Integrand size = 43, antiderivative size = 28 \[ \int \frac {1}{4} \left ((3-10 x+(3-2 x) \log (3)) \log (4)+(-4-4 \log (3)) \log (4) \log (x)+(1+\log (3)) \log (4) \log ^2(x)\right ) \, dx=x \log (4) \left (-x+\frac {1}{4} (1+\log (3)) \left (-x+(3-\log (x))^2\right )\right ) \]
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Leaf count is larger than twice the leaf count of optimal. \(67\) vs. \(2(28)=56\).
Time = 0.03 (sec) , antiderivative size = 67, normalized size of antiderivative = 2.39, number of steps used = 5, number of rules used = 3, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.070, Rules used = {12, 2332, 2333} \[ \int \frac {1}{4} \left ((3-10 x+(3-2 x) \log (3)) \log (4)+(-4-4 \log (3)) \log (4) \log (x)+(1+\log (3)) \log (4) \log ^2(x)\right ) \, dx=\frac {1}{4} x (1+\log (3)) \log (4) \log ^2(x)-\frac {\log (4) (-10 x+(3-2 x) \log (3)+3)^2}{8 (10+\log (9))}-\frac {3}{2} x (1+\log (3)) \log (4) \log (x)+\frac {3}{2} x (1+\log (3)) \log (4) \]
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Rule 12
Rule 2332
Rule 2333
Rubi steps \begin{align*} \text {integral}& = \frac {1}{4} \int \left ((3-10 x+(3-2 x) \log (3)) \log (4)+(-4-4 \log (3)) \log (4) \log (x)+(1+\log (3)) \log (4) \log ^2(x)\right ) \, dx \\ & = -\frac {(3-10 x+(3-2 x) \log (3))^2 \log (4)}{8 (10+\log (9))}+\frac {1}{4} ((1+\log (3)) \log (4)) \int \log ^2(x) \, dx-((1+\log (3)) \log (4)) \int \log (x) \, dx \\ & = x (1+\log (3)) \log (4)-\frac {(3-10 x+(3-2 x) \log (3))^2 \log (4)}{8 (10+\log (9))}-x (1+\log (3)) \log (4) \log (x)+\frac {1}{4} x (1+\log (3)) \log (4) \log ^2(x)-\frac {1}{2} ((1+\log (3)) \log (4)) \int \log (x) \, dx \\ & = \frac {3}{2} x (1+\log (3)) \log (4)-\frac {(3-10 x+(3-2 x) \log (3))^2 \log (4)}{8 (10+\log (9))}-\frac {3}{2} x (1+\log (3)) \log (4) \log (x)+\frac {1}{4} x (1+\log (3)) \log (4) \log ^2(x) \\ \end{align*}
Time = 0.04 (sec) , antiderivative size = 43, normalized size of antiderivative = 1.54 \[ \int \frac {1}{4} \left ((3-10 x+(3-2 x) \log (3)) \log (4)+(-4-4 \log (3)) \log (4) \log (x)+(1+\log (3)) \log (4) \log ^2(x)\right ) \, dx=\frac {1}{4} \log (4) \left (-\frac {1}{2} x^2 (10+\log (9))+x (9+\log (19683))-6 x (1+\log (3)) \log (x)+x (1+\log (3)) \log ^2(x)\right ) \]
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Leaf count of result is larger than twice the leaf count of optimal. \(61\) vs. \(2(27)=54\).
