Watson's integral. Seemingly uninteresting question: calculate
$$W_S=\frac{1}{\pi^3}\int\limits_0^\pi\int\limits_0^\pi\int\limits_0^\pi
\frac{dx\,dy\,dz}{3-\cos{x}-\cos{y}-\cos{z}},$$
produces truly amazing answer:
$$W_S=\frac{\sqrt{6}}{96\pi^3}\Gamma\left(\frac{1}{24}\right)\Gamma\left(\frac{5}{24}\right)\Gamma\left(\frac{7}{24}\right)\Gamma\left(\frac{11}{24}\right)=
\frac{\sqrt{3}-1}{96\pi^3}\left[\Gamma\left(\frac{1}{24}\right)
\Gamma\left(\frac{11}{24}\right)\right]^2.$$
See http://link.springer.com/article/10.1007%2Fs10955-011-0273-0 (70+ Years of the Watson Integrals, by I. J. Zucker).