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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?

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$\int\frac{\log(a+bx)}{x}\mathrm dx$ is easily transformed so that it looks like the definition of the dilogarithm, but I'm not sure that what you have is as amenable to such recognition... – J. M. Jul 13 '12 at 16:46
@J.M. - I realize that, but say I never heard of the dilogarithm, how would I know that this integral cannot be expressed in terms of elementary functions? "You can't" would also be a valid answer I guess.. – nbubis Jul 13 '12 at 16:50
up vote 1 down vote accepted

I can refer you to this paper :

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It will take me a while to figure this out specifically for the dilogarithm, but I see the general direction. Thanks! – nbubis Jul 14 '12 at 9:39

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