Inexact Newton based Lifted Implicit Integrators for fast Nonlinear MPC**This research was supported by Research Council KUL: PFV/10/002 Optimization in Engineering Center OPTEC; Eurostars SMART; Belgian Federal Science Policy Office: IUAP P7 (DYSCO, Dynamical systems, control and optimization, 2012-2017); EU: FP7-TEMPO (MCITN-607957), ERC HIGHWIND (259166), H2020-ITN AWESCO (642682). R. Quirynen holds a PhD fellowship of the Research Foundation - Flanders (FWO).