Definite integral on $[0,\pi]$ How to calculate the following integral if $\varepsilon \in (0,1)$:
$$\int \limits_{0}^{\pi}\frac{d\varphi}{(1+\varepsilon\cos \varphi)^2}$$

Hint:
Use the substitution
$$
s=\tan{\frac{\varphi}{2}}, \quad \sin{\varphi}=\frac{2s}{s^2+1}, \quad \cos{\varphi} = \frac{1-s^2}{s^2+1}.
$$
(Apparently more details are given in this answer.)
