Suppose that we have an algebraic curve $C$ in the affine plane over a field $k$ defined by $f(x,y) = 0$, and write $\overline C$ for its projective completion defined by the homogeneous equation $F(x,y,z) = 0$ (I'll usually work with $C$ for simplicity, but am interested in both cases).
By differentiating the defining equation $f(x,y) = 0$, we get $$\frac {\partial f} {\partial x} \text dx + \frac {\partial f} {\partial y} \text dy = 0 ,$$ so we can rewrite $\text dx$ and $\text dy$ in terms of each other. Thus we can write every $1$-form as $g(x,y) \rho$ for some fixed $1$-form $\rho$, and for suitable $\rho$ we can choose $g(x,y)$ to be a polynomial in $x$ and $y$.
Suppose that $\omega = g(x,y) \rho$ is exact, for some explicit $g(x,y)$ and $\rho$. How can we find a primitive for $\omega$? That is: I am looking for an algorithm to give a function $h(x,y)$ such that $\text{d}h = \omega$.
(Note: I'm aware of several answers explaining how to do this in $\mathbb R^n$ or similar, but I have a situation where my $1$-form is exact on my curve but not closed on $\mathbb R$ and haven't been able to adapt them.)
Thanks!