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I've reviewed the general theory at

and the links therein. But they do not give me sufficient traction to show the impossibility of integrating $\int \sin(\sin(x))\: \mathrm{dx}$ in closed form. Remarks from around the web say that Liouville's theorem can do the job, but that the calculations are tricky. However I've not seen a proof worked out anywhere.

Is there one, or are people just blagging?

The impossibility of integrating $\sin(\sin(x))$ in finite terms using the default functions known to Mathematica is described here (but without a proof).

As it turns out, the integral $\int \sin(\sin(x))\: \mathrm{dx}$ can in principle be represented as an infinite sum of $_2F_1$ hypergeometric functions, or as a suitably generalized Kampé de Fériet hypergeometric function of two variables.

So it seems there is, in fact, a solution to the integral as a particular kind of special function, which is different from there being a solution in "closed form," as the Wolfram folks point out. But it seems that Mathematica does not actually know how to work with Kampé de Fériet hypergeometric functions of two variables.

Can anyone here give a more satisfying answer than Mathematica, and show how to work the integral?

Note that this StackExchange post gives the integral as an infinite sum, and this one gives the Fourier expansion of the integrand. These may be heading in the direction of showing that the integral is an "infinite sum of $_2F_1$ hypergeometric functions".

Does that sound like the right idea?

For completeness, it's useful to note that AXIOM implements the Risch procedure, so the fact that it cannot do the integral could be taken as a proof that there is no elementary antiderivative, according to this usenet post by Manuel Bronstein.

if Axiom returns an unevaluated integral, then it has proven that no elementary antiderivative exists.

In other words, AXIOM certifies the proof that there is no "closed form" solution. But what about the proof itself? While I've seen some proofs that apply the Risch procedure by hand, it seems like quite a specialist skill.

Are there hints about how to proceed with a by-hand Risch proof, or hints about how to get a software system to exhibit the proof?

Joe Corneli
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    Via a roundabout way using substitution I've found the following conjecture using Mathematica 11.3: $$\int_a^b \sin(\sin(x)) ; dx=\left[\sum _{n=0}^{\infty } -\frac{(-1)^{n+1} \sin ^{2 n+2}(x) , _2F_1\left(\frac{1}{2},n+1;n+2;\sin ^2(x)\right)}{2 (n+1) (2 n+1) (2 n)!}\right]_a^b$$ which unfortunately is only valid over the interval $[-\pi/2,\pi/2]$.

    If not helpful let me know and I will delete it.

     – James Arathoon Jan 22 '19 at 14:58

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