Integrand size = 47, antiderivative size = 21 \[ \int \frac {\left (-1250-500 x+250 x^2\right ) \log ^2(2)+(10+127 x) \log ^2(2) \log (x)+(3+x) \log ^2(2) \log ^2(x)}{125 x} \, dx=\log ^2(2) (5+x+\log (x)) \left (-10+x+\frac {\log ^2(x)}{125}\right ) \]
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Leaf count is larger than twice the leaf count of optimal. \(69\) vs. \(2(21)=42\).
Time = 0.07 (sec) , antiderivative size = 69, normalized size of antiderivative = 3.29, number of steps used = 13, number of rules used = 8, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.170, Rules used = {12, 14, 2388, 2338, 2332, 2339, 30, 2333} \[ \int \frac {\left (-1250-500 x+250 x^2\right ) \log ^2(2)+(10+127 x) \log ^2(2) \log (x)+(3+x) \log ^2(2) \log ^2(x)}{125 x} \, dx=x^2 \log ^2(2)+\frac {1}{125} x \log ^2(2) \log ^2(x)+\frac {1}{25} \log ^2(2) \log ^2(x)+x \log ^2(2) \log (x)-10 \log ^2(2) \log (x)-5 x \log ^2(2)+\frac {1}{125} \log ^2(2) \log ^3(x) \]
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Rule 12
Rule 14
Rule 30
Rule 2332
Rule 2333
Rule 2338
Rule 2339
Rule 2388
Rubi steps \begin{align*} \text {integral}& = \frac {1}{125} \int \frac {\left (-1250-500 x+250 x^2\right ) \log ^2(2)+(10+127 x) \log ^2(2) \log (x)+(3+x) \log ^2(2) \log ^2(x)}{x} \, dx \\ & = \frac {1}{125} \int \left (\frac {250 \left (-5-2 x+x^2\right ) \log ^2(2)}{x}+\frac {(10+127 x) \log ^2(2) \log (x)}{x}+\frac {(3+x) \log ^2(2) \log ^2(x)}{x}\right ) \, dx \\ & = \frac {1}{125} \log ^2(2) \int \frac {(10+127 x) \log (x)}{x} \, dx+\frac {1}{125} \log ^2(2) \int \frac {(3+x) \log ^2(x)}{x} \, dx+\left (2 \log ^2(2)\right ) \int \frac {-5-2 x+x^2}{x} \, dx \\ & = \frac {1}{125} \log ^2(2) \int \log ^2(x) \, dx+\frac {1}{125} \left (3 \log ^2(2)\right ) \int \frac {\log ^2(x)}{x} \, dx+\frac {1}{25} \left (2 \log ^2(2)\right ) \int \frac {\log (x)}{x} \, dx+\frac {1}{125} \left (127 \log ^2(2)\right ) \int \log (x) \, dx+\left (2 \log ^2(2)\right ) \int \left (-2-\frac {5}{x}+x\right ) \, dx \\ & = -\frac {627}{125} x \log ^2(2)+x^2 \log ^2(2)-10 \log ^2(2) \log (x)+\frac {127}{125} x \log ^2(2) \log (x)+\frac {1}{25} \log ^2(2) \log ^2(x)+\frac {1}{125} x \log ^2(2) \log ^2(x)-\frac {1}{125} \left (2 \log ^2(2)\right ) \int \log (x) \, dx+\frac {1}{125} \left (3 \log ^2(2)\right ) \text {Subst}\left (\int x^2 \, dx,x,\log (x)\right ) \\ & = -5 x \log ^2(2)+x^2 \log ^2(2)-10 \log ^2(2) \log (x)+x \log ^2(2) \log (x)+\frac {1}{25} \log ^2(2) \log ^2(x)+\frac {1}{125} x \log ^2(2) \log ^2(x)+\frac {1}{125} \log ^2(2) \log ^3(x) \\ \end{align*}
Time = 0.03 (sec) , antiderivative size = 42, normalized size of antiderivative = 2.00 \[ \int \frac {\left (-1250-500 x+250 x^2\right ) \log ^2(2)+(10+127 x) \log ^2(2) \log (x)+(3+x) \log ^2(2) \log ^2(x)}{125 x} \, dx=\frac {1}{125} \log ^2(2) \left (-625 x+125 x^2-1250 \log (x)+125 x \log (x)+5 \log ^2(x)+x \log ^2(x)+\log ^3(x)\right ) \]
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Leaf count of result is larger than twice the leaf count of optimal. \(61\) vs. \(2(19)=38\).
