We provide an analytic formula for the (rescaled) one-loop scalar hexagon integral $tildePhi_6$ 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 $tildePhi_6$, while another first-order differential operator applied to $tildePhi_6$ 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.
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