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

Optimal. Leaf size=56 \[ -\frac{2 \sqrt{-\cos (c+d x)} F\left (\sin ^{-1}\left (\frac{\sin (c+d x)}{1-\cos (c+d x)}\right )|\frac{1}{5}\right )}{\sqrt{5} d \sqrt{\cos (c+d x)}} \]

[Out]

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

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Rubi [A]  time = 0.116976, antiderivative size = 56, 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} \[ -\frac{2 \sqrt{-\cos (c+d x)} F\left (\sin ^{-1}\left (\frac{\sin (c+d x)}{1-\cos (c+d x)}\right )|\frac{1}{5}\right )}{\sqrt{5} d \sqrt{\cos (c+d x)}} \]

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])], 1/5])/(Sqrt[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 (\sin ^{-1}\left (\frac{\sin (c+d x)}{1-\cos (c+d x)}\right )|\frac{1}{5}\right )}{\sqrt{5} d \sqrt{\cos (c+d x)}}\\ \end{align*}

Mathematica [B]  time = 1.05145, size = 143, normalized size = 2.55 \[ -\frac{4 \sin ^4\left (\frac{1}{2} (c+d x)\right ) \sqrt{\cot ^2\left (\frac{1}{2} (c+d x)\right )} \csc (c+d x) \sqrt{(2-3 \cos (c+d x)) \csc ^2\left (\frac{1}{2} (c+d x)\right )} \sqrt{\cos (c+d x) \csc ^2\left (\frac{1}{2} (c+d x)\right )} F\left (\left .\sin ^{-1}\left (\frac{1}{2} \sqrt{\cos (c+d x) \csc ^2\left (\frac{1}{2} (c+d x)\right )}\right )\right |-4\right )}{d \sqrt{2-3 \cos (c+d x)} \sqrt{\cos (c+d x)}} \]

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

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Maple [B]  time = 0.499, size = 119, normalized size = 2.1 \begin{align*} 2\,{\frac{\sqrt{2-3\,\cos \left ( dx+c \right ) } \left ( \sin \left ( dx+c \right ) \right ) ^{4}}{d \left ( \cos \left ( dx+c \right ) \right ) ^{3/2} \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 ) ^{3/2}\sqrt{{\frac{-2+3\,\cos \left ( dx+c \right ) }{1+\cos \left ( dx+c \right ) }}}{\it EllipticF} \left ({\frac{-1+\cos \left ( dx+c \right ) }{\sin \left ( dx+c \right ) }},\sqrt{5} \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]

2/d*(cos(d*x+c)/(1+cos(d*x+c)))^(3/2)*(2-3*cos(d*x+c))^(1/2)*((-2+3*cos(d*x+c))/(1+cos(d*x+c)))^(1/2)*Elliptic
F((-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{2 - 3 \cos{\left (c + d x \right )}} \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(2 - 3*cos(c + d*x))*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)