無理関数の原始関数の一覧
本項は、無理関数の原始関数 の一覧である。さらに完全な原始関数の一覧は、原始関数の一覧 を参照のこと。本項で、積分定数は簡便のために省略している。
r
=
x
2
+
a
2
{\displaystyle r={\sqrt {x^{2}+a^{2}
∫
r
d
x
=
1
2
(
x
r
+
a
2
ln
(
x
+
r
)
)
{\displaystyle \int r\;dx={\frac {1}{2}\left(xr+a^{2}\,\ln \left(x+r\right)\right)}
∫
r
3
d
x
=
1
4
x
r
3
+
3
8
a
2
x
r
+
3
8
a
4
ln
(
x
+
r
)
{\displaystyle \int r^{3}\;dx={\frac {1}{4}xr^{3}+{\frac {3}{8}a^{2}xr+{\frac {3}{8}a^{4}\ln \left(x+r\right)}
∫
r
5
d
x
=
1
6
x
r
5
+
5
24
a
2
x
r
3
+
5
16
a
4
x
r
+
5
16
a
6
ln
(
x
+
r
)
{\displaystyle \int r^{5}\;dx={\frac {1}{6}xr^{5}+{\frac {5}{24}a^{2}xr^{3}+{\frac {5}{16}a^{4}xr+{\frac {5}{16}a^{6}\ln \left(x+r\right)}
∫
x
r
d
x
=
r
3
3
{\displaystyle \int xr\;dx={\frac {r^{3}{3}
∫
x
r
3
d
x
=
r
5
5
{\displaystyle \int xr^{3}\;dx={\frac {r^{5}{5}
∫
x
r
2
n
+
1
d
x
=
r
2
n
+
3
2
n
+
3
{\displaystyle \int xr^{2n+1}\;dx={\frac {r^{2n+3}{2n+3}
∫
x
2
r
d
x
=
x
r
3
4
−
a
2
x
r
8
−
a
4
8
ln
(
x
+
r
)
{\displaystyle \int x^{2}r\;dx={\frac {xr^{3}{4}-{\frac {a^{2}xr}{8}-{\frac {a^{4}{8}\ln \left(x+r\right)}
∫
x
2
r
3
d
x
=
x
r
5
6
−
a
2
x
r
3
24
−
a
4
x
r
16
−
a
6
16
ln
(
x
+
r
)
{\displaystyle \int x^{2}r^{3}\;dx={\frac {xr^{5}{6}-{\frac {a^{2}xr^{3}{24}-{\frac {a^{4}xr}{16}-{\frac {a^{6}{16}\ln \left(x+r\right)}
∫
x
3
r
d
x
=
r
5
5
−
a
2
r
3
3
{\displaystyle \int x^{3}r\;dx={\frac {r^{5}{5}-{\frac {a^{2}r^{3}{3}
∫
x
3
r
3
d
x
=
r
7
7
−
a
2
r
5
5
{\displaystyle \int x^{3}r^{3}\;dx={\frac {r^{7}{7}-{\frac {a^{2}r^{5}{5}
∫
x
3
r
2
n
+
1
d
x
=
r
2
n
+
5
2
n
+
5
−
a
3
r
2
n
+
3
2
n
+
3
{\displaystyle \int x^{3}r^{2n+1}\;dx={\frac {r^{2n+5}{2n+5}-{\frac {a^{3}r^{2n+3}{2n+3}
∫
x
4
r
d
x
=
x
3
r
3
6
−
a
2
x
r
3
8
+
a
4
x
r
16
+
a
6
16
ln
(
x
+
r
)
{\displaystyle \int x^{4}r\;dx={\frac {x^{3}r^{3}{6}-{\frac {a^{2}xr^{3}{8}+{\frac {a^{4}xr}{16}+{\frac {a^{6}{16}\ln \left(x+r\right)}
∫
x
4
r
3
d
x
=
x
3
r
5
8
−
a
2
x
r
5
16
+
a
4
x
r
3
64
+
3
a
6
x
r
128
+
3
a
8
128
ln
(
x
+
r
)
{\displaystyle \int x^{4}r^{3}\;dx={\frac {x^{3}r^{5}{8}-{\frac {a^{2}xr^{5}{16}+{\frac {a^{4}xr^{3}{64}+{\frac {3a^{6}xr}{128}+{\frac {3a^{8}{128}\ln \left(x+r\right)}
