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

  • To: mathgroup at smc.vnet.net
  • Subject: [mg90780] Re: Can't integrate sqrt(a+b*cos(t)+c*cos(2t))
  • From: Grischika at mail.ru
  • Date: Thu, 24 Jul 2008 04:51:37 -0400 (EDT)
  • References: <g6710s$sb6$1@smc.vnet.net>

On 23 =C9=C0=CC, 13:26, Valeri Astanoff <astan... at gmail.com> 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]:= =9A(1/5^(3/4))*(Sqrt[2]*(10*EllipticE[(1/10)*(5 - Sqrt[5])] -
> =9A =9A =9A =9A 10*EllipticK[(1/10)*(5 - Sqrt[5])] + (5 + 3*Sqrt[5])*
> =9A =9A =9A =9A 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

Or even beter
eq = Integrate[Sqrt[5 - 4*Cos[t] + Cos[2*t]], {t, 0, Pi/2}] +
  Integrate[Sqrt[5 - 4*Cos[t] + Cos[2*t]], {t, Pi/2, Pi}];

FullSimplify[eq]

Out:
(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]))/
 5

In[]=N@%
Out:
6.72288+ 1.05693*10^-14* I

Moreover, you can replace Cos[2t]->Cos[t]^2-Sin[t]^2

so

Integrate[
 Sqrt[5 - 4*Cos[t] + Cos[2*t]] /. Cos[2 t] -> Cos[t]^2 - Sin[t]^2, {t,
   0, Pi}];

gives result as above
6.72288+ 8.76799*10^-15 I

And the last solution is may be the best one:

Integrate[ TrigExpand/@Sqrt[5 - 4*Cos[t] + Cos[2*t]], {t,0, Pi}];


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