Answer
$\frac{\tan^62x}{12} + \frac{\tan^82x}{16} +C$
Work Step by Step
Find the indefinite integral
$\int tan^52x sec^42x dx$
$\int tan^52xsec^22xsec^22xdx$, Split up the secant into two terms
$\int tan^52x(1+tan^22x) sec^22x dx$
$\int tan^52x sec^22xdx + \int tan^72x sec^22xdx$, Distribute into two integrals
Use U-substitution, let $u=tan2x$, $du=2sec^22x$
$\frac{1}{2}\int u^5du + \frac{1}{2}\int u^7du$, Integrate
$\frac{u^6}{12} + \frac{u^8}{16} +C$
$\frac{\tan^62x}{12} + \frac{\tan^82x}{16} +C$
