Write a value

let $\tan x=t$

Differentiating on both sides we get,

$\sec ^{2} x d x=d t$

Substituting above equation in $\int \tan ^{3} x \sec ^{2} x d x$ we get,

$=\int t^{6} d t$

$=\frac{t^{7}}{7}+c$

$=\frac{\tan ^{7} x}{7}+c$