Integrand size = 141, antiderivative size = 29 \[ \int \frac {640 x-480 x^2-1640 x^3+1190 x^4+810 x^5-490 x^6-190 x^7+60 x^8+20 x^9+e^{\frac {16}{16-8 x-7 x^2+2 x^3+x^4}} \left (-1280-1920 x+5360 x^2+4540 x^3-1330 x^4-390 x^5+170 x^6+50 x^7\right )}{-64 x^3+48 x^4+36 x^5-23 x^6-9 x^7+3 x^8+x^9} \, dx=(1+5 x) \left (4+\frac {10 \left (-e^{\frac {16}{\left (-4+x+x^2\right )^2}}+x\right )}{x^2}\right ) \] Output:
(1+5*x)*(10*(x-exp(16/(x^2+x-4)^2))/x^2+4)
Time = 0.69 (sec) , antiderivative size = 33, normalized size of antiderivative = 1.14 \[ \int \frac {640 x-480 x^2-1640 x^3+1190 x^4+810 x^5-490 x^6-190 x^7+60 x^8+20 x^9+e^{\frac {16}{16-8 x-7 x^2+2 x^3+x^4}} \left (-1280-1920 x+5360 x^2+4540 x^3-1330 x^4-390 x^5+170 x^6+50 x^7\right )}{-64 x^3+48 x^4+36 x^5-23 x^6-9 x^7+3 x^8+x^9} \, dx=10 \left (e^{\frac {16}{\left (-4+x+x^2\right )^2}} \left (-\frac {1}{x^2}-\frac {5}{x}\right )+\frac {1}{x}+2 x\right ) \] Input:
Integrate[(640*x - 480*x^2 - 1640*x^3 + 1190*x^4 + 810*x^5 - 490*x^6 - 190 *x^7 + 60*x^8 + 20*x^9 + E^(16/(16 - 8*x - 7*x^2 + 2*x^3 + x^4))*(-1280 - 1920*x + 5360*x^2 + 4540*x^3 - 1330*x^4 - 390*x^5 + 170*x^6 + 50*x^7))/(-6 4*x^3 + 48*x^4 + 36*x^5 - 23*x^6 - 9*x^7 + 3*x^8 + x^9),x]
Output:
10*(E^(16/(-4 + x + x^2)^2)*(-x^(-2) - 5/x) + x^(-1) + 2*x)
Below are the steps used by Rubi to obtain the solution. The rule number used for the transformation is given above next to the arrow. The rules definitions used are listed below.
| \(\displaystyle \int \frac {20 x^9+60 x^8-190 x^7-490 x^6+810 x^5+1190 x^4-1640 x^3-480 x^2+e^{\frac {16}{x^4+2 x^3-7 x^2-8 x+16}} \left (50 x^7+170 x^6-390 x^5-1330 x^4+4540 x^3+5360 x^2-1920 x-1280\right )+640 x}{x^9+3 x^8-9 x^7-23 x^6+36 x^5+48 x^4-64 x^3} \, dx\) |
| \(\Big \downarrow \) 2026 |
| \(\displaystyle \int \frac {20 x^9+60 x^8-190 x^7-490 x^6+810 x^5+1190 x^4-1640 x^3-480 x^2+e^{\frac {16}{x^4+2 x^3-7 x^2-8 x+16}} \left (50 x^7+170 x^6-390 x^5-1330 x^4+4540 x^3+5360 x^2-1920 x-1280\right )+640 x}{x^3 \left (x^6+3 x^5-9 x^4-23 x^3+36 x^2+48 x-64\right )}dx\) |
| \(\Big \downarrow \) 2463 |
