Answer
$$\frac{1}{3}{\left( {\operatorname{sech} x} \right)^3} + C$$
Work Step by Step
$$\eqalign{
& \int {{{\operatorname{sech} }^3}x\tanh x} dx \cr
& {\text{Split, recall that }}{a^m}{a^n} = {a^{m + n}} \cr
& \int {{{\operatorname{sech} }^3}x\tanh x} dx = \int {{{\operatorname{sech} }^2}x\operatorname{sech} x\tanh x} dx \cr
& {\text{Let }}u = \operatorname{sech} x,{\text{ }}du = \operatorname{sech} x\tanh xdx \cr
& {\text{Substituting}} \cr
& \int {{{\operatorname{sech} }^2}x\operatorname{sech} x\tanh x} dx = \int {{u^2}} du \cr
& {\text{Integrating}} \cr
& {\text{ = }}\frac{1}{3}{u^3} + C \cr
& {\text{Write in terms of }}x \cr
& = \frac{1}{3}{\left( {\operatorname{sech} x} \right)^3} + C \cr} $$
