We compared several families of algorithms for recursive integral time-extrapolation (RITE) algorithms for waves in isotropic and anisotropic media. These methods allow simulating accurate wave extrapolation with little numerical dispersion even when using larger time steps than are usually possible for conventional finite-difference methods. These various RITE algorithms all share the use of mixed space/wavenumber-domain operators derived from Fourier integral solutions of single-mode wave equations. We evaluated a taxonomy for RITE methods based on how they approximated the influence of medium heterogeneity. One family of methods uses mixed-domain series expansions to provide accurate approximations to heterogeneous extrapolators even for large time steps. We compared several methods for deriving coefficients for such series approximations. Another family of methods uses interpolation between different homogeneous extrapolations to approximate heterogeneous time extrapolation. Such methods can be based on interpolating either the extrapolators themselves or interpolating between reference wavefields extrapolated using different homogeneous parameters. Interpolation methods work well for smooth media, but can suffer from oscillatory artifacts at large velocity discontinuities unless the time step is small. We tested numerical examples of the various families of RITE algorithms to determine their relative strengths and limitations.