  In this article we present an improved exact inversion formula for the 3D
cone beam transform of vector fields. It is well known that only the solenoidal
part of a vector field can be determined by the longitudinal ray transform of a
vector field in cone beam geometry. The exact inversion formula, as it was
developed in A. Katsevich and T. Schuster, 'An exact inversion formula for cone
beam vector tomography', Inverse Problems 29 (2013), consists of two parts. the
first part is of filtered backprojection type, whereas the second part is a
costly 4D integration and very inefficient. In this article we tackle this
second term and achieve an improvement which is easily to implement and saves
one order of integration. The theory says that the first part contains all
information about the curl of the field, whereas the second part presumably has
information about the boundary values. This suggestion is supported by the fact
that the second part vanishes if the exact field is divergence free and
tangential at the boundary. A number of numerical tests, that are also subject
of this article, confirm the theoretical results and the exactness of the
formula.
