-\int \left(1+\frac{1}{u^3-1}\right)^{-\frac{1}{3}}\left(u^3-1\right)^{-\frac{4}{3}}u^2\text{du}=-\int \left(\frac{u^3}{u^3-1}\right)^{-\frac{1}{3}}\left(u^3-1\right)^{-\frac{4}{3}}u^2\text{du}=-\int u^{-1}\left(u^3-1\right)^{\frac{1}{3}}\left(u^3-1\right)^{-\frac{4}{3}}u^2\text{du}=-\int \left(u^3-1\right)^{-1}u \text{du}=-\int \frac{u}{u^3-1}\text{du}