§4.40 Integrals

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Keywords:: integrals, inverse hyperbolic functions
Permalink:: http://dlmf.nist.gov/4.40
See also:: Annotations for Ch.4

§4.40(i) Introduction

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Permalink:: http://dlmf.nist.gov/4.40.i
See also:: Annotations for §4.40 and Ch.4

Throughout this section the variables are assumed to be real. The results in §§4.40(ii) and 4.40(iv) can be extended to the complex plane by using continuous branches and avoiding singularities.

§4.40(ii) Indefinite Integrals

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Keywords:: hyperbolic functions, indefinite, integrals
Notes:: These results may be verified by differentiation.
Referenced by:: §1.4(iv), §4.40(i)
Permalink:: http://dlmf.nist.gov/4.40.ii
See also:: Annotations for §4.40 and Ch.4

4.40.1	$\displaystyle\int\sinh x\,\mathrm{d}x$	$\displaystyle=\cosh x,$
ⓘ Symbols: $\,\mathrm{d}\NVar{x}$ : differential of $x$ , $\cosh\NVar{z}$ : hyperbolic cosine function, $\sinh\NVar{z}$ : hyperbolic sine function, $\int$ : integral and $x$ : real variable A&S Ref: 4.5.77 Permalink: http://dlmf.nist.gov/4.40.E1 Encodings: TeX, pMML, png See also: Annotations for §4.40(ii), §4.40 and Ch.4
4.40.2	$\displaystyle\int\cosh x\,\mathrm{d}x$	$\displaystyle=\sinh x,$
ⓘ Symbols: $\,\mathrm{d}\NVar{x}$ : differential of $x$ , $\cosh\NVar{z}$ : hyperbolic cosine function, $\sinh\NVar{z}$ : hyperbolic sine function, $\int$ : integral and $x$ : real variable A&S Ref: 4.5.78 Permalink: http://dlmf.nist.gov/4.40.E2 Encodings: TeX, pMML, png See also: Annotations for §4.40(ii), §4.40 and Ch.4
4.40.3	$\displaystyle\int\tanh x\,\mathrm{d}x$	$\displaystyle=\ln\left(\cosh x\right).$
ⓘ Symbols: $\,\mathrm{d}\NVar{x}$ : differential of $x$ , $\cosh\NVar{z}$ : hyperbolic cosine function, $\tanh\NVar{z}$ : hyperbolic tangent function, $\int$ : integral, $\ln\NVar{z}$ : principal branch of logarithm function and $x$ : real variable A&S Ref: 4.5.79 Permalink: http://dlmf.nist.gov/4.40.E3 Encodings: TeX, pMML, png See also: Annotations for §4.40(ii), §4.40 and Ch.4
4.40.4	$\displaystyle\int\operatorname{csch}x\,\mathrm{d}x$	$\displaystyle=\ln\left(\tanh\left(\tfrac{1}{2}x\right)\right),$
$0<x<\infty$ .
ⓘ Symbols: $\,\mathrm{d}\NVar{x}$ : differential of $x$ , $\operatorname{csch}\NVar{z}$ : hyperbolic cosecant function, $\tanh\NVar{z}$ : hyperbolic tangent function, $\int$ : integral, $\ln\NVar{z}$ : principal branch of logarithm function and $x$ : real variable A&S Ref: 4.5.80 Permalink: http://dlmf.nist.gov/4.40.E4 Encodings: TeX, pMML, png See also: Annotations for §4.40(ii), §4.40 and Ch.4
4.40.5	$\displaystyle\int\operatorname{sech}x\,\mathrm{d}x$	$\displaystyle=\operatorname{gd}\left(x\right).$
ⓘ Symbols: $\operatorname{gd}\NVar{x}$ : Gudermannian function, $\,\mathrm{d}\NVar{x}$ : differential of $x$ , $\operatorname{sech}\NVar{z}$ : hyperbolic secant function, $\int$ : integral and $x$ : real variable A&S Ref: 4.5.81 Permalink: http://dlmf.nist.gov/4.40.E5 Encodings: TeX, pMML, png See also: Annotations for §4.40(ii), §4.40 and Ch.4

For the right-hand side see (4.23.39) and (4.23.40).

