Optimal. Leaf size=39 \[ -\frac{\tan ^{-1}\left (\frac{1-2 x^4}{\sqrt{3}}\right )}{4 \sqrt{3}}-\frac{1}{8} \log \left (x^8-x^4+1\right ) \]
[Out]
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Rubi [A] time = 0.0923199, antiderivative size = 39, normalized size of antiderivative = 1., number of steps used = 5, number of rules used = 5, integrand size = 23, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.217 \[ -\frac{\tan ^{-1}\left (\frac{1-2 x^4}{\sqrt{3}}\right )}{4 \sqrt{3}}-\frac{1}{8} \log \left (x^8-x^4+1\right ) \]
Antiderivative was successfully verified.
[In] Int[(x^3*(1 - x^4))/(1 - x^4 + x^8),x]
[Out]
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Rubi in Sympy [A] time = 14.0539, size = 34, normalized size = 0.87 \[ - \frac{\log{\left (x^{8} - x^{4} + 1 \right )}}{8} + \frac{\sqrt{3} \operatorname{atan}{\left (\sqrt{3} \left (\frac{2 x^{4}}{3} - \frac{1}{3}\right ) \right )}}{12} \]
Verification of antiderivative is not currently implemented for this CAS.
[In] rubi_integrate(x**3*(-x**4+1)/(x**8-x**4+1),x)
[Out]
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Mathematica [A] time = 0.0199397, size = 39, normalized size = 1. \[ \frac{\tan ^{-1}\left (\frac{2 x^4-1}{\sqrt{3}}\right )}{4 \sqrt{3}}-\frac{1}{8} \log \left (x^8-x^4+1\right ) \]
Antiderivative was successfully verified.
[In] Integrate[(x^3*(1 - x^4))/(1 - x^4 + x^8),x]
[Out]
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Maple [A] time = 0.005, size = 33, normalized size = 0.9 \[ -{\frac{\ln \left ({x}^{8}-{x}^{4}+1 \right ) }{8}}+{\frac{\sqrt{3}}{12}\arctan \left ({\frac{ \left ( 2\,{x}^{4}-1 \right ) \sqrt{3}}{3}} \right ) } \]
Verification of antiderivative is not currently implemented for this CAS.
[In] int(x^3*(-x^4+1)/(x^8-x^4+1),x)
[Out]
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Maxima [A] time = 0.825658, size = 43, normalized size = 1.1 \[ \frac{1}{12} \, \sqrt{3} \arctan \left (\frac{1}{3} \, \sqrt{3}{\left (2 \, x^{4} - 1\right )}\right ) - \frac{1}{8} \, \log \left (x^{8} - x^{4} + 1\right ) \]
Verification of antiderivative is not currently implemented for this CAS.
[In] integrate(-(x^4 - 1)*x^3/(x^8 - x^4 + 1),x, algorithm="maxima")
[Out]
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Fricas [A] time = 0.253155, size = 49, normalized size = 1.26 \[ -\frac{1}{24} \, \sqrt{3}{\left (\sqrt{3} \log \left (x^{8} - x^{4} + 1\right ) - 2 \, \arctan \left (\frac{1}{3} \, \sqrt{3}{\left (2 \, x^{4} - 1\right )}\right )\right )} \]
Verification of antiderivative is not currently implemented for this CAS.
[In] integrate(-(x^4 - 1)*x^3/(x^8 - x^4 + 1),x, algorithm="fricas")
[Out]
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Sympy [A] time = 0.329236, size = 37, normalized size = 0.95 \[ - \frac{\log{\left (x^{8} - x^{4} + 1 \right )}}{8} + \frac{\sqrt{3} \operatorname{atan}{\left (\frac{2 \sqrt{3} x^{4}}{3} - \frac{\sqrt{3}}{3} \right )}}{12} \]
Verification of antiderivative is not currently implemented for this CAS.
[In] integrate(x**3*(-x**4+1)/(x**8-x**4+1),x)
[Out]
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GIAC/XCAS [A] time = 0.273112, size = 43, normalized size = 1.1 \[ \frac{1}{12} \, \sqrt{3} \arctan \left (\frac{1}{3} \, \sqrt{3}{\left (2 \, x^{4} - 1\right )}\right ) - \frac{1}{8} \,{\rm ln}\left (x^{8} - x^{4} + 1\right ) \]
Verification of antiderivative is not currently implemented for this CAS.
[In] integrate(-(x^4 - 1)*x^3/(x^8 - x^4 + 1),x, algorithm="giac")
[Out]