Integrand size = 141, antiderivative size = 28 \[ \int \frac {-32 x^4 \log \left (\frac {1}{3} x \log (3)\right )-32 x^4 \log ^2\left (\frac {1}{3} x \log (3)\right )+\left (-1792 x^3+256 e^2 x^3\right ) \log ^3\left (\frac {1}{3} x \log (3)\right )}{-x^3+\left (-84 x^2+12 e^2 x^2\right ) \log \left (\frac {1}{3} x \log (3)\right )+\left (-2352 x+672 e^2 x-48 e^4 x\right ) \log ^2\left (\frac {1}{3} x \log (3)\right )+\left (-21952+9408 e^2-1344 e^4+64 e^6\right ) \log ^3\left (\frac {1}{3} x \log (3)\right )} \, dx=\frac {x^4}{\left (7-e^2+\frac {x}{4 \log \left (\frac {1}{3} x \log (3)\right )}\right )^2} \]
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\[ \int \frac {-32 x^4 \log \left (\frac {1}{3} x \log (3)\right )-32 x^4 \log ^2\left (\frac {1}{3} x \log (3)\right )+\left (-1792 x^3+256 e^2 x^3\right ) \log ^3\left (\frac {1}{3} x \log (3)\right )}{-x^3+\left (-84 x^2+12 e^2 x^2\right ) \log \left (\frac {1}{3} x \log (3)\right )+\left (-2352 x+672 e^2 x-48 e^4 x\right ) \log ^2\left (\frac {1}{3} x \log (3)\right )+\left (-21952+9408 e^2-1344 e^4+64 e^6\right ) \log ^3\left (\frac {1}{3} x \log (3)\right )} \, dx=\int \frac {-32 x^4 \log \left (\frac {1}{3} x \log (3)\right )-32 x^4 \log ^2\left (\frac {1}{3} x \log (3)\right )+\left (-1792 x^3+256 e^2 x^3\right ) \log ^3\left (\frac {1}{3} x \log (3)\right )}{-x^3+\left (-84 x^2+12 e^2 x^2\right ) \log \left (\frac {1}{3} x \log (3)\right )+\left (-2352 x+672 e^2 x-48 e^4 x\right ) \log ^2\left (\frac {1}{3} x \log (3)\right )+\left (-21952+9408 e^2-1344 e^4+64 e^6\right ) \log ^3\left (\frac {1}{3} x \log (3)\right )} \, dx \]
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Rubi steps \begin{align*} \text {integral}& = \int \frac {32 x^3 \log \left (\frac {1}{3} x \log (3)\right ) \left (x+x \log \left (\frac {1}{3} x \log (3)\right )-8 \left (-7+e^2\right ) \log ^2\left (\frac {1}{3} x \log (3)\right )\right )}{\left (x-4 \left (-7+e^2\right ) \log \left (\frac {1}{3} x \log (3)\right )\right )^3} \, dx \\ & = 32 \int \frac {x^3 \log \left (\frac {1}{3} x \log (3)\right ) \left (x+x \log \left (\frac {1}{3} x \log (3)\right )-8 \left (-7+e^2\right ) \log ^2\left (\frac {1}{3} x \log (3)\right )\right )}{\left (x-4 \left (-7+e^2\right ) \log \left (\frac {1}{3} x \log (3)\right )\right )^3} \, dx \\ & = 32 \int \left (\frac {x^3}{8 \left (-7+e^2\right )^2}+\frac {5 x^4}{16 \left (7-e^2\right )^2 \left (-x-28 \left (1-\frac {e^2}{7}\right ) \log \left (\frac {1}{3} x \log (3)\right )\right )}+\frac {\left (-28+4 e^2-x\right ) x^5}{16 \left (7-e^2\right )^2 \left (x+28 \left (1-\frac {e^2}{7}\right ) \log \left (\frac {1}{3} x \log (3)\right )\right )^3}+\frac {x^4 \left (7-e^2+x\right )}{4 \left (7-e^2\right )^2 \left (x+28 \left (1-\frac {e^2}{7}\right ) \log \left (\frac {1}{3} x \log (3)\right )\right )^2}\right ) \, dx \\ & = \frac {x^4}{\left (7-e^2\right )^2}+\frac {2 \int \frac {\left (-28+4 e^2-x\right ) x^5}{\left (x+28 \left (1-\frac {e^2}{7}\right ) \log \left (\frac {1}{3} x \log (3)\right )\right )^3} \, dx}{\left (7-e^2\right )^2}+\frac {8 \int \frac {x^4 \left (7-e^2+x\right )}{\left (x+28 \left (1-\frac {e^2}{7}\right ) \log \left (\frac {1}{3} x \log (3)\right )\right )^2} \, dx}{\left (7-e^2\right )^2}+\frac {10 \int \frac {x^4}{-x-28 \left (1-\frac {e^2}{7}\right ) \log \left (\frac {1}{3} x \log (3)\right )} \, dx}{\left (7-e^2\right )^2} \\ & = \frac {x^4}{\left (7-e^2\right )^2}+\frac {2 \int \left (\frac {x^6}{\left (-x-28 \left (1-\frac {e^2}{7}\right ) \log \left (\frac {1}{3} x \log (3)\right )\right )^3}+\frac {4 \left (-7+e^2\right ) x^5}{\left (x+28 \left (1-\frac {e^2}{7}\right ) \log \left (\frac {1}{3} x \log (3)\right )\right )^3}\right ) \, dx}{\left (7-e^2\right )^2}+\frac {8 \int \left (\frac {\left (7-e^2\right ) x^4}{\left (x+28 \left (1-\frac {e^2}{7}\right ) \log \left (\frac {1}{3} x \log (3)\right )\right )^2}+\frac {x^5}{\left (x+28 \left (1-\frac {e^2}{7}\right ) \log \left (\frac {1}{3} x \log (3)\right )\right )^2}\right ) \, dx}{\left (7-e^2\right )^2}+\frac {10 \int \frac {x^4}{-x-28 \left (1-\frac {e^2}{7}\right ) \log \left (\frac {1}{3} x \log (3)\right )} \, dx}{\left (7-e^2\right )^2} \\ & = \frac {x^4}{\left (7-e^2\right )^2}+\frac {2 \int \frac {x^6}{\left (-x-28 \left (1-\frac {e^2}{7}\right ) \log \left (\frac {1}{3} x \log (3)\right )\right )^3} \, dx}{\left (7-e^2\right )^2}+\frac {8 \int \frac {x^5}{\left (x+28 \left (1-\frac {e^2}{7}\right ) \log \left (\frac {1}{3} x \log (3)\right )\right )^2} \, dx}{\left (7-e^2\right )^2}+\frac {10 \int \frac {x^4}{-x-28 \left (1-\frac {e^2}{7}\right ) \log \left (\frac {1}{3} x \log (3)\right )} \, dx}{\left (7-e^2\right )^2}-\frac {8 \int \frac {x^5}{\left (x+28 \left (1-\frac {e^2}{7}\right ) \log \left (\frac {1}{3} x \log (3)\right )\right )^3} \, dx}{7-e^2}+\frac {8 \int \frac {x^4}{\left (x+28 \left (1-\frac {e^2}{7}\right ) \log \left (\frac {1}{3} x \log (3)\right )\right )^2} \, dx}{7-e^2} \\ \end{align*}
Time = 7.43 (sec) , antiderivative size = 34, normalized size of antiderivative = 1.21 \[ \int \frac {-32 x^4 \log \left (\frac {1}{3} x \log (3)\right )-32 x^4 \log ^2\left (\frac {1}{3} x \log (3)\right )+\left (-1792 x^3+256 e^2 x^3\right ) \log ^3\left (\frac {1}{3} x \log (3)\right )}{-x^3+\left (-84 x^2+12 e^2 x^2\right ) \log \left (\frac {1}{3} x \log (3)\right )+\left (-2352 x+672 e^2 x-48 e^4 x\right ) \log ^2\left (\frac {1}{3} x \log (3)\right )+\left (-21952+9408 e^2-1344 e^4+64 e^6\right ) \log ^3\left (\frac {1}{3} x \log (3)\right )} \, dx=\frac {16 x^4 \log ^2\left (\frac {1}{3} x \log (3)\right )}{\left (x-4 \left (-7+e^2\right ) \log \left (\frac {1}{3} x \log (3)\right )\right )^2} \]
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Time = 1.47 (sec) , antiderivative size = 38, normalized size of antiderivative = 1.36
method | result | size |
