3.152 \(\int \frac{x^8}{3+4 x^3+x^6} \, dx\)

Optimal. Leaf size=28 \[ \frac{x^3}{3}+\frac{1}{6} \log \left (x^3+1\right )-\frac{3}{2} \log \left (x^3+3\right ) \]

[Out]

x^3/3 + Log[1 + x^3]/6 - (3*Log[3 + x^3])/2

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Rubi [A]  time = 0.0469661, antiderivative size = 28, normalized size of antiderivative = 1., number of steps used = 5, number of rules used = 4, integrand size = 16, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.25 \[ \frac{x^3}{3}+\frac{1}{6} \log \left (x^3+1\right )-\frac{3}{2} \log \left (x^3+3\right ) \]

Antiderivative was successfully verified.

[In]  Int[x^8/(3 + 4*x^3 + x^6),x]

[Out]

x^3/3 + Log[1 + x^3]/6 - (3*Log[3 + x^3])/2

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Rubi in Sympy [A]  time = 11.1333, size = 22, normalized size = 0.79 \[ \frac{x^{3}}{3} + \frac{\log{\left (x^{3} + 1 \right )}}{6} - \frac{3 \log{\left (x^{3} + 3 \right )}}{2} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  rubi_integrate(x**8/(x**6+4*x**3+3),x)

[Out]

x**3/3 + log(x**3 + 1)/6 - 3*log(x**3 + 3)/2

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Mathematica [A]  time = 0.00915247, size = 28, normalized size = 1. \[ \frac{x^3}{3}+\frac{1}{6} \log \left (x^3+1\right )-\frac{3}{2} \log \left (x^3+3\right ) \]

Antiderivative was successfully verified.

[In]  Integrate[x^8/(3 + 4*x^3 + x^6),x]

[Out]

x^3/3 + Log[1 + x^3]/6 - (3*Log[3 + x^3])/2

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Maple [A]  time = 0.007, size = 23, normalized size = 0.8 \[{\frac{{x}^{3}}{3}}+{\frac{\ln \left ({x}^{3}+1 \right ) }{6}}-{\frac{3\,\ln \left ({x}^{3}+3 \right ) }{2}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  int(x^8/(x^6+4*x^3+3),x)

[Out]

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

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Maxima [A]  time = 0.779735, size = 30, normalized size = 1.07 \[ \frac{1}{3} \, x^{3} - \frac{3}{2} \, \log \left (x^{3} + 3\right ) + \frac{1}{6} \, \log \left (x^{3} + 1\right ) \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate(x^8/(x^6 + 4*x^3 + 3),x, algorithm="maxima")

[Out]

1/3*x^3 - 3/2*log(x^3 + 3) + 1/6*log(x^3 + 1)

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Fricas [A]  time = 0.263143, size = 30, normalized size = 1.07 \[ \frac{1}{3} \, x^{3} - \frac{3}{2} \, \log \left (x^{3} + 3\right ) + \frac{1}{6} \, \log \left (x^{3} + 1\right ) \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate(x^8/(x^6 + 4*x^3 + 3),x, algorithm="fricas")

[Out]

1/3*x^3 - 3/2*log(x^3 + 3) + 1/6*log(x^3 + 1)

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Sympy [A]  time = 0.266127, size = 22, normalized size = 0.79 \[ \frac{x^{3}}{3} + \frac{\log{\left (x^{3} + 1 \right )}}{6} - \frac{3 \log{\left (x^{3} + 3 \right )}}{2} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate(x**8/(x**6+4*x**3+3),x)

[Out]

x**3/3 + log(x**3 + 1)/6 - 3*log(x**3 + 3)/2

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GIAC/XCAS [A]  time = 0.277353, size = 32, normalized size = 1.14 \[ \frac{1}{3} \, x^{3} - \frac{3}{2} \,{\rm ln}\left ({\left | x^{3} + 3 \right |}\right ) + \frac{1}{6} \,{\rm ln}\left ({\left | x^{3} + 1 \right |}\right ) \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate(x^8/(x^6 + 4*x^3 + 3),x, algorithm="giac")

[Out]

1/3*x^3 - 3/2*ln(abs(x^3 + 3)) + 1/6*ln(abs(x^3 + 1))