Another oddity, integral in home

[parisse]

From CAS, you get the same, with a warning displayed.
You can get a more precise result like this:
a:=∫( ( √( TAN(x) ) - √( SIN(x)) )/x^3, x, 1e-4, 1);
b:=series(( √( TAN(x) ) - √( SIN(x)) )/x^3, x=0,10,polynom);
c:=int(b,x,0,1e-4);
a+c;
returns
0.52629190882