After some modelling of my data I came to the following integral: $$ \int_0^{1}\dfrac{exp{\left(-\dfrac{\left(x-\mu\right)^2}{2\,\sigma^2}\right)}}{\sqrt{-\log{(1-x)}}} $$ I cannot solve it, and neither can mathematica, maxima or sympy. I would like to know how to solve such integrals in general and I will pursue any pointers. I know that often such integrals can be solved by expressing the function in forms of a confluent hypergeometric function or something even more general, like the meijer G function, but I do not know how to do that.
I want to calculate this integral because I want to optimize the likelihood of the measurements, modelled as gaussian distributions, given the distribution. The overall distribution arises as the distribution of function values of a gaussian distribution where the values of x are randomly distributed.
Note that the distribution itself can be integrated: $$ \dfrac{d\,\mathrm{erf}\left({\sqrt{-\log{\left(1-x\right)}}}\right)}{dx} = \dfrac{1}{\sqrt{-log\left(1-x\right)}} $$