Evaluate
$$ \int_{-\infty}^{\infty} \frac{\log \Gamma(0.25 + 0.5it) + \log \Gamma(0.25 - 0.5it) }{0.25 + t^2} \mathrm{d}t$$
where $\Gamma$ is the Gamma function. In my attempt, i applied the well-known Stirling approximation for $\Gamma$, but i'm not quite sure if the approach is correct.