Integration of $|f(x)|$ or $sqrt{f(x)^2}$ or $left(sqrt{f(x)}right)^2$ as $f(x)$ tends towards chaotic

Assuming $f(x)$ is a Real function of a Real variable for each "$\sqrt{f(x)^2}$", then as $f(x)$ tends from no crossings of $f(x) = 0$ to chaotic & dense crossings of $f(x) = 0$, does computation of a definite integral remain simple or does it become intractable as the chaos/interval/density increase?

I am looking to better understand how both indefinite and definite integrals of e.g. $(\text{Real})\sqrt{(\cos(1/x) - 1/2)^2}$ are/can be calculated, but I am having difficulty finding any information relating to this type of calculus/tractability research. There are many integral calculators online that return integrals for Real functions such as $\sqrt{(\cos(1/x) - 1/2)^2}$, though they say "steps not shown: steps require too much time" or whatnot, but I'm hoping someone could suggest source & reference material, if not directly answer the opening/title question. What determines whether or not such a function as in the title is integrable i.e. has an indefinite integral?