Hyperbolic Integral 
Integrate the following:$$\int(\tanh x)^3 dx$$

I tried to solve it like a regular tan function and it did not work for me.
Could you please walk me through the process?

$$\int\tanh^3 x dx =\int \dfrac{\sinh^3 x}{\cosh^3 x} dx =-\int \dfrac{\sinh x(1-\cosh^2 x)}{\cosh^3 x} dx.$$
Take $u =\cosh x$.
$$\int\tanh^3 x \, dx=-\int\frac{1-u^2}{u^3}du=\int\left(\frac{1}{u}-\frac{1}{u^3}\right)du=\ln u+\frac{1}{2u^2}+c.$$
Back substitute to get the integral,
$$\ln u+\frac{1}{2u^2}+c=\ln(\cosh x)+\frac{1}{2 \cosh^2 x}+c.$$
