I'm dealing with these two integrals: I 1 = ∫ − π π cos ⁡ ( x ) cos ⁡ ( n x ) ( 1 + e cos ⁡ ( x ) ) 3 d x , I 2 = ∫ − π π sin ⁡ ( x ) sin ⁡ ( n x ) ( 1 + e cos ⁡ ( x ) ) 3 d xis there a way to reduce them to a Hypergeometric form or to solve them analytically?

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I&#039;m dealing with these two integrals:      I    1    =      ∫          −      π              π                  cos      ⁡      (      x      )      cos      ⁡      (      n      x      )              (      1      +      e      cos      ⁡      (      x      )              )        3                  d    x  ,        I    2    =      ∫          −      π              π                  sin      ⁡      (      x      )      sin      ⁡      (      n      x      )              (      1      +      e      cos      ⁡      (      x      )              )        3                  d    xis there a way to reduce them to a Hypergeometric form or to solve them analytically?

bideanbarrenaf5 · Accepted Answer

In terms of regularized generalized hypergeometric functions      I    1    =            2      π              (      1      −      e              )        3                        [                3                      F        ~              2        (          3      2        ,    2    ,    3    ;    2    −    n    ,    n    +    2    ;    −                  2        e                    1        −        e              )    −          3                      F        ~              2        (          1      2        ,    1    ,    3    ;    1    −    n    ,    n    +    1    ;    −                  2        e                    1        −        e              )              ]      , provided that   ℑ  (  e  )  ≠  0  ∨  0  ≤  ℜ  (  e  )  &amp;lt;  1There is no restriction about n.

I'm dealing with these two integrals: I 1 </msub> = <msubsup> ∫<!-- ∫

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