3.154 \(\int \frac{x^2}{3+4 x^3+x^6} \, dx\)

Optimal. Leaf size=10 \[ -\frac{1}{3} \tanh ^{-1}\left (x^3+2\right ) \]

[Out]

-ArcTanh[2 + x^3]/3

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Rubi [B]  time = 0.0295754, antiderivative size = 21, normalized size of antiderivative = 2.1, number of steps used = 4, number of rules used = 3, integrand size = 16, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.188 \[ \frac{1}{6} \log \left (x^3+1\right )-\frac{1}{6} \log \left (x^3+3\right ) \]

Antiderivative was successfully verified.

[In]  Int[x^2/(3 + 4*x^3 + x^6),x]

[Out]

Log[1 + x^3]/6 - Log[3 + x^3]/6

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Rubi in Sympy [A]  time = 5.73303, size = 15, normalized size = 1.5 \[ \frac{\log{\left (x^{3} + 1 \right )}}{6} - \frac{\log{\left (x^{3} + 3 \right )}}{6} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  rubi_integrate(x**2/(x**6+4*x**3+3),x)

[Out]

log(x**3 + 1)/6 - log(x**3 + 3)/6

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Mathematica [B]  time = 0.00635518, size = 21, normalized size = 2.1 \[ \frac{1}{6} \log \left (x^3+1\right )-\frac{1}{6} \log \left (x^3+3\right ) \]

Antiderivative was successfully verified.

[In]  Integrate[x^2/(3 + 4*x^3 + x^6),x]

[Out]

Log[1 + x^3]/6 - Log[3 + x^3]/6

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Maple [B]  time = 0.007, size = 18, normalized size = 1.8 \[{\frac{\ln \left ({x}^{3}+1 \right ) }{6}}-{\frac{\ln \left ({x}^{3}+3 \right ) }{6}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  int(x^2/(x^6+4*x^3+3),x)

[Out]

1/6*ln(x^3+1)-1/6*ln(x^3+3)

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Maxima [A]  time = 0.782302, size = 23, normalized size = 2.3 \[ -\frac{1}{6} \, \log \left (x^{3} + 3\right ) + \frac{1}{6} \, \log \left (x^{3} + 1\right ) \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate(x^2/(x^6 + 4*x^3 + 3),x, algorithm="maxima")

[Out]

-1/6*log(x^3 + 3) + 1/6*log(x^3 + 1)

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Fricas [A]  time = 0.253812, size = 23, normalized size = 2.3 \[ -\frac{1}{6} \, \log \left (x^{3} + 3\right ) + \frac{1}{6} \, \log \left (x^{3} + 1\right ) \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate(x^2/(x^6 + 4*x^3 + 3),x, algorithm="fricas")

[Out]

-1/6*log(x^3 + 3) + 1/6*log(x^3 + 1)

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Sympy [A]  time = 0.215654, size = 15, normalized size = 1.5 \[ \frac{\log{\left (x^{3} + 1 \right )}}{6} - \frac{\log{\left (x^{3} + 3 \right )}}{6} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate(x**2/(x**6+4*x**3+3),x)

[Out]

log(x**3 + 1)/6 - log(x**3 + 3)/6

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GIAC/XCAS [A]  time = 0.261742, size = 26, normalized size = 2.6 \[ -\frac{1}{6} \,{\rm ln}\left ({\left | x^{3} + 3 \right |}\right ) + \frac{1}{6} \,{\rm ln}\left ({\left | x^{3} + 1 \right |}\right ) \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate(x^2/(x^6 + 4*x^3 + 3),x, algorithm="giac")

[Out]

-1/6*ln(abs(x^3 + 3)) + 1/6*ln(abs(x^3 + 1))