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

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

[Out]

(-2*Log[1 - x])/3 + Log[1 + x + x^2]/3

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Rubi [A]  time = 0.0226266, antiderivative size = 22, normalized size of antiderivative = 1., number of steps used = 3, number of rules used = 3, integrand size = 13, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.231 \[ \frac{1}{3} \log \left (x^2+x+1\right )-\frac{2}{3} \log (1-x) \]

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 = 6.72534, 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 = 0.00754296, size = 22, normalized size = 1. \[ \frac{1}{3} \log \left (x^2+x+1\right )-\frac{2}{3} \log (1-x) \]

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 = 0.007, size = 17, normalized size = 0.8 \[{\frac{\ln \left ({x}^{2}+x+1 \right ) }{3}}-{\frac{2\,\ln \left ( -1+x \right ) }{3}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  int((1+x)/(-x^3+1),x)

[Out]

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

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Maxima [A]  time = 1.51777, 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 = 0.207835, 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 = 0.095287, 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 = 0.209055, 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))