Integrand size = 200, antiderivative size = 32 \[ \int \frac {4 x^3+2 e^6 x^3+e^{\frac {625+250 x-25 x^2-10 x^3+x^4}{x^2}} \left (10000+12000 x+4500 x^2+580 x^3+64 x^4+4 x^5-4 x^6+e^{12} \left (2500+500 x+20 x^3-4 x^4\right )+e^6 \left (10000+7000 x+1000 x^2+80 x^3+24 x^4-8 x^5\right )\right )}{-2 x^4-e^6 x^4-x^5+e^{\frac {625+250 x-25 x^2-10 x^3+x^4}{x^2}} \left (4 x^3+e^{12} x^3+4 x^4+x^5+e^6 \left (4 x^3+2 x^4\right )\right )} \, dx=\log \left (\frac {5}{\left (-e^{\left (x-\frac {5 (5+x)}{x}\right )^2}+\frac {x}{2+e^6+x}\right )^2}\right ) \]
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\[ \int \frac {4 x^3+2 e^6 x^3+e^{\frac {625+250 x-25 x^2-10 x^3+x^4}{x^2}} \left (10000+12000 x+4500 x^2+580 x^3+64 x^4+4 x^5-4 x^6+e^{12} \left (2500+500 x+20 x^3-4 x^4\right )+e^6 \left (10000+7000 x+1000 x^2+80 x^3+24 x^4-8 x^5\right )\right )}{-2 x^4-e^6 x^4-x^5+e^{\frac {625+250 x-25 x^2-10 x^3+x^4}{x^2}} \left (4 x^3+e^{12} x^3+4 x^4+x^5+e^6 \left (4 x^3+2 x^4\right )\right )} \, dx=\int \frac {4 x^3+2 e^6 x^3+e^{\frac {625+250 x-25 x^2-10 x^3+x^4}{x^2}} \left (10000+12000 x+4500 x^2+580 x^3+64 x^4+4 x^5-4 x^6+e^{12} \left (2500+500 x+20 x^3-4 x^4\right )+e^6 \left (10000+7000 x+1000 x^2+80 x^3+24 x^4-8 x^5\right )\right )}{-2 x^4-e^6 x^4-x^5+e^{\frac {625+250 x-25 x^2-10 x^3+x^4}{x^2}} \left (4 x^3+e^{12} x^3+4 x^4+x^5+e^6 \left (4 x^3+2 x^4\right )\right )} \, dx \]
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Rubi steps \begin{align*} \text {integral}& = \int \frac {4 x^3+2 e^6 x^3+e^{\frac {625+250 x-25 x^2-10 x^3+x^4}{x^2}} \left (10000+12000 x+4500 x^2+580 x^3+64 x^4+4 x^5-4 x^6+e^{12} \left (2500+500 x+20 x^3-4 x^4\right )+e^6 \left (10000+7000 x+1000 x^2+80 x^3+24 x^4-8 x^5\right )\right )}{\left (-2-e^6\right ) x^4-x^5+e^{\frac {625+250 x-25 x^2-10 x^3+x^4}{x^2}} \left (4 x^3+e^{12} x^3+4 x^4+x^5+e^6 \left (4 x^3+2 x^4\right )\right )} \, dx \\ & = \int \frac {\left (4+2 e^6\right ) x^3+e^{\frac {625+250 x-25 x^2-10 x^3+x^4}{x^2}} \left (10000+12000 x+4500 x^2+580 x^3+64 x^4+4 x^5-4 x^6+e^{12} \left (2500+500 x+20 x^3-4 x^4\right )+e^6 \left (10000+7000 x+1000 x^2+80 x^3+24 x^4-8 x^5\right )\right )}{\left (-2-e^6\right ) x^4-x^5+e^{\frac {625+250 x-25 x^2-10 x^3+x^4}{x^2}} \left (4 x^3+e^{12} x^3+4 x^4+x^5+e^6 \left (4 x^3+2 x^4\right )\right )} \, dx \\ & = \int \frac {-2 \left (2+e^6\right ) x^3+4 e^{\frac {\left (-25-5 x+x^2\right )^2}{x^2}} \left (2+e^6+x\right )^2 \left (-625-125 x-5 x^3+x^4\right )}{x^3 \left (\left (2+e^6\right ) x+x^2-e^{\frac {\left (-25-5 x+x^2\right )^2}{x^2}} \left (2+e^6+x\right )^2\right )} \, dx \\ & = \int \left (-\frac {4 \left (25+x^2\right ) \left (-25-5 x+x^2\right )}{x^3}+\frac {2 \left (2500 \left (1+\frac {e^6}{2}\right )+1750 \left (1+\frac {e^6}{7}\right ) x+252 \left (1+\frac {e^6}{252}\right ) x^2+20 \left (1+\frac {e^6}{2}\right ) x^3+6 \left (1-\frac {e^6}{3}\right ) x^4-2 