So I am evaluating the integral $\int_0^\infty\frac{z^2dz}{(z^2-4)(z^2+9)}$.
The residues for the four poles at $\pm2,\,\pm3i$ give$$\operatorname{Res}_{\pm2}(f)=\pm\frac{1}{13},\,\operatorname{Res}_{\pm3i}=\frac{\mp3i}{26}.$$
From here I am unsure of how to continue or if I have set this up correctly, I do know the expected result is $\frac{3\pi}{26}$ however I am unsure of how to get there.
Thank you in advance