JBar[z_]=-1/(16 Pi^2) Integrate[Log[1-x(1-x)z],{x,0,1}] // FullSimplify

Jbar2[z_]=-((-2+(Sqrt[((-4+z)/z)]) (2 ArcTanh[(1/(Sqrt[((-4+z)/z)]))]+Log[-(1/(Sqrt[((-4+z)/z)]))]+Log[-(Sqrt[-4+z]) (Sqrt[z])]-Log[z]))/(16 (π^2)))

rho[s_]=Sqrt[(s-4)/s]

A[s_,t_,u_]=(s-1)/F^2

T0[s_,t_,u_]=3 A[s,t,u]+A[t,u,s]+A[u,s,t] // Simplify

f00[s_]=-2/(s-4) Integrate[T0[s,t,4-s-t],{t,0,(4-s)/2}]/(32 Pi) // Simplify

Sqrt[(s-4)/s] f00[s]^2

T1[s_,t_,u_]=A[t,u,s]-A[u,s,t] // Simplify

f11[s_]= -2/(s-4) Integrate[T1[s,t,4-s-t] (1+2 t/(s-4)),{t,0,(4-s)/2}]/(32 Pi) // Simplify

Sqrt[(s-4)/s] f11[s]^2

T2[s_,t_,u_]=A[t,u,s]+A[u,s,t] // Simplify

f20[s_]=- 2/(s-4) Integrate[T2[s,t,4-s-t] ,{t,0,(4-s)/2}]/(32 Pi) // Simplify

Sqrt[(s-4)/s] f20[s]^2

a00=T0[4,0,0] /(32 Pi)

a20=T2[4,0,0]/(32 Pi)

b00=(f00[4 qsq+4] -a00)/qsq // Simplify

a20=(f20[4 qsq+4]-a20)/qsq  // Expand

a11=f11[4 qsq+4] /qsq  // Simplify

Integrate[(((Sqrt[((-4+s)/s)]) (((-1+2 s))^2))/(1024 (F^4) (π^4)))/(s^3(s-z)),{s,4,Infinity}]

((((120+z (-490+519 z))/(60 (z^3)))-((2 (((1-2 z))^2) (Sqrt[4-z]) ArcTan[((Sqrt[z])/(Sqrt[4-z]))])/(z^(7/2))))/(1024 (F^4) (π^4))) //  FullSimplify

(((-1+2 z))^2)Integrate[((Sqrt[((-4+s)/s)])/(1024 (F^4) (π^4)))/(s(s-z)),{s,4,Infinity}]

(((((-1+2 z))^2) ((2/z)-((2 (Sqrt[4-z]) ArcSec[(2/(Sqrt[4-z]))])/(z^(3/2)))))/(1024 (F^4) (π^4))) // FullSimplify