Integrate $\int x^2\sin^{1}\left ( \frac{\sqrt{a^2x^2} }{b} \right ) dx$
I have already done some progress for this integral at mathexchange
The objective is to compute the following definite integral:
$\int^w_0 x^2\sin^{1}\left ( \frac{\sqrt{a^2x^2} }{b} \right ) dx$
Once the indefinite one is done you can easily proceed with the limits. The constants $a$ and $b$ are nonzero positive, and $x<a$.

I do not want a complex form such as the one from Wolfram Alpha. I need the working out of it too.
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Please leave a comment if you need any clarifications.

Thank you Philip for the response. Sorry in advance if I sound ignorant (I am haven't got much knowledge on these elliptic integrals). Can the result $I(x)$ from the Wolfram Alpha page be plotted in a graph? (provided a and b).

Yes, you may graph them by Matlab: https://www.mathworks.com/help/matlab/ref/ellipke.html

All functions appearing in the formula are basic, all you need is to be able to graph the elliptic integrals of first and second kind.
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