In the integrable case, the toric height can be expressed as an alternating sum of mixed integrals as follows. Let ${\overline{L}}_{i}=(L_{i},\|\cdot\|_{i})$, $i=0,\dots,n$, be toric line bundles on $X_{\Sigma}$ equipped with integrable toric metrics and set ${\overline{L}}_{i}={\overline{L}}_{i,+}\otimes({\overline{L}}_{i,-})^{-1}$ for some approachable metrized toric line bundles ${\overline{L}}_{i,+}$, ${\overline{L}}_{i,-}$. Choose a toric section for each line bundle and write $\vartheta_{i,+}$ and $\vartheta_{i,-}$ for the corresponding roof functions. Then