Integrand size = 255, antiderivative size = 22 \[ \int \frac {-2 x-e^5 x-x^2+e^x \left (-6 e^5 x^3-6 x^4\right )+e^{2 x} \left (-6 e^5 x^3-6 x^4\right )+\left (e^5+x+e^x \left (-18 e^5 x^2-18 x^3\right )+e^{2 x} \left (-18 e^5 x^2-18 x^3\right )\right ) \log \left (e^5+x\right )+\left (e^x \left (-18 e^5 x-18 x^2\right )+e^{2 x} \left (-18 e^5 x-18 x^2\right )\right ) \log ^2\left (e^5+x\right )+\left (e^x \left (-6 e^5-6 x\right )+e^{2 x} \left (-6 e^5-6 x\right )\right ) \log ^3\left (e^5+x\right )}{e^5 x^3+x^4+\left (3 e^5 x^2+3 x^3\right ) \log \left (e^5+x\right )+\left (3 e^5 x+3 x^2\right ) \log ^2\left (e^5+x\right )+\left (e^5+x\right ) \log ^3\left (e^5+x\right )} \, dx=-3 \left (1+e^x\right )^2+\frac {x}{\left (x+\log \left (e^5+x\right )\right )^2} \]
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\[ \int \frac {-2 x-e^5 x-x^2+e^x \left (-6 e^5 x^3-6 x^4\right )+e^{2 x} \left (-6 e^5 x^3-6 x^4\right )+\left (e^5+x+e^x \left (-18 e^5 x^2-18 x^3\right )+e^{2 x} \left (-18 e^5 x^2-18 x^3\right )\right ) \log \left (e^5+x\right )+\left (e^x \left (-18 e^5 x-18 x^2\right )+e^{2 x} \left (-18 e^5 x-18 x^2\right )\right ) \log ^2\left (e^5+x\right )+\left (e^x \left (-6 e^5-6 x\right )+e^{2 x} \left (-6 e^5-6 x\right )\right ) \log ^3\left (e^5+x\right )}{e^5 x^3+x^4+\left (3 e^5 x^2+3 x^3\right ) \log \left (e^5+x\right )+\left (3 e^5 x+3 x^2\right ) \log ^2\left (e^5+x\right )+\left (e^5+x\right ) \log ^3\left (e^5+x\right )} \, dx=\int \frac {-2 x-e^5 x-x^2+e^x \left (-6 e^5 x^3-6 x^4\right )+e^{2 x} \left (-6 e^5 x^3-6 x^4\right )+\left (e^5+x+e^x \left (-18 e^5 x^2-18 x^3\right )+e^{2 x} \left (-18 e^5 x^2-18 x^3\right )\right ) \log \left (e^5+x\right )+\left (e^x \left (-18 e^5 x-18 x^2\right )+e^{2 x} \left (-18 e^5 x-18 x^2\right )\right ) \log ^2\left (e^5+x\right )+\left (e^x \left (-6 e^5-6 x\right )+e^{2 x} \left (-6 e^5-6 x\right )\right ) \log ^3\left (e^5+x\right )}{e^5 x^3+x^4+\left (3 e^5 x^2+3 x^3\right ) \log \left (e^5+x\right )+\left (3 e^5 x+3 x^2\right ) \log ^2\left (e^5+x\right )+\left (e^5+x\right ) \log ^3\left (e^5+x\right )} \, dx \]
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Rubi steps \begin{align*} \text {integral}& = \int \frac {\left (-2-e^5\right ) x-x^2+e^x \left (-6 e^5 x^3-6 x^4\right )+e^{2 x} \left (-6 e^5 x^3-6 x^4\right )+\left (e^5+x+e^x \left (-18 e^5 x^2-18 x^3\right )+e^{2 x} \left (-18 e^5 x^2-18 x^3\right )\right ) \log \left (e^5+x\right )+\left (e^x \left (-18 e^5 x-18 x^2\right )+e^{2 x} \left (-18 e^5 x-18 x^2\right )\right ) \log ^2\left (e^5+x\right )+\left (e^x \left (-6 e^5-6 x\right )+e^{2 x} \left (-6 e^5-6 x\right )\right ) \log ^3\left (e^5+x\right )}{e^5 x^3+x^4+\left (3 e^5 x^2+3 x^3\right ) \log \left (e^5+x\right )+\left (3 e^5 x+3 x^2\right ) \log ^2\left (e^5+x\right )+\left (e^5+x\right ) \log ^3\left (e^5+x\right )} \, dx \\ & = \int \frac {-x \left (2+e^5+x+6 e^{5+x} x^2+6 e^{5+2 x} x^2+6 e^x x^3+6 e^{2 x} x^3\right )-\left (e^5+x\right ) \left (-1+18 e^x x^2+18 e^{2 x} x^2\right ) \log \left (e^5+x\right )-18 e^x \left (1+e^x\right ) x \left (e^5+x\right ) \log ^2\left (e^5+x\right )-6 e^x \left (1+e^x\right ) \left (e^5+x\right ) \log ^3\left (e^5+x\right )}{\left (e^5+x\right ) \left (x+\log \left (e^5+x\right )\right )^3} \, dx \\ & = \int \left (-6 e^x-6 e^{2 x}-\frac {2 \left (1+\frac {e^5}{2}\right ) x}{\left (e^5+x\right ) \left (x+\log \left (e^5+x\right )\right )^3}-\frac {x^2}{\left (e^5+x\right ) \left (x+\log \left (e^5+x\right )\right )^3}+\frac {\log \left (e^5+x\right )}{\left (x+\log \left (e^5+x\right )\right )^3}\right ) \, dx \\ & = -\left (6 \int e^x \, dx\right )-6 \int e^{2 x} \, dx-\left (2+e^5\right ) \int \frac {x}{\left (e^5+x\right ) \left (x+\log \left (e^5+x\right )\right )^3} \, dx-\int \frac {x^2}{\left (e^5+x\right ) \left (x+\log \left (e^5+x\right )\right )^3} \, dx+\int \frac {\log \left (e^5+x\right )}{\left (x+\log \left (e^5+x\right )\right )^3} \, dx \\ & = -6 e^x-3 e^{2 x}-\left (2+e^5\right ) \int \left (\frac {1}{\left (x+\log \left (e^5+x\right )\right )^3}-\frac {e^5}{\left (e^5+x\right ) \left (x+\log \left (e^5+x\right )\right )^3}\right ) \, dx-\int \left (-\frac {e^5}{\left (x+\log \left (e^5+x\right )\right )^3}+\frac {x}{\left (x+\log \left (e^5+x\right )\right )^3}+\frac {e^{10}}{\left (e^5+x\right ) \left (x+\log \left (e^5+x\right )\right )^3}\right ) \, dx+\int \left (-\frac {x}{\left (x+\log \left (e^5+x\right )\right )^3}+\frac {1}{\left (x+\log \left (e^5+x\right )\right )^2}\right ) \, dx \\ & = -6 e^x-3 e^{2 x}+e^5 \int \frac {1}{\left (x+\log \left (e^5+x\right )\right )^3} \, dx-e^{10} \int \frac {1}{\left (e^5+x\right ) \left (x+\log \left (e^5+x\right )\right )^3} \, dx-\left (2+e^5\right ) \int \frac {1}{\left (x+\log \left (e^5+x\right )\right )^3} \, dx+\left (e^5 \left (2+e^5\right )\right ) \int \frac {1}{\left (e^5+x\right ) \left (x+\log \left (e^5+x\right )\right )^3} \, dx-2 \int \frac {x}{\left (x+\log \left (e^5+x\right )\right )^3} \, dx+\int \frac {1}{\left (x+\log \left (e^5+x\right )\right )^2} \, dx \\ \end{align*}
Time = 0.13 (sec) , antiderivative size = 25, normalized size of antiderivative = 1.14 \[ \int \frac {-2 x-e^5 x-x^2+e^x \left (-6 e^5 x^3-6 x^4\right )+e^{2 x} \left (-6 e^5 x^3-6 x^4\right )+\left (e^5+x+e^x \left (-18 e^5 x^2-18 x^3\right )+e^{2 x} \left (-18 e^5 x^2-18 x^3\right )\right ) \log \left (e^5+x\right )+\left (e^x \left (-18 e^5 x-18 x^2\right )+e^{2 x} \left (-18 e^5 x-18 x^2\right )\right ) \log ^2\left (e^5+x\right )+\left (e^x \left (-6 e^5-6 x\right )+e^{2 x} \left (-6 e^5-6 x\right )\right ) \log ^3\left (e^5+x\right )}{e^5 x^3+x^4+\left (3 e^5 x^2+3 x^3\right ) \log \left (e^5+x\right )+\left (3 e^5 x+3 x^2\right ) \log ^2\left (e^5+x\right )+\left (e^5+x\right ) \log ^3\left (e^5+x\right )} \, dx=-6 e^x-3 e^{2 x}+\frac {x}{\left (x+\log \left (e^5+x\right )\right )^2} \]
