Analytical Integration of Arccosech x

We can integrate  
\[cosech^{-1} x\]
  by parts by writing  
\[cosech^{-1}x=1 \times cosech^{-1}x\]
.
Let  
\[u=cosech^{-1}x \rightarrow cosechu=x \rightarrow -cosechucothu \frac{du}{dx}= 1\]
  then
\[\frac{du}{dx}= - \frac{1}{cosechucothu} =-\frac{1}{x \sqrt{cosech^2 u+1}}=-\frac{1}{x \sqrt{x^2+1}}\]
.
\[\frac{dv}{dx}=1 \rightarrow v=x\]

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

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