I want to calculate the integral
$$ I = \int_{-\infty}^\infty dz \dfrac{1}{(z^2-a^2)^2} \dfrac{1}{(z-a)^2+b^2} $$
where $a$ and $b$ are real, positive numbers.
The poles are at $z=a$ (multiplicity 2), $z=-a$ (multiplicity 2), and $z=a \pm ib$. When plotting these on the complex plane, the $z=\pm a$ poles sit on the real axis. I have two questions:
I'm not sure how to draw the closed semicircle contour to enclose the poles on the upper/lower part of the complex plane. Do they include the $z=\pm a$ poles or not?
How do I deal with the multiplicity of the $z=\pm a$ poles? I'm familiar with residue theorem but don't know how to deal with multiplicity greater than 1.