# Integrate and NIntegrate give different results

Good morning, i computed the following integral using Integrate and i get a different result from the one obtained with NIntegrate. Here is the integrand

 expr = (b (b - 2 d + Sqrt[4 + b^2 - 4 b d]) (c + I x) x (3 + b^2 c^2 + 2 I b^2 c x - (-6 + b^2) x^2 + 3 x^4) (-1 + a x (-2 I c + a x)))/((2 + b^2 + 2 d (d - Sqrt[4 + b^2 - 4 b d]) + b (-4 d + Sqrt[4 + b^2 - 4 b d])) \[Pi]^3 (-I c + a x) (1 - b c -I b x + x^2)^2 (1 + b c + I b x + x^2)^2 (1 + a^2 x^2));


I use Integrate

result = Integrate[expr, {x, -Infinity, Infinity}, Assumptions -> 1/2 (b + Sqrt[4 + b^2 - 4 b d]) > -1 && 1/2 (b + Sqrt[4 + b^2 - 4 b d]) < 1 && c > -1 && c < 1 && d > 1 &&
a >= 0]


and then i compare the results

 result/.{a->2,b->0.65,c->0.25,d->1.7} 0.341887 +0. I NIntegrate[expr/.{a->2,b->0.65,c->0.25,d->1.7},{x,-Infinity,Infinity}] -0.00340441-8.67362*10^-19 I


As you can see i get different results even if the assumptions are satisfied. Why?

• Your numerical result is correct. Use FullSimplify, remove the constant pre-factor and use the Rubi package in order to integrate analytically. Jul 28, 2021 at 16:36
• @yarchik: Can you elaborate your directions? In particular, can you ground "Your numerical result is correct"? TIA. Jul 28, 2021 at 16:50
• @user64494 See below, but it takes a while Jul 28, 2021 at 17:50

You get the right analytical result quite fast, if you impose the addidional condition b<1 . Maybe Integrate then finds an other branch.

expr = (b (b - 2 d + Sqrt[4 + b^2 - 4 b d]) (c + I x) x (3 +
b^2 c^2 + 2 I b^2 c x - (-6 + b^2) x^2 + 3 x^4) (-1 +
a x (-2 I c + a x)))/((2 + b^2 +
2 d (d - Sqrt[4 + b^2 - 4 b d]) +
b (-4 d + Sqrt[4 + b^2 - 4 b d])) \[Pi]^3 (-I c +
a x) (1 - b c - I b x + x^2)^2 (1 + b c + I b x + x^2)^2 (1 +
a^2 x^2));

asump = 1/2 (b + Sqrt[4 + b^2 - 4 b d]) > -1 &&
1/2 (b + Sqrt[4 + b^2 - 4 b d]) < 1 && c > -1 && c < 1 && d > 1 &&
a >= 0;

red = Reduce[asump && b < 1, {a, b, c, d}, Reals]

result = Integrate[expr, {x, -Infinity, Infinity}, Assumptions -> red]

result /. {a -> 2, b -> 0.65, c -> 0.25, d -> 1.7}

(*   -0.00340441 + 0. I   *)

NIntegrate[
expr /. {a -> 2, b -> 0.65, c -> 0.25, d -> 1.7}, {x, -Infinity,
Infinity}]


The same !

result  (*   version 8.0   *)