∫
x
5
r
d
x
=
r
7
7
−
2
a
2
r
5
5
+
a
4
r
3
3
{\displaystyle \int x^{5}r\;dx={\frac {r^{7}{7}-{\frac {2a^{2}r^{5}{5}+{\frac {a^{4}r^{3}{3}
∫
x
5
r
3
d
x
=
r
9
9
−
2
a
2
r
7
7
+
a
4
r
5
5
{\displaystyle \int x^{5}r^{3}\;dx={\frac {r^{9}{9}-{\frac {2a^{2}r^{7}{7}+{\frac {a^{4}r^{5}{5}
∫
x
5
r
2
n
+
1
d
x
=
r
2
n
+
7
2
n
+
7
−
2
a
2
r
2
n
+
5
2
n
+
5
+
a
4
r
2
n
+
3
2
n
+
3
{\displaystyle \int x^{5}r^{2n+1}\;dx={\frac {r^{2n+7}{2n+7}-{\frac {2a^{2}r^{2n+5}{2n+5}+{\frac {a^{4}r^{2n+3}{2n+3}
∫
r
d
x
x
=
r
−
a
ln
|
a
+
r
x
|
=
r
−
a
arsinh
a
x
{\displaystyle \int {\frac {r\;dx}{x}=r-a\ln \left|{\frac {a+r}{x}\right|=r-a\,\operatorname {arsinh} {\frac {a}{x}
∫
r
3
d
x
x
=
r
3
3
+
a
2
r
−
a
3
ln
|
a
+
r
x
|
{\displaystyle \int {\frac {r^{3}\;dx}{x}={\frac {r^{3}{3}+a^{2}r-a^{3}\ln \left|{\frac {a+r}{x}\right|}
∫
r
5
d
x
x
=
r
5
5
+
a
2
r
3
3
+
a
4
r
−
a
5
ln
|
a
+
r
x
|
{\displaystyle \int {\frac {r^{5}\;dx}{x}={\frac {r^{5}{5}+{\frac {a^{2}r^{3}{3}+a^{4}r-a^{5}\ln \left|{\frac {a+r}{x}\right|}
∫
r
7
d
x
x
=
r
7
7
+
a
2
r
5
5
+
a
4
r
3
3
+
a
6
r
−
a
7
ln
|
a
+
r
x
|
{\displaystyle \int {\frac {r^{7}\;dx}{x}={\frac {r^{7}{7}+{\frac {a^{2}r^{5}{5}+{\frac {a^{4}r^{3}{3}+a^{6}r-a^{7}\ln \left|{\frac {a+r}{x}\right|}
∫
d
x
r
=
arsinh
x
a
=
ln
(
x
+
r
a
)
{\displaystyle \int {\frac {dx}{r}=\operatorname {arsinh} {\frac {x}{a}=\ln \left({\frac {x+r}{a}\right)}
∫
d
x
r
3
=
x
a
2
r
{\displaystyle \int {\frac {dx}{r^{3}={\frac {x}{a^{2}r}
∫
x
d
x
r
=
r
{\displaystyle \int {\frac {x\,dx}{r}=r}
∫
x
d
x
r
3
=
−
1
r
{\displaystyle \int {\frac {x\,dx}{r^{3}=-{\frac {1}{r}
∫
x
2
d
x
r
=
x
2
r
−
a
2
2
arsinh
x
a
=
x
2
r
−
a
2
2
ln
(
x
+
r
a
)
{\displaystyle \int {\frac {x^{2}\;dx}{r}={\frac {x}{2}r-{\frac {a^{2}{2}\,\operatorname {arsinh} {\frac {x}{a}={\frac {x}{2}r-{\frac {a^{2}{2}\ln \left({\frac {x+r}{a}\right)}
∫
d
x
x
r
=
−
1
a
arsinh
a
x
=
−
1
a
ln
|
a
+
r
x
|
