Finally, let's take B[[3]]

Bz4[x_, t_, x0_, y0_, z0_, W_, 
  T_] := -I2[T/2 + z0, t*W - y0 - W/2, x - x0] + 
  I2[z0 - T/2, t*W + y0 - W/2, x - x0]
Bz2[x_, x0_, y0_, z0_, W_, T_] := 
 Bz4[x, 1, x0, y0, z0, W, T] - Bz4[x, 0, x0, y0, z0, W, T]
Bz[x0_, y0_, z0_, W_, T_, 
  L_] := (Bz2[L/2, x0, y0, z0, W, T] - 
    Bz2[-L/2, x0, y0, z0, W, T])/(4*Pi)
Plot3D[-Bz[x, y, 20, 50, 20, 50], {x, -100, 100}, {y, -100, 100}, 
 Mesh -> None, ColorFunction -> "TemperatureMap", 
 PlotLegends -> Automatic, 
 AxesLabel -> {"x", "y", 
   "\!\(\*SubscriptBox[\(H\), \(z\)]\)/\!\(\*SubscriptBox[\(H\), \
\(c\)]\)"}, PlotRange -> All, PlotPoints -> 50, Exclusions -> None]

Attachments:

Mathematica is not a mathematician, it is a system with its limitations. Therefore, direct calculations often do not work. We must use the system to solve simple problems. But we must simplify the task ourselves. I have already solved analogous problems with vector integration. I know that a solution can be expressed in this form. Let's take B [[2]]

By4[x_, t_, x0_, y0_, z0_, W_, 
  T_] := -I2[W/2 + y0, t*T + z0 - T/2, x - x0] + 
  I2[y0 - W/2, t*T - z0 - T/2, x - x0]
By2[x_, x0_, y0_, z0_, W_, T_] := 
 By4[x, 1, x0, y0, z0, W, T] - By4[x, 0, x0, y0, z0, W, T]
By[x0_, y0_, z0_, W_, T_, 
  L_] := (By2[L/2, x0, y0, z0, W, T] - 
    By2[-L/2, x0, y0, z0, W, T])/(4*Pi)
Plot3D[-By[x, y, 20, 50, 20, 50], {x, -100, 100}, {y, -100, 100}, 
         Mesh -> None, ColorFunction -> "TemperatureMap", 
         PlotLegends -> Automatic, 
         AxesLabel -> {"x", "y", 
           "\!\(\*SubscriptBox[\(H\), \(y\)]\)/\!\(\*SubscriptBox[\(H\), \
        \(c\)]\)"}, PlotRange -> All, PlotPoints -> 50, Exclusions -> None]
![Y component ][1]

Attachments:

This looks really nice and cool. A questions though: How did you chose the functions for $I1$ and $I2$ in lines:

In[23]:= Integrate[Integrate[1/(p^2 + u^2 + v^2)^(3/2), u], v]
In[9]:= Integrate[Integrate[v/(p^2 + u^2 + v^2)^(3/2), u], v]

I kind of see that there is a $u$ function depends on $t$ (not in numerator of Sum) and $v$ function depends on $x$ (in the numerator Sum) but would like to know how did you define them to be $ - \tanh^{-1} [ \sqrt{p^2 + u^2 + v^2}/u]$ and $\tan^{-1}[\frac{\frac{u v}{p \sqrt{p^2 + u^2 + v^2}}}{p}]$ , and how did you pick coefficients for Bx4

In[17]:= Bx4[x_, t_, x0_, y0_, z0_, W_, 
  T_] := -(W/2 + y0)*
   I1[W/2 + y0, t*T + z0 - T/2, x - x0] + (y0 - W/2)*
   I1[y0 - W/2, t*T - z0 - T/2, x - x0] - (z0 + T/2)*
   I1[z0 + T/2, t*W - y0 - W/2, x - x0] + (z0 - T/2)*
   I1[z0 - T/2, t*W + y0 - W/2, x - x0]

This is good to know because I want to evaluate $H(r)$ with respect to $y$ and $z$ as well. Thanks a lot.

Let's just say I want to learn and want to follow exactly similar analytical solution. I was going to do triple integration too as you see according to my first attempt, I don't get a satisfying result from integration.