I'm interested in finding in closed-form $$PV\left(\int _0^1\frac{\ln ^3\left(t\right)}{\left(1-t\right)\sqrt{2t-1}}\:dt\right)\approx-0.304615808 + 7.286201516 i$$ but I'm not sure how to evaluate it. I tried to rewrite the expression in the following way $$PV\left(2\int _0^1\frac{\ln ^3\left(t\right)}{\sqrt{2t-1}}\:dt+\int _0^1\frac{\ln ^3\left(t\right)\sqrt{2t-1}}{1-t}\:dt\right)$$ Yet I can't think of a way to evaluate either.
I managed to find that the first expression is equivalent to $$\int _0^{\infty }\frac{\ln ^3\left(\frac{i}{i+x}\right)}{x\sqrt{1+x^2}}\:dx$$ However it's sill seems very difficult.