Accreting millisecond X-ray pulsars are known to provide a wealth of physical information during their successive states of outburst and quiescence. Based on the observed spin-up and spin-down rates of these objects it is possible, among other things, to infer the stellar magnetic field strength and test models of accretion disc flow. In this paper we consider the three accreting X-ray pulsars (XTE J1751-305, IGR J00291+5934, and SAX J1808.4-3658) with the best available timing data, and model their observed spin-up rates with the help of a collection of standard torque models that describe a magnetically-threaded accretion disc truncated at the magnetospheric radius. Whilst none of these models are able to explain the observational data, we find that the inclusion of the physically motivated phenomenological parameter $xi$, which controls the uncertainty in the location of the magnetospheric radius, leads to an enhanced disc-integrated accretion torque. These new torque models are compatible with the observed spin-up rates as well as the inferred magnetic fields of these objects provided that $xi approx 0.1-0.5$. Our results are supplemented with a discussion of the relevance of additional physics effects that include the presence of a multipolar magnetic field and general-relativistic gravity.
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