Hot Integral - 30

$\large \int_{0}^{\frac13} \log\left( \Gamma (x) \right) \mathrm{d}x = \frac{\log\left( \frac{a\pi^b}{c} \right)}{d} - \frac{\pi}{e\sqrt f} + \frac{\psi '\left( \frac{g}{h} \right)}{\pi i\sqrt j} + \frac{k}{l}\log A$

The equation above holds true for positive integers $a$ , $b$ , $c$ , $d$ , $e$ , $f$ , $g$ , $h$ , $i$ , $j$ , $k$ , and $l$ , where $\gcd(a,c)=\gcd(g,h)=\gcd(k,l)=1$ and $f$ and $j$ are square free.

Evaluate $a+b+c+d+e+f+g+h+i+j+k+l$ .

Notations:

• $\Gamma (\cdot)$ denotes the gamma function .
• $\psi (\cdot)$ denotes the polygamma function .
• $A$ denotes the Glaisher's constant .