| \(\displaystyle \int \left (-\frac {12 \left (20 x^9+60 x^8-190 x^7-490 x^6+810 x^5+1190 x^4-1640 x^3-480 x^2+e^{\frac {16}{x^4+2 x^3-7 x^2-8 x+16}} \left (50 x^7+170 x^6-390 x^5-1330 x^4+4540 x^3+5360 x^2-1920 x-1280\right )+640 x\right )}{289 \sqrt {17} x^3 \left (2 x+\sqrt {17}+1\right )}-\frac {12 \left (20 x^9+60 x^8-190 x^7-490 x^6+810 x^5+1190 x^4-1640 x^3-480 x^2+e^{\frac {16}{x^4+2 x^3-7 x^2-8 x+16}} \left (50 x^7+170 x^6-390 x^5-1330 x^4+4540 x^3+5360 x^2-1920 x-1280\right )+640 x\right )}{289 x^3 \left (2 x+\sqrt {17}+1\right )^2}-\frac {12 \left (20 x^9+60 x^8-190 x^7-490 x^6+810 x^5+1190 x^4-1640 x^3-480 x^2+e^{\frac {16}{x^4+2 x^3-7 x^2-8 x+16}} \left (50 x^7+170 x^6-390 x^5-1330 x^4+4540 x^3+5360 x^2-1920 x-1280\right )+640 x\right )}{289 \sqrt {17} \left (-2 x+\sqrt {17}-1\right ) x^3}-\frac {12 \left (20 x^9+60 x^8-190 x^7-490 x^6+810 x^5+1190 x^4-1640 x^3-480 x^2+e^{\frac {16}{x^4+2 x^3-7 x^2-8 x+16}} \left (50 x^7+170 x^6-390 x^5-1330 x^4+4540 x^3+5360 x^2-1920 x-1280\right )+640 x\right )}{289 \left (-2 x+\sqrt {17}-1\right )^2 x^3}-\frac {8 \left (20 x^9+60 x^8-190 x^7-490 x^6+810 x^5+1190 x^4-1640 x^3-480 x^2+e^{\frac {16}{x^4+2 x^3-7 x^2-8 x+16}} \left (50 x^7+170 x^6-390 x^5-1330 x^4+4540 x^3+5360 x^2-1920 x-1280\right )+640 x\right )}{17 \sqrt {17} \left (-2 x+\sqrt {17}-1\right )^3 x^3}-\frac {8 \left (20 x^9+60 x^8-190 x^7-490 x^6+810 x^5+1190 x^4-1640 x^3-480 x^2+e^{\frac {16}{x^4+2 x^3-7 x^2-8 x+16}} \left (50 x^7+170 x^6-390 x^5-1330 x^4+4540 x^3+5360 x^2-1920 x-1280\right )+640 x\right )}{17 \sqrt {17} x^3 \left (2 x+\sqrt {17}+1\right )^3}\right )dx\) |
| \(\Big \downarrow \) 7239 |
| \(\displaystyle \int \frac {10 \left (x \left (2 x^2-1\right )+\frac {e^{\frac {16}{\left (x^2+x-4\right )^2}} \left (5 x^7+17 x^6-39 x^5-133 x^4+454 x^3+536 x^2-192 x-128\right )}{\left (x^2+x-4\right )^3}\right )}{x^3}dx\) |
| \(\Big \downarrow \) 27 |
| \(\displaystyle 10 \int -\frac {x \left (1-2 x^2\right )-\frac {e^{\frac {16}{\left (-x^2-x+4\right )^2}} \left (-5 x^7-17 x^6+39 x^5+133 x^4-454 x^3-536 x^2+192 x+128\right )}{\left (-x^2-x+4\right )^3}}{x^3}dx\) |
| \(\Big \downarrow \) 25 |
| \(\displaystyle -10 \int \frac {x \left (1-2 x^2\right )-\frac {e^{\frac {16}{\left (-x^2-x+4\right )^2}} \left (-5 x^7-17 x^6+39 x^5+133 x^4-454 x^3-536 x^2+192 x+128\right )}{\left (-x^2-x+4\right )^3}}{x^3}dx\) |
