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

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

[Out]

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

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Rubi [A]  time = 0.0387461, antiderivative size = 24, 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 = {2805} \[ \frac{2 \Pi \left (2;\frac{1}{2} (c+d x)|\frac{8}{7}\right )}{\sqrt{7} d} \]

Antiderivative was successfully verified.

[In]

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

[Out]

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

Rule 2805

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+d x)|\frac{8}{7}\right )}{\sqrt{7} d}\\ \end{align*}

Mathematica [A]  time = 0.0501663, size = 24, normalized size = 1. \[ \frac{2 \Pi \left (2;\frac{1}{2} (c+d x)|\frac{8}{7}\right )}{\sqrt{7} d} \]

Antiderivative was successfully verified.

[In]

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

[Out]

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

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Maple [B]  time = 1.839, size = 138, normalized size = 5.8 \begin{align*} 2\,{\frac{\sqrt{ \left ( 8\, \left ( \cos \left ( 1/2\,dx+c/2 \right ) \right ) ^{2}-1 \right ) \left ( \sin \left ( 1/2\,dx+c/2 \right ) \right ) ^{2}}\sqrt{ \left ( \sin \left ( 1/2\,dx+c/2 \right ) \right ) ^{2}}\sqrt{-8\, \left ( \cos \left ( 1/2\,dx+c/2 \right ) \right ) ^{2}+1}{\it EllipticPi} \left ( \cos \left ( 1/2\,dx+c/2 \right ) ,2,2\,\sqrt{2} \right ) }{\sqrt{-8\, \left ( \sin \left ( 1/2\,dx+c/2 \right ) \right ) ^{4}+7\, \left ( \sin \left ( 1/2\,dx+c/2 \right ) \right ) ^{2}}\sin \left ( 1/2\,dx+c/2 \right ) \sqrt{8\, \left ( \cos \left ( 1/2\,dx+c/2 \right ) \right ) ^{2}-1}d}} \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*((8*cos(1/2*d*x+1/2*c)^2-1)*sin(1/2*d*x+1/2*c)^2)^(1/2)*(sin(1/2*d*x+1/2*c)^2)^(1/2)*(-8*cos(1/2*d*x+1/2*c)^
2+1)^(1/2)/(-8*sin(1/2*d*x+1/2*c)^4+7*sin(1/2*d*x+1/2*c)^2)^(1/2)*EllipticPi(cos(1/2*d*x+1/2*c),2,2*2^(1/2))/s
in(1/2*d*x+1/2*c)/(8*cos(1/2*d*x+1/2*c)^2-1)^(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{\sec \left (d x + c\right )}{\sqrt{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(sec(d*x + c)/sqrt(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{4 \cos{\left (c + d x \right )} + 3}}\, 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(4*cos(c + d*x) + 3), 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)