Rectangular contour for an exponential beta integral (${\int }_{-\infty }^{\infty }\frac{e^{ax}}{1+e^x}dx=\frac{\pi }{\sin (\pi a)}$)

For $0<a<1$, integrate $e^{az}/(1+e^z)$ around a rectangle of imaginary height $2\pi$. Its single pole at $\pi i$ has residue $-e^{\pi ia}$, the top-edge integral is $-e^{2\pi ia}$ times the bottom-edge integral, and both vertical sides vanish at large real width. The residue theorem gives ${\int }_{\mathbb{R}}{e}^{ax}/(1+e^x)dx=\pi /\sin (\pi a)$. Substitution $t=e^x$ gives the equivalent integral ${\int }_0^{\infty }{t}^{a-1}/(1+t)dt$.