List of integrals of inverse hyperbolic functions

The following is a list of indefinite integrals (antiderivatives) of expressions involving the inverse hyperbolic functions. For a complete list of integral formulas, see lists of integrals.

• In all formulas the constant a is assumed to be nonzero, and C denotes the constant of integration.
• For each inverse hyperbolic integration formula below there is a corresponding formula in the list of integrals of inverse trigonometric functions.
• The ISO 80000-2 standard uses the prefix "ar-" rather than "arc-" for the inverse hyperbolic functions; we do that here.

$${\displaystyle \int \operatorname {arsinh} (ax)\,dx=x\operatorname {arsinh} (ax)-{\frac {\sqrt {a^{2}x^{2}+1}{a}+C}$$

$${\displaystyle \int x\operatorname {arsinh} (ax)\,dx={\frac {x^{2}\operatorname {arsinh} (ax)}{2}+{\frac {\operatorname {arsinh} (ax)}{4a^{2}-{\frac {x{\sqrt {a^{2}x^{2}+1}{4a}+C}$$

$${\displaystyle \int x^{2}\operatorname {arsinh} (ax)\,dx={\frac {x^{3}\operatorname {arsinh} (ax)}{3}-{\frac {\left(a^{2}x^{2}-2\right){\sqrt {a^{2}x^{2}+1}{9a^{3}+C}$$

$${\displaystyle \int x^{m}\operatorname {arsinh} (ax)\,dx={\frac {x^{m+1}\operatorname {arsinh} (ax)}{m+1}-{\frac {a}{m+1}\int {\frac {x^{m+1}{\sqrt {a^{2}x^{2}+1}\,dx\quad (m\neq -1)}$$

$${\displaystyle \int \operatorname {arsinh} (ax)^{2}\,dx=2x+x\operatorname {arsinh} (ax)^{2}-{\frac {2{\sqrt {a^{2}x^{2}+1}\operatorname {arsinh} (ax)}{a}+C}$$

$${\displaystyle \int \operatorname {arsinh} (ax)^{n}\,dx=x\operatorname {arsinh} (ax)^{n}-{\frac {n{\sqrt {a^{2}x^{2}+1}\operatorname {arsinh} (ax)^{n-1}{a}+n(n-1)\int \operatorname {arsinh} (ax)^{n-2}\,dx}$$

$${\displaystyle \int \operatorname {arsinh} (ax)^{n}\,dx=-{\frac {x\operatorname {arsinh} (ax)^{n+2}{(n+1)(n+2)}+{\frac {\sqrt {a^{2}x^{2}+1}\operatorname {arsinh} (ax)^{n+1}{a(n+1)}+{\frac {1}{(n+1)(n+2)}\int \operatorname {arsinh} (ax)^{n+2}\,dx\quad (n\neq -1,-2)}$$

$${\displaystyle \int \operatorname {arcosh} (ax)\,dx=x\operatorname {arcosh} (ax)-{\frac {\sqrt {ax+1}{\sqrt {ax-1}{a}+C}$$

$${\displaystyle \int x\operatorname {arcosh} (ax)\,dx={\frac {x^{2}\operatorname {arcosh} (ax)}{2}-{\frac {\operatorname {arcosh} (ax)}{4a^{2}-{\frac {x{\sqrt {ax+1}{\sqrt {ax-1}{4a}+C}$$

$${\displaystyle \int x^{2}\operatorname {arcosh} (ax)\,dx={\frac {x^{3}\operatorname {arcosh} (ax)}{3}-{\frac {\left(a^{2}x^{2}+2\right){\sqrt {ax+1}{\sqrt {ax-1}{9a^{3}+C}$$

$${\displaystyle \int x^{m}\operatorname {arcosh} (ax)\,dx={\frac {x^{m+1}\operatorname {arcosh} (ax)}{m+1}-{\frac {a}{m+1}\int {\frac {x^{m+1}{\sqrt {ax+1}{\sqrt {ax-1}\,dx\quad (m\neq -1)}$$

$${\displaystyle \int \operatorname {arcosh} (ax)^{2}\,dx=2x+x\operatorname {arcosh} (ax)^{2}-{\frac {2{\sqrt {ax+1}{\sqrt {ax-1}\operatorname {arcosh} (ax)}{a}+C}$$

$${\displaystyle \int \operatorname {arcosh} (ax)^{n}\,dx=x\operatorname {arcosh} (ax)^{n}-{\frac {n{\sqrt {ax+1}{\sqrt {ax-1}\operatorname {arcosh} (ax)^{n-1}{a}+n(n-1)\int \operatorname {arcosh} (ax)^{n-2}\,dx}$$

$${\displaystyle \int \operatorname {arcosh} (ax)^{n}\,dx=-{\frac {x\operatorname {arcosh} (ax)^{n+2}{(n+1)(n+2)}+{\frac {\sqrt {ax+1}{\sqrt {ax-1}\operatorname {arcosh} (ax)^{n+1}{a(n+1)}+{\frac {1}{(n+1)(n+2)}\int \operatorname {arcosh} (ax)^{n+2}\,dx\quad (n\neq -1,-2)}$$

