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

Optimal. Leaf size=142 \[ -\frac{3-4 x}{236196 \left (4 x^2-6 x+9\right )}+\frac{5 \log \left (4 x^2-6 x+9\right )}{2834352}+\frac{\log \left (4 x^2+6 x+9\right )}{944784}+\frac{1}{157464 (3-2 x)}-\frac{1}{472392 (2 x+3)}-\frac{\log (3-2 x)}{118098}+\frac{\log (2 x+3)}{354294}-\frac{\tan ^{-1}\left (\frac{3-4 x}{3 \sqrt{3}}\right )}{52488 \sqrt{3}}+\frac{\tan ^{-1}\left (\frac{4 x+3}{3 \sqrt{3}}\right )}{472392 \sqrt{3}} \]

[Out]

1/(157464*(3 - 2*x)) - 1/(472392*(3 + 2*x)) - (3 - 4*x)/(236196*(9 - 6*x + 4*x^2
)) - ArcTan[(3 - 4*x)/(3*Sqrt[3])]/(52488*Sqrt[3]) + ArcTan[(3 + 4*x)/(3*Sqrt[3]
)]/(472392*Sqrt[3]) - Log[3 - 2*x]/118098 + Log[3 + 2*x]/354294 + (5*Log[9 - 6*x
+ 4*x^2])/2834352 + Log[9 + 6*x + 4*x^2]/944784

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Rubi [A]  time = 0.262404, antiderivative size = 142, normalized size of antiderivative = 1., number of steps used = 14, number of rules used = 7, integrand size = 20, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.35 \[ -\frac{3-4 x}{236196 \left (4 x^2-6 x+9\right )}+\frac{5 \log \left (4 x^2-6 x+9\right )}{2834352}+\frac{\log \left (4 x^2+6 x+9\right )}{944784}+\frac{1}{157464 (3-2 x)}-\frac{1}{472392 (2 x+3)}-\frac{\log (3-2 x)}{118098}+\frac{\log (2 x+3)}{354294}-\frac{\tan ^{-1}\left (\frac{3-4 x}{3 \sqrt{3}}\right )}{52488 \sqrt{3}}+\frac{\tan ^{-1}\left (\frac{4 x+3}{3 \sqrt{3}}\right )}{472392 \sqrt{3}} \]

Antiderivative was successfully verified.

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

[Out]

1/(157464*(3 - 2*x)) - 1/(472392*(3 + 2*x)) - (3 - 4*x)/(236196*(9 - 6*x + 4*x^2
)) - ArcTan[(3 - 4*x)/(3*Sqrt[3])]/(52488*Sqrt[3]) + ArcTan[(3 + 4*x)/(3*Sqrt[3]
)]/(472392*Sqrt[3]) - Log[3 - 2*x]/118098 + Log[3 + 2*x]/354294 + (5*Log[9 - 6*x
+ 4*x^2])/2834352 + Log[9 + 6*x + 4*x^2]/944784

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Rubi in Sympy [A]  time = 55.4866, size = 117, normalized size = 0.82 \[ - \frac{- 8 x + 6}{472392 \left (4 x^{2} - 6 x + 9\right )} - \frac{\log{\left (- 2 x + 3 \right )}}{118098} + \frac{\log{\left (2 x + 3 \right )}}{354294} + \frac{5 \log{\left (4 x^{2} - 6 x + 9 \right )}}{2834352} + \frac{\log{\left (4 x^{2} + 6 x + 9 \right )}}{944784} + \frac{\sqrt{3} \operatorname{atan}{\left (\sqrt{3} \left (\frac{4 x}{9} - \frac{1}{3}\right ) \right )}}{157464} + \frac{\sqrt{3} \operatorname{atan}{\left (\sqrt{3} \left (\frac{4 x}{9} + \frac{1}{3}\right ) \right )}}{1417176} - \frac{1}{472392 \left (2 x + 3\right )} + \frac{1}{157464 \left (- 2 x + 3\right )} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

-(-8*x + 6)/(472392*(4*x**2 - 6*x + 9)) - log(-2*x + 3)/118098 + log(2*x + 3)/35
4294 + 5*log(4*x**2 - 6*x + 9)/2834352 + log(4*x**2 + 6*x + 9)/944784 + sqrt(3)*
atan(sqrt(3)*(4*x/9 - 1/3))/157464 + sqrt(3)*atan(sqrt(3)*(4*x/9 + 1/3))/1417176
- 1/(472392*(2*x + 3)) + 1/(157464*(-2*x + 3))

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Mathematica [A]  time = 0.10452, size = 111, normalized size = 0.78 \[ \frac{5 \log \left (4 x^2-6 x+9\right )+3 \log \left (4 x^2+6 x+9\right )+\frac{648 x}{-16 x^4+24 x^3-54 x+81}-24 \log (3-2 x)+8 \log (2 x+3)+18 \sqrt{3} \tan ^{-1}\left (\frac{4 x-3}{3 \sqrt{3}}\right )+2 \sqrt{3} \tan ^{-1}\left (\frac{4 x+3}{3 \sqrt{3}}\right )}{2834352} \]

Antiderivative was successfully verified.

