could you propose a way to simplify or approximate (under some assumptions) $\bar{\eta}$ defined as below?
$$ \bar{\eta} = \frac{\int f(t)dt}{\int\frac{f(t)}{\eta{(t)}}dt} $$
The $f(x)$ and $\eta(t)$ are generally unknown functions over time $t$ and $\bar{\eta}$ can be understood as overall $\eta(t)$.
Note that $\bar{\eta}$ is not average of $\eta{(t)}$ unless $\eta(t)$ is constant.
I feel that one would need more structure to the problem in order to find a solution or good approximation, feel free to propose any assumptions you find fitting.
Thank you
I think you really need more information about $\bar\eta(t)$ here. If you make the assumption that $\bar\eta(t)$ is constant (not sure how reasonable that is without context), then of course you have
$\bar\eta=\frac{\int f(t)dt}{\frac{1}{\bar\eta(t)}\int f(t)dt}=\bar\eta(t)$.
The other problem is that since you don't really know anything about $\bar\eta(t)$, it could be zero at some point on the domain, in which case $\bar\eta$ isn't even defined.
Other than that, I don't think there is much else to do without more context. Hope that helps.