Optimal. Leaf size=18 \[ -4+\frac {6}{2-x}+x+\log \left (-x^2\right ) \]
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Rubi [A] time = 0.04, antiderivative size = 15, normalized size of antiderivative = 0.83, number of steps used = 4, number of rules used = 3, integrand size = 28, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.107, Rules used = {1594, 27, 1620} \begin {gather*} x+\frac {6}{2-x}+2 \log (x) \end {gather*}
Antiderivative was successfully verified.
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Rule 27
Rule 1594
Rule 1620
Rubi steps
\begin {gather*} \begin {aligned} \text {integral} &=\int \frac {8+2 x-2 x^2+x^3}{x \left (4-4 x+x^2\right )} \, dx\\ &=\int \frac {8+2 x-2 x^2+x^3}{(-2+x)^2 x} \, dx\\ &=\int \left (1+\frac {6}{(-2+x)^2}+\frac {2}{x}\right ) \, dx\\ &=\frac {6}{2-x}+x+2 \log (x)\\ \end {aligned} \end {gather*}
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Mathematica [A] time = 0.01, size = 13, normalized size = 0.72 \begin {gather*} -\frac {6}{-2+x}+x+2 \log (x) \end {gather*}
Antiderivative was successfully verified.
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fricas [A] time = 0.61, size = 21, normalized size = 1.17 \begin {gather*} \frac {x^{2} + 2 \, {\left (x - 2\right )} \log \relax (x) - 2 \, x - 6}{x - 2} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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giac [A] time = 0.12, size = 14, normalized size = 0.78 \begin {gather*} x - \frac {6}{x - 2} + 2 \, \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 = 0.02, size = 14, normalized size = 0.78
method | result | size |
default | \(x +2 \ln \relax (x )-\frac {6}{x -2}\) | \(14\) |
risch | \(x +2 \ln \relax (x )-\frac {6}{x -2}\) | \(14\) |
norman | \(\frac {x^{2}-10}{x -2}+2 \ln \relax (x )\) | \(17\) |
Verification of antiderivative is not currently implemented for this CAS.
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maxima [A] time = 0.38, size = 13, normalized size = 0.72 \begin {gather*} x - \frac {6}{x - 2} + 2 \, \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 = 13, normalized size = 0.72 \begin {gather*} x+2\,\ln \relax (x)-\frac {6}{x-2} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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sympy [A] time = 0.09, size = 10, normalized size = 0.56 \begin {gather*} x + 2 \log {\relax (x )} - \frac {6}{x - 2} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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