Blank[]]]], Optional[Pattern[b, Blank[]]]]], Optional[Pattern[p, Blank[]]]], Power[Times[Optional[Pattern[f, Blank[]]], Pattern[x, Blank[]]], Pattern[m, Blank[]]], Power[Plus[Pattern[d, Blank[]], Times[Optional[Pattern[e, Blank[]]], Pattern[x, Blank[]]]], -1]], Pattern[x, Blank[Symbol]]] := Dist[1/d, Int[(f*x)^m*(a + b*ArcCoth[c*x])^p, x], x] - Dist[e/(d*f), Int[((f*x)^(m + 1)*(a + b*ArcCoth[c*x])^p)/(d + e*x), x], x] /; FreeQ[{a, b, c, d, e, f}, x] && IGtQ[p, 0] && EqQ[c^2*d^2 - e^2, 0] && LtQ[m, -1]
  5936. Int[Times[Plus[Optional[Pattern[a, Blank[]]], Times[ArcTanh[Times[Optional[Pattern[c, Blank[]]], Pattern[x, Blank[]]]], Optional[Pattern[b, Blank[]]]]], Power[Times[Optional[Pattern[f, Blank[]]], Pattern[x, Blank[]]], Optional[Pattern[m, Blank[]]]], Power[Plus[Optional[Pattern[d, Blank[]]], Times[Optional[Pattern[e, Blank[]]], Pattern[x, Blank[]]]], Optional[Pattern[q, Blank[]]]]], Pattern[x, Blank[Symbol]]] := With[{u = IntHide[(f*x)^m*(d + e*x)^q, x]}, Dist[a + b*ArcTanh[c*x], u, x] - Dist[b*c, Int[SimplifyIntegrand[u/(1 - c^2*x^2), x], x], x]] /; FreeQ[{a, b, c, d, e, f, q}, x] && NeQ[q, -1] && IntegerQ[2*m] && ((IGtQ[m, 0] && IGtQ[q, 0]) || (ILtQ[m + q + 1, 0] && LtQ[m*q, 0]))
  5937. Int[Times[Plus[Optional[Pattern[a, Blank[]]], Times[ArcCoth[Times[Optional[Pattern[c, Blank[]]], Pattern[x, Blank[]]]], Optional[Pattern[b, Blank[]]]]], Power[Times[Optional[Pattern[f, Blank[]]], Pattern[x, Blank[]]], Optional[Pattern[m, Blank[]]]], Power[Plus[Optional[Pattern[d, Blank[]]], Times[Optional[Pattern[e, Blank[]]], Pattern[x, Blank[]]]], Optional[Pattern[q, Blank[]]]]], Pattern[x, Blank[Symbol]]] := With[{u = IntHide[(f*x)^m*(d + e*x)^q, x]}, Dist[a + b*ArcCoth[c*x], u, x] - Dist[b*c, Int[SimplifyIntegrand[u/(1 - c^2*x^2), x], x], x]] /; FreeQ[{a, b, c, d, e, f, q}, x] && NeQ[q, -1] && IntegerQ[2*m] && ((IGtQ[m, 0] && IGtQ[q, 0]) || (ILtQ[m + q + 1, 0] && LtQ[m*q, 0]))
  5938. Int[Times[Power[Plus[Optional[Pattern[a, Blank[]]], Times[ArcTanh[Times[Optional[Pattern[c, Blank[]]], Pattern[x, Blank[]]]], Optional[Pattern[b, Blank[]]]]], Pattern[p, Blank[]]], Power[Times[Optional[Pattern[f, Blank[]]], Pattern[x, Blank[]]], Optional[Pattern[m, Blank[]]]], Power[Plus[Optional[Pattern[d, Blank[]]], Times[Optional[Pattern[e, Blank[]]], Pattern[x, Blank[]]]], Pattern[q, Blank[]]]], Pattern[x, Blank[Symbol]]] := With[{u = IntHide[(f*x)^m*(d + e*x)^q, x]}, Dist[(a + b*ArcTanh[c*x])^p, u, x] - Dist[b*c*p, Int[ExpandIntegrand[(a + b*ArcTanh[c*x])^(p - 1), u/(1 - c^2*x^2), x], x], x]] /; FreeQ[{a, b, c, d, e, f, q}, x] && IGtQ[p, 1] && EqQ[c^2*d^2 - e^2, 0] && IntegersQ[m, q] && NeQ[m, -1] && NeQ[q, -1] && ILtQ[m + q + 1, 0] && LtQ[m*q, 0]
  5939. Int[Times[Power[Plus[Optional[Pattern[a, Blank[]]], Times[ArcCoth[Times[Optional[Pattern[c, Blank[]]], Pattern[x, Blank[]]]], Optional[Pattern[b, Blank[]]]]], Pattern[p, Blank[]]], Power[Times[Optional[Pattern[f, Blank[]]], Pattern[x, Blank[]]], Optional[Pattern[m, Blank[]]]], Power[Plus[Optional[Pattern[d, Blank[]]], Times[Optional[Pattern[e, Blank[]]], Pattern[x, Blank[]]]], Pattern[q, Blank[]]]], Pattern[x, Blank[Symbol]]] := With[{u = IntHide[(f*x)^m*(d + e*x)^q, x]}, Dist[(a + b*ArcCoth[c*x])^p, u, x] - Dist[b*c*p, Int[ExpandIntegrand[(a + b*ArcCoth[c*x])^(p - 1), u/(1 - c^2*x^2), x], x], x]] /; FreeQ[{a, b, c, d, e, f, q}, x] && IGtQ[p, 1] && EqQ[c^2*d^2 - e^2, 0] && IntegersQ[m, q] && NeQ[m, -1] && NeQ[q, -1] && ILtQ[m + q + 1, 0] && LtQ[m*q, 0]
  5940. Int[Times[Power[Plus[Optional[Pattern[a, Blank[]]], Times[ArcTanh[Times[Optional[Pattern[c, Blank[]]], Pattern[x, Blank[]]]], Optional[Pattern[b, Blank[]]]]], Optional[Pattern[p, Blank[]]]], Power[Times[Optional[Pattern[f, Blank[]]], Pattern[x, Blank[]]], Optional[Pattern[m, Blank[]]]], Power[Plus[Pattern[d, Blank[]], Times[Optional[Pattern[e, Blank[]]], Pattern[x, Blank[]]]], Optional[Pattern[q, Blank[]]]]], Pattern[x, Blank[Symbol]]] := Int[ExpandIntegrand[(a + b*ArcTanh[c*x])^p, (f*x)^m*(d + e*x)^q, x], x] /; FreeQ[{a, b, c, d, e, f, m}, x] && IGtQ[p, 0] && IntegerQ[q] && (GtQ[q, 0] || NeQ[a, 0] || IntegerQ[m])
  5941. Int[Times[Power[Plus[Optional[Pattern[a, Blank[]]], Times[ArcCoth[Times[Optional[Pattern[c, Blank[]]], Pattern[x, Blank[]]]], Optional[Pattern[b, Blank[]]]]], Optional[Pattern[p, Blank[]]]], Power[Times[Optional[Pattern[f, Blank[]]], Pattern[x, Blank[]]], Optional[Pattern[m, Blank[]]]], Power[Plus[Pattern[d, Blank[]], Times[Optional[Pattern[e, Blank[]]], Pattern[x, Blank[]]]], Optional[Pattern[q, Blank[]]]]], Pattern[x, Blank[Symbol]]] := Int[ExpandIntegrand[(a + b*ArcCoth[c*x])^p, (f*x)^m*(d + e*x)^q, x], x] /; FreeQ[{a, b, c, d, e, f, m}, x] && IGtQ[p, 0] && IntegerQ[q] && (GtQ[q, 0] || NeQ[a, 0] || IntegerQ[m])
  5942. Int[Times[Plus[Optional[Pattern[a, Blank[]]], Times[ArcTanh[Times[Optional[Pattern[c, Blank[]]], Pattern[x, Blank[]]]], Optional[Pattern[b, Blank[]]]]], Power[Plus[Pattern[d, Blank[]], Times[Optional[Pattern[e, Blank[]]], Power[Pattern[x, Blank[]], 2]]], Optional[Pattern[q, Blank[]]]]], Pattern[x, Blank[Symbol]]] := Simp[(b*(d + e*x^2)^q)/(2*c*q*(2*q + 1)), x] + (Dist[(2*d*q)/(2*q + 1), Int[(d + e*x^2)^(q - 1)*(a + b*ArcTanh[c*x]), x], x] + Simp[(x*(d + e*x^2)^q*(a + b*ArcTanh[c*x]))/(2*q + 1), x]) /; FreeQ[{a, b, c, d, e}, x] && EqQ[c^2*d + e, 0] && GtQ[q, 0]
  5943. Int[Times[Plus[Optional[Pattern[a, Blank[]]], Times[ArcCoth[Times[Optional[Pattern[c, Blank[]]], Pattern[x, Blank[]]]], Optional[Pattern[b, Blank[]]]]], Power[Plus[Pattern[d, Blank[]], Times[Optional[Pattern[e, Blank[]]], Power[Pattern[x, Blank[]], 2]]], Optional[Pattern[q, Blank[]]]]], Pattern[x, Blank[Symbol]]] := Simp[(b*(d + e*x^2)^q)/(2*c*q*(2*q + 1)), x] + (Dist[(2*d*q)/(2*q + 1), Int[(d + e*x^2)^(q - 1)*(a + b*ArcCoth[c*x]), x], x] + Simp[(x*(d + e*x^2)^q*(a + b*ArcCoth[c*x]))/(2*q + 1), x]) /; FreeQ[{a, b, c, d, e}, x] && EqQ[c^2*d + e, 0] && GtQ[q, 0]
  5944. Int[Times[Power[Plus[Optional[Pattern[a, Blank[]]], Times[ArcTanh[Times[Optional[Pattern[c, Blank[]]], Pattern[x, Blank[]]]], Optional[Pattern[b, Blank[]]]]], Pattern[p, Blank[]]], Power[Plus[Pattern[d, Blank[]], Times[Optional[Pattern[e, Blank[]]], Power[Pattern[x, Blank[]], 2]]], Optional[Pattern[q, Blank[]]]]], Pattern[x, Blank[Symbol]]] := Simp[(b*p*(d + e*x^2)^q*(a + b*ArcTanh[c*x])^(p - 1))/(2*c*q*(2*q + 1)), x] + (Dist[(2*d*q)/(2*q + 1), Int[(d + e*x^2)^(q - 1)*(a + b*ArcTanh[c*x])^p, x], x] - Dist[(b^2*d*p*(p - 1))/(2*q*(2*q + 1)), Int[(d + e*x^2)^(q - 1)*(a + b*ArcTanh[c*x])^(p - 2), x], x] + Simp[(x*(d + e*x^2)^q*(a + b*ArcTanh[c*x])^p)/(2*q + 1), x]) /; FreeQ[{a, b, c, d, e}, x] && EqQ[c^2*d + e, 0] && GtQ[q, 0] && GtQ[p, 1]
  5945. Int[Times[Power[Plus[Optional[Pattern[a, Blank[]]], Times[ArcCoth[Times[Optional[Pattern[c, Blank[]]], Pattern[x, Blank[]]]], Optional[Pattern[b, Blank[]]]]], Pattern[p, Blank[]]], Power[Plus[Pattern[d, Blank[]], Times[Optional[Pattern[e, Blank[]]], Power[Pattern[x, Blank[]], 2]]], Optional[Pattern[q, Blank[]]]]], Pattern[x, Blank[Symbol]]] := Simp[(b*p*(d + e*x^2)^q*(a + b*ArcCoth[c*x])^(p - 1))/(2*c*q*(2*q + 1)), x] + (Dist[(2*d*q)/(2*q + 1), Int[(d + e*x^2)^(q - 1)*(a + b*ArcCoth[c*x])^p, x], x] - Dist[(b^2*d*p*(p - 1))/(2*q*(2*q + 1)), Int[(d + e*x^2)^(q - 1)*(a + b*ArcCoth[c*x])^(p - 2), x], x] + Simp[(x*(d + e*x^2)^q*(a + b*ArcCoth[c*x])^p)/(2*q + 1), x]) /; FreeQ[{a, b, c, d, e}, x] && EqQ[c^2*d + e, 0] && GtQ[q, 0] && GtQ[p, 1]
  5946. Int[Times[Power[Plus[Optional[Pattern[a, Blank[]]], Times[ArcTanh[Times[Optional[Pattern[c, Blank[]]], Pattern[x, Blank[]]]], Optional[Pattern[b, Blank[]]]]], -1], Power[Plus[Pattern[d, Blank[]], Times[Optional[Pattern[e, Blank[]]], Power[Pattern[x, Blank[]], 2]]], -1]], Pattern[x, Blank[Symbol]]] := Simp[Log[RemoveContent[a + b*ArcTanh[c*x], x]]/(b*c*d), x] /; FreeQ[{a, b, c, d, e}, x] && EqQ[c^2*d + e, 0]
  5947. Int[Times[Power[Plus[Optional[Pattern[a, Blank[]]], Times[ArcCoth[Times[Optional[Pattern[c, Blank[]]], Pattern[x, Blank[]]]], Optional[Pattern[b, Blank[]]]]], -1], Power[Plus[Pattern[d, Blank[]], Times[Optional[Pattern[e, Blank[]]], Power[Pattern[x, Blank[]], 2]]], -1]], Pattern[x, Blank[Symbol]]] := Simp[Log[RemoveContent[a + b*ArcCoth[c*x], x]]/(b*c*d), x] /; FreeQ[{a, b, c, d, e}, x] && EqQ[c^2*d + e, 0]
  5948. Int[Times[Power[Plus[Optional[Pattern[a, Blank[]]], Times[ArcTanh[Times[Optional[Pattern[c, Blank[]]], Pattern[x, Blank[]]]], Optional[Pattern[b, Blank[]]]]], Optional[Pattern[p, Blank[]]]], Power[Plus[Pattern[d, Blank[]], Times[Optional[Pattern[e, Blank[]]], Power[Pattern[x, Blank[]], 2]]], -1]], Pattern[x, Blank[Symbol]]] := Simp[(a + b*ArcTanh[c*x])^(p + 1)/(b*c*d*(p + 1)), x] /; FreeQ[{a, b, c, d, e, p}, x] && EqQ[c^2*d + e, 0] && NeQ[p, -1]
  5949. Int[Times[Power[Plus[Optional[Pattern[a, Blank[]]], Times[ArcCoth[Times[Optional[Pattern[c, Blank[]]], Pattern[x, Blank[]]]], Optional[Pattern[b, Blank[]]]]], Optional[Pattern[p, Blank[]]]], Power[Plus[Pattern[d, Blank[]], Times[Optional[Pattern[e, Blank[]]], Power[Pattern[x, Blank[]], 2]]], -1]], Pattern[x, Blank[Symbol]]] := Simp[(a + b*ArcCoth[c*x])^(p + 1)/(b*c*d*(p + 1)), x] /; FreeQ[{a, b, c, d, e, p}, x] && EqQ[c^2*d + e, 0] && NeQ[p, -1]
  5950. Int[Times[Plus[Optional[Pattern[a, Blank[]]], Times[ArcTanh[Times[Optional[Pattern[c, Blank[]]], Pattern[x, Blank[]]]], Optional[Pattern[b, Blank[]]]]], Power[Plus[Pattern[d, Blank[]], Times[Optional[Pattern[e, Blank[]]], Power[Pattern[x, Blank[]], 2]]], Rational[-1, 2]]], Pattern[x, Blank[Symbol]]] := Simp[(-2*(a + b*ArcTanh[c*x])*ArcTan[Sqrt[1 - c*x]/Sqrt[1 + c*x]])/(c*Sqrt[d]), x] + (-Simp[(I*b*PolyLog[2, -((I*Sqrt[1 - c*x])/Sqrt[1 + c*x])])/(c*Sqrt[d]), x] + Simp[(I*b*PolyLog[2, (I*Sqrt[1 - c*x])/Sqrt[1 + c*x]])/(c*Sqrt[d]), x]) /; FreeQ[{a, b, c, d, e}, x] && EqQ[c^2*d + e, 0] && GtQ[d, 0]
  5951. Int[Times[Plus[Optional[Pattern[a, Blank[]]], Times[ArcCoth[Times[Optional[Pattern[c, Blank[]]], Pattern[x, Blank[]]]], Optional[Pattern[b, Blank[]]]]], Power[Plus[Pattern[d, Blank[]], Times[Optional[Pattern[e, Blank[]]], Power[Pattern[x, Blank[]], 2]]], Rational[-1, 2]]], Pattern[x, Blank[Symbol]]] := Simp[(-2*(a + b*ArcCoth[c*x])*ArcTan[Sqrt[1 - c*x]/Sqrt[1 + c*x]])/(c*Sqrt[d]), x] + (-Simp[(I*b*PolyLog[2, -((I*Sqrt[1 - c*x])/Sqrt[1 + c*x])])/(c*Sqrt[d]), x] + Simp[(I*b*PolyLog[2, (I*Sqrt[1 - c*x])/Sqrt[1 + c*x]])/(c*Sqrt[d]), x]) /; FreeQ[{a, b, c, d, e}, x] && EqQ[c^2*d + e, 0] && GtQ[d, 0]
  5952. Int[Times[Power[Plus[Optional[Pattern[a, Blank[]]], Times[ArcTanh[Times[Optional[Pattern[c, Blank[]]], Pattern[x, Blank[]]]], Optional[Pattern[b, Blank[]]]]], Optional[Pattern[p, Blank[]]]], Power[Plus[Pattern[d, Blank[]], Times[Optional[Pattern[e, Blank[]]], Power[Pattern[x, Blank[]], 2]]], Rational[-1, 2]]], Pattern[x, Blank[Symbol]]] := Dist[1/(c*Sqrt[d]), Subst[Int[(a + b*x)^p*Sech[x], x], x, ArcTanh[c*x]], x] /; FreeQ[{a, b, c, d, e}, x] && EqQ[c^2*d + e, 0] && IGtQ[p, 0] && GtQ[d, 0]
  5953. Int[Times[Power[Plus[Optional[Pattern[a, Blank[]]], Times[ArcCoth[Times[Optional[Pattern[c, Blank[]]], Pattern[x, Blank[]]]], Optional[Pattern[b, Blank[]]]]], Optional[Pattern[p, Blank[]]]], Power[Plus[Pattern[d, Blank[]], Times[Optional[Pattern[e, Blank[]]], Power[Pattern[x, Blank[]], 2]]], Rational[-1, 2]]], Pattern[x, Blank[Symbol]]] := -Dist[(x*Sqrt[1 - 1/(c^2*x^2)])/Sqrt[d + e*x^2], Subst[Int[(a + b*x)^p*Csch[x], x], x, ArcCoth[c*x]], x] /; FreeQ[{a, b, c, d, e}, x] && EqQ[c^2*d + e, 0] && IGtQ[p, 0] && GtQ[d, 0]
  5954. Int[Times[Power[Plus[Optional[Pattern[a, Blank[]]], Times[ArcTanh[Times[Optional[Pattern[c, Blank[]]], Pattern[x, Blank[]]]], Optional[Pattern[b, Blank[]]]]], Optional[Pattern[p, Blank[]]]], Power[Plus[Pattern[d, Blank[]], Times[Optional[Pattern[e, Blank[]]], Power[Pattern[x, Blank[]], 2]]], Rational[-1, 2]]], Pattern[x, Blank[Symbol]]] := Dist[Sqrt[1 - c^2*x^2]/Sqrt[d + e*x^2], Int[(a + b*ArcTanh[c*x])^p/Sqrt[1 - c^2*x^2], x], x] /; FreeQ[{a, b, c, d, e}, x] && EqQ[c^2*d + e, 0] && IGtQ[p, 0] && !GtQ[d, 0]
  5955. Int[Times[Power[Plus[Optional[Pattern[a, Blank[]]], Times[ArcCoth[Times[Optional[Pattern[c, Blank[]]], Pattern[x, Blank[]]]], Optional[Pattern[b, Blank[]]]]], Optional[Pattern[p, Blank[]]]], Power[Plus[Pattern[d, Blank[]], Times[Optional[Pattern[e, Blank[]]], Power[Pattern[x, Blank[]], 2]]], Rational[-1, 2]]], Pattern[x, Blank[Symbol]]] := Dist[Sqrt[1 - c^2*x^2]/Sqrt[d + e*x^2], Int[(a + b*ArcCoth[c*x])^p/Sqrt[1 - c^2*x^2], x], x] /; FreeQ[{a, b, c, d, e}, x] && EqQ[c^2*d + e, 0] && IGtQ[p, 0] && !GtQ[d, 0]
  5956. Int[Times[Power[Plus[Optional[Pattern[a, Blank[]]], Times[ArcTanh[Times[Optional[Pattern[c, Blank[]]], Pattern[x, Blank[]]]], Optional[Pattern[b, Blank[]]]]], Optional[Pattern[p, Blank[]]]], Power[Plus[Pattern[d, Blank[]], Times[Optional[Pattern[e, Blank[]]], Power[Pattern[x, Blank[]], 2]]], -2]], Pattern[x, Blank[Symbol]]] := Simp[(x*(a + b*ArcTanh[c*x])^p)/(2*d*(d + e*x^2)), x] + (-Dist[(b*c*p)/2, Int[(x*(a + b*ArcTanh[c*x])^(p - 1))/(d + e*x^2)^2, x], x] + Simp[(a + b*ArcTanh[c*x])^(p + 1)/(2*b*c*d^2*(p + 1)), x]) /; FreeQ[{a, b, c, d, e}, x] && EqQ[c^2*d + e, 0] && GtQ[p, 0]
  5957. Int[Times[Power[Plus[Optional[Pattern[a, Blank[]]], Times[ArcCoth[Times[Optional[Pattern[c, Blank[]]], Pattern[x, Blank[]]]], Optional[Pattern[b, Blank[]]]]], Optional[Pattern[p, Blank[]]]], Power[Plus[Pattern[d, Blank[]], Times[Optional[Pattern[e, Blank[]]], Power[Pattern[x, Blank[]], 2]]], -2]], Pattern[x, Blank[Symbol]]] := Simp[(x*(a + b*ArcCoth[c*x])^p)/(2*d*(d + e*x^2)), x] + (-Dist[(b*c*p)/2, Int[(x*(a + b*ArcCoth[c*x])^(p - 1))/(d + e*x^2)^2, x], x] + Simp[(a + b*ArcCoth[c*x])^(p + 1)/(2*b*c*d^2*(p + 1)), x]) /; FreeQ[{a, b, c, d, e}, x] && EqQ[c^2*d + e, 0] && GtQ[p, 0]
  5958. Int[Times[Plus[Optional[Pattern[a, Blank[]]], Times[ArcTanh[Times[Optional[Pattern[c, Blank[]]], Pattern[x, Blank[]]]], Optional[Pattern[b, Blank[]]]]], Power[Plus[Pattern[d, Blank[]], Times[Optional[Pattern[e, Blank[]]], Power[Pattern[x, Blank[]], 2]]], Rational[-3, 2]]], Pattern[x, Blank[Symbol]]] := -Simp[b/(c*d*Sqrt[d + e*x^2]), x] + Simp[(x*(a + b*ArcTanh[c*x]))/(d*Sqrt[d + e*x^2]), x] /; FreeQ[{a, b, c, d, e}, x] && EqQ[c^2*d + e, 0]
  5959. Int[Times[Plus[Optional[Pattern[a, Blank[]]], Times[ArcCoth[Times[Optional[Pattern[c, Blank[]]], Pattern[x, Blank[]]]], Optional[Pattern[b, Blank[]]]]], Power[Plus[Pattern[d, Blank[]], Times[Optional[Pattern[e, Blank[]]], Power[Pattern[x, Blank[]], 2]]], Rational[-3, 2]]], Pattern[x, Blank[Symbol]]] := -Simp[b/(c*d*Sqrt[d + e*x^2]), x] + Simp[(x*(a + b*ArcCoth[c*x]))/(d*Sqrt[d + e*x^2]), x] /; FreeQ[{a, b, c, d, e}, x] && EqQ[c^2*d + e, 0]
  5960. Int[Times[Plus[Optional[Pattern[a, Blank[]]], Times[ArcTanh[Times[Optional[Pattern[c, Blank[]]], Pattern[x, Blank[]]]], Optional[Pattern[b, Blank[]]]]], Power[Plus[Pattern[d, Blank[]], Times[Optional[Pattern[e, Blank[]]], Power[Pattern[x, Blank[]], 2]]], Pattern[q, Blank[]]]], Pattern[x, Blank[Symbol]]] := -Simp[(b*(d + e*x^2)^(q + 1))/(4*c*d*(q + 1)^2), x] + (Dist[(2*q + 3)/(2*d*(q + 1)), Int[(d + e*x^2)^(q + 1)*(a + b*ArcTanh[c*x]), x], x] - Simp[(x*(d + e*x^2)^(q + 1)*(a + b*ArcTanh[c*x]))/(2*d*(q + 1)), x]) /; FreeQ[{a, b, c, d, e}, x] && EqQ[c^2*d + e, 0] && LtQ[q, -1] && NeQ[q, -3/2]
  5961. Int[Times[Plus[Optional[Pattern[a, Blank[]]], Times[ArcCoth[Times[Optional[Pattern[c, Blank[]]], Pattern[x, Blank[]]]], Optional[Pattern[b, Blank[]]]]], Power[Plus[Pattern[d, Blank[]], Times[Optional[Pattern[e, Blank[]]], Power[Pattern[x, Blank[]], 2]]], Pattern[q, Blank[]]]], Pattern[x, Blank[Symbol]]] := -Simp[(b*(d + e*x^2)^(q + 1))/(4*c*d*(q + 1)^2), x] + (Dist[(2*q + 3)/(2*d*(q + 1)), Int[(d + e*x^2)^(q + 1)*(a + b*ArcCoth[c*x]), x], x] - Simp[(x*(d + e*x^2)^(q + 1)*(a + b*ArcCoth[c*x]))/(2*d*(q + 1)), x]) /; FreeQ[{a, b, c, d, e}, x] && EqQ[c^2*d + e, 0] && LtQ[q, -1] && NeQ[q, -3/2]
  5962. Int[Times[Power[Plus[Optional[Pattern[a, Blank[]]], Times[ArcTanh[Times[Optional[Pattern[c, Blank[]]], Pattern[x, Blank[]]]], Optional[Pattern[b, Blank[]]]]], Pattern[p, Blank[]]], Power[Plus[Pattern[d, Blank[]], Times[Optional[Pattern[e, Blank[]]], Power[Pattern[x, Blank[]], 2]]], Rational[-3, 2]]], Pattern[x, Blank[Symbol]]] := -Simp[(b*p*(a + b*ArcTanh[c*x])^(p - 1))/(c*d*Sqrt[d + e*x^2]), x] + (Dist[b^2*p*(p - 1), Int[(a + b*ArcTanh[c*x])^(p - 2)/(d + e*x^2)^(3/2), x], x] + Simp[(x*(a + b*ArcTanh[c*x])^p)/(d*Sqrt[d + e*x^2]), x]) /; FreeQ[{a, b, c, d, e}, x] && EqQ[c^2*d + e, 0] && GtQ[p, 1]
  5963. Int[Times[Power[Plus[Optional[Pattern[a, Blank[]]], Times[ArcCoth[Times[Optional[Pattern[c, Blank[]]], Pattern[x, Blank[]]]], Optional[Pattern[b, Blank[]]]]], Pattern[p, Blank[]]], Power[Plus[Pattern[d, Blank[]], Times[Optional[Pattern[e, Blank[]]], Power[Pattern[x, Blank[]], 2]]], Rational[-3, 2]]], Pattern[x, Blank[Symbol]]] := -Simp[(b*p*(a + b*ArcCoth[c*x])^(p - 1))/(c*d*Sqrt[d + e*x^2]), x] + (Dist[b^2*p*(p - 1), Int[(a + b*ArcCoth[c*x])^(p - 2)/(d + e*x^2)^(3/2), x], x] + Simp[(x*(a + b*ArcCoth[c*x])^p)/(d*Sqrt[d + e*x^2]), x]) /; FreeQ[{a, b, c, d, e}, x] && EqQ[c^2*d + e, 0] && GtQ[p, 1]
  5964. Int[Times[Power[Plus[Optional[Pattern[a, Blank[]]], Times[ArcTanh[Times[Optional[Pattern[c, Blank[]]], Pattern[x, Blank[]]]], Optional[Pattern[b, Blank[]]]]], Pattern[p, Blank[]]], Power[Plus[Pattern[d, Blank[]], Times[Optional[Pattern[e, Blank[]]], Power[Pattern[x, Blank[]], 2]]], Pattern[q, Blank[]]]], Pattern[x, Blank[Symbol]]] := -Simp[(b*p*(d + e*x^2)^(q + 1)*(a + b*ArcTanh[c*x])^(p - 1))/(4*c*d*(q + 1)^2), x] + (Dist[(2*q + 3)/(2*d*(q + 1)), Int[(d + e*x^2)^(q + 1)*(a + b*ArcTanh[c*x])^p, x], x] + Dist[(b^2*p*(p - 1))/(4*(q + 1)^2), Int[(d + e*x^2)^q*(a + b*ArcTanh[c*x])^(p - 2), x], x] - Simp[(x*(d + e*x^2)^(q + 1)*(a + b*ArcTanh[c*x])^p)/(2*d*(q + 1)), x]) /; FreeQ[{a, b, c, d, e}, x] && EqQ[c^2*d + e, 0] && LtQ[q, -1] && GtQ[p, 1] && NeQ[q, -3/2]
  5965. Int[Times[Power[Plus[Optional[Pattern[a, Blank[]]], Times[ArcCoth[Times[Optional[Pattern[c, Blank[]]], Pattern[x, Blank[]]]], Optional[Pattern[b, Blank[]]]]], Pattern[p, Blank[]]], Power[Plus[Pattern[d, Blank[]], Times[Optional[Pattern[e, Blank[]]], Power[Pattern[x, Blank[]], 2]]], Pattern[q, Blank[]]]], Pattern[x, Blank[Symbol]]] := -Simp[(b*p*(d + e*x^2)^(q + 1)*(a + b*ArcCoth[c*x])^(p - 1))/(4*c*d*(q + 1)^2), x] + (Dist[(2*q + 3)/(2*d*(q + 1)), Int[(d + e*x^2)^(q + 1)*(a + b*ArcCoth[c*x])^p, x], x] + Dist[(b^2*p*(p - 1))/(4*(q + 1)^2), Int[(d + e*x^2)^q*(a + b*ArcCoth[c*x])^(p - 2), x], x] - Simp[(x*(d + e*x^2)^(q + 1)*(a + b*ArcCoth[c*x])^p)/(2*d*(q + 1)), x]) /; FreeQ[{a, b, c, d, e}, x] && EqQ[c^2*d + e, 0] && LtQ[q, -1] && GtQ[p, 1] && NeQ[q, -3/2]
  5966. Int[Times[Power[Plus[Optional[Pattern[a, Blank[]]], Times[ArcTanh[Times[Optional[Pattern[c, Blank[]]], Pattern[x, Blank[]]]], Optional[Pattern[b, Blank[]]]]], Pattern[p, Blank[]]], Power[Plus[Pattern[d, Blank[]], Times[Optional[Pattern[e, Blank[]]], Power[Pattern[x, Blank[]], 2]]], Pattern[q, Blank[]]]], Pattern[x, Blank[Symbol]]] := Simp[((d + e*x^2)^(q + 1)*(a + b*ArcTanh[c*x])^(p + 1))/(b*c*d*(p + 1)), x] + Dist[(2*c*(q + 1))/(b*(p + 1)), Int[x*(d + e*x^2)^q*(a + b*ArcTanh[c*x])^(p + 1), x], x] /; FreeQ[{a, b, c, d, e}, x] && EqQ[c^2*d + e, 0] && LtQ[q, -1] && LtQ[p, -1]
  5967. Int[Times[Power[Plus[Optional[Pattern[a, Blank[]]], Times[ArcCoth[Times[Optional[Pattern[c, Blank[]]], Pattern[x, Blank[]]]], Optional[Pattern[b, Blank[]]]]], Pattern[p, Blank[]]], Power[Plus[Pattern[d, Blank[]], Times[Optional[Pattern[e, Blank[]]], Power[Pattern[x, Blank[]], 2]]], Pattern[q, Blank[]]]], Pattern[x, Blank[Symbol]]] := Simp[((d + e*x^2)^(q + 1)*(a + b*ArcCoth[c*x])^(p + 1))/(b*c*d*(p + 1)), x] + Dist[(2*c*(q + 1))/(b*(p + 1)), Int[x*(d + e*x^2)^q*(a + b*ArcCoth[c*x])^(p + 1), x], x] /; FreeQ[{a, b, c, d, e}, x] && EqQ[c^2*d + e, 0] && LtQ[q, -1] && LtQ[p, -1]
  5968. Int[Times[Power[Plus[Optional[Pattern[a, Blank[]]], Times[ArcTanh[Times[Optional[Pattern[c, Blank[]]], Pattern[x, Blank[]]]], Optional[Pattern[b, Blank[]]]]], Optional[Pattern[p, Blank[]]]], Power[Plus[Pattern[d, Blank[]], Times[Optional[Pattern[e, Blank[]]], Power[Pattern[x, Blank[]], 2]]], Pattern[q, Blank[]]]], Pattern[x, Blank[Symbol]]] := Dist[d^q/c, Subst[Int[(a + b*x)^p/Cosh[x]^(2*(q + 1)), x], x, ArcTanh[c*x]], x] /; FreeQ[{a, b, c, d, e, p}, x] && EqQ[c^2*d + e, 0] && ILtQ[2*(q + 1), 0] && (IntegerQ[q] || GtQ[d, 0])
  5969. Int[Times[Power[Plus[Optional[Pattern[a, Blank[]]], Times[ArcTanh[Times[Optional[Pattern[c, Blank[]]], Pattern[x, Blank[]]]], Optional[Pattern[b, Blank[]]]]], Optional[Pattern[p, Blank[]]]], Power[Plus[Pattern[d, Blank[]], Times[Optional[Pattern[e, Blank[]]], Power[Pattern[x, Blank[]], 2]]], Pattern[q, Blank[]]]], Pattern[x, Blank[Symbol]]] := Dist[(d^(q + 1/2)*Sqrt[1 - c^2*x^2])/Sqrt[d + e*x^2], Int[(1 - c^2*x^2)^q*(a + b*ArcTanh[c*x])^p, x], x] /; FreeQ[{a, b, c, d, e, p}, x] && EqQ[c^2*d + e, 0] && ILtQ[2*(q + 1), 0] && !(IntegerQ[q] || GtQ[d, 0])
  5970. Int[Times[Power[Plus[Optional[Pattern[a, Blank[]]], Times[ArcCoth[Times[Optional[Pattern[c, Blank[]]], Pattern[x, Blank[]]]], Optional[Pattern[b, Blank[]]]]], Optional[Pattern[p, Blank[]]]], Power[Plus[Pattern[d, Blank[]], Times[Optional[Pattern[e, Blank[]]], Power[Pattern[x, Blank[]], 2]]], Pattern[q, Blank[]]]], Pattern[x, Blank[Symbol]]] := -Dist[(-d)^q/c, Subst[Int[(a + b*x)^p/Sinh[x]^(2*(q + 1)), x], x, ArcCoth[c*x]], x] /; FreeQ[{a, b, c, d, e, p}, x] && EqQ[c^2*d + e, 0] && ILtQ[2*(q + 1), 0] && IntegerQ[q]
  5971. Int[Times[Power[Plus[Optional[Pattern[a, Blank[]]], Times[ArcCoth[Times[Optional[Pattern[c, Blank[]]], Pattern[x, Blank[]]]], Optional[Pattern[b, Blank[]]]]], Optional[Pattern[p, Blank[]]]], Power[Plus[Pattern[d, Blank[]], Times[Optional[Pattern[e, Blank[]]], Power[Pattern[x, Blank[]], 2]]], Pattern[q, Blank[]]]], Pattern[x, Blank[Symbol]]] := -Dist[((-d)^(q + 1/2)*x*Sqrt[(c^2*x^2 - 1)/(c^2*x^2)])/Sqrt[d + e*x^2], Subst[Int[(a + b*x)^p/Sinh[x]^(2*(q + 1)), x], x, ArcCoth[c*x]], x] /; FreeQ[{a, b, c, d, e, p}, x] && EqQ[c^2*d + e, 0] && ILtQ[2*(q + 1), 0] && !IntegerQ[q]
  5972. Int[Times[ArcTanh[Times[Optional[Pattern[c, Blank[]]], Pattern[x, Blank[]]]], Power[Plus[Optional[Pattern[d, Blank[]]], Times[Optional[Pattern[e, Blank[]]], Power[Pattern[x, Blank[]], 2]]], -1]], Pattern[x, Blank[Symbol]]] := Dist[1/2, Int[Log[1 + c*x]/(d + e*x^2), x], x] - Dist[1/2, Int[Log[1 - c*x]/(d + e*x^2), x], x] /; FreeQ[{c, d, e}, x]
  5973. Int[Times[ArcCoth[Times[Optional[Pattern[c, Blank[]]], Pattern[x, Blank[]]]], Power[Plus[Optional[Pattern[d, Blank[]]], Times[Optional[Pattern[e, Blank[]]], Power[Pattern[x, Blank[]], 2]]], -1]], Pattern[x, Blank[Symbol]]] := Dist[1/2, Int[Log[1 + 1/(c*x)]/(d + e*x^2), x], x] - Dist[1/2, Int[Log[1 - 1/(c*x)]/(d + e*x^2), x], x] /; FreeQ[{c, d, e}, x]
  5974. Int[Times[Plus[Times[ArcTanh[Times[Optional[Pattern[c, Blank[]]], Pattern[x, Blank[]]]], Optional[Pattern[b, Blank[]]]], Pattern[a, Blank[]]], Power[Plus[Optional[Pattern[d, Blank[]]], Times[Optional[Pattern[e, Blank[]]], Power[Pattern[x, Blank[]], 2]]], -1]], Pattern[x, Blank[Symbol]]] := Dist[a, Int[1/(d + e*x^2), x], x] + Dist[b, Int[ArcTanh[c*x]/(d + e*x^2), x], x] /; FreeQ[{a, b, c, d, e}, x]
  5975. Int[Times[Plus[Times[ArcCoth[Times[Optional[Pattern[c, Blank[]]], Pattern[x, Blank[]]]], Optional[Pattern[b, Blank[]]]], Pattern[a, Blank[]]], Power[Plus[Optional[Pattern[d, Blank[]]], Times[Optional[Pattern[e, Blank[]]], Power[Pattern[x, Blank[]], 2]]], -1]], Pattern[x, Blank[Symbol]]] := Dist[a, Int[1/(d + e*x^2), x], x] + Dist[b, Int[ArcCoth[c*x]/(d + e*x^2), x], x] /; FreeQ[{a, b, c, d, e}, x]
  5976. Int[Times[Plus[Optional[Pattern[a, Blank[]]], Times[ArcTanh[Times[Optional[Pattern[c, Blank[]]], Pattern[x, Blank[]]]], Optional[Pattern[b, Blank[]]]]], Power[Plus[Optional[Pattern[d, Blank[]]], Times[Optional[Pattern[e, Blank[]]], Power[Pattern[x, Blank[]], 2]]], Optional[Pattern[q, Blank[]]]]], Pattern[x, Blank[Symbol]]] := With[{u = IntHide[(d + e*x^2)^q, x]}, Dist[a + b*ArcTanh[c*x], u, x] - Dist[b*c, Int[u/(1 - c^2*x^2), x], x]] /; FreeQ[{a, b, c, d, e}, x] && (IntegerQ[q] || ILtQ[q + 1/2, 0])
  5977. Int[Times[Plus[Optional[Pattern[a, Blank[]]], Times[ArcCoth[Times[Optional[Pattern[c, Blank[]]], Pattern[x, Blank[]]]], Optional[Pattern[b, Blank[]]]]], Power[Plus[Optional[Pattern[d, Blank[]]], Times[Optional[Pattern[e, Blank[]]], Power[Pattern[x, Blank[]], 2]]], Optional[Pattern[q, Blank[]]]]], Pattern[x, Blank[Symbol]]] := With[{u = IntHide[(d + e*x^2)^q, x]}, Dist[a + b*ArcCoth[c*x], u, x] - Dist[b*c, Int[u/(1 - c^2*x^2), x], x]] /; FreeQ[{a, b, c, d, e}, x] && (IntegerQ[q] || ILtQ[q + 1/2, 0])
  5978. Int[Times[Power[Plus[Optional[Pattern[a, Blank[]]], Times[ArcTanh[Times[Optional[Pattern[c, Blank[]]], Pattern[x, Blank[]]]], Optional[Pattern[b, Blank[]]]]], Optional[Pattern[p, Blank[]]]], Power[Plus[Pattern[d, Blank[]], Times[Optional[Pattern[e, Blank[]]], Power[Pattern[x, Blank[]], 2]]], Optional[Pattern[q, Blank[]]]]], Pattern[x, Blank[Symbol]]] := Int[ExpandIntegrand[(a + b*ArcTanh[c*x])^p, (d + e*x^2)^q, x], x] /; FreeQ[{a, b, c, d, e}, x] && IntegerQ[q] && IGtQ[p, 0]
  5979. Int[Times[Power[Plus[Optional[Pattern[a, Blank[]]], Times[ArcCoth[Times[Optional[Pattern[c, Blank[]]], Pattern[x, Blank[]]]], Optional[Pattern[b, Blank[]]]]], Optional[Pattern[p, Blank[]]]], Power[Plus[Pattern[d, Blank[]], Times[Optional[Pattern[e, Blank[]]], Power[Pattern[x, Blank[]], 2]]], Optional[Pattern[q, Blank[]]]]], Pattern[x, Blank[Symbol]]] := Int[ExpandIntegrand[(a + b*ArcCoth[c*x])^p, (d + e*x^2)^q, x], x] /; FreeQ[{a, b, c, d, e}, x] && IntegerQ[q] && IGtQ[p, 0]
  5980. Int[Times[Power[Plus[Optional[Pattern[a, Blank[]]], Times[ArcTanh[Times[Optional[Pattern[c, Blank[]]], Pattern[x, Blank[]]]], Optional[Pattern[b, Blank[]]]]], Optional[Pattern[p, Blank[]]]], Power[Times[Optional[Pattern[f, Blank[]]], Pattern[x, Blank[]]], Pattern[m, Blank[]]], Power[Plus[Pattern[d, Blank[]], Times[Optional[Pattern[e, Blank[]]], Power[Pattern[x, Blank[]], 2]]], -1]], Pattern[x, Blank[Symbol]]] := Dist[f^2/e, Int[(f*x)^(m - 2)*(a + b*ArcTanh[c*x])^p, x], x] - Dist[(d*f^2)/e, Int[((f*x)^(m - 2)*(a + b*ArcTanh[c*x])^p)/(d + e*x^2), x], x] /; FreeQ[{a, b, c, d, e, f}, x] && GtQ[p, 0] && GtQ[m, 1]
  5981. Int[Times[Power[Plus[Optional[Pattern[a, Blank[]]], Times[ArcCoth[Times[Optional[Pattern[c, Blank[]]], Pattern[x, Blank[]]]], Optional[Pattern[b, Blank[]]]]], Optional[Pattern[p, Blank[]]]], Power[Times[Optional[Pattern[f, Blank[]]], Pattern[x, Blank[]]], Pattern[m, Blank[]]], Power[Plus[Pattern[d, Blank[]], Times[Optional[Pattern[e, Blank[]]], Power[Pattern[x, Blank[]], 2]]], -1]], Pattern[x, Blank[Symbol]]] := Dist[f^2/e, Int[(f*x)^(m - 2)*(a + b*ArcCoth[c*x])^p, x], x] - Dist[(d*f^2)/e, Int[((f*x)^(m - 2)*(a + b*ArcCoth[c*x])^p)/(d + e*x^2), x], x] /; FreeQ[{a, b, c, d, e, f}, x] && GtQ[p, 0] && GtQ[m, 1]
  5982. Int[Times[Power[Plus[Optional[Pattern[a, Blank[]]], Times[ArcTanh[Times[Optional[Pattern[c, Blank[]]], Pattern[x, Blank[]]]], Optional[Pattern[b, Blank[]]]]], Optional[Pattern[p, Blank[]]]], Power[Times[Optional[Pattern[f, Blank[]]], Pattern[x, Blank[]]], Pattern[m, Blank[]]], Power[Plus[Pattern[d, Blank[]], Times[Optional[Pattern[e, Blank[]]], Power[Pattern[x, Blank[]], 2]]], -1]], Pattern[x, Blank[Symbol]]] := Dist[1/d, Int[(f*x)^m*(a + b*ArcTanh[c*x])^p, x], x] - Dist[e/(d*f^2), Int[((f*x)^(m + 2)*(a + b*ArcTanh[c*x])^p)/(d + e*x^2), x], x] /; FreeQ[{a, b, c, d, e, f}, x] && GtQ[p, 0] && LtQ[m, -1]
  5983. Int[Times[Power[Plus[Optional[Pattern[a, Blank[]]], Times[ArcCoth[Times[Optional[Pattern[c, Blank[]]], Pattern[x, Blank[]]]], Optional[Pattern[b, Blank[]]]]], Optional[Pattern[p, Blank[]]]], Power[Times[Optional[Pattern[f, Blank[]]], Pattern[x, Blank[]]], Pattern[m, Blank[]]], Power[Plus[Pattern[d, Blank[]], Times[Optional[Pattern[e, Blank[]]], Power[Pattern[x, Blank[]], 2]]], -1]], Pattern[x, Blank[Symbol]]] := Dist[1/d, Int[(f*x)^m*(a + b*ArcCoth[c*x])^p, x], x] - Dist[e/(d*f^2), Int[((f*x)^(m + 2)*(a + b*ArcCoth[c*x])^p)/(d + e*x^2), x], x] /; FreeQ[{a, b, c, d, e, f}, x] && GtQ[p, 0] && LtQ[m, -1]
  5984. Int[Times[Power[Plus[Optional[Pattern[a, Blank[]]], Times[ArcTanh[Times[Optional[Pattern[c, Blank[]]], Pattern[x, Blank[]]]], Optional[Pattern[b, Blank[]]]]], Optional[Pattern[p, Blank[]]]], Pattern[x, Blank[]], Power[Plus[Pattern[d, Blank[]], Times[Optional[Pattern[e, Blank[]]], Power[Pattern[x, Blank[]], 2]]], -1]], Pattern[x, Blank[Symbol]]] := Simp[(a + b*ArcTanh[c*x])^(p + 1)/(b*e*(p + 1)), x] + Dist[1/(c*d), Int[(a + b*ArcTanh[c*x])^p/(1 - c*x), x], x] /; FreeQ[{a, b, c, d, e}, x] && EqQ[c^2*d + e, 0] && IGtQ[p, 0]
  5985. Int[Times[Power[Plus[Optional[Pattern[a, Blank[]]], Times[ArcCoth[Times[Optional[Pattern[c, Blank[]]], Pattern[x, Blank[]]]], Optional[Pattern[b, Blank[]]]]], Optional[Pattern[p, Blank[]]]], Pattern[x, Blank[]], Power[Plus[Pattern[d, Blank[]], Times[Optional[Pattern[e, Blank[]]], Power[Pattern[x, Blank[]], 2]]], -1]], Pattern[x, Blank[Symbol]]] := Simp[(a + b*ArcCoth[c*x])^(p + 1)/(b*e*(p + 1)), x] + Dist[1/(c*d), Int[(a + b*ArcCoth[c*x])^p/(1 - c*x), x], x] /; FreeQ[{a, b, c, d, e}, x] && EqQ[c^2*d + e, 0] && IGtQ[p, 0]
  5986. Int[Times[Power[Plus[Optional[Pattern[a, Blank[]]], Times[ArcTanh[Times[Optional[Pattern[c, Blank[]]], Pattern[x, Blank[]]]], Optional[Pattern[b, Blank[]]]]], Pattern[p, Blank[]]], Pattern[x, Blank[]], Power[Plus[Pattern[d, Blank[]], Times[Optional[Pattern[e, Blank[]]], Power[Pattern[x, Blank[]], 2]]], -1]], Pattern[x, Blank[Symbol]]] := Simp[(x*(a + b*ArcTanh[c*x])^(p + 1))/(b*c*d*(p + 1)), x] - Dist[1/(b*c*d*(p + 1)), Int[(a + b*ArcTanh[c*x])^(p + 1), x], x] /; FreeQ[{a, b, c, d, e}, x] && EqQ[c^2*d + e, 0] && !IGtQ[p, 0] && NeQ[p, -1]
  5987. Int[Times[Power[Plus[Optional[Pattern[a, Blank[]]], Times[ArcCoth[Times[Optional[Pattern[c, Blank[]]], Pattern[x, Blank[]]]], Optional[Pattern[b, Blank[]]]]], Pattern[p, Blank[]]], Pattern[x, Blank[]], Power[Plus[Pattern[d, Blank[]], Times[Optional[Pattern[e, Blank[]]], Power[Pattern[x, Blank[]], 2]]], -1]], Pattern[x, Blank[Symbol]]] := -Simp[(x*(a + b*ArcCoth[c*x])^(p + 1))/(b*c*d*(p + 1)), x] - Dist[1/(b*c*d*(p + 1)), Int[(a + b*ArcCoth[c*x])^(p + 1), x], x] /; FreeQ[{a, b, c, d, e}, x] && EqQ[c^2*d + e, 0] && !IGtQ[p, 0] && NeQ[p, -1]
  5988. Int[Times[Power[Plus[Optional[Pattern[a, Blank[]]], Times[ArcTanh[Times[Optional[Pattern[c, Blank[]]], Pattern[x, Blank[]]]], Optional[Pattern[b, Blank[]]]]], Optional[Pattern[p, Blank[]]]], Power[Pattern[x, Blank[]], -1], Power[Plus[Pattern[d, Blank[]], Times[Optional[Pattern[e, Blank[]]], Power[Pattern[x, Blank[]], 2]]], -1]], Pattern[x, Blank[Symbol]]] := Simp[(a + b*ArcTanh[c*x])^(p + 1)/(b*d*(p + 1)), x] + Dist[1/d, Int[(a + b*ArcTanh[c*x])^p/(x*(1 + c*x)), x], x] /; FreeQ[{a, b, c, d, e}, x] && EqQ[c^2*d + e, 0] && GtQ[p, 0]
  5989. Int[Times[Power[Plus[Optional[Pattern[a, Blank[]]], Times[ArcCoth[Times[Optional[Pattern[c, Blank[]]], Pattern[x, Blank[]]]], Optional[Pattern[b, Blank[]]]]], Optional[Pattern[p, Blank[]]]], Power[Pattern[x, Blank[]], -1], Power[Plus[Pattern[d, Blank[]], Times[Optional[Pattern[e, Blank[]]], Power[Pattern[x, Blank[]], 2]]], -1]], Pattern[x, Blank[Symbol]]] := Simp[(a + b*ArcCoth[c*x])^(p + 1)/(b*d*(p + 1)), x] + Dist[1/d, Int[(a + b*ArcCoth[c*x])^p/(x*(1 + c*x)), x], x] /; FreeQ[{a, b, c, d, e}, x] && EqQ[c^2*d + e, 0] && GtQ[p, 0]
  5990. Int[Times[Power[Plus[Optional[Pattern[a, Blank[]]], Times[ArcTanh[Times[Optional[Pattern[c, Blank[]]], Pattern[x, Blank[]]]], Optional[Pattern[b, Blank[]]]]], Pattern[p, Blank[]]], Power[Times[Optional[Pattern[f, Blank[]]], Pattern[x, Blank[]]], Pattern[m, Blank[]]], Power[Plus[Pattern[d, Blank[]], Times[Optional[Pattern[e, Blank[]]], Power[Pattern[x, Blank[]], 2]]], -1]], Pattern[x, Blank[Symbol]]] := Simp[((f*x)^m*(a + b*ArcTanh[c*x])^(p + 1))/(b*c*d*(p + 1)), x] - Dist[(f*m)/(b*c*d*(p + 1)), Int[(f*x)^(m - 1)*(a + b*ArcTanh[c*x])^(p + 1), x], x] /; FreeQ[{a, b, c, d, e, f, m}, x] && EqQ[c^2*d + e, 0] && LtQ[p, -1]
  5991. Int[Times[Power[Plus[Optional[Pattern[a, Blank[]]], Times[ArcCoth[Times[Optional[Pattern[c, Blank[]]], Pattern[x, Blank[]]]], Optional[Pattern[b, Blank[]]]]], Pattern[p, Blank[]]], Power[Times[Optional[Pattern[f, Blank[]]], Pattern[x, Blank[]]], Pattern[m, Blank[]]], Power[Plus[Pattern[d, Blank[]], Times[Optional[Pattern[e, Blank[]]], Power[Pattern[x, Blank[]], 2]]], -1]], Pattern[x, Blank[Symbol]]] := Simp[((f*x)^m*(a + b*ArcCoth[c*x])^(p + 1))/(b*c*d*(p + 1)), x] - Dist[(f*m)/(b*c*d*(p + 1)), Int[(f*x)^(m - 1)*(a + b*ArcCoth[c*x])^(p + 1), x], x] /; FreeQ[{a, b, c, d, e, f, m}, x] && EqQ[c^2*d + e, 0] && LtQ[p, -1]
  5992. Int[Times[Plus[Optional[Pattern[a, Blank[]]], Times[ArcTanh[Times[Optional[Pattern[c, Blank[]]], Pattern[x, Blank[]]]], Optional[Pattern[b, Blank[]]]]], Power[Pattern[x, Blank[]], Optional[Pattern[m, Blank[]]]], Power[Plus[Pattern[d, Blank[]], Times[Optional[Pattern[e, Blank[]]], Power[Pattern[x, Blank[]], 2]]], -1]], Pattern[x, Blank[Symbol]]] := Int[ExpandIntegrand[a + b*ArcTanh[c*x], x^m/(d + e*x^2), x], x] /; FreeQ[{a, b, c, d, e}, x] && IntegerQ[m] && !(EqQ[m, 1] && NeQ[a, 0])
  5993. Int[Times[Plus[Optional[Pattern[a, Blank[]]], Times[ArcCoth[Times[Optional[Pattern[c, Blank[]]], Pattern[x, Blank[]]]], Optional[Pattern[b, Blank[]]]]], Power[Pattern[x, Blank[]], Optional[Pattern[m, Blank[]]]], Power[Plus[Pattern[d, Blank[]], Times[Optional[Pattern[e, Blank[]]], Power[Pattern[x, Blank[]], 2]]], -1]], Pattern[x, Blank[Symbol]]] := Int[ExpandIntegrand[a + b*ArcCoth[c*x], x^m/(d + e*x^2), x], x] /; FreeQ[{a, b, c, d, e}, x] && IntegerQ[m] && !(EqQ[m, 1] && NeQ[a, 0])
  5994. Int[Times[Power[Plus[Optional[Pattern[a, Blank[]]], Times[ArcTanh[Times[Optional[Pattern[c, Blank[]]], Pattern[x, Blank[]]]], Optional[Pattern[b, Blank[]]]]], Optional[Pattern[p, Blank[]]]], Pattern[x, Blank[]], Power[Plus[Pattern[d, Blank[]], Times[Optional[Pattern[e, Blank[]]], Power[Pattern[x, Blank[]], 2]]], Optional[Pattern[q, Blank[]]]]], Pattern[x, Blank[Symbol]]] := Simp[((d + e*x^2)^(q + 1)*(a + b*ArcTanh[c*x])^p)/(2*e*(q + 1)), x] + Dist[(b*p)/(2*c*(q + 1)), Int[(d + e*x^2)^q*(a + b*ArcTanh[c*x])^(p - 1), x], x] /; FreeQ[{a, b, c, d, e, q}, x] && EqQ[c^2*d + e, 0] && GtQ[p, 0] && NeQ[q, -1]
  5995. Int[Times[Power[Plus[Optional[Pattern[a, Blank[]]], Times[ArcCoth[Times[Optional[Pattern[c, Blank[]]], Pattern[x, Blank[]]]], Optional[Pattern[b, Blank[]]]]], Optional[Pattern[p, Blank[]]]], Pattern[x, Blank[]], Power[Plus[Pattern[d, Blank[]], Times[Optional[Pattern[e, Blank[]]], Power[Pattern[x, Blank[]], 2]]], Optional[Pattern[q, Blank[]]]]], Pattern[x, Blank[Symbol]]] := Simp[((d + e*x^2)^(q + 1)*(a + b*ArcCoth[c*x])^p)/(2*e*(q + 1)), x] + Dist[(b*p)/(2*c*(q + 1)), Int[(d + e*x^2)^q*(a + b*ArcCoth[c*x])^(p - 1), x], x] /; FreeQ[{a, b, c, d, e, q}, x] && EqQ[c^2*d + e, 0] && GtQ[p, 0] && NeQ[q, -1]
  5996. Int[Times[Power[Plus[Optional[Pattern[a, Blank[]]], Times[ArcTanh[Times[Optional[Pattern[c, Blank[]]], Pattern[x, Blank[]]]], Optional[Pattern[b, Blank[]]]]], Pattern[p, Blank[]]], Pattern[x, Blank[]], Power[Plus[Pattern[d, Blank[]], Times[Optional[Pattern[e, Blank[]]], Power[Pattern[x, Blank[]], 2]]], -2]], Pattern[x, Blank[Symbol]]] := Simp[(x*(a + b*ArcTanh[c*x])^(p + 1))/(b*c*d*(p + 1)*(d + e*x^2)), x] + (Dist[4/(b^2*(p + 1)*(p + 2)), Int[(x*(a + b*ArcTanh[c*x])^(p + 2))/(d + e*x^2)^2, x], x] + Simp[((1 + c^2*x^2)*(a + b*ArcTanh[c*x])^(p + 2))/(b^2*e*(p + 1)*(p + 2)*(d + e*x^2)), x]) /; FreeQ[{a, b, c, d, e}, x] && EqQ[c^2*d + e, 0] && LtQ[p, -1] && NeQ[p, -2]
  5997. Int[Times[Power[Plus[Optional[Pattern[a, Blank[]]], Times[ArcCoth[Times[Optional[Pattern[c, Blank[]]], Pattern[x, Blank[]]]], Optional[Pattern[b, Blank[]]]]], Pattern[p, Blank[]]], Pattern[x, Blank[]], Power[Plus[Pattern[d, Blank[]], Times[Optional[Pattern[e, Blank[]]], Power[Pattern[x, Blank[]], 2]]], -2]], Pattern[x, Blank[Symbol]]] := Simp[(x*(a + b*ArcCoth[c*x])^(p + 1))/(b*c*d*(p + 1)*(d + e*x^2)), x] + (Dist[4/(b^2*(p + 1)*(p + 2)), Int[(x*(a + b*ArcCoth[c*x])^(p + 2))/(d + e*x^2)^2, x], x] + Simp[((1 + c^2*x^2)*(a + b*ArcCoth[c*x])^(p + 2))/(b^2*e*(p + 1)*(p + 2)*(d + e*x^2)), x]) /; FreeQ[{a, b, c, d, e}, x] && EqQ[c^2*d + e, 0] && LtQ[p, -1] && NeQ[p, -2]
  5998. Int[Times[Plus[Optional[Pattern[a, Blank[]]], Times[ArcTanh[Times[Optional[Pattern[c, Blank[]]], Pattern[x, Blank[]]]], Optional[Pattern[b, Blank[]]]]], Power[Pattern[x, Blank[]], 2], Power[Plus[Pattern[d, Blank[]], Times[Optional[Pattern[e, Blank[]]], Power[Pattern[x, Blank[]], 2]]], Pattern[q, Blank[]]]], Pattern[x, Blank[Symbol]]] := -Simp[(b*(d + e*x^2)^(q + 1))/(4*c^3*d*(q + 1)^2), x] + (Dist[1/(2*c^2*d*(q + 1)), Int[(d + e*x^2)^(q + 1)*(a + b*ArcTanh[c*x]), x], x] - Simp[(x*(d + e*x^2)^(q + 1)*(a + b*ArcTanh[c*x]))/(2*c^2*d*(q + 1)), x]) /; FreeQ[{a, b, c, d, e}, x] && EqQ[c^2*d + e, 0] && LtQ[q, -1] && NeQ[q, -5/2]
  5999. Int[Times[Plus[Optional[Pattern[a, Blank[]]], Times[ArcCoth[Times[Optional[Pattern[c, Blank[]]], Pattern[x, Blank[]]]], Optional[Pattern[b, Blank[]]]]], Power[Pattern[x, Blank[]], 2], Power[Plus[Pattern[d, Blank[]], Times[Optional[Pattern[e, Blank[]]], Power[Pattern[x, Blank[]], 2]]], Pattern[q, Blank[]]]], Pattern[x, Blank[Symbol]]] := -Simp[(b*(d + e*x^2)^(q + 1))/(4*c^3*d*(q + 1)^2), x] + (Dist[1/(2*c^2*d*(q + 1)), Int[(d + e*x^2)^(q + 1)*(a + b*ArcCoth[c*x]), x], x] - Simp[(x*(d + e*x^2)^(q + 1)*(a + b*ArcCoth[c*x]))/(2*c^2*d*(q + 1)), x]) /; FreeQ[{a, b, c, d, e}, x] && EqQ[c^2*d + e, 0] && LtQ[q, -1] && NeQ[q, -5/2]
  6000. Int[Times[Power[Plus[Optional[Pattern[a, Blank[]]], Times[ArcTanh[Times[Optional[Pattern[c, Blank[]]], Pattern[x, Blank[]]]], Optional[Pattern[b, Blank[]]]]], Optional[Pattern[p, Blank[]]]], Power[Pattern[x, Blank[]], 2], Power[Plus[Pattern[d, Blank[]], Times[Optional[Pattern[e, Blank[]]], Power[Pattern[x, Blank[]], 2]]], -2]], Pattern[x, Blank[Symbol]]] := -Simp[(a + b*ArcTanh[c*x])^(p + 1)/(2*b*c^3*d^2*(p + 1)), x] + (-Dist[(b*p)/(2*c), Int[(x*(a + b*ArcTanh[c*x])^(p - 1))/(d + e*x^2)^2, x], x] + Simp[(x*(a + b*ArcTanh[c*x])^p)/(2*c^2*d*(d + e*x^2)), x]) /; FreeQ[{a, b, c, d, e}, x] && EqQ[c^2*d + e, 0] && GtQ[p, 0]
  6001. Int[Times[Power[Plus[Optional[Pattern[a, Blank[]]], Times[ArcCoth[Times[Optional[Pattern[c, Blank[]]], Pattern[x, Blank[]]]], Optional[Pattern[b, Blank[]]]]], Optional[Pattern[p, Blank[]]]], Power[Pattern[x, Blank[]], 2], Power[Plus[Pattern[d, Blank[]], Times[Optional[Pattern[e, Blank[]]], Power[Pattern[x, Blank[]], 2]]], -2]], Pattern[x, Blank[Symbol]]] := -Simp[(a + b*ArcCoth[c*x])^(p + 1)/(2*b*c^3*d^2*(p + 1)), x] + (-Dist[(b*p)/(2*c), Int[(x*(a + b*ArcCoth[c*x])^(p - 1))/(d + e*x^2)^2, x], x] + Simp[(x*(a + b*ArcCoth[c*x])^p)/(2*c^2*d*(d + e*x^2)), x]) /; FreeQ[{a, b, c, d, e}, x] && EqQ[c^2*d + e, 0] && GtQ[p, 0]
  6002. Int[Times[Plus[Optional[Pattern[a, Blank[]]], Times[ArcTanh[Times[Optional[Pattern[c, Blank[]]], Pattern[x, Blank[]]]], Optional[Pattern[b, Blank[]]]]], Power[Times[Optional[Pattern[f, Blank[]]], Pattern[x, Blank[]]], Pattern[m, Blank[]]], Power[Plus[Pattern[d, Blank[]], Times[Optional[Pattern[e, Blank[]]], Power[Pattern[x, Blank[]], 2]]], Pattern[q, Blank[]]]], Pattern[x, Blank[Symbol]]] := -Simp[(b*(f*x)^m*(d + e*x^2)^(q + 1))/(c*d*m^2), x] + (-Dist[(f^2*(m - 1))/(c^2*d*m), Int[(f*x)^(m - 2)*(d + e*x^2)^(q + 1)*(a + b*ArcTanh[c*x]), x], x] + Simp[(f*(f*x)^(m - 1)*(d + e*x^2)^(q + 1)*(a + b*ArcTanh[c*x]))/(c^2*d*m), x]) /; FreeQ[{a, b, c, d, e, f}, x] && EqQ[c^2*d + e, 0] && EqQ[m + 2*q + 2, 0] && LtQ[q, -1]
  6003. Int[Times[Plus[Optional[Pattern[a, Blank[]]], Times[ArcCoth[Times[Optional[Pattern[c, Blank[]]], Pattern[x, Blank[]]]], Optional[Pattern[b, Blank[]]]]], Power[Times[Optional[Pattern[f, Blank[]]], Pattern[x, Blank[]]], Pattern[m, Blank[]]], Power[Plus[Pattern[d, Blank[]], Times[Optional[Pattern[e, Blank[]]], Power[Pattern[x, Blank[]], 2]]], Pattern[q, Blank[]]]], Pattern[x, Blank[Symbol]]] := -Simp[(b*(f*x)^m*(d + e*x^2)^(q + 1))/(c*d*m^2), x] + (-Dist[(f^2*(m - 1))/(c^2*d*m), Int[(f*x)^(m - 2)*(d + e*x^2)^(q + 1)*(a + b*ArcCoth[c*x]), x], x] + Simp[(f*(f*x)^(m - 1)*(d + e*x^2)^(q + 1)*(a + b*ArcCoth[c*x]))/(c^2*d*m), x]) /; FreeQ[{a, b, c, d, e, f}, x] && EqQ[c^2*d + e, 0] && EqQ[m + 2*q + 2, 0] && LtQ[q, -1]
  6004. Int[Times[Power[Plus[Optional[Pattern[a, Blank[]]], Times[ArcTanh[Times[Optional[Pattern[c, Blank[]]], Pattern[x, Blank[]]]], Optional[Pattern[b, Blank[]]]]], Pattern[p, Blank[]]], Power[Times[Optional[Pattern[f, Blank[]]], Pattern[x, Blank[]]], Pattern[m, Blank[]]], Power[Plus[Pattern[d, Blank[]], Times[Optional[Pattern[e, Blank[]]], Power[Pattern[x, Blank[]], 2]]], Pattern[q, Blank[]]]], Pattern[x, Blank[Symbol]]] := -Simp[(b*p*(f*x)^m*(d + e*x^2)^(q + 1)*(a + b*ArcTanh[c*x])^(p - 1))/(c*d*m^2), x] + (-Dist[(f^2*(m - 1))/(c^2*d*m), Int[(f*x)^(m - 2)*(d + e*x^2)^(q + 1)*(a + b*ArcTanh[c*x])^p, x], x] + Dist[(b^2*p*(p - 1))/m^2, Int[(f*x)^m*(d + e*x^2)^q*(a + b*ArcTanh[c*x])^(p - 2), x], x] + Simp[(f*(f*x)^(m - 1)*(d + e*x^2)^(q + 1)*(a + b*ArcTanh[c*x])^p)/(c^2*d*m), x]) /; FreeQ[{a, b, c, d, e, f, m}, x] && EqQ[c^2*d + e, 0] && EqQ[m + 2*q + 2, 0] && LtQ[q, -1] && GtQ[p, 1]
  6005. Int[Times[Power[Plus[Optional[Pattern[a, Blank[]]], Times[ArcCoth[Times[Optional[Pattern[c, Blank[]]], Pattern[x, Blank[]]]], Optional[Pattern[b, Blank[]]]]], Pattern[p, Blank[]]], Power[Times[Optional[Pattern[f, Blank[]]], Pattern[x, Blank[]]], Pattern[m, Blank[]]], Power[Plus[Pattern[d, Blank[]], Times[Optional[Pattern[e, Blank[]]], Power[Pattern[x, Blank[]], 2]]], Pattern[q, Blank[]]]], Pattern[x, Blank[Symbol]]] := -Simp[(b*p*(f*x)^m*(d + e*x^2)^(q + 1)*(a + b*ArcCoth[c*x])^(p - 1))/(c*d*m^2), x] + (-Dist[(f^2*(m - 1))/(c^2*d*m), Int[(f*x)^(m - 2)*(d + e*x^2)^(q + 1)*(a + b*ArcCoth[c*x])^p, x], x] + Dist[(b^2*p*(p - 1))/m^2, Int[(f*x)^m*(d + e*x^2)^q*(a + b*ArcCoth[c*x])^(p - 2), x], x] + Simp[(f*(f*x)^(m - 1)*(d + e*x^2)^(q + 1)*(a + b*ArcCoth[c*x])^p)/(c^2*d*m), x]) /; FreeQ[{a, b, c, d, e, f, m}, x] && EqQ[c^2*d + e, 0] && EqQ[m + 2*q + 2, 0] && LtQ[q, -1] && GtQ[p, 1]
  6006. Int[Times[Power[Plus[Optional[Pattern[a, Blank[]]], Times[ArcTanh[Times[Optional[Pattern[c, Blank[]]], Pattern[x, Blank[]]]], Optional[Pattern[b, Blank[]]]]], Pattern[p, Blank[]]], Power[Times[Optional[Pattern[f, Blank[]]], Pattern[x, Blank[]]], Optional[Pattern[m, Blank[]]]], Power[Plus[Pattern[d, Blank[]], Times[Optional[Pattern[e, Blank[]]], Power[Pattern[x, Blank[]], 2]]], Optional[Pattern[q, Blank[]]]]], Pattern[x, Blank[Symbol]]] := Simp[((f*x)^m*(d + e*x^2)^(q + 1)*(a + b*ArcTanh[c*x])^(p + 1))/(b*c*d*(p + 1)), x] - Dist[(f*m)/(b*c*(p + 1)), Int[(f*x)^(m - 1)*(d + e*x^2)^q*(a + b*ArcTanh[c*x])^(p + 1), x], x] /; FreeQ[{a, b, c, d, e, f, m, q}, x] && EqQ[c^2*d + e, 0] && EqQ[m + 2*q + 2, 0] && LtQ[p, -1]
  6007. Int[Times[Power[Plus[Optional[Pattern[a, Blank[]]], Times[ArcCoth[Times[Optional[Pattern[c, Blank[]]], Pattern[x, Blank[]]]], Optional[Pattern[b, Blank[]]]]], Pattern[p, Blank[]]], Power[Times[Optional[Pattern[f, Blank[]]], Pattern[x, Blank[]]], Optional[Pattern[m, Blank[]]]], Power[Plus[Pattern[d, Blank[]], Times[Optional[Pattern[e, Blank[]]], Power[Pattern[x, Blank[]], 2]]], Optional[Pattern[q, Blank[]]]]], Pattern[x, Blank[Symbol]]] := Simp[((f*x)^m*(d + e*x^2)^(q + 1)*(a + b*ArcCoth[c*x])^(p + 1))/(b*c*d*(p + 1)), x] - Dist[(f*m)/(b*c*(p + 1)), Int[(f*x)^(m - 1)*(d + e*x^2)^q*(a + b*ArcCoth[c*x])^(p + 1), x], x] /; FreeQ[{a, b, c, d, e, f, m, q}, x] && EqQ[c^2*d + e, 0] && EqQ[m + 2*q + 2, 0] && LtQ[p, -1]
  6008. Int[Times[Power[Plus[Optional[Pattern[a, Blank[]]], Times[ArcTanh[Times[Optional[Pattern[c, Blank[]]], Pattern[x, Blank[]]]], Optional[Pattern[b, Blank[]]]]], Optional[Pattern[p, Blank[]]]], Power[Times[Optional[Pattern[f, Blank[]]], Pattern[x, Blank[]]], Optional[Pattern[m, Blank[]]]], Power[Plus[Pattern[d, Blank[]], Times[Optional[Pattern[e, Blank[]]], Power[Pattern[x, Blank[]], 2]]], Optional[Pattern[q, Blank[]]]]], Pattern[x, Blank[Symbol]]] := Simp[((f*x)^(m + 1)*(d + e*x^2)^(q + 1)*(a + b*ArcTanh[c*x])^p)/(d*(m + 1)), x] - Dist[(b*c*p)/(m + 1), Int[(f*x)^(m + 1)*(d + e*x^2)^q*(a + b*ArcTanh[c*x])^(p - 1), x], x] /; FreeQ[{a, b, c, d, e, f, m, q}, x] && EqQ[c^2*d + e, 0] && EqQ[m + 2*q + 3, 0] && GtQ[p, 0] && NeQ[m, -1]
  6009. Int[Times[Power[Plus[Optional[Pattern[a, Blank[]]], Times[ArcCoth[Times[Optional[Pattern[c, Blank[]]], Pattern[x, Blank[]]]], Optional[Pattern[b, Blank[]]]]], Optional[Pattern[p, Blank[]]]], Power[Times[Optional[Pattern[f, Blank[]]], Pattern[x, Blank[]]], Optional[Pattern[m, Blank[]]]], Power[Plus[Pattern[d, Blank[]], Times[Optional[Pattern[e, Blank[]]], Power[Pattern[x, Blank[]], 2]]], Optional[Pattern[q, Blank[]]]]], Pattern[x, Blank[Symbol]]] := Simp[((f*x)^(m + 1)*(d + e*x^2)^(q + 1)*(a + b*ArcCoth[c*x])^p)/(d*f*(m + 1)), x] - Dist[(b*c*p)/(f*(m + 1)), Int[(f*x)^(m + 1)*(d + e*x^2)^q*(a + b*ArcCoth[c*x])^(p - 1), x], x] /; FreeQ[{a, b, c, d, e, f, m, q}, x] && EqQ[c^2*d + e, 0] && EqQ[m + 2*q + 3, 0] && GtQ[p, 0] && NeQ[m, -1]
  6010. Int[Times[Plus[Optional[Pattern[a, Blank[]]], Times[ArcTanh[Times[Optional[Pattern[c, Blank[]]], Pattern[x, Blank[]]]], Optional[Pattern[b, Blank[]]]]], Power[Times[Optional[Pattern[f, Blank[]]], Pattern[x, Blank[]]], Pattern[m, Blank[]]], Power[Plus[Pattern[d, Blank[]], Times[Optional[Pattern[e, Blank[]]], Power[Pattern[x, Blank[]], 2]]], Rational[1, 2]]], Pattern[x, Blank[Symbol]]] := Simp[((f*x)^(m + 1)*Sqrt[d + e*x^2]*(a + b*ArcTanh[c*x]))/(f*(m + 2)), x] + (Dist[d/(m + 2), Int[((f*x)^m*(a + b*ArcTanh[c*x]))/Sqrt[d + e*x^2], x], x] - Dist[(b*c*d)/(f*(m + 2)), Int[(f*x)^(m + 1)/Sqrt[d + e*x^2], x], x]) /; FreeQ[{a, b, c, d, e, f, m}, x] && EqQ[c^2*d + e, 0] && NeQ[m, -2]
  6011. Int[Times[Plus[Optional[Pattern[a, Blank[]]], Times[ArcCoth[Times[Optional[Pattern[c, Blank[]]], Pattern[x, Blank[]]]], Optional[Pattern[b, Blank[]]]]], Power[Times[Optional[Pattern[f, Blank[]]], Pattern[x, Blank[]]], Pattern[m, Blank[]]], Power[Plus[Pattern[d, Blank[]], Times[Optional[Pattern[e, Blank[]]], Power[Pattern[x, Blank[]], 2]]], Rational[1, 2]]], Pattern[x, Blank[Symbol]]] := Simp[((f*x)^(m + 1)*Sqrt[d + e*x^2]*(a + b*ArcCoth[c*x]))/(f*(m + 2)), x] + (Dist[d/(m + 2), Int[((f*x)^m*(a + b*ArcCoth[c*x]))/Sqrt[d + e*x^2], x], x] - Dist[(b*c*d)/(f*(m + 2)), Int[(f*x)^(m + 1)/Sqrt[d + e*x^2], x], x]) /; FreeQ[{a, b, c, d, e, f, m}, x] && EqQ[c^2*d + e, 0] && NeQ[m, -2]
  6012. Int[Times[Power[Plus[Optional[Pattern[a, Blank[]]], Times[ArcTanh[Times[Optional[Pattern[c, Blank[]]], Pattern[x, Blank[]]]], Optional[Pattern[b, Blank[]]]]], Optional[Pattern[p, Blank[]]]], Power[Times[Optional[Pattern[f, Blank[]]], Pattern[x, Blank[]]], Pattern[m, Blank[]]], Power[Plus[Pattern[d, Blank[]], Times[Optional[Pattern[e, Blank[]]], Power[Pattern[x, Blank[]], 2]]], Pattern[q, Blank[]]]], Pattern[x, Blank[Symbol]]] := Int[ExpandIntegrand[(f*x)^m*(d + e*x^2)^q*(a + b*ArcTanh[c*x])^p, x], x] /; FreeQ[{a, b, c, d, e, f, m}, x] && EqQ[c^2*d + e, 0] && IGtQ[p, 0] && IGtQ[q, 1]
  6013. Int[Times[Power[Plus[Optional[Pattern[a, Blank[]]], Times[ArcCoth[Times[Optional[Pattern[c, Blank[]]], Pattern[x, Blank[]]]], Optional[Pattern[b, Blank[]]]]], Optional[Pattern[p, Blank[]]]], Power[Times[Optional[Pattern[f, Blank[]]], Pattern[x, Blank[]]], Pattern[m, Blank[]]], Power[Plus[Pattern[d, Blank[]], Times[Optional[Pattern[e, Blank[]]], Power[Pattern[x, Blank[]], 2]]], Pattern[q, Blank[]]]], Pattern[x, Blank[Symbol]]] := Int[ExpandIntegrand[(f*x)^m*(d + e*x^2)^q*(a + b*ArcCoth[c*x])^p, x], x] /; FreeQ[{a, b, c, d, e, f, m}, x] && EqQ[c^2*d + e, 0] && IGtQ[p, 0] && IGtQ[q, 1]
  6014. Int[Times[Power[Plus[Optional[Pattern[a, Blank[]]], Times[ArcTanh[Times[Optional[Pattern[c, Blank[]]], Pattern[x, Blank[]]]], Optional[Pattern[b, Blank[]]]]], Optional[Pattern[p, Blank[]]]], Power[Times[Optional[Pattern[f, Blank[]]], Pattern[x, Blank[]]], Pattern[m, Blank[]]], Power[Plus[Pattern[d, Blank[]], Times[Optional[Pattern[e, Blank[]]], Power[Pattern[x, Blank[]], 2]]], Optional[Pattern[q, Blank[]]]]], Pattern[x, Blank[Symbol]]] := Dist[d, Int[(f*x)^m*(d + e*x^2)^(q - 1)*(a + b*ArcTanh[c*x])^p, x], x] - Dist[(c^2*d)/f^2, Int[(f*x)^(m + 2)*(d + e*x^2)^(q - 1)*(a + b*ArcTanh[c*x])^p, x], x] /; FreeQ[{a, b, c, d, e, f, m}, x] && EqQ[c^2*d + e, 0] && GtQ[q, 0] && IGtQ[p, 0] && (RationalQ[m] || (EqQ[p, 1] && IntegerQ[q]))
  6015. Int[Times[Power[Plus[Optional[Pattern[a, Blank[]]], Times[ArcCoth[Times[Optional[Pattern[c, Blank[]]], Pattern[x, Blank[]]]], Optional[Pattern[b, Blank[]]]]], Optional[Pattern[p, Blank[]]]], Power[Times[Optional[Pattern[f, Blank[]]], Pattern[x, Blank[]]], Pattern[m, Blank[]]], Power[Plus[Pattern[d, Blank[]], Times[Optional[Pattern[e, Blank[]]], Power[Pattern[x, Blank[]], 2]]], Optional[Pattern[q, Blank[]]]]], Pattern[x, Blank[Symbol]]] := Dist[d, Int[(f*x)^m*(d + e*x^2)^(q - 1)*(a + b*ArcCoth[c*x])^p, x], x] - Dist[(c^2*d)/f^2, Int[(f*x)^(m + 2)*(d + e*x^2)^(q - 1)*(a + b*ArcCoth[c*x])^p, x], x] /; FreeQ[{a, b, c, d, e, f, m}, x] && EqQ[c^2*d + e, 0] && GtQ[q, 0] && IGtQ[p, 0] && (RationalQ[m] || (EqQ[p, 1] && IntegerQ[q]))
  6016. Int[Times[Power[Plus[Optional[Pattern[a, Blank[]]], Times[ArcTanh[Times[Optional[Pattern[c, Blank[]]], Pattern[x, Blank[]]]], Optional[Pattern[b, Blank[]]]]], Optional[Pattern[p, Blank[]]]], Power[Times[Optional[Pattern[f, Blank[]]], Pattern[x, Blank[]]], Pattern[m, Blank[]]], Power[Plus[Pattern[d, Blank[]], Times[Optional[Pattern[e, Blank[]]], Power[Pattern[x, Blank[]], 2]]], Rational[-1, 2]]], Pattern[x, Blank[Symbol]]] := -Simp[(f*(f*x)^(m - 1)*Sqrt[d + e*x^2]*(a + b*ArcTanh[c*x])^p)/(c^2*d*m), x] + (Dist[(b*f*p)/(c*m), Int[((f*x)^(m - 1)*(a + b*ArcTanh[c*x])^(p - 1))/Sqrt[d + e*x^2], x], x] + Dist[(f^2*(m - 1))/(c^2*m), Int[((f*x)^(m - 2)*(a + b*ArcTanh[c*x])^p)/Sqrt[d + e*x^2], x], x]) /; FreeQ[{a, b, c, d, e, f}, x] && EqQ[c^2*d + e, 0] && GtQ[p, 0] && GtQ[m, 1]
