Integrand size = 310, antiderivative size = 25 \[ \int \frac {-15-5 e^x+\left (-960-30 x+110592 x^4+3456 x^5+27 x^6+e^{2 x} \left (12288 x^4+384 x^5+3 x^6\right )+e^x \left (-320-330 x-5 x^2+73728 x^4+2304 x^5+18 x^6\right )\right ) \log (x)+\left (-15+3456 x^4+54 x^5+e^{2 x} \left (384 x^4+6 x^5\right )+e^x \left (-5-5 x+2304 x^4+36 x^5\right )\right ) \log (x) \log (\log (x))+\left (27 x^4+18 e^x x^4+3 e^{2 x} x^4\right ) \log (x) \log ^2(\log (x))}{\left (36864 x^2+1152 x^3+9 x^4+e^{2 x} \left (4096 x^2+128 x^3+x^4\right )+e^x \left (24576 x^2+768 x^3+6 x^4\right )\right ) \log (x)+\left (1152 x^2+18 x^3+e^{2 x} \left (128 x^2+2 x^3\right )+e^x \left (768 x^2+12 x^3\right )\right ) \log (x) \log (\log (x))+\left (9 x^2+6 e^x x^2+e^{2 x} x^2\right ) \log (x) \log ^2(\log (x))} \, dx=-2+x^3+\frac {5}{\left (3+e^x\right ) x (64+x+\log (\log (x)))} \]
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\[ \int \frac {-15-5 e^x+\left (-960-30 x+110592 x^4+3456 x^5+27 x^6+e^{2 x} \left (12288 x^4+384 x^5+3 x^6\right )+e^x \left (-320-330 x-5 x^2+73728 x^4+2304 x^5+18 x^6\right )\right ) \log (x)+\left (-15+3456 x^4+54 x^5+e^{2 x} \left (384 x^4+6 x^5\right )+e^x \left (-5-5 x+2304 x^4+36 x^5\right )\right ) \log (x) \log (\log (x))+\left (27 x^4+18 e^x x^4+3 e^{2 x} x^4\right ) \log (x) \log ^2(\log (x))}{\left (36864 x^2+1152 x^3+9 x^4+e^{2 x} \left (4096 x^2+128 x^3+x^4\right )+e^x \left (24576 x^2+768 x^3+6 x^4\right )\right ) \log (x)+\left (1152 x^2+18 x^3+e^{2 x} \left (128 x^2+2 x^3\right )+e^x \left (768 x^2+12 x^3\right )\right ) \log (x) \log (\log (x))+\left (9 x^2+6 e^x x^2+e^{2 x} x^2\right ) \log (x) \log ^2(\log (x))} \, dx=\int \frac {-15-5 e^x+\left (-960-30 x+110592 x^4+3456 x^5+27 x^6+e^{2 x} \left (12288 x^4+384 x^5+3 x^6\right )+e^x \left (-320-330 x-5 x^2+73728 x^4+2304 x^5+18 x^6\right )\right ) \log (x)+\left (-15+3456 x^4+54 x^5+e^{2 x} \left (384 x^4+6 x^5\right )+e^x \left (-5-5 x+2304 x^4+36 x^5\right )\right ) \log (x) \log (\log (x))+\left (27 x^4+18 e^x x^4+3 e^{2 x} x^4\right ) \log (x) \log ^2(\log (x))}{\left (36864 x^2+1152 x^3+9 x^4+e^{2 x} \left (4096 x^2+128 x^3+x^4\right )+e^x \left (24576 x^2+768 x^3+6 x^4\right )\right ) \log (x)+\left (1152 x^2+18 x^3+e^{2 x} \left (128 x^2+2 x^3\right )+e^x \left (768 x^2+12 x^3\right )\right ) \log (x) \log (\log (x))+\left (9 x^2+6 e^x x^2+e^{2 x} x^2\right ) \log (x) \log ^2(\log (x))} \, dx \]
