Integrand size = 141, antiderivative size = 29 \[ \int \frac {-9 x^5+3 x^6+\left (60 x^2-280 x^3+80 x^4\right ) \log \left (\frac {e^{4 x}}{-3 x+x^2}\right )+\left (60 x^2-20 x^3\right ) \log ^2\left (\frac {e^{4 x}}{-3 x+x^2}\right )+\left (36-168 x+48 x^2\right ) \log ^3\left (\frac {e^{4 x}}{-3 x+x^2}\right )+(36-12 x) \log ^4\left (\frac {e^{4 x}}{-3 x+x^2}\right )}{-9 x^5+3 x^6} \, dx=x+\left (\frac {5}{3}+\frac {\log ^2\left (\frac {e^{4 x}}{(-3+x) x}\right )}{x^2}\right )^2 \]
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\[ \int \frac {-9 x^5+3 x^6+\left (60 x^2-280 x^3+80 x^4\right ) \log \left (\frac {e^{4 x}}{-3 x+x^2}\right )+\left (60 x^2-20 x^3\right ) \log ^2\left (\frac {e^{4 x}}{-3 x+x^2}\right )+\left (36-168 x+48 x^2\right ) \log ^3\left (\frac {e^{4 x}}{-3 x+x^2}\right )+(36-12 x) \log ^4\left (\frac {e^{4 x}}{-3 x+x^2}\right )}{-9 x^5+3 x^6} \, dx=\int \frac {-9 x^5+3 x^6+\left (60 x^2-280 x^3+80 x^4\right ) \log \left (\frac {e^{4 x}}{-3 x+x^2}\right )+\left (60 x^2-20 x^3\right ) \log ^2\left (\frac {e^{4 x}}{-3 x+x^2}\right )+\left (36-168 x+48 x^2\right ) \log ^3\left (\frac {e^{4 x}}{-3 x+x^2}\right )+(36-12 x) \log ^4\left (\frac {e^{4 x}}{-3 x+x^2}\right )}{-9 x^5+3 x^6} \, dx \]
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Rubi steps \begin{align*} \text {integral}& = \int \frac {-9 x^5+3 x^6+\left (60 x^2-280 x^3+80 x^4\right ) \log \left (\frac {e^{4 x}}{-3 x+x^2}\right )+\left (60 x^2-20 x^3\right ) \log ^2\left (\frac {e^{4 x}}{-3 x+x^2}\right )+\left (36-168 x+48 x^2\right ) \log ^3\left (\frac {e^{4 x}}{-3 x+x^2}\right )+(36-12 x) \log ^4\left (\frac {e^{4 x}}{-3 x+x^2}\right )}{x^5 (-9+3 x)} \, dx \\ & = \int \left (1+\frac {20 \left (3-14 x+4 x^2\right ) \log \left (\frac {e^{4 x}}{(-3+x) x}\right )}{3 (-3+x) x^3}-\frac {20 \log ^2\left (\frac {e^{4 x}}{(-3+x) x}\right )}{3 x^3}+\frac {4 \left (3-14 x+4 x^2\right ) \log ^3\left (\frac {e^{4 x}}{(-3+x) x}\right )}{(-3+x) x^5}-\frac {4 \log ^4\left (\frac {e^{4 x}}{(-3+x) x}\right )}{x^5}\right ) \, dx \\ & = x+4 \int \frac {\left (3-14 x+4 x^2\right ) \log ^3\left (\frac {e^{4 x}}{(-3+x) x}\right )}{(-3+x) x^5} \, dx-4 \int \frac {\log ^4\left (\frac {e^{4 x}}{(-3+x) x}\right )}{x^5} \, dx+\frac {20}{3} \int \frac {\left (3-14 x+4 x^2\right ) \log \left (\frac {e^{4 x}}{(-3+x) x}\right )}{(-3+x) x^3} \, dx-\frac {20}{3} \int \frac {\log ^2\left (\frac {e^{4 x}}{(-3+x) x}\right )}{x^3} \, dx \\ & = x-4 \int \frac {\log ^4\left (\frac {e^{4 x}}{(-3+x) x}\right )}{x^5} \, dx+4 \int \left (-\frac {\log ^3\left (\frac {e^{4 x}}{(-3+x) x}\right )}{81 (-3+x)}-\frac {\log ^3\left (\frac {e^{4 x}}{(-3+x) x}\right )}{x^5}+\frac {13 \log ^3\left (\frac {e^{4 x}}{(-3+x) x}\right )}{3 x^4}+\frac {\log ^3\left (\frac {e^{4 x}}{(-3+x) x}\right )}{9 x^3}+\frac {\log ^3\left (\frac {e^{4 x}}{(-3+x) x}\right )}{27 x^2}+\frac {\log ^3\left (\frac {e^{4 x}}{(-3+x) x}\right )}{81 x}\right ) \, dx-\frac {20}{3} \int \frac {\log ^2\left (\frac {e^{4 x}}{(-3+x) x}\right )}{x^3} \, dx+\frac {20}{3} \int \left (-\frac {\log \left (\frac {e^{4 x}}{(-3+x) x}\right )}{9 (-3+x)}-\frac {\log \left (\frac {e^{4 x}}{(-3+x) x}\right )}{x^3}+\frac {13 \log \left (\frac {e^{4 x}}{(-3+x) x}\right )}{3 x^2}+\frac {\log \left (\frac {e^{4 x}}{(-3+x) x}\right )}{9 x}\right ) \, dx \\ & = x-\frac {4}{81} \int \frac {\log ^3\left (\frac {e^{4 x}}{(-3+x) x}\right )}{-3+x} \, dx+\frac {4}{81} \int \frac {\log ^3\left (\frac {e^{4 x}}{(-3+x) x}\right )}{x} \, dx+\frac {4}{27} \int \frac {\log ^3\left (\frac {e^{4 x}}{(-3+x) x}\right )}{x^2} \, dx+\frac {4}{9} \int \frac {\log ^3\left (\frac {e^{4 x}}{(-3+x) x}\right )}{x^3} \, dx-\frac {20}{27} \int \frac {\log \left (\frac {e^{4 x}}{(-3+x) x}\right )}{-3+x} \, dx+\frac {20}{27} \int \frac {\log \left (\frac {e^{4 x}}{(-3+x) x}\right )}{x} \, dx-4 \int \frac {\log ^3\left (\frac {e^{4 x}}{(-3+x) x}\right )}{x^5} \, dx-4 \int \frac {\log ^4\left (\frac {e^{4 x}}{(-3+x) x}\right )}{x^5} \, dx-\frac {20}{3} \int \frac {\log \left (\frac {e^{4 x}}{(-3+x) x}\right )}{x^3} \, dx-\frac {20}{3} \int \frac {\log ^2\left (\frac {e^{4 x}}{(-3+x) x}\right )}{x^3} \, dx+\frac {52}{3} \int \frac {\log ^3\left (\frac {e^{4 x}}{(-3+x) x}\right )}{x^4} \, dx+\frac {260}{9} \int \frac {\log \left (\frac {e^{4 x}}{(-3+x) x}\right )}{x^2} \, dx \\ & = x+\frac {10 \log \left (-\frac {e^{4 x}}{(3-x) x}\right )}{3 x^2}-\frac {260 \log \left (-\frac {e^{4 x}}{(3-x) x}\right )}{9 x}-\frac {4}{81} \int \frac {\log ^3\left (\frac {e^{4 x}}{(-3+x) x}\right )}{-3+x} \, dx+\frac {4}{81} \int \frac {\log ^3\left (\frac {e^{4 x}}{(-3+x) x}\right )}{x} \, dx+\frac {4}{27} \int \frac {\log ^3\left (\frac {e^{4 x}}{(-3+x) x}\right )}{x^2} \, dx+\frac {4}{9} \int \frac {\log ^3\left (\frac {e^{4 x}}{(-3+x) x}\right )}{x^3} \, dx-\frac {20}{27} \int \frac {\log \left (\frac {e^{4 x}}{(-3+x) x}\right )}{-3+x} \, dx+\frac {20}{27} \int \frac {\log \left (\frac {e^{4 x}}{(-3+x) x}\right )}{x} \, dx-\frac {10}{3} \int \frac {-3+14 x-4 x^2}{(3-x) x^3} \, dx-4 \int \frac {\log ^3\left (\frac {e^{4 x}}{(-3+x) x}\right )}{x^5} \, dx-4 \int \frac {\log ^4\left (\frac {e^{4 x}}{(-3+x) x}\right )}{x^5} \, dx-\frac {20}{3} \int \frac {\log ^2\left (\frac {e^{4 x}}{(-3+x) x}\right )}{x^3} \, dx+\frac {52}{3} \int \frac {\log ^3\left (\frac {e^{4 x}}{(-3+x) x}\right )}{x^4} \, dx+\frac {260}{9} \int \frac {-3+14 x-4 x^2}{(3-x) x^2} \, dx \\ & = x+\frac {10 \log \left (-\frac {e^{4 x}}{(3-x) x}\right )}{3 x^2}-\frac {260 \log \left (-\frac {e^{4 x}}{(3-x) x}\right )}{9 x}-\frac {4}{81} \int \frac {\log ^3\left (\frac {e^{4 x}}{(-3+x) x}\right )}{-3+x} \, dx+\frac {4}{81} \int \frac {\log ^3\left (\frac {e^{4 x}}{(-3+x) x}\right )}{x} \, dx+\frac {4}{27} \int \frac {\log ^3\left (\frac {e^{4 x}}{(-3+x) x}\right )}{x^2} \, dx+\frac {4}{9} \int \frac {\log ^3\left (\frac {e^{4 x}}{(-3+x) x}\right )}{x^3} \, dx-\frac {20}{27} \int \frac {\log \left (\frac {e^{4 x}}{(-3+x) x}\right )}{-3+x} \, dx+\frac {20}{27} \int \frac {\log \left (\frac {e^{4 x}}{(-3+x) x}\right )}{x} \, dx-\frac {10}{3} \int \left (-\frac {1}{9 (-3+x)}-\frac {1}{x^3}+\frac {13}{3 x^2}+\frac {1}{9 x}\right ) \, dx-4 \int \frac {\log ^3\left (\frac {e^{4 x}}{(-3+x) x}\right )}{x^5} \, dx-4 \int \frac {\log ^4\left (\frac {e^{4 x}}{(-3+x) x}\right )}{x^5} \, dx-\frac {20}{3} \int \frac {\log ^2\left (\frac {e^{4 x}}{(-3+x) x}\right )}{x^3} \, dx+\frac {52}{3} \int \frac {\log ^3\left (\frac {e^{4 x}}{(-3+x) x}\right )}{x^4} \, dx+\frac {260}{9} \int \left (-\frac {1}{3 (-3+x)}-\frac {1}{x^2}+\frac {13}{3 x}\right ) \, dx \\ & = -\frac {5}{3 x^2}+\frac {130}{3 x}+x-\frac {250}{27} \log (3-x)+\frac {10 \log \left (-\frac {e^{4 x}}{(3-x) x}\right )}{3 x^2}-\frac {260 \log \left (-\frac {e^{4 x}}{(3-x) x}\right )}{9 x}+\frac {3370 \log (x)}{27}-\frac {4}{81} \int \frac {\log ^3\left (\frac {e^{4 x}}{(-3+x) x}\right )}{-3+x} \, dx+\frac {4}{81} \int \frac {\log ^3\left (\frac {e^{4 x}}{(-3+x) x}\right )}{x} \, dx+\frac {4}{27} \int \frac {\log ^3\left (\frac {e^{4 x}}{(-3+x) x}\right )}{x^2} \, dx+\frac {4}{9} \int \frac {\log ^3\left (\frac {e^{4 x}}{(-3+x) x}\right )}{x^3} \, dx-\frac {20}{27} \int \frac {\log \left (\frac {e^{4 x}}{(-3+x) x}\right )}{-3+x} \, dx+\frac {20}{27} \int \frac {\log \left (\frac {e^{4 x}}{(-3+x) x}\right )}{x} \, dx-4 \int \frac {\log ^3\left (\frac {e^{4 x}}{(-3+x) x}\right )}{x^5} \, dx-4 \int \frac {\log ^4\left (\frac {e^{4 x}}{(-3+x) x}\right )}{x^5} \, dx-\frac {20}{3} \int \frac {\log ^2\left (\frac {e^{4 x}}{(-3+x) x}\right )}{x^3} \, dx+\frac {52}{3} \int \frac {\log ^3\left (\frac {e^{4 x}}{(-3+x) x}\right )}{x^4} \, dx \\ \end{align*}
