We present AKARI 2.5-5um spectra of 145 local luminous infrared galaxies in the Great Observatories All-sky LIRG Survey. In all of the spectra, we measure the line fluxes and EQWs of the polycyclic aromatic hydrocarbon (PAH) at 3.3um and the hydrogen recombination line Br-alpha, with apertures matched to the slit sizes of the Spitzer spectrograph and with an aperture covering ~95% of the total flux in the AKARI 2D spectra. The star formation rates (SFRs) derived from Br-alpha measured in the latter aperture agree well with SFRs(LIR), when the dust extinction correction is adopted based on the 9.7um absorption feature. Together with the Spitzer spectra, we are able to compare the 3.3 and 6.2um PAH features, the two most commonly used near/mid-IR indicators of starburst (SB) or active galactic nucleus (AGN) dominated galaxies. We find that the 3.3 and 6.2um PAH EQWs do not follow a linear correlation and at least 1/3 of galaxies classified as AGN-dominated using 3.3um PAH are classified as starbursts based on 6.2um PAH. These galaxies have a bluer continuum slope than galaxies that are indicated to be SB-dominated by both PAH features. The bluer continuum emission suggests that their continuum is dominated by stellar emission rather than hot dust. We also find that the median Spitzer spectra of these sources are remarkably similar to the pure SB-dominated sources indicated by high PAH EQWs in both 3.3 and 6.2um. We propose a revised SB/AGN diagnostic diagram using 2-5um data. We also use the AKARI and Spitzer spectra to examine the performance of our new diagnostics and to estimate 3.3um PAH fluxes using the JWST photometric bands in 0<z<5. Of the known PAH features and mid-IR high ionization emission lines used as SB/AGN indicators, only the 3.3um PAH feature is observable with JWST at z>3.5, because the rest of the features at longer wavelengths fall outside the JWST wavelength coverage.
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