  We present an algorithm for the rapid numerical integration of a
time-periodic ODE with a small dissipation term that is $C^1$ in the velocity.
Such an ODE arises as a model of spin-orbit coupling in a star/planet system,
and the motivation for devising a fast algorithm for its solution comes from
the desire to estimate probability of capture in various solutions, via Monte
Carlo simulation: the integration times are very long, since we are interested
in phenomena occurring on times similar to the formation time of the planets.
The proposed algorithm is based on the High-order Euler Method (HEM) which was
described in~\cite{hem}, and it requires computer algebra to set up the code
for its implementation. The pay-off is an overall increase in speed by a factor
of about $7.5$ compared to standard numerical methods. Means for accelerating
the purely numerical computation are also discussed