  6017. Int[Times[Power[Plus[Optional[Pattern[a, Blank[]]], Times[ArcCoth[Times[Optional[Pattern[c, Blank[]]], Pattern[x, Blank[]]]], Optional[Pattern[b, Blank[]]]]], Optional[Pattern[p, Blank[]]]], Power[Times[Optional[Pattern[f, Blank[]]], Pattern[x, Blank[]]], Pattern[m, Blank[]]], Power[Plus[Pattern[d, Blank[]], Times[Optional[Pattern[e, Blank[]]], Power[Pattern[x, Blank[]], 2]]], Rational[-1, 2]]], Pattern[x, Blank[Symbol]]] := -Simp[(f*(f*x)^(m - 1)*Sqrt[d + e*x^2]*(a + b*ArcCoth[c*x])^p)/(c^2*d*m), x] + (Dist[(b*f*p)/(c*m), Int[((f*x)^(m - 1)*(a + b*ArcCoth[c*x])^(p - 1))/Sqrt[d + e*x^2], x], x] + Dist[(f^2*(m - 1))/(c^2*m), Int[((f*x)^(m - 2)*(a + b*ArcCoth[c*x])^p)/Sqrt[d + e*x^2], x], x]) /; FreeQ[{a, b, c, d, e, f}, x] && EqQ[c^2*d + e, 0] && GtQ[p, 0] && GtQ[m, 1]
  6018. Int[Times[Plus[Optional[Pattern[a, Blank[]]], Times[ArcTanh[Times[Optional[Pattern[c, Blank[]]], Pattern[x, Blank[]]]], Optional[Pattern[b, Blank[]]]]], Power[Pattern[x, Blank[]], -1], Power[Plus[Pattern[d, Blank[]], Times[Optional[Pattern[e, Blank[]]], Power[Pattern[x, Blank[]], 2]]], Rational[-1, 2]]], Pattern[x, Blank[Symbol]]] := Simp[(-2*(a + b*ArcTanh[c*x])*ArcTanh[Sqrt[1 - c*x]/Sqrt[1 + c*x]])/Sqrt[d], x] + (Simp[(b*PolyLog[2, -(Sqrt[1 - c*x]/Sqrt[1 + c*x])])/Sqrt[d], x] - Simp[(b*PolyLog[2, Sqrt[1 - c*x]/Sqrt[1 + c*x]])/Sqrt[d], x]) /; FreeQ[{a, b, c, d, e}, x] && EqQ[c^2*d + e, 0] && GtQ[d, 0]
  6019. Int[Times[Plus[Optional[Pattern[a, Blank[]]], Times[ArcCoth[Times[Optional[Pattern[c, Blank[]]], Pattern[x, Blank[]]]], Optional[Pattern[b, Blank[]]]]], Power[Pattern[x, Blank[]], -1], Power[Plus[Pattern[d, Blank[]], Times[Optional[Pattern[e, Blank[]]], Power[Pattern[x, Blank[]], 2]]], Rational[-1, 2]]], Pattern[x, Blank[Symbol]]] := Simp[(-2*(a + b*ArcCoth[c*x])*ArcTanh[Sqrt[1 - c*x]/Sqrt[1 + c*x]])/Sqrt[d], x] + (Simp[(b*PolyLog[2, -(Sqrt[1 - c*x]/Sqrt[1 + c*x])])/Sqrt[d], x] - Simp[(b*PolyLog[2, Sqrt[1 - c*x]/Sqrt[1 + c*x]])/Sqrt[d], x]) /; FreeQ[{a, b, c, d, e}, x] && EqQ[c^2*d + e, 0] && GtQ[d, 0]
  6020. Int[Times[Power[Plus[Optional[Pattern[a, Blank[]]], Times[ArcTanh[Times[Optional[Pattern[c, Blank[]]], Pattern[x, Blank[]]]], Optional[Pattern[b, Blank[]]]]], Pattern[p, Blank[]]], Power[Pattern[x, Blank[]], -1], Power[Plus[Pattern[d, Blank[]], Times[Optional[Pattern[e, Blank[]]], Power[Pattern[x, Blank[]], 2]]], Rational[-1, 2]]], Pattern[x, Blank[Symbol]]] := Dist[1/Sqrt[d], Subst[Int[(a + b*x)^p*Csch[x], x], x, ArcTanh[c*x]], x] /; FreeQ[{a, b, c, d, e}, x] && EqQ[c^2*d + e, 0] && IGtQ[p, 0] && GtQ[d, 0]
  6021. Int[Times[Power[Plus[Optional[Pattern[a, Blank[]]], Times[ArcCoth[Times[Optional[Pattern[c, Blank[]]], Pattern[x, Blank[]]]], Optional[Pattern[b, Blank[]]]]], Pattern[p, Blank[]]], Power[Pattern[x, Blank[]], -1], Power[Plus[Pattern[d, Blank[]], Times[Optional[Pattern[e, Blank[]]], Power[Pattern[x, Blank[]], 2]]], Rational[-1, 2]]], Pattern[x, Blank[Symbol]]] := -Dist[(c*x*Sqrt[1 - 1/(c^2*x^2)])/Sqrt[d + e*x^2], Subst[Int[(a + b*x)^p*Sech[x], x], x, ArcCoth[c*x]], x] /; FreeQ[{a, b, c, d, e}, x] && EqQ[c^2*d + e, 0] && IGtQ[p, 0] && GtQ[d, 0]
  6022. Int[Times[Power[Plus[Optional[Pattern[a, Blank[]]], Times[ArcTanh[Times[Optional[Pattern[c, Blank[]]], Pattern[x, Blank[]]]], Optional[Pattern[b, Blank[]]]]], Optional[Pattern[p, Blank[]]]], Power[Pattern[x, Blank[]], -1], Power[Plus[Pattern[d, Blank[]], Times[Optional[Pattern[e, Blank[]]], Power[Pattern[x, Blank[]], 2]]], Rational[-1, 2]]], Pattern[x, Blank[Symbol]]] := Dist[Sqrt[1 - c^2*x^2]/Sqrt[d + e*x^2], Int[(a + b*ArcTanh[c*x])^p/(x*Sqrt[1 - c^2*x^2]), x], x] /; FreeQ[{a, b, c, d, e}, x] && EqQ[c^2*d + e, 0] && IGtQ[p, 0] && !GtQ[d, 0]
  6023. Int[Times[Power[Plus[Optional[Pattern[a, Blank[]]], Times[ArcCoth[Times[Optional[Pattern[c, Blank[]]], Pattern[x, Blank[]]]], Optional[Pattern[b, Blank[]]]]], Optional[Pattern[p, Blank[]]]], Power[Pattern[x, Blank[]], -1], Power[Plus[Pattern[d, Blank[]], Times[Optional[Pattern[e, Blank[]]], Power[Pattern[x, Blank[]], 2]]], Rational[-1, 2]]], Pattern[x, Blank[Symbol]]] := Dist[Sqrt[1 - c^2*x^2]/Sqrt[d + e*x^2], Int[(a + b*ArcCoth[c*x])^p/(x*Sqrt[1 - c^2*x^2]), x], x] /; FreeQ[{a, b, c, d, e}, x] && EqQ[c^2*d + e, 0] && IGtQ[p, 0] && !GtQ[d, 0]
  6024. Int[Times[Power[Plus[Optional[Pattern[a, Blank[]]], Times[ArcTanh[Times[Optional[Pattern[c, Blank[]]], Pattern[x, Blank[]]]], Optional[Pattern[b, Blank[]]]]], Optional[Pattern[p, Blank[]]]], Power[Pattern[x, Blank[]], -2], Power[Plus[Pattern[d, Blank[]], Times[Optional[Pattern[e, Blank[]]], Power[Pattern[x, Blank[]], 2]]], Rational[-1, 2]]], Pattern[x, Blank[Symbol]]] := -Simp[(Sqrt[d + e*x^2]*(a + b*ArcTanh[c*x])^p)/(d*x), x] + Dist[b*c*p, Int[(a + b*ArcTanh[c*x])^(p - 1)/(x*Sqrt[d + e*x^2]), x], x] /; FreeQ[{a, b, c, d, e}, x] && EqQ[c^2*d + e, 0] && GtQ[p, 0]
  6025. Int[Times[Power[Plus[Optional[Pattern[a, Blank[]]], Times[ArcCoth[Times[Optional[Pattern[c, Blank[]]], Pattern[x, Blank[]]]], Optional[Pattern[b, Blank[]]]]], Optional[Pattern[p, Blank[]]]], Power[Pattern[x, Blank[]], -2], Power[Plus[Pattern[d, Blank[]], Times[Optional[Pattern[e, Blank[]]], Power[Pattern[x, Blank[]], 2]]], Rational[-1, 2]]], Pattern[x, Blank[Symbol]]] := -Simp[(Sqrt[d + e*x^2]*(a + b*ArcCoth[c*x])^p)/(d*x), x] + Dist[b*c*p, Int[(a + b*ArcCoth[c*x])^(p - 1)/(x*Sqrt[d + e*x^2]), x], x] /; FreeQ[{a, b, c, d, e}, x] && EqQ[c^2*d + e, 0] && GtQ[p, 0]
  6026. Int[Times[Power[Plus[Optional[Pattern[a, Blank[]]], Times[ArcTanh[Times[Optional[Pattern[c, Blank[]]], Pattern[x, Blank[]]]], Optional[Pattern[b, Blank[]]]]], Optional[Pattern[p, Blank[]]]], Power[Times[Optional[Pattern[f, Blank[]]], Pattern[x, Blank[]]], Pattern[m, Blank[]]], Power[Plus[Pattern[d, Blank[]], Times[Optional[Pattern[e, Blank[]]], Power[Pattern[x, Blank[]], 2]]], Rational[-1, 2]]], Pattern[x, Blank[Symbol]]] := Simp[((f*x)^(m + 1)*Sqrt[d + e*x^2]*(a + b*ArcTanh[c*x])^p)/(d*f*(m + 1)), x] + (-Dist[(b*c*p)/(f*(m + 1)), Int[((f*x)^(m + 1)*(a + b*ArcTanh[c*x])^(p - 1))/Sqrt[d + e*x^2], x], x] + Dist[(c^2*(m + 2))/(f^2*(m + 1)), Int[((f*x)^(m + 2)*(a + b*ArcTanh[c*x])^p)/Sqrt[d + e*x^2], x], x]) /; FreeQ[{a, b, c, d, e, f}, x] && EqQ[c^2*d + e, 0] && GtQ[p, 0] && LtQ[m, -1] && NeQ[m, -2]
  6027. Int[Times[Power[Plus[Optional[Pattern[a, Blank[]]], Times[ArcCoth[Times[Optional[Pattern[c, Blank[]]], Pattern[x, Blank[]]]], Optional[Pattern[b, Blank[]]]]], Optional[Pattern[p, Blank[]]]], Power[Times[Optional[Pattern[f, Blank[]]], Pattern[x, Blank[]]], Pattern[m, Blank[]]], Power[Plus[Pattern[d, Blank[]], Times[Optional[Pattern[e, Blank[]]], Power[Pattern[x, Blank[]], 2]]], Rational[-1, 2]]], Pattern[x, Blank[Symbol]]] := Simp[((f*x)^(m + 1)*Sqrt[d + e*x^2]*(a + b*ArcCoth[c*x])^p)/(d*f*(m + 1)), x] + (-Dist[(b*c*p)/(f*(m + 1)), Int[((f*x)^(m + 1)*(a + b*ArcCoth[c*x])^(p - 1))/Sqrt[d + e*x^2], x], x] + Dist[(c^2*(m + 2))/(f^2*(m + 1)), Int[((f*x)^(m + 2)*(a + b*ArcCoth[c*x])^p)/Sqrt[d + e*x^2], x], x]) /; FreeQ[{a, b, c, d, e, f}, x] && EqQ[c^2*d + e, 0] && GtQ[p, 0] && LtQ[m, -1] && NeQ[m, -2]
  6028. Int[Times[Power[Plus[Optional[Pattern[a, Blank[]]], Times[ArcTanh[Times[Optional[Pattern[c, Blank[]]], Pattern[x, Blank[]]]], Optional[Pattern[b, Blank[]]]]], Optional[Pattern[p, Blank[]]]], Power[Pattern[x, Blank[]], Pattern[m, Blank[]]], Power[Plus[Pattern[d, Blank[]], Times[Optional[Pattern[e, Blank[]]], Power[Pattern[x, Blank[]], 2]]], Pattern[q, Blank[]]]], Pattern[x, Blank[Symbol]]] := Dist[1/e, Int[x^(m - 2)*(d + e*x^2)^(q + 1)*(a + b*ArcTanh[c*x])^p, x], x] - Dist[d/e, Int[x^(m - 2)*(d + e*x^2)^q*(a + b*ArcTanh[c*x])^p, x], x] /; FreeQ[{a, b, c, d, e}, x] && EqQ[c^2*d + e, 0] && IntegersQ[p, 2*q] && LtQ[q, -1] && IGtQ[m, 1] && NeQ[p, -1]
  6029. Int[Times[Power[Plus[Optional[Pattern[a, Blank[]]], Times[ArcCoth[Times[Optional[Pattern[c, Blank[]]], Pattern[x, Blank[]]]], Optional[Pattern[b, Blank[]]]]], Optional[Pattern[p, Blank[]]]], Power[Pattern[x, Blank[]], Pattern[m, Blank[]]], Power[Plus[Pattern[d, Blank[]], Times[Optional[Pattern[e, Blank[]]], Power[Pattern[x, Blank[]], 2]]], Pattern[q, Blank[]]]], Pattern[x, Blank[Symbol]]] := Dist[1/e, Int[x^(m - 2)*(d + e*x^2)^(q + 1)*(a + b*ArcCoth[c*x])^p, x], x] - Dist[d/e, Int[x^(m - 2)*(d + e*x^2)^q*(a + b*ArcCoth[c*x])^p, x], x] /; FreeQ[{a, b, c, d, e}, x] && EqQ[c^2*d + e, 0] && IntegersQ[p, 2*q] && LtQ[q, -1] && IGtQ[m, 1] && NeQ[p, -1]
  6030. Int[Times[Power[Plus[Optional[Pattern[a, Blank[]]], Times[ArcTanh[Times[Optional[Pattern[c, Blank[]]], Pattern[x, Blank[]]]], Optional[Pattern[b, Blank[]]]]], Optional[Pattern[p, Blank[]]]], Power[Pattern[x, Blank[]], Pattern[m, Blank[]]], Power[Plus[Pattern[d, Blank[]], Times[Optional[Pattern[e, Blank[]]], Power[Pattern[x, Blank[]], 2]]], Pattern[q, Blank[]]]], Pattern[x, Blank[Symbol]]] := Dist[1/d, Int[x^m*(d + e*x^2)^(q + 1)*(a + b*ArcTanh[c*x])^p, x], x] - Dist[e/d, Int[x^(m + 2)*(d + e*x^2)^q*(a + b*ArcTanh[c*x])^p, x], x] /; FreeQ[{a, b, c, d, e}, x] && EqQ[c^2*d + e, 0] && IntegersQ[p, 2*q] && LtQ[q, -1] && ILtQ[m, 0] && NeQ[p, -1]
  6031. Int[Times[Power[Plus[Optional[Pattern[a, Blank[]]], Times[ArcCoth[Times[Optional[Pattern[c, Blank[]]], Pattern[x, Blank[]]]], Optional[Pattern[b, Blank[]]]]], Optional[Pattern[p, Blank[]]]], Power[Pattern[x, Blank[]], Pattern[m, Blank[]]], Power[Plus[Pattern[d, Blank[]], Times[Optional[Pattern[e, Blank[]]], Power[Pattern[x, Blank[]], 2]]], Pattern[q, Blank[]]]], Pattern[x, Blank[Symbol]]] := Dist[1/d, Int[x^m*(d + e*x^2)^(q + 1)*(a + b*ArcCoth[c*x])^p, x], x] - Dist[e/d, Int[x^(m + 2)*(d + e*x^2)^q*(a + b*ArcCoth[c*x])^p, x], x] /; FreeQ[{a, b, c, d, e}, x] && EqQ[c^2*d + e, 0] && IntegersQ[p, 2*q] && LtQ[q, -1] && ILtQ[m, 0] && NeQ[p, -1]
  6032. Int[Times[Power[Plus[Optional[Pattern[a, Blank[]]], Times[ArcTanh[Times[Optional[Pattern[c, Blank[]]], Pattern[x, Blank[]]]], Optional[Pattern[b, Blank[]]]]], Optional[Pattern[p, Blank[]]]], Power[Pattern[x, Blank[]], Optional[Pattern[m, Blank[]]]], Power[Plus[Pattern[d, Blank[]], Times[Optional[Pattern[e, Blank[]]], Power[Pattern[x, Blank[]], 2]]], Pattern[q, Blank[]]]], Pattern[x, Blank[Symbol]]] := Simp[(x^m*(d + e*x^2)^(q + 1)*(a + b*ArcTanh[c*x])^(p + 1))/(b*c*d*(p + 1)), x] + (Dist[(c*(m + 2*q + 2))/(b*(p + 1)), Int[x^(m + 1)*(d + e*x^2)^q*(a + b*ArcTanh[c*x])^(p + 1), x], x] - Dist[m/(b*c*(p + 1)), Int[x^(m - 1)*(d + e*x^2)^q*(a + b*ArcTanh[c*x])^(p + 1), x], x]) /; FreeQ[{a, b, c, d, e}, x] && EqQ[c^2*d + e, 0] && IntegerQ[m] && LtQ[q, -1] && LtQ[p, -1] && NeQ[m + 2*q + 2, 0]
  6033. Int[Times[Power[Plus[Optional[Pattern[a, Blank[]]], Times[ArcCoth[Times[Optional[Pattern[c, Blank[]]], Pattern[x, Blank[]]]], Optional[Pattern[b, Blank[]]]]], Optional[Pattern[p, Blank[]]]], Power[Pattern[x, Blank[]], Optional[Pattern[m, Blank[]]]], Power[Plus[Pattern[d, Blank[]], Times[Optional[Pattern[e, Blank[]]], Power[Pattern[x, Blank[]], 2]]], Pattern[q, Blank[]]]], Pattern[x, Blank[Symbol]]] := Simp[(x^m*(d + e*x^2)^(q + 1)*(a + b*ArcCoth[c*x])^(p + 1))/(b*c*d*(p + 1)), x] + (Dist[(c*(m + 2*q + 2))/(b*(p + 1)), Int[x^(m + 1)*(d + e*x^2)^q*(a + b*ArcCoth[c*x])^(p + 1), x], x] - Dist[m/(b*c*(p + 1)), Int[x^(m - 1)*(d + e*x^2)^q*(a + b*ArcCoth[c*x])^(p + 1), x], x]) /; FreeQ[{a, b, c, d, e}, x] && EqQ[c^2*d + e, 0] && IntegerQ[m] && LtQ[q, -1] && LtQ[p, -1] && NeQ[m + 2*q + 2, 0]
  6034. Int[Times[Power[Plus[Optional[Pattern[a, Blank[]]], Times[ArcTanh[Times[Optional[Pattern[c, Blank[]]], Pattern[x, Blank[]]]], Optional[Pattern[b, Blank[]]]]], Optional[Pattern[p, Blank[]]]], Power[Pattern[x, Blank[]], Optional[Pattern[m, Blank[]]]], Power[Plus[Pattern[d, Blank[]], Times[Optional[Pattern[e, Blank[]]], Power[Pattern[x, Blank[]], 2]]], Pattern[q, Blank[]]]], Pattern[x, Blank[Symbol]]] := Dist[d^q/c^(m + 1), Subst[Int[((a + b*x)^p*Sinh[x]^m)/Cosh[x]^(m + 2*(q + 1)), x], x, ArcTanh[c*x]], x] /; FreeQ[{a, b, c, d, e, p}, x] && EqQ[c^2*d + e, 0] && IGtQ[m, 0] && ILtQ[m + 2*q + 1, 0] && (IntegerQ[q] || GtQ[d, 0])
  6035. Int[Times[Power[Plus[Optional[Pattern[a, Blank[]]], Times[ArcTanh[Times[Optional[Pattern[c, Blank[]]], Pattern[x, Blank[]]]], Optional[Pattern[b, Blank[]]]]], Optional[Pattern[p, Blank[]]]], Power[Pattern[x, Blank[]], Optional[Pattern[m, Blank[]]]], Power[Plus[Pattern[d, Blank[]], Times[Optional[Pattern[e, Blank[]]], Power[Pattern[x, Blank[]], 2]]], Pattern[q, Blank[]]]], Pattern[x, Blank[Symbol]]] := Dist[(d^(q + 1/2)*Sqrt[1 - c^2*x^2])/Sqrt[d + e*x^2], Int[x^m*(1 - c^2*x^2)^q*(a + b*ArcTanh[c*x])^p, x], x] /; FreeQ[{a, b, c, d, e, p}, x] && EqQ[c^2*d + e, 0] && IGtQ[m, 0] && ILtQ[m + 2*q + 1, 0] && !(IntegerQ[q] || GtQ[d, 0])
  6036. Int[Times[Power[Plus[Optional[Pattern[a, Blank[]]], Times[ArcCoth[Times[Optional[Pattern[c, Blank[]]], Pattern[x, Blank[]]]], Optional[Pattern[b, Blank[]]]]], Optional[Pattern[p, Blank[]]]], Power[Pattern[x, Blank[]], Optional[Pattern[m, Blank[]]]], Power[Plus[Pattern[d, Blank[]], Times[Optional[Pattern[e, Blank[]]], Power[Pattern[x, Blank[]], 2]]], Pattern[q, Blank[]]]], Pattern[x, Blank[Symbol]]] := -Dist[(-d)^q/c^(m + 1), Subst[Int[((a + b*x)^p*Cosh[x]^m)/Sinh[x]^(m + 2*(q + 1)), x], x, ArcCoth[c*x]], x] /; FreeQ[{a, b, c, d, e, p}, x] && EqQ[c^2*d + e, 0] && IGtQ[m, 0] && ILtQ[m + 2*q + 1, 0] && IntegerQ[q]
  6037. Int[Times[Power[Plus[Optional[Pattern[a, Blank[]]], Times[ArcCoth[Times[Optional[Pattern[c, Blank[]]], Pattern[x, Blank[]]]], Optional[Pattern[b, Blank[]]]]], Optional[Pattern[p, Blank[]]]], Power[Pattern[x, Blank[]], Optional[Pattern[m, Blank[]]]], Power[Plus[Pattern[d, Blank[]], Times[Optional[Pattern[e, Blank[]]], Power[Pattern[x, Blank[]], 2]]], Pattern[q, Blank[]]]], Pattern[x, Blank[Symbol]]] := -Dist[((-d)^(q + 1/2)*x*Sqrt[(c^2*x^2 - 1)/(c^2*x^2)])/(c^m*Sqrt[d + e*x^2]), Subst[Int[((a + b*x)^p*Cosh[x]^m)/Sinh[x]^(m + 2*(q + 1)), x], x, ArcCoth[c*x]], x] /; FreeQ[{a, b, c, d, e, p}, x] && EqQ[c^2*d + e, 0] && IGtQ[m, 0] && ILtQ[m + 2*q + 1, 0] && !IntegerQ[q]
  6038. Int[Times[Plus[Optional[Pattern[a, Blank[]]], Times[ArcTanh[Times[Optional[Pattern[c, Blank[]]], Pattern[x, Blank[]]]], Optional[Pattern[b, Blank[]]]]], Pattern[x, Blank[]], Power[Plus[Optional[Pattern[d, Blank[]]], Times[Optional[Pattern[e, Blank[]]], Power[Pattern[x, Blank[]], 2]]], Optional[Pattern[q, Blank[]]]]], Pattern[x, Blank[Symbol]]] := Simp[((d + e*x^2)^(q + 1)*(a + b*ArcTanh[c*x]))/(2*e*(q + 1)), x] - Dist[(b*c)/(2*e*(q + 1)), Int[(d + e*x^2)^(q + 1)/(1 - c^2*x^2), x], x] /; FreeQ[{a, b, c, d, e, q}, x] && NeQ[q, -1]
  6039. Int[Times[Plus[Optional[Pattern[a, Blank[]]], Times[ArcCoth[Times[Optional[Pattern[c, Blank[]]], Pattern[x, Blank[]]]], Optional[Pattern[b, Blank[]]]]], Pattern[x, Blank[]], Power[Plus[Optional[Pattern[d, Blank[]]], Times[Optional[Pattern[e, Blank[]]], Power[Pattern[x, Blank[]], 2]]], Optional[Pattern[q, Blank[]]]]], Pattern[x, Blank[Symbol]]] := Simp[((d + e*x^2)^(q + 1)*(a + b*ArcCoth[c*x]))/(2*e*(q + 1)), x] - Dist[(b*c)/(2*e*(q + 1)), Int[(d + e*x^2)^(q + 1)/(1 - c^2*x^2), x], x] /; FreeQ[{a, b, c, d, e, q}, x] && NeQ[q, -1]
  6040. Int[Times[Plus[Optional[Pattern[a, Blank[]]], Times[ArcTanh[Times[Optional[Pattern[c, Blank[]]], Pattern[x, Blank[]]]], Optional[Pattern[b, Blank[]]]]], Power[Times[Optional[Pattern[f, Blank[]]], Pattern[x, Blank[]]], Optional[Pattern[m, Blank[]]]], Power[Plus[Optional[Pattern[d, Blank[]]], Times[Optional[Pattern[e, Blank[]]], Power[Pattern[x, Blank[]], 2]]], Optional[Pattern[q, Blank[]]]]], Pattern[x, Blank[Symbol]]] := With[{u = IntHide[(f*x)^m*(d + e*x^2)^q, x]}, Dist[a + b*ArcTanh[c*x], u, x] - Dist[b*c, Int[SimplifyIntegrand[u/(1 - c^2*x^2), x], x], x]] /; FreeQ[{a, b, c, d, e, f, m, q}, x] && ((IGtQ[q, 0] && !(ILtQ[(m - 1)/2, 0] && GtQ[m + 2*q + 3, 0])) || (IGtQ[(m + 1)/2, 0] && !(ILtQ[q, 0] && GtQ[m + 2*q + 3, 0])) || (ILtQ[(m + 2*q + 1)/2, 0] && !ILtQ[(m - 1)/2, 0]))
  6041. Int[Times[Plus[Optional[Pattern[a, Blank[]]], Times[ArcCoth[Times[Optional[Pattern[c, Blank[]]], Pattern[x, Blank[]]]], Optional[Pattern[b, Blank[]]]]], Power[Times[Optional[Pattern[f, Blank[]]], Pattern[x, Blank[]]], Optional[Pattern[m, Blank[]]]], Power[Plus[Optional[Pattern[d, Blank[]]], Times[Optional[Pattern[e, Blank[]]], Power[Pattern[x, Blank[]], 2]]], Optional[Pattern[q, Blank[]]]]], Pattern[x, Blank[Symbol]]] := With[{u = IntHide[(f*x)^m*(d + e*x^2)^q, x]}, Dist[a + b*ArcCoth[c*x], u, x] - Dist[b*c, Int[SimplifyIntegrand[u/(1 - c^2*x^2), x], x], x]] /; FreeQ[{a, b, c, d, e, f, m, q}, x] && ((IGtQ[q, 0] && !(ILtQ[(m - 1)/2, 0] && GtQ[m + 2*q + 3, 0])) || (IGtQ[(m + 1)/2, 0] && !(ILtQ[q, 0] && GtQ[m + 2*q + 3, 0])) || (ILtQ[(m + 2*q + 1)/2, 0] && !ILtQ[(m - 1)/2, 0]))
  6042. Int[Times[Power[Plus[Optional[Pattern[a, Blank[]]], Times[ArcTanh[Times[Optional[Pattern[c, Blank[]]], Pattern[x, Blank[]]]], Optional[Pattern[b, Blank[]]]]], Optional[Pattern[p, Blank[]]]], Pattern[x, Blank[]], Power[Plus[Pattern[d, Blank[]], Times[Optional[Pattern[e, Blank[]]], Power[Pattern[x, Blank[]], 2]]], -2]], Pattern[x, Blank[Symbol]]] := Dist[1/(4*d^2*Rt[-(e/d), 2]), Int[(a + b*ArcTanh[c*x])^p/(1 - Rt[-(e/d), 2]*x)^2, x], x] - Dist[1/(4*d^2*Rt[-(e/d), 2]), Int[(a + b*ArcTanh[c*x])^p/(1 + Rt[-(e/d), 2]*x)^2, x], x] /; FreeQ[{a, b, c, d, e}, x] && IGtQ[p, 0]
  6043. Int[Times[Power[Plus[Optional[Pattern[a, Blank[]]], Times[ArcCoth[Times[Optional[Pattern[c, Blank[]]], Pattern[x, Blank[]]]], Optional[Pattern[b, Blank[]]]]], Optional[Pattern[p, Blank[]]]], Pattern[x, Blank[]], Power[Plus[Pattern[d, Blank[]], Times[Optional[Pattern[e, Blank[]]], Power[Pattern[x, Blank[]], 2]]], -2]], Pattern[x, Blank[Symbol]]] := Dist[1/(4*d^2*Rt[-(e/d), 2]), Int[(a + b*ArcCoth[c*x])^p/(1 - Rt[-(e/d), 2]*x)^2, x], x] - Dist[1/(4*d^2*Rt[-(e/d), 2]), Int[(a + b*ArcCoth[c*x])^p/(1 + Rt[-(e/d), 2]*x)^2, x], x] /; FreeQ[{a, b, c, d, e}, x] && IGtQ[p, 0]
  6044. Int[Times[Power[Plus[Optional[Pattern[a, Blank[]]], Times[ArcTanh[Times[Optional[Pattern[c, Blank[]]], Pattern[x, Blank[]]]], Optional[Pattern[b, Blank[]]]]], Optional[Pattern[p, Blank[]]]], Power[Times[Optional[Pattern[f, Blank[]]], Pattern[x, Blank[]]], Optional[Pattern[m, Blank[]]]], Power[Plus[Pattern[d, Blank[]], Times[Optional[Pattern[e, Blank[]]], Power[Pattern[x, Blank[]], 2]]], Optional[Pattern[q, Blank[]]]]], Pattern[x, Blank[Symbol]]] := With[{u = ExpandIntegrand[(a + b*ArcTanh[c*x])^p, (f*x)^m*(d + e*x^2)^q, x]}, Int[u, x] /; SumQ[u]] /; FreeQ[{a, b, c, d, e, f, m}, x] && IntegerQ[q] && IGtQ[p, 0] && (GtQ[q, 0] || IntegerQ[m])
  6045. Int[Times[Power[Plus[Optional[Pattern[a, Blank[]]], Times[ArcCoth[Times[Optional[Pattern[c, Blank[]]], Pattern[x, Blank[]]]], Optional[Pattern[b, Blank[]]]]], Optional[Pattern[p, Blank[]]]], Power[Times[Optional[Pattern[f, Blank[]]], Pattern[x, Blank[]]], Optional[Pattern[m, Blank[]]]], Power[Plus[Pattern[d, Blank[]], Times[Optional[Pattern[e, Blank[]]], Power[Pattern[x, Blank[]], 2]]], Optional[Pattern[q, Blank[]]]]], Pattern[x, Blank[Symbol]]] := With[{u = ExpandIntegrand[(a + b*ArcCoth[c*x])^p, (f*x)^m*(d + e*x^2)^q, x]}, Int[u, x] /; SumQ[u]] /; FreeQ[{a, b, c, d, e, f, m}, x] && IntegerQ[q] && IGtQ[p, 0] && (GtQ[q, 0] || IntegerQ[m])
  6046. Int[Times[Plus[Times[ArcTanh[Times[Optional[Pattern[c, Blank[]]], Pattern[x, Blank[]]]], Optional[Pattern[b, Blank[]]]], Pattern[a, Blank[]]], Power[Times[Optional[Pattern[f, Blank[]]], Pattern[x, Blank[]]], Optional[Pattern[m, Blank[]]]], Power[Plus[Pattern[d, Blank[]], Times[Optional[Pattern[e, Blank[]]], Power[Pattern[x, Blank[]], 2]]], Optional[Pattern[q, Blank[]]]]], Pattern[x, Blank[Symbol]]] := Dist[a, Int[(f*x)^m*(d + e*x^2)^q, x], x] + Dist[b, Int[(f*x)^m*(d + e*x^2)^q*ArcTanh[c*x], x], x] /; FreeQ[{a, b, c, d, e, f, m, q}, x]
  6047. Int[Times[Plus[Times[ArcCoth[Times[Optional[Pattern[c, Blank[]]], Pattern[x, Blank[]]]], Optional[Pattern[b, Blank[]]]], Pattern[a, Blank[]]], Power[Times[Optional[Pattern[f, Blank[]]], Pattern[x, Blank[]]], Optional[Pattern[m, Blank[]]]], Power[Plus[Pattern[d, Blank[]], Times[Optional[Pattern[e, Blank[]]], Power[Pattern[x, Blank[]], 2]]], Optional[Pattern[q, Blank[]]]]], Pattern[x, Blank[Symbol]]] := Dist[a, Int[(f*x)^m*(d + e*x^2)^q, x], x] + Dist[b, Int[(f*x)^m*(d + e*x^2)^q*ArcCoth[c*x], x], x] /; FreeQ[{a, b, c, d, e, f, m, q}, x]
  6048. Int[Times[Power[Plus[Optional[Pattern[a, Blank[]]], Times[ArcTanh[Times[Optional[Pattern[c, Blank[]]], Pattern[x, Blank[]]]], Optional[Pattern[b, Blank[]]]]], Optional[Pattern[p, Blank[]]]], Power[Plus[Pattern[f, Blank[]], Times[Optional[Pattern[g, Blank[]]], Pattern[x, Blank[]]]], Optional[Pattern[m, Blank[]]]], Power[Plus[Pattern[d, Blank[]], Times[Optional[Pattern[e, Blank[]]], Power[Pattern[x, Blank[]], 2]]], -1]], Pattern[x, Blank[Symbol]]] := Int[ExpandIntegrand[(a + b*ArcTanh[c*x])^p/(d + e*x^2), (f + g*x)^m, x], x] /; FreeQ[{a, b, c, d, e, f, g}, x] && IGtQ[p, 0] && EqQ[c^2*d + e, 0] && IGtQ[m, 0]
  6049. Int[Times[Power[Plus[Optional[Pattern[a, Blank[]]], Times[ArcCoth[Times[Optional[Pattern[c, Blank[]]], Pattern[x, Blank[]]]], Optional[Pattern[b, Blank[]]]]], Optional[Pattern[p, Blank[]]]], Power[Plus[Pattern[f, Blank[]], Times[Optional[Pattern[g, Blank[]]], Pattern[x, Blank[]]]], Optional[Pattern[m, Blank[]]]], Power[Plus[Pattern[d, Blank[]], Times[Optional[Pattern[e, Blank[]]], Power[Pattern[x, Blank[]], 2]]], -1]], Pattern[x, Blank[Symbol]]] := Int[ExpandIntegrand[(a + b*ArcCoth[c*x])^p/(d + e*x^2), (f + g*x)^m, x], x] /; FreeQ[{a, b, c, d, e, f, g}, x] && IGtQ[p, 0] && EqQ[c^2*d + e, 0] && IGtQ[m, 0]
  6050. Int[Times[ArcTanh[Pattern[u, Blank[]]], Power[Plus[Optional[Pattern[a, Blank[]]], Times[ArcTanh[Times[Optional[Pattern[c, Blank[]]], Pattern[x, Blank[]]]], Optional[Pattern[b, Blank[]]]]], Optional[Pattern[p, Blank[]]]], Power[Plus[Pattern[d, Blank[]], Times[Optional[Pattern[e, Blank[]]], Power[Pattern[x, Blank[]], 2]]], -1]], Pattern[x, Blank[Symbol]]] := Dist[1/2, Int[(Log[1 + u]*(a + b*ArcTanh[c*x])^p)/(d + e*x^2), x], x] - Dist[1/2, Int[(Log[1 - u]*(a + b*ArcTanh[c*x])^p)/(d + e*x^2), x], x] /; FreeQ[{a, b, c, d, e}, x] && IGtQ[p, 0] && EqQ[c^2*d + e, 0] && EqQ[u^2 - (1 - 2/(1 + c*x))^2, 0]
  6051. Int[Times[ArcCoth[Pattern[u, Blank[]]], Power[Plus[Optional[Pattern[a, Blank[]]], Times[ArcCoth[Times[Optional[Pattern[c, Blank[]]], Pattern[x, Blank[]]]], Optional[Pattern[b, Blank[]]]]], Optional[Pattern[p, Blank[]]]], Power[Plus[Pattern[d, Blank[]], Times[Optional[Pattern[e, Blank[]]], Power[Pattern[x, Blank[]], 2]]], -1]], Pattern[x, Blank[Symbol]]] := Dist[1/2, Int[(Log[SimplifyIntegrand[1 + 1/u, x]]*(a + b*ArcCoth[c*x])^p)/(d + e*x^2), x], x] - Dist[1/2, Int[(Log[SimplifyIntegrand[1 - 1/u, x]]*(a + b*ArcCoth[c*x])^p)/(d + e*x^2), x], x] /; FreeQ[{a, b, c, d, e}, x] && IGtQ[p, 0] && EqQ[c^2*d + e, 0] && EqQ[u^2 - (1 - 2/(1 + c*x))^2, 0]
  6052. Int[Times[ArcTanh[Pattern[u, Blank[]]], Power[Plus[Optional[Pattern[a, Blank[]]], Times[ArcTanh[Times[Optional[Pattern[c, Blank[]]], Pattern[x, Blank[]]]], Optional[Pattern[b, Blank[]]]]], Optional[Pattern[p, Blank[]]]], Power[Plus[Pattern[d, Blank[]], Times[Optional[Pattern[e, Blank[]]], Power[Pattern[x, Blank[]], 2]]], -1]], Pattern[x, Blank[Symbol]]] := Dist[1/2, Int[(Log[1 + u]*(a + b*ArcTanh[c*x])^p)/(d + e*x^2), x], x] - Dist[1/2, Int[(Log[1 - u]*(a + b*ArcTanh[c*x])^p)/(d + e*x^2), x], x] /; FreeQ[{a, b, c, d, e}, x] && IGtQ[p, 0] && EqQ[c^2*d + e, 0] && EqQ[u^2 - (1 - 2/(1 - c*x))^2, 0]
  6053. Int[Times[ArcCoth[Pattern[u, Blank[]]], Power[Plus[Optional[Pattern[a, Blank[]]], Times[ArcCoth[Times[Optional[Pattern[c, Blank[]]], Pattern[x, Blank[]]]], Optional[Pattern[b, Blank[]]]]], Optional[Pattern[p, Blank[]]]], Power[Plus[Pattern[d, Blank[]], Times[Optional[Pattern[e, Blank[]]], Power[Pattern[x, Blank[]], 2]]], -1]], Pattern[x, Blank[Symbol]]] := Dist[1/2, Int[(Log[SimplifyIntegrand[1 + 1/u, x]]*(a + b*ArcCoth[c*x])^p)/(d + e*x^2), x], x] - Dist[1/2, Int[(Log[SimplifyIntegrand[1 - 1/u, x]]*(a + b*ArcCoth[c*x])^p)/(d + e*x^2), x], x] /; FreeQ[{a, b, c, d, e}, x] && IGtQ[p, 0] && EqQ[c^2*d + e, 0] && EqQ[u^2 - (1 - 2/(1 - c*x))^2, 0]
  6054. Int[Times[Log[Plus[Pattern[f, Blank[]], Times[Optional[Pattern[g, Blank[]]], Pattern[x, Blank[]]]]], Power[Plus[Optional[Pattern[a, Blank[]]], Times[ArcTanh[Times[Optional[Pattern[c, Blank[]]], Pattern[x, Blank[]]]], Optional[Pattern[b, Blank[]]]]], Optional[Pattern[p, Blank[]]]], Power[Plus[Pattern[d, Blank[]], Times[Optional[Pattern[e, Blank[]]], Power[Pattern[x, Blank[]], 2]]], -1]], Pattern[x, Blank[Symbol]]] := Simp[((a + b*ArcTanh[c*x])^(p + 1)*Log[f + g*x])/(b*c*d*(p + 1)), x] - Dist[g/(b*c*d*(p + 1)), Int[(a + b*ArcTanh[c*x])^(p + 1)/(f + g*x), x], x] /; FreeQ[{a, b, c, d, e, f, g}, x] && IGtQ[p, 0] && EqQ[c^2*d + e, 0] && EqQ[c^2*f^2 - g^2, 0]
  6055. Int[Times[Log[Plus[Pattern[f, Blank[]], Times[Optional[Pattern[g, Blank[]]], Pattern[x, Blank[]]]]], Power[Plus[Optional[Pattern[a, Blank[]]], Times[ArcCoth[Times[Optional[Pattern[c, Blank[]]], Pattern[x, Blank[]]]], Optional[Pattern[b, Blank[]]]]], Optional[Pattern[p, Blank[]]]], Power[Plus[Pattern[d, Blank[]], Times[Optional[Pattern[e, Blank[]]], Power[Pattern[x, Blank[]], 2]]], -1]], Pattern[x, Blank[Symbol]]] := Simp[((a + b*ArcCoth[c*x])^(p + 1)*Log[f + g*x])/(b*c*d*(p + 1)), x] - Dist[g/(b*c*d*(p + 1)), Int[(a + b*ArcCoth[c*x])^(p + 1)/(f + g*x), x], x] /; FreeQ[{a, b, c, d, e, f, g}, x] && IGtQ[p, 0] && EqQ[c^2*d + e, 0] && EqQ[c^2*f^2 - g^2, 0]
  6056. Int[Times[Log[Pattern[u, Blank[]]], Power[Plus[Optional[Pattern[a, Blank[]]], Times[ArcTanh[Times[Optional[Pattern[c, Blank[]]], Pattern[x, Blank[]]]], Optional[Pattern[b, Blank[]]]]], Optional[Pattern[p, Blank[]]]], Power[Plus[Pattern[d, Blank[]], Times[Optional[Pattern[e, Blank[]]], Power[Pattern[x, Blank[]], 2]]], -1]], Pattern[x, Blank[Symbol]]] := Simp[((a + b*ArcTanh[c*x])^p*PolyLog[2, 1 - u])/(2*c*d), x] - Dist[(b*p)/2, Int[((a + b*ArcTanh[c*x])^(p - 1)*PolyLog[2, 1 - u])/(d + e*x^2), x], x] /; FreeQ[{a, b, c, d, e}, x] && IGtQ[p, 0] && EqQ[c^2*d + e, 0] && EqQ[(1 - u)^2 - (1 - 2/(1 + c*x))^2, 0]
  6057. Int[Times[Log[Pattern[u, Blank[]]], Power[Plus[Optional[Pattern[a, Blank[]]], Times[ArcCoth[Times[Optional[Pattern[c, Blank[]]], Pattern[x, Blank[]]]], Optional[Pattern[b, Blank[]]]]], Optional[Pattern[p, Blank[]]]], Power[Plus[Pattern[d, Blank[]], Times[Optional[Pattern[e, Blank[]]], Power[Pattern[x, Blank[]], 2]]], -1]], Pattern[x, Blank[Symbol]]] := Simp[((a + b*ArcCoth[c*x])^p*PolyLog[2, 1 - u])/(2*c*d), x] - Dist[(b*p)/2, Int[((a + b*ArcCoth[c*x])^(p - 1)*PolyLog[2, 1 - u])/(d + e*x^2), x], x] /; FreeQ[{a, b, c, d, e}, x] && IGtQ[p, 0] && EqQ[c^2*d + e, 0] && EqQ[(1 - u)^2 - (1 - 2/(1 + c*x))^2, 0]
  6058. Int[Times[Log[Pattern[u, Blank[]]], Power[Plus[Optional[Pattern[a, Blank[]]], Times[ArcTanh[Times[Optional[Pattern[c, Blank[]]], Pattern[x, Blank[]]]], Optional[Pattern[b, Blank[]]]]], Optional[Pattern[p, Blank[]]]], Power[Plus[Pattern[d, Blank[]], Times[Optional[Pattern[e, Blank[]]], Power[Pattern[x, Blank[]], 2]]], -1]], Pattern[x, Blank[Symbol]]] := -Simp[((a + b*ArcTanh[c*x])^p*PolyLog[2, 1 - u])/(2*c*d), x] + Dist[(b*p)/2, Int[((a + b*ArcTanh[c*x])^(p - 1)*PolyLog[2, 1 - u])/(d + e*x^2), x], x] /; FreeQ[{a, b, c, d, e}, x] && IGtQ[p, 0] && EqQ[c^2*d + e, 0] && EqQ[(1 - u)^2 - (1 - 2/(1 - c*x))^2, 0]
  6059. Int[Times[Log[Pattern[u, Blank[]]], Power[Plus[Optional[Pattern[a, Blank[]]], Times[ArcCoth[Times[Optional[Pattern[c, Blank[]]], Pattern[x, Blank[]]]], Optional[Pattern[b, Blank[]]]]], Optional[Pattern[p, Blank[]]]], Power[Plus[Pattern[d, Blank[]], Times[Optional[Pattern[e, Blank[]]], Power[Pattern[x, Blank[]], 2]]], -1]], Pattern[x, Blank[Symbol]]] := -Simp[((a + b*ArcCoth[c*x])^p*PolyLog[2, 1 - u])/(2*c*d), x] + Dist[(b*p)/2, Int[((a + b*ArcCoth[c*x])^(p - 1)*PolyLog[2, 1 - u])/(d + e*x^2), x], x] /; FreeQ[{a, b, c, d, e}, x] && IGtQ[p, 0] && EqQ[c^2*d + e, 0] && EqQ[(1 - u)^2 - (1 - 2/(1 - c*x))^2, 0]
  6060. Int[Times[Power[Plus[Optional[Pattern[a, Blank[]]], Times[ArcTanh[Times[Optional[Pattern[c, Blank[]]], Pattern[x, Blank[]]]], Optional[Pattern[b, Blank[]]]]], Optional[Pattern[p, Blank[]]]], Power[Plus[Pattern[d, Blank[]], Times[Optional[Pattern[e, Blank[]]], Power[Pattern[x, Blank[]], 2]]], -1], PolyLog[Pattern[k, Blank[]], Pattern[u, Blank[]]]], Pattern[x, Blank[Symbol]]] := -Simp[((a + b*ArcTanh[c*x])^p*PolyLog[k + 1, u])/(2*c*d), x] + Dist[(b*p)/2, Int[((a + b*ArcTanh[c*x])^(p - 1)*PolyLog[k + 1, u])/(d + e*x^2), x], x] /; FreeQ[{a, b, c, d, e, k}, x] && IGtQ[p, 0] && EqQ[c^2*d + e, 0] && EqQ[u^2 - (1 - 2/(1 + c*x))^2, 0]
  6061. Int[Times[Power[Plus[Optional[Pattern[a, Blank[]]], Times[ArcCoth[Times[Optional[Pattern[c, Blank[]]], Pattern[x, Blank[]]]], Optional[Pattern[b, Blank[]]]]], Optional[Pattern[p, Blank[]]]], Power[Plus[Pattern[d, Blank[]], Times[Optional[Pattern[e, Blank[]]], Power[Pattern[x, Blank[]], 2]]], -1], PolyLog[Pattern[k, Blank[]], Pattern[u, Blank[]]]], Pattern[x, Blank[Symbol]]] := -Simp[((a + b*ArcCoth[c*x])^p*PolyLog[k + 1, u])/(2*c*d), x] + Dist[(b*p)/2, Int[((a + b*ArcCoth[c*x])^(p - 1)*PolyLog[k + 1, u])/(d + e*x^2), x], x] /; FreeQ[{a, b, c, d, e, k}, x] && IGtQ[p, 0] && EqQ[c^2*d + e, 0] && EqQ[u^2 - (1 - 2/(1 + c*x))^2, 0]
  6062. Int[Times[Power[Plus[Optional[Pattern[a, Blank[]]], Times[ArcTanh[Times[Optional[Pattern[c, Blank[]]], Pattern[x, Blank[]]]], Optional[Pattern[b, Blank[]]]]], Optional[Pattern[p, Blank[]]]], Power[Plus[Pattern[d, Blank[]], Times[Optional[Pattern[e, Blank[]]], Power[Pattern[x, Blank[]], 2]]], -1], PolyLog[Pattern[k, Blank[]], Pattern[u, Blank[]]]], Pattern[x, Blank[Symbol]]] := Simp[((a + b*ArcTanh[c*x])^p*PolyLog[k + 1, u])/(2*c*d), x] - Dist[(b*p)/2, Int[((a + b*ArcTanh[c*x])^(p - 1)*PolyLog[k + 1, u])/(d + e*x^2), x], x] /; FreeQ[{a, b, c, d, e, k}, x] && IGtQ[p, 0] && EqQ[c^2*d + e, 0] && EqQ[u^2 - (1 - 2/(1 - c*x))^2, 0]
  6063. Int[Times[Power[Plus[Optional[Pattern[a, Blank[]]], Times[ArcCoth[Times[Optional[Pattern[c, Blank[]]], Pattern[x, Blank[]]]], Optional[Pattern[b, Blank[]]]]], Optional[Pattern[p, Blank[]]]], Power[Plus[Pattern[d, Blank[]], Times[Optional[Pattern[e, Blank[]]], Power[Pattern[x, Blank[]], 2]]], -1], PolyLog[Pattern[k, Blank[]], Pattern[u, Blank[]]]], Pattern[x, Blank[Symbol]]] := Simp[((a + b*ArcCoth[c*x])^p*PolyLog[k + 1, u])/(2*c*d), x] - Dist[(b*p)/2, Int[((a + b*ArcCoth[c*x])^(p - 1)*PolyLog[k + 1, u])/(d + e*x^2), x], x] /; FreeQ[{a, b, c, d, e, k}, x] && IGtQ[p, 0] && EqQ[c^2*d + e, 0] && EqQ[u^2 - (1 - 2/(1 - c*x))^2, 0]
  6064. Int[Times[Power[Plus[Optional[Pattern[a, Blank[]]], Times[ArcCoth[Times[Optional[Pattern[c, Blank[]]], Pattern[x, Blank[]]]], Optional[Pattern[b, Blank[]]]]], -1], Power[Plus[Optional[Pattern[a, Blank[]]], Times[ArcTanh[Times[Optional[Pattern[c, Blank[]]], Pattern[x, Blank[]]]], Optional[Pattern[b, Blank[]]]]], -1], Power[Plus[Pattern[d, Blank[]], Times[Optional[Pattern[e, Blank[]]], Power[Pattern[x, Blank[]], 2]]], -1]], Pattern[x, Blank[Symbol]]] := Simp[(-Log[a + b*ArcCoth[c*x]] + Log[a + b*ArcTanh[c*x]])/(b^2*c*d*(ArcCoth[c*x] - ArcTanh[c*x])), x] /; FreeQ[{a, b, c, d, e}, x] && EqQ[c^2*d + e, 0]
  6065. Int[Times[Power[Plus[Optional[Pattern[a, Blank[]]], Times[ArcCoth[Times[Optional[Pattern[c, Blank[]]], Pattern[x, Blank[]]]], Optional[Pattern[b, Blank[]]]]], Optional[Pattern[m, Blank[]]]], Power[Plus[Optional[Pattern[a, Blank[]]], Times[ArcTanh[Times[Optional[Pattern[c, Blank[]]], Pattern[x, Blank[]]]], Optional[Pattern[b, Blank[]]]]], Optional[Pattern[p, Blank[]]]], Power[Plus[Pattern[d, Blank[]], Times[Optional[Pattern[e, Blank[]]], Power[Pattern[x, Blank[]], 2]]], -1]], Pattern[x, Blank[Symbol]]] := Simp[((a + b*ArcCoth[c*x])^(m + 1)*(a + b*ArcTanh[c*x])^p)/(b*c*d*(m + 1)), x] - Dist[p/(m + 1), Int[((a + b*ArcCoth[c*x])^(m + 1)*(a + b*ArcTanh[c*x])^(p - 1))/(d + e*x^2), x], x] /; FreeQ[{a, b, c, d, e}, x] && EqQ[c^2*d + e, 0] && IGtQ[p, 0] && IGeQ[m, p]
  6066. Int[Times[Power[Plus[Optional[Pattern[a, Blank[]]], Times[ArcCoth[Times[Optional[Pattern[c, Blank[]]], Pattern[x, Blank[]]]], Optional[Pattern[b, Blank[]]]]], Optional[Pattern[p, Blank[]]]], Power[Plus[Optional[Pattern[a, Blank[]]], Times[ArcTanh[Times[Optional[Pattern[c, Blank[]]], Pattern[x, Blank[]]]], Optional[Pattern[b, Blank[]]]]], Optional[Pattern[m, Blank[]]]], Power[Plus[Pattern[d, Blank[]], Times[Optional[Pattern[e, Blank[]]], Power[Pattern[x, Blank[]], 2]]], -1]], Pattern[x, Blank[Symbol]]] := Simp[((a + b*ArcTanh[c*x])^(m + 1)*(a + b*ArcCoth[c*x])^p)/(b*c*d*(m + 1)), x] - Dist[p/(m + 1), Int[((a + b*ArcTanh[c*x])^(m + 1)*(a + b*ArcCoth[c*x])^(p - 1))/(d + e*x^2), x], x] /; FreeQ[{a, b, c, d, e}, x] && EqQ[c^2*d + e, 0] && IGtQ[p, 0] && IGtQ[m, p]
  6067. Int[Times[ArcTanh[Times[Optional[Pattern[a, Blank[]]], Pattern[x, Blank[]]]], Power[Plus[Pattern[c, Blank[]], Times[Optional[Pattern[d, Blank[]]], Power[Pattern[x, Blank[]], Optional[Pattern[n, Blank[]]]]]], -1]], Pattern[x, Blank[Symbol]]] := Dist[1/2, Int[Log[1 + a*x]/(c + d*x^n), x], x] - Dist[1/2, Int[Log[1 - a*x]/(c + d*x^n), x], x] /; FreeQ[{a, c, d}, x] && IntegerQ[n] && !(EqQ[n, 2] && EqQ[a^2*c + d, 0])
  6068. Int[Times[ArcCoth[Times[Optional[Pattern[a, Blank[]]], Pattern[x, Blank[]]]], Power[Plus[Pattern[c, Blank[]], Times[Optional[Pattern[d, Blank[]]], Power[Pattern[x, Blank[]], Optional[Pattern[n, Blank[]]]]]], -1]], Pattern[x, Blank[Symbol]]] := Dist[1/2, Int[Log[1 + 1/(a*x)]/(c + d*x^n), x], x] - Dist[1/2, Int[Log[1 - 1/(a*x)]/(c + d*x^n), x], x] /; FreeQ[{a, c, d}, x] && IntegerQ[n] && !(EqQ[n, 2] && EqQ[a^2*c + d, 0])
  6069. Int[Times[ArcTanh[Times[Optional[Pattern[c, Blank[]]], Power[Pattern[x, Blank[]], Optional[Pattern[n, Blank[]]]]]], Log[Times[Optional[Pattern[d, Blank[]]], Power[Pattern[x, Blank[]], Optional[Pattern[m, Blank[]]]]]], Power[Pattern[x, Blank[]], -1]], Pattern[x, Blank[Symbol]]] := Dist[1/2, Int[(Log[d*x^m]*Log[1 + c*x^n])/x, x], x] - Dist[1/2, Int[(Log[d*x^m]*Log[1 - c*x^n])/x, x], x] /; FreeQ[{c, d, m, n}, x]
  6070. Int[Times[ArcCoth[Times[Optional[Pattern[c, Blank[]]], Power[Pattern[x, Blank[]], Optional[Pattern[n, Blank[]]]]]], Log[Times[Optional[Pattern[d, Blank[]]], Power[Pattern[x, Blank[]], Optional[Pattern[m, Blank[]]]]]], Power[Pattern[x, Blank[]], -1]], Pattern[x, Blank[Symbol]]] := Dist[1/2, Int[(Log[d*x^m]*Log[1 + 1/(c*x^n)])/x, x], x] - Dist[1/2, Int[(Log[d*x^m]*Log[1 - 1/(c*x^n)])/x, x], x] /; FreeQ[{c, d, m, n}, x]
  6071. Int[Times[Log[Times[Optional[Pattern[d, Blank[]]], Power[Pattern[x, Blank[]], Optional[Pattern[m, Blank[]]]]]], Plus[Times[ArcTanh[Times[Optional[Pattern[c, Blank[]]], Power[Pattern[x, Blank[]], Optional[Pattern[n, Blank[]]]]]], Optional[Pattern[b, Blank[]]]], Pattern[a, Blank[]]], Power[Pattern[x, Blank[]], -1]], Pattern[x, Blank[Symbol]]] := Dist[a, Int[Log[d*x^m]/x, x], x] + Dist[b, Int[(Log[d*x^m]*ArcTanh[c*x^n])/x, x], x] /; FreeQ[{a, b, c, d, m, n}, x]
  6072. Int[Times[Log[Times[Optional[Pattern[d, Blank[]]], Power[Pattern[x, Blank[]], Optional[Pattern[m, Blank[]]]]]], Plus[Times[ArcCoth[Times[Optional[Pattern[c, Blank[]]], Power[Pattern[x, Blank[]], Optional[Pattern[n, Blank[]]]]]], Optional[Pattern[b, Blank[]]]], Pattern[a, Blank[]]], Power[Pattern[x, Blank[]], -1]], Pattern[x, Blank[Symbol]]] := Dist[a, Int[Log[d*x^m]/x, x], x] + Dist[b, Int[(Log[d*x^m]*ArcCoth[c*x^n])/x, x], x] /; FreeQ[{a, b, c, d, m, n}, x]
  6073. Int[Times[Plus[Optional[Pattern[a, Blank[]]], Times[ArcTanh[Times[Optional[Pattern[c, Blank[]]], Pattern[x, Blank[]]]], Optional[Pattern[b, Blank[]]]]], Plus[Optional[Pattern[d, Blank[]]], Times[Log[Plus[Optional[Pattern[f, Blank[]]], Times[Optional[Pattern[g, Blank[]]], Power[Pattern[x, Blank[]], 2]]]], Optional[Pattern[e, Blank[]]]]]], Pattern[x, Blank[Symbol]]] := Simp[x*(d + e*Log[f + g*x^2])*(a + b*ArcTanh[c*x]), x] + (-Dist[b*c, Int[(x*(d + e*Log[f + g*x^2]))/(1 - c^2*x^2), x], x] - Dist[2*e*g, Int[(x^2*(a + b*ArcTanh[c*x]))/(f + g*x^2), x], x]) /; FreeQ[{a, b, c, d, e, f, g}, x]
  6074. Int[Times[Plus[Optional[Pattern[a, Blank[]]], Times[ArcCoth[Times[Optional[Pattern[c, Blank[]]], Pattern[x, Blank[]]]], Optional[Pattern[b, Blank[]]]]], Plus[Optional[Pattern[d, Blank[]]], Times[Log[Plus[Optional[Pattern[f, Blank[]]], Times[Optional[Pattern[g, Blank[]]], Power[Pattern[x, Blank[]], 2]]]], Optional[Pattern[e, Blank[]]]]]], Pattern[x, Blank[Symbol]]] := Simp[x*(d + e*Log[f + g*x^2])*(a + b*ArcCoth[c*x]), x] + (-Dist[b*c, Int[(x*(d + e*Log[f + g*x^2]))/(1 - c^2*x^2), x], x] - Dist[2*e*g, Int[(x^2*(a + b*ArcCoth[c*x]))/(f + g*x^2), x], x]) /; FreeQ[{a, b, c, d, e, f, g}, x]
  6075. Int[Times[ArcTanh[Times[Optional[Pattern[c, Blank[]]], Pattern[x, Blank[]]]], Log[Plus[Optional[Pattern[f, Blank[]]], Times[Optional[Pattern[g, Blank[]]], Power[Pattern[x, Blank[]], 2]]]], Power[Pattern[x, Blank[]], -1]], Pattern[x, Blank[Symbol]]] := Dist[Log[f + g*x^2] - Log[1 - c*x] - Log[1 + c*x], Int[ArcTanh[c*x]/x, x], x] + (-Dist[1/2, Int[Log[1 - c*x]^2/x, x], x] + Dist[1/2, Int[Log[1 + c*x]^2/x, x], x]) /; FreeQ[{c, f, g}, x] && EqQ[c^2*f + g, 0]
  6076. Int[Times[ArcCoth[Times[Optional[Pattern[c, Blank[]]], Pattern[x, Blank[]]]], Log[Plus[Optional[Pattern[f, Blank[]]], Times[Optional[Pattern[g, Blank[]]], Power[Pattern[x, Blank[]], 2]]]], Power[Pattern[x, Blank[]], -1]], Pattern[x, Blank[Symbol]]] := Dist[Log[f + g*x^2] - Log[-(c^2*x^2)] - Log[1 - 1/(c*x)] - Log[1 + 1/(c*x)], Int[ArcCoth[c*x]/x, x], x] + (-Dist[1/2, Int[Log[1 - 1/(c*x)]^2/x, x], x] + Dist[1/2, Int[Log[1 + 1/(c*x)]^2/x, x], x] + Int[(Log[-(c^2*x^2)]*ArcCoth[c*x])/x, x]) /; FreeQ[{c, f, g}, x] && EqQ[c^2*f + g, 0]
  6077. Int[Times[Log[Plus[Optional[Pattern[f, Blank[]]], Times[Optional[Pattern[g, Blank[]]], Power[Pattern[x, Blank[]], 2]]]], Plus[Times[ArcTanh[Times[Optional[Pattern[c, Blank[]]], Pattern[x, Blank[]]]], Optional[Pattern[b, Blank[]]]], Pattern[a, Blank[]]], Power[Pattern[x, Blank[]], -1]], Pattern[x, Blank[Symbol]]] := Dist[a, Int[Log[f + g*x^2]/x, x], x] + Dist[b, Int[(Log[f + g*x^2]*ArcTanh[c*x])/x, x], x] /; FreeQ[{a, b, c, f, g}, x]
  6078. Int[Times[Log[Plus[Optional[Pattern[f, Blank[]]], Times[Optional[Pattern[g, Blank[]]], Power[Pattern[x, Blank[]], 2]]]], Plus[Times[ArcCoth[Times[Optional[Pattern[c, Blank[]]], Pattern[x, Blank[]]]], Optional[Pattern[b, Blank[]]]], Pattern[a, Blank[]]], Power[Pattern[x, Blank[]], -1]], Pattern[x, Blank[Symbol]]] := Dist[a, Int[Log[f + g*x^2]/x, x], x] + Dist[b, Int[(Log[f + g*x^2]*ArcCoth[c*x])/x, x], x] /; FreeQ[{a, b, c, f, g}, x]
  6079. Int[Times[Plus[Optional[Pattern[a, Blank[]]], Times[ArcTanh[Times[Optional[Pattern[c, Blank[]]], Pattern[x, Blank[]]]], Optional[Pattern[b, Blank[]]]]], Plus[Times[Log[Plus[Optional[Pattern[f, Blank[]]], Times[Optional[Pattern[g, Blank[]]], Power[Pattern[x, Blank[]], 2]]]], Optional[Pattern[e, Blank[]]]], Pattern[d, Blank[]]], Power[Pattern[x, Blank[]], -1]], Pattern[x, Blank[Symbol]]] := Dist[d, Int[(a + b*ArcTanh[c*x])/x, x], x] + Dist[e, Int[(Log[f + g*x^2]*(a + b*ArcTanh[c*x]))/x, x], x] /; FreeQ[{a, b, c, d, e, f, g}, x]
  6080. Int[Times[Plus[Optional[Pattern[a, Blank[]]], Times[ArcCoth[Times[Optional[Pattern[c, Blank[]]], Pattern[x, Blank[]]]], Optional[Pattern[b, Blank[]]]]], Plus[Times[Log[Plus[Optional[Pattern[f, Blank[]]], Times[Optional[Pattern[g, Blank[]]], Power[Pattern[x, Blank[]], 2]]]], Optional[Pattern[e, Blank[]]]], Pattern[d, Blank[]]], Power[Pattern[x, Blank[]], -1]], Pattern[x, Blank[Symbol]]] := Dist[d, Int[(a + b*ArcCoth[c*x])/x, x], x] + Dist[e, Int[(Log[f + g*x^2]*(a + b*ArcCoth[c*x]))/x, x], x] /; FreeQ[{a, b, c, d, e, f, g}, x]
  6081. Int[Times[Plus[Optional[Pattern[a, Blank[]]], Times[ArcTanh[Times[Optional[Pattern[c, Blank[]]], Pattern[x, Blank[]]]], Optional[Pattern[b, Blank[]]]]], Plus[Optional[Pattern[d, Blank[]]], Times[Log[Plus[Optional[Pattern[f, Blank[]]], Times[Optional[Pattern[g, Blank[]]], Power[Pattern[x, Blank[]], 2]]]], Optional[Pattern[e, Blank[]]]]], Power[Pattern[x, Blank[]], Optional[Pattern[m, Blank[]]]]], Pattern[x, Blank[Symbol]]] := Simp[(x^(m + 1)*(d + e*Log[f + g*x^2])*(a + b*ArcTanh[c*x]))/(m + 1), x] + (-Dist[(b*c)/(m + 1), Int[(x^(m + 1)*(d + e*Log[f + g*x^2]))/(1 - c^2*x^2), x], x] - Dist[(2*e*g)/(m + 1), Int[(x^(m + 2)*(a + b*ArcTanh[c*x]))/(f + g*x^2), x], x]) /; FreeQ[{a, b, c, d, e, f, g}, x] && ILtQ[m/2, 0]
  6082. Int[Times[Plus[Optional[Pattern[a, Blank[]]], Times[ArcCoth[Times[Optional[Pattern[c, Blank[]]], Pattern[x, Blank[]]]], Optional[Pattern[b, Blank[]]]]], Plus[Optional[Pattern[d, Blank[]]], Times[Log[Plus[Optional[Pattern[f, Blank[]]], Times[Optional[Pattern[g, Blank[]]], Power[Pattern[x, Blank[]], 2]]]], Optional[Pattern[e, Blank[]]]]], Power[Pattern[x, Blank[]], Optional[Pattern[m, Blank[]]]]], Pattern[x, Blank[Symbol]]] := Simp[(x^(m + 1)*(d + e*Log[f + g*x^2])*(a + b*ArcCoth[c*x]))/(m + 1), x] + (-Dist[(b*c)/(m + 1), Int[(x^(m + 1)*(d + e*Log[f + g*x^2]))/(1 - c^2*x^2), x], x] - Dist[(2*e*g)/(m + 1), Int[(x^(m + 2)*(a + b*ArcCoth[c*x]))/(f + g*x^2), x], x]) /; FreeQ[{a, b, c, d, e, f, g}, x] && ILtQ[m/2, 0]
  6083. Int[Times[Plus[Optional[Pattern[a, Blank[]]], Times[ArcTanh[Times[Optional[Pattern[c, Blank[]]], Pattern[x, Blank[]]]], Optional[Pattern[b, Blank[]]]]], Plus[Optional[Pattern[d, Blank[]]], Times[Log[Plus[Optional[Pattern[f, Blank[]]], Times[Optional[Pattern[g, Blank[]]], Power[Pattern[x, Blank[]], 2]]]], Optional[Pattern[e, Blank[]]]]], Power[Pattern[x, Blank[]], Optional[Pattern[m, Blank[]]]]], Pattern[x, Blank[Symbol]]] := With[{u = IntHide[x^m*(d + e*Log[f + g*x^2]), x]}, Dist[a + b*ArcTanh[c*x], u, x] - Dist[b*c, Int[ExpandIntegrand[u/(1 - c^2*x^2), x], x], x]] /; FreeQ[{a, b, c, d, e, f, g}, x] && IGtQ[(m + 1)/2, 0]
  6084. Int[Times[Plus[Optional[Pattern[a, Blank[]]], Times[ArcCoth[Times[Optional[Pattern[c, Blank[]]], Pattern[x, Blank[]]]], Optional[Pattern[b, Blank[]]]]], Plus[Optional[Pattern[d, Blank[]]], Times[Log[Plus[Optional[Pattern[f, Blank[]]], Times[Optional[Pattern[g, Blank[]]], Power[Pattern[x, Blank[]], 2]]]], Optional[Pattern[e, Blank[]]]]], Power[Pattern[x, Blank[]], Optional[Pattern[m, Blank[]]]]], Pattern[x, Blank[Symbol]]] := With[{u = IntHide[x^m*(d + e*Log[f + g*x^2]), x]}, Dist[a + b*ArcCoth[c*x], u, x] - Dist[b*c, Int[ExpandIntegrand[u/(1 - c^2*x^2), x], x], x]] /; FreeQ[{a, b, c, d, e, f, g}, x] && IGtQ[(m + 1)/2, 0]
  6085. Int[Times[Plus[Optional[Pattern[a, Blank[]]], Times[ArcTanh[Times[Optional[Pattern[c, Blank[]]], Pattern[x, Blank[]]]], Optional[Pattern[b, Blank[]]]]], Plus[Optional[Pattern[d, Blank[]]], Times[Log[Plus[Optional[Pattern[f, Blank[]]], Times[Optional[Pattern[g, Blank[]]], Power[Pattern[x, Blank[]], 2]]]], Optional[Pattern[e, Blank[]]]]], Power[Pattern[x, Blank[]], Optional[Pattern[m, Blank[]]]]], Pattern[x, Blank[Symbol]]] := With[{u = IntHide[x^m*(a + b*ArcTanh[c*x]), x]}, Dist[d + e*Log[f + g*x^2], u, x] - Dist[2*e*g, Int[ExpandIntegrand[(x*u)/(f + g*x^2), x], x], x]] /; FreeQ[{a, b, c, d, e, f, g}, x] && IntegerQ[m] && NeQ[m, -1]
  6086. Int[Times[Plus[Optional[Pattern[a, Blank[]]], Times[ArcCoth[Times[Optional[Pattern[c, Blank[]]], Pattern[x, Blank[]]]], Optional[Pattern[b, Blank[]]]]], Plus[Optional[Pattern[d, Blank[]]], Times[Log[Plus[Optional[Pattern[f, Blank[]]], Times[Optional[Pattern[g, Blank[]]], Power[Pattern[x, Blank[]], 2]]]], Optional[Pattern[e, Blank[]]]]], Power[Pattern[x, Blank[]], Optional[Pattern[m, Blank[]]]]], Pattern[x, Blank[Symbol]]] := With[{u = IntHide[x^m*(a + b*ArcCoth[c*x]), x]}, Dist[d + e*Log[f + g*x^2], u, x] - Dist[2*e*g, Int[ExpandIntegrand[(x*u)/(f + g*x^2), x], x], x]] /; FreeQ[{a, b, c, d, e, f, g}, x] && IntegerQ[m] && NeQ[m, -1]
  6087. Int[Times[Power[Plus[Optional[Pattern[a, Blank[]]], Times[ArcTanh[Times[Optional[Pattern[c, Blank[]]], Pattern[x, Blank[]]]], Optional[Pattern[b, Blank[]]]]], 2], Plus[Optional[Pattern[d, Blank[]]], Times[Log[Plus[Pattern[f, Blank[]], Times[Optional[Pattern[g, Blank[]]], Power[Pattern[x, Blank[]], 2]]]], Optional[Pattern[e, Blank[]]]]], Pattern[x, Blank[]]], Pattern[x, Blank[Symbol]]] := Simp[((f + g*x^2)*(d + e*Log[f + g*x^2])*(a + b*ArcTanh[c*x])^2)/(2*g), x] + (Dist[b/c, Int[(d + e*Log[f + g*x^2])*(a + b*ArcTanh[c*x]), x], x] + Dist[b*c*e, Int[(x^2*(a + b*ArcTanh[c*x]))/(1 - c^2*x^2), x], x] - Simp[(e*x^2*(a + b*ArcTanh[c*x])^2)/2, x]) /; FreeQ[{a, b, c, d, e, f, g}, x] && EqQ[c^2*f + g, 0]
  6088. Int[Times[Power[Plus[Optional[Pattern[a, Blank[]]], Times[ArcCoth[Times[Optional[Pattern[c, Blank[]]], Pattern[x, Blank[]]]], Optional[Pattern[b, Blank[]]]]], 2], Plus[Optional[Pattern[d, Blank[]]], Times[Log[Plus[Pattern[f, Blank[]], Times[Optional[Pattern[g, Blank[]]], Power[Pattern[x, Blank[]], 2]]]], Optional[Pattern[e, Blank[]]]]], Pattern[x, Blank[]]], Pattern[x, Blank[Symbol]]] := Simp[((f + g*x^2)*(d + e*Log[f + g*x^2])*(a + b*ArcCoth[c*x])^2)/(2*g), x] + (Dist[b/c, Int[(d + e*Log[f + g*x^2])*(a + b*ArcCoth[c*x]), x], x] + Dist[b*c*e, Int[(x^2*(a + b*ArcCoth[c*x]))/(1 - c^2*x^2), x], x] - Simp[(e*x^2*(a + b*ArcCoth[c*x])^2)/2, x]) /; FreeQ[{a, b, c, d, e, f, g}, x] && EqQ[c^2*f + g, 0]
  6089. Int[Times[Power[Plus[Optional[Pattern[a, Blank[]]], Times[ArcTanh[Times[Optional[Pattern[c, Blank[]]], Pattern[x, Blank[]]]], Optional[Pattern[b, Blank[]]]]], Optional[Pattern[p, Blank[]]]], Optional[Pattern[u, Blank[]]]], Pattern[x, Blank[Symbol]]] := Unintegrable[u*(a + b*ArcTanh[c*x])^p, x] /; FreeQ[{a, b, c, p}, x] && (EqQ[u, 1] || MatchQ[u, ((d_.) + (e_.)*x)^(q_.) /; FreeQ[{d, e, q}, x]] || MatchQ[u, ((f_.)*x)^(m_.)*((d_.) + (e_.)*x)^(q_.) /; FreeQ[{d, e, f, m, q}, x]] || MatchQ[u, ((d_.) + (e_.)*x^2)^(q_.) /; FreeQ[{d, e, q}, x]] || MatchQ[u, ((f_.)*x)^(m_.)*((d_.) + (e_.)*x^2)^(q_.) /; FreeQ[{d, e, f, m, q}, x]])
  6090. Int[Times[Power[Plus[Optional[Pattern[a, Blank[]]], Times[ArcCoth[Times[Optional[Pattern[c, Blank[]]], Pattern[x, Blank[]]]], Optional[Pattern[b, Blank[]]]]], Optional[Pattern[p, Blank[]]]], Optional[Pattern[u, Blank[]]]], Pattern[x, Blank[Symbol]]] := Unintegrable[u*(a + b*ArcCoth[c*x])^p, x] /; FreeQ[{a, b, c, p}, x] && (EqQ[u, 1] || MatchQ[u, ((d_.) + (e_.)*x)^(q_.) /; FreeQ[{d, e, q}, x]] || MatchQ[u, ((f_.)*x)^(m_.)*((d_.) + (e_.)*x)^(q_.) /; FreeQ[{d, e, f, m, q}, x]] || MatchQ[u, ((d_.) + (e_.)*x^2)^(q_.) /; FreeQ[{d, e, q}, x]] || MatchQ[u, ((f_.)*x)^(m_.)*((d_.) + (e_.)*x^2)^(q_.) /; FreeQ[{d, e, f, m, q}, x]])
  6091. Int[ArcTanh[Times[Optional[Pattern[c, Blank[]]], Power[Pattern[x, Blank[]], Pattern[n, Blank[]]]]], Pattern[x, Blank[Symbol]]] := Simp[x*ArcTanh[c*x^n], x] - Dist[c*n, Int[x^n/(1 - c^2*x^(2*n)), x], x] /; FreeQ[{c, n}, x]
  6092. Int[ArcCoth[Times[Optional[Pattern[c, Blank[]]], Power[Pattern[x, Blank[]], Pattern[n, Blank[]]]]], Pattern[x, Blank[Symbol]]] := Simp[x*ArcCoth[c*x^n], x] - Dist[c*n, Int[x^n/(1 - c^2*x^(2*n)), x], x] /; FreeQ[{c, n}, x]
  6093. Int[Power[Plus[Optional[Pattern[a, Blank[]]], Times[ArcTanh[Times[Optional[Pattern[c, Blank[]]], Power[Pattern[x, Blank[]], Pattern[n, Blank[]]]]], Optional[Pattern[b, Blank[]]]]], Optional[Pattern[p, Blank[]]]], Pattern[x, Blank[Symbol]]] := Int[ExpandIntegrand[(a + (b*Log[1 + c*x^n])/2 - (b*Log[1 - c*x^n])/2)^p, x], x] /; FreeQ[{a, b, c, n}, x] && IGtQ[p, 0] && IntegerQ[n]
  6094. Int[Power[Plus[Optional[Pattern[a, Blank[]]], Times[ArcCoth[Times[Optional[Pattern[c, Blank[]]], Power[Pattern[x, Blank[]], Pattern[n, Blank[]]]]], Optional[Pattern[b, Blank[]]]]], Optional[Pattern[p, Blank[]]]], Pattern[x, Blank[Symbol]]] := Int[ExpandIntegrand[(a + (b*Log[1 + 1/(x^n*c)])/2 - (b*Log[1 - 1/(x^n*c)])/2)^p, x], x] /; FreeQ[{a, b, c, n}, x] && IGtQ[p, 0] && IntegerQ[n]
  6095. Int[Times[Power[Plus[Optional[Pattern[a, Blank[]]], Times[ArcTanh[Times[Optional[Pattern[c, Blank[]]], Power[Pattern[x, Blank[]], Pattern[n, Blank[]]]]], Optional[Pattern[b, Blank[]]]]], Optional[Pattern[p, Blank[]]]], Power[Pattern[x, Blank[]], -1]], Pattern[x, Blank[Symbol]]] := Dist[1/n, Subst[Int[(a + b*ArcTanh[c*x])^p/x, x], x, x^n], x] /; FreeQ[{a, b, c, n}, x] && IGtQ[p, 0]
  6096. Int[Times[Power[Plus[Optional[Pattern[a, Blank[]]], Times[ArcCoth[Times[Optional[Pattern[c, Blank[]]], Power[Pattern[x, Blank[]], Pattern[n, Blank[]]]]], Optional[Pattern[b, Blank[]]]]], Optional[Pattern[p, Blank[]]]], Power[Pattern[x, Blank[]], -1]], Pattern[x, Blank[Symbol]]] := Dist[1/n, Subst[Int[(a + b*ArcCoth[c*x])^p/x, x], x, x^n], x] /; FreeQ[{a, b, c, n}, x] && IGtQ[p, 0]
  6097. Int[Times[Plus[Optional[Pattern[a, Blank[]]], Times[ArcTanh[Times[Optional[Pattern[c, Blank[]]], Power[Pattern[x, Blank[]], Pattern[n, Blank[]]]]], Optional[Pattern[b, Blank[]]]]], Power[Times[Optional[Pattern[d, Blank[]]], Pattern[x, Blank[]]], Optional[Pattern[m, Blank[]]]]], Pattern[x, Blank[Symbol]]] := Simp[((d*x)^(m + 1)*(a + b*ArcTanh[c*x^n]))/(d*(m + 1)), x] - Dist[(b*c*n)/(d*(m + 1)), Int[(x^(n - 1)*(d*x)^(m + 1))/(1 - c^2*x^(2*n)), x], x] /; FreeQ[{a, b, c, d, m, n}, x] && NeQ[m, -1]
  6098. Int[Times[Plus[Optional[Pattern[a, Blank[]]], Times[ArcCoth[Times[Optional[Pattern[c, Blank[]]], Power[Pattern[x, Blank[]], Pattern[n, Blank[]]]]], Optional[Pattern[b, Blank[]]]]], Power[Times[Optional[Pattern[d, Blank[]]], Pattern[x, Blank[]]], Optional[Pattern[m, Blank[]]]]], Pattern[x, Blank[Symbol]]] := Simp[((d*x)^(m + 1)*(a + b*ArcCoth[c*x^n]))/(d*(m + 1)), x] - Dist[(b*c*n)/(d*(m + 1)), Int[(x^(n - 1)*(d*x)^(m + 1))/(1 - c^2*x^(2*n)), x], x] /; FreeQ[{a, b, c, d, m, n}, x] && NeQ[m, -1]
  6099. Int[Times[Power[Plus[Optional[Pattern[a, Blank[]]], Times[ArcTanh[Times[Optional[Pattern[c, Blank[]]], Power[Pattern[x, Blank[]], Pattern[n, Blank[]]]]], Optional[Pattern[b, Blank[]]]]], Optional[Pattern[p, Blank[]]]], Power[Times[Optional[Pattern[d, Blank[]]], Pattern[x, Blank[]]], Optional[Pattern[m, Blank[]]]]], Pattern[x, Blank[Symbol]]] := Int[ExpandIntegrand[(d*x)^m*(a + (b*Log[1 + c*x^n])/2 - (b*Log[1 - c*x^n])/2)^p, x], x] /; FreeQ[{a, b, c, d, m, n}, x] && IGtQ[p, 0] && IntegerQ[m] && IntegerQ[n]
