3.546 \(\int \frac{81+54 x-24 x^3-16 x^4}{729-64 x^6} \, dx\)

Optimal. Leaf size=24 \[ -\frac{\tan ^{-1}\left (\frac{3-4 x}{3 \sqrt{3}}\right )}{3 \sqrt{3}} \]

[Out]

-ArcTan[(3 - 4*x)/(3*Sqrt[3])]/(3*Sqrt[3])

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Rubi [A]  time = 0.0345201, antiderivative size = 24, normalized size of antiderivative = 1., number of steps used = 3, number of rules used = 3, integrand size = 25, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.12 \[ -\frac{\tan ^{-1}\left (\frac{3-4 x}{3 \sqrt{3}}\right )}{3 \sqrt{3}} \]

Antiderivative was successfully verified.

[In]  Int[(81 + 54*x - 24*x^3 - 16*x^4)/(729 - 64*x^6),x]

[Out]

-ArcTan[(3 - 4*x)/(3*Sqrt[3])]/(3*Sqrt[3])

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Rubi in Sympy [A]  time = 6.38822, size = 22, normalized size = 0.92 \[ - \frac{\sqrt{3} \operatorname{atan}{\left (\sqrt{3} \left (- \frac{4 x}{9} + \frac{1}{3}\right ) \right )}}{9} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  rubi_integrate((-16*x**4-24*x**3+54*x+81)/(-64*x**6+729),x)

[Out]

-sqrt(3)*atan(sqrt(3)*(-4*x/9 + 1/3))/9

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Mathematica [A]  time = 0.01145, size = 24, normalized size = 1. \[ \frac{\tan ^{-1}\left (\frac{4 x-3}{3 \sqrt{3}}\right )}{3 \sqrt{3}} \]

Antiderivative was successfully verified.

[In]  Integrate[(81 + 54*x - 24*x^3 - 16*x^4)/(729 - 64*x^6),x]

[Out]

ArcTan[(-3 + 4*x)/(3*Sqrt[3])]/(3*Sqrt[3])

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Maple [A]  time = 0.006, size = 17, normalized size = 0.7 \[{\frac{\sqrt{3}}{9}\arctan \left ({\frac{ \left ( 8\,x-6 \right ) \sqrt{3}}{18}} \right ) } \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  int((-16*x^4-24*x^3+54*x+81)/(-64*x^6+729),x)

[Out]

1/9*3^(1/2)*arctan(1/18*(8*x-6)*3^(1/2))

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Maxima [A]  time = 1.54475, size = 22, normalized size = 0.92 \[ \frac{1}{9} \, \sqrt{3} \arctan \left (\frac{1}{9} \, \sqrt{3}{\left (4 \, x - 3\right )}\right ) \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate((16*x^4 + 24*x^3 - 54*x - 81)/(64*x^6 - 729),x, algorithm="maxima")

[Out]

1/9*sqrt(3)*arctan(1/9*sqrt(3)*(4*x - 3))

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Fricas [A]  time = 0.203516, size = 22, normalized size = 0.92 \[ \frac{1}{9} \, \sqrt{3} \arctan \left (\frac{1}{9} \, \sqrt{3}{\left (4 \, x - 3\right )}\right ) \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate((16*x^4 + 24*x^3 - 54*x - 81)/(64*x^6 - 729),x, algorithm="fricas")

[Out]

1/9*sqrt(3)*arctan(1/9*sqrt(3)*(4*x - 3))

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Sympy [A]  time = 0.128778, size = 24, normalized size = 1. \[ \frac{\sqrt{3} \operatorname{atan}{\left (\frac{4 \sqrt{3} x}{9} - \frac{\sqrt{3}}{3} \right )}}{9} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate((-16*x**4-24*x**3+54*x+81)/(-64*x**6+729),x)

[Out]

sqrt(3)*atan(4*sqrt(3)*x/9 - sqrt(3)/3)/9

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GIAC/XCAS [A]  time = 0.215871, size = 22, normalized size = 0.92 \[ \frac{1}{9} \, \sqrt{3} \arctan \left (\frac{1}{9} \, \sqrt{3}{\left (4 \, x - 3\right )}\right ) \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate((16*x^4 + 24*x^3 - 54*x - 81)/(64*x^6 - 729),x, algorithm="giac")

[Out]

1/9*sqrt(3)*arctan(1/9*sqrt(3)*(4*x - 3))