Integrand size = 233, antiderivative size = 31 \[ \int \frac {\left (-6-60 x+24 x^2+e^x \left (-12 x-12 x^2\right )\right ) \log ^2(x)+\left (\left (2+20 x-8 x^2+e^x \left (4 x+4 x^2\right )\right ) \log (x)+\left (\left (60 x+12 e^x x-12 x^2\right ) \log (x)+6 \log ^2(x)\right ) \log \left (10 x+2 e^x x-2 x^2+\log (x)\right )\right ) \log \left (\log \left (10 x+2 e^x x-2 x^2+\log (x)\right )\right )+\left (-20 x-4 e^x x+4 x^2-2 \log (x)\right ) \log \left (10 x+2 e^x x-2 x^2+\log (x)\right ) \log ^2\left (\log \left (10 x+2 e^x x-2 x^2+\log (x)\right )\right )}{\left (\left (10 x^2+2 e^x x^2-2 x^3\right ) \log ^3(x)+x \log ^4(x)\right ) \log \left (10 x+2 e^x x-2 x^2+\log (x)\right )} \, dx=1+\left (3-\frac {\log \left (\log \left (\left (4+2 \left (3+e^x-x\right )\right ) x+\log (x)\right )\right )}{\log (x)}\right )^2 \]
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Time = 4.26 (sec) , antiderivative size = 46, normalized size of antiderivative = 1.48, number of steps used = 4, number of rules used = 3, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.013, Rules used = {6820, 12, 6844} \[ \int \frac {\left (-6-60 x+24 x^2+e^x \left (-12 x-12 x^2\right )\right ) \log ^2(x)+\left (\left (2+20 x-8 x^2+e^x \left (4 x+4 x^2\right )\right ) \log (x)+\left (\left (60 x+12 e^x x-12 x^2\right ) \log (x)+6 \log ^2(x)\right ) \log \left (10 x+2 e^x x-2 x^2+\log (x)\right )\right ) \log \left (\log \left (10 x+2 e^x x-2 x^2+\log (x)\right )\right )+\left (-20 x-4 e^x x+4 x^2-2 \log (x)\right ) \log \left (10 x+2 e^x x-2 x^2+\log (x)\right ) \log ^2\left (\log \left (10 x+2 e^x x-2 x^2+\log (x)\right )\right )}{\left (\left (10 x^2+2 e^x x^2-2 x^3\right ) \log ^3(x)+x \log ^4(x)\right ) \log \left (10 x+2 e^x x-2 x^2+\log (x)\right )} \, dx=\frac {\log ^2\left (\log \left (2 \left (-x+e^x+5\right ) x+\log (x)\right )\right )}{\log ^2(x)}-\frac {6 \log \left (\log \left (2 \left (-x+e^x+5\right ) x+\log (x)\right )\right )}{\log (x)} \]
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Rule 12
Rule 6820
Rule 6844
Rubi steps \begin{align*} \text {integral}& = \int \frac {2 \left (3 \log (x)-\log \left (\log \left (-2 x \left (-5-e^x+x\right )+\log (x)\right )\right )\right ) \left (2 x \left (-5-e^x+x\right ) \log \left (-2 x \left (-5-e^x+x\right )+\log (x)\right ) \log \left (\log \left (-2 x \left (-5-e^x+x\right )+\log (x)\right )\right )+\log (x) \left (1+2 \left (5+e^x\right ) x+2 \left (-2+e^x\right ) x^2-\log \left (-2 x \left (-5-e^x+x\right )+\log (x)\right ) \log \left (\log \left (-2 x \left (-5-e^x+x\right )+\log (x)\right )\right )\right )\right )}{x \left (2 x \left (-5-e^x+x\right )-\log (x)\right ) \log ^3(x) \log \left (-2 x \left (-5-e^x+x\right )+\log (x)\right )} \, dx \\ & = 2 \int \frac {\left (3 \log (x)-\log \left (\log \left (-2 x \left (-5-e^x+x\right )+\log (x)\right )\right )\right ) \left (2 x \left (-5-e^x+x\right ) \log \left (-2 x \left (-5-e^x+x\right )+\log (x)\right ) \log \left (\log \left (-2 x \left (-5-e^x+x\right )+\log (x)\right )\right )+\log (x) \left (1+2 \left (5+e^x\right ) x+2 \left (-2+e^x\right ) x^2-\log \left (-2 x \left (-5-e^x+x\right )+\log (x)\right ) \log \left (\log \left (-2 x \left (-5-e^x+x\right )+\log (x)\right )\right )\right )\right )}{x \left (2 x \left (-5-e^x+x\right )-\log (x)\right ) \log ^3(x) \log \left (-2 x \left (-5-e^x+x\right )+\log (x)\right )} \, dx \\ & = -\left (2 \text {Subst}\left (\int (3-x) \, dx,x,\frac {\log \left (\log \left (-2 x \left (-5-e^x+x\right )+\log (x)\right )\right )}{\log (x)}\right )\right ) \\ & = -\frac {6 \log \left (\log \left (2 \left (5+e^x-x\right ) x+\log (x)\right )\right )}{\log (x)}+\frac {\log ^2\left (\log \left (2 \left (5+e^x-x\right ) x+\log (x)\right )\right )}{\log ^2(x)} \\ \end{align*}
