Integrand size = 99, antiderivative size = 30 \[ \int \frac {-15-12 x+3 x^2+e^4 \left (-4-2 x^2+10 x^3-2 x^4\right )+\left (-3+e^4 \left (-2 x+2 x^2\right )\right ) \log \left (3+e^4 \left (2 x-2 x^2\right )\right )}{-12+12 x-3 x^2+e^4 \left (-8 x+16 x^2-10 x^3+2 x^4\right )} \, dx=-x+\frac {9+\log \left (3+e^4 \left (2 x-2 x^2\right )\right )}{2-x} \]
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Time = 0.52 (sec) , antiderivative size = 38, normalized size of antiderivative = 1.27, number of steps used = 18, number of rules used = 11, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.111, Rules used = {6873, 6820, 6860, 1642, 648, 632, 212, 642, 2605, 12, 814} \[ \int \frac {-15-12 x+3 x^2+e^4 \left (-4-2 x^2+10 x^3-2 x^4\right )+\left (-3+e^4 \left (-2 x+2 x^2\right )\right ) \log \left (3+e^4 \left (2 x-2 x^2\right )\right )}{-12+12 x-3 x^2+e^4 \left (-8 x+16 x^2-10 x^3+2 x^4\right )} \, dx=\frac {\log \left (-2 e^4 x^2+2 e^4 x+3\right )}{2-x}-x+\frac {9}{2-x} \]
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Rule 12
Rule 212
Rule 632
Rule 642
Rule 648
Rule 814
Rule 1642
Rule 2605
Rule 6820
Rule 6860
Rule 6873
Rubi steps \begin{align*} \text {integral}& = \int \frac {15+12 x-3 x^2-e^4 \left (-4-2 x^2+10 x^3-2 x^4\right )-\left (-3+e^4 \left (-2 x+2 x^2\right )\right ) \log \left (3+e^4 \left (2 x-2 x^2\right )\right )}{12-4 \left (3-2 e^4\right ) x+\left (3-16 e^4\right ) x^2+10 e^4 x^3-2 e^4 x^4} \, dx \\ & = \int \frac {-3 \left (-5-4 x+x^2\right )+2 e^4 \left (2+x^2-5 x^3+x^4\right )-\left (-3+2 e^4 (-1+x) x\right ) \log \left (3-2 e^4 (-1+x) x\right )}{(2-x)^2 \left (3+2 e^4 x-2 e^4 x^2\right )} \, dx \\ & = \int \left (\frac {15+4 e^4+12 x-\left (3-2 e^4\right ) x^2-10 e^4 x^3+2 e^4 x^4}{(2-x)^2 \left (3+2 e^4 x-2 e^4 x^2\right )}+\frac {\log \left (3+2 e^4 x-2 e^4 x^2\right )}{(2-x)^2}\right ) \, dx \\ & = \int \frac {15+4 e^4+12 x-\left (3-2 e^4\right ) x^2-10 e^4 x^3+2 e^4 x^4}{(2-x)^2 \left (3+2 e^4 x-2 e^4 x^2\right )} \, dx+\int \frac {\log \left (3+2 e^4 x-2 e^4 x^2\right )}{(2-x)^2} \, dx \\ & = \frac {\log \left (3+2 e^4 x-2 e^4 x^2\right )}{2-x}-\int \frac {2 e^4 (1-2 x)}{(2-x) \left (3+2 e^4 x-2 e^4 x^2\right )} \, dx+\int \left (-1+\frac {9}{(-2+x)^2}-\frac {6 e^4}{\left (-3+4 e^4\right ) (-2+x)}+\frac {4 e^4 \left (3-e^4+3 e^4 x\right )}{\left (-3+4 e^4\right ) \left (-3-2 e^4 x+2 e^4 x^2\right )}\right ) \, dx \\ & = \frac {9}{2-x}-x+\frac {6 e^4 \log (2-x)}{3-4 e^4}+\frac {\log \left (3+2 e^4 x-2 e^4 x^2\right )}{2-x}-\left (2 e^4\right ) \int \frac {1-2 x}{(2-x) \left (3+2 e^4 x-2 e^4 x^2\right )} \, dx-\frac {\left (4 