Integrand size = 66, antiderivative size = 32 \[ \int \frac {e^e (10-2 x)+2 e^e \log (x)+\left (5-x^2+x \log (x)\right ) \log ^2\left (\frac {x^2}{9}\right )}{\left (15 x-3 x^2+3 x \log (x)\right ) \log ^2\left (\frac {x^2}{9}\right )} \, dx=\frac {1}{3} \left (2+x-\frac {e^e}{\log \left (\frac {x^2}{9}\right )}+5 \log (5-x+\log (x))\right ) \]
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Time = 0.76 (sec) , antiderivative size = 35, normalized size of antiderivative = 1.09, number of steps used = 9, number of rules used = 6, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.091, Rules used = {6873, 12, 6874, 6816, 2339, 30} \[ \int \frac {e^e (10-2 x)+2 e^e \log (x)+\left (5-x^2+x \log (x)\right ) \log ^2\left (\frac {x^2}{9}\right )}{\left (15 x-3 x^2+3 x \log (x)\right ) \log ^2\left (\frac {x^2}{9}\right )} \, dx=-\frac {e^e}{3 \log \left (\frac {x^2}{9}\right )}+\frac {x}{3}+\frac {5}{3} \log (-x+\log (x)+5) \]
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Rule 12
Rule 30
Rule 2339
Rule 6816
Rule 6873
Rule 6874
Rubi steps \begin{align*} \text {integral}& = \int \frac {e^e (10-2 x)+2 e^e \log (x)+\left (5-x^2+x \log (x)\right ) \log ^2\left (\frac {x^2}{9}\right )}{3 x (5-x+\log (x)) \log ^2\left (\frac {x^2}{9}\right )} \, dx \\ & = \frac {1}{3} \int \frac {e^e (10-2 x)+2 e^e \log (x)+\left (5-x^2+x \log (x)\right ) \log ^2\left (\frac {x^2}{9}\right )}{x (5-x+\log (x)) \log ^2\left (\frac {x^2}{9}\right )} \, dx \\ & = \frac {1}{3} \int \left (\frac {-5+x^2-x \log (x)}{x (-5+x-\log (x))}+\frac {2 e^e}{x \log ^2\left (\frac {x^2}{9}\right )}\right ) \, dx \\ & = \frac {1}{3} \int \frac {-5+x^2-x \log (x)}{x (-5+x-\log (x))} \, dx+\frac {1}{3} \left (2 e^e\right ) \int \frac {1}{x \log ^2\left (\frac {x^2}{9}\right )} \, dx \\ & = \frac {1}{3} \int \left (1+\frac {5 (-1+x)}{x (-5+x-\log (x))}\right ) \, dx+\frac {1}{3} e^e \text {Subst}\left (\int \frac {1}{x^2} \, dx,x,\log \left (\frac {x^2}{9}\right )\right ) \\ & = \frac {x}{3}-\frac {e^e}{3 \log \left (\frac {x^2}{9}\right )}+\frac {5}{3} \int \frac {-1+x}{x (-5+x-\log (x))} \, dx \\ & = \frac {x}{3}-\frac {e^e}{3 \log \left (\frac {x^2}{9}\right )}+\frac {5}{3} \log (5-x+\log (x)) \\ \end{align*}
Time = 0.43 (sec) , antiderivative size = 31, normalized size of antiderivative = 0.97 \[ \int \frac {e^e (10-2 x)+2 e^e \log (x)+\left (5-x^2+x \log (x)\right ) \log ^2\left (\frac {x^2}{9}\right )}{\left (15 x-3 x^2+3 x \log (x)\right ) \log ^2\left (\frac {x^2}{9}\right )} \, dx=\frac {1}{3} \left (x-\frac {e^e}{\log \left (\frac {x^2}{9}\right )}+5 \log (5-x+\log (x))\right ) \]
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Time = 0.83 (sec) , antiderivative size = 41, normalized size of antiderivative = 1.28
method | result | size |
parallelrisch | \(\frac {5 \ln \left (-\ln \left (x \right )+x -5\right ) \ln \left (\frac {x^{2}}{9}\right )+\ln \left (\frac {x^{2}}{9}\right ) x -{\mathrm e}^{{\mathrm e}}}{3 \ln \left (\frac {x^{2}}{9}\right )}\) | \(41\) |
default | \(\frac {\left (2 \ln \left (3\right )-\ln \left (x^{2}\right )+2 \ln \left (x \right )\right ) x -2 x \ln \left (x \right )+{\mathrm e}^{{\mathrm e}}}{6 \ln \left (3\right )-3 \ln \left (x^{2}\right )}+\frac {5 \ln \left (\ln \left (x \right )-x +5\right )}{3}\) | \(53\) |
