Integrand size = 47, antiderivative size = 19 \[ \int \frac {3-2 x^2+12 x^5+5 x^6+e^{2 x} \left (1-2 x-4 x^2+4 x^5+2 x^6\right )}{x^2} \, dx=\left (3+e^{2 x}+x\right ) \left (-2-\frac {1}{x}+x^4\right ) \]
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Leaf count is larger than twice the leaf count of optimal. \(43\) vs. \(2(19)=38\).
Time = 0.12 (sec) , antiderivative size = 43, normalized size of antiderivative = 2.26, number of steps used = 19, number of rules used = 6, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.128, Rules used = {14, 2230, 2225, 2208, 2209, 2207} \[ \int \frac {3-2 x^2+12 x^5+5 x^6+e^{2 x} \left (1-2 x-4 x^2+4 x^5+2 x^6\right )}{x^2} \, dx=x^5+e^{2 x} x^4+3 x^4-2 x-2 e^{2 x}-\frac {e^{2 x}}{x}-\frac {3}{x} \]
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Rule 14
Rule 2207
Rule 2208
Rule 2209
Rule 2225
Rule 2230
Rubi steps \begin{align*} \text {integral}& = \int \left (\frac {e^{2 x} \left (1-2 x-4 x^2+4 x^5+2 x^6\right )}{x^2}+\frac {3-2 x^2+12 x^5+5 x^6}{x^2}\right ) \, dx \\ & = \int \frac {e^{2 x} \left (1-2 x-4 x^2+4 x^5+2 x^6\right )}{x^2} \, dx+\int \frac {3-2 x^2+12 x^5+5 x^6}{x^2} \, dx \\ & = \int \left (-2+\frac {3}{x^2}+12 x^3+5 x^4\right ) \, dx+\int \left (-4 e^{2 x}+\frac {e^{2 x}}{x^2}-\frac {2 e^{2 x}}{x}+4 e^{2 x} x^3+2 e^{2 x} x^4\right ) \, dx \\ & = -\frac {3}{x}-2 x+3 x^4+x^5-2 \int \frac {e^{2 x}}{x} \, dx+2 \int e^{2 x} x^4 \, dx-4 \int e^{2 x} \, dx+4 \int e^{2 x} x^3 \, dx+\int \frac {e^{2 x}}{x^2} \, dx \\ & = -2 e^{2 x}-\frac {3}{x}-\frac {e^{2 x}}{x}-2 x+2 e^{2 x} x^3+3 x^4+e^{2 x} x^4+x^5-2 \operatorname {ExpIntegralEi}(2 x)+2 \int \frac {e^{2 x}}{x} \, dx-4 \int e^{2 x} x^3 \, dx-6 \int e^{2 x} x^2 \, dx \\ & = -2 e^{2 x}-\frac {3}{x}-\frac {e^{2 x}}{x}-2 x-3 e^{2 x} x^2+3 x^4+e^{2 x} x^4+x^5+6 \int e^{2 x} x \, dx+6 \int e^{2 x} x^2 \, dx \\ & = -2 e^{2 x}-\frac {3}{x}-\frac {e^{2 x}}{x}-2 x+3 e^{2 x} x+3 x^4+e^{2 x} x^4+x^5-3 \int e^{2 x} \, dx-6 \int e^{2 x} x \, dx \\ & = -\frac {7 e^{2 x}}{2}-\frac {3}{x}-\frac {e^{2 x}}{x}-2 x+3 x^4+e^{2 x} x^4+x^5+3 \int e^{2 x} \, dx \\ & = -2 e^{2 x}-\frac {3}{x}-\frac {e^{2 x}}{x}-2 x+3 x^4+e^{2 x} x^4+x^5 \\ \end{align*}
Time = 1.23 (sec) , antiderivative size = 33, normalized size of antiderivative = 1.74 \[ \int \frac {3-2 x^2+12 x^5+5 x^6+e^{2 x} \left (1-2 x-4 x^2+4 x^5+2 x^6\right )}{x^2} \, dx=\frac {-3-2 x^2+3 x^5+x^6+e^{2 x} \left (-1-2 x+x^5\right )}{x} \]
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Time = 0.10 (sec) , antiderivative size = 34, normalized size of antiderivative = 1.79
method | result | size |
risch | \(x^{5}+3 x^{4}-2 x -\frac {3}{x}+\frac {\left (x^{5}-2 x -1\right ) {\mathrm e}^{2 x}}{x}\) | \(34\) |
