How to treat a nuisance parameter in an extended likelihood

I’m in the following situation:
I have two samples A and B, each consisting of a background and a signal.
The number of events in signal_A is cross-sectionefficiency_A
The number of events in signal_B is cross-section
background_A and background_B are not related.
The goal is the determination of the signal cross section.
I’ve managed to set up a simultaneous pdf for the categories A and B extending it to account for the signal and background normalisations.
The nuissance paramter here is the efficiency ratio efficiency_A/efficiency_B. I’m assuming that the efficiency ratio has a known pdf itself from some auxialliary measurements.

So far I have successfully:

  1. Set the efficiency-ratio to a fixed value => works fine
  2. have the efficiency ratio as a free parameter => works fine
  3. Fit the efficiency ratio with an external constraint => works probably (the results look reasonable at first glance but I should have a closer look)

I have not managed to:
4) integrate over the efficiency ratio (using its pdf) => ???
I’m stuck here because this entails the integration of the extension-term in the likelihood function, which I cannot easily access.
How should I proceed?