I want to convert $$ \int_{-\pi}^\pi \frac {d\theta}{(1+\sin^2\theta)} $$ to a contour integral. I know that I can use the substitution $z=\cos\theta + i\sin\theta = e^{i\theta}$ to get $\sin\theta = \frac {(z^2 -1)}{2zi}$, and $d\theta = \frac{dz}{zi}$
What I cannot understand is how to choose the contour curve to do the integration. Some help will be very useful.