I'm trying to find the integral of a rational function f(x):
$$\int_{-\infty}^{\infty}f(x)dx=\int_{-\infty}^{\infty}\frac{1}{x-\lambda x^2}\left[a+\frac{b}{c-d\cos(sx)}\right]\mathrm{d}x$$
Here, $a$, $b$, and $c$ are constants such that $c-d\cos(sx)>0$. Therefore, the singularities are $x = 0$ and $x = 1/\lambda$ on real axis.
Let:
$$P(x) = a+\frac{b}{c-d\cos(sx)}$$
and
$$Q(x) = x-\lambda x^2$$
For singularity $x = 0$, we have:
$$\underset{x=0}{\mathrm{Res}}f(x)=\left.\frac{P(x)}{Q'(x)}\right|_{x=0} = \left.\frac{a+\frac{b}{c-d\cos(sx)}}{1-2\lambda x}\right|_{x=0} =a+\frac{b}{c - d}$$
For singularity $x = \frac{1}{\lambda}$, we have:
$$\underset{x=\frac{1}{\lambda}}{\mathrm{Res}}f(x)=\left.\frac{P(x)}{Q'(x)}\right|_{x=\frac{1}{\lambda}}= \left.\frac{a+\frac{b}{c-d\cos(sx)}}{1-2\lambda x}\right|_{x=\frac1\lambda}=-\left[a+\frac{b}{c - d\cos{(s/\lambda)}}\right]$$
Using residue theory, we have:
$$\int_{-\infty}^{\infty}f(x)\mathrm{d}x={\pi}i \sum \text{Res} f(x)={\pi}i\left[\frac{b}{c-d} - \frac{b}{c-d\cos{(s/\lambda)}}\right]$$
Now, it becomes a complex function instead of a rational function. I think something must be wrong with these calculations. Any help to find out the mistake would be appreciated.