The following is a list of integrals (anti-derivative functions) of hyperbolic functions . For a complete list of integral functions, see list of integrals .
In all formulas the constant a is assumed to be nonzero, and C
denotes the constant of integration .
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Integral of Hyperbolic Functions
Formulas for the Derivatives and Integrals of the Hyperbolic Functions
Integration of Hyperbolic Functions (3) | ExamSolutions
Integration of Hyperbolic Functions (7) | ExamSolutions
Integration of Hyperbolic Functions (4) | ExamSolutions
Integrals involving only hyperbolic sine functions
∫
sinh
a
x
d
x
=
1
a
cosh
a
x
+
C
{\displaystyle \int \sinh ax\,dx={\frac {1}{a}}\cosh ax+C}
∫
sinh
2
a
x
d
x
=
1
4
a
sinh
2
a
x
−
x
2
+
C
{\displaystyle \int \sinh ^{2}ax\,dx={\frac {1}{4a}}\sinh 2ax-{\frac {x}{2}}+C}
∫
sinh
n
a
x
d
x
=
{
1
a
n
(
sinh
n
−
1
a
x
)
(
cosh
a
x
)
−
n
−
1
n
∫
sinh
n
−
2
a
x
d
x
,
n
>
0
1
a
(
n
+
1
)
(
sinh
n
+
1
a
x
)
(
cosh
a
x
)
−
n
+
2
n
+
1
∫
sinh
n
+
2
a
x
d
x
,
n
<
0
,
n
≠
−
1
{\displaystyle \int \sinh ^{n}ax\,dx={\begin{cases}{\frac {1}{an}}(\sinh ^{n-1}ax)(\cosh ax)-{\frac {n-1}{n}}\displaystyle \int \sinh ^{n-2}ax\,dx,&n>0\\{\frac {1}{a(n+1)}}(\sinh ^{n+1}ax)(\cosh ax)-{\frac {n+2}{n+1}}\displaystyle \int \sinh ^{n+2}ax\,dx,&n<0,n\neq -1\end{cases}}}
∫
d
x
sinh
a
x
=
1
a
ln
|
tanh
a
x
2
|
+
C
=
1
a
ln
|
cosh
a
x
+
1
sinh
a
x
|
+
C
=
1
a
ln
|
sinh
a
x
cosh
a
x
+
1
|
+
C
=
1
2
a
ln
|
cosh
a
x
−
1
cosh
a
x
+
1
|
+
C
{\displaystyle {\begin{aligned}\int {\frac {dx}{\sinh ax}}&={\frac {1}{a}}\ln \left|\tanh {\frac {ax}{2}}\right|+C\\&={\frac {1}{a}}\ln \left|{\frac {\cosh ax+1}{\sinh ax}}\right|+C\\&={\frac {1}{a}}\ln \left|{\frac {\sinh ax}{\cosh ax+1}}\right|+C\\&={\frac {1}{2a}}\ln \left|{\frac {\cosh ax-1}{\cosh ax+1}}\right|+C\end{aligned}}}
∫
d
x
sinh
n
a
x
=
−
cosh
a
x
a
(
n
−
1
)
sinh
n
−
1
a
x
−
n
−
2
n
−
1
∫
d
x
sinh
n
−
2
a
x
(for
n
≠
1
)
{\displaystyle \int {\frac {dx}{\sinh ^{n}ax}}=-{\frac {\cosh ax}{a(n-1)\sinh ^{n-1}ax}}-{\frac {n-2}{n-1}}\int {\frac {dx}{\sinh ^{n-2}ax}}\qquad {\mbox{(for }}n\neq 1{\mbox{)}}}
∫
x
sinh
a
x
d
x
=
1
a
x
cosh
a
x
−
1
a
2
sinh
a
x
+
C
{\displaystyle \int x\sinh ax\,dx={\frac {1}{a}}x\cosh ax-{\frac {1}{a^{2}}}\sinh ax+C}
∫
(
sinh
a
x
)
(
sinh
b
x
)
d
x
=
1
a
2
−
b
2
(
a
(
sinh
b
x
