Find $\int_\gamma \frac{dz}{(z-\frac{1}{2}-i)(z-1-\frac{3i}{2})(z-1-\frac{i}{2})(z-\frac{3}{2}-i)}\, $ Let $f(z)=\frac{1}{[(z-\frac{1}{2}-i)(z-1-\frac{3i}{2})(z-1-\frac{i}{2})(z-\frac{3}{2}-i)]}$ and let $\gamma$ be the polygon $[0,2,2+2i,2i,0]$. Find $\int_{\gamma}^{} f$ .
I'm trying to use the partial fractions decomposition method, but it's getting too long and I'm lost in the accounts. I do not know if the author expects me to do so. Can anybody help me?
Conway, pg. 96, prob., 7.

By the residue theorem: $\int_{\gamma}f(z)dz=2\pi i\sum_i\textrm{res}_{z_i}$. So the problem essentially is to evaluate the residue of each pole. The poles are at:
$1/2+i$, $1+3i/2$, $1+i/2$, $3/2+i$.
Simply check which are in your boundary and evaluate the residues.
