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]
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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]
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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]
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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]
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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]
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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]
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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]
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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]
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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]