3.551 \(\int \frac{27-8 x^3}{729-64 x^6} \, dx\)

Optimal. Leaf size=50 \[ -\frac{1}{108} \log \left (4 x^2-6 x+9\right )+\frac{1}{54} \log (2 x+3)-\frac{\tan ^{-1}\left (\frac{3-4 x}{3 \sqrt{3}}\right )}{18 \sqrt{3}} \]

[Out]

-ArcTan[(3 - 4*x)/(3*Sqrt[3])]/(18*Sqrt[3]) + Log[3 + 2*x]/54 - Log[9 - 6*x + 4*
x^2]/108

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Rubi [A]  time = 0.0523294, antiderivative size = 50, normalized size of antiderivative = 1., number of steps used = 7, number of rules used = 7, integrand size = 17, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.412 \[ -\frac{1}{108} \log \left (4 x^2-6 x+9\right )+\frac{1}{54} \log (2 x+3)-\frac{\tan ^{-1}\left (\frac{3-4 x}{3 \sqrt{3}}\right )}{18 \sqrt{3}} \]

Antiderivative was successfully verified.

[In]  Int[(27 - 8*x^3)/(729 - 64*x^6),x]

[Out]

-ArcTan[(3 - 4*x)/(3*Sqrt[3])]/(18*Sqrt[3]) + Log[3 + 2*x]/54 - Log[9 - 6*x + 4*
x^2]/108

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Rubi in Sympy [A]  time = 7.66706, size = 42, normalized size = 0.84 \[ \frac{\log{\left (2 x + 3 \right )}}{54} - \frac{\log{\left (4 x^{2} - 6 x + 9 \right )}}{108} + \frac{\sqrt{3} \operatorname{atan}{\left (\sqrt{3} \left (\frac{4 x}{9} - \frac{1}{3}\right ) \right )}}{54} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  rubi_integrate((-8*x**3+27)/(-64*x**6+729),x)

[Out]

log(2*x + 3)/54 - log(4*x**2 - 6*x + 9)/108 + sqrt(3)*atan(sqrt(3)*(4*x/9 - 1/3)
)/54

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Mathematica [A]  time = 0.00931375, size = 50, normalized size = 1. \[ -\frac{1}{108} \log \left (4 x^2-6 x+9\right )+\frac{1}{54} \log (2 x+3)+\frac{\tan ^{-1}\left (\frac{4 x-3}{3 \sqrt{3}}\right )}{18 \sqrt{3}} \]

Antiderivative was successfully verified.

[In]  Integrate[(27 - 8*x^3)/(729 - 64*x^6),x]

[Out]

ArcTan[(-3 + 4*x)/(3*Sqrt[3])]/(18*Sqrt[3]) + Log[3 + 2*x]/54 - Log[9 - 6*x + 4*
x^2]/108

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Maple [A]  time = 0.008, size = 39, normalized size = 0.8 \[{\frac{\ln \left ( 2\,x+3 \right ) }{54}}-{\frac{\ln \left ( 4\,{x}^{2}-6\,x+9 \right ) }{108}}+{\frac{\sqrt{3}}{54}\arctan \left ({\frac{ \left ( 8\,x-6 \right ) \sqrt{3}}{18}} \right ) } \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  int((-8*x^3+27)/(-64*x^6+729),x)

[Out]

1/54*ln(2*x+3)-1/108*ln(4*x^2-6*x+9)+1/54*3^(1/2)*arctan(1/18*(8*x-6)*3^(1/2))

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Maxima [A]  time = 1.51589, size = 51, normalized size = 1.02 \[ \frac{1}{54} \, \sqrt{3} \arctan \left (\frac{1}{9} \, \sqrt{3}{\left (4 \, x - 3\right )}\right ) - \frac{1}{108} \, \log \left (4 \, x^{2} - 6 \, x + 9\right ) + \frac{1}{54} \, \log \left (2 \, x + 3\right ) \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate((8*x^3 - 27)/(64*x^6 - 729),x, algorithm="maxima")

[Out]

1/54*sqrt(3)*arctan(1/9*sqrt(3)*(4*x - 3)) - 1/108*log(4*x^2 - 6*x + 9) + 1/54*l
og(2*x + 3)

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Fricas [A]  time = 0.20985, size = 61, normalized size = 1.22 \[ -\frac{1}{324} \, \sqrt{3}{\left (\sqrt{3} \log \left (4 \, x^{2} - 6 \, x + 9\right ) - 2 \, \sqrt{3} \log \left (2 \, x + 3\right ) - 6 \, \arctan \left (\frac{1}{9} \, \sqrt{3}{\left (4 \, x - 3\right )}\right )\right )} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate((8*x^3 - 27)/(64*x^6 - 729),x, algorithm="fricas")

[Out]

-1/324*sqrt(3)*(sqrt(3)*log(4*x^2 - 6*x + 9) - 2*sqrt(3)*log(2*x + 3) - 6*arctan
(1/9*sqrt(3)*(4*x - 3)))

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Sympy [A]  time = 0.204576, size = 48, normalized size = 0.96 \[ \frac{\log{\left (x + \frac{3}{2} \right )}}{54} - \frac{\log{\left (x^{2} - \frac{3 x}{2} + \frac{9}{4} \right )}}{108} + \frac{\sqrt{3} \operatorname{atan}{\left (\frac{4 \sqrt{3} x}{9} - \frac{\sqrt{3}}{3} \right )}}{54} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate((-8*x**3+27)/(-64*x**6+729),x)

[Out]

log(x + 3/2)/54 - log(x**2 - 3*x/2 + 9/4)/108 + sqrt(3)*atan(4*sqrt(3)*x/9 - sqr
t(3)/3)/54

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GIAC/XCAS [A]  time = 0.219643, size = 47, normalized size = 0.94 \[ \frac{1}{54} \, \sqrt{3} \arctan \left (\frac{1}{9} \, \sqrt{3}{\left (4 \, x - 3\right )}\right ) - \frac{1}{108} \,{\rm ln}\left (x^{2} - \frac{3}{2} \, x + \frac{9}{4}\right ) + \frac{1}{54} \,{\rm ln}\left ({\left | x + \frac{3}{2} \right |}\right ) \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate((8*x^3 - 27)/(64*x^6 - 729),x, algorithm="giac")

[Out]

1/54*sqrt(3)*arctan(1/9*sqrt(3)*(4*x - 3)) - 1/108*ln(x^2 - 3/2*x + 9/4) + 1/54*
ln(abs(x + 3/2))