The inverse form of a hyperbolic function is called the inverse hyperbolic function.
Hyperbolic functions are six types. So, the inverse hyperbolic functions are also six types. Each hyperbolic function is defined in exponential functions form. So, each inverse hyperbolic function is defined in logarithmic function form.
Here is the list of six inverse hyperbolic functions in logarithmic functions form with proofs for beginners.
$\large \sinh^{-1}{x} \,=\, \log_{e}{(x+\sqrt{x^2+1})}$
$\large \cosh^{-1}{x} \,=\, \log_{e}{(x+\sqrt{x^2-1})}$
$\large \tanh^{-1}{x} \,=\, \dfrac{1}{2}\log_{e}{\Bigg(\dfrac{1+x}{1-x}\Bigg)}$
$\large \coth^{-1}{x} \,=\, \dfrac{1}{2}\log_{e}{\Bigg(\dfrac{x+1}{x-1}\Bigg)}$
$\large {\mathop{\rm sech}\nolimits}^{-1}{x} = \log{\Bigg(\dfrac{1+\sqrt{1-x^2}}{x}\Bigg)}$
${\mathop{\rm csch}\nolimits}^{-1}{x} = \log{\Bigg(\dfrac{1-\sqrt{1+x^2}}{x}\Bigg)}$
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