)
(
cosh
a
x
)
−
b
(
cosh
b
x
)
(
sinh
a
x
)
)
+
C
(for
a
2
≠
b
2
)
{\displaystyle \int (\sinh ax)(\sinh bx)\,dx={\frac {1}{a^{2}-b^{2}}}{\big (}a(\sinh bx)(\cosh ax)-b(\cosh bx)(\sinh ax){\big )}+C\qquad {\mbox{(for }}a^{2}\neq b^{2}{\mbox{)}}}
Integrals involving only hyperbolic cosine functions
∫
cosh
a
x
d
x
=
1
a
sinh
a
x
+
C
{\displaystyle \int \cosh ax\,dx={\frac {1}{a}}\sinh ax+C}
∫
cosh
2
a
x
d
x
=
1
4
a
sinh
2
a
x
+
x
2
+
C
{\displaystyle \int \cosh ^{2}ax\,dx={\frac {1}{4a}}\sinh 2ax+{\frac {x}{2}}+C}
∫
cosh
n
a
x
d
x
=
{
1
a
n
(
sinh
a
x
)
(
cosh
n
−
1
a
x
)
+
n
−
1
n
∫
cosh
n
−
2
a
x
d
x
,
n
>
0
−
1
a
(
n
+
1
)
(
sinh
a
x
)
(
cosh
n
+
1
a
x
)
+
n
+
2
n
+
1
∫
cosh
n
+
2
a
x
d
x
,
n
<
0
,
n
≠
−
1
{\displaystyle \int \cosh ^{n}ax\,dx={\begin{cases}{\frac {1}{an}}(\sinh ax)(\cosh ^{n-1}ax)+{\frac {n-1}{n}}\displaystyle \int \cosh ^{n-2}ax\,dx,&n>0\\-{\frac {1}{a(n+1)}}(\sinh ax)(\cosh ^{n+1}ax)+{\frac {n+2}{n+1}}\displaystyle \int \cosh ^{n+2}ax\,dx,&n<0,n\neq -1\end{cases}}}
∫
d
x
cosh
a
x
=
2
a
arctan
e
a
x
+
C
=
1
a
arctan
(
sinh
a
x
)
+
C
{\displaystyle {\begin{aligned}\int {\frac {dx}{\cosh ax}}&={\frac {2}{a}}\arctan e^{ax}+C\\&={\frac {1}{a}}\arctan(\sinh ax)+C\end{aligned}}}
∫
d
x
cosh
n
a
x
=
sinh
a
x
a
(
n
−
1
)
cosh
n
−
1
a
x
+
n
−
2
n
−
1
∫
d
x
cosh
n
−
2
a
x
(for
n
≠
1
)
{\displaystyle \int {\frac {dx}{\cosh ^{n}ax}}={\frac {\sinh ax}{a(n-1)\cosh ^{n-1}ax}}+{\frac {n-2}{n-1}}\int {\frac {dx}{\cosh ^{n-2}ax}}\qquad {\mbox{(for }}n\neq 1{\mbox{)}}}
∫
x
cosh
a
x
d
x
=
1
a
x
sinh
a
x
−
1
a
2
cosh
a
x
+
C
{\displaystyle \int x\cosh ax\,dx={\frac {1}{a}}x\sinh ax-{\frac {1}{a^{2}}}\cosh ax+C}
∫
x
2
cosh
a
x
d
x
=
−
2
x
cosh
a
x
a
2
+
(
x
2
a
+
2
a
3
)
sinh
a
x
+
C
{\displaystyle \int x^{2}\cosh ax\,dx=-{\frac {2x\cosh ax}{a^{2}}}+\left({\frac {x^{2}}{a}}+{\frac {2}{a^{3}}}\right)\sinh ax+C}
∫
(
cosh
a
x
)
(
cosh
b
x
)
d
x
=
1
a
2
−
b
2
(
a
(
sinh
a
x
)
(
cosh
b
x
)
−
b
(
sinh
b
x
)
(
cosh
a
x
)
)
+
C
(for
a
2
≠
b
2
)
{\displaystyle \int (\cosh ax)(\cosh bx)\,dx={\frac {1}{a^{2}-b^{2}}}{\big (}a(\sinh ax)(\cosh bx)-b(\sinh bx)(\cosh ax){\big )}+C\qquad {\mbox{(for }}a^{2}\neq b^{2}{\mbox{)}}}
∫
d
x
1
+
cosh
(
a
x
)
=
2
a
1
1
+
e
−
a
x
+
C
{\displaystyle \int {\frac {dx}{1+\cosh(ax)}}={\frac {2}{a}}{\frac {1}{1+e^{-ax}}}+C\quad }
2
a
{\displaystyle {\frac {2}{a}}}
