I want to show that $$ \int_{-\infty}^{\infty} \frac{\sinh ax}{\sinh \pi x} \, \cos bx \, dx = \frac{\sin a}{\cos a + \cosh b} \, , \quad -\pi < a < \pi . $$
I tried integrating $f \displaystyle (z) = e^{ibz} \, \frac{\sinh az}{\sinh \pi z} $ around an indented rectangular contour with vertices at $\pm R, \pm R+ i$.
I ended up with the equation $$ (1+ e^{-b} \cos a) \int_{-\infty}^{\infty} \frac{\sinh ax}{\sinh \pi x} \, \cos bx \, dx - e^{-b}\sin a \int_{-\infty}^{\infty} \frac{\cosh ax}{\sinh \pi x} \, \sin bx \ dx = e^{-b} \sin a . $$
But I don't see how to extract the value of the integral from this equation.
Did I integrate the wrong function?
