Question: BUG in numerical integration? with SphericalY's

Hi, 

     I'm computing some difficult integrals, involving Spherical Harmonics. I've proved that the answer must be a real number, and yet Maple returns a complex number. Also Mathematica returns purely real numbers, so I wonder if there is a bug in Maple. I found if I break the integrand into real and imaginary parts, and add them I get the right answer...usually.  Anyway's here's the code and the proof, I suspect they could be a bug somewhere, since there was a bug in plotting SphericalY's found earlier ... Unfortunatly the previous trick, Y:=(l,m,theta, phi)->simplify(convert( SphericalY(l,m,theta, phi),elementary)): does not fix this issue....

restart;
Digits:=3:
chi:=Pi/2-theta:
Vsi:=-(1/sqrt( 1+epsilon - cos(chi)*cos(phi-Pi/3)) + 1/sqrt( 1 +epsilon- cos(chi)*cos(phi-Pi))+ 1/sqrt(1+epsilon-cos(chi)*cos(phi+Pi/3))):
epsilon:=10^(-7):

SP1:=SphericalY(3,-3,theta,phi)*conjugate(SphericalY(0,0,theta,phi))*sin(theta):         
convert(SP1,LegendreP):         
SP2:=simplify(%);
SP3:=simplify(%) assuming real;        
SPr,SPi:=op(evalc([Re,Im](SP2))):

#Maple won't do this integral
#Ans:=evalf(Int(Vsi*(SP1), [theta=0..Pi,phi=0..2*Pi]));

#Wrong answer   
Ans:=evalf(Int(Vsi*(SP2), [theta=0..Pi,phi=0..2*Pi])); 
#Wrong answer   
Ans:=evalf(Int(Vsi*(SP3), [theta=0..Pi,phi=0..2*Pi])); 

#Correct Answer
Ans:=evalf(Int(Vsi*(SPr+SPi), [theta=0..Pi,phi=0..2*Pi])); 

and the proof...

     Also, maybe it's just Google Chrome, but I can't add multiple tags to my post :p even when I try seperating them by a space. Maybe a small bug w MaplePrimes/Chrome.