We present the results of a stacking analysis performed on Spitzer/Infrared Spectrograph high-resolution mid-infrared (mid-IR) spectra of luminous infrared galaxies (LIRGs) in the Great Observatories All-Sky LIRG Survey. By binning in relation to mid-IR active galactic nucleus (AGN) fraction and stacking spectra, we detect bright emission lines [NeII] and [NeIII], which trace star formation, and fainter emission lines [NeV] and [OIV], which trace AGN activity, throughout the sample. We find that the [NeII] luminosity is fairly constant across all AGN fraction bins, while the [OIV] and [NeV] luminosities increase by over an order of magnitude. Our measured average line ratios, [NeV]/[NeII] and [OIV]/[NeII], at low AGN fraction are similar to HII galaxies, while the line ratios at high AGN fraction are similar to LINERs and Seyferts. We decompose the [OIV] luminosity into star formation and AGN components by fitting the [OIV] luminosity as a function of the [NeII] luminosity and the mid-IR AGN fraction. The [OIV] luminosity in LIRGs is dominated by star formation for mid-IR AGN fractions <~0.3. With the corrected [OIV] luminosity, we calculate black hole accretion rates (BHARs) ranging from 10^-5^M_{sun}_/yr at low AGN fractions to 0.2M_{sun}_/yr at the highest AGN fractions. We find that using the [OIV] luminosity, without correcting for star formation, can lead to overestimation of the BHAR by up to a factor of 30 in starburst-dominated LIRGs. Finally, we show that the BHAR/star formation rate ratio increases by more than three orders of magnitude as a function of mid-IR AGN fraction in LIRGs.