x
n
{\displaystyle {\sqrt[{n}]{x}}}
Diese Tabelle von Ableitungs- und Stammfunktionen (Integraltafel ) gibt eine Übersicht über Ableitungsfunktionen und Stammfunktionen , die in der Differential- und Integralrechnung benötigt werden.
Diese Tabelle ist zweispaltig aufgebaut. In der linken Spalte steht eine Funktion , in der rechten Spalte eine Stammfunktion dieser Funktion. Die Funktion in der linken Spalte ist somit die Ableitung der Funktion in der rechten Spalte.
Hinweise:
Wenn
F
{\displaystyle F}
f
{\displaystyle f}
C
{\displaystyle C}
F
(
x
)
+
C
{\displaystyle F(x)+C}
f
{\displaystyle f}
F
(
x
)
=
1
2
x
2
+
5
{\displaystyle F(x)={\tfrac {1}{2}}x^{2}+5}
f
(
x
)
=
x
{\displaystyle f(x)=x}
f
{\displaystyle f}
f
{\displaystyle f}
C
{\displaystyle C}
Weiterhin gilt: Falls
F
(
x
)
{\displaystyle F(x)}
f
(
x
)
{\displaystyle f(x)}
Linearität des Integrals
a
⋅
F
(
x
)
{\displaystyle a\cdot F(x)}
a
⋅
f
(
x
)
{\displaystyle a\cdot f(x)}
Ebenso gilt: Sind
F
(
x
)
{\displaystyle F(x)}
G
(
x
)
{\displaystyle G(x)}
f
(
x
)
{\displaystyle f(x)}
g
(
x
)
{\displaystyle g(x)}
F
(
x
)
+
G
(
x
)
{\displaystyle F(x)+G(x)}
f
(
x
)
+
g
(
x
)
{\displaystyle f(x)+g(x)}
Funktion
f
(
x
)
{\displaystyle f(x)}
Stammfunktion
F
(
x
)
{\displaystyle F(x)}
0
{\displaystyle 0}
0
{\displaystyle 0}
k
(
k
∈
R
)
{\displaystyle k\quad (k\in \mathbb {R} )}
k
x
{\displaystyle kx}
x
n
{\displaystyle x^{n}}
{
1
n
+
1
x
n
+
1
,
wenn
n
≠
−
1
,
ln
|
x
|
,
wenn
n
=
−
1.
{\displaystyle {\begin{cases}{\frac {1}{n+1}}x^{n+1},&{\text{wenn }}n\neq -1,\\\ln |x|,&{\text{wenn }}n=-1.\end{cases}}}
n
x
n
−
1
{\displaystyle nx^{n-1}}
x
n
{\displaystyle x^{n}}
x
{\displaystyle x}
1
2
x
2
{\displaystyle {\tfrac {1}{2}}x^{2}}
2
x
{\displaystyle 2x}
x
2
{\displaystyle x^{2}}
x
2
{\displaystyle x^{2}}
1
3
x
3
{\displaystyle {\tfrac {1}{3}}x^{3}}
3
x
2
{\displaystyle 3x^{2}}
x
3
{\displaystyle x^{3}}
x
{\displaystyle {\sqrt {x}}}
2
3
x
3
2
{\displaystyle {\tfrac {2}{3}}x^{\tfrac {3}{2}}}
x
n
{\displaystyle {\sqrt[{n}]{x}}}
n
n
+
1
(
x
n
)
n
+
1
(
n
≠
−
1
)
{\displaystyle {\frac {n}{n+1}}({\sqrt[{n}]{x}})^{n+1}\quad (n\neq -1)}
1
x
{\displaystyle {\frac {1}{\sqrt {x}}}}
2
x
{\displaystyle 2{\sqrt {x}}}
1
n
(
x
n
−
1
n
)
{\displaystyle {\frac {1}{n({\sqrt[{n}]{x^{n-1}}})}}}
x
n
{\displaystyle {\sqrt[{n}]{x}}}
−
2
x
3
{\displaystyle -{\frac {2}{x^{3}}}}
1
x
2
{\displaystyle {\frac {1}{x^{2}}}}
−
1
x
2
{\displaystyle -{\frac {1}{x^{2}}}}
1
x
{\displaystyle {\frac {1}{x}}}
Funktion
f
(
x
)
{\displaystyle f(x)}
Stammfunktion
F
(
x
)
{\displaystyle F(x)}
e
x
{\displaystyle \mathrm {e} ^{x}}
e
x
{\displaystyle \mathrm {e} ^{x}}
e
k
x
{\displaystyle \mathrm {e} ^{kx}}
1
k
e
k
x
{\displaystyle {\frac {1}{k}}\mathrm {e} ^{kx}}
a
x
ln
a
(
a
>
0
)
{\displaystyle a^{x}\ln a\quad (a>0)}
a
x
{\displaystyle a^{x}}
a
x
{\displaystyle a^{x}}
a
x
ln
a
{\displaystyle {\frac {a^{x}}{\ln a}}}
x
x
(
1
+
ln
(
x
)
)
{\displaystyle x^{x}(1+\ln(x))}
x
x
(
x
>
0
)
{\displaystyle x^{x}\quad (x>0)}
e
x
ln
|
x
|
(
ln
|
x
|
+
1
)
{\displaystyle \mathrm {e} ^{x\ln \left|x\right|}(\ln \left|x\right|+1)}
|
x
|
x
=
e
x
ln
|
x
|
(
x
≠
0
)
{\displaystyle \left|x\right|^{x}=\mathrm {e} ^{x\ln \left|x\right|}\quad (x\neq 0)}
1
x
{\displaystyle {\frac {1}{x}}}
ln
|
x
|
{\displaystyle \ln \left|x\right|}
[ A 1]
ln
x
{\displaystyle \ln x}
x
ln
(
x
)
−
x
{\displaystyle x\ln(x)-x}
x
n
ln
x
{\displaystyle x^{n}\ln x}
x
n
+
1
n
+
1
(
ln
x
−
1
n
+
1
)
(
n
≥
0
)
{\displaystyle {\frac {x^{n+1}}{n+1}}\left(\ln x-{\frac {1}{n+1}}\right)\quad (n\geq 0)}
u
′
(
x
)
ln
u
(
x
)
{\displaystyle u'(x)\ln u(x)}
u
(
x
)
ln
u
(
x
)
−
u
(
x
)
{\displaystyle u(x)\ln u(x)-u(x)}
1
x
ln
n
x
(
n
≠
−
1
)
{\displaystyle {\frac {1}{x}}\ln ^{n}x\quad (n\neq -1)}
1
n
+
1
ln
n
+
1
x
{\displaystyle {\frac {1}{n+1}}\ln ^{n+1}x}
1
x
ln
x
n
(
n
≠
0
)
{\displaystyle {\frac {1}{x}}\ln {x^{n}}\quad (n\neq 0)}
1
2
n
ln
2
x
n
=
n
2
ln
2
x
{\displaystyle {\frac {1}{2n}}\ln ^{2}{x^{n}}={\frac {n}{2}}\ln ^{2}x}
1
x
1
ln
a
{\displaystyle {\frac {1}{x}}{\frac {1}{\ln a}}}
log
a
x
{\displaystyle \log _{a}x}
1
x
ln
x
{\displaystyle {\frac {1}{x\ln x}}}
ln
|
ln
x
|
(
x
>
0
,
x
≠
1
)
{\displaystyle \ln \left|\ln x\right|\quad (x>0,x\neq 1)}
log
a
x
{\displaystyle \log _{a}x}
1
ln
a
(
x
ln
x
−
x
)
{\displaystyle {\frac {1}{\ln a}}(x\ln x-x)}
Anmerkung:
↑ Sonderfall von
x
n
{\displaystyle x^{n}}
n
=
−
1
{\displaystyle n=-1}
Potenz- und Wurzelfunktionen “
Funktion
f
(
x
)
{\displaystyle f(x)}
Stammfunktion
F
(
x
)
{\displaystyle F(x)}
sin
(
x
)
{\displaystyle \sin(x)}
−
cos
(
x
)
{\displaystyle -\cos(x)}
cos
(
x
)
{\displaystyle \cos(x)}
sin
(
x
)
{\displaystyle \sin(x)}
tan
(
x
)
=
sin
(
x
)
/
cos
(
x
)
{\displaystyle \tan(x)=\sin(x)/\cos(x)}
ln
[
sec
(
x
)
]
{\displaystyle \ln {\bigl [}\sec(x){\bigr ]}}
cot
(
x
)
=
cos
(
x
)
/
sin
(
x
)
{\displaystyle \cot(x)=\cos(x)/\sin(x)}
−
ln
[
csc
(
x
)
]
{\displaystyle -\ln {\bigl [}\csc(x){\bigr ]}}
sec
(
x
)
=
1
/
cos
(
x
)
{\displaystyle \sec(x)=1/\cos(x)}
artanh
[
sin
(
x
)
]
{\displaystyle \operatorname {artanh} {\bigl [}\sin(x){\bigr ]}}
csc
(
x
)
=
1
/
sin
(
x
)
{\displaystyle \csc(x)=1/\sin(x)}
−
artanh
[
cos
(
x
)
]
{\displaystyle -\operatorname {artanh} {\bigl [}\cos(x){\bigr ]}}
sec
2
(
x
)
=
1
+
tan
2
(
x
)
{\displaystyle \operatorname {sec} ^{2}(x)=1+\tan ^{2}(x)}
tan
(
x
)
{\displaystyle \tan(x)}
−
csc
2
(
x
)
=
−
[
1
+
cot
2
(
x
)
]
{\displaystyle -\operatorname {csc} ^{2}(x)=-{\bigl [}1+\cot ^{2}(x){\bigr ]}}
cot
(
x
)
{\displaystyle \cot(x)}
sin
2
(
x
)
{\displaystyle \sin ^{2}(x)}
1
2
[
x
−
sin
(
x
)
cos
(
x
)
]
=
1
2
x
−
1
4
sin
(
2
x
)
{\displaystyle {\tfrac {1}{2}}{\bigl [}x-\sin(x)\cos(x){\bigr ]}={\tfrac {1}{2}}x-{\tfrac {1}{4}}\sin(2x)}
cos
2
(
x
)
{\displaystyle \cos ^{2}(x)}
1
2
[
x
+
sin
(
x
)
cos
(
x
)
]
=
1
2
x
+
1
4
sin
(
2
x
)
{\displaystyle {\tfrac {1}{2}}{\bigl [}x+\sin(x)\cos(x){\bigr ]}={\tfrac {1}{2}}x+{\tfrac {1}{4}}\sin(2x)}
sin
(
k
x
)
⋅
cos
(
k
x
)
{\displaystyle \sin(kx)\cdot \cos(kx)}
−
1
4
k
cos
(
2
k
x
)
{\displaystyle -{\frac {1}{4k}}\cos(2kx)}
sin
(
k
x
)
⋅
cos
(
k
x
)
{\displaystyle \sin(kx)\cdot \cos(kx)}
1
2
k
sin
2
(
k
x
)
{\displaystyle {\frac {1}{2k}}\sin ^{2}(kx)}
sin
(
a
x
)
exp
(
b
x
)
{\displaystyle {\frac {\sin(ax)}{\exp(bx)}}}
a
exp
(
b
x
)
−
a
cos
(
a
x
)
−
b
sin
(
a
x
)
(
a
2
+
b
2
)
exp
(
b
x
)
{\displaystyle {\frac {a\exp(bx)-a\cos(ax)-b\sin(ax)}{(a^{2}+b^{2})\exp(bx)}}}
cos
(
a
x
)
exp
(
b
x
)
{\displaystyle {\frac {\cos(ax)}{\exp(bx)}}}
a
sin
(
a
x
)
−
b
cos
(
a
x
)
+
b
exp
(
b
x
)
(
a
2
+
b
2
)
exp
(
b
x
)
{\displaystyle {\frac {a\sin(ax)-b\cos(ax)+b\exp(bx)}{(a^{2}+b^{2})\exp(bx)}}}
arcsin
x
{\displaystyle \arcsin x}
x
arcsin
x
+
1
−
x
2
{\displaystyle x\arcsin x+{\sqrt {1-x^{2}}}}
arccos
x
{\displaystyle \arccos x}
x
arccos
x
−
1
−
x
2
{\displaystyle x\arccos x-{\sqrt {1-x^{2}}}}
arctan
x
{\displaystyle \arctan x}
x
arctan
x
−
1
2
ln
(
1
+
x
2
)
