3.671 \(\int \frac{\sqrt{-\cos (c+d x)}}{\sqrt{-2-3 \cos (c+d x)}} \, dx\)

Optimal. Leaf size=79 \[ -\frac{4 \cot (c+d x) \sqrt{-\sec (c+d x)-1} \sqrt{1-\sec (c+d x)} \Pi \left (\frac{5}{3};\left .\sin ^{-1}\left (\frac{\sqrt{-3 \cos (c+d x)-2}}{\sqrt{5} \sqrt{-\cos (c+d x)}}\right )\right |5\right )}{3 d} \]

[Out]

(-4*Cot[c + d*x]*EllipticPi[5/3, ArcSin[Sqrt[-2 - 3*Cos[c + d*x]]/(Sqrt[5]*Sqrt[-Cos[c + d*x]])], 5]*Sqrt[-1 -
Sec[c + d*x]]*Sqrt[1 - Sec[c + d*x]])/(3*d)

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Rubi [A]  time = 0.0536822, antiderivative size = 79, normalized size of antiderivative = 1., number of steps used = 1, number of rules used = 1, integrand size = 27, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.037, Rules used = {2809} \[ -\frac{4 \cot (c+d x) \sqrt{-\sec (c+d x)-1} \sqrt{1-\sec (c+d x)} \Pi \left (\frac{5}{3};\left .\sin ^{-1}\left (\frac{\sqrt{-3 \cos (c+d x)-2}}{\sqrt{5} \sqrt{-\cos (c+d x)}}\right )\right |5\right )}{3 d} \]

Antiderivative was successfully verified.

[In]

Int[Sqrt[-Cos[c + d*x]]/Sqrt[-2 - 3*Cos[c + d*x]],x]

[Out]

(-4*Cot[c + d*x]*EllipticPi[5/3, ArcSin[Sqrt[-2 - 3*Cos[c + d*x]]/(Sqrt[5]*Sqrt[-Cos[c + d*x]])], 5]*Sqrt[-1 -
Sec[c + d*x]]*Sqrt[1 - Sec[c + d*x]])/(3*d)

Rule 2809

Int[Sqrt[(b_.)*sin[(e_.) + (f_.)*(x_)]]/Sqrt[(c_) + (d_.)*sin[(e_.) + (f_.)*(x_)]], x_Symbol] :> Simp[(2*b*Tan
[e + f*x]*Rt[(c + d)/b, 2]*Sqrt[(c*(1 + Csc[e + f*x]))/(c - d)]*Sqrt[(c*(1 - Csc[e + f*x]))/(c + d)]*EllipticP
i[(c + d)/d, ArcSin[Sqrt[c + d*Sin[e + f*x]]/(Sqrt[b*Sin[e + f*x]]*Rt[(c + d)/b, 2])], -((c + d)/(c - d))])/(d
*f), x] /; FreeQ[{b, c, d, e, f}, x] && NeQ[c^2 - d^2, 0] && PosQ[(c + d)/b]

Rubi steps

\begin{align*} \int \frac{\sqrt{-\cos (c+d x)}}{\sqrt{-2-3 \cos (c+d x)}} \, dx &=-\frac{4 \cot (c+d x) \Pi \left (\frac{5}{3};\left .\sin ^{-1}\left (\frac{\sqrt{-2-3 \cos (c+d x)}}{\sqrt{5} \sqrt{-\cos (c+d x)}}\right )\right |5\right ) \sqrt{-1-\sec (c+d x)} \sqrt{1-\sec (c+d x)}}{3 d}\\ \end{align*}

Mathematica [B]  time = 0.622784, size = 194, normalized size = 2.46 \[ \frac{4 \sin ^4\left (\frac{1}{2} (c+d x)\right ) \sqrt{\cot ^2\left (\frac{1}{2} (c+d x)\right )} \csc (c+d x) \sqrt{-\cos (c+d x) \csc ^2\left (\frac{1}{2} (c+d x)\right )} \sqrt{(3 \cos (c+d x)+2) \csc ^2\left (\frac{1}{2} (c+d x)\right )} \left (3 F\left (\left .\sin ^{-1}\left (\frac{1}{2} \sqrt{(3 \cos (c+d x)+2) \csc ^2\left (\frac{1}{2} (c+d x)\right )}\right )\right |-4\right )-5 \Pi \left (-\frac{2}{3};\left .\sin ^{-1}\left (\frac{1}{2} \sqrt{(3 \cos (c+d x)+2) \csc ^2\left (\frac{1}{2} (c+d x)\right )}\right )\right |-4\right )\right )}{3 d \sqrt{-3 \cos (c+d x)-2} \sqrt{-\cos (c+d x)}} \]

Antiderivative was successfully verified.

