What are the closed-form inverses of $x+\sinh(x)$, $x+\cosh(x)$?

Question

What are the closed-form inverses of the injective pieces of $x+\sinh(x)$, $x+\cosh(x)$?

I assume these functions don't have inverses that are elementary functions.

Can the inverses be represented with help of Lambert W?

Closed-form inverses can give hints for properties and calculation of the inverses.

I ask here because I want to present in my answer the closed-form representations of the inverses I found.

Further answers and methods are welcome.

For $x+\sinh x$ you can use power series inversion. $x+\cosh x$ is not invertible — Ninad Munshi, Dec 22 '22 at 18:04
@NinadMunshi $x + \cosh(x)$ is not one-to-one, but its restriction to $[-\ln(1+\sqrt{2}), \infty)$ is. The inverse has a power series around $1$, $$(t - 1) - \frac12(t - 1)^2 + \frac12 (t - 1)^3 - \frac23 (t - 1)^4 + (t - 1)^5 - \frac{289}{180} (t - 1)^6 + \frac{27}{10} (t - 1)^7 + \ldots$$ — Robert Israel, Dec 22 '22 at 18:14

IV_ · Accepted Answer · 2022-12-28T12:43:15.773

Elementary functions:

The function terms of $\sinh(x)$ and $\cosh(x)$ are irreducible polynomials of both $e^x$ and $e^{-x}$: Wikipedia: Hyperbolic functions - Exponential definitions. Liouville and Ritt proved that such kind of functions (over a complex domain without isolated points) don't have inverses that are elementary functions: How can we show that $A(z,e^z)$ and $A(\ln (z),z)$ have no elementary inverse?

Lambert W, Generalized Lambert W:

The defining equations for the inverse $x+\sinh(x)=y$ and $x+\cosh(x)=y$ can be rearranged to polynomial equations of both $x$ and $e^x$ which are quadratic for $e^x$. These equations are not in a form to apply Lambert W or Generalized Lambert W.

"Leal-functions":

The inverses of the functions mentioned in the question are presented in [Vazquez-Leal et al. 2020].

We can take them as closed-form functions because some of their algebraic properties and their applicability for some other kinds of equations are presented in the cited article.
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[Vazquez-Leal et al. 2020] Vazquez-Leal, H.; Sandoval-Hernandez, M. A.; Filobello-Ninoa, U.: The novel family of transcendental Leal-functions with applications to science and engineering. Heliyon 6 (2020) (11) e05418

What are the closed-form inverses of $x+\sinh(x)$, $x+\cosh(x)$?

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