  Calculation of hadronization, decay or scattering processes at non-zero
temperatures and densities within the Nambu-Jona-Lasinio-like models requires
some techniques for computation of Feynmann diagrams. Decomposition of Feynman
diagrams at the one loop level leads to the appearance of elementary integrals
with one, two, three, and four fermion lines. For example, evaluation of the
$\pi\pi$ scattering amplitude requires calculating a box diagram with four
fermion lines. In this work, the real and imaginary parts of the box integral
at the one loop level are provided in the form suitable for numerical
evaluation. The obtained expressions are applicable to any value of
temperature, particle mass, and chemical potential. We pay special attention to
the conditions for the existence of the appearing improper integrals and
correct the results \cite{Klevansky} for the three fermion lines. As a result,
we have obtained constraints on possible values of particle momenta. Among the
expressions for the box integral, the general formulas for the integral with an
arbitrary number of lines are derived for the case with zero or collinear
fermion momenta.
