3.555 \(\int \frac{81+36 x^2+16 x^4}{\left (729-64 x^6\right )^2} \, dx\)

Optimal. Leaf size=81 \[ \frac{1}{17496 (3-2 x)}-\frac{1}{17496 (2 x+3)}-\frac{\tan ^{-1}\left (\frac{3-4 x}{3 \sqrt{3}}\right )}{13122 \sqrt{3}}+\frac{\tan ^{-1}\left (\frac{4 x+3}{3 \sqrt{3}}\right )}{13122 \sqrt{3}}+\frac{\tanh ^{-1}\left (\frac{2 x}{3}\right )}{8748} \]

[Out]

1/(17496*(3 - 2*x)) - 1/(17496*(3 + 2*x)) - ArcTan[(3 - 4*x)/(3*Sqrt[3])]/(13122
*Sqrt[3]) + ArcTan[(3 + 4*x)/(3*Sqrt[3])]/(13122*Sqrt[3]) + ArcTanh[(2*x)/3]/874
8

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Rubi [A]  time = 0.126895, antiderivative size = 81, normalized size of antiderivative = 1., number of steps used = 8, number of rules used = 5, integrand size = 22, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.227 \[ \frac{1}{17496 (3-2 x)}-\frac{1}{17496 (2 x+3)}-\frac{\tan ^{-1}\left (\frac{3-4 x}{3 \sqrt{3}}\right )}{13122 \sqrt{3}}+\frac{\tan ^{-1}\left (\frac{4 x+3}{3 \sqrt{3}}\right )}{13122 \sqrt{3}}+\frac{\tanh ^{-1}\left (\frac{2 x}{3}\right )}{8748} \]

Antiderivative was successfully verified.

[In]  Int[(81 + 36*x^2 + 16*x^4)/(729 - 64*x^6)^2,x]

[Out]

1/(17496*(3 - 2*x)) - 1/(17496*(3 + 2*x)) - ArcTan[(3 - 4*x)/(3*Sqrt[3])]/(13122
*Sqrt[3]) + ArcTan[(3 + 4*x)/(3*Sqrt[3])]/(13122*Sqrt[3]) + ArcTanh[(2*x)/3]/874
8

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Rubi in Sympy [A]  time = 21.9605, size = 65, normalized size = 0.8 \[ \frac{\sqrt{3} \operatorname{atan}{\left (\sqrt{3} \left (\frac{4 x}{9} - \frac{1}{3}\right ) \right )}}{39366} + \frac{\sqrt{3} \operatorname{atan}{\left (\sqrt{3} \left (\frac{4 x}{9} + \frac{1}{3}\right ) \right )}}{39366} + \frac{\operatorname{atanh}{\left (\frac{2 x}{3} \right )}}{8748} - \frac{1}{17496 \left (2 x + 3\right )} + \frac{1}{17496 \left (- 2 x + 3\right )} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  rubi_integrate((16*x**4+36*x**2+81)/(-64*x**6+729)**2,x)

[Out]

sqrt(3)*atan(sqrt(3)*(4*x/9 - 1/3))/39366 + sqrt(3)*atan(sqrt(3)*(4*x/9 + 1/3))/
39366 + atanh(2*x/3)/8748 - 1/(17496*(2*x + 3)) + 1/(17496*(-2*x + 3))

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Mathematica [C]  time = 0.771274, size = 122, normalized size = 1.51 \[ \frac{\frac{36 x}{9-4 x^2}-9 \log (3-2 x)+9 \log (2 x+3)+3 \sqrt{3} \tan ^{-1}\left (\frac{1}{3} \left (\sqrt{3}-i\right ) x\right )+4 i \sqrt{3} \tanh ^{-1}\left (\frac{1}{3} \left (1-i \sqrt{3}\right ) x\right )+\left (-3+\frac{2}{\sqrt{\frac{1}{6} \left (1+i \sqrt{3}\right )}}\right ) \tanh ^{-1}\left (\frac{1}{3} \left (x+i \sqrt{3} x\right )\right )}{157464} \]

Warning: Unable to verify antiderivative.

[In]  Integrate[(81 + 36*x^2 + 16*x^4)/(729 - 64*x^6)^2,x]

[Out]

((36*x)/(9 - 4*x^2) + 3*Sqrt[3]*ArcTan[((-I + Sqrt[3])*x)/3] + (4*I)*Sqrt[3]*Arc
Tanh[((1 - I*Sqrt[3])*x)/3] + (-3 + 2/Sqrt[(1 + I*Sqrt[3])/6])*ArcTanh[(x + I*Sq
rt[3]*x)/3] - 9*Log[3 - 2*x] + 9*Log[3 + 2*x])/157464

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Maple [A]  time = 0.017, size = 68, normalized size = 0.8 \[ -{\frac{1}{34992\,x+52488}}+{\frac{\ln \left ( 2\,x+3 \right ) }{17496}}-{\frac{1}{-52488+34992\,x}}-{\frac{\ln \left ( -3+2\,x \right ) }{17496}}+{\frac{\sqrt{3}}{39366}\arctan \left ({\frac{ \left ( 8\,x+6 \right ) \sqrt{3}}{18}} \right ) }+{\frac{\sqrt{3}}{39366}\arctan \left ({\frac{ \left ( 8\,x-6 \right ) \sqrt{3}}{18}} \right ) } \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  int((16*x^4+36*x^2+81)/(-64*x^6+729)^2,x)

