\displaystyle\int_{C}\frac{\left(\frac{z^2+1}{2z}\right)^2}
{\frac{5}{4}+\frac{z^2-1}{2iz}}\cdot\frac{dz}{iz}=
\int_{C}\frac{(z^2+1)^2}
{z^2(z+2i)(2z+i)}dz