아래 목록은 역쌍곡선함수의 부정적분이다.

아래의 적분식들에서 상수 *a*는 0이 아니며, *C*는 적분상수이다. 각 역쌍곡선함수의 적분식은 그에 대응하는 역삼각함수 적분식이 존재한다.

역쌍곡사인함수의 적분
-----------



∫
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}

{\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}

{\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}

{\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)}

{\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}

{\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}

{\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)}

{\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)}

역쌍곡코사인함수의 적분
------------



∫
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}

{\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}

{\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}

{\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)}

{\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}

{\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}

{\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)}

{\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)}

역쌍곡탄젠트함수의 적분
------------



∫
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}

{\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}

{\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}

{\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)}

{\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)}

역쌍곡코시컨트함수의 적분
-------------



∫
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}

{\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}

{\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}

{\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)}

{\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)}

역쌍곡시컨트함수의 적분
------------



∫
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}

{\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}

{\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}

{\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)}

{\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)}

역쌍곡코탄젠트함수의 적분
-------------



∫
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}

{\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}

{\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}

{\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)}

{\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)}