I'm trying to compue the following integral on the interval $[0,\pi]$ on the $\theta$ variable.
(1/(4 Sqrt[2] r0^4 (Cos[\[Theta]]^6 + Sin[\[Theta]]^6)^(
13/6)))(5 + 3 Cos[4 \[Theta]])^(
1/6) (4 (Cos[\[Theta]]^6 +
Sin[\[Theta]]^6) (a15 r0^5 Cos[\[Theta]]^6 +
r0^4 Cos[\[Theta]]^5 ((a16 + b15) r0 Sin[\[Theta]] +
a10 (Cos[\[Theta]]^6 + Sin[\[Theta]]^6)^(1/6)) +
r0^3 Cos[\[Theta]]^4 ((a17 + b16) r0^2 Sin[\[Theta]]^2 + (a11 +
b10) r0 Sin[\[Theta]] (Cos[\[Theta]]^6 +
Sin[\[Theta]]^6)^(1/6) +
a6 (Cos[\[Theta]]^6 + Sin[\[Theta]]^6)^(1/3)) +
r0^2 Cos[\[Theta]]^3 ((a18 + b17) r0^3 Sin[\[Theta]]^3 + (a12 +
b11) r0^2 Sin[\[Theta]]^2 (Cos[\[Theta]]^6 +
Sin[\[Theta]]^6)^(
1/6) + (a7 + b6) r0 Sin[\[Theta]] (Cos[\[Theta]]^6 +
Sin[\[Theta]]^6)^(1/3) +
a3 Sqrt[Cos[\[Theta]]^6 + Sin[\[Theta]]^6]) +
r0 Cos[\[Theta]]^2 ((a19 + b18) r0^4 Sin[\[Theta]]^4 + (a13 +
b12) r0^3 Sin[\[Theta]]^3 (Cos[\[Theta]]^6 +
Sin[\[Theta]]^6)^(
1/6) + (a8 + b7) r0^2 Sin[\[Theta]]^2 (Cos[\[Theta]]^6 +
Sin[\[Theta]]^6)^(1/3) + (a4 + b3) r0 Sin[\[Theta]] Sqrt[
Cos[\[Theta]]^6 + Sin[\[Theta]]^6] +
a1 (Cos[\[Theta]]^6 + Sin[\[Theta]]^6)^(2/3)) +
Cos[\[Theta]] ((a20 + b19) r0^5 Sin[\[Theta]]^5 + (a14 +
b13) r0^4 Sin[\[Theta]]^4 (Cos[\[Theta]]^6 +
Sin[\[Theta]]^6)^(
1/6) + (a9 + b8) r0^3 Sin[\[Theta]]^3 (Cos[\[Theta]]^6 +
Sin[\[Theta]]^6)^(
1/3) + (a5 + b4) r0^2 Sin[\[Theta]]^2 Sqrt[
Cos[\[Theta]]^6 +
Sin[\[Theta]]^6] + (a2 +
b1) r0 Sin[\[Theta]] (Cos[\[Theta]]^6 + Sin[\[Theta]]^6)^(
2/3) + a0 (Cos[\[Theta]]^6 + Sin[\[Theta]]^6)^(5/6)) +
Sin[\[Theta]] (b20 r0^5 Sin[\[Theta]]^5 +
b14 r0^4 Sin[\[Theta]]^4 (Cos[\[Theta]]^6 +
Sin[\[Theta]]^6)^(1/6) +
b9 r0^3 Sin[\[Theta]]^3 (Cos[\[Theta]]^6 + Sin[\[Theta]]^6)^(
1/3) + b5 r0^2 Sin[\[Theta]]^2 Sqrt[
Cos[\[Theta]]^6 + Sin[\[Theta]]^6] +
b2 r0 Sin[\[Theta]] (Cos[\[Theta]]^6 + Sin[\[Theta]]^6)^(
2/3) + b0 (Cos[\[Theta]]^6 + Sin[\[Theta]]^6)^(
