Hello,

Please, I used blow code to calculate Ur, which is derivative of U with respect to y. I got the result. The problem that when I try to integrate the result Ur with respect to r Mathematic failed to integrate. I tried to check and I integrated the result Ur with respect to y which is basically U but Mathematic failed also. Why Mathematica failed to integrate equation? and how I can integrated to get the result?

Clear[r, R, y, Y, k, A, B, U, Ur, Uy]
e = ((4*R*r)/((y + Y)^2 + (r + R)^2));
e1 = ((4*R*r)/((y - Y)^2 + (r + R)^2));
A = EllipticK[e];
B = EllipticE[e];
C1 = EllipticK[e1];
D1 = EllipticE[e1];
U = (((((R*r)^(1/2))/(2*Pi*(e1^(1/2)))) )*(((2 - e1)*C1) - (2* D1))) - (((((R*r)^(1/2))/(2*Pi*(e^(1/2)))) )*(((2 - e)* A) - (2*B)));
Ur = D[U, y] // FullSimplify
Ur1 = Integrate[ Ur, r] // FullSimplify
Ur2 = Integrate[ Ur, y] // FullSimplify

This is Ur which is derivative of U with respect to y.

(1/(4 \[Pi] Sqrt[ r R]))(-(( 2 (r^2 + R^2 + (y - Y)^2) Sqrt[( r R)/((r + R)^2 + (y - Y)^2)] (y - Y) EllipticE[(
 4 r R)/((r + R)^2 + (y - Y)^2)])/((r - R)^2 + (y - Y)^2)) + ( 1/((r - R)^2 + (y + Y)^2))
  2 ((y + Y) (r^2 + R^2 + (y + Y)^2) Sqrt[( r R)/((r + R)^2 + (y + Y)^2)] EllipticE[(
  4 r R)/((r + R)^2 + (y + Y)^2)] + ((r - R)^2 + (y +  Y)^2) (Sqrt[( r R)/((r + R)^2 + (y - Y)^2)] (y - Y) EllipticK[(
  4 r R)/((r + R)^2 + (y - Y)^2)] - (y + Y) Sqrt[( r R)/((r + R)^2 + (y + Y)^2)] EllipticK[(4 r R)/((r + R)^2 + (y + Y)^2)])))

the result which is the integral of Ur with respect to r (failed)

(1/(4 \[Pi]))(\[Integral](1/Sqrt[r R])(-(( 2 (r^2 + R^2 + (y - Y)^2) Sqrt[(r R)/((r + R)^2 + (y - Y)^2)] (y - Y) EllipticE[(
 4 r R)/((r + R)^2 + (y - Y)^2)])/((r - R)^2 + (y - Y)^2)) + (1/((r - R)^2 + (y + Y)^2))
 2 ((y + Y) (r^2 + R^2 + (y + Y)^2) Sqrt[(r R)/((r + R)^2 + (y + Y)^2)] EllipticE[(4 r R)/((r + R)^2 + (y + Y)^2)] + ((r - R)^2 + (y + 
  Y)^2) (Sqrt[(r R)/((r + R)^2 + (y - Y)^2)] (y - Y) EllipticK[(4 r R)/((r + R)^2 + (y - Y)^2)] - (y + Y) Sqrt[(
  r R)/((r + R)^2 + (y + Y)^2)]  EllipticK[(4 r R)/((r + R)^2 + (y + Y)^2)]))) \[DifferentialD]r)

the result which is the integral of Ur with respect to y which is basically U but Mathematic failed also. I already have U and I took the derivative with respect to y but when I tried to integrate again with respect to y again I failed.

(1/(4 \[Pi] Sqrt[r R]))(\[Integral](-((2 (r^2 + R^2 + (y - Y)^2) Sqrt[(r R)/((r + R)^2 + (y - Y)^2)] (y - Y) EllipticE[(
 4 r R)/((r + R)^2 + (y - Y)^2)])/((r - R)^2 + (y - Y)^2)) + (1/((r - R)^2 + (y + Y)^2))
 2 ((y + Y) (r^2 + R^2 + (y + Y)^2) Sqrt[(r R)/((r + R)^2 + (y + Y)^2)] EllipticE[(
 4 r R)/((r + R)^2 + (y + Y)^2)] + ((r - R)^2 + (y +Y)^2) (Sqrt[(r R)/((r + R)^2 + (y - Y)^2)] (y - Y) EllipticK[(
 4 r R)/((r + R)^2 + (y - Y)^2)] - (y + Y) Sqrt[(r R)/((r + R)^2 + (y + Y)^2)]
 EllipticK[(4 r R)/((r + R)^2 + (y + Y)^2)]))) \[DifferentialD]y)

Thanks