risch | \(\frac {x}{3}-\frac {2 i {\mathrm e}^{{\mathrm e}}}{3 \left (\pi \operatorname {csgn}\left (i x \right )^{2} \operatorname {csgn}\left (i x^{2}\right )-2 \pi \,\operatorname {csgn}\left (i x \right ) \operatorname {csgn}\left (i x^{2}\right )^{2}+\pi \operatorname {csgn}\left (i x^{2}\right )^{3}-4 i \ln \left (3\right )+4 i \ln \left (x \right )\right )}+\frac {5 \ln \left (\ln \left (x \right )-x +5\right )}{3}\) | \(78\) |
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Time = 0.27 (sec) , antiderivative size = 42, normalized size of antiderivative = 1.31 \[ \int \frac {e^e (10-2 x)+2 e^e \log (x)+\left (5-x^2+x \log (x)\right ) \log ^2\left (\frac {x^2}{9}\right )}{\left (15 x-3 x^2+3 x \log (x)\right ) \log ^2\left (\frac {x^2}{9}\right )} \, dx=\frac {2 \, x \log \left (3\right ) - 2 \, x \log \left (x\right ) + 10 \, {\left (\log \left (3\right ) - \log \left (x\right )\right )} \log \left (-x + \log \left (x\right ) + 5\right ) + e^{e}}{6 \, {\left (\log \left (3\right ) - \log \left (x\right )\right )}} \]
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Time = 0.12 (sec) , antiderivative size = 29, normalized size of antiderivative = 0.91 \[ \int \frac {e^e (10-2 x)+2 e^e \log (x)+\left (5-x^2+x \log (x)\right ) \log ^2\left (\frac {x^2}{9}\right )}{\left (15 x-3 x^2+3 x \log (x)\right ) \log ^2\left (\frac {x^2}{9}\right )} \, dx=\frac {x}{3} + \frac {5 \log {\left (- x + \log {\left (x \right )} + 5 \right )}}{3} - \frac {e^{e}}{6 \log {\left (x \right )} - 6 \log {\left (3 \right )}} \]
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Time = 0.32 (sec) , antiderivative size = 36, normalized size of antiderivative = 1.12 \[ \int \frac {e^e (10-2 x)+2 e^e \log (x)+\left (5-x^2+x \log (x)\right ) \log ^2\left (\frac {x^2}{9}\right )}{\left (15 x-3 x^2+3 x \log (x)\right ) \log ^2\left (\frac {x^2}{9}\right )} \, dx=\frac {2 \, x \log \left (3\right ) - 2 \, x \log \left (x\right ) + e^{e}}{6 \, {\left (\log \left (3\right ) - \log \left (x\right )\right )}} + \frac {5}{3} \, \log \left (-x + \log \left (x\right ) + 5\right ) \]
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Time = 0.32 (sec) , antiderivative size = 49, normalized size of antiderivative = 1.53 \[ \int \frac {e^e (10-2 x)+2 e^e \log (x)+\left (5-x^2+x \log (x)\right ) \log ^2\left (\frac {x^2}{9}\right )}{\left (15 x-3 x^2+3 x \log (x)\right ) \log ^2\left (\frac {x^2}{9}\right )} \, dx=\frac {2 \, x \log \left (3\right ) - 2 \, x \log \left (x\right ) + 10 \, \log \left (3\right ) \log \left (-x + \log \left (x\right ) + 5\right ) - 10 \, \log \left (x\right ) \log \left (-x + \log \left (x\right ) + 5\right ) + e^{e}}{6 \, {\left (\log \left (3\right ) - \log \left (x\right )\right )}} \]
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Timed out. \[ \int \frac {e^e (10-2 x)+2 e^e \log (x)+\left (5-x^2+x \log (x)\right ) \log ^2\left (\frac {x^2}{9}\right )}{\left (15 x-3 x^2+3 x \log (x)\right ) \log ^2\left (\frac {x^2}{9}\right )} \, dx=\int \frac {\left (x\,\ln \left (x\right )-x^2+5\right )\,{\ln \left (\frac {x^2}{9}\right )}^2+2\,{\mathrm {e}}^{\mathrm {e}}\,\ln \left (x\right )-{\mathrm {e}}^{\mathrm {e}}\,\left (2\,x-10\right )}{{\ln \left (\frac {x^2}{9}\right )}^2\,\left (15\,x+3\,x\,\ln \left (x\right )-3\,x^2\right )} \,d x \]
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