Finding a closed form for $\int_{0}^{1}\frac{\ln\left ( 1-x^{2} \right )\arcsin ^{2}x}{x^{2}}\mathrm{d}x\approx -0.939332$ [duplicate]

I'm attempting to find a closed form for

$$\int_{0}^{1}\frac{\ln\left ( 1-x^{2} \right )\arcsin ^{2}x}{x^{2}}\mathrm{d}x\approx -0.939332$$

I tried to use $$\displaystyle \arcsin^{2}x=\frac{1}{2}\sum_{k=1}^{\infty }\frac{\left ( 2x \right )^{2k}}{k^{2}\dbinom{2k}{k}}$$ but it didn't work and became even more complicated. Any help would be appreciated.

Solution 1:

Using series, I get $$ -2\sum _{k=0}^{\infty }{\frac {{4}^{k} \left( \Psi \left( k+3/2 \right) +\gamma \right) \left( k! \right) ^{2}}{ \left( 1+2\,k \right) \left( 2\,k+2 \right) !}} $$ But I don't expect there to be a closed form.