Integrand size = 145, antiderivative size = 27 \[ \int \frac {-15-10 x+x^2+e^{e^{-10+6 e^{10}}-2 e^{-5+3 e^{10}} x+x^2} \left (3+2 x-6 x^2-2 x^3+e^{-5+3 e^{10}} \left (6 x+2 x^2\right )\right )}{25+e^{2 e^{-10+6 e^{10}}-4 e^{-5+3 e^{10}} x+2 x^2}-10 x+x^2+e^{e^{-10+6 e^{10}}-2 e^{-5+3 e^{10}} x+x^2} (-10+2 x)} \, dx=\frac {x (3+x)}{-5+e^{\left (-e^{-5+3 e^{10}}+x\right )^2}+x} \]
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\[ \int \frac {-15-10 x+x^2+e^{e^{-10+6 e^{10}}-2 e^{-5+3 e^{10}} x+x^2} \left (3+2 x-6 x^2-2 x^3+e^{-5+3 e^{10}} \left (6 x+2 x^2\right )\right )}{25+e^{2 e^{-10+6 e^{10}}-4 e^{-5+3 e^{10}} x+2 x^2}-10 x+x^2+e^{e^{-10+6 e^{10}}-2 e^{-5+3 e^{10}} x+x^2} (-10+2 x)} \, dx=\int \frac {-15-10 x+x^2+e^{e^{-10+6 e^{10}}-2 e^{-5+3 e^{10}} x+x^2} \left (3+2 x-6 x^2-2 x^3+e^{-5+3 e^{10}} \left (6 x+2 x^2\right )\right )}{25+\exp \left (2 e^{-10+6 e^{10}}-4 e^{-5+3 e^{10}} x+2 x^2\right )-10 x+x^2+e^{e^{-10+6 e^{10}}-2 e^{-5+3 e^{10}} x+x^2} (-10+2 x)} \, dx \]
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Rubi steps \begin{align*} \text {integral}& = \int \frac {e^{4 e^{-5+3 e^{10}} x} \left (-15-10 x+x^2+e^{e^{-10+6 e^{10}}-2 e^{-5+3 e^{10}} x+x^2} \left (3+2 x-6 x^2-2 x^3+e^{-5+3 e^{10}} \left (6 x+2 x^2\right )\right )\right )}{\left (5 e^{2 e^{-5+3 e^{10}} x}-e^{e^{-10+6 e^{10}}+x^2}-e^{2 e^{-5+3 e^{10}} x} x\right )^2} \, dx \\ & = \int \left (\frac {e^{-5+4 e^{-5+3 e^{10}} x} x (3+x) \left (-e^5+10 e^{3 e^{10}}-2 \left (5 e^5+e^{3 e^{10}}\right ) x+2 e^5 x^2\right )}{\left (5 e^{2 e^{-5+3 e^{10}} x}-e^{e^{-10+6 e^{10}}+x^2}-e^{2 e^{-5+3 e^{10}} x} x\right )^2}+\frac {e^{-5+2 e^{-5+3 e^{10}} x} \left (-3 e^5-2 \left (e^5+3 e^{3 e^{10}}\right ) x+2 \left (3 e^5-e^{3 e^{10}}\right ) x^2+2 e^5 x^3\right )}{5 e^{2 e^{-5+3 e^{10}} x}-e^{e^{-10+6 e^{10}}+x^2}-e^{2 e^{-5+3 e^{10}} x} x}\right ) \, dx \\ & = \int \frac {e^{-5+4 e^{-5+3 e^{10}} x} x (3+x) \left (-e^5+10 e^{3 e^{10}}-2 \left (5 e^5+e^{3 e^{10}}\right ) x+2 e^5 x^2\right )}{\left (5 e^{2 e^{-5+3 e^{10}} x}-e^{e^{-10+6 e^{10}}+x^2}-e^{2 e^{-5+3 e^{10}} x} x\right )^2} \, dx+\int \frac {e^{-5+2 e^{-5+3 e^{10}} x} \left (-3 e^5-2 \left (e^5+3 e^{3 e^{10}}\right ) x+2 \left (3 e^5-e^{3 e^{10}}\right ) x^2+2 e^5 x^3\right )}{5 e^{2 e^{-5+3 e^{10}} x}-e^{e^{-10+6 e^{10}}+x^2}-e^{2 e^{-5+3 e^{10}} x} x} \, dx \\ & = \int \frac {e^{-5+4 e^{-5+3 e^{10}} x} x (3+x) \left (-e^5+10 e^{3 e^{10}}-2 \left (5 e^5+e^{3 e^{10}}\right ) x+2 e^5 x^2\right )}{\left (e^{e^{-10+6 e^{10}}+x^2}+e^{2 e^{-5+3 e^{10}} x} (-5+x)\right )^2} \, dx+\int \frac {-3 e^5-2 \left (e^5+3 e^{3 e^{10}}\right ) x+2 \left (3 e^5-e^{3 e^{10}}\right ) x^2+2 e^5 x^3}{e^5 \left (5-e^{\frac {\left (e^{3 e^{10}}-e^5 x\right )^2}{e^{10}}}-x\right )} \, dx \\ & = \frac {\int \frac {-3 e^5-2 \left (e^5+3 e^{3 e^{10}}\right ) x+2 \left (3 e^5-e^{3 e^{10}}\right ) x^2+2 e^5 x^3}{5-e^{\frac {\left (e^{3 e^{10}}-e^5 x\right )^2}{e^{10}}}-x} \, dx}{e^5}+\int \left (\frac {3 e^{-5+4 e^{-5+3 e^{10}} x} \left (-e^5+10 e^{3 e^{10}}\right ) x}{\left (-5 e^{2 e^{-5+3 e^{10}} x}+e^{e^{-10+6 e^{10}}+x^2}+e^{2 e^{-5+3 e^{10}} x} x\right )^2}+\frac {e^{-5+4 e^{-5+3 e^{10}} x} \left (-31 e^5+4 e^{3 e^{10}}\right ) x^2}{\left (-5 e^{2 e^{-5+3 e^{10}} x}+e^{e^{-10+6 e^{10}}+x^2}+e^{2 e^{-5+3 e^{10}} x} x\right )^2}-\frac {2 e^{-5+4 e^{-5+3 e^{10}} x} \left (2 e^5+e^{3 e^{10}}\right ) x^3}{\left (-5 e^{2 e^{-5+3 e^{10}} x}+e^{e^{-10+6 e^{10}}+x^2}+e^{2 e^{-5+3 e^{10}} x} x\right )^2}+\frac {2 e^{4 e^{-5+3 e^{10}} x} x^4}{\left (-5 e^{2 e^{-5+3 e^{10}} x}+e^{e^{-10+6 e^{10}}+x^2}+e^{2 e^{-5+3 e^{10}} x} x\right )^2}\right ) \, dx \\ & = 2 \int \frac {e^{4 e^{-5+3 e^{10}} x} x^4}{\left (-5 e^{2 e^{-5+3 e^{10}} x}+e^{e^{-10+6 e^{10}}+x^2}+e^{2 e^{-5+3 e^{10}} x} x\right )^2} \, dx+\frac {\int \left (\frac {3 e^5}{-5+e^{\frac {\left (e^{3 e^{10}}-e^5 x\right )^2}{e^{10}}}+x}+\frac {2 \left (e^5+3 e^{3 e^{10}}\right ) x}{-5+e^{\frac {\left (e^{3 e^{10}}-e^5 x\right )^2}{e^{10}}}+x}+\frac {2 \left (-3 e^5+e^{3 e^{10}}\right ) x^2}{-5+e^{\frac {\left (e^{3 e^{10}}-e^5 x\right )^2}{e^{10}}}+x}-\frac {2 e^5 x^3}{-5+e^{\frac {\left (e^{3 e^{10}}-e^5 x\right )^2}{e^{10}}}+x}\right ) \, dx}{e^5}-\left (3 \left (e^5-10 e^{3 e^{10}}\right )\right ) \int \frac {e^{-5+4 e^{-5+3 e^{10}} x} x}{\left (-5 e^{2 e^{-5+3 e^{10}} x}+e^{e^{-10+6 e^{10}}+x^2}+e^{2 e^{-5+3 e^{10}} x} x\right )^2} \, dx-\left (2 \left (2 e^5+e^{3 e^{10}}\right )\right ) \int \frac {e^{-5+4 e^{-5+3 e^{10}} x} x^3}{\left (-5 e^{2 e^{-5+3 e^{10}} x}+e^{e^{-10+6 e^{10}}+x^2}+e^{2 e^{-5+3 e^{10}} x} x\right )^2} \, dx+\left (-31 e^5+4 e^{3 e^{10}}\right ) \int \frac {e^{-5+4 e^{-5+3 e^{10}} x} x^2}{\left (-5 e^{2 e^{-5+3 e^{10}} x}+e^{e^{-10+6 e^{10}}+x^2}+e^{2 e^{-5+3 e^{10}} x} x\right )^2} \, dx \\ & = 2 \int \frac {e^{4 e^{-5+3 e^{10}} x} x^4}{\left (e^{e^{-10+6 e^{10}}+x^2}+e^{2 e^{-5+3 e^{10}} x} (-5+x)\right )^2} \, dx-2 \int \frac {x^3}{-5+e^{\frac {\left (e^{3 e^{10}}-e^5 x\right )^2}{e^{10}}}+x} \, dx+3 \int \frac {1}{-5+e^{\frac {\left (e^{3 e^{10}}-e^5 x\right )^2}{e^{10}}}+x} \, dx-\left (3 \left (e^5-10 e^{3 e^{10}}\right )\right ) \int \frac {e^{-5+4 e^{-5+3 e^{10}} x} x}{\left (e^{e^{-10+6 e^{10}}+x^2}+e^{2 e^{-5+3 e^{10}} x} (-5+x)\right )^2} \, dx-\left (2 \left (2 e^5+e^{3 e^{10}}\right )\right ) \int \frac {e^{-5+4 e^{-5+3 e^{10}} x} x^3}{\left (e^{e^{-10+6 e^{10}}+x^2}+e^{2 e^{-5+3 e^{10}} x} (-5+x)\right )^2} \, dx+\left (-31 e^5+4 e^{3 e^{10}}\right ) \int \frac {e^{-5+4 e^{-5+3 e^{10}} x} x^2}{\left (e^{e^{-10+6 e^{10}}+x^2}+e^{2 e^{-5+3 e^{10}} x} (-5+x)\right )^2} \, dx-\left (2 \left (3-e^{-5+3 e^{10}}\right )\right ) \int \frac {x^2}{-5+e^{\frac {\left (e^{3 e^{10}}-e^5 x\right )^2}{e^{10}}}+x} \, dx+\left (2 \left (1+3 e^{-5+3 e^{10}}\right )\right ) \int \frac {x}{-5+e^{\frac {\left (e^{3 e^{10}}-e^5 x\right )^2}{e^{10}}}+x} \, dx \\ \end{align*}
Leaf count is larger than twice the leaf count of optimal. \(55\) vs. \(2(27)=54\).
Time = 7.10 (sec) , antiderivative size = 55, normalized size of antiderivative = 2.04 \[ \int \frac {-15-10 x+x^2+e^{e^{-10+6 e^{10}}-2 e^{-5+3 e^{10}} x+x^2} \left (3+2 x-6 x^2-2 x^3+e^{-5+3 e^{10}} \left (6 x+2 x^2\right )\right )}{25+e^{2 e^{-10+6 e^{10}}-4 e^{-5+3 e^{10}} x+2 x^2}-10 x+x^2+e^{e^{-10+6 e^{10}}-2 e^{-5+3 e^{10}} x+x^2} (-10+2 x)} \, dx=\frac {e^{2 e^{-5+3 e^{10}} x} x (3+x)}{e^{e^{-10+6 e^{10}}+x^2}+e^{2 e^{-5+3 e^{10}} x} (-5+x)} \]
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Time = 0.22 (sec) , antiderivative size = 33, normalized size of antiderivative = 1.22
method | result | size |
risch | \(\frac {\left (3+x \right ) x}{x +{\mathrm e}^{{\mathrm e}^{6 \,{\mathrm e}^{10}-10}-2 x \,{\mathrm e}^{3 \,{\mathrm e}^{10}-5}+x^{2}}-5}\) | \(33\) |
parallelrisch | \(\frac {x^{2}+3 x}{x +{\mathrm e}^{{\mathrm e}^{6 \,{\mathrm e}^{10}-10}-2 x \,{\mathrm e}^{3 \,{\mathrm e}^{10}-5}+x^{2}}-5}\) | \(42\) |
norman | \(\frac {x^{2}-3 \,{\mathrm e}^{{\mathrm e}^{6 \,{\mathrm e}^{10}-10}-2 x \,{\mathrm e}^{3 \,{\mathrm e}^{10}-5}+x^{2}}+15}{x +{\mathrm e}^{{\mathrm e}^{6 \,{\mathrm e}^{10}-10}-2 x \,{\mathrm e}^{3 \,{\mathrm e}^{10}-5}+x^{2}}-5}\) | \(70\) |
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Time = 0.24 (sec) , antiderivative size = 35, normalized size of antiderivative = 1.30 \[ \int \frac {-15-10 x+x^2+e^{e^{-10+6 e^{10}}-2 e^{-5+3 e^{10}} x+x^2} \left (3+2 x-6 x^2-2 x^3+e^{-5+3 e^{10}} \left (6 x+2 x^2\right )\right )}{25+e^{2 e^{-10+6 e^{10}}-4 e^{-5+3 e^{10}} x+2 x^2}-10 x+x^2+e^{e^{-10+6 e^{10}}-2 e^{-5+3 e^{10}} x+x^2} (-10+2 x)} \, dx=\frac {x^{2} + 3 \, x}{x + e^{\left (x^{2} - 2 \, x e^{\left (3 \, e^{10} - 5\right )} + e^{\left (6 \, e^{10} - 10\right )}\right )} - 5} \]
