List of integrals of hyperbolic functions

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