TITLE: Prove an integral is not solvable in elementary functions
QUESTION [5 upvotes]: I need to prove that this integral
$$A = \int \frac{e^{x^2}}{x} dx $$ is not solvable in elementary functions
What do I have as theory in my hands?

I know 5 or 6 integrals for which it has been proven they are not solvable. One of them is this one

$$B = \int \frac{e^{x}}{x} dx $$

I also know that if you transform an integral $A_1$ to $B_1$ with a substitution which involves just elementary functions, and if $B_1$ is not solvable, then $A_1$ is also not solvable.

These two results are stated in my real analysis book without proof. So I can just use them, this is my impression. I don't want to prove these two.
So to prove that $A$ is not solvable I do the substitution $x=\sqrt{t}$. This way I arrive at an integral which is $$\frac{1}{2} \cdot B$$
And since I know $B$ is not solvable, I conclude $A$ is also not solvable.
Is this a valid proof? I am not quite convinced if the substitution I did $x = \sqrt{t}$ is elementary.

REPLY [3 votes]: Here is a result that follows from Liouville's theorem (see https://math.stackexchange.com/a/3772327/589)

Let $f, g$ be rational functions, $g$ not constant.  The indefinite integral
$$
\int f(x)e^{g(x)}\;dx
$$
is elementary if and only if there is a rational function $h$ such that
$f=h'+hg'\;$.

Take $f(x) = 1/x$, $g(x) = x^2$.
Is there a rational function $h$ such that $f=h'+hg'$, that is, $
1 = xh'+2x^2 h$?
Write $h=p/q$ for coprime polynomials $p$ and $q$. Then
$$
q^2=x(p'q-pq')+2x^2pq
$$
and so $q$ divides $pq'$. Since $p$ and $q$ are coprime, $q$ divides $q'$ and so $q$ is constant. Therefore, $h$ is a polynomial. But then $1 = xh'+2x^2 h=x(h'+2xh)$ cannot happen for degree reasons.