  6100. Int[Times[Power[Plus[Optional[Pattern[a, Blank[]]], Times[ArcCoth[Times[Optional[Pattern[c, Blank[]]], Power[Pattern[x, Blank[]], Pattern[n, Blank[]]]]], Optional[Pattern[b, Blank[]]]]], Optional[Pattern[p, Blank[]]]], Power[Times[Optional[Pattern[d, Blank[]]], Pattern[x, Blank[]]], Optional[Pattern[m, Blank[]]]]], Pattern[x, Blank[Symbol]]] := Int[ExpandIntegrand[(d*x)^m*(a + (b*Log[1 + 1/(x^n*c)])/2 - (b*Log[1 - 1/(x^n*c)])/2)^p, x], x] /; FreeQ[{a, b, c, d, m, n}, x] && IGtQ[p, 0] && IntegerQ[m] && IntegerQ[n]
  6101. Int[Times[Power[Plus[Optional[Pattern[a, Blank[]]], Times[ArcTanh[Times[Optional[Pattern[c, Blank[]]], Power[Pattern[x, Blank[]], Pattern[n, Blank[]]]]], Optional[Pattern[b, Blank[]]]]], Optional[Pattern[p, Blank[]]]], Optional[Pattern[u, Blank[]]]], Pattern[x, Blank[Symbol]]] := Unintegrable[u*(a + b*ArcTanh[c*x^n])^p, x] /; FreeQ[{a, b, c, n, p}, x] && (EqQ[u, 1] || MatchQ[u, ((d_.)*x)^(m_.) /; FreeQ[{d, m}, x]])
  6102. Int[Times[Power[Plus[Optional[Pattern[a, Blank[]]], Times[ArcCoth[Times[Optional[Pattern[c, Blank[]]], Power[Pattern[x, Blank[]], Pattern[n, Blank[]]]]], Optional[Pattern[b, Blank[]]]]], Optional[Pattern[p, Blank[]]]], Optional[Pattern[u, Blank[]]]], Pattern[x, Blank[Symbol]]] := Unintegrable[u*(a + b*ArcCoth[c*x^n])^p, x] /; FreeQ[{a, b, c, n, p}, x] && (EqQ[u, 1] || MatchQ[u, ((d_.)*x)^(m_.) /; FreeQ[{d, m}, x]])
  6103. Int[Power[Plus[Optional[Pattern[a, Blank[]]], Times[ArcTanh[Plus[Pattern[c, Blank[]], Times[Optional[Pattern[d, Blank[]]], Pattern[x, Blank[]]]]], Optional[Pattern[b, Blank[]]]]], Optional[Pattern[p, Blank[]]]], Pattern[x, Blank[Symbol]]] := Dist[1/d, Subst[Int[(a + b*ArcTanh[x])^p, x], x, c + d*x], x] /; FreeQ[{a, b, c, d}, x] && IGtQ[p, 0]
  6104. Int[Power[Plus[Optional[Pattern[a, Blank[]]], Times[ArcCoth[Plus[Pattern[c, Blank[]], Times[Optional[Pattern[d, Blank[]]], Pattern[x, Blank[]]]]], Optional[Pattern[b, Blank[]]]]], Optional[Pattern[p, Blank[]]]], Pattern[x, Blank[Symbol]]] := Dist[1/d, Subst[Int[(a + b*ArcCoth[x])^p, x], x, c + d*x], x] /; FreeQ[{a, b, c, d}, x] && IGtQ[p, 0]
  6105. Int[Power[Plus[Optional[Pattern[a, Blank[]]], Times[ArcTanh[Plus[Pattern[c, Blank[]], Times[Optional[Pattern[d, Blank[]]], Pattern[x, Blank[]]]]], Optional[Pattern[b, Blank[]]]]], Pattern[p, Blank[]]], Pattern[x, Blank[Symbol]]] := Unintegrable[(a + b*ArcTanh[c + d*x])^p, x] /; FreeQ[{a, b, c, d, p}, x] && !IGtQ[p, 0]
  6106. Int[Power[Plus[Optional[Pattern[a, Blank[]]], Times[ArcCoth[Plus[Pattern[c, Blank[]], Times[Optional[Pattern[d, Blank[]]], Pattern[x, Blank[]]]]], Optional[Pattern[b, Blank[]]]]], Pattern[p, Blank[]]], Pattern[x, Blank[Symbol]]] := Unintegrable[(a + b*ArcCoth[c + d*x])^p, x] /; FreeQ[{a, b, c, d, p}, x] && !IGtQ[p, 0]
  6107. Int[Times[Power[Plus[Optional[Pattern[a, Blank[]]], Times[ArcTanh[Plus[Pattern[c, Blank[]], Times[Optional[Pattern[d, Blank[]]], Pattern[x, Blank[]]]]], Optional[Pattern[b, Blank[]]]]], Optional[Pattern[p, Blank[]]]], Power[Plus[Optional[Pattern[e, Blank[]]], Times[Optional[Pattern[f, Blank[]]], Pattern[x, Blank[]]]], Optional[Pattern[m, Blank[]]]]], Pattern[x, Blank[Symbol]]] := Dist[1/d, Subst[Int[((f*x)/d)^m*(a + b*ArcTanh[x])^p, x], x, c + d*x], x] /; FreeQ[{a, b, c, d, e, f, m}, x] && EqQ[d*e - c*f, 0] && IGtQ[p, 0]
  6108. Int[Times[Power[Plus[Optional[Pattern[a, Blank[]]], Times[ArcCoth[Plus[Pattern[c, Blank[]], Times[Optional[Pattern[d, Blank[]]], Pattern[x, Blank[]]]]], Optional[Pattern[b, Blank[]]]]], Optional[Pattern[p, Blank[]]]], Power[Plus[Optional[Pattern[e, Blank[]]], Times[Optional[Pattern[f, Blank[]]], Pattern[x, Blank[]]]], Optional[Pattern[m, Blank[]]]]], Pattern[x, Blank[Symbol]]] := Dist[1/d, Subst[Int[((f*x)/d)^m*(a + b*ArcCoth[x])^p, x], x, c + d*x], x] /; FreeQ[{a, b, c, d, e, f, m}, x] && EqQ[d*e - c*f, 0] && IGtQ[p, 0]
  6109. Int[Times[Power[Plus[Optional[Pattern[a, Blank[]]], Times[ArcTanh[Plus[Pattern[c, Blank[]], Times[Optional[Pattern[d, Blank[]]], Pattern[x, Blank[]]]]], Optional[Pattern[b, Blank[]]]]], Optional[Pattern[p, Blank[]]]], Power[Plus[Optional[Pattern[e, Blank[]]], Times[Optional[Pattern[f, Blank[]]], Pattern[x, Blank[]]]], Pattern[m, Blank[]]]], Pattern[x, Blank[Symbol]]] := Simp[((e + f*x)^(m + 1)*(a + b*ArcTanh[c + d*x])^p)/(f*(m + 1)), x] - Dist[(b*d*p)/(f*(m + 1)), Int[((e + f*x)^(m + 1)*(a + b*ArcTanh[c + d*x])^(p - 1))/(1 - (c + d*x)^2), x], x] /; FreeQ[{a, b, c, d, e, f}, x] && IGtQ[p, 0] && ILtQ[m, -1]
  6110. Int[Times[Power[Plus[Optional[Pattern[a, Blank[]]], Times[ArcCoth[Plus[Pattern[c, Blank[]], Times[Optional[Pattern[d, Blank[]]], Pattern[x, Blank[]]]]], Optional[Pattern[b, Blank[]]]]], Optional[Pattern[p, Blank[]]]], Power[Plus[Optional[Pattern[e, Blank[]]], Times[Optional[Pattern[f, Blank[]]], Pattern[x, Blank[]]]], Pattern[m, Blank[]]]], Pattern[x, Blank[Symbol]]] := Simp[((e + f*x)^(m + 1)*(a + b*ArcCoth[c + d*x])^p)/(f*(m + 1)), x] - Dist[(b*d*p)/(f*(m + 1)), Int[((e + f*x)^(m + 1)*(a + b*ArcCoth[c + d*x])^(p - 1))/(1 - (c + d*x)^2), x], x] /; FreeQ[{a, b, c, d, e, f}, x] && IGtQ[p, 0] && ILtQ[m, -1]
  6111. Int[Times[Power[Plus[Optional[Pattern[a, Blank[]]], Times[ArcTanh[Plus[Pattern[c, Blank[]], Times[Optional[Pattern[d, Blank[]]], Pattern[x, Blank[]]]]], Optional[Pattern[b, Blank[]]]]], Optional[Pattern[p, Blank[]]]], Power[Plus[Optional[Pattern[e, Blank[]]], Times[Optional[Pattern[f, Blank[]]], Pattern[x, Blank[]]]], Optional[Pattern[m, Blank[]]]]], Pattern[x, Blank[Symbol]]] := Dist[1/d, Subst[Int[((d*e - c*f)/d + (f*x)/d)^m*(a + b*ArcTanh[x])^p, x], x, c + d*x], x] /; FreeQ[{a, b, c, d, e, f, m}, x] && IGtQ[p, 0]
  6112. Int[Times[Power[Plus[Optional[Pattern[a, Blank[]]], Times[ArcCoth[Plus[Pattern[c, Blank[]], Times[Optional[Pattern[d, Blank[]]], Pattern[x, Blank[]]]]], Optional[Pattern[b, Blank[]]]]], Optional[Pattern[p, Blank[]]]], Power[Plus[Optional[Pattern[e, Blank[]]], Times[Optional[Pattern[f, Blank[]]], Pattern[x, Blank[]]]], Optional[Pattern[m, Blank[]]]]], Pattern[x, Blank[Symbol]]] := Dist[1/d, Subst[Int[((d*e - c*f)/d + (f*x)/d)^m*(a + b*ArcCoth[x])^p, x], x, c + d*x], x] /; FreeQ[{a, b, c, d, e, f, m}, x] && IGtQ[p, 0]
  6113. Int[Times[Power[Plus[Optional[Pattern[a, Blank[]]], Times[ArcTanh[Plus[Pattern[c, Blank[]], Times[Optional[Pattern[d, Blank[]]], Pattern[x, Blank[]]]]], Optional[Pattern[b, Blank[]]]]], Pattern[p, Blank[]]], Power[Plus[Optional[Pattern[e, Blank[]]], Times[Optional[Pattern[f, Blank[]]], Pattern[x, Blank[]]]], Optional[Pattern[m, Blank[]]]]], Pattern[x, Blank[Symbol]]] := Unintegrable[(e + f*x)^m*(a + b*ArcTanh[c + d*x])^p, x] /; FreeQ[{a, b, c, d, e, f, m, p}, x] && !IGtQ[p, 0]
  6114. Int[Times[Power[Plus[Optional[Pattern[a, Blank[]]], Times[ArcCoth[Plus[Pattern[c, Blank[]], Times[Optional[Pattern[d, Blank[]]], Pattern[x, Blank[]]]]], Optional[Pattern[b, Blank[]]]]], Pattern[p, Blank[]]], Power[Plus[Optional[Pattern[e, Blank[]]], Times[Optional[Pattern[f, Blank[]]], Pattern[x, Blank[]]]], Optional[Pattern[m, Blank[]]]]], Pattern[x, Blank[Symbol]]] := Unintegrable[(e + f*x)^m*(a + b*ArcCoth[c + d*x])^p, x] /; FreeQ[{a, b, c, d, e, f, m, p}, x] && !IGtQ[p, 0]
  6115. Int[Times[ArcTanh[Plus[Pattern[c, Blank[]], Times[Optional[Pattern[d, Blank[]]], Pattern[x, Blank[]]]]], Power[Plus[Pattern[e, Blank[]], Times[Optional[Pattern[f, Blank[]]], Power[Pattern[x, Blank[]], Optional[Pattern[n, Blank[]]]]]], -1]], Pattern[x, Blank[Symbol]]] := Dist[1/2, Int[Log[1 + c + d*x]/(e + f*x^n), x], x] - Dist[1/2, Int[Log[1 - c - d*x]/(e + f*x^n), x], x] /; FreeQ[{c, d, e, f}, x] && RationalQ[n]
  6116. Int[Times[ArcCoth[Plus[Pattern[c, Blank[]], Times[Optional[Pattern[d, Blank[]]], Pattern[x, Blank[]]]]], Power[Plus[Pattern[e, Blank[]], Times[Optional[Pattern[f, Blank[]]], Power[Pattern[x, Blank[]], Optional[Pattern[n, Blank[]]]]]], -1]], Pattern[x, Blank[Symbol]]] := Dist[1/2, Int[Log[(1 + c + d*x)/(c + d*x)]/(e + f*x^n), x], x] - Dist[1/2, Int[Log[(-1 + c + d*x)/(c + d*x)]/(e + f*x^n), x], x] /; FreeQ[{c, d, e, f}, x] && RationalQ[n]
  6117. Int[Times[ArcTanh[Plus[Pattern[c, Blank[]], Times[Optional[Pattern[d, Blank[]]], Pattern[x, Blank[]]]]], Power[Plus[Pattern[e, Blank[]], Times[Optional[Pattern[f, Blank[]]], Power[Pattern[x, Blank[]], Pattern[n, Blank[]]]]], -1]], Pattern[x, Blank[Symbol]]] := Unintegrable[ArcTanh[c + d*x]/(e + f*x^n), x] /; FreeQ[{c, d, e, f, n}, x] && !RationalQ[n]
  6118. Int[Times[ArcCoth[Plus[Pattern[c, Blank[]], Times[Optional[Pattern[d, Blank[]]], Pattern[x, Blank[]]]]], Power[Plus[Pattern[e, Blank[]], Times[Optional[Pattern[f, Blank[]]], Power[Pattern[x, Blank[]], Pattern[n, Blank[]]]]], -1]], Pattern[x, Blank[Symbol]]] := Unintegrable[ArcCoth[c + d*x]/(e + f*x^n), x] /; FreeQ[{c, d, e, f, n}, x] && !RationalQ[n]
  6119. Int[Times[Power[Plus[Optional[Pattern[a, Blank[]]], Times[ArcTanh[Plus[Pattern[c, Blank[]], Times[Optional[Pattern[d, Blank[]]], Pattern[x, Blank[]]]]], Optional[Pattern[b, Blank[]]]]], Optional[Pattern[p, Blank[]]]], Power[Plus[Optional[Pattern[A, Blank[]]], Times[Optional[Pattern[B, Blank[]]], Pattern[x, Blank[]]], Times[Optional[Pattern[C, Blank[]]], Power[Pattern[x, Blank[]], 2]]], Optional[Pattern[q, Blank[]]]]], Pattern[x, Blank[Symbol]]] := Dist[1/d, Subst[Int[(-(C/d^2) + (C*x^2)/d^2)^q*(a + b*ArcTanh[x])^p, x], x, c + d*x], x] /; FreeQ[{a, b, c, d, A, B, C, p, q}, x] && EqQ[B*(1 - c^2) + 2*A*c*d, 0] && EqQ[2*c*C - B*d, 0]
  6120. Int[Times[Power[Plus[Optional[Pattern[a, Blank[]]], Times[ArcCoth[Plus[Pattern[c, Blank[]], Times[Optional[Pattern[d, Blank[]]], Pattern[x, Blank[]]]]], Optional[Pattern[b, Blank[]]]]], Optional[Pattern[p, Blank[]]]], Power[Plus[Optional[Pattern[A, Blank[]]], Times[Optional[Pattern[B, Blank[]]], Pattern[x, Blank[]]], Times[Optional[Pattern[C, Blank[]]], Power[Pattern[x, Blank[]], 2]]], Optional[Pattern[q, Blank[]]]]], Pattern[x, Blank[Symbol]]] := Dist[1/d, Subst[Int[(C/d^2 + (C*x^2)/d^2)^q*(a + b*ArcCoth[x])^p, x], x, c + d*x], x] /; FreeQ[{a, b, c, d, A, B, C, p, q}, x] && EqQ[B*(1 - c^2) + 2*A*c*d, 0] && EqQ[2*c*C - B*d, 0]
  6121. Int[Times[Power[Plus[Optional[Pattern[a, Blank[]]], Times[ArcTanh[Plus[Pattern[c, Blank[]], Times[Optional[Pattern[d, Blank[]]], Pattern[x, Blank[]]]]], Optional[Pattern[b, Blank[]]]]], Optional[Pattern[p, Blank[]]]], Power[Plus[Optional[Pattern[e, Blank[]]], Times[Optional[Pattern[f, Blank[]]], Pattern[x, Blank[]]]], Optional[Pattern[m, Blank[]]]], Power[Plus[Optional[Pattern[A, Blank[]]], Times[Optional[Pattern[B, Blank[]]], Pattern[x, Blank[]]], Times[Optional[Pattern[C, Blank[]]], Power[Pattern[x, Blank[]], 2]]], Optional[Pattern[q, Blank[]]]]], Pattern[x, Blank[Symbol]]] := Dist[1/d, Subst[Int[((d*e - c*f)/d + (f*x)/d)^m*(-(C/d^2) + (C*x^2)/d^2)^q*(a + b*ArcTanh[x])^p, x], x, c + d*x], x] /; FreeQ[{a, b, c, d, e, f, A, B, C, m, p, q}, x] && EqQ[B*(1 - c^2) + 2*A*c*d, 0] && EqQ[2*c*C - B*d, 0]
  6122. Int[Times[Power[Plus[Optional[Pattern[a, Blank[]]], Times[ArcCoth[Plus[Pattern[c, Blank[]], Times[Optional[Pattern[d, Blank[]]], Pattern[x, Blank[]]]]], Optional[Pattern[b, Blank[]]]]], Optional[Pattern[p, Blank[]]]], Power[Plus[Optional[Pattern[e, Blank[]]], Times[Optional[Pattern[f, Blank[]]], Pattern[x, Blank[]]]], Optional[Pattern[m, Blank[]]]], Power[Plus[Optional[Pattern[A, Blank[]]], Times[Optional[Pattern[B, Blank[]]], Pattern[x, Blank[]]], Times[Optional[Pattern[C, Blank[]]], Power[Pattern[x, Blank[]], 2]]], Optional[Pattern[q, Blank[]]]]], Pattern[x, Blank[Symbol]]] := Dist[1/d, Subst[Int[((d*e - c*f)/d + (f*x)/d)^m*(-(C/d^2) + (C*x^2)/d^2)^q*(a + b*ArcCoth[x])^p, x], x, c + d*x], x] /; FreeQ[{a, b, c, d, e, f, A, B, C, m, p, q}, x] && EqQ[B*(1 - c^2) + 2*A*c*d, 0] && EqQ[2*c*C - B*d, 0]
  6123. Int[Power[E, Times[ArcTanh[Times[Optional[Pattern[a, Blank[]]], Pattern[x, Blank[]]]], Optional[Pattern[n, Blank[]]]]], Pattern[x, Blank[Symbol]]] := Int[(1 + a*x)^((n + 1)/2)/((1 - a*x)^((n - 1)/2)*Sqrt[1 - a^2*x^2]), x] /; FreeQ[a, x] && IntegerQ[(n - 1)/2]
  6124. Int[Times[Power[E, Times[ArcTanh[Times[Optional[Pattern[a, Blank[]]], Pattern[x, Blank[]]]], Optional[Pattern[n, Blank[]]]]], Power[Pattern[x, Blank[]], Optional[Pattern[m, Blank[]]]]], Pattern[x, Blank[Symbol]]] := Int[x^m*((1 + a*x)^((n + 1)/2)/((1 - a*x)^((n - 1)/2)*Sqrt[1 - a^2*x^2])), x] /; FreeQ[{a, m}, x] && IntegerQ[(n - 1)/2]
  6125. Int[Power[E, Times[ArcTanh[Times[Optional[Pattern[a, Blank[]]], Pattern[x, Blank[]]]], Pattern[n, Blank[]]]], Pattern[x, Blank[Symbol]]] := Int[(1 + a*x)^(n/2)/(1 - a*x)^(n/2), x] /; FreeQ[{a, n}, x] && !IntegerQ[(n - 1)/2]
  6126. Int[Times[Power[E, Times[ArcTanh[Times[Optional[Pattern[a, Blank[]]], Pattern[x, Blank[]]]], Pattern[n, Blank[]]]], Power[Pattern[x, Blank[]], Optional[Pattern[m, Blank[]]]]], Pattern[x, Blank[Symbol]]] := Int[(x^m*(1 + a*x)^(n/2))/(1 - a*x)^(n/2), x] /; FreeQ[{a, m, n}, x] && !IntegerQ[(n - 1)/2]
  6127. Int[Times[Power[E, Times[ArcTanh[Times[Optional[Pattern[a, Blank[]]], Pattern[x, Blank[]]]], Optional[Pattern[n, Blank[]]]]], Power[Plus[Pattern[c, Blank[]], Times[Optional[Pattern[d, Blank[]]], Pattern[x, Blank[]]]], Optional[Pattern[p, Blank[]]]]], Pattern[x, Blank[Symbol]]] := Dist[c^n, Int[(c + d*x)^(p - n)*(1 - a^2*x^2)^(n/2), x], x] /; FreeQ[{a, c, d, p}, x] && EqQ[a*c + d, 0] && IntegerQ[(n - 1)/2] && IntegerQ[2*p]
  6128. Int[Times[Power[E, Times[ArcTanh[Times[Optional[Pattern[a, Blank[]]], Pattern[x, Blank[]]]], Optional[Pattern[n, Blank[]]]]], Power[Plus[Pattern[c, Blank[]], Times[Optional[Pattern[d, Blank[]]], Pattern[x, Blank[]]]], Optional[Pattern[p, Blank[]]]], Power[Plus[Optional[Pattern[e, Blank[]]], Times[Optional[Pattern[f, Blank[]]], Pattern[x, Blank[]]]], Optional[Pattern[m, Blank[]]]]], Pattern[x, Blank[Symbol]]] := Dist[c^n, Int[(e + f*x)^m*(c + d*x)^(p - n)*(1 - a^2*x^2)^(n/2), x], x] /; FreeQ[{a, c, d, e, f, m, p}, x] && EqQ[a*c + d, 0] && IntegerQ[(n - 1)/2] && (IntegerQ[p] || EqQ[p, n/2] || EqQ[p - n/2 - 1, 0]) && IntegerQ[2*p]
  6129. Int[Times[Power[E, Times[ArcTanh[Times[Optional[Pattern[a, Blank[]]], Pattern[x, Blank[]]]], Optional[Pattern[n, Blank[]]]]], Optional[Pattern[u, Blank[]]], Power[Plus[Pattern[c, Blank[]], Times[Optional[Pattern[d, Blank[]]], Pattern[x, Blank[]]]], Optional[Pattern[p, Blank[]]]]], Pattern[x, Blank[Symbol]]] := Dist[c^p, Int[(u*(1 + (d*x)/c)^p*(1 + a*x)^(n/2))/(1 - a*x)^(n/2), x], x] /; FreeQ[{a, c, d, n, p}, x] && EqQ[a^2*c^2 - d^2, 0] && (IntegerQ[p] || GtQ[c, 0])
  6130. Int[Times[Power[E, Times[ArcTanh[Times[Optional[Pattern[a, Blank[]]], Pattern[x, Blank[]]]], Optional[Pattern[n, Blank[]]]]], Optional[Pattern[u, Blank[]]], Power[Plus[Pattern[c, Blank[]], Times[Optional[Pattern[d, Blank[]]], Pattern[x, Blank[]]]], Optional[Pattern[p, Blank[]]]]], Pattern[x, Blank[Symbol]]] := Int[(u*(c + d*x)^p*(1 + a*x)^(n/2))/(1 - a*x)^(n/2), x] /; FreeQ[{a, c, d, n, p}, x] && EqQ[a^2*c^2 - d^2, 0] && !(IntegerQ[p] || GtQ[c, 0])
  6131. Int[Times[Power[E, Times[ArcTanh[Times[Optional[Pattern[a, Blank[]]], Pattern[x, Blank[]]]], Optional[Pattern[n, Blank[]]]]], Optional[Pattern[u, Blank[]]], Power[Plus[Pattern[c, Blank[]], Times[Optional[Pattern[d, Blank[]]], Power[Pattern[x, Blank[]], -1]]], Optional[Pattern[p, Blank[]]]]], Pattern[x, Blank[Symbol]]] := Dist[d^p, Int[(u*(1 + (c*x)/d)^p*E^(n*ArcTanh[a*x]))/x^p, x], x] /; FreeQ[{a, c, d, n}, x] && EqQ[c^2 - a^2*d^2, 0] && IntegerQ[p]
  6132. Int[Times[Power[E, Times[ArcTanh[Times[Optional[Pattern[a, Blank[]]], Pattern[x, Blank[]]]], Pattern[n, Blank[]]]], Optional[Pattern[u, Blank[]]], Power[Plus[Pattern[c, Blank[]], Times[Optional[Pattern[d, Blank[]]], Power[Pattern[x, Blank[]], -1]]], Pattern[p, Blank[]]]], Pattern[x, Blank[Symbol]]] := Dist[(-1)^(n/2)*c^p, Int[(u*(1 + d/(c*x))^p*(1 + 1/(a*x))^(n/2))/(1 - 1/(a*x))^(n/2), x], x] /; FreeQ[{a, c, d, p}, x] && EqQ[c^2 - a^2*d^2, 0] && !IntegerQ[p] && IntegerQ[n/2] && GtQ[c, 0]
  6133. Int[Times[Power[E, Times[ArcTanh[Times[Optional[Pattern[a, Blank[]]], Pattern[x, Blank[]]]], Pattern[n, Blank[]]]], Optional[Pattern[u, Blank[]]], Power[Plus[Pattern[c, Blank[]], Times[Optional[Pattern[d, Blank[]]], Power[Pattern[x, Blank[]], -1]]], Pattern[p, Blank[]]]], Pattern[x, Blank[Symbol]]] := Int[(u*(c + d/x)^p*(1 + a*x)^(n/2))/(1 - a*x)^(n/2), x] /; FreeQ[{a, c, d, p}, x] && EqQ[c^2 - a^2*d^2, 0] && !IntegerQ[p] && IntegerQ[n/2] && !GtQ[c, 0]
  6134. Int[Times[Power[E, Times[ArcTanh[Times[Optional[Pattern[a, Blank[]]], Pattern[x, Blank[]]]], Optional[Pattern[n, Blank[]]]]], Optional[Pattern[u, Blank[]]], Power[Plus[Pattern[c, Blank[]], Times[Optional[Pattern[d, Blank[]]], Power[Pattern[x, Blank[]], -1]]], Pattern[p, Blank[]]]], Pattern[x, Blank[Symbol]]] := Dist[(x^p*(c + d/x)^p)/(1 + (c*x)/d)^p, Int[(u*(1 + (c*x)/d)^p*E^(n*ArcTanh[a*x]))/x^p, x], x] /; FreeQ[{a, c, d, n, p}, x] && EqQ[c^2 - a^2*d^2, 0] && !IntegerQ[p]
  6135. Int[Times[Power[E, Times[ArcTanh[Times[Optional[Pattern[a, Blank[]]], Pattern[x, Blank[]]]], Pattern[n, Blank[]]]], Power[Plus[Pattern[c, Blank[]], Times[Optional[Pattern[d, Blank[]]], Power[Pattern[x, Blank[]], 2]]], Rational[-3, 2]]], Pattern[x, Blank[Symbol]]] := Simp[((n - a*x)*E^(n*ArcTanh[a*x]))/(a*c*(n^2 - 1)*Sqrt[c + d*x^2]), x] /; FreeQ[{a, c, d, n}, x] && EqQ[a^2*c + d, 0] && !IntegerQ[n]
  6136. Int[Times[Power[E, Times[ArcTanh[Times[Optional[Pattern[a, Blank[]]], Pattern[x, Blank[]]]], Pattern[n, Blank[]]]], Power[Plus[Pattern[c, Blank[]], Times[Optional[Pattern[d, Blank[]]], Power[Pattern[x, Blank[]], 2]]], Pattern[p, Blank[]]]], Pattern[x, Blank[Symbol]]] := Simp[((n + 2*a*(p + 1)*x)*(c + d*x^2)^(p + 1)*E^(n*ArcTanh[a*x]))/(a*c*(n^2 - 4*(p + 1)^2)), x] - Dist[(2*(p + 1)*(2*p + 3))/(c*(n^2 - 4*(p + 1)^2)), Int[(c + d*x^2)^(p + 1)*E^(n*ArcTanh[a*x]), x], x] /; FreeQ[{a, c, d, n}, x] && EqQ[a^2*c + d, 0] && LtQ[p, -1] && !IntegerQ[n] && NeQ[n^2 - 4*(p + 1)^2, 0] && IntegerQ[2*p]
  6137. Int[Times[Power[E, Times[ArcTanh[Times[Optional[Pattern[a, Blank[]]], Pattern[x, Blank[]]]], Optional[Pattern[n, Blank[]]]]], Power[Plus[Pattern[c, Blank[]], Times[Optional[Pattern[d, Blank[]]], Power[Pattern[x, Blank[]], 2]]], -1]], Pattern[x, Blank[Symbol]]] := Simp[E^(n*ArcTanh[a*x])/(a*c*n), x] /; FreeQ[{a, c, d, n}, x] && EqQ[a^2*c + d, 0] && !IntegerQ[n/2]
  6138. Int[Times[Power[E, Times[ArcTanh[Times[Optional[Pattern[a, Blank[]]], Pattern[x, Blank[]]]], Optional[Pattern[n, Blank[]]]]], Power[Plus[Pattern[c, Blank[]], Times[Optional[Pattern[d, Blank[]]], Power[Pattern[x, Blank[]], 2]]], Optional[Pattern[p, Blank[]]]]], Pattern[x, Blank[Symbol]]] := Dist[c^p, Int[(1 - a^2*x^2)^(p - n/2)*(1 + a*x)^n, x], x] /; FreeQ[{a, c, d, p}, x] && EqQ[a^2*c + d, 0] && IntegerQ[p] && IGtQ[(n + 1)/2, 0] && !IntegerQ[p - n/2]
  6139. Int[Times[Power[E, Times[ArcTanh[Times[Optional[Pattern[a, Blank[]]], Pattern[x, Blank[]]]], Pattern[n, Blank[]]]], Power[Plus[Pattern[c, Blank[]], Times[Optional[Pattern[d, Blank[]]], Power[Pattern[x, Blank[]], 2]]], Optional[Pattern[p, Blank[]]]]], Pattern[x, Blank[Symbol]]] := Dist[c^p, Int[(1 - a^2*x^2)^(p + n/2)/(1 - a*x)^n, x], x] /; FreeQ[{a, c, d, p}, x] && EqQ[a^2*c + d, 0] && IntegerQ[p] && ILtQ[(n - 1)/2, 0] && !IntegerQ[p - n/2]
  6140. Int[Times[Power[E, Times[ArcTanh[Times[Optional[Pattern[a, Blank[]]], Pattern[x, Blank[]]]], Optional[Pattern[n, Blank[]]]]], Power[Plus[Pattern[c, Blank[]], Times[Optional[Pattern[d, Blank[]]], Power[Pattern[x, Blank[]], 2]]], Optional[Pattern[p, Blank[]]]]], Pattern[x, Blank[Symbol]]] := Dist[c^p, Int[(1 - a*x)^(p - n/2)*(1 + a*x)^(p + n/2), x], x] /; FreeQ[{a, c, d, n, p}, x] && EqQ[a^2*c + d, 0] && (IntegerQ[p] || GtQ[c, 0])
  6141. Int[Times[Power[E, Times[ArcTanh[Times[Optional[Pattern[a, Blank[]]], Pattern[x, Blank[]]]], Pattern[n, Blank[]]]], Power[Plus[Pattern[c, Blank[]], Times[Optional[Pattern[d, Blank[]]], Power[Pattern[x, Blank[]], 2]]], Optional[Pattern[p, Blank[]]]]], Pattern[x, Blank[Symbol]]] := Dist[c^(n/2), Int[(c + d*x^2)^(p - n/2)*(1 + a*x)^n, x], x] /; FreeQ[{a, c, d, p}, x] && EqQ[a^2*c + d, 0] && !(IntegerQ[p] || GtQ[c, 0]) && IGtQ[n/2, 0]
  6142. Int[Times[Power[E, Times[ArcTanh[Times[Optional[Pattern[a, Blank[]]], Pattern[x, Blank[]]]], Pattern[n, Blank[]]]], Power[Plus[Pattern[c, Blank[]], Times[Optional[Pattern[d, Blank[]]], Power[Pattern[x, Blank[]], 2]]], Optional[Pattern[p, Blank[]]]]], Pattern[x, Blank[Symbol]]] := Dist[1/c^(n/2), Int[(c + d*x^2)^(p + n/2)/(1 - a*x)^n, x], x] /; FreeQ[{a, c, d, p}, x] && EqQ[a^2*c + d, 0] && !(IntegerQ[p] || GtQ[c, 0]) && ILtQ[n/2, 0]
  6143. Int[Times[Power[E, Times[ArcTanh[Times[Optional[Pattern[a, Blank[]]], Pattern[x, Blank[]]]], Optional[Pattern[n, Blank[]]]]], Power[Plus[Pattern[c, Blank[]], Times[Optional[Pattern[d, Blank[]]], Power[Pattern[x, Blank[]], 2]]], Pattern[p, Blank[]]]], Pattern[x, Blank[Symbol]]] := Dist[(c^IntPart[p]*(c + d*x^2)^FracPart[p])/(1 - a^2*x^2)^FracPart[p], Int[(1 - a^2*x^2)^p*E^(n*ArcTanh[a*x]), x], x] /; FreeQ[{a, c, d, n, p}, x] && EqQ[a^2*c + d, 0] && !(IntegerQ[p] || GtQ[c, 0])
  6144. Int[Times[Power[E, Times[ArcTanh[Times[Optional[Pattern[a, Blank[]]], Pattern[x, Blank[]]]], Pattern[n, Blank[]]]], Pattern[x, Blank[]], Power[Plus[Pattern[c, Blank[]], Times[Optional[Pattern[d, Blank[]]], Power[Pattern[x, Blank[]], 2]]], Rational[-3, 2]]], Pattern[x, Blank[Symbol]]] := Simp[((1 - a*n*x)*E^(n*ArcTanh[a*x]))/(d*(n^2 - 1)*Sqrt[c + d*x^2]), x] /; FreeQ[{a, c, d, n}, x] && EqQ[a^2*c + d, 0] && !IntegerQ[n]
  6145. Int[Times[Power[E, Times[ArcTanh[Times[Optional[Pattern[a, Blank[]]], Pattern[x, Blank[]]]], Pattern[n, Blank[]]]], Pattern[x, Blank[]], Power[Plus[Pattern[c, Blank[]], Times[Optional[Pattern[d, Blank[]]], Power[Pattern[x, Blank[]], 2]]], Pattern[p, Blank[]]]], Pattern[x, Blank[Symbol]]] := Simp[((c + d*x^2)^(p + 1)*E^(n*ArcTanh[a*x]))/(2*d*(p + 1)), x] - Dist[(a*c*n)/(2*d*(p + 1)), Int[(c + d*x^2)^p*E^(n*ArcTanh[a*x]), x], x] /; FreeQ[{a, c, d, n}, x] && EqQ[a^2*c + d, 0] && LtQ[p, -1] && !IntegerQ[n] && IntegerQ[2*p]
  6146. Int[Times[Power[E, Times[ArcTanh[Times[Optional[Pattern[a, Blank[]]], Pattern[x, Blank[]]]], Pattern[n, Blank[]]]], Power[Pattern[x, Blank[]], 2], Power[Plus[Pattern[c, Blank[]], Times[Optional[Pattern[d, Blank[]]], Power[Pattern[x, Blank[]], 2]]], Optional[Pattern[p, Blank[]]]]], Pattern[x, Blank[Symbol]]] := Simp[((1 - a*n*x)*(c + d*x^2)^(p + 1)*E^(n*ArcTanh[a*x]))/(a*d*n*(n^2 - 1)), x] /; FreeQ[{a, c, d, n}, x] && EqQ[a^2*c + d, 0] && EqQ[n^2 + 2*(p + 1), 0] && !IntegerQ[n]
  6147. Int[Times[Power[E, Times[ArcTanh[Times[Optional[Pattern[a, Blank[]]], Pattern[x, Blank[]]]], Pattern[n, Blank[]]]], Power[Pattern[x, Blank[]], 2], Power[Plus[Pattern[c, Blank[]], Times[Optional[Pattern[d, Blank[]]], Power[Pattern[x, Blank[]], 2]]], Pattern[p, Blank[]]]], Pattern[x, Blank[Symbol]]] := -Simp[((n + 2*(p + 1)*a*x)*(c + d*x^2)^(p + 1)*E^(n*ArcTanh[a*x]))/(a*d*(n^2 - 4*(p + 1)^2)), x] + Dist[(n^2 + 2*(p + 1))/(d*(n^2 - 4*(p + 1)^2)), Int[(c + d*x^2)^(p + 1)*E^(n*ArcTanh[a*x]), x], x] /; FreeQ[{a, c, d, n}, x] && EqQ[a^2*c + d, 0] && LtQ[p, -1] && !IntegerQ[n] && NeQ[n^2 - 4*(p + 1)^2, 0] && IntegerQ[2*p]
  6148. Int[Times[Power[E, Times[ArcTanh[Times[Optional[Pattern[a, Blank[]]], Pattern[x, Blank[]]]], Optional[Pattern[n, Blank[]]]]], Power[Pattern[x, Blank[]], Optional[Pattern[m, Blank[]]]], Power[Plus[Pattern[c, Blank[]], Times[Optional[Pattern[d, Blank[]]], Power[Pattern[x, Blank[]], 2]]], Optional[Pattern[p, Blank[]]]]], Pattern[x, Blank[Symbol]]] := Dist[c^p, Int[x^m*(1 - a^2*x^2)^(p - n/2)*(1 + a*x)^n, x], x] /; FreeQ[{a, c, d, m, p}, x] && EqQ[a^2*c + d, 0] && (IntegerQ[p] || GtQ[c, 0]) && IGtQ[(n + 1)/2, 0] && !IntegerQ[p - n/2]
  6149. Int[Times[Power[E, Times[ArcTanh[Times[Optional[Pattern[a, Blank[]]], Pattern[x, Blank[]]]], Pattern[n, Blank[]]]], Power[Pattern[x, Blank[]], Optional[Pattern[m, Blank[]]]], Power[Plus[Pattern[c, Blank[]], Times[Optional[Pattern[d, Blank[]]], Power[Pattern[x, Blank[]], 2]]], Optional[Pattern[p, Blank[]]]]], Pattern[x, Blank[Symbol]]] := Dist[c^p, Int[(x^m*(1 - a^2*x^2)^(p + n/2))/(1 - a*x)^n, x], x] /; FreeQ[{a, c, d, m, p}, x] && EqQ[a^2*c + d, 0] && (IntegerQ[p] || GtQ[c, 0]) && ILtQ[(n - 1)/2, 0] && !IntegerQ[p - n/2]
  6150. Int[Times[Power[E, Times[ArcTanh[Times[Optional[Pattern[a, Blank[]]], Pattern[x, Blank[]]]], Optional[Pattern[n, Blank[]]]]], Power[Pattern[x, Blank[]], Optional[Pattern[m, Blank[]]]], Power[Plus[Pattern[c, Blank[]], Times[Optional[Pattern[d, Blank[]]], Power[Pattern[x, Blank[]], 2]]], Optional[Pattern[p, Blank[]]]]], Pattern[x, Blank[Symbol]]] := Dist[c^p, Int[x^m*(1 - a*x)^(p - n/2)*(1 + a*x)^(p + n/2), x], x] /; FreeQ[{a, c, d, m, n, p}, x] && EqQ[a^2*c + d, 0] && (IntegerQ[p] || GtQ[c, 0])
  6151. Int[Times[Power[E, Times[ArcTanh[Times[Optional[Pattern[a, Blank[]]], Pattern[x, Blank[]]]], Pattern[n, Blank[]]]], Power[Pattern[x, Blank[]], Optional[Pattern[m, Blank[]]]], Power[Plus[Pattern[c, Blank[]], Times[Optional[Pattern[d, Blank[]]], Power[Pattern[x, Blank[]], 2]]], Optional[Pattern[p, Blank[]]]]], Pattern[x, Blank[Symbol]]] := Dist[c^(n/2), Int[x^m*(c + d*x^2)^(p - n/2)*(1 + a*x)^n, x], x] /; FreeQ[{a, c, d, m, p}, x] && EqQ[a^2*c + d, 0] && !(IntegerQ[p] || GtQ[c, 0]) && IGtQ[n/2, 0]
  6152. Int[Times[Power[E, Times[ArcTanh[Times[Optional[Pattern[a, Blank[]]], Pattern[x, Blank[]]]], Pattern[n, Blank[]]]], Power[Pattern[x, Blank[]], Optional[Pattern[m, Blank[]]]], Power[Plus[Pattern[c, Blank[]], Times[Optional[Pattern[d, Blank[]]], Power[Pattern[x, Blank[]], 2]]], Optional[Pattern[p, Blank[]]]]], Pattern[x, Blank[Symbol]]] := Dist[1/c^(n/2), Int[(x^m*(c + d*x^2)^(p + n/2))/(1 - a*x)^n, x], x] /; FreeQ[{a, c, d, m, p}, x] && EqQ[a^2*c + d, 0] && !(IntegerQ[p] || GtQ[c, 0]) && ILtQ[n/2, 0]
  6153. Int[Times[Power[E, Times[ArcTanh[Times[Optional[Pattern[a, Blank[]]], Pattern[x, Blank[]]]], Optional[Pattern[n, Blank[]]]]], Power[Pattern[x, Blank[]], Optional[Pattern[m, Blank[]]]], Power[Plus[Pattern[c, Blank[]], Times[Optional[Pattern[d, Blank[]]], Power[Pattern[x, Blank[]], 2]]], Pattern[p, Blank[]]]], Pattern[x, Blank[Symbol]]] := Dist[(c^IntPart[p]*(c + d*x^2)^FracPart[p])/(1 - a^2*x^2)^FracPart[p], Int[x^m*(1 - a^2*x^2)^p*E^(n*ArcTanh[a*x]), x], x] /; FreeQ[{a, c, d, m, n, p}, x] && EqQ[a^2*c + d, 0] && !(IntegerQ[p] || GtQ[c, 0]) && !IntegerQ[n/2]
  6154. Int[Times[Power[E, Times[ArcTanh[Times[Optional[Pattern[a, Blank[]]], Pattern[x, Blank[]]]], Optional[Pattern[n, Blank[]]]]], Pattern[u, Blank[]], Power[Plus[Pattern[c, Blank[]], Times[Optional[Pattern[d, Blank[]]], Power[Pattern[x, Blank[]], 2]]], Optional[Pattern[p, Blank[]]]]], Pattern[x, Blank[Symbol]]] := Dist[c^p, Int[u*(1 - a*x)^(p - n/2)*(1 + a*x)^(p + n/2), x], x] /; FreeQ[{a, c, d, n, p}, x] && EqQ[a^2*c + d, 0] && (IntegerQ[p] || GtQ[c, 0])
  6155. Int[Times[Power[E, Times[ArcTanh[Times[Optional[Pattern[a, Blank[]]], Pattern[x, Blank[]]]], Pattern[n, Blank[]]]], Pattern[u, Blank[]], Power[Plus[Pattern[c, Blank[]], Times[Optional[Pattern[d, Blank[]]], Power[Pattern[x, Blank[]], 2]]], Pattern[p, Blank[]]]], Pattern[x, Blank[Symbol]]] := Dist[(c^IntPart[p]*(c + d*x^2)^FracPart[p])/((1 - a*x)^FracPart[p]*(1 + a*x)^FracPart[p]), Int[u*(1 - a*x)^(p - n/2)*(1 + a*x)^(p + n/2), x], x] /; FreeQ[{a, c, d, n, p}, x] && EqQ[a^2*c + d, 0] && !(IntegerQ[p] || GtQ[c, 0]) && IntegerQ[n/2]
  6156. Int[Times[Power[E, Times[ArcTanh[Times[Optional[Pattern[a, Blank[]]], Pattern[x, Blank[]]]], Optional[Pattern[n, Blank[]]]]], Pattern[u, Blank[]], Power[Plus[Pattern[c, Blank[]], Times[Optional[Pattern[d, Blank[]]], Power[Pattern[x, Blank[]], 2]]], Pattern[p, Blank[]]]], Pattern[x, Blank[Symbol]]] := Dist[(c^IntPart[p]*(c + d*x^2)^FracPart[p])/(1 - a^2*x^2)^FracPart[p], Int[u*(1 - a^2*x^2)^p*E^(n*ArcTanh[a*x]), x], x] /; FreeQ[{a, c, d, n, p}, x] && EqQ[a^2*c + d, 0] && !(IntegerQ[p] || GtQ[c, 0]) && !IntegerQ[n/2]
  6157. Int[Times[Power[E, Times[ArcTanh[Times[Optional[Pattern[a, Blank[]]], Pattern[x, Blank[]]]], Optional[Pattern[n, Blank[]]]]], Optional[Pattern[u, Blank[]]], Power[Plus[Pattern[c, Blank[]], Times[Optional[Pattern[d, Blank[]]], Power[Pattern[x, Blank[]], -2]]], Optional[Pattern[p, Blank[]]]]], Pattern[x, Blank[Symbol]]] := Dist[d^p, Int[(u*(1 - a^2*x^2)^p*E^(n*ArcTanh[a*x]))/x^(2*p), x], x] /; FreeQ[{a, c, d, n}, x] && EqQ[c + a^2*d, 0] && IntegerQ[p]
  6158. Int[Times[Power[E, Times[ArcTanh[Times[Optional[Pattern[a, Blank[]]], Pattern[x, Blank[]]]], Pattern[n, Blank[]]]], Optional[Pattern[u, Blank[]]], Power[Plus[Pattern[c, Blank[]], Times[Optional[Pattern[d, Blank[]]], Power[Pattern[x, Blank[]], -2]]], Pattern[p, Blank[]]]], Pattern[x, Blank[Symbol]]] := Dist[c^p, Int[u*(1 - 1/(a*x))^p*(1 + 1/(a*x))^p*E^(n*ArcTanh[a*x]), x], x] /; FreeQ[{a, c, d, p}, x] && EqQ[c + a^2*d, 0] && !IntegerQ[p] && IntegerQ[n/2] && GtQ[c, 0]
  6159. Int[Times[Power[E, Times[ArcTanh[Times[Optional[Pattern[a, Blank[]]], Pattern[x, Blank[]]]], Pattern[n, Blank[]]]], Optional[Pattern[u, Blank[]]], Power[Plus[Pattern[c, Blank[]], Times[Optional[Pattern[d, Blank[]]], Power[Pattern[x, Blank[]], -2]]], Pattern[p, Blank[]]]], Pattern[x, Blank[Symbol]]] := Dist[(x^(2*p)*(c + d/x^2)^p)/((1 - a*x)^p*(1 + a*x)^p), Int[(u*(1 - a*x)^p*(1 + a*x)^p*E^(n*ArcTanh[a*x]))/x^(2*p), x], x] /; FreeQ[{a, c, d, n, p}, x] && EqQ[c + a^2*d, 0] && !IntegerQ[p] && IntegerQ[n/2] && !GtQ[c, 0]
  6160. Int[Times[Power[E, Times[ArcTanh[Times[Optional[Pattern[a, Blank[]]], Pattern[x, Blank[]]]], Optional[Pattern[n, Blank[]]]]], Optional[Pattern[u, Blank[]]], Power[Plus[Pattern[c, Blank[]], Times[Optional[Pattern[d, Blank[]]], Power[Pattern[x, Blank[]], -2]]], Pattern[p, Blank[]]]], Pattern[x, Blank[Symbol]]] := Dist[(x^(2*p)*(c + d/x^2)^p)/(1 + (c*x^2)/d)^p, Int[(u*(1 + (c*x^2)/d)^p*E^(n*ArcTanh[a*x]))/x^(2*p), x], x] /; FreeQ[{a, c, d, n, p}, x] && EqQ[c + a^2*d, 0] && !IntegerQ[p] && !IntegerQ[n/2]
  6161. Int[Power[E, Times[ArcTanh[Times[Optional[Pattern[c, Blank[]]], Plus[Pattern[a, Blank[]], Times[Optional[Pattern[b, Blank[]]], Pattern[x, Blank[]]]]]], Optional[Pattern[n, Blank[]]]]], Pattern[x, Blank[Symbol]]] := Int[(1 + a*c + b*c*x)^(n/2)/(1 - a*c - b*c*x)^(n/2), x] /; FreeQ[{a, b, c, n}, x]
  6162. Int[Times[Power[E, Times[ArcTanh[Times[Optional[Pattern[c, Blank[]]], Plus[Pattern[a, Blank[]], Times[Optional[Pattern[b, Blank[]]], Pattern[x, Blank[]]]]]], Pattern[n, Blank[]]]], Power[Pattern[x, Blank[]], Pattern[m, Blank[]]]], Pattern[x, Blank[Symbol]]] := Dist[4/(n*b^(m + 1)*c^(m + 1)), Subst[Int[(x^(2/n)*(-1 - a*c + (1 - a*c)*x^(2/n))^m)/(1 + x^(2/n))^(m + 2), x], x, (1 + c*(a + b*x))^(n/2)/(1 - c*(a + b*x))^(n/2)], x] /; FreeQ[{a, b, c}, x] && ILtQ[m, 0] && LtQ[-1, n, 1]
  6163. Int[Times[Power[E, Times[ArcTanh[Times[Optional[Pattern[c, Blank[]]], Plus[Pattern[a, Blank[]], Times[Optional[Pattern[b, Blank[]]], Pattern[x, Blank[]]]]]], Optional[Pattern[n, Blank[]]]]], Power[Plus[Optional[Pattern[d, Blank[]]], Times[Optional[Pattern[e, Blank[]]], Pattern[x, Blank[]]]], Optional[Pattern[m, Blank[]]]]], Pattern[x, Blank[Symbol]]] := Int[((d + e*x)^m*(1 + a*c + b*c*x)^(n/2))/(1 - a*c - b*c*x)^(n/2), x] /; FreeQ[{a, b, c, d, e, m, n}, x]
  6164. Int[Times[Power[E, Times[ArcTanh[Plus[Pattern[a, Blank[]], Times[Optional[Pattern[b, Blank[]]], Pattern[x, Blank[]]]]], Optional[Pattern[n, Blank[]]]]], Optional[Pattern[u, Blank[]]], Power[Plus[Pattern[c, Blank[]], Times[Optional[Pattern[d, Blank[]]], Pattern[x, Blank[]]], Times[Optional[Pattern[e, Blank[]]], Power[Pattern[x, Blank[]], 2]]], Optional[Pattern[p, Blank[]]]]], Pattern[x, Blank[Symbol]]] := Dist[(c/(1 - a^2))^p, Int[u*(1 - a - b*x)^(p - n/2)*(1 + a + b*x)^(p + n/2), x], x] /; FreeQ[{a, b, c, d, e, n, p}, x] && EqQ[b*d - 2*a*e, 0] && EqQ[b^2*c + e*(1 - a^2), 0] && (IntegerQ[p] || GtQ[c/(1 - a^2), 0])
  6165. Int[Times[Power[E, Times[ArcTanh[Plus[Pattern[a, Blank[]], Times[Optional[Pattern[b, Blank[]]], Pattern[x, Blank[]]]]], Optional[Pattern[n, Blank[]]]]], Optional[Pattern[u, Blank[]]], Power[Plus[Pattern[c, Blank[]], Times[Optional[Pattern[d, Blank[]]], Pattern[x, Blank[]]], Times[Optional[Pattern[e, Blank[]]], Power[Pattern[x, Blank[]], 2]]], Optional[Pattern[p, Blank[]]]]], Pattern[x, Blank[Symbol]]] := Dist[(c + d*x + e*x^2)^p/(1 - a^2 - 2*a*b*x - b^2*x^2)^p, Int[u*(1 - a^2 - 2*a*b*x - b^2*x^2)^p*E^(n*ArcTanh[a*x]), x], x] /; FreeQ[{a, b, c, d, e, n, p}, x] && EqQ[b*d - 2*a*e, 0] && EqQ[b^2*c + e*(1 - a^2), 0] && !(IntegerQ[p] || GtQ[c/(1 - a^2), 0])
  6166. Int[Times[Power[E, Times[ArcTanh[Times[Optional[Pattern[c, Blank[]]], Power[Plus[Optional[Pattern[a, Blank[]]], Times[Optional[Pattern[b, Blank[]]], Pattern[x, Blank[]]]], -1]]], Optional[Pattern[n, Blank[]]]]], Optional[Pattern[u, Blank[]]]], Pattern[x, Blank[Symbol]]] := Int[u*E^(n*ArcCoth[a/c + (b*x)/c]), x] /; FreeQ[{a, b, c, n}, x]
  6167. Int[Times[Power[E, Times[ArcCoth[Times[Optional[Pattern[a, Blank[]]], Pattern[x, Blank[]]]], Pattern[n, Blank[]]]], Optional[Pattern[u, Blank[]]]], Pattern[x, Blank[Symbol]]] := Dist[(-1)^(n/2), Int[u*E^(n*ArcTanh[a*x]), x], x] /; FreeQ[a, x] && IntegerQ[n/2]
  6168. Int[Power[E, Times[ArcCoth[Times[Optional[Pattern[a, Blank[]]], Pattern[x, Blank[]]]], Optional[Pattern[n, Blank[]]]]], Pattern[x, Blank[Symbol]]] := -Subst[Int[(1 + x/a)^((n + 1)/2)/(x^2*(1 - x/a)^((n - 1)/2)*Sqrt[1 - x^2/a^2]), x], x, 1/x] /; FreeQ[a, x] && IntegerQ[(n - 1)/2]
  6169. Int[Times[Power[E, Times[ArcCoth[Times[Optional[Pattern[a, Blank[]]], Pattern[x, Blank[]]]], Optional[Pattern[n, Blank[]]]]], Power[Pattern[x, Blank[]], Optional[Pattern[m, Blank[]]]]], Pattern[x, Blank[Symbol]]] := -Subst[Int[(1 + x/a)^((n + 1)/2)/(x^(m + 2)*(1 - x/a)^((n - 1)/2)*Sqrt[1 - x^2/a^2]), x], x, 1/x] /; FreeQ[a, x] && IntegerQ[(n - 1)/2] && IntegerQ[m]
  6170. Int[Power[E, Times[ArcCoth[Times[Optional[Pattern[a, Blank[]]], Pattern[x, Blank[]]]], Pattern[n, Blank[]]]], Pattern[x, Blank[Symbol]]] := -Subst[Int[(1 + x/a)^(n/2)/(x^2*(1 - x/a)^(n/2)), x], x, 1/x] /; FreeQ[{a, n}, x] && !IntegerQ[n]
  6171. Int[Times[Power[E, Times[ArcCoth[Times[Optional[Pattern[a, Blank[]]], Pattern[x, Blank[]]]], Pattern[n, Blank[]]]], Power[Pattern[x, Blank[]], Optional[Pattern[m, Blank[]]]]], Pattern[x, Blank[Symbol]]] := -Subst[Int[(1 + x/a)^(n/2)/(x^(m + 2)*(1 - x/a)^(n/2)), x], x, 1/x] /; FreeQ[{a, n}, x] && !IntegerQ[n] && IntegerQ[m]
  6172. Int[Times[Power[E, Times[ArcCoth[Times[Optional[Pattern[a, Blank[]]], Pattern[x, Blank[]]]], Optional[Pattern[n, Blank[]]]]], Power[Pattern[x, Blank[]], Pattern[m, Blank[]]]], Pattern[x, Blank[Symbol]]] := -Dist[x^m*(1/x)^m, Subst[Int[(1 + x/a)^((n + 1)/2)/(x^(m + 2)*(1 - x/a)^((n - 1)/2)*Sqrt[1 - x^2/a^2]), x], x, 1/x], x] /; FreeQ[{a, m}, x] && IntegerQ[(n - 1)/2] && !IntegerQ[m]
  6173. Int[Times[Power[E, Times[ArcCoth[Times[Optional[Pattern[a, Blank[]]], Pattern[x, Blank[]]]], Pattern[n, Blank[]]]], Power[Pattern[x, Blank[]], Pattern[m, Blank[]]]], Pattern[x, Blank[Symbol]]] := -Dist[x^m*(1/x)^m, Subst[Int[(1 + x/a)^(n/2)/(x^(m + 2)*(1 - x/a)^(n/2)), x], x, 1/x], x] /; FreeQ[{a, m, n}, x] && !IntegerQ[n] && !IntegerQ[m]
  6174. Int[Times[Power[E, Times[ArcCoth[Times[Optional[Pattern[a, Blank[]]], Pattern[x, Blank[]]]], Optional[Pattern[n, Blank[]]]]], Power[Plus[Pattern[c, Blank[]], Times[Optional[Pattern[d, Blank[]]], Pattern[x, Blank[]]]], Optional[Pattern[p, Blank[]]]]], Pattern[x, Blank[Symbol]]] := Simp[((1 + a*x)*(c + d*x)^p*E^(n*ArcCoth[a*x]))/(a*(p + 1)), x] /; FreeQ[{a, c, d, n, p}, x] && EqQ[a*c + d, 0] && EqQ[p, n/2] && !IntegerQ[n/2]
  6175. Int[Times[Power[E, Times[ArcCoth[Times[Optional[Pattern[a, Blank[]]], Pattern[x, Blank[]]]], Optional[Pattern[n, Blank[]]]]], Optional[Pattern[u, Blank[]]], Power[Plus[Pattern[c, Blank[]], Times[Optional[Pattern[d, Blank[]]], Pattern[x, Blank[]]]], Optional[Pattern[p, Blank[]]]]], Pattern[x, Blank[Symbol]]] := Dist[d^p, Int[u*x^p*(1 + c/(d*x))^p*E^(n*ArcCoth[a*x]), x], x] /; FreeQ[{a, c, d, n}, x] && EqQ[a^2*c^2 - d^2, 0] && !IntegerQ[n/2] && IntegerQ[p]
  6176. Int[Times[Power[E, Times[ArcCoth[Times[Optional[Pattern[a, Blank[]]], Pattern[x, Blank[]]]], Optional[Pattern[n, Blank[]]]]], Optional[Pattern[u, Blank[]]], Power[Plus[Pattern[c, Blank[]], Times[Optional[Pattern[d, Blank[]]], Pattern[x, Blank[]]]], Pattern[p, Blank[]]]], Pattern[x, Blank[Symbol]]] := Dist[(c + d*x)^p/(x^p*(1 + c/(d*x))^p), Int[u*x^p*(1 + c/(d*x))^p*E^(n*ArcCoth[a*x]), x], x] /; FreeQ[{a, c, d, n, p}, x] && EqQ[a^2*c^2 - d^2, 0] && !IntegerQ[n/2] && !IntegerQ[p]
  6177. Int[Times[Power[E, Times[ArcCoth[Times[Optional[Pattern[a, Blank[]]], Pattern[x, Blank[]]]], Optional[Pattern[n, Blank[]]]]], Power[Plus[Pattern[c, Blank[]], Times[Optional[Pattern[d, Blank[]]], Power[Pattern[x, Blank[]], -1]]], Optional[Pattern[p, Blank[]]]]], Pattern[x, Blank[Symbol]]] := -Dist[c^n, Subst[Int[((c + d*x)^(p - n)*(1 - x^2/a^2)^(n/2))/x^2, x], x, 1/x], x] /; FreeQ[{a, c, d, p}, x] && EqQ[c + a*d, 0] && IntegerQ[(n - 1)/2] && (IntegerQ[p] || EqQ[p, n/2] || EqQ[p, n/2 + 1]) && IntegerQ[2*p]
  6178. Int[Times[Power[E, Times[ArcCoth[Times[Optional[Pattern[a, Blank[]]], Pattern[x, Blank[]]]], Optional[Pattern[n, Blank[]]]]], Power[Plus[Pattern[c, Blank[]], Times[Optional[Pattern[d, Blank[]]], Power[Pattern[x, Blank[]], -1]]], Optional[Pattern[p, Blank[]]]], Power[Pattern[x, Blank[]], Optional[Pattern[m, Blank[]]]]], Pattern[x, Blank[Symbol]]] := -Dist[c^n, Subst[Int[((c + d*x)^(p - n)*(1 - x^2/a^2)^(n/2))/x^(m + 2), x], x, 1/x], x] /; FreeQ[{a, c, d, p}, x] && EqQ[c + a*d, 0] && IntegerQ[(n - 1)/2] && IntegerQ[m] && (IntegerQ[p] || EqQ[p, n/2] || EqQ[p, n/2 + 1] || LtQ[-5, m, -1]) && IntegerQ[2*p]
  6179. Int[Times[Power[E, Times[ArcCoth[Times[Optional[Pattern[a, Blank[]]], Pattern[x, Blank[]]]], Optional[Pattern[n, Blank[]]]]], Power[Plus[Pattern[c, Blank[]], Times[Optional[Pattern[d, Blank[]]], Power[Pattern[x, Blank[]], -1]]], Optional[Pattern[p, Blank[]]]]], Pattern[x, Blank[Symbol]]] := -Dist[c^p, Subst[Int[((1 + (d*x)/c)^p*(1 + x/a)^(n/2))/(x^2*(1 - x/a)^(n/2)), x], x, 1/x], x] /; FreeQ[{a, c, d, n, p}, x] && EqQ[c^2 - a^2*d^2, 0] && !IntegerQ[n/2] && (IntegerQ[p] || GtQ[c, 0])
  6180. Int[Times[Power[E, Times[ArcCoth[Times[Optional[Pattern[a, Blank[]]], Pattern[x, Blank[]]]], Optional[Pattern[n, Blank[]]]]], Power[Plus[Pattern[c, Blank[]], Times[Optional[Pattern[d, Blank[]]], Power[Pattern[x, Blank[]], -1]]], Optional[Pattern[p, Blank[]]]], Power[Pattern[x, Blank[]], Optional[Pattern[m, Blank[]]]]], Pattern[x, Blank[Symbol]]] := -Dist[c^p, Subst[Int[((1 + (d*x)/c)^p*(1 + x/a)^(n/2))/(x^(m + 2)*(1 - x/a)^(n/2)), x], x, 1/x], x] /; FreeQ[{a, c, d, n, p}, x] && EqQ[c^2 - a^2*d^2, 0] && !IntegerQ[n/2] && (IntegerQ[p] || GtQ[c, 0]) && IntegerQ[m]
  6181. Int[Times[Power[E, Times[ArcCoth[Times[Optional[Pattern[a, Blank[]]], Pattern[x, Blank[]]]], Optional[Pattern[n, Blank[]]]]], Power[Plus[Pattern[c, Blank[]], Times[Optional[Pattern[d, Blank[]]], Power[Pattern[x, Blank[]], -1]]], Optional[Pattern[p, Blank[]]]], Power[Pattern[x, Blank[]], Pattern[m, Blank[]]]], Pattern[x, Blank[Symbol]]] := -Dist[c^p*x^m*(1/x)^m, Subst[Int[((1 + (d*x)/c)^p*(1 + x/a)^(n/2))/(x^(m + 2)*(1 - x/a)^(n/2)), x], x, 1/x], x] /; FreeQ[{a, c, d, m, n, p}, x] && EqQ[c^2 - a^2*d^2, 0] && !IntegerQ[n/2] && (IntegerQ[p] || GtQ[c, 0]) && !IntegerQ[m]
  6182. Int[Times[Power[E, Times[ArcCoth[Times[Optional[Pattern[a, Blank[]]], Pattern[x, Blank[]]]], Optional[Pattern[n, Blank[]]]]], Optional[Pattern[u, Blank[]]], Power[Plus[Pattern[c, Blank[]], Times[Optional[Pattern[d, Blank[]]], Power[Pattern[x, Blank[]], -1]]], Pattern[p, Blank[]]]], Pattern[x, Blank[Symbol]]] := Dist[(c + d/x)^p/(1 + d/(c*x))^p, Int[u*(1 + d/(c*x))^p*E^(n*ArcCoth[a*x]), x], x] /; FreeQ[{a, c, d, n, p}, x] && EqQ[c^2 - a^2*d^2, 0] && !IntegerQ[n/2] && !(IntegerQ[p] || GtQ[c, 0])
  6183. Int[Times[Power[E, Times[ArcCoth[Times[Optional[Pattern[a, Blank[]]], Pattern[x, Blank[]]]], Optional[Pattern[n, Blank[]]]]], Power[Plus[Pattern[c, Blank[]], Times[Optional[Pattern[d, Blank[]]], Power[Pattern[x, Blank[]], 2]]], -1]], Pattern[x, Blank[Symbol]]] := Simp[E^(n*ArcCoth[a*x])/(a*c*n), x] /; FreeQ[{a, c, d, n}, x] && EqQ[a^2*c + d, 0] && !IntegerQ[n/2]
  6184. Int[Times[Power[E, Times[ArcCoth[Times[Optional[Pattern[a, Blank[]]], Pattern[x, Blank[]]]], Pattern[n, Blank[]]]], Power[Plus[Pattern[c, Blank[]], Times[Optional[Pattern[d, Blank[]]], Power[Pattern[x, Blank[]], 2]]], Rational[-3, 2]]], Pattern[x, Blank[Symbol]]] := Simp[((n - a*x)*E^(n*ArcCoth[a*x]))/(a*c*(n^2 - 1)*Sqrt[c + d*x^2]), x] /; FreeQ[{a, c, d, n}, x] && EqQ[a^2*c + d, 0] && !IntegerQ[n]
  6185. Int[Times[Power[E, Times[ArcCoth[Times[Optional[Pattern[a, Blank[]]], Pattern[x, Blank[]]]], Optional[Pattern[n, Blank[]]]]], Power[Plus[Pattern[c, Blank[]], Times[Optional[Pattern[d, Blank[]]], Power[Pattern[x, Blank[]], 2]]], Pattern[p, Blank[]]]], Pattern[x, Blank[Symbol]]] := Simp[((n + 2*a*(p + 1)*x)*(c + d*x^2)^(p + 1)*E^(n*ArcCoth[a*x]))/(a*c*(n^2 - 4*(p + 1)^2)), x] - Dist[(2*(p + 1)*(2*p + 3))/(c*(n^2 - 4*(p + 1)^2)), Int[(c + d*x^2)^(p + 1)*E^(n*ArcCoth[a*x]), x], x] /; FreeQ[{a, c, d, n}, x] && EqQ[a^2*c + d, 0] && !IntegerQ[n/2] && LtQ[p, -1] && NeQ[p, -3/2] && NeQ[n^2 - 4*(p + 1)^2, 0] && (IntegerQ[p] || !IntegerQ[n])
  6186. Int[Times[Power[E, Times[ArcCoth[Times[Optional[Pattern[a, Blank[]]], Pattern[x, Blank[]]]], Pattern[n, Blank[]]]], Pattern[x, Blank[]], Power[Plus[Pattern[c, Blank[]], Times[Optional[Pattern[d, Blank[]]], Power[Pattern[x, Blank[]], 2]]], Rational[-3, 2]]], Pattern[x, Blank[Symbol]]] := -Simp[((1 - a*n*x)*E^(n*ArcCoth[a*x]))/(a^2*c*(n^2 - 1)*Sqrt[c + d*x^2]), x] /; FreeQ[{a, c, d, n}, x] && EqQ[a^2*c + d, 0] && !IntegerQ[n]
  6187. Int[Times[Power[E, Times[ArcCoth[Times[Optional[Pattern[a, Blank[]]], Pattern[x, Blank[]]]], Optional[Pattern[n, Blank[]]]]], Pattern[x, Blank[]], Power[Plus[Pattern[c, Blank[]], Times[Optional[Pattern[d, Blank[]]], Power[Pattern[x, Blank[]], 2]]], Pattern[p, Blank[]]]], Pattern[x, Blank[Symbol]]] := Simp[((2*(p + 1) + a*n*x)*(c + d*x^2)^(p + 1)*E^(n*ArcCoth[a*x]))/(a^2*c*(n^2 - 4*(p + 1)^2)), x] - Dist[(n*(2*p + 3))/(a*c*(n^2 - 4*(p + 1)^2)), Int[(c + d*x^2)^(p + 1)*E^(n*ArcCoth[a*x]), x], x] /; FreeQ[{a, c, d, n}, x] && EqQ[a^2*c + d, 0] && !IntegerQ[n/2] && LeQ[p, -1] && NeQ[p, -3/2] && NeQ[n^2 - 4*(p + 1)^2, 0] && (IntegerQ[p] || !IntegerQ[n])
  6188. Int[Times[Power[E, Times[ArcCoth[Times[Optional[Pattern[a, Blank[]]], Pattern[x, Blank[]]]], Optional[Pattern[n, Blank[]]]]], Power[Pattern[x, Blank[]], 2], Power[Plus[Pattern[c, Blank[]], Times[Optional[Pattern[d, Blank[]]], Power[Pattern[x, Blank[]], 2]]], Optional[Pattern[p, Blank[]]]]], Pattern[x, Blank[Symbol]]] := -Simp[((n + 2*(p + 1)*a*x)*(c + d*x^2)^(p + 1)*E^(n*ArcCoth[a*x]))/(a^3*c*n^2*(n^2 - 1)), x] /; FreeQ[{a, c, d, n}, x] && EqQ[a^2*c + d, 0] && !IntegerQ[n/2] && EqQ[n^2 + 2*(p + 1), 0] && NeQ[n^2, 1]
  6189. Int[Times[Power[E, Times[ArcCoth[Times[Optional[Pattern[a, Blank[]]], Pattern[x, Blank[]]]], Optional[Pattern[n, Blank[]]]]], Power[Pattern[x, Blank[]], 2], Power[Plus[Pattern[c, Blank[]], Times[Optional[Pattern[d, Blank[]]], Power[Pattern[x, Blank[]], 2]]], Pattern[p, Blank[]]]], Pattern[x, Blank[Symbol]]] := Simp[((n + 2*(p + 1)*a*x)*(c + d*x^2)^(p + 1)*E^(n*ArcCoth[a*x]))/(a^3*c*(n^2 - 4*(p + 1)^2)), x] - Dist[(n^2 + 2*(p + 1))/(a^2*c*(n^2 - 4*(p + 1)^2)), Int[(c + d*x^2)^(p + 1)*E^(n*ArcCoth[a*x]), x], x] /; FreeQ[{a, c, d, n}, x] && EqQ[a^2*c + d, 0] && !IntegerQ[n/2] && LeQ[p, -1] && NeQ[n^2 + 2*(p + 1), 0] && NeQ[n^2 - 4*(p + 1)^2, 0] && (IntegerQ[p] || !IntegerQ[n])
  6190. Int[Times[Power[E, Times[ArcCoth[Times[Optional[Pattern[a, Blank[]]], Pattern[x, Blank[]]]], Optional[Pattern[n, Blank[]]]]], Power[Pattern[x, Blank[]], Optional[Pattern[m, Blank[]]]], Power[Plus[Pattern[c, Blank[]], Times[Optional[Pattern[d, Blank[]]], Power[Pattern[x, Blank[]], 2]]], Pattern[p, Blank[]]]], Pattern[x, Blank[Symbol]]] := -Dist[(-c)^p/a^(m + 1), Subst[Int[(E^(n*x)*Coth[x]^(m + 2*(p + 1)))/Cosh[x]^(2*(p + 1)), x], x, ArcCoth[a*x]], x] /; FreeQ[{a, c, d, n}, x] && EqQ[a^2*c + d, 0] && !IntegerQ[n/2] && IntegerQ[m] && LeQ[3, m, -2*(p + 1)] && IntegerQ[p]
  6191. Int[Times[Power[E, Times[ArcCoth[Times[Optional[Pattern[a, Blank[]]], Pattern[x, Blank[]]]], Optional[Pattern[n, Blank[]]]]], Optional[Pattern[u, Blank[]]], Power[Plus[Pattern[c, Blank[]], Times[Optional[Pattern[d, Blank[]]], Power[Pattern[x, Blank[]], 2]]], Optional[Pattern[p, Blank[]]]]], Pattern[x, Blank[Symbol]]] := Dist[d^p, Int[u*x^(2*p)*(1 - 1/(a^2*x^2))^p*E^(n*ArcCoth[a*x]), x], x] /; FreeQ[{a, c, d, n}, x] && EqQ[a^2*c + d, 0] && !IntegerQ[n/2] && IntegerQ[p]
  6192. Int[Times[Power[E, Times[ArcCoth[Times[Optional[Pattern[a, Blank[]]], Pattern[x, Blank[]]]], Optional[Pattern[n, Blank[]]]]], Optional[Pattern[u, Blank[]]], Power[Plus[Pattern[c, Blank[]], Times[Optional[Pattern[d, Blank[]]], Power[Pattern[x, Blank[]], 2]]], Pattern[p, Blank[]]]], Pattern[x, Blank[Symbol]]] := Dist[(c + d*x^2)^p/(x^(2*p)*(1 - 1/(a^2*x^2))^p), Int[u*x^(2*p)*(1 - 1/(a^2*x^2))^p*E^(n*ArcCoth[a*x]), x], x] /; FreeQ[{a, c, d, n, p}, x] && EqQ[a^2*c + d, 0] && !IntegerQ[n/2] && !IntegerQ[p]
  6193. Int[Times[Power[E, Times[ArcCoth[Times[Optional[Pattern[a, Blank[]]], Pattern[x, Blank[]]]], Optional[Pattern[n, Blank[]]]]], Optional[Pattern[u, Blank[]]], Power[Plus[Pattern[c, Blank[]], Times[Optional[Pattern[d, Blank[]]], Power[Pattern[x, Blank[]], -2]]], Optional[Pattern[p, Blank[]]]]], Pattern[x, Blank[Symbol]]] := Dist[c^p/a^(2*p), Int[(u*(-1 + a*x)^(p - n/2)*(1 + a*x)^(p + n/2))/x^(2*p), x], x] /; FreeQ[{a, c, d, n, p}, x] && EqQ[c + a^2*d, 0] && !IntegerQ[n/2] && (IntegerQ[p] || GtQ[c, 0]) && IntegersQ[2*p, p + n/2]
  6194. Int[Times[Power[E, Times[ArcCoth[Times[Optional[Pattern[a, Blank[]]], Pattern[x, Blank[]]]], Optional[Pattern[n, Blank[]]]]], Power[Plus[Pattern[c, Blank[]], Times[Optional[Pattern[d, Blank[]]], Power[Pattern[x, Blank[]], -2]]], Optional[Pattern[p, Blank[]]]]], Pattern[x, Blank[Symbol]]] := -Dist[c^p, Subst[Int[((1 - x/a)^(p - n/2)*(1 + x/a)^(p + n/2))/x^2, x], x, 1/x], x] /; FreeQ[{a, c, d, n, p}, x] && EqQ[c + a^2*d, 0] && !IntegerQ[n/2] && (IntegerQ[p] || GtQ[c, 0]) && !IntegersQ[2*p, p + n/2]
  6195. Int[Times[Power[E, Times[ArcCoth[Times[Optional[Pattern[a, Blank[]]], Pattern[x, Blank[]]]], Optional[Pattern[n, Blank[]]]]], Power[Plus[Pattern[c, Blank[]], Times[Optional[Pattern[d, Blank[]]], Power[Pattern[x, Blank[]], -2]]], Optional[Pattern[p, Blank[]]]], Power[Pattern[x, Blank[]], Optional[Pattern[m, Blank[]]]]], Pattern[x, Blank[Symbol]]] := -Dist[c^p, Subst[Int[((1 - x/a)^(p - n/2)*(1 + x/a)^(p + n/2))/x^(m + 2), x], x, 1/x], x] /; FreeQ[{a, c, d, n, p}, x] && EqQ[c + a^2*d, 0] && !IntegerQ[n/2] && (IntegerQ[p] || GtQ[c, 0]) && !IntegersQ[2*p, p + n/2] && IntegerQ[m]
  6196. Int[Times[Power[E, Times[ArcCoth[Times[Optional[Pattern[a, Blank[]]], Pattern[x, Blank[]]]], Optional[Pattern[n, Blank[]]]]], Power[Plus[Pattern[c, Blank[]], Times[Optional[Pattern[d, Blank[]]], Power[Pattern[x, Blank[]], -2]]], Optional[Pattern[p, Blank[]]]], Power[Pattern[x, Blank[]], Pattern[m, Blank[]]]], Pattern[x, Blank[Symbol]]] := -Dist[c^p*x^m*(1/x)^m, Subst[Int[((1 - x/a)^(p - n/2)*(1 + x/a)^(p + n/2))/x^(m + 2), x], x, 1/x], x] /; FreeQ[{a, c, d, m, n, p}, x] && EqQ[c + a^2*d, 0] && !IntegerQ[n/2] && (IntegerQ[p] || GtQ[c, 0]) && !IntegersQ[2*p, p + n/2] && !IntegerQ[m]
  6197. Int[Times[Power[E, Times[ArcCoth[Times[Optional[Pattern[a, Blank[]]], Pattern[x, Blank[]]]], Optional[Pattern[n, Blank[]]]]], Optional[Pattern[u, Blank[]]], Power[Plus[Pattern[c, Blank[]], Times[Optional[Pattern[d, Blank[]]], Power[Pattern[x, Blank[]], -2]]], Pattern[p, Blank[]]]], Pattern[x, Blank[Symbol]]] := Dist[(c^IntPart[p]*(c + d/x^2)^FracPart[p])/(1 - 1/(a^2*x^2))^FracPart[p], Int[u*(1 - 1/(a^2*x^2))^p*E^(n*ArcCoth[a*x]), x], x] /; FreeQ[{a, c, d, n, p}, x] && EqQ[c + a^2*d, 0] && !IntegerQ[n/2] && !(IntegerQ[p] || GtQ[c, 0])
  6198. Int[Times[Power[E, Times[ArcCoth[Times[Optional[Pattern[c, Blank[]]], Plus[Pattern[a, Blank[]], Times[Optional[Pattern[b, Blank[]]], Pattern[x, Blank[]]]]]], Pattern[n, Blank[]]]], Optional[Pattern[u, Blank[]]]], Pattern[x, Blank[Symbol]]] := Dist[(-1)^(n/2), Int[u*E^(n*ArcTanh[c*(a + b*x)]), x], x] /; FreeQ[{a, b, c}, x] && IntegerQ[n/2]
  6199. Int[Power[E, Times[ArcCoth[Times[Optional[Pattern[c, Blank[]]], Plus[Pattern[a, Blank[]], Times[Optional[Pattern[b, Blank[]]], Pattern[x, Blank[]]]]]], Optional[Pattern[n, Blank[]]]]], Pattern[x, Blank[Symbol]]] := Dist[((c*(a + b*x))^(n/2)*(1 + 1/(c*(a + b*x)))^(n/2))/(1 + a*c + b*c*x)^(n/2), Int[(1 + a*c + b*c*x)^(n/2)/(-1 + a*c + b*c*x)^(n/2), x], x] /; FreeQ[{a, b, c, n}, x] && !IntegerQ[n/2]
  6200. Int[Times[Power[E, Times[ArcCoth[Times[Optional[Pattern[c, Blank[]]], Plus[Pattern[a, Blank[]], Times[Optional[Pattern[b, Blank[]]], Pattern[x, Blank[]]]]]], Pattern[n, Blank[]]]], Power[Pattern[x, Blank[]], Pattern[m, Blank[]]]], Pattern[x, Blank[Symbol]]] := Dist[-4/(n*b^(m + 1)*c^(m + 1)), Subst[Int[(x^(2/n)*(1 + a*c + (1 - a*c)*x^(2/n))^m)/(-1 + x^(2/n))^(m + 2), x], x, (1 + 1/(c*(a + b*x)))^(n/2)/(1 - 1/(c*(a + b*x)))^(n/2)], x] /; FreeQ[{a, b, c}, x] && ILtQ[m, 0] && LtQ[-1, n, 1]
  6201. Int[Times[Power[E, Times[ArcCoth[Times[Optional[Pattern[c, Blank[]]], Plus[Pattern[a, Blank[]], Times[Optional[Pattern[b, Blank[]]], Pattern[x, Blank[]]]]]], Optional[Pattern[n, Blank[]]]]], Power[Plus[Optional[Pattern[d, Blank[]]], Times[Optional[Pattern[e, Blank[]]], Pattern[x, Blank[]]]], Optional[Pattern[m, Blank[]]]]], Pattern[x, Blank[Symbol]]] := Dist[((c*(a + b*x))^(n/2)*(1 + 1/(c*(a + b*x)))^(n/2))/(1 + a*c + b*c*x)^(n/2), Int[((d + e*x)^m*(1 + a*c + b*c*x)^(n/2))/(-1 + a*c + b*c*x)^(n/2), x], x] /; FreeQ[{a, b, c, d, e, m, n}, x] && !IntegerQ[n/2]