$${\displaystyle \int \operatorname {artanh} (ax)\,dx=x\operatorname {artanh} (ax)+{\frac {\ln \left(1-a^{2}x^{2}\right)}{2a}+C}$$

$${\displaystyle \int x\operatorname {artanh} (ax)\,dx={\frac {x^{2}\operatorname {artanh} (ax)}{2}-{\frac {\operatorname {artanh} (ax)}{2a^{2}+{\frac {x}{2a}+C}$$

$${\displaystyle \int x^{2}\operatorname {artanh} (ax)\,dx={\frac {x^{3}\operatorname {artanh} (ax)}{3}+{\frac {\ln \left(1-a^{2}x^{2}\right)}{6a^{3}+{\frac {x^{2}{6a}+C}$$

$${\displaystyle \int x^{m}\operatorname {artanh} (ax)\,dx={\frac {x^{m+1}\operatorname {artanh} (ax)}{m+1}-{\frac {a}{m+1}\int {\frac {x^{m+1}{1-a^{2}x^{2}\,dx\quad (m\neq -1)}$$

$${\displaystyle \int \operatorname {arcoth} (ax)\,dx=x\operatorname {arcoth} (ax)+{\frac {\ln \left(a^{2}x^{2}-1\right)}{2a}+C}$$

$${\displaystyle \int x\operatorname {arcoth} (ax)\,dx={\frac {x^{2}\operatorname {arcoth} (ax)}{2}-{\frac {\operatorname {arcoth} (ax)}{2a^{2}+{\frac {x}{2a}+C}$$

$${\displaystyle \int x^{2}\operatorname {arcoth} (ax)\,dx={\frac {x^{3}\operatorname {arcoth} (ax)}{3}+{\frac {\ln \left(a^{2}x^{2}-1\right)}{6a^{3}+{\frac {x^{2}{6a}+C}$$

$${\displaystyle \int x^{m}\operatorname {arcoth} (ax)\,dx={\frac {x^{m+1}\operatorname {arcoth} (ax)}{m+1}+{\frac {a}{m+1}\int {\frac {x^{m+1}{a^{2}x^{2}-1}\,dx\quad (m\neq -1)}$$

$${\displaystyle \int \operatorname {arsech} (ax)\,dx=x\operatorname {arsech} (ax)-{\frac {2}{a}\operatorname {arctan} {\sqrt {\frac {1-ax}{1+ax}+C}$$

$${\displaystyle \int x\operatorname {arsech} (ax)\,dx={\frac {x^{2}\operatorname {arsech} (ax)}{2}-{\frac {(1+ax)}{2a^{2}{\sqrt {\frac {1-ax}{1+ax}+C}$$

$${\displaystyle \int x^{2}\operatorname {arsech} (ax)\,dx={\frac {x^{3}\operatorname {arsech} (ax)}{3}-{\frac {1}{3a^{3}\operatorname {arctan} {\sqrt {\frac {1-ax}{1+ax}-{\frac {x(1+ax)}{6a^{2}{\sqrt {\frac {1-ax}{1+ax}+C}$$

$${\displaystyle \int x^{m}\operatorname {arsech} (ax)\,dx={\frac {x^{m+1}\operatorname {arsech} (ax)}{m+1}+{\frac {1}{m+1}\int {\frac {x^{m}{(1+ax){\sqrt {\frac {1-ax}{1+ax}\,dx\quad (m\neq -1)}$$

$${\displaystyle \int \operatorname {arcsch} (ax)\,dx=x\operatorname {arcsch} (ax)+{\frac {1}{a}\operatorname {arcoth} {\sqrt {\frac {1}{a^{2}x^{2}+1}+C}$$

$${\displaystyle \int x\operatorname {arcsch} (ax)\,dx={\frac {x^{2}\operatorname {arcsch} (ax)}{2}+{\frac {x}{2a}{\sqrt {\frac {1}{a^{2}x^{2}+1}+C}$$

$${\displaystyle \int x^{2}\operatorname {arcsch} (ax)\,dx={\frac {x^{3}\operatorname {arcsch} (ax)}{3}-{\frac {1}{6a^{3}\operatorname {arcoth} {\sqrt {\frac {1}{a^{2}x^{2}+1}+{\frac {x^{2}{6a}{\sqrt {\frac {1}{a^{2}x^{2}+1}+C}$$

$${\displaystyle \int x^{m}\operatorname {arcsch} (ax)\,dx={\frac {x^{m+1}\operatorname {arcsch} (ax)}{m+1}+{\frac {1}{a(m+1)}\int {\frac {x^{m-1}{\sqrt {\frac {1}{a^{2}x^{2}+1}\,dx\quad (m\neq -1)}$$