  In this work, we study active particles with prescribed surface velocities in
non-Newtonian fluids. We employ the reciprocal theorem to obtain the velocity
of an active spherical particle with an arbitrary axisymmetric slip-velocity in
an otherwise quiescent second-order fluid. We then determine how the motion of
a diffusiophoretic Janus particle is affected by complex fluid rheology, namely
viscoelasticity and shear-thinning viscosity, compared to a Newtonian fluid,
assuming a fixed slip-velocity. We find that a Janus particle may go faster or
slower in a viscoelastic fluid, but is always slower in a shear-thinning fluid
as compared to a Newtonian fluid.
