Integrand size = 79, antiderivative size = 24 \[ \int \frac {e^3 \left (-3 x^2-2 x^3\right )+e^3 (12+8 x) \log ^4(2)+\left (-6 x+e^3 \left (-6 x^2-2 x^3\right )\right ) \log \left (3+e^3 \left (3 x+x^2\right )\right )}{3+e^3 \left (3 x+x^2\right )} \, dx=\left (-x^2+4 \log ^4(2)\right ) \log \left (3+e^3 x (3+x)\right ) \]
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Leaf count is larger than twice the leaf count of optimal. \(90\) vs. \(2(24)=48\).
Time = 0.29 (sec) , antiderivative size = 90, normalized size of antiderivative = 3.75, number of steps used = 16, number of rules used = 9, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.114, Rules used = {6820, 1642, 648, 632, 212, 642, 2605, 12, 814} \[ \int \frac {e^3 \left (-3 x^2-2 x^3\right )+e^3 (12+8 x) \log ^4(2)+\left (-6 x+e^3 \left (-6 x^2-2 x^3\right )\right ) \log \left (3+e^3 \left (3 x+x^2\right )\right )}{3+e^3 \left (3 x+x^2\right )} \, dx=\frac {\left (6-e^3 \left (9-8 \log ^4(2)\right )\right ) \log \left (e^3 x^2+3 e^3 x+3\right )}{2 e^3}+x^2 \left (-\log \left (e^3 x^2+3 e^3 x+3\right )\right )-\frac {3 \left (2-3 e^3\right ) \log \left (e^3 x^2+3 e^3 x+3\right )}{2 e^3} \]
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Rule 12
Rule 212
Rule 632
Rule 642
Rule 648
Rule 814
Rule 1642
Rule 2605
Rule 6820
Rubi steps \begin{align*} \text {integral}& = \int \left (-\frac {e^3 (3+2 x) \left (x^2-4 \log ^4(2)\right )}{3+3 e^3 x+e^3 x^2}-2 x \log \left (3+3 e^3 x+e^3 x^2\right )\right ) \, dx \\ & = -\left (2 \int x \log \left (3+3 e^3 x+e^3 x^2\right ) \, dx\right )-e^3 \int \frac {(3+2 x) \left (x^2-4 \log ^4(2)\right )}{3+3 e^3 x+e^3 x^2} \, dx \\ & = -x^2 \log \left (3+3 e^3 x+e^3 x^2\right )-e^3 \int \left (-\frac {3}{e^3}+\frac {2 x}{e^3}-\frac {-3 \left (3-4 e^3 \log ^4(2)\right )+x \left (6-e^3 \left (9-8 \log ^4(2)\right )\right )}{e^3 \left (3+3 e^3 x+e^3 x^2\right )}\right ) \, dx+\int \frac {e^3 x^2 (3+2 x)}{3+3 e^3 x+e^3 x^2} \, dx \\ & = 3 x-x^2-x^2 \log \left (3+3 e^3 x+e^3 x^2\right )+e^3 \int \frac {x^2 (3+2 x)}{3+3 e^3 x+e^3 x^2} \, dx+\int \frac {-3 \left (3-4 e^3 \log ^4(2)\right )+x \left (6-e^3 \left (9-8 \log ^4(2)\right )\right )}{3+3 e^3 x+e^3 x^2} \, dx \\ & = 3 x-x^2-x^2 \log \left (3+3 e^3 x+e^3 x^2\right )+e^3 \int \left (-\frac {3}{e^3}+\frac {2 x}{e^3}+\frac {3 \left (3-\left (2-3 e^3\right ) x\right )}{e^3 \left (3+3 e^3 x+e^3 x^2\right )}\right ) \, dx-\frac {1}{2} \left (9 \left (4-3 e^3\right )\right ) \int \frac {1}{3+3 e^3 x+e^3 x^2} \, dx+\frac {\left (6-e^3 \left (9-8 \log ^4(2)\right )\right ) \int \frac {3 e^3+2 e^3 x}{3+3 e^3 x+e^3 