$\int \tan^{4}x\, dx = \int \tan^{2}x \cdot \tan^{2}x\, dx$
$= \int (\tan^{2}x)(\sec^{2}x-1)\,dx$
$= \int \tan^{2}x \cdot \sec^{2}x \,dx - \int \tan^{2}x\, dx$
$= \int u^{2}\,du - \int (\sec^{2}x-1)\, dx$
$= \frac{u^{3}}{3} + \int 1\, dx - \int \sec^{2}x\, dx$
$= \frac{\tan^{3}x}{3} + x - \tan x + C$
$= \frac{\tan^{3}x}{3} - \tan x + x + C.$