Problem:: $\int_{\Gamma} \dfrac{z^2+z^{-2}}{(z^*-r_1)(r_2-z^*)}dz$ where $\Gamma= \{ z: |z|=r \}$ ($r_2>r>r_1>0$).
Question: How to solve this integral?
My attempt: My first idea is to use residue theorem but this function is not holomorphic (so I can not use residue theorem). If I try to simplify this integral i get: $$ \int_{\Gamma} \dfrac{z^4+1}{z^2(z^*-r_1)(r_2-z^*)}dz $$ but I can not get anything useful from this (as far as I can see). I tried to parametrize this integral with $z=re^{i\theta}$ and I get: $$\dfrac{i}{r} \int_{0}^{2\pi} \frac{(r^4e^{4i\theta}+1)e^{i\theta} d\theta }{e^{2i\theta}(re^{-i\theta}-r_1)(r_2-re^{-i\theta})} $$ and this is not useful (as far as I can see). I have no idea how to proceed to solve the problem.
Thank you for any help.
Hints: first replace $z^{*} by \frac {r^{2}} z$ (on $\Gamma$, $zz^{*}=r^{2}$). Once you do this the given function takes a new form which has poles at $r^{2} /r_1$ and $r^{2}/r_2$. Find the residue at these points and apply Residue Theorem.