Time = 0.12 (sec) , antiderivative size = 62, normalized size of antiderivative = 2.95
method | result | size |
risch | \(\frac {\ln \left (2\right )^{2} \ln \left (x \right )^{3}}{125}+\frac {\left (x \ln \left (2\right )^{2}+5 \ln \left (2\right )^{2}\right ) \ln \left (x \right )^{2}}{125}+\ln \left (x \right ) \ln \left (2\right )^{2} x +x^{2} \ln \left (2\right )^{2}-10 \ln \left (x \right ) \ln \left (2\right )^{2}-5 x \ln \left (2\right )^{2}\) | \(62\) |
norman | \(x^{2} \ln \left (2\right )^{2}-10 \ln \left (x \right ) \ln \left (2\right )^{2}+\ln \left (x \right ) \ln \left (2\right )^{2} x -5 x \ln \left (2\right )^{2}+\frac {\ln \left (2\right )^{2} \ln \left (x \right )^{2}}{25}+\frac {\ln \left (2\right )^{2} \ln \left (x \right )^{3}}{125}+\frac {x \ln \left (2\right )^{2} \ln \left (x \right )^{2}}{125}\) | \(64\) |
parallelrisch | \(x^{2} \ln \left (2\right )^{2}-10 \ln \left (x \right ) \ln \left (2\right )^{2}+\ln \left (x \right ) \ln \left (2\right )^{2} x -5 x \ln \left (2\right )^{2}+\frac {\ln \left (2\right )^{2} \ln \left (x \right )^{2}}{25}+\frac {\ln \left (2\right )^{2} \ln \left (x \right )^{3}}{125}+\frac {x \ln \left (2\right )^{2} \ln \left (x \right )^{2}}{125}\) | \(64\) |
parts | \(2 \ln \left (2\right )^{2} \left (-2 x +\frac {x^{2}}{2}-5 \ln \left (x \right )\right )+\frac {\ln \left (2\right )^{2} \left (127 x \ln \left (x \right )-127 x +5 \ln \left (x \right )^{2}\right )}{125}+\frac {\ln \left (2\right )^{2} \left (x \ln \left (x \right )^{2}-2 x \ln \left (x \right )+2 x +\ln \left (x \right )^{3}\right )}{125}\) | \(67\) |
default | \(\frac {\ln \left (2\right )^{2} \left (x \ln \left (x \right )^{2}-2 x \ln \left (x \right )+2 x \right )}{125}+\frac {\ln \left (2\right )^{2} \ln \left (x \right )^{3}}{125}+\frac {127 \ln \left (2\right )^{2} \left (x \ln \left (x \right )-x \right )}{125}+x^{2} \ln \left (2\right )^{2}+\frac {\ln \left (2\right )^{2} \ln \left (x \right )^{2}}{25}-4 x \ln \left (2\right )^{2}-10 \ln \left (x \right ) \ln \left (2\right )^{2}\) | \(80\) |
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Leaf count of result is larger than twice the leaf count of optimal. 46 vs. \(2 (20) = 40\).