{\displaystyle \int {\frac {dx}{xr}=-{\frac {1}{a}\,\operatorname {arsinh} {\frac {a}{x}=-{\frac {1}{a}\ln \left|{\frac {a+r}{x}\right|}
s
=
x
2
−
a
2
{\displaystyle s={\sqrt {x^{2}-a^{2}
(
x
2
>
a
2
)
{\displaystyle (x^{2}>a^{2})}
(
x
2
<
a
2
)
{\displaystyle (x^{2}<a^{2})}
∫
s
d
x
=
1
2
(
x
s
−
a
2
ln
(
x
+
s
)
)
{\displaystyle \int s\;dx={\frac {1}{2}\left(xs-a^{2}\ln(x+s)\right)}
∫
x
s
d
x
=
1
3
s
3
{\displaystyle \int xs\;dx={\frac {1}{3}s^{3}
∫
s
d
x
x
=
s
−
a
arccos
|
a
x
|
{\displaystyle \int {\frac {s\;dx}{x}=s-a\arccos \left|{\frac {a}{x}\right|}
∫
d
x
s
=
ln
|
x
+
s
a
|
{\displaystyle \int {\frac {dx}{s}=\ln \left|{\frac {x+s}{a}\right|}
ここで、
ln
|
x
+
s
a
|
=
s
g
n
(
x
)
arcosh
|
x
a
|
=
1
2
ln
(
x
+
s
x
−
s
)
{\displaystyle \ln \left|{\frac {x+s}{a}\right|=\mathrm {sgn} (x)\,\operatorname {arcosh} \left|{\frac {x}{a}\right|={\frac {1}{2}\ln \left({\frac {x+s}{x-s}\right)}
arcosh
|
x
a
|
{\displaystyle \operatorname {arcosh} \left|{\frac {x}{a}\right|}
∫
x
d
x
s
=
s
{\displaystyle \int {\frac {x\;dx}{s}=s}
∫
x
d
x
s
3
=
−
1
s
{\displaystyle \int {\frac {x\;dx}{s^{3}=-{\frac {1}{s}
∫
x
d
x
s
5
=
−
1
3
s
3
{\displaystyle \int {\frac {x\;dx}{s^{5}=-{\frac {1}{3s^{3}
∫
x
d
x
s
7
=
−
1
5
s
5
{\displaystyle \int {\frac {x\;dx}{s^{7}=-{\frac {1}{5s^{5}
∫
x
d
x
s
2
n
+
1
=
−
1
(
2
n
−
1
)
s
2
n
−
1
{\displaystyle \int {\frac {x\;dx}{s^{2n+1}=-{\frac {1}{(2n-1)s^{2n-1}
∫
x
2
m
d
x
s
2
n
+
1
=
−
1
2
n
−
1
x
2
m
−
1
s
2
n
−
1
+
2
m
−
1
2
n
−
1
∫
x
2
m
−
2
d
x
s
2
n
−
1
{\displaystyle \int {\frac {x^{2m}\;dx}{s^{2n+1}=-{\frac {1}{2n-1}{\frac {x^{2m-1}{s^{2n-1}+{\frac {2m-1}{2n-1}\int {\frac {x^{2m-2}\;dx}{s^{2n-1}
∫
x
2
d
x
s
=
x
s
2
+
a
2
2
ln
|
x
+
s
a
|
{\displaystyle \int {\frac {x^{2}\;dx}{s}={\frac {xs}{2}+{\frac {a^{2}{2}\ln \left|{\frac {x+s}{a}\right|}
∫
x
2
d
x
s
3
=
−
x
s
+
ln
|
x
+
s
a
|
{\displaystyle \int {\frac {x^{2}\;dx}{s^{3}=-{\frac {x}{s}+\ln \left|{\frac {x+s}{a}\right|}
∫
x
4
d
x
s
=
x
3
s
4
+
3
8
a
2
x
s
+
3
8
a
4
ln
|
x
+
s
a
|
{\displaystyle \int {\frac {x^{4}\;dx}{s}={\frac {x^{3}s}{4}+{\frac {3}{8}a^{2}xs+{\frac {3}{8}a^{4}\ln \left|{\frac {x+s}{a}\right|}
