Analytical Integration of Arctanh x

We can integrate  
\[tanh^{-1} x\]
  by parts by writing  
\[tanh^{-1}x=1 \times tanh^{-1}x\]
.
Let  
\[u=tanh^{-1}x \rightarrow tanhu=x \rightarrow sech^2u \frac{du}{dx}=1\]
  then
\[\frac{du}{dx}= \frac{1}{sech^2u} =\frac{1}{1-tanh^2u}=\frac{1}{1-x^2}\]
.
\[\frac{dv}{dx}=1 \rightarrow v=x\]

\[\begin{equation} \begin{aligned} \int 1 \times tanh^{-1}xdx &= x tanh^{-1}x - \int x \times \frac{1}{1-x^2}dx \\ &= xtanh^{-1}x+ \frac{1}{2} ln(1-x^2)+c \end{aligned} \end{equation}\]

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