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Another Integrate error

This is a bivariate normal probability density:

In[1]:= f[x_,y_,r_] = Exp[(x^2 + y^2 - 2*r*x*y)/(r^2 - 1)] / 
                      (Pi Sqrt[1 - r^2]);

It should integrate to 1:

In[2]:= Integrate[f[x,y,r],{x,-Infinity,Infinity},{y,-Infinity,Infinity}]
Out[2]= 0

In[3]:= Integrate[f[x,y,r],{x,-Infinity,Infinity},{y,-Infinity,Infinity}, 
                  Assumptions -> {Element[r,Reals], -1 < r < 1}]
Out[3]= 1

In this case I happened to know that the integal should be 1,
but what about more obscure cases? 
When can I believe a result from Integrate? 

In[4]:= $Version
Out[4]= 5.0 for Mac OS X (November 19, 2003)

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