∫
x
4
d
x
s
3
=
x
s
2
−
a
2
x
s
+
3
2
a
2
ln
|
x
+
s
a
|
{\displaystyle \int {\frac {x^{4}\;dx}{s^{3}={\frac {xs}{2}-{\frac {a^{2}x}{s}+{\frac {3}{2}a^{2}\ln \left|{\frac {x+s}{a}\right|}
∫
x
4
d
x
s
5
=
−
x
s
−
1
3
x
3
s
3
+
ln
|
x
+
s
a
|
{\displaystyle \int {\frac {x^{4}\;dx}{s^{5}=-{\frac {x}{s}-{\frac {1}{3}{\frac {x^{3}{s^{3}+\ln \left|{\frac {x+s}{a}\right|}
∫
x
2
m
d
x
s
2
n
+
1
=
(
−
1
)
n
−
m
1
a
2
(
n
−
m
)
∑
i
=
0
n
−
m
−
1
1
2
(
m
+
i
)
+
1
(
n
−
m
−
1
i
)
x
2
(
m
+
i
)
+
1
s
2
(
m
+
i
)
+
1
(
n
>
m
≥
0
)
{\displaystyle \int {\frac {x^{2m}\;dx}{s^{2n+1}=(-1)^{n-m}{\frac {1}{a^{2(n-m)}\sum _{i=0}^{n-m-1}{\frac {1}{2(m+i)+1}{n-m-1 \choose i}{\frac {x^{2(m+i)+1}{s^{2(m+i)+1}\qquad {\mbox{(}n>m\geq 0{\mbox{)}
∫
d
x
s
3
=
−
1
a
2
x
s
{\displaystyle \int {\frac {dx}{s^{3}=-{\frac {1}{a^{2}{\frac {x}{s}
∫
d
x
s
5
=
1
a
4
[
x
s
−
1
3
x
3
s
3
]
{\displaystyle \int {\frac {dx}{s^{5}={\frac {1}{a^{4}\left[{\frac {x}{s}-{\frac {1}{3}{\frac {x^{3}{s^{3}\right]}
∫
d
x
s
7
=
−
1
a
6
[
x
s
−
2
3
x
3
s
3
+
1
5
x
5
s
5
]
{\displaystyle \int {\frac {dx}{s^{7}=-{\frac {1}{a^{6}\left[{\frac {x}{s}-{\frac {2}{3}{\frac {x^{3}{s^{3}+{\frac {1}{5}{\frac {x^{5}{s^{5}\right]}
∫
d
x
s
9
=
1
a
8
[
x
s
−
3
3
x
3
s
3
+
3
5
x
5
s
5
−
1
7
x
7
s
7
]
{\displaystyle \int {\frac {dx}{s^{9}={\frac {1}{a^{8}\left[{\frac {x}{s}-{\frac {3}{3}{\frac {x^{3}{s^{3}+{\frac {3}{5}{\frac {x^{5}{s^{5}-{\frac {1}{7}{\frac {x^{7}{s^{7}\right]}
∫
x
2
d
x
s
5
=
−
1
a
2
x
3
3
s
3
{\displaystyle \int {\frac {x^{2}\;dx}{s^{5}=-{\frac {1}{a^{2}{\frac {x^{3}{3s^{3}
∫
x
2
d
x
s
7
=
1
a
4
[
1
3
x
3
s
3
−
1
5
x
5
s
5
]
{\displaystyle \int {\frac {x^{2}\;dx}{s^{7}={\frac {1}{a^{4}\left[{\frac {1}{3}{\frac {x^{3}{s^{3}-{\frac {1}{5}{\frac {x^{5}{s^{5}\right]}
∫
x
2
d
x
s
9
=
−
1
a
6
[
1
3
x
3
s
3
−
2
5
x
5
s
5
+
1
7
x
7
s
7
]
{\displaystyle \int {\frac {x^{2}\;dx}{s^{9}=-{\frac {1}{a^{6}\left[{\frac {1}{3}{\frac {x^{3}{s^{3}-{\frac {2}{5}{\frac {x^{5}{s^{5}+{\frac {1}{7}{\frac {x^{7}{s^{7}\right]}