(*   ConditionalExpression[(2 I b ((8192 (20 a b^3 + 32 a^3 b^3 -
36 a^5 b^3 + 15 a b^5 + 200 a^3 b^5 - 111 a^5 b^5 -
a b^7 + 208 a^3 b^7 - 78 a^5 b^7 - a b^9 + 72 a^3 b^9 -
9 a^5 b^9 + 8 a^3 b^11 + 5 a^5 b^11 + a^5 b^13 + 8 b^2 c +
24 a b^2 c + 16 a^2 b^2 c - 24 a^4 b^2 c - 24 a^5 b^2 c +
30 b^4 c + 82 a b^4 c + 212 a^2 b^4 c + 512 a^3 b^4 c -
154 a^4 b^4 c - 330 a^5 b^4 c + 12 b^6 c + 10 a b^6 c +
258 a^2 b^6 c + 1208 a^3 b^6 c - 104 a^4 b^6 c -
516 a^5 b^6 c + b^8 c - 8 a b^8 c + 92 a^2 b^8 c +
672 a^3 b^8 c + 32 a^4 b^8 c - 162 a^5 b^8 c +
10 a^2 b^10 c + 104 a^3 b^10 c + 32 a^4 b^10 c +
30 a^5 b^10 c + 5 a^4 b^12 c + 12 a^5 b^12 c +
84 b^3 c^2 + 156 a b^3 c^2 + 360 a^2 b^3 c^2 +
136 a^3 b^3 c^2 - 316 a^4 b^3 c^2 - 180 a^5 b^3 c^2 +
87 b^5 c^2 + 193 a b^5 c^2 + 1278 a^2 b^5 c^2 +
1870 a^3 b^5 c^2 - 665 a^4 b^5 c^2 - 915 a^5 b^5 c^2 +
13 b^7 c^2 + 34 a b^7 c^2 + 798 a^2 b^7 c^2 +
2124 a^3 b^7 c^2 + 10 a^4 b^7 c^2 - 684 a^5 b^7 c^2 +
6 a b^9 c^2 + 126 a^2 b^9 c^2 + 518 a^3 b^9 c^2 +
259 a^4 b^9 c^2 + 15 a^5 b^9 c^2 + 2 a^3 b^11 c^2 +
61 a^4 b^11 c^2 + 54 a^5 b^11 c^2 + 24 b^2 c^3 +
56 a b^2 c^3 + 48 a^2 b^2 c^3 - 48 a^3 b^2 c^3 -
72 a^4 b^2 c^3 - 8 a^5 b^2 c^3 + 162 b^4 c^3 +
578 a b^4 c^3 + 1524 a^2 b^4 c^3 + 460 a^3 b^4 c^3 -
934 a^4 b^4 c^3 - 430 a^5 b^4 c^3 + 54 b^6 c^3 +
404 a b^6 c^3 + 2232 a^2 b^6 c^3 + 2448 a^3 b^6 c^3 -
474 a^4 b^6 c^3 - 972 a^5 b^6 c^3 + 72 a b^8 c^3 +
582 a^2 b^8 c^3 + 1188 a^3 b^8 c^3 + 698 a^4 b^8 c^3 -
178 a^5 b^8 c^3 + 20 a^3 b^10 c^3 + 284 a^4 b^10 c^3 +
112 a^5 b^10 c^3 + 60 b^3 c^4 + 368 a b^3 c^4 +
312 a^2 b^3 c^4 - 232 a^3 b^3 c^4 - 244 a^4 b^3 c^4 -
24 a^5 b^3 c^4 + 81 b^5 c^4 + 964 a b^5 c^4 +
2178 a^2 b^5 c^4 + 474 a^3 b^5 c^4 - 763 a^4 b^5 c^4 -
414 a^5 b^5 c^4 + 304 a b^7 c^4 + 1188 a^2 b^7 c^4 +
1140 a^3 b^7 c^4 + 706 a^4 b^7 c^4 - 315 a^5 b^7 c^4 +
70 a^3 b^9 c^4 + 631 a^4 b^9 c^4 + 105 a^5 b^9 c^4 +
16 a b^2 c^5 - 16 a^3 b^2 c^5 + 32 b^4 c^5 +
636 a b^4 c^5 + 504 a^2 b^4 c^5 - 348 a^3 b^4 c^5 -
224 a^4 b^4 c^5 - 24 a^5 b^4 c^5 + 528 a b^6 c^5 +
1002 a^2 b^6 c^5 + 192 a^3 b^6 c^5 + 204 a^4 b^6 c^5 -
138 a^5 b^6 c^5 + 100 a^3 b^8 c^5 + 683 a^4 b^8 c^5 +
36 a^5 b^8 c^5 + 48 a b^3 c^6 - 32 a^3 b^3 c^6 +
324 a b^5 c^6 + 240 a^2 b^5 c^6 - 160 a^3 b^5 c^6 -
4 a^4 b^5 c^6 - 8 a^5 b^5 c^6 + 50 a^3 b^7 c^6 +
325 a^4 b^7 c^6 + 2 a^5 b^7 c^6 + 32 a b^4 c^7 -
16 a^3 b^4 c^7 + 4 a^3 b^6 c^7 + 48 a^4 b^6 c^7 +
4 I a b^2 Sqrt[-4 - b^2 - 4 b c] -
4 I a^5 b^2 Sqrt[-4 - b^2 - 4 b c] +
11 I a b^4 Sqrt[-4 - b^2 - 4 b c] +
56 I a^3 b^4 Sqrt[-4 - b^2 - 4 b c] -
39 I a^5 b^4 Sqrt[-4 - b^2 - 4 b c] +
I a b^6 Sqrt[-4 - b^2 - 4 b c] +