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Rubi steps \begin{align*} \text {integral}& = \int \frac {-5 \left (3+e^x\right )+\log (x) \left (3 e^{2 x} x^4 (64+x)^2+3 \left (-320-10 x+36864 x^4+1152 x^5+9 x^6\right )+e^x \left (-320-330 x-5 x^2+73728 x^4+2304 x^5+18 x^6\right )+\left (-15+3456 x^4+54 x^5+6 e^{2 x} x^4 (64+x)+e^x \left (-5-5 x+2304 x^4+36 x^5\right )\right ) \log (\log (x))+3 \left (3+e^x\right )^2 x^4 \log ^2(\log (x))\right )}{\left (3+e^x\right )^2 x^2 \log (x) (64+x+\log (\log (x)))^2} \, dx \\ & = \int \left (3 x^2+\frac {15}{\left (3+e^x\right )^2 x (64+x+\log (\log (x)))}-\frac {5 \left (1+64 \log (x)+66 x \log (x)+x^2 \log (x)+\log (x) \log (\log (x))+x \log (x) \log (\log (x))\right )}{\left (3+e^x\right ) x^2 \log (x) (64+x+\log (\log (x)))^2}\right ) \, dx \\ & = x^3-5 \int \frac {1+64 \log (x)+66 x \log (x)+x^2 \log (x)+\log (x) \log (\log (x))+x \log (x) \log (\log (x))}{\left (3+e^x\right ) x^2 \log (x) (64+x+\log (\log (x)))^2} \, dx+15 \int \frac {1}{\left (3+e^x\right )^2 x (64+x+\log (\log (x)))} \, dx \\ & = x^3-5 \int \frac {1+\log (x) \left (64+66 x+x^2+(1+x) \log (\log (x))\right )}{\left (3+e^x\right ) x^2 \log (x) (64+x+\log (\log (x)))^2} \, dx+15 \int \frac {1}{\left (3+e^x\right )^2 x (64+x+\log (\log (x)))} \, dx \\ & = x^3-5 \int \left (\frac {1}{\left (3+e^x\right ) (64+x+\log (\log (x)))^2}+\frac {64}{\left (3+e^x\right ) x^2 (64+x+\log (\log (x)))^2}+\frac {66}{\left (3+e^x\right ) x (64+x+\log (\log (x)))^2}+\frac {1}{\left (3+e^x\right ) x^2 \log (x) (64+x+\log (\log (x)))^2}+\frac {\log (\log (x))}{\left (3+e^x\right ) x^2 (64+x+\log (\log (x)))^2}+\frac {\log (\log (x))}{\left (3+e^x\right ) x (64+x+\log (\log (x)))^2}\right ) \, dx+15 \int \frac {1}{\left (3+e^x\right )^2 x (64+x+\log (\log (x)))} \, dx \\ & = x^3-5 \int \frac {1}{\left (3+e^x\right ) (64+x+\log (\log (x)))^2} \, dx-5 \int \frac {1}{\left (3+e^x\right ) x^2 \log (x) (64+x+\log (\log (x)))^2} \, dx-5 \int \frac {\log (\log (x))}{\left (3+e^x\right ) x^2 (64+x+\log (\log (x)))^2} \, dx-5 \int \frac {\log (\log (x))}{\left (3+e^x\right ) x (64+x+\log (\log (x)))^2} \, dx+15 \int \frac {1}{\left (3+e^x\right )^2 x (64+x+\log (\log (x)))} \, dx-320 \int \frac {1}{\left (3+e^x\right ) x^2 (64+x+\log (\log (x)))^2} \, dx-330 \int \frac {1}{\left (3+e^x\right ) x (64+x+\log (\log (x)))^2} \, dx \\ \end{align*}
