3.563 \(\int \frac{x \left (27-2 x^3\right )}{729-64 x^6} \, dx\)

Optimal. Leaf size=99 \[ \frac{5}{576} \log \left (4 x^2-6 x+9\right )+\frac{1}{192} \log \left (4 x^2+6 x+9\right )-\frac{1}{96} \log (3-2 x)-\frac{5}{288} \log (2 x+3)-\frac{5 \tan ^{-1}\left (\frac{3-4 x}{3 \sqrt{3}}\right )}{96 \sqrt{3}}-\frac{\tan ^{-1}\left (\frac{4 x+3}{3 \sqrt{3}}\right )}{32 \sqrt{3}} \]

[Out]

(-5*ArcTan[(3 - 4*x)/(3*Sqrt[3])])/(96*Sqrt[3]) - ArcTan[(3 + 4*x)/(3*Sqrt[3])]/
(32*Sqrt[3]) - Log[3 - 2*x]/96 - (5*Log[3 + 2*x])/288 + (5*Log[9 - 6*x + 4*x^2])
/576 + Log[9 + 6*x + 4*x^2]/192

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Rubi [A]  time = 0.136423, antiderivative size = 99, normalized size of antiderivative = 1., number of steps used = 13, number of rules used = 7, integrand size = 18, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.389 \[ \frac{5}{576} \log \left (4 x^2-6 x+9\right )+\frac{1}{192} \log \left (4 x^2+6 x+9\right )-\frac{1}{96} \log (3-2 x)-\frac{5}{288} \log (2 x+3)-\frac{5 \tan ^{-1}\left (\frac{3-4 x}{3 \sqrt{3}}\right )}{96 \sqrt{3}}-\frac{\tan ^{-1}\left (\frac{4 x+3}{3 \sqrt{3}}\right )}{32 \sqrt{3}} \]

Antiderivative was successfully verified.

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

[Out]

(-5*ArcTan[(3 - 4*x)/(3*Sqrt[3])])/(96*Sqrt[3]) - ArcTan[(3 + 4*x)/(3*Sqrt[3])]/
(32*Sqrt[3]) - Log[3 - 2*x]/96 - (5*Log[3 + 2*x])/288 + (5*Log[9 - 6*x + 4*x^2])
/576 + Log[9 + 6*x + 4*x^2]/192

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Rubi in Sympy [A]  time = 19.2437, size = 92, normalized size = 0.93 \[ - \frac{\log{\left (- 2 x + 3 \right )}}{96} - \frac{5 \log{\left (2 x + 3 \right )}}{288} + \frac{5 \log{\left (16 x^{2} - 24 x + 36 \right )}}{576} + \frac{\log{\left (16 x^{2} + 24 x + 36 \right )}}{192} + \frac{5 \sqrt{3} \operatorname{atan}{\left (\sqrt{3} \left (\frac{4 x}{9} - \frac{1}{3}\right ) \right )}}{288} - \frac{\sqrt{3} \operatorname{atan}{\left (\sqrt{3} \left (\frac{4 x}{9} + \frac{1}{3}\right ) \right )}}{96} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

-log(-2*x + 3)/96 - 5*log(2*x + 3)/288 + 5*log(16*x**2 - 24*x + 36)/576 + log(16
*x**2 + 24*x + 36)/192 + 5*sqrt(3)*atan(sqrt(3)*(4*x/9 - 1/3))/288 - sqrt(3)*ata
n(sqrt(3)*(4*x/9 + 1/3))/96

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Mathematica [A]  time = 0.0317212, size = 91, normalized size = 0.92 \[ \frac{1}{576} \left (5 \log \left (4 x^2-6 x+9\right )+3 \log \left (4 x^2+6 x+9\right )-6 \log (3-2 x)-10 \log (2 x+3)+10 \sqrt{3} \tan ^{-1}\left (\frac{4 x-3}{3 \sqrt{3}}\right )-6 \sqrt{3} \tan ^{-1}\left (\frac{4 x+3}{3 \sqrt{3}}\right )\right ) \]

Antiderivative was successfully verified.

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

[Out]

(10*Sqrt[3]*ArcTan[(-3 + 4*x)/(3*Sqrt[3])] - 6*Sqrt[3]*ArcTan[(3 + 4*x)/(3*Sqrt[
3])] - 6*Log[3 - 2*x] - 10*Log[3 + 2*x] + 5*Log[9 - 6*x + 4*x^2] + 3*Log[9 + 6*x
+ 4*x^2])/576

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Maple [A]  time = 0.013, size = 76, normalized size = 0.8 \[ -{\frac{5\,\ln \left ( 2\,x+3 \right ) }{288}}-{\frac{\ln \left ( -3+2\,x \right ) }{96}}+{\frac{\ln \left ( 4\,{x}^{2}+6\,x+9 \right ) }{192}}-{\frac{\sqrt{3}}{96}\arctan \left ({\frac{ \left ( 8\,x+6 \right ) \sqrt{3}}{18}} \right ) }+{\frac{5\,\ln \left ( 4\,{x}^{2}-6\,x+9 \right ) }{576}}+{\frac{5\,\sqrt{3}}{288}\arctan \left ({\frac{ \left ( 8\,x-6 \right ) \sqrt{3}}{18}} \right ) } \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

