Integrand size = 167, antiderivative size = 28 \[ \int \frac {e^{\frac {-1-e^5}{-3 e^5 x+e^{5+e^4} x}} \left (-1+e^5 (-1-3 x)+e^{5+e^4} x\right )}{-3 e^5 x^3+e^{5+e^4} x^3+e^{\frac {2 \left (-1-e^5\right )}{-3 e^5 x+e^{5+e^4} x}} \left (-3 e^5 x+e^{5+e^4} x\right )+e^{\frac {-1-e^5}{-3 e^5 x+e^{5+e^4} x}} \left (-6 e^5 x^2+2 e^{5+e^4} x^2\right )} \, dx=\frac {x}{e^{\frac {1+\frac {1}{e^5}}{\left (3-e^{e^4}\right ) x}}+x} \]
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Time = 4.07 (sec) , antiderivative size = 31, normalized size of antiderivative = 1.11, number of steps used = 4, number of rules used = 4, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.024, Rules used = {6, 6820, 12, 6819} \[ \int \frac {e^{\frac {-1-e^5}{-3 e^5 x+e^{5+e^4} x}} \left (-1+e^5 (-1-3 x)+e^{5+e^4} x\right )}{-3 e^5 x^3+e^{5+e^4} x^3+e^{\frac {2 \left (-1-e^5\right )}{-3 e^5 x+e^{5+e^4} x}} \left (-3 e^5 x+e^{5+e^4} x\right )+e^{\frac {-1-e^5}{-3 e^5 x+e^{5+e^4} x}} \left (-6 e^5 x^2+2 e^{5+e^4} x^2\right )} \, dx=\frac {x}{x+e^{\frac {1+e^5}{e^5 \left (3-e^{e^4}\right ) x}}} \]
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Rule 6
Rule 12
Rule 6819
Rule 6820
Rubi steps \begin{align*} \text {integral}& = \int \frac {e^{\frac {-1-e^5}{-3 e^5 x+e^{5+e^4} x}} \left (-1+e^5 (-1-3 x)+e^{5+e^4} x\right )}{\left (-3 e^5+e^{5+e^4}\right ) x^3+e^{\frac {2 \left (-1-e^5\right )}{-3 e^5 x+e^{5+e^4} x}} \left (-3 e^5 x+e^{5+e^4} x\right )+e^{\frac {-1-e^5}{-3 e^5 x+e^{5+e^4} x}} \left (-6 e^5 x^2+2 e^{5+e^4} x^2\right )} \, dx \\ & = \int \frac {\exp \left (\frac {1+e^5-5 e^5 \left (3-e^{e^4}\right ) x}{e^5 \left (3-e^{e^4}\right ) x}\right ) \left (1+e^5+e^5 \left (3-e^{e^4}\right ) x\right )}{\left (3-e^{e^4}\right ) x \left (e^{\frac {1+e^5}{3 e^5 x-e^{5+e^4} x}}+x\right )^2} \, dx \\ & = \frac {\int \frac {\exp \left (\frac {1+e^5-5 e^5 \left (3-e^{e^4}\right ) x}{e^5 \left (3-e^{e^4}\right ) x}\right ) \left (1+e^5+e^5 \left (3-e^{e^4}\right ) x\right )}{x \left (e^{\frac {1+e^5}{3 e^5 x-e^{5+e^4} x}}+x\right )^2} \, dx}{3-e^{e^4}} \\ & = \frac {x}{e^{\frac {1+e^5}{e^5 \left (3-e^{e^4}\right ) x}}+x} \\ \end{align*}
Time = 0.29 (sec) , antiderivative size = 33, normalized size of antiderivative = 1.18 \[ \int \frac {e^{\frac {-1-e^5}{-3 e^5 x+e^{5+e^4} x}} \left (-1+e^5 (-1-3 x)+e^{5+e^4} x\right )}{-3 e^5 x^3+e^{5+e^4} x^3+e^{\frac {2 \left (-1-e^5\right )}{-3 e^5 x+e^{5+e^4} x}} \left (-3 e^5 x+e^{5+e^4} x\right )+e^{\frac {-1-e^5}{-3 e^5 x+e^{5+e^4} x}} \left (-6 e^5 x^2+2 e^{5+e^4} x^2\right )} \, dx=\frac {x}{e^{\frac {1+e^5}{3 e^5 x-e^{5+e^4} x}}+x} \]
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Time = 0.68 (sec) , antiderivative size = 30, normalized size of antiderivative = 1.07
method | result | size |
risch | \(\frac {x}{x +{\mathrm e}^{\frac {{\mathrm e}^{5}+1}{x \left (3 \,{\mathrm e}^{5}-{\mathrm e}^{5+{\mathrm e}^{4}}\right )}}}\) | \(30\) |
parallelrisch | \(\frac {\left (x \,{\mathrm e}^{5} {\mathrm e}^{{\mathrm e}^{4}}-3 x \,{\mathrm e}^{5}\right ) {\mathrm e}^{-5}}{\left ({\mathrm e}^{{\mathrm e}^{4}}-3\right ) \left ({\mathrm e}^{-\frac {\left ({\mathrm e}^{5}+1\right ) {\mathrm e}^{-5}}{x \left ({\mathrm e}^{{\mathrm e}^{4}}-3\right )}}+x \right )}\) | \(51\) |
norman | \(-\frac {{\mathrm e}^{\frac {-{\mathrm e}^{5}-1}{x \,{\mathrm e}^{5} {\mathrm e}^{{\mathrm e}^{4}}-3 x \,{\mathrm e}^{5}}}}{x +{\mathrm e}^{\frac {-{\mathrm e}^{5}-1}{x \,{\mathrm e}^{5} {\mathrm e}^{{\mathrm e}^{4}}-3 x \,{\mathrm e}^{5}}}}\) | \(53\) |
