Integrand size = 175, antiderivative size = 35 \[ \int \frac {-x^6+120 x^7+1112 x^8+3210 x^9+3504 x^{10}+1300 x^{11}+150 x^{12}+e^9 \left (-1+50 x^2+780 x^3+2904 x^4+1300 x^5+150 x^6\right )+e^6 \left (-3 x^2+90 x^3+1374 x^4+6390 x^5+9612 x^6+3900 x^7+450 x^8\right )+e^3 \left (33 x^4+570 x^5+3336 x^6+8820 x^7+10212 x^8+3900 x^9+450 x^{10}\right )}{e^9 x+3 e^6 x^3+3 e^3 x^5+x^7} \, dx=\left (x+5 x^2\right )^2 \left (5+x+\frac {3}{\frac {e^3}{x}+x}\right )^2-\log (-x) \]
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Leaf count is larger than twice the leaf count of optimal. \(181\) vs. \(2(35)=70\).
Time = 0.21 (sec) , antiderivative size = 181, normalized size of antiderivative = 5.17, number of steps used = 8, number of rules used = 4, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.023, Rules used = {2099, 653, 205, 209} \[ \int \frac {-x^6+120 x^7+1112 x^8+3210 x^9+3504 x^{10}+1300 x^{11}+150 x^{12}+e^9 \left (-1+50 x^2+780 x^3+2904 x^4+1300 x^5+150 x^6\right )+e^6 \left (-3 x^2+90 x^3+1374 x^4+6390 x^5+9612 x^6+3900 x^7+450 x^8\right )+e^3 \left (33 x^4+570 x^5+3336 x^6+8820 x^7+10212 x^8+3900 x^9+450 x^{10}\right )}{e^9 x+3 e^6 x^3+3 e^3 x^5+x^7} \, dx=30 e^{3/2} \left (7-27 e^3\right ) \arctan \left (\frac {x}{e^{3/2}}\right )-15 e^{3/2} \left (23-54 e^3\right ) \arctan \left (\frac {x}{e^{3/2}}\right )+135 e^{3/2} \arctan \left (\frac {x}{e^{3/2}}\right )+25 x^6+260 x^5+876 x^4+1070 x^3+2 \left (278-75 e^3\right ) x^2+\frac {135 e^3 x}{x^2+e^3}-\frac {3 e^3 \left (5 \left (23-54 e^3\right ) x+50 e^6-327 e^3+6\right )}{x^2+e^3}+\frac {9 e^6 \left (10 x-25 e^3+1\right )}{\left (x^2+e^3\right )^2}+30 \left (4-27 e^3\right ) x-\log (x) \]
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Rule 205
Rule 209
Rule 653
Rule 2099
Rubi steps \begin{align*} \text {integral}& = \int \left (-30 \left (-4+27 e^3\right )-\frac {1}{x}-4 \left (-278+75 e^3\right ) x+3210 x^2+3504 x^3+1300 x^4+150 x^5+\frac {36 \left (10 e^9-e^6 \left (1-25 e^3\right ) x\right )}{\left (e^3+x^2\right )^3}+\frac {6 \left (-5 e^6 \left (23-54 e^3\right )+e^3 \left (6-327 e^3+50 e^6\right ) x\right )}{\left (e^3+x^2\right )^2}-\frac {30 e^3 \left (-7+27 e^3\right )}{e^3+x^2}\right ) \, dx \\ & = 30 \left (4-27 e^3\right ) x+2 \left (278-75 e^3\right ) x^2+1070 x^3+876 x^4+260 x^5+25 x^6-\log (x)+6 \int \frac {-5 e^6 \left (23-54 e^3\right )+e^3 \left (6-327 e^3+50 e^6\right ) x}{\left (e^3+x^2\right )^2} \, dx+36 \int \frac {10 e^9-e^6 \left (1-25 e^3\right ) x}{\left (e^3+x^2\right )^3} \, dx+\left (30 e^3 \left (7-27 e^3\right )\right ) \int \frac {1}{e^3+x^2} \, dx \\ & = 30 \left (4-27 e^3\right ) x+2 \left (278-75 e^3\right ) x^2+1070 x^3+876 x^4+260 x^5+25 x^6+\frac {9 e^6 \left (1-25 e^3+10 x\right )}{\left (e^3+x^2\right )^2}-\frac {3 e^3 \left (6-327 e^3+50 e^6+5 \left (23-54 e^3\right ) x\right )}{e^3+x^2}+30 e^{3/2} \left (7-27 e^3\right ) \arctan \left (\frac {x}{e^{3/2}}\right )-\log (x)+\left (270 e^6\right ) \int \frac {1}{\left (e^3+x^2\right )^2} \, dx-\left (15 e^3 \left (23-54 e^3\right )\right ) \int \frac {1}{e^3+x^2} \, dx \\ & = 30 \left (4-27 e^3\right ) x+2 \left (278-75 e^3\right ) x^2+1070 x^3+876 x^4+260 x^5+25 x^6+\frac {9 e^6 \left (1-25 e^3+10 x\right )}{\left (e^3+x^2\right )^2}+\frac {135 e^3 x}{e^3+x^2}-\frac {3 e^3 \left (6-327 e^3+50 e^6+5 \left (23-54 e^3\right ) x\right )}{e^3+x^2}-15 e^{3/2} \left (23-54 e^3\right ) \arctan \left (\frac {x}{e^{3/2}}\right )+30 e^{3/2} \left (7-27 e^3\right ) \arctan \left (\frac {x}{e^{3/2}}\right )-\log (x)+\left (135 e^3\right ) \int \frac {1}{e^3+x^2} \, dx \\ & = 30 \left (4-27 e^3\right ) x+2 \left (278-75 e^3\right ) x^2+1070 x^3+876 x^4+260 x^5+25 x^6+\frac {9 e^6 \left (1-25 e^3+10 x\right )}{\left (e^3+x^2\right )^2}+\frac {135 e^3 x}{e^3+x^2}-\frac {3 e^3 \left (6-327 e^3+50 e^6+5 \left (23-54 e^3\right ) x\right )}{e^3+x^2}+135 e^{3/2} \arctan \left (\frac {x}{e^{3/2}}\right )-15 e^{3/2} \left (23-54 e^3\right ) \arctan \left (\frac {x}{e^{3/2}}\right )+30 e^{3/2} \left (7-27 e^3\right ) \arctan \left (\frac {x}{e^{3/2}}\right )-\log (x) \\ \end{align*}
Leaf count is larger than twice the leaf count of optimal. \(105\) vs. \(2(35)=70\).
Time = 0.09 (sec) , antiderivative size = 105, normalized size of antiderivative = 3.00 \[ \int \frac {-x^6+120 x^7+1112 x^8+3210 x^9+3504 x^{10}+1300 x^{11}+150 x^{12}+e^9 \left (-1+50 x^2+780 x^3+2904 x^4+1300 x^5+150 x^6\right )+e^6 \left (-3 x^2+90 x^3+1374 x^4+6390 x^5+9612 x^6+3900 x^7+450 x^8\right )+e^3 \left (33 x^4+570 x^5+3336 x^6+8820 x^7+10212 x^8+3900 x^9+450 x^{10}\right )}{e^9 x+3 e^6 x^3+3 e^3 x^5+x^7} \, dx=-30 \left (-4+27 e^3\right ) x-2 \left (-278+75 e^3\right ) x^2+1070 x^3+876 x^4+260 x^5+25 x^6+\frac {9 e^6 \left (1-25 e^3+10 x\right )}{\left (e^3+x^2\right )^2}-\frac {3 e^3 \left (6+50 e^6+70 x-3 e^3 (109+90 x)\right )}{e^3+x^2}-\log (x) \]
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Leaf count of result is larger than twice the leaf count of optimal. \(115\) vs. \(2(34)=68\).
Time = 0.25 (sec) , antiderivative size = 116, normalized size of antiderivative = 3.31
method | result | size |
risch | \(25 x^{6}+260 x^{5}+876 x^{4}-150 x^{2} {\mathrm e}^{3}+1070 x^{3}-810 x \,{\mathrm e}^{3}+556 x^{2}+120 x +\frac {30 \,{\mathrm e}^{3} \left (27 \,{\mathrm e}^{3}-7\right ) x^{3}+\left (-150 \,{\mathrm e}^{9}+981 \,{\mathrm e}^{6}-18 \,{\mathrm e}^{3}\right ) x^{2}+\left (810 \,{\mathrm e}^{9}-120 \,{\mathrm e}^{6}\right ) x -150 \,{\mathrm e}^{12}+756 \,{\mathrm e}^{9}-9 \,{\mathrm e}^{6}}{x^{4}+2 x^{2} {\mathrm e}^{3}+{\mathrm e}^{6}}-\ln \left (x \right )\) | \(116\) |
norman | \(\frac {\left (260 \,{\mathrm e}^{6}+30 \,{\mathrm e}^{3}\right ) x^{3}+\left (50 \,{\mathrm e}^{3}+876\right ) x^{8}+\left (520 \,{\mathrm e}^{3}+1070\right ) x^{7}+\left (25 \,{\mathrm e}^{6}+1602 \,{\mathrm e}^{3}+556\right ) x^{6}+\left (260 \,{\mathrm e}^{6}+1330 \,{\mathrm e}^{3}+120\right ) x^{5}+\left (-1452 \,{\mathrm e}^{9}-687 \,{\mathrm e}^{6}-18 \,{\mathrm e}^{3}\right ) x^{2}+260 x^{9}+25 x^{10}-726 \,{\mathrm e}^{12}-356 \,{\mathrm e}^{9}-9 \,{\mathrm e}^{6}}{\left (x^{2}+{\mathrm e}^{3}\right )^{2}}-\ln \left (x \right )\) | \(132\) |
