Integrand size = 56, antiderivative size = 21 \[ \int \frac {e^{-x+x^{\frac {e^3+4 x}{x}}} \left (x^2+x^{\frac {e^3+4 x}{x}} \left (-e^3-4 x+e^3 \log (x)\right )\right )}{x^2} \, dx=-e^{-x+x^{\frac {e^3+4 x}{x}}} \]
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\[ \int \frac {e^{-x+x^{\frac {e^3+4 x}{x}}} \left (x^2+x^{\frac {e^3+4 x}{x}} \left (-e^3-4 x+e^3 \log (x)\right )\right )}{x^2} \, dx=\int \frac {e^{-x+x^{\frac {e^3+4 x}{x}}} \left (x^2+x^{\frac {e^3+4 x}{x}} \left (-e^3-4 x+e^3 \log (x)\right )\right )}{x^2} \, dx \]
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Rubi steps \begin{align*} \text {integral}& = \int \left (e^{-x+x^{\frac {e^3+4 x}{x}}}-e^{-x+x^{\frac {e^3+4 x}{x}}} x^{2+\frac {e^3}{x}} \left (e^3+4 x-e^3 \log (x)\right )\right ) \, dx \\ & = \int e^{-x+x^{\frac {e^3+4 x}{x}}} \, dx-\int e^{-x+x^{\frac {e^3+4 x}{x}}} x^{2+\frac {e^3}{x}} \left (e^3+4 x-e^3 \log (x)\right ) \, dx \\ & = \int e^{-x+x^{\frac {e^3+4 x}{x}}} \, dx-\int \left (e^{3-x+x^{\frac {e^3+4 x}{x}}} x^{2+\frac {e^3}{x}}+4 e^{-x+x^{\frac {e^3+4 x}{x}}} x^{3+\frac {e^3}{x}}-e^{3-x+x^{\frac {e^3+4 x}{x}}} x^{2+\frac {e^3}{x}} \log (x)\right ) \, dx \\ & = -\left (4 \int e^{-x+x^{\frac {e^3+4 x}{x}}} x^{3+\frac {e^3}{x}} \, dx\right )+\int e^{-x+x^{\frac {e^3+4 x}{x}}} \, dx-\int e^{3-x+x^{\frac {e^3+4 x}{x}}} x^{2+\frac {e^3}{x}} \, dx+\int e^{3-x+x^{\frac {e^3+4 x}{x}}} x^{2+\frac {e^3}{x}} \log (x) \, dx \\ & = -\left (4 \int e^{-x+x^{\frac {e^3+4 x}{x}}} x^{3+\frac {e^3}{x}} \, dx\right )+\log (x) \int e^{3-x+x^{\frac {e^3+4 x}{x}}} x^{2+\frac {e^3}{x}} \, dx+\int e^{-x+x^{\frac {e^3+4 x}{x}}} \, dx-\int e^{3-x+x^{\frac {e^3+4 x}{x}}} x^{2+\frac {e^3}{x}} \, dx-\int \frac {\int e^{3-x+x^{4+\frac {e^3}{x}}} x^{2+\frac {e^3}{x}} \, dx}{x} \, dx \\ \end{align*}
Time = 3.30 (sec) , antiderivative size = 19, normalized size of antiderivative = 0.90 \[ \int \frac {e^{-x+x^{\frac {e^3+4 x}{x}}} \left (x^2+x^{\frac {e^3+4 x}{x}} \left (-e^3-4 x+e^3 \log (x)\right )\right )}{x^2} \, dx=-e^{-x+x^{4+\frac {e^3}{x}}} \]
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Time = 0.94 (sec) , antiderivative size = 20, normalized size of antiderivative = 0.95
method | result | size |
risch | \(-{\mathrm e}^{x^{\frac {{\mathrm e}^{3}+4 x}{x}}-x}\) | \(20\) |
parallelrisch | \(-{\mathrm e}^{{\mathrm e}^{\frac {\left ({\mathrm e}^{3}+4 x \right ) \ln \left (x \right )}{x}}-x}\) | \(21\) |
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Time = 0.24 (sec) , antiderivative size = 19, normalized size of antiderivative = 0.90 \[ \int \frac {e^{-x+x^{\frac {e^3+4 x}{x}}} \left (x^2+x^{\frac {e^3+4 x}{x}} \left (-e^3-4 x+e^3 \log (x)\right )\right )}{x^2} \, dx=-e^{\left (x^{\frac {4 \, x + e^{3}}{x}} - x\right )} \]
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Time = 0.41 (sec) , antiderivative size = 17, normalized size of antiderivative = 0.81 \[ \int \frac {e^{-x+x^{\frac {e^3+4 x}{x}}} \left (x^2+x^{\frac {e^3+4 x}{x}} \left (-e^3-4 x+e^3 \log (x)\right )\right )}{x^2} \, dx=- e^{- x + e^{\frac {\left (4 x + e^{3}\right ) \log {\left (x \right )}}{x}}} \]
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Time = 0.26 (sec) , antiderivative size = 19, normalized size of antiderivative = 0.90 \[ \int \frac {e^{-x+x^{\frac {e^3+4 x}{x}}} \left (x^2+x^{\frac {e^3+4 x}{x}} \left (-e^3-4 x+e^3 \log (x)\right )\right )}{x^2} \, dx=-e^{\left (x^{4} x^{\frac {e^{3}}{x}} - x\right )} \]
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\[ \int \frac {e^{-x+x^{\frac {e^3+4 x}{x}}} \left (x^2+x^{\frac {e^3+4 x}{x}} \left (-e^3-4 x+e^3 \log (x)\right )\right )}{x^2} \, dx=\int { \frac {{\left ({\left (e^{3} \log \left (x\right ) - 4 \, x - e^{3}\right )} x^{\frac {4 \, x + e^{3}}{x}} + x^{2}\right )} e^{\left (x^{\frac {4 \, x + e^{3}}{x}} - x\right )}}{x^{2}} \,d x } \]
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Time = 13.16 (sec) , antiderivative size = 17, normalized size of antiderivative = 0.81 \[ \int \frac {e^{-x+x^{\frac {e^3+4 x}{x}}} \left (x^2+x^{\frac {e^3+4 x}{x}} \left (-e^3-4 x+e^3 \log (x)\right )\right )}{x^2} \, dx=-{\mathrm {e}}^{x^{\frac {{\mathrm {e}}^3}{x}+4}}\,{\mathrm {e}}^{-x} \]
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