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

[Out]

((648*x)/(81 - 54*x + 24*x^3 - 16*x^4) + 18*Sqrt[3]*ArcTan[(-3 + 4*x)/(3*Sqrt[3]
)] + 2*Sqrt[3]*ArcTan[(3 + 4*x)/(3*Sqrt[3])] - 24*Log[3 - 2*x] + 8*Log[3 + 2*x]
+ 5*Log[9 - 6*x + 4*x^2] + 3*Log[9 + 6*x + 4*x^2])/2834352

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Maple [A]  time = 0.021, size = 111, normalized size = 0.8 \[ -{\frac{1}{944784\,x+1417176}}+{\frac{\ln \left ( 2\,x+3 \right ) }{354294}}-{\frac{1}{-472392+314928\,x}}-{\frac{\ln \left ( -3+2\,x \right ) }{118098}}+{\frac{\ln \left ( 4\,{x}^{2}+6\,x+9 \right ) }{944784}}+{\frac{\sqrt{3}}{1417176}\arctan \left ({\frac{ \left ( 8\,x+6 \right ) \sqrt{3}}{18}} \right ) }+{\frac{1}{708588} \left ( 3\,x-{\frac{9}{4}} \right ) \left ({x}^{2}-{\frac{3\,x}{2}}+{\frac{9}{4}} \right ) ^{-1}}+{\frac{5\,\ln \left ( 16\,{x}^{2}-24\,x+36 \right ) }{2834352}}+{\frac{\sqrt{3}}{157464}\arctan \left ({\frac{ \left ( 32\,x-24 \right ) \sqrt{3}}{72}} \right ) } \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

-1/472392/(2*x+3)+1/354294*ln(2*x+3)-1/157464/(-3+2*x)-1/118098*ln(-3+2*x)+1/944
784*ln(4*x^2+6*x+9)+1/1417176*3^(1/2)*arctan(1/18*(8*x+6)*3^(1/2))+1/708588*(3*x
-9/4)/(x^2-3/2*x+9/4)+5/2834352*ln(16*x^2-24*x+36)+1/157464*3^(1/2)*arctan(1/72*
(32*x-24)*3^(1/2))

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Maxima [A]  time = 1.51937, size = 128, normalized size = 0.9 \[ \frac{1}{1417176} \, \sqrt{3} \arctan \left (\frac{1}{9} \, \sqrt{3}{\left (4 \, x + 3\right )}\right ) + \frac{1}{157464} \, \sqrt{3} \arctan \left (\frac{1}{9} \, \sqrt{3}{\left (4 \, x - 3\right )}\right ) - \frac{x}{4374 \,{\left (16 \, x^{4} - 24 \, x^{3} + 54 \, x - 81\right )}} + \frac{1}{944784} \, \log \left (4 \, x^{2} + 6 \, x + 9\right ) + \frac{5}{2834352} \, \log \left (4 \, x^{2} - 6 \, x + 9\right ) + \frac{1}{354294} \, \log \left (2 \, x + 3\right ) - \frac{1}{118098} \, \log \left (2 \, x - 3\right ) \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

1/1417176*sqrt(3)*arctan(1/9*sqrt(3)*(4*x + 3)) + 1/157464*sqrt(3)*arctan(1/9*sq
rt(3)*(4*x - 3)) - 1/4374*x/(16*x^4 - 24*x^3 + 54*x - 81) + 1/944784*log(4*x^2 +
6*x + 9) + 5/2834352*log(4*x^2 - 6*x + 9) + 1/354294*log(2*x + 3) - 1/118098*lo
g(2*x - 3)

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Fricas [A]  time = 0.215037, size = 269, normalized size = 1.89 \[ \frac{\sqrt{3}{\left (3 \, \sqrt{3}{\left (16 \, x^{4} - 24 \, x^{3} + 54 \, x - 81\right )} \log \left (4 \, x^{2} + 6 \, x + 9\right ) + 5 \, \sqrt{3}{\left (16 \, x^{4} - 24 \, x^{3} + 54 \, x - 81\right )} \log \left (4 \, x^{2} - 6 \, x + 9\right ) + 8 \, \sqrt{3}{\left (16 \, x^{4} - 24 \, x^{3} + 54 \, x - 81\right )} \log \left (2 \, x + 3\right ) - 24 \, \sqrt{3}{\left (16 \, x^{4} - 24 \, x^{3} + 54 \, x - 81\right )} \log \left (2 \, x - 3\right ) + 6 \,{\left (16 \, x^{4} - 24 \, x^{3} + 54 \, x - 81\right )} \arctan \left (\frac{1}{9} \, \sqrt{3}{\left (4 \, x + 3\right )}\right ) + 54 \,{\left (16 \, x^{4} - 24 \, x^{3} + 54 \, x - 81\right )} \arctan \left (\frac{1}{9} \, \sqrt{3}{\left (4 \, x - 3\right )}\right ) - 648 \, \sqrt{3} x\right )}}{8503056 \,{\left (16 \, x^{4} - 24 \, x^{3} + 54 \, x - 81\right )}} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

