Integrand size = 94, antiderivative size = 24 \[ \int \frac {-4+\left (4 x-4 e x-6 x^2\right ) \log \left (x^2\right )+\left (-2 x+2 e x+6 x^2\right ) \log ^2\left (x^2\right )+\left (2 x^2+2 e^2 x^2-9 x^3+9 x^4+e \left (-4 x^2+9 x^3\right )\right ) \log ^3\left (x^2\right )}{x \log ^3\left (x^2\right )} \, dx=x^2 \left (-1+e+\frac {3 x}{2}+\frac {1}{x \log \left (x^2\right )}\right )^2 \]
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Leaf count is larger than twice the leaf count of optimal. \(55\) vs. \(2(24)=48\).
Time = 0.19 (sec) , antiderivative size = 55, normalized size of antiderivative = 2.29, number of steps used = 21, number of rules used = 10, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.106, Rules used = {6820, 14, 2339, 30, 2357, 2367, 2337, 2209, 2344, 2335} \[ \int \frac {-4+\left (4 x-4 e x-6 x^2\right ) \log \left (x^2\right )+\left (-2 x+2 e x+6 x^2\right ) \log ^2\left (x^2\right )+\left (2 x^2+2 e^2 x^2-9 x^3+9 x^4+e \left (-4 x^2+9 x^3\right )\right ) \log ^3\left (x^2\right )}{x \log ^3\left (x^2\right )} \, dx=\frac {9 x^4}{4}-3 (1-e) x^3+(1-e)^2 x^2+\frac {1}{\log ^2\left (x^2\right )}-\frac {(2 (1-e)-3 x) x}{\log \left (x^2\right )} \]
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Rule 14
Rule 30
Rule 2209
Rule 2335
Rule 2337
Rule 2339
Rule 2344
Rule 2357
Rule 2367
Rule 6820
Rubi steps \begin{align*} \text {integral}& = \int \left (x \left (2 (1-e)^2-9 (1-e) x+9 x^2\right )-\frac {4}{x \log ^3\left (x^2\right )}+\frac {4-4 e-6 x}{\log ^2\left (x^2\right )}+\frac {2 (-1+e+3 x)}{\log \left (x^2\right )}\right ) \, dx \\ & = 2 \int \frac {-1+e+3 x}{\log \left (x^2\right )} \, dx-4 \int \frac {1}{x \log ^3\left (x^2\right )} \, dx+\int x \left (2 (1-e)^2-9 (1-e) x+9 x^2\right ) \, dx+\int \frac {4-4 e-6 x}{\log ^2\left (x^2\right )} \, dx \\ & = -\frac {(2 (1-e)-3 x) x}{\log \left (x^2\right )}+2 \int \left (\frac {-1+e}{\log \left (x^2\right )}+\frac {3 x}{\log \left (x^2\right )}\right ) \, dx-2 \text {Subst}\left (\int \frac {1}{x^3} \, dx,x,\log \left (x^2\right )\right )-(2 (1-e)) \int \frac {1}{\log \left (x^2\right )} \, dx+\int \left (2 (-1+e)^2 x+9 (-1+e) x^2+9 x^3\right ) \, dx+\int \frac {4-4 e-6 x}{\log \left (x^2\right )} \, dx \\ & = (1-e)^2 x^2-3 (1-e) x^3+\frac {9 x^4}{4}+\frac {1}{\log ^2\left (x^2\right )}-\frac {(2 (1-e)-3 x) x}{\log \left (x^2\right )}+6 \int \frac {x}{\log \left (x^2\right )} \, dx-(2 (1-e)) \int \frac {1}{\log \left (x^2\right )} \, dx-\frac {((1-e) x) \text {Subst}\left (\int \frac {e^{x/2}}{x} \, dx,x,\log \left (x^2\right )\right )}{\sqrt {x^2}}+\int \left (\frac {4 (1-e)}{\log \left (x^2\right )}-\frac {6 x}{\log \left (x^2\right )}\right ) \, dx \\ & = (1-e)^2 x^2-3 (1-e) x^3+\frac {9 x^4}{4}-\frac {(1-e) x \operatorname {ExpIntegralEi}\left (\frac {\log \left (x^2\right )}{2}\right )}{\sqrt {x^2}}+\frac {1}{\log ^2\left (x^2\right )}-\frac {(2 (1-e)-3 x) x}{\log \left (x^2\right )}+3 \text {Subst}\left (\int \frac {1}{\log (x)} \, dx,x,x^2\right )-6 \int \frac {x}{\log \left (x^2\right )} \, dx+(4 (1-e)) \int \frac {1}{\log \left (x^2\right )} \, dx-\frac {((1-e) x) \text {Subst}\left (\int \frac {e^{x/2}}{x} \, dx,x,\log \left (x^2\right )\right )}{\sqrt {x^2}} \\ & = (1-e)^2 x^2-3 (1-e) x^3+\frac {9 x^4}{4}-\frac {2 (1-e) x \operatorname {ExpIntegralEi}\left (\frac {\log \left (x^2\right )}{2}\right )}{\sqrt {x^2}}+\frac {1}{\log ^2\left (x^2\right )}-\frac {(2 (1-e)-3 x) x}{\log \left (x^2\right )}+3 \operatorname {LogIntegral}\left (x^2\right )-3 \text {Subst}\left (\int \frac {1}{\log (x)} \, dx,x,x^2\right )+\frac {(2 (1-e) x) \text {Subst}\left (\int \frac {e^{x/2}}{x} \, dx,x,\log \left (x^2\right )\right )}{\sqrt {x^2}} \\ & = (1-e)^2 x^2-3 (1-e) x^3+\frac {9 x^4}{4}+\frac {1}{\log ^2\left (x^2\right )}-\frac {(2 (1-e)-3 x) x}{\log \left (x^2\right )} \\ \end{align*}
Leaf count is larger than twice the leaf count of optimal. \(71\) vs. \(2(24)=48\).
