Optimal. Leaf size=82 \[ -\frac{1}{4} \tan ^{-1}\left (\sqrt{3}-2 x^2\right )+\frac{1}{4} \tan ^{-1}\left (2 x^2+\sqrt{3}\right )-\frac{\log \left (x^4-\sqrt{3} x^2+1\right )}{8 \sqrt{3}}+\frac{\log \left (x^4+\sqrt{3} x^2+1\right )}{8 \sqrt{3}} \]
[Out]
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Rubi [A] time = 0.108086, antiderivative size = 82, normalized size of antiderivative = 1., number of steps used = 10, number of rules used = 6, integrand size = 14, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.429 \[ -\frac{1}{4} \tan ^{-1}\left (\sqrt{3}-2 x^2\right )+\frac{1}{4} \tan ^{-1}\left (2 x^2+\sqrt{3}\right )-\frac{\log \left (x^4-\sqrt{3} x^2+1\right )}{8 \sqrt{3}}+\frac{\log \left (x^4+\sqrt{3} x^2+1\right )}{8 \sqrt{3}} \]
Antiderivative was successfully verified.
[In] Int[x/(1 - x^4 + x^8),x]
[Out]
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Rubi in Sympy [A] time = 18.366, size = 70, normalized size = 0.85 \[ - \frac{\sqrt{3} \log{\left (x^{4} - \sqrt{3} x^{2} + 1 \right )}}{24} + \frac{\sqrt{3} \log{\left (x^{4} + \sqrt{3} x^{2} + 1 \right )}}{24} + \frac{\operatorname{atan}{\left (2 x^{2} - \sqrt{3} \right )}}{4} + \frac{\operatorname{atan}{\left (2 x^{2} + \sqrt{3} \right )}}{4} \]
Verification of antiderivative is not currently implemented for this CAS.
[In] rubi_integrate(x/(x**8-x**4+1),x)
[Out]
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Mathematica [C] time = 0.104553, size = 83, normalized size = 1.01 \[ \frac{i \left (\sqrt{-1-i \sqrt{3}} \tan ^{-1}\left (\frac{1}{2} \left (1-i \sqrt{3}\right ) x^2\right )-\sqrt{-1+i \sqrt{3}} \tan ^{-1}\left (\frac{1}{2} \left (1+i \sqrt{3}\right ) x^2\right )\right )}{2 \sqrt{6}} \]
Warning: Unable to verify antiderivative.
[In] Integrate[x/(1 - x^4 + x^8),x]
[Out]
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Maple [A] time = 0.009, size = 65, normalized size = 0.8 \[{\frac{\arctan \left ( 2\,{x}^{2}-\sqrt{3} \right ) }{4}}+{\frac{\arctan \left ( 2\,{x}^{2}+\sqrt{3} \right ) }{4}}-{\frac{\ln \left ( 1+{x}^{4}-{x}^{2}\sqrt{3} \right ) \sqrt{3}}{24}}+{\frac{\ln \left ( 1+{x}^{4}+{x}^{2}\sqrt{3} \right ) \sqrt{3}}{24}} \]
Verification of antiderivative is not currently implemented for this CAS.
[In] int(x/(x^8-x^4+1),x)
[Out]
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Maxima [F] time = 0., size = 0, normalized size = 0. \[ \int \frac{x}{x^{8} - x^{4} + 1}\,{d x} \]
Verification of antiderivative is not currently implemented for this CAS.
[In] integrate(x/(x^8 - x^4 + 1),x, algorithm="maxima")
[Out]
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Fricas [A] time = 0.265392, size = 159, normalized size = 1.94 \[ -\frac{1}{24} \, \sqrt{3}{\left (4 \, \sqrt{3} \arctan \left (\frac{\sqrt{3}}{2 \, \sqrt{3} x^{2} + 2 \, \sqrt{3} \sqrt{x^{4} + \sqrt{3} x^{2} + 1} + 3}\right ) + 4 \, \sqrt{3} \arctan \left (\frac{\sqrt{3}}{2 \, \sqrt{3} x^{2} + 2 \, \sqrt{3} \sqrt{x^{4} - \sqrt{3} x^{2} + 1} - 3}\right ) - \log \left (x^{4} + \sqrt{3} x^{2} + 1\right ) + \log \left (x^{4} - \sqrt{3} x^{2} + 1\right )\right )} \]
Verification of antiderivative is not currently implemented for this CAS.
[In] integrate(x/(x^8 - x^4 + 1),x, algorithm="fricas")
[Out]
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Sympy [A] time = 0.617644, size = 70, normalized size = 0.85 \[ - \frac{\sqrt{3} \log{\left (x^{4} - \sqrt{3} x^{2} + 1 \right )}}{24} + \frac{\sqrt{3} \log{\left (x^{4} + \sqrt{3} x^{2} + 1 \right )}}{24} + \frac{\operatorname{atan}{\left (2 x^{2} - \sqrt{3} \right )}}{4} + \frac{\operatorname{atan}{\left (2 x^{2} + \sqrt{3} \right )}}{4} \]
Verification of antiderivative is not currently implemented for this CAS.
[In] integrate(x/(x**8-x**4+1),x)
[Out]
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GIAC/XCAS [F] time = 0., size = 0, normalized size = 0. \[ \int \frac{x}{x^{8} - x^{4} + 1}\,{d x} \]
Verification of antiderivative is not currently implemented for this CAS.
[In] integrate(x/(x^8 - x^4 + 1),x, algorithm="giac")
[Out]