3.155 \(\int \frac{x^3}{1-2 x^2+3 x^4} \, dx\)

Optimal. Leaf size=41 \[ \frac{1}{12} \log \left (3 x^4-2 x^2+1\right )-\frac{\tan ^{-1}\left (\frac{1-3 x^2}{\sqrt{2}}\right )}{6 \sqrt{2}} \]

[Out]

-ArcTan[(1 - 3*x^2)/Sqrt[2]]/(6*Sqrt[2]) + Log[1 - 2*x^2 + 3*x^4]/12

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Rubi [A]  time = 0.079102, antiderivative size = 41, normalized size of antiderivative = 1., number of steps used = 5, number of rules used = 5, integrand size = 18, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.278 \[ \frac{1}{12} \log \left (3 x^4-2 x^2+1\right )-\frac{\tan ^{-1}\left (\frac{1-3 x^2}{\sqrt{2}}\right )}{6 \sqrt{2}} \]

Antiderivative was successfully verified.

[In]  Int[x^3/(1 - 2*x^2 + 3*x^4),x]

[Out]

-ArcTan[(1 - 3*x^2)/Sqrt[2]]/(6*Sqrt[2]) + Log[1 - 2*x^2 + 3*x^4]/12

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Rubi in Sympy [A]  time = 4.95828, size = 37, normalized size = 0.9 \[ \frac{\log{\left (3 x^{4} - 2 x^{2} + 1 \right )}}{12} + \frac{\sqrt{2} \operatorname{atan}{\left (\sqrt{2} \left (\frac{3 x^{2}}{2} - \frac{1}{2}\right ) \right )}}{12} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  rubi_integrate(x**3/(3*x**4-2*x**2+1),x)

[Out]

log(3*x**4 - 2*x**2 + 1)/12 + sqrt(2)*atan(sqrt(2)*(3*x**2/2 - 1/2))/12

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Mathematica [A]  time = 0.0167207, size = 38, normalized size = 0.93 \[ \frac{1}{12} \left (\sqrt{2} \tan ^{-1}\left (\frac{3 x^2-1}{\sqrt{2}}\right )+\log \left (3 x^4-2 x^2+1\right )\right ) \]

Antiderivative was successfully verified.

[In]  Integrate[x^3/(1 - 2*x^2 + 3*x^4),x]

[Out]

(Sqrt[2]*ArcTan[(-1 + 3*x^2)/Sqrt[2]] + Log[1 - 2*x^2 + 3*x^4])/12

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Maple [A]  time = 0.008, size = 35, normalized size = 0.9 \[{\frac{\ln \left ( 3\,{x}^{4}-2\,{x}^{2}+1 \right ) }{12}}+{\frac{\sqrt{2}}{12}\arctan \left ({\frac{ \left ( 6\,{x}^{2}-2 \right ) \sqrt{2}}{4}} \right ) } \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  int(x^3/(3*x^4-2*x^2+1),x)

[Out]

1/12*ln(3*x^4-2*x^2+1)+1/12*2^(1/2)*arctan(1/4*(6*x^2-2)*2^(1/2))

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Maxima [A]  time = 1.56406, size = 46, normalized size = 1.12 \[ \frac{1}{12} \, \sqrt{2} \arctan \left (\frac{1}{2} \, \sqrt{2}{\left (3 \, x^{2} - 1\right )}\right ) + \frac{1}{12} \, \log \left (3 \, x^{4} - 2 \, x^{2} + 1\right ) \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate(x^3/(3*x^4 - 2*x^2 + 1),x, algorithm="maxima")

[Out]

1/12*sqrt(2)*arctan(1/2*sqrt(2)*(3*x^2 - 1)) + 1/12*log(3*x^4 - 2*x^2 + 1)

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Fricas [A]  time = 0.231784, size = 51, normalized size = 1.24 \[ \frac{1}{24} \, \sqrt{2}{\left (\sqrt{2} \log \left (3 \, x^{4} - 2 \, x^{2} + 1\right ) + 2 \, \arctan \left (\frac{1}{2} \, \sqrt{2}{\left (3 \, x^{2} - 1\right )}\right )\right )} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate(x^3/(3*x^4 - 2*x^2 + 1),x, algorithm="fricas")

[Out]

1/24*sqrt(2)*(sqrt(2)*log(3*x^4 - 2*x^2 + 1) + 2*arctan(1/2*sqrt(2)*(3*x^2 - 1))
)

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Sympy [A]  time = 0.122649, size = 42, normalized size = 1.02 \[ \frac{\log{\left (x^{4} - \frac{2 x^{2}}{3} + \frac{1}{3} \right )}}{12} + \frac{\sqrt{2} \operatorname{atan}{\left (\frac{3 \sqrt{2} x^{2}}{2} - \frac{\sqrt{2}}{2} \right )}}{12} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate(x**3/(3*x**4-2*x**2+1),x)

[Out]

log(x**4 - 2*x**2/3 + 1/3)/12 + sqrt(2)*atan(3*sqrt(2)*x**2/2 - sqrt(2)/2)/12

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GIAC/XCAS [A]  time = 0.200715, size = 46, normalized size = 1.12 \[ \frac{1}{12} \, \sqrt{2} \arctan \left (\frac{1}{2} \, \sqrt{2}{\left (3 \, x^{2} - 1\right )}\right ) + \frac{1}{12} \,{\rm ln}\left (3 \, x^{4} - 2 \, x^{2} + 1\right ) \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate(x^3/(3*x^4 - 2*x^2 + 1),x, algorithm="giac")

[Out]

1/12*sqrt(2)*arctan(1/2*sqrt(2)*(3*x^2 - 1)) + 1/12*ln(3*x^4 - 2*x^2 + 1)