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Try it again, Mathematica! Was: Can't integrate sqrt(a+b*cos(t)+c*cos(2t))

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  • Subject: [mg90810] Try it again, Mathematica! Was: Can't integrate sqrt(a+b*cos(t)+c*cos(2t))
  • From: Alois Steindl <Alois.Steindl at tuwien.ac.at>
  • Date: Fri, 25 Jul 2008 06:13:10 -0400 (EDT)
  • Organization: Inst. f. Mechanics II, TU Vienna
  • References: <g6710s$sb6$1@smc.vnet.net>

Hello, 
I played around a little bit and found the following interesting
behaviour:

In[1]:= Integrate[Sqrt[5 - 4*Cos[t] + Cos[2*t]], {t, 0, Pi}]
Out[1]:= \!\(
\*SubsuperscriptBox[\(\[Integral]\), \(0\), \(\[Pi]\)]\(
SqrtBox[\(5 - 4\ Cos[t] + Cos[2\ t]\)] \[DifferentialD]t\)\)

In[2]:= Integrate[Sqrt[5 - 4*Cos[t] + Cos[2*t]], {t, 0, Pi}]
Out[2]:= 1/5 Sqrt[
 2 + 4 I] (-5 I EllipticE[-(3/5) - (4 I)/5] + (2 + I) Sqrt[5]
     EllipticE[-(3/5) + (4 I)/5] - (12 - 4 I) EllipticK[-(3/5) - (
      4 I)/5] + (6 - 2 I) Sqrt[5] EllipticK[8/5 - (4 I)/5] + 
   4 I Sqrt[5]
     EllipticPi[
     1/5 + (2 I)/5, -(3/5) + (4 I)/5] + (8 + 4 I) EllipticPi[
     1 - 2 I, -(3/5) - (4 I)/5])

So it seems, it helps to ask Mathematica twice. 

Best wishes, 
Alois


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