I'm not sure how to solve this integral:
$$ \int_\gamma \frac{dz}{(e^z+4)(z-\pi i)}, $$
where $ \gamma$ is the region $ \|z-\sqrt7i\|+\|z+\sqrt7i\|=8$.
I know that region is the ellipse $\frac{x^2}{9}+\frac{y^2}{16}=1 $ but I don't know how to solve the integral. I tried to use Taylor series but got nothing.