  In this work, we elucidate the mathematical structure of the integral that
arises when computing the electron-ion temperature equilibration time for a
homogeneous weakly-coupled plasma from the Lenard-Balescu equation. With some
minor approximations, we derive an exact formula, requiring no input Coulomb
logarithm, for the equilibration rate that is valid for moderate electron-ion
temperature ratios and arbitrary electron degeneracy. For large temperature
ratios, we derive the necessary correction to account for the coupled-mode
effect, which can be evaluated very efficiently using ordinary Gaussian
quadrature.