u
=
a
2
−
x
2
{\displaystyle u={\sqrt {a^{2}-x^{2}
∫
u
d
x
=
1
2
(
x
u
+
a
2
arcsin
x
a
)
(
|
x
|
≤
|
a
|
)
{\displaystyle \int u\;dx={\frac {1}{2}\left(xu+a^{2}\arcsin {\frac {x}{a}\right)\qquad {\mbox{(}|x|\leq |a|{\mbox{)}
∫
x
u
d
x
=
−
1
3
u
3
(
|
x
|
≤
|
a
|
)
{\displaystyle \int xu\;dx=-{\frac {1}{3}u^{3}\qquad {\mbox{(}|x|\leq |a|{\mbox{)}
∫
x
2
u
d
x
=
−
x
4
u
3
+
a
2
8
(
x
u
+
a
2
arcsin
x
a
)
(
|
x
|
≤
|
a
|
)
{\displaystyle \int x^{2}u\;dx=-{\frac {x}{4}u^{3}+{\frac {a^{2}{8}(xu+a^{2}\arcsin {\frac {x}{a})\qquad {\mbox{(}|x|\leq |a|{\mbox{)}
∫
u
d
x
x
=
u
−
a
ln
|
a
+
u
x
|
(
|
x
|
≤
|
a
|
)
{\displaystyle \int {\frac {u\;dx}{x}=u-a\ln \left|{\frac {a+u}{x}\right|\qquad {\mbox{(}|x|\leq |a|{\mbox{)}
∫
d
x
u
=
arcsin
x
a
(
|
x
|
≤
|
a
|
)
{\displaystyle \int {\frac {dx}{u}=\arcsin {\frac {x}{a}\qquad {\mbox{(}|x|\leq |a|{\mbox{)}
∫
x
2
d
x
u
=
1
2
(
−
x
u
+
a
2
arcsin
x
a
)
(
|
x
|
≤
|
a
|
)
{\displaystyle \int {\frac {x^{2}\;dx}{u}={\frac {1}{2}\left(-xu+a^{2}\arcsin {\frac {x}{a}\right)\qquad {\mbox{(}|x|\leq |a|{\mbox{)}
∫
u
d
x
=
1
2
(
x
u
−
sgn
x
arcosh
|
x
a
|
)
(for
|
x
|
≥
|
a
|
)
{\displaystyle \int u\;dx={\frac {1}{2}\left(xu-\operatorname {sgn} x\,\operatorname {arcosh} \left|{\frac {x}{a}\right|\right)\qquad {\mbox{(for }|x|\geq |a|{\mbox{)}
∫
x
u
d
x
=
−
u
(
|
x
|
≤
|
a
|
)
{\displaystyle \int {\frac {x}{u}\;dx=-u\qquad {\mbox{(}|x|\leq |a|{\mbox{)}
R
=
a
x
2
+
b
x
+
c
{\displaystyle R={\sqrt {ax^{2}+bx+c}
(ax 2 + bx + c )は、任意のp , q に対して(px + q )2 より小さくなることはないということを前提とする。
∫
d
x
R
=
1
a
ln
|
2
a
R
+
2
a
x
+
b
|
(for
a
>
0
)
{\displaystyle \int {\frac {dx}{R}={\frac {1}{\sqrt {a}\ln \left|2{\sqrt {a}R+2ax+b\right|\qquad {\mbox{(for }a>0{\mbox{)}
∫
d
x
R
=
1
a
arsinh
2
a
x
+
b
4
a
c
−
b
2
(for
a
>
0
,
4
a
c
−
b
2
>
0
)
{\displaystyle \int {\frac {dx}{R}={\frac {1}{\sqrt {a}\,\operatorname {arsinh} {\frac {2ax+b}{\sqrt {4ac-b^{2}\qquad {\mbox{(for }a>0{\mbox{, }4ac-b^{2}>0{\mbox{)}
∫
d
x
R
=
1
a
ln
|
2
a
x
+