x^2} \, dx}{2 e^3} \\ & = -x^2 \log \left (3+3 e^3 x+e^3 x^2\right )+\frac {\left (6-e^3 \left (9-8 \log ^4(2)\right )\right ) \log \left (3+3 e^3 x+e^3 x^2\right )}{2 e^3}+3 \int \frac {3-\left (2-3 e^3\right ) x}{3+3 e^3 x+e^3 x^2} \, dx+\left (9 \left (4-3 e^3\right )\right ) \text {Subst}\left (\int \frac {1}{-3 e^3 \left (4-3 e^3\right )-x^2} \, dx,x,3 e^3+2 e^3 x\right ) \\ & = -\frac {3 \sqrt {3 \left (-4+3 e^3\right )} \text {arctanh}\left (\frac {e^{3/2} (3+2 x)}{\sqrt {3 \left (-4+3 e^3\right )}}\right )}{e^{3/2}}-x^2 \log \left (3+3 e^3 x+e^3 x^2\right )+\frac {\left (6-e^3 \left (9-8 \log ^4(2)\right )\right ) \log \left (3+3 e^3 x+e^3 x^2\right )}{2 e^3}+\frac {1}{2} \left (9 \left (4-3 e^3\right )\right ) \int \frac {1}{3+3 e^3 x+e^3 x^2} \, dx+\frac {\left (3 \left (-2+3 e^3\right )\right ) \int \frac {3 e^3+2 e^3 x}{3+3 e^3 x+e^3 x^2} \, dx}{2 e^3} \\ & = -\frac {3 \sqrt {3 \left (-4+3 e^3\right )} \text {arctanh}\left (\frac {e^{3/2} (3+2 x)}{\sqrt {3 \left (-4+3 e^3\right )}}\right )}{e^{3/2}}-\frac {3 \left (2-3 e^3\right ) \log \left (3+3 e^3 x+e^3 x^2\right )}{2 e^3}-x^2 \log \left (3+3 e^3 x+e^3 x^2\right )+\frac {\left (6-e^3 \left (9-8 \log ^4(2)\right )\right ) \log \left (3+3 e^3 x+e^3 x^2\right )}{2 e^3}-\left (9 \left (4-3 e^3\right )\right ) \text {Subst}\left (\int \frac {1}{-3 e^3 \left (4-3 e^3\right )-x^2} \, dx,x,3 e^3+2 e^3 x\right ) \\ & = -\frac {3 \left (2-3 e^3\right ) \log \left (3+3 e^3 x+e^3 x^2\right )}{2 e^3}-x^2 \log \left (3+3 e^3 x+e^3 x^2\right )+\frac {\left (6-e^3 \left (9-8 \log ^4(2)\right )\right ) \log \left (3+3 e^3 x+e^3 x^2\right )}{2 e^3} \\ \end{align*}
Time = 0.11 (sec) , antiderivative size = 23, normalized size of antiderivative = 0.96 \[ \int \frac {e^3 \left (-3 x^2-2 x^3\right )+e^3 (12+8 x) \log ^4(2)+\left (-6 x+e^3 \left (-6 x^2-2 x^3\right )\right ) \log \left (3+e^3 \left (3 x+x^2\right )\right )}{3+e^3 \left (3 x+x^2\right )} \, dx=-\left (\left (x^2-4 \log ^4(2)\right ) \log \left (3+e^3 x (3+x)\right )\right ) \]
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Time = 1.28 (sec) , antiderivative size = 39, normalized size of antiderivative = 1.62
method | result | size |
norman | \(4 \ln \left (2\right )^{4} \ln \left (\left (x^{2}+3 x \right ) {\mathrm e}^{3}+3\right )-x^{2} \ln \left (\left (x^{2}+3 x \right ) {\mathrm e}^{3}+3\right )\) | \(39\) |
risch | \(-x^{2} \ln \left (\left (x^{2}+3 x \right ) {\mathrm e}^{3}+3\right )+4 \ln \left (2\right )^{4} \ln \left (x^{2} {\mathrm e}^{3}+3 x \,{\mathrm e}^{3}+3\right )\) | \(40\) |
