  We consider the analytic calculation of a two-loop non-planar three-point
function which contributes to the two-loop amplitudes for $t \bar{t}$
production and $\gamma \gamma$ production in gluon fusion through a massive
top-quark loop. All subtopology integrals can be written in terms of multiple
polylogarithms over an irrational alphabet and we employ a new method for the
integration of the differential equations which does not rely on the
rationalization of the latter. The top topology integrals, instead, in spite of
the absence of a massive three-particle cut, cannot be evaluated in terms of
multiple polylogarithms and require the introduction of integrals over complete
elliptic integrals and polylogarithms. We provide one-fold integral
representations for the solutions and continue them analytically to all
relevant regions of the phase space in terms of real function, extracting all
imaginary parts explicitly. The numerical evaluation of our expressions becomes
straightforward in this way.
