My friend offered to solve this integral.
$$I=\int_0^{\pi/2}x^2\frac{\ln(\sin x)}{\cos x}dx=\frac{\pi^4}{32}-{4G^2} $$ Where G is the Catalan's constant.
$$I=\int _0^{\infty }\frac{\arctan ^2\left(u\right)\ln \left(\frac{u}{\sqrt{1+u^2}}\right)}{\sqrt{1+u^2}}\:du\;.$$ Put $$\tan{x}=u$$ Inability to split into two integrals, they are divergent.