Polygamma sum problem

Question

I have a problem evaluating the following sum, $$\sum_{n=1}^{+\infty}\frac{4nx(3\pi ^{2}(n+1)^{2}+x^{2})}{(x^{2}-\pi ^{2}(n+1)^{2})^{3}}$$ The sum obviously is of the form of a polygamma function. What i think is the right path is to do partial fractions and write it in the form $$\frac{4nx(3\pi ^{2}(n+1)^{2}+x^{2})}{(x^{2}-\pi ^{2}(n+1)^{2})^{3}}=\frac{A}{(x-\pi (n+1)))^{3}}+\frac{B}{(x+\pi (n+1))^{3}}$$ But funnily enough i have a problem finding what A and B are. Also im not 100% sure if this is the right way, but i think it is. Please help me if you can!

Answer 1

Hint

Write $$4nx(3\pi ^{2}(n+1)^{2}+x^{2})=A (x+\pi (n+1))^3+B (x-\pi (n+1))^3$$ Make $x=-\pi (n+1)$ to get $$-16 \pi ^3 n (n+1)^3=-8 \pi ^3 (n+1)^3 B$$ which does not look too difficult.

Do the same with $x=\pi (n+1)$ to get $A$