| \(\Big \downarrow \) 2010 |
| \(\displaystyle -10 \int \left (\frac {1-2 x^2}{x^2}-\frac {e^{\frac {16}{\left (x^2+x-4\right )^2}} \left (5 x^7+17 x^6-39 x^5-133 x^4+454 x^3+536 x^2-192 x-128\right )}{x^3 \left (x^2+x-4\right )^3}\right )dx\) |
| \(\Big \downarrow \) 2009 |
| \(\displaystyle -10 \left (\frac {232}{289} \left (17-\sqrt {17}\right ) \int \frac {e^{\frac {16}{\left (x^2+x-4\right )^2}}}{\left (-2 x+\sqrt {17}-1\right )^3}dx+\frac {3088 \int \frac {e^{\frac {16}{\left (x^2+x-4\right )^2}}}{\left (-2 x+\sqrt {17}-1\right )^3}dx}{17 \sqrt {17}}-\frac {116}{289} \left (3-\sqrt {17}\right ) \int \frac {e^{\frac {16}{\left (x^2+x-4\right )^2}}}{\left (-2 x+\sqrt {17}-1\right )^2}dx-\frac {30}{17} \left (1-\sqrt {17}\right ) \int \frac {e^{\frac {16}{\left (x^2+x-4\right )^2}}}{\left (-2 x+\sqrt {17}-1\right )^2}dx+\frac {5788}{289} \int \frac {e^{\frac {16}{\left (x^2+x-4\right )^2}}}{\left (-2 x+\sqrt {17}-1\right )^2}dx+\frac {290 \int \frac {e^{\frac {16}{\left (x^2+x-4\right )^2}}}{-2 x+\sqrt {17}-1}dx}{17 \sqrt {17}}-\frac {9}{2} \int \frac {e^{\frac {16}{\left (x^2+x-4\right )^2}}}{x^2}dx+\frac {31}{8} \int \frac {e^{\frac {16}{\left (x^2+x-4\right )^2}}}{x}dx-\frac {1}{136} \left (527+39 \sqrt {17}\right ) \int \frac {e^{\frac {16}{\left (x^2+x-4\right )^2}}}{2 x-\sqrt {17}+1}dx-\frac {232}{289} \left (17+\sqrt {17}\right ) \int \frac {e^{\frac {16}{\left (x^2+x-4\right )^2}}}{\left (2 x+\sqrt {17}+1\right )^3}dx+\frac {3088 \int \frac {e^{\frac {16}{\left (x^2+x-4\right )^2}}}{\left (2 x+\sqrt {17}+1\right )^3}dx}{17 \sqrt {17}}-\frac {116}{289} \left (3+\sqrt {17}\right ) \int \frac {e^{\frac {16}{\left (x^2+x-4\right )^2}}}{\left (2 x+\sqrt {17}+1\right )^2}dx-\frac {30}{17} \left (1+\sqrt {17}\right ) \int \frac {e^{\frac {16}{\left (x^2+x-4\right )^2}}}{\left (2 x+\sqrt {17}+1\right )^2}dx+\frac {5788}{289} \int \frac {e^{\frac {16}{\left (x^2+x-4\right )^2}}}{\left (2 x+\sqrt {17}+1\right )^2}dx-\frac {1}{136} \left (527-39 \sqrt {17}\right ) \int \frac {e^{\frac {16}{\left (x^2+x-4\right )^2}}}{2 x+\sqrt {17}+1}dx+\frac {290 \int \frac {e^{\frac {16}{\left (x^2+x-4\right )^2}}}{2 x+\sqrt {17}+1}dx}{17 \sqrt {17}}-2 \int \frac {e^{\frac {16}{\left (x^2+x-4\right )^2}}}{x^3}dx-2 x-\frac {1}{x}\right )\) |
