integrate(log(x) * asin(2*x),x) returns an expression with a integrate nounform. But an extra evaluation eliminates the nounform:
(%i14) i : integrate(log(x) * asin(2*x),x)$
(%i15) freeof('integrate,i);
(%o15) false
(%i16) ev(i,integrate)$
(%i17) freeof(integrate,%);
(%o17) true
Also, (%o16) is too complex-- I think an antiderivative is
(log((2*sqrt(1-4*x^2))/abs(x)+2/abs(x))+(2*x*log(x)-2*x)*asin(2*x)+sqrt(1-4*x^2)*log(x)-2*sqrt(1-4*x^2))/2
No Li[2] functions needed.