How to evaluate the integral Can some one provide me hint to evaluate  the following integral.
$$
\int\csc^{2}\left(x\right)
\ln\left(\cos\left(x\right) - \sqrt{\vphantom{\largeA}\,\cos\left(2x\right)\,}\,\right)
\,{\rm d}x
$$. 
 A: In fact with some trigonometry, we can simplify the integrand, in the formula of David H. (after the integration by parts step):
$$
\cot x \cdot\frac{-\sin x+\frac{\sin 2x }{\sqrt{\cos 2x }}}{\cos{ x}-\sqrt{\cos 2x }}=
\frac{\cos x}{\sin^2x\cdot\sqrt{1-2\sin^2 x}}+\cot^2x
$$
This yields the following
$$
\int \csc^2(x)\log(\cos{ x}-\sqrt{\cos 2x }) dx=-\cot x\cdot\log(\cos{x}-\sqrt{\cos 2x })-\cot x-x-\frac{\sqrt{\cos 2x}}{\sin x}
$$
A: Hint: integration by parts is a great candidate for a first step here. Using $\int\csc^2{x}\,dx=-\cot{x}$ and $$\frac{d}{dx}\log{\left(\cos{x}-\sqrt{\cos{2x}}\right)}=\frac{-\sin{x}+\frac{\sin{2x}}{\sqrt{\cos{2x}}}}{\cos{x}-\sqrt{\cos{2x}}},$$
we arrive at:
$$\int\csc^2{x}\log{\left(\cos{x}-\sqrt{\cos{2x}}\right)}dx\\
=-\cot{x}\log{\left(\cos{x}-\sqrt{\cos{2x}}\right)}-\int\cot{x}\cdot\frac{\sin{x}-\frac{\sin{2x}}{\sqrt{\cos{2x}}}}{\cos{x}-\sqrt{\cos{2x}}}dx.$$
A: MAPLE returns
$$-\cot(x)\ln(\cos(x)-\sqrt{\cos(2x)})-\cot \left( x \right)  +x-\sqrt { \left( -2\,
 \left( \cos \left( x \right)  \right) ^{2}+1 \right)  \left(  \left( 
\cos \left( x \right)  \right) ^{2}-1 \right) } \left( -1/2\,{\frac {-
2\, \left( \cos \left( x \right)  \right) ^{3}+2\, \left( \cos \left( 
x \right)  \right) ^{2}+\cos \left( x \right) -1}{\sqrt { \left( -\cos
 \left( x \right) -1 \right)  \left( 2\, \left( \cos \left( x \right) 
 \right) ^{3}-2\, \left( \cos \left( x \right)  \right) ^{2}-\cos
 \left( x \right) +1 \right) }}}+1/2\,{\frac {-2\, \left( \cos \left( 
x \right)  \right) ^{3}-2\, \left( \cos \left( x \right)  \right) ^{2}
+\cos \left( x \right) +1}{\sqrt { \left( 1-\cos \left( x \right) 
 \right)  \left( 2\, \left( \cos \left( x \right)  \right) ^{3}+2\,
 \left( \cos \left( x \right)  \right) ^{2}-\cos \left( x \right) -1
 \right) }}} \right)  \left( \sin \left( x \right)  \right) ^{-1}{
\frac {1}{\sqrt {2\, \left( \cos \left( x \right)  \right) ^{2}-1}}}$$
It seems you have a tricky integral on your hands. If I find a substitution that returns this I'll post
