Help with an integral involving the logarithm of $(q;q)_{\infty}$

Question

We have the integral : $$\int_{1}^{q}(z-1)\frac{d}{dz}\log\phi\left(z^{-s} \right )dz$$ Where : $$\phi(q)=\prod_{k=1}^{\infty}(1-q^{k})$$ And $s$ is a complex variable in the right half plane. I tried using : $$\log\phi(z^{-s})=-\sum_{k=1}^{\infty}\frac{1}{k}\frac{1}{z^{ks}-1}$$ But that didn't help much. I also tried : $$\log\phi(z^{-s})=-\sum_{k=1}^{\infty}\frac{\sigma(k)}{k}z^{-ks}$$ $\sigma(k)$ being the divisor function, but that didn't help either. Any insights are appreciated.