{\displaystyle x\arctan x-{\tfrac {1}{2}}\ln \left(1+x^{2}\right)}
arccot
x
{\displaystyle \operatorname {arccot} x}
x
arccot
x
+
1
2
ln
(
1
+
x
2
)
{\displaystyle x\operatorname {arccot} x+{\tfrac {1}{2}}\ln \left(1+x^{2}\right)}
1
1
−
x
2
{\displaystyle {\frac {1}{\sqrt {1-x^{2}}}}}
arcsin
x
{\displaystyle \arcsin x}
−
1
1
−
x
2
{\displaystyle {\frac {-1}{\sqrt {1-x^{2}}}}}
arccos
x
{\displaystyle \arccos x}
1
x
2
+
1
{\displaystyle {\frac {1}{x^{2}+1}}}
arctan
x
{\displaystyle \arctan x}
x
2
x
2
+
1
{\displaystyle {\frac {x^{2}}{x^{2}+1}}}
x
−
arctan
x
{\displaystyle x-\arctan x}
1
(
x
2
+
1
)
2
{\displaystyle {\frac {1}{(x^{2}+1)^{2}}}}
1
2
(
x
x
2
+
1
+
arctan
x
)
{\displaystyle {\frac {1}{2}}\left({\frac {x}{x^{2}+1}}+\arctan x\right)}
a
2
−
x
2
{\displaystyle {\sqrt {a^{2}-x^{2}}}}
a
2
2
arcsin
(
x
a
)
+
x
2
a
2
−
x
2
{\displaystyle {\frac {a^{2}}{2}}\arcsin \left({\frac {x}{a}}\right)+{\frac {x}{2}}{\sqrt {a^{2}-x^{2}}}}
1
a
x
2
+
b
x
+
c
{\displaystyle {\frac {1}{ax^{2}+bx+c}}}
2
4
a
c
−
b
2
arctan
(
2
a
x
+
b
4
a
c
−
b
2
)
{\displaystyle {\frac {2}{\sqrt {4ac-b^{2}}}}\arctan {\biggl (}{\frac {2ax+b}{\sqrt {4ac-b^{2}}}}{\biggr )}}
Funktion
f
(
x
)
{\displaystyle f(x)}
Stammfunktion
F
(
x
)
{\displaystyle F(x)}
sinh
(
x
)
{\displaystyle \sinh(x)}
cosh
(
x
)
{\displaystyle \cosh(x)}
cosh
(
x
)
{\displaystyle \cosh(x)}
sinh
(
x
)
{\displaystyle \sinh(x)}
tanh
(
x
)
{\displaystyle \tanh(x)}
ln
[
cosh
(
x
)
]
{\displaystyle \ln {\bigl [}\cosh(x){\bigr ]}}
coth
(
x
)
{\displaystyle \coth(x)}
ln
|
sinh
(
x
)
|
{\displaystyle \ln |{\sinh(x)}|}
sech
(
x
)
{\displaystyle \operatorname {sech} (x)}
gd
(
x
)
=
arctan
[
sinh
(
x
)
]
{\displaystyle \operatorname {gd} (x)=\arctan {\bigl [}\sinh(x){\bigr ]}}
csch
(
x
)
{\displaystyle \operatorname {csch} (x)}
−
arcoth
[
cosh
(
x
)
]
{\displaystyle -\operatorname {arcoth} {\bigl [}\cosh(x){\bigr ]}}
1
cosh
2
x
=
1
−
tanh
2
x
{\displaystyle {\frac {1}{\cosh ^{2}x}}=1-\tanh ^{2}x}
tanh
x
{\displaystyle \tanh x}
−
1
sinh
2
x
=
1
−
coth
2
x
{\displaystyle {\frac {-1}{\sinh ^{2}x}}=1-\coth ^{2}x}
coth
x
{\displaystyle \coth x}
arsinh
x
{\displaystyle \operatorname {arsinh} x}
x
arsinh
x
−
x
2
+
1
{\displaystyle x\operatorname {arsinh} x-{\sqrt {x^{2}+1}}}
arcosh
x
{\displaystyle \operatorname {arcosh} x}
x
arcosh
x
−
x
2
−
1
{\displaystyle x\operatorname {arcosh} x-{\sqrt {x^{2}-1}}}
artanh
x
{\displaystyle \operatorname {artanh} x}
x
artanh
x
+
1
2
ln
(
1
−
x
2
)
{\displaystyle x\operatorname {artanh} x+{\frac {1}{2}}\ln {\left(1-x^{2}\right)}}
arcoth
x
{\displaystyle \operatorname {arcoth} x}
x
arcoth
x
+
1
2
ln
(
x
2
−
1
)
{\displaystyle x\operatorname {arcoth} x+{\frac {1}{2}}\ln {\left(x^{2}-1\right)}}
1
x
2
+
1
{\displaystyle {\frac {1}{\sqrt {x^{2}+1}}}}
arsinh
x
{\displaystyle \operatorname {arsinh} x}
1
x
2
−
1
(
x
>
1
)
{\displaystyle {\frac {1}{\sqrt {x^{2}-1}}}\quad (x>1)}
arcosh
x
{\displaystyle \operatorname {arcosh} x}
a
2
+
x
2
{\displaystyle {\sqrt {a^{2}+x^{2}}}}
a
2
2
arsinh
(
x
a
)
+
x
2
a
2
+
x
2
{\displaystyle {\frac {a^{2}}{2}}\operatorname {arsinh} \left({\frac {x}{a}}\right)+{\frac {x}{2}}{\sqrt {a^{2}+x^{2}}}}
1
a
x
2
+
b
x
+
c
{\displaystyle {\frac {1}{\sqrt {ax^{2}+bx+c}}}}
1
a
arsinh
(
2
a
x
+
b
4
a
c
−
b
2
)
{\displaystyle {\frac {1}{\sqrt {a}}}\operatorname {arsinh} {\biggl (}{\frac {2ax+b}{\sqrt {4ac-b^{2}}}}{\biggr )}}
1
1
−
x
2
(
|
x
|
<
1
)
{\displaystyle {\frac {1}{1-x^{2}}}\quad (\left|x\right|<1)}
artanh
x
{\displaystyle \operatorname {artanh} x}
1
1
−
x
2
(
|
x
|
>
1
)
{\displaystyle {\frac {1}{1-x^{2}}}\quad (\left|x\right|>1)}
arcoth
x
{\displaystyle \operatorname {arcoth} x}
Viele Stammfunktionen von algebraischen Funktionen können nicht elementar dargestellt werden. Für die Darstellung von den Stammfunktionen dieser algebraischen Funktionen genügen für die Darstellung nicht die Kreisbogenmaßfunktionen, die Hyperbelflächenmaßfunktionen, die Logarithmen und die algebraischen Funktionen alleine. Diese nicht elementar darstellbaren Integrale von den genannten algebraischen Funktionen werden elliptische Integrale genannt. Ihre Umkehrfunktionen werden als elliptische Funktionen bezeichnet. Diejenigen elliptischen Integrale, welche den Definitionsbereich der betroffenen algebraischen Funktion komplett abschließen, werden vollständige elliptische Integrale genannt. Der Quotient des vollständigen elliptischen Integrals erster Art vom Pythagoräisch komplementären Modul dividiert durch das vollständige elliptische Integral erster Art vom betroffenen Modul selbst wird als reelles Halbperiodenverhältnis oder als reelles Periodenverhältnis bezeichnet. Das elliptische Nomen ist die Exponentialfunktion aus dem negativen Produkt der Kreiszahl und des reellen Periodenverhältnisses. Die Jacobischen Thetafunktionen ordnen das elliptische Nomen den algebraischen Vielfachen von der Quadratwurzel des vollständigen elliptischen Integrals erster Art zu. Ebenso werden diejenigen Funktionen als elliptische Funktionen bezeichnet, welche als algebraische Kombinationen aus den Jacobischen Thetafunktionen hervorgehen.
Funktion
f
(
x
)
{\displaystyle f(x)}
Stammfunktion
F
(
x
)
{\displaystyle F(x)}
[
1
−
k
2
sin
(
x
)
2
]
−
1
/
2
{\displaystyle {\bigl [}1-k^{2}\sin(x)^{2}{\bigr ]}^{-1/2}}
F
(
x
;
k
)
{\displaystyle F(x;k)}
[
1
−
k
2
sin
(
x
)
2
]
1
/
2
{\displaystyle {\bigl [}1-k^{2}\sin(x)^{2}{\bigr ]}^{1/2}}
E
(
x
;
k
)
{\displaystyle E(x;k)}
1
1
−
x
4
{\displaystyle {\frac {1}{\sqrt {1-x^{4}}}}}
arcsl
(
x
)
=
1
2
2
K
(
1
2
2
)
−
1
2
2
F
[
arccos
(
x
)
;
1
2
2
]
{\displaystyle \operatorname {arcsl} (x)={\frac {1}{2}}{\sqrt {2}}\,K{\bigl (}{\frac {1}{2}}{\sqrt {2}}{\bigr )}-{\frac {1}{2}}{\sqrt {2}}\,F{\biggl [}\arccos(x);{\frac {1}{2}}{\sqrt {2}}{\biggr ]}}
x
2
+
1
1
−
x
4
{\displaystyle {\frac {x^{2}+1}{\sqrt {1-x^{4}}}}}
2
E
(
1
2
2
)
−
2
E
[
arccos
(
x
)
;
1
2
2
]
{\displaystyle {\sqrt {2}}\,E{\bigl (}{\frac {1}{2}}{\sqrt {2}}{\bigr )}-{\sqrt {2}}\,E{\biggl [}\arccos(x);{\frac {1}{2}}{\sqrt {2}}{\biggr ]}}
x
2
1
−
x
4
{\displaystyle {\frac {x^{2}}{\sqrt {1-x^{4}}}}}
1
arcsl
(
x
)
∫
0
1
y
2
+
1
2
y
2
[
artanh
(
y
2
)
−
artanh
(
1
−
x
4
y
2
1
−
x
4
y
4
)
]
d
y
{\displaystyle {\frac {1}{\operatorname {arcsl} (x)}}\int _{0}^{1}{\frac {y^{2}+1}{2y^{2}}}{\biggl [}{\text{artanh}}{\bigl (}y^{2}{\bigr )}-{\text{artanh}}{\biggl (}{\frac {{\sqrt {1-x^{4}}}\,y^{2}}{\sqrt {1-x^{4}y^{4}}}}{\biggr )}{\biggr ]}\mathrm {d} y}
1
x
4
+
1
{\displaystyle {\frac {1}{\sqrt {x^{4}+1}}}}
2
arcsl
[
x
(
x
4
+
1
+
1
)
−
1
/
2
]
{\displaystyle {\sqrt {2}}\,\operatorname {arcsl} \left[x({\sqrt {x^{4}+1}}+1)^{-1/2}\right]}
1
1
−
x
6
{\displaystyle {\frac {1}{\sqrt {1-x^{6}}}}}
1
6
27
4
F
[
2
arctan
(
3
4
x
1
−
x
2
)
;
sin
(
π
12
)
]
{\displaystyle {\frac {1}{6}}{\sqrt[{4}]{27}}\,F{\biggl [}2\arctan {\biggl (}{\frac {{\sqrt[{4}]{3}}\,x}{\sqrt {1-x^{2}}}}{\biggr )};\sin {\bigl (}{\frac {\pi }{12}}{\bigr )}{\biggr ]}}
