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

Optimal. Leaf size=14 \[ x+2 \log (1-x)-\log (x) \]

[Out]

x + 2*Log[1 - x] - Log[x]

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Rubi [A]  time = 0.0319346, antiderivative size = 14, normalized size of antiderivative = 1., number of steps used = 3, number of rules used = 2, integrand size = 15, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.133 \[ x+2 \log (1-x)-\log (x) \]

Antiderivative was successfully verified.

[In]  Int[(1 + x^2)/(-x + x^2),x]

[Out]

x + 2*Log[1 - x] - Log[x]

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Rubi in Sympy [A]  time = 2.18017, size = 10, normalized size = 0.71 \[ x - \log{\left (x \right )} + 2 \log{\left (- x + 1 \right )} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

x - log(x) + 2*log(-x + 1)

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Mathematica [A]  time = 0.00492774, size = 14, normalized size = 1. \[ x+2 \log (1-x)-\log (x) \]

Antiderivative was successfully verified.

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

[Out]

x + 2*Log[1 - x] - Log[x]

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Maple [A]  time = 0.009, size = 13, normalized size = 0.9 \[ x-\ln \left ( x \right ) +2\,\ln \left ( -1+x \right ) \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  int((x^2+1)/(x^2-x),x)

[Out]

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

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Maxima [A]  time = 1.35165, size = 16, normalized size = 1.14 \[ x + 2 \, \log \left (x - 1\right ) - \log \left (x\right ) \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

x + 2*log(x - 1) - log(x)

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Fricas [A]  time = 0.195945, size = 16, normalized size = 1.14 \[ x + 2 \, \log \left (x - 1\right ) - \log \left (x\right ) \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

x + 2*log(x - 1) - log(x)

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Sympy [A]  time = 0.095633, size = 10, normalized size = 0.71 \[ x - \log{\left (x \right )} + 2 \log{\left (x - 1 \right )} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate((x**2+1)/(x**2-x),x)

[Out]

x - log(x) + 2*log(x - 1)

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GIAC/XCAS [A]  time = 0.205721, size = 19, normalized size = 1.36 \[ x + 2 \,{\rm ln}\left ({\left | x - 1 \right |}\right ) -{\rm ln}\left ({\left | x \right |}\right ) \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

x + 2*ln(abs(x - 1)) - ln(abs(x))