This integral cannot be expressed as a combination of a finite number of usual functions. The analytical solving involves special functions, such as Ei(x) and will lead to a wery complicated formula.
Of course it is possible to obtain this formula, but it would take a lot of time. Probably, this time would be lost without benefit, because it would be too difficult to use the big formula for further work.
That is the reason why I suggest to treat your problem with numerical calculus methods or with approximative analytical methods (limited series expansions depending on the physical range)