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I came across the following integral $$\int_{0}^{1}\ln^2\Gamma(x)\ln\Gamma(1-x)\ dx$$

and I have no idea where to even start.

I know you can interchange the $x$ and $1-x$ by king's property, and that $\int_{0}^{1}\ln^2\Gamma(x)dx$ has a nice closed form but otherwise I don't know what to do.

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  • $\begingroup$ Numerically this is roughly 0.1775090486962202. I doubt it has a closed form but I guess we'll see. $\endgroup$
    – Max0815
    Feb 5 at 22:15
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    $\begingroup$ where did you find it? $\endgroup$
    – Sine of the Time
    Feb 5 at 22:41
  • $\begingroup$ If you use the reflection formula you might get some mileage out of the results in this paper, particularly Sec. 6. $\endgroup$
    – A rural reader
    Feb 5 at 23:22
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    $\begingroup$ see also here $\endgroup$
    – alexjo
    Feb 5 at 23:28

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