Is it possible to double integrate my function over real numbers?

Question

I have a fairly complicated function:

fxsbzs[x_, y_, z_, xs_, b_, zs_, A_] = 
     (y - b) (((x - xs)^2 + (y - b)^2 + (z - zs + A)^2)^(-(3/2)) 
            - ((x - xs)^2 + (y - b)^2 + (z - zs - A)^2)^(-(3/2))) 
   - (y + b) (((x - xs)^2 + (y + b)^2 + (z - zs + A)^2)^(-(3/2)) 
            - ((x - xs)^2 + (y + b)^2 + (z - zs - A)^2)^(-(3/2)));`

And I would like to have the definite symbolic double integral of fxsbzs with respect to zs and xs:

fabc[x_, y_, z_, a_, b_, c_, A_] = 
  Integrate[fxsbzs[x, y, z, xs, b, zs, A], {xs, -a, a}, {zs, -c, c}]

Is this possible (with x, y, z, a, b, c, A restricted to reals and a, b, c, A > 0) ?

---

I can solve the first integral with Assumptions:

fxsbc[x_, y_, z_, xs_, b_, c_, A_] = Integrate[fxsbzs[x, y, z, xs, b, zs, A], 
    {zs, -c, c}, 
    Assumptions -> {Element[{x, y, z, xs, b, c, zs, A}, Reals],
        (b + y) > 0 && (A + z - zs) > 0 && (-A + z - zs) > 0 && (-b + y) > 0 &&
        (x - xs) > 0 && a > 0 && c > 0 && b > 0 && A > 0}]

The result is a long function, which I can't integrate again with respect to xs. Every summand of fxsbc has basically this structure:

u/((xs + xs^2 + v) Sqrt[xs + xs^2 + w])

Since I can't solve the definite integral of this either, I think it's the crucial point:

ClearAll[x, y, z, u, v, w, a]
Integrate[u/((xs + xs^2 + v) Sqrt[xs + xs^2 + w]), {xs, -a, a}, 
    Assumptions -> {Element[{x, y, z, u, v, w, a}, Reals], 
    xs + xs^2 + w > 0 &&  u > 0 && v > 0 && w > 0 && a > 0}]

The indefinite integral gives a complex function:

(1/(Sqrt[1 - 4 v] Sqrt[
 v - w]))I u (Log[(I + I Sqrt[1 - 4 v] - 4 I Sqrt[1 - 4 v] w + 
   2 I xs - 4 I v (1 + 2 xs) - 
   4 Sqrt[1 - 4 v] Sqrt[v - w] Sqrt[w + xs + xs^2])/(2 Sqrt[
   v - w] (1 + Sqrt[1 - 4 v] + 2 xs))] - 
 Log[(2 Sqrt[1 - 4 v] Sqrt[w + xs + xs^2])/(
 1 - Sqrt[1 - 4 v] + 2 xs) + (
 I (-1 + Sqrt[1 - 4 v] - 4 Sqrt[1 - 4 v] w - 2 xs + 
    v (4 + 8 xs)))/(2 Sqrt[v - w] (-1 + Sqrt[1 - 4 v] - 2 xs))])

Is it not possible to solve this integral in the real domain?

Any help would be greatly appreciated!

Answer 1

Install the Rule-based_Integrator of http://www.apmaths.uwo.ca/~arich/ for Mathematica. You will get a simpler structured Integral for your typical terms:

In[64]:= intR2 = Int[u/((xs + xs^2 + v) Sqrt[xs + xs^2 + w]), xs]

Out[64]= -((
 2 u ArcTan[(Sqrt[v - w] (1 + 2 xs))/(
   Sqrt[1 - 4 v] Sqrt[w + xs + xs^2])])/(Sqrt[1 - 4 v] Sqrt[v - w]))