Time = 0.15 (sec) , antiderivative size = 71, normalized size of antiderivative = 2.96 \[ \int \frac {-4+\left (4 x-4 e x-6 x^2\right ) \log \left (x^2\right )+\left (-2 x+2 e x+6 x^2\right ) \log ^2\left (x^2\right )+\left (2 x^2+2 e^2 x^2-9 x^3+9 x^4+e \left (-4 x^2+9 x^3\right )\right ) \log ^3\left (x^2\right )}{x \log ^3\left (x^2\right )} \, dx=x^2-2 e x^2+e^2 x^2-3 x^3+3 e x^3+\frac {9 x^4}{4}+\frac {1}{\log ^2\left (x^2\right )}-\frac {2 x}{\log \left (x^2\right )}+\frac {2 e x}{\log \left (x^2\right )}+\frac {3 x^2}{\log \left (x^2\right )} \]
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Leaf count of result is larger than twice the leaf count of optimal. \(68\) vs. \(2(23)=46\).
Time = 0.90 (sec) , antiderivative size = 69, normalized size of antiderivative = 2.88
method | result | size |
risch | \(\frac {9 x^{4}}{4}+3 x^{3} {\mathrm e}-3 x^{3}+x^{2} {\mathrm e}^{2}-2 x^{2} {\mathrm e}+x^{2}+\frac {2 \,{\mathrm e} \ln \left (x^{2}\right ) x +3 x^{2} \ln \left (x^{2}\right )-2 x \ln \left (x^{2}\right )+1}{\ln \left (x^{2}\right )^{2}}\) | \(69\) |
norman | \(\frac {1+\left (-3+3 \,{\mathrm e}\right ) x^{3} \ln \left (x^{2}\right )^{2}+\left (2 \,{\mathrm e}-2\right ) x \ln \left (x^{2}\right )+\left ({\mathrm e}^{2}-2 \,{\mathrm e}+1\right ) x^{2} \ln \left (x^{2}\right )^{2}+3 x^{2} \ln \left (x^{2}\right )+\frac {9 x^{4} \ln \left (x^{2}\right )^{2}}{4}}{\ln \left (x^{2}\right )^{2}}\) | \(78\) |
parallelrisch | \(\frac {12 \,{\mathrm e} x^{3} \ln \left (x^{2}\right )^{2}+9 x^{4} \ln \left (x^{2}\right )^{2}+4 \,{\mathrm e}^{2} x^{2} \ln \left (x^{2}\right )^{2}-8 x^{2} {\mathrm e} \ln \left (x^{2}\right )^{2}-12 x^{3} \ln \left (x^{2}\right )^{2}+4 x^{2} \ln \left (x^{2}\right )^{2}+8 \,{\mathrm e} \ln \left (x^{2}\right ) x +12 x^{2} \ln \left (x^{2}\right )-8 x \ln \left (x^{2}\right )+4}{4 \ln \left (x^{2}\right )^{2}}\) | \(110\) |
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Leaf count of result is larger than twice the leaf count of optimal. 75 vs. \(2 (27) = 54\).
