Nucleonic models at finite temperature with in-medium effective fields

We perform a calculation of dense and hot nuclear matter where the mean interaction between nucleons is described by in-medium effective fields and where we employ analytical approximations of the Fermi integrals. We generalize a previous work [Dutra et al., Astrophys. J. 952, 5 (2023)] where we have addressed the case of the Fermi gas model with in-medium effective mass. In the present work we fully treat the in-medium interaction by considering both its contribution to the in-medium effective fields, which can be subsumed by the mass in some cases, and to the potential term. Our formalism is general and could be applied to relativistic and nonrelativistic approaches. It is illustrated for different popular models---Skyrme, nonlinear, and density-dependent relativistic mean-field models---but also for the metamodel, and it provides a clear understanding of the in-medium correction to the pressure, which is present in the case of the Skyrme and metamodel but is not for the relativistic ones. For the Fermi integrals, we compare the analytical approximation to the so-called ``exact'' numerical calculations in order to quantitatively estimate the accuracy of the approximation.