1
x
6
+
1
{\displaystyle {\frac {1}{\sqrt {x^{6}+1}}}}
1
6
27
4
F
[
2
arctan
(
3
4
x
x
2
+
1
)
;
cos
(
π
12
)
]
{\displaystyle {\frac {1}{6}}{\sqrt[{4}]{27}}\,F{\biggl [}2\arctan {\biggl (}{\frac {{\sqrt[{4}]{3}}\,x}{\sqrt {x^{2}+1}}}{\biggr )};\cos {\bigl (}{\frac {\pi }{12}}{\bigr )}{\biggr ]}}
1
−
x
2
x
8
+
1
{\displaystyle {\frac {1-x^{2}}{\sqrt {x^{8}+1}}}}
1
2
sec
(
π
8
)
F
{
arcsin
[
2
cos
(
π
/
8
)
x
x
2
+
1
]
;
tan
(
π
8
)
}
{\displaystyle {\frac {1}{2}}\sec \left({\frac {\pi }{8}}\right)F\left\{\arcsin \left[{\frac {2\cos(\pi /8)x}{x^{2}+1}}\right];\tan \left({\frac {\pi }{8}}\right)\right\}}
x
2
+
1
x
8
+
1
{\displaystyle {\frac {x^{2}+1}{\sqrt {x^{8}+1}}}}
1
2
sec
(
π
8
)
F
{
2
arctan
[
2
cos
(
π
/
8
)
x
x
4
+
2
x
2
+
1
−
x
2
+
1
]
;
2
2
4
sin
(
π
8
)
}
{\displaystyle {\frac {1}{2}}\sec \left({\frac {\pi }{8}}\right)F\left\{2\arctan \left[{\frac {2\cos(\pi /8)x}{{\sqrt {x^{4}+{\sqrt {2}}x^{2}+1}}-x^{2}+1}}\right];2{\sqrt[{4}]{2}}\sin \left({\frac {\pi }{8}}\right)\right\}}
1
−
(
2
+
1
)
x
2
1
−
x
8
{\displaystyle {\frac {1-({\sqrt {2}}+1)\,x^{2}}{\sqrt {1-x^{8}}}}}
F
[
arcsin
(
x
1
−
x
2
1
+
x
2
)
;
tan
(
π
8
)
]
{\displaystyle F{\biggl [}\arcsin {\biggl (}{\frac {x{\sqrt {1-x^{2}}}}{\sqrt {1+x^{2}}}}{\biggr )};\tan {\bigl (}{\frac {\pi }{8}}{\bigr )}{\biggr ]}}
(
2
+
1
)
x
2
+
1
1
−
x
8
{\displaystyle {\frac {({\sqrt {2}}+1)\,x^{2}+1}{\sqrt {1-x^{8}}}}}
F
[
arctan
(
x
1
+
x
2
1
−
x
2
)
;
2
2
4
sin
(
π
8
)
]
{\displaystyle F{\biggl [}\arctan {\biggl (}{\frac {x{\sqrt {1+x^{2}}}}{\sqrt {1-x^{2}}}}{\biggr )};2{\sqrt[{4}]{2}}\sin {\bigl (}{\frac {\pi }{8}}{\bigr )}{\biggr ]}}
1
(
a
x
2
+
b
x
+
c
)
3
4
{\displaystyle {\frac {1}{\sqrt[{4}]{(ax^{2}+bx+c)^{3}}}}}
2
2
4
a
2
c
−
a
b
2
4
arcsl
[
2
a
x
+
b
4
a
(
a
x
2
+
b
x
+
c
)
+
4
a
c
−
b
2
]
{\displaystyle {\frac {2{\sqrt {2}}}{\sqrt[{4}]{4a^{2}c-ab^{2}}}}\operatorname {arcsl} \left[{\frac {2ax+b}{{\sqrt {4a(ax^{2}+bx+c)}}+{\sqrt {4ac-b^{2}}}}}\right]}
1
(
x
2
+
2
v
x
+
1
)
(
x
2
+
2
w
x
+
1
)
{\displaystyle {\frac {1}{\sqrt {(x^{2}+2vx+1)(x^{2}+2wx+1)}}}}
2
[
(
1
−
v
2
)
(
1
−
w
2
)
−
v
w
+
1
]
−
1
/
2
×
{\displaystyle {\sqrt {2}}\,{\bigl [}{\sqrt {(1-v^{2})(1-w^{2})}}-v\,w+1{\bigr ]}^{-1/2}\times }
×
F
{
arcsin
[
1
−
w
2
(
x
+
v
)
+
1
−
v
2
(
x
+
w
)
1
−
w
2
x
2
+
2
v
x
+
1
+
1
−
v
2
x
2
+
2
w
x
+
1
]
;
v
−
w
(
1
−
v
2
)
(
1
−
w
2
)
−
v
w
+
1
}
{\displaystyle \times F{\biggl \{}\arcsin {\biggl [}{\tfrac {{\sqrt {1-w^{2}}}(x+v)+{\sqrt {1-v^{2}}}(x+w)}{{\sqrt {1-w^{2}}}{\sqrt {x^{2}+2vx+1}}+{\sqrt {1-v^{2}}}{\sqrt {x^{2}+2wx+1}}}}{\biggr ]};{\tfrac {v-w}{{\sqrt {(1-v^{2})(1-w^{2})}}-v\,w+1}}{\biggr \}}}
Funktion
f
(
x
)
{\displaystyle f(x)}
Stammfunktion
F
(
x
)
{\displaystyle F(x)}
1
x
(
1
−
x
2
)
[
E
(
x
)
−
(
1
−
x
2
)
K
(
x
)
]
{\displaystyle {\frac {1}{x(1-x^{2})}}[E(x)-(1-x^{2})K(x)]}
K
(
x
)
=
∫
0
π
/
2
1
1
−
x
2
sin
(
y
)
2
d
y
=
∑
n
=
0
∞
CBC
(
n
)
2
16
n
x
2
n
{\displaystyle K(x)=\int _{0}^{\pi /2}{\frac {1}{\sqrt {1-x^{2}\sin(y)^{2}}}}\,\mathrm {d} y=\sum _{n=0}^{\infty }{\frac {\operatorname {CBC} (n)^{2}}{16^{n}}}x^{2n}}
1
x
[
E
(
x
)
−
K
(
x
)
]
{\displaystyle {\frac {1}{x}}[E(x)-K(x)]}
E
(
x
)
=
∫
0
π
/
2
1
−
x
2
sin
(
y
)
2
d
y
=
∑
n
=
0
∞
CBC
(
n
)
2
16
n
(
1
−
2
n
)
x
2
n
{\displaystyle E(x)=\int _{0}^{\pi /2}{\sqrt {1-x^{2}\sin(y)^{2}}}\,\mathrm {d} y=\sum _{n=0}^{\infty }{\frac {\operatorname {CBC} (n)^{2}}{16^{n}(1-2n)}}x^{2n}}
K
(
x
)
{\displaystyle K(x)}
∫
0
1
arcsin
(
x
y
)
y
1
−
y
2
d
y
{\displaystyle \int _{0}^{1}{\frac {\arcsin(xy)}{y{\sqrt {1-y^{2}}}}}\,\mathrm {d} y}
E
(
x
)
{\displaystyle E(x)}
∫
0
1
[
arcsin
(
x
y
)
2
y
1
−
y
2
+
x
1
−
x
2
y
2
2
1
−
y
2
]
d
y
{\displaystyle \int _{0}^{1}\left[{\frac {\arcsin(xy)}{2y{\sqrt {1-y^{2}}}}}+{\frac {x{\sqrt {1-x^{2}y^{2}}}}{2{\sqrt {1-y^{2}}}}}\right]\,\mathrm {d} y}
K
(
x
2
)
{\displaystyle K(x^{2})}
∫
0
1
2
arcsl
(
x
y
)
1
−
y
4
d
y
{\displaystyle \int _{0}^{1}{\frac {2\operatorname {arcsl} (xy)}{\sqrt {1-y^{4}}}}\,\mathrm {d} y}
E
(
x
2
)
{\displaystyle E(x^{2})}
∫
0
1
4
arcsl
(
x
y
)
3
1
−
y
4
+
2
x
y
1
−
x
4
y
4
3
1
−
y
4
d
y
{\displaystyle \int _{0}^{1}{\frac {4\operatorname {arcsl} (xy)}{3{\sqrt {1-y^{4}}}}}+{\frac {2xy{\sqrt {1-x^{4}y^{4}}}}{3{\sqrt {1-y^{4}}}}}\,\mathrm {d} y}
π
2
q
(
x
)
2
x
(
1
−
x
2
)
K
(
x
)
2
{\displaystyle {\frac {\pi ^{2}q(x)}{2x(1-x^{2})K(x)^{2}}}}
q
(
x
)
=
exp
[
−
π
K
(
1
−
x
2
)
/
K
(
x
)
]
{\displaystyle q(x)=\exp {\bigl [}-\pi \,K({\sqrt {1-x^{2}}})/K(x){\bigr ]}}
Funktion
f
(
x
)
{\displaystyle f(x)}
Stammfunktion
F
(
x
)
{\displaystyle F(x)}
sl
(
x
)
{\displaystyle \operatorname {sl} (x)}
−
arctan
[
cl
(
x
)
]
{\displaystyle -\arctan[\operatorname {cl} (x)]}
cl
(
x
)
{\displaystyle \operatorname {cl} (x)}
arctan
[
sl
(
x
)
]
{\displaystyle \arctan[\operatorname {sl} (x)]}
sn
(
x
;
k
)
=
sin
[
am
(
x
;
k
)
]
{\displaystyle \operatorname {sn} (x;k)=\sin {\bigl [}\operatorname {am} (x;k){\bigr ]}}
−
1
k
artanh
[
k
cd
(
x
;
k
)
]
{\displaystyle -{\frac {1}{k}}\operatorname {artanh} [k\,\operatorname {cd} (x;k)]}
cn
(
x
;
k
)
=
cos
[
am
(
x
;
k
)
]
{\displaystyle \operatorname {cn} (x;k)=\cos {\bigl [}\operatorname {am} (x;k){\bigr ]}}
1
k
arcsin
[
k
sn
(
x
;
k
)
]
{\displaystyle {\frac {1}{k}}\operatorname {arcsin} [k\,\operatorname {sn} (x;k)]}
dn
(
x
;
k
)
{\displaystyle \operatorname {dn} (x;k)}
am
(
x
;
k
)
{\displaystyle \operatorname {am} (x;k)}
cn
(
x
;
k
)
dn
(
x
;
k
)
{\displaystyle \operatorname {cn} (x;k)\operatorname {dn} (x;k)}
sn
(
x
;
k
)
{\displaystyle \operatorname {sn} (x;k)}
−
sn
(
x
;
k
)
dn
(
x
;
k
)
{\displaystyle -\operatorname {sn} (x;k)\operatorname {dn} (x;k)}
cn
(
x
;
k
)
{\displaystyle \operatorname {cn} (x;k)}
−
k
2
sn
(
x
;
k
)
cn
(
x
;
k
)
{\displaystyle -k^{2}\operatorname {sn} (x;k)\operatorname {cn} (x;k)}
dn
(
x
;
k
)
{\displaystyle \operatorname {dn} (x;k)}
zn
(
x
;
k
)
=
E
[
am
(
x
;
k
)
;
k
]
−
E
(
k
)
x
K
(
k
)
{\displaystyle \operatorname {zn} (x;k)=E{\bigl [}\operatorname {am} (x;k);k{\bigr ]}-{\frac {E(k)\,x}{K(k)}}}
ln
{
ϑ
01
[
π
2
K
(
k
)
−
1
x
;
q
(
k
)
]
}
{\displaystyle \ln {\bigl \{}\vartheta _{01}{\bigl [}{\frac {\pi }{2}}\,K(k)^{-1}x;q(k){\bigr ]}{\bigr \}}}
Die nicht elementaren Stammfunktionen von transzendenten Funktionen logarithmischer und arkusfunktionaler Art sowie die stammfunktionale Verkettung dieser Stammfunktionen werden als Polylogarithmen Legendresche Chifunktionen genannt. Die Bestandteile der Stammfunktionskette von den Kreisbogenmaßfunktionen werden als Arkusfunktionsintegrale wie beispielsweise als Arkustangensintegrale und Arkussinusintegrale bezeichnet. Die imaginären Gegenstücke zu den Legendreschen Chifunktionen werden akkurat durch die Arkustangensintegrale der Standardform gebildet. Die Polylogarithmen aus Exponentialfunktionsausdrücken werden Debyesche Funktionen genannt und spielen bei der statistischen Thermodynamik die essentielle Hauptrolle unter den Funktionen.