  6202. Int[Times[Power[E, Times[ArcCoth[Plus[Pattern[a, Blank[]], Times[Optional[Pattern[b, Blank[]]], Pattern[x, Blank[]]]]], Optional[Pattern[n, Blank[]]]]], Optional[Pattern[u, Blank[]]], Power[Plus[Pattern[c, Blank[]], Times[Optional[Pattern[d, Blank[]]], Pattern[x, Blank[]]], Times[Optional[Pattern[e, Blank[]]], Power[Pattern[x, Blank[]], 2]]], Optional[Pattern[p, Blank[]]]]], Pattern[x, Blank[Symbol]]] := Dist[(c/(1 - a^2))^p*((a + b*x)/(1 + a + b*x))^(n/2)*((1 + a + b*x)/(a + b*x))^(n/2)*((1 - a - b*x)^(n/2)/(-1 + a + b*x)^(n/2)), Int[u*(1 - a - b*x)^(p - n/2)*(1 + a + b*x)^(p + n/2), x], x] /; FreeQ[{a, b, c, d, e, n, p}, x] && !IntegerQ[n/2] && EqQ[b*d - 2*a*e, 0] && EqQ[b^2*c + e*(1 - a^2), 0] && (IntegerQ[p] || GtQ[c/(1 - a^2), 0])
  6203. Int[Times[Power[E, Times[ArcCoth[Plus[Pattern[a, Blank[]], Times[Optional[Pattern[b, Blank[]]], Pattern[x, Blank[]]]]], Optional[Pattern[n, Blank[]]]]], Optional[Pattern[u, Blank[]]], Power[Plus[Pattern[c, Blank[]], Times[Optional[Pattern[d, Blank[]]], Pattern[x, Blank[]]], Times[Optional[Pattern[e, Blank[]]], Power[Pattern[x, Blank[]], 2]]], Optional[Pattern[p, Blank[]]]]], Pattern[x, Blank[Symbol]]] := Dist[(c + d*x + e*x^2)^p/(1 - a^2 - 2*a*b*x - b^2*x^2)^p, Int[u*(1 - a^2 - 2*a*b*x - b^2*x^2)^p*E^(n*ArcCoth[a*x]), x], x] /; FreeQ[{a, b, c, d, e, n, p}, x] && !IntegerQ[n/2] && EqQ[b*d - 2*a*e, 0] && EqQ[b^2*c + e*(1 - a^2), 0] && !(IntegerQ[p] || GtQ[c/(1 - a^2), 0])
  6204. Int[Times[Power[E, Times[ArcCoth[Times[Optional[Pattern[c, Blank[]]], Power[Plus[Optional[Pattern[a, Blank[]]], Times[Optional[Pattern[b, Blank[]]], Pattern[x, Blank[]]]], -1]]], Optional[Pattern[n, Blank[]]]]], Optional[Pattern[u, Blank[]]]], Pattern[x, Blank[Symbol]]] := Int[u*E^(n*ArcTanh[a/c + (b*x)/c]), x] /; FreeQ[{a, b, c, n}, x]
  6205. Int[ArcTanh[Plus[Pattern[a, Blank[]], Times[Optional[Pattern[b, Blank[]]], Power[Pattern[x, Blank[]], Pattern[n, Blank[]]]]]], Pattern[x, Blank[Symbol]]] := Simp[x*ArcTanh[a + b*x^n], x] - Dist[b*n, Int[x^n/(1 - a^2 - 2*a*b*x^n - b^2*x^(2*n)), x], x] /; FreeQ[{a, b, n}, x]
  6206. Int[ArcCoth[Plus[Pattern[a, Blank[]], Times[Optional[Pattern[b, Blank[]]], Power[Pattern[x, Blank[]], Pattern[n, Blank[]]]]]], Pattern[x, Blank[Symbol]]] := Simp[x*ArcCoth[a + b*x^n], x] - Dist[b*n, Int[x^n/(1 - a^2 - 2*a*b*x^n - b^2*x^(2*n)), x], x] /; FreeQ[{a, b, n}, x]
  6207. Int[Times[ArcTanh[Plus[Optional[Pattern[a, Blank[]]], Times[Optional[Pattern[b, Blank[]]], Power[Pattern[x, Blank[]], Optional[Pattern[n, Blank[]]]]]]], Power[Pattern[x, Blank[]], -1]], Pattern[x, Blank[Symbol]]] := Dist[1/2, Int[Log[1 + a + b*x^n]/x, x], x] - Dist[1/2, Int[Log[1 - a - b*x^n]/x, x], x] /; FreeQ[{a, b, n}, x]
  6208. Int[Times[ArcCoth[Plus[Optional[Pattern[a, Blank[]]], Times[Optional[Pattern[b, Blank[]]], Power[Pattern[x, Blank[]], Optional[Pattern[n, Blank[]]]]]]], Power[Pattern[x, Blank[]], -1]], Pattern[x, Blank[Symbol]]] := Dist[1/2, Int[Log[1 + 1/(a + b*x^n)]/x, x], x] - Dist[1/2, Int[Log[1 - 1/(a + b*x^n)]/x, x], x] /; FreeQ[{a, b, n}, x]
  6209. Int[Times[ArcTanh[Plus[Pattern[a, Blank[]], Times[Optional[Pattern[b, Blank[]]], Power[Pattern[x, Blank[]], Pattern[n, Blank[]]]]]], Power[Pattern[x, Blank[]], Optional[Pattern[m, Blank[]]]]], Pattern[x, Blank[Symbol]]] := Simp[(x^(m + 1)*ArcTanh[a + b*x^n])/(m + 1), x] - Dist[(b*n)/(m + 1), Int[x^(m + n)/(1 - a^2 - 2*a*b*x^n - b^2*x^(2*n)), x], x] /; FreeQ[{a, b}, x] && RationalQ[m, n] && NeQ[m, -1] && NeQ[m + 1, n]
  6210. Int[Times[ArcCoth[Plus[Pattern[a, Blank[]], Times[Optional[Pattern[b, Blank[]]], Power[Pattern[x, Blank[]], Pattern[n, Blank[]]]]]], Power[Pattern[x, Blank[]], Optional[Pattern[m, Blank[]]]]], Pattern[x, Blank[Symbol]]] := Simp[(x^(m + 1)*ArcCoth[a + b*x^n])/(m + 1), x] - Dist[(b*n)/(m + 1), Int[x^(m + n)/(1 - a^2 - 2*a*b*x^n - b^2*x^(2*n)), x], x] /; FreeQ[{a, b}, x] && RationalQ[m, n] && NeQ[m, -1] && NeQ[m + 1, n]
  6211. Int[ArcTanh[Plus[Optional[Pattern[a, Blank[]]], Times[Optional[Pattern[b, Blank[]]], Power[Pattern[f, Blank[]], Plus[Optional[Pattern[c, Blank[]]], Times[Optional[Pattern[d, Blank[]]], Pattern[x, Blank[]]]]]]]], Pattern[x, Blank[Symbol]]] := Dist[1/2, Int[Log[1 + a + b*f^(c + d*x)], x], x] - Dist[1/2, Int[Log[1 - a - b*f^(c + d*x)], x], x] /; FreeQ[{a, b, c, d, f}, x]
  6212. Int[ArcCoth[Plus[Optional[Pattern[a, Blank[]]], Times[Optional[Pattern[b, Blank[]]], Power[Pattern[f, Blank[]], Plus[Optional[Pattern[c, Blank[]]], Times[Optional[Pattern[d, Blank[]]], Pattern[x, Blank[]]]]]]]], Pattern[x, Blank[Symbol]]] := Dist[1/2, Int[Log[1 + 1/(a + b*f^(c + d*x))], x], x] - Dist[1/2, Int[Log[1 - 1/(a + b*f^(c + d*x))], x], x] /; FreeQ[{a, b, c, d, f}, x]
  6213. Int[Times[ArcTanh[Plus[Optional[Pattern[a, Blank[]]], Times[Optional[Pattern[b, Blank[]]], Power[Pattern[f, Blank[]], Plus[Optional[Pattern[c, Blank[]]], Times[Optional[Pattern[d, Blank[]]], Pattern[x, Blank[]]]]]]]], Power[Pattern[x, Blank[]], Optional[Pattern[m, Blank[]]]]], Pattern[x, Blank[Symbol]]] := Dist[1/2, Int[x^m*Log[1 + a + b*f^(c + d*x)], x], x] - Dist[1/2, Int[x^m*Log[1 - a - b*f^(c + d*x)], x], x] /; FreeQ[{a, b, c, d, f}, x] && IGtQ[m, 0]
  6214. Int[Times[ArcCoth[Plus[Optional[Pattern[a, Blank[]]], Times[Optional[Pattern[b, Blank[]]], Power[Pattern[f, Blank[]], Plus[Optional[Pattern[c, Blank[]]], Times[Optional[Pattern[d, Blank[]]], Pattern[x, Blank[]]]]]]]], Power[Pattern[x, Blank[]], Optional[Pattern[m, Blank[]]]]], Pattern[x, Blank[Symbol]]] := Dist[1/2, Int[x^m*Log[1 + 1/(a + b*f^(c + d*x))], x], x] - Dist[1/2, Int[x^m*Log[1 - 1/(a + b*f^(c + d*x))], x], x] /; FreeQ[{a, b, c, d, f}, x] && IGtQ[m, 0]
  6215. Int[Times[Power[ArcTanh[Times[Optional[Pattern[c, Blank[]]], Power[Plus[Optional[Pattern[a, Blank[]]], Times[Optional[Pattern[b, Blank[]]], Power[Pattern[x, Blank[]], Optional[Pattern[n, Blank[]]]]]], -1]]], Optional[Pattern[m, Blank[]]]], Optional[Pattern[u, Blank[]]]], Pattern[x, Blank[Symbol]]] := Int[u*ArcCoth[a/c + (b*x^n)/c]^m, x] /; FreeQ[{a, b, c, n, m}, x]
  6216. Int[Times[Power[ArcCoth[Times[Optional[Pattern[c, Blank[]]], Power[Plus[Optional[Pattern[a, Blank[]]], Times[Optional[Pattern[b, Blank[]]], Power[Pattern[x, Blank[]], Optional[Pattern[n, Blank[]]]]]], -1]]], Optional[Pattern[m, Blank[]]]], Optional[Pattern[u, Blank[]]]], Pattern[x, Blank[Symbol]]] := Int[u*ArcTanh[a/c + (b*x^n)/c]^m, x] /; FreeQ[{a, b, c, n, m}, x]
  6217. Int[ArcTanh[Times[Optional[Pattern[c, Blank[]]], Pattern[x, Blank[]], Power[Plus[Optional[Pattern[a, Blank[]]], Times[Optional[Pattern[b, Blank[]]], Power[Pattern[x, Blank[]], 2]]], Rational[-1, 2]]]], Pattern[x, Blank[Symbol]]] := Simp[x*ArcTanh[(c*x)/Sqrt[a + b*x^2]], x] - Dist[c, Int[x/Sqrt[a + b*x^2], x], x] /; FreeQ[{a, b, c}, x] && EqQ[b, c^2]
  6218. Int[ArcCoth[Times[Optional[Pattern[c, Blank[]]], Pattern[x, Blank[]], Power[Plus[Optional[Pattern[a, Blank[]]], Times[Optional[Pattern[b, Blank[]]], Power[Pattern[x, Blank[]], 2]]], Rational[-1, 2]]]], Pattern[x, Blank[Symbol]]] := Simp[x*ArcCoth[(c*x)/Sqrt[a + b*x^2]], x] - Dist[c, Int[x/Sqrt[a + b*x^2], x], x] /; FreeQ[{a, b, c}, x] && EqQ[b, c^2]
  6219. Int[Times[ArcTanh[Times[Optional[Pattern[c, Blank[]]], Pattern[x, Blank[]], Power[Plus[Optional[Pattern[a, Blank[]]], Times[Optional[Pattern[b, Blank[]]], Power[Pattern[x, Blank[]], 2]]], Rational[-1, 2]]]], Power[Pattern[x, Blank[]], -1]], Pattern[x, Blank[Symbol]]] := Simp[ArcTanh[(c*x)/Sqrt[a + b*x^2]]*Log[x], x] - Dist[c, Int[Log[x]/Sqrt[a + b*x^2], x], x] /; FreeQ[{a, b, c}, x] && EqQ[b, c^2]
  6220. Int[Times[ArcCoth[Times[Optional[Pattern[c, Blank[]]], Pattern[x, Blank[]], Power[Plus[Optional[Pattern[a, Blank[]]], Times[Optional[Pattern[b, Blank[]]], Power[Pattern[x, Blank[]], 2]]], Rational[-1, 2]]]], Power[Pattern[x, Blank[]], -1]], Pattern[x, Blank[Symbol]]] := Simp[ArcCoth[(c*x)/Sqrt[a + b*x^2]]*Log[x], x] - Dist[c, Int[Log[x]/Sqrt[a + b*x^2], x], x] /; FreeQ[{a, b, c}, x] && EqQ[b, c^2]
  6221. Int[Times[ArcTanh[Times[Optional[Pattern[c, Blank[]]], Pattern[x, Blank[]], Power[Plus[Optional[Pattern[a, Blank[]]], Times[Optional[Pattern[b, Blank[]]], Power[Pattern[x, Blank[]], 2]]], Rational[-1, 2]]]], Power[Times[Optional[Pattern[d, Blank[]]], Pattern[x, Blank[]]], Optional[Pattern[m, Blank[]]]]], Pattern[x, Blank[Symbol]]] := Simp[((d*x)^(m + 1)*ArcTanh[(c*x)/Sqrt[a + b*x^2]])/(d*(m + 1)), x] - Dist[c/(d*(m + 1)), Int[(d*x)^(m + 1)/Sqrt[a + b*x^2], x], x] /; FreeQ[{a, b, c, d, m}, x] && EqQ[b, c^2] && NeQ[m, -1]
  6222. Int[Times[ArcCoth[Times[Optional[Pattern[c, Blank[]]], Pattern[x, Blank[]], Power[Plus[Optional[Pattern[a, Blank[]]], Times[Optional[Pattern[b, Blank[]]], Power[Pattern[x, Blank[]], 2]]], Rational[-1, 2]]]], Power[Times[Optional[Pattern[d, Blank[]]], Pattern[x, Blank[]]], Optional[Pattern[m, Blank[]]]]], Pattern[x, Blank[Symbol]]] := Simp[((d*x)^(m + 1)*ArcCoth[(c*x)/Sqrt[a + b*x^2]])/(d*(m + 1)), x] - Dist[c/(d*(m + 1)), Int[(d*x)^(m + 1)/Sqrt[a + b*x^2], x], x] /; FreeQ[{a, b, c, d, m}, x] && EqQ[b, c^2] && NeQ[m, -1]
  6223. Int[Times[Power[ArcTanh[Times[Optional[Pattern[c, Blank[]]], Pattern[x, Blank[]], Power[Plus[Optional[Pattern[a, Blank[]]], Times[Optional[Pattern[b, Blank[]]], Power[Pattern[x, Blank[]], 2]]], Rational[-1, 2]]]], -1], Power[Plus[Optional[Pattern[a, Blank[]]], Times[Optional[Pattern[b, Blank[]]], Power[Pattern[x, Blank[]], 2]]], Rational[-1, 2]]], Pattern[x, Blank[Symbol]]] := Simp[(1*Log[ArcTanh[(c*x)/Sqrt[a + b*x^2]]])/c, x] /; FreeQ[{a, b, c}, x] && EqQ[b, c^2]
  6224. Int[Times[Power[ArcCoth[Times[Optional[Pattern[c, Blank[]]], Pattern[x, Blank[]], Power[Plus[Optional[Pattern[a, Blank[]]], Times[Optional[Pattern[b, Blank[]]], Power[Pattern[x, Blank[]], 2]]], Rational[-1, 2]]]], -1], Power[Plus[Optional[Pattern[a, Blank[]]], Times[Optional[Pattern[b, Blank[]]], Power[Pattern[x, Blank[]], 2]]], Rational[-1, 2]]], Pattern[x, Blank[Symbol]]] := -Simp[Log[ArcCoth[(c*x)/Sqrt[a + b*x^2]]]/c, x] /; FreeQ[{a, b, c}, x] && EqQ[b, c^2]
  6225. Int[Times[Power[ArcTanh[Times[Optional[Pattern[c, Blank[]]], Pattern[x, Blank[]], Power[Plus[Optional[Pattern[a, Blank[]]], Times[Optional[Pattern[b, Blank[]]], Power[Pattern[x, Blank[]], 2]]], Rational[-1, 2]]]], Optional[Pattern[m, Blank[]]]], Power[Plus[Optional[Pattern[a, Blank[]]], Times[Optional[Pattern[b, Blank[]]], Power[Pattern[x, Blank[]], 2]]], Rational[-1, 2]]], Pattern[x, Blank[Symbol]]] := Simp[ArcTanh[(c*x)/Sqrt[a + b*x^2]]^(m + 1)/(c*(m + 1)), x] /; FreeQ[{a, b, c, m}, x] && EqQ[b, c^2] && NeQ[m, -1]
  6226. Int[Times[Power[ArcCoth[Times[Optional[Pattern[c, Blank[]]], Pattern[x, Blank[]], Power[Plus[Optional[Pattern[a, Blank[]]], Times[Optional[Pattern[b, Blank[]]], Power[Pattern[x, Blank[]], 2]]], Rational[-1, 2]]]], Optional[Pattern[m, Blank[]]]], Power[Plus[Optional[Pattern[a, Blank[]]], Times[Optional[Pattern[b, Blank[]]], Power[Pattern[x, Blank[]], 2]]], Rational[-1, 2]]], Pattern[x, Blank[Symbol]]] := -Simp[ArcCoth[(c*x)/Sqrt[a + b*x^2]]^(m + 1)/(c*(m + 1)), x] /; FreeQ[{a, b, c, m}, x] && EqQ[b, c^2] && NeQ[m, -1]
  6227. Int[Times[Power[ArcTanh[Times[Optional[Pattern[c, Blank[]]], Pattern[x, Blank[]], Power[Plus[Optional[Pattern[a, Blank[]]], Times[Optional[Pattern[b, Blank[]]], Power[Pattern[x, Blank[]], 2]]], Rational[-1, 2]]]], Optional[Pattern[m, Blank[]]]], Power[Plus[Optional[Pattern[d, Blank[]]], Times[Optional[Pattern[e, Blank[]]], Power[Pattern[x, Blank[]], 2]]], Rational[-1, 2]]], Pattern[x, Blank[Symbol]]] := Dist[Sqrt[a + b*x^2]/Sqrt[d + e*x^2], Int[ArcTanh[(c*x)/Sqrt[a + b*x^2]]^m/Sqrt[a + b*x^2], x], x] /; FreeQ[{a, b, c, d, e, m}, x] && EqQ[b, c^2] && EqQ[b*d - a*e, 0]
  6228. Int[Times[Power[ArcCoth[Times[Optional[Pattern[c, Blank[]]], Pattern[x, Blank[]], Power[Plus[Optional[Pattern[a, Blank[]]], Times[Optional[Pattern[b, Blank[]]], Power[Pattern[x, Blank[]], 2]]], Rational[-1, 2]]]], Optional[Pattern[m, Blank[]]]], Power[Plus[Optional[Pattern[d, Blank[]]], Times[Optional[Pattern[e, Blank[]]], Power[Pattern[x, Blank[]], 2]]], Rational[-1, 2]]], Pattern[x, Blank[Symbol]]] := Dist[Sqrt[a + b*x^2]/Sqrt[d + e*x^2], Int[ArcCoth[(c*x)/Sqrt[a + b*x^2]]^m/Sqrt[a + b*x^2], x], x] /; FreeQ[{a, b, c, d, e, m}, x] && EqQ[b, c^2] && EqQ[b*d - a*e, 0]
  6229. Int[Times[Pattern[u, Blank[]], Power[Pattern[v, Blank[]], Optional[Pattern[n, Blank[]]]]], Pattern[x, Blank[Symbol]]] := With[{tmp = InverseFunctionOfLinear[u, x]}, Dist[(-(Discriminant[v, x]/(4*Coefficient[v, x, 2])))^n/Coefficient[tmp[[1]], x, 1], Subst[Int[SimplifyIntegrand[SubstForInverseFunction[u, tmp, x]*Sech[x]^(2*(n + 1)), x], x], x, tmp], x] /; !FalseQ[tmp] && EqQ[Head[tmp], ArcTanh] && EqQ[Discriminant[v, x]*tmp[[1]]^2 - D[v, x]^2, 0]] /; QuadraticQ[v, x] && ILtQ[n, 0] && PosQ[Discriminant[v, x]] && MatchQ[u, (r_.)*(f_)^(w_) /; FreeQ[f, x]]
  6230. Int[Times[Pattern[u, Blank[]], Power[Pattern[v, Blank[]], Optional[Pattern[n, Blank[]]]]], Pattern[x, Blank[Symbol]]] := With[{tmp = InverseFunctionOfLinear[u, x]}, Dist[(-(Discriminant[v, x]/(4*Coefficient[v, x, 2])))^n/Coefficient[tmp[[1]], x, 1], Subst[Int[SimplifyIntegrand[SubstForInverseFunction[u, tmp, x]*(-Csch[x]^2)^(n + 1), x], x], x, tmp], x] /; !FalseQ[tmp] && EqQ[Head[tmp], ArcCoth] && EqQ[Discriminant[v, x]*tmp[[1]]^2 - D[v, x]^2, 0]] /; QuadraticQ[v, x] && ILtQ[n, 0] && PosQ[Discriminant[v, x]] && MatchQ[u, (r_.)*(f_)^(w_) /; FreeQ[f, x]]
  6231. Int[ArcTanh[Plus[Optional[Pattern[c, Blank[]]], Times[Optional[Pattern[d, Blank[]]], Tanh[Plus[Optional[Pattern[a, Blank[]]], Times[Optional[Pattern[b, Blank[]]], Pattern[x, Blank[]]]]]]]], Pattern[x, Blank[Symbol]]] := Simp[x*ArcTanh[c + d*Tanh[a + b*x]], x] + Dist[b, Int[x/(c - d + c*E^(2*a + 2*b*x)), x], x] /; FreeQ[{a, b, c, d}, x] && EqQ[(c - d)^2, 1]
  6232. Int[ArcCoth[Plus[Optional[Pattern[c, Blank[]]], Times[Optional[Pattern[d, Blank[]]], Tanh[Plus[Optional[Pattern[a, Blank[]]], Times[Optional[Pattern[b, Blank[]]], Pattern[x, Blank[]]]]]]]], Pattern[x, Blank[Symbol]]] := Simp[x*ArcCoth[c + d*Tanh[a + b*x]], x] + Dist[b, Int[x/(c - d + c*E^(2*a + 2*b*x)), x], x] /; FreeQ[{a, b, c, d}, x] && EqQ[(c - d)^2, 1]
  6233. Int[ArcTanh[Plus[Optional[Pattern[c, Blank[]]], Times[Coth[Plus[Optional[Pattern[a, Blank[]]], Times[Optional[Pattern[b, Blank[]]], Pattern[x, Blank[]]]]], Optional[Pattern[d, Blank[]]]]]], Pattern[x, Blank[Symbol]]] := Simp[x*ArcTanh[c + d*Coth[a + b*x]], x] + Dist[b, Int[x/(c - d - c*E^(2*a + 2*b*x)), x], x] /; FreeQ[{a, b, c, d}, x] && EqQ[(c - d)^2, 1]
  6234. Int[ArcCoth[Plus[Optional[Pattern[c, Blank[]]], Times[Coth[Plus[Optional[Pattern[a, Blank[]]], Times[Optional[Pattern[b, Blank[]]], Pattern[x, Blank[]]]]], Optional[Pattern[d, Blank[]]]]]], Pattern[x, Blank[Symbol]]] := Simp[x*ArcCoth[c + d*Coth[a + b*x]], x] + Dist[b, Int[x/(c - d - c*E^(2*a + 2*b*x)), x], x] /; FreeQ[{a, b, c, d}, x] && EqQ[(c - d)^2, 1]
  6235. Int[ArcTanh[Plus[Optional[Pattern[c, Blank[]]], Times[Optional[Pattern[d, Blank[]]], Tanh[Plus[Optional[Pattern[a, Blank[]]], Times[Optional[Pattern[b, Blank[]]], Pattern[x, Blank[]]]]]]]], Pattern[x, Blank[Symbol]]] := Simp[x*ArcTanh[c + d*Tanh[a + b*x]], x] + (Dist[b*(1 - c - d), Int[(x*E^(2*a + 2*b*x))/(1 - c + d + (1 - c - d)*E^(2*a + 2*b*x)), x], x] - Dist[b*(1 + c + d), Int[(x*E^(2*a + 2*b*x))/(1 + c - d + (1 + c + d)*E^(2*a + 2*b*x)), x], x]) /; FreeQ[{a, b, c, d}, x] && NeQ[(c - d)^2, 1]
  6236. Int[ArcCoth[Plus[Optional[Pattern[c, Blank[]]], Times[Optional[Pattern[d, Blank[]]], Tanh[Plus[Optional[Pattern[a, Blank[]]], Times[Optional[Pattern[b, Blank[]]], Pattern[x, Blank[]]]]]]]], Pattern[x, Blank[Symbol]]] := Simp[x*ArcCoth[c + d*Tanh[a + b*x]], x] + (Dist[b*(1 - c - d), Int[(x*E^(2*a + 2*b*x))/(1 - c + d + (1 - c - d)*E^(2*a + 2*b*x)), x], x] - Dist[b*(1 + c + d), Int[(x*E^(2*a + 2*b*x))/(1 + c - d + (1 + c + d)*E^(2*a + 2*b*x)), x], x]) /; FreeQ[{a, b, c, d}, x] && NeQ[(c - d)^2, 1]
  6237. Int[ArcTanh[Plus[Optional[Pattern[c, Blank[]]], Times[Coth[Plus[Optional[Pattern[a, Blank[]]], Times[Optional[Pattern[b, Blank[]]], Pattern[x, Blank[]]]]], Optional[Pattern[d, Blank[]]]]]], Pattern[x, Blank[Symbol]]] := Simp[x*ArcTanh[c + d*Coth[a + b*x]], x] + (-Dist[b*(1 - c - d), Int[(x*E^(2*a + 2*b*x))/(1 - c + d - (1 - c - d)*E^(2*a + 2*b*x)), x], x] + Dist[b*(1 + c + d), Int[(x*E^(2*a + 2*b*x))/(1 + c - d - (1 + c + d)*E^(2*a + 2*b*x)), x], x]) /; FreeQ[{a, b, c, d}, x] && NeQ[(c - d)^2, 1]
  6238. Int[ArcCoth[Plus[Optional[Pattern[c, Blank[]]], Times[Coth[Plus[Optional[Pattern[a, Blank[]]], Times[Optional[Pattern[b, Blank[]]], Pattern[x, Blank[]]]]], Optional[Pattern[d, Blank[]]]]]], Pattern[x, Blank[Symbol]]] := Simp[x*ArcCoth[c + d*Coth[a + b*x]], x] + (-Dist[b*(1 - c - d), Int[(x*E^(2*a + 2*b*x))/(1 - c + d - (1 - c - d)*E^(2*a + 2*b*x)), x], x] + Dist[b*(1 + c + d), Int[(x*E^(2*a + 2*b*x))/(1 + c - d - (1 + c + d)*E^(2*a + 2*b*x)), x], x]) /; FreeQ[{a, b, c, d}, x] && NeQ[(c - d)^2, 1]
  6239. Int[Times[ArcTanh[Plus[Optional[Pattern[c, Blank[]]], Times[Optional[Pattern[d, Blank[]]], Tanh[Plus[Optional[Pattern[a, Blank[]]], Times[Optional[Pattern[b, Blank[]]], Pattern[x, Blank[]]]]]]]], Power[Plus[Optional[Pattern[e, Blank[]]], Times[Optional[Pattern[f, Blank[]]], Pattern[x, Blank[]]]], Optional[Pattern[m, Blank[]]]]], Pattern[x, Blank[Symbol]]] := Simp[((e + f*x)^(m + 1)*ArcTanh[c + d*Tanh[a + b*x]])/(f*(m + 1)), x] + Dist[b/(f*(m + 1)), Int[(e + f*x)^(m + 1)/(c - d + c*E^(2*a + 2*b*x)), x], x] /; FreeQ[{a, b, c, d, e, f}, x] && IGtQ[m, 0] && EqQ[(c - d)^2, 1]
  6240. Int[Times[ArcCoth[Plus[Optional[Pattern[c, Blank[]]], Times[Optional[Pattern[d, Blank[]]], Tanh[Plus[Optional[Pattern[a, Blank[]]], Times[Optional[Pattern[b, Blank[]]], Pattern[x, Blank[]]]]]]]], Power[Plus[Optional[Pattern[e, Blank[]]], Times[Optional[Pattern[f, Blank[]]], Pattern[x, Blank[]]]], Optional[Pattern[m, Blank[]]]]], Pattern[x, Blank[Symbol]]] := Simp[((e + f*x)^(m + 1)*ArcCoth[c + d*Tanh[a + b*x]])/(f*(m + 1)), x] + Dist[b/(f*(m + 1)), Int[(e + f*x)^(m + 1)/(c - d + c*E^(2*a + 2*b*x)), x], x] /; FreeQ[{a, b, c, d, e, f}, x] && IGtQ[m, 0] && EqQ[(c - d)^2, 1]
  6241. Int[Times[ArcTanh[Plus[Optional[Pattern[c, Blank[]]], Times[Coth[Plus[Optional[Pattern[a, Blank[]]], Times[Optional[Pattern[b, Blank[]]], Pattern[x, Blank[]]]]], Optional[Pattern[d, Blank[]]]]]], Power[Plus[Optional[Pattern[e, Blank[]]], Times[Optional[Pattern[f, Blank[]]], Pattern[x, Blank[]]]], Optional[Pattern[m, Blank[]]]]], Pattern[x, Blank[Symbol]]] := Simp[((e + f*x)^(m + 1)*ArcTanh[c + d*Coth[a + b*x]])/(f*(m + 1)), x] + Dist[b/(f*(m + 1)), Int[(e + f*x)^(m + 1)/(c - d - c*E^(2*a + 2*b*x)), x], x] /; FreeQ[{a, b, c, d, e, f}, x] && IGtQ[m, 0] && EqQ[(c - d)^2, 1]
  6242. Int[Times[ArcCoth[Plus[Optional[Pattern[c, Blank[]]], Times[Coth[Plus[Optional[Pattern[a, Blank[]]], Times[Optional[Pattern[b, Blank[]]], Pattern[x, Blank[]]]]], Optional[Pattern[d, Blank[]]]]]], Power[Plus[Optional[Pattern[e, Blank[]]], Times[Optional[Pattern[f, Blank[]]], Pattern[x, Blank[]]]], Optional[Pattern[m, Blank[]]]]], Pattern[x, Blank[Symbol]]] := Simp[((e + f*x)^(m + 1)*ArcCoth[c + d*Coth[a + b*x]])/(f*(m + 1)), x] + Dist[b/(f*(m + 1)), Int[(e + f*x)^(m + 1)/(c - d - c*E^(2*a + 2*b*x)), x], x] /; FreeQ[{a, b, c, d, e, f}, x] && IGtQ[m, 0] && EqQ[(c - d)^2, 1]
  6243. Int[Times[ArcTanh[Plus[Optional[Pattern[c, Blank[]]], Times[Optional[Pattern[d, Blank[]]], Tanh[Plus[Optional[Pattern[a, Blank[]]], Times[Optional[Pattern[b, Blank[]]], Pattern[x, Blank[]]]]]]]], Power[Plus[Optional[Pattern[e, Blank[]]], Times[Optional[Pattern[f, Blank[]]], Pattern[x, Blank[]]]], Optional[Pattern[m, Blank[]]]]], Pattern[x, Blank[Symbol]]] := Simp[((e + f*x)^(m + 1)*ArcTanh[c + d*Tanh[a + b*x]])/(f*(m + 1)), x] + (Dist[(b*(1 - c - d))/(f*(m + 1)), Int[((e + f*x)^(m + 1)*E^(2*a + 2*b*x))/(1 - c + d + (1 - c - d)*E^(2*a + 2*b*x)), x], x] - Dist[(b*(1 + c + d))/(f*(m + 1)), Int[((e + f*x)^(m + 1)*E^(2*a + 2*b*x))/(1 + c - d + (1 + c + d)*E^(2*a + 2*b*x)), x], x]) /; FreeQ[{a, b, c, d, e, f}, x] && IGtQ[m, 0] && NeQ[(c - d)^2, 1]
  6244. Int[Times[ArcCoth[Plus[Optional[Pattern[c, Blank[]]], Times[Optional[Pattern[d, Blank[]]], Tanh[Plus[Optional[Pattern[a, Blank[]]], Times[Optional[Pattern[b, Blank[]]], Pattern[x, Blank[]]]]]]]], Power[Plus[Optional[Pattern[e, Blank[]]], Times[Optional[Pattern[f, Blank[]]], Pattern[x, Blank[]]]], Optional[Pattern[m, Blank[]]]]], Pattern[x, Blank[Symbol]]] := Simp[((e + f*x)^(m + 1)*ArcCoth[c + d*Tanh[a + b*x]])/(f*(m + 1)), x] + (Dist[(b*(1 - c - d))/(f*(m + 1)), Int[((e + f*x)^(m + 1)*E^(2*a + 2*b*x))/(1 - c + d + (1 - c - d)*E^(2*a + 2*b*x)), x], x] - Dist[(b*(1 + c + d))/(f*(m + 1)), Int[((e + f*x)^(m + 1)*E^(2*a + 2*b*x))/(1 + c - d + (1 + c + d)*E^(2*a + 2*b*x)), x], x]) /; FreeQ[{a, b, c, d, e, f}, x] && IGtQ[m, 0] && NeQ[(c - d)^2, 1]
  6245. Int[Times[ArcTanh[Plus[Optional[Pattern[c, Blank[]]], Times[Coth[Plus[Optional[Pattern[a, Blank[]]], Times[Optional[Pattern[b, Blank[]]], Pattern[x, Blank[]]]]], Optional[Pattern[d, Blank[]]]]]], Power[Plus[Optional[Pattern[e, Blank[]]], Times[Optional[Pattern[f, Blank[]]], Pattern[x, Blank[]]]], Optional[Pattern[m, Blank[]]]]], Pattern[x, Blank[Symbol]]] := Simp[((e + f*x)^(m + 1)*ArcTanh[c + d*Coth[a + b*x]])/(f*(m + 1)), x] + (-Dist[(b*(1 - c - d))/(f*(m + 1)), Int[((e + f*x)^(m + 1)*E^(2*a + 2*b*x))/(1 - c + d - (1 - c - d)*E^(2*a + 2*b*x)), x], x] + Dist[(b*(1 + c + d))/(f*(m + 1)), Int[((e + f*x)^(m + 1)*E^(2*a + 2*b*x))/(1 + c - d - (1 + c + d)*E^(2*a + 2*b*x)), x], x]) /; FreeQ[{a, b, c, d, e, f}, x] && IGtQ[m, 0] && NeQ[(c - d)^2, 1]
  6246. Int[Times[ArcCoth[Plus[Optional[Pattern[c, Blank[]]], Times[Coth[Plus[Optional[Pattern[a, Blank[]]], Times[Optional[Pattern[b, Blank[]]], Pattern[x, Blank[]]]]], Optional[Pattern[d, Blank[]]]]]], Power[Plus[Optional[Pattern[e, Blank[]]], Times[Optional[Pattern[f, Blank[]]], Pattern[x, Blank[]]]], Optional[Pattern[m, Blank[]]]]], Pattern[x, Blank[Symbol]]] := Simp[((e + f*x)^(m + 1)*ArcCoth[c + d*Coth[a + b*x]])/(f*(m + 1)), x] + (-Dist[(b*(1 - c - d))/(f*(m + 1)), Int[((e + f*x)^(m + 1)*E^(2*a + 2*b*x))/(1 - c + d - (1 - c - d)*E^(2*a + 2*b*x)), x], x] + Dist[(b*(1 + c + d))/(f*(m + 1)), Int[((e + f*x)^(m + 1)*E^(2*a + 2*b*x))/(1 + c - d - (1 + c + d)*E^(2*a + 2*b*x)), x], x]) /; FreeQ[{a, b, c, d, e, f}, x] && IGtQ[m, 0] && NeQ[(c - d)^2, 1]
  6247. Int[ArcTanh[Tan[Plus[Optional[Pattern[a, Blank[]]], Times[Optional[Pattern[b, Blank[]]], Pattern[x, Blank[]]]]]], Pattern[x, Blank[Symbol]]] := Simp[x*ArcTanh[Tan[a + b*x]], x] - Dist[b, Int[x*Sec[2*a + 2*b*x], x], x] /; FreeQ[{a, b}, x]
  6248. Int[ArcCoth[Tan[Plus[Optional[Pattern[a, Blank[]]], Times[Optional[Pattern[b, Blank[]]], Pattern[x, Blank[]]]]]], Pattern[x, Blank[Symbol]]] := Simp[x*ArcCoth[Tan[a + b*x]], x] - Dist[b, Int[x*Sec[2*a + 2*b*x], x], x] /; FreeQ[{a, b}, x]
  6249. Int[ArcTanh[Cot[Plus[Optional[Pattern[a, Blank[]]], Times[Optional[Pattern[b, Blank[]]], Pattern[x, Blank[]]]]]], Pattern[x, Blank[Symbol]]] := Simp[x*ArcTanh[Cot[a + b*x]], x] - Dist[b, Int[x*Sec[2*a + 2*b*x], x], x] /; FreeQ[{a, b}, x]
  6250. Int[ArcCoth[Cot[Plus[Optional[Pattern[a, Blank[]]], Times[Optional[Pattern[b, Blank[]]], Pattern[x, Blank[]]]]]], Pattern[x, Blank[Symbol]]] := Simp[x*ArcCoth[Cot[a + b*x]], x] - Dist[b, Int[x*Sec[2*a + 2*b*x], x], x] /; FreeQ[{a, b}, x]
  6251. Int[Times[ArcTanh[Tan[Plus[Optional[Pattern[a, Blank[]]], Times[Optional[Pattern[b, Blank[]]], Pattern[x, Blank[]]]]]], Power[Plus[Optional[Pattern[e, Blank[]]], Times[Optional[Pattern[f, Blank[]]], Pattern[x, Blank[]]]], Optional[Pattern[m, Blank[]]]]], Pattern[x, Blank[Symbol]]] := Simp[((e + f*x)^(m + 1)*ArcTanh[Tan[a + b*x]])/(f*(m + 1)), x] - Dist[b/(f*(m + 1)), Int[(e + f*x)^(m + 1)*Sec[2*a + 2*b*x], x], x] /; FreeQ[{a, b, e, f}, x] && IGtQ[m, 0]
  6252. Int[Times[ArcCoth[Tan[Plus[Optional[Pattern[a, Blank[]]], Times[Optional[Pattern[b, Blank[]]], Pattern[x, Blank[]]]]]], Power[Plus[Optional[Pattern[e, Blank[]]], Times[Optional[Pattern[f, Blank[]]], Pattern[x, Blank[]]]], Optional[Pattern[m, Blank[]]]]], Pattern[x, Blank[Symbol]]] := Simp[((e + f*x)^(m + 1)*ArcCoth[Tan[a + b*x]])/(f*(m + 1)), x] - Dist[b/(f*(m + 1)), Int[(e + f*x)^(m + 1)*Sec[2*a + 2*b*x], x], x] /; FreeQ[{a, b, e, f}, x] && IGtQ[m, 0]
  6253. Int[Times[ArcTanh[Cot[Plus[Optional[Pattern[a, Blank[]]], Times[Optional[Pattern[b, Blank[]]], Pattern[x, Blank[]]]]]], Power[Plus[Optional[Pattern[e, Blank[]]], Times[Optional[Pattern[f, Blank[]]], Pattern[x, Blank[]]]], Optional[Pattern[m, Blank[]]]]], Pattern[x, Blank[Symbol]]] := Simp[((e + f*x)^(m + 1)*ArcTanh[Cot[a + b*x]])/(f*(m + 1)), x] - Dist[b/(f*(m + 1)), Int[(e + f*x)^(m + 1)*Sec[2*a + 2*b*x], x], x] /; FreeQ[{a, b, e, f}, x] && IGtQ[m, 0]
  6254. Int[Times[ArcCoth[Cot[Plus[Optional[Pattern[a, Blank[]]], Times[Optional[Pattern[b, Blank[]]], Pattern[x, Blank[]]]]]], Power[Plus[Optional[Pattern[e, Blank[]]], Times[Optional[Pattern[f, Blank[]]], Pattern[x, Blank[]]]], Optional[Pattern[m, Blank[]]]]], Pattern[x, Blank[Symbol]]] := Simp[((e + f*x)^(m + 1)*ArcCoth[Cot[a + b*x]])/(f*(m + 1)), x] - Dist[b/(f*(m + 1)), Int[(e + f*x)^(m + 1)*Sec[2*a + 2*b*x], x], x] /; FreeQ[{a, b, e, f}, x] && IGtQ[m, 0]
  6255. Int[ArcTanh[Plus[Optional[Pattern[c, Blank[]]], Times[Optional[Pattern[d, Blank[]]], Tan[Plus[Optional[Pattern[a, Blank[]]], Times[Optional[Pattern[b, Blank[]]], Pattern[x, Blank[]]]]]]]], Pattern[x, Blank[Symbol]]] := Simp[x*ArcTanh[c + d*Tan[a + b*x]], x] + Dist[I*b, Int[x/(c + I*d + c*E^(2*I*a + 2*I*b*x)), x], x] /; FreeQ[{a, b, c, d}, x] && EqQ[(c + I*d)^2, 1]
  6256. Int[ArcCoth[Plus[Optional[Pattern[c, Blank[]]], Times[Optional[Pattern[d, Blank[]]], Tan[Plus[Optional[Pattern[a, Blank[]]], Times[Optional[Pattern[b, Blank[]]], Pattern[x, Blank[]]]]]]]], Pattern[x, Blank[Symbol]]] := Simp[x*ArcCoth[c + d*Tan[a + b*x]], x] + Dist[I*b, Int[x/(c + I*d + c*E^(2*I*a + 2*I*b*x)), x], x] /; FreeQ[{a, b, c, d}, x] && EqQ[(c + I*d)^2, 1]
  6257. Int[ArcTanh[Plus[Optional[Pattern[c, Blank[]]], Times[Cot[Plus[Optional[Pattern[a, Blank[]]], Times[Optional[Pattern[b, Blank[]]], Pattern[x, Blank[]]]]], Optional[Pattern[d, Blank[]]]]]], Pattern[x, Blank[Symbol]]] := Simp[x*ArcTanh[c + d*Cot[a + b*x]], x] + Dist[I*b, Int[x/(c - I*d - c*E^(2*I*a + 2*I*b*x)), x], x] /; FreeQ[{a, b, c, d}, x] && EqQ[(c - I*d)^2, 1]
  6258. Int[ArcCoth[Plus[Optional[Pattern[c, Blank[]]], Times[Cot[Plus[Optional[Pattern[a, Blank[]]], Times[Optional[Pattern[b, Blank[]]], Pattern[x, Blank[]]]]], Optional[Pattern[d, Blank[]]]]]], Pattern[x, Blank[Symbol]]] := Simp[x*ArcCoth[c + d*Cot[a + b*x]], x] + Dist[I*b, Int[x/(c - I*d - c*E^(2*I*a + 2*I*b*x)), x], x] /; FreeQ[{a, b, c, d}, x] && EqQ[(c - I*d)^2, 1]
  6259. Int[ArcTanh[Plus[Optional[Pattern[c, Blank[]]], Times[Optional[Pattern[d, Blank[]]], Tan[Plus[Optional[Pattern[a, Blank[]]], Times[Optional[Pattern[b, Blank[]]], Pattern[x, Blank[]]]]]]]], Pattern[x, Blank[Symbol]]] := Simp[x*ArcTanh[c + d*Tan[a + b*x]], x] + (-Dist[I*b*(1 + c - I*d), Int[(x*E^(2*I*a + 2*I*b*x))/(1 + c + I*d + (1 + c - I*d)*E^(2*I*a + 2*I*b*x)), x], x] + Dist[I*b*(1 - c + I*d), Int[(x*E^(2*I*a + 2*I*b*x))/(1 - c - I*d + (1 - c + I*d)*E^(2*I*a + 2*I*b*x)), x], x]) /; FreeQ[{a, b, c, d}, x] && NeQ[(c + I*d)^2, 1]
  6260. Int[ArcCoth[Plus[Optional[Pattern[c, Blank[]]], Times[Optional[Pattern[d, Blank[]]], Tan[Plus[Optional[Pattern[a, Blank[]]], Times[Optional[Pattern[b, Blank[]]], Pattern[x, Blank[]]]]]]]], Pattern[x, Blank[Symbol]]] := Simp[x*ArcCoth[c + d*Tan[a + b*x]], x] + (-Dist[I*b*(1 + c - I*d), Int[(x*E^(2*I*a + 2*I*b*x))/(1 + c + I*d + (1 + c - I*d)*E^(2*I*a + 2*I*b*x)), x], x] + Dist[I*b*(1 - c + I*d), Int[(x*E^(2*I*a + 2*I*b*x))/(1 - c - I*d + (1 - c + I*d)*E^(2*I*a + 2*I*b*x)), x], x]) /; FreeQ[{a, b, c, d}, x] && NeQ[(c + I*d)^2, 1]
  6261. Int[ArcTanh[Plus[Optional[Pattern[c, Blank[]]], Times[Cot[Plus[Optional[Pattern[a, Blank[]]], Times[Optional[Pattern[b, Blank[]]], Pattern[x, Blank[]]]]], Optional[Pattern[d, Blank[]]]]]], Pattern[x, Blank[Symbol]]] := Simp[x*ArcTanh[c + d*Cot[a + b*x]], x] + (-Dist[I*b*(1 - c - I*d), Int[(x*E^(2*I*a + 2*I*b*x))/(1 - c + I*d - (1 - c - I*d)*E^(2*I*a + 2*I*b*x)), x], x] + Dist[I*b*(1 + c + I*d), Int[(x*E^(2*I*a + 2*I*b*x))/(1 + c - I*d - (1 + c + I*d)*E^(2*I*a + 2*I*b*x)), x], x]) /; FreeQ[{a, b, c, d}, x] && NeQ[(c - I*d)^2, 1]
  6262. Int[ArcCoth[Plus[Optional[Pattern[c, Blank[]]], Times[Cot[Plus[Optional[Pattern[a, Blank[]]], Times[Optional[Pattern[b, Blank[]]], Pattern[x, Blank[]]]]], Optional[Pattern[d, Blank[]]]]]], Pattern[x, Blank[Symbol]]] := Simp[x*ArcCoth[c + d*Cot[a + b*x]], x] + (-Dist[I*b*(1 - c - I*d), Int[(x*E^(2*I*a + 2*I*b*x))/(1 - c + I*d - (1 - c - I*d)*E^(2*I*a + 2*I*b*x)), x], x] + Dist[I*b*(1 + c + I*d), Int[(x*E^(2*I*a + 2*I*b*x))/(1 + c - I*d - (1 + c + I*d)*E^(2*I*a + 2*I*b*x)), x], x]) /; FreeQ[{a, b, c, d}, x] && NeQ[(c - I*d)^2, 1]
  6263. Int[Times[ArcTanh[Plus[Optional[Pattern[c, Blank[]]], Times[Optional[Pattern[d, Blank[]]], Tan[Plus[Optional[Pattern[a, Blank[]]], Times[Optional[Pattern[b, Blank[]]], Pattern[x, Blank[]]]]]]]], Power[Plus[Optional[Pattern[e, Blank[]]], Times[Optional[Pattern[f, Blank[]]], Pattern[x, Blank[]]]], Optional[Pattern[m, Blank[]]]]], Pattern[x, Blank[Symbol]]] := Simp[((e + f*x)^(m + 1)*ArcTanh[c + d*Tan[a + b*x]])/(f*(m + 1)), x] + Dist[(I*b)/(f*(m + 1)), Int[(e + f*x)^(m + 1)/(c + I*d + c*E^(2*I*a + 2*I*b*x)), x], x] /; FreeQ[{a, b, c, d, e, f}, x] && IGtQ[m, 0] && EqQ[(c + I*d)^2, 1]
  6264. Int[Times[ArcCoth[Plus[Optional[Pattern[c, Blank[]]], Times[Optional[Pattern[d, Blank[]]], Tan[Plus[Optional[Pattern[a, Blank[]]], Times[Optional[Pattern[b, Blank[]]], Pattern[x, Blank[]]]]]]]], Power[Plus[Optional[Pattern[e, Blank[]]], Times[Optional[Pattern[f, Blank[]]], Pattern[x, Blank[]]]], Optional[Pattern[m, Blank[]]]]], Pattern[x, Blank[Symbol]]] := Simp[((e + f*x)^(m + 1)*ArcCoth[c + d*Tan[a + b*x]])/(f*(m + 1)), x] + Dist[(I*b)/(f*(m + 1)), Int[(e + f*x)^(m + 1)/(c + I*d + c*E^(2*I*a + 2*I*b*x)), x], x] /; FreeQ[{a, b, c, d, e, f}, x] && IGtQ[m, 0] && EqQ[(c + I*d)^2, 1]
  6265. Int[Times[ArcTanh[Plus[Optional[Pattern[c, Blank[]]], Times[Cot[Plus[Optional[Pattern[a, Blank[]]], Times[Optional[Pattern[b, Blank[]]], Pattern[x, Blank[]]]]], Optional[Pattern[d, Blank[]]]]]], Power[Plus[Optional[Pattern[e, Blank[]]], Times[Optional[Pattern[f, Blank[]]], Pattern[x, Blank[]]]], Optional[Pattern[m, Blank[]]]]], Pattern[x, Blank[Symbol]]] := Simp[((e + f*x)^(m + 1)*ArcTanh[c + d*Cot[a + b*x]])/(f*(m + 1)), x] + Dist[(I*b)/(f*(m + 1)), Int[(e + f*x)^(m + 1)/(c - I*d - c*E^(2*I*a + 2*I*b*x)), x], x] /; FreeQ[{a, b, c, d, e, f}, x] && IGtQ[m, 0] && EqQ[(c - I*d)^2, 1]
  6266. Int[Times[ArcCoth[Plus[Optional[Pattern[c, Blank[]]], Times[Cot[Plus[Optional[Pattern[a, Blank[]]], Times[Optional[Pattern[b, Blank[]]], Pattern[x, Blank[]]]]], Optional[Pattern[d, Blank[]]]]]], Power[Plus[Optional[Pattern[e, Blank[]]], Times[Optional[Pattern[f, Blank[]]], Pattern[x, Blank[]]]], Optional[Pattern[m, Blank[]]]]], Pattern[x, Blank[Symbol]]] := Simp[((e + f*x)^(m + 1)*ArcCoth[c + d*Cot[a + b*x]])/(f*(m + 1)), x] + Dist[(I*b)/(f*(m + 1)), Int[(e + f*x)^(m + 1)/(c - I*d - c*E^(2*I*a + 2*I*b*x)), x], x] /; FreeQ[{a, b, c, d, e, f}, x] && IGtQ[m, 0] && EqQ[(c - I*d)^2, 1]
  6267. Int[Times[ArcTanh[Plus[Optional[Pattern[c, Blank[]]], Times[Optional[Pattern[d, Blank[]]], Tan[Plus[Optional[Pattern[a, Blank[]]], Times[Optional[Pattern[b, Blank[]]], Pattern[x, Blank[]]]]]]]], Power[Plus[Optional[Pattern[e, Blank[]]], Times[Optional[Pattern[f, Blank[]]], Pattern[x, Blank[]]]], Optional[Pattern[m, Blank[]]]]], Pattern[x, Blank[Symbol]]] := Simp[((e + f*x)^(m + 1)*ArcTanh[c + d*Tan[a + b*x]])/(f*(m + 1)), x] + (-Dist[(I*b*(1 + c - I*d))/(f*(m + 1)), Int[((e + f*x)^(m + 1)*E^(2*I*a + 2*I*b*x))/(1 + c + I*d + (1 + c - I*d)*E^(2*I*a + 2*I*b*x)), x], x] + Dist[(I*b*(1 - c + I*d))/(f*(m + 1)), Int[((e + f*x)^(m + 1)*E^(2*I*a + 2*I*b*x))/(1 - c - I*d + (1 - c + I*d)*E^(2*I*a + 2*I*b*x)), x], x]) /; FreeQ[{a, b, c, d, e, f}, x] && IGtQ[m, 0] && NeQ[(c + I*d)^2, 1]
  6268. Int[Times[ArcCoth[Plus[Optional[Pattern[c, Blank[]]], Times[Optional[Pattern[d, Blank[]]], Tan[Plus[Optional[Pattern[a, Blank[]]], Times[Optional[Pattern[b, Blank[]]], Pattern[x, Blank[]]]]]]]], Power[Plus[Optional[Pattern[e, Blank[]]], Times[Optional[Pattern[f, Blank[]]], Pattern[x, Blank[]]]], Optional[Pattern[m, Blank[]]]]], Pattern[x, Blank[Symbol]]] := Simp[((e + f*x)^(m + 1)*ArcCoth[c + d*Tan[a + b*x]])/(f*(m + 1)), x] + (-Dist[(I*b*(1 + c - I*d))/(f*(m + 1)), Int[((e + f*x)^(m + 1)*E^(2*I*a + 2*I*b*x))/(1 + c + I*d + (1 + c - I*d)*E^(2*I*a + 2*I*b*x)), x], x] + Dist[(I*b*(1 - c + I*d))/(f*(m + 1)), Int[((e + f*x)^(m + 1)*E^(2*I*a + 2*I*b*x))/(1 - c - I*d + (1 - c + I*d)*E^(2*I*a + 2*I*b*x)), x], x]) /; FreeQ[{a, b, c, d, e, f}, x] && IGtQ[m, 0] && NeQ[(c + I*d)^2, 1]
  6269. Int[Times[ArcTanh[Plus[Optional[Pattern[c, Blank[]]], Times[Cot[Plus[Optional[Pattern[a, Blank[]]], Times[Optional[Pattern[b, Blank[]]], Pattern[x, Blank[]]]]], Optional[Pattern[d, Blank[]]]]]], Power[Plus[Optional[Pattern[e, Blank[]]], Times[Optional[Pattern[f, Blank[]]], Pattern[x, Blank[]]]], Optional[Pattern[m, Blank[]]]]], Pattern[x, Blank[Symbol]]] := Simp[((e + f*x)^(m + 1)*ArcTanh[c + d*Cot[a + b*x]])/(f*(m + 1)), x] + (-Dist[(I*b*(1 - c - I*d))/(f*(m + 1)), Int[((e + f*x)^(m + 1)*E^(2*I*a + 2*I*b*x))/(1 - c + I*d - (1 - c - I*d)*E^(2*I*a + 2*I*b*x)), x], x] + Dist[(I*b*(1 + c + I*d))/(f*(m + 1)), Int[((e + f*x)^(m + 1)*E^(2*I*a + 2*I*b*x))/(1 + c - I*d - (1 + c + I*d)*E^(2*I*a + 2*I*b*x)), x], x]) /; FreeQ[{a, b, c, d, e, f}, x] && IGtQ[m, 0] && NeQ[(c - I*d)^2, 1]
  6270. Int[Times[ArcCoth[Plus[Optional[Pattern[c, Blank[]]], Times[Cot[Plus[Optional[Pattern[a, Blank[]]], Times[Optional[Pattern[b, Blank[]]], Pattern[x, Blank[]]]]], Optional[Pattern[d, Blank[]]]]]], Power[Plus[Optional[Pattern[e, Blank[]]], Times[Optional[Pattern[f, Blank[]]], Pattern[x, Blank[]]]], Optional[Pattern[m, Blank[]]]]], Pattern[x, Blank[Symbol]]] := Simp[((e + f*x)^(m + 1)*ArcCoth[c + d*Cot[a + b*x]])/(f*(m + 1)), x] + (-Dist[(I*b*(1 - c - I*d))/(f*(m + 1)), Int[((e + f*x)^(m + 1)*E^(2*I*a + 2*I*b*x))/(1 - c + I*d - (1 - c - I*d)*E^(2*I*a + 2*I*b*x)), x], x] + Dist[(I*b*(1 + c + I*d))/(f*(m + 1)), Int[((e + f*x)^(m + 1)*E^(2*I*a + 2*I*b*x))/(1 + c - I*d - (1 + c + I*d)*E^(2*I*a + 2*I*b*x)), x], x]) /; FreeQ[{a, b, c, d, e, f}, x] && IGtQ[m, 0] && NeQ[(c - I*d)^2, 1]
  6271. Int[ArcTanh[Pattern[u, Blank[]]], Pattern[x, Blank[Symbol]]] := Simp[x*ArcTanh[u], x] - Int[SimplifyIntegrand[(x*D[u, x])/(1 - u^2), x], x] /; InverseFunctionFreeQ[u, x]
  6272. Int[ArcCoth[Pattern[u, Blank[]]], Pattern[x, Blank[Symbol]]] := Simp[x*ArcCoth[u], x] - Int[SimplifyIntegrand[(x*D[u, x])/(1 - u^2), x], x] /; InverseFunctionFreeQ[u, x]
  6273. Int[Times[Plus[Optional[Pattern[a, Blank[]]], Times[ArcTanh[Pattern[u, Blank[]]], Optional[Pattern[b, Blank[]]]]], Power[Plus[Optional[Pattern[c, Blank[]]], Times[Optional[Pattern[d, Blank[]]], Pattern[x, Blank[]]]], Optional[Pattern[m, Blank[]]]]], Pattern[x, Blank[Symbol]]] := Simp[((c + d*x)^(m + 1)*(a + b*ArcTanh[u]))/(d*(m + 1)), x] - Dist[b/(d*(m + 1)), Int[SimplifyIntegrand[((c + d*x)^(m + 1)*D[u, x])/(1 - u^2), x], x], x] /; FreeQ[{a, b, c, d, m}, x] && NeQ[m, -1] && InverseFunctionFreeQ[u, x] && !FunctionOfQ[(c + d*x)^(m + 1), u, x] && FalseQ[PowerVariableExpn[u, m + 1, x]]
  6274. Int[Times[Plus[Optional[Pattern[a, Blank[]]], Times[ArcCoth[Pattern[u, Blank[]]], Optional[Pattern[b, Blank[]]]]], Power[Plus[Optional[Pattern[c, Blank[]]], Times[Optional[Pattern[d, Blank[]]], Pattern[x, Blank[]]]], Optional[Pattern[m, Blank[]]]]], Pattern[x, Blank[Symbol]]] := Simp[((c + d*x)^(m + 1)*(a + b*ArcCoth[u]))/(d*(m + 1)), x] - Dist[b/(d*(m + 1)), Int[SimplifyIntegrand[((c + d*x)^(m + 1)*D[u, x])/(1 - u^2), x], x], x] /; FreeQ[{a, b, c, d, m}, x] && NeQ[m, -1] && InverseFunctionFreeQ[u, x] && !FunctionOfQ[(c + d*x)^(m + 1), u, x] && FalseQ[PowerVariableExpn[u, m + 1, x]]
  6275. Int[Times[Plus[Optional[Pattern[a, Blank[]]], Times[ArcTanh[Pattern[u, Blank[]]], Optional[Pattern[b, Blank[]]]]], Pattern[v, Blank[]]], Pattern[x, Blank[Symbol]]] := With[{w = IntHide[v, x]}, Dist[a + b*ArcTanh[u], w, x] - Dist[b, Int[SimplifyIntegrand[(w*D[u, x])/(1 - u^2), x], x], x] /; InverseFunctionFreeQ[w, x]] /; FreeQ[{a, b}, x] && InverseFunctionFreeQ[u, x] && !MatchQ[v, ((c_.) + (d_.)*x)^(m_.) /; FreeQ[{c, d, m}, x]] && FalseQ[FunctionOfLinear[v*(a + b*ArcTanh[u]), x]]
  6276. Int[Times[Plus[Optional[Pattern[a, Blank[]]], Times[ArcCoth[Pattern[u, Blank[]]], Optional[Pattern[b, Blank[]]]]], Pattern[v, Blank[]]], Pattern[x, Blank[Symbol]]] := With[{w = IntHide[v, x]}, Dist[a + b*ArcCoth[u], w, x] - Dist[b, Int[SimplifyIntegrand[(w*D[u, x])/(1 - u^2), x], x], x] /; InverseFunctionFreeQ[w, x]] /; FreeQ[{a, b}, x] && InverseFunctionFreeQ[u, x] && !MatchQ[v, ((c_.) + (d_.)*x)^(m_.) /; FreeQ[{c, d, m}, x]] && FalseQ[FunctionOfLinear[v*(a + b*ArcCoth[u]), x]]
  6277. Int[ArcSech[Times[Optional[Pattern[c, Blank[]]], Pattern[x, Blank[]]]], Pattern[x, Blank[Symbol]]] := Simp[x*ArcSech[c*x], x] + Dist[Sqrt[1 + c*x]*Sqrt[1/(1 + c*x)], Int[1/Sqrt[1 - c^2*x^2], x], x] /; FreeQ[c, x]
  6278. Int[ArcCsch[Times[Optional[Pattern[c, Blank[]]], Pattern[x, Blank[]]]], Pattern[x, Blank[Symbol]]] := Simp[x*ArcCsch[c*x], x] + Dist[1/c, Int[1/(x*Sqrt[1 + 1/(c^2*x^2)]), x], x] /; FreeQ[c, x]
  6279. Int[Power[Plus[Optional[Pattern[a, Blank[]]], Times[ArcSech[Times[Optional[Pattern[c, Blank[]]], Pattern[x, Blank[]]]], Optional[Pattern[b, Blank[]]]]], Pattern[n, Blank[]]], Pattern[x, Blank[Symbol]]] := -Dist[c^(-1), Subst[Int[(a + b*x)^n*Sech[x]*Tanh[x], x], x, ArcSech[c*x]], x] /; FreeQ[{a, b, c, n}, x] && IGtQ[n, 0]
  6280. Int[Power[Plus[Optional[Pattern[a, Blank[]]], Times[ArcCsch[Times[Optional[Pattern[c, Blank[]]], Pattern[x, Blank[]]]], Optional[Pattern[b, Blank[]]]]], Pattern[n, Blank[]]], Pattern[x, Blank[Symbol]]] := -Dist[c^(-1), Subst[Int[(a + b*x)^n*Csch[x]*Coth[x], x], x, ArcCsch[c*x]], x] /; FreeQ[{a, b, c, n}, x] && IGtQ[n, 0]
  6281. Int[Times[Plus[Optional[Pattern[a, Blank[]]], Times[ArcSech[Times[Optional[Pattern[c, Blank[]]], Pattern[x, Blank[]]]], Optional[Pattern[b, Blank[]]]]], Power[Pattern[x, Blank[]], -1]], Pattern[x, Blank[Symbol]]] := -Subst[Int[(a + b*ArcCosh[x/c])/x, x], x, 1/x] /; FreeQ[{a, b, c}, x]
  6282. Int[Times[Plus[Optional[Pattern[a, Blank[]]], Times[ArcCsch[Times[Optional[Pattern[c, Blank[]]], Pattern[x, Blank[]]]], Optional[Pattern[b, Blank[]]]]], Power[Pattern[x, Blank[]], -1]], Pattern[x, Blank[Symbol]]] := -Subst[Int[(a + b*ArcSinh[x/c])/x, x], x, 1/x] /; FreeQ[{a, b, c}, x]
  6283. Int[Times[Plus[Optional[Pattern[a, Blank[]]], Times[ArcSech[Times[Optional[Pattern[c, Blank[]]], Pattern[x, Blank[]]]], Optional[Pattern[b, Blank[]]]]], Power[Times[Optional[Pattern[d, Blank[]]], Pattern[x, Blank[]]], Optional[Pattern[m, Blank[]]]]], Pattern[x, Blank[Symbol]]] := Simp[((d*x)^(m + 1)*(a + b*ArcSech[c*x]))/(d*(m + 1)), x] + Dist[(b*Sqrt[1 + c*x]*Sqrt[1/(1 + c*x)])/(m + 1), Int[(d*x)^m/(Sqrt[1 - c*x]*Sqrt[1 + c*x]), x], x] /; FreeQ[{a, b, c, d, m}, x] && NeQ[m, -1]
  6284. Int[Times[Plus[Optional[Pattern[a, Blank[]]], Times[ArcCsch[Times[Optional[Pattern[c, Blank[]]], Pattern[x, Blank[]]]], Optional[Pattern[b, Blank[]]]]], Power[Times[Optional[Pattern[d, Blank[]]], Pattern[x, Blank[]]], Optional[Pattern[m, Blank[]]]]], Pattern[x, Blank[Symbol]]] := Simp[((d*x)^(m + 1)*(a + b*ArcCsch[c*x]))/(d*(m + 1)), x] + Dist[(b*d)/(c*(m + 1)), Int[(d*x)^(m - 1)/Sqrt[1 + 1/(c^2*x^2)], x], x] /; FreeQ[{a, b, c, d, m}, x] && NeQ[m, -1]
  6285. Int[Times[Power[Plus[Optional[Pattern[a, Blank[]]], Times[ArcSech[Times[Optional[Pattern[c, Blank[]]], Pattern[x, Blank[]]]], Optional[Pattern[b, Blank[]]]]], Pattern[n, Blank[]]], Power[Pattern[x, Blank[]], Optional[Pattern[m, Blank[]]]]], Pattern[x, Blank[Symbol]]] := -Dist[(c^(m + 1))^(-1), Subst[Int[(a + b*x)^n*Sech[x]^(m + 1)*Tanh[x], x], x, ArcSech[c*x]], x] /; FreeQ[{a, b, c}, x] && IntegerQ[n] && IntegerQ[m] && (GtQ[n, 0] || LtQ[m, -1])
  6286. Int[Times[Power[Plus[Optional[Pattern[a, Blank[]]], Times[ArcCsch[Times[Optional[Pattern[c, Blank[]]], Pattern[x, Blank[]]]], Optional[Pattern[b, Blank[]]]]], Pattern[n, Blank[]]], Power[Pattern[x, Blank[]], Optional[Pattern[m, Blank[]]]]], Pattern[x, Blank[Symbol]]] := -Dist[(c^(m + 1))^(-1), Subst[Int[(a + b*x)^n*Csch[x]^(m + 1)*Coth[x], x], x, ArcCsch[c*x]], x] /; FreeQ[{a, b, c}, x] && IntegerQ[n] && IntegerQ[m] && (GtQ[n, 0] || LtQ[m, -1])
  6287. Int[Times[Plus[Optional[Pattern[a, Blank[]]], Times[ArcSech[Times[Optional[Pattern[c, Blank[]]], Pattern[x, Blank[]]]], Optional[Pattern[b, Blank[]]]]], Power[Plus[Optional[Pattern[d, Blank[]]], Times[Optional[Pattern[e, Blank[]]], Pattern[x, Blank[]]]], -1]], Pattern[x, Blank[Symbol]]] := Simp[((a + b*ArcSech[c*x])*Log[1 + (e - Sqrt[-(c^2*d^2) + e^2])/(c*d*E^ArcSech[c*x])])/e, x] + (Dist[b/e, Int[(Sqrt[(1 - c*x)/(1 + c*x)]*Log[1 + (e - Sqrt[-(c^2*d^2) + e^2])/(c*d*E^ArcSech[c*x])])/(x*(1 - c*x)), x], x] + Dist[b/e, Int[(Sqrt[(1 - c*x)/(1 + c*x)]*Log[1 + (e + Sqrt[-(c^2*d^2) + e^2])/(c*d*E^ArcSech[c*x])])/(x*(1 - c*x)), x], x] - Dist[b/e, Int[(Sqrt[(1 - c*x)/(1 + c*x)]*Log[1 + 1/E^(2*ArcSech[c*x])])/(x*(1 - c*x)), x], x] + Simp[((a + b*ArcSech[c*x])*Log[1 + (e + Sqrt[-(c^2*d^2) + e^2])/(c*d*E^ArcSech[c*x])])/e, x] - Simp[((a + b*ArcSech[c*x])*Log[1 + 1/E^(2*ArcSech[c*x])])/e, x]) /; FreeQ[{a, b, c, d, e}, x]
  6288. Int[Times[Plus[Optional[Pattern[a, Blank[]]], Times[ArcSech[Times[Optional[Pattern[c, Blank[]]], Pattern[x, Blank[]]]], Optional[Pattern[b, Blank[]]]]], Power[Plus[Optional[Pattern[d, Blank[]]], Times[Optional[Pattern[e, Blank[]]], Pattern[x, Blank[]]]], Optional[Pattern[m, Blank[]]]]], Pattern[x, Blank[Symbol]]] := Simp[((d + e*x)^(m + 1)*(a + b*ArcSech[c*x]))/(e*(m + 1)), x] + Dist[(b*Sqrt[1 + c*x]*Sqrt[1/(1 + c*x)])/(e*(m + 1)), Int[(d + e*x)^(m + 1)/(x*Sqrt[1 - c^2*x^2]), x], x] /; FreeQ[{a, b, c, d, e, m}, x] && NeQ[m, -1]
  6289. Int[Times[Plus[Optional[Pattern[a, Blank[]]], Times[ArcCsch[Times[Optional[Pattern[c, Blank[]]], Pattern[x, Blank[]]]], Optional[Pattern[b, Blank[]]]]], Power[Plus[Optional[Pattern[d, Blank[]]], Times[Optional[Pattern[e, Blank[]]], Pattern[x, Blank[]]]], -1]], Pattern[x, Blank[Symbol]]] := Simp[((a + b*ArcCsch[c*x])*Log[1 - ((e - Sqrt[c^2*d^2 + e^2])*E^ArcCsch[c*x])/(c*d)])/e, x] + (Dist[b/(c*e), Int[Log[1 - ((e - Sqrt[c^2*d^2 + e^2])*E^ArcCsch[c*x])/(c*d)]/(x^2*Sqrt[1 + 1/(c^2*x^2)]), x], x] + Dist[b/(c*e), Int[Log[1 - ((e + Sqrt[c^2*d^2 + e^2])*E^ArcCsch[c*x])/(c*d)]/(x^2*Sqrt[1 + 1/(c^2*x^2)]), x], x] - Dist[b/(c*e), Int[Log[1 - E^(2*ArcCsch[c*x])]/(x^2*Sqrt[1 + 1/(c^2*x^2)]), x], x] + Simp[((a + b*ArcCsch[c*x])*Log[1 - ((e + Sqrt[c^2*d^2 + e^2])*E^ArcCsch[c*x])/(c*d)])/e, x] - Simp[((a + b*ArcCsch[c*x])*Log[1 - E^(2*ArcCsch[c*x])])/e, x]) /; FreeQ[{a, b, c, d, e}, x]
  6290. Int[Times[Plus[Optional[Pattern[a, Blank[]]], Times[ArcCsch[Times[Optional[Pattern[c, Blank[]]], Pattern[x, Blank[]]]], Optional[Pattern[b, Blank[]]]]], Power[Plus[Optional[Pattern[d, Blank[]]], Times[Optional[Pattern[e, Blank[]]], Pattern[x, Blank[]]]], Optional[Pattern[m, Blank[]]]]], Pattern[x, Blank[Symbol]]] := Simp[((d + e*x)^(m + 1)*(a + b*ArcCsch[c*x]))/(e*(m + 1)), x] + Dist[b/(c*e*(m + 1)), Int[(d + e*x)^(m + 1)/(x^2*Sqrt[1 + 1/(c^2*x^2)]), x], x] /; FreeQ[{a, b, c, d, e, m}, x] && NeQ[m, -1]
  6291. Int[Times[Plus[Optional[Pattern[a, Blank[]]], Times[ArcSech[Times[Optional[Pattern[c, Blank[]]], Pattern[x, Blank[]]]], Optional[Pattern[b, Blank[]]]]], Power[Plus[Optional[Pattern[d, Blank[]]], Times[Optional[Pattern[e, Blank[]]], Power[Pattern[x, Blank[]], 2]]], Optional[Pattern[p, Blank[]]]]], Pattern[x, Blank[Symbol]]] := With[{u = IntHide[(d + e*x^2)^p, x]}, Dist[a + b*ArcSech[c*x], u, x] + Dist[b*Sqrt[1 + c*x]*Sqrt[1/(1 + c*x)], Int[SimplifyIntegrand[u/(x*Sqrt[1 - c*x]*Sqrt[1 + c*x]), x], x], x]] /; FreeQ[{a, b, c, d, e}, x] && (IGtQ[p, 0] || ILtQ[p + 1/2, 0])
  6292. Int[Times[Plus[Optional[Pattern[a, Blank[]]], Times[ArcCsch[Times[Optional[Pattern[c, Blank[]]], Pattern[x, Blank[]]]], Optional[Pattern[b, Blank[]]]]], Power[Plus[Optional[Pattern[d, Blank[]]], Times[Optional[Pattern[e, Blank[]]], Power[Pattern[x, Blank[]], 2]]], Optional[Pattern[p, Blank[]]]]], Pattern[x, Blank[Symbol]]] := With[{u = IntHide[(d + e*x^2)^p, x]}, Dist[a + b*ArcCsch[c*x], u, x] - Dist[(b*c*x)/Sqrt[-(c^2*x^2)], Int[SimplifyIntegrand[u/(x*Sqrt[-1 - c^2*x^2]), x], x], x]] /; FreeQ[{a, b, c, d, e}, x] && (IGtQ[p, 0] || ILtQ[p + 1/2, 0])
  6293. Int[Times[Power[Plus[Optional[Pattern[a, Blank[]]], Times[ArcSech[Times[Optional[Pattern[c, Blank[]]], Pattern[x, Blank[]]]], Optional[Pattern[b, Blank[]]]]], Optional[Pattern[n, Blank[]]]], Power[Plus[Optional[Pattern[d, Blank[]]], Times[Optional[Pattern[e, Blank[]]], Power[Pattern[x, Blank[]], 2]]], Optional[Pattern[p, Blank[]]]]], Pattern[x, Blank[Symbol]]] := -Subst[Int[((e + d*x^2)^p*(a + b*ArcCosh[x/c])^n)/x^(2*(p + 1)), x], x, 1/x] /; FreeQ[{a, b, c, d, e, n}, x] && IGtQ[n, 0] && IntegerQ[p]
  6294. Int[Times[Power[Plus[Optional[Pattern[a, Blank[]]], Times[ArcCsch[Times[Optional[Pattern[c, Blank[]]], Pattern[x, Blank[]]]], Optional[Pattern[b, Blank[]]]]], Optional[Pattern[n, Blank[]]]], Power[Plus[Optional[Pattern[d, Blank[]]], Times[Optional[Pattern[e, Blank[]]], Power[Pattern[x, Blank[]], 2]]], Optional[Pattern[p, Blank[]]]]], Pattern[x, Blank[Symbol]]] := -Subst[Int[((e + d*x^2)^p*(a + b*ArcSinh[x/c])^n)/x^(2*(p + 1)), x], x, 1/x] /; FreeQ[{a, b, c, d, e, n}, x] && IGtQ[n, 0] && IntegerQ[p]
  6295. Int[Times[Power[Plus[Optional[Pattern[a, Blank[]]], Times[ArcSech[Times[Optional[Pattern[c, Blank[]]], Pattern[x, Blank[]]]], Optional[Pattern[b, Blank[]]]]], Optional[Pattern[n, Blank[]]]], Power[Plus[Optional[Pattern[d, Blank[]]], Times[Optional[Pattern[e, Blank[]]], Power[Pattern[x, Blank[]], 2]]], Pattern[p, Blank[]]]], Pattern[x, Blank[Symbol]]] := -Dist[Sqrt[x^2]/x, Subst[Int[((e + d*x^2)^p*(a + b*ArcCosh[x/c])^n)/x^(2*(p + 1)), x], x, 1/x], x] /; FreeQ[{a, b, c, d, e, n}, x] && IGtQ[n, 0] && EqQ[c^2*d + e, 0] && IntegerQ[p + 1/2] && GtQ[e, 0] && LtQ[d, 0]
  6296. Int[Times[Power[Plus[Optional[Pattern[a, Blank[]]], Times[ArcCsch[Times[Optional[Pattern[c, Blank[]]], Pattern[x, Blank[]]]], Optional[Pattern[b, Blank[]]]]], Optional[Pattern[n, Blank[]]]], Power[Plus[Optional[Pattern[d, Blank[]]], Times[Optional[Pattern[e, Blank[]]], Power[Pattern[x, Blank[]], 2]]], Pattern[p, Blank[]]]], Pattern[x, Blank[Symbol]]] := -Dist[Sqrt[x^2]/x, Subst[Int[((e + d*x^2)^p*(a + b*ArcSinh[x/c])^n)/x^(2*(p + 1)), x], x, 1/x], x] /; FreeQ[{a, b, c, d, e, n}, x] && IGtQ[n, 0] && EqQ[e - c^2*d, 0] && IntegerQ[p + 1/2] && GtQ[e, 0] && LtQ[d, 0]
  6297. Int[Times[Power[Plus[Optional[Pattern[a, Blank[]]], Times[ArcSech[Times[Optional[Pattern[c, Blank[]]], Pattern[x, Blank[]]]], Optional[Pattern[b, Blank[]]]]], Optional[Pattern[n, Blank[]]]], Power[Plus[Optional[Pattern[d, Blank[]]], Times[Optional[Pattern[e, Blank[]]], Power[Pattern[x, Blank[]], 2]]], Pattern[p, Blank[]]]], Pattern[x, Blank[Symbol]]] := -Dist[Sqrt[d + e*x^2]/(x*Sqrt[e + d/x^2]), Subst[Int[((e + d*x^2)^p*(a + b*ArcCosh[x/c])^n)/x^(2*(p + 1)), x], x, 1/x], x] /; FreeQ[{a, b, c, d, e, n}, x] && IGtQ[n, 0] && EqQ[c^2*d + e, 0] && IntegerQ[p + 1/2] && !(GtQ[e, 0] && LtQ[d, 0])
  6298. Int[Times[Power[Plus[Optional[Pattern[a, Blank[]]], Times[ArcCsch[Times[Optional[Pattern[c, Blank[]]], Pattern[x, Blank[]]]], Optional[Pattern[b, Blank[]]]]], Optional[Pattern[n, Blank[]]]], Power[Plus[Optional[Pattern[d, Blank[]]], Times[Optional[Pattern[e, Blank[]]], Power[Pattern[x, Blank[]], 2]]], Pattern[p, Blank[]]]], Pattern[x, Blank[Symbol]]] := -Dist[Sqrt[d + e*x^2]/(x*Sqrt[e + d/x^2]), Subst[Int[((e + d*x^2)^p*(a + b*ArcSinh[x/c])^n)/x^(2*(p + 1)), x], x, 1/x], x] /; FreeQ[{a, b, c, d, e, n}, x] && IGtQ[n, 0] && EqQ[e - c^2*d, 0] && IntegerQ[p + 1/2] && !(GtQ[e, 0] && LtQ[d, 0])
  6299. Int[Times[Plus[Optional[Pattern[a, Blank[]]], Times[ArcSech[Times[Optional[Pattern[c, Blank[]]], Pattern[x, Blank[]]]], Optional[Pattern[b, Blank[]]]]], Pattern[x, Blank[]], Power[Plus[Optional[Pattern[d, Blank[]]], Times[Optional[Pattern[e, Blank[]]], Power[Pattern[x, Blank[]], 2]]], Optional[Pattern[p, Blank[]]]]], Pattern[x, Blank[Symbol]]] := Simp[((d + e*x^2)^(p + 1)*(a + b*ArcSech[c*x]))/(2*e*(p + 1)), x] + Dist[(b*Sqrt[1 + c*x]*Sqrt[1/(1 + c*x)])/(2*e*(p + 1)), Int[(d + e*x^2)^(p + 1)/(x*Sqrt[1 - c*x]*Sqrt[1 + c*x]), x], x] /; FreeQ[{a, b, c, d, e, p}, x] && NeQ[p, -1]
  6300. Int[Times[Plus[Optional[Pattern[a, Blank[]]], Times[ArcCsch[Times[Optional[Pattern[c, Blank[]]], Pattern[x, Blank[]]]], Optional[Pattern[b, Blank[]]]]], Pattern[x, Blank[]], Power[Plus[Optional[Pattern[d, Blank[]]], Times[Optional[Pattern[e, Blank[]]], Power[Pattern[x, Blank[]], 2]]], Optional[Pattern[p, Blank[]]]]], Pattern[x, Blank[Symbol]]] := Simp[((d + e*x^2)^(p + 1)*(a + b*ArcCsch[c*x]))/(2*e*(p + 1)), x] - Dist[(b*c*x)/(2*e*(p + 1)*Sqrt[-(c^2*x^2)]), Int[(d + e*x^2)^(p + 1)/(x*Sqrt[-1 - c^2*x^2]), x], x] /; FreeQ[{a, b, c, d, e, p}, x] && NeQ[p, -1]
  6301. Int[Times[Plus[Optional[Pattern[a, Blank[]]], Times[ArcSech[Times[Optional[Pattern[c, Blank[]]], Pattern[x, Blank[]]]], Optional[Pattern[b, Blank[]]]]], Power[Times[Optional[Pattern[f, Blank[]]], Pattern[x, Blank[]]], Optional[Pattern[m, Blank[]]]], Power[Plus[Optional[Pattern[d, Blank[]]], Times[Optional[Pattern[e, Blank[]]], Power[Pattern[x, Blank[]], 2]]], Optional[Pattern[p, Blank[]]]]], Pattern[x, Blank[Symbol]]] := With[{u = IntHide[(f*x)^m*(d + e*x^2)^p, x]}, Dist[a + b*ArcSech[c*x], u, x] + Dist[b*Sqrt[1 + c*x]*Sqrt[1/(1 + c*x)], Int[SimplifyIntegrand[u/(x*Sqrt[1 - c*x]*Sqrt[1 + c*x]), x], x], x]] /; FreeQ[{a, b, c, d, e, f, m, p}, x] && ((IGtQ[p, 0] && !(ILtQ[(m - 1)/2, 0] && GtQ[m + 2*p + 3, 0])) || (IGtQ[(m + 1)/2, 0] && !(ILtQ[p, 0] && GtQ[m + 2*p + 3, 0])) || (ILtQ[(m + 2*p + 1)/2, 0] && !ILtQ[(m - 1)/2, 0]))
