Find the integral of $\tan^4(x) \sec(x)$ I want to know if there is a shorter way to find integral of $\tan^4(x) \sec(x)$
without using reduction formula of $\sec(x)$
because it's really takes a long time. 

Thanks all
Write it as
$$
\int\frac{\sin^4x}{\cos^5x}\,dx=
\int\frac{\sin^4{x}}{\cos^6x}\cos x\,dx=
\int\frac{\sin^4{x}}{(1-\sin^2x)^3}\cos x\,dx
$$
Can you see an obvious substitution?
What about $(\tan x)^3 \cdot \dfrac{1}{(\cos x)^2} \cdot \sin x$ and make integration by parts?