Solving an integral for a characteristic function

Question

For $L>0, H>0,\alpha>0,\sigma>0,$ $$f(t)=\int_L^H \frac{ e^{i t x} \alpha H \left(\frac{\sigma -H \log \left(\frac{H-x}{H-L}\right)}{\sigma }\right)^{-\alpha -1}}{\sigma (H-x)} \, \mathrm{d}x$$

$\textbf{Background}$ This is the characteristic function of a nonstandard probability distribution.