  6302. Int[Times[Plus[Optional[Pattern[a, Blank[]]], Times[ArcCsch[Times[Optional[Pattern[c, Blank[]]], Pattern[x, Blank[]]]], Optional[Pattern[b, Blank[]]]]], Power[Times[Optional[Pattern[f, Blank[]]], Pattern[x, Blank[]]], Optional[Pattern[m, Blank[]]]], Power[Plus[Optional[Pattern[d, Blank[]]], Times[Optional[Pattern[e, Blank[]]], Power[Pattern[x, Blank[]], 2]]], Optional[Pattern[p, Blank[]]]]], Pattern[x, Blank[Symbol]]] := With[{u = IntHide[(f*x)^m*(d + e*x^2)^p, x]}, Dist[a + b*ArcCsch[c*x], u, x] - Dist[(b*c*x)/Sqrt[-(c^2*x^2)], Int[SimplifyIntegrand[u/(x*Sqrt[-1 - c^2*x^2]), x], x], x]] /; FreeQ[{a, b, c, d, e, f, m, p}, x] && ((IGtQ[p, 0] && !(ILtQ[(m - 1)/2, 0] && GtQ[m + 2*p + 3, 0])) || (IGtQ[(m + 1)/2, 0] && !(ILtQ[p, 0] && GtQ[m + 2*p + 3, 0])) || (ILtQ[(m + 2*p + 1)/2, 0] && !ILtQ[(m - 1)/2, 0]))
  6303. Int[Times[Power[Plus[Optional[Pattern[a, Blank[]]], Times[ArcSech[Times[Optional[Pattern[c, Blank[]]], Pattern[x, Blank[]]]], Optional[Pattern[b, Blank[]]]]], Optional[Pattern[n, Blank[]]]], Power[Pattern[x, Blank[]], Optional[Pattern[m, Blank[]]]], Power[Plus[Optional[Pattern[d, Blank[]]], Times[Optional[Pattern[e, Blank[]]], Power[Pattern[x, Blank[]], 2]]], Optional[Pattern[p, Blank[]]]]], Pattern[x, Blank[Symbol]]] := -Subst[Int[((e + d*x^2)^p*(a + b*ArcCosh[x/c])^n)/x^(m + 2*(p + 1)), x], x, 1/x] /; FreeQ[{a, b, c, d, e, n}, x] && IGtQ[n, 0] && IntegersQ[m, p]
  6304. Int[Times[Power[Plus[Optional[Pattern[a, Blank[]]], Times[ArcCsch[Times[Optional[Pattern[c, Blank[]]], Pattern[x, Blank[]]]], Optional[Pattern[b, Blank[]]]]], Optional[Pattern[n, Blank[]]]], Power[Pattern[x, Blank[]], Optional[Pattern[m, Blank[]]]], Power[Plus[Optional[Pattern[d, Blank[]]], Times[Optional[Pattern[e, Blank[]]], Power[Pattern[x, Blank[]], 2]]], Optional[Pattern[p, Blank[]]]]], Pattern[x, Blank[Symbol]]] := -Subst[Int[((e + d*x^2)^p*(a + b*ArcSinh[x/c])^n)/x^(m + 2*(p + 1)), x], x, 1/x] /; FreeQ[{a, b, c, d, e, n}, x] && IGtQ[n, 0] && IntegersQ[m, p]
  6305. Int[Times[Power[Plus[Optional[Pattern[a, Blank[]]], Times[ArcSech[Times[Optional[Pattern[c, Blank[]]], Pattern[x, Blank[]]]], Optional[Pattern[b, Blank[]]]]], Optional[Pattern[n, Blank[]]]], Power[Pattern[x, Blank[]], Optional[Pattern[m, Blank[]]]], Power[Plus[Optional[Pattern[d, Blank[]]], Times[Optional[Pattern[e, Blank[]]], Power[Pattern[x, Blank[]], 2]]], Pattern[p, Blank[]]]], Pattern[x, Blank[Symbol]]] := -Dist[Sqrt[x^2]/x, Subst[Int[((e + d*x^2)^p*(a + b*ArcCosh[x/c])^n)/x^(m + 2*(p + 1)), x], x, 1/x], x] /; FreeQ[{a, b, c, d, e, n}, x] && IGtQ[n, 0] && EqQ[c^2*d + e, 0] && IntegerQ[m] && IntegerQ[p + 1/2] && GtQ[e, 0] && LtQ[d, 0]
  6306. Int[Times[Power[Plus[Optional[Pattern[a, Blank[]]], Times[ArcCsch[Times[Optional[Pattern[c, Blank[]]], Pattern[x, Blank[]]]], Optional[Pattern[b, Blank[]]]]], Optional[Pattern[n, Blank[]]]], Power[Pattern[x, Blank[]], Optional[Pattern[m, Blank[]]]], Power[Plus[Optional[Pattern[d, Blank[]]], Times[Optional[Pattern[e, Blank[]]], Power[Pattern[x, Blank[]], 2]]], Pattern[p, Blank[]]]], Pattern[x, Blank[Symbol]]] := -Dist[Sqrt[x^2]/x, Subst[Int[((e + d*x^2)^p*(a + b*ArcSinh[x/c])^n)/x^(m + 2*(p + 1)), x], x, 1/x], x] /; FreeQ[{a, b, c, d, e, n}, x] && IGtQ[n, 0] && EqQ[e - c^2*d, 0] && IntegerQ[m] && IntegerQ[p + 1/2] && GtQ[e, 0] && LtQ[d, 0]
  6307. Int[Times[Power[Plus[Optional[Pattern[a, Blank[]]], Times[ArcSech[Times[Optional[Pattern[c, Blank[]]], Pattern[x, Blank[]]]], Optional[Pattern[b, Blank[]]]]], Optional[Pattern[n, Blank[]]]], Power[Pattern[x, Blank[]], Optional[Pattern[m, Blank[]]]], Power[Plus[Optional[Pattern[d, Blank[]]], Times[Optional[Pattern[e, Blank[]]], Power[Pattern[x, Blank[]], 2]]], Pattern[p, Blank[]]]], Pattern[x, Blank[Symbol]]] := -Dist[Sqrt[d + e*x^2]/(x*Sqrt[e + d/x^2]), Subst[Int[((e + d*x^2)^p*(a + b*ArcCosh[x/c])^n)/x^(m + 2*(p + 1)), x], x, 1/x], x] /; FreeQ[{a, b, c, d, e, n}, x] && IGtQ[n, 0] && EqQ[c^2*d + e, 0] && IntegerQ[m] && IntegerQ[p + 1/2] && !(GtQ[e, 0] && LtQ[d, 0])
  6308. Int[Times[Power[Plus[Optional[Pattern[a, Blank[]]], Times[ArcCsch[Times[Optional[Pattern[c, Blank[]]], Pattern[x, Blank[]]]], Optional[Pattern[b, Blank[]]]]], Optional[Pattern[n, Blank[]]]], Power[Pattern[x, Blank[]], Optional[Pattern[m, Blank[]]]], Power[Plus[Optional[Pattern[d, Blank[]]], Times[Optional[Pattern[e, Blank[]]], Power[Pattern[x, Blank[]], 2]]], Pattern[p, Blank[]]]], Pattern[x, Blank[Symbol]]] := -Dist[Sqrt[d + e*x^2]/(x*Sqrt[e + d/x^2]), Subst[Int[((e + d*x^2)^p*(a + b*ArcSinh[x/c])^n)/x^(m + 2*(p + 1)), x], x, 1/x], x] /; FreeQ[{a, b, c, d, e, n}, x] && IGtQ[n, 0] && EqQ[e - c^2*d, 0] && IntegerQ[m] && IntegerQ[p + 1/2] && !(GtQ[e, 0] && LtQ[d, 0])
  6309. Int[Times[Plus[Optional[Pattern[a, Blank[]]], Times[ArcSech[Times[Optional[Pattern[c, Blank[]]], Pattern[x, Blank[]]]], Optional[Pattern[b, Blank[]]]]], Pattern[u, Blank[]]], Pattern[x, Blank[Symbol]]] := With[{v = IntHide[u, x]}, Dist[a + b*ArcSech[c*x], v, x] + Dist[(b*Sqrt[1 - c^2*x^2])/(c*x*Sqrt[-1 + 1/(c*x)]*Sqrt[1 + 1/(c*x)]), Int[SimplifyIntegrand[v/(x*Sqrt[1 - c^2*x^2]), x], x], x] /; InverseFunctionFreeQ[v, x]] /; FreeQ[{a, b, c}, x]
  6310. Int[Times[Plus[Optional[Pattern[a, Blank[]]], Times[ArcCsch[Times[Optional[Pattern[c, Blank[]]], Pattern[x, Blank[]]]], Optional[Pattern[b, Blank[]]]]], Pattern[u, Blank[]]], Pattern[x, Blank[Symbol]]] := With[{v = IntHide[u, x]}, Dist[a + b*ArcCsch[c*x], v, x] + Dist[b/c, Int[SimplifyIntegrand[v/(x^2*Sqrt[1 + 1/(c^2*x^2)]), x], x], x] /; InverseFunctionFreeQ[v, x]] /; FreeQ[{a, b, c}, x]
  6311. Int[Times[Power[Plus[Optional[Pattern[a, Blank[]]], Times[ArcSech[Times[Optional[Pattern[c, Blank[]]], Pattern[x, Blank[]]]], Optional[Pattern[b, Blank[]]]]], Optional[Pattern[n, Blank[]]]], Optional[Pattern[u, Blank[]]]], Pattern[x, Blank[Symbol]]] := Unintegrable[u*(a + b*ArcSech[c*x])^n, x] /; FreeQ[{a, b, c, n}, x]
  6312. Int[Times[Power[Plus[Optional[Pattern[a, Blank[]]], Times[ArcCsch[Times[Optional[Pattern[c, Blank[]]], Pattern[x, Blank[]]]], Optional[Pattern[b, Blank[]]]]], Optional[Pattern[n, Blank[]]]], Optional[Pattern[u, Blank[]]]], Pattern[x, Blank[Symbol]]] := Unintegrable[u*(a + b*ArcCsch[c*x])^n, x] /; FreeQ[{a, b, c, n}, x]
  6313. Int[ArcSech[Plus[Pattern[c, Blank[]], Times[Optional[Pattern[d, Blank[]]], Pattern[x, Blank[]]]]], Pattern[x, Blank[Symbol]]] := Simp[((c + d*x)*ArcSech[c + d*x])/d, x] + Int[Sqrt[(1 - c - d*x)/(1 + c + d*x)]/(1 - c - d*x), x] /; FreeQ[{c, d}, x]
  6314. Int[ArcCsch[Plus[Pattern[c, Blank[]], Times[Optional[Pattern[d, Blank[]]], Pattern[x, Blank[]]]]], Pattern[x, Blank[Symbol]]] := Simp[((c + d*x)*ArcCsch[c + d*x])/d, x] + Int[1/((c + d*x)*Sqrt[1 + 1/(c + d*x)^2]), x] /; FreeQ[{c, d}, x]
  6315. Int[Power[Plus[Optional[Pattern[a, Blank[]]], Times[ArcSech[Plus[Pattern[c, Blank[]], Times[Optional[Pattern[d, Blank[]]], Pattern[x, Blank[]]]]], Optional[Pattern[b, Blank[]]]]], Optional[Pattern[p, Blank[]]]], Pattern[x, Blank[Symbol]]] := Dist[1/d, Subst[Int[(a + b*ArcSech[x])^p, x], x, c + d*x], x] /; FreeQ[{a, b, c, d}, x] && IGtQ[p, 0]
  6316. Int[Power[Plus[Optional[Pattern[a, Blank[]]], Times[ArcCsch[Plus[Pattern[c, Blank[]], Times[Optional[Pattern[d, Blank[]]], Pattern[x, Blank[]]]]], Optional[Pattern[b, Blank[]]]]], Optional[Pattern[p, Blank[]]]], Pattern[x, Blank[Symbol]]] := Dist[1/d, Subst[Int[(a + b*ArcCsch[x])^p, x], x, c + d*x], x] /; FreeQ[{a, b, c, d}, x] && IGtQ[p, 0]
  6317. Int[Power[Plus[Optional[Pattern[a, Blank[]]], Times[ArcSech[Plus[Pattern[c, Blank[]], Times[Optional[Pattern[d, Blank[]]], Pattern[x, Blank[]]]]], Optional[Pattern[b, Blank[]]]]], Pattern[p, Blank[]]], Pattern[x, Blank[Symbol]]] := Unintegrable[(a + b*ArcSech[c + d*x])^p, x] /; FreeQ[{a, b, c, d, p}, x] && !IGtQ[p, 0]
  6318. Int[Power[Plus[Optional[Pattern[a, Blank[]]], Times[ArcCsch[Plus[Pattern[c, Blank[]], Times[Optional[Pattern[d, Blank[]]], Pattern[x, Blank[]]]]], Optional[Pattern[b, Blank[]]]]], Pattern[p, Blank[]]], Pattern[x, Blank[Symbol]]] := Unintegrable[(a + b*ArcCsch[c + d*x])^p, x] /; FreeQ[{a, b, c, d, p}, x] && !IGtQ[p, 0]
  6319. Int[Times[Power[Plus[Optional[Pattern[a, Blank[]]], Times[ArcSech[Plus[Pattern[c, Blank[]], Times[Optional[Pattern[d, Blank[]]], Pattern[x, Blank[]]]]], Optional[Pattern[b, Blank[]]]]], Optional[Pattern[p, Blank[]]]], Power[Plus[Optional[Pattern[e, Blank[]]], Times[Optional[Pattern[f, Blank[]]], Pattern[x, Blank[]]]], Optional[Pattern[m, Blank[]]]]], Pattern[x, Blank[Symbol]]] := Dist[1/d, Subst[Int[((f*x)/d)^m*(a + b*ArcSech[x])^p, x], x, c + d*x], x] /; FreeQ[{a, b, c, d, e, f, m}, x] && EqQ[d*e - c*f, 0] && IGtQ[p, 0]
  6320. Int[Times[Power[Plus[Optional[Pattern[a, Blank[]]], Times[ArcCsch[Plus[Pattern[c, Blank[]], Times[Optional[Pattern[d, Blank[]]], Pattern[x, Blank[]]]]], Optional[Pattern[b, Blank[]]]]], Optional[Pattern[p, Blank[]]]], Power[Plus[Optional[Pattern[e, Blank[]]], Times[Optional[Pattern[f, Blank[]]], Pattern[x, Blank[]]]], Optional[Pattern[m, Blank[]]]]], Pattern[x, Blank[Symbol]]] := Dist[1/d, Subst[Int[((f*x)/d)^m*(a + b*ArcCsch[x])^p, x], x, c + d*x], x] /; FreeQ[{a, b, c, d, e, f, m}, x] && EqQ[d*e - c*f, 0] && IGtQ[p, 0]
  6321. Int[Times[Power[Plus[Optional[Pattern[a, Blank[]]], Times[ArcSech[Plus[Pattern[c, Blank[]], Times[Optional[Pattern[d, Blank[]]], Pattern[x, Blank[]]]]], Optional[Pattern[b, Blank[]]]]], Optional[Pattern[p, Blank[]]]], Power[Plus[Optional[Pattern[e, Blank[]]], Times[Optional[Pattern[f, Blank[]]], Pattern[x, Blank[]]]], Optional[Pattern[m, Blank[]]]]], Pattern[x, Blank[Symbol]]] := -Dist[(d^(m + 1))^(-1), Subst[Int[(a + b*x)^p*Sech[x]*Tanh[x]*(d*e - c*f + f*Sech[x])^m, x], x, ArcSech[c + d*x]], x] /; FreeQ[{a, b, c, d, e, f}, x] && IGtQ[p, 0] && IntegerQ[m]
  6322. Int[Times[Power[Plus[Optional[Pattern[a, Blank[]]], Times[ArcCsch[Plus[Pattern[c, Blank[]], Times[Optional[Pattern[d, Blank[]]], Pattern[x, Blank[]]]]], Optional[Pattern[b, Blank[]]]]], Optional[Pattern[p, Blank[]]]], Power[Plus[Optional[Pattern[e, Blank[]]], Times[Optional[Pattern[f, Blank[]]], Pattern[x, Blank[]]]], Optional[Pattern[m, Blank[]]]]], Pattern[x, Blank[Symbol]]] := -Dist[(d^(m + 1))^(-1), Subst[Int[(a + b*x)^p*Csch[x]*Coth[x]*(d*e - c*f + f*Csch[x])^m, x], x, ArcCsch[c + d*x]], x] /; FreeQ[{a, b, c, d, e, f}, x] && IGtQ[p, 0] && IntegerQ[m]
  6323. Int[Times[Power[Plus[Optional[Pattern[a, Blank[]]], Times[ArcSech[Plus[Pattern[c, Blank[]], Times[Optional[Pattern[d, Blank[]]], Pattern[x, Blank[]]]]], Optional[Pattern[b, Blank[]]]]], Optional[Pattern[p, Blank[]]]], Power[Plus[Optional[Pattern[e, Blank[]]], Times[Optional[Pattern[f, Blank[]]], Pattern[x, Blank[]]]], Optional[Pattern[m, Blank[]]]]], Pattern[x, Blank[Symbol]]] := Dist[1/d, Subst[Int[((d*e - c*f)/d + (f*x)/d)^m*(a + b*ArcSech[x])^p, x], x, c + d*x], x] /; FreeQ[{a, b, c, d, e, f, m}, x] && IGtQ[p, 0]
  6324. Int[Times[Power[Plus[Optional[Pattern[a, Blank[]]], Times[ArcCsch[Plus[Pattern[c, Blank[]], Times[Optional[Pattern[d, Blank[]]], Pattern[x, Blank[]]]]], Optional[Pattern[b, Blank[]]]]], Optional[Pattern[p, Blank[]]]], Power[Plus[Optional[Pattern[e, Blank[]]], Times[Optional[Pattern[f, Blank[]]], Pattern[x, Blank[]]]], Optional[Pattern[m, Blank[]]]]], Pattern[x, Blank[Symbol]]] := Dist[1/d, Subst[Int[((d*e - c*f)/d + (f*x)/d)^m*(a + b*ArcCsch[x])^p, x], x, c + d*x], x] /; FreeQ[{a, b, c, d, e, f, m}, x] && IGtQ[p, 0]
  6325. Int[Times[Power[Plus[Optional[Pattern[a, Blank[]]], Times[ArcSech[Plus[Pattern[c, Blank[]], Times[Optional[Pattern[d, Blank[]]], Pattern[x, Blank[]]]]], Optional[Pattern[b, Blank[]]]]], Pattern[p, Blank[]]], Power[Plus[Optional[Pattern[e, Blank[]]], Times[Optional[Pattern[f, Blank[]]], Pattern[x, Blank[]]]], Optional[Pattern[m, Blank[]]]]], Pattern[x, Blank[Symbol]]] := Unintegrable[(e + f*x)^m*(a + b*ArcSech[c + d*x])^p, x] /; FreeQ[{a, b, c, d, e, f, m, p}, x] && !IGtQ[p, 0]
  6326. Int[Times[Power[Plus[Optional[Pattern[a, Blank[]]], Times[ArcCsch[Plus[Pattern[c, Blank[]], Times[Optional[Pattern[d, Blank[]]], Pattern[x, Blank[]]]]], Optional[Pattern[b, Blank[]]]]], Pattern[p, Blank[]]], Power[Plus[Optional[Pattern[e, Blank[]]], Times[Optional[Pattern[f, Blank[]]], Pattern[x, Blank[]]]], Optional[Pattern[m, Blank[]]]]], Pattern[x, Blank[Symbol]]] := Unintegrable[(e + f*x)^m*(a + b*ArcCsch[c + d*x])^p, x] /; FreeQ[{a, b, c, d, e, f, m, p}, x] && !IGtQ[p, 0]
  6327. Int[Times[Power[ArcSech[Times[Optional[Pattern[c, Blank[]]], Power[Plus[Optional[Pattern[a, Blank[]]], Times[Optional[Pattern[b, Blank[]]], Power[Pattern[x, Blank[]], Optional[Pattern[n, Blank[]]]]]], -1]]], Optional[Pattern[m, Blank[]]]], Optional[Pattern[u, Blank[]]]], Pattern[x, Blank[Symbol]]] := Int[u*ArcCosh[a/c + (b*x^n)/c]^m, x] /; FreeQ[{a, b, c, n, m}, x]
  6328. Int[Times[Power[ArcCsch[Times[Optional[Pattern[c, Blank[]]], Power[Plus[Optional[Pattern[a, Blank[]]], Times[Optional[Pattern[b, Blank[]]], Power[Pattern[x, Blank[]], Optional[Pattern[n, Blank[]]]]]], -1]]], Optional[Pattern[m, Blank[]]]], Optional[Pattern[u, Blank[]]]], Pattern[x, Blank[Symbol]]] := Int[u*ArcSinh[a/c + (b*x^n)/c]^m, x] /; FreeQ[{a, b, c, n, m}, x]
  6329. Int[Power[E, ArcSech[Times[Optional[Pattern[a, Blank[]]], Pattern[x, Blank[]]]]], Pattern[x, Blank[Symbol]]] := Simp[x*E^ArcSech[a*x], x] + (Dist[1/a, Int[(1*Sqrt[(1 - a*x)/(1 + a*x)])/(x*(1 - a*x)), x], x] + Simp[Log[x]/a, x]) /; FreeQ[a, x]
  6330. Int[Power[E, ArcSech[Times[Optional[Pattern[a, Blank[]]], Power[Pattern[x, Blank[]], Pattern[p, Blank[]]]]]], Pattern[x, Blank[Symbol]]] := Simp[x*E^ArcSech[a*x^p], x] + (Dist[p/a, Int[1/x^p, x], x] + Dist[(p*Sqrt[1 + a*x^p]*Sqrt[1/(1 + a*x^p)])/a, Int[1/(x^p*Sqrt[1 + a*x^p]*Sqrt[1 - a*x^p]), x], x]) /; FreeQ[{a, p}, x]
  6331. Int[Power[E, ArcCsch[Times[Optional[Pattern[a, Blank[]]], Power[Pattern[x, Blank[]], Optional[Pattern[p, Blank[]]]]]]], Pattern[x, Blank[Symbol]]] := Dist[1/a, Int[1/x^p, x], x] + Int[Sqrt[1 + 1/(a^2*x^(2*p))], x] /; FreeQ[{a, p}, x]
  6332. Int[Power[E, Times[ArcSech[Pattern[u, Blank[]]], Optional[Pattern[n, Blank[]]]]], Pattern[x, Blank[Symbol]]] := Int[(1/u + Sqrt[(1 - u)/(1 + u)] + (1*Sqrt[(1 - u)/(1 + u)])/u)^n, x] /; IntegerQ[n]
  6333. Int[Power[E, Times[ArcCsch[Pattern[u, Blank[]]], Optional[Pattern[n, Blank[]]]]], Pattern[x, Blank[Symbol]]] := Int[(1/u + Sqrt[1 + 1/u^2])^n, x] /; IntegerQ[n]
  6334. Int[Times[Power[E, ArcSech[Times[Optional[Pattern[a, Blank[]]], Power[Pattern[x, Blank[]], Optional[Pattern[p, Blank[]]]]]]], Power[Pattern[x, Blank[]], -1]], Pattern[x, Blank[Symbol]]] := -Simp[(a*p*x^p)^(-1), x] + Dist[(Sqrt[1 + a*x^p]*Sqrt[1/(1 + a*x^p)])/a, Int[(Sqrt[1 + a*x^p]*Sqrt[1 - a*x^p])/x^(p + 1), x], x] /; FreeQ[{a, p}, x]
  6335. Int[Times[Power[E, ArcSech[Times[Optional[Pattern[a, Blank[]]], Power[Pattern[x, Blank[]], Optional[Pattern[p, Blank[]]]]]]], Power[Pattern[x, Blank[]], Optional[Pattern[m, Blank[]]]]], Pattern[x, Blank[Symbol]]] := Simp[(x^(m + 1)*E^ArcSech[a*x^p])/(m + 1), x] + (Dist[p/(a*(m + 1)), Int[x^(m - p), x], x] + Dist[(p*Sqrt[1 + a*x^p]*Sqrt[1/(1 + a*x^p)])/(a*(m + 1)), Int[x^(m - p)/(Sqrt[1 + a*x^p]*Sqrt[1 - a*x^p]), x], x]) /; FreeQ[{a, m, p}, x] && NeQ[m, -1]
  6336. Int[Times[Power[E, ArcCsch[Times[Optional[Pattern[a, Blank[]]], Power[Pattern[x, Blank[]], Optional[Pattern[p, Blank[]]]]]]], Power[Pattern[x, Blank[]], Optional[Pattern[m, Blank[]]]]], Pattern[x, Blank[Symbol]]] := Dist[1/a, Int[x^(m - p), x], x] + Int[x^m*Sqrt[1 + 1/(a^2*x^(2*p))], x] /; FreeQ[{a, m, p}, x]
  6337. Int[Times[Power[E, Times[ArcSech[Pattern[u, Blank[]]], Optional[Pattern[n, Blank[]]]]], Power[Pattern[x, Blank[]], Optional[Pattern[m, Blank[]]]]], Pattern[x, Blank[Symbol]]] := Int[x^m*(1/u + Sqrt[(1 - u)/(1 + u)] + (1*Sqrt[(1 - u)/(1 + u)])/u)^n, x] /; FreeQ[m, x] && IntegerQ[n]
  6338. Int[Times[Power[E, Times[ArcCsch[Pattern[u, Blank[]]], Optional[Pattern[n, Blank[]]]]], Power[Pattern[x, Blank[]], Optional[Pattern[m, Blank[]]]]], Pattern[x, Blank[Symbol]]] := Int[x^m*(1/u + Sqrt[1 + 1/u^2])^n, x] /; FreeQ[m, x] && IntegerQ[n]
  6339. Int[Times[Power[E, ArcSech[Times[Optional[Pattern[c, Blank[]]], Pattern[x, Blank[]]]]], Power[Plus[Pattern[a, Blank[]], Times[Optional[Pattern[b, Blank[]]], Power[Pattern[x, Blank[]], 2]]], -1]], Pattern[x, Blank[Symbol]]] := Dist[1/(a*c), Int[Sqrt[1/(1 + c*x)]/(x*Sqrt[1 - c*x]), x], x] + Dist[1/c, Int[1/(x*(a + b*x^2)), x], x] /; FreeQ[{a, b, c}, x] && EqQ[b + a*c^2, 0]
  6340. Int[Times[Power[E, ArcCsch[Times[Optional[Pattern[c, Blank[]]], Pattern[x, Blank[]]]]], Power[Plus[Pattern[a, Blank[]], Times[Optional[Pattern[b, Blank[]]], Power[Pattern[x, Blank[]], 2]]], -1]], Pattern[x, Blank[Symbol]]] := Dist[1/(a*c^2), Int[1/(x^2*Sqrt[1 + 1/(c^2*x^2)]), x], x] + Dist[1/c, Int[1/(x*(a + b*x^2)), x], x] /; FreeQ[{a, b, c}, x] && EqQ[b - a*c^2, 0]
  6341. Int[Times[Power[E, ArcSech[Times[Optional[Pattern[c, Blank[]]], Pattern[x, Blank[]]]]], Power[Times[Optional[Pattern[d, Blank[]]], Pattern[x, Blank[]]], Optional[Pattern[m, Blank[]]]], Power[Plus[Pattern[a, Blank[]], Times[Optional[Pattern[b, Blank[]]], Power[Pattern[x, Blank[]], 2]]], -1]], Pattern[x, Blank[Symbol]]] := Dist[d/(a*c), Int[((d*x)^(m - 1)*Sqrt[1/(1 + c*x)])/Sqrt[1 - c*x], x], x] + Dist[d/c, Int[(d*x)^(m - 1)/(a + b*x^2), x], x] /; FreeQ[{a, b, c, d, m}, x] && EqQ[b + a*c^2, 0]
  6342. Int[Times[Power[E, ArcCsch[Times[Optional[Pattern[c, Blank[]]], Pattern[x, Blank[]]]]], Power[Times[Optional[Pattern[d, Blank[]]], Pattern[x, Blank[]]], Optional[Pattern[m, Blank[]]]], Power[Plus[Pattern[a, Blank[]], Times[Optional[Pattern[b, Blank[]]], Power[Pattern[x, Blank[]], 2]]], -1]], Pattern[x, Blank[Symbol]]] := Dist[d^2/(a*c^2), Int[(d*x)^(m - 2)/Sqrt[1 + 1/(c^2*x^2)], x], x] + Dist[d/c, Int[(d*x)^(m - 1)/(a + b*x^2), x], x] /; FreeQ[{a, b, c, d, m}, x] && EqQ[b - a*c^2, 0]
  6343. Int[ArcSech[Pattern[u, Blank[]]], Pattern[x, Blank[Symbol]]] := Simp[x*ArcSech[u], x] + Dist[Sqrt[1 - u^2]/(u*Sqrt[-1 + 1/u]*Sqrt[1 + 1/u]), Int[SimplifyIntegrand[(x*D[u, x])/(u*Sqrt[1 - u^2]), x], x], x] /; InverseFunctionFreeQ[u, x] && !FunctionOfExponentialQ[u, x]
  6344. Int[ArcCsch[Pattern[u, Blank[]]], Pattern[x, Blank[Symbol]]] := Simp[x*ArcCsch[u], x] - Dist[u/Sqrt[-u^2], Int[SimplifyIntegrand[(x*D[u, x])/(u*Sqrt[-1 - u^2]), x], x], x] /; InverseFunctionFreeQ[u, x] && !FunctionOfExponentialQ[u, x]
  6345. Int[Times[Plus[Optional[Pattern[a, Blank[]]], Times[ArcSech[Pattern[u, Blank[]]], Optional[Pattern[b, Blank[]]]]], Power[Plus[Optional[Pattern[c, Blank[]]], Times[Optional[Pattern[d, Blank[]]], Pattern[x, Blank[]]]], Optional[Pattern[m, Blank[]]]]], Pattern[x, Blank[Symbol]]] := Simp[((c + d*x)^(m + 1)*(a + b*ArcSech[u]))/(d*(m + 1)), x] + Dist[(b*Sqrt[1 - u^2])/(d*(m + 1)*u*Sqrt[-1 + 1/u]*Sqrt[1 + 1/u]), Int[SimplifyIntegrand[((c + d*x)^(m + 1)*D[u, x])/(u*Sqrt[1 - u^2]), x], x], x] /; FreeQ[{a, b, c, d, m}, x] && NeQ[m, -1] && InverseFunctionFreeQ[u, x] && !FunctionOfQ[(c + d*x)^(m + 1), u, x] && !FunctionOfExponentialQ[u, x]
  6346. Int[Times[Plus[Optional[Pattern[a, Blank[]]], Times[ArcCsch[Pattern[u, Blank[]]], Optional[Pattern[b, Blank[]]]]], Power[Plus[Optional[Pattern[c, Blank[]]], Times[Optional[Pattern[d, Blank[]]], Pattern[x, Blank[]]]], Optional[Pattern[m, Blank[]]]]], Pattern[x, Blank[Symbol]]] := Simp[((c + d*x)^(m + 1)*(a + b*ArcCsch[u]))/(d*(m + 1)), x] - Dist[(b*u)/(d*(m + 1)*Sqrt[-u^2]), Int[SimplifyIntegrand[((c + d*x)^(m + 1)*D[u, x])/(u*Sqrt[-1 - u^2]), x], x], x] /; FreeQ[{a, b, c, d, m}, x] && NeQ[m, -1] && InverseFunctionFreeQ[u, x] && !FunctionOfQ[(c + d*x)^(m + 1), u, x] && !FunctionOfExponentialQ[u, x]
  6347. Int[Times[Plus[Optional[Pattern[a, Blank[]]], Times[ArcSech[Pattern[u, Blank[]]], Optional[Pattern[b, Blank[]]]]], Pattern[v, Blank[]]], Pattern[x, Blank[Symbol]]] := With[{w = IntHide[v, x]}, Dist[a + b*ArcSech[u], w, x] + Dist[(b*Sqrt[1 - u^2])/(u*Sqrt[-1 + 1/u]*Sqrt[1 + 1/u]), Int[SimplifyIntegrand[(w*D[u, x])/(u*Sqrt[1 - u^2]), x], x], x] /; InverseFunctionFreeQ[w, x]] /; FreeQ[{a, b}, x] && InverseFunctionFreeQ[u, x] && !MatchQ[v, ((c_.) + (d_.)*x)^(m_.) /; FreeQ[{c, d, m}, x]]
  6348. Int[Times[Plus[Optional[Pattern[a, Blank[]]], Times[ArcCsch[Pattern[u, Blank[]]], Optional[Pattern[b, Blank[]]]]], Pattern[v, Blank[]]], Pattern[x, Blank[Symbol]]] := With[{w = IntHide[v, x]}, Dist[a + b*ArcCsch[u], w, x] - Dist[(b*u)/Sqrt[-u^2], Int[SimplifyIntegrand[(w*D[u, x])/(u*Sqrt[-1 - u^2]), x], x], x] /; InverseFunctionFreeQ[w, x]] /; FreeQ[{a, b}, x] && InverseFunctionFreeQ[u, x] && !MatchQ[v, ((c_.) + (d_.)*x)^(m_.) /; FreeQ[{c, d, m}, x]]
  6349. Int[Erf[Plus[Optional[Pattern[a, Blank[]]], Times[Optional[Pattern[b, Blank[]]], Pattern[x, Blank[]]]]], Pattern[x, Blank[Symbol]]] := Simp[((a + b*x)*Erf[a + b*x])/b, x] + Simp[1/(b*Sqrt[Pi]*E^(a + b*x)^2), x] /; FreeQ[{a, b}, x]
  6350. Int[Erfc[Plus[Optional[Pattern[a, Blank[]]], Times[Optional[Pattern[b, Blank[]]], Pattern[x, Blank[]]]]], Pattern[x, Blank[Symbol]]] := Simp[((a + b*x)*Erfc[a + b*x])/b, x] - Simp[1/(b*Sqrt[Pi]*E^(a + b*x)^2), x] /; FreeQ[{a, b}, x]
  6351. Int[Erfi[Plus[Optional[Pattern[a, Blank[]]], Times[Optional[Pattern[b, Blank[]]], Pattern[x, Blank[]]]]], Pattern[x, Blank[Symbol]]] := Simp[((a + b*x)*Erfi[a + b*x])/b, x] - Simp[E^(a + b*x)^2/(b*Sqrt[Pi]), x] /; FreeQ[{a, b}, x]
  6352. Int[Power[Erf[Plus[Optional[Pattern[a, Blank[]]], Times[Optional[Pattern[b, Blank[]]], Pattern[x, Blank[]]]]], 2], Pattern[x, Blank[Symbol]]] := Simp[((a + b*x)*Erf[a + b*x]^2)/b, x] - Dist[4/Sqrt[Pi], Int[((a + b*x)*Erf[a + b*x])/E^(a + b*x)^2, x], x] /; FreeQ[{a, b}, x]
  6353. Int[Power[Erfc[Plus[Optional[Pattern[a, Blank[]]], Times[Optional[Pattern[b, Blank[]]], Pattern[x, Blank[]]]]], 2], Pattern[x, Blank[Symbol]]] := Simp[((a + b*x)*Erfc[a + b*x]^2)/b, x] + Dist[4/Sqrt[Pi], Int[((a + b*x)*Erfc[a + b*x])/E^(a + b*x)^2, x], x] /; FreeQ[{a, b}, x]
  6354. Int[Power[Erfi[Plus[Optional[Pattern[a, Blank[]]], Times[Optional[Pattern[b, Blank[]]], Pattern[x, Blank[]]]]], 2], Pattern[x, Blank[Symbol]]] := Simp[((a + b*x)*Erfi[a + b*x]^2)/b, x] - Dist[4/Sqrt[Pi], Int[(a + b*x)*E^(a + b*x)^2*Erfi[a + b*x], x], x] /; FreeQ[{a, b}, x]
  6355. Int[Power[Erf[Plus[Optional[Pattern[a, Blank[]]], Times[Optional[Pattern[b, Blank[]]], Pattern[x, Blank[]]]]], Pattern[n, Blank[]]], Pattern[x, Blank[Symbol]]] := Unintegrable[Erf[a + b*x]^n, x] /; FreeQ[{a, b, n}, x] && NeQ[n, 1] && NeQ[n, 2]
  6356. Int[Power[Erfc[Plus[Optional[Pattern[a, Blank[]]], Times[Optional[Pattern[b, Blank[]]], Pattern[x, Blank[]]]]], Pattern[n, Blank[]]], Pattern[x, Blank[Symbol]]] := Unintegrable[Erfc[a + b*x]^n, x] /; FreeQ[{a, b, n}, x] && NeQ[n, 1] && NeQ[n, 2]
  6357. Int[Power[Erfi[Plus[Optional[Pattern[a, Blank[]]], Times[Optional[Pattern[b, Blank[]]], Pattern[x, Blank[]]]]], Pattern[n, Blank[]]], Pattern[x, Blank[Symbol]]] := Unintegrable[Erfi[a + b*x]^n, x] /; FreeQ[{a, b, n}, x] && NeQ[n, 1] && NeQ[n, 2]
  6358. Int[Times[Erf[Times[Optional[Pattern[b, Blank[]]], Pattern[x, Blank[]]]], Power[Pattern[x, Blank[]], -1]], Pattern[x, Blank[Symbol]]] := Simp[(2*b*x*HypergeometricPFQ[{1/2, 1/2}, {3/2, 3/2}, -(b^2*x^2)])/Sqrt[Pi], x] /; FreeQ[b, x]
  6359. Int[Times[Erfc[Times[Optional[Pattern[b, Blank[]]], Pattern[x, Blank[]]]], Power[Pattern[x, Blank[]], -1]], Pattern[x, Blank[Symbol]]] := Simp[Log[x], x] - Int[Erf[b*x]/x, x] /; FreeQ[b, x]
  6360. Int[Times[Erfi[Times[Optional[Pattern[b, Blank[]]], Pattern[x, Blank[]]]], Power[Pattern[x, Blank[]], -1]], Pattern[x, Blank[Symbol]]] := Simp[(2*b*x*HypergeometricPFQ[{1/2, 1/2}, {3/2, 3/2}, b^2*x^2])/Sqrt[Pi], x] /; FreeQ[b, x]
  6361. Int[Times[Erf[Plus[Optional[Pattern[a, Blank[]]], Times[Optional[Pattern[b, Blank[]]], Pattern[x, Blank[]]]]], Power[Plus[Optional[Pattern[c, Blank[]]], Times[Optional[Pattern[d, Blank[]]], Pattern[x, Blank[]]]], Optional[Pattern[m, Blank[]]]]], Pattern[x, Blank[Symbol]]] := Simp[((c + d*x)^(m + 1)*Erf[a + b*x])/(d*(m + 1)), x] - Dist[(2*b)/(Sqrt[Pi]*d*(m + 1)), Int[(c + d*x)^(m + 1)/E^(a + b*x)^2, x], x] /; FreeQ[{a, b, c, d, m}, x] && NeQ[m, -1]
  6362. Int[Times[Erfc[Plus[Optional[Pattern[a, Blank[]]], Times[Optional[Pattern[b, Blank[]]], Pattern[x, Blank[]]]]], Power[Plus[Optional[Pattern[c, Blank[]]], Times[Optional[Pattern[d, Blank[]]], Pattern[x, Blank[]]]], Optional[Pattern[m, Blank[]]]]], Pattern[x, Blank[Symbol]]] := Simp[((c + d*x)^(m + 1)*Erfc[a + b*x])/(d*(m + 1)), x] + Dist[(2*b)/(Sqrt[Pi]*d*(m + 1)), Int[(c + d*x)^(m + 1)/E^(a + b*x)^2, x], x] /; FreeQ[{a, b, c, d, m}, x] && NeQ[m, -1]
  6363. Int[Times[Erfi[Plus[Optional[Pattern[a, Blank[]]], Times[Optional[Pattern[b, Blank[]]], Pattern[x, Blank[]]]]], Power[Plus[Optional[Pattern[c, Blank[]]], Times[Optional[Pattern[d, Blank[]]], Pattern[x, Blank[]]]], Optional[Pattern[m, Blank[]]]]], Pattern[x, Blank[Symbol]]] := Simp[((c + d*x)^(m + 1)*Erfi[a + b*x])/(d*(m + 1)), x] - Dist[(2*b)/(Sqrt[Pi]*d*(m + 1)), Int[(c + d*x)^(m + 1)*E^(a + b*x)^2, x], x] /; FreeQ[{a, b, c, d, m}, x] && NeQ[m, -1]
  6364. Int[Times[Power[Erf[Times[Optional[Pattern[b, Blank[]]], Pattern[x, Blank[]]]], 2], Power[Pattern[x, Blank[]], Optional[Pattern[m, Blank[]]]]], Pattern[x, Blank[Symbol]]] := Simp[(x^(m + 1)*Erf[b*x]^2)/(m + 1), x] - Dist[(4*b)/(Sqrt[Pi]*(m + 1)), Int[(x^(m + 1)*Erf[b*x])/E^(b^2*x^2), x], x] /; FreeQ[b, x] && (IGtQ[m, 0] || ILtQ[(m + 1)/2, 0])
  6365. Int[Times[Power[Erfc[Times[Optional[Pattern[b, Blank[]]], Pattern[x, Blank[]]]], 2], Power[Pattern[x, Blank[]], Optional[Pattern[m, Blank[]]]]], Pattern[x, Blank[Symbol]]] := Simp[(x^(m + 1)*Erfc[b*x]^2)/(m + 1), x] + Dist[(4*b)/(Sqrt[Pi]*(m + 1)), Int[(x^(m + 1)*Erfc[b*x])/E^(b^2*x^2), x], x] /; FreeQ[b, x] && (IGtQ[m, 0] || ILtQ[(m + 1)/2, 0])
  6366. Int[Times[Power[Erfi[Times[Optional[Pattern[b, Blank[]]], Pattern[x, Blank[]]]], 2], Power[Pattern[x, Blank[]], Optional[Pattern[m, Blank[]]]]], Pattern[x, Blank[Symbol]]] := Simp[(x^(m + 1)*Erfi[b*x]^2)/(m + 1), x] - Dist[(4*b)/(Sqrt[Pi]*(m + 1)), Int[x^(m + 1)*E^(b^2*x^2)*Erfi[b*x], x], x] /; FreeQ[b, x] && (IGtQ[m, 0] || ILtQ[(m + 1)/2, 0])
  6367. Int[Times[Power[Erf[Plus[Pattern[a, Blank[]], Times[Optional[Pattern[b, Blank[]]], Pattern[x, Blank[]]]]], 2], Power[Plus[Optional[Pattern[c, Blank[]]], Times[Optional[Pattern[d, Blank[]]], Pattern[x, Blank[]]]], Optional[Pattern[m, Blank[]]]]], Pattern[x, Blank[Symbol]]] := Dist[1/b^(m + 1), Subst[Int[ExpandIntegrand[Erf[x]^2, (b*c - a*d + d*x)^m, x], x], x, a + b*x], x] /; FreeQ[{a, b, c, d}, x] && IGtQ[m, 0]
  6368. Int[Times[Power[Erfc[Plus[Pattern[a, Blank[]], Times[Optional[Pattern[b, Blank[]]], Pattern[x, Blank[]]]]], 2], Power[Plus[Optional[Pattern[c, Blank[]]], Times[Optional[Pattern[d, Blank[]]], Pattern[x, Blank[]]]], Optional[Pattern[m, Blank[]]]]], Pattern[x, Blank[Symbol]]] := Dist[1/b^(m + 1), Subst[Int[ExpandIntegrand[Erfc[x]^2, (b*c - a*d + d*x)^m, x], x], x, a + b*x], x] /; FreeQ[{a, b, c, d}, x] && IGtQ[m, 0]
  6369. Int[Times[Power[Erfi[Plus[Pattern[a, Blank[]], Times[Optional[Pattern[b, Blank[]]], Pattern[x, Blank[]]]]], 2], Power[Plus[Optional[Pattern[c, Blank[]]], Times[Optional[Pattern[d, Blank[]]], Pattern[x, Blank[]]]], Optional[Pattern[m, Blank[]]]]], Pattern[x, Blank[Symbol]]] := Dist[1/b^(m + 1), Subst[Int[ExpandIntegrand[Erfi[x]^2, (b*c - a*d + d*x)^m, x], x], x, a + b*x], x] /; FreeQ[{a, b, c, d}, x] && IGtQ[m, 0]
  6370. Int[Times[Power[Erf[Plus[Optional[Pattern[a, Blank[]]], Times[Optional[Pattern[b, Blank[]]], Pattern[x, Blank[]]]]], Optional[Pattern[n, Blank[]]]], Power[Plus[Optional[Pattern[c, Blank[]]], Times[Optional[Pattern[d, Blank[]]], Pattern[x, Blank[]]]], Optional[Pattern[m, Blank[]]]]], Pattern[x, Blank[Symbol]]] := Unintegrable[(c + d*x)^m*Erf[a + b*x]^n, x] /; FreeQ[{a, b, c, d, m, n}, x]
  6371. Int[Times[Power[Erfc[Plus[Optional[Pattern[a, Blank[]]], Times[Optional[Pattern[b, Blank[]]], Pattern[x, Blank[]]]]], Optional[Pattern[n, Blank[]]]], Power[Plus[Optional[Pattern[c, Blank[]]], Times[Optional[Pattern[d, Blank[]]], Pattern[x, Blank[]]]], Optional[Pattern[m, Blank[]]]]], Pattern[x, Blank[Symbol]]] := Unintegrable[(c + d*x)^m*Erfc[a + b*x]^n, x] /; FreeQ[{a, b, c, d, m, n}, x]
  6372. Int[Times[Power[Erfi[Plus[Optional[Pattern[a, Blank[]]], Times[Optional[Pattern[b, Blank[]]], Pattern[x, Blank[]]]]], Optional[Pattern[n, Blank[]]]], Power[Plus[Optional[Pattern[c, Blank[]]], Times[Optional[Pattern[d, Blank[]]], Pattern[x, Blank[]]]], Optional[Pattern[m, Blank[]]]]], Pattern[x, Blank[Symbol]]] := Unintegrable[(c + d*x)^m*Erfi[a + b*x]^n, x] /; FreeQ[{a, b, c, d, m, n}, x]
  6373. Int[Times[Power[E, Plus[Optional[Pattern[c, Blank[]]], Times[Optional[Pattern[d, Blank[]]], Power[Pattern[x, Blank[]], 2]]]], Power[Erf[Times[Optional[Pattern[b, Blank[]]], Pattern[x, Blank[]]]], Optional[Pattern[n, Blank[]]]]], Pattern[x, Blank[Symbol]]] := Dist[(E^c*Sqrt[Pi])/(2*b), Subst[Int[x^n, x], x, Erf[b*x]], x] /; FreeQ[{b, c, d, n}, x] && EqQ[d, -b^2]
  6374. Int[Times[Power[E, Plus[Optional[Pattern[c, Blank[]]], Times[Optional[Pattern[d, Blank[]]], Power[Pattern[x, Blank[]], 2]]]], Power[Erfc[Times[Optional[Pattern[b, Blank[]]], Pattern[x, Blank[]]]], Optional[Pattern[n, Blank[]]]]], Pattern[x, Blank[Symbol]]] := -Dist[(E^c*Sqrt[Pi])/(2*b), Subst[Int[x^n, x], x, Erfc[b*x]], x] /; FreeQ[{b, c, d, n}, x] && EqQ[d, -b^2]
  6375. Int[Times[Power[E, Plus[Optional[Pattern[c, Blank[]]], Times[Optional[Pattern[d, Blank[]]], Power[Pattern[x, Blank[]], 2]]]], Power[Erfi[Times[Optional[Pattern[b, Blank[]]], Pattern[x, Blank[]]]], Optional[Pattern[n, Blank[]]]]], Pattern[x, Blank[Symbol]]] := Dist[(E^c*Sqrt[Pi])/(2*b), Subst[Int[x^n, x], x, Erfi[b*x]], x] /; FreeQ[{b, c, d, n}, x] && EqQ[d, b^2]
  6376. Int[Times[Power[E, Plus[Optional[Pattern[c, Blank[]]], Times[Optional[Pattern[d, Blank[]]], Power[Pattern[x, Blank[]], 2]]]], Erf[Times[Optional[Pattern[b, Blank[]]], Pattern[x, Blank[]]]]], Pattern[x, Blank[Symbol]]] := Simp[(b*E^c*x^2*HypergeometricPFQ[{1, 1}, {3/2, 2}, b^2*x^2])/Sqrt[Pi], x] /; FreeQ[{b, c, d}, x] && EqQ[d, b^2]
  6377. Int[Times[Power[E, Plus[Optional[Pattern[c, Blank[]]], Times[Optional[Pattern[d, Blank[]]], Power[Pattern[x, Blank[]], 2]]]], Erfc[Times[Optional[Pattern[b, Blank[]]], Pattern[x, Blank[]]]]], Pattern[x, Blank[Symbol]]] := Int[E^(c + d*x^2), x] - Int[E^(c + d*x^2)*Erf[b*x], x] /; FreeQ[{b, c, d}, x] && EqQ[d, b^2]
  6378. Int[Times[Power[E, Plus[Optional[Pattern[c, Blank[]]], Times[Optional[Pattern[d, Blank[]]], Power[Pattern[x, Blank[]], 2]]]], Erfi[Times[Optional[Pattern[b, Blank[]]], Pattern[x, Blank[]]]]], Pattern[x, Blank[Symbol]]] := Simp[(b*E^c*x^2*HypergeometricPFQ[{1, 1}, {3/2, 2}, -(b^2*x^2)])/Sqrt[Pi], x] /; FreeQ[{b, c, d}, x] && EqQ[d, -b^2]
  6379. Int[Times[Power[E, Plus[Optional[Pattern[c, Blank[]]], Times[Optional[Pattern[d, Blank[]]], Power[Pattern[x, Blank[]], 2]]]], Power[Erf[Plus[Optional[Pattern[a, Blank[]]], Times[Optional[Pattern[b, Blank[]]], Pattern[x, Blank[]]]]], Optional[Pattern[n, Blank[]]]]], Pattern[x, Blank[Symbol]]] := Unintegrable[E^(c + d*x^2)*Erf[a + b*x]^n, x] /; FreeQ[{a, b, c, d, n}, x]
  6380. Int[Times[Power[E, Plus[Optional[Pattern[c, Blank[]]], Times[Optional[Pattern[d, Blank[]]], Power[Pattern[x, Blank[]], 2]]]], Power[Erfc[Plus[Optional[Pattern[a, Blank[]]], Times[Optional[Pattern[b, Blank[]]], Pattern[x, Blank[]]]]], Optional[Pattern[n, Blank[]]]]], Pattern[x, Blank[Symbol]]] := Unintegrable[E^(c + d*x^2)*Erfc[a + b*x]^n, x] /; FreeQ[{a, b, c, d, n}, x]
  6381. Int[Times[Power[E, Plus[Optional[Pattern[c, Blank[]]], Times[Optional[Pattern[d, Blank[]]], Power[Pattern[x, Blank[]], 2]]]], Power[Erfi[Plus[Optional[Pattern[a, Blank[]]], Times[Optional[Pattern[b, Blank[]]], Pattern[x, Blank[]]]]], Optional[Pattern[n, Blank[]]]]], Pattern[x, Blank[Symbol]]] := Unintegrable[E^(c + d*x^2)*Erfi[a + b*x]^n, x] /; FreeQ[{a, b, c, d, n}, x]
  6382. Int[Times[Power[E, Plus[Optional[Pattern[c, Blank[]]], Times[Optional[Pattern[d, Blank[]]], Power[Pattern[x, Blank[]], 2]]]], Erf[Plus[Optional[Pattern[a, Blank[]]], Times[Optional[Pattern[b, Blank[]]], Pattern[x, Blank[]]]]], Pattern[x, Blank[]]], Pattern[x, Blank[Symbol]]] := Simp[(E^(c + d*x^2)*Erf[a + b*x])/(2*d), x] - Dist[b/(d*Sqrt[Pi]), Int[E^(-a^2 + c - 2*a*b*x - (b^2 - d)*x^2), x], x] /; FreeQ[{a, b, c, d}, x]
  6383. Int[Times[Power[E, Plus[Optional[Pattern[c, Blank[]]], Times[Optional[Pattern[d, Blank[]]], Power[Pattern[x, Blank[]], 2]]]], Erfc[Plus[Optional[Pattern[a, Blank[]]], Times[Optional[Pattern[b, Blank[]]], Pattern[x, Blank[]]]]], Pattern[x, Blank[]]], Pattern[x, Blank[Symbol]]] := Simp[(E^(c + d*x^2)*Erfc[a + b*x])/(2*d), x] + Dist[b/(d*Sqrt[Pi]), Int[E^(-a^2 + c - 2*a*b*x - (b^2 - d)*x^2), x], x] /; FreeQ[{a, b, c, d}, x]
  6384. Int[Times[Power[E, Plus[Optional[Pattern[c, Blank[]]], Times[Optional[Pattern[d, Blank[]]], Power[Pattern[x, Blank[]], 2]]]], Erfi[Plus[Optional[Pattern[a, Blank[]]], Times[Optional[Pattern[b, Blank[]]], Pattern[x, Blank[]]]]], Pattern[x, Blank[]]], Pattern[x, Blank[Symbol]]] := Simp[(E^(c + d*x^2)*Erfi[a + b*x])/(2*d), x] - Dist[b/(d*Sqrt[Pi]), Int[E^(a^2 + c + 2*a*b*x + (b^2 + d)*x^2), x], x] /; FreeQ[{a, b, c, d}, x]
  6385. Int[Times[Power[E, Plus[Optional[Pattern[c, Blank[]]], Times[Optional[Pattern[d, Blank[]]], Power[Pattern[x, Blank[]], 2]]]], Erf[Plus[Optional[Pattern[a, Blank[]]], Times[Optional[Pattern[b, Blank[]]], Pattern[x, Blank[]]]]], Power[Pattern[x, Blank[]], Pattern[m, Blank[]]]], Pattern[x, Blank[Symbol]]] := Simp[(x^(m - 1)*E^(c + d*x^2)*Erf[a + b*x])/(2*d), x] + (-Dist[(m - 1)/(2*d), Int[x^(m - 2)*E^(c + d*x^2)*Erf[a + b*x], x], x] - Dist[b/(d*Sqrt[Pi]), Int[x^(m - 1)*E^(-a^2 + c - 2*a*b*x - (b^2 - d)*x^2), x], x]) /; FreeQ[{a, b, c, d}, x] && IGtQ[m, 1]
  6386. Int[Times[Power[E, Plus[Optional[Pattern[c, Blank[]]], Times[Optional[Pattern[d, Blank[]]], Power[Pattern[x, Blank[]], 2]]]], Erfc[Plus[Optional[Pattern[a, Blank[]]], Times[Optional[Pattern[b, Blank[]]], Pattern[x, Blank[]]]]], Power[Pattern[x, Blank[]], Pattern[m, Blank[]]]], Pattern[x, Blank[Symbol]]] := Simp[(x^(m - 1)*E^(c + d*x^2)*Erfc[a + b*x])/(2*d), x] + (-Dist[(m - 1)/(2*d), Int[x^(m - 2)*E^(c + d*x^2)*Erfc[a + b*x], x], x] + Dist[b/(d*Sqrt[Pi]), Int[x^(m - 1)*E^(-a^2 + c - 2*a*b*x - (b^2 - d)*x^2), x], x]) /; FreeQ[{a, b, c, d}, x] && IGtQ[m, 1]
  6387. Int[Times[Power[E, Plus[Optional[Pattern[c, Blank[]]], Times[Optional[Pattern[d, Blank[]]], Power[Pattern[x, Blank[]], 2]]]], Erfi[Plus[Optional[Pattern[a, Blank[]]], Times[Optional[Pattern[b, Blank[]]], Pattern[x, Blank[]]]]], Power[Pattern[x, Blank[]], Pattern[m, Blank[]]]], Pattern[x, Blank[Symbol]]] := Simp[(x^(m - 1)*E^(c + d*x^2)*Erfi[a + b*x])/(2*d), x] + (-Dist[(m - 1)/(2*d), Int[x^(m - 2)*E^(c + d*x^2)*Erfi[a + b*x], x], x] - Dist[b/(d*Sqrt[Pi]), Int[x^(m - 1)*E^(a^2 + c + 2*a*b*x + (b^2 + d)*x^2), x], x]) /; FreeQ[{a, b, c, d}, x] && IGtQ[m, 1]
  6388. Int[Times[Power[E, Plus[Optional[Pattern[c, Blank[]]], Times[Optional[Pattern[d, Blank[]]], Power[Pattern[x, Blank[]], 2]]]], Erf[Times[Optional[Pattern[b, Blank[]]], Pattern[x, Blank[]]]], Power[Pattern[x, Blank[]], -1]], Pattern[x, Blank[Symbol]]] := Simp[(2*b*E^c*x*HypergeometricPFQ[{1/2, 1}, {3/2, 3/2}, b^2*x^2])/Sqrt[Pi], x] /; FreeQ[{b, c, d}, x] && EqQ[d, b^2]
  6389. Int[Times[Power[E, Plus[Optional[Pattern[c, Blank[]]], Times[Optional[Pattern[d, Blank[]]], Power[Pattern[x, Blank[]], 2]]]], Erfc[Times[Optional[Pattern[b, Blank[]]], Pattern[x, Blank[]]]], Power[Pattern[x, Blank[]], -1]], Pattern[x, Blank[Symbol]]] := Int[E^(c + d*x^2)/x, x] - Int[(E^(c + d*x^2)*Erf[b*x])/x, x] /; FreeQ[{b, c, d}, x] && EqQ[d, b^2]
  6390. Int[Times[Power[E, Plus[Optional[Pattern[c, Blank[]]], Times[Optional[Pattern[d, Blank[]]], Power[Pattern[x, Blank[]], 2]]]], Erfi[Times[Optional[Pattern[b, Blank[]]], Pattern[x, Blank[]]]], Power[Pattern[x, Blank[]], -1]], Pattern[x, Blank[Symbol]]] := Simp[(2*b*E^c*x*HypergeometricPFQ[{1/2, 1}, {3/2, 3/2}, -(b^2*x^2)])/Sqrt[Pi], x] /; FreeQ[{b, c, d}, x] && EqQ[d, -b^2]
  6391. Int[Times[Power[E, Plus[Optional[Pattern[c, Blank[]]], Times[Optional[Pattern[d, Blank[]]], Power[Pattern[x, Blank[]], 2]]]], Erf[Plus[Optional[Pattern[a, Blank[]]], Times[Optional[Pattern[b, Blank[]]], Pattern[x, Blank[]]]]], Power[Pattern[x, Blank[]], Pattern[m, Blank[]]]], Pattern[x, Blank[Symbol]]] := Simp[(x^(m + 1)*E^(c + d*x^2)*Erf[a + b*x])/(m + 1), x] + (-Dist[(2*d)/(m + 1), Int[x^(m + 2)*E^(c + d*x^2)*Erf[a + b*x], x], x] - Dist[(2*b)/((m + 1)*Sqrt[Pi]), Int[x^(m + 1)*E^(-a^2 + c - 2*a*b*x - (b^2 - d)*x^2), x], x]) /; FreeQ[{a, b, c, d}, x] && ILtQ[m, -1]
  6392. Int[Times[Power[E, Plus[Optional[Pattern[c, Blank[]]], Times[Optional[Pattern[d, Blank[]]], Power[Pattern[x, Blank[]], 2]]]], Erfc[Plus[Optional[Pattern[a, Blank[]]], Times[Optional[Pattern[b, Blank[]]], Pattern[x, Blank[]]]]], Power[Pattern[x, Blank[]], Pattern[m, Blank[]]]], Pattern[x, Blank[Symbol]]] := Simp[(x^(m + 1)*E^(c + d*x^2)*Erfc[a + b*x])/(m + 1), x] + (-Dist[(2*d)/(m + 1), Int[x^(m + 2)*E^(c + d*x^2)*Erfc[a + b*x], x], x] + Dist[(2*b)/((m + 1)*Sqrt[Pi]), Int[x^(m + 1)*E^(-a^2 + c - 2*a*b*x - (b^2 - d)*x^2), x], x]) /; FreeQ[{a, b, c, d}, x] && ILtQ[m, -1]
  6393. Int[Times[Power[E, Plus[Optional[Pattern[c, Blank[]]], Times[Optional[Pattern[d, Blank[]]], Power[Pattern[x, Blank[]], 2]]]], Erfi[Plus[Optional[Pattern[a, Blank[]]], Times[Optional[Pattern[b, Blank[]]], Pattern[x, Blank[]]]]], Power[Pattern[x, Blank[]], Pattern[m, Blank[]]]], Pattern[x, Blank[Symbol]]] := Simp[(x^(m + 1)*E^(c + d*x^2)*Erfi[a + b*x])/(m + 1), x] + (-Dist[(2*d)/(m + 1), Int[x^(m + 2)*E^(c + d*x^2)*Erfi[a + b*x], x], x] - Dist[(2*b)/((m + 1)*Sqrt[Pi]), Int[x^(m + 1)*E^(a^2 + c + 2*a*b*x + (b^2 + d)*x^2), x], x]) /; FreeQ[{a, b, c, d}, x] && ILtQ[m, -1]
  6394. Int[Times[Power[E, Plus[Optional[Pattern[c, Blank[]]], Times[Optional[Pattern[d, Blank[]]], Power[Pattern[x, Blank[]], 2]]]], Power[Erf[Plus[Optional[Pattern[a, Blank[]]], Times[Optional[Pattern[b, Blank[]]], Pattern[x, Blank[]]]]], Optional[Pattern[n, Blank[]]]], Power[Times[Optional[Pattern[e, Blank[]]], Pattern[x, Blank[]]], Optional[Pattern[m, Blank[]]]]], Pattern[x, Blank[Symbol]]] := Unintegrable[E^(c + d*x^2)*(e*x)^m*Erf[a + b*x]^n, x] /; FreeQ[{a, b, c, d, e, m, n}, x]
  6395. Int[Times[Power[E, Plus[Optional[Pattern[c, Blank[]]], Times[Optional[Pattern[d, Blank[]]], Power[Pattern[x, Blank[]], 2]]]], Power[Erfc[Plus[Optional[Pattern[a, Blank[]]], Times[Optional[Pattern[b, Blank[]]], Pattern[x, Blank[]]]]], Optional[Pattern[n, Blank[]]]], Power[Times[Optional[Pattern[e, Blank[]]], Pattern[x, Blank[]]], Optional[Pattern[m, Blank[]]]]], Pattern[x, Blank[Symbol]]] := Unintegrable[E^(c + d*x^2)*(e*x)^m*Erfc[a + b*x]^n, x] /; FreeQ[{a, b, c, d, e, m, n}, x]
  6396. Int[Times[Power[E, Plus[Optional[Pattern[c, Blank[]]], Times[Optional[Pattern[d, Blank[]]], Power[Pattern[x, Blank[]], 2]]]], Power[Erfi[Plus[Optional[Pattern[a, Blank[]]], Times[Optional[Pattern[b, Blank[]]], Pattern[x, Blank[]]]]], Optional[Pattern[n, Blank[]]]], Power[Times[Optional[Pattern[e, Blank[]]], Pattern[x, Blank[]]], Optional[Pattern[m, Blank[]]]]], Pattern[x, Blank[Symbol]]] := Unintegrable[E^(c + d*x^2)*(e*x)^m*Erfi[a + b*x]^n, x] /; FreeQ[{a, b, c, d, e, m, n}, x]
  6397. Int[Erf[Times[Plus[Optional[Pattern[a, Blank[]]], Times[Log[Times[Optional[Pattern[c, Blank[]]], Power[Pattern[x, Blank[]], Optional[Pattern[n, Blank[]]]]]], Optional[Pattern[b, Blank[]]]]], Optional[Pattern[d, Blank[]]]]], Pattern[x, Blank[Symbol]]] := Simp[x*Erf[d*(a + b*Log[c*x^n])], x] - Dist[(2*b*d*n)/Sqrt[Pi], Int[1/E^(d*(a + b*Log[c*x^n]))^2, x], x] /; FreeQ[{a, b, c, d, n}, x]
  6398. Int[Erfc[Times[Plus[Optional[Pattern[a, Blank[]]], Times[Log[Times[Optional[Pattern[c, Blank[]]], Power[Pattern[x, Blank[]], Optional[Pattern[n, Blank[]]]]]], Optional[Pattern[b, Blank[]]]]], Optional[Pattern[d, Blank[]]]]], Pattern[x, Blank[Symbol]]] := Simp[x*Erfc[d*(a + b*Log[c*x^n])], x] + Dist[(2*b*d*n)/Sqrt[Pi], Int[1/E^(d*(a + b*Log[c*x^n]))^2, x], x] /; FreeQ[{a, b, c, d, n}, x]
  6399. Int[Erfi[Times[Plus[Optional[Pattern[a, Blank[]]], Times[Log[Times[Optional[Pattern[c, Blank[]]], Power[Pattern[x, Blank[]], Optional[Pattern[n, Blank[]]]]]], Optional[Pattern[b, Blank[]]]]], Optional[Pattern[d, Blank[]]]]], Pattern[x, Blank[Symbol]]] := Simp[x*Erfi[d*(a + b*Log[c*x^n])], x] - Dist[(2*b*d*n)/Sqrt[Pi], Int[E^(d*(a + b*Log[c*x^n]))^2, x], x] /; FreeQ[{a, b, c, d, n}, x]
  6400. Int[Times[Power[Pattern[x, Blank[]], -1], Pattern[F, Blank[]][Times[Plus[Optional[Pattern[a, Blank[]]], Times[Log[Times[Optional[Pattern[c, Blank[]]], Power[Pattern[x, Blank[]], Optional[Pattern[n, Blank[]]]]]], Optional[Pattern[b, Blank[]]]]], Optional[Pattern[d, Blank[]]]]]], Pattern[x, Blank[Symbol]]] := Dist[1/n, Subst[F[d*(a + b*x)], x, Log[c*x^n]], x] /; FreeQ[{a, b, c, d, n}, x] && MemberQ[{Erf, Erfc, Erfi}, F]
  6401. Int[Times[Erf[Times[Plus[Optional[Pattern[a, Blank[]]], Times[Log[Times[Optional[Pattern[c, Blank[]]], Power[Pattern[x, Blank[]], Optional[Pattern[n, Blank[]]]]]], Optional[Pattern[b, Blank[]]]]], Optional[Pattern[d, Blank[]]]]], Power[Times[Optional[Pattern[e, Blank[]]], Pattern[x, Blank[]]], Optional[Pattern[m, Blank[]]]]], Pattern[x, Blank[Symbol]]] := Simp[((e*x)^(m + 1)*Erf[d*(a + b*Log[c*x^n])])/(e*(m + 1)), x] - Dist[(2*b*d*n)/(Sqrt[Pi]*(m + 1)), Int[(e*x)^m/E^(d*(a + b*Log[c*x^n]))^2, x], x] /; FreeQ[{a, b, c, d, e, m, n}, x] && NeQ[m, -1]
  6402. Int[Times[Erfc[Times[Plus[Optional[Pattern[a, Blank[]]], Times[Log[Times[Optional[Pattern[c, Blank[]]], Power[Pattern[x, Blank[]], Optional[Pattern[n, Blank[]]]]]], Optional[Pattern[b, Blank[]]]]], Optional[Pattern[d, Blank[]]]]], Power[Times[Optional[Pattern[e, Blank[]]], Pattern[x, Blank[]]], Optional[Pattern[m, Blank[]]]]], Pattern[x, Blank[Symbol]]] := Simp[((e*x)^(m + 1)*Erfc[d*(a + b*Log[c*x^n])])/(e*(m + 1)), x] + Dist[(2*b*d*n)/(Sqrt[Pi]*(m + 1)), Int[(e*x)^m/E^(d*(a + b*Log[c*x^n]))^2, x], x] /; FreeQ[{a, b, c, d, e, m, n}, x] && NeQ[m, -1]
  6403. Int[Times[Erfi[Times[Plus[Optional[Pattern[a, Blank[]]], Times[Log[Times[Optional[Pattern[c, Blank[]]], Power[Pattern[x, Blank[]], Optional[Pattern[n, Blank[]]]]]], Optional[Pattern[b, Blank[]]]]], Optional[Pattern[d, Blank[]]]]], Power[Times[Optional[Pattern[e, Blank[]]], Pattern[x, Blank[]]], Optional[Pattern[m, Blank[]]]]], Pattern[x, Blank[Symbol]]] := Simp[((e*x)^(m + 1)*Erfi[d*(a + b*Log[c*x^n])])/(e*(m + 1)), x] - Dist[(2*b*d*n)/(Sqrt[Pi]*(m + 1)), Int[(e*x)^m*E^(d*(a + b*Log[c*x^n]))^2, x], x] /; FreeQ[{a, b, c, d, e, m, n}, x] && NeQ[m, -1]
  6404. Int[Times[Erf[Times[Optional[Pattern[b, Blank[]]], Pattern[x, Blank[]]]], Sin[Plus[Optional[Pattern[c, Blank[]]], Times[Optional[Pattern[d, Blank[]]], Power[Pattern[x, Blank[]], 2]]]]], Pattern[x, Blank[Symbol]]] := Dist[I/2, Int[E^(-(I*c) - I*d*x^2)*Erf[b*x], x], x] - Dist[I/2, Int[E^(I*c + I*d*x^2)*Erf[b*x], x], x] /; FreeQ[{b, c, d}, x] && EqQ[d^2, -b^4]
  6405. Int[Times[Erfc[Times[Optional[Pattern[b, Blank[]]], Pattern[x, Blank[]]]], Sin[Plus[Optional[Pattern[c, Blank[]]], Times[Optional[Pattern[d, Blank[]]], Power[Pattern[x, Blank[]], 2]]]]], Pattern[x, Blank[Symbol]]] := Dist[I/2, Int[E^(-(I*c) - I*d*x^2)*Erfc[b*x], x], x] - Dist[I/2, Int[E^(I*c + I*d*x^2)*Erfc[b*x], x], x] /; FreeQ[{b, c, d}, x] && EqQ[d^2, -b^4]
  6406. Int[Times[Erfi[Times[Optional[Pattern[b, Blank[]]], Pattern[x, Blank[]]]], Sin[Plus[Optional[Pattern[c, Blank[]]], Times[Optional[Pattern[d, Blank[]]], Power[Pattern[x, Blank[]], 2]]]]], Pattern[x, Blank[Symbol]]] := Dist[I/2, Int[E^(-(I*c) - I*d*x^2)*Erfi[b*x], x], x] - Dist[I/2, Int[E^(I*c + I*d*x^2)*Erfi[b*x], x], x] /; FreeQ[{b, c, d}, x] && EqQ[d^2, -b^4]
  6407. Int[Times[Cos[Plus[Optional[Pattern[c, Blank[]]], Times[Optional[Pattern[d, Blank[]]], Power[Pattern[x, Blank[]], 2]]]], Erf[Times[Optional[Pattern[b, Blank[]]], Pattern[x, Blank[]]]]], Pattern[x, Blank[Symbol]]] := Dist[1/2, Int[E^(-(I*c) - I*d*x^2)*Erf[b*x], x], x] + Dist[1/2, Int[E^(I*c + I*d*x^2)*Erf[b*x], x], x] /; FreeQ[{b, c, d}, x] && EqQ[d^2, -b^4]
  6408. Int[Times[Cos[Plus[Optional[Pattern[c, Blank[]]], Times[Optional[Pattern[d, Blank[]]], Power[Pattern[x, Blank[]], 2]]]], Erfc[Times[Optional[Pattern[b, Blank[]]], Pattern[x, Blank[]]]]], Pattern[x, Blank[Symbol]]] := Dist[1/2, Int[E^(-(I*c) - I*d*x^2)*Erfc[b*x], x], x] + Dist[1/2, Int[E^(I*c + I*d*x^2)*Erfc[b*x], x], x] /; FreeQ[{b, c, d}, x] && EqQ[d^2, -b^4]
  6409. Int[Times[Cos[Plus[Optional[Pattern[c, Blank[]]], Times[Optional[Pattern[d, Blank[]]], Power[Pattern[x, Blank[]], 2]]]], Erfi[Times[Optional[Pattern[b, Blank[]]], Pattern[x, Blank[]]]]], Pattern[x, Blank[Symbol]]] := Dist[1/2, Int[E^(-(I*c) - I*d*x^2)*Erfi[b*x], x], x] + Dist[1/2, Int[E^(I*c + I*d*x^2)*Erfi[b*x], x], x] /; FreeQ[{b, c, d}, x] && EqQ[d^2, -b^4]
  6410. Int[Times[Erf[Times[Optional[Pattern[b, Blank[]]], Pattern[x, Blank[]]]], Sinh[Plus[Optional[Pattern[c, Blank[]]], Times[Optional[Pattern[d, Blank[]]], Power[Pattern[x, Blank[]], 2]]]]], Pattern[x, Blank[Symbol]]] := Dist[1/2, Int[E^(c + d*x^2)*Erf[b*x], x], x] - Dist[1/2, Int[E^(-c - d*x^2)*Erf[b*x], x], x] /; FreeQ[{b, c, d}, x] && EqQ[d^2, b^4]
  6411. Int[Times[Erfc[Times[Optional[Pattern[b, Blank[]]], Pattern[x, Blank[]]]], Sinh[Plus[Optional[Pattern[c, Blank[]]], Times[Optional[Pattern[d, Blank[]]], Power[Pattern[x, Blank[]], 2]]]]], Pattern[x, Blank[Symbol]]] := Dist[1/2, Int[E^(c + d*x^2)*Erfc[b*x], x], x] - Dist[1/2, Int[E^(-c - d*x^2)*Erfc[b*x], x], x] /; FreeQ[{b, c, d}, x] && EqQ[d^2, b^4]
  6412. Int[Times[Erfi[Times[Optional[Pattern[b, Blank[]]], Pattern[x, Blank[]]]], Sinh[Plus[Optional[Pattern[c, Blank[]]], Times[Optional[Pattern[d, Blank[]]], Power[Pattern[x, Blank[]], 2]]]]], Pattern[x, Blank[Symbol]]] := Dist[1/2, Int[E^(c + d*x^2)*Erfi[b*x], x], x] - Dist[1/2, Int[E^(-c - d*x^2)*Erfi[b*x], x], x] /; FreeQ[{b, c, d}, x] && EqQ[d^2, b^4]
  6413. Int[Times[Cosh[Plus[Optional[Pattern[c, Blank[]]], Times[Optional[Pattern[d, Blank[]]], Power[Pattern[x, Blank[]], 2]]]], Erf[Times[Optional[Pattern[b, Blank[]]], Pattern[x, Blank[]]]]], Pattern[x, Blank[Symbol]]] := Dist[1/2, Int[E^(c + d*x^2)*Erf[b*x], x], x] + Dist[1/2, Int[E^(-c - d*x^2)*Erf[b*x], x], x] /; FreeQ[{b, c, d}, x] && EqQ[d^2, b^4]
  6414. Int[Times[Cosh[Plus[Optional[Pattern[c, Blank[]]], Times[Optional[Pattern[d, Blank[]]], Power[Pattern[x, Blank[]], 2]]]], Erfc[Times[Optional[Pattern[b, Blank[]]], Pattern[x, Blank[]]]]], Pattern[x, Blank[Symbol]]] := Dist[1/2, Int[E^(c + d*x^2)*Erfc[b*x], x], x] + Dist[1/2, Int[E^(-c - d*x^2)*Erfc[b*x], x], x] /; FreeQ[{b, c, d}, x] && EqQ[d^2, b^4]
  6415. Int[Times[Cosh[Plus[Optional[Pattern[c, Blank[]]], Times[Optional[Pattern[d, Blank[]]], Power[Pattern[x, Blank[]], 2]]]], Erfi[Times[Optional[Pattern[b, Blank[]]], Pattern[x, Blank[]]]]], Pattern[x, Blank[Symbol]]] := Dist[1/2, Int[E^(c + d*x^2)*Erfi[b*x], x], x] + Dist[1/2, Int[E^(-c - d*x^2)*Erfi[b*x], x], x] /; FreeQ[{b, c, d}, x] && EqQ[d^2, b^4]
  6416. Int[Pattern[F, Blank[]][Times[Plus[Optional[Pattern[a, Blank[]]], Times[Log[Times[Optional[Pattern[c, Blank[]]], Power[Plus[Pattern[d, Blank[]], Times[Optional[Pattern[e, Blank[]]], Pattern[x, Blank[]]]], Optional[Pattern[n, Blank[]]]]]], Optional[Pattern[b, Blank[]]]]], Optional[Pattern[f, Blank[]]]]], Pattern[x, Blank[Symbol]]] := Dist[1/e, Subst[Int[F[f*(a + b*Log[c*x^n])], x], x, d + e*x], x] /; FreeQ[{a, b, c, d, e, f, n}, x] && MemberQ[{Erf, Erfc, Erfi, FresnelS, FresnelC, ExpIntegralEi, SinIntegral, CosIntegral, SinhIntegral, CoshIntegral}, F]
  6417. Int[Times[Power[Plus[Pattern[g, Blank[]], Times[Optional[Pattern[h, Blank[]]], Pattern[x, Blank[]]]], Optional[Pattern[m, Blank[]]]], Pattern[F, Blank[]][Times[Plus[Optional[Pattern[a, Blank[]]], Times[Log[Times[Optional[Pattern[c, Blank[]]], Power[Plus[Pattern[d, Blank[]], Times[Optional[Pattern[e, Blank[]]], Pattern[x, Blank[]]]], Optional[Pattern[n, Blank[]]]]]], Optional[Pattern[b, Blank[]]]]], Optional[Pattern[f, Blank[]]]]]], Pattern[x, Blank[Symbol]]] := Dist[1/e, Subst[Int[((g*x)/d)^m*F[f*(a + b*Log[c*x^n])], x], x, d + e*x], x] /; FreeQ[{a, b, c, d, e, f, g, m, n}, x] && EqQ[e*f - d*g, 0] && MemberQ[{Erf, Erfc, Erfi, FresnelS, FresnelC, ExpIntegralEi, SinIntegral, CosIntegral, SinhIntegral, CoshIntegral}, F]
  6418. Int[FresnelS[Plus[Optional[Pattern[a, Blank[]]], Times[Optional[Pattern[b, Blank[]]], Pattern[x, Blank[]]]]], Pattern[x, Blank[Symbol]]] := Simp[((a + b*x)*FresnelS[a + b*x])/b, x] + Simp[Cos[(Pi*(a + b*x)^2)/2]/(b*Pi), x] /; FreeQ[{a, b}, x]
  6419. Int[FresnelC[Plus[Optional[Pattern[a, Blank[]]], Times[Optional[Pattern[b, Blank[]]], Pattern[x, Blank[]]]]], Pattern[x, Blank[Symbol]]] := Simp[((a + b*x)*FresnelC[a + b*x])/b, x] - Simp[Sin[(Pi*(a + b*x)^2)/2]/(b*Pi), x] /; FreeQ[{a, b}, x]
  6420. Int[Power[FresnelS[Plus[Optional[Pattern[a, Blank[]]], Times[Optional[Pattern[b, Blank[]]], Pattern[x, Blank[]]]]], 2], Pattern[x, Blank[Symbol]]] := Simp[((a + b*x)*FresnelS[a + b*x]^2)/b, x] - Dist[2, Int[(a + b*x)*Sin[(Pi*(a + b*x)^2)/2]*FresnelS[a + b*x], x], x] /; FreeQ[{a, b}, x]
  6421. Int[Power[FresnelC[Plus[Optional[Pattern[a, Blank[]]], Times[Optional[Pattern[b, Blank[]]], Pattern[x, Blank[]]]]], 2], Pattern[x, Blank[Symbol]]] := Simp[((a + b*x)*FresnelC[a + b*x]^2)/b, x] - Dist[2, Int[(a + b*x)*Cos[(Pi*(a + b*x)^2)/2]*FresnelC[a + b*x], x], x] /; FreeQ[{a, b}, x]
  6422. Int[Power[FresnelS[Plus[Optional[Pattern[a, Blank[]]], Times[Optional[Pattern[b, Blank[]]], Pattern[x, Blank[]]]]], Pattern[n, Blank[]]], Pattern[x, Blank[Symbol]]] := Unintegrable[FresnelS[a + b*x]^n, x] /; FreeQ[{a, b, n}, x] && NeQ[n, 1] && NeQ[n, 2]
  6423. Int[Power[FresnelC[Plus[Optional[Pattern[a, Blank[]]], Times[Optional[Pattern[b, Blank[]]], Pattern[x, Blank[]]]]], Pattern[n, Blank[]]], Pattern[x, Blank[Symbol]]] := Unintegrable[FresnelC[a + b*x]^n, x] /; FreeQ[{a, b, n}, x] && NeQ[n, 1] && NeQ[n, 2]
  6424. Int[Times[FresnelS[Times[Optional[Pattern[b, Blank[]]], Pattern[x, Blank[]]]], Power[Pattern[x, Blank[]], -1]], Pattern[x, Blank[Symbol]]] := Dist[(1 + I)/4, Int[Erf[(Sqrt[Pi]*(1 + I)*b*x)/2]/x, x], x] + Dist[(1 - I)/4, Int[Erf[(Sqrt[Pi]*(1 - I)*b*x)/2]/x, x], x] /; FreeQ[b, x]