[Out]

-1/17496/(2*x+3)+1/17496*ln(2*x+3)-1/17496/(-3+2*x)-1/17496*ln(-3+2*x)+1/39366*3
^(1/2)*arctan(1/18*(8*x+6)*3^(1/2))+1/39366*3^(1/2)*arctan(1/18*(8*x-6)*3^(1/2))

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Maxima [A]  time = 1.50652, size = 82, normalized size = 1.01 \[ \frac{1}{39366} \, \sqrt{3} \arctan \left (\frac{1}{9} \, \sqrt{3}{\left (4 \, x + 3\right )}\right ) + \frac{1}{39366} \, \sqrt{3} \arctan \left (\frac{1}{9} \, \sqrt{3}{\left (4 \, x - 3\right )}\right ) - \frac{x}{4374 \,{\left (4 \, x^{2} - 9\right )}} + \frac{1}{17496} \, \log \left (2 \, x + 3\right ) - \frac{1}{17496} \, \log \left (2 \, x - 3\right ) \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate((16*x^4 + 36*x^2 + 81)/(64*x^6 - 729)^2,x, algorithm="maxima")

[Out]

1/39366*sqrt(3)*arctan(1/9*sqrt(3)*(4*x + 3)) + 1/39366*sqrt(3)*arctan(1/9*sqrt(
3)*(4*x - 3)) - 1/4374*x/(4*x^2 - 9) + 1/17496*log(2*x + 3) - 1/17496*log(2*x -
3)

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Fricas [A]  time = 0.205496, size = 131, normalized size = 1.62 \[ \frac{\sqrt{3}{\left (3 \, \sqrt{3}{\left (4 \, x^{2} - 9\right )} \log \left (2 \, x + 3\right ) - 3 \, \sqrt{3}{\left (4 \, x^{2} - 9\right )} \log \left (2 \, x - 3\right ) + 4 \,{\left (4 \, x^{2} - 9\right )} \arctan \left (\frac{4}{81} \, \sqrt{3}{\left (2 \, x^{3} + 9 \, x\right )}\right ) + 4 \,{\left (4 \, x^{2} - 9\right )} \arctan \left (\frac{2}{9} \, \sqrt{3} x\right ) - 12 \, \sqrt{3} x\right )}}{157464 \,{\left (4 \, x^{2} - 9\right )}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate((16*x^4 + 36*x^2 + 81)/(64*x^6 - 729)^2,x, algorithm="fricas")

[Out]

1/157464*sqrt(3)*(3*sqrt(3)*(4*x^2 - 9)*log(2*x + 3) - 3*sqrt(3)*(4*x^2 - 9)*log
(2*x - 3) + 4*(4*x^2 - 9)*arctan(4/81*sqrt(3)*(2*x^3 + 9*x)) + 4*(4*x^2 - 9)*arc
tan(2/9*sqrt(3)*x) - 12*sqrt(3)*x)/(4*x^2 - 9)

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Sympy [A]  time = 0.271751, size = 70, normalized size = 0.86 \[ - \frac{x}{17496 x^{2} - 39366} + \frac{\sqrt{3} \left (2 \operatorname{atan}{\left (\frac{2 \sqrt{3} x}{9} \right )} + 2 \operatorname{atan}{\left (\frac{8 \sqrt{3} x^{3}}{81} + \frac{4 \sqrt{3} x}{9} \right )}\right )}{78732} - \frac{\log{\left (x - \frac{3}{2} \right )}}{17496} + \frac{\log{\left (x + \frac{3}{2} \right )}}{17496} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate((16*x**4+36*x**2+81)/(-64*x**6+729)**2,x)

[Out]

-x/(17496*x**2 - 39366) + sqrt(3)*(2*atan(2*sqrt(3)*x/9) + 2*atan(8*sqrt(3)*x**3
/81 + 4*sqrt(3)*x/9))/78732 - log(x - 3/2)/17496 + log(x + 3/2)/17496

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GIAC/XCAS [A]  time = 0.21981, size = 85, normalized size = 1.05 \[ \frac{1}{39366} \, \sqrt{3} \arctan \left (\frac{1}{9} \, \sqrt{3}{\left (4 \, x + 3\right )}\right ) + \frac{1}{39366} \, \sqrt{3} \arctan \left (\frac{1}{9} \, \sqrt{3}{\left (4 \, x - 3\right )}\right ) - \frac{x}{4374 \,{\left (4 \, x^{2} - 9\right )}} + \frac{1}{17496} \,{\rm ln}\left ({\left | 2 \, x + 3 \right |}\right ) - \frac{1}{17496} \,{\rm ln}\left ({\left | 2 \, x - 3 \right |}\right ) \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate((16*x^4 + 36*x^2 + 81)/(64*x^6 - 729)^2,x, algorithm="giac")

[Out]

1/39366*sqrt(3)*arctan(1/9*sqrt(3)*(4*x + 3)) + 1/39366*sqrt(3)*arctan(1/9*sqrt(
3)*(4*x - 3)) - 1/4374*x/(4*x^2 - 9) + 1/17496*ln(abs(2*x + 3)) - 1/17496*ln(abs
(2*x - 3))