3.1489 \(\int \frac{x^7}{1+x^8} \, dx\)

Optimal. Leaf size=10 \[ \frac{1}{8} \log \left (x^8+1\right ) \]

[Out]

Log[1 + x^8]/8

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Rubi [A]  time = 0.0072393, antiderivative size = 10, normalized size of antiderivative = 1., number of steps used = 1, number of rules used = 1, integrand size = 11, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.091 \[ \frac{1}{8} \log \left (x^8+1\right ) \]

Antiderivative was successfully verified.

[In]  Int[x^7/(1 + x^8),x]

[Out]

Log[1 + x^8]/8

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Rubi in Sympy [A]  time = 1.70303, size = 7, normalized size = 0.7 \[ \frac{\log{\left (x^{8} + 1 \right )}}{8} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  rubi_integrate(x**7/(x**8+1),x)

[Out]

log(x**8 + 1)/8

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Mathematica [A]  time = 0.00410698, size = 10, normalized size = 1. \[ \frac{1}{8} \log \left (x^8+1\right ) \]

Antiderivative was successfully verified.

[In]  Integrate[x^7/(1 + x^8),x]

[Out]

Log[1 + x^8]/8

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Maple [A]  time = 0.001, size = 9, normalized size = 0.9 \[{\frac{\ln \left ({x}^{8}+1 \right ) }{8}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  int(x^7/(x^8+1),x)

[Out]

1/8*ln(x^8+1)

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Maxima [A]  time = 1.44319, size = 11, normalized size = 1.1 \[ \frac{1}{8} \, \log \left (x^{8} + 1\right ) \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate(x^7/(x^8 + 1),x, algorithm="maxima")

[Out]

1/8*log(x^8 + 1)

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Fricas [A]  time = 0.208027, size = 11, normalized size = 1.1 \[ \frac{1}{8} \, \log \left (x^{8} + 1\right ) \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate(x^7/(x^8 + 1),x, algorithm="fricas")

[Out]

1/8*log(x^8 + 1)

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Sympy [A]  time = 0.213706, size = 7, normalized size = 0.7 \[ \frac{\log{\left (x^{8} + 1 \right )}}{8} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate(x**7/(x**8+1),x)

[Out]

log(x**8 + 1)/8

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GIAC/XCAS [A]  time = 0.237823, size = 11, normalized size = 1.1 \[ \frac{1}{8} \,{\rm ln}\left (x^{8} + 1\right ) \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate(x^7/(x^8 + 1),x, algorithm="giac")

[Out]

1/8*ln(x^8 + 1)