Do we have a general form for this integral? Is there a general formula or recursion for this integral?
$$\int_0^1\left(\frac{\arcsin x}{x}\right)^n\text{d}x,\ \ n\in\mathbb{N}$$
 A: The generating function would be
$$ g(t) = \sum_{n=0}^\infty \int_0^1 \left( \frac{\arcsin(x)}{x} t \right)^n \ dx = 
\int_0^1 \dfrac{x}{x - t\; \arcsin(x)}\ dt$$
but I doubt that this has a closed form.  The first few terms $n = 0$ to $6$ are
$$  \eqalign{a_0 =&1\cr a_1 =& \frac12\,\pi \,\ln  \left( 2 \right) \cr a_2 =&
4\,{\it Catalan}-\frac14\,{\pi }^{2}\cr a_3 = &-\frac1{16}\,\pi \, \left( {\pi }^{2}-24\,\ln  \left( 2 \right)  \right)\cr a_4 =
&-\frac1{48}\,{\pi }^{4}-\frac12\,{\pi }^{2}+8\,{\it Catalan}+{\pi }^{2}{\it 
Catalan}-8\,{\it Im} \left( {\it polylog} \left( 4,i \right)  \right) 
\cr a_5 =& -{\frac {1}{384}}\,\pi \, \left( 3\,{\pi }^{4}-160\,{\pi }^{2}\ln 
 \left( 2 \right) +40\,{\pi }^{2}+720\,\zeta  \left( 3 \right) -960\,
\ln  \left( 2 \right)  \right)\cr a_6 =&{\frac {9}{32}}\,{\pi }^{4}{\it 
Catalan}-{\frac {1}{320}}\,{\pi }^{6}+15\,{\pi }^{2}{\it Catalan}-{
\frac {11}{64}}\,{\pi }^{4}+12\,{\it Catalan}-\frac34\,{\pi }^{2}\cr &-\frac32\,{
\it Im} \left( 9\,{\pi }^{2}{\it polylog} \left( 4,i \right) -72\,{
\it polylog} \left( 6,i \right) +80\,{\it polylog} \left( 4,i \right) 
 \right) }$$
It doesn't look promising.
