I am attempting to solve the following definite integral $$\int_{-\infty}^{+\infty}\frac{\sin{(\cosh{x})}\cosh{x}}{1+\cosh^{2}{x}}dx.$$
I was trying passing to the complex plane and use the residue theorem, but I did not manage to find a satisfying contour. The half-circle seem to not work because the poles accumulate at infinity, while a rectangle does not because the integrand diverges on the vertical sides.
Which one is the right approach to tackle this problem?