Integral with sine, cosine, and rational function

1. Nov 17, 2015

Ravendark

1. The problem statement, all variables and given/known data
I would like to compute the following integral:
$$I = \int\limits_0^\pi \mathrm{d}\theta \, \frac{\sin^2 \theta}{a^2 + b^2 - 2 \sqrt{ab} \cos \theta}$$
where $a,b \in \mathbb{R}_+$.

2. The attempt at a solution
Substitution $x = \cos \theta$ yields
$$I = \int\limits_{-1}^1 \mathrm{d}x \, \frac{\sqrt{1 - x^2}}{a^2 + b^2 - 2 \sqrt{ab} \, x} \; .$$
Now I dont know how to proceed. I have in mind to use the residue theorem somehow, but I dont know if this is applicable here. Can someone give me a hint, please?

2. Nov 17, 2015

Ray Vickson

Assuming that a,b > 0, Maple 11 gets the integral as