inverse hyperbolic functions

Differentiation

$$
y=\sinh^{-1}x\quad\Longrightarrow\quad x=\sinh y\\
\frac{{\rm d}x}{{\rm d}y}=\cosh y=\sqrt{(\sinh y)^2+1}=\sqrt{x^2+1}\\[6pt]
y=\cosh^{-1}x\quad\Longrightarrow\quad x=\cosh y\\
\frac{{\rm d}x}{{\rm d}y}=\sinh y=\sqrt{(\cosh y)^2-1}=\sqrt{x^2-1}\\[6pt]
$$

Integration

$$
\begin{aligned}
\int\sinh^{-1}x{\rm\ d}x&=x\sinh^{-1}x-\int x{\rm\ d}\sinh^{-1}x\\
&=x\sinh^{-1}x-\int\frac{x}{\sqrt{1+x^2}}{\rm\ d}x\\
&=x\sinh^{-1}x-\int\frac{1}{2}(1+x^2)^{-\frac{1}{2}}{\rm\ d}(1+x^2)\\
&=x\sinh^{-1}x-\sqrt{1+x^2}+c\\[6pt]
\int\cosh^{-1}x{\rm\ d}x&=x\cosh^{-1}x-\int x{\rm\ d}\cosh^{-1}x\\
&=x\cosh^{-1}x-\int\frac{x}{\sqrt{x^2-1}}{\rm\ d}x\\
&=x\cosh^{-1}x-\int\frac{1}{2}(x^2-1)^{-\frac{1}{2}}{\rm\ d}(x^2-1)\\
&=x\cosh^{-1}x-\sqrt{x^2-1}+c\\[6pt]
\int\tanh^{-1}x{\rm\ d}x&=x\tanh^{-1}x-\int x{\rm\ d}\tanh^{-1}x\\
&=x\tanh^{-1}x-\int\frac{x}{1-x^2}{\rm\ d}x\\
&=x\tanh^{-1}x+\frac{1}{2}\int\frac{1}{x^2-1}{\rm\ d}(x^2-1)\\
&=x\tanh^{-1}x+\frac{1}{2}\ln|x^2-1|+c\\[6pt]
\end{aligned}
$$

other related

$$
\begin{aligned}
\int\sqrt{x^2+1}{\rm\ d}x&=\int\cosh y{\rm\ d}\sinh y=\int\cosh^2y{\rm\ d}y\\
&=\int\frac{\cosh2y+1}{2}{\rm\ d}y=\frac{\sinh2y}{4}+\frac{y}{2}+c\\
&=\frac{\sinh y\cosh y}{2}+\frac{y}{2}+c=\frac{x\sqrt{x^2+1}}{2}+\frac{\sinh^{-1}x}{2}+c\\
&=\frac{x\sqrt{x^2+1}}{2}+\frac{\ln\left|x+\sqrt{x^2+1}\right|}{2}+c\\[6pt]
\int\sqrt{x^2-1}{\rm\ d}x&=\int\sinh y{\rm\ d}\cosh y=\int\sinh^2y{\rm\ d}y\\
&=\int\frac{\cosh2y-1}{2}{\rm\ d}y=\frac{\sinh2y}{2}-\frac{y}{2}+c\\
&=\frac{\sinh y\cosh y}{2}-\frac{y}{2}+c=\frac{x\sqrt{x^2-1}}{2}-\frac{\cosh^{-1}x}{2}+c\\
&=\frac{\sinh y\cosh y}{2}-\frac{y}{2}+c=\frac{x\sqrt{x^2-1}}{2}-\frac{\ln\left|x+\sqrt{x^2-1}\right|}{2}+c\\[6pt]
\end{aligned}
$$