3.553 \(\int \frac{243-162 x+108 x^2-72 x^3+48 x^4-32 x^5}{\left (729-64 x^6\right )^2} \, dx\)

Optimal. Leaf size=110 \[ -\frac{\log \left (4 x^2-6 x+9\right )}{17496}-\frac{\log \left (4 x^2+6 x+9\right )}{17496}-\frac{1}{2916 (2 x+3)}-\frac{\log (3-2 x)}{17496}+\frac{5 \log (2 x+3)}{17496}-\frac{\tan ^{-1}\left (\frac{3-4 x}{3 \sqrt{3}}\right )}{8748 \sqrt{3}}+\frac{\tan ^{-1}\left (\frac{4 x+3}{3 \sqrt{3}}\right )}{2916 \sqrt{3}} \]

[Out]

-1/(2916*(3 + 2*x)) - ArcTan[(3 - 4*x)/(3*Sqrt[3])]/(8748*Sqrt[3]) + ArcTan[(3 +
4*x)/(3*Sqrt[3])]/(2916*Sqrt[3]) - Log[3 - 2*x]/17496 + (5*Log[3 + 2*x])/17496
- Log[9 - 6*x + 4*x^2]/17496 - Log[9 + 6*x + 4*x^2]/17496

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Rubi [A]  time = 0.218052, antiderivative size = 110, normalized size of antiderivative = 1., number of steps used = 11, number of rules used = 6, integrand size = 35, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.171 \[ -\frac{\log \left (4 x^2-6 x+9\right )}{17496}-\frac{\log \left (4 x^2+6 x+9\right )}{17496}-\frac{1}{2916 (2 x+3)}-\frac{\log (3-2 x)}{17496}+\frac{5 \log (2 x+3)}{17496}-\frac{\tan ^{-1}\left (\frac{3-4 x}{3 \sqrt{3}}\right )}{8748 \sqrt{3}}+\frac{\tan ^{-1}\left (\frac{4 x+3}{3 \sqrt{3}}\right )}{2916 \sqrt{3}} \]

Antiderivative was successfully verified.

[In]  Int[(243 - 162*x + 108*x^2 - 72*x^3 + 48*x^4 - 32*x^5)/(729 - 64*x^6)^2,x]

[Out]

-1/(2916*(3 + 2*x)) - ArcTan[(3 - 4*x)/(3*Sqrt[3])]/(8748*Sqrt[3]) + ArcTan[(3 +
4*x)/(3*Sqrt[3])]/(2916*Sqrt[3]) - Log[3 - 2*x]/17496 + (5*Log[3 + 2*x])/17496
- Log[9 - 6*x + 4*x^2]/17496 - Log[9 + 6*x + 4*x^2]/17496

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Rubi in Sympy [F(-1)]  time = 0., size = 0, normalized size = 0. \[ \text{Timed out} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  rubi_integrate((-32*x**5+48*x**4-72*x**3+108*x**2-162*x+243)/(-64*x**6+729)**2,x)

[Out]

Timed out

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Mathematica [A]  time = 0.172915, size = 100, normalized size = 0.91 \[ \frac{-3 \log \left (4 x^2-6 x+9\right )-3 \log \left (4 x^2+6 x+9\right )-\frac{18}{2 x+3}-3 \log (3-2 x)+15 \log (2 x+3)+2 \sqrt{3} \tan ^{-1}\left (\frac{4 x-3}{3 \sqrt{3}}\right )+6 \sqrt{3} \tan ^{-1}\left (\frac{4 x+3}{3 \sqrt{3}}\right )}{52488} \]

Antiderivative was successfully verified.

