One Integral — Two Results

Question

I compared Mathematica's default integral evaluation to what you get by computing the indefinite integral and then plugging in the limits.

(I did this because the latter approach was much faster for some integrals.)

I got two different results:

integral=-Log[-a+b+2 a x+2 a Sqrt[(-1+x) x]]+Log[-a+b-2 b x+2 b Sqrt[(-1+x) x]]

integralDefault=Assuming[a>0&&b>0,Integrate[integral,{x,0,1}]]//Timing
(*I Pi/2*)

integralInDef[x]=Assuming[a>0&&b>0,Integrate[integral,x]]//Timing;
integralDef=Assuming[a>0&&b>0,Limit[integralInDef[x][[2]],x->1]-Limit[integralInDef[x][[2]],x->0]]//Timing
(*-I (a^2-8 a b+b^2) Pi/(4 a b)*)

The first result agrees with the numeric value.

Is this a known bug?

Answer 1

I think it's useful to manipulate the integrand before integration; notice you have a negative number under square root :

simplifiedIntegrand = Simplify[ComplexExpand[integral, TargetFunctions -> {Re, Im}], Assumptions -> {a > 0, b > 0, 0 < x < 1}]
(* I (-ArcTan[-a + b + 2 a x, 2 a Sqrt[(1 - x) x]] + 
       ArcTan[-a + b - 2 b x, 2 b Sqrt[(1 - x) x]]) *)

defIntegral =  Integrate[simplifiedIntegrand, {x, 0, 1}, Assumptions -> {a > 0, b > 0}]
(* ConditionalExpression[I Pi/2, a <= b] *)

indefIntegral = Integrate[simplifiedIntegrand, x, Assumptions -> {a > 0, b > 0}] ;

defIntegral2 =  Assuming[a > 0 && b > 0, Limit[indefIntegral, x -> 1, Direction -> +1] - Limit[indefIntegral, x -> 0, Direction -> -1]]
(* (I Pi)/2 *)

As pointed out by @Simon :

Integrate[simplifiedIntegrand, {x, 0, 1}, Assumptions -> {a > b > 0}]
(* (I Pi)/2 *)

which would establish the identity integral = I Pi /2 for any positive a, b.

However

general = Integrate[simplifiedIntegrand, {x, 0, 1}, Assumptions -> {a \[Element] Reals, b \[Element] Reals}]
(* ConditionalExpression[-(1/2) I (\[Pi] - 2 ArcTan[-a - b, 0] + 2 Arg[a + b]), 
    (b < 0 && (b <= a < 0 || 0 < a <= -b)) || (b > 0 && (-b <= a < 0 || 0 < a <= b))] *)

and

Simplify[general, Assumptions -> {0 < a < b}]
(* (I Pi)/2 *)

but

Simplify[general, Assumptions -> {a > b > 0}]
(* Undefined *)