3.556 \(\int \frac{81+54 x-24 x^3-16 x^4}{\left (729-64 x^6\right )^2} \, dx\)

Optimal. Leaf size=92 \[ \frac{x}{4374 \left (4 x^2-6 x+9\right )}-\frac{\log \left (4 x^2-6 x+9\right )}{157464}+\frac{\log \left (4 x^2+6 x+9\right )}{52488}-\frac{\log (3-2 x)}{26244}+\frac{\log (2 x+3)}{78732}-\frac{\tan ^{-1}\left (\frac{3-4 x}{3 \sqrt{3}}\right )}{4374 \sqrt{3}} \]

[Out]

x/(4374*(9 - 6*x + 4*x^2)) - ArcTan[(3 - 4*x)/(3*Sqrt[3])]/(4374*Sqrt[3]) - Log[
3 - 2*x]/26244 + Log[3 + 2*x]/78732 - Log[9 - 6*x + 4*x^2]/157464 + Log[9 + 6*x
+ 4*x^2]/52488

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Rubi [A]  time = 0.207291, antiderivative size = 92, normalized size of antiderivative = 1., number of steps used = 11, number of rules used = 7, integrand size = 25, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.28 \[ \frac{x}{4374 \left (4 x^2-6 x+9\right )}-\frac{\log \left (4 x^2-6 x+9\right )}{157464}+\frac{\log \left (4 x^2+6 x+9\right )}{52488}-\frac{\log (3-2 x)}{26244}+\frac{\log (2 x+3)}{78732}-\frac{\tan ^{-1}\left (\frac{3-4 x}{3 \sqrt{3}}\right )}{4374 \sqrt{3}} \]

Antiderivative was successfully verified.

[In]  Int[(81 + 54*x - 24*x^3 - 16*x^4)/(729 - 64*x^6)^2,x]

[Out]

x/(4374*(9 - 6*x + 4*x^2)) - ArcTan[(3 - 4*x)/(3*Sqrt[3])]/(4374*Sqrt[3]) - Log[
3 - 2*x]/26244 + Log[3 + 2*x]/78732 - Log[9 - 6*x + 4*x^2]/157464 + Log[9 + 6*x
+ 4*x^2]/52488

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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((-16*x**4-24*x**3+54*x+81)/(-64*x**6+729)**2,x)

[Out]

Timed out

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Mathematica [A]  time = 0.0593725, size = 84, normalized size = 0.91 \[ \frac{\frac{36 x}{4 x^2-6 x+9}-\log \left (4 x^2-6 x+9\right )+3 \log \left (4 x^2+6 x+9\right )-6 \log (3-2 x)+2 \log (2 x+3)+12 \sqrt{3} \tan ^{-1}\left (\frac{4 x-3}{3 \sqrt{3}}\right )}{157464} \]

Antiderivative was successfully verified.

[In]  Integrate[(81 + 54*x - 24*x^3 - 16*x^4)/(729 - 64*x^6)^2,x]

[Out]

((36*x)/(9 - 6*x + 4*x^2) + 12*Sqrt[3]*ArcTan[(-3 + 4*x)/(3*Sqrt[3])] - 6*Log[3
- 2*x] + 2*Log[3 + 2*x] - Log[9 - 6*x + 4*x^2] + 3*Log[9 + 6*x + 4*x^2])/157464

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Maple [A]  time = 0.019, size = 73, normalized size = 0.8 \[{\frac{\ln \left ( 2\,x+3 \right ) }{78732}}-{\frac{\ln \left ( -3+2\,x \right ) }{26244}}+{\frac{\ln \left ( 4\,{x}^{2}+6\,x+9 \right ) }{52488}}+{\frac{x}{17496} \left ({x}^{2}-{\frac{3\,x}{2}}+{\frac{9}{4}} \right ) ^{-1}}-{\frac{\ln \left ( 16\,{x}^{2}-24\,x+36 \right ) }{157464}}+{\frac{\sqrt{3}}{13122}\arctan \left ({\frac{ \left ( 32\,x-24 \right ) \sqrt{3}}{72}} \right ) } \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  int((-16*x^4-24*x^3+54*x+81)/(-64*x^6+729)^2,x)

[Out]

1/78732*ln(2*x+3)-1/26244*ln(-3+2*x)+1/52488*ln(4*x^2+6*x+9)+1/17496*x/(x^2-3/2*
x+9/4)-1/157464*ln(16*x^2-24*x+36)+1/13122*3^(1/2)*arctan(1/72*(32*x-24)*3^(1/2)
)

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Maxima [A]  time = 1.58063, size = 100, normalized size = 1.09 \[ \frac{1}{13122} \, \sqrt{3} \arctan \left (\frac{1}{9} \, \sqrt{3}{\left (4 \, x - 3\right )}\right ) + \frac{x}{4374 \,{\left (4 \, x^{2} - 6 \, x + 9\right )}} + \frac{1}{52488} \, \log \left (4 \, x^{2} + 6 \, x + 9\right ) - \frac{1}{157464} \, \log \left (4 \, x^{2} - 6 \, x + 9\right ) + \frac{1}{78732} \, \log \left (2 \, x + 3\right ) - \frac{1}{26244} \, \log \left (2 \, x - 3\right ) \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate(-(16*x^4 + 24*x^3 - 54*x - 81)/(64*x^6 - 729)^2,x, algorithm="maxima")

