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.

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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