Example of integration over path on Riemann surface

The path is homotopic in the complement of the branch points $1,j,j^2$ to a large circle $|z|=R$, parametrized in the mathematically positive sense. Since the homotopy does not pass through branch points, it can be lifted to the surface $X$. Since the form is holomorphic, it is closed, so the integral only depends on the homotopy class, and it is equal to $$ \int_{|z|=R} \frac{dz}{\sqrt[3]{1-z^3}} = \int_{|z|=R} \frac{dz}{z\sqrt[3]{z^{-3}-1}} = \frac{1}{e^{\frac{i \pi}{3}}} \int_{|z|=R} \frac{dz}{z \sqrt[3]{1-z^{-3}}}$$ where I hope I got the branch of the third root right, and in the last expression it is the branch that maps $1$ to $1$. Expanding this out in a power series and using the fact that $\int\limits_{|z|=R} z^n \, dz = 0$ for $n \ne -1$ and $\int\limits_{|z|=R} z^{-1} \, dz =2\pi i$ (basically using the residue theorem at $\infty$) gives $$ \frac{1}{e^{\frac{i \pi}{3}}} \int_{|z|=R} \frac{1}z\left(1-\frac13z^{-3}\pm\ldots\right) dz = 2\pi ie^{-\frac{i \pi}{3}}$$ and I think this is the same result you got.