Over the last decade, dual-energy CT scanners have gone from prototypes to clinically available machines, and spectral photon counting CT scanners are following. They require a specific reconstruction process, consisting of two steps: material decomposition and tomographic reconstruction. Image-based methods perform reconstruction, then decomposition, while projection-based methods perform decomposition first, and then reconstruction. As an alternative, 'one-step inversion' methods have been proposed, which perform decomposition and reconstruction simultaneously. Unfortunately, one-step methods are typically slower than their two-step counterparts, and in most CT applications, reconstruction time is critical. This paper therefore proposes to compare the convergence speeds of five one-step algorithms. We adapted all these algorithms to solve the same problem: spectral photon-counting CT reconstruction from five energy bins, using a three materials decomposition basis and spatial regularization. The paper compares a Bayesian method which uses non-linear conjugate gradient for minimization (Cai et al 2013 Med. Phys. 40 111916-31), three methods based on quadratic surrogates (Long and Fessler 2014 IEEE Trans. Med. Imaging 33 1614-26, Weidinger et al 2016 Int. J. Biomed. Imaging 2016 1-15, Mechlem et al 2018 IEEE Trans. Med. Imaging 37 68-80), and a primal-dual method based on MOCCA, a modified Chambolle-Pock algorithm (Barber et al 2016 Phys. Med. Biol. 61 3784). Some of these methods have been accelerated by using μ-preconditioning, i.e. by performing all internal computations not with the actual materials the object is made of, but with carefully chosen linear combinations of those. In this paper, we also evaluated the impact of three different μ-preconditioners on convergence speed. Our experiments on simulated data revealed vast differences in the number of iterations required to reach a common image quality objective: Mechlem et al (2018 IEEE Trans. Med. Imaging 37 68-80) needed ten iterations, Cai et al (2013 Med. Phys. 40 111916-31), Long and Fessler (2014 IEEE Trans. Med. Imaging 33 1614-26) and Weidinger et al (2016 Int. J. Biomed. Imaging 2016 1-15) several hundreds, and Barber et al (2016 Phys. Med. Biol. 61 3784) several thousands. We also sum up other practical aspects, like memory footprint and the need to tune extra parameters.