Optimal. Leaf size=113 \[ -\frac{1}{12} \log \left (x^2-x+1\right )+\frac{1}{4} \sqrt [3]{3} \log \left (x^2-\sqrt [3]{3} x+3^{2/3}\right )+x+\frac{1}{6} \log (x+1)-\frac{1}{2} \sqrt [3]{3} \log \left (x+\sqrt [3]{3}\right )-\frac{\tan ^{-1}\left (\frac{1-2 x}{\sqrt{3}}\right )}{2 \sqrt{3}}+\frac{1}{2} 3^{5/6} \tan ^{-1}\left (\frac{\sqrt [3]{3}-2 x}{3^{5/6}}\right ) \]
[Out]
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Rubi [A] time = 0.146049, antiderivative size = 113, normalized size of antiderivative = 1., number of steps used = 14, number of rules used = 9, integrand size = 16, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.562 \[ -\frac{1}{12} \log \left (x^2-x+1\right )+\frac{1}{4} \sqrt [3]{3} \log \left (x^2-\sqrt [3]{3} x+3^{2/3}\right )+x+\frac{1}{6} \log (x+1)-\frac{1}{2} \sqrt [3]{3} \log \left (x+\sqrt [3]{3}\right )-\frac{\tan ^{-1}\left (\frac{1-2 x}{\sqrt{3}}\right )}{2 \sqrt{3}}+\frac{1}{2} 3^{5/6} \tan ^{-1}\left (\frac{\sqrt [3]{3}-2 x}{3^{5/6}}\right ) \]
Antiderivative was successfully verified.
[In] Int[x^6/(3 + 4*x^3 + x^6),x]
[Out]
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Rubi in Sympy [A] time = 28.8504, size = 105, normalized size = 0.93 \[ x + \frac{\log{\left (x + 1 \right )}}{6} - \frac{\sqrt [3]{3} \log{\left (x + \sqrt [3]{3} \right )}}{2} - \frac{\log{\left (x^{2} - x + 1 \right )}}{12} + \frac{\sqrt [3]{3} \log{\left (x^{2} - \sqrt [3]{3} x + 3^{\frac{2}{3}} \right )}}{4} + \frac{\sqrt{3} \operatorname{atan}{\left (\sqrt{3} \left (\frac{2 x}{3} - \frac{1}{3}\right ) \right )}}{6} + \frac{3^{\frac{5}{6}} \operatorname{atan}{\left (\sqrt{3} \left (- \frac{2 \cdot 3^{\frac{2}{3}} x}{9} + \frac{1}{3}\right ) \right )}}{2} \]
Verification of antiderivative is not currently implemented for this CAS.
[In] rubi_integrate(x**6/(x**6+4*x**3+3),x)
[Out]
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Mathematica [A] time = 0.0505266, size = 111, normalized size = 0.98 \[ \frac{1}{12} \left (-\log \left (x^2-x+1\right )+3 \sqrt [3]{3} \log \left (\sqrt [3]{3} x^2-3^{2/3} x+3\right )+12 x+2 \log (x+1)-6 \sqrt [3]{3} \log \left (3^{2/3} x+3\right )+6\ 3^{5/6} \tan ^{-1}\left (\frac{\sqrt [3]{3}-2 x}{3^{5/6}}\right )+2 \sqrt{3} \tan ^{-1}\left (\frac{2 x-1}{\sqrt{3}}\right )\right ) \]
Antiderivative was successfully verified.
[In] Integrate[x^6/(3 + 4*x^3 + x^6),x]
[Out]
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Maple [A] time = 0.009, size = 85, normalized size = 0.8 \[ x-{\frac{\sqrt [3]{3}\ln \left ( \sqrt [3]{3}+x \right ) }{2}}+{\frac{\sqrt [3]{3}\ln \left ({3}^{{\frac{2}{3}}}-\sqrt [3]{3}x+{x}^{2} \right ) }{4}}-{\frac{{3}^{{\frac{5}{6}}}}{2}\arctan \left ({\frac{\sqrt{3}}{3} \left ({\frac{2\,{3}^{2/3}x}{3}}-1 \right ) } \right ) }+{\frac{\ln \left ( 1+x \right ) }{6}}-{\frac{\ln \left ({x}^{2}-x+1 \right ) }{12}}+{\frac{\sqrt{3}}{6}\arctan \left ({\frac{ \left ( 2\,x-1 \right ) \sqrt{3}}{3}} \right ) } \]
Verification of antiderivative is not currently implemented for this CAS.
