  Consider a superelliptic integral $I=\int P/(Q S^{1/k}) dx$ with
$\mathbb{K}=\mathbb{Q}(\xi)$, $\xi$ a primitive $k$th root of unity,
$P,Q,S\in\mathbb{K}[x]$ and $S$ has simple roots and degree coprime with $k$.
Note $d$ the maximum of the degree of $P,Q,S$, $h$ the logarithmic height of
the coefficients and $g$ the genus of $y^k-S(x)$. We present an algorithm which
solves the elementary integration problem of $I$ generically in
$O((kd)^{\omega+2g+1} h^{g+1})$ operations.
