Optimal. Leaf size=14 \[ x+2 \log (1-x)-\log (x) \]
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Rubi [A]
time = 0.01, antiderivative size = 14, normalized size of antiderivative = 1.00, number of steps
used = 3, number of rules used = 2, integrand size = 15, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.133, Rules used = {1607, 908}
\begin {gather*} x+2 \log (1-x)-\log (x) \end {gather*}
Antiderivative was successfully verified.
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Rule 908
Rule 1607
Rubi steps
\begin {align*} \int \frac {1+x^2}{-x+x^2} \, dx &=\int \frac {1+x^2}{(-1+x) x} \, dx\\ &=\int \left (1+\frac {2}{-1+x}-\frac {1}{x}\right ) \, dx\\ &=x+2 \log (1-x)-\log (x)\\ \end {align*}
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Mathematica [A]
time = 0.00, size = 14, normalized size = 1.00 \begin {gather*} x+2 \log (1-x)-\log (x) \end {gather*}
Antiderivative was successfully verified.
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Maple [A]
time = 0.20, size = 13, normalized size = 0.93
method | result | size |
default | \(x +2 \ln \left (-1+x \right )-\ln \left (x \right )\) | \(13\) |
norman | \(x +2 \ln \left (-1+x \right )-\ln \left (x \right )\) | \(13\) |
risch | \(x +2 \ln \left (-1+x \right )-\ln \left (x \right )\) | \(13\) |
meijerg | \(2 \ln \left (1-x \right )-\ln \left (x \right )-i \pi +x\) | \(19\) |
Verification of antiderivative is not currently implemented for this CAS.
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Maxima [A]
time = 0.26, size = 12, normalized size = 0.86 \begin {gather*} x + 2 \, \log \left (x - 1\right ) - \log \left (x\right ) \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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Fricas [A]
time = 0.40, size = 12, normalized size = 0.86 \begin {gather*} x + 2 \, \log \left (x - 1\right ) - \log \left (x\right ) \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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Sympy [A]
time = 0.03, size = 10, normalized size = 0.71 \begin {gather*} x - \log {\left (x \right )} + 2 \log {\left (x - 1 \right )} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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Giac [A]
time = 9.07, size = 14, normalized size = 1.00 \begin {gather*} x + 2 \, \log \left ({\left | x - 1 \right |}\right ) - \log \left ({\left | x \right |}\right ) \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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Mupad [B]
time = 0.04, size = 12, normalized size = 0.86 \begin {gather*} x+2\,\ln \left (x-1\right )-\ln \left (x\right ) \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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