The Logistic Function
Other integrals Integrals of hyperbolic tangent, cotangent, secant, cosecant functions
∫
tanh
x
d
x
=
ln
cosh
x
+
C
{\displaystyle \int \tanh x\,dx=\ln \cosh x+C}
∫
tanh
2
a
x
d
x
=
x
−
tanh
a
x
a
+
C
{\displaystyle \int \tanh ^{2}ax\,dx=x-{\frac {\tanh ax}{a}}+C}
∫
tanh
n
a
x
d
x
=
−
1
a
(
n
−
1
)
tanh
n
−
1
a
x
+
∫
tanh
n
−
2
a
x
d
x
(for
n
≠
1
)
{\displaystyle \int \tanh ^{n}ax\,dx=-{\frac {1}{a(n-1)}}\tanh ^{n-1}ax+\int \tanh ^{n-2}ax\,dx\qquad {\mbox{(for }}n\neq 1{\mbox{)}}}
∫
coth
x
d
x
=
ln
|
sinh
x
|
+
C
,
for
x
≠
0
{\displaystyle \int \coth x\,dx=\ln |\sinh x|+C,{\text{ for }}x\neq 0}
∫
coth
n
a
x
d
x
=
−
1
a
(
n
−
1
)
coth
n
−
1
a
x
+
∫
coth
n
−
2
a
x
d
x
(for
n
≠
1
)
{\displaystyle \int \coth ^{n}ax\,dx=-{\frac {1}{a(n-1)}}\coth ^{n-1}ax+\int \coth ^{n-2}ax\,dx\qquad {\mbox{(for }}n\neq 1{\mbox{)}}}
∫
sech
x
d
x
=
arctan
(
sinh
x
)
+
C
{\displaystyle \int \operatorname {sech} \,x\,dx=\arctan \,(\sinh x)+C}
∫
csch
x
d
x
=
ln
|
tanh
x
2
|
+
C
=
ln
|
coth
x
−
csch
x
|
+
C
,
for
x
≠
0
{\displaystyle \int \operatorname {csch} \,x\,dx=\ln \left|\tanh {x \over 2}\right|+C=\ln \left|\coth {x}-\operatorname {csch} {x}\right|+C,{\text{ for }}x\neq 0}
Integrals involving hyperbolic sine and cosine functions
∫
(
cosh
a
x
)
(
sinh
b
x
)
d
x
=
1
a
2
−
b
2
(
a
(
sinh
a
x
)
(
sinh
b
x
)
−
b
(
cosh
a
x
)
(
cosh
b
x
)
)
+
C
(for
a
2
≠
b
2
)
{\displaystyle \int (\cosh ax)(\sinh bx)\,dx={\frac {1}{a^{2}-b^{2}}}{\big (}a(\sinh ax)(\sinh bx)-b(\cosh ax)(\cosh bx){\big )}+C\qquad {\mbox{(for }}a^{2}\neq b^{2}{\mbox{)}}}
∫
cosh
n
a
x
sinh
m
a
x
d
x
=
cosh
n
−
1
a
x
a
(
n
−
m
)
sinh
m
−
1
a
x
+
n
−
1
n
−
m
∫
cosh
n
−
2
a
x
sinh
m
a
x
d
x
(for
m
≠
n
)
=
−
cosh
n
+
1
a
x
a
(
m
−
1
)
sinh
m
−
1
a
x
+
n
−
m
+
2
m
−
1
∫
cosh
n
a
x
sinh
m
−
2
a
x
d
x
(for
m
≠
1
)
=
−
cosh
n
−
1
a
x
a
(
m
−
1
)
sinh
m
−
1
a
x
+
n
−
1
m
−
1
∫
cosh
n
−
2
a
x
sinh
m
−
2
a
x
d
x
(for
m
≠
1
)
{\displaystyle {\begin{aligned}\int {\frac {\cosh ^{n}ax}{\sinh ^{m}ax}}\,dx&={\frac {\cosh ^{n-1}ax}{a(n-m)\sinh ^{m-1}ax}}+{\frac {n-1}{n-m}}\int {\frac {\cosh ^{n-2}ax}{\sinh ^{m}ax}}\,dx\qquad {\mbox{(for }}m\neq n{\mbox{)}}\\&=-{\frac {\cosh ^{n+1}ax}{a(m-1)\sinh ^{m-1}ax}}+{\frac {n-m+2}{m-1}}\int {\frac {\cosh ^{n}ax}{\sinh ^{m-2}ax}}\,dx\qquad {\mbox{(for }}m\neq 1{\mbox{)}}\\&=-{\frac {\cosh ^{n-1}ax}{a(m-1)\sinh ^{m-1}ax}}+{\frac {n-1}{m-1}}\int {\frac {\cosh ^{n-2}ax}{\sinh ^{m-2}ax}}\,dx\qquad {\mbox{(for }}m\neq 1{\mbox{)}}\end{aligned}}}