[In]

Integrate[Sqrt[-Cos[c + d*x]]/Sqrt[-2 - 3*Cos[c + d*x]],x]

[Out]

(4*Sqrt[Cot[(c + d*x)/2]^2]*Sqrt[-(Cos[c + d*x]*Csc[(c + d*x)/2]^2)]*Sqrt[(2 + 3*Cos[c + d*x])*Csc[(c + d*x)/2
]^2]*Csc[c + d*x]*(3*EllipticF[ArcSin[Sqrt[(2 + 3*Cos[c + d*x])*Csc[(c + d*x)/2]^2]/2], -4] - 5*EllipticPi[-2/
3, ArcSin[Sqrt[(2 + 3*Cos[c + d*x])*Csc[(c + d*x)/2]^2]/2], -4])*Sin[(c + d*x)/2]^4)/(3*d*Sqrt[-2 - 3*Cos[c +
d*x]]*Sqrt[-Cos[c + d*x]])

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Maple [B]  time = 0.408, size = 164, normalized size = 2.1 \begin{align*}{\frac{\sqrt{10}\sqrt{2} \left ( \sin \left ( dx+c \right ) \right ) ^{2}}{5\,d \left ( 3\, \left ( \cos \left ( dx+c \right ) \right ) ^{2}-\cos \left ( dx+c \right ) -2 \right ) \cos \left ( dx+c \right ) } \left ({\it EllipticF} \left ({\frac{-1+\cos \left ( dx+c \right ) }{\sin \left ( dx+c \right ) }},{\frac{\sqrt{5}}{5}} \right ) -2\,{\it EllipticPi} \left ({\frac{-1+\cos \left ( dx+c \right ) }{\sin \left ( dx+c \right ) }},-1,1/5\,\sqrt{5} \right ) \right ) \sqrt{{\frac{2+3\,\cos \left ( dx+c \right ) }{1+\cos \left ( dx+c \right ) }}}\sqrt{{\frac{\cos \left ( dx+c \right ) }{1+\cos \left ( dx+c \right ) }}}\sqrt{-2-3\,\cos \left ( dx+c \right ) }\sqrt{-\cos \left ( dx+c \right ) }} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

int((-cos(d*x+c))^(1/2)/(-2-3*cos(d*x+c))^(1/2),x)

[Out]

1/5/d*10^(1/2)*2^(1/2)*(EllipticF((-1+cos(d*x+c))/sin(d*x+c),1/5*5^(1/2))-2*EllipticPi((-1+cos(d*x+c))/sin(d*x
+c),-1,1/5*5^(1/2)))*((2+3*cos(d*x+c))/(1+cos(d*x+c)))^(1/2)*(cos(d*x+c)/(1+cos(d*x+c)))^(1/2)*(-2-3*cos(d*x+c
))^(1/2)*(-cos(d*x+c))^(1/2)*sin(d*x+c)^2/(3*cos(d*x+c)^2-cos(d*x+c)-2)/cos(d*x+c)

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Maxima [F]  time = 0., size = 0, normalized size = 0. \begin{align*} \int \frac{\sqrt{-\cos \left (d x + c\right )}}{\sqrt{-3 \, \cos \left (d x + c\right ) - 2}}\,{d x} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((-cos(d*x+c))^(1/2)/(-2-3*cos(d*x+c))^(1/2),x, algorithm="maxima")

[Out]

integrate(sqrt(-cos(d*x + c))/sqrt(-3*cos(d*x + c) - 2), x)

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Fricas [F]  time = 0., size = 0, normalized size = 0. \begin{align*}{\rm integral}\left (-\frac{\sqrt{-\cos \left (d x + c\right )} \sqrt{-3 \, \cos \left (d x + c\right ) - 2}}{3 \, \cos \left (d x + c\right ) + 2}, x\right ) \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((-cos(d*x+c))^(1/2)/(-2-3*cos(d*x+c))^(1/2),x, algorithm="fricas")

[Out]

integral(-sqrt(-cos(d*x + c))*sqrt(-3*cos(d*x + c) - 2)/(3*cos(d*x + c) + 2), x)

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Sympy [F]  time = 0., size = 0, normalized size = 0. \begin{align*} \int \frac{\sqrt{- \cos{\left (c + d x \right )}}}{\sqrt{- 3 \cos{\left (c + d x \right )} - 2}}\, dx \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((-cos(d*x+c))**(1/2)/(-2-3*cos(d*x+c))**(1/2),x)

[Out]

Integral(sqrt(-cos(c + d*x))/sqrt(-3*cos(c + d*x) - 2), x)

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Giac [F]  time = 0., size = 0, normalized size = 0. \begin{align*} \int \frac{\sqrt{-\cos \left (d x + c\right )}}{\sqrt{-3 \, \cos \left (d x + c\right ) - 2}}\,{d x} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((-cos(d*x+c))^(1/2)/(-2-3*cos(d*x+c))^(1/2),x, algorithm="giac")

[Out]

integrate(sqrt(-cos(d*x + c))/sqrt(-3*cos(d*x + c) - 2), x)