parallelrisch | \(-\frac {-520 x^{7} {\mathrm e}^{3}-260 x^{5} {\mathrm e}^{6}-25 x^{6} {\mathrm e}^{6}+x^{4} \ln \left (x \right )-260 x^{3} {\mathrm e}^{6}-30 x^{3} {\mathrm e}^{3}+687 x^{2} {\mathrm e}^{6}-1330 x^{5} {\mathrm e}^{3}+18 x^{2} {\mathrm e}^{3}+356 \,{\mathrm e}^{9}+726 \,{\mathrm e}^{12}-1602 x^{6} {\mathrm e}^{3}+9 \,{\mathrm e}^{6}-25 x^{10}-260 x^{9}-1070 x^{7}-876 x^{8}-556 x^{6}-120 x^{5}+1452 x^{2} {\mathrm e}^{9}-50 \,{\mathrm e}^{3} x^{8}+\ln \left (x \right ) {\mathrm e}^{6}+2 \ln \left (x \right ) {\mathrm e}^{3} x^{2}}{x^{4}+2 x^{2} {\mathrm e}^{3}+{\mathrm e}^{6}}\) | \(178\) |
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Leaf count of result is larger than twice the leaf count of optimal. 149 vs. \(2 (34) = 68\).
Time = 0.24 (sec) , antiderivative size = 149, normalized size of antiderivative = 4.26 \[ \int \frac {-x^6+120 x^7+1112 x^8+3210 x^9+3504 x^{10}+1300 x^{11}+150 x^{12}+e^9 \left (-1+50 x^2+780 x^3+2904 x^4+1300 x^5+150 x^6\right )+e^6 \left (-3 x^2+90 x^3+1374 x^4+6390 x^5+9612 x^6+3900 x^7+450 x^8\right )+e^3 \left (33 x^4+570 x^5+3336 x^6+8820 x^7+10212 x^8+3900 x^9+450 x^{10}\right )}{e^9 x+3 e^6 x^3+3 e^3 x^5+x^7} \, dx=\frac {25 \, x^{10} + 260 \, x^{9} + 876 \, x^{8} + 1070 \, x^{7} + 556 \, x^{6} + 120 \, x^{5} - 12 \, {\left (25 \, x^{2} - 63\right )} e^{9} + {\left (25 \, x^{6} + 260 \, x^{5} + 576 \, x^{4} + 260 \, x^{3} + 1537 \, x^{2} - 9\right )} e^{6} + 2 \, {\left (25 \, x^{8} + 260 \, x^{7} + 801 \, x^{6} + 665 \, x^{5} + 556 \, x^{4} + 15 \, x^{3} - 9 \, x^{2}\right )} e^{3} - {\left (x^{4} + 2 \, x^{2} e^{3} + e^{6}\right )} \log \left (x\right ) - 150 \, e^{12}}{x^{4} + 2 \, x^{2} e^{3} + e^{6}} \]
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Leaf count of result is larger than twice the leaf count of optimal. 116 vs. \(2 (26) = 52\).
Time = 2.03 (sec) , antiderivative size = 116, normalized size of antiderivative = 3.31 \[ \int \frac {-x^6+120 x^7+1112 x^8+3210 x^9+3504 x^{10}+1300 x^{11}+150 x^{12}+e^9 \left (-1+50 x^2+780 x^3+2904 x^4+1300 x^5+150 x^6\right )+e^6 \left (-3 x^2+90 x^3+1374 x^4+6390 x^5+9612 x^6+3900 x^7+450 x^8\right )+e^3 \left (33 x^4+570 x^5+3336 x^6+8820 x^7+10212 x^8+3900 x^9+450 x^{10}\right )}{e^9 x+3 e^6 x^3+3 e^3 x^5+x^7} \, dx=25 x^{6} + 260 x^{5} + 876 x^{4} + 1070 x^{3} + x^{2} \cdot \left (556 - 150 e^{3}\right ) + x \left (120 - 810 e^{3}\right ) - \log {\left (x \right )} + \frac {x^{3} \left (- 210 e^{3} + 810 e^{6}\right ) + x^{2} \left (- 150 e^{9} - 18 e^{3} + 981 e^{6}\right ) + x \left (- 120 e^{6} + 810 e^{9}\right ) - 150 e^{12} - 9 e^{6} + 756 e^{9}}{x^{4} + 2 x^{2} e^{3} + e^{6}} \]
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Leaf count of result is larger than twice the leaf count of optimal. 119 vs. \(2 (34) = 68\).
