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

Optimal. Leaf size=27 \[ \frac{x}{4 \left (1-x^4\right )}-\frac{1}{8} \tan ^{-1}(x)-\frac{1}{8} \tanh ^{-1}(x) \]

[Out]

x/(4*(1 - x^4)) - ArcTan[x]/8 - ArcTanh[x]/8

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Rubi [A]  time = 0.0197442, antiderivative size = 27, normalized size of antiderivative = 1., number of steps used = 5, number of rules used = 5, integrand size = 16, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.312 \[ \frac{x}{4 \left (1-x^4\right )}-\frac{1}{8} \tan ^{-1}(x)-\frac{1}{8} \tanh ^{-1}(x) \]

Antiderivative was successfully verified.

[In]  Int[x^4/(1 - 2*x^4 + x^8),x]

[Out]

x/(4*(1 - x^4)) - ArcTan[x]/8 - ArcTanh[x]/8

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Rubi in Sympy [A]  time = 4.34331, size = 17, normalized size = 0.63 \[ \frac{x}{4 \left (- x^{4} + 1\right )} - \frac{\operatorname{atan}{\left (x \right )}}{8} - \frac{\operatorname{atanh}{\left (x \right )}}{8} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

x/(4*(-x**4 + 1)) - atan(x)/8 - atanh(x)/8

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Mathematica [A]  time = 0.0214261, size = 31, normalized size = 1.15 \[ \frac{1}{16} \left (-\frac{4 x}{x^4-1}+\log (1-x)-\log (x+1)-2 \tan ^{-1}(x)\right ) \]

Antiderivative was successfully verified.

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

[Out]

((-4*x)/(-1 + x^4) - 2*ArcTan[x] + Log[1 - x] - Log[1 + x])/16

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Maple [A]  time = 0.018, size = 42, normalized size = 1.6 \[ -{\frac{1}{-16+16\,x}}+{\frac{\ln \left ( -1+x \right ) }{16}}-{\frac{1}{16+16\,x}}-{\frac{\ln \left ( 1+x \right ) }{16}}+{\frac{x}{8\,{x}^{2}+8}}-{\frac{\arctan \left ( x \right ) }{8}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  int(x^4/(x^8-2*x^4+1),x)

[Out]

-1/16/(-1+x)+1/16*ln(-1+x)-1/16/(1+x)-1/16*ln(1+x)+1/8*x/(x^2+1)-1/8*arctan(x)

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Maxima [A]  time = 0.860397, size = 36, normalized size = 1.33 \[ -\frac{x}{4 \,{\left (x^{4} - 1\right )}} - \frac{1}{8} \, \arctan \left (x\right ) - \frac{1}{16} \, \log \left (x + 1\right ) + \frac{1}{16} \, \log \left (x - 1\right ) \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

-1/4*x/(x^4 - 1) - 1/8*arctan(x) - 1/16*log(x + 1) + 1/16*log(x - 1)

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Fricas [A]  time = 0.282568, size = 58, normalized size = 2.15 \[ -\frac{2 \,{\left (x^{4} - 1\right )} \arctan \left (x\right ) +{\left (x^{4} - 1\right )} \log \left (x + 1\right ) -{\left (x^{4} - 1\right )} \log \left (x - 1\right ) + 4 \, x}{16 \,{\left (x^{4} - 1\right )}} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

-1/16*(2*(x^4 - 1)*arctan(x) + (x^4 - 1)*log(x + 1) - (x^4 - 1)*log(x - 1) + 4*x
)/(x^4 - 1)

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Sympy [A]  time = 0.435699, size = 26, normalized size = 0.96 \[ - \frac{x}{4 x^{4} - 4} + \frac{\log{\left (x - 1 \right )}}{16} - \frac{\log{\left (x + 1 \right )}}{16} - \frac{\operatorname{atan}{\left (x \right )}}{8} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

-x/(4*x**4 - 4) + log(x - 1)/16 - log(x + 1)/16 - atan(x)/8

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GIAC/XCAS [A]  time = 0.282249, size = 39, normalized size = 1.44 \[ -\frac{x}{4 \,{\left (x^{4} - 1\right )}} - \frac{1}{8} \, \arctan \left (x\right ) - \frac{1}{16} \,{\rm ln}\left ({\left | x + 1 \right |}\right ) + \frac{1}{16} \,{\rm ln}\left ({\left | x - 1 \right |}\right ) \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

-1/4*x/(x^4 - 1) - 1/8*arctan(x) - 1/16*ln(abs(x + 1)) + 1/16*ln(abs(x - 1))