Integrand size = 151, antiderivative size = 24 \[ \int \frac {-84 x-10 x^2+2 e^6 x^2+6 x^3+6 x^4+2 x^5+e^4 \left (16 x+10 x^2+6 x^3\right )+e^2 \left (14+46 x+30 x^2+16 x^3+6 x^4\right )+\left (-26 x+2 e^4 x-2 x^2+e^2 \left (2+6 x+2 x^2\right )\right ) \log \left (25 e^x x\right )-2 x \log ^2\left (25 e^x x\right )}{e^6 x+3 e^4 x^2+3 e^2 x^3+x^4} \, dx=\left (x+\frac {7+2 x+\log \left (25 e^x x\right )}{e^2+x}\right )^2 \]
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\[ \int \frac {-84 x-10 x^2+2 e^6 x^2+6 x^3+6 x^4+2 x^5+e^4 \left (16 x+10 x^2+6 x^3\right )+e^2 \left (14+46 x+30 x^2+16 x^3+6 x^4\right )+\left (-26 x+2 e^4 x-2 x^2+e^2 \left (2+6 x+2 x^2\right )\right ) \log \left (25 e^x x\right )-2 x \log ^2\left (25 e^x x\right )}{e^6 x+3 e^4 x^2+3 e^2 x^3+x^4} \, dx=\int \frac {-84 x-10 x^2+2 e^6 x^2+6 x^3+6 x^4+2 x^5+e^4 \left (16 x+10 x^2+6 x^3\right )+e^2 \left (14+46 x+30 x^2+16 x^3+6 x^4\right )+\left (-26 x+2 e^4 x-2 x^2+e^2 \left (2+6 x+2 x^2\right )\right ) \log \left (25 e^x x\right )-2 x \log ^2\left (25 e^x x\right )}{e^6 x+3 e^4 x^2+3 e^2 x^3+x^4} \, dx \]
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Rubi steps \begin{align*} \text {integral}& = \int \frac {-84 x+\left (-10+2 e^6\right ) x^2+6 x^3+6 x^4+2 x^5+e^4 \left (16 x+10 x^2+6 x^3\right )+e^2 \left (14+46 x+30 x^2+16 x^3+6 x^4\right )+\left (-26 x+2 e^4 x-2 x^2+e^2 \left (2+6 x+2 x^2\right )\right ) \log \left (25 e^x x\right )-2 x \log ^2\left (25 e^x x\right )}{e^6 x+3 e^4 x^2+3 e^2 x^3+x^4} \, dx \\ & = \int \frac {-84 x+\left (-10+2 e^6\right ) x^2+6 x^3+6 x^4+2 x^5+e^4 \left (16 x+10 x^2+6 x^3\right )+e^2 \left (14+46 x+30 x^2+16 x^3+6 x^4\right )+\left (-26 x+2 e^4 x-2 x^2+e^2 \left (2+6 x+2 x^2\right )\right ) \log \left (25 e^x x\right )-2 x \log ^2\left (25 e^x x\right )}{x \left (e^6+3 e^4 x+3 e^2 x^2+x^3\right )} \, dx \\ & = \int \left (-\frac {84}{\left (e^2+x\right )^3}+\frac {2 \left (-5+e^6\right ) x}{\left (e^2+x\right )^3}+\frac {6 x^2}{\left (e^2+x\right )^3}+\frac {6 x^3}{\left (e^2+x\right )^3}+\frac {2 x^4}{\left (e^2+x\right )^3}+\frac {2 e^4 \left (8+5 x+3 x^2\right )}{\left (e^2+x\right )^3}+\frac {2 e^2 \left (7+23 x+15 x^2+8 x^3+3 x^4\right )}{x \left (e^2+x\right )^3}+\frac {2 \left (e^2-\left (13-3 e^2-e^4\right ) x-\left (1-e^2\right ) x^2\right ) \log \left (25 e^x x\right )}{x \left (e^2+x\right )^3}-\frac {2 \log ^2\left (25 e^x x\right )}{\left (e^2+x\right )^3}\right ) \, dx \\ & = \frac {42}{\left (e^2+x\right )^2}+2 \int \frac {x^4}{\left (e^2+x\right )^3} \, dx+2 \int \frac {\left (e^2-\left (13-3 e^2-e^4\right ) x-\left (1-e^2\right ) x^2\right ) \log \left (25 e^x x\right )}{x \left (e^2+x\right )^3} \, dx-2 \int \frac {\log ^2\left (25 e^x x\right )}{\left (e^2+x\right )^3} \, dx+6 \int \frac {x^2}{\left (e^2+x\right )^3} \, dx+6 \int \frac {x^3}{\left (e^2+x\right )^3} \, dx+\left (2 e^2\right ) \int \frac {7+23 x+15 x^2+8 x^3+3 x^4}{x \left (e^2+x\right )^3} \, dx+\left (2 e^4\right ) \int \frac {8+5 x+3 x^2}{\left (e^2+x\right )^3} \, dx-\left (2 \left (5-e^6\right )\right ) \int \frac {x}{\left (e^2+x\right )^3} \, dx \\ & = \frac {42}{\left (e^2+x\right )^2}-\frac {\left (5-e^6\right ) x^2}{e^2 \left (e^2+x\right )^2}+2 \int \left (-3 e^2+x+\frac {e^8}{\left (e^2+x\right )^3}-\frac {4 e^6}{\left (e^2+x\right )^2}+\frac {6 e^4}{e^2+x}\right ) \, dx-2 \int \frac {\log ^2\left (25 e^x x\right )}{\left (e^2+x\right )^3} \, dx+2 \int \left (\frac {\log \left (25 e^x x\right )}{e^4 x}+\frac {2 \left (-7+2 e^2\right ) \log \left (25 e^x x\right )}{\left (e^2+x\right )^3}+\frac {\left (-1-e^2+e^4\right ) \log \left (25 e^x x\right )}{e^2 \left (e^2+x\right )^2}-\frac {\log \left (25 e^x x\right )}{e^4 \left (e^2+x\right )}\right ) \, dx+6 \int \left (\frac {e^4}{\left (e^2+x\right )^3}-\frac {2 e^2}{\left (e^2+x\right )^2}+\frac {1}{e^2+x}\right ) \, dx+6 \int \left (1-\frac {e^6}{\left (e^2+x\right )^3}+\frac {3 e^4}{\left (e^2+x\right )^2}-\frac {3 e^2}{e^2+x}\right ) \, dx+\left (2 e^2\right ) \int \left (3+\frac {7}{e^6 x}+\frac {-7+23 e^2-15 e^4+8 e^6-3 e^8}{e^2 \left (e^2+x\right )^3}+\frac {-7+15 e^4-16 e^6+9 e^8}{e^4 \left (e^2+x\right )^2}+\frac {-7+8 e^6-9 e^8}{e^6 \left (e^2+x\right )}\right ) \, dx+\left (2 e^4\right ) \int \left (\frac {8-5 e^2+3 e^4}{\left (e^2+x\right )^3}+\frac {5-6 e^2}{\left (e^2+x\right )^2}+\frac {3}{e^2+x}\right ) \, dx \\ & = 6 x+x^2+\frac {42}{\left (e^2+x\right )^2}-\frac {3 e^4}{\left (e^2+x\right )^2}+\frac {3 e^6}{\left (e^2+x\right )^2}-\frac {e^8}{\left (e^2+x\right )^2}-\frac {e^4 \left (8-5 e^2+3 e^4\right )}{\left (e^2+x\right )^2}+\frac {7-23 e^2+15 e^4-8 e^6+3 e^8}{\left (e^2+x\right )^2}-\frac {\left (5-e^6\right ) x^2}{e^2 \left (e^2+x\right )^2}+\frac {12 e^2}{e^2+x}-\frac {18 e^4}{e^2+x}+\frac {8 e^6}{e^2+x}-\frac {2 e^4 \left (5-6 e^2\right )}{e^2+x}+\frac {2 \left (7-15 e^4+16 e^6-9 e^8\right )}{e^2 \left (e^2+x\right )}+\frac {14 \log (x)}{e^4}+6 \log \left (e^2+x\right )-18 e^2 \log \left (e^2+x\right )+18 e^4 \log \left (e^2+x\right )-\frac {2 \left (7-8 e^6+9 e^8\right ) \log \left (e^2+x\right )}{e^4}-2 \int \frac {\log ^2\left (25 e^x x\right )}{\left (e^2+x\right )^3} \, dx+\frac {2 \int \frac {\log \left (25 e^x x\right )}{x} \, dx}{e^4}-\frac {2 \int \frac {\log \left (25 e^x x\right )}{e^2+x} \, dx}{e^4}-\left (4 \left (7-2 e^2\right )\right ) \int \frac {\log \left (25 e^x x\right )}{\left (e^2+x\right )^3} \, dx+\frac {\left (2 \left (-1-e^2+e^4\right )\right ) \int \frac {\log \left (25 e^x x\right )}{\left (e^2+x\right )^2} \, dx}{e^2} \\ & = 6 x+x^2+\frac {42}{\left (e^2+x\right )^2}-\frac {3 e^4}{\left (e^2+x\right )^2}+\frac {3 e^6}{\left (e^2+x\right )^2}-\frac {e^8}{\left (e^2+x\right )^2}-\frac {e^4 \left (8-5 e^2+3 e^4\right )}{\left (e^2+x\right )^2}+\frac {7-23 e^2+15 e^4-8 e^6+3 e^8}{\left (e^2+x\right )^2}-\frac {\left (5-e^6\right ) x^2}{e^2 \left (e^2+x\right )^2}+\frac {12 e^2}{e^2+x}-\frac {18 e^4}{e^2+x}+\frac {8 e^6}{e^2+x}-\frac {2 e^4 \left (5-6 e^2\right )}{e^2+x}+\frac {2 \left (7-15 e^4+16 e^6-9 e^8\right )}{e^2 \left (e^2+x\right )}+\frac {14 \log (x)}{e^4}+\frac {2 \left (7-2 e^2\right ) \log \left (25 e^x x\right )}{\left (e^2+x\right )^2}+\frac {2 \left (1+e^2-e^4\right ) \log \left (25 e^x x\right )}{e^2 \left (e^2+x\right )}+6 \log \left (e^2+x\right )-18 e^2 \log \left (e^2+x\right )+18 e^4 \log \left (e^2+x\right )-\frac {2 \left (7-8 e^6+9 e^8\right ) \log \left (e^2+x\right )}{e^4}-2 \int \frac {\log ^2\left (25 e^x x\right )}{\left (e^2+x\right )^3} \, dx+\frac {2 \int \frac {\log \left (25 e^x x\right )}{x} \, dx}{e^4}-\frac {2 \int \frac {\log \left (25 e^x x\right )}{e^2+x} \, dx}{e^4}-\left (2 \left (7-2 e^2\right )\right ) \int \frac {1+x}{x \left (e^2+x\right )^2} \, dx+\frac {\left (2 \left (-1-e^2+e^4\right )\right ) \int \frac {1+x}{x \left (e^2+x\right )} \, dx}{e^2} \\ & = 6 x+x^2+\frac {42}{\left (e^2+x\right )^2}-\frac {3 e^4}{\left (e^2+x\right )^2}+\frac {3 e^6}{\left (e^2+x\right )^2}-\frac {e^8}{\left (e^2+x\right )^2}-\frac {e^4 \left (8-5 e^2+3 e^4\right )}{\left (e^2+x\right )^2}+\frac {7-23 e^2+15 e^4-8 e^6+3 e^8}{\left (e^2+x\right )^2}-\frac {\left (5-e^6\right ) x^2}{e^2 \left (e^2+x\right )^2}+\frac {12 e^2}{e^2+x}-\frac {18 e^4}{e^2+x}+\frac {8 e^6}{e^2+x}-\frac {2 e^4 \left (5-6 e^2\right )}{e^2+x}+\frac {2 \left (7-15 