2 deleted 1 characters in body
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It's a bug, of course. Mathematica gets dizzy by the "hanging" integration order (I believe).

As simple as it is, it was tough to find out, but it gives the right result if you just change the order of limits:

$Assumptions = {ρ > 0, L > 0};

limits = Sequence[{ux, 0, L}, {uy, 0, ux}, {vx, 0, L}, {vy, 0, vx}];
(* instead of 
limits = Sequence[{ux, 0, L}, {vx, 0, L}, {uy, 0, ux}, {vy, 0, vx}]
*)

N1 = Integrate[a1 = Exp[-ρ (ux - uy) (vx - vy)],                            limits];
N2 = Integrate[a2 = Exp[-ρ (ux - uy) (vx - vy)] (ρ (ux - uy) (vx - vy)),    limits];
N3 = Integrate[a3 = Exp[-ρ (ux - uy) (vx - vy)] (ρ (ux - uy) (vx - vy))^2,  limits];
N4 = Integrate[a4 = Exp[-ρ (ux - uy) (vx - vy)] (ρ (ux - uy) (vx - vy))^3,  limits];

int1 = Integrate[4 ρ (a1 - 7/2 a2 + 2 a3 - 1/4 a4), limits]

int2 = FullSimplify[4 ρ (N1 - 7/2 N2 + 2 N3 - 1/4 N4)]

(*
 (1 - E^(-L^2 ρ)) L^2
 (1 - E^(-L^2 ρ)) L^2
*)

It's a bug, of course. Mathematica gets dizzy by the "hanging" integration order (I believe).

As simple as it is, it was tough to find out, but it gives the right result if you just change the order of limits:

$Assumptions = {ρ > 0, L > 0};

limits = Sequence[{ux, 0, L}, {uy, 0, ux}, {vx, 0, L}, {vy, 0, vx}];
(* instead of 
limits = Sequence[{ux, 0, L}, {vx, 0, L}, {uy, 0, ux}, {vy, 0, vx}]
*)

N1 = Integrate[a1 = Exp[-ρ (ux - uy) (vx - vy)],                            limits];
N2 = Integrate[a2 = Exp[-ρ (ux - uy) (vx - vy)] (ρ (ux - uy) (vx - vy)),    limits];
N3 = Integrate[a3 = Exp[-ρ (ux - uy) (vx - vy)] (ρ (ux - uy) (vx - vy))^2,  limits];
N4 = Integrate[a4 = Exp[-ρ (ux - uy) (vx - vy)] (ρ (ux - uy) (vx - vy))^3,  limits];

int1 = Integrate[4 ρ (a1 - 7/2 a2 + 2 a3 - 1/4 a4), limits]

int2 = FullSimplify[4 ρ (N1 - 7/2 N2 + 2 N3 - 1/4 N4)]

(*
 (1 - E^(-L^2 ρ)) L^2
 (1 - E^(-L^2 ρ)) L^2
*)

It's a bug, of course. Mathematica gets dizzy by the "hanging" integration order (I believe).

As simple as it is, it was tough to find out but it gives the right result if you just change the order of limits:

$Assumptions = {ρ > 0, L > 0};

limits = Sequence[{ux, 0, L}, {uy, 0, ux}, {vx, 0, L}, {vy, 0, vx}];
(* instead of 
limits = Sequence[{ux, 0, L}, {vx, 0, L}, {uy, 0, ux}, {vy, 0, vx}]
*)

N1 = Integrate[a1 = Exp[-ρ (ux - uy) (vx - vy)],                            limits];
N2 = Integrate[a2 = Exp[-ρ (ux - uy) (vx - vy)] (ρ (ux - uy) (vx - vy)),    limits];
N3 = Integrate[a3 = Exp[-ρ (ux - uy) (vx - vy)] (ρ (ux - uy) (vx - vy))^2,  limits];
N4 = Integrate[a4 = Exp[-ρ (ux - uy) (vx - vy)] (ρ (ux - uy) (vx - vy))^3,  limits];

int1 = Integrate[4 ρ (a1 - 7/2 a2 + 2 a3 - 1/4 a4), limits]

int2 = FullSimplify[4 ρ (N1 - 7/2 N2 + 2 N3 - 1/4 N4)]

(*
 (1 - E^(-L^2 ρ)) L^2
 (1 - E^(-L^2 ρ)) L^2
*)
1
source | link

It's a bug, of course. Mathematica gets dizzy by the "hanging" integration order (I believe).

As simple as it is, it was tough to find out, but it gives the right result if you just change the order of limits:

$Assumptions = {ρ > 0, L > 0};

limits = Sequence[{ux, 0, L}, {uy, 0, ux}, {vx, 0, L}, {vy, 0, vx}];
(* instead of 
limits = Sequence[{ux, 0, L}, {vx, 0, L}, {uy, 0, ux}, {vy, 0, vx}]
*)

N1 = Integrate[a1 = Exp[-ρ (ux - uy) (vx - vy)],                            limits];
N2 = Integrate[a2 = Exp[-ρ (ux - uy) (vx - vy)] (ρ (ux - uy) (vx - vy)),    limits];
N3 = Integrate[a3 = Exp[-ρ (ux - uy) (vx - vy)] (ρ (ux - uy) (vx - vy))^2,  limits];
N4 = Integrate[a4 = Exp[-ρ (ux - uy) (vx - vy)] (ρ (ux - uy) (vx - vy))^3,  limits];

int1 = Integrate[4 ρ (a1 - 7/2 a2 + 2 a3 - 1/4 a4), limits]

int2 = FullSimplify[4 ρ (N1 - 7/2 N2 + 2 N3 - 1/4 N4)]

(*
 (1 - E^(-L^2 ρ)) L^2
 (1 - E^(-L^2 ρ)) L^2
*)