Prove $\sum_{n=1}^{\infty}\sum_{m=1}^{\infty}\frac{m^2-n^2}{(m^2+n^2)^2}=\frac{\pi}{4}$

Question

Prove $\sum_{n=1}^{\infty}\sum_{m=1}^{\infty}\frac{m^2-n^2}{(m^2+n^2)^2}=\frac{\pi}{4}$

can anyone show me how to prove this since it seems to not be able to swap the summation because $\sum_{n=1}^{\infty}\sum_{m=1}^{\infty}|\frac{m^2-n^2}{(m^2+n^2)^2}| $may not converge.