b
|
(for
a
>
0
,
4
a
c
−
b
2
=
0
)
{\displaystyle \int {\frac {dx}{R}={\frac {1}{\sqrt {a}\ln |2ax+b|\quad {\mbox{(for }a>0{\mbox{, }4ac-b^{2}=0{\mbox{)}
∫
d
x
R
=
−
1
−
a
arcsin
2
a
x
+
b
b
2
−
4
a
c
(for
a
<
0
,
4
a
c
−
b
2
<
0
,
|
2
a
x
+
b
|
<
b
2
−
4
a
c
)
{\displaystyle \int {\frac {dx}{R}=-{\frac {1}{\sqrt {-a}\arcsin {\frac {2ax+b}{\sqrt {b^{2}-4ac}\qquad {\mbox{(for }a<0{\mbox{, }4ac-b^{2}<0{\mbox{, }\left|2ax+b\right|<{\sqrt {b^{2}-4ac}{\mbox{)}
∫
d
x
R
3
=
4
a
x
+
2
b
(
4
a
c
−
b
2
)
R
{\displaystyle \int {\frac {dx}{R^{3}={\frac {4ax+2b}{(4ac-b^{2})R}
∫
d
x
R
5
=
4
a
x
+
2
b
3
(
4
a
c
−
b
2
)
R
(
1
R
2
+
8
a
4
a
c
−
b
2
)
{\displaystyle \int {\frac {dx}{R^{5}={\frac {4ax+2b}{3(4ac-b^{2})R}\left({\frac {1}{R^{2}+{\frac {8a}{4ac-b^{2}\right)}
∫
d
x
R
2
n
+
1
=
2
(
2
n
−
1
)
(
4
a
c
−
b
2
)
(
2
a
x
+
b
R
2
n
−
1
+
4
a
(
n
−
1
)
∫
d
x
R
2
n
−
1
)
{\displaystyle \int {\frac {dx}{R^{2n+1}={\frac {2}{(2n-1)(4ac-b^{2})}\left({\frac {2ax+b}{R^{2n-1}+4a(n-1)\int {\frac {dx}{R^{2n-1}\right)}
∫
x
R
d
x
=
R
a
−
b
2
a
∫
d
x
R
{\displaystyle \int {\frac {x}{R}\;dx={\frac {R}{a}-{\frac {b}{2a}\int {\frac {dx}{R}
∫
x
R
3
d
x
=
−
2
b
x
+
4
c
(
4
a
c
−
b
2
)
R
{\displaystyle \int {\frac {x}{R^{3}\;dx=-{\frac {2bx+4c}{(4ac-b^{2})R}
∫
x
R
2
n
+
1
d
x
=
−
1
(
2
n
−
1
)
a
R
2
n
−
1
−
b
2
a
∫
d
x
R
2
n
+
1
{\displaystyle \int {\frac {x}{R^{2n+1}\;dx=-{\frac {1}{(2n-1)aR^{2n-1}-{\frac {b}{2a}\int {\frac {dx}{R^{2n+1}
∫
d
x
x
R
=
−
1
c
ln
(
2
c
R
+
b
x
+
2
c
x
)
{\displaystyle \int {\frac {dx}{xR}=-{\frac {1}{\sqrt {c}\ln \left({\frac {2{\sqrt {c}R+bx+2c}{x}\right)}
∫
d
x
x
R
=
−
1
c
arsinh
(
b
x
+
2
c
|
x
|
4
a
c
−
b
2
)
{\displaystyle \int {\frac {dx}{xR}=-{\frac {1}{\sqrt {c}\operatorname {arsinh} \left({\frac {bx+2c}{|x|{\sqrt {4ac-b^{2}\right)}
S
=
a
x
+
b
{\displaystyle S={\sqrt {ax+b}
∫
S
d
x
=
2
S
3
3
a
{\displaystyle \int S{dx}={\frac {2S^{3}{3a}
∫
d
x
S
=
2
S
a
{\displaystyle \int {\frac {dx}{S}={\frac {2S}{a}
∫
d
x
x
S
=
{
−
2
b
a
r
c
o
t
h
(
S
b
)
(for
b
>
0
,
a
x
>
0
)
−
