In my old writes I've found the following formula, where ${_{}^2}x$ is tetration:
$$\int_0^1 {_{}^2}x \ dx = \sum\limits_{i=1}^\infty \frac {(-1)^{i+1}} {{_{}^2}i} \approx 0.783430511\ldots$$
Now I am interested in series of generalized case of tetration:
$$\int_0^1 {_{}^n}x \ dx = ?$$
Could anybody find out it with an explanation?
Let $n \in \mathbb{Z}^+$, $x>0$ and
$$a_{n,k}= \begin{cases} 1 & \quad \text{if $k=0$}\\ \dfrac{1}{k!} & \quad \text{if $n=1$}\\ \displaystyle \frac{1}{k}\sum_{j=1}^k ja_{n,k-j}a_{n-1,j-1} & \quad \text{otherwise.}\\ \end{cases} $$
Then
$$ \int {}^n x\, dx= \sum_{k=0}^n \frac{(-1)^k (k+1)^{k-1}\Gamma(k+1, -\log x)}{k!} + \sum_{k=n+1}^\infty (-1)^k a_{n,k} \Gamma(k+1, -\log x) + C. $$
Source: I.N. Galidakis, On an Application of Lambert’s W Function to Infinite Exponentials, Corollary 10.9.
