3.557 \(\int \frac{1}{\sqrt{3-4 \cos (c+d x)}} \, dx\)

Optimal. Leaf size=24 \[ \frac{2 F\left (\frac{1}{2} (c+d x+\pi )|\frac{8}{7}\right )}{\sqrt{7} d} \]

[Out]

(2*EllipticF[(c + Pi + d*x)/2, 8/7])/(Sqrt[7]*d)

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Rubi [A]  time = 0.0119922, antiderivative size = 24, normalized size of antiderivative = 1., number of steps used = 1, number of rules used = 1, integrand size = 14, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.071, Rules used = {2662} \[ \frac{2 F\left (\frac{1}{2} (c+d x+\pi )|\frac{8}{7}\right )}{\sqrt{7} d} \]

Antiderivative was successfully verified.

[In]

Int[1/Sqrt[3 - 4*Cos[c + d*x]],x]

[Out]

(2*EllipticF[(c + Pi + d*x)/2, 8/7])/(Sqrt[7]*d)

Rule 2662

Int[1/Sqrt[(a_) + (b_.)*sin[(c_.) + (d_.)*(x_)]], x_Symbol] :> Simp[(2*EllipticF[(1*(c + Pi/2 + d*x))/2, (-2*b
)/(a - b)])/(d*Sqrt[a - b]), x] /; FreeQ[{a, b, c, d}, x] && NeQ[a^2 - b^2, 0] && GtQ[a - b, 0]

Rubi steps

\begin{align*} \int \frac{1}{\sqrt{3-4 \cos (c+d x)}} \, dx &=\frac{2 F\left (\frac{1}{2} (c+\pi +d x)|\frac{8}{7}\right )}{\sqrt{7} d}\\ \end{align*}

Mathematica [A]  time = 0.034354, size = 44, normalized size = 1.83 \[ \frac{2 \sqrt{4 \cos (c+d x)-3} F\left (\left .\frac{1}{2} (c+d x)\right |8\right )}{d \sqrt{3-4 \cos (c+d x)}} \]

Antiderivative was successfully verified.

[In]

Integrate[1/Sqrt[3 - 4*Cos[c + d*x]],x]

[Out]

(2*Sqrt[-3 + 4*Cos[c + d*x]]*EllipticF[(c + d*x)/2, 8])/(d*Sqrt[3 - 4*Cos[c + d*x]])

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Maple [C]  time = 0.163, size = 54, normalized size = 2.3 \begin{align*} 2\,{\frac{\sqrt{8\, \left ( \cos \left ( 1/2\,dx+c/2 \right ) \right ) ^{2}-7}{\it InverseJacobiAM} \left ( 1/2\,dx+c/2,2\,\sqrt{2} \right ) }{d\sqrt{-8\, \left ( \cos \left ( 1/2\,dx+c/2 \right ) \right ) ^{2}+7}}} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

int(1/(3-4*cos(d*x+c))^(1/2),x)

[Out]

2/d/(-8*cos(1/2*d*x+1/2*c)^2+7)^(1/2)*(8*cos(1/2*d*x+1/2*c)^2-7)^(1/2)*InverseJacobiAM(1/2*d*x+1/2*c,2*2^(1/2)
)

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Maxima [F]  time = 0., size = 0, normalized size = 0. \begin{align*} \int \frac{1}{\sqrt{-4 \, \cos \left (d x + c\right ) + 3}}\,{d x} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(1/(3-4*cos(d*x+c))^(1/2),x, algorithm="maxima")

[Out]

integrate(1/sqrt(-4*cos(d*x + c) + 3), x)

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Fricas [F]  time = 0., size = 0, normalized size = 0. \begin{align*}{\rm integral}\left (-\frac{\sqrt{-4 \, \cos \left (d x + c\right ) + 3}}{4 \, \cos \left (d x + c\right ) - 3}, x\right ) \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(1/(3-4*cos(d*x+c))^(1/2),x, algorithm="fricas")

[Out]

integral(-sqrt(-4*cos(d*x + c) + 3)/(4*cos(d*x + c) - 3), x)

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Sympy [F]  time = 0., size = 0, normalized size = 0. \begin{align*} \int \frac{1}{\sqrt{3 - 4 \cos{\left (c + d x \right )}}}\, dx \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(1/(3-4*cos(d*x+c))**(1/2),x)

[Out]

Integral(1/sqrt(3 - 4*cos(c + d*x)), x)

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Giac [F]  time = 0., size = 0, normalized size = 0. \begin{align*} \int \frac{1}{\sqrt{-4 \, \cos \left (d x + c\right ) + 3}}\,{d x} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(1/(3-4*cos(d*x+c))^(1/2),x, algorithm="giac")

[Out]

integrate(1/sqrt(-4*cos(d*x + c) + 3), x)