Answer
See the result below.
Work Step by Step
We need to verify both sides of the expression. In order to do this, we will differentiate both sides.
$\dfrac{d}{dx} ( \int x tanh^{-1} x dx)=\dfrac{d}{dx} ( x \tanh^{-1} x +\dfrac{1}{2} \ln (1-x^2) + C)$
or, $ tanh^{-1} x = x(\dfrac{1}{1-x^2})+(\tanh^{-1} x) (1)+\dfrac{1}{2}(\dfrac{-2x}{1-x^2})+(0)$
or, $tanh^{-1} x = tanh^{-1} x $
Hence, the result has been verified.