Prove whether there exists a nonconstant function $f(x)$ such that $\displaystyle\int (\ln x)^{f(x)}dx$ has a closed form (i.e. it can be written using function composition, basic arithmetic operations, exponentiation operators, logarithmic operations, etc.). For instance, $\displaystyle\int \frac{\sin x}{x}dx$ does not have a closed form.
I know that if I find one counterexample, I'll be able to disprove this statement. However, I am having trouble finding one. I know that $(\ln x)^{f(x)} = e^{f(x)\ln (\ln x)}$
Let $f(x) = \frac{1}{\ln \ln x}$. Then $$ (\ln x) ^ {f(x)} = \exp(f(x) \ln \ln x) = \exp (\frac{1}{\ln \ln x} \ln \ln x) = e. $$ which is nicely integrable. If you'd rather it was nonconstant, you can pick $$ f(x) = \frac{x}{\ln \ln x} $$ to make the integrand be $e^x$.