Integrate: slow and wrong results

Question

Very often I struggle to understand what is happening when I try to integrate stuff in Mathematica. Generally, it deals very well with ugly symbolic integrals. But I have an example in which it deals very badly: it is slow and wrong. I have a semi-numeric calculation; a 9x9 matrix whose elements are all of the form,

$$M_{ij}=(a_{ij}+i b_{ij}) e^{(c_{ij}+i d_{ij})s},$$

with $a_{ij}, b_{ij}, c_{ij}, d_{ij}$ real numbers (numeric values). I want to compute $\int_0^\tau M ds$, which I need symbolically, i.e., $s$ is symbolic and $\tau$ too.

Notably, after performing the integral, I obtain a matrix function of $\tau$ and it must be the zero matrix whenever I set $\tau=0$. However, in my case it doesn't, I found the problem was in some particular elements of the matrix; see below one of them:

(-0.149566 + 0.149181 I) \[Epsilon]^2 + (0.0000637042 + 
    0.00011343 I) E^((-0.16522 + 
     0.661222 I) s) \[Epsilon]^2 + (0.0615629 + 
    0.00935082 I) E^((-0.16522 - 
     0.161252 I) s) \[Epsilon]^2 + (0.086871 - 
    0.141669 I) E^((-0.16522 - 
     0.13798 I) s) \[Epsilon]^2 + (0.00124649 - 
    0.0179924 I) E^((-0.16522 - 
     1.65846 I) s) \[Epsilon]^2 + (0.0000790824 + 
    0.0000814583 I) E^((-0.161286 - 
     0.860535 I) s) \[Epsilon]^2 - (0.00423634 - 
    0.00658894 I) E^((-0.161286 - 
     1.68301 I) s) \[Epsilon]^2 + (0.0139537 - 
    0.0131108 I) E^((-0.161286 - 
     1.65974 I) s) \[Epsilon]^2 - (0.0000300511 + 
    8.17237*10^-8 I) E^((-0.161286 - 
     3.18022 I) s) \[Epsilon]^2 - (0.000349407 - 
    0.00152877 I) E^((-0.00393366 + 
     1.52176 I) s) \[Epsilon]^2 - (0.00959497 - 
    0.0059272 I) E^((0.00393366 - 1.52176 I) s) \[Epsilon]^2

Using simply

Integrate[%,{s,0,\tau}]/.\tau->0

I get as a result

(-0.531902 + 0.461049 I) \[Epsilon]^2

instead of 0. I find it a bit frustrating that the computation is also slow, and we are only dealing with a bunch of exponentials.

I have found a workaround that I usually keep in mind when I deal with Integrate. It consists in finding the primitive and then evaluating the boundaries concerned

integrate[f_,s_,xi_,xf_]:=Module[{Ii,If, adv},
    adv = Assuming[assumptions,Integrate[f,s]];
    Ii =adv/.s->xi;
    If =adv/.s->xf;
    Return[If-Ii]
    ] 

Using the above, the sanity check $\tau \to0$ gives zero for the integral as expected.

I really wanted to understand what is "wrong" with Integrate[], how and why I should use it. It feels really weird that such a nice function doesn't recognize simple symbolic integrals and cannot decide the correct/most efficient way of integrating automatically. What am my missing here?

Answer 1

In version 13 on Windows 10

f= Rationalize[(-0.149566 + 0.149181 I) \[Epsilon]^2 + (0.0000637042 + 
   0.00011343 I) E^((-0.16522 + 
      0.661222 I) s) \[Epsilon]^2 + (0.0615629 + 
   0.00935082 I) E^((-0.16522 - 
      0.161252 I) s) \[Epsilon]^2 + (0.086871 - 
   0.141669 I) E^((-0.16522 - 
      0.13798 I) s) \[Epsilon]^2 + (0.00124649 - 
   0.0179924 I) E^((-0.16522 - 
      1.65846 I) s) \[Epsilon]^2 + (0.0000790824 + 
   0.0000814583 I) E^((-0.161286 - 
      0.860535 I) s) \[Epsilon]^2 - (0.00423634 - 
   0.00658894 I) E^((-0.161286 - 
      1.68301 I) s) \[Epsilon]^2 + (0.0139537 - 
   0.0131108 I) E^((-0.161286 - 
      1.65974 I) s) \[Epsilon]^2 - (0.0000300511 + 
   8.17237*10^-8 I) E^((-0.161286 - 
      3.18022 I) s) \[Epsilon]^2 - (0.000349407 - 
   0.00152877 I) E^((-0.00393366 + 
      1.52176 I) s) \[Epsilon]^2 - (0.00959497 - 
   0.0059272 I) E^((0.00393366 - 1.52176 I) s) \[Epsilon]^2, 0];
Integrate[f, {s, 0, \[Tau]}]) /. \[Tau] -> 0

0

Answer 2

To investigate what's happening during the integration try using EvaluatioMonitor or StepMonitor.