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.

Integrals involving only hyperbolic sine functions

$\int \sinh ax\,dx={\frac {1}{a))\cosh ax+C$
$\int \sinh ^{2}ax\,dx={\frac {1}{4a))\sinh 2ax-{\frac {x}{2))+C$
$\int \sinh ^{n}ax\,dx={\begin{cases}{\frac {1}{an))(\sinh ^{n-1}ax)(\cosh ax)-{\frac {n-1}{n))\displaystyle \int \sinh ^{n-2}ax\,dx,&n>0\\{\frac {1}{a(n+1)))(\sinh ^{n+1}ax)(\cosh ax)-{\frac {n+2}{n+1))\displaystyle \int \sinh ^{n+2}ax\,dx,&n<0,n\neq -1\end{cases))$
${\begin{aligned}\int {\frac {dx}{\sinh ax))&={\frac {1}{a))\ln \left|\tanh {\frac {ax}{2))\right|+C\\&={\frac {1}{a))\ln \left|{\frac {\cosh ax+1}{\sinh ax))\right|+C\\&={\frac {1}{a))\ln \left|{\frac {\sinh ax}{\cosh ax+1))\right|+C\\&={\frac {1}{2a))\ln \left|{\frac {\cosh ax-1}{\cosh ax+1))\right|+C\end{aligned))$
$\int {\frac {dx}{\sinh ^{n}ax))=-{\frac {\cosh ax}{a(n-1)\sinh ^{n-1}ax))-{\frac {n-2}{n-1))\int {\frac {dx}{\sinh ^{n-2}ax))\qquad {\mbox{(for ))n\neq 1{\mbox{)))$
$\int x\sinh ax\,dx={\frac {1}{a))x\cosh ax-{\frac {1}{a^{2))}\sinh ax+C$
$\int (\sinh ax)(\sinh bx)\,dx={\frac {1}{a^{2}-b^{2))}{\big (}a(\sinh bx)(\cosh ax)-b(\cosh bx)(\sinh ax){\big )}+C\qquad {\mbox{(for ))a^{2}\neq b^{2}{\mbox{)))$

Integrals involving only hyperbolic cosine functions

$\int \cosh ax\,dx={\frac {1}{a))\sinh ax+C$
$\int \cosh ^{2}ax\,dx={\frac {1}{4a))\sinh 2ax+{\frac {x}{2))+C$
$\int \cosh ^{n}ax\,dx={\begin{cases}{\frac {1}{an))(\sinh ax)(\cosh ^{n-1}ax)+{\frac {n-1}{n))\displaystyle \int \cosh ^{n-2}ax\,dx,&n>0\\-{\frac {1}{a(n+1)))(\sinh ax)(\cosh ^{n+1}ax)+{\frac {n+2}{n+1))\displaystyle \int \cosh ^{n+2}ax\,dx,&n<0,n\neq -1\end{cases))$
${\begin{aligned}\int {\frac {dx}{\cosh ax))&={\frac {2}{a))\arctan e^{ax}+C\\&={\frac {1}{a))\arctan(\sinh ax)+C\end{aligned))$
$\int {\frac {dx}{\cosh ^{n}ax))={\frac {\sinh ax}{a(n-1)\cosh ^{n-1}ax))+{\frac {n-2}{n-1))\int {\frac {dx}{\cosh ^{n-2}ax))\qquad {\mbox{(for ))n\neq 1{\mbox{)))$
$\int x\cosh ax\,dx={\frac {1}{a))x\sinh ax-{\frac {1}{a^{2))}\cosh ax+C$
$\int x^{2}\cosh ax\,dx=-{\frac {2x\cosh ax}{a^{2))}+\left({\frac {x^{2)){a))+{\frac {2}{a^{3))}\right)\sinh ax+C$
$\int (\cosh ax)(\cosh bx)\,dx={\frac {1}{a^{2}-b^{2))}{\big (}a(\sinh ax)(\cosh bx)-b(\sinh bx)(\cosh ax){\big )}+C\qquad {\mbox{(for ))a^{2}\neq b^{2}{\mbox{)))$
$\int {\frac {dx}{1+\cosh(ax)))={\frac {2}{a)){\frac {1}{1+e^{-ax))}+C\quad$ or ${\frac {2}{a))$ times The Logistic Function

