Optimal. Leaf size=88 \[ -\frac{\log \left (x^2-\sqrt{3} x+1\right )}{4 \sqrt{3}}+\frac{\log \left (x^2+\sqrt{3} x+1\right )}{4 \sqrt{3}}-\frac{\tan ^{-1}\left (\frac{1-2 x}{\sqrt{3}}\right )}{2 \sqrt{3}}+\frac{\tan ^{-1}\left (\frac{2 x+1}{\sqrt{3}}\right )}{2 \sqrt{3}} \]
[Out]
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Rubi [A] time = 0.0986079, antiderivative size = 88, normalized size of antiderivative = 1., number of steps used = 9, number of rules used = 6, integrand size = 10, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.6 \[ -\frac{\log \left (x^2-\sqrt{3} x+1\right )}{4 \sqrt{3}}+\frac{\log \left (x^2+\sqrt{3} x+1\right )}{4 \sqrt{3}}-\frac{\tan ^{-1}\left (\frac{1-2 x}{\sqrt{3}}\right )}{2 \sqrt{3}}+\frac{\tan ^{-1}\left (\frac{2 x+1}{\sqrt{3}}\right )}{2 \sqrt{3}} \]
Antiderivative was successfully verified.
[In] Int[(1 + x^4 + x^8)^(-1),x]
[Out]
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Rubi in Sympy [A] time = 21.1603, size = 83, normalized size = 0.94 \[ - \frac{\sqrt{3} \log{\left (x^{2} - \sqrt{3} x + 1 \right )}}{12} + \frac{\sqrt{3} \log{\left (x^{2} + \sqrt{3} x + 1 \right )}}{12} + \frac{\sqrt{3} \operatorname{atan}{\left (\sqrt{3} \left (\frac{2 x}{3} - \frac{1}{3}\right ) \right )}}{6} + \frac{\sqrt{3} \operatorname{atan}{\left (\sqrt{3} \left (\frac{2 x}{3} + \frac{1}{3}\right ) \right )}}{6} \]
Verification of antiderivative is not currently implemented for this CAS.
[In] rubi_integrate(1/(x**8+x**4+1),x)
[Out]
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Mathematica [A] time = 0.0350676, size = 68, normalized size = 0.77 \[ \frac{-\log \left (-x^2+\sqrt{3} x-1\right )+\log \left (x^2+\sqrt{3} x+1\right )+2 \tan ^{-1}\left (\frac{2 x-1}{\sqrt{3}}\right )+2 \tan ^{-1}\left (\frac{2 x+1}{\sqrt{3}}\right )}{4 \sqrt{3}} \]
Antiderivative was successfully verified.
[In] Integrate[(1 + x^4 + x^8)^(-1),x]
[Out]
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Maple [A] time = 0.015, size = 67, normalized size = 0.8 \[{\frac{\sqrt{3}}{6}\arctan \left ({\frac{ \left ( 1+2\,x \right ) \sqrt{3}}{3}} \right ) }-{\frac{\ln \left ( 1+{x}^{2}-x\sqrt{3} \right ) \sqrt{3}}{12}}+{\frac{\ln \left ( 1+{x}^{2}+x\sqrt{3} \right ) \sqrt{3}}{12}}+{\frac{\sqrt{3}}{6}\arctan \left ({\frac{ \left ( 2\,x-1 \right ) \sqrt{3}}{3}} \right ) } \]
Verification of antiderivative is not currently implemented for this CAS.
[In] int(1/(x^8+x^4+1),x)
[Out]
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Maxima [F] time = 0., size = 0, normalized size = 0. \[ \frac{1}{6} \, \sqrt{3} \arctan \left (\frac{1}{3} \, \sqrt{3}{\left (2 \, x + 1\right )}\right ) + \frac{1}{6} \, \sqrt{3} \arctan \left (\frac{1}{3} \, \sqrt{3}{\left (2 \, x - 1\right )}\right ) - \frac{1}{2} \, \int \frac{x^{2} - 1}{x^{4} - x^{2} + 1}\,{d x} \]
Verification of antiderivative is not currently implemented for this CAS.
[In] integrate(1/(x^8 + x^4 + 1),x, algorithm="maxima")
[Out]
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Fricas [A] time = 0.269096, size = 90, normalized size = 1.02 \[ \frac{1}{12} \, \sqrt{3}{\left (2 \, \arctan \left (\frac{1}{3} \, \sqrt{3}{\left (x^{3} + 2 \, x\right )}\right ) + 2 \, \arctan \left (\frac{1}{3} \, \sqrt{3} x\right ) + \log \left (\frac{6 \, x^{3} + \sqrt{3}{\left (x^{4} + 5 \, x^{2} + 1\right )} + 6 \, x}{x^{4} - x^{2} + 1}\right )\right )} \]
Verification of antiderivative is not currently implemented for this CAS.
[In] integrate(1/(x^8 + x^4 + 1),x, algorithm="fricas")
[Out]
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Sympy [A] time = 0.515138, size = 82, normalized size = 0.93 \[ \frac{\sqrt{3} \left (2 \operatorname{atan}{\left (\frac{\sqrt{3} x}{3} \right )} + 2 \operatorname{atan}{\left (\frac{\sqrt{3} x^{3}}{3} + \frac{2 \sqrt{3} x}{3} \right )}\right )}{12} - \frac{\sqrt{3} \log{\left (x^{2} - \sqrt{3} x + 1 \right )}}{12} + \frac{\sqrt{3} \log{\left (x^{2} + \sqrt{3} x + 1 \right )}}{12} \]
Verification of antiderivative is not currently implemented for this CAS.
[In] integrate(1/(x**8+x**4+1),x)
[Out]
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GIAC/XCAS [A] time = 0.300524, size = 97, normalized size = 1.1 \[ \frac{1}{6} \, \sqrt{3} \arctan \left (\frac{1}{3} \, \sqrt{3}{\left (2 \, x + 1\right )}\right ) + \frac{1}{6} \, \sqrt{3} \arctan \left (\frac{1}{3} \, \sqrt{3}{\left (2 \, x - 1\right )}\right ) - \frac{1}{12} \, \sqrt{3}{\rm ln}\left (\frac{{\left | 2 \, x - 2 \, \sqrt{3} + \frac{2}{x} \right |}}{{\left | 2 \, x + 2 \, \sqrt{3} + \frac{2}{x} \right |}}\right ) \]
Verification of antiderivative is not currently implemented for this CAS.
[In] integrate(1/(x^8 + x^4 + 1),x, algorithm="giac")
[Out]