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

• To: mathgroup at smc.vnet.net
• Subject: [mg90794] Re: Can't integrate sqrt(a+b*cos(t)+c*cos(2t))
• From: "David W.Cantrell" <DWCantrell at sigmaxi.net>
• Date: Thu, 24 Jul 2008 04:54:18 -0400 (EDT)
• References: <g6710s\$sb6\$1@smc.vnet.net>

```Valeri Astanoff <astanoff at gmail.com> wrote:
> 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}]

Actually, we can use Mathematica 6 to integrate that.

In:= indef = Integrate[Sqrt[5 - 4*Cos[t] + Cos[2*t]], t]

Out= ((2/5 + (4*I)/5)*Sqrt[5 - 4*Cos[t] + Cos[2*t]]*((2 + I)*Sqrt[1 -
2*I]* EllipticE[I*ArcSinh[Sqrt[1 - 2*I]*Tan[t/2]], -(3/5) + (4*I)/5]*(1 +
Tan[t/2]^2)*Sqrt[1 + (1 - 2*I)*Tan[t/2]^2]*Sqrt[1 + (1 + 2*I)*Tan[t/2]^2] -
I*((6 - 2*I)*Sqrt[1 - 2*I]*EllipticF[I*ArcSinh[Sqrt[1 - 2*I]*Tan[t/2]],
-(3/5) + (4*I)/5]*(1 + Tan[t/2]^2)*Sqrt[1 + (1 - 2*I)*Tan[t/2]^2]*Sqrt[1 +
(1 + 2*I)*Tan[t/2]^2] - 4*Sqrt[1 - 2*I]*EllipticPi[1/5 + (2*I)/5,
I*ArcSinh[Sqrt[1 - 2*I]*Tan[t/2]], -(3/5) + (4*I)/5]*(1 +
Tan[t/2]^2)*Sqrt[1 + (1 - 2*I)*Tan[t/2]^2]*Sqrt[1 + (1 + 2*I)*Tan[t/2]^2] +
(2 + I)*(Tan[t/2] + 2*Tan[t/2]^3 + 5*Tan[t/2]^5))))/((1 + Cos[t])*Sqrt[(5 -
4*Cos[t] + Cos[2*t])/(1 + Cos[t])^2]*(1 + Tan[t/2]^2)*Sqrt[2 + 4*Tan[t/2]^2
+ 10*Tan[t/2]^4])

In:= FullSimplify[
Limit[indef, t -> Pi, Direction -> 1] - (indef /. t -> 0)]

Out= (1/5)*Sqrt[2 + 4*I]*(-5*I*EllipticE[-(3/5) - (4*I)/5] +
(2 + I)*Sqrt*EllipticE[-(3/5) + (4*I)/5] -
(12 - 4*I)*EllipticK[-(3/5) - (4*I)/5] +
(6 - 2*I)*Sqrt*EllipticK[8/5 - (4*I)/5] +
4*I*Sqrt*EllipticPi[1/5 + (2*I)/5, -(3/5) + (4*I)/5] +
(8 + 4*I)*EllipticPi[1 - 2*I, -(3/5) - (4*I)/5])

In:= N[%]

Out= 6.722879723440325 + 1.0534455252564358*^-14*I

Of course I readily agree that Out is not as nice in appearance as your
In below. (Nobody who works much with Mathematica and elliptic integrals
would be surprised by that.) Nonetheless, Out is a correct answer.

Best regards,
David W. Cantrell

> 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

```

• Prev by Date: Suggestions for selling a copy of Mathematica V6 wanted
• Next by Date: Re: Interval arithmetic bug
• Previous by thread: Re: Can't integrate sqrt(a+b*cos(t)+c*cos(2t))
• Next by thread: Re: Can't integrate sqrt(a+b*cos(t)+c*cos(2t))