Find the Taylor Series expansion of $e^{a\arcsin x}$ and hence deduce, the Taylor series expansion of $\arcsin x$.
I could find the Taylor Series expansion of $e^{a\arcsin x}$ as $$1+ax+\frac{a^2x^2}{2}+\frac{a^3+a}{6}x^3+\cdots $$
However, I have no idea how to deduce the series for $\arcsin x$ from the above expansion. I tried writing, $e^{a\arcsin x}$ as $$1+a\arcsin x+\frac{(a\arcsin x)^2}{2!}+\cdots$$ and comparing the coefficients with the one above, but it was not of any help because , I only got the inference that, $x=\arcsin x$ which was quite obvious. Any help regarding solving this issue will be highly appreciated.