We present theoretical calculations of higher-order QCD and electroweak corrections for the associated production of a top-antitop quark pair and a $W$ boson ($t{\bar t}W$ production) at LHC energies. We show predictions for cross sections at approximate N$^3$LO (aN$^3$LO) which include second-order and third-order soft-gluon corrections added to the exact NLO QCD+electroweak result. We compare our results to recent measurements from the LHC, and we find that our predictions provide improved agreement with the data. We also calculate the top-quark transverse momentum and rapidity distributions in $t{\bar t}W$ production. We find significant enhancements from the higher-order corrections to the total and differential cross sections for this process.