3.655 \(\int \frac{1}{\sqrt{-2-3 \cos (c+d x)} \sqrt{-\cos (c+d x)}} \, dx\)

Optimal. Leaf size=27 \[ -\frac{2 F\left (\left .\sin ^{-1}\left (\frac{\sin (c+d x)}{1-\cos (c+d x)}\right )\right |5\right )}{d} \]

[Out]

(-2*EllipticF[ArcSin[Sin[c + d*x]/(1 - Cos[c + d*x])], 5])/d

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Rubi [A]  time = 0.0553474, antiderivative size = 27, 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 = {2813} \[ -\frac{2 F\left (\left .\sin ^{-1}\left (\frac{\sin (c+d x)}{1-\cos (c+d x)}\right )\right |5\right )}{d} \]

Antiderivative was successfully verified.

[In]

Int[1/(Sqrt[-2 - 3*Cos[c + d*x]]*Sqrt[-Cos[c + d*x]]),x]

[Out]

(-2*EllipticF[ArcSin[Sin[c + d*x]/(1 - Cos[c + d*x])], 5])/d

Rule 2813

Int[1/(Sqrt[(d_.)*sin[(e_.) + (f_.)*(x_)]]*Sqrt[(a_) + (b_.)*sin[(e_.) + (f_.)*(x_)]]), x_Symbol] :> Simp[(-2*
d*EllipticF[ArcSin[Cos[e + f*x]/(1 + d*Sin[e + f*x])], -((a - b*d)/(a + b*d))])/(f*Sqrt[a + b*d]), x] /; FreeQ
[{a, b, d, e, f}, x] && LtQ[a^2 - b^2, 0] && EqQ[d^2, 1] && GtQ[b*d, 0]

Rubi steps

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

Mathematica [B]  time = 0.471152, size = 155, normalized size = 5.74 \[ \frac{4 \sin ^4\left (\frac{1}{2} (c+d x)\right ) \sqrt{\cot ^2\left (\frac{1}{2} (c+d x)\right )} \csc (c+d x) \sqrt{-\cos (c+d x) \csc ^2\left (\frac{1}{2} (c+d x)\right )} \sqrt{(3 \cos (c+d x)+2) \csc ^2\left (\frac{1}{2} (c+d x)\right )} F\left (\sin ^{-1}\left (\sqrt{\frac{5}{2}} \sqrt{\frac{\cos (c+d x)}{\cos (c+d x)-1}}\right )|\frac{4}{5}\right )}{\sqrt{5} d \sqrt{-3 \cos (c+d x)-2} \sqrt{-\cos (c+d x)}} \]

Antiderivative was successfully verified.

[In]

Integrate[1/(Sqrt[-2 - 3*Cos[c + d*x]]*Sqrt[-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]*EllipticF[ArcSin[Sqrt[5/2]*Sqrt[Cos[c + d*x]/(-1 + Cos[c + d*x])]], 4/5]*Sin[(c + d*x)/2]^4)
/(Sqrt[5]*d*Sqrt[-2 - 3*Cos[c + d*x]]*Sqrt[-Cos[c + d*x]])

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Maple [B]  time = 0.467, size = 129, normalized size = 4.8 \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 ) }{\it EllipticF} \left ({\frac{-1+\cos \left ( dx+c \right ) }{\sin \left ( dx+c \right ) }},{\frac{\sqrt{5}}{5}} \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 ) }{\frac{1}{\sqrt{-\cos \left ( dx+c \right ) }}}} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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Maxima [F]  time = 0., size = 0, normalized size = 0. \begin{align*} \int \frac{1}{\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(1/(-2-3*cos(d*x+c))^(1/2)/(-cos(d*x+c))^(1/2),x, algorithm="maxima")

[Out]

integrate(1/(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} + 2 \, \cos \left (d x + c\right )}, x\right ) \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

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Sympy [F]  time = 0., size = 0, normalized size = 0. \begin{align*} \int \frac{1}{\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(1/(-2-3*cos(d*x+c))**(1/2)/(-cos(d*x+c))**(1/2),x)

[Out]

Integral(1/(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{1}{\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(1/(-2-3*cos(d*x+c))^(1/2)/(-cos(d*x+c))^(1/2),x, algorithm="giac")

[Out]

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