e^4+16 e^6-9 e^8\right )}{e^2 \left (e^2+x\right )}+\frac {14 \log (x)}{e^4}+\frac {2 \left (7-2 e^2\right ) \log \left (25 e^x x\right )}{\left (e^2+x\right )^2}+\frac {2 \left (1+e^2-e^4\right ) \log \left (25 e^x x\right )}{e^2 \left (e^2+x\right )}+6 \log \left (e^2+x\right )-18 e^2 \log \left (e^2+x\right )+18 e^4 \log \left (e^2+x\right )-\frac {2 \left (7-8 e^6+9 e^8\right ) \log \left (e^2+x\right )}{e^4}-2 \int \frac {\log ^2\left (25 e^x x\right )}{\left (e^2+x\right )^3} \, dx+\frac {2 \int \frac {\log \left (25 e^x x\right )}{x} \, dx}{e^4}-\frac {2 \int \frac {\log \left (25 e^x x\right )}{e^2+x} \, dx}{e^4}-\left (2 \left (7-2 e^2\right )\right ) \int \left (\frac {1}{e^4 x}+\frac {-1+e^2}{e^2 \left (e^2+x\right )^2}-\frac {1}{e^4 \left (e^2+x\right )}\right ) \, dx+\frac {\left (2 \left (-1-e^2+e^4\right )\right ) \int \left (\frac {1}{e^2 x}+\frac {-1+e^2}{e^2 \left (e^2+x\right )}\right ) \, dx}{e^2} \\ & = 6 x+x^2+\frac {42}{\left (e^2+x\right )^2}-\frac {3 e^4}{\left (e^2+x\right )^2}+\frac {3 e^6}{\left (e^2+x\right )^2}-\frac {e^8}{\left (e^2+x\right )^2}-\frac {e^4 \left (8-5 e^2+3 e^4\right )}{\left (e^2+x\right )^2}+\frac {7-23 e^2+15 e^4-8 e^6+3 e^8}{\left (e^2+x\right )^2}-\frac {\left (5-e^6\right ) x^2}{e^2 \left (e^2+x\right )^2}+\frac {12 e^2}{e^2+x}-\frac {18 e^4}{e^2+x}+\frac {8 e^6}{e^2+x}-\frac {2 e^4 \left (5-6 e^2\right )}{e^2+x}+\frac {2 \left (1-\frac {1}{e^2}\right ) \left (7-2 e^2\right )}{e^2+x}+\frac {2 \left (7-15 e^4+16 e^6-9 e^8\right )}{e^2 \left (e^2+x\right )}+\frac {14 \log (x)}{e^4}-\frac {2 \left (7-2 e^2\right ) \log (x)}{e^4}-\frac {2 \left (1+e^2-e^4\right ) \log (x)}{e^4}+\frac {2 \left (7-2 e^2\right ) \log \left (25 e^x x\right )}{\left (e^2+x\right )^2}+\frac {2 \left (1+e^2-e^4\right ) \log \left (25 e^x x\right )}{e^2 \left (e^2+x\right )}+6 \log \left (e^2+x\right )-18 e^2 \log \left (e^2+x\right )+18 e^4 \log \left (e^2+x\right )+\frac {2 \left (7-2 e^2\right ) \log \left (e^2+x\right )}{e^4}+\frac {2 \left (1-2 e^4+e^6\right ) \log \left (e^2+x\right )}{e^4}-\frac {2 \left (7-8 e^6+9 e^8\right ) \log \left (e^2+x\right )}{e^4}-2 \int \frac {\log ^2\left (25 e^x x\right )}{\left (e^2+x\right )^3} \, dx+\frac {2 \int \frac {\log \left (25 e^x x\right )}{x} \, dx}{e^4}-\frac {2 \int \frac {\log \left (25 e^x x\right )}{e^2+x} \, dx}{e^4} \\ \end{align*}
Leaf count is larger than twice the leaf count of optimal. \(113\) vs. \(2(24)=48\).
