Integrand size = 108, antiderivative size = 28 \[ \int \frac {e^{-x} \left (-4 e^x x+e^{x^3} \left (-81-18 x-x^2+e^x \left (-81 x-18 x^2+728 x^3-81 x^4-45 x^5-3 x^6\right )\right )+e^{x^3} \left (81 x+18 x^2-242 x^3-54 x^4-3 x^5\right ) \log (x)\right )}{81 x+18 x^2+x^3} \, dx=\frac {4}{9+x}+e^{x^3} \left (3-x-e^{-x} \log (x)\right ) \]
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Result contains higher order function than in optimal. Order 4 vs. order 3 in optimal.
Time = 0.80 (sec) , antiderivative size = 73, normalized size of antiderivative = 2.61, number of steps used = 10, number of rules used = 9, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.083, Rules used = {1608, 27, 6820, 2258, 2239, 2240, 2250, 6838, 2634} \[ \int \frac {e^{-x} \left (-4 e^x x+e^{x^3} \left (-81-18 x-x^2+e^x \left (-81 x-18 x^2+728 x^3-81 x^4-45 x^5-3 x^6\right )\right )+e^{x^3} \left (81 x+18 x^2-242 x^3-54 x^4-3 x^5\right ) \log (x)\right )}{81 x+18 x^2+x^3} \, dx=3 e^{x^3}-e^{x^3-x} \log (x)+\frac {x \Gamma \left (\frac {1}{3},-x^3\right )}{3 \sqrt [3]{-x^3}}+\frac {x^4 \Gamma \left (\frac {4}{3},-x^3\right )}{\left (-x^3\right )^{4/3}}+\frac {4}{x+9} \]
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Rule 27
Rule 1608
Rule 2239
Rule 2240
Rule 2250
Rule 2258
Rule 2634
Rule 6820
Rule 6838
Rubi steps \begin{align*} \text {integral}& = \int \frac {e^{-x} \left (-4 e^x x+e^{x^3} \left (-81-18 x-x^2+e^x \left (-81 x-18 x^2+728 x^3-81 x^4-45 x^5-3 x^6\right )\right )+e^{x^3} \left (81 x+18 x^2-242 x^3-54 x^4-3 x^5\right ) \log (x)\right )}{x \left (81+18 x+x^2\right )} \, dx \\ & = \int \frac {e^{-x} \left (-4 e^x x+e^{x^3} \left (-81-18 x-x^2+e^x \left (-81 x-18 x^2+728 x^3-81 x^4-45 x^5-3 x^6\right )\right )+e^{x^3} \left (81 x+18 x^2-242 x^3-54 x^4-3 x^5\right ) \log (x)\right )}{x (9+x)^2} \, dx \\ & = \int \left (-\frac {e^{-x+x^3}}{x}-\frac {4}{(9+x)^2}+e^{x^3} \left (-1+9 x^2-3 x^3\right )+e^{-x+x^3} \left (1-3 x^2\right ) \log (x)\right ) \, dx \\ & = \frac {4}{9+x}-\int \frac {e^{-x+x^3}}{x} \, dx+\int e^{x^3} \left (-1+9 x^2-3 x^3\right ) \, dx+\int e^{-x+x^3} \left (1-3 x^2\right ) \log (x) \, dx \\ & = \frac {4}{9+x}-e^{-x+x^3} \log (x)+\int \left (-e^{x^3}+9 e^{x^3} x^2-3 e^{x^3} x^3\right ) \, dx \\ & = \frac {4}{9+x}-e^{-x+x^3} \log (x)-3 \int e^{x^3} x^3 \, dx+9 \int e^{x^3} x^2 \, dx-\int e^{x^3} \, dx \\ & = 3 e^{x^3}+\frac {4}{9+x}+\frac {x \Gamma \left (\frac {1}{3},-x^3\right )}{3 \sqrt [3]{-x^3}}+\frac {x^4 \Gamma \left (\frac {4}{3},-x^3\right )}{\left (-x^3\right )^{4/3}}-e^{-x+x^3} \log (x) \\ \end{align*}
Time = 1.35 (sec) , antiderivative size = 31, normalized size of antiderivative = 1.11 \[ \int \frac {e^{-x} \left (-4 e^x x+e^{x^3} \left (-81-18 x-x^2+e^x \left (-81 x-18 x^2+728 x^3-81 x^4-45 x^5-3 x^6\right )\right )+e^{x^3} \left (81 x+18 x^2-242 x^3-54 x^4-3 x^5\right ) \log (x)\right )}{81 x+18 x^2+x^3} \, dx=-e^{x^3} (-3+x)+\frac {4}{9+x}-e^{-x+x^3} \log (x) \]
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Time = 1.41 (sec) , antiderivative size = 45, normalized size of antiderivative = 1.61
method | result | size |
risch | \(-\ln \left (x \right ) {\mathrm e}^{x \left (-1+x \right ) \left (1+x \right )}-\frac {x^{2} {\mathrm e}^{x^{3}}+6 \,{\mathrm e}^{x^{3}} x -27 \,{\mathrm e}^{x^{3}}-4}{x +9}\) | \(45\) |