  6425. Int[Times[FresnelC[Times[Optional[Pattern[b, Blank[]]], Pattern[x, Blank[]]]], Power[Pattern[x, Blank[]], -1]], Pattern[x, Blank[Symbol]]] := Dist[(1 - I)/4, Int[Erf[(Sqrt[Pi]*(1 + I)*b*x)/2]/x, x], x] + Dist[(1 + I)/4, Int[Erf[(Sqrt[Pi]*(1 - I)*b*x)/2]/x, x], x] /; FreeQ[b, x]
  6426. Int[Times[FresnelS[Times[Optional[Pattern[b, Blank[]]], Pattern[x, Blank[]]]], Power[Times[Optional[Pattern[d, Blank[]]], Pattern[x, Blank[]]], Optional[Pattern[m, Blank[]]]]], Pattern[x, Blank[Symbol]]] := Simp[((d*x)^(m + 1)*FresnelS[b*x])/(d*(m + 1)), x] - Dist[b/(d*(m + 1)), Int[(d*x)^(m + 1)*Sin[(Pi*b^2*x^2)/2], x], x] /; FreeQ[{b, d, m}, x] && NeQ[m, -1]
  6427. Int[Times[FresnelC[Times[Optional[Pattern[b, Blank[]]], Pattern[x, Blank[]]]], Power[Times[Optional[Pattern[d, Blank[]]], Pattern[x, Blank[]]], Optional[Pattern[m, Blank[]]]]], Pattern[x, Blank[Symbol]]] := Simp[((d*x)^(m + 1)*FresnelC[b*x])/(d*(m + 1)), x] - Dist[b/(d*(m + 1)), Int[(d*x)^(m + 1)*Cos[(Pi*b^2*x^2)/2], x], x] /; FreeQ[{b, d, m}, x] && NeQ[m, -1]
  6428. Int[Times[FresnelS[Plus[Optional[Pattern[a, Blank[]]], Times[Optional[Pattern[b, Blank[]]], Pattern[x, Blank[]]]]], Power[Plus[Optional[Pattern[c, Blank[]]], Times[Optional[Pattern[d, Blank[]]], Pattern[x, Blank[]]]], Optional[Pattern[m, Blank[]]]]], Pattern[x, Blank[Symbol]]] := Simp[((c + d*x)^(m + 1)*FresnelS[a + b*x])/(d*(m + 1)), x] - Dist[b/(d*(m + 1)), Int[(c + d*x)^(m + 1)*Sin[(Pi*(a + b*x)^2)/2], x], x] /; FreeQ[{a, b, c, d}, x] && IGtQ[m, 0]
  6429. Int[Times[FresnelC[Plus[Optional[Pattern[a, Blank[]]], Times[Optional[Pattern[b, Blank[]]], Pattern[x, Blank[]]]]], Power[Plus[Optional[Pattern[c, Blank[]]], Times[Optional[Pattern[d, Blank[]]], Pattern[x, Blank[]]]], Optional[Pattern[m, Blank[]]]]], Pattern[x, Blank[Symbol]]] := Simp[((c + d*x)^(m + 1)*FresnelC[a + b*x])/(d*(m + 1)), x] - Dist[b/(d*(m + 1)), Int[(c + d*x)^(m + 1)*Cos[(Pi*(a + b*x)^2)/2], x], x] /; FreeQ[{a, b, c, d}, x] && IGtQ[m, 0]
  6430. Int[Times[Power[FresnelS[Times[Optional[Pattern[b, Blank[]]], Pattern[x, Blank[]]]], 2], Power[Pattern[x, Blank[]], Optional[Pattern[m, Blank[]]]]], Pattern[x, Blank[Symbol]]] := Simp[(x^(m + 1)*FresnelS[b*x]^2)/(m + 1), x] - Dist[(2*b)/(m + 1), Int[x^(m + 1)*Sin[(Pi*b^2*x^2)/2]*FresnelS[b*x], x], x] /; FreeQ[b, x] && IntegerQ[m] && NeQ[m, -1]
  6431. Int[Times[Power[FresnelC[Times[Optional[Pattern[b, Blank[]]], Pattern[x, Blank[]]]], 2], Power[Pattern[x, Blank[]], Optional[Pattern[m, Blank[]]]]], Pattern[x, Blank[Symbol]]] := Simp[(x^(m + 1)*FresnelC[b*x]^2)/(m + 1), x] - Dist[(2*b)/(m + 1), Int[x^(m + 1)*Cos[(Pi*b^2*x^2)/2]*FresnelC[b*x], x], x] /; FreeQ[b, x] && IntegerQ[m] && NeQ[m, -1]
  6432. Int[Times[Power[FresnelS[Plus[Pattern[a, Blank[]], Times[Optional[Pattern[b, Blank[]]], Pattern[x, Blank[]]]]], 2], Power[Plus[Optional[Pattern[c, Blank[]]], Times[Optional[Pattern[d, Blank[]]], Pattern[x, Blank[]]]], Optional[Pattern[m, Blank[]]]]], Pattern[x, Blank[Symbol]]] := Dist[1/b^(m + 1), Subst[Int[ExpandIntegrand[FresnelS[x]^2, (b*c - a*d + d*x)^m, x], x], x, a + b*x], x] /; FreeQ[{a, b, c, d}, x] && IGtQ[m, 0]
  6433. Int[Times[Power[FresnelC[Plus[Pattern[a, Blank[]], Times[Optional[Pattern[b, Blank[]]], Pattern[x, Blank[]]]]], 2], Power[Plus[Optional[Pattern[c, Blank[]]], Times[Optional[Pattern[d, Blank[]]], Pattern[x, Blank[]]]], Optional[Pattern[m, Blank[]]]]], Pattern[x, Blank[Symbol]]] := Dist[1/b^(m + 1), Subst[Int[ExpandIntegrand[FresnelC[x]^2, (b*c - a*d + d*x)^m, x], x], x, a + b*x], x] /; FreeQ[{a, b, c, d}, x] && IGtQ[m, 0]
  6434. Int[Times[Power[FresnelS[Plus[Optional[Pattern[a, Blank[]]], Times[Optional[Pattern[b, Blank[]]], Pattern[x, Blank[]]]]], Optional[Pattern[n, Blank[]]]], Power[Plus[Optional[Pattern[c, Blank[]]], Times[Optional[Pattern[d, Blank[]]], Pattern[x, Blank[]]]], Optional[Pattern[m, Blank[]]]]], Pattern[x, Blank[Symbol]]] := Unintegrable[(c + d*x)^m*FresnelS[a + b*x]^n, x] /; FreeQ[{a, b, c, d, m, n}, x]
  6435. Int[Times[Power[FresnelC[Plus[Optional[Pattern[a, Blank[]]], Times[Optional[Pattern[b, Blank[]]], Pattern[x, Blank[]]]]], Optional[Pattern[n, Blank[]]]], Power[Plus[Optional[Pattern[c, Blank[]]], Times[Optional[Pattern[d, Blank[]]], Pattern[x, Blank[]]]], Optional[Pattern[m, Blank[]]]]], Pattern[x, Blank[Symbol]]] := Unintegrable[(c + d*x)^m*FresnelC[a + b*x]^n, x] /; FreeQ[{a, b, c, d, m, n}, x]
  6436. Int[Times[Power[E, Plus[Optional[Pattern[c, Blank[]]], Times[Optional[Pattern[d, Blank[]]], Power[Pattern[x, Blank[]], 2]]]], FresnelS[Times[Optional[Pattern[b, Blank[]]], Pattern[x, Blank[]]]]], Pattern[x, Blank[Symbol]]] := Dist[(1 + I)/4, Int[E^(c + d*x^2)*Erf[(Sqrt[Pi]*(1 + I)*b*x)/2], x], x] + Dist[(1 - I)/4, Int[E^(c + d*x^2)*Erf[(Sqrt[Pi]*(1 - I)*b*x)/2], x], x] /; FreeQ[{b, c, d}, x] && EqQ[d^2, -((Pi^2*b^4)/4)]
  6437. Int[Times[Power[E, Plus[Optional[Pattern[c, Blank[]]], Times[Optional[Pattern[d, Blank[]]], Power[Pattern[x, Blank[]], 2]]]], FresnelC[Times[Optional[Pattern[b, Blank[]]], Pattern[x, Blank[]]]]], Pattern[x, Blank[Symbol]]] := Dist[(1 - I)/4, Int[E^(c + d*x^2)*Erf[(Sqrt[Pi]*(1 + I)*b*x)/2], x], x] + Dist[(1 + I)/4, Int[E^(c + d*x^2)*Erf[(Sqrt[Pi]*(1 - I)*b*x)/2], x], x] /; FreeQ[{b, c, d}, x] && EqQ[d^2, -((Pi^2*b^4)/4)]
  6438. Int[Times[Power[E, Plus[Optional[Pattern[c, Blank[]]], Times[Optional[Pattern[d, Blank[]]], Power[Pattern[x, Blank[]], 2]]]], Power[FresnelS[Plus[Optional[Pattern[a, Blank[]]], Times[Optional[Pattern[b, Blank[]]], Pattern[x, Blank[]]]]], Optional[Pattern[n, Blank[]]]]], Pattern[x, Blank[Symbol]]] := Unintegrable[E^(c + d*x^2)*FresnelS[a + b*x]^n, x] /; FreeQ[{a, b, c, d, n}, x]
  6439. Int[Times[Power[E, Plus[Optional[Pattern[c, Blank[]]], Times[Optional[Pattern[d, Blank[]]], Power[Pattern[x, Blank[]], 2]]]], Power[FresnelC[Plus[Optional[Pattern[a, Blank[]]], Times[Optional[Pattern[b, Blank[]]], Pattern[x, Blank[]]]]], Optional[Pattern[n, Blank[]]]]], Pattern[x, Blank[Symbol]]] := Unintegrable[E^(c + d*x^2)*FresnelC[a + b*x]^n, x] /; FreeQ[{a, b, c, d, n}, x]
  6440. Int[Times[Power[FresnelS[Times[Optional[Pattern[b, Blank[]]], Pattern[x, Blank[]]]], Optional[Pattern[n, Blank[]]]], Sin[Times[Optional[Pattern[d, Blank[]]], Power[Pattern[x, Blank[]], 2]]]], Pattern[x, Blank[Symbol]]] := Dist[(Pi*b)/(2*d), Subst[Int[x^n, x], x, FresnelS[b*x]], x] /; FreeQ[{b, d, n}, x] && EqQ[d^2, (Pi^2*b^4)/4]
  6441. Int[Times[Cos[Times[Optional[Pattern[d, Blank[]]], Power[Pattern[x, Blank[]], 2]]], Power[FresnelC[Times[Optional[Pattern[b, Blank[]]], Pattern[x, Blank[]]]], Optional[Pattern[n, Blank[]]]]], Pattern[x, Blank[Symbol]]] := Dist[(Pi*b)/(2*d), Subst[Int[x^n, x], x, FresnelC[b*x]], x] /; FreeQ[{b, d, n}, x] && EqQ[d^2, (Pi^2*b^4)/4]
  6442. Int[Times[FresnelS[Times[Optional[Pattern[b, Blank[]]], Pattern[x, Blank[]]]], Sin[Plus[Pattern[c, Blank[]], Times[Optional[Pattern[d, Blank[]]], Power[Pattern[x, Blank[]], 2]]]]], Pattern[x, Blank[Symbol]]] := Dist[Sin[c], Int[Cos[d*x^2]*FresnelS[b*x], x], x] + Dist[Cos[c], Int[Sin[d*x^2]*FresnelS[b*x], x], x] /; FreeQ[{b, c, d}, x] && EqQ[d^2, (Pi^2*b^4)/4]
  6443. Int[Times[Cos[Plus[Pattern[c, Blank[]], Times[Optional[Pattern[d, Blank[]]], Power[Pattern[x, Blank[]], 2]]]], FresnelC[Times[Optional[Pattern[b, Blank[]]], Pattern[x, Blank[]]]]], Pattern[x, Blank[Symbol]]] := Dist[Cos[c], Int[Cos[d*x^2]*FresnelC[b*x], x], x] - Dist[Sin[c], Int[Sin[d*x^2]*FresnelC[b*x], x], x] /; FreeQ[{b, c, d}, x] && EqQ[d^2, (Pi^2*b^4)/4]
  6444. Int[Times[Power[FresnelS[Plus[Optional[Pattern[a, Blank[]]], Times[Optional[Pattern[b, Blank[]]], Pattern[x, Blank[]]]]], Optional[Pattern[n, Blank[]]]], Sin[Plus[Optional[Pattern[c, Blank[]]], Times[Optional[Pattern[d, Blank[]]], Power[Pattern[x, Blank[]], 2]]]]], Pattern[x, Blank[Symbol]]] := Unintegrable[FresnelS[a + b*x]^n*Sin[c + d*x^2], x] /; FreeQ[{a, b, c, d, n}, x]
  6445. Int[Times[Cos[Plus[Optional[Pattern[c, Blank[]]], Times[Optional[Pattern[d, Blank[]]], Power[Pattern[x, Blank[]], 2]]]], Power[FresnelC[Plus[Optional[Pattern[a, Blank[]]], Times[Optional[Pattern[b, Blank[]]], Pattern[x, Blank[]]]]], Optional[Pattern[n, Blank[]]]]], Pattern[x, Blank[Symbol]]] := Unintegrable[Cos[c + d*x^2]*FresnelC[a + b*x]^n, x] /; FreeQ[{a, b, c, d, n}, x]
  6446. Int[Times[Cos[Times[Optional[Pattern[d, Blank[]]], Power[Pattern[x, Blank[]], 2]]], FresnelS[Times[Optional[Pattern[b, Blank[]]], Pattern[x, Blank[]]]]], Pattern[x, Blank[Symbol]]] := Simp[(FresnelC[b*x]*FresnelS[b*x])/(2*b), x] + (-Simp[(1*I*b*x^2*HypergeometricPFQ[{1, 1}, {3/2, 2}, -((I*b^2*Pi*x^2)/2)])/8, x] + Simp[(1*I*b*x^2*HypergeometricPFQ[{1, 1}, {3/2, 2}, (1*I*b^2*Pi*x^2)/2])/8, x]) /; FreeQ[{b, d}, x] && EqQ[d^2, (Pi^2*b^4)/4]
  6447. Int[Times[FresnelC[Times[Optional[Pattern[b, Blank[]]], Pattern[x, Blank[]]]], Sin[Times[Optional[Pattern[d, Blank[]]], Power[Pattern[x, Blank[]], 2]]]], Pattern[x, Blank[Symbol]]] := Simp[(b*Pi*FresnelC[b*x]*FresnelS[b*x])/(4*d), x] + (Simp[(1*I*b*x^2*HypergeometricPFQ[{1, 1}, {3/2, 2}, -(I*d*x^2)])/8, x] - Simp[(1*I*b*x^2*HypergeometricPFQ[{1, 1}, {3/2, 2}, I*d*x^2])/8, x]) /; FreeQ[{b, d}, x] && EqQ[d^2, (Pi^2*b^4)/4]
  6448. Int[Times[Cos[Plus[Pattern[c, Blank[]], Times[Optional[Pattern[d, Blank[]]], Power[Pattern[x, Blank[]], 2]]]], FresnelS[Times[Optional[Pattern[b, Blank[]]], Pattern[x, Blank[]]]]], Pattern[x, Blank[Symbol]]] := Dist[Cos[c], Int[Cos[d*x^2]*FresnelS[b*x], x], x] - Dist[Sin[c], Int[Sin[d*x^2]*FresnelS[b*x], x], x] /; FreeQ[{b, c, d}, x] && EqQ[d^2, (Pi^2*b^4)/4]
  6449. Int[Times[FresnelC[Times[Optional[Pattern[b, Blank[]]], Pattern[x, Blank[]]]], Sin[Plus[Pattern[c, Blank[]], Times[Optional[Pattern[d, Blank[]]], Power[Pattern[x, Blank[]], 2]]]]], Pattern[x, Blank[Symbol]]] := Dist[Sin[c], Int[Cos[d*x^2]*FresnelC[b*x], x], x] + Dist[Cos[c], Int[Sin[d*x^2]*FresnelC[b*x], x], x] /; FreeQ[{b, c, d}, x] && EqQ[d^2, (Pi^2*b^4)/4]
  6450. Int[Times[Cos[Plus[Optional[Pattern[c, Blank[]]], Times[Optional[Pattern[d, Blank[]]], Power[Pattern[x, Blank[]], 2]]]], Power[FresnelS[Plus[Optional[Pattern[a, Blank[]]], Times[Optional[Pattern[b, Blank[]]], Pattern[x, Blank[]]]]], Optional[Pattern[n, Blank[]]]]], Pattern[x, Blank[Symbol]]] := Unintegrable[Cos[c + d*x^2]*FresnelS[a + b*x]^n, x] /; FreeQ[{a, b, c, d, n}, x]
  6451. Int[Times[Power[FresnelC[Plus[Optional[Pattern[a, Blank[]]], Times[Optional[Pattern[b, Blank[]]], Pattern[x, Blank[]]]]], Optional[Pattern[n, Blank[]]]], Sin[Plus[Optional[Pattern[c, Blank[]]], Times[Optional[Pattern[d, Blank[]]], Power[Pattern[x, Blank[]], 2]]]]], Pattern[x, Blank[Symbol]]] := Unintegrable[FresnelC[a + b*x]^n*Sin[c + d*x^2], x] /; FreeQ[{a, b, c, d, n}, x]
  6452. Int[Times[FresnelS[Times[Optional[Pattern[b, Blank[]]], Pattern[x, Blank[]]]], Pattern[x, Blank[]], Sin[Times[Optional[Pattern[d, Blank[]]], Power[Pattern[x, Blank[]], 2]]]], Pattern[x, Blank[Symbol]]] := -Simp[(Cos[d*x^2]*FresnelS[b*x])/(2*d), x] + Dist[1/(2*b*Pi), Int[Sin[2*d*x^2], x], x] /; FreeQ[{b, d}, x] && EqQ[d^2, (Pi^2*b^4)/4]
  6453. Int[Times[Cos[Times[Optional[Pattern[d, Blank[]]], Power[Pattern[x, Blank[]], 2]]], FresnelC[Times[Optional[Pattern[b, Blank[]]], Pattern[x, Blank[]]]], Pattern[x, Blank[]]], Pattern[x, Blank[Symbol]]] := Simp[(Sin[d*x^2]*FresnelC[b*x])/(2*d), x] - Dist[b/(4*d), Int[Sin[2*d*x^2], x], x] /; FreeQ[{b, d}, x] && EqQ[d^2, (Pi^2*b^4)/4]
  6454. Int[Times[FresnelS[Times[Optional[Pattern[b, Blank[]]], Pattern[x, Blank[]]]], Power[Pattern[x, Blank[]], Pattern[m, Blank[]]], Sin[Times[Optional[Pattern[d, Blank[]]], Power[Pattern[x, Blank[]], 2]]]], Pattern[x, Blank[Symbol]]] := -Simp[(x^(m - 1)*Cos[d*x^2]*FresnelS[b*x])/(2*d), x] + (Dist[(m - 1)/(2*d), Int[x^(m - 2)*Cos[d*x^2]*FresnelS[b*x], x], x] + Dist[1/(2*b*Pi), Int[x^(m - 1)*Sin[2*d*x^2], x], x]) /; FreeQ[{b, d}, x] && EqQ[d^2, (Pi^2*b^4)/4] && IGtQ[m, 1]
  6455. Int[Times[Cos[Times[Optional[Pattern[d, Blank[]]], Power[Pattern[x, Blank[]], 2]]], FresnelC[Times[Optional[Pattern[b, Blank[]]], Pattern[x, Blank[]]]], Power[Pattern[x, Blank[]], Pattern[m, Blank[]]]], Pattern[x, Blank[Symbol]]] := Simp[(x^(m - 1)*Sin[d*x^2]*FresnelC[b*x])/(2*d), x] + (-Dist[(m - 1)/(2*d), Int[x^(m - 2)*Sin[d*x^2]*FresnelC[b*x], x], x] - Dist[b/(4*d), Int[x^(m - 1)*Sin[2*d*x^2], x], x]) /; FreeQ[{b, d}, x] && EqQ[d^2, (Pi^2*b^4)/4] && IGtQ[m, 1]
  6456. Int[Times[FresnelS[Times[Optional[Pattern[b, Blank[]]], Pattern[x, Blank[]]]], Power[Pattern[x, Blank[]], Pattern[m, Blank[]]], Sin[Times[Optional[Pattern[d, Blank[]]], Power[Pattern[x, Blank[]], 2]]]], Pattern[x, Blank[Symbol]]] := Simp[(x^(m + 1)*Sin[d*x^2]*FresnelS[b*x])/(m + 1), x] + (-Dist[(2*d)/(m + 1), Int[x^(m + 2)*Cos[d*x^2]*FresnelS[b*x], x], x] + Dist[d/(Pi*b*(m + 1)), Int[x^(m + 1)*Cos[2*d*x^2], x], x] - Simp[(d*x^(m + 2))/(Pi*b*(m + 1)*(m + 2)), x]) /; FreeQ[{b, d}, x] && EqQ[d^2, (Pi^2*b^4)/4] && ILtQ[m, -2]
  6457. Int[Times[Cos[Times[Optional[Pattern[d, Blank[]]], Power[Pattern[x, Blank[]], 2]]], FresnelC[Times[Optional[Pattern[b, Blank[]]], Pattern[x, Blank[]]]], Power[Pattern[x, Blank[]], Pattern[m, Blank[]]]], Pattern[x, Blank[Symbol]]] := Simp[(x^(m + 1)*Cos[d*x^2]*FresnelC[b*x])/(m + 1), x] + (Dist[(2*d)/(m + 1), Int[x^(m + 2)*Sin[d*x^2]*FresnelC[b*x], x], x] - Dist[b/(2*(m + 1)), Int[x^(m + 1)*Cos[2*d*x^2], x], x] - Simp[(b*x^(m + 2))/(2*(m + 1)*(m + 2)), x]) /; FreeQ[{b, d}, x] && EqQ[d^2, (Pi^2*b^4)/4] && ILtQ[m, -2]
  6458. Int[Times[Power[FresnelS[Plus[Optional[Pattern[a, Blank[]]], Times[Optional[Pattern[b, Blank[]]], Pattern[x, Blank[]]]]], Optional[Pattern[n, Blank[]]]], Power[Times[Optional[Pattern[e, Blank[]]], Pattern[x, Blank[]]], Optional[Pattern[m, Blank[]]]], Sin[Plus[Optional[Pattern[c, Blank[]]], Times[Optional[Pattern[d, Blank[]]], Power[Pattern[x, Blank[]], 2]]]]], Pattern[x, Blank[Symbol]]] := Unintegrable[(e*x)^m*FresnelS[a + b*x]^n*Sin[c + d*x^2], x] /; FreeQ[{a, b, c, d, e, m, n}, x]
  6459. Int[Times[Cos[Plus[Optional[Pattern[c, Blank[]]], Times[Optional[Pattern[d, Blank[]]], Power[Pattern[x, Blank[]], 2]]]], Power[FresnelC[Plus[Optional[Pattern[a, Blank[]]], Times[Optional[Pattern[b, Blank[]]], Pattern[x, Blank[]]]]], Optional[Pattern[n, Blank[]]]], Power[Times[Optional[Pattern[e, Blank[]]], Pattern[x, Blank[]]], Optional[Pattern[m, Blank[]]]]], Pattern[x, Blank[Symbol]]] := Unintegrable[(e*x)^m*Cos[c + d*x^2]*FresnelC[a + b*x]^n, x] /; FreeQ[{a, b, c, d, e, m, n}, x]
  6460. Int[Times[Cos[Times[Optional[Pattern[d, Blank[]]], Power[Pattern[x, Blank[]], 2]]], FresnelS[Times[Optional[Pattern[b, Blank[]]], Pattern[x, Blank[]]]], Pattern[x, Blank[]]], Pattern[x, Blank[Symbol]]] := Simp[(Sin[d*x^2]*FresnelS[b*x])/(2*d), x] - Dist[1/(Pi*b), Int[Sin[d*x^2]^2, x], x] /; FreeQ[{b, d}, x] && EqQ[d^2, (Pi^2*b^4)/4]
  6461. Int[Times[FresnelC[Times[Optional[Pattern[b, Blank[]]], Pattern[x, Blank[]]]], Pattern[x, Blank[]], Sin[Times[Optional[Pattern[d, Blank[]]], Power[Pattern[x, Blank[]], 2]]]], Pattern[x, Blank[Symbol]]] := -Simp[(Cos[d*x^2]*FresnelC[b*x])/(2*d), x] + Dist[b/(2*d), Int[Cos[d*x^2]^2, x], x] /; FreeQ[{b, d}, x] && EqQ[d^2, (Pi^2*b^4)/4]
  6462. Int[Times[Cos[Times[Optional[Pattern[d, Blank[]]], Power[Pattern[x, Blank[]], 2]]], FresnelS[Times[Optional[Pattern[b, Blank[]]], Pattern[x, Blank[]]]], Power[Pattern[x, Blank[]], Pattern[m, Blank[]]]], Pattern[x, Blank[Symbol]]] := Simp[(x^(m - 1)*Sin[d*x^2]*FresnelS[b*x])/(2*d), x] + (-Dist[1/(Pi*b), Int[x^(m - 1)*Sin[d*x^2]^2, x], x] - Dist[(m - 1)/(2*d), Int[x^(m - 2)*Sin[d*x^2]*FresnelS[b*x], x], x]) /; FreeQ[{b, d}, x] && EqQ[d^2, (Pi^2*b^4)/4] && IGtQ[m, 1]
  6463. Int[Times[FresnelC[Times[Optional[Pattern[b, Blank[]]], Pattern[x, Blank[]]]], Power[Pattern[x, Blank[]], Pattern[m, Blank[]]], Sin[Times[Optional[Pattern[d, Blank[]]], Power[Pattern[x, Blank[]], 2]]]], Pattern[x, Blank[Symbol]]] := -Simp[(x^(m - 1)*Cos[d*x^2]*FresnelC[b*x])/(2*d), x] + (Dist[(m - 1)/(2*d), Int[x^(m - 2)*Cos[d*x^2]*FresnelC[b*x], x], x] + Dist[b/(2*d), Int[x^(m - 1)*Cos[d*x^2]^2, x], x]) /; FreeQ[{b, d}, x] && EqQ[d^2, (Pi^2*b^4)/4] && IGtQ[m, 1]
  6464. Int[Times[Cos[Times[Optional[Pattern[d, Blank[]]], Power[Pattern[x, Blank[]], 2]]], FresnelS[Times[Optional[Pattern[b, Blank[]]], Pattern[x, Blank[]]]], Power[Pattern[x, Blank[]], Pattern[m, Blank[]]]], Pattern[x, Blank[Symbol]]] := Simp[(x^(m + 1)*Cos[d*x^2]*FresnelS[b*x])/(m + 1), x] + (Dist[(2*d)/(m + 1), Int[x^(m + 2)*Sin[d*x^2]*FresnelS[b*x], x], x] - Dist[d/(Pi*b*(m + 1)), Int[x^(m + 1)*Sin[2*d*x^2], x], x]) /; FreeQ[{b, d}, x] && EqQ[d^2, (Pi^2*b^4)/4] && ILtQ[m, -1]
  6465. Int[Times[FresnelC[Times[Optional[Pattern[b, Blank[]]], Pattern[x, Blank[]]]], Power[Pattern[x, Blank[]], Pattern[m, Blank[]]], Sin[Times[Optional[Pattern[d, Blank[]]], Power[Pattern[x, Blank[]], 2]]]], Pattern[x, Blank[Symbol]]] := Simp[(x^(m + 1)*Sin[d*x^2]*FresnelC[b*x])/(m + 1), x] + (-Dist[(2*d)/(m + 1), Int[x^(m + 2)*Cos[d*x^2]*FresnelC[b*x], x], x] - Dist[b/(2*(m + 1)), Int[x^(m + 1)*Sin[2*d*x^2], x], x]) /; FreeQ[{b, d}, x] && EqQ[d^2, (Pi^2*b^4)/4] && ILtQ[m, -1]
  6466. Int[Times[Cos[Plus[Optional[Pattern[c, Blank[]]], Times[Optional[Pattern[d, Blank[]]], Power[Pattern[x, Blank[]], 2]]]], Power[FresnelS[Plus[Optional[Pattern[a, Blank[]]], Times[Optional[Pattern[b, Blank[]]], Pattern[x, Blank[]]]]], Optional[Pattern[n, Blank[]]]], Power[Times[Optional[Pattern[e, Blank[]]], Pattern[x, Blank[]]], Optional[Pattern[m, Blank[]]]]], Pattern[x, Blank[Symbol]]] := Unintegrable[(e*x)^m*Cos[c + d*x^2]*FresnelS[a + b*x]^n, x] /; FreeQ[{a, b, c, d, e, m, n}, x]
  6467. Int[Times[Power[FresnelC[Plus[Optional[Pattern[a, Blank[]]], Times[Optional[Pattern[b, Blank[]]], Pattern[x, Blank[]]]]], Optional[Pattern[n, Blank[]]]], Power[Times[Optional[Pattern[e, Blank[]]], Pattern[x, Blank[]]], Optional[Pattern[m, Blank[]]]], Sin[Plus[Optional[Pattern[c, Blank[]]], Times[Optional[Pattern[d, Blank[]]], Power[Pattern[x, Blank[]], 2]]]]], Pattern[x, Blank[Symbol]]] := Unintegrable[(e*x)^m*FresnelC[a + b*x]^n*Sin[c + d*x^2], x] /; FreeQ[{a, b, c, d, e, m, n}, x]
  6468. Int[FresnelS[Times[Plus[Optional[Pattern[a, Blank[]]], Times[Log[Times[Optional[Pattern[c, Blank[]]], Power[Pattern[x, Blank[]], Optional[Pattern[n, Blank[]]]]]], Optional[Pattern[b, Blank[]]]]], Optional[Pattern[d, Blank[]]]]], Pattern[x, Blank[Symbol]]] := Simp[x*FresnelS[d*(a + b*Log[c*x^n])], x] - Dist[b*d*n, Int[Sin[(Pi*(d*(a + b*Log[c*x^n]))^2)/2], x], x] /; FreeQ[{a, b, c, d, n}, x]
  6469. Int[FresnelC[Times[Plus[Optional[Pattern[a, Blank[]]], Times[Log[Times[Optional[Pattern[c, Blank[]]], Power[Pattern[x, Blank[]], Optional[Pattern[n, Blank[]]]]]], Optional[Pattern[b, Blank[]]]]], Optional[Pattern[d, Blank[]]]]], Pattern[x, Blank[Symbol]]] := Simp[x*FresnelC[d*(a + b*Log[c*x^n])], x] - Dist[b*d*n, Int[Cos[(Pi*(d*(a + b*Log[c*x^n]))^2)/2], x], x] /; FreeQ[{a, b, c, d, n}, x]
  6470. Int[Times[Power[Pattern[x, Blank[]], -1], Pattern[F, Blank[]][Times[Plus[Optional[Pattern[a, Blank[]]], Times[Log[Times[Optional[Pattern[c, Blank[]]], Power[Pattern[x, Blank[]], Optional[Pattern[n, Blank[]]]]]], Optional[Pattern[b, Blank[]]]]], Optional[Pattern[d, Blank[]]]]]], Pattern[x, Blank[Symbol]]] := Dist[1/n, Subst[F[d*(a + b*x)], x, Log[c*x^n]], x] /; FreeQ[{a, b, c, d, n}, x] && MemberQ[{FresnelS, FresnelC}, F]
  6471. Int[Times[FresnelS[Times[Plus[Optional[Pattern[a, Blank[]]], Times[Log[Times[Optional[Pattern[c, Blank[]]], Power[Pattern[x, Blank[]], Optional[Pattern[n, Blank[]]]]]], Optional[Pattern[b, Blank[]]]]], Optional[Pattern[d, Blank[]]]]], Power[Times[Optional[Pattern[e, Blank[]]], Pattern[x, Blank[]]], Optional[Pattern[m, Blank[]]]]], Pattern[x, Blank[Symbol]]] := Simp[((e*x)^(m + 1)*FresnelS[d*(a + b*Log[c*x^n])])/(e*(m + 1)), x] - Dist[(b*d*n)/(m + 1), Int[(e*x)^m*Sin[(Pi*(d*(a + b*Log[c*x^n]))^2)/2], x], x] /; FreeQ[{a, b, c, d, e, m, n}, x] && NeQ[m, -1]
  6472. Int[Times[FresnelC[Times[Plus[Optional[Pattern[a, Blank[]]], Times[Log[Times[Optional[Pattern[c, Blank[]]], Power[Pattern[x, Blank[]], Optional[Pattern[n, Blank[]]]]]], Optional[Pattern[b, Blank[]]]]], Optional[Pattern[d, Blank[]]]]], Power[Times[Optional[Pattern[e, Blank[]]], Pattern[x, Blank[]]], Optional[Pattern[m, Blank[]]]]], Pattern[x, Blank[Symbol]]] := Simp[((e*x)^(m + 1)*FresnelC[d*(a + b*Log[c*x^n])])/(e*(m + 1)), x] - Dist[(b*d*n)/(m + 1), Int[(e*x)^m*Cos[(Pi*(d*(a + b*Log[c*x^n]))^2)/2], x], x] /; FreeQ[{a, b, c, d, e, m, n}, x] && NeQ[m, -1]
  6473. Int[ExpIntegralE[Pattern[n, Blank[]], Plus[Optional[Pattern[a, Blank[]]], Times[Optional[Pattern[b, Blank[]]], Pattern[x, Blank[]]]]], Pattern[x, Blank[Symbol]]] := -Simp[ExpIntegralE[n + 1, a + b*x]/b, x] /; FreeQ[{a, b, n}, x]
  6474. Int[Times[ExpIntegralE[Pattern[n, Blank[]], Times[Optional[Pattern[b, Blank[]]], Pattern[x, Blank[]]]], Power[Pattern[x, Blank[]], Optional[Pattern[m, Blank[]]]]], Pattern[x, Blank[Symbol]]] := -Simp[(x^m*ExpIntegralE[n + 1, b*x])/b, x] + Dist[m/b, Int[x^(m - 1)*ExpIntegralE[n + 1, b*x], x], x] /; FreeQ[b, x] && EqQ[m + n, 0] && IGtQ[m, 0]
  6475. Int[Times[ExpIntegralE[1, Times[Optional[Pattern[b, Blank[]]], Pattern[x, Blank[]]]], Power[Pattern[x, Blank[]], -1]], Pattern[x, Blank[Symbol]]] := Simp[b*x*HypergeometricPFQ[{1, 1, 1}, {2, 2, 2}, -(b*x)], x] + (-Simp[EulerGamma*Log[x], x] - Simp[(1*Log[b*x]^2)/2, x]) /; FreeQ[b, x]
  6476. Int[Times[ExpIntegralE[Pattern[n, Blank[]], Times[Optional[Pattern[b, Blank[]]], Pattern[x, Blank[]]]], Power[Pattern[x, Blank[]], Pattern[m, Blank[]]]], Pattern[x, Blank[Symbol]]] := Simp[(x^(m + 1)*ExpIntegralE[n, b*x])/(m + 1), x] + Dist[b/(m + 1), Int[x^(m + 1)*ExpIntegralE[n - 1, b*x], x], x] /; FreeQ[b, x] && EqQ[m + n, 0] && ILtQ[m, -1]
  6477. Int[Times[ExpIntegralE[Pattern[n, Blank[]], Times[Optional[Pattern[b, Blank[]]], Pattern[x, Blank[]]]], Power[Times[Optional[Pattern[d, Blank[]]], Pattern[x, Blank[]]], Pattern[m, Blank[]]]], Pattern[x, Blank[Symbol]]] := Simp[((d*x)^m*Gamma[m + 1]*Log[x])/(b*(b*x)^m), x] - Simp[((d*x)^(m + 1)*HypergeometricPFQ[{m + 1, m + 1}, {m + 2, m + 2}, -(b*x)])/(d*(m + 1)^2), x] /; FreeQ[{b, d, m, n}, x] && EqQ[m + n, 0] && !IntegerQ[m]
  6478. Int[Times[ExpIntegralE[Pattern[n, Blank[]], Times[Optional[Pattern[b, Blank[]]], Pattern[x, Blank[]]]], Power[Times[Optional[Pattern[d, Blank[]]], Pattern[x, Blank[]]], Optional[Pattern[m, Blank[]]]]], Pattern[x, Blank[Symbol]]] := Simp[((d*x)^(m + 1)*ExpIntegralE[n, b*x])/(d*(m + n)), x] - Simp[((d*x)^(m + 1)*ExpIntegralE[-m, b*x])/(d*(m + n)), x] /; FreeQ[{b, d, m, n}, x] && NeQ[m + n, 0]
  6479. Int[Times[ExpIntegralE[Pattern[n, Blank[]], Plus[Pattern[a, Blank[]], Times[Optional[Pattern[b, Blank[]]], Pattern[x, Blank[]]]]], Power[Plus[Optional[Pattern[c, Blank[]]], Times[Optional[Pattern[d, Blank[]]], Pattern[x, Blank[]]]], Optional[Pattern[m, Blank[]]]]], Pattern[x, Blank[Symbol]]] := -Simp[((c + d*x)^m*ExpIntegralE[n + 1, a + b*x])/b, x] + Dist[(d*m)/b, Int[(c + d*x)^(m - 1)*ExpIntegralE[n + 1, a + b*x], x], x] /; FreeQ[{a, b, c, d, m, n}, x] && (IGtQ[m, 0] || ILtQ[n, 0] || (GtQ[m, 0] && LtQ[n, -1]))
  6480. Int[Times[ExpIntegralE[Pattern[n, Blank[]], Plus[Pattern[a, Blank[]], Times[Optional[Pattern[b, Blank[]]], Pattern[x, Blank[]]]]], Power[Plus[Optional[Pattern[c, Blank[]]], Times[Optional[Pattern[d, Blank[]]], Pattern[x, Blank[]]]], Optional[Pattern[m, Blank[]]]]], Pattern[x, Blank[Symbol]]] := Simp[((c + d*x)^(m + 1)*ExpIntegralE[n, a + b*x])/(d*(m + 1)), x] + Dist[b/(d*(m + 1)), Int[(c + d*x)^(m + 1)*ExpIntegralE[n - 1, a + b*x], x], x] /; FreeQ[{a, b, c, d, m, n}, x] && (IGtQ[n, 0] || (LtQ[m, -1] && GtQ[n, 0])) && NeQ[m, -1]
  6481. Int[Times[ExpIntegralE[Pattern[n, Blank[]], Plus[Pattern[a, Blank[]], Times[Optional[Pattern[b, Blank[]]], Pattern[x, Blank[]]]]], Power[Plus[Optional[Pattern[c, Blank[]]], Times[Optional[Pattern[d, Blank[]]], Pattern[x, Blank[]]]], Optional[Pattern[m, Blank[]]]]], Pattern[x, Blank[Symbol]]] := Unintegrable[(c + d*x)^m*ExpIntegralE[n, a + b*x], x] /; FreeQ[{a, b, c, d, m, n}, x]
  6482. Int[ExpIntegralEi[Plus[Optional[Pattern[a, Blank[]]], Times[Optional[Pattern[b, Blank[]]], Pattern[x, Blank[]]]]], Pattern[x, Blank[Symbol]]] := Simp[((a + b*x)*ExpIntegralEi[a + b*x])/b, x] - Simp[E^(a + b*x)/b, x] /; FreeQ[{a, b}, x]
  6483. Int[Times[ExpIntegralEi[Times[Optional[Pattern[b, Blank[]]], Pattern[x, Blank[]]]], Power[Pattern[x, Blank[]], -1]], Pattern[x, Blank[Symbol]]] := Simp[Log[x]*(ExpIntegralEi[b*x] + ExpIntegralE[1, -(b*x)]), x] - Int[ExpIntegralE[1, -(b*x)]/x, x] /; FreeQ[b, x]
  6484. Int[Times[ExpIntegralEi[Plus[Optional[Pattern[a, Blank[]]], Times[Optional[Pattern[b, Blank[]]], Pattern[x, Blank[]]]]], Power[Plus[Optional[Pattern[c, Blank[]]], Times[Optional[Pattern[d, Blank[]]], Pattern[x, Blank[]]]], -1]], Pattern[x, Blank[Symbol]]] := Unintegrable[ExpIntegralEi[a + b*x]/(c + d*x), x] /; FreeQ[{a, b, c, d}, x]
  6485. Int[Times[ExpIntegralEi[Plus[Optional[Pattern[a, Blank[]]], Times[Optional[Pattern[b, Blank[]]], Pattern[x, Blank[]]]]], Power[Plus[Optional[Pattern[c, Blank[]]], Times[Optional[Pattern[d, Blank[]]], Pattern[x, Blank[]]]], Optional[Pattern[m, Blank[]]]]], Pattern[x, Blank[Symbol]]] := Simp[((c + d*x)^(m + 1)*ExpIntegralEi[a + b*x])/(d*(m + 1)), x] - Dist[b/(d*(m + 1)), Int[((c + d*x)^(m + 1)*E^(a + b*x))/(a + b*x), x], x] /; FreeQ[{a, b, c, d, m}, x] && NeQ[m, -1]
  6486. Int[Power[ExpIntegralEi[Plus[Optional[Pattern[a, Blank[]]], Times[Optional[Pattern[b, Blank[]]], Pattern[x, Blank[]]]]], 2], Pattern[x, Blank[Symbol]]] := Simp[((a + b*x)*ExpIntegralEi[a + b*x]^2)/b, x] - Dist[2, Int[E^(a + b*x)*ExpIntegralEi[a + b*x], x], x] /; FreeQ[{a, b}, x]
  6487. Int[Times[Power[ExpIntegralEi[Times[Optional[Pattern[b, Blank[]]], Pattern[x, Blank[]]]], 2], Power[Pattern[x, Blank[]], Optional[Pattern[m, Blank[]]]]], Pattern[x, Blank[Symbol]]] := Simp[(x^(m + 1)*ExpIntegralEi[b*x]^2)/(m + 1), x] - Dist[2/(m + 1), Int[x^m*E^(b*x)*ExpIntegralEi[b*x], x], x] /; FreeQ[b, x] && IGtQ[m, 0]
  6488. Int[Times[Power[ExpIntegralEi[Plus[Pattern[a, Blank[]], Times[Optional[Pattern[b, Blank[]]], Pattern[x, Blank[]]]]], 2], Power[Pattern[x, Blank[]], Optional[Pattern[m, Blank[]]]]], Pattern[x, Blank[Symbol]]] := Simp[(x^(m + 1)*ExpIntegralEi[a + b*x]^2)/(m + 1), x] + (-Dist[2/(m + 1), Int[x^m*E^(a + b*x)*ExpIntegralEi[a + b*x], x], x] - Dist[(a*m)/(b*(m + 1)), Int[x^(m - 1)*ExpIntegralEi[a + b*x]^2, x], x] + Simp[(a*x^m*ExpIntegralEi[a + b*x]^2)/(b*(m + 1)), x]) /; FreeQ[{a, b}, x] && IGtQ[m, 0]
  6489. Int[Times[Power[E, Plus[Optional[Pattern[a, Blank[]]], Times[Optional[Pattern[b, Blank[]]], Pattern[x, Blank[]]]]], ExpIntegralEi[Plus[Optional[Pattern[c, Blank[]]], Times[Optional[Pattern[d, Blank[]]], Pattern[x, Blank[]]]]]], Pattern[x, Blank[Symbol]]] := Simp[(E^(a + b*x)*ExpIntegralEi[c + d*x])/b, x] - Dist[d/b, Int[E^(a + c + (b + d)*x)/(c + d*x), x], x] /; FreeQ[{a, b, c, d}, x]
  6490. Int[Times[Power[E, Plus[Optional[Pattern[a, Blank[]]], Times[Optional[Pattern[b, Blank[]]], Pattern[x, Blank[]]]]], ExpIntegralEi[Plus[Optional[Pattern[c, Blank[]]], Times[Optional[Pattern[d, Blank[]]], Pattern[x, Blank[]]]]], Power[Pattern[x, Blank[]], Optional[Pattern[m, Blank[]]]]], Pattern[x, Blank[Symbol]]] := Simp[(x^m*E^(a + b*x)*ExpIntegralEi[c + d*x])/b, x] + (-Dist[d/b, Int[(x^m*E^(a + c + (b + d)*x))/(c + d*x), x], x] - Dist[m/b, Int[x^(m - 1)*E^(a + b*x)*ExpIntegralEi[c + d*x], x], x]) /; FreeQ[{a, b, c, d}, x] && IGtQ[m, 0]
  6491. Int[Times[Power[E, Plus[Optional[Pattern[a, Blank[]]], Times[Optional[Pattern[b, Blank[]]], Pattern[x, Blank[]]]]], ExpIntegralEi[Plus[Optional[Pattern[c, Blank[]]], Times[Optional[Pattern[d, Blank[]]], Pattern[x, Blank[]]]]], Power[Pattern[x, Blank[]], Pattern[m, Blank[]]]], Pattern[x, Blank[Symbol]]] := Simp[(x^(m + 1)*E^(a + b*x)*ExpIntegralEi[c + d*x])/(m + 1), x] + (-Dist[b/(m + 1), Int[x^(m + 1)*E^(a + b*x)*ExpIntegralEi[c + d*x], x], x] - Dist[d/(m + 1), Int[(x^(m + 1)*E^(a + c + (b + d)*x))/(c + d*x), x], x]) /; FreeQ[{a, b, c, d}, x] && ILtQ[m, -1]
  6492. Int[ExpIntegralEi[Times[Plus[Optional[Pattern[a, Blank[]]], Times[Log[Times[Optional[Pattern[c, Blank[]]], Power[Pattern[x, Blank[]], Optional[Pattern[n, Blank[]]]]]], Optional[Pattern[b, Blank[]]]]], Optional[Pattern[d, Blank[]]]]], Pattern[x, Blank[Symbol]]] := Simp[x*ExpIntegralEi[d*(a + b*Log[c*x^n])], x] - Dist[b*n*E^(a*d), Int[(c*x^n)^(b*d)/(a + b*Log[c*x^n]), x], x] /; FreeQ[{a, b, c, d, n}, x]
  6493. Int[Times[ExpIntegralEi[Times[Plus[Optional[Pattern[a, Blank[]]], Times[Log[Times[Optional[Pattern[c, Blank[]]], Power[Pattern[x, Blank[]], Optional[Pattern[n, Blank[]]]]]], Optional[Pattern[b, Blank[]]]]], Optional[Pattern[d, Blank[]]]]], Power[Pattern[x, Blank[]], -1]], Pattern[x, Blank[Symbol]]] := Dist[1/n, Subst[ExpIntegralEi[d*(a + b*x)], x, Log[c*x^n]], x] /; FreeQ[{a, b, c, d, n}, x]
  6494. Int[Times[ExpIntegralEi[Times[Plus[Optional[Pattern[a, Blank[]]], Times[Log[Times[Optional[Pattern[c, Blank[]]], Power[Pattern[x, Blank[]], Optional[Pattern[n, Blank[]]]]]], Optional[Pattern[b, Blank[]]]]], Optional[Pattern[d, Blank[]]]]], Power[Times[Optional[Pattern[e, Blank[]]], Pattern[x, Blank[]]], Optional[Pattern[m, Blank[]]]]], Pattern[x, Blank[Symbol]]] := Simp[((e*x)^(m + 1)*ExpIntegralEi[d*(a + b*Log[c*x^n])])/(e*(m + 1)), x] - Dist[(b*n*E^(a*d)*(c*x^n)^(b*d))/((m + 1)*(e*x)^(b*d*n)), Int[(e*x)^(m + b*d*n)/(a + b*Log[c*x^n]), x], x] /; FreeQ[{a, b, c, d, e, m, n}, x] && NeQ[m, -1]
  6495. Int[LogIntegral[Plus[Optional[Pattern[a, Blank[]]], Times[Optional[Pattern[b, Blank[]]], Pattern[x, Blank[]]]]], Pattern[x, Blank[Symbol]]] := Simp[((a + b*x)*LogIntegral[a + b*x])/b, x] - Simp[ExpIntegralEi[2*Log[a + b*x]]/b, x] /; FreeQ[{a, b}, x]
  6496. Int[Times[LogIntegral[Times[Optional[Pattern[b, Blank[]]], Pattern[x, Blank[]]]], Power[Pattern[x, Blank[]], -1]], Pattern[x, Blank[Symbol]]] := -Simp[b*x, x] + Simp[Log[b*x]*LogIntegral[b*x], x] /; FreeQ[b, x]
  6497. Int[Times[LogIntegral[Plus[Optional[Pattern[a, Blank[]]], Times[Optional[Pattern[b, Blank[]]], Pattern[x, Blank[]]]]], Power[Plus[Optional[Pattern[c, Blank[]]], Times[Optional[Pattern[d, Blank[]]], Pattern[x, Blank[]]]], -1]], Pattern[x, Blank[Symbol]]] := Unintegrable[LogIntegral[a + b*x]/(c + d*x), x] /; FreeQ[{a, b, c, d}, x]
  6498. Int[Times[LogIntegral[Plus[Optional[Pattern[a, Blank[]]], Times[Optional[Pattern[b, Blank[]]], Pattern[x, Blank[]]]]], Power[Plus[Optional[Pattern[c, Blank[]]], Times[Optional[Pattern[d, Blank[]]], Pattern[x, Blank[]]]], Optional[Pattern[m, Blank[]]]]], Pattern[x, Blank[Symbol]]] := Simp[((c + d*x)^(m + 1)*LogIntegral[a + b*x])/(d*(m + 1)), x] - Dist[b/(d*(m + 1)), Int[(c + d*x)^(m + 1)/Log[a + b*x], x], x] /; FreeQ[{a, b, c, d, m}, x] && NeQ[m, -1]
  6499. Int[SinIntegral[Plus[Optional[Pattern[a, Blank[]]], Times[Optional[Pattern[b, Blank[]]], Pattern[x, Blank[]]]]], Pattern[x, Blank[Symbol]]] := Simp[((a + b*x)*SinIntegral[a + b*x])/b, x] + Simp[Cos[a + b*x]/b, x] /; FreeQ[{a, b}, x]
  6500. Int[CosIntegral[Plus[Optional[Pattern[a, Blank[]]], Times[Optional[Pattern[b, Blank[]]], Pattern[x, Blank[]]]]], Pattern[x, Blank[Symbol]]] := Simp[((a + b*x)*CosIntegral[a + b*x])/b, x] - Simp[Sin[a + b*x]/b, x] /; FreeQ[{a, b}, x]
  6501. Int[Times[Power[Pattern[x, Blank[]], -1], SinIntegral[Times[Optional[Pattern[b, Blank[]]], Pattern[x, Blank[]]]]], Pattern[x, Blank[Symbol]]] := Simp[(1*b*x*HypergeometricPFQ[{1, 1, 1}, {2, 2, 2}, -(I*b*x)])/2, x] + Simp[(1*b*x*HypergeometricPFQ[{1, 1, 1}, {2, 2, 2}, I*b*x])/2, x] /; FreeQ[b, x]
  6502. Int[Times[CosIntegral[Times[Optional[Pattern[b, Blank[]]], Pattern[x, Blank[]]]], Power[Pattern[x, Blank[]], -1]], Pattern[x, Blank[Symbol]]] := -Simp[(I*b*x*HypergeometricPFQ[{1, 1, 1}, {2, 2, 2}, -(I*b*x)])/2, x] + (Simp[(1*I*b*x*HypergeometricPFQ[{1, 1, 1}, {2, 2, 2}, I*b*x])/2, x] + Simp[EulerGamma*Log[x], x] + Simp[(1*Log[b*x]^2)/2, x]) /; FreeQ[b, x]
  6503. Int[Times[Power[Plus[Optional[Pattern[c, Blank[]]], Times[Optional[Pattern[d, Blank[]]], Pattern[x, Blank[]]]], Optional[Pattern[m, Blank[]]]], SinIntegral[Plus[Optional[Pattern[a, Blank[]]], Times[Optional[Pattern[b, Blank[]]], Pattern[x, Blank[]]]]]], Pattern[x, Blank[Symbol]]] := Simp[((c + d*x)^(m + 1)*SinIntegral[a + b*x])/(d*(m + 1)), x] - Dist[b/(d*(m + 1)), Int[((c + d*x)^(m + 1)*Sin[a + b*x])/(a + b*x), x], x] /; FreeQ[{a, b, c, d, m}, x] && NeQ[m, -1]
  6504. Int[Times[CosIntegral[Plus[Optional[Pattern[a, Blank[]]], Times[Optional[Pattern[b, Blank[]]], Pattern[x, Blank[]]]]], Power[Plus[Optional[Pattern[c, Blank[]]], Times[Optional[Pattern[d, Blank[]]], Pattern[x, Blank[]]]], Optional[Pattern[m, Blank[]]]]], Pattern[x, Blank[Symbol]]] := Simp[((c + d*x)^(m + 1)*CosIntegral[a + b*x])/(d*(m + 1)), x] - Dist[b/(d*(m + 1)), Int[((c + d*x)^(m + 1)*Cos[a + b*x])/(a + b*x), x], x] /; FreeQ[{a, b, c, d, m}, x] && NeQ[m, -1]
  6505. Int[Power[SinIntegral[Plus[Optional[Pattern[a, Blank[]]], Times[Optional[Pattern[b, Blank[]]], Pattern[x, Blank[]]]]], 2], Pattern[x, Blank[Symbol]]] := Simp[((a + b*x)*SinIntegral[a + b*x]^2)/b, x] - Dist[2, Int[Sin[a + b*x]*SinIntegral[a + b*x], x], x] /; FreeQ[{a, b}, x]
  6506. Int[Power[CosIntegral[Plus[Optional[Pattern[a, Blank[]]], Times[Optional[Pattern[b, Blank[]]], Pattern[x, Blank[]]]]], 2], Pattern[x, Blank[Symbol]]] := Simp[((a + b*x)*CosIntegral[a + b*x]^2)/b, x] - Dist[2, Int[Cos[a + b*x]*CosIntegral[a + b*x], x], x] /; FreeQ[{a, b}, x]
  6507. Int[Times[Power[Pattern[x, Blank[]], Optional[Pattern[m, Blank[]]]], Power[SinIntegral[Times[Optional[Pattern[b, Blank[]]], Pattern[x, Blank[]]]], 2]], Pattern[x, Blank[Symbol]]] := Simp[(x^(m + 1)*SinIntegral[b*x]^2)/(m + 1), x] - Dist[2/(m + 1), Int[x^m*Sin[b*x]*SinIntegral[b*x], x], x] /; FreeQ[b, x] && IGtQ[m, 0]
  6508. Int[Times[Power[CosIntegral[Times[Optional[Pattern[b, Blank[]]], Pattern[x, Blank[]]]], 2], Power[Pattern[x, Blank[]], Optional[Pattern[m, Blank[]]]]], Pattern[x, Blank[Symbol]]] := Simp[(x^(m + 1)*CosIntegral[b*x]^2)/(m + 1), x] - Dist[2/(m + 1), Int[x^m*Cos[b*x]*CosIntegral[b*x], x], x] /; FreeQ[b, x] && IGtQ[m, 0]
  6509. Int[Times[Power[Plus[Optional[Pattern[c, Blank[]]], Times[Optional[Pattern[d, Blank[]]], Pattern[x, Blank[]]]], Optional[Pattern[m, Blank[]]]], Power[SinIntegral[Plus[Pattern[a, Blank[]], Times[Optional[Pattern[b, Blank[]]], Pattern[x, Blank[]]]]], 2]], Pattern[x, Blank[Symbol]]] := Simp[((a + b*x)*(c + d*x)^m*SinIntegral[a + b*x]^2)/(b*(m + 1)), x] + (-Dist[2/(m + 1), Int[(c + d*x)^m*Sin[a + b*x]*SinIntegral[a + b*x], x], x] + Dist[((b*c - a*d)*m)/(b*(m + 1)), Int[(c + d*x)^(m - 1)*SinIntegral[a + b*x]^2, x], x]) /; FreeQ[{a, b, c, d}, x] && IGtQ[m, 0]
  6510. Int[Times[Power[CosIntegral[Plus[Pattern[a, Blank[]], Times[Optional[Pattern[b, Blank[]]], Pattern[x, Blank[]]]]], 2], Power[Plus[Optional[Pattern[c, Blank[]]], Times[Optional[Pattern[d, Blank[]]], Pattern[x, Blank[]]]], Optional[Pattern[m, Blank[]]]]], Pattern[x, Blank[Symbol]]] := Simp[((a + b*x)*(c + d*x)^m*CosIntegral[a + b*x]^2)/(b*(m + 1)), x] + (-Dist[2/(m + 1), Int[(c + d*x)^m*Cos[a + b*x]*CosIntegral[a + b*x], x], x] + Dist[((b*c - a*d)*m)/(b*(m + 1)), Int[(c + d*x)^(m - 1)*CosIntegral[a + b*x]^2, x], x]) /; FreeQ[{a, b, c, d}, x] && IGtQ[m, 0]
  6511. Int[Times[Sin[Plus[Optional[Pattern[a, Blank[]]], Times[Optional[Pattern[b, Blank[]]], Pattern[x, Blank[]]]]], SinIntegral[Plus[Optional[Pattern[c, Blank[]]], Times[Optional[Pattern[d, Blank[]]], Pattern[x, Blank[]]]]]], Pattern[x, Blank[Symbol]]] := -Simp[(Cos[a + b*x]*SinIntegral[c + d*x])/b, x] + Dist[d/b, Int[(Cos[a + b*x]*Sin[c + d*x])/(c + d*x), x], x] /; FreeQ[{a, b, c, d}, x]
  6512. Int[Times[Cos[Plus[Optional[Pattern[a, Blank[]]], Times[Optional[Pattern[b, Blank[]]], Pattern[x, Blank[]]]]], CosIntegral[Plus[Optional[Pattern[c, Blank[]]], Times[Optional[Pattern[d, Blank[]]], Pattern[x, Blank[]]]]]], Pattern[x, Blank[Symbol]]] := Simp[(Sin[a + b*x]*CosIntegral[c + d*x])/b, x] - Dist[d/b, Int[(Sin[a + b*x]*Cos[c + d*x])/(c + d*x), x], x] /; FreeQ[{a, b, c, d}, x]
  6513. Int[Times[Power[Plus[Optional[Pattern[e, Blank[]]], Times[Optional[Pattern[f, Blank[]]], Pattern[x, Blank[]]]], Optional[Pattern[m, Blank[]]]], Sin[Plus[Optional[Pattern[a, Blank[]]], Times[Optional[Pattern[b, Blank[]]], Pattern[x, Blank[]]]]], SinIntegral[Plus[Optional[Pattern[c, Blank[]]], Times[Optional[Pattern[d, Blank[]]], Pattern[x, Blank[]]]]]], Pattern[x, Blank[Symbol]]] := -Simp[((e + f*x)^m*Cos[a + b*x]*SinIntegral[c + d*x])/b, x] + (Dist[d/b, Int[((e + f*x)^m*Cos[a + b*x]*Sin[c + d*x])/(c + d*x), x], x] + Dist[(f*m)/b, Int[(e + f*x)^(m - 1)*Cos[a + b*x]*SinIntegral[c + d*x], x], x]) /; FreeQ[{a, b, c, d, e, f}, x] && IGtQ[m, 0]
  6514. Int[Times[Cos[Plus[Optional[Pattern[a, Blank[]]], Times[Optional[Pattern[b, Blank[]]], Pattern[x, Blank[]]]]], CosIntegral[Plus[Optional[Pattern[c, Blank[]]], Times[Optional[Pattern[d, Blank[]]], Pattern[x, Blank[]]]]], Power[Plus[Optional[Pattern[e, Blank[]]], Times[Optional[Pattern[f, Blank[]]], Pattern[x, Blank[]]]], Optional[Pattern[m, Blank[]]]]], Pattern[x, Blank[Symbol]]] := Simp[((e + f*x)^m*Sin[a + b*x]*CosIntegral[c + d*x])/b, x] + (-Dist[d/b, Int[((e + f*x)^m*Sin[a + b*x]*Cos[c + d*x])/(c + d*x), x], x] - Dist[(f*m)/b, Int[(e + f*x)^(m - 1)*Sin[a + b*x]*CosIntegral[c + d*x], x], x]) /; FreeQ[{a, b, c, d, e, f}, x] && IGtQ[m, 0]
  6515. Int[Times[Power[Plus[Optional[Pattern[e, Blank[]]], Times[Optional[Pattern[f, Blank[]]], Pattern[x, Blank[]]]], Pattern[m, Blank[]]], Sin[Plus[Optional[Pattern[a, Blank[]]], Times[Optional[Pattern[b, Blank[]]], Pattern[x, Blank[]]]]], SinIntegral[Plus[Optional[Pattern[c, Blank[]]], Times[Optional[Pattern[d, Blank[]]], Pattern[x, Blank[]]]]]], Pattern[x, Blank[Symbol]]] := Simp[((e + f*x)^(m + 1)*Sin[a + b*x]*SinIntegral[c + d*x])/(f*(m + 1)), x] + (-Dist[b/(f*(m + 1)), Int[(e + f*x)^(m + 1)*Cos[a + b*x]*SinIntegral[c + d*x], x], x] - Dist[d/(f*(m + 1)), Int[((e + f*x)^(m + 1)*Sin[a + b*x]*Sin[c + d*x])/(c + d*x), x], x]) /; FreeQ[{a, b, c, d, e, f}, x] && ILtQ[m, -1]
  6516. Int[Times[Cos[Plus[Optional[Pattern[a, Blank[]]], Times[Optional[Pattern[b, Blank[]]], Pattern[x, Blank[]]]]], CosIntegral[Plus[Optional[Pattern[c, Blank[]]], Times[Optional[Pattern[d, Blank[]]], Pattern[x, Blank[]]]]], Power[Plus[Optional[Pattern[e, Blank[]]], Times[Optional[Pattern[f, Blank[]]], Pattern[x, Blank[]]]], Optional[Pattern[m, Blank[]]]]], Pattern[x, Blank[Symbol]]] := Simp[((e + f*x)^(m + 1)*Cos[a + b*x]*CosIntegral[c + d*x])/(f*(m + 1)), x] + (Dist[b/(f*(m + 1)), Int[(e + f*x)^(m + 1)*Sin[a + b*x]*CosIntegral[c + d*x], x], x] - Dist[d/(f*(m + 1)), Int[((e + f*x)^(m + 1)*Cos[a + b*x]*Cos[c + d*x])/(c + d*x), x], x]) /; FreeQ[{a, b, c, d, e, f}, x] && ILtQ[m, -1]
  6517. Int[Times[Cos[Plus[Optional[Pattern[a, Blank[]]], Times[Optional[Pattern[b, Blank[]]], Pattern[x, Blank[]]]]], SinIntegral[Plus[Optional[Pattern[c, Blank[]]], Times[Optional[Pattern[d, Blank[]]], Pattern[x, Blank[]]]]]], Pattern[x, Blank[Symbol]]] := Simp[(Sin[a + b*x]*SinIntegral[c + d*x])/b, x] - Dist[d/b, Int[(Sin[a + b*x]*Sin[c + d*x])/(c + d*x), x], x] /; FreeQ[{a, b, c, d}, x]
  6518. Int[Times[CosIntegral[Plus[Optional[Pattern[c, Blank[]]], Times[Optional[Pattern[d, Blank[]]], Pattern[x, Blank[]]]]], Sin[Plus[Optional[Pattern[a, Blank[]]], Times[Optional[Pattern[b, Blank[]]], Pattern[x, Blank[]]]]]], Pattern[x, Blank[Symbol]]] := -Simp[(Cos[a + b*x]*CosIntegral[c + d*x])/b, x] + Dist[d/b, Int[(Cos[a + b*x]*Cos[c + d*x])/(c + d*x), x], x] /; FreeQ[{a, b, c, d}, x]
  6519. Int[Times[Cos[Plus[Optional[Pattern[a, Blank[]]], Times[Optional[Pattern[b, Blank[]]], Pattern[x, Blank[]]]]], Power[Plus[Optional[Pattern[e, Blank[]]], Times[Optional[Pattern[f, Blank[]]], Pattern[x, Blank[]]]], Optional[Pattern[m, Blank[]]]], SinIntegral[Plus[Optional[Pattern[c, Blank[]]], Times[Optional[Pattern[d, Blank[]]], Pattern[x, Blank[]]]]]], Pattern[x, Blank[Symbol]]] := Simp[((e + f*x)^m*Sin[a + b*x]*SinIntegral[c + d*x])/b, x] + (-Dist[d/b, Int[((e + f*x)^m*Sin[a + b*x]*Sin[c + d*x])/(c + d*x), x], x] - Dist[(f*m)/b, Int[(e + f*x)^(m - 1)*Sin[a + b*x]*SinIntegral[c + d*x], x], x]) /; FreeQ[{a, b, c, d, e, f}, x] && IGtQ[m, 0]
  6520. Int[Times[CosIntegral[Plus[Optional[Pattern[c, Blank[]]], Times[Optional[Pattern[d, Blank[]]], Pattern[x, Blank[]]]]], Power[Plus[Optional[Pattern[e, Blank[]]], Times[Optional[Pattern[f, Blank[]]], Pattern[x, Blank[]]]], Optional[Pattern[m, Blank[]]]], Sin[Plus[Optional[Pattern[a, Blank[]]], Times[Optional[Pattern[b, Blank[]]], Pattern[x, Blank[]]]]]], Pattern[x, Blank[Symbol]]] := -Simp[((e + f*x)^m*Cos[a + b*x]*CosIntegral[c + d*x])/b, x] + (Dist[d/b, Int[((e + f*x)^m*Cos[a + b*x]*Cos[c + d*x])/(c + d*x), x], x] + Dist[(f*m)/b, Int[(e + f*x)^(m - 1)*Cos[a + b*x]*CosIntegral[c + d*x], x], x]) /; FreeQ[{a, b, c, d, e, f}, x] && IGtQ[m, 0]
  6521. Int[Times[Cos[Plus[Optional[Pattern[a, Blank[]]], Times[Optional[Pattern[b, Blank[]]], Pattern[x, Blank[]]]]], Power[Plus[Optional[Pattern[e, Blank[]]], Times[Optional[Pattern[f, Blank[]]], Pattern[x, Blank[]]]], Optional[Pattern[m, Blank[]]]], SinIntegral[Plus[Optional[Pattern[c, Blank[]]], Times[Optional[Pattern[d, Blank[]]], Pattern[x, Blank[]]]]]], Pattern[x, Blank[Symbol]]] := Simp[((e + f*x)^(m + 1)*Cos[a + b*x]*SinIntegral[c + d*x])/(f*(m + 1)), x] + (Dist[b/(f*(m + 1)), Int[(e + f*x)^(m + 1)*Sin[a + b*x]*SinIntegral[c + d*x], x], x] - Dist[d/(f*(m + 1)), Int[((e + f*x)^(m + 1)*Cos[a + b*x]*Sin[c + d*x])/(c + d*x), x], x]) /; FreeQ[{a, b, c, d, e, f}, x] && ILtQ[m, -1]
  6522. Int[Times[CosIntegral[Plus[Optional[Pattern[c, Blank[]]], Times[Optional[Pattern[d, Blank[]]], Pattern[x, Blank[]]]]], Power[Plus[Optional[Pattern[e, Blank[]]], Times[Optional[Pattern[f, Blank[]]], Pattern[x, Blank[]]]], Pattern[m, Blank[]]], Sin[Plus[Optional[Pattern[a, Blank[]]], Times[Optional[Pattern[b, Blank[]]], Pattern[x, Blank[]]]]]], Pattern[x, Blank[Symbol]]] := Simp[((e + f*x)^(m + 1)*Sin[a + b*x]*CosIntegral[c + d*x])/(f*(m + 1)), x] + (-Dist[b/(f*(m + 1)), Int[(e + f*x)^(m + 1)*Cos[a + b*x]*CosIntegral[c + d*x], x], x] - Dist[d/(f*(m + 1)), Int[((e + f*x)^(m + 1)*Sin[a + b*x]*Cos[c + d*x])/(c + d*x), x], x]) /; FreeQ[{a, b, c, d, e, f}, x] && ILtQ[m, -1]
  6523. Int[SinIntegral[Times[Plus[Optional[Pattern[a, Blank[]]], Times[Log[Times[Optional[Pattern[c, Blank[]]], Power[Pattern[x, Blank[]], Optional[Pattern[n, Blank[]]]]]], Optional[Pattern[b, Blank[]]]]], Optional[Pattern[d, Blank[]]]]], Pattern[x, Blank[Symbol]]] := Simp[x*SinIntegral[d*(a + b*Log[c*x^n])], x] - Dist[b*d*n, Int[Sin[d*(a + b*Log[c*x^n])]/(d*(a + b*Log[c*x^n])), x], x] /; FreeQ[{a, b, c, d, n}, x]
  6524. Int[CosIntegral[Times[Plus[Optional[Pattern[a, Blank[]]], Times[Log[Times[Optional[Pattern[c, Blank[]]], Power[Pattern[x, Blank[]], Optional[Pattern[n, Blank[]]]]]], Optional[Pattern[b, Blank[]]]]], Optional[Pattern[d, Blank[]]]]], Pattern[x, Blank[Symbol]]] := Simp[x*CosIntegral[d*(a + b*Log[c*x^n])], x] - Dist[b*d*n, Int[Cos[d*(a + b*Log[c*x^n])]/(d*(a + b*Log[c*x^n])), x], x] /; FreeQ[{a, b, c, d, n}, x]
  6525. Int[Times[Power[Pattern[x, Blank[]], -1], Pattern[F, Blank[]][Times[Plus[Optional[Pattern[a, Blank[]]], Times[Log[Times[Optional[Pattern[c, Blank[]]], Power[Pattern[x, Blank[]], Optional[Pattern[n, Blank[]]]]]], Optional[Pattern[b, Blank[]]]]], Optional[Pattern[d, Blank[]]]]]], Pattern[x, Blank[Symbol]]] := Dist[1/n, Subst[F[d*(a + b*x)], x, Log[c*x^n]], x] /; FreeQ[{a, b, c, d, n}, x] && MemberQ[{SinIntegral, CosIntegral}, x]
  6526. Int[Times[Power[Times[Optional[Pattern[e, Blank[]]], Pattern[x, Blank[]]], Optional[Pattern[m, Blank[]]]], SinIntegral[Times[Plus[Optional[Pattern[a, Blank[]]], Times[Log[Times[Optional[Pattern[c, Blank[]]], Power[Pattern[x, Blank[]], Optional[Pattern[n, Blank[]]]]]], Optional[Pattern[b, Blank[]]]]], Optional[Pattern[d, Blank[]]]]]], Pattern[x, Blank[Symbol]]] := Simp[((e*x)^(m + 1)*SinIntegral[d*(a + b*Log[c*x^n])])/(e*(m + 1)), x] - Dist[(b*d*n)/(m + 1), Int[((e*x)^m*Sin[d*(a + b*Log[c*x^n])])/(d*(a + b*Log[c*x^n])), x], x] /; FreeQ[{a, b, c, d, e, m, n}, x] && NeQ[m, -1]
  6527. Int[Times[CosIntegral[Times[Plus[Optional[Pattern[a, Blank[]]], Times[Log[Times[Optional[Pattern[c, Blank[]]], Power[Pattern[x, Blank[]], Optional[Pattern[n, Blank[]]]]]], Optional[Pattern[b, Blank[]]]]], Optional[Pattern[d, Blank[]]]]], Power[Times[Optional[Pattern[e, Blank[]]], Pattern[x, Blank[]]], Optional[Pattern[m, Blank[]]]]], Pattern[x, Blank[Symbol]]] := Simp[((e*x)^(m + 1)*CosIntegral[d*(a + b*Log[c*x^n])])/(e*(m + 1)), x] - Dist[(b*d*n)/(m + 1), Int[((e*x)^m*Cos[d*(a + b*Log[c*x^n])])/(d*(a + b*Log[c*x^n])), x], x] /; FreeQ[{a, b, c, d, e, m, n}, x] && NeQ[m, -1]
  6528. Int[SinhIntegral[Plus[Optional[Pattern[a, Blank[]]], Times[Optional[Pattern[b, Blank[]]], Pattern[x, Blank[]]]]], Pattern[x, Blank[Symbol]]] := Simp[((a + b*x)*SinhIntegral[a + b*x])/b, x] - Simp[Cosh[a + b*x]/b, x] /; FreeQ[{a, b}, x]
  6529. Int[CoshIntegral[Plus[Optional[Pattern[a, Blank[]]], Times[Optional[Pattern[b, Blank[]]], Pattern[x, Blank[]]]]], Pattern[x, Blank[Symbol]]] := Simp[((a + b*x)*CoshIntegral[a + b*x])/b, x] - Simp[Sinh[a + b*x]/b, x] /; FreeQ[{a, b}, x]
  6530. Int[Times[Power[Pattern[x, Blank[]], -1], SinhIntegral[Times[Optional[Pattern[b, Blank[]]], Pattern[x, Blank[]]]]], Pattern[x, Blank[Symbol]]] := Simp[(1*b*x*HypergeometricPFQ[{1, 1, 1}, {2, 2, 2}, -(b*x)])/2, x] + Simp[(1*b*x*HypergeometricPFQ[{1, 1, 1}, {2, 2, 2}, b*x])/2, x] /; FreeQ[b, x]
  6531. Int[Times[CoshIntegral[Times[Optional[Pattern[b, Blank[]]], Pattern[x, Blank[]]]], Power[Pattern[x, Blank[]], -1]], Pattern[x, Blank[Symbol]]] := -Simp[(b*x*HypergeometricPFQ[{1, 1, 1}, {2, 2, 2}, -(b*x)])/2, x] + (Simp[(1*b*x*HypergeometricPFQ[{1, 1, 1}, {2, 2, 2}, b*x])/2, x] + Simp[EulerGamma*Log[x], x] + Simp[(1*Log[b*x]^2)/2, x]) /; FreeQ[b, x]
  6532. Int[Times[Power[Plus[Optional[Pattern[c, Blank[]]], Times[Optional[Pattern[d, Blank[]]], Pattern[x, Blank[]]]], Optional[Pattern[m, Blank[]]]], SinhIntegral[Plus[Optional[Pattern[a, Blank[]]], Times[Optional[Pattern[b, Blank[]]], Pattern[x, Blank[]]]]]], Pattern[x, Blank[Symbol]]] := Simp[((c + d*x)^(m + 1)*SinhIntegral[a + b*x])/(d*(m + 1)), x] - Dist[b/(d*(m + 1)), Int[((c + d*x)^(m + 1)*Sinh[a + b*x])/(a + b*x), x], x] /; FreeQ[{a, b, c, d, m}, x] && NeQ[m, -1]
  6533. Int[Times[CoshIntegral[Plus[Optional[Pattern[a, Blank[]]], Times[Optional[Pattern[b, Blank[]]], Pattern[x, Blank[]]]]], Power[Plus[Optional[Pattern[c, Blank[]]], Times[Optional[Pattern[d, Blank[]]], Pattern[x, Blank[]]]], Optional[Pattern[m, Blank[]]]]], Pattern[x, Blank[Symbol]]] := Simp[((c + d*x)^(m + 1)*CoshIntegral[a + b*x])/(d*(m + 1)), x] - Dist[b/(d*(m + 1)), Int[((c + d*x)^(m + 1)*Cosh[a + b*x])/(a + b*x), x], x] /; FreeQ[{a, b, c, d, m}, x] && NeQ[m, -1]
  6534. Int[Power[SinhIntegral[Plus[Optional[Pattern[a, Blank[]]], Times[Optional[Pattern[b, Blank[]]], Pattern[x, Blank[]]]]], 2], Pattern[x, Blank[Symbol]]] := Simp[((a + b*x)*SinhIntegral[a + b*x]^2)/b, x] - Dist[2, Int[Sinh[a + b*x]*SinhIntegral[a + b*x], x], x] /; FreeQ[{a, b}, x]
  6535. Int[Power[CoshIntegral[Plus[Optional[Pattern[a, Blank[]]], Times[Optional[Pattern[b, Blank[]]], Pattern[x, Blank[]]]]], 2], Pattern[x, Blank[Symbol]]] := Simp[((a + b*x)*CoshIntegral[a + b*x]^2)/b, x] - Dist[2, Int[Cosh[a + b*x]*CoshIntegral[a + b*x], x], x] /; FreeQ[{a, b}, x]
  6536. Int[Times[Power[Pattern[x, Blank[]], Optional[Pattern[m, Blank[]]]], Power[SinhIntegral[Times[Optional[Pattern[b, Blank[]]], Pattern[x, Blank[]]]], 2]], Pattern[x, Blank[Symbol]]] := Simp[(x^(m + 1)*SinhIntegral[b*x]^2)/(m + 1), x] - Dist[2/(m + 1), Int[x^m*Sinh[b*x]*SinhIntegral[b*x], x], x] /; FreeQ[b, x] && IGtQ[m, 0]
  6537. Int[Times[Power[CoshIntegral[Times[Optional[Pattern[b, Blank[]]], Pattern[x, Blank[]]]], 2], Power[Pattern[x, Blank[]], Optional[Pattern[m, Blank[]]]]], Pattern[x, Blank[Symbol]]] := Simp[(x^(m + 1)*CoshIntegral[b*x]^2)/(m + 1), x] - Dist[2/(m + 1), Int[x^m*Cosh[b*x]*CoshIntegral[b*x], x], x] /; FreeQ[b, x] && IGtQ[m, 0]
  6538. Int[Times[Power[Plus[Optional[Pattern[c, Blank[]]], Times[Optional[Pattern[d, Blank[]]], Pattern[x, Blank[]]]], Optional[Pattern[m, Blank[]]]], Power[SinhIntegral[Plus[Pattern[a, Blank[]], Times[Optional[Pattern[b, Blank[]]], Pattern[x, Blank[]]]]], 2]], Pattern[x, Blank[Symbol]]] := Simp[((a + b*x)*(c + d*x)^m*SinhIntegral[a + b*x]^2)/(b*(m + 1)), x] + (-Dist[2/(m + 1), Int[(c + d*x)^m*Sinh[a + b*x]*SinhIntegral[a + b*x], x], x] + Dist[((b*c - a*d)*m)/(b*(m + 1)), Int[(c + d*x)^(m - 1)*SinhIntegral[a + b*x]^2, x], x]) /; FreeQ[{a, b, c, d}, x] && IGtQ[m, 0]
  6539. Int[Times[Power[CoshIntegral[Plus[Pattern[a, Blank[]], Times[Optional[Pattern[b, Blank[]]], Pattern[x, Blank[]]]]], 2], Power[Plus[Optional[Pattern[c, Blank[]]], Times[Optional[Pattern[d, Blank[]]], Pattern[x, Blank[]]]], Optional[Pattern[m, Blank[]]]]], Pattern[x, Blank[Symbol]]] := Simp[((a + b*x)*(c + d*x)^m*CoshIntegral[a + b*x]^2)/(b*(m + 1)), x] + (-Dist[2/(m + 1), Int[(c + d*x)^m*Cosh[a + b*x]*CoshIntegral[a + b*x], x], x] + Dist[((b*c - a*d)*m)/(b*(m + 1)), Int[(c + d*x)^(m - 1)*CoshIntegral[a + b*x]^2, x], x]) /; FreeQ[{a, b, c, d}, x] && IGtQ[m, 0]
  6540. Int[Times[Sinh[Plus[Optional[Pattern[a, Blank[]]], Times[Optional[Pattern[b, Blank[]]], Pattern[x, Blank[]]]]], SinhIntegral[Plus[Optional[Pattern[c, Blank[]]], Times[Optional[Pattern[d, Blank[]]], Pattern[x, Blank[]]]]]], Pattern[x, Blank[Symbol]]] := Simp[(Cosh[a + b*x]*SinhIntegral[c + d*x])/b, x] - Dist[d/b, Int[(Cosh[a + b*x]*Sinh[c + d*x])/(c + d*x), x], x] /; FreeQ[{a, b, c, d}, x]
  6541. Int[Times[Cosh[Plus[Optional[Pattern[a, Blank[]]], Times[Optional[Pattern[b, Blank[]]], Pattern[x, Blank[]]]]], CoshIntegral[Plus[Optional[Pattern[c, Blank[]]], Times[Optional[Pattern[d, Blank[]]], Pattern[x, Blank[]]]]]], Pattern[x, Blank[Symbol]]] := Simp[(Sinh[a + b*x]*CoshIntegral[c + d*x])/b, x] - Dist[d/b, Int[(Sinh[a + b*x]*Cosh[c + d*x])/(c + d*x), x], x] /; FreeQ[{a, b, c, d}, x]
