$\int \sqrt{\frac{\cos x-{\cos }^{3}x}{1-{\cos }^{3}x}}dx$ is equal to 

Solution:

We have,  

$\begin{array}{l}I=\int \sqrt{\frac{\cos x-{\cos }^{3}x}{1-{\cos }^{3}x}}dx=\int \sqrt{\frac{\cos x}{1-{\cos }^{3}x}}\sin xdx\\ \Rightarrow I=-\frac{2}{3}\int \frac{1}{\sqrt{1^2-{\left({\cos }^{3/2}x\right)}^2}}d\left({\cos }^{3/2}x\right)\\ \Rightarrow I=\frac{2}{3}{\cos }^{-1}\left({\cos }^{3/2}x\right)+C\end{array}$