Optimal. Leaf size=18 \[ 2+\frac {8}{-1+x}-x+\log (x)+\log \left (x^2\right ) \]
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Rubi [A] time = 0.04, antiderivative size = 17, normalized size of antiderivative = 0.94, 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 {8}{1-x}+3 \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 {3-15 x+5 x^2-x^3}{x \left (1-2 x+x^2\right )} \, dx\\ &=\int \frac {3-15 x+5 x^2-x^3}{(-1+x)^2 x} \, dx\\ &=\int \left (-1-\frac {8}{(-1+x)^2}+\frac {3}{x}\right ) \, dx\\ &=-\frac {8}{1-x}-x+3 \log (x)\\ \end {aligned} \end {gather*}
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Mathematica [A] time = 0.01, size = 15, normalized size = 0.83 \begin {gather*} \frac {8}{-1+x}-x+3 \log (x) \end {gather*}
Antiderivative was successfully verified.
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fricas [A] time = 1.11, size = 22, normalized size = 1.22 \begin {gather*} -\frac {x^{2} - 3 \, {\left (x - 1\right )} \log \relax (x) - x - 8}{x - 1} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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giac [A] time = 0.22, size = 16, normalized size = 0.89 \begin {gather*} -x + \frac {8}{x - 1} + 3 \, \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 = 16, normalized size = 0.89
method | result | size |
default | \(-x +3 \ln \relax (x )+\frac {8}{x -1}\) | \(16\) |
risch | \(-x +3 \ln \relax (x )+\frac {8}{x -1}\) | \(16\) |
norman | \(\frac {-x^{2}+9}{x -1}+3 \ln \relax (x )\) | \(19\) |
Verification of antiderivative is not currently implemented for this CAS.
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maxima [A] time = 0.64, size = 15, normalized size = 0.83 \begin {gather*} -x + \frac {8}{x - 1} + 3 \, \log \relax (x) \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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mupad [B] time = 0.05, size = 15, normalized size = 0.83 \begin {gather*} 3\,\ln \relax (x)-x+\frac {8}{x-1} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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sympy [A] time = 0.08, size = 10, normalized size = 0.56 \begin {gather*} - x + 3 \log {\relax (x )} + \frac {8}{x - 1} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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