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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?

Improve this question
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3
  • $\begingroup$ Your numerical result is correct. Use FullSimplify, remove the constant pre-factor and use the Rubi package in order to integrate analytically. $\endgroup$
    – yarchik
    Jul 28, 2021 at 16:36
  • $\begingroup$ @yarchik: Can you elaborate your directions? In particular, can you ground "Your numerical result is correct"? TIA. $\endgroup$
    – user64494
    Jul 28, 2021 at 16:50
  • $\begingroup$ @user64494 See below, but it takes a while $\endgroup$
    – yarchik
    Jul 28, 2021 at 17:50

3 Answers 3

Reset to default
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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.

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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.

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11
  • $\begingroup$ First, the limits of the integration are {x, -Infinity, Infinity}, not {x,0, Infinity}. Second, can you kindly explain Re[u*res]? TIA. $\endgroup$
    – user64494
    Jul 28, 2021 at 17:56
  • $\begingroup$ @user64494 see edits $\endgroup$
    – yarchik
    Jul 28, 2021 at 18:34
  • $\begingroup$ +1. A good work. $\endgroup$
    – user64494
    Jul 29, 2021 at 4:24
  • $\begingroup$ @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 $\endgroup$
    – Andreas
    Jul 29, 2021 at 8:04
  • $\begingroup$ @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] $\endgroup$
    – yarchik
    Jul 29, 2021 at 8:16
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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

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