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

Optimal. Leaf size=25 \[ -\frac{2 \Pi \left (2;\frac{1}{2} (c+d x+\pi )|\frac{8}{7}\right )}{\sqrt{7} d} \]

[Out]

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

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Rubi [A]  time = 0.0385874, antiderivative size = 25, normalized size of antiderivative = 1., number of steps used = 1, number of rules used = 1, integrand size = 21, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.048, Rules used = {2806} \[ -\frac{2 \Pi \left (2;\frac{1}{2} (c+d x+\pi )|\frac{8}{7}\right )}{\sqrt{7} d} \]

Antiderivative was successfully verified.

[In]

Int[Sec[c + d*x]/Sqrt[3 - 4*Cos[c + d*x]],x]

[Out]

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

Rule 2806

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

Rubi steps

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

Mathematica [A]  time = 0.0593641, size = 45, normalized size = 1.8 \[ \frac{2 \sqrt{4 \cos (c+d x)-3} \Pi \left (2;\left .\frac{1}{2} (c+d x)\right |8\right )}{d \sqrt{3-4 \cos (c+d x)}} \]

Antiderivative was successfully verified.

[In]

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

[Out]

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

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Maple [B]  time = 2.503, size = 139, normalized size = 5.6 \begin{align*}{\frac{2}{7\,d}\sqrt{- \left ( 8\, \left ( \cos \left ( 1/2\,dx+c/2 \right ) \right ) ^{2}-7 \right ) \left ( \sin \left ({\frac{dx}{2}}+{\frac{c}{2}} \right ) \right ) ^{2}}\sqrt{ \left ( \sin \left ({\frac{dx}{2}}+{\frac{c}{2}} \right ) \right ) ^{2}}\sqrt{56\, \left ( \sin \left ( 1/2\,dx+c/2 \right ) \right ) ^{2}-7}{\it EllipticPi} \left ( \cos \left ({\frac{dx}{2}}+{\frac{c}{2}} \right ) ,2,{\frac{2\,\sqrt{14}}{7}} \right ){\frac{1}{\sqrt{8\, \left ( \sin \left ( 1/2\,dx+c/2 \right ) \right ) ^{4}- \left ( \sin \left ({\frac{dx}{2}}+{\frac{c}{2}} \right ) \right ) ^{2}}}} \left ( \sin \left ({\frac{dx}{2}}+{\frac{c}{2}} \right ) \right ) ^{-1}{\frac{1}{\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(sec(d*x+c)/(3-4*cos(d*x+c))^(1/2),x)

[Out]

2/7*(-(8*cos(1/2*d*x+1/2*c)^2-7)*sin(1/2*d*x+1/2*c)^2)^(1/2)*(sin(1/2*d*x+1/2*c)^2)^(1/2)*(56*sin(1/2*d*x+1/2*
c)^2-7)^(1/2)/(8*sin(1/2*d*x+1/2*c)^4-sin(1/2*d*x+1/2*c)^2)^(1/2)*EllipticPi(cos(1/2*d*x+1/2*c),2,2/7*14^(1/2)
)/sin(1/2*d*x+1/2*c)/(-8*cos(1/2*d*x+1/2*c)^2+7)^(1/2)/d

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Maxima [F]  time = 0., size = 0, normalized size = 0. \begin{align*} \int \frac{\sec \left (d x + c\right )}{\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(sec(d*x+c)/(3-4*cos(d*x+c))^(1/2),x, algorithm="maxima")

[Out]

integrate(sec(d*x + c)/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} \sec \left (d x + c\right )}{4 \, \cos \left (d x + c\right ) - 3}, x\right ) \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

Integral(sec(c + d*x)/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{\sec \left (d x + c\right )}{\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(sec(d*x+c)/(3-4*cos(d*x+c))^(1/2),x, algorithm="giac")

[Out]

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