Integrand size = 66, antiderivative size = 33 \[ \int \frac {-12+3 e^5+6 x-x^2+e^{2 x} \left (-64+104 x-52 x^2+8 x^3+e^5 \left (64-104 x+52 x^2-8 x^3\right )\right )}{9-6 x+x^2} \, dx=-4-x+\frac {\left (-1+e^5\right ) \left (e^{2 x} (4-2 x)^2+x\right )}{3-x} \]
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Leaf count is larger than twice the leaf count of optimal. \(70\) vs. \(2(33)=66\).
Time = 0.30 (sec) , antiderivative size = 70, normalized size of antiderivative = 2.12, number of steps used = 14, number of rules used = 9, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.136, Rules used = {27, 6873, 6874, 697, 2230, 2225, 2208, 2209, 2207} \[ \int \frac {-12+3 e^5+6 x-x^2+e^{2 x} \left (-64+104 x-52 x^2+8 x^3+e^5 \left (64-104 x+52 x^2-8 x^3\right )\right )}{9-6 x+x^2} \, dx=4 \left (1-e^5\right ) e^{2 x} x-x-\frac {4 \left (1-e^5\right ) e^{2 x}}{3-x}-\frac {3 \left (1-e^5\right )}{3-x}-4 \left (1-e^5\right ) e^{2 x} \]
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Rule 27
Rule 697
Rule 2207
Rule 2208
Rule 2209
Rule 2225
Rule 2230
Rule 6873
Rule 6874
Rubi steps \begin{align*} \text {integral}& = \int \frac {-12+3 e^5+6 x-x^2+e^{2 x} \left (-64+104 x-52 x^2+8 x^3+e^5 \left (64-104 x+52 x^2-8 x^3\right )\right )}{(-3+x)^2} \, dx \\ & = \int \frac {-12 \left (1-\frac {e^5}{4}\right )+6 x-x^2+e^{2 x} \left (-64+104 x-52 x^2+8 x^3+e^5 \left (64-104 x+52 x^2-8 x^3\right )\right )}{(3-x)^2} \, dx \\ & = \int \left (\frac {-3 \left (4-e^5\right )+6 x-x^2}{(3-x)^2}-\frac {4 (-1+e) e^{2 x} \left (1+e+e^2+e^3+e^4\right ) (-2+x) \left (8-9 x+2 x^2\right )}{(-3+x)^2}\right ) \, dx \\ & = \left (4 \left (1-e^5\right )\right ) \int \frac {e^{2 x} (-2+x) \left (8-9 x+2 x^2\right )}{(-3+x)^2} \, dx+\int \frac {-3 \left (4-e^5\right )+6 x-x^2}{(3-x)^2} \, dx \\ & = \left (4 \left (1-e^5\right )\right ) \int \left (-e^{2 x}-\frac {e^{2 x}}{(-3+x)^2}+\frac {2 e^{2 x}}{-3+x}+2 e^{2 x} x\right ) \, dx+\int \left (-1+\frac {3 \left (-1+e^5\right )}{(-3+x)^2}\right ) \, dx \\ & = -\frac {3 \left (1-e^5\right )}{3-x}-x-\left (4 \left (1-e^5\right )\right ) \int e^{2 x} \, dx-\left (4 \left (1-e^5\right )\right ) \int \frac {e^{2 x}}{(-3+x)^2} \, dx+\left (8 \left (1-e^5\right )\right ) \int \frac {e^{2 x}}{-3+x} \, dx+\left (8 \left (1-e^5\right )\right ) \int e^{2 x} x \, dx \\ & = -2 e^{2 x} \left (1-e^5\right )-\frac {3 \left (1-e^5\right )}{3-x}-\frac {4 e^{2 x} \left (1-e^5\right )}{3-x}-x+4 e^{2 x} \left (1-e^5\right ) x+8 e^6 \left (1-e^5\right ) \operatorname {ExpIntegralEi}(-2 (3-x))-\left (4 \left (1-e^5\right )\right ) \int e^{2 x} \, dx-\left (8 \left (1-e^5\right )\right ) \int \frac {e^{2 x}}{-3+x} \, dx \\ & = -4 e^{2 x} \left (1-e^5\right )-\frac {3 \left (1-e^5\right )}{3-x}-\frac {4 e^{2 x} \left (1-e^5\right )}{3-x}-x+4 e^{2 x} \left (1-e^5\right ) x \\ \end{align*}
