A trigo integration How to evaluate $\displaystyle\int_{-1}^{1}\frac{1}{1-e^x\sqrt{1-x^2}}dx$
I have to tried $x=\sin t$ or $\cos t$
but I can't solve $\displaystyle\int_{0}^{\pi}\frac{1}{1-e^{\cos t}}dt$
can anyone give me some hints or steps ?
Thanks for your help! ^^

Near $x=0$,
$$e^x \sqrt{1-x^2} \sim 1+x \qquad (x\to 0)$$
The integrand has a singularity that behaves as $1/x$ near $x=0$.  Thus, the integral diverges.