Time = 0.25 (sec) , antiderivative size = 46, normalized size of antiderivative = 2.19 \[ \int \frac {\left (-1250-500 x+250 x^2\right ) \log ^2(2)+(10+127 x) \log ^2(2) \log (x)+(3+x) \log ^2(2) \log ^2(x)}{125 x} \, dx=\frac {1}{125} \, {\left (x + 5\right )} \log \left (2\right )^{2} \log \left (x\right )^{2} + \frac {1}{125} \, \log \left (2\right )^{2} \log \left (x\right )^{3} + {\left (x - 10\right )} \log \left (2\right )^{2} \log \left (x\right ) + {\left (x^{2} - 5 \, x\right )} \log \left (2\right )^{2} \]
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Leaf count of result is larger than twice the leaf count of optimal. 68 vs. \(2 (20) = 40\).
Time = 0.14 (sec) , antiderivative size = 68, normalized size of antiderivative = 3.24 \[ \int \frac {\left (-1250-500 x+250 x^2\right ) \log ^2(2)+(10+127 x) \log ^2(2) \log (x)+(3+x) \log ^2(2) \log ^2(x)}{125 x} \, dx=x^{2} \log {\left (2 \right )}^{2} + x \log {\left (2 \right )}^{2} \log {\left (x \right )} - 5 x \log {\left (2 \right )}^{2} + \left (\frac {x \log {\left (2 \right )}^{2}}{125} + \frac {\log {\left (2 \right )}^{2}}{25}\right ) \log {\left (x \right )}^{2} + \frac {\log {\left (2 \right )}^{2} \log {\left (x \right )}^{3}}{125} - 10 \log {\left (2 \right )}^{2} \log {\left (x \right )} \]
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Leaf count of result is larger than twice the leaf count of optimal. 75 vs. \(2 (20) = 40\).
Time = 0.18 (sec) , antiderivative size = 75, normalized size of antiderivative = 3.57 \[ \int \frac {\left (-1250-500 x+250 x^2\right ) \log ^2(2)+(10+127 x) \log ^2(2) \log (x)+(3+x) \log ^2(2) \log ^2(x)}{125 x} \, dx=\frac {1}{125} \, \log \left (2\right )^{2} \log \left (x\right )^{3} + \frac {1}{125} \, {\left (\log \left (x\right )^{2} - 2 \, \log \left (x\right ) + 2\right )} x \log \left (2\right )^{2} + x^{2} \log \left (2\right )^{2} + \frac {1}{25} \, \log \left (2\right )^{2} \log \left (x\right )^{2} + \frac {127}{125} \, {\left (x \log \left (x\right ) - x\right )} \log \left (2\right )^{2} - 4 \, x \log \left (2\right )^{2} - 10 \, \log \left (2\right )^{2} \log \left (x\right ) \]
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Leaf count of result is larger than twice the leaf count of optimal. 61 vs. \(2 (20) = 40\).
Time = 0.28 (sec) , antiderivative size = 61, normalized size of antiderivative = 2.90 \[ \int \frac {\left (-1250-500 x+250 x^2\right ) \log ^2(2)+(10+127 x) \log ^2(2) \log (x)+(3+x) \log ^2(2) \log ^2(x)}{125 x} \, dx=\frac {1}{125} \, \log \left (2\right )^{2} \log \left (x\right )^{3} + x^{2} \log \left (2\right )^{2} + x \log \left (2\right )^{2} \log \left (x\right ) - 5 \, x \log \left (2\right )^{2} - 10 \, \log \left (2\right )^{2} \log \left (x\right ) + \frac {1}{125} \, {\left (x \log \left (2\right )^{2} + 5 \, \log \left (2\right )^{2}\right )} \log \left (x\right )^{2} \]
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Time = 12.15 (sec) , antiderivative size = 40, normalized size of antiderivative = 1.90 \[ \int \frac {\left (-1250-500 x+250 x^2\right ) \log ^2(2)+(10+127 x) \log ^2(2) \log (x)+(3+x) \log ^2(2) \log ^2(x)}{125 x} \, dx=\frac {{\ln \left (2\right )}^2\,\left (125\,x^2+x\,{\ln \left (x\right )}^2+125\,x\,\ln \left (x\right )-625\,x+{\ln \left (x\right )}^3+5\,{\ln \left (x\right )}^2-1250\,\ln \left (x\right )\right )}{125} \]
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