[In] int(x^6/(x^6+4*x^3+3),x)
[Out]
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Maxima [A] time = 0.852332, size = 115, normalized size = 1.02 \[ -\frac{1}{2} \cdot 3^{\frac{5}{6}} \arctan \left (\frac{1}{3} \cdot 3^{\frac{1}{6}}{\left (2 \, x - 3^{\frac{1}{3}}\right )}\right ) + \frac{1}{6} \, \sqrt{3} \arctan \left (\frac{1}{3} \, \sqrt{3}{\left (2 \, x - 1\right )}\right ) + \frac{1}{4} \cdot 3^{\frac{1}{3}} \log \left (x^{2} - 3^{\frac{1}{3}} x + 3^{\frac{2}{3}}\right ) - \frac{1}{2} \cdot 3^{\frac{1}{3}} \log \left (x + 3^{\frac{1}{3}}\right ) + x - \frac{1}{12} \, \log \left (x^{2} - x + 1\right ) + \frac{1}{6} \, \log \left (x + 1\right ) \]
Verification of antiderivative is not currently implemented for this CAS.
[In] integrate(x^6/(x^6 + 4*x^3 + 3),x, algorithm="maxima")
[Out]
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Fricas [A] time = 0.265852, size = 142, normalized size = 1.26 \[ -\frac{1}{36} \, \sqrt{3}{\left (3 \, \sqrt{3} \left (-3\right )^{\frac{1}{3}} \log \left (x^{2} + \left (-3\right )^{\frac{1}{3}} x + \left (-3\right )^{\frac{2}{3}}\right ) - 6 \, \sqrt{3} \left (-3\right )^{\frac{1}{3}} \log \left (x - \left (-3\right )^{\frac{1}{3}}\right ) - 12 \, \sqrt{3} x + \sqrt{3} \log \left (x^{2} - x + 1\right ) - 2 \, \sqrt{3} \log \left (x + 1\right ) + 18 \, \left (-3\right )^{\frac{1}{3}} \arctan \left (-\frac{1}{9} \, \sqrt{3} \left (-3\right )^{\frac{2}{3}}{\left (2 \, x + \left (-3\right )^{\frac{1}{3}}\right )}\right ) - 6 \, \arctan \left (\frac{1}{3} \, \sqrt{3}{\left (2 \, x - 1\right )}\right )\right )} \]
Verification of antiderivative is not currently implemented for this CAS.
[In] integrate(x^6/(x^6 + 4*x^3 + 3),x, algorithm="fricas")
[Out]
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Sympy [A] time = 2.21808, size = 126, normalized size = 1.12 \[ x + \frac{\log{\left (x + 1 \right )}}{6} + \left (- \frac{1}{12} - \frac{\sqrt{3} i}{12}\right ) \log{\left (x - \frac{121}{246} - \frac{121 \sqrt{3} i}{246} + \frac{864 \left (- \frac{1}{12} - \frac{\sqrt{3} i}{12}\right )^{4}}{41} \right )} + \left (- \frac{1}{12} + \frac{\sqrt{3} i}{12}\right ) \log{\left (x - \frac{121}{246} + \frac{864 \left (- \frac{1}{12} + \frac{\sqrt{3} i}{12}\right )^{4}}{41} + \frac{121 \sqrt{3} i}{246} \right )} + \operatorname{RootSum}{\left (8 t^{3} + 3, \left ( t \mapsto t \log{\left (\frac{864 t^{4}}{41} + \frac{242 t}{41} + x \right )} \right )\right )} \]
Verification of antiderivative is not currently implemented for this CAS.
[In] integrate(x**6/(x**6+4*x**3+3),x)
[Out]
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GIAC/XCAS [F(-2)] time = 0., size = 0, normalized size = 0. \[ \text{Exception raised: NotImplementedError} \]
Verification of antiderivative is not currently implemented for this CAS.
[In] integrate(x^6/(x^6 + 4*x^3 + 3),x, algorithm="giac")
[Out]