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Showing content from https://dlmf.nist.gov/4.40 below:

§4.40 Integrals ‣ Hyperbolic Functions ‣ Chapter 4 Elementary Functions

§4.40 Integrals Contents
  1. §4.40(i) Introduction
  2. §4.40(ii) Indefinite Integrals
  3. §4.40(iii) Definite Integrals
  4. §4.40(iv) Inverse Hyperbolic Functions
  5. §4.40(v) Compendia
§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 ∫ sinh ⁡ x ⁢ d x = cosh ⁡ x , 4.40.2 ∫ cosh ⁡ x ⁢ d x = sinh ⁡ x , 4.40.3 ∫ tanh ⁡ x ⁢ d x = ln ⁡ ( cosh ⁡ x ) . 4.40.4 ∫ csch ⁡ x ⁢ d x = ln ⁡ ( tanh ⁡ ( 1 2 ⁢ x ) ) , 0 < x < ∞ . 4.40.5 ∫ sech ⁡ x ⁢ d x = gd ⁡ ( x ) .

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

§4.40(iii) Definite Integrals 4.40.7 ∫ 0 ∞ e − x ⁢ sin ⁡ ( a ⁢ x ) sinh ⁡ x ⁢ d x = 1 2 ⁢ π ⁢ coth ⁡ ( 1 2 ⁢ π ⁢ a ) − 1 a , a ≠ 0 , 4.40.8 ∫ 0 ∞ sinh ⁡ ( a ⁢ x ) sinh ⁡ ( π ⁢ x ) ⁢ d x = 1 2 ⁢ tan ⁡ ( 1 2 ⁢ a ) , − π < a < π , 4.40.9 ∫ − ∞ ∞ e a ⁢ x ( cosh ⁡ ( 1 2 ⁢ x ) ) 2 ⁢ d x = 4 ⁢ π ⁢ a sin ⁡ ( π ⁢ a ) , − 1 < a < 1 , §4.40(iv) Inverse Hyperbolic Functions §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).

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