Calculate $\int_{|z|=2}\frac{1}{\sqrt{z^4+4z+1}}$

Question

Calculate $\int_{|z|=2}\frac{1}{\sqrt{z^4+4z+1}}$ when $\sqrt{25}=5$
So i know that when i look at $\sqrt{z^4+4z+1}=z^2\sqrt{1+4/z^3+1/z^4}$ the transformation under the root is moving $z$ to a circle with center at $z=1$ and radius less than $1$ and i can take the main branch for the log.
I got stuck after it at calculating the integral any help please?

Answer 1

Note that all roots of the quadric have modulus less than $2$. Now either expand the path of integration to $\vert z \rvert = R$ for $ R\geq 2$ and consider $R\to \infty$ or make the substitution $z \leftarrow z^{-1}$ and verify that the integrand is then holomorphic on the disc $\lvert z \rvert \leq \tfrac12$.