Integrand size = 204, antiderivative size = 31 \[ \int \frac {-100 x+40 x^2+e^{12 x^4} \left (-100 x+20 x^2+2400 x^5-960 x^6+96 x^7\right )+e^{6 x^4} \left (-200 x+60 x^2+2400 x^5-1440 x^6+192 x^7\right )}{625-1000 x+350 x^2+40 x^3+x^4+e^{6 x^4} \left (2500-3500 x+1300 x^2-100 x^3-8 x^4\right )+e^{24 x^4} \left (625-500 x+150 x^2-20 x^3+x^4\right )+e^{18 x^4} \left (2500-2500 x+900 x^2-140 x^3+8 x^4\right )+e^{12 x^4} \left (3750-4500 x+1700 x^2-260 x^3+14 x^4\right )} \, dx=\frac {2}{5-\left (-3+\frac {5-e^{6 x^4} (-5+x)+x}{x}\right )^2} \]
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\[ \int \frac {-100 x+40 x^2+e^{12 x^4} \left (-100 x+20 x^2+2400 x^5-960 x^6+96 x^7\right )+e^{6 x^4} \left (-200 x+60 x^2+2400 x^5-1440 x^6+192 x^7\right )}{625-1000 x+350 x^2+40 x^3+x^4+e^{6 x^4} \left (2500-3500 x+1300 x^2-100 x^3-8 x^4\right )+e^{24 x^4} \left (625-500 x+150 x^2-20 x^3+x^4\right )+e^{18 x^4} \left (2500-2500 x+900 x^2-140 x^3+8 x^4\right )+e^{12 x^4} \left (3750-4500 x+1700 x^2-260 x^3+14 x^4\right )} \, dx=\int \frac {-100 x+40 x^2+e^{12 x^4} \left (-100 x+20 x^2+2400 x^5-960 x^6+96 x^7\right )+e^{6 x^4} \left (-200 x+60 x^2+2400 x^5-1440 x^6+192 x^7\right )}{625-1000 x+350 x^2+40 x^3+x^4+e^{6 x^4} \left (2500-3500 x+1300 x^2-100 x^3-8 x^4\right )+e^{24 x^4} \left (625-500 x+150 x^2-20 x^3+x^4\right )+e^{18 x^4} \left (2500-2500 x+900 x^2-140 x^3+8 x^4\right )+e^{12 x^4} \left (3750-4500 x+1700 x^2-260 x^3+14 x^4\right )} \, dx \]
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Rubi steps \begin{align*} \text {integral}& = \int \frac {4 x \left (5 (-5+2 x)+e^{12 x^4} \left (-25+5 x+600 x^4-240 x^5+24 x^6\right )+e^{6 x^4} \left (-50+15 x+600 x^4-360 x^5+48 x^6\right )\right )}{\left (25+e^{12 x^4} (-5+x)^2-20 x-x^2+e^{6 x^4} \left (50-30 x+4 x^2\right )\right )^2} \, dx \\ & = 4 \int \frac {x \left (5 (-5+2 x)+e^{12 x^4} \left (-25+5 x+600 x^4-240 x^5+24 x^6\right )+e^{6 x^4} \left (-50+15 x+600 x^4-360 x^5+48 x^6\right )\right )}{\left (25+e^{12 x^4} (-5+x)^2-20 x-x^2+e^{6 x^4} \left (50-30 x+4 x^2\right )\right )^2} \, dx \\ & = 4 \int \left (\frac {x \left (5-120 x^4+24 x^5\right )}{(-5+x) \left (25+50 e^{6 x^4}+25 e^{12 x^4}-20 x-30 e^{6 x^4} x-10 e^{12 x^4} x-x^2+4 e^{6 x^4} x^2+e^{12 x^4} x^2\right )}-\frac {x^2 \left (-25-25 e^{6 x^4}-15 x+5 e^{6 x^4} x-3000 x^3-3000 e^{6 x^4} x^3+3000 x^4+2400 e^{6 x^4} x^4-360 x^5-600 e^{6 x^4} x^5-24 x^6+48 e^{6 x^4} x^6\right )}{(-5+x) \left (25+50 e^{6 x^4}+25 e^{12 x^4}-20 x-30 e^{6 x^4} x-10 e^{12 x^4} x-x^2+4 e^{6 x^4} x^2+e^{12 x^4} x^2\right )^2}\right ) \, dx \\ & = 4 \int \frac {x \left (5-120 x^4+24 x^5\right )}{(-5+x) \left (25+50 e^{6 x^4}+25 e^{12 x^4}-20 x-30 e^{6 x^4} x-10 e^{12 x^4} x-x^2+4 e^{6 x^4} x^2+e^{12 x^4} x^2\right )} \, dx-4 \int \frac {x^2 \left (-25-25 e^{6 x^4}-15 x+5 e^{6 x^4} x-3000 x^3-3000 e^{6 x^4} x^3+3000 x^4+2400 e^{6 x^4} x^4-360 x^5-600 e^{6 x^4} x^5-24 x^6+48 e^{6 x^4} x^6\right )}{(-5+x) \left (25+50 e^{6 x^4}+25 e^{12 x^4}-20 x-30 e^{6 x^4} x-10 e^{12 x^4} x-x^2+4 e^{6 x^4} x^2+e^{12 x^4} x^2\right )^2} \, dx \\ & = 4 \int \frac {x \left (-5+120 x^4-24 x^5\right )}{(5-x) \left (25+e^{12 x^4} (-5+x)^2-20 x-x^2+e^{6 x^4} \left (50-30 x+4 x^2\right )\right )} \, dx-4 \int \frac {x^2 \left (25+15 x+3000 x^3-3000 x^4+360 x^5+24 x^6-e^{6 x^4} \left (-25+5 x-3000 x^3+2400 x^4-600 x^5+48 x^6\right )\right )}{(5-x) \left (25+e^{12 x^4} (-5+x)^2-20 x-x^2+e^{6 x^4} \left (50-30 x+4 x^2\right )\right )^2} \, dx \\ & = 4 \int \left (\frac {5}{25+50 e^{6 x^4}+25 e^{12 x^4}-20 x-30 e^{6 x^4} x-10 e^{12 x^4} x-x^2+4 e^{6 x^4} x^2+e^{12 x^4} x^2}+\frac {25}{(-5+x) \left (25+50 e^{6 x^4}+25 e^{12 x^4}-20 x-30 e^{6 x^4} x-10 e^{12 x^4} x-x^2+4 e^{6 x^4} x^2+e^{12 x^4} x^2\right )}+\frac {24 x^5}{25+50 e^{6 x^4}+25 e^{12 x^4}-20 x-30 e^{6 x^4} x-10 e^{12 x^4} x-x^2+4 e^{6 x^4} x^2+e^{12 x^4} x^2}\right ) \, dx-4 \int \left (\frac {5 \left (-25-25 e^{6 x^4}-15 x+5 e^{6 x^4} x-3000 x^3-3000 e^{6 x^4} x^3+3000 x^4+2400 e^{6 x^4} x^4-360 x^5-600 e^{6 x^4} x^5-24 x^6+48 e^{6 x^4} x^6\right )}{\left (25+50 e^{6 x^4}+25 e^{12 x^4}-20 x-30 e^{6 x^4} x-10 e^{12 x^4} x-x^2+4 e^{6 x^4} x^2+e^{12 x^4} x^2\right )^2}+\frac {25 \left (-25-25 e^{6 x^4}-15 x+5 e^{6 x^4} x-3000 x^3-3000 e^{6 x^4} x^3+3000 x^4+2400 e^{6 x^4} x^4-360 x^5-600 e^{6 x^4} x^5-24 x^6+48 e^{6 x^4} x^6\right )}{(-5+x) \left (25+50 e^{6 x^4}+25 e^{12 x^4}-20 x-30 e^{6 x^4} x-10 e^{12 x^4} x-x^2+4 e^{6 x^4} x^2+e^{12 x^4} x^2\right )^2}+\frac {x \left (-25-25 e^{6 x^4}-15 x+5 e^{6 x^4} x-3000 x^3-3000 e^{6 x^4} x^3+3000 x^4+2400 e^{6 x^4} x^4-360 x^5-600 e^{6 x^4} x^5-24 x^6+48 e^{6 x^4} x^6\right )}{\left (25+50 e^{6 x^4}+25 e^{12 x^4}-20 x-30 e^{6 x^4} x-10 e^{12 x^4} x-x^2+4 e^{6 x^4} x^2+e^{12 x^4} x^2\right )^2}\right ) \, dx \\ & = -\left (4 \int \frac {x \left (-25-25 e^{6 x^4}-15 x+5 e^{6 x^4} x-3000 x^3-3000 e^{6 x^4} x^3+3000 x^4+2400 e^{6 x^4} x^4-360 x^5-600 e^{6 x^4} x^5-24 x^6+48 e^{6 x^4} x^6\right )}{\left (25+50 e^{6 x^4}+25 e^{12 x^4}-20 x-30 e^{6 x^4} x-10 e^{12 x^4} x-x^2+4 e^{6 x^4} x^2+e^{12 x^4} x^2\right )^2} \, dx\right )+20 \int \frac {1}{25+50 e^{6 x^4}+25 e^{12 x^4}-20 x-30 e^{6 x^4} x-10 e^{12 x^4} x-x^2+4 e^{6 x^4} x^2+e^{12 x^4} x^2} \, dx-20 \int \frac {-25-25 e^{6 x^4}-15 x+5 e^{6 