Is there any way to derive this identity?$$\int_0^1 \left(\ln(\Gamma(z))\right)^2dz=\frac{\gamma^2}{12}+\frac{\pi^2}{48}+\frac{\gamma\ln(2\pi)}{6}+\frac{(\ln(2\pi))^2}{3}-(\gamma+\ln(2\pi))\frac{\zeta'(2)}{\pi^2}+\frac{\zeta''(2)}{2\pi^2}$$
I have a suspicion that it could be related to the integral $$\int_0^1 \ln(\Gamma(z))dz=\frac{\ln(2\pi)}{2}$$ but I am not sure on how to approach deriving this relation. If anyone could let me know any methods on how to derive this integral I would be very grateful.