In[]:=
func=(-1)*(x+l/2)/((x+l/2)^2+(y-y1)^2+(z-z1)^2)^(3/2)
Out[]=
-
l
2
+x
3/2
2
l
2
+x
+
2
(y-y1)
+
2
(z-z1)
In[]:=
F=Integrate[func,y1]
Out[]=
-
8
l
2
+x(-y+y1)
(
2
l
+4lx+4(
2
x
+
2
(z-z1)
))
2
l
+4lx+4(
2
x
+
2
y
-2yy1+
2
y1
+
2
z
-2zz1+
2
z1
)
In[]:=
func2=(F/.y1->2)-(F/.y1->1)//Simplify//Expand
Out[]=
-
8l
(
2
l
+4lx+4(
2
x
+
2
(z-z1)
))
2
l
+4lx+4(4+
2
x
-4y+
2
y
+
2
z
-2zz1+
2
z1
)
-
16x
(
2
l
+4lx+4(
2
x
+
2
(z-z1)
))
2
l
+4lx+4(4+
2
x
-4y+
2
y
+
2
z
-2zz1+
2
z1
)
+
4ly
(
2
l
+4lx+4(
2
x
+
2
(z-z1)
))
2
l
+4lx+4(4+
2
x
-4y+
2
y
+
2
z
-2zz1+
2
z1
)
+
8xy
(
2
l
+4lx+4(
2
x
+
2
(z-z1)
))
2
l
+4lx+4(4+
2
x
-4y+
2
y
+
2
z
-2zz1+
2
z1
)
+
4l
(
2
l
+4lx+4(
2
x
+
2
(z-z1)
))
2
l
+4lx+4(1+
2
x
-2y+
2
y
+
2
z
-2zz1+
2
z1
)
+
8x
(
2
l
+4lx+4(
2
x
+
2
(z-z1)
))
2
l
+4lx+4(1+
2
x
-2y+
2
y
+
2
z
-2zz1+
2
z1
)
-
4ly
(
2
l
+4lx+4(
2
x
+
2
(z-z1)
))
2
l
+4lx+4(1+
2
x
-2y+
2
y
+
2
z
-2zz1+
2
z1
)
-
8xy
(
2
l
+4lx+4(
2
x
+
2
(z-z1)
))
2
l
+4lx+4(1+
2
x
-2y+
2
y
+
2
z
-2zz1+
2
z1
)
In[]:=
F2=Integrate[func2,z1]
Out[]=
-
1
2
Log
4(l+2x)
2
l
+4lx+4
2
x
+2l(z-z1)+4x(z-z1)+2(-2+y)-4+2y+
2
l
+4lx+4(4+
2
x
-4y+
2
y
+
2
z
-2zz1+
2
z1
)

2
(-2+y)
(l+2x+2(z-z1))
-Log
4(l+2x)
2
l
+4lx+4
2
x
+2l(-z+z1)+4x(-z+z1)+2(-2+y)-4+2y+
2
l
+4lx+4(4+
2
x
-4y+
2
y
+
2
z
-2zz1+
2
z1
)

2
(-2+y)
(l+2x+2(-z+z1))
-Log-
4(l+2x)
2
l
+4lx+4
2
x
+2l(z-z1)+4x(z-z1)+2(-1+y)-2+2y+
2
l
+4lx+4(1+
2
x
-2y+
2
y
+
2
z
-2zz1+
2
z1
)

2
(-1+y)
(l+2x+2(z-z1))
+Log-
4(l+2x)
2
l
+4lx+4
2
x
+2l(-z+z1)+4x(-z+z1)+2(-1+y)-2+2y+
2
l
+4lx+4(1+
2
x
-2y+
2
y
+
2
z
-2zz1+
2
z1
)

2
(-1+y)
(l+2x+2(-z+z1))

func3=(F2/.z1->4)-(F2/.z1->3)//Simplify(**Solution**)
Out[]=
1
2
Log
4(l+2x)
2
l
+4lx+4
2
x
-2l(-4+z)-4x(-4+z)+2(-2+y)-4+2y+
2
l
+4lx+4(20+
2
x
-4y+
2
y
-8z+
2
z
)

2
(-2+y)
(8+l+2x-2z)
-Log
4(l+2x)
2
l
+4lx+4
2
x
+2l(-4+z)+4x(-4+z)+2(-2+y)-4+2y+
2
l
+4lx+4(20+
2
x
-4y+
2
y
-8z+
2
z
)

2
(-2+y)
(l+2x+2(-4+z))
-Log-
4(l+2x)
2
l
+4lx+4
2
x
-2l(-4+z)-4x(-4+z)+2(-1+y)-2+2y+
2
l
+4lx+4(17+
2
x
-2y+
2
y
-8z+
2
z
)

2
(-1+y)
(8+l+2x-2z)
+Log-
4(l+2x)
2
l
+4lx+4
2
x
+2l(-4+z)+4x(-4+z)+2(-1+y)-2+2y+
2
l
+4lx+4(17+
2
x
-2y+
2
y
-8z+
2
z
)

2
(-1+y)
(l+2x+2(-4+z))
-Log
4(l+2x)
2
l
+4lx+4
2
x
-2l(-3+z)-4x(-3+z)+2(-2+y)-4+2y+
2
l
+4lx+4(13+
2
x
-4y+
2
y
-6z+
2
z
)

2
(-2+y)
(6+l+2x-2z)
+Log
4(l+2x)
2
l
+4lx+4
2
x
+2l(-3+z)+4x(-3+z)+2(-2+y)-4+2y+
2
l
+4lx+4(13+
2
x
-4y+
2
y
-6z+
2
z
)

2
(-2+y)
(l+2x+2(-3+z))
+Log-
4(l+2x)
2
l
+4lx+4
2
x
-2l(-3+z)-4x(-3+z)+2(-1+y)-2+2y+
2
l
+4lx+4(10+
2
x
-2y+
2
y
-6z+
2
z
)

2
(-1+y)
(6+l+2x-2z)
-Log-
4(l+2x)
2
l
+4lx+4
2
x
+2l(-3+z)+4x(-3+z)+2(-1+y)-2+2y+
2
l
+4lx+4(10+
2
x
-2y+
2
y
-6z+
2
z
)

2
(-1+y)
(l+2x+2(-3+z))

In[]:=
g[x_,y_,z_,l_]:=NIntegrate-
l
2
+x
3/2
2
l
2
+x
+
2
(y-y1)
+
2
(z-z1)
,{y1,1,2},{z1,3,4}
In[]:=
g[1,-1,-1,1]
Out[]=
-0.00984229
In[]:=
func3/.x->1/.y->-1/.z->-1/.l->1//N(*CheckOK*)
Out[]=
-0.00984229+0.