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

Optimal. Leaf size=18 \[ -\log \left (x^2+x+1\right )-2 \log (1-x) \]

[Out]

-2*Log[1 - x] - Log[1 + x + x^2]

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Rubi [A]  time = 0.0329192, antiderivative size = 18, normalized size of antiderivative = 1., number of steps used = 3, number of rules used = 3, integrand size = 18, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.167 \[ -\log \left (x^2+x+1\right )-2 \log (1-x) \]

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 = 10.0078, size = 15, normalized size = 0.83 \[ - 2 \log{\left (- x + 1 \right )} - \log{\left (x^{2} + x + 1 \right )} \]

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 = 0.00953197, size = 18, 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 = 0.007, size = 17, normalized size = 0.9 \[ -\ln \left ({x}^{2}+x+1 \right ) -2\,\ln \left ( -1+x \right ) \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  int((4*x^2+x+1)/(-x^3+1),x)

[Out]

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

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Maxima [A]  time = 1.57186, size = 22, normalized size = 1.22 \[ -\log \left (x^{2} + x + 1\right ) - 2 \, \log \left (x - 1\right ) \]

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 = 0.22663, size = 22, normalized size = 1.22 \[ -\log \left (x^{2} + x + 1\right ) - 2 \, \log \left (x - 1\right ) \]

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 = 0.105992, size = 15, normalized size = 0.83 \[ - 2 \log{\left (x - 1 \right )} - \log{\left (x^{2} + x + 1 \right )} \]

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 = 0.218783, size = 23, normalized size = 1.28 \[ -{\rm ln}\left (x^{2} + x + 1\right ) - 2 \,{\rm ln}\left ({\left | x - 1 \right |}\right ) \]

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))