112 I a^3 b^6 Sqrt[-4 - b^2 - 4 b c] -
50 I a^5 b^6 Sqrt[-4 - b^2 - 4 b c] -
I a b^8 Sqrt[-4 - b^2 - 4 b c] +
56 I a^3 b^8 Sqrt[-4 - b^2 - 4 b c] -
13 I a^5 b^8 Sqrt[-4 - b^2 - 4 b c] +
8 I a^3 b^10 Sqrt[-4 - b^2 - 4 b c] +
3 I a^5 b^10 Sqrt[-4 - b^2 - 4 b c] +
I a^5 b^12 Sqrt[-4 - b^2 - 4 b c] +
12 I b^3 c Sqrt[-4 - b^2 - 4 b c] +
32 I a b^3 c Sqrt[-4 - b^2 - 4 b c] +
48 I a^2 b^3 c Sqrt[-4 - b^2 - 4 b c] +
64 I a^3 b^3 c Sqrt[-4 - b^2 - 4 b c] -
44 I a^4 b^3 c Sqrt[-4 - b^2 - 4 b c] -
64 I a^5 b^3 c Sqrt[-4 - b^2 - 4 b c] +
10 I b^5 c Sqrt[-4 - b^2 - 4 b c] +
16 I a b^5 c Sqrt[-4 - b^2 - 4 b c] +
134 I a^2 b^5 c Sqrt[-4 - b^2 - 4 b c] +
456 I a^3 b^5 c Sqrt[-4 - b^2 - 4 b c] -
76 I a^4 b^5 c Sqrt[-4 - b^2 - 4 b c] -
224 I a^5 b^5 c Sqrt[-4 - b^2 - 4 b c] +
I b^7 c Sqrt[-4 - b^2 - 4 b c] -
6 I a b^7 c Sqrt[-4 - b^2 - 4 b c] +
72 I a^2 b^7 c Sqrt[-4 - b^2 - 4 b c] +
416 I a^3 b^7 c Sqrt[-4 - b^2 - 4 b c] -
2 I a^4 b^7 c Sqrt[-4 - b^2 - 4 b c] -
132 I a^5 b^7 c Sqrt[-4 - b^2 - 4 b c] +
10 I a^2 b^9 c Sqrt[-4 - b^2 - 4 b c] +
88 I a^3 b^9 c Sqrt[-4 - b^2 - 4 b c] +
22 I a^4 b^9 c Sqrt[-4 - b^2 - 4 b c] +
8 I a^5 b^9 c Sqrt[-4 - b^2 - 4 b c] +
5 I a^4 b^11 c Sqrt[-4 - b^2 - 4 b c] +
10 I a^5 b^11 c Sqrt[-4 - b^2 - 4 b c] +
12 I b^2 c^2 Sqrt[-4 - b^2 - 4 b c] +
20 I a b^2 c^2 Sqrt[-4 - b^2 - 4 b c] +
24 I a^2 b^2 c^2 Sqrt[-4 - b^2 - 4 b c] -
8 I a^3 b^2 c^2 Sqrt[-4 - b^2 - 4 b c] -
36 I a^4 b^2 c^2 Sqrt[-4 - b^2 - 4 b c] -
12 I a^5 b^2 c^2 Sqrt[-4 - b^2 - 4 b c] +
49 I b^4 c^2 Sqrt[-4 - b^2 - 4 b c] +
99 I a b^4 c^2 Sqrt[-4 - b^2 - 4 b c] +
426 I a^2 b^4 c^2 Sqrt[-4 - b^2 - 4 b c] +
386 I a^3 b^4 c^2 Sqrt[-4 - b^2 - 4 b c] -
279 I a^4 b^4 c^2 Sqrt[-4 - b^2 - 4 b c] -
237 I a^5 b^4 c^2 Sqrt[-4 - b^2 - 4 b c] +
11 I b^6 c^2 Sqrt[-4 - b^2 - 4 b c] +
32 I a b^6 c^2 Sqrt[-4 - b^2 - 4 b c] +
482 I a^2 b^6 c^2 Sqrt[-4 - b^2 - 4 b c] +
956 I a^3 b^6 c^2 Sqrt[-4 - b^2 - 4 b c] -
122 I a^4 b^6 c^2 Sqrt[-4 - b^2 - 4 b c] -
360 I a^5 b^6 c^2 Sqrt[-4 - b^2 - 4 b c] +
6 I a b^8 c^2 Sqrt[-4 - b^2 - 4 b c] +
106 I a^2 b^8 c^2 Sqrt[-4 - b^2 - 4 b c] +
354 I a^3 b^8 c^2 Sqrt[-4 - b^2 - 4 b c] +
133 I a^4 b^8 c^2 Sqrt[-4 - b^2 - 4 b c] -
39 I a^5 b^8 c^2 Sqrt[-4 - b^2 - 4 b c] +
2 I a^3 b^10 c^2 Sqrt[-4 - b^2 - 4 b c] +
51 I a^4 b^10 c^2 Sqrt[-4 - b^2 - 4 b c] +
36 I a^5 b^10 c^2 Sqrt[-4 - b^2 - 4 b c] +
48 I b^3 c^3 Sqrt[-4 - b^2 - 4 b c] +
144 I a b^3 c^3 Sqrt[-4 - b^2 - 4 b c] +
240 I a^2 b^3 c^3 Sqrt[-4 - b^2 - 4 b c] -
32 I a^3 b^3 c^3 Sqrt[-4 - b^2 - 4 b c] -
192 I a^4 b^3 c^3 Sqrt[-4 - b^2 - 4 b c] -
48 I a^5 b^3 c^3 Sqrt[-4 - b^2 - 4 b c] +
34 I b^5 c^3 Sqrt[-4 - b^2 - 4 b c] +
