Date: Feb 2, 2013 1:15 AM
Author: Alexey
Subject: Integrate bug in v 9.0.0

In version 9.0.0 the following integral is reported as divergent:

In[71]:= Integrate[x^2/(x^2 - x0^2), {x, x1, x2},
Assumptions -> x0 > x2 > x1 >= 0]

During evaluation of In[71]:= Integrate::idiv: Integral of x^2/(x^2-x0^2)
does not converge on {x1,x2}. >>

Out[71]= Integrate[x^2/(x^2 - x0^2), {x, x1, x2},
Assumptions -> x0 > x2 > x1 >= 0]

Versions 8.0.4 and 5.2 give equivalent expressions:

In[2]:= Integrate[x^2/(x^2 - x0^2), {x, x1, x2},
Assumptions -> x0 > x2 > x1 >= 0]

Out[2]= -x1 + x2 + x0 ArcTanh[(x0 (x1 - x2))/(x0^2 - x1 x2)]

In[9]:=
Integrate[x^2/(x^2-x0^2),{x,x1,x2},Assumptions->x0>x2>x1>=0]
Out[9]=
-x1+x2+1/2 x0 (Log[x0+x1]+Log[x0-x2]-Log[(x0-x1) (x0+x2)])

In[8]:= FullSimplify[-x1 + x2 +
1/2 x0 (Log[x0 + x1] + Log[x0 - x2] -
Log[(x0 - x1) (x0 + x2)]) == -x1 + x2 +
x0 ArcTanh[(x0 (x1 - x2))/(x0^2 - x1 x2)],
Assumptions -> x0 > x2 > x1 >= 0]

Out[8]= True

Alexey