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I wounder whether there are any Integrals that are known to have a solution in terms of elementary functions only (or that it is unknown whether they have an elementary solution), but couldn't be solved until now?

I have read about Risch algorithm already which can in principle answer this question, but there are cases where the algorithm fails because of the constant problem.

So, the question is, in these cases where the algorithm fails, is there (always) some way to get the result at least manually/approximatly? Or are there any integrals which haven't been solved where its unknown whether they have an elementary antiderivative?

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  • $\begingroup$ I'm no expert, but based on the linked pages: If $\alpha$ is some real expression for which "$\alpha = 0$" is unknown (and Difficult to Decide), the question of whether or not $f(x) = e^{\alpha x^{2}}$ has an elementary antiderivative is Equally Difficult. $\endgroup$ – Andrew D. Hwang Oct 25 '16 at 11:52
  • $\begingroup$ Hm ok, that sounds reasonable... do you know some $\alpha(x)$ for which this is really hard to decide (or better: unknown)? $\endgroup$ – SampleTime Oct 25 '16 at 18:10
  • $\begingroup$ @SampleTime A cheeky answer: Define $\alpha$ to be the number of nontrivial zeroes of the Riemann Zeta function that lie off the critical line (or $\pi$ if there are infinitely many). If the Riemann Hypothesis is true, then $\alpha=0$, otherwise, $\alpha>0$. $\endgroup$ – Mark S. Mar 21 '17 at 1:32
  • $\begingroup$ @MarkS. It is know that if RH is false, then there are infinitely many (in fact a positive fraction) zeros off the critical line. So $\alpha=0$ or $\alpha=\pi$. $\endgroup$ – wythagoras Aug 30 '17 at 19:52

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