How to evaluate the integral Can some one provide me hint to evaluate  the following integral.
$$
\int\csc^{2}\left(x\right)
\ln\left(\cos\left(x\right) - \sqrt{\vphantom{\largeA}\,\cos\left(2x\right)\,}\,\right)
\,{\rm d}x
$$. 

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
