Evaluating ${\Large\int} _{-\frac{\pi}{2}}^{\frac{\pi}{2}}\frac{\sin(x)+\cos(x)}{\sin^4(x)-4}dx$ This integral is giving me hard times, could anyone "prompt" a strategy about? I tried, resultless, parameterization and some change of variables.
$${\Large\int} _{-\frac{\pi}{2}}^{\frac{\pi}{2}}\frac{\sin(x)+\cos(x)}{\sin^4(x)-4}dx$$

Remove the odd part and then set $t = \sin x$.
$$\int_{-\frac\pi2}^{\frac\pi2}\frac{\cos x}{\sin^4(x)-4}\,dx = \int_{-1}^1 \frac{dt}{t^4-4} = \frac14\int_{-1}^1 \frac{dt}{t^2-2} - \frac14\int_{-1}^1 \frac{dt}{t^2+2}$$
Can you finish?
