This is follow up to this question which you will have to see for context:
which had the closed form of $$\mathrm{ta^{{ta}^t}-\frac{Wi\left(a^{{ta}^t}\right)}{ln(a)}+C,Wi(x)=\int _1^xW(ln(x))dx\mathop =^{x\to ln(x)}\int_0^{ln(x)} W(x)e^x dx}$$
This is a good creative function made up by @Arjun Vyavaharkar and this is the Sophomore’s Dream function created by @JJacquelin which seem like cop outs for a closed form. The “Wi” W-Lambert/Product Logarithm Integral function may possibly be represented as shown here. These are great ideas, but they are simply another invention.
In order to find a closed form for the linked question, and possibly other similar questions, one must find. Note that the integrals are indefinite unlike the definition of Wi(x).
$$\mathrm{\int \,^2\left(a^t\right)dt=\int a^{ta^t} dt =C+t+\frac1{ln(a)}\sum_{n=0}^\infty \frac{(-1)^n Q(n+1,-nt\,ln(a))}{n^{n+1}}=C+t-t\sum_{n=0}^\infty \frac{(t\,ln(a))^n E_{-n}(-nt\,ln(a))}{n!}=ta^{{ta}^t}-\frac{Wi\left(ta^{{ta}^t}\right)}{ln(a)}+C\implies Wi\left(ta^{{ta}^t}\right) =ta^{{ta}^t} ln(a) -t\, ln(a)-\sum_{n=0}^\infty \frac{(-1)^n Q(n+1,-nt\,ln(a))}{n^{n+1}}= ln(a)ta^{{ta}^t} -ln(a)\int (a^t)^{(a^t)}dt= ta^{{ta}^t} ln(a)-t\,ln(a)+t\,ln(a)\sum_{n=0}^\infty \frac{(t\,ln(a))^n E_{-n}(-nt\,ln(a)))}{n!}}$$
This representation clearly works and this representation which converges based on desmos, as the graph does not have special functions yet, but also diverges according to Wolfram Alpha, with the evaluated functions. It may be that a sum form is possible with the Regularized Lower Gamma function:
$$\mathrm{Wi(x)\mathop =^?\sum_{n=1}^\infty n^{n-1}P(n+1,-ln(x))}$$
Some useful Lambert-W function identities. In short, what is a closed form of Wi(x) using any academically accepted functions? I need this to find closed forms of other tetration-like integrals; this closed form will also help others. Here is the definition of Wi(x) as proposed by @Arjun. This uses
$$\mathrm{Wi(x)=\int_1^x W(ln(x))dx\mathop =^{x\to ln(x)}\int _0^{ln(x)} W(x)e^x dx+}$$
An equivalent integral is of the inverse integrand. Note the original question already found this integral. I just applied the inverse function integration theorem.This is the gamma sum verification and exponential integral sum verification. The second representation is seen from the Exponential Integral function:
$$\mathrm{y=\int e^{xe^x}dx\implies Wi(x)=\int_1^x W(ln(x))dx =W(ln(x))(x-1)+\sum_{n=1}^\infty\frac{(-1)^n P(n+1,-n\,W(ln(x))}{n^{n+1}}= W(ln(x))(x-1)+\sum_{n=1}^\infty\frac{(-1)^n}{n^{n+1}} +W(ln(x))\sum_{n=1}^\infty \frac{W^n(ln(x))E_{-n}(-n\,W(ln(x))}{n!}}$$
Also take a look at these generalized Hypergeometric functions which probably give a closed form for Wi(x).
I already have sum and integral representations of the function, so now I need a closed form to solve more tetration integrals. The Wi function will help find closed forms of these and similar problems as well. Please correct me and give me feedback!
Here are a few functions which may help give a closed form. The first is from this arxiv article. It is a Mittag-Leffler function and a Regularized Lower Incomplete Gamma function. Remember that P(a,z)+Q(a,z)=1: $$\mathrm{e^{-z}E_{\frac1n}\left(z^\frac1n\right)-1=\sum_{k=1}^{n-1}P\left(1-\frac kn,z\right)}$$
There are also 2 useful sums representations using DLMF 8.15:
If there are no closed form in terms of any known special functions, then please prove it. This series reminds me of the Marcum Q function for m$\not \in \Bbb Z^-$: $$\mathrm{Q_m(a,b)=1-e^{-\frac{a^2}2}\sum_{n=0}^\infty\left(\frac{a^2}{2}\right)^n\frac{P\left(m+n,\frac{b^2}2\right)}{n!}}$$
