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

Optimal. Leaf size=49 \[ -\frac{2 \sqrt{-\cos (c+d x)} F\left (\left .\sin ^{-1}\left (\frac{\sin (c+d x)}{1-\cos (c+d x)}\right )\right |5\right )}{d \sqrt{\cos (c+d x)}} \]

[Out]

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

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Rubi [A]  time = 0.110361, antiderivative size = 49, normalized size of antiderivative = 1., number of steps used = 2, number of rules used = 2, integrand size = 25, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.08, Rules used = {2814, 2813} \[ -\frac{2 \sqrt{-\cos (c+d x)} F\left (\left .\sin ^{-1}\left (\frac{\sin (c+d x)}{1-\cos (c+d x)}\right )\right |5\right )}{d \sqrt{\cos (c+d x)}} \]

Antiderivative was successfully verified.

[In]

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

[Out]

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

Rule 2814

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

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

Mathematica [B]  time = 1.4557, size = 153, normalized size = 3.12 \[ \frac{4 \sin ^4\left (\frac{1}{2} (c+d x)\right ) \sqrt{\cot ^2\left (\frac{1}{2} (c+d x)\right )} \csc (c+d x) \sqrt{-\cos (c+d x) \csc ^2\left (\frac{1}{2} (c+d x)\right )} \sqrt{(3 \cos (c+d x)+2) \csc ^2\left (\frac{1}{2} (c+d x)\right )} F\left (\sin ^{-1}\left (\sqrt{\frac{5}{2}} \sqrt{\frac{\cos (c+d x)}{\cos (c+d x)-1}}\right )|\frac{4}{5}\right )}{\sqrt{5} d \sqrt{-3 \cos (c+d x)-2} \sqrt{\cos (c+d x)}} \]

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 = 0.447, size = 132, normalized size = 2.7 \begin{align*}{\frac{\sqrt{5}\sqrt{2}\sqrt{10} \left ( \sin \left ( dx+c \right ) \right ) ^{4}}{5\,d \left ( 2+3\,\cos \left ( dx+c \right ) \right ) \left ( -1+\cos \left ( dx+c \right ) \right ) ^{2}} \left ({\frac{\cos \left ( dx+c \right ) }{1+\cos \left ( dx+c \right ) }} \right ) ^{{\frac{3}{2}}}\sqrt{-2-3\,\cos \left ( dx+c \right ) }\sqrt{{\frac{2+3\,\cos \left ( dx+c \right ) }{1+\cos \left ( dx+c \right ) }}}{\it EllipticF} \left ({\frac{\sqrt{5} \left ( -1+\cos \left ( dx+c \right ) \right ) }{5\,\sin \left ( dx+c \right ) }},\sqrt{5} \right ) \left ( \cos \left ( dx+c \right ) \right ) ^{-{\frac{3}{2}}}} \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*5^(1/2)*2^(1/2)*(cos(d*x+c)/(1+cos(d*x+c)))^(3/2)*(-2-3*cos(d*x+c))^(1/2)*10^(1/2)*((2+3*cos(d*x+c))/(1+
cos(d*x+c)))^(1/2)*EllipticF(1/5*5^(1/2)*(-1+cos(d*x+c))/sin(d*x+c),5^(1/2))*sin(d*x+c)^4/cos(d*x+c)^(3/2)/(2+
3*cos(d*x+c))/(-1+cos(d*x+c))^2

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

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

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