I'm trying to find the analytical form of the following integral:
$\int_0^{2\pi}\left|\frac{(1-i\cos(\phi))}{(1+a-i\cos(\phi))}\right|^2d\phi$
I've tried in Wolfram Alpha, which ran out of free calculation time, and in Mathematica I get the original expression for the indefinite integral and the program stays in the "Running..." state for several minutes, when evaluating the definite integral, and I stopped it.
However, I am reading a paper, which says there is an analytical expression for this integral, and also to me it seems rather straightforward. Any hints?