Time = 0.09 (sec) , antiderivative size = 113, normalized size of antiderivative = 4.71 \[ \int \frac {-84 x-10 x^2+2 e^6 x^2+6 x^3+6 x^4+2 x^5+e^4 \left (16 x+10 x^2+6 x^3\right )+e^2 \left (14+46 x+30 x^2+16 x^3+6 x^4\right )+\left (-26 x+2 e^4 x-2 x^2+e^2 \left (2+6 x+2 x^2\right )\right ) \log \left (25 e^x x\right )-2 x \log ^2\left (25 e^x x\right )}{e^6 x+3 e^4 x^2+3 e^2 x^3+x^4} \, dx=\frac {49-23 e^4+6 e^6+28 x-32 e^2 x+14 e^4 x-5 x^2+14 e^2 x^2+e^4 x^2+6 x^3+2 e^2 x^3+x^4+2 \left (e^2+x\right )^2 \log (x)-2 \left (-7+e^4-2 x+e^2 x\right ) \log \left (25 e^x x\right )+\log ^2\left (25 e^x x\right )}{\left (e^2+x\right )^2} \]
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Leaf count of result is larger than twice the leaf count of optimal. \(123\) vs. \(2(22)=44\).
Time = 0.82 (sec) , antiderivative size = 124, normalized size of antiderivative = 5.17
method | result | size |
parallelrisch | \(-\frac {-49-2 \ln \left (25 \,{\mathrm e}^{x} x \right ) {\mathrm e}^{2} x -2 \ln \left (25 \,{\mathrm e}^{x} x \right ) x^{2}-4 \ln \left (25 \,{\mathrm e}^{x} x \right ) x -28 x -2 x^{3} {\mathrm e}^{2}-14 \ln \left (25 \,{\mathrm e}^{x} x \right )+2 x \,{\mathrm e}^{6}+4 \,{\mathrm e}^{6}+{\mathrm e}^{8}+8 x \,{\mathrm e}^{4}+18 \,{\mathrm e}^{4}-x^{4}-4 x^{3}+22 \,{\mathrm e}^{2} x -\ln \left (25 \,{\mathrm e}^{x} x \right )^{2}}{{\mathrm e}^{4}+2 \,{\mathrm e}^{2} x +x^{2}}\) | \(124\) |
risch | \(\text {Expression too large to display}\) | \(940\) |
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Leaf count of result is larger than twice the leaf count of optimal. 86 vs. \(2 (22) = 44\).
Time = 0.26 (sec) , antiderivative size = 86, normalized size of antiderivative = 3.58 \[ \int \frac {-84 x-10 x^2+2 e^6 x^2+6 x^3+6 x^4+2 x^5+e^4 \left (16 x+10 x^2+6 x^3\right )+e^2 \left (14+46 x+30 x^2+16 x^3+6 x^4\right )+\left (-26 x+2 e^4 x-2 x^2+e^2 \left (2+6 x+2 x^2\right )\right ) \log \left (25 e^x x\right )-2 x \log ^2\left (25 e^x x\right )}{e^6 x+3 e^4 x^2+3 e^2 x^3+x^4} \, dx=\frac {x^{4} + 4 \, x^{3} + {\left (x^{2} + 8 \, x - 18\right )} e^{4} + 2 \, {\left (x^{3} + 4 \, x^{2} - 11 \, x\right )} e^{2} + 2 \, {\left (x^{2} + x e^{2} + 2 \, x + 7\right )} \log \left (25 \, x e^{x}\right ) + \log \left (25 \, x e^{x}\right )^{2} + 28 \, x + 4 \, e^{6} + 49}{x^{2} + 2 \, x e^{2} + e^{4}} \]
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Leaf count of result is larger than twice the leaf count of optimal. 112 vs. \(2 (20) = 40\).
Time = 0.77 (sec) , antiderivative size = 112, normalized size of antiderivative = 4.67 \[ \int \frac {-84 x-10 x^2+2 e^6 x^2+6 x^3+6 x^4+2 x^5+e^4 \left (16 x+10 x^2+6 x^3\right )+e^2 \left (14+46 x+30 x^2+16 x^3+6 x^4\right )+\left (-26 x+2 e^4 x-2 x^2+e^2 \left (2+6 x+2 x^2\right )\right ) \log \left (25 e^x x\right )-2 x \log ^2\left (25 e^x x\right )}{e^6 x+3 e^4 x^2+3 e^2 x^3+x^4} \, dx=x^{2} + 6 x + 2 \log {\left (x \right )} + \frac {x \left (- 22 e^{2} + 28 + 4 e^{4}\right ) - 18 e^{4} + 49 + 4 e^{6}}{x^{2} + 2 x e^{2} + e^{4}} + \frac {\left (- 2 x e^{2} + 4 x - 2 e^{4} + 14\right ) \log {\left (25 x e^{x} \right )}}{x^{2} + 2 x e^{2} + e^{4}} + \frac {\log {\left (25 x e^{x} \right )}^{2}}{x^{2} + 2 x e^{2} + e^{4}} \]
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Leaf count of result is larger than twice the leaf count of optimal. 548 vs. \(2 (22) = 44\).
