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a bug in Integrate

• To: mathgroup at smc.vnet.net
• Subject: [mg122247] a bug in Integrate
• From: dimitris <dimmechan at yahoo.com>
• Date: Sat, 22 Oct 2011 06:07:15 -0400 (EDT)
• Delivered-to: l-mathgroup@mail-archive0.wolfram.com

The following has a connection with the recent post of mine called
Simplification.

http://groups.google.com/group/comp.soft-sys.math.mathematica/browse_thread/thread/b6e8fc7f70a0f20f#

Let

int = 1/((x^2 + x + 1) Sqrt[x^2 - x + 1]);

Then Mathematica's result for the definite integral in the range
[-1,0] is

In[74]:= Integrate[int, {x, -1, 0}]

Out[74]= -((1/(4* Sqrt[3]))*(I*(Sqrt[
         1 + I*Sqrt[3]]*(-2*I* ArcTan[(6 + 3*I*Sqrt[3])/(3*I - Sqrt[3]
+
               3*Sqrt[1 - I*Sqrt[3]])] +
          2*ArcTanh[(3*Sqrt[3])/(3*I + Sqrt[3] -
               Sqrt[3 - 3*I*Sqrt[3]])] -
          Log[11*I + 4*Sqrt[3] + 10*I*Sqrt[1 - I*Sqrt[3]]] +
          Log[3*(13*I + 4*Sqrt[3] + 6*I*Sqrt[3 - 3*I*Sqrt[3]])]) +
       Sqrt[1 - I*Sqrt[3]]*(2*ArcTanh[(3*(2*I + Sqrt[3]))/(3*I +
Sqrt[3] - 3*Sqrt[1 + I*Sqrt[3]])] +
         2*ArcTanh[(3*Sqrt[3])/(3*I - Sqrt[3] + Sqrt[3 +
3*I*Sqrt[3]])] +
          Log[(1/16)*(-11*I + 4*Sqrt[3] - 10*I*Sqrt[1 + I*Sqrt[3]])]
-
          Log[(3/16)*(-13*I + 4*Sqrt[3] - 6*I*Sqrt[3 +
3*I*Sqrt[3]])]))))

which is incorect. Indeed:

In[75]:= N[%]//Chop

Out[75]= -1.294232744953466

In[76]:= NIntegrate[int, {x, -1, 0}]

Out[76]= 0.9272087241257184

I think the reason is that Mathematica firsts evaluates an
antiderivative:

In[78]:= intMath = Integrate[int, x]
Out[78]= (1/(4*Sqrt[3]))*(-2*Sqrt[1 + I*Sqrt[3]]*
    ArcTan[(3*(1 - x + x^2)*(I*Sqrt[3] - x + x^2))/
            (3*I + Sqrt[3] - 2*Sqrt[3]*x^4 -
        Sqrt[3 - 3*I*Sqrt[3]]*Sqrt[1 - x + x^2] +
        2*x^3*(3*I + Sqrt[3 - 3*I*Sqrt[3]]*Sqrt[1 - x + x^2]) +
        x*(3*I - 3*Sqrt[3] + Sqrt[3 - 3*I*Sqrt[3]]*Sqrt[1 - x + x^2])
+
        x^2*(-3*I + Sqrt[3] +
           Sqrt[3 - 3*I*Sqrt[3]]*Sqrt[1 - x + x^2]))] -
      2*Sqrt[1 - I*Sqrt[3]]*
    ArcTan[(3*(I*Sqrt[3] + x - x^2)*(1 - x + x^2))/(3*I - Sqrt[3] +
        2*Sqrt[3]*x^4 +
               Sqrt[3 + 3*I*Sqrt[3]]*Sqrt[1 - x + x^2] +
        x^3*(6*I - 2*Sqrt[3 + 3*I*Sqrt[3]]*Sqrt[1 - x + x^2])
+
        x*(3*I + 3*Sqrt[3] -
           Sqrt[3 + 3*I*Sqrt[3]]*Sqrt[1 - x + x^2]) -
        x^2*(3*I + Sqrt[3] +
           Sqrt[3 + 3*I*Sqrt[3]]*Sqrt[1 - x + x^2]))] +
      I*((Sqrt[1 - I*Sqrt[3]] - Sqrt[1 + I*Sqrt[3]])*
       Log[16*(1 + x + x^2)^2] +
      Sqrt[1 + I*Sqrt[3]]*
       Log[(1 + x + x^2)*(11*I + 4*Sqrt[3] + (11*I + 4*Sqrt[3])*x^2 +
           10*I*Sqrt[1 - I*Sqrt[3]]*
                      Sqrt[1 - x + x^2] -
           x*(17*I + 4*Sqrt[3] +
              8*I*Sqrt[1 - I*Sqrt[3]]*Sqrt[1 - x + x^2]))] -
      Sqrt[1 - I*Sqrt[3]]*
       Log[(1 + x + x^2)*(-11*I +
           4*Sqrt[3] + (-11*I + 4*Sqrt[3])*x^2 -
           10*I*Sqrt[1 + I*Sqrt[3]]*
                      Sqrt[1 - x + x^2] +
           x*(17*I - 4*Sqrt[3] +
              8*I*Sqrt[1 + I*Sqrt[3]]*Sqrt[1 - x + x^2]))]))

