Optimal. Leaf size=14 \[ x+2 \log (1-x)-\log (x) \]
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Rubi [A] time = 0.017737, 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, Rules used = {1593, 894} \[ x+2 \log (1-x)-\log (x) \]
Antiderivative was successfully verified.
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Rule 1593
Rule 894
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*}
Mathematica [A] time = 0.0031274, size = 14, normalized size = 1. \[ x+2 \log (1-x)-\log (x) \]
Antiderivative was successfully verified.
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Maple [A] time = 0.004, size = 13, normalized size = 0.9 \begin{align*} x-\ln \left ( x \right ) +2\,\ln \left ( -1+x \right ) \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Maxima [A] time = 0.926839, size = 16, normalized size = 1.14 \begin{align*} x + 2 \, \log \left (x - 1\right ) - \log \left (x\right ) \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Fricas [A] time = 1.67985, size = 36, normalized size = 2.57 \begin{align*} x + 2 \, \log \left (x - 1\right ) - \log \left (x\right ) \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Sympy [A] time = 0.08969, size = 10, normalized size = 0.71 \begin{align*} x - \log{\left (x \right )} + 2 \log{\left (x - 1 \right )} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Giac [A] time = 1.05026, size = 19, normalized size = 1.36 \begin{align*} x + 2 \, \log \left ({\left | x - 1 \right |}\right ) - \log \left ({\left | x \right |}\right ) \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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