2
b
a
r
t
a
n
h
(
S
b
)
(for
b
>
0
,
a
x
<
0
)
2
−
b
arctan
(
S
−
b
)
(for
b
<
0
)
{\displaystyle \int {\frac {dx}{xS}={\begin{cases}-{\frac {2}{\sqrt {b}\mathrm {arcoth} \left({\frac {S}{\sqrt {b}\right)&{\mbox{(for }b>0,\quad ax>0{\mbox{)}\\-{\frac {2}{\sqrt {b}\mathrm {artanh} \left({\frac {S}{\sqrt {b}\right)&{\mbox{(for }b>0,\quad ax<0{\mbox{)}\\{\frac {2}{\sqrt {-b}\arctan \left({\frac {S}{\sqrt {-b}\right)&{\mbox{(for }b<0{\mbox{)}\\\end{cases}
∫
S
x
d
x
=
{
2
(
S
−
b
a
r
c
o
t
h
(
S
b
)
)
(for
b
>
0
,
a
x
>
0
)
2
(
S
−
b
a
r
t
a
n
h
(
S
b
)
)
(for
b
>
0
,
a
x
<
0
)
2
(
S
−
−
b
arctan
(
S
−
b
)
)
(for
b
<
0
)
{\displaystyle \int {\frac {S}{x}\,dx={\begin{cases}2\left(S-{\sqrt {b}\,\mathrm {arcoth} \left({\frac {S}{\sqrt {b}\right)\right)&{\mbox{(for }b>0,\quad ax>0{\mbox{)}\\2\left(S-{\sqrt {b}\,\mathrm {artanh} \left({\frac {S}{\sqrt {b}\right)\right)&{\mbox{(for }b>0,\quad ax<0{\mbox{)}\\2\left(S-{\sqrt {-b}\arctan \left({\frac {S}{\sqrt {-b}\right)\right)&{\mbox{(for }b<0{\mbox{)}\\\end{cases}
∫
x
n
S
d
x
=
2
a
(
2
n
+
1
)
(
x
n
S
−
b
n
∫
x
n
−
1
S
d
x
)
{\displaystyle \int {\frac {x^{n}{S}dx={\frac {2}{a(2n+1)}\left(x^{n}S-bn\int {\frac {x^{n-1}{S}dx\right)}
∫
x
n
S
d
x
=
2
a
(
2
n
+
3
)
(
x
n
S
3
−
n
b
∫
x
n
−
1
S
d
x
)
{\displaystyle \int x^{n}Sdx={\frac {2}{a(2n+3)}\left(x^{n}S^{3}-nb\int x^{n-1}Sdx\right)}
∫
1
x
n
S
d
x
=
−
1
b
(
n
−
1
)
(
S
x
n
−
1
+
(
n
−
3
2
)
a
∫
d
x
x
n
−
1
S
)
{\displaystyle \int {\frac {1}{x^{n}S}dx=-{\frac {1}{b(n-1)}\left({\frac {S}{x^{n-1}+\left(n-{\frac {3}{2}\right)a\int {\frac {dx}{x^{n-1}S}\right)}
S. Gradshteyn (И.С. Градштейн), I.M. Ryzhik (И.М. Рыжик); Alan Jeffrey, Daniel Zwillinger, editors. Table of Integrals, Series, and Products, seventh edition. Academic Press, 2007. ISBN 978-0-12-373637-6 . Errata. (Several previous editions as well.)
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A link to the original article can be found here and attribution parties here
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