3.549 \(\int \frac{9-6 x+4 x^2}{729-64 x^6} \, dx\)

Optimal. Leaf size=60 \[ -\frac{1}{324} \log \left (4 x^2+6 x+9\right )-\frac{1}{324} \log (3-2 x)+\frac{1}{108} \log (2 x+3)+\frac{\tan ^{-1}\left (\frac{4 x+3}{3 \sqrt{3}}\right )}{54 \sqrt{3}} \]

[Out]

ArcTan[(3 + 4*x)/(3*Sqrt[3])]/(54*Sqrt[3]) - Log[3 - 2*x]/324 + Log[3 + 2*x]/108
- Log[9 + 6*x + 4*x^2]/324

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Rubi [A]  time = 0.0884097, antiderivative size = 60, normalized size of antiderivative = 1., number of steps used = 7, number of rules used = 6, integrand size = 20, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.3 \[ -\frac{1}{324} \log \left (4 x^2+6 x+9\right )-\frac{1}{324} \log (3-2 x)+\frac{1}{108} \log (2 x+3)+\frac{\tan ^{-1}\left (\frac{4 x+3}{3 \sqrt{3}}\right )}{54 \sqrt{3}} \]

Antiderivative was successfully verified.

[In]  Int[(9 - 6*x + 4*x^2)/(729 - 64*x^6),x]

[Out]

ArcTan[(3 + 4*x)/(3*Sqrt[3])]/(54*Sqrt[3]) - Log[3 - 2*x]/324 + Log[3 + 2*x]/108
- Log[9 + 6*x + 4*x^2]/324

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Rubi in Sympy [F]  time = 0., size = 0, normalized size = 0. \[ \int \frac{1}{- 16 x^{4} - 24 x^{3} + 54 x + 81}\, dx \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  rubi_integrate((4*x**2-6*x+9)/(-64*x**6+729),x)

[Out]

Integral(1/(-16*x**4 - 24*x**3 + 54*x + 81), x)

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Mathematica [A]  time = 0.0199033, size = 56, normalized size = 0.93 \[ \frac{1}{324} \left (-\log \left (4 x^2+6 x+9\right )-\log (3-2 x)+3 \log (2 x+3)+2 \sqrt{3} \tan ^{-1}\left (\frac{4 x+3}{3 \sqrt{3}}\right )\right ) \]

Antiderivative was successfully verified.

[In]  Integrate[(9 - 6*x + 4*x^2)/(729 - 64*x^6),x]

[Out]

(2*Sqrt[3]*ArcTan[(3 + 4*x)/(3*Sqrt[3])] - Log[3 - 2*x] + 3*Log[3 + 2*x] - Log[9
+ 6*x + 4*x^2])/324

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Maple [A]  time = 0.01, size = 47, normalized size = 0.8 \[{\frac{\ln \left ( 2\,x+3 \right ) }{108}}-{\frac{\ln \left ( -3+2\,x \right ) }{324}}-{\frac{\ln \left ( 4\,{x}^{2}+6\,x+9 \right ) }{324}}+{\frac{\sqrt{3}}{162}\arctan \left ({\frac{ \left ( 8\,x+6 \right ) \sqrt{3}}{18}} \right ) } \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  int((4*x^2-6*x+9)/(-64*x^6+729),x)

[Out]

1/108*ln(2*x+3)-1/324*ln(-3+2*x)-1/324*ln(4*x^2+6*x+9)+1/162*3^(1/2)*arctan(1/18
*(8*x+6)*3^(1/2))

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Maxima [A]  time = 1.52214, size = 62, normalized size = 1.03 \[ \frac{1}{162} \, \sqrt{3} \arctan \left (\frac{1}{9} \, \sqrt{3}{\left (4 \, x + 3\right )}\right ) - \frac{1}{324} \, \log \left (4 \, x^{2} + 6 \, x + 9\right ) + \frac{1}{108} \, \log \left (2 \, x + 3\right ) - \frac{1}{324} \, \log \left (2 \, x - 3\right ) \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate(-(4*x^2 - 6*x + 9)/(64*x^6 - 729),x, algorithm="maxima")

[Out]

1/162*sqrt(3)*arctan(1/9*sqrt(3)*(4*x + 3)) - 1/324*log(4*x^2 + 6*x + 9) + 1/108
*log(2*x + 3) - 1/324*log(2*x - 3)

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Fricas [A]  time = 0.212307, size = 74, normalized size = 1.23 \[ -\frac{1}{972} \, \sqrt{3}{\left (\sqrt{3} \log \left (4 \, x^{2} + 6 \, x + 9\right ) - 3 \, \sqrt{3} \log \left (2 \, x + 3\right ) + \sqrt{3} \log \left (2 \, x - 3\right ) - 6 \, \arctan \left (\frac{1}{9} \, \sqrt{3}{\left (4 \, x + 3\right )}\right )\right )} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate(-(4*x^2 - 6*x + 9)/(64*x^6 - 729),x, algorithm="fricas")

[Out]

-1/972*sqrt(3)*(sqrt(3)*log(4*x^2 + 6*x + 9) - 3*sqrt(3)*log(2*x + 3) + sqrt(3)*
log(2*x - 3) - 6*arctan(1/9*sqrt(3)*(4*x + 3)))

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Sympy [A]  time = 0.284135, size = 56, normalized size = 0.93 \[ - \frac{\log{\left (x - \frac{3}{2} \right )}}{324} + \frac{\log{\left (x + \frac{3}{2} \right )}}{108} - \frac{\log{\left (x^{2} + \frac{3 x}{2} + \frac{9}{4} \right )}}{324} + \frac{\sqrt{3} \operatorname{atan}{\left (\frac{4 \sqrt{3} x}{9} + \frac{\sqrt{3}}{3} \right )}}{162} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate((4*x**2-6*x+9)/(-64*x**6+729),x)

[Out]

-log(x - 3/2)/324 + log(x + 3/2)/108 - log(x**2 + 3*x/2 + 9/4)/324 + sqrt(3)*ata
n(4*sqrt(3)*x/9 + sqrt(3)/3)/162

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GIAC/XCAS [A]  time = 0.220924, size = 65, normalized size = 1.08 \[ \frac{1}{162} \, \sqrt{3} \arctan \left (\frac{1}{9} \, \sqrt{3}{\left (4 \, x + 3\right )}\right ) - \frac{1}{324} \,{\rm ln}\left (4 \, x^{2} + 6 \, x + 9\right ) + \frac{1}{108} \,{\rm ln}\left ({\left | 2 \, x + 3 \right |}\right ) - \frac{1}{324} \,{\rm ln}\left ({\left | 2 \, x - 3 \right |}\right ) \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate(-(4*x^2 - 6*x + 9)/(64*x^6 - 729),x, algorithm="giac")

[Out]

1/162*sqrt(3)*arctan(1/9*sqrt(3)*(4*x + 3)) - 1/324*ln(4*x^2 + 6*x + 9) + 1/108*
ln(abs(2*x + 3)) - 1/324*ln(abs(2*x - 3))