Input:
Int[(640*x - 480*x^2 - 1640*x^3 + 1190*x^4 + 810*x^5 - 490*x^6 - 190*x^7 + 60*x^8 + 20*x^9 + E^(16/(16 - 8*x - 7*x^2 + 2*x^3 + x^4))*(-1280 - 1920*x + 5360*x^2 + 4540*x^3 - 1330*x^4 - 390*x^5 + 170*x^6 + 50*x^7))/(-64*x^3 + 48*x^4 + 36*x^5 - 23*x^6 - 9*x^7 + 3*x^8 + x^9),x]
Output:
$Aborted
Time = 4.39 (sec) , antiderivative size = 31, normalized size of antiderivative = 1.07
| method | result | size |
| risch | \(20 x +\frac {10}{x}-\frac {10 \left (1+5 x \right ) {\mathrm e}^{\frac {16}{\left (x^{2}+x -4\right )^{2}}}}{x^{2}}\) | \(31\) |
| parallelrisch | \(\frac {2000 x^{3}+210 x^{2}-5000 \,{\mathrm e}^{\frac {16}{x^{4}+2 x^{3}-7 x^{2}-8 x +16}} x +1000 x -1000 \,{\mathrm e}^{\frac {16}{x^{4}+2 x^{3}-7 x^{2}-8 x +16}}}{100 x^{2}}\) | \(71\) |
| parts | \(20 x +\frac {10}{x}+\frac {-720 \,{\mathrm e}^{\frac {16}{x^{4}+2 x^{3}-7 x^{2}-8 x +16}} x +470 \,{\mathrm e}^{\frac {16}{x^{4}+2 x^{3}-7 x^{2}-8 x +16}} x^{2}+330 \,{\mathrm e}^{\frac {16}{x^{4}+2 x^{3}-7 x^{2}-8 x +16}} x^{3}-110 \,{\mathrm e}^{\frac {16}{x^{4}+2 x^{3}-7 x^{2}-8 x +16}} x^{4}-50 \,{\mathrm e}^{\frac {16}{x^{4}+2 x^{3}-7 x^{2}-8 x +16}} x^{5}-160 \,{\mathrm e}^{\frac {16}{x^{4}+2 x^{3}-7 x^{2}-8 x +16}}}{x^{2} \left (x^{2}+x -4\right )^{2}}\) | \(186\) |
| norman | \(\frac {-720 x^{2}-210 x^{5}+140 x^{4}+570 x^{3}+160 x +20 x^{7}-720 \,{\mathrm e}^{\frac {16}{x^{4}+2 x^{3}-7 x^{2}-8 x +16}} x +470 \,{\mathrm e}^{\frac {16}{x^{4}+2 x^{3}-7 x^{2}-8 x +16}} x^{2}+330 \,{\mathrm e}^{\frac {16}{x^{4}+2 x^{3}-7 x^{2}-8 x +16}} x^{3}-110 \,{\mathrm e}^{\frac {16}{x^{4}+2 x^{3}-7 x^{2}-8 x +16}} x^{4}-50 \,{\mathrm e}^{\frac {16}{x^{4}+2 x^{3}-7 x^{2}-8 x +16}} x^{5}-160 \,{\mathrm e}^{\frac {16}{x^{4}+2 x^{3}-7 x^{2}-8 x +16}}}{x^{2} \left (x^{2}+x -4\right )^{2}}\) | \(205\) |
| orering | \(\frac {2 x \left (25 x^{15}-1224 x^{13}-1739 x^{12}+21896 x^{11}+7835 x^{10}-255684 x^{9}-92613 x^{8}+1174043 x^{7}+151085 x^{6}-2344176 x^{5}-905784 x^{4}+1699392 x^{3}+87424 x^{2}-446464 x +20480\right ) \left (\left (50 x^{7}+170 x^{6}-390 x^{5}-1330 x^{4}+4540 x^{3}+5360 x^{2}-1920 x -1280\right ) {\mathrm e}^{\frac {16}{x^{4}+2 x^{3}-7 x^{2}-8 x +16}}+20 x^{9}+60 x^{8}-190 x^{7}-490 