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I was recently asked to evaluate the following integral:

$$\int_0^x e^t \sqrt{2 + \sin(2t)} \, dt$$

It was beyond the ken of WolframAlpha, which I find quite discouraging.

Does anyone have an idea of how to tackle this problem?

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    Maple couldn't find the antiderivative either, and also couldn't integrate from $-\infty$ to $0$.2012-11-10
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    It means there is closed form for this integral?2012-11-10
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    It has a nice power series expression. Simply write-out the power series expansions of $e^t$, $\sqrt{1+t}$, re-scale, compose and multiply accordingly, then you take the anti-derivative.2012-11-14
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    Looks hard. Already, $\int\sqrt{2+\sin\,2u}\mathrm du$ is an elliptic integral; the additional exponential term makes me think a simple closed form is terribly unlikely.2012-11-17
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    Is this really what you were asked? The fact that you are integrating from 0 to $x$ makes me wonder if you were asked to compute the derivative of this expression with respect to $x$.2012-11-17
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    @JimConant That is really what I was asked.2012-11-18
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    @B.D. I just wanted to throw that out there just to make sure.2012-11-18
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    Not an elementary function for sure. What does the person who asked you about the integral want to know about it?2012-11-21
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    @fedja I highly suspect there was a transcription error somewhere along the way, because I am also quite convinced this is not an elementary function. Nonetheless, I made an attempt (somewhat feebly) at evaluating it, only to come up empty-handed. I would settle for a proof (probably using the well-known theorem of Liouville) that there is no such answer using only elementary functions. If there is another sneaky way of getting to the answer (through some unlikely manipulation of infinite series) that would be great too.2012-11-21
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    ---I would settle for a proof (probably using the well-known theorem of Liouville) that there is no such answer using only elementary functions--- The only little problem with that is that one needs a full day to carry out such an exercise by hand and to write a random formula to integrate takes under 20 seconds, so I'm not really inclined to enter this race on the "proving impossibility" side though I've done it a few times on AoPS. As to the series representation, mike4ty4 has already provided it :).2012-11-21
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    @fedja Okay; noted.2012-11-21

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