Indefinite integral $ \int \cos x \sqrt{\cos2x}dx$

Question

$$ \int \cos x \sqrt{\cos2x}dx $$

I know that $\cos 2x = 1-2\sin^2x,$ but not sure if it will help me?

Integrate by parts?

$$ \sin x \sqrt{1-2\sin^{2}x}-\int \sin x \frac{-\sin2x}{\sqrt{\cos2x}}dx $$ This doesn't seem to get me anywhere?

Answer 1

Hint: write $\cos2x = 1 - 2\sin^2 x$, and let $u = \sin xdx$, then $du = \cos xdx$. You can take it from here.