how to calculate the value of $\int_{-\infty}^\infty \frac{e^{ax}}{\cosh x}\,dx$

The usual trick for this integral is to take a rectangular contour with vertices $\pm R$ and $\pm R+\pi i$ and let $R\to\infty$. This contains only one pole, simple at $z=\pi i/2$, and the integral over the top edge is closely related to that over the bottom edge (the one you care about).