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

Optimal. Leaf size=113 \[ \frac{x}{4374 \left (8 x^3+27\right )}-\frac{7 \log \left (4 x^2-6 x+9\right )}{944784}+\frac{\log \left (4 x^2+6 x+9\right )}{314928}-\frac{\log (3-2 x)}{157464}+\frac{7 \log (2 x+3)}{472392}-\frac{7 \tan ^{-1}\left (\frac{3-4 x}{3 \sqrt{3}}\right )}{157464 \sqrt{3}}+\frac{\tan ^{-1}\left (\frac{4 x+3}{3 \sqrt{3}}\right )}{52488 \sqrt{3}} \]

[Out]

x/(4374*(27 + 8*x^3)) - (7*ArcTan[(3 - 4*x)/(3*Sqrt[3])])/(157464*Sqrt[3]) + Arc
Tan[(3 + 4*x)/(3*Sqrt[3])]/(52488*Sqrt[3]) - Log[3 - 2*x]/157464 + (7*Log[3 + 2*
x])/472392 - (7*Log[9 - 6*x + 4*x^2])/944784 + Log[9 + 6*x + 4*x^2]/314928

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Rubi [A]  time = 0.167001, antiderivative size = 113, normalized size of antiderivative = 1., number of steps used = 15, number of rules used = 9, integrand size = 17, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.529 \[ \frac{x}{4374 \left (8 x^3+27\right )}-\frac{7 \log \left (4 x^2-6 x+9\right )}{944784}+\frac{\log \left (4 x^2+6 x+9\right )}{314928}-\frac{\log (3-2 x)}{157464}+\frac{7 \log (2 x+3)}{472392}-\frac{7 \tan ^{-1}\left (\frac{3-4 x}{3 \sqrt{3}}\right )}{157464 \sqrt{3}}+\frac{\tan ^{-1}\left (\frac{4 x+3}{3 \sqrt{3}}\right )}{52488 \sqrt{3}} \]

Antiderivative was successfully verified.

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

[Out]

x/(4374*(27 + 8*x^3)) - (7*ArcTan[(3 - 4*x)/(3*Sqrt[3])])/(157464*Sqrt[3]) + Arc
Tan[(3 + 4*x)/(3*Sqrt[3])]/(52488*Sqrt[3]) - Log[3 - 2*x]/157464 + (7*Log[3 + 2*
x])/472392 - (7*Log[9 - 6*x + 4*x^2])/944784 + Log[9 + 6*x + 4*x^2]/314928

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Rubi in Sympy [A]  time = 25.8493, size = 100, normalized size = 0.88 \[ \frac{x}{4374 \left (8 x^{3} + 27\right )} - \frac{\log{\left (- 2 x + 3 \right )}}{157464} + \frac{7 \log{\left (2 x + 3 \right )}}{472392} - \frac{7 \log{\left (4 x^{2} - 6 x + 9 \right )}}{944784} + \frac{\log{\left (4 x^{2} + 6 x + 9 \right )}}{314928} + \frac{7 \sqrt{3} \operatorname{atan}{\left (\sqrt{3} \left (\frac{4 x}{9} - \frac{1}{3}\right ) \right )}}{472392} + \frac{\sqrt{3} \operatorname{atan}{\left (\sqrt{3} \left (\frac{4 x}{9} + \frac{1}{3}\right ) \right )}}{157464} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  rubi_integrate((-8*x**3+27)/(-64*x**6+729)**2,x)

[Out]

x/(4374*(8*x**3 + 27)) - log(-2*x + 3)/157464 + 7*log(2*x + 3)/472392 - 7*log(4*
x**2 - 6*x + 9)/944784 + log(4*x**2 + 6*x + 9)/314928 + 7*sqrt(3)*atan(sqrt(3)*(
4*x/9 - 1/3))/472392 + sqrt(3)*atan(sqrt(3)*(4*x/9 + 1/3))/157464

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Mathematica [A]  time = 0.0930914, size = 103, normalized size = 0.91 \[ \frac{\frac{216 x}{8 x^3+27}-7 \log \left (4 x^2-6 x+9\right )+3 \log \left (4 x^2+6 x+9\right )-6 \log (3-2 x)+14 \log (2 x+3)+14 \sqrt{3} \tan ^{-1}\left (\frac{4 x-3}{3 \sqrt{3}}\right )+6 \sqrt{3} \tan ^{-1}\left (\frac{4 x+3}{3 \sqrt{3}}\right )}{944784} \]

Antiderivative was successfully verified.

