I am completely stuck with the evaluation of the following integral :
$I = \int_{-\infty}^{\infty} \frac{\sinh(x)}{\sinh(ax)}dx, a>1.$
I am supposed to use a rectangle, such that the bounds are $-R < Re(z) < R$ and $0< Im(z)<A$, with an indentation.
I have found that there are poles at $z_k = \frac{i \pi k}{a}$, where $k$ is an integer. I am, however, stuck.