and then simply does (intMath /. x -> 0) - (intMath /. x -> -1).

In[84]:= N[(intMath /. x -> 0) - (intMath /. x -> -1)]//Chop

Out[84]= -1.2942327449534659

However the evaluated antiderivative possesses a jump singularity at
near x = -0.725
and Mathematica fails unfortunately to take it into acount.

In[85]:= Plot[intMath, {x, -3, 3}]

Nevertheless, there are and some good news. The antiderivative is
correct in the whole complex plane.

In[89]:= FullSimplify[D[intMath, x] - int]

Out[89]= 0

If the integration range does not include the jump, Mathematica will
give correct result. E.g.

In[13]:= Integrate[int, {x, 0, Infinity}] (*after several minutes*)

Out[13]= (1/(4* Sqrt[3]))*(I*(Sqrt[
       1 + I*Sqrt[3]]*(2*ArcTanh[(3*Sqrt[3])/(3*I + Sqrt[3] -
             Sqrt[3 - 3*I*Sqrt[3]])] +  Log[11*I + 4*Sqrt[3] -
8*I*Sqrt[1 - I*Sqrt[3]]] -
        Log[11*I + 4*Sqrt[3] + 10*I*Sqrt[1 - I*Sqrt[3]]] -
        Log[(-2 - I*Sqrt[3] + 2*Sqrt[1 - I*Sqrt[3]])/(-1 +
            Sqrt[1 - I*Sqrt[3]])] +
        Log[(-2 + I*Sqrt[3] + 2*Sqrt[1 - I*Sqrt[3]])/(-1 +
            Sqrt[1 - I*Sqrt[3]])]) + Sqrt[1 - I*Sqrt[3]]*(2*
         ArcTanh[(3*Sqrt[3])/(3*I - Sqrt[3] +
             Sqrt[3 + 3*I*Sqrt[3]])] -
              Log[-11*I + 4*Sqrt[3] + 8*I*Sqrt[1 + I*Sqrt[3]]] +
        Log[-11*I + 4*Sqrt[3] - 10*I*Sqrt[1 + I*Sqrt[3]]] -
        Log[(-2 - I*Sqrt[3] + 2*Sqrt[1 + I*Sqrt[3]])/(-1 +
            Sqrt[1 + I*Sqrt[3]])] +  Log[(-2 + I*Sqrt[3] + 2*Sqrt[1 +
I*Sqrt[3]])/(-1 +
            Sqrt[1 + I*Sqrt[3]])])))

In[14]:= N[%]

Out[14]= 0.9358813101035703 + 0.*I

In[15]:= NIntegrate[int, {x, 0, Infinity}]

Out[15]= 0.9358813101035696

At first glance integrand 1/((x^2 + x + 1) Sqrt[x^2 - x + 1]) may look
simple.
This is far from true.

Dimitris