Time = 0.26 (sec) , antiderivative size = 24, normalized size of antiderivative = 0.96 \[ \int \frac {-15-5 e^x+\left (-960-30 x+110592 x^4+3456 x^5+27 x^6+e^{2 x} \left (12288 x^4+384 x^5+3 x^6\right )+e^x \left (-320-330 x-5 x^2+73728 x^4+2304 x^5+18 x^6\right )\right ) \log (x)+\left (-15+3456 x^4+54 x^5+e^{2 x} \left (384 x^4+6 x^5\right )+e^x \left (-5-5 x+2304 x^4+36 x^5\right )\right ) \log (x) \log (\log (x))+\left (27 x^4+18 e^x x^4+3 e^{2 x} x^4\right ) \log (x) \log ^2(\log (x))}{\left (36864 x^2+1152 x^3+9 x^4+e^{2 x} \left (4096 x^2+128 x^3+x^4\right )+e^x \left (24576 x^2+768 x^3+6 x^4\right )\right ) \log (x)+\left (1152 x^2+18 x^3+e^{2 x} \left (128 x^2+2 x^3\right )+e^x \left (768 x^2+12 x^3\right )\right ) \log (x) \log (\log (x))+\left (9 x^2+6 e^x x^2+e^{2 x} x^2\right ) \log (x) \log ^2(\log (x))} \, dx=x^3+\frac {5}{\left (3+e^x\right ) x (64+x+\log (\log (x)))} \]
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Time = 47.20 (sec) , antiderivative size = 24, normalized size of antiderivative = 0.96
method | result | size |
risch | \(x^{3}+\frac {5}{x \left (\ln \left (\ln \left (x \right )\right )+64+x \right ) \left (3+{\mathrm e}^{x}\right )}\) | \(24\) |
parallelrisch | \(\frac {{\mathrm e}^{x} \ln \left (\ln \left (x \right )\right ) x^{4}+x^{5} {\mathrm e}^{x}+64 \,{\mathrm e}^{x} x^{4}+3 x^{4} \ln \left (\ln \left (x \right )\right )+3 x^{5}+192 x^{4}+5}{x \left (\ln \left (\ln \left (x \right )\right )+64+x \right ) \left (3+{\mathrm e}^{x}\right )}\) | \(61\) |
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Leaf count of result is larger than twice the leaf count of optimal. 74 vs. \(2 (24) = 48\).
Time = 0.26 (sec) , antiderivative size = 74, normalized size of antiderivative = 2.96 \[ \int \frac {-15-5 e^x+\left (-960-30 x+110592 x^4+3456 x^5+27 x^6+e^{2 x} \left (12288 x^4+384 x^5+3 x^6\right )+e^x \left (-320-330 x-5 x^2+73728 x^4+2304 x^5+18 x^6\right )\right ) \log (x)+\left (-15+3456 x^4+54 x^5+e^{2 x} \left (384 x^4+6 x^5\right )+e^x \left (-5-5 x+2304 x^4+36 x^5\right )\right ) \log (x) \log (\log (x))+\left (27 x^4+18 e^x x^4+3 e^{2 x} x^4\right ) \log (x) \log ^2(\log (x))}{\left (36864 x^2+1152 x^3+9 x^4+e^{2 x} \left (4096 x^2+128 x^3+x^4\right )+e^x \left (24576 x^2+768 x^3+6 x^4\right )\right ) \log (x)+\left (1152 x^2+18 x^3+e^{2 x} \left (128 x^2+2 x^3\right )+e^x \left (768 x^2+12 x^3\right )\right ) \log (x) \log (\log (x))+\left (9 x^2+6 e^x x^2+e^{2 x} x^2\right ) \log (x) \log ^2(\log (x))} \, dx=\frac {3 \, x^{5} + 192 \, x^{4} + {\left (x^{5} + 64 \, x^{4}\right )} e^{x} + {\left (x^{4} e^{x} + 3 \, x^{4}\right )} \log \left (\log \left (x\right )\right ) + 5}{3 \, x^{2} + {\left (x^{2} + 64 \, x\right )} e^{x} + {\left (x e^{x} + 3 \, x\right )} \log \left (\log \left (x\right )\right ) + 192 \, x} \]
