Line-intensity mapping (LIM) provides a promising way to probe cosmology, reionization and galaxy evolution. However, its sensitivity to cosmology and astrophysics at the same time is also a nuisance. Here we develop a comprehensive framework for modelling the LIM power spectrum, which includes redshift space distortions and the Alcock-Paczynski effect. We then identify and isolate degeneracies with astrophysics so that they can be marginalized over. We study the gains of using the multipole expansion of the anisotropic power spectrum, providing an accurate analytic expression for their covariance, and find a 10%-60% increase in the precision of the baryon acoustic oscillation scale measurements when including the hexadecapole in the analysis. We discuss different observational strategies when targeting other cosmological parameters, such as the sum of neutrino masses or primordial non-Gaussianity, finding that fewer and wider bins are typically more optimal. Overall, our formalism facilitates an optimal extraction of cosmological constraints robust to astrophysics.
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