3.406 \(\int \frac{x^{11}}{2+3 x^4+x^8} \, dx\)

Optimal. Leaf size=26 \[ \frac{x^4}{4}+\frac{1}{4} \log \left (x^4+1\right )-\log \left (x^4+2\right ) \]

[Out]

x^4/4 + Log[1 + x^4]/4 - Log[2 + x^4]

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Rubi [A]  time = 0.0448069, antiderivative size = 26, 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 \[ \frac{x^4}{4}+\frac{1}{4} \log \left (x^4+1\right )-\log \left (x^4+2\right ) \]

Antiderivative was successfully verified.

[In]  Int[x^11/(2 + 3*x^4 + x^8),x]

[Out]

x^4/4 + Log[1 + x^4]/4 - Log[2 + x^4]

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Rubi in Sympy [A]  time = 11.1046, size = 19, normalized size = 0.73 \[ \frac{x^{4}}{4} + \frac{\log{\left (x^{4} + 1 \right )}}{4} - \log{\left (x^{4} + 2 \right )} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  rubi_integrate(x**11/(x**8+3*x**4+2),x)

[Out]

x**4/4 + log(x**4 + 1)/4 - log(x**4 + 2)

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Mathematica [A]  time = 0.00697979, size = 26, normalized size = 1. \[ \frac{x^4}{4}+\frac{1}{4} \log \left (x^4+1\right )-\log \left (x^4+2\right ) \]

Antiderivative was successfully verified.

[In]  Integrate[x^11/(2 + 3*x^4 + x^8),x]

[Out]

x^4/4 + Log[1 + x^4]/4 - Log[2 + x^4]

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Maple [A]  time = 0.008, size = 23, normalized size = 0.9 \[{\frac{{x}^{4}}{4}}+{\frac{\ln \left ({x}^{4}+1 \right ) }{4}}-\ln \left ({x}^{4}+2 \right ) \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  int(x^11/(x^8+3*x^4+2),x)

[Out]

1/4*x^4+1/4*ln(x^4+1)-ln(x^4+2)

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Maxima [A]  time = 0.744052, size = 30, normalized size = 1.15 \[ \frac{1}{4} \, x^{4} - \log \left (x^{4} + 2\right ) + \frac{1}{4} \, \log \left (x^{4} + 1\right ) \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate(x^11/(x^8 + 3*x^4 + 2),x, algorithm="maxima")

[Out]

1/4*x^4 - log(x^4 + 2) + 1/4*log(x^4 + 1)

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Fricas [A]  time = 0.251848, size = 30, normalized size = 1.15 \[ \frac{1}{4} \, x^{4} - \log \left (x^{4} + 2\right ) + \frac{1}{4} \, \log \left (x^{4} + 1\right ) \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate(x^11/(x^8 + 3*x^4 + 2),x, algorithm="fricas")

[Out]

1/4*x^4 - log(x^4 + 2) + 1/4*log(x^4 + 1)

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Sympy [A]  time = 0.298764, size = 19, normalized size = 0.73 \[ \frac{x^{4}}{4} + \frac{\log{\left (x^{4} + 1 \right )}}{4} - \log{\left (x^{4} + 2 \right )} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate(x**11/(x**8+3*x**4+2),x)

[Out]

x**4/4 + log(x**4 + 1)/4 - log(x**4 + 2)

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GIAC/XCAS [A]  time = 0.281172, size = 30, normalized size = 1.15 \[ \frac{1}{4} \, x^{4} -{\rm ln}\left (x^{4} + 2\right ) + \frac{1}{4} \,{\rm ln}\left (x^{4} + 1\right ) \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate(x^11/(x^8 + 3*x^4 + 2),x, algorithm="giac")

[Out]

1/4*x^4 - ln(x^4 + 2) + 1/4*ln(x^4 + 1)