236 I a b^5 c^3 Sqrt[-4 - b^2 - 4 b c] +
936 I a^2 b^5 c^3 Sqrt[-4 - b^2 - 4 b c] +
632 I a^3 b^5 c^3 Sqrt[-4 - b^2 - 4 b c] -
366 I a^4 b^5 c^3 Sqrt[-4 - b^2 - 4 b c] -
308 I a^5 b^5 c^3 Sqrt[-4 - b^2 - 4 b c] +
60 I a b^7 c^3 Sqrt[-4 - b^2 - 4 b c] +
390 I a^2 b^7 c^3 Sqrt[-4 - b^2 - 4 b c] +
600 I a^3 b^7 c^3 Sqrt[-4 - b^2 - 4 b c] +
236 I a^4 b^7 c^3 Sqrt[-4 - b^2 - 4 b c] -
144 I a^5 b^7 c^3 Sqrt[-4 - b^2 - 4 b c] +
16 I a^3 b^9 c^3 Sqrt[-4 - b^2 - 4 b c] +
192 I a^4 b^9 c^3 Sqrt[-4 - b^2 - 4 b c] +
56 I a^5 b^9 c^3 Sqrt[-4 - b^2 - 4 b c] +
4 I b^2 c^4 Sqrt[-4 - b^2 - 4 b c] +
24 I a b^2 c^4 Sqrt[-4 - b^2 - 4 b c] +
8 I a^2 b^2 c^4 Sqrt[-4 - b^2 - 4 b c] -
24 I a^3 b^2 c^4 Sqrt[-4 - b^2 - 4 b c] -
12 I a^4 b^2 c^4 Sqrt[-4 - b^2 - 4 b c] +
31 I b^4 c^4 Sqrt[-4 - b^2 - 4 b c] +
346 I a b^4 c^4 Sqrt[-4 - b^2 - 4 b c] +
510 I a^2 b^4 c^4 Sqrt[-4 - b^2 - 4 b c] -
82 I a^3 b^4 c^4 Sqrt[-4 - b^2 - 4 b c] -
245 I a^4 b^4 c^4 Sqrt[-4 - b^2 - 4 b c] -
60 I a^5 b^4 c^4 Sqrt[-4 - b^2 - 4 b c] +
196 I a b^6 c^4 Sqrt[-4 - b^2 - 4 b c] +
580 I a^2 b^6 c^4 Sqrt[-4 - b^2 - 4 b c] +
332 I a^3 b^6 c^4 Sqrt[-4 - b^2 - 4 b c] +
110 I a^4 b^6 c^4 Sqrt[-4 - b^2 - 4 b c] -
125 I a^5 b^6 c^4 Sqrt[-4 - b^2 - 4 b c] +
42 I a^3 b^8 c^4 Sqrt[-4 - b^2 - 4 b c] +
329 I a^4 b^8 c^4 Sqrt[-4 - b^2 - 4 b c] +
35 I a^5 b^8 c^4 Sqrt[-4 - b^2 - 4 b c] +
4 I b^3 c^5 Sqrt[-4 - b^2 - 4 b c] +
96 I a b^3 c^5 Sqrt[-4 - b^2 - 4 b c] +
32 I a^2 b^3 c^5 Sqrt[-4 - b^2 - 4 b c] -
64 I a^3 b^3 c^5 Sqrt[-4 - b^2 - 4 b c] -
20 I a^4 b^3 c^5 Sqrt[-4 - b^2 - 4 b c] +
232 I a b^5 c^5 Sqrt[-4 - b^2 - 4 b c] +
298 I a^2 b^5 c^5 Sqrt[-4 - b^2 - 4 b c] -
56 I a^3 b^5 c^5 Sqrt[-4 - b^2 - 4 b c] -
14 I a^4 b^5 c^5 Sqrt[-4 - b^2 - 4 b c] -
24 I a^5 b^5 c^5 Sqrt[-4 - b^2 - 4 b c] +
40 I a^3 b^7 c^5 Sqrt[-4 - b^2 - 4 b c] +
255 I a^4 b^7 c^5 Sqrt[-4 - b^2 - 4 b c] +
6 I a^5 b^7 c^5 Sqrt[-4 - b^2 - 4 b c] +
72 I a b^4 c^6 Sqrt[-4 - b^2 - 4 b c] +
24 I a^2 b^4 c^6 Sqrt[-4 - b^2 - 4 b c] -
40 I a^3 b^4 c^6 Sqrt[-4 - b^2 - 4 b c] -
4 I a^4 b^4 c^6 Sqrt[-4 - b^2 - 4 b c] +
10 I a^3 b^6 c^6 Sqrt[-4 - b^2 - 4 b c] +
75 I a^4 b^6 c^6 Sqrt[-4 - b^2 - 4 b c] +
4 I a^4 b^5 c^7 Sqrt[-4 - b^2 - 4 b c]))/(Sqrt[-4 - b^2 -
4 b c] (4 + b^2 + 4 b c) (-2 I - I a b +
a Sqrt[-4 - b^2 - 4 b c])^2 (2 I - I a b +
a Sqrt[-4 - b^2 - 4 b c])^2 (-I a b - 2 I c +
a Sqrt[-4 - b^2 - 4 b c])^2 (-2 I b +
Sqrt[-4 - b^2 - 4 b c] -
Sqrt[-4 - b^2 + 4 b c])^3 (-2 I b +
Sqrt[-4 - b^2 - 4 b c] +
Sqrt[-4 - b^2 + 4 b c])^3) + (256 I a (-1 + a c) (3 -
6 a^2 + 3 a^4 + a^2 b^2 - 2 a^3 b^2 c + a^4 b^2 c^2))/((2 +
a b - I a Sqrt[-4 - b^2 - 4 b c])^2 (2 + a b +