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Time = 21.04 (sec) , antiderivative size = 23, normalized size of antiderivative = 1.05
method | result | size |
risch | \(-3 \,{\mathrm e}^{2 x}-6 \,{\mathrm e}^{x}+\frac {x}{{\left (\ln \left ({\mathrm e}^{5}+x \right )+x \right )}^{2}}\) | \(23\) |
parallelrisch | \(\frac {-3 \,{\mathrm e}^{2 x} x^{2}-6 \ln \left ({\mathrm e}^{5}+x \right ) {\mathrm e}^{2 x} x -3 \ln \left ({\mathrm e}^{5}+x \right )^{2} {\mathrm e}^{2 x}-6 \,{\mathrm e}^{x} x^{2}-12 x \,{\mathrm e}^{x} \ln \left ({\mathrm e}^{5}+x \right )-6 \,{\mathrm e}^{x} \ln \left ({\mathrm e}^{5}+x \right )^{2}+x}{x^{2}+2 x \ln \left ({\mathrm e}^{5}+x \right )+\ln \left ({\mathrm e}^{5}+x \right )^{2}}\) | \(87\) |
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Leaf count of result is larger than twice the leaf count of optimal. 80 vs. \(2 (20) = 40\).
Time = 0.27 (sec) , antiderivative size = 80, normalized size of antiderivative = 3.64 \[ \int \frac {-2 x-e^5 x-x^2+e^x \left (-6 e^5 x^3-6 x^4\right )+e^{2 x} \left (-6 e^5 x^3-6 x^4\right )+\left (e^5+x+e^x \left (-18 e^5 x^2-18 x^3\right )+e^{2 x} \left (-18 e^5 x^2-18 x^3\right )\right ) \log \left (e^5+x\right )+\left (e^x \left (-18 e^5 x-18 x^2\right )+e^{2 x} \left (-18 e^5 x-18 x^2\right )\right ) \log ^2\left (e^5+x\right )+\left (e^x \left (-6 e^5-6 x\right )+e^{2 x} \left (-6 e^5-6 x\right )\right ) \log ^3\left (e^5+x\right )}{e^5 x^3+x^4+\left (3 e^5 x^2+3 x^3\right ) \log \left (e^5+x\right )+\left (3 e^5 x+3 x^2\right ) \log ^2\left (e^5+x\right )+\left (e^5+x\right ) \log ^3\left (e^5+x\right )} \, dx=-\frac {3 \, x^{2} e^{\left (2 \, x\right )} + 6 \, x^{2} e^{x} + 3 \, {\left (e^{\left (2 \, x\right )} + 2 \, e^{x}\right )} \log \left (x + e^{5}\right )^{2} + 6 \, {\left (x e^{\left (2 \, x\right )} + 2 \, x e^{x}\right )} \log \left (x + e^{5}\right ) - x}{x^{2} + 2 \, x \log \left (x + e^{5}\right ) + \log \left (x + e^{5}\right )^{2}} \]
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Time = 0.19 (sec) , antiderivative size = 34, normalized size of antiderivative = 1.55 \[ \int \frac {-2 x-e^5 x-x^2+e^x \left (-6 e^5 x^3-6 x^4\right )+e^{2 x} \left (-6 e^5 x^3-6 x^4\right )+\left (e^5+x+e^x \left (-18 e^5 x^2-18 x^3\right )+e^{2 x} \left (-18 e^5 x^2-18 x^3\right )\right ) \log \left (e^5+x\right )+\left (e^x \left (-18 e^5 x-18 x^2\right )+e^{2 x} \left (-18 e^5 x-18 x^2\right )\right ) \log ^2\left (e^5+x\right )+\left (e^x \left (-6 e^5-6 x\right )+e^{2 x} \left (-6 e^5-6 x\right )\right ) \log ^3\left (e^5+x\right )}{e^5 x^3+x^4+\left (3 e^5 x^2+3 x^3\right ) \log \left (e^5+x\right )+\left (3 e^5 x+3 x^2\right ) \log ^2\left (e^5+x\right )+\left (e^5+x\right ) \log ^3\left (e^5+x\right )} \, dx=\frac {x}{x^{2} + 2 x \log {\left (x + e^{5} \right )} + \log {\left (x + e^{5} \right )}^{2}} - 3 e^{2 x} - 6 e^{x} \]
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Leaf count of result is larger than twice the leaf count of optimal. 80 vs. \(2 (20) = 40\).