∫
sinh
m
a
x
cosh
n
a
x
d
x
=
sinh
m
−
1
a
x
a
(
m
−
n
)
cosh
n
−
1
a
x
+
m
−
1
n
−
m
∫
sinh
m
−
2
a
x
cosh
n
a
x
d
x
(for
m
≠
n
)
=
sinh
m
+
1
a
x
a
(
n
−
1
)
cosh
n
−
1
a
x
+
m
−
n
+
2
n
−
1
∫
sinh
m
a
x
cosh
n
−
2
a
x
d
x
(for
n
≠
1
)
=
−
sinh
m
−
1
a
x
a
(
n
−
1
)
cosh
n
−
1
a
x
+
m
−
1
n
−
1
∫
sinh
m
−
2
a
x
cosh
n
−
2
a
x
d
x
(for
n
≠
1
)
{\displaystyle {\begin{aligned}\int {\frac {\sinh ^{m}ax}{\cosh ^{n}ax}}\,dx&={\frac {\sinh ^{m-1}ax}{a(m-n)\cosh ^{n-1}ax}}+{\frac {m-1}{n-m}}\int {\frac {\sinh ^{m-2}ax}{\cosh ^{n}ax}}\,dx\qquad {\mbox{(for }}m\neq n{\mbox{)}}\\&={\frac {\sinh ^{m+1}ax}{a(n-1)\cosh ^{n-1}ax}}+{\frac {m-n+2}{n-1}}\int {\frac {\sinh ^{m}ax}{\cosh ^{n-2}ax}}\,dx\qquad {\mbox{(for }}n\neq 1{\mbox{)}}\\&=-{\frac {\sinh ^{m-1}ax}{a(n-1)\cosh ^{n-1}ax}}+{\frac {m-1}{n-1}}\int {\frac {\sinh ^{m-2}ax}{\cosh ^{n-2}ax}}\,dx\qquad {\mbox{(for }}n\neq 1{\mbox{)}}\end{aligned}}}
Integrals involving hyperbolic and trigonometric functions
∫
sinh
(
a
x
+
b
)
sin
(
c
x
+
d
)
d
x
=
a
a
2
+
c
2
cosh
(
a
x
+
b
)
sin
(
c
x
+
d
)
−
c
a
2
+
c
2
sinh
(
a
x
+
b
)
cos
(
c
x
+
d
)
+
C
{\displaystyle \int \sinh(ax+b)\sin(cx+d)\,dx={\frac {a}{a^{2}+c^{2}}}\cosh(ax+b)\sin(cx+d)-{\frac {c}{a^{2}+c^{2}}}\sinh(ax+b)\cos(cx+d)+C}
∫
sinh
(
a
x
+
b
)
cos
(
c
x
+
d
)
d
x
=
a
a
2
+
c
2
cosh
(
a
x
+
b
)
cos
(
c
x
+
d
)
+
c
a
2
+
c
2
sinh
(
a
x
+
b
)
sin
(
c
x
+
d
)
+
C
{\displaystyle \int \sinh(ax+b)\cos(cx+d)\,dx={\frac {a}{a^{2}+c^{2}}}\cosh(ax+b)\cos(cx+d)+{\frac {c}{a^{2}+c^{2}}}\sinh(ax+b)\sin(cx+d)+C}
∫
cosh
(
a
x
+
b
)
sin
(
c
x
+
d
)
d
x
=
a
a
2
+
c
2
sinh
(
a
x
+
b
)
sin
(
c
x
+
d
)
−
c
a
2
+
c
2
cosh
(
a
x
+
b
)
cos
(
c
x
+
d
)
+
C
{\displaystyle \int \cosh(ax+b)\sin(cx+d)\,dx={\frac {a}{a^{2}+c^{2}}}\sinh(ax+b)\sin(cx+d)-{\frac {c}{a^{2}+c^{2}}}\cosh(ax+b)\cos(cx+d)+C}
∫
cosh
(
a
x
+
b
)
cos
(
c
x
+
d
)
d
x
=
a
a
2
+
c
2
sinh
(
a
x
+
b
)
cos
(
c
x
+
d
)
+
c
a
2
+
c
2
cosh
(
a
x
+
b
)
sin
(
c
x
+
d
)
+
C
{\displaystyle \int \cosh(ax+b)\cos(cx+d)\,dx={\frac {a}{a^{2}+c^{2}}}\sinh(ax+b)\cos(cx+d)+{\frac {c}{a^{2}+c^{2}}}\cosh(ax+b)\sin(cx+d)+C}
This page was last edited on 18 December 2023, at 16:57
This page was last edited on 18 December 2023, at 16:57