  Building upon recent work by the author, we prove that multivariate
integration in the following subspace of the Wiener algebra over $[0,1)^d$ is
strongly polynomially tractable: \[ F_d:=\left\{ f\in C([0,1)^d)\:\middle| \:
\|f\|:=\sum_{\boldsymbol{k}\in
\mathbb{Z}^{d}}|\hat{f}(\boldsymbol{k})|\max\left(\mathrm{width}(\mathrm{supp}(\boldsymbol{k})),\min_{j\in
\mathrm{supp}(\boldsymbol{k})}\log |k_j|\right)<\infty \right\},\] with
$\hat{f}(\boldsymbol{k})$ being the $\boldsymbol{k}$-th Fourier coefficient of
$f$, $\mathrm{supp}(\boldsymbol{k}):=\{j\in \{1,\ldots,d\}\mid k_j\neq 0\}$,
and $\mathrm{width}: 2^{\{1,\ldots,d\}}\to \{1,\ldots,d\}$ being defined by \[
\mathrm{width}(u):=\max_{j\in u}j-\min_{j\in u}j+1,\] for non-empty subset
$u\subseteq \{1,\ldots,d\}$ and $\mathrm{width}(\emptyset):=1$. Strong
polynomial tractability is achieved by an explicit quasi-Monte Carlo rule using
a multiset union of Korobov's $p$-sets. We also show that, if we replace
$\mathrm{width}(\mathrm{supp}(\boldsymbol{k}))$ with 1 for all
$\boldsymbol{k}\in \mathbb{Z}^d$ in the above definition of norm, multivariate
integration is polynomially tractable but not strongly polynomially tractable.
