Integrand size = 50, antiderivative size = 24 \[ \int \frac {e^{-2 x} \left (e^{2 x}+6 e^{8 x} x^2+162 x^3-162 x^4+e^x \left (18 x^2-18 x^3\right )\right )}{x^2} \, dx=-7+e^{6 x}-\frac {1}{x}+\left (1+9 e^{-x} x\right )^2 \]
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Time = 0.22 (sec) , antiderivative size = 40, normalized size of antiderivative = 1.67, number of steps used = 12, number of rules used = 4, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.080, Rules used = {6820, 2225, 2207, 2227} \[ \int \frac {e^{-2 x} \left (e^{2 x}+6 e^{8 x} x^2+162 x^3-162 x^4+e^x \left (18 x^2-18 x^3\right )\right )}{x^2} \, dx=81 e^{-2 x} x^2+18 e^{-x}+e^{6 x}-18 e^{-x} (1-x)-\frac {1}{x} \]
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Rule 2207
Rule 2225
Rule 2227
Rule 6820
Rubi steps \begin{align*} \text {integral}& = \int \left (6 e^{6 x}-18 e^{-x} (-1+x)+\frac {1}{x^2}-162 e^{-2 x} (-1+x) x\right ) \, dx \\ & = -\frac {1}{x}+6 \int e^{6 x} \, dx-18 \int e^{-x} (-1+x) \, dx-162 \int e^{-2 x} (-1+x) x \, dx \\ & = e^{6 x}-18 e^{-x} (1-x)-\frac {1}{x}-18 \int e^{-x} \, dx-162 \int \left (-e^{-2 x} x+e^{-2 x} x^2\right ) \, dx \\ & = 18 e^{-x}+e^{6 x}-18 e^{-x} (1-x)-\frac {1}{x}+162 \int e^{-2 x} x \, dx-162 \int e^{-2 x} x^2 \, dx \\ & = 18 e^{-x}+e^{6 x}-18 e^{-x} (1-x)-\frac {1}{x}-81 e^{-2 x} x+81 e^{-2 x} x^2+81 \int e^{-2 x} \, dx-162 \int e^{-2 x} x \, dx \\ & = -\frac {81}{2} e^{-2 x}+18 e^{-x}+e^{6 x}-18 e^{-x} (1-x)-\frac {1}{x}+81 e^{-2 x} x^2-81 \int e^{-2 x} \, dx \\ & = 18 e^{-x}+e^{6 x}-18 e^{-x} (1-x)-\frac {1}{x}+81 e^{-2 x} x^2 \\ \end{align*}
Time = 0.06 (sec) , antiderivative size = 29, normalized size of antiderivative = 1.21 \[ \int \frac {e^{-2 x} \left (e^{2 x}+6 e^{8 x} x^2+162 x^3-162 x^4+e^x \left (18 x^2-18 x^3\right )\right )}{x^2} \, dx=e^{6 x}-\frac {1}{x}+18 e^{-x} x+81 e^{-2 x} x^2 \]
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Time = 0.06 (sec) , antiderivative size = 27, normalized size of antiderivative = 1.12
method | result | size |
default | \(-\frac {1}{x}+{\mathrm e}^{6 x}+18 x \,{\mathrm e}^{-x}+81 x^{2} {\mathrm e}^{-2 x}\) | \(27\) |
risch | \(-\frac {1}{x}+{\mathrm e}^{6 x}+18 x \,{\mathrm e}^{-x}+81 x^{2} {\mathrm e}^{-2 x}\) | \(27\) |
parts | \(-\frac {1}{x}+{\mathrm e}^{6 x}+18 x \,{\mathrm e}^{-x}+81 x^{2} {\mathrm e}^{-2 x}\) | \(29\) |
parallelrisch | \(-\frac {\left (-x \,{\mathrm e}^{2 x} {\mathrm e}^{6 x}-81 x^{3}-18 \,{\mathrm e}^{x} x^{2}+{\mathrm e}^{2 x}\right ) {\mathrm e}^{-2 x}}{x}\) | \(40\) |
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Time = 0.25 (sec) , antiderivative size = 33, normalized size of antiderivative = 1.38 \[ \int \frac {e^{-2 x} \left (e^{2 x}+6 e^{8 x} x^2+162 x^3-162 x^4+e^x \left (18 x^2-18 x^3\right )\right )}{x^2} \, dx=\frac {{\left (81 \, x^{3} + 18 \, x^{2} e^{x} + x e^{\left (8 \, x\right )} - e^{\left (2 \, x\right )}\right )} e^{\left (-2 \, x\right )}}{x} \]
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Time = 0.07 (sec) , antiderivative size = 24, normalized size of antiderivative = 1.00 \[ \int \frac {e^{-2 x} \left (e^{2 x}+6 e^{8 x} x^2+162 x^3-162 x^4+e^x \left (18 x^2-18 x^3\right )\right )}{x^2} \, dx=81 x^{2} e^{- 2 x} + 18 x e^{- x} + e^{6 x} - \frac {1}{x} \]
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Leaf count of result is larger than twice the leaf count of optimal. 52 vs. \(2 (22) = 44\).
Time = 0.19 (sec) , antiderivative size = 52, normalized size of antiderivative = 2.17 \[ \int \frac {e^{-2 x} \left (e^{2 x}+6 e^{8 x} x^2+162 x^3-162 x^4+e^x \left (18 x^2-18 x^3\right )\right )}{x^2} \, dx=18 \, {\left (x + 1\right )} e^{\left (-x\right )} + \frac {81}{2} \, {\left (2 \, x^{2} + 2 \, x + 1\right )} e^{\left (-2 \, x\right )} - \frac {81}{2} \, {\left (2 \, x + 1\right )} e^{\left (-2 \, x\right )} - \frac {1}{x} + e^{\left (6 \, x\right )} - 18 \, e^{\left (-x\right )} \]
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Time = 0.25 (sec) , antiderivative size = 30, normalized size of antiderivative = 1.25 \[ \int \frac {e^{-2 x} \left (e^{2 x}+6 e^{8 x} x^2+162 x^3-162 x^4+e^x \left (18 x^2-18 x^3\right )\right )}{x^2} \, dx=\frac {81 \, x^{3} e^{\left (-2 \, x\right )} + 18 \, x^{2} e^{\left (-x\right )} + x e^{\left (6 \, x\right )} - 1}{x} \]
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Time = 7.84 (sec) , antiderivative size = 26, normalized size of antiderivative = 1.08 \[ \int \frac {e^{-2 x} \left (e^{2 x}+6 e^{8 x} x^2+162 x^3-162 x^4+e^x \left (18 x^2-18 x^3\right )\right )}{x^2} \, dx={\mathrm {e}}^{6\,x}+18\,x\,{\mathrm {e}}^{-x}+81\,x^2\,{\mathrm {e}}^{-2\,x}-\frac {1}{x} \]
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