I was wondering why $\int{e^{x^2}dx}$ was not integrable using elementary functions and looking for a proof. I found this video by Michael Penn which explains that these two statements are equivalent:
$f(x)e^{g(x)}$ cannot be integrated using elementary functions $\iff$ there exists a rational function $r(x)$ such that the equation $r'(x)+g'(x)r(x)=f(x)$ holds for all $x$.
Later in the video he uses this to prove by contradiction that for $f(x)=1$ and $g(x)=x^2$ there exists no $r(x)$ and so $e^{x^2}$ has no elementary antiderivative.
Why is this equivalence true? Is there a name for this theorem?