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.

Inverse hyperbolic sine integration formulas

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

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

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

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

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

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

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

Inverse hyperbolic cosine integration formulas

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

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

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

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

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

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

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

Inverse hyperbolic tangent integration formulas

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

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

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

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

Inverse hyperbolic cotangent integration formulas

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

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

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

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

Inverse hyperbolic secant integration formulas

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

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

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

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

Inverse hyperbolic cosecant integration formulas

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

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

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

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