Time = 0.24 (sec) , antiderivative size = 80, normalized size of antiderivative = 3.64 \[ \int \frac {-2 x-e^5 x-x^2+e^x \left (-6 e^5 x^3-6 x^4\right )+e^{2 x} \left (-6 e^5 x^3-6 x^4\right )+\left (e^5+x+e^x \left (-18 e^5 x^2-18 x^3\right )+e^{2 x} \left (-18 e^5 x^2-18 x^3\right )\right ) \log \left (e^5+x\right )+\left (e^x \left (-18 e^5 x-18 x^2\right )+e^{2 x} \left (-18 e^5 x-18 x^2\right )\right ) \log ^2\left (e^5+x\right )+\left (e^x \left (-6 e^5-6 x\right )+e^{2 x} \left (-6 e^5-6 x\right )\right ) \log ^3\left (e^5+x\right )}{e^5 x^3+x^4+\left (3 e^5 x^2+3 x^3\right ) \log \left (e^5+x\right )+\left (3 e^5 x+3 x^2\right ) \log ^2\left (e^5+x\right )+\left (e^5+x\right ) \log ^3\left (e^5+x\right )} \, dx=-\frac {3 \, x^{2} e^{\left (2 \, x\right )} + 6 \, x^{2} e^{x} + 3 \, {\left (e^{\left (2 \, x\right )} + 2 \, e^{x}\right )} \log \left (x + e^{5}\right )^{2} + 6 \, {\left (x e^{\left (2 \, x\right )} + 2 \, x e^{x}\right )} \log \left (x + e^{5}\right ) - x}{x^{2} + 2 \, x \log \left (x + e^{5}\right ) + \log \left (x + e^{5}\right )^{2}} \]
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Leaf count of result is larger than twice the leaf count of optimal. 301 vs. \(2 (20) = 40\).
Time = 0.36 (sec) , antiderivative size = 301, normalized size of antiderivative = 13.68 \[ \int \frac {-2 x-e^5 x-x^2+e^x \left (-6 e^5 x^3-6 x^4\right )+e^{2 x} \left (-6 e^5 x^3-6 x^4\right )+\left (e^5+x+e^x \left (-18 e^5 x^2-18 x^3\right )+e^{2 x} \left (-18 e^5 x^2-18 x^3\right )\right ) \log \left (e^5+x\right )+\left (e^x \left (-18 e^5 x-18 x^2\right )+e^{2 x} \left (-18 e^5 x-18 x^2\right )\right ) \log ^2\left (e^5+x\right )+\left (e^x \left (-6 e^5-6 x\right )+e^{2 x} \left (-6 e^5-6 x\right )\right ) \log ^3\left (e^5+x\right )}{e^5 x^3+x^4+\left (3 e^5 x^2+3 x^3\right ) \log \left (e^5+x\right )+\left (3 e^5 x+3 x^2\right ) \log ^2\left (e^5+x\right )+\left (e^5+x\right ) \log ^3\left (e^5+x\right )} \, dx=-\frac {3 \, {\left (x + e^{5}\right )}^{2} e^{\left (2 \, x + 3 \, e^{5} + 5\right )} + 6 \, {\left (x + e^{5}\right )}^{2} e^{\left (x + 3 \, e^{5} + 5\right )} + 6 \, {\left (x + e^{5}\right )} e^{\left (2 \, x + 3 \, e^{5} + 5\right )} \log \left (x + e^{5}\right ) + 12 \, {\left (x + e^{5}\right )} e^{\left (x + 3 \, e^{5} + 5\right )} \log \left (x + e^{5}\right ) + 3 \, e^{\left (2 \, x + 3 \, e^{5} + 5\right )} \log \left (x + e^{5}\right )^{2} + 6 \, e^{\left (x + 3 \, e^{5} + 5\right )} \log \left (x + e^{5}\right )^{2} - 6 \, {\left (x + e^{5}\right )} e^{\left (2 \, x + 3 \, e^{5} + 10\right )} - 12 \, {\left (x + e^{5}\right )} e^{\left (x + 3 \, e^{5} + 10\right )} - {\left (x + e^{5}\right )} e^{\left (3 \, e^{5} + 5\right )} - 6 \, e^{\left (2 \, x + 3 \, e^{5} + 10\right )} \log \left (x + e^{5}\right ) - 12 \, e^{\left (x + 3 \, e^{5} + 10\right )} \log \left (x + e^{5}\right ) + 3 \, e^{\left (2 \, x + 3 \, e^{5} + 15\right )} + 6 \, e^{\left (x + 3 \, e^{5} + 15\right )} + e^{\left (3 \, e^{5} + 10\right )}}{{\left (x + e^{5}\right )}^{2} e^{\left (3 \, e^{5} + 5\right )} + 2 \, {\left (x + e^{5}\right )} e^{\left (3 \, e^{5} + 5\right )} \log \left (x + e^{5}\right ) + e^{\left (3 \, e^{5} + 5\right )} \log \left (x + e^{5}\right )^{2} - 2 \, {\left (x + e^{5}\right )} e^{\left (3 \, e^{5} + 10\right )} - 2 \, e^{\left (3 \, e^{5} + 10\right )} \log \left (x + e^{5}\right ) + e^{\left (3 \, e^{5} + 15\right )}} \]
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Time = 10.14 (sec) , antiderivative size = 181, normalized size of antiderivative = 8.23 \[ \int \frac {-2 x-e^5 x-x^2+e^x \left (-6 e^5 x^3-6 x^4\right )+e^{2 x} \left (-6 e^5 x^3-6 x^4\right )+\left (e^5+x+e^x \left (-18 e^5 x^2-18 x^3\right )+e^{2 x} \left (-18 e^5 x^2-18 x^3\right )\right ) \log \left (e^5+x\right )+\left (e^x \left (-18 e^5 x-18 x^2\right )+e^{2 x} \left (-18 e^5 x-18 x^2\right )\right ) \log ^2\left (e^5+x\right )+\left (e^x \left (-6 e^5-6 x\right )+e^{2 x} \left (-6 e^5-6 x\right )\right ) \log ^3\left (e^5+x\right )}{e^5 x^3+x^4+\left (3 e^5 x^2+3 x^3\right ) \log \left (e^5+x\right )+\left (3 e^5 x+3 x^2\right ) \log ^2\left (e^5+x\right )+\left (e^5+x\right ) \log ^3\left (e^5+x\right )} \, dx=\frac {\frac {x\,\left (x+{\mathrm {e}}^5+2\right )}{2\,\left (x+{\mathrm {e}}^5+1\right )}-\frac {\ln \left (x+{\mathrm {e}}^5\right )\,\left (x+{\mathrm {e}}^5\right )}{2\,\left (x+{\mathrm {e}}^5+1\right )}}{x^2+2\,x\,\ln \left (x+{\mathrm {e}}^5\right )+{\ln \left (x+{\mathrm {e}}^5\right )}^2}-6\,{\mathrm {e}}^x-3\,{\mathrm {e}}^{2\,x}+\frac {\frac {\left (x+{\mathrm {e}}^5\right )\,\left (x+2\,{\mathrm {e}}^5+{\mathrm {e}}^{10}+2\,x\,{\mathrm {e}}^5+x^2+1\right )}{2\,{\left (x+{\mathrm {e}}^5+1\right )}^3}-\frac {\ln \left (x+{\mathrm {e}}^5\right )\,\left (x+{\mathrm {e}}^5\right )}{2\,{\left (x+{\mathrm {e}}^5+1\right )}^3}}{x+\ln \left (x+{\mathrm {e}}^5\right )}+\frac {x+{\mathrm {e}}^5}{2\,x^3+\left (6\,{\mathrm {e}}^5+6\right )\,x^2+\left (12\,{\mathrm {e}}^5+6\,{\mathrm {e}}^{10}+6\right )\,x+6\,{\mathrm {e}}^5+6\,{\mathrm {e}}^{10}+2\,{\mathrm {e}}^{15}+2} \]
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