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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