  We compute the six-dimensional hexagon integral with three non-adjacent
external masses analytically. After a simple rescaling, it is given by a
function of six dual conformally invariant cross-ratios. The result can be
expressed as a sum of 24 terms involving only one basic function, which is a
simple linear combination of logarithms, dilogarithms, and trilogarithms of
uniform degree three transcendentality. Our method uses differential equations
to determine the symbol of the function, and an algorithm to reconstruct the
latter from its symbol. It is known that six-dimensional hexagon integrals are
closely related to scattering amplitudes in N=4 super Yang-Mills theory, and we
therefore expect our result to be helpful for understanding the structure of
scattering amplitudes in this theory, in particular at two loops.
