Inelastic x-ray scattering (IXS) with very high (meV) energy resolution has become a valuable spectroscopic tool, complementing the well established coherent inelastic neutron scattering (INS) technique for phonon dispersion investigations. In the study of crystalline systems IXS is a viable alternative to INS, especially in cases where only small samples are available. Using IXS, we have measured the phonon dispersion of Nd1.86Ce0.14CuO4+δ along the [ξ, 0, 0] and [ξ, ξ, 0] inplane directions. Compared to the undoped parent compound, the two highest longitudinal optical (LO) phonon branches are shifted to lower energies because of Coulomb-screening effects brought about by the doped charge carriers. An additional anomalous softening of the highest branch is observed around q = (0-2, 0, 0). This anomalous softening, akin to what has been observed in other compounds, provides evidence for a strong electron-phonon coupling in the electron-doped high-temperature superconductors.