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Time = 0.18 (sec) , antiderivative size = 37, normalized size of antiderivative = 1.48 \[ \int \frac {-15-5 e^x+\left (-960-30 x+110592 x^4+3456 x^5+27 x^6+e^{2 x} \left (12288 x^4+384 x^5+3 x^6\right )+e^x \left (-320-330 x-5 x^2+73728 x^4+2304 x^5+18 x^6\right )\right ) \log (x)+\left (-15+3456 x^4+54 x^5+e^{2 x} \left (384 x^4+6 x^5\right )+e^x \left (-5-5 x+2304 x^4+36 x^5\right )\right ) \log (x) \log (\log (x))+\left (27 x^4+18 e^x x^4+3 e^{2 x} x^4\right ) \log (x) \log ^2(\log (x))}{\left (36864 x^2+1152 x^3+9 x^4+e^{2 x} \left (4096 x^2+128 x^3+x^4\right )+e^x \left (24576 x^2+768 x^3+6 x^4\right )\right ) \log (x)+\left (1152 x^2+18 x^3+e^{2 x} \left (128 x^2+2 x^3\right )+e^x \left (768 x^2+12 x^3\right )\right ) \log (x) \log (\log (x))+\left (9 x^2+6 e^x x^2+e^{2 x} x^2\right ) \log (x) \log ^2(\log (x))} \, dx=x^{3} + \frac {5}{3 x^{2} + 3 x \log {\left (\log {\left (x \right )} \right )} + 192 x + \left (x^{2} + x \log {\left (\log {\left (x \right )} \right )} + 64 x\right ) e^{x}} \]
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Leaf count of result is larger than twice the leaf count of optimal. 74 vs. \(2 (24) = 48\).
Time = 0.35 (sec) , antiderivative size = 74, normalized size of antiderivative = 2.96 \[ \int \frac {-15-5 e^x+\left (-960-30 x+110592 x^4+3456 x^5+27 x^6+e^{2 x} \left (12288 x^4+384 x^5+3 x^6\right )+e^x \left (-320-330 x-5 x^2+73728 x^4+2304 x^5+18 x^6\right )\right ) \log (x)+\left (-15+3456 x^4+54 x^5+e^{2 x} \left (384 x^4+6 x^5\right )+e^x \left (-5-5 x+2304 x^4+36 x^5\right )\right ) \log (x) \log (\log (x))+\left (27 x^4+18 e^x x^4+3 e^{2 x} x^4\right ) \log (x) \log ^2(\log (x))}{\left (36864 x^2+1152 x^3+9 x^4+e^{2 x} \left (4096 x^2+128 x^3+x^4\right )+e^x \left (24576 x^2+768 x^3+6 x^4\right )\right ) \log (x)+\left (1152 x^2+18 x^3+e^{2 x} \left (128 x^2+2 x^3\right )+e^x \left (768 x^2+12 x^3\right )\right ) \log (x) \log (\log (x))+\left (9 x^2+6 e^x x^2+e^{2 x} x^2\right ) \log (x) \log ^2(\log (x))} \, dx=\frac {3 \, x^{5} + 192 \, x^{4} + {\left (x^{5} + 64 \, x^{4}\right )} e^{x} + {\left (x^{4} e^{x} + 3 \, x^{4}\right )} \log \left (\log \left (x\right )\right ) + 5}{3 \, x^{2} + {\left (x^{2} + 64 \, x\right )} e^{x} + {\left (x e^{x} + 3 \, x\right )} \log \left (\log \left (x\right )\right ) + 192 \, x} \]
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Leaf count of result is larger than twice the leaf count of optimal. 78 vs. \(2 (24) = 48\).
