Timing Six Energetic Rotation-powered X-Ray Pulsars, Including the Fast-spinning Young PSR J0058-7218 and Big Glitcher PSR J0537-6910

Measuring a pulsar's rotational evolution is crucial to understanding the nature of the pulsar. Here, we provide updated timing models for the rotational evolution of six pulsars, five of which are rotation phase-connected using primarily NICER X-ray data. For the newly discovered fast energetic young pulsar, PSR J0058-7218, we increase the baseline of its timing model from 1.4 days to 8 months and not only measure more precisely its spin-down rate (nu) over dot=(-6.2324 +/- 0.0001)x10(-11)Hzs(-1) (nu) over dot=(4.2 +/- 0.2)x10(-21)Hzs(-2). For the fastest and most energetic young pulsar, PSR J0537-6910 (with a 16 ms spin period), we detect four more glitches, for a total of 15 glitches over 4.5 yr of NICER monitoring, and show that its spin-down behavior continues to set this pulsar apart from all others, including a long-term braking index n = -1.234 +/- 0.009 and interglitch braking indices that asymptote to less than or similar to 7 for long times after a glitch. For PSR J1101-6101, we measure a much more accurate spin-down rate that agrees with a previous value measured without phase connection. For PSR J1412+7922 (also known as Calvera), we extend the baseline of its timing model from our previous 1 yr model to 4.4 yr, and for PSR J1849-0001, we extend the baseline from 1.5 to 4.7 yr. We also present a long-term timing model of the energetic pulsar PSR J1813-1749, by fitting previous radio and X-ray spin frequencies from 2009-2019 and new ones measured here using 2018 NuSTAR and 2021 Chandra data.