I'm having trouble finding the definite integral:
$$\int_{-\infty}^{\infty}\cos(x)\operatorname{sech}(x)dx$$
I know the answer is $\pi\operatorname{sech}(\frac{\pi}{2})$. There isn't an indefinite integral in terms of elementary functions. The indefinite integral given by Wolfram Alpha is in terms of hypergeometric functions, but I wanted to know if you could get the answer without that. Maybe some differentiating under the integral jazz? Or maybe series. Not too sure where to start, but I would appreciate an explanation. Also, I personally don't know Complex Analysis, but if that's the only way, that wouldn't be a bad answer for others on this site.
