I want to evaluate $$\oint _{|z|=1} \frac{1}{z(2 \overline z^2 +3)^2 (\overline z ^4 +2)^3} dz$$ . I initially tried using parametrization $u=e^{i \theta}$ and this led me to:
$$\int _{-\pi}^{\pi} \frac{i}{z(2 e^{-2iu} +3)^2 (e^{-4iu}+2)^3} du$$ but I'm not sure how to proceed from here.
Alternatively, I tried to see if I could somehow implement the Cauchy-Integral formula. Using $\overline z =\frac{1}{z}$, I get $$ \oint_{|z|=1} \frac{z^{15}}{(2+3z^2)^2 (1+2z^4)^3}$$ but I cannot use Cauchy Integral formula since there is no single $(z-a)$ in the denominator... I also wondered if the Cauchy-Goursat theorem is applicable but there is a singularity at $z=i\frac{2}{3^{1/2}}$ which I'm not sure how to deal with.
Any help would be greatly appreciated.