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

Optimal. Leaf size=12 \[ x+\log (1-x)-\log (x) \]

[Out]

x + Log[1 - x] - Log[x]

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Rubi [A]  time = 0.0463124, antiderivative size = 12, normalized size of antiderivative = 1., number of steps used = 4, number of rules used = 3, integrand size = 15, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.2 \[ x+\log (1-x)-\log (x) \]

Antiderivative was successfully verified.

[In]  Int[(1 + x^3)/(-x + x^3),x]

[Out]

x + Log[1 - x] - Log[x]

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Rubi in Sympy [A]  time = 7.05429, size = 8, normalized size = 0.67 \[ x - \log{\left (x \right )} + \log{\left (- x + 1 \right )} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

x - log(x) + log(-x + 1)

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Mathematica [A]  time = 0.00727673, size = 12, normalized size = 1. \[ x+\log (1-x)-\log (x) \]

Antiderivative was successfully verified.

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

[Out]

x + Log[1 - x] - Log[x]

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Maple [A]  time = 0.008, size = 11, normalized size = 0.9 \[ x+\ln \left ( -1+x \right ) -\ln \left ( x \right ) \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  int((x^3+1)/(x^3-x),x)

[Out]

x+ln(-1+x)-ln(x)

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Maxima [A]  time = 0.805075, size = 14, normalized size = 1.17 \[ x + \log \left (x - 1\right ) - \log \left (x\right ) \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

x + log(x - 1) - log(x)

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Fricas [A]  time = 0.258767, size = 14, normalized size = 1.17 \[ x + \log \left (x - 1\right ) - \log \left (x\right ) \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

x + log(x - 1) - log(x)

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Sympy [A]  time = 0.186061, size = 8, normalized size = 0.67 \[ x - \log{\left (x \right )} + \log{\left (x - 1 \right )} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate((x**3+1)/(x**3-x),x)

[Out]

x - log(x) + log(x - 1)

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GIAC/XCAS [A]  time = 0.259719, size = 16, normalized size = 1.33 \[ x +{\rm ln}\left ({\left | x - 1 \right |}\right ) -{\rm ln}\left ({\left | x \right |}\right ) \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

x + ln(abs(x - 1)) - ln(abs(x))