Recent measurements of the proton-proton total cross section $sigma_{tot}$ at 7 and 8 TeV by the TOTEM and ATLAS Collaborations are characterized by some discrepant values: the TOTEM data suggest a rise of the cross section with the energy faster than the ATLAS data. Attempting to quantify these different behaviors, we develop new analytical fits to $sigma_{tot}$ and $rho$ data from $pp$ and $bar{p}p$ scattering in the energy region 5 GeV - 8 TeV. The dataset comprises all the accelerator data below 7 TeV and we consider three ensembles by adding: either only the TOTEM data (T), or only the ATLAS data (A), or both sets (T+A). For the purposes, we use our previous RRPL$gamma$ parametrization for $sigma_{tot}(s)$, consisting of two Reggeons (RR), one critical Pomeron (P) and a leading log-raised-to-gamma (L$gamma$) contribution (with $gamma$ as a free fit parameter), analytically connected to $rho(s)$ through singly-subtracted derivative dispersion relations and energy scale fixed at the physical threshold. The data reductions with ensembles T and A present good agreement with the experimental data analyzed and cannot be distinguished on statistical grounds. The quality of the fit is not as good with ensemble T+A. The fit results provide $gamma sim 2.3 pm 0.1$ (T), $2.0 pm 0.2$ (A), $2.1 pm 0.2$ (T+A), with $chi^2/mathrm{DOF} sim 1.07$ (T), $1.09$ (A), $1.14$ (T+A), suggesting extrema bounds for $gamma$ given by 1.8 and 2.4. Fits with $gamma = 2$ (fixed) are also developed and discussed.
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