What is the period of the composition of hyperbolic tangent and hyperbolic arcsine?

Question

Consider the following indefinite integral: $ \int \mathrm d s = \tanh \left ( \mathrm {arcsinh} \frac{\beta}{\alpha} \right ) \, $ where $ 0 < \alpha \in \mathbb R \; $ is constant and $ \beta \in \mathbb R \; $ varies . How can I evaluate the period of $ \int \mathrm d s \, $ ?

Answer 1

The hyperbolic functions are not perodic.

But what you have can be simplified.

$y = \sinh x = \frac 12 (e^x - e^{-x})$

let $z = e^{x}$

$2y = z - \frac {1}{z}\\ z^2 - 2yz - 1 = 0\\ z = y\pm \sqrt {y^2+1}\\ x = \arctan y = \ln z$

If $z = y + \sqrt {y^2+1}$ then $\frac {1}{z} = y - \sqrt {y^2+1}$

$\tanh x = \frac {\sinh x}{\cosh x}\\ \tanh (\text{arcsinh} x) = \frac {x}{\cosh (\text{arcsinh} x)}\\ \tanh (\text{arcsinh} x) = \frac {x}{e^{\text{arcsinh} x} +e^{-\text {arcsinh} x} }\\ \tanh (\text{arcsinh} x) = \frac {x}{z + \frac {1}{z}}\\ \tanh (\text{arcsinh} x) = \frac {x}{\sqrt {x^2+1}}\\ \tanh (\text{arcsinh} \frac {b}{a}) = \frac {b}{\sqrt {a^2+b^2}}\\ $