Other integrals

Integrals of hyperbolic tangent, cotangent, secant, cosecant functions

$\int \tanh x\,dx=\ln \cosh x+C$
$\int \tanh ^{2}ax\,dx=x-{\frac {\tanh ax}{a))+C$
$\int \tanh ^{n}ax\,dx=-{\frac {1}{a(n-1)))\tanh ^{n-1}ax+\int \tanh ^{n-2}ax\,dx\qquad {\mbox{(for ))n\neq 1{\mbox{)))$
$\int \coth x\,dx=\ln |\sinh x|+C,{\text{ for ))x\neq 0$
$\int \coth ^{n}ax\,dx=-{\frac {1}{a(n-1)))\coth ^{n-1}ax+\int \coth ^{n-2}ax\,dx\qquad {\mbox{(for ))n\neq 1{\mbox{)))$
$\int \operatorname {sech} \,x\,dx=\arctan \,(\sinh x)+C$
$\int \operatorname {csch} \,x\,dx=\ln \left|\tanh {x \over 2}\right|+C=\ln \left|\coth {x}-\operatorname {csch} {x}\right|+C,{\text{ for ))x\neq 0$

Integrals involving hyperbolic sine and cosine functions

$\int (\cosh ax)(\sinh bx)\,dx={\frac {1}{a^{2}-b^{2))}{\big (}a(\sinh ax)(\sinh bx)-b(\cosh ax)(\cosh bx){\big )}+C\qquad {\mbox{(for ))a^{2}\neq b^{2}{\mbox{)))$
${\begin{aligned}\int {\frac {\cosh ^{n}ax}{\sinh ^{m}ax))\,dx&={\frac {\cosh ^{n-1}ax}{a(n-m)\sinh ^{m-1}ax))+{\frac {n-1}{n-m))\int {\frac {\cosh ^{n-2}ax}{\sinh ^{m}ax))\,dx\qquad {\mbox{(for ))m\neq n{\mbox{)))\\&=-{\frac {\cosh ^{n+1}ax}{a(m-1)\sinh ^{m-1}ax))+{\frac {n-m+2}{m-1))\int {\frac {\cosh ^{n}ax}{\sinh ^{m-2}ax))\,dx\qquad {\mbox{(for ))m\neq 1{\mbox{)))\\&=-{\frac {\cosh ^{n-1}ax}{a(m-1)\sinh ^{m-1}ax))+{\frac {n-1}{m-1))\int {\frac {\cosh ^{n-2}ax}{\sinh ^{m-2}ax))\,dx\qquad {\mbox{(for ))m\neq 1{\mbox{)))\end{aligned))$
${\begin{aligned}\int {\frac {\sinh ^{m}ax}{\cosh ^{n}ax))\,dx&={\frac {\sinh ^{m-1}ax}{a(m-n)\cosh ^{n-1}ax))+{\frac {m-1}{n-m))\int {\frac {\sinh ^{m-2}ax}{\cosh ^{n}ax))\,dx\qquad {\mbox{(for ))m\neq n{\mbox{)))\\&={\frac {\sinh ^{m+1}ax}{a(n-1)\cosh ^{n-1}ax))+{\frac {m-n+2}{n-1))\int {\frac {\sinh ^{m}ax}{\cosh ^{n-2}ax))\,dx\qquad {\mbox{(for ))n\neq 1{\mbox{)))\\&=-{\frac {\sinh ^{m-1}ax}{a(n-1)\cosh ^{n-1}ax))+{\frac {m-1}{n-1))\int {\frac {\sinh ^{m-2}ax}{\cosh ^{n-2}ax))\,dx\qquad {\mbox{(for ))n\neq 1{\mbox{)))\end{aligned))$

Integrals involving hyperbolic and trigonometric functions

$\int \sinh(ax+b)\sin(cx+d)\,dx={\frac {a}{a^{2}+c^{2))}\cosh(ax+b)\sin(cx+d)-{\frac {c}{a^{2}+c^{2))}\sinh(ax+b)\cos(cx+d)+C$
$\int \sinh(ax+b)\cos(cx+d)\,dx={\frac {a}{a^{2}+c^{2))}\cosh(ax+b)\cos(cx+d)+{\frac {c}{a^{2}+c^{2))}\sinh(ax+b)\sin(cx+d)+C$
$\int \cosh(ax+b)\sin(cx+d)\,dx={\frac {a}{a^{2}+c^{2))}\sinh(ax+b)\sin(cx+d)-{\frac {c}{a^{2}+c^{2))}\cosh(ax+b)\cos(cx+d)+C$
$\int \cosh(ax+b)\cos(cx+d)\,dx={\frac {a}{a^{2}+c^{2))}\sinh(ax+b)\cos(cx+d)+{\frac {c}{a^{2}+c^{2))}\cosh(ax+b)\sin(cx+d)+C$

v t e Lists of integrals
Rational functions Irrational functions Trigonometric functions Inverse trigonometric functions Hyperbolic functions Inverse hyperbolic functions Exponential functions Logarithmic functions Gaussian functions Definite integrals

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