Evaluate the integral $\int x^{\frac{-4}{3}}(-x^{\frac{2}{3}}+1)^{\frac{1}{2}}\mathrm dx$

As suggested in RecklessReckoner's comment, let us first change variable $x=u^3$ then $dx=3u^2du$. So, $$I=\int \frac{\sqrt{1-x^{2/3}}}{x^{4/3}}dx=3\int\frac{ \sqrt{1-u^2}}{u^2}du$$ Now, $u=\sin(t)$, $du=\cos(t)\,dt$ makes $$I=3\int \cot ^2(t)\,dt=3\int \frac{1-\sin^2(t)}{\sin^2(t)}dt =3\Big(\int \frac{dt}{\sin^2(t)}-\int dt \Big)=-3 \big(t+\cot (t)\big)$$ For sure, we could have saved a step with a single change of variable $x=\sin^3(t)$.