norman | \(\frac {16 x^{4} \ln \left (\frac {x \ln \left (3\right )}{3}\right )^{2}}{\left (4 \ln \left (\frac {x \ln \left (3\right )}{3}\right ) {\mathrm e}^{2}-28 \ln \left (\frac {x \ln \left (3\right )}{3}\right )-x \right )^{2}}\) | \(38\) |
derivativedivides | \(\frac {16 \ln \left (3\right )^{2} \ln \left (\frac {x \ln \left (3\right )}{3}\right )^{2} x^{4}}{\left (4 \ln \left (\frac {x \ln \left (3\right )}{3}\right ) \ln \left (3\right ) {\mathrm e}^{2}-28 \ln \left (\frac {x \ln \left (3\right )}{3}\right ) \ln \left (3\right )-x \ln \left (3\right )\right )^{2}}\) | \(48\) |
default | \(\frac {16 \ln \left (3\right )^{2} \ln \left (\frac {x \ln \left (3\right )}{3}\right )^{2} x^{4}}{\left (4 \ln \left (\frac {x \ln \left (3\right )}{3}\right ) \ln \left (3\right ) {\mathrm e}^{2}-28 \ln \left (\frac {x \ln \left (3\right )}{3}\right ) \ln \left (3\right )-x \ln \left (3\right )\right )^{2}}\) | \(48\) |
risch | \(\frac {x^{4}}{-14 \,{\mathrm e}^{2}+{\mathrm e}^{4}+49}+\frac {x^{5} \left (8 \ln \left (\frac {x \ln \left (3\right )}{3}\right ) {\mathrm e}^{2}-x -56 \ln \left (\frac {x \ln \left (3\right )}{3}\right )\right )}{\left (4 \ln \left (\frac {x \ln \left (3\right )}{3}\right ) {\mathrm e}^{2}-28 \ln \left (\frac {x \ln \left (3\right )}{3}\right )-x \right )^{2} \left ({\mathrm e}^{2}-7\right )^{2}}\) | \(72\) |
parallelrisch | \(\frac {16 x^{4} \ln \left (\frac {x \ln \left (3\right )}{3}\right )^{2}}{16 \ln \left (\frac {x \ln \left (3\right )}{3}\right )^{2} {\mathrm e}^{4}-8 \ln \left (\frac {x \ln \left (3\right )}{3}\right ) {\mathrm e}^{2} x -224 \ln \left (\frac {x \ln \left (3\right )}{3}\right )^{2} {\mathrm e}^{2}+x^{2}+56 x \ln \left (\frac {x \ln \left (3\right )}{3}\right )+784 \ln \left (\frac {x \ln \left (3\right )}{3}\right )^{2}}\) | \(76\) |
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Leaf count of result is larger than twice the leaf count of optimal. 53 vs. \(2 (23) = 46\).
Time = 0.25 (sec) , antiderivative size = 53, normalized size of antiderivative = 1.89 \[ \int \frac {-32 x^4 \log \left (\frac {1}{3} x \log (3)\right )-32 x^4 \log ^2\left (\frac {1}{3} x \log (3)\right )+\left (-1792 x^3+256 e^2 x^3\right ) \log ^3\left (\frac {1}{3} x \log (3)\right )}{-x^3+\left (-84 x^2+12 e^2 x^2\right ) \log \left (\frac {1}{3} x \log (3)\right )+\left (-2352 x+672 e^2 x-48 e^4 x\right ) \log ^2\left (\frac {1}{3} x \log (3)\right )+\left (-21952+9408 e^2-1344 e^4+64 e^6\right ) \log ^3\left (\frac {1}{3} x \log (3)\right )} \, dx=\frac {16 \, x^{4} \log \left (\frac {1}{3} \, x \log \left (3\right )\right )^{2}}{16 \, {\left (e^{4} - 14 \, e^{2} + 49\right )} \log \left (\frac {1}{3} \, x \log \left (3\right )\right )^{2} + x^{2} - 8 \, {\left (x e^{2} - 7 \, x\right )} \log \left (\frac {1}{3} \, x \log \left (3\right )\right )} \]
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Leaf count of result is larger than twice the leaf count of optimal. 122 vs. \(2 (20) = 40\).