I a Sqrt[-4 - b^2 - 4 b c])^2 (2 I - I a b +
a Sqrt[-4 - b^2 + 4 b c])^2 (-2 I + I a b +
a Sqrt[-4 - b^2 + 4 b c])^2) - (256 I (-1 +
a) a c^2 (3 a^4 - 6 a^2 c^2 + a^2 b^2 c^2 - 2 a^3 b^2 c^2 +
a^4 b^2 c^2 + 3 c^4))/((a b + 2 c -
I a Sqrt[-4 - b^2 - 4 b c])^2 (a b + 2 c +
I a Sqrt[-4 - b^2 - 4 b c])^2 (I a b - 2 I c +
a Sqrt[-4 - b^2 + 4 b c])^2 (-I a b + 2 I c +
a Sqrt[-4 - b^2 + 4 b c])^2) + (8192 (-20 a b^3 -
32 a^3 b^3 + 36 a^5 b^3 - 15 a b^5 - 200 a^3 b^5 +
111 a^5 b^5 + a b^7 - 208 a^3 b^7 + 78 a^5 b^7 + a b^9 -
72 a^3 b^9 + 9 a^5 b^9 - 8 a^3 b^11 - 5 a^5 b^11 -
a^5 b^13 + 8 b^2 c + 24 a b^2 c + 16 a^2 b^2 c -
24 a^4 b^2 c - 24 a^5 b^2 c + 30 b^4 c + 82 a b^4 c +
212 a^2 b^4 c + 512 a^3 b^4 c - 154 a^4 b^4 c -
330 a^5 b^4 c + 12 b^6 c + 10 a b^6 c + 258 a^2 b^6 c +
1208 a^3 b^6 c - 104 a^4 b^6 c - 516 a^5 b^6 c + b^8 c -
8 a b^8 c + 92 a^2 b^8 c + 672 a^3 b^8 c + 32 a^4 b^8 c -
162 a^5 b^8 c + 10 a^2 b^10 c + 104 a^3 b^10 c +
32 a^4 b^10 c + 30 a^5 b^10 c + 5 a^4 b^12 c +
12 a^5 b^12 c - 84 b^3 c^2 - 156 a b^3 c^2 -
360 a^2 b^3 c^2 - 136 a^3 b^3 c^2 + 316 a^4 b^3 c^2 +
180 a^5 b^3 c^2 - 87 b^5 c^2 - 193 a b^5 c^2 -
1278 a^2 b^5 c^2 - 1870 a^3 b^5 c^2 + 665 a^4 b^5 c^2 +
915 a^5 b^5 c^2 - 13 b^7 c^2 - 34 a b^7 c^2 -
798 a^2 b^7 c^2 - 2124 a^3 b^7 c^2 - 10 a^4 b^7 c^2 +
684 a^5 b^7 c^2 - 6 a b^9 c^2 - 126 a^2 b^9 c^2 -
518 a^3 b^9 c^2 - 259 a^4 b^9 c^2 - 15 a^5 b^9 c^2 -
2 a^3 b^11 c^2 - 61 a^4 b^11 c^2 - 54 a^5 b^11 c^2 +
24 b^2 c^3 + 56 a b^2 c^3 + 48 a^2 b^2 c^3 -
48 a^3 b^2 c^3 - 72 a^4 b^2 c^3 - 8 a^5 b^2 c^3 +
162 b^4 c^3 + 578 a b^4 c^3 + 1524 a^2 b^4 c^3 +
460 a^3 b^4 c^3 - 934 a^4 b^4 c^3 - 430 a^5 b^4 c^3 +
54 b^6 c^3 + 404 a b^6 c^3 + 2232 a^2 b^6 c^3 +
2448 a^3 b^6 c^3 - 474 a^4 b^6 c^3 - 972 a^5 b^6 c^3 +
72 a b^8 c^3 + 582 a^2 b^8 c^3 + 1188 a^3 b^8 c^3 +
698 a^4 b^8 c^3 - 178 a^5 b^8 c^3 + 20 a^3 b^10 c^3 +
284 a^4 b^10 c^3 + 112 a^5 b^10 c^3 - 60 b^3 c^4 -
368 a b^3 c^4 - 312 a^2 b^3 c^4 + 232 a^3 b^3 c^4 +
244 a^4 b^3 c^4 + 24 a^5 b^3 c^4 - 81 b^5 c^4 -
964 a b^5 c^4 - 2178 a^2 b^5 c^4 - 474 a^3 b^5 c^4 +
763 a^4 b^5 c^4 + 414 a^5 b^5 c^4 - 304 a b^7 c^4 -
1188 a^2 b^7 c^4 - 1140 a^3 b^7 c^4 - 706 a^4 b^7 c^4 +
315 a^5 b^7 c^4 - 70 a^3 b^9 c^4 - 631 a^4 b^9 c^4 -
105 a^5 b^9 c^4 + 16 a b^2 c^5 - 16 a^3 b^2 c^5 +
32 b^4 c^5 + 636 a b^4 c^5 + 504 a^2 b^4 c^5 -
348 a^3 b^4 c^5 - 224 a^4 b^4 c^5 - 24 a^5 b^4 c^5 +
528 a b^6 c^5 + 1002 a^2 b^6 c^5 + 192 a^3 b^6 c^5 +
204 a^4 b^6 c^5 - 138 a^5 b^6 c^5 + 100 a^3 b^8 c^5 +
683 a^4 b^8 c^5 + 36 a^5 b^8 c^5 - 48 a b^3 c^6 +
32 a^3 b^3 c^6 - 324 a b^5 c^6 - 240 a^2 b^5 c^6 +
