  The Ginzburg-Landau (GL) effective theory is a useful tool to study a
superconductivity or superfluidity near the critical temperature, and usually
the expansion up to the 4th order in terms of order parameters is sufficient
for the description of the second-order phase transition. In this paper, we
discuss the GL equation for the neutron $^{3}P_{2}$ superfluidity relevant for
interior of neutron stars. We derive the GL expansion up to the 8th order in
the condensates and find that this order is necessary for the system to have
the unique ground state, unlike the ordinary cases. Starting from the $LS$
potential, which provides the dominant attraction between two neutrons at the
high density, we derive the GL equation in the path-integral formalism, where
the auxiliary field method and the Nambu-Gor'kov representation are used. We
present the detailed description for the trace calculation necessary in the
derivation of the GL equation. As numerical results, we show the phase diagram
of the neutron $^{3}P_{2}$ superfluidity on the plane spanned by the
temperature and magnetic field, and find that the 8th order terms lead to a
first-order phase transition, whose existence was predicted in the
Bogoliubov-de Gennes equation but has not been found thus far within the
framework of the GL expansion up to the 6th order.The first-order phase
transition will affect the interior structures inside the neutron stars.
