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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.
Inverse hyperbolic sine integration formulas[edit]
![{\displaystyle \int \operatorname {arsinh} (ax)\,dx=x\operatorname {arsinh} (ax)-{\frac {\sqrt {a^{2}x^{2}+1}}{a}}+C}](https://wikimedia.org/api/rest_v1/media/math/render/svg/df59af76478ffbbc096b1e6b76c2713d811077cf)
![{\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}](https://wikimedia.org/api/rest_v1/media/math/render/svg/d99b23755b4814e41dbc2da6972e75b2f7e88205)
![{\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}](https://wikimedia.org/api/rest_v1/media/math/render/svg/12a51102d8524bd0ddac1c4d94eba3601efe658f)
![{\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)}](https://wikimedia.org/api/rest_v1/media/math/render/svg/d085f1a5370f8b7150f16ec54dafe025c262f825)
![{\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}](https://wikimedia.org/api/rest_v1/media/math/render/svg/b9268edad064cbd29c068c0582d7c7a67a45e0fe)
![{\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}](https://wikimedia.org/api/rest_v1/media/math/render/svg/07753337d7a3207747b1316249c782678e51bc5e)
![{\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)}](https://wikimedia.org/api/rest_v1/media/math/render/svg/caf68cf8f97dfa27ad584d5c9065f9b75bc56b70)
Inverse hyperbolic cosine integration formulas[edit]
![{\displaystyle \int \operatorname {arcosh} (ax)\,dx=x\operatorname {arcosh} (ax)-{\frac {{\sqrt {ax+1}}{\sqrt {ax-1}}}{a}}+C}](https://wikimedia.org/api/rest_v1/media/math/render/svg/88c9750a57762b31c0edad8b19f0e837a09b46c4)
![{\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}](https://wikimedia.org/api/rest_v1/media/math/render/svg/7d4a00beb3b399dfb574e2f53c4ae119811e9053)
![{\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}](https://wikimedia.org/api/rest_v1/media/math/render/svg/318d75b3987e8a3e8d3fa2545d046e0c825d5cbf)
![{\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)}](https://wikimedia.org/api/rest_v1/media/math/render/svg/baf3e65b3da93ad346412fc9f119fb447ba7fe6f)
![{\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}](https://wikimedia.org/api/rest_v1/media/math/render/svg/275ba566534fa755b2a955113281ef8fe5787e91)
![{\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}](https://wikimedia.org/api/rest_v1/media/math/render/svg/d8f514708faa6063d9c317d1864fa2f19e50085f)
![{\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)}](https://wikimedia.org/api/rest_v1/media/math/render/svg/27697891edae7bf67b13db79c11544565e6cc0c0)
Inverse hyperbolic tangent integration formulas[edit]
![{\displaystyle \int \operatorname {artanh} (ax)\,dx=x\operatorname {artanh} (ax)+{\frac {\ln \left(1-a^{2}x^{2}\right)}{2a}}+C}](https://wikimedia.org/api/rest_v1/media/math/render/svg/429c4c5c198ab806a1df350ce1420f901d1f0c07)
![{\displaystyle \int x\operatorname {artanh} (ax)\,dx={\frac {x^{2}\operatorname {artanh} (ax)}{2}}-{\frac {\operatorname {artanh} (ax)}{2a^{2}}}+{\frac {x}{2a}}+C}](https://wikimedia.org/api/rest_v1/media/math/render/svg/64158d791b14d1555a3f37b5923d4dee90614ef9)
![{\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}](https://wikimedia.org/api/rest_v1/media/math/render/svg/e484f53949099545ee5a141888d51f24ef0da58f)
![{\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)}](https://wikimedia.org/api/rest_v1/media/math/render/svg/4e0b86c16560c96f4b09c727cfab1e24a6b6938e)
Inverse hyperbolic cotangent integration formulas[edit]
![{\displaystyle \int \operatorname {arcoth} (ax)\,dx=x\operatorname {arcoth} (ax)+{\frac {\ln \left(a^{2}x^{2}-1\right)}{2a}}+C}](https://wikimedia.org/api/rest_v1/media/math/render/svg/176687f69b5ef7045e77c394074d50762f924def)
![{\displaystyle \int x\operatorname {arcoth} (ax)\,dx={\frac {x^{2}\operatorname {arcoth} (ax)}{2}}-{\frac {\operatorname {arcoth} (ax)}{2a^{2}}}+{\frac {x}{2a}}+C}](https://wikimedia.org/api/rest_v1/media/math/render/svg/814dd60335e902957e9973b525b3d2e05d5b04ab)
![{\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}](https://wikimedia.org/api/rest_v1/media/math/render/svg/2b2366611871834061321bb583bdf09e25d65070)
![{\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)}](https://wikimedia.org/api/rest_v1/media/math/render/svg/1bf8a184814f589c62c53bec6a9f4e497e974c09)
Inverse hyperbolic secant integration formulas[edit]
![{\displaystyle \int \operatorname {arsech} (ax)\,dx=x\operatorname {arsech} (ax)-{\frac {2}{a}}\operatorname {arctan} {\sqrt {\frac {1-ax}{1+ax}}}+C}](https://wikimedia.org/api/rest_v1/media/math/render/svg/d28f025510c6dd01ee8e40dc8a1c8ab57ca7f503)
![{\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}](https://wikimedia.org/api/rest_v1/media/math/render/svg/b4ca292f9f805a5cd294114a8b971bae6b675447)
![{\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}](https://wikimedia.org/api/rest_v1/media/math/render/svg/28837298325757792f1ea35356718503e0a1a0f6)
![{\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)}](https://wikimedia.org/api/rest_v1/media/math/render/svg/aba672848408312d020a88fd557b46d622c27bd6)
Inverse hyperbolic cosecant integration formulas[edit]
![{\displaystyle \int \operatorname {arcsch} (ax)\,dx=x\operatorname {arcsch} (ax)+{\frac {1}{a}}\operatorname {arcoth} {\sqrt {{\frac {1}{a^{2}x^{2}}}+1}}+C}](https://wikimedia.org/api/rest_v1/media/math/render/svg/760a004d822e47a2921dfa7b27805704b2974ad5)
![{\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}](https://wikimedia.org/api/rest_v1/media/math/render/svg/2d3ffb71724aa9b2ae58a3e905afd964b600fe06)
![{\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}](https://wikimedia.org/api/rest_v1/media/math/render/svg/047fc2437cce4208433d59ba332eb31ce3ffa45a)
![{\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)}](https://wikimedia.org/api/rest_v1/media/math/render/svg/e1dd8d5f54cb3cf28cd5b6e194c3693aa8008fac)