  6542. Int[Times[Power[Plus[Optional[Pattern[e, Blank[]]], Times[Optional[Pattern[f, Blank[]]], Pattern[x, Blank[]]]], Optional[Pattern[m, Blank[]]]], Sinh[Plus[Optional[Pattern[a, Blank[]]], Times[Optional[Pattern[b, Blank[]]], Pattern[x, Blank[]]]]], SinhIntegral[Plus[Optional[Pattern[c, Blank[]]], Times[Optional[Pattern[d, Blank[]]], Pattern[x, Blank[]]]]]], Pattern[x, Blank[Symbol]]] := Simp[((e + f*x)^m*Cosh[a + b*x]*SinhIntegral[c + d*x])/b, x] + (-Dist[d/b, Int[((e + f*x)^m*Cosh[a + b*x]*Sinh[c + d*x])/(c + d*x), x], x] - Dist[(f*m)/b, Int[(e + f*x)^(m - 1)*Cosh[a + b*x]*SinhIntegral[c + d*x], x], x]) /; FreeQ[{a, b, c, d, e, f}, x] && IGtQ[m, 0]
  6543. Int[Times[Cosh[Plus[Optional[Pattern[a, Blank[]]], Times[Optional[Pattern[b, Blank[]]], Pattern[x, Blank[]]]]], CoshIntegral[Plus[Optional[Pattern[c, Blank[]]], Times[Optional[Pattern[d, Blank[]]], Pattern[x, Blank[]]]]], Power[Plus[Optional[Pattern[e, Blank[]]], Times[Optional[Pattern[f, Blank[]]], Pattern[x, Blank[]]]], Optional[Pattern[m, Blank[]]]]], Pattern[x, Blank[Symbol]]] := Simp[((e + f*x)^m*Sinh[a + b*x]*CoshIntegral[c + d*x])/b, x] + (-Dist[d/b, Int[((e + f*x)^m*Sinh[a + b*x]*Cosh[c + d*x])/(c + d*x), x], x] - Dist[(f*m)/b, Int[(e + f*x)^(m - 1)*Sinh[a + b*x]*CoshIntegral[c + d*x], x], x]) /; FreeQ[{a, b, c, d, e, f}, x] && IGtQ[m, 0]
  6544. Int[Times[Power[Plus[Optional[Pattern[e, Blank[]]], Times[Optional[Pattern[f, Blank[]]], Pattern[x, Blank[]]]], Pattern[m, Blank[]]], Sinh[Plus[Optional[Pattern[a, Blank[]]], Times[Optional[Pattern[b, Blank[]]], Pattern[x, Blank[]]]]], SinhIntegral[Plus[Optional[Pattern[c, Blank[]]], Times[Optional[Pattern[d, Blank[]]], Pattern[x, Blank[]]]]]], Pattern[x, Blank[Symbol]]] := Simp[((e + f*x)^(m + 1)*Sinh[a + b*x]*SinhIntegral[c + d*x])/(f*(m + 1)), x] + (-Dist[b/(f*(m + 1)), Int[(e + f*x)^(m + 1)*Cosh[a + b*x]*SinhIntegral[c + d*x], x], x] - Dist[d/(f*(m + 1)), Int[((e + f*x)^(m + 1)*Sinh[a + b*x]*Sinh[c + d*x])/(c + d*x), x], x]) /; FreeQ[{a, b, c, d, e, f}, x] && ILtQ[m, -1]
  6545. Int[Times[Cosh[Plus[Optional[Pattern[a, Blank[]]], Times[Optional[Pattern[b, Blank[]]], Pattern[x, Blank[]]]]], CoshIntegral[Plus[Optional[Pattern[c, Blank[]]], Times[Optional[Pattern[d, Blank[]]], Pattern[x, Blank[]]]]], Power[Plus[Optional[Pattern[e, Blank[]]], Times[Optional[Pattern[f, Blank[]]], Pattern[x, Blank[]]]], Optional[Pattern[m, Blank[]]]]], Pattern[x, Blank[Symbol]]] := Simp[((e + f*x)^(m + 1)*Cosh[a + b*x]*CoshIntegral[c + d*x])/(f*(m + 1)), x] + (-Dist[b/(f*(m + 1)), Int[(e + f*x)^(m + 1)*Sinh[a + b*x]*CoshIntegral[c + d*x], x], x] - Dist[d/(f*(m + 1)), Int[((e + f*x)^(m + 1)*Cosh[a + b*x]*Cosh[c + d*x])/(c + d*x), x], x]) /; FreeQ[{a, b, c, d, e, f}, x] && ILtQ[m, -1]
  6546. Int[Times[Cosh[Plus[Optional[Pattern[a, Blank[]]], Times[Optional[Pattern[b, Blank[]]], Pattern[x, Blank[]]]]], SinhIntegral[Plus[Optional[Pattern[c, Blank[]]], Times[Optional[Pattern[d, Blank[]]], Pattern[x, Blank[]]]]]], Pattern[x, Blank[Symbol]]] := Simp[(Sinh[a + b*x]*SinhIntegral[c + d*x])/b, x] - Dist[d/b, Int[(Sinh[a + b*x]*Sinh[c + d*x])/(c + d*x), x], x] /; FreeQ[{a, b, c, d}, x]
  6547. Int[Times[CoshIntegral[Plus[Optional[Pattern[c, Blank[]]], Times[Optional[Pattern[d, Blank[]]], Pattern[x, Blank[]]]]], Sinh[Plus[Optional[Pattern[a, Blank[]]], Times[Optional[Pattern[b, Blank[]]], Pattern[x, Blank[]]]]]], Pattern[x, Blank[Symbol]]] := Simp[(Cosh[a + b*x]*CoshIntegral[c + d*x])/b, x] - Dist[d/b, Int[(Cosh[a + b*x]*Cosh[c + d*x])/(c + d*x), x], x] /; FreeQ[{a, b, c, d}, x]
  6548. Int[Times[Cosh[Plus[Optional[Pattern[a, Blank[]]], Times[Optional[Pattern[b, Blank[]]], Pattern[x, Blank[]]]]], Power[Plus[Optional[Pattern[e, Blank[]]], Times[Optional[Pattern[f, Blank[]]], Pattern[x, Blank[]]]], Optional[Pattern[m, Blank[]]]], SinhIntegral[Plus[Optional[Pattern[c, Blank[]]], Times[Optional[Pattern[d, Blank[]]], Pattern[x, Blank[]]]]]], Pattern[x, Blank[Symbol]]] := Simp[((e + f*x)^m*Sinh[a + b*x]*SinhIntegral[c + d*x])/b, x] + (-Dist[d/b, Int[((e + f*x)^m*Sinh[a + b*x]*Sinh[c + d*x])/(c + d*x), x], x] - Dist[(f*m)/b, Int[(e + f*x)^(m - 1)*Sinh[a + b*x]*SinhIntegral[c + d*x], x], x]) /; FreeQ[{a, b, c, d, e, f}, x] && IGtQ[m, 0]
  6549. Int[Times[CoshIntegral[Plus[Optional[Pattern[c, Blank[]]], Times[Optional[Pattern[d, Blank[]]], Pattern[x, Blank[]]]]], Power[Plus[Optional[Pattern[e, Blank[]]], Times[Optional[Pattern[f, Blank[]]], Pattern[x, Blank[]]]], Optional[Pattern[m, Blank[]]]], Sinh[Plus[Optional[Pattern[a, Blank[]]], Times[Optional[Pattern[b, Blank[]]], Pattern[x, Blank[]]]]]], Pattern[x, Blank[Symbol]]] := Simp[((e + f*x)^m*Cosh[a + b*x]*CoshIntegral[c + d*x])/b, x] + (-Dist[d/b, Int[((e + f*x)^m*Cosh[a + b*x]*Cosh[c + d*x])/(c + d*x), x], x] - Dist[(f*m)/b, Int[(e + f*x)^(m - 1)*Cosh[a + b*x]*CoshIntegral[c + d*x], x], x]) /; FreeQ[{a, b, c, d, e, f}, x] && IGtQ[m, 0]
  6550. Int[Times[Cosh[Plus[Optional[Pattern[a, Blank[]]], Times[Optional[Pattern[b, Blank[]]], Pattern[x, Blank[]]]]], Power[Plus[Optional[Pattern[e, Blank[]]], Times[Optional[Pattern[f, Blank[]]], Pattern[x, Blank[]]]], Optional[Pattern[m, Blank[]]]], SinhIntegral[Plus[Optional[Pattern[c, Blank[]]], Times[Optional[Pattern[d, Blank[]]], Pattern[x, Blank[]]]]]], Pattern[x, Blank[Symbol]]] := Simp[((e + f*x)^(m + 1)*Cosh[a + b*x]*SinhIntegral[c + d*x])/(f*(m + 1)), x] + (-Dist[b/(f*(m + 1)), Int[(e + f*x)^(m + 1)*Sinh[a + b*x]*SinhIntegral[c + d*x], x], x] - Dist[d/(f*(m + 1)), Int[((e + f*x)^(m + 1)*Cosh[a + b*x]*Sinh[c + d*x])/(c + d*x), x], x]) /; FreeQ[{a, b, c, d, e, f}, x] && ILtQ[m, -1]
  6551. Int[Times[CoshIntegral[Plus[Optional[Pattern[c, Blank[]]], Times[Optional[Pattern[d, Blank[]]], Pattern[x, Blank[]]]]], Power[Plus[Optional[Pattern[e, Blank[]]], Times[Optional[Pattern[f, Blank[]]], Pattern[x, Blank[]]]], Pattern[m, Blank[]]], Sinh[Plus[Optional[Pattern[a, Blank[]]], Times[Optional[Pattern[b, Blank[]]], Pattern[x, Blank[]]]]]], Pattern[x, Blank[Symbol]]] := Simp[((e + f*x)^(m + 1)*Sinh[a + b*x]*CoshIntegral[c + d*x])/(f*(m + 1)), x] + (-Dist[b/(f*(m + 1)), Int[(e + f*x)^(m + 1)*Cosh[a + b*x]*CoshIntegral[c + d*x], x], x] - Dist[d/(f*(m + 1)), Int[((e + f*x)^(m + 1)*Sinh[a + b*x]*Cosh[c + d*x])/(c + d*x), x], x]) /; FreeQ[{a, b, c, d, e, f}, x] && ILtQ[m, -1]
  6552. Int[SinhIntegral[Times[Plus[Optional[Pattern[a, Blank[]]], Times[Log[Times[Optional[Pattern[c, Blank[]]], Power[Pattern[x, Blank[]], Optional[Pattern[n, Blank[]]]]]], Optional[Pattern[b, Blank[]]]]], Optional[Pattern[d, Blank[]]]]], Pattern[x, Blank[Symbol]]] := Simp[x*SinhIntegral[d*(a + b*Log[c*x^n])], x] - Dist[b*d*n, Int[Sinh[d*(a + b*Log[c*x^n])]/(d*(a + b*Log[c*x^n])), x], x] /; FreeQ[{a, b, c, d, n}, x]
  6553. Int[CoshIntegral[Times[Plus[Optional[Pattern[a, Blank[]]], Times[Log[Times[Optional[Pattern[c, Blank[]]], Power[Pattern[x, Blank[]], Optional[Pattern[n, Blank[]]]]]], Optional[Pattern[b, Blank[]]]]], Optional[Pattern[d, Blank[]]]]], Pattern[x, Blank[Symbol]]] := Simp[x*CoshIntegral[d*(a + b*Log[c*x^n])], x] - Dist[b*d*n, Int[Cosh[d*(a + b*Log[c*x^n])]/(d*(a + b*Log[c*x^n])), x], x] /; FreeQ[{a, b, c, d, n}, x]
  6554. Int[Times[Power[Pattern[x, Blank[]], -1], Pattern[F, Blank[]][Times[Plus[Optional[Pattern[a, Blank[]]], Times[Log[Times[Optional[Pattern[c, Blank[]]], Power[Pattern[x, Blank[]], Optional[Pattern[n, Blank[]]]]]], Optional[Pattern[b, Blank[]]]]], Optional[Pattern[d, Blank[]]]]]], Pattern[x, Blank[Symbol]]] := Dist[1/n, Subst[F[d*(a + b*x)], x, Log[c*x^n]], x] /; FreeQ[{a, b, c, d, n}, x] && MemberQ[{SinhIntegral, CoshIntegral}, x]
  6555. Int[Times[Power[Times[Optional[Pattern[e, Blank[]]], Pattern[x, Blank[]]], Optional[Pattern[m, Blank[]]]], SinhIntegral[Times[Plus[Optional[Pattern[a, Blank[]]], Times[Log[Times[Optional[Pattern[c, Blank[]]], Power[Pattern[x, Blank[]], Optional[Pattern[n, Blank[]]]]]], Optional[Pattern[b, Blank[]]]]], Optional[Pattern[d, Blank[]]]]]], Pattern[x, Blank[Symbol]]] := Simp[((e*x)^(m + 1)*SinhIntegral[d*(a + b*Log[c*x^n])])/(e*(m + 1)), x] - Dist[(b*d*n)/(m + 1), Int[((e*x)^m*Sinh[d*(a + b*Log[c*x^n])])/(d*(a + b*Log[c*x^n])), x], x] /; FreeQ[{a, b, c, d, e, m, n}, x] && NeQ[m, -1]
  6556. Int[Times[CoshIntegral[Times[Plus[Optional[Pattern[a, Blank[]]], Times[Log[Times[Optional[Pattern[c, Blank[]]], Power[Pattern[x, Blank[]], Optional[Pattern[n, Blank[]]]]]], Optional[Pattern[b, Blank[]]]]], Optional[Pattern[d, Blank[]]]]], Power[Times[Optional[Pattern[e, Blank[]]], Pattern[x, Blank[]]], Optional[Pattern[m, Blank[]]]]], Pattern[x, Blank[Symbol]]] := Simp[((e*x)^(m + 1)*CoshIntegral[d*(a + b*Log[c*x^n])])/(e*(m + 1)), x] - Dist[(b*d*n)/(m + 1), Int[((e*x)^m*Cosh[d*(a + b*Log[c*x^n])])/(d*(a + b*Log[c*x^n])), x], x] /; FreeQ[{a, b, c, d, e, m, n}, x] && NeQ[m, -1]
  6557. Int[Gamma[Pattern[n, Blank[]], Plus[Optional[Pattern[a, Blank[]]], Times[Optional[Pattern[b, Blank[]]], Pattern[x, Blank[]]]]], Pattern[x, Blank[Symbol]]] := Simp[((a + b*x)*Gamma[n, a + b*x])/b, x] - Simp[Gamma[n + 1, a + b*x]/b, x] /; FreeQ[{a, b, n}, x]
  6558. Int[Times[Gamma[0, Times[Optional[Pattern[b, Blank[]]], Pattern[x, Blank[]]]], Power[Pattern[x, Blank[]], -1]], Pattern[x, Blank[Symbol]]] := Simp[b*x*HypergeometricPFQ[{1, 1, 1}, {2, 2, 2}, -(b*x)], x] + (-Simp[EulerGamma*Log[x], x] - Simp[(1*Log[b*x]^2)/2, x]) /; FreeQ[b, x]
  6559. Int[Times[Gamma[Pattern[n, Blank[]], Times[Optional[Pattern[b, Blank[]]], Pattern[x, Blank[]]]], Power[Pattern[x, Blank[]], -1]], Pattern[x, Blank[Symbol]]] := -Simp[Gamma[n - 1, b*x], x] + Dist[n - 1, Int[Gamma[n - 1, b*x]/x, x], x] /; FreeQ[b, x] && IGtQ[n, 1]
  6560. Int[Times[Gamma[Pattern[n, Blank[]], Times[Optional[Pattern[b, Blank[]]], Pattern[x, Blank[]]]], Power[Pattern[x, Blank[]], -1]], Pattern[x, Blank[Symbol]]] := Simp[Gamma[n, b*x]/n, x] + Dist[1/n, Int[Gamma[n + 1, b*x]/x, x], x] /; FreeQ[b, x] && ILtQ[n, 0]
  6561. Int[Times[Gamma[Pattern[n, Blank[]], Times[Optional[Pattern[b, Blank[]]], Pattern[x, Blank[]]]], Power[Pattern[x, Blank[]], -1]], Pattern[x, Blank[Symbol]]] := Simp[Gamma[n]*Log[x], x] - Simp[((b*x)^n*HypergeometricPFQ[{n, n}, {1 + n, 1 + n}, -(b*x)])/n^2, x] /; FreeQ[{b, n}, x] && !IntegerQ[n]
  6562. Int[Times[Gamma[Pattern[n, Blank[]], Times[Optional[Pattern[b, Blank[]]], Pattern[x, Blank[]]]], Power[Times[Optional[Pattern[d, Blank[]]], Pattern[x, Blank[]]], Optional[Pattern[m, Blank[]]]]], Pattern[x, Blank[Symbol]]] := Simp[((d*x)^(m + 1)*Gamma[n, b*x])/(d*(m + 1)), x] - Simp[((d*x)^m*Gamma[m + n + 1, b*x])/(b*(m + 1)*(b*x)^m), x] /; FreeQ[{b, d, m, n}, x] && NeQ[m, -1]
  6563. Int[Times[Gamma[Pattern[n, Blank[]], Plus[Pattern[a, Blank[]], Times[Optional[Pattern[b, Blank[]]], Pattern[x, Blank[]]]]], Power[Plus[Pattern[c, Blank[]], Times[Optional[Pattern[d, Blank[]]], Pattern[x, Blank[]]]], Optional[Pattern[m, Blank[]]]]], Pattern[x, Blank[Symbol]]] := Dist[1/b, Subst[Int[((d*x)/b)^m*Gamma[n, x], x], x, a + b*x], x] /; FreeQ[{a, b, c, d, m, n}, x] && EqQ[b*c - a*d, 0]
  6564. Int[Times[Gamma[Pattern[n, Blank[]], Plus[Optional[Pattern[a, Blank[]]], Times[Optional[Pattern[b, Blank[]]], Pattern[x, Blank[]]]]], Power[Plus[Optional[Pattern[c, Blank[]]], Times[Optional[Pattern[d, Blank[]]], Pattern[x, Blank[]]]], -1]], Pattern[x, Blank[Symbol]]] := Int[(a + b*x)^(n - 1)/((c + d*x)*E^(a + b*x)), x] + Dist[n - 1, Int[Gamma[n - 1, a + b*x]/(c + d*x), x], x] /; FreeQ[{a, b, c, d}, x] && IGtQ[n, 1]
  6565. Int[Times[Gamma[Pattern[n, Blank[]], Plus[Optional[Pattern[a, Blank[]]], Times[Optional[Pattern[b, Blank[]]], Pattern[x, Blank[]]]]], Power[Plus[Optional[Pattern[c, Blank[]]], Times[Optional[Pattern[d, Blank[]]], Pattern[x, Blank[]]]], Optional[Pattern[m, Blank[]]]]], Pattern[x, Blank[Symbol]]] := Block[{False = True}, Simp[((c + d*x)^(m + 1)*Gamma[n, a + b*x])/(d*(m + 1)), x] + Dist[b/(d*(m + 1)), Int[((c + d*x)^(m + 1)*(a + b*x)^(n - 1))/E^(a + b*x), x], x]] /; FreeQ[{a, b, c, d, m, n}, x] && (IGtQ[m, 0] || IGtQ[n, 0] || IntegersQ[m, n]) && NeQ[m, -1]
  6566. Int[Times[Gamma[Pattern[n, Blank[]], Plus[Optional[Pattern[a, Blank[]]], Times[Optional[Pattern[b, Blank[]]], Pattern[x, Blank[]]]]], Power[Plus[Optional[Pattern[c, Blank[]]], Times[Optional[Pattern[d, Blank[]]], Pattern[x, Blank[]]]], Optional[Pattern[m, Blank[]]]]], Pattern[x, Blank[Symbol]]] := Unintegrable[(c + d*x)^m*Gamma[n, a + b*x], x] /; FreeQ[{a, b, c, d, m, n}, x]
  6567. Int[LogGamma[Plus[Optional[Pattern[a, Blank[]]], Times[Optional[Pattern[b, Blank[]]], Pattern[x, Blank[]]]]], Pattern[x, Blank[Symbol]]] := Simp[PolyGamma[-2, a + b*x]/b, x] /; FreeQ[{a, b}, x]
  6568. Int[Times[LogGamma[Plus[Optional[Pattern[a, Blank[]]], Times[Optional[Pattern[b, Blank[]]], Pattern[x, Blank[]]]]], Power[Plus[Optional[Pattern[c, Blank[]]], Times[Optional[Pattern[d, Blank[]]], Pattern[x, Blank[]]]], Optional[Pattern[m, Blank[]]]]], Pattern[x, Blank[Symbol]]] := Simp[((c + d*x)^m*PolyGamma[-2, a + b*x])/b, x] - Dist[(d*m)/b, Int[(c + d*x)^(m - 1)*PolyGamma[-2, a + b*x], x], x] /; FreeQ[{a, b, c, d}, x] && IGtQ[m, 0]
  6569. Int[Times[LogGamma[Plus[Optional[Pattern[a, Blank[]]], Times[Optional[Pattern[b, Blank[]]], Pattern[x, Blank[]]]]], Power[Plus[Optional[Pattern[c, Blank[]]], Times[Optional[Pattern[d, Blank[]]], Pattern[x, Blank[]]]], Optional[Pattern[m, Blank[]]]]], Pattern[x, Blank[Symbol]]] := Unintegrable[(c + d*x)^m*LogGamma[a + b*x], x] /; FreeQ[{a, b, c, d, m}, x]
  6570. Int[PolyGamma[Pattern[n, Blank[]], Plus[Optional[Pattern[a, Blank[]]], Times[Optional[Pattern[b, Blank[]]], Pattern[x, Blank[]]]]], Pattern[x, Blank[Symbol]]] := Simp[PolyGamma[n - 1, a + b*x]/b, x] /; FreeQ[{a, b, n}, x]
  6571. Int[Times[Power[Plus[Optional[Pattern[c, Blank[]]], Times[Optional[Pattern[d, Blank[]]], Pattern[x, Blank[]]]], Optional[Pattern[m, Blank[]]]], PolyGamma[Pattern[n, Blank[]], Plus[Optional[Pattern[a, Blank[]]], Times[Optional[Pattern[b, Blank[]]], Pattern[x, Blank[]]]]]], Pattern[x, Blank[Symbol]]] := Simp[((c + d*x)^m*PolyGamma[n - 1, a + b*x])/b, x] - Dist[(d*m)/b, Int[(c + d*x)^(m - 1)*PolyGamma[n - 1, a + b*x], x], x] /; FreeQ[{a, b, c, d, n}, x] && GtQ[m, 0]
  6572. Int[Times[Power[Plus[Optional[Pattern[c, Blank[]]], Times[Optional[Pattern[d, Blank[]]], Pattern[x, Blank[]]]], Optional[Pattern[m, Blank[]]]], PolyGamma[Pattern[n, Blank[]], Plus[Optional[Pattern[a, Blank[]]], Times[Optional[Pattern[b, Blank[]]], Pattern[x, Blank[]]]]]], Pattern[x, Blank[Symbol]]] := Simp[((c + d*x)^(m + 1)*PolyGamma[n, a + b*x])/(d*(m + 1)), x] - Dist[b/(d*(m + 1)), Int[(c + d*x)^(m + 1)*PolyGamma[n + 1, a + b*x], x], x] /; FreeQ[{a, b, c, d, n}, x] && LtQ[m, -1]
  6573. Int[Times[Power[Plus[Optional[Pattern[c, Blank[]]], Times[Optional[Pattern[d, Blank[]]], Pattern[x, Blank[]]]], Optional[Pattern[m, Blank[]]]], PolyGamma[Pattern[n, Blank[]], Plus[Optional[Pattern[a, Blank[]]], Times[Optional[Pattern[b, Blank[]]], Pattern[x, Blank[]]]]]], Pattern[x, Blank[Symbol]]] := Unintegrable[(c + d*x)^m*PolyGamma[n, a + b*x], x] /; FreeQ[{a, b, c, d, m, n}, x]
  6574. Int[Times[Power[Gamma[Plus[Optional[Pattern[a, Blank[]]], Times[Optional[Pattern[b, Blank[]]], Pattern[x, Blank[]]]]], Optional[Pattern[n, Blank[]]]], PolyGamma[0, Plus[Optional[Pattern[a, Blank[]]], Times[Optional[Pattern[b, Blank[]]], Pattern[x, Blank[]]]]]], Pattern[x, Blank[Symbol]]] := Simp[Gamma[a + b*x]^n/(b*n), x] /; FreeQ[{a, b, n}, x]
  6575. Int[Times[Power[Factorial[Plus[Optional[Pattern[a, Blank[]]], Times[Optional[Pattern[b, Blank[]]], Pattern[x, Blank[]]]]], Optional[Pattern[n, Blank[]]]], PolyGamma[0, Plus[Optional[Pattern[c, Blank[]]], Times[Optional[Pattern[b, Blank[]]], Pattern[x, Blank[]]]]]], Pattern[x, Blank[Symbol]]] := Simp[(a + b*x)!^n/(b*n), x] /; FreeQ[{a, b, c, n}, x] && EqQ[c, a + 1]
  6576. Int[Gamma[Pattern[p, Blank[]], Times[Plus[Optional[Pattern[a, Blank[]]], Times[Log[Times[Optional[Pattern[c, Blank[]]], Power[Pattern[x, Blank[]], Optional[Pattern[n, Blank[]]]]]], Optional[Pattern[b, Blank[]]]]], Optional[Pattern[d, Blank[]]]]], Pattern[x, Blank[Symbol]]] := Simp[x*Gamma[p, d*(a + b*Log[c*x^n])], x] + Dist[(b*d*n)/E^(a*d), Int[(d*(a + b*Log[c*x^n]))^(p - 1)/(c*x^n)^(b*d), x], x] /; FreeQ[{a, b, c, d, n, p}, x]
  6577. Int[Times[Gamma[Pattern[p, Blank[]], Times[Plus[Optional[Pattern[a, Blank[]]], Times[Log[Times[Optional[Pattern[c, Blank[]]], Power[Pattern[x, Blank[]], Optional[Pattern[n, Blank[]]]]]], Optional[Pattern[b, Blank[]]]]], Optional[Pattern[d, Blank[]]]]], Power[Pattern[x, Blank[]], -1]], Pattern[x, Blank[Symbol]]] := Dist[1/n, Subst[Gamma[p, d*(a + b*x)], x, Log[c*x^n]], x] /; FreeQ[{a, b, c, d, n, p}, x]
  6578. Int[Times[Gamma[Pattern[p, Blank[]], Times[Plus[Optional[Pattern[a, Blank[]]], Times[Log[Times[Optional[Pattern[c, Blank[]]], Power[Pattern[x, Blank[]], Optional[Pattern[n, Blank[]]]]]], Optional[Pattern[b, Blank[]]]]], Optional[Pattern[d, Blank[]]]]], Power[Times[Optional[Pattern[e, Blank[]]], Pattern[x, Blank[]]], Optional[Pattern[m, Blank[]]]]], Pattern[x, Blank[Symbol]]] := Simp[((e*x)^(m + 1)*Gamma[p, d*(a + b*Log[c*x^n])])/(e*(m + 1)), x] + Dist[(b*d*n*(e*x)^(b*d*n))/(E^(a*d)*((m + 1)*(c*x^n)^(b*d))), Int[(e*x)^(m - b*d*n)*(d*(a + b*Log[c*x^n]))^(p - 1), x], x] /; FreeQ[{a, b, c, d, e, m, n, p}, x] && NeQ[m, -1]
  6579. Int[Gamma[Pattern[p, Blank[]], Times[Plus[Optional[Pattern[a, Blank[]]], Times[Log[Times[Optional[Pattern[c, Blank[]]], Power[Plus[Pattern[d, Blank[]], Times[Optional[Pattern[e, Blank[]]], Pattern[x, Blank[]]]], Optional[Pattern[n, Blank[]]]]]], Optional[Pattern[b, Blank[]]]]], Optional[Pattern[f, Blank[]]]]], Pattern[x, Blank[Symbol]]] := Dist[1/e, Subst[Int[Gamma[p, f*(a + b*Log[c*x^n])], x], x, d + e*x], x] /; FreeQ[{a, b, c, d, e, f, n, p}, x]
  6580. Int[Times[Gamma[Pattern[p, Blank[]], Times[Plus[Optional[Pattern[a, Blank[]]], Times[Log[Times[Optional[Pattern[c, Blank[]]], Power[Plus[Pattern[d, Blank[]], Times[Optional[Pattern[e, Blank[]]], Pattern[x, Blank[]]]], Optional[Pattern[n, Blank[]]]]]], Optional[Pattern[b, Blank[]]]]], Optional[Pattern[f, Blank[]]]]], Power[Plus[Pattern[g, Blank[]], Times[Optional[Pattern[h, Blank[]]], Pattern[x, Blank[]]]], Optional[Pattern[m, Blank[]]]]], Pattern[x, Blank[Symbol]]] := Dist[1/e, Subst[Int[((g*x)/d)^m*Gamma[p, f*(a + b*Log[c*x^n])], x], x, d + e*x], x] /; FreeQ[{a, b, c, d, e, f, g, h, m, n, p}, x] && EqQ[e*g - d*h, 0]
  6581. Int[Zeta[2, Plus[Optional[Pattern[a, Blank[]]], Times[Optional[Pattern[b, Blank[]]], Pattern[x, Blank[]]]]], Pattern[x, Blank[Symbol]]] := Int[PolyGamma[1, a + b*x], x] /; FreeQ[{a, b}, x]
  6582. Int[Zeta[Pattern[s, Blank[]], Plus[Optional[Pattern[a, Blank[]]], Times[Optional[Pattern[b, Blank[]]], Pattern[x, Blank[]]]]], Pattern[x, Blank[Symbol]]] := -Simp[Zeta[s - 1, a + b*x]/(b*(s - 1)), x] /; FreeQ[{a, b, s}, x] && NeQ[s, 1] && NeQ[s, 2]
  6583. Int[Times[Power[Plus[Optional[Pattern[c, Blank[]]], Times[Optional[Pattern[d, Blank[]]], Pattern[x, Blank[]]]], Optional[Pattern[m, Blank[]]]], Zeta[2, Plus[Optional[Pattern[a, Blank[]]], Times[Optional[Pattern[b, Blank[]]], Pattern[x, Blank[]]]]]], Pattern[x, Blank[Symbol]]] := Int[(c + d*x)^m*PolyGamma[1, a + b*x], x] /; FreeQ[{a, b, c, d}, x] && RationalQ[m]
  6584. Int[Times[Power[Plus[Optional[Pattern[c, Blank[]]], Times[Optional[Pattern[d, Blank[]]], Pattern[x, Blank[]]]], Optional[Pattern[m, Blank[]]]], Zeta[Pattern[s, Blank[]], Plus[Optional[Pattern[a, Blank[]]], Times[Optional[Pattern[b, Blank[]]], Pattern[x, Blank[]]]]]], Pattern[x, Blank[Symbol]]] := -Simp[((c + d*x)^m*Zeta[s - 1, a + b*x])/(b*(s - 1)), x] + Dist[(d*m)/(b*(s - 1)), Int[(c + d*x)^(m - 1)*Zeta[s - 1, a + b*x], x], x] /; FreeQ[{a, b, c, d, s}, x] && NeQ[s, 1] && NeQ[s, 2] && GtQ[m, 0]
  6585. Int[Times[Power[Plus[Optional[Pattern[c, Blank[]]], Times[Optional[Pattern[d, Blank[]]], Pattern[x, Blank[]]]], Optional[Pattern[m, Blank[]]]], Zeta[Pattern[s, Blank[]], Plus[Optional[Pattern[a, Blank[]]], Times[Optional[Pattern[b, Blank[]]], Pattern[x, Blank[]]]]]], Pattern[x, Blank[Symbol]]] := Simp[((c + d*x)^(m + 1)*Zeta[s, a + b*x])/(d*(m + 1)), x] + Dist[(b*s)/(d*(m + 1)), Int[(c + d*x)^(m + 1)*Zeta[s + 1, a + b*x], x], x] /; FreeQ[{a, b, c, d, s}, x] && NeQ[s, 1] && NeQ[s, 2] && LtQ[m, -1]
  6586. Int[PolyLog[Pattern[n, Blank[]], Times[Optional[Pattern[a, Blank[]]], Power[Times[Optional[Pattern[b, Blank[]]], Power[Pattern[x, Blank[]], Optional[Pattern[p, Blank[]]]]], Optional[Pattern[q, Blank[]]]]]], Pattern[x, Blank[Symbol]]] := Simp[x*PolyLog[n, a*(b*x^p)^q], x] - Dist[p*q, Int[PolyLog[n - 1, a*(b*x^p)^q], x], x] /; FreeQ[{a, b, p, q}, x] && GtQ[n, 0]
  6587. Int[PolyLog[Pattern[n, Blank[]], Times[Optional[Pattern[a, Blank[]]], Power[Times[Optional[Pattern[b, Blank[]]], Power[Pattern[x, Blank[]], Optional[Pattern[p, Blank[]]]]], Optional[Pattern[q, Blank[]]]]]], Pattern[x, Blank[Symbol]]] := Simp[(x*PolyLog[n + 1, a*(b*x^p)^q])/(p*q), x] - Dist[1/(p*q), Int[PolyLog[n + 1, a*(b*x^p)^q], x], x] /; FreeQ[{a, b, p, q}, x] && LtQ[n, -1]
  6588. Int[PolyLog[Pattern[n, Blank[]], Times[Optional[Pattern[a, Blank[]]], Power[Times[Optional[Pattern[b, Blank[]]], Power[Pattern[x, Blank[]], Optional[Pattern[p, Blank[]]]]], Optional[Pattern[q, Blank[]]]]]], Pattern[x, Blank[Symbol]]] := Unintegrable[PolyLog[n, a*(b*x^p)^q], x] /; FreeQ[{a, b, n, p, q}, x]
  6589. Int[Times[Power[Plus[Optional[Pattern[d, Blank[]]], Times[Optional[Pattern[e, Blank[]]], Pattern[x, Blank[]]]], -1], PolyLog[Pattern[n, Blank[]], Times[Optional[Pattern[c, Blank[]]], Power[Plus[Optional[Pattern[a, Blank[]]], Times[Optional[Pattern[b, Blank[]]], Pattern[x, Blank[]]]], Optional[Pattern[p, Blank[]]]]]]], Pattern[x, Blank[Symbol]]] := Simp[PolyLog[n + 1, c*(a + b*x)^p]/(e*p), x] /; FreeQ[{a, b, c, d, e, n, p}, x] && EqQ[b*d, a*e]
  6590. Int[Times[Power[Pattern[x, Blank[]], -1], PolyLog[Pattern[n, Blank[]], Times[Optional[Pattern[a, Blank[]]], Power[Times[Optional[Pattern[b, Blank[]]], Power[Pattern[x, Blank[]], Optional[Pattern[p, Blank[]]]]], Optional[Pattern[q, Blank[]]]]]]], Pattern[x, Blank[Symbol]]] := Simp[PolyLog[n + 1, a*(b*x^p)^q]/(p*q), x] /; FreeQ[{a, b, n, p, q}, x]
  6591. Int[Times[Power[Times[Optional[Pattern[d, Blank[]]], Pattern[x, Blank[]]], Optional[Pattern[m, Blank[]]]], PolyLog[Pattern[n, Blank[]], Times[Optional[Pattern[a, Blank[]]], Power[Times[Optional[Pattern[b, Blank[]]], Power[Pattern[x, Blank[]], Optional[Pattern[p, Blank[]]]]], Optional[Pattern[q, Blank[]]]]]]], Pattern[x, Blank[Symbol]]] := Simp[((d*x)^(m + 1)*PolyLog[n, a*(b*x^p)^q])/(d*(m + 1)), x] - Dist[(p*q)/(m + 1), Int[(d*x)^m*PolyLog[n - 1, a*(b*x^p)^q], x], x] /; FreeQ[{a, b, d, m, p, q}, x] && NeQ[m, -1] && GtQ[n, 0]
  6592. Int[Times[Power[Times[Optional[Pattern[d, Blank[]]], Pattern[x, Blank[]]], Optional[Pattern[m, Blank[]]]], PolyLog[Pattern[n, Blank[]], Times[Optional[Pattern[a, Blank[]]], Power[Times[Optional[Pattern[b, Blank[]]], Power[Pattern[x, Blank[]], Optional[Pattern[p, Blank[]]]]], Optional[Pattern[q, Blank[]]]]]]], Pattern[x, Blank[Symbol]]] := Simp[((d*x)^(m + 1)*PolyLog[n + 1, a*(b*x^p)^q])/(d*p*q), x] - Dist[(m + 1)/(p*q), Int[(d*x)^m*PolyLog[n + 1, a*(b*x^p)^q], x], x] /; FreeQ[{a, b, d, m, p, q}, x] && NeQ[m, -1] && LtQ[n, -1]
  6593. Int[Times[Power[Times[Optional[Pattern[d, Blank[]]], Pattern[x, Blank[]]], Optional[Pattern[m, Blank[]]]], PolyLog[Pattern[n, Blank[]], Times[Optional[Pattern[a, Blank[]]], Power[Times[Optional[Pattern[b, Blank[]]], Power[Pattern[x, Blank[]], Optional[Pattern[p, Blank[]]]]], Optional[Pattern[q, Blank[]]]]]]], Pattern[x, Blank[Symbol]]] := Unintegrable[(d*x)^m*PolyLog[n, a*(b*x^p)^q], x] /; FreeQ[{a, b, d, m, n, p, q}, x]
  6594. Int[Times[Power[Log[Times[Optional[Pattern[c, Blank[]]], Power[Pattern[x, Blank[]], Optional[Pattern[m, Blank[]]]]]], Optional[Pattern[r, Blank[]]]], Power[Pattern[x, Blank[]], -1], PolyLog[Pattern[n, Blank[]], Times[Optional[Pattern[a, Blank[]]], Power[Times[Optional[Pattern[b, Blank[]]], Power[Pattern[x, Blank[]], Optional[Pattern[p, Blank[]]]]], Optional[Pattern[q, Blank[]]]]]]], Pattern[x, Blank[Symbol]]] := Simp[(Log[c*x^m]^r*PolyLog[n + 1, a*(b*x^p)^q])/(p*q), x] - Dist[(m*r)/(p*q), Int[(Log[c*x^m]^(r - 1)*PolyLog[n + 1, a*(b*x^p)^q])/x, x], x] /; FreeQ[{a, b, c, m, n, q, r}, x] && GtQ[r, 0]
  6595. Int[PolyLog[Pattern[n, Blank[]], Times[Optional[Pattern[c, Blank[]]], Power[Plus[Optional[Pattern[a, Blank[]]], Times[Optional[Pattern[b, Blank[]]], Pattern[x, Blank[]]]], Optional[Pattern[p, Blank[]]]]]], Pattern[x, Blank[Symbol]]] := Simp[x*PolyLog[n, c*(a + b*x)^p], x] + (-Dist[p, Int[PolyLog[n - 1, c*(a + b*x)^p], x], x] + Dist[a*p, Int[PolyLog[n - 1, c*(a + b*x)^p]/(a + b*x), x], x]) /; FreeQ[{a, b, c, p}, x] && GtQ[n, 0]
  6596. Int[Times[Power[Plus[Optional[Pattern[d, Blank[]]], Times[Optional[Pattern[e, Blank[]]], Pattern[x, Blank[]]]], -1], PolyLog[2, Times[Optional[Pattern[c, Blank[]]], Plus[Optional[Pattern[a, Blank[]]], Times[Optional[Pattern[b, Blank[]]], Pattern[x, Blank[]]]]]]], Pattern[x, Blank[Symbol]]] := Simp[(Log[1 - a*c - b*c*x]*PolyLog[2, c*(a + b*x)])/e, x] + Dist[b/e, Int[Log[1 - a*c - b*c*x]^2/(a + b*x), x], x] /; FreeQ[{a, b, c, d, e}, x] && EqQ[c*(b*d - a*e) + e, 0]
  6597. Int[Times[Power[Plus[Optional[Pattern[d, Blank[]]], Times[Optional[Pattern[e, Blank[]]], Pattern[x, Blank[]]]], -1], PolyLog[2, Times[Optional[Pattern[c, Blank[]]], Plus[Optional[Pattern[a, Blank[]]], Times[Optional[Pattern[b, Blank[]]], Pattern[x, Blank[]]]]]]], Pattern[x, Blank[Symbol]]] := Simp[(Log[d + e*x]*PolyLog[2, c*(a + b*x)])/e, x] + Dist[b/e, Int[(Log[d + e*x]*Log[1 - a*c - b*c*x])/(a + b*x), x], x] /; FreeQ[{a, b, c, d, e}, x] && NeQ[c*(b*d - a*e) + e, 0]
  6598. Int[Times[Power[Plus[Optional[Pattern[d, Blank[]]], Times[Optional[Pattern[e, Blank[]]], Pattern[x, Blank[]]]], Optional[Pattern[m, Blank[]]]], PolyLog[2, Times[Optional[Pattern[c, Blank[]]], Plus[Optional[Pattern[a, Blank[]]], Times[Optional[Pattern[b, Blank[]]], Pattern[x, Blank[]]]]]]], Pattern[x, Blank[Symbol]]] := Simp[((d + e*x)^(m + 1)*PolyLog[2, c*(a + b*x)])/(e*(m + 1)), x] + Dist[b/(e*(m + 1)), Int[((d + e*x)^(m + 1)*Log[1 - a*c - b*c*x])/(a + b*x), x], x] /; FreeQ[{a, b, c, d, e, m}, x] && NeQ[m, -1]
  6599. Int[Times[Power[Pattern[x, Blank[]], Optional[Pattern[m, Blank[]]]], PolyLog[Pattern[n, Blank[]], Times[Optional[Pattern[c, Blank[]]], Power[Plus[Optional[Pattern[a, Blank[]]], Times[Optional[Pattern[b, Blank[]]], Pattern[x, Blank[]]]], Optional[Pattern[p, Blank[]]]]]]], Pattern[x, Blank[Symbol]]] := -Simp[((a^(m + 1) - b^(m + 1)*x^(m + 1))*PolyLog[n, c*(a + b*x)^p])/((m + 1)*b^(m + 1)), x] + Dist[p/((m + 1)*b^m), Int[ExpandIntegrand[PolyLog[n - 1, c*(a + b*x)^p], (a^(m + 1) - b^(m + 1)*x^(m + 1))/(a + b*x), x], x], x] /; FreeQ[{a, b, c, p}, x] && GtQ[n, 0] && IntegerQ[m] && NeQ[m, -1]
  6600. Int[Times[Plus[Optional[Pattern[g, Blank[]]], Times[Log[Times[Optional[Pattern[f, Blank[]]], Power[Plus[Optional[Pattern[d, Blank[]]], Times[Optional[Pattern[e, Blank[]]], Pattern[x, Blank[]]]], Optional[Pattern[n, Blank[]]]]]], Optional[Pattern[h, Blank[]]]]], PolyLog[2, Times[Optional[Pattern[c, Blank[]]], Plus[Optional[Pattern[a, Blank[]]], Times[Optional[Pattern[b, Blank[]]], Pattern[x, Blank[]]]]]]], Pattern[x, Blank[Symbol]]] := Simp[x*(g + h*Log[f*(d + e*x)^n])*PolyLog[2, c*(a + b*x)], x] + (Dist[b, Int[(g + h*Log[f*(d + e*x)^n])*Log[1 - a*c - b*c*x]*ExpandIntegrand[x/(a + b*x), x], x], x] - Dist[e*h*n, Int[PolyLog[2, c*(a + b*x)]*ExpandIntegrand[x/(d + e*x), x], x], x]) /; FreeQ[{a, b, c, d, e, f, g, h, n}, x]
  6601. Int[Times[Log[Plus[1, Times[Optional[Pattern[e, Blank[]]], Pattern[x, Blank[]]]]], Power[Pattern[x, Blank[]], -1], PolyLog[2, Times[Optional[Pattern[c, Blank[]]], Pattern[x, Blank[]]]]], Pattern[x, Blank[Symbol]]] := -Simp[PolyLog[2, c*x]^2/2, x] /; FreeQ[{c, e}, x] && EqQ[c + e, 0]
  6602. Int[Times[Plus[Times[Log[Plus[1, Times[Optional[Pattern[e, Blank[]]], Pattern[x, Blank[]]]]], Optional[Pattern[h, Blank[]]]], Pattern[g, Blank[]]], Power[Pattern[x, Blank[]], -1], PolyLog[2, Times[Optional[Pattern[c, Blank[]]], Pattern[x, Blank[]]]]], Pattern[x, Blank[Symbol]]] := Dist[g, Int[PolyLog[2, c*x]/x, x], x] + Dist[h, Int[(Log[1 + e*x]*PolyLog[2, c*x])/x, x], x] /; FreeQ[{c, e, g, h}, x] && EqQ[c + e, 0]
  6603. Int[Times[Plus[Optional[Pattern[g, Blank[]]], Times[Log[Times[Optional[Pattern[f, Blank[]]], Power[Plus[Optional[Pattern[d, Blank[]]], Times[Optional[Pattern[e, Blank[]]], Pattern[x, Blank[]]]], Optional[Pattern[n, Blank[]]]]]], Optional[Pattern[h, Blank[]]]]], Power[Pattern[x, Blank[]], Optional[Pattern[m, Blank[]]]], PolyLog[2, Times[Optional[Pattern[c, Blank[]]], Plus[Optional[Pattern[a, Blank[]]], Times[Optional[Pattern[b, Blank[]]], Pattern[x, Blank[]]]]]]], Pattern[x, Blank[Symbol]]] := Simp[(x^(m + 1)*(g + h*Log[f*(d + e*x)^n])*PolyLog[2, c*(a + b*x)])/(m + 1), x] + (Dist[b/(m + 1), Int[ExpandIntegrand[(g + h*Log[f*(d + e*x)^n])*Log[1 - a*c - b*c*x], x^(m + 1)/(a + b*x), x], x], x] - Dist[(e*h*n)/(m + 1), Int[ExpandIntegrand[PolyLog[2, c*(a + b*x)], x^(m + 1)/(d + e*x), x], x], x]) /; FreeQ[{a, b, c, d, e, f, g, h, n}, x] && IntegerQ[m] && NeQ[m, -1]
  6604. Int[Times[Plus[Optional[Pattern[g, Blank[]]], Times[Log[Times[Optional[Pattern[f, Blank[]]], Power[Plus[Optional[Pattern[d, Blank[]]], Times[Optional[Pattern[e, Blank[]]], Pattern[x, Blank[]]]], Optional[Pattern[n, Blank[]]]]]], Optional[Pattern[h, Blank[]]]]], Pattern[Px, Blank[]], PolyLog[2, Times[Optional[Pattern[c, Blank[]]], Plus[Optional[Pattern[a, Blank[]]], Times[Optional[Pattern[b, Blank[]]], Pattern[x, Blank[]]]]]]], Pattern[x, Blank[Symbol]]] := With[{u = IntHide[Px, x]}, Simp[u*(g + h*Log[f*(d + e*x)^n])*PolyLog[2, c*(a + b*x)], x] + (Dist[b, Int[ExpandIntegrand[(g + h*Log[f*(d + e*x)^n])*Log[1 - a*c - b*c*x], u/(a + b*x), x], x], x] - Dist[e*h*n, Int[ExpandIntegrand[PolyLog[2, c*(a + b*x)], u/(d + e*x), x], x], x])] /; FreeQ[{a, b, c, d, e, f, g, h, n}, x] && PolyQ[Px, x]
  6605. Int[Times[Plus[Optional[Pattern[g, Blank[]]], Times[Log[Plus[1, Times[Optional[Pattern[e, Blank[]]], Pattern[x, Blank[]]]]], Optional[Pattern[h, Blank[]]]]], Pattern[Px, Blank[]], Power[Pattern[x, Blank[]], Pattern[m, Blank[]]], PolyLog[2, Times[Optional[Pattern[c, Blank[]]], Pattern[x, Blank[]]]]], Pattern[x, Blank[Symbol]]] := Dist[Coeff[Px, x, -m - 1], Int[((g + h*Log[1 + e*x])*PolyLog[2, c*x])/x, x], x] + Int[x^m*(Px - Coeff[Px, x, -m - 1]*x^(-m - 1))*(g + h*Log[1 + e*x])*PolyLog[2, c*x], x] /; FreeQ[{c, e, g, h}, x] && PolyQ[Px, x] && ILtQ[m, 0] && EqQ[c + e, 0] && NeQ[Coeff[Px, x, -m - 1], 0]
  6606. Int[Times[Plus[Optional[Pattern[g, Blank[]]], Times[Log[Times[Optional[Pattern[f, Blank[]]], Power[Plus[Optional[Pattern[d, Blank[]]], Times[Optional[Pattern[e, Blank[]]], Pattern[x, Blank[]]]], Optional[Pattern[n, Blank[]]]]]], Optional[Pattern[h, Blank[]]]]], Pattern[Px, Blank[]], Power[Pattern[x, Blank[]], Optional[Pattern[m, Blank[]]]], PolyLog[2, Times[Optional[Pattern[c, Blank[]]], Plus[Optional[Pattern[a, Blank[]]], Times[Optional[Pattern[b, Blank[]]], Pattern[x, Blank[]]]]]]], Pattern[x, Blank[Symbol]]] := With[{u = IntHide[x^m*Px, x]}, Simp[u*(g + h*Log[f*(d + e*x)^n])*PolyLog[2, c*(a + b*x)], x] + (Dist[b, Int[ExpandIntegrand[(g + h*Log[f*(d + e*x)^n])*Log[1 - a*c - b*c*x], u/(a + b*x), x], x], x] - Dist[e*h*n, Int[ExpandIntegrand[PolyLog[2, c*(a + b*x)], u/(d + e*x), x], x], x])] /; FreeQ[{a, b, c, d, e, f, g, h, n}, x] && PolyQ[Px, x] && IntegerQ[m]
  6607. Int[Times[Plus[Optional[Pattern[g, Blank[]]], Times[Log[Times[Optional[Pattern[f, Blank[]]], Power[Plus[Optional[Pattern[d, Blank[]]], Times[Optional[Pattern[e, Blank[]]], Pattern[x, Blank[]]]], Optional[Pattern[n, Blank[]]]]]], Optional[Pattern[h, Blank[]]]]], Optional[Pattern[Px, Blank[]]], Power[Pattern[x, Blank[]], Pattern[m, Blank[]]], PolyLog[2, Times[Optional[Pattern[c, Blank[]]], Plus[Optional[Pattern[a, Blank[]]], Times[Optional[Pattern[b, Blank[]]], Pattern[x, Blank[]]]]]]], Pattern[x, Blank[Symbol]]] := Unintegrable[Px*x^m*(g + h*Log[f*(d + e*x)^n])*PolyLog[2, c*(a + b*x)], x] /; FreeQ[{a, b, c, d, e, f, g, h, m, n}, x] && PolyQ[Px, x]
  6608. Int[PolyLog[Pattern[n, Blank[]], Times[Optional[Pattern[d, Blank[]]], Power[Power[Pattern[F, Blank[]], Times[Optional[Pattern[c, Blank[]]], Plus[Optional[Pattern[a, Blank[]]], Times[Optional[Pattern[b, Blank[]]], Pattern[x, Blank[]]]]]], Optional[Pattern[p, Blank[]]]]]], Pattern[x, Blank[Symbol]]] := Simp[PolyLog[n + 1, d*(F^(c*(a + b*x)))^p]/(b*c*p*Log[F]), x] /; FreeQ[{F, a, b, c, d, n, p}, x]
  6609. Int[Times[Power[Plus[Optional[Pattern[e, Blank[]]], Times[Optional[Pattern[f, Blank[]]], Pattern[x, Blank[]]]], Optional[Pattern[m, Blank[]]]], PolyLog[Pattern[n, Blank[]], Times[Optional[Pattern[d, Blank[]]], Power[Power[Pattern[F, Blank[]], Times[Optional[Pattern[c, Blank[]]], Plus[Optional[Pattern[a, Blank[]]], Times[Optional[Pattern[b, Blank[]]], Pattern[x, Blank[]]]]]], Optional[Pattern[p, Blank[]]]]]]], Pattern[x, Blank[Symbol]]] := Simp[((e + f*x)^m*PolyLog[n + 1, d*(F^(c*(a + b*x)))^p])/(b*c*p*Log[F]), x] - Dist[(f*m)/(b*c*p*Log[F]), Int[(e + f*x)^(m - 1)*PolyLog[n + 1, d*(F^(c*(a + b*x)))^p], x], x] /; FreeQ[{F, a, b, c, d, e, f, n, p}, x] && GtQ[m, 0]
  6610. Int[Times[Pattern[u, Blank[]], PolyLog[Pattern[n, Blank[]], Pattern[v, Blank[]]]], Pattern[x, Blank[Symbol]]] := With[{w = DerivativeDivides[v, u*v, x]}, Simp[w*PolyLog[n + 1, v], x] /; !FalseQ[w]] /; FreeQ[n, x]
  6611. Int[Times[Log[Pattern[w, Blank[]]], Pattern[u, Blank[]], PolyLog[Pattern[n, Blank[]], Pattern[v, Blank[]]]], Pattern[x, Blank[Symbol]]] := With[{z = DerivativeDivides[v, u*v, x]}, Simp[z*Log[w]*PolyLog[n + 1, v], x] - Int[SimplifyIntegrand[(z*D[w, x]*PolyLog[n + 1, v])/w, x], x] /; !FalseQ[z]] /; FreeQ[n, x] && InverseFunctionFreeQ[w, x]
  6612. Int[Power[Times[Optional[Pattern[c, Blank[]]], ProductLog[Plus[Optional[Pattern[a, Blank[]]], Times[Optional[Pattern[b, Blank[]]], Pattern[x, Blank[]]]]]], Pattern[p, Blank[]]], Pattern[x, Blank[Symbol]]] := Simp[((a + b*x)*(c*ProductLog[a + b*x])^p)/(b*(p + 1)), x] + Dist[p/(c*(p + 1)), Int[(c*ProductLog[a + b*x])^(p + 1)/(1 + ProductLog[a + b*x]), x], x] /; FreeQ[{a, b, c}, x] && LtQ[p, -1]
  6613. Int[Power[Times[Optional[Pattern[c, Blank[]]], ProductLog[Plus[Optional[Pattern[a, Blank[]]], Times[Optional[Pattern[b, Blank[]]], Pattern[x, Blank[]]]]]], Optional[Pattern[p, Blank[]]]], Pattern[x, Blank[Symbol]]] := Simp[((a + b*x)*(c*ProductLog[a + b*x])^p)/b, x] - Dist[p, Int[(c*ProductLog[a + b*x])^p/(1 + ProductLog[a + b*x]), x], x] /; FreeQ[{a, b, c}, x] && !LtQ[p, -1]
  6614. Int[Times[Power[Plus[Optional[Pattern[e, Blank[]]], Times[Optional[Pattern[f, Blank[]]], Pattern[x, Blank[]]]], Optional[Pattern[m, Blank[]]]], Power[Times[Optional[Pattern[c, Blank[]]], ProductLog[Plus[Pattern[a, Blank[]], Times[Optional[Pattern[b, Blank[]]], Pattern[x, Blank[]]]]]], Optional[Pattern[p, Blank[]]]]], Pattern[x, Blank[Symbol]]] := Dist[1/b^(m + 1), Subst[Int[ExpandIntegrand[(c*ProductLog[x])^p, (b*e - a*f + f*x)^m, x], x], x, a + b*x], x] /; FreeQ[{a, b, c, e, f, p}, x] && IGtQ[m, 0]
  6615. Int[Power[Times[Optional[Pattern[c, Blank[]]], ProductLog[Times[Optional[Pattern[a, Blank[]]], Power[Pattern[x, Blank[]], Pattern[n, Blank[]]]]]], Optional[Pattern[p, Blank[]]]], Pattern[x, Blank[Symbol]]] := Simp[x*(c*ProductLog[a*x^n])^p, x] - Dist[n*p, Int[(c*ProductLog[a*x^n])^p/(1 + ProductLog[a*x^n]), x], x] /; FreeQ[{a, c, n, p}, x] && (EqQ[n*(p - 1), -1] || (IntegerQ[p - 1/2] && EqQ[n*(p - 1/2), -1]))
  6616. Int[Power[Times[Optional[Pattern[c, Blank[]]], ProductLog[Times[Optional[Pattern[a, Blank[]]], Power[Pattern[x, Blank[]], Pattern[n, Blank[]]]]]], Optional[Pattern[p, Blank[]]]], Pattern[x, Blank[Symbol]]] := Simp[(x*(c*ProductLog[a*x^n])^p)/(n*p + 1), x] + Dist[(n*p)/(c*(n*p + 1)), Int[(c*ProductLog[a*x^n])^(p + 1)/(1 + ProductLog[a*x^n]), x], x] /; FreeQ[{a, c, n}, x] && ((IntegerQ[p] && EqQ[n*(p + 1), -1]) || (IntegerQ[p - 1/2] && EqQ[n*(p + 1/2), -1]))
  6617. Int[Power[Times[Optional[Pattern[c, Blank[]]], ProductLog[Times[Optional[Pattern[a, Blank[]]], Power[Pattern[x, Blank[]], Pattern[n, Blank[]]]]]], Optional[Pattern[p, Blank[]]]], Pattern[x, Blank[Symbol]]] := -Subst[Int[(c*ProductLog[a/x^n])^p/x^2, x], x, 1/x] /; FreeQ[{a, c, p}, x] && ILtQ[n, 0]
  6618. Int[Times[Power[Pattern[x, Blank[]], Optional[Pattern[m, Blank[]]]], Power[Times[Optional[Pattern[c, Blank[]]], ProductLog[Times[Optional[Pattern[a, Blank[]]], Power[Pattern[x, Blank[]], Optional[Pattern[n, Blank[]]]]]]], Optional[Pattern[p, Blank[]]]]], Pattern[x, Blank[Symbol]]] := Simp[(x^(m + 1)*(c*ProductLog[a*x^n])^p)/(m + 1), x] - Dist[(n*p)/(m + 1), Int[(x^m*(c*ProductLog[a*x^n])^p)/(1 + ProductLog[a*x^n]), x], x] /; FreeQ[{a, c, m, n, p}, x] && NeQ[m, -1] && ((IntegerQ[p - 1/2] && IGtQ[2*Simplify[p + (m + 1)/n], 0]) || ( !IntegerQ[p - 1/2] && IGtQ[Simplify[p + (m + 1)/n] + 1, 0]))
  6619. Int[Times[Power[Pattern[x, Blank[]], Optional[Pattern[m, Blank[]]]], Power[Times[Optional[Pattern[c, Blank[]]], ProductLog[Times[Optional[Pattern[a, Blank[]]], Power[Pattern[x, Blank[]], Optional[Pattern[n, Blank[]]]]]]], Optional[Pattern[p, Blank[]]]]], Pattern[x, Blank[Symbol]]] := Simp[(x^(m + 1)*(c*ProductLog[a*x^n])^p)/(m + n*p + 1), x] + Dist[(n*p)/(c*(m + n*p + 1)), Int[(x^m*(c*ProductLog[a*x^n])^(p + 1))/(1 + ProductLog[a*x^n]), x], x] /; FreeQ[{a, c, m, n, p}, x] && (EqQ[m, -1] || (IntegerQ[p - 1/2] && ILtQ[Simplify[p + (m + 1)/n] - 1/2, 0]) || ( !IntegerQ[p - 1/2] && ILtQ[Simplify[p + (m + 1)/n], 0]))
  6620. Int[Times[Power[Pattern[x, Blank[]], Optional[Pattern[m, Blank[]]]], Power[Times[Optional[Pattern[c, Blank[]]], ProductLog[Times[Optional[Pattern[a, Blank[]]], Pattern[x, Blank[]]]]], Optional[Pattern[p, Blank[]]]]], Pattern[x, Blank[Symbol]]] := Int[(x^m*(c*ProductLog[a*x])^p)/(1 + ProductLog[a*x]), x] + Dist[1/c, Int[(x^m*(c*ProductLog[a*x])^(p + 1))/(1 + ProductLog[a*x]), x], x] /; FreeQ[{a, c, m}, x]
  6621. Int[Times[Power[Pattern[x, Blank[]], Optional[Pattern[m, Blank[]]]], Power[Times[Optional[Pattern[c, Blank[]]], ProductLog[Times[Optional[Pattern[a, Blank[]]], Power[Pattern[x, Blank[]], Pattern[n, Blank[]]]]]], Optional[Pattern[p, Blank[]]]]], Pattern[x, Blank[Symbol]]] := -Subst[Int[(c*ProductLog[a/x^n])^p/x^(m + 2), x], x, 1/x] /; FreeQ[{a, c, p}, x] && ILtQ[n, 0] && IntegerQ[m] && NeQ[m, -1]
  6622. Int[Power[Plus[Pattern[d, Blank[]], Times[Optional[Pattern[d, Blank[]]], ProductLog[Plus[Optional[Pattern[a, Blank[]]], Times[Optional[Pattern[b, Blank[]]], Pattern[x, Blank[]]]]]]], -1], Pattern[x, Blank[Symbol]]] := Simp[(a + b*x)/(b*d*ProductLog[a + b*x]), x] /; FreeQ[{a, b, d}, x]
  6623. Int[Times[ProductLog[Plus[Optional[Pattern[a, Blank[]]], Times[Optional[Pattern[b, Blank[]]], Pattern[x, Blank[]]]]], Power[Plus[Pattern[d, Blank[]], Times[Optional[Pattern[d, Blank[]]], ProductLog[Plus[Optional[Pattern[a, Blank[]]], Times[Optional[Pattern[b, Blank[]]], Pattern[x, Blank[]]]]]]], -1]], Pattern[x, Blank[Symbol]]] := Simp[d*x, x] - Int[1/(d + d*ProductLog[a + b*x]), x] /; FreeQ[{a, b, d}, x]
  6624. Int[Times[Power[Times[Optional[Pattern[c, Blank[]]], ProductLog[Plus[Optional[Pattern[a, Blank[]]], Times[Optional[Pattern[b, Blank[]]], Pattern[x, Blank[]]]]]], Pattern[p, Blank[]]], Power[Plus[Pattern[d, Blank[]], Times[Optional[Pattern[d, Blank[]]], ProductLog[Plus[Optional[Pattern[a, Blank[]]], Times[Optional[Pattern[b, Blank[]]], Pattern[x, Blank[]]]]]]], -1]], Pattern[x, Blank[Symbol]]] := Simp[(c*(a + b*x)*(c*ProductLog[a + b*x])^(p - 1))/(b*d), x] - Dist[c*p, Int[(c*ProductLog[a + b*x])^(p - 1)/(d + d*ProductLog[a + b*x]), x], x] /; FreeQ[{a, b, c, d}, x] && GtQ[p, 0]
  6625. Int[Times[Power[ProductLog[Plus[Optional[Pattern[a, Blank[]]], Times[Optional[Pattern[b, Blank[]]], Pattern[x, Blank[]]]]], -1], Power[Plus[Pattern[d, Blank[]], Times[Optional[Pattern[d, Blank[]]], ProductLog[Plus[Optional[Pattern[a, Blank[]]], Times[Optional[Pattern[b, Blank[]]], Pattern[x, Blank[]]]]]]], -1]], Pattern[x, Blank[Symbol]]] := Simp[ExpIntegralEi[ProductLog[a + b*x]]/(b*d), x] /; FreeQ[{a, b, d}, x]
  6626. Int[Times[Power[Times[Optional[Pattern[c, Blank[]]], ProductLog[Plus[Optional[Pattern[a, Blank[]]], Times[Optional[Pattern[b, Blank[]]], Pattern[x, Blank[]]]]]], Rational[-1, 2]], Power[Plus[Pattern[d, Blank[]], Times[Optional[Pattern[d, Blank[]]], ProductLog[Plus[Optional[Pattern[a, Blank[]]], Times[Optional[Pattern[b, Blank[]]], Pattern[x, Blank[]]]]]]], -1]], Pattern[x, Blank[Symbol]]] := Simp[(Rt[Pi*c, 2]*Erfi[Sqrt[c*ProductLog[a + b*x]]/Rt[c, 2]])/(b*c*d), x] /; FreeQ[{a, b, c, d}, x] && PosQ[c]
  6627. Int[Times[Power[Times[Optional[Pattern[c, Blank[]]], ProductLog[Plus[Optional[Pattern[a, Blank[]]], Times[Optional[Pattern[b, Blank[]]], Pattern[x, Blank[]]]]]], Rational[-1, 2]], Power[Plus[Pattern[d, Blank[]], Times[Optional[Pattern[d, Blank[]]], ProductLog[Plus[Optional[Pattern[a, Blank[]]], Times[Optional[Pattern[b, Blank[]]], Pattern[x, Blank[]]]]]]], -1]], Pattern[x, Blank[Symbol]]] := Simp[(Rt[-(Pi*c), 2]*Erf[Sqrt[c*ProductLog[a + b*x]]/Rt[-c, 2]])/(b*c*d), x] /; FreeQ[{a, b, c, d}, x] && NegQ[c]
  6628. Int[Times[Power[Times[Optional[Pattern[c, Blank[]]], ProductLog[Plus[Optional[Pattern[a, Blank[]]], Times[Optional[Pattern[b, Blank[]]], Pattern[x, Blank[]]]]]], Pattern[p, Blank[]]], Power[Plus[Pattern[d, Blank[]], Times[Optional[Pattern[d, Blank[]]], ProductLog[Plus[Optional[Pattern[a, Blank[]]], Times[Optional[Pattern[b, Blank[]]], Pattern[x, Blank[]]]]]]], -1]], Pattern[x, Blank[Symbol]]] := Simp[((a + b*x)*(c*ProductLog[a + b*x])^p)/(b*d*(p + 1)), x] - Dist[1/(c*(p + 1)), Int[(c*ProductLog[a + b*x])^(p + 1)/(d + d*ProductLog[a + b*x]), x], x] /; FreeQ[{a, b, c, d}, x] && LtQ[p, -1]
  6629. Int[Times[Power[Times[Optional[Pattern[c, Blank[]]], ProductLog[Plus[Optional[Pattern[a, Blank[]]], Times[Optional[Pattern[b, Blank[]]], Pattern[x, Blank[]]]]]], Optional[Pattern[p, Blank[]]]], Power[Plus[Pattern[d, Blank[]], Times[Optional[Pattern[d, Blank[]]], ProductLog[Plus[Optional[Pattern[a, Blank[]]], Times[Optional[Pattern[b, Blank[]]], Pattern[x, Blank[]]]]]]], -1]], Pattern[x, Blank[Symbol]]] := Simp[(Gamma[p + 1, -ProductLog[a + b*x]]*(c*ProductLog[a + b*x])^p)/(b*d*(-ProductLog[a + b*x])^p), x] /; FreeQ[{a, b, c, d, p}, x]
  6630. Int[Times[Power[Plus[Optional[Pattern[e, Blank[]]], Times[Optional[Pattern[f, Blank[]]], Pattern[x, Blank[]]]], Optional[Pattern[m, Blank[]]]], Power[Plus[Pattern[d, Blank[]], Times[Optional[Pattern[d, Blank[]]], ProductLog[Plus[Pattern[a, Blank[]], Times[Optional[Pattern[b, Blank[]]], Pattern[x, Blank[]]]]]]], -1]], Pattern[x, Blank[Symbol]]] := Dist[1/b^(m + 1), Subst[Int[ExpandIntegrand[1/(d + d*ProductLog[x]), (b*e - a*f + f*x)^m, x], x], x, a + b*x], x] /; FreeQ[{a, b, d, e, f}, x] && IGtQ[m, 0]
  6631. Int[Times[Power[Plus[Optional[Pattern[e, Blank[]]], Times[Optional[Pattern[f, Blank[]]], Pattern[x, Blank[]]]], Optional[Pattern[m, Blank[]]]], Power[Times[Optional[Pattern[c, Blank[]]], ProductLog[Plus[Pattern[a, Blank[]], Times[Optional[Pattern[b, Blank[]]], Pattern[x, Blank[]]]]]], Optional[Pattern[p, Blank[]]]], Power[Plus[Pattern[d, Blank[]], Times[Optional[Pattern[d, Blank[]]], ProductLog[Plus[Pattern[a, Blank[]], Times[Optional[Pattern[b, Blank[]]], Pattern[x, Blank[]]]]]]], -1]], Pattern[x, Blank[Symbol]]] := Dist[1/b^(m + 1), Subst[Int[ExpandIntegrand[(c*ProductLog[x])^p/(d + d*ProductLog[x]), (b*e - a*f + f*x)^m, x], x], x, a + b*x], x] /; FreeQ[{a, b, c, d, e, f, p}, x] && IGtQ[m, 0]
  6632. Int[Power[Plus[Pattern[d, Blank[]], Times[Optional[Pattern[d, Blank[]]], ProductLog[Times[Optional[Pattern[a, Blank[]]], Power[Pattern[x, Blank[]], Pattern[n, Blank[]]]]]]], -1], Pattern[x, Blank[Symbol]]] := -Subst[Int[1/(x^2*(d + d*ProductLog[a/x^n])), x], x, 1/x] /; FreeQ[{a, d}, x] && ILtQ[n, 0]
  6633. Int[Times[Power[Times[Optional[Pattern[c, Blank[]]], ProductLog[Times[Optional[Pattern[a, Blank[]]], Power[Pattern[x, Blank[]], Optional[Pattern[n, Blank[]]]]]]], Optional[Pattern[p, Blank[]]]], Power[Plus[Pattern[d, Blank[]], Times[Optional[Pattern[d, Blank[]]], ProductLog[Times[Optional[Pattern[a, Blank[]]], Power[Pattern[x, Blank[]], Optional[Pattern[n, Blank[]]]]]]]], -1]], Pattern[x, Blank[Symbol]]] := Simp[(c*x*(c*ProductLog[a*x^n])^(p - 1))/d, x] /; FreeQ[{a, c, d, n, p}, x] && EqQ[n*(p - 1), -1]
  6634. Int[Times[Power[ProductLog[Times[Optional[Pattern[a, Blank[]]], Power[Pattern[x, Blank[]], Optional[Pattern[n, Blank[]]]]]], Optional[Pattern[p, Blank[]]]], Power[Plus[Pattern[d, Blank[]], Times[Optional[Pattern[d, Blank[]]], ProductLog[Times[Optional[Pattern[a, Blank[]]], Power[Pattern[x, Blank[]], Optional[Pattern[n, Blank[]]]]]]]], -1]], Pattern[x, Blank[Symbol]]] := Simp[(a^p*ExpIntegralEi[-(p*ProductLog[a*x^n])])/(d*n), x] /; FreeQ[{a, d}, x] && IntegerQ[p] && EqQ[n*p, -1]
  6635. Int[Times[Power[Times[Optional[Pattern[c, Blank[]]], ProductLog[Times[Optional[Pattern[a, Blank[]]], Power[Pattern[x, Blank[]], Optional[Pattern[n, Blank[]]]]]]], Pattern[p, Blank[]]], Power[Plus[Pattern[d, Blank[]], Times[Optional[Pattern[d, Blank[]]], ProductLog[Times[Optional[Pattern[a, Blank[]]], Power[Pattern[x, Blank[]], Optional[Pattern[n, Blank[]]]]]]]], -1]], Pattern[x, Blank[Symbol]]] := Simp[(Rt[Pi*c*n, 2]*Erfi[Sqrt[c*ProductLog[a*x^n]]/Rt[c*n, 2]])/(d*n*a^(1/n)*c^(1/n)), x] /; FreeQ[{a, c, d}, x] && IntegerQ[1/n] && EqQ[p, 1/2 - 1/n] && PosQ[c*n]
  6636. Int[Times[Power[Times[Optional[Pattern[c, Blank[]]], ProductLog[Times[Optional[Pattern[a, Blank[]]], Power[Pattern[x, Blank[]], Optional[Pattern[n, Blank[]]]]]]], Pattern[p, Blank[]]], Power[Plus[Pattern[d, Blank[]], Times[Optional[Pattern[d, Blank[]]], ProductLog[Times[Optional[Pattern[a, Blank[]]], Power[Pattern[x, Blank[]], Optional[Pattern[n, Blank[]]]]]]]], -1]], Pattern[x, Blank[Symbol]]] := Simp[(Rt[-(Pi*c*n), 2]*Erf[Sqrt[c*ProductLog[a*x^n]]/Rt[-(c*n), 2]])/(d*n*a^(1/n)*c^(1/n)), x] /; FreeQ[{a, c, d}, x] && IntegerQ[1/n] && EqQ[p, 1/2 - 1/n] && NegQ[c*n]
  6637. Int[Times[Power[Times[Optional[Pattern[c, Blank[]]], ProductLog[Times[Optional[Pattern[a, Blank[]]], Power[Pattern[x, Blank[]], Optional[Pattern[n, Blank[]]]]]]], Optional[Pattern[p, Blank[]]]], Power[Plus[Pattern[d, Blank[]], Times[Optional[Pattern[d, Blank[]]], ProductLog[Times[Optional[Pattern[a, Blank[]]], Power[Pattern[x, Blank[]], Optional[Pattern[n, Blank[]]]]]]]], -1]], Pattern[x, Blank[Symbol]]] := Simp[(c*x*(c*ProductLog[a*x^n])^(p - 1))/d, x] - Dist[c*(n*(p - 1) + 1), Int[(c*ProductLog[a*x^n])^(p - 1)/(d + d*ProductLog[a*x^n]), x], x] /; FreeQ[{a, c, d}, x] && GtQ[n, 0] && GtQ[n*(p - 1) + 1, 0]
  6638. Int[Times[Power[Times[Optional[Pattern[c, Blank[]]], ProductLog[Times[Optional[Pattern[a, Blank[]]], Power[Pattern[x, Blank[]], Optional[Pattern[n, Blank[]]]]]]], Optional[Pattern[p, Blank[]]]], Power[Plus[Pattern[d, Blank[]], Times[Optional[Pattern[d, Blank[]]], ProductLog[Times[Optional[Pattern[a, Blank[]]], Power[Pattern[x, Blank[]], Optional[Pattern[n, Blank[]]]]]]]], -1]], Pattern[x, Blank[Symbol]]] := Simp[(x*(c*ProductLog[a*x^n])^p)/(d*(n*p + 1)), x] - Dist[1/(c*(n*p + 1)), Int[(c*ProductLog[a*x^n])^(p + 1)/(d + d*ProductLog[a*x^n]), x], x] /; FreeQ[{a, c, d}, x] && GtQ[n, 0] && LtQ[n*p + 1, 0]
  6639. Int[Times[Power[Times[Optional[Pattern[c, Blank[]]], ProductLog[Times[Optional[Pattern[a, Blank[]]], Power[Pattern[x, Blank[]], Pattern[n, Blank[]]]]]], Optional[Pattern[p, Blank[]]]], Power[Plus[Pattern[d, Blank[]], Times[Optional[Pattern[d, Blank[]]], ProductLog[Times[Optional[Pattern[a, Blank[]]], Power[Pattern[x, Blank[]], Pattern[n, Blank[]]]]]]], -1]], Pattern[x, Blank[Symbol]]] := -Subst[Int[(c*ProductLog[a/x^n])^p/(x^2*(d + d*ProductLog[a/x^n])), x], x, 1/x] /; FreeQ[{a, c, d, p}, x] && ILtQ[n, 0]
  6640. Int[Times[Power[Pattern[x, Blank[]], Optional[Pattern[m, Blank[]]]], Power[Plus[Pattern[d, Blank[]], Times[Optional[Pattern[d, Blank[]]], ProductLog[Times[Optional[Pattern[a, Blank[]]], Pattern[x, Blank[]]]]]], -1]], Pattern[x, Blank[Symbol]]] := Simp[x^(m + 1)/(d*(m + 1)*ProductLog[a*x]), x] - Dist[m/(m + 1), Int[x^m/(ProductLog[a*x]*(d + d*ProductLog[a*x])), x], x] /; FreeQ[{a, d}, x] && GtQ[m, 0]
  6641. Int[Times[Power[Pattern[x, Blank[]], -1], Power[Plus[Pattern[d, Blank[]], Times[Optional[Pattern[d, Blank[]]], ProductLog[Times[Optional[Pattern[a, Blank[]]], Pattern[x, Blank[]]]]]], -1]], Pattern[x, Blank[Symbol]]] := Simp[Log[ProductLog[a*x]]/d, x] /; FreeQ[{a, d}, x]
  6642. Int[Times[Power[Pattern[x, Blank[]], Optional[Pattern[m, Blank[]]]], Power[Plus[Pattern[d, Blank[]], Times[Optional[Pattern[d, Blank[]]], ProductLog[Times[Optional[Pattern[a, Blank[]]], Pattern[x, Blank[]]]]]], -1]], Pattern[x, Blank[Symbol]]] := Simp[x^(m + 1)/(d*(m + 1)), x] - Int[(x^m*ProductLog[a*x])/(d + d*ProductLog[a*x]), x] /; FreeQ[{a, d}, x] && LtQ[m, -1]
  6643. Int[Times[Power[Pattern[x, Blank[]], Optional[Pattern[m, Blank[]]]], Power[Plus[Pattern[d, Blank[]], Times[Optional[Pattern[d, Blank[]]], ProductLog[Times[Optional[Pattern[a, Blank[]]], Pattern[x, Blank[]]]]]], -1]], Pattern[x, Blank[Symbol]]] := Simp[(x^m*Gamma[m + 1, -((m + 1)*ProductLog[a*x])])/(a*d*(m + 1)*E^(m*ProductLog[a*x])*(-((m + 1)*ProductLog[a*x]))^m), x] /; FreeQ[{a, d, m}, x] && !IntegerQ[m]
  6644. Int[Times[Power[Pattern[x, Blank[]], -1], Power[Plus[Pattern[d, Blank[]], Times[Optional[Pattern[d, Blank[]]], ProductLog[Times[Optional[Pattern[a, Blank[]]], Power[Pattern[x, Blank[]], Optional[Pattern[n, Blank[]]]]]]]], -1]], Pattern[x, Blank[Symbol]]] := Simp[Log[ProductLog[a*x^n]]/(d*n), x] /; FreeQ[{a, d, n}, x]
  6645. Int[Times[Power[Pattern[x, Blank[]], Optional[Pattern[m, Blank[]]]], Power[Plus[Pattern[d, Blank[]], Times[Optional[Pattern[d, Blank[]]], ProductLog[Times[Optional[Pattern[a, Blank[]]], Power[Pattern[x, Blank[]], Pattern[n, Blank[]]]]]]], -1]], Pattern[x, Blank[Symbol]]] := -Subst[Int[1/(x^(m + 2)*(d + d*ProductLog[a/x^n])), x], x, 1/x] /; FreeQ[{a, d}, x] && IntegerQ[m] && ILtQ[n, 0] && NeQ[m, -1]
