I need to solve the following integral:
$I_1 = \int_0^{2\pi} d \varphi \ \cos^m(\varphi) \frac{J_4\left(A\sqrt{a^2+b^2+2ab \cos \varphi}\right)}{a^2+b^2+2ab \cos \varphi}$ with $m$ positive integer and $a$, $b$ and $A$ positive real.
Looking through Gradshteyn (7th ed.) the closest one I could find was 6.684.1. (which sadly is not very close...).
Are you aware/have intuition of wheather there might be an analytic solution to/simpler form of $I_1$?
Thank you!