x^{6}+810 x^{5}+1190 x^{4}-1640 x^{3}-480 x^{2}+640 x \right )}{5 \left (10 x^{15}+66 x^{14}-54 x^{13}-1055 x^{12}+1784 x^{11}+14975 x^{10}-5118 x^{9}-72681 x^{8}+12170 x^{7}+161447 x^{6}+42568 x^{5}-119232 x^{4}-34560 x^{3}+26112 x^{2}+8192 x -4096\right ) \left (x^{9}+3 x^{8}-9 x^{7}-23 x^{6}+36 x^{5}+48 x^{4}-64 x^{3}\right )}+\frac {\left (100 x^{9}-1932 x^{7}+9 x^{6}+14903 x^{5}-21889 x^{4}-38531 x^{3}+12052 x^{2}+8848 x -320\right ) x^{2} \left (x^{2}+x -4\right )^{3} \left (\frac {\left (350 x^{6}+1020 x^{5}-1950 x^{4}-5320 x^{3}+13620 x^{2}+10720 x -1920\right ) {\mathrm e}^{\frac {16}{x^{4}+2 x^{3}-7 x^{2}-8 x +16}}-\frac {16 \left (50 x^{7}+170 x^{6}-390 x^{5}-1330 x^{4}+4540 x^{3}+5360 x^{2}-1920 x -1280\right ) \left (4 x^{3}+6 x^{2}-14 x -8\right ) {\mathrm e}^{\frac {16}{x^{4}+2 x^{3}-7 x^{2}-8 x +16}}}{\left (x^{4}+2 x^{3}-7 x^{2}-8 x +16\right )^{2}}+180 x^{8}+480 x^{7}-1330 x^{6}-2940 x^{5}+4050 x^{4}+4760 x^{3}-4920 x^{2}-960 x +640}{x^{9}+3 x^{8}-9 x^{7}-23 x^{6}+36 x^{5}+48 x^{4}-64 x^{3}}-\frac {\left (\left (50 x^{7}+170 x^{6}-390 x^{5}-1330 x^{4}+4540 x^{3}+5360 x^{2}-1920 x -1280\right ) {\mathrm e}^{\frac {16}{x^{4}+2 x^{3}-7 x^{2}-8 x +16}}+20 x^{9}+60 x^{8}-190 x^{7}-490 x^{6}+810 x^{5}+1190 x^{4}-1640 x^{3}-480 x^{2}+640 x \right ) \left (9 x^{8}+24 x^{7}-63 x^{6}-138 x^{5}+180 x^{4}+192 x^{3}-192 x^{2}\right )}{\left (x^{9}+3 x^{8}-9 x^{7}-23 x^{6}+36 x^{5}+48 x^{4}-64 x^{3}\right )^{2}}\right )}{100 x^{15}+660 x^{14}-540 x^{13}-10550 x^{12}+17840 x^{11}+149750 x^{10}-51180 x^{9}-726810 x^{8}+121700 x^{7}+1614470 x^{6}+425680 x^{5}-1192320 x^{4}-345600 x^{3}+261120 x^{2}+81920 x -40960}\) | \(825\) |
Input:
int(((50*x^7+170*x^6-390*x^5-1330*x^4+4540*x^3+5360*x^2-1920*x-1280)*exp(1 6/(x^4+2*x^3-7*x^2-8*x+16))+20*x^9+60*x^8-190*x^7-490*x^6+810*x^5+1190*x^4 -1640*x^3-480*x^2+640*x)/(x^9+3*x^8-9*x^7-23*x^6+36*x^5+48*x^4-64*x^3),x,m ethod=_RETURNVERBOSE)
Output:
20*x+10/x-10*(1+5*x)/x^2*exp(16/(x^2+x-4)^2)
Time = 0.08 (sec) , antiderivative size = 42, normalized size of antiderivative = 1.45 \[ \int \frac {640 x-480 x^2-1640 x^3+1190 x^4+810 x^5-490 x^6-190 x^7+60 x^8+20 x^9+e^{\frac {16}{16-8 x-7 x^2+2 x^3+x^4}} \left (-1280-1920 x+5360 x^2+4540 x^3-1330 x^4-390 x^5+170 x^6+50 x^7\right )}{-64 x^3+48 x^4+36 x^5-23 x^6-9 x^7+3 x^8+x^9} \, dx=\frac {10 \, {\left (2 \, x^{3} - {\left (5 \, x + 1\right )} e^{\left (\frac {16}{x^{4} + 2 \, x^{3} - 7 \, x^{2} - 8 \, x + 16}\right )} + x\right )}}{x^{2}} \] Input:
