Exact values of the equation $\ln (x+1)=\frac{x}{4-x}$ I'm asking for a closed form (an exact value) of the equation solved for $x$
$$\ln (x+1)=\frac{x}{4-x}$$
$0$ is trivial but there is another solution (approximately 2,2...).
I've tried with Lambert's W Function or with Regularized Gamma Functions but I didn't get so far.

As said in comments, there is no closed form for the solution.
However, we can generate closed forms of very good approximations starting with as estimate $x_0$ written as a rational number and apply one single iteration of Newton method to get $x_1$ given, for the case of this function, by $$x_1=\frac{(16-9 x_0) x_0-(x_0-4)^2 (x_0+1) \log (x_0+1)}{(x_0-12) x_0+12}$$ Using $x_0=\frac {11}5$ as you gave in the post this will lead to $$x_1=\frac{1045+1296 \log \left(\frac{16}{5}\right)}{1195}\approx 2.135935941$$ Using this result, make $x_0=\frac {2136}{1000}$ which leads to $$x_1=\frac{13450125+21281288 \log \left(\frac{392}{125}\right)}{17713875}\approx 2.132426289$$ and you could repeat this process as long as you wish.
