How To Integrate 1/1-cos x

 

$$\begin{array}{l}\int \frac{1}{1-cosx}\mathrm{d}x\\ \\ \int \frac{1}{1-\cos x}\ast \frac{1+\cos x}{1+\cos x}\mathrm{d}x\\ \int \frac{1+\cos x}{1-{\cos }^{2}x}\mathrm{d}x\\ \int \frac{1+\cos x}{{\sin }^{2}x}\mathrm{d}x\\ \int ({\csc }^{2}x+\frac{\cos x}{{\sin }^{2}x})\mathrm{d}x\\ \int {\csc }^{2}x\mathrm{d}x+\int \cot x\csc xdx\\ \Rightarrow -\cot x-\csc x+c\end{array}$$