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- expr = c14 (a0+a1 Cos[n x]+a3 Cos[2 n x]+a5 Cos[3 n x]+a7 Cos[4 n x]+a2 Sin[n x]+a4 Sin[2 n x]+a6 Sin[3 n x]+a8 Sin[4 n x]) (e1 Cos[n x]+e3 Cos[2 n x]+e5 Cos[3 n x]+e7 Cos[4 n x]+e2 Sin[n x]+e4 Sin[2 n x]+e6 Sin[3 n x]+e8 Sin[4 n x])+c44 (e1 Cos[n x]+e3 Cos[2 n x]+e5 Cos[3 n x]+e7 Cos[4 n x]+e2 Sin[n x]+e4 Sin[2 n x]+e6 Sin[3 n x]+e8 Sin[4 n x])^2+c66 (j0+j1 Cos[n x]+j3 Cos[2 n x]+j5 Cos[3 n x]+j7 Cos[4 n x]+j2 Sin[n x]+j4 Sin[2 n x]+j6 Sin[3 n x]+j8 Sin[4 n x])^2+c11 (1/2 (h0+h1 Cos[n x]+h3 Cos[2 n x]+h2 Sin[n x]+h4 Sin[2 n x])^2+3/2 (a0+a1 Cos[n x]+a3 Cos[2 n x]+a5 Cos[3 n x]+a7 Cos[4 n x]+a2 Sin[n x]+a4 Sin[2 n x]+a6 Sin[3 n x]+a8 Sin[4 n x])^2+1/2 (d0+d1 Cos[n x]+d3 Cos[2 n x]+d5 Cos[3 n x]+d7 Cos[4 n x]+d2 Sin[n x]+d4 Sin[2 n x]+d6 Sin[3 n x]+d8 Sin[4 n x])^2)+c14 ((b0+b1 Cos[n x]+b3 Cos[2 n x]+b5 Cos[3 n x]+b7 Cos[4 n x]+b2 Sin[n x]+b4 Sin[2 n x]+b6 Sin[3 n x]+b8 Sin[4 n x]) (d0+d1 Cos[n x]+d3 Cos[2 n x]+d5 Cos[3 n x]+d7 Cos[4 n x]+d2 Sin[n x]+d4 Sin[2 n x]+d6 Sin[3 n x]+d8 Sin[4 n x])+2 (a0+a1 Cos[n x]+a3 Cos[2 n x]+a5 Cos[3 n x]+a7 Cos[4 n x]+a2 Sin[n x]+a4 Sin[2 n x]+a6 Sin[3 n x]+a8 Sin[4 n x]) (e1 Cos[n x]+e3 Cos[2 n x]+e5 Cos[3 n x]+e7 Cos[4 n x]+e2 Sin[n x]+e4 Sin[2 n x]+e6 Sin[3 n x]+e8 Sin[4 n x])+(h0+h1 Cos[n x]+h3 Cos[2 n x]+h2 Sin[n x]+h4 Sin[2 n x]) (g0+g1 Cos[n x]+g3 Cos[2 n x]+g5 Cos[3 n x]+g7 Cos[4 n x]+g2 Sin[n x]+g4 Sin[2 n x]+g6 Sin[3 n x]+g8 Sin[4 n x]))+c12 (1/2 (b0+b1 Cos[n x]+b3 Cos[2 n x]+b5 Cos[3 n x]+b7 Cos[4 n x]+b2 Sin[n x]+b4 Sin[2 n x]+b6 Sin[3 n x]+b8 Sin[4 n x])^2+1/2 (e1 Cos[n x]+e3 Cos[2 n x]+e5 Cos[3 n x]+e7 Cos[4 n x]+e2 Sin[n x]+e4 Sin[2 n x]+e6 Sin[3 n x]+e8 Sin[4 n x])^2+1/2 (g0+g1 Cos[n x]+g3 Cos[2 n x]+g5 Cos[3 n x]+g7 Cos[4 n x]+g2 Sin[n x]+g4 Sin[2 n x]+g6 Sin[3 n x]+g8 Sin[4 n x])^2)+c13 (1/2 (c0+c1 Cos[n x]+c3 Cos[2 n x]+c2 Sin[n x]+c4 Sin[2 n x])^2+1/2 (f0+f1 Cos[n x]+f3 Cos[2 n x]+f5 Cos[3 n x]+f7 Cos[4 n x]+f2 Sin[n x]+f4 Sin[2 n x]+f6 Sin[3 n x]+f8 Sin[4 n x])^2+1/2 (j0+j1 Cos[n x]+j3 Cos[2 n x]+j5 Cos[3 n x]+j7 Cos[4 n x]+j2 Sin[n x]+j4 Sin[2 n x]+j6 Sin[3 n x]+j8 Sin[4 n x])^2);
