3.132 \(\int \frac{1+x}{-1+x^3} \, dx\)

Optimal. Leaf size=22 \[ \frac{2}{3} \log (1-x)-\frac{1}{3} \log \left (x^2+x+1\right ) \]

[Out]

(2*Log[1 - x])/3 - Log[1 + x + x^2]/3

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Rubi [A]  time = 0.0184451, antiderivative size = 22, normalized size of antiderivative = 1., number of steps used = 3, number of rules used = 3, integrand size = 11, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.273 \[ \frac{2}{3} \log (1-x)-\frac{1}{3} \log \left (x^2+x+1\right ) \]

Antiderivative was successfully verified.

[In]  Int[(1 + x)/(-1 + x^3),x]

[Out]

(2*Log[1 - x])/3 - Log[1 + x + x^2]/3

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Rubi in Sympy [A]  time = 2.67986, size = 17, normalized size = 0.77 \[ \frac{2 \log{\left (- x + 1 \right )}}{3} - \frac{\log{\left (x^{2} + x + 1 \right )}}{3} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

2*log(-x + 1)/3 - log(x**2 + x + 1)/3

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Mathematica [A]  time = 0.00457224, size = 22, normalized size = 1. \[ \frac{2}{3} \log (1-x)-\frac{1}{3} \log \left (x^2+x+1\right ) \]

Antiderivative was successfully verified.

[In]  Integrate[(1 + x)/(-1 + x^3),x]

[Out]

(2*Log[1 - x])/3 - Log[1 + x + x^2]/3

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Maple [A]  time = 0.009, size = 17, normalized size = 0.8 \[{\frac{2\,\ln \left ( -1+x \right ) }{3}}-{\frac{\ln \left ({x}^{2}+x+1 \right ) }{3}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  int((1+x)/(x^3-1),x)

[Out]

2/3*ln(-1+x)-1/3*ln(x^2+x+1)

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Maxima [A]  time = 1.51585, size = 22, normalized size = 1. \[ -\frac{1}{3} \, \log \left (x^{2} + x + 1\right ) + \frac{2}{3} \, \log \left (x - 1\right ) \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

-1/3*log(x^2 + x + 1) + 2/3*log(x - 1)

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Fricas [A]  time = 0.188249, size = 22, normalized size = 1. \[ -\frac{1}{3} \, \log \left (x^{2} + x + 1\right ) + \frac{2}{3} \, \log \left (x - 1\right ) \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

-1/3*log(x^2 + x + 1) + 2/3*log(x - 1)

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Sympy [A]  time = 0.078172, size = 17, normalized size = 0.77 \[ \frac{2 \log{\left (x - 1 \right )}}{3} - \frac{\log{\left (x^{2} + x + 1 \right )}}{3} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate((1+x)/(x**3-1),x)

[Out]

2*log(x - 1)/3 - log(x**2 + x + 1)/3

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GIAC/XCAS [A]  time = 0.213335, size = 23, normalized size = 1.05 \[ -\frac{1}{3} \,{\rm ln}\left (x^{2} + x + 1\right ) + \frac{2}{3} \,{\rm ln}\left ({\left | x - 1 \right |}\right ) \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

-1/3*ln(x^2 + x + 1) + 2/3*ln(abs(x - 1))