Can one use contour integration to evaluate $\int^{\pi}_{0} \frac{1}{1-\rho*sin(\theta)}d\theta$ for $0<\rho<1$? This would be trivial if the upper limit were $2\pi$ as we could let $z=e^{i\theta}$, and do the usual integral around the full unit circle. Here, integral is just over a semi-circle, and due to asymmetry, we cant simply do $\frac{1}{2}\int^{\pi}_{0}f(\theta)d\theta$.

Note: Based on another approach (using properties of multivariate normal distribution), I have reason to believe that this integral equals $\frac{\pi+2\tan^{-1}(\sqrt{\frac{\rho}{1-\rho^2}})}{\sqrt{1-\rho^2}}$, but I would like to compute this from first principles.