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