Biot-Savart Integral on a curved path

Question

I am calculating the following Biot-Savart integral according to the suggestion of @Ulrich Neumann in a previous post called Biot-Savart Integral:

(*p[\[Theta]] represents the path to map the curved portion of the \
saddle coil*)

p[\[Theta]_] := {d/2 Cos[\[Pi]/3 - \[Theta]], 0, 
   d/2 Sin[\[Pi]/3 - \[Theta]]};

(*FP is the final point at which I want to calculate the field.*)

FP = {x, y, z};
R = FP - p[\[Theta]];

(*The unit vector from R*)
ar = R/Sqrt[R.R];
(*Bx, By,Bz*)
{Bx, By, Bz} = \[Mu]0/(4 \[Pi])*
  Integrate[
   Cross[D[p[\[Theta]], \[Theta]], ar]/(R.R), {\[Theta], 0, 
    2/3 \[Pi]}, GenerateConditions -> False]

The formula implemented by the code is:

where the vector R connects the element of current to the final point:

(the \theta angle in figure is not the same present in the code) If FP is for example the point of coordinates FP={0,h/2,0} (centre of the saddle coil), the code returns the solution quite fast. However, If I set FP={x,y,z} to find the magnetic field at a generic point of coordinates {x,y,z} it runs for several minutes (~>10) and then returns the following answer:

{(1/(4 \[Pi]))\[Mu]0 Integrate[(d y Sin[\[Pi]/6 + \[Theta]])/(
   2 (y^2 + (z - 1/2 d Cos[\[Pi]/6 + \[Theta]])^2 + (x - 
        1/2 d Sin[\[Pi]/6 + \[Theta]])^2)^(3/2)), {\[Theta], 0, (
    2 \[Pi])/3}, GenerateConditions -> False],....}

which makes me thinking that Mathematica performs the derivative D function but it is not doing the integral. Is there a way in which I can perform the integral? p.s.: The code runs very smoothly and produces the correct answer even for FP={x,y,z} for the case for which the path is a linear segment of the form d[s]:=(1-s)A+sB rather than acurved path p[[theta]] as reported above.