I have to solve $$A(t)=\int_t x(t)\frac{\dot{y}(t)}{y(t)}\,dt$$
where $x$ and $y$ are both functions of time, and $\dot{y}(t)/{y(t)}$ denotes the growth rate of $y$ over time. I also know that $x(t)$ and $y(t)$ are not independent: they are related through a single, unique relationship in which a third variable $z(t)$ appears:
$$x(t)=\frac{y(t)}{y(t)+z(t)}\quad \iff \quad y(t)=\frac{z(t)\cdot x(t)}{1-x(t)}$$
I do not know what the functions $x(t)$, $y(t)$ and $z(t)$ are.
Question: Solve the integral $A(t)$
I'm going around in circles; no matter what I do (substitution or integration by parts), there's always an integral I can't solve. Is this to be expected???