I was interested in an integral that I known from [1], it is

$$\int_0^1 \log(x!)dx.$$

I tried to get such closed-form using myself ideas and symbolic calculations, also with the help of Wolfram Alpha online calculator. But I don't know how get the sum of certain series involving a special function.

My first step was to invoke the formula $(3.13)$ from [2], taking the logarithm and integrating one has that $$\int_0^1 \log(x!)dx$$ is equals to $$\log 2+\frac{1}{24}\sum_{n=1}^\infty\left(-2^{n}(6(6\log A)+\log \pi)-24\psi^{(-2)}(\frac{1}{2}+2^{-n})+\log(32))-12\log\pi\right),$$ where $A$ is the Glaisher-Kinkelin constant and $\psi^{(n)}(s)$ denotes the $n^{th}$ derivative of the digamma function.

See the first comment that you can to evaluate in Wolfram Alpha online calculator. I've some computational evidence, for example the following code (it is a line)

`sum 1/24 (-2^n (6 (6 log(Glaisher) + log(π)) - 24 polygamma(-2, 1/2 + 2^(-n)) + log(32)) - 12 log(π)), from n=1 to 100`

and

`log(2)-0.7742086473552725676369-(1/2log(2pi)-1)`

but I can not to prove the corresponding closed-form. Of course I know how to get the sum of geometric series but the problem here is different.

Question.Please, prove that $$\log 2+\frac{1}{24}\sum_{n=1}^\infty\left(-2^{n}(6(6\log A)+\log \pi)-24\psi^{(-2)}(\frac{1}{2}+2^{-n})+\log(32))-12\log\pi\right)$$ is equals to the closed-form for $\int_0^1 \log(x!)dx$ given in [1]. What I ask is if you can analyze the series to calculate its sum. I've deduced/considered previous expression and I would like to know if it is possible to prove that previous expression equals to $\frac{1}{2}\log(2\pi )-1$ analizing the series to get its sums (without invoking that it is equals to $\int_0^1 \log(x!)dx$).Many thanks.

I have no intuition/knowledges to know if it is easy to get the sum of the series.

## References:

[1] *Muliplicative integral of* $\Gamma(x)$, this MathOverflow (July of 2010).

[2] Manuel Benito, Luis M. Navas and Juan Luis Varona, *Möbius inversion from the point of view of arithmetical semigroup flows*, Biblioteca de la Revista Matemática Iberoamericana, Proceedings of the "Segundas Jornadas de Teoría de Números", (2008), pages 61-81.

`int log gamma(1/2+x/2^n)-log(sqrt(pi))dx, from x=0 to 1`

$\endgroup$ – user142929 Aug 2 at 16:13