how to integrate this

(sec x)(tan x)^2

\int sec(x)tan^{2}(x)dx

Rewrite:

\int (\underbrace{sec^{2}(x)-1}_{\text{tan^2(x)}})sec(x)dx

\int sec^{3}(x)dx-\int sec(x)dx

Now, integrate each one. The first one can be done by just using the reduction formula. The second is just ln|sec(x)+tan(x)|+C, and can be derived by letting u=sec(x)+tan(x), \;\ du=(sec^{2}(x)+sec(x)tan(x))dx

Reduction formula:

\int sec^{n}(x)dx=\frac{sec^{n-2}(x)tan(x)}{n-1}+\frac{n-2}{n-1}\int sec^{n-2}(x)dx

The reduction formula can be derived by using integration by parts.