Time = 0.28 (sec) , antiderivative size = 75, normalized size of antiderivative = 3.12 \[ \int \frac {-4+\left (4 x-4 e x-6 x^2\right ) \log \left (x^2\right )+\left (-2 x+2 e x+6 x^2\right ) \log ^2\left (x^2\right )+\left (2 x^2+2 e^2 x^2-9 x^3+9 x^4+e \left (-4 x^2+9 x^3\right )\right ) \log ^3\left (x^2\right )}{x \log ^3\left (x^2\right )} \, dx=\frac {{\left (9 \, x^{4} - 12 \, x^{3} + 4 \, x^{2} e^{2} + 4 \, x^{2} + 4 \, {\left (3 \, x^{3} - 2 \, x^{2}\right )} e\right )} \log \left (x^{2}\right )^{2} + 4 \, {\left (3 \, x^{2} + 2 \, x e - 2 \, x\right )} \log \left (x^{2}\right ) + 4}{4 \, \log \left (x^{2}\right )^{2}} \]
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Leaf count of result is larger than twice the leaf count of optimal. 58 vs. \(2 (22) = 44\).
Time = 0.06 (sec) , antiderivative size = 58, normalized size of antiderivative = 2.42 \[ \int \frac {-4+\left (4 x-4 e x-6 x^2\right ) \log \left (x^2\right )+\left (-2 x+2 e x+6 x^2\right ) \log ^2\left (x^2\right )+\left (2 x^2+2 e^2 x^2-9 x^3+9 x^4+e \left (-4 x^2+9 x^3\right )\right ) \log ^3\left (x^2\right )}{x \log ^3\left (x^2\right )} \, dx=\frac {9 x^{4}}{4} + x^{3} \left (-3 + 3 e\right ) + x^{2} \left (- 2 e + 1 + e^{2}\right ) + \frac {\left (3 x^{2} - 2 x + 2 e x\right ) \log {\left (x^{2} \right )} + 1}{\log {\left (x^{2} \right )}^{2}} \]
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Leaf count of result is larger than twice the leaf count of optimal. 59 vs. \(2 (27) = 54\).
Time = 0.24 (sec) , antiderivative size = 59, normalized size of antiderivative = 2.46 \[ \int \frac {-4+\left (4 x-4 e x-6 x^2\right ) \log \left (x^2\right )+\left (-2 x+2 e x+6 x^2\right ) \log ^2\left (x^2\right )+\left (2 x^2+2 e^2 x^2-9 x^3+9 x^4+e \left (-4 x^2+9 x^3\right )\right ) \log ^3\left (x^2\right )}{x \log ^3\left (x^2\right )} \, dx=\frac {9}{4} \, x^{4} + 3 \, x^{3} e - 3 \, x^{3} + x^{2} e^{2} - 2 \, x^{2} e + x^{2} + \frac {3 \, x^{2} + 2 \, x {\left (e - 1\right )}}{2 \, \log \left (x\right )} + \frac {1}{\log \left (x^{2}\right )^{2}} \]
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Leaf count of result is larger than twice the leaf count of optimal. 107 vs. \(2 (27) = 54\).
Time = 0.31 (sec) , antiderivative size = 107, normalized size of antiderivative = 4.46 \[ \int \frac {-4+\left (4 x-4 e x-6 x^2\right ) \log \left (x^2\right )+\left (-2 x+2 e x+6 x^2\right ) \log ^2\left (x^2\right )+\left (2 x^2+2 e^2 x^2-9 x^3+9 x^4+e \left (-4 x^2+9 x^3\right )\right ) \log ^3\left (x^2\right )}{x \log ^3\left (x^2\right )} \, dx=\frac {9 \, x^{4} \log \left (x^{2}\right )^{2} + 12 \, x^{3} e \log \left (x^{2}\right )^{2} - 12 \, x^{3} \log \left (x^{2}\right )^{2} + 4 \, x^{2} e^{2} \log \left (x^{2}\right )^{2} - 8 \, x^{2} e \log \left (x^{2}\right )^{2} + 4 \, x^{2} \log \left (x^{2}\right )^{2} + 12 \, x^{2} \log \left (x^{2}\right ) + 8 \, x e \log \left (x^{2}\right ) - 8 \, x \log \left (x^{2}\right ) + 4}{4 \, \log \left (x^{2}\right )^{2}} \]
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Time = 13.38 (sec) , antiderivative size = 47, normalized size of antiderivative = 1.96 \[ \int \frac {-4+\left (4 x-4 e x-6 x^2\right ) \log \left (x^2\right )+\left (-2 x+2 e x+6 x^2\right ) \log ^2\left (x^2\right )+\left (2 x^2+2 e^2 x^2-9 x^3+9 x^4+e \left (-4 x^2+9 x^3\right )\right ) \log ^3\left (x^2\right )}{x \log ^3\left (x^2\right )} \, dx=\frac {{\left (2\,x\,\mathrm {e}-2\,x+3\,x^2\right )}^2}{4}+\frac {\ln \left (x^2\right )\,\left (2\,x\,\mathrm {e}-2\,x+3\,x^2\right )+1}{{\ln \left (x^2\right )}^2} \]
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