wolfram-mathematicaintegral

Long definite integral


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.


Solution

  • 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.