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 $\sum \frac{1}{(n^2+1)^2}$?



$\sum \frac{n^2-1}{(n^2+1)^2}=\frac{1}{2}(1-\frac{\pi^2}{\sinh(\pi)^2})$
This is a special case of
$$\sum\_{n=-\infty}^\infty f(n)=-\sum\_{poles\ of\ f} Res(\pi\cot(\pi z)f(z)).$$
The formula is true for rational functions for which the series converges. Proofs can be found in complex analysis texts


