OK, here goes:
Let \(\displaystyle u=sec(x)\),\(\displaystyle dv=sec^{2}(x)dx\), \(\displaystyle du=sec(x)tan
(x)dx\), \(\displaystyle v=tan(x)\).
\(\displaystyle \L\int{sec^{3}(x)}dx=sec(x)tan(x)-\int{sec(x)tan^{2}(x)dx}\)
=\(\displaystyle \L\sec(x)tan(x)-\int{sec(x)(sec^{2}(x)-1)dx\)
=\(\displaystyle \L\sec(x)tan(x)-\int{sec^{3}(x)dx}+\int{sec(x)dx\),
\(\displaystyle \L\2\int{sec^{3}(x)dx=sec(x)tan(x)+\int{sec(x)dx\),
Therefore,
\(\displaystyle \L\int{sec^{3}(x)}dx=\frac{1}{2}sec(x)tan(x)+\frac{1}{2}\int{sec(x)dx\)
=\(\displaystyle \L\int{sec^{3}(x)dx=\frac{1}{2}sec(x)tan(x)+\frac{1}{2}ln(|sec(x)+tan(x)|)+C\)
