Optimal. Leaf size=22 \[ \frac {-5+x^2+\log (x)}{3 (4+4 x \log (3 x))} \]
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Rubi [F]
time = 0.88, 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+5 x+2 x^2-x^3-x \log (x)+\left (6 x+x^3-x \log (x)\right ) \log (3 x)}{12 x+24 x^2 \log (3 x)+12 x^3 \log ^2(3 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+5 x+2 x^2-x^3-x \log (x)+\left (6 x+x^3-x \log (x)\right ) \log (3 x)}{12 x (1+x \log (3 x))^2} \, dx\\ &=\frac {1}{12} \int \frac {1+5 x+2 x^2-x^3-x \log (x)+\left (6 x+x^3-x \log (x)\right ) \log (3 x)}{x (1+x \log (3 x))^2} \, dx\\ &=\frac {1}{12} \int \left (-\frac {(-1+x) \left (-5+x^2+\log (x)\right )}{x (1+x \log (3 x))^2}+\frac {6+x^2-\log (x)}{x (1+x \log (3 x))}\right ) \, dx\\ &=-\left (\frac {1}{12} \int \frac {(-1+x) \left (-5+x^2+\log (x)\right )}{x (1+x \log (3 x))^2} \, dx\right )+\frac {1}{12} \int \frac {6+x^2-\log (x)}{x (1+x \log (3 x))} \, dx\\ &=-\left (\frac {1}{12} \int \left (\frac {5-x^2-\log (x)}{x (1+x \log (3 x))^2}+\frac {-5+x^2+\log (x)}{(1+x \log (3 x))^2}\right ) \, dx\right )+\frac {1}{12} \int \left (\frac {6}{x (1+x \log (3 x))}+\frac {x}{1+x \log (3 x)}-\frac {\log (x)}{x (1+x \log (3 x))}\right ) \, dx\\ &=-\left (\frac {1}{12} \int \frac {5-x^2-\log (x)}{x (1+x \log (3 x))^2} \, dx\right )-\frac {1}{12} \int \frac {-5+x^2+\log (x)}{(1+x \log (3 x))^2} \, dx+\frac {1}{12} \int \frac {x}{1+x \log (3 x)} \, dx-\frac {1}{12} \int \frac {\log (x)}{x (1+x \log (3 x))} \, dx+\frac {1}{2} \int \frac {1}{x (1+x \log (3 x))} \, dx\\ &=\frac {1}{12} \int \frac {x}{1+x \log (3 x)} \, dx-\frac {1}{12} \int \frac {\log (x)}{x (1+x \log (3 x))} \, dx-\frac {1}{12} \int \left (-\frac {5}{(1+x \log (3 x))^2}+\frac {x^2}{(1+x \log (3 x))^2}+\frac {\log (x)}{(1+x \log (3 x))^2}\right ) \, dx-\frac {1}{12} \int \left (\frac {5}{x (1+x \log (3 x))^2}-\frac {x}{(1+x \log (3 x))^2}-\frac {\log (x)}{x (1+x \log (3 x))^2}\right ) \, dx+\frac {1}{2} \int \frac {1}{x (1+x \log (3 x))} \, dx\\ &=\frac {1}{12} \int \frac {x}{(1+x \log (3 x))^2} \, dx-\frac {1}{12} \int \frac {x^2}{(1+x \log (3 x))^2} \, dx-\frac {1}{12} \int \frac {\log (x)}{(1+x \log (3 x))^2} \, dx+\frac {1}{12} \int \frac {\log (x)}{x (1+x \log (3 x))^2} \, dx+\frac {1}{12} \int \frac {x}{1+x \log (3 x)} \, dx-\frac {1}{12} \int \frac {\log (x)}{x (1+x \log (3 x))} \, dx+\frac {5}{12} \int \frac {1}{(1+x \log (3 x))^2} \, dx-\frac {5}{12} \int \frac {1}{x (1+x \log (3 x))^2} \, dx+\frac {1}{2} \int \frac {1}{x (1+x \log (3 x))} \, dx\\ \end {aligned} \end {gather*}
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Mathematica [A]
time = 0.11, size = 21, normalized size = 0.95 \begin {gather*} \frac {-5+x^2+\log (x)}{12 (1+x \log (3 x))} \end {gather*}
Antiderivative was successfully verified.
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Maple [A]
time = 2.35, size = 26, normalized size = 1.18
method | result | size |
default | \(-\frac {5-\ln \left (x \right )-x^{2}}{12 \left (x \ln \left (3\right )+x \ln \left (x \right )+1\right )}\) | \(26\) |
risch | \(\frac {1}{12 x}+\frac {-2+2 x^{3}-2 x \ln \left (3\right )-10 x}{12 x \left (2+2 x \ln \left (3\right )+2 x \ln \left (x \right )\right )}\) | \(41\) |
Verification of antiderivative is not currently implemented for this CAS.
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Maxima [A]
time = 0.52, size = 21, normalized size = 0.95 \begin {gather*} \frac {x^{2} + \log \left (x\right ) - 5}{12 \, {\left (x \log \left (3\right ) + x \log \left (x\right ) + 1\right )}} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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Fricas [A]
time = 0.42, size = 21, normalized size = 0.95 \begin {gather*} \frac {x^{2} + \log \left (x\right ) - 5}{12 \, {\left (x \log \left (3\right ) + x \log \left (x\right ) + 1\right )}} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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Sympy [B] Leaf count of result is larger than twice the leaf count of optimal. 37 vs.
\(2 (17) = 34\).
time = 0.09, size = 37, normalized size = 1.68 \begin {gather*} \frac {x^{3} - 5 x - x \log {\left (3 \right )} - 1}{12 x^{2} \log {\left (x \right )} + 12 x^{2} \log {\left (3 \right )} + 12 x} + \frac {1}{12 x} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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Giac [A]
time = 0.41, size = 37, normalized size = 1.68 \begin {gather*} \frac {x^{3} - x \log \left (3\right ) - 5 \, x - 1}{12 \, {\left (x^{2} \log \left (3\right ) + x^{2} \log \left (x\right ) + x\right )}} + \frac {1}{12 \, x} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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Mupad [B]
time = 4.47, size = 19, normalized size = 0.86 \begin {gather*} \frac {\ln \left (x\right )+x^2-5}{12\,\left (x\,\ln \left (3\,x\right )+1\right )} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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