To evaluate higher-order derivatives may be both time consuming and delicate from a numerical point of view, since there is always the risk of loosing precision when calculating derivatives numerically. Unless we have an analytic expression for \( f(u+x) \) and can evaluate the derivatives in a closed form, the above approach is not the preferred one.

Rather, we show here how to use the Gauss-Legendre method to compute Eq. (19). Let us first introduce a new variable \( s=u/\Delta \) and rewrite Eq. (19) as $$ \begin{equation} I_{\Delta}(x)={\cal P}\int_{-1}^{+1}ds\frac{f(\Delta s+x)}{s}. \tag{20} \end{equation} $$