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I would like to know the underlying reason for different outcomes for the two integration operation below. One of them includes a few assumptions, otherwise both have the same integrand:

Clear[a, b, c, d]
Integrate[(1/(a q^2 + b q Cos[θ] - c) - 1/(a q^2 + b q Cos[θ] + c)), {θ, 0, 2 π}, 
  Assumptions -> {a, c, b} ∈ Reals]
ConditionalExpression[-((2 Log[-((c - b q + a q^2)/Sqrt[-c^2 + b^2 q^2 - 2 a c q^2 - a^2 q^4])])/ Sqrt[-c^2 + b^2 q^2 - 2 a c q^2 - a^2 q^4]) + (2 Log[(c - b q + a q^2)/ Sqrt[-c^2 + b^2 q^2 - 2 a c q^2 - a^2 q^4]])/Sqrt[-c^2 + b^2 q^2 - 2 a c q^2 - a^2 q^4] + (2 (-Log[-((c + q (b - a q))/Sqrt[-c^2 + b^2 q^2 + 2 a c q^2 - a^2 q^4])] ....
Integrate[(1/(a q^2 + b q Cos[θ] - c) - 1/(a q^2 + b q Cos[θ] + c)), {θ, 0, 2 π}]
-((2 π Sqrt[1 - (2 b q)/(-c + q (b + a q))])/( c + q (b - a q))) - 
 (2 π)/((c + q (b + a q)) Sqrt[1 - (2 b q)/(c + q (b + a q))])
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Well, in the first one you tell Mathematica that a, b and c are real whereas in the second Mathematica treats them as complex as that's the default assumption. –  RunnyKine Aug 20 at 4:58
    
@RunnyKine So the second outcome should reduce to the first (more lengthy) output when a,b,c real? –  thils Aug 20 at 5:08
    
Well, the first one appears lengthy because it's a ConditionalExpression. So it's giving you various answers that correspond to different combinations of a, b and c in the Reals. –  RunnyKine Aug 20 at 5:10
    
The second outcome is applicable to more general conditions, and should reduce to the first with application of those a,b,c combinations, how do you show this? –  thils Aug 20 at 5:47
1  
Probably not as simple as it seems, running Assuming[{a, c, b} \[Element] Reals, FullSimplify[ eq1 == eq2]] does not simplify to True (where eq1 and eq2 are the respective integral outputs) –  rhermans Aug 20 at 10:32

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