3.256 \(\int \frac{x}{-1+x^4} \, dx\)

Optimal. Leaf size=8 \[ -\frac{1}{2} \tanh ^{-1}\left (x^2\right ) \]

[Out]

-ArcTanh[x^2]/2

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Rubi [A]  time = 0.00911792, antiderivative size = 8, normalized size of antiderivative = 1., number of steps used = 2, number of rules used = 2, integrand size = 9, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.222 \[ -\frac{1}{2} \tanh ^{-1}\left (x^2\right ) \]

Antiderivative was successfully verified.

[In]  Int[x/(-1 + x^4),x]

[Out]

-ArcTanh[x^2]/2

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Rubi in Sympy [A]  time = 1.10944, size = 7, normalized size = 0.88 \[ - \frac{\operatorname{atanh}{\left (x^{2} \right )}}{2} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  rubi_integrate(x/(x**4-1),x)

[Out]

-atanh(x**2)/2

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Mathematica [B]  time = 0.00402059, size = 23, normalized size = 2.88 \[ \frac{1}{4} \log \left (1-x^2\right )-\frac{1}{4} \log \left (x^2+1\right ) \]

Antiderivative was successfully verified.

[In]  Integrate[x/(-1 + x^4),x]

[Out]

Log[1 - x^2]/4 - Log[1 + x^2]/4

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Maple [B]  time = 0.001, size = 22, normalized size = 2.8 \[{\frac{\ln \left ( -1+x \right ) }{4}}+{\frac{\ln \left ( 1+x \right ) }{4}}-{\frac{\ln \left ({x}^{2}+1 \right ) }{4}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  int(x/(x^4-1),x)

[Out]

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

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Maxima [A]  time = 1.31853, size = 23, normalized size = 2.88 \[ -\frac{1}{4} \, \log \left (x^{2} + 1\right ) + \frac{1}{4} \, \log \left (x^{2} - 1\right ) \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate(x/(x^4 - 1),x, algorithm="maxima")

[Out]

-1/4*log(x^2 + 1) + 1/4*log(x^2 - 1)

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Fricas [A]  time = 0.197185, size = 23, normalized size = 2.88 \[ -\frac{1}{4} \, \log \left (x^{2} + 1\right ) + \frac{1}{4} \, \log \left (x^{2} - 1\right ) \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate(x/(x^4 - 1),x, algorithm="fricas")

[Out]

-1/4*log(x^2 + 1) + 1/4*log(x^2 - 1)

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Sympy [A]  time = 0.08554, size = 15, normalized size = 1.88 \[ \frac{\log{\left (x^{2} - 1 \right )}}{4} - \frac{\log{\left (x^{2} + 1 \right )}}{4} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate(x/(x**4-1),x)

[Out]

log(x**2 - 1)/4 - log(x**2 + 1)/4

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GIAC/XCAS [A]  time = 0.20203, size = 24, normalized size = 3. \[ -\frac{1}{4} \,{\rm ln}\left (x^{2} + 1\right ) + \frac{1}{4} \,{\rm ln}\left ({\left | x^{2} - 1 \right |}\right ) \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate(x/(x^4 - 1),x, algorithm="giac")

[Out]

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