How would one compute the following integral using a contour in the complex plane? I know how to perform this integral by other means such as differentiation under the integration and series expansion.
$$\int_0^\infty \frac{\ln x}{(1+e^{x})^2}dx$$
One may try the full circle with a slit along the x-axis. However, it is hard to argue for the vanishing of the integral on the left semicircle at infinity.