3.6.50 \(\int \frac {1}{\sqrt {3+4 \cos (c+d x)}} \, dx\) [550]

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

[Out]

2/7*(cos(1/2*d*x+1/2*c)^2)^(1/2)/cos(1/2*d*x+1/2*c)*EllipticF(sin(1/2*d*x+1/2*c),2/7*14^(1/2))/d*7^(1/2)

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Rubi [A]
time = 0.01, antiderivative size = 23, normalized size of antiderivative = 1.00, 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 = {2740} \begin {gather*} \frac {2 F\left (\frac {1}{2} (c+d x)|\frac {8}{7}\right )}{\sqrt {7} d} \end {gather*}

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 2740

Int[1/Sqrt[(a_) + (b_.)*sin[(c_.) + (d_.)*(x_)]], x_Symbol] :> Simp[(2/(d*Sqrt[a + b]))*EllipticF[(1/2)*(c - P
i/2 + d*x), 2*(b/(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*}

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Mathematica [A]
time = 0.04, size = 23, normalized size = 1.00 \begin {gather*} \frac {2 F\left (\frac {1}{2} (c+d x)|\frac {8}{7}\right )}{\sqrt {7} d} \end {gather*}

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] Result contains higher order function than in optimal. Order 9 vs. order 4.
time = 0.05, size = 23, normalized size = 1.00

method	result	size
default	\(\frac {2 \sqrt {7}\, \mathrm {am}^{-1}\left (\frac {d x}{2}+\frac {c}{2}\bigg | \frac {2 \sqrt {14}}{7}\right )}{7 d}\)	\(23\)

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[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 = 0.00, size = 0, normalized size = 0.00 \begin {gather*} \text {Failed to integrate} \end {gather*}

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 [C] Result contains higher order function than in optimal. Order 9 vs. order 4.
time = 0.10, size = 54, normalized size = 2.35 \begin {gather*} \frac {-i \, \sqrt {2} {\rm weierstrassPInverse}\left (-1, 1, \cos \left (d x + c\right ) + i \, \sin \left (d x + c\right ) + \frac {1}{2}\right ) + i \, \sqrt {2} {\rm weierstrassPInverse}\left (-1, 1, \cos \left (d x + c\right ) - i \, \sin \left (d x + c\right ) + \frac {1}{2}\right )}{2 \, d} \end {gather*}

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]

1/2*(-I*sqrt(2)*weierstrassPInverse(-1, 1, cos(d*x + c) + I*sin(d*x + c) + 1/2) + I*sqrt(2)*weierstrassPInvers
e(-1, 1, cos(d*x + c) - I*sin(d*x + c) + 1/2))/d

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Sympy [F]
time = 0.00, size = 0, normalized size = 0.00 \begin {gather*} \int \frac {1}{\sqrt {4 \cos {\left (c + d x \right )} + 3}}\, dx \end {gather*}

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 = 0.00, size = 0, normalized size = 0.00 \begin {gather*} \text {could not integrate} \end {gather*}

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)

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Mupad [B]
time = 0.57, size = 39, normalized size = 1.70 \begin {gather*} \frac {2\,\sqrt {\frac {4\,\cos \left (c+d\,x\right )}{7}+\frac {3}{7}}\,\mathrm {F}\left (\frac {c}{2}+\frac {d\,x}{2}\middle |\frac {8}{7}\right )}{d\,\sqrt {4\,\cos \left (c+d\,x\right )+3}} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

int(1/(4*cos(c + d*x) + 3)^(1/2),x)

[Out]

(2*((4*cos(c + d*x))/7 + 3/7)^(1/2)*ellipticF(c/2 + (d*x)/2, 8/7))/(d*(4*cos(c + d*x) + 3)^(1/2))

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