I have stumbled across the following integral and have struck a dead end...
$$\int\_0^1 \ln{\left(\Gamma(x)\right)}\cos^2{(\pi x)} \; {\mathrm{d}x}$$
Where $\Gamma(x)$ is the Gamma function.

I tried expressing $\Gamma(x)$ as $(x-1)!$ then using log properties to split the integral. Maybe there should be a summation in combination with the integral?? I believe this integral has a closed form but I would like help finding it.