Integration of certain real functions using Euler's Formula.

Question

I've heard about using Euler's formula $$e^{ix}=\cos(x)+i\sin(x)$$ to transform rational functions of sine and cosine into computable indefinite integrals. However, upon attempting to apply this myself, I ran into some trouble. I'm asking for someone to give a run-down of the method and idea, specifically I need help with the following integral $$\int \frac{a+\sin(x)}{b+\cos(x)} {dx}$$ My attempt to use THIS method to solve this integral is as follows. First, I recognized that $$\sin(x)=\frac{e^{ix}-e^{-ix}}{2i}=-i\sinh(ix)$$ and $$\cos(x)=\frac{e^{ix}+e^{-ix}}{2}=\cosh(ix)$$ then I set $u=e^{ix}$ so ${du}=ie^{ix} \ {dx}=iu \ {dx}$ and finally arrived at the following expression in terms of u $$-\int\frac{u^2+2aiu-1}{u^3+2bu^2+u}{du}$$ I am not sure how to proceed from here, my only idea is polynomial long division and I hesitate to undertake such a messy path if there was a better way to do this.