  Typically, in the description of active Brownian particles, a constant
effective propulsion force is assumed, which is then subjected to fluctuations
in orientation and translation leading to a persistent random walk with an
enlarged long-time diffusion coefficient. Here, we generalize previous results
for the swimming path statistics to a time-dependent and thus in many
situations more realistic propulsion which is a prescribed input. We
analytically calculate both the noise-free and the noise-averaged trajectories
for time-periodic propulsion under the action of an additional torque. In the
deterministic case, such an oscillatory microswimmer moves on closed paths that
can be highly more complicated than the commonly observed straight lines and
circles. When exposed to random fluctuations, the mean trajectories turn out to
be self-similar curves which bear the characteristics of their noise-free
counterparts. Furthermore, we consider a propulsion force which scales in time
$t$ as $\propto \! t^\alpha$ (with $\alpha=0,1,2,\ldots$) and analyze the
resulting superdiffusive behaviour. Our predictions are verifiable for
diffusiophoretic artificial microswimmers with prescribed propulsion protocols.
