Optimal. Leaf size=17 \[ 2 x-\frac {1}{3-x+\frac {1}{\log (x)}} \]
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Rubi [F]
time = 0.69, antiderivative size = 0, normalized size of antiderivative = 0.00, number of steps
used = 0, number of rules used = 0, integrand size = 0, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.000, Rules used = {}
\begin {gather*} \int \frac {-1+2 x+\left (12 x-4 x^2\right ) \log (x)+\left (17 x-12 x^2+2 x^3\right ) \log ^2(x)}{x+\left (6 x-2 x^2\right ) \log (x)+\left (9 x-6 x^2+x^3\right ) \log ^2(x)} \, dx \end {gather*}
Verification is not applicable to the result.
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Rubi steps
\begin {gather*} \begin {aligned} \text {integral} &=\int \frac {-1+2 x-4 (-3+x) x \log (x)+x \left (17-12 x+2 x^2\right ) \log ^2(x)}{x (1-(-3+x) \log (x))^2} \, dx\\ &=\int \left (\frac {17-12 x+2 x^2}{(-3+x)^2}+\frac {-9+5 x-x^2}{(-3+x)^2 x (-1-3 \log (x)+x \log (x))^2}-\frac {2}{(-3+x)^2 (-1-3 \log (x)+x \log (x))}\right ) \, dx\\ &=-\left (2 \int \frac {1}{(-3+x)^2 (-1-3 \log (x)+x \log (x))} \, dx\right )+\int \frac {17-12 x+2 x^2}{(-3+x)^2} \, dx+\int \frac {-9+5 x-x^2}{(-3+x)^2 x (-1-3 \log (x)+x \log (x))^2} \, dx\\ &=-\left (2 \int \frac {1}{(-3+x)^2 (-1-3 \log (x)+x \log (x))} \, dx\right )+\int \left (2-\frac {1}{(-3+x)^2}\right ) \, dx+\int \left (-\frac {1}{(-3+x)^2 (-1-3 \log (x)+x \log (x))^2}-\frac {1}{x (-1-3 \log (x)+x \log (x))^2}\right ) \, dx\\ &=\frac {1}{-3+x}+2 x-2 \int \frac {1}{(-3+x)^2 (-1-3 \log (x)+x \log (x))} \, dx-\int \frac {1}{(-3+x)^2 (-1-3 \log (x)+x \log (x))^2} \, dx-\int \frac {1}{x (-1-3 \log (x)+x \log (x))^2} \, dx\\ \end {aligned} \end {gather*}
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Mathematica [A]
time = 0.08, size = 27, normalized size = 1.59 \begin {gather*} \frac {1}{-3+x}+2 x+\frac {1}{(-3+x) (-1-3 \log (x)+x \log (x))} \end {gather*}
Antiderivative was successfully verified.
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Maple [A]
time = 1.46, size = 30, normalized size = 1.76
method | result | size |
default | \(\frac {-17 \ln \left (x \right )+2 x^{2} \ln \left (x \right )-2 x -6}{x \ln \left (x \right )-3 \ln \left (x \right )-1}\) | \(30\) |
norman | \(\frac {-17 \ln \left (x \right )+2 x^{2} \ln \left (x \right )-2 x -6}{x \ln \left (x \right )-3 \ln \left (x \right )-1}\) | \(30\) |
risch | \(\frac {2 x^{2}-6 x +1}{x -3}+\frac {1}{\left (x -3\right ) \left (x \ln \left (x \right )-3 \ln \left (x \right )-1\right )}\) | \(36\) |
Verification of antiderivative is not currently implemented for this CAS.
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Maxima [A]
time = 0.30, size = 28, normalized size = 1.65 \begin {gather*} \frac {{\left (2 \, x^{2} - 6 \, x + 1\right )} \log \left (x\right ) - 2 \, x}{{\left (x - 3\right )} \log \left (x\right ) - 1} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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Fricas [A]
time = 0.35, size = 28, normalized size = 1.65 \begin {gather*} \frac {{\left (2 \, x^{2} - 6 \, x + 1\right )} \log \left (x\right ) - 2 \, x}{{\left (x - 3\right )} \log \left (x\right ) - 1} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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Sympy [A]
time = 0.07, size = 24, normalized size = 1.41 \begin {gather*} 2 x + \frac {1}{- x + \left (x^{2} - 6 x + 9\right ) \log {\left (x \right )} + 3} + \frac {1}{x - 3} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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Giac [B] Leaf count of result is larger than twice the leaf count of optimal. 31 vs.
\(2 (15) = 30\).
time = 0.43, size = 31, normalized size = 1.82 \begin {gather*} 2 \, x + \frac {1}{x^{2} \log \left (x\right ) - 6 \, x \log \left (x\right ) - x + 9 \, \log \left (x\right ) + 3} + \frac {1}{x - 3} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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Mupad [B]
time = 5.54, size = 21, normalized size = 1.24 \begin {gather*} 2\,x-\frac {\ln \left (x\right )}{3\,\ln \left (x\right )-x\,\ln \left (x\right )+1} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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