Optimal. Leaf size=80 \[ -\frac{\log \left (x^2-\sqrt{3} x+1\right )}{4 \sqrt{3}}+\frac{\log \left (x^2+\sqrt{3} x+1\right )}{4 \sqrt{3}}-\frac{1}{6} \tan ^{-1}\left (\sqrt{3}-2 x\right )+\frac{1}{3} \tan ^{-1}(x)+\frac{1}{6} \tan ^{-1}\left (2 x+\sqrt{3}\right ) \]
[Out]
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Rubi [A] time = 0.322, antiderivative size = 80, normalized size of antiderivative = 1., number of steps used = 10, number of rules used = 6, integrand size = 7, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.857 \[ -\frac{\log \left (x^2-\sqrt{3} x+1\right )}{4 \sqrt{3}}+\frac{\log \left (x^2+\sqrt{3} x+1\right )}{4 \sqrt{3}}-\frac{1}{6} \tan ^{-1}\left (\sqrt{3}-2 x\right )+\frac{1}{3} \tan ^{-1}(x)+\frac{1}{6} \tan ^{-1}\left (2 x+\sqrt{3}\right ) \]
Antiderivative was successfully verified.
[In] Int[(1 + x^6)^(-1),x]
[Out]
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Rubi in Sympy [A] time = 58.4862, size = 68, normalized size = 0.85 \[ - \frac{\sqrt{3} \log{\left (x^{2} - \sqrt{3} x + 1 \right )}}{12} + \frac{\sqrt{3} \log{\left (x^{2} + \sqrt{3} x + 1 \right )}}{12} + \frac{\operatorname{atan}{\left (x \right )}}{3} + \frac{\operatorname{atan}{\left (2 x - \sqrt{3} \right )}}{6} + \frac{\operatorname{atan}{\left (2 x + \sqrt{3} \right )}}{6} \]
Verification of antiderivative is not currently implemented for this CAS.
[In] rubi_integrate(1/(x**6+1),x)
[Out]
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Mathematica [A] time = 0.0194892, size = 73, normalized size = 0.91 \[ \frac{1}{12} \left (-\sqrt{3} \log \left (x^2-\sqrt{3} x+1\right )+\sqrt{3} \log \left (x^2+\sqrt{3} x+1\right )-2 \tan ^{-1}\left (\sqrt{3}-2 x\right )+4 \tan ^{-1}(x)+2 \tan ^{-1}\left (2 x+\sqrt{3}\right )\right ) \]
Antiderivative was successfully verified.
[In] Integrate[(1 + x^6)^(-1),x]
[Out]
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Maple [A] time = 0.017, size = 61, normalized size = 0.8 \[{\frac{\arctan \left ( x \right ) }{3}}+{\frac{\arctan \left ( 2\,x-\sqrt{3} \right ) }{6}}+{\frac{\arctan \left ( 2\,x+\sqrt{3} \right ) }{6}}-{\frac{\ln \left ( 1+{x}^{2}-x\sqrt{3} \right ) \sqrt{3}}{12}}+{\frac{\ln \left ( 1+{x}^{2}+x\sqrt{3} \right ) \sqrt{3}}{12}} \]
Verification of antiderivative is not currently implemented for this CAS.
[In] int(1/(x^6+1),x)
[Out]
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Maxima [A] time = 1.58691, size = 81, normalized size = 1.01 \[ \frac{1}{12} \, \sqrt{3} \log \left (x^{2} + \sqrt{3} x + 1\right ) - \frac{1}{12} \, \sqrt{3} \log \left (x^{2} - \sqrt{3} x + 1\right ) + \frac{1}{6} \, \arctan \left (2 \, x + \sqrt{3}\right ) + \frac{1}{6} \, \arctan \left (2 \, x - \sqrt{3}\right ) + \frac{1}{3} \, \arctan \left (x\right ) \]
Verification of antiderivative is not currently implemented for this CAS.
[In] integrate(1/(x^6 + 1),x, algorithm="maxima")
[Out]
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Fricas [A] time = 0.233131, size = 126, normalized size = 1.58 \[ \frac{1}{12} \, \sqrt{3} \log \left (x^{2} + \sqrt{3} x + 1\right ) - \frac{1}{12} \, \sqrt{3} \log \left (x^{2} - \sqrt{3} x + 1\right ) + \frac{1}{3} \, \arctan \left (x\right ) - \frac{1}{3} \, \arctan \left (\frac{1}{2 \, x + \sqrt{3} + 2 \, \sqrt{x^{2} + \sqrt{3} x + 1}}\right ) - \frac{1}{3} \, \arctan \left (\frac{1}{2 \, x - \sqrt{3} + 2 \, \sqrt{x^{2} - \sqrt{3} x + 1}}\right ) \]
Verification of antiderivative is not currently implemented for this CAS.
[In] integrate(1/(x^6 + 1),x, algorithm="fricas")
[Out]
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Sympy [A] time = 0.665653, size = 68, normalized size = 0.85 \[ - \frac{\sqrt{3} \log{\left (x^{2} - \sqrt{3} x + 1 \right )}}{12} + \frac{\sqrt{3} \log{\left (x^{2} + \sqrt{3} x + 1 \right )}}{12} + \frac{\operatorname{atan}{\left (x \right )}}{3} + \frac{\operatorname{atan}{\left (2 x - \sqrt{3} \right )}}{6} + \frac{\operatorname{atan}{\left (2 x + \sqrt{3} \right )}}{6} \]
Verification of antiderivative is not currently implemented for this CAS.
[In] integrate(1/(x**6+1),x)
[Out]
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GIAC/XCAS [A] time = 0.226511, size = 81, normalized size = 1.01 \[ \frac{1}{12} \, \sqrt{3}{\rm ln}\left (x^{2} + \sqrt{3} x + 1\right ) - \frac{1}{12} \, \sqrt{3}{\rm ln}\left (x^{2} - \sqrt{3} x + 1\right ) + \frac{1}{6} \, \arctan \left (2 \, x + \sqrt{3}\right ) + \frac{1}{6} \, \arctan \left (2 \, x - \sqrt{3}\right ) + \frac{1}{3} \, \arctan \left (x\right ) \]
Verification of antiderivative is not currently implemented for this CAS.
[In] integrate(1/(x^6 + 1),x, algorithm="giac")
[Out]