5/6))) - (b15 r0^5 Cos[\[Theta]]^6 +
r0^4 Cos[\[Theta]]^5 ((-a15 + b16) r0 Sin[\[Theta]] +
b10 (Cos[\[Theta]]^6 + Sin[\[Theta]]^6)^(1/6)) +
r0^3 Cos[\[Theta]]^4 ((-a16 +
b17) r0^2 Sin[\[Theta]]^2 + (-a10 +
b11) r0 Sin[\[Theta]] (Cos[\[Theta]]^6 +
Sin[\[Theta]]^6)^(1/6) +
b6 (Cos[\[Theta]]^6 + Sin[\[Theta]]^6)^(1/3)) +
r0^2 Cos[\[Theta]]^3 ((-a17 +
b18) r0^3 Sin[\[Theta]]^3 + (-a11 +
b12) r0^2 Sin[\[Theta]]^2 (Cos[\[Theta]]^6 +
Sin[\[Theta]]^6)^(
1/6) + (-a6 + b7) r0 Sin[\[Theta]] (Cos[\[Theta]]^6 +
Sin[\[Theta]]^6)^(1/3) +
b3 Sqrt[Cos[\[Theta]]^6 + Sin[\[Theta]]^6]) +
r0 Cos[\[Theta]]^2 ((-a18 + b19) r0^4 Sin[\[Theta]]^4 + (-a12 +
b13) r0^3 Sin[\[Theta]]^3 (Cos[\[Theta]]^6 +
Sin[\[Theta]]^6)^(
1/6) + (-a7 + b8) r0^2 Sin[\[Theta]]^2 (Cos[\[Theta]]^6 +
Sin[\[Theta]]^6)^(1/3) + (-a3 + b4) r0 Sin[\[Theta]] Sqrt[
Cos[\[Theta]]^6 + Sin[\[Theta]]^6] +
b1 (Cos[\[Theta]]^6 + Sin[\[Theta]]^6)^(2/3)) -
Sin[\[Theta]] (a20 r0^5 Sin[\[Theta]]^5 +
a14 r0^4 Sin[\[Theta]]^4 (Cos[\[Theta]]^6 +
Sin[\[Theta]]^6)^(1/6) +
a9 r0^3 Sin[\[Theta]]^3 (Cos[\[Theta]]^6 + Sin[\[Theta]]^6)^(
1/3) + a5 r0^2 Sin[\[Theta]]^2 Sqrt[
Cos[\[Theta]]^6 + Sin[\[Theta]]^6] +
a2 r0 Sin[\[Theta]] (Cos[\[Theta]]^6 + Sin[\[Theta]]^6)^(
2/3) + a0 (Cos[\[Theta]]^6 + Sin[\[Theta]]^6)^(5/6)) +
Cos[\[Theta]] ((-a19 + b20) r0^5 Sin[\[Theta]]^5 + (-a13 +
b14) r0^4 Sin[\[Theta]]^4 (Cos[\[Theta]]^6 +
Sin[\[Theta]]^6)^(
1/6) + (-a8 + b9) r0^3 Sin[\[Theta]]^3 (Cos[\[Theta]]^6 +
Sin[\[Theta]]^6)^(
1/3) + (-a4 + b5) r0^2 Sin[\[Theta]]^2 Sqrt[
Cos[\[Theta]]^6 +
Sin[\[Theta]]^6] + (-a1 +
b2) r0 Sin[\[Theta]] (Cos[\[Theta]]^6 + Sin[\[Theta]]^6)^(
2/3) + b0 (Cos[\[Theta]]^6 + Sin[\[Theta]]^6)^(5/6))) Sin[
4 \[Theta]])
But mathematica doesn't answer, I try to split the expresion, but also doesn't works. So could anyboydy give a hint to compute this integral?
