How to prove this?!🤔

\[\displaystyle\int_\tfrac{1}{2}^1\dfrac{\psi(x)}{1+\Gamma^2(x)}dx=\dfrac{1}{2}\ln\left(\dfrac{1+\pi}{2\pi}\right)\]

where \(\Gamma(x)\) and \(\psi(x)\) are the gamma and digamma functions respectively.

#Integral #definiteintegral #gammafunction #digammafunction #pi #logarithm