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

Optimal. Leaf size=17 \[ x+\frac{1}{x}+2 \log (1-x)-\log (x) \]

[Out]

x^(-1) + x + 2*Log[1 - x] - Log[x]

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Rubi [A]  time = 0.0332264, antiderivative size = 17, normalized size of antiderivative = 1., number of steps used = 3, number of rules used = 2, integrand size = 17, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.118 \[ x+\frac{1}{x}+2 \log (1-x)-\log (x) \]

Antiderivative was successfully verified.

[In]  Int[(1 + x^3)/(-x^2 + x^3),x]

[Out]

x^(-1) + x + 2*Log[1 - x] - Log[x]

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Rubi in Sympy [A]  time = 3.49449, size = 14, normalized size = 0.82 \[ x - \log{\left (x \right )} + 2 \log{\left (- x + 1 \right )} + \frac{1}{x} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

x - log(x) + 2*log(-x + 1) + 1/x

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Mathematica [A]  time = 0.0061008, size = 17, normalized size = 1. \[ x+\frac{1}{x}+2 \log (1-x)-\log (x) \]

Antiderivative was successfully verified.

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

[Out]

x^(-1) + x + 2*Log[1 - x] - Log[x]

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Maple [A]  time = 0.011, size = 16, normalized size = 0.9 \[ x+{x}^{-1}-\ln \left ( x \right ) +2\,\ln \left ( -1+x \right ) \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  int((x^3+1)/(x^3-x^2),x)

[Out]

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

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Maxima [A]  time = 1.37391, size = 20, normalized size = 1.18 \[ x + \frac{1}{x} + 2 \, \log \left (x - 1\right ) - \log \left (x\right ) \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

x + 1/x + 2*log(x - 1) - log(x)

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Fricas [A]  time = 0.247433, size = 28, normalized size = 1.65 \[ \frac{x^{2} + 2 \, x \log \left (x - 1\right ) - x \log \left (x\right ) + 1}{x} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

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

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Sympy [A]  time = 0.106666, size = 14, normalized size = 0.82 \[ x - \log{\left (x \right )} + 2 \log{\left (x - 1 \right )} + \frac{1}{x} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

x - log(x) + 2*log(x - 1) + 1/x

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GIAC/XCAS [A]  time = 0.208645, size = 23, normalized size = 1.35 \[ x + \frac{1}{x} + 2 \,{\rm ln}\left ({\left | x - 1 \right |}\right ) -{\rm ln}\left ({\left | x \right |}\right ) \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

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