I was interested in solving for $\eta'(1)$ where $\eta(s)$ is Dirichlet eta function, and I am now able to expand it like this based on the fact that $\displaystyle\Gamma(s)\eta(s)=\int_0^\infty{t^{s-1}\over e^t+1}\mathrm{d}t$:
$$ \eta'(1)=\gamma\ln(2)+\int_0^\infty{\ln(t)\over e^t+1}\mathrm{d}t $$
I would like to evaluate the remaining integral via contour integration, and I have failed to find a proper contour for it (I have tried half circle and branch cuts but failed), so could anybody in MSE provide me some hints for finding the integral.
NOTE: In fact, I have read $\int_0^{\infty } \frac{\log (x)}{e^x+1} \, dx = -\frac{1}{2} \log ^2(2)$ How to show? before asking this question. Instead of series method, I am looking for an approach that solves this integral by contour integration, so I wonder if anybody could come up some contours that solve this.
