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

Optimal. Leaf size=131 \[ -\frac{3-2 x}{26244 \left (4 x^2-6 x+9\right )}+\frac{17 \log \left (4 x^2-6 x+9\right )}{944784}+\frac{\log \left (4 x^2+6 x+9\right )}{314928}+\frac{1}{26244 (3-2 x)}-\frac{7 \log (3-2 x)}{157464}+\frac{\log (2 x+3)}{472392}-\frac{11 \tan ^{-1}\left (\frac{3-4 x}{3 \sqrt{3}}\right )}{157464 \sqrt{3}}-\frac{\tan ^{-1}\left (\frac{4 x+3}{3 \sqrt{3}}\right )}{157464 \sqrt{3}} \]

[Out]

1/(26244*(3 - 2*x)) - (3 - 2*x)/(26244*(9 - 6*x + 4*x^2)) - (11*ArcTan[(3 - 4*x)
/(3*Sqrt[3])])/(157464*Sqrt[3]) - ArcTan[(3 + 4*x)/(3*Sqrt[3])]/(157464*Sqrt[3])
- (7*Log[3 - 2*x])/157464 + Log[3 + 2*x]/472392 + (17*Log[9 - 6*x + 4*x^2])/944
784 + Log[9 + 6*x + 4*x^2]/314928

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Rubi [A]  time = 0.263878, antiderivative size = 131, normalized size of antiderivative = 1., number of steps used = 14, number of rules used = 7, integrand size = 25, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.28 \[ -\frac{3-2 x}{26244 \left (4 x^2-6 x+9\right )}+\frac{17 \log \left (4 x^2-6 x+9\right )}{944784}+\frac{\log \left (4 x^2+6 x+9\right )}{314928}+\frac{1}{26244 (3-2 x)}-\frac{7 \log (3-2 x)}{157464}+\frac{\log (2 x+3)}{472392}-\frac{11 \tan ^{-1}\left (\frac{3-4 x}{3 \sqrt{3}}\right )}{157464 \sqrt{3}}-\frac{\tan ^{-1}\left (\frac{4 x+3}{3 \sqrt{3}}\right )}{157464 \sqrt{3}} \]

Antiderivative was successfully verified.

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

[Out]

1/(26244*(3 - 2*x)) - (3 - 2*x)/(26244*(9 - 6*x + 4*x^2)) - (11*ArcTan[(3 - 4*x)
/(3*Sqrt[3])])/(157464*Sqrt[3]) - ArcTan[(3 + 4*x)/(3*Sqrt[3])]/(157464*Sqrt[3])
- (7*Log[3 - 2*x])/157464 + Log[3 + 2*x]/472392 + (17*Log[9 - 6*x + 4*x^2])/944
784 + Log[9 + 6*x + 4*x^2]/314928

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

[Out]

Timed out

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Mathematica [A]  time = 0.110413, size = 111, normalized size = 0.85 \[ \frac{17 \log \left (4 x^2-6 x+9\right )+3 \log \left (4 x^2+6 x+9\right )+\frac{216 x}{-8 x^3+24 x^2-36 x+27}-42 \log (3-2 x)+2 \log (2 x+3)+22 \sqrt{3} \tan ^{-1}\left (\frac{4 x-3}{3 \sqrt{3}}\right )-2 \sqrt{3} \tan ^{-1}\left (\frac{4 x+3}{3 \sqrt{3}}\right )}{944784} \]

Antiderivative was successfully verified.

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

[Out]

((216*x)/(27 - 36*x + 24*x^2 - 8*x^3) + 22*Sqrt[3]*ArcTan[(-3 + 4*x)/(3*Sqrt[3])
] - 2*Sqrt[3]*ArcTan[(3 + 4*x)/(3*Sqrt[3])] - 42*Log[3 - 2*x] + 2*Log[3 + 2*x] +
17*Log[9 - 6*x + 4*x^2] + 3*Log[9 + 6*x + 4*x^2])/944784

