  We report the effect of the cosmological constant and the internal energy
density of a cosmic string on the deflection angle of light in the spacetime of
a rotating cosmic string with internal structure. We first revisit the
deflection angle by a rotating cosmic string and then provide a generalization
using the geodesic equations and the Gauss-Bonnet Theorem. We show there is an
agreement between the two methods when employing higher order terms of the
linear mass density of the cosmic string. By modifying the integration domain
for the global conical topology, we resolve the inconsistency between these two
methods previously reported in the literature. We show that the deflection
angle is not affected by the rotation of the cosmic string; however, the
cosmological constant $\Lambda$ strongly affects the deflection angle, which
generalizes the well-known result.