Time = 2.87 (sec) , antiderivative size = 47, normalized size of antiderivative = 1.42 \[ \int \frac {-12+3 e^5+6 x-x^2+e^{2 x} \left (-64+104 x-52 x^2+8 x^3+e^5 \left (64-104 x+52 x^2-8 x^3\right )\right )}{9-6 x+x^2} \, dx=\frac {3-3 e^5+4 e^{2 x} (-2+x)^2-4 e^{5+2 x} (-2+x)^2+3 x-x^2}{-3+x} \]
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Time = 0.16 (sec) , antiderivative size = 55, normalized size of antiderivative = 1.67
method | result | size |
norman | \(\frac {\left (16-16 \,{\mathrm e}^{5}\right ) {\mathrm e}^{2 x}+\left (-16+16 \,{\mathrm e}^{5}\right ) x \,{\mathrm e}^{2 x}+\left (-4 \,{\mathrm e}^{5}+4\right ) x^{2} {\mathrm e}^{2 x}-x^{2}+12-3 \,{\mathrm e}^{5}}{-3+x}\) | \(55\) |
risch | \(-x +\frac {3}{-3+x}-\frac {3 \,{\mathrm e}^{5}}{-3+x}-\frac {4 \left (x^{2} {\mathrm e}^{5}-4 x \,{\mathrm e}^{5}-x^{2}+4 \,{\mathrm e}^{5}+4 x -4\right ) {\mathrm e}^{2 x}}{-3+x}\) | \(57\) |
parallelrisch | \(-\frac {4 \,{\mathrm e}^{5} {\mathrm e}^{2 x} x^{2}-16 \,{\mathrm e}^{5} {\mathrm e}^{2 x} x -4 \,{\mathrm e}^{2 x} x^{2}+16 \,{\mathrm e}^{5} {\mathrm e}^{2 x}+16 x \,{\mathrm e}^{2 x}+x^{2}-12-16 \,{\mathrm e}^{2 x}+3 \,{\mathrm e}^{5}}{-3+x}\) | \(67\) |
parts | \(-x -\frac {-3+3 \,{\mathrm e}^{5}}{-3+x}+\frac {4 \,{\mathrm e}^{2 x}}{-3+x}+64 \,{\mathrm e}^{5} \left (-\frac {{\mathrm e}^{2 x}}{-3+x}-2 \,{\mathrm e}^{6} \operatorname {Ei}_{1}\left (6-2 x \right )\right )-4 \,{\mathrm e}^{2 x}+4 x \,{\mathrm e}^{2 x}-104 \,{\mathrm e}^{5} \left (-7 \,{\mathrm e}^{6} \operatorname {Ei}_{1}\left (6-2 x \right )-\frac {3 \,{\mathrm e}^{2 x}}{-3+x}\right )+52 \,{\mathrm e}^{5} \left (\frac {{\mathrm e}^{2 x}}{2}-24 \,{\mathrm e}^{6} \operatorname {Ei}_{1}\left (6-2 x \right )-\frac {9 \,{\mathrm e}^{2 x}}{-3+x}\right )-8 \,{\mathrm e}^{5} \left (\frac {11 \,{\mathrm e}^{2 x}}{4}+\frac {x \,{\mathrm e}^{2 x}}{2}-81 \,{\mathrm e}^{6} \operatorname {Ei}_{1}\left (6-2 x \right )-\frac {27 \,{\mathrm e}^{2 x}}{-3+x}\right )\) | \(169\) |
default | \(\frac {3}{-3+x}-x -\frac {3 \,{\mathrm e}^{5}}{-3+x}+\frac {4 \,{\mathrm e}^{2 x}}{-3+x}+64 \,{\mathrm e}^{5} \left (-\frac {{\mathrm e}^{2 x}}{-3+x}-2 \,{\mathrm e}^{6} \operatorname {Ei}_{1}\left (6-2 x \right )\right )-4 \,{\mathrm e}^{2 x}+4 x \,{\mathrm e}^{2 x}-104 \,{\mathrm e}^{5} \left (-7 \,{\mathrm e}^{6} \operatorname {Ei}_{1}\left (6-2 x \right )-\frac {3 \,{\mathrm e}^{2 x}}{-3+x}\right )+52 \,{\mathrm e}^{5} \left (\frac {{\mathrm e}^{2 x}}{2}-24 \,{\mathrm e}^{6} \operatorname {Ei}_{1}\left (6-2 x \right )-\frac {9 \,{\mathrm e}^{2 x}}{-3+x}\right )-8 \,{\mathrm e}^{5} \left (\frac {11 \,{\mathrm e}^{2 x}}{4}+\frac {x \,{\mathrm e}^{2 x}}{2}-81 \,{\mathrm e}^{6} \operatorname {Ei}_{1}\left (6-2 x \right )-\frac {27 \,{\mathrm e}^{2 x}}{-3+x}\right )\) | \(172\) |
