Integrand size = 45, antiderivative size = 31 \[ \int \frac {2-8 e^4 x+2 e^9 x+e^{4+x} \left (-6 x+6 x^2+2 x^3\right )-4 \log (x)}{e^4 x^3} \, dx=\frac {2 \left (4-e^5+e^x (3+x)+\frac {x+\frac {\log (x)}{x}}{e^4}\right )}{x} \]
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Time = 0.08 (sec) , antiderivative size = 56, normalized size of antiderivative = 1.81, number of steps used = 14, number of rules used = 9, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.200, Rules used = {6, 12, 14, 2230, 2225, 2208, 2209, 37, 2341} \[ \int \frac {2-8 e^4 x+2 e^9 x+e^{4+x} \left (-6 x+6 x^2+2 x^3\right )-4 \log (x)}{e^4 x^3} \, dx=-\frac {\left (1-e^4 \left (4-e^5\right ) x\right )^2}{e^4 x^2}+\frac {1}{e^4 x^2}+\frac {2 \log (x)}{e^4 x^2}+2 e^x+\frac {6 e^x}{x} \]
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Rule 6
Rule 12
Rule 14
Rule 37
Rule 2208
Rule 2209
Rule 2225
Rule 2230
Rule 2341
Rubi steps \begin{align*} \text {integral}& = \int \frac {2+\left (-8 e^4+2 e^9\right ) x+e^{4+x} \left (-6 x+6 x^2+2 x^3\right )-4 \log (x)}{e^4 x^3} \, dx \\ & = \frac {\int \frac {2+\left (-8 e^4+2 e^9\right ) x+e^{4+x} \left (-6 x+6 x^2+2 x^3\right )-4 \log (x)}{x^3} \, dx}{e^4} \\ & = \frac {\int \left (\frac {2 e^{4+x} \left (-3+3 x+x^2\right )}{x^2}+\frac {2 \left (1-4 e^4 \left (1-\frac {e^5}{4}\right ) x-2 \log (x)\right )}{x^3}\right ) \, dx}{e^4} \\ & = \frac {2 \int \frac {e^{4+x} \left (-3+3 x+x^2\right )}{x^2} \, dx}{e^4}+\frac {2 \int \frac {1-4 e^4 \left (1-\frac {e^5}{4}\right ) x-2 \log (x)}{x^3} \, dx}{e^4} \\ & = \frac {2 \int \left (e^{4+x}-\frac {3 e^{4+x}}{x^2}+\frac {3 e^{4+x}}{x}\right ) \, dx}{e^4}+\frac {2 \int \left (\frac {1-e^4 \left (4-e^5\right ) x}{x^3}-\frac {2 \log (x)}{x^3}\right ) \, dx}{e^4} \\ & = \frac {2 \int e^{4+x} \, dx}{e^4}+\frac {2 \int \frac {1-e^4 \left (4-e^5\right ) x}{x^3} \, dx}{e^4}-\frac {4 \int \frac {\log (x)}{x^3} \, dx}{e^4}-\frac {6 \int \frac {e^{4+x}}{x^2} \, dx}{e^4}+\frac {6 \int \frac {e^{4+x}}{x} \, dx}{e^4} \\ & = 2 e^x+\frac {1}{e^4 x^2}+\frac {6 e^x}{x}-\frac {\left (1-e^4 \left (4-e^5\right ) x\right )^2}{e^4 x^2}+6 \text {Ei}(x)+\frac {2 \log (x)}{e^4 x^2}-\frac {6 \int \frac {e^{4+x}}{x} \, dx}{e^4} \\ & = 2 e^x+\frac {1}{e^4 x^2}+\frac {6 e^x}{x}-\frac {\left (1-e^4 \left (4-e^5\right ) x\right )^2}{e^4 x^2}+\frac {2 \log (x)}{e^4 x^2} \\ \end{align*}
Time = 0.06 (sec) , antiderivative size = 30, normalized size of antiderivative = 0.97 \[ \int \frac {2-8 e^4 x+2 e^9 x+e^{4+x} \left (-6 x+6 x^2+2 x^3\right )-4 \log (x)}{e^4 x^3} \, dx=\frac {2 \left (-e^4 x \left (-4+e^5-e^x (3+x)\right )+\log (x)\right )}{e^4 x^2} \]
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Time = 0.84 (sec) , antiderivative size = 29, normalized size of antiderivative = 0.94
method | result | size |
risch | \(\frac {2 \,{\mathrm e}^{-4} \ln \left (x \right )}{x^{2}}-\frac {2 \left ({\mathrm e}^{5}-{\mathrm e}^{x} x -3 \,{\mathrm e}^{x}-4\right )}{x}\) | \(29\) |
