Optimal. Leaf size=34 \[ \frac{x^5}{4 \left (1-x^4\right )}+\frac{5 x}{4}-\frac{5}{8} \tan ^{-1}(x)-\frac{5}{8} \tanh ^{-1}(x) \]
[Out]
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Rubi [A] time = 0.0264783, antiderivative size = 34, normalized size of antiderivative = 1., number of steps used = 6, number of rules used = 6, integrand size = 16, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.375 \[ \frac{x^5}{4 \left (1-x^4\right )}+\frac{5 x}{4}-\frac{5}{8} \tan ^{-1}(x)-\frac{5}{8} \tanh ^{-1}(x) \]
Antiderivative was successfully verified.
[In] Int[x^8/(1 - 2*x^4 + x^8),x]
[Out]
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Rubi in Sympy [A] time = 6.27079, size = 27, normalized size = 0.79 \[ \frac{x^{5}}{4 \left (- x^{4} + 1\right )} + \frac{5 x}{4} - \frac{5 \operatorname{atan}{\left (x \right )}}{8} - \frac{5 \operatorname{atanh}{\left (x \right )}}{8} \]
Verification of antiderivative is not currently implemented for this CAS.
[In] rubi_integrate(x**8/(x**8-2*x**4+1),x)
[Out]
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Mathematica [A] time = 0.0253772, size = 38, normalized size = 1.12 \[ -\frac{x}{4 \left (x^4-1\right )}+x+\frac{5}{16} \log (1-x)-\frac{5}{16} \log (x+1)-\frac{5}{8} \tan ^{-1}(x) \]
Antiderivative was successfully verified.
[In] Integrate[x^8/(1 - 2*x^4 + x^8),x]
[Out]
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Maple [A] time = 0.02, size = 43, normalized size = 1.3 \[ x-{\frac{1}{-16+16\,x}}+{\frac{5\,\ln \left ( -1+x \right ) }{16}}-{\frac{1}{16+16\,x}}-{\frac{5\,\ln \left ( 1+x \right ) }{16}}+{\frac{x}{8\,{x}^{2}+8}}-{\frac{5\,\arctan \left ( x \right ) }{8}} \]
Verification of antiderivative is not currently implemented for this CAS.
[In] int(x^8/(x^8-2*x^4+1),x)
[Out]
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Maxima [A] time = 0.845837, size = 38, normalized size = 1.12 \[ x - \frac{x}{4 \,{\left (x^{4} - 1\right )}} - \frac{5}{8} \, \arctan \left (x\right ) - \frac{5}{16} \, \log \left (x + 1\right ) + \frac{5}{16} \, \log \left (x - 1\right ) \]
Verification of antiderivative is not currently implemented for this CAS.
[In] integrate(x^8/(x^8 - 2*x^4 + 1),x, algorithm="maxima")
[Out]
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Fricas [A] time = 0.264062, size = 66, normalized size = 1.94 \[ \frac{16 \, x^{5} - 10 \,{\left (x^{4} - 1\right )} \arctan \left (x\right ) - 5 \,{\left (x^{4} - 1\right )} \log \left (x + 1\right ) + 5 \,{\left (x^{4} - 1\right )} \log \left (x - 1\right ) - 20 \, x}{16 \,{\left (x^{4} - 1\right )}} \]
Verification of antiderivative is not currently implemented for this CAS.
[In] integrate(x^8/(x^8 - 2*x^4 + 1),x, algorithm="fricas")
[Out]
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Sympy [A] time = 0.45463, size = 32, normalized size = 0.94 \[ x - \frac{x}{4 x^{4} - 4} + \frac{5 \log{\left (x - 1 \right )}}{16} - \frac{5 \log{\left (x + 1 \right )}}{16} - \frac{5 \operatorname{atan}{\left (x \right )}}{8} \]
Verification of antiderivative is not currently implemented for this CAS.
[In] integrate(x**8/(x**8-2*x**4+1),x)
[Out]
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GIAC/XCAS [A] time = 0.304627, size = 41, normalized size = 1.21 \[ x - \frac{x}{4 \,{\left (x^{4} - 1\right )}} - \frac{5}{8} \, \arctan \left (x\right ) - \frac{5}{16} \,{\rm ln}\left ({\left | x + 1 \right |}\right ) + \frac{5}{16} \,{\rm ln}\left ({\left | x - 1 \right |}\right ) \]
Verification of antiderivative is not currently implemented for this CAS.
[In] integrate(x^8/(x^8 - 2*x^4 + 1),x, algorithm="giac")
[Out]