How does G function appear?

If the integral $\displaystyle \int_0^{\frac{\pi}{24}}\ln(\tan x)\, dx$ can be expressed as $\pi\ln\left(\frac{G\left(\frac{a}{24}\right)G\left(\frac{b}{24}\right)}{ G\left(\frac{c}{24}\right)G\left(\frac{d}{24}\right)}\right)-\frac{\pi}{24}\ln\left(\frac{4096\pi^{12}}{\sqrt{2-\sqrt{3}}\left(2+\sqrt{2+\sqrt{5}}\right)^5}\right)$ where $a,b,c$ and $d$ are positive integers with $c,d$ being primes and $a$ being perfect square number. Find the value of $a+b+c+d$ .

Notation: $G$ denotes Barnes G-function .

This is an original and inspired problem.