# §4.40 Integrals

## §4.40(i) Introduction

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

 4.40.1 $\displaystyle\int\sinh x\,\mathrm{d}x$ $\displaystyle=\cosh x,$ 4.40.2 $\displaystyle\int\cosh x\,\mathrm{d}x$ $\displaystyle=\sinh x,$ 4.40.3 $\displaystyle\int\tanh x\,\mathrm{d}x$ $\displaystyle=\ln\left(\cosh x\right).$ 4.40.4 $\displaystyle\int\operatorname{csch}x\,\mathrm{d}x$ $\displaystyle=\ln\left(\tanh\left(\tfrac{1}{2}x\right)\right),$ $0. 4.40.5 $\displaystyle\int\operatorname{sech}x\,\mathrm{d}x$ $\displaystyle=\operatorname{gd}\left(x\right).$

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.

## §4.40(iii) Definite Integrals

 4.40.7 $\int_{0}^{\infty}e^{-x}\frac{\sin\left(ax\right)}{\sinh x}\,\mathrm{d}x=\tfrac% {1}{2}\pi\coth\left(\tfrac{1}{2}\pi a\right)-\frac{1}{a},$ $a\neq 0$,
 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,
 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,
 4.40.10 ${\int_{0}^{\infty}\frac{\tanh\left(ax\right)-\tanh\left(bx\right)}{x}\,\mathrm% {d}x=\ln\left(\frac{a}{b}\right)},$ $a>0$, $b>0$.

## §4.40(iv) Inverse Hyperbolic Functions

 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, ⓘ 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,
 4.40.14 $\int\operatorname{arccsch}x\,\mathrm{d}x=x\operatorname{arccsch}x+% \operatorname{arcsinh}x,$ $0, ⓘ 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, ⓘ 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.

## §4.40(v) Compendia

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).