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Time = 0.25 (sec) , antiderivative size = 45, normalized size of antiderivative = 1.36 \[ \int \frac {-12+3 e^5+6 x-x^2+e^{2 x} \left (-64+104 x-52 x^2+8 x^3+e^5 \left (64-104 x+52 x^2-8 x^3\right )\right )}{9-6 x+x^2} \, dx=-\frac {x^{2} - 4 \, {\left (x^{2} - {\left (x^{2} - 4 \, x + 4\right )} e^{5} - 4 \, x + 4\right )} e^{\left (2 \, x\right )} - 3 \, x + 3 \, e^{5} - 3}{x - 3} \]
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Leaf count of result is larger than twice the leaf count of optimal. 49 vs. \(2 (24) = 48\).
Time = 0.11 (sec) , antiderivative size = 49, normalized size of antiderivative = 1.48 \[ \int \frac {-12+3 e^5+6 x-x^2+e^{2 x} \left (-64+104 x-52 x^2+8 x^3+e^5 \left (64-104 x+52 x^2-8 x^3\right )\right )}{9-6 x+x^2} \, dx=- x + \frac {\left (- 4 x^{2} e^{5} + 4 x^{2} - 16 x + 16 x e^{5} - 16 e^{5} + 16\right ) e^{2 x}}{x - 3} - \frac {-3 + 3 e^{5}}{x - 3} \]
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\[ \int \frac {-12+3 e^5+6 x-x^2+e^{2 x} \left (-64+104 x-52 x^2+8 x^3+e^5 \left (64-104 x+52 x^2-8 x^3\right )\right )}{9-6 x+x^2} \, dx=\int { -\frac {x^{2} - 4 \, {\left (2 \, x^{3} - 13 \, x^{2} - {\left (2 \, x^{3} - 13 \, x^{2} + 26 \, x - 16\right )} e^{5} + 26 \, x - 16\right )} e^{\left (2 \, x\right )} - 6 \, x - 3 \, e^{5} + 12}{x^{2} - 6 \, x + 9} \,d x } \]
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Leaf count of result is larger than twice the leaf count of optimal. 70 vs. \(2 (29) = 58\).
Time = 0.28 (sec) , antiderivative size = 70, normalized size of antiderivative = 2.12 \[ \int \frac {-12+3 e^5+6 x-x^2+e^{2 x} \left (-64+104 x-52 x^2+8 x^3+e^5 \left (64-104 x+52 x^2-8 x^3\right )\right )}{9-6 x+x^2} \, dx=\frac {4 \, x^{2} e^{\left (2 \, x\right )} - 4 \, x^{2} e^{\left (2 \, x + 5\right )} - x^{2} - 16 \, x e^{\left (2 \, x\right )} + 16 \, x e^{\left (2 \, x + 5\right )} + 3 \, x - 3 \, e^{5} + 16 \, e^{\left (2 \, x\right )} - 16 \, e^{\left (2 \, x + 5\right )} + 3}{x - 3} \]
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Time = 8.48 (sec) , antiderivative size = 55, normalized size of antiderivative = 1.67 \[ \int \frac {-12+3 e^5+6 x-x^2+e^{2 x} \left (-64+104 x-52 x^2+8 x^3+e^5 \left (64-104 x+52 x^2-8 x^3\right )\right )}{9-6 x+x^2} \, dx=-\frac {x\,\left ({\mathrm {e}}^5-4\right )+{\mathrm {e}}^{2\,x}\,\left (16\,{\mathrm {e}}^5-16\right )+x^2-x\,{\mathrm {e}}^{2\,x}\,\left (16\,{\mathrm {e}}^5-16\right )+x^2\,{\mathrm {e}}^{2\,x}\,\left (4\,{\mathrm {e}}^5-4\right )}{x-3} \]
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