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

  • To: mathgroup at smc.vnet.net
  • Subject: [mg90784] Re: Can't integrate sqrt(a+b*cos(t)+c*cos(2t))
  • From: "Kevin J. McCann" <Kevin.McCann at umbc.edu>
  • Date: Thu, 24 Jul 2008 04:52:22 -0400 (EDT)
  • Organization: University System of Maryland
  • References: <g6710s$sb6$1@smc.vnet.net>

I actually get:

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])

which  evaluates to your numerical answer below,

Valeri Astanoff wrote:
> 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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