  We derive solutions to transit light curves of exoplanets orbiting
rapidly-rotating stars. These stars exhibit significant oblateness and gravity
darkening, a phenomenon where the poles of the star have a higher temperature
and luminosity than the equator. Light curves for exoplanets transiting these
stars can exhibit deviations from those of slowly-rotating stars, even
displaying significantly asymmetric transits depending on the system's
spin-orbit angle. As such, these phenomena can be used as a protractor to
measure the spin-orbit alignment of the system. In this paper, we introduce a
novel semi-analytic method for generating model light curves for
gravity-darkened and oblate stars with transiting exoplanets. We implement the
model within the code package starry and demonstrate several orders of
magnitude improvement in speed and precision over existing methods. We test the
model on a TESS light curve of WASP-33, whose host star displays rapid rotation
($v \sin i_* = 86.4$ km/s). We subtract the host's $\delta$-Scuti pulsations
from the light curve, finding an asymmetric transit characteristic of gravity
darkening. We find the projected spin orbit angle is consistent with Doppler
tomography and constrain the true spin-orbit angle of the system as
$\varphi=108.3^{+19.0}_{-15.4}$~$^{\circ}$. We demonstrate the method's uses in
constraining spin-orbit inclinations of such systems photometrically with
posterior inference. Lastly, we note the use of such a method for inferring the
dynamical history of thousands of such systems discovered by TESS.