x^5\right )}{x^2 \left (2+e^6+x\right ) \left (2 e^{\frac {\left (-25-5 x+x^2\right )^2}{x^2}} \left (1+\frac {e^6}{2}\right )-x+e^{\frac {\left (-25-5 x+x^2\right )^2}{x^2}} x\right )}\right ) \, dx \\ & = 2 \int \frac {2500 \left (1+\frac {e^6}{2}\right )+1750 \left (1+\frac {e^6}{7}\right ) x+252 \left (1+\frac {e^6}{252}\right ) x^2+20 \left (1+\frac {e^6}{2}\right ) x^3+6 \left (1-\frac {e^6}{3}\right ) x^4-2 x^5}{x^2 \left (2+e^6+x\right ) \left (2 e^{\frac {\left (-25-5 x+x^2\right )^2}{x^2}} \left (1+\frac {e^6}{2}\right )-x+e^{\frac {\left (-25-5 x+x^2\right )^2}{x^2}} x\right )} \, dx-4 \int \frac {\left (25+x^2\right ) \left (-25-5 x+x^2\right )}{x^3} \, dx \\ & = -\frac {2 \left (25+5 x-x^2\right )^2}{x^2}+2 \int \frac {2500+1750 x+252 x^2+20 x^3+6 x^4-2 x^5+e^6 \left (1250+250 x+x^2+10 x^3-2 x^4\right )}{x^2 \left (2+e^6+x\right ) \left (e^{6+\frac {\left (-25-5 x+x^2\right )^2}{x^2}}-x+e^{\frac {\left (-25-5 x+x^2\right )^2}{x^2}} (2+x)\right )} \, dx \\ & = -\frac {2 \left (25+5 x-x^2\right )^2}{x^2}+2 \int \left (\frac {2 x^2}{-2 e^{\frac {\left (-25-5 x+x^2\right )^2}{x^2}} \left (1+\frac {e^6}{2}\right )+x-e^{\frac {\left (-25-5 x+x^2\right )^2}{x^2}} x}+\frac {1250}{x^2 \left (2 e^{\frac {\left (-25-5 x+x^2\right )^2}{x^2}} \left (1+\frac {e^6}{2}\right )-x+e^{\frac {\left (-25-5 x+x^2\right )^2}{x^2}} x\right )}+\frac {250}{x \left (2 e^{\frac {\left (-25-5 x+x^2\right )^2}{x^2}} \left (1+\frac {e^6}{2}\right )-x+e^{\frac {\left (-25-5 x+x^2\right )^2}{x^2}} x\right )}+\frac {10 x}{2 e^{\frac {\left (-25-5 x+x^2\right )^2}{x^2}} \left (1+\frac {e^6}{2}\right )-x+e^{\frac {\left (-25-5 x+x^2\right )^2}{x^2}} x}+\frac {2+e^6}{\left (2+e^6+x\right ) \left (2 e^{\frac {\left (-25-5 x+x^2\right )^2}{x^2}} \left (1+\frac {e^6}{2}\right )-x+e^{\frac {\left (-25-5 x+x^2\right )^2}{x^2}} x\right )}\right ) \, dx \\ & = -\frac {2 \left (25+5 x-x^2\right )^2}{x^2}+4 \int \frac {x^2}{-2 e^{\frac {\left (-25-5 x+x^2\right )^2}{x^2}} \left (1+\frac {e^6}{2}\right )+x-e^{\frac {\left (-25-5 x+x^2\right )^2}{x^2}} x} \, dx+20 \int \frac {x}{2 e^{\frac {\left (-25-5 x+x^2\right )^2}{x^2}} \left (1+\frac {e^6}{2}\right )-x+e^{\frac {\left (-25-5 x+x^2\right )^2}{x^2}} x} \, dx+500 \int \frac {1}{x \left (2 e^{\frac {\left (-25-5 x+x^2\right )^2}{x^2}} \left (1+\frac {e^6}{2}\right )-x+e^{\frac {\left (-25-5 x+x^2\right )^2}{x^2}} x\right )} \, dx+2500 \int \frac {1}{x^2 \left (2 e^{\frac {\left (-25-5 x+x^2\right )^2}{x^2}} \left (1+\frac {e^6}{2}\right )-x+e^{\frac {\left (-25-5 x+x^2\right )^2}{x^2}} x\right )} \, dx+\left (2 \left (2+e^6\right )\right ) \int \frac {1}{\left (2+e^6+x\right ) \left (2 e^{\frac {\left (-25-5 x+x^2\right )^2}{x^2}} \left (1+\frac {e^6}{2}\right )-x+e^{\frac {\left (-25-5 x+x^2\right )^2}{x^2}} x\right )} \, dx \\ \end{align*}
Leaf count is larger than twice the leaf count of optimal. \(82\) vs. \(2(32)=64\).
