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Getting Integrate to produce the simple analytic result

Integrate[Sin[1.5 M+a] Cos[1.5 M+b] Sin[1.5 M+b] , {M, 0, 4 Pi}]  produces -1.22465 x 10^{-16}Cos[a-2.b]+1.22465 x 10^{-16}Cos[a+2.b]+0. Sin[a-2b] + 0.Sin[a+2.b]. 

But the integral is rigorously zero with rather trivial trigonometric integrations.  One could interpret  the result of Integrate as zero, but why does Mathematica not give the simple answser?  Is there a way to use Integrate on this product  to yield the rigporous analytic result?  Simplify [%] does not eliminate the terms with zero factor.
POSTED BY: Stanton Peale
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Oh, I forgot:
Sin[SetPrecision[1.5, Infinity] M + a] Cos[
   SetPrecision[1.5, Infinity] M + b] Sin[
   SetPrecision[1.5 , Infinity] M + b], {M, 0, 4 Pi}]
would also work...
POSTED BY: Marco Thiel
Dear Stanton,

please try
Integrate[Sin[3/2 M + a] Cos[3/2 M + b] Sin[3/2 M + b], {M, 0, 4 Pi}]

Mathematica assume that 1.5 etc in your inpute are finite precision numbers. 

POSTED BY: Marco Thiel
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