[In]  Integrate[(243 - 162*x + 108*x^2 - 72*x^3 + 48*x^4 - 32*x^5)/(729 - 64*x^6)^2,x]

[Out]

(-18/(3 + 2*x) + 2*Sqrt[3]*ArcTan[(-3 + 4*x)/(3*Sqrt[3])] + 6*Sqrt[3]*ArcTan[(3
+ 4*x)/(3*Sqrt[3])] - 3*Log[3 - 2*x] + 15*Log[3 + 2*x] - 3*Log[9 - 6*x + 4*x^2]
- 3*Log[9 + 6*x + 4*x^2])/52488

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Maple [A]  time = 0.016, size = 85, normalized size = 0.8 \[ -{\frac{1}{5832\,x+8748}}+{\frac{5\,\ln \left ( 2\,x+3 \right ) }{17496}}-{\frac{\ln \left ( -3+2\,x \right ) }{17496}}-{\frac{\ln \left ( 4\,{x}^{2}+6\,x+9 \right ) }{17496}}+{\frac{\sqrt{3}}{8748}\arctan \left ({\frac{ \left ( 8\,x+6 \right ) \sqrt{3}}{18}} \right ) }-{\frac{\ln \left ( 4\,{x}^{2}-6\,x+9 \right ) }{17496}}+{\frac{\sqrt{3}}{26244}\arctan \left ({\frac{ \left ( 8\,x-6 \right ) \sqrt{3}}{18}} \right ) } \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  int((-32*x^5+48*x^4-72*x^3+108*x^2-162*x+243)/(-64*x^6+729)^2,x)

[Out]

-1/2916/(2*x+3)+5/17496*ln(2*x+3)-1/17496*ln(-3+2*x)-1/17496*ln(4*x^2+6*x+9)+1/8
748*3^(1/2)*arctan(1/18*(8*x+6)*3^(1/2))-1/17496*ln(4*x^2-6*x+9)+1/26244*3^(1/2)
*arctan(1/18*(8*x-6)*3^(1/2))

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Maxima [A]  time = 1.51212, size = 113, normalized size = 1.03 \[ \frac{1}{8748} \, \sqrt{3} \arctan \left (\frac{1}{9} \, \sqrt{3}{\left (4 \, x + 3\right )}\right ) + \frac{1}{26244} \, \sqrt{3} \arctan \left (\frac{1}{9} \, \sqrt{3}{\left (4 \, x - 3\right )}\right ) - \frac{1}{2916 \,{\left (2 \, x + 3\right )}} - \frac{1}{17496} \, \log \left (4 \, x^{2} + 6 \, x + 9\right ) - \frac{1}{17496} \, \log \left (4 \, x^{2} - 6 \, x + 9\right ) + \frac{5}{17496} \, \log \left (2 \, x + 3\right ) - \frac{1}{17496} \, \log \left (2 \, x - 3\right ) \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate(-(32*x^5 - 48*x^4 + 72*x^3 - 108*x^2 + 162*x - 243)/(64*x^6 - 729)^2,x, algorithm="maxima")

[Out]

1/8748*sqrt(3)*arctan(1/9*sqrt(3)*(4*x + 3)) + 1/26244*sqrt(3)*arctan(1/9*sqrt(3
)*(4*x - 3)) - 1/2916/(2*x + 3) - 1/17496*log(4*x^2 + 6*x + 9) - 1/17496*log(4*x
^2 - 6*x + 9) + 5/17496*log(2*x + 3) - 1/17496*log(2*x - 3)

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Fricas [A]  time = 0.214244, size = 169, normalized size = 1.54 \[ -\frac{\sqrt{3}{\left (\sqrt{3}{\left (2 \, x + 3\right )} \log \left (4 \, x^{2} + 6 \, x + 9\right ) + \sqrt{3}{\left (2 \, x + 3\right )} \log \left (4 \, x^{2} - 6 \, x + 9\right ) - 5 \, \sqrt{3}{\left (2 \, x + 3\right )} \log \left (2 \, x + 3\right ) + \sqrt{3}{\left (2 \, x + 3\right )} \log \left (2 \, x - 3\right ) - 6 \,{\left (2 \, x + 3\right )} \arctan \left (\frac{1}{9} \, \sqrt{3}{\left (4 \, x + 3\right )}\right ) - 2 \,{\left (2 \, x + 3\right )} \arctan \left (\frac{1}{9} \, \sqrt{3}{\left (4 \, x - 3\right )}\right ) + 6 \, \sqrt{3}\right )}}{52488 \,{\left (2 \, x + 3\right )}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate(-(32*x^5 - 48*x^4 + 72*x^3 - 108*x^2 + 162*x - 243)/(64*x^6 - 729)^2,x, algorithm="fricas")

