Optimal. Leaf size=37 \[ \frac{1}{2} \sqrt{3} \tan ^{-1}\left (\frac{2 x^2+1}{\sqrt{3}}\right )+\frac{1}{4} \log \left (x^4+x^2+1\right ) \]
[Out]
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Rubi [A] time = 0.0727849, antiderivative size = 37, normalized size of antiderivative = 1., number of steps used = 5, number of rules used = 5, integrand size = 17, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.294 \[ \frac{1}{2} \sqrt{3} \tan ^{-1}\left (\frac{2 x^2+1}{\sqrt{3}}\right )+\frac{1}{4} \log \left (x^4+x^2+1\right ) \]
Antiderivative was successfully verified.
[In] Int[(x*(2 + x^2))/(1 + x^2 + x^4),x]
[Out]
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Rubi in Sympy [A] time = 11.2756, size = 34, normalized size = 0.92 \[ \frac{\log{\left (x^{4} + x^{2} + 1 \right )}}{4} + \frac{\sqrt{3} \operatorname{atan}{\left (\sqrt{3} \left (\frac{2 x^{2}}{3} + \frac{1}{3}\right ) \right )}}{2} \]
Verification of antiderivative is not currently implemented for this CAS.
[In] rubi_integrate(x*(x**2+2)/(x**4+x**2+1),x)
[Out]
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Mathematica [A] time = 0.0157249, size = 37, normalized size = 1. \[ \frac{1}{2} \sqrt{3} \tan ^{-1}\left (\frac{2 x^2+1}{\sqrt{3}}\right )+\frac{1}{4} \log \left (x^4+x^2+1\right ) \]
Antiderivative was successfully verified.
[In] Integrate[(x*(2 + x^2))/(1 + x^2 + x^4),x]
[Out]
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Maple [A] time = 0.004, size = 31, normalized size = 0.8 \[{\frac{\ln \left ({x}^{4}+{x}^{2}+1 \right ) }{4}}+{\frac{\sqrt{3}}{2}\arctan \left ({\frac{ \left ( 2\,{x}^{2}+1 \right ) \sqrt{3}}{3}} \right ) } \]
Verification of antiderivative is not currently implemented for this CAS.
[In] int(x*(x^2+2)/(x^4+x^2+1),x)
[Out]
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Maxima [A] time = 0.784581, size = 41, normalized size = 1.11 \[ \frac{1}{2} \, \sqrt{3} \arctan \left (\frac{1}{3} \, \sqrt{3}{\left (2 \, x^{2} + 1\right )}\right ) + \frac{1}{4} \, \log \left (x^{4} + x^{2} + 1\right ) \]
Verification of antiderivative is not currently implemented for this CAS.
[In] integrate((x^2 + 2)*x/(x^4 + x^2 + 1),x, algorithm="maxima")
[Out]
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Fricas [A] time = 0.254209, size = 41, normalized size = 1.11 \[ \frac{1}{2} \, \sqrt{3} \arctan \left (\frac{1}{3} \, \sqrt{3}{\left (2 \, x^{2} + 1\right )}\right ) + \frac{1}{4} \, \log \left (x^{4} + x^{2} + 1\right ) \]
Verification of antiderivative is not currently implemented for this CAS.
[In] integrate((x^2 + 2)*x/(x^4 + x^2 + 1),x, algorithm="fricas")
[Out]
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Sympy [A] time = 0.24943, size = 37, normalized size = 1. \[ \frac{\log{\left (x^{4} + x^{2} + 1 \right )}}{4} + \frac{\sqrt{3} \operatorname{atan}{\left (\frac{2 \sqrt{3} x^{2}}{3} + \frac{\sqrt{3}}{3} \right )}}{2} \]
Verification of antiderivative is not currently implemented for this CAS.
[In] integrate(x*(x**2+2)/(x**4+x**2+1),x)
[Out]
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GIAC/XCAS [A] time = 0.273147, size = 41, normalized size = 1.11 \[ \frac{1}{2} \, \sqrt{3} \arctan \left (\frac{1}{3} \, \sqrt{3}{\left (2 \, x^{2} + 1\right )}\right ) + \frac{1}{4} \,{\rm ln}\left (x^{4} + x^{2} + 1\right ) \]
Verification of antiderivative is not currently implemented for this CAS.
[In] integrate((x^2 + 2)*x/(x^4 + x^2 + 1),x, algorithm="giac")
[Out]