3.310 \(\int \frac{1-x}{x^3 \left (1+x^3\right )} \, dx\)

Optimal. Leaf size=32 \[ -\frac{1}{2 x^2}+\frac{1}{3} \log \left (x^2-x+1\right )+\frac{1}{x}-\frac{2}{3} \log (x+1) \]

[Out]

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

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Rubi [A]  time = 0.0623125, antiderivative size = 32, normalized size of antiderivative = 1., number of steps used = 3, number of rules used = 2, integrand size = 16, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.125 \[ -\frac{1}{2 x^2}+\frac{1}{3} \log \left (x^2-x+1\right )+\frac{1}{x}-\frac{2}{3} \log (x+1) \]

Antiderivative was successfully verified.

[In]  Int[(1 - x)/(x^3*(1 + x^3)),x]

[Out]

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

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Rubi in Sympy [A]  time = 12.3339, size = 27, normalized size = 0.84 \[ - \frac{2 \log{\left (x + 1 \right )}}{3} + \frac{\log{\left (x^{2} - x + 1 \right )}}{3} + \frac{1}{x} - \frac{1}{2 x^{2}} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

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

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Mathematica [A]  time = 0.00754776, size = 32, normalized size = 1. \[ -\frac{1}{2 x^2}+\frac{1}{3} \log \left (x^2-x+1\right )+\frac{1}{x}-\frac{2}{3} \log (x+1) \]

Antiderivative was successfully verified.

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

[Out]

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

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Maple [A]  time = 0.011, size = 27, normalized size = 0.8 \[ -{\frac{1}{2\,{x}^{2}}}+{x}^{-1}-{\frac{2\,\ln \left ( 1+x \right ) }{3}}+{\frac{\ln \left ({x}^{2}-x+1 \right ) }{3}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  int((1-x)/x^3/(x^3+1),x)

[Out]

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

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Maxima [A]  time = 1.52555, size = 38, normalized size = 1.19 \[ \frac{2 \, x - 1}{2 \, x^{2}} + \frac{1}{3} \, \log \left (x^{2} - x + 1\right ) - \frac{2}{3} \, \log \left (x + 1\right ) \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

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

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Fricas [A]  time = 0.22255, size = 45, normalized size = 1.41 \[ \frac{2 \, x^{2} \log \left (x^{2} - x + 1\right ) - 4 \, x^{2} \log \left (x + 1\right ) + 6 \, x - 3}{6 \, x^{2}} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

1/6*(2*x^2*log(x^2 - x + 1) - 4*x^2*log(x + 1) + 6*x - 3)/x^2

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Sympy [A]  time = 0.131823, size = 27, normalized size = 0.84 \[ - \frac{2 \log{\left (x + 1 \right )}}{3} + \frac{\log{\left (x^{2} - x + 1 \right )}}{3} + \frac{2 x - 1}{2 x^{2}} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

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

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GIAC/XCAS [A]  time = 0.21133, size = 39, normalized size = 1.22 \[ \frac{2 \, x - 1}{2 \, x^{2}} + \frac{1}{3} \,{\rm ln}\left (x^{2} - x + 1\right ) - \frac{2}{3} \,{\rm ln}\left ({\left | x + 1 \right |}\right ) \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

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