[Out]

1/13122*sqrt(3)*arctan(1/9*sqrt(3)*(4*x - 3)) + 1/4374*x/(4*x^2 - 6*x + 9) + 1/5
2488*log(4*x^2 + 6*x + 9) - 1/157464*log(4*x^2 - 6*x + 9) + 1/78732*log(2*x + 3)
- 1/26244*log(2*x - 3)

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Fricas [A]  time = 0.21308, size = 190, normalized size = 2.07 \[ \frac{\sqrt{3}{\left (3 \, \sqrt{3}{\left (4 \, x^{2} - 6 \, x + 9\right )} \log \left (4 \, x^{2} + 6 \, x + 9\right ) - \sqrt{3}{\left (4 \, x^{2} - 6 \, x + 9\right )} \log \left (4 \, x^{2} - 6 \, x + 9\right ) + 2 \, \sqrt{3}{\left (4 \, x^{2} - 6 \, x + 9\right )} \log \left (2 \, x + 3\right ) - 6 \, \sqrt{3}{\left (4 \, x^{2} - 6 \, x + 9\right )} \log \left (2 \, x - 3\right ) + 36 \,{\left (4 \, x^{2} - 6 \, x + 9\right )} \arctan \left (\frac{1}{9} \, \sqrt{3}{\left (4 \, x - 3\right )}\right ) + 36 \, \sqrt{3} x\right )}}{472392 \,{\left (4 \, x^{2} - 6 \, x + 9\right )}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate(-(16*x^4 + 24*x^3 - 54*x - 81)/(64*x^6 - 729)^2,x, algorithm="fricas")

[Out]

1/472392*sqrt(3)*(3*sqrt(3)*(4*x^2 - 6*x + 9)*log(4*x^2 + 6*x + 9) - sqrt(3)*(4*
x^2 - 6*x + 9)*log(4*x^2 - 6*x + 9) + 2*sqrt(3)*(4*x^2 - 6*x + 9)*log(2*x + 3) -
6*sqrt(3)*(4*x^2 - 6*x + 9)*log(2*x - 3) + 36*(4*x^2 - 6*x + 9)*arctan(1/9*sqrt
(3)*(4*x - 3)) + 36*sqrt(3)*x)/(4*x^2 - 6*x + 9)

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Sympy [A]  time = 0.462992, size = 82, normalized size = 0.89 \[ \frac{x}{17496 x^{2} - 26244 x + 39366} - \frac{\log{\left (x - \frac{3}{2} \right )}}{26244} + \frac{\log{\left (x + \frac{3}{2} \right )}}{78732} - \frac{\log{\left (x^{2} - \frac{3 x}{2} + \frac{9}{4} \right )}}{157464} + \frac{\log{\left (4 x^{2} + 6 x + 9 \right )}}{52488} + \frac{\sqrt{3} \operatorname{atan}{\left (\frac{4 \sqrt{3} x}{9} - \frac{\sqrt{3}}{3} \right )}}{13122} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate((-16*x**4-24*x**3+54*x+81)/(-64*x**6+729)**2,x)

[Out]

x/(17496*x**2 - 26244*x + 39366) - log(x - 3/2)/26244 + log(x + 3/2)/78732 - log
(x**2 - 3*x/2 + 9/4)/157464 + log(4*x**2 + 6*x + 9)/52488 + sqrt(3)*atan(4*sqrt(
3)*x/9 - sqrt(3)/3)/13122

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GIAC/XCAS [A]  time = 0.219609, size = 103, normalized size = 1.12 \[ \frac{1}{13122} \, \sqrt{3} \arctan \left (\frac{1}{9} \, \sqrt{3}{\left (4 \, x - 3\right )}\right ) + \frac{x}{4374 \,{\left (4 \, x^{2} - 6 \, x + 9\right )}} + \frac{1}{52488} \,{\rm ln}\left (4 \, x^{2} + 6 \, x + 9\right ) - \frac{1}{157464} \,{\rm ln}\left (4 \, x^{2} - 6 \, x + 9\right ) + \frac{1}{78732} \,{\rm ln}\left ({\left | 2 \, x + 3 \right |}\right ) - \frac{1}{26244} \,{\rm ln}\left ({\left | 2 \, x - 3 \right |}\right ) \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate(-(16*x^4 + 24*x^3 - 54*x - 81)/(64*x^6 - 729)^2,x, algorithm="giac")

[Out]

1/13122*sqrt(3)*arctan(1/9*sqrt(3)*(4*x - 3)) + 1/4374*x/(4*x^2 - 6*x + 9) + 1/5
2488*ln(4*x^2 + 6*x + 9) - 1/157464*ln(4*x^2 - 6*x + 9) + 1/78732*ln(abs(2*x + 3
)) - 1/26244*ln(abs(2*x - 3))