linguistic:translation
Inverse functions as logarithms

arsinh (x) = ln (x + x 2 + 1) arcosh (x) = ln (x + x 2 − 1) x ≥ 1 artanh (x) = 1 2 ln (1 + x 1 − x) | x | < 1 arcoth (x) = 1 2 ln (x + 1 x − 1) | x | > 1 arsech (x) = ln (1 x + 1 x 2 − 1) = ln (1 + 1 − x 2 x) 0 < x ≤ 1 arcsch (x) = ln (1 x + 1 x 2 + 1) x ≠ 0 {\displaystyle {\begin{aligned}\operatorname {arsinh} (x)&=\ln \left(x+{\sqrt {x^{2}+1}}\right)\\\operatorname {arcosh} (x)&=\ln \left(x+{\sqrt {x^{2}-1}}\right)&&x\geq 1\\\operatorname {artanh} (x)&={\frac {1}{2}}\ln \left({\frac {1+x}{1-x}}\right)&&|x|<1\\\operatorname {arcoth} (x)&={\frac {1}{2}}\ln \left({\frac {x+1}{x-1}}\right)&&|x|>1\\\operatorname {arsech} (x)&=\ln \left({\frac {1}{x}}+{\sqrt {{\frac {1}{x^{2}}}-1}}\right)=\ln \left({\frac {1+{\sqrt {1-x^{2}}}}{x}}\right)&&0<x\leq 1\\\operatorname {arcsch} (x)&=\ln \left({\frac {1}{x}}+{\sqrt {{\frac {1}{x^{2}}}+1}}\right)&&x\neq 0\end{aligned}}}