How about this example?
How can a double angle formula be used in this situation? hint?
This is easiest done using a Pythagorean Identity:
$\displaystyle \begin{align*} \int{ \tan^2{(5x)}\,\mathrm{d}x} &= \int{ \sec^2{(5x)} - 1 \, \mathrm{d}x} \end{align*}$
You should be able to go from here...