- Integrate[expr, {x, 0, 2 Pi}]
- a12 = Cos[n x] (c14 (a0+a1 Cos[n x]+a3 Cos[2 n x]+a5 Cos[3 n x]+a7 Cos[4 n x]+a2 Sin[n x]+a4 Sin[2 n x]+a6 Sin[3 n x]+a8 Sin[4 n x]) (e1 Cos[n x]+e3 Cos[2 n x]+e5 Cos[3 n x]+e7 Cos[4 n x]+e2 Sin[n x]+e4 Sin[2 n x]+e6 Sin[3 n x]+e8 Sin[4 n x])+c44 (e1 Cos[n x]+e3 Cos[2 n x]+e5 Cos[3 n x]+e7 Cos[4 n x]+e2 Sin[n x]+e4 Sin[2 n x]+e6 Sin[3 n x]+e8 Sin[4 n x])^2+c66 (j0+j1 Cos[n x]+j3 Cos[2 n x]+j5 Cos[3 n x]+j7 Cos[4 n x]+j2 Sin[n x]+j4 Sin[2 n x]+j6 Sin[3 n x]+j8 Sin[4 n x])^2+c11 (1/2 (h0+h1 Cos[n x]+h3 Cos[2 n x]+h2 Sin[n x]+h4 Sin[2 n x])^2+3/2 (a0+a1 Cos[n x]+a3 Cos[2 n x]+a5 Cos[3 n x]+a7 Cos[4 n x]+a2 Sin[n x]+a4 Sin[2 n x]+a6 Sin[3 n x]+a8 Sin[4 n x])^2+1/2 (d0+d1 Cos[n x]+d3 Cos[2 n x]+d5 Cos[3 n x]+d7 Cos[4 n x]+d2 Sin[n x]+d4 Sin[2 n x]+d6 Sin[3 n x]+d8 Sin[4 n x])^2)+c14 ((b0+b1 Cos[n x]+b3 Cos[2 n x]+b5 Cos[3 n x]+b7 Cos[4 n x]+b2 Sin[n x]+b4 Sin[2 n x]+b6 Sin[3 n x]+b8 Sin[4 n x]) (d0+d1 Cos[n x]+d3 Cos[2 n x]+d5 Cos[3 n x]+d7 Cos[4 n x]+d2 Sin[n x]+d4 Sin[2 n x]+d6 Sin[3 n x]+d8 Sin[4 n x])+2 (a0+a1 Cos[n x]+a3 Cos[2 n x]+a5 Cos[3 n x]+a7 Cos[4 n x]+a2 Sin[n x]+a4 Sin[2 n x]+a6 Sin[3 n x]+a8 Sin[4 n x]) (e1 Cos[n x]+e3 Cos[2 n x]+e5 Cos[3 n x]+e7 Cos[4 n x]+e2 Sin[n x]+e4 Sin[2 n x]+e6 Sin[3 n x]+e8 Sin[4 n x])+(h0+h1 Cos[n x]+h3 Cos[2 n x]+h2 Sin[n x]+h4 Sin[2 n x]) (g0+g1 Cos[n x]+g3 Cos[2 n x]+g5 Cos[3 n x]+g7 Cos[4 n x]+g2 Sin[n x]+g4 Sin[2 n x]+g6 Sin[3 n x]+g8 Sin[4 n x]))+c12 (1/2 (b0+b1 Cos[n x]+b3 Cos[2 n x]+b5 Cos[3 n x]+b7 Cos[4 n x]+b2 Sin[n x]+b4 Sin[2 n x]+b6 Sin[3 n x]+b8 Sin[4 n x])^2+1/2 (e1 Cos[n x]+e3 Cos[2 n x]+e5 Cos[3 n x]+e7 Cos[4 n x]+e2 Sin[n x]+e4 Sin[2 n x]+e6 Sin[3 n x]+e8 Sin[4 n x])^2+1/2 (g0+g1 Cos[n x]+g3 Cos[2 n x]+g5 Cos[3 n x]+g7 Cos[4 n x]+g2 Sin[n x]+g4 Sin[2 