To Frank: I didn't like your idea, because in the end I would like to have the result: B(x0,y0,z0) if my x0,y0,z0 are the constants I am talking about, but I tried it anyway. (see below).
To David: Yours doesn't work either.
I still get a complex result. Here is the code:
In[2]:= Elpolje = Simplify[-Grad[Koncna[x, y], {x, y}]]
Out[2]= {(
4 a A (a^2 - x^2 + y^2))/((a^2 - 2 a x + x^2 + y^2) (a^2 + 2 a x +
x^2 + y^2)), -((
8 a A x y)/((a^2 - 2 a x + x^2 + y^2) (a^2 + 2 a x + x^2 + y^2)))}
The integral
In[8]:= a = 20
y0 = 1
x0 = 1
z0 = 1
Out[8]= 20
Out[9]= 1
Out[10]= 1
Out[11]= 1
In[12]:= Integrate[(Elpolje[[2]]*
z0)/((aA) ((x0 - x)^2 + (y0 - y)^2 + (z0 - z)^2)^(3/2)), x]
Out[12]= -(1/aA)
160 A y ((-482379 + 157585 x + 326374 y + 1628 x y - 168785 y^2 -
20 x y^2 + 4796 y^3 + 8 x y^3 - 1600 y^4 + 326374 z +
1628 x z - 6408 y z - 24 x y z + 4812 y^2 z + 8 x y^2 z -
8 y^3 z - 166391 z^2 - 826 x z^2 + 3228 y z^2 + 12 x y z^2 -
2410 y^2 z^2 - 4 x y^2 z^2 + 4 y^3 z^2 + 3212 z^3 + 12 x z^3 -
24 y z^3 + 4 y^2 z^3 - 813 z^4 - 3 x z^4 + 6 y z^4 - y^2 z^4 +
6 z^5 - z^6)/(Sqrt[
3 - 2 x + x^2 - 2 y + y^2 - 2 z +
z^2] (51719068962 - 52755965394 y + 27935849089 y^2 -
1568132376 y^3 + 535095200 y^4 - 10253120 y^5 +
2560064 y^6 - 52755965394 z + 2089332928 y z -
1583724024 y^2 z + 28405040 y^3 z - 10330368 y^4 z +
25728 y^5 z + 27422649161 z^2 - 1068053936 y z^2 +
806167828 y^2 z^2 - 14318520 y^3 z^2 + 5178112 y^4 z^2 -
12864 y^5 z^2 - 1052462288 z^3 + 23490768 y z^3 -
14344696 y^2 z^3 + 116064 y^3 z^3 - 12928 y^4 z^3 +
272886176 z^4 - 6001972 y z^4 + 3634790 y^2 z^4 -
29096 y^3 z^4 + 3232 y^4 z^4 - 5898548 z^5 + 77792 y z^5 -
29192 y^2 z^5 + 48 y^3 z^5 + 1013294 z^6 - 13152 y z^6 +
4884 y^2 z^6 - 8 y^3 z^6 - 12992 z^7 + 80 y z^7 -
8 y^2 z^7 + 1654 z^8 - 10 y z^8 + y^2 z^8 - 10 z^9 + z^10)) +
Log[-((320 y (-363 I + (38 + 2 I) y + 2 I z - I z^2) Sqrt[
3 - 2 x + x^2 - 2 y + y^2 - 2 z + z^2])/(-20 + x - I y)) + (
320 (6171 y -
6897 x y + (692 + 1009 I) y^2 + (38 - 1085 I) x y^2 - (405 -
74 I) y^3 + (38 + 2 I) x y^3 + (2 - 38 I) y^4 + 692 y z +
38 x y z - (8 - 74 I) y^2 z + 2 I x y^2 z + 2 y^3 z -
350 y z^2 - 19 x y z^2 + (4 - 37 I) y^2 z^2 - I x y^2 z^2 -
y^3 z^2 + 4 y z^3 - y z^4))/((-20 I + I x + y) Sqrt[
363 - (2 - 38 I) y - 2 z + z^2])]/(
160 y (-363 I + (38 + 2 I) y + 2 I z - I z^2) Sqrt[
363 - (2 - 38 I) y - 2 z + z^2]) + (
I Log[(
320 I y (-443 + (2 + 42 I) y + 2 z - z^2) Sqrt[
3 - 2 x + x^2 - 2 y + y^2 - 2 z + z^2])/(20 + x - I y) - (
320 (-10189 y +
9303 x y + (932 + 1409 I) y^2 - (42 +
1325 I) x y^2 - (405 + 86 I) y^3 - (42 -
2 I) x y^3 + (2 + 42 I) y^4 + 932 y z -
42 x y z - (8 + 86 I) y^2 z + 2 I x y^2 z + 2 y^3 z -
470 y z^2 + 21 x y z^2 + (4 + 43 I) y^2 z^2 - I x y^2 z^2 -
y^3 z^2 + 4 y z^3 - y z^4))/((20 I + I x + y) Sqrt[
443 - (2 + 42 I) y - 2 z + z^2])])/(
160 y (-443 + (2 + 42 I) y + 2 z - z^2) Sqrt[
443 - (2 + 42 I) y - 2 z + z^2]) + (
I Log[(320 I y (-363 + (2 + 38 I) y + 2 z - z^2) Sqrt[
3 - 2 x + x^2 - 2 y + y^2 - 2 z + z^2])/(-20 + x + I y) + (
320 (6171 y -
6897 x y + (692 - 1009 I) y^2 + (38 +
1085 I) x y^2 - (405 + 74 I) y^3 + (38 -
2 I) x y^3 + (2 + 38 I) y^4 + 692 y z +
38 x y z - (8 + 74 I) y^2 z - 2 I x y^2 z + 2 y^3 z -
350 y z^2 - 19 x y z^2 + (4 + 37 I) y^2 z^2 + I x y^2 z^2 -
y^3 z^2 + 4 y z^3 - y z^4))/((20 I - I x + y) Sqrt[
363 - (2 + 38 I) y - 2 z + z^2])])/(
160 y (-363 + (2 + 38 I) y + 2 z - z^2) Sqrt[
363 - (2 + 38 I) y - 2 z + z^2]) +
Log[-((320 y (-443 I + (42 + 2 I) y + 2 I z - I z^2) Sqrt[
3 - 2 x + x^2 - 2 y + y^2 - 2 z + z^2])/(20 + x + I y)) + (
320 (10189 y -
9303 x y - (932 - 1409 I) y^2 + (42 - 1325 I) x y^2 + (405 -
86 I) y^3 + (42 + 2 I) x y^3 - (2 - 42 I) y^4 - 932 y z +
42 x y z + (8 - 86 I) y^2 z + 2 I x y^2 z - 2 y^3 z +
470 y z^2 - 21 x y z^2 - (4 - 43 I) y^2 z^2 - I x y^2 z^2 +
y^3 z^2 - 4 y z^3 + y z^4))/((-20 I - I x + y) Sqrt[
443 - (2 - 42 I) y - 2 z + z^2])]/(
160 y (-443 I + (42 + 2 I) y + 2 I z - I z^2) Sqrt[
443 - (2 - 42 I) y - 2 z + z^2]))
I ignored the whole result as soon as I say those imaginary parts in it.... -.-