How about this example?
$\displaystyle \int \tan 5x dx$
How can a double angle formula be used in this situation? hint?
I would use u-sub u= 5x and write $\displaystyle tan(u)$ as $\displaystyle \frac{sin(u)}{cos(u)}$
the double angle would be $\displaystyle \displaystyle \int \frac{2tan(\frac{5}{2}x)}{1-tan^2(\frac{5}{2}x)}dx$ I'm not interested in solving it this way.
One more time: u= 5x gives a very simple integral.
(If your problem had been $\displaystyle \int sin^2(x) dx$, $\displaystyle \int cos^4(x) dx$, and $\displaystyle \int tan^5(x) dx$ that would be a different matter.)
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...