Funktion
f
(
x
)
{\displaystyle f(x)}
Stammfunktion
F
(
x
)
{\displaystyle F(x)}
1
1
−
x
{\displaystyle {\frac {1}{1-x}}}
Li
1
(
x
)
=
ln
(
1
1
−
x
)
{\displaystyle \operatorname {Li} _{1}(x)=\ln {\bigl (}{\frac {1}{1-x}}{\bigr )}}
1
x
ln
(
1
1
−
x
)
{\displaystyle {\frac {1}{x}}\ln {\bigl (}{\frac {1}{1-x}}{\bigr )}}
Li
2
(
x
)
{\displaystyle \operatorname {Li} _{2}(x)}
1
x
Li
2
(
x
)
{\displaystyle {\frac {1}{x}}\operatorname {Li} _{2}(x)}
Li
3
(
x
)
{\displaystyle \operatorname {Li} _{3}(x)}
1
x
Li
n
(
x
)
{\displaystyle {\frac {1}{x}}\operatorname {Li} _{n}(x)}
Li
n
+
1
(
x
)
{\displaystyle \operatorname {Li} _{n+1}(x)}
1
x
ln
(
1
1
−
x
)
2
{\displaystyle {\frac {1}{x}}\ln {\bigl (}{\frac {1}{1-x}}{\bigr )}^{2}}
2
Li
3
(
x
)
+
2
Li
3
(
x
x
−
1
)
+
2
Li
1
(
x
)
Li
2
(
x
)
+
1
3
Li
1
(
x
)
3
{\displaystyle 2\operatorname {Li} _{3}(x)+2\operatorname {Li} _{3}{\bigl (}{\frac {x}{x-1}}{\bigr )}+2\operatorname {Li} _{1}(x)\operatorname {Li} _{2}(x)+{\frac {1}{3}}\operatorname {Li} _{1}(x)^{3}}
1
x
artanh
(
x
)
{\displaystyle {\frac {1}{x}}\operatorname {artanh} (x)}
χ
2
(
x
)
=
Li
2
(
x
)
−
1
4
Li
2
(
x
2
)
=
∫
0
1
arcsin
(
x
y
)
1
−
y
2
d
y
{\displaystyle \chi _{2}(x)=\operatorname {Li} _{2}(x)-{\frac {1}{4}}\operatorname {Li} _{2}(x^{2})=\int _{0}^{1}{\frac {\arcsin(xy)}{\sqrt {1-y^{2}}}}\,\mathrm {d} y}
1
x
χ
2
(
x
)
{\displaystyle {\frac {1}{x}}\chi _{2}(x)}
χ
3
(
x
)
=
Li
3
(
x
)
−
1
8
Li
3
(
x
2
)
{\displaystyle \chi _{3}(x)=\operatorname {Li} _{3}(x)-{\frac {1}{8}}\operatorname {Li} _{3}(x^{2})}
ln
(
t
x
+
u
)
v
x
+
w
{\displaystyle {\frac {\ln(tx+u)}{vx+w}}}
1
v
ln
(
u
v
−
t
w
v
)
ln
(
v
x
+
w
)
−
1
v
Li
2
(
−
t
v
x
+
w
u
v
−
t
w
)
{\displaystyle {\frac {1}{v}}\ln {\biggl (}{\frac {uv-tw}{v}}{\biggr )}\ln(vx+w)-{\frac {1}{v}}\operatorname {Li} _{2}{\biggl (}-t\,{\frac {vx+w}{uv-tw}}{\biggr )}}
t
>
0
∩
v
>
0
∩
t
w
−
u
v
<
0
{\displaystyle t>0\,\cap \,v>0\,\cap \,tw-uv<0}
ln
(
t
x
+
u
)
v
x
+
w
{\displaystyle {\frac {\ln(tx+u)}{vx+w}}}
1
v
ln
(
t
v
x
+
w
t
w
−
u
v
)
ln
(
t
x
+
u
)
+
1
v
Li
2
(
−
v
t
x
+
u
t
w
−
u
v
)
{\displaystyle {\frac {1}{v}}\ln {\biggl (}t\,{\frac {vx+w}{tw-uv}}{\biggr )}\ln(tx+u)+{\frac {1}{v}}\operatorname {Li} _{2}{\biggl (}-v\,{\frac {tx+u}{tw-uv}}{\biggr )}}
t
>
0
∩
v
>
0
∩
t
w
−
u
v
>
0
{\displaystyle t>0\,\cap \,v>0\,\cap \,tw-uv>0}
artanh
(
x
)
x
1
−
x
2
{\displaystyle {\frac {\operatorname {artanh} (x)}{x{\sqrt {1-x^{2}}}}}}
2
Li
2
(
x
1
+
1
−
x
2
)
−
1
2
Li
2
(
1
−
1
−
x
2
1
+
1
−
x
2
)
{\displaystyle 2\operatorname {Li} _{2}{\biggl (}{\frac {x}{1+{\sqrt {1-x^{2}}}}}{\biggr )}-{\frac {1}{2}}\operatorname {Li} _{2}{\biggl (}{\frac {1-{\sqrt {1-x^{2}}}}{1+{\sqrt {1-x^{2}}}}}{\biggr )}}
artanh
(
x
)
x
(
1
−
x
2
)
{\displaystyle {\frac {\operatorname {artanh} (x)}{x(1-x^{2})}}}
1
2
Li
2
(
2
x
x
+
1
)
+
1
2
artanh
(
x
)
2
{\displaystyle {\frac {1}{2}}\operatorname {Li} _{2}{\biggl (}{\frac {2x}{x+1}}{\biggr )}+{\frac {1}{2}}\operatorname {artanh} (x)^{2}}
1
x
arsinh
(
x
)
{\displaystyle {\frac {1}{x}}\operatorname {arsinh} (x)}
1
2
Li
2
[
1
−
(
x
2
+
1
−
x
)
2
]
+
1
2
arsinh
(
x
)
2
{\displaystyle {\frac {1}{2}}\operatorname {Li} _{2}[1-({\sqrt {x^{2}+1}}-x)^{2}]+{\frac {1}{2}}\operatorname {arsinh} (x)^{2}}
1
x
arsinh
(
x
)
2
{\displaystyle {\frac {1}{x}}\operatorname {arsinh} (x)^{2}}
1
2
Li
3
[
1
−
(
x
2
+
1
−
x
)
2
]
+
1
2
Li
3
[
1
−
(
x
2
+
1
+
x
)
2
]
+
arsinh
(
x
)
Li
2
[
1
−
(
x
2
+
1
−
x
)
2
]
+
arsinh
(
x
)
3
{\displaystyle {\frac {1}{2}}\operatorname {Li} _{3}[1-({\sqrt {x^{2}+1}}-x)^{2}]+{\frac {1}{2}}\operatorname {Li} _{3}[1-({\sqrt {x^{2}+1}}+x)^{2}]+\operatorname {arsinh} (x)\operatorname {Li} _{2}[1-({\sqrt {x^{2}+1}}-x)^{2}]+\operatorname {arsinh} (x)^{3}}
arsinh
(
x
)
x
x
2
+
1
{\displaystyle {\frac {\operatorname {arsinh} (x)}{x{\sqrt {x^{2}+1}}}}}
2
Li
2
(
x
x
2
+
1
+
1
)
−
1
2
Li
2
(
x
2
+
1
−
1
x
2
+
1
+
1
)
=
∫
0
1
arctan
(
x
1
−
y
2
)
1
−
y
2
d
y
{\displaystyle 2\operatorname {Li} _{2}{\biggl (}{\frac {x}{{\sqrt {x^{2}+1}}+1}}{\biggr )}-{\frac {1}{2}}\operatorname {Li} _{2}{\biggl (}{\frac {{\sqrt {x^{2}+1}}-1}{{\sqrt {x^{2}+1}}+1}}{\biggr )}=\int _{0}^{1}{\frac {\arctan(x{\sqrt {1-y^{2}}})}{\sqrt {1-y^{2}}}}\,\mathrm {d} y}
arsinh
(
x
)
x
(
x
2
+
1
)
{\displaystyle {\frac {\operatorname {arsinh} (x)}{x(x^{2}+1)}}}
Li
2
(
x
x
2
+
1
)
−
1
4
Li
2
(
x
2
x
2
+
1
)
=
Li
2
[
1
−
(
x
2
+
1
−
x
)
2
]
−
1
4
Li
2
[
1
−
(
x
2
+
1
−
x
)
4
]
{\displaystyle \operatorname {Li} _{2}{\biggl (}{\frac {x}{\sqrt {x^{2}+1}}}{\biggr )}-{\frac {1}{4}}\operatorname {Li} _{2}{\biggl (}{\frac {x^{2}}{x^{2}+1}}{\biggr )}=\operatorname {Li} _{2}[1-({\sqrt {x^{2}+1}}-x)^{2}]-{\frac {1}{4}}\operatorname {Li} _{2}[1-({\sqrt {x^{2}+1}}-x)^{4}]}
Funktion
f
(
x
)
{\displaystyle f(x)}
Stammfunktion
F
(
x
)
{\displaystyle F(x)}
1
x
arctan
(
x
)
{\displaystyle {\frac {1}{x}}\arctan(x)}
Ti
2
(
x
)
{\displaystyle \operatorname {Ti} _{2}(x)}
1
x
Ti
2
(
x
)
{\displaystyle {\frac {1}{x}}\operatorname {Ti} _{2}(x)}
Ti
3
(
x
)
{\displaystyle \operatorname {Ti} _{3}(x)}
arctan
(
x
)
x
x
2
+
1
{\displaystyle {\frac {\arctan(x)}{x{\sqrt {x^{2}+1}}}}}
2
Ti
2
(
x
x
2
+
1
+
1
)
{\displaystyle 2\operatorname {Ti} _{2}{\biggl (}{\frac {x}{{\sqrt {x^{2}+1}}+1}}{\biggr )}}
arctan
(
x
)
x
(
x
2
+
1
)
{\displaystyle {\frac {\arctan(x)}{x(x^{2}+1)}}}
∫
0
1
arctan
(
x
)
−
1
−
y
2
arctan
(
x
1
−
y
2
)
y
1
−
y
2
d
y
{\displaystyle \int _{0}^{1}{\frac {\arctan(x)-{\sqrt {1-y^{2}}}\arctan(x{\sqrt {1-y^{2}}})}{y{\sqrt {1-y^{2}}}}}\,\mathrm {d} y}
Funktion
f
(
x
)
{\displaystyle f(x)}
Stammfunktion
F
(
x
)
{\displaystyle F(x)}
x
n
exp
(
x
)
−
1
{\displaystyle {\frac {x^{n}}{\exp(x)-1}}}
1
n
x
n
D
n
(
x
)
{\displaystyle {\frac {1}{n}}x^{n}\,\mathrm {D} _{n}(x)}
x
exp
(
x
)
−
1
{\displaystyle {\frac {x}{\exp(x)-1}}}
x
D
1
(
x
)
=
Li
2
[
1
−
exp
(
−
x
)
]
{\displaystyle x\,\mathrm {D} _{1}(x)=\operatorname {Li} _{2}[1-\exp(-x)]}
x
2
exp
(
x
)
−
1
{\displaystyle {\frac {x^{2}}{\exp(x)-1}}}
1
2
x
2
D
2
(
x
)
=
2
Li
3
[
1
−
exp
(
−
x
)
]
+
2
Li
3
[
1
−
exp
(
x
)
]
+
2
x
Li
2
[
1
−
exp
(
−
x
)
]
+
1
3
x
3
{\displaystyle {\frac {1}{2}}x^{2}\,\mathrm {D} _{2}(x)=2\operatorname {Li} _{3}[1-\exp(-x)]+2\operatorname {Li} _{3}[1-\exp(x)]+2x\operatorname {Li} _{2}[1-\exp(-x)]+{\frac {1}{3}}x^{3}}
1
x
arsinh
(
x
)
n
{\displaystyle {\frac {1}{x}}\operatorname {arsinh} (x)^{n}}
1
n
arsinh
(
x
)
n
D
n
[
2
arsinh
(
x
)
]
+
1
n
+
1
arsinh
(
x
)
n
+
1
{\displaystyle {\frac {1}{n}}\operatorname {arsinh} (x)^{n}\,\mathrm {D} _{n}{\bigl [}2\operatorname {arsinh} (x){\bigr ]}+{\frac {1}{n+1}}\operatorname {arsinh} (x)^{n+1}}
1
x
artanh
(
x
)
n
{\displaystyle {\frac {1}{x}}\operatorname {artanh} (x)^{n}}
1
n
artanh
(
x
)
n
{
2
D
n
[
2
artanh
(
x
)
]
−
D
n
[
4
artanh
(
x
)
]
}
{\displaystyle {\frac {1}{n}}\operatorname {artanh} (x)^{n}{\bigl \{}2\,\mathrm {D} _{n}{\bigl [}2\operatorname {artanh} (x){\bigr ]}-\mathrm {D} _{n}{\bigl [}4\operatorname {artanh} (x){\bigr ]}{\bigr \}}}
Funktion
f
(
x
)
{\displaystyle f(x)}