Time = 0.21 (sec) , antiderivative size = 82, normalized size of antiderivative = 2.56 \[ \int \frac {4 x^3+2 e^6 x^3+e^{\frac {625+250 x-25 x^2-10 x^3+x^4}{x^2}} \left (10000+12000 x+4500 x^2+580 x^3+64 x^4+4 x^5-4 x^6+e^{12} \left (2500+500 x+20 x^3-4 x^4\right )+e^6 \left (10000+7000 x+1000 x^2+80 x^3+24 x^4-8 x^5\right )\right )}{-2 x^4-e^6 x^4-x^5+e^{\frac {625+250 x-25 x^2-10 x^3+x^4}{x^2}} \left (4 x^3+e^{12} x^3+4 x^4+x^5+e^6 \left (4 x^3+2 x^4\right )\right )} \, dx=2 \log \left (2+e^6+x\right )-2 \log \left (2 e^{\frac {625}{x^2}+\frac {250}{x}-10 x+x^2}+e^{6+\frac {625}{x^2}+\frac {250}{x}-10 x+x^2}-e^{25} x+e^{\frac {625}{x^2}+\frac {250}{x}-10 x+x^2} x\right ) \]
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Leaf count of result is larger than twice the leaf count of optimal. \(70\) vs. \(2(32)=64\).
Time = 1.01 (sec) , antiderivative size = 71, normalized size of antiderivative = 2.22
method | result | size |
risch | \(-2 x^{2}+20 x +\frac {-500 x -1250}{x^{2}}+\frac {2 x^{4}-20 x^{3}-50 x^{2}+500 x +1250}{x^{2}}-2 \ln \left ({\mathrm e}^{\frac {\left (x^{2}-5 x -25\right )^{2}}{x^{2}}}-\frac {x}{{\mathrm e}^{6}+2+x}\right )\) | \(71\) |
norman | \(2 \ln \left ({\mathrm e}^{6}+2+x \right )-2 \ln \left ({\mathrm e}^{\frac {x^{4}-10 x^{3}-25 x^{2}+250 x +625}{x^{2}}} {\mathrm e}^{6}+{\mathrm e}^{\frac {x^{4}-10 x^{3}-25 x^{2}+250 x +625}{x^{2}}} x -x +2 \,{\mathrm e}^{\frac {x^{4}-10 x^{3}-25 x^{2}+250 x +625}{x^{2}}}\right )\) | \(97\) |
parallelrisch | \(2 \ln \left ({\mathrm e}^{6}+2+x \right )-2 \ln \left ({\mathrm e}^{\frac {x^{4}-10 x^{3}-25 x^{2}+250 x +625}{x^{2}}} {\mathrm e}^{6}+{\mathrm e}^{\frac {x^{4}-10 x^{3}-25 x^{2}+250 x +625}{x^{2}}} x -x +2 \,{\mathrm e}^{\frac {x^{4}-10 x^{3}-25 x^{2}+250 x +625}{x^{2}}}\right )\) | \(97\) |
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Time = 0.24 (sec) , antiderivative size = 44, normalized size of antiderivative = 1.38 \[ \int \frac {4 x^3+2 e^6 x^3+e^{\frac {625+250 x-25 x^2-10 x^3+x^4}{x^2}} \left (10000+12000 x+4500 x^2+580 x^3+64 x^4+4 x^5-4 x^6+e^{12} \left (2500+500 x+20 x^3-4 x^4\right )+e^6 \left (10000+7000 x+1000 x^2+80 x^3+24 x^4-8 x^5\right )\right )}{-2 x^4-e^6 x^4-x^5+e^{\frac {625+250 x-25 x^2-10 x^3+x^4}{x^2}} \left (4 x^3+e^{12} x^3+4 x^4+x^5+e^6 \left (4 x^3+2 x^4\right )\right )} \, dx=-2 \, \log \left (\frac {{\left (x + e^{6} + 2\right )} e^{\left (\frac {x^{4} - 10 \, x^{3} - 25 \, x^{2} + 250 \, x + 625}{x^{2}}\right )} - x}{x + e^{6} + 2}\right ) \]
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Time = 0.52 (sec) , antiderivative size = 36, normalized size of antiderivative = 1.12 \[ \int \frac {4 x^3+2 e^6 x^3+e^{\frac {625+250 x-25 x^2-10 x^3+x^4}{x^2}} \left (10000+12000 x+4500 x^2+580 x^3+64 x^4+4 x^5-4 x^6+e^{12} \left (2500+500 x+20 x^3-4 x^4\right )+e^6 \left (10000+7000 x+1000 x^2+80 x^3+24 x^4-8 x^5\right )\right )}{-2 x^4-e^6 x^4-x^5+e^{\frac {625+250 x-25 x^2-10 x^3+x^4}{x^2}} \left (4 x^3+e^{12} x^3+4 x^4+x^5+e^6 \left (4 x^3+2 x^4\right )\right )} \, dx=- 2 \log {\left (- \frac {x}{x + 2 + e^{6}} + e^{\frac {x^{4} - 10 x^{3} - 25 x^{2} + 250 x + 625}{x^{2}}} \right )} \]
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Leaf count of result is larger than twice the leaf count of optimal. 70 vs. \(2 (30) = 60\).