160 a^3 b^5 c^6 + 4 a^4 b^5 c^6 + 8 a^5 b^5 c^6 -
50 a^3 b^7 c^6 - 325 a^4 b^7 c^6 - 2 a^5 b^7 c^6 +
32 a b^4 c^7 - 16 a^3 b^4 c^7 + 4 a^3 b^6 c^7 +
48 a^4 b^6 c^7 + 4 I a b^2 Sqrt[-4 - b^2 + 4 b c] -
4 I a^5 b^2 Sqrt[-4 - b^2 + 4 b c] +
11 I a b^4 Sqrt[-4 - b^2 + 4 b c] +
56 I a^3 b^4 Sqrt[-4 - b^2 + 4 b c] -
39 I a^5 b^4 Sqrt[-4 - b^2 + 4 b c] +
I a b^6 Sqrt[-4 - b^2 + 4 b c] +
112 I a^3 b^6 Sqrt[-4 - b^2 + 4 b c] -
50 I a^5 b^6 Sqrt[-4 - b^2 + 4 b c] -
I a b^8 Sqrt[-4 - b^2 + 4 b c] +
56 I a^3 b^8 Sqrt[-4 - b^2 + 4 b c] -
13 I a^5 b^8 Sqrt[-4 - b^2 + 4 b c] +
8 I a^3 b^10 Sqrt[-4 - b^2 + 4 b c] +
3 I a^5 b^10 Sqrt[-4 - b^2 + 4 b c] +
I a^5 b^12 Sqrt[-4 - b^2 + 4 b c] -
12 I b^3 c Sqrt[-4 - b^2 + 4 b c] -
32 I a b^3 c Sqrt[-4 - b^2 + 4 b c] -
48 I a^2 b^3 c Sqrt[-4 - b^2 + 4 b c] -
64 I a^3 b^3 c Sqrt[-4 - b^2 + 4 b c] +
44 I a^4 b^3 c Sqrt[-4 - b^2 + 4 b c] +
64 I a^5 b^3 c Sqrt[-4 - b^2 + 4 b c] -
10 I b^5 c Sqrt[-4 - b^2 + 4 b c] -
16 I a b^5 c Sqrt[-4 - b^2 + 4 b c] -
134 I a^2 b^5 c Sqrt[-4 - b^2 + 4 b c] -
456 I a^3 b^5 c Sqrt[-4 - b^2 + 4 b c] +
76 I a^4 b^5 c Sqrt[-4 - b^2 + 4 b c] +
224 I a^5 b^5 c Sqrt[-4 - b^2 + 4 b c] -
I b^7 c Sqrt[-4 - b^2 + 4 b c] +
6 I a b^7 c Sqrt[-4 - b^2 + 4 b c] -
72 I a^2 b^7 c Sqrt[-4 - b^2 + 4 b c] -
416 I a^3 b^7 c Sqrt[-4 - b^2 + 4 b c] +
2 I a^4 b^7 c Sqrt[-4 - b^2 + 4 b c] +
132 I a^5 b^7 c Sqrt[-4 - b^2 + 4 b c] -
10 I a^2 b^9 c Sqrt[-4 - b^2 + 4 b c] -
88 I a^3 b^9 c Sqrt[-4 - b^2 + 4 b c] -
22 I a^4 b^9 c Sqrt[-4 - b^2 + 4 b c] -
8 I a^5 b^9 c Sqrt[-4 - b^2 + 4 b c] -
5 I a^4 b^11 c Sqrt[-4 - b^2 + 4 b c] -
10 I a^5 b^11 c Sqrt[-4 - b^2 + 4 b c] +
12 I b^2 c^2 Sqrt[-4 - b^2 + 4 b c] +
20 I a b^2 c^2 Sqrt[-4 - b^2 + 4 b c] +
24 I a^2 b^2 c^2 Sqrt[-4 - b^2 + 4 b c] -
8 I a^3 b^2 c^2 Sqrt[-4 - b^2 + 4 b c] -
36 I a^4 b^2 c^2 Sqrt[-4 - b^2 + 4 b c] -
12 I a^5 b^2 c^2 Sqrt[-4 - b^2 + 4 b c] +
49 I b^4 c^2 Sqrt[-4 - b^2 + 4 b c] +
99 I a b^4 c^2 Sqrt[-4 - b^2 + 4 b c] +
426 I a^2 b^4 c^2 Sqrt[-4 - b^2 + 4 b c] +
386 I a^3 b^4 c^2 Sqrt[-4 - b^2 + 4 b c] -
279 I a^4 b^4 c^2 Sqrt[-4 - b^2 + 4 b c] -
237 I a^5 b^4 c^2 Sqrt[-4 - b^2 + 4 b c] +
11 I b^6 c^2 Sqrt[-4 - b^2 + 4 b c] +
32 I a b^6 c^2 Sqrt[-4 - b^2 + 4 b c] +
482 I a^2 b^6 c^2 Sqrt[-4 - b^2 + 4 b c] +
956 I a^3 b^6 c^2 Sqrt[-4 - b^2 + 4 b c] -
122 I a^4 b^6 c^2 Sqrt[-4 - b^2 + 4 b c] -
360 I a^5 b^6 c^2 Sqrt[-4 - b^2 + 4 b c] +
6 I a b^8 c^2 Sqrt[-4 - b^2 + 4 b c] +
106 I a^2 b^8 c^2 Sqrt[-4 - b^2 + 4 b c] +
354 I a^3 b^8 c^2 Sqrt[-4 - b^2 + 4 b c] +
133 I a^4 b^8 c^2 Sqrt[-4 - b^2 + 4 b c] -
39 I a^5 b^8 c^2 Sqrt[-4 - b^2 + 4 b c] +
2 I a^3 b^10 c^2 Sqrt[-4 - b^2 + 4 b c] +
