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Integration Bug

I am finding some weird bugs when Mathematica integrates
EXP*TRIG/Polynomial to infinity.

$Assumptions = {0 < t, 0 < Ï?, 0 < γ< Ω};

Ï?1 = +Ω + i γ;
Ï?2 = +Ω - i γ;
Ï?3 = -Ω + i γ;
Ï?4 = -Ω - i γ;

this integral is correct

Integrate[Ï? Cos[Ï? t] /
((Ï?-Ï?1)(Ï?-Ï?1)(Ï?-Ï?1)(Ï?-Ï?1)),{Ï?,0,Infinity}] //FullSimplify

Although the answer it gives is correct, I want answers in Ei form and
not Ci and Si

so I substituted Cos[x] for 1/2(Exp[i x]+Exp[-i x])

Integrate[Ï? (Exp[i Ï? t]+Exp[-i Ï? t])/2 /
((Ï?-Ï?1)(Ï?-Ï?1)(Ï?-Ï?1)(Ï?-Ï?1)),{Ï?,0,Infinity}] //FullSimplify

and this answer is in the form of Ei, but it is completely wrong

similar things happen with other integrals of this kind. Fortunately I
know how to do these by hand.

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