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

[Out]

((216*x)/(27 + 8*x^3) + 14*Sqrt[3]*ArcTan[(-3 + 4*x)/(3*Sqrt[3])] + 6*Sqrt[3]*Ar
cTan[(3 + 4*x)/(3*Sqrt[3])] - 6*Log[3 - 2*x] + 14*Log[3 + 2*x] - 7*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 = 0.9 \[ -{\frac{1}{157464\,x+236196}}+{\frac{7\,\ln \left ( 2\,x+3 \right ) }{472392}}-{\frac{\ln \left ( -3+2\,x \right ) }{157464}}+{\frac{\ln \left ( 4\,{x}^{2}+6\,x+9 \right ) }{314928}}+{\frac{\sqrt{3}}{157464}\arctan \left ({\frac{ \left ( 8\,x+6 \right ) \sqrt{3}}{18}} \right ) }-{\frac{1}{118098} \left ( -{\frac{3\,x}{4}}-{\frac{9}{8}} \right ) \left ({x}^{2}-{\frac{3\,x}{2}}+{\frac{9}{4}} \right ) ^{-1}}-{\frac{7\,\ln \left ( 16\,{x}^{2}-24\,x+36 \right ) }{944784}}+{\frac{7\,\sqrt{3}}{472392}\arctan \left ({\frac{ \left ( 32\,x-24 \right ) \sqrt{3}}{72}} \right ) } \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

-1/78732/(2*x+3)+7/472392*ln(2*x+3)-1/157464*ln(-3+2*x)+1/314928*ln(4*x^2+6*x+9)
+1/157464*3^(1/2)*arctan(1/18*(8*x+6)*3^(1/2))-1/118098*(-3/4*x-9/8)/(x^2-3/2*x+
9/4)-7/944784*ln(16*x^2-24*x+36)+7/472392*3^(1/2)*arctan(1/72*(32*x-24)*3^(1/2))

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Maxima [A]  time = 1.52106, size = 117, normalized size = 1.04 \[ \frac{1}{157464} \, \sqrt{3} \arctan \left (\frac{1}{9} \, \sqrt{3}{\left (4 \, x + 3\right )}\right ) + \frac{7}{472392} \, \sqrt{3} \arctan \left (\frac{1}{9} \, \sqrt{3}{\left (4 \, x - 3\right )}\right ) + \frac{x}{4374 \,{\left (8 \, x^{3} + 27\right )}} + \frac{1}{314928} \, \log \left (4 \, x^{2} + 6 \, x + 9\right ) - \frac{7}{944784} \, \log \left (4 \, x^{2} - 6 \, x + 9\right ) + \frac{7}{472392} \, \log \left (2 \, x + 3\right ) - \frac{1}{157464} \, \log \left (2 \, x - 3\right ) \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate(-(8*x^3 - 27)/(64*x^6 - 729)^2,x, algorithm="maxima")

[Out]

1/157464*sqrt(3)*arctan(1/9*sqrt(3)*(4*x + 3)) + 7/472392*sqrt(3)*arctan(1/9*sqr
t(3)*(4*x - 3)) + 1/4374*x/(8*x^3 + 27) + 1/314928*log(4*x^2 + 6*x + 9) - 7/9447
84*log(4*x^2 - 6*x + 9) + 7/472392*log(2*x + 3) - 1/157464*log(2*x - 3)

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Fricas [A]  time = 0.21153, size = 193, normalized size = 1.71 \[ \frac{\sqrt{3}{\left (3 \, \sqrt{3}{\left (8 \, x^{3} + 27\right )} \log \left (4 \, x^{2} + 6 \, x + 9\right ) - 7 \, \sqrt{3}{\left (8 \, x^{3} + 27\right )} \log \left (4 \, x^{2} - 6 \, x + 9\right ) + 14 \, \sqrt{3}{\left (8 \, x^{3} + 27\right )} \log \left (2 \, x + 3\right ) - 6 \, \sqrt{3}{\left (8 \, x^{3} + 27\right )} \log \left (2 \, x - 3\right ) + 18 \,{\left (8 \, x^{3} + 27\right )} \arctan \left (\frac{1}{9} \, \sqrt{3}{\left (4 \, x + 3\right )}\right ) + 42 \,{\left (8 \, x^{3} + 27\right )} \arctan \left (\frac{1}{9} \, \sqrt{3}{\left (4 \, x - 3\right )}\right ) + 216 \, \sqrt{3} x\right )}}{2834352 \,{\left (8 \, x^{3} + 27\right )}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate(-(8*x^3 - 27)/(64*x^6 - 729)^2,x, algorithm="fricas")

