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

Optimal. Leaf size=21 \[ \frac{1}{4} \log \left (x^4+1\right )-\frac{1}{4} \log \left (x^4+2\right ) \]

[Out]

Log[1 + x^4]/4 - Log[2 + x^4]/4

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Rubi [A]  time = 0.0294496, antiderivative size = 21, normalized size of antiderivative = 1., number of steps used = 4, number of rules used = 3, integrand size = 16, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.188 \[ \frac{1}{4} \log \left (x^4+1\right )-\frac{1}{4} \log \left (x^4+2\right ) \]

Antiderivative was successfully verified.

[In]  Int[x^3/(2 + 3*x^4 + x^8),x]

[Out]

Log[1 + x^4]/4 - Log[2 + x^4]/4

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Rubi in Sympy [A]  time = 5.65434, size = 15, normalized size = 0.71 \[ \frac{\log{\left (x^{4} + 1 \right )}}{4} - \frac{\log{\left (x^{4} + 2 \right )}}{4} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

log(x**4 + 1)/4 - log(x**4 + 2)/4

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Mathematica [A]  time = 0.00633758, size = 21, normalized size = 1. \[ \frac{1}{4} \log \left (x^4+1\right )-\frac{1}{4} \log \left (x^4+2\right ) \]

Antiderivative was successfully verified.

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

[Out]

Log[1 + x^4]/4 - Log[2 + x^4]/4

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Maple [A]  time = 0.009, size = 18, normalized size = 0.9 \[{\frac{\ln \left ({x}^{4}+1 \right ) }{4}}-{\frac{\ln \left ({x}^{4}+2 \right ) }{4}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  int(x^3/(x^8+3*x^4+2),x)

[Out]

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

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Maxima [A]  time = 0.746434, size = 23, normalized size = 1.1 \[ -\frac{1}{4} \, \log \left (x^{4} + 2\right ) + \frac{1}{4} \, \log \left (x^{4} + 1\right ) \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

-1/4*log(x^4 + 2) + 1/4*log(x^4 + 1)

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Fricas [A]  time = 0.254498, size = 23, normalized size = 1.1 \[ -\frac{1}{4} \, \log \left (x^{4} + 2\right ) + \frac{1}{4} \, \log \left (x^{4} + 1\right ) \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

-1/4*log(x^4 + 2) + 1/4*log(x^4 + 1)

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Sympy [A]  time = 0.255413, size = 15, normalized size = 0.71 \[ \frac{\log{\left (x^{4} + 1 \right )}}{4} - \frac{\log{\left (x^{4} + 2 \right )}}{4} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

log(x**4 + 1)/4 - log(x**4 + 2)/4

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GIAC/XCAS [A]  time = 0.281482, size = 23, normalized size = 1.1 \[ -\frac{1}{4} \,{\rm ln}\left (x^{4} + 2\right ) + \frac{1}{4} \,{\rm ln}\left (x^{4} + 1\right ) \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

-1/4*ln(x^4 + 2) + 1/4*ln(x^4 + 1)