I am failing to integrate
$$ \int \log {\bigg(\frac{a}{x}\frac{x-c}{a-c}\bigg)^{s-1}} \log{\bigg(\frac{b}{x}\bigg)}\bigg(\frac{c}{x}\frac{1}{x-c}\bigg) dx $$
for a positive integer $s$, and real numbers $b, c$.
I have noticed that
$$ \frac{d}{dx} \frac{a}{x}\frac{x-c}{a-c} = \frac{c}{x}\frac{1}{x-c}$$
but I don't know how to use that. I can't see what to substitute here, or how to approach it at all.