Time = 0.34 (sec) , antiderivative size = 548, normalized size of antiderivative = 22.83 \[ \int \frac {-84 x-10 x^2+2 e^6 x^2+6 x^3+6 x^4+2 x^5+e^4 \left (16 x+10 x^2+6 x^3\right )+e^2 \left (14+46 x+30 x^2+16 x^3+6 x^4\right )+\left (-26 x+2 e^4 x-2 x^2+e^2 \left (2+6 x+2 x^2\right )\right ) \log \left (25 e^x x\right )-2 x \log ^2\left (25 e^x x\right )}{e^6 x+3 e^4 x^2+3 e^2 x^3+x^4} \, dx=2 \, {\left (e^{6} - 3 \, e^{4} + 7\right )} e^{\left (-4\right )} \log \left (x + e^{2}\right ) + x^{2} + 3 \, {\left (\frac {4 \, x e^{2} + 3 \, e^{4}}{x^{2} + 2 \, x e^{2} + e^{4}} + 2 \, \log \left (x + e^{2}\right )\right )} e^{4} - 3 \, {\left (6 \, e^{2} \log \left (x + e^{2}\right ) - 2 \, x + \frac {6 \, x e^{4} + 5 \, e^{6}}{x^{2} + 2 \, x e^{2} + e^{4}}\right )} e^{2} - 7 \, {\left (2 \, e^{\left (-6\right )} \log \left (x + e^{2}\right ) - 2 \, e^{\left (-6\right )} \log \left (x\right ) - \frac {2 \, x + 3 \, e^{2}}{x^{2} e^{4} + 2 \, x e^{6} + e^{8}}\right )} e^{2} - 6 \, x e^{2} + 8 \, {\left (\frac {4 \, x e^{2} + 3 \, e^{4}}{x^{2} + 2 \, x e^{2} + e^{4}} + 2 \, \log \left (x + e^{2}\right )\right )} e^{2} + 12 \, e^{4} \log \left (x + e^{2}\right ) - 18 \, e^{2} \log \left (x + e^{2}\right ) + 6 \, x - \frac {{\left (2 \, x + e^{2}\right )} e^{6}}{x^{2} + 2 \, x e^{2} + e^{4}} - \frac {5 \, {\left (2 \, x + e^{2}\right )} e^{4}}{x^{2} + 2 \, x e^{2} + e^{4}} - \frac {15 \, {\left (2 \, x + e^{2}\right )} e^{2}}{x^{2} + 2 \, x e^{2} + e^{4}} + \frac {e^{4} \log \left (x\right )^{2} - 2 \, {\left ({\left (2 \, \log \left (5\right ) + 7\right )} e^{6} - 2 \, {\left (3 \, \log \left (5\right ) + 8\right )} e^{4} - e^{8} + 7 \, e^{2}\right )} x - {\left (4 \, \log \left (5\right ) + 9\right )} e^{8} + 2 \, {\left (2 \, \log \left (5\right )^{2} + 14 \, \log \left (5\right ) - 7\right )} e^{4} + 2 \, {\left (x^{2} {\left (e^{4} - 7\right )} + x {\left (e^{6} + 3 \, e^{4} - 14 \, e^{2}\right )} + 2 \, e^{4} \log \left (5\right )\right )} \log \left (x\right ) + 2 \, e^{10} + 18 \, e^{6}}{x^{2} e^{4} + 2 \, x e^{6} + e^{8}} + \frac {8 \, x e^{6} + 7 \, e^{8}}{x^{2} + 2 \, x e^{2} + e^{4}} - \frac {3 \, {\left (6 \, x e^{4} + 5 \, e^{6}\right )}}{x^{2} + 2 \, x e^{2} + e^{4}} + \frac {3 \, {\left (4 \, x e^{2} + 3 \, e^{4}\right )}}{x^{2} + 2 \, x e^{2} + e^{4}} + \frac {5 \, {\left (2 \, x + e^{2}\right )}}{x^{2} + 2 \, x e^{2} + e^{4}} - \frac {8 \, e^{4}}{x^{2} + 2 \, x e^{2} + e^{4}} - \frac {23 \, e^{2}}{x^{2} + 2 \, x e^{2} + e^{4}} + \frac {42}{x^{2} + 2 \, x e^{2} + e^{4}} + 6 \, \log \left (x + e^{2}\right ) \]
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Leaf count of result is larger than twice the leaf count of optimal. 126 vs. \(2 (22) = 44\).
