I have been trying to solve a research problem for a while now and in doing so, I stumbled upon the following integral: $$\int_0^{\infty } r \frac{2^{r-1} \log (2) e^{-\frac{\sqrt{2^r-1}}{b}} \left(2^r-1\right)^{\frac{d}{2}-1}}{b^d \Gamma (d)} \, dr.$$ However, I've got no idea how to solve that. Therefore, I'd like to figure out the solution for this integral.