If $∫\sqrt{\frac{cosx-cos^3x}{(1-cos^3x)}}dx= f (x) + c$, then f(x) is equal to

$\frac{2}{3}cos^{-1}(cos^{\frac{3}{2}}x)$

Let $I=∫\sqrt{\frac{cosx-cos^3x}{(1-cos^3x)}}dx=∫\frac{\sqrt{cosx}(\sqrt{1-cos^2x})}{\sqrt{1-(cos^{\frac{3}{2}x})^2}}dx$

$=∫\frac{\sqrt{cosx}(sinx)}{\sqrt{1-(cos^{3/2}x)^2}}dx$

If $cos^{\frac{3}{2}}x=p$, then $(-\frac{3}{2}cos^{\frac{1}{2}}x\,sin\,x)dx=dp$

$I = -\frac{2}{3}∫\frac{dp}{\sqrt{1-p^2}}=-\frac{2}{3}sin^{-1}(cos^{\frac{3}{2}}x)+c$

Hence (C) and (D) are the correct answers.