3.309 \(\int \frac{1-x}{x^2 \left (1+x^3\right )} \, dx\)

Optimal. Leaf size=49 \[ \frac{1}{6} \log \left (x^2-x+1\right )-\frac{1}{x}-\log (x)+\frac{2}{3} \log (x+1)+\frac{\tan ^{-1}\left (\frac{1-2 x}{\sqrt{3}}\right )}{\sqrt{3}} \]

[Out]

-x^(-1) + ArcTan[(1 - 2*x)/Sqrt[3]]/Sqrt[3] - Log[x] + (2*Log[1 + x])/3 + Log[1
- x + x^2]/6

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Rubi [A]  time = 0.102185, antiderivative size = 49, normalized size of antiderivative = 1., number of steps used = 6, number of rules used = 5, integrand size = 16, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.312 \[ \frac{1}{6} \log \left (x^2-x+1\right )-\frac{1}{x}-\log (x)+\frac{2}{3} \log (x+1)+\frac{\tan ^{-1}\left (\frac{1-2 x}{\sqrt{3}}\right )}{\sqrt{3}} \]

Antiderivative was successfully verified.

[In]  Int[(1 - x)/(x^2*(1 + x^3)),x]

[Out]

-x^(-1) + ArcTan[(1 - 2*x)/Sqrt[3]]/Sqrt[3] - Log[x] + (2*Log[1 + x])/3 + Log[1
- x + x^2]/6

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Rubi in Sympy [A]  time = 14.7888, size = 46, normalized size = 0.94 \[ - \log{\left (x \right )} + \frac{2 \log{\left (x + 1 \right )}}{3} + \frac{\log{\left (x^{2} - x + 1 \right )}}{6} - \frac{\sqrt{3} \operatorname{atan}{\left (\sqrt{3} \left (\frac{2 x}{3} - \frac{1}{3}\right ) \right )}}{3} - \frac{1}{x} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  rubi_integrate((1-x)/x**2/(x**3+1),x)

[Out]

-log(x) + 2*log(x + 1)/3 + log(x**2 - x + 1)/6 - sqrt(3)*atan(sqrt(3)*(2*x/3 - 1
/3))/3 - 1/x

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Mathematica [A]  time = 0.0284103, size = 60, normalized size = 1.22 \[ \frac{1}{3} \log \left (x^3+1\right )-\frac{1}{6} \log \left (x^2-x+1\right )-\frac{1}{x}-\log (x)+\frac{1}{3} \log (x+1)-\frac{\tan ^{-1}\left (\frac{2 x-1}{\sqrt{3}}\right )}{\sqrt{3}} \]

Antiderivative was successfully verified.

[In]  Integrate[(1 - x)/(x^2*(1 + x^3)),x]

[Out]

-x^(-1) - ArcTan[(-1 + 2*x)/Sqrt[3]]/Sqrt[3] - Log[x] + Log[1 + x]/3 - Log[1 - x
+ x^2]/6 + Log[1 + x^3]/3

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Maple [A]  time = 0.011, size = 44, normalized size = 0.9 \[ -{x}^{-1}-\ln \left ( x \right ) +{\frac{\ln \left ({x}^{2}-x+1 \right ) }{6}}-{\frac{\sqrt{3}}{3}\arctan \left ({\frac{ \left ( 2\,x-1 \right ) \sqrt{3}}{3}} \right ) }+{\frac{2\,\ln \left ( 1+x \right ) }{3}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  int((1-x)/x^2/(x^3+1),x)

[Out]

-1/x-ln(x)+1/6*ln(x^2-x+1)-1/3*3^(1/2)*arctan(1/3*(2*x-1)*3^(1/2))+2/3*ln(1+x)

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Maxima [A]  time = 1.5259, size = 58, normalized size = 1.18 \[ -\frac{1}{3} \, \sqrt{3} \arctan \left (\frac{1}{3} \, \sqrt{3}{\left (2 \, x - 1\right )}\right ) - \frac{1}{x} + \frac{1}{6} \, \log \left (x^{2} - x + 1\right ) + \frac{2}{3} \, \log \left (x + 1\right ) - \log \left (x\right ) \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate(-(x - 1)/((x^3 + 1)*x^2),x, algorithm="maxima")

[Out]

-1/3*sqrt(3)*arctan(1/3*sqrt(3)*(2*x - 1)) - 1/x + 1/6*log(x^2 - x + 1) + 2/3*lo
g(x + 1) - log(x)

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Fricas [A]  time = 0.217175, size = 81, normalized size = 1.65 \[ \frac{\sqrt{3}{\left (\sqrt{3} x \log \left (x^{2} - x + 1\right ) + 4 \, \sqrt{3} x \log \left (x + 1\right ) - 6 \, \sqrt{3} x \log \left (x\right ) - 6 \, x \arctan \left (\frac{1}{3} \, \sqrt{3}{\left (2 \, x - 1\right )}\right ) - 6 \, \sqrt{3}\right )}}{18 \, x} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate(-(x - 1)/((x^3 + 1)*x^2),x, algorithm="fricas")

[Out]

1/18*sqrt(3)*(sqrt(3)*x*log(x^2 - x + 1) + 4*sqrt(3)*x*log(x + 1) - 6*sqrt(3)*x*
log(x) - 6*x*arctan(1/3*sqrt(3)*(2*x - 1)) - 6*sqrt(3))/x

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Sympy [A]  time = 0.279235, size = 49, normalized size = 1. \[ - \log{\left (x \right )} + \frac{2 \log{\left (x + 1 \right )}}{3} + \frac{\log{\left (x^{2} - x + 1 \right )}}{6} - \frac{\sqrt{3} \operatorname{atan}{\left (\frac{2 \sqrt{3} x}{3} - \frac{\sqrt{3}}{3} \right )}}{3} - \frac{1}{x} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate((1-x)/x**2/(x**3+1),x)

[Out]

-log(x) + 2*log(x + 1)/3 + log(x**2 - x + 1)/6 - sqrt(3)*atan(2*sqrt(3)*x/3 - sq
rt(3)/3)/3 - 1/x

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GIAC/XCAS [A]  time = 0.210805, size = 61, normalized size = 1.24 \[ -\frac{1}{3} \, \sqrt{3} \arctan \left (\frac{1}{3} \, \sqrt{3}{\left (2 \, x - 1\right )}\right ) - \frac{1}{x} + \frac{1}{6} \,{\rm ln}\left (x^{2} - x + 1\right ) + \frac{2}{3} \,{\rm ln}\left ({\left | x + 1 \right |}\right ) -{\rm ln}\left ({\left | x \right |}\right ) \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate(-(x - 1)/((x^3 + 1)*x^2),x, algorithm="giac")

[Out]

-1/3*sqrt(3)*arctan(1/3*sqrt(3)*(2*x - 1)) - 1/x + 1/6*ln(x^2 - x + 1) + 2/3*ln(
abs(x + 1)) - ln(abs(x))