Consider the smooth function : $$f(x)=\sin(\pi x)\left(\frac{d}{dx}\log\left(x;j^{-1}\right)\right)$$ $\left(z;q\right)$ being the Q-Pochhammer symbol, and $j$ is a positive integer.
We want to compute the Mellin transform of $f(x)$, that is : $$\int_{0}^{\infty}f(x)x^{s-1}dx$$ Any good idea on how to do this integral ?