Integrand size = 63, antiderivative size = 22 \[ \int \frac {4 x+12 x^2+\left (4 x+8 x^2\right ) \log (x)+3 x \log ^2(x)+(1+x) \log ^3(x)+\left (8 x+4 x \log (x)+3 \log ^2(x)\right ) \log (3 x)}{4 x} \, dx=\left (x+\log (x) \left (x+\frac {\log ^2(x)}{4}\right )\right ) (x+\log (3 x)) \]
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Leaf count is larger than twice the leaf count of optimal. \(63\) vs. \(2(22)=44\).
Time = 0.17 (sec) , antiderivative size = 63, normalized size of antiderivative = 2.86, number of steps used = 25, number of rules used = 11, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.175, Rules used = {12, 14, 2350, 2333, 2332, 2388, 2339, 30, 6874, 2408, 2413} \[ \int \frac {4 x+12 x^2+\left (4 x+8 x^2\right ) \log (x)+3 x \log ^2(x)+(1+x) \log ^3(x)+\left (8 x+4 x \log (x)+3 \log ^2(x)\right ) \log (3 x)}{4 x} \, dx=-\frac {x^2}{2}+x^2 \log (x)+\frac {1}{6} (3 x+1)^2-x+\frac {1}{4} x \log ^3(x)+\frac {1}{4} \log (3 x) \log ^3(x)+x \log (3 x) \log (x)+x \log (3 x) \]
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Rule 12
Rule 14
Rule 30
Rule 2332
Rule 2333
Rule 2339
Rule 2350
Rule 2388
Rule 2408
Rule 2413
Rule 6874
Rubi steps \begin{align*} \text {integral}& = \frac {1}{4} \int \frac {4 x+12 x^2+\left (4 x+8 x^2\right ) \log (x)+3 x \log ^2(x)+(1+x) \log ^3(x)+\left (8 x+4 x \log (x)+3 \log ^2(x)\right ) \log (3 x)}{x} \, dx \\ & = \frac {1}{4} \int \left (\frac {4 x+12 x^2+4 x \log (x)+8 x^2 \log (x)+3 x \log ^2(x)+\log ^3(x)+x \log ^3(x)}{x}+\frac {\left (8 x+4 x \log (x)+3 \log ^2(x)\right ) \log (3 x)}{x}\right ) \, dx \\ & = \frac {1}{4} \int \frac {4 x+12 x^2+4 x \log (x)+8 x^2 \log (x)+3 x \log ^2(x)+\log ^3(x)+x \log ^3(x)}{x} \, dx+\frac {1}{4} \int \frac {\left (8 x+4 x \log (x)+3 \log ^2(x)\right ) \log (3 x)}{x} \, dx \\ & = \frac {1}{4} \int \left (4 (1+3 x)+4 (1+2 x) \log (x)+3 \log ^2(x)+\frac {(1+x) \log ^3(x)}{x}\right ) \, dx+\frac {1}{4} \int \left (8 \log (3 x)+4 \log (x) \log (3 x)+\frac {3 \log ^2(x) \log (3 x)}{x}\right ) \, dx \\ & = \frac {1}{6} (1+3 x)^2+\frac {1}{4} \int \frac {(1+x) \log ^3(x)}{x} \, dx+\frac {3}{4} \int \log ^2(x) \, dx+\frac {3}{4} \int \frac {\log ^2(x) \log (3 x)}{x} \, dx+2 \int \log (3 x) \, dx+\int (1+2 x) \log (x) \, dx+\int \log (x) \log (3 x) \, dx \\ & = -2 x+\frac {1}{6} (1+3 x)^2+x \log (x)+x^2 \log (x)+\frac {3}{4} x \log ^2(x)+x \log (3 x)+x \log (x) \log (3 x)+\frac {1}{4} \log ^3(x) \log (3 x)+\frac {1}{4} \int \log ^3(x) \, dx+\frac {1}{4} \int \frac {\log ^3(x)}{x} \, dx-\frac {3}{4} \int \frac {\log ^3(x)}{3 x} \, dx-\frac {3}{2} \int \log (x) \, dx-\int (1+x) \, dx-\int (-1+\log (x)) \, dx \\ & = -\frac {x}{2}-\frac {x^2}{2}+\frac {1}{6} (1+3 x)^2-\frac {1}{2} x \log (x)+x^2 \log (x)+\frac {3}{4} x \log ^2(x)+\frac {1}{4} x \log ^3(x)+x \log (3 x)+x \log (x) \log (3 x)+\frac {1}{4} \log ^3(x) \log (3 x)-\frac {1}{4} \int \frac {\log ^3(x)}{x} \, dx+\frac {1}{4} \text {Subst}\left (\int x^3 \, dx,x,\log (x)\right )-\frac {3}{4} \int \log ^2(x) \, dx-\int \log (x) \, dx \\ & = \frac {x}{2}-\frac {x^2}{2}+\frac {1}{6} (1+3 x)^2-\frac {3}{2} x \log (x)+x^2 \log (x)+\frac {1}{4} x \log ^3(x)+\frac {\log ^4(x)}{16}+x \log (3 x)+x \log (x) \log (3 x)+\frac {1}{4} \log ^3(x) \log (3 x)-\frac {1}{4} \text {Subst}\left (\int x^3 \, dx,x,\log (x)\right )+\frac {3}{2} \int \log (x) \, dx \\ & = -x-\frac {x^2}{2}+\frac {1}{6} (1+3 x)^2+x^2 \log (x)+\frac {1}{4} x \log ^3(x)+x \log (3 x)+x \log (x) \log (3 x)+\frac {1}{4} \log ^3(x) \log (3 x) \\ \end{align*}
