Evaluating $\int_{0}^{\infty} \frac{W^{-n}}{1+\exp\left(\frac{W-\alpha}{\beta}\right)} dW$ with $0I'm trying to integrate the function $$\frac{W^{-n}}{1+\exp\left(\frac{W-\alpha}{\beta}\right)}$$ with $0<n<1$ and limits between $0$ and $\infty$
I've initially tried plotting the function for two different values of $\alpha$ and $n=1/8$, to see if I could find a easily integrable function which fits the integrand but unsuccessful in this approach.
Next I'm trying to approximate the numerator and denominator separately with function which I could integrate. This is where I'm stuck at the moment.
It would be helpful in guiding me towards some books or techiniques to approach this integral.
Thank you.

$$I=\int_{0}^{\infty}\frac{x^{-n} dx}{(1+ge^{x/b})}, c=a/b, g=e^{-c} \implies I=\int_{0}^{\infty} \frac{ x^{-n} e^{-x/b}dx}{1+g^{-1} e^{-x/b}}.$$
Let $y=x/b$, then $$I=- b^{1-n}\int_{0}^{\infty} \sum_{k=1}^{\infty} y^{-n} (-g^{-1})^ke^{-ky} dy= -b^{1-n} \Gamma(1-n) \sum_{k=1}^{\infty} \frac{(-g^{-1})^k }{k^{1-n}}.$$ $$\implies I =-b^{1-n}~ \Gamma(1-n) ~Li_{(1-n)}(-e^{a/b})~~if~~ n <1. $$
Here $Li_{p}(z)$ is polylog function.