1/8503056*sqrt(3)*(3*sqrt(3)*(16*x^4 - 24*x^3 + 54*x - 81)*log(4*x^2 + 6*x + 9)
+ 5*sqrt(3)*(16*x^4 - 24*x^3 + 54*x - 81)*log(4*x^2 - 6*x + 9) + 8*sqrt(3)*(16*x
^4 - 24*x^3 + 54*x - 81)*log(2*x + 3) - 24*sqrt(3)*(16*x^4 - 24*x^3 + 54*x - 81)
*log(2*x - 3) + 6*(16*x^4 - 24*x^3 + 54*x - 81)*arctan(1/9*sqrt(3)*(4*x + 3)) +
54*(16*x^4 - 24*x^3 + 54*x - 81)*arctan(1/9*sqrt(3)*(4*x - 3)) - 648*sqrt(3)*x)/
(16*x^4 - 24*x^3 + 54*x - 81)

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Sympy [A]  time = 0.833748, size = 116, normalized size = 0.82 \[ - \frac{x}{69984 x^{4} - 104976 x^{3} + 236196 x - 354294} - \frac{\log{\left (x - \frac{3}{2} \right )}}{118098} + \frac{\log{\left (x + \frac{3}{2} \right )}}{354294} + \frac{5 \log{\left (x^{2} - \frac{3 x}{2} + \frac{9}{4} \right )}}{2834352} + \frac{\log{\left (x^{2} + \frac{3 x}{2} + \frac{9}{4} \right )}}{944784} + \frac{\sqrt{3} \operatorname{atan}{\left (\frac{4 \sqrt{3} x}{9} - \frac{\sqrt{3}}{3} \right )}}{157464} + \frac{\sqrt{3} \operatorname{atan}{\left (\frac{4 \sqrt{3} x}{9} + \frac{\sqrt{3}}{3} \right )}}{1417176} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

-x/(69984*x**4 - 104976*x**3 + 236196*x - 354294) - log(x - 3/2)/118098 + log(x
+ 3/2)/354294 + 5*log(x**2 - 3*x/2 + 9/4)/2834352 + log(x**2 + 3*x/2 + 9/4)/9447
84 + sqrt(3)*atan(4*sqrt(3)*x/9 - sqrt(3)/3)/157464 + sqrt(3)*atan(4*sqrt(3)*x/9
+ sqrt(3)/3)/1417176

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GIAC/XCAS [A]  time = 0.221159, size = 143, normalized size = 1.01 \[ \frac{1}{1417176} \, \sqrt{3} \arctan \left (\frac{1}{9} \, \sqrt{3}{\left (4 \, x + 3\right )}\right ) + \frac{1}{157464} \, \sqrt{3} \arctan \left (\frac{1}{9} \, \sqrt{3}{\left (4 \, x - 3\right )}\right ) - \frac{x}{4374 \,{\left (4 \, x^{2} - 6 \, x + 9\right )}{\left (2 \, x + 3\right )}{\left (2 \, x - 3\right )}} + \frac{1}{944784} \,{\rm ln}\left (4 \, x^{2} + 6 \, x + 9\right ) + \frac{5}{2834352} \,{\rm ln}\left (4 \, x^{2} - 6 \, x + 9\right ) + \frac{1}{354294} \,{\rm ln}\left ({\left | 2 \, x + 3 \right |}\right ) - \frac{1}{118098} \,{\rm ln}\left ({\left | 2 \, x - 3 \right |}\right ) \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

1/1417176*sqrt(3)*arctan(1/9*sqrt(3)*(4*x + 3)) + 1/157464*sqrt(3)*arctan(1/9*sq
rt(3)*(4*x - 3)) - 1/4374*x/((4*x^2 - 6*x + 9)*(2*x + 3)*(2*x - 3)) + 1/944784*l
n(4*x^2 + 6*x + 9) + 5/2834352*ln(4*x^2 - 6*x + 9) + 1/354294*ln(abs(2*x + 3)) -
1/118098*ln(abs(2*x - 3))