calculus - Evaluate $int_0^1 xarcsinleft(sinleft(frac 1xright)right) dx$

Evaluate $$\int_0^1 x\arcsin\left(\sin\left(\frac 1x\right)\right) dx=\int_1^{\infty} \frac {\arcsin(\sin x)}{x^3}dx$$

I tried to use Feynman's trick in the second one as

$$I(a)=\int_1^{\infty} \frac {\arcsin(\sin ax)}{x^3}dx$$

We have $I(0)=0$

Now differentiating both sides leads me to an absurd integral as $$I'(a)=\int_1^{\infty} \frac {\cos ax }{x^2\vert \cos ax \vert}dx$$

I am now unable proceed further.

Also If I keep in mind that I need value of integral at $a=1$ and so break the integral in intervals as $$\left(1,\frac {\pi}{2}\right) ;\left(\frac {\pi}{2},\frac {3\pi}{2}\right) ; \left(\frac {3\pi}{2},\frac {5\pi}{2}\right)$$
and so on then this gives me the value of integral as 0. But I suspect it isn't correct because one online software suggests it's value to be $\frac {1}{2}$

Moreover the signum function is always reminding me of the Grandi's series which also in some definition equals $1/2$

Any help would be greatly appreciated.