Is there a simple proof that functions of the form $f(x)^{g(x)}$ in general don't have integrals in standard functions?

Question

My attempt: Proof by contradiction: Restriction: for any function $h(x)$ inside $g(x)$, and for any function $j(x)$ inside $f(x)$, $j(x)$ or $f(x)$ may not be inverse functions of $h(x)$ or $g(x)$. $$f(x)^{g(x)}=e^{g(x)\ln(f(x))}\rightarrow\int e^{g(x)\ln(f(x))}dx=$$I then realized it was impossible to use the same method as is used to find the derivative, to prove that it's impossible to find the integral. I now tried to turn $f(x)^{g(x)}$ into a Taylor series, to somehow prove that it wasn't integrable. However as the derivative is:$$f(x)^{g(x)}\left(f'(x)\frac{g(x)}{f(x)}+g'(x)\ln(f(x))\right)$$ I quickly realized that this method, too, is unviable. Please Help!