I've tried differentiating/integrating it, too, in the hope of finding an easy series, but had no luck whatsoever; there is still a lingering product of two terms with distinct series. sad
Perhaps I should discuss the motivation of this problem; yes, indeed, it came from an integral that popped into my head on Christmas Eve.
The integral was
This integral is doable without said series expansion. Sketch of a solution: Use a partial fraction decomposition to break it into two equal integrals, and each of these integrals can be done with a bit of Maclaurin series manipulation. :3
However, an alternate approach is to use the Maclaurin series of log(1 - y), where y = x(1 - x); what you get is a linear combination of Beta integrals that conveniently sums up to a multiple of the given series evaluated at x = 1/2.
Thing is, while also I got it using this formula, I saw this said formula on some paper that I can't find anymore (I saved it to my PC years ago, then my hard disk crashed), so I can't find the proof.
You can get the series for (asin(x))^2 by [differentiating and then multiplying by sqrt(1-x^2)] twice. That gives you 2 on the left and some series on the right which you can extract a recurrence relation from by matching coefficients. No series multiplication needed.