  A recently introduced approach to the gravitational dynamics of binary
systems involves intricate integrals, linked to nonlocal-in-time interactions
arising at the 5-loop level of classical gravitational scattering. We complete
the analytical evaluation of classical gravitational scattering at the sixth
order in Newton's constant, $G$, and at the sixth post-Newtonian accuracy. We
use computing techniques developed for the evaluation of multi-loop Feynman
integrals to obtain our results in two ways: high-precision arithmetic,
yielding reconstructed analytic expressions, and direct integration {\it via}
Harmonic Polylogarithms. The analytic expression of the tail contribution to
the scattering involve transcendental constants up to weight four.
