3.273 \(\int \frac{3-x+3 x^2-2 x^3+x^4}{3 x-2 x^2+x^3} \, dx\)

Optimal. Leaf size=23 \[ \frac{x^2}{2}-\frac{1}{2} \log \left (x^2-2 x+3\right )+\log (x) \]

[Out]

x^2/2 + Log[x] - Log[3 - 2*x + x^2]/2

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Rubi [A]  time = 0.0636577, antiderivative size = 23, normalized size of antiderivative = 1., number of steps used = 4, number of rules used = 3, integrand size = 33, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.091 \[ \frac{x^2}{2}-\frac{1}{2} \log \left (x^2-2 x+3\right )+\log (x) \]

Antiderivative was successfully verified.

[In]  Int[(3 - x + 3*x^2 - 2*x^3 + x^4)/(3*x - 2*x^2 + x^3),x]

[Out]

x^2/2 + Log[x] - Log[3 - 2*x + x^2]/2

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Rubi in Sympy [A]  time = 30.1732, size = 19, normalized size = 0.83 \[ \frac{x^{2}}{2} + \log{\left (x \right )} - \frac{\log{\left (x^{2} - 2 x + 3 \right )}}{2} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  rubi_integrate((x**4-2*x**3+3*x**2-x+3)/(x**3-2*x**2+3*x),x)

[Out]

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

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Mathematica [A]  time = 0.0131068, size = 23, normalized size = 1. \[ \frac{x^2}{2}-\frac{1}{2} \log \left (x^2-2 x+3\right )+\log (x) \]

Antiderivative was successfully verified.

[In]  Integrate[(3 - x + 3*x^2 - 2*x^3 + x^4)/(3*x - 2*x^2 + x^3),x]

[Out]

x^2/2 + Log[x] - Log[3 - 2*x + x^2]/2

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Maple [A]  time = 0.007, size = 20, normalized size = 0.9 \[{\frac{{x}^{2}}{2}}+\ln \left ( x \right ) -{\frac{\ln \left ({x}^{2}-2\,x+3 \right ) }{2}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  int((x^4-2*x^3+3*x^2-x+3)/(x^3-2*x^2+3*x),x)

[Out]

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

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Maxima [A]  time = 0.801188, size = 26, normalized size = 1.13 \[ \frac{1}{2} \, x^{2} - \frac{1}{2} \, \log \left (x^{2} - 2 \, x + 3\right ) + \log \left (x\right ) \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate((x^4 - 2*x^3 + 3*x^2 - x + 3)/(x^3 - 2*x^2 + 3*x),x, algorithm="maxima")

[Out]

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

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Fricas [A]  time = 0.24787, size = 26, normalized size = 1.13 \[ \frac{1}{2} \, x^{2} - \frac{1}{2} \, \log \left (x^{2} - 2 \, x + 3\right ) + \log \left (x\right ) \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate((x^4 - 2*x^3 + 3*x^2 - x + 3)/(x^3 - 2*x^2 + 3*x),x, algorithm="fricas")

[Out]

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

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Sympy [A]  time = 0.21191, size = 19, normalized size = 0.83 \[ \frac{x^{2}}{2} + \log{\left (x \right )} - \frac{\log{\left (x^{2} - 2 x + 3 \right )}}{2} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate((x**4-2*x**3+3*x**2-x+3)/(x**3-2*x**2+3*x),x)

[Out]

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

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GIAC/XCAS [A]  time = 0.261477, size = 27, normalized size = 1.17 \[ \frac{1}{2} \, x^{2} - \frac{1}{2} \,{\rm ln}\left (x^{2} - 2 \, x + 3\right ) +{\rm ln}\left ({\left | x \right |}\right ) \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate((x^4 - 2*x^3 + 3*x^2 - x + 3)/(x^3 - 2*x^2 + 3*x),x, algorithm="giac")

[Out]

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