Time = 0.21 (sec) , antiderivative size = 122, normalized size of antiderivative = 4.36 \[ \int \frac {-32 x^4 \log \left (\frac {1}{3} x \log (3)\right )-32 x^4 \log ^2\left (\frac {1}{3} x \log (3)\right )+\left (-1792 x^3+256 e^2 x^3\right ) \log ^3\left (\frac {1}{3} x \log (3)\right )}{-x^3+\left (-84 x^2+12 e^2 x^2\right ) \log \left (\frac {1}{3} x \log (3)\right )+\left (-2352 x+672 e^2 x-48 e^4 x\right ) \log ^2\left (\frac {1}{3} x \log (3)\right )+\left (-21952+9408 e^2-1344 e^4+64 e^6\right ) \log ^3\left (\frac {1}{3} x \log (3)\right )} \, dx=\frac {x^{4}}{- 14 e^{2} + 49 + e^{4}} + \frac {- x^{6} + \left (- 56 x^{5} + 8 x^{5} e^{2}\right ) \log {\left (\frac {x \log {\left (3 \right )}}{3} \right )}}{- 14 x^{2} e^{2} + 49 x^{2} + x^{2} e^{4} + \left (- 1176 x e^{2} - 8 x e^{6} + 2744 x + 168 x e^{4}\right ) \log {\left (\frac {x \log {\left (3 \right )}}{3} \right )} + \left (- 448 e^{6} - 21952 e^{2} + 38416 + 16 e^{8} + 4704 e^{4}\right ) \log {\left (\frac {x \log {\left (3 \right )}}{3} \right )}^{2}} \]
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Leaf count of result is larger than twice the leaf count of optimal. 199 vs. \(2 (23) = 46\).
Time = 0.40 (sec) , antiderivative size = 199, normalized size of antiderivative = 7.11 \[ \int \frac {-32 x^4 \log \left (\frac {1}{3} x \log (3)\right )-32 x^4 \log ^2\left (\frac {1}{3} x \log (3)\right )+\left (-1792 x^3+256 e^2 x^3\right ) \log ^3\left (\frac {1}{3} x \log (3)\right )}{-x^3+\left (-84 x^2+12 e^2 x^2\right ) \log \left (\frac {1}{3} x \log (3)\right )+\left (-2352 x+672 e^2 x-48 e^4 x\right ) \log ^2\left (\frac {1}{3} x \log (3)\right )+\left (-21952+9408 e^2-1344 e^4+64 e^6\right ) \log ^3\left (\frac {1}{3} x \log (3)\right )} \, dx=-\frac {16 \, {\left (2 \, x^{4} {\left (\log \left (3\right ) - \log \left (\log \left (3\right )\right )\right )} \log \left (x\right ) - x^{4} \log \left (x\right )^{2} - {\left (\log \left (3\right )^{2} - 2 \, \log \left (3\right ) \log \left (\log \left (3\right )\right ) + \log \left (\log \left (3\right )\right )^{2}\right )} x^{4}\right )}}{16 \, {\left (e^{4} - 14 \, e^{2} + 49\right )} \log \left (x\right )^{2} + 8 \, {\left ({\left (\log \left (3\right ) - \log \left (\log \left (3\right )\right )\right )} e^{2} - 7 \, \log \left (3\right ) + 7 \, \log \left (\log \left (3\right )\right )\right )} x + x^{2} + 16 \, {\left (\log \left (3\right )^{2} - 2 \, \log \left (3\right ) \log \left (\log \left (3\right )\right ) + \log \left (\log \left (3\right )\right )^{2}\right )} e^{4} - 224 \, {\left (\log \left (3\right )^{2} - 2 \, \log \left (3\right ) \log \left (\log \left (3\right )\right ) + \log \left (\log \left (3\right )\right )^{2}\right )} e^{2} + 784 \, \log \left (3\right )^{2} - 8 \, {\left (x {\left (e^{2} - 7\right )} + 4 \, {\left (\log \left (3\right ) - \log \left (\log \left (3\right )\right )\right )} e^{4} - 56 \, {\left (\log \left (3\right ) - \log \left (\log \left (3\right )\right )\right )} e^{2} + 196 \, \log \left (3\right ) - 196 \, \log \left (\log \left (3\right )\right )\right )} \log \left (x\right ) - 1568 \, \log \left (3\right ) \log \left (\log \left (3\right )\right ) + 784 \, \log \left (\log \left (3\right )\right )^{2}} \]
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Leaf count of result is larger than twice the leaf count of optimal. 154 vs. \(2 (23) = 46\).