Time = 0.27 (sec) , antiderivative size = 70, normalized size of antiderivative = 2.19 \[ \int \frac {4 x^3+2 e^6 x^3+e^{\frac {625+250 x-25 x^2-10 x^3+x^4}{x^2}} \left (10000+12000 x+4500 x^2+580 x^3+64 x^4+4 x^5-4 x^6+e^{12} \left (2500+500 x+20 x^3-4 x^4\right )+e^6 \left (10000+7000 x+1000 x^2+80 x^3+24 x^4-8 x^5\right )\right )}{-2 x^4-e^6 x^4-x^5+e^{\frac {625+250 x-25 x^2-10 x^3+x^4}{x^2}} \left (4 x^3+e^{12} x^3+4 x^4+x^5+e^6 \left (4 x^3+2 x^4\right )\right )} \, dx=-\frac {2 \, {\left (x^{3} - 10 \, x^{2} + 250\right )}}{x} - 2 \, \log \left (\frac {{\left ({\left (x + e^{6} + 2\right )} e^{\left (x^{2} + \frac {250}{x} + \frac {625}{x^{2}}\right )} - x e^{\left (10 \, x + 25\right )}\right )} e^{\left (-x^{2} - \frac {250}{x}\right )}}{x + e^{6} + 2}\right ) \]
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Timed out. \[ \int \frac {4 x^3+2 e^6 x^3+e^{\frac {625+250 x-25 x^2-10 x^3+x^4}{x^2}} \left (10000+12000 x+4500 x^2+580 x^3+64 x^4+4 x^5-4 x^6+e^{12} \left (2500+500 x+20 x^3-4 x^4\right )+e^6 \left (10000+7000 x+1000 x^2+80 x^3+24 x^4-8 x^5\right )\right )}{-2 x^4-e^6 x^4-x^5+e^{\frac {625+250 x-25 x^2-10 x^3+x^4}{x^2}} \left (4 x^3+e^{12} x^3+4 x^4+x^5+e^6 \left (4 x^3+2 x^4\right )\right )} \, dx=\text {Timed out} \]
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Time = 9.65 (sec) , antiderivative size = 95, normalized size of antiderivative = 2.97 \[ \int \frac {4 x^3+2 e^6 x^3+e^{\frac {625+250 x-25 x^2-10 x^3+x^4}{x^2}} \left (10000+12000 x+4500 x^2+580 x^3+64 x^4+4 x^5-4 x^6+e^{12} \left (2500+500 x+20 x^3-4 x^4\right )+e^6 \left (10000+7000 x+1000 x^2+80 x^3+24 x^4-8 x^5\right )\right )}{-2 x^4-e^6 x^4-x^5+e^{\frac {625+250 x-25 x^2-10 x^3+x^4}{x^2}} \left (4 x^3+e^{12} x^3+4 x^4+x^5+e^6 \left (4 x^3+2 x^4\right )\right )} \, dx=-2\,\ln \left (\frac {2\,{\mathrm {e}}^{x^2}\,{\mathrm {e}}^{250/x}\,{\mathrm {e}}^{\frac {625}{x^2}}-x\,{\mathrm {e}}^{10\,x}\,{\mathrm {e}}^{25}+{\mathrm {e}}^{x^2}\,{\mathrm {e}}^6\,{\mathrm {e}}^{250/x}\,{\mathrm {e}}^{\frac {625}{x^2}}+x\,{\mathrm {e}}^{x^2}\,{\mathrm {e}}^{250/x}\,{\mathrm {e}}^{\frac {625}{x^2}}}{2\,{\mathrm {e}}^{10\,x}\,{\mathrm {e}}^{25}+{\mathrm {e}}^{10\,x}\,{\mathrm {e}}^{31}+x\,{\mathrm {e}}^{10\,x}\,{\mathrm {e}}^{25}}\right ) \]
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