  In this work I develop a new framework for anisotropic hydrodynamics that
generalizes the leading order of the hydrodynamic expansion to the full
(3+1)-dimensional anisotropic massive case. Following previous works, my
considerations are based on the Boltzmann kinetic equation with the collisional
term treated in the relaxation time approximation. The momentum anisotropy is
included explicitly in the leading term, allowing for a large difference
between the longitudinal and transverse pressures as well as for non trivial
transverse dynamics. Energy and momentum conservation is expressed by the first
moment of the Boltzmann equation. The system of equations is closed by using
the zeroth and second moments of the Boltzmann equation. The
close-to-equilibrium matching with second-order viscous hydrodynamics is
demonstrated. In particular, I show that the coupling between shear and bulk
pressure corrections, recently proved to be important for an accurate
description of momentum anisotropy and bulk viscous dynamics, does not vanish
in the close-to-equilibrium limit.
