Need help in solving this indefinite integral.

Question

I tried solving $$\int{\sqrt{\frac{\cos(x) - \cos^3 (x)}{1 - \cos^3(x)}}}\, dx$$ And was able to reduce it to $$\int{\sqrt{\frac{\cos (x) + \cos^2(x)}{1 + \cos(x) + \cos^2(x)}}}\, dx$$ I guess i got myself landed on another planet! Thankful if anyone helps.

Answer 1

Hint: Put $\cos x=t$

You'll obtain, after simplification, $$\int{\sqrt{\frac{t}{1-t^3}}}dt$$

Now, put $t^{3/2}=a$ and proceed to obtain an expression in $\sin^{-1}$.