I have this integral:
$$\int_{|z|=2}\frac{\cosh z}{(z+1)^3(z-1)}dz$$
Both singularities $z=1,z=-1$ are inside the circle. I have already solve this using partial fractions, and I don't have much problem with that (I got $-\frac{\pi i}e$). Now I want to solve it with the deformation theorem, but I'm having trouble with it. I say the hypothesis are satisfied, if we consider $\Gamma$ as the circle $|z|=2$, and $f(z)=\frac{\cosh z}{(z+1)^3(z-1)}$ where $f$ is analytic in $\Bbb C\setminus \{-1,1\}$, so how do I find $C^1$ curves that work? I have considered the curves $|z-1|=1/4$ and $|z+1|=1/4$ to isolate the singularities, but what next?