3.293 \(\int \frac{x^5}{1-2 x^4+x^8} \, dx\)

Optimal. Leaf size=25 \[ \frac{x^2}{4 \left (1-x^4\right )}-\frac{1}{4} \tanh ^{-1}\left (x^2\right ) \]

[Out]

x^2/(4*(1 - x^4)) - ArcTanh[x^2]/4

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Rubi [A]  time = 0.0289943, antiderivative size = 25, normalized size of antiderivative = 1., number of steps used = 4, number of rules used = 4, integrand size = 16, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.25 \[ \frac{x^2}{4 \left (1-x^4\right )}-\frac{1}{4} \tanh ^{-1}\left (x^2\right ) \]

Antiderivative was successfully verified.

[In]  Int[x^5/(1 - 2*x^4 + x^8),x]

[Out]

x^2/(4*(1 - x^4)) - ArcTanh[x^2]/4

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Rubi in Sympy [A]  time = 5.9401, size = 15, normalized size = 0.6 \[ \frac{x^{2}}{4 \left (- x^{4} + 1\right )} - \frac{\operatorname{atanh}{\left (x^{2} \right )}}{4} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  rubi_integrate(x**5/(x**8-2*x**4+1),x)

[Out]

x**2/(4*(-x**4 + 1)) - atanh(x**2)/4

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Mathematica [A]  time = 0.0177655, size = 33, normalized size = 1.32 \[ \frac{1}{8} \left (\log \left (1-x^2\right )-\log \left (x^2+1\right )-\frac{2 x^2}{x^4-1}\right ) \]

Antiderivative was successfully verified.

[In]  Integrate[x^5/(1 - 2*x^4 + x^8),x]

[Out]

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

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Maple [A]  time = 0.013, size = 36, normalized size = 1.4 \[ -{\frac{1}{8\,{x}^{2}-8}}+{\frac{\ln \left ({x}^{2}-1 \right ) }{8}}-{\frac{1}{8\,{x}^{2}+8}}-{\frac{\ln \left ({x}^{2}+1 \right ) }{8}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  int(x^5/(x^8-2*x^4+1),x)

[Out]

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

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Maxima [A]  time = 0.778889, size = 39, normalized size = 1.56 \[ -\frac{x^{2}}{4 \,{\left (x^{4} - 1\right )}} - \frac{1}{8} \, \log \left (x^{2} + 1\right ) + \frac{1}{8} \, \log \left (x^{2} - 1\right ) \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate(x^5/(x^8 - 2*x^4 + 1),x, algorithm="maxima")

[Out]

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

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Fricas [A]  time = 0.251481, size = 54, normalized size = 2.16 \[ -\frac{2 \, x^{2} +{\left (x^{4} - 1\right )} \log \left (x^{2} + 1\right ) -{\left (x^{4} - 1\right )} \log \left (x^{2} - 1\right )}{8 \,{\left (x^{4} - 1\right )}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate(x^5/(x^8 - 2*x^4 + 1),x, algorithm="fricas")

[Out]

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

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Sympy [A]  time = 0.273902, size = 26, normalized size = 1.04 \[ - \frac{x^{2}}{4 x^{4} - 4} + \frac{\log{\left (x^{2} - 1 \right )}}{8} - \frac{\log{\left (x^{2} + 1 \right )}}{8} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate(x**5/(x**8-2*x**4+1),x)

[Out]

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

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GIAC/XCAS [A]  time = 0.282924, size = 41, normalized size = 1.64 \[ -\frac{x^{2}}{4 \,{\left (x^{4} - 1\right )}} - \frac{1}{8} \,{\rm ln}\left (x^{2} + 1\right ) + \frac{1}{8} \,{\rm ln}\left ({\left | x^{2} - 1 \right |}\right ) \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate(x^5/(x^8 - 2*x^4 + 1),x, algorithm="giac")

[Out]

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