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

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

∫ x arsinh ⁡ ( a x ) d x = x 2 arsinh ⁡ ( a x ) 2 + arsinh ⁡ ( a x ) 4 a 2 − x a 2 x 2 + 1 4 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}

∫ x 2 arsinh ⁡ ( a x ) d x = x 3 arsinh ⁡ ( a x ) 3 − ( a 2 x 2 − 2 ) a 2 x 2 + 1 9 a 3 + 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}

∫ x m arsinh ⁡ ( a x ) d x = x m + 1 arsinh ⁡ ( a x ) m + 1 − a m + 1 ∫ x m + 1 a 2 x 2 + 1 d x ( m ≠ − 1 ) {\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)}

∫ arsinh ⁡ ( a x ) 2 d x = 2 x + x arsinh ⁡ ( a x ) 2 − 2 a 2 x 2 + 1 arsinh ⁡ ( a x ) a + C {\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}

∫ arsinh ⁡ ( a x ) n d x = x arsinh ⁡ ( a x ) n − n a 2 x 2 + 1 arsinh ⁡ ( a x ) n − 1 a + n ( n − 1 ) ∫ arsinh ⁡ ( a x ) n − 2 d x {\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}

∫ arsinh ⁡ ( a x ) n d x = − x arsinh ⁡ ( a x ) n + 2 ( n + 1 ) ( n + 2 ) + a 2 x 2 + 1 arsinh ⁡ ( a x ) n + 1 a ( n + 1 ) + 1 ( n + 1 ) ( n + 2 ) ∫ arsinh ⁡ ( a x ) n + 2 d x ( n ≠ − 1 , − 2 ) {\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

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

∫ x arcosh ⁡ ( a x ) d x = x 2 arcosh ⁡ ( a x ) 2 − arcosh ⁡ ( a x ) 4 a 2 − x a x + 1 a x − 1 4 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}

∫ x 2 arcosh ⁡ ( a x ) d x = x 3 arcosh ⁡ ( a x ) 3 − ( a 2 x 2 + 2 ) a x + 1 a x − 1 9 a 3 + 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}

∫ x m arcosh ⁡ ( a x ) d x = x m + 1 arcosh ⁡ ( a x ) m + 1 − a m + 1 ∫ x m + 1 a x + 1 a x − 1 d x ( m ≠ − 1 ) {\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)}

∫ arcosh ⁡ ( a x ) 2 d x = 2 x + x arcosh ⁡ ( a x ) 2 − 2 a x + 1 a x − 1 arcosh ⁡ ( a x ) a + C {\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}

∫ arcosh ⁡ ( a x ) n d x = x arcosh ⁡ ( a x ) n − n a x + 1 a x − 1 arcosh ⁡ ( a x ) n − 1 a + n ( n − 1 ) ∫ arcosh ⁡ ( a x ) n − 2 d x {\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}

∫ arcosh ⁡ ( a x ) n d x = − x arcosh ⁡ ( a x ) n + 2 ( n + 1 ) ( n + 2 ) + a x + 1 a x − 1 arcosh ⁡ ( a x ) n + 1 a ( n + 1 ) + 1 ( n + 1 ) ( n + 2 ) ∫ arcosh ⁡ ( a x ) n + 2 d x ( n ≠ − 1 , − 2 ) {\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

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

∫ x artanh ⁡ ( a x ) d x = x 2 artanh ⁡ ( a x ) 2 − artanh ⁡ ( a x ) 2 a 2 + x 2 a + 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}

∫ x 2 artanh ⁡ ( a x ) d x = x 3 artanh ⁡ ( a x ) 3 + ln ⁡ ( 1 − a 2 x 2 ) 6 a 3 + x 2 6 a + 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}

∫ x m artanh ⁡ ( a x ) d x = x m + 1 artanh ⁡ ( a x ) m + 1 − a m + 1 ∫ x m + 1 1 − a 2 x 2 d x ( m ≠ − 1 ) {\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

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

∫ x arcoth ⁡ ( a x ) d x = x 2 arcoth ⁡ ( a x ) 2 − arcoth ⁡ ( a x ) 2 a 2 + x 2 a + 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}

∫ x 2 arcoth ⁡ ( a x ) d x = x 3 arcoth ⁡ ( a x ) 3 + ln ⁡ ( a 2 x 2 − 1 ) 6 a 3 + x 2 6 a + 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}

∫ x m arcoth ⁡ ( a x ) d x = x m + 1 arcoth ⁡ ( a x ) m + 1 + a m + 1 ∫ x m + 1 a 2 x 2 − 1 d x ( m ≠ − 1 ) {\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

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

∫ x arsech ⁡ ( a x ) d x = x 2 arsech ⁡ ( a x ) 2 − ( 1 + a x ) 2 a 2 1 − a x 1 + a x + 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}

∫ x 2 arsech ⁡ ( a x ) d x = x 3 arsech ⁡ ( a x ) 3 − 1 3 a 3 arctan ⁡ 1 − a x 1 + a x − x ( 1 + a x ) 6 a 2 1 − a x 1 + a x + 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}

∫ x m arsech ⁡ ( a x ) d x = x m + 1 arsech ⁡ ( a x ) m + 1 + 1 m + 1 ∫ x m ( 1 + a x ) 1 − a x 1 + a x d x ( m ≠ − 1 ) {\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

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

∫ x arcsch ⁡ ( a x ) d x = x 2 arcsch ⁡ ( a x ) 2 + x 2 a 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}

∫ x 2 arcsch ⁡ ( a x ) d x = x 3 arcsch ⁡ ( a x ) 3 − 1 6 a 3 arcoth ⁡ 1 a 2 x 2 + 1 + x 2 6 a 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}

∫ x m arcsch ⁡ ( a x ) d x = x m + 1 arcsch ⁡ ( a x ) m + 1 + 1 a ( m + 1 ) ∫ x m − 1 1 a 2 x 2 + 1 d x ( m ≠ − 1 ) {\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)}