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Maple [A]  time = 0.019, size = 102, normalized size = 0.8 \[{\frac{\ln \left ( 2\,x+3 \right ) }{472392}}-{\frac{1}{-78732+52488\,x}}-{\frac{7\,\ln \left ( -3+2\,x \right ) }{157464}}+{\frac{\ln \left ( 4\,{x}^{2}+6\,x+9 \right ) }{314928}}-{\frac{\sqrt{3}}{472392}\arctan \left ({\frac{ \left ( 8\,x+6 \right ) \sqrt{3}}{18}} \right ) }+{\frac{1}{118098} \left ({\frac{9\,x}{4}}-{\frac{27}{8}} \right ) \left ({x}^{2}-{\frac{3\,x}{2}}+{\frac{9}{4}} \right ) ^{-1}}+{\frac{17\,\ln \left ( 16\,{x}^{2}-24\,x+36 \right ) }{944784}}+{\frac{11\,\sqrt{3}}{472392}\arctan \left ({\frac{ \left ( 32\,x-24 \right ) \sqrt{3}}{72}} \right ) } \]

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)^2,x)

[Out]

1/472392*ln(2*x+3)-1/26244/(-3+2*x)-7/157464*ln(-3+2*x)+1/314928*ln(4*x^2+6*x+9)
-1/472392*3^(1/2)*arctan(1/18*(8*x+6)*3^(1/2))+1/118098*(9/4*x-27/8)/(x^2-3/2*x+
9/4)+17/944784*ln(16*x^2-24*x+36)+11/472392*3^(1/2)*arctan(1/72*(32*x-24)*3^(1/2
))

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Maxima [A]  time = 1.54312, size = 128, normalized size = 0.98 \[ -\frac{1}{472392} \, \sqrt{3} \arctan \left (\frac{1}{9} \, \sqrt{3}{\left (4 \, x + 3\right )}\right ) + \frac{11}{472392} \, \sqrt{3} \arctan \left (\frac{1}{9} \, \sqrt{3}{\left (4 \, x - 3\right )}\right ) - \frac{x}{4374 \,{\left (8 \, x^{3} - 24 \, x^{2} + 36 \, x - 27\right )}} + \frac{1}{314928} \, \log \left (4 \, x^{2} + 6 \, x + 9\right ) + \frac{17}{944784} \, \log \left (4 \, x^{2} - 6 \, x + 9\right ) + \frac{1}{472392} \, \log \left (2 \, x + 3\right ) - \frac{7}{157464} \, \log \left (2 \, x - 3\right ) \]

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)^2,x, algorithm="maxima")

[Out]

-1/472392*sqrt(3)*arctan(1/9*sqrt(3)*(4*x + 3)) + 11/472392*sqrt(3)*arctan(1/9*s
qrt(3)*(4*x - 3)) - 1/4374*x/(8*x^3 - 24*x^2 + 36*x - 27) + 1/314928*log(4*x^2 +
6*x + 9) + 17/944784*log(4*x^2 - 6*x + 9) + 1/472392*log(2*x + 3) - 7/157464*lo
g(2*x - 3)

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Fricas [A]  time = 0.213829, size = 269, normalized size = 2.05 \[ \frac{\sqrt{3}{\left (3 \, \sqrt{3}{\left (8 \, x^{3} - 24 \, x^{2} + 36 \, x - 27\right )} \log \left (4 \, x^{2} + 6 \, x + 9\right ) + 17 \, \sqrt{3}{\left (8 \, x^{3} - 24 \, x^{2} + 36 \, x - 27\right )} \log \left (4 \, x^{2} - 6 \, x + 9\right ) + 2 \, \sqrt{3}{\left (8 \, x^{3} - 24 \, x^{2} + 36 \, x - 27\right )} \log \left (2 \, x + 3\right ) - 42 \, \sqrt{3}{\left (8 \, x^{3} - 24 \, x^{2} + 36 \, x - 27\right )} \log \left (2 \, x - 3\right ) - 6 \,{\left (8 \, x^{3} - 24 \, x^{2} + 36 \, x - 27\right )} \arctan \left (\frac{1}{9} \, \sqrt{3}{\left (4 \, x + 3\right )}\right ) + 66 \,{\left (8 \, x^{3} - 24 \, x^{2} + 36 \, x - 27\right )} \arctan \left (\frac{1}{9} \, \sqrt{3}{\left (4 \, x - 3\right )}\right ) - 216 \, \sqrt{3} x\right )}}{2834352 \,{\left (8 \, x^{3} - 24 \, x^{2} + 36 \, x - 27\right )}} \]