Leaf count is larger than twice the leaf count of optimal. \(70\) vs. \(2(22)=44\).
Time = 0.03 (sec) , antiderivative size = 70, normalized size of antiderivative = 3.18 \[ \int \frac {4 x+12 x^2+\left (4 x+8 x^2\right ) \log (x)+3 x \log ^2(x)+(1+x) \log ^3(x)+\left (8 x+4 x \log (x)+3 \log ^2(x)\right ) \log (3 x)}{4 x} \, dx=x^2-\frac {1}{4} x \log (81)+\frac {1}{4} x \log (6561)+x \log (x)+x^2 \log (x)+\frac {1}{4} x \log (81) \log (x)+x \log ^2(x)+\frac {1}{4} x \log ^3(x)+\frac {1}{12} \log (27) \log ^3(x)+\frac {\log ^4(x)}{4} \]
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Leaf count of result is larger than twice the leaf count of optimal. \(41\) vs. \(2(20)=40\).
Time = 0.13 (sec) , antiderivative size = 42, normalized size of antiderivative = 1.91
method | result | size |
parallelrisch | \(\frac {x \ln \left (x \right )^{3}}{4}+\frac {\ln \left (3 x \right ) \ln \left (x \right )^{3}}{4}+x^{2} \ln \left (x \right )+\ln \left (x \right ) \ln \left (3 x \right ) x +x^{2}+x \ln \left (3 x \right )\) | \(42\) |
risch | \(\frac {\ln \left (x \right )^{4}}{4}+\frac {\left (\ln \left (3\right )+x \right ) \ln \left (x \right )^{3}}{4}+x \ln \left (x \right )^{2}+\frac {x \left (2+2 \ln \left (3\right )+2 x \right ) \ln \left (x \right )}{2}+\frac {\left (2 \ln \left (3\right )+2 x \right ) x}{2}\) | \(49\) |
default | \(\frac {x \ln \left (x \right )^{3}}{4}+x \ln \left (x \right )^{2}+x \ln \left (x \right )+\frac {\ln \left (x \right )^{4}}{4}+\frac {\ln \left (3\right ) \ln \left (x \right )^{3}}{4}+\ln \left (3\right ) \left (x \ln \left (x \right )-x \right )+x^{2} \ln \left (x \right )+x^{2}+2 x \ln \left (3\right )\) | \(58\) |
parts | \(x \ln \left (x \right )^{2}-x \ln \left (x \right )+\frac {\ln \left (3\right ) \ln \left (x \right )^{3}}{4}+\frac {\ln \left (x \right )^{4}}{4}+\ln \left (3\right ) \left (x \ln \left (x \right )-x \right )+x^{2} \ln \left (x \right )+x^{2}+\frac {x \ln \left (x \right )^{3}}{4}+2 x \ln \left (3 x \right )\) | \(61\) |
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Time = 0.26 (sec) , antiderivative size = 42, normalized size of antiderivative = 1.91 \[ \int \frac {4 x+12 x^2+\left (4 x+8 x^2\right ) \log (x)+3 x \log ^2(x)+(1+x) \log ^3(x)+\left (8 x+4 x \log (x)+3 \log ^2(x)\right ) \log (3 x)}{4 x} \, dx=\frac {1}{4} \, {\left (x + \log \left (3\right )\right )} \log \left (x\right )^{3} + \frac {1}{4} \, \log \left (x\right )^{4} + x \log \left (x\right )^{2} + x^{2} + x \log \left (3\right ) + {\left (x^{2} + x \log \left (3\right ) + x\right )} \log \left (x\right ) \]
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Leaf count of result is larger than twice the leaf count of optimal. 48 vs. \(2 (19) = 38\).