e^4\right ) \int \frac {3-e^4+3 e^4 x}{-3-2 e^4 x+2 e^4 x^2} \, dx}{3-4 e^4} \\ & = \frac {9}{2-x}-x+\frac {6 e^4 \log (2-x)}{3-4 e^4}+\frac {\log \left (3+2 e^4 x-2 e^4 x^2\right )}{2-x}-\left (2 e^4\right ) \int \left (-\frac {3}{\left (-3+4 e^4\right ) (-2+x)}+\frac {2 \left (3-e^4+3 e^4 x\right )}{\left (-3+4 e^4\right ) \left (-3-2 e^4 x+2 e^4 x^2\right )}\right ) \, dx-\frac {\left (3 e^4\right ) \int \frac {-2 e^4+4 e^4 x}{-3-2 e^4 x+2 e^4 x^2} \, dx}{3-4 e^4}-\frac {\left (2 e^4 \left (6+e^4\right )\right ) \int \frac {1}{-3-2 e^4 x+2 e^4 x^2} \, dx}{3-4 e^4} \\ & = \frac {9}{2-x}-x-\frac {3 e^4 \log \left (3+2 e^4 x-2 e^4 x^2\right )}{3-4 e^4}+\frac {\log \left (3+2 e^4 x-2 e^4 x^2\right )}{2-x}+\frac {\left (4 e^4\right ) \int \frac {3-e^4+3 e^4 x}{-3-2 e^4 x+2 e^4 x^2} \, dx}{3-4 e^4}+\frac {\left (4 e^4 \left (6+e^4\right )\right ) \text {Subst}\left (\int \frac {1}{4 e^4 \left (6+e^4\right )-x^2} \, dx,x,-2 e^4+4 e^4 x\right )}{3-4 e^4} \\ & = \frac {9}{2-x}-x-\frac {2 e^2 \sqrt {6+e^4} \text {arctanh}\left (\frac {e^2 (1-2 x)}{\sqrt {6+e^4}}\right )}{3-4 e^4}-\frac {3 e^4 \log \left (3+2 e^4 x-2 e^4 x^2\right )}{3-4 e^4}+\frac {\log \left (3+2 e^4 x-2 e^4 x^2\right )}{2-x}+\frac {\left (3 e^4\right ) \int \frac {-2 e^4+4 e^4 x}{-3-2 e^4 x+2 e^4 x^2} \, dx}{3-4 e^4}+\frac {\left (2 e^4 \left (6+e^4\right )\right ) \int \frac {1}{-3-2 e^4 x+2 e^4 x^2} \, dx}{3-4 e^4} \\ & = \frac {9}{2-x}-x-\frac {2 e^2 \sqrt {6+e^4} \text {arctanh}\left (\frac {e^2 (1-2 x)}{\sqrt {6+e^4}}\right )}{3-4 e^4}+\frac {\log \left (3+2 e^4 x-2 e^4 x^2\right )}{2-x}-\frac {\left (4 e^4 \left (6+e^4\right )\right ) \text {Subst}\left (\int \frac {1}{4 e^4 \left (6+e^4\right )-x^2} \, dx,x,-2 e^4+4 e^4 x\right )}{3-4 e^4} \\ & = \frac {9}{2-x}-x+\frac {\log \left (3+2 e^4 x-2 e^4 x^2\right )}{2-x} \\ \end{align*}
Time = 0.23 (sec) , antiderivative size = 26, normalized size of antiderivative = 0.87 \[ \int \frac {-15-12 x+3 x^2+e^4 \left (-4-2 x^2+10 x^3-2 x^4\right )+\left (-3+e^4 \left (-2 x+2 x^2\right )\right ) \log \left (3+e^4 \left (2 x-2 x^2\right )\right )}{-12+12 x-3 x^2+e^4 \left (-8 x+16 x^2-10 x^3+2 x^4\right )} \, dx=-x+\frac {-9-\log \left (3-2 e^4 (-1+x) x\right )}{-2+x} \]
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Time = 1.00 (sec) , antiderivative size = 31, normalized size of antiderivative = 1.03
method | result | size |
norman | \(\frac {-\ln \left (\left (-2 x^{2}+2 x \right ) {\mathrm e}^{4}+3\right )-x^{2}-5}{-2+x}\) | \(31\) |
parallelrisch | \(\frac {-18 x^{2}-45 x -18 \ln \left (\left (-2 x^{2}+2 x \right ) {\mathrm e}^{4}+3\right )}{-36+18 x}\) | \(34\) |