parallelrisch | \(-\frac {\left (9 \,{\mathrm e}^{x} {\mathrm e}^{x^{3}} x^{2}+9 x \,{\mathrm e}^{x^{3}} \ln \left (x \right )+54 \,{\mathrm e}^{x} {\mathrm e}^{x^{3}} x +81 \,{\mathrm e}^{x^{3}} \ln \left (x \right )-243 \,{\mathrm e}^{x} {\mathrm e}^{x^{3}}-36 \,{\mathrm e}^{x}\right ) {\mathrm e}^{-x}}{9 \left (x +9\right )}\) | \(62\) |
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Time = 0.24 (sec) , antiderivative size = 41, normalized size of antiderivative = 1.46 \[ \int \frac {e^{-x} \left (-4 e^x x+e^{x^3} \left (-81-18 x-x^2+e^x \left (-81 x-18 x^2+728 x^3-81 x^4-45 x^5-3 x^6\right )\right )+e^{x^3} \left (81 x+18 x^2-242 x^3-54 x^4-3 x^5\right ) \log (x)\right )}{81 x+18 x^2+x^3} \, dx=-\frac {{\left ({\left (x + 9\right )} e^{\left (x^{3}\right )} \log \left (x\right ) + {\left (x^{2} + 6 \, x - 27\right )} e^{\left (x^{3} + x\right )} - 4 \, e^{x}\right )} e^{\left (-x\right )}}{x + 9} \]
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Time = 0.16 (sec) , antiderivative size = 26, normalized size of antiderivative = 0.93 \[ \int \frac {e^{-x} \left (-4 e^x x+e^{x^3} \left (-81-18 x-x^2+e^x \left (-81 x-18 x^2+728 x^3-81 x^4-45 x^5-3 x^6\right )\right )+e^{x^3} \left (81 x+18 x^2-242 x^3-54 x^4-3 x^5\right ) \log (x)\right )}{81 x+18 x^2+x^3} \, dx=\left (- x e^{x} + 3 e^{x} - \log {\left (x \right )}\right ) e^{- x} e^{x^{3}} + \frac {4}{x + 9} \]
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Time = 0.24 (sec) , antiderivative size = 27, normalized size of antiderivative = 0.96 \[ \int \frac {e^{-x} \left (-4 e^x x+e^{x^3} \left (-81-18 x-x^2+e^x \left (-81 x-18 x^2+728 x^3-81 x^4-45 x^5-3 x^6\right )\right )+e^{x^3} \left (81 x+18 x^2-242 x^3-54 x^4-3 x^5\right ) \log (x)\right )}{81 x+18 x^2+x^3} \, dx=-{\left ({\left (x - 3\right )} e^{x} + \log \left (x\right )\right )} e^{\left (x^{3} - x\right )} + \frac {4}{x + 9} \]
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Leaf count of result is larger than twice the leaf count of optimal. 54 vs. \(2 (24) = 48\).
Time = 0.29 (sec) , antiderivative size = 54, normalized size of antiderivative = 1.93 \[ \int \frac {e^{-x} \left (-4 e^x x+e^{x^3} \left (-81-18 x-x^2+e^x \left (-81 x-18 x^2+728 x^3-81 x^4-45 x^5-3 x^6\right )\right )+e^{x^3} \left (81 x+18 x^2-242 x^3-54 x^4-3 x^5\right ) \log (x)\right )}{81 x+18 x^2+x^3} \, dx=-\frac {x^{2} e^{\left (x^{3}\right )} + x e^{\left (x^{3} - x\right )} \log \left (x\right ) + 6 \, x e^{\left (x^{3}\right )} + 9 \, e^{\left (x^{3} - x\right )} \log \left (x\right ) - 27 \, e^{\left (x^{3}\right )} - 4}{x + 9} \]
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Timed out. \[ \int \frac {e^{-x} \left (-4 e^x x+e^{x^3} \left (-81-18 x-x^2+e^x \left (-81 x-18 x^2+728 x^3-81 x^4-45 x^5-3 x^6\right )\right )+e^{x^3} \left (81 x+18 x^2-242 x^3-54 x^4-3 x^5\right ) \log (x)\right )}{81 x+18 x^2+x^3} \, dx=\int -\frac {{\mathrm {e}}^{-x}\,\left ({\mathrm {e}}^{x^3}\,\left (18\,x+{\mathrm {e}}^x\,\left (3\,x^6+45\,x^5+81\,x^4-728\,x^3+18\,x^2+81\,x\right )+x^2+81\right )+4\,x\,{\mathrm {e}}^x+{\mathrm {e}}^{x^3}\,\ln \left (x\right )\,\left (3\,x^5+54\,x^4+242\,x^3-18\,x^2-81\,x\right )\right )}{x^3+18\,x^2+81\,x} \,d x \]
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