Integrand size = 45, antiderivative size = 26 \[ \int \frac {1-36 x-3 x^2+e^x \left (-18-18 x-3 x^2\right )}{36+12 x+x^2+e \left (36+12 x+x^2\right )} \, dx=4-\frac {x \left (\frac {1}{x}+3 \left (e^x+x\right )\right )}{(1+e) (6+x)} \]
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Time = 0.17 (sec) , antiderivative size = 46, normalized size of antiderivative = 1.77, number of steps used = 13, number of rules used = 9, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.200, Rules used = {6873, 27, 12, 6874, 697, 2230, 2225, 2208, 2209} \[ \int \frac {1-36 x-3 x^2+e^x \left (-18-18 x-3 x^2\right )}{36+12 x+x^2+e \left (36+12 x+x^2\right )} \, dx=-\frac {3 x}{1+e}+\frac {18 e^x}{(1+e) (x+6)}-\frac {109}{(1+e) (x+6)}-\frac {3 e^x}{1+e} \]
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Rule 12
Rule 27
Rule 697
Rule 2208
Rule 2209
Rule 2225
Rule 2230
Rule 6873
Rule 6874
Rubi steps \begin{align*} \text {integral}& = \int \frac {1-36 x-3 x^2+e^x \left (-18-18 x-3 x^2\right )}{36 (1+e)+12 (1+e) x+(1+e) x^2} \, dx \\ & = \int \frac {1-36 x-3 x^2+e^x \left (-18-18 x-3 x^2\right )}{(1+e) (6+x)^2} \, dx \\ & = \frac {\int \frac {1-36 x-3 x^2+e^x \left (-18-18 x-3 x^2\right )}{(6+x)^2} \, dx}{1+e} \\ & = \frac {\int \left (\frac {1-36 x-3 x^2}{(6+x)^2}-\frac {3 e^x \left (6+6 x+x^2\right )}{(6+x)^2}\right ) \, dx}{1+e} \\ & = \frac {\int \frac {1-36 x-3 x^2}{(6+x)^2} \, dx}{1+e}-\frac {3 \int \frac {e^x \left (6+6 x+x^2\right )}{(6+x)^2} \, dx}{1+e} \\ & = \frac {\int \left (-3+\frac {109}{(6+x)^2}\right ) \, dx}{1+e}-\frac {3 \int \left (e^x+\frac {6 e^x}{(6+x)^2}-\frac {6 e^x}{6+x}\right ) \, dx}{1+e} \\ & = -\frac {3 x}{1+e}-\frac {109}{(1+e) (6+x)}-\frac {3 \int e^x \, dx}{1+e}-\frac {18 \int \frac {e^x}{(6+x)^2} \, dx}{1+e}+\frac {18 \int \frac {e^x}{6+x} \, dx}{1+e} \\ & = -\frac {3 e^x}{1+e}-\frac {3 x}{1+e}-\frac {109}{(1+e) (6+x)}+\frac {18 e^x}{(1+e) (6+x)}+\frac {18 \text {Ei}(6+x)}{e^6 (1+e)}-\frac {18 \int \frac {e^x}{6+x} \, dx}{1+e} \\ & = -\frac {3 e^x}{1+e}-\frac {3 x}{1+e}-\frac {109}{(1+e) (6+x)}+\frac {18 e^x}{(1+e) (6+x)} \\ \end{align*}
Time = 0.18 (sec) , antiderivative size = 27, normalized size of antiderivative = 1.04 \[ \int \frac {1-36 x-3 x^2+e^x \left (-18-18 x-3 x^2\right )}{36+12 x+x^2+e \left (36+12 x+x^2\right )} \, dx=-\frac {109+3 \left (6+e^x\right ) x+3 x^2}{(1+e) (6+x)} \]
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Time = 2.84 (sec) , antiderivative size = 26, normalized size of antiderivative = 1.00
method | result | size |
parallelrisch | \(-\frac {3 x^{2}+3 \,{\mathrm e}^{x} x +1}{\left (1+{\mathrm e}\right ) \left (6+x \right )}\) | \(26\) |
norman | \(\frac {-\frac {3 x^{2}}{1+{\mathrm e}}-\frac {3 x \,{\mathrm e}^{x}}{1+{\mathrm e}}-\frac {1}{1+{\mathrm e}}}{6+x}\) | \(38\) |
