The path integral methodology for computation of the equilibrium isotope effect is described and applied to three [1,5] sigmatropic hydrogen shift reactions. An efficient estimator for the derivative of the free energy is used, which shortens the computation time by the factor of about 60. Calculations reveal that the correction beyond the usual harmonic approximation for nuclear motion amount up to 30 % of the symmetry reduced reaction free energy. The numerical results are compared with recent experiments of Doering and coworkers, confirming the accuracy of the most recent measurement as well as concerns about compromised accuracy, due to side reactions, of another measurement.