Integrand size = 29, antiderivative size = 29 \[ \int \frac {-6-12 x-3 x^2+e^4 (6+10 x)}{-6+6 e^4} \, dx=\frac {1}{6} \left (x+\frac {1}{4} x (4+4 x) \left (5+\frac {x}{1-e^4}\right )\right ) \]
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Leaf count is larger than twice the leaf count of optimal. \(64\) vs. \(2(29)=58\).
Time = 0.01 (sec) , antiderivative size = 64, normalized size of antiderivative = 2.21, number of steps used = 2, number of rules used = 1, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.034, Rules used = {12} \[ \int \frac {-6-12 x-3 x^2+e^4 (6+10 x)}{-6+6 e^4} \, dx=\frac {x^3}{6 \left (1-e^4\right )}+\frac {x^2}{1-e^4}+\frac {x}{1-e^4}-\frac {e^4 (5 x+3)^2}{30 \left (1-e^4\right )} \]
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Rule 12
Rubi steps \begin{align*} \text {integral}& = -\frac {\int \left (-6-12 x-3 x^2+e^4 (6+10 x)\right ) \, dx}{6 \left (1-e^4\right )} \\ & = \frac {x}{1-e^4}+\frac {x^2}{1-e^4}+\frac {x^3}{6 \left (1-e^4\right )}-\frac {e^4 (3+5 x)^2}{30 \left (1-e^4\right )} \\ \end{align*}
Time = 0.00 (sec) , antiderivative size = 39, normalized size of antiderivative = 1.34 \[ \int \frac {-6-12 x-3 x^2+e^4 (6+10 x)}{-6+6 e^4} \, dx=\frac {-6 x+6 e^4 x-6 x^2+5 e^4 x^2-x^3}{6 \left (-1+e^4\right )} \]
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Time = 0.02 (sec) , antiderivative size = 29, normalized size of antiderivative = 1.00
method | result | size |
gosper | \(\frac {x \left (5 x \,{\mathrm e}^{4}-x^{2}+6 \,{\mathrm e}^{4}-6 x -6\right )}{6 \,{\mathrm e}^{4}-6}\) | \(29\) |
norman | \(x -\frac {x^{3}}{6 \left ({\mathrm e}^{4}-1\right )}+\frac {\left (5 \,{\mathrm e}^{4}-6\right ) x^{2}}{6 \,{\mathrm e}^{4}-6}\) | \(31\) |
default | \(\frac {5 x^{2} {\mathrm e}^{4}-x^{3}+6 x \,{\mathrm e}^{4}-6 x^{2}-6 x}{6 \,{\mathrm e}^{4}-6}\) | \(35\) |
parallelrisch | \(\frac {5 x^{2} {\mathrm e}^{4}-x^{3}+6 x \,{\mathrm e}^{4}-6 x^{2}-6 x}{6 \,{\mathrm e}^{4}-6}\) | \(36\) |
risch | \(\frac {5 x^{2} {\mathrm e}^{4}}{6 \,{\mathrm e}^{4}-6}-\frac {x^{3}}{6 \,{\mathrm e}^{4}-6}+\frac {6 x \,{\mathrm e}^{4}}{6 \,{\mathrm e}^{4}-6}-\frac {6 x^{2}}{6 \,{\mathrm e}^{4}-6}-\frac {6 x}{6 \,{\mathrm e}^{4}-6}\) | \(67\) |
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Time = 0.23 (sec) , antiderivative size = 33, normalized size of antiderivative = 1.14 \[ \int \frac {-6-12 x-3 x^2+e^4 (6+10 x)}{-6+6 e^4} \, dx=-\frac {x^{3} + 6 \, x^{2} - {\left (5 \, x^{2} + 6 \, x\right )} e^{4} + 6 \, x}{6 \, {\left (e^{4} - 1\right )}} \]
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Time = 0.02 (sec) , antiderivative size = 27, normalized size of antiderivative = 0.93 \[ \int \frac {-6-12 x-3 x^2+e^4 (6+10 x)}{-6+6 e^4} \, dx=- \frac {x^{3}}{-6 + 6 e^{4}} + \frac {x^{2} \left (-6 + 5 e^{4}\right )}{-6 + 6 e^{4}} + x \]
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Time = 0.17 (sec) , antiderivative size = 33, normalized size of antiderivative = 1.14 \[ \int \frac {-6-12 x-3 x^2+e^4 (6+10 x)}{-6+6 e^4} \, dx=-\frac {x^{3} + 6 \, x^{2} - {\left (5 \, x^{2} + 6 \, x\right )} e^{4} + 6 \, x}{6 \, {\left (e^{4} - 1\right )}} \]
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Time = 0.28 (sec) , antiderivative size = 33, normalized size of antiderivative = 1.14 \[ \int \frac {-6-12 x-3 x^2+e^4 (6+10 x)}{-6+6 e^4} \, dx=-\frac {x^{3} + 6 \, x^{2} - {\left (5 \, x^{2} + 6 \, x\right )} e^{4} + 6 \, x}{6 \, {\left (e^{4} - 1\right )}} \]
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Time = 9.04 (sec) , antiderivative size = 33, normalized size of antiderivative = 1.14 \[ \int \frac {-6-12 x-3 x^2+e^4 (6+10 x)}{-6+6 e^4} \, dx=-\frac {x^3}{6\,{\mathrm {e}}^4-6}+\frac {\left (10\,{\mathrm {e}}^4-12\right )\,x^2}{2\,\left (6\,{\mathrm {e}}^4-6\right )}+x \]
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