3.120 \(\int \frac{1+x+4 x^2}{-1+x^3} \, dx\)

Optimal. Leaf size=16 \[ \log \left (x^2+x+1\right )+2 \log (1-x) \]

[Out]

2*Log[1 - x] + Log[1 + x + x^2]

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Rubi [A]  time = 0.027795, antiderivative size = 16, normalized size of antiderivative = 1., number of steps used = 3, number of rules used = 3, integrand size = 16, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.188 \[ \log \left (x^2+x+1\right )+2 \log (1-x) \]

Antiderivative was successfully verified.

[In]  Int[(1 + x + 4*x^2)/(-1 + x^3),x]

[Out]

2*Log[1 - x] + Log[1 + x + x^2]

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Rubi in Sympy [A]  time = 3.87229, size = 14, normalized size = 0.88 \[ 2 \log{\left (- x + 1 \right )} + \log{\left (x^{2} + x + 1 \right )} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  rubi_integrate((4*x**2+x+1)/(x**3-1),x)

[Out]

2*log(-x + 1) + log(x**2 + x + 1)

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Mathematica [A]  time = 0.00724154, size = 16, normalized size = 1. \[ \log \left (x^2+x+1\right )+2 \log (1-x) \]

Antiderivative was successfully verified.

[In]  Integrate[(1 + x + 4*x^2)/(-1 + x^3),x]

[Out]

2*Log[1 - x] + Log[1 + x + x^2]

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Maple [A]  time = 0.008, size = 15, normalized size = 0.9 \[ 2\,\ln \left ( -1+x \right ) +\ln \left ({x}^{2}+x+1 \right ) \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  int((4*x^2+x+1)/(x^3-1),x)

[Out]

2*ln(-1+x)+ln(x^2+x+1)

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Maxima [A]  time = 1.62193, size = 19, normalized size = 1.19 \[ \log \left (x^{2} + x + 1\right ) + 2 \, \log \left (x - 1\right ) \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate((4*x^2 + x + 1)/(x^3 - 1),x, algorithm="maxima")

[Out]

log(x^2 + x + 1) + 2*log(x - 1)

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Fricas [A]  time = 0.192281, size = 19, normalized size = 1.19 \[ \log \left (x^{2} + x + 1\right ) + 2 \, \log \left (x - 1\right ) \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate((4*x^2 + x + 1)/(x^3 - 1),x, algorithm="fricas")

[Out]

log(x^2 + x + 1) + 2*log(x - 1)

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Sympy [A]  time = 0.085473, size = 14, normalized size = 0.88 \[ 2 \log{\left (x - 1 \right )} + \log{\left (x^{2} + x + 1 \right )} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate((4*x**2+x+1)/(x**3-1),x)

[Out]

2*log(x - 1) + log(x**2 + x + 1)

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GIAC/XCAS [A]  time = 0.21679, size = 20, normalized size = 1.25 \[{\rm ln}\left (x^{2} + x + 1\right ) + 2 \,{\rm ln}\left ({\left | x - 1 \right |}\right ) \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate((4*x^2 + x + 1)/(x^3 - 1),x, algorithm="giac")

[Out]

ln(x^2 + x + 1) + 2*ln(abs(x - 1))