Stammfunktion
F
(
x
)
{\displaystyle F(x)}
ζ
(
x
)
−
x
+
1
2
x
−
2
{\displaystyle \zeta (x)-{\frac {x+1}{2x-2}}}
∫
0
∞
4
(
y
2
+
1
)
x
/
2
arctan
(
y
)
−
4
arctan
(
y
)
cos
[
x
arctan
(
y
)
]
−
2
ln
(
y
2
+
1
)
sin
[
x
arctan
(
y
)
]
(
y
2
+
1
)
x
/
2
[
ln
(
y
2
+
1
)
2
+
4
arctan
(
y
)
2
]
exp
(
π
y
)
sinh
(
π
y
)
d
y
{\displaystyle \int _{0}^{\infty }{\frac {4(y^{2}+1)^{x/2}\arctan(y)-4\arctan(y)\cos[x\arctan(y)]-2\ln(y^{2}+1)\sin[x\arctan(y)]}{(y^{2}+1)^{x/2}[\ln(y^{2}+1)^{2}+4\arctan(y)^{2}]\exp(\pi y)\sinh(\pi y)}}\,\mathrm {d} y}
λ
(
x
)
−
x
2
x
−
2
{\displaystyle \lambda (x)-{\frac {x}{2x-2}}}
∫
0
∞
2
(
y
2
+
1
)
x
/
2
arctan
(
y
)
−
2
arctan
(
y
)
cos
[
x
arctan
(
y
)
]
−
ln
(
y
2
+
1
)
sin
[
x
arctan
(
y
)
]
(
y
2
+
1
)
x
/
2
[
ln
(
y
2
+
1
)
2
+
4
arctan
(
y
)
2
]
exp
(
π
y
/
2
)
sinh
(
π
y
/
2
)
d
y
{\displaystyle \int _{0}^{\infty }{\frac {2(y^{2}+1)^{x/2}\arctan(y)-2\arctan(y)\cos[x\arctan(y)]-\ln(y^{2}+1)\sin[x\arctan(y)]}{(y^{2}+1)^{x/2}[\ln(y^{2}+1)^{2}+4\arctan(y)^{2}]\exp(\pi y/2)\sinh(\pi y/2)}}\,\mathrm {d} y}
η
(
x
)
−
1
2
{\displaystyle \eta (x)-{\frac {1}{2}}}
∫
0
∞
4
(
y
2
+
1
)
x
/
2
arctan
(
y
)
−
4
arctan
(
y
)
cos
[
x
arctan
(
y
)
]
−
2
ln
(
y
2
+
1
)
sin
[
x
arctan
(
y
)
]
(
y
2
+
1
)
x
/
2
[
ln
(
y
2
+
1
)
2
+
4
arctan
(
y
)
2
]
sinh
(
π
y
)
d
y
{\displaystyle \int _{0}^{\infty }{\frac {4(y^{2}+1)^{x/2}\arctan(y)-4\arctan(y)\cos[x\arctan(y)]-2\ln(y^{2}+1)\sin[x\arctan(y)]}{(y^{2}+1)^{x/2}[\ln(y^{2}+1)^{2}+4\arctan(y)^{2}]\sinh(\pi y)}}\,\mathrm {d} y}
β
(
x
)
−
1
2
{\displaystyle \beta (x)-{\frac {1}{2}}}
∫
0
∞
2
(
y
2
+
1
)
x
/
2
arctan
(
y
)
−
2
arctan
(
y
)
cos
[
x
arctan
(
y
)
]
−
ln
(
y
2
+
1
)
sin
[
x
arctan
(
y
)
]
(
y
2
+
1
)
x
/
2
[
ln
(
y
2
+
1
)
2
+
4
arctan
(
y
)
2
]
sinh
(
π
y
/
2
)
d
y
{\displaystyle \int _{0}^{\infty }{\frac {2(y^{2}+1)^{x/2}\arctan(y)-2\arctan(y)\cos[x\arctan(y)]-\ln(y^{2}+1)\sin[x\arctan(y)]}{(y^{2}+1)^{x/2}[\ln(y^{2}+1)^{2}+4\arctan(y)^{2}]\sinh(\pi y/2)}}\,\mathrm {d} y}
Funktion
f
(
x
)
{\displaystyle f(x)}
Stammfunktion
F
(
x
)
{\displaystyle F(x)}
u
′
(
x
)
u
(
x
)
{\displaystyle {\frac {u'(x)}{u(x)}}}
ln
|
u
(
x
)
|
{\displaystyle \ln \left|u(x)\right|}
u
′
(
x
)
⋅
u
(
x
)
{\displaystyle u'(x)\cdot u(x)}
1
2
(
u
(
x
)
)
2
{\displaystyle {\tfrac {1}{2}}(u(x))^{2}}
u
′
(
x
)
⋅
(
u
(
x
)
)
n
{\displaystyle u'(x)\cdot (u(x))^{n}}
1
n
+
1
(
u
(
x
)
)
n
+
1
{\displaystyle {\frac {1}{n+1}}(u(x))^{n+1}}
Funktion
f
(
x
)
{\displaystyle f(x)}
Stammfunktion
F
(
x
)
{\displaystyle F(x)}
1
exp
[
W
(
x
)
]
+
x
{\displaystyle {\frac {1}{\exp[W(x)]+x}}}
W
(
x
)
{\displaystyle W(x)}
W
(
x
)
=
1
π
∫
−
∞
∞
1
y
2
+
1
ln
{
1
+
x
exp
[
y
arccot
(
y
)
]
y
2
+
1
arccot
(
y
)
}
d
y
{\displaystyle W(x)={\frac {1}{\pi }}\int _{-\infty }^{\infty }{\frac {1}{y^{2}+1}}\ln {\biggl \{}1+{\frac {x\exp {\bigl [}y\operatorname {arccot}(y){\bigr ]}}{{\sqrt {y^{2}+1}}\,\operatorname {arccot}(y)}}{\biggr \}}\,\mathrm {d} y}
exp
[
W
(
x
)
]
[
W
(
x
)
2
−
W
(
x
)
+
1
]
−
1
{\displaystyle \exp {\bigl [}W(x){\bigr ]}{\bigl [}W(x)^{2}-W(x)+1{\bigr ]}-1}
1
W
(
x
)
+
1
=
∫
−
∞
∞
1
[
x
exp
(
y
)
−
y
]
2
+
π
2
d
y
{\displaystyle {\frac {1}{W(x)+1}}=\int _{-\infty }^{\infty }{\frac {1}{[x\exp(y)-y]^{2}+\pi ^{2}}}\mathrm {d} y}
exp
[
W
(
x
)
]
=
x
W
(
x
)
{\displaystyle \exp {\bigl [}W(x){\bigr ]}={\frac {x}{W(x)}}}
L
⟨
−
1
⟩
(
x
)
2
1
−
L
⟨
−
1
⟩
(
x
)
2
csch
[
L
⟨
−
1
⟩
(
x
)
]
2
{\displaystyle {\frac {L^{\langle -1\rangle }(x)^{2}}{1-L^{\langle -1\rangle }(x)^{2}\operatorname {csch} {\bigl [}L^{\langle -1\rangle }(x){\bigr ]}^{2}}}}
L
⟨
−
1
⟩
(
x
)
{\displaystyle L^{\langle -1\rangle }(x)}
Die Integralexponentialfunktion und der Integrallogarithmus sind nicht elementar lösbar. Deswegen wird in den Stammfunktionen zusätzlich die Reihenentwicklung angegeben. Die als Integrationskonstante auftretende Konstante
γ
{\displaystyle \gamma }
Euler-Mascheroni-Konstante .
Funktion
f
(
x
)
{\displaystyle f(x)}
Stammfunktion
F
(
x
)
{\displaystyle F(x)}
1
x
e
x
{\displaystyle {\frac {1}{x}}e^{x}}
Ei
(
x
)
=
γ
+
ln
|
x
|
+
∑
k
=
1
∞
x
k
k
!
⋅
k
{\displaystyle \operatorname {Ei} (x)=\gamma +\ln \left|x\right|+\sum _{k=1}^{\infty }{\frac {x^{k}}{k!\cdot k}}}
1
x
n
e
x
(
n
∈
N
)
{\displaystyle {\frac {1}{x^{n}}}e^{x}\quad (n\in \mathbb {N} )}
(
−
∑
k
=
1
n
−
1
(
k
−
1
)
!
(
n
−
1
)
!
x
−
k
)
⋅
e
x
+
Ei
(
x
)
=
−
∑
k
=
1
n
−
1
x
−
(
n
−
k
)
(
n
−
k
)
⋅
(
k
−
1
)
!
+
ln
|
x
|
(
n
−
1
)
!
+
∑
k
=
0
∞
x
k
+
1
(
k
+
1
)
⋅
(
n
+
k
)
!
{\displaystyle \left(-\sum _{k=1}^{n-1}{\frac {(k-1)!}{(n-1)!}}x^{-k}\right)\cdot e^{x}+\operatorname {Ei} (x)=-\sum _{k=1}^{n-1}{\frac {x^{-(n-k)}}{(n-k)\cdot (k-1)!}}+{\frac {\ln |x|}{(n-1)!}}+\sum _{k=0}^{\infty }{\frac {x^{k+1}}{(k+1)\cdot (n+k)!}}}
log
x
(
e
)
=
1
ln
(
x
)
{\displaystyle \log _{x}(e)={\frac {1}{\ln(x)}}}
li
(
x
)
=
Ei
(
ln
(
x
)
)
=
γ
+
ln
|
ln
x
|
+
∑
k
=
1
∞
(
ln
x
)
k
k
⋅
k
!
{\displaystyle \operatorname {li} (x)=\operatorname {Ei} (\ln(x))=\gamma +\ln \left|\ln x\right|+\sum _{k=1}^{\infty }{\frac {(\ln x)^{k}}{k\cdot k!}}}
log
x
(
a
)
=
ln
(
a
)
ln
(
x
)
{\displaystyle \log _{x}(a)={\frac {\ln(a)}{\ln(x)}}}
ln
(
a
)
⋅
li
(
x
)
{\displaystyle \ln(a)\cdot \operatorname {li} (x)}
ln
(
x
)
⋅
e
x
(
x
>
0
)
{\displaystyle \ln(x)\cdot e^{x}\quad (x>0)}
ln
(
x
)
⋅
e
x
−
Ei
(
x
)
{\displaystyle \ln(x)\cdot e^{x}-\operatorname {Ei} (x)}
ln
|
ln
x
|
(
x
>
0
,
x
≠
1
)
{\displaystyle \ln \left|\ln x\right|\quad (x>0,x\neq 1)}
ln
|
ln
x
|
−
li
(
x
)
{\displaystyle \ln \left|\ln x\right|-\operatorname {li} (x)}
Funktion
f
(
x
)
{\displaystyle f(x)}
Stammfunktion
F
(
x
)
{\displaystyle F(x)}
1
x
sin
(
x
)
{\displaystyle {\frac {1}{x}}\sin(x)}
Si
(
x
)
=
π
2
−
cos
(
x
)
∫
0
∞
exp
(
−
x
y
)
y
2
+
1
d
y
−
sin
(
x
)
∫
0
∞
y
exp
(
−
x
y
)
y
2
+
1
d
y
{\displaystyle \operatorname {Si} (x)={\frac {\pi }{2}}-\cos(x)\int _{0}^{\infty }{\frac {\exp(-xy)}{y^{2}+1}}\,\mathrm {d} y-\sin(x)\int _{0}^{\infty }{\frac {y\exp(-xy)}{y^{2}+1}}\,\mathrm {d} y}
1
x
cos
(
x
)
{\displaystyle {\frac {1}{x}}\cos(x)}
Ci
(
x
)
=
sin
(
x
)
∫
0
∞
exp
(
−
x
y
)
y
2
+
1
d
y
−
cos
(
x
)
∫
0
∞
y
exp
(
−
x
y
)
y
2
+
1
d
y
{\displaystyle \operatorname {Ci} (x)=\sin(x)\int _{0}^{\infty }{\frac {\exp(-xy)}{y^{2}+1}}\,\mathrm {d} y-\cos(x)\int _{0}^{\infty }{\frac {y\exp(-xy)}{y^{2}+1}}\,\mathrm {d} y}
e
−
x
2
{\displaystyle \mathrm {e} ^{-x^{2}}}
π
2
erf
(
x
)
=
2
π
erf
(
x
)
∫
0
1
1
−
exp
[
−
x
2
(
y
2
+
1
)
]
y
2
+
1
d
y
{\displaystyle {\frac {\sqrt {\pi }}{2}}\operatorname {erf} (x)={\frac {2}{{\sqrt {\pi }}\operatorname {erf} (x)}}\int _{0}^{1}{\frac {1-\exp[-x^{2}(y^{2}+1)]}{y^{2}+1}}\,\mathrm {d} y}
[ B 1]
e
−
a
x
2
+
b
x
+
c
{\displaystyle \mathrm {e} ^{-ax^{2}+bx+c}}
π
2
a
e
b
2
4
a
+
c
erf
(
a
x
−
b
2
a
)
{\displaystyle {\frac {\sqrt {\pi }}{2{\sqrt {a}}}}\mathrm {e} ^{{\frac {b^{2}}{4a}}+c}\operatorname {erf} \left({\sqrt {a}}x-{\frac {b}{2{\sqrt {a}}}}\right)}
[ B 1]
Funktion
f
(
x
)
{\displaystyle f(x)}
Stammfunktion
F
(
x
)
{\displaystyle F(x)}
Π
(
x
)
=
Γ
(
x
+
1
)
=
x
!
