We present experimental constraints on the spin-dependent WIMP-nucleon elastic cross sections from the total 129.5  kg yr exposure acquired by the Large Underground Xenon experiment (LUX), operating at the Sanford Underground Research Facility in Lead, South Dakota (USA). A profile likelihood ratio analysis allows 90% C.L. upper limits to be set on the WIMP-neutron (WIMP-proton) cross section of σn=1.6×10−41  cm2 (σp=5×10−40  cm2) at 35  GeV c−2, almost a sixfold improvement over the previous LUX spin-dependent results. The spin-dependent WIMP-neutron limit is the most sensitive constraint to date.