Optimal. Leaf size=14 \[ x+2 \log (1-x)-\log (x) \]
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Rubi [A] time = 0.02, 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 = {1593, 894} \begin {gather*} x+2 \log (1-x)-\log (x) \end {gather*}
Antiderivative was successfully verified.
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Rule 894
Rule 1593
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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IntegrateAlgebraic [F] time = 0.00, size = 0, normalized size = 0.00 \begin {gather*} \int \frac {1+x^2}{-x+x^2} \, dx \end {gather*}
Verification is not applicable to the result.
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fricas [A] time = 1.35, size = 12, normalized size = 0.86 \begin {gather*} x + 2 \, \log \left (x - 1\right ) - \log \relax (x) \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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giac [A] time = 0.26, 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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maple [A] time = 0.00, size = 13, normalized size = 0.93 \begin {gather*} x -\ln \relax (x )+2 \ln \left (x -1\right ) \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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maxima [A] time = 0.91, size = 12, normalized size = 0.86 \begin {gather*} x + 2 \, \log \left (x - 1\right ) - \log \relax (x) \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 \relax (x) \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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sympy [A] time = 0.10, size = 10, normalized size = 0.71 \begin {gather*} x - \log {\relax (x )} + 2 \log {\left (x - 1 \right )} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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