risch | \(-\frac {\ln \left (\left (-2 x^{2}+2 x \right ) {\mathrm e}^{4}+3\right )}{-2+x}-\frac {x^{2}-2 x +9}{-2+x}\) | \(39\) |
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Time = 0.26 (sec) , antiderivative size = 29, normalized size of antiderivative = 0.97 \[ \int \frac {-15-12 x+3 x^2+e^4 \left (-4-2 x^2+10 x^3-2 x^4\right )+\left (-3+e^4 \left (-2 x+2 x^2\right )\right ) \log \left (3+e^4 \left (2 x-2 x^2\right )\right )}{-12+12 x-3 x^2+e^4 \left (-8 x+16 x^2-10 x^3+2 x^4\right )} \, dx=-\frac {x^{2} - 2 \, x + \log \left (-2 \, {\left (x^{2} - x\right )} e^{4} + 3\right ) + 9}{x - 2} \]
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Time = 0.14 (sec) , antiderivative size = 26, normalized size of antiderivative = 0.87 \[ \int \frac {-15-12 x+3 x^2+e^4 \left (-4-2 x^2+10 x^3-2 x^4\right )+\left (-3+e^4 \left (-2 x+2 x^2\right )\right ) \log \left (3+e^4 \left (2 x-2 x^2\right )\right )}{-12+12 x-3 x^2+e^4 \left (-8 x+16 x^2-10 x^3+2 x^4\right )} \, dx=- x - \frac {\log {\left (\left (- 2 x^{2} + 2 x\right ) e^{4} + 3 \right )}}{x - 2} - \frac {9}{x - 2} \]
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Leaf count of result is larger than twice the leaf count of optimal. 1175 vs. \(2 (27) = 54\).
Time = 0.35 (sec) , antiderivative size = 1175, normalized size of antiderivative = 39.17 \[ \int \frac {-15-12 x+3 x^2+e^4 \left (-4-2 x^2+10 x^3-2 x^4\right )+\left (-3+e^4 \left (-2 x+2 x^2\right )\right ) \log \left (3+e^4 \left (2 x-2 x^2\right )\right )}{-12+12 x-3 x^2+e^4 \left (-8 x+16 x^2-10 x^3+2 x^4\right )} \, dx=\text {Too large to display} \]
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Time = 0.31 (sec) , antiderivative size = 30, normalized size of antiderivative = 1.00 \[ \int \frac {-15-12 x+3 x^2+e^4 \left (-4-2 x^2+10 x^3-2 x^4\right )+\left (-3+e^4 \left (-2 x+2 x^2\right )\right ) \log \left (3+e^4 \left (2 x-2 x^2\right )\right )}{-12+12 x-3 x^2+e^4 \left (-8 x+16 x^2-10 x^3+2 x^4\right )} \, dx=-\frac {x^{2} - 2 \, x + \log \left (-2 \, x^{2} e^{4} + 2 \, x e^{4} + 3\right ) + 9}{x - 2} \]
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Time = 14.63 (sec) , antiderivative size = 30, normalized size of antiderivative = 1.00 \[ \int \frac {-15-12 x+3 x^2+e^4 \left (-4-2 x^2+10 x^3-2 x^4\right )+\left (-3+e^4 \left (-2 x+2 x^2\right )\right ) \log \left (3+e^4 \left (2 x-2 x^2\right )\right )}{-12+12 x-3 x^2+e^4 \left (-8 x+16 x^2-10 x^3+2 x^4\right )} \, dx=-\frac {\ln \left ({\mathrm {e}}^4\,\left (2\,x-2\,x^2\right )+3\right )-2\,x+x^2+9}{x-2} \]
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