Integrating $\frac{dy}{dx} = a^{-x^2}$ [duplicate]

The classic book Calculus made easy includes the following statement:

No one, even today, is able to find the general integral of the expression,

$\frac{dy}{dx} = a^{-x^2}$,

because $a^{-x^2}$ has never yet been found to result from differentiating anything else.

Is anyone aware of a proof for this statement?

A much better phrasing of the idea that your textbook is trying to convey is that $a^{-x^2}$ is not the derivative of any sum, multiplication, or composition of rational functions over the complex numbers, radical functions over the real numbers, trigonometric functions over the real numbers, logarithmic functions over the real numbers, or inverses thereof. Also, it is not that no such function with $a^{-x^2}$ as its derivatives is know. Rather, it is more that it has been proven that no such function can have $a^{-x^2}$ as its derivatives.