Time = 1.13 (sec) , antiderivative size = 50, normalized size of antiderivative = 1.72 \[ \int \frac {-9 x^5+3 x^6+\left (60 x^2-280 x^3+80 x^4\right ) \log \left (\frac {e^{4 x}}{-3 x+x^2}\right )+\left (60 x^2-20 x^3\right ) \log ^2\left (\frac {e^{4 x}}{-3 x+x^2}\right )+\left (36-168 x+48 x^2\right ) \log ^3\left (\frac {e^{4 x}}{-3 x+x^2}\right )+(36-12 x) \log ^4\left (\frac {e^{4 x}}{-3 x+x^2}\right )}{-9 x^5+3 x^6} \, dx=-\frac {928}{3}+x+\frac {10 \log ^2\left (\frac {e^{4 x}}{(-3+x) x}\right )}{3 x^2}+\frac {\log ^4\left (\frac {e^{4 x}}{(-3+x) x}\right )}{x^4} \]
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Leaf count of result is larger than twice the leaf count of optimal. \(55\) vs. \(2(26)=52\).
Time = 28.13 (sec) , antiderivative size = 56, normalized size of antiderivative = 1.93
method | result | size |
parallelrisch | \(-\frac {-18 x^{5}-27 x^{4}-60 \ln \left (\frac {{\mathrm e}^{4 x}}{x \left (-3+x \right )}\right )^{2} x^{2}-18 \ln \left (\frac {{\mathrm e}^{4 x}}{x \left (-3+x \right )}\right )^{4}}{18 x^{4}}\) | \(56\) |
risch | \(\text {Expression too large to display}\) | \(489742\) |
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Time = 0.27 (sec) , antiderivative size = 52, normalized size of antiderivative = 1.79 \[ \int \frac {-9 x^5+3 x^6+\left (60 x^2-280 x^3+80 x^4\right ) \log \left (\frac {e^{4 x}}{-3 x+x^2}\right )+\left (60 x^2-20 x^3\right ) \log ^2\left (\frac {e^{4 x}}{-3 x+x^2}\right )+\left (36-168 x+48 x^2\right ) \log ^3\left (\frac {e^{4 x}}{-3 x+x^2}\right )+(36-12 x) \log ^4\left (\frac {e^{4 x}}{-3 x+x^2}\right )}{-9 x^5+3 x^6} \, dx=\frac {3 \, x^{5} + 10 \, x^{2} \log \left (\frac {e^{\left (4 \, x\right )}}{x^{2} - 3 \, x}\right )^{2} + 3 \, \log \left (\frac {e^{\left (4 \, x\right )}}{x^{2} - 3 \, x}\right )^{4}}{3 \, x^{4}} \]
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Time = 0.17 (sec) , antiderivative size = 41, normalized size of antiderivative = 1.41 \[ \int \frac {-9 x^5+3 x^6+\left (60 x^2-280 x^3+80 x^4\right ) \log \left (\frac {e^{4 x}}{-3 x+x^2}\right )+\left (60 x^2-20 x^3\right ) \log ^2\left (\frac {e^{4 x}}{-3 x+x^2}\right )+\left (36-168 x+48 x^2\right ) \log ^3\left (\frac {e^{4 x}}{-3 x+x^2}\right )+(36-12 x) \log ^4\left (\frac {e^{4 x}}{-3 x+x^2}\right )}{-9 x^5+3 x^6} \, dx=x + \frac {10 \log {\left (\frac {e^{4 x}}{x^{2} - 3 x} \right )}^{2}}{3 x^{2}} + \frac {\log {\left (\frac {e^{4 x}}{x^{2} - 3 x} \right )}^{4}}{x^{4}} \]
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Leaf count of result is larger than twice the leaf count of optimal. 118 vs. \(2 (29) = 58\).
