One of several ways to evaluate $$\int_{0}^{\infty} \frac{\sinh (ax)}{\sinh (\pi x)} \, \cos (bx) \, dx \, , \quad \, |a|< \pi,$$ is to sum the residues of $$ f(z) = \frac{\sinh (az)}{\sinh (\pi z)} \,e^{ibz}$$ in the upper half-plane.
But if you restrict $b$ to positive values, how do show that $\int f(z) \, dz$ vanishes along the right, left, and upper sides of a rectangle with vertices at $\pm N, \pm N + i\left(N+\frac{1}{2} \right)$ as $N \to \infty$ through the positive integers?
I think we can use the M-L inequality (in combination with the triangle and reverse triangle inequalities) to show that that integral vanishes along the vertical sides of the rectangle.
But showing that the integral vanishes along the top of the rectangle seems a bit tricky.