The following series evaluation

$\sum \frac{n^2-1}{(n^2+1)^2}=\frac{1}{2}(1-\frac{\pi^2}{\sinh(\pi)^2})$

seems attractive to me, and has a proof related to the evaluation of $\zeta(2)$.

Does this identity (and/or it's many variants) occur in the literature? If so, in what context?

Also, does there exist a closed form for $\sum \frac{1}{(n^2+1)^2}$?