Time = 0.25 (sec) , antiderivative size = 118, normalized size of antiderivative = 4.07 \[ \int \frac {-9 x^5+3 x^6+\left (60 x^2-280 x^3+80 x^4\right ) \log \left (\frac {e^{4 x}}{-3 x+x^2}\right )+\left (60 x^2-20 x^3\right ) \log ^2\left (\frac {e^{4 x}}{-3 x+x^2}\right )+\left (36-168 x+48 x^2\right ) \log ^3\left (\frac {e^{4 x}}{-3 x+x^2}\right )+(36-12 x) \log ^4\left (\frac {e^{4 x}}{-3 x+x^2}\right )}{-9 x^5+3 x^6} \, dx=x - \frac {12 \, {\left (4 \, x - \log \left (x\right )\right )} \log \left (x - 3\right )^{3} - 3 \, \log \left (x - 3\right )^{4} + 848 \, x^{3} \log \left (x\right ) - 298 \, x^{2} \log \left (x\right )^{2} + 48 \, x \log \left (x\right )^{3} - 3 \, \log \left (x\right )^{4} - 2 \, {\left (149 \, x^{2} - 72 \, x \log \left (x\right ) + 9 \, \log \left (x\right )^{2}\right )} \log \left (x - 3\right )^{2} + 4 \, {\left (212 \, x^{3} - 149 \, x^{2} \log \left (x\right ) + 36 \, x \log \left (x\right )^{2} - 3 \, \log \left (x\right )^{3}\right )} \log \left (x - 3\right )}{3 \, x^{4}} \]
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Leaf count of result is larger than twice the leaf count of optimal. 59 vs. \(2 (29) = 58\).
Time = 1.70 (sec) , antiderivative size = 59, normalized size of antiderivative = 2.03 \[ \int \frac {-9 x^5+3 x^6+\left (60 x^2-280 x^3+80 x^4\right ) \log \left (\frac {e^{4 x}}{-3 x+x^2}\right )+\left (60 x^2-20 x^3\right ) \log ^2\left (\frac {e^{4 x}}{-3 x+x^2}\right )+\left (36-168 x+48 x^2\right ) \log ^3\left (\frac {e^{4 x}}{-3 x+x^2}\right )+(36-12 x) \log ^4\left (\frac {e^{4 x}}{-3 x+x^2}\right )}{-9 x^5+3 x^6} \, dx=x - \frac {848 \, \log \left (x^{2} - 3 \, x\right )}{3 \, x} + \frac {298 \, \log \left (x^{2} - 3 \, x\right )^{2}}{3 \, x^{2}} - \frac {16 \, \log \left (x^{2} - 3 \, x\right )^{3}}{x^{3}} + \frac {\log \left (x^{2} - 3 \, x\right )^{4}}{x^{4}} \]
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Time = 9.86 (sec) , antiderivative size = 53, normalized size of antiderivative = 1.83 \[ \int \frac {-9 x^5+3 x^6+\left (60 x^2-280 x^3+80 x^4\right ) \log \left (\frac {e^{4 x}}{-3 x+x^2}\right )+\left (60 x^2-20 x^3\right ) \log ^2\left (\frac {e^{4 x}}{-3 x+x^2}\right )+\left (36-168 x+48 x^2\right ) \log ^3\left (\frac {e^{4 x}}{-3 x+x^2}\right )+(36-12 x) \log ^4\left (\frac {e^{4 x}}{-3 x+x^2}\right )}{-9 x^5+3 x^6} \, dx=\frac {x^5+\frac {10\,x^2\,{\ln \left (-\frac {{\mathrm {e}}^{4\,x}}{3\,x-x^2}\right )}^2}{3}+{\ln \left (-\frac {{\mathrm {e}}^{4\,x}}{3\,x-x^2}\right )}^4}{x^4} \]
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