Elementary functions:
The function terms of the hyperbolic functions ($\sinh(x)$, $\cosh(x)$, $\tanh(x)$, $\text{sech}(x)$, $\coth(x)$, $\text{csch}(x)$) are irreducible rational expressions of $e^x$: Wikipedia: Hyperbolic functions - Exponential definitions. The function terms $x\ \text{hyp}(x)$, wherein $\text{hyp}$ is one of the hyperbolic functions, are irreducible rational expressions of both $x$ and $e^x$ therefore.
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:
The defining equations for the inverse $x\ \text{hyp}(x)=y$ 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.
Generalized Lambert W:
The equations $x\tanh(x)=y$ and $x\coth(x)$ can be solved in terms of Generalized Lambert W of [Mezö 2017], [Mezö/Baricz 2017], [Castle 2018]:
\begin{matrix}
x\tanh(x)=y & & x\coth(x)=y\\
x\frac{e^{2x}-1}{e^{2x}+1}=y & & x\frac{e^{2x}+1}{e^{2x}-1}=y\\
\frac{x-y}{x+y}e^{2x}=1 & & \frac{x-y}{(x+y)}e^{2x}=-1\\
x=\frac{1}{2}W(^{+2y}_{-2y};1) & & x=\frac{1}{2}W(^{+2y}_{-2y};-1)\\
\end{matrix}
"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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[Castle 2018] Castle, P.: Taylor series for generalized Lambert W functions. 2018
[Mezö 2017] Mezö, I.: On the structure of the solution set of a generalized Euler-Lambert equation. J. Math. Anal. Appl. 455 (2017) (1) 538-553
[Mezö/Baricz 2017] Mezö, I.; Baricz, A.: On the generalization of the Lambert W function. Transact. Amer. Math. Soc. 369 (2017) (11) 7917–7934 (On the generalization of the Lambert W function with applications in theoretical physics. 2015)
[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