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Time = 0.25 (sec) , antiderivative size = 28, normalized size of antiderivative = 1.00 \[ \int \frac {e^{\frac {-1-e^5}{-3 e^5 x+e^{5+e^4} x}} \left (-1+e^5 (-1-3 x)+e^{5+e^4} x\right )}{-3 e^5 x^3+e^{5+e^4} x^3+e^{\frac {2 \left (-1-e^5\right )}{-3 e^5 x+e^{5+e^4} x}} \left (-3 e^5 x+e^{5+e^4} x\right )+e^{\frac {-1-e^5}{-3 e^5 x+e^{5+e^4} x}} \left (-6 e^5 x^2+2 e^{5+e^4} x^2\right )} \, dx=\frac {x}{x + e^{\left (\frac {e^{5} + 1}{3 \, x e^{5} - x e^{\left (e^{4} + 5\right )}}\right )}} \]
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Time = 0.12 (sec) , antiderivative size = 27, normalized size of antiderivative = 0.96 \[ \int \frac {e^{\frac {-1-e^5}{-3 e^5 x+e^{5+e^4} x}} \left (-1+e^5 (-1-3 x)+e^{5+e^4} x\right )}{-3 e^5 x^3+e^{5+e^4} x^3+e^{\frac {2 \left (-1-e^5\right )}{-3 e^5 x+e^{5+e^4} x}} \left (-3 e^5 x+e^{5+e^4} x\right )+e^{\frac {-1-e^5}{-3 e^5 x+e^{5+e^4} x}} \left (-6 e^5 x^2+2 e^{5+e^4} x^2\right )} \, dx=\frac {x}{x + e^{\frac {- e^{5} - 1}{- 3 x e^{5} + x e^{5} e^{e^{4}}}}} \]
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Time = 0.42 (sec) , antiderivative size = 40, normalized size of antiderivative = 1.43 \[ \int \frac {e^{\frac {-1-e^5}{-3 e^5 x+e^{5+e^4} x}} \left (-1+e^5 (-1-3 x)+e^{5+e^4} x\right )}{-3 e^5 x^3+e^{5+e^4} x^3+e^{\frac {2 \left (-1-e^5\right )}{-3 e^5 x+e^{5+e^4} x}} \left (-3 e^5 x+e^{5+e^4} x\right )+e^{\frac {-1-e^5}{-3 e^5 x+e^{5+e^4} x}} \left (-6 e^5 x^2+2 e^{5+e^4} x^2\right )} \, dx=-\frac {1}{x e^{\left (-\frac {1}{x {\left (3 \, e^{5} - e^{\left (e^{4} + 5\right )}\right )}} + \frac {1}{x {\left (e^{\left (e^{4}\right )} - 3\right )}}\right )} + 1} \]
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\[ \int \frac {e^{\frac {-1-e^5}{-3 e^5 x+e^{5+e^4} x}} \left (-1+e^5 (-1-3 x)+e^{5+e^4} x\right )}{-3 e^5 x^3+e^{5+e^4} x^3+e^{\frac {2 \left (-1-e^5\right )}{-3 e^5 x+e^{5+e^4} x}} \left (-3 e^5 x+e^{5+e^4} x\right )+e^{\frac {-1-e^5}{-3 e^5 x+e^{5+e^4} x}} \left (-6 e^5 x^2+2 e^{5+e^4} x^2\right )} \, dx=\int { \frac {{\left ({\left (3 \, x + 1\right )} e^{5} - x e^{\left (e^{4} + 5\right )} + 1\right )} e^{\left (\frac {e^{5} + 1}{3 \, x e^{5} - x e^{\left (e^{4} + 5\right )}}\right )}}{3 \, x^{3} e^{5} - x^{3} e^{\left (e^{4} + 5\right )} + {\left (3 \, x e^{5} - x e^{\left (e^{4} + 5\right )}\right )} e^{\left (\frac {2 \, {\left (e^{5} + 1\right )}}{3 \, x e^{5} - x e^{\left (e^{4} + 5\right )}}\right )} + 2 \, {\left (3 \, x^{2} e^{5} - x^{2} e^{\left (e^{4} + 5\right )}\right )} e^{\left (\frac {e^{5} + 1}{3 \, x e^{5} - x e^{\left (e^{4} + 5\right )}}\right )}} \,d x } \]
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Time = 13.21 (sec) , antiderivative size = 44, normalized size of antiderivative = 1.57 \[ \int \frac {e^{\frac {-1-e^5}{-3 e^5 x+e^{5+e^4} x}} \left (-1+e^5 (-1-3 x)+e^{5+e^4} x\right )}{-3 e^5 x^3+e^{5+e^4} x^3+e^{\frac {2 \left (-1-e^5\right )}{-3 e^5 x+e^{5+e^4} x}} \left (-3 e^5 x+e^{5+e^4} x\right )+e^{\frac {-1-e^5}{-3 e^5 x+e^{5+e^4} x}} \left (-6 e^5 x^2+2 e^{5+e^4} x^2\right )} \, dx=\frac {x}{x+{\mathrm {e}}^{\frac {{\mathrm {e}}^5}{3\,x\,{\mathrm {e}}^5-x\,{\mathrm {e}}^5\,{\mathrm {e}}^{{\mathrm {e}}^4}}}\,{\mathrm {e}}^{\frac {1}{3\,x\,{\mathrm {e}}^5-x\,{\mathrm {e}}^5\,{\mathrm {e}}^{{\mathrm {e}}^4}}}} \]
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