Time = 0.15 (sec) , antiderivative size = 62, normalized size of antiderivative = 2.21
method | result | size |
norman | \(\left (-\frac {\ln \left (2\right ) \ln \left (3\right )}{2}-\frac {5 \ln \left (2\right )}{2}\right ) x^{2}+\left (\frac {9 \ln \left (2\right ) \ln \left (3\right )}{2}+\frac {9 \ln \left (2\right )}{2}\right ) x +\left (-3 \ln \left (2\right ) \ln \left (3\right )-3 \ln \left (2\right )\right ) x \ln \left (x \right )+\left (\frac {\ln \left (2\right ) \ln \left (3\right )}{2}+\frac {\ln \left (2\right )}{2}\right ) x \ln \left (x \right )^{2}\) | \(62\) |
parts | \(-\frac {\ln \left (2\right ) \left (x^{2} \ln \left (3\right )-3 x \ln \left (3\right )+5 x^{2}-3 x \right )}{2}-2 \left (\ln \left (3\right )+1\right ) \ln \left (2\right ) \left (x \ln \left (x \right )-x \right )+\frac {\left (\ln \left (3\right )+1\right ) \ln \left (2\right ) \left (x \ln \left (x \right )^{2}-2 x \ln \left (x \right )+2 x \right )}{2}\) | \(65\) |
risch | \(\frac {\ln \left (2\right ) x \ln \left (x \right )^{2} \ln \left (3\right )}{2}+\frac {\ln \left (2\right ) x \ln \left (x \right )^{2}}{2}-3 \ln \left (2\right ) x \ln \left (x \right ) \ln \left (3\right )-3 x \ln \left (2\right ) \ln \left (x \right )+\frac {9 x \ln \left (2\right ) \ln \left (3\right )}{2}+\frac {9 x \ln \left (2\right )}{2}-\frac {\ln \left (2\right ) \ln \left (3\right ) x^{2}}{2}-\frac {5 x^{2} \ln \left (2\right )}{2}\) | \(66\) |
parallelrisch | \(\frac {\ln \left (2\right ) x \ln \left (x \right )^{2} \ln \left (3\right )}{2}+\frac {\ln \left (2\right ) x \ln \left (x \right )^{2}}{2}-3 \ln \left (2\right ) x \ln \left (x \right ) \ln \left (3\right )-3 x \ln \left (2\right ) \ln \left (x \right )+\frac {9 x \ln \left (2\right ) \ln \left (3\right )}{2}+\frac {9 x \ln \left (2\right )}{2}-\frac {\ln \left (2\right ) \ln \left (3\right ) x^{2}}{2}-\frac {5 x^{2} \ln \left (2\right )}{2}\) | \(66\) |
default | \(\frac {\ln \left (2\right ) \left (-x^{2} \ln \left (3\right )+3 x \ln \left (3\right )-5 x^{2}+3 x \right )}{2}+2 \left (-\ln \left (3\right )-1\right ) \ln \left (2\right ) \left (x \ln \left (x \right )-x \right )+\frac {\left (\ln \left (3\right )+1\right ) \ln \left (2\right ) \left (x \ln \left (x \right )^{2}-2 x \ln \left (x \right )+2 x \right )}{2}\) | \(68\) |
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Leaf count of result is larger than twice the leaf count of optimal. 59 vs. \(2 (24) = 48\).
Time = 0.24 (sec) , antiderivative size = 59, normalized size of antiderivative = 2.11 \[ \int \frac {1}{4} \left ((3-10 x+(3-2 x) \log (3)) \log (4)+(-4-4 \log (3)) \log (4) \log (x)+(1+\log (3)) \log (4) \log ^2(x)\right ) \, dx=-\frac {1}{2} \, {\left (x^{2} - 9 \, x\right )} \log \left (3\right ) \log \left (2\right ) + \frac {1}{2} \, {\left (x \log \left (3\right ) \log \left (2\right ) + x \log \left (2\right )\right )} \log \left (x\right )^{2} - \frac {1}{2} \, {\left (5 \, x^{2} - 9 \, x\right )} \log \left (2\right ) - 3 \, {\left (x \log \left (3\right ) \log \left (2\right ) + x \log \left (2\right )\right )} \log \left (x\right ) \]
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Leaf count of result is larger than twice the leaf count of optimal. 82 vs. \(2 (24) = 48\).