x^4} x-3000 x^3-3000 e^{6 x^4} x^3+3000 x^4+2400 e^{6 x^4} x^4-360 x^5-600 e^{6 x^4} x^5-24 x^6+48 e^{6 x^4} x^6}{\left (25+50 e^{6 x^4}+25 e^{12 x^4}-20 x-30 e^{6 x^4} x-10 e^{12 x^4} x-x^2+4 e^{6 x^4} x^2+e^{12 x^4} x^2\right )^2} \, dx+96 \int \frac {x^5}{25+50 e^{6 x^4}+25 e^{12 x^4}-20 x-30 e^{6 x^4} x-10 e^{12 x^4} x-x^2+4 e^{6 x^4} x^2+e^{12 x^4} x^2} \, dx+100 \int \frac {1}{(-5+x) \left (25+50 e^{6 x^4}+25 e^{12 x^4}-20 x-30 e^{6 x^4} x-10 e^{12 x^4} x-x^2+4 e^{6 x^4} x^2+e^{12 x^4} x^2\right )} \, dx-100 \int \frac {-25-25 e^{6 x^4}-15 x+5 e^{6 x^4} x-3000 x^3-3000 e^{6 x^4} x^3+3000 x^4+2400 e^{6 x^4} x^4-360 x^5-600 e^{6 x^4} x^5-24 x^6+48 e^{6 x^4} x^6}{(-5+x) \left (25+50 e^{6 x^4}+25 e^{12 x^4}-20 x-30 e^{6 x^4} x-10 e^{12 x^4} x-x^2+4 e^{6 x^4} x^2+e^{12 x^4} x^2\right )^2} \, dx \\ & = -\left (4 \int \frac {x \left (-25-15 x-3000 x^3+3000 x^4-360 x^5-24 x^6+e^{6 x^4} \left (-25+5 x-3000 x^3+2400 x^4-600 x^5+48 x^6\right )\right )}{\left (25+e^{12 x^4} (-5+x)^2-20 x-x^2+e^{6 x^4} \left (50-30 x+4 x^2\right )\right )^2} \, dx\right )+20 \int \frac {1}{25+e^{12 x^4} (-5+x)^2-20 x-x^2+e^{6 x^4} \left (50-30 x+4 x^2\right )} \, dx-20 \int \frac {-25-15 x-3000 x^3+3000 x^4-360 x^5-24 x^6+e^{6 x^4} \left (-25+5 x-3000 x^3+2400 x^4-600 x^5+48 x^6\right )}{\left (25+e^{12 x^4} (-5+x)^2-20 x-x^2+e^{6 x^4} \left (50-30 x+4 x^2\right )\right )^2} \, dx+96 \int \frac {x^5}{25+e^{12 x^4} (-5+x)^2-20 x-x^2+e^{6 x^4} \left (50-30 x+4 x^2\right )} \, dx+100 \int \frac {1}{(-5+x) \left (25+e^{12 x^4} (-5+x)^2-20 x-x^2+e^{6 x^4} \left (50-30 x+4 x^2\right )\right )} \, dx-100 \int \frac {25+15 x+3000 x^3-3000 x^4+360 x^5+24 x^6-e^{6 x^4} \left (-25+5 x-3000 x^3+2400 x^4-600 x^5+48 x^6\right )}{(5-x) \left (25+e^{12 x^4} (-5+x)^2-20 x-x^2+e^{6 x^4} \left (50-30 x+4 x^2\right )\right )^2} \, dx \\ & = \text {Too large to display} \\ \end{align*}
Time = 10.10 (sec) , antiderivative size = 48, normalized size of antiderivative = 1.55 \[ \int \frac {-100 x+40 x^2+e^{12 x^4} \left (-100 x+20 x^2+2400 x^5-960 x^6+96 x^7\right )+e^{6 x^4} \left (-200 x+60 x^2+2400 x^5-1440 x^6+192 x^7\right )}{625-1000 x+350 x^2+40 x^3+x^4+e^{6 x^4} \left (2500-3500 x+1300 x^2-100 x^3-8 x^4\right )+e^{24 x^4} \left (625-500 x+150 x^2-20 x^3+x^4\right )+e^{18 x^4} \left (2500-2500 x+900 x^2-140 x^3+8 x^4\right )+e^{12 x^4} \left (3750-4500 x+1700 x^2-260 x^3+14 x^4\right )} \, dx=-\frac {2 x^2}{25+e^{12 x^4} (-5+x)^2-20 x-x^2+e^{6 x^4} \left (50-30 x+4 x^2\right )} \]
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Leaf count of result is larger than twice the leaf count of optimal. \(72\) vs. \(2(30)=60\).