Time = 0.27 (sec) , antiderivative size = 126, normalized size of antiderivative = 5.25 \[ \int \frac {-84 x-10 x^2+2 e^6 x^2+6 x^3+6 x^4+2 x^5+e^4 \left (16 x+10 x^2+6 x^3\right )+e^2 \left (14+46 x+30 x^2+16 x^3+6 x^4\right )+\left (-26 x+2 e^4 x-2 x^2+e^2 \left (2+6 x+2 x^2\right )\right ) \log \left (25 e^x x\right )-2 x \log ^2\left (25 e^x x\right )}{e^6 x+3 e^4 x^2+3 e^2 x^3+x^4} \, dx=\frac {x^{4} + 2 \, x^{3} e^{2} + 6 \, x^{3} + x^{2} e^{4} + 12 \, x^{2} e^{2} - 4 \, x e^{2} \log \left (5\right ) + 2 \, x^{2} \log \left (x\right ) + 2 \, x e^{2} \log \left (x\right ) + 12 \, x e^{4} - 32 \, x e^{2} + 12 \, x \log \left (5\right ) - 4 \, e^{4} \log \left (5\right ) + 4 \, \log \left (5\right )^{2} + 6 \, x \log \left (x\right ) + 4 \, \log \left (5\right ) \log \left (x\right ) + \log \left (x\right )^{2} + 42 \, x + 6 \, e^{6} - 23 \, e^{4} + 28 \, \log \left (5\right ) + 14 \, \log \left (x\right ) + 49}{x^{2} + 2 \, x e^{2} + e^{4}} \]
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Time = 13.54 (sec) , antiderivative size = 113, normalized size of antiderivative = 4.71 \[ \int \frac {-84 x-10 x^2+2 e^6 x^2+6 x^3+6 x^4+2 x^5+e^4 \left (16 x+10 x^2+6 x^3\right )+e^2 \left (14+46 x+30 x^2+16 x^3+6 x^4\right )+\left (-26 x+2 e^4 x-2 x^2+e^2 \left (2+6 x+2 x^2\right )\right ) \log \left (25 e^x x\right )-2 x \log ^2\left (25 e^x x\right )}{e^6 x+3 e^4 x^2+3 e^2 x^3+x^4} \, dx=6\,x+2\,\ln \left (x\right )+\frac {{\ln \left (25\,x\,{\mathrm {e}}^x\right )}^2}{x^2+2\,{\mathrm {e}}^2\,x+{\mathrm {e}}^4}+\frac {4\,{\mathrm {e}}^6-18\,{\mathrm {e}}^4+x\,\left (4\,{\mathrm {e}}^4-22\,{\mathrm {e}}^2+28\right )+49}{x^2+2\,{\mathrm {e}}^2\,x+{\mathrm {e}}^4}+x^2-\frac {\ln \left (25\,x\,{\mathrm {e}}^x\right )\,\left (2\,{\mathrm {e}}^2+{\mathrm {e}}^4+{\mathrm {e}}^2\,\left ({\mathrm {e}}^2-2\right )+x\,\left (2\,{\mathrm {e}}^2-4\right )-14\right )}{x^2+2\,{\mathrm {e}}^2\,x+{\mathrm {e}}^4} \]
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