Time = 0.10 (sec) , antiderivative size = 48, normalized size of antiderivative = 2.18 \[ \int \frac {4 x+12 x^2+\left (4 x+8 x^2\right ) \log (x)+3 x \log ^2(x)+(1+x) \log ^3(x)+\left (8 x+4 x \log (x)+3 \log ^2(x)\right ) \log (3 x)}{4 x} \, dx=x^{2} + x \log {\left (x \right )}^{2} + x \log {\left (3 \right )} + \left (\frac {x}{4} + \frac {\log {\left (3 \right )}}{4}\right ) \log {\left (x \right )}^{3} + \left (x^{2} + x + x \log {\left (3 \right )}\right ) \log {\left (x \right )} + \frac {\log {\left (x \right )}^{4}}{4} \]
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Leaf count of result is larger than twice the leaf count of optimal. 108 vs. \(2 (23) = 46\).
Time = 0.26 (sec) , antiderivative size = 108, normalized size of antiderivative = 4.91 \[ \int \frac {4 x+12 x^2+\left (4 x+8 x^2\right ) \log (x)+3 x \log ^2(x)+(1+x) \log ^3(x)+\left (8 x+4 x \log (x)+3 \log ^2(x)\right ) \log (3 x)}{4 x} \, dx=\frac {1}{16} \, \log \left (3 \, x\right )^{4} - \frac {1}{4} \, \log \left (3 \, x\right )^{3} \log \left (x\right ) + \frac {3}{8} \, \log \left (3 \, x\right )^{2} \log \left (x\right )^{2} + \frac {1}{16} \, \log \left (x\right )^{4} + x^{2} \log \left (x\right ) + \frac {1}{4} \, {\left (\log \left (x\right )^{3} - 3 \, \log \left (x\right )^{2} + 6 \, \log \left (x\right ) - 6\right )} x + \frac {3}{4} \, {\left (\log \left (x\right )^{2} - 2 \, \log \left (x\right ) + 2\right )} x + x^{2} - x {\left (\log \left (3\right ) - 2\right )} + 2 \, x \log \left (3 \, x\right ) + {\left (x \log \left (3 \, x\right ) - x\right )} \log \left (x\right ) - 2 \, x \]
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Time = 0.26 (sec) , antiderivative size = 43, normalized size of antiderivative = 1.95 \[ \int \frac {4 x+12 x^2+\left (4 x+8 x^2\right ) \log (x)+3 x \log ^2(x)+(1+x) \log ^3(x)+\left (8 x+4 x \log (x)+3 \log ^2(x)\right ) \log (3 x)}{4 x} \, dx=\frac {1}{4} \, {\left (x + \log \left (3\right )\right )} \log \left (x\right )^{3} + \frac {1}{4} \, \log \left (x\right )^{4} + x \log \left (x\right )^{2} + x^{2} + x \log \left (3\right ) + {\left (x^{2} + x {\left (\log \left (3\right ) + 1\right )}\right )} \log \left (x\right ) \]
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Time = 8.25 (sec) , antiderivative size = 21, normalized size of antiderivative = 0.95 \[ \int \frac {4 x+12 x^2+\left (4 x+8 x^2\right ) \log (x)+3 x \log ^2(x)+(1+x) \log ^3(x)+\left (8 x+4 x \log (x)+3 \log ^2(x)\right ) \log (3 x)}{4 x} \, dx=\frac {\left (x+\ln \left (3\right )+\ln \left (x\right )\right )\,\left ({\ln \left (x\right )}^3+4\,x\,\ln \left (x\right )+4\,x\right )}{4} \]
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