  Various challenges are faced when animalcules such as bacteria, protozoa,
algae, or sperms move autonomously in aqueous media at low Reynolds number.
These active agents are subject to strong stochastic fluctuations, that compete
with the directed motion. So far most studies consider the lowest order moments
of the displacements only, while more general spatio-temporal information on
the stochastic motion is provided in scattering experiments. Here we derive
analytically exact expressions for the directly measurable intermediate
scattering function for a mesoscopic model of a single, anisotropic active
Brownian particle in three dimensions. The mean-square displacement and the
non-Gaussian parameter of the stochastic process are obtained as derivatives of
the intermediate scattering function. These display different temporal regimes
dominated by effective diffusion and directed motion due to the interplay of
translational and rotational diffusion which is rationalized within the theory.
The most prominent feature of the intermediate scattering function is an
oscillatory behavior at intermediate wavenumbers reflecting the persistent
swimming motion, whereas at small length scales bare translational and at large
length scales an enhanced effective diffusion emerges. We anticipate that our
characterization of the motion of active agents will serve as a reference for
more realistic models and experimental observations.