integrate(((50*x^7+170*x^6-390*x^5-1330*x^4+4540*x^3+5360*x^2-1920*x-1280) *exp(16/(x^4+2*x^3-7*x^2-8*x+16))+20*x^9+60*x^8-190*x^7-490*x^6+810*x^5+11 90*x^4-1640*x^3-480*x^2+640*x)/(x^9+3*x^8-9*x^7-23*x^6+36*x^5+48*x^4-64*x^ 3),x, algorithm="fricas")
Output:
10*(2*x^3 - (5*x + 1)*e^(16/(x^4 + 2*x^3 - 7*x^2 - 8*x + 16)) + x)/x^2
Time = 0.10 (sec) , antiderivative size = 37, normalized size of antiderivative = 1.28 \[ \int \frac {640 x-480 x^2-1640 x^3+1190 x^4+810 x^5-490 x^6-190 x^7+60 x^8+20 x^9+e^{\frac {16}{16-8 x-7 x^2+2 x^3+x^4}} \left (-1280-1920 x+5360 x^2+4540 x^3-1330 x^4-390 x^5+170 x^6+50 x^7\right )}{-64 x^3+48 x^4+36 x^5-23 x^6-9 x^7+3 x^8+x^9} \, dx=20 x + \frac {10}{x} + \frac {\left (- 50 x - 10\right ) e^{\frac {16}{x^{4} + 2 x^{3} - 7 x^{2} - 8 x + 16}}}{x^{2}} \] Input:
integrate(((50*x**7+170*x**6-390*x**5-1330*x**4+4540*x**3+5360*x**2-1920*x -1280)*exp(16/(x**4+2*x**3-7*x**2-8*x+16))+20*x**9+60*x**8-190*x**7-490*x* *6+810*x**5+1190*x**4-1640*x**3-480*x**2+640*x)/(x**9+3*x**8-9*x**7-23*x** 6+36*x**5+48*x**4-64*x**3),x)
Output:
20*x + 10/x + (-50*x - 10)*exp(16/(x**4 + 2*x**3 - 7*x**2 - 8*x + 16))/x** 2
Leaf count of result is larger than twice the leaf count of optimal. 377 vs. \(2 (29) = 58\).
Time = 0.19 (sec) , antiderivative size = 377, normalized size of antiderivative = 13.00 \[ \int \frac {640 x-480 x^2-1640 x^3+1190 x^4+810 x^5-490 x^6-190 x^7+60 x^8+20 x^9+e^{\frac {16}{16-8 x-7 x^2+2 x^3+x^4}} \left (-1280-1920 x+5360 x^2+4540 x^3-1330 x^4-390 x^5+170 x^6+50 x^7\right )}{-64 x^3+48 x^4+36 x^5-23 x^6-9 x^7+3 x^8+x^9} \, dx=20 \, x + \frac {10 \, {\left (567 \, x^{4} + 1284 \, x^{3} - 3013 \, x^{2} - 4466 \, x + 4624\right )}}{289 \, {\left (x^{5} + 2 \, x^{4} - 7 \, x^{3} - 8 \, x^{2} + 16 \, x\right )}} - \frac {10 \, {\left (4134 \, x^{3} - 1891 \, x^{2} - 17512 \, x + 17232\right )}}{289 \, {\left (x^{4} + 2 \, x^{3} - 7 \, x^{2} - 8 \, x + 16\right )}} + \frac {30 \, {\left (924 \, x^{3} - 1793 \, x^{2} - 