I split the expresion on the following way
1/(4 Sqrt[2] r0^4 (Cos[\[Theta]]^6 + Sin[\[Theta]]^6)^(
13/6)) (5 + 3 Cos[4 \[Theta]])^(
1/6) (4 (Cos[\[Theta]]^6 +
Sin[\[Theta]]^6) (a15 r0^5 Cos[\[Theta]]^6 +
r0^4 Cos[\[Theta]]^5 ((a16 + b15) r0 Sin[\[Theta]] +
a10 (Cos[\[Theta]]^6 + Sin[\[Theta]]^6)^(1/6)) +
r0^3 Cos[\[Theta]]^4 ((a17 + b16) r0^2 Sin[\[Theta]]^2 + (a11 +
b10) r0 Sin[\[Theta]] (Cos[\[Theta]]^6 +
Sin[\[Theta]]^6)^(1/6) +
a6 (Cos[\[Theta]]^6 + Sin[\[Theta]]^6)^(1/3)) +
r0^2 Cos[\[Theta]]^3 ((a18 + b17) r0^3 Sin[\[Theta]]^3 + (a12 +
b11) r0^2 Sin[\[Theta]]^2 (Cos[\[Theta]]^6 +
Sin[\[Theta]]^6)^(
1/6) + (a7 + b6) r0 Sin[\[Theta]] (Cos[\[Theta]]^6 +
Sin[\[Theta]]^6)^(1/3) +
a3 Sqrt[Cos[\[Theta]]^6 + Sin[\[Theta]]^6]) +
r0 Cos[\[Theta]]^2 ((a19 + b18) r0^4 Sin[\[Theta]]^4 + (a13 +
b12) r0^3 Sin[\[Theta]]^3 (Cos[\[Theta]]^6 +
Sin[\[Theta]]^6)^(
1/6) + (a8 + b7) r0^2 Sin[\[Theta]]^2 (Cos[\[Theta]]^6 +
Sin[\[Theta]]^6)^(1/3) + (a4 + b3) r0 Sin[\[Theta]] Sqrt[
Cos[\[Theta]]^6 + Sin[\[Theta]]^6] +
a1 (Cos[\[Theta]]^6 + Sin[\[Theta]]^6)^(2/3)) +
Cos[\[Theta]] ((a20 + b19) r0^5 Sin[\[Theta]]^5 + (a14 +
b13) r0^4 Sin[\[Theta]]^4 (Cos[\[Theta]]^6 +
Sin[\[Theta]]^6)^(
1/6) + (a9 + b8) r0^3 Sin[\[Theta]]^3 (Cos[\[Theta]]^6 +
Sin[\[Theta]]^6)^(
1/3) + (a5 + b4) r0^2 Sin[\[Theta]]^2 Sqrt[
Cos[\[Theta]]^6 +
Sin[\[Theta]]^6] + (a2 +
b1) r0 Sin[\[Theta]] (Cos[\[Theta]]^6 + Sin[\[Theta]]^6)^(
2/3) + a0 (Cos[\[Theta]]^6 + Sin[\[Theta]]^6)^(5/6)) +
Sin[\[Theta]] (b20 r0^5 Sin[\[Theta]]^5 +
b14 r0^4 Sin[\[Theta]]^4 (Cos[\[Theta]]^6 +
Sin[\[Theta]]^6)^(1/6) +
b9 r0^3 Sin[\[Theta]]^3 (Cos[\[Theta]]^6 + Sin[\[Theta]]^6)^(
1/3) + b5 r0^2 Sin[\[Theta]]^2 Sqrt[
Cos[\[Theta]]^6 + Sin[\[Theta]]^6] +
b2 r0 Sin[\[Theta]] (Cos[\[Theta]]^6 + Sin[\[Theta]]^6)^(
2/3) + b0 (Cos[\[Theta]]^6 + Sin[\[Theta]]^6)^(5/6))) -
(b15 r0^5 Cos[\[Theta]]^6 +
r0^4 Cos[\[Theta]]^5 ((-a15 + b16) r0 Sin[\[Theta]] +
b10 (Cos[\[Theta]]^6 + Sin[\[Theta]]^6)^(1/6)) +
r0^3 Cos[\[Theta]]^4 ((-a16 +
b17) r0^2 Sin[\[Theta]]^2 + (-a10 +
b11) r0 Sin[\[Theta]] (Cos[\[Theta]]^6 +
Sin[\[Theta]]^6)^(1/6) +
b6 (Cos[\[Theta]]^6 + Sin[\[Theta]]^6)^(1/3)) +
r0^2 Cos[\[Theta]]^3 ((-a17 +
b18) r0^3 Sin[\[Theta]]^3 + (-a11 +
b12) r0^2 Sin[\[Theta]]^2 (Cos[\[Theta]]^6 +
Sin[\[Theta]]^6)^(
1/6) + (-a6 + b7) r0 Sin[\[Theta]] (Cos[\[Theta]]^6 +
Sin[\[Theta]]^6)^(1/3) +
b3 Sqrt[Cos[\[Theta]]^6 + Sin[\[Theta]]^6]) +
r0 Cos[\[Theta]]^2 ((-a18 + b19) r0^4 Sin[\[Theta]]^4 + (-a12 +