default | \(-x^{2} \ln \left (x^{2} {\mathrm e}^{3}+3 x \,{\mathrm e}^{3}+3\right )+4 \ln \left (2\right )^{4} \ln \left (x^{2} {\mathrm e}^{3}+3 x \,{\mathrm e}^{3}+3\right )\) | \(41\) |
parallelrisch | \(4 \ln \left (2\right )^{4} \ln \left (\left (x^{2} {\mathrm e}^{3}+3 x \,{\mathrm e}^{3}+3\right ) {\mathrm e}^{-3}\right )-x^{2} \ln \left (\left (x^{2}+3 x \right ) {\mathrm e}^{3}+3\right )\) | \(45\) |
parts | \({\mathrm e}^{3} \left (-{\mathrm e}^{-3} \left (x^{2}-3 x \right )+{\mathrm e}^{-3} \left (\frac {\left (8 \,{\mathrm e}^{3} \ln \left (2\right )^{4}-9 \,{\mathrm e}^{3}+6\right ) {\mathrm e}^{-3} \ln \left (x^{2} {\mathrm e}^{3}+3 x \,{\mathrm e}^{3}+3\right )}{2}-\frac {2 \left (\frac {27 \,{\mathrm e}^{3}}{2}-18\right ) \operatorname {arctanh}\left (\frac {2 x \,{\mathrm e}^{3}+3 \,{\mathrm e}^{3}}{\sqrt {-12 \,{\mathrm e}^{3}+9 \,{\mathrm e}^{6}}}\right )}{\sqrt {-12 \,{\mathrm e}^{3}+9 \,{\mathrm e}^{6}}}\right )\right )-x^{2} \ln \left (x^{2} {\mathrm e}^{3}+3 x \,{\mathrm e}^{3}+3\right )+{\mathrm e}^{3} \left ({\mathrm e}^{-3} \left (x^{2}-3 x \right )+3 \,{\mathrm e}^{-3} \left (\frac {\left (3 \,{\mathrm e}^{3}-2\right ) {\mathrm e}^{-3} \ln \left (x^{2} {\mathrm e}^{3}+3 x \,{\mathrm e}^{3}+3\right )}{2}-\frac {2 \left (-\frac {9 \,{\mathrm e}^{3}}{2}+6\right ) \operatorname {arctanh}\left (\frac {2 x \,{\mathrm e}^{3}+3 \,{\mathrm e}^{3}}{\sqrt {-12 \,{\mathrm e}^{3}+9 \,{\mathrm e}^{6}}}\right )}{\sqrt {-12 \,{\mathrm e}^{3}+9 \,{\mathrm e}^{6}}}\right )\right )\) | \(215\) |
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Time = 0.27 (sec) , antiderivative size = 26, normalized size of antiderivative = 1.08 \[ \int \frac {e^3 \left (-3 x^2-2 x^3\right )+e^3 (12+8 x) \log ^4(2)+\left (-6 x+e^3 \left (-6 x^2-2 x^3\right )\right ) \log \left (3+e^3 \left (3 x+x^2\right )\right )}{3+e^3 \left (3 x+x^2\right )} \, dx={\left (4 \, \log \left (2\right )^{4} - x^{2}\right )} \log \left ({\left (x^{2} + 3 \, x\right )} e^{3} + 3\right ) \]
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Time = 0.20 (sec) , antiderivative size = 39, normalized size of antiderivative = 1.62 \[ \int \frac {e^3 \left (-3 x^2-2 x^3\right )+e^3 (12+8 x) \log ^4(2)+\left (-6 x+e^3 \left (-6 x^2-2 x^3\right )\right ) \log \left (3+e^3 \left (3 x+x^2\right )\right )}{3+e^3 \left (3 x+x^2\right )} \, dx=- x^{2} \log {\left (\left (x^{2} + 3 x\right ) e^{3} + 3 \right )} + 4 \log {\left (2 \right )}^{4} \log {\left (x^{2} e^{3} + 3 x e^{3} + 3 \right )} \]
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Leaf count of result is larger than twice the leaf count of optimal. 493 vs. \(2 (23) = 46\).