risch | \(-\frac {3 x}{1+{\mathrm e}}-\frac {109 \,{\mathrm e}}{\left (1+{\mathrm e}\right ) \left (x \,{\mathrm e}+6 \,{\mathrm e}+x +6\right )}-\frac {109}{\left (1+{\mathrm e}\right ) \left (x \,{\mathrm e}+6 \,{\mathrm e}+x +6\right )}-\frac {3 x \,{\mathrm e}^{x}}{\left (1+{\mathrm e}\right ) \left (6+x \right )}\) | \(71\) |
parts | \(-\frac {3 x +\frac {109}{6+x}}{1+{\mathrm e}}-\frac {3 \,{\mathrm e}^{x}}{1+{\mathrm e}}-\frac {18 \left (-\frac {{\mathrm e}^{x}}{6+x}-{\mathrm e}^{-6} \operatorname {Ei}_{1}\left (-x -6\right )\right )}{1+{\mathrm e}}-\frac {18 \,{\mathrm e}^{-6} \operatorname {Ei}_{1}\left (-x -6\right )}{1+{\mathrm e}}\) | \(77\) |
default | \(-\frac {1}{\left (1+{\mathrm e}\right ) \left (6+x \right )}-\frac {36 \left (\frac {6}{6+x}+\ln \left (6+x \right )\right )}{1+{\mathrm e}}-\frac {3 \left (x -\frac {36}{6+x}-12 \ln \left (6+x \right )\right )}{1+{\mathrm e}}-\frac {18 \left (-\frac {{\mathrm e}^{x}}{6+x}-{\mathrm e}^{-6} \operatorname {Ei}_{1}\left (-x -6\right )\right )}{1+{\mathrm e}}-\frac {18 \,{\mathrm e}^{-6} \operatorname {Ei}_{1}\left (-x -6\right )}{1+{\mathrm e}}-\frac {3 \,{\mathrm e}^{x}}{1+{\mathrm e}}\) | \(114\) |
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Time = 0.25 (sec) , antiderivative size = 28, normalized size of antiderivative = 1.08 \[ \int \frac {1-36 x-3 x^2+e^x \left (-18-18 x-3 x^2\right )}{36+12 x+x^2+e \left (36+12 x+x^2\right )} \, dx=-\frac {3 \, x^{2} + 3 \, x e^{x} + 18 \, x + 109}{{\left (x + 6\right )} e + x + 6} \]
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Time = 0.16 (sec) , antiderivative size = 44, normalized size of antiderivative = 1.69 \[ \int \frac {1-36 x-3 x^2+e^x \left (-18-18 x-3 x^2\right )}{36+12 x+x^2+e \left (36+12 x+x^2\right )} \, dx=- \frac {3 x}{1 + e} - \frac {3 x e^{x}}{x + e x + 6 + 6 e} - \frac {109}{x \left (1 + e\right ) + 6 + 6 e} \]
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\[ \int \frac {1-36 x-3 x^2+e^x \left (-18-18 x-3 x^2\right )}{36+12 x+x^2+e \left (36+12 x+x^2\right )} \, dx=\int { -\frac {3 \, x^{2} + 3 \, {\left (x^{2} + 6 \, x + 6\right )} e^{x} + 36 \, x - 1}{x^{2} + {\left (x^{2} + 12 \, x + 36\right )} e + 12 \, x + 36} \,d x } \]
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Time = 0.25 (sec) , antiderivative size = 30, normalized size of antiderivative = 1.15 \[ \int \frac {1-36 x-3 x^2+e^x \left (-18-18 x-3 x^2\right )}{36+12 x+x^2+e \left (36+12 x+x^2\right )} \, dx=-\frac {3 \, x^{2} + 3 \, x e^{x} + 18 \, x + 109}{x e + x + 6 \, e + 6} \]
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Time = 0.23 (sec) , antiderivative size = 23, normalized size of antiderivative = 0.88 \[ \int \frac {1-36 x-3 x^2+e^x \left (-18-18 x-3 x^2\right )}{36+12 x+x^2+e \left (36+12 x+x^2\right )} \, dx=-\frac {x\,\left (18\,x+18\,{\mathrm {e}}^x-1\right )}{6\,\left (\mathrm {e}+1\right )\,\left (x+6\right )} \]
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