Optimal. Leaf size=29 \[ \frac{x^3}{4 \left (1-x^4\right )}-\frac{1}{8} \tan ^{-1}(x)+\frac{1}{8} \tanh ^{-1}(x) \]
[Out]
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Rubi [A] time = 0.0249913, antiderivative size = 29, 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 \[ \frac{x^3}{4 \left (1-x^4\right )}-\frac{1}{8} \tan ^{-1}(x)+\frac{1}{8} \tanh ^{-1}(x) \]
Antiderivative was successfully verified.
[In] Int[x^2/(1 - 2*x^4 + x^8),x]
[Out]
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Rubi in Sympy [A] time = 5.77625, size = 19, normalized size = 0.66 \[ \frac{x^{3}}{4 \left (- x^{4} + 1\right )} - \frac{\operatorname{atan}{\left (x \right )}}{8} + \frac{\operatorname{atanh}{\left (x \right )}}{8} \]
Verification of antiderivative is not currently implemented for this CAS.
[In] rubi_integrate(x**2/(x**8-2*x**4+1),x)
[Out]
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Mathematica [A] time = 0.019038, size = 33, normalized size = 1.14 \[ \frac{1}{16} \left (-\frac{4 x^3}{x^4-1}-\log (1-x)+\log (x+1)-2 \tan ^{-1}(x)\right ) \]
Antiderivative was successfully verified.
[In] Integrate[x^2/(1 - 2*x^4 + x^8),x]
[Out]
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Maple [A] time = 0.016, size = 42, normalized size = 1.5 \[ -{\frac{1}{-16+16\,x}}-{\frac{\ln \left ( -1+x \right ) }{16}}-{\frac{1}{16+16\,x}}+{\frac{\ln \left ( 1+x \right ) }{16}}-{\frac{x}{8\,{x}^{2}+8}}-{\frac{\arctan \left ( x \right ) }{8}} \]
Verification of antiderivative is not currently implemented for this CAS.
[In] int(x^2/(x^8-2*x^4+1),x)
[Out]
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Maxima [A] time = 0.851383, size = 39, normalized size = 1.34 \[ -\frac{x^{3}}{4 \,{\left (x^{4} - 1\right )}} - \frac{1}{8} \, \arctan \left (x\right ) + \frac{1}{16} \, \log \left (x + 1\right ) - \frac{1}{16} \, \log \left (x - 1\right ) \]
Verification of antiderivative is not currently implemented for this CAS.
[In] integrate(x^2/(x^8 - 2*x^4 + 1),x, algorithm="maxima")
[Out]
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Fricas [A] time = 0.257829, size = 61, normalized size = 2.1 \[ -\frac{4 \, x^{3} + 2 \,{\left (x^{4} - 1\right )} \arctan \left (x\right ) -{\left (x^{4} - 1\right )} \log \left (x + 1\right ) +{\left (x^{4} - 1\right )} \log \left (x - 1\right )}{16 \,{\left (x^{4} - 1\right )}} \]
Verification of antiderivative is not currently implemented for this CAS.
[In] integrate(x^2/(x^8 - 2*x^4 + 1),x, algorithm="fricas")
[Out]
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Sympy [A] time = 0.437867, size = 27, normalized size = 0.93 \[ - \frac{x^{3}}{4 x^{4} - 4} - \frac{\log{\left (x - 1 \right )}}{16} + \frac{\log{\left (x + 1 \right )}}{16} - \frac{\operatorname{atan}{\left (x \right )}}{8} \]
Verification of antiderivative is not currently implemented for this CAS.
[In] integrate(x**2/(x**8-2*x**4+1),x)
[Out]
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GIAC/XCAS [A] time = 0.274123, size = 42, normalized size = 1.45 \[ -\frac{x^{3}}{4 \,{\left (x^{4} - 1\right )}} - \frac{1}{8} \, \arctan \left (x\right ) + \frac{1}{16} \,{\rm ln}\left ({\left | x + 1 \right |}\right ) - \frac{1}{16} \,{\rm ln}\left ({\left | x - 1 \right |}\right ) \]
Verification of antiderivative is not currently implemented for this CAS.
[In] integrate(x^2/(x^8 - 2*x^4 + 1),x, algorithm="giac")
[Out]