Given a square $\gamma$ with vertices at $\pm 2,\pm2i$, evaluate the following integral
$$\int_{\gamma}\frac{z}{\sqrt{1+z+z^4}}dz$$
I know that I can look at a circle with radius 2 and center at 0 which encloses the square and I want to see what the transformation $(1+1/z^3+1/z^4)$ does to the circle.