  A $k$-antipower (for $k \ge 2$) is a concatenation of $k$ pairwise distinct
words of the same length. The study of fragments of a word being antipowers was
initiated by Fici et al. (ICALP 2016) and first algorithms for computing such
fragments were presented by Badkobeh et al. (Inf. Process. Lett., 2018). We
address two open problems posed by Badkobeh et al. We propose efficient
algorithms for counting and reporting fragments of a word which are
$k$-antipowers. They work in $\mathcal{O}(nk \log k)$ time and $\mathcal{O}(nk
\log k + C)$ time, respectively, where $C$ is the number of reported fragments.
For $k=o(\sqrt{n/\log n})$, this improves the time complexity of
$\mathcal{O}(n^2/k)$ of the solution by Badkobeh et al. We also show that the
number of different $k$-antipower factors of a word of length $n$ can be
computed in $\mathcal{O}(nk^4 \log k \log n)$ time. Our main algorithmic tools
are runs and gapped repeats. Finally we present an improved data structure that
checks, for a given fragment of a word and an integer $k$, if the fragment is a
$k$-antipower. This is a full and extended version of a paper from LATA 2019.
In particular, all results about counting different antipowers factors are
completely new compared with the LATA proceedings version.
