Optimal. Leaf size=50 \[ \frac{\log \left (x^4+\sqrt{3} x^2+1\right )}{4 \sqrt{3}}-\frac{\log \left (x^4-\sqrt{3} x^2+1\right )}{4 \sqrt{3}} \]
[Out]
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Rubi [A] time = 0.0802674, antiderivative size = 50, normalized size of antiderivative = 1., number of steps used = 4, number of rules used = 3, integrand size = 21, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.143 \[ \frac{\log \left (x^4+\sqrt{3} x^2+1\right )}{4 \sqrt{3}}-\frac{\log \left (x^4-\sqrt{3} x^2+1\right )}{4 \sqrt{3}} \]
Antiderivative was successfully verified.
[In] Int[(x*(1 - x^4))/(1 - x^4 + x^8),x]
[Out]
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Rubi in Sympy [A] time = 20.2581, size = 42, normalized size = 0.84 \[ - \frac{\sqrt{3} \log{\left (x^{4} - \sqrt{3} x^{2} + 1 \right )}}{12} + \frac{\sqrt{3} \log{\left (x^{4} + \sqrt{3} x^{2} + 1 \right )}}{12} \]
Verification of antiderivative is not currently implemented for this CAS.
[In] rubi_integrate(x*(-x**4+1)/(x**8-x**4+1),x)
[Out]
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Mathematica [A] time = 0.0259039, size = 44, normalized size = 0.88 \[ \frac{\log \left (x^4+\sqrt{3} x^2+1\right )-\log \left (-x^4+\sqrt{3} x^2-1\right )}{4 \sqrt{3}} \]
Antiderivative was successfully verified.
[In] Integrate[(x*(1 - x^4))/(1 - x^4 + x^8),x]
[Out]
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Maple [A] time = 0.015, size = 39, normalized size = 0.8 \[ -{\frac{\ln \left ( 1+{x}^{4}-{x}^{2}\sqrt{3} \right ) \sqrt{3}}{12}}+{\frac{\ln \left ( 1+{x}^{4}+{x}^{2}\sqrt{3} \right ) \sqrt{3}}{12}} \]
Verification of antiderivative is not currently implemented for this CAS.
[In] int(x*(-x^4+1)/(x^8-x^4+1),x)
[Out]
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Maxima [F] time = 0., size = 0, normalized size = 0. \[ -\int \frac{{\left (x^{4} - 1\right )} x}{x^{8} - x^{4} + 1}\,{d x} \]
Verification of antiderivative is not currently implemented for this CAS.
[In] integrate(-(x^4 - 1)*x/(x^8 - x^4 + 1),x, algorithm="maxima")
[Out]
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Fricas [A] time = 0.24828, size = 59, normalized size = 1.18 \[ \frac{1}{12} \, \sqrt{3} \log \left (\frac{6 \, x^{6} + 6 \, x^{2} + \sqrt{3}{\left (x^{8} + 5 \, x^{4} + 1\right )}}{x^{8} - x^{4} + 1}\right ) \]
Verification of antiderivative is not currently implemented for this CAS.
[In] integrate(-(x^4 - 1)*x/(x^8 - x^4 + 1),x, algorithm="fricas")
[Out]
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Sympy [A] time = 0.288936, size = 42, normalized size = 0.84 \[ - \frac{\sqrt{3} \log{\left (x^{4} - \sqrt{3} x^{2} + 1 \right )}}{12} + \frac{\sqrt{3} \log{\left (x^{4} + \sqrt{3} x^{2} + 1 \right )}}{12} \]
Verification of antiderivative is not currently implemented for this CAS.
[In] integrate(x*(-x**4+1)/(x**8-x**4+1),x)
[Out]
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GIAC/XCAS [A] time = 0.286236, size = 42, normalized size = 0.84 \[ -\frac{1}{12} \, \sqrt{3}{\rm ln}\left (\frac{x^{2} - \sqrt{3} + \frac{1}{x^{2}}}{x^{2} + \sqrt{3} + \frac{1}{x^{2}}}\right ) \]
Verification of antiderivative is not currently implemented for this CAS.
[In] integrate(-(x^4 - 1)*x/(x^8 - x^4 + 1),x, algorithm="giac")
[Out]