Measurement of deep inelastic scattering with leading protons or large rapidity gaps with ZEUS at HERA The dissociation of virtual photons, gamma*p -> Xp, was observed in events in which the leading proton was either measured directly in the leading proton spectrometer of the ZEUS detector or which had a large rapidity gap (LRG) between X and the outgoing proton. The data cover photon virtualities Q2 > 2 GeV2, photon-proton centre-of-mass energies 40 < W < 240 GeV and masses of the hadronic final state, X, M_X > 2GeV. The data are discussed in terms of the reduced diffractive cross sections, sigma_r^D(3) and sigma_r^D(4). The cross section is also presented as a function of t, the squared four-momentum transfer at the proton vertex and phi, the azimuthal angle between the positron and the proton scattering planes.