Answer
$$ \displaystyle\int \dfrac{1}{{\left(\sin\left(x\right)\right)}^{3}}\, \mathrm d x = -{{\cos ^2x\,\ln \left(\cos x+1\right)-\ln \left(\cos x+1\right)- \cos ^2x\,\ln \left(\cos x-1\right)+\ln \left(\cos x-1\right)-2\, \cos x}\over{4\,\left(\cos x-1\right)\,\left(\cos x+1\right)}} $$
Explanation
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