It can be shown that
\[ \sum_{n=1}^\infty \frac{\coth^2 \pi n}{n^2} \; = \; \frac{P}{Q}G + \frac{R}{S}\pi^T \]
where
\[G \;=\; \sum_{n=0}^\infty \frac{(-1)^n}{(2n+1)^2} \;=\; 0.915965594\ldots \]
is Catalan's constant, and where \(P,Q,R,S\) and \(T\) are all positive integers with \(P,Q\) and \(R,S\) coprime pairs. Find \(P+Q+R+S+T\).

Inspiration.