I recently encountered an integral of the form: $$\int{\frac{\log(a+bx+\sqrt{x^2+c})}{x}}dx$$ The result involves the dilogarithm function, but I was wondering if there is a fast way of showing that the integral cannot be expressed in terms of elementary functions. I am aware of the existence of the Risch Algorithm, but I thought there may be a more intuitive way.
Is there such a method?
I can refer you to this paper : http://www.claymath.org/programs/outreach/academy/LectureNotes05/Conrad.pdf.