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Consider the following function in which * (super star) stands for complex conjugate (I have used "Quantum package" which recognizes *):

f=E^((-0.1 - 1.67033 I) t + (0.1 - 
 1.67033 I) u) (((0.5 - 
     0.329276 I) E^((0. + 3.34066 I) t) + (0.5 + 
     0.329276 I) E^((0. + 3.34066 I) u)) SuperStar[((0.5 - 
     0.329276 I) E^((-0.1 + 1.67033 I) t + (0.1 - 
       1.67033 I) u) + (0.5 + 
     0.329276 I) E^((-0.1 - 1.67033 I) t + (0.1 + 
       1.67033 I) u))] + (-0.598684 E^((0. + 3.34066 I) t) + 
  0.598684 E^((0. + 
      3.34066 I) u)) SuperStar[(-0.598684 E^((-0.1 + 
       1.67033 I) t + (0.1 - 1.67033 I) u) + 
  0.598684 E^((-0.1 - 1.67033 I) t + (0.1 + 1.67033 I) u))])

I want to integrate over this function with the assumption that t and u are real parameters. I have tried the followings, but none of them worked:

Integrate[f , {u, 0, t}, 
Assumptions -> Element[u | t, Reals]]

Integrate[f , {u, 0, t}, 
Assumptions -> {t \[Element] Reals, u \[Element] Reals}]

Integrate[f, {u, 0, t}, 
Assumptions -> t \[Element] Reals && u \[Element] Reals]
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1 Answer 1

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Changing SuperStar to Conjugate and FullSimplifying f:

f = E^((-0.1 - 1.67033 I) t + (0.1 - 
      1.67033 I) u) (((0.5 - 
        0.329276 I) E^((0. + 3.34066 I) t) + (0.5 + 
        0.329276 I) E^((0. + 3.34066 I) u)) SuperStar[((0.5 - 
         0.329276 I) E^((-0.1 + 1.67033 I) t + (0.1 - 
             1.67033 I) u) + (0.5 + 
         0.329276 I) E^((-0.1 - 1.67033 I) t + (0.1 + 
             1.67033 I) u))] + (-0.598684 E^((0. + 
           3.34066 I) t) + 
     0.598684 E^((0. + 
           3.34066 I) u)) SuperStar[(-0.598684 E^((-0.1 + 
             1.67033 I) t + (0.1 - 1.67033 I) u) + 
      0.598684 E^((-0.1 - 1.67033 I) t + (0.1 + 
             1.67033 I) u))]) /. SuperStar -> Conjugate // 
FullSimplify

Integrate[f, {u, 0, t}, Assumptions -> {t ∈ Reals,u ∈ Reals}]//TeXForm

$-(7.16845\, +\text{1.1102230246251565$\grave{ }$*${}^{\wedge}$-15} i) e^{-0.2 t}-(0.0943419\, +0.070559 i) e^{(-0.2-3.34066 i) t}-(0.0943419\, -0.070559 i) e^{(-0.2+3.34066 i) t}+(7.35714\, +\text{1.1102230246251565$\grave{ }$*${}^{\wedge}$-15} i)$

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