  The non-Gaussian part of the covariance matrix of the galaxy power spectrum
involves the connected four-point correlation in Fourier space, i.e.
trispectrum. This paper introduces a fast method to compute the non-Gaussian
part of the covariance matrix of the galaxy power spectrum multipoles in
redshift space at tree-level standard perturbation theory. For the tree-level
galaxy trispectrum, the angular integral between two wavevectors can be
evaluated analytically by employing an FFTLog. The new implementation computes
the non-Gaussian covariance of the power spectrum monopole, quadrupole,
hexadecapole and their cross-covariance in $O(10)$ seconds, for an effectively
arbitrary number of instances of cosmological and galaxy bias parameters and
redshift, without any parallelization or acceleration. It is a large advantage
over conventional numerical integration. We demonstrate that the computation of
the covariance at $k = 0.005 - 0.4\,h\,\mathrm{Mpc}^{-1}$ gives results with
$0.1 - 1\%$ accuracy. The efficient computation of the analytic covariance can
be useful for future galaxy surveys, especially utilizing multi-tracer
analysis.
