Why is the following integral not computed

 Integrate[(Exp[-t] - 1)/
 Sqrt[2 + Exp[-2 t] - Exp[-t] - 
   Exp[t (-(1/2) - I a) - Exp[t (-(1/2) + I a)]]], {t, 0, 1}]?
  • 2
    $\begingroup$ Rubi cannot do it either. Some integrals are just too hard - some even have no closed form in elementary functions - and your double exponential under a square root is probably what makes it so difficult. You will need to resort to numerics with NIntegrate, but to do that you must give a some value - it cannot remain a symbol but you could make a function: intg[a_?NumericQ]:= NIntegrate[...] $\endgroup$
    – flinty
    May 30, 2021 at 10:18
  • $\begingroup$ Thanks @flinty. Actually I need to plot the result with respect to $a$. Don't know how to proceed... $\endgroup$
    – Mike
    May 30, 2021 at 10:30
  • 2
    $\begingroup$ From comment of user @flinty, try: Plot[intg[a] // ReIm // Evaluate, {a, -2, 2}] $\endgroup$ May 30, 2021 at 10:54
  • $\begingroup$ You also could try using ParametricNDSolveValue with a as the parameter. It may be faster than NIntegrate, provided a is not too large. $\endgroup$
    – bbgodfrey
    May 30, 2021 at 12:38
  • $\begingroup$ Personal opinion: I think any similar question should be closed unless the asker can convincingly explain why they believe that Mathematica should be able to solve the integral, differential equation, or any other hard problem. $\endgroup$
    – Szabolcs
    May 30, 2021 at 13:24