Primitives of Hyperbolic Functions

Theorem

This page gathers together primitives of hyperbolic functions.

In the below, $C$ is an arbitrary constant throughout.

Primitive of Hyperbolic Sine Function

$\ds \int \sinh x \rd x = \cosh x + C$

Primitive of Hyperbolic Cosine Function

$\ds \int \cosh x \rd x = \sinh x + C$

Primitive of Hyperbolic Tangent Function

$\ds \int \tanh x \rd x = \map \ln {\cosh x} + C$

Primitive of Hyperbolic Cotangent Function

$\ds \int \coth x \rd x = \ln \size {\sinh x} + C$

where $\sinh x \ne 0$.

Hyperbolic Secant Function: Arcsine Form

$\ds \int \sech x \rd x = \map \arcsin {\tanh x} + C$

Hyperbolic Secant Function: Arctangent of Exponential Form

$\ds \int \sech x \rd x = 2 \map \arctan {e^x} + C$

Primitive of Hyperbolic Cosecant Function: Logarithm Form

$\ds \int \csch x \rd x = -\ln \size {\csch x + \coth x} + C$

where $\csch x + \coth x \ne 0$.

Primitive of Hyperbolic Cosecant Function: Hyperbolic Tangent Form

$\ds \int \csch x \rd x = \ln \size {\tanh \frac x 2} + C$

where $\tanh \dfrac x 2 \ne 0$.

Primitive of Hyperbolic Cosecant Function: Inverse Hyperbolic Cotangent Form

$\ds \int \csch x \rd x = -2 \map {\coth^{-1} } {e^x} + C$

Primitive of Hyperbolic Cosecant Function: Inverse Hyperbolic Cotangent of Hyperbolic Cosine Form

$\ds \int \csch x \rd x = -\map {\coth^{-1} } {\cosh x} + C$