Csu={{1/3,-1,5/3},{-1/3,1/2,5/6},{1/3,1/2,1/6}};

Ctu={{1,0,0},{0,-1,0},{0,0,1}};

Cst={{1/3,1,5/3},{1/3,1/2,-5/6},{1/3,-1/2,1/6}};

g1[s_,t_]=((s IdentityMatrix[3]+(u) Csu+t Cst) /. u-> 4 - s - t);

g2[s_,t_,x_]=(-x t/(x-4)(u Cst+s Cst.Ctu).(IdentityMatrix[3]/(x-t)+(Csu)/(x-(u+s))) /. u-> 4-s-t);

g3[s_,t_,x_]=(-x s u/(x-(s+u))(IdentityMatrix[3]/(x-s)+Csu/(x-u)) /. u-> 4-s-t);

tz[s_,z_]=-1/2(s-4)(1-z);

(*(z/(Pi x^2))(  g2[s,tz[s,z],x][[2]][[2]]+ g3[s,tz[s,z],x][[2]][[2]] (1+2tz[s,z]/(x-4))) // Simplify;*)

Integrate[%,{z,0,1}]

(*F[s_,t_,u_]=1/Pi 1/(x (x-4) )((s(s-4)/(x-s)IdentityMatrix[3]+t(t-4)/(x-t) Cst+u(u-4)/(x-u)Csu).{Imf0,0,Imf2}+3(s(s-4)/(x-s) (1+ 2 t/(s-4)) IdentityMatrix[3]+t(t-4)/(x-t)(1+2 s/(t-4))Cst +u(u-4)/(x-u) (1+2 t/(u-4))Csu).{0,Imf1,0});*)

F[s_,t_,u_]=1/Pi 1/(x (x-4) )((s(s-4)/(x-s)IdentityMatrix[3]+t(t-4)/(x-t) Cst+u(u-4)/(x-u)Csu).{Imf0,0,Imf2}+3(s(t-u)/(x-s)  IdentityMatrix[3]+t(s-u)/(x-t)Cst +u(t-s)/(x-u) Csu).{0,Imf1,0})

Imf0=0; Imf2=0;

1/(s-4)Integrate[(1+2 t/(s-4))*F[s,t,4-s-t][[2]],{t,(4-s),0}]

L0=(3 Imf1 (1/(3 (-4+s) (s-x))x (144 s-72 (s^2)+9 (s^3)-48 x-96 s x+39 (s^2) x+72 (x^2)-6 s (x^2)-16 (x^3)+(6 (s^3)+9 (s^2) (-4+x)+s (48-9 (x^2))-6 x (8-6 x+(x^2))) Log[4-s-x]+(-6 (s^3)-9 (s^2) (-4+x)+6 x (8-6 x+(x^2))+s (-48+9 (x^2))) Log[x])-1/(3 (-4+s) (s-x))(-192 s+144 (s^2)-36 (s^3)+3 (s^4)+64 x+288 s x-204 (s^2) x+32 (s^3) x-240 (x^2)+27 (s^2) (x^2)+120 (x^3)-18 s (x^3)-16 (x^4)+3 x (2 (s^3)+3 (s^2) (-4+x)+s (16-3 (x^2))-2 x (8-6 x+(x^2))) Log[-x]+3 x (-2 (s^3)-3 (s^2) (-4+x)+2 x (8-6 x+(x^2))+s (-16+3 (x^2))) Log[-4+s+x])))/(2 π (-4+s) (-4+x) x) // FullSimplify;

L1= L0 /. Log[__]->0 // Simplify

P0=-3((-4+s) (3 (s^3)+(s^2) (-24+23 x)-4 x (4-12 x+3 (x^2))-4 s (-12+10 x+3 (x^2)))+6 x (2 (s^3)+3 (s^2) (-4+x)+s (16-3 (x^2))-2 x (8-6 x+(x^2))) Log[-2 x]-6 x (2 (s^3)+3 (s^2) (-4+x)+s (16-3 (x^2))-2 x (8-6 x+(x^2))) Log[-2 (-4+s+x)])/(6 π (((-4+s))^2) (s-x) (-4+x) x) // FullSimplify;

P1= Imf1 * P0 /. Log[__]->0 // Simplify

1/(s-4)Integrate[F[s,t,4-s-t][[1]],{t,(4-s),0}]

S0=(1/(π (-4+s) (-4+x) x))3 Imf1 (-16+16 s-3 (s^2)+12 x-4 s x-((3 (x^2))/2)-x (-4+2 s+x) Log[-x]+(1/2) x (-8+4 s+3 x+2 (-4+2 s+x) Log[4-s-x]-2 (-4+2 s+x) Log[x])+x (-4+2 s+x) Log[-4+s+x]) // Simplify

S1=S0 /. Log[__]->0 // Simplify

S0-S1 // Simplify