I try to show $$\sum _{k=1}^{\infty } \frac{e^{-2 k} k}{e^{-2 k}+1}=\frac{\pi ^2}{48}-\frac{\pi ^2-6 \left(\psi _{e^{\frac{\pi ^2}{2}}}^{(1)}(1)+\psi _{e^{\frac{\pi ^2}{2}}}^{(1)}\left(\frac{-2 i+\pi }{\pi }\right)-\psi _{e^{\frac{\pi ^2}{2}}}^{(1)}\left(\frac{-i+\pi }{\pi }\right)-\psi _{e^{\frac{\pi ^2}{2}}}^{(1)}\left(\frac{i+\pi }{\pi }\right)\right)}{24 \pi ^2}$$ numerically seem to fit maybe using mellin transform