Closed form solution for $\int_{0}^{2\pi}|\cos^N x||\sin^M x|\cos^n x\sin^m x dx$ I am trying to calculate Fourier series coefficients (by hand) and the integrals I need to solve are of the following type
$$I(N,M,n,m)=\int_{0}^{2\pi}|\cos^N x||\sin^M x|\cos^n x\sin^m x dx,$$
in which $N,M,n,m \in \{0,1,2,3\}$. I tried to use WolframAlpha / Maple to come up with a general formula because doing all the $4^4$ cases would make it impossible for me to work with that list but both didn't give a result.

It would be great if there was a way to obtain a simple closed form solution. If that is not possible is there a way to get very accurate approximation for $I(N,M,n,m)$?


Hint: An easy way is to use the following identity which is quite easy to prove

which is an easy consequence of Integration, trigonometry, gamma/beta functions
