Hello everyone! I need to calculate the following integral:

$\int\int_{\sqrt{x^2+y^2}\leq a}\cos^2(v_x x+v_y y)dxdy$

First step, I convert to polar coordinates:

$\int_0^a\int_0^{2\pi}\cos^2(v_x \rho \cos\theta+v_y \rho\sin\theta)\rho d \rho d\theta$

Then...what? Any suggestions?

Mathematica manages to find a closed form solution, but it is very ugly (4-5 terms involving Hypergeometric Regularised functions and Struve functions). I am wondering if a more elegant solution exists, given the apparent simplicity of the integral.

Notice that I can also use $|\cos(v_x x+v_y y)|$ instead of $\cos^2(v_x x+v_y y)$, but my guess is that it will only get more complicated. If needed, I can also change the domain of integration, as long as the integral does not change with rotation of the vector $[v_x,v_y]$ (a condition that is satisfied by my integral above).

Cheers!