4.40.6		$\int\coth x\,\mathrm{d}x=\ln\left(\sinh x\right),$
$0<x<\infty$ .
ⓘ Symbols: $\,\mathrm{d}\NVar{x}$ : differential of $x$ , $\coth\NVar{z}$ : hyperbolic cotangent function, $\sinh\NVar{z}$ : hyperbolic sine function, $\int$ : integral, $\ln\NVar{z}$ : principal branch of logarithm function and $x$ : real variable A&S Ref: 4.5.82 Permalink: http://dlmf.nist.gov/4.40.E6 Encodings: TeX, pMML, png See also: Annotations for §4.40(ii), §4.40 and Ch.4

§4.40(iii) Definite Integrals

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Keywords:: definite, hyperbolic functions, integrals
Notes:: See Copson (1935, p. 155).
Permalink:: http://dlmf.nist.gov/4.40.iii
See also:: Annotations for §4.40 and Ch.4

4.40.7		$\int_{0}^{\infty}e^{-x}\frac{\sin\left(ax\right)}{\sinh x}\,\mathrm{d}x=\tfrac% {1}{2}\pi\coth\left(\tfrac{1}{2}\pi a\right)-\frac{1}{a},$
$a\neq 0$ ,
ⓘ Symbols: $\pi$ : the ratio of the circumference of a circle to its diameter, $\,\mathrm{d}\NVar{x}$ : differential of $x$ , $\mathrm{e}$ : base of natural logarithm, $\coth\NVar{z}$ : hyperbolic cotangent function, $\sinh\NVar{z}$ : hyperbolic sine function, $\int$ : integral, $\sin\NVar{z}$ : sine function, $a$ : real or complex constant and $x$ : real variable Permalink: http://dlmf.nist.gov/4.40.E7 Encodings: TeX, pMML, png See also: Annotations for §4.40(iii), §4.40 and Ch.4

4.40.8		$\int_{0}^{\infty}\frac{\sinh\left(ax\right)}{\sinh\left(\pi x\right)}\,\mathrm% {d}x=\tfrac{1}{2}\tan\left(\tfrac{1}{2}a\right),$
$-\pi<a<\pi$ ,
ⓘ Symbols: $\pi$ : the ratio of the circumference of a circle to its diameter, $\,\mathrm{d}\NVar{x}$ : differential of $x$ , $\sinh\NVar{z}$ : hyperbolic sine function, $\int$ : integral, $\tan\NVar{z}$ : tangent function, $a$ : real or complex constant and $x$ : real variable Permalink: http://dlmf.nist.gov/4.40.E8 Encodings: TeX, pMML, png See also: Annotations for §4.40(iii), §4.40 and Ch.4

4.40.9		$\int_{-\infty}^{\infty}\frac{e^{ax}}{\left(\cosh\left(\tfrac{1}{2}x\right)% \right)^{2}}\,\mathrm{d}x=\frac{4\pi a}{\sin\left(\pi a\right)},$
$-1<a<1$ ,
ⓘ Symbols: $\pi$ : the ratio of the circumference of a circle to its diameter, $\,\mathrm{d}\NVar{x}$ : differential of $x$ , $\mathrm{e}$ : base of natural logarithm, $\cosh\NVar{z}$ : hyperbolic cosine function, $\int$ : integral, $\sin\NVar{z}$ : sine function, $a$ : real or complex constant and $x$ : real variable Permalink: http://dlmf.nist.gov/4.40.E9 Encodings: TeX, pMML, png See also: Annotations for §4.40(iii), §4.40 and Ch.4

4.40.10		${\int_{0}^{\infty}\frac{\tanh\left(ax\right)-\tanh\left(bx\right)}{x}\,\mathrm% {d}x=\ln\left(\frac{a}{b}\right)},$
$a>0$ , $b>0$ .
ⓘ Symbols: $\,\mathrm{d}\NVar{x}$ : differential of $x$ , $\tanh\NVar{z}$ : hyperbolic tangent function, $\int$ : integral, $\ln\NVar{z}$ : principal branch of logarithm function, $a$ : real or complex constant and $x$ : real variable Permalink: http://dlmf.nist.gov/4.40.E10 Encodings: TeX, pMML, png See also: Annotations for §4.40(iii), §4.40 and Ch.4

§4.40(iv) Inverse Hyperbolic Functions

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Notes:: These results may be verified by differentiation.
Referenced by:: §1.4(iv), §4.40(i)
Permalink:: http://dlmf.nist.gov/4.40.iv
See also:: Annotations for §4.40 and Ch.4

