  Peters' formula is an analytical estimate of the time-scale of gravitational
wave (GW)-induced coalescence of binary systems. It is used in countless
applications, where the convenience of a simple formula outweighs the need for
precision. However, many promising sources of the Laser Interferometer Space
Antenna (LISA), such as supermassive black hole binaries and extreme mass-ratio
inspirals (EMRIs), are expected to enter the LISA band with highly eccentric
($e \gtrsim$ 0.9) and highly relativistic orbits. These are exactly the two
limits in which Peters' estimate performs the worst. In this work, we expand
upon previous results and give simple analytical fits to quantify how the
inspiral time-scale is affected by the 1.5 post-Newtonian (PN) hereditary
fluxes and spin-orbit couplings. We discuss several cases that demand a more
accurate GW time-scale. We show how this can have a major influence on
quantities that are relevant for LISA event-rate estimates, such as the EMRI
critical semi-major axis. We further discuss two types of environmental
perturbations that can play a role in the inspiral phase: the gravitational
interaction with a third massive body and the energy loss due to dynamical
friction and torques from a surrounding gas phase ubiquitous in galactic
nuclei. With the aid of PN corrections to the time-scale in vacuum, we find
simple analytical expressions for the regions of phase space in which
environmental perturbations are of comparable strength to the effects of any
particular PN order, being able to qualitatively reproduce the results of much
more sophisticated analyses.