4696 \, x + 7536\right )}}{289 \, {\left (x^{4} + 2 \, x^{3} - 7 \, x^{2} - 8 \, x + 16\right )}} + \frac {95 \, {\left (386 \, x^{3} + x^{2} - 1096 \, x + 656\right )}}{289 \, {\left (x^{4} + 2 \, x^{3} - 7 \, x^{2} - 8 \, x + 16\right )}} - \frac {30 \, {\left (29 \, x^{3} + 188 \, x^{2} + 9 \, x - 942\right )}}{289 \, {\left (x^{4} + 2 \, x^{3} - 7 \, x^{2} - 8 \, x + 16\right )}} - \frac {245 \, {\left (24 \, x^{3} - 253 \, x^{2} - 152 \, x + 496\right )}}{289 \, {\left (x^{4} + 2 \, x^{3} - 7 \, x^{2} - 8 \, x + 16\right )}} - \frac {405 \, {\left (14 \, x^{3} + 21 \, x^{2} + 104 \, x - 96\right )}}{289 \, {\left (x^{4} + 2 \, x^{3} - 7 \, x^{2} - 8 \, x + 16\right )}} - \frac {820 \, {\left (12 \, x^{3} + 18 \, x^{2} - 76 \, x - 41\right )}}{289 \, {\left (x^{4} + 2 \, x^{3} - 7 \, x^{2} - 8 \, x + 16\right )}} - \frac {35 \, {\left (6 \, x^{3} + 9 \, x^{2} - 38 \, x + 124\right )}}{17 \, {\left (x^{4} + 2 \, x^{3} - 7 \, x^{2} - 8 \, x + 16\right )}} - \frac {10 \, {\left (5 \, x + 1\right )} e^{\left (\frac {16}{x^{4} + 2 \, x^{3} - 7 \, x^{2} - 8 \, x + 16}\right )}}{x^{2}} \] Input:
integrate(((50*x^7+170*x^6-390*x^5-1330*x^4+4540*x^3+5360*x^2-1920*x-1280) *exp(16/(x^4+2*x^3-7*x^2-8*x+16))+20*x^9+60*x^8-190*x^7-490*x^6+810*x^5+11 90*x^4-1640*x^3-480*x^2+640*x)/(x^9+3*x^8-9*x^7-23*x^6+36*x^5+48*x^4-64*x^ 3),x, algorithm="maxima")
Output:
20*x + 10/289*(567*x^4 + 1284*x^3 - 3013*x^2 - 4466*x + 4624)/(x^5 + 2*x^4 - 7*x^3 - 8*x^2 + 16*x) - 10/289*(4134*x^3 - 1891*x^2 - 17512*x + 17232)/ (x^4 + 2*x^3 - 7*x^2 - 8*x + 16) + 30/289*(924*x^3 - 1793*x^2 - 4696*x + 7 536)/(x^4 + 2*x^3 - 7*x^2 - 8*x + 16) + 95/289*(386*x^3 + x^2 - 1096*x + 6 56)/(x^4 + 2*x^3 - 7*x^2 - 8*x + 16) - 30/289*(29*x^3 + 188*x^2 + 9*x - 94 2)/(x^4 + 2*x^3 - 7*x^2 - 8*x + 16) - 245/289*(24*x^3 - 253*x^2 - 152*x + 496)/(x^4 + 2*x^3 - 7*x^2 - 8*x + 16) - 405/289*(14*x^3 + 21*x^2 + 104*x - 96)/(x^4 + 2*x^3 - 7*x^2 - 8*x + 16) - 820/289*(12*x^3 + 18*x^2 - 76*x - 41)/(x^4 + 2*x^3 - 7*x^2 - 8*x + 16) - 35/17*(6*x^3 + 9*x^2 - 38*x + 124)/ (x^4 + 2*x^3 - 7*x^2 - 8*x + 16) - 10*(5*x + 1)*e^(16/(x^4 + 2*x^3 - 7*x^2 - 8*x + 16))/x^2
Leaf count of result is larger than twice the leaf count of optimal. 101 vs. \(2 (29) = 58\).
