  We provably compute the full set of rational points on 1403 Picard curves
defined over $\mathbb{Q}$ with Jacobians of Mordell-Weil rank $1$ using the
Chabauty-Coleman method. To carry out this computation, we extend Magma code of
Balakrishnan and Tuitman for Coleman integration. The new code computes
$p$-adic (Coleman) integrals on curves to points defined over number fields
where the prime $p$ splits completely and implements effective Chabauty for
curves whose Jacobians have infinite order points that are not the image of a
rational point under the Abel-Jacobi map. We discuss several interesting
examples of curves where the Chabauty-Coleman set contains points defined over
number fields.