{\displaystyle \Pi (x)=\Gamma (x+1)=x!}
∫
0
∞
y
x
−
1
ln
(
y
)
exp
(
y
)
d
y
{\displaystyle \int _{0}^{\infty }{\frac {y^{x}-1}{\ln(y)\exp(y)}}\,\mathrm {d} y}
ln
[
Π
(
x
)
]
=
ln
[
Γ
(
x
+
1
)
]
{\displaystyle \ln {\bigl [}\Pi (x){\bigr ]}=\ln {\bigl [}\Gamma (x+1){\bigr ]}}
ln
[
hf
(
x
)
]
−
x
2
[
x
+
1
−
ln
(
2
π
)
]
=
(
x
+
1
)
ln
[
Π
(
x
)
]
−
ln
[
sf
(
x
)
]
−
x
2
[
x
+
1
−
ln
(
2
π
)
]
{\displaystyle \ln {\bigl [}\operatorname {hf} (x){\bigr ]}-{\frac {x}{2}}{\bigl [}x+1-\ln(2\,\pi ){\bigr ]}=(x+1)\ln {\bigl [}\Pi (x){\bigr ]}-\ln {\bigl [}\operatorname {sf} (x){\bigr ]}-{\frac {x}{2}}{\bigl [}x+1-\ln(2\,\pi ){\bigr ]}}
H
(
x
)
=
γ
+
ψ
(
x
+
1
)
{\displaystyle \mathrm {H} (x)=\gamma +\psi (x+1)}
[ B 2]
γ
x
+
ln
[
Π
(
x
)
]
=
∑
n
=
1
∞
[
x
n
−
ln
(
1
+
x
n
)
]
=
∫
0
∞
exp
(
−
x
y
)
+
x
y
−
1
y
[
exp
(
y
)
−
1
]
d
y
{\displaystyle \gamma x+\ln {\bigl [}\Pi (x){\bigr ]}=\sum _{n=1}^{\infty }{\biggl [}{\frac {x}{n}}-\ln {\biggl (}1+{\frac {x}{n}}{\biggr )}{\biggr ]}=\int _{0}^{\infty }{\frac {\exp(-xy)+xy-1}{y{\bigl [}\exp(y)-1{\bigr ]}}}\,\mathrm {d} y}
ψ
1
(
x
+
1
)
=
∑
n
=
1
∞
1
(
x
+
n
)
2
{\displaystyle \psi _{1}(x+1)=\sum _{n=1}^{\infty }{\frac {1}{(x+n)^{2}}}}
H
(
x
)
{\displaystyle \mathrm {H} (x)}
Funktion
f
(
x
)
{\displaystyle f(x)}
Stammfunktion
F
(
x
)
{\displaystyle F(x)}
I
0
(
x
)
=
∑
n
=
0
∞
x
2
n
4
n
(
n
!
)
2
{\displaystyle \mathrm {I} _{0}(x)=\sum _{n=0}^{\infty }{\frac {x^{2n}}{4^{n}(n!)^{2}}}}
∫
0
π
1
π
csc
(
y
)
sinh
[
x
sin
(
y
)
]
d
y
{\displaystyle \int _{0}^{\pi }{\frac {1}{\pi }}\csc(y)\sinh {\bigl [}x\sin(y){\bigr ]}\,\mathrm {d} y}
J
0
(
x
)
=
∑
n
=
0
∞
(
−
1
)
n
x
2
n
4
n
(
n
!
)
2
{\displaystyle \mathrm {J} _{0}(x)=\sum _{n=0}^{\infty }{\frac {(-1)^{n}x^{2n}}{4^{n}(n!)^{2}}}}
∫
0
π
1
π
csc
(
y
)
sin
[
x
sin
(
y
)
]
d
y
{\displaystyle \int _{0}^{\pi }{\frac {1}{\pi }}\csc(y)\sin {\bigl [}x\sin(y){\bigr ]}\,\mathrm {d} y}
A
i
(
x
)
{\displaystyle \mathrm {Ai} (x)}
∫
0
∞
1
π
y
[
sin
(
1
3
y
3
+
x
y
)
−
sin
(
1
3
y
3
)
]
d
y
{\displaystyle \int _{0}^{\infty }{\frac {1}{\pi \,y}}{\bigl [}\sin {\bigl (}{\frac {1}{3}}y^{3}+xy{\bigr )}-\sin {\bigl (}{\frac {1}{3}}y^{3}{\bigr )}{\bigr ]}\,\mathrm {d} y}
B
i
(
x
)
{\displaystyle \mathrm {Bi} (x)}
∫
0
∞
1
π
y
[
exp
(
−
1
3
y
3
+
x
y
)
−
exp
(
−
1
3
y
3
)
−
cos
(
1
3
y
3
+
x
y
)
+
cos
(
1
3
y
3
)
]
d
y
{\displaystyle \int _{0}^{\infty }{\frac {1}{\pi \,y}}{\bigl [}\exp {\bigl (}-{\frac {1}{3}}y^{3}+xy{\bigr )}-\exp {\bigl (}-{\frac {1}{3}}y^{3}{\bigr )}-\cos {\bigl (}{\frac {1}{3}}y^{3}+xy{\bigr )}+\cos {\bigl (}{\frac {1}{3}}y^{3}{\bigr )}{\bigr ]}\,\mathrm {d} y}
↑ a b
erf
{\displaystyle \operatorname {erf} }
Fehlerfunktion ↑
H
{\displaystyle \mathrm {H} }
Harmonische Reihe
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{\displaystyle \int {\frac {1}{(x^{2}+1)^{n}}}\,\mathrm {d} x={\frac {1}{2n-2}}\cdot {\frac {x}{(x^{2}+1)^{n-1}}}+{\frac {2n-3}{2n-2}}\cdot \int {\frac {1}{(x^{2}+1)^{n-1}}}\,\mathrm {d} x,\quad n\geq 2}
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{\displaystyle \int \sin ^{n}(x)\,\mathrm {d} x={\frac {n-1}{n}}\cdot \int \sin ^{n-2}(x)\,\mathrm {d} x-{\frac {1}{n}}\cdot \cos(x)\cdot \sin ^{n-1}(x),\quad n\geq 2}
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{\displaystyle \int \cos ^{n}(x)\,\mathrm {d} x={\frac {n-1}{n}}\cdot \int \cos ^{n-2}(x)\,\mathrm {d} x+{\frac {1}{n}}\cdot \sin(x)\cdot \cos ^{n-1}(x),\quad n\geq 2}
Für die Multiplikation zweier Stammfunktionen kann der Satz von Fubini in Kombination mit der Produktregel angewendet werden:
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{\displaystyle {\biggl [}\int _{0}^{w}f(x)\,\mathrm {d} x{\biggr ]}{\biggl [}\int _{0}^{w}g(x)\,\mathrm {d} x{\biggr ]}=\int _{0}^{1}\int _{0}^{w}xf(x)g(xy)+xg(x)f(xy)\,\mathrm {d} x\,\mathrm {d} y}
![{\displaystyle {\sqrt[{n}]{x}}}](https://wikimedia.org/api/rest_v1/media/math/render/svg/7b3ba2638d05cd9ed8dafae7e34986399e48ea99)
![{\displaystyle {\frac {n}{n+1}}({\sqrt[{n}]{x}})^{n+1}\quad (n\neq -1)}](https://wikimedia.org/api/rest_v1/media/math/render/svg/4a4acf00f99778c6eec9565031e9b1c0e4e5f616)
![{\displaystyle {\frac {1}{n({\sqrt[{n}]{x^{n-1}}})}}}](https://wikimedia.org/api/rest_v1/media/math/render/svg/c3c64448de2a4e1aaa693ff3719d698c402c2199)
![{\displaystyle \ln {\bigl [}\sec(x){\bigr ]}}](https://wikimedia.org/api/rest_v1/media/math/render/svg/4cf7a42b0fa21c7421578626774319047bc38ad8)
![{\displaystyle -\ln {\bigl [}\csc(x){\bigr ]}}](https://wikimedia.org/api/rest_v1/media/math/render/svg/ea64c6a3b0edfb2f6f1f07024ba0c5c0b2d68624)
![{\displaystyle \operatorname {artanh} {\bigl [}\sin(x){\bigr ]}}](https://wikimedia.org/api/rest_v1/media/math/render/svg/e855fd9f66bbb75ff095d81b30152229901005ab)
![{\displaystyle -\operatorname {artanh} {\bigl [}\cos(x){\bigr ]}}](https://wikimedia.org/api/rest_v1/media/math/render/svg/e387cb5cd5df4e15105b4e3faba5e0196ecf2f7e)
![{\displaystyle -\operatorname {csc} ^{2}(x)=-{\bigl [}1+\cot ^{2}(x){\bigr ]}}](https://wikimedia.org/api/rest_v1/media/math/render/svg/2c2fcce379978bf33c86154a5402d93c48fd254f)
![{\displaystyle {\tfrac {1}{2}}{\bigl [}x-\sin(x)\cos(x){\bigr ]}={\tfrac {1}{2}}x-{\tfrac {1}{4}}\sin(2x)}](https://wikimedia.org/api/rest_v1/media/math/render/svg/45b0b0b51c411bd56a9b36079a3112e60b6c9875)
![{\displaystyle {\tfrac {1}{2}}{\bigl [}x+\sin(x)\cos(x){\bigr ]}={\tfrac {1}{2}}x+{\tfrac {1}{4}}\sin(2x)}](https://wikimedia.org/api/rest_v1/media/math/render/svg/3aa464d3e3fc25332b7cbc405b2535f3d5ef6635)
![{\displaystyle \ln {\bigl [}\cosh(x){\bigr ]}}](https://wikimedia.org/api/rest_v1/media/math/render/svg/1ced3aca5a9b24ea73bc2604adb65670c1c0ca96)
![{\displaystyle \operatorname {gd} (x)=\arctan {\bigl [}\sinh(x){\bigr ]}}](https://wikimedia.org/api/rest_v1/media/math/render/svg/bf151b0da37f0636e4b2fa97e87ab18f30e1c3d8)
![{\displaystyle -\operatorname {arcoth} {\bigl [}\cosh(x){\bigr ]}}](https://wikimedia.org/api/rest_v1/media/math/render/svg/48c65be6e81a5ab3600bb7bf3d921537ee7292fb)
![{\displaystyle {\bigl [}1-k^{2}\sin(x)^{2}{\bigr ]}^{-1/2}}](https://wikimedia.org/api/rest_v1/media/math/render/svg/06e84b65b9c31d1da663c5c6e229a5d85faea5b9)
![{\displaystyle {\bigl [}1-k^{2}\sin(x)^{2}{\bigr ]}^{1/2}}](https://wikimedia.org/api/rest_v1/media/math/render/svg/5f36300b027227b17f8136f9baa8125a5d0f3ad2)
![{\displaystyle \operatorname {arcsl} (x)={\frac {1}{2}}{\sqrt {2}}\,K{\bigl (}{\frac {1}{2}}{\sqrt {2}}{\bigr )}-{\frac {1}{2}}{\sqrt {2}}\,F{\biggl [}\arccos(x);{\frac {1}{2}}{\sqrt {2}}{\biggr ]}}](https://wikimedia.org/api/rest_v1/media/math/render/svg/265d422abfd3e0431e95c3368b3b2cc72456c58d)
![{\displaystyle {\sqrt {2}}\,E{\bigl (}{\frac {1}{2}}{\sqrt {2}}{\bigr )}-{\sqrt {2}}\,E{\biggl [}\arccos(x);{\frac {1}{2}}{\sqrt {2}}{\biggr ]}}](https://wikimedia.org/api/rest_v1/media/math/render/svg/4838a7af35f8e04e2fdcae11b14e0ba1bfc0d795)
![{\displaystyle {\frac {1}{\operatorname {arcsl} (x)}}\int _{0}^{1}{\frac {y^{2}+1}{2y^{2}}}{\biggl [}{\text{artanh}}{\bigl (}y^{2}{\bigr )}-{\text{artanh}}{\biggl (}{\frac {{\sqrt {1-x^{4}}}\,y^{2}}{\sqrt {1-x^{4}y^{4}}}}{\biggr )}{\biggr ]}\mathrm {d} y}](https://wikimedia.org/api/rest_v1/media/math/render/svg/ec9f2ce0793f1a1b08f9f9a8c08e0a561d8935f5)
