# List of integrals of hyperbolic functions

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The following is a list of integrals (antiderivative 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.

$\int\sinh ax\,dx = \frac{1}{a}\cosh ax+C\,$
$\int\cosh ax\,dx = \frac{1}{a}\sinh ax+C\,$
$\int\sinh^2 ax\,dx = \frac{1}{4a}\sinh 2ax - \frac{x}{2}+C\,$
$\int\cosh^2 ax\,dx = \frac{1}{4a}\sinh 2ax + \frac{x}{2}+C\,$
$\int\tanh^2 ax\,dx = x - \frac{\tanh ax}{a}+C\,$
$\int\sinh^n ax\,dx = \frac{1}{an}\sinh^{n-1} ax\cosh ax - \frac{n-1}{n}\int\sinh^{n-2} ax\,dx \qquad\mbox{(for }n>0\mbox{)}\,$
also: $\int\sinh^n ax\,dx = \frac{1}{a(n+1)}\sinh^{n+1} ax\cosh ax - \frac{n+2}{n+1}\int\sinh^{n+2}ax\,dx \qquad\mbox{(for }n<0\mbox{, }n\neq -1\mbox{)}\,$
$\int\cosh^n ax\,dx = \frac{1}{an}\sinh ax\cosh^{n-1} ax + \frac{n-1}{n}\int\cosh^{n-2} ax\,dx \qquad\mbox{(for }n>0\mbox{)}\,$
also: $\int\cosh^n ax\,dx = -\frac{1}{a(n+1)}\sinh ax\cosh^{n+1} ax + \frac{n+2}{n+1}\int\cosh^{n+2}ax\,dx \qquad\mbox{(for }n<0\mbox{, }n\neq -1\mbox{)}\,$
$\int\frac{dx}{\sinh ax} = \frac{1}{a} \ln\left|\tanh\frac{ax}{2}\right|+C\,$
also: $\int\frac{dx}{\sinh ax} = \frac{1}{a} \ln\left|\frac{\cosh ax - 1}{\sinh ax}\right|+C\,$
also: $\int\frac{dx}{\sinh ax} = \frac{1}{a} \ln\left|\frac{\sinh ax}{\cosh ax + 1}\right|+C\,$
also: $\int\frac{dx}{\sinh ax} = \frac{1}{2a} \ln\left|\frac{\cosh ax - 1}{\cosh ax + 1}\right|+C\,$
$\int\frac{dx}{\cosh ax} = \frac{2}{a} \arctan e^{ax}+C\,$
also: $\int\frac{dx}{\cosh ax} = \frac{1}{a} \arctan (\sinh ax)+C\,$
$\int\frac{dx}{\sinh^n ax} = -\frac{\cosh ax}{a(n-1)\sinh^{n-1} ax}-\frac{n-2}{n-1}\int\frac{dx}{\sinh^{n-2} ax} \qquad\mbox{(for }n\neq 1\mbox{)}\,$
$\int\frac{dx}{\cosh^n ax} = \frac{\sinh ax}{a(n-1)\cosh^{n-1} ax}+\frac{n-2}{n-1}\int\frac{dx}{\cosh^{n-2} ax} \qquad\mbox{(for }n\neq 1\mbox{)}\,$
$\int\frac{\cosh^n ax}{\sinh^m ax} dx = \frac{\cosh^{n-1} ax}{a(n-m)\sinh^{m-1} ax} + \frac{n-1}{n-m}\int\frac{\cosh^{n-2} ax}{\sinh^m ax} dx \qquad\mbox{(for }m\neq n\mbox{)}\,$
also: $\int\frac{\cosh^n ax}{\sinh^m ax} dx = -\frac{\cosh^{n+1} ax}{a(m-1)\sinh^{m-1} ax} + \frac{n-m+2}{m-1}\int\frac{\cosh^n ax}{\sinh^{m-2} ax} dx \qquad\mbox{(for }m\neq 1\mbox{)}\,$