[Out]

1/2834352*sqrt(3)*(3*sqrt(3)*(8*x^3 + 27)*log(4*x^2 + 6*x + 9) - 7*sqrt(3)*(8*x^
3 + 27)*log(4*x^2 - 6*x + 9) + 14*sqrt(3)*(8*x^3 + 27)*log(2*x + 3) - 6*sqrt(3)*
(8*x^3 + 27)*log(2*x - 3) + 18*(8*x^3 + 27)*arctan(1/9*sqrt(3)*(4*x + 3)) + 42*(
8*x^3 + 27)*arctan(1/9*sqrt(3)*(4*x - 3)) + 216*sqrt(3)*x)/(8*x^3 + 27)

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Sympy [A]  time = 0.679236, size = 110, normalized size = 0.97 \[ \frac{x}{34992 x^{3} + 118098} - \frac{\log{\left (x - \frac{3}{2} \right )}}{157464} + \frac{7 \log{\left (x + \frac{3}{2} \right )}}{472392} - \frac{7 \log{\left (x^{2} - \frac{3 x}{2} + \frac{9}{4} \right )}}{944784} + \frac{\log{\left (x^{2} + \frac{3 x}{2} + \frac{9}{4} \right )}}{314928} + \frac{7 \sqrt{3} \operatorname{atan}{\left (\frac{4 \sqrt{3} x}{9} - \frac{\sqrt{3}}{3} \right )}}{472392} + \frac{\sqrt{3} \operatorname{atan}{\left (\frac{4 \sqrt{3} x}{9} + \frac{\sqrt{3}}{3} \right )}}{157464} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate((-8*x**3+27)/(-64*x**6+729)**2,x)

[Out]

x/(34992*x**3 + 118098) - log(x - 3/2)/157464 + 7*log(x + 3/2)/472392 - 7*log(x*
*2 - 3*x/2 + 9/4)/944784 + log(x**2 + 3*x/2 + 9/4)/314928 + 7*sqrt(3)*atan(4*sqr
t(3)*x/9 - sqrt(3)/3)/472392 + sqrt(3)*atan(4*sqrt(3)*x/9 + sqrt(3)/3)/157464

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GIAC/XCAS [A]  time = 0.219819, size = 120, normalized size = 1.06 \[ \frac{1}{157464} \, \sqrt{3} \arctan \left (\frac{1}{9} \, \sqrt{3}{\left (4 \, x + 3\right )}\right ) + \frac{7}{472392} \, \sqrt{3} \arctan \left (\frac{1}{9} \, \sqrt{3}{\left (4 \, x - 3\right )}\right ) + \frac{x}{4374 \,{\left (8 \, x^{3} + 27\right )}} + \frac{1}{314928} \,{\rm ln}\left (4 \, x^{2} + 6 \, x + 9\right ) - \frac{7}{944784} \,{\rm ln}\left (4 \, x^{2} - 6 \, x + 9\right ) + \frac{7}{472392} \,{\rm ln}\left ({\left | 2 \, x + 3 \right |}\right ) - \frac{1}{157464} \,{\rm ln}\left ({\left | 2 \, x - 3 \right |}\right ) \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate(-(8*x^3 - 27)/(64*x^6 - 729)^2,x, algorithm="giac")

[Out]

1/157464*sqrt(3)*arctan(1/9*sqrt(3)*(4*x + 3)) + 7/472392*sqrt(3)*arctan(1/9*sqr
t(3)*(4*x - 3)) + 1/4374*x/(8*x^3 + 27) + 1/314928*ln(4*x^2 + 6*x + 9) - 7/94478
4*ln(4*x^2 - 6*x + 9) + 7/472392*ln(abs(2*x + 3)) - 1/157464*ln(abs(2*x - 3))