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

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

[Out]

(2*Log[1 + x])/3 - Log[1 - x + x^2]/3

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Rubi [A]  time = 0.0228269, 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 \[ \frac{2}{3} \log (x+1)-\frac{1}{3} \log \left (x^2-x+1\right ) \]

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 = 6.92861, 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 = 0.00694171, size = 22, normalized size = 1. \[ \frac{2}{3} \log (x+1)-\frac{1}{3} \log \left (x^2-x+1\right ) \]

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 = 0.007, size = 19, normalized size = 0.9 \[{\frac{2\,\ln \left ( 1+x \right ) }{3}}-{\frac{\ln \left ({x}^{2}-x+1 \right ) }{3}} \]

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 = 1.51038, size = 24, normalized size = 1.09 \[ -\frac{1}{3} \, \log \left (x^{2} - x + 1\right ) + \frac{2}{3} \, \log \left (x + 1\right ) \]

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 = 0.210305, size = 24, normalized size = 1.09 \[ -\frac{1}{3} \, \log \left (x^{2} - x + 1\right ) + \frac{2}{3} \, \log \left (x + 1\right ) \]

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 = 0.088051, 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 = 0.210853, size = 26, normalized size = 1.18 \[ -\frac{1}{3} \,{\rm ln}\left (x^{2} - x + 1\right ) + \frac{2}{3} \,{\rm ln}\left ({\left | x + 1 \right |}\right ) \]

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))