# Strange behavior of Integrate

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?

• It's even more unfortunate that Integrate[Abs[Sin[b + x]], {x, 0, 2 Pi}, Assumptions -> {b \[Element] Reals}] returns 0 for me. May 4, 2013 at 14:38
• @b.gatessucks. I tried your way, it returns 0, too. Is this a BUG? May 4, 2013 at 14:48
• Yes, I think so. May 4, 2013 at 14:56
• It does seem to be one. I've tagged appropriately. (For fun, see what happens if you use assumptions like b > 0 or b < 0, or some finite interval.) May 4, 2013 at 15:17

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

(*
4
*)

In:= \$Version

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

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

Out= {81.073720, 4}

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

(*4*)

Out= {1.404009, 4}