Time = 0.41 (sec) , antiderivative size = 154, normalized size of antiderivative = 5.50 \[ \int \frac {-32 x^4 \log \left (\frac {1}{3} x \log (3)\right )-32 x^4 \log ^2\left (\frac {1}{3} x \log (3)\right )+\left (-1792 x^3+256 e^2 x^3\right ) \log ^3\left (\frac {1}{3} x \log (3)\right )}{-x^3+\left (-84 x^2+12 e^2 x^2\right ) \log \left (\frac {1}{3} x \log (3)\right )+\left (-2352 x+672 e^2 x-48 e^4 x\right ) \log ^2\left (\frac {1}{3} x \log (3)\right )+\left (-21952+9408 e^2-1344 e^4+64 e^6\right ) \log ^3\left (\frac {1}{3} x \log (3)\right )} \, dx=\frac {16 \, {\left (x^{4} \log \left (3\right )^{2} - 2 \, x^{4} \log \left (3\right ) \log \left (x \log \left (3\right )\right ) + x^{4} \log \left (x \log \left (3\right )\right )^{2}\right )}}{8 \, x e^{2} \log \left (3\right ) + 16 \, e^{4} \log \left (3\right )^{2} - 224 \, e^{2} \log \left (3\right )^{2} - 8 \, x e^{2} \log \left (x \log \left (3\right )\right ) - 32 \, e^{4} \log \left (3\right ) \log \left (x \log \left (3\right )\right ) + 448 \, e^{2} \log \left (3\right ) \log \left (x \log \left (3\right )\right ) + 16 \, e^{4} \log \left (x \log \left (3\right )\right )^{2} - 224 \, e^{2} \log \left (x \log \left (3\right )\right )^{2} + x^{2} - 56 \, x \log \left (3\right ) + 784 \, \log \left (3\right )^{2} + 56 \, x \log \left (x \log \left (3\right )\right ) - 1568 \, \log \left (3\right ) \log \left (x \log \left (3\right )\right ) + 784 \, \log \left (x \log \left (3\right )\right )^{2}} \]
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Timed out. \[ \int \frac {-32 x^4 \log \left (\frac {1}{3} x \log (3)\right )-32 x^4 \log ^2\left (\frac {1}{3} x \log (3)\right )+\left (-1792 x^3+256 e^2 x^3\right ) \log ^3\left (\frac {1}{3} x \log (3)\right )}{-x^3+\left (-84 x^2+12 e^2 x^2\right ) \log \left (\frac {1}{3} x \log (3)\right )+\left (-2352 x+672 e^2 x-48 e^4 x\right ) \log ^2\left (\frac {1}{3} x \log (3)\right )+\left (-21952+9408 e^2-1344 e^4+64 e^6\right ) \log ^3\left (\frac {1}{3} x \log (3)\right )} \, dx=-\int \frac {32\,x^4\,{\ln \left (\frac {x\,\ln \left (3\right )}{3}\right )}^2-{\ln \left (\frac {x\,\ln \left (3\right )}{3}\right )}^3\,\left (256\,x^3\,{\mathrm {e}}^2-1792\,x^3\right )+32\,x^4\,\ln \left (\frac {x\,\ln \left (3\right )}{3}\right )}{{\ln \left (\frac {x\,\ln \left (3\right )}{3}\right )}^3\,\left (9408\,{\mathrm {e}}^2-1344\,{\mathrm {e}}^4+64\,{\mathrm {e}}^6-21952\right )-{\ln \left (\frac {x\,\ln \left (3\right )}{3}\right )}^2\,\left (2352\,x-672\,x\,{\mathrm {e}}^2+48\,x\,{\mathrm {e}}^4\right )+\ln \left (\frac {x\,\ln \left (3\right )}{3}\right )\,\left (12\,x^2\,{\mathrm {e}}^2-84\,x^2\right )-x^3} \,d x \]
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