b13) r0^3 Sin[\[Theta]]^3 (Cos[\[Theta]]^6 +
Sin[\[Theta]]^6)^(
1/6) + (-a7 + b8) r0^2 Sin[\[Theta]]^2 (Cos[\[Theta]]^6 +
Sin[\[Theta]]^6)^(1/3) + (-a3 + b4) r0 Sin[\[Theta]] Sqrt[
Cos[\[Theta]]^6 + Sin[\[Theta]]^6] +
b1 (Cos[\[Theta]]^6 + Sin[\[Theta]]^6)^(2/3)) -
Sin[\[Theta]] (a20 r0^5 Sin[\[Theta]]^5 +
a14 r0^4 Sin[\[Theta]]^4 (Cos[\[Theta]]^6 +
Sin[\[Theta]]^6)^(1/6) +
a9 r0^3 Sin[\[Theta]]^3 (Cos[\[Theta]]^6 + Sin[\[Theta]]^6)^(
1/3) + a5 r0^2 Sin[\[Theta]]^2 Sqrt[
Cos[\[Theta]]^6 + Sin[\[Theta]]^6] +
a2 r0 Sin[\[Theta]] (Cos[\[Theta]]^6 + Sin[\[Theta]]^6)^(
2/3) + a0 (Cos[\[Theta]]^6 + Sin[\[Theta]]^6)^(5/6)) +
Cos[\[Theta]] ((-a19 + b20) r0^5 Sin[\[Theta]]^5 + (-a13 +
b14) r0^4 Sin[\[Theta]]^4 (Cos[\[Theta]]^6 +
Sin[\[Theta]]^6)^(
1/6) + (-a8 + b9) r0^3 Sin[\[Theta]]^3 (Cos[\[Theta]]^6 +
Sin[\[Theta]]^6)^(
1/3) + (-a4 + b5) r0^2 Sin[\[Theta]]^2 Sqrt[
Cos[\[Theta]]^6 +
Sin[\[Theta]]^6] + (-a1 +
b2) r0 Sin[\[Theta]] (Cos[\[Theta]]^6 + Sin[\[Theta]]^6)^(
2/3) + b0 (Cos[\[Theta]]^6 + Sin[\[Theta]]^6)^(5/6))) Sin[
4 \[Theta]])
And compute each integral, but doesn't work. for instance in the second block the answer that mathematica give me is in gray and not in black Why? I don't know.
Here is a partial, maybe even a complete, solution.
v=Expand[...your huge expression...]
That turns your huge expression into the sum of 126 p/q smaller simpler expressions.
Then
Map[Integrate[#,{\[Theta],0,Pi}]&,Take[v,20]]
will happily fairly quickly integrate the sum of the first 20 p/q expressions and
Map[Integrate[#,{\[Theta],0,Pi}]&,Take[v,-6]]
will try to integrate the sum of the last six p/q expressions.
You can look at the 49th p/q expression using
v[[49]]
or look at the sum of the first 20 p/q expressions using
Take[v,20]
Some of those p/q are far easier and faster to integrate than others.
But maybe, if you wait long enough, and you are very very lucky then
Map[Integrate[#,{\[Theta],0,Pi}]&,v]
will give you the the integral of your entire expression, but at least you can see some progress on this if you integrate subsets of your complete expression.
I am guessing that the reason your attempt at splitting your integrand into three parts didn't work is that the ( before the 4 near the top of your third line that begins with 1/6) (4..., that ( only matches the final ) which is at the end of the last line in your third part. Does this make any sense?
Good luck. And test this very carefully to try to make sure I haven't made any mistakes.