Optimal. Leaf size=26 \[ \frac {x^2}{2}+2 x+\frac {1}{1-x}+3 \log (1-x) \]
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Rubi [A] time = 0.01, antiderivative size = 26, normalized size of antiderivative = 1.00, number of steps used = 3, number of rules used = 2, integrand size = 14, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.143, Rules used = {27, 43} \begin {gather*} \frac {x^2}{2}+2 x+\frac {1}{1-x}+3 \log (1-x) \end {gather*}
Antiderivative was successfully verified.
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Rule 27
Rule 43
Rubi steps
\begin {align*} \int \frac {x^3}{1-2 x+x^2} \, dx &=\int \frac {x^3}{(-1+x)^2} \, dx\\ &=\int \left (2+\frac {1}{(-1+x)^2}+\frac {3}{-1+x}+x\right ) \, dx\\ &=\frac {1}{1-x}+2 x+\frac {x^2}{2}+3 \log (1-x)\\ \end {align*}
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Mathematica [A] time = 0.01, size = 25, normalized size = 0.96 \begin {gather*} \frac {1}{2} \left (x^2+4 x-\frac {2}{x-1}+6 \log (x-1)-5\right ) \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 {x^3}{1-2 x+x^2} \, dx \end {gather*}
Verification is not applicable to the result.
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fricas [A] time = 0.40, size = 29, normalized size = 1.12 \begin {gather*} \frac {x^{3} + 3 \, x^{2} + 6 \, {\left (x - 1\right )} \log \left (x - 1\right ) - 4 \, x - 2}{2 \, {\left (x - 1\right )}} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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giac [A] time = 0.16, size = 23, normalized size = 0.88 \begin {gather*} \frac {1}{2} \, x^{2} + 2 \, x - \frac {1}{x - 1} + 3 \, \log \left ({\left | x - 1 \right |}\right ) \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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maple [A] time = 0.08, size = 23, normalized size = 0.88 \begin {gather*} \frac {x^{2}}{2}+2 x +3 \ln \left (x -1\right )-\frac {1}{x -1} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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maxima [A] time = 0.90, size = 22, normalized size = 0.85 \begin {gather*} \frac {1}{2} \, x^{2} + 2 \, x - \frac {1}{x - 1} + 3 \, \log \left (x - 1\right ) \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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mupad [B] time = 0.04, size = 22, normalized size = 0.85 \begin {gather*} 2\,x+3\,\ln \left (x-1\right )-\frac {1}{x-1}+\frac {x^2}{2} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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sympy [A] time = 0.08, size = 19, normalized size = 0.73 \begin {gather*} \frac {x^{2}}{2} + 2 x + 3 \log {\left (x - 1 \right )} - \frac {1}{x - 1} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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