3.1472 \(\int \frac{x^3}{1-x^8} \, dx\)

Optimal. Leaf size=8 \[ \frac{1}{4} \tanh ^{-1}\left (x^4\right ) \]

[Out]

ArcTanh[x^4]/4

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Rubi [A]  time = 0.0153998, antiderivative size = 8, normalized size of antiderivative = 1., number of steps used = 2, number of rules used = 2, integrand size = 13, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.154 \[ \frac{1}{4} \tanh ^{-1}\left (x^4\right ) \]

Antiderivative was successfully verified.

[In]  Int[x^3/(1 - x^8),x]

[Out]

ArcTanh[x^4]/4

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Rubi in Sympy [A]  time = 2.95082, size = 5, normalized size = 0.62 \[ \frac{\operatorname{atanh}{\left (x^{4} \right )}}{4} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  rubi_integrate(x**3/(-x**8+1),x)

[Out]

atanh(x**4)/4

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Mathematica [B]  time = 0.0063379, size = 23, normalized size = 2.88 \[ \frac{1}{8} \log \left (x^4+1\right )-\frac{1}{8} \log \left (1-x^4\right ) \]

Antiderivative was successfully verified.

[In]  Integrate[x^3/(1 - x^8),x]

[Out]

-Log[1 - x^4]/8 + Log[1 + x^4]/8

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Maple [B]  time = 0.005, size = 30, normalized size = 3.8 \[ -{\frac{\ln \left ( -1+x \right ) }{8}}-{\frac{\ln \left ( 1+x \right ) }{8}}-{\frac{\ln \left ({x}^{2}+1 \right ) }{8}}+{\frac{\ln \left ({x}^{4}+1 \right ) }{8}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  int(x^3/(-x^8+1),x)

[Out]

-1/8*ln(-1+x)-1/8*ln(1+x)-1/8*ln(x^2+1)+1/8*ln(x^4+1)

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Maxima [A]  time = 1.41653, size = 23, normalized size = 2.88 \[ \frac{1}{8} \, \log \left (x^{4} + 1\right ) - \frac{1}{8} \, \log \left (x^{4} - 1\right ) \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate(-x^3/(x^8 - 1),x, algorithm="maxima")

[Out]

1/8*log(x^4 + 1) - 1/8*log(x^4 - 1)

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Fricas [A]  time = 0.216382, size = 23, normalized size = 2.88 \[ \frac{1}{8} \, \log \left (x^{4} + 1\right ) - \frac{1}{8} \, \log \left (x^{4} - 1\right ) \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate(-x^3/(x^8 - 1),x, algorithm="fricas")

[Out]

1/8*log(x^4 + 1) - 1/8*log(x^4 - 1)

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Sympy [A]  time = 0.279371, size = 15, normalized size = 1.88 \[ - \frac{\log{\left (x^{4} - 1 \right )}}{8} + \frac{\log{\left (x^{4} + 1 \right )}}{8} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate(x**3/(-x**8+1),x)

[Out]

-log(x**4 - 1)/8 + log(x**4 + 1)/8

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GIAC/XCAS [A]  time = 0.217299, size = 24, normalized size = 3. \[ \frac{1}{8} \,{\rm ln}\left (x^{4} + 1\right ) - \frac{1}{8} \,{\rm ln}\left ({\left | x^{4} - 1 \right |}\right ) \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate(-x^3/(x^8 - 1),x, algorithm="giac")

[Out]

1/8*ln(x^4 + 1) - 1/8*ln(abs(x^4 - 1))