3.1471 \(\int \frac{x^5}{1-x^8} \, dx\)

Optimal. Leaf size=17 \[ \frac{1}{4} \tanh ^{-1}\left (x^2\right )-\frac{1}{4} \tan ^{-1}\left (x^2\right ) \]

[Out]

-ArcTan[x^2]/4 + ArcTanh[x^2]/4

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Rubi [A]  time = 0.0298768, antiderivative size = 17, normalized size of antiderivative = 1., number of steps used = 4, number of rules used = 4, integrand size = 13, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.308 \[ \frac{1}{4} \tanh ^{-1}\left (x^2\right )-\frac{1}{4} \tan ^{-1}\left (x^2\right ) \]

Antiderivative was successfully verified.

[In]  Int[x^5/(1 - x^8),x]

[Out]

-ArcTan[x^2]/4 + ArcTanh[x^2]/4

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Rubi in Sympy [A]  time = 5.12585, size = 12, normalized size = 0.71 \[ - \frac{\operatorname{atan}{\left (x^{2} \right )}}{4} + \frac{\operatorname{atanh}{\left (x^{2} \right )}}{4} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  rubi_integrate(x**5/(-x**8+1),x)

[Out]

-atan(x**2)/4 + atanh(x**2)/4

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Mathematica [A]  time = 0.00868914, size = 31, normalized size = 1.82 \[ -\frac{1}{8} \log \left (1-x^2\right )+\frac{1}{8} \log \left (x^2+1\right )+\frac{1}{4} \tan ^{-1}\left (\frac{1}{x^2}\right ) \]

Antiderivative was successfully verified.

[In]  Integrate[x^5/(1 - x^8),x]

[Out]

ArcTan[x^(-2)]/4 - Log[1 - x^2]/8 + Log[1 + x^2]/8

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Maple [B]  time = 0.008, size = 28, normalized size = 1.7 \[ -{\frac{\ln \left ( -1+x \right ) }{8}}-{\frac{\ln \left ( 1+x \right ) }{8}}+{\frac{\ln \left ({x}^{2}+1 \right ) }{8}}-{\frac{\arctan \left ({x}^{2} \right ) }{4}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  int(x^5/(-x^8+1),x)

[Out]

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

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Maxima [A]  time = 1.5845, size = 31, normalized size = 1.82 \[ -\frac{1}{4} \, \arctan \left (x^{2}\right ) + \frac{1}{8} \, \log \left (x^{2} + 1\right ) - \frac{1}{8} \, \log \left (x^{2} - 1\right ) \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate(-x^5/(x^8 - 1),x, algorithm="maxima")

[Out]

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

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Fricas [A]  time = 0.219865, size = 31, normalized size = 1.82 \[ -\frac{1}{4} \, \arctan \left (x^{2}\right ) + \frac{1}{8} \, \log \left (x^{2} + 1\right ) - \frac{1}{8} \, \log \left (x^{2} - 1\right ) \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate(-x^5/(x^8 - 1),x, algorithm="fricas")

[Out]

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

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Sympy [A]  time = 0.438275, size = 22, normalized size = 1.29 \[ - \frac{\log{\left (x^{2} - 1 \right )}}{8} + \frac{\log{\left (x^{2} + 1 \right )}}{8} - \frac{\operatorname{atan}{\left (x^{2} \right )}}{4} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate(x**5/(-x**8+1),x)

[Out]

-log(x**2 - 1)/8 + log(x**2 + 1)/8 - atan(x**2)/4

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GIAC/XCAS [A]  time = 0.219613, size = 74, normalized size = 4.35 \[ \frac{1}{4} \, \arctan \left (\frac{1}{2} \, \sqrt{2}{\left (2 \, x + \sqrt{2}\right )}\right ) - \frac{1}{4} \, \arctan \left (\frac{1}{2} \, \sqrt{2}{\left (2 \, x - \sqrt{2}\right )}\right ) + \frac{1}{8} \,{\rm ln}\left (x^{2} + 1\right ) - \frac{1}{8} \,{\rm ln}\left ({\left | x + 1 \right |}\right ) - \frac{1}{8} \,{\rm ln}\left ({\left | x - 1 \right |}\right ) \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate(-x^5/(x^8 - 1),x, algorithm="giac")

[Out]

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