Simplify this Gamma monster!

$\begin{aligned} X & = \sqrt[3]{\frac{33554432\ \Gamma\left(\frac {13}{12}\right)^2\Gamma\left(\frac {17}{12}\right)^2\Gamma\left(\frac 74 \right)^8}{225\pi^4 \ \Gamma\left(\frac 7{12}\right)^2\Gamma\left(\frac {11}{12}\right)^2}} \\ & = \sqrt[3]{\frac{33554432\ \blue{\Gamma\left(\frac 7{12}+\frac 12 \right)^2\Gamma\left(\frac {11}{12}+\frac 12 \right)^2} \Gamma \left( \frac 74 \right)^8}{225\pi^4 \ \Gamma\left(\frac 7{12}\right)^2\Gamma\left(\frac {11}{12}\right)^2}} & \small \blue{\text{By Legendre duplication formula}} \\ & = \sqrt[3]{\frac{\blue{8388608\ \Gamma\left(\frac 76\right)^2\Gamma\left(\frac {11}6 \right)^2} \Gamma \left( \frac 74 \right)^8}{225\pi^{\blue 2} \ \Gamma\left(\frac 7{12}\right)^{\blue 4}\Gamma\left(\frac {11}{12}\right)^{\blue 4}}} & \small \blue{\text{By }\Gamma (1+z) = z \Gamma(z)} \\ & = \sqrt[3]{\frac{\blue{524288\ \Gamma\left(\frac 16\right)^2\Gamma\left(\frac 56 \right)^2} \Gamma \left( \frac 74 \right)^8}{\blue{729}\pi^2 \ \red{\Gamma\left(\frac 14 + \frac 13 \right)^4\Gamma\left(\frac 14 + \frac 23 \right)^4}}} & \small \red{\text{By triplication formula}} \\ & = \sqrt[3]{\frac{\red{32768}\ \blue{(2\pi)^2} \red{\Gamma\left(\frac 14\right)^4} \Gamma \left( \frac 74 \right)^8}{\red{243}\pi^{\red 6} \ \red{\Gamma\left(\frac 34\right)^4}}} & \small \blue{\text{By }\Gamma(z)\Gamma(1-z) = \frac \pi{\sin (\pi z)}} \\ & = \sqrt[3]{\frac{\blue{131072}\ \Gamma\left(\frac 14\right)^4 \left(\frac 34\Gamma \left( \frac 34 \right) \right)^8}{243 \pi^{\blue 4} \ \Gamma\left(\frac 34\right)^4}} \\ & = \sqrt[3]{\frac {54}{\pi^4}\Gamma \left(\frac 14\right)^4\Gamma \left(\frac 34\right)^4} \\ & = \sqrt[3]{\frac {54}{\pi^4} \left(\frac \pi{\sin \frac \pi 4}\right)^4} = \sqrt[3]{216} = \boxed 6 \end{aligned}$

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References:

• Gamma function
• Legendre duplication formula : $\Gamma (2z) = \dfrac {2^{2z-\frac 12}}{\sqrt{2\pi}}\Gamma (z) \Gamma \left(z+\frac 12\right)$
• Triplication formula (equation 51) : $\Gamma (3z) = \dfrac {3^{3z-\frac 12}}{\sqrt{2\pi}}\Gamma (z) \Gamma \left(z+\frac 13\right) \Gamma \left(z+\frac 23\right)$