Time = 0.36 (sec) , antiderivative size = 493, normalized size of antiderivative = 20.54 \[ \int \frac {e^3 \left (-3 x^2-2 x^3\right )+e^3 (12+8 x) \log ^4(2)+\left (-6 x+e^3 \left (-6 x^2-2 x^3\right )\right ) \log \left (3+e^3 \left (3 x+x^2\right )\right )}{3+e^3 \left (3 x+x^2\right )} \, dx=4 \, {\left (e^{\left (-3\right )} \log \left (x^{2} e^{3} + 3 \, x e^{3} + 3\right ) - \frac {\sqrt {3} e^{\left (-\frac {3}{2}\right )} \log \left (\frac {2 \, x e^{3} - \sqrt {3} \sqrt {3 \, e^{3} - 4} e^{\frac {3}{2}} + 3 \, e^{3}}{2 \, x e^{3} + \sqrt {3} \sqrt {3 \, e^{3} - 4} e^{\frac {3}{2}} + 3 \, e^{3}}\right )}{\sqrt {3 \, e^{3} - 4}}\right )} e^{3} \log \left (2\right )^{4} + \frac {4 \, \sqrt {3} e^{\frac {3}{2}} \log \left (2\right )^{4} \log \left (\frac {2 \, x e^{3} - \sqrt {3} \sqrt {3 \, e^{3} - 4} e^{\frac {3}{2}} + 3 \, e^{3}}{2 \, x e^{3} + \sqrt {3} \sqrt {3 \, e^{3} - 4} e^{\frac {3}{2}} + 3 \, e^{3}}\right )}{\sqrt {3 \, e^{3} - 4}} - \frac {3}{2} \, \sqrt {3} \sqrt {3 \, e^{3} - 4} e^{\left (-\frac {3}{2}\right )} \log \left (\frac {2 \, x e^{3} - \sqrt {3} \sqrt {3 \, e^{3} - 4} e^{\frac {3}{2}} + 3 \, e^{3}}{2 \, x e^{3} + \sqrt {3} \sqrt {3 \, e^{3} - 4} e^{\frac {3}{2}} + 3 \, e^{3}}\right ) - {\left (3 \, {\left (3 \, e^{3} - 1\right )} e^{\left (-6\right )} \log \left (x^{2} e^{3} + 3 \, x e^{3} + 3\right ) - \frac {9 \, \sqrt {3} {\left (e^{3} - 1\right )} e^{\left (-\frac {9}{2}\right )} \log \left (\frac {2 \, x e^{3} - \sqrt {3} \sqrt {3 \, e^{3} - 4} e^{\frac {3}{2}} + 3 \, e^{3}}{2 \, x e^{3} + \sqrt {3} \sqrt {3 \, e^{3} - 4} e^{\frac {3}{2}} + 3 \, e^{3}}\right )}{\sqrt {3 \, e^{3} - 4}} + {\left (x^{2} - 6 \, x\right )} e^{\left (-3\right )}\right )} e^{3} - \frac {3}{2} \, {\left (\frac {\sqrt {3} {\left (3 \, e^{3} - 2\right )} e^{\left (-\frac {9}{2}\right )} \log \left (\frac {2 \, x e^{3} - \sqrt {3} \sqrt {3 \, e^{3} - 4} e^{\frac {3}{2}} + 3 \, e^{3}}{2 \, x e^{3} + \sqrt {3} \sqrt {3 \, e^{3} - 4} e^{\frac {3}{2}} + 3 \, e^{3}}\right )}{\sqrt {3 \, e^{3} - 4}} + 2 \, x e^{\left (-3\right )} - 3 \, e^{\left (-3\right )} \log \left (x^{2} e^{3} + 3 \, x e^{3} + 3\right )\right )} e^{3} + \frac {1}{2} \, {\left (2 \, x^{2} e^{3} - 6 \, x e^{3} - {\left (2 \, x^{2} e^{3} - 9 \, e^{3} + 6\right )} \log \left (x^{2} e^{3} + 3 \, x e^{3} + 3\right )\right )} e^{\left (-3\right )} \]
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Time = 0.32 (sec) , antiderivative size = 40, normalized size of antiderivative = 1.67 \[ \int \frac {e^3 \left (-3 x^2-2 x^3\right )+e^3 (12+8 x) \log ^4(2)+\left (-6 x+e^3 \left (-6 x^2-2 x^3\right )\right ) \log \left (3+e^3 \left (3 x+x^2\right )\right )}{3+e^3 \left (3 x+x^2\right )} \, dx=4 \, \log \left (2\right )^{4} \log \left (x^{2} e^{3} + 3 \, x e^{3} + 3\right ) - x^{2} \log \left (x^{2} e^{3} + 3 \, x e^{3} + 3\right ) \]
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Time = 0.57 (sec) , antiderivative size = 27, normalized size of antiderivative = 1.12 \[ \int \frac {e^3 \left (-3 x^2-2 x^3\right )+e^3 (12+8 x) \log ^4(2)+\left (-6 x+e^3 \left (-6 x^2-2 x^3\right )\right ) \log \left (3+e^3 \left (3 x+x^2\right )\right )}{3+e^3 \left (3 x+x^2\right )} \, dx=\ln \left ({\mathrm {e}}^3\,x^2+3\,{\mathrm {e}}^3\,x+3\right )\,\left (4\,{\ln \left (2\right )}^4-x^2\right ) \]
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