First, I would like to give a disclaimer that I do not know whether or not a closed form actually exists.
Also, I am trying to find a closed form that fits all integer values for $x$.
I was messing around with Desmos and came up with this interesting function. My computer is too slow to run it properly so I had to get around that by plotting integer values for $x$ for all $0<x\le200$ as shown here: https://www.desmos.com/calculator/yzvgfh3qwb. Using Desmos, I was able to see that there could be an improper integral from $1$ to $\infty$.
Here is what I tried by hand: $\frac{\sum\limits_{n=1}^{x}\frac{1}{\int_{0}^{\infty}e^{-t^{n}}dt}}{x}-1=\frac{\sum\limits_{n=1}^{x}\frac{1}{\int_{0}^{\infty}e^{-t^{n}}dt}-x}{x}=\frac{\sum\limits_{n=1}^{x}\left(\frac{1}{\int_{0}^{\infty}e^{-t^{n}}dt}-1\right)}{x}=\frac{1}{x}\cdot{\sum\limits_{n=1}^{x}\left(\frac{1}{\int_{0}^{\infty}e^{-t^{n}}dt}-1\right)}$
I don't know how to simplify from here much less find the previously mentioned improper integral from here.