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Can't integrate sqrt(a+b*cos(t)+c*cos(2t))

  • To: mathgroup at smc.vnet.net
  • Subject: [mg90775] Can't integrate sqrt(a+b*cos(t)+c*cos(2t))
  • From: Valeri Astanoff <astanoff at gmail.com>
  • Date: Wed, 23 Jul 2008 06:27:06 -0400 (EDT)

Good day,

Neither Mathematica 6 nor anyone here can integrate this:

In[1]:= Integrate[Sqrt[5 - 4*Cos[t] + Cos[2*t]], {t, 0, Pi}]
Out[1]= Integrate[Sqrt[5 - 4*Cos[t] + Cos[2*t]], {t, 0, Pi}]

In[2]:= NIntegrate[Sqrt[5 - 4*Cos[t] + Cos[2*t]], {t, 0, Pi}]
Out[2]= 6.72288

I know the exact result:

In[3]:=  (1/5^(3/4))*(Sqrt[2]*(10*EllipticE[(1/10)*(5 - Sqrt[5])] -
    	10*EllipticK[(1/10)*(5 - Sqrt[5])] + (5 + 3*Sqrt[5])*
     	EllipticPi[(1/10)*(5 - 3*Sqrt[5]), (1/10)*(5 - Sqrt[5])]))//N
Out[3]= 6.72288

but I would like to prove it.

Thanks in advance to the samaritan experts...


V.Astanoff


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