Time = 0.20 (sec) , antiderivative size = 51, normalized size of antiderivative = 1.65 \[ \int \frac {\left (-6-60 x+24 x^2+e^x \left (-12 x-12 x^2\right )\right ) \log ^2(x)+\left (\left (2+20 x-8 x^2+e^x \left (4 x+4 x^2\right )\right ) \log (x)+\left (\left (60 x+12 e^x x-12 x^2\right ) \log (x)+6 \log ^2(x)\right ) \log \left (10 x+2 e^x x-2 x^2+\log (x)\right )\right ) \log \left (\log \left (10 x+2 e^x x-2 x^2+\log (x)\right )\right )+\left (-20 x-4 e^x x+4 x^2-2 \log (x)\right ) \log \left (10 x+2 e^x x-2 x^2+\log (x)\right ) \log ^2\left (\log \left (10 x+2 e^x x-2 x^2+\log (x)\right )\right )}{\left (\left (10 x^2+2 e^x x^2-2 x^3\right ) \log ^3(x)+x \log ^4(x)\right ) \log \left (10 x+2 e^x x-2 x^2+\log (x)\right )} \, dx=2 \left (-\frac {3 \log \left (\log \left (-2 x \left (-5-e^x+x\right )+\log (x)\right )\right )}{\log (x)}+\frac {\log ^2\left (\log \left (-2 x \left (-5-e^x+x\right )+\log (x)\right )\right )}{2 \log ^2(x)}\right ) \]
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Time = 240.28 (sec) , antiderivative size = 51, normalized size of antiderivative = 1.65
method | result | size |
risch | \(\frac {{\ln \left (\ln \left (\ln \left (x \right )+2 \,{\mathrm e}^{x} x -2 x^{2}+10 x \right )\right )}^{2}}{\ln \left (x \right )^{2}}-\frac {6 \ln \left (\ln \left (\ln \left (x \right )+2 \,{\mathrm e}^{x} x -2 x^{2}+10 x \right )\right )}{\ln \left (x \right )}\) | \(51\) |
parallelrisch | \(-\frac {24 \ln \left (x \right ) \ln \left (\ln \left (\ln \left (x \right )+2 \,{\mathrm e}^{x} x -2 x^{2}+10 x \right )\right )-4 {\ln \left (\ln \left (\ln \left (x \right )+2 \,{\mathrm e}^{x} x -2 x^{2}+10 x \right )\right )}^{2}}{4 \ln \left (x \right )^{2}}\) | \(52\) |
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Time = 0.33 (sec) , antiderivative size = 51, normalized size of antiderivative = 1.65 \[ \int \frac {\left (-6-60 x+24 x^2+e^x \left (-12 x-12 x^2\right )\right ) \log ^2(x)+\left (\left (2+20 x-8 x^2+e^x \left (4 x+4 x^2\right )\right ) \log (x)+\left (\left (60 x+12 e^x x-12 x^2\right ) \log (x)+6 \log ^2(x)\right ) \log \left (10 x+2 e^x x-2 x^2+\log (x)\right )\right ) \log \left (\log \left (10 x+2 e^x x-2 x^2+\log (x)\right )\right )+\left (-20 x-4 e^x x+4 x^2-2 \log (x)\right ) \log \left (10 x+2 e^x x-2 x^2+\log (x)\right ) \log ^2\left (\log \left (10 x+2 e^x x-2 x^2+\log (x)\right )\right )}{\left (\left (10 x^2+2 e^x x^2-2 x^3\right ) \log ^3(x)+x \log ^4(x)\right ) \log \left (10 x+2 e^x x-2 x^2+\log (x)\right )} \, dx=-\frac {6 \, \log \left (x\right ) \log \left (\log \left (-2 \, x^{2} + 2 \, x e^{x} + 10 \, x + \log \left (x\right )\right )\right ) - \log \left (\log \left (-2 \, x^{2} + 2 \, x e^{x} + 10 \, x + \log \left (x\right )\right )\right )^{2}}{\log \left (x\right )^{2}} \]
