Multiple methods for The integral of an implicit hyperbolic sec function

Question

I already know that $\int \operatorname{sech} x\text{d}x$ is $2\tan^{-1} e^{x} + C$

I derived it with the exponential definition of the $\operatorname{sech} x$ and I've looked it up to find the same.

$\int \frac {2} {e^{x}+e^{-x}} dx = \int \frac {2e^{x}}{e^{2x}-1} dx = 2 tan^{-1}e^{x} +C$

are there any other methods? noting that i have attempted to with $\int \frac{1}{\operatorname{cosh}x}\text{d}x$ but went nowhere.

Answer 1

HINT: use that $$\cosh(x)^2-\sinh(x)^2=1$$

Answer 2

Following Dr. Sonnhard's hint, multiply both top and bottom by $\cosh x$ and then you will get $\int\frac{\cosh x}{\cosh^2 x}\text{d}x=\int\frac{\cosh x}{1+\sinh^2 x}\text{d}x\overbrace{=}^{u=\sinh x}\int\frac{1}{1+u^2}\text{d}u=2\tan^{-1}(\sinh x)+C$