Htykuut

I have used the Kummer representation series of loggamma function but does not look promissing to tackle this integral. Any idea to calculate this integral in closed-form ?

$$int_{0}^{1}ln(x)lnGamma(x)dx$$

I am quite skeptical about a possible closed form of this integral.

For an approximation, I should use the expansion
$$log (Gamma (x))=-log (x)-gamma x+frac{pi ^2 x^2}{12}+frac{x^3 psi ^{(2)}(1)}{6}+frac{pi ^4
x^4}{360}+frac{x^5 psi ^{(4)}(1)}{120}+frac{pi ^6
x^6}{5670}+Oleft(x^7right)$$ and integrate termwise to end with
$$int_0^1 log(x)log (Gamma (x))=-2+frac{gamma }{4}-frac{pi ^2}{108}-frac{pi ^4}{9000}-frac{pi
^6}{277830}-frac{psi ^{(2)}(1)}{96}-frac{psi ^{(4)}(1)}{4320}$$ which is $approx -1.93056$ while numerical integration leads to $approx -1.92922$.

Expanding $log (Gamma (x))$ to $Oleft(x^{10}right)$ would lead to
$$int_0^1 log(x)log (Gamma (x))=-2+frac{gamma }{4}-frac{pi ^2}{108}-frac{pi ^4}{9000}-frac{pi
^6}{277830}-frac{pi ^8}{6123600}-frac{pi ^{10}}{113201550}-frac{psi
^{(2)}(1)}{96}-frac{psi ^{(4)}(1)}{4320}-frac{psi
^{(6)}(1)}{322560}-frac{psi ^{(8)}(1)}{36288000}$$ which is $approx -1.92922$.