I mean the Laurent series at $s=1$.
I want to do it by proving $\displaystyle \int_0^\infty \frac{2t}{(t^2+1)(e^{\pi t}+1)} dt = \ln 2 - \gamma$,
based on the integral formula given in Wikipedia. But I cannot solve this integral except by using Mathematica. Tried complex analytic ways but no luck. Any suggestions? Thanks for your attention!