Let $\mathcal{HK}(I)$ denote the Henstock-Kurzweil integrable functions on $I$. By mimicking the case for Lebesgue integral I've proven the following:
Theorem $1$. Let $F$ be an indefinite integral of $f\in \mathcal{HK}(J)$, and $\Phi:I\to J$ be such that $\Phi'$ exists a.e. Then $F\circ \Phi$ is an indefinite integral of gauge integrable function iff $(f\circ \Phi)\cdot \Phi'\in \mathcal{HK}(I)$ and \begin{equation}\int_\alpha^\beta (f\circ \Phi)\cdot \Phi' = \int_{\Phi(\alpha)}^{\Phi(\beta)} f \tag{1}\end{equation} for all $\alpha, \beta\in I$.
I'm wondering about if the following theorem holds:
Conjecture $1$. Let $f\in\mathcal{HK}(J)$ and $\Phi:I\to J$ be a monotone function that is an indefinite integral of a Henstock-Kurzweil integrable function. Then $(f\circ \Phi)\cdot \Phi'\in \mathcal{HK}(I)$ and equation $(1)$ is true.
Some special cases that are known to me:
Theorem $2$. Let $f\in\mathcal{HK}(J)$, $\Phi:I\to J$ a strictly monotone, continuous function such that $\Phi'$ exists except for a countable set. Then $(f\circ \Phi)\cdot \Phi'\in \mathcal{HK}(I)$ and equation $(1)$ is true.
Theorem $3$. Let $f\in\mathcal{L}^1(J)$ and $\Phi:I\to J$ be monotone and absolutely continuous. Then $(f\circ \Phi)\cdot \Phi'\in \mathcal{L}^1(I)$ and equation $(1)$ is true.
Compare theorem $3$ to conjecture $1$.