integrate((atan(4*x)-atan(2*x))/x, (x, 0, oo)) is returned unevaluated

[oscargus]

From #14078:

integrate((atan(4*x)-atan(2*x))/x, (x, 0, oo)) is returned unevaluated, its value is pi*log(2)/2

[ethankward]

It can be evaluated if you rewrite as log and expand, but unfortunately it gives the answer as 0:

>>> from sympy import *
>>> from sympy.abc import x
>>> e = (atan(4*x)-atan(2*x))/x
>>> integrate(expand(e.rewrite(log)), (x,0,oo))
0