Inverse of $f(x)=\sin(x)+x$ What is the inverse of 
$$f(x)=\sin(x)+x.$$
I thought about it for a while but I couldn't figure it out and I couldn't find the answer on the internet.
What about
$$f(x)=\sin(a \cdot x)+x$$
where $a$ is a known real constant.
Thank you for taking the time to read this question!
Sorry if this has been asked before...
 A: Why not give another “closed form” since @GEdgar’s solution for the Kepler equation technically has $E(a,M)$ as the solution in the Wikipedia link. Here is a solution to the inverse of $y=\sin(x)+x$ using mathematica function using Inverse Beta Regularized which is a standard function introduced in $1996$. The answer is from:

Closed form of $x$ for $x=\cos(x)$: Intuition for why the Dottie number is an inverse sine of the median of a Beta distribution.

where

$$M=E-e\sin(E)\iff x=y-a\sin(y)\implies y=\text E(a,x)$$

and

$$\text E(-1,x)=2\sin^{-1}\sqrt{\text I^{-1}_{\frac x\pi}\left(\frac12,\frac32\right)}=\text{hav}^{-1}\left(\text I^{-1}_{\frac x\pi}\left(\frac12,\frac32\right)\right)$$

Here is a numerical tester for the inverse function
Here is a plot of:
$$\boxed{x=\sin(y)+y\implies y=\text E(-1,x) = \text{hav}^{-1}\left(\text I^{-1}_{\frac x\pi}\left(\frac12,\frac32\right)\right),0\le x\le \pi}: $$

The code is inversehaversine(inversebetaregularized(x/pi,1/2,3/2)) 
where appears the Inverse Haversine, a transformed inverse sine function.
The differential equation for the inverse of $y=\sin(x)+x$ is:
$$y’\text{vercos}(y))=1$$
where vercos is the versed cosine, a transformed cosine function. The inverse function’s domain is extended by using a series expansion of $\text I^{-1}_z(a,b)$. Please correct me and give me feedback!
Unfortunately, it seems that no more solutions of the inverse of $x-a\sin(x)=y$ can be solved for using this method. The only exception is that the inverse of $x-\sin(x)$ can be given in closed form by switching the $\frac32$ and
$\frac12$ in the boxed answer.
A: Kepler's equation  ... its solution is known not to be an elementary function.
A: The best simple solution I found is using the fixed-point technique.
$y = x - \sin(y)$
$y = x - \sin(x - \sin(y))$
$y = x - \sin(x - \sin(x - \sin(y)))$
…
$y = x - \sin(x - \sin(x - \sin(x - \sin(x - ... \sin(y) ...))))$
That you can put any number for the $y$ on the right side because the effect is removed after too many $sin$ operations.
The function plots are (the $z$ slider is the $y$ value on the right side of the equation):
sin(x) + x inverse
As you can see, the source and inverse functions are symmetric according to the $y = x$ line.
A: That's a kind of Kepler's equation. Already Liouville proved that the inverse isn't an elementary function.
1) Lambert W, Genralized Lambert W
$$\sin(ax)+x=y$$
$$-\frac{1}{2}i\left(e^{aix}-e^{-aix}\right)+x=+y$$
$$-\frac{1}{2}i\left(e^{aix}\right)^2+\frac{1}{2}i+e^{aix}x-e^{aix}y=0$$
$$e^{aix}=-ix\pm\sqrt{-x^2+2yx-y^2+1}+iy$$
We see, the inverse cannot be represented in closed form in terms of Lambert W either, and not by generalized Lambert W of Mezö et. al. But possibly by other generalized Lambert W - see the references below.
2) "Leal-functions"
$$\sin(x)+x=y$$
because $\sin(it)=i\sinh(t)$: $x\to it$:
$$\sin(it)+it=y$$
$$i(\sinh(t)+t)=y$$
$$\sinh(t)+t=-iy$$
$$t=\text{Lsin}_2(-iy)$$
$$x=i\ \text{Lsin}_2(-iy)$$
[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)
[Castle 2018] Castle, P.: Taylor series for generalized Lambert W functions. 2018
[Stoutemyer 2022] Stoutemyer, D. R.: Inverse spherical Bessel functions generalize Lambert W and solve similar equations containing trigonometric or hyperbolic subexpressions or their inverses. 2022
[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
