Consider a superelliptic integral $I=\int P/(Q S^1/k ) dx$ with $\mathbbK =\mathbbQ (ξ)$, ξ a primitive kth root of unity, $P,Q,S\in\mathbbK [x]$ 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 $O((kd)^ømega+2g+1 h^g+1 )$ operations.