The proposed equality is true.
Details: To find
$$l:=\int_0^1\left(\frac{\arcsin x}x\right)^3\,dx
=-\frac{1}{16} \pi \left(\pi ^2-24 \ln2\right),
$$
make the substitution $t=\arcsin x$ and repeatedly integrate by parts to kill the powers of $t$ and reduce this integral to
$$\int_0^{\pi/2}\ln\sin t\,dt=-\frac\pi2\,\ln2,
$$
by
formula 4.225.3, page 531, of Gradshteyn--Ryzhik.
To find
$$r_1:=\int_0^1 2\,\text{arctanh}\, x\,dx
=\ln4,
$$
integrate by parts to find an antiderivative of $\text{arctanh}$. Alternatively, expand $\text{arctanh}\, x$ into the Maclaurin series (using $\text{arctanh}'x=\frac1{1-u}=1+u+u^2+\dots$ with $u=x^2$) and integrate termwise, to get
$$r_1/2=\sum_1^\infty\frac1{(2j-1)2j}
=\sum_1^\infty\Big(\frac1{2j-1}-\frac1{2j}\Big)
=\sum_1^\infty\frac{(-1)^{k-1}}k=\ln2.
$$
To find
$$r_2:=\int_0^1 \frac{\ln(1-x^2)}x\,dx
=-\frac12\,\sum_1^\infty\frac1{j^2}=-\frac{\pi^2}{12},
$$
expand $\ln(1-x^2)$ into the Maclaurin series (by using the Maclaurin series for $\ln(1-u)$) and integrate termwise.
Now one can see that $l=\frac{3\pi}4\,(r_1+r_2)$, that is, the equality holds.
Best Answer
I have proved this equality by means of Cauchy’s Theorem applied to an adequate function. Since my solution is too long to post it here, I posted it in arXiv:
The function $$G(z)=\frac{\log(1+(1+i)\,f(z)\,)}z$$ where $$f(x)=\frac{\operatorname{arctanh}(x)-\arctan(x)}{\pi}$$ extended analytically.