Integration Problem with a Trig substitution Okay I am a little stuck on this problem.
$$\int \tan^5(x)\sqrt{\sec(x)} \; dx$$
What should be my first step for a u sub or a trig sub? I have tried to use $u=\sec(x)$ and then $u=\tan(x)$, but I get stuck.  A little help?

Let $u=\sec x$. Then $du=\sec x\tan x\,dx$.  Substitute. With part of the substitution done, we are looking at
$$\int \frac{\tan^5 x}{\sec x\tan x}u^{1/2}\,du.$$
Using the fact that $\tan^2 x=\sec^2 x-1$, we end up with 
$$\int \frac{(u^2-1)^2}{u^{1/2}}\,du,$$
that is,
$$\int (u^{7/2}-2u^{3/2}+u^{-1/2})\,du.$$
