  Consider an algebraic equation $P(x,y)=0$ where $P\in \mathbb C[x,y] $ (or
$\mathbb F[x,y]$ with $\mathbb F\subset \mathbb C$ a subfield) is a bivariate
polynomial, it defines a plane algebraic curve. We provide an efficient method
for computing integrals of the type $ \int_\gamma R(x,y)dx $ where $R(x,y)\in
\mathbb C(x,y) $ is any rational fraction, and $y$ is solution of $P(x,y)=0$,
and $\gamma$ any Jordan arc open or closed on the plane algebraic curve. The
method uses only algebraic and combinatorial manipulations, it rests on the
combinatorics of the Newton's polygon. We illustrate it with many practical
examples.
