If integration sends the space of all rational functions to the same space union $\ln x$, where does integration send $\mathbb Q[x,\ln x]$?
For example, if I have any rational function, which can be written as a sum
$$Q(x)=\sum_n \left(\frac{C_n}{x-\rho_n}+a_nx^n\right),$$ I'm bound to get another rational function, or in the specific case of $(1/x)$, some $\ln x$ term. Once I have the logarithm, I can construction functions in the space $\mathbb Q[x,\ln x],$ in the sense that if any $F$ belonging to the space can be written
$$F(x)=\frac{\sum_n a_nx^n+b_n(\ln x)^n}{\sum_k c_kx^k+d_k(\ln x)^k},$$ which I can then integrate and obtain a new novel function which needs closed form $$\int\frac{1}{\ln x}dx,$$ and so on. Am I right to think that there is a closed form for any $n$th integral of a rational function if I include the following sequence of functions in my space:
$$\int \frac{1}{x}dx, \ \int\frac{1}{\int \frac{1}{x}dx}dx, \ \int \frac{1}{\int \frac{1}{\int \frac{1}{x}dx}dx}dx, \text{etc} ?$$
I want to clarify a few points for the commenters:
I apologize for not specifying, I am aware. By integrating rational functions we obtain $\ln x$ up to linear substitutions, which I don't care much for. I want to think about this in the sense that "I integrated rational functions, and I found out about this new function $\ln$, which I can compose my rational functions with. Now I wonder how many more functions I need to write $n$th integrals of $\mathbb Q[x]$ in closed form."
Also, what I'm asking is, for example, can we integrate an arbitrary function in $\mathbb Q[x,\ln x]$, in the sense above, and write it in rational functions of
$$x,\ \lambda_1(x)=\ln |x|,\ \lambda_2(x) =\int^x\frac{1}{\ln |\chi|}d\chi$$