How To Integrate 1/1-cos x

 

\begin{array}{l} \smallint \frac{1}{{1 - cosx}}{\rm{d}}x\\\\ \smallint \frac{1}{{1 - \cos x}}\; * \;\frac{{1 + \cos x}}{{1 + \cos x}}{\rm{d}}x\\ \smallint \;\frac{{1 + \cos x}}{{1 - {{\cos }^2}x}}{\rm{d}}x\\ \smallint \;\frac{{1 + \cos x}}{{{{\sin }^2}x}}{\rm{d}}x\\ \smallint (\;\;{\csc ^2}x + \frac{{\cos x}}{{{{\sin }^2}x}}\;\;){\rm{d}}x\\ \smallint \;\;{\csc ^2}x\;\;{\rm{d}}x + \smallint \cot x\csc xdx\\ \Rightarrow - \cot x - \csc x + c \end{array}