norman | \(\frac {\left (-2 \,{\mathrm e}^{5}+8\right ) x +6 \,{\mathrm e}^{x} x +2 \,{\mathrm e}^{x} x^{2}+2 \,{\mathrm e}^{-4} \ln \left (x \right )}{x^{2}}\) | \(34\) |
default | \({\mathrm e}^{-4} \left (-\frac {2 \left (-4 \,{\mathrm e}^{4}+{\mathrm e}^{9}\right )}{x}+\frac {2 \ln \left (x \right )}{x^{2}}+2 \,{\mathrm e}^{4} \left (\frac {3 \,{\mathrm e}^{x}}{x}+{\mathrm e}^{x}\right )\right )\) | \(40\) |
parallelrisch | \(-\frac {{\mathrm e}^{-4} \left (-2 x^{2} {\mathrm e}^{4} {\mathrm e}^{x}+2 x \,{\mathrm e}^{4} {\mathrm e}^{5}-6 x \,{\mathrm e}^{4} {\mathrm e}^{x}-8 x \,{\mathrm e}^{4}-2 \ln \left (x \right )\right )}{x^{2}}\) | \(43\) |
parts | \(\frac {2 \,{\mathrm e}^{-4} \ln \left (x \right )}{x^{2}}+\frac {{\mathrm e}^{-4}}{x^{2}}+\frac {6 \,{\mathrm e}^{x}}{x}+2 \,{\mathrm e}^{x}+2 \,{\mathrm e}^{-4} \left (-\frac {-4 \,{\mathrm e}^{4}+{\mathrm e}^{9}}{x}-\frac {1}{2 x^{2}}\right )\) | \(56\) |
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Time = 0.24 (sec) , antiderivative size = 34, normalized size of antiderivative = 1.10 \[ \int \frac {2-8 e^4 x+2 e^9 x+e^{4+x} \left (-6 x+6 x^2+2 x^3\right )-4 \log (x)}{e^4 x^3} \, dx=-\frac {2 \, {\left (x e^{9} - 4 \, x e^{4} - {\left (x^{2} + 3 \, x\right )} e^{\left (x + 4\right )} - \log \left (x\right )\right )} e^{\left (-4\right )}}{x^{2}} \]
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Time = 0.13 (sec) , antiderivative size = 29, normalized size of antiderivative = 0.94 \[ \int \frac {2-8 e^4 x+2 e^9 x+e^{4+x} \left (-6 x+6 x^2+2 x^3\right )-4 \log (x)}{e^4 x^3} \, dx=\frac {\left (2 x + 6\right ) e^{x}}{x} - \frac {-8 + 2 e^{5}}{x} + \frac {2 \log {\left (x \right )}}{x^{2} e^{4}} \]
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Result contains higher order function than in optimal. Order 4 vs. order 3.
Time = 0.23 (sec) , antiderivative size = 44, normalized size of antiderivative = 1.42 \[ \int \frac {2-8 e^4 x+2 e^9 x+e^{4+x} \left (-6 x+6 x^2+2 x^3\right )-4 \log (x)}{e^4 x^3} \, dx=2 \, {\left (3 \, {\rm Ei}\left (x\right ) e^{4} - 3 \, e^{4} \Gamma \left (-1, -x\right ) - \frac {e^{9}}{x} + \frac {4 \, e^{4}}{x} + \frac {\log \left (x\right )}{x^{2}} + e^{\left (x + 4\right )}\right )} e^{\left (-4\right )} \]
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Time = 0.27 (sec) , antiderivative size = 35, normalized size of antiderivative = 1.13 \[ \int \frac {2-8 e^4 x+2 e^9 x+e^{4+x} \left (-6 x+6 x^2+2 x^3\right )-4 \log (x)}{e^4 x^3} \, dx=\frac {2 \, {\left (x^{2} e^{\left (x + 4\right )} - x e^{9} + 4 \, x e^{4} + 3 \, x e^{\left (x + 4\right )} + \log \left (x\right )\right )} e^{\left (-4\right )}}{x^{2}} \]
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Time = 8.68 (sec) , antiderivative size = 28, normalized size of antiderivative = 0.90 \[ \int \frac {2-8 e^4 x+2 e^9 x+e^{4+x} \left (-6 x+6 x^2+2 x^3\right )-4 \log (x)}{e^4 x^3} \, dx=2\,{\mathrm {e}}^x+\frac {x\,\left (6\,{\mathrm {e}}^x-2\,{\mathrm {e}}^5+8\right )+2\,{\mathrm {e}}^{-4}\,\ln \left (x\right )}{x^2} \]
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