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]

-2/7*(sin(1/2*d*x+1/2*c)^2)^(1/2)/sin(1/2*d*x+1/2*c)*EllipticF(cos(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 = 24, 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 = {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*}

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Mathematica [A]  time = 0.04, 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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fricas [F]  time = 1.20, size = 0, normalized size = 0.00 \[ {\rm integral}\left (-\frac {\sqrt {-4 \, \cos \left (d x + c\right ) + 3}}{4 \, \cos \left (d x + c\right ) - 3}, x\right ) \]

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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giac [F]  time = 0.00, size = 0, normalized size = 0.00 \[ \int \frac {1}{\sqrt {-4 \, \cos \left (d x + c\right ) + 3}}\,{d x} \]

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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maple [C]  time = 0.07, size = 54, normalized size = 2.25 \[ \frac {2 \sqrt {8 \left (\cos ^{2}\left (\frac {d x}{2}+\frac {c}{2}\right )\right )-7}\, \mathrm {am}^{-1}\left (\frac {d x}{2}+\frac {c}{2}| 2 \sqrt {2}\right )}{d \sqrt {-8 \left (\cos ^{2}\left (\frac {d x}{2}+\frac {c}{2}\right )\right )+7}} \]

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 = 0.00, size = 0, normalized size = 0.00 \[ \int \frac {1}{\sqrt {-4 \, \cos \left (d x + c\right ) + 3}}\,{d x} \]

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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mupad [B]  time = 0.57, size = 39, normalized size = 1.62 \[ \frac {2\,\sqrt {4\,\cos \left (c+d\,x\right )-3}\,\mathrm {F}\left (\frac {c}{2}+\frac {d\,x}{2}\middle |8\right )}{d\,\sqrt {3-4\,\cos \left (c+d\,x\right )}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

int(1/(3 - 4*cos(c + d*x))^(1/2),x)

[Out]

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

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

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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