List of integrals of hyperbolic functions

The following is a list of Integrals (Antiderivative functions) of hyperbolic functions. For a complete list of Integral functions, please see Table of Integrals and List of integrals.

<math>\int\sinh cx\,dx = \frac{1}{c}\cosh cx</math>

<math>\int\cosh cx\,dx = \frac{1}{c}\sinh cx</math>

<math>\int\sinh^2 cx\,dx = \frac{1}{4c}\sinh 2cx - \frac{x}{2}</math>

<math>\int\cosh^2 cx\,dx = \frac{1}{4c}\sinh 2cx + \frac{x}{2}</math>

<math>\int\sinh^n cx\,dx = \frac{1}{cn}\sinh^{n-1} cx\cosh cx - \frac{n-1}{n}\int\sinh^{n-2} cx\,dx \qquad\mbox{(for }n>0\mbox{)}</math>

also: <math>\int\sinh^n cx\,dx = \frac{1}{c(n+1)}\sinh^{n+1} cx\cosh cx - \frac{n+2}{n+1}\int\sinh^{n+2}cx\,dx \qquad\mbox{(for }n<0\mbox{, }n\neq -1\mbox{)}</math>

<math>\int\cosh^n cx\,dx = \frac{1}{cn}\sinh cx\cosh^{n-1} cx + \frac{n-1}{n}\int\cosh^{n-2} cx\,dx \qquad\mbox{(for }n>0\mbox{)}</math>

also: <math>\int\cosh^n cx\,dx = -\frac{1}{c(n+1)}\sinh cx\cosh^{n+1} cx - \frac{n+2}{n+1}\int\cosh^{n+2}cx\,dx \qquad\mbox{(for }n<0\mbox{, }n\neq -1\mbox{)}</math>

<math>\int\frac{dx}{\sinh cx} = \frac{1}{c} \ln\left|\tanh\frac{cx}{2}\right|</math>

also: <math>\int\frac{dx}{\sinh cx} = \frac{1}{c} \ln\left|\frac{\cosh cx - 1}{\sinh cx}\right|</math>

also: <math>\int\frac{dx}{\sinh cx} = \frac{1}{c} \ln\left|\frac{\sinh cx}{\cosh cx + 1}\right|</math>

also: <math>\int\frac{dx}{\sinh cx} = \frac{1}{c} \ln\left|\frac{\cosh cx - 1}{\cosh cx + 1}\right|</math>

<math>\int\frac{dx}{\cosh cx} = \frac{2}{c} \arctan e^{cx}</math>

<math>\int\frac{dx}{\sinh^n cx} = \frac{\cosh cx}{c(n-1)\sinh^{n-1} cx}-\frac{n-2}{n-1}\int\frac{dx}{\sinh^{n-2} cx} \qquad\mbox{(for }n\neq 1\mbox{)}</math>

<math>\int\frac{dx}{\cosh^n cx} = \frac{\sinh cx}{c(n-1)\cosh^{n-1} cx}+\frac{n-2}{n-1}\int\frac{dx}{\cosh^{n-2} cx} \qquad\mbox{(for }n\neq 1\mbox{)}</math>

<math>\int\frac{\cosh^n cx}{\sinh^m cx} dx = \frac{\cosh^{n-1} cx}{c(n-m)\sinh^{m-1} cx} + \frac{n-1}{n-m}\int\frac{\cosh^{n-2} cx}{\sinh^m cx} dx \qquad\mbox{(for }m\neq n\mbox{)}</math>

also: <math>\int\frac{\cosh^n cx}{\sinh^m cx} dx = -\frac{\cosh^{n+1} cx}{c(m-1)\sinh^{m-1} cx} + \frac{n-m+2}{m-1}\int\frac{\cosh^n cx}{\sinh^{m-2} cx} dx \qquad\mbox{(for }m\neq 1\mbox{)}</math>

also: <math>\int\frac{\cosh^n cx}{\sinh^m cx} dx = -\frac{\cosh^{n-1} cx}{c(m-1)\sinh^{m-1} cx} + \frac{n-1}{m-1}\int\frac{\cosh^{n-2} cx}{\sinh^{m-2} cx} dx \qquad\mbox{(for }m\neq 1\mbox{)}</math>

<math>\int\frac{\sinh^m cx}{\cosh^n cx} dx = \frac{\sinh^{m-1} cx}{c(m-n)\cosh^{n-1} cx} + \frac{m-1}{m-n}\int\frac{\sinh^{m-2} cx}{\cosh^n cx} dx \qquad\mbox{(for }m\neq n\mbox{)}</math>

also: <math>\int\frac{\sinh^m cx}{\cosh^n cx} dx = \frac{\sinh^{m+1} cx}{c(n-1)\cosh^{n-1} cx} + \frac{m-n+2}{n-1}\int\frac{\sinh^m cx}{\cosh^{n-2} cx} dx \qquad\mbox{(for }n\neq 1\mbox{)}</math>

also: <math>\int\frac{\sinh^m cx}{\cosh^n cx} dx = -\frac{\sinh^{m-1} cx}{c(n-1)\cosh^{n-1} cx} + \frac{m-1}{n-1}\int\frac{\sinh^{m -2} cx}{\cosh^{n-2} cx} dx \qquad\mbox{(for }n\neq 1\mbox{)}</math>

<math>\int x\sinh cx\,dx = \frac{1}{c} x\cosh cx - \frac{1}{c^2}\sinh cx</math>

<math>\int x\cosh cx\,dx = \frac{1}{c} x\sinh cx - \frac{1}{c^2}\cosh cx</math>

<math>\int \tanh cx\,dx = \frac{1}{c}\ln|\cosh cx|</math>

<math>\int \coth cx\,dx = \frac{1}{c}\ln|\sinh cx|</math>

<math>\int \tanh^n cx\,dx = -\frac{1}{c(n-1)}\tanh^{n-1} cx+\int\tanh^{n-2} cx\,dx \qquad\mbox{(for }n\neq 1\mbox{)}</math>

<math>\int \coth^n cx\,dx = -\frac{1}{c(n-1)}\coth^{n-1} cx+\int\coth^{n-2} cx\,dx \qquad\mbox{(for }n\neq 1\mbox{)}</math>

<math>\int \sinh bx \sinh cx\,dx = \frac{1}{b^2-c^2} (b\sinh cx \cosh bx - c\cosh cx \sinh bx) \qquad\mbox{(for }b^2\neq c^2\mbox{)}</math>

<math>\int \cosh bx \cosh cx\,dx = \frac{1}{b^2-c^2} (b\sinh bx \cosh cx - c\sinh cx \cosh bx) \qquad\mbox{(for }b^2\neq c^2\mbox{)}</math>

<math>\int \cosh bx \sinh cx\,dx = \frac{1}{b^2-c^2} (b\sinh bx \sinh cx - c\cosh bx \cosh cx) \qquad\mbox{(for }b^2\neq c^2\mbox{)}</math>

<math>\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)</math>

<math>\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)</math>

<math>\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)</math>

<math>\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)</math>