also: $\int\frac{\cosh^n ax}{\sinh^m ax} dx = -\frac{\cosh^{n-1} ax}{a(m-1)\sinh^{m-1} ax} + \frac{n-1}{m-1}\int\frac{\cosh^{n-2} ax}{\sinh^{m-2} ax} dx \qquad\mbox{(for }m\neq 1\mbox{)}\,$
$\int\frac{\sinh^m ax}{\cosh^n ax} dx = \frac{\sinh^{m-1} ax}{a(m-n)\cosh^{n-1} ax} + \frac{m-1}{n-m}\int\frac{\sinh^{m-2} ax}{\cosh^n ax} dx \qquad\mbox{(for }m\neq n\mbox{)}\,$
also: $\int\frac{\sinh^m ax}{\cosh^n ax} dx = \frac{\sinh^{m+1} ax}{a(n-1)\cosh^{n-1} ax} + \frac{m-n+2}{n-1}\int\frac{\sinh^m ax}{\cosh^{n-2} ax} dx \qquad\mbox{(for }n\neq 1\mbox{)}\,$
also: $\int\frac{\sinh^m ax}{\cosh^n ax} dx = -\frac{\sinh^{m-1} ax}{a(n-1)\cosh^{n-1} ax} + \frac{m-1}{n-1}\int\frac{\sinh^{m -2} ax}{\cosh^{n-2} ax} dx \qquad\mbox{(for }n\neq 1\mbox{)}\,$
$\int x\sinh ax\,dx = \frac{1}{a} x\cosh ax - \frac{1}{a^2}\sinh ax+C\,$
$\int x\cosh ax\,dx = \frac{1}{a} x\sinh ax - \frac{1}{a^2}\cosh ax+C\,$
$\int x^2 \cosh ax\,dx = -\frac{2x \cosh ax}{a^2} + \left(\frac{x^2}{a}+\frac{2}{a^3}\right) \sinh ax+C\,$
$\int \tanh ax\,dx = \frac{1}{a}\ln\cosh ax+C\,$
$\int \coth ax\,dx = \frac{1}{a}\ln|\sinh ax|+C\,$
$\int \tanh^n ax\,dx = -\frac{1}{a(n-1)}\tanh^{n-1} ax+\int\tanh^{n-2} ax\,dx \qquad\mbox{(for }n\neq 1\mbox{)}\,$
$\int \coth^n ax\,dx = -\frac{1}{a(n-1)}\coth^{n-1} ax+\int\coth^{n-2} ax\,dx \qquad\mbox{(for }n\neq 1\mbox{)}\,$
$\int \sinh ax \sinh bx\,dx = \frac{1}{a^2-b^2} (a\sinh bx \cosh ax - b\cosh bx \sinh ax)+C \qquad\mbox{(for }a^2\neq b^2\mbox{)}\,$
$\int \cosh ax \cosh bx\,dx = \frac{1}{a^2-b^2} (a\sinh ax \cosh bx - b\sinh bx \cosh ax)+C \qquad\mbox{(for }a^2\neq b^2\mbox{)}\,$
$\int \cosh ax \sinh bx\,dx = \frac{1}{a^2-b^2} (a\sinh ax \sinh bx - b\cosh ax \cosh bx)+C \qquad\mbox{(for }a^2\neq b^2\mbox{)}\,$
$\int \sinh (ax+b)\sin (cx+d)\,dx = \frac{a}{a^2+c^2}\cosh(ax+b)\sin(cx+d)-\frac{c}{a^2+c^2}\sinh(ax+b)\cos(cx+d)+C\,$
$\int \sinh (ax+b)\cos (cx+d)\,dx = \frac{a}{a^2+c^2}\cosh(ax+b)\cos(cx+d)+\frac{c}{a^2+c^2}\sinh(ax+b)\sin(cx+d)+C\,$
$\int \cosh (ax+b)\sin (cx+d)\,dx = \frac{a}{a^2+c^2}\sinh(ax+b)\sin(cx+d)-\frac{c}{a^2+c^2}\cosh(ax+b)\cos(cx+d)+C\,$
$\int \cosh (ax+b)\cos (cx+d)\,dx = \frac{a}{a^2+c^2}\sinh(ax+b)\cos(cx+d)+\frac{c}{a^2+c^2}\cosh(ax+b)\sin(cx+d)+C\,$