Optimal. Leaf size=36 \[ \frac {1}{3} x \left (-e^3+9 \left (5+\left (2+\log \left (\frac {x}{\log \left (\frac {x^2}{3-x}\right )}\right )\right )^2\right )\right ) \]
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Rubi [F]
time = 0.53, 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 {216-36 x+\left (-351+e^3 (3-x)+117 x\right ) \log \left (-\frac {x^2}{-3+x}\right )+\left (108-18 x+(-162+54 x) \log \left (-\frac {x^2}{-3+x}\right )\right ) \log \left (\frac {x}{\log \left (-\frac {x^2}{-3+x}\right )}\right )+(-27+9 x) \log \left (-\frac {x^2}{-3+x}\right ) \log ^2\left (\frac {x}{\log \left (-\frac {x^2}{-3+x}\right )}\right )}{(-9+3 x) \log \left (-\frac {x^2}{-3+x}\right )} \, dx \end {gather*}
Verification is not applicable to the result.
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Rubi steps
\begin {gather*} \begin {aligned} \text {integral} &=\int \left (39-\frac {e^3}{3}+18 \log \left (\frac {x}{\log \left (-\frac {x^2}{-3+x}\right )}\right )+3 \log ^2\left (\frac {x}{\log \left (-\frac {x^2}{-3+x}\right )}\right )-\frac {6 (-6+x) \left (2+\log \left (\frac {x}{\log \left (-\frac {x^2}{-3+x}\right )}\right )\right )}{(-3+x) \log \left (-\frac {x^2}{-3+x}\right )}\right ) \, dx\\ &=\frac {1}{3} \left (117-e^3\right ) x+3 \int \log ^2\left (\frac {x}{\log \left (-\frac {x^2}{-3+x}\right )}\right ) \, dx-6 \int \frac {(-6+x) \left (2+\log \left (\frac {x}{\log \left (-\frac {x^2}{-3+x}\right )}\right )\right )}{(-3+x) \log \left (-\frac {x^2}{-3+x}\right )} \, dx+18 \int \log \left (\frac {x}{\log \left (-\frac {x^2}{-3+x}\right )}\right ) \, dx\\ &=\frac {1}{3} \left (117-e^3\right ) x+18 x \log \left (\frac {x}{\log \left (\frac {x^2}{3-x}\right )}\right )+3 \int \log ^2\left (\frac {x}{\log \left (-\frac {x^2}{-3+x}\right )}\right ) \, dx-6 \int \left (\frac {2 (-6+x)}{(-3+x) \log \left (-\frac {x^2}{-3+x}\right )}+\frac {(-6+x) \log \left (\frac {x}{\log \left (-\frac {x^2}{-3+x}\right )}\right )}{(-3+x) \log \left (-\frac {x^2}{-3+x}\right )}\right ) \, dx-18 \int \frac {6-x+(-3+x) \log \left (-\frac {x^2}{-3+x}\right )}{(-3+x) \log \left (-\frac {x^2}{-3+x}\right )} \, dx\\ &=\frac {1}{3} \left (117-e^3\right ) x+18 x \log \left (\frac {x}{\log \left (\frac {x^2}{3-x}\right )}\right )+3 \int \log ^2\left (\frac {x}{\log \left (-\frac {x^2}{-3+x}\right )}\right ) \, dx-6 \int \frac {(-6+x) \log \left (\frac {x}{\log \left (-\frac {x^2}{-3+x}\right )}\right )}{(-3+x) \log \left (-\frac {x^2}{-3+x}\right )} \, dx-12 \int \frac {-6+x}{(-3+x) \log \left (-\frac {x^2}{-3+x}\right )} \, dx-18 \int \left (1+\frac {6-x}{(-3+x) \log \left (-\frac {x^2}{-3+x}\right )}\right ) \, dx\\ &=-18 x+\frac {1}{3} \left (117-e^3\right ) x+18 x \log \left (\frac {x}{\log \left (\frac {x^2}{3-x}\right )}\right )+3 \int \log ^2\left (\frac {x}{\log \left (-\frac {x^2}{-3+x}\right )}\right ) \, dx-6 \int \left (\frac {\log \left (\frac {x}{\log \left (-\frac {x^2}{-3+x}\right )}\right )}{\log \left (-\frac {x^2}{-3+x}\right )}-\frac {3 \log \left (\frac {x}{\log \left (-\frac {x^2}{-3+x}\right )}\right )}{(-3+x) \log \left (-\frac {x^2}{-3+x}\right )}\right ) \, dx-12 \int \frac {-6+x}{(-3+x) \log \left (-\frac {x^2}{-3+x}\right )} \, dx-18 \int \frac {6-x}{(-3+x) \log \left (-\frac {x^2}{-3+x}\right )} \, dx\\ &=-18 x+\frac {1}{3} \left (117-e^3\right ) x+18 x \log \left (\frac {x}{\log \left (\frac {x^2}{3-x}\right )}\right )+3 \int \log ^2\left (\frac {x}{\log \left (-\frac {x^2}{-3+x}\right )}\right ) \, dx-6 \int \frac {\log \left (\frac {x}{\log \left (-\frac {x^2}{-3+x}\right )}\right )}{\log \left (-\frac {x^2}{-3+x}\right )} \, dx-12 \int \frac {-6+x}{(-3+x) \log \left (-\frac {x^2}{-3+x}\right )} \, dx-18 \int \frac {6-x}{(-3+x) \log \left (-\frac {x^2}{-3+x}\right )} \, dx+18 \int \frac {\log \left (\frac {x}{\log \left (-\frac {x^2}{-3+x}\right )}\right )}{(-3+x) \log \left (-\frac {x^2}{-3+x}\right )} \, dx\\ \end {aligned} \end {gather*}
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Mathematica [A]
time = 0.15, size = 51, normalized size = 1.42 \begin {gather*} -\frac {1}{3} \left (-81+e^3\right ) x+12 x \log \left (\frac {x}{\log \left (-\frac {x^2}{-3+x}\right )}\right )+3 x \log ^2\left (\frac {x}{\log \left (-\frac {x^2}{-3+x}\right )}\right ) \end {gather*}
Antiderivative was successfully verified.
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Maple [A]
time = 7.19, size = 50, normalized size = 1.39
method | result | size |
norman | \(\left (-\frac {{\mathrm e}^{3}}{3}+27\right ) x +3 x \ln \left (\frac {x}{\ln \left (-\frac {x^{2}}{x -3}\right )}\right )^{2}+12 \ln \left (\frac {x}{\ln \left (-\frac {x^{2}}{x -3}\right )}\right ) x\) | \(50\) |
Verification of antiderivative is not currently implemented for this CAS.
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Maxima [B] Leaf count of result is larger than twice the leaf count of optimal. 63 vs.
\(2 (31) = 62\).
time = 0.31, size = 63, normalized size = 1.75 \begin {gather*} 3 \, x \log \left (x\right )^{2} + 3 \, x \log \left (2 \, \log \left (x\right ) - \log \left (-x + 3\right )\right )^{2} - \frac {1}{3} \, x {\left (e^{3} - 81\right )} + 12 \, x \log \left (x\right ) - 6 \, {\left (x \log \left (x\right ) + 2 \, x\right )} \log \left (2 \, \log \left (x\right ) - \log \left (-x + 3\right )\right ) \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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Fricas [A]
time = 0.44, size = 49, normalized size = 1.36 \begin {gather*} 3 \, x \log \left (\frac {x}{\log \left (-\frac {x^{2}}{x - 3}\right )}\right )^{2} - \frac {1}{3} \, x e^{3} + 12 \, x \log \left (\frac {x}{\log \left (-\frac {x^{2}}{x - 3}\right )}\right ) + 27 \, x \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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Sympy [A]
time = 0.34, size = 42, normalized size = 1.17 \begin {gather*} 3 x \log {\left (\frac {x}{\log {\left (- \frac {x^{2}}{x - 3} \right )}} \right )}^{2} + 12 x \log {\left (\frac {x}{\log {\left (- \frac {x^{2}}{x - 3} \right )}} \right )} + x \left (27 - \frac {e^{3}}{3}\right ) \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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Giac [F]
time = 0.00, size = 0, normalized size = 0.00 \begin {gather*} \text {could not integrate} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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Mupad [B]
time = 2.69, size = 50, normalized size = 1.39 \begin {gather*} 3\,x\,{\ln \left (\frac {x}{\ln \left (-\frac {x^2}{x-3}\right )}\right )}^2+12\,x\,\ln \left (\frac {x}{\ln \left (-\frac {x^2}{x-3}\right )}\right )-x\,\left (\frac {{\mathrm {e}}^3}{3}-27\right ) \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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