By cross-correlating AKARI infrared (IR) sources with the SDSS galaxies, we identified 2357 infrared galaxies with a spectroscopic redshift. This is not just one of the largest samples of local IR galaxies, but AKARI provides crucial FIR bands (9, 18, 65, 90, 140, and 160um) in accurately measuring galaxy SED across the peak of the dust emission at ~100um. By fitting modern IR SED models to the AKARI photometry, we measured the total infrared luminosity (L_IR) of individual galaxies more accurately. Using this L_IR, we constructed luminosity functions of infrared galaxies at a median redshift of z=0.031, with 4 times larger sample than previous work. The LF agrees well with that at z=0.0082 (RBGS), showing smooth and continuous evolution toward higher redshift LFs measured in the AKARI NEP deep field. The derived local cosmic IR luminosity density is Omega_IR=3.8x10^8 LsunMpc^-3. We separate galaxies into AGN, star-forming, and composite by using the [NII]/Ha vs [OIII]/Hb line ratios. The fraction of AGN shows a continuous increase with increasing L_IR from 25% to 90% at 9<log L_IR<12.5. The SFR_Ha and L_[OIII] show good correlations with L_IR for SFG (star-forming galaxies) and AGN, respectively. The self-absorption corrected Ha/Hb ratio shows a weak increase with L_IR with a substantial scatter. When we separate IR LFs into contributions from AGN and SFG, the AGN contribution becomes dominant at L_IR>10^11Lsun, coinciding the break of the both SFG and AGN IR LFs. At L_IR<10^11Lsun, SFG dominates IR Lfs. Only 1.1% of Omega_IR is produced by LIRG, and only 0.03% is by ULIRG in the local Universe. This work also provides the most accurate infrared luminosity density of the local Universe to date. Compared with high redshift results from the AKARI NEP deep survey, we observed a strong evolution of Omega_IR^SFG ~(1+z)^4.1+-0.4 and Omega_IR^AGN ~(1+z)^4.1+-0.5 (abridged).
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