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

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

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

I know the exact result:

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

but I would like to prove it.

Thanks in advance to the samaritan experts...

V.Astanoff

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