default | \(-2 x -\frac {{\mathrm e}^{2 x}}{x}-\frac {3}{x}+3 x^{4}+x^{5}-2 \,{\mathrm e}^{2 x}+{\mathrm e}^{2 x} x^{4}\) | \(41\) |
norman | \(\frac {x^{5} {\mathrm e}^{2 x}+x^{6}+3 x^{5}-2 x \,{\mathrm e}^{2 x}-2 x^{2}-{\mathrm e}^{2 x}-3}{x}\) | \(41\) |
parallelrisch | \(\frac {x^{5} {\mathrm e}^{2 x}+x^{6}+3 x^{5}-2 x \,{\mathrm e}^{2 x}-2 x^{2}-{\mathrm e}^{2 x}-3}{x}\) | \(41\) |
parts | \(-2 x -\frac {{\mathrm e}^{2 x}}{x}-\frac {3}{x}+3 x^{4}+x^{5}-2 \,{\mathrm e}^{2 x}+{\mathrm e}^{2 x} x^{4}\) | \(41\) |
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none
Time = 0.24 (sec) , antiderivative size = 32, normalized size of antiderivative = 1.68 \[ \int \frac {3-2 x^2+12 x^5+5 x^6+e^{2 x} \left (1-2 x-4 x^2+4 x^5+2 x^6\right )}{x^2} \, dx=\frac {x^{6} + 3 \, x^{5} - 2 \, x^{2} + {\left (x^{5} - 2 \, x - 1\right )} e^{\left (2 \, x\right )} - 3}{x} \]
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Time = 0.06 (sec) , antiderivative size = 29, normalized size of antiderivative = 1.53 \[ \int \frac {3-2 x^2+12 x^5+5 x^6+e^{2 x} \left (1-2 x-4 x^2+4 x^5+2 x^6\right )}{x^2} \, dx=x^{5} + 3 x^{4} - 2 x + \frac {\left (x^{5} - 2 x - 1\right ) e^{2 x}}{x} - \frac {3}{x} \]
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Result contains higher order function than in optimal. Order 4 vs. order 3.
Time = 0.24 (sec) , antiderivative size = 83, normalized size of antiderivative = 4.37 \[ \int \frac {3-2 x^2+12 x^5+5 x^6+e^{2 x} \left (1-2 x-4 x^2+4 x^5+2 x^6\right )}{x^2} \, dx=x^{5} + 3 \, x^{4} + \frac {1}{2} \, {\left (2 \, x^{4} - 4 \, x^{3} + 6 \, x^{2} - 6 \, x + 3\right )} e^{\left (2 \, x\right )} + \frac {1}{2} \, {\left (4 \, x^{3} - 6 \, x^{2} + 6 \, x - 3\right )} e^{\left (2 \, x\right )} - 2 \, x - \frac {3}{x} - 2 \, {\rm Ei}\left (2 \, x\right ) - 2 \, e^{\left (2 \, x\right )} + 2 \, \Gamma \left (-1, -2 \, x\right ) \]
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Leaf count of result is larger than twice the leaf count of optimal. 40 vs. \(2 (18) = 36\).
Time = 0.27 (sec) , antiderivative size = 40, normalized size of antiderivative = 2.11 \[ \int \frac {3-2 x^2+12 x^5+5 x^6+e^{2 x} \left (1-2 x-4 x^2+4 x^5+2 x^6\right )}{x^2} \, dx=\frac {x^{6} + x^{5} e^{\left (2 \, x\right )} + 3 \, x^{5} - 2 \, x^{2} - 2 \, x e^{\left (2 \, x\right )} - e^{\left (2 \, x\right )} - 3}{x} \]
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Time = 12.40 (sec) , antiderivative size = 34, normalized size of antiderivative = 1.79 \[ \int \frac {3-2 x^2+12 x^5+5 x^6+e^{2 x} \left (1-2 x-4 x^2+4 x^5+2 x^6\right )}{x^2} \, dx=x^4\,\left ({\mathrm {e}}^{2\,x}+3\right )-2\,{\mathrm {e}}^{2\,x}-\frac {{\mathrm {e}}^{2\,x}+3}{x}-2\,x+x^5 \]
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