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.
Categories: math.AG