51 I a^4 b^10 c^2 Sqrt[-4 - b^2 + 4 b c] +
36 I a^5 b^10 c^2 Sqrt[-4 - b^2 + 4 b c] -
48 I b^3 c^3 Sqrt[-4 - b^2 + 4 b c] -
144 I a b^3 c^3 Sqrt[-4 - b^2 + 4 b c] -
240 I a^2 b^3 c^3 Sqrt[-4 - b^2 + 4 b c] +
32 I a^3 b^3 c^3 Sqrt[-4 - b^2 + 4 b c] +
192 I a^4 b^3 c^3 Sqrt[-4 - b^2 + 4 b c] +
48 I a^5 b^3 c^3 Sqrt[-4 - b^2 + 4 b c] -
34 I b^5 c^3 Sqrt[-4 - b^2 + 4 b c] -
236 I a b^5 c^3 Sqrt[-4 - b^2 + 4 b c] -
936 I a^2 b^5 c^3 Sqrt[-4 - b^2 + 4 b c] -
632 I a^3 b^5 c^3 Sqrt[-4 - b^2 + 4 b c] +
366 I a^4 b^5 c^3 Sqrt[-4 - b^2 + 4 b c] +
308 I a^5 b^5 c^3 Sqrt[-4 - b^2 + 4 b c] -
60 I a b^7 c^3 Sqrt[-4 - b^2 + 4 b c] -
390 I a^2 b^7 c^3 Sqrt[-4 - b^2 + 4 b c] -
600 I a^3 b^7 c^3 Sqrt[-4 - b^2 + 4 b c] -
236 I a^4 b^7 c^3 Sqrt[-4 - b^2 + 4 b c] +
144 I a^5 b^7 c^3 Sqrt[-4 - b^2 + 4 b c] -
16 I a^3 b^9 c^3 Sqrt[-4 - b^2 + 4 b c] -
192 I a^4 b^9 c^3 Sqrt[-4 - b^2 + 4 b c] -
56 I a^5 b^9 c^3 Sqrt[-4 - b^2 + 4 b c] +
4 I b^2 c^4 Sqrt[-4 - b^2 + 4 b c] +
24 I a b^2 c^4 Sqrt[-4 - b^2 + 4 b c] +
8 I a^2 b^2 c^4 Sqrt[-4 - b^2 + 4 b c] -
24 I a^3 b^2 c^4 Sqrt[-4 - b^2 + 4 b c] -
12 I a^4 b^2 c^4 Sqrt[-4 - b^2 + 4 b c] +
31 I b^4 c^4 Sqrt[-4 - b^2 + 4 b c] +
346 I a b^4 c^4 Sqrt[-4 - b^2 + 4 b c] +
510 I a^2 b^4 c^4 Sqrt[-4 - b^2 + 4 b c] -
82 I a^3 b^4 c^4 Sqrt[-4 - b^2 + 4 b c] -
245 I a^4 b^4 c^4 Sqrt[-4 - b^2 + 4 b c] -
60 I a^5 b^4 c^4 Sqrt[-4 - b^2 + 4 b c] +
196 I a b^6 c^4 Sqrt[-4 - b^2 + 4 b c] +
580 I a^2 b^6 c^4 Sqrt[-4 - b^2 + 4 b c] +
332 I a^3 b^6 c^4 Sqrt[-4 - b^2 + 4 b c] +
110 I a^4 b^6 c^4 Sqrt[-4 - b^2 + 4 b c] -
125 I a^5 b^6 c^4 Sqrt[-4 - b^2 + 4 b c] +
42 I a^3 b^8 c^4 Sqrt[-4 - b^2 + 4 b c] +
329 I a^4 b^8 c^4 Sqrt[-4 - b^2 + 4 b c] +
35 I a^5 b^8 c^4 Sqrt[-4 - b^2 + 4 b c] -
4 I b^3 c^5 Sqrt[-4 - b^2 + 4 b c] -
96 I a b^3 c^5 Sqrt[-4 - b^2 + 4 b c] -
32 I a^2 b^3 c^5 Sqrt[-4 - b^2 + 4 b c] +
64 I a^3 b^3 c^5 Sqrt[-4 - b^2 + 4 b c] +
20 I a^4 b^3 c^5 Sqrt[-4 - b^2 + 4 b c] -
232 I a b^5 c^5 Sqrt[-4 - b^2 + 4 b c] -
298 I a^2 b^5 c^5 Sqrt[-4 - b^2 + 4 b c] +
56 I a^3 b^5 c^5 Sqrt[-4 - b^2 + 4 b c] +
14 I a^4 b^5 c^5 Sqrt[-4 - b^2 + 4 b c] +
24 I a^5 b^5 c^5 Sqrt[-4 - b^2 + 4 b c] -
40 I a^3 b^7 c^5 Sqrt[-4 - b^2 + 4 b c] -
255 I a^4 b^7 c^5 Sqrt[-4 - b^2 + 4 b c] -
6 I a^5 b^7 c^5 Sqrt[-4 - b^2 + 4 b c] +
72 I a b^4 c^6 Sqrt[-4 - b^2 + 4 b c] +
24 I a^2 b^4 c^6 Sqrt[-4 - b^2 + 4 b c] -
40 I a^3 b^4 c^6 Sqrt[-4 - b^2 + 4 b c] -
4 I a^4 b^4 c^6 Sqrt[-4 - b^2 + 4 b c] +
10 I a^3 b^6 c^6 Sqrt[-4 - b^2 + 4 b c] +
75 I a^4 b^6 c^6 Sqrt[-4 - b^2 + 4 b c] -
4 I a^4 b^5 c^7 Sqrt[-4 - b^2 + 4 b c]))/((4 + b^2 -
4 b c) Sqrt[-4 - b^2 +

