  We provide an analytic formula for the (rescaled) one-loop scalar hexagon
integral $\tilde\Phi_6$ with all external legs massless, in terms of classical
polylogarithms. We show that this integral is closely connected to two
integrals appearing in one- and two-loop amplitudes in planar $\\mathcal{N}=4$
super-Yang-Mills theory, $\Omega^{(1)}$ and $\Omega^{(2)}$. The derivative of
$\Omega^{(2)}$ with respect to one of the conformal invariants yields
$\tilde\Phi_6$, while another first-order differential operator applied to
$\tilde\Phi_6$ yields $\Omega^{(1)}$. We also introduce some kinematic
variables that rationalize the arguments of the polylogarithms, making it easy
to verify the latter differential equation. We also give a further example of a
six-dimensional integral relevant for amplitudes in $\\mathcal{N}=4$
super-Yang-Mills.