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)^2,x, algorithm="fricas")

[Out]

1/2834352*sqrt(3)*(3*sqrt(3)*(8*x^3 - 24*x^2 + 36*x - 27)*log(4*x^2 + 6*x + 9) +
17*sqrt(3)*(8*x^3 - 24*x^2 + 36*x - 27)*log(4*x^2 - 6*x + 9) + 2*sqrt(3)*(8*x^3
- 24*x^2 + 36*x - 27)*log(2*x + 3) - 42*sqrt(3)*(8*x^3 - 24*x^2 + 36*x - 27)*lo
g(2*x - 3) - 6*(8*x^3 - 24*x^2 + 36*x - 27)*arctan(1/9*sqrt(3)*(4*x + 3)) + 66*(
8*x^3 - 24*x^2 + 36*x - 27)*arctan(1/9*sqrt(3)*(4*x - 3)) - 216*sqrt(3)*x)/(8*x^
3 - 24*x^2 + 36*x - 27)

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Sympy [A]  time = 0.82442, size = 119, normalized size = 0.91 \[ - \frac{x}{34992 x^{3} - 104976 x^{2} + 157464 x - 118098} - \frac{7 \log{\left (x - \frac{3}{2} \right )}}{157464} + \frac{\log{\left (x + \frac{3}{2} \right )}}{472392} + \frac{17 \log{\left (x^{2} - \frac{3 x}{2} + \frac{9}{4} \right )}}{944784} + \frac{\log{\left (x^{2} + \frac{3 x}{2} + \frac{9}{4} \right )}}{314928} + \frac{11 \sqrt{3} \operatorname{atan}{\left (\frac{4 \sqrt{3} x}{9} - \frac{\sqrt{3}}{3} \right )}}{472392} - \frac{\sqrt{3} \operatorname{atan}{\left (\frac{4 \sqrt{3} x}{9} + \frac{\sqrt{3}}{3} \right )}}{472392} \]

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)**2,x)

[Out]

-x/(34992*x**3 - 104976*x**2 + 157464*x - 118098) - 7*log(x - 3/2)/157464 + log(
x + 3/2)/472392 + 17*log(x**2 - 3*x/2 + 9/4)/944784 + log(x**2 + 3*x/2 + 9/4)/31
4928 + 11*sqrt(3)*atan(4*sqrt(3)*x/9 - sqrt(3)/3)/472392 - sqrt(3)*atan(4*sqrt(3
)*x/9 + sqrt(3)/3)/472392

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GIAC/XCAS [A]  time = 0.219799, size = 134, normalized size = 1.02 \[ -\frac{1}{472392} \, \sqrt{3} \arctan \left (\frac{1}{9} \, \sqrt{3}{\left (4 \, x + 3\right )}\right ) + \frac{11}{472392} \, \sqrt{3} \arctan \left (\frac{1}{9} \, \sqrt{3}{\left (4 \, x - 3\right )}\right ) - \frac{x}{4374 \,{\left (4 \, x^{2} - 6 \, x + 9\right )}{\left (2 \, x - 3\right )}} + \frac{1}{314928} \,{\rm ln}\left (4 \, x^{2} + 6 \, x + 9\right ) + \frac{17}{944784} \,{\rm ln}\left (4 \, x^{2} - 6 \, x + 9\right ) + \frac{1}{472392} \,{\rm ln}\left ({\left | 2 \, x + 3 \right |}\right ) - \frac{7}{157464} \,{\rm ln}\left ({\left | 2 \, x - 3 \right |}\right ) \]

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)^2,x, algorithm="giac")

[Out]

-1/472392*sqrt(3)*arctan(1/9*sqrt(3)*(4*x + 3)) + 11/472392*sqrt(3)*arctan(1/9*s
qrt(3)*(4*x - 3)) - 1/4374*x/((4*x^2 - 6*x + 9)*(2*x - 3)) + 1/314928*ln(4*x^2 +
6*x + 9) + 17/944784*ln(4*x^2 - 6*x + 9) + 1/472392*ln(abs(2*x + 3)) - 7/157464
*ln(abs(2*x - 3))