Evaluation of the indefinite integral $\int \frac{\operatorname{sech}^2(x)}{ \operatorname{sech} ^2(x)+1} \,dx$

Question

$\newcommand{\sech}{\operatorname{sech}}$I have simplied $$\int \frac{\sech^2(x)}{ \sech ^2(x)+1} \, dx$$ to $$\int\frac 2 {\cosh(2x)+3} \,dx$$ and am wondering if this is the most efficient way of answering the question and where to go from here.

Answer 1

hint

using the substitution $$t=\tanh (x) $$ with $$dt=(1-t^2)dx $$ and $$\cosh(2x)=\frac {1+t^2}{1-t^2} $$

the integrale becomes $$\int \frac {dt}{2-t^2} $$ to finish, put $t=u\sqrt {2} $ and observe that $$\frac{2}{1-u^2}=\frac {1}{1-u}+\frac {1}{1+u} $$