Time = 1.39 (sec) , antiderivative size = 73, normalized size of antiderivative = 2.35
method | result | size |
risch | \(-\frac {2 x^{2}}{{\mathrm e}^{12 x^{4}} x^{2}-10 \,{\mathrm e}^{12 x^{4}} x +4 \,{\mathrm e}^{6 x^{4}} x^{2}+25 \,{\mathrm e}^{12 x^{4}}-30 \,{\mathrm e}^{6 x^{4}} x -x^{2}+50 \,{\mathrm e}^{6 x^{4}}-20 x +25}\) | \(73\) |
parallelrisch | \(-\frac {2 x^{2}}{{\mathrm e}^{12 x^{4}} x^{2}-10 \,{\mathrm e}^{12 x^{4}} x +4 \,{\mathrm e}^{6 x^{4}} x^{2}+25 \,{\mathrm e}^{12 x^{4}}-30 \,{\mathrm e}^{6 x^{4}} x -x^{2}+50 \,{\mathrm e}^{6 x^{4}}-20 x +25}\) | \(79\) |
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Time = 0.27 (sec) , antiderivative size = 49, normalized size of antiderivative = 1.58 \[ \int \frac {-100 x+40 x^2+e^{12 x^4} \left (-100 x+20 x^2+2400 x^5-960 x^6+96 x^7\right )+e^{6 x^4} \left (-200 x+60 x^2+2400 x^5-1440 x^6+192 x^7\right )}{625-1000 x+350 x^2+40 x^3+x^4+e^{6 x^4} \left (2500-3500 x+1300 x^2-100 x^3-8 x^4\right )+e^{24 x^4} \left (625-500 x+150 x^2-20 x^3+x^4\right )+e^{18 x^4} \left (2500-2500 x+900 x^2-140 x^3+8 x^4\right )+e^{12 x^4} \left (3750-4500 x+1700 x^2-260 x^3+14 x^4\right )} \, dx=\frac {2 \, x^{2}}{x^{2} - {\left (x^{2} - 10 \, x + 25\right )} e^{\left (12 \, x^{4}\right )} - 2 \, {\left (2 \, x^{2} - 15 \, x + 25\right )} e^{\left (6 \, x^{4}\right )} + 20 \, x - 25} \]
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Leaf count of result is larger than twice the leaf count of optimal. 46 vs. \(2 (20) = 40\).
