10.1.14 Integral of a complex variable

Problem: Calculate the integral of a complex variable along a closed loop: \[ \int_{|z+5|=2}\left(z \sin \frac{1}{z+5}+\frac{2 \operatorname{ch}\left(\frac{\pi i z}{4}\right)}{(z+4)^{2}(z+2)}\right) d z . \]