We have developed a von Neumann stability and dispersion analysis of two time-integration techniques in the framework of Fourier pseudospectral (PS) discretizations of the second-order wave equation. The first technique is a rapid expansion method (REM) that uses Chebyshev matrix polynomials to approximate the continuous solution operator of the discrete wave equation. The second technique is a Lax-Wendroff method (LWM) that replaces time derivatives in the Taylor expansion of the solution wavefield with their equivalent spatial PS differentiations. In both time-integration schemes, each expansion term J results in an extra application of the spatial differentiation operator; thus, both methods are similar in terms of their implementation and the freedom to arbitrarily increase accuracy by using more expansion terms. Nevertheless, their limiting Courant-Friedrichs-Lewy stability number S and dispersion inaccuracies behave differently as J varies. We establish the S bounds for both methods in cases of practical use, J≤10, and we confirm the results by numerical simulations. For both schemes, we explore the dispersion dependence on modeling parameters J and S on the wavenumber domain, through a new error metric. This norm weights errors by the source spectrum to adequately measure the accuracy differences. Then, we compare the theoretical computational costs of LWM and REM simulations to attain the same accuracy target by using the efficiency metric J/S. In particular, we find optimal (J,S) pairs that ensure a certain accuracy at a minimal computational cost. We also extend our dispersion analysis to heterogeneous media and find the LWM accuracy to be significantly better for representative J values. Moreover, we perform 2D wave simulations on the SEG/EAGE Salt Model, in which larger REM inaccuracies are clearly observed on waveform comparisons in the range J≤3.