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Exception generated. \[ \int \frac {\left (-6-60 x+24 x^2+e^x \left (-12 x-12 x^2\right )\right ) \log ^2(x)+\left (\left (2+20 x-8 x^2+e^x \left (4 x+4 x^2\right )\right ) \log (x)+\left (\left (60 x+12 e^x x-12 x^2\right ) \log (x)+6 \log ^2(x)\right ) \log \left (10 x+2 e^x x-2 x^2+\log (x)\right )\right ) \log \left (\log \left (10 x+2 e^x x-2 x^2+\log (x)\right )\right )+\left (-20 x-4 e^x x+4 x^2-2 \log (x)\right ) \log \left (10 x+2 e^x x-2 x^2+\log (x)\right ) \log ^2\left (\log \left (10 x+2 e^x x-2 x^2+\log (x)\right )\right )}{\left (\left (10 x^2+2 e^x x^2-2 x^3\right ) \log ^3(x)+x \log ^4(x)\right ) \log \left (10 x+2 e^x x-2 x^2+\log (x)\right )} \, dx=\text {Exception raised: TypeError} \]
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Time = 0.29 (sec) , antiderivative size = 51, normalized size of antiderivative = 1.65 \[ \int \frac {\left (-6-60 x+24 x^2+e^x \left (-12 x-12 x^2\right )\right ) \log ^2(x)+\left (\left (2+20 x-8 x^2+e^x \left (4 x+4 x^2\right )\right ) \log (x)+\left (\left (60 x+12 e^x x-12 x^2\right ) \log (x)+6 \log ^2(x)\right ) \log \left (10 x+2 e^x x-2 x^2+\log (x)\right )\right ) \log \left (\log \left (10 x+2 e^x x-2 x^2+\log (x)\right )\right )+\left (-20 x-4 e^x x+4 x^2-2 \log (x)\right ) \log \left (10 x+2 e^x x-2 x^2+\log (x)\right ) \log ^2\left (\log \left (10 x+2 e^x x-2 x^2+\log (x)\right )\right )}{\left (\left (10 x^2+2 e^x x^2-2 x^3\right ) \log ^3(x)+x \log ^4(x)\right ) \log \left (10 x+2 e^x x-2 x^2+\log (x)\right )} \, dx=-\frac {6 \, \log \left (x\right ) \log \left (\log \left (-2 \, x^{2} + 2 \, x e^{x} + 10 \, x + \log \left (x\right )\right )\right ) - \log \left (\log \left (-2 \, x^{2} + 2 \, x e^{x} + 10 \, x + \log \left (x\right )\right )\right )^{2}}{\log \left (x\right )^{2}} \]
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Exception generated. \[ \int \frac {\left (-6-60 x+24 x^2+e^x \left (-12 x-12 x^2\right )\right ) \log ^2(x)+\left (\left (2+20 x-8 x^2+e^x \left (4 x+4 x^2\right )\right ) \log (x)+\left (\left (60 x+12 e^x x-12 x^2\right ) \log (x)+6 \log ^2(x)\right ) \log \left (10 x+2 e^x x-2 x^2+\log (x)\right )\right ) \log \left (\log \left (10 x+2 e^x x-2 x^2+\log (x)\right )\right )+\left (-20 x-4 e^x x+4 x^2-2 \log (x)\right ) \log \left (10 x+2 e^x x-2 x^2+\log (x)\right ) \log ^2\left (\log \left (10 x+2 e^x x-2 x^2+\log (x)\right )\right )}{\left (\left (10 x^2+2 e^x x^2-2 x^3\right ) \log ^3(x)+x \log ^4(x)\right ) \log \left (10 x+2 e^x x-2 x^2+\log (x)\right )} \, dx=\text {Exception raised: TypeError} \]
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Time = 13.81 (sec) , antiderivative size = 46, normalized size of antiderivative = 1.48 \[ \int \frac {\left (-6-60 x+24 x^2+e^x \left (-12 x-12 x^2\right )\right ) \log ^2(x)+\left (\left (2+20 x-8 x^2+e^x \left (4 x+4 x^2\right )\right ) \log (x)+\left (\left (60 x+12 e^x x-12 x^2\right ) \log (x)+6 \log ^2(x)\right ) \log \left (10 x+2 e^x x-2 x^2+\log (x)\right )\right ) \log \left (\log \left (10 x+2 e^x x-2 x^2+\log (x)\right )\right )+\left (-20 x-4 e^x x+4 x^2-2 \log (x)\right ) \log \left (10 x+2 e^x x-2 x^2+\log (x)\right ) \log ^2\left (\log \left (10 x+2 e^x x-2 x^2+\log (x)\right )\right )}{\left (\left (10 x^2+2 e^x x^2-2 x^3\right ) \log ^3(x)+x \log ^4(x)\right ) \log \left (10 x+2 e^x x-2 x^2+\log (x)\right )} \, dx=\frac {\ln \left (\ln \left (10\,x+\ln \left (x\right )+2\,x\,{\mathrm {e}}^x-2\,x^2\right )\right )\,\left (\ln \left (\ln \left (10\,x+\ln \left (x\right )+2\,x\,{\mathrm {e}}^x-2\,x^2\right )\right )-6\,\ln \left (x\right )\right )}{{\ln \left (x\right )}^2} \]
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