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

${\displaystyle \int \sinh ax\,dx={\frac {1}{a))\cosh ax+C}$

${\displaystyle \int \sinh ^{2}ax\,dx={\frac {1}{4a))\sinh 2ax-{\frac {x}{2))+C}$

${\displaystyle \int \sinh ^{n}ax\,dx={\frac {1}{an))(\sinh ^{n-1}ax)(\cosh ax)-{\frac {n-1}{n))\int \sinh ^{n-2}ax\,dx\qquad {\mbox{(for ))n>0{\mbox{)))}$

also: ${\displaystyle \int \sinh ^{n}ax\,dx={\frac {1}{a(n+1)))(\sinh ^{n+1}ax)(\cosh ax)-{\frac {n+2}{n+1))\int \sinh ^{n+2}ax\,dx\qquad {\mbox{(for ))n<0{\mbox{, ))n\neq -1{\mbox{)))}$

${\displaystyle \int {\frac {dx}{\sinh ax))={\frac {1}{a))\ln \left|\tanh {\frac {ax}{2))\right|+C}$

also: ${\displaystyle \int {\frac {dx}{\sinh ax))={\frac {1}{a))\ln \left|{\frac {\cosh ax-1}{\sinh ax))\right|+C}$

${\displaystyle \int {\frac {dx}{\sinh ax))={\frac {1}{a))\ln \left|{\frac {\sinh ax}{\cosh ax+1))\right|+C}$

${\displaystyle \int {\frac {dx}{\sinh ax))={\frac {1}{2a))\ln \left|{\frac {\cosh ax-1}{\cosh ax+1))\right|+C}$

${\displaystyle \int {\frac {dx}{\sinh ^{n}ax))=-{\frac {\cosh ax}{a(n-1)\sinh ^{n-1}ax))-{\frac {n-2}{n-1))\int {\frac {dx}{\sinh ^{n-2}ax))\qquad {\mbox{(for ))n\neq 1{\mbox{)))}$

${\displaystyle \int x\sinh ax\,dx={\frac {1}{a))x\cosh ax-{\frac {1}{a^{2))}\sinh ax+C}$

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

Integrals involving only hyperbolic cosine functions

${\displaystyle \int \cosh ax\,dx={\frac {1}{a))\sinh ax+C}$

${\displaystyle \int \cosh ^{2}ax\,dx={\frac {1}{4a))\sinh 2ax+{\frac {x}{2))+C}$

${\displaystyle \int \cosh ^{n}ax\,dx={\frac {1}{an))(\sinh ax)(\cosh ^{n-1}ax)+{\frac {n-1}{n))\int \cosh ^{n-2}ax\,dx\qquad {\mbox{(for ))n>0{\mbox{)))}$

also: ${\displaystyle \int \cosh ^{n}ax\,dx=-{\frac {1}{a(n+1)))(\sinh ax)(\cosh ^{n+1}ax)+{\frac {n+2}{n+1))\int \cosh ^{n+2}ax\,dx\qquad {\mbox{(for ))n<0{\mbox{, ))n\neq -1{\mbox{)))}$

${\displaystyle \int {\frac {dx}{\cosh ax))={\frac {2}{a))\arctan e^{ax}+C}$

also: ${\displaystyle \int {\frac {dx}{\cosh ax))={\frac {1}{a))\arctan(\sinh ax)+C}$

${\displaystyle \int {\frac {dx}{\cosh ^{n}ax))={\frac {\sinh ax}{a(n-1)\cosh ^{n-1}ax))+{\frac {n-2}{n-1))\int {\frac {dx}{\cosh ^{n-2}ax))\qquad {\mbox{(for ))n\neq 1{\mbox{)))}$

${\displaystyle \int x\cosh ax\,dx={\frac {1}{a))x\sinh ax-{\frac {1}{a^{2))}\cosh ax+C}$

${\displaystyle \int x^{2}\cosh ax\,dx=-{\frac {2x\cosh ax}{a^{2))}+\left({\frac {x^{2)){a))+{\frac {2}{a^{3))}\right)\sinh ax+C}$

