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

Optimal. Leaf size=23 \[ \frac{2 F\left (\frac{1}{2} (c+d x)|\frac{8}{7}\right )}{\sqrt{7} d} \]

[Out]

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

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Rubi [A]  time = 0.0121487, antiderivative size = 23, 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 = {2661} \[ \frac{2 F\left (\frac{1}{2} (c+d x)|\frac{8}{7}\right )}{\sqrt{7} d} \]

Antiderivative was successfully verified.

[In]

Int[1/Sqrt[3 + 4*Cos[c + d*x]],x]

[Out]

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

Rule 2661

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

Mathematica [A]  time = 0.0277494, size = 23, normalized size = 1. \[ \frac{2 F\left (\frac{1}{2} (c+d x)|\frac{8}{7}\right )}{\sqrt{7} d} \]

Antiderivative was successfully verified.

[In]

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

[Out]

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

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Maple [C]  time = 0.046, size = 23, normalized size = 1. \begin{align*}{\frac{2\,\sqrt{7}}{7\,d}{\it InverseJacobiAM} \left ({\frac{dx}{2}}+{\frac{c}{2}},{\frac{2\,\sqrt{14}}{7}} \right ) } \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]

2/7/d*7^(1/2)*InverseJacobiAM(1/2*d*x+1/2*c,2/7*14^(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{1}{\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(1/(3+4*cos(d*x+c))^(1/2),x, algorithm="fricas")

[Out]

integral(1/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{1}{\sqrt{4 \cos{\left (c + d x \right )} + 3}}\, 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(4*cos(c + d*x) + 3), 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)