3.552 \(\int \frac{27+36 x+24 x^2+8 x^3}{729-64 x^6} \, dx\)

Optimal. Leaf size=50 \[ \frac{1}{36} \log \left (4 x^2-6 x+9\right )-\frac{1}{18} \log (3-2 x)-\frac{\tan ^{-1}\left (\frac{3-4 x}{3 \sqrt{3}}\right )}{18 \sqrt{3}} \]

[Out]

-ArcTan[(3 - 4*x)/(3*Sqrt[3])]/(18*Sqrt[3]) - Log[3 - 2*x]/18 + Log[9 - 6*x + 4*
x^2]/36

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Rubi [A]  time = 0.072656, antiderivative size = 50, normalized size of antiderivative = 1., number of steps used = 7, number of rules used = 6, integrand size = 25, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.24 \[ \frac{1}{36} \log \left (4 x^2-6 x+9\right )-\frac{1}{18} \log (3-2 x)-\frac{\tan ^{-1}\left (\frac{3-4 x}{3 \sqrt{3}}\right )}{18 \sqrt{3}} \]

Antiderivative was successfully verified.

[In]  Int[(27 + 36*x + 24*x^2 + 8*x^3)/(729 - 64*x^6),x]

[Out]

-ArcTan[(3 - 4*x)/(3*Sqrt[3])]/(18*Sqrt[3]) - Log[3 - 2*x]/18 + Log[9 - 6*x + 4*
x^2]/36

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Rubi in Sympy [A]  time = 20.4172, size = 44, normalized size = 0.88 \[ - \frac{\log{\left (- 2 x + 3 \right )}}{18} + \frac{\log{\left (2 x + 4 \left (x - 1\right )^{2} + 5 \right )}}{36} + \frac{\sqrt{3} \operatorname{atan}{\left (\sqrt{3} \left (\frac{4 x}{9} - \frac{1}{3}\right ) \right )}}{54} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  rubi_integrate((8*x**3+24*x**2+36*x+27)/(-64*x**6+729),x)

[Out]

-log(-2*x + 3)/18 + log(2*x + 4*(x - 1)**2 + 5)/36 + sqrt(3)*atan(sqrt(3)*(4*x/9
- 1/3))/54

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Mathematica [A]  time = 0.0168301, size = 50, normalized size = 1. \[ \frac{1}{36} \log \left (4 x^2-6 x+9\right )-\frac{1}{18} \log (3-2 x)+\frac{\tan ^{-1}\left (\frac{4 x-3}{3 \sqrt{3}}\right )}{18 \sqrt{3}} \]

Antiderivative was successfully verified.

[In]  Integrate[(27 + 36*x + 24*x^2 + 8*x^3)/(729 - 64*x^6),x]

[Out]

ArcTan[(-3 + 4*x)/(3*Sqrt[3])]/(18*Sqrt[3]) - Log[3 - 2*x]/18 + Log[9 - 6*x + 4*
x^2]/36

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Maple [A]  time = 0.008, size = 39, normalized size = 0.8 \[ -{\frac{\ln \left ( -3+2\,x \right ) }{18}}+{\frac{\ln \left ( 4\,{x}^{2}-6\,x+9 \right ) }{36}}+{\frac{\sqrt{3}}{54}\arctan \left ({\frac{ \left ( 8\,x-6 \right ) \sqrt{3}}{18}} \right ) } \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  int((8*x^3+24*x^2+36*x+27)/(-64*x^6+729),x)

[Out]

-1/18*ln(-3+2*x)+1/36*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 = 1.52891, size = 51, normalized size = 1.02 \[ \frac{1}{54} \, \sqrt{3} \arctan \left (\frac{1}{9} \, \sqrt{3}{\left (4 \, x - 3\right )}\right ) + \frac{1}{36} \, \log \left (4 \, x^{2} - 6 \, x + 9\right ) - \frac{1}{18} \, \log \left (2 \, x - 3\right ) \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate(-(8*x^3 + 24*x^2 + 36*x + 27)/(64*x^6 - 729),x, algorithm="maxima")

[Out]

1/54*sqrt(3)*arctan(1/9*sqrt(3)*(4*x - 3)) + 1/36*log(4*x^2 - 6*x + 9) - 1/18*lo
g(2*x - 3)

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Fricas [A]  time = 0.210262, size = 61, normalized size = 1.22 \[ \frac{1}{108} \, \sqrt{3}{\left (\sqrt{3} \log \left (4 \, x^{2} - 6 \, x + 9\right ) - 2 \, \sqrt{3} \log \left (2 \, x - 3\right ) + 2 \, \arctan \left (\frac{1}{9} \, \sqrt{3}{\left (4 \, x - 3\right )}\right )\right )} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate(-(8*x^3 + 24*x^2 + 36*x + 27)/(64*x^6 - 729),x, algorithm="fricas")

[Out]

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

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Sympy [A]  time = 0.206795, size = 48, normalized size = 0.96 \[ - \frac{\log{\left (x - \frac{3}{2} \right )}}{18} + \frac{\log{\left (x^{2} - \frac{3 x}{2} + \frac{9}{4} \right )}}{36} + \frac{\sqrt{3} \operatorname{atan}{\left (\frac{4 \sqrt{3} x}{9} - \frac{\sqrt{3}}{3} \right )}}{54} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate((8*x**3+24*x**2+36*x+27)/(-64*x**6+729),x)

[Out]

-log(x - 3/2)/18 + log(x**2 - 3*x/2 + 9/4)/36 + sqrt(3)*atan(4*sqrt(3)*x/9 - sqr
t(3)/3)/54

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GIAC/XCAS [A]  time = 0.218279, size = 53, normalized size = 1.06 \[ \frac{1}{54} \, \sqrt{3} \arctan \left (\frac{1}{9} \, \sqrt{3}{\left (4 \, x - 3\right )}\right ) + \frac{1}{36} \,{\rm ln}\left (4 \, x^{2} - 6 \, x + 9\right ) - \frac{1}{18} \,{\rm ln}\left ({\left | 2 \, x - 3 \right |}\right ) \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate(-(8*x^3 + 24*x^2 + 36*x + 27)/(64*x^6 - 729),x, algorithm="giac")

[Out]

1/54*sqrt(3)*arctan(1/9*sqrt(3)*(4*x - 3)) + 1/36*ln(4*x^2 - 6*x + 9) - 1/18*ln(
abs(2*x - 3))