-5/288*ln(2*x+3)-1/96*ln(-3+2*x)+1/192*ln(4*x^2+6*x+9)-1/96*3^(1/2)*arctan(1/18*
(8*x+6)*3^(1/2))+5/576*ln(4*x^2-6*x+9)+5/288*3^(1/2)*arctan(1/18*(8*x-6)*3^(1/2)
)

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Maxima [A]  time = 1.52351, size = 101, normalized size = 1.02 \[ -\frac{1}{96} \, \sqrt{3} \arctan \left (\frac{1}{9} \, \sqrt{3}{\left (4 \, x + 3\right )}\right ) + \frac{5}{288} \, \sqrt{3} \arctan \left (\frac{1}{9} \, \sqrt{3}{\left (4 \, x - 3\right )}\right ) + \frac{1}{192} \, \log \left (4 \, x^{2} + 6 \, x + 9\right ) + \frac{5}{576} \, \log \left (4 \, x^{2} - 6 \, x + 9\right ) - \frac{5}{288} \, \log \left (2 \, x + 3\right ) - \frac{1}{96} \, \log \left (2 \, x - 3\right ) \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

-1/96*sqrt(3)*arctan(1/9*sqrt(3)*(4*x + 3)) + 5/288*sqrt(3)*arctan(1/9*sqrt(3)*(
4*x - 3)) + 1/192*log(4*x^2 + 6*x + 9) + 5/576*log(4*x^2 - 6*x + 9) - 5/288*log(
2*x + 3) - 1/96*log(2*x - 3)

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Fricas [A]  time = 0.213124, size = 116, normalized size = 1.17 \[ \frac{1}{1728} \, \sqrt{3}{\left (3 \, \sqrt{3} \log \left (4 \, x^{2} + 6 \, x + 9\right ) + 5 \, \sqrt{3} \log \left (4 \, x^{2} - 6 \, x + 9\right ) - 10 \, \sqrt{3} \log \left (2 \, x + 3\right ) - 6 \, \sqrt{3} \log \left (2 \, x - 3\right ) - 18 \, \arctan \left (\frac{1}{9} \, \sqrt{3}{\left (4 \, x + 3\right )}\right ) + 30 \, \arctan \left (\frac{1}{9} \, \sqrt{3}{\left (4 \, x - 3\right )}\right )\right )} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

1/1728*sqrt(3)*(3*sqrt(3)*log(4*x^2 + 6*x + 9) + 5*sqrt(3)*log(4*x^2 - 6*x + 9)
- 10*sqrt(3)*log(2*x + 3) - 6*sqrt(3)*log(2*x - 3) - 18*arctan(1/9*sqrt(3)*(4*x
+ 3)) + 30*arctan(1/9*sqrt(3)*(4*x - 3)))

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Sympy [A]  time = 0.573778, size = 102, normalized size = 1.03 \[ - \frac{\log{\left (x - \frac{3}{2} \right )}}{96} - \frac{5 \log{\left (x + \frac{3}{2} \right )}}{288} + \frac{5 \log{\left (x^{2} - \frac{3 x}{2} + \frac{9}{4} \right )}}{576} + \frac{\log{\left (x^{2} + \frac{3 x}{2} + \frac{9}{4} \right )}}{192} + \frac{5 \sqrt{3} \operatorname{atan}{\left (\frac{4 \sqrt{3} x}{9} - \frac{\sqrt{3}}{3} \right )}}{288} - \frac{\sqrt{3} \operatorname{atan}{\left (\frac{4 \sqrt{3} x}{9} + \frac{\sqrt{3}}{3} \right )}}{96} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

-log(x - 3/2)/96 - 5*log(x + 3/2)/288 + 5*log(x**2 - 3*x/2 + 9/4)/576 + log(x**2
+ 3*x/2 + 9/4)/192 + 5*sqrt(3)*atan(4*sqrt(3)*x/9 - sqrt(3)/3)/288 - sqrt(3)*at
an(4*sqrt(3)*x/9 + sqrt(3)/3)/96

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GIAC/XCAS [A]  time = 0.220171, size = 93, normalized size = 0.94 \[ -\frac{1}{96} \, \sqrt{3} \arctan \left (\frac{1}{9} \, \sqrt{3}{\left (4 \, x + 3\right )}\right ) + \frac{5}{288} \, \sqrt{3} \arctan \left (\frac{1}{9} \, \sqrt{3}{\left (4 \, x - 3\right )}\right ) + \frac{1}{192} \,{\rm ln}\left (x^{2} + \frac{3}{2} \, x + \frac{9}{4}\right ) + \frac{5}{576} \,{\rm ln}\left (x^{2} - \frac{3}{2} \, x + \frac{9}{4}\right ) - \frac{5}{288} \,{\rm ln}\left ({\left | x + \frac{3}{2} \right |}\right ) - \frac{1}{96} \,{\rm ln}\left ({\left | x - \frac{3}{2} \right |}\right ) \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

-1/96*sqrt(3)*arctan(1/9*sqrt(3)*(4*x + 3)) + 5/288*sqrt(3)*arctan(1/9*sqrt(3)*(
4*x - 3)) + 1/192*ln(x^2 + 3/2*x + 9/4) + 5/576*ln(x^2 - 3/2*x + 9/4) - 5/288*ln
(abs(x + 3/2)) - 1/96*ln(abs(x - 3/2))