Elementary Integration of Superelliptic Integrals

Consider a superelliptic integral $I=\int P\mathrm{/}(Q{S}^{1\mathrm{/}k})dx$ 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.