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Solving a double integral in spherical coordinates of the projected cosine of a spherical cap, the following integral is an "interesting" sub-problem.

It seems more difficult than it looks. Any ideas?

$$ \int{ \sin(x) \sqrt{ 1-x^2 } } dx$$

assuming x is between 0 and 1.

Partial integration and substitution were not helpful because the sine as well as the root do not have easy derivatives or antiderivatives.

Neither Mathematica nor Maple nor Rubi are able to solve it.

I know about StruveH for the definite integral but that's not the question.

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    $\begingroup$ Yes it has one. You can plot it numerically. $\endgroup$ – G. S. Aug 29 '17 at 16:50
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    $\begingroup$ I meant not by elementary functions. $\endgroup$ – 高田航 Aug 29 '17 at 16:50
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    $\begingroup$ The substitution $x= \cos(u)$ doesn't work here because because$\frac{du}{dx} = -\sin(u)$ This does not cancel out the $\sin(x)$ as it is with a different variable. $$\int \sin(x) \sqrt{1-x^2}dx = \int \sin(\cos(u))*\sqrt{1-\cos^2(u)}*\frac{1}{-\sin(u)}du$$ $\endgroup$ – John Lou Aug 29 '17 at 16:53
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    $\begingroup$ I don't think there's any easy antiderivative, WolframAlpha exceeds computation time:wolframalpha.com/input/?i=integrate+%7Bsin(x)*sqrt(1-x%5E2)%7D $\endgroup$ – John Lou Aug 29 '17 at 16:54
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    $\begingroup$ Mathematica outputs $\frac{\pi}{2} H_1(1)$ where $H_n$ is the Struve function. $\endgroup$ – anderstood Aug 29 '17 at 16:55

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