List of integrals of hyperbolic functions

Short description: Wikipedia list article

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.

## Contents

* 1 Integrals involving only hyperbolic sine functions
* 2 Integrals involving only hyperbolic cosine functions
* 3 Other integrals
* 3.1 Integrals of hyperbolic tangent, cotangent, secant, cosecant functions
* 3.2 Integrals involving hyperbolic sine and cosine functions
* 3.3 Integrals involving hyperbolic and trigonometric functions

## Integrals involving only hyperbolic sine functions

∫sinh⁡axdx=1acosh⁡ax+C

∫sinh2axdx=14asinh⁡2ax−x2+C

∫sinhnaxdx=1an(sinhn−1ax)(cosh⁡ax)−n−1n∫sinhn−2axdx(for n>0)

also: ∫sinhnaxdx=1a(n+1)(sinhn+1ax)(cosh⁡ax)−n+2n+1∫sinhn+2axdx(for n<0, n≠−1)

∫dxsinh⁡ax=1aln⁡|tanh⁡ax2|+C

also: ∫dxsinh⁡ax=1aln⁡|cosh⁡ax−1sinh⁡ax|+C

∫dxsinh⁡ax=1aln⁡|sinh⁡axcosh⁡ax+1|+C

∫dxsinh⁡ax=12aln⁡|cosh⁡ax−1cosh⁡ax+1|+C

∫dxsinhnax=−cosh⁡axa(n−1)sinhn−1ax−n−2n−1∫dxsinhn−2ax(for n≠1)

∫xsinh⁡axdx=1axcosh⁡ax−1a2sinh⁡ax+C

∫(sinh⁡ax)(sinh⁡bx)dx=1a2−b2(a(sinh⁡bx)(cosh⁡ax)−b(cosh⁡bx)(sinh⁡ax))+C(for a2≠b2)

## Integrals involving only hyperbolic cosine functions

∫cosh⁡axdx=1asinh⁡ax+C

∫cosh2axdx=14asinh⁡2ax+x2+C

∫coshnaxdx=1an(sinh⁡ax)(coshn−1ax)+n−1n∫coshn−2axdx(for n>0)

also: ∫coshnaxdx=−1a(n+1)(sinh⁡ax)(coshn+1ax)+n+2n+1∫coshn+2axdx(for n<0, n≠−1)

∫dxcosh⁡ax=2aarctan⁡eax+C

also: ∫dxcosh⁡ax=1aarctan⁡(sinh⁡ax)+C

∫dxcoshnax=sinh⁡axa(n−1)coshn−1ax+n−2n−1∫dxcoshn−2ax(for n≠1)

∫xcosh⁡axdx=1axsinh⁡ax−1a2cosh⁡ax+C

∫x2cosh⁡axdx=−2xcosh⁡axa2+(x2a+2a3)sinh⁡ax+C

∫(cosh⁡ax)(cosh⁡bx)dx=1a2−b2(a(sinh⁡ax)(cosh⁡bx)−b(sinh⁡bx)(cosh⁡ax))+C(for a2≠b2)

∫dx1+cosh⁡(ax)=2a11+e−ax+C or 2a times The Logistic Function

## Other integrals

### Integrals of hyperbolic tangent, cotangent, secant, cosecant functions

∫tanh⁡xdx=ln⁡cosh⁡x+C

∫tanh2axdx=x−tanh⁡axa+C

∫tanhnaxdx=−1a(n−1)tanhn−1ax+∫tanhn−2axdx(for n≠1)

∫coth⁡xdx=ln⁡|sinh⁡x|+C, for x≠0

∫cothnaxdx=−1a(n−1)cothn−1ax+∫cothn−2axdx(for n≠1)

∫sech⁡xdx=arctan⁡(sinh⁡x)+C

∫csch⁡xdx=ln⁡|tanh⁡x2|+C=ln⁡|coth⁡x−csch⁡x|+C, for x≠0

### Integrals involving hyperbolic sine and cosine functions

∫(cosh⁡ax)(sinh⁡bx)dx=1a2−b2(a(sinh⁡ax)(sinh⁡bx)−b(cosh⁡ax)(cosh⁡bx))+C(for a2≠b2)

∫coshnaxsinhmaxdx=coshn−1axa(n−m)sinhm−1ax+n−1n−m∫coshn−2axsinhmaxdx(for m≠n)

also: ∫coshnaxsinhmaxdx=−coshn+1axa(m−1)sinhm−1ax+n−m+2m−1∫coshnaxsinhm−2axdx(for m≠1)

∫coshnaxsinhmaxdx=−coshn−1axa(m−1)sinhm−1ax+n−1m−1∫coshn−2axsinhm−2axdx(for m≠1)

∫sinhmaxcoshnaxdx=sinhm−1axa(m−n)coshn−1ax+m−1n−m∫sinhm−2axcoshnaxdx(for m≠n)

∫sinhmaxcoshnaxdx=sinhm+1axa(n−1)coshn−1ax+m−n+2n−1∫sinhmaxcoshn−2axdx(for n≠1)

∫sinhmaxcoshnaxdx=−sinhm−1axa(n−1)coshn−1ax+m−1n−1∫sinhm−2axcoshn−2axdx(for n≠1)

### Integrals involving hyperbolic and trigonometric functions

∫sinh⁡(ax+b)sin⁡(cx+d)dx=aa2+c2cosh⁡(ax+b)sin⁡(cx+d)−ca2+c2sinh⁡(ax+b)cos⁡(cx+d)+C

∫sinh⁡(ax+b)cos⁡(cx+d)dx=aa2+c2cosh⁡(ax+b)cos⁡(cx+d)+ca2+c2sinh⁡(ax+b)sin⁡(cx+d)+C

∫cosh⁡(ax+b)sin⁡(cx+d)dx=aa2+c2sinh⁡(ax+b)sin⁡(cx+d)−ca2+c2cosh⁡(ax+b)cos⁡(cx+d)+C

∫cosh⁡(ax+b)cos⁡(cx+d)dx=aa2+c2sinh⁡(ax+b)cos⁡(cx+d)+ca2+c2cosh⁡(ax+b)sin⁡(cx+d)+C

* v
* t
* e

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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