cotsum[x_] /; QuadraticIrrationalQ[x] :=
 Module[{a1 = x, list, l, r, i, nlist, f, solution, output, equation,
   string}, list = ContinuedFraction[a1];
  l = Length[list] - 1;
  r = Length[list[[l + 1]]];
  i = 1; string = "{";
  While[i < l + r + 1, string = string <> "x" <> ToString[i] <> ",";
   i++]; string = StringTake[string, StringLength[string] - 1] <> "}";
  Do[Evaluate[ToExpression["a" <> ToString[i + 1]]] =
    FromContinuedFraction[Drop[list, i]], {i, 1, l - 1}];
  nlist = list[[l + 1]];
  Do[Evaluate[ToExpression["a" <> ToString[i + l + 1]]] =
    FromContinuedFraction[{Flatten[
       Append[Drop[nlist, i], Take[nlist, i]]]}], {i, 0, r - 1}];
  f[a_] := (1 - 5 a^2 + a^4)/90/a;
  equation =
   Table[ToExpression[
     "x" <> ToString[i + 1] <> "+a" <> ToString[i + 1] <> "^2*x" <>
      ToString[i] <> "==f[a" <> ToString[i + 1] <> "]"], {i, 1,
     r + l - 1}];
  equation =
   Append[equation,
    ToExpression[
     "x" <> ToString[l + 1] <> "+a" <> ToString[l + 1] <> "^2*x" <>
      ToString[r + l] <> "==f[a" <> ToString[l + 1] <> "]"]];
  solution = Solve[equation, ToExpression[string]]; Clear["a*"];
  output = (ToExpression[string][[1]] /. Flatten[solution])*Pi^3;
  FullSimplify[output]]