[ohsonice]

I don't think it makes sense to the same degree. Simple compositions of elementary functions, e.g. exp(x^2), do not have indefinite integrals.

Autodiff provides benefit since differentiation can be expensive numerically while easier symbolically. Integration is, in some ways, the opposite. Integrals with no symbolic representation can be estimated with quadrature methods.

[JadeNB]

> I don't think it makes sense to the same degree. Simple compositions of elementary functions, e.g. exp(x^2), do not have indefinite integrals.

While it's clear what you meant, it's mathematically important to note that functions like yours, and, indeed, all continuous functions, definitely do have indefinite integrals—that of `exp(x^2)` even has a name (https://mathworld.wolfram.com/Erfi.html); those integrals just aren't elementary, in the same technical sense in which you are using the word (https://en.wikipedia.org/wiki/Elementary_function).

[ohsonice]

Your first link is not working for me on mobile. Thanks for the clarification though.

Exp(x^2) is a finite composition of a polynomial and an exponential function so from what you posted, it does fall into the Wikipedia definition of elementary function.

[brewmarche]

The first link of your parent should've been https://mathworld.wolfram.com/Erfi.html

They were arguing that the antiderivative F of f: x ↦ exp(x²) is not elementary, whereas f itself is elementary. But both F and f are well-defined functions, which means that f has an indefinite integral, it just happens to be not elementary.

[JadeNB]

> The first link of your parent should've been https://mathworld.wolfram.com/Erfi.html

Sometimes Markdown eats following parentheses and sometimes it doesn't, and I've never figured out why. When I first viewed the post it showed me the parenthesis not being eaten, but then it did it anyway when I viewed it from the main page (too late for me to edit). Thanks for fixing it up!