  In this paper, we derive a set of equations of motions for binaries on
eccentric orbits undergoing spin-induced precession that can efficiently be
integrated on the radiation-reaction timescale. We find a family of solutions
with a computation cost improved by a factor $10$ - $50$ down to $\sim 10$ ms
per waveform evaluation compared to waveforms obtained by directly integrating
the precession equations, that maintain a mismatch of the order $10^{-4}$ -
$10^{-6}$ for waveforms lasting a million orbital cycles and a thousand
spin-induced precession cycles. We express it in terms of parameters that make
the solution regular in the equal-mass limit, thus bypassing a problem of
previous similar solutions. We point to ways in which the solution presented in
this paper can be perturbed to take into account effects such as general
quadrupole momenta and post-Newtonian corrections to the precession equations.
This new waveform, with its improved efficiency and its accuracy, makes
possible Bayesian parameter estimation using the full spin and eccentricity
parameter volume for long lasting inspiralling signals such as stellar-origin
black hole binaries observed by LISA.
