Integrand size = 120, antiderivative size = 26 \[ \int \frac {3 x+2 x^2+\left (4 e^{2 e}+e^e \left (9 x+6 x^2\right )\right ) \log (x)+e^{2 e} \left (-2+9 x+6 x^2\right ) \log ^2(x)+e^{3 e} \left (-2+3 x+2 x^2\right ) \log ^3(x)}{2 x^2+6 e^e x^2 \log (x)+6 e^{2 e} x^2 \log ^2(x)+2 e^{3 e} x^2 \log ^3(x)} \, dx=2+x+\frac {x}{\left (x+\frac {e^{-e} x}{\log (x)}\right )^2}+\frac {3 \log (x)}{2} \]
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Result contains higher order function than in optimal. Order 4 vs. order 3 in optimal.
Time = 0.91 (sec) , antiderivative size = 162, normalized size of antiderivative = 6.23, number of steps used = 15, number of rules used = 7, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.058, Rules used = {6820, 12, 6874, 14, 2343, 2346, 2209} \[ \int \frac {3 x+2 x^2+\left (4 e^{2 e}+e^e \left (9 x+6 x^2\right )\right ) \log (x)+e^{2 e} \left (-2+9 x+6 x^2\right ) \log ^2(x)+e^{3 e} \left (-2+3 x+2 x^2\right ) \log ^3(x)}{2 x^2+6 e^e x^2 \log (x)+6 e^{2 e} x^2 \log ^2(x)+2 e^{3 e} x^2 \log ^3(x)} \, dx=e^{e^{-e}-2 e} \left (1-2 e^e\right ) \operatorname {ExpIntegralEi}\left (-e^{-e} \left (e^e \log (x)+1\right )\right )+2 e^{e^{-e}-e} \operatorname {ExpIntegralEi}\left (-e^{-e} \left (e^e \log (x)+1\right )\right )-e^{e^{-e}-2 e} \operatorname {ExpIntegralEi}\left (-e^{-e} \left (e^e \log (x)+1\right )\right )+x+\frac {1}{x}+\frac {3 \log (x)}{2}-\frac {2-e^{-e}}{x \left (e^e \log (x)+1\right )}-\frac {e^{-e}}{x \left (e^e \log (x)+1\right )}+\frac {1}{x \left (e^e \log (x)+1\right )^2} \]
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Rule 12
Rule 14
Rule 2209
Rule 2343
Rule 2346
Rule 6820
Rule 6874
Rubi steps \begin{align*} \text {integral}& = \int \frac {x (3+2 x)+e^e \left (4 e^e+3 x (3+2 x)\right ) \log (x)+e^{2 e} \left (-2+9 x+6 x^2\right ) \log ^2(x)+e^{3 e} \left (-2+3 x+2 x^2\right ) \log ^3(x)}{2 x^2 \left (1+e^e \log (x)\right )^3} \, dx \\ & = \frac {1}{2} \int \frac {x (3+2 x)+e^e \left (4 e^e+3 x (3+2 x)\right ) \log (x)+e^{2 e} \left (-2+9 x+6 x^2\right ) \log ^2(x)+e^{3 e} \left (-2+3 x+2 x^2\right ) \log ^3(x)}{x^2 \left (1+e^e \log (x)\right )^3} \, dx \\ & = \frac {1}{2} \int \left (\frac {-2+3 x+2 x^2}{x^2}-\frac {4 e^e}{x^2 \left (1+e^e \log (x)\right )^3}+\frac {2 \left (-1+2 e^e\right )}{x^2 \left (1+e^e \log (x)\right )^2}+\frac {4}{x^2 \left (1+e^e \log (x)\right )}\right ) \, dx \\ & = \frac {1}{2} \int \frac {-2+3 x+2 x^2}{x^2} \, dx+2 \int \frac {1}{x^2 \left (1+e^e \log (x)\right )} \, dx-\left (2 e^e\right ) \int \frac {1}{x^2 \left (1+e^e \log (x)\right )^3} \, dx+\left (-1+2 e^e\right ) \int \frac {1}{x^2 \left (1+e^e \log (x)\right )^2} \, dx \\ & = \frac {1}{x \left (1+e^e \log (x)\right )^2}+\frac {e^{-e} \left (1-2 e^e\right )}{x \left (1+e^e \log (x)\right )}+\frac {1}{2} \int \left (2-\frac {2}{x^2}+\frac {3}{x}\right ) \, dx+2 \text {Subst}\left (\int \frac {e^{-x}}{1+e^e x} \, dx,x,\log (x)\right )+\left (-2+e^{-e}\right ) \int \frac {1}{x^2 \left (1+e^e \log (x)\right )} \, dx+\int \frac {1}{x^2 \left (1+e^e \log (x)\right )^2} \, dx \\ & = \frac {1}{x}+x+2 e^{-e+e^{-e}} \text {Ei}\left (-e^{-e} \left (1+e^e \log (x)\right )\right )+\frac {3 \log (x)}{2}+\frac {1}{x \left (1+e^e \log (x)\right )^2}-\frac {e^{-e}}{x \left (1+e^e \log (x)\right )}+\frac {e^{-e} \left (1-2 e^e\right )}{x \left (1+e^e \log (x)\right )}-e^{-e} \int \frac {1}{x^2 \left (1+e^e \log (x)\right )} \, dx+\left (-2+e^{-e}\right ) \text {Subst}\left (\int \frac {e^{-x}}{1+e^e x} \, dx,x,\log (x)\right ) \\ & = \frac {1}{x}+x+2 e^{-e+e^{-e}} \text {Ei}\left (-e^{-e} \left (1+e^e \log (x)\right )\right )+e^{-2 e+e^{-e}} \left (1-2 e^e\right ) \text {Ei}\left (-e^{-e} \left (1+e^e \log (x)\right )\right )+\frac {3 \log (x)}{2}+\frac {1}{x \left (1+e^e \log (x)\right )^2}-\frac {e^{-e}}{x \left (1+e^e \log (x)\right )}+\frac {e^{-e} \left (1-2 e^e\right )}{x \left (1+e^e \log (x)\right )}-e^{-e} \text {Subst}\left (\int \frac {e^{-x}}{1+e^e x} \, dx,x,\log (x)\right ) \\ & = \frac {1}{x}+x-e^{-2 e+e^{-e}} \text {Ei}\left (-e^{-e} \left (1+e^e \log (x)\right )\right )+2 e^{-e+e^{-e}} \text {Ei}\left (-e^{-e} \left (1+e^e \log (x)\right )\right )+e^{-2 e+e^{-e}} \left (1-2 e^e\right ) \text {Ei}\left (-e^{-e} \left (1+e^e \log (x)\right )\right )+\frac {3 \log (x)}{2}+\frac {1}{x \left (1+e^e \log (x)\right )^2}-\frac {e^{-e}}{x \left (1+e^e \log (x)\right )}+\frac {e^{-e} \left (1-2 e^e\right )}{x \left (1+e^e \log (x)\right )} \\ \end{align*}
Time = 0.14 (sec) , antiderivative size = 42, normalized size of antiderivative = 1.62 \[ \int \frac {3 x+2 x^2+\left (4 e^{2 e}+e^e \left (9 x+6 x^2\right )\right ) \log (x)+e^{2 e} \left (-2+9 x+6 x^2\right ) \log ^2(x)+e^{3 e} \left (-2+3 x+2 x^2\right ) \log ^3(x)}{2 x^2+6 e^e x^2 \log (x)+6 e^{2 e} x^2 \log ^2(x)+2 e^{3 e} x^2 \log ^3(x)} \, dx=\frac {1}{2} \left (3 \log (x)+2 \left (x-\frac {2}{x+e^e x \log (x)}+\frac {1+\frac {1}{\left (1+e^e \log (x)\right )^2}}{x}\right )\right ) \]
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Time = 0.36 (sec) , antiderivative size = 43, normalized size of antiderivative = 1.65
method | result | size |
risch | \(\frac {3 x \ln \left (x \right )+2 x^{2}+2}{2 x}-\frac {2 \,{\mathrm e}^{{\mathrm e}} \ln \left (x \right )+1}{x \left ({\mathrm e}^{{\mathrm e}} \ln \left (x \right )+1\right )^{2}}\) | \(43\) |
