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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 [ ]
∫
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)}
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