List of integrals of inverse hyperbolic functions

Short description: Wikipedia list article

The following is a list of indefinite integrals (antiderivatives) of expressions involving the inverse hyperbolic functions. For a complete list of integral formulas, see lists of integrals.

* In all formulas the constant a is assumed to be nonzero, and C denotes the constant of integration.
* For each inverse hyperbolic integration formula below there is a corresponding formula in the list of integrals of inverse trigonometric functions.

## Contents

* 1 Inverse hyperbolic sine integration formulas
* 2 Inverse hyperbolic cosine integration formulas
* 3 Inverse hyperbolic tangent integration formulas
* 4 Inverse hyperbolic cotangent integration formulas
* 5 Inverse hyperbolic secant integration formulas
* 6 Inverse hyperbolic cosecant integration formulas

## Inverse hyperbolic sine integration formulas

∫arsinh⁡(ax)dx=xarsinh⁡(ax)−a2x2+1a+C

∫xarsinh⁡(ax)dx=x2arsinh⁡(ax)2+arsinh⁡(ax)4a2−xa2x2+14a+C

∫x2arsinh⁡(ax)dx=x3arsinh⁡(ax)3−(a2x2−2)a2x2+19a3+C

∫xmarsinh⁡(ax)dx=xm+1arsinh⁡(ax)m+1−am+1∫xm+1a2x2+1dx(m≠−1)

∫arsinh⁡(ax)2dx=2x+xarsinh⁡(ax)2−2a2x2+1arsinh⁡(ax)a+C

∫arsinh⁡(ax)ndx=xarsinh⁡(ax)n−na2x2+1arsinh⁡(ax)n−1a+n(n−1)∫arsinh⁡(ax)n−2dx

∫arsinh⁡(ax)ndx=−xarsinh⁡(ax)n+2(n+1)(n+2)+a2x2+1arsinh⁡(ax)n+1a(n+1)+1(n+1)(n+2)∫arsinh⁡(ax)n+2dx(n≠−1,−2)

## Inverse hyperbolic cosine integration formulas

∫arcosh⁡(ax)dx=xarcosh⁡(ax)−ax+1ax−1a+C

∫xarcosh⁡(ax)dx=x2arcosh⁡(ax)2−arcosh⁡(ax)4a2−xax+1ax−14a+C

∫x2arcosh⁡(ax)dx=x3arcosh⁡(ax)3−(a2x2+2)ax+1ax−19a3+C

∫xmarcosh⁡(ax)dx=xm+1arcosh⁡(ax)m+1−am+1∫xm+1ax+1ax−1dx(m≠−1)

∫arcosh⁡(ax)2dx=2x+xarcosh⁡(ax)2−2ax+1ax−1arcosh⁡(ax)a+C

∫arcosh⁡(ax)ndx=xarcosh⁡(ax)n−nax+1ax−1arcosh⁡(ax)n−1a+n(n−1)∫arcosh⁡(ax)n−2dx

∫arcosh⁡(ax)ndx=−xarcosh⁡(ax)n+2(n+1)(n+2)+ax+1ax−1arcosh⁡(ax)n+1a(n+1)+1(n+1)(n+2)∫arcosh⁡(ax)n+2dx(n≠−1,−2)

## Inverse hyperbolic tangent integration formulas

∫artanh⁡(ax)dx=xartanh⁡(ax)+ln⁡(1−a2x2)2a+C

∫xartanh⁡(ax)dx=x2artanh⁡(ax)2−artanh⁡(ax)2a2+x2a+C

∫x2artanh⁡(ax)dx=x3artanh⁡(ax)3+ln⁡(1−a2x2)6a3+x26a+C

∫xmartanh⁡(ax)dx=xm+1artanh⁡(ax)m+1−am+1∫xm+11−a2x2dx(m≠−1)

## Inverse hyperbolic cotangent integration formulas

∫arcoth⁡(ax)dx=xarcoth⁡(ax)+ln⁡(a2x2−1)2a+C

∫xarcoth⁡(ax)dx=x2arcoth⁡(ax)2−arcoth⁡(ax)2a2+x2a+C

∫x2arcoth⁡(ax)dx=x3arcoth⁡(ax)3+ln⁡(a2x2−1)6a3+x26a+C

∫xmarcoth⁡(ax)dx=xm+1arcoth⁡(ax)m+1+am+1∫xm+1a2x2−1dx(m≠−1)

## Inverse hyperbolic secant integration formulas

∫arsech⁡(ax)dx=xarsech⁡(ax)−2aarctan⁡1−ax1+ax+C

∫xarsech⁡(ax)dx=x2arsech⁡(ax)2−(1+ax)2a21−ax1+ax+C

∫x2arsech⁡(ax)dx=x3arsech⁡(ax)3−13a3arctan⁡1−ax1+ax−x(1+ax)6a21−ax1+ax+C

∫xmarsech⁡(ax)dx=xm+1arsech⁡(ax)m+1+1m+1∫xm(1+ax)1−ax1+axdx(m≠−1)

## Inverse hyperbolic cosecant integration formulas

∫arcsch⁡(ax)dx=xarcsch⁡(ax)+1aarcoth⁡1a2x2+1+C

∫xarcsch⁡(ax)dx=x2arcsch⁡(ax)2+x2a1a2x2+1+C

∫x2arcsch⁡(ax)dx=x3arcsch⁡(ax)3−16a3arcoth⁡1a2x2+1+x26a1a2x2+1+C

∫xmarcsch⁡(ax)dx=xm+1arcsch⁡(ax)m+1+1a(m+1)∫xm−11a2x2+1dx(m≠−1)

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Lists of integrals

* Rational functions
* Irrational functions
* Trigonometric functions
* Inverse trigonometric functions
* Hyperbolic functions
* Inverse hyperbolic functions
* Exponential functions
* Logarithmic functions
* Gaussian functions

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