Computationally predicting reverse intersystem crossing (RISC) rates is important for designing new thermally activated delayed fluorescence (TADF) materials. We report a method that can quantitatively predict RISC rates by explicitly considering the spin-vibronic coupling mechanism. The coupling element of the spin-vibronic Hamiltonian is obtained by expanding the spin-orbit and the non-Born-Oppenheimer terms to second order and is then brought into the Golden Rule rate under the Condon approximation. The rate equation is solved directly in the time domain using a correlation function approach. The contributions of the first-order direct spin-orbit coupling and the second-order spin-vibronic coupling to an RISC rate can be quantitatively analyzed in a separate manner. We demonstrate the utility of the method by applying it to a representative TADF system, where we observe that the spin-vibronic portion is substantial but not dominant especially with a relatively small triplet-singlet energy gap. Likewise, our method may elucidate the physical background of efficient nonradiative transitions from the lowest triplet to a higher lying singlet in other purely organic TADF systems, and it will be of great utility toward designing new such molecules.