Say, for example, I take a reasonably-complicated function $f(x)=\tanh[\ln(x^x)]$, and differentiate it to get $$f'(x)=\frac{4x^{2x} [1+\ln(x)]}{(x^{2x}+1)^2}.$$
Now, to integrate this, I imagine, would be very difficult and time-consuming.
My question is: does there exist a function $f$ whose derivative $f'$ we can't integrate (without differentiating $f$ in the first place), using substitution, parts and/or partial fractions (or other integration methods)?
My motivation for this is that it's very easy to differentiate even a ridiculously-complicated function (using the chain and/or product rule), but I've often wondered whether we could get back to the original function by integrating this derivative.
If I'm not articulating myself clearly enough, please ask me to explain further.
Thanks!
Edit
I know, from the Fundamental Theorem of Calculus, that such a function can be integrated, but, other than knowing the fact that this is the derivative of a suitable function, could it be impossible to reverse-engineer the problem, to get $f'$ back to $f$ (using only methods of integration)?