norman | \(\frac {x^{2}+{\mathrm e}^{2 \,{\mathrm e}} \ln \left (x \right )^{2}+x^{2} {\mathrm e}^{2 \,{\mathrm e}} \ln \left (x \right )^{2}+\frac {3 \,{\mathrm e}^{2 \,{\mathrm e}} x \ln \left (x \right )^{3}}{2}+\frac {3 x \ln \left (x \right )}{2}+3 \,{\mathrm e}^{{\mathrm e}} x \ln \left (x \right )^{2}+2 x^{2} {\mathrm e}^{{\mathrm e}} \ln \left (x \right )}{x \left ({\mathrm e}^{{\mathrm e}} \ln \left (x \right )+1\right )^{2}}\) | \(79\) |
parallelrisch | \(\frac {3 \,{\mathrm e}^{2 \,{\mathrm e}} x \ln \left (x \right )^{3}+2 x^{2} {\mathrm e}^{2 \,{\mathrm e}} \ln \left (x \right )^{2}+2 \,{\mathrm e}^{2 \,{\mathrm e}} \ln \left (x \right )^{2}+6 \,{\mathrm e}^{{\mathrm e}} x \ln \left (x \right )^{2}+4 x^{2} {\mathrm e}^{{\mathrm e}} \ln \left (x \right )+3 x \ln \left (x \right )+2 x^{2}}{2 x \left ({\mathrm e}^{2 \,{\mathrm e}} \ln \left (x \right )^{2}+2 \,{\mathrm e}^{{\mathrm e}} \ln \left (x \right )+1\right )}\) | \(95\) |
default | \(\text {Expression too large to display}\) | \(960\) |
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Leaf count of result is larger than twice the leaf count of optimal. 82 vs. \(2 (24) = 48\).
Time = 0.25 (sec) , antiderivative size = 82, normalized size of antiderivative = 3.15 \[ \int \frac {3 x+2 x^2+\left (4 e^{2 e}+e^e \left (9 x+6 x^2\right )\right ) \log (x)+e^{2 e} \left (-2+9 x+6 x^2\right ) \log ^2(x)+e^{3 e} \left (-2+3 x+2 x^2\right ) \log ^3(x)}{2 x^2+6 e^e x^2 \log (x)+6 e^{2 e} x^2 \log ^2(x)+2 e^{3 e} x^2 \log ^3(x)} \, dx=\frac {3 \, x e^{\left (2 \, e\right )} \log \left (x\right )^{3} + 2 \, {\left ({\left (x^{2} + 1\right )} e^{\left (2 \, e\right )} + 3 \, x e^{e}\right )} \log \left (x\right )^{2} + 2 \, x^{2} + {\left (4 \, x^{2} e^{e} + 3 \, x\right )} \log \left (x\right )}{2 \, {\left (x e^{\left (2 \, e\right )} \log \left (x\right )^{2} + 2 \, x e^{e} \log \left (x\right ) + x\right )}} \]
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Leaf count of result is larger than twice the leaf count of optimal. 51 vs. \(2 (24) = 48\).
Time = 0.09 (sec) , antiderivative size = 51, normalized size of antiderivative = 1.96 \[ \int \frac {3 x+2 x^2+\left (4 e^{2 e}+e^e \left (9 x+6 x^2\right )\right ) \log (x)+e^{2 e} \left (-2+9 x+6 x^2\right ) \log ^2(x)+e^{3 e} \left (-2+3 x+2 x^2\right ) \log ^3(x)}{2 x^2+6 e^e x^2 \log (x)+6 e^{2 e} x^2 \log ^2(x)+2 e^{3 e} x^2 \log ^3(x)} \, dx=x + \frac {- 2 e^{e} \log {\left (x \right )} - 1}{x e^{2 e} \log {\left (x \right )}^{2} + 2 x e^{e} \log {\left (x \right )} + x} + \frac {3 \log {\left (x \right )}}{2} + \frac {1}{x} \]
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Leaf count of result is larger than twice the leaf count of optimal. 108 vs. \(2 (24) = 48\).
