  I present a highly efficient integration method for scattering calculations,
which can reduce the evaluation time of particularly challenging analyses by a
factor of $10^7$. By projecting each item in the integrand onto vector spaces
spanned by simple basis functions, a multidimensional numerical integral is
replaced by a much easier exercise in matrix multiplication. This method can be
generalized to any integrand that depends linearly on multiple input functions.
In this paper I apply the method to dark matter (DM) direct detection with
anisotropic target materials, where the DM scattering rate depends not only on
the DM particle model, the astrophysical DM velocity distribution, and the
properties of the target material, but now also on the $SO(3)$ orientation of
the detector. The vector space calculation factorizes the astrophysics, the DM
particle model, and the SM material properties from each other, while
describing detector rotations by simple matrix multiplication acting on the
basis functions. Difficult analyses, which otherwise would take decades or
centuries on a computing cluster, can now be performed in a few hours on a
laptop.