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Time = 5.59 (sec) , antiderivative size = 34, normalized size of antiderivative = 1.26 \[ \int \frac {-15-10 x+x^2+e^{e^{-10+6 e^{10}}-2 e^{-5+3 e^{10}} x+x^2} \left (3+2 x-6 x^2-2 x^3+e^{-5+3 e^{10}} \left (6 x+2 x^2\right )\right )}{25+e^{2 e^{-10+6 e^{10}}-4 e^{-5+3 e^{10}} x+2 x^2}-10 x+x^2+e^{e^{-10+6 e^{10}}-2 e^{-5+3 e^{10}} x+x^2} (-10+2 x)} \, dx=\frac {x^{2} + 3 x}{x + e^{x^{2} - 2 x e^{-5 + 3 e^{10}} + e^{-10 + 6 e^{10}}} - 5} \]
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Leaf count of result is larger than twice the leaf count of optimal. 66 vs. \(2 (24) = 48\).
Time = 0.28 (sec) , antiderivative size = 66, normalized size of antiderivative = 2.44 \[ \int \frac {-15-10 x+x^2+e^{e^{-10+6 e^{10}}-2 e^{-5+3 e^{10}} x+x^2} \left (3+2 x-6 x^2-2 x^3+e^{-5+3 e^{10}} \left (6 x+2 x^2\right )\right )}{25+e^{2 e^{-10+6 e^{10}}-4 e^{-5+3 e^{10}} x+2 x^2}-10 x+x^2+e^{e^{-10+6 e^{10}}-2 e^{-5+3 e^{10}} x+x^2} (-10+2 x)} \, dx=\frac {{\left (x^{2} - 5 \, x + 40\right )} e^{\left (2 \, x e^{\left (3 \, e^{10} - 5\right )}\right )} - 8 \, e^{\left (x^{2} + e^{\left (6 \, e^{10} - 10\right )}\right )}}{{\left (x - 5\right )} e^{\left (2 \, x e^{\left (3 \, e^{10} - 5\right )}\right )} + e^{\left (x^{2} + e^{\left (6 \, e^{10} - 10\right )}\right )}} \]
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Time = 0.45 (sec) , antiderivative size = 35, normalized size of antiderivative = 1.30 \[ \int \frac {-15-10 x+x^2+e^{e^{-10+6 e^{10}}-2 e^{-5+3 e^{10}} x+x^2} \left (3+2 x-6 x^2-2 x^3+e^{-5+3 e^{10}} \left (6 x+2 x^2\right )\right )}{25+e^{2 e^{-10+6 e^{10}}-4 e^{-5+3 e^{10}} x+2 x^2}-10 x+x^2+e^{e^{-10+6 e^{10}}-2 e^{-5+3 e^{10}} x+x^2} (-10+2 x)} \, dx=\frac {x^{2} + 3 \, x}{x + e^{\left (x^{2} - 2 \, x e^{\left (3 \, e^{10} - 5\right )} + e^{\left (6 \, e^{10} - 10\right )}\right )} - 5} \]
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Time = 81.53 (sec) , antiderivative size = 36, normalized size of antiderivative = 1.33 \[ \int \frac {-15-10 x+x^2+e^{e^{-10+6 e^{10}}-2 e^{-5+3 e^{10}} x+x^2} \left (3+2 x-6 x^2-2 x^3+e^{-5+3 e^{10}} \left (6 x+2 x^2\right )\right )}{25+e^{2 e^{-10+6 e^{10}}-4 e^{-5+3 e^{10}} x+2 x^2}-10 x+x^2+e^{e^{-10+6 e^{10}}-2 e^{-5+3 e^{10}} x+x^2} (-10+2 x)} \, dx=\frac {x^2+3\,x}{x+{\mathrm {e}}^{x^2-2\,{\mathrm {e}}^{3\,{\mathrm {e}}^{10}}\,{\mathrm {e}}^{-5}\,x+{\mathrm {e}}^{6\,{\mathrm {e}}^{10}}\,{\mathrm {e}}^{-10}}-5} \]
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