Eq. (\[fEq\]) $$\begin{aligned}
 \dot{R} &= - \frac{f'}{f''} H + \frac{1}{6H} \frac{f' R - f}{f''} + \frac{1}{f''H} \frac{\rho}{3 \Mp^2} \, , \label{dotR} \\
\ddot{R} &= - 3H \dot{R} - \frac{f'''}{f''} \dot{R}^2 + \frac{ R f' - 2f}{3 f''} + \frac{1}{f''} \frac{\rho - 3P}{3 \Mp^2} \, , \label{ddotR} \end{aligned}$$ where $\rho$ and $P$ are the energy density and the pressure of some species, respectively. If one adopts the specific form of $f(R)$ given by Eq.(\[SR2lnR\]), then one obtains $$\begin{aligned}
 & \dot{R} + \left(\frac{M^2 }{2\alpha+3\beta+2\beta \ln x} \right) H + \left(\frac{2\alpha+\beta+2\beta \ln x }{2\alpha+3\beta+2\beta \ln x} \right) H R \nonumber \\ 
&- \left(\frac{ \alpha+\beta+\beta \ln x }{2\alpha+3\beta+2\beta \ln x} \right) \frac{R^2}{6H} = \left(\frac{M^2}{2\alpha+3\beta+2\beta \ln x} \right)  \frac{\rho}{3 \bMp^2 H} \, , \label{dotR2} \\
& \ddot{R} + 3H \dot{R} + \left(\frac{2\beta}{2\alpha+3\beta+2\beta \ln x} \right)\frac{\dot{R}^2}{R} + \frac{M^2}{3} \left(\frac{1-\beta x}{2\alpha+3\beta+2\beta \ln x} \right) R \nonumber \\ 
&= \left(\frac{M^2}{2\alpha+3\beta+2\beta \ln x} \right)\frac{\rho - 3P}{3 \bMp^2} \, . \label{ddotR2} \end{aligned}$$ One can use the relation between the Ricci scalar and the Hubble parameter, $R = 6 \left( \dot{H} + 2 H^2 \right)$ to investigate the evolution of $H$. Then, the equation (\[dotR\]) is rewritten as $$\begin{aligned}
 & \ddot{H} - \left( \frac{\alpha+\beta+\beta \ln x}{2\alpha+3\beta+2\beta \ln x} \right) \frac{\dot{H}^2}{H} + 