If $f$ has antiderivative, must $|f|$ also have antiderivative?
Note the following quote from the wikipedia article on the Henstock–Kurzweil integral (under Properties)
In general, every Henstock–Kurzweil integrable function is measurable, and f is Lebesgue integrable if and only if both f and |f| are Henstock–Kurzweil integrable. This means that the Henstock–Kurzweil integral can be thought of as a "non-absolutely convergent version of Lebesgue integral". It also implies that the Henstock–Kurzweil integral satisfies appropriate versions of the monotone convergence theorem (without requiring the functions to be nonnegative) and dominated convergence theorem (where the condition of dominance is loosened to g(x) ≤ fn(x) ≤ h(x) for some integrable g, h).
If F is differentiable everywhere (or with countable many exceptions), the derivative F′ is Henstock–Kurzweil integrable, and its indefinite Henstock–Kurzweil integral is F.
From the highlighted sentences (and the Henstock–Kurzweil differentiation theorem), it follows that if $f$ has an anti-derivative, then $|f|$ will also have an anti-derivative if and only if $f$ is Lesbegue integrable.