n x]+g6 Sin[3 n x]+g8 Sin[4 n x])^2)+c13 (1/2 (c0+c1 Cos[n x]+c3 Cos[2 n x]+c2 Sin[n x]+c4 Sin[2 n x])^2+1/2 (f0+f1 Cos[n x]+f3 Cos[2 n x]+f5 Cos[3 n x]+f7 Cos[4 n x]+f2 Sin[n x]+f4 Sin[2 n x]+f6 Sin[3 n x]+f8 Sin[4 n x])^2+1/2 (j0+j1 Cos[n x]+j3 Cos[2 n x]+j5 Cos[3 n x]+j7 Cos[4 n x]+j2 Sin[n x]+j4 Sin[2 n x]+j6 Sin[3 n x]+j8 Sin[4 n x])^2))
- Clear[a, b];
- iRules = {(Cos | Sin)[_]^2 -> 1/2, (Cos | Sin)[_] -> 0};
- ia11 = 2 Pi Expand[a11] /. iRules; // AbsoluteTiming
- (* {0.010274, Null} *)
- evalRules = { (* assumes n is an Integer *)
- Cos[(a_Integer: 1) n Pi] :> (-1)^a,
- Sin[(b_Integer: 1) n Pi] :> 0
- };
- ja11 = Integrate[a11, {x, 0, 2 Pi}] /. evalRules; // AbsoluteTiming
- (* {177.270803, Null} *)
- ia11 == Expand[ja11] // Simplify
- (* True *)
- ka11 = 2 Pi Expand[TrigToExp@a11] /. E^_ -> 0; // AbsoluteTiming
- (* {0.205734, Null} *)
- ka11 == Expand[ja11 /. evalRules] // Simplify
- (* True *)
- 2 Pi TrigReduce[a11] /. iRules; // AbsoluteTiming
- (* {0.086483, Null} *)
- iRules =(*Dispatch@*)
- {Cos[(a_Integer: 1) n x] Sin[(a_Integer: 1) n x] ->
- Integrate[Cos[a n x] Sin[a n x], {x, 0, 2 Pi}]/(2 Pi),
- Cos[(a_Integer: 1) n x] Sin[(b_Integer: 1) n x] /; a == -b ->
- Integrate[Cos[a n x] Sin[-a n x], {x, 0, 2 Pi}]/(2 Pi),
- Cos[(a_Integer: 1) n x] Sin[(b_Integer: 1) n x] ->
- Integrate[Cos[a n x] Sin[b n x], {x, 0, 2 Pi}]/(2 Pi),
- Sin[(a_Integer: 1) n x] Sin[(b_Integer: 1) n x] ->
- Integrate[Cos[a n x] Sin[b n x], {x, 0, 2 Pi}]/(2 Pi),
- Cos[(a_Integer: 1) n x] Cos[(b_Integer: 1) n x] ->
- Integrate[Cos[a n x] Sin[b n x], {x, 0, 2 Pi}]/(2 Pi),
- Cos[(a_Integer: 1) n x]^2 ->
- Integrate[Cos[a n x]^2, {x, 0, 2 Pi}]/(2 Pi),
- Sin[(b_Integer: 1) n x]^2 ->
- Integrate[Sin[b n x]^2, {x, 0, 2 Pi}]/(2 Pi),
- Cos[(a_Integer: 1) n x] ->
- Integrate[Cos[a n x], {x, 0, 2 Pi}]/(2 Pi),
- Sin[(b_Integer: 1) n x] ->
- Integrate[Sin[b n x], {x, 0, 2 Pi}]/(2 Pi)};
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