Table of Integrals, Series, and Products, 8th Edition

2.46 Algebraic functions of hyperbolic functions

2.461

1. 

${\displaystyle \int \sqrt{\text{tanh}x\text{d}x}=\text{archtanh}}\sqrt{\text{tanh}x}-\text{archtan}\sqrt{\text{tanh}x}$

MZ 221

2. 

${\displaystyle \int \sqrt{\text{coth}x}\text{d}x=\text{arccoth}\sqrt{\text{coth}x}-\text{arctan}\sqrt{\text{coth}x}}$

2.462

1. 

$\begin{array}{lll}{\displaystyle \int \frac{\text{sinh}x\text{d}x}{\sqrt{a^2+{\text{sinh}}^{2}x}}}\hfill & =\text{arcsinh}\frac{\text{cosh}x}{\sqrt{a^2-1}}=ln\left(\text{cosh}x+\sqrt{a^2+{\text{sinh}}^{2}x}\right)\hfill & \left[a^2>1\right]\hfill \\ =\text{arccosh}\frac{\text{cosh}x}{\sqrt{1-a^2}}=ln\left(\text{cosh}x+\sqrt{a^2+{\text{sinh}}^{2}x}\right)\hfill & \left[a^2<1\right]\hfill \\ =ln\text{cosh}x\hfill & \left[a^2=1\right]\hfill \end{array}$

2. 

${\displaystyle \int \frac{\text{sinh}x\text{d}x}{\sqrt{a^2-{\text{sinh}}^{2}x}}=\text{arcsin}\frac{\text{cosh}x}{\sqrt{a^2+1}}\left[{\text{sinh}}^{2}x<a^2\right]}$

3. 

$\begin{array}{l}{\displaystyle \int \frac{\text{sinh}x\text{d}x}{\sqrt{{\text{sinh}}^{2}x-a^2}}=\text{arccosh}\frac{\text{cosh}x}{\sqrt{a^2+1}}=ln\left(\text{cosh}x+\sqrt{{\text{sinh}}^{2}x-a^2}\right)}\hfill \\ \hfill \end{array}$