Nice Expression Made Ugly!

$\begin{aligned} \displaystyle \int_0^{\infty} \int_0^{\infty} \frac{1}{(x^5y)^{1/6} e^{(x+y)}}\,dx\,dy & =\left(\int_0^{\infty} x^{-5/6}e^{-x}\,dx \right)\left(\int_0^{\infty} y^{-1/6}e^{-y}\,dy\right)\\ &=\left(\int_0^{\infty} x^{1/6-1}e^{-x}\,dx\right)\left(\int_0^{\infty} y^{5/6-1}e^{-y}\,dy\right)\\ &=\Gamma \left(\frac{1}{6}\right) \Gamma \left(\frac{5}{6}\right) \\ \end{aligned}$

Since

$\Gamma(x)\Gamma(1-x)=\frac{\pi}{\sin(\pi x)}$

Hence, with $x=1/6$ ,

$\Gamma \left(\frac{1}{6}\right) \Gamma \left(\frac{5}{6}\right)=\frac{\pi}{\sin(\pi/6)}=\boxed{2\pi}$