Time = 0.23 (sec) , antiderivative size = 108, normalized size of antiderivative = 4.15 \[ \int \frac {3 x+2 x^2+\left (4 e^{2 e}+e^e \left (9 x+6 x^2\right )\right ) \log (x)+e^{2 e} \left (-2+9 x+6 x^2\right ) \log ^2(x)+e^{3 e} \left (-2+3 x+2 x^2\right ) \log ^3(x)}{2 x^2+6 e^e x^2 \log (x)+6 e^{2 e} x^2 \log ^2(x)+2 e^{3 e} x^2 \log ^3(x)} \, dx=\frac {8 \, x^{2} e^{\left (2 \, e\right )} \log \left (x\right ) + 4 \, x^{2} e^{e} + 4 \, {\left (x^{2} e^{\left (3 \, e\right )} + e^{\left (3 \, e\right )}\right )} \log \left (x\right )^{2} + 3 \, x}{4 \, {\left (x e^{\left (3 \, e\right )} \log \left (x\right )^{2} + 2 \, x e^{\left (2 \, e\right )} \log \left (x\right ) + x e^{e}\right )}} - \frac {3}{4 \, {\left (e^{\left (3 \, e\right )} \log \left (x\right )^{2} + 2 \, e^{\left (2 \, e\right )} \log \left (x\right ) + e^{e}\right )}} + \frac {3}{2} \, \log \left (x\right ) \]
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Leaf count of result is larger than twice the leaf count of optimal. 93 vs. \(2 (24) = 48\).
Time = 0.28 (sec) , antiderivative size = 93, normalized size of antiderivative = 3.58 \[ \int \frac {3 x+2 x^2+\left (4 e^{2 e}+e^e \left (9 x+6 x^2\right )\right ) \log (x)+e^{2 e} \left (-2+9 x+6 x^2\right ) \log ^2(x)+e^{3 e} \left (-2+3 x+2 x^2\right ) \log ^3(x)}{2 x^2+6 e^e x^2 \log (x)+6 e^{2 e} x^2 \log ^2(x)+2 e^{3 e} x^2 \log ^3(x)} \, dx=\frac {2 \, x^{2} e^{\left (2 \, e\right )} \log \left (x\right )^{2} + 3 \, x e^{\left (2 \, e\right )} \log \left (x\right )^{3} + 4 \, x^{2} e^{e} \log \left (x\right ) + 6 \, x e^{e} \log \left (x\right )^{2} + 2 \, e^{\left (2 \, e\right )} \log \left (x\right )^{2} + 2 \, x^{2} + 3 \, x \log \left (x\right )}{2 \, {\left (x e^{\left (2 \, e\right )} \log \left (x\right )^{2} + 2 \, x e^{e} \log \left (x\right ) + x\right )}} \]
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Time = 10.89 (sec) , antiderivative size = 178, normalized size of antiderivative = 6.85 \[ \int \frac {3 x+2 x^2+\left (4 e^{2 e}+e^e \left (9 x+6 x^2\right )\right ) \log (x)+e^{2 e} \left (-2+9 x+6 x^2\right ) \log ^2(x)+e^{3 e} \left (-2+3 x+2 x^2\right ) \log ^3(x)}{2 x^2+6 e^e x^2 \log (x)+6 e^{2 e} x^2 \log ^2(x)+2 e^{3 e} x^2 \log ^3(x)} \, dx=x+\frac {3\,\ln \left (x\right )}{2}-\frac {\frac {{\mathrm {e}}^{-3\,\mathrm {e}}}{2\,x}+\frac {{\mathrm {e}}^{-\mathrm {e}}\,{\ln \left (x\right )}^2}{x}+\frac {{\mathrm {e}}^{-2\,\mathrm {e}}\,\ln \left (x\right )\,\left (2\,{\mathrm {e}}^{\mathrm {e}}+3\right )}{2\,x}}{{\ln \left (x\right )}^2+2\,{\mathrm {e}}^{-\mathrm {e}}\,\ln \left (x\right )+{\mathrm {e}}^{-2\,\mathrm {e}}}-\frac {\frac {{\mathrm {e}}^{-3\,\mathrm {e}}\,\left (2\,{\mathrm {e}}^{2\,\mathrm {e}}+3\,{\mathrm {e}}^{\mathrm {e}}-1\right )}{2\,x}-\frac {{\mathrm {e}}^{-\mathrm {e}}\,{\ln \left (x\right )}^2}{x}+\frac {{\mathrm {e}}^{-2\,\mathrm {e}}\,\ln \left (x\right )\,\left (2\,{\mathrm {e}}^{\mathrm {e}}-3\right )}{2\,x}}{{\mathrm {e}}^{-\mathrm {e}}+\ln \left (x\right )}-\frac {{\mathrm {e}}^{-\mathrm {e}}\,\ln \left (x\right )}{x}+\frac {{\mathrm {e}}^{-2\,\mathrm {e}}\,\left (2\,{\mathrm {e}}^{2\,\mathrm {e}}+4\,{\mathrm {e}}^{\mathrm {e}}-1\right )}{2\,x} \]
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