  6646. Int[Times[Power[Pattern[x, Blank[]], -1], Power[Times[Optional[Pattern[c, Blank[]]], ProductLog[Times[Optional[Pattern[a, Blank[]]], Power[Pattern[x, Blank[]], Optional[Pattern[n, Blank[]]]]]]], Optional[Pattern[p, Blank[]]]], Power[Plus[Pattern[d, Blank[]], Times[Optional[Pattern[d, Blank[]]], ProductLog[Times[Optional[Pattern[a, Blank[]]], Power[Pattern[x, Blank[]], Optional[Pattern[n, Blank[]]]]]]]], -1]], Pattern[x, Blank[Symbol]]] := Simp[(c*ProductLog[a*x^n])^p/(d*n*p), x] /; FreeQ[{a, c, d, n, p}, x]
  6647. Int[Times[Power[Pattern[x, Blank[]], Optional[Pattern[m, Blank[]]]], Power[Times[Optional[Pattern[c, Blank[]]], ProductLog[Times[Optional[Pattern[a, Blank[]]], Power[Pattern[x, Blank[]], Optional[Pattern[n, Blank[]]]]]]], Optional[Pattern[p, Blank[]]]], Power[Plus[Pattern[d, Blank[]], Times[Optional[Pattern[d, Blank[]]], ProductLog[Times[Optional[Pattern[a, Blank[]]], Power[Pattern[x, Blank[]], Optional[Pattern[n, Blank[]]]]]]]], -1]], Pattern[x, Blank[Symbol]]] := Simp[(c*x^(m + 1)*(c*ProductLog[a*x^n])^(p - 1))/(d*(m + 1)), x] /; FreeQ[{a, c, d, m, n, p}, x] && NeQ[m, -1] && EqQ[m + n*(p - 1), -1]
  6648. Int[Times[Power[Pattern[x, Blank[]], Optional[Pattern[m, Blank[]]]], Power[ProductLog[Times[Optional[Pattern[a, Blank[]]], Power[Pattern[x, Blank[]], Optional[Pattern[n, Blank[]]]]]], Optional[Pattern[p, Blank[]]]], Power[Plus[Pattern[d, Blank[]], Times[Optional[Pattern[d, Blank[]]], ProductLog[Times[Optional[Pattern[a, Blank[]]], Power[Pattern[x, Blank[]], Optional[Pattern[n, Blank[]]]]]]]], -1]], Pattern[x, Blank[Symbol]]] := Simp[(a^p*ExpIntegralEi[-(p*ProductLog[a*x^n])])/(d*n), x] /; FreeQ[{a, d, m, n}, x] && IntegerQ[p] && EqQ[m + n*p, -1]
  6649. Int[Times[Power[Pattern[x, Blank[]], Optional[Pattern[m, Blank[]]]], Power[Times[Optional[Pattern[c, Blank[]]], ProductLog[Times[Optional[Pattern[a, Blank[]]], Power[Pattern[x, Blank[]], Optional[Pattern[n, Blank[]]]]]]], Pattern[p, Blank[]]], Power[Plus[Pattern[d, Blank[]], Times[Optional[Pattern[d, Blank[]]], ProductLog[Times[Optional[Pattern[a, Blank[]]], Power[Pattern[x, Blank[]], Optional[Pattern[n, Blank[]]]]]]]], -1]], Pattern[x, Blank[Symbol]]] := Simp[(a^(p - 1/2)*c^(p - 1/2)*Rt[(Pi*c)/(p - 1/2), 2]*Erf[Sqrt[c*ProductLog[a*x^n]]/Rt[c/(p - 1/2), 2]])/(d*n), x] /; FreeQ[{a, c, d, m, n}, x] && NeQ[m, -1] && IntegerQ[p - 1/2] && EqQ[m + n*(p - 1/2), -1] && PosQ[c/(p - 1/2)]
  6650. Int[Times[Power[Pattern[x, Blank[]], Optional[Pattern[m, Blank[]]]], Power[Times[Optional[Pattern[c, Blank[]]], ProductLog[Times[Optional[Pattern[a, Blank[]]], Power[Pattern[x, Blank[]], Optional[Pattern[n, Blank[]]]]]]], Pattern[p, Blank[]]], Power[Plus[Pattern[d, Blank[]], Times[Optional[Pattern[d, Blank[]]], ProductLog[Times[Optional[Pattern[a, Blank[]]], Power[Pattern[x, Blank[]], Optional[Pattern[n, Blank[]]]]]]]], -1]], Pattern[x, Blank[Symbol]]] := Simp[(a^(p - 1/2)*c^(p - 1/2)*Rt[-((Pi*c)/(p - 1/2)), 2]*Erfi[Sqrt[c*ProductLog[a*x^n]]/Rt[-(c/(p - 1/2)), 2]])/(d*n), x] /; FreeQ[{a, c, d, m, n}, x] && NeQ[m, -1] && IntegerQ[p - 1/2] && EqQ[m + n*(p - 1/2), -1] && NegQ[c/(p - 1/2)]
  6651. Int[Times[Power[Pattern[x, Blank[]], Optional[Pattern[m, Blank[]]]], Power[Times[Optional[Pattern[c, Blank[]]], ProductLog[Times[Optional[Pattern[a, Blank[]]], Power[Pattern[x, Blank[]], Optional[Pattern[n, Blank[]]]]]]], Optional[Pattern[p, Blank[]]]], Power[Plus[Pattern[d, Blank[]], Times[Optional[Pattern[d, Blank[]]], ProductLog[Times[Optional[Pattern[a, Blank[]]], Power[Pattern[x, Blank[]], Optional[Pattern[n, Blank[]]]]]]]], -1]], Pattern[x, Blank[Symbol]]] := Simp[(c*x^(m + 1)*(c*ProductLog[a*x^n])^(p - 1))/(d*(m + 1)), x] - Dist[(c*(m + n*(p - 1) + 1))/(m + 1), Int[(x^m*(c*ProductLog[a*x^n])^(p - 1))/(d + d*ProductLog[a*x^n]), x], x] /; FreeQ[{a, c, d, m, n, p}, x] && NeQ[m, -1] && GtQ[Simplify[p + (m + 1)/n], 1]
  6652. Int[Times[Power[Pattern[x, Blank[]], Optional[Pattern[m, Blank[]]]], Power[Times[Optional[Pattern[c, Blank[]]], ProductLog[Times[Optional[Pattern[a, Blank[]]], Power[Pattern[x, Blank[]], Optional[Pattern[n, Blank[]]]]]]], Optional[Pattern[p, Blank[]]]], Power[Plus[Pattern[d, Blank[]], Times[Optional[Pattern[d, Blank[]]], ProductLog[Times[Optional[Pattern[a, Blank[]]], Power[Pattern[x, Blank[]], Optional[Pattern[n, Blank[]]]]]]]], -1]], Pattern[x, Blank[Symbol]]] := Simp[(x^(m + 1)*(c*ProductLog[a*x^n])^p)/(d*(m + n*p + 1)), x] - Dist[(m + 1)/(c*(m + n*p + 1)), Int[(x^m*(c*ProductLog[a*x^n])^(p + 1))/(d + d*ProductLog[a*x^n]), x], x] /; FreeQ[{a, c, d, m, n, p}, x] && NeQ[m, -1] && LtQ[Simplify[p + (m + 1)/n], 0]
  6653. Int[Times[Power[Pattern[x, Blank[]], Optional[Pattern[m, Blank[]]]], Power[Times[Optional[Pattern[c, Blank[]]], ProductLog[Times[Optional[Pattern[a, Blank[]]], Pattern[x, Blank[]]]]], Optional[Pattern[p, Blank[]]]], Power[Plus[Pattern[d, Blank[]], Times[Optional[Pattern[d, Blank[]]], ProductLog[Times[Optional[Pattern[a, Blank[]]], Pattern[x, Blank[]]]]]], -1]], Pattern[x, Blank[Symbol]]] := Simp[(x^m*Gamma[m + p + 1, -((m + 1)*ProductLog[a*x])]*(c*ProductLog[a*x])^p)/(a*d*(m + 1)*E^(m*ProductLog[a*x])*(-((m + 1)*ProductLog[a*x]))^(m + p)), x] /; FreeQ[{a, c, d, m, p}, x] && NeQ[m, -1]
  6654. Int[Times[Power[Pattern[x, Blank[]], Optional[Pattern[m, Blank[]]]], Power[Times[Optional[Pattern[c, Blank[]]], ProductLog[Times[Optional[Pattern[a, Blank[]]], Power[Pattern[x, Blank[]], Optional[Pattern[n, Blank[]]]]]]], Optional[Pattern[p, Blank[]]]], Power[Plus[Pattern[d, Blank[]], Times[Optional[Pattern[d, Blank[]]], ProductLog[Times[Optional[Pattern[a, Blank[]]], Power[Pattern[x, Blank[]], Optional[Pattern[n, Blank[]]]]]]]], -1]], Pattern[x, Blank[Symbol]]] := -Subst[Int[(c*ProductLog[a/x^n])^p/(x^(m + 2)*(d + d*ProductLog[a/x^n])), x], x, 1/x] /; FreeQ[{a, c, d, p}, x] && NeQ[m, -1] && IntegerQ[m] && LtQ[n, 0]
  6655. Int[Pattern[u, Blank[]], Pattern[x, Blank[Symbol]]] := Subst[Int[SimplifyIntegrand[(x + 1)*E^x*SubstFor[ProductLog[x], u, x], x], x], x, ProductLog[x]] /; FunctionOfQ[ProductLog[x], u, x]
  6656. Int[Derivative[Pattern[n, Blank[]]][Pattern[f, Blank[]]][Pattern[x, Blank[]]], Pattern[x, Blank[Symbol]]] := Simp[Derivative[n - 1][f][x], x] /; FreeQ[{f, n}, x]
  6657. Int[Times[Power[Times[Optional[Pattern[c, Blank[]]], Power[Pattern[F, Blank[]], Plus[Optional[Pattern[a, Blank[]]], Times[Optional[Pattern[b, Blank[]]], Pattern[x, Blank[]]]]]], Optional[Pattern[p, Blank[]]]], Derivative[Pattern[n, Blank[]]][Pattern[f, Blank[]]][Pattern[x, Blank[]]]], Pattern[x, Blank[Symbol]]] := Simp[(c*F^(a + b*x))^p*Derivative[n - 1][f][x], x] - Dist[b*p*Log[F], Int[(c*F^(a + b*x))^p*Derivative[n - 1][f][x], x], x] /; FreeQ[{a, b, c, f, F, p}, x] && IGtQ[n, 0]
  6658. Int[Times[Power[Times[Optional[Pattern[c, Blank[]]], Power[Pattern[F, Blank[]], Plus[Optional[Pattern[a, Blank[]]], Times[Optional[Pattern[b, Blank[]]], Pattern[x, Blank[]]]]]], Optional[Pattern[p, Blank[]]]], Derivative[Pattern[n, Blank[]]][Pattern[f, Blank[]]][Pattern[x, Blank[]]]], Pattern[x, Blank[Symbol]]] := Simp[((c*F^(a + b*x))^p*Derivative[n][f][x])/(b*p*Log[F]), x] - Dist[1/(b*p*Log[F]), Int[(c*F^(a + b*x))^p*Derivative[n + 1][f][x], x], x] /; FreeQ[{a, b, c, f, F, p}, x] && ILtQ[n, 0]
  6659. Int[Times[Sin[Plus[Optional[Pattern[a, Blank[]]], Times[Optional[Pattern[b, Blank[]]], Pattern[x, Blank[]]]]], Derivative[Pattern[n, Blank[]]][Pattern[f, Blank[]]][Pattern[x, Blank[]]]], Pattern[x, Blank[Symbol]]] := Simp[Sin[a + b*x]*Derivative[n - 1][f][x], x] - Dist[b, Int[Cos[a + b*x]*Derivative[n - 1][f][x], x], x] /; FreeQ[{a, b, f}, x] && IGtQ[n, 0]
  6660. Int[Times[Cos[Plus[Optional[Pattern[a, Blank[]]], Times[Optional[Pattern[b, Blank[]]], Pattern[x, Blank[]]]]], Derivative[Pattern[n, Blank[]]][Pattern[f, Blank[]]][Pattern[x, Blank[]]]], Pattern[x, Blank[Symbol]]] := Simp[Cos[a + b*x]*Derivative[n - 1][f][x], x] + Dist[b, Int[Sin[a + b*x]*Derivative[n - 1][f][x], x], x] /; FreeQ[{a, b, f}, x] && IGtQ[n, 0]
  6661. Int[Times[Sin[Plus[Optional[Pattern[a, Blank[]]], Times[Optional[Pattern[b, Blank[]]], Pattern[x, Blank[]]]]], Derivative[Pattern[n, Blank[]]][Pattern[f, Blank[]]][Pattern[x, Blank[]]]], Pattern[x, Blank[Symbol]]] := -Simp[(Cos[a + b*x]*Derivative[n][f][x])/b, x] + Dist[1/b, Int[Cos[a + b*x]*Derivative[n + 1][f][x], x], x] /; FreeQ[{a, b, f}, x] && ILtQ[n, 0]
  6662. Int[Times[Cos[Plus[Optional[Pattern[a, Blank[]]], Times[Optional[Pattern[b, Blank[]]], Pattern[x, Blank[]]]]], Derivative[Pattern[n, Blank[]]][Pattern[f, Blank[]]][Pattern[x, Blank[]]]], Pattern[x, Blank[Symbol]]] := Simp[(Sin[a + b*x]*Derivative[n][f][x])/b, x] - Dist[1/b, Int[Sin[a + b*x]*Derivative[n + 1][f][x], x], x] /; FreeQ[{a, b, f}, x] && ILtQ[n, 0]
  6663. Int[Times[Pattern[u, Blank[]], Derivative[Pattern[n, Blank[]]][Pattern[f, Blank[]]][Pattern[x, Blank[]]]], Pattern[x, Blank[Symbol]]] := Subst[Int[SimplifyIntegrand[SubstFor[Derivative[n - 1][f][x], u, x], x], x], x, Derivative[n - 1][f][x]] /; FreeQ[{f, n}, x] && FunctionOfQ[Derivative[n - 1][f][x], u, x]
  6664. Int[Times[Pattern[u, Blank[]], Plus[Times[Optional[Pattern[a, Blank[]]], Pattern[g, Blank[]][Pattern[x, Blank[]]], Derivative[1][Pattern[f, Blank[]]][Pattern[x, Blank[]]]], Times[Optional[Pattern[a, Blank[]]], Pattern[f, Blank[]][Pattern[x, Blank[]]], Derivative[1][Pattern[g, Blank[]]][Pattern[x, Blank[]]]]]], Pattern[x, Blank[Symbol]]] := Dist[a, Subst[Int[SimplifyIntegrand[SubstFor[f[x]*g[x], u, x], x], x], x, f[x]*g[x]], x] /; FreeQ[{a, f, g}, x] && FunctionOfQ[f[x]*g[x], u, x]
  6665. Int[Times[Pattern[u, Blank[]], Plus[Times[Optional[Pattern[a, Blank[]]], Pattern[g, Blank[]][Pattern[x, Blank[]]], Derivative[Pattern[m, Blank[]]][Pattern[f, Blank[]]][Pattern[x, Blank[]]]], Times[Optional[Pattern[a, Blank[]]], Derivative[1][Pattern[g, Blank[]]][Pattern[x, Blank[]]], Derivative[Pattern[m1, Blank[]]][Pattern[f, Blank[]]][Pattern[x, Blank[]]]]]], Pattern[x, Blank[Symbol]]] := Dist[a, Subst[Int[SimplifyIntegrand[SubstFor[Derivative[m - 1][f][x]*g[x], u, x], x], x], x, Derivative[m - 1][f][x]*g[x]], x] /; FreeQ[{a, f, g, m}, x] && EqQ[m1, m - 1] && FunctionOfQ[Derivative[m - 1][f][x]*g[x], u, x]
  6666. Int[Times[Pattern[u, Blank[]], Plus[Times[Optional[Pattern[a, Blank[]]], Derivative[Pattern[m1, Blank[]]][Pattern[f, Blank[]]][Pattern[x, Blank[]]], Derivative[Pattern[n, Blank[]]][Pattern[g, Blank[]]][Pattern[x, Blank[]]]], Times[Optional[Pattern[a, Blank[]]], Derivative[Pattern[m, Blank[]]][Pattern[f, Blank[]]][Pattern[x, Blank[]]], Derivative[Pattern[n1, Blank[]]][Pattern[g, Blank[]]][Pattern[x, Blank[]]]]]], Pattern[x, Blank[Symbol]]] := Dist[a, Subst[Int[SimplifyIntegrand[SubstFor[Derivative[m - 1][f][x]*Derivative[n - 1][g][x], u, x], x], x], x, Derivative[m - 1][f][x]*Derivative[n - 1][g][x]], x] /; FreeQ[{a, f, g, m, n}, x] && EqQ[m1, m - 1] && EqQ[n1, n - 1] && FunctionOfQ[Derivative[m - 1][f][x]*Derivative[n - 1][g][x], u, x]
  6667. Int[Times[Pattern[u, Blank[]], Power[Pattern[f, Blank[]][Pattern[x, Blank[]]], Optional[Pattern[p, Blank[]]]], Plus[Times[Optional[Pattern[a, Blank[]]], Pattern[g, Blank[]][Pattern[x, Blank[]]], Derivative[1][Pattern[f, Blank[]]][Pattern[x, Blank[]]]], Times[Optional[Pattern[b, Blank[]]], Pattern[f, Blank[]][Pattern[x, Blank[]]], Derivative[1][Pattern[g, Blank[]]][Pattern[x, Blank[]]]]]], Pattern[x, Blank[Symbol]]] := Dist[b, Subst[Int[SimplifyIntegrand[SubstFor[f[x]^(p + 1)*g[x], u, x], x], x], x, f[x]^(p + 1)*g[x]], x] /; FreeQ[{a, b, f, g, p}, x] && EqQ[a, b*(p + 1)] && FunctionOfQ[f[x]^(p + 1)*g[x], u, x]
  6668. Int[Times[Pattern[u, Blank[]], Power[Derivative[Pattern[m1, Blank[]]][Pattern[f, Blank[]]][Pattern[x, Blank[]]], Optional[Pattern[p, Blank[]]]], Plus[Times[Optional[Pattern[a, Blank[]]], Pattern[g, Blank[]][Pattern[x, Blank[]]], Derivative[Pattern[m, Blank[]]][Pattern[f, Blank[]]][Pattern[x, Blank[]]]], Times[Optional[Pattern[b, Blank[]]], Derivative[1][Pattern[g, Blank[]]][Pattern[x, Blank[]]], Derivative[Pattern[m1, Blank[]]][Pattern[f, Blank[]]][Pattern[x, Blank[]]]]]], Pattern[x, Blank[Symbol]]] := Dist[b, Subst[Int[SimplifyIntegrand[SubstFor[Derivative[m - 1][f][x]^(p + 1)*g[x], u, x], x], x], x, Derivative[m - 1][f][x]^(p + 1)*g[x]], x] /; FreeQ[{a, b, f, g, m, p}, x] && EqQ[m1, m - 1] && EqQ[a, b*(p + 1)] && FunctionOfQ[Derivative[m - 1][f][x]^(p + 1)*g[x], u, x]
  6669. Int[Times[Pattern[u, Blank[]], Power[Pattern[g, Blank[]][Pattern[x, Blank[]]], Optional[Pattern[q, Blank[]]]], Plus[Times[Optional[Pattern[a, Blank[]]], Pattern[g, Blank[]][Pattern[x, Blank[]]], Derivative[Pattern[m, Blank[]]][Pattern[f, Blank[]]][Pattern[x, Blank[]]]], Times[Optional[Pattern[b, Blank[]]], Derivative[1][Pattern[g, Blank[]]][Pattern[x, Blank[]]], Derivative[Pattern[m1, Blank[]]][Pattern[f, Blank[]]][Pattern[x, Blank[]]]]]], Pattern[x, Blank[Symbol]]] := Dist[a, Subst[Int[SimplifyIntegrand[SubstFor[Derivative[m - 1][f][x]*g[x]^(q + 1), u, x], x], x], x, Derivative[m - 1][f][x]*g[x]^(q + 1)], x] /; FreeQ[{a, b, f, g, m, q}, x] && EqQ[m1, m - 1] && EqQ[a*(q + 1), b] && FunctionOfQ[Derivative[m - 1][f][x]*g[x]^(q + 1), u, x]
  6670. Int[Times[Pattern[u, Blank[]], Power[Derivative[Pattern[m1, Blank[]]][Pattern[f, Blank[]]][Pattern[x, Blank[]]], Optional[Pattern[p, Blank[]]]], Plus[Times[Optional[Pattern[b, Blank[]]], Derivative[Pattern[m1, Blank[]]][Pattern[f, Blank[]]][Pattern[x, Blank[]]], Derivative[Pattern[n, Blank[]]][Pattern[g, Blank[]]][Pattern[x, Blank[]]]], Times[Optional[Pattern[a, Blank[]]], Derivative[Pattern[m, Blank[]]][Pattern[f, Blank[]]][Pattern[x, Blank[]]], Derivative[Pattern[n1, Blank[]]][Pattern[g, Blank[]]][Pattern[x, Blank[]]]]]], Pattern[x, Blank[Symbol]]] := Dist[b, Subst[Int[SimplifyIntegrand[SubstFor[Derivative[m - 1][f][x]^(p + 1)*Derivative[n - 1][g][x], u, x], x], x], x, Derivative[m - 1][f][x]^(p + 1)*Derivative[n - 1][g][x]], x] /; FreeQ[{a, b, f, g, m, n, p}, x] && EqQ[m1, m - 1] && EqQ[n1, n - 1] && EqQ[a, b*(p + 1)] && FunctionOfQ[Derivative[m - 1][f][x]^(p + 1)*Derivative[n - 1][g][x], u, x]
  6671. Int[Times[Pattern[u, Blank[]], Power[Pattern[f, Blank[]][Pattern[x, Blank[]]], Optional[Pattern[p, Blank[]]]], Power[Pattern[g, Blank[]][Pattern[x, Blank[]]], Optional[Pattern[q, Blank[]]]], Plus[Times[Optional[Pattern[a, Blank[]]], Pattern[g, Blank[]][Pattern[x, Blank[]]], Derivative[1][Pattern[f, Blank[]]][Pattern[x, Blank[]]]], Times[Optional[Pattern[b, Blank[]]], Pattern[f, Blank[]][Pattern[x, Blank[]]], Derivative[1][Pattern[g, Blank[]]][Pattern[x, Blank[]]]]]], Pattern[x, Blank[Symbol]]] := Dist[a/(p + 1), Subst[Int[SimplifyIntegrand[SubstFor[f[x]^(p + 1)*g[x]^(q + 1), u, x], x], x], x, f[x]^(p + 1)*g[x]^(q + 1)], x] /; FreeQ[{a, b, f, g, p, q}, x] && EqQ[a*(q + 1), b*(p + 1)] && FunctionOfQ[f[x]^(p + 1)*g[x]^(q + 1), u, x]
  6672. Int[Times[Pattern[u, Blank[]], Power[Pattern[g, Blank[]][Pattern[x, Blank[]]], Optional[Pattern[q, Blank[]]]], Power[Derivative[Pattern[m1, Blank[]]][Pattern[f, Blank[]]][Pattern[x, Blank[]]], Optional[Pattern[p, Blank[]]]], Plus[Times[Optional[Pattern[a, Blank[]]], Pattern[g, Blank[]][Pattern[x, Blank[]]], Derivative[Pattern[m, Blank[]]][Pattern[f, Blank[]]][Pattern[x, Blank[]]]], Times[Optional[Pattern[b, Blank[]]], Derivative[1][Pattern[g, Blank[]]][Pattern[x, Blank[]]], Derivative[Pattern[m1, Blank[]]][Pattern[f, Blank[]]][Pattern[x, Blank[]]]]]], Pattern[x, Blank[Symbol]]] := Dist[a/(p + 1), Subst[Int[SimplifyIntegrand[SubstFor[Derivative[m - 1][f][x]^(p + 1)*g[x]^(q + 1), u, x], x], x], x, Derivative[m - 1][f][x]^(p + 1)*g[x]^(q + 1)], x] /; FreeQ[{a, b, f, g, m, p, q}, x] && EqQ[m1, m - 1] && EqQ[a*(q + 1), b*(p + 1)] && FunctionOfQ[Derivative[m - 1][f][x]^(p + 1)*g[x]^(q + 1), u, x]
  6673. Int[Times[Pattern[u, Blank[]], Power[Derivative[Pattern[m1, Blank[]]][Pattern[f, Blank[]]][Pattern[x, Blank[]]], Optional[Pattern[p, Blank[]]]], Power[Derivative[Pattern[n1, Blank[]]][Pattern[g, Blank[]]][Pattern[x, Blank[]]], Optional[Pattern[q, Blank[]]]], Plus[Times[Optional[Pattern[b, Blank[]]], Derivative[Pattern[m1, Blank[]]][Pattern[f, Blank[]]][Pattern[x, Blank[]]], Derivative[Pattern[n, Blank[]]][Pattern[g, Blank[]]][Pattern[x, Blank[]]]], Times[Optional[Pattern[a, Blank[]]], Derivative[Pattern[m, Blank[]]][Pattern[f, Blank[]]][Pattern[x, Blank[]]], Derivative[Pattern[n1, Blank[]]][Pattern[g, Blank[]]][Pattern[x, Blank[]]]]]], Pattern[x, Blank[Symbol]]] := Dist[a/(p + 1), Subst[Int[SimplifyIntegrand[SubstFor[Derivative[m - 1][f][x]^(p + 1)*Derivative[n - 1][g][x]^(q + 1), u, x], x], x], x, Derivative[m - 1][f][x]^(p + 1)*Derivative[n - 1][g][x]^(q + 1)], x] /; FreeQ[{a, b, f, g, m, n, p, q}, x] && EqQ[m1, m - 1] && EqQ[n1, n - 1] && EqQ[a*(q + 1), b*(p + 1)] && FunctionOfQ[Derivative[m - 1][f][x]^(p + 1)*Derivative[n - 1][g][x]^(q + 1), u, x]
  6674. Int[Plus[Times[Pattern[g, Blank[]][Pattern[x, Blank[]]], Derivative[1][Pattern[f, Blank[]]][Pattern[x, Blank[]]]], Times[Pattern[f, Blank[]][Pattern[x, Blank[]]], Derivative[1][Pattern[g, Blank[]]][Pattern[x, Blank[]]]]], Pattern[x, Blank[Symbol]]] := Simp[f[x]*g[x], x] /; FreeQ[{f, g}, x]
  6675. Int[Times[Power[Pattern[g, Blank[]][Pattern[x, Blank[]]], -2], Plus[Times[Pattern[g, Blank[]][Pattern[x, Blank[]]], Derivative[1][Pattern[f, Blank[]]][Pattern[x, Blank[]]]], Times[-1, Pattern[f, Blank[]][Pattern[x, Blank[]]], Derivative[1][Pattern[g, Blank[]]][Pattern[x, Blank[]]]]]], Pattern[x, Blank[Symbol]]] := Simp[f[x]/g[x], x] /; FreeQ[{f, g}, x]
  6676. Int[Times[Power[Pattern[f, Blank[]][Pattern[x, Blank[]]], -1], Power[Pattern[g, Blank[]][Pattern[x, Blank[]]], -1], Plus[Times[Pattern[g, Blank[]][Pattern[x, Blank[]]], Derivative[1][Pattern[f, Blank[]]][Pattern[x, Blank[]]]], Times[-1, Pattern[f, Blank[]][Pattern[x, Blank[]]], Derivative[1][Pattern[g, Blank[]]][Pattern[x, Blank[]]]]]], Pattern[x, Blank[Symbol]]] := Simp[Log[f[x]/g[x]], x] /; FreeQ[{f, g}, x]
  6677. Int[Times[Pattern[u, Blank[]], Power[Times[Optional[Pattern[c, Blank[]]], Power[Plus[Optional[Pattern[a, Blank[]]], Times[Optional[Pattern[b, Blank[]]], Pattern[x, Blank[]]]], Pattern[n, Blank[]]]], Pattern[p, Blank[]]]], Pattern[x, Blank[Symbol]]] := Dist[(c^IntPart[p]*(c*(a + b*x)^n)^FracPart[p])/(a + b*x)^(n*FracPart[p]), Int[u*(a + b*x)^(n*p), x], x] /; FreeQ[{a, b, c, n, p}, x] && !IntegerQ[p] && !MatchQ[u, x^(n1_.)*(v_.) /; EqQ[n, n1 + 1]]
  6678. Int[Times[Optional[Pattern[u, Blank[]]], Power[Times[Optional[Pattern[c, Blank[]]], Power[Times[Pattern[d, Blank[]], Plus[Optional[Pattern[a, Blank[]]], Times[Optional[Pattern[b, Blank[]]], Pattern[x, Blank[]]]]], Pattern[p, Blank[]]]], Pattern[q, Blank[]]]], Pattern[x, Blank[Symbol]]] := Dist[(c*(d*(a + b*x))^p)^q/(a + b*x)^(p*q), Int[u*(a + b*x)^(p*q), x], x] /; FreeQ[{a, b, c, d, p, q}, x] && !IntegerQ[p] && !IntegerQ[q]
  6679. Int[Times[Optional[Pattern[u, Blank[]]], Power[Times[Optional[Pattern[c, Blank[]]], Power[Times[Optional[Pattern[d, Blank[]]], Power[Plus[Optional[Pattern[a, Blank[]]], Times[Optional[Pattern[b, Blank[]]], Pattern[x, Blank[]]]], Pattern[n, Blank[]]]], Pattern[p, Blank[]]]], Pattern[q, Blank[]]]], Pattern[x, Blank[Symbol]]] := Dist[(c*(d*(a + b*x)^n)^p)^q/(a + b*x)^(n*p*q), Int[u*(a + b*x)^(n*p*q), x], x] /; FreeQ[{a, b, c, d, n, p, q}, x] && !IntegerQ[p] && !IntegerQ[q]
  6680. Int[Times[Power[Plus[Optional[Pattern[A, Blank[]]], Times[Optional[Pattern[B, Blank[]]], Pattern[x, Blank[]]], Times[Optional[Pattern[C, Blank[]]], Power[Pattern[x, Blank[]], 2]]], -1], Power[Plus[Optional[Pattern[a, Blank[]]], Times[Optional[Pattern[b, Blank[]]], Pattern[F, Blank[]][Times[Optional[Pattern[c, Blank[]]], Power[Plus[Optional[Pattern[d, Blank[]]], Times[Optional[Pattern[e, Blank[]]], Pattern[x, Blank[]]]], Rational[1, 2]], Power[Plus[Optional[Pattern[f, Blank[]]], Times[Optional[Pattern[g, Blank[]]], Pattern[x, Blank[]]]], Rational[-1, 2]]]]]], Optional[Pattern[n, Blank[]]]]], Pattern[x, Blank[Symbol]]] := Dist[(2*e*g)/(C*(e*f - d*g)), Subst[Int[(a + b*F[c*x])^n/x, x], x, Sqrt[d + e*x]/Sqrt[f + g*x]], x] /; FreeQ[{a, b, c, d, e, f, g, A, B, C, F}, x] && EqQ[C*d*f - A*e*g, 0] && EqQ[B*e*g - C*(e*f + d*g), 0] && IGtQ[n, 0]
  6681. Int[Times[Power[Plus[Optional[Pattern[A, Blank[]]], Times[Optional[Pattern[C, Blank[]]], Power[Pattern[x, Blank[]], 2]]], -1], Power[Plus[Optional[Pattern[a, Blank[]]], Times[Optional[Pattern[b, Blank[]]], Pattern[F, Blank[]][Times[Optional[Pattern[c, Blank[]]], Power[Plus[Optional[Pattern[d, Blank[]]], Times[Optional[Pattern[e, Blank[]]], Pattern[x, Blank[]]]], Rational[1, 2]], Power[Plus[Optional[Pattern[f, Blank[]]], Times[Optional[Pattern[g, Blank[]]], Pattern[x, Blank[]]]], Rational[-1, 2]]]]]], Optional[Pattern[n, Blank[]]]]], Pattern[x, Blank[Symbol]]] := Dist[(2*e*g)/(C*(e*f - d*g)), Subst[Int[(a + b*F[c*x])^n/x, x], x, Sqrt[d + e*x]/Sqrt[f + g*x]], x] /; FreeQ[{a, b, c, d, e, f, g, A, C, F}, x] && EqQ[C*d*f - A*e*g, 0] && EqQ[e*f + d*g, 0] && IGtQ[n, 0]
  6682. Int[Times[Power[Plus[Optional[Pattern[A, Blank[]]], Times[Optional[Pattern[B, Blank[]]], Pattern[x, Blank[]]], Times[Optional[Pattern[C, Blank[]]], Power[Pattern[x, Blank[]], 2]]], -1], Power[Plus[Optional[Pattern[a, Blank[]]], Times[Optional[Pattern[b, Blank[]]], Pattern[F, Blank[]][Times[Optional[Pattern[c, Blank[]]], Power[Plus[Optional[Pattern[d, Blank[]]], Times[Optional[Pattern[e, Blank[]]], Pattern[x, Blank[]]]], Rational[1, 2]], Power[Plus[Optional[Pattern[f, Blank[]]], Times[Optional[Pattern[g, Blank[]]], Pattern[x, Blank[]]]], Rational[-1, 2]]]]]], Pattern[n, Blank[]]]], Pattern[x, Blank[Symbol]]] := Unintegrable[(a + b*F[(c*Sqrt[d + e*x])/Sqrt[f + g*x]])^n/(A + B*x + C*x^2), x] /; FreeQ[{a, b, c, d, e, f, g, A, B, C, F, n}, x] && EqQ[C*d*f - A*e*g, 0] && EqQ[B*e*g - C*(e*f + d*g), 0] && !IGtQ[n, 0]
  6683. Int[Times[Power[Plus[Pattern[A, Blank[]], Times[Optional[Pattern[C, Blank[]]], Power[Pattern[x, Blank[]], 2]]], -1], Power[Plus[Optional[Pattern[a, Blank[]]], Times[Optional[Pattern[b, Blank[]]], Pattern[F, Blank[]][Times[Optional[Pattern[c, Blank[]]], Power[Plus[Optional[Pattern[d, Blank[]]], Times[Optional[Pattern[e, Blank[]]], Pattern[x, Blank[]]]], Rational[1, 2]], Power[Plus[Optional[Pattern[f, Blank[]]], Times[Optional[Pattern[g, Blank[]]], Pattern[x, Blank[]]]], Rational[-1, 2]]]]]], Pattern[n, Blank[]]]], Pattern[x, Blank[Symbol]]] := Unintegrable[(a + b*F[(c*Sqrt[d + e*x])/Sqrt[f + g*x]])^n/(A + C*x^2), x] /; FreeQ[{a, b, c, d, e, f, g, A, C, F, n}, x] && EqQ[C*d*f - A*e*g, 0] && EqQ[e*f + d*g, 0] && !IGtQ[n, 0]
  6684. Int[Times[Pattern[u, Blank[]], Power[Pattern[y, Blank[]], -1]], Pattern[x, Blank[Symbol]]] := With[{q = DerivativeDivides[y, u, x]}, Simp[q*Log[RemoveContent[y, x]], x] /; !FalseQ[q]]
  6685. Int[Times[Pattern[u, Blank[]], Power[Pattern[w, Blank[]], -1], Power[Pattern[y, Blank[]], -1]], Pattern[x, Blank[Symbol]]] := With[{q = DerivativeDivides[y*w, u, x]}, Simp[q*Log[RemoveContent[y*w, x]], x] /; !FalseQ[q]]
  6686. Int[Times[Pattern[u, Blank[]], Power[Pattern[y, Blank[]], Optional[Pattern[m, Blank[]]]]], Pattern[x, Blank[Symbol]]] := With[{q = DerivativeDivides[y, u, x]}, Simp[(q*y^(m + 1))/(m + 1), x] /; !FalseQ[q]] /; FreeQ[m, x] && NeQ[m, -1]
  6687. Int[Times[Pattern[u, Blank[]], Power[Pattern[y, Blank[]], Optional[Pattern[m, Blank[]]]], Power[Pattern[z, Blank[]], Optional[Pattern[n, Blank[]]]]], Pattern[x, Blank[Symbol]]] := With[{q = DerivativeDivides[y*z, u*z^(n - m), x]}, Simp[(q*y^(m + 1)*z^(m + 1))/(m + 1), x] /; !FalseQ[q]] /; FreeQ[{m, n}, x] && NeQ[m, -1]
  6688. Int[Pattern[u, Blank[]], Pattern[x, Blank[Symbol]]] := With[{v = SimplifyIntegrand[u, x]}, Int[v, x] /; SimplerIntegrandQ[v, u, x]]
  6689. Int[Times[Optional[Pattern[u, Blank[]]], Power[Plus[Times[Optional[Pattern[e, Blank[]]], Power[Plus[Optional[Pattern[a, Blank[]]], Times[Optional[Pattern[b, Blank[]]], Power[Pattern[x, Blank[]], Optional[Pattern[n, Blank[]]]]]], Rational[1, 2]]], Times[Optional[Pattern[f, Blank[]]], Power[Plus[Optional[Pattern[c, Blank[]]], Times[Optional[Pattern[d, Blank[]]], Power[Pattern[x, Blank[]], Optional[Pattern[n, Blank[]]]]]], Rational[1, 2]]]], Pattern[m, Blank[]]]], Pattern[x, Blank[Symbol]]] := Dist[(a*e^2 - c*f^2)^m, Int[ExpandIntegrand[u/(e*Sqrt[a + b*x^n] - f*Sqrt[c + d*x^n])^m, x], x], x] /; FreeQ[{a, b, c, d, e, f, n}, x] && ILtQ[m, 0] && EqQ[b*e^2 - d*f^2, 0]
  6690. Int[Times[Optional[Pattern[u, Blank[]]], Power[Plus[Times[Optional[Pattern[e, Blank[]]], Power[Plus[Optional[Pattern[a, Blank[]]], Times[Optional[Pattern[b, Blank[]]], Power[Pattern[x, Blank[]], Optional[Pattern[n, Blank[]]]]]], Rational[1, 2]]], Times[Optional[Pattern[f, Blank[]]], Power[Plus[Optional[Pattern[c, Blank[]]], Times[Optional[Pattern[d, Blank[]]], Power[Pattern[x, Blank[]], Optional[Pattern[n, Blank[]]]]]], Rational[1, 2]]]], Pattern[m, Blank[]]]], Pattern[x, Blank[Symbol]]] := Dist[(b*e^2 - d*f^2)^m, Int[ExpandIntegrand[(u*x^(m*n))/(e*Sqrt[a + b*x^n] - f*Sqrt[c + d*x^n])^m, x], x], x] /; FreeQ[{a, b, c, d, e, f, n}, x] && ILtQ[m, 0] && EqQ[a*e^2 - c*f^2, 0]
  6691. Int[Times[Power[Pattern[u, Blank[]], Optional[Pattern[m, Blank[]]]], Power[Plus[Times[Optional[Pattern[a, Blank[]]], Power[Pattern[u, Blank[]], Pattern[n, Blank[]]]], Pattern[v, Blank[]]], Optional[Pattern[p, Blank[]]]], Pattern[w, Blank[]]], Pattern[x, Blank[Symbol]]] := Int[u^(m + n*p)*(a + v/u^n)^p*w, x] /; FreeQ[{a, m, n}, x] && IntegerQ[p] && !GtQ[n, 0] && !FreeQ[v, x]
  6692. Int[Times[Pattern[u, Blank[]], Power[Plus[Optional[Pattern[c, Blank[]]], Times[Optional[Pattern[d, Blank[]]], Pattern[v, Blank[]]]], Optional[Pattern[n, Blank[]]]], Power[Plus[Optional[Pattern[a, Blank[]]], Times[Optional[Pattern[b, Blank[]]], Pattern[y, Blank[]]]], Optional[Pattern[m, Blank[]]]]], Pattern[x, Blank[Symbol]]] := With[{q = DerivativeDivides[y, u, x]}, Dist[q, Subst[Int[(a + b*x)^m*(c + d*x)^n, x], x, y], x] /; !FalseQ[q]] /; FreeQ[{a, b, c, d, m, n}, x] && EqQ[v, y]
  6693. Int[Times[Pattern[u, Blank[]], Power[Plus[Optional[Pattern[c, Blank[]]], Times[Optional[Pattern[d, Blank[]]], Pattern[v, Blank[]]]], Optional[Pattern[n, Blank[]]]], Power[Plus[Optional[Pattern[e, Blank[]]], Times[Optional[Pattern[f, Blank[]]], Pattern[w, Blank[]]]], Optional[Pattern[p, Blank[]]]], Power[Plus[Optional[Pattern[a, Blank[]]], Times[Optional[Pattern[b, Blank[]]], Pattern[y, Blank[]]]], Optional[Pattern[m, Blank[]]]]], Pattern[x, Blank[Symbol]]] := With[{q = DerivativeDivides[y, u, x]}, Dist[q, Subst[Int[(a + b*x)^m*(c + d*x)^n*(e + f*x)^p, x], x, y], x] /; !FalseQ[q]] /; FreeQ[{a, b, c, d, e, f, m, n, p}, x] && EqQ[v, y] && EqQ[w, y]
  6694. Int[Times[Pattern[u, Blank[]], Power[Plus[Optional[Pattern[c, Blank[]]], Times[Optional[Pattern[d, Blank[]]], Pattern[v, Blank[]]]], Optional[Pattern[n, Blank[]]]], Power[Plus[Optional[Pattern[e, Blank[]]], Times[Optional[Pattern[f, Blank[]]], Pattern[w, Blank[]]]], Optional[Pattern[p, Blank[]]]], Power[Plus[Optional[Pattern[a, Blank[]]], Times[Optional[Pattern[b, Blank[]]], Pattern[y, Blank[]]]], Optional[Pattern[m, Blank[]]]], Power[Plus[Optional[Pattern[g, Blank[]]], Times[Optional[Pattern[h, Blank[]]], Pattern[z, Blank[]]]], Optional[Pattern[q, Blank[]]]]], Pattern[x, Blank[Symbol]]] := With[{r = DerivativeDivides[y, u, x]}, Dist[r, Subst[Int[(a + b*x)^m*(c + d*x)^n*(e + f*x)^p*(g + h*x)^q, x], x, y], x] /; !FalseQ[r]] /; FreeQ[{a, b, c, d, e, f, g, h, m, n, p, q}, x] && EqQ[v, y] && EqQ[w, y] && EqQ[z, y]
  6695. Int[Times[Optional[Pattern[u, Blank[]]], Plus[Pattern[a, Blank[]], Times[Optional[Pattern[b, Blank[]]], Power[Pattern[y, Blank[]], Pattern[n, Blank[]]]]]], Pattern[x, Blank[Symbol]]] := With[{q = DerivativeDivides[y, u, x]}, Dist[a, Int[u, x], x] + Dist[b*q, Subst[Int[x^n, x], x, y], x] /; !FalseQ[q]] /; FreeQ[{a, b, n}, x]
  6696. Int[Times[Optional[Pattern[u, Blank[]]], Power[Plus[Optional[Pattern[a, Blank[]]], Times[Optional[Pattern[b, Blank[]]], Power[Pattern[y, Blank[]], Pattern[n, Blank[]]]]], Pattern[p, Blank[]]]], Pattern[x, Blank[Symbol]]] := With[{q = DerivativeDivides[y, u, x]}, Dist[q, Subst[Int[(a + b*x^n)^p, x], x, y], x] /; !FalseQ[q]] /; FreeQ[{a, b, n, p}, x]
  6697. Int[Times[Optional[Pattern[u, Blank[]]], Power[Pattern[v, Blank[]], Optional[Pattern[m, Blank[]]]], Power[Plus[Optional[Pattern[a, Blank[]]], Times[Optional[Pattern[b, Blank[]]], Power[Pattern[y, Blank[]], Pattern[n, Blank[]]]]], Optional[Pattern[p, Blank[]]]]], Pattern[x, Blank[Symbol]]] := Module[{q, r}, Dist[q*r, Subst[Int[x^m*(a + b*x^n)^p, x], x, y], x] /; !FalseQ[r = Divides[y^m, v^m, x]] && !FalseQ[q = DerivativeDivides[y, u, x]]] /; FreeQ[{a, b, m, n, p}, x]
  6698. Int[Times[Optional[Pattern[u, Blank[]]], Power[Plus[Optional[Pattern[a, Blank[]]], Times[Optional[Pattern[c, Blank[]]], Power[Pattern[v, Blank[]], Optional[Pattern[n2, Blank[]]]]], Times[Optional[Pattern[b, Blank[]]], Power[Pattern[y, Blank[]], Pattern[n, Blank[]]]]], Pattern[p, Blank[]]]], Pattern[x, Blank[Symbol]]] := With[{q = DerivativeDivides[y, u, x]}, Dist[q, Subst[Int[(a + b*x^n + c*x^(2*n))^p, x], x, y], x] /; !FalseQ[q]] /; FreeQ[{a, b, c, n, p}, x] && EqQ[n2, 2*n] && EqQ[v, y]
  6699. Int[Times[Optional[Pattern[u, Blank[]]], Power[Plus[Optional[Pattern[a, Blank[]]], Times[Optional[Pattern[b, Blank[]]], Power[Pattern[v, Blank[]], Pattern[n, Blank[]]]], Times[Optional[Pattern[c, Blank[]]], Power[Pattern[w, Blank[]], Optional[Pattern[n2, Blank[]]]]]], Optional[Pattern[p, Blank[]]]], Plus[Pattern[A, Blank[]], Times[Optional[Pattern[B, Blank[]]], Power[Pattern[y, Blank[]], Pattern[n, Blank[]]]]]], Pattern[x, Blank[Symbol]]] := With[{q = DerivativeDivides[y, u, x]}, Dist[q, Subst[Int[(A + B*x^n)*(a + b*x^n + c*x^(2*n))^p, x], x, y], x] /; !FalseQ[q]] /; FreeQ[{a, b, c, A, B, n, p}, x] && EqQ[n2, 2*n] && EqQ[v, y] && EqQ[w, y]
  6700. Int[Times[Optional[Pattern[u, Blank[]]], Power[Plus[Optional[Pattern[a, Blank[]]], Times[Optional[Pattern[c, Blank[]]], Power[Pattern[w, Blank[]], Optional[Pattern[n2, Blank[]]]]]], Optional[Pattern[p, Blank[]]]], Plus[Pattern[A, Blank[]], Times[Optional[Pattern[B, Blank[]]], Power[Pattern[y, Blank[]], Pattern[n, Blank[]]]]]], Pattern[x, Blank[Symbol]]] := With[{q = DerivativeDivides[y, u, x]}, Dist[q, Subst[Int[(A + B*x^n)*(a + c*x^(2*n))^p, x], x, y], x] /; !FalseQ[q]] /; FreeQ[{a, c, A, B, n, p}, x] && EqQ[n2, 2*n] && EqQ[w, y]
  6701. Int[Times[Optional[Pattern[u, Blank[]]], Power[Pattern[v, Blank[]], Optional[Pattern[m, Blank[]]]], Power[Plus[Optional[Pattern[a, Blank[]]], Times[Optional[Pattern[c, Blank[]]], Power[Pattern[w, Blank[]], Optional[Pattern[n2, Blank[]]]]], Times[Optional[Pattern[b, Blank[]]], Power[Pattern[y, Blank[]], Pattern[n, Blank[]]]]], Optional[Pattern[p, Blank[]]]]], Pattern[x, Blank[Symbol]]] := Module[{q, r}, Dist[q*r, Subst[Int[x^m*(a + b*x^n + c*x^(2*n))^p, x], x, y], x] /; !FalseQ[r = Divides[y^m, v^m, x]] && !FalseQ[q = DerivativeDivides[y, u, x]]] /; FreeQ[{a, b, c, m, n, p}, x] && EqQ[n2, 2*n] && EqQ[w, y]
  6702. Int[Times[Optional[Pattern[u, Blank[]]], Power[Plus[Optional[Pattern[a, Blank[]]], Times[Optional[Pattern[b, Blank[]]], Power[Pattern[v, Blank[]], Pattern[n, Blank[]]]], Times[Optional[Pattern[c, Blank[]]], Power[Pattern[w, Blank[]], Optional[Pattern[n2, Blank[]]]]]], Optional[Pattern[p, Blank[]]]], Plus[Pattern[A, Blank[]], Times[Optional[Pattern[B, Blank[]]], Power[Pattern[y, Blank[]], Pattern[n, Blank[]]]]], Power[Pattern[z, Blank[]], Optional[Pattern[m, Blank[]]]]], Pattern[x, Blank[Symbol]]] := Module[{q, r}, Dist[q*r, Subst[Int[x^m*(A + B*x^n)*(a + b*x^n + c*x^(2*n))^p, x], x, y], x] /; !FalseQ[r = Divides[y^m, z^m, x]] && !FalseQ[q = DerivativeDivides[y, u, x]]] /; FreeQ[{a, b, c, A, B, m, n, p}, x] && EqQ[n2, 2*n] && EqQ[v, y] && EqQ[w, y]
  6703. Int[Times[Optional[Pattern[u, Blank[]]], Power[Plus[Optional[Pattern[a, Blank[]]], Times[Optional[Pattern[c, Blank[]]], Power[Pattern[w, Blank[]], Optional[Pattern[n2, Blank[]]]]]], Optional[Pattern[p, Blank[]]]], Plus[Pattern[A, Blank[]], Times[Optional[Pattern[B, Blank[]]], Power[Pattern[y, Blank[]], Pattern[n, Blank[]]]]], Power[Pattern[z, Blank[]], Optional[Pattern[m, Blank[]]]]], Pattern[x, Blank[Symbol]]] := Module[{q, r}, Dist[q*r, Subst[Int[x^m*(A + B*x^n)*(a + c*x^(2*n))^p, x], x, y], x] /; !FalseQ[r = Divides[y^m, z^m, x]] && !FalseQ[q = DerivativeDivides[y, u, x]]] /; FreeQ[{a, c, A, B, m, n, p}, x] && EqQ[n2, 2*n] && EqQ[w, y]
  6704. Int[Times[Optional[Pattern[u, Blank[]]], Power[Plus[Optional[Pattern[c, Blank[]]], Times[Optional[Pattern[d, Blank[]]], Power[Pattern[v, Blank[]], Pattern[n, Blank[]]]]], Optional[Pattern[p, Blank[]]]], Power[Plus[Optional[Pattern[a, Blank[]]], Times[Optional[Pattern[b, Blank[]]], Power[Pattern[y, Blank[]], Pattern[n, Blank[]]]]], Optional[Pattern[m, Blank[]]]]], Pattern[x, Blank[Symbol]]] := With[{q = DerivativeDivides[y, u, x]}, Dist[q, Subst[Int[(a + b*x^n)^m*(c + d*x^n)^p, x], x, y], x] /; !FalseQ[q]] /; FreeQ[{a, b, c, d, m, n, p}, x] && EqQ[v, y]
  6705. Int[Times[Optional[Pattern[u, Blank[]]], Power[Plus[Optional[Pattern[c, Blank[]]], Times[Optional[Pattern[d, Blank[]]], Power[Pattern[v, Blank[]], Pattern[n, Blank[]]]]], Optional[Pattern[p, Blank[]]]], Power[Plus[Optional[Pattern[e, Blank[]]], Times[Optional[Pattern[f, Blank[]]], Power[Pattern[w, Blank[]], Pattern[n, Blank[]]]]], Optional[Pattern[q, Blank[]]]], Power[Plus[Optional[Pattern[a, Blank[]]], Times[Optional[Pattern[b, Blank[]]], Power[Pattern[y, Blank[]], Pattern[n, Blank[]]]]], Optional[Pattern[m, Blank[]]]]], Pattern[x, Blank[Symbol]]] := With[{r = DerivativeDivides[y, u, x]}, Dist[r, Subst[Int[(a + b*x^n)^m*(c + d*x^n)^p*(e + f*x^n)^q, x], x, y], x] /; !FalseQ[r]] /; FreeQ[{a, b, c, d, e, f, m, n, p, q}, x] && EqQ[v, y] && EqQ[w, y]
  6706. Int[Times[Power[Pattern[F, Blank[]], Pattern[v, Blank[]]], Pattern[u, Blank[]]], Pattern[x, Blank[Symbol]]] := With[{q = DerivativeDivides[v, u, x]}, Simp[(q*F^v)/Log[F], x] /; !FalseQ[q]] /; FreeQ[F, x]
  6707. Int[Times[Power[Pattern[F, Blank[]], Pattern[v, Blank[]]], Pattern[u, Blank[]], Power[Pattern[w, Blank[]], Optional[Pattern[m, Blank[]]]]], Pattern[x, Blank[Symbol]]] := With[{q = DerivativeDivides[v, u, x]}, Dist[q, Subst[Int[x^m*F^x, x], x, v], x] /; !FalseQ[q]] /; FreeQ[{F, m}, x] && EqQ[w, v]
  6708. Int[Times[Pattern[u, Blank[]], Power[Plus[Pattern[a, Blank[]], Times[Optional[Pattern[b, Blank[]]], Power[Pattern[v, Blank[]], Optional[Pattern[p, Blank[]]]], Power[Pattern[w, Blank[]], Optional[Pattern[p, Blank[]]]]]], Optional[Pattern[m, Blank[]]]]], Pattern[x, Blank[Symbol]]] := With[{c = Simplify[u/(w*D[v, x] + v*D[w, x])]}, Dist[c, Subst[Int[(a + b*x^p)^m, x], x, v*w], x] /; FreeQ[c, x]] /; FreeQ[{a, b, m, p}, x] && IntegerQ[p]
  6709. Int[Times[Pattern[u, Blank[]], Power[Pattern[v, Blank[]], Optional[Pattern[r, Blank[]]]], Power[Plus[Pattern[a, Blank[]], Times[Optional[Pattern[b, Blank[]]], Power[Pattern[v, Blank[]], Optional[Pattern[p, Blank[]]]], Power[Pattern[w, Blank[]], Optional[Pattern[q, Blank[]]]]]], Optional[Pattern[m, Blank[]]]]], Pattern[x, Blank[Symbol]]] := With[{c = Simplify[u/(p*w*D[v, x] + q*v*D[w, x])]}, Dist[(c*p)/(r + 1), Subst[Int[(a + b*x^(p/(r + 1)))^m, x], x, v^(r + 1)*w], x] /; FreeQ[c, x]] /; FreeQ[{a, b, m, p, q, r}, x] && EqQ[p, q*(r + 1)] && NeQ[r, -1] && IntegerQ[p/(r + 1)]
  6710. Int[Times[Pattern[u, Blank[]], Power[Pattern[v, Blank[]], Optional[Pattern[r, Blank[]]]], Power[Pattern[w, Blank[]], Optional[Pattern[s, Blank[]]]], Power[Plus[Pattern[a, Blank[]], Times[Optional[Pattern[b, Blank[]]], Power[Pattern[v, Blank[]], Optional[Pattern[p, Blank[]]]], Power[Pattern[w, Blank[]], Optional[Pattern[q, Blank[]]]]]], Optional[Pattern[m, Blank[]]]]], Pattern[x, Blank[Symbol]]] := With[{c = Simplify[u/(p*w*D[v, x] + q*v*D[w, x])]}, Dist[(c*p)/(r + 1), Subst[Int[(a + b*x^(p/(r + 1)))^m, x], x, v^(r + 1)*w^(s + 1)], x] /; FreeQ[c, x]] /; FreeQ[{a, b, m, p, q, r, s}, x] && EqQ[p*(s + 1), q*(r + 1)] && NeQ[r, -1] && IntegerQ[p/(r + 1)]
  6711. Int[Times[Pattern[u, Blank[]], Power[Plus[Times[Optional[Pattern[a, Blank[]]], Power[Pattern[v, Blank[]], Optional[Pattern[p, Blank[]]]]], Times[Optional[Pattern[b, Blank[]]], Power[Pattern[w, Blank[]], Optional[Pattern[q, Blank[]]]]]], Optional[Pattern[m, Blank[]]]]], Pattern[x, Blank[Symbol]]] := With[{c = Simplify[u/(p*w*D[v, x] - q*v*D[w, x])]}, Dist[c*p, Subst[Int[(b + a*x^p)^m, x], x, v*w^(m*q + 1)], x] /; FreeQ[c, x]] /; FreeQ[{a, b, m, p, q}, x] && EqQ[p + q*(m*p + 1), 0] && IntegerQ[p] && IntegerQ[m]
  6712. Int[Times[Pattern[u, Blank[]], Power[Pattern[v, Blank[]], Optional[Pattern[r, Blank[]]]], Power[Plus[Times[Optional[Pattern[a, Blank[]]], Power[Pattern[v, Blank[]], Optional[Pattern[p, Blank[]]]]], Times[Optional[Pattern[b, Blank[]]], Power[Pattern[w, Blank[]], Optional[Pattern[q, Blank[]]]]]], Optional[Pattern[m, Blank[]]]]], Pattern[x, Blank[Symbol]]] := With[{c = Simplify[u/(p*w*D[v, x] - q*v*D[w, x])]}, -Dist[c*q, Subst[Int[(a + b*x^q)^m, x], x, v^(m*p + r + 1)*w], x] /; FreeQ[c, x]] /; FreeQ[{a, b, m, p, q, r}, x] && EqQ[p + q*(m*p + r + 1), 0] && IntegerQ[q] && IntegerQ[m]
  6713. Int[Times[Pattern[u, Blank[]], Power[Pattern[w, Blank[]], Optional[Pattern[s, Blank[]]]], Power[Plus[Times[Optional[Pattern[a, Blank[]]], Power[Pattern[v, Blank[]], Optional[Pattern[p, Blank[]]]]], Times[Optional[Pattern[b, Blank[]]], Power[Pattern[w, Blank[]], Optional[Pattern[q, Blank[]]]]]], Optional[Pattern[m, Blank[]]]]], Pattern[x, Blank[Symbol]]] := With[{c = Simplify[u/(p*w*D[v, x] - q*v*D[w, x])]}, -Dist[(c*q)/(s + 1), Subst[Int[(a + b*x^(q/(s + 1)))^m, x], x, v^(m*p + 1)*w^(s + 1)], x] /; FreeQ[c, x]] /; FreeQ[{a, b, m, p, q, s}, x] && EqQ[p*(s + 1) + q*(m*p + 1), 0] && NeQ[s, -1] && IntegerQ[q/(s + 1)] && IntegerQ[m]
  6714. Int[Times[Pattern[u, Blank[]], Power[Pattern[v, Blank[]], Optional[Pattern[r, Blank[]]]], Power[Pattern[w, Blank[]], Optional[Pattern[s, Blank[]]]], Power[Plus[Times[Optional[Pattern[a, Blank[]]], Power[Pattern[v, Blank[]], Optional[Pattern[p, Blank[]]]]], Times[Optional[Pattern[b, Blank[]]], Power[Pattern[w, Blank[]], Optional[Pattern[q, Blank[]]]]]], Optional[Pattern[m, Blank[]]]]], Pattern[x, Blank[Symbol]]] := With[{c = Simplify[u/(p*w*D[v, x] - q*v*D[w, x])]}, -Dist[(c*q)/(s + 1), Subst[Int[(a + b*x^(q/(s + 1)))^m, x], x, v^(m*p + r + 1)*w^(s + 1)], x] /; FreeQ[c, x]] /; FreeQ[{a, b, m, p, q, r, s}, x] && EqQ[p*(s + 1) + q*(m*p + r + 1), 0] && NeQ[s, -1] && IntegerQ[q/(s + 1)] && IntegerQ[m]
  6715. Int[Times[Pattern[u, Blank[]], Power[Pattern[x, Blank[]], Optional[Pattern[m, Blank[]]]]], Pattern[x, Blank[Symbol]]] := Dist[1/(m + 1), Subst[Int[SubstFor[x^(m + 1), u, x], x], x, x^(m + 1)], x] /; FreeQ[m, x] && NeQ[m, -1] && FunctionOfQ[x^(m + 1), u, x]
  6716. Int[Pattern[u, Blank[]], Pattern[x, Blank[Symbol]]] := With[{lst = SubstForFractionalPowerOfLinear[u, x]}, Dist[lst[[2]]*lst[[4]], Subst[Int[lst[[1]], x], x, lst[[3]]^(1/lst[[2]])], x] /; !FalseQ[lst] && SubstForFractionalPowerQ[u, lst[[3]], x]]
  6717. Int[Pattern[u, Blank[]], Pattern[x, Blank[Symbol]]] := With[{lst = SubstForFractionalPowerOfQuotientOfLinears[u, x]}, Dist[lst[[2]]*lst[[4]], Subst[Int[lst[[1]], x], x, lst[[3]]^(1/lst[[2]])], x] /; !FalseQ[lst]]
  6718. Int[Times[Optional[Pattern[u, Blank[]]], Power[Times[Optional[Pattern[a, Blank[]]], Power[Pattern[v, Blank[]], Optional[Pattern[m, Blank[]]]], Power[Pattern[w, Blank[]], Optional[Pattern[n, Blank[]]]], Power[Pattern[z, Blank[]], Optional[Pattern[q, Blank[]]]]], Pattern[p, Blank[]]]], Pattern[x, Blank[Symbol]]] := Dist[(a^IntPart[p]*(a*v^m*w^n*z^q)^FracPart[p])/(v^(m*FracPart[p])*w^(n*FracPart[p])*z^(q*FracPart[p])), Int[u*v^(m*p)*w^(n*p)*z^(p*q), x], x] /; FreeQ[{a, m, n, p, q}, x] && !IntegerQ[p] && !FreeQ[v, x] && !FreeQ[w, x] && !FreeQ[z, x]
  6719. Int[Times[Optional[Pattern[u, Blank[]]], Power[Times[Optional[Pattern[a, Blank[]]], Power[Pattern[v, Blank[]], Optional[Pattern[m, Blank[]]]], Power[Pattern[w, Blank[]], Optional[Pattern[n, Blank[]]]]], Pattern[p, Blank[]]]], Pattern[x, Blank[Symbol]]] := Dist[(a^IntPart[p]*(a*v^m*w^n)^FracPart[p])/(v^(m*FracPart[p])*w^(n*FracPart[p])), Int[u*v^(m*p)*w^(n*p), x], x] /; FreeQ[{a, m, n, p}, x] && !IntegerQ[p] && !FreeQ[v, x] && !FreeQ[w, x]
  6720. Int[Times[Optional[Pattern[u, Blank[]]], Power[Times[Optional[Pattern[a, Blank[]]], Power[Pattern[v, Blank[]], Optional[Pattern[m, Blank[]]]]], Pattern[p, Blank[]]]], Pattern[x, Blank[Symbol]]] := Dist[(a^IntPart[p]*(a*v^m)^FracPart[p])/v^(m*FracPart[p]), Int[u*v^(m*p), x], x] /; FreeQ[{a, m, p}, x] && !IntegerQ[p] && !FreeQ[v, x] && !(EqQ[a, 1] && EqQ[m, 1]) && !(EqQ[v, x] && EqQ[m, 1])
  6721. Int[Times[Optional[Pattern[u, Blank[]]], Power[Plus[Optional[Pattern[a, Blank[]]], Times[Optional[Pattern[b, Blank[]]], Power[Pattern[x, Blank[]], Pattern[n, Blank[]]]]], Pattern[p, Blank[]]]], Pattern[x, Blank[Symbol]]] := Dist[(b^IntPart[p]*(a + b*x^n)^FracPart[p])/(x^(n*FracPart[p])*(1 + a/(x^n*b))^FracPart[p]), Int[u*x^(n*p)*(1 + a/(x^n*b))^p, x], x] /; FreeQ[{a, b, p}, x] && !IntegerQ[p] && ILtQ[n, 0] && !RationalFunctionQ[u, x] && IntegerQ[p + 1/2]
  6722. Int[Times[Optional[Pattern[u, Blank[]]], Power[Plus[Optional[Pattern[a, Blank[]]], Times[Optional[Pattern[b, Blank[]]], Power[Pattern[v, Blank[]], Pattern[n, Blank[]]]]], Pattern[p, Blank[]]]], Pattern[x, Blank[Symbol]]] := Dist[(a + b*v^n)^FracPart[p]/(v^(n*FracPart[p])*(b + a/v^n)^FracPart[p]), Int[u*v^(n*p)*(b + a/v^n)^p, x], x] /; FreeQ[{a, b, p}, x] && !IntegerQ[p] && ILtQ[n, 0] && BinomialQ[v, x] && !LinearQ[v, x]
  6723. Int[Times[Optional[Pattern[u, Blank[]]], Power[Plus[Optional[Pattern[a, Blank[]]], Times[Optional[Pattern[b, Blank[]]], Power[Pattern[v, Blank[]], Pattern[n, Blank[]]], Power[Pattern[x, Blank[]], Optional[Pattern[m, Blank[]]]]]], Pattern[p, Blank[]]]], Pattern[x, Blank[Symbol]]] := Dist[(a + b*x^m*v^n)^FracPart[p]/(v^(n*FracPart[p])*(b*x^m + a/v^n)^FracPart[p]), Int[u*v^(n*p)*(b*x^m + a/v^n)^p, x], x] /; FreeQ[{a, b, m, p}, x] && !IntegerQ[p] && ILtQ[n, 0] && BinomialQ[v, x]
  6724. Int[Times[Optional[Pattern[u, Blank[]]], Power[Plus[Times[Optional[Pattern[a, Blank[]]], Power[Pattern[x, Blank[]], Optional[Pattern[r, Blank[]]]]], Times[Optional[Pattern[b, Blank[]]], Power[Pattern[x, Blank[]], Optional[Pattern[s, Blank[]]]]]], Pattern[m, Blank[]]]], Pattern[x, Blank[Symbol]]] := With[{v = (a*x^r + b*x^s)^FracPart[m]/(x^(r*FracPart[m])*(a + b*x^(s - r))^FracPart[m])}, Dist[v, Int[u*x^(m*r)*(a + b*x^(s - r))^m, x], x] /; NeQ[Simplify[v], 1]] /; FreeQ[{a, b, m, r, s}, x] && !IntegerQ[m] && PosQ[s - r]
  6725. Int[Times[Pattern[u, Blank[]], Power[Plus[Pattern[a, Blank[]], Times[Optional[Pattern[b, Blank[]]], Power[Pattern[x, Blank[]], Pattern[n, Blank[]]]]], -1]], Pattern[x, Blank[Symbol]]] := With[{v = RationalFunctionExpand[u/(a + b*x^n), x]}, Int[v, x] /; SumQ[v]] /; FreeQ[{a, b}, x] && IGtQ[n, 0]
  6726. Int[Times[Pattern[u, Blank[]], Power[Plus[Optional[Pattern[a, Blank[]]], Times[Optional[Pattern[b, Blank[]]], Power[Pattern[x, Blank[]], Optional[Pattern[n, Blank[]]]]], Times[Optional[Pattern[c, Blank[]]], Power[Pattern[x, Blank[]], Optional[Pattern[n2, Blank[]]]]]], Optional[Pattern[p, Blank[]]]]], Pattern[x, Blank[Symbol]]] := Dist[1/(4^p*c^p), Int[u*(b + 2*c*x^n)^(2*p), x], x] /; FreeQ[{a, b, c, n}, x] && EqQ[n2, 2*n] && EqQ[b^2 - 4*a*c, 0] && IntegerQ[p] && !AlgebraicFunctionQ[u, x]
  6727. Int[Times[Pattern[u, Blank[]], Power[Plus[Optional[Pattern[a, Blank[]]], Times[Optional[Pattern[b, Blank[]]], Power[Pattern[x, Blank[]], Optional[Pattern[n, Blank[]]]]], Times[Optional[Pattern[c, Blank[]]], Power[Pattern[x, Blank[]], Optional[Pattern[n2, Blank[]]]]]], Pattern[p, Blank[]]]], Pattern[x, Blank[Symbol]]] := Dist[(a + b*x^n + c*x^(2*n))^p/(b + 2*c*x^n)^(2*p), Int[u*(b + 2*c*x^n)^(2*p), x], x] /; FreeQ[{a, b, c, n, p}, x] && EqQ[n2, 2*n] && EqQ[b^2 - 4*a*c, 0] && !IntegerQ[p] && !AlgebraicFunctionQ[u, x]
  6728. Int[Times[Pattern[u, Blank[]], Power[Plus[Optional[Pattern[a, Blank[]]], Times[Optional[Pattern[b, Blank[]]], Power[Pattern[x, Blank[]], Optional[Pattern[n, Blank[]]]]], Times[Optional[Pattern[c, Blank[]]], Power[Pattern[x, Blank[]], Optional[Pattern[n2, Blank[]]]]]], -1]], Pattern[x, Blank[Symbol]]] := With[{v = RationalFunctionExpand[u/(a + b*x^n + c*x^(2*n)), x]}, Int[v, x] /; SumQ[v]] /; FreeQ[{a, b, c}, x] && EqQ[n2, 2*n] && IGtQ[n, 0]
  6729. Int[Times[Optional[Pattern[u, Blank[]]], Power[Plus[Times[Optional[Pattern[a, Blank[]]], Power[Pattern[x, Blank[]], Optional[Pattern[m, Blank[]]]]], Times[Optional[Pattern[b, Blank[]]], Power[Times[Optional[Pattern[c, Blank[]]], Power[Pattern[x, Blank[]], Pattern[n, Blank[]]]], Rational[1, 2]]]], -1]], Pattern[x, Blank[Symbol]]] := Int[(u*(a*x^m - b*Sqrt[c*x^n]))/(a^2*x^(2*m) - b^2*c*x^n), x] /; FreeQ[{a, b, c, m, n}, x]
  6730. Int[Pattern[u, Blank[]], Pattern[x, Blank[Symbol]]] := With[{lst = FunctionOfLinear[u, x]}, Dist[1/lst[[3]], Subst[Int[lst[[1]], x], x, lst[[2]] + lst[[3]]*x], x] /; !FalseQ[lst]]
  6731. Int[Times[Pattern[u, Blank[]], Power[Pattern[x, Blank[]], -1]], Pattern[x, Blank[Symbol]]] := With[{lst = PowerVariableExpn[u, 0, x]}, Dist[1/lst[[2]], Subst[Int[NormalizeIntegrand[Simplify[lst[[1]]/x], x], x], x, (lst[[3]]*x)^lst[[2]]], x] /; !FalseQ[lst] && NeQ[lst[[2]], 0]] /; NonsumQ[u] && !RationalFunctionQ[u, x]
  6732. Int[Times[Pattern[u, Blank[]], Power[Pattern[x, Blank[]], Optional[Pattern[m, Blank[]]]]], Pattern[x, Blank[Symbol]]] := With[{lst = PowerVariableExpn[u, m + 1, x]}, Dist[1/lst[[2]], Subst[Int[NormalizeIntegrand[Simplify[lst[[1]]/x], x], x], x, (lst[[3]]*x)^lst[[2]]], x] /; !FalseQ[lst] && NeQ[lst[[2]], m + 1]] /; IntegerQ[m] && NeQ[m, -1] && NonsumQ[u] && (GtQ[m, 0] || !AlgebraicFunctionQ[u, x])
  6733. Int[Times[Pattern[u, Blank[]], Power[Pattern[x, Blank[]], Pattern[m, Blank[]]]], Pattern[x, Blank[Symbol]]] := With[{k = Denominator[m]}, Dist[k, Subst[Int[x^(k*(m + 1) - 1)*(u /. x -> x^k), x], x, x^(1/k)], x]] /; FractionQ[m]
  6734. Int[Pattern[u, Blank[]], Pattern[x, Blank[Symbol]]] := With[{lst = FunctionOfSquareRootOfQuadratic[u, x]}, Dist[2, Subst[Int[lst[[1]], x], x, lst[[2]]], x] /; !FalseQ[lst] && EqQ[lst[[3]], 1]] /; EulerIntegrandQ[u, x]
  6735. Int[Pattern[u, Blank[]], Pattern[x, Blank[Symbol]]] := With[{lst = FunctionOfSquareRootOfQuadratic[u, x]}, Dist[2, Subst[Int[lst[[1]], x], x, lst[[2]]], x] /; !FalseQ[lst] && EqQ[lst[[3]], 2]] /; EulerIntegrandQ[u, x]
  6736. Int[Pattern[u, Blank[]], Pattern[x, Blank[Symbol]]] := With[{lst = FunctionOfSquareRootOfQuadratic[u, x]}, Dist[2, Subst[Int[lst[[1]], x], x, lst[[2]]], x] /; !FalseQ[lst] && EqQ[lst[[3]], 3]] /; EulerIntegrandQ[u, x]
  6737. Int[Power[Plus[Pattern[a, Blank[]], Times[Optional[Pattern[b, Blank[]]], Power[Pattern[v, Blank[]], 2]]], -1], Pattern[x, Blank[Symbol]]] := Dist[1/(2*a), Int[Together[1/(1 - v/Rt[-(a/b), 2])], x], x] + Dist[1/(2*a), Int[Together[1/(1 + v/Rt[-(a/b), 2])], x], x] /; FreeQ[{a, b}, x]
  6738. Int[Power[Plus[Pattern[a, Blank[]], Times[Optional[Pattern[b, Blank[]]], Power[Pattern[v, Blank[]], Pattern[n, Blank[]]]]], -1], Pattern[x, Blank[Symbol]]] := Dist[2/(a*n), Sum[Int[Together[1/(1 - v^2/((-1)^((4*k)/n)*Rt[-(a/b), n/2]))], x], {k, 1, n/2}], x] /; FreeQ[{a, b}, x] && IGtQ[n/2, 1]
  6739. Int[Power[Plus[Pattern[a, Blank[]], Times[Optional[Pattern[b, Blank[]]], Power[Pattern[v, Blank[]], Pattern[n, Blank[]]]]], -1], Pattern[x, Blank[Symbol]]] := Dist[1/(a*n), Sum[Int[Together[1/(1 - v/((-1)^((2*k)/n)*Rt[-(a/b), n]))], x], {k, 1, n}], x] /; FreeQ[{a, b}, x] && IGtQ[(n - 1)/2, 0]
  6740. Int[Times[Power[Plus[Pattern[a, Blank[]], Times[Optional[Pattern[b, Blank[]]], Power[Pattern[u, Blank[]], Optional[Pattern[n, Blank[]]]]]], -1], Pattern[v, Blank[]]], Pattern[x, Blank[Symbol]]] := Int[ExpandIntegrand[PolynomialInSubst[v, u, x]/(a + b*x^n), x] /. x -> u, x] /; FreeQ[{a, b}, x] && IGtQ[n, 0] && PolynomialInQ[v, u, x]
  6741. Int[Pattern[u, Blank[]], Pattern[x, Blank[Symbol]]] := With[{v = NormalizeIntegrand[u, x]}, Int[v, x] /; v =!= u]
  6742. Int[Pattern[u, Blank[]], Pattern[x, Blank[Symbol]]] := With[{v = ExpandIntegrand[u, x]}, Int[v, x] /; SumQ[v]]
  6743. Int[Times[Optional[Pattern[u, Blank[]]], Power[Plus[Optional[Pattern[a, Blank[]]], Times[Optional[Pattern[b, Blank[]]], Power[Pattern[x, Blank[]], Optional[Pattern[m, Blank[]]]]]], Optional[Pattern[p, Blank[]]]], Power[Plus[Optional[Pattern[c, Blank[]]], Times[Optional[Pattern[d, Blank[]]], Power[Pattern[x, Blank[]], Optional[Pattern[n, Blank[]]]]]], Optional[Pattern[q, Blank[]]]]], Pattern[x, Blank[Symbol]]] := Dist[((a + b*x^m)^p*(c + d*x^n)^q)/x^(m*p), Int[u*x^(m*p), x], x] /; FreeQ[{a, b, c, d, m, n, p, q}, x] && EqQ[a + d, 0] && EqQ[b + c, 0] && EqQ[m + n, 0] && EqQ[p + q, 0]
  6744. Int[Times[Pattern[u, Blank[]], Power[Plus[Pattern[a, Blank[]], Times[Optional[Pattern[b, Blank[]]], Power[Pattern[x, Blank[]], Optional[Pattern[n, Blank[]]]]], Times[Optional[Pattern[c, Blank[]]], Power[Pattern[x, Blank[]], Optional[Pattern[n2, Blank[]]]]]], Pattern[p, Blank[]]]], Pattern[x, Blank[Symbol]]] := Dist[Sqrt[a + b*x^n + c*x^(2*n)]/((4*c)^(p - 1/2)*(b + 2*c*x^n)), Int[u*(b + 2*c*x^n)^(2*p), x], x] /; FreeQ[{a, b, c, n, p}, x] && EqQ[n2, 2*n] && EqQ[b^2 - 4*a*c, 0] && IntegerQ[p - 1/2]
  6745. Int[Pattern[u, Blank[]], Pattern[x, Blank[Symbol]]] := With[{lst = SubstForFractionalPowerOfLinear[u, x]}, Dist[lst[[2]]*lst[[4]], Subst[Int[lst[[1]], x], x, lst[[3]]^(1/lst[[2]])], x] /; !FalseQ[lst]]
  6746. Int[Pattern[u, Blank[]], List[Pattern[x, Blank[Symbol]], Pattern[a, Blank[]], Pattern[b, Blank[]]]] := With[{result = Int[u, x]}, Limit[result, x -> b] - Limit[result, x -> a]]
  6747. Int[List[Pattern[u, BlankSequence[]]], Pattern[x, Blank[Symbol]]] := (Int[#1, x] & ) /@ {u}
  6748. Int[Pattern[u, Blank[]], Pattern[x, Blank[]]] := CannotIntegrate[u, x]
  6749. Int[Pattern[e, Blank[]], Pattern[x, Blank[]], Pattern[flag, Alternatives[Stats, Step, Steps]]] := flag[Int[e, x]] /; (Message[Int::oldFlag, flag]; True)
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