The Maclaurin series of $\arcsin^2$ and $\arcsin^4$ are fairly well-known,
$$ \arcsin^2(x) = \sum_{n\geq 1}\frac{(2x)^{2n}}{2n^2\binom{2n}{n}},\qquad \arcsin^4(x)=3\sum_{n\geq 1}\frac{H_{n-1}^{(2)}(2x)^{2n}}{2n^2\binom{2n}{n}} $$
but in order to deal with some logarithmic integrals I need the Maclaurin series of $\arcsin^3(x)$.
Mr. Wolfram states this is a result of Ramanujan, but I have not been able to find it in his notebooks, so I would like some help. Any derivation from scratch is clearly just as welcome.
Best Answer
See Ramanujan's Notebooks. Part 1 at page 263
For a more general result see also the paper "Integer Powers of Arcsin" by J. M. Borwein and M. Chamberland.