Optimal. Leaf size=41 \[ -\frac{\tan ^{-1}\left (\frac{1-2 x^4}{\sqrt{3}}\right )}{4 \sqrt{3}}-\frac{1}{8} \log \left (x^8-x^4+1\right )+\log (x) \]
[Out]
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Rubi [A] time = 0.0718375, antiderivative size = 41, normalized size of antiderivative = 1., number of steps used = 7, number of rules used = 7, integrand size = 16, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.438 \[ -\frac{\tan ^{-1}\left (\frac{1-2 x^4}{\sqrt{3}}\right )}{4 \sqrt{3}}-\frac{1}{8} \log \left (x^8-x^4+1\right )+\log (x) \]
Antiderivative was successfully verified.
[In] Int[1/(x*(1 - x^4 + x^8)),x]
[Out]
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Rubi in Sympy [A] time = 13.4691, size = 41, normalized size = 1. \[ \frac{\log{\left (x^{4} \right )}}{4} - \frac{\log{\left (x^{8} - x^{4} + 1 \right )}}{8} + \frac{\sqrt{3} \operatorname{atan}{\left (\sqrt{3} \left (\frac{2 x^{4}}{3} - \frac{1}{3}\right ) \right )}}{12} \]
Verification of antiderivative is not currently implemented for this CAS.
[In] rubi_integrate(1/x/(x**8-x**4+1),x)
[Out]
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Mathematica [C] time = 0.0209464, size = 55, normalized size = 1.34 \[ \log (x)-\frac{1}{4} \text{RootSum}\left [\text{$\#$1}^8-\text{$\#$1}^4+1\&,\frac{\text{$\#$1}^4 \log (x-\text{$\#$1})-\log (x-\text{$\#$1})}{2 \text{$\#$1}^4-1}\&\right ] \]
Antiderivative was successfully verified.
[In] Integrate[1/(x*(1 - x^4 + x^8)),x]
[Out]
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Maple [A] time = 0.011, size = 35, normalized size = 0.9 \[ \ln \left ( x \right ) -{\frac{\ln \left ({x}^{8}-{x}^{4}+1 \right ) }{8}}+{\frac{\sqrt{3}}{12}\arctan \left ({\frac{ \left ( 2\,{x}^{4}-1 \right ) \sqrt{3}}{3}} \right ) } \]
Verification of antiderivative is not currently implemented for this CAS.
[In] int(1/x/(x^8-x^4+1),x)
[Out]
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Maxima [A] time = 0.827508, size = 51, normalized size = 1.24 \[ \frac{1}{12} \, \sqrt{3} \arctan \left (\frac{1}{3} \, \sqrt{3}{\left (2 \, x^{4} - 1\right )}\right ) - \frac{1}{8} \, \log \left (x^{8} - x^{4} + 1\right ) + \frac{1}{4} \, \log \left (x^{4}\right ) \]
Verification of antiderivative is not currently implemented for this CAS.
[In] integrate(1/((x^8 - x^4 + 1)*x),x, algorithm="maxima")
[Out]
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Fricas [A] time = 0.256825, size = 58, normalized size = 1.41 \[ -\frac{1}{24} \, \sqrt{3}{\left (\sqrt{3} \log \left (x^{8} - x^{4} + 1\right ) - 8 \, \sqrt{3} \log \left (x\right ) - 2 \, \arctan \left (\frac{1}{3} \, \sqrt{3}{\left (2 \, x^{4} - 1\right )}\right )\right )} \]
Verification of antiderivative is not currently implemented for this CAS.
[In] integrate(1/((x^8 - x^4 + 1)*x),x, algorithm="fricas")
[Out]
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Sympy [A] time = 0.383117, size = 41, normalized size = 1. \[ \log{\left (x \right )} - \frac{\log{\left (x^{8} - x^{4} + 1 \right )}}{8} + \frac{\sqrt{3} \operatorname{atan}{\left (\frac{2 \sqrt{3} x^{4}}{3} - \frac{\sqrt{3}}{3} \right )}}{12} \]
Verification of antiderivative is not currently implemented for this CAS.
[In] integrate(1/x/(x**8-x**4+1),x)
[Out]
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GIAC/XCAS [A] time = 0.275213, size = 51, normalized size = 1.24 \[ \frac{1}{12} \, \sqrt{3} \arctan \left (\frac{1}{3} \, \sqrt{3}{\left (2 \, x^{4} - 1\right )}\right ) - \frac{1}{8} \,{\rm ln}\left (x^{8} - x^{4} + 1\right ) + \frac{1}{4} \,{\rm ln}\left (x^{4}\right ) \]
Verification of antiderivative is not currently implemented for this CAS.
[In] integrate(1/((x^8 - x^4 + 1)*x),x, algorithm="giac")
[Out]