Time = 0.47 (sec) , antiderivative size = 78, normalized size of antiderivative = 3.12 \[ \int \frac {-15-5 e^x+\left (-960-30 x+110592 x^4+3456 x^5+27 x^6+e^{2 x} \left (12288 x^4+384 x^5+3 x^6\right )+e^x \left (-320-330 x-5 x^2+73728 x^4+2304 x^5+18 x^6\right )\right ) \log (x)+\left (-15+3456 x^4+54 x^5+e^{2 x} \left (384 x^4+6 x^5\right )+e^x \left (-5-5 x+2304 x^4+36 x^5\right )\right ) \log (x) \log (\log (x))+\left (27 x^4+18 e^x x^4+3 e^{2 x} x^4\right ) \log (x) \log ^2(\log (x))}{\left (36864 x^2+1152 x^3+9 x^4+e^{2 x} \left (4096 x^2+128 x^3+x^4\right )+e^x \left (24576 x^2+768 x^3+6 x^4\right )\right ) \log (x)+\left (1152 x^2+18 x^3+e^{2 x} \left (128 x^2+2 x^3\right )+e^x \left (768 x^2+12 x^3\right )\right ) \log (x) \log (\log (x))+\left (9 x^2+6 e^x x^2+e^{2 x} x^2\right ) \log (x) \log ^2(\log (x))} \, dx=\frac {x^{5} e^{x} + x^{4} e^{x} \log \left (\log \left (x\right )\right ) + 3 \, x^{5} + 64 \, x^{4} e^{x} + 3 \, x^{4} \log \left (\log \left (x\right )\right ) + 192 \, x^{4} + 5}{x^{2} e^{x} + x e^{x} \log \left (\log \left (x\right )\right ) + 3 \, x^{2} + 64 \, x e^{x} + 3 \, x \log \left (\log \left (x\right )\right ) + 192 \, x} \]
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Time = 12.73 (sec) , antiderivative size = 191, normalized size of antiderivative = 7.64 \[ \int \frac {-15-5 e^x+\left (-960-30 x+110592 x^4+3456 x^5+27 x^6+e^{2 x} \left (12288 x^4+384 x^5+3 x^6\right )+e^x \left (-320-330 x-5 x^2+73728 x^4+2304 x^5+18 x^6\right )\right ) \log (x)+\left (-15+3456 x^4+54 x^5+e^{2 x} \left (384 x^4+6 x^5\right )+e^x \left (-5-5 x+2304 x^4+36 x^5\right )\right ) \log (x) \log (\log (x))+\left (27 x^4+18 e^x x^4+3 e^{2 x} x^4\right ) \log (x) \log ^2(\log (x))}{\left (36864 x^2+1152 x^3+9 x^4+e^{2 x} \left (4096 x^2+128 x^3+x^4\right )+e^x \left (24576 x^2+768 x^3+6 x^4\right )\right ) \log (x)+\left (1152 x^2+18 x^3+e^{2 x} \left (128 x^2+2 x^3\right )+e^x \left (768 x^2+12 x^3\right )\right ) \log (x) \log (\log (x))+\left (9 x^2+6 e^x x^2+e^{2 x} x^2\right ) \log (x) \log ^2(\log (x))} \, dx=\frac {15}{x\,\left ({\mathrm {e}}^{2\,x}+6\,{\mathrm {e}}^x+9\right )}+x^3+\frac {\frac {5\,\left ({\mathrm {e}}^x+192\,\ln \left (x\right )+64\,{\mathrm {e}}^x\,\ln \left (x\right )+6\,x\,\ln \left (x\right )+66\,x\,{\mathrm {e}}^x\,\ln \left (x\right )+x^2\,{\mathrm {e}}^x\,\ln \left (x\right )+3\right )}{x\,\left (x\,\ln \left (x\right )+1\right )\,{\left ({\mathrm {e}}^x+3\right )}^2}+\frac {5\,\ln \left (\ln \left (x\right )\right )\,\ln \left (x\right )\,\left ({\mathrm {e}}^x+x\,{\mathrm {e}}^x+3\right )}{x\,\left (x\,\ln \left (x\right )+1\right )\,{\left ({\mathrm {e}}^x+3\right )}^2}}{x+\ln \left (\ln \left (x\right )\right )+64}-\frac {5\,\left (x^2+x\right )}{x^3\,\left ({\mathrm {e}}^x+3\right )}+\frac {5\,\left (9\,x-{\mathrm {e}}^{2\,x}-6\,{\mathrm {e}}^x+3\,x^2\,{\mathrm {e}}^x+x^2\,{\mathrm {e}}^{2\,x}+3\,x\,{\mathrm {e}}^x-9\right )}{x^2\,\left (x\,\ln \left (x\right )+1\right )\,{\left ({\mathrm {e}}^x+3\right )}^3\,\left (x-1\right )} \]
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