4 b c] (2 I b - Sqrt[-4 - b^2 - 4 b c] +
Sqrt[-4 - b^2 + 4 b c])^3 (2 I b + Sqrt[-4 - b^2 - 4 b c] +
Sqrt[-4 - b^2 + 4 b c])^3 (-2 I + I a b +
a Sqrt[-4 - b^2 + 4 b c])^2 (2 I + I a b +
a Sqrt[-4 - b^2 + 4 b c])^2 (I a b - 2 I c +
a Sqrt[-4 - b^2 + 4 b c])^2)) (b - 2 d + Sqrt[
4 + b^2 - 4 b d]))/((2 + b^2 + 2 d^2 -
2 d Sqrt[4 + b^2 - 4 b d] +
b (-4 d + Sqrt[4 + b^2 - 4 b d])) \[Pi]^2), c > 0 && a > 0]
*)


Edit

For b>1  you also get the right analytical (complex) result.

Separate $$x$$-independent prefactor:

u = (b (b - 2 d + Sqrt[4 + b^2 - 4 b d]))/((2 + b^2 + 2 d (d - Sqrt[4 + b^2 - 4 b d]) + b (-4 d + Sqrt[4 + b^2 - 4 b d])) (\[Pi]^3) );
y = ((c + I x) x (-I + a x (2 c + I a x)) (b^2 (c + I x)^2 + 3 (1 + x^2)^2))/((c + I a x) (1 + a^2 x^2) (-b^2 (c + I x)^2 + (1 + x^2)^2)^2);
FullSimplify[expr - u y]
(* 0 *)


Use an external package

<< Rubi
Assuming[c > -1 && c < 1 && d > 1 && a >= 0, yy = Int[y, x]];
res = Limit[yy /. {a -> 2, b -> 0.65, c -> 0.25, d -> 1.7}, x -> Infinity]
- Limit[yy /. {a -> 2, b -> 0.65, c -> 0.25, d -> 1.7}, x -> 0];
r = 2 Re[u res] /. {a -> 2, b -> 0.65, c -> 0.25, d -> 1.7}
(* -0.00340441 *)


This is the same result as NIntegrate gives. One can also plot expr and the result of integration to make sure they are continuous.

The integrand, let us call it $$f(x)$$ fulfills $$f(-x) = f^*(x).$$

In order to get a meaningful result I further assumed that the improper integral is defined in symmetric limits, i.e., $$I=\lim_{a\rightarrow \infty}\int_{-a}^{a}f(x)dx=\Re\left(\lim_{a\rightarrow \infty}\int_{0}^{a}f(x)dx\right).$$

Disclaimer

• Since the original question presents no further details, my answer only focuses on the integral in this narrow sense.

• I also do not attempt to answer the original question "Why?".

• The intention of this post is to demonstrate that under the additional assumption one can get the numerical value by an analytical approach.

• First, the limits of the integration are {x, -Infinity, Infinity}, not {x,0, Infinity}. Second, can you kindly explain Re[u*res]? TIA. Jul 28, 2021 at 17:56
• @user64494 see edits Jul 28, 2021 at 18:34
• +1. A good work. Jul 29, 2021 at 4:24
• @yarchik thank you for the amazing answer, however i need an analytic expression for the definite integral. Does your result work for any values of a,b,c,d in my original assumption? Is your solution continuous in all cases? Also, why does mathematica give the wrong result? I thought that could happen only when you compute a definite integral with the fundamental theorem of calculus with discontinuous function Jul 29, 2021 at 8:04
• @Andreas The answer is too long to paste here. Can you run the commands above? The analytical result is given by 2 Re[u res] Jul 29, 2021 at 8:16

Too long for a comment. The command

result1 = Integrate[expr, {x, -Infinity, Infinity}, Assumptions -> c > -1 && c < 1 && d > 1 &&
a >= 0, GenerateConditions -> True]


performs a huge output under the conditions Im[Sqrt[-4 - b^2 + 4 b c]] < Re[b] && Re[b] < Im[Sqrt[-4 - b^2 - 4 b c]] && Im[Sqrt[-4 - b^2 - 4 b c]] + Re[b] < 0 && a > 0 && c > 0. Now

Im[Sqrt[-4 - b^2 + 4 b c]] < Re[b] &&  Re[b] < Im[Sqrt[-4 - b^2 - 4 b c]] &&
Im[Sqrt[-4 - b^2 - 4 b c]] + Re[b] < 0 && a > 0 && c > 0 /. {a -> 2,  b -> 0.65, c -> 0.25}


False