Optimal. Leaf size=31 \[ \log \left (x^2+x+1\right )+\log (x+1)-\frac{2 \tan ^{-1}\left (\frac{2 x+1}{\sqrt{3}}\right )}{\sqrt{3}} \]
[Out]
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Rubi [A] time = 0.0885131, antiderivative size = 31, normalized size of antiderivative = 1., number of steps used = 6, number of rules used = 5, integrand size = 26, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.192 \[ \log \left (x^2+x+1\right )+\log (x+1)-\frac{2 \tan ^{-1}\left (\frac{2 x+1}{\sqrt{3}}\right )}{\sqrt{3}} \]
Antiderivative was successfully verified.
[In] Int[(1 + 3*x + 3*x^2)/(1 + 2*x + 2*x^2 + x^3),x]
[Out]
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Rubi in Sympy [F(-1)] time = 0., size = 0, normalized size = 0. \[ \text{Timed out} \]
Verification of antiderivative is not currently implemented for this CAS.
[In] rubi_integrate((3*x**2+3*x+1)/(x**3+2*x**2+2*x+1),x)
[Out]
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Mathematica [A] time = 0.0216145, size = 31, normalized size = 1. \[ \log \left (x^2+x+1\right )+\log (x+1)-\frac{2 \tan ^{-1}\left (\frac{2 x+1}{\sqrt{3}}\right )}{\sqrt{3}} \]
Antiderivative was successfully verified.
[In] Integrate[(1 + 3*x + 3*x^2)/(1 + 2*x + 2*x^2 + x^3),x]
[Out]
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Maple [A] time = 0.008, size = 29, normalized size = 0.9 \[ \ln \left ( 1+x \right ) +\ln \left ({x}^{2}+x+1 \right ) -{\frac{2\,\sqrt{3}}{3}\arctan \left ({\frac{ \left ( 1+2\,x \right ) \sqrt{3}}{3}} \right ) } \]
Verification of antiderivative is not currently implemented for this CAS.
[In] int((3*x^2+3*x+1)/(x^3+2*x^2+2*x+1),x)
[Out]
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Maxima [A] time = 0.882593, size = 38, normalized size = 1.23 \[ -\frac{2}{3} \, \sqrt{3} \arctan \left (\frac{1}{3} \, \sqrt{3}{\left (2 \, x + 1\right )}\right ) + \log \left (x^{2} + x + 1\right ) + \log \left (x + 1\right ) \]
Verification of antiderivative is not currently implemented for this CAS.
[In] integrate((3*x^2 + 3*x + 1)/(x^3 + 2*x^2 + 2*x + 1),x, algorithm="maxima")
[Out]
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Fricas [A] time = 0.258849, size = 51, normalized size = 1.65 \[ \frac{1}{3} \, \sqrt{3}{\left (\sqrt{3} \log \left (x^{2} + x + 1\right ) + \sqrt{3} \log \left (x + 1\right ) - 2 \, \arctan \left (\frac{1}{3} \, \sqrt{3}{\left (2 \, x + 1\right )}\right )\right )} \]
Verification of antiderivative is not currently implemented for this CAS.
[In] integrate((3*x^2 + 3*x + 1)/(x^3 + 2*x^2 + 2*x + 1),x, algorithm="fricas")
[Out]
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Sympy [A] time = 0.270119, size = 3, normalized size = 0.1 \[ \log{\left (x + 1 \right )} \]
Verification of antiderivative is not currently implemented for this CAS.
[In] integrate((3*x**2+3*x+1)/(x**3+2*x**2+2*x+1),x)
[Out]
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GIAC/XCAS [A] time = 0.261863, size = 39, normalized size = 1.26 \[ -\frac{2}{3} \, \sqrt{3} \arctan \left (\frac{1}{3} \, \sqrt{3}{\left (2 \, x + 1\right )}\right ) +{\rm ln}\left (x^{2} + x + 1\right ) +{\rm ln}\left ({\left | x + 1 \right |}\right ) \]
Verification of antiderivative is not currently implemented for this CAS.
[In] integrate((3*x^2 + 3*x + 1)/(x^3 + 2*x^2 + 2*x + 1),x, algorithm="giac")
[Out]