Strange behavior of Integrate

Question

I'm trying to calculate the following integral using Mathematica 9.0.1.0

 a11=Integrate[Abs[Sin[b+x]],{x,0,2*\[Pi]}]

This should be a simple problem; however, it took Mathematica a very long time to obtain the answer, which is 4. I asked a colleague to do it with Maple, the answer is obtained instantly. I thought may be this has something to do with the fact that Mathematica treats everthing as Complex value, so I rewrote the programm as

a1=Integrate[Abs[Sin[b+x]],{x,0,2*\[Pi]},Assumptions->{Element[{b,x},Reals]}]

Unfortunately, it took Mathematica even more time to run this command. What's more unfortunate is that the obtained answer is wrong:

4 Abs[Sin[b]] Cot[b]

Does anybody know what's wrong here?

Answer 1

FullSimplify@ Integrate[Abs@Sin[b + x] , {x, 0, 2 Pi},   Assumptions -> {0 < b < 2 Pi}]

(*
 4
*)

Answer 2

In[77]:= $Version

Out[77]= "9.0 for Microsoft Windows (64-bit) (January 25, 2013)"

In[73]:= Timing[All = Integrate[Abs[Sin[b + x]], {x, 0, 2*\[Pi]}]]


Out[73]= {81.073720, 4}

In[75]:= Timing[
 FullSimplify@
  Integrate[Abs@Sin[b + x], {x, 0, 2 Pi}, 
   Assumptions -> {0 < b < 2 Pi}]]

(*4*)


Out[75]= {1.404009, 4}