I want to find $$\int_0^1\frac{1}{(1-x^3)^{1/3}}$$ using contour integral where the contour is obtained by connecting three small circles around $\zeta^0,\zeta,\zeta^2$ ($\zeta$ is the 3rd root of unity) with straight lines and oriented positively. This contour looks like a fidget spinner containing the three branch cuts that are line segments from $\zeta^i$ to $0$.
I don't know which branch I should choose to compute the contour integral. Also, the integral from $0$ to $1$ will cancel the integral from $1$ to $0$. What's the remedy for that?