Optimal. Leaf size=81 \[ \frac{\log \left (x^2-\sqrt{3} x+1\right )}{4 \sqrt{3}}-\frac{\log \left (x^2+\sqrt{3} x+1\right )}{4 \sqrt{3}}+x+\frac{1}{6} \tan ^{-1}\left (\sqrt{3}-2 x\right )-\frac{1}{3} \tan ^{-1}(x)-\frac{1}{6} \tan ^{-1}\left (2 x+\sqrt{3}\right ) \]
[Out]
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Rubi [A] time = 0.334236, antiderivative size = 81, normalized size of antiderivative = 1., number of steps used = 11, number of rules used = 7, integrand size = 11, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.636 \[ \frac{\log \left (x^2-\sqrt{3} x+1\right )}{4 \sqrt{3}}-\frac{\log \left (x^2+\sqrt{3} x+1\right )}{4 \sqrt{3}}+x+\frac{1}{6} \tan ^{-1}\left (\sqrt{3}-2 x\right )-\frac{1}{3} \tan ^{-1}(x)-\frac{1}{6} \tan ^{-1}\left (2 x+\sqrt{3}\right ) \]
Antiderivative was successfully verified.
[In] Int[x^6/(1 + x^6),x]
[Out]
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Rubi in Sympy [A] time = 57.6371, size = 70, normalized size = 0.86 \[ x + \frac{\sqrt{3} \log{\left (x^{2} - \sqrt{3} x + 1 \right )}}{12} - \frac{\sqrt{3} \log{\left (x^{2} + \sqrt{3} x + 1 \right )}}{12} - \frac{\operatorname{atan}{\left (x \right )}}{3} - \frac{\operatorname{atan}{\left (2 x - \sqrt{3} \right )}}{6} - \frac{\operatorname{atan}{\left (2 x + \sqrt{3} \right )}}{6} \]
Verification of antiderivative is not currently implemented for this CAS.
[In] rubi_integrate(x**6/(x**6+1),x)
[Out]
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Mathematica [A] time = 0.0230964, size = 76, normalized size = 0.94 \[ \frac{1}{12} \left (\sqrt{3} \log \left (x^2-\sqrt{3} x+1\right )-\sqrt{3} \log \left (x^2+\sqrt{3} x+1\right )+12 x+2 \tan ^{-1}\left (\sqrt{3}-2 x\right )-4 \tan ^{-1}(x)-2 \tan ^{-1}\left (2 x+\sqrt{3}\right )\right ) \]
Antiderivative was successfully verified.
[In] Integrate[x^6/(1 + x^6),x]
[Out]
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Maple [A] time = 0.032, size = 62, normalized size = 0.8 \[ x-{\frac{\arctan \left ( x \right ) }{3}}-{\frac{\arctan \left ( 2\,x-\sqrt{3} \right ) }{6}}-{\frac{\arctan \left ( 2\,x+\sqrt{3} \right ) }{6}}+{\frac{\ln \left ( 1+{x}^{2}-x\sqrt{3} \right ) \sqrt{3}}{12}}-{\frac{\ln \left ( 1+{x}^{2}+x\sqrt{3} \right ) \sqrt{3}}{12}} \]
Verification of antiderivative is not currently implemented for this CAS.
[In] int(x^6/(x^6+1),x)
[Out]
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Maxima [A] time = 1.59198, size = 82, normalized size = 1.01 \[ -\frac{1}{12} \, \sqrt{3} \log \left (x^{2} + \sqrt{3} x + 1\right ) + \frac{1}{12} \, \sqrt{3} \log \left (x^{2} - \sqrt{3} x + 1\right ) + x - \frac{1}{6} \, \arctan \left (2 \, x + \sqrt{3}\right ) - \frac{1}{6} \, \arctan \left (2 \, x - \sqrt{3}\right ) - \frac{1}{3} \, \arctan \left (x\right ) \]
Verification of antiderivative is not currently implemented for this CAS.
[In] integrate(x^6/(x^6 + 1),x, algorithm="maxima")
[Out]
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Fricas [A] time = 0.236685, size = 127, normalized size = 1.57 \[ -\frac{1}{12} \, \sqrt{3} \log \left (x^{2} + \sqrt{3} x + 1\right ) + \frac{1}{12} \, \sqrt{3} \log \left (x^{2} - \sqrt{3} x + 1\right ) + x - \frac{1}{3} \, \arctan \left (x\right ) + \frac{1}{3} \, \arctan \left (\frac{1}{2 \, x + \sqrt{3} + 2 \, \sqrt{x^{2} + \sqrt{3} x + 1}}\right ) + \frac{1}{3} \, \arctan \left (\frac{1}{2 \, x - \sqrt{3} + 2 \, \sqrt{x^{2} - \sqrt{3} x + 1}}\right ) \]
Verification of antiderivative is not currently implemented for this CAS.
[In] integrate(x^6/(x^6 + 1),x, algorithm="fricas")
[Out]
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Sympy [A] time = 0.653377, size = 70, normalized size = 0.86 \[ x + \frac{\sqrt{3} \log{\left (x^{2} - \sqrt{3} x + 1 \right )}}{12} - \frac{\sqrt{3} \log{\left (x^{2} + \sqrt{3} x + 1 \right )}}{12} - \frac{\operatorname{atan}{\left (x \right )}}{3} - \frac{\operatorname{atan}{\left (2 x - \sqrt{3} \right )}}{6} - \frac{\operatorname{atan}{\left (2 x + \sqrt{3} \right )}}{6} \]
Verification of antiderivative is not currently implemented for this CAS.
[In] integrate(x**6/(x**6+1),x)
[Out]
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GIAC/XCAS [F] time = 0., size = 0, normalized size = 0. \[ \int \frac{x^{6}}{x^{6} + 1}\,{d x} \]
Verification of antiderivative is not currently implemented for this CAS.
[In] integrate(x^6/(x^6 + 1),x, algorithm="giac")
[Out]