![{\displaystyle {\sqrt {2}}\,\operatorname {arcsl} \left[x({\sqrt {x^{4}+1}}+1)^{-1/2}\right]}](https://wikimedia.org/api/rest_v1/media/math/render/svg/18cea007c7919bf5406d2e09d38f6b314554378c)
![{\displaystyle {\frac {1}{6}}{\sqrt[{4}]{27}}\,F{\biggl [}2\arctan {\biggl (}{\frac {{\sqrt[{4}]{3}}\,x}{\sqrt {1-x^{2}}}}{\biggr )};\sin {\bigl (}{\frac {\pi }{12}}{\bigr )}{\biggr ]}}](https://wikimedia.org/api/rest_v1/media/math/render/svg/9e9fdbfadc779207ca28f231864d43dddfc6be9c)
![{\displaystyle {\frac {1}{6}}{\sqrt[{4}]{27}}\,F{\biggl [}2\arctan {\biggl (}{\frac {{\sqrt[{4}]{3}}\,x}{\sqrt {x^{2}+1}}}{\biggr )};\cos {\bigl (}{\frac {\pi }{12}}{\bigr )}{\biggr ]}}](https://wikimedia.org/api/rest_v1/media/math/render/svg/00eaf5b1d5740c9b05e1e900cf18c2e12eb4355d)
![{\displaystyle {\frac {1}{2}}\sec \left({\frac {\pi }{8}}\right)F\left\{\arcsin \left[{\frac {2\cos(\pi /8)x}{x^{2}+1}}\right];\tan \left({\frac {\pi }{8}}\right)\right\}}](https://wikimedia.org/api/rest_v1/media/math/render/svg/9760827d672fded60935b8dd662810b72b4fc318)
![{\displaystyle {\frac {1}{2}}\sec \left({\frac {\pi }{8}}\right)F\left\{2\arctan \left[{\frac {2\cos(\pi /8)x}{{\sqrt {x^{4}+{\sqrt {2}}x^{2}+1}}-x^{2}+1}}\right];2{\sqrt[{4}]{2}}\sin \left({\frac {\pi }{8}}\right)\right\}}](https://wikimedia.org/api/rest_v1/media/math/render/svg/73d2a9849e2a82f085d31f494f1d725e96f7a5b9)
![{\displaystyle F{\biggl [}\arcsin {\biggl (}{\frac {x{\sqrt {1-x^{2}}}}{\sqrt {1+x^{2}}}}{\biggr )};\tan {\bigl (}{\frac {\pi }{8}}{\bigr )}{\biggr ]}}](https://wikimedia.org/api/rest_v1/media/math/render/svg/e046aff15a3e914a46a2487c2d53e30b5679f971)
![{\displaystyle F{\biggl [}\arctan {\biggl (}{\frac {x{\sqrt {1+x^{2}}}}{\sqrt {1-x^{2}}}}{\biggr )};2{\sqrt[{4}]{2}}\sin {\bigl (}{\frac {\pi }{8}}{\bigr )}{\biggr ]}}](https://wikimedia.org/api/rest_v1/media/math/render/svg/4fa35d185bb2825fd4da38b3c12ce7a33fc7683e)
![{\displaystyle {\frac {1}{\sqrt[{4}]{(ax^{2}+bx+c)^{3}}}}}](https://wikimedia.org/api/rest_v1/media/math/render/svg/f5fd42a23e65c4b18f14fb111924bfcbd8b56600)
![{\displaystyle {\frac {2{\sqrt {2}}}{\sqrt[{4}]{4a^{2}c-ab^{2}}}}\operatorname {arcsl} \left[{\frac {2ax+b}{{\sqrt {4a(ax^{2}+bx+c)}}+{\sqrt {4ac-b^{2}}}}}\right]}](https://wikimedia.org/api/rest_v1/media/math/render/svg/dd06527bcb73c06572d551483e5b134c7adc7099)
![{\displaystyle {\sqrt {2}}\,{\bigl [}{\sqrt {(1-v^{2})(1-w^{2})}}-v\,w+1{\bigr ]}^{-1/2}\times }](https://wikimedia.org/api/rest_v1/media/math/render/svg/286ef7bf95ef81b2d8b451068a8d32b747e1ce5a)
![{\displaystyle \times F{\biggl \{}\arcsin {\biggl [}{\tfrac {{\sqrt {1-w^{2}}}(x+v)+{\sqrt {1-v^{2}}}(x+w)}{{\sqrt {1-w^{2}}}{\sqrt {x^{2}+2vx+1}}+{\sqrt {1-v^{2}}}{\sqrt {x^{2}+2wx+1}}}}{\biggr ]};{\tfrac {v-w}{{\sqrt {(1-v^{2})(1-w^{2})}}-v\,w+1}}{\biggr \}}}](https://wikimedia.org/api/rest_v1/media/math/render/svg/3169e1ffa4ebce8ad5362824d9da380aa9aab657)
![{\displaystyle {\frac {1}{x(1-x^{2})}}[E(x)-(1-x^{2})K(x)]}](https://wikimedia.org/api/rest_v1/media/math/render/svg/160ee2df30e63e5edc5bf6b7c1be0c1f68bbf6d0)
![{\displaystyle {\frac {1}{x}}[E(x)-K(x)]}](https://wikimedia.org/api/rest_v1/media/math/render/svg/35b78b897062fe6daa6d80da6a3d1bb96018a709)
![{\displaystyle \int _{0}^{1}\left[{\frac {\arcsin(xy)}{2y{\sqrt {1-y^{2}}}}}+{\frac {x{\sqrt {1-x^{2}y^{2}}}}{2{\sqrt {1-y^{2}}}}}\right]\,\mathrm {d} y}](https://wikimedia.org/api/rest_v1/media/math/render/svg/1efabd3c79eb9d5add996a38cd453ca3ae0a669a)
![{\displaystyle q(x)=\exp {\bigl [}-\pi \,K({\sqrt {1-x^{2}}})/K(x){\bigr ]}}](https://wikimedia.org/api/rest_v1/media/math/render/svg/568e3f2e81f4b55d25119a46c9dae4c8024794ac)
![{\displaystyle -\arctan[\operatorname {cl} (x)]}](https://wikimedia.org/api/rest_v1/media/math/render/svg/77d5e11dafbe96b26e27d90e3c55923c9c5c4f81)
![{\displaystyle \arctan[\operatorname {sl} (x)]}](https://wikimedia.org/api/rest_v1/media/math/render/svg/c2db75ddaa0f9d2342b564d68191dbc408dc690f)
![{\displaystyle \operatorname {sn} (x;k)=\sin {\bigl [}\operatorname {am} (x;k){\bigr ]}}](https://wikimedia.org/api/rest_v1/media/math/render/svg/563a180829b46e30c940aaaedd64e9a647b48e3e)
![{\displaystyle -{\frac {1}{k}}\operatorname {artanh} [k\,\operatorname {cd} (x;k)]}](https://wikimedia.org/api/rest_v1/media/math/render/svg/5cb99e886bdda458cd4fd85283ed0e5d39ca49bd)
![{\displaystyle \operatorname {cn} (x;k)=\cos {\bigl [}\operatorname {am} (x;k){\bigr ]}}](https://wikimedia.org/api/rest_v1/media/math/render/svg/ad121f47862c66fb0d051fa7a056abf568de9ac4)
![{\displaystyle {\frac {1}{k}}\operatorname {arcsin} [k\,\operatorname {sn} (x;k)]}](https://wikimedia.org/api/rest_v1/media/math/render/svg/b753221eaeb18cf8989e63cf25275ab500cc1c05)
![{\displaystyle \operatorname {zn} (x;k)=E{\bigl [}\operatorname {am} (x;k);k{\bigr ]}-{\frac {E(k)\,x}{K(k)}}}](https://wikimedia.org/api/rest_v1/media/math/render/svg/e03b452e59e1fdd328f864a4a241fa0874e5bfb2)
![{\displaystyle \ln {\bigl \{}\vartheta _{01}{\bigl [}{\frac {\pi }{2}}\,K(k)^{-1}x;q(k){\bigr ]}{\bigr \}}}](https://wikimedia.org/api/rest_v1/media/math/render/svg/6eaf199113a2ef7483d3f14013fc734499252961)
![{\displaystyle {\frac {1}{2\pi x}}\vartheta _{10}(x)\vartheta _{00}(x)^{2}E\left[{\frac {\vartheta _{10}(x)^{2}}{\vartheta _{00}(x)^{2}}}\right]}](https://wikimedia.org/api/rest_v1/media/math/render/svg/aba14177c6bdfb28342f672cc8b40b743aca827f)
![{\displaystyle \vartheta _{00}(x)[\vartheta _{00}(x)^{2}+\vartheta _{01}(x)^{2}]\times }](https://wikimedia.org/api/rest_v1/media/math/render/svg/fd451f072df8da0c00f5e79ade738f90ddcf56ec)
![{\displaystyle \times \left\{{\frac {1}{2\pi x}}E\left[{\frac {\vartheta _{00}(x)^{2}-\vartheta _{01}(x)^{2}}{\vartheta _{00}(x)^{2}+\vartheta _{01}(x)^{2}}}\right]-{\frac {\vartheta _{01}(x)^{2}}{4x}}\right\}}](https://wikimedia.org/api/rest_v1/media/math/render/svg/5e722c141afbf88c453e932bbfe12b447dd9667d)
![{\displaystyle \vartheta _{01}(x)[\vartheta _{00}(x)^{2}+\vartheta _{01}(x)^{2}]\times }](https://wikimedia.org/api/rest_v1/media/math/render/svg/080892919dd256fc1b1dabae61fa6070638fac11)
![{\displaystyle \times \left\{{\frac {1}{2\pi x}}E\left[{\frac {\vartheta _{00}(x)^{2}-\vartheta _{01}(x)^{2}}{\vartheta _{00}(x)^{2}+\vartheta _{01}(x)^{2}}}\right]-{\frac {\vartheta _{00}(x)^{2}}{4x}}\right\}}](https://wikimedia.org/api/rest_v1/media/math/render/svg/96b98f88d18c3f548d6655e4481d162e5eb009b6)
![{\displaystyle \vartheta _{00}{\bigl [}\exp(-x){\bigr ]}}](https://wikimedia.org/api/rest_v1/media/math/render/svg/b843558a021a81cf692ac62278d2101b6e1faa92)
![{\displaystyle x+\sum _{n=1}^{\infty }{\frac {2}{n^{2}}}{\bigl [}1-\exp(-n^{2}x){\bigr ]}}](https://wikimedia.org/api/rest_v1/media/math/render/svg/330ecb71178abee54c1c749f81b5d724e0192bf3)
![{\displaystyle \vartheta _{01}{\bigl [}\exp(-x){\bigr ]}}](https://wikimedia.org/api/rest_v1/media/math/render/svg/f804a31314aa1765cfb5dd86e65763084863a360)
![{\displaystyle x+\sum _{n=1}^{\infty }{\frac {2}{n^{2}}}(-1)^{n}{\bigl [}1-\exp(-n^{2}x){\bigr ]}}](https://wikimedia.org/api/rest_v1/media/math/render/svg/647351630901089af0992609bad58813ac697d44)
![{\displaystyle \vartheta _{01}{\bigl [}\exp(-x){\bigr ]}^{2}}](https://wikimedia.org/api/rest_v1/media/math/render/svg/1f2d4403e47a20e48765e1175359723fddbd574d)
