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strange Integrate[] behavior


Integrate[] seems unable to split my integrand into two integrable pieces.
Am I simply doing something wrong?


In[23]:=  integrand


            -(1/2) - x1 - x1 /2         -(1/2) + x1 - x1 /2
           E                    x1     E                    x1
Out[23]= -(-----------------------) +  -----------------------
                Sqrt[2 Pi]                   Sqrt[2 Pi]
 

In[24]:= Integrate[%, {x1, -1, 1}]

                                      2                          2
                      -(1/2) - x1 - x1 /2        -(1/2) + x1 - x1 /2
                     E                    x1    E                    x1
Out[24]= Integrate[-(-----------------------) + -----------------------, 
                           Sqrt[2 Pi]                 Sqrt[2 Pi]
 
>    {x1, -1, 1}]





There doesn't appear to be much search done by the Integrate[] function; it
returns within a second.  Yet I can integrate both terms individually:


In[25]:= InputForm[%]

Out[25]//InputForm= Integrate[-((E^(-1/2 - x1 - x1^2/2)*x1)/(2*Pi)^(1/2)) + 
     (E^(-1/2 + x1 - x1^2/2)*x1)/(2*Pi)^(1/2), {x1, -1, 1}]



In[26]:=  Integrate[-((E^(-1/2 - x1 - x1^2/2)*x1)/(2*Pi)^(1/2)), {x1, -1, 1}]

                         2             2                  1
         2 Sqrt[Pi] - 2 E  Sqrt[Pi] + E  Sqrt[2 Pi] Gamma[-, 0, 2]
                                                          2
Out[26]= ---------------------------------------------------------
                                 3/2  2
                                2    E  Pi


In[27]:= Integrate[(E^(-1/2 + x1 - x1^2/2)*x1)/(2*Pi)^(1/2), {x1, -1, 1}]

               -2        Pi
         -1 + E   + Sqrt[--] Erf[Sqrt[2]]
                         2
Out[27]= --------------------------------
                    Sqrt[2 Pi]




Why does the Integrate[] function seem unable to split the pieces?

Thanks for any help.








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