I try this numerical summation (in two parts)

 a = NSum[(HarmonicNumber[2 m])/m^3, {m, 1, Infinity}, 
 WorkingPrecision -> 100, PrecisionGoal -> 100];
 N[Pi^3/24 Log[2]^2 + Log[2] Pi/16 Zeta[3] - Pi^5/960 - Pi/16 a, 100]

and this numerical integration

 NIntegrate[x^2 Log[Sin[x]] Log[Cos[x]], {x, 0, Pi/2}, 
 WorkingPrecision -> 100, PrecisionGoal -> 100]

which are supposed to give the same result, and they do, but only to 25 places.

Obviously, at least one of the results is off. How can I increase the precision so that these agree past 25 places? If that can't be done, which of these is the more accurate?

If I try to evaluate the first quantity symbolically, I get

 b = Sum[(HarmonicNumber[2 m])/m^3, {m, 1, Infinity}]

 (* 1/144 (π^4+72 (EulerGamma+Log[4]) Zeta[3]-36 Sqrt[π] 
 (HypergeometricPFQRegularized^({0,0,0,0,0},{0,0,0,1},0))[{1,1,1,1,3/2},{2,2,2,3/2},1]) *)

however, `N[b,20]` never returns. The problem seems to be the evaluation of the derivative of `HypergeometricPFQRegularized`.