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

Optimal. Leaf size=30 \[ -\frac{x^2}{2}+\frac{1}{3} \log \left (x^2-x+1\right )+x-\frac{2}{3} \log (x+1) \]

[Out]

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

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Rubi [A]  time = 0.0687791, antiderivative size = 30, normalized size of antiderivative = 1., number of steps used = 5, number of rules used = 4, integrand size = 16, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.25 \[ -\frac{x^2}{2}+\frac{1}{3} \log \left (x^2-x+1\right )+x-\frac{2}{3} \log (x+1) \]

Antiderivative was successfully verified.

[In]  Int[((1 - x)*x^3)/(1 + x^3),x]

[Out]

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

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Rubi in Sympy [F]  time = 0., size = 0, normalized size = 0. \[ x - \frac{2 \log{\left (x + 1 \right )}}{3} + \frac{\log{\left (x^{2} - x + 1 \right )}}{3} - \int x\, dx \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

x - 2*log(x + 1)/3 + log(x**2 - x + 1)/3 - Integral(x, x)

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Mathematica [A]  time = 0.00712826, size = 30, normalized size = 1. \[ -\frac{x^2}{2}+\frac{1}{3} \log \left (x^2-x+1\right )+x-\frac{2}{3} \log (x+1) \]

Antiderivative was successfully verified.

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

[Out]

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

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Maple [A]  time = 0.007, size = 25, normalized size = 0.8 \[ x-{\frac{{x}^{2}}{2}}-{\frac{2\,\ln \left ( 1+x \right ) }{3}}+{\frac{\ln \left ({x}^{2}-x+1 \right ) }{3}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  int((1-x)*x^3/(x^3+1),x)

[Out]

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

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Maxima [A]  time = 1.52492, size = 32, normalized size = 1.07 \[ -\frac{1}{2} \, x^{2} + x + \frac{1}{3} \, \log \left (x^{2} - x + 1\right ) - \frac{2}{3} \, \log \left (x + 1\right ) \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

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

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Fricas [A]  time = 0.208356, size = 32, normalized size = 1.07 \[ -\frac{1}{2} \, x^{2} + x + \frac{1}{3} \, \log \left (x^{2} - x + 1\right ) - \frac{2}{3} \, \log \left (x + 1\right ) \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

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

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Sympy [A]  time = 0.094271, size = 24, normalized size = 0.8 \[ - \frac{x^{2}}{2} + x - \frac{2 \log{\left (x + 1 \right )}}{3} + \frac{\log{\left (x^{2} - x + 1 \right )}}{3} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

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

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GIAC/XCAS [A]  time = 0.210913, size = 34, normalized size = 1.13 \[ -\frac{1}{2} \, x^{2} + x + \frac{1}{3} \,{\rm ln}\left (x^{2} - x + 1\right ) - \frac{2}{3} \,{\rm ln}\left ({\left | x + 1 \right |}\right ) \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

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