Time = 0.14 (sec) , antiderivative size = 101, normalized size of antiderivative = 3.48 \[ \int \frac {640 x-480 x^2-1640 x^3+1190 x^4+810 x^5-490 x^6-190 x^7+60 x^8+20 x^9+e^{\frac {16}{16-8 x-7 x^2+2 x^3+x^4}} \left (-1280-1920 x+5360 x^2+4540 x^3-1330 x^4-390 x^5+170 x^6+50 x^7\right )}{-64 x^3+48 x^4+36 x^5-23 x^6-9 x^7+3 x^8+x^9} \, dx=\frac {10 \, {\left (2 \, x^{3} - 5 \, x e^{\left (-\frac {x^{4} + 2 \, x^{3} - 7 \, x^{2} - 8 \, x}{x^{4} + 2 \, x^{3} - 7 \, x^{2} - 8 \, x + 16} + 1\right )} + x - e^{\left (-\frac {x^{4} + 2 \, x^{3} - 7 \, x^{2} - 8 \, x}{x^{4} + 2 \, x^{3} - 7 \, x^{2} - 8 \, x + 16} + 1\right )}\right )}}{x^{2}} \] Input:
integrate(((50*x^7+170*x^6-390*x^5-1330*x^4+4540*x^3+5360*x^2-1920*x-1280) *exp(16/(x^4+2*x^3-7*x^2-8*x+16))+20*x^9+60*x^8-190*x^7-490*x^6+810*x^5+11 90*x^4-1640*x^3-480*x^2+640*x)/(x^9+3*x^8-9*x^7-23*x^6+36*x^5+48*x^4-64*x^ 3),x, algorithm="giac")
Output:
10*(2*x^3 - 5*x*e^(-(x^4 + 2*x^3 - 7*x^2 - 8*x)/(x^4 + 2*x^3 - 7*x^2 - 8*x + 16) + 1) + x - e^(-(x^4 + 2*x^3 - 7*x^2 - 8*x)/(x^4 + 2*x^3 - 7*x^2 - 8 *x + 16) + 1))/x^2
Time = 3.30 (sec) , antiderivative size = 42, normalized size of antiderivative = 1.45 \[ \int \frac {640 x-480 x^2-1640 x^3+1190 x^4+810 x^5-490 x^6-190 x^7+60 x^8+20 x^9+e^{\frac {16}{16-8 x-7 x^2+2 x^3+x^4}} \left (-1280-1920 x+5360 x^2+4540 x^3-1330 x^4-390 x^5+170 x^6+50 x^7\right )}{-64 x^3+48 x^4+36 x^5-23 x^6-9 x^7+3 x^8+x^9} \, dx=20\,x+\frac {10}{x}-\frac {{\mathrm {e}}^{\frac {16}{x^4+2\,x^3-7\,x^2-8\,x+16}}\,\left (50\,x+10\right )}{x^2} \] Input:
int((640*x - exp(16/(2*x^3 - 7*x^2 - 8*x + x^4 + 16))*(1920*x - 5360*x^2 - 4540*x^3 + 1330*x^4 + 390*x^5 - 170*x^6 - 50*x^7 + 1280) - 480*x^2 - 1640 *x^3 + 1190*x^4 + 810*x^5 - 490*x^6 - 190*x^7 + 60*x^8 + 20*x^9)/(48*x^4 - 64*x^3 + 36*x^5 - 23*x^6 - 9*x^7 + 3*x^8 + x^9),x)
Output:
20*x + 10/x - (exp(16/(2*x^3 - 7*x^2 - 8*x + x^4 + 16))*(50*x + 10))/x^2
Time = 0.20 (sec) , antiderivative size = 65, normalized size of antiderivative = 2.24 \[ \int \frac {640 x-480 x^2-1640 x^3+1190 x^4+810 x^5-490 x^6-190 x^7+60 x^8+20 x^9+e^{\frac {16}{16-8 x-7 x^2+2 x^3+x^4}} \left (-1280-1920 x+5360 x^2+4540 x^3-1330 x^4-390 x^5+170 x^6+50 x^7\right )}{-64 x^3+48 x^4+36 x^5-23 x^6-9 x^7+3 x^8+x^9} \, dx=\frac {-50 e^{\frac {16}{x^{4}+2 x^{3}-7 x^{2}-8 x +16}} x -10 e^{\frac {16}{x^{4}+2 x^{3}-7 x^{2}-8 x +16}}+20 x^{3}+10 x}{x^{2}} \] Input:
int(((50*x^7+170*x^6-390*x^5-1330*x^4+4540*x^3+5360*x^2-1920*x-1280)*exp(1 6/(x^4+2*x^3-7*x^2-8*x+16))+20*x^9+60*x^8-190*x^7-490*x^6+810*x^5+1190*x^4 -1640*x^3-480*x^2+640*x)/(x^9+3*x^8-9*x^7-23*x^6+36*x^5+48*x^4-64*x^3),x)
Output:
(10*( - 5*e**(16/(x**4 + 2*x**3 - 7*x**2 - 8*x + 16))*x - e**(16/(x**4 + 2 *x**3 - 7*x**2 - 8*x + 16)) + 2*x**3 + x))/x**2