Time = 0.10 (sec) , antiderivative size = 82, normalized size of antiderivative = 2.93 \[ \int \frac {1}{4} \left ((3-10 x+(3-2 x) \log (3)) \log (4)+(-4-4 \log (3)) \log (4) \log (x)+(1+\log (3)) \log (4) \log ^2(x)\right ) \, dx=x^{2} \left (- \frac {5 \log {\left (2 \right )}}{2} - \frac {\log {\left (2 \right )} \log {\left (3 \right )}}{2}\right ) + x \left (\frac {9 \log {\left (2 \right )}}{2} + \frac {9 \log {\left (2 \right )} \log {\left (3 \right )}}{2}\right ) + \left (\frac {x \log {\left (2 \right )}}{2} + \frac {x \log {\left (2 \right )} \log {\left (3 \right )}}{2}\right ) \log {\left (x \right )}^{2} + \left (- 3 x \log {\left (2 \right )} \log {\left (3 \right )} - 3 x \log {\left (2 \right )}\right ) \log {\left (x \right )} \]
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Leaf count of result is larger than twice the leaf count of optimal. 59 vs. \(2 (24) = 48\).
Time = 0.21 (sec) , antiderivative size = 59, normalized size of antiderivative = 2.11 \[ \int \frac {1}{4} \left ((3-10 x+(3-2 x) \log (3)) \log (4)+(-4-4 \log (3)) \log (4) \log (x)+(1+\log (3)) \log (4) \log ^2(x)\right ) \, dx=\frac {1}{2} \, {\left (\log \left (x\right )^{2} - 2 \, \log \left (x\right ) + 2\right )} x {\left (\log \left (3\right ) + 1\right )} \log \left (2\right ) - 2 \, {\left (x \log \left (x\right ) - x\right )} {\left (\log \left (3\right ) + 1\right )} \log \left (2\right ) - \frac {1}{2} \, {\left (5 \, x^{2} + {\left (x^{2} - 3 \, x\right )} \log \left (3\right ) - 3 \, x\right )} \log \left (2\right ) \]
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Leaf count of result is larger than twice the leaf count of optimal. 63 vs. \(2 (24) = 48\).
Time = 0.26 (sec) , antiderivative size = 63, normalized size of antiderivative = 2.25 \[ \int \frac {1}{4} \left ((3-10 x+(3-2 x) \log (3)) \log (4)+(-4-4 \log (3)) \log (4) \log (x)+(1+\log (3)) \log (4) \log ^2(x)\right ) \, dx=\frac {1}{2} \, {\left (x \log \left (x\right )^{2} - 2 \, x \log \left (x\right ) + 2 \, x\right )} {\left (\log \left (3\right ) + 1\right )} \log \left (2\right ) - 2 \, {\left (x \log \left (x\right ) - x\right )} {\left (\log \left (3\right ) + 1\right )} \log \left (2\right ) - \frac {1}{2} \, {\left (5 \, x^{2} + {\left (x^{2} - 3 \, x\right )} \log \left (3\right ) - 3 \, x\right )} \log \left (2\right ) \]
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Time = 13.73 (sec) , antiderivative size = 55, normalized size of antiderivative = 1.96 \[ \int \frac {1}{4} \left ((3-10 x+(3-2 x) \log (3)) \log (4)+(-4-4 \log (3)) \log (4) \log (x)+(1+\log (3)) \log (4) \log ^2(x)\right ) \, dx=x\,\left (\frac {9\,\ln \left (2\right )\,\left (\ln \left (3\right )+1\right )}{2}-\frac {\ln \left (x\right )\,\left (6\,\ln \left (2\right )+6\,\ln \left (2\right )\,\ln \left (3\right )\right )}{2}+\frac {\ln \left (2\right )\,{\ln \left (x\right )}^2\,\left (\ln \left (3\right )+1\right )}{2}\right )-x^2\,\left (\frac {\ln \left (32\right )}{2}+\frac {\ln \left (2\right )\,\ln \left (3\right )}{2}\right ) \]
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