Answer
$$ \displaystyle\int \dfrac{1+\cos\left(x\right)}{1-\cos\left(x\right)}\, \mathrm d x = -{{2\,\left(\sin x\,\arctan \left({{\sin x}\over{\cos x+1}}\right)+ \cos x+1\right)}\over{\sin x}} $$
Explanation
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