![{\displaystyle {\frac {1}{2}}\operatorname {Li} _{2}[1-({\sqrt {x^{2}+1}}-x)^{2}]+{\frac {1}{2}}\operatorname {arsinh} (x)^{2}}](https://wikimedia.org/api/rest_v1/media/math/render/svg/0806e19444ac268fbdaa7a91c710b28520615f85)
![{\displaystyle {\frac {1}{2}}\operatorname {Li} _{3}[1-({\sqrt {x^{2}+1}}-x)^{2}]+{\frac {1}{2}}\operatorname {Li} _{3}[1-({\sqrt {x^{2}+1}}+x)^{2}]+\operatorname {arsinh} (x)\operatorname {Li} _{2}[1-({\sqrt {x^{2}+1}}-x)^{2}]+\operatorname {arsinh} (x)^{3}}](https://wikimedia.org/api/rest_v1/media/math/render/svg/8438b4abf617acded18b0e6d9fe06d34123f0696)
![{\displaystyle \operatorname {Li} _{2}{\biggl (}{\frac {x}{\sqrt {x^{2}+1}}}{\biggr )}-{\frac {1}{4}}\operatorname {Li} _{2}{\biggl (}{\frac {x^{2}}{x^{2}+1}}{\biggr )}=\operatorname {Li} _{2}[1-({\sqrt {x^{2}+1}}-x)^{2}]-{\frac {1}{4}}\operatorname {Li} _{2}[1-({\sqrt {x^{2}+1}}-x)^{4}]}](https://wikimedia.org/api/rest_v1/media/math/render/svg/aa48354aeac0ec307247a85e917df214d57c245f)
![{\displaystyle x\,\mathrm {D} _{1}(x)=\operatorname {Li} _{2}[1-\exp(-x)]}](https://wikimedia.org/api/rest_v1/media/math/render/svg/f60fa9771564a59a7cfc73b4c3778ee01bc5fd43)
![{\displaystyle {\frac {1}{2}}x^{2}\,\mathrm {D} _{2}(x)=2\operatorname {Li} _{3}[1-\exp(-x)]+2\operatorname {Li} _{3}[1-\exp(x)]+2x\operatorname {Li} _{2}[1-\exp(-x)]+{\frac {1}{3}}x^{3}}](https://wikimedia.org/api/rest_v1/media/math/render/svg/cba36615bb3d7626b1bd6c258b31b3ce241d822f)
![{\displaystyle {\frac {1}{n}}\operatorname {arsinh} (x)^{n}\,\mathrm {D} _{n}{\bigl [}2\operatorname {arsinh} (x){\bigr ]}+{\frac {1}{n+1}}\operatorname {arsinh} (x)^{n+1}}](https://wikimedia.org/api/rest_v1/media/math/render/svg/40ed05240524c8c9cb4708bd6aab868d5ed2b219)
![{\displaystyle {\frac {1}{n}}\operatorname {artanh} (x)^{n}{\bigl \{}2\,\mathrm {D} _{n}{\bigl [}2\operatorname {artanh} (x){\bigr ]}-\mathrm {D} _{n}{\bigl [}4\operatorname {artanh} (x){\bigr ]}{\bigr \}}}](https://wikimedia.org/api/rest_v1/media/math/render/svg/549c57002070b86ef540298be0c81071bfeff079)
![{\displaystyle \int _{0}^{\infty }{\frac {4(y^{2}+1)^{x/2}\arctan(y)-4\arctan(y)\cos[x\arctan(y)]-2\ln(y^{2}+1)\sin[x\arctan(y)]}{(y^{2}+1)^{x/2}[\ln(y^{2}+1)^{2}+4\arctan(y)^{2}]\exp(\pi y)\sinh(\pi y)}}\,\mathrm {d} y}](https://wikimedia.org/api/rest_v1/media/math/render/svg/0c76f3495cafe2fc649932f8d15bdc09e7fe8e31)
![{\displaystyle \int _{0}^{\infty }{\frac {2(y^{2}+1)^{x/2}\arctan(y)-2\arctan(y)\cos[x\arctan(y)]-\ln(y^{2}+1)\sin[x\arctan(y)]}{(y^{2}+1)^{x/2}[\ln(y^{2}+1)^{2}+4\arctan(y)^{2}]\exp(\pi y/2)\sinh(\pi y/2)}}\,\mathrm {d} y}](https://wikimedia.org/api/rest_v1/media/math/render/svg/1dc57ef8bbb6737279eb37d3559056c7c1f855ee)
![{\displaystyle \int _{0}^{\infty }{\frac {4(y^{2}+1)^{x/2}\arctan(y)-4\arctan(y)\cos[x\arctan(y)]-2\ln(y^{2}+1)\sin[x\arctan(y)]}{(y^{2}+1)^{x/2}[\ln(y^{2}+1)^{2}+4\arctan(y)^{2}]\sinh(\pi y)}}\,\mathrm {d} y}](https://wikimedia.org/api/rest_v1/media/math/render/svg/2c1b4a439e2bc4cf9d7335f80174dc3a58b6c38f)
![{\displaystyle \int _{0}^{\infty }{\frac {2(y^{2}+1)^{x/2}\arctan(y)-2\arctan(y)\cos[x\arctan(y)]-\ln(y^{2}+1)\sin[x\arctan(y)]}{(y^{2}+1)^{x/2}[\ln(y^{2}+1)^{2}+4\arctan(y)^{2}]\sinh(\pi y/2)}}\,\mathrm {d} y}](https://wikimedia.org/api/rest_v1/media/math/render/svg/cb67e93405f0ba8720a45a0e6b3587239979e8f0)
![{\displaystyle {\frac {1}{\exp[W(x)]+x}}}](https://wikimedia.org/api/rest_v1/media/math/render/svg/fbc344c7cb2b88ed6bde5438e4213930529ac761)
![{\displaystyle W(x)={\frac {1}{\pi }}\int _{-\infty }^{\infty }{\frac {1}{y^{2}+1}}\ln {\biggl \{}1+{\frac {x\exp {\bigl [}y\operatorname {arccot}(y){\bigr ]}}{{\sqrt {y^{2}+1}}\,\operatorname {arccot}(y)}}{\biggr \}}\,\mathrm {d} y}](https://wikimedia.org/api/rest_v1/media/math/render/svg/0684830ba8e14dd56948bcfcae9bf736ad7b258b)
![{\displaystyle \exp {\bigl [}W(x){\bigr ]}{\bigl [}W(x)^{2}-W(x)+1{\bigr ]}-1}](https://wikimedia.org/api/rest_v1/media/math/render/svg/7f53c7e193a532a76b9aebe4daee871a44ca0d28)
![{\displaystyle {\frac {1}{W(x)+1}}=\int _{-\infty }^{\infty }{\frac {1}{[x\exp(y)-y]^{2}+\pi ^{2}}}\mathrm {d} y}](https://wikimedia.org/api/rest_v1/media/math/render/svg/e36ae9c0b5d1d1b2a44e38924cd77d2043ce51cb)
![{\displaystyle \exp {\bigl [}W(x){\bigr ]}={\frac {x}{W(x)}}}](https://wikimedia.org/api/rest_v1/media/math/render/svg/de1a5250a24f5f0bdefbaff254a0529fbe3f331c)
![{\displaystyle {\frac {L^{\langle -1\rangle }(x)^{2}}{1-L^{\langle -1\rangle }(x)^{2}\operatorname {csch} {\bigl [}L^{\langle -1\rangle }(x){\bigr ]}^{2}}}}](https://wikimedia.org/api/rest_v1/media/math/render/svg/f1ce44f5ef8e9c97667680df5d2ba30cfbecf89b)
![{\displaystyle {\frac {\sqrt {\pi }}{2}}\operatorname {erf} (x)={\frac {2}{{\sqrt {\pi }}\operatorname {erf} (x)}}\int _{0}^{1}{\frac {1-\exp[-x^{2}(y^{2}+1)]}{y^{2}+1}}\,\mathrm {d} y}](https://wikimedia.org/api/rest_v1/media/math/render/svg/07e8d033ce175d105f585e7b7eda39b57666756e)
![{\displaystyle \ln {\bigl [}\Pi (x){\bigr ]}=\ln {\bigl [}\Gamma (x+1){\bigr ]}}](https://wikimedia.org/api/rest_v1/media/math/render/svg/dd03f8895eed2d0ae009ee6f5386c334a878d7d4)
![{\displaystyle \ln {\bigl [}\operatorname {hf} (x){\bigr ]}-{\frac {x}{2}}{\bigl [}x+1-\ln(2\,\pi ){\bigr ]}=(x+1)\ln {\bigl [}\Pi (x){\bigr ]}-\ln {\bigl [}\operatorname {sf} (x){\bigr ]}-{\frac {x}{2}}{\bigl [}x+1-\ln(2\,\pi ){\bigr ]}}](https://wikimedia.org/api/rest_v1/media/math/render/svg/19eb96afc5d16913bd95cd50bb59bd73d713db3d)
![{\displaystyle \gamma x+\ln {\bigl [}\Pi (x){\bigr ]}=\sum _{n=1}^{\infty }{\biggl [}{\frac {x}{n}}-\ln {\biggl (}1+{\frac {x}{n}}{\biggr )}{\biggr ]}=\int _{0}^{\infty }{\frac {\exp(-xy)+xy-1}{y{\bigl [}\exp(y)-1{\bigr ]}}}\,\mathrm {d} y}](https://wikimedia.org/api/rest_v1/media/math/render/svg/ec8ac20b23b9e2baada6e91be23ee56583dea756)
![{\displaystyle \int _{0}^{\pi }{\frac {1}{\pi }}\csc(y)\sinh {\bigl [}x\sin(y){\bigr ]}\,\mathrm {d} y}](https://wikimedia.org/api/rest_v1/media/math/render/svg/34638d93252c9b0fc33148ef6cce598e3d66308f)
![{\displaystyle \int _{0}^{\pi }{\frac {1}{\pi }}\csc(y)\sin {\bigl [}x\sin(y){\bigr ]}\,\mathrm {d} y}](https://wikimedia.org/api/rest_v1/media/math/render/svg/83cd554b4b8cb03b9e3c2943444699057f7aec30)
![{\displaystyle \int _{0}^{\infty }{\frac {1}{\pi \,y}}{\bigl [}\sin {\bigl (}{\frac {1}{3}}y^{3}+xy{\bigr )}-\sin {\bigl (}{\frac {1}{3}}y^{3}{\bigr )}{\bigr ]}\,\mathrm {d} y}](https://wikimedia.org/api/rest_v1/media/math/render/svg/2e010ffd70939aa0d94a4743e50d4cb7f337a1f8)
![{\displaystyle \int _{0}^{\infty }{\frac {1}{\pi \,y}}{\bigl [}\exp {\bigl (}-{\frac {1}{3}}y^{3}+xy{\bigr )}-\exp {\bigl (}-{\frac {1}{3}}y^{3}{\bigr )}-\cos {\bigl (}{\frac {1}{3}}y^{3}+xy{\bigr )}+\cos {\bigl (}{\frac {1}{3}}y^{3}{\bigr )}{\bigr ]}\,\mathrm {d} y}](https://wikimedia.org/api/rest_v1/media/math/render/svg/d5bbe4c206ef899f0a10ce7789605cfa6659dda0)
![{\displaystyle {\biggl [}\int _{0}^{w}f(x)\,\mathrm {d} x{\biggr ]}{\biggl [}\int _{0}^{w}g(x)\,\mathrm {d} x{\biggr ]}=\int _{0}^{1}\int _{0}^{w}xf(x)g(xy)+xg(x)f(xy)\,\mathrm {d} x\,\mathrm {d} y}](https://wikimedia.org/api/rest_v1/media/math/render/svg/85c933a91689b1ba3b23e5046ab7b0d5b5451433)
