I just stumbled upon a (german) article that features the following formula to compute integrals by differentiating:
$$\def\e{\varepsilon}\int_a^b\!\! f(x)dx\ =\ \lim_{\e\to0}f\left(\frac d{d\e}\right)\frac{e^{\e b}-e^{\e a}}{\e} \tag{*}$$
As the article states, the method was initially developed to calculate / approximate path integrals in Quantum Field Theory. The new (from 2014) approach may have it's merits in QFT, but the article also states that it can make computation of integrals over $\Bbb R$ more efficient is some cases.
It took me some seconds to decode the meaning of the right hand side, though. It means that $d/d \e$ is expanded by $f$ which has to be analytic, and then apply that operator to $g_{a,b}(\e) = (e^{\e b}-e^{\e a})/\e$ and finally let $\e\to 0$.
My question: If $f$ cannot be integrated easily or symbolically, why is the right hand side of $(*)$ easier to compute or to approximate than the left hand side? As $f$ is analytic, it's expansion has to be known in order to apply $f$ to $d/d\e$. Hence computing $n$ term of the RHS looks way more complicated than integrating $n$ terms of the power-series expansion of $f$ on the LHS. Does the RHS converge faster? That would basically mean that the RHS can "guess" expansion coefficients of $f$.
The artical also states that:
The computer algebra system Maple imlpemented the approach in 2019 and could achieve considerable speed-ups of some calculations.
So it appears useful in practice, even if the title "Revolution in Calculus" might be 90% click-bait.
It there some rule-of-thumb when that approach might be advantageous?