The reheating dynamics after the inflation induced by Starobinsky model have been investigated both in Einstein frame and in Jordan frame [@09091737; @12041472; @12124466; @160604346].

The layout of this manuscript proceeds as follows. In the next section, we briefly review the equation of motions of the $R + \alpha R^2 + \beta R^2 \ln R$ inflation model both in Jordan frame and in Einstein frame. One needs to be careful when $\beta$ is negative. The approximate analysis of the inflationary behaviors was investigated in the reference [@14047349]. We scrutinize the exact inflation in this model for the different values of $\alpha$ and $\beta$ to be consistent with the Planck results. We investigate the oscillatory epoch and the reheating period of this model in section 3. Finally, we conclude in section 4.

$f(R)$ Inflation
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$f(R)$ inflation models are conventionally analyzed in the Einstein frame after the conformal transformation of the original action in the Jordan frame. In this section, we briefly review this process and provide the analytic relations formulae between the general form of $f(R)$ and the cosmological parameters obtained from the observations. We also investigate the specific form of $f(R)$ model obtained from the quantum correction of the 