4.40.11		$\int\operatorname{arcsinh}x\,\mathrm{d}x=x\operatorname{arcsinh}x-(1+x^{2})^{1% /2}.$
ⓘ Symbols: $\,\mathrm{d}\NVar{x}$ : differential of $x$ , $\operatorname{arcsinh}\NVar{z}$ : inverse hyperbolic sine function, $\int$ : integral and $x$ : real variable A&S Ref: 4.6.43 (modified) Permalink: http://dlmf.nist.gov/4.40.E11 Encodings: TeX, pMML, png See also: Annotations for §4.40(iv), §4.40 and Ch.4

4.40.12		$\int\operatorname{arccosh}x\,\mathrm{d}x=x\operatorname{arccosh}x-(x^{2}-1)^{1% /2},$
$1<x<\infty$ ,
ⓘ Symbols: $\,\mathrm{d}\NVar{x}$ : differential of $x$ , $\operatorname{arccosh}\NVar{z}$ : inverse hyperbolic cosine function, $\int$ : integral and $x$ : real variable A&S Ref: 4.6.44 (modified) Permalink: http://dlmf.nist.gov/4.40.E12 Encodings: TeX, pMML, png See also: Annotations for §4.40(iv), §4.40 and Ch.4

4.40.13		$\int\operatorname{arctanh}x\,\mathrm{d}x=x\operatorname{arctanh}x+\tfrac{1}{2}% \ln\left(1-x^{2}\right),$
$-1<x<1$ ,
ⓘ Symbols: $\,\mathrm{d}\NVar{x}$ : differential of $x$ , $\operatorname{arctanh}\NVar{z}$ : inverse hyperbolic tangent function, $\int$ : integral, $\ln\NVar{z}$ : principal branch of logarithm function and $x$ : real variable A&S Ref: 4.6.45 (modified) Permalink: http://dlmf.nist.gov/4.40.E13 Encodings: TeX, pMML, png See also: Annotations for §4.40(iv), §4.40 and Ch.4

4.40.14		$\int\operatorname{arccsch}x\,\mathrm{d}x=x\operatorname{arccsch}x+% \operatorname{arcsinh}x,$
$0<x<\infty$ ,
ⓘ Symbols: $\,\mathrm{d}\NVar{x}$ : differential of $x$ , $\operatorname{arccsch}\NVar{z}$ : inverse hyperbolic cosecant function, $\operatorname{arcsinh}\NVar{z}$ : inverse hyperbolic sine function, $\int$ : integral and $x$ : real variable A&S Ref: 4.6.46 (modified. The first printing had an error.) Permalink: http://dlmf.nist.gov/4.40.E14 Encodings: TeX, pMML, png See also: Annotations for §4.40(iv), §4.40 and Ch.4

4.40.15		$\int\operatorname{arcsech}x\,\mathrm{d}x=x\operatorname{arcsech}x+% \operatorname{arcsin}x,$
$0<x<1$ ,
ⓘ Symbols: $\,\mathrm{d}\NVar{x}$ : differential of $x$ , $\operatorname{arcsech}\NVar{z}$ : inverse hyperbolic secant function, $\int$ : integral, $\operatorname{arcsin}\NVar{z}$ : arcsine function and $x$ : real variable A&S Ref: 4.6.47 (modified. The first printing had an error.) Permalink: http://dlmf.nist.gov/4.40.E15 Encodings: TeX, pMML, png See also: Annotations for §4.40(iv), §4.40 and Ch.4

4.40.16		$\int\operatorname{arccoth}x\,\mathrm{d}x=x\operatorname{arccoth}x+\tfrac{1}{2}% \ln\left(x^{2}-1\right),$
$1<x<\infty$ .
ⓘ Symbols: $\,\mathrm{d}\NVar{x}$ : differential of $x$ , $\operatorname{arccoth}\NVar{z}$ : inverse hyperbolic cotangent function, $\int$ : integral, $\ln\NVar{z}$ : principal branch of logarithm function and $x$ : real variable A&S Ref: 4.6.48 (modified) Permalink: http://dlmf.nist.gov/4.40.E16 Encodings: TeX, pMML, png See also: Annotations for §4.40(iv), §4.40 and Ch.4

§4.40(v) Compendia

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Permalink:: http://dlmf.nist.gov/4.40.v
See also:: Annotations for §4.40 and Ch.4

Extensive compendia of indefinite and definite integrals of hyperbolic functions include Apelblat (1983, pp. 96–109), Bierens de Haan (1939), Gröbner and Hofreiter (1949, pp. 139–160), Gröbner and Hofreiter (1950, pp. 160–167), Gradshteyn and Ryzhik (2015, Chapters 2–4), and Prudnikov et al. (1986a, §§1.4, 1.8, 2.4, 2.8).