Integrand size = 65, antiderivative size = 27 \[ \int \frac {9 x^3+e^{\frac {2+x+2 x^2}{x}} \left (-2-x+2 x^2\right )+e^{\frac {2+x+2 x^2}{2 x}} \left (-6 x+6 x^3\right )}{4 x^3} \, dx=\frac {1}{4} \left (3+\frac {e^{x+\frac {2+x}{2 x}}}{x}\right )^2 x \]
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Leaf count is larger than twice the leaf count of optimal. \(55\) vs. \(2(27)=54\).
Time = 0.12 (sec) , antiderivative size = 55, normalized size of antiderivative = 2.04, number of steps used = 5, number of rules used = 4, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.062, Rules used = {12, 14, 6838, 2326} \[ \int \frac {9 x^3+e^{\frac {2+x+2 x^2}{x}} \left (-2-x+2 x^2\right )+e^{\frac {2+x+2 x^2}{2 x}} \left (-6 x+6 x^3\right )}{4 x^3} \, dx=-\frac {e^{2 x+\frac {2}{x}+1} \left (1-x^2\right )}{4 \left (1-\frac {1}{x^2}\right ) x^3}+\frac {9 x}{4}+\frac {3}{2} e^{x+\frac {1}{x}+\frac {1}{2}} \]
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Rule 12
Rule 14
Rule 2326
Rule 6838
Rubi steps \begin{align*} \text {integral}& = \frac {1}{4} \int \frac {9 x^3+e^{\frac {2+x+2 x^2}{x}} \left (-2-x+2 x^2\right )+e^{\frac {2+x+2 x^2}{2 x}} \left (-6 x+6 x^3\right )}{x^3} \, dx \\ & = \frac {1}{4} \int \left (9+\frac {6 e^{\frac {1}{2}+\frac {1}{x}+x} \left (-1+x^2\right )}{x^2}+\frac {e^{1+\frac {2}{x}+2 x} \left (-2-x+2 x^2\right )}{x^3}\right ) \, dx \\ & = \frac {9 x}{4}+\frac {1}{4} \int \frac {e^{1+\frac {2}{x}+2 x} \left (-2-x+2 x^2\right )}{x^3} \, dx+\frac {3}{2} \int \frac {e^{\frac {1}{2}+\frac {1}{x}+x} \left (-1+x^2\right )}{x^2} \, dx \\ & = \frac {3}{2} e^{\frac {1}{2}+\frac {1}{x}+x}+\frac {9 x}{4}-\frac {e^{1+\frac {2}{x}+2 x} \left (1-x^2\right )}{4 \left (1-\frac {1}{x^2}\right ) x^3} \\ \end{align*}
Time = 0.11 (sec) , antiderivative size = 36, normalized size of antiderivative = 1.33 \[ \int \frac {9 x^3+e^{\frac {2+x+2 x^2}{x}} \left (-2-x+2 x^2\right )+e^{\frac {2+x+2 x^2}{2 x}} \left (-6 x+6 x^3\right )}{4 x^3} \, dx=\frac {1}{4} \left (6 e^{\frac {1}{2}+\frac {1}{x}+x}+\frac {e^{1+\frac {2}{x}+2 x}}{x}+9 x\right ) \]
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Time = 4.46 (sec) , antiderivative size = 39, normalized size of antiderivative = 1.44
method | result | size |
risch | \(\frac {9 x}{4}+\frac {{\mathrm e}^{\frac {2 x^{2}+x +2}{x}}}{4 x}+\frac {3 \,{\mathrm e}^{\frac {2 x^{2}+x +2}{2 x}}}{2}\) | \(39\) |
parts | \(\frac {9 x}{4}+\frac {{\mathrm e}^{\frac {2 x^{2}+x +2}{x}}}{4 x}+\frac {3 \,{\mathrm e}^{\frac {2 x^{2}+x +2}{2 x}}}{2}\) | \(42\) |
parallelrisch | \(\frac {6 x \,{\mathrm e}^{\frac {2 x^{2}+x +2}{2 x}}+9 x^{2}+{\mathrm e}^{\frac {2 x^{2}+x +2}{x}}}{4 x}\) | \(45\) |
norman | \(\frac {\frac {9 x^{3}}{4}+\frac {3 \,{\mathrm e}^{\frac {2 x^{2}+x +2}{2 x}} x^{2}}{2}+\frac {{\mathrm e}^{\frac {2 x^{2}+x +2}{x}} x}{4}}{x^{2}}\) | \(49\) |
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Time = 0.25 (sec) , antiderivative size = 41, normalized size of antiderivative = 1.52 \[ \int \frac {9 x^3+e^{\frac {2+x+2 x^2}{x}} \left (-2-x+2 x^2\right )+e^{\frac {2+x+2 x^2}{2 x}} \left (-6 x+6 x^3\right )}{4 x^3} \, dx=\frac {9 \, x^{2} + 6 \, x e^{\left (\frac {2 \, x^{2} + x + 2}{2 \, x}\right )} + e^{\left (\frac {2 \, x^{2} + x + 2}{x}\right )}}{4 \, x} \]
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Leaf count of result is larger than twice the leaf count of optimal. 36 vs. \(2 (17) = 34\).
Time = 0.08 (sec) , antiderivative size = 36, normalized size of antiderivative = 1.33 \[ \int \frac {9 x^3+e^{\frac {2+x+2 x^2}{x}} \left (-2-x+2 x^2\right )+e^{\frac {2+x+2 x^2}{2 x}} \left (-6 x+6 x^3\right )}{4 x^3} \, dx=\frac {9 x}{4} + \frac {12 x e^{\frac {x^{2} + \frac {x}{2} + 1}{x}} + 2 e^{\frac {2 \left (x^{2} + \frac {x}{2} + 1\right )}{x}}}{8 x} \]
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Time = 0.24 (sec) , antiderivative size = 31, normalized size of antiderivative = 1.15 \[ \int \frac {9 x^3+e^{\frac {2+x+2 x^2}{x}} \left (-2-x+2 x^2\right )+e^{\frac {2+x+2 x^2}{2 x}} \left (-6 x+6 x^3\right )}{4 x^3} \, dx=\frac {9}{4} \, x + \frac {6 \, x e^{\left (x + \frac {1}{x} + \frac {1}{2}\right )} + e^{\left (2 \, x + \frac {2}{x} + 1\right )}}{4 \, x} \]
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Time = 0.27 (sec) , antiderivative size = 41, normalized size of antiderivative = 1.52 \[ \int \frac {9 x^3+e^{\frac {2+x+2 x^2}{x}} \left (-2-x+2 x^2\right )+e^{\frac {2+x+2 x^2}{2 x}} \left (-6 x+6 x^3\right )}{4 x^3} \, dx=\frac {9 \, x^{2} + 6 \, x e^{\left (\frac {2 \, x^{2} + x + 2}{2 \, x}\right )} + e^{\left (\frac {2 \, x^{2} + x + 2}{x}\right )}}{4 \, x} \]
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Time = 9.42 (sec) , antiderivative size = 18, normalized size of antiderivative = 0.67 \[ \int \frac {9 x^3+e^{\frac {2+x+2 x^2}{x}} \left (-2-x+2 x^2\right )+e^{\frac {2+x+2 x^2}{2 x}} \left (-6 x+6 x^3\right )}{4 x^3} \, dx=\frac {{\left (3\,x+{\mathrm {e}}^{x+\frac {1}{x}+\frac {1}{2}}\right )}^2}{4\,x} \]
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