I need to calculate the following integral
$$\int_{-1}^1\frac{(1+z)^{1/4}(1-z)^{3/4}}{1+z}\, dz$$
I know how to do it by real-analysis method but I am supposed to use Cauchy theorem. But the residues at $z=-1$ and $\infty$ are zero. Am I doing something wrong with the branches?