[Out]

-1/52488*sqrt(3)*(sqrt(3)*(2*x + 3)*log(4*x^2 + 6*x + 9) + sqrt(3)*(2*x + 3)*log
(4*x^2 - 6*x + 9) - 5*sqrt(3)*(2*x + 3)*log(2*x + 3) + sqrt(3)*(2*x + 3)*log(2*x
- 3) - 6*(2*x + 3)*arctan(1/9*sqrt(3)*(4*x + 3)) - 2*(2*x + 3)*arctan(1/9*sqrt(
3)*(4*x - 3)) + 6*sqrt(3))/(2*x + 3)

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Sympy [A]  time = 0.594917, size = 105, normalized size = 0.95 \[ - \frac{\log{\left (x - \frac{3}{2} \right )}}{17496} + \frac{5 \log{\left (x + \frac{3}{2} \right )}}{17496} - \frac{\log{\left (x^{2} - \frac{3 x}{2} + \frac{9}{4} \right )}}{17496} - \frac{\log{\left (x^{2} + \frac{3 x}{2} + \frac{9}{4} \right )}}{17496} + \frac{\sqrt{3} \operatorname{atan}{\left (\frac{4 \sqrt{3} x}{9} - \frac{\sqrt{3}}{3} \right )}}{26244} + \frac{\sqrt{3} \operatorname{atan}{\left (\frac{4 \sqrt{3} x}{9} + \frac{\sqrt{3}}{3} \right )}}{8748} - \frac{1}{5832 x + 8748} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate((-32*x**5+48*x**4-72*x**3+108*x**2-162*x+243)/(-64*x**6+729)**2,x)

[Out]

-log(x - 3/2)/17496 + 5*log(x + 3/2)/17496 - log(x**2 - 3*x/2 + 9/4)/17496 - log
(x**2 + 3*x/2 + 9/4)/17496 + sqrt(3)*atan(4*sqrt(3)*x/9 - sqrt(3)/3)/26244 + sqr
t(3)*atan(4*sqrt(3)*x/9 + sqrt(3)/3)/8748 - 1/(5832*x + 8748)

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GIAC/XCAS [A]  time = 0.218589, size = 116, normalized size = 1.05 \[ \frac{1}{8748} \, \sqrt{3} \arctan \left (\frac{1}{9} \, \sqrt{3}{\left (4 \, x + 3\right )}\right ) + \frac{1}{26244} \, \sqrt{3} \arctan \left (\frac{1}{9} \, \sqrt{3}{\left (4 \, x - 3\right )}\right ) - \frac{1}{2916 \,{\left (2 \, x + 3\right )}} - \frac{1}{17496} \,{\rm ln}\left (4 \, x^{2} + 6 \, x + 9\right ) - \frac{1}{17496} \,{\rm ln}\left (4 \, x^{2} - 6 \, x + 9\right ) + \frac{5}{17496} \,{\rm ln}\left ({\left | 2 \, x + 3 \right |}\right ) - \frac{1}{17496} \,{\rm ln}\left ({\left | 2 \, x - 3 \right |}\right ) \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate(-(32*x^5 - 48*x^4 + 72*x^3 - 108*x^2 + 162*x - 243)/(64*x^6 - 729)^2,x, algorithm="giac")

[Out]

1/8748*sqrt(3)*arctan(1/9*sqrt(3)*(4*x + 3)) + 1/26244*sqrt(3)*arctan(1/9*sqrt(3
)*(4*x - 3)) - 1/2916/(2*x + 3) - 1/17496*ln(4*x^2 + 6*x + 9) - 1/17496*ln(4*x^2
- 6*x + 9) + 5/17496*ln(abs(2*x + 3)) - 1/17496*ln(abs(2*x - 3))