{a, b, c} = RandomReal[{0, 1}, 3, WorkingPrecision -> 50];
NIntegrate[rho[xi] Erf[a xi] OwenT[ b xi, c], {xi, 0, Infinity},
 WorkingPrecision -> 20]

xx = (I b c 2)/Sqrt[
 1 + b^2]; V =  (Sqrt[1 + 2 a^2 + b^2] - Sqrt[2] a)/Sqrt[1 + b^2];
vs = {-(Sqrt[-I + b]/Sqrt[I + b]), Sqrt[-I + b]/Sqrt[
   I + b], -(Sqrt[I + b]/Sqrt[-I + b]), Sqrt[I + b]/Sqrt[-I + b]};
ws = {1/2 (xx - Sqrt[-4 + xx^2]), 1/2 (-xx - Sqrt[-4 + xx^2]),
   1/2 (xx + Sqrt[-4 + xx^2]), 1/2 (-xx + Sqrt[-4 + xx^2])};

1/(Sqrt[2] Pi^2) (ArcTan[Sqrt[2] a]/Sqrt[2] ArcTan[ c] + (
    Sqrt[2] b I)/(1 +
      b^2) Sum[((Log[
            V - vs[[i]]] (Log[V - ws[[j]]] -
             Log[(V - ws[[j]])/(vs[[i]] - ws[[j]])]) -
          PolyLog[2, (-V + vs[[i]])/(
           vs[[i]] - ws[[j]])]) - (Log[
            1 - vs[[i]]] (Log[1 - ws[[j]]] -
             Log[(1 - ws[[j]])/(vs[[i]] - ws[[j]])]) -
          PolyLog[2, (-1 + vs[[i]])/(vs[[i]] - ws[[j]])])) (
      vs[[i]] (-1)^(j + 1))/(
      Product[vs[[i]] - vs[[l]], {l, 1, i - 1}] Product[
        vs[[i]] - vs[[l]], {l, i + 1, 4}]), {i, 1, 4}, {j, 1, 4}] )