Time = 0.32 (sec) , antiderivative size = 46, normalized size of antiderivative = 1.48 \[ \int \frac {-100 x+40 x^2+e^{12 x^4} \left (-100 x+20 x^2+2400 x^5-960 x^6+96 x^7\right )+e^{6 x^4} \left (-200 x+60 x^2+2400 x^5-1440 x^6+192 x^7\right )}{625-1000 x+350 x^2+40 x^3+x^4+e^{6 x^4} \left (2500-3500 x+1300 x^2-100 x^3-8 x^4\right )+e^{24 x^4} \left (625-500 x+150 x^2-20 x^3+x^4\right )+e^{18 x^4} \left (2500-2500 x+900 x^2-140 x^3+8 x^4\right )+e^{12 x^4} \left (3750-4500 x+1700 x^2-260 x^3+14 x^4\right )} \, dx=- \frac {2 x^{2}}{- x^{2} - 20 x + \left (x^{2} - 10 x + 25\right ) e^{12 x^{4}} + \left (4 x^{2} - 30 x + 50\right ) e^{6 x^{4}} + 25} \]
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Time = 0.25 (sec) , antiderivative size = 49, normalized size of antiderivative = 1.58 \[ \int \frac {-100 x+40 x^2+e^{12 x^4} \left (-100 x+20 x^2+2400 x^5-960 x^6+96 x^7\right )+e^{6 x^4} \left (-200 x+60 x^2+2400 x^5-1440 x^6+192 x^7\right )}{625-1000 x+350 x^2+40 x^3+x^4+e^{6 x^4} \left (2500-3500 x+1300 x^2-100 x^3-8 x^4\right )+e^{24 x^4} \left (625-500 x+150 x^2-20 x^3+x^4\right )+e^{18 x^4} \left (2500-2500 x+900 x^2-140 x^3+8 x^4\right )+e^{12 x^4} \left (3750-4500 x+1700 x^2-260 x^3+14 x^4\right )} \, dx=\frac {2 \, x^{2}}{x^{2} - {\left (x^{2} - 10 \, x + 25\right )} e^{\left (12 \, x^{4}\right )} - 2 \, {\left (2 \, x^{2} - 15 \, x + 25\right )} e^{\left (6 \, x^{4}\right )} + 20 \, x - 25} \]
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Leaf count of result is larger than twice the leaf count of optimal. 72 vs. \(2 (29) = 58\).
Time = 0.67 (sec) , antiderivative size = 72, normalized size of antiderivative = 2.32 \[ \int \frac {-100 x+40 x^2+e^{12 x^4} \left (-100 x+20 x^2+2400 x^5-960 x^6+96 x^7\right )+e^{6 x^4} \left (-200 x+60 x^2+2400 x^5-1440 x^6+192 x^7\right )}{625-1000 x+350 x^2+40 x^3+x^4+e^{6 x^4} \left (2500-3500 x+1300 x^2-100 x^3-8 x^4\right )+e^{24 x^4} \left (625-500 x+150 x^2-20 x^3+x^4\right )+e^{18 x^4} \left (2500-2500 x+900 x^2-140 x^3+8 x^4\right )+e^{12 x^4} \left (3750-4500 x+1700 x^2-260 x^3+14 x^4\right )} \, dx=-\frac {4 \, x^{2}}{x^{2} e^{\left (12 \, x^{4}\right )} + 4 \, x^{2} e^{\left (6 \, x^{4}\right )} - x^{2} - 10 \, x e^{\left (12 \, x^{4}\right )} - 30 \, x e^{\left (6 \, x^{4}\right )} - 20 \, x + 25 \, e^{\left (12 \, x^{4}\right )} + 50 \, e^{\left (6 \, x^{4}\right )} + 25} \]
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Timed out. \[ \int \frac {-100 x+40 x^2+e^{12 x^4} \left (-100 x+20 x^2+2400 x^5-960 x^6+96 x^7\right )+e^{6 x^4} \left (-200 x+60 x^2+2400 x^5-1440 x^6+192 x^7\right )}{625-1000 x+350 x^2+40 x^3+x^4+e^{6 x^4} \left (2500-3500 x+1300 x^2-100 x^3-8 x^4\right )+e^{24 x^4} \left (625-500 x+150 x^2-20 x^3+x^4\right )+e^{18 x^4} \left (2500-2500 x+900 x^2-140 x^3+8 x^4\right )+e^{12 x^4} \left (3750-4500 x+1700 x^2-260 x^3+14 x^4\right )} \, dx=\int \frac {{\mathrm {e}}^{12\,x^4}\,\left (96\,x^7-960\,x^6+2400\,x^5+20\,x^2-100\,x\right )-100\,x+{\mathrm {e}}^{6\,x^4}\,\left (192\,x^7-1440\,x^6+2400\,x^5+60\,x^2-200\,x\right )+40\,x^2}{{\mathrm {e}}^{24\,x^4}\,\left (x^4-20\,x^3+150\,x^2-500\,x+625\right )-1000\,x+{\mathrm {e}}^{18\,x^4}\,\left (8\,x^4-140\,x^3+900\,x^2-2500\,x+2500\right )-{\mathrm {e}}^{6\,x^4}\,\left (8\,x^4+100\,x^3-1300\,x^2+3500\,x-2500\right )+{\mathrm {e}}^{12\,x^4}\,\left (14\,x^4-260\,x^3+1700\,x^2-4500\,x+3750\right )+350\,x^2+40\,x^3+x^4+625} \,d x \]
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