Time = 0.19 (sec) , antiderivative size = 119, normalized size of antiderivative = 3.40 \[ \int \frac {-x^6+120 x^7+1112 x^8+3210 x^9+3504 x^{10}+1300 x^{11}+150 x^{12}+e^9 \left (-1+50 x^2+780 x^3+2904 x^4+1300 x^5+150 x^6\right )+e^6 \left (-3 x^2+90 x^3+1374 x^4+6390 x^5+9612 x^6+3900 x^7+450 x^8\right )+e^3 \left (33 x^4+570 x^5+3336 x^6+8820 x^7+10212 x^8+3900 x^9+450 x^{10}\right )}{e^9 x+3 e^6 x^3+3 e^3 x^5+x^7} \, dx=25 \, x^{6} + 260 \, x^{5} + 876 \, x^{4} + 1070 \, x^{3} - 2 \, x^{2} {\left (75 \, e^{3} - 278\right )} - 30 \, x {\left (27 \, e^{3} - 4\right )} + \frac {3 \, {\left (10 \, x^{3} {\left (27 \, e^{6} - 7 \, e^{3}\right )} - x^{2} {\left (50 \, e^{9} - 327 \, e^{6} + 6 \, e^{3}\right )} + 10 \, x {\left (27 \, e^{9} - 4 \, e^{6}\right )} - 50 \, e^{12} + 252 \, e^{9} - 3 \, e^{6}\right )}}{x^{4} + 2 \, x^{2} e^{3} + e^{6}} - \log \left (x\right ) \]
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Leaf count of result is larger than twice the leaf count of optimal. 114 vs. \(2 (34) = 68\).
Time = 0.26 (sec) , antiderivative size = 114, normalized size of antiderivative = 3.26 \[ \int \frac {-x^6+120 x^7+1112 x^8+3210 x^9+3504 x^{10}+1300 x^{11}+150 x^{12}+e^9 \left (-1+50 x^2+780 x^3+2904 x^4+1300 x^5+150 x^6\right )+e^6 \left (-3 x^2+90 x^3+1374 x^4+6390 x^5+9612 x^6+3900 x^7+450 x^8\right )+e^3 \left (33 x^4+570 x^5+3336 x^6+8820 x^7+10212 x^8+3900 x^9+450 x^{10}\right )}{e^9 x+3 e^6 x^3+3 e^3 x^5+x^7} \, dx=25 \, x^{6} + 260 \, x^{5} + 876 \, x^{4} + 1070 \, x^{3} - 150 \, x^{2} e^{3} + 556 \, x^{2} - 810 \, x e^{3} + 120 \, x + \frac {3 \, {\left (270 \, x^{3} e^{6} - 70 \, x^{3} e^{3} - 50 \, x^{2} e^{9} + 327 \, x^{2} e^{6} - 6 \, x^{2} e^{3} + 270 \, x e^{9} - 40 \, x e^{6} - 50 \, e^{12} + 252 \, e^{9} - 3 \, e^{6}\right )}}{{\left (x^{2} + e^{3}\right )}^{2}} - \log \left ({\left | x \right |}\right ) \]
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Time = 0.31 (sec) , antiderivative size = 116, normalized size of antiderivative = 3.31 \[ \int \frac {-x^6+120 x^7+1112 x^8+3210 x^9+3504 x^{10}+1300 x^{11}+150 x^{12}+e^9 \left (-1+50 x^2+780 x^3+2904 x^4+1300 x^5+150 x^6\right )+e^6 \left (-3 x^2+90 x^3+1374 x^4+6390 x^5+9612 x^6+3900 x^7+450 x^8\right )+e^3 \left (33 x^4+570 x^5+3336 x^6+8820 x^7+10212 x^8+3900 x^9+450 x^{10}\right )}{e^9 x+3 e^6 x^3+3 e^3 x^5+x^7} \, dx=1070\,x^3-x^2\,\left (150\,{\mathrm {e}}^3-556\right )-\frac {\left (210\,{\mathrm {e}}^3-810\,{\mathrm {e}}^6\right )\,x^3+\left (18\,{\mathrm {e}}^3-981\,{\mathrm {e}}^6+150\,{\mathrm {e}}^9\right )\,x^2+\left (120\,{\mathrm {e}}^6-810\,{\mathrm {e}}^9\right )\,x+9\,{\mathrm {e}}^6-756\,{\mathrm {e}}^9+150\,{\mathrm {e}}^{12}}{x^4+2\,{\mathrm {e}}^3\,x^2+{\mathrm {e}}^6}-\ln \left (x\right )+876\,x^4+260\,x^5+25\,x^6-x\,\left (810\,{\mathrm {e}}^3-120\right ) \]
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