To express the antiderivatives of $\frac{1}{x}$, we cannot apply the formula $\int x^n dx=\frac{x^{n+1}}{n+1}+C$ and we need to introduce a new function, the logarithm.
But how can we prove that there is not a complicated way to express $\int\frac{dx}{x}$ using classical functions like polynomials, nth roots, trigonometric functions and their inverse, (exponential not allowed) and any finite combination of sums, products or compositions of functions?
If we exclude trig and inverse trig functions of our list, the question comes down to prove that $\ln$ is not algebraic, i.e there is no non zero polynomial $P$ such that $P(\ln(x))=0$ for any $x>0$. This is quite simple (if $P$ satisfies the conditions, so does $P'$; so we obtain a contradiction by assuming $P$ of minimal degree).
But I cannot see any proof when trig and inverse trig functions join the party. Any idea is welcomed. For example, $\frac{1}{1+x^2}$ is a rational function like $\frac{1}{x}$ and its antiderivatives are $\arctan(x)+C$.
EDIT: From comments, I think I should precise my question. We know that sometimes, the composition of trigonometric and inverse trigonometric functions is algebraic. For example, $\cot(arctan(x))$ is nothing more than $\frac{1}{x}$. I am wondering if there is a function written as a combination (sum, product, composition) of classical functions (trig and inverse trig, as well as polynomial and $n^{th}$ root functions,..) whose derivative is $\frac{1}{x}$. Does this make any sense?
For example, one could argue that introducing $argsinh$ as no interest except convenience since $argsinh(x)=ln(x+\sqrt{x^2+1})$ can be written using already known functions.