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

${\displaystyle \int {\frac {dx}{1+\cosh(ax)))={\frac {2}{a)){\frac {1}{1+e^{-ax))}+C}$ or ${\displaystyle {\frac {2}{a))}$ times The Logistic Function

Other integrals

Integrals of hyperbolic tangent, cotangent, secant, cosecant functions

${\displaystyle \int \tanh x\,dx=\ln \cosh x+C}$

${\displaystyle \int \tanh ^{2}ax\,dx=x-{\frac {\tanh ax}{a))+C}$

${\displaystyle \int \tanh ^{n}ax\,dx=-{\frac {1}{a(n-1)))\tanh ^{n-1}ax+\int \tanh ^{n-2}ax\,dx\qquad {\mbox{(for ))n\neq 1{\mbox{)))}$

${\displaystyle \int \coth x\,dx=\ln |\sinh x|+C,{\text{ for ))x\neq 0}$

${\displaystyle \int \coth ^{n}ax\,dx=-{\frac {1}{a(n-1)))\coth ^{n-1}ax+\int \coth ^{n-2}ax\,dx\qquad {\mbox{(for ))n\neq 1{\mbox{)))}$

${\displaystyle \int \operatorname {sech} \,x\,dx=\arctan \,(\sinh x)+C}$

${\displaystyle \int \operatorname {csch} \,x\,dx=\ln \left|\tanh {x \over 2}\right|+C=\ln \left|\coth {x}-\operatorname {csch} {x}\right|+C,{\text{ for ))x\neq 0}$

Integrals involving hyperbolic sine and cosine functions

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

${\displaystyle \int {\frac {\cosh ^{n}ax}{\sinh ^{m}ax))dx={\frac {\cosh ^{n-1}ax}{a(n-m)\sinh ^{m-1}ax))+{\frac {n-1}{n-m))\int {\frac {\cosh ^{n-2}ax}{\sinh ^{m}ax))dx\qquad {\mbox{(for ))m\neq n{\mbox{)))}$

also: ${\displaystyle \int {\frac {\cosh ^{n}ax}{\sinh ^{m}ax))dx=-{\frac {\cosh ^{n+1}ax}{a(m-1)\sinh ^{m-1}ax))+{\frac {n-m+2}{m-1))\int {\frac {\cosh ^{n}ax}{\sinh ^{m-2}ax))dx\qquad {\mbox{(for ))m\neq 1{\mbox{)))}$

${\displaystyle \int {\frac {\cosh ^{n}ax}{\sinh ^{m}ax))dx=-{\frac {\cosh ^{n-1}ax}{a(m-1)\sinh ^{m-1}ax))+{\frac {n-1}{m-1))\int {\frac {\cosh ^{n-2}ax}{\sinh ^{m-2}ax))dx\qquad {\mbox{(for ))m\neq 1{\mbox{)))}$

${\displaystyle \int {\frac {\sinh ^{m}ax}{\cosh ^{n}ax))dx={\frac {\sinh ^{m-1}ax}{a(m-n)\cosh ^{n-1}ax))+{\frac {m-1}{n-m))\int {\frac {\sinh ^{m-2}ax}{\cosh ^{n}ax))dx\qquad {\mbox{(for ))m\neq n{\mbox{)))}$

${\displaystyle \int {\frac {\sinh ^{m}ax}{\cosh ^{n}ax))dx={\frac {\sinh ^{m+1}ax}{a(n-1)\cosh ^{n-1}ax))+{\frac {m-n+2}{n-1))\int {\frac {\sinh ^{m}ax}{\cosh ^{n-2}ax))dx\qquad {\mbox{(for ))n\neq 1{\mbox{)))}$

${\displaystyle \int {\frac {\sinh ^{m}ax}{\cosh ^{n}ax))dx=-{\frac {\sinh ^{m-1}ax}{a(n-1)\cosh ^{n-1}ax))+{\frac {m-1}{n-1))\int {\frac {\sinh ^{m-2}ax}{\cosh ^{n-2}ax))dx\qquad {\mbox{(for ))n\neq 1{\mbox{)))}$

Integrals involving hyperbolic and trigonometric functions

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

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

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

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