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

$\int\sinh ax\,dx = \frac\cosh ax+C$

$\int\sinh^2 ax\,dx = \frac\sinh 2ax - \frac+C$

$\int\sinh^n ax\,dx = \frac(\sinh^ ax)(\cosh ax) - \frac\int\sinh^ ax\,dx \qquad\mboxn>0\mbox$

: also: $\int\sinh^n ax\,dx = \frac(\sinh^ ax)(\cosh ax) - \frac\int\sinh^ax\,dx \qquad\mboxn<0\mboxn\neq -1\mbox$

$\int\frac = \frac \ln\left, \tanh\frac\+C$

: also: $\int\frac = \frac \ln\left, \frac\+C$

: $\int\frac = \frac \ln\left, \frac\+C$

: $\int\frac = \frac \ln\left, \frac\+C$

$\int\frac = -\frac-\frac\int\frac \qquad\mboxn\neq 1\mbox$

$\int x\sinh ax\,dx = \frac x\cosh ax - \frac\sinh ax+C$

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

Integrals involving only hyperbolic cosine functions

$\int\cosh ax\,dx = \frac\sinh ax+C$

$\int\cosh^2 ax\,dx = \frac\sinh 2ax + \frac+C$

$\int\cosh^n ax\,dx = \frac(\sinh ax)(\cosh^ ax) + \frac\int\cosh^ ax\,dx \qquad\mboxn>0\mbox$

: also: $\int\cosh^n ax\,dx = -\frac(\sinh ax)(\cosh^ ax) + \frac\int\cosh^ax\,dx \qquad\mboxn<0\mboxn\neq -1\mbox$

$\int\frac = \frac \arctan e^+C$

: also: $\int\frac = \frac \arctan (\sinh ax)+C$

$\int\frac = \frac+\frac\int\frac \qquad\mboxn\neq 1\mbox$

$\int x\cosh ax\,dx = \frac x\sinh ax - \frac\cosh ax+C$

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

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

$\int \frac = \frac \frac+C$ or $\frac$ times The Logistic Function

Other integrals

Integrals of hyperbolic tangent, cotangent, secant, cosecant functions

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

$\int\tanh^2 ax\,dx = x - \frac+C$

$\int \tanh^n ax\,dx = -\frac\tanh^ ax+\int\tanh^ ax\,dx \qquad\mboxn\neq 1\mbox$

$\int \coth x \, dx = \ln, \sinh x , + C , \text x \neq 0$

$\int \coth^n ax\,dx = -\frac\coth^ ax+\int\coth^ ax\,dx \qquad\mboxn\neq 1\mbox$

$\int \operatorname\,x \, dx = \arctan\,(\sinh x) + C$

$\int \operatorname\,x \, dx = \ln\left, \tanh \ + C = \ln\left, \coth-\operatorname\+C, \text x \neq 0$

Integrals involving hyperbolic sine and cosine functions

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

$\int\frac dx = \frac + \frac\int\frac dx \qquad\mboxm\neq n\mbox$

: also: $\int\frac dx = -\frac + \frac\int\frac dx \qquad\mboxm\neq 1\mbox$

: $\int\frac dx = -\frac + \frac\int\frac dx \qquad\mboxm\neq 1\mbox$

: $\int\frac dx = \frac + \frac\int\frac dx \qquad\mboxm\neq n\mbox$

: $\int\frac dx = \frac + \frac\int\frac dx \qquad\mboxn\neq 1\mbox$

: $\int\frac dx = -\frac + \frac\int\frac dx \qquad\mboxn\neq 1\mbox$

Integrals involving hyperbolic and trigonometric functions

$\int \sinh (ax+b)\sin (cx+d)\,dx = \frac\cosh(ax+b)\sin(cx+d)-\frac\sinh(ax+b)\cos(cx+d)+C$

$\int \sinh (ax+b)\cos (cx+d)\,dx = \frac\cosh(ax+b)\cos(cx+d)+\frac\sinh(ax+b)\sin(cx+d)+C$

$\int \cosh (ax+b)\sin (cx+d)\,dx = \frac\sinh(ax+b)\sin(cx+d)-\frac\cosh(ax+b)\cos(cx+d)+C$

$\int \cosh (ax+b)\cos (cx+d)\,dx = \frac\sinh(ax+b)\cos(cx+d)+\frac\cosh(ax+b)\sin(cx+d)+C$

{{DEFAULTSORT:Integrals of hyperbolic functions
Exponentials
Hyperbolic functions